Many-body quantum phases of
dipolar gases– DISSERTATION –byClaudia Politi, MSc.

submitted to the Faculty of Mathematics, Computer Science, andPhysics of the University of Innsbruckin partial fulfillment of the requirements
for the degree of Doctor of Philosophy (PhD)
advisors:
Univ. - Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences
Univ. - Prof. Dr. Helmut Ritsch,
Institute for Theoretical Physics, University of InnsbruckInnsbruck, December 2022

To my family and to the game of basketball,for teaching me the values of perseverance,
respect, teamwork, and the importance of"rest at the end and not in the middle".

Declaration

I hereby declare that I have written this dissertation independently and have not used
any sources other than those specified. All passages that were taken from the specified
sources, either verbatim or in terms of content, are marked as such. This dissertation
has not yet been submitted as a scientific work in the same or a similar form in this,or any other university.
Claudia Politi, MSc.

Innsbruck, December 2022

Synopsis
Since the realization of the first Bose-Einstein condensate in 1995, the number ofquantum-gas experiments grew enormously, involving different atomic species ranging
from alkali metals, such as rubidium, to triel elements, with the new-come indium.
This huge interest is motivated by the high control that quantum gases offer in termsof interatomic interactions, dimensionality of the system, and the possibility of adding
complexity in a controlled manner. For this reason, in the past decades, ultracold
atoms revealed to be ideal platforms for simulating many-body phenomena, linked tovarious fields as condensed-matter physics, high-energy physics, and quantum optics.
Ultracold gases interact usually with a short-range and isotropic contact-type
interaction. The achievement of degenerate gases with highly-magnetic atoms, suchas chromium, erbium, dysprosium, and just recently thulium and europium, which
possess a permanent magnetic moment in the ground state, led to richer interactions,
showing a long-range and anisotropic nature. The addition of this new ingredient to
the quantum-gas tool box brought to the discovery of interesting many-body phases,
as quantum droplets, and supersolid states, showing both superfluid and crystalline
order.
The work presented in this thesis focuses on two main topics. The first part
reports on the investigation of the interspecies interactions between the two highly
magnetic lanthanides, erbium and dysprosium, with a focus on the role played by the
dipole-dipole interaction. It presents an alternative method to estimate the interspecies
scattering length from the in-trap clouds displacement, and it shows how tuning theinterspecies repulsion can lead to binary supersolid states.
The second part focuses on the creation of a dipolar supersolid state with dysprosium
atoms. It first gives insights on the role played by finite temperatures in the superfluidto-supersolid phase transition. It then presents the first realization of two-dimensional
supersolid states, first in a zig-zag pattern, and then in a hexagon pattern. The
realization of two-dimensional supersolidity opens the door to many research directions,viiisuch as the investigation of the excitation modes, quantized vortices, and persistentcurrents.

Zusammenfassung
Seit der Realisierung des ersten Bose-Einstein-Kondensats im Jahr 1995 hat die Zahl der
Quantengasexperimente enorm zugenommen, wobei verschiedene Atomarten von Alkalimetallen wie Rubidium bis zu Trielelementen mit dem neu hinzugekommenen Indium
einbezogen wurden. Dieses enorme Interesse wird durch die hohe Kontrolle motiviert,
die Quantengase in Bezug auf interatomare Wechselwirkungen, die Dimensionalität desSystems und die Möglichkeit, Komplexität auf kontrollierte Weise hinzuzufügen, bieten.
Aus diesem Grund haben sich ultrakalte Atome in den vergangenen Jahrzehnten als
ideale Plattformen für die Simulation von Vielteilchenphänomenen erwiesen, die mit verschiedenen Bereichen wie der Physik der kondensierten Materie, der Hochenergiephysik
und der Quantenoptik verknüpft sind.
Ultrakalte Gase interagieren normalerweise mit einer kurzreichweitigen und isotropen
Wechselwirkung, einer sogenannten Kontaktwechselwirkung. Das Erreichen entarteter
Gase mit hochmagnetischen Atomen wie Chrom, Erbium, Dysprosium und erst kürzlich Thulium und Europium, die im Grundzustand ein permanentes magnetisches
Moment besitzen, ermöglichte die Nutzung von Wechselwirkungen, die eine langreichweitige und anisotrope Natur zeigen. Die Aufnahme dieses neuen Inhaltsstoffs in die
Quantengas-Werkzeugkiste führte zur Entdeckung exotischer Vielteilchenphasen wie
Quantentröpfchen und suprasolide Zustände, die sowohl eine supraflüssige als aucheine kristalline Ordnung zeigen.
Diese Dissertation konzentriert sich auf zwei Hauptthemen. Der erste Teil berichtetüber die Untersuchung der Interspezies-Wechselwirkungen zwischen den beiden hochmagnetischen Lanthanoiden Erbium und Dysprosium, wobei der Schwerpunkt auf derRolle der Dipol-Dipol-Wechselwirkung liegt. Wir beschreiben eine alternative Methodezur Abschätzung der Interspezies-Streulänge aus der Verschiebung der Wolken innerhalb der Falle und zeigen, wie die Abstimmung der Interspezies-Abstoßung zu binären
suprasoliden Zuständen führen kann.
Der zweite Teil konzentriert sich auf die Erzeugung eines dipolaren Suprasolids mit
Dysprosiumatomen. Zunächst wird diskutiert, welche Rolle endliche Temperaturen

xbeim Phasenübergang von suprafluid zu suprasolid spielen. Danach präsentieren wir
die erste Realisierung von zweidimensionalen Suprasoliden, zuerst in einem Zick-ZackMuster und dann in einem Hexagon-Muster. Die Realisierung der zweidimensionalen
Supersolidität öffnet die Tür zu vielen Forschungsrichtungen, wie der Untersuchungvon Anregungsmoden, quantisierten Wirbeln und Dauerströmen.

Contents
Synopsis

viii

Zusammenfassungx

Introduction

1

1 Supersolidity in dipolar quantum gases
1.1 Brief historical overview . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Supersolidity in dipolar gases: main ingredients . . . . . . . . . . . . .
1.2.1 Crystallization mechanism . . . . . . . . . . . . . . . . . . . . .
1.2.2 Stabilization mechanism . . . . . . . . . . . . . . . . . . . . . .
1.3 Many-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Thermal fluctuations . . . . . . . . . . . . . . . . . . . . . . . .

7
8
10
11
12
14
18

2 Er-Dy experiment
2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Optical dipole traps . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Imaging systems . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Quantum gas microscope . . . . . . . . . . . . . . . . . . . . . .

21
21
22
25
27
29

3 Interspecies interaction in an ultracold dipolar mixture
3.1 Hetero-nuclear mixtures: an overview . . . . . . . . . . . . . . . . . . .
3.2 Cross-species thermalization measurements . . . . . . . . . . . . . . . .
3.2.1 Feshbach spectroscopy . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Interspecies scattering length from mean-field shift . . . . . . . . . . . .
3.3.1 Extended Gross-Pitaevskii equation . . . . . . . . . . . . . . . .
3.3.2 Experimental sequence . . . . . . . . . . . . . . . . . . . . . . .

31
31
33
35
35
35
38
39

xii

Contents

3.4
3.5

3.3.3 Evaluation of the systematic errors . . . . . . . . . . . . . . . . 39
Outlook: induced supersolidity in dipolar mixture . . . . . . . . . . . . 41
Publication I:
Interspecies interactions in an ultracold dipolar mixture . . . . . . . . . 43

4 Two-dimensional supersolidity in dysprosium
53
4.1 Direct evaporation into the supersolid state . . . . . . . . . . . . . . . . 54
4.1.1 Probing supersolidity in the experiment . . . . . . . . . . . . . . 55
4.1.2 The role of finite temperature . . . . . . . . . . . . . . . . . . . 58
4.2 From one to two-dimensional supersolidity . . . . . . . . . . . . . . . . 60
4.2.1 Supersolidity in a circular trap . . . . . . . . . . . . . . . . . . . 63
4.3 Excitations in two-dimensional supersolids . . . . . . . . . . . . . . . . 64
4.3.1 Scissors mode and angular oscillations . . . . . . . . . . . . . . 65
4.3.2 Outlook: vortices in dipolar supersolids . . . . . . . . . . . . . . 67
4.4 Publication II:
Birth, life, and death of a dipolar supersolid . . . . . . . . . . . . . . . 69
4.5 Publication III:Two-dimensional supersolidity in a dipolar quantum gas . . . . . . . . . 79
4.6 Publication IV:
Maintaining supersolidity in one and two dimensions . . . . . . . . . . 88
4.7 Publication V:
Two-dimensional supersolid formation in dipolar condensates . . . . . . 98
4.8 Publication VI:
Can angular oscillations probe superfluidity in dipolar supersolids? . . . 111
4.9 Publication VII:
Heating a quantum dipolar fluid into a solid . . . . . . . . . . . . . . . 123
Conclusions

131

Bibliography

135

A Additional publications
151
A.1 Publication:
Dipolar quantum mixtures of erbium and dysprosium atoms . . . . . . 152
A.2 Publication:
Long-lived and transient supersolid behaviors in dipolar quantum gases 159
A.3 Publication:
Feshbach resonances in an erbium-dysprosium dipolar mixture . . . . . 180

Contents

xiii

A.4 Publication:
Phase coherence in out-of-equilibrium supersolid states of ultracold
dipolar atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.5 Publication:
Observation of vortices and vortex stripes in a dipolar Bose-Einsteincondensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Acknowledgments

219

Introduction
Since the first experimental realization of a Bose-Einstein condensate (BEC) [5, 35, 48]
and a degenerate Fermi gas (dFg) [53], ultracold gases have demonstrated to be ideal
platforms to study many-body quantum phenomena. Prominent examples are theobservation of long-range phase coherence [28] and quantized vortices [1, 121, 124],
common to superconductivity and superfluidity. Furthermore, ultracold gases featureseveral attractive properties, as the tunability of the atomic interactions via Feshbachresonances [39], and the possibility of changing the dimensionality of the system andrealizing ideal strong periodic potentials, through the use of far off-resonant laser
light. These features led to the investigation of strongly correlated regimes, as thesuperfluid-to-Mott insulator transition [70] and Bardeen–Cooper–Schrieffer (BCS)
pairing [13, 33, 164]. It is clear now that quantum gases represent a powerful tool
for simulating and understanding complex condensed-matter problems, as well as
engineering novel quantum phases that have not been observed yet.
In ultracold gases, the form of the atomic interactions strongly affects the quantum
phases and the system’s dynamics. For this reason, in the last 20 years, a lot ofexperimental effort was spent in creating ultracold gases with richer interactions, e.g.
long-range and anisotropic dipole-dipole interactions. The addition of this ingredient toultracold systems led to interesting predictions of many-body phases in optical lattices,such as charged-density wave and supersolid phases, where superfluidity and crystalline
order coexist [11]. Another way of enriching the atomic interactions is through atomic
mixtures. In fact, by tuning the interspecies interactions, miscible and immiscible
quantum phases have been experimentally observed in alkali mixtures made of two
spin states [178], two different isotopes of the same species [142], or two species [126].
Theoretical works predict that dipole-dipole interactions add interesting features to
miscible and immiscible phases [69, 101, 195]. Moreover, exotic supersolid phases are
predicted to appear due to the competition between intra- and interspecies interactions
both in optical lattices [36, 186] and in the bulk [26, 169].

2

Introduction

Long-range interaction can be achieved in different ways. In ultracold-gas experiments, it can arise from strong magnetic or electric dipole moments, as in magnetic
lanthanides, and hetero-nuclear molecules and Rydberg atoms [106], respectively. Moreover, by placing the atomic cloud in an optical resonator, long-range interactions can beachieved by coupling the atoms to a single or multiple modes of the cavity [14, 123, 131].
Finally, long-range interactions can be engineered via atom-light interactions, see e.g.
Refs. [68, 140]. Each of these platforms is characterized by a different strength of thelong-range interactions, which makes them unique for investigating various phenomena.
For instance, in magnetic lanthanides, such as erbium and dysprosium, short-range
and long-range interactions have similar strength and can compete with each othergiving rise to exotic quantum phases, as discussed in this thesis.
In 2003, theoretical works [139, 168] predicted that this competition leads to theappearance of a roton-maxon excitation spectrum, a precursor of crystallization, which
is a fundamental ingredient for supersolidity. Experimentally, the investigation ofquantum gases possessing a large permanent magnetic moment started in 2005 withthe realization of a Bose-Einstein condensate of chromium atoms [71]. Chromium wasfollowed, in 2011 and 2012, by dysprosium [119] and erbium [3], respectively. The listof Bose-condensed magnetic lanthanides continues to grow, and more recently includes
thulium [49] as well as europium [129]. Breakthrough experiments, as the observationof the roton mode in erbium [44] and the realization of a density-modulated state withdysprosium [91], in a regime in which the mean-field physics predicts a collapse, led tothe discovery of the supersolid state in 2019 [32, 42, 182].
The Er-Dy experiment in Innsbruck, which is the focus of this work, links together
the world of hetero-nuclear mixtures with the one of magnetic lanthanides with longrange interactions. This enables to perform experiments with erbium, or dysprosium,
but also combining the two. Since the first realization of double BECs of erbium anddysprosium, and the creation of a single-component long-lived supersolid state withdysprosium, the experiment has improved the knowledge of dipolar gases, including
the recent works presented in this thesis, such as the first experimental realization of
two-dimensional supersolidity.
During my PhD in the Er-Dy experiment, I contributed to two main research
topics: the investigation of erbium-dysprosium quantum mixtures, and the study ofsupersolidity in dipolar quantum gases of dysprosium atoms. This resulted in a totalof 12 publications, among which 7 constitute the core of this thesis.

Introduction

3

Erbium-dysprosium quantum mixtures. When I started my PhD in January
2018, the team was working on the realization of the first dipolar quantum mixturesof erbium and dysprosium. This study revealed a first hint of repulsive interaction
between the species, from a shift and a deformation of the atomic clouds. This brought
to one of the main results of this thesis. In fact, in collaboration with M. Modugno
from the University of the Basque Country in Bilbao (Spain), we investigated theinterspecies interactions in an erbium-dysprosium quantum mixture, with a focus on
the role played by the dipole-dipole interactions. Furthermore, by probing the in-trap
clouds displacement as a function of the magnetic-field orientation, and by comparing
our experimental results to ground-state calculations, we were able to estimate theinterspecies scattering length. This study is reported in the publication below:• Interspecies interactions in an ultracold dipolar mixture.
C. Politi, A. Trautmann, P. Ilzhöfer, G. Durastante, M. J. Mark, M. Modugno,
and F. Ferlaino,
Phys. Rev. A, 105, 023304 (2022)
Supersolidity with dysprosium atoms. Another topic on which I contributed
during my PhD is the realization and the investigation of supersolidity in dipolar
quantum gases of dysprosium atoms. This subject can be divided in two more branches:
one-dimensional supersolid states, where the system consists of a linear chain ofquantum droplets linked to each other via a superfluid background, and two-dimensional
supersolid states, where the supersolid properties are extended along two directions.
Regarding the first research branch, we investigated the formation of a supersolidstate via evaporative cooling from a thermal cloud, from the birth until its disappearance.
Moreover, in collaboration with the group of T. Pohl at Aarhus University (Denmark),
we proved the important role played by thermal fluctuations in the formation ofsupersolid states. These works are presented in the following publications:• Birth, life, and death of a dipolar supersolid.
M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark, M. A. Norcia, and F.
Ferlaino,
Phys. Rev. Lett.,126, 233401 (2021)• Heating a quantum dipolar fluid into a solid.
J. Sanchez-Baena, C. Politi, F. Maucher, F. Ferlaino, and T. Pohl,
arXiv:2209.00335 (2022)

4

Introduction

In the second research branch, we were able to extend, for the first time, thesupersolid properties from one to two dimensions. In particular, we performed astructural phase transition from a linear supersolid state to, first, a supersolid madeof quantum droplets arranged in a zig-zag pattern, and then in an isotropic hexagon
configuration. The realization of such supersolid states opened many research directions.
For instance, we investigated the excitations of two-dimensional supersolids, including
angular oscillations and their role as a potential probe of the superfluid fraction in
dipolar supersolids. The publications related to this research topic are listed below:• Two-dimensional supersolidity in a dipolar quantum gas.
M. A. Norcia*, C. Politi*, L. Klaus, E. Poli, M. Sohmen, M. J. Mark, R. Bisset,
L. Santos, and F. Ferlaino,
Nature 596, 357 (2021)*These authors contributed equally.
• Maintaining supersolidity in one and two dimensions.
E. Poli, T. Bland, C. Politi, L. Klaus, M. A. Norcia, F. Ferlaino, R. N. Bisset,
and L. Santos.,
Phys. Rev. A, 104, 063307 (2021)• Two-dimensional supersolid formation in dipolar condensates.
T. Bland, E. Poli, C. Politi, L. Klaus, M. A. Norcia, F. Ferlaino, L. Santos, and
R. N. Bisset,
Phys. Rev. Lett., 128, 195302 (2022)• Can angular oscillations probe superfluidity in dipolar supersolids?
M. A. Norcia, E. Poli, C. Politi, L. Klaus, T. Bland, M. J. Mark, L. Santos, R.
N. Bisset, and F. Ferlaino,
Phys. Rev. Lett., 129, 040403 (2022)

Thesis overview
This thesis work consists of four chapters, which are briefly introduced below.
Chapter 1 gives an overview on supersolidity, mentioning the experimental efforts
in solid helium and the results achieved with ultracold gases. It then introduces themain ingredients necessary to realize supersolidity in ultracold dipolar gases. Finally,
it describes the extended Gross-Pitaevskii equation with the beyond mean-field term,

Introduction

5

which constitutes the main theoretical tool to represent various quantum phases of adipolar gas.
Chapter 2 describes the basic properties of erbium and dysprosium atoms, as theelectronic configuration and the optical transitions used to cool, manipulate and image
the atoms. It then gives a brief overview of the experimental apparatus with a focus on
the optical-dipole-trap setup used to confine the atomic cloud and produce supersolidstates.
Chapter 3 focuses on the first main result of this thesis: the study of the interspecies
interactions in an erbium-dysprosium mixture. First, it gives an overview on heteronuclear mixture experiments and describes the experimental tools commonly used todetermine the interspecies scattering lengths. It then introduces the method developedin our experiment to estimate the interspecies scattering length from the in-trap
displacement between the clouds. Finally, it gives an outlook on the interesting
quantum phases achievable by tuning the erbium-dysprosium interactions.
Chapter 4 is dedicated to the investigation of supersolidity in a cloud of dysprosium
atoms. This chapter first presents the technique developed in our experiment to create
supersolid states by directly evaporating from a thermal cloud and describes the tools
we use to probe such a state. It then focuses on the role played by finite temperatures.
Finally, it describes how to create two-dimensional supersolid states and it gives an
overview of the many research directions that this state offers, with the investigationof excitation modes and quantized vortices.

Additional publications
Beside the above listed publications, in the past five years, I also contributed toother publications which are not discussed in this thesis work. These publications arepresented in the Appendix A and listed below:• Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms.
A. Trautmann*, P. Ilzhöfer*, G. Durastante, C. Politi, M. Sohmen, M. J. Mark,and F. Ferlaino,
Phys. Rev. Lett., 121, 213601 (2018)*These authors contributed equally.

6

Introduction• Long-lived and transient supersolid behaviors in dipolar quantum gases.
L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F.
Ferlaino,
Phys. Rev. X, 9, 021012 (2019)• Feshbach resonances in an erbium-dysprosium dipolar mixture.
G. Durastante, C. Politi, M. Sohmen, P. Ilzhöfer, M. J. Mark, M. A. Norcia, and
F. Ferlaino,
Phys. Rev. A, 102, 033330 (2020)• Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms.
P. Ilzhöfer*, M. Sohmen*, G. Durastante, C. Politi, A. Trautmann, G. Natale, G.
Morpurgo, T. Giamarchi, L. Chomaz, M. J. Mark, and F. Ferlaino,
Nature Physics 17, 356-361 (2021)*These authors contributed equally.
• Observation of vortices and vortex stripes in a dipolar Bose-Einstein condensate.
L. Klaus*, T. Bland*, E. Poli, C. Politi, G. Lamporesi, E. Casotti, R. N. Bisset,
M. J. Mark, and F. Ferlaino,
Nature Physics (2022)*These authors contributed equally.

Chapter 1
Supersolidity in dipolar quantum
gasesSupersolidity is a counterintuitive state of matter, combining both density modulationand superfluid properties. Initially predicted in the late ’50s by E. P. Gross for a system
of interacting bosons [73], it was intensively searched in consensed-matter systems,with solid helium as the prime candidate. However, the limited control of interactions
and imperfections in solid-state systems have pushed the theoretical investigations
towards ultracold atoms. It was then about a decade ago that the supersolid physics
became a topic of growing interest for the ultracold-gas community, triggered by thehigh degree of control over many-body systems, e.g. over the interatomic interactions,geometry, and dimensionality. In 2010, and later in 2017, two experimental groups
reported the observation of supersolid properties using two different settings: BECs
of rubidium atoms with cavity-mediated interactions [14, 112], and spin mixturesof sodium atoms coupled via a two-photon Raman transition [114]. More recently,
experiments with dipolar gases of erbium and dysprosium, including the ones in ourgroup, demonstrated the existence of a supersolid state intrinsically arising from thecompetition between the contact interactions and the long-range anisotropic dipoledipole interactions (DDI) [32, 42, 182]. This chapter gives a brief historical overview
of supersolidity and introduces the key ingredients to achieve simultaneous density
modulation and global phase coherence in dipolar gases. Finally, the last section recalls
the Gross-Pitaevskii equation and the relevant energy terms necessary to describe an
ultracold gas of dipolar atoms.

8

1.1

Supersolidity in dipolar quantum gases

Brief historical overview

Supersolidity was for the first time predicted in 1957 by Gross [72, 73]. In his works,
Gross considered a system of bosonic particles in a box, interacting via a two-bodypotential, and demonstrated that, together with the standard uniform solution, forstrong enough attractive interactions, a ground state showing periodic density modulation exists. In these seminal papers, Gross investigated the excitations of such asystem and showed that at long wavelengths they have a phonon character, whileat short wavelengths they have a band character. These works unify for the firsttime the theory of superfluidity and solids. Such a prediction have been object oftheoretical controversies with notable works by Penrose and Onsager [146], who pointed
out that localization and superfluidity are competing orders. However, over the years,
the existence of supersolidity as a possible quantum state gathered momentum, withseveral theoretical works on this topic [6, 38, 111, 199]. Among them, Leggett suggested
an experimental measurement to determine the presence of superfluidity in solid He,
through the detection of a reduction of the moment of inertia with respect to theclassical value, below the critical temperature for superfluidity [111].
Experimental efforts in helium. Helium has always been considered a prime
candidate for observing supersolidity. This arises from its special quantum properties
with respect to other solids. In fact, its light mass and weak interatomic potential leads
to an exceptional high zero-point energy with respect to the depth of the molecularpotential wells for helium atoms, leading to an exchange of atoms via tunneling, whichmakes the Bose (4 He) or Fermi (3 He) statistics relevant even in the solid phase [16]. As
known, helium has two liquid phases, with a superfluid phase at temperatures below
the λ transition at about 2K. A fundamental property arising from the large zero-point
energy and that distinguishes liquid helium from other liquids, is the possibility ofreaching the solid phase only by applying a pressure greater than about 2MPa.
Following Leggett’s suggestion to determine the superfluid fraction, several experiments used a torsional oscillator to investigate the properties of solid helium. The
torsional oscillator consists of a cell filled with He connected to a base via a torsionalrod. The resonant frequency of this oscillator q
can be directly connected to the momentof inertia I through the relation fres = (1/2π) K/I, where K is the torsional stiffness.
At constant K, a reduction of the moment of inertia can be directly connected to anincrease of fres , suggesting a transition to a supersolid phase, with the superfluid part
decoupling from the rest of the oscillator [15].

1.1 Brief historical overview

9

In 2004, Kim and Chan performed a measurement of the resonant frequency andobserved an increase when lowering the temperature below 200mK [95, 96]. They
attributed this behaviour to a possible transition to a supersolid state. Several
experimental works tried to replicate Kim and Chan’s observation, with most of themshowing qualitative agreement. However, in 2007, Day and Beamish reported on theexperimental determination of the shear modulus of solid helium as a function of thetemperature [50]. The authors measured an increase of stiffness when lowering thetemperature below a critical value, which was consistent with the critical temperature
to observe the increase of the resonant frequency in the torsional-oscillator experiment.
This behaviour was confirmed by a later work [82] and it is attributed to the motionof crystal defects, which changes with temperature.
The temperature dependence of the shear modulus started to pose doubts on theobservation of supersolidity claimed by Kim and Chan, and in 2012 this claim wasdisproved, see Ref. [94]. It is now clear that the increase in the resonant frequency
is related to a change in the shear modulus µshear , and therefore a change in the
stiffness K, rather than to a variation in the superfluid fraction. This was confirmed
in 2016, when a simultaneous measurement of the torsional oscillation frequency andthe shear modulus showed that the two quantities exhibited identical dependences ontemperature, driving frequency, and 3 He impurities. A different behaviour with thedriving amplitude was observed, but this was confirmed to be induced by a change inthe elastic properties of solid 4 He [172].
Ultracold quantum gases. The fast development of new techniques for controllingand detecting ultracold atoms, makes these systems an ideal platform for investigating
many-body phenomena and overcome the issues related to imperfections and complexityof condensed matter systems. For these reasons, in the quest for supersolidity, a lotof theoretical effort moved in the direction of ultracold atoms. In 2010 and in 2017,
states with supersolid properties were observed in experiments with Bose-Einsteincondensates (BECs) coupled to light [14, 112, 114]. In particular, two different methods
were used. In one experiment, in 2010, rubidium atoms were coupled to the light fieldof an optical cavity and a phase transition to a modulated state was observed [14].
Some years later, the supersolid properties were confirmed in similar settings [112].
In a second experiment, in 2017, the supersolid properties were achieved by inducing
spin-orbit coupling in an ultracold spin mixture of sodium atoms [114].
In the meantime, ultracold systems with dipolar interaction started to draw great
attention in the community, because of the momentum dependence of the interactions.

10Supersolidity in dipolar quantum gases

Theoretical works predicted a supersolid phase for ultracold systems with a soft-corepotential, achievable in the experiment via Rydberg excitation [46, 83, 203]. This idea
followed the prediction by Andreev and Lifshitz [6], and Chester [38] of a solid phase
where particles can flow without friction due to the presence of zero-point defects.
A system interacts via a soft-core potential if the interatomic potential assumes aconstant value Vsoft > 0 for distances r smaller than the particle diameter and zerootherwise. This means that, as the density increases, it is more energetically favourable
for the system to cluster some particles together, leaving empty cells. In a lattice, thistranslates in the possibility of tunnelling between different lattice sites and defects
delocalization, favouring superfluid motion inside the crystal [47].
Other promising candidates for supersolidity were polar molecules and ultracoldgases with magnetic dipole-dipole interactions [120]. An important turning point in thequest of supersolidity in the ultracold realm was the observation of the roton mode andits softening with erbium atoms in experiments in Innsbruck [45]. Later on, the same
experiment demonstrated the existence of a roton-maxon excitation spectrum [150].
More breakthrough experiments with dysprosium experimentally proved the transition
to a density modulated state in a parameter regime where the mean-field theory predicts
a collapse [91]. However, the proof of global phase coherence had to wait until 2019,
when supersolidity in dipolar gases was simultaneously proved in three experiments,
including ours, by tuning the interactions [32, 42, 182]. Moreover, in our experiment,
we were able to prove direct evaporative cooling into a supersolid state and achieve
lifetime on the order of seconds [42, 177].

1.2

Supersolidity in dipolar gases: main ingredients

Supersolidity is a phase of matter whose properties challenge our imagination due
to the antithetical behaviours of solids and superfluids. There are different ways toaddress this challenge. One can consider a solid and ask the question "Can a solid besuperfluid?", such as for example the quest of supersolidity in solid He. For ultracold
dipolar gases, which are superfluids, the question is reversed and becomes "Can asuperfluid be solid?". The following section answers this question and focuses on
the main ingredients necessary to achieve a supersolid state with ultracold dipolargases, namely a source for the development of density modulation and a stabilization
mechanism.

1.2 Supersolidity in dipolar gases: main ingredients

1.2.1

11

Crystallization mechanism

The knowledge of the excitation spectrum is crucial to understand whether an external
perturbation with energy ϵ and momentum p can create an excitation in the system.
One of the most important theoretical predictions in the quest for supersolidity, isthe emergence of a minimum in the excitation spectrum at large momenta, which isconsidered a precursor for crystallization [97, 157]. In superfluid He, the existenceof this minimum was originally predicted by Landau [108], it relates to its nature ofstrongly correlated liquid, and it was thought to be linked to a local vorticity. In
Ref [138], Nozières addresses the roton in a different way, as a soft mode, and states:“a superfluid close to a charged density-wave instability has a dip in its excitation
spectrum near the incipient Bragg spot.”.
This minimum is usually not present in ultracold bosonic gases, whose isotropic
and short-range interactions1 are generally weak. In these systems, by following the
Bogoliubov approach, the excited states can be described in terms of non-interacting
quasi-particles of energy ϵ(p), which follows the Bogoliubov dispersion relation. For
small momenta p, the dispersion has a linear phonon-like behaviour (∝ p), while,for larger momenta, it follows a free-particle behaviour (∝ q
p2 ). The transition point
between the two regimes is given by the healing length ξ = h̄2 /2mgn [153].
In 2003, two important theoretical proposals appeared in the ultracold-gas community. In Ref. [139], O’Dell et al. predicted the formation of a roton minimum at
finite momenta in the excitation spectrum, for a BEC with long-range interactions
induced by illuminating the atomic cloud with far off-resonant laser light. The positionof the roton minimum and the roton energy gap are tunable by changing the radialwidth of the BEC, and the intensity and the wavelength of the laser light. In Ref. [168],
Santos et al. predicted that for an ultracold dipolar gas, when the contribution of thelong-range and anisotropic dipolar interaction is dominant over the contact interaction,
the spectrum of excitations develops a roton minimum at finite momenta krot with anenergy gap ∆rot . In this work, the authors considered an infinite pancake trap withthe dipoles aligned along the tight confinement.
This behaviour relates to the momentum dependence of the DDI, which in real
space has the following form:
Udd (r) =i
4π h̄2 add h
1 − 3(r̂ · µ̂m )2 ,m

1We refere to those interactions as contact interactions.

(1.1)

12

Supersolidity in dipolar quantum gases

where m is the atomic mass, add = µ0 µ2m m/12πh̄2 is the dipolar length2 , and µm is themagnetic dipole moment (see Fig. 1.1a). For atoms confined in traps, if the dipoles
are aligned along the tight direction of the trap, the total mean-field interaction has
a negative contribution at large momenta, causing the dispersion relation to soften.
Figure 1.1b displays the effective 1D mean-field interaction Ũ , from Ref. [141], which
includes contact interactions and DDI and it is obtained from a variational theory; see
Refs [23, 24]. The effective interaction Ũ is shown as a function of the momentum k,
for 164 Dy atoms and for two values of the scattering length, as = (115, 140) a0 . For
as > add , Ũ becomes negative for k ≳ 1/lz . The appearance of a roton minimum atlarge momenta can be also understood by looking at the excitations. Figure 1.1cshows the case of an infinite cigar-shaped harmonic trap with the dipoles oriented
along the direction othogonal to the tight confinement, of size lz . For small momenta
k << 1/lz , and therefore long wavelengths, the excitations have a two-dimensional
character and the dipoles, which sit side-by-side, mainly repel each other. Whereas,
for large momenta k >> 1/lz , and therefore short wavelengths, the excitations havea three-dimensional character and the dipoles mainly sit head-to-tail attracting eachother and reducing the interparticle interaction.
The energy gap ∆rot depends on the density and on the interactions. For example,
by reducing the scattering length as via Feshbach resonaces, it is possible to decrease∆rot until, when ∆rot becomes imaginary, in the mean-field picture, all the populationis transferred in ±krot . At this point, it is energetically free for the system to develop
a density modulation with periodicity ∼ 2π/krot . Experimentally, the roton mode
was first observed in erbium via a softening of the excitation spectrum [45] and few
years later the same experiment measured the full spectrum of excitations via Bragg
spectroscopy and observed the appearance of a roton minimum when increasing the
dipolar strength [150]. More recent experiments with Dy have inferred the spectrumof excitations by looking at the static structure factor determined from insitu density
fluctuations in a 1D cigar-shaped trap [85] and later in a oblate trap [170]. The
existence of a roton minimum at large momenta lays the basis for the first ingredientnecessary for a supersolid state: a crystallization mechanism.

1.2.2

Stabilization mechanism

In a dipolar gas, when reducing the strength of the repulsive contact interaction as
via Feshbach resonances, below a certain threshold, the mean-field theory predicts a2a
164 Dy.
dd = 130 a0 for13

1.2 Supersolidity in dipolar gases: main ingredientsa. repulsive

attractive c.

a

/2

x

ñ (a.u.)

z

1.0

0.8
0.6

y

0.4

kx

0.2

/4

b

0

0

attraction
repulsionkydEnergy, ‫(׻‬k)

3 /4

c

as

kx

krot

Momentum, ky

7 /4

5 /4

b1

ky

b2
z

3 /2
y

dominantly repulsive DDIb.

dominantly attractive DDI

1.5

1

0.5

0

-0.5

0

1

2

3

4

5

6

Figure 1.1 (a) DDI at fixed distance r as a function of the angle θ. (b) Effective
mean-field interactions as a function of the momentum k, for 164 Dy atoms, where
add = 130 a0 , and for two values of the scattering length: as = 140 a0 (black), andas = 115 a0 (orange). For as = 115 a0 , as < add and the effective interaction becomes
negative at momenta approximately larger than 1/lz (grey dashed line). Parameters
and analytic formula taken from Ref. [141]. (c) Excitation spectrum for a dipolar BEC
in an infinite cigar-shaped trap. By decreasing the scattering length the spectrum
develops a roton minimum at a finite momentum krot . This can be understood bylooking at the excitations, where a change of the DDI from mainly repulsive at small
momenta to mainly attractive at large momenta happens. Figure adapted from [45].
collapse. This is due to the attractive contribution of the DDI. First studies of the14

Supersolidity in dipolar quantum gases

dipolar collapse were performed with chromium [107, 127]. Here, the collapse showeda pattern given by the d-wave anisotropy of the DDI. However, precursor experimentswith Dy showed that the system, instead of collapsing, splits into multiple peaks,
which are known as quantum droplets [91], leading to a density modulated state. The
behaviour of these quantum droplets is reminiscent of ordinary liquid droplets. For
high enough atom numbers, they show a self-bound nature and the system stays boundeven without any trapping confinement [40, 171].
This observation triggered a lot of theoretical effort in understanding the origin of
the stabilization mechanism, with a three-body conservative repulsion3 [21, 198] andquantum fluctuations as possible candidates. The first experimental work aimed at
elucidating the stabilization mechanism was performed with Dy and pointed at thequantum fluctuations [61]. Few months later, experiments with Er confirmed that thetransition to a droplet state was driven by the quantum fluctuations. Furthermore,thanks to the unique control and knowledge of the scattering length, the erbium
experiments revealed the role of quantum fluctuations on the system’s properties, suchas e.g. atom losses, expansion dynamics, and collective modes [41]. These experimental
observations were promptly confirmed by theoretical works [22, 189, 190]. The beyond
mean-field effects have not been seen in chromium due to the weak dipolar length(15 a0 ). In fact, in chromium, the losses due to three-body recombinations, overcome
the beyond mean-field effects, which can be neglected [43].
The effect of quantum fluctuations can be well described by the first-order correction
to the mean-field energy, namely the LHY term. A more detailed description of thisterm will be given in the next section (Sec. 1.3). When the repulsive contact interactions
and the attractive dipole-dipole interactions almost balance each other, the beyond
mean-field term, which is positive and scales with a higher power of the densitywith respect to the mean-field energy, becomes relevant preventing the system fromcollapsing. The quantum fluctuations constitute the second ingredient necessary for a
supersolid state: a stabilization mechanism.

1.3

Many-body interactions

In standard conditions in which the overall interactions are repulsive, ultracold dipolargases are generally well pictured by mean-field theory, where the field operator Ψ̂ =3 Earlier on, Ref. [120] proposed three-body interactions as stabilization mechanism for a supersolidstates in two-dimensional dipolar bosons.

15

1.3 Many-body interactions

ψ(r, t) + δψ can be safely replaced with its mean value ψ(r, t). This section briefly
recalls the basic steps necessary to derive the Gross-Pitaevskii equation (GPE) [153].
The time evolution of the field operator Ψ̂ follows the Heisenberg equation:i∂ Ψ̂(r, t) hh̄2 ∇2i h̄= Ψ̂(r, t), Ĥ = −+ V (r)
∂t
2m
"

+

Z

†

(1.2)

#

′

dr Ψ̂ (r, t)U (r − r )Ψ̂(r, t) Ψ̂(r, t),where the Hamiltonian operator Ĥ includes the one-particle operators describing
kinetic and trapping potential, and the two-particle operator describing the two-bodyinteraction.
Under the assumption of zero temperature and that most of the atoms occupy
the ground state of the system, the effect of quantum fluctuations, as the quantum
depletion or a shift of the ground state energy, can be usually neglected. This means
ignoring the term δψ. Furthermore, the real interatomic potential can be replaced by asoft pseudopotential, which gives the same scattering length as as the exact one. The
system’s behaviour can be then described through the GPE:Zh̄2 ∇2∂ψ(r, t)= −+ V (r) + dr′ U (r − r′ )n(r′ ) ψ(r, t),
i h̄
∂t
2m
"

#

(1.3)

with n(r′ ) = |ψ(r′ , t)|2 . Equation 1.3 includes the kinetic term, the trapping potential
V and the mean-field interaction U . For what concern the phenomena investigated inthis thesis, the trapping term V is described by a harmonic potential with trapping
frequencies (ωx , ωy , ωz ):
X
1
V = m
ω 2 x2 .
2 xi =x,y,z xi i

(1.4)

The mean-field interaction term U includes the contact interaction, characterized by
the s-wave scattering length as , and the long-range dipole-dipole interaction, as follows:
3ϵdd 1 − 3 cos2 θU (r) = g δ(r) +
,
4π
|r|3
!

(1.5)

where g = 4π h̄2 as /m is the coupling constant and ϵdd = add /as is the relative
dipolar strength. The angle θ is the angle between the dipoles orientation and theinterparticle separation vector r.

16Supersolidity in dipolar quantum gases

The beyond-mean field term δψ can become relevant when energy terms with
opposite sign compete, enahancing the effect of quantum fluctuations in the system.
The important role of quantum fluctuations in stabilizing the system against thecollapse was originally proposed by Petrov for Bose-Bose mixtures [149], where beyond
mean-field effects can be enhanced by tuning the competition between intra- and interspecies interactions close to the mean-field instability condition4 . For dipolar gases,
when as is below a critical value, the repulsive contact interaction can not overcome
the attractive dipole-dipole interaction, inducing the collapse at the mean-field level.
Note that the exact critical scattering length depends on the trap geometry.
a.

b. 8

1

0.8

6

0.6
4
0.4

2

0.2
0.5

1

1.5

2

0.5

1

1.5

2

Figure 1.2 (a) Beyond mean-field quantum fluctuation term γQF of Eq. 1.6 as a functionof the dipolar strength with fixed add , and (b) Re(Q5 ) of Eq. 1.7 as a function of thedipolar strength ϵdd .
The effect of quantum fluctuations can be described by the first-order correction
to the mean-field energy, known as Lee-Huang-Yang (LHY) correction. This term,
originally formulated for weakly interacting ultracold Bose gases [110], was thenextended to ultracold dipolar gases [115, 116] and has the expression:
128 h̄2 q 5
πas Re{Q5 (ϵdd )},3m
where Q5 is a monotonic function of ϵdd and reads as below:
γQF =Q5 (ϵdd ) =Z 1
0(1 − ϵdd + 3ϵdd u2 )5/2 du.

(1.6)

(1.7)

4 In Bose-Bose mixtures, the instability happens when the interspecies attraction overcomes thegeometrical average of the intraspecies repulsion.

17

1.3 Many-body interactions

Figure 1.2 displays γQF and Re(Q5 ) as a function of ϵdd . By including the first-ordercorrection, the GPE takes the form:Z∂ψ(r, t)h̄2 ∇2i h̄= −+ V (r) + dr′ U (r − r′ )n(r′ )
∂t
2m
"

3

(1.8)

#

+ γQF |ψ(r, t)| ψ(r, t).

z
B

y
x
10−4

10−3

10−2

10−1

1

L

30
25
ID

N (103)

20

SSP

15
10

ID

SSP

BEC

5
BEC

80

85

90
as (a0 )95

100

Figure 1.3 Ground-state phase diagram for a 164 Dy gas confined in a cigar-shaped trapwith trapping frequencies ωx,y,z = 2π × (229, 37, 135)s−1 . At high as , the system is in
the standard dBEC phase (grey region). At low as , the ground state consists of a chainof quantum droplets with independent phases (blue region). In a narrow range of as ,where the link L is significant, the ground state is a supersolid (red region) showing
density modulation and global phase coherence. Figure adapted from [89].
So far only two-body collisions are taken into account but, for high densities, theapproximation of considering only two-body interactions breaks down and inelastic
three-body collisions can have a big impact on the system. Three-body losses can be
included in Eq. 1.8, by adding the following term:

18Supersolidity in dipolar quantum gasesi h̄
L3 n2 (r)ψ(r, t),(1.9)
2
where L3 is the three-body collision coefficient.
By minimizing the energy functional of Eq. 1.8, one can find the ground-state of thesystem and develop a phase diagram as a function of atom number and scattering length.
Figure 1.3 shows the phase diagram for 164 Dy in a cigar-shaped trap with the magnetic
field aligned along the tight axis. From the phase diagram the following phases canbe identified. At high scattering lengths, the ground-state of the system is a standard
dipolar BEC. By reducing the scattering length, the system enters the supersolid regime(SSP), in which a density modulation develops in form of quantum droplets linked
to each other by a superfluid background, allowing global phase coherence. Whenlowering the scattering length further, the link between the droplets decreases until it
becomes neglegible and the system enters the so-called isolated-droplet regime (ID)and the global phase coherence is lost. The link strength is encoded in the color map
and it is defined as L = 1 − (nmax − nmin )/(nmax + nmin ), where nmax and nmin are
the density maximum and minimum in the center region of the density distribution,
respectively. The experimental tools to realize a long-lived linear and two-dimensional
supersolid are described in more details in Chapter 4.
−

1.3.1

Thermal fluctuations

So far, this section underlined the important role of quantum fluctuations in stabilizing
a dipolar gas and avoiding the collapse when the dipolar strength ϵdd ≳ 1. The GPE in
Eq. 1.8, which includes the beyond mean-field term describing the quantum fluctuations,assumes a dipolar quantum gas at zero temperature. However, in experiments, thetemperature of the system is finite and the condensed cloud usually coexists with athermal component, which can be a non-negligible fraction of the total atoms, leading
to a considerable interaction between condensed atoms and thermal excitations. In
this case, the fluctuation term δψ includes quantum as well as thermal fluctuations. In
this regard, later on in this thesis, we demonstrate how not only quantum fluctuations,but also thermal fluctuations have a determinant role in the supersolid formation; see
Chapter 4.
In fact, in dipolar gases, finite temperatures can have an important effect on theproperties of the system. For instance, theoretical works reported on the role of thermal
fluctuations in stabilizing quantum droplets and changing their density profiles [7],even at temperatures as low as 100 nK. Moreover, our works reported in Sec. 4.4 and1.3 Many-body interactions

19

Sec. 4.9 show how thermal fluctuations shift the transition to a supersolid state towards
lower dipolar strengths, favouring the appearance of density modulation [180]. In thelatter, a temperature-dependent GPE, obtained by minimizing the grand-canonicalpotential with respect to the wave function ψ(r) [67], well captures the phase diagramof our dysprosium cloud at finite temperature. The total grand potential includes
the grand-canonical zero-temperature energy and the finite-temperature energy term
resulting from Bogoliubov theory. By minimizing this potential, one obtains the wave
function in thermal equilibrium. For more details, see Ref. [180].
The theory mentioned above is valid for a high fraction of condensed atoms andrepresents properties of the system at thermal equilibrium. Therefore, for describing
dynamical processes, such as the supersolid formation, a more appropriate theory is thestochastic GPE (SGPE) formalism [25, 117], whose validity is in a broader temperature
range and does not require the major occupation of the lowest energy state, as when
applying Bogoliubov theory. In the publication shown in Sec. 4.7, the SGPE with thebeyond mean-field quantum fluctuations is used to describe the process developed inour group to experimentally achieve a supersolid state by direct evaporative coolingfrom a thermal cloud [27].

Chapter 2
Er-Dy experiment
The previous chapter showed how the long-range dipole-dipole interactions can lead tonovel quantum phases of matter. Among neutral atoms, long-range and anisotropic
DDI can be achieved in several ways. The systems showing the strongest DDI are
Rydberg atoms with ϵdd larger than 106 . Hetero-nuclear molecules follow with ϵdd ∼ 102 .
Magnetic lanthanides, such as erbium and dysprosium, with their not completely filled
submerged f shell, present a permanent magnetic moment which can be 10 times larger
than alkali, giving rise to an ϵdd ∼ 1. Long-range interactions can also be induced
through atom-light coupling. To date four elements from the magnetic lanthanides
have been Bose condensed: dysprosium [119] (µm = 10µB ), erbium [3] (µm = 7µB ),thulium [49] (µm = 4µB ), and very recently europium [129] (µm = 7µB ). Althoughmagnetic atoms have an ϵdd much smaller than the Rydberg atoms and hetero-nuclear
molecules, they feature several interesting properties, such as a rich manifold of internal
states, many electronic transitions with various linewidths, and the presence of two
main competing interactions (contact interaction and DDI), which are independently
tunable. This chapter briefly recalls the basic properties of erbium and dysprosium.
It then describes the experimental apparatus and the protocol used to create erbiumdysprosium ultracold gases.

2.1

Basic properties

Erbium and dysprosium are part of the magnetic lanthanides with a permanent
magnetic moment of 7µB and 10µB , respectively. Dysprosium was the first to be
Bose-condensed in 2011 [119], with erbium following one year later [3]. Few years after,
degenerate Fermi gases of dysprosium [118] and erbium [2] were realized via universal
dipolar scattering. Since then a lot of theoretical and experimental works brought to the22

Er-Dy experimentdiscovery of novel phenomena arising from the long-range and anisotropic dipole-dipole
interaction [43, 135]. Just recently the family of Bose-condensed magnetic lanthanides
grew with the addition of thulium [49] and europium [129].
The large permanent magnetic moment arises from the electronic configuration,[Xe]4f m 6s2 , where m=10 and 12 for dysprosium and erbium, respectively, which ischaracterized by an unfilled 4f shell surrounded by a completely filled s shell. The
unpaired electrons in the f shell leads to a large angular momentum quantum number
L, which has a big impact on the scattering properties (see discussion in Sec. 3.2) andon the atomic polarizability (see Sec. 3.3.3). The submerged shell also leads to a richenergy spectrum with various electronic transitions with linewidths ranging from µHz
to hundreds of MHz [10, 59].
In our experiment, the broad transitions at 401 nm and 421 nm, with a linewidth of
Γ401 /2π = 29.4 MHz and Γ421 /2π = 32.2 MHz, for erbium and dysprosium, respectively,
are used for a first stage of laser cooling and for imaging. The intercombination lines at
583 nm and 626 nm with linewidths of Γ583 /2π = 186 kHz and Γ626 /2π = 135 kHz areused for the Magneto-Optical trap (MOT) for erbium and dysprosium, respectively. The
narrow-line transitions at 741 nm and 841 nm with linewidths of Γ741 /2π = 2 kHz andΓ841 /2π = 8 kHz can be used for an additional stage of laser cooling, as done for example
in the group of M. Greiner at Harvard [152]. The Hz transition at 1299 nm, recently
measured in erbium [145], and the 1001 nm in dysprosium [147] can be a powerfultool for optical manipulation and control of the atoms. Two additional narrow-line
transitions are at 631 nm and 598 nm for erbium and dysprosium, respectively andtheir calculated linewidths are Γ631 /2π = 28 kHz and Γ598 /2π = 12 kHz. The openshell nature of magnetic lanthanides makes, on the one hand, the study of scattering
properties and atom-light interaction rather complex but, on the other hand, leads torich energy spectra, which offer many possibilities for manipulating the atoms. The
next section describes the experimental apparatus and the electronic transitions usedduring the experimental sequence to produce ultracold gases of erbium and dysprosium.

2.2

Experimental apparatus

A detailed description of the experimental apparatus can be found in Ref. [88]. In brief,the experimental apparatus is designed to produce degenerate quantum gases of erbium
and dysprosium, both in single and double-species operation. Figure 2.1 displays theexperimental apparatus. It is composed by three main modules: the atomic beamsource, the atomic beam shutter and the main chamber.

23

2.2 Experimental apparatus

dual-filament
effusion cellmain chamber

Zeeman slower

atomic beam
shuttertransversal cooling

Figure 2.1 Experimental apparatus for the creation of erbium and dysprosium quantum
gases. An effusion cell generates the atomic beam, which travels through several stepsof collimations (apertures and 2D transversal cooling). After crossing the atomic-beam
shutter section, the atoms are slowed down by a Zeeman slower and captured in themain chamber by a two-species five-beam MOT. The atoms are transferred in a crossed
optical dipole trap, where quantum degeneracy is reached through evaporative cooling.
Credits: Philipp Ilzhöfer
The atomic beam source consists of a dual-filament effusion cell1 and a transversal
cooling (TC) section. An ion pump keeps the whole module at a pressure of 10−10 mbar.
The effusion cell includes a tantalum crucible with two units. One unit contains the
erbium and dysprosium material, cut in small pieces, and its operating temperature isusually set to 1100 °C2 . The second unit is set to a higher temperature3 in order tocreate a temperature gradient and avoid condensation of material inside the crucible.
This unit is partially filled with only erbium material in order to compensate for the
lower vapor pressure of erbium. In total, the first unit is filled with about 12 g, 9 g of
Dy and 3 g of Er4 , whereas the second unit is filled with approximately 2 g of Er. Three
apertures inside the effusion-cell section provide a collimated atomic beam source. The
atomic beam is further collimated in the TC units, where two orthogonal laser beams
cool the atomic beam along the transversal directions.
The second module consists of an atomic beam shutter made of an in-vacuum
stainless steel plate that is controlled by a servo and, after the loading of the Magneto1 Createc Fischer & Co. GmbH, DFC-40-10-WK-2B.

2A detailed description of the procedure followed to refill the effusion cell can be found in Ref. [65].
3The standard operating temperature is 1200 °C.

4The larger amount of Dy in the first unit compensates for the lower melting temperature.

24

Er-Dy experiment

Optical-Trap (MOT), prevents the atomic beam from reaching the next module. In
addition, two more pumps complete the module: a titanium sublimation pump, which
is usually flashed only after a replacement of the effusion cell, and an ion pump, whichallows pressure in the whole module down to 10−11 mbar.
The final module consists of a spin-flip Zeeman slower (ZS)5 . The ZS slows down
the erbium and dysprosium atoms, which are finally captured by a MOT in the mainchamber. Another ion pump keeps the pressure at 10−11 mbar.
The MOT operates on the intercombination lines at 583 nm and 626 nm in erbium
and dysprosium, respectively. The narrow-line character of the optical transition andthe effect of gravity lead to a smile-shaped MOT, which sits below the zero of themagnetic quadrupole field [64, 92]. Therefore, for large enough detunings, the atoms
mainly absorb light from the σ − -polarized beam coming from the bottom. As a result,the atoms are spin-polarized in the lowest Zeeman sublevel and the top MOT beam canbe removed to leave space for a high resolution objective. Details about the operationof our two-species five-beam MOT can be found in Refs. [87, 88]. The atoms are
typically loaded in the MOT for 5 s. A compression phase follows by decreasing themagnetic field gradient, and the laser detuning and power, leading to typical atom
numbers and temperatures of ∼ 108 and ∼ 10 µK. At this step, the atoms are loaded
into an optical dipole trap (ODT) made from a laser beam operating at 1064 nm to
evaporatively cool the atomic clouds down to quantum degeneracy.
The power of this beam is controlled by an acousto-optic deflector (AOD)6 , whose
central frequency is modulated with an arccos function. For fast enough modulation
frequencies7 , the atoms see a time-averaged potential, whose shape is the one of a
Gaussian beam with a horizontal waist that can be actively tuned to maximize theoverlap with the MOT volume. Moreover, this tunability allows us to vary the trapping
geometry from a cigar-shape to a pancake trap, a fundamental tool for the transition
from one to two-dimensional supersolidity (see publications in Sec. 4.5 and Sec. 4.7).
The aspect ratio (AR) of the trapping beam can be tuned from 1 to a maximum of
10. We named this optical dipole trap scanning ODT. An important element, that
influences the performance of the scanning system, is the modulation bandwidth of
5 In such a configuration, the total magnetic field changes sign over the ZS length. In particular,
first it decreases, reaches zero and then increases as a function of the distance from the first coil (seeRef. [88]).
6 Gooch & Housego AODF, 4075-2, center frequency 75 MHz, bandwidth 32 MHz.
7The modulation frequencies need to be much higher than the trap frequencies.

2.2 Experimental apparatus

25

the voltage-controlled oscillator (VCO)8 used to set the frequency of the AOD. Indeed,
the modulation bandwidth affects both the maximum width and the shape of theradio-frequency signal. More details can be found in Refs. [9, 156].
After few seconds, two more ODTs propagating horizontally are switched on inorder to reach the optimal density condition for elastic scattering. In contrast to thescanning ODT, the waists of these beams are fixed. We named these optical dipole
traps static 1 and static 2 ODTs. The evaporative cooling last for about 5 s to reachthe degenerate quantum gas regime. In 2018, our experiment, produced double BECsof erbium and dysprosium atoms with five different bosonic isotope combinations, aswell as a Bose-Fermi mixture9 . Typical total atom numbers are about 105 with a BEC
fraction above 50%.

2.2.1

Optical dipole traps

The evaporative cooling of erbium and dysprosium clouds takes place in a crossed
optical dipole trap made by three laser beams at λ = 1064 nm. The three beams comefrom a commercial master oscillator power amplifier (MOPA)10 with a maximum
output power of 55 W.
Figure 2.2 shows the optical setup used to distribute the power from the laser headto the three optical dipole traps. From the laser head the beam goes through a telescope
before it is divided into three paths, which are coupled into three Large-Mode-Area(LMA) photonic crystal fibers11 , one for each ODT. The fiber holders12 consist of
two parts: a fiber collimator with a multi-element focusing lens13 and a water-cooled
copper block which cools down the fiber connector. Before splitting the laser beam, aniris masks spurious modes from the main peak, which would not be coupled into thefibers. NTC thermistors and photodiodes interface with an Arduino® microcontroller
to monitor the temperature of the fibers and the fiber coupling, respectively. The
power is distributed in the following way: about 22 W are sent to the scanning ODTpath and approximately 10 W are sent to each of the static ODTs.
8The VCO implemented in the experiment is the model MiniCircuits ZX95-100 with a 3dB

modulation bandwidth of 180 kHz. This will be exchanged with the fast VCO DRFA10Y - B – 0 50.110 from AAOptoElectronic, with a modulation bandwidth higher than 1 MHz.
9At this early stage of the experiment, the crossed ODT was made by one horizontal beam and a
vertical beam propagating along gravity. This setting is also the one used for the publication shownin Sec. 3.5.
10 Coherent Inc., Mephisto MOPA 55 W.
11 NKT Photonics LMA-PM-15.
12 Home-built inspired by the Greiner’s lab in Harvard.
13 OptoSigma HFTLSQ-20-30PF1.
14Thorlabs AC254-f -C, with f the focal length of the lens.

26

Er-Dy experiment

MirrorPolarizing beam
splitter cube

scanning ODTstatic ODT 2

static ODT 1

f f

Lens with focal
length f in mm

λ / 2 -plate
Iris
Mephisto 1064 nm 55 W
PD
Fiber incoupler
with cooling plate

PD
Optical isolator
Beam dump
100-50

Figure 2.2 Optical setup used to distribute the power from a MOPA, operating at
1064 nm with a maximum output power of 55 W, to the three optical dipole traps used
to trap the atomic clouds. A first telescope collimates and shapes the laser beam for an
optimal fiber coupling into the LMA photonic crystal fibers. An iris follows to get rid of
spurious modes from the main peak. Photodiodes (PDs) and NTC thermistors placedon the fiber connector monitor the incoming power and the temperature, respectively.
Figure 2.3 shows the scanning-ODT laser path to the main chamber. From theoutcoupler, the laser beam goes through an AOD, which modulates and intensity
stabilizes the beam. Four lenses are used to shape the laser beam before being focused
onto the atoms. In particular, f1 converts the diffraction into a parallel displacementof the beam. The distance between f3 and f4 is set by the position of two mirrors,
which can be tuned by moving a linear stage15 . This leads to a shift of the focus at thechamber without affecting the waist size (see Fig. 2.4). Finally, the last lens f5 focuses
the light onto the atoms with a final waist of approximately 18 µm. At the position of
the f4 lens, the beam is close to the full aperture of the lens. This makes the alignment
quite challenging and leads, in some cases, to an astigmatic beam at the focus position
onto the atoms. For this reason, the lenses f3 and f4 are mounted on kinematic mirror
mounts. Furthermore, the five lenses need to be in a 10f imaging system configuration.
Deviations from this setting lead to a change in the focus position when varying themodulation amplitude of the AOD, i.e. the AR of the scanning ODT beam. Figure 2.4
shows calculations of the beam waist as a function of the propagation distance fromthe AOD for three different positions of the f4 lens. By translating the lens by ±5 mm
15 OWIS GmbH MT 60.

27

2.2 Experimental apparatus
y
x

f f

Lens with focal
length f in mm
Mirrorz

Polarizing beam
splitter cube
scanning ODT

f1
100
75MHz

Fiber
outcoupler

AOD

f2

λ / 2 -plate

200

f3
f5
300

50

f4
250
Linear stage

Figure 2.3 Optical setup for the scanning-ODT path next to the main chamber. The
AOD modulates and intensity stabilizes the beam. The lenses used in this setup are
achromatic doublets14 . A linear stage shifts two mirrors, which translates in changing
the distance between f3 and f4 and therefore moving the focus at the atoms without
affecting the final waist size. Figure adapted from [88].
the focus moves by ±7 mm without variations in its size. Furthermore, modulating thecentral frequency of the AOD does not affect the focus position and the beam size.
Figure 2.5 shows the static-ODTs laser paths to the main chamber. From theoutcoupler, the laser beam goes through an acousto-optic modulator (AOM)16 , which
intensity stabilizes the beam. Similarly to the scanning ODT, two lenses shape thelaser beam before focusing the light onto the atoms. In the same way, the beam waist
can be translated by changing the distance between f1 and f2 , which is done by moving
two mirrors on a linear stage. For the two static-ODTs, the final waist is about 60 µm.
Note that the AOD and the AOMs used for the optical dipole traps are mounted on awater-cooled block in order to improve the beam-pointing stability.

2.2.2

Imaging systems

The erbium-dysprosium experiment has two imaging systems. One propagates horizontally at 45° with respect to the y direction, overlapping with the static-ODT 1. Thisimaging system is described in details in Ref [88] and is used to perform absorption
imaging of the atomic cloud after Time-of-Flight (TOF) expansion. The pixel size
16 Gooch & Housego I-M080-2C10G-4-AM3.

28

Er-Dy experiment

Scanning distance (µm)

Beam waist (mm)

100

50

200

250

300

6

0.06

4

0.04

2

0.02

0

0

200
50

100
0

0

-100

-50

-200
0

500

1000

1500

2000

1790

1800

1810

Propagation distance (mm)Figure 2.4 Beam waist as a function of the propagation distance from the AOD (upperrow) and horizontal displacement of the laser beam for a modulation frequency of
10 MHz (bottom row). When changing the distance between f3 and f4 by ±5 mm thebeam waist moves by ±7 mm without variations in its size. The waist size and position
do not change with the scanning amplitude.
of the camera at the position of the atoms is approximately 2.1 µm. This horizontal
imaging beam is used in the experiment to get information on the atom number andtemperature. In particular, we record the momentum distribution after TOF, whichshows, for an unmodulated BEC, the standard profile with a broad contribution givenby the thermal component and a narrower peak given by the condensed atoms. We fit
a bimodal distribution to the 1D profile and extract atom number and temperature.
For a supersolid state, due to the presence of the density modulation resulting in aseries of high-density peaks, a different analysis is performed (see e.g. the publications
shown in Sec. 4.4 and in Sec. 4.5).
The second imaging system allows both insitu and TOF imaging. The laser beam
propagates vertically along the z direction and consists of an objective with a maximum
numerical aperture (NA) of 0.45 and a measured resolution of about 700 nm, at 401 nm
and 421 nm. The pixel size of the camera at the position of the atoms is approximately
0.4 µm. A detailed description can be found in Ref. [176]. To get information on
the insitu density distribution, we perform phase-contrast imaging [34, 193] with29

2.2 Experimental apparatus
f1

y

AOM
100

x
z

f3

f f

f2 80MHz

Lens with focal
length f in mm
Mirror500

Polarizing beam
splitter cube

static ODT 1

Linear stage50

λ / 2 -plate

0

Fiber
outcoupler

static ODT 2

Linear stage

f1
100

50
0

80MHz

AOM

f3
f2
500

Figure 2.5 Optical setup for the two static-ODTs next to the main chamber. The
AOMs intensity stabilize the laser beams. The RF frequency that drives the AOM
for the staticODT 1 is shifted by few MHz from the central frequency to not interferewith the scanning ODT when the modulation is on. The lenses used in this setup are
achromatic doublets. A linear stage shifts two mirrors, which translates in changing
the distance between f1 and f2 and therefore moving the focus without affecting thefinal waist size. Figure adapted from [88].
far-detuned light17 and linearly polarized light. Whereas, to get information on the
momentum distribution, we perform standard absorption imaging after TOF expansion
with resonant light and circular polarization. A stepper motor allows to move theobjective vertically and switch from insitu and TOF imaging. This imaging system isfundamental to probe density modulation and global phase-coherence of a supersolidstate (see Chapter 4).

2.2.3

Quantum gas microscope

In July 2021 a new element was added to the experimental apparatus, namely a glass
cell for quantum gas microscopy of erbium and dysprosium atoms. The quantum gas
miscroscope consists of an octogonal glass cell18 and an in-vacuum objective19 witha maximum NA of approximately 0.89 and a measured resolution of about 300 nm,
at 401 nm and 421 nm. The details on the design of the glass cell and the objective,
together with its performances can be found in Ref. [176]. The quantum gas microscope
17The detuning is set to ∆ ∼ 36Γ.

18 Manifactured by Precision Glassblowing of Colorado, Inc.
19 Manifactured by Special Optics, Inc.

30

Er-Dy experimentis a powerful tool that will open many research directions based on the investigation of
new quantum phases in optical lattices arising from the long-range interaction.

Chapter 3
Interspecies interaction in an
ultracold dipolar mixture
The previous chapter described the features of our experiment and how erbiumdysprosium ultracold mixtures can be created with different isotope combinations.
This chapter focuses on the first study of the interspecies interactions and the determination of the interspecies scattering length a12 , done by tracking the in-trap clouds
displacement generated by the mean-field interspecies interactions. In particular, thefirst section gives a general overview of hetero-nuclear mixtures. The second section
introduces the methods that are generally used in ultracold experiments to determine
a12 . The third section introduces the technique developed in our experiment andgives details on the experimental sequence and on the theoretical model based on
the Gross-Pitaevskii equation with the energy terms arising from the interspecies
interactions. Finally, the last section is dedicated to the numerous possibilities thathetero-nuclear dipolar mixtures offer for studying new quantum phases.

3.1

Hetero-nuclear mixtures: an overview

In the past decades, ultracold homo- and hetero-nuclear mixtures have been realized
all around the world driven by multifold reasons. One of the first ultracold mixtures
was realized by merging two fermionic spin states of potassium atoms [53] in order toovercome the limitation given by identical fermions in reaching quantum degeneracy.
Indeed, reaching quantum degeneracy relies on direct evaporative cooling and therefore
on elastic collisions. In the regime of low temperatures, the main contribution to
the scattering cross section is given by s-wave states. However, for identical fermions
with short-range interactions, the requirement of an antisymmetric wave function

32Interspecies interaction in an ultracold dipolar mixture

prohibits s-wave scattering. Therefore only distinguishable fermions can interact atthese low temperatures. This changes for dipolar fermions interacting via long-range
and anisotropic dipole-dipole interactions, where not only s-waves but all partial
waves contribute to the scattering cross section [2, 12]. For a similar purpose, BoseFermi mixtures have been realized to use the bosonic species as a thermal bath tosympathetically cool fermions [78, 188]. Furthermore, quantum mixtures can be used
to study polaron physics, where impurity atoms are immersed in a bosonic or fermionic
quantum degenerate bath [86, 90]. Long-range attractive mediated interactions havebeen studied in Bose-Fermi mixtures, where a trapped BEC is surrounded by adegenerate Fermi gas, leading to an instability of the condensate and, in a quasi 1D
trap, to a transition to a Bose-Fermi soliton train [55]. Another important driving
motivation for studying hetero-nuclear mixtures is the realization of ultracold molecules,
which possess a large electric dipole moment [128]. Finally, by tuning the interspecies
interaction to negative values a stable mixture of BECs can perform a phase transition
to quantum droplets [37, 54, 149].
All the hetero-nuclear mixtures mentioned so far have been realized by combining
alkali-metal atoms, with the advantage to make use of the wide knowledge developed
over many years of research on single-species experiments. Ultracold experiments havealso combined alkali-metal with alkaline-earth-metal atoms [143, 202] and alkali-metal
with alkaline-earth-like atoms [80, 81, 194] motivated by the possibility of realizing
molecules having both magnetic and electric dipole moments. In the last few years, thefamily of hetero-nuclear mixtures has grown including also dipolar atoms. In particular,in our experiment, the first double BECs of magnetic atoms were realized with several
isotope combinations of erbium and dysprosium [187]. In the same year, a degenerate
Fermi-Fermi mixture with dysprosium and potassium atoms was realized in Innsbruckas well [163].
The presence of long-range and anisotropic interactions is expected to enrich
the already fascinating quantum phases predicted with contact-interacting mixtures.
Theoretical works addressed the formation of vortex lattices in rotating dipolar mixtures [102, 103], binary quantum droplets [20, 174, 175], pattern formation [197], double
supersolid states [169], and domain supersolids [26]. All the phenomena mentioned
so far are highly sensitive on the interspecies interaction strength and to be observed
require a fine and controlled tuning of a12 . In fact, the interaction between the species
strongly affects the in-trap density distribution of the clouds, leading in some cases tophase separation [101]. For dipolar mixtures, whether the two species overlap, namely
they are miscible, or they are pushed aside, namely they are immiscible, depends on

3.2 Cross-species thermalization measurements

33

both the contact and dipolar intra- and interspecies scattering lengths. Whereas foralkali atoms, collisional models can predict the scattering lengths, for dipolar atoms,
the complexity of the electronic structure and the anisotropic interactions make atheoretical determination quite challenging [100, 148]. Therefore the determinationof the intra- and interspecies scattering lengths relies on experimental measurements.
The next section recalls the most common experimental methods for determining
the interspecies scattering length, while Sec. 3.3 describes the one developed in our
experiment and presented in the publication shown in Sec. 3.5.

3.2

Cross-species thermalization measurements

One of the methods used in ultracold mixtures to infer the interspecies scattering
length is the cross-species thermalization technique. This approach consists of driving
the system out of thermodynamic equilibrium, e.g. by selectively heating one of thespecies and then registering the evolution of the temperature towards the equilibrium,
through elastic collisions. One can relate the thermalization time τ to the elastic crosssection and finally to the interspecies scattering length a12 . By following Refs. [4, 76],
one can model the cross thermalization through a set of rate equations, as below:Ṅ1,2 = 0,(3.1)

ξ Γ12 (T2 − T1 )
,
3N1,2

(3.2)

Ṫ1,2 = ±where Ni , Ti are the atom number and temperature of the i species, respectively. Γ12
is the interspecies collisional rate and ξ = 4 m1 m2 /(m1 + m2 )2 , with m1 (m2 ) the massof the 1(2) species. The factor ξ takes into account the reduced energy transfer causedby different masses. Furthermore, the factor 3 at the denominator, which is usuallyknown as α, considers that about three collisions are needed for thermalization [130].
This factor changes whether the collisions are s-wave or p-wave, where α is about
4 [52]. In the case of dipolar atoms, due to the anisotropic nature of the interactions,the parameter α depends on the direction of the quantization axis, generally givenby the magnetic field, with respect to the gravity axis, as well as on the scattering
length [30, 191] (see the experimental work done with erbium atoms [144]).
Equations 3.1 and 3.2 neglect the terms coming from evaporative cooling, three-bodylosses, losses due to collisions with background gas, and heating due to the trapping34

Interspecies interaction in an ultracold dipolar mixture

laser. From Eq. 3.2 one finds:
Ṫ2 − Ṫ1 =ξΓ12 1
1(+)(T2 − T1 ).
3 N1 N2(3.3)

By assuming that the thermalization time of the temperature difference is the same as
the individual temperatures, τ can be defined as:1ξΓ12=(N1 + N2 ).
τ
3N1 N2(3.4)

The thermalization rate can be linked to the elastic cross section σ12 from therelation:Γ12 = n̄12 σ12 v̄12 ,(3.5)
where n̄12 and v̄12 are the mean spatial overlap and the mean collisional velocity,respectively. These two quantities have the following form:
3/2 3/2

N1 N2 m1 ω̄1n̄12 =,[(2πkB )(T1 + β 2 T2 )]3/2
sv̄12 =

8kB T1
T2(+),
π m1 m2

(3.6)

(3.7)

with ω̄i=1,2 the geometric average of the trapping frequencies and β 2 = m2 ω̄22 /m1 ω̄12 .
By combining Eq. 3.4, Eq. 3.6 and Eq. 3.7, one can write:
1 ξ(N1 + N2 )=n̄12 σ12 v̄12 .
τ
3N1 N2(3.8)

By measuring τ , Ni=1,2 and Ti=1,2 it is possible to estimate σ12 . The dependence of αon the polarization angle and on a12 makes the determination of the scattering length
challenging and dependent on theoretical predictions. However, a rough estimation
can be inferred from the relation:
σ12 = 4πa212 +

64π 2
a
,
45 dd,12(3.9)

where add,12 is the interspecies dipolar length. This thermalization model was usedin the publication presented in A.3 to determine the scattering cross section in thevicinity of an interspecies Feshbach resonance and get insight into the strength andwidth of the resonance.

35

3.3 Interspecies scattering length from mean-field shift

3.2.1

Feshbach spectroscopy

Due to the quadratic dependence on a12 of the elastic cross-section, the cross-species
thermalization method allows determining only the modulus of the interspecies scattering length and not the sign. In order to determine the sign, thermalization measurements can be combined with Feshbach spectroscopy to map out the dependenceof a12 with the magnetic field B and estimate the background scattering length abg
12 .
Once an interspecies Feshbach resonance is identified, one can perform cross-species
thermalization measurements at different B around the resonance and, by using themodel described in Sec. 3.2, determine a12 . The magnetic-field dependence of a12 canbe represented by the following relation [39]:!

a(B) = abg∆
1−,
B − B0(3.10)

where B0 is the resonance position and ∆ the resonance width1 . This method is widelyused in our community and was adopted to determine the interspecies scattering length
in various systems as hetero-nuclear mixtures of alkali atoms [51, 173], in cesiumytterbium mixtures [77] and lately in dysprosium-potassium mixtures [162, 201].

3.2.2

Other methods

Other methods include the determination of the interspecies scattering length fromcollective-excitation spectroscopy. The frequencies of excitation modes, e.g. breathingmodes, are predicted to be dependent on a12 [60, 158]. Theoretical works proposed
also the possibility of inferring a12 from the insitu density profiles [104]. Finally, amore demanding techinique is lattice-modulation spectroscopy (see Sec. 3.3.3), which
was used in ultracold fermionic erbium atoms to determine the interspecies scattering
length between the two lowest spin states [8].

3.3Interspecies scattering length from mean-fieldshift

The technique developed in our group and presented in Sec. 3.5 consists of a joined
theoretical and experimental work to study the effect that dipole-dipole interactions
1 For dipolar atoms, the presence of a highly dense Feshbach spectrum requires the use of theformula for overlapping resonances: a(B) = abg

Q 
i

∆i
1 − B−B0,i[109].

36

Interspecies interaction in an ultracold dipolar mixture

have on the total mean-field interspecies interactions and to estimate a12 from tracing
the in-trap cloud displacement as a function of the magnetic field orientation. Let’s
consider a non-interacting ultracold mixture of Er and Dy atoms trapped in an elongated1064-nm optical dipole trap. Due to the slight difference in mass and polarizability thetwo clouds are displaced in the vertical direction by what is known as gravitationalsag, ∆zgrav = g(1/ωz21 − 1/ωz22 ), where g is the gravitational acceleration and ωzi thetrap frequency along the gravity direction for the i species. This effect combineswith a repulsive or attractive shift coming from the interspecies interactions; see later
discussion. Figure 3.1(a) illustrates the geometry of our system with the two atomic
clouds vertically displaced and the magnetic field orientation, which is given by theangles θ and ϕ, can be arbitrarily oriented along any desired direction. Figure 3.1(b)shows the ground state of the system calculated from the binary Gross-Pitaevskii
equation (see Sec. 3.3.1) for the non-interacting case, when the magnetic field points
along the gravity direction (z-axis). The 2D column density shows that the center-ofmasses (COM) are vertically shifted by the gravitational sag, but the atomic clouds
merge in the overlapping region.
When switching the interactions on, the clouds displacement is modified by anotherterm, namely the mean-field shift ∆zMF . This term includes the mean-field shift due
to the contact and dipole-dipole interactions and can push the two clouds closer orfurther apart, whether the total mean-field interactions are attractive or repulsive,
respectively. Figure 3.1(c) shows the ground state of the system for the same parameters
as Fig. 3.1(b), but with a repulsive interaction characterized by a12 = 100 a0 . Dueto the interactions, the two clouds repel each other leading to a larger clouds COM
displacement. It becomes clear that the in-trap displacement with respect to the
non-interacting position is a measure of the strength of the interactions, namely a12 .
For repulsive interactions, the stronger a12 the larger the in-trap displacement gets.
An effect that plays a big role in dipolar gases and distinguishes them from theshort-range interacting ones is magnetostriction [179], which is an elongation of theatomic cloud along the magnetic field direction. Figure 3.1(d) shows the ground stateof the system with the same parameters as Fig. 3.1(c), but with the magnetic field
pointing along the x-axis. Whereas for a magnetic field pointing along the z-axis, themagnetostriction leads to an increase of the vertical overlap between the clouds, whenthe magnetic field points along the x-axis, the atomic clouds elongate horizontally,hence reducing the vertical density overlap. This in turn reduces the interspecies
repulsion and the two clouds approach the non-interacting clouds displacement, fixed

37

a.

4

164 Dy166 Erb.

z ( µm)

2

B

θ

0

−2
φ

y

−4

x
4

− 20
164 Dy166 Er

c.

12

B ||Z

a 12 = 0, Cdd = 0
− 15

− 10

−5

0
y ( µm)5

10

15

z ( µm)

2
0

−2
−4
− 20

a 12 = 100 a 0
− 15

− 10

−5

0
y ( µm)

5

10

15

B ||X

a 12 = 100 a 0

B ||Z
20 − 20

− 15

− 10

−5

0

5

10

15

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

20

164 Dy166 Er

d.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

column density (arb. units)zcolumn density (arb. units)

3.3 Interspecies scattering length from mean-field shift20

y ( µm)Figure 3.1 (a) Geometry of our dipolar mixture with 164 Dy and 166 Er shown as red andblue ellipse, respectively. The angles ϕ and θ define the orientation of the magnetic field.
(b-d) Ground-state calculation of the column density for a mixture with imbalanced
atom numbers: NDy = 1.3 × 104 , NEr = 4.9 × 104 . Iso-density contour levels for Dy(dashed lines) and Er (filled lines). z=0 corresponds to the center of the gravitationalsag. The magnetic field points along the z-axis with no interspecies interactions (b)and a12 = 100 a0 (c). The magnetic field is aligned along the x-axis, and a12 = 100 a0(d). Figure adapted from [155]. (c,d) Credits: Michele Modugno.
by the gravitational sag. Hence, the maximum value of the COM shift occurs when themagnetic field points along the gravity direction, which maximizes the density overlap.
Another knob that can be tuned in dipolar gases is the magnetic field direction. In
fact, in an asymmetric trap, for a fixed modulus of the magnetic field, the contributionof the dipolar interaction to the total mean-field interaction can be tuned from attractive
to repulsive by varying the magnetic-field orientation. Figure 3.2 shows the dipole
potential created by the dysprosium cloud and felt by the erbium atoms for two different
magnetic field orientations. On the one hand, when the magnetic field points along thevertical direction, see Fig. 3.2(a), the interspecies interaction is mainly attractive due tothe dipoles aligned predominantly head-to-tail. On the other hand, when the magnetic
field points along the x-axis, see Fig. 3.2(b), the dipoles mainly sit side-by-side giving
rise to a dipole potential felt by erbium atoms which is mainly repulsive (see Eq. 1.1).

38

Interspecies interaction in an ultracold dipolar mixture

6ṼDy (arb. units)
0.003

a.

4

0.002

2

0.001

ṼDy (arb. units)
0.002

b.

0.001

z ( µm)000Er− 0.001−4− 0.002

B ||Z−6− 20 − 15 − 10 − 5

0

5

10

15

− 0.001

Er

−2

20

− 0.003y ( µm)− 0.002
B ||X− 20 − 15 − 10 − 5

0

5

10

15

20

− 0.003y ( µm)Figure 3.2R Heat map of the dipole potential produced by the Dy condensate
VeDy (r) ≡ Vdd (r − r′ )nDy (r′ )dr′ , in the x = 0-plane. Iso-density contour levels of
the Er component (dotted-dashed lines). The magnetic field is oriented along the z-axis(a) and along the x-axis (b). In the former case, the potential produced by Dy on Er in
the overlapping region is negative, indicating that the interspecies dipolar interactions
are predominantly attractive. Whereas, for the latter, the potential is positive, indicating that the interspecies dipolar interactions are predominantly repulsive. Figure
adapted from [155]. (b) Credits: Michele Modugno.

3.3.1

Extended Gross-Pitaevskii equation

Before giving details on the experimental sequence used to probe the interspecies
interactions, the following section briefly recalls the theoretical model developed torepresent our system. To describe our mixture of dipolar erbium and dysprosium
condensates, the GPE presented in Sec. 1.3 needs to be modified to include the terms
arising from the interspecies interactions. These terms include the interspecies contact
and dipolar interaction, whose energy functional can be expressed as below:
c
E12
= g12
dd
E12
= µ0 µ1 µ2ZZ

n1 (r)n2 (r)drn1 (r)Udd (r − r′ )n2 (r′ )drdr′ ,(3.11)(3.12)

where ni (r) = |ψi (r)|2 is the density of each condensate, g12 = 2πh̄2 a12 (m1 + m2 )/m1 m2
is the interspecies coupling constant, µ1 (µ2 ) is the magnetic moment of species 1(2),and Udd is the interspecies dipole-dipole interaction. Since the regimes investigatedin the experiment and presented in Sec. 3.5 are stable against mean-field collapse, forthe results presented in this thesis only the single species beyond-mean field terms, as3.3 Interspecies scattering length from mean-field shift

39

in Eq. 1.6, have been taken into account. The interspecies LHY becomes crucial forthe stability of the system when, for attractive interspecies dipolar interactions, a12 isreduced below a critical value leading to a collapse of both species at the mean-fieldlevel. A theoretical description including the interspecies LHY term has been done inRefs [20, 174]. After minimizing the energy functional including all the relevant terms,
we determine the ground state of the system, as shown in Fig. 3.1 and Fig. 3.2, andget the calculated in-trap displacement for a fixed value of a12 .

3.3.2

Experimental sequence

The experimental sequence used to probe the in-trap cloud displacement is describedin details in the publication reported in Sec. 3.5. In brief, the experiment consistsof preparing a double BEC of erbium and dysprosium atoms, as described in theprevious chapter. The two atomic clouds repel each other due to repulsive interspecies
interactions. One of the species is then selectively removed, by shining resonant light.
This removal put the remaining species out of equilibrium and induces center-of-mass
oscillations around the new equilibrium position, fixed by the minimum of the trap
and the gravitational sag. The amplitude of this oscillation can be connected to thestrength of the interspecies repulsion (see publication in Sec. 3.5).
After performing this sequence for one magnetic field orientation, we repeat theprocedure for various θ and ϕ. As a result, we obtain the mean-field shift as a function of
the magnetic field orientation, where the maximum displacement appears for a magnetic
field direction oriented along the gravity direction, which maximizes the density overlapof the two clouds due to magnetostriction and therefore the interspecies repulsion.
By performing ground-state calculations at different interspecies scattering lengths
and for different magnetic field orientations, it is possible to infer the a12 that best
represents the experimental data, which we estimate to be a12 = 105[−65, +162] a0 .
The experimental sequence described above was also used to probe other isotope
mixtures (162 Dy-168 Er, 162 Dy-170 Er) with all showing a qualitatively similar behaviour
with total mean-field repulsive interactions at the operating magnetic field.

3.3.3

Evaluation of the systematic errors

The calculated mean-field shift qualitatively agrees with the experimental one with thelargest deviation when the magnetic field points along the horizontal plane. Below, few
factors that might be the cause of the disagreement are mentioned. In our theory model,
the trap frequencies are assumed to be constant and independent on the magnetic field40

Interspecies interaction in an ultracold dipolar mixture

orientation. However, unlike alkali atoms, for erbium and dysprosium, the polarizability
features a tensorial character [17, 113]. The light shift has a dependence on the angle
between the light polarization and the magnetic field orientation (θp ) and the angle
between the light propagation and the magnetic field orientation (θk ). Following
Ref. [113], the total light shift can be expressed as below:

"

I(r)
mJ
Ushift (ω) = −αs (ω) + |u∗ × u| cos θkαv (ω)
2ϵ0 c
2J#3m2J − J(J + 1) 3 cos2 θp − 1+
αt (ω) ,J(2J − 1)
2(3.13)

where I is the light intensity, ϵ0 is the vacuum permittivity, c is the speed of light, u isthe light polarization vector, θk (θp ) is the angle between the magnetic field direction andthe light propagation(polarization). αs , αv , and αt are the polarizability coefficientsfor the scalar, vector, and tensor part, respectively. For our experiments, we trap
the atomic cloud in a 1064-nm laser beam, which is linearly polarized. This makes
the vectorial term in Eq. 3.13 vanish. Regarding the tensorial term, when tilting themagnetic field from the z-axis into the xy-plane, θp can maximally change from θp = 90◦
to θp = 0◦ . This means that the polarization-dependent factor can change from -1/2
to +1. Taking the values of αs = 184.4(2.4) a.u. and αt = 1.7(6) a.u. reported inRef. [161], the contribution of the tensorial term varies from 0.5% up to 1% of the√scalar one. The trap frequencies scale as ∝ αtot , where the polarizability αtot takes
into account all the scalar, vector, and tensor parts. Therefore there would be only asmall contribution of the tensorial part to the change in trap frequencies, which weneglect.
The trap frequencies are also affected by the presence of residual magnetic field
gradients. To quantify those gradients we performed measurements of the COM position
Zi (tTOF ) for three different orientations of the magnetic field, along the three axes (x,y, z), and as a function of tTOF . By fitting the function y = a + b tTOF + 12 c t2TOF toZi (tTOF ), it is possible to estimate the correction to the gravitational acceleration due
to the presence of residual magnetic field gradients. We measure a higher acceleration
by 2% and 1% for dysprosium and erbium, respectively, when the magnetic field points
along the z-axis. These gradients, which point along the same direction of gravity,slightly decrease the trap frequencies by few % when going from B along the xy-plane
to B along z.

3.4 Outlook: induced supersolidity in dipolar mixture

41

Another important input for the theoretical model is the value of the intraspecies
scattering length. For erbium, the scattering length was accurately determined using
lattice-modulation spectroscopy [144]. This technique is based on the measurement of
the on-site interaction between atoms trapped in an optical lattice, from which as canbe determined. The on-site interaction is measured by modulating the lattice depth
and by looking at the induced heating caused by the doubly occupied lattice sites atthe specific resonance frequency. Therefore, for the magnetic field value relevant forErus, aErs was set to as = 83(3) a0 [40]. However, such an accurate determination of
the scattering length is still missing for dysprosium and various experiments reported
different values [62, 181]. We set aDys = 95 a0 , which gives an unmodulated BEC for
our atom numbers and trap geometry.

3.4

Outlook: induced supersolidity in dipolar mixture

In Chapter 1 it was shown that erbium and dysprosium can make a phase transition toa supersolid state and develop density modulation and maintain global phase coherence
in a narrow range of scattering lengths, where ϵdd ≳ 1.3. From our ground-statecalculations of a double BEC made of erbium and dysprosium atoms, we found thatwhen the scattering length of dysprosium is just above the aforementioned range,
whereas erbium is in the contact-dominated regime with ϵEr
dd < 1, by increasing therepulsive interactions a12 between the species, it is possible to induce the onset ofsupersolidity in dysprosium. Furthermore, when pushing the two clouds closer bytightening the trap, the density modulation present in dysprosium can be transferredon erbium.
After this work, several theoretical proposals focused on the different ground-statephases that can be realized by merging two dipolar condensates. In particular, inRef. [169], the authors study a dipolar mixture with (µ2 /µ1 ≃ 1) in the miscible regime,
where µ1 (µ2 ) is the magnetic moment of species 1(2), and N2 /N = [0, 0.5] with N
the total atom number and N2 the atom number of species 2. The authors showthat a binary condensate can be described as a single-component condensate with aneffective dipolar strength, which can be tuned by changing the interspecies scattering
length and the atom number of the second species. In this work, a peculiar phasewith a double supersolid formed by immiscible droplets is predicted. In Ref. [26], theauthors investigate the formation of domain supersolids formed after a spin roton
instability [195]. Unlike supersolids induced by roton instability, where to sustain42Interspecies interaction in an ultracold dipolar mixture

supersolidity a certain density need to be maintained [154], for domain supersolids,
such a stringent condition on the density is not present. This leads to extended systems
with atom numbers per domain an order of magnitude lower than the case of droplet
supersolids, leading to a potentially longer lifetime due to reduced three-body losses.
The aforementioned interesting regimes can be reached by varying the magnetic
field value in proximity of an interspecies Feshbach resonance (FR) in order to tune
the sign and the strength of a12 . In the publication shown in A.3, we experimentallyobserved interspecies FR in two bosonic isotope mixtures of erbium and dysprosiumwith a width larger than 1 G, whose interspecies character was probed by looking at
the scattering cross-section in proximity of the loss feature, as described in Sec. 3.2.
These Feshbach resonances are located in a reasonably low magnetic field region2 andare promising tools to achieve interesting many-body phases in erbium-dysprosium
mixtures.

2The isotope mixture 164 Dy-168 Er presents an interspecies FR at around 13 G, while 164 Dy-166 Er

at around 34 G.

3.5 Publication I:
Interspecies interactions in an ultracold dipolar mixture

3.5

43

Publication I:

Interspecies interactions in an ultracold dipolar mixture
The following publication has appeared in
Physical Review A 105, 023304 (2022)∗
submitted 19 Oct 2021; published 3 Feb 2022
DOI: https://doi.org/10.1103/PhysRevA.105.023304
C. Politi,1,2 A. Trautmann,1 P. Ilzhöfer,1,2 G. Durastante,1,2 M. J. Mark,1,2 M. Modugno,3,4
and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain
4 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain∗The author of the present thesis performed the measurements together with G. D., analysed theexperimental data, wrote the manuscript together with M. M. and F. F. and contributed in interpreting
the results together with all the authors.

PHYSICAL REVIEW A 105, 023304 (2022)
Editors’ Suggestion

Interspecies interactions in an ultracold dipolar mixture
C. Politi,1,2 A. Trautmann ,1,* P. Ilzhöfer,1,2,† G. Durastante,1,2 M. J. Mark ,1,2 M. Modugno ,3,4 and F. Ferlaino1,2,‡
1

Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain
4
IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain(Received 19 October 2021; accepted 10 January 2022; published 3 February 2022)
We experimentally and theoretically investigate the influence of the dipole-dipole interactions (DDIs) on
the total interspecies interaction in an erbium-dysprosium mixture. By rotating the dipole orientation we areable to tune the effect of the long-range and anisotropic DDI, and therefore the in-trap displacements of the
erbium and dysprosium clouds. We present a theoretical description for our binary system based on an extended
Gross-Pitaevskii theory, including the single-species beyond mean-field terms, and we predict a lower andan upper bound for the interspecies scattering length a12 = 105[−65, +162]a0 . Our work is a step towards
the investigation of the experimentally unexplored dipolar miscibility-immiscibility phase diagram and therealization of quantum droplets and supersolid states with heteronuclear dipolar mixtures.
DOI: 10.1103/PhysRevA.105.023304
I. INTRODUCTION

The ability to tune the interparticle interactions, the geometry and dimensionality of the system, and the possibility ofadding complexity in a controlled manner, has made ultracold atomic gases a great platform for studying a plethoraof physical phenomena that would be otherwise hard toachieve [1]. Combining two atomic species gives even further
opportunities for investigating the effects arising from theinterplay between the intra- and interspecies interactions, aspolarons [2,3], heteronuclear quantum droplets [4–6], solitons [7], and ultracold molecules [8].
Heteronuclear mixtures are typically realized by combining contact-interacting atomic species (see alkali-alkali
mixtures [9–15], alkali-alkaline-earth mixtures [16], andalkali-alkaline-earth-like mixtures [17–19]). Recently, experiments were able to produce novel types of ultracold mixtures
where either one or both mixture components are long-range
interacting (lanthanide) atomic species [20,21]. In particular,
the realization of Er-Dy dipolar quantum mixtures is attracting great interest, driven by the possibility of creating newquantum phases even more exotic than the one achieved incontact-interacting mixtures [1] or in single-species dipolargases [22]. Several theoretical works reported on the studyof miscibility in dipolar condensates [23–26], vortex lattice
formation [27,28], and on binary quantum droplets realizedwith dipolar mixtures [29–31].
In heteronuclear dipolar Bose-Bose mixtures, the phenomena mentioned above rely quite strongly on the miscibilityimmiscibility conditions. These conditions define whether

the two components mix together with the center of masses
overlapping at the trap center or whether they are in a phaseseparated state where the two centers of mass are pushed awayfrom each other. The miscibility-immiscibility phase diagram
depends on the contact intraspecies scattering lengths a11 , a22
and dipolar lengths add,1 , add,2 , and the interspecies scattering
lengths a12 and dipolar lengths add,12 . While add,12 can becalculated analytically, a12 is unknown and its determination
relies on experimental measurements.
In this paper, we prepare ultracold degenerate mixturesof erbium and dysprosium, and experimentally investigate
the effect of the mean-field dipole-dipole interactions on
the total interspecies interaction by tracing the center-ofmass displacement for different dipole orientations. We
present a theoretical description for our system, including
the single-species beyond mean-field terms, which reproduces
qualitatively well the experiment. By matching theory andexperiment, we define a lower and upper bound for the interspecies scattering length a12 .
II. THEORY

Here, we consider a binary mixture of dipolar condensatesof 164 Dy and 166 Er atoms confined in a harmonic potential, in
the presence of a magnetic field B aligned along an arbitrary
direction in space. The system can be described in terms ofan extended Gross-Pitaevskii energy functional E = EMF +Edd + ELHY with.

2   2h̄|∇ψi (r)|2 + Vi (r)|ψi (r)|2 dr
EMF =
2mi
i=1

*

Present address: Physikalisches Institut, Auf der Morgenstelle 14(D-Bau), 72076 Tübingen, Germany.
†
Present address: 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany.
‡
Corresponding author: francesca.ferlaino@uibk.ac.at2469-9926/2022/105(2)/023304(9)023304-1

+


2

gi j
2
dd  

ni (r)n j (r)dr,(1)

ni (r)Vdd (r − r )n j (r )drdr ,(2)i, j=1

Edd =

2

Ci j
i, j=1

2©2022 American Physical Society

C. POLITI et al.

PHYSICAL REVIEW A 105, 023304 (2022)

and the single-species Lee-Huang-Yang (LHY) correction for
the two components
√


2
3 2
256 π  h̄2 aii5/2
1 + dd,ini (r)5/2 dr, (3)ELHY =
15 i=1 mi
2
where ni (r) = |ψi (r)|2 represents
the density of each con
2 2densate, Vi (r) = (mi /2) α=x,y,z ωα,irα + mi gz includes the
harmonic trapping and gravity potentials, gi j = 2π h̄2 ai j (mi +m j )/(mi m j ) is the contact interaction strength, Vdd (r) = (1 −3 cos2 θ̄ )/(4π r 3 ) the (bare) dipole-dipole potential, Ciddj ≡μ0 di d j its strength, di the modulus of the dipole moment d iof each species, dd,i = μ0 di2 /3gii [32], r the distance between the dipoles, and θ̄ the angle between the vector r andthe dipole axis, cos θ̄ = d · r/(dr) [33]. In the following we
identify species 1 with the Er condensate, and species 2 with
the Dy condensate (we have also omitted the reference to themass number, for ease of notations). As described later on, theorientation of the magnetic dipoles is varied along arbitrary
directions through an external magnetic field B.
Then, for each set of parameters the ground state of
the system is obtained by minimizing the energy functional
E [ψ1 , ψ2 ] by means of a conjugate algorithm (see, e.g.,
Refs. [33–35]). In the numerical code the double integral
appearing in Eq. (2) is mapped into Fourier space where it
can be conveniently computed using fast Fourier transform(FFT) algorithms, after regularization (see Appendix B). The
LHY correction in Eq. (3) is obtained from the expression
for homogeneous three-dimensional (3D) dipolar condensates
under the local-density approximation [32,36]. For the sakeof simplicity, here we do not include the interspecies LHY
correction as it would require a much more complicated treatment [29,30], which is not essential when the system is stableagainst the mean-field collapse driven by the interspecies interactions, as in the present analysis.
Finally, we remark that the intraspecies scattering lengths
are given as the input to the theory and, whereas the value for
Er has been measured with high accuracy to be a11 = 83(3)a0
at the magnetic field we are working at [37], the scattering
length for Dy, a22 , still lacks an accurate determination. Several works have reported different values ranging from 60a0
to 100a0 [38,39]. As in the present work no signs of supersolid or droplet states have been observed [40], we set a22 to
the minimal value for which Dy is stable against mean-field
collapse (without LHY), namely a22 = 95a0 . This guarantees
that the ground state is an unmodulated BEC, for our atom
numbers and trap frequencies. Note that, since a11 > add,1
while a22  add,2 , with add,i = Ciidd mi /(12π h̄2 ) the dipolar
length, we expect Eq. (3) to be more relevant for Dy than for
Er. Indeed, when dropping this term for Er, we observe nochanges in the behavior. Instead, for Dy, the system would
collapse for a22 < 95a0 for B perpendicular to the gravity
direction.
III. EXPERIMENT

Our experiment starts with a degenerate mixture of 166 Er
and 164 Dy, similar to Ref. [21]. In brief, after cooling theatomic clouds in a dual-species magneto-optical trap [41], we
start the evaporative cooling by loading the mixture into a

(a)

(c)

(b)

FIG. 1. Trap geometry, ground-state column density, and dipole
potential. (a) Illustration of the geometry of our 164 Dy (red ellipse)
and 166 Er (blue ellipse) mixture. The orientation of the magnetic field
is defined by the angles φ and θ. The imaging beam propagatesin the horizontal plane, at an angle of 45◦ with respect to the yaxis (not shown). (b) Ground-state column density for an imbalanced mixture with NDy = 1.3 × 104 , NEr = 4.9 × 104 , a12 = 100a0 .
Dashed and solid lines show the isodensity contour levels for Dy
and Er, respectively. For comparison, the in-trap displacement due
to the gravitational sag for a noninteracting mixture is also shown(black dashed lines). We set z = 0 at the center of the gravitationalsag. (c) Heat map of the dipole potential produced by the Dy condensate (parameters below), VDy (r) ≡ Vdd (r − r )nDy (r )dr , in thex = 0 plane. Here, the magnetic field points along the z axis. The
dotted-dashed lines represent the isodensity contour levels of the Ercomponent, indicating that in this regime the interspecies dipolarinteraction is predominantly attractive.

single-beam optical dipole trap at 1064 nm, which propagateshorizontally (y axis); see the reference frame in Fig. 1(a).
After about 600 ms, the power of a second trapping beam,
propagating vertically along the direction of gravity (z axis),
is linearly ramped up to form a crossed optical dipole trap(cODT). Here, the evaporation further proceeds for about
5 s. We perform the evaporation at a magnetic field of B =2.028 G, pointing along the z axis, which allows an efficientcooling of both species.
The final harmonic trap has a cigarlike shape,axially elongated along the y axis, with frequencies
ωx,y,z = 2π × [96(1), 18(1), 150(5)] s−1 ,andωx,y,z =2π × [104(1), 18(1), 165(5)] s−1 for Er and Dy, respectively.
The trapping frequencies of the two species slightly differ.
This is due to the small difference in their mass and atomic
polarizability [42,43]. In a harmonic trap, each species
experiences a shift of its center-of-mass (COM) position
along the z axis due to gravity. This effect is known as

023304-2

INTERSPECIES INTERACTIONS IN AN ULTRACOLD …

PHYSICAL REVIEW A 105, 023304 (2022)

(a)

(b)

FIG. 2. Experimental protocol. After preparing our Er-Dy mixture with B  z, the magnetic field is rotated to an arbitrary direction,
defined by θ and φ, in 120 ms. The atomic clouds are held in thetrap for 50 ms to reach equilibrium, before either of the species
is removed with resonant light. The remaining cloud is held for avariable hold time th . The cloud is then released from the trap andimaged with standard absorption imaging after a TOF expansion oftTOF = 26 ms. We prepare imbalanced mixtures with condensed atom
numbers NC in the range [1–3] × 104 and [4–6] × 104 for Dy and Er,
respectively (see Appendix A).

gravitational sag [44–46]. For mixtures, the differential
gravitational sag between the components is given by
2
2
zgrav = g(1/ωz1− 1/ωz2), which for our Er-Dy mixtureis zgrav = 1.9(1) μm with Er shifted downwards morethan Dy; see Fig. 1(a). Such gravitational sag favors phaseseparation along the z axis, reducing the interspecies overlap
density. In the presence of interspecies interactions, thevertical distance of the clouds’ centers is not only determinedby the gravitational sag but also by their mutual mean-field
attraction or repulsion [13,15,47–49], quantified by thezMF ismean-field shift
zMF . For dipolar mixtures,
determined by the interplay between the dipolar and contact
interspecies interactions, as we will discuss later. The total
vertical in-trap displacement is thus z = zgrav + zMF .
Figure 1(b) shows exemplary calculations of the 2D
ground-state column density of an imbalanced mixture forB  z and a12 = 100a0 . In this configuration, a COM shift isclearly visible, which exceeds the gravitational sag, indicatinga total repulsive mean-field interaction between the components. To understand the role of the DDI, it is interesting tocalculate the effective potential generated by one species (e.g.,Dy), VDy (r) ≡ Vdd (r − r )nDy (r )dr , felt by the other species(e.g., Er). Such effective potentials are most relevant in theregion where the two species overlap (beside a long-range tailfrom the DDI). As shown in Fig. 1(c), for our trap geometry
and dipole orientation, Er experiences a dominant attractive
DDI generated by Dy, which is however weaker than therepulsive interspecies contact interaction for a12 = 100a0 .
To experimentally study the interspecies mean-field shift,
we selectively remove either one of the two species and follow
the COM dynamics of the remaining species towards its new
equilibrium position in the trap [21]. Figure 2 illustrates our
protocol. After preparing our trapped Bose-Bose Er-Dy mixture with B  z, we first adiabatically rotate the magnetic field

FIG. 3. COM oscillations after removal of either one of thespecies. (a) Vertical COM position of Dy after removing Er and(b) vice versa. The vertical position Zi is recorded after a TOF
expansion of 26 ms, as a function of the holding time. The offset
zoff has been subtracted to facilitate comparison. The measurements
are repeated for two magnetic-field orientations: B  z, θ = 0◦ , φ =0◦ (circles) and B ∈ xy, θ = 90◦ , φ = 15◦ (diamonds). The atom
numbers are NDy = 1.3(2) × 104 , NEr = 4.9(7) × 104 and NDy =3.1(5) × 104 , NEr = 4.7(5) × 104 for B  z and B ∈ xy, respectively.
The error bars reported represent the standard error on the mean overthree experimental trials, and are mostly smaller than the markers.
We fit Eq. (4) to the data for B  z (solid lines) and B ∈ xy (dottedlines).

in 120 ms to the desired orientation (i.e., changing θ and φ)and let the mixture equilibrate for 50 ms. We then selectively
remove either Er or Dy by shining a resonant light pulse, operating on either of the two strong atomic transitions [401 nm(421 nm) for Er (Dy)]. We have checked that this resonant
pulse of 3-ms duration does not affect the remaining species.
Finally, we hold the remaining species in trap for a variable
time th and probe the system with standard absorption imaging
after a time-of-flight (TOF) expansion of tTOF = 26 ms.
After the selective removal of either of the two species,
the remaining species is out of equilibrium and the cloudCOM starts to oscillate around its new equilibrium position, given by the dipole-trap minimum in the presence ofgravity. Figure 3(a) [Fig. 3(b)] shows the vertical COM position Zi (see Appendix A), measured after TOF, for Dy(Er) after removing Er (Dy) and for two different dipole
orientations.
The amplitude of the observed oscillation is directly
connected to the interspecies mean-field shift experienced
by the atoms in the trap. Within the assumption ofballistic expansion, which is justified in the weakly interact-023304-3

C. POLITI et al.

PHYSICAL REVIEW A 105, 023304 (2022)

2
ing regime, Zi (th , tTOF ) = zi (th ) + żi (th )tTOF + gtTOF/2, wherezi (th ) = zMF,i cos(ωi th ) + zgrav is the in-trap COM position. The oscillation frequency ωi is the trap frequency along
the z axis.
By combining the previous equations, one gets the following expression,

Zi (th , tTOF ) = zMF,i cos(ωi th ) −zMF,i ωi sin(ωi th )tTOF +zoff ,(4)2
where zoff = zgrav + gtTOF/2. We fit Eq. (4) to the experimental data for the two magnetic-field orientations withthe mean-field shift zMF,i , ωi , and zoff being free fitting
parameters.

(a)

(b)

(c)

IV. RESULTS

Figure 3 shows important information on the interspecies
interactions. First, by comparing the dynamics of the two
species, we observe that the oscillations are counterphase. The
Dy cloud starts moving downwards towards the trap center,
whereas the Er one moves upwards, confirming a total repulsive interspecies interaction for this geometry. Second, we see
a clear difference in the oscillation amplitude between Dy and
Er. This is due to the fact that the mixture is imbalanced with
Er being the majority species, and therefore the mean-field
shift caused by Er on Dy is larger. Finally, for each species,
the oscillation amplitude strongly depends on the magneticfield orientation. This behavior cannot be simply explainedby the anisotropy of the DDI. For B  z, the DDI is more
attractive over the interspecies overlap region than for B in thexy plane, B ∈ xy. Hence, one would expect zMF,z < zMF,xy ,contrasting the observations.
The additional effect to account for is the magnetostriction [50] of each species, i.e., a cloud elongation along themagnetization direction caused by the single-species DDI.
For B  z, the two clouds elongate along the z axis, thusincreasing the interspecies overlap density; see Fig. 1(b). This
increased overlap activates a backaction on the strength of therepulsive contact interaction, which acquires a larger weight,
leading to an increased repulsion between the clouds. On thecontrary, for B ∈ xy, the clouds elongate horizontally, thereby
minimizing the overlap density and therefore the interspecies
repulsion. The slight difference in frequency observed for thetwo magnetic-field orientations is due to the presence of small
residual magnetic-field gradients (see Appendix C).
To get further insight into the anisotropy of the interspecies
interactions, we repeat the above measurement for various
dipole orientations, set by the angles θ and φ. As before,
we perform two sets of measurements: We probe the outof-equilibrium dynamics of Dy after removing Er and viceversa. To enhance the amplitude of the COM oscillations of
one species (Dy), we perform measurements with imbalanced
mixtures, where Er is the majority species with condensed
atom numbers in the range [4–6] × 104 , while the Dy cloudcontains about [1–3] × 104 (see Appendix A).
Figure 4 summarizes our results. It shows both the measured and calculated mean-field shift zMF,i for each plane ofrotation for Dy (red points) and Er (blue points). We observethat zMF,i has a maximum for B  z and decreases when approaching the horizontal plane. The gray lines show the theory

FIG. 4. Mean-field displacement and theory prediction. Experimental estimation of the mean-field displacement zMF,i for Dy(red points) and Er (blue points), as a function of the magnetic-fieldorientation. (a) θ = [0◦ , 90◦ ], φ = 0◦ . (b) φ = [0◦ , 90◦ ], θ = 90◦ .
(c) θ = [0◦ , 90◦ ], φ = 90◦ . Theory prediction for an interspecies
scattering length a12 = 100a0 (gray lines). The gray shaded area
takes into account the experimental uncertainty on the estimationof the atom number. The error bars in zMF,i correspond to thestatistical uncertainty from the fit. The mismatch between the data
points at θ = 0◦ in (a) and (c) is due to different atom numbers (seeAppendix A).

results for an interspecies scattering length a12 = 100a0 andfor our experimental parameters, i.e., atom numbers and trap
frequencies. We chose a12 = 100a0 as it describes best theexperimental data. The gray shaded area takes into accountthe experimental uncertainty on the estimation of the atom
number.
The theory curves agree qualitatively with the experimental
observations. In particular, experiment and theory are in good
agreement for B  z, while they start to deviate for B ∈ xy.
The small mismatch can be due to the presence of residual
vertical magnetic-field gradients, which are not taken into
account in the theory. These can cause a systematic shiftof the trap frequencies to higher values when going fromB  z to B ∈ xy, thereby reducing the gravitational sag (seeAppendix C). Furthermore, while our Dy ground-state calculations predict the transition to a macrodroplet at a22 = 95a0
for B  y, and a further reduction of the overlap density, inthe experiment we observe a stable Dy BEC. Previous works
have also shown a quantitative mismatch between theory and023304-4

INTERSPECIES INTERACTIONS IN AN ULTRACOLD …
0.9

PHYSICAL REVIEW A 105, 023304 (2022)(a)mean field shift,z MF (µm)

0.8
0.7
0.6

B || z-0.1

B || z-0.12(b)column density (arb. units)

-0.14
0

200

400

600

800

1000

creases with a12 owing to the fact that Dy [Fig. 5(a)] is pushed
away from Er [Fig. 5(b)]. Figure 5(c) shows the Dy (red)and Er (blue) density cuts along y = 0, for a12 = 30a0 (solidlines), a12 = 100a0 (dashed lines), and a12 = 200a0 (dottedlines). The repulsive interaction between the species leads toa decrease of the density overlap when going to higher a12 . We
compare the theory results with the experimentally measured
mean-field shift at [θ = 0◦ , φ = 90◦ ], and by performing a
χ 2 analysis we are able to estimate the interspecies scattering length to be a12 = 105[−65, +162]a0 (see Appendix D).
From our ground-state calculations, when choosing a12 <30a0 the repulsive contribution of the contact interactions to
the mean-field shift is not enough to overcome the attractive
contribution from the DDI [see Fig. 1(c)] leading to a collapseof both species. In this regime, it might be necessary to include
the interspecies LHY term as done in Refs. [29,30].

interspecies scattering length, a12 (units of a 0 )0.04 (c)V. CONCLUSIONS AND OUTLOOK

0.02

0
-4

-2

0

2

4

z (µm)
FIG. 5. Calculated mean-field displacement as a function of a12 .
Calculated mean-field displacement for (a) Dy and (b) Er as a function of the interspecies scattering length a12 . The red dashed line
and the red shaded area in (a) represent the Dy experimental meanfield displacement and its error, respectively. The magnetic field isoriented along the z axis. (c) In-trap density cut along y = 0 for Dy(red) and Er (blue), for a12 = 30a0 (solid lines), a12 = 100a0 (dashedlines), and a12 = 200a0 (dotted lines). Here, NDy = 0.8 × 104 andNEr = 5.9 × 104 .

experiment in predicting the macrodroplet transition, suggesting the need for refined models and an accurate determinationof a22 [36,37,51].
The overall behavior shown in Fig. 4 can be explained by
the effect of the magnetostriction on the interspecies overlap density. In fact, as discussed earlier, for magnetic-fieldorientations in the horizontal plane, the clouds are elongatedhorizontally along the direction of B, thereby minimizing thedensity overlap and the interspecies repulsion, whereas whenorienting the magnetic field along the vertical direction, themagnetostriction leads to an increase of the density overlap
and therefore of the interspecies repulsion, which overcomesthe attractive DDI. The system undergoes a transition to a state
where the two components are pushed aside, maximizing thein-trap displacement [see Fig. 1(b)].
To study the behavior of the mean-field shift as a functionof a12 , we consider a specific magnetic-field orientation. In
particular, for B  z, we perform ground-state calculations
varying the interspecies scattering length a12 and calculate
the Er-Dy mean-field displacement as a function of a12 . The
results are shown in Fig. 5. The mean-field displacement in-

In conclusion, we have experimentally investigated theeffect of the DDI on the total interspecies interaction by tracing the mean-field in-trap displacement between the species.
We have presented a theoretical description for our Er-Dy
mixture, including the single-species beyond mean-field corrections, which qualitatively describes well our system andallows us to predict an interspecies scattering length on theorder of a12 = 100a0 . By changing the magnetic-field orientation from the horizontal plane to the vertical direction,
we were able to observe a transition to a state in which thetwo components are pushed apart by the dominant mean-field
repulsive interaction. Future studies will focus on the use ofinterspecies Feshbach resonances, recently reported in ourgroup [52], to reach the conditions in which one or bothcomponents exhibit a phase transition to a quantum droplet or
supersolid regime. As an example, Fig. 6 shows that the onsetof a supersolid phase in the Dy component can be induced byincreasing the interspecies contact scattering length a12 .

ACKNOWLEDGMENTS

We thank Maximilian Sohmen for experimental support and for valuable discussions. We thank Rick van
Bijnen for theoretical support. We also thank Lauriane
Chomaz, Matthew Norcia, Lauritz Klaus, and the Innsbruck Erbium team for fruitful discussions. This work
is financially supported through an ERC Consolidator
Grant (RARE, No. 681432), the QuantERA grant MAQS
by the Austrian Science Fund FWF No. I4391-N, the
DFG/FWF via FOR 2247/I 4317-N36, and the FWF Grant
No. I4426 RSF/Russia 2019. M.M. acknowledges support through Grant No. PGC2018-101355-B-I00 funded by
MCIN/AEI/10.13039/501100011033 and by ERDF A way
of making Europe, and by the Basque Government throughGrant No. IT986-16. We also acknowledge the Innsbruck
Laser Core Facility, financed by the Austrian Federal Ministryof Science, Research and Economy.

023304-5

C. POLITI et al.

PHYSICAL REVIEW A 105, 023304 (2022)

z

(a)

z

z
B

B

y

x
6.5

(b)

FIG. 6. Calculation of an interaction-induced supersolidity.
Ground-state configurations for an imbalanced dipolar mixture withthe magnetic field pointing along the x axis, for two different values of the interspecies scattering length: (a) a12 = 100a0 , (b)a12 =
150a0 . Dashed and solid lines show the isodensity contour levels
for Dy and Er, respectively. Here, NDy = 1.2 × 104 , NEr = 6 × 104 ,a11 = 83a0 , and a22 = 95a0 .
APPENDIX A: ATOM NUMBER AND VERTICALCOM POSITION

After each experimental sequence—described in Fig. 2—
we release the clouds and perform absorption imaging after
a TOF expansion of 26 ms. We measure the condensed atom
number for each species after subtracting the thermal part byfitting a symmetric 2D Gaussian to the wings of the density distribution. We then fit an asymmetric Gaussian to theremaining density distribution to extract the vertical COM
position Zi . Figure 7 shows the measured condensed atom
numbers NC of Dy (red points) and Er (blue points), related
to the results presented in Fig. 4 of the main text. These atom
numbers are given as input to the theory for each value of θand φ.
APPENDIX B: FOURIER REPRESENTATION AND
REGULARIZATION OF THE DIPOLAR ENERGY

Here, we outline the method used for calculating the
double integral in Eq. (2), following the standard approach introduced in Ref. [33]. As anticipated, we start by rewriting theabove integral in Fourier space. In particular, we make use of
the Parseval theorem [34], g(x)h∗ (x)dx = g̃(k)h̃∗ (k)dk,
where g̃(k) ≡ FT[g](k) and h̃∗ (k) ≡ {FT[h](k)}∗ . Then, by
defining f ≡ h∗ and recalling that FT[ f ∗ ](k) = f˜∗ (−k), we
have h̃∗ (k) = {FT[ f ∗ ](k)}∗ = f˜(−k), so that

1
(B1)
Edd =
ñi∗ (k)Vdd (k)ñ j (k)dk,
2
where we have used the fact that ni (r) is real, which impliesñi (−k) = ñi∗ (k) [53]. At this point it is worth recalling thatthe use of the FT implicitly entails a periodic system, andCondensed atom number Nc

6

B

y

y

x

x

(b)

(c)

104

(a)

5.5
5
4.5
4

104

3
2
1
0

45

(deg)

90

0

45

(deg)

90

-90

-45

0

(deg)

FIG. 7. Condensed atom numbers as a function of the magneticfield orientation. Measured atom numbers in the BEC for Dy (redpoints) and Er (blue points) related to the measurement shown in
Fig. 4 of the main text. The magnetic field is oriented (a) in the
X Z plane, (b) in the XY plane, and (c) in the Y Z plane. The error
bars reported represent the standard error on the mean over threeexperimental trials.

this brings along an unwanted effect: The long-range dipolarinteractions can couple the system to virtual periodic replica.
Such a coupling is obviously unphysical, and it can be cured
by limiting the range of the dipolar interaction within a sphereof radius R (contained inside the computational box of sizeL), namely multiplying Vdd (r) by the Heaviside step function(R − r), with R  L/2. The corresponding FT is [33]

cos(Rk)sin(Rk)1
cut
2
cos α −.
Vdd (k) = 4π 1 + 3 2 2 − 3 3 3
R k
R k
3
(B2)
APPENDIX C: ESTIMATION OF THE RESIDUALMAGNETIC-FIELD GRADIENT

To evaluate the residual magnetic-field gradient we measure the COM position of Er and Dy as a function of the
TOF and for different values of the magnetic field. In thisway, we are able to extract the correction to the gravitational
acceleration g due to residual magnetic-field gradients. When
B is oriented along the z axis we measure an increase in g ofabout 2% for Dy and 1% for Er. The presence of these residual
magnetic-field gradients along the direction of gravity leads
to a slight decrease of the trap frequencies (see Fig. 3 in themain text) when orienting B from the XY plane to the z axis.
The tensorial polarizability [42,54] could also cause a shift023304-6

INTERSPECIES INTERACTIONS IN AN ULTRACOLD …

PHYSICAL REVIEW A 105, 023304 (2022)of the trap frequencies when changing the orientation of themagnetic field, but these are negligible in our case.

3

APPENDIX D: ESTIMATION OF THE INTERSPECIESSCATTERING LENGTH

2

FIG. 8. χ 2 distribution of the mean-field shift. χ 2 distribution for
the Dy mean-field shift (black points). We estimate the interspecies
scattering length to be a12 = 105[−65, +162]a0 , by locally doing a
Gaussian fit around the minimum of the distribution (gray line) andby defining the lower and upper bounds as the values at which χ 2 = 1(black dashed lines).

From the calculated mean-field shift as a function of theinterspecies scattering length, shown in Fig. 5 of the main
text, we can estimate the value of a12 that best represents
the experimental results and its confidence interval. Since Er
is the majority species, its mean-field displacement is lesssensitive to the change in interspecies scattering length [asshown in Fig. 5(b)]. In particular, the change in mean-field
shift is within the experimental error. Therefore, we only take
Dy into account for our analysis. We perform a χ 2 analysis
for the Dy mean-field shift at [θ = 0◦ , φ = 90◦ ], with χ 2 =thth( zMF,2 − zMF,2)2 /σ 2 zMF,2 , where zMF,2is the theoreticallycalculated in-trap mean-field displacement, and zMF,2 andσ zMF,2 the experimental value and its statistical error. By
fitting a Gaussian around the minimum of the distribution and
by defining its confidence interval as the range in which χ 2 <1 [55], we estimate a12 = 105[−65, +162]a0 , see Fig. 8.

[1] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics withultracold gases, Rev. Mod. Phys. 80, 885 (2008).
[2] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A.
Cornell, and D. S. Jin, Bose Polarons in the Strongly Interacting
Regime, Phys. Rev. Lett. 117, 055301 (2016).
[3] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M. Parish,
J. Levinsen, R. S. Christensen, G. M. Bruun, and J. J.
Arlt, Observation of Attractive and Repulsive Polarons ina Bose-Einstein Condensate, Phys. Rev. Lett. 117, 055302(2016).
[4] D. S. Petrov, Quantum Mechanical Stabilization of a Collapsing
Bose-Bose Mixture, Phys. Rev. Lett. 115, 155302 (2015).
[5] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas,
P. Cheiney, and L. Tarruell, Quantum liquid droplets ina mixture of Bose-Einstein condensates, Science 359, 301(2018).
[6] C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F.
Ancilotto, M. Modugno, F. Minardi, and C. Fort, Observationof quantum droplets in a heteronuclear bosonic mixture, Phys.
Rev. Research 1, 033155 (2019).
[7] B. J. DeSalvo, K. Patel, G. Cai, and C. Chin, Observation offermion-mediated interactions between bosonic atoms, Nature(London) 568, 61 (2019).
[8] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B.
Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin,
and J. Ye, A high phase-space-density gas of polar molecules,
Science 322, 231 (2008).
[9] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and
M. Inguscio, Bose-Einstein condensation of potassium atoms
by sympathetic cooling, Science 294, 1320 (2001).
[10] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W.
Zwierlein, A. Görlitz, and W. Ketterle, Two-Species Mixture of
Quantum Degenerate Bose and Fermi Gases, Phys. Rev. Lett.
88, 160401 (2002).

[11] M. Mudrich, S. Kraft, K. Singer, R. Grimm, A. Mosk,
and M. Weidemüller, Sympathetic Cooling with Two Atomic
Species in an Optical Trap, Phys. Rev. Lett. 88, 253001(2002).
[12] C. Silber, S. Günther, C. Marzok, B. Deh, P. W. Courteille, andC. Zimmermann, Quantum-Degenerate Mixture of Fermionic
Lithium and Bosonic Rubidium Gases, Phys. Rev. Lett. 95,
170408 (2005).
[13] D. J. McCarron, H. W. Cho, D. L. Jenkin, M. P. Köppinger, andS. L. Cornish, Dual-species Bose-Einstein condensate of 87 Rb
and 133 Cs, Phys. Rev. A 84, 011603(R) (2011).
[14] J. W. Park, C.-H. Wu, I. Santiago, T. G. Tiecke, S. Will, P.
Ahmadi, and M. W. Zwierlein, Quantum degenerate BoseFermi mixture of chemically different atomic species withwidely tunable interactions, Phys. Rev. A 85, 051602(R)(2012).
[15] L. Wacker, N. B. Jørgensen, D. Birkmose, R. Horchani, W.
Ertmer, C. Klempt, N. Winter, J. Sherson, and J. J. Arlt, Tunabledual-species Bose-Einstein condensates of 39 K and 87 Rb, Phys.
Rev. A 92, 053602 (2015).
[16] B. Pasquiou, A. Bayerle, S. M. Tzanova, S. Stellmer, J.
Szczepkowski, M. Parigger, R. Grimm, and F. Schreck, Quantum degenerate mixtures of strontium and rubidium atoms,
Phys. Rev. A 88, 023601 (2013).
[17] A. Khramov, A. Hansen, W. Dowd, R. J. Roy, C. Makrides, A.
Petrov, S. Kotochigova, and S. Gupta, Ultracold Heteronuclear
Mixture of Ground and Excited State Atoms, Phys. Rev. Lett.
112, 033201 (2014).
[18] V. D. Vaidya, J. Tiamsuphat, S. L. Rolston, and J. V. Porto,
Degenerate Bose-Fermi mixtures of rubidium and ytterbium,
Phys. Rev. A 92, 043604 (2015).
[19] K. E. Wilson, A. Guttridge, J. Segal, and S. L. Cornish, Quantum degenerate mixtures of Cs and Yb, Phys. Rev. A 103,
033306 (2021).

2

4

1
0
0

200

400

600

800

1000

interspecies scattering length a12 (units of a0 )023304-7

C. POLITI et al.

PHYSICAL REVIEW A 105, 023304 (2022)[20] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, E. Kirilov,
and R. Grimm, Production of a degenerate Fermi-Fermi mixtureof dysprosium and potassium atoms, Phys. Rev. A 98, 063624(2018).
[21] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M. Sohmen,
M. J. Mark, and F. Ferlaino, Dipolar Quantum Mixtures of
Erbium and Dysprosium Atoms, Phys. Rev. Lett. 121, 213601(2018).
[22] M. A. Norcia and F. Ferlaino, Developments in atomic control using ultracold magnetic lanthanides, Nat. Phys. 17, 1349
(2021).
[23] R. M. Wilson, C. Ticknor, J. L. Bohn, and E. Timmermans, Roton immiscibility in a two-component dipolar Bose gas, Phys.
Rev. A 86, 033606 (2012).
[24] R. K. Kumar, P. Muruganandam, L. Tomio, and A. Gammal,
Miscibility in coupled dipolar and non-dipolar Bose–Einsteincondensates, J. Phys. Commun. 1, 035012 (2017).
[25] K.-T. Xi, T. Byrnes, and H. Saito, Fingering instabilities andpattern formation in a two-component dipolar Bose-Einsteincondensate, Phys. Rev. A 97, 023625 (2018).
[26] R. K. Kumar, L. Tomio, and A. Gammal, Spatial separation ofrotating binary Bose-Einstein condensates by tuning the dipolarinteractions, Phys. Rev. A 99, 043606 (2019).
[27] R. K. Kumar, L. Tomio, B. A. Malomed, and A. Gammal,
Vortex lattices in binary Bose-Einstein condensates with dipoledipole interactions, Phys. Rev. A 96, 063624 (2017).
[28] R. K. Kumar, L. Tomio, and A. Gammal, Vortex patterns in rotating dipolar Bose–Einstein condensate mixtures with squared
optical lattices, J. Phys. B: At., Mol. Opt. Phys. 52, 025302(2018).
[29] R. N. Bisset, L. A. Pena Ardila, and L. Santos, Quantum
Droplets of Dipolar Mixtures, Phys. Rev. Lett. 126, 025301(2021).
[30] J. C. Smith, D. Baillie, and P. B. Blakie, Quantum Droplet
States of a Binary Magnetic Gas, Phys. Rev. Lett. 126, 025302(2021).
[31] J. C. Smith, P. B. Blakie, and D. Baillie, Approximate theories for binary magnetic quantum droplets, Phys. Rev. A 104,
053316 (2021).
[32] F. Wächtler and L. Santos, Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einsteincondensates, Phys. Rev. A 94, 043618 (2016).
[33] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Bogoliubov
modes of a dipolar condensate in a cylindrical trap, Phys. Rev.
A 74, 013623 (2006).
[34] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.
Flannery, Numerical Recipes: The Art of Scientific Computing,
3rd ed. (Cambridge University Press, New York, 2007).
[35] M. Modugno, L. Pricoupenko, and Y. Castin, Bose-Einsteincondensates with a bent vortex in rotating traps, Eur. Phys. J.
D 22, 235 (2003).
[36] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T.
Pfau, Self-bound droplets of a dilute magnetic quantum liquid,
Nature (London) 539, 259 (2016).
[37] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6, 041039(2016).

[38] Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and B. L. Lev,s-wave scattering lengths of the strongly dipolar bosons 162 Dy
and 164 Dy, Phys. Rev. A 92, 022703 (2015).
[39] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen, M. Isoard,
S. Stringari, and T. Pfau, Scissors Mode of Dipolar Quantum
Droplets of Dysprosium Atoms, Phys. Rev. Lett. 120, 160402(2018).
[40] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C.
Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M.
Sohmen, M. J. Mark, and F. Ferlaino, Long-Lived and TransientSupersolid Behaviors in Dipolar Quantum Gases, Phys. Rev. X
9, 021012 (2019).
[41] P. Ilzhöfer, G. Durastante, A. Patscheider, A. Trautmann, M. J.
Mark, and F. Ferlaino, Two-species five-beam magneto-opticaltrap for erbium and dysprosium, Phys. Rev. A 97, 023633(2018).
[42] J. H. Becher, S. Baier, K. Aikawa, M. Lepers, J.-F. Wyart, O.
Dulieu, and F. Ferlaino, Anisotropic polarizability of erbium
atoms, Phys. Rev. A 97, 012509 (2018).
[43] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, S. Tzanova, E.
Kirilov, and R. Grimm, Accurate Determination of the Dynamical Polarizability of Dysprosium, Phys. Rev. Lett. 120, 223001(2018).
[44] M. Modugno, F. Ferlaino, F. Riboli, G. Roati, G. Modugno,
and M. Inguscio, Mean-field analysis of the stability of a K-Rb
Fermi-Bose mixture, Phys. Rev. A 68, 043626 (2003).
[45] A. H. Hansen, A. Y. Khramov, W. H. Dowd, A. O. Jamison,
B. Plotkin-Swing, R. J. Roy, and S. Gupta, Production ofquantum-degenerate mixtures of ytterbium and lithium with
controllable interspecies overlap, Phys. Rev. A 87, 013615(2013).
[46] F. Wang, X. Li, D. Xiong, and D. Wang, A double species 23 Na
and 87 Rb Bose–Einstein condensate with tunable miscibility via
an interspecies Feshbach resonance, J. Phys. B: At., Mol. Opt.
Phys. 49, 015302 (2015).
[47] S. B. Papp, J. M. Pino, and C. E. Wieman, Tunable Miscibility
in a Dual-Species Bose-Einstein Condensate, Phys. Rev. Lett.
101, 040402 (2008).
[48] K. L. Lee, N. B. Jørgensen, I.-K. Liu, L. Wacker, J. J.
Arlt, and N. P. Proukakis, Phase separation and dynamics oftwo-component Bose-Einstein condensates, Phys. Rev. A 94,
013602 (2016).
[49] K. E. Wilson, A. Guttridge, I.-K. Liu, J. Segal, T. P. Billam,
N. G. Parker, N. P. Proukakis, and S. L. Cornish, Dynamics ofa degenerate Cs-Yb mixture with attractive interspecies interactions, Phys. Rev. Research 3, 033096 (2021).
[50] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S.
Giovanazzi, P. Pedri, and L. Santos, Observation of DipoleDipole Interaction in a Degenerate Quantum Gas, Phys. Rev.
Lett. 95, 150406 (2005).
[51] F. Böttcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Langen,
I. Ferrier-Barbut, T. Pfau, R. Bombín, J. Sánchez-Baena, J.
Boronat, and F. Mazzanti, Dilute dipolar quantum droplets
beyond the extended Gross-Pitaevskii equation, Phys. Rev.
Research 1, 033088 (2019).
[52] G. Durastante, C. Politi, M. Sohmen, P. Ilzhöfer, M. J. Mark,M. A. Norcia, and F. Ferlaino, Feshbach resonances in an
erbium-dysprosium dipolar mixture, Phys. Rev. A 102, 033330(2020).

023304-8

INTERSPECIES INTERACTIONS IN AN ULTRACOLD …

PHYSICAL REVIEW A 105, 023304 (2022)[53] We notice that the expression in Eq. (A8) of Ref. [33] canbe used only if n(r) is even under parity in the three spatial
directions, namely when ñ(−k) = ñ(k).
[54] H. Li, J.-F. Wyart, O. Dulieu, and M. Lepers, Anisotropic
optical trapping as a manifestation of the complex elec-

tronic structure of ultracold lanthanide atoms: The example ofholmium, Phys. Rev. A 95, 062508 (2017).
[55] P. Young, Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask (Springer, Berlin,2015).

023304-9

Chapter 4
Two-dimensional supersolidity indysprosium
In 2017 the ERBIUM experiment in Innsbruck observed the roton instability in a
dipolar quantum gas [45]. Such a find provided first hints of the existence of a novel
state of matter, namely a supersolid state, in analogy with the state predicted inhelium. Later in 2019, in our experiment and in the ERBIUM experiment, the first
dipolar supersolids were realized with 164 Dy and 166 Er, respectively [42]. Simultaneous
results where achieved in the group of Giovanni Modugno [182] in Pisa (Italy), andin the group of Tilman Pfau [32] in Stuttgart (Germany). The realization of adipolar supersolid triggered many novel theoretical, as well as experimental works,
e.g. low-lying excitation modes (Goldstone and Higgs modes) [75, 84, 184], excitation
spectrum [132, 151], superfluid fraction [136, 165, 183], out-of-equilibrium dynamics [89],and many more. One method to realize a dipolar supersolid is by first preparing a
Bose-Einstein condensate at as > add , with ϵdd < 1, and then reach the supersolid
regime by ramping the scattering length to lower values, in order to have ϵdd > 1 (seeChapter 1).
This chapter introduces a different preparation scheme to achieve supersolidity
by direct evaporative cooling from a thermal cloud. The first supersolid states wereformed by three or four droplets aligned along a single direction. The second sectionof this chapter points out the fundamental steps necessary to extend the supersolidityfrom one to two dimensions, first in a zig-zag pattern and finally in a hexagonal pattern.
Lastly, the third section describes the possibility offered by two-dimensional states ofstudying interesting excitation modes, as well as vortices in supersolid states.

54

4.1

Two-dimensional supersolidity in dysprosium

Direct evaporation into the supersolid stateErbium and dysprosium present many stable isotopes with relatively high abundance,
whose interaction properties are different, offering a powerful knob to tune in theexperiment. In contrast to 166 Er and 162 Dy (used in Stuttgart [32] and in Pisa [182]),
to prepare a supersolid state we loaded our atomic cloud with the isotope 164 Dy. The
special property of this isotope is the background scattering length abg , which is smaller
than add , where for 164 Dy is add = 131 a0 . Figure 4.1 shows an illustration of how
the dipolar strength changes with the magnetic field. Figure 4.1(a) shows the 164 Dy
case, in which the scattering length can be tuned away from the Feshbach resonanceto reach the supersolid regime with ϵdd > 1. Figure 4.1(b) shows instead the 166 Er
and the 162 Dy case where, to reach the dipolar dominated regime, the magnetic field
has to be tuned close to the Feshbach resonance. This caused big losses and a short
lifetime of the supersolid state due to three-body recombination losses. In fact, while
the other groups, using 166 Er and 162 Dy, observed a lifetime on the order of tens of
ms, the 164 Dy supersolid survived for several hundreds of ms [42], when prepared via
interaction ramp.
The long lifetime motivated us to develop a new preparation scheme, in which
we performed direct evaporative cooling into the supersolid state from a thermal
cloud [42, 177]. In this way, instead of moving horizontally in the T = 0 phase diagram(see Sec. 1.3), we cross the supersolid phase at fixed scattering length from the third
dimension, given by the temperature. Although, for linear supersolids, where thedroplets are aligned along a single direction, the interaction-ramp protocol can be
weakly first order or continuous [18, 23] and therefore causing only small excitations, thedirect-evaporative cooling protocol proved to be essential in realizing two-dimensional
supersolidity and allowed us to reach lifetime on the order of seconds (see Sec. 4.7).
The formation of a supersolid state requires the breaking of two continuous symmetries: the phase invariance of the superfluid and the translational invariance of thecrystal. When accessing the supersolid state from a thermal cloud, the initial stateof the system has neither phase coherence nor density modulation. Therefore, it isnatural to ask whether the two symmetries are broken simultaneously and, if not, which
one occurs first. In our publication in Sec. 4.4, by following the evaporative-coolingprocess of the atomic cloud, from several hundreds of nK down to tens of nK, we could
gain information on the development of the supersolid state. Our first experimental
observation pointed at the initial formation of a quasi-condensate crystal, where phasecoherence is localized within the droplets, and later, when lowering the temperature,
the creation of the supersolid state with the coexistence of density modulation and55

4.1 Direct evaporation into the supersolid state
2

a.

b.

dd

1.5
1
0.5

B (G)

B (G)

Figure 4.1 Illustration of the dipolar strength ϵdd as a function of the magnetic field
B, for 164 Dy (a) and for 166 Er and 162 Dy (b). In the former case, the condition of
abg < add allows us to achieve a supersolid state by moving away from the Feshbachresonance. On the other hand, in the latter case, abg > add . Therefore, to reach the
dipolar dominated regime, the magnetic field is tuned closer to the Feshbach resonancecausing large three-body recombination losses.
global phase coherence [177]. Furthermore, when looking at the modulation contrast
for similar atom number but different temperatures, we observed a higher degree ofmodulation for higher temperatures, underlining the important role that finite temperature plays in the supersolid formation. These findings triggered many theoreticalworks aimed at improving the understanding of the finite-temperature phase diagramof dipolar gases. Section 4.1.2 and 4.9 show how, not only quantum fluctuations, butalso thermal fluctuations have a fundamental role in the formation of a supersolid state.
In particular, higher temperatures lower the roton gap in the excitation spectrum andfavour the appearance of density modulation.

4.1.1

Probing supersolidity in the experiment

As evidence of supersolidity in the system, both density modulation and global phasecoherence need to be proven. In this regard, we use two different imaging techniques:
one probing the insitu density distribution, from which we determine the density
modulation and therefore the crystalline nature of the state, and a second one probing
the momentum distribution after a TOF expansion, from which we determine theglobal phase coherence and therefore the superfluid nature of the state1 .
1The recorded images taken after TOF expansion give also information on the amount of density

modulation. More details will be given later in this section.

56

Two-dimensional supersolidity in dysprosium

Insitu imaging. To probe the insitu density distribution, we shine far-detuned lightonto the atoms and perform phase-contrast imaging (see Chapter 2). Fig. 4.2a shows
the insitu density distribution n(x, y) of a linear supersolid state made of 164 Dy atoms.
By integrating n(x, y) along the direction orthogonal to the modulation, we obtain the
integrated density profile ny (x) (Fig. 4.2b), which shows a crystalline structure withperiodicity lx . The periodicity depends on the trap geometry and on the scattering
1/4
length through the relation 1/lx ∝ κ ϵdd /lz , where lz is the harmonic oscillator length
along the dipole orientation2 and κ is a geometrical factor only dependent on fz /fy [45].
To gain quantitative information on the amount of modulation present in the system,
we perform the Fourier transform of ny (x). This profile shows peaks in the momentum
space at ±k ∗ ≃ 2π/lx . By taking the ratio of the Fourier component at k ∗ and theone at zero momentum, we determine the insitu density modulation M (see Fig. 4.2c).
Note that, at the moment, the insitu imaging does not allow us to clearly resolve
the low-density region between the droplets and extract direct information on the
superfluid link. The low-density area is indeed more sensitive to fluctuations of the
background, which can lead in some cases to unphysical negative values. Futhermore,
for large detunings, the atomic susceptibility can assume a small non-zero value, whichmakes the dense atomic cloud behave as a lens. This effect is strongly reduced inphase-contrast imaging, but getting rid of it completely remains challenging due to thecoupled dependence on detuning and intensity of the light. For this reason, to probe
the superfluidity of the system, we look at the matter-wave interference pattern after
TOF expansion.
Time-of-Flight imaging. To probe the global phase coherence of the state, we
image the cloud after a TOF expansion by shining resonant light and performing
standard absorption imaging (see Chapter 2). For an unmodulated dipolar BEC, therecorded picture shows a standard bimodal distribution, with a dense peak given by
the condensed atoms and a broad contribution given by the thermal component. For
a modulated system, the dense peak develops a pattern arising from the interference
between the expanding droplets, with maxima at ±k ∗ ≃ 2π/lx . Fig. 4.3 displays an
example of absorption picture taken after 35ms TOF for a linear modulated state, inthe supersolid (a) and in the isolated-droplet (b) regimes. By recording the interference
pattern for several runs under the same experimental parameters, it is possible todetermine whether the system has global phase coherence. Indeed, on the one hand, if
each droplet shared the same phase in trap, the interference pattern is reproducible
2 1/l = 2πzpmνz /h.

57

4.1 Direct evaporation into the supersolid state
a. -5

c.
1

0

0.8

0.6

5

b.

100
80

0.4

60
40

0.2

20
0

-10

-8

-6

-4

-2

0

2

4

6

8

10

0

-6

-4

-2

0

2

4

6

Figure 4.2 (a) In-trap density distribution n(x, y) of a linear supersolid measured
from phase-contrast imaging. The low-density link can not be reliably resolved (seediscussion in the insitu imaging paragraph). (b) 1D integrated profile ny (x). (c)Absolute value of the Fourier transform of ny (x) showing a central peak at k = 0 andsecondary peaks at ±k ∗ .
from shot to shot, and survives after averaging the pictures over several experimental
runs. On the other hand, if the droplets had all different phases, the interference
pattern is not reproducible, exhibiting peaks with random amplitudes from shot toshot, and gets washed out in the average [79] (see Fig. 4.3(a,b)).
To quantify the amount of global phase coherence, as for the insitu pictures,
we integrate the momentum distribution n(kx , ky ) along the direction orthogonal tothe modulation and perform the Fourier transform (see Fig. 4.3(c-f)). The Fouriertransform ñy (x) holds the information on both the degree of modulation and global
phase coherence of the system, through the complex phasors Pi = ρi exp(−iΦi ), with
ρi and Φi its magnitude and phase, respectively. We define the phasor amplitudes
at k ∗ , AM = ⟨|Pi |⟩ and AΦ = |⟨Pi ⟩|, which carry the information on the amount ofdensity modulation and global phase coherence, respectively. For an unmodulated state
AM = 0, whereas, for a modulated and fully coherent state AM = AΦ . Fig. 4.3(e,f)
show ñy (x) in the supersolid and isolated droplets case, respectively. Additionally,by recording the phase Φi for each experimental run, it is possible to get a further
estimation of the amount of global phase coherence. The insets in Fig. 4.3(e,f) show
the polar scatter plots of the phase for two cases: supersolid state and isolated droplet
state, respectively. In the former, all the points cluster in a small region of the polar
plot whereas, for the latter, the points are spred all over the 2π region.

58

Two-dimensional supersolidity in dysprosium
a.

c.

20

b. 20

10

10

0

0

-10

-10

-20

-20

0.8

d.

0.6

0.6

0.4

0.4

0.2

0.2

0
-40

e.

0.8

-20

0

20

0
-40

40

0.6

f.

0.5

0.3

0.3

0.2

0.2

0.1

0.1
-0.5

0

0.5

1

1.5

20

40

0.5
0.4

-1

0

0.6

0.4

0
-1.5

-20

0
-1.5

-1

-0.5

0

0.5

1

1.5

Figure 4.3 (a,b) Absorption images taken after 35ms TOF and averaged over 100
shots, for a modulated state as in Fig. 4.2a, in the supersolid and isolated droplet
regime, respectively. (c,d) Integrated density profiles for several runs under the same
experimental parameters (gray lines) and the average (black line) for a supersolid state
and an isolated droplet array, respectively. (e,f) Fourier transform of the integrated
profiles (gray lines) and nM = ⟨|ñy (x)|⟩ (red line) and nΦ = |⟨ñy (x)⟩| (blue line) for a
supersolid state and an isolated droplet array, respectively. The insets show the phaseΦi for each experimental run.

4.1.2

The role of finite temperature

The role played by the temperature in the phase transition to a crystalline phasehas drawn particular interest in the community to bridge theory and experiment.
Indeed, in the experiment, the temperature of the condensed cloud has a finite value,
while theoretical predictions are usually done assuming zero temperature. Recently,
we carried out studies in collaboration with the group of Thomas Pohl at Aarhus
University on the role of thermal fluctuations in dipolar systems. Our findings reveal

4.1 Direct evaporation into the supersolid state

59

that the interaction between condensed atoms and thermal excitations can have afocusing effect on the density of condensed atoms, facilitating the appearance of density
modulation [180]. This behaviour can be understood by looking at how the relevantenergy terms scale with the condensate density and interactions. In particular, theenergy term describing the quantum fluctuations (Eq. 1.6) is positive, increases withthe density of condensed atoms and with the interactions and stabilizes the systemagainst the collapse. The interaction between the condensed atoms and the thermal
excitations is also positive and can be described by the following energy term:
Eth =

Z

dk
τk
,
Ṽ
(k)f
(r)
k
(2π)3
ϵk (r)(4.1)

where Ṽ (k) is the Fourier transform of the total interaction potential, including contact interaction and long-range DDI, fk = 1/(eϵk /kB T − 1) is the Bose distribution
2 2of theq thermal excitations, τk = h̄ k /(2m) is the kinetic energy of the atoms, andϵk = τk (τk + 2|ψ(r)|2 Ṽ (k)) is the dispersion relation of the excitations. A more detailed description of the theoretical formalism can be found in the publication in Sec. 4.9.
This energy term decreases when the density of condensed atoms increases, because
higher density of condensed atoms means less thermal excitations. Therefore at finite
temperatures, in seeking the lowest energy possible, higher densities are preferred [7]leading to a decrease in the roton energy gap ∆rot , supporting density modulation.
Another way of understanding this behaviour is by looking at the population of theexcitation modes. At finite temperatures, the roton mode has a higher probability to
be occupied. A population of this mode is associated with a modulation of the density
in real space at the roton wavelength. This modulation locally increases the density,
which in turn lowers the roton gap.
These theoretical findings agree well with our experimental observations and shedsome light on the role of finite temperatures on the superfluid-supersolid phase transition. Figure 4.4a shows the average integrated profiles ny (x), for a dysprosium cloud
prepared via evaporative cooling at fixed scattering length, with similar condensed
atom numbers but different temperatures. The cloud at higher temperature shows astronger modulation, which is also evident in the Fourier transform (Fig. 4.4b).
A decrease of the roton gap for higher temperatures was also observed in liquid
helium [56, 93], where the elementary excitation spectrum is experimentally addressed
via inelastic neutron scattering. Furthermore, theoretical predictions on the softeningof the roton mode with increasing temperatures were done in BECs with light-induced
dipole-dipole interactions, where long-range interactions can be engineered by irradi-60

Two-dimensional supersolidity in dysprosium

ating a cigar-shaped BEC with far off-resonant laser light [125]. Finally, theoreticalworks on Bose gases in reduced dimensions underlined the importance of thermal
fluctuations and predicted the appearance of a thermally induced roton in contact
interacting two-dimensional gases [134], and a decrease and a shift of the roton mode
for increasing temperatures [200].
100

103

6

a.

T = 107(4)nK
T = 77(8)nKb.

T = 107(4)nK
T = 77(8)nK

5

80

4
60
3
40

2

20

0
-10

1

-5

0

5

10

0

0

1

2

3

4

5

Figure 4.4 (a) Averaged integrated density profile of a 164 Dy cloud measured at two
different temperatures: T = 107(4) nK (red line) and T = 77(8) nK (blue dash-dottedline). The condensed and the total atom number are 1.42(4) × 104 and ∼ 7 × 104 ,respectively, for the profile at higher temperature, and 1.34(7)×104 and ∼ 4×104 for the
profile at lower temperature. The trap geometry is cigar shaped with trap frequencies
ωx,y,z = 2π × (36, 88, 141)s−1 (b) Fourier transform of the averaged integrated profiles,
showing a peak at the modulation wavevector for the system at higher temperature.

4.2

From one to two-dimensional supersolidity

The previous section described the main tools used in our experiment to create - direct
evaporative cooling - and probe - phase-contrast and absorption imaging - supersolidity.
As shown in Fig. 4.2a, the early supersolid states consisted of few droplets – threeor four – aligned along a single direction, i.e. the elongated axis of a cigar-shaped
trap. The clear further step to realize a supersolid state closer to the predicted bulk
supersolids [31] was to extend the supersolid properties along a second direction andachieve two-dimensional supersolidity with a larger number of droplets.
The first two-dimensional states showing density modulation were realized in2016 [91], but the system did not show any global phase coherence. It was not clear,from the beginning, whether it was possible to realize periodic structure and global
phase coherence in systems with atom numbers that where realistically achievable in4.2 From one to two-dimensional supersolidity

61

the actual experiments. Most theoretical works predicted two-dimensional supersolidity
for unconfined systems [99], systems confined only along the dipole orientation [204],or for atom numbers larger than the ones reported in the experiments [165]. The firstsupersolids were indeed realized with ∼ 2 × 104 atoms [42, 89, 177]. It was clear thatto achieve density modulation and global phase coherence along two directions, and alarger system’s size, a higher number of atoms was needed to build up the superfluid
bath necessary for the particles to flow between droplets, and allow the developmentof global phase coherence.
In particular, the quantity that plays a key role in the transition from 1D to 2D
supersolid states is the averaged 2D density ρ = Nc fx fy [154]. In fact, for a fixed atom
number, when loosening the trap confinement, e.g. fy , the supersolidity is lost and theground state of the system is a standard dipolar BEC. To maintain supersolidity, the2D density needs to be kept constant, by either increasing the atom number Nc or fx .
Since in the experiment Nc can not be increased indefinitely, to achieve a transition toa 2D supersolid state, Nc can be kept constant while varying fx and fy accordingly tokeep ρ constant. More details can be found in the publication shown in Sec. 4.6.
In the experiment, after improving the condensed atom number up to ∼ 7 × 104 ,we were able to produce states with up to 10 droplets. To reach supersolidity along
two dimensions, we employed a similar approach already seen with ion crystals, andperformed a structural phase transition from a linear chain of droplets to an arrange ofdroplets in a zig-zag pattern. In crystals made of singly-charged ions, the location of theions and therefore the system’s structure, arises from the competition between the longrange Coulomb repulsive interaction and the harmonic trapping confinement, usuallyprovided by a Paul or a Penning trap. By tuning the trap frequencies independently,
or by fixing the trap and increasing the ion number, the ions can arrange in a linear,
zig-zag or helical configuration [19, 160].
In our system, the DDI plays the role of the Coulomb interaction and it is the competition between the contact and the long-range interaction and the trap confinementthat induces the transition to a zig-zag supersolid state. In particular, by changing
the power of the laser beams that generate our crossed optical dipole trap3 , we wereable to prepare the supersolid state at different trap frequencies, while weakening theconfinement orthogonal to the droplet direction and increasing the atom number, until
the system performed a structural transition from a linear chain to a zig-zag pattern.
Figure 4.5 shows the structural phase transition from a linear to a zig-zag pattern for asystem made of ion crystals and for our dipolar supersolid made of 164 Dy atoms. While
3 See Chapter 2.2.1 for details on the optical dipole trap.

62

Two-dimensional supersolidity in dysprosiumin the former case releasing the transverse confinement is sufficient to achieve a zig-zag
pattern, for a dipolar gas, to maintain supersolidity, while lowering the transverse trap
frequency, the atom number needs to be increased.
ion crystals

dipolar gasese.

b.
10 4 )

6

Nc (lower transverse
confinement

a.

c.

4

d.
2

0.3

0.34

0.38

Trap aspect ratio

0.42t

Figure 4.5 Structural phase transition from a linear chain to a zig-zag pattern in ionic
crystals (a,c) and in a ultracold dipolar gas of 164 Dy atoms (b,d). In the former case, thetransition to the zig-zag pattern is obtained by releasing the transverse confinement. In
the latter case, to maintain supersolidity, the trap aspect ratio changes from αt = 0.32(b) to αt = 0.43 (d), while the condensed atom number increases from 3.1(3) × 104 (b)to 6.7(3) × 104 (d). The trap frequencies along the other two directions are fixed tofx = 33(2)Hz and fz = 167(1)Hz. (e) Condensed atom number as a function of thetrap aspect ratio, αt = fx /fy . The blue and green data points correspond to the insitu
density distribution shown in (b) and (d), respectively. Figures (a,c) adapted from[159]. Figures (b,d) adapted from [137].
As mentioned in the previous section, to evaluate the global phase coherence ofthe supersolid state, we switch off the trap and let the droplet fall. While expanding,they interfere with each other and the resulting intereference pattern is reminiscentof the in-trap geometry. Figure 4.6 shows the interference pattern calculated forthree different in-trap droplets arrangements: a triangular, a square and a diamondgeometry. The interference pattern are obtained by considering free expanding Gaussian
droplets with a phase randomly chosen in the narrow range 0.1 × [0, 2π] to emulate
the supersolid regime, and averaging over 50 trials, as done in the measurement
shown in the publication in Sec. 4.5. The peaks distance and the configuration of
the interference pattern resemble the in-trap droplets arrangement, with the hexagon
pattern reproducing quantitatively well the experimental one. The amount of phasecoherence changes the sharpness of the peaks and blurs the average pattern. Figure 4.7
shows the interference pattern for an insitu triangular geometry (as in the experiment)
and for three different phase distributions: 0.1 × [0, 2π], 0.5 × [0, 2π], and 1 × [0, 2π],
where the first and the last represent the case of supersolid and isolated droplets,
respectively.

63

4.2 From one to two-dimensional supersolidity
-20

a.

b.

c.

-10
0
10
20
-20
-40

-10

0

10

d.

20 -20-10

010

20 -20

e.

-10

0

10

20

-20

0

20

40

f.

-20
0
20
40
-40

-20

0

20

40 -40-20

020

40 -40Figure 4.6 Calculated average interference pattern for three different in-trap geometrical
droplets configurations: triangular (a,d), square (b,e) and diamond (c,f). (a-c) In-trap
droplet configuration. (d-f) Calculated interference pattern for the in-trap configurationof (a-c), respectively. Each droplet is pictured as an expanding Gaussian with a random
phase in the narrow range 0.1 × [0, 2π]. The resulting interference patterns are averaged
over 50 trials, as in the experiment.
-40

a.

b.

c.

-20
0
20
40
-40

-20

0

20

40 -40-20

020

40 -40

-20

0

20

40

Figure 4.7 Calculated average interference pattern for three different droplets phase
distributions: (a) 0.1 × [0, 2π], (b) 0.5 × [0, 2π], and (c) 1 × [0, 2π]. Orange boxes:
experimental interference pattern obtained after 36ms TOF from absorption imaging
for a supersolid (left) and an isolated droplet state (right). Figures in orange boxes
adapted from [137].

4.2.1

Supersolidity in a circular trap

After the experimental realization of two-dimensional supersolidity, the natural next
step was to go from a zig-zag state towards an isotropic state in a cicular trap. At
this stage, in the experiment, the atomic cloud was trapped in a crossed optical dipole
trap made of two laser beams crossing at an angle of 45°. One of this laser beam is

64

Two-dimensional supersolidity in dysprosium

modulated to create a time-averaged potential with a Gaussian shape, whose effective
waist in one direction can be tuned. The second dipole trap has instead a fixed waist.
Due to this geometrical configuration, the maximum trap aspect ratio was limited toαt = 0.43. To be able to increase the aspect ratio to 1 and reach a circular trap, we
implemented a third optical dipole trap (see Chapter 2) with similar specifications to
the optical dipole trap with fixed waist and that crosses this one at an angle of 90°.
In this way, we have an individual control over the trap frequencies by changing thepower of the three laser beams independently.
In the publication presented in Sec. 4.7, we could realize a hexagon state made of
7 droplets in a circular trap, as predicted from theory. These two-dimensional states
have a lifetime beyond one second. The process of directly evaporating from a thermal
cloud has a fundamental role in realizing robust supersolid states in pancake traps.
By comparing the dynamic formation of a circular supersolid via an interaction ramp
with a temperature ramp, it can be shown that the former leads to a highly excitedstate, preventing global phase coherence at realistic timescales. Whereas, the latterallows the system to achieve global phase coherence soon after the temperature ramp.
Indeed, while for linear systems one roton mode can connect the superfluid state to
the supersolid state with a continuous phase transition4 , in 2D systems the transition
is predicted to be discontinuous leading to strong excitations, which interfere withthe development of global phase coherence across the system [27]. Our theoreticalsimulations of the interaction ramp from a dipolar superfluid to a supersolid show that
this process follows a crystal-growth type of formation, where the central droplets form
and are then succeeded by an outer ring of droplets, suggesting that the dipolar BEC
cannot be linked to the supersolid by a single roton mode as in the 1D case. Moredetails about the theoretical model are reported in the publication shown in Sec. 4.7.

4.3

Excitations in two-dimensional supersolids

The realization of the hexagon state opens up a whole new experimental research on the
excitations of two-dimensional supersolids. In this regard, we investigated the role thatangular oscillations can play in determining the superfluid fraction (see publicationin Sec. 4.8). Furthermore, supersolids in circular traps pave the way for the study ofvortices [165] and persistent currents [185].
4 Note that, in 1D systems, the phase transition from a superfluid to a supersolid can be continuousor discontinuous depending on the density [18, 23].

4.3 Excitations in two-dimensional supersolids

4.3.1

65

Scissors mode and angular oscillations

In the quest for supersolidity, one of the major result was to be able to prove thesuperfluid nature of the system. In our group, we probe the global phase coherence ofthe supersolid state by performing matter-wave interference experiments and looking
at the resulting pattern (see Sec. 4.1.1). Another way of quantifying the superfluidityof the system can be through the excitation of the scissors mode [74]. The scissors
mode is a counter rotation of the atomic cloud with respect to the axes of the trap.
It has been widely employed to infer the superfluidity in several systems as trapped
BEC [122], Fermi gases [196], 2D Bose gases [167], dipolar quantum droplets [63], andnulcei [29], where protons and neutrons can oscillate out of phase. In the experimentit can be excited by a sudden rotation of the trap.
Given the trap ellipticity ϵ = (ωx2 −ωy2 )/(ωx2 +ωy2 ), where ωx,y are the trap frequencies
in the plane of rotation, and the atomic cloud ellipticity β = ⟨x2 − y 2 ⟩/⟨x2 + y 2 ⟩, theangular oscillation frequency ωsc can be related to the moment of inertia as below [183]:Θ = Θc ϵβωx2 + ωy2
,
2
ωsc

(4.2)

where Θ is the moment of inertia and Θc is its classical value. The angular oscillation
frequency strongly depends on whether the system is superfluid or not. Following
Leggett’s definition for the superfluid fraction fs , Θ = (1 − fs )Θc [111], which for non
isotropic trap can be written as:
Θ = (1 − fs )Θc + fs β 2 Θc ,(4.3)

and using Eq. 4.2, one can derive the angular oscillation frequency for a fully superfluid
system (fs = 1) and for a normal fluid performing rigid body rotation (fs = 0). These
are given by
SF
ωsc
=

q

(ωy2 − ωx2 )/β,

RBR
ωsc
=

(ωy2 − ωx2 )β,

q

(4.4)

for superfluid and normal fluid, respectively. Due to the density modulation, a supersolid
RBR andshould respond to a scissors-mode excitation with a frequency ωsc between ωsc
SF .
ωsc
From the relations above, it is visible that both a change in the superfluid fraction
and in the atomic ellipticity lead to a change in the moment of inertia and therefore

66

Two-dimensional supersolidity in dysprosiumin ωsc , with no possibility of distinction between superfluid and normal fluid for very
elongated systems where β ≃ 1 [165]. For that reason, exciting the scissors mode to
extract information on the superfluid fraction of a linear supersolid has some limitations.
Indeed, the transition from an unmodulated dBEC to a supersolid state is accompanied
by a dramatic change in the atomic ellipticity β. In addition, the supersolid stateis characterized by interconnected droplets surrounded by a superfluid bath, which
we refer to as "halo". This halo feeds the droplets and the low density link between
them, which is responsible for the phase coherence of the system. When crossing
the transition, there is a change in the density of the halo and of the interdroplet
connection. Furthermore, in linear systems, the scissors mode is always orthogonal tothe droplet axis, which weakens any conclusion on the exact value of the superfluid
link between the droplets.
In this regard, the possibility of realizing two-dimensional supersolids and having
control on β allows us to investigate this aspect deeper. Indeed, in two dimensional
systems, when crossing the transition from a dBEC to a supersolid state, β stays
almost constant. Furthermore, the angular oscillation involves motion that partiallyhappens along the droplet axis. In the publication shown in Sec. 4.8, surprisingly, by
exciting the scissors mode with a sudden rotation of the trap, we observe an angularoscillation frequency that is always close to the case of fs = 1, for several scattering
lengths corresponding to a standard dBEC, a supersolid and an isolated droplet regime,
and for different values of β. These experimental findings are well reproduced by realtime simulations based on the eGPE, where a similar protocol for the excitation of thescissors mode and a similar analysis of the oscillation dynamics are performed.
A possible explanation for these observations can be given by the multi-frequency
response of the system to the trap excitation. Equation 4.4 is in fact valid under
the assumption of shape preserving modes and single-frequency response and, inthe case of many frequencies, ω SF and ω RBR are only an upper bound for ωsc . In
the experiment the high frequency angular oscillation modes dominate over the low
frequency ones. Therefore, to infer the superfluid fraction, a different experimental
protocol able to address the scissors mode needs to be implemented [166]. Furthernore,such a low frequency can not easily be probed in the actual experiment, due to e.g. alimited lifetime of the supersolid state. Due to the aforementioned limitations, different
techniques might be more reliable to probe the superfluid nature of the supersolid, as
the observation of quantized vortices.

4.3 Excitations in two-dimensional supersolids

4.3.2

67

Outlook: vortices in dipolar supersolids

Quantized vortices represent a strong evidence of superfluidity and they have been
observed in different ultracold systems, as BECs [1, 121, 124] and Fermi gases [205]. In
dipolar supersolids, the presence of high density peaks connected by low density regions
is predicted to modify the properties of the vortices, e.g. critical angular velocity,
angular momentum per particle, and vortex lattice geometry [66]. In particular, thecritical angular velocity to nucleate vortices, which follows the relation Ωc = ωq /2, with
ωq the frequency of the quadrupolar mode, decreases in the supersolid regime. Thisis given by the low density inter-droplet regions that make the nucleation of vortices
energetically more favorable. The angular momentum per particle carried by the vortex
is predicted to be smaller than h̄, due to the reduction of superfluidity in the supersolidstate. Furthermore, for large angular velocities, the presence of high density peaks,
arranged in a triangular pattern, modifies the vortex lattice. The vortices are pinned
in low density regions between the droplets and form a honeycomb lattice [206].
Recently, in our experiment we demonstrated the first realization of quantizedvortices in a dipolar ultracold gas made of dysprosium atoms (see Appendix A.5). So
far, vortices have been created by rotating a slightly anisotropic trap [1], by stirring thesystem with a focused laser beam [121], by rapidly crossing the atomic sample with an
obstacle (e.g. blue detuned laser beam) [105, 133], or by Kibble-Zurek mechanisms [192],
as topological defects arising from a rapid quench of the system. In dipolar systems,
the magnetostriction - dipole alignment along the magnetic field direction - offers analternative way of deforming the atomic cloud, through the anisotropic nature of the
DDI and impart angular momentum into the system. Indeed, by rotating the magnetic
field at a constant angular velocity, above the critical one Ωc , vortices appear in the
dysprosium cloud.
By using a similar technique, but preparing the cloud at a lower scattering length
during evaporative cooling, it could be possible to nucleate vortices in the circularsupersolid state. The major limitation is given by the ability of detecting vortices in
the low density region between the high density peaks. A recent theoretical proposal
suggested a protocol to nucleate and detect vortices in supersolid states based ona interaction ramp from a superfluid to a supersolid state and then back to thesuperfluid [206]. The starting condition is a rotating dipolar BEC at an angularvelocity below the critical one. By reducing the scattering length, the system enters
the supersolid state. Due to the lower value of the critical angular velocity, a vortex
can be nucleated in the low density region. For detection, the scattering length can be

68

Two-dimensional supersolidity in dysprosium

increased again, melting the supersolid into an unmodulated dBEC with a vortex at
the center, which can now be detected.

4.4 Publication II:
Birth, life, and death of a dipolar supersolid

4.4

69

Publication II:

Birth, life, and death of a dipolar supersolid
The following publication has appeared in
Physical Review Letters 126, 233401 (2021)∗
submitted 19 Jan 2021; published 7 Jun 2021
DOI: https://doi.org/10.1103/PhysRevLett.126.233401
M. Sohmen,1,2 C. Politi,1,2 L. Klaus,1 L. Chomaz,1 M. J. Mark,1,2 M. A. Norcia,1and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria∗The author of the present thesis performed the measurements together with M. S., M. A. N., andL. K., analysed the data with M. S. and M. A. N., and contributed in writing the manuscript andinterpreting the results together with all the authors.

PHYSICAL REVIEW LETTERS 126, 233401 (2021)
Editors' Suggestion

Featured in Physics

Birth, Life, and Death of a Dipolar Supersolid

1

Maximilian Sohmen ,1,2 Claudia Politi ,1,2 Lauritz Klaus ,1,2 Lauriane Chomaz ,2,† Manfred J. Mark ,1,2Matthew A. Norcia ,1 and Francesca Ferlaino 1,2,*

Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria(Received 19 January 2021; accepted 19 April 2021; published 7 June 2021)
In the short time since the first observation of supersolid states of ultracold dipolar atoms, substantial
progress has been made in understanding the zero-temperature phase diagram and low-energy excitationsof these systems. Less is known, however, about their finite-temperature properties, particularly relevant forsupersolids formed by cooling through direct evaporation. Here, we explore this realm by characterizing
the evaporative formation and subsequent decay of a dipolar supersolid by combining high-resolution intrap imaging with time-of-flight observables. As our atomic system cools toward quantum degeneracy, it
first undergoes a transition from thermal gas to a crystalline state with the appearance of periodic density
modulation. This is followed by a transition to a supersolid state with the emergence of long-range phasecoherence. Further, we explore the role of temperature in the development of the modulated state.
DOI: 10.1103/PhysRevLett.126.233401

Supersolid states, which exhibit both global phasecoherence and periodic spatial modulation [1–7], haverecently been demonstrated and studied in ultracold gasesof dipolar atoms [8–10]. These states are typically accessedby starting with an unmodulated Bose-Einstein condensate(BEC) and then quenching the strength of interatomic
interactions to a value that favors a density-modulated state.
In this production scheme, the superfluidity (or global
phase coherence) of the supersolid is inherited from thepreexisting condensate. However, a dipolar supersolid state
can also be reached by direct evaporation from a thermal
gas with fixed interactions, as demonstrated in Ref. [10].
A thermal gas at temperatures well above condensation
has neither phase coherence nor modulation, so both must
emerge during the evaporative formation process. This
leads one to question whether these two features appear
simultaneously, or if not, which comes first. Further,because this transition explicitly takes place at finite
temperature T, thermal excitations may play an importantrole in the formation of the supersolid, presenting achallenging situation for theory. Moreover, in the case ofa dipolar supersolid, the nonmonotonic dispersion relationand the spontaneous formation of periodic density modulation lead to important new length and energy scales notpresent in contact-interacting systems, which dramaticallymodify the evaporative formation process.
While the ground state and dynamics of a zero-temperature dipolar quantum gas can be computed by solving an
extended Gross-Pitaevskii equation [8,11–17] [see also
Fig. 1(a)], similar treatments are currently lacking for finite
temperatures in the supersolid regime. In principle, effectsof finite temperature can be taken into account by perturbatively including the thermal population of excited modes.
0031-9007=21=126(23)=233401(6)This can be done either coherently, by adding them in asingle classical field that abides the Gross-Pitaevskii
equation, as in Refs. [18–20], or incoherently, by iteratively
computing mode populations via a set of coupled HartreeFock-Bogoliubov equations [9,21,22]. In order to accurately describe dynamical processes occurring at temperatures approaching the critical temperature, both coherent
excitations and incoherent interactions with the backgroundthermal gas must be accounted for, requiring either moreadvanced c-field [18] or quantum Monte Carlo [23–27]
techniques. So far, theories with realistic experimental
parameters have not been developed to unveil the finitetemperature dipolar phase diagram and to determine theproperties of the thermal-to-supersolid phase transition.
In this Letter, we experimentally study the evaporativetransition into and out of a supersolid state in a dilute gas ofdysprosium atoms. As the atoms cool down to quantum
degeneracy, the number of condensed atoms increases,
giving birth to the supersolid state. Continued evaporationand collisional loss lead to a reduction of atom number and,eventually, the death of the supersolid. Such an evaporation
trajectory, as illustrated in Fig. 1(a), passes through thelittle-understood finite-temperature portion of the supersolid phase diagram. During the evaporative birth of thesupersolid, we discover that the system first establishes
strong periodic density modulation of locally coherent
atoms and only later acquires long-range phase coherence.
When comparing the birth and death of the supersolid,
which occur at different temperatures, we observe higher
levels of modulation during the birth, suggesting that
thermal fluctuations may play an important role in theformation of density modulation.

233401-1© 2021 American Physical Society

PHYSICAL REVIEW LETTERS 126, 233401 (2021)
((a))

(b)

(c)

FIG. 1. Evaporation trajectory through the finite-temperaturephase diagram. (a) At T ¼ 0 (bottom plane), the phase diagram
for a gas of dipolar atoms is spanned by the s-wave scattering
length as and the condensate atom number N c. In an elongatedtrap, it features a BEC (white) and independent droplet (ID,black) phases, separated in places by a supersolid state (SSS, grayscale). The plotted lightness in the T ¼ 0 phase diagram
represents the droplet link strength across the system (cf.
Ref. [16]). Away from T ¼ 0, the phase diagram is not known.
We explore this region through evaporation into [red, near (i)] andout of [blue, near (ii)] the SSS, along a trajectory represented
schematically by the colored arrow. (b) Single-shot image of the
optical density (OD) of the sample in trap. Here, a system of four
“droplets” within the SSS region is shown, together with its
projected density profile. (c) Single-shot matter-wave interference pattern after 35 ms TOF expansion (OD) and the corresponding projected profile. The color scale is truncated for visualclarity. The background clouds of thermal atoms present are notvisible in the color scales of (b) or (c); for 35 ms TOF and around
50 nK [as in (c)], the thermal atoms show an approximately
isotropic 2D Gaussian distribution of mean width σ̄ ∼ 55 μm.

For our experiments, we first prepare an opticallytrapped gas of approximately 105 dysprosium atoms (isotope 164Dy), precooled via forced evaporation to temperatures of several hundred nanokelvin, at which point the gas
remains thermal. From here, we can apply further evaporation either by a nearly adiabatic ramp-down of the trap
depth (“slow ramp”) or by a rapid reduction of the trap
depth followed by a hold time at fixed depth (“fast ramp”)
to further lower the temperature and induce condensation
into the supersolid state. The slow ramp protocol yields ahigher number of condensed atoms (N c ∼ 2 × 104 ; see next
paragraph for definition) and lower shot-to-shot atom
number fluctuations, whereas the fast ramp protocol(N c ∼ 104 ) allows to follow the evolution of the system
in a constant trap, disentangling the system dynamics fromvarying trap parameters. In contrast to protocols based on
quenching the interactions in a BEC [8–10], we hold themagnetic field (and hence the contact interaction strength)
fixed during the entire evaporation process at 17.92 G,
where the system ground state at our N c is a supersolid[scattering length ∼ 85ð5Þ a0 ].
For the present Letter, we have implemented in-situ
Faraday phase contrast imaging [28,29], which allows us toprobe the in-trap density of our quantum gas at micronscale resolution. During the formation of the densitymodulated state, the translation symmetry is broken along
the long (axial) direction of our cigar-shaped trap [30],typically giving rise to a chain of 3–6 density peaks, which
we call droplets. These droplets have a spacing of roughly
3 μm, clearly visible in our in-situ images [Fig. 1(b)]. As in
our previous works [10,16], we also image the sample after
a time-of-flight (TOF) expansion using standard absorption
imaging. These TOF images include a spatially broad
contribution that we attribute to thermal atoms, whose
number N th and temperature T we estimate by 2D fitting ofa Bose-enhanced Gaussian function [31], excluding thecloud center. Surplus atoms at the cloud center (comparedto the broad Gaussian) are at least locally coherent, or“(quasi-)condensed” in the sense of Refs. [32–34]. With thetotal number of atoms N measured by pixel count, we
define N c ¼ N − N th to be the number of these (at least
locally) coherent atoms. During TOF, matter-wave interference between the expanding droplets gives rise to acharacteristic interference pattern [Fig. 1(c)]. The highcontrast of the interference pattern is visible in single
TOF images and indicates that each individual droplet is byitself a phase-coherent many-body object. The stability of
the interference fringes within the envelope over multiple
experimental realizations encodes the degree of phasecoherence between droplets (cf. Refs. [10,16] and discussion below). The combination of in situ and TOF diagnostics provides complementary information, allowing us
to measure both density modulation and its spatial extent(number of droplets), as well as phase coherence.
Figure 2 shows the birth of the supersolid. Starting froma thermal sample, we apply the fast ramp (225 ms)
evaporation protocol to the desired final trap depth, too
fast for the cloud to follow adiabatically and intermediately
resulting in a nonthermalized, noncondensed sample.
Simply holding the sample at constant trap depth for atime th , collisions and plain evaporation lead to thermalization and cooling. In Fig. 2(a), we plot the average axial
in-situ density profile [cf. Fig. 1(b)] versus th , for about 20
images per time step without any image recentering. At earlyth the atoms are primarily thermal and show up as a broad,low-density background in our images. For th ≲ 150 ms,
inspection of single-shot images reveals an increasing,though substantially fluctuating, number of droplets appearing out of the thermal cloud. After this time, the droplet
number stabilizes to its final value. We observe that the
droplet formation happens on the same timescale as theequilibration of N c and T (see Supplemental Material [35]).
This timescale is set by the rate of evaporation, which in turndepends on the thermalization rate and hence on the elastic
collision rate 1=τel . For our experimental parameters, weestimate for two thermal atoms τel ≈ 3 ms (τel ≈ 10 ms)
before (after) our last evaporation ramp. Once the droplets
have formed, other timescales might be relevant in

233401-2

PHYSICAL REVIEW LETTERS 126, 233401 (2021)
(a)

(b)

(c)

FIG. 2. Growth and spread of density modulation during
evaporation. (a) Averaged in-situ density profiles (no recentering,approximately 20 shots per time step) along the long trap axis as afunction of hold time th after the fast ramp reduction of trap depth(see main text). (b) The density correlator C0 ðdÞ (solid black line)
is fitted by a cosine-modulated Gaussian function (dashed redline) to extract the correlation length L. Gray regions are strongly
influenced by imaging noise and excluded from fits. Correlators
are displayed for th ¼ 50 ms (upper) and th ¼ 300 ms (lower).
(c) Density-density correlation length L versus N c , for the sametime steps shown in (a). Horizontal error bars are the standard
deviation over repetitive shots, vertical error bars reflect thecorrelator fit uncertainty, red points correspond to the correlatorsof (b). The dashed line indicates the simple atom number
scaling of the Thomas-Fermi radius of a harmonically trapped
BEC, ∝ N 1=5
c .

determining the equilibration rate of their relative positions
and phases; the details of this possibility remain an open
question [16].
To better quantify the growth of the modulated state, we
consider the density-density correlator C0 ðdÞ for the in-situ
density profiles over distances d [35]. We find that C0 ðdÞ iswell described by a cosine-modulated Gaussian and define
the density correlation length L [Fig. 2(b)] as its fitted
width. This method provides a way to determine the extent
over which density modulation has formed. Figure 2(c)shows L for the dataset of Fig. 2(a) versus the number ofcoherent atoms N c , which we extract from TOF absorption
images in separate experimental trials with identicalparameters. Interestingly, despite the strongly modulated

structure of the supersolid state, the density correlation
length L closely follows a scaling ∝ N 1=5
c , just as theThomas-Fermi radius of a harmonically trapped BEC,
suggesting a dominant role of interactions over kinetic
energy.
While in-situ images provide information about density
modulation (diagonal long-range order), they do not carrydirect information about phase coherence (off-diagonal
long-range order), either within or between droplets. For
this, we use TOF imaging and address the question ofwhether the formation of density modulation precedes
global (i.e., interdroplet) phase coherence during theevaporative formation of the supersolid, or the otherway round.
For this study, we perform a slow (500 ms) final forced
evaporation ramp of constant slope that is nearly adiabaticwith respect to N c and T (though not necessarily withrespect to excitations of droplet positions and phase) andterminate the ramp at selected crop times tc [38]. After tc,
we immediately release the atoms and perform TOF
imaging. Figure 3(a) shows the observed evolution of
the total (N) and (quasi-)condensed (N c ) atom number,
as well as the sample temperature (T) versus tc . We expand
on the observed evolution by measuring coherence properties. Following Refs. [10,16], for each measurement i we
extract a rescaled complex phasor Pi ¼ ρi exp ð−iΦi Þ, i.e.,
the Fourier component corresponding to the modulation
wavelength in the TOF interference profile (seeSupplemental Material [35]). For systems with a small
number of droplets (but at least 2), the magnitude of thephasor ρi encodes the modulation strength and also the(local) degree of coherence within each of the individual
droplets. Meanwhile, the phase Φi depends primarily on
the relative phase between the droplets (cf. [39]).
We plot the phasors for different evaporation times on thepolar plane in Fig. 3(b), where two effects become
apparent. First, the modulus of the phasors grows during
the evaporation, indicating that the degree of modulation
increases. Second, the distribution of phases Φi is initially
uniform and then narrows down over tc. To determine thetime sequence of these two effects, we calculate theincoherent and coherent amplitude means AM ¼ hjPi jii ,encoding modulation strength and local phase coherence,and AΦ ¼ jhPi ii j, encoding the degree of global phasecoherence across the system [10,16]. Plotting AM and AΦ
against tc [Fig. 3(c)], we notice a time lag of around 40 ms
between the increase of AM and AΦ , indicating that during
evaporation into a supersolid the translational and the phasesymmetry are not broken simultaneously [40]. Rather,
density modulation and local phase coherence appear
before global phase coherence, consistent with predictions
from Monte Carlo simulations (cf., e.g., Ref. [27]).
A similar effect is observed in the fast ramp protocol [35].
This observation suggests the transient formation of aquasicondensate crystal—a state with local but not

233401-3

PHYSICAL REVIEW LETTERS 126, 233401 (2021)

150

2

100

1

50

0
0.1

0.2

0.3

0.4

0.5

0

75

100

125

150

2
0.6

0.6
1.5

0.5 0.4(b)0

T (nK)
500

0.4 0.2
0.3

1

0
0

0.2

1

2

0.5

N C (10 4 )

0.1

0

(c)
(arb. u.)1

N C (10 4 )3modulation M

T (nK)200

N c (10 4 ), N (10 5 )

(a)

0

AM
A

10-3

10-1

100

101

-0.5

hold time t h (s)

0.5

0
0.1

0.2

0.3

0.4

0.5

crop time tc (s)FIG. 3. Development of modulation and coherence whileevaporating into the supersolid state. (a) Sample temperature T(left ordinate, bullets), total (N, right ordinate, dashed red line),and coherent atom number (N c , solid red line) as a function of theramp crop time tc . The shadings reflect the respective confidence
intervals. (b) The phasors Pi (black dots), representing themagnitude and phase coherence of modulation for selected tc(dotted lines; same radial scale for all polar plots). The red
shading reflects mean and variance of the distribution. (c) Evolution of the Fourier amplitude means AM (filled markers) and AΦ
(open markers).

long-range coherence [32–34], whose increased compressibility relative to a thermal gas allows for the formation ofdensity modulation [41]—prior to the formation of asupersolid with phase coherence between droplets. The
lack of global phase coherence could be attributed to a
Kibble-Zurek-type mechanism [42], in which different
regions of the sample condense independently, to excitationof modes involving the motion or phase of the droplets
during the evaporation process, or to the thermal populationof collective modes (which reduce long-range coherence) at
finite temperature. As the evaporation process does notallow independent control of temperature and condensation
rate without also changing density or trap geometry, we
cannot reliably determine the relative importance of theseeffects (or others) from the experiment. Dedicated theoretical studies at finite temperature will thus be needed toelucidate the impact of these types of processes and tounderstand the exact formation process.
After the birth of the supersolid state, both density
modulation and global phase coherence persist for remarkably long times, exceeding 1 s. Figure 4 shows theevolution of the coherent atom number N c and temperature

FIG. 4. Life cycle of a supersolid state. Density modulation M(from in-situ images) during the evaporation process (leftordinate, bullets; the vertical error bars reflect the propagateduncertainty returned by the fitting routine). The sample temperature decreases during the hold time th and is encoded by thecolor filling. N c (from TOF images) is the number of coherent
atoms over th (right ordinate, red line; the light red shadingreflects the measurement standard deviation). At two times whereN c ∼ 1.1 × 104 (vertical dashed lines), but at which the atoms
have different temperatures, M differs substantially. The corresponding averaged in-situ images below confirm a higher level ofmodulation at earlier th. Inset: the observed modulation M plottedversus N c .

T at long hold times under conditions similar to Fig. 2—thesame fast ramp followed immediately by hold time th .
Evaporative cooling first increases the coherent atom
number until, at long th ≥ 1 s, atom losses become dominant and lead to a continuous decrease of N c , eventuallyleading to the disappearance of the modulated state.
However, this death of the supersolid is not a mere time
reversal of the birth. N c decreases, i.e., evolves in the
opposite direction, but more slowly and at lower temperature than for the birth. Furthermore, phase coherence
appears to outlive modulation and to be maintained untilthe very end [35]. Thus, a comparison between the birthand death process provides us with important clues to theimpact of temperature on the supersolid.
We contrast the birth and death of the supersolid in Fig. 4
by also plotting the observed in-situ density modulation M,
which is calculated by Fourier transforming the in-situ
density profiles and normalizing the Fourier component
corresponding to the modulation wavelength to the zerofrequency Fourier component. By comparing M between

233401-4

PHYSICAL REVIEW LETTERS 126, 233401 (2021)
times that have similar N c during the birth and the death of
the supersolid, respectively, we find that the degree ofmodulation is higher during the birth of the supersolid thanduring the death. Because the sample is hotter at shorter
hold times, this suggests that the observed modulation isincreased at higher temperature, perhaps due to thermal
population of collective modes or due to finite-temperature
modifications to the dispersion relation [43], as predicted inRef. [22]. Again, further development of finite-temperature
theory will be needed to conclusively determine theimportance of such effects.
The role of finite temperature in the formation ofmodulation, as well as the mechanism by which phasevariations across the modulated state arise and thenultimately disappear, represent important future directions
for theoretical investigations of dipolar supersolids awayfrom the relatively well understood T ¼ 0 limit.
Experimentally, it would be of great interest to study theevaporative formation process in a larger and more uniform
system, where distinct domains may be observed to form
and a broader separation of length scales may be exploredin correlation measurements. Such measurements, alongwith improved finite-temperature theory, could enable more
precise statements as to the nature of the supersolid phasetransition away from zero temperature.
We thank Russell Bisset, Thomas Bland, the Innsbruck
Erbium team, Massimo Boninsegni, Philippe Chomaz,
Thomas Pohl, Nikolay Prokof’ev, and Boris Svistunovfor insightful discussions and Gianmaria Durastante,
Philipp Ilzhöfer, and Arno Trautmann for early contributions. This work is financially supported through an ERC
Consolidator Grant (RARE, No. 681432), a NFRI Grant(MIRARE, No. ÖAW0600) of the Austrian Academy of
Science, the QuantERA Grant MAQS by the Austrian
Science Fund FWF No. I4391-N, and the DFG/FWF via
FOR 2247/I4317-N36. M. S. acknowledges support by theAustrian Science Fund FWF within the DK-ALM(No. W1259-N27). M. A. N. has received funding froman ESQ Postdoctoral Fellowship from the European
Union’s Horizon 2020 Research and Innovation
Programme under the Marie Skłodowska Curie Grant
Agreement No. 801110 and the Austrian Federal
Ministry of Education, Science and Research (BMBWF).
M. J. M. acknowledges support through an ESQ Discovery
Grant by the Austrian Academy of Sciences. L. C.
acknowledges support through the FWF Elise Richter
Fellowship No. V792. We also acknowledge theInnsbruck Laser Core Facility, financed by the AustrianFederal Ministry of Science, Research, and Economy.

*

Corresponding author.
Francesca.Ferlaino@uibk.ac.at†
Present address: Physikalisches Institut der Universität
Heidelberg, Germany.

[1] E. P. Gross, Unified theory of interacting bosons, Phys. Rev.
106, 161 (1957).
[2] E. P. Gross, Classical theory of boson wave fields, Ann.
Phys. (N.Y.) 4, 57 (1958).
[3] A. F. Andreev and I. M. Lifshitz, Quantum theory of defects
in crystals, Sov. Phys. JETP 29, 1107 (1969), http://jetp.ac.ru/cgi-bin/dn/e_029_06_1107.pdf.
[4] G. V. Chester, Speculations on Bose-Einstein condensationand quantum crystals, Phys. Rev. A 2, 256 (1970).
[5] A. J. Leggett, Can a Solid be “Superfluid"? Phys. Rev. Lett.
25, 1543 (1970).
[6] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. Ç.
Top, A. O. Jamison, and W. Ketterle, A stripe phase withsupersolid properties in spin-orbit-coupled Bose-Einsteincondensates, Nature (London) 543, 91 (2017).
[7] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T.
Donner, Supersolid formation in a quantum gas breaking acontinuous translational symmetry, Nature (London) 543,
87 (2017).
[8] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo,T. Langen, and T. Pfau, Transient Supersolid Properties in
an Array of Dipolar Quantum Droplets, Phys. Rev. X 9,011051 (2019).
[9] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C.
Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,
Observation of a Dipolar Quantum Gas with Metastable
Supersolid Properties, Phys. Rev. Lett. 122, 130405 (2019).
[10] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann,
C. Politi, G. Durastante, R. M. W. van Bijnen, A.
Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,Long-Lived and Transient Supersolid Behaviors in Dipolar
Quantum Gases, Phys. Rev. X 9, 021012 (2019).
[11] F. Wächtler and L. Santos, Quantum filaments in dipolar
Bose-Einstein condensates, Phys. Rev. A 93, 061603(R)(2016).
[12] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L.
Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a
Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6,041039 (2016).
[13] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,
Ground-state phase diagram of a dipolar condensate withquantum fluctuations, Phys. Rev. A 94, 033619 (2016).
[14] F. Wächtler and L. Santos, Ground-state properties andelementary excitations of quantum droplets in dipolar BoseEinstein condensates, Phys. Rev. A 94, 043618 (2016).
[15] D. Baillie and P. B. Blakie, Droplet Crystal Ground States ofa Dipolar Bose Gas, Phys. Rev. Lett. 121, 195301 (2018).
[16] P. Ilzhöfer, M. Sohmen, G. Durastante, C. Politi, A.
Trautmann, G. Natale, G. Morpurgo, T. Giamarchi, L.
Chomaz, M. J. Mark, and F. Ferlaino, Phase coherence inout-of-equilibrium supersolid states of ultracold dipolar
atoms, Nat. Phys. 17, 356 (2021).
[17] S. M. Roccuzzo and F. Ancilotto, Supersolid behavior of adipolar Bose-Einstein condensate confined in a tube, Phys.
Rev. A 99, 041601(R) (2019).
[18] P. Blakie, A. Bradley, M. Davis, R. Ballagh, and C.
Gardiner, Dynamics and statistical mechanics of ultra-cold
Bose gases using c-field techniques, Adv. Phys. 57, 363(2008).

233401-5

PHYSICAL REVIEW LETTERS 126, 233401 (2021)[19] D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. A.
Baranov, R. v. Bijnen, S. M. Roccuzzo, A. Recati, B. Blakie,
D. Baillie, L. Chomaz, and F. Ferlaino, High-energy Bragg
scattering measurements of a dipolar supersolid, arXiv:
2005.02213.
[20] J. Hertkorn, J.-N. Schmidt, F. Böttcher, M. Guo, M.
Schmidt, K. S. H. Ng, S. D. Graham, H. P. Büchler, T.
Langen, M. Zwierlein, and T. Pfau, Density Fluctuations
across the Superfluid-Supersolid Phase Transition in a
Dipolar Quantum Gas, Phys. Rev. X 11, 011037 (2021).
[21] S. Ronen and J. L. Bohn, Dipolar Bose-Einstein condensates at finite temperature, Phys. Rev. A 76, 043607 (2007).
[22] E. Aybar and M. O. Oktel, Temperature-dependent density
profiles of dipolar droplets, Phys. Rev. A 99, 013620 (2019).
[23] F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G.
Pupillo, Supersolid Droplet Crystal in a Dipole-Blockaded
Gas, Phys. Rev. Lett. 105, 135301 (2010).
[24] H. Saito, Path-integral Monte Carlo study on a droplet of adipolar Bose-Einstein condensate stabilized by quantum
fluctuation, J. Phys. Soc. Jpn. 85, 053001 (2016).
[25] A. Macia, J. Sánchez-Baena, J. Boronat, and F. Mazzanti,
Droplets of Trapped Quantum Dipolar Bosons, Phys. Rev.
Lett. 117, 205301 (2016).
[26] F. Cinti and M. Boninsegni, Classical and quantum filaments in the ground state of trapped dipolar Bose gases,
Phys. Rev. A 96, 013627 (2017).
[27] Y. Kora and M. Boninsegni, Patterned supersolids in dipolar
Bose systems, J. Low Temp. Phys. 197, 337 (2019).
[28] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Bose-Einstein
Condensation of Lithium: Observation of Limited Condensate Number, Phys. Rev. Lett. 78, 985 (1997).
[29] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I.
Ferrier-Barbut, and T. Pfau, Observing the Rosensweig
instability of a quantum ferrofluid, Nature (London) 530,
194 (2016).
[30] At the end of evaporation, we typically use trap frequencies
around ωx;y;z ¼ 2π × ð36; 88; 141Þ s−1 , with the tightest
trap direction oriented parallel to gravity and to an applied
uniform magnetic field, cf. Ref. [10].
[31] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, Making, Probing and Understanding Bose-Einstein Condensates, Proceedings of the International School of Physics“Enrico Fermi,” arXiv:cond-mat/9904034.

[32] D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven,
Phase-Fluctuating 3D Bose-Einstein Condensates in Elongated Traps, Phys. Rev. Lett. 87, 050404 (2001).
[33] S. Dettmer, D. Hellweg, P. Ryytty, J. J. Arlt, W. Ertmer, K.
Sengstock, D. S. Petrov, G. V. Shlyapnikov, H. Kreutzmann,
L. Santos, and M. Lewenstein, Observation of Phase
Fluctuations in Elongated Bose-Einstein Condensates,
Phys. Rev. Lett. 87, 160406 (2001).
[34] N. Prokof’ev, O. Ruebenacker, and B. Svistunov, Weaklyinteracting Bose gas in the vicinity of the normal-fluid–
superfluid transition, Phys. Rev. A 69, 053625 (2004).
[35] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.126.233401 for formal
definitions, supporting datasets, and technical information,
which includes Refs. [16,31,36,37].
[36] S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke,and I. Bloch, Spatial quantum noise interferometry inexpanding ultracold atom clouds, Nature (London) 434,
481 (2005).
[37] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S.
Baier, J.-H. Becher, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino, Observation of roton mode population in a
dipolar quantum gas, Nat. Phys. 14, 442 (2018).
[38] While for this protocol the trap parameters do changeslightly versus tc , this protocol proved more robust to
shot-to-shot fluctuations in atom number than the fast rampof Fig. 2, which is important for this measurement.
[39] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J.
Dalibard, Interference of an Array of Independent
Bose-Einstein Condensates, Phys. Rev. Lett. 93, 180403(2004).
[40] We find that observing this time lag does not require aparticular fine-tuning of the experimental parameters (starting atom number, B field, trap frequencies, ramp speed) to ahigher degree than other experiments with supersolids. This
indicates that this time lag between AM and AΦ is a rather
general feature of the evaporation into the supersolid.
[41] N. Prokof’ev and B. Svistunov, University of Massachusetts, Amherst (private communication).
[42] B. Damski and W. H. Zurek, Soliton Creation during a
Bose-Einstein Condensation, Phys. Rev. Lett. 104, 160404(2010).
[43] Th. Pohl, University of Aarhus (private communication).

233401-6

Supplementary Material:
Birth, life, and death of a dipolar supersolid
Maximilian Sohmen,1, 2 Claudia Politi,1, 2 Lauritz Klaus,1, 2 Lauriane Chomaz,2Manfred J. Mark,1, 2 Matthew A. Norcia,1 and Francesca Ferlaino1, 2, ∗1

Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Austria(Dated: March 12, 2021)

CALCULATION OF DENSITY-DENSITYCORRELATOR

We define our correlator as
C(d) = h∫ n(x) n(x + d) dxi ,(1)

where n(x) is the projected density at position x along
our cigar-shaped trap, and the expectation value h. . .i iscalculated over different runs of the experiment. In practice, we follow a standard procedure (e. g., Ref. [1]) andcalculate the correlator by computing the square of theFourier transform of each image to obtain its power spectral density, then Fourier transform again to obtain its
autocorrelation function. The autocorrelation functions
for the different images in the sample are then averaged
to obtain C(d). Note that we do not normalize this as is
typical for a noise correlator, as we are interested in the
structure of the density profile and not specifically in its
fluctuations. To extract the correlation length, we first
subtract off a slowly varying background that represents
the envelope of our density profile from C(d) to obtainC 0 (d), shown in Fig. 2b of the main text. We then fit
the product of a Gaussian and a cosine with spatial frequency km = 2π/xm corresponding to the in-trap modulation wavelength xm , i. e. cos(km x) exp(−x2 /2L2 ), anddefine the correlation length as L.
CALCULATION OF COHERENCE QUANTITIESAs described in the main text, we evaluate the coherence of our droplet array by imaging the sample after
TOF expansion and Fourier transformation (F ) of the
projected density profile n(x0 ) (cf. Fig. 1c in main text),
where in-situ distances x and the corresponding transF
TOF
forms are denoted as x −→ x0 −→ x00 [2]. For eachexperimental repetition i this yields a phasorP̃i (x00 ) = F {n(x0 )}x00 .

Pi (x00 ) = R

P̃i (x00 )|P̃i (x00 )|dx00(4)

mentioned in the main text. The amplitude means corresponding to the in-trap modulation at wavelength xm
are then given by
AM = h|Pi (x00m )|ii

and AΦ = |hPi (x00m )ii |.

(5)

SUPPLEMENTARY DATA FOR FIG. 2

The data of Fig. 2 of the main text is obtained fromin-situ images of samples created via the ‘fast ramp’ evaporation procedure. From corresponding TOF images,
taken after the data of Fig. 2, we can study the time
evolution of ÃM (x00 ) and ÃΦ (x00 ) over the hold time th .
After about a hundred milliseconds a sidepeak has developped in ÃM (x00 ), corresponding to the in-trap density
modulation at xm ∼ 3.5 µm wavelength. A corresponding peak develops in ÃΦ (x00 ), signalling growing coherence between the droplets. In Fig. S1 we plot a directcomparison of the rescaled Fourier amplitude means, AM
and AΦ , calculated at x00m = 3.5 µm. We see that both
AM and AΦ increase with th and the increase of modulation strength starts before the development of phasecoherence.
For reference, we plot in Fig. S2 the evolution of thetotal (N ) and coherent (Nc ) atom number for the data
set of Fig. 2 of the main text, obtained using the ‘fast
ramp’ evaporation protocol.

(2)

We can calculate the incoherent and coherent means of
the Fourier amplitudes over the experimental repetitionsi, writingÃM (x00 ) = h|P̃i (x00 )|ii

respectively.
The quantities AM and AΦ from the main text areclosely connected to ÃM (x00 ) and ÃΦ (x00 ). To disentangle
the spectral amplitude from the coherent atom number(i. e., the area under the density profile), we calculate therescaled phasors

and ÃΦ (x00 ) = |hP̃i (x00 )ii |, (3)

SUPPLEMENTARY DATA FOR FIG. 4

Fig. 4 of the main text shows the death of the supersolid over long hold times th . Here we compare in Fig. S3
the evolution of AM and AΦ , calculated at the sidepeak

2
ration of this process AM and AΦ (which is bounded byAM ) evolve closely together, suggesting that coherence ismaintained in the sample throughout the life and death.

(arb. u.)

1

0.5
AM

1

AM

0
0

0.05

0.1

0.15

0.2

0.25

(arb. u.)A

0.3

t h (s)FIG. S1. Development of coherence after the ‘fast
ramp’ evaporation. Evolution of the means AM and AΦ
from the TOF profiles during the hold time th . The grayshading marks the region where due to low overall signal therescaling of the phasors (Eq. 4) is dominated by noise.

aatom number2

Nc (10 4)
N (10 5)

T (nK)

0.5

0
10-1

100

101

t h (s)FIG. S3. Coherence properties during the death of thesupersolid. Evolution of the means AM and AΦ from the
TOF profiles during th after a ‘slow ramp’ evaporation. The
gray shading marks the region from when on the sidepeaks
in ÃM corresponding to in-trap modulation have disappearedand AM is not a good measure for modulation anymore.

1

EVAPORATION RAMPS

0

b

A

100

75

50

0

0.05

0.1

0.15

0.2

0.25

0.3

hold time t h (s)FIG. S2. Atom number and temperature after the‘fast ramp’ evaporation. a. Evolution of total (N , dashed)
and condensed atom number (Nc , solid line). b. Evolution of
the temperature, as extracted by a Bose-enhanced Gaussian
fit to the background cloud of thermal atoms [3].

In the experiment, once the atoms are loaded into ourcrossed optical dipole trap (ODT), we perform a nearexponential evaporation ramp of trap power, approximated by piecewise linear ramp sections. The trap fre0=
quencies after the penultimate ramp are around ωx,y,z−1
2π×(39, 178, 174) s , where we typically have aroundN = 3×105 atoms at around 200 nK. From here, we ramp
the power of the ODT linearly down to the final value,
giving around ωx,y,z = 2π×(36, 88, 141) s−1 . This procedure yields the atom numbers and temperatures presented in Figs. 3–4 of the main text and Fig. S2.

IMAGING SPECIFICATIONS

in ÃM at x00 = 3.5 µm, during the death of the supersolid. We start our discussion looking at AM , the measure for in-trap modulation. AM stays roughly constantfor around 600 ms before it starts to decay. At above∼ 1.1 s (gray shading in Fig. S3), the sidepeak in ÃM
around x00 = 3.5 µm has vanished. However, AM does
not go straight to zero, since when the modulation disappears, the fundamental peak (around x00 = 0) broadens to x00 > 3.5 µm because the condensate size becomescomparable to the (former) droplet spacing. From thispoint onwards, AM cannot be used anymore as a measure for modulation. Recall that in the in-situ analysis(Fig. 4 of the main text) a very similar behaviour wasobserved, with maximal modulation until ∼ 600 ms andmodulation having disappeared by ∼ 1.1 s.
Now turning to AΦ , we note that over the full du-

The images shown in this work have been recorded
using a new imaging system recently installed in our experiment. The direction of view of the new system isvertical (counter-directed to gravity).
Images from our imaging along the horizontal direction (as in our earlier works, see, e. g., Refs [4, 5]), in
contrast, suffer from the fact that the line of sight is
at 45° with respect to the axis connecting the droplets,
leading to a small apparent fringe spacing and to the interference peaks partially hiding each other; additionally
the interference peaks do not lie in a single focus plane.
These drawbacks were eliminated with the vertical imaging setup, which is why the images are much clearer tointerpret now.
The fundamental resolution of this imaging system,
applicable to in-trap images and characterised by the3Rayleigh criterion, has been measured offline to be approximately 700 nm. We report micron-scale resolution
as a conservative claim that accounts for possible alignment imperfections in the finally installed condition, anda reduction of the imaging aperture to increase depthof field. The pixel size of our camera is smaller than
the imaging resolution, corresponding to approximately
400 nm at the location of the atoms. Additionally, theimages displayed have been supersampled to allow them
to be rotated while maintaining resolution.

∗

Correspondence
shouldbe
addressed
toFrancesca.Ferlaino@uibk.ac.at[1] S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke,and I. Bloch, Spatial quantum noise interferometry in expanding ultracold atom clouds, Nature 434, 481 (2005).

[2] We avoid identifying the positions of atoms after TOF
with the in-trap momenta (i. e. k ≡ x0 ) because we know
from simulation and experiment that in the first few milliseconds of TOF the droplets can still be loosely boundand the expansion is not purely ballistic.
[3] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, Making, probing and understanding Bose–Einstein condensates, Proceedings of the International School of Physics“Enrico Fermi” (1999).
[4] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
S. Baier, J.-H. Becher, M. J. Mark, F. Wächtler, L. Santos,
and F. Ferlaino, Observation of roton mode population ina dipolar quantum gas, Nat. Phys. 14, 442 (2018).
[5] P. Ilzhöfer, M. Sohmen, G. Durastante, C. Politi,
A. Trautmann, G. Natale, G. Morpurgo, T. Giamarchi,
L. Chomaz, M. J. Mark, and F. Ferlaino, Phase coherence in out-of-equilibrium supersolid states of ultracold
dipolar atoms, Nature Physics 10.1038/s41567-020-011003 (2021).

4.5 Publication III:Two-dimensional supersolidity in a dipolar quantum gas

4.5

79

Publication III:Two-dimensional supersolidity in a dipolar quantumgas
The following publication has appeared in
Nature 596, 357-361 (2021)†‡submitted 10 Feb 2021; published 18 Aug 2021
DOI: https://doi.org/10.1038/s41586-021-03725-7
M. A. Norcia,1,4 C. Politi,1,2,4 L. Klaus,1 E. Poli,2 M. Sohmen,1,2 M. J. Mark,1,2
R. N. Bisset,2 L. Santos,3 and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 Institut für Theoretische Physik, Universität Hannover, Hannover, Germany
4 These authors contributed equally: M. A. N. and C. P.

†The author of the present thesis performed the measurements together with M. A. N.,

and
L. K., analysed the experimental data with M. A. N., and contributed in writing the manuscript andinterpreting the results together with all the authors.
‡ Here, the preprint version is attached.

Two-dimensional supersolidity in a dipolar quantum gas
Matthew A. Norcia,1, ∗ Claudia Politi,1, 2, ∗ Lauritz Klaus,1, 2 Elena Poli,2 Maximilian
Sohmen,1, 2 Manfred J. Mark,1, 2 Russell Bisset,2 Luis Santos,3 and Francesca Ferlaino1, 2, †1

Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Innsbruck, Austria

arXiv:2102.05555v1 [cond-mat.quant-gas] 10 Feb 2021

2
3

Institut für Experimentalphysik, Universität Innsbruck, Austria

Institut für Theoretische Physik, Leibniz Universität Hannover, Germany(Dated: February 11, 2021)

Supersolidity — a quantum-mechanical phenomenon characterized by the presence of both superfluidity and crystalline order — was initially envisioned in the context of bulk solid helium, as
a possible answer to the question of whether a solid could have superfluid properties [1–5]. While
supersolidity has not been observed in solid helium (despite much effort)[6], ultracold atomic gases
have provided a fundamentally new approach, recently enabling the observation and study of supersolids with dipolar atoms [7–16]. However, unlike the proposed phenomena in helium, these gaseous
systems have so far only shown supersolidity along a single direction. By crossing a structural phasetransition similar to those occurring in ionic chains [17–20], quantum wires [21, 22], and theoretically in chains of individual dipolar particles [23, 24], we demonstrate the extension of supersolid
properties into two dimensions, providing an important step closer to the bulk situation envisioned
in helium. This opens the possibility of studying rich excitation properties [25–28], including vortex
formation [29–31], as well as ground-state phases with varied geometrical structure [7, 32] in a highly
flexible and controllable system.

Ultracold atoms have recently offered a fundamentallynew direction for the creation of supersolids — rather
than looking for superfluid properties in a solid system
like 4 He, ultracold atoms allow one to induce a crystalline structure in a gaseous superfluid, a system which
provides far greater opportunity for control and observation. This new perspective has enabled supersolid
properties to be observed in systems with spin-orbit coupling [33] or long-range cavity-mediated interactions [34],though in these cases the crystalline structure is externally imposed, yielding an incompressible state. In contrast, dipolar quantum gases of highly magnetic atoms
can spontaneously form crystalline structure due to intrinsic interactions [11–13], allowing for a supersolid withboth crystalline and superfluid excitations [14–16]. In
these demonstrations, supersolid properties have only
been observed along a single dimension, as a linear chainof phase-coherent “droplets”, i.e. regions of high density
connected by low-density bridges of condensed atoms,
confined within an elongated optical trap.
The extension of supersolidity into two dimensions is a
key step towards creating an ultracold gas supersolid thatis closer to the states envisioned in solid helium. Compared to previous studies of incoherent two-dimensional
dipolar droplet crystals [8, 35], we work with both a sub-stantially higher atom number N and relatively strong repulsive contact interactions between atoms. This leads tothe formation of large numbers of loosely bound droplets,
enabling us to establish phase coherence in two dimensions. In our system, the repulsive dipolar interactions
between droplets facilitate a structural transition froma linear to a two-dimensional array, analogous to the
Coulomb-interaction-mediated structural phase transitions observed with ions [17–20]. Unlike ions however,our droplets are compressible and result from the spontaneous formation of a density wave, allowing for dynamical variation in both droplet number and size. Further,
the exchange of particles between droplets enables the
spontaneous synchronization of the internal phase of each
droplet across the system, and the associated superfluid
excitations [14–16].
Dipolar quantum gases exhibit a rich set of groundand excited-state phenomena due to the competition
between many energetic contributions. These include
mean-field interactions of both contact and dipolar nature, quantum fluctuations, and external confinement,
parameterized by potentially anisotropic trapping frequencies fx,y,z . Such systems can be described withgreat accuracy by using an extended Gross–Pitaevskii
equation (eGPE) [36–39]. Even a fine variation of the

2

x (µm)

a. 10

0

-10

-5

b.

0
5
y (µm)-50
5y (µm)

05
y (µm)

10 4
7

6
Atom number N

-5

2D8
2D7

1D6

5

2D6
1D5

4

3

2D9

1D4
0.34

2D5

1D3
0.36
0.38
Trap aspect ratio t

0.4

FIG. 1. Calculated phases of dipolar droplet array. a.
In-trap ground-state density profiles calculated using eGPE
for atom numbers N ∈ [3.3, 4.4, 5.8] × 104 in the droplets
and trap aspect ratios αt = fx /fy ∈ [0.33, 0.35, 0.39] (leftto right). The scattering length a = 88 a0 , where a0 is theBohr radius. Green dots depict the droplet positions obtained
from the variational model, assuming the same N and droplet
number ND as the eGPE. Stars connect to experimentallyobserved density profiles in Fig. 2b. b. Phase diagram, obtained from our variational model, as a function of N and αtfor fx = 33 Hz, fz = 167 Hz. Linear (two-dimensional) phaseswith ND droplets are labelled as 1DND (2DND ).

strength of these energetic contributions can lead to dramatic qualitative changes in the state of the system, forexample enabling a transition from a uniform condensate to a supersolid, or in our present case, from a linear
supersolid to a two-dimensional one.
Fig. 1a shows ground-state density profiles calculated
across this transition using the eGPE at zero temperature. These profiles feature arrays of high-density

droplets, immersed in a low-density coherent “halo” that
establishes phase-coherence across the system. As thetrap becomes more round, the initially linear chain ofdroplets acquires greater transverse structure, eventuallyforming a zig-zag state consisting of two offset linear arrays.
Although the eGPE has remarkable predictive power,
full simulations in three dimensions are numericallyintensive, making a global survey of the array properties
as a function of our experimental parameters difficult.
To overcome this limitation, we employ a variational
ansatz that captures the key behavior of the system, andallows us to disentangle the competing energetic contributions. In this approach, we describe
array of ND
PNanDψj (r), where
droplets by the wavefunction ψ(r) = j=1
the j–th droplet is assumed
rρ,j  to be ofthe form:
rz,j ψj (r) ∝p|ρ−ρj |
1 |z−zj |Nj exp − 12exp−
, inσρ,j2
σz,jterpolating between a Gaussian and a flat-top profile
characteristic of quantum droplets [40]. For a given total
number of atoms N and droplet number ND , energy
minimization provides the atom number Nj in each
droplet, as well as their widths σρ(z),j , exponents rρ(z),j ,and positions ρj = (xj , yj ). Repeating this energy
minimization as a function of ND gives the optimal
number of droplets. This model provides a good qualitative description of the overall phase diagram (Fig. 1b),
revealing that the interplay between intra-droplet
physics and inter-droplet interaction results in a richlandscape of structural transitions as a function of theatom number and the trap aspect ratio αt = fx /fy .
Several trends are immediately visible from the phase
diagram. Larger N and higher αt generally produce
states with larger numbers of droplets. Further, as withions, a large number of droplets favors a 2D configuration,
while tighter transverse confinement (small αt ) favors 1D[17–20]. A transition from 1D to 2D is thus expected
when moving towards larger N or to higher αt . In stark
contrast to the case of ions, the number of droplets typically increases across the 1D to 2D transition, implying afirst-order nature, while only narrow regions in the phase
diagram may allow for a 1D-to-2D transition at constant
droplet number.
The variational results are in excellent agreement with
our eGPE numerics, in terms of predicting the qualitative
structure of droplet array patterns, as shown in Fig. 1a.
Slight discrepancies exist between the two theories regarding the predicted droplet positions and the locationof the 1D-to-2D transition. This is likely because of thepresence of the halo in the eGPE simulation (and presumably in the experiment), visible in Fig. 1a, which is
not accounted for in the variational model. This halo appears to accumulate at the ends of the trap, pushing thedroplets toward the trap center and likely increasing theeffective trap aspect ratio experienced by the droplets.
To explore the 1D to 2D transition experimentally, we

3

a.

c.
z
B

d.
0.44

y
x

0.42
0.4
0.38
atomic aspect ratio

0.36
0.34

b.

0.32
2

4

6

0.3
8

10
Number of droplets

0.28

lower transverse confinement

FIG. 2. Linear to zig-zag transition in an anisotropic trap. a. We confine and condense dipolar 164 Dy atoms within
an anisotropic optical dipole trap (ODT) formed by the intersection of two laser beams. By tuning the aspect ratio of thetrap in the x-y plane (αt ), perpendicular to an applied magnetic field B, we induce a transition between linear and zig-zag
configurations of droplets. b. Single-trial images of the in-trap density profile of atoms at different αt , showing structuraltransition from linear to zig-zag states, as well as an increase in droplet number for higher αt . Stars indicate values αt and N
corresponding to the eGPE calculations of Fig. 1a. c. Atomic aspect ratio αa versus trap aspect ratio αt . αa is the ratio ofminor to major axes of a two-dimensional Gaussian fit to the imaged in-trap density profile (inset). For the supersolid droplet
array (black markers) we see an abrupt change in αa at the critical trap aspect ratio αt∗ , extracted from the fit (gray line, see
methods). The shape of the transition agrees well with eGPE prediction (green diamonds, see methods). For an unmodulatedcondensate (white markers), no abrupt change is evident. d. Distribution of droplet number versus αt , showing a distinct
increase in droplet number at the transition of linear to zig-zag configurations.

use a condensate of highly magnetic 164 Dy atoms confined within an anisotropic optical dipole trap with independently tunable trap frequencies fx,y,z . The trap,
shown in Fig. 2a, is shaped like a surf-board with thetight axis along gravity and along a uniform magnetic
field that orients the atomic dipoles and allows tuningof the contact interaction strength. Typically, we perform evaporation directly into our state of interest at
our desired final interaction strength, as demonstrated
in Refs. [13, 41]. A combination of in-trap and time-offlight (TOF) imaging provides us with complementaryprobes of the density profile of our atomic states, andthe phase coherence across the system.
We begin by studying the transition from one to twodimensions by changing the strength of transverse confinement provided by the trap. Our optical setup allows
us to tune fy from roughly 75 to 120 Hz, while leaving
fx , fz nearly constant at 33(2), 167(1) Hz, and thus tovary the trap aspect ratio αt in the plane perpendicular to the applied magnetic field and our imaging axis.

For small αt , the atoms are tightly squeezed transversely,
and form a linear-chain supersolid (as seen in in-trap images of Fig. 2b). As we increase αt above a critical valueαt∗ = 0.34(2), we observe a structural phase transition
to a two-dimensional (2D) state with two side-by-side
droplets in the center of the chain. By further increasing
αt , the 2D structure extends to two offset lines of droplets
in a zig-zag configuration. The observed patterns match
well with the ground-state predictions from the eGPE
calculations when we globally fix the scattering length to
88a0 .
We obtain higher atom numbers in the more oblate
traps (higher αt ), giving N = 6.5(5) × 104 at αt = 0.44
and N = 2.5(4) × 104 at αt = 0.28. This further facilitates the crossing of the 1D to 2D transition, by favoring states with larger numbers of droplets in the broader
traps. In the zig-zag regime, two-dimensional modulation is clearly visible for durations beyond one second.
Further, the droplet configuration patterns are fairly repeatable, with clear structure visible in averaged images

-20
0
20

b.

-40
-20
0
20

40

-40 -200

20 40

y position (µm)

40-40 -200

20 40

y position (µm)

c.

-40

x position (µm)-40

x position(µm)

a.

-20
0
20
40

-40 -200x position (µm)

4

20 40

y position (µm)

FIG. 3. Coherence in linear and zig-zag states. Upper panels show averaged images of experimental TOF interference
patterns, along with projections along horizontal and vertical directions of average (solid black lines) and individual images(gray lines). The vertical projection is calculated between the dashed lines. Lower panels show interference patterns calculated
for the pictured in-trap droplet configurations (green outlines). a. Linear chain of phase-coherent droplets, showing uniaxial
modulation persisting in averaged image (26 trials). b. Zig-zag configuration of phase-coherent droplets, showing modulation
along two directions that persists in averaged image (51 trials), and hexagonal structure. The spacing of rows in the simulation
was adjusted to approximate the observed aspect ratio of TOF image. The image outlined in blue shows the average momentum
distribution calculated from a series of 20 variational calculations converging to slightly different droplet configurations, showing
the tendency of such fluctuations to broaden features in the interference pattern while maintaining the underlying structure.
c. Zig-zag configuration of phase-incoherent droplets. Modulation remains in single images, as evidenced by the spread of graytraces in projection, but washes out in average (43 trials).

as shown in the inset of Fig. 2c, which is an average of 23
trials taken over roughly two hours.
The transition from 1D to 2D is immediately visible
when plotting the atomic aspect ratio αa versus αt , asshown in Fig. 2c. We find that αa undergoes a rapidchange at αt∗ , as the single linear chain develops twodimensional structure. For comparison, we plot αa measured for an unmodulated BEC, formed at a different
magnetic field, which does not feature the sharp kinkpresent for the supersolid state.
In Fig. 2d, we show the number of droplets present fordifferent αt . In the 1D regime, we typically see between
five and six droplets. This number abruptly jumps up by
approximately one droplet for 2D states near the transition point, and then increases up to an average valueof eight droplets as αt is further increased. The change
in droplet number indicates that the transition that we
observe is not of simple structural nature, but is also

accompanied by a reconfiguration of atoms within thedroplets, as expected from theory (see Fig. 1).
The measurements of in-trap density presented above
inform us about the structural nature of the transition,
but not about phase coherence, which is the key distinguishing feature between an incoherent droplet crystal
and a supersolid. Previous observations of 2D droplet
arrays [35] were performed in traps where the groundstate is a single droplet [8], and the observed droplet
crystal was likely a metastable state lacking inter-droplet
phase coherence. In contrast, we expect from our theoretical calculations that the 2D array is the ground stateof our surfboard-shaped trap (for αt > αt∗ ), facilitating
the formation of a phase-coherent, and therefore supersolid state for our experimental parameters.
We experimentally demonstrate the supersolid natureof our 2D modulated state using a matter-wave interference measurement, as previously used in linear supersolid

5chains [11–13], (Fig. 3a). In this measurement, an arrayof uniformly spaced droplets creates an interference pattern with spatial period proportional to the inverse of thein-trap droplet spacing. The relative internal phase of thedroplets determines both the contrast and spatial phaseof the interference pattern [42]. When averaging overmany interference patterns, obtained on separate runsof the experiment, clear periodic modulation persists forphase-coherent droplets, but averages out if the relative
droplet phases vary between experimental trials. Thus,
the presence of periodic modulation in an average TOF
image provides a clear signature of supersolidity in oursystem, as it indicates both periodic density modulationand phase coherence.
Figure 3a shows an example of such an averaged interference pattern for a linear chain. Uniaxial modulation
is clearly present along the direction of the chain, indicating a high degree of phase coherence. For comparison,
we also show the expected interference pattern calculated
for a linear array of four droplets from free-expansion calculations, showing similar structure.
For conditions where in-trap imaging shows a 2D zigzag structure, the averaged interference pattern exhibits
clear hexagonal symmetry (Fig. 3b). This is consistent
with our expectation, and is indicative of the triangularstructure of the underlying state. To confirm that theobserved modulation is not present without phase coherence, we repeat the measurement of Fig. 3b at a magnetic field corresponding to independent droplets, andalso compute averaged interference pattern for a zig-zag
state with the phases of the individual droplets randomized between simulated trials (Fig. 3c). In both cases, theaveraged image does not show clear periodic modulation.
By exploiting the transition between linear and zigzag states, we have accessed a regime where the supersolid properties of periodic density modulation and phasecoherence exist along two separate dimensions. Future
work will focus on further understanding the spectrumof collective excitations in the full two-dimensional system [26–28, 43], where both the crystalline structure andthe exchange of particles between droplets will play animportant role. Further investigations may elucidate inmore detail the nature of the phase transitions and expected configurations in a wider range of trap aspect ratios, as well as the role that defects play in the 2D system,
either as phase-slips in the zig-zag patterns [44, 45], or as
vortices trapped between droplets of the array [29–31].
We thank the Innsbruck Erbium team and Blair Blakie
for discussions. We acknowledge R. M. W. van Bijnen fordeveloping the code for our eGPE ground-state simulations.
Author Contributions: M.A.N, C.P., L.K., M.S.,
M.J.M and F.F. contributed experimental work. E.P
and R.B. performed eGPE calculations. L.S. contributed
variational model. All authors contributed to interpretation of results and preparation of manuscript.

Funding: The experimental team is financially supported through an ERC Consolidator Grant (RARE,
No. 681432), an NFRI grant (MIRARE, No. ÖAW0600)
of the Austrian Academy of Science, the QuantERA
grant MAQS by the Austrian Science Fund FWF
No I4391-N. L.S and F.F. acknowledge the DFG/FWF
via FOR 2247/PI2790. M.S. acknowledges support by
the Austrian Science Fund FWF within the DK-ALM(No. W1259-N27). L.S. thanks the funding by theDeutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy– EXC-2123 QuantumFrontiers – 390837967. M.A.N. hasreceived funding as an ESQ Postdoctoral Fellow fromthe European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No. 801110 and the Austrian Federal Ministry of Education, Science and Research (BMBWF).
M.J.M. acknowledges support through an ESQ Discovery Grant by the Austrian Academy of Sciences. We also
acknowledge the Innsbruck Laser Core Facility, financedby the Austrian Federal Ministry of Science, Researchand Economy. Part of the computational results presented have been achieved using the HPC infrastructure
LEO of the University of Innsbruck.

∗

M. A. N. and C. P. contributed equally to this work.

†

Correspondence

shouldbe

addressed

toFrancesca.Ferlaino@uibk.ac.at[1] E. P. Gross, Unified theory of interacting bosons, Phys.
Rev. 106, 161 (1957).
[2] E. P. Gross, Classical theory of boson wave fields, Annalsof Physics 4, 57 (1958).
[3] A. F. Andreev and I. M. Lifshitz, Quantum theory ofdefects in crystals, Sov. Phys. JETP 29, 1107 (1969).
[4] G. V. Chester, Speculations on Bose–Einstein condensation and quantum crystals, Phys. Rev. A 2, 256 (1970).
[5] A. J. Leggett, Can a solid be “Superfluid”?, Phys. Rev.
Lett. 25, 1543 (1970).
[6] M. H.-W. Chan, R. Hallock, and L. Reatto, Overview onsolid 4 he and the issue of supersolidity, Journal of Low
Temperature Physics 172, 317 (2013).
[7] Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov, Stable dilute supersolid of two-dimensional dipolar bosons,Phys. Rev. Lett. 115, 075303 (2015).
[8] D. Baillie and P. B. Blakie, Droplet crystal ground statesof a dipolar bose gas, Phys. Rev. Lett. 121, 195301

6(2018).
[9] S. M. Roccuzzo and F. Ancilotto, Supersolid behaviorof a dipolar bose-einstein condensate confined in a tube,
Phys. Rev. A 99, 041601 (2019).
[10] M. Boninsegni and N. V. Prokof’ev, Colloquium: Supersolids: What and where are they?, Rev. Mod. Phys. 84,
759 (2012).
[11] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti,
C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,

zigzag transition in ion chains, Phys. Rev. Lett. 106,
010401 (2011).
[21] W. K. Hew, K. J. Thomas, M. Pepper, I. Farrer, D. Anderson, G. A. C. Jones, and D. A. Ritchie, Incipient formation of an electron lattice in a weakly confined quantum wire, Phys. Rev. Lett. 102, 056804 (2009).
[22] A. C. Mehta, C. J. Umrigar, J. S. Meyer, and H. U.
Baranger, Zigzag phase transition in quantum wires,
Phys. Rev. Lett. 110, 246802 (2013).

Observation of a dipolar quantum gas with metastable[23] G. E. Astrakharchik, G. Morigi, G. De Chiara, andsupersolid properties, Phys. Rev. Lett. 122, 130405

J. Boronat, Ground state of low-dimensional dipolar(2019).

gases: Linear and zigzag chains, Phys. Rev. A 78, 063622[12] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn,(2008).

M. Guo, T. Langen, and T. Pfau, Transient supersolid[24] J. Ruhman, E. G. Dalla Torre, S. D. Huber, and E. Alt-

properties in an array of dipolar quantum droplets, Phys.

man, Nonlocal order in elongated dipolar gases, Phys.

Rev. X 9, 011051 (2019).

Rev. B 85, 125121 (2012).

[13] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Traut-[25] L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Roton-mann, C. Politi, G. Durastante, R. M. W. van Bijnen,

maxon spectrum and stability of trapped dipolar bose-

A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,

einstein condensates, Phys. Rev. Lett. 90, 250403 (2003).

Long-lived and transient supersolid behaviors in dipolar[26] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Radialquantum gases, Phys. Rev. X 9, 021012 (2019).
[14] M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt,

and angular rotons in trapped dipolar gases, Phys. Rev.
Lett. 98, 030406 (2007).

M. Wenzel, H. P. Büchler, T. Langen, and T. Pfau, The[27] R. M. Wilson, S. Ronen, J. L. Bohn, and H. Pu, Mani-

low-energy goldstone mode in a trapped dipolar super-

festations of the roton mode in dipolar bose-einstein con-solid, Nature 574, 386 (2019).

densates, Phys. Rev. Lett. 100, 245302 (2008).

[15] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Pet-[28] R. N. Bisset, D. Baillie, and P. B. Blakie, Roton excita-ter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation

tions in a trapped dipolar bose-einstein condensate, Phys.

spectrum of a trapped dipolar supersolid and its experimental evidence, Phys. Rev. Lett. 123, 050402 (2019).
[16] L. Tanzi, S. Roccuzzo, E. Lucioni, F. Famà, A. Fioretti,

Rev. A 88, 043606 (2013).
[29] A. Gallemı́, S. M. Roccuzzo, S. Stringari, and A. Recati,
Quantized vortices in dipolar supersolid bose-einstein-

C. Gabbanini, G. Modugno, A. Recati, and S. Stringari,condensed gases, Phys. Rev. A 102, 023322 (2020).

Supersolid symmetry breaking from compressional oscil-[30] S. M. Roccuzzo, A. Gallemı́, A. Recati, and S. Stringari,lations in a dipolar quantum gas, Nature 574, 382 (2019).
[17] G. Birkl, S. Kassner, and H. Walther, Multiple-shellstructures of laser-cooled 24 mg+ ions in a quadrupole
storage ring, Nature 357, 310 (1992).
[18] M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M.
Itano, and D. J. Wineland, Ionic crystals in a linear paultrap, Phys. Rev. A 45, 6493 (1992).
[19] S. Fishman, G. De Chiara, T. Calarco, and G. Morigi,
Structural phase transitions in low-dimensional ion crystals, Phys. Rev. B 77, 064111 (2008).
[20] E. Shimshoni, G. Morigi, and S. Fishman, Quantum

Rotating a supersolid dipolar gas, Phys. Rev. Lett. 124,045702 (2020).
[31] F. Ancilotto, M. Barranco, M. Pi, and L. Reatto,
Vortex properties in the extended supersolid phaseof dipolar bose-einstein condensates, arXiv preprint
arXiv:2012.15157 (2020).
[32] Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidity
around a critical point in dipolar bose-einstein condensates, Phys. Rev. Lett. 123, 015301 (2019).
[33] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,
F. Ç. Top, A. O. Jamison, and W. Ketterle, A stripe

7phase with supersolid properties in spin–orbit-coupled

spontaneous symmetry breaking in ion coulomb crystals,

Bose-–Einstein condensates, Nature 543, 91 (2017).

Nature communications 4, 1 (2013).

[34] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and[45] S. Ulm,

J. Roßnagel,

G. Jacob,

C. Degünther,T. Donner, Supersolid formation in a quantum gas break-S. Dawkins, U. Poschinger, R. Nigmatullin, A. Retzker,

ing a continuous translational symmetry, Nature 543, 87

M. Plenio, F. Schmidt-Kaler, et al., Observation of the(2017).

kibble–zurek scaling law for defect formation in ion crys-[35] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier,tals, Nature communications 4, 1 (2013).

I. Ferrier-Barbut, and T. Pfau, Observing the rosensweig[46] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi,

instability of a quantum ferrofluid, Nature 530, 194

M. Sohmen, M. J. Mark, and F. Ferlaino, Dipolar quan-(2016).

tum mixtures of erbium and dysprosium atoms, Phys.

[36] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, andT. Pfau, Observation of quantum droplets in a strongly
dipolar bose gas, Phys. Rev. Lett. 116, 215301 (2016).
[37] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Quantum-fluctuation-driven
crossover from a dilute bose-einstein condensate to a
macrodroplet in a dipolar quantum fluid, Phys. Rev. X
6, 041039 (2016).
[38] F. Wächtler and L. Santos, Quantum filaments in dipolar bose-einstein condensates, Phys. Rev. A 93, 061603(2016).
[39] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,
Ground-state phase diagram of a dipolar condensate withquantum fluctuations, Phys. Rev. A 94, 033619 (2016).
[40] L. Lavoine and T. Bourdel, 1d to 3d beyond-meanfield dimensional crossover in mixture quantum droplets(2020), arXiv:2011.12394 [cond-mat.quant-gas].
[41] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark,M. A. Norcia, and F. Ferlaino, Birth, life, and deathof a dipolar supersolid, arXiv preprint arXiv:2101.06975(2021).
[42] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and
J. Dalibard, Interference of an array of independent
Bose–Einstein condensates, Phys. Rev. Lett. 93, 180403(2004).
[43] J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher,
M. Schmidt, K. S. Ng, S. D. Graham, T. Langen,
M. Zwierlein, and T. Pfau, Roton excitations in an oblate
dipolar quantum gas, arXiv preprint arXiv:2102.01461(2021).
[44] K. Pyka, J. Keller, H. Partner, R. Nigmatullin, T. Burgermeister, D. Meier, K. Kuhlmann, A. Retzker, M. B.
Plenio, W. Zurek, et al., Topological defect formation andRev. Lett. 121, 213601 (2018).

8
Methods

Experimental apparatus and protocols: Our experimental apparatus has been described in detail in
Ref. [46]. Here, we evaporatively prepare up to N =6.5(5) × 104 condensed 164 Dy atoms in a crossed optical dipole trap formed at the intersection of two beams
derived from the same 1064 nm laser, although detuned
in frequency to avoid interference. One beam (the static
ODT) has an approximately 60 µm waist. The second(the scanning ODT) has an 18 µm waist, whose position
can be rapidly scanned horizontally at 250 kHz to create a variably anisotropic time-averaged potential. Bytuning the power in each beam, and the scanning rangeof the scanning ODT, we gain independent control of thetrap frequencies in all three directions. The two trapping
beams propagate in a plane perpendicular to gravity, andcross at a 45° angle, which leads to the rotation of thezig-zag state at high αt visible in Fig. 2b.
We apply a uniform magnetic field oriented along gravity and perpendicular to the intersecting dipole traps,
with which we can tune the strength of contact interactions between atoms. This allows us to create unmodulated Bose-Einstein condensates, supersolid states, orstates consisting of independent droplets at fields of B =23.2 G, 17.92 G, and 17.78 G, respectively.
Details of our imaging setup are provided in Ref. [41].
In-trap and TOF images are performed along the verticaldirection (along B and gravity), using standard phasecontrast and absorption techniques, respectively. The
resolution of our in-trap images is approximately one micron. We use a 36 ms TOF duration for imaging interference patterns.
Atom number: We extract the condensed atom number
N from absorption imaging performed along a horizontal
direction in a separate set of experimental trials underotherwise identical experimental conditions. This allows
for a larger field of view, and better fitting of thermal
atoms. N is determined by subtracting the fitted thermal
component from the total absorption signal.
For comparison between experiment and theory, andbetween the variational and eGPE theory methods, weassociate N with the number of atoms in the droplets,
and not in the diffuse halo that surrounds the droplets.
From simulation of TOF expansion, we find that the halo

is repelled at early expansion times, and is likely indistinguishable from the thermal cloud in our TOF measurements. While it is possible that some of the halo iscounted in N , we neglect this possibility and assume that
N includes only atoms within droplets.
Scattering length: The positions of phase boundaries
between different droplet configurations are quite sensitive to the scattering length a, which is not known withhigh precision in our range of magnetic fields. For all
theory, we use a value of a = 88 a0 , where a0 is the Bohrradius, as this value provides good agreement between
experiment and theory for the 1D-to-2D transition point.
Extracting critical aspect ratio: The critical aspectratio αt∗ is extracted
p from fit to the function αa = α0 for
αt < αt∗ , αa = α02 + b(αt − αt∗ )2 for αt > αt∗ , where
αt∗ , α0 , and b are fit parameters. The error bars reported
in Fig. 2c represent the standard error on the mean, andare smaller than the markers on most points.
Interference patterns: The predicted interference patterns of Fig. 3 are calculated by assuming free expansionof Gaussian droplets. In reality, the droplets are probably not Gaussian, and interactions during TOF expansion may modify the interference pattern. However, thedroplet shape primarily effects the envelope of the interference pattern, which is not our primary interest here,
and from eGPE simulations, we expect the effects of interactions to be minor, provided that the droplets become unbound in a time short compared to the TOF,
which we verify by both looking at shorter TOFs andcomparing the fringe spacing observed in TOF with that
expected from the in-trap droplet spacing. The positions
and size of the droplets are tuned to provide illustrativeinterference patterns.
Droplet number: We extract the droplet number from
our in-trap images using a peak-finding algorithm applied to smoothed images. The algorithm finds the local
maxima above a threshold, which is chosen to be 40%of the overall peak value. Each in-trap density distribution is classified as linear array or 2D zig-zag based on
the atomic aspect ratio. Finally, the counts with a given
droplet number are normalized by the total number oftrials to get the probability shown in Fig. 2d. Fluctuations in the number of atoms in a given trial can pushdroplets above or below the threshold value, contributing
to the spread in extracted droplet number for a given αt .

88

4.6

Two-dimensional supersolidity in dysprosium

Publication IV:

Maintaining supersolidity in one and two dimensions
The following publication has appeared in
Physical Review A 104, 063307 (2021)§
submitted 5 Aug 2021; published 9 Dec 2021
DOI: https://doi.org/10.1103/PhysRevA.104.063307
E. Poli,2 T. Bland,1 C. Politi,1,2 L. Klaus,1 M. A. Norcia,1 F. Ferlaino,1,2 R. N. Bisset,2and L. Santos3
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 Institut für Theoretische Physik, Universität Hannover, Hannover, Germany§The author of the present thesis contributed in writing the manuscript and interpreting the results

together with all the authors.

PHYSICAL REVIEW A 104, 063307 (2021)

Maintaining supersolidity in one and two dimensions
E. Poli ,1 T. Bland ,2,* C. Politi ,2,1 L. Klaus ,2,1 M. A. Norcia ,2 F. Ferlaino ,2,1 R. N. Bisset,1 and L. Santos

3

1

2

Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, 30167 Hannover, Germany(Received 5 August 2021; accepted 19 November 2021; published 9 December 2021)
We theoretically investigate supersolidity in three-dimensional dipolar Bose-Einstein condensates. We focus
on the role of trap geometry in determining the dimensionality of the resulting droplet arrays, which range
from one-dimensional to zigzag, through to two-dimensional supersolids in circular traps. Supersolidity is wellestablished in one-dimensional arrays, and may be just as favorable in two-dimensional arrays provided that one
appropriately scales the atom number to the trap volume. We develop a tractable variational model—which webenchmark against full numerical simulations—and use it to study droplet crystals and their excitations. We also
outline how exotic ring and stripe states may be created with experimentally feasible parameters. Our work paves
the way for future studies of two-dimensional dipolar supersolids in realistic settings.
DOI: 10.1103/PhysRevA.104.063307
I. INTRODUCTION

A supersolid concurrently exhibits both superfluidity andcrystalline order [1–6]. Although predicted over half a century
ago, supersolidity was only recently realized in experiments:
a feat made possible by the flexibility and high degree ofcontrol afforded by quantum gas systems. While supersolid
properties were observed in experiments with cavity-mediatedinteractions [7] and spin-orbit coupling [8,9], those platforms
produce rigid lattices that are impervious to the usual excitations expected of crystals. In contrast, supersolids with
deformable crystals have now been realized in dipolar BoseEinstein condensates [10–12], in which genuine crystal andsuperfluid excitations have been observed [13–15].
Dipolar Bose-Einstein condensates (BECs) can be obtained from highly magnetic atoms such as chromium [16],dysprosium [17], and erbium [18]. It was already predicted
in 2003 that dipolar BECs could undergo a roton instability [19]—where the unstable excitations occur at finite
momenta—as observed in cigar-shaped Er BECs [13,20] and,more recently, in a pancake-shaped Dy BEC [21]. However, it
was also expected from theory that the ensuing periodic density modulations would undergo a runaway collapse, and theregions of high local density would invoke three-body losses
that rapidly destroy the underlying BEC. Indeed, a similar
process was observed with the implosion of entire chromium
BECs, driven by the attractive head-to-tail dipolar interactions[22]. From the perspective of supersolidity, the missing ingredient was a mechanism to stabilize against such implosions,and the answer came from the experimental discovery of
dipolar droplets in Dy [23,24] and Er [25] BECs. Intriguingly,
the stabilization mechanism is well described by including
the leading-order effects of quantum fluctuations, resulting ina theory now known as the extended Gross-Pitaevskii equation (eGPE) [25–28]. These beyond-mean-field effects are*

Corresponding author: thomas.bland@uibk.ac.at2469-9926/2021/104(6)/063307(9)especially important for the highly magnetic Er and Dy atoms.
With this knowledge in hand, the first dipolar supersolids werecreated by crossing the roton instability from the BEC regime
to the droplet array regime [10–12], or directly by evaporative
cooling into the supersolid phase [12]. The supersolid groundstate region exists close to this phase transition, where thedroplets overlap enough for the superfluid to globally conduct
throughout the crystal.
While almost all dipolar supersolids have been experimentally realized as one-dimensional (1D) droplet arrays (see, forexample, Refs. [10–15]), two recent experiments have created
two-dimensional (2D) supersolids [29,30], thus opening an
exciting frontier. An early theoretical study in 2D predicted
a rich phase diagram determined by competing metastable
crystal configurations [31]. More recent works in 2D havepredicted supersolid edge phases [32], intriguing manifestations of quantum vortices and persistent currents [33–36],
honeycomb supersolids [37], as well as ring and stripe
phases [38,39].
Associated with this rich physics, dipolar supersolids havea large number of control parameters and their effects on the
ground-state phase diagram interplay in a complicated way.
Furthermore, the supersolid regime typically lies only withina small range of parameters, located between the ordinary
unmodulated BEC and a crystal of isolated droplets. It istherefore paramount to develop strategies for maintaining supersolidity while exploring phase space. From a theoreticalperspective, it is also necessary to develop tractable and accurate descriptions to supplement the computationally intensiveeGPE.
In this work, we study supersolidity in three-dimensional(3D) dipolar BECs. We systematically explore 1D and 2D
droplet arrays, identifying the crucial role that the average 2D density plays to maintain supersolidity for various
trap geometries and atom numbers. We implement an eGPE
formalism—and develop a tractable variational model—to
examine the phase diagram from linear supersolids in elongated traps to 2D supersolids in circular traps, passing through063307-1©2021 American Physical Society

E. POLI et al.

PHYSICAL REVIEW A 104, 063307 (2021)

zigzag and multirow elliptical phases along the way. We findthat 2D supersolids may be just as favorable as their 1D
counterparts, provided that one fixes the average 2D density.
Through increasing the average 2D density, we show how to
observe the exotic ring and stripe phases [38,39] with realistic
experimental parameters. Finally, we extend our variationalmodel to study 2D crystal excitations and benchmark thisagainst full numerical calculations.
The paper is structured as follows. In Sec. II, we outline oursystem and the eGPE, while Sec. III introduces the concept of
the average 2D density and uses it to theoretically build a 1D2D supersolid phase diagram. We also introduce our droplet
crystal variational model. Section IV examines increasing the
average 2D density to access the exotic ring and stripe phases.
In Sec. V, we present some exemplary 2D crystal excitations,
before concluding with Sec. VI.
II. FORMALISM

We consider 3D dipolar BECs under harmonic confinement and we use the eGPE, given by [25–28]h̄2 ∇ 2
1 ∂(x, t )= −+ m ωx2 x 2 + ωy2 y2 + ωz2 z2
i h̄
∂t
2m
2

+ d 3 x U (x − x )|(x , t )|2+ γQF |(x, t )|3 (x, t ),(1)

where m is the mass and ωi = 2π fi are the harmonic trap
frequencies. The wave
 function  is normalized to the total
atom number N = d 3 x||2 . For dilute gases, two-body interactions are well described by the pseudopotential,
3h̄ add 1 − 3 cos θ
4π h̄ asδ(r) +,mm
r3
2

U (r) =

2

2

(2)

with the first term describing the short-range interactions governed by the s-wave scattering length as . The second term
represents the anisotropic and long-range dipole-dipole interactions, characterized by dipole length add = μ0 μ2m m/12π h̄2 ,with magnetic moment μm and vacuum permeability μ0 . We
take the dipoles to be polarized along z, and θ is the angle
between the polarization axis and the vector pointing from oneof the interacting particles to the other. We always consider
164
Dy, such that add = 130.8a0 , where a0 is the Bohr radius.
The final term in (1) is the dipolar Lee-Huang-Yang correction
arising from quantum fluctuations [40], having the coefficient128h̄2γQF =π as5 Re{Q5 (εdd )},
(3)
3m
1
where Q5 (εdd ) = 0 du (1 − εdd + 3u2 εdd )5/2 is the auxiliary function, and the relative dipole strength is given by
εdd = add /as . Note that Q5 can be calculated analytically(Appendix A), but this is just a monotonically increasing
function that is of the order of unity for the regimes that
we consider here. Ground-state and metastable solutions of
Eq. (1) are calculated by minimizing the energy functionalcorresponding to the eGPE using a conjugate-gradients technique [41].

III. TWO-DIMENSIONAL SUPERSOLIDITY
A. Average 2D density

In dipolar gases, the strong interplay between the confinement geometry and the long-range and anisotropic dipoledipole interactions means that the ground-state phase diagram
is complex and the relevant parameter space to consider ishuge. This may conceal the identification of the most important control parameters. For example, it was demonstrated
in Refs. [10–12,31,39,42] that varying as and fz dramatically affects the supersolid ground state, with supersolidity
easily being lost. In what follows, we identify an important
control parameter for moving between or within the various
supersolid regimes, as well as maintaining supersolidity while
progressing from 1D to 2D droplet arrays.
Dipolar supersolids require tight confinement along thedirection of dipole polarization, and the precise choice of fz
determines the narrow range of as over which supersolidity
occurs. For this reason, we take both fz and as to be fixed in thefollowing argument. We propose that the average 2D density
acts as an important control parameter. This can be thoughtof as an average over the droplet and interdroplet regions, andonly the 2D density is considered because fz is fixed. A simple
yet powerful estimate for how the average 2D density scales isfurnished by the Thomas-Fermi approximation, where kinetic
energy is neglected, and the x and y radii of a BEC scale ∼1/ fx
and ∼1/ fy , respectively, giving a BEC area scaling ∼1/ fx fy .
The key point is then to realize that the average 2D density
scales approximately with the parameter = N fx fy . In thenext section, we explore the consequences of varying versus
keeping it fixed.
B. From 1D to 2D

In order to illustrate the utility of the average 2D density—
characterized by —the first two columns of Fig. 1 explore the
1D-2D transition for two different phase-space trajectories:
first by allowing to vary and second by fixing . For both, we
consider fixed interactions while moving from a cigar-shaped
trap (top row) to a pancake-shaped trap (bottom row). The
key difference between the trajectories is that column 1 has
a fixed atom number—and hence
decreases as the traploosens—while column 2 instead fixes , with N increasing
to compensate for the widening of the trap. Crucially, thereduction of in the first column leads to a loss of the supersolid phase, replaced by an unmodulated BEC, while fixing
allows us to loosen the trap while remaining in the supersolid
regime, eventually resulting in a large, 19-droplet supersolid
for the circular trap [Fig. 1(j)]. We have theoretically verified
in other work that this large 2D supersolid state is robust
against thermal fluctuations [30].
C. Droplet variational theory

Although direct simulations of the eGPE have a remarkablepredictive power, they are numerically intensive and hinder
a thorough overview. We develop a variational model that
permits a much simpler determination of the available droplet
phases, while presenting an excellent qualitative, and largely
quantitative, agreement with our eGPE calculations.

063307-2

MAINTAINING SUPERSOLIDITY IN ONE AND TWO …

(a)

(f)

(k)

(b)

(g)

(l)

(c)

(h)

(m)

(d)

(i)

(n)

(e)


B

(j)

PHYSICAL REVIEW A 104, 063307 (2021)

gies in the system using well-known properties of the Gamma
function.
Our general strategy is to first numerically minimize thesingle-droplet problem for a range of possible parameters
to build interpolation functions for the variational widths
σρ,z (N ) and exponents rρ,z (N ). These functions are then used
to solve the many-droplet problem.
For a single droplet, Ansätze (4) and (5) can be used tominimize the eGPE energy functional,
Esd (N ) = Ekin + Etrap + Esr + Edd + Eqf ,where these quantities are the kinetic, trap, short-range interaction, dipole-dipole interaction, and quantum fluctuation
contributions, respectively. The evaluation of these terms isdetailed in Appendix A.
Now consider a droplet array with ND droplets, with N j
atoms in the jth droplet. Within the variational model, theenergy of the droplet array is then given by

(o)

y

ND

10 μm

x

(6)

E=
j=1

D
 
m 2 2
ωx x j + ωy2 y2j N j +E j j ,2
j=1 j  > j

N

Esd (N j ) +(7)

FIG. 1. Opening up the trap from 1D to 2D for 164 Dy atoms
with as = 88a0 and add = 130.8a0 . In each panel, we fix ( fx , fz ) =(33, 167) Hz and decrease fy ∈ {110, 84.6, 60, 40, 33} Hz, from top
to bottom, showing the integrated column density. Column 1: eGPE
result with constant N = 6.3 × 104 . Column 2: eGPE with constant average 2D density, increasing N to fix = N fx fy with N ∈{6.3, 8.19, 11.55, 17.325, 21} × 104 . Column 3: same as column 2,
but the variational model. The atom number in the variational model
is chosen to match the droplet atom number of the eGPE (see text).
We always take the dipoles to be polarized by magnetic field B
along z.

Inspired by recent work with nondipolar droplets [43], we
assume the following Ansatz for a dipolar droplet:√(4)
(x) = N φ(ρ)ψ (z),with N the number of particles and ρ = x 2 + y2 . We
again consider dipoles polarized along the z axis, and thedroplets are cylindrically symmetric, which we have confirmed as a good approximation by comparing with full eGPE
calculations. The radial and axial functions take the form,
respectively,φ(ρ) =rrρ
−1 ρ ρ
e 2 ( σρ ) ,2
2π (2/rρ )σρψ (z) =

1 |z| rzrz
e− 2 ( σz ) ,2(1/rz )σz(5)with (x) being the Gamma function. The widths σρ,z andthe exponents rρ,z are variational parameters. Note that thisfunction permits the interpolation between a Gaussian (r = 2)
and a flat-top (r  1) profile in a natural way. Furthermore,this Ansatz allows for a simple evaluation of the various ener-

where E j j  is the interdroplet interaction, detailed inAppendix B. By solving the single- then multidroplet
problems separately, we effectively reduce the number of variational parameters from 7ND − 1 to 3ND − 1jj({σρ,z, rρ,z, N j , x j , y j } → {N j , x j , y j }), wherethe −1 arises
from fixing the total atom number, N = j N j .
It is worth noting that important early work employed apurely Gaussian variational model (i.e., rρ = rz = 2) to explore crystal and supersolid configurations [31]. Our model
goes a step further by allowing for the possibility of dropletswith flat-top density profiles, which partially acts to shieldinterdroplet repulsion in the supersolid regime where thedroplets are tightly packed together.
Example solutions of our variational Ansatz are shown in
Fig. 1 (column 3), displaying excellent agreement with thecorresponding eGPE results (column 2). It should be notedthat for the eGPE solutions, a sizable number of atoms exist
outside the droplets in an outer ring, which we term the “halo.”
To make direct comparisons between the variational andeGPE methods, we estimate the total number of atoms in thedroplets alone from the eGPE and use this to set the total atom
number for the corresponding variational calculation. For reference, the variational to eGPE atom number ratio varies from
Nvar = 0.84NeGPE for the linear chain [Figs. 1(f) and 1(k)] to
Nvar = 0.58NeGPE for the circular crystal [Figs. 1(j) and 1(o)].
Small deviations in the droplet positions occur between themodels due to repulsion between the droplets and the halo in
the eGPE, whereas the halo is absent in the variational model.
In general, the halo leads to a slight compression of the crystal.
Additionally, because the halo density is nonuniform aroundthe perimeter of the droplet array (in some cases forming
nearly-droplet-like regions of higher density), its presence canalso qualitatively modify the structure and the symmetry ofthe droplet array in certain situations [cf. Figs. 1(h) and 1(m)].

063307-3

E. POLI et al.

PHYSICAL REVIEW A 104, 063307 (2021)/

Following the solutions from bottom left to top right in
Fig. 2, there are two distinct
 jumps in the average transversal spread (y = 1/ND Nj D |y j − ȳ|, for the y position of
the jth droplet y j , and mean y position ȳ), marked as whitedashed lines in Fig. 2. These signify the transition from linear[Fig. 2 ( )] to zigzag [Fig. 2 ( )] configurations, and then2D solutions with three [Fig. 2 ()] or more [Fig. 2 (×)]
rows of droplets. The first three of these highlighted solutions
contain the same number of droplets for a fixed atom number,
until αt ≈ 1, where the ground-state configuration consists of
12 droplets. Intriguingly, these jumps in y are also usuallyassociated with a change in the ground-state droplet number. It
is interesting to note that in the 1D regime, the regions of constant ND slope downwards to the left. This can be understoodby considering a horizontal trajectory, for which both N and
are constant. As we move left along this trajectory, increasing
fy can no longer force the droplets closer together—since thearray is already 1D—while the decreasing fx provides more
space for longer droplet arrays, with larger ND .

−IV. INCREASING AVERAGE 2D DENSITY

−

−
−

−

−

FIG. 2. Crystal phase diagram for 164 Dy atoms from 1D (left) tocircular trap regime (right) using Ansätze (4) and (5). Color indicatesground-state droplet number vs total atom number N and aspectratio αt = fx / fy . A constant average 2D density (controlled by fixing= N fx fy ) is used throughout, which means the trap tightens fromfx fy = 43 Hz (top) to fx fy = 114 Hz (bottom). White lines separate the 1D, zigzag (ZZ), and 2D regions. Example configurations
for fixed N = 5.4 × 104 are shown below. Parameters fz = 167 Hz
and as = 88a0 remain constant.

D. Crystal phase diagram

Here, with the variational model, we seek to explore thefull phase diagram of droplet crystal configurations while
maintaining a fixed average 2D density, which we control bykeeping constant. Figure 2 shows the droplet configurationsof the ground state as a function of the trap aspect ratio
αt = fx / fy and atom number. Since is held fixed throughout,the bottom of the phase diagram corresponds to N = 104
and fx fy = 114 Hz, while the top reaches N = 7 × 104
and fx fy = 43 Hz. Traversing right on the phase diagram
equates to increasing fx and decreasing fy , hence moving tomore circular configurations.
Several trends are apparent from this phase diagram.
Larger N corresponds to ground states with a larger number ofdroplets. If the configuration is linear (left in Fig. 2), then thedroplet number increases incrementally one droplet at a time;
however, for large αt ∼ 1 (right in Fig. 2), there are occasional
jumps of two or more droplets—within the resolution of ourphase diagram—corresponding to preferential triangular configurations of the lattice in 2D. For example, we find that forαt = 1, the ground state jumps from ND = 8 to the ND = 12
state shown in Fig. 2 (×), with only a very narrow range of N
corresponding to a 10-droplet configuration in between.

Previous theoretical works have found exotic twodimensional supersolid states with either large atom numbers(∼106 ) or tight trapping (∼1 kHz) [37–39]. Notably, honeycomb ground states have been predicted [37] with crystal
arrays of holes rather than droplets. Such states are appealing
due to their predicted strong superfluid conductance across the
crystal, without relying on low-density connections between
droplets. Also predicted are intriguing stripe and ring states[38], as well as labyrinthine instabilities [39] familiar in classical ferrofluids [44].
Using the eGPE, we investigate the feasibility of creatingthese exotic supersolids by increasing the average 2D density
through tightening the radial trap frequencies, without relying
on pushing the parameters to unrealistically large values. Figures 3(a)–3(f) show how the solution changes by increasingfx = fy ∈ {30, 50, 80, 90, 100, 150} Hz, respectively, while
increases. This
holding fixed N = 1.4 × 105 , and hence
trajectory through phase space takes us from an unmodulated
BEC [Fig. 3(a)] to a hexagonal supersolid [Fig. 3(b)], a stripe
supersolid [Fig. 3(d)], through to a ring state [Fig. 3(e)], and,
finally, a macrodroplet [Fig. 3(f)]. Interestingly, while thepeak density of the BEC phase is about 1.5 × 1020 m−3 , for all
droplet and supersolid phases it is roughly constant at ∼1.5 ×
1021 m−3 , suggesting that the atom losses from inelastic threebody collisions—and hence also the lifetimes—of these exotic
states may be comparable to that for the current generation ofsupersolid experiments.
V. EXCITATIONS OF A 2D SUPERSOLID

Following the recent experimental observation of a sevendroplet hexagon supersolid [30], we further investigate theexcitations of this state in a circular trap using the eGPE [seeFig. 4(a1)] and variational model [see Fig. 4(b1)].
We find excitations in the Bogoliubov–de Gennes (BdG)
framework, which consists of a linearization of the eGPE
around the stationary solution ψ0 with perturbations of
the form δψ = ue−it/h̄ + v ∗ eit/h̄ [45]. To visualize the063307-4

MAINTAINING SUPERSOLIDITY IN ONE AND TWO …

PHYSICAL REVIEW A 104, 063307 (2021)

FIG. 3. Increasing the average 2D density. The radial trap
frequency is increased from (a)–(f), respectively, as fx = fy ∈{30, 50, 80, 90, 100, 150} Hz, while N = 1.4 × 105 is held fixed.
Density isosurfaces are shown at the 5%, 0.1%, and 0.01% of the
maximum density level. Shadow shows the 2D integrated density.
Other parameters: fz = 167 Hz and as = 88 a0 .

excitations,
we
plot
the
density
perturbationψ = (u + v ∗ )|ψ0 | for several exemplary excitations in
Figs. 4(a2)–4(a5) (arbitrary normalization). The arrows
represent the droplet displacement vectors (with arbitrary
global scaling), calculated from the shift in density peaks
caused by adding a small amount of excitation to theground-state wave function. These results are compared withthe corresponding excitations calculated with the variationalmodel [Figs. 4(b2)–4(b5)], with droplet displacement vectors
obtained through linearizing perturbations to the droplet
positions (see Appendix C). Since these modes exist
in the variational model—which does not account forsuperfluid flow between droplets—we can classify them aspredominantly crystalline in nature.
Due to rotational symmetry, there is a zero-energy rotational mode [Figs. 4(a2) and 4(b2)], which is unique tocircular trap supersolids. As expected, there are two degenerate Kohn modes at the radial trap frequency, one of which isshown in Figs. 4(a3) and 4(b3). Also plotted are quadrupole
excitations [Figs. 4(a4) and 4(b4)], as well as an example
surface crystal mode [Figs. 4(a5) and 4(b5)], which is a uniquefeature of 2D supersolids highlighting the rich tapestry of excitations. In the last two examples, the mode energy obtainedin the BdG framework and the variational models differs.
The energies are E /h = 54 Hz [Fig. 4(a4)] and E /h = 72 Hz[Fig. 4(a5)] from the BdG calculations and E /h = 65 Hz[Fig. 4(b4)] and E /h = 69 Hz [Fig. 4(b5)] from the variational

FIG. 4. Crystal excitations. (a1),(b1) Seven-droplet crystal stateand corresponding excitations from the (a2)–(a5) eGPE-BdG calculations and (b2)–(b5) variational model. Arrows indicate relative
droplet motion (see main text). Parameters: as = 90a0 , fx,y,z =(52.83, 52.83, 167) Hz, N = 9.5 × 104 . (c) Exemplary excitations
for the 19-droplet state from the variational model shown in Fig. 1(o).

model. These deviations point to a measurable role played by
the superfluid connection between the droplets, and the effectof the surrounding halo, which are not accounted for by thevariational model. Such comparisons between models provide
an excellent platform to distinguish contributions from thecrystal and the superfluid surrounding and connecting thedroplets.
The computational cost of obtaining modes from BdG
linearization is high, requiring the diagonalization of large
dense matrices consisting of the total number of position space
grid points squared; in our case, ∼106 × 106 . We achieve
this using an eigensolver based on the implicitly restarted
Arnoldi method. We also find that the linearization is slower
when there is no appreciable superfluid connection between
the droplets, making excitations in the isolated droplet regime
difficult to obtain. However, in this regime, the variationalmodel agrees well with the BdG calculations, and the former
only requires the diagonalization of a 2ND × 2ND matrix [i.e.,
the total number of (x j , y j ) pairs]. This allows us to explore
excitations of larger crystals.
In Figs. 4(c1)–4(c3), we show excitations of the 19-droplet
crystal [Fig. 1(o)] using the variational model, a state that

063307-5

E. POLI et al.

PHYSICAL REVIEW A 104, 063307 (2021)

would require months of computational time to obtain excitations within the eGPE-BdG framework. This configurationconsists of two concentric hexagons with a single droplet in
the middle, where the inner hexagon consists of six droplets
and the outer hexagon has 12. In Fig. 4(c1), we highlight aninteresting mode in which the two outer hexagons counterrotate. We also find a quadrupole mode [Fig. 4(c2)] and, inFig. 4(c3), we show an analog of the surface crystal mode that
we saw for the seven-droplet hexagon [Figs. 4(a5) and 4(b5)].

tiers, Grant No. 390837967. M.A.N. has received funding
as an ESQ Postdoctoral Fellow from the European Union’s
Horizon 2020 research and innovation programme under theMarie Sklodowska-Curie Grant Agreement No. 801110 and
the Austrian Federal Ministry of Education, Science and Research (BMBWF). We also acknowledge the Innsbruck Laser
Core Facility, financed by the Austrian Federal Ministry of
Science, Research and Economy.
APPENDIX A: SINGLE-DROPLET VARIATIONAL MODEL

VI. CONCLUSIONS

We have investigated the scope and feasibility of 2D supersolidity in harmonically trapped dipolar Bose gases, identifying the crucial role of the average 2D density in maintaining
both the crystal structure and global superfluidity while varying the dimensionality and size of the droplet array. By developing a variational multidroplet model, we explored the phase
diagram of crystal configurations for a wide range of atom
numbers and aspect ratios for a fixed 2D density, identifying
the transition from one- to two-dimensional droplet arrays.
We theoretically explored how increasing the average 2Ddensity may provide a route for creating exotic stripe andring supersolids under experimentally realistic conditions. We
also extended our variational model to explore crystal excitations, verified by direct comparison to the BdG analysis. Thismethod allows for the investigation of crystal modes in large
2D supersolids, where exact diagonalization of the eGPE isdemanding.
Future work will further explore the potential of the variational model. Implementing a system of Hamilton equations
would allow for dynamics of the droplet arrays, and further open up the study of excitations in two-dimensional
supersolid crystals. While we have revealed how to vary an
important triplet of coupled parameters, i.e., N, and the two
trapping frequencies perpendicular to the direction of dipole
polarization, fx and fy , enabling the exploration of supersolidsof various shapes and sizes, future studies will seek an easy
determination for how best to vary other control parameters,such as the coupling between the interaction strengths and theremaining trap frequency, fz .
ACKNOWLEDGMENTS

We gratefully acknowledge useful discussions with Manfred Mark and the Innsbruck Erbium team. We acknowledge
R. M. W. van Bijnen for developing the code for our eGPE and
BdG simulations. Part of the computational results presented
here have been achieved using the HPC infrastructure LEO of
the University of Innsbruck. The experimental team is financially supported through an ERC Consolidator grant (RARE,
Grant No. 681432), an NFRI grant (MIRARE, Grant No.
OAW0600) of the Austrian Academy of Science, the QuantERA grant MAQS by the Austrian Science Fund FWF (Grant
No I4391-N). L.S. and F.F. acknowledge the DFG/FWF(Grant No. FOR 2247/I4317-N36) and a joint-project grantfrom the FWF (Grant No. I4426, RSF/Russland 2019). L.S.
acknowledges the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy–EXC-2123 QuantumFron-

Here we detail the individual contributions to the singledroplet energy functional for N atoms,
Esd (N ) = Ekin + Etrap + Esr + Edd + Eqf .

These terms are given byh̄2Ekin = −d 3 x  ∗ ∇ 2 ,2mm
Etrap =
ωi2 xi2 ,
d 3x  ∗
2
i

1
Esr =
d 3 x  ∗ g||2 ,
2



gdd
d 3 k 3kz2
Edd =− 1 |ñ(k)|2 ,2(2π )3 k 2
2
Eqf = γQF d 3 x ||5 ,
5

(A1)

(A2)

corresponding to the kinetic, trap, short-range interaction,
dipole-dipole interaction, and quantum fluctuation contributions, respectively. Here, the short-range interaction
coefficient is g = 4π h̄2 as /m, and the quantum fluctuaa3

tion coefficient is γQF = 32
g πs Q5 (εdd ), where Q5 (εdd ) =3
12
Re[ 0 du (1 − εddεdd )5/2 ], and the density in Fourier +3 3u −ik·x|(x)|2 . The integral Q5 (εdd ) can
space is ñ(k) = d x e
be evaluated as
√(3εdd )5/2
(8 + 26 + 33 2 ) 1 + 
Q5 (εdd ) = Re
48
√


1+ 1+
3+ 15 ln
,
(A3)
√


where  = (1 − εdd )/(3εdd ). Note that when using this definition, care should
be taken for the special cases Q5 (0) = 1 and√Q5 (1) = 3 3/2.
These integrals√ are evaluated upon substitution of theAnsatz (x) = N φ(ρ)ψ (z) [43] [see main text, Eq. (4)],with N the number of particles in the droplet. The radial andaxial functions, normalized to one, are assumed to be of theform
φ(ρ) =rrρ
−1 ρ ρ
e 2 ( σρ ) ,2
2π (2/rρ )σρψ (z) =

1 |z| rzrz
e− 2 ( σz ) ,2(1/rz )σz(A4)with (x) the Gamma function. The widths σρ,z and the exponents rρ,z are variational parameters. Substituting the Ansätze

063307-6

MAINTAINING SUPERSOLIDITY IN ONE AND TWO …

PHYSICAL REVIEW A 104, 063307 (2021)

given by Eqs. (A4) into the energy contributions given by
Eqs. (A2) gives the following results. The kinetic energy ofthe droplet is of the formrρ2h̄2h̄2 rz fK (rz )
Ekin
=
+
,
2
N
2mσρ 4(2/rρ ) 2mσz2 2(1/rz )(A5)with fK (rz ) = (rz − 1)(1 − 1/rz ) − r2z (2 − 1/rz ). The trapenergy is σρ2 (4/rρ )
Etrap
m 2
m 2 σz2 (3/rz )2=
ω + ωy+ ωz.
N2 x2(2/rρ )2
(1/rz )(A6)
Short-range interactions lead to an energy contribution,
Esr
gN
rρ rz
=
,
N
8π σρ2 σz 22/rρ +1/rz (2/rρ )(1/rz )(A7)whereas quantum fluctuations result in the Lee-Huang-Yang
correction: 2 1
Eqf
64Q5 (εdd ) 2 rρ + rz=
gn0 n0 a3 ,
(A8)
√
N
5
15 π

replace fx,y → 110 Hz to simulate the effect of interdroplet
interactions on a given droplet’s shape; then, to get the energy,
we use the fx,y of the actual trap.
For all minimization procedures related to variational
calculations, we use the sequential quadratic programmingalgorithm implemented in the MATLAB function fmincon.
APPENDIX B: INTERDROPLET INTERACTION ENERGY

Let us consider two droplets with N1 and N2 atoms, respectively, which are sufficiently separated, such that we canneglect any overlapping. The center of mass of the droplets isplaced at r j=1,2 = (x j , y j , 0), i.e., we permit displacements on
the xy plane, but assume that z j = 0. As for the single-droplet
dipolar energy, the interdroplet dipole-dipole interaction isbest calculated in momentum space,
 2


kz
d 3k
3
−
1
ñ1∗ (k)ñ2 (k),(B1)
E12 = gdd N1 N2(2π )3 k 2
where we can approximate the Fourier transform of the density profile of the droplets asñ j (k)  e−kρ ρ (N j ) /4 e−kz z (N j ) /4 eikρ (x j cos φ+y j sin φ) .
2

Nr r

ρ z
is the central density.
where n0 = 4π(2/rρ )(1/r2z )σρ σz
The dipolar energy is best evaluated in momentum space.
The Ansatz density in Fourier space can be decomposed asñ(k) = ñρ (kρ )ñz (kz ), with
 ∞
rρ
rρ
ñρ (kρ ) =dρ ρe−ρ J0 (kρ σρ ρ),(2/rρ ) 0
(A9)
 ∞
rz
−zrz
ñz (kz ) =dz ecos(kz σz z),
(1/rz ) 0

where J0 is the first Bessel function of the first kind.
Interestingly, these functions can be very closely approx2
imated by Gaussians: ñρ (kρ )  e−αρ (rρ )(kρ σρ ) and ñz (kz ) 2e−αz (rz )(kz σz ) , where αρ (ρ) and αz (z) are functions foundthrough numerical fitting to Eqs. (A9) prior to variational
minimization. The dipole-dipole interaction can then be easily
expressed asEdd
gεdd N f (ρ /z )=,N
2(2π )3/2 2ρ z

2

(B2)

where 2¯2ρ,z = ρ,z (N1 )2 + ρ,z (N2 )2 ,  = ¯z /¯ρ , and
√

(A10)

Our approach is to first minimize the single-droplet energy (A1) for a suitable range of atom numbers. Thus, in
preparation for solving the multidroplet problem, we generate interpolating functions Esd (N ), σρ,z (N ), and rρ,z (N ),furnishing a library of single-droplet solutions for a given trap
and interaction parameters.
Employing this two-step method reduces the numberof variational parameters from seven per droplet to threejj({σρ,z, rρ,z, N j , x j , y j } → {N j , x j , y j }). Note that the final
populations ofthe droplets are constrained by the total atom
number N = j N j . The effect of interdroplet repulsion is
not accounted for in calculating the shape of the droplets. We

2

The phase φ is accumulated due to the central position of
the droplets being different from the origin and plays norole in the energy calculation. We can then evaluate theinteraction energy E12 as a function of the distance r12 =(x1 − x2 )2 + (y1 − y2 )2 between the droplets:√
gεdd N1 N2 2
E12 (r12 ) =¯2ρ ¯z π 2

 2
 1
(2 +2)u2 − 2
2r12 (1−u2 ),×duG(1−2 )u2 + 2
¯2ρ
0
(B3)

G(x) =

2
where 2ρ,z = 4αρ,z (rρ,z )σρ,z
, and
√


2
1
2
2 arctan( κ − 1)2κ + 1 − 3κ. (A11)f (κ ) = 2√
κ −1
κ2 − 12

 x 
π −x/8   x  x  x 
e
+ I1
− I0
,
I0
4
8
4
8
8

(B4)

with In (x) the modified Bessel function. The interaction energy (B3) is attractive at short distances, a spurious effect up
to the radial size of a droplet. In order to prevent the dropletsfrom “piling up”â in this inner region, we instead approximate
the interdroplet potential as
E j j  (r j j  ) 

V0 (N j , N j  )N j N j ,[r j j  + r0 (N j , N j  )]3(B5)

for any two droplets j and j  , where V0 and r0 are determinedby fitting to Eq. (B3). This term is the last contribution to
Eq. (7) and is utilized in the phase diagram given in Fig. 2.
By considering a range of particle number pairs between
droplets, we determine the interpolating functions V0 (N , N  )and r0 (N , N  ) prior to solving the full many-droplet problem.
Note that the shift r0 , which results from the z extension of thedroplet, is relevant because typical interdroplet distances arecomparable to the z size of the droplets.

063307-7

E. POLI et al.

PHYSICAL REVIEW A 104, 063307 (2021)

with

APPENDIX C: EXCITATIONS OF THE VARIATIONAL
MODEL

Expanding around the equilibrium positions R j = (x j , y j ),r j = R j +  j , the energy of the array becomes, up to second order in the displacement  j = (x; j , y; j ), of the form
E = E0 + E (2) (the first-order contribution cancels because
we move from an energy minimum), with E0 the ground-stateenergy, andNDE (2) =

Tj · Â j ·  j −

where



B̂ j j  = V0 (N j , N j  ) N j N j Â j =B̂ j j  ·  j  ,(C1)

j  = j

j=1

β j j  + γ j j  X j2j 
γ j j X j j Y j j
mN j ωx2
0
2


0
+
ωy2



γ j j X j j Y j j,
β j j  + γ j j  Y j2j (C2)
B̂ j j  ,(C3)

j = j[1] E. P. Gross, Unified theory of interacting bosons, Phys. Rev.
106, 161 (1957).
[2] A. F. Andreev and I. M. Lifshitz, Quantum theory of defects incrystals, Zh. Eksp. Teor. Fiz. 56, 2057 (1969) [Sov. Phys. JETP
29, 1107 (1969)].
[3] D. Thouless, The flow of a dense superfluid, Ann. Phys. 52, 403(1969).
[4] G. Chester, Speculations on Bose-Einstein condensation andquantum crystals, Phys. Rev. A 2, 256 (1970).
[5] A. J. Leggett, Can a Solid Be” Superfluid”?, Phys. Rev. Lett.
25, 1543 (1970).
[6] M. Boninsegni and N. V. Prokofâev, Colloquium: Supersolids:
What and where are they?, Rev. Mod. Phys. 84, 759 (2012).
[7] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T.
Donner, Supersolid formation in a quantum gas breaking acontinuous translational symmetry, Nature (London) 543, 87(2017).
[8] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. Ç.
Top, A. O. Jamison, and W. Ketterle, A stripe phase withsupersolid properties in spin–orbit-coupled Bose-Einstein condensates, Nature (London) 543, 91 (2017).
[9] T. M. Bersano, J. Hou, S. Mossman, V. Gokhroo, X.-W.
Luo, K. Sun, C. Zhang, and P. Engels, Experimental realization of a long-lived striped Bose-Einstein condensate inducedby momentum-space hopping, Phys. Rev. A 99, 051602(R)(2019).
[10] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C.
Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a Dipolar Quantum Gas with Metastable Supersolid
Properties, Phys. Rev. Lett. 122, 130405 (2019).
[11] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo,T. Langen, and T. Pfau, Transient Supersolid Properties in an
Array of Dipolar Quantum Droplets, Phys. Rev. X 9, 011051(2019).
[12] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C.
Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M.

β j j =γ j j =−3,
2R (R j j  + r0, j j  )4

(C4)

j j

3
2R3j j  (R j j  + r0, j j  )4+

6
R2j j  (R j j  + r0, j j  )5

, (C5)and the separation matrices X j j  = x j − x j  , Y j j  = y j − y j  ,and R j j  = |r j − r j  |.
 T · M̂ · ,
 =with 
We can write E (2) = (x,1 , y,1 , . . . x,ND y,ND ). Now, we can diagonalize M toobtain the eigenvalues λν , which provide
the excitation√
frequencies of the droplet array, ν = 2λν . Note that thisis an expansion around the equilibrium positions only, andnot a perturbation of the individual droplet shape or atom
number, so other shape excitations, such as droplet breathingmodes, will not be captured by this method. Some example
excitations are shown in Figs. 4(b2)–4(b5) and 4(c1)–4(c3),where the arrow indicates the vector between R j and r j foreach droplet j.

Sohmen, M. J. Mark, and F. Ferlaino, Long-Lived And Transient Supersolid Behaviors in Dipolar Quantum Gases, Phys.
Rev. X 9, 021012 (2019).
[13] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter,
M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation Spectrumof a Trapped Dipolar Supersolid and Its Experimental Evidence,Phys. Rev. Lett. 123, 050402 (2019).
[14] L. Tanzi, S. Roccuzzo, E. Lucioni, F. Famà, A. Fioretti, C.
Gabbanini, G. Modugno, A. Recati, and S. Stringari, Supersolid
symmetry breaking from compressional oscillations in a dipolar
quantum gas, Nature (London) 574, 382 (2019).
[15] M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt, M. Wenzel,
H. P. Büchler, T. Langen, and T. Pfau, The low-energy goldstone
mode in a trapped dipolar supersolid, Nature (London) 574, 386(2019).
[16] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau,
Bose-Einstein Condensation of Chromium, Phys. Rev. Lett. 94,
160401 (2005).
[17] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Strongly
Dipolar Bose-Einstein Condensate of Dysprosium, Phys. Rev.
Lett. 107, 190401 (2011).
[18] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R.
Grimm, and F. Ferlaino, Bose-Einstein Condensation of Erbium, Phys. Rev. Lett. 108, 210401 (2012).
[19] L. Santos, G. V. Shlyapnikov, and M. Lewenstein,
Roton-Maxon Spectrum and Stability of Trapped Dipolar
Bose-Einstein Condensates, Phys. Rev. Lett. 90, 250403(2003).
[20] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S. Baier,J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino,
Observation of roton mode population in a dipolar quantum gas,Nat. Phys. 14, 442 (2018).
[21] J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher, M. Schmidt,
K. S. H. Ng, S. D. Graham, T. Langen, M. Zwierlein, and T.
Pfau, Roton Excitations in an Oblate Dipolar Quantum Gas,
Phys. Rev. Lett. 126, 193002 (2021).

063307-8

MAINTAINING SUPERSOLIDITY IN ONE AND TWO …

PHYSICAL REVIEW A 104, 063307 (2021)[22] T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A.
Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda,
d-Wave Collapse and Explosion of a Dipolar Bose-Einstein
Condensate, Phys. Rev. Lett. 101, 080401 (2008).
[23] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I.
Ferrier-Barbut, and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid, Nature (London) 530, 194(2016).
[24] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T.
Pfau, Self-bound droplets of a dilute magnetic quantum liquid,
Nature (London) 539, 259 (2016).
[25] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6, 041039(2016).
[26] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T.
Pfau, Observation of Quantum Droplets in a Strongly Dipolar
Bose Gas, Phys. Rev. Lett. 116, 215301 (2016).
[27] F. Wächtler and L. Santos, Quantum filaments in dipolar Bose-Einstein condensates, Phys. Rev. A 93, 061603(R)(2016).
[28] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Groundstate phase diagram of a dipolar condensate with quantum
fluctuations, Phys. Rev. A 94, 033619 (2016).
[29] M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen, M. J.
Mark, R. N. Bisset, L. Santos, and F. Ferlaino, Two-dimensional
supersolidity in a dipolar quantum gas, Nature (London) 596,
357 (2021).
[30] T. Bland, E. Poli, C. Politi, L. Klaus, M. Norcia, F. Ferlaino,
L. Santos, and R. Bisset, Two-dimensional supersolidity in acircular trap, arXiv:2107.06680.
[31] D. Baillie and P. B. Blakie, Droplet Crystal Ground Statesof a Dipolar Bose Gas, Phys. Rev. Lett. 121, 195301(2018).
[32] S. Roccuzzo, S. Stringari, and A. Recati, Supersolid edge
and bulk phases of a dipolar quantum gas in a box,
arXiv:2104.01068.

[33] A. Gallemí, S. M. Roccuzzo, S. Stringari, and A. Recati,
Quantized vortices in dipolar supersolid Bose-Einsteincondensed gases, Phys. Rev. A 102, 023322 (2020).
[34] S. M. Roccuzzo, A. Gallemí, A. Recati, and S. Stringari, Rotating a Supersolid Dipolar Gas, Phys. Rev. Lett. 124, 045702(2020).
[35] M. N. Tengstrand, D. Boholm, R. Sachdeva, J. Bengtsson, andS. M. Reimann, Persistent currents in toroidal dipolar supersolids, Phys. Rev. A 103, 013313 (2021).
[36] F. Ancilotto, M. Barranco, M. Pi, and L. Reatto, Vortex properties in the extended supersolid phase of dipolar Bose-Einsteincondensates, Phys. Rev. A 103, 033314 (2021).
[37] Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidity Around a
Critical Point in Dipolar Bose-Einstein Condensates, Phys. Rev.
Lett. 123, 015301 (2019).
[38] Y.-C. Zhang, T. Pohl, and F. Maucher, Phases of supersolids inconfined dipolar Bose-Einstein condensates, Phys. Rev. A 104,
013310 (2021).
[39] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H. Ng,
S. D. Graham, P. Uerlings, T. Langen, M. Zwierlein, and T.
Pfau, Pattern formation in quantum ferrofluids: From supersolids to superglasses, Phys. Rev. Research 3, 033125 (2021).
[40] A. R. P. Lima and A. Pelster, Quantum fluctuations in dipolar
Bose gases, Phys. Rev. A 84, 041604(R) (2011).
[41] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Bogoliubov
modes of a dipolar condensate in a cylindrical trap, Phys. Rev.
A 74, 013623 (2006).
[42] S. M. Roccuzzo and F. Ancilotto, Supersolid behavior of adipolar Bose-Einstein condensate confined in a tube, Phys. Rev.
A 99, 041601(R) (2019).
[43] L. Lavoine and T. Bourdel, Beyond-mean-field crossover fromone dimension to three dimensions in quantum droplets of binary mixtures, Phys. Rev. A 103, 033312 (2021).
[44] A. J. Dickstein, S. Erramilli, R. E. Goldstein, D. P. Jackson, and
S. A. Langer, Labyrinthine pattern formation in magnetic fluids,
Science 261, 1012 (1993).
[45] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation andSuperfluidity (Oxford University Press, Oxford, 2016).

063307-9

98

4.7

Two-dimensional supersolidity in dysprosium

Publication V:

Two-dimensional supersolid formation in dipolar condensates
The following publication has appeared in
Physical Review Letters 128, 195302 (2022)¶
submitted 14 Jul 2021; published 13 May 2022
DOI: https://doi.org/10.1103/PhysRevLett.128.195302
T. Bland,1 E. Poli,2 C. Politi,1,2 L. Klaus,1 M. A. Norcia,1 F. Ferlaino,1,2 L. Santos,3and R. N. Bisset2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 Institut für Theoretische Physik, Universität Hannover, Hannover, Germany¶The author of the present thesis performed the measurements together with M. A. N.,

and
L. K., analysed the experimental data with M. A. N., and contributed in writing the manuscript andinterpreting the results together with all the authors.

PHYSICAL REVIEW LETTERS 128, 195302 (2022)
Featured in Physics

Two-Dimensional Supersolid Formation in Dipolar Condensates
T. Bland ,1 E. Poli ,2 C. Politi ,1,2 L. Klaus ,1,2 M. A. Norcia ,1 F. Ferlaino ,1,2 L. Santos ,3 and R. N. Bisset

2,*

1

Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck 6020, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Innsbruck 6020, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, Hannover 30167, Germany(Received 14 July 2021; revised 19 January 2022; accepted 11 April 2022; published 13 May 2022)
Dipolar condensates have recently been coaxed to form the long-sought supersolid phase. While onedimensional supersolids may be prepared by triggering a roton instability, we find that such a procedurein two dimensions (2D) leads to a loss of both global phase coherence and crystalline order. Unlike in 1D,
the 2D roton modes have little in common with the supersolid configuration. We develop a finitetemperature stochastic Gross-Pitaevskii theory that includes beyond-mean-field effects to explore theformation process in 2D and find that evaporative cooling directly into the supersolid phase—hence
bypassing the first-order roton instability—can produce a robust supersolid in a circular trap.
Importantly, the resulting supersolid is stable at the final nonzero temperature. We then experimentally
produce a 2D supersolid in a near-circular trap through such an evaporative procedure. Our work
provides insight into the process of supersolid formation in 2D and defines a realistic path to theformation of large two-dimensional supersolid arrays.
DOI: 10.1103/PhysRevLett.128.195302

The supersolid phase was predicted to simultaneously
exhibit crystalline order and superfluidity [1–6]. While itremains elusive in helium, recent developments in ultracold
quantum gases have finally made supersolidity a reality,
providing an excellent platform for the control and observation of these states. Important early advances were madein systems with spin-orbit coupling [7,8] and cavitymediated interactions [9], where supersolid properties wereobserved in rigid crystal configurations. Bose-Einsteincondensates (BECs) with dipole-dipole interactions havenow been observed in a supersolid state with deformable
crystals [10–13], with their lattices genuinely arising fromthe atom-atom interactions [14–16].
In the first dipolar supersolid experiments, translational
symmetry was broken only along one axis, giving rise to a
one-dimensional (1D) density wave, commonly referred toas a 1D droplet array [10–12]. A more recent experimenthas created the first states with two-dimensional (2D)supersolidity in elongated traps of variable aspect ratio[13]. This opens the door to study vortices and persistentcurrents [17–20], as well as exotic ground state phases
predicted for large atom numbers [21–24].
It is still an open question whether 2D arrays provide asfavorable conditions for supersolidity as 1D arrays do. In
1D, following an interaction quench from an unmodulated
to modulated BEC, the density pattern induced by a roton
instability [14,25–28] can smoothly connect with the final
supersolid array [10–12]. This transition, hence, has aweakly first-order character or is even continuous [29,30],and such quenches through the transition cause only small
excitations of the resulting supersolid [10–12]. While it has
0031-9007=22=128(19)=195302(7)been predicted that a similar procedure may lead tocoherence between three droplets in a triangular configuration [31], earlier work with nondipolar superfluids
suggests that such symmetry-breaking quenches may be
unfavorable for supersolid formation in 2D and 3D [32,33].
An alternative method exists to experimentally produce
dipolar supersolids. Instead of quenching the interactions
to trigger a roton instability, it is possible to cool athermal sample directly into the supersolid state using
evaporative cooling techniques [12,34]. Crucially, this is
the only known method for producing 2D supersolids todate [13]. While a dynamic interaction quench may bedescribed by the extended Gross-Pitaevskii equation(eGPE) [35–38], we are not aware of any available theory
to model the required evaporation process. From a theoretical perspective, much remains unknown about evaporative supersolid formation. Is it a general feature that the
droplets form before global phase coherence develops, asreported in Ref. [34]? Under what conditions do defects
persist? Such answers will be paramount in the quest forever-larger 2D supersolids, as well as for the observation ofvortices embedded within them.
In this Letter, we explore the formation of large 2D
supersolids in circular-shaped traps. We develop a finitetemperature Stochastic eGPE (SeGPE) theory to model theentire evaporative cooling process. Importantly, our theory
includes the beyond-mean-field quantum fluctuations
responsible for stabilizing the individual droplets.
We compare the evaporative cooling formation dynamics
with those resulting from an interaction quench, finding
striking differences between the two protocols. Following

195302-1© 2022 American Physical Society

PHYSICAL REVIEW LETTERS 128, 195302 (2022)
(a)

(b)

(c)

FIG. 1. (a) Crystal preparation from interaction quench, evolved with the eGPE, for N ≈ 2.1 × 105 Dy atoms [quench (i)]. Isosurfaces
are at 5% max density, with color indicating phase. Insets: z column densities normalized to max value from the entire simulation.
(b) Dynamic structure factor for an unmodulated BEC (as ¼ 92a0 ) in energy-momentum space, normalized to peak value. The lowestenergy roton modes are indicated, and the ground state with an m ¼ 2 roton mode added is shown, revealing the localized nature of therotons. Parameters are otherwise the same as in (a). (c) Crystal preparation from temperature quench (evaporative cooling) evolved with
the SeGPE [quench (ii)]. The temperature decreases as the chemical
ppotential
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and condensate number rise, with scattering length fixed at
as ¼ 88a0 . For all subplots f x;y;z ¼ ð33; 33; 167Þ Hz, and lz ¼ ℏ=mωz .

an interaction quench, the 2D crystal grows nonlinearlywith the droplets developing sequentially, producing configurations that are unrelated to any roton mode combination of the original unmodulated BEC. The resulting crystal
is substantially excited and lacks global phase coherence.
Alternatively, by directly cooling into the supersolid
regime, our SeGPE theory predicts the formation of large
2D supersolids in circular traps, with global phase coherence that remains robust at finite temperature. To benchmark our theory—as well as to test the direct cooling
protocol for pancake-shaped trapping geometries—weperform experiments and observe a 7-droplet hexagonal
supersolid in a near-circular trap.
Formalism.—We are interested in ultracold, dipolar
Bose gases harmonically confined in 3D with trapping
frequencies ωx;y;z ¼ 2πf x;y;z . Two-body contact interactions and the long-ranged, anisotropic dipole-dipole
interactions are well described by a pseudopotential,UðrÞ ¼ ð4πℏ2 as =mÞδðrÞ þ ð3ℏ2 add =mÞ½ð1 − 3cos2 θÞ=r3 ,with as being the s-wave scattering length and add ¼μ0 μ2m m=12πℏ2 the dipole length, with magnetic moment
μm , and θ is the angle between the polarization axis (z axis)
and the vector joining two particles. The ratio ϵdd ¼ add =as(for as > 0) is convenient to keep in mind, since for ϵdd ≤ 1
the ground state will be an unmodulated BEC, whereas
for the dipole-dominated regime ϵdd > 1 the unmodulated
BEC may become unstable [39]. Here, we always consider
164
Dy, with add ¼ 131a0 . The eGPE has been described
elsewhere [35–38], and its details have been deferred toSupplemental Material [40].
We phenomenologically introduce a finite-temperaturesimple growth SeGPE theory [55]. This describes the“classical” field, Ψðr; tÞ, of all highly populatedmodes up to an energy cutoff. The dynamics are governedby [56]iℏ∂Ψ¼ P̂fð1 − iγÞðL½Ψ − μÞΨ þ ηg:∂t

ð1ÞHere, L is the eGPE operator defined in Ref. [40],and γ describes the coupling of the classical field modes
to the high-lying modes. We find that γ ¼ 7.5 × 10−3
gives good agreement to the condensate number growthrate of a recent experiment under comparable conditions[34] (see also Ref. [40]). The dynamical noise term η,
subject to noise correlations given by hη ðr; tÞηðr0 ; t0 Þi ¼
2ℏγkB Tδðt − t0 Þδðr − r0 Þ, means that each simulation run
is unique. Finally, P̂ is a projector which constrains the
dynamics of the system up to energy cutoff ϵcut ðμÞ ¼ 2μ—consistent with previous treatments [57,58]—where we use
the final μ after evaporative cooling.
Supersolid formation simulations.—With these two theories in hand, we perform two kinds of dynamic quench
simulations in a pancake-shaped trap, where in bothcases the ground state for the final parameters would bea 19-droplet supersolid:(i) An interaction quench from an unmodulated BEC tothe supersolid regime using the eGPE [Fig. 1(a)]. Noise is
first added to the BEC ground state [59], and this is evolved
for a 20 ms equilibration time before the interaction
strength is linearly ramped over the next 30 ms fromas ¼ 95a0 to as ¼ 88a0 —crossing the roton phase transition to the supersolid regime—and then held constantagain for the remainder of the simulation.
(ii) A temperature quench from a thermal cloud to thesupersolid phase using the SeGPE [Fig. 1(c)]. Each simulation begins with a 200 ms equilibration time at fixedhigh temperature T ¼ 150 nK to generate a thermal cloud.
To simulate the evaporative cooling process, the chemical
potential and temperature are then linearly ramped
over 100 ms, from ðμ; TÞ ¼ ð−12.64ℏωz ; 150 nKÞ to195302-2

PHYSICAL REVIEW LETTERS 128, 195302 (2022)
ð12.64ℏωz ; 30 nKÞ, mimicking the growing condensate
number observed in experiments [60,61], while the scattering length is always held fixed at as ¼ 88a0 .
Focusing first on the interaction quench, the density
isosurfaces in Fig. 1(a) represent snapshots at various times
for a single simulation run, revealing intriguing formation
dynamics. Initial droplets are seeded through unstable rotonmodes, but staggered droplet formation reveals a processof nonlinear crystal growth, as highlighted by the column
densities shown as insets in Fig. 1(a). In Fig. 1(a2), twocentral droplets have already attained their final peakdensity, while a secondary ring of droplets is only justbeginning to form. Then, in Fig. 1(a3), eight droplets
have fully matured, and the process continues radiallyoutward until a 19-droplet crystal is approximately
attained. Similar droplet formation dynamics have been
predicted in optical media [62].
The colors on the density isosurfaces in Fig. 1(a)represent the wave function phase. The color scale isrecentered in each subplot, and an ideal phase coherent
solution would have a uniform color everywhere.
Importantly, the crystal growth process disrupts the global
phase coherence, as evidenced by the various colors in
Fig. 1(a4), leaving an excited crystal in which some outerdroplets dissolve and reemerge from the halo. Note that thesituation does not qualitatively change for reduced initialnoise or gentler interaction ramps, suggesting that thestrong excitations result from a first-order character ofthe roton instability in 2D.
We explain the interaction quench dynamics by
calculating the elementary excitations of the unmodulated
BEC close to the roton instability, i.e., for as ¼ 92a0.
These results are displayed in Fig. 1(b) as the dynamic
structure factor Sðk; ωÞ, which predicts the system
response to perturbations of momentum ℏk and energyℏω [28,63–65] (also see Ref. [40]). A roton minimum canbe seen at kx lz ≈ 1.1, and we plot the lowest roton modes
corresponding to m ¼ 0, 1, 2, with m being the angularquantum number in the z direction [66]. On the top right isthe density obtained by adding an m ¼ 2 roton mode to the
BEC wave function, revealing how rotons are confined tohigh-density regions [67,68]. This reveals a qualitative
difference between the 1D and 2D situations, since, from a
simple geometric standpoint, in 2D the high-density region
inherently encompasses a smaller proportion of the total
atom number. Thus, the roton-induced droplet number isonly a small fraction of the final droplet number, meaning
the droplets appear sequentially for 2D.
Another qualitative difference between 1D and 2D is akind of frustration. First, note that our target supersolid
ground state for the final quench parameters is a 19-droplet
crystal, with a central droplet [see the inset of Fig. 2(b)].
Only an m ¼ 0 roton mode [see Fig. 1(b)] could directly
trigger the formation of a central droplet, but then only
concentric rings could form further out. Thus, unlike for

(a)

(b)

FIG. 2. Supersolid quality. (a) Global phase coherence Cp overtime for interaction quenches [quench (i)] into linear chain (blue)
and pancake crystal (red) and temperature quenches [quench (ii)]into the pancake crystal (black). Diamonds link to example
frames in Figs. 1(a) and 1(c). Each curve is averaged over 3–5
runs with an error band marking one standard deviation. Timet ¼ 0 corresponds to when the crystals first fully mature.
(b) Density overlap Cd between the time-dependent and groundstate densities. Parameters are the same as Fig. 1, but for linear
chain fx;y;z ¼ ð33; 110; 167Þ Hz and N ¼ 82 × 103 .

1D, no single roton mode can smoothly connect theunmodulated BEC to the 2D supersolid ground state.
Next, we analyze the finite-temperature quench results.
Figure 1(c) shows snapshots of the condensate growth,demonstrating that both the crystal structure and theglobal phase coherence—evidenced by the uniform color
in Fig. 1(c4)—develop soon after the quench. Note thattimescales will be quantified shortly. It is also an importantresult in itself that we predict such a large 2D supersolid
to be stable against thermal fluctuations (recall that
T final ¼ 30 nK). As they form, each droplet individually
has a uniform phase that may be different from that of its
neighbors, sometimes creating vortex pairs between droplets of different phase. In this scenario, the droplets do
not form as a result of a roton instability, and the partial
phase coherence continues to improve after the crystal
has formed, consistent with earlier observations [34].
Occasionally, long-lived isolated vortices remain near thecenter of the supersolid. Simulation videos are provided in
Supplemental Material [40].
Supersolid quality.—We seek to quantify the resulting
supersolid quality for both quench protocols. We startby analyzing the phase excitations, taking the phasecoherence Cp with a similar measure presented inRef. [10]. A value of Cp ¼ 1ð0Þ implies global phase

195302-3

PHYSICAL REVIEW LETTERS 128, 195302 (2022)
coherence (incoherence) [69]. In Fig. 2(a), we plot thisquantity for interaction quenches into the pancake supersolid regime (red) and linear supersolid regime (blue) andtemperature quenches into the pancake supersolid (black).
The time t ¼ 0 indicates when the droplet number has
approximately stabilized and the crystal has first matured[70]. For the linear chain, the system remains coherent(high Cp ≈ 0.8), indicating a stable supersolid. However,quenching into the pancake geometry is qualitatively
different, with strong incoherence (Cp ≈ 0.3) soon after
crystal formation, recovering a high value at around 150 ms
after the crystal forms. During evaporative cooling, theglobal phase coherence is predicted by the high value of
Cp ≈ 0.8 around 50 ms after the crystal forms, withqualitatively similar values to the interaction quench
simulations for the linear supersolid case.
We quantify the quality of the supersolid crystal bymeasuring the density overlap Cd between the ground statetarget solution and the time-dependent wave function [71].
We find the maximal value of Cd after applying translations
and rotations to the state, noting that perfect overlap would
give Cd ¼ 1. In Fig. 2(b), this quantity is presented for thetwo geometries, with the ground state solutions shown as
insets. For the linear chain, once the droplets have formed,
the density overlap rapidly attains Cd > 0.9 and remains
there, consistent with the interaction quenched state being
close to the ground state supersolid. However, the pancake
case shows weak overlap after the droplets are formed,
which only recovers slowly—after around 300 ms—tovalues comparable with the linear chain. Primarily, this isdue to the sensitivity of droplet positions of Cd andindicates that there are many excited supersolid modes
present after the droplets form [40]. Direct evaporative
cooling for the pancake case, however, shows that after the
droplets have formed they rapidly settle into the expected
crystal pattern (Cd ≈ 0.95).
Finally, it is important to note that for the pancake
interaction quench, while the phase coherence is restored
by around t ¼ 150 ms after the droplets are formed, thecrystal remains highly excited until around 300 ms. On
these timescales, three-body losses become significant, and
it is unlikely that a large supersolid would be observed. In
contrast, direct evaporative cooling may lead to a robust
supersolid within around 50 ms of the crystal first appearing, a timescale that we find to be weakly dependent on thevalue of γ [40].
Experimental observation.—While experiments have
evaporatively cooled directly into the supersolid phasefor linear and elongated 2D configurations [12,13,34],
this could prove an optimal method in circular traps foravoiding the excitations associated with crossing theroton instability. We confirm this by producing a 7-droplet
hexagon supersolid in a near-circular trap, as shown in
Fig. 3. The experimental apparatus and procedure is similar
to that described previously [13], but new modifications in

FIG. 3. Experimental realization of a 7-droplet hexagon state.
(a) Exemplary in situ image of the density profile. (b) Image after
36 ms time-of-flight (TOF) expansion, averaged over 68 trials of
the experiment. Hexagonal modulation structure is clearly present
in the averaged image. Note the rotation of the hexagon between
in situ and TOF images. (c),(d) Corresponding simulations forthe same trap, and with as ¼ 90a0 and ≈4.4 × 104 atoms within
the droplets.

the optical dipole trap setup have enabled us to tune
between anisotropic and round traps. The current opticaltrap consists of three 1064 nm wavelength trapping beams,
each propagating in the plane perpendicular to gravity. Twoof the beams, which cross perpendicularly, have approximately 60 μm waists and define the horizontal trapping
frequencies. The third, crossing at a roughly 45° angle fromthe others, has a waist of approximately 18 μm and is
rapidly scanned to create a time-averaged light sheet thatdefines the vertical confinement.
In a harmonic trap with frequencies f x;y;z ¼ ½47ð1Þ;
43ð1Þ; 133ð5Þ Hz, we observe in trap a 7-droplet state
consisting of a hexagon with a central droplet, with acondensate atom number of N ∼ 4 × 104 [Fig. 3(a)]. To
confirm that this state is phase coherent, we release theatoms from the trap and image the interference pattern after
36 ms time of flight [Fig. 3(b)]. The presence of clear
modulation in the interference pattern averaged over
68 runs of the experiment indicates a well-defined andreproducible relative phase between the droplets and isconsistent with our expectations for a phase-coherent state
undergoing expansion [Fig. 3(d)], obtained through 3D
dynamic simulations starting from the eGPE ground state[Fig. 3(c)]. Even rounder traps are possible, but the slightanisotropy orients the state, helping to observe the reproducible interference pattern.
Summary.—We have theoretically explored the formation of large 2D supersolids using both an interaction

195302-4

PHYSICAL REVIEW LETTERS 128, 195302 (2022)
quench from an unmodulated BEC and a temperature
quench from a thermal cloud. For the latter, we developed
a finite-temperature stochastic Gross-Pitaevskii theory thatcan simulate evaporative cooling directly into the supersolid regime. Our simulations predict that a temperature
quench provides a robust path for creating 2D supersolids
in circular traps, and we confirm this experimentally byusing this method to create a reproducible hexagonal
7-droplet supersolid.
In contrast, the interaction quench results in highly
excited crystals that lack global phase coherence in the
period following their formation. Interestingly, droplets
appear sequentially rather than simultaneously, with thefinal crystal structure being unrelated to the roton modes
that seeded the instability. This is in contrast to the situationfor 1D arrays, where an interaction quench through a roton
instability can smoothly connect an unmodulated BEC tothe supersolid ground state.
Our finite-temperature theory is broadly applicablefor future studies on topics such as formation dynamics,
supersolid vortices, improved quench protocols to produce
large 2D supersolids, and thermal resilience, as well as
dipolar droplets in general.
We thank Manfred Mark and the Innsbruck Erbium teamfor valuable discussions and thank Péter Juhász for carefully reading the manuscript. We acknowledge R. M. W.
van Bijnen for developing the code for our eGPE and BdG
simulations. Part of the computational results presented
have been achieved using the HPC infrastructure LEO of
the University of Innsbruck. The experimental team isfinancially supported through an ERC Consolidator Grant(RARE, No. 681432), an NFRI grant (MIRARE,
No. OAW0600) of the Austrian Academy of Science,and the QuantERA grant MAQS by the Austrian
Science Fund FWF No. I4391-N. L. S. and F. F. acknowledge the DFG/FWF (Grant No. FOR 2247/I4317-N36) anda joint-project grant from the FWF (Grant No. I4426, RSF/
Russland 2019). L. S. thanks the funding by the Deutsche
Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy—
EXC-2123 QuantumFrontiers—390837967. M. A. N. hasreceived funding as an ESQ postdoctoral fellow from theEuropean Unions Horizon 2020 research and innovation
program under the Marie Sklodowska-Curie Grant
Agreement No. 801110 and the Austrian Federal
Ministry of Education, Science and Research (BMBWF).
We also acknowledge the Innsbruck Laser Core Facility,
financed by the Austrian Federal Ministry of Science,
Research and Economy.

*

Corresponding author.
russell.bisset@uibk.ac.at[1] E. P. Gross, Unified theory of interacting bosons, Phys. Rev.
106, 161 (1957).

[2] A. Andreev and I. Lifshitz, Quantum theory of defects incrystals, Zh. Eksp. Teor. Fiz. 56, 2057 (1969) [JETP 29,1107 (1969)].
[3] D. Thouless, The flow of a dense superfluid, Ann. Phys.
(N.Y.) 52, 403 (1969).
[4] G. Chester, Speculations on Bose-Einstein condensationand quantum crystals, Phys. Rev. A 2, 256 (1970).
[5] A. J. Leggett, Can a Solid Be “Superfluid”?, Phys. Rev. Lett.
25, 1543 (1970).
[6] M. Boninsegni and N. V. Prokof'ev, Colloquium: Supersolids:
What and where are they?, Rev. Mod. Phys. 84, 759 (2012).
[7] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. Ç.
Top, A. O. Jamison, and W. Ketterle, A stripe phase withsupersolid properties in spin–orbit-coupled Bose–Einsteincondensates, Nature (London) 543, 91 (2017).
[8] T. M. Bersano, J. Hou, S. Mossman, V. Gokhroo, X.-W.
Luo, K. Sun, C. Zhang, and P. Engels, Experimental
realization of a long-lived striped Bose-Einstein condensate
induced by momentum-space hopping, Phys. Rev. A 99,
051602(R) (2019).
[9] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T.
Donner, Supersolid formation in a quantum gas breaking acontinuous translational symmetry, Nature (London) 543,
87 (2017).
[10] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C.
Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,
Observation of a Dipolar Quantum Gas with Metastable
Supersolid Properties, Phys. Rev. Lett. 122, 130405 (2019).
[11] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo,T. Langen, and T. Pfau, Transient Supersolid Properties in
an Array of Dipolar Quantum Droplets, Phys. Rev. X 9,011051 (2019).
[12] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann,
C. Politi, G. Durastante, R. M. W. van Bijnen, A.
Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,Long-Lived and Transient Supersolid Behaviors in Dipolar
Quantum Gases, Phys. Rev. X 9, 021012 (2019).
[13] M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen,
M. J. Mark, R. N. Bisset, L. Santos, and F. Ferlaino, Twodimensional supersolidity in a dipolar quantum gas, Nature(London) 596, 357 (2021).
[14] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter,
M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation Spectrum of a Trapped Dipolar Supersolid and Its Experimental
Evidence, Phys. Rev. Lett. 123, 050402 (2019).
[15] L. Tanzi, S. Roccuzzo, E. Lucioni, F. Famà, A. Fioretti,
C. Gabbanini, G. Modugno, A. Recati, and S. Stringari,
Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas, Nature (London) 574, 382(2019).
[16] M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt, M. Wenzel,
H. P. Büchler, T. Langen, and T. Pfau, The low-energy
goldstone mode in a trapped dipolar supersolid, Nature(London) 574, 386 (2019).
[17] A. Gallemí, S. M. Roccuzzo, S. Stringari, and A. Recati,
Quantized vortices in dipolar supersolid Bose-Einsteincondensed gases, Phys. Rev. A 102, 023322 (2020).
[18] S. M. Roccuzzo, A. Gallemí, A. Recati, and S. Stringari,
Rotating a Supersolid Dipolar Gas, Phys. Rev. Lett. 124,045702 (2020).

195302-5

PHYSICAL REVIEW LETTERS 128, 195302 (2022)[19] M. N. Tengstrand, D. Boholm, R. Sachdeva, J. Bengtsson,
and S. M. Reimann, Persistent currents in toroidal dipolar
supersolids, Phys. Rev. A 103, 013313 (2021).
[20] F. Ancilotto, M. Barranco, M. Pi, and L. Reatto, Vortex
properties in the extended supersolid phase of dipolar
Bose-Einstein condensates, Phys. Rev. A 103, 033314(2021).
[21] D. Baillie and P. B. Blakie, Droplet Crystal Ground States ofa Dipolar Bose Gas, Phys. Rev. Lett. 121, 195301 (2018).
[22] Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidity Arounda Critical Point in Dipolar Bose-Einstein Condensates, Phys.
Rev. Lett. 123, 015301 (2019).
[23] Y.-C. Zhang, T. Pohl, and F. Maucher, Phases of supersolids
in confined dipolar Bose-Einstein condensates, Phys. Rev.
A 104, 013310 (2021).
[24] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H.
Ng, S. D. Graham, P. Uerlings, T. Langen, M. Zwierlein,
and T. Pfau, Pattern formation in quantum ferrofluids: Fromsupersolids to superglasses, Phys. Rev. Research 3, 033125(2021).
[25] L. Santos, G. V. Shlyapnikov, and M. Lewenstein, RotonMaxon Spectrum and Stability of Trapped Dipolar BoseEinstein Condensates, Phys. Rev. Lett. 90, 250403 (2003).
[26] S. Giovanazzi and D. H. J. O’Dell, Instabilities and the roton
spectrum of a quasi-1d Bose-Einstein condensed gas with
dipole-dipole interactions, Eur. Phys. J. D 31, 439 (2004).
[27] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider,
M. J. Mark, L. Chomaz, and F. Ferlaino, Probing the Roton
Excitation Spectrum of a Stable Dipolar Bose Gas, Phys.
Rev. Lett. 122, 183401 (2019).
[28] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S.
Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino, Observation of roton mode population in a
dipolar quantum gas, Nat. Phys. 14, 442 (2018).
[29] P. B. Blakie, D. Baillie, L. Chomaz, and F. Ferlaino,
Supersolidity in an elongated dipolar condensate, Phys.
Rev. Research 2, 043318 (2020).
[30] G. Biagioni, N. Antolini, A. Alaa, M. Modugno, A. Fioretti,
C. Gabbanini, L. Tanzi, and G. Modugno, Dimensional
Crossover in the Superfluid-Supersolid Quantum Phase
Transition, Phys. Rev. X 12, 021019 (2016).
[31] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H.
Ng, S. D. Graham, P. Uerlings, H. P. Büchler, T. Langen, M.
Zwierlein, and T. Pfau, Supersolidity in Two-Dimensional
Trapped Dipolar Droplet Arrays, Phys. Rev. Lett. 127,
155301 (2021).
[32] T. Macrì, F. Maucher, F. Cinti, and T. Pohl, Elementary
excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition, Phys. Rev. A 87, 061602(R) (2013).
[33] N. Henkel, R. Nath, and T. Pohl, Three-Dimensional Roton
Excitations and Supersolid Formation in Rydberg-Excited
Bose-Einstein Condensates, Phys. Rev. Lett. 104, 195302(2010).
[34] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark,M. A. Norcia, and F. Ferlaino, Birth, Life, and Death of a
Dipolar Supersolid, Phys. Rev. Lett. 126, 233401 (2021).
[35] F. Wächtler and L. Santos, Quantum filaments in dipolar
Bose-Einstein condensates, Phys. Rev. A 93, 061603(R)(2016).

[36] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,
Ground-state phase diagram of a dipolar condensate withquantum fluctuations, Phys. Rev. A 94, 033619 (2016).
[37] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, andT. Pfau, Observation of Quantum Droplets in a Strongly
Dipolar Bose Gas, Phys. Rev. Lett. 116, 215301 (2016).
[38] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a
Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6,041039 (2016).
[39] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T.
Pfau, The physics of dipolar bosonic quantum gases, Rep.
Prog. Phys. 72, 126401 (2009).
[40] See Supplemental Material http://link.aps.org/supplemental/
10.1103/PhysRevLett.128.195302 for a detailed analysis of
our finite-temperature theory parameters and discussionof the excitations of a two-dimensional supersolid, which
includes additional Refs. [41–54].
[41] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Bogoliubov
modes of a dipolar condensate in a cylindrical trap, Phys.
Rev. A 74, 013623 (2006).
[42] A. R. P. Lima and A. Pelster, Quantum fluctuations in
dipolar bose gases, Phys. Rev. A 84, 041604(R) (2011).
[43] S. P. Cockburn and N. P. Proukakis, Ab initio methods forfinite-temperature two-dimensional bose gases, Phys. Rev.
A 86, 033610 (2012).
[44] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K.
Campbell, A Rapidly Expanding Bose-Einstein Condensate: An Expanding Universe in the Lab, Phys. Rev. X 8,021021 (2018).
[45] M. Ota, F. Larcher, F. Dalfovo, L. Pitaevskii, N. P.
Proukakis, and S. Stringari, Collisionless Sound in a
Uniform Two-Dimensional Bose Gas, Phys. Rev. Lett.
121, 145302 (2018).
[46] T. Bland, Q. Marolleau, P. Comaron, B. Malomed, and N.
Proukakis, Persistent current formation in double-ringgeometries, J. Phys. B 53, 115301 (2020).
[47] S. De, D. L. Campbell, R. M. Price, A. Putra, B. M.
Anderson, and I. B. Spielman, Quenched binary BoseEinstein condensates: Spin-domain formation and coarsening, Phys. Rev. A 89, 033631 (2014).
[48] C. W. Gardiner and M. J. Davis, The stochastic GrossPitaevskii equation:II, J. Phys. B 36, 4731 (2003).
[49] S. J. Rooney, P. B. Blakie, and A. S. Bradley, Stochastic
projected Gross-Pitaevskii equation, Phys. Rev. A 86,
053634 (2012).
[50] R. Kubo, The fluctuation-dissipation theorem, Rep. Prog.
Phys. 29, 255 (1966).
[51] E. B. Linscott and P. B. Blakie, Thermally activated local
collapse of a flattened dipolar condensate, Phys. Rev. A 90,
053605 (2014).
[52] D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. A.
Baranov, R. van Bijnen, S. M. Roccuzzo, A. Recati, B.
Blakie, D. Baillie, L. Chomaz, and F. Ferlaino, Bragg
scattering of an ultracold dipolar gas across the phasetransition from Bose-Einstein condensate to supersolid in
the free-particle regime, Phys. Rev. A 104, L011302 (2021).
[53] E. Poli, T. Bland, C. Politi, L. Klaus, M. A. Norcia,
F. Ferlaino, R. N. Bisset, and L. Santos, Maintaining

195302-6

PHYSICAL REVIEW LETTERS 128, 195302 (2022)
supersolidity in one and two dimensions, Phys. Rev. A 104,
063307 (2021).
[54] J. Goldstone, Field theories with superconductor solutions,Nuovo Cimento (1955–1965) 19, 154 (1961).
[55] This method has been applied previously in a quasi-twodimensional setting without quantum fluctuations [51].
[56] P. Blakie, A. Bradley, M. Davis, R. Ballagh, and C. Gardiner,
Dynamics and statistical mechanics of ultra-cold Bose gases
using c-field techniques, Adv. Phys. 57, 363 (2008).
[57] S. J. Rooney, T. W. Neely, B. P. Anderson, and A. S.
Bradley, Persistent-current formation in a high-temperature
Bose-Einstein condensate: An experimental test forclassical-field theory, Phys. Rev. A 88, 063620 (2013).
[58] R. G. McDonald, P. S. Barnett, F. Atayee, and A. S. Bradley,
Dynamics of hot Bose-Einstein condensates: Stochastic
ehrenfest relations for number and energy damping, SciPost
Phys. 8, 029 (2020).
P[59] Our initial state is ψðr; 0Þ ¼ ψ 0 ðrÞ þ 0n αn ϕn ðrÞ, where
ϕn are the single-particle states, αn is a complex Gaussian
random variable with hjαn j2 i ¼ ðeϵn =kB T − 1Þ−1 þ 12, and thesum is restricted to modes with ϵn ≤ 2kB T, with T ¼ 30 nK.
[60] C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley,
M. J. Davis, and B. P. Anderson, Spontaneous vortices in theformation of Bose-Einstein condensates, Nature (London)
455, 948 (2008).
[61] I.-K. Liu, S. Donadello, G. Lamporesi, G. Ferrari, S.-C.
Gou, F. Dalfovo, and N. Proukakis, Dynamical equilibration
across a quenched phase transition in a trapped quantum
gas, Commun. Phys. 1, 1 (2018).
[62] F. Maucher, T. Pohl, S. Skupin, and W. Krolikowski, SelfOrganization of Light in Optical Media with Competing
Nonlinearities, Phys. Rev. Lett. 116, 163902 (2016).
[63] F. Zambelli, L. Pitaevskii, D. M. Stamper-Kurn, and S.
Stringari, Dynamic structure factor and momentum distribution of a trapped bose gas, Phys. Rev. A 61, 063608 (2000).

[64] P. B. Blakie, R. J. Ballagh, and C. W. Gardiner, Theory ofcoherent bragg spectroscopy of a trapped Bose-Einsteincondensate, Phys. Rev. A 65, 033602 (2002).
[65] P. B. Blakie, D. Baillie, and R. N. Bisset, Roton
spectroscopy in a harmonically trapped dipolar BoseEinstein condensate, Phys. Rev. A 86, 021604(R)(2012).
[66] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Radial andAngular Rotons in Trapped Dipolar Gases, Phys. Rev. Lett.
98, 030406 (2007).
[67] M. Jona-Lasinio, K. Łakomy, and L. Santos, Roton confinement in trapped dipolar Bose-Einstein condensates, Phys.
Rev. A 88, 013619 (2013).
[68] R. N. Bisset, D. Baillie, and P. B. Blakie, Roton excitations
in a trapped dipolar Bose-Einstein condensate, Phys. Rev. A
88, 043606 (2013).
R[69] We measure phase coherence
as Cp ¼ 1 − ð2=πÞ C dxdyjRψðx; yÞj2 jθðx; yÞ − βj= C dxdyjψðx; yÞj2 , where θðx; yÞ isthe phase of ψðx; yÞ in the z ¼ 0 plane and β is a fitting
parameter to maximize Cp at each time. The integration
region C encompasses the droplets.
[70] Relative to the crystal maturation time, each quench
ramp ended at 5, −13, and −90 ms for the linear
interaction quench, pancake interaction quench, andpancake temperature quench, respectively. Note that
evaporative cooling continues after the fμ; Tg quench
ends, taking approximately a further 100 ms until theclassical field modes have equilibrated with the highenergy modes [40].
R[71] The density
is given by Cd ¼ d3 xnðx − x0 ;R 3overlap
2
ϕ0 ; tÞnGS = d xnGS , with the ground state density andtime-dependent
density, respectively, normalized as
R 3
R
d xnGS ¼ d3 xnðx − x0 ; ϕ0 ; tÞ ¼ 1. The optimization
parameters x0 and ϕ0 are translations and a rotation,
respectively, applied to the wave function to maximize Cd .

195302-7

Supplemental material: Two-dimensional supersolid formation in dipolar condensates
T. Bland,1 E. Poli,2 C. Politi,1, 2 L. Klaus,1, 2 M. A. Norcia,1 F. Ferlaino,1, 2 L. Santos,3 and R. N. Bisset2
1
Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, Germany(Dated: April 13, 2022)

FORMALISMCHOICE OF γ FOR STOCHASTIC EXTENDED
GPE THEORY (EQ. 1 OF MAIN TEXT)

We utilize dynamic and ground state calculations of
the extended Gross-Pitaevskii equation (eGPE), given byi~ψt = L[ψ]ψ, where the eGPE operator is [1–4]

1
~2 ∇2
+ m ωx2 x2 + ωy2 y 2 + ωz2 z 2(S1)
L[ψ] = −2m2Z+ d3 x0 U (x − x0 )|ψ(x0 , t)|2 + γQF |ψ(x, t)|3 ,m is the mass and ωx,y,z = 2πfx,y,z are the externaltrapping frequencies. Two-body contact interactions andthe long-ranged, anisotropic dipole-dipole interactions
are described by the pseudo-potential

U (r) =

4π~2 as3~2 add 1 − 3 cos2 θδ(r) +
,
m
m
r3

(S2)

respectively, with θ being the angle between the polarization axis (z axis) and the vector joining two particles. This is characterized by s-wave scattering length
as and dipole length add = µ0 µ2m m/12π~2 , with magnetic moment µm . To find the ground state we employ a conjugate-gradients technique minimizing the corresponding energy functional [5]. The last term appearing in Eq. (S1) represents quantum fluctuations in theform of a dipolar Lee-Huang-Yang correction [6], γQF =
p
R1
128~2
πa5s Re {Q5 (εdd )}, where Q5 (εdd ) = 0 du (1 −3m
εdd + 3u2 εdd )5/2 is the auxiliary function, which can besolved analytically, and εdd = add /as .
Primarily, we use the eGPE to simulate the formation
dynamics of a supersolid through an interaction quenchof the scattering length, constructing an initial state byadding non-interacting noise to an unmodulated BEC
ground state (far from the roton P
instability). Thus, our0initial state is ψ(r, 0) = ψ0 (r) + n αn φn (r), where φn
are the single-particle states, αn is a complex Gaussian
random variable with h|αn |2 i = (en /kB T − 1)−1 + 12 withtemperature T and the sum is restricted to modes withn ≤ 2kB T . On average, this adds about 1000 atoms
when T = 30nK.

Stochastic Gross-Pitaevskii equations have been
benchmarked against numerous Bose gas experiments invarious geometries [7–13], including Bose-Bose mixtures[14]. These comparisons to experimental data include direct modeling of the evaporative cooling process [7, 8]. In
these works, γ is approximated by fitting the condensate
atom number growth rate to experimental observations.
However, there is also an approximate analytic solution
appropriate for near-equilibrium solutions that depends
on the chemical potential, energy cut-off, temperature,and interaction strength [15]. One comparison found γ
extracted from condensate growth data is an order ofmagnitude larger than the analytic approximation [8].
The choice of γ does not affect the equilibrium properties of the system [16] (due to the fluctuation-dissipation
theorem [17, 18]), however it affects many observables
during equilibration.
To the best of our knowledge there does not exist any
analytic prediction for γ for the dipolar system [18]. We
approximate γ based on direct experiment-theory comparisons with the condensate growth rate in Ref. [19].
For that case we have the relevant experimental data
available. There, the supersolid formation was studied
in detail for a dysprosium supersolid in a cigar-shaped
geometry, and a value of γ = 0.0075 was found to givequantitatively similar growth behavior as observed experimentally. Here, for a qualitatively similar regime [20] we
initially assume the same value.
In Fig. S1 we investigate the γ dependency on theevaporative cooling protocol presented in the main text,
namely a 164 Dy gas in a pancake trap with (fx , fy , fz ) =(33, 33, 167)Hz and as = 88a0 . Further details are givenin the main text. We consider both half and double theinitial value, i.e. γ = (0.00375, 0.0075, 0.015). Interestingly, several observables are sensitive to the choice of γ.
This includes: atom number versus time, onset time ofglobal phase coherence versus onset time of crystal structure, and the number of free vortices trapped within the
crystal. For our simulations we calculate the phase coherence C p and density overlap C d . Despite the c-field
atom number increasing faster with larger γ, the growthof phase coherence does not appear as clearly dependenton γ. This is possibly due to the number of vortices gen-2
quenches. Even in the presence of a free vortex the final state can still be considered as a coherent supersolid,
with C d ∼ 1, however C p < 1. Future improvements tothis measure could involve finding the vortex centre andmultiplying the phase by the opposite circulation. This
is not easy however due to the nonlinear azimuthal phaseprofile of a vortex in a supersolid [21, 22].
The effect of varying γ is most obvious in the overlap
between the simulation density and ground state density, C d [Fig. S1(c)]. Larger γ forces the c-field atom
number, and hence density, to rapidly increase, forcing
the fast production of droplets. We believe that comparing a spectrum of observables such as these will provide
important benchmarks to fine tune the simulations, andthat this will also provide an important test in the future
for the development of analytic theories that can predictγ. As supersolid production in 2D becomes more routine,
direct comparison between the condensate atom number
and droplet number growth rates in particular will become crucial in determining the appropriate choice of γ.
It is worth noting that even if the supersolid formation time was a few 100ms longer than the data presented here, the whole process would still be faster thanevaporatively cooling into the BEC state, quenching theinteractions and then waiting for the phase coherence toreappear. In this latter scenario, significant three-bodylosses play a negative role. Previous works in 1D havemaximized phase coherence by increasing the final as ,and hence increasing the superfluid connection between
droplets [23], and decreasing the droplet peak density.
However, the droplet number is strongly dependent on as ,and we find that such a strategy significantly decreases
the number of droplets.

THE ROLE OF ENERGY DURING SUPERSOLIDFORMATION
FIG. S1. Effect of varying γ. Five simulations are shownwith the same color for each γ. (a) Energy per particle versus
time. The average final value is independent of γ, as expected.
Inset shows the same data over a longer time range, wheret = 0 corresponds to the beginning of the 100ms temperature
quench. (b) Phase coherence for all data. Curves with a lower
opacity correspond to simulations with isolated vortices. (c)Density overlap with the ground state 19 droplet supersolid.
In all plots the dashed vertical line indicates t = 0 in Fig. 2of the main text.

erated through the quench, which have been shown toappear more readily with increasing γ, although there isalso evidence they are damped quicker too [7]. Curves
for simulations with a long-lived single vortex are shownwith a lower opacity in Fig. S1(b), as this greatly influences C p , and these simulations are not included in the
averages shown in the main paper. It is also worth noting that we do not see free vortices after the interaction

It is instructive to investigate the role of energy during
the formation of 2D droplet arrays, via both interaction
quenches and the temperature quenches. In Fig. S2 (a),we show the energy versus time for the five interaction
quench simulations considered in Fig. 1(a) of the maintext. We have also marked the energy of the ground state
BEC
before the interaction quench (EGS), and the energy ofSSthe ground state following the interaction quench (EGS),
with the superscript indicating that the ground state is
initially in the BEC phase, then later in the supersolid
phase. An estimate for the energy added by crossing theSSphase transition can be evaluated as [E(tfinal ) − EGS]−
BEC[E(tinitial ) − EGS] ≈ 0.35~ωx . Note that E(tinitial ) −BEC
EGS
does not equal zero due to the random noise added
to the initial state.
It is interesting at this point to compare the final energies of the eGPE simulations in Fig. S2 (a) with the final
energies of the c-field simulations following the evapo-

3
the 7-droplet supersolid rather than the 19-droplet one,such low-energy Goldstone modes are a general feature ofdipolar supersolids [24, 25]. Since these modes inherently
cause both phase and crystal excitations, they directly
act to disrupt the supersolid. Furthermore, we also seesome vortex pairs after the interaction quench, and these
also play a role. Interestingly, even in the interactionquench simulations, a supersolid is obtained in the longtime limit (although too long to be useful for current experiments due to lifetime limitations), even though thetotal energy is conserved, thanks to a damping of thesephase and crystal excitations.
SUPERSOLID EXCITATIONS

We investigate the 7-droplet hexagon supersolid, thesame configuration as shown in Fig. 3 of the main text,
using the extended Gross-Pitaevskii equation (eGPE), focusing here on its excitations. We perform a Bogoliubovde Gennes linearization and present results in the formof the dynamic structure factor,
S(k, ω) =X Z
lFIG. S2. Energy versus time for the pancake quench simulation runs considered in Fig. 2 of the main text. (a) Interaction quenches simulated using the eGPE with N ≈ 2.1 × 105 .
Dashed lines show the ground state energy for the initial paBEC, and the final paremeters corresponding torameters, EGS
SS. (b) Temperature quenches simuthe target supersolid, EGS
lated with the SeGPE. All curves are with fixed γ = 0.0075.
For both subplots fx,y,z = (33, 33, 167) Hz.

rative cooling quench, which are shown in Fig. S2 (b),which differ by more than 30~ωx . From such a comparison we can deduce that the important factor for disrupting supersolid formation following an interacting quench
is not so much the total energy injected into the system
by crossing the first-order phase transition [in Fig. S2(a)] but, rather, which modes become excited. This large
disparity in energy, however, tells us that at much longer
time scales than shown here, the eGPE states may settle down to a better quality supersolid than the SeGPE[see e.g. Fig. 2 of the main text], but this could be on
the order of seconds, much larger than the supersolid life
time.
From our dynamic structure factor calculations shownin Fig. S3 (which will be discussed shortly), one can seethat the out-of-phase Goldstone modes (the low-energy
modes at finite momentum that show up as red ovals)
are particularly vulnerable to excitation by the interaction quench. Note that although this figure is for2

d3 x [u∗l (r) + vl∗ (x)]eik·x ψ0 (x) δ(ω − ωl ) ,(S3)Here, ψ0 is the
R ground state wavefunction normalized
to unity, i.e. d3 x|ψ0 (x)|2 = 1, and {ul (x), vl (x)} are
the quasiparticle excitations with energy ωl [26, 27]. The
dynamic structure factor along two orthogonal directions
is displayed in Fig. S3. Note, the asymmetry along kx andky is due to the triangular configuration of the crystal,
which can be seen clearly in Fig. S4(b).
To explore the role of dimensionality,
Fig. S4 comparesRthe static structure factor, S(k) = dω S(k, ω), for bothlinear and 7-droplet hexagon supersolids. These results
are converged within the dashedellipses, set by ensurR
ing that the f -sum rule, dω ωS(k, ω) = ~2 k 2 /2m, issatisfied, and should be ignored outside. Convergenceis limited by the number of BdG modes, for which weuse 512 modes for both cases. In order to make a faircomparison between a 1D and 2D supersolids we choose
to approximately match the average 2D trap density by
fixing % = N fx fy [28]. As previously reported, the structure factor for the linear case [Fig. S4(a)] has peaks corresponding to the average inter-droplet spacing (2.67µm),
kx lz ≈ 1.43, and subsequent peaks at integer multiples ofthis [24]. We find that the dominant contributing modes
to the structure factor peaks are low energy out-of-phase
Goldstone modes [24, 25, 29], where the superfluid current and crystal oscillate out-of-phase with one another.
Note that for possible comparison with experiments our
spectrum in Fig. S4 was energy broadened with a Gaussian of width σ = 0.008 ~ωz , note that Fig. 1(b) of themain text was similarly broadened by σ = 0.004 ~ωz .

4FIG. S3. Dynamic structure factor (DSF), Eq. (S3), normalized to peak value, in energy-momentum space for a 164 Dy 7-droplet
hexagon supersolid. Left: DSF along kx with ky = kz = 0. Right: DSF along ky with kx = kz = 0. Parameters: as = 90a0 ,
fx,y,z = (52.83, 52.83, 167) Hz, N = 9.5 × 104 .

FIG. S4. Static structure factors for (a) linear and (b) 7droplet hexagon supersolids. Convergence is achieved within
the dashed ellipses (see text). Dotted lines in (b) correspond to the trajectories shown in Fig. S3, integrated overenergy. Parameters for linear chain: as = 90a0 , fx,y,z =(52.83, 130, 167) Hz, N = 4 × 104 . For 2D crystal: as = 90a0 ,
fx,y,z = (52.83, 52.83, 167) Hz, N = 9.5 × 104 .

For the 2D supersolid, Fig. S4(b) displays peaks situatedqat kρ lz ≈ 1.43 every 60◦ azimuthally, wherekρ = kx2 + ky2 . These peaks reflect the hexagonal structure of the ground state, however this value does notdirectly reflect the inter-droplet spacing (3.05µm, which
would correspond to kρ lz ≈ 1.25), but rather the spacingof lattice planes between droplets. Crucially, the six inner
momentum peaks are rotated compared to the droplet
crystal, analogous to what we observed experimentallyin the TOF images. Similar to the 1D chain, we findthat the out-of-phase Goldstone modes–a manifestationof superfluidity–contribute to the majority of the peaksignal.

[1] F. Wächtler and L. Santos, Quantum filaments in dipolar bose-einstein condensates, Phys. Rev. A 93, 061603(2016).
[2] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,
Ground-state phase diagram of a dipolar condensate withquantum fluctuations, Phys. Rev. A 94, 033619 (2016).
[3] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, andT. Pfau, Observation of quantum droplets in a strongly
dipolar bose gas, Phys. Rev. Lett. 116, 215301 (2016).
[4] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Quantum-fluctuation-driven
crossover from a dilute bose-einstein condensate to a
macrodroplet in a dipolar quantum fluid, Phys. Rev. X
6, 041039 (2016).
[5] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Bogoliubov
modes of a dipolar condensate in a cylindrical trap, Phys.
Rev. A 74, 013623 (2006).
[6] A. R. P. Lima and A. Pelster, Quantum fluctuations in
dipolar bose gases, Phys. Rev. A 84, 041604 (2011).
[7] C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley,
M. J. Davis, and B. P. Anderson, Spontaneous vortices in
the formation of Bose-Einstein condensates, Nature 455,
948 (2004).
[8] I.-K. Liu, S. Donadello, G. Lamporesi, G. Ferrari, S.-C.
Gou, F. Dalfovo, and N. Proukakis, Dynamical equilibration across a quenched phase transition in a trapped
quantum gas, Communications Physics 1, 1 (2018).
[9] S. Cockburn and N. Proukakis, Ab initio methods forfinite-temperature two-dimensional bose gases, PhysicalReview A 86, 033610 (2012).
[10] S. J. Rooney, T. W. Neely, B. P. Anderson, and
A. S. Bradley, Persistent-current formation in a hightemperature bose-einstein condensate: An experimental
test for classical-field theory, Phys. Rev. A 88, 063620(2013).
[11] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, andG. K. Campbell, A rapidly expanding bose-einstein condensate: an expanding universe in the lab, Physical Review X 8, 021021 (2018).
[12] M. Ota, F. Larcher, F. Dalfovo, L. Pitaevskii, N. P.

5Proukakis, and S. Stringari, Collisionless sound in a uniform two-dimensional bose gas, Physical review letters
121, 145302 (2018).
[13] T. Bland, Q. Marolleau, P. Comaron, B. Malomed, and
N. Proukakis, Persistent current formation in double-ringgeometries, Journal of Physics B: Atomic, Molecular andOptical Physics 53, 115301 (2020).
[14] S. De, D. Campbell, R. Price, A. Putra, B. M. Anderson,
and I. Spielman, Quenched binary bose-einstein condensates: Spin-domain formation and coarsening, PhysicalReview A 89, 033631 (2014).
[15] C. W. Gardiner and M. J. Davis, The stochastic GrossPitaevskii equation:II, J Phys. B 36, 4731 (2003).
[16] S. Rooney, P. Blakie, and A. Bradley, Stochastic projected gross-pitaevskii equation, Physical Review A 86,
053634 (2012).
[17] R. Kubo, The fluctuation-dissipation theorem, Reports
on progress in physics 29, 255 (1966).
[18] E. B. Linscott and P. B. Blakie, Thermally activated local
collapse of a flattened dipolar condensate, Phys. Rev. A
90, 053605 (2014).
[19] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark,M. A. Norcia, and F. Ferlaino, Birth, life, and death of adipolar supersolid, Phys. Rev. Lett. 126, 233401 (2021).
[20] Despite the different trap geometries, (fx , fy , fz ) =(36, 88, 141)Hz in Ref. [19] compared to (fx , fy , fz ) =(33, 33, 167)Hz, the peak densities are similar npeak =1.6 × 1021 µm−3 compared to npeak = 1.8 × 1021 µm−3
and thus will have similar quality supersolids.
[21] F. Ancilotto, M. Barranco, M. Pi, and L. Reatto, Vortex properties in the extended supersolid phase of dipolar bose-einstein condensates, Physical Review A 103,

033314 (2021).
[22] S. Roccuzzo, A. Gallemı́, A. Recati, and S. Stringari,
Rotating a supersolid dipolar gas, Physical review letters
124, 045702 (2020).
[23] D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. A.
Baranov, R. van Bijnen, S. M. Roccuzzo, A. Recati,
B. Blakie, D. Baillie, L. Chomaz, and F. Ferlaino, Bragg
scattering of an ultracold dipolar gas across the phasetransition from bose-einstein condensate to supersolidin the free-particle regime, Phys. Rev. A 104, L011302(2021).
[24] G. Natale, R. van Bijnen, A. Patscheider, D. Petter,
M. Mark, L. Chomaz, and F. Ferlaino, Excitation spectrum of a trapped dipolar supersolid and its experimental
evidence, Physical review letters 123, 050402 (2019).
[25] M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt,
M. Wenzel, H. P. Büchler, T. Langen, and T. Pfau, The
low-energy goldstone mode in a trapped dipolar supersolid, Nature 574, 386 (2019).
[26] F. Zambelli, L. Pitaevskii, D. M. Stamper-Kurn, and
S. Stringari, Dynamic structure factor and momentum
distribution of a trapped bose gas, Phys. Rev. A 61,
063608 (2000).
[27] P. Blakie, R. Ballagh, and C. Gardiner, Theory of coherent bragg spectroscopy of a trapped bose-einstein condensate, Physical Review A 65, 033602 (2002).
[28] E. Poli, T. Bland, C. Politi, L. Klaus, M. A. Norcia,
F. Ferlaino, R. N. Bisset, and L. Santos, Maintaining
supersolidity in one and two dimensions, Phys. Rev. A
104, 063307 (2021).
[29] J. Goldstone, Field theories with superconductor solutions, Il Nuovo Cimento (1955-1965) 19, 154 (1961).

4.8 Publication VI:
Can angular oscillations probe superfluidity in dipolar supersolids?

4.8

111

Publication VI:

Can angular oscillations probe superfluidity in dipolar supersolids?
The following publication has appeared in
Physical Review Letters 129, 040403 (2022)‖
submitted 17 Nov 2021; published 22 July 2022
DOI: https://doi.org/10.1103/PhysRevLett.128.195302
M. A. Norcia,1 E. Poli,2 C. Politi,1,2 L. Klaus,1 T. Bland,1 M. J. Mark,1,2 L. Santos,3
R. N. Bisset,2 and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 Institut für Theoretische Physik, Universität Hannover, Hannover, Germany‖The author of the present thesis performed the measurements together with M. A. N.,

and
L. K., analysed the experimental data with M. A. N., and contributed in writing the manuscript andinterpreting the results together with all the authors.

PHYSICAL REVIEW LETTERS 129, 040403 (2022)

Can Angular Oscillations Probe Superfluidity in Dipolar Supersolids?
Matthew A. Norcia ,1 Elena Poli ,2 Claudia Politi,1,2 Lauritz Klaus ,1,2 Thomas Bland ,1,2 Manfred J. Mark ,1,2
Luis Santos ,3 Russell N. Bisset,2 and Francesca Ferlaino1,2,*
1

Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck 6020, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Innsbruck 6020, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, 30167 Hannover, Germany(Received 17 November 2021; revised 3 March 2022; accepted 6 June 2022; published 22 July 2022)
Angular oscillations can provide a useful probe of the superfluid properties of a system. Suchmeasurements have recently been applied to dipolar supersolids, which exhibit both density modulationand phase coherence, and for which robust probes of superfluidity are particularly interesting. So far, theseinvestigations have been confined to linear droplet arrays, which feature relatively simple excitation
spectra, but limited sensitivity to the effects of superfluidity. Here, we explore angular oscillations insystems with 2D structure which, in principle, have greater sensitivity to superfluidity. In both experimentand simulation, we find that the interplay of superfluid and crystalline excitations leads to a frequency ofangular oscillations that remains nearly unchanged even when the superfluidity of the system is altereddramatically. This indicates that angular oscillation measurements do not always provide a robust
experimental probe of superfluidity with typical experimental protocols.
DOI: 10.1103/PhysRevLett.129.040403

Some of the most distinctive manifestations of superfluidity in ultracold quantum gases relate to their behavior
under rotation. These include the presence of quantizedvortices [1–3] and persistent currents in ring traps [4], as
well as shape-preserving angular oscillations associatedwith a “scissors” mode [5]. Measurements of the scissors
mode frequency have long been used to illuminate thesuperfluid properties of a variety of systems [6–11]. With
the recent advent of dipolar supersolids [12–18]—statesthat possess both the global phase coherence of a superfluid
and the spatial density modulation of a solid—the scissors
mode provides a tempting way to quantify changes insuperfluidity across the superfluid-supersolid transition[19,20]. Angular oscillations have also been used to searchfor superfluid properties in solid helium [21]. In this case,however, a change in oscillation frequency initially attributed to superfluidity was eventually traced, instead, to otherreasons [22]. In this Letter, we study more deeply theconnection between angular oscillations and superfluidity
in dipolar supersolids to determine the extent to which such
experiments can inform our understanding of superfluidity
in these systems.
The goal of these angular oscillation measurements is toinfer the flow patterns allowed for a given fluid. A superfluid is constrained by the single-valued nature of its wave
function to irrotational flow (IF), while a nonsuperfluid
system faces no such constraint and, in certain situations,
may be expected to undergo rigid-body rotation (RBR).
Prototypical velocity fields for angular oscillations underIF (⃗v ∝ ∇xy) and RBR (⃗v ∝ rθ̂) are depicted in Figs. 1(a)and 1(b), respectively. The velocity field associated with
0031-9007=22=129(4)=040403(6)angular rotation is related to the moment of inertia of thesystem and, thus, the frequency of angular oscillations.
The ability to distinguish between RBR and IF (and,thus, in principle, between a classical and superfluid
(a)

(b)

(c) y

x

θ

(d)

FIG. 1. Characteristic velocity profiles for irrotational flow (a)and rigid-body rotation (b). A wide atomic state (light turquoiseoval) samples a region of space where the two differ significantly,
while a highly elongated state (dark turquoise oval) samples aregion where the two patterns are nearly indistinguishable. (c) We
excite oscillations in the angle θ of our atomic gas by rapidly
rotating the anisotropic trap (dashed oval), then returning it to its
original orientation and observing the subsequent dynamics.
(d) Typical example of experimental angular oscillation for the
zigzag modulated state shown on the right (image averaged overnine iterations). In this case, the errors from the fit to the state angle
are smaller than the markers. The red line is a damped sinusoidal fit
used to extract the angular oscillation frequency fosc.

040403-1© 2022 American Physical Society

PHYSICAL REVIEW LETTERS 129, 040403 (2022)
system) depends critically on the geometry of the system,
and is sensitive only to the character of the flow patternwhere the atomic density is appreciable. As illustrated in
Figs. 1(a) and 1(b), highly elongated states sample only theregion along the weak axis of the trap (near x ¼ 0) where IF
and RBR are identical for small rotations (dark turquoise
regions), while rounder states (light turquoise regions)sample regions of space where the flow patterns differ
significantly and, thus, are far more sensitive to theirrotational constraint. Recent works have focused on
systems that form a short linear chain of about two“droplets” [23] in the supersolid regime [19,20].
In this Letter, we study angular oscillations in systems
with linear and two-dimensional modulation to disentangle
the effect of three important contributions: (i) a narrowingof the aspect ratio of the gas (geometrical change), (ii) a
reduction in the population of the low-density superfluid“halo” that occupies the outer regions of the trap, and (iii) a
reduction in the density of the interdroplet connection thatenables the exchange of atoms between droplets, which is
key to the superfluid nature of supersolid systems. We findthat, in linear systems, contributions (i) and (ii) dominate
the change in oscillation frequency associated with theonset of modulation, while (iii) has a negligible effect.
In dipolar condensates with two-dimensional structure,
which have been a focus of recent work [24–28], the effects
of geometry and superfluidity may be disentangled, andone may expect to observe a direct link between a change ofthe superfluid fraction and a modification of the angularoscillation frequency. However, we find that the physics atplay is much more complex. Indeed, not only does theoscillation frequency fail to approach its rigid-body value
for states with a vanishing superfluid connection, but it
remains very close to the value predicted for a superfluid
state. We extensively investigate the system behavior as afunction of geometry and interaction parameters, revealinga unique multimode response of the dipolar supersolid.
Experimentally, we use a dipolar quantum gas of 164Dy
atoms (up to approximately 5 × 104 condensed atoms),
confined within an optical dipole trap (ODT) of tunablegeometry, formed at the intersection of three laser beams[25,27,29]. The trap geometry and particle number at theend of the evaporative cooling sequence determine thecharacter of the modulated ground state, which can form
linear, zigzag, or triangular lattice configurations [28]. By
varying the applied magnetic field in the vicinity ofFeshbach resonances near 18–23 G, we can access scattering lengths that correspond to either unmodulated BECs or
modulated states. In past works, we have demonstrated thatmodulated states created at the corresponding field haveglobal phase coherence [25,27]. In this Letter, we expect
the same to be true, but refer to these experimental states
simply as modulated, as we do not repeat the characterization for every trap condition used. We excite angularoscillations by using the well-established protocol of

(a)

(b)

FIG. 2. Normalized oscillation frequencies f osc from experiment (a) and simulation (b). Blue points represent unmodulated
BECs, red points represent modulated states (expt.) and supersolid states (sim.), and green points represent independent droplet
arrays. Solid lines are predictions for irrotational flow firr. Dashedlines are predictions for rigid body rotation frig . The trap frequencies used in the simulation, from left to right, are ðfx ; fy Þ ¼½ð43; 53Þ; ð40; 57Þ; ð37; 62Þ; ð32; 70Þ; ð26; 87Þ Hz. fz ¼ 122 Hz
for all cases. A similar range is used in the experiment.

applying a sudden small rotation of the trap, by varying
the relative powers in the ODT beams for 6 ms before
returning them to their original values [Fig. 1(c)]. Using our
high-resolution imaging [30], we observe the in-trap
density profile at a variable time from the excitation, andextract the angle of the major and minor axes using a
two-dimensional Gaussian fit to the state [31].
A typical angular oscillation is shown in Fig. 1(d), for a“zigzag” modulated state [25]. From such oscillation traces,
we extract the dominant oscillatory frequency f osc using a
fit to an exponentially damped sinusoid. Typically, thestatistical error on our measurements of f osc is on the subHertz level, better than our knowledge of the trap frequencies, due to drifts between calibrations. We perform suchmeasurements for trap geometries ranging from an elongated cigar shape to pancake shaped, and for different
scattering lengths, as summarized in Fig. 2(a).
Within a single-mode approximation, the angular oscillation frequency f osc can be predicted using either a sumrule based approach [19,34], or considerations based onhydrodynamic flow [5]. For q
RBR,
the angular oscillation
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðf 2y − f 2x Þβ, whereas for IF,
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
the predicted value is f irr ¼ ðf 2y − f 2x Þ=β [19,20]. Here,frequency is given by f rig ¼

f x;y are the trap frequencies along directions x and y. β ¼hx2 − y2 i=hx2 þ y2 i is a geometrical factor that quantifies
the degree of elongation of the atomic cloud (but carries no040403-2

PHYSICAL REVIEW LETTERS 129, 040403 (2022)
information about the superfluid fraction). As shown in
Fig. 2, f rig and f rig are more distinct for smaller values of β.
Remarkably, independent of trap geometry or the presenceof modulation, we observe f osc close to the IF prediction
and far from the RBR prediction when the two predictions
differ appreciably.
To gain a deeper understanding of our observations, wetheoretically study the oscillation dynamics using a realtime simulation of the extended Gross-Pitaevskii equation(EGPE) [35–37]. To compare to the experimental observations of Fig. 2(a), first, we calculate the ground state for agiven trap, scattering length, and atom number. Then, weapply a 0.5° rotation of the trap for 6 ms (we haveconfirmed that the character and frequency of the response
do not change for much larger excitations), and then let thestate evolve for 50 ms. Then, we perform the same fitting
procedure as used in the experiment to extract f osc . For thesimulation, we calculate β directly for the ground state (weconfirm that the exact value of β agrees with that extracted
from a Gaussian fit at the 5% level). For simulations
performed on states ranging from the unmodulated BEC tosupersolid (SS) to independent droplet (ID) regimes, withvanishing superfluid connection between droplets, weagain find that f osc is always very close to f irr , in very
good agreement with the experimental data. For isolated
droplet states in particular, f osc can actually be even higher
than the expected value for irrotational flow, indicating thatthe oscillation frequency is not necessarily in between
the irrotational and rigid body values.
To further illuminate the dependence f osc on superfluidity, we analyze the results of the simulation as afunction of the s-wave scattering length as (Fig. 3).
Scattering lengths of 85a0 yield arrays of (nearly) independent droplets, while as ¼ 97a0 produces an unmodulated BEC. In between, we find supersolid states, with lowdensity connections between droplets. Inspired by theformulation of Leggett [38], we quantify the
R degree ofinterdroplet density connection as C ¼ ½ dx=ρðxÞ−1 ,where ρðxÞ is the projected atomic density, evaluated overthe interdroplet connection [Fig. 3(a)] [39].
As shown in Fig. 3, despite the rapid reduction of C withas , the simulated f osc exhibits a rather constant behavior
with a value always close to the purely irrotational
predictions, f irr , for both a linear (1D) and hexagon state(2D). This observation indicates that (i) the degree ofinterdroplet connection is not actually a major determinantof the angular oscillation frequency and (ii) that the systemdoes not undergo RBR even for vanishingly small interdroplet density connection. The latter conclusion is particularly evident for hexagon states, where the rigid-body
prediction substantially departs from the irrotational one.
For the linear array, the elongated geometry means that thef rig and f irr differ only slightly; see Supplemental Material
for further discussion [31].
At this point, we can clearly see the geometricallimitations of the linear systems. In linear systems, the

(a)

ID

SS

BEC

(b)

FIG. 3. Impact of scattering length on simulated scissors mode
frequencies. (a) Interdroplet connection C (defined in text) versus
scattering length for different trap geometries. The calculated
ground state in each trap is shown on the right, with corresponding border colors. (b) Scissors mode frequency versus scattering
length. Solid lines are predictions for irrotational flow firr.
Dashed lines are predictions for rigid body rotation frig . β ranges
from 0.93 to 0.99, and 0.27 to 0.31 in the linear and hexagonalcases, respectively.

narrowing of the atomic density distribution that occurs
with the onset of modulation causes the dominant contribution to a modification in oscillation frequency as well
as a reduction in sensitivity of the oscillation frequency tosuperfluidity. Simultaneously, the transfer of atoms fromthe halo to the droplets leads to a reduction of the superfluidity of the composite halo-droplet system, which isaccompanied by a small change in the oscillation frequency. However, because the motion induced by rotation
in a linear system is perpendicular to the interdroplet axis,
these effects should not be interpreted as a result of theweakening superfluid connection along the interdroplet
axis. In contrast, systems with two-dimensional structure
maintain a relatively round aspect ratio in the modulated
regime, and the rotational motion does orient along certain
interdroplet axes.
To better understand the nonrigid nature of the angularoscillations, we employ a method to extract the character ofthe system’s response by analyzing our experimental and
EGPE simulation dynamics in the frequency domain withrespect to time, but in the position domain with respect tothe spatial coordinates. A similar technique has been
applied along one dimension to understand the mode
structure of an elongated condensate [40]. This technique,which for convenience we refer to as “Fourier transform
image analysis” (FTIA) [31], allows us to extract a power
spectrum of density fluctuations driven by the angular
excitation, as well as the spatial form of the density
fluctuations at each frequency. For comparison, we also

040403-3

PHYSICAL REVIEW LETTERS 129, 040403 (2022)
(b)

(c)

spectral power(a)FIG. 4. Analysis of mode shapes and response due to angular excitation. Solid lines are the power spectrum obtained from therotational signal (θ in the experiment and hxyi in the simulation), and dashed lines are obtained from FTIA (see text, Supplemental
Material [31] for description). Inset panels show the mode shapes for selected modes. Red and blue indicate out-of-phase changes indensity, overlaid onto the average density profile in the panels corresponding to simulation (gray to white). Solid and dashed vertical red
lines represent f irr and f rig , respectively. (a) Responses in elongated traps from simulation (top) and experiment (bottom), for anunmodulated BEC (left) and a zigzag droplet state (right). Trap frequencies are f x;y ¼ ½31ð1Þ; 73ð1Þ; 128ð1Þ Hz, and f x;y;z ¼½32; 70; 122 Hz for the experiment and theory, respectively. (b) Simulated response of supersolid hexagon state (as ¼ 92a0 ).
(c) Simulated response of droplet crystal hexagon state (as ¼ 85a0 ). Note that the ground state has a different orientation for the two
scattering lengths in this trap. Trap frequencies are fx;y;z ¼ ½43; 53; 122 Hz for (b) and (c).

calculate the spectral power of our rotational signal througha Fourier transform. For computational robustness, we use
the fitted angle θ in the experimental case, and hxyi for thesimulations. To enhance our frequency resolution, we
analyze simulations with longer durations than are accessible in the experiment (160 to 290 ms).
We apply the FTIA to both simulation and experimental
images in Fig. 4(a). For a BEC, the FTIA gives a dominantpeak in both simulation and experiment, whose frequency
and shape are consistent with a scissors mode oscillation at
the frequency observed from the angular response. For azigzag modulated state, we again predominantly observe asingle peak in the FTIA spectrum at the frequency of theangular oscillation. In the simulation, we can see that themode corresponds to the motion of the different droplets ina pattern reminiscent of IF in an unmodulated superfluid,and clearly distinct from RBR. In the experiment, theresponse of individual droplets is not visible due to shot-toshot fluctuations in the exact number and position of
the droplets, but the overall structure is similar to thesimulation.
For hexagonal supersolid [Fig. 4(b)] and isolated droplet[Fig. 4(c)] states, the FTIA reveals a clear multifrequency
response. For the supersolid, we observe the excitation ofmodes near 3 and 25 Hz that do not contribute strongly to
hxyi. The droplet motion associated with the 3 Hz mode isapproximately (but not exactly) shape preserving, and thefrequency is much lower than would be expected for asingle-mode RBR response. For the isolated droplet array,we again observe a nearly shape-preserving low-frequency
response from FTIA, as well as a dominant angularresponse that is split into two frequencies, both above
the scissors mode frequency f irr expected for a superfluid with the same geometry. In the experiment, thecombination of nonangular excitations associated with
our method used to rotate the trap and relatively rapid
damping of the oscillation prevent us from observing
meaningful mode profiles for small β.
Importantly, the FTIA reveals that, even in cases where
we observe an apparently single-frequency response intypical rotational observables like θ or hxyi [as in Figs. 4(a)and 4(b)], the response of the system may, in fact, bemultimode in nature, breaking the single-mode approximation used to analytically extract f irr and f rig [19,34]. In
the case of a multifrequency response, f irr and f rig , instead,provide an upper bound for the frequency of the lowest
energy excitation—an excitation that is difficult to see withexperimentally accessible observables. Features of these
subdominant modes, including the lack of a strong rotational signal in the low-frequency oscillations and theapparent similarity between the droplet motion (the motionof the halo is quite different) near 25 Hz to that of thedominant rotational mode, remain interesting topics for
future investigation.
As we have noted, not only does the dominant angularresponse frequency fail to approach the rigid-body value inthe isolated droplet regime, but it also stays near to theirrotational prediction. A possible intuitive explanation forthis observation is that the flow pattern of Fig. 1(a)resembles that of a quadrupolar surface mode, and it iswell known that, for sufficiently strong interactions, thefrequency of such modes is predominantly determined by
the trap parameters, rather than the details of the interparticle interactions [34].
In conclusion, measurements of angular oscillation
frequencies offer a simple way to demonstrate superfluidity
in certain conditions. However, care must be taken when
making and interpreting such measurements—geometrical

040403-4

PHYSICAL REVIEW LETTERS 129, 040403 (2022)
changes can mask the effects of changing superfluidity, andusual predictions to which one might compare rely on theassumption of a single-frequency response of the lowest
energy rotational mode. While the moment of inertia of thesystem is defined as the angular momentum of a system in
response to a shape-preserving, steady-state drive, oscillation measurements involve a time-localized change in therotation rate of the trap, which may excite modes that do notmeet this criterion. In small, linear systems, the simple excitation spectra means that approximately shape-preserving
oscillations can still be excited [31]. However, we find thata supersolid with 2D structure, which one might expect tobe an ideal candidate for such measurements, can exhibit anapparently single-frequency response associated with amode that is not the lowest in energy. Further, this excitation frequency is typically very close to that of a purely
superfluid system, even for systems where the effects ofsuperfluidity are minimal. Therefore, such measurements
do not provide a robust indicator of superfluidity formodulated systems. In the future, it may be possible toextract information about superfluidity using a modified
excitation scheme to preferentially excite the lower energy
modes and a more comprehensive analysis scheme suitable
for multifrequency response [41]. However, such techniques would require detailed knowledge of the exact
excitation applied and measurement of response amplitudes, both of which are considerably more challenging
in an experiment than measuring the frequency of an
oscillation.
Finally, we note that, even in the case of single-frequency
response, where the frequency of angular oscillations has adirect connection to the moment of inertia of the system,
making a clear connection between the moment of inertia
and quantities like a superfluid fraction can be problematic.
Past works have predicted that a system which is partiallysuperfluid should have a moment of inertia in between the
RBR and IF predictions, linearly interpolated according to
a superfluid fraction [20,38]. While this interpretation may
be valid for systems featuring a rigid crystalline structure
and a uniform distribution of crystalline and superfluid
components, as in [38], it is not necessarily valid for oursmall dipolar supersolids, which, in addition to coupled
superfluid-crystalline excitations, feature a nonuniform
degree of modulation across the system.
We thank Sandro Stringari and Alessio Recati for useful
discussions. We acknowledge R. M. W. van Bijnen fordeveloping the code for our EGPE ground-state simulations.
The experimental team is financially supported throughan ERC Consolidator Grant (RARE, Grant No. 681432),
an NFRI grant (MIRARE, Grant No. ÖAW0600) of theAustrian Academy of Science, the QuantERA grant MAQS
by the Austrian Science Fund FWF Grant No. I4391-N. L. S.
and F. F. acknowledge the DFG/FWF via Grant
No. FOR 2247/PI2790. L. S. acknowledges funding by
the Deutsche Forschungsgemeinschaft (DFG, German

Research Foundation) under Germany’s Excellence
Strategy—Grant No. EXC-2123 QuantumFrontiers—
390837967. M. A. N. has received funding as an ESQ
Postdoctoral Fellow from the European Union’s Horizon
2020 Research and Innovation Programme under the Marie
Skłodowska Curie Grant Agreement No. 801110 and
the Austrian Federal Ministry of Education, Scienceand Research (BMBWF). M. J. M. acknowledges support
through an ESQ Discovery Grant by the Austrian Academyof Sciences. We also acknowledge the Innsbruck Laser Core
Facility, financed by the Austrian Federal Ministry of
Science, Research, and Economy. Part of the computational
results presented have been achieved using the HPC infrastructure LEO of the University of Innsbruck.

*

Corresponding author.
Francesca.Ferlaino@uibk.ac.at[1] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall,
C. E. Wieman, and E. A. Cornell, Vortices in a BoseEinstein Condensate, Phys. Rev. Lett. 83, 2498 (1999).
[2] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard,
Vortex Formation in a Stirred Bose-Einstein Condensate,
Phys. Rev. Lett. 84, 806 (2000).
[3] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.
Schunck, and W. Ketterle, Vortices and superfluidity in astrongly interacting Fermi gas, Nature (London) 435, 1047
(2005).
[4] A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T.
Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K.
Campbell, Superflow in a Toroidal Bose-Einstein Condensate: An Atom Circuit with a Tunable Weak Link, Phys.
Rev. Lett. 106, 130401 (2011).
[5] D. Guéry-Odelin and S. Stringari, Scissors Mode andSuperfluidity of a Trapped Bose-Einstein Condensed Gas,
Phys. Rev. Lett. 83, 4452 (1999).
[6] D. Bohle, A. Richter, W. Steffen, A. Dieperink, N. L. Iudice,
F. Palumbo, and O. Scholten, New magnetic dipole excitation mode studied in the heavy deformed nucleus 156GD
by inelastic electron scattering, Phys. Lett. 137B, 27 (1984).
[7] O. M. Maragò, S. A. Hopkins, J. Arlt, E. Hodby, G.
Hechenblaikner, and C. J. Foot, Observation of the Scissors
Mode and Evidence for Superfluidity of a Trapped
Bose-Einstein Condensed Gas, Phys. Rev. Lett. 84, 2056
(2000).
[8] M. J. Wright, S. Riedl, A. Altmeyer, C. Kohstall, E. R.
Sanchez Guajardo, J. H. Denschlag, and R. Grimm, FiniteTemperature Collective Dynamics of a Fermi Gas in the
BEC-BCS Crossover, Phys. Rev. Lett. 99, 150403 (2007).
[9] R. M. W. van Bijnen, N. G. Parker, S. J. J. M. F.
Kokkelmans, A. M. Martin, and D. H. J. O’Dell, Collective
excitation frequencies and stationary states of trapped
dipolar Bose-Einstein condensates in the Thomas-Fermi
regime, Phys. Rev. A 82, 033612 (2010).
[10] C. D. Rossi, R. Dubessy, K. Merloti, M. de Goër de Herve,
T. Badr, A. Perrin, L. Longchambon, and H. Perrin, The
scissors oscillation of a quasi two-dimensional Bose gas as alocal signature of superfluidity, J. Phys. Conf. Ser. 793,
012023 (2017).

040403-5

PHYSICAL REVIEW LETTERS 129, 040403 (2022)[11] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen, M.
Isoard, S. Stringari, and T. Pfau, Scissors Mode of Dipolar
Quantum Droplets of Dysprosium Atoms, Phys. Rev. Lett.
120, 160402 (2018).
[12] M. Boninsegni and N. V. Prokof’ev, Colloquium: Supersolids: What and where are they?, Rev. Mod. Phys. 84, 759(2012).
[13] Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov, Stable
Dilute Supersolid of Two-Dimensional Dipolar Bosons,Phys. Rev. Lett. 115, 075303 (2015).
[14] D. Baillie and P. B. Blakie, Droplet Crystal Ground States ofa Dipolar Bose Gas, Phys. Rev. Lett. 121, 195301 (2018).
[15] S. M. Roccuzzo and F. Ancilotto, Supersolid behavior of adipolar Bose-Einstein condensate confined in a tube, Phys.
Rev. A 99, 041601(R) (2019).
[16] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C.
Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,
Observation of a Dipolar Quantum Gas with Metastable
Supersolid Properties, Phys. Rev. Lett. 122, 130405 (2019).
[17] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo,T. Langen, and T. Pfau, Transient Supersolid Properties in
an Array of Dipolar Quantum Droplets, Phys. Rev. X 9,011051 (2019).
[18] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann,
C. Politi, G. Durastante, R. M. W. van Bijnen, A.
Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,Long-Lived and Transient Supersolid Behaviors in Dipolar
Quantum Gases, Phys. Rev. X 9, 021012 (2019).
[19] S. M. Roccuzzo, A. Gallemí, A. Recati, and S. Stringari,
Rotating a Supersolid Dipolar Gas, Phys. Rev. Lett. 124,045702 (2020).
[20] L. Tanzi, J. Maloberti, G. Biagioni, A. Fioretti, C.
Gabbanini, and G. Modugno, Evidence of superfluidity
in a dipolar supersolid from nonclassical rotational inertia,
Science 371, 1162 (2021).
[21] E. Kim and M. H.-W. Chan, Probable observation of asupersolid helium phase, Nature (London) 427, 225 (2004).
[22] D. Y. Kim and M. H. W. Chan, Absence of Supersolidity in
Solid Helium in Porous Vycor Glass, Phys. Rev. Lett. 109,155301 (2012).
[23] Here, the word droplet refers to a high-density region, which
is not necessarily self-bound.
[24] J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher, M.
Schmidt, K. S. H. Ng, S. D. Graham, T. Langen, M.
Zwierlein, and T. Pfau, Roton Excitations in an Oblate
Dipolar Quantum Gas, Phys. Rev. Lett. 126, 193002 (2021).
[25] M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen,
M. J. Mark, R. Bisset, L. Santos, and F. Ferlaino, Twodimensional supersolidity in a dipolar quantum gas, Nature(London) 596, 357 (2021).
[26] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H.
Ng, S. D. Graham, P. Uerlings, H. P. Büchler, T. Langen, M.
Zwierlein, and T. Pfau, Supersolidity in Two-Dimensional
Trapped Dipolar Droplet Arrays, Phys. Rev. Lett. 127,
155301 (2021).

[27] T. Bland, E. Poli, C. Politi, L. Klaus, M. A. Norcia, F.
Ferlaino, L. Santos, and R. N. Bisset, Two-Dimensional
Supersolid Formation in Dipolar Condensates, Phys. Rev.
Lett. 128, 195302 (2022).
[28] E. Poli, T. Bland, C. Politi, L. Klaus, M. A. Norcia, F.
Ferlaino, R. N. Bisset, and L. Santos, Maintaining supersolidity in one and two dimensions, Phys. Rev. A 104,
063307 (2021).
[29] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M.
Sohmen, M. J. Mark, and F. Ferlaino, Dipolar Quantum
Mixtures of Erbium and Dysprosium Atoms, Phys. Rev.
Lett. 121, 213601 (2018).
[30] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J.
Mark, M. A. Norcia, and F. Ferlaino, Birth, Life, and Deathof a Dipolar Supersolid, Phys. Rev. Lett. 126, 233401(2021).
[31] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.129.040403 for more
information, which contains refs. [32,33].
[32] I. T. Jolliffe, Principal Component Analysis, Springer Seriesin Statistics (Springer, New York, 2002).
[33] R. Dubessy, C. D. Rossi, T. Badr, L. Longchambon, and H.
Perrin, Imaging the collective excitations of an ultracold gas
using statistical correlations, New J. Phys. 16, 122001(2014).
[34] L. Pitaevskii and S. Stringari, Bose-Einstein Condensationand Superfluidity (Oxford University Press, New York,
2016), Vol. 164.
[35] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel,
and T. Pfau, Observation of Quantum Droplets in a
Strongly Dipolar Bose Gas, Phys. Rev. Lett. 116, 215301(2016).
[36] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L.
Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a
Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6,041039 (2016).
[37] F. Wächtler and L. Santos, Quantum filaments in dipolar
Bose–Einstein condensates, Phys. Rev. A 93, 061603(R)(2016).
[38] A. J. Leggett, Can a Solid be “Superfluid”?, Phys. Rev. Lett.
25, 1543 (1970).
[39] We note that the geometry of our system is very different
from that considered in Ref. [38], and so, we do not expect
this quantity to have a direct connection to the nonclassicalmoment of inertia of the system. Rather, we simply use it asa convenient way to quantify overlap between droplets.
[40] Edmundo R. Sanchez Guajardo, M. K. Tey, L. A.
Sidorenkov, and R. Grimm, Higher-nodal collective modes
in a resonantly interacting Fermi gas, Phys. Rev. A 87,
063601 (2013).
[41] S. M. Roccuzzo, A. Recati, and S. Stringari, Moment ofinertia and dynamical rotational response of a supersolid
dipolar gas, Phys. Rev. A 105, 023316 (2022).

040403-6

Supplemental materials for: Can angular oscillations probe superfluidity in dipolar
supersolids?
Matthew A. Norcia,1 Elena Poli,2 Claudia Politi,1, 2 Lauritz Klaus,1, 2 Thomas Bland,1, 2
Manfred J. Mark,1, 2 Luis Santos,3 Russell N. Bisset,2 and Francesca Ferlaino1, 2, ∗1

Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, Germany

EXCITATION PROTOCOL

In both experiment and simulation, we excite theatoms by suddenly rotating the trap, holding for 6 ms,
then returning it to its initial orientation. This was important in the experiment, as the trap frequencies generally change slightly as the trap is rotated, and we want toobserve the evolution of a state that is equilibrated to thetrap prior to the rotation. To explore whether the exact
excitation protocol influences our results, we performed
additional simulations where the trap angle was rotatedand held in the new orientation, but not rotated back.
We find that the same modes are excited in this case,and the frequency of their responses are the same. For
some parameters the relative contributions of the modes
to the spectrum of hxyi can differ between the two protocols, but for the parameters we explore the frequency ofthe peak response remains unchanged. In particular, forthe droplet crystal hexagon shown in Fig. 4c of the main
text, the contribution of the low-frequency mode to thehxyi power spectrum becomes appreciable, though is still
smaller than the contribution of the modes near 60 Hz.
Thus, the multimode response appears to be a genericfeature of possible schemes to excite angular oscillations.
While the spectral content of the excitation may differ,influencing the relative amplitudes of different modes, thefrequency and character of the modes is determined by
the system, not the drive.
We have also performed excitation in the simulation bydirectly imprinting a small phase variation αxy onto theground-state wavefunction. This protocol produces qualitatively similar results to those described above. Again,
the same modes are excited and respond with the same
frequencies, though sometimes with different amplitudes.
The dominant mode excited is the same as the rotateand-return protocol for all cases investigated.
EXTRACTING ANGULAR POWER SPECTRUM

Several methods can be used to extract the angularresponse of our system. For the experiment, we perform
a two-dimensional Gaussian fit to the in-trap image, andrecord the angle of the major and minor axes as a function of time. For the simulation, we report the angu-

lar response obtained using one of two observables. For
direct comparison to the experiment, we use the state
angle extracted from a 2D Gaussian fit, as in the experiment. For more detailed spectral analysis, we use
the quantity hxyi, as this is expected to have a strong
response to a rapid rotation of the trap and we find it
to be numerically more robust. We have confirmed thatthese and other similar observables, such as the directions of maximal and minimal variance, provide consistent results (up to overall normalization). In some cases,
the Fourier spectrum of hL̂z i (though not experimentally
accessible) shows different relative response amplitudes
between modes compared to hxyi, particularly for thosemodes at low frequencies.
FOURIER TRANSFORM IMAGE ANALYSIS

Time

Time

Subtract
mean
image

Frequency

FFT alongtime axis

In-trap images(averaged for eachtimestep)

FIG. S1. Procedure for Fourier transform image analysis(FTIA). See text for description.

The goal of our Fourier transform image analysis(FTIA) protocol is to visualize the density response of our
atomic system in real-space with respect to position, but
in frequency space with respect to time. This provides
a simple way to extract the spatial profile of excitation
modes. The process is illustrated in Fig. S1. To performthe FTIA, we assemble images of projected density profiles corresponding to single time-steps (directly from thesimulation, or averaged over several in-trap images fromthe experiment), then subtract the average (over all timesteps) image from each. We then Fourier transform theresults along the time axis. The output is then a sequenceof real-space images, showing the fluctuation pattern at
a given frequency. Because each pixel is now represented

2
by a complex number (encoding the amplitude and phaseof the density variations at that location), we plot withrespect to the global phase for each frequency that shows
maximum variation, thus plotting the in-phase quadrature of the oscillation.
In order to obtain a power spectrum (useful for locating the frequencies of excited modes), we compute thesum of the absolute square of the fluctuations over a region of interest containing the atomic cloud for each frequency. This power spectrum can be used to identify thefrequency and spatial character of modes, but is not expressed in physically meaningful units, and so should notnecessarily be used to compare the strength of differentmode responses.
We note that there are some similarities between the
FTIA method and principal component analysis (PCA)[1, 2]. Both provide a model-free way of extracting theform of excitations present in a system. PCA does so
by finding correlated patterns of fluctuations within aset of images, with no prior information about the timesequence of the images. This makes it well-suited torevealing modes that are excited incoherently, for example by thermal or quantum noise. In contrast, our
FTIA method explicitly incorporates the time-domain
information associated with the images. This makes it
well-suited to extracting modes that are coherently excited (FTIA, as we apply it, would not work for incoherently excited modes). In practice, we find that the
FTIA is more robust than PCA at extracting fluctuation
patterns that each exhibit a single-frequency response.
While PCA often returns components whose weights varywith multiple frequencies (indicating that they actuallycorrespond to a linear combination of eigenmodes), FTIA
by construction returns a fluctuation pattern associatedwith a single frequency. We find that this feature makes
it more robust for identifying eigenmodes of a system
subject to a coherent drive.
SPECTRA/TABLE FOR ALL PARAMS

Excitation power spectra from simulation for a range oftraps and scattering lengths used in the main manuscript
can be found in Fig. S2.

Here, Θ is the moment of inertia associated with steadystate rotation.
The numerator of Eq. 1 can be interpreted as a rotational spring constant: kτ = −τ /θ, where τ is thetorque exerted on a state whose major and minor axes
y and x are rotated relative to the their equilibrium position in the trap by an angle θ. To see this, consider
a mass element m at position (x, y) in a trapping potential V = (mωx x2 + mωy y 2 )/2, which exerts a torqueτ = xFy − yFx = xym(ωx2 − ωy2 ). We can then calculate kτ = −∂τ /∂θ = −m(y∂x/∂θ + x∂y/∂θ)(ωx2 − ωy2 ) =m(y 2 − x2 )(ωx2 − ωy2 ). Summing over mass elements provides the numerator of Eq. 1. This highlights that thenumerator of this expression is purely geometrical, independent of whether the state is superfluid or classical. In
the case of multi-frequency response, Eq. 1 (as defined by
the sum rule) becomes an inequality, defining the upperbound for the lowest frequency angular excitation in thesystem [4].

BETA VERSUS SCATTERING LENGTH FOR 1D
AND 2DIn Fig. S3, we show the change in the anisotropy of theatomic state in response to a change in scattering length
for a variety of traps, featuring both linear and 2D array
modulated configurations. Here, we consider the quantity β 2 = (hx2 − y 2 i/hx2 + y 2 i)2 , as this quantity gives
the expected change in moment of inertia between irrotational flow (IF) and rigid-body rotation (RBR). As β 2approached unity, the difference between the two vanishes, so such states can exhibit minimal sensitivity tosuperfluidity.
States in more elongated traps generally have values ofβ 2 closer to one than their rounder counterparts. However, even in relatively round traps, such as those ofRefs. [5, 6], low atom numbers can lead to the formationof linear arrays, which are highly elongated. In thesecases, the sensitivity of the state to superfluidity is dramatically reduced upon entering the modulated regime.
In contrast, combinations of trap parameters and atom
number that lead to a 2D modulated state typically maintain values of β 2 substantially different from one even in
the low scattering length, independent droplet regime.

PREDICTIONS FOR ROTATIONAL MODE
FREQUENCIESLINEAR CASE

The rotational response of a gas can be calculated using hydrodynamic equations [3] or a sum-rule approach[4, 5]. From the sum-rule approach, an expression canbe derived for the rotational oscillation frequency, under
the assumption that the response is single-frequency:
ω2 =mhy 2 − x2 i(ωx2 − ωy2 )Θ

(1)In Fig. S4, we explore the parameters of refs [5, 6],
where a change in scattering length induces a transition from an unmodulated BEC to a linear array of two
droplets. This transition is accompanied by a dramaticchange in the aspect ratio of the atomic state, as evident
in the near convergence of the predictions for rigid bodyNormalized spectral power

3

Fourier frequency (Hz)FIG. S2. Response sepctra extracted from simulations for different trap parameters (rows) and scattering lengths (columns).
Upper rows correspond to more elongated traps, while lower rows correspond to more round ones. From top to bottom,(fx , fy ) = [(26, 87), (32, 70), (37, 62), (40, 57), (43, 53)] Hz. fz = 122 Hz for all cases. Red vertical dashed and solid lines
correspond to the rigid-body rotation and irrotational flow predictions, respectively. Gray traces are power spectra extracted
from FTIA, while black traces are from hxyi. In all cases, as = 97a0 corresponds to an unmodulated BEC, while lower scattering
lengths correspond to modulated states, with the overlap between droplets decreasing with scattering length. (fx , fy ) = (26, 87)
is a linear droplet chain for all scattering lengths that produce a modulated state. All other modulated states have transverse
structure, increasing in prevalence as the trap becomes more round.

and irrotational flow (frig and firr ) at lower scattering
lengths, corresponding to the droplet state.
∗

We see that the dominant frequency of angular response is between frig and firr , indicating a change inthe level of superfluidity in the system. We find thatthe angular response in the supersolid regime (as = 90
or 92 a0 ) has two clear frequency components, though inthis case the dominant frequency observed matches theprediction from the sum rule (with moment of inertia
calculated under static rotation). Because of the geometry of the system, rotation does not lead to a significant
transfer of mass between the two droplets. Thus, we attribute the change in superfluidity to the low-density halothat surrounds the droplets, rather than the inter-droplet
connection itself.

Correspondence
shouldbe
addressed
toFrancesca.Ferlaino@uibk.ac.at[1] I. T. Jolliffe, Principal Component Analysis (Springer Series in Statistics. Springer, New York, NY, 2002).
[2] R. Dubessy, C. D. Rossi, T. Badr, L. Longchambon, and
H. Perrin, Imaging the collective excitations of an ultracold gas using statistical correlations, New Journal of
Physics 16, 122001 (2014).
[3] D. Guéry-Odelin and S. Stringari, Scissors mode and superfluidity of a trapped bose-einstein condensed gas, Phys.
Rev. Lett. 83, 4452 (1999).
[4] L. Pitaevskii and S. Stringari, Bose-Einstein condensation and superfluidity, Vol. 164 (Oxford University Press,
2016).
[5] S. M. Roccuzzo, A. Gallemı́, A. Recati, and S. Stringari,
Rotating a supersolid dipolar gas, Phys. Rev. Lett. 124,045702 (2020).
[6] L. Tanzi, J. Maloberti, G. Biagioni, A. Fioretti, C. Gab-

4banini, and G. Modugno, Evidence of superfluidity in a
dipolar supersolid from nonclassical rotational inertia, Science 371, 1162 (2021).

1.0(fRBR/fIF)2 = 2

0.8
0.6
0.4
0.2
0.0

86

88

90

92

94

scattering length as (a0)96

FIG. S3. Difference in moment of inertia between IF and
RBR, β 2 for different traps and scattering lengths. Traps andatom numbers that correspond to linear arrays for modulated
states are indicated by round markers, while those that correspond to 2D modulated arrays are indicated by triangles. All
points except those at as = 97a0 are modulated, interdroplet
connection decreasing with scattering length. The states explored in Fig. S2 are shown in grey-scale, with lighter saturation corresponding to rounder traps, while the conditions
similar to those of Refs. [5, 6] are shown in red.

5

a.

b.

FIG. S4. Analysis of linear two-droplet arrays of [5, 6]. a.
Dominant angular oscillation frequency (markers) extracted
from simulations versus scattering length, through transition
from BEC (right) to isolated droplets (left). The irrotationaland rigid-body predictions firr and frig are shown as solid anddashed lines, respectively. b. The Fourier spectrum of hxyi
for the point near as = 92a0 exhibits a response with dominant and sub-dominant mode contributions. The fluctuation
profiles associated with these two frequencies are shown in theinsets. Solid and dashed vertical red lines represent firr andfrig , respectively.

4.9 Publication VII:
Heating a quantum dipolar fluid into a solid

4.9

123

Publication VII:

Heating a quantum dipolar fluid into a solid
The following publication has appeared inarXiv:2209.00335 (2022)∗∗
DOI: https://arxiv.org/abs/2209.00335
J. Sánchez-Baena,3,4 C. Politi,1,2 F. Maucher,3,5 F. Ferlaino,1,2 and T. Pohl3
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus

University, DK-8000 Aarhus C, Denmark
4 Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, 08034

Barcelona, Spain
5 Departament de Física, Universitat de les Illes Balears & IAC-3, Campus UIB, E-07122

Palma de Mallorca, Spain∗∗The author of the present thesis performed the measurements, analysed the data with J. S., andcontributed in writing the manuscript and interpreting the results together with all the authors.

Heating a quantum dipolar fluid into a solid
J. Sánchez-Baena,1, 2, ∗ Claudia Politi,3, 4 F. Maucher,5, 1 Francesca Ferlaino,3, 4 and T. Pohl1

arXiv:2209.00335v1 [cond-mat.quant-gas] 1 Sep 2022

1

Center for Complex Quantum Systems, Department of Physics and Astronomy,
Aarhus University, DK-8000 Aarhus C, Denmark2
Departament de Fı́sica, Universitat Politècnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain
3
Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Innsbruck, Austria
4Institut für Experimentalphysik, Universität Innsbruck, Austria
5Departament de Fı́sica, Universitat de les Illes Balears & IAC-3,
Campus UIB, E-07122 Palma de Mallorca, Spain(Dated: September 2, 2022)
Raising the temperature of a material enhances the thermal motion of particles. Such an increasein thermal energy commonly leads to the melting of a solid into a fluid and eventually vaporises
the liquid into a gaseous phase of matter. Here, we study the finite-temperature physics of dipolar
quantum fluids and find surprising deviations from this general phenomenology. In particular, we
describe how heating a dipolar superfluid from near-zero temperatures can induce a phase transition to a supersolid state with a broken translational symmetry. The predicted effect agrees withexperimental measurements on ultracold dysprosium atoms, which opens the door for exploring theunusual thermodynamics of dipolar quantum fluids.

A supersolid is an exotic phase of matter in which particles develop regular spatial order and simultaneously
support the frictionless flow of a superfluid. Having
evaded experimental verification for several decades [1],supersolidity can now be observed in Bose-Einstein condensates of ultracold atoms with finite-range interactions[2–6]. Spontaneous symmetry breaking in these systems
occurs in the form of regular periodic patterns of thecondensate density as first predicted by Gross in 1957
[7]. One would thus expect the lowest possible temperatures to provide optimal conditions for supersolidity byensuring a high degree of phase coherence and maximal
population of the Bose-Einstein condensate. On the contrary, we demonstrate here that thermal fluctuations in
dipolar condensates do not merely diminish global phasecoherence but can instead facilitate the formation of periodic modulations of the condensate density. This finding
sheds light on recent experimental observations and reveals an unusual fluid-solid phase transition, whereby asupersolid state of matter emerges upon increasing thetemperature.
As we shall see below, this surprising behaviour arises
from the anisotropic nature of the dipole-dipole interaction
Vdd (r) =C3 1 − 3 cos2 θ
,
4π
r3

(1)

which has repulsive as well as attractive contributions,depending on the angle θ between the atomic dipoles
and the distance vector r of two interacting atoms. Theinteraction strength C3 and the atomic mass m definea length scale ad = mC3 /(12π~2 ) that competes withthe scattering length a of the short-range interaction between the atoms. This competition between ad and a > 0∗ jsbaena@phys.au.dk

(b)

(a)
<latexit sha1_base64="oqEWvdLOtNlnEHkT4qAP19m2x9k=">AAAB8nicjVDLSgNBEOyNrxhfUY9eBoPgKeyKqMeAHjxGMDGQLGF2djYZMo9lplcISz7DiwdFvPo13vwbN4+DioIFDUVVN91dUSqFQ9//8EpLyyura+X1ysbm1vZOdXev7UxmGW8xI43tRNRxKTRvoUDJO6nlVEWS30Wjy6l/d8+tE0bf4jjloaIDLRLBKBZSN+9ZRZyRIp70q7Wg7s9A/iY1WKDZr773YsMyxTUySZ3rBn6KYU4tCib5pNLLHE8pG9EB7xZUU8VdmM9OnpCjQolJYmxRGslM/TqRU+XcWEVFp6I4dD+9qfib180wuQhzodMMuWbzRUkmCRoy/Z/EwnKGclwQyqwobiVsSC1lWKRU+V8I7ZN6cFYPbk5rjatFHGU4gEM4hgDOoQHX0IQWMDDwAE/w7KH36L14r/PWkreY2Ydv8N4+AXWMkWE=</latexit><latexit sha1_base64="oqEWvdLOtNlnEHkT4qAP19m2x9k=">AAAB8nicjVDLSgNBEOyNrxhfUY9eBoPgKeyKqMeAHjxGMDGQLGF2djYZMo9lplcISz7DiwdFvPo13vwbN4+DioIFDUVVN91dUSqFQ9//8EpLyyura+X1ysbm1vZOdXev7UxmGW8xI43tRNRxKTRvoUDJO6nlVEWS30Wjy6l/d8+tE0bf4jjloaIDLRLBKBZSN+9ZRZyRIp70q7Wg7s9A/iY1WKDZr773YsMyxTUySZ3rBn6KYU4tCib5pNLLHE8pG9EB7xZUU8VdmM9OnpCjQolJYmxRGslM/TqRU+XcWEVFp6I4dD+9qfib180wuQhzodMMuWbzRUkmCRoy/Z/EwnKGclwQyqwobiVsSC1lWKRU+V8I7ZN6cFYPbk5rjatFHGU4gEM4hgDOoQHX0IQWMDDwAE/w7KH36L14r/PWkreY2Ydv8N4+AXWMkWE=</latexit>

solid

solid

<latexit sha1_base64="oPnBQ/KfxmAeXcuUmmpgiyPtriU=">AAAB8nicjVDLSgNBEOyNrxhfUY9eBoPgKeyKqMeAHjxGMDGQLGF2djYZMjuzzPQKYclnePGgiFe/xpt/4+RxUFGwoKGo6qa7K8qksOj7H15paXllda28XtnY3Nreqe7uta3ODeMtpqU2nYhaLoXiLRQoeScznKaR5HfR6HLq391zY4VWtzjOeJjSgRKJYBSd1C16JiWJzEU86VdrQd2fgfxNarBAs19978Wa5SlXyCS1thv4GYYFNSiY5JNKL7c8o2xEB7zrqKIpt2ExO3lCjpwSk0QbVwrJTP06UdDU2nEauc6U4tD+9Kbib143x+QiLITKcuSKzRcluSSoyfR/EgvDGcqxI5QZ4W4lbEgNZehSqvwvhPZJPTirBzentcbVIo4yHMAhHEMA59CAa2hCCxhoeIAnePbQe/RevNd5a8lbzOzDN3hvn2q+kVo=</latexit><latexit sha1_base64="oPnBQ/KfxmAeXcuUmmpgiyPtriU=">AAAB8nicjVDLSgNBEOyNrxhfUY9eBoPgKeyKqMeAHjxGMDGQLGF2djYZMjuzzPQKYclnePGgiFe/xpt/4+RxUFGwoKGo6qa7K8qksOj7H15paXllda28XtnY3Nreqe7uta3ODeMtpqU2nYhaLoXiLRQoeScznKaR5HfR6HLq391zY4VWtzjOeJjSgRKJYBSd1C16JiWJzEU86VdrQd2fgfxNarBAs19978Wa5SlXyCS1thv4GYYFNSiY5JNKL7c8o2xEB7zrqKIpt2ExO3lCjpwSk0QbVwrJTP06UdDU2nEauc6U4tD+9Kbib143x+QiLITKcuSKzRcluSSoyfR/EgvDGcqxI5QZ4W4lbEgNZehSqvwvhPZJPTirBzentcbVIo4yHMAhHEMA59CAa2hCCxhoeIAnePbQe/RevNd5a8lbzOzDN3hvn2q+kVo=</latexit>

fluid

fluid

(c)

sha1_base64="zHhL/pltdH3JrwkkI8/roknWmWY=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XwVBIR9VjQg8cK9gPaUDbbabt0Nwm7E6XE/hQvHhTx6i/x5r9x2+agrQ8GHu/NMDMvTKQw6Hnfzsrq2vrGZmGruL2zu7fvlg4aJk41hzqPZaxbITMgRQR1FCihlWhgKpTQDEfXU7/5ANqIOLrHcQKBYoNI9AVnaKWuW8o6WlEElYBmmGqYdN2yV/FmoMvEz0mZ5Kh13a9OL+apggi5ZMa0fS/BIGMaBZcwKXZSAwnjIzaAtqURU2CCbHb6hJ5YpUf7sbYVIZ2pvycypowZq9B2KoZDs+hNxf+8dor9qyATUZIiRHy+qJ9KijGd5kB7QgNHObaEcS3srZQPmWYcbVpFG4K/+PIyaZxV/IuKf3dert7kcRTIETkmp8Qnl6RKbkmN1Aknj+SZvJI358l5cd6dj3nripPPHJI/cD5/AM69lF4=</latexit><latexitt e mp e

ratureFIG. 1.
Heating a dipolar quantum fluid can lead tothe emergence of a supersolid phase of matter. (a) Thisis demonstrated in the thermodynamic phase diagram for
an infinitely elongated Bose-Einstein condensate in a radial
harmonic trap with no axial confinement, as illustrated inpanel (c). In between the superfluid (blue) and supersolid(red) region, both phases coexist (purple region) as characteristic for a first-order phase transition. The calculations
were performed for a fixed chemical potential µ/d = 1,
where d = ~2 /((12π)2 ma2d ) parametrizes the characteristicenergy scale of the dipole-dipole interactions. Our experiments in an ultracold gas of dysprosium atoms demonstrate
the temperature-driven emergence of supersolidity. The measured contrast of axial density modulations is shown by thecolored dots in panel (b). The observations show that supersolidity at higher temperatures indeed occurs for smaller
atom numbers, Nc in the condensate, in agreement with thetheoretical transition line (purple line).

can cause the condensate to collapse when the stabilizing short-range repulsion is not sufficient to overcome the2attractive part of the dipole-dipole interaction between
the atoms [8–10]. Subsequent experiments [11–13] have
however found a higher level of stability, which arises
from quantum fluctuations [14, 15] that prevent the otherwise inevitable collapse of the condensate [16–18]. In
fact, the balance of attraction and repulsion effectively
enhances the role of quantum fluctuations [17] beyond
the semiclassical mean-field physics of weakly interacting quantum gases. This yields a unique setting that hasrevealed rich physics and a host of new quantum states,from self-bound quantum droplets [12, 13, 19] and supersolid phases [3–5, 20, 21] to complex patterns in twodimensional fluids [22, 23].
Given this striking role of quantum fluctuations in
dipolar Bose-Einstein condensates, one may also anticipate significant effects of thermal fluctuations despite theultralow temperatures that are required to reach quantum degeneracy. To address this question we start fromthe grand canonical potential Ω of the system at a finite
temperature T . For a weakly interacting gas with a highfraction of atoms in the condensate, one can use Bogoliubov theory to determine Ω. This yields simple expressions for infinitely extended homogeneous systems [24]that can be applied to describe trapped inhomogeneous
gases within a local density approximation. Hereby, one
determines the Bogoliubov excitation spectrum and all
relevant observables for a homogeneous particle density
ρ, which is then identified as ρ ≡ |ψ(r)|2 with the local
condensate wave function ψ(r) at a given position r. Thispermits to express the grand canonical potential as


Z
Z
ε (r)
kB T
− kk TBΩ = E0 +
dr
dk
ln
1
−
e
,
(2)
(2π)3
where kB denotes the Boltzmann constant and E0
is the zero-temperature grand canonical energy that
contains the mean-field interaction energy and leading order corrections due to quantum fluctuations [14],i.e. small occupations of excited states above theformedBose-Einstein condensate. The dispersion εk =qτk (τk + 2|ψ(r)|2 Ṽ (k)) of these excitations is determined by the kinetic energy τk = ~2 k 2 /(2m) of the atoms
2aand the Fourier transform Ṽ (k) = 4π~
m + Ṽdd (k) of their
total interaction potential.
Minimizing Ω with respect to ψ(r) then yields a nonlinear wave equation that accounts for quantum as wellas thermal fluctuations (see Methods Section). At zerotemperature, it describes the mean-field physics of thecondensate and captures leading-order effects of quantum fluctuations through an effective density-dependent
potential Hqu [17] that increases the energy of the system.
The second term in Eq.(2) yields an additional potentialZdkτk
Hth (r) =Ṽ (k)fk (r)
,
(3)
(2π)3
εk (r)
that accounts for finite-temperature effects. It describes
the interaction between the condensate and thermally

(a)

(b)
(c)
Hqu
<latexit sha1_base64="OLND3kVNjqktaJrqlaQqvBziLQ8=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9RjwkpNEMA9MljA7mU2GzGOdmRXCkr/w4kERr/6NN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9HoZuq3nqg2TMl7O05oKPBAspgRbJ30UOtlXS3QYzrplcp+xZ8BLZMgJ2XIUe+Vvrp9RVJBpSUcG9MJ/MSGGdaWEU4nxW5qaILJCA9ox1GJBTVhNrt4gk6d0kex0q6kRTP190SGhTFjEblOge3QLHpT8T+vk9r4OsyYTFJLJZkvilOOrELT91GfaUosHzuCiWbuVkSGWGNiXUhFF0Kw+PIyaZ5XgstKcHdRrt7mcRTgGE7gDAK4girUoA4NICDhGV7hzTPei/fufcxbV7x85gj+wPv8AXLMkNA=</latexit>

Hth<latexit sha1_base64="RmRDUqphsCqP0C/v+2EXp3NFE5M=">AAAB8XicdVDLSgMxFM3UV62vqks3wSK4GjKDtrMsuOlKKtgHtkPJpJk2NMkMSUYoQ//CjQtF3Po37vwb04egogcuHM65l3vviVLOtEHowymsrW9sbhW3Szu7e/sH5cOjtk4yRWiLJDxR3QhrypmkLcMMp91UUSwiTjvR5Grud+6p0iyRt2aa0lDgkWQxI9hY6a4xyPtKQDOeDcoV5AYIXfpViFzfD5YE1TwvqEHPRQtUwArNQfm9P0xIJqg0hGOtex5KTZhjZRjhdFbqZ5qmmEzwiPYslVhQHeaLi2fwzCpDGCfKljRwoX6fyLHQeioi2ymwGevf3lz8y+tlJg7CnMk0M1SS5aI449AkcP4+HDJFieFTSzBRzN4KyRgrTIwNqWRD+PoU/k/avutVXe/molK/XsVRBCfgFJwDD9RAHTRAE7QAARI8gCfw7Gjn0XlxXpetBWc1cwx+wHn7BMi1kQw=</latexit>(d)FIG. 2. Raising the temperature of a dipolar quantum fluid
can induce a pronounced roton-maxon spectrum of its collective excitations, as shown in panel (a) for an infinitely
elongated condensate along the z-axis [see Fig.1(c)]. Heating the fluid tends to lower the energy of the roton minimum
and eventually softens the roton excitation as the temperature increases. This effect can be traced back to the density
dependence of the energy correction caused by fluctuations,shown in panel (b). While quantum fluctuations yield anenergy Hqu (dashed line) that increases with a rising condensate density ρ = |ψ|2 , the contribution Hth from thermal fluctuations decreases (solid lines). The thermal energy
correction Hth (r), therefore, acts as a focusing nonlinearitythat supports the formation of regular density modulations.
This is illustratedR in panel (c,d), where we show the axial
density ρ|| (z)R = dxdyρ(r) along with the axial potential
H̄th = ρ−1dxdyρ(r)H̄th (r), respectively. The calculations||
are performed for a/ad = 0.7 and µ = εd .

created excitations that populate Bogoliubov modes according to the Bose distribution fk = 1/(eεk /kB T − 1).
The resulting form of the finite-temperature extended
Gross-Pitaevskii equation (TeGPE) agrees with the result of Hartree-Fock Bogoliubov theory [24, 25], and includes relevant fluctuation terms that are commonly neglected within the Popov approximation [26] (see Methods Section).
Let us first use this framework to study an elongatedatomic gas that is confined harmonically in the x − yplane and extends infinitely in the z-direction without
confinement along the z-axis. Figure 1(a) shows the thermodynamic phase diagram obtained by simulating theimaginary time evolution of the TeGPE at a fixed chemical potential µ (see Methods Section). At zero temperature, we find a superfluid-supersolid quantum phasetransition, with a co-existence region that is expected
for a first-order phase transition [27]. While increasing
the temperature may generally be expected to melt the3

FIG. 3. Our measured axial density |ψ|| (z)|2 (filled area), observed for T = 76.5 nK and Nc = 13400 condensate atoms,
demonstrates the formation of a supersolid state and agrees
well with the numerical simulation of the TeGPE (red line).
On the other hand, equivalent simulations at zero temperature (blue line) disagree qualitatively and instead yield an
unmodulated superfluid phase.

supersolid phase [28], we find instead that it shifts thetransition towards weaker dipole-dipole interactions. As
a result, heating the system effectively drives a phasetransition from a fluid into a solid phase.
We can understand this effect from the excitation spectrum of the condensate in the superfluid phase. To thisend, we solve the time-dependent TeGPE within linear response theory to find the excitation spectrum ωkz
for periodic plane-wave excitations along the z-direction.
As shown in Fig.2, the obtained dispersion exhibits the
expected roton-maxon form [29–32], known from lowtemperature helium [33] and Bose-Einstein condensateswith finite-range interactions [34–37]. The local minimum at finite momenta supports the formation of roton
quasiparticles, which were introduced by Landau as elementary vortices to describe superfluidity in 4 He [33].
Experiments show that the roton minimum in helium
decreases with increasing temperature [38] due to rotonroton scattering [39]. Yet, the roton energy remains sizable at the transition to a normal-fluid phase [38], beyond which it only varies weakly with temperature. The
presence of a Bose-Einstein condensate in dilute dipolar superfluids, however, enhances the effect of thermal
fluctuations due to the larger energy scale of the interaction between Bogoliubov excitations and the condensate.
A similar effect is found for atoms with light-induced
interactions and predicted to lower the roton minimum
and cause enhanced condensate depletion [40]. In thepresent case, we find a thermal softening of the rotonmode that can drive an instability of the superfluid andthereby cause the formation of a supersolid phase withincreasing temperature.
We can gain further intuition about the underlying
mechanism by closer inspection of the two fluctuationenergies Hqu and Hth that both contribute a local nonlinearity to the wave equation for ψ(r). Hqu > 0 is
the Lee-Huang-Yang correction to the equation of state[14, 15], and raises the ground state energy due to thesmall condensate depletion caused by the atomic interactions. It therefore increases for higher particle densities
and stronger interactions, as shown in Fig. 2(b). Consequently, Hqu generates an effective repulsion that stabilizes the condensate against collapse [17], and shifts theroton instability towards higher densities and stronger
dipole-dipole interactions. On the contrary, Hth increases
as we lower the density of the condensate [see Fig.2(b)].
This behaviour is readily understood as follows. Decreasing the condensate density increases the fraction of thermally excited, non-condensed atoms [25]. In the limit
where this fraction remains small, such an increase implies a larger potential energy due to interactions withthe thermal atoms. It therefore contributes a positive energy correction that decreases upon increasing the density ρ = |ψ|2 of the condensate. As a result, thermal
fluctuations energetically favour higher condensate densities, such that Hth acts as a focusing nonlinearity which
lowers the roton energy and facilitates the formation ofa density-modulated phase, as illustrated in Fig.2c.
We recently observed experimental signatures of thiseffect by studying the cooling-heating lifecycle of bosonic
dysprosium atoms at ultralow temperatures [41]. The experiment starts from a thermal cloud of 105 atoms in acigar-shaped optical dipole trap and traces the time evolution of the gas as it is cooled evaporatively to quantum
degeneracy by lowering the depth of the trap. During thecontinual cooling and thermalization we observed the expected emergence of supersolidity, and studied the equilibrium states of the quantum fluid across the supersolid
phase transition. The measured density profiles indicate
a higher degree of modulation at higher temperatures.
While this has cast mystery on the origin of the observations, they can now be used to corroborate and benchmark our theoretical understanding . Figure 1b shows
our measured contrast of the axial density modulations
for different temperatures and condensed-atom numbers(see Methods Section). The results confirm the formation of a supersolid phase with increasing temperature ingood agreement with the theoretical transition line obtained numerically by the TeGPE. Moreover, Fig. 3 compares our measured axial density, ρ|| (z), to the theoreticalprediction. The predicted zero-temperature ground statecorresponds to an unstructured superfluid and deviates
qualitatively from the observed supersolid state. The result of our finite-temperature TeGPE simulation, however, agrees with the experiment and reproduces quantitatively the period and amplitude of the measured density modulations. This remarkable level of agreement
offers strong indication that the observed supersolid has
indeed been generated by the finite temperature of theatoms.
The possibility to make detailed comparisons between
theory and experiments opens up several directions for
exploring the surprising thermodynamic behaviour ofquantum ferrofluids. Already, the ground state phase
diagram exhibits a rich structure, including first-order as4well as second-order quantum phase transitions in one
and two-dimensional systems [27, 42, 43]. This offers apromising starting point for investigating how thermal
fluctuations influence the nature of the fluid-solid phasetransition and may affect the physics of higher dimensional supersolids [44, 45], which can come in a diverse
range of complex patterns [22, 23, 42]. Our present findings motivate future experiments and first-principle simulations [46, 47] to expand the phase diagram of Fig.1
into the high-temperature domain and draw a direct connection to the more familiar physics of liquid-solid phasetransitions in the absence of superfluidity. Such numerical approaches may also reveal how the present phenomenology extends into the regime of strong interactions, which is becoming accessible in experiments withultracold polar molecules [48–50]. Equally important, an
improved understanding of finite-temperature effects in
dipolar quantum fluids could help resolving current questions about quantitative discrepancies between measurements and theory [3, 31].
We thank Yongchang Zhang, Georg Bruun, and Jordi
Boronat for valuable discussions and the Er-Dy team in
Innsbruck for experimental support. This work was supported by the DNRF through the Center of Excellence”CCQ” (Grant agreement no.: DNRF156) and the Carlsberg Foundation through the ’Semper Ardens’ Research
Project QCooL. JB acknowledges funding by the European Union, the Spanish Ministry of Universities and theRecovery, Transformation and Resilience Plan through agrant from Universitat Politècnica de Catalunya. FF and
CP acknowledge support through an ERC Consolidator
Grant (RARE, No. 681432), the QuantERA grant MAQS
(No I4391-N) and the FOR grant (2247/PI2790) by theAustrian Science Fund FWF.

Appendix A: The nonlinear wave equation

The grand canonical potential is minimal in equilibrium such that we can minimize Eq.(2) with respect tothe condensate wave function ψ(r). This yields the nonlinear wave equation
 2 2
~ ∇
4π~2 a2+ U (r) +|ψ(r)|(A1)µψ(r) = −
2m
m

Z
2
+ dr0 Vdd (r − r0 )|ψ(r0 )| + Hqu (r) + Hth (r) ψ(r),which determines the equilibrium state of the condensate
for a given chemical potential µ. We consider a harmonic2
2
2
trapping potential U (r) = m
2 (ωx x + ωy y + ωz z ), withtrapping frequencies ωx,y,z along the three cartesian axes.
The first four terms correspond to the Gross-Pitaevskii
equation that describes the mean-field physics of the condensate at zero temperature. The next term is given by
Hqu (r) = γqu |ψ(r)|3 and accounts for leading order effects of quantum fluctuations with a strength γqu that
increases with a and ad [14, 15, 17]. Finite-temperatureeffects are captured by the last term as given in Eq.(3).
We note here that the applied local-density approximation can cause an infrared divergence of the momentum
integral in Eq.(3). However, the finite system size oftrapped systems yields a natural momentum cutoff that
ensures converged results. Indeed, we find that our calculated condensate wave functions are not sensitive tothe precise choice of the momentum cutoff for relevant
trap geometries.
Appendix B: Finite-temperature simulations

We have calculated the condensate wave function at
finite temperatures by simulating the imaginary time
evolution of the wave equation (A1). More concretely,
we replace µψ by −∂t ψ in Eq.((A1)) and simulate thetime evolution until ψ(r,R t) reaches a steady state fora given norm Nc = dr|ψ(r, t)|2 . Nc corresponds tothe number of condensate atoms under 3D confinement
as considered in Figs.1(b) and 3, and yields the axial
R
RL
density Nc /L = L−1 dxdy 0 dz|ψ(r, t)|2 for a given
length L of the periodic simulation box as consideredin Figs.1(a) and 2. Finally, we determine the chemical
potential from Eq.(A1) in order to construct the thermodynamic phase diagram shown in Fig.1(a). The results
shown in Figs.1(b) and 3 are obtained for the experimental trap parameters ωx /2π = 88Hz, ωy /2π = 141Hz, andωz /2π = 36Hz, and a scattering length of 89a0 , where a0
is the Bohr radius. The simulations of Figs.1(a) and 2
have been performed for wx = 0.0717εd , wy = 0.142εd ,and wz = 0. In all cases, the dipoles are considered to
be polarized along the y-axis .
Appendix C: Experimental determination of theaverage axial density and the density contrast

We probe the emergence of a supersolid state via timeof-flight measurements and in-situ Faraday phase contrast imaging [41]. The former provides information on
the global phase coherence and the latter allows us to
extract the modulation contrast from the in-situ atomic
density. In particular, to obtain the modulation contrastof the experimental points in Fig. 1c, we proceed as follows. For each {Nc , T }, we record the in-trap density
distribution and integrate along the direction orthogonal to the droplet chain to get a 1D density profile ρ|| (z).
We repeat the measurement under the same experimental
conditions for 10-20 times. For each profile,
R we then calculate its Fourier Transform, ρ̃|| (k) = e−ikz z ρ|| (z)dz.
Finally, we determine the average of |ρ̃|| (k)| and obtainthe modulation contrast as the ratio between the Fouriercomponent at the modulation wavelength and the density|ρ̃|| (0)| at kz = 0.
The experimental profile shown in Fig. 3 is obtainedby averaging the density distributions from the repeatedmeasurements. The central maximum in each profile is5shifted to the origin, z = 0, to correct for unavoidable
center-of-mass fluctuations in the experiment. Moreover,

we remove negative density contributions in each density
image, which are caused by small but inevitable misalignments of the imaging objective.

[1] M. H. W. Chan, R. B. Hallock, and Reatto L., “Overview
on solid 4 He and the issue of supersolidity,” Journal of
Low Temperature Physics 172, 317 (2013).
[2] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and
T. Donner, “Supersolid formation in a quantum gas
breaking a continuous translational symmetry,” Nature
543, 87 (2017).
[3] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn,
M. Guo, T. Langen, and T. Pfau, “Transient supersolid properties in an array of dipolar quantum droplets,”
Phys. Rev. X 9, 011051 (2019).
[4] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen,
A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, “Long-lived and transient supersolid behaviors in
dipolar quantum gases,” Phys. Rev. X 9, 021012 (2019).
[5] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti,
C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,“Observation of a dipolar quantum gas with metastablesupersolid properties,” Phys. Rev. Lett. 122, 130405(2019).
[6] S. C. Schuster, P. Wolf, S. Ostermann, S. Slama,
and C. Zimmermann, “Supersolid properties of a BoseEinstein condensate in a ring resonator,” Phys. Rev. Lett.
124, 143602 (2020).
[7] Eugene P. Gross, “Unified theory of interacting bosons,”
Phys. Rev. 106, 161–162 (1957).
[8] L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose-Einstein condensation in trapped dipolargases,” Phys. Rev. Lett. 85, 1791 (2000).
[9] D. H. J. O’Dell, S. Giovanazzi, and C. Eberlein, “Exacthydrodynamics of a trapped dipolar Bose-Einstein condensate,” Phys. Rev. Lett. 92, 250401 (2004).
[10] D. C. E. Bortolotti, S. Ronen, J. L. Bohn, and D. Blume,“Scattering length instability in dipolar Bose-Einsteincondensates,” Phys. Rev. Lett. 97, 160402 (2006).
[11] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier,
I. Ferrier-Barbut,
and T. Pfau, “Observing the
Rosensweig instability of a quantum ferrofluid,” Nature
530, 194 (2016).
[12] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut,
and T. Pfau, “Self-bound droplets of a dilute magnetic
quantum liquid,” Nature 539, 259 (2016).
[13] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, “Quantum-fluctuation-driven
crossover from a dilute Bose-Einstein condensate to a
macrodroplet in a dipolar quantum fluid,” Phys. Rev.
X 6, 041039 (2016).
[14] Aristeu R. P. Lima and Axel Pelster, “Quantum fluctuations in dipolar bose gases,” Phys. Rev. A 84, 041604(2011).
[15] A. R. P. Lima and A. Pelster, “Beyond mean-field lowlying excitations of dipolar Bose gases,” Phys. Rev. A 86,
063609 (2012).
[16] D. S. Petrov, “Quantum mechanical stabilization of acollapsing Bose-Bose mixture,” Phys. Rev. Lett. 115,155302 (2015).
[17] F. Wächtler and L. Santos, “Quantum filaments in dipolar Bose-Einstein condensates,” Phys. Rev. A 93, 061603(2016).
[18] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas,
P. Cheiney, and L. Tarruell, “Quantum liquid droplets
in a mixture of Bose-Einstein condensates,” Science 359,
301 (2018).
[19] F. Böttcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Langen, I. Ferrier-Barbut, T. Pfau, R. Bombı́n, J. SánchezBaena, J. Boronat, and F. Mazzanti, “Dilute dipolar
quantum droplets beyond the extended Gross-Pitaevskii
equation,” Phys. Rev. Research 1, 033088 (2019).
[20] L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Famà,A. Fioretti, C. Gabbanini, G. Modugno, A. Recati, and
S. Stringari, “Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas,” Nature
574, 382 (2019).
[21] M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt,
M. Wenzel, H. P. Büchler, T. Langen, and T. Pfau,“The low-energy goldstone mode in a trapped dipolar
supersolid,” Nature 574, 386 (2019).
[22] Y.-C. Zhang, T. Pohl, and F. Maucher, “Phases of supersolids in confined dipolar Bose-Einstein condensates,”
Phys. Rev. A 104, 013310 (2021).
[23] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H.
Ng, S. D. Graham, P. Uerlings, T. Langen, M. Zwierlein,
and T. Pfau, “Pattern formation in quantum ferrofluids:
From supersolids to superglasses,” Phys. Rev. Research
3, 033125 (2021).
[24] S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Thermodynamics of a trapped Bose-condensed gas,” Journal of
Low Temperature Physics 109, 309 (1997).
[25] E. Aybar and M. Ö. Oktel, “Temperature-dependent
density profiles of dipolar droplets,” Phys. Rev. A 99,
013620 (2019).
[26] A. Griffin, “Conserving and gapless approximations for
an inhomogeneous Bose gas at finite temperatures,”Phys. Rev. B 53, 9341 (1996).
[27] P. B. Blakie, D. Baillie, L. Chomaz, and F. Ferlaino, “Supersolidity in an elongated dipolar condensate,” Phys.
Rev. Research 2, 043318 (2020).
[28] F. Cinti, M. Boninsegni, and T. Pohl, “Exchangeinduced crystallization of soft-core Bosons,” New Journalof Physics 16, 033038 (2014).
[29] L. Santos, G. V. Shlyapnikov, and M. Lewenstein,“Roton-maxon spectrum and stability of trapped dipolar
Bose-Einstein condensates,” Phys. Rev. Lett. 90, 250403(2003).
[30] R. M. Wilson, S. Ronen, J. L. Bohn, and H. Pu, “Manifestations of the roton mode in dipolar Bose-Einsteincondensates,” Phys. Rev. Lett. 100, 245302 (2008).
[31] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz, and F. Ferlaino, “Probing
the roton excitation spectrum of a stable dipolar bosegas,” Phys. Rev. Lett. 122, 183401 (2019).

6[32] J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher,
M. Schmidt, K. S. H. Ng, S. D. Graham, T. Langen,
M. Zwierlein, and T. Pfau, “Roton excitations in an
oblate dipolar quantum gas,” Phys. Rev. Lett. 126,
193002 (2021).
[33] L. Landau, “On the theory of superfluidity,” Phys. Rev.
75, 884 (1949).
[34] D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose-Einstein condensates irradiated bya laser,” Phys. Rev. Lett. 90, 110402 (2003).
[35] N. Henkel, R. Nath, and T. Pohl, “Three-dimensionalroton excitations and supersolid formation in Rydbergexcited Bose-Einstein condensates,” Phys. Rev. Lett.
104, 195302 (2010).
[36] R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening ina quantum gas with cavity-mediated long-range interactions,” Science 336, 1570 (2012).
[37] Y.-C. Zhang, V. Walther, and T. Pohl, “Long-range
interactions and symmetry breaking in quantum gases
through optical feedback,” Phys. Rev. Lett. 121, 073604(2018).
[38] O. W. Dietrich, E. H. Graf, C. H. Huang, and L. Passell,“Neutron scattering by rotons in liquid helium,” Phys.
Rev. A 5, 1377 (1972).
[39] T. Kebukawa, “The Temperature Dependence of Phonon
Velocity and Roton Minimum in Liquid He II,” Progressof Theoretical Physics 49, 388. (1973).
[40] I. E. Mazets, D. H. J. O'Dell, G. Kurizki, Davidson
N., and W. P. Schleich, “Depletion of a Bose-Einsteincondensate by laser-induced dipole-dipole interactions,”
Journal of Physics B: Atomic, Molecular and Optical
Physics 37, S155 (2004).
[41] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark,M. A. Norcia, and F. Ferlaino, “Birth, life, and deathof a dipolar supersolid,” Phys. Rev. Lett. 126, 233401(2021).
[42] Y.-C. Zhang, F. Maucher, and T. Pohl, “Supersolidity
around a critical point in dipolar bose-einstein condensates,” Phys. Rev. Lett. 123, 015301 (2019).
[43] G. Biagioni, N. Antolini, A. Alaña, M. Modugno,
A. Fioretti, C. Gabbanini, L. Tanzi, and G. Modugno, “Dimensional crossover in the superfluid-supersolid
quantum phase transition,” Phys. Rev. X 12, 021019(2022).
[44] M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen,
M. J. Mark, R. N. Bisset, L. Santos, and . Ferlaino,“Two-dimensional supersolidity in a dipolar quantumgas,” Nature 596, 357 (2021).
[45] L. Tanzi, J. G. Maloberti, G. Biagioni, A. Fioretti,
C. Gabbanini, and G. Modugno, “Evidence of superfluidity in a dipolar supersolid from nonclassical rotational
inertia,” Science 371, 1162 (2021).
[46] A. Macia, J. Sánchez-Baena, J. Boronat, and F. Mazzanti, “Droplets of trapped quantum dipolar bosons,”
Phys. Rev. Lett. 117, 205301 (2016).
[47] F. Cinti, A. Cappellaro, L. Salasnich, and T. Macrı̀,“Superfluid filaments of dipolar bosons in free space,”
Phys. Rev. Lett. 119, 215302 (2017).
[48] L Anderegg, S. Burchesky, Y. Bao, S. S. Yu, T. Karman,
E. Chae, K.-K. Ni, W. Ketterle, and J. M. Doyle, “Observation of microwave shielding of ultracold molecules,”Science 373, 779 (2021).
[49] A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda,
T. Karman, I. Bloch, and X.-Y. Luo, “Evaporation of
microwave-shielded polar molecules to quantum degeneracy,” (2022), arXiv:2201.05143 [cond-mat.quant-gas].
[50] M. Schmidt, L. Lassablière, G. Quéméner, and T. Langen, “Self-bound dipolar droplets and supersolids inmolecular Bose-Einstein condensates,” Phys. Rev. Research 4, 013235 (2022).

Conclusions
The work discussed in this thesis can be divided into two main topics: the study of theinterspecies interactions in an ultracold dipolar mixture of erbium and dysprosium,and the investigation of supersolidity with dysprosium atoms, with the phase transition
from one to two-dimensional supersolidity as one of the main results.
Er-Dy interspecies interactions. My journey as a PhD student in the Er-Dyexperiment in Innsbruck started in 2018. In the same year, our group merged forthe first time the physics of dipolar gases with the one of hetero-nuclear mixtures bycombining degenerate quantum gases of erbium and dysprosium with different isotope
combinations, among them also a Bose-Fermi mixture [187]. Each of these isotope
combinations interact differently due to the different scattering properties. Focusing
on one of the mixtures, 166 Er-164 Dy, already from the first measurements, by looking
at the atomic clouds after time-of-flight expansion, a clear deformation of the clouds
shape arising from the repulsive interaction between erbium and dysprosium atoms was
visible. This observation led to one of the main results of this thesis: the investigationof the interspecies interaction, by measuring the in-trap displacement for different
orientations of the magnetic field [155].
In fact, for dipolar mixtures, the in-trap displacement does not only depend on theinterspecies contact interactions, but also on the dipole-dipole interactions. At fixedmagnetic field, and therefore at fixed interspecies contact interaction, the contributionof the dipole-dipole interaction to the interspecies repulsion and therefore to the in-trap
displacement can be tuned by changing the orientation of the magnetic field. In
collaboration with M. Modugno from the University of the Basque Country in Bilbao(Spain), we developed a model for our hetero-nuclear mixture. By calculating thein-trap displacement for different interspecies scattering lengths, we could estimate
the value that best fitted the experimental results, which we predicted to be abouta12 ∼ 100 a0 . Furthermore, the theoretical calculations revealed interesting regimes inwhich supersolidity, a state combining density modulation and global phase coherence,

132

Conclusions

is induced in one of the species by increasing the interspecies repulsive interactions [155].
This regime could be easily achieved in the experiment by tuning the magnetic field in
proximity of an interspecies Feshbach resonance.
In 2020, we performed extensive studies of interspecies Feshbach resonances fordifferent isotope combinations [58]. More details can be found in Ref. [57]. This project
had as main result the discovery, through atomic-loss spectrocopy, of two Gauss-broadinterspecies Feshbach resonances for two isotope mixtures. The interspecies nature ofthose resonances was probed via cross-species thermalization measurements. These
results will allow us in the future to tune the interaction between erbium and dysprosium
and reach interesting many-body phases as the one mentioned above.
Dipolar supersolids. The second topic of this thesis regards the study of supersolidstates in a system of dysprosium atoms. This project started in 2019, when our groupjoined the quest for supersolidity and, in collaboration with the ERBIUM experimentin Innsbruck, we demonstrated the existence of a supersolid state in form of quantum
droplets linked to each other via a superfluid background [42]. This state, which was
found in a regime where dipolar interactions are dominant and the mean-field physics
predicts a collapse, is stabilized by quantum fluctuations. Simultaneous results wereachieved in the group of G. Modugno in Pisa (Italy) [182], and in the group of T.
Pfau in Stuttgart (Germany) [32]. What distinguished the supersolid realized in our
experiment from the others was the choice of the isotope. In fact, we created supersolidstates with a cloud of 164 Dy atoms. 164 Dy has the special property that the backgroundscattering length is lower than the dipolar length add . For this reason, to reach the
dipolar-dominated regime, we could tune our scattering length away from Feshbachresonances and observe supersolidity for times longer than 1s. Furthermore, instead ofpreparing the supersolid state via the standard interaction ramp from a dBEC adopted
by the other groups, we developed a new scheme to create a dipolar supersolid bydirectly evaporating from a thermal cloud. This technique proved to be a fundamental
tool in the realization of two-dimensional supersolid states.
Soon after the realization of the first supersolid, we upgraded the experiment witha vertical imaging characterized by an objective, whose resolution of 700nm at theatoms allows us to probe the supersolid state in trap [176]. This tool, together withthe possibility of directly evaporating from a thermal cloud, gave us the opportunityof studying the full life-cycle of the supersolid [177]. We observed that, during theformation of the supersolid via direct evaporative cooling, the system develops firsta density modulation with phase coherence only localized within the droplets, andConclusions

133

then acquires global phase coherence. Another important result regards the roleof finite temperatures. In particular, our findings show that higher temperatures
favour the appearance of density modulation. This underlines that not only quantum
fluctuations, but also thermal fluctuations play a significant role in the formation ofa supersolid state. In a separate work in collaboration with the group of T. Pohl
at Aarhus University (Denmark), we revealed that indeed higher temperatures do
not melt the supersolid state but rather help in developing density modulation. Thiscounterintuitive behaviour relates to the excitation spectrum of a dipolar BEC, whichsoftens for higher temperatures [180].
Two-dimensional supersolidity. All the results on supersolidity mentioned above
regard a system made of three or four droplets aligned along the elongated axis of acigar-shaped trap. At the end of the year 2020, improvements in the experimental
apparatus allowed us to gain in condensed atoms and therefore also in number ofdroplets. At the beginning of 2021, we managed for the first time to realize twodimensional supersolidity, by performing a structural phase transition from a linear
chain of droplets to a supersolid in a zig-zag pattern, with up to 10 droplets [137]. This
work was done in collaboration with L. Santos (Hannover) and R. Bisset (Innsbruck)and represents one of the main results of this thesis. Soon after, we could extend
supersolidity from the zig-zag state to a hexagon state in a circular trap [27]. The
realization of this state opened the door towards many research directions, e.g. study
the excitation modes [136, 154]. Furthermore, the ability of producing states in an
isotropic circular trap allowed recently our group to observe, for the first time, quantizedvortices in a dipolar BEC by using the anisotropic nature of the dipole-dipole interaction
to impart angular momentum into the system [98].
Outlook. The Er-Dy experiment had so far really productive and exciting years.
This was also thanks to the many collaborations we had with external groups as the oneof T. Giamarchi (Geneva), L. Santos (Hannover), R. Bisset (Innsbruck), M. Modugno(Bilbao), G. Lamporesi (Trento), and T. Pohl (Aarhus), but also thanks to our theory
group with T. Bland and E. Poli, who helped us with daily stimulating discussions andtheoretical support. I expect the next years to be as exciting. Next research directions
involve the nucleation of vortices in supersolid states, by following a similar approach
used in our group to impart angular momentum in a dipolar BEC. Furthermore, theknowledge we gained in the past years on the interspecies interactions and Feshbachresonances opens the door to the realization of binary supersolid states with erbium

134

Conclusions

and dysprosium. Moreover, in July 2021 we upgraded our experimental apparatus
with a second chamber, namely a glass cell for quantum gas microscopy of erbium anddysprosium atoms [176], which will enable us to study interesting many body phases
in optical lattices arising from the long-range dipole-dipole interaction.

Bibliography[1] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle. Observation ofvortex lattices in Bose-Einstein condensates. Science, 292(5516):476–479, 2001.
[2] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm, and F. Ferlaino. Reachingfermi degeneracy via universal dipolar scattering. Phys. Rev. Lett., 112:010404,
Jan 2014.
[3] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino.
Bose-einstein condensation of erbium. Phys. Rev. Lett., 108:210401, May 2012.
[4] M. Anderlini, D. Ciampini, D. Cossart, E. Courtade, M. Cristiani, C. Sias,
O. Morsch, and E. Arimondo. Model for collisions in ultracold-atom mixtures.
Phys. Rev. A, 72:033408, Sep 2005.
[5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell.
Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science,
269:198–201, 1995.
[6] A. F. Andreev and I. M. Lifshitz. Quantum theory of defects in crystals. Sov.
Phys. JETP, 29:1107, 1969.
[7] E. Aybar and M. Ö. Oktel. Temperature-dependent density profiles of dipolar
droplets. Phys. Rev. A, 99:013620, Jan 2019.
[8] S. Baier, D. Petter, J. H. Becher, A. Patscheider, G. Natale, L. Chomaz, M. J.
Mark, and F. Ferlaino. Realization of a strongly interacting fermi gas of dipolar
atoms. Phys. Rev. Lett., 121:093602, Aug 2018.
[9] Simon Baier. An optical dipole trap for Erbium with tunable geometry. Master’s
thesis, Faculty of Mathematics, Computer Science and Physics; University of
Innsbruck, 2012.
[10] H Y Ban, M Jacka, J L Hanssen, J Reader, J J McClelland, C Y Chen, Y M Li,
K Bailey, W R Anderson, and C C Bradley. Laser cooling transitions in atomic
erbium. J. Phys. D – Appl. Phys, 36:335–385, 2000.
[11] M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller. Condensed mattertheory of dipolar quantum gases. Chemical Reviews, 112:5012–5061, 2012.
[12] M.A. Baranov. Theoretical progress in many-body physics with ultracold dipolargases. Physics Reports, 464(3):71–111, 2008.

136

Bibliography[13] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Hecker Denschlag,
and R. Grimm. Crossover from a molecular bose-einstein condensate to adegenerate fermi gas. Phys. Rev. Lett., 92:120401, Mar 2004.
[14] Kristian Baumann, Christine Guerlin, Ferdinand Brennecke, and Tilman
Esslinger. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature, 415:1301–1306, 2010.
[15] J. R. Beamish, A. D. Fefferman, A. Haziot, X. Rojas, and S. Balibar. Elastic
effects in torsional oscillators containing solid helium. Phys. Rev. B, 85:180501,
May 2012.
[16] John Beamish and Sébastien Balibar. Mechanical behavior of solid helium:
Elasticity, plasticity, and defects. Rev. Mod. Phys., 92:045002, Oct 2020.
[17] J. H. Becher, S. Baier, K. Aikawa, M. Lepers, J.-F. Wyart, O. Dulieu, and
F. Ferlaino. Anisotropic polarizability of erbium atoms. Phys. Rev. A, 97:012509,
Jan 2018.
[18] Giulio Biagioni, Nicolò Antolini, Aitor Alaña, Michele Modugno, Andrea Fioretti,
Carlo Gabbanini, Luca Tanzi, and Giovanni Modugno. Dimensional crossover inthe superfluid-supersolid quantum phase transition. Phys. Rev. X, 12:021019,
Apr 2022.
[19] Gerhard Birkl, Sven Kassner, and Herbert Walther. Multiple-shell structures oflaser-cooled 24 mg+ ions in a quadrupole storage ring. Nature, 357(6376):310–313,
1992.
[20] R. N. Bisset, L. A. Peña Ardila, and L. Santos. Quantum droplets of dipolar
mixtures. Phys. Rev. Lett., 126:025301, Jan 2021.
[21] R. N. Bisset and P. B. Blakie. Crystallization of a dilute atomic dipolar condensate.
Phys. Rev. A, 92:061603, Dec 2015.
[22] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie. Ground-state phase
diagram of a dipolar condensate with quantum fluctuations. Phys. Rev. A,
94:033619, Sep 2016.
[23] P. B. Blakie, D. Baillie, L. Chomaz, and F. Ferlaino. Supersolidity in an elongateddipolar condensate. Phys. Rev. Research, 2:043318, Dec 2020.
[24] P Blair Blakie, D Baillie, and Sukla Pal. Variational theory for the ground stateand collective excitations of an elongated dipolar condensate. Communicationsin Theoretical Physics, 72(8):085501, jul 2020.
[25] P.B. Blakie, A.S. Bradley, M.J. Davis, R.J. Ballagh, and C.W. Gardiner. Dynamics and statistical mechanics of ultra-cold bose gases using c-field techniques.
Advances in Physics, 57(5):363–455, 2008.
[26] T. Bland, E. Poli, L. A. Pe na Ardila, L. Santos, F. Ferlaino, and R. N. Bisset.
Domain supersolids in binary dipolar condensates. arXiv:2203.11119, Mar 2022.

Bibliography137[27] T. Bland, E. Poli, C. Politi, L. Klaus, M. A. Norcia, F. Ferlaino, L. Santos, and
R. N. Bisset. Two-dimensional supersolid formation in dipolar condensates. Phys.
Rev. Lett., 128:195302, May 2022.
[28] I. Bloch, T. W. Hänsch, and T. Esslinger. Measurement of the spatial coherenceof a trapped Bose gas at the phase transition. Nature, 403:166–170, 2000.
[29] D Bohle, A Richter, W Steffen, AEL Dieperink, N Lo Iudice, F Palumbo, andO Scholten. New magnetic dipole excitation mode studied in the heavy deformed
nucleus 156gd by inelastic electron scattering. Physics Letters B, 137(1-2):27–31,
1984.
[30] John L. Bohn and Deborah S. Jin. Differential scattering and rethermalization
in ultracold dipolar gases. Phys. Rev. A, 89:022702, Feb 2014.
[31] Massimo Boninsegni and Nikolay V. Prokof’ev. Colloquium: Supersolids: What
and where are they? Rev. Mod. Phys., 84:759–776, May 2012.
[32] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, andT. Pfau. Transient Supersolid Properties in an Array of Dipolar Quantum
Droplets. Phys. Rev. X, 9:011051, Mar 2019.
[33] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann,
L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon. Experimental study ofthe bec-bcs crossover region in lithium 6. Phys. Rev. Lett., 93:050401, Jul 2004.
[34] C. C. Bradley, C. A. Sackett, and R. G. Hulet. Bose-einstein condensation oflithium: Observation of limited condensate number. Phys. Rev. Lett., 78:985–989,
Feb 1997.
[35] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of boseeinstein condensation in an atomic gas with attractive interactions. Phys. Rev.
Lett., 75:1687–1690, Aug 1995.
[36] H. P. Büchler and G. Blatter. Supersolid versus phase separation in atomic
bose-fermi mixtures. Phys. Rev. Lett., 91:130404, Sep 2003.
[37] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and
L. Tarruell. Quantum liquid droplets in a mixture of bose-einstein condensates.
Science, 359(6373):301–304, 2018.
[38] G. V. Chester. Speculations on Bose–Einstein condensation and quantum crystals.
Phys. Rev. A, 2:256–258, Jul 1970.
[39] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach resonances in ultracold gases. Reviews of Modern Physics, 82:1225–1286, 2010.
[40] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino. Quantum-Fluctuation-driven crossover from a dilute Bose-Einsteincondensate to a macrodroplet in a dipolar quantum fluid. Physical Review X, 6,
2016.

138

Bibliography[41] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino. Quantum-fluctuation-driven crossover from a dilute bose-einsteincondensate to a macrodroplet in a dipolar quantum fluid. Phys. Rev. X, 6:041039,
Nov 2016.
[42] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and
F. Ferlaino. Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum
Gases. Phys. Rev. X, 9:021012, Apr 2019.
[43] Lauriane Chomaz, Igor Ferrier-Barbut, Francesca Ferlaino, Bruno Laburthe-Tolra,
Benjamin L. Lev, and Tilman Pfau. Dipolar physics: A review of experimentswith magnetic quantum gases. arXiv:2201.02672, 2022.
[44] Lauriane Chomaz, Rick M W van Bijnen, Daniel Petter, Giulia Faraoni, Simon
Baier, J Hendrik Becher, Manfred J Mark, Falk Wächtler, Luis Santos, andFrancesca Ferlaino. Observation of roton mode population in a dipolar quantumgas. Nat. Phys., 14(5):442, 2018.
[45] Lauriane Chomaz, Rick M W van Bijnen, Daniel Petter, Giulia Faraoni, Simon
Baier, Jan-Hendrik Becher, Manfred J Mark, Falk Wächtler, Luis Santos, andFrancesca Ferlaino. Observation of roton mode population in a dipolar quantumgas. Nat. Phys., 14(5):442–446, 2018.
[46] F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G. Pupillo. Supersolid
droplet crystal in a dipole-blockaded gas. Phys. Rev. Lett., 105:135301, Sep 2010.
[47] F. Cinti, T. Macrì, W. Lechner, G. Pupillo, and T. Pohl. Defect-induced
supersolidity with soft-core bosons. Nature Communications, 5(3235), 2014.
[48] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. Van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodium
atoms. Physical Review Letters, 75:3969–3973, 1995.
[49] E. T. Davletov, V. V. Tsyganok, V. A. Khlebnikov, D. A. Pershin, D. V. Shaykin,
and A. V. Akimov. Machine learning for achieving bose-einstein condensation ofthulium atoms. Phys. Rev. A, 102:011302, Jul 2020.
[50] James Day and John Beamish. Low-temperature shear modulus changes in solid
4He and connection to supersolidity. Nature, 450(7171):853–856, 2007.
[51] L. De Sarlo, P. Maioli, G. Barontini, J. Catani, F. Minardi, and M. Inguscio.
Collisional properties of sympathetically cooled 39 K. Phys. Rev. A, 75:022715,
Feb 2007.
[52] B. DeMarco, J. L. Bohn, J. P. Burke, M. Holland, and D. S. Jin. Measurementof p-wave threshold law using evaporatively cooled fermionic atoms. Phys. Rev.
Lett., 82:4208–4211, May 1999.
[53] B. DeMarco and D. S. Jin. Onset of Fermi Degeneracy in a Trapped Atomic Gas.
Science, 285:1703–1706, 1999.

Bibliography139[54] C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno,
F. Minardi, and C. Fort. Observation of quantum droplets in a heteronuclear
bosonic mixture. Phys. Rev. Research, 1:033155, Dec 2019.
[55] B. J. DeSalvo, Krutik Patel, Geyue Cai, and Cheng Chin. Observation offermion-mediated interactions between bosonic atoms. Nature, 568:61–64, Apr
2019.
[56] O. W. Dietrich, E. H. Graf, C. H. Huang, and L. Passell. Neutron scattering by
rotons in liquid helium. Phys. Rev. A, 5:1377–1391, Mar 1972.
[57] Gianmaria Durastante. Creation of Erbium-Dysprosium Dipolar Quantum Mixtures and Their Interspecies Feshbach Resonances. PhD thesis, University of
Innsbruck, 2020.
[58] Gianmaria Durastante, Claudia Politi, Maximilian Sohmen, Philipp Ilzhöfer,Manfred J. Mark, Matthew A. Norcia, and Francesca Ferlaino. Feshbach resonances in an erbium-dysprosium dipolar mixture. Phys. Rev. A, 102:033330, Sep
2020.
[59] V. A. Dzuba and V. V. Flambaum. Theoretical study of some experimentally
relevant states of dysprosium. Phys. Rev. A, 81:052515, May 2010.
[60] M. Egorov, B. Opanchuk, P. Drummond, B. V. Hall, P. Hannaford, and A. I.
Sidorov. Measurement of s-wave scattering lengths in a two-component boseeinstein condensate. Phys. Rev. A, 87:053614, May 2013.
[61] Igor Ferrier-Barbut, Holger Kadau, Matthias Schmitt, Matthias Wenzel, andTilman Pfau. Observation of quantum droplets in a strongly dipolar bose gas.
Phys. Rev. Lett., 116:215301, May 2016.
[62] Igor Ferrier-Barbut, Matthias Wenzel, Fabian Böttcher, Tim Langen, Mathieu
Isoard, Sandro Stringari, and Tilman Pfau. Scissors mode of dipolar quantum
droplets of dysprosium atoms. Phys. Rev. Lett., 120:160402, Apr 2018.
[63] Igor Ferrier-Barbut, Matthias Wenzel, Fabian Böttcher, Tim Langen, Mathieu
Isoard, Sandro Stringari, and Tilman Pfau. Scissors mode of dipolar quantum
droplets of dysprosium atoms. Phys. Rev. Lett., 120:160402, Apr 2018.
[64] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler, E. Zupanič, R. Grimm,
and F. Ferlaino. Narrow-line magneto-optical trap for erbium. Phys. Rev. A,
85:051401, May 2012.
[65] Albert Frisch. Dipolar Quantum Gases of Erbium. PhD thesis, University of
Innsbruck, 2014.
[66] A. Gallemí, S. M. Roccuzzo, S. Stringari, and A. Recati. Quantized vortices in
dipolar supersolid bose-einstein-condensed gases. Phys. Rev. A, 102:023322, Aug
2020.
[67] Christopher Gaul and Cord A. Müller. A grand-canonical approach to thedisordered bose gas. Applied Physics B, 117:775–784, Dec 2014.

140

Bibliography[68] S. Giovanazzi, D. O’Dell, and G. Kurizki. Self-binding transition in bose condensates with laser-induced “gravitation”. Phys. Rev. A, 63:031603, Feb 2001.
[69] Goran Gligorić, Aleksandra Maluckov, Milutin Stepić, Ljupč Hadžievski, andBoris A. Malomed. Transition to miscibility in linearly coupled binary dipolarbose-einstein condensates. Phys. Rev. A, 82:033624, Sep 2010.
[70] Markus Greiner, Olaf Mandel, Tilman Esslinger, T. W. Hänsch, and Immanuel
Bloch. Quantum phase transition from a superfluid to a Mott insulator in a gasof ultracold atoms. Nature, 415:39–44, 2002.
[71] Axel Griesmaier, Jörg Werner, Sven Hensler, Jürgen Stuhler, and Tilman Pfau.
Bose-Einstein condensation of chromium. Physical Review Letters, 94, 2005.
[72] E.P Gross. Classical theory of boson wave fields. Annals of Physics, 4(1):57–74,
1958.
[73] Eugene P. Gross. Unified theory of interacting bosons. Phys. Rev., 106(1):161–162,
apr 1957.
[74] D. Guéry-Odelin and S. Stringari. Scissors mode and superfluidity of a trappedbose-einstein condensed gas. Phys. Rev. Lett., 83:4452–4455, Nov 1999.
[75] Mingyang Guo, Fabian Böttcher, Jens Hertkorn, Jan-Niklas Schmidt, Matthias
Wenzel, Hans Peter Büchler, Tim Langen, and Tilman Pfau. The low-energy
Goldstone mode in a trapped dipolar supersolid. Nature, 574(7778):386–389,
2019.
[76] A. Guttridge, S. A. Hopkins, S. L. Kemp, Matthew D. Frye, Jeremy M. Hutson,
and Simon L. Cornish. Interspecies thermalization in an ultracold mixture of cs
and yb in an optical trap. Phys. Rev. A, 96:012704, Jul 2017.
[77] A. Guttridge, S. A. Hopkins, S. L. Kemp, Matthew D. Frye, Jeremy M. Hutson,
and Simon L. Cornish. Interspecies thermalization in an ultracold mixture of cs
and yb in an optical trap. Phys. Rev. A, 96:012704, Jul 2017.
[78] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Görlitz,
and W. Ketterle. Two-species mixture of quantum degenerate bose and fermi
gases. Phys. Rev. Lett., 88:160401, Apr 2002.
[79] Zoran Hadzibabic, Sabine Stock, Baptiste Battelier, Vincent Bretin, and Jean
Dalibard. Interference of an array of independent bose-einstein condensates.
Phys. Rev. Lett., 93:180403, Oct 2004.
[80] Anders H. Hansen, Alexander Khramov, William H. Dowd, Alan O. Jamison,
Vladyslav V. Ivanov, and Subhadeep Gupta. Quantum degenerate mixture ofytterbium and lithium atoms. Phys. Rev. A, 84:011606, Jul 2011.
[81] Hideaki Hara, Yosuke Takasu, Yoshifumi Yamaoka, John M. Doyle, and Yoshiro
Takahashi. Quantum degenerate mixtures of alkali and alkaline-earth-like atoms.
Phys. Rev. Lett., 106:205304, May 2011.

Bibliography141[82] Ariel Haziot, Xavier Rojas, Andrew D. Fefferman, John R. Beamish, and Sébastien
Balibar. Giant plasticity of a quantum crystal. Phys. Rev. Lett., 110:035301, Jan
2013.
[83] N. Henkel, R. Nath, and T. Pohl. Three-dimensional roton excitations andsupersolid formation in rydberg-excited bose-einstein condensates. Phys. Rev.
Lett., 104:195302, May 2010.
[84] J. Hertkorn, F. Böttcher, M. Guo, J. N. Schmidt, T. Langen, H. P. Büchler, andT. Pfau. Fate of the Amplitude Mode in a Trapped Dipolar Supersolid. Phys.
Rev. Lett., 123:193002, Nov 2019.
[85] J. Hertkorn, J.-N. Schmidt, F. Böttcher, M. Guo, M. Schmidt, K. S. H. Ng,
S. D. Graham, H. P. Büchler, T. Langen, M. Zwierlein, and T. Pfau. Density
fluctuations across the superfluid-supersolid phase transition in a dipolar quantumgas. Phys. Rev. X, 11:011037, Feb 2021.
[86] Ming-Guang Hu, Michael J. Van de Graaff, Dhruv Kedar, John P. Corson, Eric A.
Cornell, and Deborah S. Jin. Bose polarons in the strongly interacting regime.
Phys. Rev. Lett., 117:055301, Jul 2016.
[87] P. Ilzhöfer, G. Durastante, A. Patscheider, A. Trautmann, M. J. Mark, and F. Ferlaino. Two-species five-beam magneto-optical trap for erbium and dysprosium.
Phys. Rev. A, 97:023633, Feb 2018.
[88] Philipp Ilzhöfer. Creation of Dipolar Quantum Mixtures of Erbium and Dysprosium. PhD thesis, University of Innsbruck, 2020.
[89] Philipp Ilzhöfer, Maximilian Sohmen, Gianmaria Durastante, Claudia Politi,
Arno Trautmann, G Natale, G Morpurgo, T Giamarchi, L Chomaz, MJ Mark,et al. Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar
atoms. Nature Physics, 17(3):356–361, 2021.
[90] Nils B. Jørgensen, Lars Wacker, Kristoffer T. Skalmstang, Meera M. Parish,
Jesper Levinsen, Rasmus S. Christensen, Georg M. Bruun, and Jan J. Arlt.
Observation of attractive and repulsive polarons in a bose-einstein condensate.
Phys. Rev. Lett., 117:055302, Jul 2016.
[91] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, andT. Pfau. Observing the Rosensweig instability of a quantum ferrofluid. Nature,
530(7589):194–197, 02 2016.
[92] Hidetoshi Katori, Tetsuya Ido, Yoshitomo Isoya, and Makoto Kuwata-Gonokami.
Magneto-optical trapping and cooling of strontium atoms down to the photon
recoil temperature. Phys. Rev. Lett., 82:1116–1119, Feb 1999.
[93] Takeji Kebukawa. The Temperature Dependence of Phonon Velocity and Roton
Minimum in Liquid He II. Progress of Theoretical Physics, 49(2):388–400, 02
1973.
[94] Duk Y. Kim and Moses H. W. Chan. Absence of supersolidity in solid helium inporous vycor glass. Phys. Rev. Lett., 109:155301, Oct 2012.

142

Bibliography[95] E. Kim and M. H. W. Chan. Observation of superflow in solid helium. Science,
305(5692):1941–1944, 2004.
[96] E. Kim and M. H. W. Chan. Probable observation of a supersolid helium phase.
Nature, 427(6971):225–227, 2004.
[97] D. A. Kirzhnits and Y. A. Nepomnyashchii. Coherent crystallization of quantum
liquid. Sov. Phys. JETP, 32(6), 1971.
[98] Lauritz Klaus, Thomas Bland, Elena Poli, Claudia Politi, Giacomo Lamporesi,
Eva Casotti, Russell N. Bisset, Manfred M. Mark, and Francesca Ferlaino.
Observation of vortices and vortex stripes in a dipolar Bose-Einstein condensate.
Nature Physics, 2022.
[99] Youssef Kora and Massimo Boninsegni. Patterned supersolids in dipolar Bosesystems. J. Low Temp. Phys., 197(5):337–347, Dec 2019.
[100] Svetlana Kotochigova. Controlling interactions between highly magnetic atoms
with Feshbach resonances. Reports on Progress in Physics, 77:093901, 2014.
[101] Ramavarmaraja Kishor Kumar, Paulsamy Muruganandam, Lauro Tomio, andArnaldo Gammal. Miscibility in coupled dipolar and non-dipolar bose–einsteincondensates. Journal of Physics Communications, 1(3):035012, nov 2017.
[102] Ramavarmaraja Kishor Kumar, Lauro Tomio, and Arnaldo Gammal. Vortex
patterns in rotating dipolar bose–einstein condensate mixtures with squared
optical lattices. Journal of Physics B: Atomic, Molecular and Optical Physics,
52(2):025302, dec 2018.
[103] Ramavarmaraja Kishor Kumar, Lauro Tomio, Boris A. Malomed, and Arnaldo
Gammal. Vortex lattices in binary bose-einstein condensates with dipole-dipole
interactions. Phys. Rev. A, 96:063624, Dec 2017.
[104] P Kuopanportti, Y M Liu, Y Z He, and C G Bao. Effect of the interspecies interaction on the density profiles of miscible two-species bose–einstein condensates.
Journal of Physics B: Atomic, Molecular and Optical Physics, 52(1):015001, nov
2018.
[105] W. J. Kwon, G. Del Pace, K. Xhani, L. Galantucci, A. Muzi Falconi, M. Inguscio,
F. Scazza, and G. Roati. Sound emission and annihilations in a programmablequantum vortex collider. Nature (London), 600(7887):64–69, 2021.
[106] T Lahaye, C Menotti, L Santos, M Lewenstein, and T Pfau. The physics ofdipolar bosonic quantum gases. Reports on Progress in Physics, 72:126401, 2009.
[107] Thierry Lahaye, Jonas Metz, Bernd Fröhlich, and Tobias Koc. d-wave collapse
and explosion of a dipolar Bose-Einstein condensate. Phys. Rev. Lett., 101:080401,
2008.
[108] L. Landau. Theory of the superfluidity of helium ii. Phys. Rev., 60:356–358, Aug1941.

Bibliography143[109] A. D. Lange, K. Pilch, A. Prantner, F. Ferlaino, B. Engeser, H.-C. Nägerl,
R. Grimm, and C. Chin. Determination of atomic scattering lengths frommeasurements of molecular binding energies near feshbach resonances. Phys. Rev.
A, 79:013622, Jan 2009.
[110] T. D. Lee, K. Huang, and C. N. Yang. Eigenvalues and Eigenfunctions of a
Bose System of Hard Spheres and its Low-Temperature Properties. Phys. Rev.,
106:1135–1145, 1957.
[111] A. J. Leggett. Can a solid be “Superfluid"? Phys. Rev. Lett., 25:1543–1546, Nov
1970.
[112] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner. Supersolid
formation in a quantum gas breaking a continuous translational symmetry.
Nature, 543(7643):87–90, 2017.
[113] Hui Li, Jean-Fran çois Wyart, Olivier Dulieu, and Maxence Lepers. Anisotropic
optical trapping as a manifestation of the complex electronic structure of ultracold
lanthanide atoms: The example of holmium. Phys. Rev. A, 95:062508, Jun 2017.
[114] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. Çağrı Top, A. O.
Jamison, and W. Ketterle. A stripe phase with supersolid properties in spin–orbitcoupled Bose–Einstein condensates. Nature, 543(7643):91–94, 2017.
[115] A. R. P. Lima and A. Pelster. Beyond mean-field low-lying excitations of dipolarbose gases. Phys. Rev. A, 86:063609, Dec 2012.
[116] Aristeu R. P. Lima and Axel Pelster. Quantum fluctuations in dipolar bose gases.
Phys. Rev. A, 84:041604, Oct 2011.
[117] E. B. Linscott and P. B. Blakie. Thermally activated local collapse of a flattened
dipolar condensate. Phys. Rev. A, 90:053605, Nov 2014.
[118] Mingwu Lu, Nathaniel Q. Burdick, and Benjamin L. Lev. Quantum degenerate
dipolar fermi gas. Phys. Rev. Lett., 108:215301, May 2012.
[119] Mingwu Lu, Nathaniel Q. Burdick, Seo Ho Youn, and Benjamin L. Lev. Strongly
dipolar Bose-Einstein condensate of dysprosium. Physical Review Letters, 107,2011.
[120] Zhen-Kai Lu, Yun Li, D. S. Petrov, and G. V. Shlyapnikov. Stable dilute
supersolid of two-dimensional dipolar bosons. Phys. Rev. Lett., 115:075303, Aug
2015.
[121] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard. Vortex formation ina stirred bose-einstein condensate. Phys. Rev. Lett., 84:806–809, Jan 2000.
[122] O. M. Maragò, S. A. Hopkins, J. Arlt, E. Hodby, G. Hechenblaikner, and C. J.
Foot. Observation of the scissors mode and evidence for superfluidity of a trappedbose-einstein condensed gas. Phys. Rev. Lett., 84:2056–2059, Mar 2000.

144

Bibliography[123] Christoph Maschler and Helmut Ritsch. Cold atom dynamics in a quantum
optical lattice potential. Phys. Rev. Lett., 95:260401, Dec 2005.
[124] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman,and E. A. Cornell. Vortices in a bose-einstein condensate. Phys. Rev. Lett.,
83:2498–2501, Sep 1999.
[125] I E Mazets, D H J O'Dell, G Kurizki, N Davidson, and W P Schleich. Depletionof a bose–einstein condensate by laser-induced dipole–dipole interactions. Journalof Physics B: Atomic, Molecular and Optical Physics, 37(7):S155–S164, mar 2004.
[126] D. J. McCarron, H. W. Cho, D. L. Jenkin, M. P. Köppinger, and S. L. Cornish.
Dual-species bose-einstein condensate of 87 Rb and 133 Cs. Phys. Rev. A, 84:011603,
Jul 2011.
[127] Jonas Metz, Thierry Lahaye, B Fröhlich, Axel Griesmaier, Tilman Pfau, Hiroki Saito, Y Kawaguchi, and Masahito Ueda. Coherent collapses of dipolar Bose–Einstein condensates for different trap geometries. New J. Phys.,
11(5):055032, ma 2009.
[128] D. Mitra, K. H. Leung, and T. Zelevinsky. Quantum control of molecules for
fundamental physics. Phys. Rev. A, 105:040101, Apr 2022.
[129] Yuki Miyazawa, Ryotaro Inoue, Hiroki Matsui, Gyohei Nomura, and Mikio
Kozuma. Bose-einstein condensation of europium. arXiv:2207.11692, 2022.
[130] A. Mosk, S. Kraft, M. Mudrich, K. Singer, W. Wohlleben, R. Grimm, and
M. Weidemüller. Mixture of ultracold lithium and cesium atoms in an optical
dipole trap. Appl. Phys. B, 73(8):791, 2001.
[131] P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe. Observation of cavity-mediated long-range light forces between strongly coupled atoms.
Phys. Rev. Lett., 84:4068–4071, May 2000.
[132] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter, M. J. Mark,L. Chomaz, and F. Ferlaino. Excitation Spectrum of a Trapped Dipolar Supersolid and Its Experimental Evidence. Phys. Rev. Lett., 123:050402, Aug
2019.
[133] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis, and B. P. Anderson.
Observation of vortex dipoles in an oblate Bose-Einstein condensate. Phys. Rev.
Lett., 104:160401, Apr 2010.
[134] Flavio S. Nogueira and Hagen Kleinert. Thermally induced rotons in twodimensional dilute bose gases. Phys. Rev. B, 73:104515, Mar 2006.
[135] Matthew A. Norcia and Francesca Ferlaino. Developments in atomic control
using ultracold magnetic lanthanides. Nat. Phys., 17(12):1349–1357, 2021.
[136] Matthew A. Norcia, Elena Poli, Claudia Politi, Lauritz Klaus, Thomas Bland,
Manfred J. Mark, Luis Santos, Russell N. Bisset, and Francesca Ferlaino. Can
angular oscillations probe superfluidity in dipolar supersolids? Phys. Rev. Lett.,
129:040403, Jul 2022.

Bibliography145[137] Matthew A. Norcia, Claudia Politi, Lauritz Klaus, Elena Poli, Maximilian
Sohmen, Manfred J. Mark, Russell N. Bisset, Luis Santos, and Francesca Ferlaino.
Two-dimensional supersolidity in a dipolar quantum gas. Nature, 596(7872):357–
361, Aug 2021.
[138] P. Nozières. Is the roton in superfluid 4he the ghost of a bragg spot? J. Low
Temp. Phys., 137(1):45–67, Oct 2004.
[139] D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki. Rotons in gaseous bose-einsteincondensates irradiated by a laser. Phys. Rev. Lett., 90:110402, Mar 2003.
[140] S. Ostermann, F. Piazza, and H. Ritsch. Spontaneous crystallization of light andultracold atoms. Phys. Rev. X, 6:021026, May 2016.
[141] Sukla Pal, D. Baillie, and P. B. Blakie. Excitations and number fluctuations in
an elongated dipolar bose-einstein condensate. Phys. Rev. A, 102:043306, Oct
2020.
[142] S. B. Papp, J. M. Pino, and C. E. Wieman. Tunable miscibility in a dual-species
bose-einstein condensate. Phys. Rev. Lett., 101:040402, Jul 2008.
[143] Benjamin Pasquiou, Alex Bayerle, Slava M. Tzanova, Simon Stellmer, Jacek
Szczepkowski, Mark Parigger, Rudolf Grimm, and Florian Schreck. Quantum
degenerate mixtures of strontium and rubidium atoms. Phys. Rev. A, 88:023601,
Aug 2013.
[144] A. Patscheider, L. Chomaz, G. Natale, D. Petter, M. J. Mark, S. Baier, B. Yang,
R. R. W. Wang, J. L. Bohn, and F. Ferlaino. Determination of the scattering
length of erbium atoms. Phys. Rev. A, 105:063307, Jun 2022.
[145] A. Patscheider, B. Yang, G. Natale, D. Petter, L. Chomaz, M. J. Mark, G. Hovhannesyan, M. Lepers, and F. Ferlaino. Observation of a narrow inner-shellorbital transition in atomic erbium at 1299 nm. Phys. Rev. Research, 3:033256,
Sep 2021.
[146] Oliver Penrose and Lars Onsager. Bose-einstein condensation and liquid helium.
Phys. Rev., 104:576–584, Nov 1956.
[147] N. Petersen, M. Trümper, and P. Windpassinger. Spectroscopy of the 1001-nm
transition in atomic dysprosium. Phys. Rev. A, 101:042502, Apr 2020.
[148] Alexander Petrov, Eite Tiesinga, and Svetlana Kotochigova. Anisotropy-induced
feshbach resonances in a quantum dipolar gas of highly magnetic atoms. Phys.
Rev. Lett., 109:103002, Sep 2012.
[149] D. S. Petrov. Quantum mechanical stabilization of a collapsing bose-bose mixture.
Phys. Rev. Lett., 115:155302, Oct 2015.
[150] D. Petter, G. Natale, R. M.W. Van Bijnen, A. Patscheider, M. J. Mark,L. Chomaz, and F. Ferlaino. Probing the Roton Excitation Spectrum of a
Stable Dipolar Bose Gas. Physical Review Letters, 122(18):183401, 2019.

146

Bibliography[151] D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. A. Baranov, R. van Bijnen,
S. M. Roccuzzo, A. Recati, B. Blakie, D. Baillie, L. Chomaz, and F. Ferlaino.
Bragg scattering of an ultracold dipolar gas across the phase transition from
Bose-Einstein condensate to supersolid in the free-particle regime. Phys. Rev. A,
104:L011302, Jul 2021.
[152] Gregory A. Phelps, Anne Hébert, Aaron Krahn, Susannah Dickerson, FurkanÖztürk, Sepehr Ebadi, Lin Su, and Markus Greiner. Sub-second production of aquantum degenerate gas. arXiv:2007.10807, 2020.
[153] Lev Pitaevskii and Sandro Stringari. Bose-Einstein condensation and superfluidity, volume 164. Oxford University Press, 2016.
[154] E. Poli, T. Bland, C. Politi, L. Klaus, M. A. Norcia, F. Ferlaino, R. N. Bisset,
and L. Santos. Maintaining supersolidity in one and two dimensions. Phys. Rev.
A, 104:063307, Dec 2021.
[155] C. Politi, A. Trautmann, P. Ilzhöfer, G. Durastante, M. J. Mark, M. Modugno,
and F. Ferlaino. Interspecies interactions in an ultracold dipolar mixture. Phys.
Rev. A, 105:023304, Feb 2022.
[156] Claudia Politi. Optical dipole trap for an erbium and dysprosium mixture. Master’s
thesis, Dipartimento di Fisica "E. Fermi", Università di Pisa, 2017.
[157] Yves Pomeau and Sergio Rica. Dynamics of a model of supersolid. Phys. Rev.
Lett., 72:2426–2429, Apr 1994.
[158] H. Pu and N. P. Bigelow. Collective excitations, metastability, and nonlinear
response of a trapped two-species bose-einstein condensate. Phys. Rev. Lett.,
80:1134–1137, Feb 1998.
[159] K Pyka, J Keller, HL Partner, R Nigmatullin, T Burgermeister, DM Meier,
K Kuhlmann, A Retzker, Martin B Plenio, WH Zurek, et al. Topological defectformation and spontaneous symmetry breaking in ion coulomb crystals. Naturecommunications, 4(1):1–6, 2013.
[160] M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland.
Ionic crystals in a linear paul trap. Phys. Rev. A, 45:6493–6501, May 1992.
[161] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, S. Tzanova, E. Kirilov, and
R. Grimm. Accurate determination of the dynamical polarizability of dysprosium.
Phys. Rev. Lett., 120:223001, May 2018.
[162] C. Ravensbergen, E. Soave, V. Corre, M. Kreyer, B. Huang, E. Kirilov, and
R. Grimm. Resonantly interacting fermi-fermi mixture of 161 Dy and 40 K. Phys.
Rev. Lett., 124:203402, May 2020.
[163] Cornelis Ravensbergen, Vincent Corre, Elisa Soave, Marian Kreyer, Emil Kirilov,
and Rudolf Grimm. Production of a degenerate fermi-fermi mixture of dysprosium
and potassium atoms. Phys. Rev. A, 98:063624, Dec 2018.

Bibliography147[164] C. A. Regal, M. Greiner, and D. S. Jin. Observation of resonance condensationof fermionic atom pairs. Phys. Rev. Lett., 92:040403, Jan 2004.
[165] S. M. Roccuzzo, A. Gallemí, A. Recati, and S. Stringari. Rotating a supersolid
dipolar gas. Phys. Rev. Lett., 124:045702, Jan 2020.
[166] S. M. Roccuzzo, A. Recati, and S. Stringari. Moment of inertia and dynamicalrotational response of a supersolid dipolar gas. Phys. Rev. A, 105:023316, Feb
2022.
[167] Camilla De Rossi, Romain Dubessy, Karina Merloti, Mathieu de Goër de Herve,
Thomas Badr, Aurélien Perrin, Laurent Longchambon, and Hélène Perrin. Probing superfluidity in a quasi two-dimensional bose gas through its local dynamics.
New Journal of Physics, 18(6):062001, jun 2016.
[168] L. Santos, G. V. Shlyapnikov, and M. Lewenstein. Roton-maxon spectrum andstability of trapped dipolar bose-einstein condensates. Phys. Rev. Lett., 90:250403,
Jun 2003.
[169] D. Scheiermann, L. A. Peña Ardila, T. Bland, R. N. Bisset, and L. Santos.
Catalyzation of supersolidity in binary dipolar condensates. arXiv:2202.08259,
2022.
[170] J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher, M. Schmidt, K. S. H. Ng, S. D.
Graham, T. Langen, M. Zwierlein, and T. Pfau. Roton excitations in an oblate
dipolar quantum gas. Phys. Rev. Lett., 126:193002, May 2021.
[171] Matthias Schmitt, Matthias Wenzel, Fabian Böttcher, Igor Ferrier-Barbut, andTilman Pfau. Self-bound droplets of a dilute magnetic quantum liquid. Nature,539(7628):259, 2016.
[172] Jaeho Shin, Jaewon Choi, Keiya Shirahama, and Eunseong Kim. Simultaneous
investigation of shear modulus and torsional resonance of solid 4 He. Phys. Rev.
B, 93:214512, Jun 2016.
[173] C. Silber, S. Günther, C. Marzok, B. Deh, Ph. W. Courteille, and C. Zimmermann.
Quantum-degenerate mixture of fermionic lithium and bosonic rubidium gases.
Phys. Rev. Lett., 95:170408, Oct 2005.
[174] Joseph C. Smith, D. Baillie, and P. B. Blakie. Quantum droplet states of abinary magnetic gas. Phys. Rev. Lett., 126:025302, Jan 2021.
[175] Joseph C. Smith, P. B. Blakie, and D. Baillie. Approximate theories for binary
magnetic quantum droplets. Phys. Rev. A, 104:053316, Nov 2021.
[176] Maximilian Sohmen. Supersolidity in dipolar quantum gases in and out of
equilibrium. PhD thesis, University of Innsbruck, 2021.
[177] Maximilian Sohmen, Claudia Politi, Lauritz Klaus, Lauriane Chomaz, Manfred J.
Mark, Matthew A. Norcia, and Francesca Ferlaino. Birth, life, and death of adipolar supersolid. Phys. Rev. Lett., 126:233401, Jun 2021.

148

Bibliography[178] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and
W. Ketterle. Spin domains in ground-state Bose–Einstein condensates. Nature,
396:345–348, 1998.
[179] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri,and L. Santos. Observation of dipole-dipole interaction in a degenerate quantumgas. Phys. Rev. Lett., 95:150406, Oct 2005.
[180] Juan Sánchez-Baena, Claudia Politi, Fabian Maucher, Francesca Ferlaino, andThomas Pohl. Heating a quantum dipolar fluid into a solid. arXiv:2209.00335,
2022.
[181] Yijun Tang, Andrew Sykes, Nathaniel Q. Burdick, John L. Bohn, and Benjamin L.
Lev. s-wave scattering lengths of the strongly dipolar bosons 162 Dy and 164 Dy.
Phys. Rev. A, 92:022703, Aug 2015.
[182] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset,
L. Santos, and G. Modugno. Observation of a Dipolar Quantum Gas with
Metastable Supersolid Properties. Phys. Rev. Lett., 122:130405, Apr 2019.
[183] L Tanzi, JG Maloberti, G Biagioni, A Fioretti, C Gabbanini, and G Modugno.
Evidence of superfluidity in a dipolar supersolid from nonclassical rotational
inertia. Science, 371(6534):1162–1165, 2021.
[184] L Tanzi, SM Roccuzzo, E Lucioni, F Famà, A Fioretti, C Gabbanini, G Modugno,
A Recati, and S Stringari. Supersolid symmetry breaking from compressional
oscillations in a dipolar quantum gas. Nature, 574(7778):382–385, 2019.
[185] M. Nilsson Tengstrand, D. Boholm, R. Sachdeva, J. Bengtsson, and S. M.
Reimann. Persistent currents in toroidal dipolar supersolids. Phys. Rev. A,
103:013313, Jan 2021.
[186] I. Titvinidze, M. Snoek, and W. Hofstetter. Supersolid bose-fermi mixtures in
optical lattices. Phys. Rev. Lett., 100:100401, Mar 2008.
[187] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M. Sohmen, M. J. Mark,and F. Ferlaino. Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms.
Phys. Rev. Lett., 121:213601, Nov 2018.
[188] Andrew G. Truscott, Kevin E. Strecker, William I. McAlexander, Guthrie B.
Partridge, and Randall G. Hulet. Observation of fermi pressure in a gas oftrapped atoms. Science, 291(5513):2570–2572, 2001.
[189] F. Wächtler and L. Santos. Ground-state properties and elementary excitations ofquantum droplets in dipolar bose-einstein condensates. Phys. Rev. A, 94:043618,
Oct 2016.
[190] F. Wächtler and L. Santos. Quantum filaments in dipolar bose-einstein condensates. Phys. Rev. A, 93:061603, Jun 2016.
[191] Reuben R. W. Wang and John L. Bohn. Anisotropic thermalization of dilutedipolar gases. Phys. Rev. A, 103:063320, Jun 2021.

Bibliography149[192] Chad N Weiler, Tyler W Neely, David R Scherer, Ashton S Bradley, Matthew J
Davis, and Brian P Anderson. Spontaneous vortices in the formation of boseeinstein condensates. Nature, 455(7215):948–951, 2008.
[193] Matthias Wenzel. A dysprosium quantum gas in highly controllable optical traps.
Master’s thesis, Physikalisches Institut; Universität Stuttgart, 2015.
[194] Kali E. Wilson, Alexander Guttridge, Jack Segal, and Simon L. Cornish. Quantum
degenerate mixtures of cs and yb. Phys. Rev. A, 103:033306, Mar 2021.
[195] Ryan M. Wilson, Christopher Ticknor, John L. Bohn, and Eddy Timmermans.
Roton immiscibility in a two-component dipolar bose gas. Phys. Rev. A, 86:033606,
Sep 2012.
[196] M. J. Wright, S. Riedl, A. Altmeyer, C. Kohstall, E. R. Sánchez Guajardo,
J. Hecker Denschlag, and R. Grimm. Finite-temperature collective dynamics ofa fermi gas in the bec-bcs crossover. Phys. Rev. Lett., 99:150403, Oct 2007.
[197] Kui-Tian Xi, Tim Byrnes, and Hiroki Saito. Fingering instabilities and patternformation in a two-component dipolar bose-einstein condensate. Phys. Rev. A,
97:023625, Feb 2018.
[198] Kui-Tian Xi and Hiroki Saito. Droplet formation in a bose-einstein condensate
with strong dipole-dipole interaction. Phys. Rev. A, 93:011604, Jan 2016.
[199] C. N. Yang. Concept of off-diagonal long-range order and the quantum phases ofliquid he and of superconductors. Rev. Mod. Phys., 34:694–704, Oct 1962.
[200] Heshmatollah Yavari and Forouzan Forouharmanesh. Effects of thermally induced
roton-like excitation on the superfluid density of a quasi-2d dipolar bose condensed
gas. Annalen der Physik, 532(6):1900412, 2020.
[201] Zhu-Xiong Ye, A. Canali, E. Soave, M. Kreyer, Y. Yudkin, C. Ravensbergen,
E. Kirilov, and R. Grimm. Observation of low-field feshbach resonances between
161 dy and 40 k. arXiv:2207.03407, 2022.
[202] Zhu-Xiong Ye, Li-Yang Xie, Zhen Guo, Xiao-Bin Ma, Gao-Ren Wang, Li You,and Meng Khoon Tey. Double-degenerate bose-fermi mixture of strontium andlithium. Phys. Rev. A, 102:033307, Sep 2020.
[203] Johannes Zeiher, Rick van Bijnen, Peter Schauß, Sebastian Hild, Jae-yoon Choi,
Thomas Pohl, Immanuel Bloch, and Christian Gross. Many-body interferometryof a Rydberg-dressed spin lattice. Nature Physics, 12(12):1095–1099, 2016.
[204] Yong-Chang Zhang, Fabian Maucher, and Thomas Pohl. Supersolidity around a
critical point in dipolar bose-einstein condensates. Phys. Rev. Lett., 123:015301,
Jul 2019.
[205] Martin W Zwierlein, Jamil R Abo-Shaeer, Andre Schirotzek, Christian H Schunck,and Wolfgang Ketterle. Vortices and superfluidity in a strongly interacting fermi
gas. Nature, 435(7045):1047–1051, 2005.

150

Bibliography[206] Marija Šindik, Alessio Recati, Santo Maria Roccuzzo, Luis Santos, and Sandro
Stringari. Creation and robustness of quantized vortices in a dipolar supersolidwhen crossing the superfluid-to-supersolid transition. arXiv:2206.14100, 2022.

Appendix A
Additional publications

152

A.1

Additional publications

Publication:

Dipolar quantum mixtures of erbium and dysprosium
atoms
The following publication has appeared in
Physical Review Letters 121, 213601 (2018)∗
submitted 20 July 2018; published 21 Nov 2018
DOI: https://doi.org/10.1103/PhysRevLett.121.213601
A. Trautmann,1,3 P. Ilzhöfer,1,2,3 G. Durastante,1,2 C. Politi,1,2 M. Sohmen,1,2 M. J. Mark,1,2
and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 These authors contributed equally: A. T. and P. I.

∗The author of the present thesis contributed in performing the measurements together with

A. T., P. I., G. D., and M. S., analysed the experimental data with A. T., P. I., G. D., and M. S., andcontributed in writing the manuscript and interpreting the results together with all the authors.

PHYSICAL REVIEW LETTERS 121, 213601 (2018)
Editors' Suggestion

Featured in Physics

Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms
1

A. Trautmann,1,* P. Ilzhöfer,1,2,* G. Durastante,1,2 C. Politi,1 M. Sohmen,1,2 M. J. Mark,1,2 and F. Ferlaino1,2

Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria(Received 20 July 2018; revised manuscript received 8 October 2018; published 21 November 2018)
We report on the first realization of heteronuclear dipolar quantum mixtures of highly magnetic erbium
and dysprosium atoms. With a versatile experimental setup, we demonstrate binary Bose-Einsteincondensation in five different Er-Dy isotope combinations, as well as one Er-Dy Bose-Fermi mixture.
Finally, we present first studies of the interspecies interaction between the two species for one mixture.
DOI: 10.1103/PhysRevLett.121.213601

In recent years, the field of atomic dipolar quantum gases
has witnessed an impressive expansion as researchers havemade substantial headway in using and controlling a novel
class of atoms, the highly magnetic rare-earth species.
Since the first experimental successes in creating Bose andFermi quantum gases of Dy [1,2] or Er [3,4], fascinating
many-body phenomena based on the dipole-dipole interaction (DDI) have been observed, including Fermi surface
deformation [5], quantum-stabilized droplet states [6–8],
and roton quasiparticles [9]. Remarkably, for Dy and Er, theintriguing physics within reach comes with comparatively
simple experimental approaches to achieve quantum degeneracy. Several research groups have either recently reported
on new experimental realizations of quantum gases with Dy[10,11] or Er [12] or are actively pursuing it [13,14].
An alternative route to access dipolar quantum physics isprovided by polar molecules, possessing an electric dipole
moment. Up to now, ultracold gases of polar molecules
have been created from nondipolar binary quantum mixtures of alkali atoms [15–18] and dipolar spin-exchangeinteractions have been recently observed with latticeconfined molecules [19]. Besides molecule creation, heteronuclear quantum mixtures have been used as powerful
resources to realize a broad class of many-body quantum
states (e.g., [20–27]), in which intra- and interspecies shortrange contact interactions are at play.
In the experiment described in this Letter, we merge forthe first time the physics of heteronuclear mixtures withthe one of magnetic dipolar quantum gases. Our motivations to create quantum mixtures by combining two different dipolar species, Er and Dy, are numerous. First, thecoupling between the two components acquires an anisotropic and long-range character due to the strong interspecies DDI, in contrast to purely contact-interacting
mixtures. The emergent physical richness of the system
has only begun to be uncovered by theory. Recent studies
include the prediction of anisotropic boundaries in the
dipolar immiscibility-miscibility phase diagram [28,29],0031-9007=18=121(21)=213601(6)roton immiscibility [30], vortex lattice formation [31], andimpurity physics both in dipolar quantum droplets [32] anddipolar Binary Bose-Einstein condensates (BECs) [33,34].
Moreover, the magnetic moments are large, yet different(7 bohr magneton for Er and 10 for Dy), leading to a DDI
twice as strong in Dy as in Er. Such a difference is on one
hand advantageous to deeper elucidate the complex scattering and many-body physics by performing comparative
single-species studies with Er and Dy in the same experimental environment. On the other hand, we also anticipate
(a)

(b)

(c)

(d)

(e)

FIG. 1. Binary Bose-Einstein condensation in a 166Er-164Dy
mixture. (a)–(c) Pairs of TOF absorption images at different
evaporation stages, showing (a) a thermal mixture at about
180 nK, (b) an Er cloud at the onset of condensation coexistingwith a thermal Dy gas at about 80 nK, and (c) the binary dipolar
BEC with total atom numbers N ¼ 3.4 × 104 ð2.6 × 104 Þ forEr(Dy) with condensate fractions of about 45%. x̃ denotes thehorizontal axis perpendicular to the imaging axis. (d),(e) Density
profiles integrated along z, extracted from (c). Solid lines
depict the 1D bimodal fit, the dotted lines show Gaussian fits
to the thermal components. (f)–(h) BECs with a controlled
number imbalance giving about N Er ¼ ð3.2; 6.4; 9.2Þ × 104 with
(35, 70, 85)% condensate fraction and N Dy ¼ ð3.1; 2.9; 0.9Þ ×
104 with (30, 55, 30)% condensate fraction for (f)–(h), respectively. The deformations and the relative displacement of theclouds are caused by interspecies interaction (see main text). The
color bar indicates the optical density.

213601-1© 2018 American Physical Society

PHYSICAL REVIEW LETTERS 121, 213601 (2018)
that, in mixture experiments, the imbalance in dipolar
strength, combined with the interspecies interactions,promises fascinating prospects for creating long-lived
quantum-droplet states and for accessing exotic fermionicsuperfluidity, for which the degree of deformation of the
Fermi surface is species-dependent [5].
Second, the rich, but different atomic spectra of Er andDy open promising prospects for implementing speciesdependent optical manipulations schemes, including species-selective optical potentials at magic wavelengths andcheckerboard-pattern-like lattice structures. Third, Er and
Dy feature many stable bosonic and fermionic isotopes(both elements have four isotopes with natural abundances
above 15%). Such an isotope variety allows us to create
dipolar Bose-Bose, Bose-Fermi, and Fermi-Fermi heteronuclear mixtures. Last, mixtures composed of two different
magnetic species serve as an ideal platform to produce
ground-state polar molecules with both an electric andmagnetic dipole moment, offering novel degrees of control
and competing long-range interactions [35–37].
We here report on the first experimental realization ofquantum-degenerate dipolar mixtures of Er and Dy atoms,
using an all-optical approach for trapping and cooling.
Taking advantage of the isotope richness in Er and Dy, we
produce dipolar Bose-Bose mixtures with five different
isotope combinations, as well as one Bose-Fermi mixture.
We note that, prior to this work, the production of a Dy
isotope mixture of a degenerate Fermi gas and a Bose gas
near condensation has been reported [2] and, more recently,
a doubly degenerate Fermi-Fermi mixture has been createdfrom two Er spin states [39]. Experimental efforts are also
devoted to creating Dy-K mixtures [40].
In the following, we detail the production of a double
dipolar Bose-Einstein condensate (ddBEC) of 166Er and164
Dy. The same procedure is used for the other isotope
mixtures. Our experiment starts with a double magnetooptical trap (MOT) of Er and Dy, as reported in our recent
work [41]. For both species, the MOT operates on narrowintercombination lines and yields cold and spin-polarized
samples in the absolute lowest Zeeman sublevels[10,13,41,42]. After loading the double MOT, we opticallycompress the mixture in 400 ms (cMOT phase) by reducing
the detuning and power of the MOT beams as well as themagnetic-field gradient.
We then transfer the mixture into an optical dipole trap
(ODT) by superposing it with the cMOT for 100 ms.
Initially, the ODT consists of a single laser beam at
1064 nm, propagating along the horizontal (y) axis. The
beam has a fixed vertical (z) focus of about 22 μm, whereas
the horizontal waist can be controlled via a time-averagingpotential technique (see, e.g., [43]). This leads to an elliptic
beam with variable aspect ratio (AR). Best transfer efficiency
is observed for a beam power of 32 W and an AR of 4, which
provides good spatial overlap between the cMOT and the
ODT. We then switch off the MOT beams and magnetic-field

gradient, and start a 5-sec evaporation sequence, during
which we apply a bias magnetic field Bevap along thegravity (z) axis to preserve spin polarization.
Our strategy for evaporative cooling can be divided into
three main stages. (i) During the initial 600 ms, we reduce
the AR to unity while lowering the power of the singlebeam ODT. This increases the density of the mixture at aroughly constant trap depth. (ii) We start forced evaporationin the horizontal ODT and add a vertically propagating
dipole trap beam. The vertical beam is derived from thesame laser source as the horizontal one and has a power of
15 W and a waist of 130 μm. (iii) We proceed with forced
evaporation in the crossed ODT by reducing the powers ofboth beams nearly exponentially until the mixture is close
to quantum degeneracy. In the final stage of the evaporation, we increase the AR to 5 to create a pancakelike
trapping geometry and further decrease the trap depth until
we reach double quantum degeneracy. To probe the atomic
mixture, we switch off the ODT and, after a time-of-flight(TOF) expansion of 25 ms, we perform sequential absorption imaging with a resonant light pulse at 401 nm for Er
and 500 μs later at 421 nm for Dy [3,44]; both pulses have aduration of 50 μs. The imaging light propagates horizontally with an angle of 45° with respect to the y axis.
Unlike many alkali mixtures [45–48], Er and Dy exhibit
very comparable atomic polarizabilities α because of theirsimilar atomic spectra. From single-species experiments[14,49], a ratio αDy =αEr ¼ 1.06 at 1064 nm is expected. For
our initial ODT parameters, we calculate trap frequencies
of about νEr ¼ ð490; 5; 1980Þ Hz and νDy ¼ ð505; 5;2050Þ Hz [50], corresponding to trap depths of 380 and
410 μK for Er and Dy, respectively. Although small, thedifference in trap depths has an important effect on theevaporation trajectory of the mixture. We observe thatthe more weakly trapped Er atoms act as a coolant for Dy
and are preferentially evaporated from the trap (“sympathetic losses” [47,51]). To sustain Er atom numbers highenough to achieve double quantum degeneracy, we imbalance the initial atom number in the MOT with Er as the
majority component. The atom number imbalance can beeasily controlled by individually changing the MOT loading time and beam power. This strategy is often employedin multispecies experiments, e.g., [52,53].
Figures 1(a)–1(c) show the phase transition from athermal Er-Dy mixture (a) to a ddBEC (c). The TOF
absorption images reveal the textbooklike fingerprint ofcondensation, the emergence of a bimodal density distribution, as plotted in Fig. 1(d). The condensation series[Figs. 1(a)–1(c)] is taken for an Er(Dy) MOT loading timeof 3 s (1 s), for which we transfer 8 × 106 (7 × 105 ) Er(Dy)atoms into our ODT and measure a temperature of about
35 μK; this parameter set allows us to create numberbalanced ddBECs. In agreement with the expected polarizabilities, we measure ODT trap frequencies of νEr ¼(48.6ð3Þ;29.7ð9Þ;144ð1Þ) Hz and νDy ¼ (50.6ð3Þ; 30.2ð9Þ;

213601-2

PHYSICAL REVIEW LETTERS 121, 213601 (2018)160ð1Þ) Hz [54]. The resultant gravitational sag between
the two species is 2.1ð2Þ μm. By varying the imbalance of the
MOT loading, we can produce degenerate mixtures withdifferent atom number ratios and condensate fractions, whichis exemplified in Figs. 1(f)–1(h). For large condensates, one
directly observes a deformation of the density profiles due tointerspecies interaction, as we discuss later in more detail.
To quantify the cooling efficiency, we plot the normalized phase-space density (PSD=PSD0 ) as a function of
normalized atom numbers (N=N 0 ) during the evaporation
stages ii and iii [see Fig. 2(a)]. PSD0 and N 0 are therespective initial values at stage ii. From this plot, we
extract γ ¼ −d lnðPSD=PSD0 Þ=d lnðN=N 0 Þ [55], which
captures the evaporation efficiency, via a linear fit to thedata. In the single-beam ODT (stage ii), we see similar
efficiencies both in mixture and single-species operations,with γ ≈ 1.2. In the crossed ODT (stage iii), we find γ Er ¼
2.4ð9Þ for Er in the mixture operation. This value iscomparable to state-of-the-art single-species Er experiments [56] and, as expected, little affected by a small
admixture of Dy atoms. Contrarily, the cooling efficiency
of Dy in stage (iii) strongly benefits from the sympathetic
cooling by Er: We observe a steep increase of the Dy PSD
in the mixture and extract γ Dy
sym ¼ 7ð2Þ, whereas for the

(a)

(b)

FIG. 2. (a) Evaporation trajectories: PSD=PSD0 as a function of
N=N 0 . Filled squares (circles) indicate the Er(Dy) trajectory in
mixture operation. The lines are linear fits to the data forevaporation in the single-beam (ii) and crossed-beam (iii) ODT(see main text). Open symbols show the single-species operation
for Er (squares) and for Dy with small (circles) and large(diamonds) initial atom numbers. In the latter case, Dy condenses
alone. (b) Atom numbers in the mixture of Er (red) and Dy (blue)at the onset of condensation as a function of the magnetic-fieldvalue during evaporation. Condensation is reached for atom
numbers above about 3.5 × 104 (gray region). We record the best
performance for a ddBEC around 2.075 G. Arrows indicate theposition of known single-species Feshbach resonances [57–59].

same N Dy
0 but in single-species operation, the evaporation
efficiency is considerably lower and would not suffice forcondensation. However, with higher N Dy
we can still0
produce large Dy BECs in single-species operation.
The proper choice of Bevap plays an important role forcooling magnetic rare-earth atoms and becomes even morecritical in mixture operation. It has indeed been observed insingle-species experiments [57–59] that both Er and Dy
exhibit extremely dense and temperature-dependent spectraof homonuclear Feshbach resonances. Figure 2(b) shows
the atom numbers of the 166Er-164Dy mixture at the onset ofcondensation as a function of Bevap in a small magneticfield range from 0.5 to 5 G. As expected, we find a numberof broad and narrow loss features. Some of them areconnected to known homonuclear Feshbach resonances[57–59], others we attribute to unknown high-temperatureresonances or detrimental interspecies scattering conditions. In a few narrow magnetic-field windows, we observe
atom numbers large enough for both components tocondense. Our magnetic-field stability of about 2 mG issufficient to reliably operate in most of these small
windows. The optimal value of Bevap , listed in Table I,
depends on the isotope combination.
Combining Er and Dy offers an unprecedented variety ofheteronuclear mixtures with 16 possible isotope configurations, including Bose-Bose, Bose-Fermi, and FermiFermi quantum gases (see Table I). Using the coolingand trapping procedure optimized for 166Er-164Dy, we areable to produce five ddBECs and one Bose-Fermi mixture.
Concerning the remaining combinations, we know fromprevious experiments that both 167Er and 163Dy need adifferent experimental approach since 167Er undergoes
light-induced losses in a 1064-nm ODT [4], whereas
163Dy, never brought to quantum degeneracy so far, has
an inverted hyperfine structure, requiring most probably
additional optical pumping stages. Both isotopes will be
investigated for future studies of Fermi-Fermi mixtures.
Figures 3(a)–3(d) show absorption pictures of ourdoubly degenerate isotope mixtures. We are able to
TABLE I. (Left) List of optimal Bevap and γ Dy
sym for the quantumdegenerate Er-Dy mixtures. (Right) Chart of the available isotope
mixtures: ð✓Þ realized double-degenerate mixtures, ð⨯Þ thermal
mixtures, where degeneracy is not yet reached. (…) Mixtureswith 167Er and 163Dy are not investigated here.

213601-3

PHYSICAL REVIEW LETTERS 121, 213601 (2018)
(a)
(a)

(b)

(c)

(d)

(b)

(e)
(d)
(e)

FIG. 3. Absorption pictures of the double-degenerate BoseBose mixtures (a)–(d) and the Bose-Fermi mixture (e). The
pictures are averaged over 5–10 single shots. For all combinations, degeneracy is reached with the evaporation ramp
optimized for the 166Er-164Dy mixture (cf. Fig. 1). Bevap islisted in Table I. Typical condensate fractions are around 30%,
total atom numbers range between 1 × 104 and 3.5 × 104 atoms.
For the imbalanced case, higher condensate fractions can beachieved (see Fig. 1). For the 161Dy Fermi gas, N ¼ 8 × 103 ,T=T F ≈ 0.5, and TOF ¼ 15 ms.

condense all Bose-Bose isotope mixtures with the exception of 166Er-162Dy, for which we record severe losses
during the evaporation, potentially due to a very large
interspecies scattering length. For all degenerate mixtures,
we observe sympathetic cooling of Dy by Er. The atom
numbers in the ddBECs differ significantly for the different
mixtures, while the initial atom numbers in the MOT arevery similar. This points to different intra- and interspecies
scattering properties during evaporation. The optimal Bevap
and the extracted γ Dy
sym are listed in Table I.
We also prepare one Bose-Fermi mixture [see Fig. 3(e)],in which a 168Er BEC coexists with a degenerate Fermi gasof 161Dy. Although the cooling process of spin-polarized
fermions can differ substantially from bosons, we are able
to reach Bose-Fermi degeneracy with a similar evaporation
scheme [60]. We measure a temperature of the Fermi gas of
T=T F ≈ 0.5, with the Fermi temperature T F ¼ 140 nK. We
expect that deeper degeneracy might be reached by using
smaller ODT beam waists [4].
Remarkably, in the TOF images in Figs. 1 and 3 hints ofinterspecies interactions can be spotted: in mixture operation, the center-of-mass (c.m.) position of each BEC isvertically displaced with respect to its thermal-cloud center[see also Fig. 4(a)]. The two BECs are displaced in opposite
directions, with the heavier (lighter) Er(Dy) always shifted
down (up). Contrarily, in single-species operation thecondensates and their thermal clouds are centered [seeFigs. 4(b) and 4(c)].
To confirm that the displacement after TOF originatesfrom in-trap interspecies interaction, we prepare a ddBEC,
let it equilibrate for 50 ms, and then selectively remove
either of the two species from the ODT using a resonant
light pulse [61]. After a variable hold time in the ODT, werelease the remaining cloud and record its c.m. position
after TOF. As shown in Figs. 4(d) and 4(e), we observe a(c)FIG. 4. Evidence of interspecies interactions in the 166Er-164Dy
mixture: absorption pictures of Er and Dy in mixture (a) andsingle-species (b),(c) operation. (d),(e) Filled symbols show thec.m. position along z of the Dy BEC (d) and the Er BEC (e) after
removal of the other species with resonant light. The gray region
indicates the transient time until full removal. The solid lines aredamped sine fits to the oscillations. For comparison, open
symbols show the c.m. position in a thermal mixture.

very pronounced c.m. oscillation of the remaining BEC
component with a frequency close to its bare trap frequency.
The oscillations of Er [removing Dy, Fig. 4(d)] and of Dy[removing Er, Fig. 4(e)] proceed in counterphase, as
expected from their initial separation in trap. Repeating
the same measurement with a thermal-thermal mixture,or a mixture with just one condensed component (notshown), yields negligible or significantly weaker oscillations, respectively.
The spatial separation between the two condensed
components and their oscillating behavior after removal
indicate that, for our trap geometry, the overall interspecies
interaction—contact plus dipolar—has a repulsive character. We note that the interspecies scattering length, governing the contact interaction, and its Feshbach tuning arepresently unknown and beyond reach of state-of-the-art
scattering models [57,62]. To isolate the different sourcesof interaction and determine their signs, future dedicatedexperiments studying the interplay between trap geometry,
dipole orientation, and interspecies scattering length, combined with simulations based on generalized coupled
Gross-Pitaeskvii equations are necessary. Indeed, theDDI breaks the angular symmetry of the mean-fieldinterspecies potentials and is expected to render thestrength and the sign of the overall interspecies interaction
anisotropic and trap dependent.
In conclusion, we have produced heteronuclear dipolar
quantum mixtures by combining two strongly magnetic
atomic species, Er and Dy. Their isotope variety, therichness of their interactions, the imbalance in the dipolar
strength, and simple laser-cooling schemes make Er-Dy
mixtures a powerful experimental platform to access manybody quantum phenomena, in which contact and dipolar
intra- and interspecies interactions are at play.
This work is supported by the ERC Consolidator Grant(RARE, No. 681432) and a NFRI Grant (MIRARE,

213601-4

PHYSICAL REVIEW LETTERS 121, 213601 (2018)
No. ÖAW0600) from the Austrian Academy of Science.
G. D. and M. S. acknowledge support by the Austrian
Science Fund FWF within the DK-ALM: W1259-N27.
We thank the ERBIUM team and the Dy-K team in
Innsbruck and the ERBIUM team of Markus Greiner for
fruitful discussions.

*
These authors contributed equally to this work.
[1] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Strongly
Dipolar Bose-Einstein Condensate of Dysprosium, Phys.
Rev. Lett. 107, 190401 (2011).
[2] M. Lu, N. Q. Burdick, and B. L. Lev, Quantum Degenerate
Dipolar Fermi Gas, Phys. Rev. Lett. 108, 215301 (2012).
[3] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R.
Grimm, and F. Ferlaino, Bose-Einstein Condensation of
Erbium, Phys. Rev. Lett. 108, 210401 (2012).
[4] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm, and F.
Ferlaino, Reaching Fermi Degeneracy via Universal Dipolar
Scattering, Phys. Rev. Lett. 112, 010404 (2014).
[5] K. Aikawa, S. Baier, A. Frisch, M. Mark, C. Ravensbergen,and F. Ferlaino, Observation of Fermi surface deformation
in a dipolar quantum gas, Science 345, 1484 (2014).
[6] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I.
Ferrier-Barbut, and T. Pfau, Observing the Rosensweig
instability of a quantum ferrofluid, Nature (London) 530,
194 (2016).
[7] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L.
Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a
Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6,041039 (2016).
[8] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, andT. Pfau, Self-bound droplets of a dilute magnetic quantum
liquid, Nature (London) 539, 259 (2016).
[9] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S.
Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino, Observation of roton mode population in a
dipolar quantum gas, Nat. Phys. 14, 442 (2018).
[10] T. Maier, H. Kadau, M. Schmitt, A. Griesmaier, and T. Pfau,
Narrow-line magneto-optical trap for dysprosium atoms,
Opt. Lett. 39, 3138 (2014).
[11] E. Lucioni, L. Tanzi, A. Fregosi, J. Catani, S. Gozzini, M.
Inguscio, A. Fioretti, C. Gabbanini, and G. Modugno,
Dysprosium dipolar Bose-Einstein condensate with broad
feshbach resonances, Phys. Rev. A 97, 060701 (2018).
[12] J. Ulitzsch, D. Babik, R. Roell, and M. Weitz, Bose-Einsteincondensation of erbium atoms in a quasielectrostatic optical
dipole trap, Phys. Rev. A 95, 043614 (2017).
[13] D. Dreon, L. A. Sidorenkov, C. Bouazza, W. Maineult, J.
Dalibard, and S. Nascimbene, Optical cooling and trappingof highly magnetic atoms: The benefits of a spontaneous
spin polarization, J. Phys. B 50, 065005 (2017).
[14] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, S.
Tzanova, E. Kirilov, and R. Grimm, Accurate Determinationof the Dynamical Polarizability of Dysprosium, Phys. Rev.
Lett. 120, 223001 (2018).
[15] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B.
Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S.

Jin, and J. Ye, A high phase-space-density gas of polar
molecules, Science 322, 231 (2008).
[16] T. Takekoshi, L. Reichsöllner, A. Schindewolf, J. M. Hutson,
C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm, and H.-C.
Nägerl, Ultracold Dense Samples of Dipolar RBCS Molecules in the Rovibrational and Hyperfine Ground State, Phys.
Rev. Lett. 113, 205301 (2014).
[17] P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger,
C. R. Le Sueur, C. L. Blackley, J. M. Hutson, and S. L.
Cornish, Creation of Ultracold 87Rb133Cs Molecules in the
Rovibrational Ground State, Phys. Rev. Lett. 113, 255301(2014).
[18] S. Moses, J. Covey, M. Miecnikowski, D. Jin, and J. Ye,
New frontiers for quantum gases of polar molecules, Nat.
Phys. 13, 13 (2016).
[19] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A.
Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Observation of
dipolar spin-exchange interactions with lattice-confinedpolar molecules, Nature (London) 501, 521 (2013).
[20] T.-L. Ho and V. B. Shenoy, Binary Mixtures of Bose
Condensates of Alkali Atoms, Phys. Rev. Lett. 77, 3276
(1996).
[21] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst,
K. Sengstock, and K. Bongs, Localization of BosonicAtoms by Fermionic Impurities in a Three-Dimensional
Optical Lattice, Phys. Rev. Lett. 96, 180403 (2006).
[22] S. Will, T. Best, S. Braun, U. Schneider, and I. Bloch,
Coherent Interaction of a Single Fermion with a Small
Bosonic Field, Phys. Rev. Lett. 106, 115305 (2011).
[23] J. Heinze, S. Götze, J. S. Krauser, B. Hundt, N. Fläschner,
D.-S. Lühmann, C. Becker, and K. Sengstock, Multiband
Spectroscopy of Ultracold Fermions: Observation of Reduced Tunneling in Attractive Bose-Fermi Mixtures, Phys.
Rev. Lett. 107, 135303 (2011).
[24] N. Spethmann, F. Kindermann, S. John, C. Weber, D.
Meschede, and A. Widera, Dynamics of Single Neutral
Impurity Atoms Immersed in an Ultracold Gas, Phys. Rev.
Lett. 109, 235301 (2012).
[25] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M.
Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, andJ. J. Arlt, Observation of Attractive and Repulsive Polarons
in a Bose-Einstein Condensate, Phys. Rev. Lett. 117,055302 (2016).
[26] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A.
Cornell, and D. S. Jin, Bose Polarons in the Strongly
Interacting Regime, Phys. Rev. Lett. 117, 055301 (2016).
[27] T. Rentrop, A. Trautmann, F. A. Olivares, F. Jendrzejewski,
A. Komnik, and M. K. Oberthaler, Observation of the
Phononic Lamb Shift with a Synthetic Vacuum, Phys.
Rev. X 6, 041041 (2016).
[28] G. Gligorić, A. Maluckov, M. Stepić, L. Hadžievski, and
B. A. Malomed, Transition to miscibility in linearly coupledbinary dipolar Bose-Einstein condensates, Phys. Rev. A 82,
033624 (2010).
[29] R. K. Kumar, P. Muruganandam, L. Tomio, and A. Gammal,
Miscibility in coupled dipolar and non-dipolar Boseeinsteincondensates, J. Phys. Commun. 1, 035012 (2017).
[30] R. M. Wilson, C. Ticknor, J. L. Bohn, and E. Timmermans,
Roton immiscibility in a two-component dipolar Bose gas,
Phys. Rev. A 86, 033606 (2012).

213601-5

PHYSICAL REVIEW LETTERS 121, 213601 (2018)[31] R. K. Kumar, L. Tomio, B. A. Malomed, and A. Gammal,
Vortex lattices in binary Bose-Einstein condensates with
dipole-dipole interactions, Phys. Rev. A 96, 063624 (2017).
[32] M. Wenzel, T. Pfau, and I. Ferrier-Barbut, A fermionic
impurity in a dipolar quantum droplet, Phys. Scr. 93,
104004 (2018).
[33] B. Kain and H. Y. Ling, Polarons in a dipolar condensate,
Phys. Rev. A 89, 023612 (2014).
[34] L. A. P. Ardila and T. Pohl, Ground-state properties of
dipolar bose polarons, arXiv:1804.06390.
[35] E. Abrahamsson, T. V. Tscherbul, and R. V. Krems, Quantum chaos in ultracold collisions of gas-phase erbium atoms,
J. Chem. Phys. 127, 044302 (2007).
[36] T. M. Rvachov, H. Son, A. T. Sommer, S. Ebadi, J. J. Park,
M. W. Zwierlein, W. Ketterle, and A. O. Jamison, Observation of Low-Field Fano-Feshbach Resonances in Ultracold Gases of Dysprosium, Phys. Rev. Lett. 119, 143001(2017).
[37] The electric dipole moment of ErDy is currently unknown;
Ref. [38] reports the calculation for another rare-earth
combination, EuYb, with −0.13 D. Based on electronegativity arguments, a similar value can be expected in ErDy.
[38] M. Tomza, Ab initio properties of the ground-state polar andparamagnetic europium–alkali-metal-atom and 5 europium–alkaline-earth-metal-atom molecules, Phys. Rev. A 90,
022514 (2014).
[39] S. Baier, D. Petter, J. H. Becher, A. Patscheider, G. Natale,
L. Chomaz, M. J. Mark, and F. Ferlaino, Realization of aStrongly Interacting Fermi Gas of Dipolar Atoms, Phys.
Rev. Lett. 121, 093602 (2018).
[40] R. Grimm (private communication).
[41] P. Ilzhöfer, G. Durastante, A. Patscheider, A. Trautmann,
M. J. Mark, and F. Ferlaino, Phys. Rev. A 97, 023633(2018).
[42] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler, E.
Zupanič, R. Grimm, and F. Ferlaino, Narrow-line magnetooptical trap for erbium, Phys. Rev. A 85, 051401 (2012).
[43] S. Baier, An optical dipole trap for Erbium with tunablegeometry, Master’s thesis, University of Innsbruck, 2012.
[44] M. Lu, S. H. Youn, and B. L. Lev, Trapping Ultracold
Dysprosium: A Highly Magnetic Gas for Dipolar Physics,
Phys. Rev. Lett. 104, 063001 (2010).
[45] G. Modugno, M. Modugno, F. Riboli, G. Roati, and M.
Inguscio, Two Atomic Species Superfluid, Phys. Rev. Lett.
89, 190404 (2002).
[46] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W.
Zwierlein, A. Görlitz, and W. Ketterle, Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases, Phys.
Rev. Lett. 88, 160401 (2002).
[47] M. Mudrich, S. Kraft, K. Singer, R. Grimm, A. Mosk, and
M. Weidemüller, Sympathetic Cooling with Two Atomic

Species in an Optical Trap, Phys. Rev. Lett. 88, 253001(2002).
[48] M. Gröbner, P. Weinmann, F. Meinert, K. Lauber, E. Kirilov,
and H.-C. Nägerl, A new quantum gas apparatus forultracold mixtures of K and Cs and KCs ground-state
molecules, J. Mod. Opt. 63, 1829 (2016).
[49] J. H. Becher, S. Baier, K. Aikawa, M. Lepers, J.-F. Wyart,
O. Dulieu, and F. Ferlaino, Anisotropic polarizability of
erbium atoms, Phys. Rev. A 97, 012509 (2018).
[50] We use the notation νEr;Dy ¼ ðνx ; νy ; νz Þ.
[51] A. Mosk, S. Kraft, M. Mudrich, K. Singer, W. Wohlleben,
R. Grimm, and M. Weidemüller, Mixture of ultracold
lithium and cesium atoms in an optical dipole trap, Appl.
Phys. B 73, 791 (2001).
[52] Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and B. L. Lev,s-wave scattering lengths of the strongly dipolar bosons
162
Dy and 164Dy, Phys. Rev. A 92, 022703 (2015).
[53] C.-H. Wu, I. Santiago, J. W. Park, P. Ahmadi, and M. W.
Zwierlein, Strongly interacting isotopic bose-fermi mixture
immersed in a fermi sea, Phys. Rev. A 84, 011601 (2011).
[54] The trap frequencies along the y and z axes are extracted
from center-of-mass-oscillation measurement; the frequency along x is calculated based on the beam waist,
calibrated via the z frequency, and our knowledge of thepolarizability [14,49].
[55] W. Ketterle and N. V. Druten, in Evaporative Cooling ofTrapped Atoms, edited by B. Bederson and H. Walther,
Advances in Atomic, Molecular, and Optical Physics Vol. 37(Academic Press, New York, 1996), pp. 181–236.
[56] A. Frisch, Dipolar quantum gases of erbium, Ph.D. thesis,
University of Innsbruck, 2014.
[57] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C.
Makrides, A. Petrov, and S. Kotochigova, Nature (London)
507, 475 (2014).
[58] K. Baumann, N. Q. Burdick, M. Lu, and B. L. Lev,
Observation of low-field fano-feshbach resonances in ultracold gases of dysprosium, Phys. Rev. A 89, 020701 (2014).
[59] T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. FerrierBarbut, T. Pfau, A. Frisch, S. Baier, K. Aikawa, L. Chomaz,
M. J. Mark, F. Ferlaino, C. Makrides, E. Tiesinga, A. Petrov,and S. Kotochigova, Emergence of Chaotic Scattering in
Ultracold Er and Dy, Phys. Rev. X 5, 041029 (2015).
[60] For the Bose-Fermi mixture, slightly different final trap
parameters are required, with typically νEr ¼ (95ð5Þ;
33ð1Þ; 120ð1Þ), νDy ¼ (106ð5Þ; 34ð1Þ; 142ð1Þ).
[61] The light pulse at the respective imaging transition has aduration of 1 ms. We have checked that the resonant light of
one species does not affect the other.
[62] S. Kotochigova, Controlling interactions between highly
magnetic atoms with feshbach resonances, Rep. Prog. Phys.
77, 093901 (2014).

213601-6

A.2 Publication:
Long-lived and transient supersolid behaviors in dipolar quantum gases

A.2

159

Publication:

Long-lived and transient supersolid behaviors in dipolar quantum gases
The following publication has appeared in
Physical Review X 9, 021012 (2019)†
submitted 12 Mar 2019; published 19 April 2019
DOI: https://doi.org/10.1103/PhysRevX.9.021012
L. Chomaz,2 D. Petter,2 P. Ilzhöfer,1,2 G. Natale,2 A. Trautmann,1 C. Politi,1,2 G. Durastante,1,2
R. M. W. van Bijnen,1 A. Patscheider,2 M. Sohmen,1,2 M. J. Mark,1,2 and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria†The author of the present thesis analysed the data relative to the dysprosium experiments with

M. S., and M. J. M., and contributed in writing the manuscript and interpreting the results together
with all the authors.

PHYSICAL REVIEW X 9, 021012 (2019)
Featured in Physics

Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases
L. Chomaz,1 D. Petter,1 P. Ilzhöfer,2 G. Natale,1 A. Trautmann,2 C. Politi,2 G. Durastante,1,2 R. M. W. van Bijnen,2A. Patscheider,1 M. Sohmen,1,2 M. J. Mark,1,2 and F. Ferlaino1,2,*
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria(Received 11 March 2019; published 19 April 2019; corrected 25 April 2019)

By combining theory and experiments, we demonstrate that dipolar quantum gases of both 166Er andDy support a state with supersolid properties, where a spontaneous density modulation and a global
phase coherence coexist. This paradoxical state occurs in a well-defined parameter range, separating thephases of a regular Bose-Einstein condensate and of an insulating droplet array, and is rooted in the rotonmode softening, on the one side, and in the stabilization driven by quantum fluctuations, on the other side.
Here, we identify the parameter regime for each of the three phases. In the experiment, we rely on a detailed
analysis of the interference patterns resulting from the free expansion of the gas, quantifying both its
density modulation and its global phase coherence. Reaching the phases via a slow interaction tuning,
starting from a stable condensate, we observe that 166Er and 164Dy exhibit a striking difference in the
lifetime of the supersolid properties, due to the different atom loss rates in the two systems. Indeed, while in
166
Er the supersolid behavior survives only a few tens of milliseconds, we observe coherent density
modulations for more than 150 ms in 164Dy. Building on this long lifetime, we demonstrate an alternative
path to reach the supersolid regime, relying solely on evaporative cooling starting from a thermal gas.
164

DOI: 10.1103/PhysRevX.9.021012

Subject Areas: Atomic and Molecular Physics,
Condensed Matter Physics,
Quantum Physics

I. INTRODUCTION
Supersolidity is a paradoxical quantum phase of matter
where both crystalline and superfluid order coexist [1–3].
Such a counterintuitive phase, featuring rather antithetic
properties, has been originally considered for quantum
crystals with mobile bosonic vacancies, the latter being
responsible for the superfluid order. Solid 4He has long
been considered a prime system to observe such a phenomenon [4,5]. However, after decades of theoretical andexperimental efforts, an unambiguous proof of supersolidity in solid 4He is still missing [6,7].
In search of more favorable and controllable systems,
ultracold atoms emerged as a very promising candidate,
thanks to their highly tunable interactions. Theoreticalworks point to the existence of a supersolid ground state
in different cold-atom settings, including dipolar [8]*Corresponding author.
Francesca.Ferlaino@uibk.ac.at
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI.

2160-3308=19=9(2)=021012(12)and Rydberg particles [9,10], cold atoms with a softcore potential [11], or lattice-confined systems [7].
Breakthrough experiments with Bose-Einstein condensates(BECs) coupled to light have recently demonstrated a statewith supersolid properties [12,13]. While in these systems
indeed two continuous symmetries are broken, the crystal
periodicity is set by the laser wavelength, making thesupersolid incompressible.
Another key notion concerns the close relation between apossible transition to a supersolid ground state and theexistence of a local energy minimum at large momentum
in the excitation spectrum of a nonmodulated superfluid,
known as the roton mode [14]. Since excitations corresponding to a periodic density modulation at the roton
wavelength are energetically favored, the existence of thismode indicates the system’s tendency to crystallize [15]and it is predicted to favor a transition to a supersolid
ground state [4,5,9].
Remarkably, BECs of highly magnetic atoms, in which
the particles interact through the long-range and anisotropic
dipole-dipole interaction (DDI), appear to gather several
key ingredients for realizing a supersolid phase. First,
as predicted more than 15 years ago [16,17] and recently
demonstrated in experiments [18,19], the partial attraction
in momentum space due to the DDI gives rise to a roton

021012-1

Published by the American Physical Society

L. CHOMAZ et al.

PHYS. REV. X 9, 021012 (2019)minimum. The corresponding excitation energy, i.e., theroton gap, can be tuned in the experiments down tovanishing values. Here, the excitation spectrum softens
at the roton momentum and the system becomes unstable.
Second, there is a nontrivial interplay between the trap
geometry and the phase diagram of a dipolar BEC. For
instance, our recent observations have pointed out theadvantage of axially elongated trap geometries (i.e., cigar
shaped) compared to the typically considered cylindricallysymmetric ones (i.e., pancake shaped) in enhancing thevisibility of the roton excitation in experiments. Last but
not least, while the concept of a fully softened mode istypically related to instabilities and disruption of a coherent
quantum phase, groundbreaking works in the quantum-gas
community have demonstrated that quantum fluctuations
can play a crucial role in stabilizing a dipolar BEC [20–26].
Such a stabilization mechanism enables the existence,
beyond the mean-field instability, of a variety of stableground states, from a single macrodroplet [22,24,27] tostriped phases [28], and droplet crystals [29]; see also
related works [30–33]. For multidroplet ground states,
efforts have been devoted to understanding if a phasecoherence among ground-state droplets could be established [28,29]. However, previous experiments with 164Dy
have shown the absence of phase coherence across thedroplets [28], probably due to the limited atom numbers.
Droplet ground states, quantum stabilization, and dipolar
rotons have caused a huge amount of excitement with very
recent advancements adding key pieces of information to
the supersolid scenario. The quench experiments in an
166
Er BEC at the roton instability have revealed out-ofequilibrium modulated states with an early-time phasecoherence over a timescale shorter than a quarter of theoscillation period along the weak-trap axis [18]. In the same
work, it has been suggested that the roton softening
combined with the quantum stabilization mechanism
may open a promising route towards a supersolid groundstate. A first confirmation came from a recent theoreticalwork [34], considering an Er BEC in an infinite elongatedtrap with periodic boundary conditions and tight transverse
confinement. The supersolid phase appears to exist within anarrow region in interaction strength, separating a roton
excitation with a vanishing energy and an incoherent
assembly of insulating droplets. Almost simultaneously,
experiments with 162Dy BECs in a shallow elongated trap,
performing a slow tuning of the contact interaction,
reported on the production of stripe states with phasecoherence persisting up to half of the weak trapping period[35]. More recently, such observations have been confirmed in another 162Dy experiment [36]. Here, theoretical
calculations showed the existence of a phase-coherent
droplet ground state, linking the experimental findings to
the realization of a state with supersolid properties. The
results on 162Dy show, however, transient supersolid properties whose lifetime is limited by fast inelastic losses

caused by three-body collisions [35,36]. These realizations
raise the crucial question of whether a long-lived or
stationary supersolid state can be created despite theusually non-negligble atom losses and the crossing of adiscontinuous phase transition, which inherently creates
excitations in the system.
In this work, we study both experimentally and theoretically the phase diagram of degenerate gases of highly
magnetic atoms beyond the roton softening. Our investigations are carried out using two different experimental
setups producing BECs of 166Er [22,37] and of 164Dy [38]and rely on a fine-tuning of the contact-interaction strength
in both systems. In the regime of interest, these two atomic
species have different contact-interaction scattering lengths
as , whose precise dependence on the magnetic field isknown only for Er [18,22,39], and different three-body-loss
rate coefficients. Moreover, Er and Dy possess different
magnetic moments μ and masses m, yielding the dipolar
lengths, add ¼ μ0 μ2 m=12πℏ2 , of 65.5a0 and 131a0 , respectively. Here, μ0 is the vacuum permeability, ℏ ¼ h=2π thereduced Planck constant, and a0 the Bohr radius. For both
systems, we find states showing hallmarks of supersolidity,
namely, the coexistence of density modulation and global
phase coherence. For such states, we quantify the extent of
the as parameter range for their existence and study their
lifetime. For 166Er, we find results very similar to the one
recently reported for 162Dy [35,36], both systems being
limited by strong three-body losses, which destroy thesupersolid properties in about half of a trap period.
However, for 164Dy, we have identified an advantageous
magnetic-field region where losses are very low and large
BECs can be created. In this condition, we observe that thesupersolid properties persist over a remarkably long time,
well exceeding the trap period. Based on such a highstability, we finally demonstrate a novel route to reach thesupersolid state, based on evaporative cooling from athermal gas.
II. THEORETICAL DESCRIPTIONAs a first step in our study of the supersolid phase indipolar BECs, we compute the ground-state phase diagram
for both 166Er and 164Dy quantum gases. The gases are
confined in a cigar-shaped harmonic trap, as illustrated inFig. 1(a). Our theory is based on numerical calculations of
the extended Gross-Pitaevskii equation [40], which
includes our anisotropic trapping potential, the short-range
contact and long-range dipolar interactions at a mean-fieldlevel, as well as the first-order beyond-mean-field correction in the form of a Lee-Huang-Yang (LHY) term
[18,22–24,27]. We note that, while both the exact strengthof the LHY term and its dependence on the gas characteristics are under debate [18,19,25,31,41], the importance ofsuch a term, scaling with a higher power in density, isessential for stabilizing states beyond the mean-fieldinstability [18,25,41]; see also Refs. [8,42–44].

021012-2

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
z

0

y

B

4

0

4

5
0
46

48

50

52

54

-1

2 (c)

4

n (10 µm )aS (a0)

1
0

0.8

(d)

0.4
-10 0 10y (µm)

0

(g)

8

2
N( 10 )y (µm)(b)44

Sn (10 µm )10 (f)
4x-5

0.1 0.2 0.3 0.4 0.5

-1

ID

SSP

ID

0

*

0.6
0.4
0.2

6

0

-10

0
10
y (µm)

SSP
BEC

2

166

0.2
-10 0 10y (µm)BEC

4
0.4 (e)n (10 4 µm-1)(a)PHYS. REV. X 9, 021012 (2019)45-10 0 10y (µm)50
aS (a0)

164

Er
55

90

95

Dy
100

aS (a0)

FIG. 1. Phase diagram of an 166Er and a 164Dy dipolar BEC in a cigar-shaped trap. (a) Illustration of the trap geometry with atomic
dipoles oriented along z. (b) Integrated density profile as a function of as for an 166Er ground state of N ¼ 5 × 104 . In the color bar, thedensity scale is upper limited to 4 × 104 μm−1 in order to enhance the visibility in the supersolid regime. (c)–(e) Exemplary density
profiles for an insulating droplet state (ID) at as ¼ 49a0 , for a state with supersolid properties (SSP) at 51a0 , and for a BEC at 52a0 ,respectively. (f),(g) Phase diagrams for 166Er and 164Dy for trap frequencies ωx;y;z ¼ 2π × ð227; 31.5; 151Þ and 2π × ð225; 37; 135Þ Hz,
respectively. The gray color identifies ground states with a single peak in nðyÞ of large Gaussian width, σ y > 2ly . The dark blue regionin (f) shows the region where nðyÞ exhibits a single sharp peak, σ y ≤ 2ly , and no density modulation. The red-to-blue color map shows S
in the case of a density-modulated nðyÞ. In (g) the color map is upper limited to use the same color code as in (f) and to enhance visibility
in the low-N regime. The inset in (g) shows the calculated density profile for 164Dy at N ¼ 7 × 104 and as ¼ 91a0 .

Our theoretical results are summarized in Fig. 1. By
varying the condensed-atom number N and as , the phase
diagram shows three very distinct phases. To illustratethem, we first describe the evolution of the integrated in situ
density profile nðyÞ with fixed N for varying as , Fig. 1(b).
The first phase, appearing at large as , resembles a regulardilute BEC. It corresponds to a nonmodulated density
profile of low peak density and large axial size σ y exceeding several times
the corresponding harmonic oscillator
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
length (ly ¼ ℏ=mωy ); see Fig. 1(e) and the region
denoted BEC in Figs. 1(f) and 1(g). The second phaseappears when decreasing as down to a certain critical value,as . Here, the system undergoes an abrupt transition to aperiodic density-modulated ground state, consisting of anarray of overlapping narrow droplets, each of high peakdensity. Because the droplets are coupled to each other via adensity overlap, later quantified in terms of the link strength
S, particles can tunnel from one droplet to a neighboring
one, establishing a global phase coherence across the cloud;see Fig. 1(d). Such a phase, in which periodic density
modulation and phase coherence coexist, is identified as
the supersolid (SSP) one [10,34]; see the SSP region in
Figs. 1(f) and 1(g). When further decreasing as , we observea fast reduction of the density overlap, which eventuallyvanishes; see Fig. 1(c). Here, the droplets become fully
separated. Under realistic experimental conditions, it isexpected that the phase relation between such dropletscannot be maintained; see later discussion. We identify thisthird phase as the one of an insulating droplet (ID) array[27,28,45]; see the ID region in Figs. 1(f) and 1(g). For low
N, we find a single droplet of high peak density, as inRefs. [24,27]; see dark blue region in Fig. 1(f). Generally
speaking, our calculations show that the number of dropletsin the array decreases with lowering as or N. The existenceof these three phases (BEC, SSP, ID) is consistent withrecent calculations considering an infinitely elongatedEr BEC [34] and a cigar-shaped 162Dy BEC [36], illustrating the generality of this behavior in dipolar gases.
To study the supersolid character of the density-modulated phases, we compute the average of the wave function
overlap between neighboring droplets S. As an ansatz to
extract S, we use a Gaussian function to describe the wave
function of each individual droplet. This is found to be an
appropriate description from an analysis of the density
profiles of Figs. 1(b)–1(d); see also Ref. [46]. For two
droplets at a distance d and of identical Gaussian widths σ yalong the array direction, S is simply S ¼ expð−d2 =4σ 2y Þ.
Here, we generalize the computation of the wave function
overlap to account for the difference in widths andamplitudes among neighboring droplets. This analysis
allows us to distinguish between the two types of modulated ground states, SSP and ID in Figs. 1(f) and 1(g).
Within the Josephson-junction picture [47–49], the tunneling rate of atoms between neighboring droplets depends on
the wave function overlap, and an estimate for the singleparticle tunneling rate can be derived within the Gaussian
approximation [46]; see also Ref. [40]. The ID phasecorresponds to vanishingly small values of S, yielding

021012-3

L. CHOMAZ et al.

PHYS. REV. X 9, 021012 (2019)tunneling times extremely long compared to any otherrelevant timescale. In contrast, the supersolid phase isidentified by a substantial value of S, with a correspondingly short tunneling time.
As shown in Figs. 1(f) and 1(g), a comparative analysisof the phase diagram for 166Er and 164Dy reveals similarities
between the two species (see also Ref. [36]). A supersolid
phase is found for sufficiently high N, in a narrow regionof as , upper bounded by the critical value as  ðNÞ. Forintermediate N, as increases with increasing N. We note
that, for low N, the nonmodulated BEC evolves directly
into a single droplet state for decreasing as [50]. In thiscase, no supersolid phase is found in between; see also
Refs. [24,27]. Despite the general similarities, we see thatthe supersolid phase for 164Dy appears for lower atom
number than for Er and has a larger extension in as . This ismainly due to the different add and strength of the LHY
term. We note that, at large N and for decreasing as , Dy
exhibits ground states with a density modulation appearing
first in the wings, which then progresses inwards until a
substantial modulation over the whole cloud is established[51]; see inset of Fig. 1(g). In this regime, we also observethat as decreases with increasing N. These types of states
have not been previously reported and, although challenging to access in experiments because of the large N, they
deserve further theoretical investigations.
III. EXPERIMENTAL SEQUENCE
FOR 166Er AND 164Dy
To experimentally access the above-discussed physics, we
produce dipolar BECs of either 166Er or 164Dy atoms. These
two systems are created in different setups and below wesummarize the main experimental steps; see also Ref. [40].
Erbium.—We prepare a stable 166Er BEC following
the scheme of Ref. [18]. At the end of the preparation,
the Er BEC contains about N ¼ 8 × 104 atoms at
as ¼ 64.5a0 . The sample is confined in a cigar-shaped
optical dipole trap with harmonic frequencies ωx;y;z ¼2π × ð227; 31.5; 151Þ Hz. A homogeneous magnetic field
B polarizes the sample along z and controls the value of as
via a magnetic Feshbach resonance (FR) [18,22,40]. Our
measurements start by linearly ramping down as within
20 ms and waiting an additional 15 ms so that as reaches its
target value [40]. We note that ramping times between 20
and 60 ms have been tested in the experiment and we do notrecord a significant difference in the system’s behavior.
After the 15-ms stabilization time, we then hold the sample
for a variable time th before switching off the trap. Finally,
we let the cloud expand for 30 ms and perform absorption
imaging along the z (vertical) direction, from which we
extract the density distribution of the cloud in momentum
space, nðkx ; ky Þ.
Dysprosium.—The experimental procedure to create a164
Dy BEC follows the one described in Ref. [38]; see also

Ref. [40]. Similarly to Er, the Dy BEC is also confined in acigar-shaped optical dipole trap and a homogeneous
magnetic field B sets the quantization axis along z andthe value of as . For Dy, we will discuss our results interms of magnetic field B, since the as -to-B conversion is
not well known in the magnetic-field range considered[25,40,41,52]. In a first set of measurements, we first
produce a stable BEC of about N ¼ 3.5 × 104 condensed
atoms at a magnetic field of B ¼ 2.5 G and then probe thephase diagram by tuning as . Here, before ramping themagnetic field to access the interesting as regions, we
slowly increase the power of the trapping beams within
200 ms. The final trap frequencies are ωx;y;z ¼ 2π ×ð300; 16; 222Þ Hz. After preparing a stable BEC, we ramp
B to the desired value within 20 ms and hold the sample forth [40]. In a second set of measurements, we study acompletely different approach to reach the supersolid state.
As discussed later, here we first prepare a thermal sample ata B value where supersolid properties are observed and thenfurther cool the sample until a transition to a coherent
droplet-array state is reached. In both cases, at the endof the experimental sequence, we perform absorption
imaging after typically 27 ms of time-of-flight (TOF)
expansion. The imaging beam propagates horizontally
under an angle α of ≈45° with respect to the weak axisof the trap (y). From the TOF images, we thus extractnðkY ; kz Þ with kY ¼ cosðαÞky þ sinðαÞkx .
A special property of 164Dy is that its backgroundscattering length is smaller than add . This allows us toenter the supersolid regime without the need of setting B
close to a FR, as is done for 166Er and 162Dy, which
typically causes severe atom losses due to increased threebody-loss coefficients. In contrast, in the case of 164Dy, thesupersolid regime is reached by ramping B away from the
FR pole used to produce the stable BEC via evaporative
cooling, as the as range of Fig. 1(g) lies close to thebackground as reported in Ref. [52]; see also Ref. [40]. At
the background level, three-body-loss coefficients below
1.3 × 10−41 m6 s−1 have been reported for 164Dy [25].
IV. DENSITY MODULATION AND
PHASE COHERENCE
The coexistence of density modulation and phase coherence is the key feature that characterizes the supersolid
phase and allows us to discriminate it from the BEC and ID
cases. To experimentally probe this aspect in our dipolar
quantum gases, we record their density distribution after a
TOF expansion for various values of as across the phase
diagram. As for a BEC in a weak optical lattice [53] or for
an array of BECs [54–56], the appearance of interference
patterns in the TOF images is associated with a density
modulation of the in situ atomic distribution. Moreover, theshot-to-shot reproducibility of the patterns (in amplitude
and position) and the persistence of fringes in averaged

021012-4

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …

(b)

-6
-4
-2
0
2
4
6

(c)

6
5
4
3
2
1
0

(e)

4
3
2
1
0

(f)

2.5
2
1.5
1
0.5
0

-6 -4 -2 0 2 4 6-6 -4 -2 0 2 4 6-6 -4 -2 0 2 4 6-6 -4 -2 0 2 4 6

kx (µm-1)

kx (µm-1)

kx (µm-1)

kx (µm-1)

25 x10

3

x10 4

(g)
20

5

x10 4

(j)

5

4

4

3

3

2

2

5

1

1

0

0

n (µm)

-6
-4
-2
0
2
4
6

(d)

15
10

14
12
10
8
6
4
2
0

(h)

12 (i)
108
64
2
0-6 -4 -2 0 2 4 6
y (µm)

5n (µm)

(a)

(k)

5

4

4

3
2

3
2

1

1

0
5
n (µm)ky (µm-1)

ky (µm-1)

ky (µm-1)

n (103 µm2)
-6
-4
-2
0
2
4
6

PHYS. REV. X 9, 021012 (2019)

(n)

0

(l)

5 (o)

4

4

3

3

2

2

1

1

0

(m)

-4 -2 0 2
ky (µm-1)

4

0-6 -4 -2 0 2 4 6ky (µm-1)

FIG. 2. Coherence in the interference patterns: measurement and toy model. (a)–(c) Examples of single TOF absorption images at
th ¼ 5 ms for 166Er at as ¼ f54.7ð2Þ; 53.8ð2Þ; 53.3ð2Þga0 , respectively. Corresponding average pictures for 100 images obtained under
the same experimental conditions (d)–(f) and their Fourier transform (FT) profiles (g)–(i). The gray lines show the FT norm jF ½nðyÞj ofthe individual profiles. The averages, nM (blue squares) and nΦ (red dots), are fitted to three-Gaussian functions (blue solid line andbrown dashed line, respectively). The dotted lines show the components of the total fitted function corresponding to the two side peaks
in nΦ . (j)–(l) Interference patterns from the toy-model realizations with 100 independent draws using N D ¼ 4, d ¼ 2.8 μm, σ y ¼
0.56 μm (see text) and for different ϕi distributions: (j) ϕi ¼ 0, (k) ϕi normally distributed around 0 with 0.2π standard deviation, (l) ϕi
uniformly distributed between 0 and 2π. (m)–(o) Corresponding FT profiles for the toy model, same color code as (g)–(i).

pictures, obtained from many repeated images taken under
the same experimental conditions, reveals the presence ofphase coherence across the sample [56].
Figure 2 exemplifies snapshots of the TOF distributions
for Er, measured at three different as values; see
Figs. 2(a)–2(c). Even if very close in scattering length,
the recorded nðkx ; ky Þ shows a dramatic change in behavior.
For as ¼ 54.7ð2Þa0, we observe a nonmodulated distribution with a density profile characteristic of a dilute BEC.
When lowering as to 53.8ð2Þa0 , we observe the appearanceof an interference pattern in the density distribution,
consisting of a high central peak and two almost symmetriclow-density side peaks [57]. Remarkably, the observed
pattern is very reproducible with a high shot-to-shotstability, as shown in the repeated single snapshots andin the average image [Figs. 2(b) and 2(e)]. This behavior
indicates a coexistence of density modulation and global
phase coherence in the in situ state, as expected in the
supersolid phase. This observation is consistent with
our previous quench experiments [18] and with the recent
162
Dy experiments [35,36]. When further lowering as to53.3ð2Þa0 , complicated patterns develop with fringes
varying from shot to shot in number, position, andamplitude, signaling the persistence of in situ density
modulation. However, the interference pattern is completely washed out in the averaged density profiles[Fig. 2(f)], pointing to the absence of a global phasecoherence. We identify this behavior as the one of
ID states.
Toy model—To get an intuitive understanding of theinterplay between density modulation and phase coherence
and to estimate the role of the different sources offluctuations in our experiment, we here develop a simple
toy model, which is inspired by Ref. [56]; see also
Ref. [40]. In our model, the initial state is an array of
N D droplets containing in total N atoms. Each droplet isdescribed by a one-dimensional Gaussian wave function
ψ i ðyÞ of amplitude αi , phase ϕi , width σ i , and center yi. To
account for fluctuations in the experiments, we allow αi,di ¼ yi − yi−1 , and σ i to vary by 10% around their expectation values. The spread of the phases ϕi among thedroplets is treated specially as it controls the global phasecoherence of the array. By fixing ϕi ¼ 0 for each droplet orby setting a random distribution of ϕi , we range from fullphase coherence to the incoherent cases. Therefore, thedegree of phase incoherence can be varied by changing thestandard deviation of the distribution of ϕi .
To mimic our experiment, we compute the free evolutionof each individual ψ i over 30
P ms, and then compute theaxial distribution nðy; tÞ ¼ j i ψ i ðy; tÞj2 , from which we
extract the momentum distribution nðky Þ, also accounting
for the finite imaging resolution [40]. For each computation
run, we randomly draw N D values for ϕi, as well as of σ i ,di , and αi , and extract nðky Þ. We then collect a set of nðky Þ

021012-5

L. CHOMAZ et al.

PHYS. REV. X 9, 021012 (2019)by drawing these values multiple times using the same
statistical parameters and compute the expectation value,hnðky Þi; see Figs. 2(j)–2(l). The angled brackets denote theensemble average.
The results of our toy model show large similarity withthe observed behavior in the experiment. In particular,
while for each single realization one can clearly distinguish
multipeak structures regardless of the degree of phasecoherence between the droplets, the visibility of theinterference pattern in the averaged nðky Þ survives only
if the standard deviation of the phase fluctuations between
droplets is small (roughly, below 0.3π). In the incoherent
case, we note that the shape of the patterns strongly varies
from shot to shot. Interestingly, the toy model also showsthat the visibility of the coherent peaks in the average
images is robust against the typical shot-to-shot fluctuations in droplet size, amplitude, and distance that occur inthe experiments; see Figs. 2(j) and 2(k).
Probing density modulation and phase coherence.—To
separate and quantify the information on the in situ density
modulation and its phase coherence, we analyze the measured
interference patterns in Fourier space [36,58–60]. Here, we
extract two distinct averaged density profiles, nM and nΦ .
Their structures at finite y spatial frequency (i.e., in Fourier
space) quantify the two abovementioned properties.
More precisely, we perform a Fourier transform (FT) of
the integrated momentum distributions nðky Þ denoted
F ½nðyÞ. Generally speaking, modulations in nðky Þ induce
peaks at finite spatial frequency, y ¼ y , in the FT norm,
jF ½nðyÞj; see Figs. 2(g)–2(i) and 2(m)–2(o). Following theabove discussion (see also Refs. [56,61]), such peaks in an
individual realization hence reveal a density modulation of
the corresponding in situ state, with a wavelength roughly
equal to y . Consequently, we consider the average of the
FT norm of the individual images, nM ðyÞ ¼ hjF ½nðyÞji, as
the first profile of interest. The peaks of nM at finite y thenindicate the mere existence of an in situ density modulationof roughly constant spacing within the different realizations. As the second profile of interest, we use the FT
norm of the average profile hnðky Þi, nΦ ðyÞ ¼ jF ½hniðyÞj.
Connecting to our previous discussion, the peaks of nΦ atfinite y point to the persistence of a modulation in the
average hnðky Þi, which we identified as a hallmark for a
global phase coherence within the density-modulated state.
In particular, we point out that a perfect phase coherence,
implying identical interference patterns in all the individual
realizations, yields nM ¼ nΦ and, thus, identical peaks
at finite y in both profiles. We note that, by linearity, nΦ
also matches the norm of the average of the full FTof the individual images, i.e., nΦ ðyÞ ¼ jhF ½nðyÞij; seealso Ref. [40].
Figures 2(g)–2(i) and 2(m)–2(o) demonstrate the significance of our FT analysis scheme by applying itto the momentum distributions from the experiment[Figs. 2(d)–2(f)] and the momentum distributions fromthe toy model [Figs. 2(j)–2(l)], respectively. As expected,
for the BEC case, both nM and nΦ show a single peak atzero spatial frequency, y ¼ 0, characterizing the absence ofdensity modulation, Fig. 2(g). In the case of phase-coherent
droplets, Fig. 2(e), we observe that nM and nΦ aresuperimposed and both show two symmetric side peaks
at finite y, in addition to a dominant peak at y ¼ 0; see
Fig. 2(h). In the incoherent droplet case, we find that, whilenM still shows side peaks at finite y, the ones in nΦ wash
out from the averaging, Figs. 2(f), 2(i), 2(l), and 2(o). For
both coherent and incoherent droplet arrays, the toy-model
results show behaviors matching the above description,
providing a further justification of our FT analysis scheme;
see Figs. 2(j)–2(o). Our toy model additionally proves twointeresting features. First, it shows that the equalitynM ¼ nΦ , revealing the global phase coherence of adensity-modulated state, is remarkably robust to noise in
the structure of the droplet arrays; see Figs. 2(j) and 2(m).
Second, our toy model, however, shows that phase fluctuations across the droplet array on the order of 0.2π
standard deviation are already sufficient to make nΦ andnM deviate from each other; see Figs. 2(k) and 2(n). The
incoherent behavior is also associated with strong variations in the side peak amplitude of the individual realizations of jF ½nj, connecting, e.g., to the observationsof Ref. [36].
Finally, to quantify the density modulation and thephase coherence, we fit a three-Gaussian function to bothnM ðyÞ and nΦ ðyÞ and extract the amplitudes of thefinite-spatial-frequency peaks, AM and AΦ , for both distributions, respectively. Note that for a BEC, which is aphase-coherent state, AΦ will be zero since it probes
only finite-spatial-frequency peaks; see Figs. 2(g)–2(i)and 2(m)–2(o).
V. CHARACTERIZATION OF THE
SUPERSOLID STATE
We are now in the position to study two key aspects,
namely, (i) the evolution of the density modulation andphase coherence across the BEC-supersolid-ID phases and(ii) the lifetime of the coherent density-modulated state in
the supersolid regime.
Evolution of the supersolid properties across the phase
diagram.—The first type of investigation is conducted with
166
Er since, contrary to 164Dy, its scattering length anddependence on the magnetic field has been precisely
characterized [18,22]. After preparing the sample, we ramp
as to the desired value and study the density patterns as wellas their phase coherence by probing the amplitudes AM
and AΦ as a function of as after th ¼ 5 ms. As shown in
Fig. 3(a), in the BEC region (i.e., for large as ), we observethat both AM and AΦ are almost zero, evidencing the
expected absence of a density modulation in the system. As
soon as as reaches a critical value as , the system’s behavior
dramatically changes with a sharp and simultaneous

021012-6

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
x103
1

A /AM

(a)
A AM

4

0
-2

-1

0

1

as - as*(a0)

2

0
0.8

(b)

S

0.6
0.4
0.2
0
-2

-1

as - as*(a0)01FIG. 3. Supersolid behavior across the phase diagram. Measured side peak amplitudes, AΦ (circles) and AM (squares), with
their ratio in inset (a), and calculated link strength S (b) as afunction of as − as for 166Er. For nonmodulated states, we set
S ¼ 0 in theory and AΦ =AM ¼ 0 in experiment (crosses in inset).
In the inset, open and closed symbols correspond to AΦ =AM >0.8 and ≤ 0.8, respectively. In the experiments, we probe thesystem at a fixed th ¼ 5 ms. Horizontal error bars are derived
from our experimental uncertainty in B, vertical error bars
corresponding to the statistical uncertainty from the fit are smaller
than the data points. The measured and calculated criticalscattering lengths are as ¼ 54.9ð2Þa0 and 51.15a0 , respectively[62]. The numerical results are obtained for the experimental trap
frequencies and for a constant N ¼ 5 × 104 [63].

increase of both AM and AΦ . While the strength of AM
and AΦ varies with decreasing as —first increasing thendecreasing—we observe that their ratio AΦ =AM remains
constant and close to unity over a narrow as range below asof ≳1a0 width; see the inset of Fig. 3(a). This behavior
pinpoints the coexistence in the system of phase coherence
and density modulation, as predicted to occur in the
supersolid regime. For ðas − as Þ < −1a0, we observe thatthe two amplitudes depart from each other. Here, while thedensity modulation still survives with AM saturating to alower finite value, the global phase coherence is lost withAΦ =AM < 1, as expected in the insulating droplet phase.
Note that we also study the evolution of AΦ and AM in164
Dy, but as a function of B, and find a qualitatively similar
behavior.
To get a deeper insight on how our observations compare
to the phase-diagram predictions (see Fig. 1), we study thelink strength S as a function of as ; see Fig. 3(b). Since S
quantifies the density overlap between neighboring droplets and is related to the tunneling rate of atoms across thedroplet array, it thus provides information on the ability of

PHYS. REV. X 9, 021012 (2019)the system to establish or maintain a global phase coherence. In this plot, we set S ¼ 0 in the case where no
modulation is found in the ground state. At the BEC-tosupersolid transition, i.e., at as ¼ as , a density modulation
abruptly appears in the system’s ground state with S taking
a finite value. Here, S is maximal, corresponding to adensity modulation of minimal amplitude. Below thetransition, we observe a progressive decrease of S withlowering as , pointing to the gradual reduction of thetunneling rate in the droplet arrays. Close to the transition,
we estimate a large tunneling compared to all other relevanttimescales. However, we expect this rate to become vanishingly small, on the sub-Hertz level [40], when decreasing
as 1–2a0 below as. Our observation also hints at the smoothcharacter of the transition from a supersolid to an ID phase.
The general trend of S, including the extension in as
where it takes nonvanishing values, is similar to the as
behavior of AM and AΦ observed in the experiments [62].
We observe in the experiments that the as dependence at
the BEC-to-supersolid transition appears sharper than at
the supersolid-to-ID interface, potentially suggesting a
different nature of the two transitions. However, moreinvestigations are needed since atom losses, finite temperature, and finite-size effects can affect, and in particularsmoothen, the observed behavior [64–66]. Moreover,dynamical effects, induced by, e.g., excitations created at
the crossing of the phase transitions or atom losses during
the time evolution, can also play a substantial role in theexperimental observations, complicating a direct comparison with the ground-state calculations. The time dynamics
as well as a different scheme to achieve a state withsupersolid properties is the focus of the remainder ofthe paper.
Lifetime of the supersolid properties.—Having identified
the as range in which our dipolar quantum gas exhibits
supersolid properties, the next central question concerns thestability and lifetime of such a fascinating state. Recent
experiments on 162Dy have shown the transient character ofthe supersolid properties, whose lifetime is limited bythree-body losses [35,36]. In these experiments, the phasecoherence is found to survive up to 20 ms after the density
modulation has formed. This time corresponds to about halfof the weak-trap period. Stability is a key issue in the
supersolid regime, especially since the tuning of as , used toenter this regime, has a twofold consequence on the
inelastic loss rate. First, it gives rise to an increase in
the peak density [see Figs. 1(b)–1(d)] and, second, it may
lead to an enhancement of the three-body-loss coefficient.
We address this question by conducting comparative
studies on 166Er and 164Dy gases. These two species allow
us to tackle two substantially different scattering scenarios.
Indeed, the background value of as for 166Er (as well as for
162Dy) is larger than add . Thus, reaching the supersolid
regime, which occurs at add =as ≈ 1.2–1.4 in our geometry,
requires us to tune B close to the pole of a FR. This tuning

021012-7

L. CHOMAZ et al.

PHYS. REV. X 9, 021012 (2019)

also causes an increase of the three-body-loss rate. In
contrast, 164Dy realizes the opposite case with the background scattering length smaller than add . This featurebrings the important advantage of requiring tuning B awayfrom the FR pole to reach the supersolid regime. As we
describe below, this important difference in scattering
properties leads to a strikingly longer lifetime of the
164Dy supersolid properties with respect to 166Er and to
the recently observed behavior in 162Dy [35,36].
The measurements proceed as follows. For both 166Er
and 164Dy, we first prepare the quantum gas in the stable
BEC regime and then ramp as to a fixed value in the
supersolid regime for which the system exhibits a state ofcoherent droplets (i.e., AΦ =AM ≈ 1); see previous discussion. Finally, we record the TOF images after a variable th
and we extract the time evolution of both AΦ and AM .
The study of these two amplitudes will allow us to answer
the question of whether the droplet structure—i.e., thedensity modulation in space—persists in time whereas
the coherence among droplets is lost (AM > AΦ → 0) or
if the density structures themselves vanish in time
(AM ≈ AΦ → 0).
As shown in Fig. 4, for both species, we observe that AΦ
and AM decay almost synchronously with a remarkably
longer lifetime for 164Dy [Fig. 4(b)] than 166Er [Fig. 4(a)].
x103
5

(a)
A /AM

1

A AM

4
3

0

2

0

4

8
12 16
th (ms)

1
0
12

166

x10

(b)

1
A /AM

A AM

Er

2

8

0

0

60

4

th (ms)

120

164

0
0

40

80

Dy

120

Interestingly, AΦ and AM remain approximately equal
during the whole time dynamics; see insets of Figs. 4(a)and 4(b). This behavior indicates that it is the strength of thedensity modulation itself and not the phase coherence
among droplets that decays over time. Similar results havebeen found theoretically in Ref. [67]. We connect thisdecay mainly to three-body losses, especially detrimental
for 166Er, and possible excitations created while crossing
the BEC-to-supersolid phase transition [40]. For comparison, the inset of Fig. 4(a) shows also the behavior in the ID
regime for 166Er, where AΦ =AM < 1 already at short th andremains so during the time evolution [40].
To get a quantitative estimate of the survival time ofthe phase-coherent and density-modulated state, we fit asimple exponential function to AΦ and extract tΦ , definedas the 1=10 lifetime; see Fig. 4. For 166Er, we extracttΦ ¼ 38ð6Þ ms. For th > tΦ, the interference patterns
become undetectable in our experiment and we recovera signal similar to the one of a nonmodulated BEC state [asin Figs. 2(a) and 2(d)]. These results are consistent withrecent observations of transient supersolid properties in162
Dy [35]. For 164Dy, we observe that the coherent densitymodulated state is remarkably long-lived. Here, we findtΦ ¼ 152ð13Þ ms.
The striking difference in the lifetime and robustness ofthe supersolid properties between 166Er and 164Dy becomeseven more visible when studying tΦ as a function of as(B for Dy). As shown in Fig. 5, tΦ for Er remains
comparatively low in the investigated supersolid regime
and slightly varies between 20 and 40 ms. Similarly to therecent studies with 162Dy, this finding reveals the transientcharacter of the state and opens the question of whether astationary supersolid state can be reached with these
species. On the contrary, for 164Dy we observe that tΦ
first increases with B in the range from 1.8 G to about
1.98 G. Then, for B > 1.98 G, tΦ acquires a remarkably
large and almost constant value of about 150 ms over awide B range. This shows the long-lived character of thesupersolid properties in our 164Dy quantum gas. We notethat over the investigated range, as is expected to monotonously increase with B [40]. Such a large value of tΦ
exceeds not only the estimated tunneling time acrossneighboring droplets but also the weak-axis trap period,
which together set the typical timescale to achieve global
equilibrium and to study collective excitations.
VI. CREATION OF STATES WITH SUPERSOLID
PROPERTIES BY EVAPORATIVE COOLING

th (ms)FIG. 4. Time evolution of the supersolid properties. AmplitudesAΦ (circles) and AM (squares) in the supersolid regime as afunction of the holding time in trap for (a) 166Er at 54.2ð2Þa0 andfor (b) 164Dy at 2.04 G. The solid lines are exponential fits to thedata. The insets show the time evolution of AΦ =AM for the above
cases (filled triangles), and, for comparison, in the ID regime(empty triangles) for Er at as ¼ 53.1ð2Þa0 (a).

The long-lived supersolid properties in 164Dy motivate us
to explore an alternative route to cross the supersolid phasetransition, namely, by evaporative cooling instead ofinteraction tuning. For this set of experiments, we havemodified the waists of our trapping beams in order toachieve quantum degeneracy in tighter traps with respect to021012-8

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
(b)

(a)

20

(e)

10 ms

(i)

(f)

50 ms

(j)

(g)

100 ms(k)(h)

300 ms(l)n (102 µm)(a)PHYS. REV. X 9, 021012 (2019)

15

150

10

0

100

(b)

0
53.5

164

Er

54 54.5 1.8
aS (a0)

1.9

2

Dy
2.1

B (G)

FIG. 5. Survival time of the coherent density-modulated state.
tΦ in 166Er as a function of as (a) and 164Dy as a function of B (b).
The error bars refer to the statistical uncertainty from the fit. The
range of investigation corresponds to the supersolid regime forwhich phase-coherent density-modulated states are observed.
This range is particularly narrow for 166Er.

n (102 µm)

15
10
5
0

(c)

20
n (102 µm)

166

20

15
10
5
0

(d)

20
n (102 µm)

50

Evaporation time (ms)t (ms)

515
10

10 µm-1

the one used for condensation in the previous set ofmeasurements. In this way, the interference peaks in the
supersolid region are already visible without the need toapply a further compression of the trap since the sideto-central-peak distance in the momentum distribution
scales roughly as 1=lz [18]. Forced evaporative coolingis performed by reducing the power of the trapping beams
piecewise linearly in subsequent evaporation steps until a
final trap with frequencies 2π × ð225; 37; 134Þ Hz isachieved. During the whole evaporation process, which
has an overall duration of about 3 s, the magnetic field is
kept either at B ¼ 2.43 G, where we observe long-lived
interference patterns, or at B ¼ 2.55 G, where we produce
a stable nonmodulated BEC. We note that these two B
values are very close without any FR lying in between [40].
Figure 6 shows the phase transition from a thermal cloud
to a final state with supersolid properties by evaporative
cooling. In particular, we study the phase transition byvarying the duration of the last evaporation ramp, while
maintaining the initial and final trap-beam power fixed.
This procedure effectively changes the atom number andtemperature in the final trap while keeping the trap
parameters unchanged, which is important to not alter
the final ground-state phase diagram of the system. At theend of the evaporation, we let the system equilibrate andthermalize for th ¼ 100 ms, after which we switch off thetrap, let the atoms expand for 26.5 ms, and finally perform
absorption imaging. We record the TOF images for different ramp durations, i.e., for different thermalization times.
For a short ramp, too many atoms are lost such that thecritical atom number for condensation is not reached, andthe atomic distribution remains thermal; see Fig. 6(a).

02

4 6 8 10n (102 µm2)

50-15 -10 -5 0 5 10 15
kY (µm-1)

FIG. 6. Evaporative cooling to a state with supersolid properties.
164Dy absorption images showing the transition to a state withsupersolid properties at 2.43 G (a)–(d) and to a BEC state at 2.55 G(i)–(l), via different durations of the last evaporation step. Thesedurations are 10 ms (a),(i), 50 ms (b),(j), 100 ms (c),(k), and
300 ms (d),(l). The density profiles (e)–(h) are integrated over thecentral regions of the corresponding absorption images (a)–(d).
The color map indicates the atomic density in momentum space.

By increasing the ramp time, the evaporative cooling
becomes more efficient and we observe the appearance of abimodal density profile with a narrow and dense peak at thecenter, which we identify as a regular BEC; see Fig. 6(b).
By further cooling, the BEC fraction increases and thecharacteristic pattern of the supersolid state emerges; see
Figs. 6(c) and 6(d). The observed evaporation process
shows a strikingly different behavior in comparison
with the corresponding situation at B ¼ 2.55 G, where
the usual thermal-to-BEC phase transition is observed; see
Figs. 6(i)–6(l).
We finally probe the lifetime of the supersolid properties
by extracting the time evolution of both the amplitudes AΦ
and AM , as previously discussed. We use the same
experimental sequence as the one in Fig. 6(d)—i.e., 300ms duration of the last evaporation ramp and 100 ms ofequilibration time—and subsequently hold the sample in
the trap for a variable th . As shown in Fig. 7(a), we observea very long lifetime with both amplitudes staying large andalmost constant over more than 200 ms. At longer holding

021012-9

L. CHOMAZ et al.
300

PHYS. REV. X 9, 021012 (2019)

(a)

(b)

250

10
150
100
50
0
0

1

(c)

A / AM

A , AM

200
8
6
4

0
0
50

200
100

400
150

200

5 µm-1
250

300

350

400

th (ms)20n (102 µm2)

FIG. 7. Lifetime of the supersolid properties achieved via
evaporative cooling. Time evolution of the amplitudes AΦ (redcircle) and AM (square) after an evaporation time of 300 ms at
2.43 G and an equilibration time of 100 ms. The inset shows
the time evolution of AΦ =AM . At th ¼ 0 ms, the atom numberin the phase-coherent density-modulated component is N ¼
2.2ð2Þ × 104 . (b),(c) Averaged absorption images of 25 realizations after 50 and 300 ms of holding time, respectively. Notethat the thermal background has been subtracted from the images.
The color map indicates the atomic density in momentum
space.

time, we observe a slow decay of AΦ and AM , following the
decay of the atom number. Moreover, during the dynamics,
the ratio AΦ =AM stays constant. The long lifetime ofthe phase-coherent density modulation is also directly
visible in the persistence of the interference patterns in
the averaged momentum density profiles [similar to
Fig. 2(e)], both at intermediate and long times; see
Figs. 7(b) and 7(c), respectively. For even longer th, wecannot resolve anymore interference patterns in the TOF
images. Here, we recover a signal consistent with a regular
BEC of low N.
Achieving the coherent droplet phase via evaporative
cooling is a very powerful alternative path to supersolidity.
We speculate that, for instance, excitations, which might beimportant when crossing the phase transitions by interaction tuning, may be small or removed by evaporation
when reaching this state kinematically. Other interestingquestions, open to future investigations, are the nature of
the phase transition, the critical atom number, and the roleof noncondensed atoms.
VII. CONCLUSIONSFor both 166Er and 164Dy dipolar quantum gases, we haveidentified and studied states showing hallmarks of supersolidity, namely, global phase coherence and spontaneousdensity modulations. These states exist in a narrow scattering-length region, lying between a regular BEC phase and aphase of an insulating droplet array. While for 166Er,
similarly to the recently reported 162Dy case [35,36], theobserved supersolid properties fade out over a comparatively short time because of atom losses, we find that 164Dy
exhibits remarkably long-lived supersolid properties.
Moreover, we are able to directly create stationary stateswith supersolid properties by evaporative cooling, demonstrating a powerful alternative approach to interaction
tuning on a BEC. This novel technique provides prospectsof creating states with supersolid properties while avoiding
additional excitations and dynamics. The ability to produce
long-lived supersolid states paves the way for future
investigations on quantum fluctuations and many-bodycorrelations, as well as of collective excitations in suchan intriguing many-body quantum state. A central goal ofthese future investigations lies in proving the superfluid
character of this phase, beyond its global phase coherence[7,34,68–70].
ACKNOWLEDGMENTS
We thank R. Bisset, B. Blakie, M. Boninsegni, G.
Modugno, T. Pfau, and, in particular, L. Santos for many
stimulating discussions. Part of the computational resultspresented have been achieved using the HPC infrastructure
LEO of the University of Innsbruck. We acknowledge
support by the Austrian Science Fund FWF through the
DFG/FWF Forschergruppe (FOR 2247/PI2790), by the
ERC Consolidator Grant (RARE, No. 681432), and by a
NFRI Grant (MIRARE, No. ÖAW0600) from the Austrian
Academy of Science. G. D. and M. S. acknowledge support
by the Austrian Science Fund FWF within the DK-ALM(No. W1259-N27).
Note added.—Recently, we became aware of related works
reporting theoretical studies of the ground-state phase
diagram [71,72].

[1] A. F. Andreev and I. M. Lifshitz, Many-Body Problem in
Quantum Mechanics and Quantum Statistical Mechanics,
Sov. Phys. JETP 29, 1107 (1969).
[2] G. V. Chester, Speculations on Bose-Einstein Condensation
and Quantum Crystals, Phys. Rev. A 2, 256 (1970).
[3] A. J. Leggett, Can a Solid Be “Superfluid”?, Phys. Rev.
Lett. 25, 1543 (1970).
[4] D. A. Kirzhnits and Y. A. Nepomnyashchii, Coherent Crystallization of Quantum Liquid, Sov. Phys. JETP 32, 1191
(1971).
[5] T. Schneider and C. P. Enz, Theory of the Superfluid-Solid
Transition of 4He, Phys. Rev. Lett. 27, 1186 (1971).
[6] S. Balibar, The Enigma of Supersolidity, Nature (London)
464, 176 (2010).
[7] M. Boninsegni and N. V. Prokof’ev, Colloquium: Supersolids: What and Where Are They?, Rev. Mod. Phys. 84,
759 (2012).
[8] Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov, Stable
Dilute Supersolid of Two-Dimensional Dipolar Bosons,Phys. Rev. Lett. 115, 075303 (2015).
[9] N. Henkel, R. Nath, and T. Pohl, Three-Dimensional Roton
Excitations and Supersolid Formation in Rydberg-Excited
Bose-Einstein Condensates, Phys. Rev. Lett. 104, 195302(2010).

021012-10

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
[10] F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G.
Pupillo, Supersolid Droplet Crystal in a Dipole-Blockaded
Gas, Phys. Rev. Lett. 105, 135301 (2010).
[11] M. Boninsegni, Supersolid Phases of Cold Atom Assemblies, J. Low Temp. Phys. 168, 137 (2012).
[12] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T.
Donner, Supersolid Formation in a Quantum Gas Breaking
a Continuous Translational Symmetry, Nature (London)
543, 87 (2017).
[13] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. Ç.
Top, A. O. Jamison, and W. Ketterle, A Stripe Phase with
Supersolid Properties in Spin–Orbit-Coupled Bose–
Einstein Condensates, Nature (London) 543, 91 (2017).
[14] L. D. Landau, The Theory of Superfluidity of Helium II,
J. Phys. (Moscow) 5, 71 (1941).
[15] P. Nozières, Is the Roton in Superfluid 4He the Ghost of a
Bragg Spot?, J. Low Temp. Phys. 137, 45 (2004).
[16] D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, Rotons inGaseous Bose-Einstein Condensates Irradiated by a Laser,
Phys. Rev. Lett. 90, 110402 (2003).
[17] L. Santos, G. V. Shlyapnikov, and M. Lewenstein, RotonMaxon Spectrum and Stability of Trapped Dipolar BoseEinstein Condensates, Phys. Rev. Lett. 90, 250403 (2003).
[18] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S.
Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino, Observation of Roton Mode Population in a
Dipolar Quantum Gas, Nat. Phys. 14, 442 (2018).
[19] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider,
M. J. Mark, L. Chomaz, and F. Ferlaino, Probing the Roton
Excitation Spectrum of a Stable Dipolar Bose Gas, arXiv:
1811.12115.
[20] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I.
Ferrier-Barbut, and T. Pfau, Observing the Rosensweig
Instability of a Quantum Ferrofluid, Nature (London)
530, 194 (2016).
[21] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, andT. Pfau, Observation of Quantum Droplets in a Strongly
Dipolar Bose Gas, Phys. Rev. Lett. 116, 215301 (2016).
[22] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Quantum-Fluctuation-Driven
Crossover from a Dilute Bose-Einstein Condensate to a
Macrodroplet in a Dipolar Quantum Fluid, Phys. Rev. X 6,041039 (2016).
[23] F. Wächtler and L. Santos, Quantum Filaments in Dipolar
Bose-Einstein Condensates, Phys. Rev. A 93, 061603(R)(2016).
[24] F. Wächtler and L. Santos, Ground-State Properties andElementary Excitations of Quantum Droplets in Dipolar
Bose-Einstein Condensates, Phys. Rev. A 94, 043618(2016).
[25] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T.
Pfau, Self-Bound Droplets of a Dilute Magnetic Quantum
Liquid, Nature (London) 539, 259 (2016).
[26] I. Ferrier-Barbut, M. Wenzel, M. Schmitt, F. Böttcher, andT. Pfau, Onset of a Modulational Instability in Trapped
Dipolar Bose-Einstein Condensates, Phys. Rev. A 97,
011604(R) (2018).
[27] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,
Ground-State Phase Diagram of a Dipolar Condensate with
Quantum Fluctuations, Phys. Rev. A 94, 033619 (2016).

PHYS. REV. X 9, 021012 (2019)[28] M. Wenzel, F. Böttcher, T. Langen, I. Ferrier-Barbut, and T.
Pfau, Striped States in a Many-Body System of Tilted
Dipoles, Phys. Rev. A 96, 053630 (2017).
[29] D. Baillie and P. B. Blakie, Droplet Crystal Ground
States of a Dipolar Bose Gas, Phys. Rev. Lett. 121,
195301 (2018).
[30] D. S. Petrov, Quantum Mechanical Stabilization of a
Collapsing Bose-Bose Mixture, Phys. Rev. Lett. 115,155302 (2015).
[31] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P.
Cheiney, and L. Tarruell, Quantum Liquid Droplets in a
Mixture of Bose-Einstein Condensates, Science 359, 301(2018).
[32] G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L.
Wolswijk, F. Minardi, M. Modugno, G. Modugno, M.
Inguscio, and M. Fattori, Self-Bound Quantum Dropletsof Atomic Mixtures in Free Space, Phys. Rev. Lett. 120,235301 (2018).
[33] P. Cheiney, C. R. Cabrera, J. Sanz, B. Naylor, L. Tanzi, and
L. Tarruell, Bright Soliton to Quantum Droplet Transition ina Mixture of Bose-Einstein Condensates, Phys. Rev. Lett.
120, 135301 (2018).
[34] S. M. Roccuzzo and F. Ancilotto, Supersolid Behaviour of a
Dipolar Bose-Einstein Condensate Confined in a Tube,
Phys. Rev. A 99, 041601(R) (2019).
[35] L. Tanzi, E. Lucioni, F. Fama, J. Catani, A. Fioretti,
C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,
Observation of a Dipolar Quantum Gas with Metastable
Supersolid Properties, Phys. Rev. Lett. 122, 130405 (2019).
[36] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo,T. Langen, and T. Pfau, Transient Supersolid Properties in
an Array of Dipolar Quantum Droplets, Phys. Rev. X 9,011051 (2019).
[37] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R.
Grimm, and F. Ferlaino, Bose-Einstein Condensation of
Erbium, Phys. Rev. Lett. 108, 210401 (2012).
[38] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M.
Sohmen, M. J. Mark, and F. Ferlaino, Dipolar Quantum
Mixtures of Erbium and Dysprosium Atoms, Phys. Rev. Lett.
121, 213601 (2018).
[39] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z.
Cai, M. Baranov, P. Zoller, and F. Ferlaino, Extended BoseHubbard Models with Ultracold Magnetic Atoms, Science
352, 201 (2016).
[40] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevX.9.021012 for details on
the experimental setup, measurement schemes, analysis,
and theory calculations.
[41] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen, M.
Isoard, S. Stringari, and T. Pfau, Scissors Mode of Dipolar
Quantum Droplets of Dysprosium Atoms, Phys. Rev. Lett.
120, 160402 (2018).
[42] A. Gammal, T. Frederico, L. Tomio, and P. Chomaz, Atomic
Bose-Einstein Condensation with Three-Body Interactions
and Collective Excitations, J. Phys. B 33, 4053 (2000).
[43] A. Bulgac, Dilute Quantum Droplets, Phys. Rev. Lett. 89,050402 (2002).
[44] D. S. Petrov, Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture, Phys. Rev. Lett. 115, 155302(2015).

021012-11

L. CHOMAZ et al.

PHYS. REV. X 9, 021012 (2019)[45] F. Wächtler and L. Santos, Ground-State Properties andElementary Excitations of Quantum Droplets in Dipolar
Bose-Einstein Condensates, Phys. Rev. A 94, 043618(2016).
[46] M. Wenzel, F. Böttcher, J.-N. Schmidt, M. Eisenmann, T.
Langen, T. Pfau, and I. Ferrier-Barbut, Anisotropic Superfluid Behavior of a Dipolar Bose-Einstein Condensate,
Phys. Rev. Lett. 121, 030401 (2018).
[47] B. D. Josephson, Possible New Effects in Superconductive
Tunnelling, Phys. Lett. 1, 251 (1962).
[48] J. Javanainen, Oscillatory Exchange of Atoms between Traps
Containing Bose Condensates, Phys. Rev. Lett. 57, 3164
(1986).
[49] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy,
Coherent Oscillations between Two Weakly Coupled BoseEinstein Condensates: Josephson Effects, π Oscillations, andMacroscopic Quantum Self-Trapping, Phys. Rev. A 59, 620(1999).
[50] A similar behavior is also seen for 164Dy for N < 4000
[out of the scale of Fig. 1(g)].
[51] We note that a similar behavior is also found in our 166Er
phase diagram but for larger N than the one reported here.
[52] Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and B. L. Lev,
s-Wave Scattering Lengths of the Strongly Dipolar Bosons
162Dy and 164Dy, Phys. Rev. A 92, 022703 (2015).
[53] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I.
Bloch, Quantum Phase Transition from a Superfluid to a
Mott Insulator in a Gas of Ultracold Atoms, Nature (London)
415, 39 (2002).
[54] M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T.
Esslinger, Exploring Phase Coherence in a 2D Lattice of
Bose-Einstein Condensates, Phys. Rev. Lett. 87, 160405(2001).
[55] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I.
Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch,
Tonks–Girardeau Gas of Ultracold Atoms in an Optical
Lattice, Nature (London) 429, 277 (2004).
[56] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J.
Dalibard, Interference of an Array of Independent BoseEinstein Condensates, Phys. Rev. Lett. 93, 180403 (2004).
[57] In our absorption imaging a residual asymmetry can beobserved between ky . This asymmetry, providing a
sharper peak at ky > 0, stems from optical aberrations in
our imperfect optical setup. These imperfections make thequantitative analysis of our data more challenging, yet they
do not affect the quantitative interpretation. In particular, wehave checked that the peak structures are also observable
along a distinct imaging axis, nonorthogonal to y, using an
independent imaging setup.
[58] M. Takeda, H. Ina, and S. Kobayashi, Fourier-Transform
Method of Fringe-Pattern Analysis for Computer-BasedTopography and Interferometry, J. Opt. Soc. Am. 72, 156(1982).

[59] C. Kohstall, S. Riedl, E. R. S. Guajardo, L. A. Sidorenkov,
J. H. Denschlag, and R. Grimm, Observation of Interference
between Two Molecular Bose–Einstein Condensates, New
J. Phys. 13, 065027 (2011).
[60] L. Chomaz, L. Corman, T. Bienaimé, R. Desbuquois, C.
Weitenberg, S. Nascimbène, J. Beugnon, and J. Dalibard,
Emergence of Coherence via Transverse Condensation in a
Uniform Quasi-Two-Dimensional Bose Gas, Nat. Commun.
6, 6162 (2015).
[61] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J.
Schmiedmayer, Non-Equilibrium Coherence Dynamics in
One-Dimensional Bose Gases, Nature (London) 449, 324(2007).
[62] The values of as extracted from experiments and fromtheory disagree by 3.8a0. Such a mismatch is similar tothe one observed in our previous work on the roton
instability [18,19]. This behavior might be related to thepossibility that beyond-mean-field effects are not properly accounted for by the conventional LHY correction
term; see main text. In addition, other effects, such as
the dynamics and the atom losses [40], can affect theexperimental observations.
[63] This atom number corresponds to the condensed
atom number measured within the stabilization time ofas [40].
[64] M. E. Fisher and A. E. Ferdinand, Interfacial, Boundary,and Size Effects at Critical Points, Phys. Rev. Lett. 19, 169(1967).
[65] Y. Imry and D. Bergman, Critical Points and Scaling Laws
for Finite Systems, Phys. Rev. A 3, 1416 (1971).
[66] Y. Imry, Finite-Size Rounding of a First-Order Phase
Transition, Phys. Rev. B 21, 2042 (1980).
[67] See the theoretical analysis in the revised version of
Ref. [35].
[68] V. W. Scarola, E. Demler, and S. D. Sarma, Searching for a
Supersolid in Cold-Atom Optical Lattices, Phys. Rev. A 73,
051601(R) (2006).
[69] Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov, Stable
Dilute Supersolid of Two-Dimensional Dipolar Bosons,Phys. Rev. Lett. 115, 075303 (2015).
[70] F. Cinti and M. Boninsegni, Classical and Quantum
Filaments in the Ground State of Trapped Dipolar Bose
Gases, Phys. Rev. A 96, 013627 (2017).
[71] Y. Kora and M. Boninsegni, Patterned Supersolids in
Dipolar Bose Systems, arXiv:1902.08256.
[72] Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidity arounda Critical Point in Dipolar Bose Einstein Condensates,
arXiv:1903.06161.
Correction: The inadvertent omission of a marker indicating“Featured in Physics” has been fixed.

021012-12

Supplemental Material: Long-lived and transient supersolid behaviors in dipolar
quantum gases
L. Chomaz,1 D. Petter,1 P. Ilzhöfer,2 G. Natale,1 A. Trautmann,2 C. Politi,2 G. Durastante,1,2
R. M. W. van Bijnen,2 A. Patscheider,1 M. Sohmen,1,2 M. J. Mark,1,2 and F. Ferlaino1,2,∗
1

Institut für Experimentalphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria(Dated: March 27, 2019)

GROUND STATE CALCULATIONS

We perform numerical calculations of the ground state
following the procedure detailed in the supplementary
information of Ref. [1]. The calculations are based on
the conjugate-gradients technique to minimize the energy functional of an eGPE [2]. In particular, the eGPE
accounts for the effect of quantum fluctuations, by includ√ing the LHY term ∆µ[n] = 32g(nas )3/2 (1 + 32dd /2)/3 πin the system’s Hamiltonian (here g = 4πh̄2 as /m andn = |ψ|2 is the spatial density of the macroscopic stateψ). ∆µ[n] has been obtained under a local density
approximation in Refs. [3, 4]. The relevance of the
LHY correction has been demonstrated in various studies of dipolar Bose gases close to the mean-field instability [1, 5–9] as it brings an additional repulsive potential,
stabilizing the gas against mean-field collapse at large
density. We note that the exact functional form of thepotential, originating from beyond mean-field effects, hasbeen questioned by several experimental results in finitesize trapped systems [1, 9–11], calling for further theory
developments [12].
Our numerical calculations provide us with the threedimensional ground-state wavefunctions ψ(r). From this,we compute the axial in-situ
R density profile along the
trap’s weak axis, n(y) = |ψ(r)|2 dxdz and find density profiles, corresponding to the BEC, the supersolidor the ID phase, that we plot in Fig. 1. From thedensity profiles that exhibit a density modulation, we
evaluate S by performing Gaussian fits to each droplet,i. e. to n(y) with y ranging between two neighboring local density minima. From these Gaussian fits, we eval(0)uate the sets of centers {yi }i and widths {σi }i corresponding to the macroscopic Gaussian wavefunctions{ψi }i associated to the individual droplets in the array. We qthen approximate the droplet wavefunction via


(0)
(0)
ψi (y) ≈ n(y ≈ yi ) = αi exp −(y − yi )2 /2σi2 withR
αi a normalization coefficient such that |ψi (y)|2 dy = 1.
We then evaluate the wavefunction overlap Si between

the neighboring droplets i − 1 and i via:
Si ≡
=

Z

s

∗
ψi−1
(y)ψi (y)dy

(1)
(0)

(0)

(y − yi−1 )22σi σi−1
exp − i 2222 )σi + σi−12(σi + σi−1

!

.

(2)

The latter equation is obtained via an analytical evaluation of the Gaussian integral. The characteristic linkstrength defined in the paper is then computed by averaging Si over all droplet links in the array: S = hSi ii . In
our calculation, we only consider as droplets all density
peaks of at least 5 % of the global density maximum.

LINK STRENGTH AND ESTIMATE OFTUNNELING RATE

Generally speaking, the wavefunction overlap between
neighboring droplets relates to a tunneling term, whichsets a particle exchange term between two neighboring
droplets [13–16]. Following the work of Ref. [17], we perform a first estimate of the tunneling coefficient by simply considering the single-particle part of the Hamiltonian and evaluate it between two neighboring droplets.
We note that, in our particular setting where the density
modulation is not externally imposed but arises from themere interparticle interactions, the inter-droplet interaction may also play a crucial role. To perform a more
precise estimation of the tunneling between droplets, one
would certainly need to properly account for this effect.
Here, we stress that our approach simply gives a roughidea of the magnitude of tunneling while it does not aim
to be a quantitative description of it. This consideration
calls for further studies making a systematic analysis of
the full Hamiltonian and of the full phase diagram within
the Josephson junction formalism and beyond.
Generalizing the description of Ref. [17] to neighboring droplets of different sizes and amplitudes, which aredescribed by a three-dimensional wavefunction ψi (r) approximated to a three-dimensional Gaussian of widths

2

TOY MODEL FOR THE INTERFERENCEPATTERN

As described in the main text we use a simple toymodel, adapted from Ref. [18], to identify the main features of the TOF interference patterns obtained from aninsitu density-modulated state. As a quick reminder, ourmodel considers a one-dimensional array of ND Gaussian droplets, described by a single classical field, ψi ,thus neglecting quantum and thermal fluctuations. We
compute the TOF density distribution from the freeexpansion Pof the individual ψi during a time t vian(y, t) = | i ψi (y, t)|2 . In our calculations, we also account for the finite imaging resolution by convolving theresulting n(y, t) with a gaussian function of width σim .
Here we allow the characteristics of the individual ψi tofluctuate. In this aim, we introduce noise on the corresponding parameter with a normal distribution around
its expectation value and with a variable standard deviation (only φi can also have a uniform distribution). We
then perform a Monte-Carlo study and perform ensemble
averages, similar to our experimental analysis procedure.
We note that, in this simple implementation, the noiseon the different parameters – droplet amplitudes, widths
and distances – are uncorrelated.

4
3
2
1
0

c

n(µm)

F[n]

n(µm)

F[n]

4g
3
2
1
0
6k
5
4
3
2
1
0 -4 -2 0 2 4ky(µm-1)

x104

5d
4
3
2
1
0
5h
4
3
2
1
0
5l
4
3
2
1
0
-6 -4 -2 0 2 4 6
y (µm)

F[n]

x104

x104

5b
4
3
2
1
0
5f
4
3
2
1
0
5j
4
3
2
1
0
-6 -4 -2 0 2 4 6
y (µm)

n(µm)

x104
6a
5
4
3
2
1
0
5e
4
3
2
1
0
5i
4
3
2
1
0
-4 -2 0 2 4ky(µm-1)F[n]pwhere `x,y,z = h̄/mωx,y,z are the harmonic oscillator
lengths.
In general, the tunnelling coefficients set two typicalrates relevant for equilibration processes. The first one
is the bare single-particle tunneling rate, which is equal
to Ji /h, while the second accounts for the bosonic enhancement from p
the occupation of the droplet modes
and writes t̃i = Ni Ni−1 |Ji |/h where Ni is the number of atoms in droplet i. In our analysis, we then define
the average rates over the droplet arrays as characteristic
rates J/h = hJi ii /h, and t̃ = ht̃i ii ; see e.g. [18]. While
the ground state evolves from a BEC to a supersolid toan ID, the relevant timescale for achieving (global) equilibrium crosses from being set by the trap frequencies tothe above-mentioned tunneling rates.
Using our approximate model, we here give a first estimate of the rates J/h and t̃ as a function of as , for theparameters of Fig. 1(b-d) of the main text (i.e. Er quantum gas with N = 5 × 104 atoms). Here we find that,for as = a∗s , J/h ∼ 400 Hz and t̃ ∼ 10 MHz while for
as = a∗s − 2.5 a0 , J/h ∼ 10−7 Hz and t̃ ∼ 10−3 Hz.

F[n]

(3)

F[n]

+

n(µm)

#
(0)
− yi−1 )2 (σi σi−1 /`y )4 − 1,22σi σi−1
σi2 + σi−1

(0)

(yi

n(µm)

k=x,y,zIn the main text, we present results for a single set ofparameters, namely ND = 4, d ≡ hdi ii = 2.8 µm (meandroplet distance), σy ≡ hσi ii = 0.56 µm (mean dropletsize), t = 30 ms, and σim = 3 µm, typical for our experimental Er setting and the corresponding theory expectations in the supersolid regime. h·ii denotes the averageover the droplets. In this section, we have a deeper lookat the impact of the different parameters on both the
TOF signal and our FT analysis. We study both the
fully phase coherent and fully incoherent case, and theunchanged parameters are set as in Fig. 2(j,m) and (l,o).

n(µm)(σi,x , σi,y , σi,z ) with σi,y = σi , our estimate writes:2
σi,k σi−1,k
2
1
+
X
2
h̄ Si `kJi =
2 + σ2
2m
σi,k
i−1,k

FIG. S1. Toy model realizations with varying numberof droplets ND . We use 100 independent draws, and expectation values d = 2.85 µm, σy = 0.56 µm (with 10% noise) andeither φi = 0 (a,b,e,f,i,j), or φi uniformly distributed between
0 and 2π (c,d,g,h,k,l). (a–d) ND = 2, (e–h) ND = 3 and (i–l)
ND = 8. (a,c,e,g,i,k) TOF density profiles and (b,d,f,h,j,l)corresponding FT analysis of the interference patterns, samecolor code as Fig .2.

In Fig. S1, we first exemplify the TOF and FT profiles for a varying number of droplets, between 2 and 8,
which cover the range of relevant ND over the phase diagram of Fig. 1. The results remain remarkably similar
to the realization of Fig. 2 with only slight quantitativechanges. The main difference lies in the individual interference patterns obtained in the phase incoherent case.
With increasing ND , those profiles become more complex and made of a larger number of peaks (see (c,g,k)).
Yet, in this incoherent case, a similar (non-modulated)
profile is recovered in the averaged n(ky ) for all ND .
Additionally, we note that for the coherent case with
ND = 8, the side peaks in the FT analysis (see (j))
become less visible. By performing additional tests, we
attribute this behavior to the limited TOF duration, t,
used
p in our experiment yielding a typical length scale,h̄t/m (= 3.4µm), which becomes small compared tothe system size (≈ (ND − 1)d + σy ) for large ND . This
intermediate regime in the TOF expansion leads to more

3

We then investigate the evolution of the interference
patterns and their FT analysis for a varying mean droplet
size, σy , while keeping their mean distance, d, fixed. This
study is particularly relevant recalling that, within theJosephson junction formalism (see main text and corresponding section of this Supplemental Material), thekey parameter controlling the tunneling rate between thedroplets is set by the ratio σy /d, and the link strength parameter that we use to characterize the supersolid regime
scales roughly as exp(−(d/2σy )2 ). Thus, in our experiment, σy /d is intrinsically expected to decrease with thescattering length (see Fig. 3). Performing a direct estimate of the average droplet link from the initial state ofour toy model, we find S = 0.004 for the calculationsof Fig. 2(j-o), lying in an expected supersolid regime yetrather close to the supersolid-to-ID transition. Figure
S2 investigates the effect of smaller and larger values ofσy /d (and consequently of S) on the TOF and FT profiles
while independently assuming phase coherence or incoherence. Qualitatively, the features remain similar as inFig. 2(j-o). In the coherent case, side peaks are visible in

3
2
1
0
5g
4
3
2
1
0
-4 -2 0 2 4ky(µm-1)

x104
5 d
4
3
2
1
0
5 h
4
3
2
1
0
-6-4-2 0 2 4 6
y (µm)

F[n]

n(µm)

n(µm)

4
3
2
1
0
5 f
4
3
2
1
0
-6 -4 -2 0 2 4 6
y (µm)

4c

F[n]

5b

F[n]

4
3
2
1
0
5e
4
3
2
1
0
-4 -2 0 2-1 4ky(µm )

x104

x104

5a

F[n]

FIG. S2. Toy model realizations with varying σy /d.
We use 100 independent draws, with ND = 4, d = 2.85 µm(with 10% noise) and either φi = 0 (a,b,e,f,i,j), or φi uniformly distributed between 0 and 2π (c,d,g,h,k,l). For eachrealization we also compute the associated mean S. (a–d)σy /d = 0.1, yielding S = 1.8 × 10−7 (e–h)σy /d = 0.15, matching S = 1.7×10−4 and (i–l) σy /d = 0.25, matching S = 0.028.
(a,c,e,g,i,k) TOF density profiles and (b,d,f,h,j,l) Corresponding FT analysis of the interference patterns, same color code
as Fig. 2.

x104

n(µm)

F[n]

5 d
4
3
2
1
0
5 h
4
3
2
1
0
5 l
4
3
2
1
0
-6 -4 -2 0 2 4 6
y (µm)

F[n]

n(µm)

4g
3
2
1
0
6k
5
4
3
2
1
0 -4 -2 0 2 4ky(µm-1)

x104

F[n]

n(µm)

x104
6c
5
4
3
2
1
0

n(µm)

n(µm)

n(µm)

4e
3
2
1
0
5i
4
3
2
1
0
-4 -2 0 2 4ky(µm-1)

5 b
4
3
2
1
0
5 f
4
3
2
1
0
5 j
4
3
2
1
0
-6 -4 -2 0 2 4 6
y (µm)

F[n]

n(µm)

5
4
3
2
1
0

x104

F[n]

6a

F[n]

x104

the individual as well as in the mean n(ky ) (see (a,e,i))and yield side peaks in the FT profiles, with nM ≈ n (see(b,f,j)). Increasing (decreasing) σ/d mainly results in astronger (weaker) signal both in the interference pattern
and their FT analysis. Within our toy model, we findthat, already for σ/d = 0.25, the signal nearly vanishes;see (i,j). Even if, given the approximations used in our
toy model, this exact value may not fully hold for our
experimental conditions, we expect a similar trend. It isinteresting to keep in mind that this effect may limit ourcapacity of detecting an underlying supersolid state via
matter-wave interference in experiments. In the incoherent case, the effect of decreasing σy /d mainly results ina broader shape of the mean density profile, while it remains non-modulated; see (c,g,k). In the FT analysis nΦremains structure-less independently of σy /d while thestructures in nM becomes sharper with decreasing σy /d,as in the coherent case; see (d,h,l).

n(µm)

complex features, including smaller-sized motifs, in theinterference patterns. Finally, when accounting for ourimaging resolution, it yields a broadening of the structure
observed in the TOF images and less visible peaks in theFT (see (i)). We note that our experiments, because oflimited N and additional losses, should rather lie in theregime 2 ≤ ND ≤ 5; see Fig. 1(b).

FIG. S3. Toy model realizations allowing noise in
the center position. We use 100 independent draws, with
ND = 4, d = 2.85 µm (with 10% noise), σy /d = 0.15 (a–d) or σy /d = 0.2 (e–h), and either φi = 0 (a,b,e,f,i,j), or φiuniformly distributed between 0 and 2π (c,d,g,h,k,l). Center fluctuation are introduced as normal noise around 0 withstandard deviation of 2 µm−1 in situ (a,c,e,g,i,k) TOF density profiles and (b,d,f,h,j,l) corresponding FT analysis of theinterference patterns, same color code as Fig. 2.

Finally, we investigate how a possible shot-to-shotnoise on the position of the central interference peak
could affect our observables of the density modulationand phase coherence. In the experiments, such fluctuations may occur, for instance, because of beam-pointing
fluctuations or excitations of the gas. Although we compensate for such effects by recentering the individual images (see Imaging Analysis section), residual effects may
remain, in particular due to center misestimation in themere presence of the interference patterns of interest. To
investigate this aspect, we repeat our toy model calculations now including noise in the global droplet array
position and using a standard deviation of 2 µm for twovalues of σy /d; see Fig.S3. Again, qualitatively the observed features remains similar to our prediction in themain text. The main effect lies in the appearance of a4small discrepancy in the coherent case between nΦ andnM , while the structure in the incoherent case remains
similar. As the center misestimation should be the most
severe in the latter case (due to the variability of theinterference patterns observed here), our test shows therobustness of our analysis procedure against this issue.

IMAGING ANALYSIS: 164 Dy AND 166 Er

The density distributions in momentum space are extracted from the TOF images using the free-expansion
expectation. In the Dy case, the thermal component is
subtracted from the individual distribution by cuttingout the central region of the cloud and performing an
isotropic Gaussian fit on the outer region. This subtraction is beneficial because of the large thermal fraction. In the 166 Er case, such a subtraction is on thecontrary complicated because of the weak thermal component and this pre-treatment may lead to improper estimation of AM and AΦ in the later analysis. The obtained momentum density distributions are then recentered and integrated numerically along kz (kx ) between[−2.0, +2.0] µm−1 ([−1.28, +1.28] µm−1 ) to obtain n(kY )(n(ky )) for 164 Dy (166 Er). The recentering procedureuses the result a single Gauss fit to the TOF images.
The fit is performed after convoluting each image witha Gaussian function of width 0.5 µm whose purpose isto reduce the impact of the interference pattern on thecenter estimation [19].
In order to characterise the system’s state, we use theFourier transform, F [n](y) of the single density profile,n(ky ). We then compute two average profiles, nM andnΦ , relying on ensemble average over all measurements
under the same experimental conditions; see below for a
detailed discussion on nM and nΦ . In all the measurements reported in this work we use averages over typically
15 to 100 realizations.
To quantify both the existence of a density modulationand global phase coherence on top of this modulation, we
fit both nM (y) and nΦ (y) with a triple-Gaussian function,
where one Gaussian accounts for the central peak and theother Gaussians are accounting for the symmetric side
peaks. The amplitudes of the latter give AM and AΦ ,respectively. The distance between the side peaks andthe central one is allowed to vary between [2.5, 2.7] µm([2.3, 2.5] µm) in the case of 164 Dy (166 Er).

DETAILS ON THE FOURIER ANALYSIS

In our analysis we rely on two averaged profiles, namednM or nΦ , to quantify both the density modulation andits phase coherence. Here we detail the meaning of theaverage performed.

The Fourier transform (FT) of the integrated momentum distributions, n(ky ), which reads F [n](y) =|F [n](y)| exp(i arg (F [n](y))) sets the ground for ouranalysis. As stated in the main text, an in-situ density
modulation of wavelength y ∗ yields patterns in n(ky ) andconsequently induce peaks at y ≈ y ∗ , in the FT norm,|F [n](y)|, see Fig. 2(g-i) and (m-o). Spatial variations of
the phase relation within the above-mentioned density
modulation translate into phase shifts of the interference
patterns, which are stored in the FT argument at y ≈ y ∗ ,arg (F [n](y ∗ )); see also Ref. [18, 20].
The first average that we use is nM (y) = h|F [n](y)|i,i. e. the average of the FT norm of the individual images.
As the phase information contained in arg (F [n](y)) is
discarded from nM when taking the norm, the peaks
in nM probe the mere existence of an insitu density
modulation of roughly constant spacing within the different realizations. The second average of interest isnΦ (y) = |hF [n](y)i|, i. e. the average of the full FT of theindividual images. In contrast to nM , nΦ keeps the phase
information of the individual realizations contained inarg (F [n](y ∗ )). Consequently, peaks in nΦ indicate thatthe phase relation is maintained over the density modulation, in a similar way for all realizations. Their presence
thus provides information on the global phase coherenceof a density-modulated state.
EXPERIMENTAL SEQUENCE: 164 Dy AND 166 Er
166

Erbium - The BEC of 166 Er is prepared similarly toRefs. [1, 8, 21, 22]. We start from a magneto-optical trapwith 2.4 × 107 166 Er atoms at a temperature of 10µK,
spin-polarized in the lowest Zeeman sub-level. In a next
step we load about 3 × 106 atoms into a crossed optical dipole trap (ODT) operated at 1064 nm. We evaporatively cool the atomic cloud by reducing the power
and then increasing the ellipticity of one of the ODTbeams. During the whole evaporation a constant magnetic field of B = 1.9 G (as = 80 a0 ) along z is applied.
We typically achieve BEC with 1.4 × 105 atoms and acondensed fraction of 70%. In a next step the ODTis reshaped in 300 ms into the final trapping frequencies
ωx,y,z = 2π × (227, 31.5, 151) Hz. Consecutively, we ramp
B linearly to 0.62 G (64.5 a0 ) in 50 ms and obtain a BEC
with 8.5 × 104 atoms, which are surrounded by 3.5 × 104
thermal atoms. This point marks the start of the ramp
to the final as .
164
Dysprosium - For the production of a 164 Dy BEC
we closely follow the scheme presented in [23]. Startingfrom a 3 s loading phase of our 5-beam MOT in open-top
configuration [24], we overlap a 1064 nm single-beam
dipole trap with a 1/e2 -waist of about 22 µm, for 120 ms.
Eventually, we transfer typically 8×106 atoms utilizing atime averaging potential technique to increase the spatial
overlap with the MOT. After an initial 1.1 s evaporative5

To be able to condense directly into the supersolid,
we modify the dipole trap to condense at a stronger
confinement of ωx,y,z = 2π × (225, 37, 134) Hz. After a
total evaporative cooling duration of 3.1 s, we achieve
Bose-Einstein condensation at 2.55 G and reach a statewith supersolid properties at 2.43 G, keeping the magnetic field constant throughout the entire evaporation
sequence for both cases.
Time of flight and imaging for 166 Er and 164 Dy - Inorder to probe the momentum distribution of the Dy (Er)gases, we switch off the confining laser beams and let theatoms expand freely for 18 ms (15 ms), while keeping themagnetic field constant. Consecutively the amplitude of
B is increased to a fixed amplitude of 5.4 G (0.6 G). In thecase of 164 Dy, the magnetic field orientation is rotatedin order to point along the imaging axis. This ensuresconstant imaging conditions for different as . After an
additional 9 ms (15 ms) we perform a standard absorption
imaging.
TUNING THE SCATTERING LENGTH IN 166 Er
AND 164 Dy
166Erbium - All measurements start with a BEC at
64.5 a0 . In order to probe the BEC-supersolid-ID region,
we linearly ramp as to its target value in tr = 20 ms
by performing a corresponding ramp in B. Due to a
finite time delay of the magnetic field in our experimental
setup and the highly precise values of as needed for theexperiment, we let the magnetic field stabilize for another15 ms before th = 0 starts. By this, we ensure that theresidual lowering of as during the entire hold time is <∼0.3 a0 . In the main text, we always give the as at th = 0 .
Furthermore, we estimate our magnetic field uncertainty
to be ±2.5 mG, leading to a ±0.2 a0 uncertainty of as in
our experiments.
To choose the best ramping scheme, we have performedexperiments varying tr from 0.5 ms to 60 ms, ramping toa fixed as lying in the supersolid regime, and holding forth = 5 ms after a fixed 15 ms waiting time. We record theevolution of AΦ as a function of tr ; see Fig. S4. Whenincreasing tr , we first observe that AΦ increases, up totr = 20 ms, and then AΦ gradually decreases. The initial
increase can be due to diabatic effects and larger excitation when fast-crossing the phase transition. On theother hand, the slow decrease at longer tr can be explained by larger atom loss during the ramp. We thenchoose tr = 20 ms as an optimum value where a supersolid behavior develops and maintains itself over a significant time while the losses are minimal.
x10

3

5

4

AФ

cooling phase by lowering the power of the beam, we
add a vertically propagating beam, derived from thesame laser, with a 1/e2 -waist of about 130 µm to form acrossed optical dipole trap for additional confinement.
Subsequently, we proceed forced evaporative coolingto reach quantum degeneracy by nearly exponentiallydecreasing the laser powers in the two dipole-trap beams
over 3.6 s. We achieve BECs of 164 Dy with typically 105
atoms and condensate fractions of about 40%. During
the entire evaporation sequence the magnetic field is
kept constant at 2.5 G pointing along the vertical (z-)
axis.

3

2

1

0

0

10

20

30
tr (ms)

40

50

60

FIG. S4. Ramp time effect on the supersolid behavior
Measured AΦ for various durations of the scattering-length
ramp with 166 Er and a final as = 54.1(2) a0 . All measurements include a 15 ms stabilization time after tr and are performed with an additional hold of th = 5 ms.
164

Dysprosium - As the value of the background scattering, abg length for 164 Dy is still under debate [9, 10,25], we discuss the experimental settings in terms of magnetic field. Yet, to gain a better understanding of thetunability of as in our experiment, we first perform a Feshbach spectroscopy scan on a BEC at T = 60 nK. Afterevaporative cooling at B = 2.5 G, we jump to B varying
from 1 G to 7.5 G and we hold the sample for 100 ms.
Finally, we switch off the trap, let the cloud expand for
26 ms and record the total atom number as a function of
B. We then fit the observed loss features with a gaussian
fit to obtain the position B0,i and width ∆Bi of the FRs,
numbered i. We finally use the standard Feshbach resonance formula
Q to estimate the as -to-B dependence via
as (B) = abg i (1 − ∆Bi /(B − B0,i )). Here we accountfor 8 FRs located between 1.2 G and 7.2 G. Depending on
the background scattering length abg , the overall magnitude of as (B) changes. We can get an estimate of abg
from literature. In Fig. S5, we use the value of as from
Ref. [25] obtained at 1.58 G close to the B-region investigated in our experiment, as = 92(8) a0 . By reverting

6
erties of this isotope.
To quantify the role of these losses, we report here
the evolution of the number of condensed atoms, N , as afunction of the hold time in parallel to the phase coherent
character of the density modulation observed. We countN by fitting the thermal fraction of each individual image
with a two-dimensional Gaussian function. To ensurethat only the thermal atoms are fitted, we mask out thecentral region of the cloud associated with the quantumgas. Afterwards we subtract this fit from the image andperform a numerical integration of the resulting image(so called pixel count) to obtain N .
86

N (104)the as (B) formula, we set abg = 87(8) a0 . For the measurements of Figs. 4-5, we ramp B linearly from 2.5 G in
20 ms to a final value ranging from 1.8 to 2.1 G, for which
we estimate as ranging from 97(9) a0 to 105(10) a0 . We
calibrate our magnetic field using RF spectroscopy, witha stability of about 2 mG. In the Dy case, we do not apply
an additional stabilization time. This is justified becauseof the more mellow as -to-B dependence in the B-range
of interest as well as of the wider as -range of the superoslid regime (see Fig. 1) compared to the Er case. For themeasurements of Figs. 6–7, we use two B-values, namely
2.43 G and 2.55 G, at which we perform the evaporative
cooling scheme. Here we estimate as = 109(10) a0 and
as = 134(12) a0 , respectively.
160

e

b

d

f

2

120
AФ(103)

Scattering length (a0)

c

4

140

100
80

0
5
4
3
2
1
0

60
40
1.1

a

1.3

1.5

1.7

1.9
B (G)

2.1

2.3

2.5

FIG. S5. Estimated scattering length tuning in 164 Dy
Estimated dependence of as on B for 164 Dy. The FR positions and widths have been extracted from trap-loss spectroscopy measurements, the background scattering length isestimated to abg = 87(8) a0 , see text. The blue dashed line
gives an error-estimate considering only the errorbar on abg
from the mere as measurement of Ref. [25] and not accounting for uncertainty of the Feshhach scan. For Figs. 4-5, we
use B between 1.8 G and 2.1 G (red area); for Figs. 6–7, we
keep at two constant B-values, namely 2.43 G and 2.55 G (redarrows).

ATOM LOSSES IN 166 Er AND 164 Dy

As pointed out in the main text, in the time evolution of the quantum gases in both the supersolid and the
ID regime, inelastic atom losses play a crucial role. The
atom losses are increased in the above mentioned regime
as (i) higher densities are required so that a stabilization under quantum fluctuation effects becomes relevantand (ii) the magnetic field may need to be tune close toa FR pole to access the relevant regime of interaction
parameters. (i) is at play for all magnetic species but
more significant for 166 Er due to the smaller value of add .
(ii) is relevant for both 166 Er and 162 Dy but conveniently
avoided for 164 Dy thanks to the special short-range prop-

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

FIG. S6. atom number and coherence decays in 166 Er
Time evolution of N and AΦ for 166 Er at different as , including points before th = 0 ms in the experiment. The corresponding scattering lengths are 53.3(2) a0 (a,b), 54.0(2) a0(c,d), 54.2(2) a0 (e,f).
166

Erbium - In the Er case, a 15 ms stabilization timeis added to ensure that as is reached up to 0.3 a0 . During this time, i. e. for th < 0, we suspect that the timeevolution of the cloud properties is mainly dictated by
the mere evolution of the scattering length. Therefore,
in the main text, we report on the time evolution forth ≥ 0. We note that because of the narrow as -range
for the supersolid regime, the long stabilization time for
as is crucial. However, because of the significant role of
the atom losses in our system, in particular for 166 Er,
the early evolution of N and the cloud’s properties are
intimately connected. Therefore, the early time evolution at th < 0 is certainly of high importance for ourobservations at th ≥ 0.
To fully report on this behavior, we show the evolutionof N and AΦ during both the stabilization and the holding time in Fig. S6 for three different as values – either in
the ID (a, b) or supersolid regime (c-f). The time evolution shows significant atom loss, prominent already during the stabilization time, and levels off towards a remaining atom number at longer holding times in which we recover small BECs. Simultaneously, in each case reported
here, we observe that during the stabilization time AΦ
increases and a coherent density modulated state grows.

7
TABLE I. Extracted 1/10-lifetime of 166 Er atom number
decay for th ≥ 0 and remaining atom number at long holding
time for data in Fig. S6.

anism limiting the lifetime of the supersolid properties inDy.

Nr (104 )1.03(5)1.29(11)
1.7(2)tΦ (ms)
25(6)
32(9)

This density modulation starts to appear at a typical4
atom number of N >∼ 6 × 10 and consecutively decays.
For the lower as = 53.3(2) a0 case, we observe that thecoherent state does not survive the as stabilization time,and decays faster than the atoms loss; see Fig. S6 (a, b).
This behavior corresponds to the ID case discussed in
the main text. The central point of the present work is
to identify a parameter range where the coherence of thedensity modulated state survives for th > 0 and its decay
time scale is similar to the one of the atom loss. In orderto quantify a timescale for the atom number decay, we
fit an exponential decay to th ≥ 0 ms. Here we allow anoffset Nr of the fit, accounting for the BEC recovered atlong holding times. In Table I, we report on the typical
1/10-decay times of the atom number, which are up to
50 ms. These values are of the order as the extracted tΦ ,see Table I and Fig. 5 of the main text. This reveals thatin 166 Er the extracted lifetime of the coherent density
modulated states are mainly limited by atom loss.

Furthermore we note that the extracted Nr values for therecovered BECs are smaller than 2 × 104 , which is consistent with the BEC region found in the phase diagramof Fig. 1(f).
164
Dysprosium - Differently from the 166 Er case, for164
Dy, we operate in a magnetic-field range in which thethree-body collision coefficients are small and only moderate atom losses occur. This enables the observationof an unprecendented long-lived supersolid behavior. To
understand the effects limiting the supersolid lifetime, we
study the lifetime of the condensed-atom number for different B. We perform this detailed study for the data ofFig. 5 of the main text, which are obtained after preparing a stable BEC and then ramping B to the target value.
Fig. S7 shows the parallel evolution of N and AΦ for threedifferent magnetic field values 1.8 G, 2.04 G and 2.1 G.
Here we observe that, for all B values, AΦ seems to decay faster than the atom number. This suggests that the
lifetime of the density-modulated state in our 164 Dy experiment is not limited by atom losses. To confirm thisobservation, we extract the 1/10 lifetimes of both N andAΦ ; see Table II. The values confirm our observation andshows an atom number lifetime larger than tΦ at least bya factor of ≈ 5. In addition, we find that the ratio tN /tΦvaries, indicating that atom losses are not the only mech-

a

c

e

b

d

f

2
1
0
12

AФ(102)tN (ms)32(5)51(9)
46(12)

N (104)3

as (a0 )53.3(2)54.0(2)54.2(2)

8
4
0

0

50

100

th (ms)

0

50

100

th (ms)

0

50

100

th (ms)FIG. S7. atom number and coherence decays in 164 Dy
Time evolution of N and AΦ for 164 Dy at different B for thedata of Fig. 5. The corresponding magnetic fields are 1.8 G(a,b), 2.04 G (c,d), 2.1 G (e,f).

TABLE II. Extracted 1/10-lifetime of 164 Dy atom number
decay and AΦ decay for data in Fig. S7.
B (G)
1.8
2.04
2.1

tN (ms)300(12)728(34)926(36)tΦ (ms)12(5)152(13)133(25)[1] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
S. Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, “Observation of roton mode population in a dipolar quantum gas,” Nat. Phys. 14, 442(2018).
[2] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, “Bogoliubov modes of a dipolar condensate in a cylindricaltrap,” Phys. Rev. A 74, 013623 (2006).
[3] A. R. P. Lima and A. Pelster, “Quantum fluctuations in
dipolar Bose gases,” Phys. Rev. A 84, 041604 (2011).
[4] A. R. P. Lima and A. Pelster, “Beyond mean-field lowlying excitations of dipolar Bose gases,” Phys. Rev. A 86,
063609 (2012).
[5] F. Wächtler and L. Santos, “Quantum filaments in
dipolar Bose-Einstein condensates,” Phys. Rev. A 93,
061603(R) (2016).
[6] F. Wächtler and L. Santos, “Ground-state properties andelementary excitations of quantum droplets in dipolar
Bose-Einstein condensates,” Phys. Rev. A 94, 043618(2016).
[7] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,“Ground-state phase diagram of a dipolar condensate

8with quantum fluctuations,” Phys. Rev. A 94, 033619(2016).
[8] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, “Quantum-fluctuation-driven
crossover from a dilute Bose-Einstein condensate to a
macrodroplet in a dipolar quantum fluid,” Phys. Rev.
X 6, 041039 (2016).
[9] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut,
and T. Pfau, “Self-bound droplets of a dilute magnetic
quantum liquid,” Nature (London) 539, 259–262 (2016).
[10] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen,
M. Isoard, S. Stringari, and T. Pfau, “Scissors Mode of
Dipolar Quantum Droplets of Dysprosium Atoms,” Phys.
Rev. Lett. 120, 160402 (2018).
[11] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas,
P. Cheiney, and L. Tarruell, “Quantum liquid droplets
in a mixture of Bose-Einstein condensates,” Science 359,
301–304 (2018).
[12] V. Cikojevi, L. Vranje Marki, G. E. Astrakharchik, andJ. Boronat, “Universality in ultradilute liquid Bose-Bose
mixtures,” arXiv:1811.04436 (2018).
[13] B. D. Josephson, “Possible new effects in superconductive
tunnelling,” Phys. Lett. 1, 251 – 253 (1962).
[14] A. Barone and G. Patern, Physics and applications of theJosephson effect (Wiley, New York, NY, 1982).
[15] J. Javanainen, “Oscillatory exchange of atoms between
traps containing Bose condensates,” Phys. Rev. Lett. 57,
3164–3166 (1986).
[16] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy,“Coherent oscillations between two weakly coupled boseeinstein condensates: Josephson effects, π oscillations,and macroscopic quantum self-trapping,” Phys. Rev. A
59, 620–633 (1999).
[17] M. Wenzel, F. Böttcher, J.-N. Schmidt, M. Eisenmann,
T. Langen, T. Pfau, and I. Ferrier-Barbut, “Anisotropic

Superfluid Behavior of a Dipolar Bose-Einstein Condensate,” Phys. Rev. Lett. 121, 030401 (2018).
[18] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and
J. Dalibard, “Interference of an Array of Independent
Bose-Einstein Condensates,” Phys. Rev. Lett. 93, 180403(2004).
[19] We note that we have also checked our analysis without
performing the recentering step and the same features
remain. For instance, for our test data of Fig. 2, the effectbeing mainly that the side peaks in (e) are more washed
out and a slight difference occurs between nM and nΦ ,both showing still side peaks.
[20] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm,
and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature (London)
449, 324 (2007).
[21] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler,
R. Grimm, and F. Ferlaino, “Bose-Einstein condensationof Erbium,” Phys. Rev. Lett. 108, 210401 (2012).
[22] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz, and F. Ferlaino, “Probing
the roton excitation spectrum of a stable dipolar Bosegas,” arXiv:1811.12115 (2018).
[23] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi,
M. Sohmen, M. J. Mark, and F. Ferlaino, “Dipolar
Quantum Mixtures of Erbium and Dysprosium Atoms,”
Phys. Rev. Lett. 121, 213601 (2018).
[24] P. Ilzhöfer, G. Durastante, A. Patscheider, A. Trautmann, M. J. Mark, and F. Ferlaino, “Two-species fivebeam magneto-optical trap for erbium and dysprosium,”
Phys. Rev. A 97, 023633 (2018).
[25] Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and
B. L. Lev, “s-wave scattering lengths of the strongly dipolar bosons 162 Dy and 164 Dy,” Phys. Rev. A 92, 022703(2015).

180

A.3

Additional publications

Publication:

Feshbach resonances in an erbium-dysprosium dipolar mixture
The following publication has appeared in
Physical Review A 102, 033330 (2020)‡
submitted 12 June 2020; published 21 Sept 2020
DOI: https://doi.org/10.1103/PhysRevA.102.033330
G. Durastante,1,2 C. Politi,1,2 M. Sohmen,1,2 P. Ilzhöfer,1,2 M. J. Mark,1,2 M. A. Norcia,1and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria‡The author of the present thesis performed the measurements together with G. D., M. A. N.,

and M. S., analysed the experimental data with G. D., and M. A. N., and contributed in writing themanuscript and interpreting the results together with all the authors.

PHYSICAL REVIEW A 102, 033330 (2020)

Feshbach resonances in an erbium-dysprosium dipolar mixture
Gianmaria Durastante,1,2 Claudia Politi,1,2 Maximilian Sohmen ,1,2 Philipp Ilzhöfer,1,2 Manfred J. Mark ,1,2Matthew A. Norcia ,1 and Francesca Ferlaino1,2,*
12Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria(Received 12 June 2020; accepted 25 August 2020; published 21 September 2020)
We report on the observation of heteronuclear magnetic Feshbach resonances in several isotope mixtures of
the highly magnetic elements erbium and dysprosium. Among many narrow features, we identify two resonanceswith a width greater than one Gauss. We characterize one of these resonances, in a mixture of 168 Er and 164 Dy, interms of loss rates and elastic cross section, and observe a temperature dependence of the on-resonance loss ratesuggestive of a universal scaling associated with broad resonances. Our observations hold promise for the use ofsuch a resonance for tuning the interspecies scattering properties in a dipolar mixture. We further compare theprevalence of narrow resonances in an 166 Er - 164 Dy mixture to the single-species case, and observe an increased
density of resonances in the mixture.
DOI: 10.1103/PhysRevA.102.033330

I. INTRODUCTION

Ultracold quantum gases are a highly successful platform
for physics research largely because it is possible to create
simplified and controllable versions of condensed matter systems [1]. As the field has advanced, great progress has been
made by reintroducing complexity in a carefully controlledmanner. This complexity can manifest in the form of interparticle interactions [2–4], the species and statistics of theparticle under study [5–7], or in the form of the potential
landscape, control protocols, and imaging techniques applied
to the system [8,9]. In this work, we explore interspecies Feshbach resonances as a means of generating tunable interactions
between two different species of complex dipolar atoms.
Atoms with large magnetic dipole moments, such as thelanthanide series elements erbium and dysprosium, interact
in a manner that is both long-range and anisotropic. This
is in contrast to more commonly used atomic species, suchas alkali and alkaline earth metals, which primarily interact
in a short-range and isotropic way. The recent creation ofdegenerate Bose and Fermi gases of such atoms [10–13] hasenabled the observation of a wealth of phenomena including
quantum-stabilized droplet states [14–16], roton quasiparticles [17], supersolid states [18–20], and a nonisotropic Fermi
surface [21].
In a separate direction, degenerate mixtures of multiple
atomic species have also provided diverse opportunities forthe study of physical phenomena. Examples include studies
of polarons that arise when an impurity species interacts witha background gas [22–27], and the formation of heteronuclear
molecules with large electric dipole moments [28–31].
We expect that combining dipolar interactions with heteronuclear mixtures will lead to a rich set of physical*Corresponding author: Francesca.Ferlaino@uibk.ac.at2469-9926/2020/102(3)/033330(9)phenomena, the exploration of which has only recently begun. In particular, dipolar interactions are expected to havedramatic consequences for the miscibility of binary condensates [32–34], and in turn on vortex lattices that arise insuch systems [35]. Further, certain properties of polarons are
predicted to emerge when either the background [36] or bothbackground and impurity [37] particles experience dipolarinteractions [38].
Dipolar heteronuclear mixtures have recently been demonstrated [39], but so far the interspecies scattering properties
have not been explored, either experimentally or theoretically. In these complex dipolar species, scattering propertiesare dictated by both anisotropic long-range dipolar interactions, which can be tuned through a combination of systemgeometry and magnetic field angle, and by contact interactions, which can be tuned through the use of interspecies
Feshbach resonances. While scattering models and experimental demonstrations exist for mixtures of single- andtwo-valence electron atoms (which lack strong dipolar interactions) [40,41], the scenario of two multivalence electron
atoms has yet to be considered, and represents a frontier for
our understanding of ultracold scattering. In many commonlyused atomic systems, the strength, character, and location ofmagnetic Feshbach resonances can be predicted with highprecision through coupled-channel calculations [3]. However,the complexity of the internal level structure and couplingmechanisms present in lanthanide atoms lead to significant
challenges for the development of a microscopic theory withpredictive power, and so necessitate an experimental survey tofind resonances with favorable properties [42–46].
To this end, we searched for heteronuclear Feshbach resonances broad enough to provide a practical means for tuning
the interspecies interaction in Bose-Bose and Bose-Fermi
dipolar quantum mixtures. Using atomic-loss spectroscopy toidentify resonances, we perform surveys of fermionic 161 Dy
and bosonic 164 Dy together with 166 Er, 168 Er, and 170 Er over033330-1©2020 American Physical Society

GIANMARIA DURASTANTE et al.

PHYSICAL REVIEW A 102, 033330 (2020)

TABLE I. Comparatively broad resonances found in specific isotope mixtures together with estimated center positions and widths(FWHM) from Gaussian fits to atom loss spectra. Each value is anaverage between the fit values of Er and Dy.
Combination
168

Er - 164 Dy
Er - 164 Dy
166
Er - 161 Dy
168
Er - 161 Dy
170
Er - 161 Dy
166

Resonance magnetic field (G)Width (G)13.32(4)34.09(3)161.31(3)161.30(2)161.26(3)1.7(1)
1.5(1)
0.84(9)0.93(5)0.91(8)a magnetic-field range from zero to several hundred gauss(the exact range varies by isotope combination due to availability of favorable evaporation conditions). We also exploreda Fermi-Fermi mixture of 167 Er and 161 Dy, but observed nobroad resonances there. In Table I we summarize positions
and widths of these features observed in our surveys. As an
exemplary case, we present a more detailed characterizationof the resonance near 13.5 G in the 168 Er - 164 Dy Bose-Bose
mixture, through measurements of interspecies thermalizationand the dependence of atomic loss on temperature.
In addition, our dipolar mixtures host a large number ofnarrow interspecies resonances. In previous experiments withsingle species, the density and spacing of these narrow resonances has been studied to reveal a pseudorandom distributionthat can be modeled well using random matrices [43,45,46].
By performing high resolution scans over specific magneticfield ranges, we find that the average density of interspecies
resonances exceeds the combined density of intraspecies resonances, perhaps indicating the contribution of odd partial
waves or molecular states with antisymmetric electron configurations for the interspecies case, which are not present in
the scattering of identical bosons.
Finally, in each Fermi-Bose mixture involving 161 Dy weobserve a correlated loss feature between fermionic Dy andbosonic Er atoms. Strangely, the loss feature is present at
the same magnetic-field value for all three bosonic erbium
isotopes studied. Such behavior is inconsistent with a typicalinterspecies Feshbach resonance, where the magnetic field
at which the resonance occurs is strongly dependent on the
reduced mass of the atoms involved [47]. The mechanism
behind this unusual feature is as of yet unknown and calls forfurther experimental and theoretical investigations.

II. OBSERVATION OF INTERSPECIES RESONANCES

Our experimental sequence is similar to the one introduced in our previous works [39,48]. After cooling the desired
isotope combination of erbium and dysprosium atoms in adual-species magneto-optical trap (MOT), we load the atoms
into a crossed optical dipole trap (ODT) created by 1064-nm
laser light. Here we perform evaporative cooling down to thedesired sample temperature. During the whole evaporation sequence, we apply a constant and homogeneous magnetic field(Bev ), pointing along the z direction opposite to gravity. Bev
preserves the spin polarization in the lowest Zeeman sublevelof both species. We use different values of Bev to optimize
the evaporation efficiency depending on the isotope combination and on the range of the target magnetic field (BFB ) to
be investigated. The final ODT has trap frequencies ωx,y,z =2π × (222, 24, 194)s−1 . We typically obtain mixtures withatom numbers ranging from 3 × 104 to 1 × 105 atoms for eachspecies. The sample is in thermal equilibrium at about 500 nK,
which corresponds to about twice the critical temperature forcondensation. Typical densities are up to a few ×1012 cm−3for each species. After preparing the mixture, we linearlyramp the magnetic field from Bev to BFB in 5 ms, either in
an increasing or decreasing manner. The current flowing in
the coils that generate the magnetic fields can be changed on
the millisecond time scale, and the field at the position of theatoms settles to the part-per-thousand level in approximately
10 ms. We hold the mixture for a time ranging between 5 ms
and 400 ms depending on the experiment. At the end of thehold time, we release the atoms from the ODT in a 15 ms timeof-flight (TOF) expansion after which we record an image ofthe atoms using a standard low-field absorption imaging technique [12]. Note that we adjust the relative amount of erbium
and dysprosium in the final thermal mixture for the specificexperiments by independently tuning the MOT loading timefor each species between 0.5 and 5 s.
In the isotope combinations and range of magnetic fields
that we explore here, we observe two interspecies resonanceswith widths greater than 1 G (see Table I). We now turn toa more detailed characterization of a feature present in the168
Er - 164 Dy combination, for which atom loss is shown in
Fig. 1(a). We chose to focus on this feature because it isrelatively isolated from the many narrow homonuclear andheteronuclear resonances typical of lanthanides. In this experiment, the starting mixture contains 6.2 × 104 erbium and9.1 × 104 dysprosium atoms and it is prepared by evaporation at Bev = 10.9 G. In order to compensate for loss during
magnetic-field ramps and slow drifts of the atom number, wenormalize measurements performed with 200-ms hold times
at BFB to interleaved measurements at 10-ms hold time at
the same field. We further performed independent trap-loss
spectra in single-species operation to confirm the interspecies
nature of the resonance. Moreover, such scans allow us to
identify intraspecies resonances and exclude them from thefit [see empty symbols in Fig. 1(a)]. As shown in the inset for
erbium, a high-resolution scan reveals a narrow region withless loss near the center of our broad loss feature, probably due
to the influence of a second interspecies resonance—because aresonance contributes a scattering length with a different signon either side of its pole, the contributions from two nearby
resonances may counteract in between them, leading to a reduction in loss. This structure is also visible on the dysprosium
loss feature but it is not shown in the inset for ease of reading.
A Gaussian fit to the loss profiles, with known narrow
single-species resonance excluded, returns a center value of
13.31(2) and 13.33(4) G and a full width at half maximum(FWHM) value of 1.95(5) and 1.3(1) G for erbium and dysprosium, respectively. The observed difference in the fitted
width of the two species can be explained by the imbalance
in atom number: because this measurement was performed
with fewer erbium atoms than dysprosium, the fractional lossof erbium is higher than that of dysprosium, leading to a

033330-2

FESHBACH RESONANCES IN AN ERBIUM-DYSPROSIUM …
1
(a)

0.6
0.4

3

Atom
numberNormalized
atom number

0.8

0.2

104

2
1
0

0

m )

2

(b)

6

ErDy

(10

-15

Cross-section

8

4
2
0

11

12

13

14

15

16

17

18

Magnetic field (G)FIG. 1. (a) Trap loss from the 13.5 G resonance in the BoseBose mixture 168 Er - 164 Dy (red circles and blue squares points,
respectively). Empty symbols correspond to narrow single-species
resonances, which we exclude from fits. Each point is an averageover four experimental repetitions. For each magnetic field, the atom
number recorded after 200 ms of hold time is normalized to that at ashort hold time of 10 ms. The lines are the Gaussian fits to the data.
The inset shows erbium loss measured in a different dataset with
5-mG resolution, and highlights the structure present on the centerof the feature. The same structure is visible also for the dysprosium
atoms in the mixture. (b) Interspecies elastic cross section σErDy
measured across the Feshbach resonance using cross-species thermalization. Each value of σErDy is extracted from thermalization datausing a numerical model for thermalization that includes temporalvariation in atom number and temperature; see main text and the
Appendix.

greater saturation of loss and broadening of the erbium lossfeature.
III. INTERSPECIES THERMALIZATION

To get insights on its effective strength and width, we perform cross-species thermalization measurements across theresonance [see Fig. 1(b)]. Interspecies thermalization experiments are well established techniques to extract effective
thermalization cross sections, which in turn depend on the
scattering length [49–51]. While inferring a precise valueof the scattering length would require the development ofa detailed and rigorous model that accurately captures thetemperature dependence of the interspecies and anisotropic
dipolar scattering [52], and would go beyond the scope of thiswork, we are able to determine a thermally averaged scatteringcross-section from which we can estimate the width of theresonance.
In this cross-thermalization experiment, we selectively heat
dysprosium by means of a near-resonant 421-nm light pulse

PHYSICAL REVIEW A 102, 033330 (2020)

along the vertical direction. We confirmed that the light pulse
has no direct measurable effect on erbium. The magnetic fieldis then jumped to the desired value BFB and held for a variable
amount of time, during which the temperature of erbium rises
to equilibrate with dysprosium due to elastic collisions. We
record the temperature of the two species along a direction
orthogonal to the heating pulse, as the effects of center of mass
motion are less prevalent here [53], and use a numerical model
to extract a cross section from the rate of thermalization. This
simple model assumes an energy independent cross section,
an assumption which may break down near resonance where
unitarity limits on scattering may become significant.
From these thermalization measurements, we can see a dramatic increase in the scattering cross section near resonance,
as one would expect for an interspecies Feshbach resonance.
Further, we observe a significant modification of the cross section associated with the resonance over a Gauss-scale rangeof magnetic fields, similar to the width we observe in loss
measurements. For comparison, the expected contribution to
the scattering cross section due to dipolar interactions is 2.4 ×
10−16 m2 , over an order of magnitude below what we infer
near resonance. While the exact relationship between the measured cross section and scattering length is complicated by
the finite temperature of our atoms and anisotropic nature of
the interactions, an approximate value can be attained throughthe simple expression for s-wave scattering σ = 4π as2 , where
as is the s-wave scattering length [3]. Our largest measured
cross sections, near resonance at 13.5 G, imply a scattering
length of roughly 400 a0 , substantially greater than the dipolar
length aD = mμ0 μEr μDy /4π h̄2  139 a0 associated with interspecies collisions (a0 is the Bohr radius). In this expression,m = mEr mDy /(mEr + mDy ) is the reduced mass, and μEr , μDy
are the ground-state magnetic dipole moments for erbium anddysprosium, respectively.
For an isolated resonance and pure contact interactions, acommon way to characterize the resonance width is the parameter , given by the difference in magnetic field between
the pole of the resonance, at which the thermalization rate is
maximal, and the nearest zero crossing in the thermalization
rate, which would correspond to a lack of scattering [3]. In
lanthanides, the presence of anisotropic dipolar interactions
leads to a scattering cross section that does not completely
vanish. In addition, multiple narrow and overlapping resonances may be present, which may influence the interpretationof such a width measurement. However, to get a rough estimate of the width of the resonance, we can consider the
distance between the resonance pole and the apparent minimum in the thermalization rate at 17 G. This suggests a widthof   3.5 G.
IV. DEPENDENCE OF LOSS ON TEMPERATURE

The dependence of the loss feature on the cloud temperature can provide additional information on the nature of
the resonance. For broad resonances, a universal regime isexpected to emerge near resonance where the scattering crosssection and loss are dictated primarily by the atomic momentum, rather than the scattering length [54]. In this regime, thethree-body loss parameter L3 follows a nearly universal form
scaling as 1/T 2 , where T is the temperature. Such scaling has

033330-3

GIANMARIA DURASTANTE et al.

PHYSICAL REVIEW A 102, 033330 (2020)

3

Width (G)2

L 3 (10 -24 cm6 /s)

2.5
2

1
0

1.5

0.5

1

1.5

Temperature (µK)
1
0.5
0
0.5

1

1.5

2

2.5

Temperature (µK)
FIG. 2. Three-body loss coefficient L3 extracted from onresonance loss measurements at the resonance position for different
temperatures (black circles), along with a fit to a 1/T 2 scaling (blackline), as expected for universal three-body loss. The inset shows
the resonance width extracted as FWHM from Gaussian fits to the
trap-loss spectra versus cloud temperature for a different dataset. Red
circles and blue squares refer to erbium and dysprosium, respectively.
The reported temperature comes from a TOF estimation.

been observed in broad resonances of several atomic species[54–56].
We observe a temperature dependence of the loss rate nearresonance that is suggestive of such universal behavior. By
varying the final depth of the ODT reached during evaporation, we tune the temperature of the atomic mixture. For eachtemperature, we measure atom loss on resonance at 13.4 G asa function of the hold time. We then use a numerical model toextract the rate of interspecies three-body loss, and L3 .

These loss coefficients are plotted as a function of temperature in Fig. 2, along with a fit to a 1/T 2 dependence, which
provides a reasonable description of our data. The universal
temperature dependence arises from a maximum value of L3
possible at a given temperature, given by√
λ3,maxh̄5 36 3π 2.
(1)
L3,max =T2
m3 (kB T )2
Factors associated with Efimov physics can lead to a lower
value for L3 , but not higher [54,57,58]. From our fit to a
1/T 2 dependence for our data, we extract a value of λ3 =1.0(2) × 10−24 μK2 cm6 s−1 , which is compatible with the predicted bound of λ3,max = 2.4 × 10−24 μK2 cm6 s−1 .
A reduction in the peak loss rate with increasing temperature can also result from thermal broadening of the resonance,
especially for very narrow resonances [45]. This is unlikely
to be the dominant effect here, as for typical differential
magnetic moments between entrance and closed channels in
our lanthanide system [59], we would expect broadening on
the scale of a few times 10 mG for temperatures near 1 μK,
much narrower than the Gauss-scale width of our feature.
Further, suppression of peak loss is typically accompanied by
a commensurate broadening and shift of the loss feature on thescale of its width, which we do not observe (inset in Fig. 2).
V. SURVEY OF NARROW RESONANCESIn addition to the few relatively broad resonances, thelanthanides exhibit many narrow resonances, whose statistical properties have been investigated for single-species gases[43,45,46]. In this section we compare the abundance ofinterspecies resonances to single-species resonances by performing high-resolution trap-loss spectroscopy on the isotope
combination 166 Er - 164 Dy (see Fig. 3). Here, we investigate four different magnetic-field ranges, each 10 G wide,

FIG. 3. High-resolution trap-loss spectroscopy for a balanced mixture of 166 Er and 164 Dy (red and blue curves respectively), with initialatom numbers of roughly 105 per species at a temperature of 500 nK after 400 ms of interaction time. The magenta ticks indicate theheteronuclear resonance positions as extracted by our analysis (see main text). The measurement is composed of four datasets [0, 10] G,[22, 32] G, [50, 60] G, and [60, 70] G with a stepsize of 5 mG. Each point is an average over four experimental repetitions. Atom numbers arenormalized to the maximum of each dataset for ease of reading. The broad loss feature in Dy near 68.8 G was not observed in previous work[45], and may result from a technical source of loss in our experiment.
033330-4

FESHBACH RESONANCES IN AN ERBIUM-DYSPROSIUM …
100
80

(a)

(b)

60

Number of resonances

40
20
0
0
100

2

80

(c)

4

6

8

10

22

24

26

28

30

32

64

66

68

70

(d)

60
40
20
0
50

52

54

56

58

60

60

62

Magnetic field (G)FIG. 4. (a)–(d) Staircase function describing the number of Feshbach resonances as a function of the four investigated magnetic-field
ranges: [0, 10] G, [22, 32] G, [50, 60] G, and [60, 70] G respectively.
The black line shows the number of heteronuclear resonances. The
red and blue lines show the number of homonuclear resonancesfor 166 Er and 164 Dy, respectively. The shaded areas represent ourconfidence intervals (see main text).

with a resolution 40 times higher than the one used for the
exploratory surveys. To enable direct comparison with theprevious works performed on single species [43,45], we use
similar experimental conditions (isotope, atom number, temperature, and hold time).
As expected, we observe many narrow homonuclear resonances [43,45]. In addition, we also identify many narrow
heteronuclear resonances. To distinguish these two types ofresonance, we first label features with a fractional loss above
30% as resonances. We then categorize these resonances asinterspecies if erbium and dysprosium loss features occur
simultaneously within a range of ±10 mG and with a loss
amplitude ratio in the range 0.5–2. Features that do not meetboth of these criteria, are labeled either as homonuclear or ambiguous, based on comparison with separate scans performed
with single species, either within this work or from previously
published data [43,45]. The numbers of ambiguous features
define our confidence intervals.
In order to visualize the number of resonances, weconstruct the staircase function N (B), which describes
the cumulative number of resonances from the start ofa scan range up to a given magnetic field BFB . Figures 4(a)–4(d) shows N (B) for the four investigatedmagnetic-field ranges. The black lines represent heteronuclear Feshbach resonances, while the red and the blue lines
represent the homonuclear 166 Er and 164 Dy resonances, respectively. The shaded regions represent our confidence

PHYSICAL REVIEW A 102, 033330 (2020)

interval defined by the total number of ambiguous Feshbachresonances.
Our analysis results in a total number of heteronuclear
resonances of NErDy (tot) = 339(16), counting all magneticfield ranges, and a number of homonuclear resonances ofNEr (tot) = 116(16) and NDy (tot) = 144(16). Within our confidence intervals, we detect a total number of homonuclear
resonances comparable with those of previous works [43,45].
The corresponding total density of resonances ρ̄, given by
the total number of resonances divided by the total rangeof magnetic fields scanned are ρ̄ErDy = 8.5(4) G−1 , ρ̄Er =2.9(4) G−1 , and ρ̄Dy = 3.6(4) G−1 .
For our combined dataset, we find that the total numberof heteronuclear resonances exceeds the combined number ofhomonuclear resonances for the two species: ρ̄ErDy = α(ρ̄Er +ρ̄Dy ), with α = 1.3(2). We would expect that the averagedensity of heteronuclear resonances should be greater than
the sum of the two homonuclear resonance densities. Thisis because each species contributes a set of internal statesthat can be coupled to, and the heteronuclear resonances are
not subject to the same symmetrization requirements as thehomonuclear resonances. In resonances involving distinguishable particles, both gerade and ungerade Born-Oppenheimer
molecular potentials contribute, as well as both even and odd
partial waves for the entrance channel. Our data is consistent
with this expectation (α > 1). Note that we do observe a lower
number of interspecies resonances in the range 50–60 G, perhaps as a result of the nonrandom distribution of resonances as
observed in the single-species case [43,45], or to the presenceof broad homonuclear erbium resonances that could obscure
the observation of interspecies resonances.
VI. COINCIDENT LOSS FEATURE
IN DIFFERENT ISOTOPES

Finally, we have also searched for broad (Gauss-range)
resonances in Bose-Fermi mixtures consisting of fermionic161
Dy combined with different bosonic isotopes of erbium–
166
Er, 168 Er, and 170 Er, as well as Fermi-Fermi mixtures of161
Dy and 167 Er. For these combinations, we perform only
coarse scans and thus only resolve broad features. In mixtures
involving the bosonic isotopes of erbium we observe a correlated loss feature between erbium and dysprosium near 161
G (see Fig. 5). This loss feature is not present at our levelof measurement sensitivity with either species alone, or in
the mixture with the fermionic 167 Er. Surprisingly, the lossfeature is centered at the same magnetic field (to within our
resolution of 0.1 G) for all bosonic isotopes of erbium. This isquite unexpected as the magnetic-field value of the resonanceposition is typically highly sensitive to the reduced mass of
the atoms involved [47].
To shed more light on this puzzling feature, we conductedfurther investigations on the exemplary case of 168 Er - 161 Dy.
For such a mixture, we performed loss measurements with
erbium numbers ranging from 1.3 × 104 to 3 × 104 (with thenumber of dysprosium fixed at 2.5 × 104 ). We found in eachcase that the number of erbium lost was roughly equal to
the number of dysprosium lost, and that the number of total
atoms lost was roughly proportional to the number of erbium
present. We further varied the temperature of the mixture from

033330-5

GIANMARIA DURASTANTE et al.

PHYSICAL REVIEW A 102, 033330 (2020)VII. CONCLUSIONS

1

Normalized atom number

0.5

(a)

0
1
0.5

(b)

0
1
0.5

(c)

0
1
0.5
0
158

(d)
159

160

161

162

163

164

165

Magnetic field (G)FIG. 5. Trap loss spectra for fermionic 161 Dy in combination
with bosonic 166 Er, 168 Er, and 170 Er, and fermionic 167 Er (a)–(d),respectively. Red circles represent erbium, blues squares represent
dysprosium, and lines are Gaussian fits to the losses. The solid vertical gray lines highlight the peak centers from the fit over dysprosium
losses. For the plots with bosonic erbium, the atom number after
100 ms of interaction time is normalized to a short hold time of 5
ms. In the plot with 167 Er, the normalization is performed using the
maximum value in the dataset. For all panels, each point is an averageover four experimental repetitions.

600 to 2000 nK, and did not find a dramatic dependence of theloss coefficient on temperature (modeling loss as either twobody Er-Dy loss or three-body loss with equal coefficients for
Er-Er-Dy and Er-Dy-Dy).
Several physical mechanisms could be consistent with sucha feature. One possibility is that the resonance we observe isassociated with a bound state of a shallow molecular potential[60]. Mechanisms to create such potentials have been proposed for species with dipolar interactions [61,62]. However,none are obviously applicable to magnetic atoms in the lowest
energy entrance channel. Further, given the level of insensitivity to the mass of erbium, we would expect to see additional
resonances of a shallow potential in the magnetic-field range
over which we survey, which we do not. A second possibility
is that the feature we observe is not a true interspecies resonance, but rather an intraspecies resonance in dysprosium
whose loss rate is enhanced by the presence of bosonic erbium atoms. A similar effect was reported in a mixture offermionic lithium and bosonic rubidium atoms [63]. Finally,
it is possible that this feature is not a Feshbach resonance atall, but rather the result of spin-changing processes resulting
from unintentional radio-frequency tones in the laboratory, orof an interspecies photoassociation resonance. We have ruledout the most likely culprits for the last effect by varying therelative detuning between our horizontal and vertical dipole
traps and observing no change in the resonance position. We
hope that our presentation of this mysterious feature may spur
theoretical exploration of possible physical mechanisms.

In conclusion, we have reported experimental observationof heteronuclear magnetic Feshbach resonances in several
isotope mixtures of erbium and dysprosium. Among theGauss-broad features identified in our surveys, we have characterized one in the combination 168 Er - 164 Dy by means ofcross-species thermalization measurement and temperaturedependence analysis. We performed high-resolution trap-loss
spectroscopy in the combination 166 Er - 164 Dy to compare theaverage resonance density of the mixture with respect tothe single-species case. In mixtures of fermionic 161 Dy andbosonic erbium atoms, we observed a correlated loss feature
which appears to be insensitive on the erbium isotope used
but absent in dysprosium alone. Our observations pave theway to realize tunable interactions in quantum degenerate
mixtures of dipolar atoms. Knowledge of the range of tunability of these interspecies interactions is a key ingredientfor varied opportunities including studies of the miscibility ofbinary condensates, of vortex lattices, and of dipolar polarons[32–37].

ACKNOWLEDGMENTS

We thank J. Hutson, M. Frye, J. Bohn, A. Trautmann,and the Erbium and DyK teams in Innsbruck for insightful
discussions. This work is financially supported through an
ERC Consolidator Grant (RARE, No. 681432), a NFRI Grant(MIRARE, No. ÖAW0600) from the Austrian Academy of
Science, a QuantERA grant MAQS by the Austrian Science Fund FWF No. I4391-N, and a DFG/FWF (FORNo. 2247/PI2790). M.S. and G.D. acknowledge support
by the Austrian Science Fund FWF within the DK-ALM(No. W1259-N27). We also acknowledge the Innsbruck Laser
Core Facility, financed by the Austrian Federal Ministry of
Science, Research and Economy.

APPENDIX: MEASUREMENT AND ANALYSIS DETAILS
1. Cross-species thermalization

As an exemplary case, we study in more detail the resonance found in the 168 Er - 164 Dy Bose-Bose mixture near 13.5
G. To reliably quantify the value of the interspecies crosssection, we developed the following scheme for cross-species
thermalization measurements [49–51]. To avoid heating of thesample by crossing Feshbach resonances, we evaporate themixture at Bev close to resonance. Specifically, when measuring on the low(high)-field side of the feature we evaporate
at Bevap = 10.8 G(16.4 G). Once the sample is prepared aspreviously described (here we use an unbalanced mixture withtwice as much Dy as Er), we compress the trap by linearly
increasing the horizontal ODT power by a factor of 5 and thevertical ODT power by two in 500 ms to prevent any plain
evaporation. The final trap frequencies in the compressed trapare ωx,y,z = 2π × (409, 26, 391)s−1 . Subsequently, we ramp
the magnetic field in 5 ms to either 10 or 16 G. Here, a pulseof near-resonant 421-nm light propagating along the magnetic
field direction (z) is used to selectively heat dysprosium. We
fix the duration of the pulse at 5.5 ms to roughly match the trap

033330-6

FESHBACH RESONANCES IN AN ERBIUM-DYSPROSIUM …

PHYSICAL REVIEW A 102, 033330 (2020)of species 1 with atoms of species 2 given byN2 m13/2 ω̄13
T1
T2
+
σ12 ,
γ12 = 2−2
3/2
π kB (T1 + β T2 )
m1
m2

(A1)

where m1 , m2 , T1 , and T2 are the masses and temperaturesof species 1 and 2, ω̄ = (ωx ωy ωz )1/3 characterizes the frequency of the trap, β 2 = m2 ω̄22 /m1 ω̄12 , and σ12 is the effective
interspecies cross section. We assume that the energy exchanged per collision is given by E = ξ kB (T2 − T1 ) whereξ = 4m1 m2 /(m1 + m2 )2 , and that the heat capacity of each
atom is 3kB . This leads to a differential equation for thetemperature of erbium:

FIG. 6. Sample temperature traces for erbium (filled circles) anddysprosium (hollow squares) after dysprosium is heated. Purple,
green, and orange correspond to magnetic fields of 12, 13.5, and
17 G, respectively. Fit lines represent the results of the numericalintegration of Eq. (A2), which fits the temperature profile of erbium
based on its initial value and the dysprosium temperatures. Different
evaporation conditions cause the curves to have slightly different
initial and final conditions (see main text).

3/2 3
ω̄Er
ξ kB (TDy − TEr )NDy mEr
dTEr
=
dt
3π 2 kB (TEr + β −2 TDy )3/2

TDy
TEr
×
+
σErDy ,
mEr
mDy

(A2)

which we can numerically integrate using the instantaneous
values for TDy and NDy , and from this extract the scatteringcross section σErDy that yields a thermalization profile thatbest matches our data, as determined through a least-squares
difference. Examples of three such fits, for 12, 13.5, and 17 G
are shown in Fig. 6, and generally describe our thermalization
data well.
2. Temperature dependence of loss

oscillation period along this direction and set the pulse intensity to give the desired temperature increase of the dysprosium
cloud (up to 4 μK). We confirmed that the light pulse has nodirect measurable effect on erbium. Finally, with a quench
fast compared to the shortest thermalization rate, the magnetic
field is set to the desired value BFB and held for a variable
amount of time, during which the temperature of erbium rises
to equilibrate with dysprosium due to thermalizing collisions(Fig. 6). We note that in the temperature evolution of theclouds, the initial temperatures are slightly different. Thisbehavior is mainly due to different evaporation conditions
on the two sides of the resonance, the different strength in
the quench to the final BFB , and the heating caused by theresonance itself. By comparing the two species’ temperature,
we ensure that these different conditions are consistent withgeneral offsets on the single measurement thus not affecting
the final estimation of the cross-section.
To extract a scattering cross section from our cross-species
thermalization data, we use a fit to a numerical model forthe thermalization of two species. In principle, a simple exponential fit to the temperature difference between the two
species could also be used, but does not account for changes
in the atom number or average temperature of the sample that
may arise from residual evaporation during the thermalization
time. Our numerical model follows that of Ref. [49]. We treat
the scattering cross section as independent of the energy of thecolliding particles, an assumption that greatly simplifies theanalysis, but inevitably breaks down near enough to resonance
where unitarity considerations bound the scattering cross section. This assumption leads to a collision rate for each atom

We quantify the temperature dependence of three-bodyloss in terms of the interspecies three-body loss coefficient.
For a single species, the three-body loss coefficient L3 can
2
be defined by Ṅ/N = −L 33n ,3 where N is the total number
2
of atoms, and n  = d r n (r)/N represents the averagesquared density of scattering partners for an atom in the gas.
n(r) is the local density of the gas.
We define analogous quantities for our two-species mixture, containing particles denoted i and j. In this case,

Ṅi
−1
=
d 3 r (2L3i,i, j ni2 (r)n j (r) + L3j, j,i ni (r)n2j (r)).
Ni
3Ni
(A3)
Here, L3i,i, j represents the loss rate due to collisions involving
two atoms of species i and one of j.
To arrive at simple expressions, we make several assumptions and approximations. First, we treat the mass,temperature, and polarizability of the two atomic species as
equal, which is a reasonable approximation for erbium anddysprosium isotopes in our 1064-nm wavelength ODT [39].
This assumption implies equivalent spatial distributions forthe two species, which we assume to be thermal in our threedimensional harmonic trap. We next set L3i,i, j = L3j, j,i ≡ L3i
near resonance, essentially assuming that the loss process is
primarily determined by the two pairwise interactions between the minority participant and the two majority atoms.
We find this assumption leads to a model consistent with
our observed relative loss between the two species. With
these simplifications in place, we define L3i using Ṅi /Ni =033330-7

GIANMARIA DURASTANTE et al.

PHYSICAL REVIEW A 102, 033330 (2020)−L3i n2 ieff , whereand ω̄ = (ωx ωy ωz )1/3 is the geometric mean of the trap oscillation frequencies.
We extract the resonant value of L3 by measuring remaining atom number versus hold time in mixtures prepared at
different temperatures, with the magnetic field set near resonance at 13.4 G. We then fit the resulting data by numericallyintegrating Eq. (A3). Because we observe significant singlespecies loss of erbium (the majority species), we treat the
erbium atom number measured at each time step as inputs
to our fit, and extract the value of L3 that best predicts theloss of dysprosium. Here, we assume that L3i,i, j = L3j, j,i ≡ L3 .
We bound the effects of single-species loss in dysprosium by
repeating the same measurement and analysis protocol off
resonance at 11.5 G and 16.5 G. The error bars in Fig. 2of the main text include a contribution corresponding to theextracted L3 in the off-resonant condition, which containboth the effects of single-species loss and the small effectof off-resonant interspecies loss. Also included are errors associated with the observed change in temperature during theloss measurement, and relating to the approximations made inestimating the density.
In a regime where the scattering
length a exceeds the√thermal wavelength λth = h/ 2π mkB T , and thermal broadening is small compared to the width of the loss feature, we
expect roughly L3 ∝ 1/T 2 , as has been observed in several
experiments involving single atomic species [54–56]. This
picture becomes complicated somewhat in the case of a binary mixture due to stronger Efimov effects, which lead toa temperature-dependent modulation of loss relative to thesimple 1/T 2 prediction. In particular, the parameter s0 , whichcharacterizes the strength of the three-body Efimov potential, is equal to approximately 1.006 for identical bosons,but approximately 0.41 for our binary mixture [57,58]. The
fractional importance of these temperature-dependent modifications scale as e−πs0 [54], making them a minor correction
for identical bosons, but a potentially important effect in mixtures. It is possible that such effects contribute to deviationsof our data from a 1/T 2 form, but a true calculation would
require knowledge of short-range inelastic processes in oursystem.

[1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885(2008).
[2] C. Menotti, M. Lewenstein, T. Lahaye, and T. Pfau, in Dynamics
and Thermodynamics of System with Long Range Interactions:
Theory and Experiments, edited by A. Campa, A. Giansanti, G.
Morigi, and F. S. Labini, AIP Conf. Proc. No. 970 (AIP, New
York, 2008), p. 332.
[3] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev. Mod.
Phys. 82, 1225 (2010).
[4] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82,
2313 (2010).
[5] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).
[6] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W.
Zwierlein, A. Görlitz, and W. Ketterle, Phys. Rev. Lett. 88,
160401 (2002).
[7] M. Taglieber, A.-C. Voigt, T. Aoki, T. W. Hänsch, and K.
Dieckmann, Phys. Rev. Lett. 100, 010401 (2008).
[8] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner,
Nature (London) 462, 74 (2009).
[9] C. Gross and I. Bloch, Science 357, 995 (2017).
[10] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys. Rev.
Lett. 107, 190401 (2011).
[11] M. Lu, N. Q. Burdick, and B. L. Lev, Phys. Rev. Lett. 108,215301 (2012).
[12] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R.
Grimm, and F. Ferlaino, Phys. Rev. Lett. 108, 210401 (2012).
[13] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm, and F.
Ferlaino, Phys. Rev. Lett. 112, 010404 (2014).
[14] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. FerrierBarbut, and T. Pfau, Nature (London) 530, 194 (2016).
[15] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L.
Santos, and F. Ferlaino, Phys. Rev. X 6, 041039 (2016).

[16] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T.
Pfau, Nature (London) 539, 259 (2016).
[17] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S.
Baier, J.-H. Becher, M. J. Mark, F. Wächtler, L. Santos, and
F. Ferlaino, Nat. Phys. 14, 442 (2018).
[18] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T.
Langen, and T. Pfau, Phys. Rev. X 9, 011051 (2019).
[19] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C.
Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M.
Sohmen, M. J. Mark, and F. Ferlaino, Phys. Rev. X 9, 021012(2019).
[20] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C.
Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Phys. Rev.
Lett. 122, 130405 (2019).
[21] K. Aikawa, S. Baier, A. Frisch, M. Mark, C. Ravensbergen, and
F. Ferlaino, Science 345, 1484 (2014).
[22] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst, K.
Sengstock, and K. Bongs, Phys. Rev. Lett. 96, 180403 (2006).
[23] S. Will, T. Best, S. Braun, U. Schneider, and I. Bloch, Phys.
Rev. Lett. 106, 115305 (2011).
[24] J. Heinze, S. Götze, J. S. Krauser, B. Hundt, N. Fläschner, D.-S.
Lühmann, C. Becker, and K. Sengstock, Phys. Rev. Lett. 107,135303 (2011).
[25] N. Spethmann, F. Kindermann, S. John, C. Weber, D.
Meschede, and A. Widera, Phys. Rev. Lett. 109, 235301 (2012).
[26] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M. Parish, J.
Levinsen, R. S. Christensen, G. M. Bruun, and J. J. Arlt, Phys.
Rev. Lett. 117, 055302 (2016).
[27] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A.
Cornell, and D. S. Jin, Phys. Rev. Lett. 117, 055301 (2016).
[28] T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys. 78,
1311 (2006).

n2 ieff =(2Ni N j + N j2 )m3 ω̄65

3 2 8π 3 (kB T )3

(A4)033330-8

FESHBACH RESONANCES IN AN ERBIUM-DYSPROSIUM …

PHYSICAL REVIEW A 102, 033330 (2020)[29] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B.
Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S.
Jin, and J. Ye, Science 322, 231 (2008).
[30] T. Takekoshi, L. Reichsöllner, A. Schindewolf, J. M. Hutson,
C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm, and H.-C.
Nägerl, Phys. Rev. Lett. 113, 205301 (2014).
[31] P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger, C. R.
Le Sueur, C. L. Blackley, J. M. Hutson, and S. L. Cornish, Phys.
Rev. Lett. 113, 255301 (2014).
[32] G. Gligorić, A. Maluckov, M. Stepić, L. Hadžievski, and B. A.
Malomed, Phys. Rev. A 82, 033624 (2010).
[33] R. M. Wilson, C. Ticknor, J. L. Bohn, and E. Timmermans,
Phys. Rev. A 86, 033606 (2012).
[34] R. K. Kumar, P. Muruganandam, L. Tomio, and A. Gammal, J.
Phys. Commun. 1, 035012 (2017).
[35] R. K. Kumar, L. Tomio, B. A. Malomed, and A. Gammal, Phys.
Rev. A 96, 063624 (2017).
[36] B. Kain and H. Y. Ling, Phys. Rev. A 89, 023612 (2014).
[37] L. A. P. Ardila and T. Pohl, J. Phys. B 52, 015004 (2018).
[38] M. Wenzel, T. Pfau, and I. Ferrier-Barbut, Phys. Scripta 93,
104004 (2018).
[39] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M. Sohmen,
M. J. Mark, and F. Ferlaino, Phys. Rev. Lett. 121, 213601(2018).
[40] C. A. Stan, M. W. Zwierlein, C. H. Schunck, S. M. F.
Raupach, and W. Ketterle, Phys. Rev. Lett. 93, 143001(2004).
[41] S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, and
D. S. Jin, Phys. Rev. Lett. 93, 183201 (2004).
[42] S. Kotochigova and A. Petrov, Phys. Chem. Chem. Phys. 13,
19165 (2011).
[43] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C.
Makrides, A. Petrov, and S. Kotochigova, Nature (London) 507,
475 (2014).
[44] K. Baumann, N. Q. Burdick, M. Lu, and B. L. Lev, Phys. Rev.
A 89, 020701(R) (2014).
[45] T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. Ferrier-Barbut,
T. Pfau, A. Frisch, S. Baier, K. Aikawa, L. Chomaz, M. J.
Mark, F. Ferlaino, C. Makrides, E. Tiesinga, A. Petrov, and S.
Kotochigova, Phys. Rev. X 5, 041029 (2015).

[46] V. A. Khlebnikov, D. A. Pershin, V. V. Tsyganok,
E. T. Davletov, I. S. Cojocaru, E. S. Fedorova, A. A.
Buchachenko, and A. V. Akimov, Phys. Rev. Lett. 123, 213402(2019).
[47] M. D. Frye, S. L. Cornish, and J. M. Hutson, arXiv:1910.09641(2019).
[48] P. Ilzhöfer, G. Durastante, A. Patscheider, A. Trautmann, M. J.
Mark, and F. Ferlaino, Phys. Rev. A 97, 023633 (2018).
[49] M. Anderlini, D. Ciampini, D. Cossart, E. Courtade, M.
Cristiani, C. Sias, O. Morsch, and E. Arimondo, Phys. Rev. A
72, 033408 (2005).
[50] A. Guttridge, S. A. Hopkins, S. L. Kemp, M. D. Frye, J. M.
Hutson, and S. L. Cornish, Phys. Rev. A 96, 012704 (2017).
[51] C. Ravensbergen, V. Corre, E. Soave, M. Kreyer, E. Kirilov, and
R. Grimm, Phys. Rev. A 98, 063624 (2018).
[52] J. L. Bohn and D. S. Jin, Phys. Rev. A 89, 022702 (2014).
[53] Along the vertical direction z we observe center-of-mass motion
caused by the momentum transfer of the resonant light pulse.
[54] B. S. Rem, A. T. Grier, I. Ferrier-Barbut, U. Eismann, T.
Langen, N. Navon, L. Khaykovich, F. Werner, D. S. Petrov, F.
Chevy, and C. Salomon, Phys. Rev. Lett. 110, 163202 (2013).
[55] T. Maier, I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel,
C. Wink, T. Pfau, K. Jachymski, and P. S. Julienne, Phys. Rev.
A 92, 060702(R) (2015).
[56] U. Eismann, L. Khaykovich, S. Laurent, I. Ferrier-Barbut, B. S.
Rem, A. T. Grier, M. Delehaye, F. Chevy, C. Salomon, L.-C.
Ha, and C. Chin, Phys. Rev. X 6, 021025 (2016).
[57] K. Helfrich, H.-W. Hammer, and D. S. Petrov, Phys. Rev. A 81,
042715 (2010).
[58] J. P. D’Incao, J. Phys. B 51, 043001 (2018).
[59] A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm, A. Petrov,
S. Kotochigova, G. Quéméner, M. Lepers, O. Dulieu, and F.
Ferlaino, Phys. Rev. Lett. 115, 203201 (2015).
[60] J. Hutson and M. Frye (Private communication).
[61] A. V. Avdeenkov, D. C. E. Bortolotti, and J. L. Bohn, Phys. Rev.
A 69, 012710 (2004).
[62] T. Karman, M. D. Frye, J. D. Reddel, and J. M. Hutson, Phys.
Rev. A 98, 062502 (2018).
[63] B. Deh, C. Marzok, C. Zimmermann, and P. W. Courteille,
Phys. Rev. A 77, 010701(R) (2008).

033330-9

190

A.4

Additional publications

Publication:

Phase coherence in out-of-equilibrium supersolidstates of ultracold dipolar atoms
The following publication has appeared in
Nature Physics 17, 356–361 (2021)§
submitted 21 Dec 2019; published 4 Jan 2021
DOI: https://doi.org/10.1038/s41567-020-01100-3
P. Ilzhöfer,1,2 M. Sohmen,1,2 G. Durastante,1,2 C. Politi,1,2 A. Trautmann,1 G. Natale,2G. Morpurgo,3 T. Giamarchi,3 L. Chomaz,2 M. J. Mark,1,2 and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 DQMP, University of Geneva, Geneva, Switzerland
4 These authors contributed equally: P. I. and M. S.

§The author of the present thesis analysed the experimental data with M. S., and M. J. M., andcontributed in writing the manuscript and interpreting the results together with all the authors.

Articles
https://doi.org/10.1038/s41567-020-01100-3

Phase coherence in out-of-equilibrium supersolidstates of ultracold dipolar atoms
P. Ilzhöfer1,4, M. Sohmen 1,2,4, G. Durastante1,2, C. Politi 1, A. Trautmann 1, G. Natale
G. Morpurgo 3, T. Giamarchi 3, L. Chomaz 2, M. J. Mark 1,2 and F. Ferlaino 1,2 ✉,2A supersolid is a counterintuitive phase of matter that combines the global phase coherence of a superfluid with a crystal-likeself-modulation in space. Recently, such states have been experimentally realized using dipolar quantum gases. Here we investigate the response of a dipolar supersolid to an interaction quench that shatters the global phase coherence. We identify aparameter regime in which this out-of-equilibrium state rephases, indicating superfluid flow across the sample as well as anefficient dissipation mechanism. We find a crossover to a regime where the tendency to rephase gradually decreases until thesystem relaxes into an incoherent droplet array. Although a dipolar supersolid is, by its nature, ‘soft’, we capture the essential
behaviour of the de- and rephasing process within a rigid Josephson junction array model. Yet, both experiment and simulation
indicate that the interaction quench causes substantial collective mode excitations that connect to phonons in solids and affectthe phase dynamics.

T

he notion of phase coherence lies at the foundation of quantum physics. It is considered a key property in understanding
many-body quantum phenomena, ranging from superfluidity
and the Josephson effect to the more applied examples of matterwave
interference, atom lasing processes and quantum transport in mesoand macroscopic systems1,2. Although phase-coherent states arewell studied at equilibrium, understanding their out-of-equilibrium
dynamics remains an open problem at the forefront of statistical andquantum physics, especially when interactions are present3.
Rephasing dynamics of an initially incoherent many-body quantum system requires, first, the system to be conducting such thatdifferent parts can exchange energy and particles, and second, an
efficient mechanism to dissipate the phase excitations. As for the
first requirement, a famous example illustrating the inhibition ofthermalization is many-body localization4. The second ingredient—dissipation—is more subtle and multifaceted, relating, for instance,
to the growth of thermal correlations in isolated systems5, complex interaction-mediated dynamics6,7 or the exponential growthof unstable modes and topological defects in connection with theKibble–Zurek mechanism8–10.
The interplay among coherence, self-localization and relaxation
dynamics is an intriguing problem. In this respect, the recently discovered11–13 supersolid states in dipolar quantum gases can potentially provide a new twist in studying non-equilibrium quantum
phenomena, about which very little is known so far. A supersolid
combines phase coherence and periodic localization in space, properties corresponding to the spontaneous and simultaneous breakingof both gauge and translational symmetry. Intuitively, a supersolid
can be viewed as a fully coherent state, which self-establishes compressible density modulation. In this Article, we explore the evolution of a supersolid of ultracold dysprosium (Dy) atoms whenbrought out of equilibrium after an interaction quench that destroys
its global phase coherence. Due to the dynamic formation of thesupersolid, an interesting question is whether its phase dynamics
are similar to or different from comparable rigid structures, such asa Bose–Einstein condensate (BEC) spliced in an optical lattice14–16,
or if new phenomena can manifest.
In a dipolar supersolid, the particle self-arrangement in space islargely dictated by the many-body interactions17–22 and can be modified by either tuning the interatomic potentials or changing theatom number (N) in the system. Figure 1a shows the phase diagramof a cigar-shaped quantum gas of bosonic Dy atoms with transverse dipole orientation. It is constructed from the ground-state
wavefunctions obtained by numerically solving the extended
Gross–Pitaevskii equation (eGPE)11,13,21 (Methods). Three distinct
quantum phases can be accessed by changing N or the s-wave scattering length as, which parametrizes the contact interaction. For agiven N and large enough as, the ground state of the system is a
non-modulated dipolar BEC (grey region). By lowering as, the influence of the dipolar interaction increases. When reaching a criticalvalue of as, the system undergoes a phase transition to a supersolid
phase (SSP). Here density modulation at a wavelength close to theroton excitation23,24 appears in the ground-state density profile (redregion). By further lowering as, the system evolves into an array of
insulating droplets (IDs) with an exponentially vanishing density
link between them (blue region).
Our eGPE calculations, following a standard non-stochastic
approach, are inherently phase coherent and thus cannot capture
uncorrelated local phases. However, recent experiments have showna connection between the strength of the density modulation andthe coherence properties of the system, revealing a clear difference
between the SSP and ID phase11–13,25. In the SSP, the whole systemshares a global phase. In contrast, in the ID case, any fluctuation or
excitation can lead to a locally different evolution of the phase. The
absence of particle tunnelling between the droplets leads to dephasing of the system.
By performing interaction quenches and moving across thephase diagram, one can create random phase excitations (‘phasescrambling’) and thus distinguish between the different coherence
characters of the ID and SSP, as shown in Fig. 1b. Our experimental

Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck, Austria. 2Institut für Experimentalphysik
und Zentrum für Quantenoptik, Universität Innsbruck, Innsbruck, Austria. 3DQMP, University of Geneva, Geneva, Switzerland. 4These authors contributed
equally: P. Ilzhöfer, M. Sohmen. ✉e-mail: Francesca.Ferlaino@uibk.ac.at
1356

Nature Physics | VOL 17 | March 2021 | 356–361 | www.nature.com/naturephysics

Articles

NAtuRE PHYsics
10–4

10–3

10–2

10–1

c

10–0

30

π

th = 100 ms

ID

SSP

5
80

85

90
as (a0)

95

0.5
0

100

Preparation

f 1.0

d 1.0

BEC

0

50
th (ms)Phase scrambling

100

0.5
0
–π

π

0
Ф

Out-of-equilibrium dynamics

Scramble time

Change as

0

0.2

PDF

15
10

0.4

0

20

∆Ф

N (103)25beAФ, AM (a.u.)aChange as

ts
θ0

θ0

θ 1 θ2 θ 3 θ4

θ1

θ2

θ3

θ4

Fig. 1 | Phase diagram, experimental sequence and starting condition. a, Ground-state phase diagram for our cigar-shaped trapped 164Dy gas. The colour
map encodes the density modulation via the link strength L ¼ 1 � ðnmax � nmin Þ=ðnmax þ nmin Þ, where nmax (nmin) is the density maximum (minimum) in theI equals unity for a non-modulated state and zero for a fully modulated state. The grey colour
central region of the calculated in situ density distribution. L
I
indicates a non-modulated BEC, whereas the red and blue regions
correspond to the SSP and ID phase, respectively. The insets show the density profiles
along the weak axis for the different phases. b, Illustration of the phase-scrambling sequence: starting from an SSP (left) prepared at 2.43 G (as = 87.9a0),
we reduce as to enter the ID regime (middle). During tS the phases of the droplets evolve independently, leading to phase scrambling between the droplets.
Eventually, we jump as back into the supersolid regime (right), where the time evolution of the global phase coherence is studied. c,d, Amplitudes AM andAΦ (c) and ΔΦ (d) for our initial SSP (left panel in b) when simply held for time th. Each data point is derived from q ∈ [80, 90] individual experimental
realizations. The error bars (partly covered by plot symbols) are the 1σ confidence intervals from BCA bootstrapping (Methods)39. e,f, Polar scatter plot for
Pi (e) and histogram of the probability density function (PDF) (f) for Φi at th = 100 ms.

protocol starts by preparing a supersolid state by evaporative coolingfrom a thermal sample (Fig. 1b, left) (ref. 13). We then decrease as toenter into the ID regime (Fig. 1b, middle). Here the system groundstate is an array of IDs, each with a phase that is expected to evolve
independently in time. After these phases have become fully uncorrelated, we jump as back to its initial value (Fig. 1b, right). We thenstudy the time evolution of the out-of-equilibrium system. We measure phase coherence and density modulation, whose coexistence is ahallmark of supersolidity, using a matterwave interference technique(Methods and refs. 11,13,26–29). In brief, for each experimental realization i, we take an absorption image after a time-of-flight (TOF)
expansion. The recorded image exhibits an interference pattern if
in-trap density modulation is present. Via Fourier transform, we
extract the phasor Pi ¼ ρi e�i Φi with amplitude ρi and phase Φi atthe spatial frequencyI of the interference pattern (cf. Extended Data
Fig. 1a). Averaging over an ensemble of q realizations, the mean of
the phasor amplitudes, AM = 〈∣Pi∣〉, characterizes the degree of density
modulation, whereas the amplitude of the complex mean, AΦ = ∣〈Pi〉∣,
contains information about the global phase coherence. As an additional measure of coherence, we calculate the circular variance of theqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
�Pq2 �Pq
2ﬃ
cos
ð
Φ
Þ
þ
sin
ð
Φ
Þ
(ref. 30).
phase ΔΦ = 1 � 1q
i
i
i¼1
i¼1
For a perfect Isupersolid (resp. ID) state and in the limit q → ∞,one expects AΦ = AM > 0 (respectively AΦ = 0, AM > 0) and ΔΦ = 0(respectively ΔΦ = 1). In the following, we describe and characterize each step of our phase-scramble-and-rephase protocol (Fig. 1b).
We initially prepare the supersolid state (Fig. 1b, left) of 164Dy
atoms13 (N = 1.4 × 104) in an axially elongated optical dipole trap
(ODT) with the final harmonic frequencies ωx,y,z = 2π × (225, 37,165) s−1. From our interferometric analysis, we see that this initial state is long lived and has a high degree of phase coherence
Nature Physics | VOL 17 | March 2021 | 356–361 | www.nature.com/naturephysics(Fig. 1c–f). The characteristic quantities of AΦ and AM (Fig. 1c) andΔΦ (Fig. 1d) are constant over hold times th up to 100 ms. The smallmean value 〈ΔΦ〉 = 0.142(8) (the value in parentheses gives thestandard error of the last digit) over the entire range of th reveals
a constantly narrow spread in the phase distribution, as shown for
th = 100 ms in the polar plot (Fig. 1e) of the phasors Pi and the corresponding histogram (Fig. 1f) for Φi.
In the next step of our protocol (Fig. 1b, middle), namely, thephase-scrambling excitation, we tune as via magnetic Feshbachresonances by varying the external magnetic field B (Methods andExtended Data Fig. 2). From the initial supersolid state (~87.9a0,
where a0 denotes the Bohr radius), we transfer the system into the
ID regime (~76.9a0) using a 20 ms linear B-field ramp. Here theatoms are expected to spatially arrange in an array of almost isolated
droplets with exponentially small particle tunnelling between them.
We then let the system evolve for a variable scrambling time tS.
Figure 2 shows the evolution of ΔΦ, AΦ and AM with tS. Aftercompletion of the B-field ramp, ΔΦ initially keeps rapidly increasing for 20 ms and then slowly saturates to a large ΔΦ value. Herethe droplets develop uncorrelated phases, as illustrated by the polar
plot of Pi at tS = 100 ms (Fig. 2a, inset). We extract a saturation valueof 〈ΔΦ〉sat = 0.92(1) (simple mean for tS > 50 ms). We highlight thatΔΦ is not expected to reach unity because of the finite sample
size in the experiment (q ∈ [90, 100] repetitions; slight variations
are due to a post-selection by atom number). It is evident that themeasured 〈ΔΦ〉sat agrees with the expectation for a sample withthe same q and uniformly random phases (grey shading in Fig. 2a,
Extended Data Fig. 1b and Methods). As ΔΦ increases and theglobal phase coherence is lost, AΦ decreases quickly towards zero(Fig. 2b), whereas the density modulation persists as evidenced by
AM remaining large.
357

ArticlesaaRamp B Hold in ID phase

SSP

1.0

NAtuRE PHYsics

–20

0

20

40

60

80

100

∆Ф

∆Ф

0.6
π

0

0.2

0.6

AФ, AM (a.u.)AФ, AM (a.u.)

0

b
0.4
0.2
0

20

40
60
ts (ms)

80

�
2
X3
�

Nj � Nj�
J cos θjþ1 � θj ;
j¼1
j¼1 j
2C j

0.4
0.2

0

20

40

60

80

100

th (ms)

The interaction quench into the ID phase is a robust method tocreate phase-scrambling excitations. It is natural to ask how the relative phases between the individual droplets evolve once the density
links between them are restored and whether the system relaxes into
an equilibrium supersolid state. To address this question, we perform the last stage of our protocol (Fig. 1b, right). After a scramblingtime of tS = 20 ms, we couple the droplets back together by jumping
the B field and thus quenching as to 87.9a0, where the system groundstate is a supersolid. However, after the quench, our state is out of
equilibrium in terms of both phase and density distribution: since
the density-modulated states in dipolar gases are self-assembled,
they can deform after a sudden change in the many-body interactions31–33. As shown in Fig. 3a, we first observe a rapid reductionin ΔΦ and then slower dynamics towards an equilibrium value
with ΔΦ reaching 〈ΔΦ〉sat = 0.13(2) (simple mean for th > 50 ms).
Simultaneously, AΦ approaches AM on the same timescale, whereas
AM remains nearly constant. These observations show that the system efficiently rephases by dissipating the phase excitation.
Our system of multiple superfluid droplets with individual
phases, interconnected via weak links, is reminiscent of a Josephson
junction array (JJA)34. Motivated by this analogy, we investigate
whether a simple JJA model can adequately describe the observed
phase dynamics. This is a non-trivial question, since in contrast toa rigid JJA, our droplet array is ‘soft’ in the sense that the droplet
shapes and their distances change with as. We construct our modelfrom a one-dimensional (1D) array of four coupled grains. For eachgrain, the number of particles is Nj and phase is θj. The Hamiltonianof this system is
X4

0.6

0

100

Fig. 2 | Phase scrambling. a, ΔΦ as a function of tS when ramping from the
SSP to the ID phase (at 1.65 G; as = 76.9a0) within 20 ms (similar results are
obtained with a fast quench of ~1 ms). Each point is derived from q ∈ [90,100] experimental realizations. The error bars are the 1σ confidence
intervals from BCA bootstrapping (Methods)39. The grey-shaded area
indicates the theoretically expected 1σ confidence interval for ΔΦ withsame q and uniformly random phases. The inset shows the polar scatterplot of the phasors Pi at tS = 100 ms. The light-blue line shows ΔΦ fordephasing in a four-grain JJA model using q = 100 and averaging over four
simulation runs (main text and Methods). For the simulation, J is ramped
from 6,000 to 0.002 Hz in 20 ms and η = 60. b, The corresponding AΦ
(light-blue circles) and AM (dark-blue diamonds).

358

0

0.2

Ф

0

H¼

π

0.6
0.4

0.4

0

Ф

0.8

0.8

b

1.0

ð1Þ

Fig. 3 | Rephasing. a, ΔΦ as a function of th after tS = 20 ms scrambling in
the ID phase and a jump back to the SSP regime at 2.43 G (as = 87.9a0). For
each point, q ∈ [66, 74]. The error bars are the 1σ confidence intervals from
BCA bootstrapping. The inset shows the polar scatter plot of the phasors Pi
at th = 100 ms. The light-red line shows ΔΦ for rephasing in a four-grain JJA
model using q = 100 and averaging over four simulation runs (main text andMethods). For the simulation, we start with a random configuration of grain
phases and let the system rephase with J = 6,000 Hz and η = 60. b, The
corresponding AΦ (light-red circles) and AM (dark-red diamonds).

where the overline denotes an ensemble average. The first term is the‘charging’ energy of the droplet (corresponding to its mean interaction energy) with the capacitance Cj. The second term describes
the tunnelling of particles between the droplets with the Josephsonamplitude Jj. The JJA model is appropriate if the droplets are reasonably well separated in space. For simplicity, we assume that Cj and
Jj are identical for all the droplets and later denote them as C and J.
Note that this Hamiltonian describes a quantum evolution since Nj
and θj are connected via canonical commutation relations.
We describe the time evolution of the system via a Langevin formalism35,36. The phase of the droplet j followsdθj
ð2Þ
¼ J½sinðθjþ1 � θj Þ � sinðθj � θj�1 Þ þ ξj ðT; η; tÞ;dtwhere the friction parameter η is a phenomenological way to accountfor dissipative mechanisms. The temperature T is introduced via thethermal noise ξj(t), which shows Gaussian uncorrelated fluctuations
at times t and t' given by ξj ðtÞξi ðt 0 Þ ¼ 2ηkB Tδij δðt � t 0 Þ, where δij
is the Kronecker delta and δ(.)
I denotes the Dirac distribution. The
thermal energy scale kBT (for T = 150 nK in the experiment and the
Boltzmann constant kB) is much higher than the estimated capacitance effect, allowing to neglect the second-order time derivative
term related to C in the evolution of equation (2) (Methods).
This JJA model provides an intuitive understanding of thedephasing and rephasing dynamics shown in Figs. 2 and 3, respectively. It encapsulates the essential physics of a rigid droplet situation at a finite temperature T in terms of the two phenomenologicalparameters J and η. The timescale of de- and rephasing is dictated by
the dissipation η. The dissipation mechanisms include atom losses(cf. Extended Data Fig. 3), energy and particle exchange with thethermal component of the gas or with some internal degrees of freedom of the droplets, as discussed later. In contrast, the phase fluctuations in the equilibrium state, namely, the stationary value of ΔΦ,are set by the competition between J and T and are independent of η.
We develop a parameter-free theory–experiment comparison
for the rephasing dynamics by first fixing the value of J and η fromη

Nature Physics | VOL 17 | March 2021 | 356–361 | www.nature.com/naturephysics

Articles

NAtuRE PHYsics

Nature Physics | VOL 17 | March 2021 | 356–361 | www.nature.com/naturephysics

a

0.4

SSP

ts

0.3

0.2

0.1

0

th (rephase)y (µm)

5
0
–5
–10

1.0

∆Ф

–20

b

0.5

0

20

40

60

80

100

0

Residuals

independent measurements. Taking advantage of the fact that η andJ play different roles at different stages of the protocol, we fix J byreproducing the ΔΦ measured at the end of evaporation (Fig. 1d).
The parameter η is instead extracted from the dephasing dynamicsof ΔΦ during tS (Fig. 2 and Methods). We find that J = 6,000 Hz andη = 60 ± 10.
With all the values of the parameters fixed, we now compare therephasing dynamics from the JJA theory and the experiment. As
shown in Fig. 3, we observe that despite its simplicity, the model
qualitatively predicts the rephasing curve well. In particular, the plateau value at large th as well as the time to reach the plateau are consistent with the experiment. In addition, we note that in the earlytime evolution (th < 30 ms), the experimental data is systematicallyslightly above the JJA curve, namely, the system seems to rephaseslower than predicted by the model. This indicates that other phenomena beyond our rigid JJA model are important to fully capture
the out-of-equilibrium physics.
To elucidate why the observed experimental rephasing isslower than expected from the JJA model, we simulate the system’s
real-time evolution (RTE)24 in the simplified zero-temperature andzero-atom-loss case, with a quench sequence mimicking the experiment. Different from the phenomenological JJA model, the ab initio
RTE approach can additionally account for the ‘soft’ nature of thesupersolid, namely, the crystal and phase phonons31–33. As shown in
Fig. 4a, the RTE simulation shows that the collective modes are initially excited by the interaction quench. The positions and heightsof the high-density peaks evolve in time, especially during the initialtens of milliseconds, resulting in time-dependent density links. At
longer times, we observe a damping of the collective motion, which
can be attributed to an initial redistribution of the population froma few modes to many higher-lying modes over time. This suggests
a possible dissipation mechanism of the phase excitations for our
experiment. Even though the finite temperature, atom loss and presence of a normal component affect the precise dynamics and damping in the experiment, the RTE calculation strongly indicates thatthe droplet dynamics play an important role at early times. This iscompatible with the observed deviations from the rigid JJA model.
In the experiments, we do not have direct access to the in-trap
density evolution. However, by repeating our experiment in a tighter
trap, which gives more distinct side peaks in the TOF interference
patterns, we observe the indications of collective mode excitations.
In Fig. 4b, we plot the time evolution of ΔΦ following the rephasing
protocol. Here, on top of a global decrease in ΔΦ, a low-amplitude
oscillating behaviour is evident, as highlighted by plotting the residuals of an exponential fit to the data. We extract an oscillation frequency of 50(5) Hz (cf. Extended Data Fig. 4 and Methods).
By comparing the predictions of the JJA and RTE approach to ourexperimental data, we conclude that the phase dynamics is largely
described by a dissipative and ‘rigid’ JJA picture. We speculate thatthe phase excitations primarily dissipate via coupling to excitedmodes37. Moreover, the droplet dynamics seems to play an important role, affecting the rephasing efficiency. Note that one could
modify the JJA model to take the droplet dynamics into account, atleast to a certain degree (Methods). Such coupling between the two
types of degree of freedom usually leads to additional dissipation
channels38. Besides being of intrinsic theoretical interest, this could,
for example, open the perspective to using supersolids to study
similar dynamics in regular solids between electronic and phononic
degrees of freedom. This, however, goes beyond the scope of thepresent work and will be addressed in future studies.
In a final set of experiments, we investigate the role of thedensity-link strength between the droplets, namely, the Josephson
coupling, in the rephasing dynamics. After phase scrambling(Fig. 1b, middle), we quench as to different values and record ΔΦ
as a function of the hold time th (Extended Data Fig. 5a). For eachas, we quantify the strength of the density link via L, which was
I

0.2
0
–0.2

0

20

40

60

80

100

th (ms)Fig. 4 | Out-of-equilibrium dynamics. a, RTE simulation starting froma zero-temperature cloud of 2 × 104 atoms of 164Dy and performing ascramble-and-rephase sequence. We plot the 1D integrated density profile(colour code) along the weak trap axis as a function of time. Starting fromthe SSP, the scattering length is linearly ramped at 1 ms to the ID phase(first dashed line), held there for tS = 20 ms (second dashed line), ramped
back to the SSP in 1 ms (third dashed line) and then simply held for th. The
trap frequencies are ωx,y,z = 2π × (229, 37, 135) s−1. b, Experimental rephasingdata for the same trap. The upper panel shows ΔΦ calculated analogously
to that shown in Fig. 3 (black dots), as well as an exponential fit (grey line).
The lower panel shows the residuals of the fit before (black line) and after(red line) the application of a numerical low-pass filter.

determined from a ground-state calculation (Fig. 1a). We study
the short- and long-time evolution of ΔΦ by the determinationof the initial rephasing rate ∣R∣ (the slope of a linear fit to ΔΦ forth ≤ 20 ms) and the saturation value 〈ΔΦ〉sat (simple mean of ΔΦ forth > 50 ms), respectively, as shown in Fig. 5a,b (for AM and AΦ see
Extended Data Fig. 5b).
We observe different rephasing dynamics depending on L. For
extremely weak density links (L < 10–3), which we associateI withI to rephase and it remains incothe ID regime, the system is unable
herent over the whole time evolution as indicated by a low ∣R∣ andlarge 〈ΔΦ〉sat ≈ 0.9. As L slightly increases, the system shows a
I sat decreasing to about 0.5. By further
partial rephasing with 〈ΔΦ〉
increasing the density-link strength, there exists a critical value of
L above which the system recovers its full phase coherence after
aIlong time (〈ΔΦ〉sat = 0.15). In this regime, ∣R∣ ≈ 30 s−1 is large andseems relatively independent of L.
I with our experimental obserTo compare the JJA predictions
vations, we first extract J for each as. For this purpose, we look at
the long-time behaviour after scrambling and rephasing when thesystem has equilibrated, matching the theoretical and experimental 〈ΔΦ〉sat values. This is justified by the long-time agreement
observed in Fig. 3. As expected from the increasing density-linkstrength L between the droplets, J globally grows with as (Fig. 5a,
I these J values and the single calibrated value of η = 60
inset). Using(Fig. 2), we extract the rephasing rate ∣R∣JJA from the short-time
evolution of ΔΦ within our JJA framework (Fig. 5b, inset). Despite
the simplicity of the JJA approach, the ∣R∣JJA values are of the sameorder of magnitude as our experimental data and show the same
qualitative behaviour with respect to as. As observed in Fig. 3, ∣R∣JJA
359

Articles

NAtuRE PHYsics

a
10–5

0.5

J (kHz)〈∆Ф〉sat

1.0

10–3

10–1

Online content

8

Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of
author contributions and competing interests; and statements ofdata and code availability are available at https://doi.org/10.1038/
s41567-020-01100-3.

4
0

77.8

82.4

88.0

77.8

82.4

88.0

0

b
∣R∣ (s–1)

20–1∣R∣JJA (s )

150

30

100
50
0

Received: 21 December 2019; Accepted: 22 October 2020;
Published online: 4 January 2021

10
0
77.8

82.4

88.0

as (a0)Fig. 5 | Density-link dependence of rephasing dynamics. a, Saturation
value of the phase variance 〈ΔΦ〉sat as a function of the scattering length
as and the corresponding L. We calculate 〈ΔΦ〉sat and the error bars asIthe mean and standard deviationof ΔΦ over the three longest hold times(Fig. 3 and Extended Data Fig. 5). The grey-shaded rectangle indicates
the theoretical 1σ confidence interval for uniformly random phases. The
inset shows the J values for which the JJA simulations yield a 〈ΔΦ〉sat
matching the experimental value at as, and the grey shading represents
its confidence interval. b, Initial rephasing rates ∣R∣ for the different asvalues as extracted from the experimental data by using a linear fit.
The error bars are the 1σ confidence intervals of the fitted slopes.
The inset shows the theoretical ∣R∣JJA (as) expected from a linear fit to
the initial decrease in the JJA rephasing curves (theory line in Fig. 3); thegrey shading represents the confidence interval. The red filled points
correspond to the dataset shown in Fig. 3.

generally predicts rephasing faster than that measured in the experiment, suggesting the presence of non-negligible processes beyond
the rigid JJA model.
It is interesting that despite the JJA being able to consistently
model the phase dynamics in the experiment, the extracted dependence of J on as is mild in comparison to the expected ground-state
density link L (Fig. 5a). For instance, J changes only by a factor of
I changes by two orders of magnitude. Further, J seems
two whereas L
to effectively Isaturate for increasing strength of the ground-state
density link L, which is in agreement with the observed plateausof both 〈ΔΦ〉Isat and ∣R∣ for large as. Possible explanations include
the breakdown of the JJA model for a low-contrast supersolid, theimpact of finite temperature on both the equilibrium supersolidstate itself and the experimental measurement, or the role of collective dynamics.
In conclusion, we have performed a study of the out-of-equilibrium
dynamics of a dipolar supersolid after an interaction-driven phase
excitation that fully destroys its phase coherence. We have demonstrated that if the inter-droplet density links are sufficient, thisphase-scrambled system relaxes into an equilibrium phase-coherent
state. With decreasing link strength, the rephasing substantiallyslows down and eventually ceases in the ID regime. We find an overall consistency between the phase dynamics observed in the experiment and an intuitive, theoretically easily tractable rigid JJA model.
However, both ab initio simulations and experimental observations
indicate post-quench collective excitations of the droplet array,
which can affect the phase dynamics. Our study shows the evidenceof particle flow across a dipolar supersolid, connecting to its superfluidity. It also suggests the efficient dissipation of phase excitations,360

whose microscopic mechanism is still under question. Future experimental works, combined with advanced out-of-equilibrium theoretical models, will be crucial to understand the relaxation dynamics
at play in isolated and open supersolid states of quantum matter.

References

1. Cohen-Tannoudji, C. & Guéry-Odelin, D. Advances in Atomic Physics: An
Overview (World Scientific, 2011).
2. Svistunov, B. V. et al. Superfluid States of Matter (CRC Press, 2015).
3. Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of
equilibrium. Nat. Phys. 11, 124–130 (2015).
4. Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: manybody localization, thermalization, and entanglement. Rev. Mod. Phys. 91,
021001 (2019).
5. Langen, T., Geiger, R., Kuhnert, M., Rauer, B. & Schmiedmayer, J. Local
emergence of thermal correlations in an isolated quantum many-body system.
Nat. Phys. 9, 640–643 (2013).
6. Wright, E. M., Walls, D. F. & Garrison, J. C. Collapses and revivals ofBose–Einstein condensates formed in small atomic samples. Phys. Rev. Lett.
77, 2158–2161 (1996).
7. Greiner, M., Mandel, O., Hänsch, T. W. & Bloch, I. Collapse and revivalof the matter wave field of a Bose–Einstein condensate. Nature 419,
51–54 (2002).
8. Scherer, D. R., Weiler, C. N., Neely, T. W. & Anderson, B. P. Vortex formation
by merging of multiple trapped Bose–Einstein condensates. Phys. Rev. Lett.
98, 110402 (2007).
9. del Campo, A. & Zurek, W. H. Universality of phase transition dynamics:
topological defects from symmetry breaking. Int. J. Mod. Phys. A 29,
1430018 (2014).
10. Aidelsburger, M. et al. Relaxation dynamics in the merging of N independent
condensates. Phys. Rev. Lett. 119, 190403 (2017).
11. Böttcher, F. et al. Transient supersolid properties in an array of dipolar
quantum droplets. Phys. Rev. X 9, 011051 (2019).
12. Tanzi, L. et al. Observation of a dipolar quantum gas with metastablesupersolid properties. Phys. Rev. Lett. 122, 130405 (2019).
13. Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar
quantum gases. Phys. Rev. X 9, 021012 (2019).
14. Cataliotti, F. S. et al. Josephson junction arrays with Bose–Einsteincondensates. Science 293, 843–846 (2001).
15. Albiez, M. et al. Direct observation of tunneling and nonlinear
self-trapping in a single bosonic Josephson junction. Phys. Rev. Lett. 95,
010402 (2005).
16. Levy, S., Lahoud, E., Shomroni, I. & Steinhauer, J. The a.c. and d.c. Josephson
effects in a Bose–Einstein condensate. Nature 449, 579–583 (2007).
17. Lu, Z.-K., Li, Y., Petrov, D. S. & Shlyapnikov, G. V. Stable dilute
supersolid of two-dimensional dipolar bosons. Phys. Rev. Lett. 115,
075303 (2015).
18. Cinti, F. & Boninsegni, M. Classical and quantum filaments in the groundstate of trapped dipolar Bose gases. Phys. Rev. A 96, 013627 (2017).
19. Wenzel, M., Böttcher, F., Langen, T., Ferrier-Barbut, I. & Pfau, T.
Striped states in a many-body system of tilted dipoles. Phys. Rev. A 96,
053630 (2017).
20. Baillie, D. & Blakie, P. B. Droplet crystal ground states of a dipolar Bose gas.
Phys. Rev. Lett. 121, 195301 (2018).
21. Roccuzzo, S. M. & Ancilotto, F. Supersolid behavior of a dipolar Bose–
Einstein condensate confined in a tube. Phys. Rev. A 99, 041601 (2019).
22. Kora, Y. & Boninsegni, M. Patterned supersolids in dipolar Bose systems.
J. Low Temp. Phys. 197, 337–347 (2019).
23. Santos, L., Shlyapnikov, G. V. & Lewenstein, M. Roton-maxon spectrum andstability of trapped dipolar Bose–Einstein condensates. Phys. Rev. Lett. 90,
250403 (2003).
24. Chomaz, L. et al. Observation of roton mode population in a dipolar
quantum gas. Nat. Phys. 14, 442–446 (2018).
25. Petter, D. et al. High-energy Bragg scattering measurements of a dipolar
supersolid. Preprint at https://arxiv.org/abs/2005.02213 (2020).
Nature Physics | VOL 17 | March 2021 | 356–361 | www.nature.com/naturephysics

NAtuRE PHYsics
26. Takeda, M., Ina, H. & Kobayashi, S. Fourier-transform method offringe-pattern analysis for computer-based topography and interferometry.
J. Opt. Soc. Am. 72, 156–160 (1982).
27. Hadzibabic, Z., Stock, S., Battelier, B., Bretin, V. & Dalibard, J. Interference ofan array of independent Bose–Einstein condensates. Phys. Rev. Lett. 93,
180403 (2004).
28. Kohstall, C. et al. Observation of interference between two molecular
Bose–Einstein condensates. New J. Phys. 13, 065027 (2011).
29. Chomaz, L. et al. Emergence of coherence via transverse condensation in a
uniform quasi-two-dimensional Bose gas. Nat. Commun. 6, 6162 (2015).
30. Fisher, N. I. Statistical Analysis Of Circular Data (Cambridge Univ. Press, 1993).
31. Natale, G. et al. Excitation spectrum of a trapped dipolar supersolid and its
experimental evidence. Phys. Rev. Lett. 123, 050402 (2019).
32. Tanzi, L. et al. Supersolid symmetry breaking from compressional oscillations
in a dipolar quantum gas. Nature 574, 382–385 (2019).
33. Hertkorn, J. et al. Fate of the amplitude mode in a trapped dipolar supersolid.
Phys. Rev. Lett. 123, 193002 (2019).

Nature Physics | VOL 17 | March 2021 | 356–361 | www.nature.com/naturephysics

Articles
34. Fazio, R. & van der Zant, H. Quantum phase transitions and vortex dynamics
in superconducting networks. Phys. Rep. 355, 235 (2001).
35. Risken, H. The Fokker–Planck Equation (Springer, 1961).
36. Ambegaokar, V. & Halperin, B. I. Voltage due to thermal noise in the dc
Josephson effect. Phys. Rev. Lett. 22, 1364–1366 (1969).
37. Dalla Torre, E. G., Demler, E. & Polkovnikov, A. Universal rephasing
dynamics after a quantum quench via sudden coupling of two initiallyindependent condensates. Phys. Rev. Lett. 110, 090404 (2013).
38. Leggett, A. J. et al. Dynamics of the dissipative two-state system. Rev. Mod.
Phys. 59, 1–85 (1987).
39. Efron, B. Better bootstrap confidence intervals. J. Am. Stat. Assoc. 82,
171–185 (1987).
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.
© The Author(s), under exclusive licence to Springer Nature Limited 2021

361

Articles
Methods

Ground-state phase diagram and contrast. Our numerical calculations of theground-state phase diagram of a cigar-shaped 164Dy dipolar quantum gas follow
the procedure described in our earlier works13,24. In brief, the calculations are
based on minimizing the energy functional of the eGPE using the conjugate
gradient technique40. The eGPE includes the anisotropic trapping potential, theshort-range contact and long-range dipolar interactions at the mean-field level andthe first-order beyond-mean-field correction in the form of a Lee–Huang–Yang
term41–43. From the derived three-dimensional wavefunction ψ(r) at positions r,
we calculate the 1D in situ density profile as n(y) = ∫∣ψ(r)∣2dxdz. We evaluate thein situ density contrast as C = (nmax − nmin)/(nmax + nmin) for profiles that featuredensity modulations by searching the central extrema of n(y) and determining
the maximum (nmax) and minimum (nmin) values. For profiles without density
modulation (ordinary BEC), we set C = 0. We use the quantity L ¼ 1 � C to
I the tunnelling
estimate the density link between the droplets, which is related tostrength13.
RTE. We perform the RTE based on the eGPE and mimicking the experimental
parameters and sequence. In particular, in Fig. 4a, we use the parameters of theexperimental data shown in Fig. 4b, using a gas of 164Dy atoms (N = 2 × 104) in a
trap of frequencies ωx,y,z = 2π × (229, 37, 135) s−1. We start from the eGPE groundstate ψ(r) at as = 92a0 and calculate the time evolution of the wavefunction
ψ(r,t) according to the eGPE using a split-operator technique24. For thescramble-and-rephase protocol, the scattering length is linearly ramped from the
SSP (as = 92a0) to the ID regime (as = 80a0) in 1 ms, kept constant for tS = 20 ms,
ramped back to as = 92a0 in 1 ms and finally held constantfor a variable th. We
Rcompute the 1D integrated density profile nðy; tÞ ¼ jψðr; tÞj2 dxdz;, and its time
evolution is shown in Fig. 4a. Similar to theIwork in refs. 12,25, we gain further
insights into the rephasing process by extracting the global spatial variation α ofthe phase of the wavefunction along y defined asR nð0; y; 0; tÞjθð0; y; 0; tÞ � hθð0; y; 0; tÞijdy
RαðtÞ ¼ Y;
ð3Þ
Y  nð0; y; 0; tÞdywhere ψðr;R tÞ ¼ jψðr; tÞj expði θðr; tÞÞ and r = (x, y, z). Here the average
hf ðyÞi ¼I Y  f ðyÞdy for any function f and Y* denotes the inner region
YI* = [−5, 5] μm.
Extended Data Fig. 4a shows the evolution of α with th, namely, after ramping
as back to the SSP. Here, α first quickly decreases, transiently reaching a value
close to zero (green filled dot), before increasing again up to an intermediate
value (blue filled dot) and finally smoothly decreasing towards zero, setting to alow saturation value (α* = 0.027 × 2π). After long times, only small variations of α
persist (orange filled dot), indicating that an equilibrium distribution is reached.
The density and phase profiles corresponding to the coloured filled dots are shownin the inset of Extended Data Fig. 4a. Although α first reaches a low value withina short time, we observe that the corresponding density profile is strongly out of
equilibrium (green curve), differing from both the initial ground state (not shown)and the long-time equilibrium state (orange curve). The phase variations are onlytransiently suppressed here. In contrast, when α reaches the next local maximum,
the density profile appears very close to the equilibrium expectation, yet with aremaining phase pattern of large amplitude (blue curves). This opposite behaviourof the phase and density in the early time is reminiscent of the quadratureoscillations of phase and density perturbations associated with a given elementary
excitation in the linear regime31. The initial oscillatory behaviour of the RTE is
also reminiscent of the behaviour of ΔΦ from repeated instances of experimental
interference patterns observed in Fig. 4b. Here we subject α to a similar analysis asΔΦ, performing an exponential fit and calculating the residuals (Extended Data
Fig. 4b). The extracted decay time is 6.3(2) ms; from the residuals, we find the mainfrequency components in the interval of [50, 100] Hz. We note that this frequency
is similar to the low-lying mode of the SSP excitation spectrum, as computed from
Bogoliubov theory31. The observed frequency of oscillation and the time evolutionof phase and density profiles in apparent quadrature suggest an important role ofthe low-lying collective mode of the SSP in the rephasing dynamics.
Experimental sequence. We apply our phase-scrambling protocol to theevaporatively cooled SSP of 164Dy atoms13. For this, we initially load our
five-beam open-top magneto-optical trap for 3 s and apply a magneto-opticaltrap compression phase, which lasts for 400 ms (ref. 44). We then load about
8 × 106 atoms into a single-beam horizontal ODT, propagating along the y axis.
The horizontal ODT is derived from a 1,064 nm focused laser beam. Afterloading, we apply forced evaporative cooling by exponentially reducing the opticalpower in the horizontal ODT for 0.9 s. Subsequently, we switch on a second ODTbeam along the vertical z axis to form a crossed ODT and continue with the laststage of evaporative cooling for 2 s (ref. 45) until the SSP is reached. During theevaporation sequence, a vertical magnetic field of B = 2.430(4) G sets the dipole
orientation. The final trap geometry is axially elongated with harmonic frequencies
ωx,y,z = 2π × (225, 37, 165) s−1. After the initial-state preparation (SSP), we apply ourphase-scrambling protocol. For that, without any additional waiting time after theevaporative cooling, we change the B field to 1.65 G deep in the ID regime. Here

NAtuRE PHYsics
we allow the system’s global phase to freely evolve for tS = 20 ms. We have explored
two types of protocol: jumping, which results in an effective ~1 ms change in the B
field due to the finite time response of the system, and ramping within 20 ms. We
observe a similar scrambling behaviour in ΔΦ for both jump and ramp protocols.
We complete our phase-scrambling sequence by jumping the B field back to its
initial value and by letting the system evolve for a variable hold time th. Finally, we
perform a matter–wave-interference-type experiment during TOF expansion andrecord the resulting interference pattern by absorption imaging. A TOF duration of
26.5 ms ensures sufficient mapping onto the momentum space. The imaging beam
propagates in the horizontal x–y plane at an angle of ~45° with respect to the weak
trap axis y.
Tuning the scattering length. To connect the experimental B-field values tothe contact scattering length as, weQ
use� the establishedformulafor overlapping�Feshbach resonances: as ðBÞ ¼ abg i 1 � ΔBi = B � B0;i (ref. 46), where B0,idenotes the poles; ΔBi, Ithe corresponding distance from the pole to the zerocrossing; and abg, the (local) background scattering length. We determine the poles
and zero crossings in our B-field region of interest by performing loss spectroscopy
and thermalization measurements. Starting from a thermal cloud prepared at2.55 G, we first ramp the magnetic field to the final value within 5 ms and thenlower the trap depth to its final value within 50 ms; we wait an additional hold timeof about 400 ms. In the absence of Feshbach resonances, we typically end up with athermal gas of 5 × 105 atoms at about 500 nK. When scanning the magnetic field in
our region of interest, we observe several atom loss features together with peaks inthe atom cloud temperature, which we fit by Gaussians to extract the positions of
the poles B0,i and widths ΔBi.
The value of the background scattering length of 164Dy is a more subtle topic,
as several measurements have reported different values in the range between
60a0 and 100a0 (ref. 47). These measurements were performed using different
methods (for example, cross-thermalization and theory–experiment comparisonsof oscillation frequencies), for different initial states (thermal gases and quantum
droplets) and at different magnetic fields. In particular, the existence of very broadresonances at higher magnetic fields48 affects the measured local backgroundscattering lengths. Therefore, we set the value of abg in such a way that the B-to-as
conversion reproduces the calculated critical scattering length of as = 91a0 at theexperimentally estimated phase transition point between the BEC and SSP around
2.5 G. This gives a value of abg = 73a0, which lies within the error bars of the latest
published value of abg = 69(4)a0 (ref. 47). Extended Data Fig. 2 shows the resulting
calculated B-to-as conversion from which we estimate as,SSP = 88a0 at 2.43 G in the
SSP and as,ID = 76.9a0 at 1.65 G in the ID phase, as used in the experiment.
Interference pattern analysis. Our analysis is similar to the one described inref. 13. We record q ∈ [30, 100] experimental repetitions for each parameter set
P. Each recorded TOF picture i (i = 1,…,q) is processed by first subtracting thethermal background via a symmetric 2D Gaussian fit to the wings of the density
distribution. Next we recentre the image of the degenerate cloud and integrate
its central region, where the matter–wave interference signal is concentrated,along the z direction within ±2 μm−1. We obtain a momentum density profilethat we normalize by its sum. From such a momentum profile, a fast Fouriertransform yields the 1D density profile ni ð~yÞ. An in situ density modulation in
an atomic cloud leads to side peaks in niIð~yÞ, symmetrically centred around thepeak at zero. To isolate the centre of thisI specific modulation, we calculate theincoherent and coherent means of ni ð~yÞ, which are denoted as nM ð~yÞ ¼ hjni ð~yÞjiP
and nΦ ð~yÞ ¼ jhni ð~yÞiP j, respectively.I The incoherent mean (nM)Ireflects the
meanImodulation amplitude of the cloud at the respective wavelength ~y. The
coherent mean (nΦ ≲ nM) if the phases of the interference pattern among theq repetitions at the respective ~y are roughly constant, and nΦ → 0 (and hencenΦ ≪ nM) if the phases are random. Therefore, the most pronounced differencenM − nΦ is observed for the ID regime (Extended Data Fig. 1a). From the maximumof this difference, the modulation wavelength (or ‘droplet distance’) can be
determined as ~y  d. The fast Fourier transform phasors at d can be expressed
as Pi ¼ ni ðd Þ ¼I ρi e�iΦi , yielding the sets fP1 ; ¼ ; Pq gP . To characterize theI
distributionofﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
phases Φi within our sets, we
I calculate the circular varianceq
�Pq
2 �Pq
2
cos
ð
Φ
Þ
þ
sin
ðΦi Þ (ref. 30). For a phase-coherentΔΦ = 1 � 1q
i
i¼1
i¼1
sample,I and hence interference fringes stable within the envelope, ΔΦ is small,whereas for an incoherent sample, it approaches unity. To estimate the confidence
intervals of ΔΦ, we apply a bias-corrected and accelerated (BCA) bootstrapping
scheme39 for each P, resampling 106 times from the respective q experimental
values. We note that possible non-ballistic (namely, interaction-driven) evolution
during the early TOF might have a residual systematic effect on the measured
values of nM, nΦ and ΔΦ.
Effect of finite sample size. Even the circular variance ΔΦ of a sample of q angles(Φ1,…,Φq) drawn from a completely random distribution approaches unity only
when q → ∞. To estimate the fully incoherent limit of ΔΦ for our finite q, we
calculate 106 values for ΔqΦ, each for q independent draws from a theoretical,
uniform distribution in [0, 2π). The histograms of ΔqΦ are shown in Extended
Data Fig. 1b. The indicated 1σ confidence intervals are [0.77, 0.93] for q = 35 draws
and [0.86, 0.96] for q = 100 draws. We note that the histograms of ΔqΦ seem
Nature Physics | www.nature.com/naturephysics

Articles

NAtuRE PHYsics
to closely follow a beta distribution49, even if one generalizes the underlying
distribution of phases Φi to a von Mises distribution, of which the uniform
distribution is just a degenerate case.
Interference pattern quantities for a simple model of a droplet array. For
simplicity, let us consider here that the state is made of ND identical droplets. In thiscase, the full wavefunction of the system would be
XN D �
ψ ðx; y; z Þ ¼
f x; y � Rj ; z eiθj ;ð4Þ
j¼1

where Rj is the spatial coordinate of the jth droplet, θj is its phase (taken to beuniform over the droplet) and f is the wavefunction of a single droplet localizedaround y = 0. With such a wavefunction, the phasor extracted from one realization
would beZ
XN D� 
Pi ¼ dkyeiky ðRj1 �Rj2 �dÞ eiðθj1 �θj2 Þ j~f ky j2 ;ð5Þ
j ;j ¼1
1 2

where ~f is the Fourier transform of the function f and d is the distance between theneighbouring droplets d = 〈Rj+1 − Rj〉. It simplifies to
XN D
�

Pi ¼
g Rj1 � Rj2 � d eiðθj1 �θj2 Þð6Þ
j ;j ¼1
1 2

 g ð0Þ

XN D �1
j¼1

eiðθjþ1 �θj Þ ;

ð7Þ

 
where g ð yÞ is the Fourier transform of j~f ky j2 , which is thus a peak function witha width Iof the order of the droplet size. I
Equation (7) yields
AM ¼ hjPi jiP ¼ ðN D � 1Þgð0ÞlΔθ  ðN D � 1Þgð0Þ;

ð8Þ

which shows only a weak dependence on the phase relation between the droplets
and on the droplets’ shape. We note that the residual information on the droplets’phase relation is contained in AM via the length of the mean phase-difference
vector lΔθ = hjhexpðiΔθ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃjﬃÞij jiP . In the fully coherent (resp. incoherent) case, lΔθ = 1(resp. lΔθ = 1=I N D � 1 on average). The change in lΔθ with the degree of coherenceIremains limited,
especially for the relatively small ND relevant for our experimental
situation (Fig. 1). Here h:iP denotes the average over an ensemble of realizations P
and 〈.〉j denotes the average
I over the droplet array.
On the contrary, the function AΦ ¼ jhPi iP j contains the average of the phases
with
I
AΦ
’ jhheiðθjþ1 �θj Þ ij iP j:AMð9Þ

Therefore, the ratio AΦ/AM essentially measures the mean difference in the phases
between two neighbouring droplets in the array.
From equation (7), it is also evident that the phase of the phasor isΦ  hθjþ1 � θj ij. The circular variance ΔΦ for q realizations can be expressed asIXq
Xq
1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ΔΦ ¼ 1 �eiΦi1e�iΦi2 :
ð10Þ
i1 ¼1
i2 ¼1
q

For a totally phase-coherent state, Φ = 0 for all the realizations leading to ΔΦ = 0,
while for a totally phase-incoherent sample, only the diagonal terms in equation(10) survive, leading to ΔΦ = 1 � p1ﬃﬃq for q independent measurements.
I
Langevin formalism for the JJA model. To model the self-modulated dipolargas, we use the JJA whose Hamiltonian H is given in equation (1). To describe
the dynamics, we use a Langevin formalism (example in ref. 36), where the timedependence of the phases obeys
η

dθj
dH
¼�
þ ξj ðT; η; tÞ;
dθj
dt

ð11Þ

where the two sites are indexed using i and j and the thermal noise is denoted as
ξi(t), which is uncorrelated from grain to grain and at different times shows thecorrelations
ξj ðtÞξi ðt 0 Þ ¼ 2ηkB Tδij δðt � t 0 Þ;

ð12Þ

where kB is the Boltzmann constant. This thermal noise ensures that in equilibrium,
the configurations are realized with a probability proportional to the Boltzmann
weight e�H=ðkB TÞ .
I
To determine
the parameters of the JJA model, we use the procedure describedin the main text. With J fixed by the equilibrium value of ΔΦ, we determine η bycomparing to the dephasing timescale.
Discussion on the quantum part of the Hamiltonian. The JJA model contains
both quantum evolution and thermal noise. In our case, one can make the
Nature Physics | www.nature.com/naturephysics

additional simplification to neglect the term related to the capacitance leading toquantum fluctuations compared with thermal fluctuations.
To justify this approximation, we estimated the interaction energy cost with awavefunction given by a variational principle50. We obtain an energy to add one
particle to the droplet of the order of a few hertz, which is very small comparedwith the temperature of 150 nK.
This allows to drop the second-order time derivative of the phase in theLangevin equations, leading to equation (2). For a system of four droplets, wehave four equations for the phases. For the equations describing the evolution of
the droplets at the edge of the array (namely, j = 1 or j = 4), we drop the terms ofequation (2) containing the phases of non-existent neighbours (namely, θ0 or θ5).
Numerical solution of the equations. To solve our system of equations, we
discretize the time dependence with a time step adapted to the considered value ofJ. For J/h up to 6,000 Hz, we used a time step of 10−5 s. For larger J, we used a time
step of 5 × 10−6 s.
To relate the phases of the four individual droplets to the observables, equation(7) and the phasor definition, we use
X3
Φ ¼ argð j¼1 eiðθjþ1 �θj Þ Þ:
ð13Þ
To obtain a ΔΦ value that can be compared to the experimental one, we computethe evolution over q = 100 independent realizations, determine Φ for eachrealization and compute the evolution of ΔΦ from these 100 values.
For the scrambling, we initialize each of the 100 runs with random phases
and let the system evolve for 80 ms with a finite J to reach the corresponding
equilibrium state. We then ramp J down to 0.002 Hz in 20 ms and let the systemevolve for 100 ms.
For the rephasing, we initialize 100 configurations with random phases, whichcorresponds to the equilibrium state for the system with a small J (0.002 Hz). We
then let the system evolve with J = 6,000 Hz.

Modified JJA model with droplet dynamics. One can phenomenologicallymodify the JJA model to take the droplet dynamics into account. Each droplet ischaracterized by a position uj. The dynamics of the droplets could be modelled by aphonon-like Hamiltonian:
"
#
X p2j
1
2
2
;
ð14Þ
HD ¼
ðu
�
u
Þ
þ
Mω
jþ1
j
0
j 2M
2
where M denotes the mass of the droplet, pj is the momentum conjugate to uj
and ω0 is the characteristic energy depending on the interactions between thedroplets. This motion would be coupled to the degrees of freedom of the phase in
the Hamiltonian in equation (1) by introducing a uj dependence of the Josephson
coupling: if two droplets get closer, J should increase. The precise form would
need to be determined, but some exponential dependence seems plausible. If
the displacements of the droplets are very small, one can expand the Josephson
coupling as a function of the displacements of the droplets to obtain the Josephson
coupling between the droplets j and j + 1 viaJ jþ1;j ¼ J 0 � Aðujþ1 � uj Þ;

ð15Þ

where A is a positive quantity. Therefore, the two Hamiltonians in equations (14)and (1) with the coupling in equation (15) would, in principle, allow to incorporate
the coupled dynamics of the degrees of freedom of the phase and the vibrations of
the droplet ‘lattice’.
Time traces of ΔΦ and behaviour of AM and AΦ. We have studied the rephasing
dynamics at different scattering lengths. Extended Data Fig. 5a shows the full time
evolution of the experimentally measured ΔΦ as a function of the calculated L.
From this data, we also extract the initial rephasing rate ∣R∣ and the saturation I
value 〈ΔΦ〉sat, as shown in Fig. 5. For L within the ID regime, ΔΦ remains large
I(>0.5) for the entire th (blue region); therefore,
we can conclude that rephasing
is absent. As L increases, ΔΦ gradually approaches small final values (~0.15) at
I times (red region), that is, the system rephases.
longer evolution
From the time traces, we also analyse the long-time behaviour of AΦ and AM as
a function of L. As demonstrated in previous works13,25, at equilibrium, AΦ and AMI
are powerful parameters
to pinpoint the different quantum regimes in the phase
diagram. In the SSP, AΦ/AM ≈ 1, whereas in the ID phase, this ratio tends to zero
since AΦ vanishes. As shown in Extended Data Fig. 5b, the value of AM at a longer
hold time (th = 100 ms) is substantial in the whole range investigated, indicating thepersistence of density modulation in the system, and it varies only weakly with L.
Differently, AΦ has a clear dependence on the density link. Its value at th = 100 msI
nearly vanishes for L < 0.001, evidencing a final phase-incoherent state (IDI
L > 0.01, AΦ and AM are nearly equal.
regime), whereas at large
The exact dependence Iof AM on L is not completely obvious and deserves
some explanation. From(7),I we see that AM, which is expressed as
Pequation
D �1
AM ¼ hjPi ji ¼ gð0Þhj Nj¼1expði Δθj Þji, contains (1) information on theindividual
droplet density profile via g(0) and (2) residual information on
IArticles
the coherence
Pof the system via the averaged length of the vector of the phasedifference hj j expði Δθj Þji. In the fully coherent case, this length is identicallyequal
to NDI− 1, whereas in the fully incoherent case, it has an average of only
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
N D � 1. This change explains the abrupt jump in AM observed when the dropletI
arraybecomes partially incoherent. We note that the observed jump with a value
divided by ~1.5 is compatible with the predicted number of droplets in our array,
namely, ND = 4. We also note that the smooth increase in AM for decreasing as bothin the incoherent ( L < 0.01) and in the coherent ( L > 0.01) regions are expected
Ifrom the proportionality
to g(0). Here g(0) indicatesI the individual droplet peakdensity and typically scales as the inverse of the droplet size. It is thus expected
to increase for decreasing as. We highlight again, as stated in Methods, that the
information on the phase coherence of the array contained in AM is much less
vector isdominant than that in AΦ, where the length of the phase-differencePmeasured after performing the ensemble average, jh j expði Δθj Þij. Therefore, AΦ
is expectedptoﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
cancel in a fully incoherent case rather
I than simply being reduced bya factor of N D � 1.
I
Atom number evolution. When connecting the rephasing behaviour shown in
Fig. 4, where 〈ΔΦ〉sat depends on L, to the phase diagram, the atom numbers
I also show differences after the completeof the thermal and coherent part may
phase-scrambling sequence. To this end, we extract the total atom number, Ntotal,
via a pixel count from each unprocessed TOF absorption image. We then subtract
the thermal background via a 2D Gaussian fit (Methods) to obtain the atom
number in the coherent part, Ncoherent. The width of the thermal background at thegiven TOF allows to determine the temperature T. Extended Data Fig. 3 shows
Ntotal, T, Nthermal and Ncoherent as a function of the hold time th for three exemplaryrephasing B fields, namely, 1.65 G (blue), 2 G (light blue) and 2.43 G (red). These
correspond to L ’ 4 × 10–6, 4 × 10–4 and 1 × 10–1 and thus to the ID, intermediate
I respectively. Regardless of the chosen value of L, the individual
and SSP regimes,
Ntotal, Nthermal and T essentially coincide and decrease continuouslyIwith increasing thin a similar manner. This behaviour suggests the remaining plain evaporation fromthe ODT during th.
From the description of an ideal BEC, we can simply approximate 3 !
T;
ð16Þ
N coherent ¼ N total 1 �TCwhere the phase transition’s critical temperature TC mainly depends on the trapping
frequencies ωx,y,z and Ntotal. From our measured Ntotal and T, we would expect a
decrease in Ncoherent for all the rephasing B fields. However, similar to AΦ, AM, 〈ΔΦ〉satand ∣R∣ (Fig. 5d), we observe three distinct behaviours for Ncoherent depending on
L. For small L < 0.001 (ID regime) Ncoherent decreases with th, whereas for large
I > 0.01 (SSP Iregime), Ncoherent increases, with relative changes of up to 30% at
L
thI = 100 ms compared with the respective initial values. In the intermediate- LIregime, Ncoherent remains almost constant. A possible explanation for the occurrenceof these three behaviours might be the differences in their rate of three-body loss.
Although in the investigated B-field regime, the three-body loss coefficient isalmost constant, the increased peak density expected in the ID regime results in ahigher three-body loss rate, which can surpass the plain evaporation rate and thus
result in a decreasing Ncoherent. The lower peak density in the SSP, on the contrary,
could result in a lower three-body loss rate and thus lead to an increasing Ncoherent
via plain evaporation.

Data availability

NAtuRE PHYsics
41. Chomaz, L. et al. Quantum-fluctuation-driven crossover from a diluteBose–Einstein condensate to a macrodroplet in a dipolar quantum fluid. Phys.
Rev. X 6, 041039 (2016).
42. Wächtler, F. & Santos, L. Quantum filaments in dipolar Bose–Einsteincondensates. Phys. Rev. A 93, 061603 (2016).
43. Wächtler, F. & Santos, L. Ground-state properties and elementary excitationsof quantum droplets in dipolar Bose–Einstein condensates. Phys. Rev. A 94,
043618 (2016).
44. Ilzhöfer, P. et al. Two-species five-beam magneto-optical trap for erbium anddysprosium. Phys. Rev. A 97, 023633 (2018).
45. Trautmann, A. et al. Dipolar quantum mixtures of erbium and dysprosium
atoms. Phys. Rev. Lett. 121, 213601 (2018).
46. Chin, C., Grimm, R., Julienne, P. S. & Tiesinga, E. Feshbach resonances inultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010).
47. Ferrier-Barbut, I. et al. Scissors mode of dipolar quantum droplets ofdysprosium atoms. Phys. Rev. Lett. 120, 160402 (2018).
48. Maier, T. et al. Broad universal Feshbach resonances in the chaotic spectrumof dysprosium atoms. Phys. Rev. A 92, 060702 (2015).
49. Forbes, C., Evans, M., Hastings, N. & Peacock, B. Statistical Distributions 4thedn, Ch. 8, 55–61 (Wiley, 2010).
50. Blakie, P., Baillie, D., Chomaz, L. & Ferlaino, F. Supersolidity in an elongateddipolar condensate. Preprint at https://arxiv.org/abs/2004.12577 (2020).
51. Sohmen, M. Source Data—Ilzhofer et al.—Phase coherence inout-of-equilibrium supersolid states of ultracold dipolar atoms. Zenodo
https://doi.org/10.5281/zenodo.4050234 (2020).

Acknowledgements

We are grateful to S. Erne, J. Schmiedmayer and the ERBIUM team for insightful
discussions and M. A. Norcia for careful reading the manuscript. We acknowledge R.
M. W. van Bijnen for developing the code for our eGPE ground-state simulations. G.M.
and T.G. thank N. Caballero for insightful discussions on the numerical solution of theLangevin equation. This work is financially supported through an ERC Consolidator
grant (RARE, no. 681432), an NFRI grant (MIRARE, no. ÖAW0600) from the Austrian
Academy of Science and DFG/FWF (FOR 2247/PI2790) and by the Swiss National
Science Foundation under Division II. M.S. and G.D. acknowledge support by theAustrian Science Fund FWF within the DK-ALM (no. W1259-N27). A.T. acknowledges
support by the Austrian Science Fund FWF within the Lise Meitner programme (no.
M2683-N36). L.C. acknowledges support through the FWF Elise Richter Fellowship no.
V792. We also acknowledge the Innsbruck Laser Core Facility, financed by the AustrianFederal Ministry of Science, Research and Economy. Part of the computational resultspresented have been achieved using the HPC infrastructure LEO at the University of
Innsbruck.

Author contributions

P.I., G.D. and A.T. conducted the experiment and collected the experimental data. M.S.
and C.P. analysed the data. G.N., M.J.M., L.C., M.S. and C.P. performed and analysed
the eGPE simulations. G.M. and T.G. developed the JJA model and performed thecorresponding simulations. All the authors contributed to the writing of the paper. F.F.
supervised the project.

Competing interests

The authors declare no competing interests.

Source data are available for this paper51. All other data that support theplots within this paper and other findings of this study are available from thecorresponding author upon reasonable request.

Additional information

References

Peer review information Nature Physics thanks the anonymous reviewers for theircontribution to the peer review of this work.

40. Ronen, S., Bortolotti, D. C. E. & Bohn, J. L. Bogoliubov modes of a dipolarcondensate in a cylindrical trap. Phys. Rev. A 74, 013623 (2006).

Extended data is available for this paper at https://doi.org/10.1038/s41567-020-01100-3.
Correspondence and requests for materials should be addressed to F.F.

Reprints and permissions information is available at www.nature.com/reprints.

Nature Physics | www.nature.com/naturephysics

NAtuRE PHYsics

ArticlesExtended Data Fig. 1 | Wavelength of the modulation and finite-sampling effect. a, Difference between incoherent and coherent mean of the density
profiles in the ID regime (1.65 G), peaking at the modulation wavelength d ≃ ± 2 μm (dashed lines). b, Histograms of 106 realisations (each) for
calculations of ΔqΦ from uniformly random phases Φi, for q = 35 (green) and q = 100 (yellow) draws, respectively. The dashed vertical lines reflect theconfidence interval enclosing 68.3 % (‘one σ’) of the calculated values. The solid lines depict a Beta distribution with same mean and variance as thedrawn distribution of ΔqΦ (no free fit parameters).

Nature Physics | www.nature.com/naturephysics

Articles

NAtuRE PHYsics

Extended Data Fig. 2 | Estimated scattering length. Calculated B-to-as conversion for 164Dy. Red and blue shaded areas indicate the SSP and the ID region,
respectively. The grey area indicates the BEC region, while the yellow areas indicate regions around the two narrow Feshbach resonances located at2.174 G and 2.336 G where we observe increased atom loss. We estimate as,SSP = 88 a0 in the SSP at 2.43 G and as,ID = 76.9 a0 in the ID at 1.65 G.

Nature Physics | www.nature.com/naturephysics

NAtuRE PHYsics

ArticlesExtended Data Fig. 3 | Temporal evolution of the atom numbers and temperature in the ID regime and SSP. a, Total atom number Ntotal, b, temperature T
and atom numbers of c, the thermal and d, the coherent part, Nthermal and Ncoherent, as a function of the hold time th. The data sets at 1.65 G (blue) and 2.43 G(red) correspond to the ID regime and the SSP, respectively, whereas the one at 2 G (light blue) corresponds to the intermediate regime.

Nature Physics | www.nature.com/naturephysics

Articles

NAtuRE PHYsics

Extended Data Fig. 4 | The global phase variation α from the RTE simulation of a scramble-and-rephase protocol. a, Evolution of α over the hold time th.
The solid orange line depicts an exponential fit to the data. In the inset, the integrated density n and the phase profile θ are exemplarily shown for t* = [3.5,9.5, 60.5] ms (note the corresponding color filling of the plot markers). b, Residuals from the exponential fit to α.

Nature Physics | www.nature.com/naturephysics

NAtuRE PHYsics

ArticlesExtended Data Fig. 5 | Dependence of experimental rephasing dynamics on the density link strength L. a, Temporal evolution of ΔΦ (color map) at
I
different L starting from phases scrambled in the ID regime. For each th we record q≥35 individual experimental
realizations. For large L the system
I global phase coherence (ΔΦ ≃ 0), whereas for small L it does not (ΔΦ ≃ 1). b, A (circles) and A (diamonds) for the sameI data set atrecovers its
MΦlong hold time, th = 100 ms. The error bars (partly covered by plot Imarkers) are statistical standard errors of AM and AΦ. The red filled pair of symbols
corresponds to the data set presented in Fig. 3.

Nature Physics | www.nature.com/naturephysics

A.5 Publication:
Observation of vortices and vortex stripes in a dipolar Bose-Einstein condensate 205

A.5

Publication:

Observation of vortices and vortex stripes in a dipolar
Bose-Einstein condensate
The following publication has appeared in
Nature Physics (2022)¶
submitted 15 Jun 2022; published 31 Oct 2022
DOI: https://arxiv.org/abs/2206.12265
L. Klaus,1,2,4 T. Bland,1,4 E. Poli,2 C. Politi,1,2 G. Lamporesi,3 E. Casotti,1,2 R. N. Bisset,2
M. J. Mark,1,2 and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wis-

senschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3 INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123, Trento,

Italy
4 These authors contributed equally: L. K. and T. B.

¶The author of the present thesis performed the measurements together with L. K., and E. C.,

analysed the experimental data with L. K., E. C., and M. J. M., and contributed in writing themanuscript and interpreting the results together with all the authors.

nature physics
Articlehttps://doi.org/10.1038/s41567-022-01793-8

Observation of vortices and vortex stripes ina dipolar condensate
Received: 15 June 2022
Accepted: 8 September 2022

Lauritz Klaus 1,2,4, Thomas Bland 1,2,4, Elena Poli 2, Claudia Politi1,2,
Giacomo Lamporesi 3, Eva Casotti 1,2, Russell N. Bisset 2,
Manfred J. Mark 1,2 and Francesca Ferlaino 1,2

Published online: xx xx xxxx
Check for updates

Quantized vortices are a prototypical feature of superfluidity that havebeen observed in multiple quantum gas experiments. But the occurrence ofvortices in dipolar quantum gases—a class of ultracold gases characterized
by long-range anisotropic interactions—has not been reported yet. Here
we exploit the anisotropic nature of the dipole–dipole interaction ofa dysprosium Bose–Einstein condensate to induce angular symmetry
breaking in an otherwise cylindrically symmetric pancake-shaped trap.
Tilting the magnetic field towards the radial plane deforms the cloud into an
ellipsoid, which is then set into rotation. At stirring frequencies approaching
the radial trap frequency, we observe the generation of dynamicallyunstable surface excitations, which cause angular momentum to be pumped
into the system through vortices. Under continuous rotation, the vortices
arrange into a stripe configuration along the field, in close agreement withnumerical simulations.

Since the first experiments on gaseous Bose–Einstein condensates(BECs), the observation of quantized vortices has been considered
the most fundamental and defining signature of the superfluid natureof such systems. Their very existence sets a unifying concept encompassing a variety of quantum fluids from liquid helium1 to the core ofneutron stars2 and from superconductors3 to quantum fluids of light4.
Their classical counterparts have as well fascinated scientists from different epochs and fields as vortices are found in many scales of physicalsystems, from tornadoes in the atmosphere to ferrohydrodynamics.
In the quantum realm, a quantized vortex may emerge as a unique
response of a superfluid to rotation. It can be understood as a typeof topologically protected singularity with a 2π phase winding that
preserves the single-valuedness of the superfluid wave function andthe irrotational nature of its velocity field. In contact-interacting BECs,
vortical singularities have been observed experimentally in the form of
single vortices5,6, vortex–antivortex pairs7, solitonic vortices8,9, vortex
rings10 and vortex lattices6,11 using a number of different techniques.
Moreover, vortices play a fundamental role in the description of theBerezinskii–Kosterlitz–Thouless transition in two-dimensional (2D)

systems12, as well as in the evolution of quantum turbulence13,14, andhave been observed in interacting Fermi gases along the Bose-Einsteincondensate to Bardeen-Cooper-Schrieffer crossover8,15.
Recently, a new class of ultracold quantum gases are being createdin various laboratories around the world, using strongly magnetic
lanthanide atoms16,17. Such a system, providing a quantum analogueof classical ferrofluids, enables access to the physics of dipolar BECs,
in which atoms feature a strong long-range anisotropic dipole–dipole
interaction (DDI)18,19 on top of the traditional contact-type isotropic
one. This intriguing platform provided the key to observe, for example, extended Bose–Hubbard dynamics20, roton excitations21–23, thequantum version of the Rosensweig instability24 and supersolid statesof matter25–28, and is foreseen to host novel phenomena for quantum
simulation and metrology18,19.
The dipolar interaction is predicted to also intimately change theproperties of vortices in quantum gases29. For instance, theoreticalworks predict single vortices to exhibit an elliptic-shaped core for aquasi-2D setting with in-plane dipole orientation30–33 or the presenceof density oscillations around the vortex core induced by the roton

1
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck, Austria. 2Institut für Experimentalphysik,
Universität Innsbruck, Innsbruck, Austria. 3INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, Povo, Italy. 4These authors contributed
equally: Lauritz Klaus, Thomas Bland.
e-mail: Francesca.Ferlaino@uibk.ac.at

Nature Physics

Articleahttps://doi.org/10.1038/s41567-022-01793-8

i
z

ii
y

iii

iv

B=0

Ω

x

b
Ω

y
tΩ

tΩ

tΩ

x

Ω

tΩ

5 µm

c

Ω

Aspect ratio

4

3

2

tΩ

1

0

0.25

0.50

0.75

Rotation frequency Ω (ω )

1.00

┴

Fig. 1 | Magnetostirring of a Dy dipolar BEC and evolution of the cloud aspectratio. a, 3D simulations and corresponding shadow on the x-y plane of anon-dipolar (i) and dipolar BEC with B ≠ 0 (ii–iv) in a cylindrically symmetric,
oblate trap. The magnetic-field (green arrows) angle with respect to the z axis
varies from θ = 0° (ii) to θ = 35° (iii) and rotating at θ = 35° around z (iv). b, Leftpanels show the experimental sequence for the stirring procedure. The grey areas
indicate the stage during which the images in the right panels were taken. The
right panels are representative axial absorption images showing the dipolar BEC
while spinning up the magnetic field for tΩ̇ = [140, 430, 627, 692] ms (top) andsubsequent constant rotation at Ω = 2π × 36 Hz for tΩ = [0, 6, 11, 17] ms (bottom).
The rotation of the magnetic field in the x-y plane is indicated by the white line.
c, (left) Time evolution of the magnetic field rotation frequency. Ω is linearly
increased to its final value at a speed of Ω̇ = 2𝜋𝜋 × 50 Hz s−1. (right) Cloud AR for
different final rotation frequencies. To mitigate influences of trap anisotropies on
the AR, a full period at the final rotation frequency is probed. The error bars,
representing the standard error on the mean after 100 trials per point, are smaller
than the symbol size. The solid (dashed) black line shows the corresponding eGPE
simulations with a 2 s (1 s) ramp and as = 110a0, (ω⊥, ωz) = 2π × [50, 130] Hz, andN = 15,000. Different colors of the experimental point in the right panel indicate
the corresponding time during the ramp in the left panel.

minimum in the dispersion relation30–34. For vortex pairs, the anisotropic DDI is expected to alter the lifetime and dynamics33,35 and caneven suppress vortex–antivortex annihilation33. These interaction
properties are predicted to give rise to a vortex lattice structure that canfollow a triangular pattern30,34, as is typical for non-dipolar BECs11, or asquare lattice for attractive or zero contact interactions36–38 when the
DDI is isotropic (dipoles aligned with the rotation axis). A very striking
consequence of the dipoles tilted towards the plane is the formation ofvortex stripes30,39,40. Moreover, vortices could provide an unambiguous
Nature Physics

smoking gun of superfluidity in supersolid states41–43. However, despite
these intriguing predictions, vortices in dipolar quantum gases havenot been observed until now.
This Article presents the experimental realization of quantized vortices in a dipolar BEC of highly magnetic dysprosium (Dy)atoms. Following a method proposed in ref. 40, extended to arbitrary
magnetic-field angles in ref. 44, we show that the many-body phenomenon of magnetostriction45, genuinely arising from the anisotropic
DDI among atoms, provides a natural route to rotate the systems andnucleate vortices in a dipolar BEC. We carry out studies on the dynamics of the vortex formation, which agree very well with our theoreticalpredictions. Finally, we observe one of the earliest predictions for
vortices in dipolar BECs: the formation of vortex stripes in the system.
In non-dipolar gases, quantized vortices have been produced using
several conceptually different techniques, for instance, by rotating
non-symmetric optical6,11 or magnetic46 potentials, by rapidly shaking
the gas14, by traversing it with obstacles with large enough velocity7,47, by
rapidly cooling the gas across the BEC phase transition48,49 or by directly
imprinting the vortex phase pattern50. Dipolar quantum gases, whileable to form vortices with these same standard procedures29, also offer
unique opportunities that have no counterpart in contact-interacting
gases. Crucially, the DDI gives rise to the phenomenon of magnetostriction in position space45. When dipolar BECs are polarized by an external
magnetic field B—defining the dipole orientation—the DDI causes an
elongation of the cloud along the polarization direction. This is a directconsequence of the system tendency to favour head-to-tail dipole
configurations, which effectively reduces the interaction energy19.
Such a magnetostrictive effect provides a simple method to induce
an elliptic effective potential and drive rotation with a single control
parameter. This modification of the effective potential is shown in
Fig. 1a for a BEC in an oblate trap with cylindrical symmetry about thez axis. While a non-dipolar BEC takes the same shape as the confining
trap (Fig. 1a(i)), introducing dipolar interactions with polarization axis
along z stretches the cloud along this axis yet maintains cylindricalsymmetry (Fig. 1a(ii)). Tilting the magnetic field leads to a breakingof the cylindrical symmetry, resulting in an ellipsoidal deformationof the cloud shape, as seen from the density projection onto the x–yplane (Fig. 1a(iii)). Finally, under continuous rotation of the magnetic
field, which we coin ‘magnetostirring’, the condensate is predicted torotate (Fig. 1a(iv)). This unique approach to stir a dipolar condensate
can eventually lead to the nucleation of vortices40,44, realizing genuinely
interaction-driven vorticity through many-body phenomena.
We explore this protocol using a dipolar BEC of 162Dy atoms. We
create the BEC similar to our previous work51 with the distinction that
here the magnetic-field unit vector, B̂, is kept tilted at an angle of θ = 35°
with respect to the z axis both during evaporative cooling and magnetostirring (Fig. 1a(iii) and Methods). After preparation, the sample
contains about 2 × 104 condensed atoms confined within a cylindricallysymmetric optical dipole trap (ODT) with typical radial and axial trap
frequencies (ω⊥, ωz) = 2π × [50.8(2), 140(1)] Hz. Here, before stirring,the magnetostriction is expected from simulations to increase thecloud aspect ratio (AR) in the horizontal plane from 1 up to 1.03, whereas
the trap anisotropy is negligible. We use a vertical (z) absorption imaging to probe the radial (x,y) atomic distribution after a short
time-of-flight (TOF) expansion of 3 ms. The atom number is insteadmeasured using horizontal absorption imaging with a TOF of 26 ms.
Similarly to a rotation of a bucket containing superfluid helium orof a smoothly deformed ODT for non-dipolar BECs, magnetostirring
is predicted to transfer angular momentum into a dipolar BEC40,44. In
response to such an imposed rotation, the shape of an irrotational
cloud is expected to elongate with an amplitude that increases withthe rotation frequency Ω. This phenomenon is clearly visible in our
experiments, as shown in Fig. 1b. Here we first revolve the tilted B̂
around the z axis with a linearly increasing rotation frequency( Ω̇ = 2𝜋𝜋 × 50 Hz s−1) and observe that the dipolar BEC starts to rotate

Articlehttps://doi.org/10.1038/s41567-022-01793-8
207 ms

314 ms

447 ms

527 ms607 ms741 ms

848 ms981 msExperimental

SimulatedtΩ = 127 ms

5 µm

Fig. 2 | Observation of vortices in a dipolar BEC. Each column shows thesimulated (top) and experimental (bottom) images for various rotation timestΩ. For the experiment, the atoms are imaged along the z direction. In eachexperimental run, we rotate the magnetic field anticlockwise at Ω = 0.74ω⊥ fordifferent rotation times tΩ. The magnetic-field value is kept to B = 5.333(5) G. The

initial condensed atom number is N = 15,000. The decreasing size of the cloudsuggests a decrease in atom number. However, for states with vortices or spiralshapes, appearing at large tΩ, our bimodal fit to extract the atom number breaks
down. For the corresponding simulations, the parameters are as = 112a0, trap
frequencies (ω⊥, ωz) = 2π × [50, 150] Hz, N = 8,000 and Ω = 0.75ω⊥.

at the same angular speed as the field and deforms with increasing
elongation (Fig. 1b, top). We then stop the adiabatic ramp at a given
value of Ω and probe the system under continuous rotation. We now
find that the cloud continues rotating in the radial plane with an almost
constant shape (Fig. 1b, bottom). Note that B is held constant at
5.333(5) G, where we estimate a contact scattering length of about
as = 111a0, where a0 is the Bohr radius (Methods).
We further explore the response of our dipolar BEC to magnetostirring by repeating the measurements in Fig. 1b (top), but stopping
the ramp at different final values of Ω. The maximum value used for Ω
approaches ω⊥, corresponding to a ramp duration of 1 s. We quantify
the cloud elongation in terms of the aspect ratio AR = σmax/σmin, where
the cloud widths σmax and σmin are extracted by fitting a rotated 2D
Gaussian function to the density profiles. Figure 1c summarizes our
results. We observe that initially the AR slightly deviates from 1 due tomagnetostriction. It then slowly grows with increasing Ω, until a rapid
increase at around 0.6ω⊥ occurs, as this allows the angular momentum
to increase, which decreases the energy in the rotating frame52. Suddenly, at a critical rotation frequency Ωc ≈ 0.74ω⊥, the AR abruptly collapses back to AR ≈ 1, showing how the superfluid irrotational naturecompetes with the imposed rotation. This critical frequency is close to
the value found in non-dipolar gases with a rotating elliptical harmonic
trap, associated with a resonance at the quadrupole frequency53.
To substantiate our observation, we perform numerical simulations of the zero-temperature extended Gross–Pitaevskii equation(eGPE)54 (Methods). Quantum and thermal fluctuations are added tothe initial states, which are important to seed the dynamic instabilities
once they emerge at large enough Ω; see later discussion. The lines inFig. 1c show our results. The dashed line is obtained through the sameprocedure as the experiment, whereas for the solid line, we halve theramp rate, spending more time at each frequency. Both ramp procedures show quantitatively the same behaviour up to Ω = 0.8ω⊥ and are
in excellent agreement with the experimental results. The stability of
the 1 s ramp exceeds the experimentally observed critical frequency.
We partly attribute this discrepancy to asymmetries of the rotation in
the experiment that are not present in the simulations, which may leadto an effective speed-up of the dynamical instabilities. However, in allcases, the AR rapidly decreases to about 1.
The growing AR and subsequent collapse to 1 is a signature of the
dynamical instability of surface modes, known for being an importantmechanism for seeding vortices and allowing them to penetrate into
the high-density regions of rotated BECs52,53,55, as also predicted forour dipolar system40. To search for quantum vortices in our system, we
perform a new investigation where we directly set Ω close to Ωc, aiming
to trigger the instability at an earlier time when more atoms are condensed. We then hold the magnetic-field rotation fixed at this constantfrequency for a time tΩ. As shown in Fig. 2 (bottom), the cloud rapidly
elongates, and the density starts to exhibit a spiral pattern, emanatinga

Nature Physics

–0.3

0.4

5 µm

b

Vortex number

v

8
6
4
2
0

Aspect ratio

4

3

2

1

0

100

200

300

400

500

600

700

Rotation time t Ω (ms)Fig. 3 | Time evolution of the average vortex number, 𝒩𝒩v, and cloud AR. a, Left:sample image after rotating for tΩ = 474 ms. Middle: blurred reference image(σ = 2.1 μm). Right: residuals with markers (black circles) indicating the identified
vortices. b, The detected vortex number 𝒩𝒩v (top) and the AR of the cloud(bottom) after the rotation time tΩ. Data points and error bars show the mean andstandard error from about ten experimental runs. Solid lines indicate theaveraged results from ten corresponding simulations with different initial noisefor parameters as = 110a0, (ω⊥, ωz) = 2π × [50, 130] Hz, N = 10,000 and Ω = 0.75ω⊥;the shaded area gives its standard error.

from the tips of the ellipsoid. As early as tΩ = 314 ms, clear holes areobserved in the density profile, forming in the density halo around thecentre, the first clear indication of vortices in a dipolar gas. These vortices, initially nucleated at the edge of the sample, persist as we continue
to stir and eventually migrate towards the central (high-density) region.
Vortices are still visible in the experiment after 1 s of magnetostirring,
although our atom number decreases throughout this procedure.

Articlehttps://doi.org/10.1038/s41567-022-01793-8
Density

Residual

FT average-1

1 µm–1

FT

DensityFT

FTResidual

Experimentald

FT

Simulation

Simulationc

DensityFT

5 µm

Residualb

Experimental

Ground stateaeFig. 4 | Stripe nature of vortices in a dipolar BEC. a, Left: ground-state stripe
lattice solution for our experimental parameters as = 109a0, trap frequencies(ω⊥, ωz) = 2π × [50, 130] Hz, N = 10,000 and Ω = 0.75ω⊥. Middle: corresponding
residual image, found by subtracting the ground state from the blurred image,with circles showing the detected vortices. Right: Fourier transform of theresidual image. b, Left: single experimental image after 500 ms of continuous
rotation at Ω = 0.75ω⊥. Middle: the corresponding residual image. Right: Fouriertransform of the residual images, averaged over 49 runs, with example shots

shown to the right. c, Left: simulation result for the dynamic experimental
procedure in b. Middle and right: residuals (middle) and FT analysis (right)(115 temporal images) as in b. d, The same as b for 121 runs, but we rotate foran additional 100 ms and then spiral the magnetic field to θ = 0° over a further
100 ms before imaging. e, Simulation result for procedure in d. All simulation
images are rotated to have the same magnetic-field direction as the experiment,
as indicated by the white arrow in a and by the circles in d.

Our observations bear a remarkable resemblance to the simulations;Fig. 2 (top) shows the in situ column densities. Taking a fixed atom number of N = 8,000, but otherwise repeating the experimental sequence,
we observe many similar features. In the first 100 ms, the system elongates, consistent with Fig. 1, and a spiral density pattern appears before
the instability, forming two arms that are filled with vortices close to thecentral density. Next, turbulent dynamics ensue as the density surface
goes unstable and vortices emerge in the central high-density region.
For this scattering length and atom number, the relaxation timescale
to a stable vortex lattice is longer than the experimentally available(see Extended Data Fig. 3 for more images from this dataset). Notethat at angles θ deeper into the plane, more atoms align head-to-tailin the loose radial confinement direction. Thus, when performing therotation procedure, we find that the BEC is resilient to instability on
the timescales of the experiment.
The observed evolution of the system under constant rotationshows some concurrence between the appearance of vortices in theabsorption images and the formation of a round density pattern in theradial plane with AR ≈ 1 (Fig. 2). Note that the drop in AR observed inFig. 1 is concurrent with the creation of vortices, but they reside in the
low-density regions at this time, and we do not see them. To study thisdynamical evolution in more detail, we adopt an analysis protocol forboth the experiment and theory that allows us to quantitatively trackthe evolution of the average number of vortices, 𝒩𝒩v (Methods). The
result is shown in Fig. 3a. In brief, for each single image (Fig. 3a, left),
we create a blurred reference image by applying a 2D Gaussian filter56,57.
We then calculate the difference between each single image (Fig. 3a,left) and the corresponding reference (Fig. 3a, middle) to obtain theresidual image (Fig. 3a, right), from which we count 𝒩𝒩v by finding local
minima below a certain threshold.
For the experimental density profiles, which are affected by boththe limited resolution of the imaging system and the weak contrast in

the low-density zones (halo) where the vortices initially nest, we expect
𝒩𝒩v to be underestimated relative to the true value and the number
expected by theory. However, to carry out a quantitative comparison
with the simulations, we apply a blurring filter and add noise to thelatter that mimics the actual resolution in the experiment(Methods).
Figure 3b shows both the evolution of 𝒩𝒩v and cloud AR as a functionof rotation time, tΩ. Solid lines are the results from the eGPE simulations
without any adjustable parameters. For tΩ < 200 ms, 𝒩𝒩v is below 1, where
vortices, if present, are at the edge of the cloud. For longer times, 𝒩𝒩v
increases and saturates to an average value of about three and a maximum of six vortices (see Fig. 3a for an example of five vortices). The
observed saturation might be due to the decreased visibility and to theatom-loss-induced shrinking of the BEC size, which is not accounted
for in the theory. We also compare the course of the average vortex
number with the AR of the cloud. After initial large oscillations, due tothe sudden jump in rotation frequency, the AR declines towards ~1(ref. 53). This happens as the vortex number simultaneously increases.
One fascinating prediction with vortices in a strongly dipolar gas
under the influence of a rotating magnetic field relates to the structure of the resulting vortex lattice. Due to magnetostriction and theanisotropic vortex cores, the resulting vortex configuration is also
anisotropic, producing a stripe phase in the strongly dipolar regime29,30,
instead of the usual triangular lattice in non-dipolar BECs6. The groundstate stripe lattice solution for our parameters is shown in Fig. 4a,with a cloud AR = 1.08. In the vortex stripe phase, vertical planes ofhigh-density regions, parallel to the magnetic field, alternate with
low-density ones, which host rows of vertical vortex filaments. Sucha configuration promotes head-to-tail dipolar attraction within thehigh-density ridges, and this acts to lower the energy. It should be notedthat these states are distinct from the oscillating vortex sheets states,
which appear after squeezing a triangular vortex lattice58.

Nature Physics

Article
To explore this prediction, we perform two new surveys. First,
we slightly reduce the magnetic-field value, reducing the scattering
length to as ≈ 109a0 and hence making the system relatively more
dipolar. We magnetostir the BEC at a constant rotation frequency
Ω = 0.75ω⊥ for 500 ms, but during TOF, we stop the magnetic-fieldrotation and keep it in place at θ = 35°. The stripe structure is revealedin Fig. 4b (left) for a single experimental run and is clearly visible in
the residual image (Fig. 4b, middle) where the vortices align along
three stripes. The spatial structure of the residual image can beassessed through the absolute value of 2D Fourier transform (FT).
After taking the FT of each residual image, we then average the result(Fig. 4b, right), finding a clear peak at the wave number k of theinter-stripe spacing. This shows that the stripe spatial structure
survives the averaging, implying that the majority of images show
stripes with the same spacing, and they also have the same orientation as set by the magnetic field, as evidenced by the example images
shown in the right of Fig. 4. Note that these observations do not
rely on our ability to resolve individual vortices, as the stripes are
an ensemble effect of many aligned vortices. In fact, by comparing
with the numerical simulations of the dynamical procedure (Fig. 4c),
we expect there are more vortices than detected here that fill in the
stripes, forging out this structure. In general, our simulations showthat the stripes appear faster when the scattering length is lower andwhen the atom number is larger. In the long time limit of the scenariopresented in Fig. 2, we expect the stationary solution to also be thestripe state, but this is not observable on our timescales.
Remarkably, the stripe structure washes out when we subsequently tilt the magnetic-field orientation to θ = 0° (parallel to thetrap symmetry axis), as shown in Fig. 4d (left). Here, after 600 ms ofmagnetostirring, we add another step in which we spiral up the magnetic field to θ = 0° (with Ω fixed) over 100 ms, before imaging. Under
these conditions, all vortex properties are again isotropic within theplane. The non-equilibrium positioning of the vortices is arbitrary,and if we average the FT of the residuals directly, we observe a homogeneous ring in the average FT (Fig. 4d, right). Also, this behaviour isconfirmed by the simulations, as shown in Fig. 4e. The vortices survive
long after the magnetostirring has stopped (not shown), due to their
topological protection.
By exploiting magnetostirring—a novel, robust method of generating angular momentum—we have observed quantized vortices
in a dipolar quantum gas and the appearance of the vortex stripe
configuration. Future works will focus on investigations of the individual vortex shape and behaviour, such as the anisotropic natureof the vortex cores for in-plane magnetic fields30–33, the interplay
between the vortex and roton excitations30–34 and exotic vortex patterns such as square lattices29, and investigations into anisotropic turbulence59. This work also opens the door to studying more complex
matter under rotation, such as dipolar droplets60–62 and supersolidstates41–43,51. Such proposals will be challenging due to the intricate
density patterns63; however, such observations would provide conclusive evidence of superfluidity in supersolids. Rotating the magnetic field at frequencies far larger than the radial trap frequencies,
but smaller than the Larmor frequency, has been observed to tune
the sign and magnitude of the dipole–dipole interaction64—a method
also employed in nuclear magnetic resonance spectroscopy—
but there remain open questions on the stability of this procedure65,66, which if rectifiable would unlock new research directions64.
Other vortex generation methods, such as thermally activatedpairs in quasi-two dimensions to assess the Berezinskii–Kosterlitz–
Thouless transition and stochastically generated vortex tangles
through temperature quenches to assess the Kibble–Zurek mechanism, remain unexplored in dipolar gases29. The technique introduced here is also applicable to a wide range of systems governed
by long-range interactions through the manipulation of magnetic
or electric fields.
Nature Physics

https://doi.org/10.1038/s41567-022-01793-8

Online content

Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information,
acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41567-022-01793-8.

References
1.

Donnelly, R. J. Quantized Vortices in Helium II (Cambridge Univ.
Press, 1991).
2. Pines, D. & Alpar, M. A. Superfluidity in neutron stars. Nature 316,
27–32 (1985).
3. Abrikosov, A. A. Nobel Lecture: Type-II superconductors and thevortex lattice. Rev. Mod. Phys. 76, 975 (2004).
4. Lagoudakis, K. G. et al. Quantized vortices in an exciton–polariton
condensate. Nat. Phys. 4, 706–710 (2008).
5. Matthews, M. R. et al. Vortices in a Bose–Einstein condensate.
Phys. Rev. Lett. 83, 2498 (1999).
6. Madison, K. W., Chevy, F., Wohlleben, W. & Dalibard, J. Vortex
formation in a stirred Bose–Einstein condensate. Phys. Rev. Lett.
84, 806 (2000).
7. Neely, T. W., Samson, E. C., Bradley, A. S., Davis, M. J. & Anderson,
B. P. Observation of vortex dipoles in an oblate Bose–Einsteincondensate. Phys. Rev. Lett. 104, 160401 (2010).
8. Ku, M. J. H. et al. Motion of a solitonic vortex in the BEC–BCS
crossover. Phys. Rev. Lett. 113, 065301 (2014).
9. Donadello, S. et al. Observation of solitonic vortices in Bose–
Einstein condensates. Phys. Rev. Lett. 113, 065302 (2014).
10. Anderson, B. P. et al. Watching dark solitons decay into vortex
rings in a Bose–Einstein condensate. Phys. Rev. Lett. 86, 2926
(2001).
11. Abo-Shaeer, J. R., Raman, C., Vogels, J. M. & Ketterle, W.
Observation of vortex lattices in Bose–Einstein condensates.
Science 292, 476–479 (2001).
12. Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard,
J. Berezinskii–Kosterlitz–Thouless crossover in a trapped atomicgas. Nature 441, 1118–1121 (2006).
13. Neely, T. W. et al. Characteristics of two-dimensional quantum
turbulence in a compressible superfluid. Phys. Rev. Lett. 111,
235301 (2013).
14. Navon, N., Gaunt, A. L., Smith, R. P. & Hadzibabic, Z. Emergence ofa turbulent cascade in a quantum gas. Nature 539, 72–75 (2016).
15. Zwierlein, M. W., Abo-Shaeer, J. R., Schirotzek, A., Schunck, C. H.
& Ketterle, W. Vortices and superfluidity in a strongly interacting
Fermi gas. Nature 435, 1047–1051 (2005).
16. Lu, M., Burdick, N. Q., Youn, S. H. & Lev, B. L. Strongly dipolar
Bose–Einstein condensate of dysprosium. Phys. Rev. Lett. 107,
190401 (2011).
17. Aikawa, K. et al. Bose–Einstein condensation of erbium. Phys. Rev.
Lett. 108, 210401 (2012).
18. Norcia, M. A. & Ferlaino, F. Developments in atomic control using
ultracold magnetic lanthanides. Nat. Phys. 17, 1349–1357 (2021).
19. Chomaz, L. et al. Dipolar physics: a review of experimentswith magnetic quantum gases. Preprint at https://arxiv.org/
abs/2201.02672 (2022).
20. Baier, S. et al. Extended Bose–Hubbard models with ultracold
magnetic atoms. Science 352, 201–205 (2016).
21. Landau, L. Theory of the superfluidity of helium II. Phys. Rev. 60,
356 (1941).
22. Chomaz, L. et al. Observation of roton mode population in a
dipolar quantum gas. Nat. Phys. 14, 442–446 (2018).
23. Schmidt, J.-N. et al. Roton excitations in an oblate dipolar
quantum gas. Phys. Rev. Lett. 126, 193002 (2021).
24. Kadau, H. et al. Observing the Rosensweig instability of aquantum ferrofluid. Nature 530, 194–197 (2016).

Article
25. Tanzi, L. et al. Observation of a dipolar quantum gas withmetastable supersolid properties. Phys. Rev. Lett. 122, 130405(2019).
26. Böttcher, F. et al. Transient supersolid properties in an array of
dipolar quantum droplets. Phys. Rev. X 9, 011051 (2019).
27. Chomaz, L. et al. Long-lived and transient supersolid behaviors in
dipolar quantum gases. Phys. Rev. X 9, 021012 (2019).
28. Norcia, M. A. et al. Two-dimensional supersolidity in a dipolar
quantum gas. Nature 596, 357–361 (2021).
29. Martin, A. M., Marchant, N. G., O’Dell, D. H. J. & Parker, N. G.
Vortices and vortex lattices in quantum ferrofluids. J. Phys.
Condens. Matter 29, 103004 (2017).
30. Yi, S. & Pu, H. Vortex structures in dipolar condensates. Phys. Rev.
A 73, 061602 (2006).
31. Ticknor, C., Wilson, R. M. & Bohn, J. L. Anisotropic superfluidity ina dipolar Bose gas. Phys. Rev. Lett. 106, 065301 (2011).
32. Mulkerin, B. C., van Bijnen, R. M. W., O’Dell, D. H. J., Martin, A. M.
& Parker, N. G. Anisotropic and long-range vortex interactions in
two-dimensional dipolar Bose gases. Phys. Rev. Lett. 111, 170402(2013).
33. Mulkerin, B. C., O’Dell, D. H. J., Martin, A. M. & Parker, N. G.
Vortices in the two-dimensional dipolar Bose gas. J. Phys. Conf.
Ser. 497, 012025 (2014).
34. Jona-Lasinio, M., Łakomy, K. & Santos, L. Roton confinement intrapped dipolar Bose–Einstein condensates. Phys. Rev. A 88,
013619 (2013).
35. Gautam, S. Dynamics of the corotating vortices in dipolar Bose–
Einstein condensates in the presence of dissipation. J. Phys. B 47,
165301 (2014).
36. Cooper, N., Rezayi, E. & Simon, S. Vortex lattices in rotating
atomic Bose gases with dipolar interactions. Phys. Rev. Lett. 95,
200402 (2005).
37. Zhang, J. & Zhai, H. Vortex lattices in planar Bose–Einsteincondensates with dipolar interactions. Phys. Rev. Lett. 95, 200403(2005).
38. Kumar, R. K., Sriraman, T., Fabrelli, H., Muruganandam, P. &Gammal, A. Three-dimensional vortex structures in a rotating
dipolar Bose–Einstein condensate. J. Phys. B 49, 155301 (2016).
39. Cai, Y., Yuan, Y., Rosenkranz, M., Pu, H. & Bao, W. Vortex patterns
and the critical rotational frequency in rotating dipolar Bose–
Einstein condensates. Phys. Rev. A 98, 023610 (2018).
40. Prasad, S. B., Bland, T., Mulkerin, B. C., Parker, N. G. & Martin, A. M.
Vortex lattice formation in dipolar Bose–Einstein condensates via
rotation of the polarization. Phys. Rev. A 100, 023625 (2019).
41. Roccuzzo, S. M., Gallemí, A., Recati, A. & Stringari, S. Rotating a
supersolid dipolar gas. Phys. Rev. Lett. 124, 045702 (2020).
42. Gallemí, A., Roccuzzo, S. M., Stringari, S. & Recati, A. Quantizedvortices in dipolar supersolid Bose–Einstein-condensed gases.
Phys. Rev. A 102, 023322 (2020).
43. Ancilotto, F., Barranco, M., Pi, M. & Reatto, L. Vortex properties
in the extended supersolid phase of dipolar Bose–Einsteincondensates. Phys. Rev. A 103, 033314 (2021).
44. Prasad, S. B., Mulkerin, B. C. & Martin, A. M. Arbitrary-angle
rotation of the polarization of a dipolar Bose–Einstein condensate.
Phys. Rev. A 103, 033322 (2021).
45. Stuhler, J. et al. Observation of dipole–dipole interaction in a
degenerate quantum gas. Phys. Rev. Lett. 95, 150406 (2005).
46. Haljan, P. C., Coddington, I., Engels, P. & Cornell, E. A. Driving
Bose–Einstein condensate vorticity with a rotating normal cloud.
Phys. Rev. Lett. 87, 210403 (2001).
47. Kwon, W. J. et al. Sound emission and annihilations in a
programmable quantum vortex collider. Nature 600, 64–69(2021).
48. Corman, L. et al. Quench-induced supercurrents in an annular
Bose gas. Phys. Rev. Lett. 113, 135302 (2014).
Nature Physics

https://doi.org/10.1038/s41567-022-01793-8
49. Liu, I. K. et al. Dynamical equilibration across a quenched phasetransition in a trapped quantum gas. Commun. Phys. 1, 24 (2018).
50. Del Pace, G. et al. Imprinting persistent currents in tunablefermionic rings. Preprint at https://arxiv.org/abs/2204.06542(2022).
51. Norcia, M. A. et al. Can angular oscillations probe superfluidity indipolar supersolids? Phys. Rev. Lett. 129, 040403 (2022).
52. Recati, A., Zambelli, F. & Stringari, S. Overcritical rotation of atrapped Bose–Einstein condensate. Phys. Rev. Lett. 86, 377 (2001).
53. Madison, K. W., Chevy, F., Bretin, V. & Dalibard, J. Stationary
states of a rotating Bose–Einstein condensate: routes to vortex
nucleation. Phys. Rev. Lett. 86, 4443 (2001).
54. Wächtler, F. & Santos, L. Quantum filaments in dipolar Bose–
Einstein condensates. Phys. Rev. A 93, 061603 (2016).
55. Sinha, S. & Castin, Y. Dynamic instability of a rotating Bose–
Einstein condensate. Phys. Rev. Lett. 87, 190402 (2001).
56. Abo-Shaeer, J. R., Raman, C. & Ketterle, W. Formation anddecay of vortex lattices in Bose–Einstein condensates at finite
temperatures. Phys. Rev. Lett. 88, 070409 (2002).
57. Kwon, W. J., Moon, G., Choi, J.-y, Seo, S. W. & Shin, Y.-i Relaxationof superfluid turbulence in highly oblate Bose–Einsteincondensates. Phys. Rev. A 90, 063627 (2014).
58. Engels, P., Coddington, I., Haljan, P. C. & Cornell, E. A.
Nonequilibrium effects of anisotropic compression applied tovortex lattices in Bose–Einstein condensates. Phys. Rev. Lett. 89,
100403 (2002).
59. Bland, T., Stagg, G. W., Galantucci, L., Baggaley, A. W. & Parker,
N. G. Quantum ferrofluid turbulence. Phys. Rev. Lett. 121, 174501(2018).
60. Cidrim, A., dos Santos, F. E., Henn, E. A. & Macrì, T. Vortices inself-bound dipolar droplets. Phys. Rev. A 98, 023618 (2018).
61. Lee, A.-C., Baillie, D., Bisset, R. N. & Blakie, P. B. Excitations of avortex line in an elongated dipolar condensate. Phys. Rev. A 98,
063620 (2018).
62. Ferrier-Barbut, I. et al. Scissors mode of dipolar quantum dropletsof dysprosium atoms. Phys. Rev. Lett. 120, 160402 (2018).
63. Hertkorn, J. et al. Pattern formation in quantum ferrofluids: fromsupersolids to superglasses. Phys. Rev. Res. 3, 033125 (2021).
64. Tang, Y., Kao, W., Li, K.-Y. & Lev, B. L. Tuning the dipole–dipole
interaction in a quantum gas with a rotating magnetic field. Phys.
Rev. Lett. 120, 230401 (2018).
65. Prasad, S. B. et al. Instability of rotationally tuned dipolar Bose–
Einstein condensates. Phys. Rev. Lett. 122, 050401 (2019).
66. Baillie, D. & Blakie, P. B. Rotational tuning of the dipole–dipole
interaction in a Bose gas of magnetic atoms. Phys. Rev. A 101,
043606 (2020).
Publisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format,
as long as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate
if changes were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons license and your intended
use is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this license, visit http://creativecommons.
org/licenses/by/4.0/.
© The Author(s) 2022

Article

Methods

Experimental procedure
We prepare an ultracold gas of 162Dy atoms in an ODT. Three 1,064 nm
laser beams, overlapping at their foci, form the ODT. The experimental
procedure to BEC is similar to the one followed in our previous work51,
but the magnetic-field unit vector, B̂, is tilted by an angle of θ = 35° withrespect to the z-axis during the whole sequence. After preparation, thesample contains about 2 × 104 condensed atoms. The corresponding
trap frequencies are typically (ω⊥, ωz) = 2π × [50.8(2), 140(1)] Hz. For
all our measurements, the deviation of the trap AR in the x–y plane
ARtrap = ωy/ωx from 1 is always smaller than 0.6%. We evaporate the atoms
at B = 5.423(5) G and jump to the final magnetic-field value during thelast evaporation ramp. After the preparation of the BEC, the magnetic
field is rotated as described in the next section. We use standard absorption imaging to record the atomic distribution. We probe the vortices
using the vertical imaging taken along the axis of rotation (z), for which
the dark spots within the condensate correspond to the cores of individual vortices. The vertical images are taken with a short TOF of 3 ms
and a pulse duration of 3–4 μs. For the data in Figs. 1–3, we let themagnetic-field spinning during TOF, whereas for Fig. 4, we use a static
field orientation.

Control of the magnetic field

Calibration. Three pairs of coils—each oriented along a primary axis in
the laboratory frame—enable the creation of a homogeneous field with
arbitrary orientations. The absolute magnetic-field value B of each pairof coils is independently calibrated using radio frequency (RF) spectroscopy. The RF drives transitions to excited Zeeman states, leading toa resonant dip in the atom number. The long-term stability—measured
via the peak position of the RF resonance over the course of several
days—is on the order of ΔB = ±1 mG, while shot-to-shot fluctuations,
measured via the width of the RF resonance for a single calibration
set, is ΔB = ±5 mG.
Rotation. We drive the rotation of the magnetic field by sinusoidally
modulating the magnetic-field value components Bx and By with aphase difference of 90° between them. As we want to keep the absolute
magnetic-field value B constant during rotation, we measure it forvarious values of the azimuthal angle ϕ and fixed θ = 35° by performing Feshbach loss spectroscopy around 5.1 G. We find an average shiftof B of about 10 mG from the θ = 0° case, which we take into account.
We also find small deviations as a function of ϕ of ΔB < 20 mG, whichmight appear due to slightly non-orthogonal alignment of the magnetic
fields. We did not correct these deviations for the sake of simplicity.

Scattering length
The scattering length in 162Dy is currently not known with large accuracy67–70. To estimate the scattering length in the small magnetic-field
range around B = 5.3 G, relevant to this work, we use the well-known
relation as = abg∏i[1 − ΔBi/(B − B0,i)] (ref. 71), where B0,i and ΔBi are thecentre position and the width of the i-th feature of the Feshbach loss
measurement reported in ref. 70, respectively. The value of the background scattering length, abg, is empirically fixed by measuring themagnetic-field value at which the supersolid transition occurs and comparing it with the corresponding critical as predicted from simulations.
Such an approach leads to as = 111(9)a0 at B = 5.333 G. Extended Data
Fig. 1 shows the resulting scattering lengths for the relevant magnetic
fields. Although such an approach gives very good agreement between
theory and experiments, future works on a precise determination of as,
similar to the one achieved with erbium72, would be desirable.

Magnetostirring
Tilting the magnetic-field vector B away from the symmetry axis of
our cylindrical trap leads to an ellipsoidal deformation of the cloud45
and therefore to a breaking of the cylindrical symmetry. This allows
Nature Physics

https://doi.org/10.1038/s41567-022-01793-8

for the transfer of angular momentum to the sample by rotating themagnetic field (magnetostirring). In all our measurements, we use a
B tilted with respect to the z axis by 35° and a constant value B. Thatvalue is B = 5.333(5) G for the surveys in Figs. 1–3 and B = 5.323(5 )G forFig. 4. For these parameters, the deformed magnetostricted AR of thecloud is AR − 1 = 0.03. For all our measurements, the measured trap
ARtrap − 1 < 0.006 is much smaller than the deformation due to magnetostriction. Additionally, we have confirmed with simulations that evenwith trap asymmetries of up to 10%, for example, (ωx, ωy) = (55, 50) Hz,
this procedure can still generate vortices in a lattice configuration.
At the scattering lengths considered in this work, 35° is an optimal
choice to see the vortices within ~500 ms of rotation and anisotropic
enough to observe the stripe phase. From the simulations, we find thattilt angles smaller than 35° increase the timescale to vortex nucleation.
Similarly, tilting the angle further into the plane increases the numberof atoms that are aligned head to tail, making the dipolar interaction
dominantly attractive. This attractive force holds the condensate
together during the rotation, also increasing the time to vortex nucleation. From the experimental side, increasing the tilt angle reduces thecontrast of the absorption imaging, since the magnetic field is notparallel to the imaging axis. As the TOF is only 3 ms, we do not rotate
up the magnetic field before imaging to avoid undesired effects, suchas losing the anisotropy given by θ ≠ 0°.
Adiabatic frequency ramp. We employ different magnetic-field rotation sequences for the different datasets. For the dataset of Fig. 1c, therotation frequency of the magnetic field is linearly increased to different final values at a speed of Ω̇ = 2𝜋𝜋 × 50 Hz s−1 and for a duration oftΩ̇ = 0−1 s. The ramp time is much longer than the period of the rotation
Ω−1 for higher rotation frequencies Ω ≳ Ωc, and, therefore, the ramp is
adiabatic for the regimes considered, until the onset of dynamical
instabilities. After the ramp, the magnetic-field direction is rotated at
the target rotation frequency Ω for one final period (as shown in
Fig. 1b). We sample ten different final magnetic-field angles during thislast rotation, measuring the corresponding AR and averaging the result
to remove any potential biases due to latent trap anisotropies. Eachdata point is then obtained with eight to ten experimental runs.
Constant rotation frequency. For the dataset of Figs. 2, 3 and 4b, we
directly start to rotate at the final rotation frequency Ω without any
acceleration phase. The magnetic field is then rotated for a variable
time tΩ, after which the atoms are released from the trap and a verticalimage is taken.
Spiral up magnetic field. For the dataset of Fig. 4d, we employ a similar
sequence as described above. However, after constantly rotating themagnetic field at Ω = 0.75ω⊥, the magnetic field is spiralled up in 100 ms
to θ = 0° by linearly reducing θ while continuing rotating. Afterwards,
the atoms are released from the trap and a vertical image is taken.

Theoretical model
We employ an eGP formalism to model our experimental set-up. In thisscheme, the inter-particle interactions are described by the two-bodypseudo-potential
U(r) =2̂ ⋅ r)
4 ℏ2 as3ℏ2 add 1 − 3(e(t)δ(r) +
,
m
m
r3

(1)

with δ(r) being the Kronecker delta function and r = (x, y, z). The firstterm describes the short-range interactions governed by the s-wave
scattering length as, with Planck’s constant ℏ and particle mass m.
The second term represents the anisotropic and long-range dipole–
dipole interactions, characterized by dipole length add = μ0 μ2m m/12 ℏ2,with magnetic moment μm and vacuum permeability μ0. We always
consider 162Dy, such that add = 129.2a0, where a0 is the Bohr radius.

Articlehttps://doi.org/10.1038/s41567-022-01793-8

The dipoles are polarized uniformly along a time-dependent axis,
given by
̂ = (sin θ(t) cos ϕ(t), sin θ(t) sin ϕ(t), cos θ(t))
e(t)

(2)
t

with time-dependent polarization angle θ(t) and ϕ(t) = ∫0 dt′ Ω(t′ ), forrotation frequency protocol Ω(t).
Beyond-mean-field effects are treated through the inclusion of a
Lee–Huang–Yang correction term73
γQF =

128ℏ2
a5 Re {𝒬𝒬5 (ϵdd )} ,3m √ s

(3)

5/2

1

with 𝒬𝒬5 (ϵdd ) = ∫0 du (1 − ϵdd + 3u2 ϵdd ) being the auxiliary function, andthe relative dipole strength is given by ϵdd = add/as. Finally, the full eGPE
then reads54,74–76iℏ∂Ψ (x,t)∂t= [−ℏ2 ∇ 22m

1+ m (ω2x x2 + ω2y y2 + ω2z z2 )2

+ ∫ d3 x′ U(x − x′ )|Ψ (x′ , t)|2 + γQF |Ψ (x, t)|3 ] Ψ (x, t),(4)where ωx,y,z are the harmonic trap frequencies. The wave function Ψ isnormalized to the total atom number N = ∫d3x∣Ψ∣2. The stationary solution for Fig. 4a is found through the imaginary time procedure in therotating frame, introducing the usual angular momentum operatorΩ Lẑ into equation (4). The initial state Ψ(x, 0) of the real-time simulations is always obtained by adding non-interacting noise to the groundstate Ψ0(x). Given the single-particle eigenstates ϕn and the complex
1
Gaussian random variables αn sampled with ⟨|αn |2 ⟩ = (en /kB T − 1)−1 +2
for a temperature T = 20 nK and Boltzmann’s constant kB, the initialstate can be described as Ψ (x, 0) = Ψ0 (x) + ∑ ′ αn ϕn (x), where the sum isnrestricted only to the modes with ϵn ≤ 2kBT (ref. 77). Throughout, thedensity images are presented in situ, with a scaling factor to accountfor the 3 ms TOF for the experimental images.
To obtain the average residual FT images for Fig. 4c,e, we first
Fourier transform 115 frames from the simulation between 700 ms and1.1 s in the rotating frame before averaging the result.

Atom number
Extended Data Fig. 2 shows the condensed atom number Nc for themeasurement with an adiabatic ramp of the magnetic-field rotational
velocity ( Ω̇ = 2𝜋𝜋 × 50 Hz s−1 ), corresponding to the data of Fig. 1c.
Three-body losses are negligible in the low-density BEC phase, withlosses probably coming from imperfections in the rotation procedureand heating. To extract the atom number, we use the horizontal imaging
with 26 ms of TOF. About 3 ms before flashing the imaging resonant
light to the atoms, we rotate the magnetic field in the imaging planeand perform standard absorption imaging. From the absorption
images, we extract Nc from a bimodal fit up to 700 ms. At later times,
the system undergoes a dynamic instability (see discussion in the maintext), and the density profile deviates from a simple bimodal distribution. During the observation time, we see a slight decrease of Nc, andfor our theory simulations, we use a constant atom number of
Nc = 15,000. Note that in all following datasets, in which we abruptly
accelerate the magnetic-field rotation to the desired final velocity, weobserve a faster decay, and our simulations are performed with either
Nc = 8,000 or Nc = 10,000.

Vortex detection

Vortex detection algorithm. Since vortices appear as dark holes in thedensity profile of a BEC, which would otherwise have a smooth profile,
our approach to extract the number of vortices is to look at deviations
between the image and an unmodulated reference image. To extract
the vortex number from the raw images, we proceed as follows.
Nature Physics

First, we prepare the image nimg, the reference image nref and theresidual image nres. The image is normalized such that the maximum
density max(nimg ) = 1. We create the reference image by blurring theimage via applying a 2D Gaussian filter with σ = 5 pixel, corresponding
to about 2.1 μm. This blurring smoothens any structure on the lengthscale of the filter width; therefore, any holes in the density profile
wash out. We then normalize the atom number of the reference to be
the same as from the image Nref = ∫∫nref ≐ Nimg = ∫∫nimg. The residual
image is calculated as the difference between the image and the reference nres = nimg − nref . We additionally mask the region where thedensity of the reference is below a certain threshold ( nres = 0, where
nref ≤ 0.1).
Second, we identify local minima in the residual image
and determine whether they are connected to vortices. For this, wecreate a list of local minima (xmin , ymin ), defined by the condition thatthe pixel density nres (xmin , ymin ) is lower than of all surrounding pixels.
Then we remove minima with density values above zeronres (xmin , ymin ) ≥ 0 or which are within one pixel distance of the maskborder. Now we determine a local contrast for each minimum by calculating the difference between its central density value and themean of the density values ± 2 pixel values away from it
ncon (xmin , ymin ) = nres (xmin , ymin ) − mean(nres (xmin ± 2 px, ymin ± 2 px)) , andremove minima above a certain threshold ncon > −0.11. As a last step, wecheck the distance d between all remaining minima to avoid double
counting of minima too close to each other. In case d is below thethreshold d < 5 pixel, the minimum with the higher residual density
value nres is discarded.
Preparation of theory density profiles. For the direct comparison of
the vortex number shown in Fig. 3b, we apply additional steps to thedensity profiles obtained from theory. First, we reduce the resolution
by a 2 × 2 binning to make the pixel size of the theory density profiles
ntheo
img essentially the same as for the experimental images (sizes are) = 1, we apply Gaussian white
within 5%). After normalizing to max(ntheo
img
noise with zero mean and a variance of 0.01 to each pixel, recreating
the noise pattern from empty regions of experimental images. Then
we blur the image using a 2D Gaussian filter with σ = 1 pixel (~0.42 μm);
this recreates the same resolution condition as our experimental
set-up. The resulting density profile is taken as the input image for thevortex detection algorithm described above.
Benchmarking the vortex detection algorithm. As the vortex positions for the simulation images are known a priori due to the available phase map, we can derive the fidelity of the vortex detection
algorithm for simulation data. For the theory data shown in Fig. 3b in
the time frame between 600 and 700 ms, the average detected vortex
number in the simulated density profiles (applying the preparation
scheme described above) is about 9, while the real number of vortices
present in the same area of the image is about 33 in average. This mismatch is explained by the conservative choice of the thresholds forvortex detection together with the added noise, which results in only
counting clear density dips as vortices, throwing out many vortices
in the low-density region. This conservative choice of thresholds on
the other hand leads to a very high fidelity of >97%, where we define
the fidelity as the percentage of detected vortices that correspond toactual present vortices in the data. For raw simulation data (without
resolution reduction, added noise and blurring), the vortex detection algorithm would detect up to 80% of the vortices present witha fidelity of >95%.
Note that for the visualization of the vortex positions for Fig. 4,
we slightly increased the local threshold ncon > −0.08 and decreased
the minimum distance between vortices d < 3 pixel, which increases
the overall number of vortices detected. For the density distributions
obtained from theory, we additionally omit the resolution reduction,
addition of noise and blurring steps.

Article

Data availability

Data pertaining to this work can be found at https://doi.org/10.5281/
zenodo.7019859.

Code availability

The codes that support the findings of this study are available from thecorresponding author upon reasonable request.

References

67. Tang, Y., Sykes, A., Burdick, N. Q., Bohn, J. L. & Lev, B. L. s-wave
scattering lengths of the strongly dipolar bosons 162Dy and 164Dy.
Phys. Rev. A 92, 022703 (2015).
68. Tang, Y. et al. Anisotropic expansion of a thermal dipolar Bosegas. Phys. Rev. Lett. 117, 155301 (2016).
69. Lucioni, E. et al. Dysprosium dipolar Bose–Einstein condensate
with broad Feshbach resonances. Phys. Rev. A 97, 060701(2018).
70. Böttcher, F. et al. Dilute dipolar quantum droplets beyond theextended Gross–Pitaevskii equation. Phys. Rev. Res. 1, 033088(2019).
71. Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbachresonances in ultracold gases. Rev. Mod. Phys. 82, 1225
(2010).
72. Patscheider, A. et al. Determination of the scattering length of
erbium atoms. Phys. Rev. A 105, 063307 (2022).
73. Lima, A. R. P. & Pelster, A. Quantum fluctuations in dipolar Bose
gases. Phys. Rev. A 84, 041604 (2011).
74. Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau, T.
Observation of quantum droplets in a strongly dipolar Bose gas.
Phys. Rev. Lett. 116, 215301 (2016).
75. Chomaz, L. et al. Quantum-fluctuation-driven crossoverfrom a dilute Bose–Einstein condensate to a macrodroplet
in a dipolar quantum fluid. Phys. Rev. X 6, 041039(2016).
76. Bisset, R. N., Wilson, R. M., Baillie, D. & Blakie, P. B. Ground-statephase diagram of a dipolar condensate with quantum
fluctuations. Phys. Rev. A 94, 033619 (2016).
77. Blakie, P., Bradley, A., Davis, M., Ballagh, R. & Gardiner, C.
Dynamics and statistical mechanics of ultra-cold Bose
gases using c-field techniques. Adv. Phys. 57, 363–455(2008).

Nature Physics

https://doi.org/10.1038/s41567-022-01793-8

Acknowledgements

We are grateful to S. B. Prasad, M. Norcia, R. M. W. van Bijnen and L.
Santos for helpful discussions. We acknowledge M. Norcia and A.
Patscheider for experimental contributions. This study received support
from the European Research Council through the Consolidator Grant
RARE (No. 681432), the QuantERA grant MAQS by the Austrian Science
Fund FWF (No. I4391-N), and the DFG/FWF via FOR 2247/PI2790 a
joint-project grant from the FWF (No. I4426). M. J. M. acknowledges
support through an ESQ Discovery grant by the Austrian Academyof Sciences. We also acknowledge the Innsbruck Laser Core Facility,
financed by the Austrian Federal Ministry of Science, Research andEconomy. Part of the computational results presented have beenachieved using the HPC infrastructure LEO of the University of Innsbruck.
G. L. acknowledges financial support from Provincia Autonoma di Trento.

Author contributions

L.K., C.P., G.L., E.C., M.J.M. and F.F. performed the experimental workand data analysis. E.P., T.B. and R.N.B. performed the theoretical work.
All authors contributed to the interpretation of the results and the
preparation of the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Extended data is available for this paper athttps://doi.org/10.1038/s41567-022-01793-8.
Supplementary information The online version
contains supplementary material available athttps://doi.org/10.1038/s41567-022-01793-8.
Correspondence and requests for materials should be addressed toFrancesca Ferlaino.
Peer review information Nature Physics thanks Oliver Gorceix, Han
Pu and the other, anonymous, reviewer(s) for their contribution to thepeer review of this work.
Reprints and permissions information is available atwww.nature.com/reprints.

Articlehttps://doi.org/10.1038/s41567-022-01793-8

Extended Data Fig. 1 | Calculated B-to-as conversion for 162Dy. Scattering length as a function of the magnetic-field value with the background scattering lengthabg = 129(9) a0. We find as = 111(9) a0 at B = 5.333 G.

Nature Physics

Articlehttps://doi.org/10.1038/s41567-022-01793-8

Extended Data Fig. 2 | Condensed atom number Nc during magnetostirring (Fig. 1c). Condensed atom number as a function of spin-up time tΩ̇ for the same
sequence as in Fig. 1c. The condensed atom number is extracted by fitting a two-dimensional bimodal distribution of Thomas-Fermi and Gaussian function to thehorizontal density distributions.

Nature Physics

Article

Extended Data Fig. 3 | Repeatability of the vortex generation protocol. Eachrow shows the simulated image (a, b, c) and the corresponding vertical TOF
images from independent experimental runs (ai, bi, ci) for a different rotationtime: ta = 127 ms, tb = 207 ms, and tc = 741 ms. The rotation frequency is Ω = 0.74ω⊥Nature Physics

https://doi.org/10.1038/s41567-022-01793-8

with the trap frequencies being ωt = 2π × [50.7(1), 50.8(1), 129(1)] Hz. The
magnetic field value is B = 5.333(5) G. For the simulation the scattering length
used is 112 a0, the trap frequencies are (50, 50, 150) Hz, the condensed atom
number is N = 8000 and the rotation frequency is 0.75ω⊥.

Acknowledgments
I take now the opportunity to acknowledge all the people that contributed to making
this PhD thesis possible.
First, I would like to thank my supervisor Francesca Ferlaino for giving me theopportunity to do my PhD in a stimulating and productive environment, such as the
Er-Dy lab in Innsbruck. I appreciate her trust in me since I first joined the team formy master’s project in 2016. She always tried to get out the best of me, constantly
setting the bar higher, and I am really grateful for all the feedback regarding this thesis
work and my future research directions.
Such a complex experiment does not run so effectively without the help of everybody
and a strong team effort. Therefore, if I got to this point, it is also thanks to mylab colleagues. When I joined the lab for my PhD studies in January 2018, we wereat the stage of producing double degenerate mixtures of Er and Dy, and I had thegreat opportunity to learn from former PhD students and postdocs. In particular, I
would like to thank Philipp Ilzhöfer (1st PhD student) for transferring part of his greatknowledge about optics and laser systems, and the importance of TIGHTLY screwing
optics on the experimental table. I thank Gianmaria Durastante (2nd PhD student)
for teaching me the tricks on water cooling and how to dive into the experimental
security system. I thank Maximilian Sohmen (3rd PhD student) for his incredible
attitude in always finding the time to help people in the lab, and for his natural love
for science. I enjoyed the time spent together implementing the phase-contrast imaging.
I thank Arno Trautmann (1st postdoc) for sharing his knowledge about electronics,
including soldering, Arduino® , and LATEX. I also thank Matthew Norcia (2nd postdoc),
who joined the experiment in 2020, and with whom I had a really great time in thelab. He joined during the corona pandemic, which was not an easy time. Nevertheless,he did a great job at maintaining a strong team spirit, sometimes leading to outdoor
lunch in the snow with freezing temperatures. I appreciate his way of having always
an intuitive physical picture and his ability in finding the most efficient path to reachthe goal.

220

Acknowledgments

I want to thank the current PhD students: Lauritz Klaus, Eva Casotti, and the
newly arrived Clemens Ulm. I really enjoyed the cold winter months spent in the labwith Lauritz to get the first two-dimensional supersolid states, and I thank him foralways starting the day with a smile. I also thank Eva for always challenging herself
and facing the lab’s riddles without fear. I am sure the lab is in good hands andwill continue to produce fascinating results in the years to come. I thank our seniorscientist Manfred Mark for all the helpful discussions and for always finding the timeto help in the lab if needed. I also thank him for the feedback on part of this thesis. A
particular thanks goes to the theory group with Thomas Bland, and Elena Poli. Your
theoretical support and your love and dedication to science were an amazing help for
us and the results we had in the experiment would not have been possible without
your contribution. I further thank Tom for reading part of this work.
I would also like to thank the master students that worked in the Er-Dy experiment,
especially Sandra Brandstetter and Nefeli Sonnberger, and the other two labs of ourdipolar quantum gas group, namely the ERBIUM and the T-REQS, including current
and former members. In particular, I would like to thank Daniel Petter, Alexander
Patscheider, Lauriane Chomaz, Gabriele Natale, Louis Lafforgue, Arfor Houwman,
Hagai Edri, Antonio Ortu, Julián Maloberti, Samuel White, Daniel Grün, for theinteresting discussions and for the time spent outside the lab, including our cooking
evenings at Dani’s place, bouldering, mountain biking, or a simple beer in the evening.
During the past five years, we had several collaborations with external groups. In
particular, I would like to thank Michele Modugno from the University of the BasqueCountry in Bilbao (Spain), with whom I exchanged an uncountable number of emails.
I thank Thomas Pohl and Juan Sánchez Baena from Aarhus University (Denmark),
Luis Santos from the University of Hannover, and Russell Bisset from Innsbruck, for
the inspiring discussions. In addition, I would like to acknowledge the support frommy co-supervisor Prof. Helmut Ritsch.
Our dipolar quantum gas group is part of the big ultracold group in Innsbruck.
Therefore, I thank all of them and in particular, I would like to thank Emil Kirilov,
Prof. Rudolf Grimm, and Prof. Hanns-Christoph Nägerl for the interesting discussions
and questions during our Thursday group seminar.
A special thanks goes to Elisa Soave, Lorenzo Dania, Cornee Ravensbergen, Cosetta
Baroni, Deborah Capecchi, and Elvia Colella for the nice time spent inside and outside
the university. I will always remember our nice weekend dinners, and the day spent
preparing an uncountable amount of cherry jam.

Acknowledgments

221

A big thanks goes to the administrative support at the University of Innsbruck andthe Institute for Quantum Optics and Quantum Information (IQOQI). In particular,
my thanks go to Silvia Bonazza, Leonarda Garcia, and Laura Uiberreiter from theuniversity and Elisabeth Huck, and Nikolaus Falschunger from IQOQI, who help us tofocus our time on the scientific work. I would like to thank Gerhard Hendl from theelectronic workshop, and the mechanical workshop, particularly Bernhard Öttl, who
was always available for helping us with the realization and design of mechanical parts
for the experiment.
In conclusion, I would like to thank the people outside the physics world that were
for me as fresh air and helped to get my mind free and fully re-charged to start a new
day in the lab. In particular, I would like to thank my german teacher Boris frominlingua, who always brought our german lesson to life with interesting discussions anddeep questions, and all my colleagues from the course, who helped me improve mygerman. I would like to thank the 2. Bundesliga basketball team from TI Innsbruck,including present and former team members, and coaches. Since I joined the team in
2019, they soon became like a second family for me. I am grateful for their support
and for all the time spent together in and outside the basketball court. It has been agreat honor to be part of this amazing group.
A special thanks goes to my family: my parents, Dolores and Enzo, and my sister,Giulia. If I reached this goal, it is also thanks to their unconditional love. This mademe free from the fear of failing and making mistakes, which was an incredible strength
for me. I thank them, for all the support they gave me from the first day of school. At
every step of my life, they accepted and respected my choices. They did everything
possible and beyond that to satisfy my needs, even if this meant staying away fromhome. Finally, I thank Gabriele, who for more than 10 years is central part of mylife and confirms every day to be the best choice I ever made. I am thankful for ourlate-night physics discussions and how we support and motivate each other to givealways our best, from small things, such as a simple cooking recipe, to bigger goals,
such as a PhD thesis.