Competing Interactions in Dipolar
Quantum Gases of BosonicErbium Atoms– DISSERTATION –byGabriele Natale, MSc

submitted to the Faculty of Mathematics, Computer Science,
and Physics of the University of Innsbruckin partial fulfillment of the requirements
for the degree of Doctor of Philosophy (PhD)

advisor:Univ.-Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInnsbruck, December 2022

To my family.

iAbstract
Ultracold gases provide a versatile platform for trapping, controlling, and manipulating both
external and internal degrees of freedom of the atoms. In particular, the experimental
realization of Bose-Einstein condensates and degenerate Fermi gases inaugurated an exciting
era of quantum-gas experiments to perform quantum simulations or investigate interesting
quantum phases. Famous examples in our community are the superfluid to Mott-insulator
phase transition and the bright soliton. In particular, the study of exotic quantum phases is
the main topic of this thesis.
A remarkably bizarre phase is the supersolid phase, which can arise from the competition
between short and long-range interactions. In this state, two symmetries, the translational
and the gauge symmetry, are spontaneously broken. For this reason, two antithetical orders
coexist, and the system is simultaneously superfluid and crystal.
In this thesis, we use dipolar gases of erbium atoms to find the regime in which a supersolid
exists. We then ask the following question: how does this system manifest its properties? Weanswer this question by performing two different protocols to perturb this state and study
its response. A particularly relevant result is the presence in the excitation spectrum of two
branches. The first branch is related to crystal excitations, while the second to superfluid
excitations.
Furthermore, we approach the study of the quantum phases of an erbium condensate confinedin reduced dimensions. By using a one-dimensional optical lattice, we load the atoms in anarray of quasi-2D planes and study their distribution in the different lattice sites. We foundthat if the onsite dipole-dipole interaction is attractive on average, reducing the scattering
length drives a localization of the atoms in a single lattice site. We assess the role thatquantum fluctuations play in this geometry and discover a regime in which kinetic energy
and quantum fluctuations compete to stabilize the system. This geometry gives rise to soliton
or droplet solutions.

iii

Zusammenfassung
Ultrakalte Gase bieten eine vielseitige Plattform für das Einfangen, Kontrollieren und Manipulieren sowohl externer als auch interner Freiheitsgrade der Atome. Insbesondere die experimentelle Realisierung von Bose-Einstein-Kondensaten und entarteten Fermi-Gasen hat
eine aufregende Ära von Quantengasexperimenten eingeläutet, mit denen Quantensimulationen durchgeführt oder interessante Quantenphasen untersucht werden können. Berühmte
Beispiele in unserer Gemeinschaft sind der Phasenübergang vom Superfluid zum MottIsolator und das helle Soliton. Insbesondere die Untersuchung von exotischen Quantenphasen
ist das Hauptthema dieser Arbeit.
Eine bemerkenswert bizarre Phase ist die supersolide Phase, die durch den Wettbewerb
zwischen kurz- und langreichweitigen Wechselwirkungen entstehen kann. In diesem Zustandwerden zwei Symmetrien, die Translations- und die Eichsymmetrie, spontan gebrochen. Aus
diesem Grund existieren zwei gegensätzliche Ordnungen nebeneinander, und das System ist
gleichzeitig superfluid und kristallin.
In dieser Arbeit verwenden wir dipolare Gase von Erbium-Atomen, um das Regime zu finden,
in dem ein Supersolid existiert. Dann stellen wir die folgende Frage: Wie manifestiert dieses
System seine Eigenschaften? Zur Beantwortung dieser Frage führen wir zwei verschiedene
Protokolle durch, um diesen Zustand zu stören und seine Reaktion zu untersuchen. Einbesonders wichtiges Ergebnis ist das Vorhandensein von zwei Zweigen im Anregungsspektrum. Der erste Zweig steht im Zusammenhang mit Kristallanregungen, der zweite mit
superfluiden Anregungen.
Darüber hinaus nähern wir uns der Untersuchung der Quantenphasen eines Erbium-Kondensats,das in reduzierten Dimensionen. Durch die Verwendung eines eindimensionalen optischen
Gitters laden wir die Atome in eine Anordnung von Quasi-2D-Ebenen und untersuchen
ihre Verteilung auf den verschiedenen Gitterplätzen. Wir fanden heraus dass, wenn die
Dipol-Dipol-Wechselwirkung vor Ort im Durchschnitt anziehend ist, die Verringerung derStreulänge Länge zu einer Lokalisierung der Atome an einem einzigen Gitterplatz führt. Wir
bewerten die Rolle, die Quantenfluktuationen in dieser Geometrie spielen, und entdecken
ein Regime, in dem kinetische Energie und Quantenfluktuationen um die Stabilisierung desSystems konkurrieren. Diese Geometrie führt zu Soliton oder Tröpfchenlösungen.

v

Contents
Abstractiii

Zusammenfassungv

Introduction
0.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
3
4

1 Ultracold erbium quantum gas
1.1 Erbium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Electronic configuration . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Erbium experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Quantum degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 The Bragg-spectroscopy setup and the 1D optical lattice . . . . . . . . . . . .

7
7
8
9
10
12
13

2 Excitation spectrum of an unmodulated dipolar BEC
15
2.1 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Bogoliubov excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 A roton-maxon excitation spectrum . . . . . . . . . . . . . . . . . . . 19
2.2.2 Quantum depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Quantum fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Finite temperature and non-equilibrium dynamics . . . . . . . . . . . . . . . 24
2.5 Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Publication: Probing the roton excitation spectrum of a stable dipolar Bose gas 28
3 Supersolidity
3.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Supersolidity in dipolar gases . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Early hints of supersolidity . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Probing a supersolid state . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The excitation spectrum of a dipolar supersolid . . . . . . . . . . . . . . . . .
3.3.1 Exciting a dipolar supersolid state . . . . . . . . . . . . . . . . . . . .
3.3.2 Principal component analysis . . . . . . . . . . . . . . . . . . . . . . .
3.4 Other supersolids and where to find them . . . . . . . . . . . . . . . . . . . .
3.5 Publication: Long-lived and transient supersolid behaviors in dipolar quantum
gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Publication: Excitation spectrum of a trapped dipolar supersolid and its experimental evidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
43
45
46
48
49
51
52
53
54
77
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3.7

Publication: Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4 Bloch oscillations and matter-wave localization
105
4.1 One-dimensional lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.1 Single-particle physics: Bloch theorem . . . . . . . . . . . . . . . . . . 106
4.1.2 Single-particle physics: Wannier function . . . . . . . . . . . . . . . . 107
4.2 Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Many-body physics: mean-field model . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Overview of Bloch oscillations in ultracold atoms . . . . . . . . . . . . 112
4.3.2 Bloch oscillations in ultracold dipolar gases . . . . . . . . . . . . . . . 113
4.4 Quantum phases of erbium atoms in a one-dimensional lattice . . . . . . . . . 113
4.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.4.2 Single-site localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4.3 Two-dimensional bound states: solitons and droplets . . . . . . . . . . 115
4.5 Publication: Strongly dipolar gases in a one-dimensional lattice: Bloch oscillations and matter-wave localization . . . . . . . . . . . . . . . . . . . . . . . 118
5 Conclusion and outlook
131
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Additional publications
135
6.1 Publication: Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms 135
6.2 Publication: Bragg scattering of an ultracold dipolar gas across the phasetransition from Bose-Einstein condensate to supersolid in the free-particle
regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3 Publication: Observation of a narrow inner-shell orbital transition in atomic
erbium at 1299 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.4 Publication: Determination of the scattering length of erbium atoms. . . . . . 183
Bibliography

197

Acknowledgments

211

viii

Introduction
Motivation
Imagine you need to solve a challenging puzzle. The first thing you do is carefully lookat the picture shown on the box. As the next step, it is useful to collect pieces of similarcolors or the ones at the edges. After having arranged those, you can now start with themore demanding ones, showing different shades or ambiguous figures. Finally, you can startputting all those pieces together to reconstruct the big picture.
Similarly, physics research often starts with identifying a challenging phenomenon or question. To solve it, one approach is to first tackle a simpler version of that problem. As soon as
the solution becomes accepted, it is adapted to a more complicated version, where elaborate
elements are added to the problem to unravel the big question. Differently from puzzles, insome cases someone threw away the box, leaving us with no hint on what the final picture
will look like.
Ultracold gases are powerful platforms to trap, control and manipulate both external andinternal degrees of freedom of the atoms [Blo08]. Nowadays, in experiments, atoms can betrapped in arbitrary geometries and dimensionality by using far-off-resonant laser light. A
few common examples are box potentials, arrays of 1D or 2D confinements, and optical lattices. The internal states can be manipulated as well through the use of magnetic fields andelectromagnetic waves. Furthermore, it is possible to tune the atomic interaction, by changing a bias magnetic field through the use of Feshbach resonances [Chi10]. The experimental
realization of Bose-Einstein condensates (BEC) [And95, Bra95, Dav95] and degenerate Fermi
gases [DeM99, Sch01, Tru01] opened the door to a new era of quantum-gas experiments performing quantum simulations or investigating fascinating phenomena and exotic quantum
phases. The investigation of such exotic quantum phases is the main topic of this thesis.
A well-known example in the quantum-gas community is the superfluid to Mott-insulator
phase transition [Gre02]. In this case, a Bose-Einstein condensate is loaded into an optical lattice, formed by the intersection of multiple laser beams, producing a standing wave
pattern. Since an atom loaded in this lattice tends to sit in the minima or maxima of the optical potential, this system resembles the geometrical configurations of electrons in crystalline
solids. This is a highly controlled environment, where one can study solid-state behaviors.
Changing the intensity of the laser beams modifies the lattice potential depth. This results
in a change in the probability for an atom to tunnel from one lattice site to an adjacent one.
The competition between tunneling and inter-atomic interaction determines the propertiesof the system. On the one hand, if tunneling dominates, each atom is delocalized over the1

2

Contents

entire lattice, leading to the superfluid phase. On the other hand, if onsite interactions are
dominant, the atoms are localized in each lattice site, and long-range phase coherence is lost
resulting in the so-called Mott-insulator phase.
Another interesting phase is the bright soliton [Zak72, Dra89]. Originally studied to understand the propagation of waves in the oceans, solitons are self-reinforcing waves that maintaintheir shape while propagating. When colliding with other solitons, they emerge unaffected.
In the ultracold community, this phase arises from the interplay between kinetic energies (dispersive effect) and attractive interaction (focusing effect), see for e.g. Ref. [Kha02, Mal08].
The existence of solitons can depend on the dimensionality of the system. For instance, inalkali atoms, interacting via a contact-type potential, this state exists as a ground state in
one dimension independently of the atom numbers. In contrast, in two dimensions, over a
critical number of atoms the interactions dominate over the kinetic energies, and the system
collapses [Sai03].
Going back to the analogy of “solving a puzzle”. After arranging the pieces of similarcolors, the next step for solving the puzzle is trying to add the more challenging ones.
This can correspond, in the ultracold-gas community, to study: mixtures by adding anotherspecies/spin; obtaining single atom control with a quantum gas microscope; coupling thesystem to an external cavity to add infinite-range interactions; or like in our case, Bosecondense magnetic atoms that inherently experience long-range and anisotropic dipole-dipole
interaction.
A breakthrough experiment with magnetic atoms was the one in the group of Tilman Pfau in
Stuttgart, where a transition from a BEC to a crystal (droplet) phase was observed in a parameter regime in which the traditional mean-field approach predicted the collapse [Kad16].
Surprised by this experiment the interest of the community focused on the mechanism responsible for the stability of this state. Between the different theoretical proposals, the twomost rated mechanisms were the quantum fluctuations [Pet15] and the 3-body repulsive potentials [Lu15, Xi16, Bis15]. Further research in the same group and in our ERBIUM group,
before starting my PhD, confirmed the quantum fluctuations as the stabilization mechanism [FB16, Cho16]. In these works, the authors create a single or multi-droplets state bychanging the interactions in order to make the long-range contribution dominant over thecontact one. In this phase, the atoms behave similarly to a liquid droplet, and as in thesoliton phase, the system is self-bound. However, in this case, the dispersive effect arises
from the quantum fluctuations and the focusing effect from the interplay between contact
and dipole-dipole interactions.
Let me mention two other experiments from our group that are particularly inspiring andrelevant to this thesis. In the first one, erbium atoms are loaded into a three-dimensional
optical lattice realizing the so-called extended Bose-Hubbard model [Bai16a]. The effect ofdipole-dipole interaction does not only shift the condition for being in the superfluid/Mottinsulator state but also extends the interaction to the nearest neighbor due to its longrange nature. In the second one, our group confirmed the presence of a roton mode in the
excitation spectrum, which resembles the one observed in superfluid helium and connects
to the tendency of the system to crystallize [Cho18]. In this work, a cigar-shaped geometry
confines the atoms, and an external magnetic field aligns the dipoles along the radial direction

0.1 Thesis overview

3of the cigar. To excite the roton mode, the system is quenched into a dipolar-dominatedregime.
By combining the knowledge developed over the years on quantum fluctuations and theroton mode, in this thesis, we explore the quantum phase transition from a BEC to a new
fascinating state of matter: the supersolid [B1̈9a, Tan19b, Cho19]. In this phase, the atoms
exhibit intriguing properties. The density shows a spontaneous modulation like in a crystal,but the atoms can tunnel between different lattice sites without viscosity like in a superfluid.
In this thesis we address the problem: Can a dipolar supersolid state be stable? To answer
this question, we identify both theoretically and experimentally the parameter regime inwhich the supersolid state is the ground state of the system. We then investigate how
the two antithetical behaviors of crystal and superfluid reveal themselves after an external
perturbation.
Furthermore, to explore more exotic phases, we approach the study of dipolar physics in
reduced dimensions, by using a one-dimensional optical lattice to create an array of quasi2D planes. We use this confinement to trap our atoms and study their distribution in the
different lattice sites. We found that, if the onsite dipole-dipole interaction is on average
attractive, reducing the scattering length drives a localization of the atoms in a single latticesite. We assess the role quantum fluctuations play in this geometry and discover a regime inwhich kinetic energy and quantum fluctuations compete to stabilize the system, giving rise
to soliton or droplet solutions.

0.1 Thesis overview
This thesis covers fundamental research on dipolar Bose-Einstein condensates of erbium
atoms. The main results include: A study of the excitation spectrum of a dipolar Bose-Einstein condensate experimentally confirming a roton minimum.
 The discovery of the dipolar supersolid state: a state with two spontaneously broken
symmetries, the translational and the gauge symmetry.
 A study of the excitation spectrum of a dipolar supersolid confirming the presence of
two branches associated to the two broken symmetries.
 The investigation of erbium atoms in one-dimensional lattice and the realization of atransition to a state localized in a single lattice plane, driven purely by interactions.

These results are structured in next four chapters as follows:
Chapter 1 provides a description of the basic properties of erbium atoms and of the ERBIUM
experimental apparatus.

4Contents

Chapter 2 reviews the theoretical description used in this thesis and focuses on the first result:the excitation spectrum of a dipolar BEC of erbium atoms in a wide range of interactions.
Chapter 3 focuses on supersolidity and includes three publications. In the first publication,
we determine the parameter regime in which our erbium cloud performs a transition to asupersolid state. We present the ground-state phase diagram and discuss how to charaterize
the state. In the second publication, we study the excitation spectrum in the supersolid
regime, and identify two branches, relative to the superfluid and crystal nature of the state. In
the third one, we probe the superfluid tunneling between the density peaks of the supersolidwhen crossing the phase transition in and out the insulated droplet regime. The lattercorresponds to a state in which the crystal nature persists, while the global phase coherenceof the state is lost.
Chapter 4 approaches the investigation of dipolar gases in reduced dimensions. In thisstudy the system is trapped in an array of quasi two-dimensional planes, with an on average
attractive onsite dipolar interactions. We found the localization to a single lattice site whenreducing the interactions and discover that the competition between the kinetic term andthe interactions terms provide a rich phase diagram, with solitonic and droplet solutions.

0.2 List of publications
The following list contains the publications that form the main part of this thesis. They are
given in chronological order, which also reflects the thematic framework of the chapters.
 Probing the roton excitation spectrum of a stable dipolar Bose gas.
D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz,
F. Ferlaino,
Physical Review Letters, 122, 183401 (2019).
 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, F. Ferlaino,
Physical Review X, 9, 021012 (2019).
 Excitation spectrum of a trapped dipolar supersolid and its experimental
evidence.
G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter, M. J. Mark, L. Chomaz,
F. Ferlaino,
Physical Review Letters 123, 050402 (2019).
 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,

0.2 List of publications

5

T. Giamarchi, L. Chomaz, M. J. Mark, F. Ferlaino,
Nature Physics, 17, 356-361, (2021).
 Bloch oscillations and matter-wave localization of a dipolar quantum gas ina one-dimensional lattice.
G. Natale, T. Bland, S. Gschwendtner, L. Lafforgue, D. S. Grün, A. Patscheider, M. J. Mark,F. Ferlaino,
Communications Physics 5, 227 (2022).

In addition to the main publications, I also contributed extensively to a number of otherprojects within the course of this work. These further publications are included in Chapter 6.

Additional publications
 Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms.
S. Baier, D. Petter, J. H. Becher, A. Patscheider, G. Natale, L. Chomaz, M. J. Mark,F. Ferlaino,
Physical Review Letters 121, 093602 (2018).

 Bragg scattering of an ultracold dipolar gas across the phase transition from
Bose-Einstein condensate to supersolid in the free-particle regime.
D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. Baranov, R. M. W. van Bijnen,
S. M. Roccuzzo, A. Recati, B. Blakie, D. Baillie, L. Chomaz, F. Ferlaino,
Physical Review A 104, L011302 (2021).
 Observation of a narrow inner-shell orbital transition in atomic erbium at1299 nm.
A. Patscheider, B. Yang, G. Natale, D. Petter, L. Chomaz, M. J. Mark, G. Hovannesyan,
M. Lepers, and F. Ferlaino
Physical Review Research 3, 033256 (2021).
 Determination of the scattering length of erbium atoms.
A. Patscheider, L. Chomaz, G. Natale, D. Petter, M. J. Mark, S. Baier, B. Yang,
R. R. W. Wang, J. L. Bohn, F. Ferlaino
Phys. Rev. A 105, 063307 (2021).

Chapter

1

Ultracold erbium quantum gas

This chapter describes the basic properties of erbium atoms and the ERBIUM experimental
apparatus. Section 1.1 presents the electronic configuration and describes its resulting mainfeatures: a magnetic moment in the ground state and a rich and complex energy spectrum.
Section 1.2 focuses on the ERBIUM experiment, with the aim of providing an overview of thevacuum apparatus. Section 1.3 introduces the concept of phase-space density and explains
the steps necessary to reach the quantum degenerate regime. Finally, Sec. 1.4 illustrates the
Bragg-spectroscopy setup used to probe the excitation spectrum of the dipolar erbium gas,
and the one-dimensional optical lattice used to produce an array of quasi-2D clouds.

1.1 Erbium properties

Erbium (Er) is a rare-earth element, part of the lanthanides family. The attribute rare-earth
is connected to the geochemical properties and historical reasons rather than due to the abundance. Er is naturally present in the earth’s crust with an abundance of 3.4 ppm [CRC04].
This value makes erbium at the 43rd position of the most common chemical elements on
the crust. It is more abundant than gold (0.004 ppm) and silver (0.075 ppm), but less abundant than copper (60 ppm). However, the lanthanides are difficult to separate due to similar
chemical properties [Bal18]. Once Er is dissociated, there are six stable isotopes ranging inatomic weight from 162 u to 170 u. Table 1.1 summarizes the relative abundances and thequantum mechanical statistics for each isotope.
This thesis focuses on the most abundant isotope 166 Er, but the ERBIUM machine canbring to degeneracy also the other ones. The presence of several isotopes allows us to study
different regimes due to their different interaction properties [Pat22a]. Note also that theexperiment can study dipolar Fermi gases thanks to the high abundances of the fermionicisotope.
7

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1 Ultracold erbium quantum gas

Erbium isotopes
Isotopes

natural abundance

statistics

162 Er

0.14 %

bosonic

164 Er

1.6 %bosonic

166 Er

33.5 %

bosonic

167 Er

22.9 %

fermionic

168 Er

27.0 %

bosonic

170 Er

14.9 %

bosonic

Table 1.1: Erbium isotopes, relative abundaces, and quantum mechanical statistics.

1.1.1 Electronic configuration
In the ground state, the electronic configuration of erbium reads:1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 4f 12 6s2 ,1.1

where the part inside the parenthesis corresponds to the noble gas xenon, and thus refers to
the inner-shell core electrons. In the outer shells, Er has 14 valence electrons. The 4f shell is
not complete, accommodating only 12 of the 14 possible electrons, while the 6s is completed.
This configuration, where the 4f shell is submerged [Jen91], i.e. lies energetically below the
6s shell, is common in rare-earth elements.
The open f-shell and the submerged electronic configuration give rise to primary characteristics of erbium, first and foremost, the large magnetic dipole moment, µ, which leads tolong-range dipole-dipole interaction. According to Hund´s rule, the quantum numbers forthe ground state are S = 1 and L = 5. The LS coupling scheme provides the total angularmomentum quantum number J = L + S = 6; hence the ground state reads 3 H6 in the
Russell-Saunders notation. Finally, the magnetic moment is given by:µJ = −µB(gS S + gL L) ,ℏ

1.2

since for bosonic isotopes the nuclear spin I = 0. Here, µB is the Bohr magneton, gS ≈2.00232 and gL = 1 are the g-factors. The projection along the quantization axis (definedby the direction of the magnetic field) is given by:µ = −mJ gJ µB ,1.3

where gJ is the Landé g-factor, and mJ is the magnetic quantum number. Here, mJ canassume the values in the range [−J ÷ J]. Using the experimental value of the Landé gfactor [Con63] gJ = 1.163801(1), the magnetic moment is:µ = −6.982806(6)µB ,1.4

where we assume the atoms spin polarized in the lowest Zeeman sublevel [J = 6; mj = −6].

1.1 Erbium properties

10

9

4

2.5

2

1.5

1

0.5

0

Figure 1.1: Energy spectrum. Illustration of the energy levels according to their total angularmomentum quantum number J. The colors indicate the parity of the level, red (blue) for even (odd)states. The arrows indicate the electronic transitions used in the experiment to cool, manipulate andtrap the erbium atoms. Adapted from Ref. [Pat22b]. Energy levels from Ref. [Ral11].

1.1.2 Energy spectrum

Another consequence of the electronic configuration is the multitude of optical transitions
available. They result from the possibility to excite an electron from both the 6s and the 4f
shell. Figure 1.1 illustrates the energy levels according to the NIST database [Ral11] up to
2.6 × 10−4 cm. The arrows indicate the laser lights used in the ERBIUM experiment to manipulate the external and internal degrees of freedom. There is a forest of optical transitions,whose linewidths range from around 30 MHz to 2 Hz. In particular, these two linewidths correspond to the optical transitions with wavelengths 401 nm and 1299 nm, respectively, andare good examples of the different types of transitions. The 401-nm line is a transition where
an electron from the outer shell 6s is excited in the 6p shell. We use this broad transition
for laser cooling and imaging; see later discussion. The 1299-nm line is a transition where anelectron from the 4f is excited in the 5d shell. In this case, both the shells are inner shells
shielded by the presence of 2 electrons in the 6s shell. Therefore, this is a narrow transition
used in our experiment for addressing the internal degrees of freedom, e.g., controlling the
population of the Zeeman sublevels [Pat21, Pat22b]. Finally, the other two arrows in Figure 1.1 indicate the 583-nm line and the 1064-nm light. We use the 583-nm line to create amagneto-optical trap (MOT). The 1064-nm light is far detuned from any optical transition.
We use this laser light at high power (≈ 10 W) as an optical-dipole trap (ODT).

10

1 Ultracold erbium quantum gas

Ti-sub pump (hidden)z

75 l/s ion getter pumpx

20 l/s ion getter pumps

Ti-sub pumpy
200 mmfull-metal valve
port aligner

port aligner

bellowpressure gauge

atomic beam shutter

UHV section
Zeeman slower
mirrorpumping stage 2

main chamber

HV section
Zeeman slower

pumping stage 1

transversal
coolinghigh-temperature oven

Figure 1.2: Vacuum apparatus. Illustration of the vacuum parts of the ERBIUM experiment. It
is divided into two sections. The high vacuum section contains three modules: the high-temperatureoven, the transversal cooling module, and the pumping stage. The ultra-high vacuum section contains
four modules: the Zeeman slower, the main chamber, the second pumping stage, and the Zeeman
slower mirror. Figure from [Fri14].

1.2 Erbium experiment
The ERBIUM experiment brings erbium atoms to quantum degeneracy. In the case ofbosonic atoms, this condition requires trapping around 10 thousand atoms and cooling themto about 100 nK, see discussion on phase-space density in Sec. 1.3. Reaching this condition
is impossible without isolating the atomic cloud from the environment. For this reason, theatoms are located in an ultra-high vacuum chamber, where magnetic fields and laser beams
enable cooling and trapping. The experiment consists of the vacuum apparatus and the laser
systems. Fig. 1.2 illustrates the vacuum apparatus.
The first Bose-Einstein condensate of erbium was obtained in our experiment 10 years ago
in 2012. Over the years, there was no need to maintain or upgrade the vacuum apparatus,
except for the dual-filament high-temperature oven, which is used to generate the atomic
beam of erbium atoms1 . However, during the work reported in this thesis, upgrades ofthe laser systems were necessary in order to investigate unexplored directions and improve
the overall stability of the system. The following section provides a brief overview of theexperimental apparatus. More details can be found in the PhD thesis of Albert Frisch [Fri12].
As highlighted in Fig. 1.2, two sections constitute the apparatus.
The first one is a high-vacuum (HV) section. It consists of the high-temperature oven, thetransversal-cooling module, and the first pumping stage. The goal of the first section is toprovide a collimated atomic beam of erbium atoms. The first module is the commercial
1

One of the thermocouples of the oven broke after 5 years of operation. We, therefore, exchanged the ovenin 2020.

1.2 Erbium experiment11

diffusive dual-filament oven2 . It consists of two regions: the effusion cell and the hot lip.
These two regions are heated up by two independent filaments. In the effusion-cell region,
small pieces of solid erbium (≈ 10 g) are stored and heated up to 1100 ◦ C to efficiently
evaporate Er atoms3 . The hot-lip region consists of a series of apertures that collimate
the atomic flux. This region is heated up to a higher temperature (1200 ◦ C) to avoid theaccumulation of erbium that could lead to occlusion. From the operating oven temperatureand the erbium mass, one can estimate the speed of the atomic beam4 voven ≈ 450 m s−1 . The
second module is the transversal cooling section. This aims at further collimate the atomic
beam. In this section, two retro-reflected laser beams operating at 401 nm create a 2D opticalmolasses. Finally, the last module of the high-vacuum section is the pumping stage. This
module has three main aims. The first is to connect two ion pumps for maintaining theexperiment under HV. The second aim is to have the possibility to separate this part fromthe UHV section via a metal valve, e.g, for refilling the oven. Finally, a differential pumping
tube connects this HV section to the UHV avoiding pressure equilibration between the two
sections.
The second section instead is in ultra-high vacuum (UHV). It consists of the Zeeman slower(ZS), the main chamber, the second pumping stage, and a movable ZS mirror. The Zeeman
slower reduces the longitudinal velocity of the atoms using radiative pressure from a 401-nm
laser beam counterpropagating to the atomic beam. The broad linewidth combined with thesmall wavelength makes this transition ideal. The broad linewidth ensures a good number ofabsorption and spontaneous emission cycles, while the small wavelength ensures adequately
recoil in each absorption. The idea is to reduce the velocity up to the capture velocity of thenarrow-line magneto-optical trap operating at 583 nm. The latter linewidth is ∆ν = 190 kHz,
which results in a capture velocity of vcapture ≈ 5 m s−1 . The ZS is an increasing-field spinflip design with a length of 36 cm. This spin-flip design compensates the space-dependent
Doppler shift with an inhomogeneous low magnetic field. An interesting difference compared
to standard Zeeman slowers design is the presence of a tunable bias field that can optimize
the capture velocity of the ZS. After the Zeeman slower, we trap the atoms in the mainchamber. As already mentioned, the magneto-optical trap operating on the 583-nm provides
the first trap. The combination of the large mass of erbium atoms and the narrow line gives
rise to two benefits. First, the sample is automatically spin-polarized in the lowest Zeeman
sublevel. Second, the optical setup needs only five beams, leaving the vertical viewport ofthe main chamber free for optical access. In fact, since the gravitational force is on the sameorder of the confinement force, the atoms favour the absorption of the beam propagating
against gravity. In our case, the polarization of the beam is σ − , and the MOT provides apure sample in the lowest Zeeman level. The free space due to the unnecessary sixth-MOT
beam, gives us the possibility to have simultaneously: an optical-dipole trap (see Sec. 1.3), a
DMD setup (see Sec. 2.5), an optical lattice (see Chap. 4), and vertical imaging (See. [Fri12]).
In the future, we plan to use this space for high-resolution imaging (see outlook in Chap. 5).
The main elements of the last two modules are the ion pump for maintaining this section
2

3
4

The master thesis from Johannes Schindler [Sch11] provides more details on the commercial oven model:
DFC-40-10-WK-2B with control unit CU-3504-S2-DC from CreaTec Fischer & Co. GmbH.
Note the melting point for Er atoms is 1529 ◦ C.
The velocity of the atoms out of the oven follows a Maxwell-Boltzmann
distribution, see Refs. [Chi08, Fri12].
q
The maximum velocity of the distribution is given by
the mass of erbium and T is the temperature.

3kB T
.
m

Here, kB is the Boltzmann constant, m is

12

1 Ultracold erbium quantum gas

under UHV and an aluminum mirror for the Zeeman slower beam. This mirror is essential
since using the standard setup of a beam in transmission over a viewport would lead todecreasing efficiency over time. In fact, the viewport would get coated with atoms, and thehigh melting point of erbium atoms would not allow to remove the coating by heating the
viewport. This design does not block the mirror from getting coated with erbium atoms, butsince the mirror works in reflection, the coating will only partially reduce the performanceof the mirror.

1.3 Quantum degeneracy
This section briefly explains the optical-dipole-trap setup and the evaporative cooling toreach a quantum degenerate sample. The degeneracy conditions require that the distance
between the particle is on the same order of the extension of the wave packet. For thisreason, the temperature and the atom number alone does not directly tell how close thesystem is to the degeneracy condition. A better quantity is the phase-space density (PSD).

3
2
m
3
It is defined as the product of the peak density n0 = N ω̄ 2πkB T , where m is the atomic√
mass, and the de Broglie wavelength λdB = h/ 2πmkB T :PSD = N ω̄

3



ℏ
kB T

3

.

1.5

Here, N is the atom number, ω̄ the geometrical mean trap frequency, and ℏ and kB the
reduced Planck and the Boltzmann constant, respectively. One reaches the Bose-Einsteincondensation threshold when this quantity is on the order of unity [Pet02]. In cold-atomexperiments, it is a standard technique to trap the atoms in an optical-dipole trap and rely on
evaporative cooling to reach this regime. The optical-dipole trap in the ERBIUM experimentconsists of two crossing 1064-nm beams, one propagating in the horizontal direction and one
in the vertical direction. We load the atoms from the MOT by switching on the horizontal
beam 300 ms before switching off the MOT. This beam has a waist of about 20 µm, but atunable aspect ratio. The latter is obtained via a time-averaged potential by scanning themodulation frequency of an acousto-optic modulator (AOM). This technique is extensively
used in our group and explained in the master theses of Simon Baier [Bai12] and Claudia
Politi [Pol17]. The vertical beam instead has a bigger waist of ≈ 100 µm and prevents theatoms from escaping the trap from the direction of propagation of the horizontal beam.
After loading the sample in the optical trap, the depth of the potential is reduced stepwise
to perform evaporative cooling. The basic idea is to permit the hottest atoms to escape andsubsequentially allow the remaining sample to rethermalize via elastic collisions to a lower
temperature. For this reason, not only the trapping potential should be carefully chosen foreach step, but also the scattering cross-section.
In the case of bosonic erbium, the interaction potential between two atoms separated by
the distance r has two contributions, the van der Waals interactions and the long-range
dipole-dipole interactions (DDI):

1.4 The Bragg-spectroscopy setup and the 1D optical lattice

U (r) = UvdW (r) + Udd (r).

13

1.6

At low temperatures, the typical momentum of the atom k ≪ 2π/R, where R is the rangeof the interaction. In this limit, a pseudopotential fully described by a single parameter, thescattering length as , can approximate the van der Waals term. Therefore, the interaction
potential reads:
Cdd 1 − 3 cos2 θU (r) = gδ (r) +
.
1.7
4π
|r|3

Here, the first term represents the short-range and isotropic contact interaction, where g =4πℏ2 asm . The sign of as determines if the interaction is repulsive (as > 0) or attractive (as < 0).
The second term is the dipole-dipole potential in the case of an external magnetic field. In
this case, the dipoles align themselves in the direction of the field. Here, Cdd = µ0 µ2 ,where µ0 is the vacuum magnetic permeability. The many-body phase emerging from thecompetition between these two interaction terms is the core result of this thesis.

1.4 The Bragg-spectroscopy setup and the 1D optical latticeThis section aims to illustrate two principal optical setups for this thesis: the Braggspectroscopy setup and the one-dimensional optical lattice. Fig. 1.3 illustrates the opticalsetups and their integration in the ERBIUM experiment, together with the vertical ODT
and imaging setup. For simplicity, the sketch does not show the vertical MOT light beam.
As already highlighted in the previous section, our MOT consists of only five beams. It
allows us to use the upper vertical viewport to study the excitation spectrum of ultracold
erbium atoms via Bragg spectroscopy and to load the atoms in a 1D optical lattice.
We use the Bragg-spectroscopy setup to study the excitation spectrum of a dipolar BEC fromthe regime where the contact interactions dominate to the regime where the predominantinteraction is the DDI, see Chap. 2. The idea is to use a digital micromirror device (DMD) tocreate a moving lattice. In a DMD, the mirrors can assume two different positions. Hence,it allows the drawing of arbitrary potentials. In our case, the setup begins with a beam
sampler to monitor and stabilize the intensity of the light. Afterward, a telescope creates
an elliptic beam to cover the area of the DMD, where we project a grating to generate two
beams propagating with an angle. The combination of a lens and an iris removes unwantedorders. Finally, the two beams interfere at the atoms positions, thanks to the last lens. We
use this setup with the 401-nm light, but the design allows the use of the 532-nm light.
The reason for this double-wavelength design relies on the low-power efficiency of the setup.
Working close to the 401-nm line can guarantee a high enough polarizability. However, thiscomes with the drawback of higher heating, where working further detuned via a 532-nm
laser could help. Nevertheless, we found a good spot by working at a detuning of 71 GHz
from the 401-nm transition, see Sec. 2.5 and Refs. [Pet19, Pet20].
The vertical-lattice setup aims to confine the atoms in an array of pancakes. The combination of this setup with the other two horizontal lattices creates a 3D optical lattice, see

14

1 Ultracold erbium quantum gas
300

1:
99

DMD

λ/2

vertical ODT 1064 nm

2
IPG fiber laser

300

3

300

400

250
achrm.
1"
60

λ/4

iris

100

-50

PD
cylindrical
telescope1

vert. lattice
retro-reflection

DMD mirror boardcyl.

DMD controller

vertical imaging

2
1

PD

PD

-30

200

3

532 nm

AOM

1:
99

r.
ch r
di irro
m

cyl.

100

CCD

Bragg setup 401 nm

vertical lattice 1064 nm

400
100
PD

-50

λ/2
Mephistovert. im
aging

Figure 1.3: Optical setups. Sketch of the optical setups in the ERBIUM experiment. The left
side shows the Bragg-spectroscopy setup. The right side shows the vertical ODT and the vertical
optical lattice. The central part of the figure shows an illustration of the main chamber and how
the different laser beams are combined onto the atoms from the vertical viewports. Figure adaptedfrom [Fri12].

Refs. [Bai16a, Bai18, Pat20]. In this thesis, we combine the vertical-lattice with the optical
dipole traps to create quasi-2D elongated layers. For this setup, we recycle the power fromthe zero order of an AOM of the horizontally propagating ODT. This light is fiber-coupled
and sent to a breadboard above the main chamber. After the fiber, a polarizing beam splitter cleans the polarization, and a 1:2 telescope enlarges the beam waist. We use the leaking
light transmitted from a mirror to monitor and stabilize the power. Finally, a lens focuses
the beam onto the atoms, and a mirror reflects the light back to create the optical lattice.
The simultaneous presence along the vertical direction of the Bragg spectroscopy, ODTlight, imaging light, and lattice beam requires some compromises, such as for example thenecessity of having laser beams propagating at different angles with respect to gravity. In
particular, the imaging beam propagating at ≈ 15◦ with respect to the gravity axis has thelargest inclination. Also, the lattice beams have an angle of 9(1)◦ compared to the gravity
axis. This angle limits the Bloch oscillation observation time, see Chap. 4.2 and Ref. [Nat22].
Regarding the Bragg-spectroscopy setup, the size of the final lens determines the maximum
momentum imparted on the atoms. The presence of the other optical setups compromises
the size of this lens. Finally, the biggest drawback comes from the numerical aperture of theobjective, which is about 0.08, limiting the resolution of the vertical imaging.

Chapter

2

Excitation spectrum of anunmodulated dipolar BEC

This chapter has two main goals. The first one is to give an overview of the theoretical
approach used in this thesis to describe the physics of an ultracold dipolar Bose-Einsteincondensate (dBEC). The second one is to report, from theory and experiments, the observation of the excitation spectrum of a dBEC manifesting a maxon and a roton mode.
Sec. 2.1 introduces the Hamiltonian for dilute trapped quantum gases and discusses the approximations necessary to write the mean-field Gross-Pitaevskii equation (GPE). Section 2.2
uses the GPE to calculate the excitation spectrum of the system, which is fundamental tounderstand how the system responds to an external perturbation. Furthermore, it introduces the roton mode and highlights its characteristics. A key concept that emerges forimportance in the context of a dipolar BEC is quantum fluctuation. In particular, Sec. 2.3
introduces the first-order correction to the mean-field GPE: the Lee-Huang-Yang term. We
will discuss in which condition this term plays a fundamental role in determining the stabilityof the system and the formation of novel quantum phases. Section 2.4 focuses on the effectof finite temperatures, introducing the Wigner approach. Section 2.5 introduces the Bragg
spectroscopy technique. Finally, Sec. 2.6 shows the publication, in which our group studied
the excitation spectrum of dipolar erbium atoms via Bragg spectroscopy for a broad range
of interactions.

2.1 Gross-Pitaevskii equation
The Bose-Einstein condensation in atomic-gases experiment is a phenomenon achieved bycooling identical bosons trapped in an external potential below a critical temperature. Thissection introduces the typical theoretical framework used to study BECs. Before starting,it is useful to remind the experimental conditions for BECs. Let us consider, for example,
our experiment where we daily produce a BEC of erbium atoms, in a cigar-shaped trapwith frequencies ωi about ωx,y,z ≈ 2π × (150, 30, 150)Hz. The condensate temperature is15

16

2 Excitation spectrum of an unmodulated dipolar BEC

around 50 nK and we achieve around 1 × 105 atoms trapped in a volume of 50 µm3 . Table 2.1
summarizes these parameters. Note, in particular, the finite system size, atom number andExperimental parameters
Temperature

50 nK

Density

Trapping frequencies

ωx,y,z = 2π × (150, 30, 150)Hz

Trapping volume

2 × 1015 cm−3
50 µm3

Table 2.1: Typical experimental parameters for a BEC of erbium atoms investigated in this thesis.

temperature modify the fundamental concept of BEC derived in the thermodynamic limit.
Here, we will only discuss some of the consequences of finite system size and temperaturenecessary to explain the results presented in this thesis, but for a more detailed discussion,see Refs. [Pit16, Pro13].
Our starting point is a system of N interacting bosonic atoms at zero temperature. The
influence of finite temperature is described in brief in Sec. 2.4. At low density, the interactions
between more than two atoms can be neglected leading to the following Hamiltonian:

H=

N  2
X
p⃗


1 XX
+ Vtrap (⃗xi ) +V (⃗xi − ⃗xj ) .
2m
2
i

i=1

i

2.8

j̸=i

Here, the first term represents the kinetic energy with the momentum p and mass m of an
erbium atom. The second term accounts for the trapping potential at the position of theindividual atoms. Finally, the last term includes the two-body interaction. The system is
weakly interacting. This aspect becomes clear when comparing the two relevant length scales:
the interparticle spacing d and the interaction lengths (as , add ). Considering the density nof our BEC, n ≈ 1015 cm−3 , the interparticle spacing d can be calculated as d = n−1/3 .
In the work presented in this thesis, the scattering length is changed in the range as =[50 − 100] a0 ≃ [2 − 5] nm. For these conditions, the dilute requirement ads ≪ 1 → na3s ≪ 1
is satisfied.
The presence of the interaction term gives rise to quantum depletion [Bog47]. The depletionmeans that even at zero temperature not all the atoms are in the ground state. Nevertheless,the dilute condition allows us to consider the following mean-field approximation of the N body wavefunction:
Φ (r1 , r2 , . . . , rN ) ≈ ϕ (r1 ) ϕ (r2 ) · · · ϕ (rN ) .
2.9
Note that while this is only an approximation in the interacting case, it becomes an equality
in absence of interactions.
We can now define the energy functional as:
Z
E[Φ] = ⟨Φ|H|Φ⟩ = dr1 . . . drN Φ∗ (r1 , r2 , . . . , rN ) ĤΦ (r1 , r2 , . . . , rN ) .

2.10

2.2 Bogoliubov excitations

17

Minimizing this energy functional leads to the time-independent extended Gross-Pitaevskii
equation:µΨ(⃗x, t) =

ℏ2 2
∇ + Vharm (⃗x) + g|Ψ(⃗x, t)|2
2m

Z
2

′
3
3 ′
′
+ d ⃗x Udd ⃗x − ⃗x Ψ ⃗x , t + γQF |Ψ(⃗x, t)| Ψ(⃗x, t),
−

2.11

√
where Ψ = N Φ and γQF is the beyond mean-field correction in the form introduced by
Lee-Huang-Yang (LHY) [Lee57, Hua57, Lim11, Lim12].
Once the ground-state wavefunction is known, the dynamics of the dBEC can be studied via
the time-dependent Gross–Pitaevskii equation:∂Ψ(r, t)
iℏ
=
∂t



ℏ2 2
∇ + Vharm (⃗x) + g|Ψ(⃗x, t)|2
2m

Z

2
3 ′
′
′
3
+ d ⃗x Udd ⃗x − ⃗x Ψ ⃗x , t + γQF |Ψ(⃗x, t)| Ψ(r, t).
−

2.12

It is important to highlight that this expression is often also used to find the ground state
by using imaginary time evolution [Wic54].

2.2 Bogoliubov excitations
The method to calculate the elementary excitations is based on the Bogoliubov treatment of
the Gross-Pitaevskii equation, see e.g [Pit16]. The first step is to write the condensate wavefunction as the sum of the ground-state wavefunction Ψ0 and a small perturbation ηψ(r, t),
as follows:Ψ(r, t) = Ψ0 (r) + ηψ(r, t).
2.13
Here η ≪ 1. We can think about the small perturbation as a mode ”wavefunction” and in
this thesis we use this term for referring to ψ. The second step is to assume the following
decomposition for ψ:ψ = e−iµt u(r)e−iωt − v ∗ (r)eiωt ,2.14

Rwhere u, v are the Bogoliubov amplitudes fulfilling the normalization conditions |u(r)|2 −|v(r)|2 dr = 1 and ε = ℏω is the eigenenergy of the mode.
Finally, when substituting this expression in the GPE equation (Eq. 2.11) and neglecting allterms that scale as η 2 , we found two coupled differential equations, usually referred to as

2 Excitation spectrum of an unmodulated dipolar BEC
Bogoliubov amplitude |u|2, |v|2

18
5

(a)

(b)

4
3
2
1
0
0.2

0.4

0.6

0.8

1

1.2

1.4

0.5

Energy E (h z )1

1.5

Momentum ky l z

Figure 2.1: Bogoliubov amplitudes |u|2 and |v|2 . Bogoliubov amplitudes |u|2 (|v|2 ) are plottedin blue (green) as a function of the energy (a), and effective momentum (b) of the corresponding
modes. The black horizontal lines denote the single particle condition (|u|2 = 1; |v|2 = 0).

Bogoliubov-de Gennes (BdG) equations that can be written in a matrix form as follows
ĤGP [ψ0 ] + A−AA−ĤGP [ψ0 ] − A



u
v



=ε



u
v



.

2.15

Here, A is an operator that can be computed on a generic function f as follows:(Af )(r) =

Z




d′ ψ0 r′ U r − r′ f r′ ψ0 (r)
16 3/23 2+ √ gas1 + εdd |ψ0 (r)|3 f (r).
2
π

2.16

This eigenvalue problem can be written in a diagonal form following Ref. [Ron06]. The
Bogoliubov amplitude u (v) gives the eigenmodes related to positive (negative) frequencies.
Figure. 2.1 shows the typical evolution of the Bogoliubov amplitudes as a function of theenergy (a) and momentum (b). For large wavelengths (low momenta), the v component
is not negligible, and these modes correspond to collective excitations. Instead, when the
wavelength is smaller than the healing length the response has to be attributed to the single
particle rather than to the whole condensate [Bog47, Ron06, Pit16].
In absence of harmonic confinements, the spectrum is a continuous function and, for eachenergy, there is an associated momentum q ∗ . The wavefunction associated with a mode
has a characteristic density modulation 2π/q ∗ . Instead, the presence of a harmonic trap
discretizes the allowed energies and removes the possibility to describe the Bogoliubov modewith a precise single momentum/wavelength. Even if it is still possible to associate to eachmode an effective momentum [Bis13], it turned out that the dynamic structure factor provides
a useful visualization of the spectrum.
Once the Bogoliubov amplitudes are known, the zero temperature dynamic structure factor(S0 ) is computed as follows:

2.2 Bogoliubov excitations

S0 q, ω

′



=

19

X Z

2
∗

∗

iq·r

dr [u (r) + v (r)] elψ0 (r) δ (ω − ωl ) .

2.17

S0 determines the density response of the system when probed with a small perturbation
at a specific momentum and energy. Note that to evaluate this quantity one needs tocompute both the ground-state wavefunction and the Bogoliubov modes. The dynamic
structure factor not only contains information about the energy and the wavefunction of all
the Bogoliubov modes but, as shown in Eq. 2.17, gives also the strength of the coupling withthe ground state.

2.2.1 A roton-maxon excitation spectrum
For understanding the properties of 4 He, Landau introduced the concept of roton [Lan41].
This quasiparticle is responsible for a minimum in the energy spectrum at a finite momentum.
While the strong interparticle interactions are the ingredient for a roton in 4 He, in the context
of dipolar gases, the responsible term is the dipole-dipole interaction and the roton exists
even in a dilute regime [San03]. Dipolar atoms presenting a roton spectrum were studied
theoretically mainly in pancake traps [San03, Ron07, Boh09, Bla12, JL13]. In this chapter
and for the results on supersolidity, we concentrate on a cigar geometry with the dipolealigned along one of the tight axes of the trap.
We first focus on a regime of interactions where the ground state is a non-modulated BEC,
and we add a small population of a Bogoliuobov mode to understand the origin of the rotonmode in dipolar BECs. As we highlighted in the previous section, the wavefunction associated with a Bogoliubov mode presents in real space a density modulation. Due to theanisotropic nature, the effect of the DDI changes depending on the wavelength of the populated mode. If one considers a mode at low momenta, the atoms in the excitation will bealigned mainly side by side, giving rise to an additional repulsive contribution that stiffens
the energies of the mode. However, in the case of large momenta, the atoms experiencea head-to-tail alignment, that induces a softening of the modes. This picture can be analytically confirmed from Ref. [Bla20a, Bla20b, Pal20]. In these papers, the authors present
a variational model to describe an elongated cigar. The basic idea is to assume that the
wavefunction can be decomposed in a radial and axial part. It is then possible to find an analytical approximation to describe the momentum space interaction along the axial directionof the cigar. Fig. 2.2 (a,b) plots the effective interactions, which include only the mean-fieldterms, for two different orientations of the magnetic field. The dipoles in Fig. 2.2(a) arealigned along the axial direction of the trap, whereas in (b) along the radial direction.
Figure 2.2 contains informations of the ground state and of the excitation spectrum. In
particular, at q = 0 in Fig. 2.2(a), the contact and dipole-dipole interaction compensate eachother, whereas in Fig. 2.2(b) they sum up. As a consequence, in the first case the condensate
is weakly interacting and dense, while in Fig. 2.2(b) the interactions induce the size of theground state to increase. From the variational theory, it is possible to relate the effective
interaction Ũ also to the energy of the excitation spectrum via:

20

2 Excitation spectrum of an unmodulated dipolar BEC

Figure 2.2: Effective interactions and excitation spectrum for a cigar-shape configuration. Effective interactions in the case of dipoles aligned along the axial (a) and along the radial (b)direction of the cigar trap. The three colors (dark blue, light blue, green) represent three different
scattering lengths: [(75, 65, 55) a0 ]. The gray shaded area indicates attractive effective interactions.
(c,d) Excitation spectrum related to the effective interactions in (a,b).

Eqvar
=
y

r

h
i
ϵqy ϵqy + 2nŨ (qy ) ,2.18

where ϵqy = ℏ2 qy2 /2m and n is the density. This equation is valid under the same shape
approximation, in which it is assumed that the radial profile of the modes has the same
shape of the condensate, see Ref. [Bai15].
Figure 2.2(c,d) show the excitation spectrum calculated from Eq.2.18, for the two different
orientations of the magnetic field. We observe that a negative (positive) value of the effective
interaction induces a softening (stiffening) of the energy spectrum. The negative effective
interactions at large momenta, which appears when the dipoles are aligned along the radial
direction, favour a roton excitation spectrum. The presence of the roton mode depends
on the competition between DDI and contact interactions. In fact, a negative contributionof Ũ is present only for as < add , see different colors in Fig 2.2. Note that the excitation

2.2 Bogoliubov excitations

21

Figure 2.3: Energy and momentum evolution of the roton mode with as . (a) Energy
evolution of the roton mode as a function of as . (b) Momentum evolution of the roton mode as afunction of as . The three colors (gray, orange, light blue) represent three different trapping potential(νx , νy , νz )=[(150, 30, 150) Hz; (300, 30, 300) Hz; (450, 30, 450) Hz], respectively.

spectrum in Fig 2.2(c), calculated from Eq. 2.18, reproduces well the real one only for low
momenta [Pal20]. For this orientation of the magnetic field, the spectrum shows a novel
multi-band behaviour [Pal20] that will be investigated in future publications.
To gain further understanding of the roton mode, Fig. 2.3 shows the evolution of the energy
and momentum of the roton mode with as following ref. [Cho18]. The calculations are done
for three different trapping potentials to underline the role of the radial trap. Fig. 2.3 (a),displays the softening of the roton mode. Note that the maxon energy is around 0.5νz , hence
only if the energy is below this value there is a real minimum in the energy spectrum. Thishighlights that a maxon-roton energy spectrum is only present for few a0 . Fig. 2.3 (b) shows
another important characteristic of the roton mode in dBEC: the tunability with the trapping
potential and with the scattering length. In the figure, it is interesting to note that a softeningof the roton mode happens always at the same value of krot lz (this happens at different valuesof as for the different trapping potentials), underlining its universal geometrical scaling.
The analytical models used so far simplify the computations and the understanding of thedipolar roton physics, but do not fully account for the finite system size. Therefore, following
Sec. 2.2, we can compute the Bogoliubov amplitudes without additional approximations.
Fig. 2.4 plots the associated mode wavefunction for a few selected modes and visualizes
the spectrum by plotting the dynamic structure factor as a color shading. The finite size
system reveals itself via a discrete spectrum where it is not correct to associate a precise
momentum to a mode. This is particularly the case for some modes, e.g. Fig. 2.4 (e,f)
where the same mode has a response in a broad and not continuous range of momenta. Aninteresting question is how the roton wavefunction looks and compares with the other modesof the spectum. Fig. 2.4 answers this question by comparing different modes. It highlights

22

2 Excitation spectrum of an unmodulated dipolar BEC

Density(arb. unit)

1
(a)

(c)

(e)

(g)

0.5
0

-10

0

10 -100y ( m)

10 -100y ( m)

10 -100y ( m)10y ( m)

1.5
25
(b)

(d)

(f)

(h)
20

EnergyE (h z)

1

15
10

0.5

5
0
0

1

2

30

Momentum ky lz

1

2

Momentum ky lz

30

1

2

Momentum ky lz

30

1

2

3

0

Momentum ky lz

Figure 2.4: Energy spectrum, ground state, and excited modes. (a, c, e, g) The dashedline shows the integrated density of the ground state, whereas the green line shows the wavefunctionof an excited mode. (b, d, f, h) The color shading represents the dynamic structure factor. The
modes selected in the upper panels are highlighted in blue. Trapping frequencies, scattering length,
and atom numbers are: ωx,y,z = 2π × (160, 31, 250) Hz, as = 51.2a0 , N = 5 × 104 .

(Fig. 2.4(a-d)) that in the spectrum there are two degenerate roton modes. Interestingly,
they are localized at the center of the trap. This contrasts with respect to the phonon
and the single-particle excitations (Fig. 2.4(g,h) at high momenta, where the wavefunction
is delocalized over the condensate (dashed line (Fig. 2.4(a,c,e,g))).
Finally, the presence of the roton mode results in a peak of the structure factor. The dynamic
structure factor needs to satisfy the following sum rule [Str16]
Z ∞N ℏ2 q 2dω S(q, ω)ω =
ℏ2
.
2.19
2m
−∞
This sum rule is general and its validity does not depend on the interaction potential. For
various scattering lengths, the roton mode changes energy according to Fig. 2.3(a). If one
neglects the variation of the roton momentum when changing the scattering length (this does
not qualitatively change the result), then the softening of the roton mode directly results in
an increase of the structure factor, which diverges when the roton mode fully softens. This
gives rise to a stronger response of the system to a perturbation like Bragg spectroscopy.
This aspect is also experimentally measured in Fig. 4 of the publication in Sec. 2.6.

2.2.2 Quantum depletion
The quantum depletion is an effect of interactions. As described in Sec. 2.1, one of theassumption necessary to derive the GPE is a state with negligible quantum depletion. Due2.3 Quantum fluctuations

23

to the dilute nature of the condensate, this condition is usually satisfied. Nevertheless, manyof the novel dipolar phases studied in this thesis are characterized by a higher density than astandard BEC. It is for this reason important to check the amount of depleted atoms. Thisis possible if the Bogoliubov amplitudes u, v are known thanks to the following relation:
Z
X
Ñ = dr|vj (r)|2 .
2.20
jNote that all modes with momenta ky ≤ ℏ/ξ are necessary to converge the sum in Eq. 2.20.
In fact, |vj (r)|2 ≈ 1 for modes with momenta ky ≈ ℏ/ξ, where ξ is the healing length, andrapidly decays to |vj (r)|2 ≈ 0 at larger momenta.

2.3 Quantum fluctuations
In the context of dipolar ultracold atoms, quantum fluctuations are usually associated to thefirst-order beyond mean-field correction to the energy functional. This term is fundamental
to describe the experimental discoveries of different stable quantum phases of dBEC that are
in a regime where the mean-field picture predicts a collapse. This correction reads [Lim11,
Lim12]

E0
1 2
128 √ 3√na F (ϵdd ) ,2.21= gn 1 +
V
2
15 π
where

1
F (ϵdd ) =
2

Z π
0


 5
dθ sin θ 1 + ϵdd 3 cos2 θ − 1 2 .

2.22

It is important to highlight a few aspects of this correction. First, this correction scaleswith the density with a higher power compared to the mean-field term. Additionally, it is arepulsive contribution, hence can stabilize the system against a collapse.
The effect of this term was measured in the pioneering experiment [Alt07], where a shift in the
excitation frequencies, in agreement with Monte Carlo simulations, was found for a stronglyinteracting degenerate gas of 6 Li atoms in the lowest two internal states. Other remarkableapproaches to highlight the LHY contribution were: the use of a Feshbach resonance toenter the strongly interacting regime with 85 Rb [Pap08]; the use of an optical lattice toenhance the atomic interactions [Xu06]; the careful analysis and measurement of the density
profile [Shi08, Nav10]. Nevertheless, the influence of this term was never highlighted andmeasured in the condition of small mean-field interactions. It is relevant to note that there are
two regimes in which this correction term becomes important, strongly interacting gases or ina dilute system when the mean-field interactions are small. The last regime, in single species
experiment, can not be reached by simply reducing the contact interaction since also the LHY
correction gets null at zero scattering length. For this reason, a dipolar gas, where small
mean-field contributions can be obtained by competing dipole-dipole and contact interaction
is an interesting platform to exploit this term for realizing novel states. However, the firstpredicted manifestation of this quantum mechanical stabilization mechanism was for a BoseBose mixture. Also in this system, it is possible to exploit the three contact-interaction

24

2 Excitation spectrum of an unmodulated dipolar BEC

terms (two single species and one intra-species) to reduce the mean-field interaction at a
finite scattering length. In Ref. [Pet15], Petrov shows that in a regime where the meanfield theory predicts a collapse, introducing this term in the Hamiltonian provides a stableself-bound solution: the droplet state. This state was observed in Bose-Bose mixtures in
Refs. [Cab18, Sem18] and also with dipolar atoms, where depending on the trap geometry
and the direction of the dipoles it was possible to find single or multi-droplet solutions (intheoretical works, see Refs. [W1̈6a, Bis16, Bai16b] and experiments, see Refs. [Kad16, Cho16,
FB16]).
It is then natural to ask whether the dipolar multi-droplets state was a supersolid state. In
such a state, one expects the system to have a crystal structure while being superfluid. Sucha paradigmatic state is a primary investigation of this thesis, and Chapter 3 explains theproperties and the regime in which such a state exists.
Finally, it is important to highlight that to obtain Eq. 2.21 one needs different assumptions.
Together with the one already used to derive the Gross-Pitaevskii equation, one needs to use
the Local-Density Approximation (LDA). Additionally, the LHY term has an imaginary partfor ϵdd > 1 that is neglected in the theoretical simulations. For this reason, in many works inthis thesis, there is a quantitative comparison of the experimental and theoretical results. We
found that for different trapping geometries and states the level of agreements is different.
The importance of this term is indisputable, but its exact form needs further theoretical andexperimental effort, where other important effects like temperature [Bla21, SB22] are undercontrol.

2.4 Finite temperature and non-equilibrium dynamics
This section introduces the Wigner approach to consider finite temperatures in our simulations. The method here reported shows good agreement with the observable studied in theexperiment [Pet21]. Nevertheless, a comprehensive summary of the correct approach to simulate different observables at finite temperatures and studying non-equilibrium dynamics goesbeyond the aim of the current thesis and for that, we refer the reader to e.g., [Bla08, Pro08].
In our experiment, the typical conditions after the evaporation are states with a temperature
ranging from 50 to 100 nK and BEC fraction ranging from 50 to 80 %.
In brief, the simulations at finite temperature start by finding the ground state and calculating the excitation spectrum, as detailed in the previous section. The next step is torandomly seed the ground state with vacuum noise and thermal fluctuations according to
the truncated Wigner prescription. We finally time evolve these states according to Eq. 2.12.
Similarly to a real experiment, where it is important to repeat the same sequence to geta significant statistic, also for the truncated Wigner approach, it is important to repeat
the same simulation with different initial seeds, where the number necessary to reach theconvergence depends on the observable.
In more detail, once the excitation spectrum is known, we consider only the modes withenergy below 4KB T . From the Bogoliubov amplitudes u, v of these modes, we generate the2.5 Bragg spectroscopy

25

wavefunction ΨT associated with the vacuum and thermal noise as follows:ΨT = βj uj + βj∗ vj∗ .

2.23

The coefficient βj is given by:
βj =q

n̄j + 1/2



xj + iyj
√
2



,

2.24

1
where n̄j = ϵj /kB. Here, ϵj is the energy of the mode j, T is the temperature andTe−1
xj , yj are randomly distributed Gaussian variables, satisfying the following conditions on the
average and variance: < xj >= 0, var(xj ) = 1. The final wavefunction is then obtained
from the sum of the ground state Ψ0 at T=0 by Ψ = Ψ0 + ΨT .

The wavefunction is then time evolved according to Eq. 2.11. The validity of these simulations
tends to decrease the further from equilibrium the system is, and the longer the time scales
needed to follow the dynamics.

2.5 Bragg spectroscopy
Atomic interferometers are a famous application of Bragg scattering, where atomic samples are manipulated via light grating. They have been used to improve the knowledgeof fundamental quantities from the gravitational constant [Tin21] to the fine-structure constant [Cla06, Mor20], or to create quantum sensors, ranging from highly accurate gyroscopes[Fan12] used e.g., for accurate navigation of aircraft and submarines but also missiles, togravity gradient sensor [Str22] e.g., for climate research [Tay13]. In this section, we introduce the technique of Bragg spectroscopy, to study the intrinsic properties of the condensate.
In the breakthrough experiments at NIST and MIT [Ste99, SK99], this technique was exploited to determine the dynamic structure factor of the condensate for a specific wave vector
k. In this section and in the next one, this technique is used to study the excitation spectrumof a dBEC.
Bragg-spectroscopy experiments in ultracold atoms rely on the interference of two laser beams
with wave vector k1 and k2 and a frequency difference δω . The difference in the wavevector
creates an interference lattice potential of the form V0 cos(qx − ωt), with q = k1 − k2 . The
frequency difference makes the interference pattern move with a constant velocity. Thisvelocity translates into the energy of the perturbation, whereas the momentum is given by
q. Finally, the absolute frequency of the laser beams controls the potential experienced by
the atoms due to the AC-Stark shift.
In the experiments, the Bragg potential is projected on the atoms, with a square pulse of
duration τ . This results in a population of the Bogoliubov modes. The excited fraction
during the pulse can be calculated in the linear regime via perturbation theory:|cl (t)|

2

= V02

Z

2

dr [u∗ (r) + v∗ (r)] eiq·r ψ0 (r) t

sin2 [(El /ℏ − ω) t/2],
ℏ2 (El /ℏ − ω)2

2.25

26

2 Excitation spectrum of an unmodulated dipolar BEC

where Ei is the energy of the Bogoliubov mode. This expression can be related to S0 from
Eq. 2.17. More precisely, it is a Fourier-broadened version of S0 , where the resonant energies
do not change but get broadened:2
X Ze
dr [u∗ (r) + v∗ (r)] eiq·r ψ0 (r) τ sinc2 (| τ (ω − ωl ) /2) .
2.26
S0 (q, ω) =l

The duration of the pulse must be carefully chosen. There are five aspects to take into
account. First, there is an intrinsic finite lifetime of the quasiparticle due to the damping
process e.g., Landau and Beliaev damping [Gio98]. In these processes there is redistributionof the mode energy and momentum, however, energy conservation usually suppresses theseprocesses for trapped systems due to the discretization of the energies and particularly if thespectrum presents a roton minimum. Second, from Eq. 2.25, the pulse duration changes the
population of the excited modes. This depends also on the detuning and the intensity, which
are two preferable knobs to obtain the wanted modes population. Third, from Eq. 2.26, toreduce the Fourier broadening is necessary to increase the pulse duration. Fourth, the pulse
needs to be shorter than the quarter period of the trap (T /4), after which the coherent
Bragg diffraction is lost, since atoms scattered at the beginning of the pulse come to rest.
Finally, the pulse time should always be short compared to the lifetime of the state. This
last condition becomes particularly relevant when probing states like supersolids and droplets
for which the high density and the vicinity of the magnetic field to the Feshbach resonanceinduce usually a short lifetime of the sample. In our experimental regime, the shortest timescale is given by the axial trapping frequency (Bragg spectroscopy direction), which is usually
about 30 Hz, limiting the pulse duration to 7 ms.
To compare the experimental observable to Se0 from the theoretical ground state, it is necessary to satisfy the validity conditions based on the linear regime. The linear regime requires
achieving at the end of the Bragg pulse only a small transfer of population to the Bogoliubov
modes such that:X|ci (Tp )|2 ≪ 1,
2.27
i̸=0

where ci are the population of the i-excited modes. In this regime, the sum of the population
ci of all the excited modes has to be much smaller than unity. In fact, Eq. 2.25 is obtained
neglecting the interactions of the excited part. The natural experimental drawback is that,
if only a small percentage of the atoms are excited, extracting their population becomeschallenging. Bragg scattering beyond the linear regime changes the measured response after
the excitation [Bla02]. For this reason it is possible to test the linear regime by measuring
the excited fraction for different laser power. A deviation from a simple scaling with V02
demonstrates to be outside of the regime. After testing this regime in various measurements,
we found that a population of ≈ 10% of excited atoms at the end of the Bragg pulse is a goodcompromise. Note that for different wavevector q of the Bragg potential such condition is not
the same. It is therefore important that for each measurement the potential V0 is adjusted
to reach the linear regime. Once the pulse duration is chosen, intensity and detuning are theknob parameter to control the potential depth and limit the absorption rate.
The detuning of the Bragg beams control the heating induced in the sample. As for any
optical lattice the ideal condition is to use high intensity and large detuning to limit the ab-2.5 Bragg spectroscopy

27

sorption induced by the imaginary part of the polarizability. Intensity is limited for different
reasons. First, the waist of the beams should be much larger than the system size, and second the power is limited by the source and efficiency of the optical setup. In our experiment,
typical cloud sizes are on the order of 20 µm. For this reason, we use a beam waist of around
200 µm for the two Bragg beams. In this way, the potential can be consider uniform over the
cloud size. The optical setup is based on a digital micromirror device (DMD) to generate
the two interfering beams. More information about the setup can be found in Chap. 1.4 andin Refs. [Pet19, Pet20]. This setup is very versatile and can be used to probe the system at
different energies and momenta without realigning the beams onto the atoms. One drawbackof this setup is the power efficiency. The idea is to use the DMD to create a tunable grating in
Fourier space. In this configuration only few mirrors are simultaneously on during the pulse,
reducing the power efficiency of the setup to below 1%. The potential depth of our Bragg
probe can be estimated via real-time simulations suggesting that, to achieve a population ofaround 10%, V0 needs to be V0 ≈200 Hz. This potential is obtained by using all the available
power (50 mW before the DMD) for our Bragg beams at a detuning of 71(1) GHz from the
401-nm transition.

28

2 Excitation spectrum of an unmodulated dipolar BEC

2.6 Publication: Probing the roton excitation spectrum of astable dipolar Bose gas
Physical Review Letters 122, 183401 (2019)a
submitted 29 Nov 2018; published 8 May 2019
DOI: 10.1103/PhysRevLett.122.183401

D. Petter,1 G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 M. J. Mark,1,2 L. Chomaz,1and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2a

The author of the present thesis developed the experimental procedures together with D. P., L. C.,
performed the measurements together with D. P., analyzed the data, helped with the theoretical
calculations of R. M. W..̌ B., and contributed in writing the manuscript.

PHYSICAL REVIEW LETTERS 122, 183401 (2019)

Probing the Roton Excitation Spectrum of a Stable Dipolar Bose Gas
D. Petter,1 G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 M. J. Mark,1,2 L. Chomaz,1 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 29 November 2018; published 8 May 2019)

We measure the excitation spectrum of a stable dipolar Bose-Einstein condensate over a wide momentum
range via Bragg spectroscopy. We precisely control the relative strength ϵdd of the dipolar to the contact
interactions and observe that the spectrum increasingly deviates from the linear phononic behavior forincreasing ϵdd . Reaching the dipolar-dominated regime ϵdd > 1, we observe the emergence of a roton
minimum in the spectrum and its softening towards instability. We characterize how the excitation energy andthe strength of the density-density correlations at the roton momentum vary with ϵdd . Our findings are inexcellent agreement with numerical calculations based on mean-field Bogoliubov theory. When including
beyond-mean-field corrections, in the form of a Lee-Huang-Yang potential, we observe a quantitative
deviation from the experiment, questioning the validity of such a description in the roton regime.
DOI: 10.1103/PhysRevLett.122.183401

The spectrum of elementary excitations is a key concept
providing insight into the quantum behavior of many-bodysystems. An emblematic example is the one of superfluid
helium. At low momentum, the interactions among particles lead to collective-excitation modes with a linear
energy (ε)-momentum (q) dependence. Those are known asphonons, highlighting their analogy to sound waves. In
addition, the strong interactions in He induce pronounced
correlations at the mean interparticle distance d. Suchcorrelations reveal themselves in an energy minimum inthe excitation spectrum at q ≈ 1=d, termed roton [1]. Its
physical interpretation, and even its mere existence, hasbeen intensively debated for decades [2]. In today’s understanding, the roton relates to the system’s tendency toestablish a crystalline order [3], possibly providing accessto supersolid phases [4]. Between the phonon and the roton,
a local energy maximum, termed maxon, appears.
For gaseous Bose-Einstein condensates (BECs),
the excitation spectrum also embeds the many-body interacting behavior. In the weakly interacting bulk regime,
the excitation spectrum is well described within the
Bogoliubov
theory andtakes the form εðqÞ ¼
ﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
EðqÞ2 þ 2EðqÞV int ðqÞ, with EðqÞ ∝ q2 being the freeparticle energy and V int ðqÞ being the mean-field interactionenergy contribution [5]. In the case of short-range (contact)interactions, V int is independent of q and a roton minimum
is absent, as confirmed in experiments [6–10]. Deviations
from the Bogoliubov theory were observed in the stronglyinteracting regime [11,12], yet a roton minimum has
remained elusive [13].
Quantum gases with dipole-dipole interactions (DDIs),
underlying a q dependence of V int , bring a paradigm shift inthe many-body behavior [14–18]. In particular, dipolar BECs
0031-9007=19=122(18)=183401(6)(DBECs) are predicted to support a roton mode in their
Bogoliubov spectrum [19,20]. This roton spectrum requires
specific conditions, namely, (i) an anisotropic geometry,tighter along the dipole direction, and (ii) a dominant DDI
over the contact interaction. These conditions enable V int todepend and change sign with q ¼ jqj, yielding a local
minimum in εðqÞ for q along the weak confinement axes.
Conditions (i) and (ii) also dictate the roton mode’s characteristics: its momentum qrot is governed by the confinement
length along the dipoles (i), and εðqrot Þ is controlled by theratio ϵdd ¼ add =as of the dipolar (add ) and s-wave scattering(as ) lengths (ii). In particular, εðqrot Þ decreases (softens) for
increasing ϵdd and ultimately vanishes, yielding a mean-fieldinstability. The existence of dipolar rotons has been demonstrated in recent quench experiments, via the exponential
growth of the roton mode’s population when εðqrot Þ turns
imaginary, i.e., in the roton instability regime [21].
In this Letter, we directly probe the phonon-maxon-roton
excitation spectrum of a stable DBEC of ultracold erbium
atoms. By precisely controlling ϵdd (via as ), we observe theemergence of a roton minimum at large momentum andstudy in detail its softening. Our spectroscopic approach isbased on the well-established technique of Bragg spectroscopy [6–12,22–26]. For DBECs, this technique has beenpreviously applied on Cr in the regime of weak DDI [27],proving the anisotropy of εðqÞ, and consequently of the
speed of sound, recently confirmed with a different
technique with Dy [28]. Bragg spectroscopy has also been
employed to observe rotonlike minima in the dispersion
relations of hybrid systems of short-range interacting atoms
and light [29–31].
Our Bragg spectroscopy is performed using a DBEC ofstrongly magnetic 166Er atoms, prepared as in Refs. [17,21].

183401-1© 2019 American Physical Society

PHYSICAL REVIEW LETTERS 122, 183401 (2019)(a)z

Bragg beam 2

Bragg beam 1

B

l
l

y
x

2π/q
(c) 0

(b)

1

(d)

qx

n(qy )

1energy

l

0
-3.2

qy

(e)

0

3.2

qy (um-1)

6.4

‹qy2›

1

q

0

momentum

0

0.5

1

1.5

2

2.5

( z)

FIG. 1. (a) The DBEC (gray ellipsoid) is axially elongated alongy and the atomic dipoles point along z. Two Bragg beams offrequency ωl and ωl − ω (blue arrows) form a traveling grating
along y with a wave vector q and velocity ω=q (blue shading).
(b) The Bragg excitation drives a stimulated two-photon transition(dashed arrows), transferring a momentum q and an energy ℏω tothe atoms, resonant for ℏω ¼ εðqÞ (solid line). Δ ≫ ω is thedetuning from the intermediate state. (c) Example of nðqx ; qy Þafter a Bragg excitation with ðq; ωÞ ¼ ½1.74ð9Þl−1z ; 2π × 180 Hz.
(d) Corresponding nðqy Þ (dots), fitted with a multi-Gauss function(dashed line). The solid line shows the component of the fit
corresponding to the Bragg-excited atoms. (e) hq2y i vs ω for
q ¼ 0.74ð3Þl−1
z . The solid line shows a Gaussian fit used forextracting ωq and normalizing the data.

After preparation, we confine the DBEC in a cigar-shaped
optical dipole trap with harmonic frequencies ωx;y;z ¼2π × ð261; 27; 256Þ Hz. A homogeneous magnetic field
B maintains spin polarization of the sample in the lowest
Zeeman sublevel, with atomic dipoles aligned along z; see
Fig. 1(a). It also sets the value of as via a magnetic
Feshbach resonance (FR) [32], centered at about 0 G, forwhich the B-to-as conversion has been precisely extractedwith a 2a0 -wide prediction interval in the as range here
explored [17,21]. Systematic uncertainties on as are
estimated to be up to 3a0 [33]. The dipolar length,
add ¼ μ0 μ2 m=12πℏ2 ¼ 65.5a0 , results from the atomic
magnetic moment μ, and mass m of 166Er. μ0 is the vacuum
permeability and ℏ ¼ h=2π the reduced Planck constant.
After preparation, as equals 67a0 , corresponding toϵdd ≈ 1. In this geometry, a roton mode is expected toemerge along the axial (y) direction for ϵdd > 1 and softens
for increasing ϵdd . The roton minimum
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ appears at a
momentum qrot ∼ 1=lz , with lz ¼ ℏ=mωz ≈ 0.5 μm.
To reach ϵdd > 1, we decrease as to the desired value by
ramping B closer to the FR’s pole. The ramping time tr ischosen to be long enough to ensure adiabaticity withrespect to the tight trapping frequencies (tr > 1=ωx;z Þ but
short enough to avoid too strong three-body collisional
losses near the FR. For the highest ϵdd , we find an optimal
trade-off for tr ¼ 15 ms, defining our fastest ramp. Thisramp is not fully adiabatic with respect to the axial dynamics.
We observe small-amplitude breathing and sloshing modes
along y, which we account for in our spectroscopic measurements [33]. After ramping B, we hold the atoms for atime th , after which we perform Bragg spectroscopy to probe
the excitation spectrum of our DBEC of N atoms.
Our Bragg spectroscopy setup is illustrated in Fig. 1(a)and detailed in Ref. [33]. In brief, it uses two coherent laser
beams of wave vector kL ¼ 2π=λL, with λL ¼ 401 nm,
propagating in the z-y plane and intersecting each other
under an angle θ. At the cloud’s position, the beams form alight grating along y of potential depth V 0 and wave vectorq ¼ 2kL sinðθ=2Þ. The two beams have a small frequency
difference ω, causing the grating to travel at a velocity ω=q.
A key feature of our setup is the wide dynamical tunabilityof θ. This is obtained by creating the Bragg beams
using holographic gratings [33,35], generated with adigital micromirror device [30]. By uploading different
holograms, we can vary θ, and accordingly q from 0 to
1.8l−1z . Moreover, by employing hologram sequences andchanging their display rate, ω can be directly varied, up to∼2π × 1 kHz. In the experiment, we illuminate the DBEC
with a Bragg pulse of duration τ. The value of τ ¼ 7 ms ischosen to be long enough to minimize Fourier broadeningof the frequency spectrum, and yet short with respect to aquarter of the axial trap period [8,23,25]. Immediately after
the pulse, we switch off the trap and let the cloud expandfor 30 ms. We then image the atoms along z via standard
absorption imaging, from which we extract the momentum
distribution of the cloud, nðqx ; qy Þ.
The Bragg excitation can be interpreted as a stimulated
two-photon transition, imparting a well-defined momentum
q and energy ℏω to the atoms; see Fig. 1(b). In bulk
systems, for a fixed q and varying ω, atoms, initially at
qy ¼ 0, are resonantly transferred to qy ¼ q for ℏω ¼ εðqÞ[6–9]. When accounting for finite-size effects, the responseis broadened in q. The dynamic structure factor, whichquantifies the system’s response to an external perturbation,
can be related to the fraction of excited atoms during a
Bragg pulse, F ¼ N exc =ðN 0 þ N exc Þ. Here, N 0 (N exc ) is thenumber of the zero-momentum (Bragg-excited) atoms. In
the linear response regime [9,22,23],F¼

π 2 V 20 τS̃0 ðq; ωÞ;h2ð1Þ

where S̃0 ðq; ωÞ is the zero-temperature dynamic structure
factor, Fourier broadened in ω due to the finite τ, see
Ref. [33].
Figure 1(c) shows a representative nðqx ; qy Þ for a q ≳ l−1z
excitation. For this high q, the Bragg-excited atoms are well

183401-2

PHYSICAL REVIEW LETTERS 122, 183401 (2019)
dd

<1

dd

(c)

(e)

‹qy2 ›

(g)

1.5

1.5
1

/

z

2.0 (a)

>1

1.0
0.5

0.5
0
2.0 (b)

0

(d)

(f)

norm.

(h)

~
S0(q,

)

1.5

/

z

3

2

1.0

1

0.5

0

0

0.4

0.8 1.2
qlz

1.6

0

0.4

0.8 1.2
qlz

1.6 0

0.4

0.8 1.2
qlz

1.6 0

0.4

0.8 1.2
qlz

1.6

FIG. 2. Excitation spectra from ϵdd < 1 to ϵdd > 1: (a), (c), (e), (g) Measured hq2y i for varying q (columns, delineated by whitetick marks) and ω at given as . Each column is fitted with a Gaussian function and renormalized by the fitted peak amplitude.
Black columns are inaccessible to measurements [33]. (b), (d), (f), (h) Extracted εðqÞ (white dots) from (a), (c), (e), (g), respectively.
Here and throughout the Letter, the error bars denote  one standard deviation. The solid lines are guides to the eye, based on theanalytic formula from Ref. [24]. The color map shows the calculated S̃0 ðq; ωÞ, normalized by the maximum of S̃0 ðq; ωÞ at qlz ¼ 1.3
and as ¼ 82a0 . For [(a), (b); (c), (d); (e), (f); (g), (h)], N ¼ ½4.6ð5Þ; 3.9ð4Þ; 3.3ð3Þ; 2.5ð3Þ × 104 and as ¼ ðas exp ; aths Þ¼½ð80.0; 82.0Þ; ð60.5; 62.5Þ; ð55.3; 54.5Þ; ð52.5; 51.6Þa0 , respectively.

resolved as a side peak. From a multi-Gauss fit to theintegrated density nðqy Þ, we extract F ; see Fig. 1(d) [33].
For q ≲ l−1
z , the zero-momentum peak and the Braggexcited one overlap and F cannot be precisely extracted. To
access S̃R0 ðq; ωÞ for all q, we use the momentum variancehq2y i ¼ nðqy Þq2y dqy , which relates to the imparted energy
into the system. As F , hq2y i gives access to S̃0 ðq; ωÞ, but via
a more complex relation [5,22,23,33,36]. Figure 1(e)exemplifies a resonance in hq2y i when varying ω at fixedq. We extract its center frequency ωq via a Gaussian fit. By
varying q over the experimentally accessible range, weprobe the lowest-lying branch of the axial excitation
spectrum εðqÞ ¼ ℏωq [33].
Figure 2 shows the results of our Bragg measurements,
revealing how εðqÞ is modified when tuning from ϵdd < 1
to ϵdd > 1. For ϵdd < 1, εðqÞ shows a linear dependence
over the whole q range, characteristic of phonon modes,
Figs. 2(a) and 2(b). From a linear fit to εðqÞ, we estimate thesound velocity c ¼ limq→0 εðqÞ=q ¼ 1.01ð1Þ mm=s alongy. As we probe the system for increasing ϵdd > 1, we findan overall reduction of the excitation energies and increasing deviations of the spectra from the linear phonon
behavior, Figs. 2(c) and 2(d). When further increasing
ϵdd , the spectrum starts to flatten at large q, Figs. 2(e) and
2(f). Ultimately, at the highest ϵdd , we observe a local
minimum occurring at q ≈ qrot ¼ 1.27ð6Þl−1
z , providing an

unambiguous signature of the existence of the roton mode,
Figs. 2(g) and 2(h). At intermediate momenta between thephonon and roton regimes, a maxon [local maximum inεðqÞ] is also identifiable. Because of optical constraints on
our Bragg setup, the maxon regime is not fully accessible;
see black region in Figs. 2(e) and 2(g). To compare ourmeasurements with theory, we perform calculations ofS̃0 ðq; ωÞ, by calculating the Bogoliubov modes from theGross-Pitaevskii equation (GPE) linearized around equilibrium at the final as [24,33,37]. Here we explicitly do notinclude beyond-mean-field effects [33]; see later discussion. Over the entire range of ϵdd , our theory describes
the experimental data, both qualitatively and quantitatively.
In the calculations of Fig. 2, we let as vary within theprediction interval (2a0 ) of our B-to-as conversion to best
match the measured spectrum.
To get a deeper insight into the roton softening, we
perform Bragg measurements at a fixed q ¼ qrot and extractωq , denoted ωrot , as a function of as for fixed N. As shownin Fig. 3, ωrot exhibits a reduction that becomes increasingly sharp for decreasing as . Below 52a0, we observe thatthe system undergoes a roton instability, i.e., a spontaneous
population of the roton mode even without applying a
Bragg pulse; see also Ref. [21]. We find that the softeningof ωrot is well approximated by an as -power-law scaling.
By fitting the data to ωrot ðas Þ ¼ Aðas − as Þp , we extract
the critical scattering length at which ωrot vanishes,

183401-3

PHYSICAL REVIEW LETTERS 122, 183401 (2019)
dd

>1

dd

<1

dd

2.5
25

0.7

10
0

15

0.6

0.5

20

40 60
V0 /h (Hz)

45

80

1.5

1

0.5
0.4
0.3

(arb. units)

0.8

2

rot )

1

30

20

~

res (%)zres (%)50

S0(qrot ,

1.5

0

>1

10

52
53
54
55
*
a
s
50
55
60
65
70
75
80
scattering length as (a0 )0.5
50FIG. 3. Measured ℏωrot with N ¼ 2.4ð2Þ × 104 vs as (circles)
and corresponding numerical calculations excluding (solid line)
and including (dotted line) beyond-mean-field effects. The
shadings show the calculations over the prediction interval ofas . The dashed line shows the power-law fit to the experiment.
Inset: Enlargement around as comparing ℏωrot (circles) to ℏωm(triangles), respectively, with the power-law fit and a guide to theeye (dashed lines). The as region where 0 ≤ ℏωrot ≤ ℏωm ishighlighted with a white background. The error bars in ℏωrot;m(as ) are the fit’s statistical uncertainties (uncertainties fromexperimental B-field fluctuations [33]).

as ¼ 51.9ð2Þa0 , matching our instability observation. We
also observe a scaling exponent of p ¼ 0.27ð2Þ (A is ascaling coefficient). The pronounced dependence of theroton energy on the interparticle interactions, i.e., on bothas and the atomic density, makes the measurements at low
energy very sensitive to fluctuations. Indeed, small fluctuations and drifts in B and N can already drive the system
into instability, eventually preventing a reliable measurement of the spectrum for ωrot ≲ 2π × 100 Hz; e.g., seehorizontal error bar in the inset. For comparison, we
additionally probe the as dependence of the excitationenergy ωq near the maxon at q ¼ 0.74ð3Þl−1
z , denoted ωm .
We observe that ωm decreases much slower than the rotoncase. As shown in the inset of Fig. 3, the two modes’
energies cross around as ¼ 52.8a0 . For as < as < 52.8a0,
ωrot < ωm , showing the emergence of a local minimum inthe spectrum of a stable DBEC. At as ¼ 52.2ð2Þa0 , theminimum can be distinguished with a confidence level of
98% [33].
Figure 3 also shows ωrot extracted from our numericalcalculations, together with its variation within the prediction interval of as . The theory describes our observation
very well and confirms the rapid variation of the rotonenergy with as . We have also performed calculations
including beyond-mean-field effects in the form of a
Lee-Huang-Yang correction in the GPE [38–41]. Thisadditional term has proven to be crucial to understanding

52

54
56
58
scattering length as (a0 )60

FIG. 4. Measured F res with N ¼ 2.4ð2Þ × 104 and V 0 =h ¼
42 Hz (circles) and calculated S̃0 ðqrot ; ωrot Þ (solid line) vs as . The
shading shows the theory over the prediction interval of as . Inset,
measured F res (circles) vs V 0 at as ¼ 52.2a0 . A quadratic fit up
to V 0 =h ¼ 50 Hz (dashed line) shows deviations to the data forhigher V 0 (dotted line).

the behavior of a DBEC in the droplet regime [17,18,
42–44]. Interestingly, the agreement between theory andexperiment becomes worse with a discrepancy that cannot
be accounted for with the experimental as uncertainty. Sucha discrepancy can have several origins. These range fromadditional experimental uncertainties (e.g., N values,
effects of residual density-dependent dynamics [33]) tomore fundamental reasons. As speculated in Ref. [21], thismismatch could call into question the validity of standard
treatments of beyond-mean-field effects in the roton
regime. For instance, the standard inclusion of a LeeHuang-Yang term in the GPE relies on a local density
approximation and is justified for negligible quantum
depletion and higher-order corrections [11,40–51]. These
conditions might not be completely fulfilled in the roton
regime. Future theoretical efforts, combined with stringent
validity tests on experiments, are needed to shed light on
this important aspect.
The emergence of a roton minimum intrinsically connects to an increase of density-density correlations. This isquantified by the amplitude of S̃0 ðqrot ; ωrot Þ [7,23,24,33],
which is related via Eq. (1) to the fraction of excited atoms
at the Bragg resonance F res . In the experiment, we explore
this aspect by measuring F as a function of ω at q ¼ qrot
using a fixed V 0 . From a Gaussian fit to the data, we extract
F res . We repeat the experiment for various as in the ϵdd > 1
regime; see Fig. 4. When approaching the roton instability(as ), F soars, with an increase by about a factor of 3 whenchanging as by less than 15%. Such a behavior is also
confirmed by our theoretical calculations. Experimentally,
we find that the linear-response regime, i.e., the validity ofEq. (1), extends up to F res ≈ 25%; see inset of Fig. 4.

183401-4

PHYSICAL REVIEW LETTERS 122, 183401 (2019)
In conclusion, we measure the excitation spectrum of a
DBEC and its evolution from the contact-dominated regime
to the dipolar-dominated regime. In the latter regime, we
observe the emergence of a roton minimum. Comparisonswith theory reveal a good agreement with mean-field
Bogoliubov calculations and show deviations when including beyond-mean-field corrections, calling for further
studies of their effects and their treatment in the roton
regime. Similar to the cases of superfluid helium [4,52–54]
and of hybrid systems of atoms and light [55,56], the roton
minimum may provide a path for the creation of supersolidor crystalline phases in DBECs [4,28,57–59]. With theachievement of a precise knowledge and control of theroton softening, our work provides a first step in thisdirection.
We thank B. Blakie, S. Stringari, and L. Santos for
inspiring discussions and fruitful comments on our manuscript. Additionally, we acknowledge D. Baillie and
F. Dalfovo for stimulating discussions, S. Burchesky forhelp during initial offline tests, and S. Baier for theassistance during the early stage of the experiment. We
acknowledge Swarovski Optik (S. Kroess and his team) for
technical support on processing optics for the setup. Part ofthe computational results presented have been achieved
using the HPC infrastructure LEO of the University of
Innsbruck. This work is financially supported through an
ERC Consolidator Grant (RARE, No. 681432) and a DFG/FWF (FOR 2247/PI2790).

*

Corresponding author.
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183401-6

Supplemental Material: Probing the roton excitation spectrum of a stable dipolar
Bose gas
D. Petter,1 G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 M. J. Mark,1,2 L. Chomaz,1 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: April 8, 2019)

Bragg
pulse

th

14
σy,ToF (um)

PREPARATION AND CALIBRATION OF THE
dBECs

12

The number of atoms in the probed dBEC, N , is extracted from time-of-flight (TOF) measurements, performed using the same experimental sequence as for the
Bragg measurements, but, instead of applying the Bragg
pulse, simply waiting th + τ /2 before releasing the atoms
from the trap. We extract the integrated density distribution from standard absorption-imaging technique after
30 ms of TOF. We fit a two-dimensional bimodal function made of a Gaussian and an inverted parabola at thepower 3/2 to the density distribution. The values of N
reported in the main text corresponds to the number ofatoms in the parabolic peak. We note that N typicallyfluctuates by up to 10% between experimental runs. In
addition, by measuring N at th and at th + τ , we observe
three-body losses during the Bragg pulse up to 20% for
the lowest values of as . Finally, we point out that thebimodal function employed to extract N is an approximate description of a finite temperature BEC and maylead to an underestimation of N , especially at our lowest
as values. The experimentally calibrated N and ωx,y,z
are used as fixed parameters in the theory calculations;see below.

108vat (100 um/s)

A dBEC of 166 Er is prepared in the same way as described in Refs. [1, 2]. Trapping is provided by crossed
optical beams forming a harmonic potential V (r) =m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )/2 for the atoms. At the endof the preparation procedure, V (r) has a cigar-shaped
geometry with ωx,y,z = 2π × (261, 27, 256) Hz. The frequencies are measured via exciting and probing either
the center-of-mass oscillation of dBECs (for ωx and ωz )or the breathing mode of cold, thermal samples (for ωy ).
The uncertainties of the trapping frequencies are at the
few-percent level. After reshaping the trap, we ramp as
linearly from as = 67 a0 to its final value in a time tr , byperforming a corresponding ramp in B, computed fromthe calibrated B−to−as conversion [1, 2]. The ramp time
is chosen to be relatively long, tr ≥ 15 ms; see main text.
In our Bragg spectroscopy measurements, we apply the
Bragg pulse after an additional holding time th .

2
1
0
-1
-2
0

10

20

30
th(ms)

40

50

60

FIG. S1.
Example of calibration measurements of thebreathing and sloshing modes for the measurements at as =52.5 a0 of Fig. 2 (g, h) of the main text. We record the variations with th of the cloud’s size (upper panel) and position in
TOF. From the position we compute the mean atomic velocity vat (th ) (lower panel). The solid lines are sinusiodal fits tothe data. We time the Bragg pulse (orange shaded area) tobe centered on the cloud’s size maximum. The corresponding mean atomic velocity is in this case vat = −68(16)µm/s.
Similar measurements and analyses have been performed foreach measurement reported in the main text.

ACCOUNTING FOR THE BREATHING AND
SLOSHING MODES: TIMING AND DOPPLERSHIFTS

Our fastest as -ramps are not fully adiabatic with respect to the axial dynamics and may induce smallamplitude breathing and sloshing modes along y. Such
excitations could affect our Bragg measurements. The
former mode induces density oscillations and can influence the value of the roton excitation energy. The latter mode causes a sizable momentum-dependent Doppler
shift of the Bragg excitation frequency [3]. We accountfor these effects in our experiment by performing dedicated calibration measurements. In particular, we probe
the evolution of the atomic density distribution after
TOF, as a function of th . We perform such calibration
measurements for each as -ramp employed in the experi-

2

BRAGG SETUP

Our Bragg spectroscopy setup is illustrated inFig. S3 (a, b). It employs a digital micromirror device (DMD), DLP-V9500 from Vialux with 1920 ×1080 micromirrors. The DMD features a programmablemirror area, consisting of 10.8 × 10.8 µm-sized micromirrors that can be individually tilted in one of two directions. Depending on the mirror’s tilting direction, theincoming light is reflected either into the Bragg spectroscopy setup or on a beam dump. We illuminate themirror area with a single frequency laser beam of wavelength λL = 401 nm. The beam has an elliptic shapewith waists of (wz , whor ) = (10, 1) mm. Here, wz (whor )denotes the beam’s waist in the z (horizontal) direction;
see Fig. S3 (c). The beam is sent on the DMD’s mirror area under an angle of ∼ 25◦ with respect to its
perpendicular axis, fulfilling the condition for a blazed
grating. This ensures a maximum diffraction efficiency

0.9
0.8
0.7
ωq /ωzment.
The breathing excitation reveals itself in the evolutionof the axial size; see Fig. S1 (upper panel). To minimize
the impact of the breathing mode on our measurements,
we synchronize the Bragg pulse symmetrically around themoment at which the size in TOF reached its maximum.
Then, the in-situ density of the dBEC changes the leastand remains close to its highest value during probing.
The corresponding th , after which we switch the Bragg
beams on, is typically between 10 and 20 ms.
The sloshing mode reveals itself in the variation ofcenter-of-mass position of the atomic cloud. This gives
direct access to the mean velocity, vat (th ), of the atoms
in the dBEC as a function of th ; see Fig. S1 (lower panel).
By averaging over the duration of the Bragg pulse, vat =hvat (th )iτ , we extract the induced Doppler shifts for the
Bragg excitation, ωD = vat q which we then use to correct
the applied Bragg frequencies ω. To check the accuracyof our treatment, we have repeated Bragg spectrscopy
measurements at various as and q using Bragg pulses
corresponding to distinct ωD . In particular, to achieve
distinct ωD , we reversed the Bragg excitation direction
to compare measurements with ±q and used pulses starting at different th , yielding vat ≈ {−vmax , 0, vmax }, vmax
being the maximum insitu mean velocity. A set of suchmeasurements is exemplified in Fig. S2, where we show
both the uncorrected and corrected resonance frequencies ωq . The good agreement of the Doppler-correctedvalues proves the validity of our approach. All data reported in the main text are Doppler-corrected.
We stress that the value of ωD increases with q. As
an example, ωD /2π varies from 15 Hz to 40 Hz for qvarying from 0.74 lz−1 to 1.74 lz−1 in the measurementsof Fig. 2 (h). In the analysis of our data, it has thus been
important to carefully account for this effect.

0.6
0.5
0.4
0.3

- qrotvat= 133 µm/s

- qrotvat= -9.9 µm/s

+ qrotvat= 133 µm/s

FIG. S2. Bragg resonance frequencies ωq at |q| = qrot andas = 52.7 a0 measured using three different Bragg pulses,
characterized by the couple {q, vat } (abscissa’s labels). For
each measurement, both the uncorrected (diamond) andDoppler-corrected (circle) frequencies are shown. The dashedline indicates the mean of the Doppler-corrected resonancefrequenies.

of the incoming beam into the beam path of the Bragg
spectroscopy setup.
Following Ref. [4], the general idea is to use binary
holograms that represent maps of titling directions forthe micromirrors of the DMD. By placing the DMD in
the Fourier-plane of the atoms, the holograms allow for
both amplitude and phase modulation of the laser beamat the atoms’ position. They consist of an underlying
binary grating with two Gaussian envelopes separated
by a distance d on the DMD; see Fig. S3 (d). The Gaussian envelopes cut out two beams from the incoming one.
Additionally, the envelopes correct for the local intensity
inhomogeneities of the incoming beam. After the DMD,
the two beams travel parallel with a distance d between
each other before being focused in a first optical telescope; see Fig. S3 (b). Due to the binary grating structure of the holograms, each beam splits at the telescope’s
focus point into a 0th order and ±1st side orders. The focus point is used to let only the +1st order of each beampass by filtering out the other ones. The two remaining +1st order beams constitute our two Bragg beams
and have a similar Gaussian profile as well as similar intensities. After the telescope they are reflected down to
our experimental chamber where a last lens focuses themunder an angle θ onto our atomic cloud. At the focus
point, matching the atom’s position, the beams create
an interference pattern with a wavevector q along y. By
uploading a hologram with a different d one can change θand thus q in an almost continuous manner. We note that
due to optical constraints in the experiment (not shownin Fig. S3), we can not create interference patterns in therange of q = [0.4 − 0.7] lz−1 .

3incoming beamx

DMD
mirror area(a) Top view~25°

ydumped(e)filterdown to1 : 4 telescope

vacuumchamber(b) Side view
DMD
mirror area

z

25 um

+1st orders

qlz=1.65d
0th and -1st orders

y
(c)

whor

(d)
f = 200 mmwz

ω=dφ/dtΘφ
qlz=1.01

viewportdatomic cloudφ/2�=0φ/2�=3/9...
qlz=1.65 qlz=1.01 qlz=0.74

φ/2�=6/9...
qlz=0.27

FIG. S3. Setup for Bragg spectroscopy: (a) Top view of the Bragg spectroscopy setup, showing the beam path of the incoming
beam (dashed arrow), the beams travelling in the Bragg-spectroscopy setup (solid arrow), and the dumped part of the incoming
beam (dotted arrow). (b) Side view of the beams travelling in the Bragg spectroscopy setup when a holographic grating patternis displayed on the DMD (see text for details). (c) Sketch of the elliptic beam shape of the incoming laser beam on the DMD.
(d) Examples of binary holograms uploaded on the DMD that allow to create two beams in the Bragg spectroscopy setup,
travelling parallel separated by a distance d. The three smaller images on the right show a zoom of one part of a hologram witha phase shift of the underlying binary grating, resulting in three distinct phase difference, ϕ, of the Bragg beams. (e) Exampleof light interference patterns at the position of the atomic cloud, obtained from offline calibrations with a CCD camera.

Furthermore, the relative phase, ϕ, of the two beam’s
wavefronts is directly related to the phase of the appliedbinary grating on the DMD. It thus allows us to introduce a frequency difference ω = dϕ/dt by displaying a sequence of holograms during the Bragg excitation, where
the phase of one of the two Bragg beams is constantly
shifted in time; see Fig. S3 (d). Practically, we set a sequence of nine holograms, which defines a phase revolution of 2π, and display it in a loop with a fixed rate,
γ. This results in ω = 2πγ/9. The discrete phase stepsof 2π/9 are sufficient to not suffer from higher frequency
harmonics in our measurements. We note that γ is limited by the maximal refreshing rate of the DMD. Furthermore, the inherent dark time of the DMD at eachhologram update results in a decrease of the average intensity of the light grating when increasing γ. In theexperiment, we compensate for this effect by increasing
the intensity in the Bragg beams to maintain a constantV0 .
We further note, that the binary grating in our amplitude holograms take phase aberrations of the optical setup into account and corrects for them [4]. These
corrections are obtained from offline calibrations witha CCD camera and greatly improve the beam pointingof the individual Bragg beams on the atomic cloud, as
all lenses in the optical setup are spherical singlets. In
Fig. S3 (e), we show three example images of final interference patterns, obtained with a CCD camera during

4
1
0.8

ωq /ωzoffline calibrations.
As the Bragg spectroscopy setup uses amplitude holograms, typically only ∼ 0.1% of the incoming laser lightis used for the Bragg pulse. We take advantage of thestrong transition of erbium at 401 nm. First it allows awide tuning of V0 via the frequency detuning, ∆, of thelaser light from the atomic resonance. Second, its shortwavelength also leads to a higher maximum q for a fixedmaximum θ as compared to longer laser wavelengths.
In the measurements presented in the main
manuscript, the detuning ∆ = 2π × 71(1) GHz ischosen such that we achieve suitable depths of the
Bragg potential (typically V0 ∼ h × 10 − 100 Hz),while spontaneous light scattering remains negligibleon the experimental time scale. We extract V0 via the
Kapitza-Dirac-diffraction technique [5]. We note that
this approach neglects the inhomogeneity of the atomic
cloud over the wavelength of the interference pattern
and the interactions in the system.

0.6
0.4
0.2
0.8

1

1.2

qlZ

1.4

1.6

1.8

FIG. S4. Resonance frequencies at as = 52.5 a0 obtained
from hqy2 i-analysis (cirlces) and F-analysis (diamonds). In
the latter case, we do not report on values for qlz < 1.1 as F
can not be reliably extracted.

IMAGE ANALYSISTo probe the system’s response to the Bragg pulse,
we image the atomic cloud after a TOF expansion of
30 ms. As described in the main text, we perform absorption imaging along the z direction. During the first
15 ms of the TOF, the B-field is kept constant to avoid
any sudden change of the dipolar or contact interactions
when the atomic density is high. We then set the B-field
to B = 0.3 G and then rotate its direction to obtain amaximal imaging-light scattering cross-section and constant imaging conditions. Assuming ballistic expansion,
we obtain the mean momentum distribution n(qx , qy ), by
averaging typically four individual images; see Fig. 1 (c)in main text. Due to slight variations of the cloud’s position from shot to shot, each single image is recenteredby extracting the cloud’s center from a two-dimensional
Gaussian fit. In order to obtain the momentum distribution along the excitation direction of our Bragg
pulses, we numerically integrate n(qx , qy ) along qx from[−4.5, +4.5] µm−1 and obtain n(qy ) (1 pixel in our imaging corresponds to ∼ 0.32 µm−1 ). To extract information
on Se0 (q, ω) from n(qy ), we measure either the fraction of
excited atoms, F, or the momentum variance, hqy2 i, as
introduced in the main text and detailed below.
The procedure used to extract hqy2 i depends on q. For
qlz > 0.7, we use an asymmetric region of interest (ROI)
ranging from qy = [−1.9 µm−1 , c̃ q], reflecting the factthat the Bragg excited atoms occur around qy = q > 0.
The factor c̃ varies between [2.5, 4.5] in order to accountfor the change in the cloud’s momentum width with as(increasing for increasing as ). At low momenta qlz < 0.5,
the excited fraction of atoms lies completely within theunscattered peak and only a broadening of the atomic
cloud on resonance is observed. Therefore, we choose asymmetric ROI from qy = c̃phon [−1.6, +1.6] µm−1 , wherec̃phon is varied between [1, 3] for different as . We furthermore note that for qlz < 0.2 we are not able to extract a reliable signal in our measurements, as a potential
broadening of the atomic cloud on resonance can not be
resolved.
In order to extract the excited fraction F, we use a bigger ROI that includes the full thermal fraction of atoms
and fit a three-Gauss function. The individual Gaussian distributions account for the unscattered atoms inthe dBEC N0 (centered at qy ≈ 0), the excited atom inthe Bragg excitation Nexc (centered at qy ≈ q) and thebroad thermal background. The center positions of theGaussian distributions for the unscattered atoms and thethermal background are kept the same. The center for
fitting the excited fraction is limited to [0.95, 1.05] q. F
is then defined as F = Nexc /(N0 + Nexc ), thus discarding
the initial (thermal) population at q and focusing on themere fraction of atom promoted during the Bragg pulse.
We extract ωq and Fres by a Gaussian fit to the resonances in hqy2 i and F for varying ω and fixed q. For
too low ω, the discrete phase steps of our holograms
do not provide a well-defined excitation energy over thetimescale of the 7-ms Bragg excitation. Hence, in our
analysis, we discard points at 0 < ω/2π ≤ 40 Hz (corresponding here to the non-Doppler-corrected frequencies; the 0 Hz-case is the static case). Nevertheless, wenote that an inclusion of these points do not alter the
extracted resonance frequencies within their uncertainties. Comparing the hqy2 i and F analysis, we also verify
that both analysis procedures give the same resonancefrequencies within their uncertainties; see Fig. S4).

5(a)SCATTERING-LENGTH VALUE AND ITSUNCERTAINTIES

(b)

(c)

0

CONFIDENCE LEVEL OF THE EXISTENCE OF
A ROTON MINIMUM

In order to confirm the existence of a local minimumin the excitation spectrum of our dBECs, we compare
the maxon with the roton energy and extract a confidence level from a statistical analysis. First, we focus
on a scattering length value of as = 52.5(2) a0 for which

-0.1

0.5

∆ (ωz )

In our experiments we control the contact interaction
as by means of a magnetic Feshbach resonance, centeredclose to 0 G [6]. From previous lattice-spectroscopy measurements, where we probe the excitation gap in the Mott
insulator regime [1], we have obtained a precise mapping of the as -to-B conversion. In those measurements,
the statistical uncertainty on as has an average value ofs̄ = 1.8 a0 coming from the uncertainty on the resonancefrequency of the Gaussian fit to the spectroscopic data.
From a fit to the as -data, we obtained a precise B-toas conversion function for B ranging from 0 to 3 G [1].
For as ranging from 80 a0 down to 51 a0 (B from 2.1 G
to 0.21 G), our conversion function yields a confidence
interval of width c̄ varying from ± 0.9 to ± 1.3 √a0 . This
results into a prediction interval of width p̄ = c̄2 + s̄2varying from ± 1.9 a0 to ± 2.1 a0 , which defines our conversion uncertainty. In addition we estimate the systematic uncertainty to be ±3 a0 with a dominant contribution coming from the uncertainty on the depth of the lattice potential in the spectroscopic measurements (which
crucially determine the on-site Wannier function’s shapeand thus the conversion of the resonance frequency value
into as ).
Besides the conversion and systematic uncertainties,
statistical uncertainties on as arise from magnetic field
fluctuations and drifts in our experiments. For eachdataset we perform independent magnetic-field calibrations by performing radio-frequency (RF) spectroscopy
on cold thermal clouds with a 1-ms RF pulse. Here weuse the same experimental B ramp as for the Bragg measurement and apply a RF pulse of 1 ms duration, either
after holding a time th or a time th +τ . The mean value ofthese two measurements is used to extract as . Furthermore, it probes the change of B over the Bragg pulse,which can be up to 3 mG. Additionally, we have independently estimated that B fluctuates up to ±2 mG. In
total, we consider a B uncertainty of ±2.5 mG, which weconvert in an as uncertainty based on the B-to-as conversion formula. This can be up to ± 0.2 a0 , which isthe case for our lowest as . In conclusion, in the relevant
regime for this work, the statistic, conversion and systematic uncertainties on as are of ± 0.2 a0 , p̄ ∼ ±2 a0 ,and ±3.2 a0 , respectively.

ω (ωz )

0.6

-0.2

0.4
0.74 1.27
qlz

0.74 1.27
qlz

52.2 52.5
aS (a0)

FIG. S5. Analysis on the extracted resonance frequencies ωm(triangles) and ωrot (circles) for 52.5(2) a0 (a) and 52.2(2) a0(b). In (a, b) filled symbols show data from Fig. 3, whereas empty symbols show data points from Fig. 2 (h). Error
bars represent ± one standard deviation of the corresponding
Gaussian fits. (c) shows the corresponding differences ∆ (after
averaging for 52.5 a0 ) together with its uncertainties, deduced
from standard error propagation.

two sets of data are available (from Fig. 2 (h) and inset
of Fig. 3). From both resonance frequencies ωm and ωrot ,we obtain the corresponding mean values ω̃m and ω̃rot .
Calculating the difference ∆ = ω̃m − ω̃rot = −0.08(5)ωzreveals the existence of a roton minimum with a 93%
confidence level; see Fig. S5 (a,c). The existence of a roton minimum in the spectrum is even more evident byanalyzing ∆ = ωm − ωrot at 52.2(2) a0 , where the minimum is deeper. Here, we find ∆ = −0.15(7)ωz giving a
confidence level of 98% for the existence of a minimun in
the spectrum of a stable dBEC; see Fig. S5 (b,c).

THEORY

To compare our experiment with theory predictions,
we perform numerical calculations of the dynamic structure factor following the procedure detailed in the supplementary information of Ref. [2]. The calculations are
based on a Bogoliubov treatment of an extended GPEwith energy functional ĤGP [ψ], for which our 166 Er
dBEC is the ground-state. The classical field ψ describes
the macroscopic wavefunction of N atoms and is normalized to N . The dBEC state, |0i, corresponds to the
wavefunction ψ0 . The Bogoliubov analysis gives access,in second-order perturbation, to the discrete modes, |li,of the dBEC’s excitation spectrum, to their energies h̄ωl
and the Bogoliubov spatial amplitudes ul and vl .
In the present calculations, we can in principle accountfor the effect of the quantum fluctuations, by includ-6
ing the Lee-Huang-Yang term ∆µ[n] = 32g(nas )3/2 (1 +√232dd /2)/3 π in ĤGP [ψ] (here g = 4πh̄m as and n = |ψ|2 ).
∆µ[n] has been obtained under a local density approximation [7, 8] when computing the Bogoliubov modes.
The relevance of the inclusion of such a potential correction has been demonstrated in various studies of dipolar
Bose gases close to the mean field instability [1, 2, 9–12]. However, as described in the main text, a better
agreement with experimental data, close to the instability, is found instead by omitting the Lee-Huang-Yang
term. ∆µ[n] is then only included in ĤGP for computing
the dotted line and the corresponding yellow shading in
Fig. 3 of the main text. It is discarded from all othertheory calculations reported in the main text.
The knowledge of the Bogoliubov modes allows one tocompute the bare zero-temperature dynamic structure
factor S0 (q, ω), which is defined as [13, 14],X2
hl|δn̂†q |0i δ(ω − ωl ),(1)
S0 (q, ω) =lwith δn̂q being the density fluctuation operator in momentum space:Zδn̂q = dr eiq·r ψ̂ † (r)ψ̂(r) − h0|ψ̂ † (r)ψ̂(r)|0i , (2)and ψ̂ is the field operator. The matrix elements of thedensity fluctuation operator are computed asZ
hl|δn̂†q |0i = dr[u∗l (r) + vl∗ (r)]eiq·r ψ0 (r).
(3)
Considering Eq. (1), one sees that S0 (q, ω) consists of infinitely narrow peaks centered around ω = ωl . The integrated amplitude of each peak corresponds to the contribution of the mode l to the quantum density fluctuationsof the dBEC at momentum q.
For our experimental probing, the relevant quantityeish the Fourier-broadened istructure factor, S0 (q, ω) =τ sinc2 (τ ω 0 /2) ∗ S0 (q, ω 0 ) (ω), where ∗ denotes a con-volution over ω 0 . This ultimately writes
X
2
Se0 (q, ω) =hl|δn̂†q |0i τ sinc2 (τ (ω − ωl )/2).

(4)l

Se0 (q, ω) shows the same peaks in frequency as S0 (q, ω)
but broadened with a typical width 1/τ . The amplitude of Se0 (q, ω) on resonance matches the contributionof the mode |li to the quantum density fluctuations of
the dBEC at momentum q, multiplied by the Bragg pulse
duration, τ .
CONNECTION BETWEEN MEASUREDQUANTITIES AND DSF

In our experiment, we probe the dynamic structure
factor either via the fraction of atoms excited from thedBEC peak at qy = 0 to the Bragg peak at qy = q or
via the momentum variance along the y axis. Following
Refs. [13, 14], we derive the relations of our observables
to Se0 (q, ω). The occupation of each mode after the pulse
is given by (Ref. [14], Eq. (2.31)):hNl (t = τ )i − hNl (t = 0)iN
π 2 V02 τ
2hl|δn̂†q |0i τ sinc2 (τ (ω − ωl )/2),
=
2
h

Fl =

(5)
(6)

with hNl (t)i being the mean number of atoms in the model at time t (t = 0 matching the beginning of the Bragg
pulse). Equation (1) ofPthe main text is then found bysimply summing F = l Fl and using Eq. (4). We notethat in our data analysis, the thermal (i. e. initial) population at qy = q is encompassed in the broad background
Gaussian and thus excluded from the definition of F,
similarly to Eq. (5).
For the momentum variance, the situation is more
complex. Assuming a fully ballistic expansion and a linear perturbation regime, h̄2 hqy2 i/2m matches the energy
transferred during the Bragg pulse. For each mode, theenergy transferred during the pulse writes Fl N h̄ωl . Using Eq. (6), one finds
hqy2 i − hqy2 i0 =4π 3 mV02 τ N X
2
hl|δn̂†q |0i τ ωl sinc2 (τ (ω − ωl )/2). (7)3hlwhere hqy2 i0 is the value of hqy2 i for the dBEC (of N atoms)
in absence of a Bragg pulse (at t = 0) and typically depends on the value of as . Because of the multiplicationof ωl in the sum, hqy2 i can only be related to Se0 (q, ω)
approximately. When only one mode contributes significantly to Se0 (q, ω), at a given (q, ω) one can write
hqy2 i − hqy2 i0 ≈8π 4 mV02 τ N ε(q) e
S0 (q, ω).
h4(8)Note that, for a fixed q, ε(q) is a constant multiplying theoverall amplitude but not affecting the peak position in ω.
We highlight again that, in the experiment, we observe nosignificant difference in the extracted ε(q) when analyzing
F or hqy2 i; see Fig. S4.

[1] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Phys. Rev. X 6, 041039(2016).
[2] 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).
[3] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. StamperKurn, D. E. Pritchard, and W. Ketterle, Phys. Rev.
Lett. 82, 4569 (1999).

7[4] P. Zupancic, P. M. Preiss, R. Ma, A. Lukin, M. E. Tai,
M. Rispoli, R. Islam, and M. Greiner, Opt. Express 24,
13881 (2016).
[5] P. L. Gould, G. A. Ruff, and D. E. Pritchard, Phys. Rev.
Lett. 56, 827 (1986).
[6] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga,
Rev. Mod. Phys. 82, 1225 (2010).
[7] A. R. P. Lima and A. Pelster, Phys. Rev. A 84, 041604(2011).
[8] A. R. P. Lima and A. Pelster, Phys. Rev. A 86, 063609(2012).
[9] F. Wächtler and L. Santos, Phys. Rev. A 93, 061603(2016).
[10] F. Wächtler and L. Santos, Phys. Rev. A 94, 043618(2016).
[11] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,
Phys. Rev. A 94, 033619 (2016).
[12] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut,
and T. Pfau, Nature (London) 539, 259 (2016).
[13] A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari, andF. Zambelli, Phys. Rev. A 64, 063614 (2001).
[14] P. B. Blakie, R. J. Ballagh, and C. W. Gardiner, Phys.
Rev. A 65, 033602 (2002).

Chapter

3

Supersolidity

This chapter gives an overview of supersolidity in a dipolar quantum gas. In particular,
Sec. 3.1 reviews the basic properties of a supersolid state with emphasis on its first predictions, and early experimental search. Section 3.2 focuses on the dipolar case with the aimof providing the ingredients for realizing such a counter-intuitive state. Section 3.3 explains
the excitation spectrum of the supersolid state and how to measure it. Sec. 3.4 compares the
dipolar supersolid state to other supersolids. Finally, Sec. 3.5 to 3.7 present the three related
publications. In the first one, our group reported on the observation of supersolidity. In sucha work, our ERBIUM team joined the force with the Dy team in our group and reported theobservation of supersolid states with both atomic species and uncovered the phase diagram
as a function of the strength of the contact interactions. In the second publication, theexcitation spectrum of the supersolid state is theoretically investigated and probed in theexperiment. Such a work, performed with Er, shows the appearance of two distinct branchesof excitation, each one related to the spontaneous breaking of a symmetry. Finally, the third
publication studies the dephasing and rephasing process after an interaction ramp from asupersolid state to an insulating droplet crystal and back.

3.1 Historical overview
A supersolid is a state of matter with two spontaneously broken continuous symmetries:
the translational and the gauge symmetry. The broken translational symmetry signifies the
emergence of a crystalline structure in space with a defined phase of the modulation. The
gauge symmetry, which is typical of Bose-Einstein condensates, signifies a global phase coherence. Such a system is quite counterintuitive since a supersolid state combines superfluid
flow with crystal localization.
The search for supersolid states can be traced back to 1908. It was the 10th of July, when
“Leiden briefly became the coldest place on earth”[vD08]. On this date, Heike Kamerlingh
Onnes liquefied helium for the first time. The experiment produced just about “a teacup” ofliquid He at −269 ◦ C, but suddenly the path for studying superconductivity and superfluidity
was set.
43

44

3 Supersolidity

Only three years later, Omnes’s group produced a superconductor in the same laboratories
using helium as a refrigerant. They discovered that by cooling mercury at about 3 K, theresistivity of mercury drops to zero [Kam11, vD10]. This incredible discovery put a little aside
another quantum phase transition that happened during the same experiment. When cooling
helium below 2.2 K, helium suddenly stopped boiling. It highlights the transition to a phasewith high heat conductivity, hence the transition to the superfluid state. The discovery of
the properties of superfluid helium will only happen almost 20 years later. In the meantime,in 1926, Willem Hendrik Keesom (a former student of Onnes) discovered that, by increasing
the pressure to a value of at least 25 atm, liquid helium performs a phase transition to thesolid phase [Kee26]. In fact, due to the high zero-point energy, even at zero temperature, it
is not possible to solidify helium at atmospheric pressure [Bea20]. In 1938, Kapitza was able
to discover the superfluidity of He when cooling it below the lambda point [Kap38]. This
point corresponds at room pressure to 2.17 K, and the name lambda point derives from thebehaviour of the specific heat versus temperature that resembles the greek letter. At thislow temperature, the viscosity of helium was negligible. Allen and Misener independently
confirmed this result [All38]. It became clear that 4 He below the transition temperature
behaves like a different liquid: a superfluid. During the same year, London’s work connected
the superfluidity observed in 4 He with Bose-Einstein condensate [Lon38]. From this, Tisza
introduced a two-fluid hydrodynamic description based on the idea that a certain fractionof condensed particles occupies the lowest energy state [Tis38]. The fact that no lambdatransition was present in liquid 3 He and that many experiments verified the prediction of thetwo-fluid model, see for e.g., [Pes48, And71b], finally confirmed this assumption. However,
the theoretical approach from Tisza and London raised two strong objections. First, thetheoretical model considered 4 He a perfect gas, despite the fact that instead He is a stronglyinteracting fluid. Second, the mechanism that prevented the momenta exchange between thecondensed particles with the excited ones was not clear. Therefore, Landau and Bogoliubov
would have expected friction. These shortcomings led Landau to consider the elementarycollective excitations to explain the superfluidity. In 1941, Landau introduced the conceptof roton elementary excitations [Lan41]. He described the energy spectrum as formed by aphononic part at low momenta and a roton part. The phonon part has a linear dispersion
relation, ϵ = cp (c being the sound velocity), while the roton part has a quadratic dispersion
relation, ϵ = ∆ + p2 /2µ (here µ is an effective mass and ∆ the roton gap).
In 1956, Penrose extended the mathematical description of BEC to a system of interacting
particles [Pen56]. The properties and the phase diagram of 4 He became more clear. Whenchanging temperature and pressure, it was clear that there were two different liquid phases
and one solid phase. In this context, Gross attempted a unified theory of interacting bosons
to connect the vibration spectrum of the solid with the excitation spectrum of the liquid
phase [Gro57]. At the beginning of the 60s, the question that inspired the scientists in the field
was: Can a solid be superfluid? It was the beginning of the race for supersolidity. In 1970,
Leggett proposed a possible experiment to probe superfluidity in solid helium [Leg70]. This
experiment consisted in extracting the moment of inertia of a rotating solid He as a functionof the temperature. An observation of a reduction of the moment of inertia with respect tothe classical value, when lowering the temperature below the critical value for superfluidity,
would signalize the transition to the supersolid state, where the superfluid part decouples
from the rotation of the normal component. This is reminiscent of the experiment performedby Andronikashivili to determine the superfluid fraction of liquid helium [And71a].

3.2 Supersolidity in dipolar gases

45

In 2004, Kim and Chan prepared the experiment conceived by Leggett. This consisted
in probing the period of a torsional oscillator, containing solid He, as a function of thetemperature. The period relates to the momentpof inertia, I, and the torsional stiffness ofthe system, K, through the relation: Tosc = 2π I/K [Kim04]. The authors observed thepredicted reduction below the critical temperature for superfluidity. This observation wasquestioned in 2007, when Day and Beamish published a measurement of the increase of theshear modulus of solid He when decreasing the temperature [Day07]. This work, followedby another work in which the torsional oscillation period and shear modulus were probed
simultaneously, suggested a different interpretation for the observed behaviour [Kim12]. In
particular, the observed reduction of the torsional oscillation period was not related to a
reduction of the moment of inertia, but rather caused by a change in the shear modulus,
whose temperature behaviour arises from the temperature dependence of the crystal defects
motion.
In fact, the properties of solid helium are extremely different than standard crystals. Evenat zero temperature, the atoms in the crystal can tunnel between different lattice sites
and have a non negligible exchange-interaction term. This arises from the large zero-point
energy, due to the small mass and inter-particle spacing [Bea20]. Even if the theoreticalresults were promising and showed that the defects in solid 4 He could lead to superfluidity,
experimental proof of supersolid 4 He is still missing at the date. The research on solid helium
is fascinating but goes beyond the aim of the current thesis, and for that, we refer the reader
to e.g., [Bea20].
Nevertheless, supersolids were found about 50 years after Leggett’s thought experimentin ultracold quantum gases, thanks to the control and understanding reached over theyears in these platforms. To date, supersolids have been observed in different systems,
owning momentum-dependent interactions, such as systems with cavity-mediated interactions [Bau10], spin-orbit coupled BECs [Li17], and dipolar atoms [Tan19a, B1̈9b, Cho19]. In
the rest of the chapter, we will mainly focus on the latter system, and briefly comment on
its peculiarity with respect to the other platforms.

3.2 Supersolidity in dipolar gases
Even if the research on supersolids with 4 He did not give clear results, the understanding ofhelium was fundamental to the cold-atom community. In particular, Schneider associatedthe superfluid-solid transitions of 4 He to the softening of the roton mode [Sch71, Noz04].
It was confirmed by measuring the reduction of the roton gap obtained by increasing thepressure, see Ref. [Sud77]. Additionally, the softening of the roton mode was also linked toa route towards a supersolid from Kirzhnits and Pomeau [Kir71, Pom94].
Let us now consider some differences between the two systems. Superfluid 4 He and trapped
atomic gases correspond to two limiting cases of quantum fluids [Dal01]. In the case of superfluid 4 He, the system is in the liquid phase. Here, the density is about one-eighth the densityof liquid water at atmospheric pressure. The average interatomic distance (dHe ) is approximately equal to the interaction range (aHe ), dHe ≈ aHe ≈ 0.3 nm. The system is strongly

46

3 Supersolidity

interacting. Therefore, the condensate fraction is about 10% even at zero temperature. The
excitation spectrum presents a roton mode due to the strong correlations, whose wavelength
is comparable with the interparticle distance. Hence, when it softens, it provides a standard
crystallization with one atom for lattice site.
In the case of dipolar gases, an excitation spectrum developing a roton minimum was predicted in two cases (and geometries) qualitatively very different from each other. For twodimensional dipolar bosons oriented perpendicularly to the plane, with zero s-wave scattering
length, a roton minimum appears when increasing the particle density. This roton originates,
similar to the case of helium, from the strong correlations, and its wavelength is approximately the inter-particle separation, resulting in one atom per lattice site. However, even athigh density close to the solidification point, the roton energy gap is finite [Ast07, Maz09].
In the case of dipolar gases in three-dimensional traps, which is the case of interest here,
the system is dilute, and the roton modes come from the interplay between the short-range
contact interaction, the momentum-dependent long-range interaction, and the trapping geometry, see Chap.2. Additionally, the roton wavelength in the case of erbium or dysprosium
is much larger than the interparticle distance. Therefore, it leads to a dilute crystal withthousands of atoms in each lattice site. Table 3.1 summarizes those parameters, comparing
helium droplets to the typical experimental parameters for a BEC of erbium atoms investigated in this thesis.
Typical parameters for 4 He and 166 Er
Quantum fluid

4 He

166 ErInteraction range≈ 0.3 nm

≈ 3 nmInterparticle spacing≈ 0.3 nm

≈ 200 nmDiluteness na3s

≈ 0.3

≈ 2 × 10−6

≈ 0.1 nm

≈ 400 nmHealing length ξ = [8πna]−1/2

Table 3.1: Comparison between typical parameters of 4 He and a BEC of 166 Er atoms investigatedin this thesis.

We note that, theoretical works predicted the appearance of a supersolid phase also in
systems interacting via a soft-core potential, where more particles arrange in clusters. Sucha potential can be experimentally realized through Rydberg excitation [Cin10, Hen10, Anc13,
Mac14].

3.2.1 Early hints of supersolidity
The conceptual development that led to the discovery of supersolidity in magnetic atoms hasdeveloped along two parallel lines: on the one hand, the prediction and later observation of
the roton excitation in dipolar gases [San03, Cho18], and on the other hand, the identification

krot

k

8

14

(b)

12

6

(c)

10

4

3

y

Energy ϵ(k)xzn (10 m-1 )(a)

47
Renormalised roton population

3.2 Supersolidity in dipolar gases

2

8
6
4
2

0

0

1

2
3
5
4
Hold time t (ms)

6

0

-10

-5

0
5
y ( m)

10

Figure 3.1: Softening of the roton mode, growth of the roton population and groundstate. (a) Illustration of the excitation spectrum of a dipolar BEC for two different scattering
lengths. (b) Evolution of the roton population (renormalized to their 3 ms value) after a quench toas = 50a0 . The solid brown line shows the theoretical prediction from a simple analytic model not
accounting for quantum fluctuations, three-body-loss processes, and finite temperature. Those areincluded in the red-dashed line representing the results of numerical simulations, see Ref. [Cho18].
(c) Integrated ground-state density profile for the parameter of (b) showing the modulated supersolidstate for 166 Er with 6×104 atoms. Trapping frequencies and scattering length for (b,c) are: ωx,y,z =2π × (114, 32, 149) Hz, as = 50a0 . Adapted from Ref. [Cho18].

of a quantum stabilization mechanism seeded by the quantum fluctuations [FB16, Cho16].
These two lines interweaved many times and, although their link to dipolar supersolidity isclear nowadays, it was not the case during the first investigations.
In 2003, the presence of the roton mode was predicted in a BEC with dipole-dipole interactions. On the one hand, in Ref. [O’D03], the dipole-dipole interaction is engineered via
atom-light interaction by shining a far off-resonant laser beam onto the atoms. On the other
hand, in Ref. [San03], the roton arises from the momentum dependence of the long-range
anisotropic interaction of dipolar gases.
In the meantime, Bose-Einstein condensates of dysprosium [Lu11], and erbium [Aik12] became available, and in 2016, the group of Tilman Pfau in Stuttgart, using a dipolar BEC
of dysprosium atoms confined in a pancake geometry, observed that the system developed
a crystalline structure when decreasing the contact interaction [Kad16]. Surprisingly, thelifetime of this state was not compatible with the mean-field picture of a collapse. This
state, even if not showing a global phase coherence, was opening a fundamental question: Why is the system stable? Several works proposed 3-body repulsive interactions as
a possible mechanism for stabilizing the system, see Refs. [Lu15, Xi16, Bis15]. Inspired by
Ref. [Pet15], where the quantum fluctuations are proposed as a stabilization mechanism fora mixture with attractive mean-field interactions, the experiments done with Dy and Er,
in Refs. [FB16, Cho16] supported the idea of beyond mean-field interactions as responsible for the stable droplet states. Theoretical works followed soon after and confirmed theexperimental observations [W1̈6b, Bis16, W1̈6a].
Thereafter, our group using erbium atoms confined in a cigar-shape geometry, with dipolesorientated orthogonal to the cigar, studied the response of the system to a quench of thescattering length to lower values [Cho18]. The main results of this experiment include:

48

3 Supersolidity
 The first experimental observation of the roton mode.
 A study of the roton geometrical scaling and its softening.

Figure 3.1 illustrates the softening of the roton mode (a) and the associate grow of the roton
populations from Ref. [Cho18]. Interestingly, our group observed that, while it is possible
to interpret the early evolution of the roton population, after the interaction quench, as an
exponential growth dictated by the imaginary gap, the growth in populations saturates andstabilizes after a few milliseconds, see Fig. 3.1(b). This observation is not compatible with apure mean-field approach.
In a subsequent work, we mapped via Bragg spectroscopy the excitation spectrum before its
softening, see Sec. 2.6. While this work confirmed the mean-field theory predictions of theroton minimum and its related increase in density-density correlations, did not provide an
additional hint to understand why after the roton instability the system’s response was nota collapse. It seems that a fundamental piece was missing. Also, in the first version of thepaper [Cho17], the authors write: “the high shot-to-shot repeatability of the remarkable peakstructure in ñ(kx , ky ) suggests the persistence of a macroscopic phase coherence in the gas.”,and conclude: “These density modulations, providing a signature of the roton softening inreal space, are a precursor of a supersolid phase, in which a phase-coherent density modulated
ground-state would arise [Pom94]. Although the density modulation is expected to be meanfield unstable against local collapses [Kom07], quantum stabilization may prevent collapse as
for the case of recently explored quantum droplets [W1̈6b, FB16, Cho16, W1̈6a, Bis16]”.
Soon after, Roccuzzo and Ancilotto, revising this experimental paper on the roton softening,but in a tubular confinement with periodic boundary conditions, found a supersolid groundstate [Roc19]. After the publication of the manuscript, we performed the ground-state calculations for our finite-size system, and confirmed that, for some of the parameters, theground state is a supersolid. Figure 3.1(c) shows the calculation of the integrated density
profile of the ground state for the parameters of Fig. 3.1(b). The theoretical and experimental efforts exponentially increased in the following years. Finally, in 2019, by merging
the two concepts of roton softening and quantum fluctuations, the groups of Giovanni Modugno in Pisa [Tan19a], Tilman Pfau in Stuttgart [B1̈9b], and our group in Innsbruck [Cho19](with two different experiments) independently confirmed a state with supersolid properties,
namely the observation of a density-modulated state that has global-phase coherence.
Note that, while all those works claim supersolid properties, there are some differences. In
particular, we can group the work performed with Er in Ref. [Cho19] with the work performed
in Refs. [Tan19a, B1̈9b]. In all these works, the supersolid state is obtained via a ramp of thescattering length from a standard BEC phase and the lifetime observed was ≈ 20 ms. This
raised two questions: first, whether the phase coherence was reminiscent of the standard
BEC phase, and second if a long-lived stationary supersolid could be obtained. Differently,
in Ref. [Cho19], we include the work performed in our group with 164 Dy, where we reached
the supersolid via evaporative cooling from a thermal gas, and observed a lifetime above
100 ms.

3.3 The excitation spectrum of a dipolar supersolid

49

3.2.2 Probing a supersolid state
Let us now focus on how to probe a supersolid state. In particular, the properties of density modulation and global phase coherence. The presence of a density modulation can bedetected either by probing the density distribution of the gas in trap (in-situ imaging) or byprobing the momentum distribution of the cloud after a long expansion time (TOF imaging).
For the result presented here, the ERBIUM experiment did not have yet a high resolution
objective to probe a density modulation with a wavelength of about 2 to 3 µm1 . Figure 3.2(c)shows a typical ground-state density profile for an 166 Er supersolid optically confined in acigar-shaped three-dimensional trap. Here, we see about three leading density peaks withan interpeak spacing of ≈ 2.5 µm and a total sistem size of ≈ 20 µm. Figure 3.2(a), shows
instead the equivalent calculations for the ground-state density in the case of an infinite tube,
where the confinement is not applied along the modulation direction.
In our experiment, after evaporatively cooling the dipolar gas to BEC, we reshape the BEC
in a cigar-shape confinement and linearly ramp in 20 ms the scattering length to the desired
value, corresponding to a supersolid state. After an additional 10 ms of holding time, werelease the cloud from the trap and observe the momentum distribution after TOF expansionof about 30 ms. In the BEC regime, the momentum distribution shows the standard bimodal
distribution. In the modulated phase, we find an interference pattern with a central peak
and two side peaks. The separation between these peaks reflects the wavelength of the insitu
density modulation. To probe the supersolid nature of the state, we take several pictures
under the same experimental parameters. If the system has global phase coherence, thepicture, averaged over the many repetitions, preserves the interference pattern. Whereas,
if the system is in an insulating droplet regime, while the single pictures still show aninterference pattern, the latter gets washed out in the average. To get a quantitative analysisof the modulation contrast and of the coherence, we perform a Fourier-transform analysis of
the interference pattern. A detailed description of the analysis is reported in the publicationin Sec. 3.5 at the end of the chapter.
Figure 3.2(a,c) allows a first consideration of the effect of the finite system. The supersolidstate obtained in the experiments presents 3 to 6 density peaks. Due to the presence of theharmonic confinements, the ground state is not degenerate to a phase variation of the density
modulation. For each configuration, there are only two metastable states, corresponding to
the case of odd or even number of density peaks, that have similar energy. The difference
between the two configurations is the presence or not of a density peak in the center of theharmonic confinement, see e.g. Fig.1 in the publication in Sec.3.6.

3.3 The excitation spectrum of a dipolar supersolid
The excitation spectrum provides information about the properties of the system. For example, the linear dispersion at low momenta relates to the speed of sound. The Landau criteriumvc ≤ mink ε(k)ℏk gives an upper bound on the critical velocity for superfluidity. Additionally,1

A high-resolution objective has been recently installed in the experiment and will be used in the future.

50

3 Supersolidity

1

(a)

(c)

0.5

0
-4

-2

0

20

0

40

2

4

80 DSF

60

-10

-5

0

(b)

0

20

40

5

10

80 DSF

60

(d)

1.5

1

0.5

0
0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

Figure 3.2: Ground-state density and excitation spectrum for an 166 Er supersolid. Integrated ground-state density profiles for an 166 Er supersolid with 5×104 atoms for an infinite system(a), and in the case of a harmonic confinement (c). Relative excitation spectra for the infinite(b) and trapped (d) case. Blue and red dashed lines correspond to the crystal and phase branch,respectively (see discussion in Sec. 3.3). Colorbar represents the dynamic structure factor. Trapping frequencies for (a,b) are: ωx,y,z = 2π × (250, 0, 160) Hz. Trapping frequencies for (c,d) are:
ωx,y,z = 2π × (250, 31, 160) Hz. Adapted from Ref. [Pet20].

quantity like the dynamic structure factor, introduced in Sec. 2.2, directly gives informationon the response of the system to perturbations and on the strength of the density-density
correlations. Finally, the excitation spectrum reflects the broken symmetries in the system.
A spontaneous symmetry breaking reveals itself in the excitation spectrum with the appearance of an excitation branch [Gol61, Sac12, Mac13, Roc19]. In the case of a supersolid, we
observe two branches recalling the crystal and the superfluid nature of the system. Starting from a standard dipolar BEC, when reducing the contact interaction, the roton softens
and the ground state acquires a density modulation with a periodicity close to the roton
wavelength. At the phase transition, the spectrum presents an additional branch of nearly
degenerate modes. When reducing further the scattering length, the contrast of the density
modulation increases and one of the branches start to soften in energy (red-dashed line in
Fig. 3.2(d)). This branch is associated with modes involving a superfluid flow between thedensity peaks. Differently, the second branch (blue-dashed line in Fig. 3.2(d)) slightly stiffens
in energy. This branch is associated with crystal modes, with mass currents only inside the3.3 The excitation spectrum of a dipolar supersolid

51

individual density peaks, e.g. breathing or oscillation of the density peak position. Additionally, some of the modes have a mixed character, this is expected in our system since DDI
correlates the density with the position of the peaks.
In the publication presented in Sec. 3.6, we detailed the evolution of the excitation spectrum
and its experimental measurements from a standard superfluid BEC to the supersolid phaseand in the independent droplets (ID) regime, for our experimental parameters. Here, we focus
on the difference between the finite and the infinite case. Figures 3.2(b,d) show the excitation
spectrum associated with the supersolid ground-state density profile in (a,c), respectively.
In (a) the cigar is infinitely long, whereas in (b) the harmonic trap is present also along thedirection of the modulation.
Figures 3.2(b,d) show how the presence of the confinement induces a discretization in theenergy of the allowed Bogoliubov modes, as described in Sec. 2.2. Additionally, the finite size
leads to a broadening in the momentum response of the modes. This makes the two branches
less visible than in the infinite case. This effect is particularly evident in Fig. 3.2(c,d) where
the ground state has only a few density peaks and the two branches have similar energies.

3.3.1 Exciting a dipolar supersolid state
In addition to the observation of density modulation and global phase coherence, the existence of two branches are a further proof of supersolidity [And69, Pom94, Sac12, Mac13].
In our experiment, after studying the roton excitation spectrum via Bragg spectroscopy, it
could have been natural to apply the same technique to resolve the two branches of thesupersolid state. However, to resolve two branches of the excitation spectrum that arecomparatively close to each other, it is necessary to constrain the mechanisms of energy
broadening. For instance, the pulse duration of the Bragg spectroscopy induces an energy
broadening of each Bogoliubov mode (Fourier broadening in energy ≈ h × 150 Hz), see
Eq. 2.26 and discussion in Sec. 2.5. This makes it harder to resolve the two branches using
this technique. As detailed in Sec. 2.5, the pulse duration needs to be shorter than thequarter period (T/4) of the trap along the modulation axis (Bragg direction). Even reducing
the trapping confinement, to increase T/4 and hence the duration of the pulse, would not
significantly help in resolving the two branches due to the short lifetime of the erbium
supersolid states. In fact, in Er, the supersolid state is lost due to three-body losses on atime scale of 20 ms, which set an upper limit to the pulse duration. We note that in our
dysprosium experiment, we found a regime of parameters where the supersolid is stable,
living several hundreds of ms [Cho19, Soh21], opening the way to use Bragg spectroscopy toresolve the two branches in the excitation spectrum of the supersolid.
For the aforementioned reason, we used a different approach to excite the supersolid. After
preparing the state, we induce collective excitations, such as a quadrupole mode, by suddenly
changing the confinement potential.
We then probed the system in TOF after a variable holding time. For large values of thescattering length, where the system is in the BEC phase, this scheme excites a quadrupole

52

3 Supersolidity

mode. However, in the supersolid regime, several modes, of the two branches, can be excited,
providing complex dynamics. The next step was to develop a method to distinguish thedifferent excitation modes, i.e. the ones related to the crystal branch and the ones belonging
to the phase branch. To this aim, we have adapted the so-called principal component analysis(PCA) to our supersolid state, as detailed in the next section.

3.3.2 Principal component analysis
The principal component analysis is a machine-learning algorithm. It is widely used indifferent fields, see e.g., Ref. [Moo01], where it is used for Face-Recognition, or Ref. [Kot13],
where the authors classify electroencephalographic signals. In the field of ultracold atoms,
it was successfully used for processing atomic interferometric data in Ref. [Seg10]. The useof the PCA was further extended in Ref. [Dub14] to observe and classify the elementary
low-energy collective excitations of a quantum degenerate gas of 87 Rb atoms.
Similarly, in our group, we use the PCA to decompose the complex dynamics of the excited
supersolid in the modes of the Bogoliubov theory. Below, the steps for the PCA are summarized. We select a portion of our image of 71*71 pixels from the total ROI. This selection
is centered on the atomic position by a Gaussian fit. We rewrite all our images as a vectorρs (i). Here i and s are two indexes associated with the image number (i ∈ [1, M ] , M is thenumber of images), and the pixel number (s ∈ [1, P ] , P is the number of pixel), respectively.
The average images are then calculated as follows:ρ̄(s) =MX

ρi (s)/M.

3.28

i=1

From this, we can derive for each pixel the fluctuations of the image compared to the averagedone:δρi (s) = ρi (s) − ρ̄(s).

3.29

We then compute the correlations between two pixels (s, s∗ ) from our set of images:∗

Cov(s, s ) =

M
X
i=1

δρi (s)δρi (s∗ )/ (M − 1) .

3.30

This is usually referred to as the covariance matrix. We then diagonalize this matrix. The
eigenvectors are images called the principal component Cs . In our case, the dataset contains
images for different holding times. From the scalar product of a given principal component
and the images of the series, we can calculate the weight evolution of the component:
ws,i =PXs∗ =1

Cs (s∗ )ρi (s∗ ).

3.31

3.4 Other supersolids and where to find them

53

From the weight and Cs we can recover each image of the initial sample, as follows:ρi (s∗ ) = ρ̄(s∗ ) +

P
X
s=1

ws,i Cs (s∗ ).

3.32

Once the covariance matrix is diagonalized, one needs to identify the principal components.
The population of the BdG modes affects the dynamics of the state and, starting from
Eq. 2.13, we can write the time evolution for the density:Xn(r, t) ≈ n0 (r, t) + 2
ηδρl (r) cos (ωl t + ϕl ) ,3.33
lwhere we neglect quadratic terms in η. Here, the first term is the ground-state density andδρl = (ul + vl∗ ) |ψ0 | are the density fluctuations associated to the mode l.
The similarity between Eq. 3.33 and Eq. 3.32 are evident and one can apply a sine fit to
extract the oscillation frequency from the weight ws,i . By directly comparing the frequenciesof the Bogoliubov mode ωl from GPE to the one obtained from the weight, we can correctly
assign the mode.
The comparison of the principal components Cp (s) to the Fourier transform of the density
fluctuations δρ confirms the assignment. We performed the Fourier transform since, while
Eq. 3.33 considers the in-trap density distribution, the principal components Cp (s) derivefrom the experimental pictures taken after TOF expansion.

3.4 Other supersolids and where to find them
As mentioned in Sec. 3.1, the experimental search for supersolidity started in solid helium.
Differently from other crystals, the small mass together with the atomic interactions allows helium to have a large zero-point energy. This causes the atoms to have tunnelingand exchange interactions even at zero temperature [Bea20]. A fascinating prospect is theBose-Einstein condensation of defects in solid helium. At this date, unambiguous proofs ofsupersolid helium are missing, but the research in the field is very active. Recently, different groups suggested superfluid flow related to dislocations to explain their results, see e.g,
Ref. [Ray09, Che16, Hal19]. While in solid helium there is limited control on interactions
and defects in the crystal structure, such control is nowadays possible in quantum-gases
platforms.
The tunability offered by ultracold atoms made them a unique candidate to demonstrate
this exotic phase. In particular, supersolid properties were probed in three different ultracold platforms. The first observation comes from the seminal work in 2010 in the group of
Tilman Esslinger [Bau10]. In this work, a supersolid state is created in a 87 Rb BEC thanks
to the coupling to external cavities. Later on, in 2016 the group of Wolfgang Ketterle useda spin–orbit-coupled Bose-Einstein condensate to realize the supersolid state, see Ref.[Li17].

54

3 Supersolidity

This phase was also studied by the group of Ian B. Spielman, who confirmed the coherence between the spin components [Put20]. The next paragraphs briefly discusses these two
platforms.
Cavity-mediated long-range interactions can be engineered in ultracold systems using external cavities [Dom02, Nag08, Bau10]. In this system, a pump laser controls the interaction
strength that can be tuned via its power and detuning. Similar to the dipolar case, themomentum-dependent interaction can cause the softening of a roton-type mode [Mot12]. By
increasing further the interaction, the system enters into a phase with density modulationand off-diagonal long-range order [Bau10]. The breaking of the continuous translational
symmetry was experimentally observed by measuring a high degeneracy in the phase of
the crystal modulation [Léo17]. The resulting supersolid state is rigid, which leads to theabsence in the excitation spectrum of the branch corresponding to low-momentum phonon
excitations. However, even if modes associated to a change of the modulation wavelength areabsent, the one associated to a change in modulation amplitude or phase were experimentallyprobed [Lé17].
In a spin–orbit-coupled system, two components, usually two spin states, are Bose-Einsteincondensed and coupled via a Raman laser [Lin11]. This leads to a momentum-dependentinteraction that induces a roton minimum in the dispersion relation. In the supersolid phase,
also known as the stripe phase, the period of the density modulation is the wavelength of
the Raman laser with a shift that grows quadratically with Raman coupling [Mar21]. Notethat by increasing the Raman coupling one induces stiffening of the roton mode. The phaseof the modulation is set both by the relative phase between the Raman lasers and thepre-existing relative phase between spin components. While the simultaneous presence oflong-range coherence and the spatial density modulation of Spin-Orbit-Coupled Bose gases
was experimentally explored [Li17, Put20], the lattice-phonon mode remains unobserved.
However, the spectrum was studied in theory and two gapless branchs are present [Mar21,
Gei21, Gei22]. The modes in the lower branch correspond to spin excitations, while the one
in the upper branch to density modes [Gei21]. Hence, also oscillations of the fringe spacingare possible [Gei22]. Nevertheless, as pointed out in Ref. [Put20], even in absence of Raman
coupling, the system is a two-component mixture that exhibits two broken symmetries.
Therefore, no additional branch appears in the supersolid phase.
Supersolidity is not a phenomenon restricted to these platforms, but these systems sharesome common properties. For instance, the roton mode and its softening are present in allthese three platforms together with the presence in the excitation spectra of two or moreGoldstone modes. This section focused on the aforemoentioned three platforms, but othersystems are under investigation and could demonstrate in the future supersolid properties,
see e.g., [Bon12]. We note that, a roton-maxon excitation spectrum has also been observedin another platform, namely in quantum gases in shaken optical lattices [Ha15].

3.5 Publication: Long-lived and transient supersolid behaviors in dipolar quantum gases.55

3.5 Publication: Long-lived and transient supersolid behaviors in
dipolar quantum gases.
Physical Review X 9, 021012 (2019)a
submitted 11 March 2019; published 19 April 2019
DOI: 10.1103/PhysRevX.9.021012

L. Chomaz,1 D. Petter,1 P. Ilzhöfer,1,2 G. Natale,1 A. Trautmann,2 C. Politi,2G. 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
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2a

The author of the present thesis developed the experimental procedures and performed the measurements together with L. C., D. P., P. I., A. T., C. P., G. D., ran the theoretical calculations together
with R. M. W. v. B. and contributed in writing the manuscript.

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

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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].

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[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
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[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
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6k
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1
0 -4 -2 0 2 4ky(µm-1)

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5d
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5h
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5l
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y (µm)

F[n]

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x104

5b
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5f
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5j
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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
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0
5g
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-4 -2 0 2 4ky(µm-1)

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5 d
4
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y (µm)

F[n]

n(µm)

n(µm)

4
3
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5 f
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1
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-6 -4 -2 0 2 4 6
y (µm)

4c

F[n]

5b

F[n]

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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]

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-6 -4 -2 0 2 4 6
y (µm)

F[n]

n(µm)

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6k
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0 -4 -2 0 2 4ky(µm-1)

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F[n]

n(µm)

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n(µm)

n(µm)

n(µm)

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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

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3.6 Publication: Excitation spectrum of a trapped dipolar supersolid and its experimental
evidence.
77

3.6 Publication: Excitation spectrum of a trapped dipolar
supersolid and its experimental evidence.
Physical Review Letters 123, 050402 (2019)a
submitted 3 July 2019; published 1 August 2019
DOI: 10.1103/PhysRevLett.123.050402

G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 D. Petter,1 , M. J. Mark,1,2 L. Chomaz,1and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria

2a

The author of the present thesis developed the experimental procedures and performed the measurements together with A. P., D. P., ran the theoretical calculations together with R. M. W. v. B. andcontributed in writing the manuscript.

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
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Excitation Spectrum of a Trapped Dipolar Supersolid and Its Experimental Evidence
G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 D. Petter,1 M. J. Mark,1,2 L. Chomaz,1 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 3 July 2019; published 1 August 2019)

We study the spectrum of elementary excitations of a dipolar Bose gas in a three-dimensional anisotropic
trap across the superfluid-supersolid phase transition. Theoretically, we show that, when entering thesupersolid phase, two distinct excitation branches appear, respectively associated with dominantly crystal
and superfluid excitations. These results confirm infinite-system predictions, showing that finite-size
effects play only a small qualitative role, and connect the two branches to the simultaneous occurrence of
crystal and superfluid orders. Experimentally, we probe compressional excitations in an Er quantum gas
across the phase diagram. While in the Bose-Einstein condensate regime the system exhibits an ordinary
quadrupole oscillation, in the supersolid regime we observe a striking two-frequency response of thesystem, related to the two spontaneously broken symmetries.
DOI: 10.1103/PhysRevLett.123.050402

Supersolidity—a paradoxical quantum phase of matterthat combines crystal rigidity and superfluid flow—was
suggested more than half a century ago as a paradigmatic
manifestation of a state in which two continuous symmetries are simultaneously broken [1]. In a supersolid, thespontaneously broken symmetries are the gauge symmetry,associated with the phase coherence in a superfluid, and thetranslational invariance, signalizing crystalline order. The
striking aspect is that, in a supersolid of indistinguishable
bosons, the same particles are participating in developing
such two apparently antithetical, yet coexisting, orders.
Originally predicted in quantum solids with mobile bosonicvacancies [2–4], the search for supersolidity has fueled
research across different areas of quantum matter fromcondensed matter to atomic physics, including quantum
gases with nonlocal interparticle interactions [5–19].
Recent experiments have revealed that axially elongateddipolar quantum gases can undergo a phase transition froma regular Bose-Einstein condensate (BEC), possessing ahomogeneous density in the local-density-approximation
sense, to a state with supersolid properties, where density
modulation and global phase coherence coexist [15–17].
Such experiments, complementing the ones with BECs
coupled to light [20–22], have opened a whole set offundamental questions, covering the very real meaning ofsuperfluidity in a supersolid state, its shear transport, andphase rigidity.
Of particular relevance is the study of the spectrum ofelementary excitations, which governs the system responseto perturbations [23–25]. Typically, phase transitions occur
in concomitance with drastic modifications of the excitation
spectra—as in the case of the emergence of roton excitations in He II or the phononic dispersion for BECs—and
0031-9007=19=123(5)=050402(6)similar dramatic changes are expected when crossing thesuperfluid-supersolid transition. Theoretical studies of uniform (infinite) gases with periodic boundary conditions andsoft-core [26–28] or dipolar interactions [14,29,30] haveshown two distinct branches appearing in the excitation
spectrum of a supersolid state—one for each broken
symmetry. Their coexistence has been identified as an
unambiguous proof of supersolidity, being the directconsequence of the simultaneous presence of superfluidand crystalline orders [2,26,27,31].
An important issue is to understand if these trademarks
survive—and can be measured—in the experimentally
relevant regimes of a finite-size quantum gas, confinedin all three spatial dimensions. In this Letter, we addressthese points by performing full spectrum calculations andby experimentally exciting collective modes in an erbium
quantum gas. Both the theory and experiment show theexistence of two distinct classes of excitations, one connected to crystal modes and the other to phase modes,
providing the finite-size equivalent of the two-branches
spectrum for infinite systems.
In our study, we consider a three-dimensional dipolar
quantum gas confined in an axially elongated (y) harmonic
trap with transverse orientation (z) of the atomic dipoles.
These systems are well described by an extended GrossPitaevskii equation (EGPE), including nonlinear terms,
accounting for contact interactions depending on the scattering length as , the anisotropic long-range dipole-dipole
interaction (DDI), and quantum fluctuations in the form of
a Lee-Huang-Yang type of correction [12,14–17,19,32–36];
see also Ref. [37]. We calculate ground-state wave functions
ψ 0 ðrÞ by minimizing the energy functional resulting fromthe EGPE using the conjugate-gradients technique [46].

050402-1© 2019 American Physical Society

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
(a)

(c)

(e)

(g)

(b)

(d)

(f)

(h)

FIG. 1. Axial excitation spectra of a trapped dipolar quantum gas across the BEC-supersolid-ID phase transition. The trap frequencies
are 2π × ð260; 29.6; 171Þ Hz. The upper (lower) row shows calculations for a 164Dy (166Er) quantum gas of 4 × 104 (5 × 104 ) atoms in
the BEC (a),(b), supersolid (c)–(f), and ID (g),(h) regimes, together with the corresponding ground-state density profiles (insets). (a), (c),(e), and (g) correspond to as ¼ ð92; 91; 90; 81Þa0 , and (b), (d), (f), and (h) to as ¼ ð50.8; 50.5; 50; 48Þa0 , respectively. In (e) and (f), thedashed and dash-dotted lines are guides to the eyes, indicating the two excitation branches. The color map indicates the calculated DSF,
and lz is the harmonic oscillator length along the dipoles’ direction.

As shown in Fig. 1 (insets), the ground state evolves withdecreasing as from a regular BEC (a), (b) to a supersolidstate with axial density-wave modulation (c)–(f) and finallyto an insulating array of independent droplets (ID) (g), (h)[7,14,15,17,27].
The spectrum of elementary excitations is calculated bynumerically solving the Bogoliubov–de Gennes equations,which are obtained from an expansion of the macroscopic wave function as ψðr;tÞ¼½ψ 0 ðrÞþηðul e−iϵl t=ℏ þvl eiϵl t=ℏ Þe−iμt with η ≪ 1 and linearizing the EGPE aroundψ 0 [13,25,46,47]. Here, μ is the ground state’s chemical
potential. By solving the resulting eigenvalue problem,
we find a set of discrete modes, numbered by l, of energy
ϵl ¼ ℏωl and amplitudes ul and vl . We calculate thedynamic structure factor (DSF) Sðk; ωÞ, which informs
on the system’s response when its density is perturbed at agiven modulation momentum k and with an energyℏω [25,48,49]. Whereas in the absence of an external trap
the spectrum is continuous and the DSF is a δ-peakresonance at the Bogoliubov mode ðωl ; kl Þ, the confining
potential yields instead a discretization of the excitation
spectrum and a k broadening in Sðk; ωÞ. For a given energy(i.e., a single mode), finite-size effects may even yieldseveral peaks in k; see, e.g., three-peak structures at large
energy in Figs. 1(a) and 1(b). For the considered parameters, these finite-size effects are more pronounced in Er
than Dy, since the latter exhibits a larger number of maxima
in the density-modulated phases, rendering its excitation
spectrum more reminiscent of the infinite-system case;see Fig. 1.

Figure 1 shows the calculated excitation spectrum forground states in the regular BEC, the supersolid, and the ID
phases for a Dy (upper row) and Er (lower row) quantumgas. In the BEC regime close to the supersolid transition[Figs. 1(a) and 1(b)], the spectrum of excitations shows asingle excitation branch with the characteristic phononmaxon-roton dispersion of a BEC [50–54], as recently
measured [55]. When the roton fully softens (at as ¼ as  ),the ground state becomes density modulated with a wave
number close to the roton one, krot . Here, the excitation
spectrum develops additional structures, marked by theappearance of nearly degenerate modes [Figs. 1(c) and
1(d)]. When lowering as , we find that these modes start toseparate in energy, where some harden and the otherssoften, and two excitation branches become visible[Figs. 1(e) and 1(f)]. This result resembles that of infinite
systems, where the broken translational and gauge symmetry are each associated with the appearance of one
excitation branch [14,26,27]. Additionally, we observe thatthe spectrum acquires a periodic structure, reminiscent ofBrillouin zones in a crystal, with reciprocal lattice constantk ≃ krot . Modes with an energy higher than the maxon(energy maximum at k < krot ) seem to have a singledroplet-excitation character, and they will be the subjectof future investigations. When further decreasing as < as  ,the lower-lying branch decreases both in energy and in DSF
values, whereas the opposite occurs for the higher branch.
Eventually, when reaching the ID regime, the lower branch
progressively vanishes, underlying the disappearance of
global superfluidity [Figs. 1(g) and 1(h)].

050402-2

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
We focus on the properties of the excitation spectrum
in the supersolid regime. The interesting question is
how the two branches relate to the two orders in thesystems, crystal and superfluid. To gain insight, westudy the system’s dynamics when a single mode l is
iμt
excitedwith amplitude
ﬃ η ≪ 1 by writing ψðtÞe ≈pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2−iηδφl sin ωl tjψ 0 j þ 2ηδρl cos ωl te, in terms of density perturbations δρl ¼ ðul þ vl Þjψ 0 j and phase perturbations
δφl ¼ ðul − vl Þ=jψ 0 j. The subsequent time evolution of
the axial density profile is shown in Figs. 2(a)–2(c) for three
relevant cases. For simplicity, only the two extremes of themode oscillation are shown. The mode character can beunderstood by noting that phase gradients correspond tomass currents. Large gradients inside a density peak imply
motion of the density peak [e.g., Fig. 2(a)] and relate to
crystal modes. Large phase gradients between density peaks
signify a superfluid current of particles tunneling from one
density peak to another [e.g., Fig. 2(b)] and are associatedwith phase modes. However, in our system, the phase or
crystal mode classification is not strict, and we find that thesetwo characters mix; see Figs. 2(a)–2(c). Particularly, weobserve both behaviors simultaneously in Fig. 2(c). Such a
mixing is expected from the long-range nature of the DDI,
coupling density, and position of the peaks [26,27]. Note thatthe character of the mode can change with as . For instance,
the mode in Fig. 2(c) develops an almost pure crystal
character for decreasing as . To quantify a mode’s character,we plot in Fig. 2(d) the DSF spectrum at a fixed as , colored
(a)

(b)

(c)

(d)

FIG. 2. Evolution of three different even modes of the system
calculated for 5 × 104 Er atoms at as ¼ 49.8a0 : (a) fourth-,(b) second-, and (c) third-lowest-lying even modes in energy withfrequencies (67.4, 40.3, 49.8) Hz, corresponding to crystal,
phase, and mixed modes, respectively. Each panel shows n ¼jψð0; y; 0; tÞj2 for t ¼ π=2ωl and t ¼ 3π=2ωl with η ¼ 0.15 andthe corresponding δφð0; y; 0Þ. (d) DSF for the same setting as in(a)–(c), where the modes are colored according to their associatedphase (red) or crystal (blue) character via C [37].

according to the ratio C of phase variances inside, andbetween the density peaks [37]. This allows us to differentiate
the dominant character of the two branches, being phase type
for the lower branch and crystal type for the upper one.
To test our predictions, we experimentally study thecollective excitations in an erbium quantum gas across the
BEC-supersolid-ID phases. We prepare a BEC at as ¼ 64a0 .
The atoms are confined in an axially elongated opticaldipole trap of harmonic frequencies 2π × ðνx ; νy ; νz Þ ¼
2π × (259ð2Þ; 30ð1Þ; 170ð1Þ) Hz and polarized along z by
an external magnetic field; see Refs. [13,17]. To probe oursystem, we perform standard absorption imaging after 30 msof time-of-flight expansion, yielding measurements of the
momentum space density nðkx ; ky Þ [37]. Using the tunabilityof the contact interaction via magnetic Feshbach resonances[56], we can prepare the system at desired locations in thephase diagram in the BEC, supersolid, or ID phase by linearlyramping down as in 20 ms to the target value. We then allow
the system to stabilize for 10 ms. At this point, we record an
atom number of typically 5 × 104 for the supersolid regime.
We confirmed the relevant as ranges by repeating the matterwave interferometric analysis of Ref. [17]. While in the BEC
region the momentum distribution shows a regular, nearly
Gaussian single peak, in the supersolid regime the in-trap
density modulation gives rise to coherent interference
patterns along ky , consisting of a central peak with two
lower-amplitude side peaks; see Fig. 3(a).
After preparing the system in the desired phase, we
excite collective modes in the gas by suddenly reducing theaxial harmonic confinement to 10% of its initial value (i.e.,νy ≈ 3 Hz) for 1 ms, before restoring it again. The atomic
cloud is subsequently held for a variable time th , before
releasing it from the trap and recording the time evolutionof nðkx ; ky Þ. As the lifetime of the supersolid state is limited
to around 40 ms [17], we focus on th ≤ 30 ms. As
expected, in the BEC phase, we predominantly observe
an oscillation of the axial width, connected to the lowestlying quadrupole mode [25]. In the supersolid regime, thesituation is more complex; see Figs. 1(c)–1(f). Here,multiple modes, of both crystal and phase character, canbe simultaneously populated, resulting in a convoluted
dynamics of the interference pattern.
We therefore employ a model-free statistical approach,known as principal component analysis (PCA) [57], tostudy the time evolution of the measured interference
patterns at a fixed as . This method has been successfully
used to study e.g., matter-wave interference [58] andcollective excitations [59] in ultracold-gas experiments.
The PCA analyzes the correlations between pixels in a setof images, decomposes them into statistically independent
components, and orders these principal components (PCs)
according to their contributions to the overall fluctuationsin the dataset.
In a dataset probing the system dynamics after an
excitation, the PCA can identify the elementary modes

050402-3

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
(a)

(b)

(c)

(d)

(e)

(f)

FIG. 3. (a) Example of a measured mean interference pattern in
the renormalized central cut of the density distribution nðky Þ forth ¼ 5 ms in the supersolid regime at as ¼ 49.8a0 (filled circles)
and in the BEC regime at as ¼ 51.7a0 (open circles). (b)–(d)PCA results at as ¼ 49.8a0 . (b) Time evolution of the weights of
PC1 (filled circles) and PC2 (open circles) together with their sinefit. Error bars denote the standard error of the mean. (c),(d)Evolution of the partially recomposed nðky Þ accounting for the
population of PC1 (c) and PC2 (d) only. (e),(f) Calculated time
evolution of nðky Þ from excitation of the mode shown in
Figs. 2(b) and 2(c), respectively, and using η ¼ 0.15.

with the PC weights in the individual images exhibiting
oscillations at the mode frequencies [37,59]. We apply the
PCA to the time evolution of the interference patterns after
the trap excitation. Figure 3(b) shows the PCA results in the
supersolid regime at as ¼ 49.8a0 . We identify two leadingPCs, which we label as PC1 and PC2. Their weights
oscillate with different amplitudes and at distinct frequencies, namely, 41(1) Hz for PC1 and 52(5) Hz for PC2. The
comparison between the measured frequencies and thetheoretically calculated mode energies indicates that, following our trap excitation, the second- and third-lowestlying even modes are simultaneously populated. As shownin Figs. 2(b) and 2(c), these modes possess a phase and amixed character, respectively. Note that we apply an overall
shift of −4.3a0 to the as value for the experimental data; formore details, see the discussion in Refs. [55,60].
To visualize the role of each PC on the interferencepattern dynamics, we apply a partial recomposition of theimages, accounting only for the PC of interest; see
Ref. [37]. The effect of PC1 on the axial dynamics isshown in Fig. 3(c), mainly being an axial breathing of thecentral peak, accompanied by weaker in-phase breathing of
the side peaks. Instead, PC2 exhibits a dominant variation

FIG. 4. Comparison between the mode energy obtained fromthe theory calculations and the energies extracted from the PCs(circles). The gradual color code of the theory lines represents therelative strength of Rl going from strong (red) to no (gray)coupling. Error bars denote one standard deviation from the fit.
The background color indicates the BEC, supersolid, and ID
regions (see upper labels), identified using a matter-wave
interferometric analysis of the experimental data [17].

of the side-peak amplitude; see Fig. 3(d). These results
show a good agreement with the calculated time evolutionsof the interference patterns for the second and third evenmodes, shown in Figs. 3(e) and 3(f).
Finally, we study the evolution of the modes across the
BEC to supersolid and ID phases. We repeat the collective
excitation measurements for various as , and, using the
PCA, we extract the oscillation frequencies of all theleading PCs. Figure 4 shows our experimental results
together with the mode tracking from the spectrum calculations. For a give elementary mode l, we plot ωl as well as
the response amplitude Rl ¼ mω2y hljŷ2 j0i=2ℏωl , whichindicates the probability to be excited by our trap-excitation
scheme. For completeness, the figure shows both even andodd modes, although only even modes are coupled to our
trap-excitation scheme. Here, j0i and jli denote, respectively, the ground and excited states of interest, and ŷ is theaxial position operator.
In the BEC regime, besides the roton mode that
progressively softens with decreasing as , the other modes
show a regular spacing in energy and are nearly constantwith as . In both the theory and experiment, we observe thatjust one mode couples to the trap-excitation scheme. Thismode has a compressional, axial breathing character.
Experimentally, we observe that all the leading PCs
oscillate at the same frequency, suggesting that theyaccount for the same mode [37]. In this regime, both the
PC frequencies ωl and Rl remain rather constant. At thesupersolid phase transition, reached around as ¼ 50.6a0 ,the numerical calculations reveal that different modes
undergo an abrupt change and can mix with each other.

050402-4

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
Their energy and phase or crystal character exhibits astrong dependence on as . Here, several modes respond to
the trap-excitation scheme, as shown by the value of Rl .
In the PCA, we observe that the leading PCs now oscillate
at distinct frequencies and have different characters (seealso Fig. 3). One set of PCs reduces their frequency whenlowering as , indicating (at least) one phase mode thatsoftens strongly in the supersolid regime, even below thetrap frequency νy. Another set of PCs shows a frequency
that remains hard when decreasing as . Calculations of C
show that this mode changes character along the phase
diagram and eventually becomes crystal type.
In conclusion, the overall agreement between the experiment and theory confirms the calculations in the supersolid
regime, revealing two distinct branches with respective
crystal and superfluid characters. The trademarks of supersolidity expected in infinite systems thus carry over tothe finite-size ones currently available in laboratories. The
knowledge of the excitation spectrum will provide the basefor future investigations related to the superfluid properties
and phase rigidity in a supersolid state.
We thank D. Baillie, R. Bisset, B. Blakie, T. Pfau, A.
Recati, L. Santos, and P. Silvi for stimulating discussions.
We acknowledge useful conversations with the participantsof the meeting on “Perspectives for supersolidity in dipolar
droplet arrays” in Stuttgart, where we became aware of
related work by the groups of Modugno [61] and Pfau [62].
Part of the computational results presented have beenachieved using the HPC infrastructure LEO of the
University of Innsbruck. This work is financially supported
through an ERC Consolidator Grant (RARE, no. 681432)
and a DFG/FWF (FOR 2247/PI2790), and the project
PASQUANS of the EU Quantum Technology flagship.

*

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050402-6

Supplemental Material: Excitation spectrum of a trapped dipolar supersolid and its
experimental evidence
G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 D. Petter,1 M. J. Mark,1, 2 L. Chomaz,1 and F. Ferlaino1, 21Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria(Dated: July 18, 2019)

Calculation of the Bogoliubov Spectrum

Our theory is based on an extended version of theGross-Pitaevskii equation (eGPE)Z∂ψ(r, t)  h̄2 ∇2= −+ V (r) + dr0 U (r − r0 )n(r0 )
ih̄
∂t
2m

+ ∆µ[n] ψ(r, t),(1)where ψ(r, t) is the dipolar quantum-gas’ wave function
ψ(r, t). The eGPE includes the kinetic energy, external trap potential and the mean-field effect of the interactions [1, 2]. The first three terms of Eq. (1) accountfor the kinetic energy, the external harmonic trapping
potential, and the mean-field interactions, respectively.
The latter includes the contact and the dipolar interactions. In order to study the supersolid phase, it isfundamental to also include a beyond-mean-field corrections in order to stabilize the supersolid state against theroton instability. This is done by adding a term in theform of the Lee-Huang-Yang correction, ∆µ[n] [3–13]; seealso [14–17]. This is typically included as a correction to
the chemical potential obtained under the assumption of
local density approximation [18, 19]. However, recent
experimental results have raised the questions about therange of validity of such a treatment since quantitative
disagreements at a level of few % have been observed
when comparing the theory results with the experimental findings [6, 20–24]. To the best of our knowledge, thisis still an open question, which will need future additional theoretical investigations. To compensate for thiseffect, throughout this letter, we shift as by −4.3a0 . To
calculate the ground-state (GS) wave-function, ψ0 (r), wethen minimize the energy functional resulting from theeGPE using the conjugate-gradients technique [25].
In a next step, we study the Bogoliubov de Gennes(BdG) excitation spectrum of a dipolar Bose-Einsteincondensate trapped in a harmonic cigar shaped potential [1, 25]. Our calculations are obtained by expanding
the wavefunction ψ(r, t) around ψ0 (r). Here, we write:
ψ(r, t) = (ψ0 (r) + ηδψ(r, t)) e−iµt ,where η  1, µ is the chemical potential of the groundstate andδψ(r, t) = ul e−il t/h̄ + vl∗ eil t/h̄ .

The spatial modes ul and vl are oscillating in time withthe corresponding frequency ωl = l /h̄. We then linearize
the eGPE around ψ0 at first order in η. By solving theset of coupled linear equations, we obtained the discretemodes, numbered by l, of energy l and amplitudes uland vl . We define the (odd) even parity of the mode
from their amplitude ul and vl being (anti-)symmetric iny.
In order to illustrate the spectrum, we compute thedynamic structure factor (DSF), since it directly gives information about the density response of the system when
perturbed at specific energies and momenta. At T = 0
the DSF is defined as [20, 26]:0S(k, ω ) =X Z

2dr [u∗l (r) + vl∗ (r)] eik·r ψ0 (r)l× δ(ω 0 − ω),×(2)

where the sum is over the different spatial modes and kis the wave vector. In Fig. 1 and Fig. 2 we plot the DSF
of Eq. (2). For better visualization, we use an energy
broadening of 0.09 hνy and 0.12 hνy for Fig. 1 and Fig. 2,
respectively, similar to what was done in Ref. [26].
Defining the mode character

Within the Bogoliubov theory and in the linear regime,
the effect of the population of the mode l on the global
state dynamics can be studied using the following expression [1]pψ(r, t)eiµt ≈ |ψ0 (r)|2 + 2ηδρl (r) cos ωl te−iηδϕl (r) sin ωl t ,where the density fluctuations δρl = (ul + vl∗ ) |ψ0 | andphase fluctuations δϕl = (ul − vl∗ )/ |ψ0 | have been separated.
In order to evaluate the dominant character of eachmode l, we introduce the quantity C. As discussed in
the main text, the crystal and phase mode differentiate
from each other by the spatial region where δϕl varies
the most. For crystal modes, this is inside the density
peaks, resulting e. g. in a center-of-mass motion of one
individual peak, which leads to a change of the crystal
structure. Differently, for phase modes, δϕl changes the2Vin(out) =1
Nin(out)×

Nin(out)

×X
j=1h|δϕ(0, y, 0) − hδϕ(0, y, 0)iRj |2 iRj .

The mean h· · ·iRj is defined for a generic function f as
ZZ
hf (y)iRj =f (y) dy
dy.
y∈Rj

y∈Rj

The mode character is then evaluated by considering theratio C = Vin/Vout . C is large for modes with prevalent
crystal and small for the ones with dominant phase character. In Fig. 2 (d) we encode the information on C as acolor scale on the DSF spectrum. The same color map isused to illustrate the modes of the panels (a-c) in Fig. 2,
confirming their correct assignment. For completeness,
we also illustrate in Fig. S1 the modes’ character on the
spectrum of a 164 Dy supersolid, using the parameters of
Fig. 1 (e) of the main text.
Applying the principal component analysis to ourdataDataset for applying the PCA

To identify the excited modes from our experimental
data, we apply a general statistical method called principal component analysis (PCA) [27–29] to a set of measured density distributions after a time-of-flight expansion. For our trap-excitation measurement, a dataset for
the PCA is composed as follow. For each target valueof as , we record the time evolution of the density distribution for holding time, th , between 0 and 30 ms. For
each th , we record between 15 and 30 repeated images,
all together yielding a dataset of Nm >∼ 200 images. Eachexperimental run i yields a two-dimensional density distribution ni (kx , ky ). By performing a simple two dimensional Gaussian fit, we extract 71 × 71 pixels region-ofinterest (ROI) centered on the atomic cloud (the pixel’s

0

1.6

≥3.2
C x1034

E (hνy )most between neighboring peaks, signalizing a particle
exchange between peaks and thus a modification of theatom numbers in the peaks. We quantify these two typesof character by computing the spatial variance of δϕl (r)inside the density peaks, Vin , and in between them, Vout .
The quantities Vin and Vout are defined as follow.
For a given axial density cut of the GS wave function|ψ0 (0, y, 0)|2 , we first define the region inside (between)
the density peaks by identifying the different density
maxima (minima) and number them by j ∈ [1, Nin(out) ].
In a next step, we compute the mean distance d between
all density minima to their neighbouring maxima. Finally, we isolate the region Rj = [−d/3, +d/3] of spacecentered around each maxima (minima) and calculate:

2

0

0

1

ky lz

2

3

FIG. S1. Characterisation of the excitation modes for
N = 4 × 104 atoms of 164 Dy at as = 90 a0 in a trap of frequencies 2π ×(260, 29.6, 171) Hz. As in the Er case (Fig. 2 (d)of the main text), the blue color reveals the dominant crystal
character for the upper branch, whereas the red color shows
the dominant phase character for the lower branch.

width in kx,y is 0.32µm−1 ). In addition, we post-selectthe shots in which the atom number, the axial cloud size
and the transverse cloud size vary by less than 20%, 30%
and 15% than their mean values, respectively.
PCA’s working principle

To apply the PCA, we represent each ROI of a dataset
as a vector ρi (s) where s represent the index of the pixel(s ∈ [1, Np ], Np is the number of pixels in one image). We
PNm
compute the mean vector image ρ̄(s) = i=1
ρi (s)/Nmand consider the variations of the pixel values in each
vector image compared to ρ̄, δρi (s) = ρi (s) − ρ̄(s). Finally, we considerPthe covariance matrix of these variaNmtions Cov(p, s) = i=1
ρi (s)ρi (p)/(Nm −1), which is real
symmetric. By simply diagonalizing the covariance matrix, the PCA constructs a new basis of Np vector-images,
called principal components (PCs) and written C p (s) (p ∈[1, Np ]) in the original pixel basis, that are uncorrelated
one from an other. The PCs satisfy CovC p = λpC p where
λp is the eigenvalue of the covariance matrix associated to
the PC p. The original vector images
be all rewritten
PNcansin this new basis as ρi (s) = ρ̄(s) + p=1
wp,iC p (s), wherePNp
C p (s)ρi (s) is the weight of the component
wp,i = s=1
p. We note that, by converting back the pixel representation to the original two-dimensional momentum space,
the above decomposition meansni (kx , ky ) = n̄(kx , ky ) +

Ns
X
p=1

wp,iC p (kx , ky ),

(3)

3

-4

ñ(kx ,ky) (arb. u.)

a

-0.1

0

0

2

0.1

b

ky (μm-1)

-2
0
2
4
-4

-2

0

2

kx (μm-1)

4

-4

-2

kx (μm-1)

4

FIG. S2. Examples of the two leading PCs for our dataset
at as = 50 a0 . (a) PC1 reveals a dominant fluctuation of
the interference patterns in the central peak at ky ≈ 0 µm−1
(central blue region) with a slighter change of the sidepeaks
at ky ≈ ±2 µm−1 (red regions) . (b) PC2 shows fluctuationsin the interference patterns’ sidepeaks around ky ≈ 2 µm−1
and no significant change of the central peak.

where C p (kx , ky ) encompasses now the densitydistribution change induced by the PC p. The factthat the covariance matrix is diagonal in the PC basis
indicates that the PCs correspond to uncorrelated
sources of variations in the dataset. More explicitly, thecoefficients wp,i show no correlations in between different
p. This feature makes the PCA a powerful tool, e. g. to
identify and discriminate between elementary modesof different frequencies when applied to time-evolution
data, as used in Ref. [29]. An example of the obtained
two leading PCs in the supersolid region is given in
Fig. S2.

Identifying the elementary modes of a quantum gas via the
PCA

We quickly remind the working principle, of the identification of modes via the PCA. In the linear regime,
the contribution of each mode to density oscillations is
expected to decouple and separate temporal and spatial
variations as:Xn(r, t) ≈ n0 (r, t) + 2
ηδρl (r) cos (ωl t + φl ) ,(4)lwith φl an arbitrary phase for the mode l. This relation should also hold for the density distribution after
the gas’s free-expansion. If one considers that the image index i encloses a time dependence (ti ), the equations (3) and (4) have a very similar structure, associating C p (kx , ky ) and wp,i to ρl (r) and cos (ωl ti ), respectively. Thus the PCA-based identification of uncorrelated components in the time-evolution of the density
profiles should enable to identify the elementary modesof the system. The corresponding PCs’ weights are then

expected to oscillate in time at the frequency ωl of themodes. In particular, the PCA should separate themodes oscillating at different frequencies and differentiate them from other sources of fluctuations or of dynamics (e.g. dissipation). Following Ref. [29], we note thatmodes can be properly distinguished if the period associated to their beating is smaller than the total time forwhich the time-evolution is recorded, or, even for shorterprobe time, if they have different enough amplitudes ofoscillations (i.e. excitation probability).
From our dataset with repeated realizations ofeach hold time th , we thus consider, for eachPCweights at time th , Wp (th ) =P p, the meanPw/i/ti =th p,ii/ti =th 1. We then fit Wp (th ) to a sinefunction A0 + As cos (ωth + φ) and extract the PC’s frequency (ω) and amplitude As of oscillation. We thenconsider as relevant the PCs that show oscillation of amplitude As > 8 × 10−4 , frequency ν > 20 Hz, and where
the oscillation frequency can be extracted with a precision < 10%. Examples of the time evolution of Wp andof their fits are shown in Fig. 3 (b).
We note that the PCA does not always assemble in aunique PC all the correlations in the pixel values that
follow the same time dependence, and a single modecan be artificially split into several components in theanalysis process. To better understand this behavior, we
performed tests on theoretical calculations and comparethem to the experiments. Theoretically, we specifically
populate a single Bogoliubov mode on top of the groundstate, we then compute the interference patterns as afunction of the hold time th , similar to what is done in
Fig. 3 (e-f), and finally we apply the PCA. For each modeconsidered, both for regular (BEC) or density-modulated
ground-states, several leading PCs are found to oscillate. Their frequencies match the mode frequency while
their oscillation amplitude decreases with the PC’s index.
Typically the ratio in the oscillation amplitudes between
the first and the second PC is about 10, and the amplitude of the larger-index PCs are negligible. Therefore, inthe cases where the modes are the most strongly excited,
i.e. mainly in the BEC regime (see Rl scaling in Fig. 4
for the excitation amplitude), one can indeed expect thatseveral PCs are sensitive to a single mode in experiment,
matching our observation.
From those theory tests, we can also better understandthe origin of this artificial splitting of one mode in several PCs. Indeed, the oscillations of the different PCs arefound to have the same frequencies but different phases,
typically shifted by about π/2. As it treats the pixels independently, the PCA gets confused by such π/2 phaseshifts in the oscillations occurring in different regions ofspace, i.e. pixels’ values that distinctly oscillate, startingfrom their extremal or medial values. The PCA then artificially splits the oscillations occurring in these different
regions into several components while they correspond to
the same mode. Finally, this effect can be further favored4
in typically imperfect experimental settings, by the addition of experimental noise as well as other technical(e.g. imaging artifacts) or physical (e.g. dissipation) effects, which yield differences in the pixel values. Our
theory tests, however, show that those additions are not
the main reasons for the observed splitting.
Based on the conclusions of those tests, in the experiment (see discussion of Fig. 4 of the main text), we interpret as probing distinct modes only the PCs showing
different frequencies, while PCs whose frequencies matchwithin their error bars are interpreted as probing a single
elementary excitation of the system.

Partial recomposition

To isolate the effect of each PC on the complex timeevolution of the interference patterns, we use partial recomposition of the images inspired from Eq. (3). In particular we definen(p) (kx , ky , t) = n̄(kx , ky ) + Wp (t)C p (kx , ky ).

(5)This is equivalent to consider that a single PC is ”excited”, similarly to what can be done in theory for the individual excited modes of the BdG spectrum (see Fig. 2)
and its description in the main text and Supp. Mat.).
In Fig. 3 (c-d), we show examples of the axial cuts ofn(p) (kx , ky , t) for two of the leading PCs. We note that
here, as well as for all experimental data shown in thismanuscript, the axial cuts correspond to the average ofthe density distributions for |kx | < 1.6 µm−1 .

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3.7 Publication: Phase coherence in out-of-equilibrium supersolid states of ultracold
dipolar atoms

89

3.7 Publication: Phase coherence in out-of-equilibrium supersolidstates of ultracold dipolar atoms
Nature Physics 17, 356–361 (2021)a
submitted 21 December 2019; published October 2020
DOI: 10.1038/s41567-020-01100-3

P. Ilzhöfer,1,2 M. Sohmen,1,2 G. Durastante,1,2 C. Politi,2 A. Trautmann,2 G. Natale,2G. Morpurgo,3 T. Giamarchi,3 L. Chomaz,2 M. J. Mark,1,2 and F. Ferlaino1,2
1Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
3
DQMP, University of Geneva, Geneva, Switzerlanda

The author of the present thesis performed and analysed the theoretical calculations together with
M. J. M, L. C., M. S. and C. P. and contributed in writing the manuscript.

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

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0

20

40

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80

100

∆Ф

∆Ф

0.6
π

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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

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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

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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.

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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

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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
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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
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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.

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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).

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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.

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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.

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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.

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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.

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Chapter4

Bloch oscillations andmatter-wave localization

This chapter describes dipolar quantum gases of erbium atoms confined in a one-dimensional
optical lattice, creating an array of quasi-2D planes. In such a platform, the new statesarising from the stabilization effect provided by the quantum fluctuations and the softeningof Bogoliubov modes are still unexplored. As we will see, in this geometry, the beyond
mean-field effects are not the only stabilization mechanism, indeed the kinetic term can also
avoid the collapse in the attractive interaction regime. We will refer to these states as soliton
states that, like the droplet ones, are lattice-bound, i.e., they have a finite size in the plane
and would not expand even without the radial harmonic confinements.
This chapter is organized as follows. Section 4.1 summarizes the basic knowledge to describe
one-dimensional lattices, focusing on the single-particle physics. Section 4.2 introduces thefundamental properties of Bloch oscillations. Section 4.3 extends the lattice description to a
many-body system and introduces the Bloch-oscillations dephasing rate as a tool to probe theinteraction. Section 4.4 first illustrates the experimental realization of the one-dimensional
optical lattice, and then presents the phase diagram with the possible quantum phases achievable in this geometry. Finally, Sec. 4.5 shows the publication, in which our group studied thesystem using Bloch oscillations and found the localization into a single lattice site. In thiswork, we also uncover, in a phase diagram, the regime in which the ground state is radiallyself-bound and highlight the soliton and droplet regime.

4.1 One-dimensional lattice
Optical lattices are nowadays widely used in the ultracold community [Mor06, Blo08]; themain reason is that they are relatively simple to realize in the experiment and open many
possibilities. On the one hand, these systems simulate the behaviour of strongly correlated
electrons in solid-state crystals, assembling variations of Hubbard models [Lew07, Dut15].
On the other hand, optical lattices create an interferometric tool to assess the importanceof mean-field interactions. To mention few examples: lattice modulations spectroscopy is astandard technique to establish the scattering length [Kol06, Bai16a], Bragg spectroscopy is
105

106

4 Bloch oscillations and matter-wave localization

used to study the excitation spectrum, see Sec.2.5, and Bloch oscillations are used in atominterferometry [Cla15].

4.1.1 Single-particle physics: Bloch theorem
We will exploit Bloch oscillations, a single-particle effect, to understand the many-body
behavior of a dipolar quantum gas in an optical lattice. Hence, it is worth introducing
some general concepts and starting from the Hamiltonian of a single particle confined in a
one-dimensional periodic potential, which reads:
H=−

ℏ2 2
∇ + Vlatt (r),
2m

4.34

where the first term represents the kinetic energy, with m mass of the particle and Vlatt (r)the periodic potential. In the case of electrons in a solid, this potential is generated bytheir interactions with the ions, see e.g. Ref. [Ash76], whereas in our case, this is externallyimposed by the interference of a laser beam superimposed on its retroreflection. The resulting
potential reads:
Vlatt (z) = V0 sin2 (kl z),
4.35
where kl = 2π
λ is the wavenumber and λ is the wavelength of the laser beam. In this expression, we neglect the finite size of the beam. Table 4.1 defines some convenient parameters
for this chapter that are commonly used by the ultracold-gases community:
Lattice parameters
Lattice spacing
Lattice wavenumber

alatt = λ2kl = 2πλℏ2 k2→Recoil energy

ER = 2ml →Adimensional lattice depths = EVR0πkl = alattℏ π
ER = 2ma
2

2 2
latt

Table 4.1: Lattice parameters commonly used in ultracold-gases experiments.

We can use the Bloch theorem to look for solutions of the Hamiltonian 4.34 (for more details
see Refs. [Ash76, Mor06]),Hψq (z) = E(k)ψq (z),4.36
where q is the quasi-momentum parameter defined within the Brillouin zone (BZ), q ∈ BZ =[−ℏkl , ℏkl ]. The theorem states that the solutions can be written as the product of a function
uq with the periodicity of the lattice and a plane wave.
ψq(n) (z) = uq(n) (z)eiqz ,4.37

Here, n denotes the band index, since for each quasi-momentum q there are many solutions(n)corresponding to the different energy bands. This theorem is extremely useful because ψq (z)4.1 One-dimensional lattice107

is a generic function defined in real space R, whereas the solution can be now restricted to(n)the periodic function uq , which is defined inside the lattice period. Note that the real(n)wavefunction ψq (z) does not need to maintain the symmetry of the optical lattice with the(n)(n)
exception of the q = 0 case, where ψq=0 (z) = uq (z).
It is also worth noticing that we can rewrite the eigenvalue problem 4.36 as a Mathieu
differential equation by using the trigonometric rule sin2 (z) = 1/2 (1 − cos(2z)):∂2ψ(z̃) + (aM − 2qM cos(2z))ψ(z̃) = 0.
4.38
∂ z̃
Here, we used the rescaled z̃ = qz. The Mathieu parameters (aM , qM ) are equal to aM =E/ER − s/2 and qM = −s/4, respectively.
For a sinusoidal periodic potential, the eigenvalues and eigenstates can be calculated bothanalytically and numerically [Hum01]. Here, we limit our discussion to the results.
Figure 4.1 shows the free particle energy spectrum in the Brillouin zone as a function of thelattice depth. It is strongly modified by the presence of the lattice, with the following featureemerging:
 at small momenta |(ℏkl )| ≪ 1 in the lowest energy band, the spectrum is still quadratic
and the atoms feel the underlying lattice by moving with an effective mass;
 at the edge of the Brillouin zone, the symmetry of the system leads to a flat derivativeof the energies, ∂E∂q → 0; the spectrum is divided into bands with a finite gap between them.

We can now define the bandwidth BWn = max(E n )−min(E n ) of the n-band, and the energy
gap ∆n = max(E n ) − min(E n+1 ). Figures 4.2(a,c) display the evolution of the width of theband for the first 5 bands and Fig. 4.2 (b) shows the energy difference of min(E n )−max(E 1 ).
Note that for the lowest line this corresponds to the energy gap.
From the band structure, it is possible to directly visualize the tunneling term J, which gives
the intersite gain in kinetic energy. This is directly proportional to the bandwidth and givenby:max(E 0 ) − min(E 0 )
J=
.
4.39
4ER

4.1.2 Single-particle physics: Wannier function
The Bloch solutions are delocalized over the full lattice sites. However, the degree of localization of the wavepackets depends on the lattice depth. In the case of a deep lattice V0 ≫ Er ,it is convenient to introduce the Wannier function [Wan37], which is defined as the Fouriertransform of the Bloch function:

108

4 Bloch oscillations and matter-wave localization

20
15
10
5
0
1
0.5

16
0

8

-0.5
-1

0

Figure 4.1: Single-particle energy spectrum. Energy spectrum as a function of the lattice
depth in the Brillouin zone. The black lines denote the energy bands for three values of the lattice
depth ([0 , 8, 16] ER ).

1
wn (z − zj ) = √M

Z π/alatt−π/alatt

−iqzj
dq,
ϕ(n)
q (z)e4.40

where M is a normalization factor. This wavefunction is well localized at the lattice site
R in real space. In this case, we can understand that the tunneling element J represents a
localized atom hopping from the lattice site j to the adjacent one j ± 1. This is then givenby the following overlap integral:
J=

Z



ℏ2 ∂ 2dzwn (z − zj ) −+ Vlatt (z) wn (z − zj+1 ) .
2m ∂z 2

4.41

Note that, in this regime, the recoil energy is smaller than the local trapping potential
in each site ℏω0 = 2Er (V0 /Er )1/2 . Therefore, each local site supports many bound states.
In our experiment and for most of the ultracold atom experiments with optical lattices, theatomic motion is restricted to the lowest energy band thanks to an adiabatic loading. Hence,
the lattice band index n is omitted from now on. The dispersion relation of this band canbe obtained by solving the Mathieu differential equation:3
ϵ(q) = ℏω0 − 2J (cos qa) .
2

4.42

In Eq. 4.42, the amplitude of the cosine term is 2J, hence we find the result of Eq. 4.39,
which connects the energy bandwidth to the tunneling J. As a final remark, it is worth
noticing that for deep lattices the Wannier function is well approximated by a Gaussian of
the form:
−1/4 −z 2 /2l2
2latt ,
w(z) ≈ wG = πllatte
4.43√
where llatt = (kl 4 s)−1 is the harmonic oscillator length of the lattice site.

4.2 Bloch oscillations
40

109

20

(b)

30

10
0

E + W(E))

20

10

E/ (

Energy E (n) (E R)

R

E (E )

(a)

0
0

5
10
lattice depth s (E R)

15

1

(c)

0.5
0
0

5
10
lattice depth s (E R)

15

Figure 4.2: Energy band properties. (a) Energy width as a function of the lattice depth. Energy
gap between adjacent bands (b), and bandwidth (c) as a function of the lattice depth. The dashedline corresponds to the lattice depth used in the publication presented in Sec. 4.5.

4.2 Bloch oscillations
Bloch oscillations is a phenomenon, which describes the motion of electrons in a crystal
structure under the influence of an electric static field. It is now well known that, as thename suggests, this motion is oscillatory rather than a uniform accelerated motion. This
process is quite generic and was observed also with cold atoms loaded in a periodic potential
under the influence of a constant force Fext . The Hamiltonian describing the system isreferred to as the Wannier-Stark Hamiltonian, and it reads
HW = −ℏ2 2∇ + Vlatt (z) + Fext z,
2m

V (z + d) = V (z).

4.44

Differently than in Eq. 4.34, here the external force breaks the lattice symmetry since
Veff (z + d) ̸= Veff (z), where Veff = Vlatt (z) + Fext z. Note also that if E0 is an eigenvalue ofthe problem HW Ψ = EΨ, one can construct the commonly named Wannier-Stark ladder ofsolutions EnW = E0 +nW dF where nW is an arbitrary integer. Figure 4.3 gives an illustrationof the potential and of the eigenvalues of the Wannier-Stark ladder. A simple picture of the
Bloch oscillations can be obtained by a semi-classical approach that considers a wavepacket
with an initial quasi-momentum q0 prepared in the lowest energy band. In presence of an
external force we can write:dqℏ = Fext .
4.45
dt
By integrating over time, we find that the evolution of the quasi-momentum is given by:
q(t) = q0 + Fext t/ℏ.

4.46

This linear increase in momentum is a characteristic of the Bloch oscillations. We can now
use the periodicity of the BZ to find the period of the oscillation:
2klatt = Fext TBO /ℏ → TBO = 2klatt ℏ/Fext .

4.47

The Bloch period (TBO ) depends linearly on the lattice wavenumber and it is inversely
proportional to the force. When the momentum of the wavefunction reaches the edge of110

4 Bloch oscillations and matter-wave localization
sin2 (z) − Fext z−Fext zVlatt (arb. u)

1

0.5

0

−0.5
−1

0

1

2

3

4

5

6

z (a)

Figure 4.3: Wannier-Stark ladder. The sum of the lattice potential with the external force gives
rice to the Wannier-Stark ladder (black line). The dotted line represents the external force. The bluelines inside the potential show the eigenvalue solutions.

the Brillouin zone a Bragg reflection occurs. This gives rise to the famous sawtooth signal
looking at the position of the cloud after a time of flight. Before considering the many-bodycontribution to the Bloch oscillations, it is interesting to compare the Bloch oscillations of
the erbium system in an optical lattice to a standard semiconductor like GaAs. Table 4.2
summarizes the typical parameters:
Bloch oscillations parameters
ErbiumGaAs

Lattice spacing

532 nm0.56 nmTypical Force

Gravity

Electric fieldTypical Force strength

2.7 × 10−24 N

6.4 × 10−13 N

Period

0.5 ms

1.8 psScattering time

100 ms0.2 psTable 4.2: Typical parameters for Bloch oscillations of erbium atoms trapped in an optical lattice
and in a standard GaAs semiconductor.

In a cold atom experiment, a dilute gas is loaded in an optical potential realizing an ideal
platform to observe Bloch oscillations. The time scale of the Bloch period, typically ms ismuch greater than the typical collisional time. Instead, in a bulk GaAs semiconductor the
scattering rates are larger than the oscillation frequencies, hence it is not possible to observe
Bloch oscillations. Nevertheless, they were observed first in semiconductors by realizing
artificial superlattice. In this case, it was possible to reduce the period of the oscillation byengineering an effective larger lattice spacing.

4.3 Many-body physics: mean-field model

111

Bloch oscillations and more generally atom interferometry have been widely used to getan accurate determination of fundamental quantities such as the gravitational acceleration [Cla05, Fer06], and the fine structure constant α [Bat04, Cla06, Cad08]. In fact, asshown in Eq. 4.47, the Bloch-oscillations period is inversely proportional to the driving force,from which one can determine the gravitational acceleration, and directly proportional to ℏfrom which one can infer the fine structure constant. Many-body effects arising from theinter-atomic interactions can lead to a damping of the Bloch oscillations. These effects limit
the observation time, and therefore the precision of the measurements.

4.3 Many-body physics: mean-field model
In Chapter 2, we used the Gross-Pitaevskii equation to describe an ultracold Bose gas ofdipolar erbium atoms, trapped in a harmonic confinement. We will now extend this equation
to include the periodic potential. We will rely on the knowledge of single-particle physics
highlighted in the previous sections to reduce the otherwise demanding resource necessary
to solve this equation.
To describe our system, we can add the periodic potential of Eq. 4.35 to the GPE of
Eq. 2.11:
iℏ

∂
ℏ2 2
Ψ(⃗x, t) = H3D Ψ(⃗x, t) = [ −∇ + Vharm (⃗x) + Vlatt (z) + g|Ψ(⃗x, t)|2
∂t
2m
Z

2

+ d3 ⃗x′ Udd ⃗x − ⃗x′ Ψ ⃗x′ , t+γQF |Ψ(⃗x, t)|3 Ψ(⃗x, t).

4.48

The atoms are now confined by the combination of the dipole traps and the lattice potential.
Equation 4.48 exhibits various solutions depending on the dipolar orientation and the exact
strength of the harmonic and lattice trap, leading to a rich phase diagram. Moreover,resolving this equation can be computationally demanding. For this reason, we developed
a discrete effective one-dimensional eGPE, following Ref. [Bla20b]. In brief, we assume thatthe wavefunction solution can be written in the following form:2221
Ψ(⃗x, t) = Φ(x, y, l, η)ψ(z, t) ≡ √ e−(ηx +y /η)/2l ψ(z, t),
πl

4.49

where we separate the transverse and radial part of the wavefunction. We further
passumed
that the radial part can be written as an asymmetric Gaussian with size l = lx ly andaspect ratio η = ly /lRx . Here, lx and ly are the half-width of the function at the 1/e value,and Φ is normalized dx dy |Φ|2 = 1. We obtain the effective 1D eGPE by integrating along
the radial directions x, y, as follows:
Hz =

ZZ

dx dy Φ∗ H3D Φ.

4.50

Note that up to now, we did not make any assumptions about the lattice depth. In fact, this112

4 Bloch oscillations and matter-wave localization

effective one-dimensional eGPE was introduced in Ref. [Bla20a] without a lattice potential
to describe the supersolid and the droplet transition in a bulk system.
For a deep enough lattice, we can further simplify the equation by assuming the longitudinal
wavefunction as a sum of Gaussian functions, as in Eq. 4.43:√ X
ψ(z, t) = N
cj (t)wG (z − zj ) .
4.51
jHere, cj is the complex wavefunction amplitude on lattice site j. Finally, we achieve a set of
discrete effective one-dimensional eGPEs by performing the integration over z:iℏ∂cj= −J (cj+1 + cj−1 )∂t+ −Fext zj + Vharm (z) + g 1D N |cj |2 + NXk

1D 3/2dd|ck |2 + γQF
N |cj |3
U|j−k|

!

4.52
cj .

Here, Fext is the external force given in our case by gravity on the atoms. Finally, we
1D =have introduced the effective interaction parameter g 1D = g/ (2π)3/2 l2 llatt and γQF
3/2
23/2 / 5π 3/2 l2 llattγQF .

We can use this equation without the external force term Fext to find the ground statesolution and then study the dynamical evolution under the influence of the force. Further
information on the numerical approach to find the stationary state solution are given in the
supplementary material of the publication in Sec. 4.5.

4.3.1 Overview of Bloch oscillations in ultracold atoms
Several experiments with alkali atoms have used Bloch oscillations as a tool to measure forces
with high precision or to study the interaction properties of the system. Using a BEC ofcesium atoms and a Feshbach resonance to tune the contact interaction, it was possible toobserve more than 20000 oscillations by finding the zero crossing of the scattering length,
essentially realizing an ideal Bose gas [Gus08]. In that regime, the limit was associatedwith magnetic field noise resulting in effective non-zero interactions. Using sodium atoms,
in Ref. [Fat08], the dephasing rate of the Bloch oscillations was found to shift compared to
the zero crossing of the scattering length. The authors associated this shift to the presence of long-range interaction between different lattice sites. This was further confirmed bymeasurements and theoretical simulations with different polarization angles of the dipoles.
It is also important to underline that the dephasing minimum was not associated with themean-field (MF) cancellation at as = add but rather with the minimization of the varianceof the energies in the different lattice sites.
In the case of dipolar erbium atoms, the stronger dipole-dipole interaction does not only
further shift the mean-field cancellation but, as we have seen in Sec. 2.3, leads to a regime
where the beyond mean-field term (BMF) plays an important role. In particular, the Lee
5/2
Huang Yang term depends on as , becoming relevant only if the mean-field cancellation

4.4 Quantum phases of erbium atoms in a one-dimensional lattice113

happens at a large scattering length like for erbium, dysprosium, or alkaline mixtures. This
makes our experiment an ideal candidate to interferometrically assess the role of quantum
fluctuations and in particular to further test the LHY formulation.

4.3.2 Bloch oscillations in ultracold dipolar gases
The previous paragraph connects the minimum dephasing of the Bloch oscillations to vanishing mean-field interaction energies. However, in the case of LHY more caution is needed.
In fact, the speed of the wavefunction phase winding is given by the chemical potential thatdiffers in each lattice site. The minimum of the dephasing is the point with the minimum
variance of the chemical potential. We can relate the chemical potential to the energy via
the following relation:jjµj = 2EMF+ 5/2EBMF/ |cj |2 ,4.53

where cj is the population in the lattice site j. It is possible to notice that in the case where
the beyond mean-field contributions are negligible the minimum variance of the chemical
potential corresponds to the minimum variance of the energies. Instead, when the BMF
contribution is relevant, the different scaling with the density of the LHY term causes thedifferent prefactor in Eq. 4.53. In the publication in Sec. 4.5, we compare the dephasingrate obtained via an analytic model by using only the ground state to the one obtained byanalyzing the full real-time simulations.

The importance of the LHY term to describe the stabilization mechanism of a dipolar Bosegas is established in our field. However, there are still discrepancies between theory andexperiments [Cho18, Pet19]. In this regard, Bloch oscillations, observed in the regime where
the mean-field term is minimized, can be a powerful tool to shed some light on the LHY term,see Sec. 4.5. This experiment can be performed also in other platforms, such as mixturesof quantum gases, where the mean-field term can be set to zero by tuning the intra andthe interspecies interactions [Sko21]. Performing such measurements could bridge the gap
between theory and experiments.

4.4 Quantum phases of erbium atoms in a one-dimensional lattice
The interplay of the long-range dipole-dipole interactions with the contact interactions gives
rise to many new quantum phases, as we pointed out in Chapter 3. In that chapter, we also
highlighted the importance of the trapping geometry. In this section, we add to our trapping
potential an optical lattice and explore how it affects the phase diagram.

4.4.1 Geometry
To understand the phase diagram, it is important to remind the geometry. We obtained
an array of surfboard potential by combining a one-dimensional lattice with two optical

Our geometry114

4 Bloch oscillations and matter-wave localization• Array of surfboards (8 Er
z
y
x

−
→
B

Figure 4.4: Geometry illustration. Optical dipole trap propagating along the y direction. Opticallattice along the z direction. The combination of dipole traps (an additional optical dipole trap
not shown is propagating along the lattice direction) and the optical lattice creates an array ofsurboard potentials with the onsite trapping frequencies ωx,y,z = 2π × (240(3), 30(3), 6000(100))Hz.
Gabriele Natale • MAQS meeting |The magnetic field (B) is aligned along the elongated direction y.

dipole traps: the first one propagates along the y direction, and the second one propagatesalong the vertical direction. For each surfboard, the lattice beams create the following
onsite trapping frequencies: ωx,y,z = 2π × (4(1), 4(1), 6000(100))Hz, where the 4 Hz refers
to the radial confinements and the 6 kHz to the harmonic confinement along the latticedirection. Instead, the overlapping dipole traps add mainly radial confinement with thefollowing trapping frequencies: ωx,y,z = 2π × (240(1), 30(1), 217(1))Hz. The frequencies are
obtained by independent measurements. For the dipole traps, we rely on the excitation of thedipole modes with the lattice potential off, whereas for the lattice we use the Raman-Nathtechnique, see e.g. Ref. [Hec02]. Figure 4.4 shows the geometry of our system. Note that thedipoles are polarized along the elongated direction (y) of the trap.
In this geometry, since the DDI is predominantly attractive in the plane, it is possible tostudy the regime in which it compensates for the repulsive contact interaction realizing an
averaged zero or even attractive mean-field. Without the lattice, this configuration led to thediscovery of the macrodroplet. In our study, we want to shed light on the role that beyond
mean-field interactions and kinetic energy play in determining the phase diagram and thesystem stability.

4.4.2 Single-site localization
In this regime, in which repulsive contact interaction and attractive DDI balance each other,tuning the scattering length or the atom number gives rise to different ground states andin particular a different number of occupied layers. In the typical experimental condition
N = [1e3 − 5e4], as = [50 − 80] a0 , only few (1-7) lattice sites are populated. When reducing
the contact interaction or the atom number, the number of populated lattice sites reduces.
Fig. 4.5, plots the evolution of the population of the central lattice site as a function of
scattering length and atom number. We notice that for scattering lengths lower than 55 a0 ,only one lattice site is significantly populated. It is important to highlight that this is aninteraction-driven way to realize quasi-2D planes. From the cuts in Fig. 4.5(b,c), it is possible

4.4 Quantum phases of erbium atoms in a one-dimensional lattice|c 0|2

2

1.5

0.8
0.7

1

0.6
0.5

0.5

0.4
(a)

50

0.3

60

70
as (a 0)80

Population central lattice site

Atom number

0.9

115

N =1e+03

as =52.2 a0

N =5e+03

as =56.0 a0

N =1e+04

as =59.8 a0

1
0.8
0.6
(b)
0.4
50

(c)
60
70
as (a 0)

80

0.5
1
1.5
Atom number

2

Figure 4.5: Population of the central lattice site. (a) Population of the central lattice site|c0 |2 as a function of atom number and scattering length. Horizontal (b) and vertical (c) cut of (a)for fixed atom number and scattering length, respectively.

to notice that the transition to a single occupied lattice site is smooth for low atom numbers
or high scattering lengths, but becomes more abrupt in the opposite cases. To detect thistransition to a single lattice site, in our experiment, we record the momentum distributionof the cloud taken after TOF. The occupation of a single lattice plane gives rise to thedisappearance of the interference pattern. More in-depth, in the publication in Sec. 4.5, westudy the visibility of the interference pattern (contrast) C = nFT (|z ∗ |)/nFT (0) as a functionof the scattering length.

4.4.3 Two-dimensional bound states: solitons and droplets
We theoretically investigate the phase diagram as a function of the atom number and scattering length using the 1D discrete model introduced in Sec. 4.3. In the publication presented
in Sec. 4.5, we show the phase diagram for our experimental system where we include theredistribution of the atom in the different lattice sites when changing atom number or scattering length. This section details the simpler case in which only the central lattice site ispopulated. This results in studying a system trapped in a quasi-2D plane with the following
trapping frequencies ωx,y,z = 2π ×(240(3), 30(3), 6000(100))Hz. The resulting phase diagram
is shown in Fig. 4.6. Note that the phase diagram presented in Sec. 4.5 and the one of Fig. 4.6
coincide for low atom number and low scattering lengths, where all the population is in thecentral lattice site.
The presence of small or even attractive mean-field interactions allows for radially self-boundsolutions. These states are trapped by the lattice potential along the tight direction but are
self-bound radially. In such a state, if we switch off the radial trapping potential, the atoms
change size towards the new ground state, without expanding like in a typical gas phase.
Note that the condition for the self-bound in all directions coincides instead with negative
chemical potential that in our condition is obtained only for lower scattering length (∼ 40a0 )due to the presence of the tight confinement along z.
To theoretically discriminate the self-bound regime from the standard BEC case, in which the4 Bloch oscillations and matter-wave localization
2

1.5

(b)

Dropletl y (y ho )(a)atom number N (10 4)

116

BEC

4
2

(c)

l y (y ho )1

0.5

Soliton50

Trap OFF

55

60

65

scattering length as (a0)7075Trap ON421l x (x ho )21l x (x ho )21l x (x ho )2

Figure 4.6: Phase diagram and energy landscapes. (a) Phase diagram as a function of atom
number and as . The white data points indicate the experimental atom numbers from the publication in Sec. 4.5. The colored blue (green) area represents the lattice-bound regime for the soliton(droplet) case, see discussion in the main text.
(b,c ) Energy landscapes as a function of theradial widths lx and ly , in units of the radial harmonic oscillator lengths xho = 0.50(1) µm andyho = 1.42(1) µm, respectively. Top (bottom) row shows the case without (with) the radial harmonic
trap, for (right) BEC (as , N ) = (70 a0 , 1.5 × 104 ), (center) droplet (as , N ) = (65 a0 , 1.5 × 104 ) and(left) soliton (as , N ) = (51.5 a0 , 0.4 × 104 ) regimes. The darker shading indicates the energy minima.

transverse confinement(x, y) is necessary to prevent the cloud from expanding, we calculate
the energy landscape from the discrete model. We then compare the fully-trapped case to
the one in which the confinement is only along z. For an easier visualization of the result,√we plot the energy landscape as a function of lx = √lη and ly = l η.
In the phase diagram, it is possible to find three different phases. The standard BEC phaseand the self-bound solutions: soliton and droplet. The difference between the two self-boundsolutions is the stabilization mechanism. In the droplet case, the LHY is the repulsive termthat stabilizes the condensate and avoids the collapse. However, for the soliton the stability
relies on the kinetic term.
Using the 1D discrete model we can compare the two energy functional terms associatedwith kinetic energy and LHY:  5/2
2
3/2
X
ℏ
1
2
N
[Ekin [l, η]; ELHY [l, η]] = η+;γQF|cj |5  .
4.54
3/2
3/2
2
2ml2
η
5
(π l llatt )jNote that both terms can avoid the collapse of the condensate due to their scaling with theaveraged size l. In particular, the kinetic term scales as 1/l2 whereas the LHY term as 1/l3 .
For this reason, the kinetic term will usually prevail for larger system sizes, whereas LHY will
be predominant for smaller system sizes. Additionally, while the kinetic term is independentof the number of atoms, the LHY term increases with atom number. For this reason, inthe case of large atom numbers and low scattering lengths, the LHY will be predominant
whereas in the opposite case the kinetic term will be more important.
Let’s imagine that we can switch off the LHY term. Consider a point in the phase diagramin the radially bound regime and assume we increase the atom numbers. This corresponds

4.4 Quantum phases of erbium atoms in a one-dimensional lattice65as=50.5
as=50.5aa00
as=50.5
as=52.8
as=52.8aa00
as=52.84

as=54.8
as=54.8aa00
as=54.80as=57.2
as=57.2aa 0
as=57.2
as=60.000
as=60.0aa 0
as=60.0

3
l x (x ho )

117

2

00

1

0.5

1

1.5

2

atom number N (10 4)

Figure 4.7: Radial width (lx ). Radial width lx as a function of N for different scattering lengths(see legend). The double triangle indicates the transition from the soliton behaviour (decreasingradial width) to the droplet one (increasing radial width).

to move vertically in the phase diagram. In this case, the mean-field attraction increases
and tries to reduce the size of the condensate. Since the kinetic term is independent of theatom number, the wavefunction shrinks up to the new equilibrium point.
The situation is different if we switch off the kinetic term instead of the LHY contribution.
In this case, the LHY term scales faster with N compared to the mean-field interaction terms.
This will now determine an increase in size. We use this different behaviour to distinguishthe different phases in the phase diagram of Fig.4.6. Figure 4.7 plots the evolution of radialwidth lx for different scattering lengths. We observe that the critical atom number for thetransition from soliton to droplet behaviour depends on the scattering length and increases
for higher scattering lengths. This can be understood from the energy functional in Eq. 4.54.
The different behaviour of the radial width as a function of atom number is also used todifferentiate between soliton and droplet in Sec. 4.5. Nevertheless, in the regime where morethan one lattice site is occupied, this definition is less rigorous and further studies of theindividual energy contributions are needed.

118

4 Bloch oscillations and matter-wave localization

4.5 Publication: Strongly dipolar gases in a one-dimensionallattice: Bloch oscillations and matter-wave localization
Communications Physics 5, 227 (2022)a
submitted May 2022; published August 2022
DOI: 10.1038/s42005-022-01009-8

G. Natale,1,2 T.Bland,1 S. Gschwendtner,1 L. Lafforgue,1 D. S. Grün,1 A. Patscheider,1
M. J. Mark,1,2 and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2a

The author of the present thesis performed the measurements together with D. S. G. and A. P. contributed in analyzing the data with L. L., and developed the theoretical model and performing thecorresponding simulations together with T. B., S. G., M. J. M.

ARTICLE
https://doi.org/10.1038/s42005-022-01009-8

OPEN

Bloch oscillations and matter-wave localization of adipolar quantum gas in a one-dimensional lattice1234567890():,;

Gabriele Natale 1,2, Thomas Bland 2, Simon Gschwendtner2, Louis Lafforgue2, Daniel S. Grün2,
Alexander Patscheider2, Manfred J. Mark 1,2 & Francesca Ferlaino 1,2 ✉Three-dimensional quantum gases of strongly dipolar atoms can undergo a crossover from adilute gas to a dense macrodroplet, stabilized by quantum ﬂuctuations. Adding a onedimensional optical lattice creates a platform where quantum ﬂuctuations are still unexplored, and a rich variety of phases may be observable. We employ Bloch oscillations as aninterferometric tool to assess the role quantum ﬂuctuations play in an array of quasi-twodimensional Bose-Einstein condensates. Long-lived oscillations are observed when the chemical potential is balanced between sites, in a region where a macrodroplet is extended over
several lattice sites. Further, we observe a transition to a state that is localized to a single
lattice plane–driven purely by interactions–marked by the disappearance of the interference
pattern in the momentum distribution. To describe our observations, we develop a discrete
one-dimensional extended Gross-Pitaevskii theory, including quantum ﬂuctuations and avariational approach for the on-site wavefunction. This model is in quantitative agreementwith the experiment, revealing the existence of single and multisite macrodroplets, andsignatures of a two-dimensional bright soliton.

1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria. 2 Institut
für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria. ✉email: francesca.ferlaino@uibk.ac.at

COMMUNICATIONS PHYSICS | (2022)5:227 | https://doi.org/10.1038/s42005-022-01009-8 | www.nature.com/commsphys

1

ARTICLETCOMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-022-01009-8

he dipole-dipole interaction (DDI) between magnetic
atoms in an ultracold quantum gas has been key to thediscovery of supersolids1–3 and macrodroplets4,5, states ofmatter with extremely intriguing and counter-intuitive
properties6,7. Macrodroplets are macroscopic quantum statesthat behave in many ways like liquid droplets4,5,8,9. They are atleast an order of magnitude denser than normal Bose–Einsteincondensates (BECs), and can be self-bound. They exist in aparameter regime in which mean-ﬁeld theories predict the collapse of the entire system when the attractive dipolar interactions
overcome the repulsive contact interactions. Instead, the systemremains stable thanks to the so-called quantum ﬂuctuations, thus
providing one of the rare examples where beyond-mean-ﬁeld
interactions substantially change the ground state of thesystem10,11. Although the functional form of the beyond-meanﬁeld term, otherwise known as the Lee-Huang-Yang (LHY)
correction12, is still subject to intense study and debate13,14, its
importance is now undoubted. Isolating beyond-mean-ﬁeld
effects may be crucial to settle disputes on its validity, particularly in dipole dominated systems; however, it is very difﬁcult tohave access to individual interaction contributions. Though, thediffering atom number scaling between mean-ﬁeld and LHY
contributions provide a promising method to differentiate
between them.
Optical lattices enable powerful interferometric approaches to,e.g., measure with high precision the zero-crossing of the scattering length or of the mean-ﬁeld interaction with the so-called
Bloch oscillation (BO) technique15–18, and to achieve an accurate
determination of the background scattering length via lattice
spectroscopy in Hubbard models4,19,20. Moreover, the presence ofthe lattice itself may change completely the phase diagram of thesystem, as shown in seminal experiments with contact interacting
gases21–23. Unique phenomena are predicted with the addition of
long-range DDIs24,25. Experiments with lattice-conﬁned atomic
dipolar gases have already shown important results, e.g., therealization of extended Bose–Hubbard models26 and spinmodels27–30 in three-dimensional (3D) lattices. In 2D lattices,forming quasi-1D tubes, suppression of dipolar relaxation31 andthe controlled breakdown of integrability32 have been observed.
Instead, up to now, 1D lattices, forming an array of quasi-2D
layers, have been used with large wavelengths to load a single
pancake trap33, or multi-layer traps to study the role of DDI in
the stability against collapse34. Further, theoretical proposals havesuggested that the DDI between layers not only can lead to
modiﬁcations within each layer35–38 but also to inter-layer boundstates39–41. Other works predict the existence of bright-soliton
structures along the lattice42 or anisotropic on-site solitons43,44.
However, those proposals lack the important stabilization
mechanism given by the LHY term, which is known to provide
many intriguing phases in continuous systems (e.g., harmonicallytrapped), opening up many questions: What is the ground state of
an attractive dipolar gas in a 1D lattice potential? Can droplets bedelocalized over many lattice planes? Will solitonic solutions
continue to exist?
In the present work, we study an erbium dipolar gas in a 1D
optical lattice with dominantly attractive DDI. We employ BOs as
an interferometric tool to probe the interaction contributions of
the system, and to isolate the role of beyond-mean-ﬁeld effects.
We ﬁnd long-lived oscillations, associated with a minimum in thedephasing rate, close to the cancellation point between mean-ﬁeld
and beyond-mean-ﬁeld interactions, and at scattering lengths
signiﬁcantly shifted from the expected mean-ﬁeld result. We
develop a discrete effective 1D extended Gross-Pitaevskii equation (eGPE) with variational transverse widths45,46. We ﬁnd that
this minimum occurs when the chemical potentials on each siteare equal, not the energies–as has been employed successfully in
2contact interaction dominated systems17,18–due to the difference
in density scaling between the interactions. The close correspondence between theory and experiment shows the validity of
the LHY prediction, even while highly inhomogeneous densities
are expected to break the local density approximation12. Moreover, we see that for low scattering lengths the system undergoes astructural transition to a single localized 2D plane, signifying aninteraction-driven approach to generate systems in reduced
geometries. Finally, using our theoretical model we produce a fullphase diagram of the system, revealing the impact of the LHY
contribution to the predicted 2D anisotropic soliton state43,
which is instead morphed into a droplet solution at high atom
numbers. Though, promisingly, we still ﬁnd soliton-like
solutions exist.
Results and discussion
Setup and preparation. In the experiment, we prepare a degenerate dipolar gas of erbium atoms in a one-dimensional opticallattice as follows. We start with a dipolar quantum gas of 5 × 104
spin-polarized 166Er atoms conﬁned in a cigar-shaped optical
dipole trap47 elongated along y. Typical BEC fractions range from
60% to 80%. The dipolar length for 166Er is ﬁxed at add = 66.5 a0,
where a0 is the Bohr radius. We change the contact interaction
between atoms and therefore the s-wave scattering length as, via
Feshbach tuning48 using a resonance located near zero magnetic
ﬁeld20, as detailed in the “Methods” section. We ﬁx the orientation of B to be along the weak axis (y) of the trap, making the
DDI dominantly attractive4,7.
Once the harmonically trapped cloud is prepared at the desiredas, we switch on a 1D optical lattice, aligned along the gravity
direction (z); see Fig. 1(a). The vertical lattice is created by retroreﬂecting a λ = 1064 nm laser beam. We load the planes byexponentially increasing the lattice depth V0 to 8 Erec in 20 ms,
where Erec = ℏ2k2/2m = h × 10.5 kHz. Here, ℏ = h/2π is the
reduced Planck’s constant (h), m is the mass of 166Er atoms
and k = 2π/λ is the wave-vector of the lattice. The 1D lattice
forms an array of tightly conﬁned quasi-2D planes with a trap
frequency along the tight direction ωz ≃ 2π × 6 kHz, corresponding to an harmonic oscillator length zho = 100 nm. Due to thetransversal Gaussian proﬁle of the lattice beam, we estimate aresidual trap of frequencies ωx;y ¼ 2π ´ ð4; 4ÞÞ Hz. The tunnelingrate, J, between planes is about h × 33 Hz. For these 1D lattice
parameters, ℏωz > kBT and the system is kinematically 2D49.
Bloch oscillations in a one-dimensional optical lattice. We ﬁrst
aim at inducing Bloch oscillations to interferometrically assess therole of beyond-mean-ﬁeld effects and test the validity of the LHY
term. We thus suddenly switch off the dipole trap and let thesystem evolve in the combined lattice and gravitational potential
for a variable hold time th. Finally, using standard absorption
imaging after 30 ms of time-of-ﬂight (TOF), we record the evolution of the momentum distribution and extract the position of
the main peak, qmax , as a function of th. Figure 1b shows an
exemplary set of absorption images during a single Bloch period
TBO. We observe the key paradigm of BOs, i.e., the linear increaseof the mean momentum due to the acceleration and the Bragg
reﬂection occurring at the border of the Brillouin zone50, welldescribed by ﬁtting a sawtooth function to qmax .
The high sensitivity of BOs to interactions17,51 clearly appears
by tracing the evolution for two different as (see Fig. 1c, d), as theinteraction dependence is encoded into the dephasing rate. For acontact-dominated gas (add < as = 90 a0, Fig. 1c), we see that theBOs vanish within a few TBO. On the contrary, decreasing as, andthereby going into the regime where contact interactions and DDI
nearly compensate each other (as = 60 a0, Fig. 1d), we observe

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(a)

(b)

(a)
10-1
(ms-1 )

zyqmax (hk)

(d)

//

1
0
-1
1

10-2

10-3

(1)

10-4
55

//
//

-1
0

2

4

//
6
8
10
21
hold time th ( T BO )23

25

Fig. 1 Bloch oscillations of a dipolar BEC in a one-dimensional opticallattice. a Sketch of our experiment, consisting of a 1D optical lattice in thez-direction, loaded with an erbium BEC from an optical dipole trap withtrapping frequencies ωx,y,z = 2π × (240(3), 30(3), 217(1)) Hz. Gravity acts
along z. b Absorption images after time of ﬂight (TOF) showing the
momentum distributions during one Bloch cycle. c, d Evolution of the peak
position of the momentum distribution (qmax ) for as = (71.6(1.0), 59.8(1.0))
a0, respectively. A sawtooth ﬁt (solid gray) to the data yields
TBO = 0.469(4) ms, consistent with the expected value TBO = 2k/(mggrav).
The error bars represent the standard error on the mean over 4–6
repetitions.

persisting oscillations for more than 25 Bloch cycles, set by ourlimited observation time (for details on the analysis of the
momentum distribution during Bloch Oscillation, see Supplementary Note 4). To systematically study this effect, we repeat the
BO measurements for different values of as, and extract thecorresponding dephasing rate γ. As shown in Fig. 2a, we observe aresonant-type behavior with γ showing a pronounced dip with aminimum at as = 61 a0. This minimum is clearly different to thepoint as ≈ add, where the variance of the mean-ﬁeld energies
across different lattice sites cancel18, which would be expectedfrom previous observations17,51.
To get further insight on the origin of the minimum, wedevelop a discrete effective 1D eGPE, inspired by the close
correspondence between predictions from discrete models andexperimental observations in non-dipolar52,53 and weakly
dipolar18 BECs. We separate the 3D wavefunction into radial
and axial contributions, allowing for a variational anisotropic
radial width and thus maintaining the 3D character45. Along thelattice direction (z), we further decompose the wavefunction,
ψ(z, t), as a sum of Wannier functions
pﬃﬃﬃﬃw(z) of the lowest energy
band over all lattice sites: ψðz; tÞ ¼ N ∑j cj ðtÞwðz  z j Þ, where N
is the atom number and cj(t) the complex wavefunction
amplitude on lattice site j, leading to a set of discrete effective
1D eGPEs, each including mean-ﬁeld and beyond mean-ﬁeld
interactions. For the beyond-mean-ﬁeld interaction, the 3D formof the LHY still fully applies since the contact interaction energy
exceed the conﬁnement energy scale (for details on the 2D to 3D
crossover, see Supplementary Note 2)54. However, our system
may also open to further studies on the 2D to 3D crossover of the
LHY. We solve these equations coupled to a minimization of the

(2)
60

(b)
0

/h (Hz)qmax (hk)(c)

dephasing rate

B

x

(1)

(3)

(4)

65
70
scattering length as (a 0)

(2)

(3)

75

(4)

100
0

-2

0

2

-2

site j0site j2-20site j

2

-2

0

2

site jFig. 2 Dephasing rate and chemical potential distributions. a Experimental
dephasing rate γ (circles) as a function of scattering length as. The green
solid line shows the theory result, with an uncertainty region (shaded area)
accounting for 20% atom number variation. The blue dashed line shows thetheory expectation without Lee-Huang-Yang (LHY). The gray dot-dashedline gives the prediction of the semi-analytic approximation for γ. Error bars
show the 68% conﬁdence interval (see Supplementary Note 4 for details
on the analysis of the momentum distribution during Bloch Oscillation). The
statistical uncertainties on the scattering length is smaller than the symbols
size. b Chemical potential per lattice site μj extracted from the discretemodel for as = (59, 60, 65.5, 70) a0(1, 2, 3, 4). The green area depicts the
LHY contribution to μj.

energy functional with respect to the variational parameters todetermine the ground states, benchmarking them against the full
3D theory. We then perform dynamic simulations of the expected
time evolution (see the “Methods” section), giving an accuratedephasing rate (solid line) in Fig. 2a without free parameters.
In previous studies17,18, the point of minimum dephasing was
found to occur when the mean-ﬁeld interaction energies vanish orcancel. We isolate the mean-ﬁeld contribution by removing beyondmean-ﬁeld effects from our simulations (dashed line in Fig. 2a),predicting a minimum at as ≈ add. However, this is in clear
contradiction with our experimental observations by a shift of 6a0
and a different overall shape due to the different scaling of the LHY
term with the density. Without LHY, the cancellation of mean-ﬁeld
jenergies, EMF , is equivalent to the cancellation of onsite chemicaljpotentials, given by μj ¼ 2EMF =jcj j2 . Note, μj dictates the wavefunction phase winding on each site through cj ¼ jcj jeiμj t=_ . Reintroduj
jcing quantum ﬂuctuations, we obtain μj ¼ ð2EMF þ 5=2EBMF Þ=jcj j2 ,5
where the 5/2 appears due to the ∣cj∣ density scaling in the beyondj
mean-ﬁeld energy ðEBMF Þ. Figure 2b shows μj from the ground statecalculation for four scattering lengths, additionally indicating thecontribution of the LHY correction.
We observe that the point of minimal dephasing in theexperiment is close to the point where the variance of μj is
minimized. Indeed, within a semi-analytic approximation (seeSupplementary Note 3 for details on the analytic model ofdephasing), we ﬁnd a direct relationship between γ and μj, whichreads γ ∝ ∣μ1 − μ0∣ when 3 lattice sites (j = −1, 0, 1) are occupied.

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(a)

(2)

contrast C

0.6
(1)
0.4

(2)

0.2
(1)
0
45

contrast C

(b)

50

0.6

(1)

55

(1)

60
65
70
scattering length as (a 0)
(2)

75

80

(3)

0.4

(3)

0.2
0

(2)
0

10

20
30
hold time th (ms)40

50

Fig. 3 Interaction-induced localization. a Contrast of the interference
pattern after loading the lattice at different as. The green dot-dashed (blacksolid) line represents the result of the 1D discrete model (3D extended GrossPitaevskii equation, eGPE) multiplied by 0.7. The insets show the respective
density distributions along z of the 1D discrete model (bars) and 3D eGPE
(lines) and corresponding experimental averaged interference patterns after
TOF expansion (1,2). The statistical uncertainties on the scattering length issmaller than the symbols size. b Dynamic evolution of the contrast quenchingback (ﬁlled circles) or holding as (open circles); see text. The error bars
represent the standard error on the mean over 4–6 repetitions.

This model can be extended to 5 lattice sites, giving the dotdashed line (Fig. 2a) which reproduces very well the systembehavior. These results lead us to believe that measuring thedephasing rate through the chemical potential will be ubiquitousto systems with arbitrary interaction potentials, which can besubstantiated through future work. Note, the total (MF + LHY)
interaction energies over the system cancel at 57.5 a0, which does
not coincide with the minimum of the dephasing, occurring at
approximately 60 a0. Additionally, we mainly attribute theasymmetric shape of the dephasing rate in both the theory andexperiment to the change of the number of occupied sites in the
ground state, which smoothly reduces from ~5 to 3 from 75 to
60 a0, before rapidly reducing to a single occupied lattice site
at ~55 a0 (see later discussion and insets of (Fig. 3a).
Interaction-induced single site localization. By decreasing thescattering length below 57 a0, no BOs nor interference peaks arevisible anymore. We observe at the initial instant (th = 0 ms) thatthe momentum distribution is already spread over the entire ﬁrst
Brillouin zone. To quantify this, we study the contrast, C, of theinterference pattern of the initial momentum distribution as afunction of as, see Fig. 3a. We extract C, deﬁned as the amplitudeof the momentum peaks at ±2ℏk relative to the zero momentum
peak, from the Fourier analysis of the TOF images (see the“Methods” section). For large as, we observe the typical matterwave interference pattern, as expected from a coherent state
populating several lattice planes (see inset)53. As we lower as, C
ﬁrst remains fairly constant. For as below a certain critical valueas  57 a0 , we observe a sudden loss of the interference patternwith a sharp decrease of C to almost zero.
4

Additionally, we observe that this interaction-driven process isreversible. To test the restoring of the interference pattern, we
employ the following protocol (see the “Methods” section): In
brief, we ﬁrst prepare the system in the lattice at constant andlarge as (as = 69(2) a0). We then ramp down as below as(as = 56(2) a0) in 20 ms and wait until C stabilizes to a smallvalue; see Fig. 3b. Note that the interference pattern disappears
after about 10 ms, which is on the order of the tunneling time h/J
between two neighboring lattice sites. At this point, we quench as
back to its initial value and probe the time evolution of the system
towards its new equilibrium state. On a similar timescale, we
observe the reappearance of the interference pattern with anincrease of C, which then saturates to about 60% of its initialvalue. We attribute the partial recovery of the contrast to anincrease of the losses and heating. These arise from the combined
effect of an increase of the density and three-body scattering rates
occurring when tuning the magnetic ﬁeld closer to a Feshbachresonance4. For comparison, we also show the data without
inverting the ﬁeld ramp.
The observed broad distribution in reciprocal space suggeststhat the system ground state has undergone a structural change,
with the macroscopic wavefunction localized in one lattice plane.
To verify this interpretation, we calculate the ground state of thesystem as a function of as. When the repulsive contact interaction
dominates (as > add), we ﬁnd an array of BECs occupying
approximately three to ﬁve lattice planes; see insets Fig. 3a. In
contrast, when the relative strength of the attractive dipolarinteraction with respect to the other terms in the Hamiltonian isincreased, the system reaches a critical point. Here, it undergoes aphase transition to a quasi-2D state, in which all atoms are
localized into a single lattice plane to minimize their energy. Thispurely interaction-driven phase transition–somewhat reminiscentof a continuous version of a superﬂuid to Mott insulatortransition55–is stabilized by quantum ﬂuctuations (LHY), preventing the subsequent collapse of the system42,56. The predicted
critical point occurs exactly where we observe the disappearanceof the interference pattern in the experiments. We ﬁnd an overall
excellent agreement between the measured and the calculated Cfrom both the discrete 1D model and the 3D theory without any
free ﬁtting parameters, except for a rescaling factor to the contrast
amplitude to account for the thermal atoms in the experiment.
Two-dimensional bound states: solitons and droplets. The
observation of this phase transition to a quasi-2D localized statedriven by interactions points to the existence of a rich variety ofphases. The importance of the LHY correction and its peculiar
density scaling motivate us to investigate the properties of theground state as a function of as and atom number to identify
distinct phases in this unique setting. For this, we employ ourdiscrete model to derive a full phase diagram; see Fig. 4a. To
investigate the boundness of the states, we assess the impact of theradial harmonic trap on the minimum of the variational energy,
which is a function of the radial widths lx and ly (the individualenergy contributions are shown in Supplementary Note 1). At
large scattering lengths, as expected, we ﬁnd a stable delocalized
BEC phase, where the total interaction energy (mean-ﬁeld +LHY) is positive. The state is trap-bound, meaning that there isno energy minimum without the radial harmonic conﬁnement;
inset of Fig. 4a.
Reducing as, we ﬁnd an energy minimum even without theradial harmonic trap (Fig. 4b, c), highlighted as colored region in
the phase diagram. These quasi-2D self-bound solutions (thelattice still provides axial conﬁnement) are either extended over
several sites (lighter color) or localized to a single plane (darker
color). In the literature, there are two paradigmatic examples of

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1.5

DropletBEC
l y (y ho )atom number N (10 4)

(a)

1

4
2

0.5

Soliton5055

12l x (x ho )

65

70

75

scattering length as (a0)
(d)
4

1.1

2

l x (x ho )l y (y ho )l y (y ho )

(b)

(c)

60

1
2
l x (x ho )

4

1
0.9

2
1
2
l x (x ho )12l x (x ho )

0.8
0

1

2

solitons. This opens the door to future experimental investigationon the self-bound nature and properties of these 2D phases.
In conclusion, we theoretically and experimentally investigate
the behavior of a strongly dipolar quantum gas in a 1D opticallattice. We employ BOs and characterize their dephasing rate as afunction of as. We observe a minimum in the dephasing shifted
6 a0 away from the purely mean-ﬁeld prediction, providing aninterferometric measure of the beyond-mean-ﬁeld contribution.
For low enough as, the system enters into a quasi-2D state which
is localized onto a single lattice plane, providing a genuineinteraction-driven path to reach reduced dimensions in dipolargases. Using our developed discrete theory model, we derive a fullphase diagram which conﬁrms the observed localization transition. This also reveals signatures of quasi-2D self-bound dipolar
droplet solutions, and the long sought-after 2D anisotropic
dipolar soliton, ﬁrst predicted in ref. 43 (see also44,60). Our workpaves the way for future studies of the soliton-to-droplet
crossover in a dipolar gas, as observed in a Bose–Bose gas61,
and of the “solitonic” nature62 of dipolar solitary waves63–67. A
future promising prospect will be to observe the density proﬁleswith in-situ imaging such to experimentally discriminate between
the soliton/droplet regimes.

4

atom number N (10 )Fig. 4 Phase diagram and energy landscapes. a Phase diagram as afunction of as and atom number. The white region denotes a trap-bound
BEC extended over several lattice sites. The colored regions denote quasi2D self-bound solutions: a droplet (green), a soliton (blue), each either
extended over several lattice sites (lighter shade) or localized (darker
shade, >95% of the atoms are localized in the central lattice plane). Circles
show the atom number from our experimental data points in Fig. 3a. The
statistical uncertainties on the scattering length is smaller than the symbols
size. a–c Energy landscapes as a function of the radial widths lx and ly, inunits of the radial harmonic oscillator lengths xho = 0.50(1) μm andyho = 1.42(1) μm, respectively, with (left) and without (right) the radial
harmonic trap, for (a) BEC (as, N) = (70 a0, 1.5 × 104), (b) droplet(as, N) = (65 a0, 1.5 × 104) and (c) soliton (as, N) = (51.5 a0, 0.4 × 104)
regimes, with darker shading at the minima. d Radial width lx versus N for
as = 51.5 a0. The dashed line indicates the soliton-to-droplet transition
point, and the circles indicate the position of b, c.

self-bound objects with attractive mean-ﬁeld energy: droplets andsolitons. Droplets can exist in one, two or three dimensions andare stabilized through the LHY correction7. Stable bright solitons
only exist in quasi-1D systems with attractive contact interactions
and are stabilized against collapse purely by kinetic energy. In the
search for solitons in higher dimensions, theoretical studies havesuggested that the DDI could stabilize such 2D solutions43,44. To
the best of our knowledge, there have been no studies on theeffect the LHY correction has on this prediction, nor experimental observation. In the present case, where many interactions
and kinetic energy compete, a classiﬁcation of self-boundsolutions is much less straightforward. As a crucial distinction
between a droplet (Fig. 4b) and a soliton (Fig. 4c), we extract thewidth lx from the energy landscape minimum in the radiallyuntrapped system, and investigate its scaling with atom number(Fig. 4d). The soliton width (along the collapse direction) scales
inversely with increasing atom number57, while in contrast, thedroplet size increases in all directions with N58, as predicted in aquasi-1D setting59. We use this distinction to draw a boundary
between the two phases, observing a phase transition at around5000 atoms, for both single-site and multi-site solitons. The
overlaying of our measurements (Fig. 3a) onto the phase diagramsuggests that the experiments have already reached the unexplored regimes of both 2D self-bound droplet and dipolar

Methods
Theoretical model. In this work, we use an extended Gross-Pitaevskii theory fordirect comparison to our experimental results. We employ both the standard threedimensional form of the extended Gross-Pitaevskii equation (eGPE) and derive a
discrete effective one-dimensional eGPE. Starting with the three-dimensional case,our system can be described by the 3D eGPE of the form4,68–70
i_

h _2
∂ !
∇2 þ V harm ð!
xÞΨð x ; tÞ ¼ 2m∂t
x ; tÞj2
þ V latt ðzÞ  F ext z þ gjΨð!
Z
0
0
0
!
!
!
þ d3 x U dd ð x  x ÞjΨð!
x ; tÞj2i
þ γQF jΨð!
x ; tÞj3 Ψð!
x ; tÞ;ð1Þ

where the wavefunction Ψ is normalized to the total atom numberR
N ¼ d3 !
x jΨj2 . The atoms are conﬁned in a harmonic potential V harm ¼∑ξ¼x;y;z 12 mω2ξ ξ 2 with single particle mass m and trap frequencies ωξ, together withthe lattice potential V latt ¼ sErec sin2 ðkzÞ where s is the tunable lattice depth inmultiples of the recoil energy Erec and k = 2π/λ is the lattice spacing in reciprocal
space. The mean-ﬁeld interaction contributions are g = 4πℏ2as/m for the contact
interaction, governed by the s-wave scattering length as, and the long-ranged
anisotropic dipolar interaction potential U dd ð!
x Þ ¼ 3_2 add =m 1  3cos2 θ =j!
x j3 ,
2
where add ¼ μ0 μ2m m=12π_ with magnetic moment μm and θ is the angle between
the polarization axis (y-axis) and the vector between neighboring atoms. We also
include beyond-mean-ﬁeld
effects through the quantum ﬂuctuations term
qﬃﬃﬃﬃ
a3s
3 2 12γQF ¼ 32
, which depends on the relative strength between theπ 1 þ 2 εdd
3 gdipolar and short-ranged interactions εdd = add/as. Finally, Fext = ggravm denotes
the external force exerted on the system by gravity.
In this work, we employ the imaginary time-evolution technique on Eq. (1) inorder to ﬁnd stationary solutions for the wavefunction in the lattice, without
gravity. For various atom numbers and scattering lengths, we use a numerical gridof lengths (Lx, Ly, Lz) = (6, 33.3, 6) μm, with corresponding grid points
128 × 256 × 128. The dipolar term is efﬁciently calculated in momentum space, andwe use a cylindrical cut-off in order to negate the effects of aliasing from theFourier transforms71.
To derive the effective one-dimensional model, we follow ref. 45 by assuming a
wavefunction decomposition
2
2
2
Ψð!
x ; tÞ ¼ Φðx; y; l; ηÞψðz; tÞ  p1ﬃﬃπl eðηx þy =ηÞ=2l ψðz; tÞ ;

ð2Þwith variational parameters l and η representing the width of the radial
wavefunction and the anisotropy of the state, respectively. Integrating out thetransverse directions (x, y) in Eq. (1) upon substitution of the ansatz above gives thecontinuous quasi-one-dimensional eGPE, which when combined with a variational
minimization of the energy functional to ﬁnd (l, η) gives close agreement to the full
3D eGPE45. We further decompose the longitudinal wave function ψ(z, t) into asum of Wannier functions w(z) of the lowest energy band over all lattice sites
pﬃﬃﬃﬃψðz; tÞ ¼ N ∑ cj ðtÞ wðz  z j Þ ;ð3ÞjCOMMUNICATIONS PHYSICS | (2022)5:227 | https://doi.org/10.1038/s42005-022-01009-8 | www.nature.com/commsphys

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for complex amplitudes cj, and positions of lattice minima zj = (λ/2)j. For deep
enough lattices, the Wannier functions are well approximated by Gaussians of the1=4 z2 =2l2
p
ﬃﬃ 1
4
latt , with lform wðzÞ ¼ πl2lattelatt ¼ ðk sÞ . After multiplying on the leftby cj and integrating over z, we obtain a set of discrete effective one-dimensional
eGPEs
i_



∂cj
2
1D 3=2¼ Jðcjþ1 þ cj1 Þ þ F ext z j þ V harm ðzÞ þ g 1D Njcj j2 þ N ∑ U ddγQF jcj j3 cj ;jjkj jck j þ γQF N
∂t
k

ð4Þ
with the reduced effective one-dimensional parameters γ1D
QF ¼
3=2

23=2 =ð5π 3=2 l2 llatt Þ γQF and g1D = g/((2π)3/2l2llatt). Here, J denotes the tunnelingrate between two neighboring lattice sites. The dipolar interaction coefﬁcients
between lattice sites j and k depend both on the separation ∣j − k∣, and non-triviallyon the size l and anisotropy η of the transverse cloud. For the variational
minimization, we generate an interpolating function for a sensible range of (l, η)and separations up to ∣j − k∣ = 6 via
Z
n
U dd
d3 !
x jΨ0 ð!
x  z jjkj^ez ; l; ηÞj2
jjkj ðl; ηÞ ¼

Z
ð5Þ
0
0
0
´ d3 !
x U dd ð!
x !
x ÞjΨ0 ð!
x ; l; ηÞj2 ;0
where Ψ0 ð!
x ; l; ηÞ ¼ Φðx; y; l; ηÞwðzÞ [see Eqs. (2) and (3)]. This allows us tosimply look up the values of U dd
jjkj without having to recalculate for every time stepduring the energy minimization. We note that the energy contribution rapidly
declines for separations larger than 2 sites, and ﬁnd that 6 is more than sufﬁcient toquantitatively describe the physics.
To ﬁnd the stationary state solution of Eq. (4) (without gravity) we employ an
imaginary time-evolution in combination with an optimization scheme, aiming toﬁnd the state which minimizes the total energy functional

E½c; l; η ¼ E ? ½l; η þ E k ½c; l; η ;

ð6Þ

where c = (c1, c2, …, cn) for n total lattice sites. Here, E ? ½l; η gives the energy
contribution from the transverse variational wave function, which reads




_2
1
ml2 ω2x
ð7Þ
E ? ½l; η ¼
þ ηω2y :
þ
2 ηþ
4
η
η
2ml
The latter term of Eq. (6) gives the discrete energy functional for the amplitudes
cj, which includes the tunneling and all interaction terms
E k ½c; l; η ¼  ∑ Jðcjþ1 þ cj1 Þcjj11þ Ng 1D ∑ jcj j4 þ N ∑ U dd
jc j2 jc j2
ð8Þ
2
2 j;k jjkj k jj
2 3=2 1D
5
þ N γQF ∑ jcj j :5jStarting from an initial distribution of the amplitudes cj we ﬁrst determine thevariational parameters (l, η), which is done via an optimization scheme minimizing
Eq. (8). Subsequently, we evolve the amplitudes in imaginary time using Eq. (4)and repeat this process until we ﬁnd the minimum of the total energy function Eq.
(6). Once we have the ground state of the system, we employ the discrete effective
one-dimensional eGPE in real-time to simulate the Bloch oscillations in thepresence of gravity. To account for changing transverse widths, after removing theharmonic potential from the optical dipole traps, we take l and η from theharmonic oscillator length q
calculated
from the residual trapping potential of the
ﬃﬃﬃﬃﬃﬃﬃ
lattice beams, given by l ¼lx ly and η = ly/lx. This provides a good approximation

to the time averaged transverse behavior, which we have veriﬁed throughcomparison with the 3D eGPE.
Experimental protocol. We prepare a 166Er spin-polarized BEC similar to ref. 4.
The magnetic ﬁeld during the evaporation is along the z-axis with absolute value∣B∣ = Bz = 1.9 G (as = 80(1) a0), see Fig. 1a. The magnetic ﬁeld sets also the valueof the scattering length thanks to a Feshbach resonance, centered close to 0 G. For
the magnetic ﬁeld range considered in this work, the B-to-as conversion has been
precisely mapped out in previous experiments4,20. Before loading the lattice, werotate the magnetic ﬁeld direction along the y-axis in 50 ms and change its absolutevalue to set the scattering length. At this step, we typically achieve 5 × 104 atoms
with more than 60% condensed fraction in a cigar shape dipole trap with trapping
frequencies ωx,y,z = 2π (240(3), 30(3), 217(1)) Hz. For our experiments, the atoms
are then loaded in a 1D lattice by a 20 ms exponential ramp of the lattice depth.
This is the experimental protocol used in Figs. 1, 2, and 3a.
To study the reversibility of the interaction-induced transition to a single latticesite (Fig. 3b), i.e., the evolution of the contrast due to a change of the scattering
length, we employ a different protocol from the one above. In fact, in our
experiment, the magnetic ﬁeld along the y-direction can be changed on a timescaleof ≃20 ms, which is slower compared to the z-direction (≃1 ms). For this dataset,
we prepare the BEC with B = (0, 0.25, 1)G and then we load the lattice as describedabove. We then linearly ramp the ﬁeld in 20 ms to B = (0, 0.25, 0)G and record the6time evolution. In Fig. 3b, we study the contrast evolution after the ramp. For theblack dataset, the magnetic ﬁeld is quenched back to the initial value after 10 ms.
For Fig. 4, we extract the atom number condensed in the lattice by releasing thecloud from the combined ODT-lattice trap and by performing an absorption imaging
after 30 ms of TOF. We integrate the density along the lattice axis and use a double
Gaussian ﬁt on the integrated density proﬁle. We repeat the sequence 4–8 times forevery scattering length. At low scattering lengths, we ﬁnd a decreased number ofcondensed atoms, see Fig. 4. We attribute this to an increase of three-body loss in thevicinity of a Feshbach resonance4 and the increased density of the groundstate.
Contrast of the interference pattern. The density modulation that usuallycharacterizes a BEC loaded into a 1D lattice can be experimentally extracted fromthe matter-wave interferometry after a TOF expansion55. To study the transition to
one single occupied lattice site, we record the density distribution as a function ofas. In more details, for each picture we perform a Fourier transform (FT) of theintegrated momentum distribution, n(qz). In the contact dominated regime, thelattice induces two sidepeaks at ± qz in n(qz). Consequently, in the FT analysis thepeaks are at z* ≃ λlattice. The visibility of the interference pattern is then estimatedas nFT(∣z*∣)/nFT(0).

Data availability
Experimental data is available on reasonable request from the authors.

Code availability
Code used to generate the theory results is available on reasonable request from theauthors.

Received: 16 May 2022; Accepted: 31 August 2022;

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Acknowledgements
We thank R.N. Bisset, A. Houwman, L. Lavoine, R. Grimm, and L. Tarruell for stimulating discussions, and B. Yang for his support in the early stage of the experiment. This
work is ﬁnancially supported through an ERC Consolidator Grant (RARE, no. 681432)
and a DFG/FWF (FOR 2247/I4317-N36). 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 been achieved using the HPC
infrastructure LEO of the University of Innsbruck.

Author contributions
G.N., D.S.G. and A.P. conducted the experiment and collected the experimental data.
G.N. and L.L. analyzed the data. T.B., S.G., M.J.M and G.N. developed the theoreticalmodel and performed the corresponding simulations. All the authors contributed to thewriting of the paper. F.F. supervised the project.

Competing interests
The authors declare no competing interests.

Additional information
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s42005-022-01009-8.
Correspondence and requests for materials should be addressed to Francesca Ferlaino.
Peer review information Communications Physics thanks the anonymous reviewers fortheir contribution to the peer review of this work.
Reprints and permission information is available at http://www.nature.com/reprints
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional afﬁliations.

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Supplementary information: Bloch oscillations and matter-wave localization of adipolar quantum gas in a one-dimensional lattice
Gabriele Natale,1, 2 Thomas Bland,2 Simon Gschwendtner,2 Louis Lafforgue,2Daniel S. Grün,2 Alexander Patscheider,2 Manfred J. Mark,1, 2 and Francesca Ferlaino1, 21Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
2Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

SUPPLEMENTARY NOTE 1: ENERGYCONTRIBUTION

200

Energy (Hz)The rich phase diagram of dipolar atoms relies on thecompetition between the different interaction terms. In
Fig. S1 we assess the different interaction energy contributions to Eq. (6) for a range of scattering lengths. For
as > add the total interaction energy is positive, andit corresponds to a dilute BEC. Following as to smaller
values all interaction contributions are almost constant,until at around as = 60 a0 there is a phase transition
from the BEC to droplet state, as identified in Fig. 4of the main text. This sharp gradient ceases at aroundas = 55 a0 , where the atoms are localized to a single
lattice plane. Note that although the DDI offsite energy is typically only 10% of the onsite counterpart, it
constitutes a significant contribution to the total interaction energy in the system, shifting the BEC to droplet
crossover and localization transitions by a few a0 .

300

E int < 0

DD offsite

E int

DD onsite

QF

Contact

100
0
-100
-200
-300
50

55

60
65
scattering length a s (a 0)7075FIG. S1. Interaction energy contributions. Scattering
length dependency of the individual interaction contributionsof the ground state solutions from the 1D model, calculated
for N = 104 atoms.

SUPPLEMENTARY NOTE 3: ANALYTIC
MODEL OF DEPHASING
SUPPLEMENTARY NOTE 2: 2D TO 3D
CROSSOVER

The dimensionality of the system is known to highly
influence the size and even the sign of the beyond-meanfield contribution, in both Bose-Bose1–3 and dipolar4–6
gases. Here, we assess the validity of employing the full
3D LHY correction to our system. Following Ref. ,6 wedefine the dimensionless parameter ξ = gn/0 –dependent
on the contact interactions g, peak 3D density n, and2
the confinement energy scale 0 = ~2 π 2 /2mzho–that indicates which dimensionality regime our system is in. If
ξ & 1 we are safe to use the 3D LHY term, whereas if
ξ  1 the 2D solution deviates from the 3D one. Deepin the localized droplet regime, where the peak density ison the order of 1022 m−3 , we find ξ ≈ 2, and the 3D LHY
as used throughout this work is valid. Even at large
scattering lengths, where the peak density is closer to
5 × 1020 m−3 , we find ξ ≈ 0.5, which introduces an errorof less than 5% between the 2D and 3D LHY terms.6 In
this limit, the 2D LHY term may be more appropriate,however in the dilute BEC phase the impact of the LHY
is minimal.

Starting from the discrete 1D eGPE we decompose
the coefficients cj into amplitude and phase as cj =|cj | exp(−iφj ), and then integrate Eq. (4) in time to giveφj (t) =− Fext zj + g 1D N |cj |2 + NXkdd
U|j−k||ck |2

1D 3/2+ γQF
N γQF |cj |3≡ (−Fext zj + µj )

t
,
~

!

t
~
(S1)

with onsite chemical potentials µj , and where we havealso assumed that Fext d  J such that the amplitudes|cj | are frozen.
Following Ref. ,7 we write the Fourier transform of thequasi-1D wavefunction asψ(k, t) = w(k)Xj|cj | exp[−i(kzj + φj (t))] = w(k)C̃(k, t) ,
(S2)

2
0.45
0.4
0.35
0.3
0.25

0

2

4
6
hold time th ( T BO )8

10

FIG. S3. Evolution of the h|qz |i . In the figure h|qz |i asa function of time for as =58.8 a0 . The points with red edges
are the one selected for the fit. The red line corresponds tothe fit result A(t − τ ) + 0.5

Exact solutions to |C̃(0, td )|2 = α can be only found inlimiting cases. For the three lattice site case, with j =−1, 0, 1 and noting the symmetry of |cj | = |c−j | we obtain
FIG. S2. Analytic dephasing rate. (a) Evolution of thefunction C̃, with k̃ normalized to the Brillouin zone in themoving frame, and as = 60.5 a0 . Here, the solution of Eq. (S4)
is td = 0.59s. (b) Time evolution of the central point of C̃,showing when |C̃|2 crosses α = 0.5. The function C̃ is scaled
to the value at C̃(0, 0). (c) Analytic dephasing rate (γ =1/td ) obtained for the 3 lattice site approximation Eq. (S4)and the 5 lattice site approximation Eq. (S5), compared to
the numerically obtained value from a real-time simulation of
the discrete model, Eq. (4).

where w(k) is the momentum space Wannier function,and phases φj are given above. If all interactions are
set to zero this function is initially a delta function situated at k = 0 and moves in k-space as k̃ = k − Fext t/~.
Interactions broaden C̃(k, t), leading to a dephasing ofcoefficients cj . Fig. S2(a) depicts |C̃(k, t)|2 as a functionof k at different times t, normalized to |C̃(0, 0)|2 .
We extract an analytic approximation to the dephasing time by considering the temporal behaviour of thepoint |C̃(0, t)|2 , i.e. at k̃ = 0. During dephasing this point
rapidly decreases through interference between neighboring sites. This quantity is plotted in Fig. S2(b) for a
few example scattering lengths. It reaches the threshold α/C̃(0, 0) = 0.5 at the dephasing time t = td , where
many k-modes are now highly occupied. This time canbe found through the smallest positive solution of


α= 

X
j

|cj | cos






 2

 2
X
µj td µj td +|cj | sin
.
~
~
j
(S3)

td = arccos



4|c0 c1 |α + |c0 |2 − 2~,(µ1 − µ0 )(S4)This relation is expected to give an accurate prediction of
the dephasing time for all states where only 3 lattice sites
are dominant. From this equation, one can see how thedephasing time tends to infinity in the limit of equallydistributed chemical potentials, as observed in Fig. 2 ofthe main text. We can extend this to 5 sites, but it is
not as trivial. One needs to numerically solve the transcendental equation(µ0 − µ1 )td
α = 2 − |c0 |2 + 4|c0 c1 | cos~(µ0 − µ2 )td+ 4|c0 c2 | cos~(µ1 − µ2 )td,+ 8|c1 c2 | cos~(S5)

for the smallest non-zero root td . We compare the resultsfrom Eqs. (S4) and (S5) to the numerically obtained dephasing rate, γ = 1/td , in Fig. S2(c), as presented inFig. 2 of the main text, and find excellent agreement.
SUPPLEMENTARY NOTE 4: ANALYSIS OF
MOMENTUM DISTRIBUTION DURING BLOCH
OSCILLATION

When the Bloch oscillation dephases, the width of the
momentum distribution increases with time.8 To evaluate the dephasing rate we analyze the 1D momentum
distribution along z, n(qz ), as a function of the holding
time. Because of our limited vertical optical access, the

3
17

(a)

18

19

20

21

22

A (ħk.ms -1 )

0.9
0.8
0.7

(b)

3

10 -3

P( )

2

1

0

12

13

14
15
16
dephasing time (ms)17

FIG. S4. Uncertainties χ2 analysis. (a) χ2 as a functionof A and τ . The black dash lines correspond to the value
χ2min + 1. Below, (b) probability distribution of τ , given byR2numerical integration of A dAe−χ (A,τ )/2 and normalization
to 1. as =58.8 a0 . The dashed line indicates the fit result andthe shaded area the 68% confidence interval.

1D lattice is not perfectly aligned with the z (gravity) direction. We measure a tilt of 9(1)°. Such a tilt effectively
weakens the radial trapping strength, limiting our observation time to 12 ms, which anyhow allows us to observe
up to 25 BO period.
From n(qz ), we can extract the maximum position(qzmax ) and the quantity h|qz |i , given byX
h|qz |i =n(qz ) |qz − qzmax | .

This quantity is proportional to the width of the distribution. In Fig. S3, we report h|qz |i for as = 65.7(1.0) a0 .
To quantify the dephasing rate γ, we apply a linear fit toh|qz |i . For the fit, we select only the points at the centerof the Brillouin zone, up to the time when h|qz |i is reaching the fully dephased configuration, 0.5~k. Indeed, when
the cloud is at the edge of the Brillouin zone, h|qz |i is artificially increased and it does not represent the dephasing,as shown in Fig. S3. We define the dephasing rate γ as
the inverse of the time τ that the fitted function needs toreach the value 0.5 ~k. Thus, using the fit parametrization A(t−τ )+0.5, where A and τ are the fitting variables
and t is the time, we can directly extract τ and its inverseγ.
To determine the uncertainties with our non-linear
parametrization, we analyze the χ2 (A, τ ). We estimate
the uncertainties on our data points by assuming equal
statistical fluctuations around our fitting model and using the expected value χ2 = Ndata − 2. Figure S4
shows a clear asymmetric shape for χ2 , indicating asymmetric uncertainties on our fit parameters. As we are
only interested
R in the 2 uncertainties on τ , we considerPχ2 (τ ) = N1 A dAe−χ (A,τ )/2 , with N a normalization
constant. Pχ2 (τ ) corresponds to the probability distribution of τ for our fitting model. Finally, from Pχ2 (τ ),we define the 68% confident interval of our dephasingrate γ shown in Fig. S4.
In order to compare our experimental data with thetheoretical predictions, we repeat the same analysis withthe data from the 1D discrete model. Since in the experiment the condensed atom number changes with thescattering length, see Fig. 4, the atom number considered in the theoretical simulations varies accordingly. In
Fig. 2, we account for the experimental fluctuations bytaking an interval of ±20% of the BEC atoms number.
For each scattering length, we determine the extreme values of γ in the ±20% range, which we use to create theshaded area.

qz

1. Petrov, D. S. & Astrakharchik, G. E. Ultradilute lowdimensional liquids. Phys. Rev. Lett. 117, 100401 (2016).
2. Ilg, T., Kumlin, J., Santos, L., Petrov, D. S. & Büchler,
H. P. Dimensional crossover for the beyond-mean-field correction in bose gases. Physical Review A 98, 051604 (2018).
3. Lavoine, L. & Bourdel, T. Beyond-mean-field crossoverfrom one dimension to three dimensions in quantum
droplets of binary mixtures. Phys. Rev. A 103, 033312(2021).
4. Edler, D. et al. Quantum fluctuations in quasi-onedimensional dipolar bose-einstein condensates. Phys. Rev.
Lett. 119, 050403 (2017).
5. Boudjemâa, A. Two-dimensional quantum droplets in

dipolar bose gases. New Journal of Physics 21, 093027(2019).
6. Zin, P., Pylak, M., Wasak, T., Jachymski, K. & Idziaszek,
Z. Quantum droplets in a dipolar bose gas at a dimensionalcrossover. Journal of Physics B: Atomic, Molecular andOptical Physics 54, 165302 (2021).
7. Witthaut, D., Werder, M., Mossmann, S. & Korsch, H. J.
Bloch oscillations of bose-einstein condensates: Breakdown
and revival. Phys. Rev. E 71, 036625 (2005).
8. Morsch, O., Müller, J. H., Cristiani, M., Ciampini, D. &Arimondo, E. Bloch oscillations and mean-field effects of
bose-einstein condensates in 1d optical lattices. Phys. Rev.
Lett. 87, 140402 (2001).

Chapter5

Conclusion and outlook

5.1 Conclusion
This thesis presents the work I have done in the ERBIUM lab in Innsbruck on dipolar
quantum gases of erbium atoms. I joined the team for my PhD studies at the beginningof 2018, when the experiment was at the stage of being ready to implement the Bragg
setup. Since then, I worked in the experiment, as well as helped the team with theoretical simulations. These efforts resulted in 9 publications, among which five constitute thecore of this thesis. This work focuses on the regime of small mean-field interactions to
exploit the competition among kinetic energy, and the different interactions contributions(contact interaction, DDI, quantum fluctuations). While the contact interaction is local,
the dipole-dipole interaction term is long-range and anisotropic. For this reason, it is notpossible to completely compensate one interaction with the other, and the resulting interaction will be momentum dependent. However, it was thought that independently of theexact shape in momentum space of the interaction, being in the attractive regime would
lead to a collapse [San03, Kom07, Lah08]. This picture comes from neglecting the effect ofquantum fluctuations. In fact, beyond-mean field interactions can stabilize the system andlead to novel quantum phases [Pet15]. In particular, one of the main focus of this thesis is
the supersolid state, a density-modulated state that reveals global phase coherence.
In the first publication, we study the excitation spectrum of the system in the standard
BEC phase via Bragg spectroscopy. It shows the presence of a fundamental ingredient forsupersolidity: the roton mode. This mode highlights a minimum at a finite momentum of
the dispersion relation. We study the softening when reducing the contact interaction andfound an associated increase in the density-density correlation. This mode is precursor to
crystallization, which occurs when the roton mode is softened.
In the second publication, we examine the phase of the system after softening the rotonmode. In this publication, we probe the simultaneous presence of off-diagonal and diagonal
long-range order: a supersolid state. This work is in collaboration with the Er-Dy teamin Innsbruck. While in Er, we found that the state has a lifetime on the order of 20 ms, in
164 Dy, the lifetime is above 100 ms. This enabled us to carry out evaporative cooling directly
into the supersolid state.
131

132

5 Conclusion and outlook

Probing the existence of this state raised many questions. In the third publication, we address
how a supersolid state reacts to excitations. Theoretically, we identify two branches in the
supersolid’s excitation spectrum. We characterize their respective crystal and superfluid
characters, linking the trademarks of supersolidity expected in infinite systems to the finite
trapped case of laboratory systems. By using a trap excitation scheme, we experimentallyfound the presence of two branches of excitation reflecting the two broken symmetries of thesystem. The upper branch in energy is connected to the excitation of the crystal, whereas
the lower branch is related to superfluid tunneling between the different density peaks.
In the fourth work, we study the response of a dipolar 164 Dy supersolid to an interaction
quench between different phases, which is possible with dysprosium due to the long lifetime.
More specifically, we ramp the contact interactions into the insulated droplet regime, where
the absence of a link between the density peaks leads to a loss of the global phase coherence.
Afterward, we analyze the inverse transition in which the tunneling restores global phasecoherence. This study demonstrates superfluid flow through a dipolar supersolid.
Finally, we confine the atoms to quasi-2D layers where the onsite DDI is attractive. In thisregime, we study the system by performing Bloch oscillations. By using this tool, we wereable to evaluate the role that quantum fluctuations play in this geometry. We found thatthe system tends to reduce the number of occupied layers when decreasing the scattering
length, up to the point that all atoms occupy a single layer. Moreover, in this geometry, thecompetition between the different terms in the Hamiltonian makes the phase diagram rich,with self-bound phases stabilized by the kinetic energy (soliton) or quantum fluctuations(droplet).
Our studies improved the understanding of the quantum phases in dipolar BECs. The
publications of this thesis are the results of cooperation and friendly competition with manyother scientific teams, as the group of Tilman Pfau in Stuttgart, and the team led by Giovanni
Modugno in Pisa. Among the theoretical groups, we were inspired by the works done by thegroup of Luis Santos in Hannover, Blair Blakie in Otago, Alessio Recati in Trento, as wellas Rick van Bijnen and Mikhail Baranov in Innsbruck. It has been an exciting experience
to be part of this team and to investigate these fascinating quantum phases with erbium
atoms. The next section summarizes some of the future research directions in the ERBIUM
experiment.

5.2 Outlook
In this thesis, we studied 166 Er in two different trapping geometries.
In the first geometry, a cigar-shape confinement, we explored the physics of dipolar supersolidstates. However, the supersolid obtained with 166 Er possesses only a few density peaks
and a lifetime that is limited by the high density, due to three-body losses. Recently, theconcept of dipolar supersolids was extended to mixtures. As a result of their multiple sourcesof interaction, dipolar mixtures provide novel phases with spontaneous modulation that
overcome the need for quantum fluctuations. The resulting theoretically predicted supersolid

5.2 Outlook

133

phase can have similar peak densities to unmodulated BECs [Bla22], with the extension of thecrystal not limited by the density. By realizing these novel supersolids, we could understandhow the supersolid state’s properties scale from small to large systems and how they connect
to a thermodynamic limit. In our lab, we can explore this regime, by using different spinstates of erbium [Pat21].
In the second geometry, an array of quasi-2D planes, we started to explore the physics ofdipolar gases in reduced dimensions. In this system, we exploit an attractive onsite dipolarinteraction to create a single occupied layer. The possibility to realize quasi-2D states canbe exploited to prepare novel supersolid states or to study the effect that DDI plays in BKT
physics [Bom19].
Another interesting open direction is the investigation of the excitation spectrum of a dipolar
BEC in the macro-droplet regime [Pal20, Pal22]. This is a self-bound state, that is dense
but experiences strong repulsion at short wavelengths. In this state, a novel multiband
response of the dynamic structure factor is predicted. There are also many fundamental
open questions on the speed of sound and the critical velocities in this system. We aim toaddress these questions in future publications.
Supersolidity, droplets, and solitons are fascinating phases, but dipolar gases offer even moreinteresting research directions. For example, using a Fermi gas of 167 Er, it is possible toexplore the BEC to BCS crossover [Mat14].
We recently characterized also a narrow inner-shell orbital transition at 1299 nm [Pat21].
Since it is directly in the telecommunication-wavelength O-band, the transition at 1299 nm
is an interesting candidate for quantum information systems [Pat21].
During the writing of this thesis, a new objective was implemented in the experiment. This
tool gives us the possibility to image and manipulate our sample with sub-micron resolution.
Adding a DMD setup to the imaging setup would allow us to further project arbitrary
potentials, such as ring potentials, on our samples.

Chapter

6

Additional publications

6.1 Publication: Realization of a Strongly Interacting Fermi Gas
of Dipolar Atoms
Physical Review Letters 121, 093602 (2018)a
submitted 30 Mar 2018; published 29 Aug 2018
DOI: 10.1103/PhysRevLett.121.093602

S. Baier,1 D. Petter,1 J. H. Becher,1 A. Patscheider,1 G. Natale,1 L. Chomaz,1
M. J. Mark,1,2 and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2a

The author of the present thesis performed the measurements together with J. H. B., S. B., D. P.,
L. C., and A. P., and contributed in writing the manuscript.

135

PHYSICAL REVIEW LETTERS 121, 093602 (2018)

Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms
S. Baier,1 D. Petter,1 J. H. Becher,1,* A. Patscheider,1,2 G. Natale,1 L. Chomaz,1 M. J. Mark,1,2 and F. Ferlaino1,2,†
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria

2(Received 30 March 2018; published 29 August 2018)
We realize a two-component dipolar Fermi gas with tunable interactions, using erbium atoms.
Employing a lattice-protection technique, we selectively prepare deeply degenerate mixtures of thetwo lowest spin states and perform high-resolution Feshbach spectroscopy in an optical dipole trap. We
identify a comparatively broad Feshbach resonance and map the interspin scattering length in its vicinity.
The Fermi mixture shows a remarkable collisional stability in the strongly interacting regime, providing afirst step towards studies of superfluid pairing, crossing from Cooper pairs to bound molecules, in presenceof dipole-dipole interactions.
DOI: 10.1103/PhysRevLett.121.093602

The ability to prepare dipolar quantum gases of magnetic
atoms [1–6] has enabled fascinating, yet unexpected, observations, emerging from the long-range and anisotropic
character of the dipole-dipole interaction (DDI) among
particles. In bosonic systems with dominant DDI, thisincludes d-wave-patterned collapse [7], droplet stabilization[8–10], and roton quasiparticles [11]. With fermions, manybody dipolar phenomena have been investigated only in spinpolarized systems. Here, the DDI competes with the Paulipressure, rendering dipolar effects much more subtle, as, e.g.,their influence on the shape of the Fermi surface [12].
Magnetic atoms further realize high-spin systems; e.g.,fermionic Er has 20 available spin states in the lowest
hyperfine manifold. In particular, bosonic dipolar spinor
gases have been investigated in remarkable experimentswith magnetic Cr atoms [13–16], whereas the fermioniccounterpart remains rather unexplored in the quantum
regime. Scattering experiments with fermionic Dy mixturesslightly above quantum degeneracy showed a large collisional stability against inelastic dipolar relaxation [17],enabling, e.g., the production of long-lived spin-orbitcoupled gases via Raman excitations [18].
As yet, the realization of a two-component dipolar Fermi
mixture with tunable interactions has remained elusive. Sucha system can disclose fascinating phenomena, from anisotropic quantum phases of matter, e.g., anisotropic Fermi
liquids and superfluid pairing [19,20], to dipolar magnetism[21], but also extended Fermi-Hubbard models with off-site
interactions [22]. Fermionic Er and Dy are very promising
candidates for such studies, given their large magnetic
moment. However, the large density of Feshbach resonances(FRs) even in spin-polarized gases [23–25] raises thequestion of whether stable fermionic quantum mixtures withtunable interactions can be realized with lanthanides.
We here report on a powerful platform to produce a
two-component dipolar Fermi gas of pseudospin 1=2 and0031-9007=18=121(9)=093602(6)demonstrate tunability of the interspin interactions. By
using highly magnetic 167Er atoms and a three-dimensional(3D) optical lattice as a tool for spin preparation, we
perform high-resolution Feshbach spectroscopy and unambiguously identify the spin nature of the different FRs.
Among the resonances, we find a well-isolated and comparatively broad interspin FR and precisely measure theinterspin scattering length. Our Fermi mixture reveals aremarkable collisional stability in the strongly interacting
regime.
Achieving a deterministic preparation of a spin-1=2
mixture and a precise control over the interspin interactions
in highly magnetic lanthanide atoms challenges experimental schemes. Indeed, the enormous density of FRs cancause collisional losses and severe heating, limiting the
production and preparation of deeply degenerate mixturesat arbitrary magnetic fields (B), where hundreds of FRs
might need to be crossed (see, e.g., [18]). Moreover, stateselective preparation of a spin-1=2 system typicallyrequires large B values for which the quadratic Zeeman
effect lifts the degeneracy on the Zeeman splitting amongconsecutive sublevels [17,26].
For these reasons, we establish a technique for collisional protection during the spin preparation (see Fig. 1). In
a nutshell, the key production steps are as follows. We
produce a spin-polarized degenerate Fermi gas (DFG) in an
optical dipole trap (ODT) at low B [1 in Fig. 1(a)] and load
the atoms into the lowest band of a deep 3D optical lattice,which acts as a collisional shield [2 in Fig. 1(a)] [27,28].
We then sweep to high B for spin preparation and perform
radio-frequency (rf) transfer [3 in Fig. 1(a)], sweep to thedesired B, and eventually melt the lattice [4 in Fig. 1(a)].
Experimentally, we prepare a spin-polarized DFG of
167
Er atoms in a crossed-beam ODT [5,29] [1 in Fig. 1(a)].
All fermions occupy the lowest Zeeman state j↓i ≡ jF ¼
19=2; mF ¼ −19=2i of the ground-state manifold. Here,093602-1© 2018 American Physical Society

PHYSICAL REVIEW LETTERS 121, 093602 (2018)
4
rf

3
J
(b)

(c)

z

(b)

2,0
1,5

lifetime (s)

(a)

2

1

atom number (x104)

(a)

1,0
0,5

0

y

FIG. 1. Spin-1=2 dipolar fermions in a 3D optical lattice.
(a) Sketch of the four key stages of our preparation scheme;
see text. (b) Band population in the horizontal xy plane, obtainedby averaging 50 absorption images for a 12 ms time of flight(TOF). The red arrows indicate the first Brillouin zone of thelattice. (c) Spin-resolved band-mapping images after 9 ms of TOF
in the vertical zx̃ plane, where x̃ accounts for the angle between
the imaging beam and the y axis of the lattice, for population
imbalances δ ¼ 1 (left), 0.02 (middle), and −0.94 (right). The
images are averages of about 20 absorption pictures. The spinstates are separated along the z direction by a Stern-Gerlach
technique.

F is the total spin quantum number and mF its projection
along the quantization axis. A homogeneous magnetic fieldof B ¼ 0.6 G is applied along the vertical z direction todefine the quantization axis and to maintain spin polarization. The sample typically contains N ¼ 2.4 × 104 atoms at
about T ¼ 0.25T F . Note that the ODT is shaped to optimize
single-band loading of the optical lattice and yields EF ¼
kB × T F ¼ kB × 170 nK ¼ h × 3.6 kHz (see Supplemental
Material [29]). Here, T F is the Fermi temperature, h is the
Planck constant, and kB is the Boltzmann constant.
In the next step, we transfer the spin-polarized DFG into
a 3D optical lattice [2 in Fig. 1(a)]. Our lattice has a cuboid
geometry with lattice spacings ðdx ; dy ; dz Þ ¼ ð266; 266;
532Þ nm along the three orthogonal directions [29,37].
In order to pin the atoms in a one-fermion-per-lattice-site
configuration (unit filling), we use large lattice depths ofabout ðsx ; sy ; sz Þ ¼ ð20; 20; 80Þ, where si with i ∈ fx; y; zg
is given in units of the respective recoil energies, ER;x;y ¼
h × 4.2 kHz and ER;z ¼ h × 1.05 kHz. After lattice loading, we obtain a single-component fermionic band insulator(BI) of about 2.2 × 104 j↓i atoms. By melting the latticeand reloading the fermions into the ODT, we measure T ≲0.3T F with N ¼ 2.1 × 104 (T F ≈ 160 nK) and extract aheating rate in the lattice as low as T_ ¼ 0.03 T F =s.
Our system is well described by a single-band extended
Fermi-Hubbard model [22] with residual tunneling rates ofJx;y ¼ h × 10.5 Hz and Jz ¼ h × 0.001 Hz and nearestneighbor interactions on the order of h × 50 Hz [37]. We
confirm the single-band population by performing standard

30
20
10

0,0

x

40

5
10
15
holding time (s)200-1,0 -0,5 0,0 0,5 1,0
pop. imbalance δ

FIG. 2. Spin mixture of dipolar 167Er in a 3D lattice. (a) Lifetime
measurements for spin-polarized samples of j↓i (squares) with
δ ¼ 1 and of j↑i (circles) with δ ¼ −0.92 at B ¼ 3.99 G and theirrespective exponential decay (solid lines). (b) Lifetimes as afunction of δ. Constant fits extract mean lifetimes across δ ofτ̄↓ ¼ 29.9ð3Þ s and τ̄↑ ¼ 11.8ð7Þ s. All error bars indicate thestatistical uncertainty.

band-mapping measurements [38]. In the horizontal (xy)plane, we do not resolve higher-band occupation [seeFig. 1(b) and Supplemental Material [29]). Along the zaxis, we detect a residual < 5% population in the firstexcited band, resulting from the fact that EF > ER;z [39].
Because of the Pauli exclusion principle, doubly occupied
sites (doublons) in a single band are strictly forbidden for
identical particles (j↓i).
In the BI regime, the lattice is expected to provide a strong
collisional protection to the particles. As a first application,
we use the lattice-protection technique to realize a spinor
Fermi gas with pseudospin 1=2 (j↓i–j↑i), with j↑i ≡ jF ¼
19=2; mF ¼ −17=2i [3 in Fig. 1(a)]. Experimentally, we
start with a j↓i BI at B ¼ 0.6 G and then ramp B in 40 ms toa value of about 40 G, for which the quadratic Zeeman effectin 167Er is large enough to lift the degenerate coupling of theindividual spin levels [29]. After letting the field stabilize for
120 ms, we use a standard rf-sweep technique to transfer partof the atoms into the j↑i state. By tuning the rf power, we canprecisely control the population imbalance, δ ¼ ðN ↓ − N ↑ Þ=N, in the mixture, with N ↓ (N ↑ ) the number of atoms in j↓i(j↑i). Figure 1(c) shows exemplary spin-resolved absorption
images of j↓i–j↑i mixtures for various δ after B is swept
back to low values. We typically record N ¼ N ↓ þ N ↑ ¼
1.8 × 104 and T ≈ 50 nK after melting the lattice down [4 inFig. 1(a)]. For comparison, similar measurements in absenceof the lattice clearly show a much lower atom number of
N ¼ 0.6 × 104 , proving the strength of our lattice-protection
scheme to circumvent losses when cruising through theultradense Feshbach spectrum [18,23].
Figure 2 shows the high collisional stability of the latticeconfined spin mixture. In particular, we probe N ↓;↑ as afunction of the holding time in the lattice [see Fig. 2(a)].
From an exponential fit to the data, we extract long
lifetimes of τ↓ ¼ 31ð3Þ s and τ↑ ¼ 12.2ð7Þ s. The measurements are carried out at B ¼ 3.99 G, where no FRs

093602-2

PHYSICAL REVIEW LETTERS 121, 093602 (2018)atom number (x104)

2
1

(a)

0
1.0
0.5
0.0
1.0

(b)

0.5

(c)
0.0

560

580

600

620

640
660
magnetic field (mG)

680

700

720

740

FIG. 3. High-resolution Feshbach spectroscopy for three different population imbalances in an ODT (illustrations): atoms in j↓i(squares) and j↑i (circles) for δ ¼ 1 (a), −0.6 (b), and 0 (c) as a function of B. The determined width and spin nature of the FRs areindicated by the blue (j↓i), orange (j↑i), and green (j↓i–j↑i) shaded regions. Each data point is the mean of 2–4 repetitions.

occur (see Supplemental Material [29]). Interestingly,
within our error bars, we find no dependence of the lifetimeof each spin state on the population in the other state; theyremain long regardless of δ [see Fig. 2(b)].
We note that, although very long for our purpose, wealways record shorter lifetimes for a j↑i BI with respect tothe ones measured for a j↓i BI. Differently from the j↓icase, two-body relaxation processes for j↑i are allowed. At
our magnetic fields, this process converts Zeeman energy
into a large enough kinetic energy to let the atoms escapefrom the lattice [13,40] and requires the particles to collide
at short distance (on site) [17,41]. In the spin-polarizedcases (e.g., δ ¼ −1; j↑i), double occupancy necessarily
involves population in higher bands since the Pauli exclusion principle forbids doublons in the lowest band. In oursystem, a continuous transfer of a small fraction of atoms
into higher bands might be driven by intensity andfrequency noise of the lattice beams [28]. In the case ofj↑i, this would lead to subsequent fast relaxation andjustify the observed difference in the lifetimes.
With our spin-preparation method, we are now able toconduct high-precision Feshbach spectroscopy in an ODT[4 in Fig. 1(a)] in search of interspin loss features. For this,
we first prepare the spin-1=2 mixture in a deep lattice atthe desired B value. We then transfer the mixture back into
the ODT, hold the atoms for 500 ms, and finally measure
the spin populations. Figure 3 exemplifies the highprecision Feshbach spectroscopy for three values of δ withina narrow magnetic field range from B ¼ 550 to 750 mG witha resolution of 1 mG. A lower-resolution and larger-range
scan is shown in the Supplemental Material [29].
As expected [23,24], the atom-number trace as a functionof B shows a high density of resonant loss features on top ofa constant background. By controlling δ, we are able todistinguish the spin nature of each of the observed FRs.
In the excerpt shown in Fig. 3, we identify three narrowhomospin FRs in a pure j↓i sample [Fig. 3(a)] and four in aquasipure j↑i sample [Fig. 3(b)]. In the spin-polarizedcases, all FRs exhibit widths of the order of our magnetic
field stability of ≈1 mG. Thanks to our lattice-preparation
technique, the shape and the width of the FRs are not affectedby the magnetic field ramps, namely, we do not observe
neither broadening nor fictitious asymmetry in the loss
peaks. For the 50%–50% spin mixture (δ ¼ 0), we observe
five additional interspin FRs [Fig. 3(c)], where atoms in thetwo spin states are simultaneously lost. Because of thecomplicated scattering behavior of Er, standard coupledchannel methods to assign the leading partial-wave characterof the FRs are currently not available [42]. However, thewidth of the FRs can give indications on the strength ofthe coupling between open and closed channels [43].
Among the observed interspin FRs, the one at about
0.68 G stands out from the forest of narrow FRs. This FR isalmost 2 orders of magnitude broader, making it a promising
candidate for Fermi-gas experiments in the strongly interacting regime. We further investigate this FR by performing
modulation spectroscopy on the lattice-confined spin-1=2
mixture [3 in Fig. 1(a)] to extract the interspin on-site
interaction energy, U ↓↑ ¼ Uc þ U dd , given by the sum of
the interspin contact interaction U c and the DDI Udd [37].
Thanks to the precise knowledge of U dd and to its angle
dependence, we are able to directly extract the interspin
scattering length, a↓↑ ∝ Uc ¼ U↓↑ − U dd , both in amplitude
and in sign (for details, see Supplemental Material [29]).
Figure 4(a) summarizes our results, showing the tunabilityof a↓↑ from positive to negative values across the interspin
FR. As a first estimate of the B-to-a↓↑ conversion, we use
the simple single-channel formula, leading to a↓↑ ðBÞ ¼
abg ð1 − Δ=ðB − B0 Þ − Δ0 =ðB − B00 ÞÞ [43]. From the fit to
the data, we extract the background scattering lengthabg ¼ 91ð8Þa0 , the position of the comparatively broad
FR B0 ¼ 687ð1Þ mG, and its width Δ ¼ 58ð6Þ mG. Notethat our fitting function also accounts for a nearby interspin
FR at B00 ¼ 480 mG (out of range of Figs. 3 and 4) of
width Δ0 ¼ 29ð4Þ mG, whereas narrower interspin FRs are
neglected. Based on the extracted values, we can estimate an

093602-3

scattering length a (a0)

PHYSICAL REVIEW LETTERS 121, 093602 (2018)
300

(a)

200
100
0

-100
-200

initial decay rate N/N0 (s -1)

atom number (x103)

560

.

8

580

600

620 640 660 680
magnetic field (mG)

700

720

740

a = -1500(500)a0

a = 880(140)a0

6
4
2

(c)

(b)
0

10 1

0.00 0.05 0.10 0.15 0.20 0.0 0.5
holding time (s)

1.0

1.5

2.0

(d)

10 0

10-1
15

10

5
0
coupling constant (1/(kFa ))-5

FIG. 4. Interspin scattering length and collisional behavior of
the strongly interacting Fermi mixture (a) a↓↑ extracted frommodulation spectroscopy in the lattice. Error bars are smaller than
the size of the data points. Vertical dashed lines indicate theposition of narrow interspin FRs as identified in Fig. 3(c). The
solid line is a fit to the data with the fit statistical uncertainty
indicated as shaded region. (b),(c) Lifetimes of a spin mixture ofj↓i (squares) and j↑i (circles) with δ ¼ 0 in an ODT for large
positive a↓↑ at 680 mG (b) and for large negative a↓↑ at 690 mG.
_ 0 of the normalized atom numbers as a(d) Initial decay rate N=N
function of 1=ðkF a↓↑ Þ in the vicinity of the FR. Error bars on the
coupling constant are deduced via standard error propagation.

order of magnitude for the effective range of the FR, R ¼
ℏ2 =ðmEr Δabg δμ) [43]. Here, mEr is the mass of 167Er. The
differential magnetic moment between the open and closed
channel δμ is not known for the considered FR. However,taking δμ ¼ 3μB , which is the typical value measured onbosonic Er2 [42], we estimate R on the order of 1000a0 .
With this order of magnitude, our typical two-component
Fermi gases verify 1=kF R ≳ 1, with kF being the Fermi
wave vector [29]. This identifies the intermediate strength of
the FR [44], for which the gas is expected to remain stronglyinteracting at unitarity [45,46].
For strongly interacting alkali Fermi gases, the large
collisional stability in two-component mixtures has been
essential for observing the crossover from a superfluid ofdelocalized pairs along the Bardeen-Cooper-Schrieffer(BCS) mechanism to a Bose-Einstein condensate (BEC)of bound molecules [47]. As a direct consequence of thePauli principle, three-body recombination occurs primarilyon the repulsive (BEC) side of broad s-wave FRs, where aweakly bound molecular level exists [48], whereas on theattractive (BCS) side, large scattering lengths coexist with aremarkable collisional stability [49–52]. Such an asymmetry in the scattering behavior is identified as an essential
attribute of BEC-BCS physics.
We investigate this aspect in a second set of experiments.
We prepare an equally populated spin mixture (δ ¼ 0) in an
ODT [4 in Fig. 1(a)] and probe the time evolution of thespin population as a function of the holding time in thetrap for various B across the FR. Exemplary decay curves
are shown in Figs. 4(b) and 4(c). On the BEC side, ata↓↑ ¼ 880ð140Þ a0 , we observe a fast decay of both j↑i
and j↓i atoms [Fig. 4(b)]. A simple exponential fit to thedata gives lifetimes of τ1=e ≈ 150 ms. In contrast, on the
BCS side at a↓↑ ¼ −1500ð500Þ a0 [Fig. 4(d)], the spin
mixture shows a large collisional stability with lifetimes
exceeding τ1=e ¼ 1200 ms [Fig. 4(c)].
To get deeper insights, we systematically study the initial_ 0 as a function of B. We determine the rates
decay rate N=N
by using a linear fit to the data for the initial time evolution.
Figure 4(d) summarizes our results, plotted in terms of thedimensionless coupling constant 1=ðkF a↓↑ Þ. We observe
an asymmetry of the loss rate curve, indicating that the
Fermi mixture is remarkably stable in the unitary andstrongly attractive regime. We note that both the qualitative
shape and the quantitative values of the loss rates in 167Ershow strong similarities to the ones measured in 40K [51].
The existence of a comparatively broad interspin FR and
our demonstration of the interaction tuning across thisresonance make fermionic Er gases a promising system
for accessing BEC-BCS crossover physics within a distinct
scattering scenario. Indeed, our mixture adds both the DDI
and an intermediate effective range in the short-range
scattering compared to the alkali cases [46,47], paving theway for studying exotic Cooper pairs and molecular BECs
[19,44,53] and calling for new theory developments [54].
We thank R. Grimm, M. Greiner, and A. M. Rey
for fruitful discussions. We acknowledge financial support
through an ERC Consolidator Grant (RARE, No. 681432),
FET Proactive Project (RySQ, No. 640378) of the EU
H2020, and a DFG/FWF (FOR 2247). L. C. is supported
within a Marie Curie Project (DipPhase, No. 706809) of the
EU H2020.

093602-4*

Present address: Physikalisches Institut, Universität
Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg,
Germany.
†
To whom all correspondence should be addressed.
francesca.ferlaino@uibk.ac.at

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093602-5

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093602-6

Supplemental Material for ”Realization of a Strongly Interacting Fermi Gas of
Dipolar Atoms”
S. Baier1 , D. Petter1 , J. H. Becher1,† , A. Patscheider1,2 , G. Natale1 , L. Chomaz1 , M. J. Mark1,2 , F. Ferlaino1,2,∗
1

Institut für Experimentalphysik,Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria(Dated: July 18, 2018)

Spin-polarized degenerate Fermi gases

Our experimental protocol for the preparation of
deeply degenerate Fermi gases (dFgs) of 167 Er follows
the one described in ref. [1]. The experiment starts witha narrow-line magneto-optical trap operated at 583 nm toprepare spin-polarized 167 Er atoms with N = 1.2 × 107
atoms and T ≈ 10µK in the lowest hyperfine sublevel|F = 19/2, mF = –19/2i, where F is the total angular momentum quantum number and mF is its projection along the quantization axis. The atoms are thentransferred to a horizontal optical dipole trap (ODT)
formed by a laser beam at 1064 nm. The aspect ratio
AR = w⊥ /wz between the horizontal, w⊥ , and vertical, wz , waists of this beam can be tuned from 1.6 to15 via a time-averaging potential technique [2], which allows to reach a good spatial mode overlap between theatomic cloud and the ODT. Subsequentially the atomic
cloud is compressed by reducing the AR and transferred
to a tight ODT created by a laser beam at 1570 nm witha waist of about 15 µm, and counterpropagating to the1064 nm-beam such that their focii overlap. At this stage
we typically have 1 × 106 atoms. During the evaporation
procedure the atoms are further confined by an additionnal ODT at 1570 nm, formed by a beam propagating
vertically with a waist of about 32 µm. The crossed ODT
at 1570 nm is later denoted ODT1570 .
Following our previous work of ref. [1], we perform
evaporative cooling based on elastic dipolar scattering
among identical fermions. Such a cooling scheme has
been proven to be very efficient to produce samples inthe deeply quantum degenerate regime [1, 3]. At theend of the evaporation, the trap frequencies in ODT1570
are (ν⊥ , νk , νz ) = (286(3), 85(1), 255(3)) Hz with k (⊥)
corresponding to the axis along (perpendicular to) the
horizontal ODT beam and z indicating the axis of gravity. We typically obtain spin-polarized dFgs with up to
N = 6×104 atoms and temperatures of T ≤ 0.15 TF , with
TF being the Fermi temperature corresponding to the
Fermi energy EF = kB TF = hν̄(6N )1/3 , where h is the1/3
Planck constant, ν̄ = ν⊥ νk νz
the geometric meanof the trap frequencies and N the atom number. At thisstage the Fermi energy is EF = kB ×630 nK = h×13 kHz.

During the whole evaporation, the magnetic field has avalue of B = 0.6 G and is oriented along z, which sets
the quantization axis of the atomic dipoles. Here and in
the following, N and T /TF are estimated from polylogarithmic fits to the absorption images of the dFGs after
12 ms of time-of-flight (ToF) using the horizontal imaging
setup.

Preparation for lattice loading

In deeply dFgs, the atoms fill the Fermi sea up to
the Fermi energy, EF . Hence, the number of populated
bands, when the atoms are loaded to an optical lattice,
crucially depends on the initial value of EF . In first approximation, EF can be compared to the lattice recoil
energy Erec = h2 /(2mEr λ2 ), with mEr being the mass of
167
Er and λ the lattice wavelength. In particular, during
the initial increase of the lattice potential higher bands
become populated if EF > Erec [4].
To minimize the occupation of higher bands due tothe loading procedure we reduce the Fermi energy of
our sample. To this aim, we transfer the atoms back toa crossed ODT operated at 1064 nm (ODT1064 ), within
510 ms. Here, the dynamically adjustable AR of the horizontal beam allows a convenient control on ν̄ (see section
above). We optimize the ODT parameters by loweringν̄ and N while keeping a low temperature of the sample.
The best conditions for subsequent lattice loading arereached for (ν⊥ , νk , νz ) = (63(1), 36(2), 137(1)) Hz andN = 2.4 × 104 atoms with T ≤ 0.3 TF , corresponding toa Fermi energy EF = kB × 170 nK = h × 3.6 kHz. We
note that for lower νz atoms get lost due to gravity.

Three-dimensional optical lattice and its loading

The three-dimensional (3D) optical lattice in our experiment is created by two retro-reflected 532 nm laser
beams along the x- and y-axis and one retro-reflected1064 nm vertical laser beam along the z axis; see Fig. S1.
The lattice spacings are dx,y = 266 nm along the horizontal xy-plane and dz = 532 nm along the verticalz-axis [5]. With the available power, we reach maximum lattice depths of (sx , sy , sz ) = (25, 25, 120), wheresi with i ∈ {x, y, z} is given in units of the respective recoil energies, ER;x,y = h × 4.2 kHz and ER;z =h × 1.05 kHz. Typical lattice depths used in the experiment are (sx , sy , sz ) = (20, 20, 80) corresponding toband gaps of h× 32.8 kHz along x and y and h× 17.7 kHz
along z.
After preparation and transfer to the ODT1064 , we adiabatically load the spin-polarized dFg into the 3D lattice
by increasing the lattice-beam intensities exponentiallyin 150 ms to the final values. Subsequently, the ODTbeams are switched off in 10 ms and we additionally holdthe atoms for 500 ms before applying our spin preparation scheme. This holdtime enables to remove most of theresidual atoms that have been pushed to higher bands ofthe optical lattice by the Fermi pressure, through theirnatural faster decay. We note that when the atoms areloaded directly from ODT1570 we find up to 25 % of population in higher bands, which in this case get strongly depopulated within 500 ms. Despite our most careful loading procedure and our holdtime, we measure that up to
5 % of the atoms can still populate the higher band of
the vertical lattice; see main manuscript (Note that thehigher bands along z are the most tightly trapped within
our lattice geometry).

dz

2

z

x

dy

y

dx

FIG. S1. Sketch of our lattice geometry. The coordinate
system {x, y, z} and the lattice constants dx , dy , and dz are
indicated.

configuration between the imaging and horizontal latticeaxes. To the best of our detection sensitivity, we do not
observe any population in higher-bands along these axes.
Based on the higher recoil energies in these directions,
we physically expect a lower initial population of thosebands than along z, as discussed above. In addition, because of the lower lattice depths, we expect a faster lossof their population.

Higher-band populations

Zeeman energy for fermionic Er

To access the band population we perform bandmapping measurements. Here, we decrease all latticepotential to zero within 1 ms, thus mapping the quasimomentum of the band to real momentum. We thenperform ToF absorption imaging, which thus probes the
population of the different bands in directions transverseto the imaging axis. We note that, the edges of the lowest band can be smeared out because of the finite widthof the in-situ cloud and due to an imperfect adiabatic
switch-off of the lattice potentials [6], limiting the accuracy of our determination. In our setup, we obtainour best estimate of the remaining higher band populations by comparing the absorption images to the expected profiles computed from the first Brillouin zone.
In the z-direction, we observe a very small populationin higher bands, which we quantitatively estimate using
the horizontal imaging setup. Here, we additionally takeadvantage of the observed structure of the higher band
population, which systematically appears below the lowest band (along the gravity axis). This might be due to a
combined effect of residual magnetic gradient and gravity. We use this behavior to our advantage and extract
the population of the higher band in z from the topbottom asymmetry of the band-mapping images. In the
xy-directions, the estimate of the higher band population
is more subtle in particular because of the non-orthogonal

Fermionic Er exhibits a hyperfine structure resulting
from the coupling of its nuclear spin I, whose quantum
number is I = 7/2, with the total electronic angularmomentum J, which in the ground state of Er has forquantum number J = 6. The total angular momentum reads F = J + I. In the lowest hyperfine manifold (F = 19/2) there are 2F + 1 = 20 sublevels which
can be differentiated by the eigenvalues of the projection of F on the quantization axis, corresponding to thequantum number mF . Because, in our experiment, an
external magnetic field B is always applied, the degeneracy of the sublevels is lifted by the interaction of B
with the different angular momenta. In our description,
the quantization axis is chosen to be parallel to B, andthe mF sublevels are then denoted magnetic states. In
the low B limit, the magnetic states are simply shiftedin energy along EmF = mF gF µB B, corresponding toa state dependent magnetic moment, µ = mF gF µB .
In Erbium, the absolute ground state has a magnetic
moment of µ = −6.982804µB , giving the Landé factor
gF = 0.735032 [7]. Here µB is the Bohr magneton.
In a more general way, the atomic energy levels in a
uniform B-field can be calculated via an exact diagonalization of the atomic Hamiltonian [8]. In Fig. S2 we plot
the energy levels of the lowest hyperfine manifold as afunction of the magnetic field computed in such a way.

3
0

-200

differential splitting (kHz)Zeeman energy (MHz)

-100

-300

-400

0

40

∆1-∆2

30
20

∆2

10

∆1

0
0

10

5

10

20

30

40

15
20
25
magnetic field (G)

30

35

40

FIG. S2. Zeeman energy for the magnetic substates in the|F = 19/2i hyperfine manifold. For this work the energy
splitting of the lowest three spin states |–19/2i ≡ | ↓ i (blueline), |–17/2i ≡ | ↑ i (orange line) and |–15/2i (red line) is of
most relevance. Higher spin states are visualized by grey lines.
The linear Zeeman effect dominates the energy evolution suchthat ∆1 ≈ ∆2 ≈ qli while the quadratic Zeeman effect isevident in the differential splitting ∆1 − ∆2 (inset).

The dominant trend evidences the linear dependence discussed above in the low B regime. However, at large
enough B, deviations from this simple picture appear due
to the Paschen-Back effect, as J and I start to decouple.
In a perturbative description, this can be accounted via aquadratic correction to the Zeeman energies which writes
EmF − qli mF = qqu (m2F − F 2 ) with qli = gF µB B andqqu ∝ B 2 . The deviation from the linear Zeeman energy
becomes evident when considering the differential splitting ∆EZ (mF ) = (EmF − EmF +1 ) − (EmF +1 − EmF +2 ),as the linear Zeeman effect qli mF cancels out; see inset
of Fig. S2. In the folowing we define ∆EZ = ∆EZ (mF =−19/2), which is the most relevant quantity for the current study, as restricted to mF = −19/2 and −17/2.
Preparation of a spin mixture in the lattice

To achieve a deterministic spin preparation of the twolowest spin states we typically use a large enough magnetic field of B = 40.51 G for which the differential Zeeman splitting ∆EZ = 42.6 kHz is larger than the fluctuations of the Zeeman energies coming from the magnetic
field noise, corresponding to ≈ 20 kHz at this field. In thisway, the spin-spin coupling induced by a small amplitude
modulation of the magnetic field in the radio-frequency(rf) domain can be restricted to the subspace formed by
the lowest two spin states mF = −19/2 and −17/2. To
couple the two hyperfine sub-states we apply a rf-sweepby chirping the rf-frequency continuously from a value of(ν ? + 30 kHz) to (ν ? − 30 kHz) within about 10 ms, wherehν ? matches the energy difference E−19/2 − E−17/2 (∆1in Fig. S2). We can prepare a well-reproducible mixtureof | ↓ i and | ↑ i without populating the next higher spinstate. The population imbalance δ between the two spinstates can be freely controlled by varying the power of therf signal. In particular, also almost all the atoms can betransferred to | ↑ i reaching up to δ = −0.94 (see Fig. 1(c)of the main manuscript). We note that, while our preparation technique in the lattice initially leads to a coherent
superposition of the two spin states, additional measurements suggest a fast decoherence, leading to a projectionof pure states on the individual lattice sites for the experimental relevant time scales. In particular, we observethat coherently driven Rabi oscillations between the two
spin states quickly damp within a few ms.
To image the spin mixture, we perform spin-resolved
TOF absorption imaging using a Stern-Gerlach techniquewith a 1-ms pulsed magnetic field gradient at the beginning of the TOF. After an additional 7.2 ms of TOF,
the populations of each spin state are spatially separated
and we measure them by using the horizontal imaging
setup (Fig. 1(c)). N↓.↑ are then counted by integrating
the measured density distribution over well-defined regions of interest.

Lifetime of the spin mixture in a deep lattice

To conduct a clean measurement of the collisional
properties of a spin mixture in the deep optical lattice
it is important to fulfill the following requirements: (i)The spin mixture is in an insulating regime where theformation of doublons is suppressed via sufficiently large
ratios of the onsite interspin interaction energy U↑↓ to thetunneling rate J. This requires not only to use a deep lattice potential but also to sit away from any Feshbach resonance (FR) so that U↑↓ is not resonantly modified andhas a value close to its background one. (ii) The Zeeman
energies are large and do not have an equidistant spacing
so that both magnetization changing and magnetizationconserving spin-exchange processes induced by the DDI
are energetically supressed [9–12].
Due to the high density of FRs (see next section andRef. [13]), the requirement (i) is not so straightforwardly
achieved in our fermionic erbium mixture. In this system, to find a magnetic field value for which we sit stably
away from any FR is eased by a low technical magneticfield noise. Because of the different sets of coils used inour experiment, the magnetic field noise is found to be≈ 1 mG up to B = 5 G while it increases to ≈ 20 mG
when we go to higher B values. Hence, working at
B ≤ 5 G turns out to be more favorable in our setup.
On the other hand, the requirement (ii) is matched fora sufficiently large magnetic field, where the quadratic
Zeeman effect (see Fig. S2) is strong enough to not be4canceled by quadratic light shifts [14]. The best conditions for meeting the requirements (i) and (ii) are foundat B = 3.99 G, which is used for the lifetime measurements of Fig. 2. At this field, we have measured the onsite interaction energy using a similar technique as forFig. 4(a) of the main text; see also below. The extracted
value of U↑↓ = h×2.43(2) kHz exceeds by far the relevant
tunneling rates Jx,y = h × 10.5 Hz so that it lies deeply
in the insulating regime. In addition, we note that nospin-changing dynamics are observed from the measured
spin population.
We finally note that, for technical reasons, our setup
allows holding in the lattice up to 20 seconds. For longer
times, thermal effects in the high-power fiber start to prevent us from properly stabilizing the output power for ourlattice. We avoid observation in this regime where thelattice potential would not be properly controlled (which
may also lead to additional losses), as the allowed observation time is already very long for our purposes. This
restriction however limits the precision of our lifetime
measurements for the long lifetimes observed in our setup
and in particular for the longest-lived | ↓ i state.
State-resolved Feshbach spectroscopy

To identify the magnetic field regions where promisinginterspin FRs occur, we first perform a rough Feshbachscan in the 0 − 2 G region for different population imbalances δ (Fig. S3). For this set of data we do not use
our lattice-protection technique. Instead, the spin preparation, the magnetic-field ramps, and the Feshbach spectroscopy are directly performed in the ODT. As expected,
without the lattice, the loss features present broadening
and asymmetric shapes due to the mere magnetic-fieldsweeps (e. g. via losses occuring during the sweeps). Yet,the most prominent features of the scattering physics canbe identified.
Using this technique, we perform three sets of measurements, varying the composition of the mixture δ.
In a first set, we perform a Feshbach scan in a spinpolarized gas in ODT1570 (Fig. S3, upper panel). We
jump to the final magnetic field and hold for thold =70 ms before TOF imaging. The trap frequencies are(ν⊥ , νk , νz ) = (324(1), 147(5), 259(4)) Hz. The system
has an initial temperature of T = 0.18(1) TF . Similar to Ref. [13], we observe a high density of loss features, which correspond to single-component (| ↓ i) FRs
of high partial-wave character. In a second set of measurements, we repeat the magnetic-field scan in an almost pure | ↑ i sample (Fig. S3, middle panel). Here, we
use a resonant rf-pulse at 0.99 G to prepare a mixture
with mainly | ↑ i atoms. Then we jump on a purely | ↓ ihomo-spin FR located at 1.034 G to remove remaining| ↓ i atoms. The measurement is performed in the more
shallow ODT1064 to prevent too strong interspecies losses

and thold = 500 ms. For this trap, the trap frequencies
are (ν⊥ , νk , νz ) = (39(1), 37(1), 145(3)) Hz and the initial
temperature is T = 0.35(1) TF . We find new FRs, whichmainly correspond to single-component | ↑ i FRs. In a
third set of measurements, we observe the loss features
for a spin mixture prepared at 0.58 G in the same trap asfor the pure | ↓ i measurement with thold = 50 ms (Fig. S3,lower panel). Here, the initial temperature is slightly increased to T = 0.24(1) TF due to the spin mixing. The
individual homo-spin FRs are still visible while we alsofind new interspin | ↓ i–| ↑ i FRs.
We analyze the three sets of data to extract the spin
nature of the individual FRs. For several FRs, the entrance spin channel can be easily identified. In addition,
we also observe overlapping FRs. Here, an exact assignment requires a high-resolution magnetic-field scan andour lattice-protection technique; see main text. Amongthe forest of FRs recorded in the two-component mixture,
we observe a promising interspin FR at about 700 mG,
which remains rather isolated from other homo-spin FRs;
see green shading in Fig. S3.
As a second step, we focus on the magnetic-field region
around 700 mG in which the promising interspin Feshbach resonance has been identified and perform highresolution Feshbach spectroscopy, taking advantage ofthe lattice-preparation scheme, as described in the maintext. The lattice-protection technique is very powerful inremoving technical broadening and artificial asymmetryof the loss peaks, as it clearly appears from a comparison
between the atom-number traces recorded with ODTpreparation (Fig. S3) and lattice-preparation schemes(Fig. 3). We perform the measurements for Fig. 3 as follows. We prepare a spin mixture in the lattice at highB as described above and sub-sequentially ramp the field
to the desired value within 10 ms. After letting the Bfield stabilize for about 100 ms, the dipole trap beams
are ramped up within 10 ms and we unload the atoms
from the lattice back into the ODT1064 within 150 ms.
At this stage, the sample contains N ≈ 1.6 × 104 atoms
at T ≈ 0.3 TF , almost independent of δ, and the trap frequencies are (ν⊥ , νk , νz ) = (111.6(2), 35(1), 169.4(6)) Hz.
We then record the spin population after a holding timeof 500 ms. For each magnetic field value, the measurement is repeated between two to four times and the average is reported in Fig. 3.
For all the above described measurements, we note thatthe observed atom losses can be mainly attributed to resonant three-body recombination collisions in the shortrange potential. Inelastic two-body losses driven by the
spin-non-conserving dipolar interactions are, in principle,
also energetically allowed since | ↑ i atoms are in an excited Zeeman state [15]. However, we do not expect thisprocess to be enhanced at resonance.

5
3
2

atom number (x104)

1
0
0.4
0.2
0.0
3
2
1
0
0.0

0.2

0.4

0.6

0.8

1.0
magnetic field (G)

1.2

1.4

1.6

1.8

2.0

To measure the scattering length between two spinstates of 167 Er, we rely on a method similar to the
one that we have already successfully implemented with
168
Er [5] and 166 Er [2]. It is based on the measurement of
the onsite interaction energy of two atoms in a deep optical lattice. Here, after preparing a spin mixture of | ↓ i
and | ↑ i in the lattice we drive particle-hole excitationsof neighboring atoms by a resonant modulation of thehorizontal lattice depths sx,y . Note that for neighboring identical particles no single-band excitation will beobserved due to the Pauli exclusion principle and only
neighboring atoms in different spin states can be excited,allowing to effectively only probe the interspin onsite interaction U↓↑ . In our experiment, we probe the doublon
creation via the resulting increase in atom loss. We speculate that onsite dipolar relaxation is responsible for theobserved loss [16]. We note that a distinct and convenientmethod to measure double occupancies has been demonstrated using the coupling to a third spin state [17]. Close
to a molecular state of the original doublon components,
the third spin state features a smaller interspin onsite
energy and can thus be used to detect an initial double
occupancy. Yet, such a method remains to be exploredin our system.
In our experiment, we typically modulate the lattice
depth for 1 s with a sine function with a peak-to-peakamplitude of 30% and a frequency νmod . Maximum loss
occurs when νmod reaches the resonance condition νres =U↓↑ /h (see Fig. S4). Following our previous work [5], theonsite energy U↓↑ consists out of two contributions: the

1.0

0.9

U↓↑= hνres

Modulation spectroscopy with a fermionic spin
mixture in the latticenormalized total atom number

FIG. S3. Feshbach spectroscopy of a two-component spin mixture in an ODT (without the lattice-protection technique) for
different population imbalances δ: δ = 1 (upper panel), −0.54 (middle panel), 0.4 (lower panel). While the measurements forthe upper and lower panel are performed in ODT1570 with ν̄ = 231(3) Hz, the data of the middle panel is measured in ODT1064
with ν̄ = 59(1) Hz. Due to the finite resolution of the scans of 10 mG it is possible that narrower FRs are not resolved. The grey
shading shows the magnetic field region studied in the main manuscript with the green shading indicating the comparatively
broad interspin FR. Each data point is the mean of two repetitions. The shading around the data points indicates statisticaluncertainties, which are often smaller than the data points.

0.8
3

4
5
6
modulation frequency ν Mod (kHz)78

FIG. S4. Exemplary modulation spectroscopy measurementwith a spin mixture of | ↓ i and | ↑ i in the deep lattice at
B = 650 mG. The resonance condition determines νres , whichis related to the onsite energy U↓↑ (cartoon).

contact interaction Uc
Uc =4πh̄a↓↑
mEr

Z

4

dr |φ(r)| ,and the DDI Udd
ZZ2µ0 µ↓ µ↑22 1 − 3 cos θr−r0
Udd =dr dr0 |φ(r)||φ(r0 )| .
4π
|r − r0 |3
Here, φ(r) denotes the onsite Wannier function, |r − r0 | isthe interatomic distance and θr−r0 corresponds to the angle between the polarization axis of the two dipoles withrespect to their interparticle axis. The contact part depends on the interspin scattering length a↓↑ , the reduced
Plank constant h̄, and the mass mEr of a 167 Er atom,
while the DDI part is proportional to the vacuum permeability µ0 and to the magnetic moments of the two spin

6
states µ↓ and µ↑ . The contributions of nearest-neighborinteractions are minor and therefore neglected.

In the experiment, we use both our precise control andour exact knowledge of Udd to determine not only the amplitude but also the sign of the scattering length a↓↑ . For
a given magnetic field and a given lattice configuration,
we repeat our modulation spectroscopy measurements fortwo different dipole orientations: (i) when oriented alongzz, we extract the total onsite energy |U↓↑| while knowz
ing the dipolar contribution Udd, (ii) when oriented inxy
the xy-plane, we extract |U↓↑| while knowing the dipoxy
lar contribution Udd . This yields the two indepedentzz
and incommensurate relations: |U↓↑
| = |Uc + Udd
| and
xy
xy
|U↓↑ | = |Uc + Udd |. Their combination gives access toboth the magnitude and the sign of Uc , and thus of a↓↑
as reported in Fig. 4(a).
As a final test of our method we study the dependenceof the onsite energy as a function of the lattice depth sz(Fig. S5). Here, we fix the magnetic field, oriented alongz, to 650 mG and vary the depth of the z lattice. We
repeat the modulation spectroscopy for different valuesof sz and extract νres for each measurement. A comparison to our theoretical model with a↓↑ being the only freeparameter shows a good agreement, confirming the validity of our modulation spectroscopy technique. Here, thefit gives a value for a↓↑ of 225(2) a0 matching the value
extracted from an independent analysis of the individual
lattice configurations as reported in Fig. 4(a) and givinga↓↑ = 225(4) a0 .
The data presented in Fig. 4(a) shows the mean of thedifferent experimental datasets, taken with different lattice parameters, for a given magnetic field B. Table 1
summarizes all lattice parameters used in the experiment
as well as the expected values of Uc for a↓↑ = 100 a0 ,(100)zdenoted Uc, and of Udd
from our theoretical model.
Uc being proportional to a↓↑ and Udd depending only
on the lattice parameters, the interspin scattering length
can be evaluated from a given measurement of U↓↑ via(100)za↓↑ /a0 = (U↓↑ − Udd) × 100/Uc.

5.5
ν res (kHz)Both, the strength and the sign of Udd strongly dependon the dipole orientation and the anisotropy of the onsite
Wannier function. As specified in our earlier work [5], we
define the aspect ratio, AR, associated to the Wannierfunction by the ratio of the onsite harmonic oscillator
lengths perpendicular and in the xy-plane, AR = lz /lx,y .
Note that, in a deep lattice, the onsite harmonic oscillator
1/4
lengths match li = di /(πsi ) for i ∈ {x, y, z}. For ourtypical lattice parameters we find AR > 1 and hence Udd
can be tuned by rotating the atomic dipole. In particular,
Udd is negative (positive) for a dipole orientation out of(in) the xy-plane.

6.0

5.0

4.5

4.0
40

60

80

100

120

sz (ER,z)FIG. S5. Modulation resonance νres as a function of thevertical lattice power sz for sx,y = 20 at B = 650 mG. The
solid line shows a fit with our theory to extract the scattering
length a↓↑ . The shaded region accounts for the systematic
uncertainty of the scattering length of ±4 a0 at 0.65 G, which
results from our magnetic field fluctuations of ±1 mG.
(sx , sy , sz )(20, 20, 40)(20, 20, 60)(20, 20, 80)(20, 20, 100)
(20, 20, 120)
(15, 15, 80)(22, 22, 80)

AR
1.68
1.52
1.41
1.34
1.28
1.32
1.45

(100)

Uc

/h (Hz)
2029
2263
2443
2590
2717
2068
2578

z
/h (Hz)
Udd
−441
−396
−350
−307
−265
−223
−399

TABLE I. Lattice parameters for the determination of a↓↑(Fig. 4(a)). The lattice depths (sx , sy , sz ) define the onsite
Wannier function AR. From our theoretical model we eval(100)zuate the onsite energy contributions Uc
and Udd
for aninterspin scattering length of a↓↑ = 100 a0 . Here, the dipoles
are oriented along z. This values are used to extract the interspin scattering length from the measured total onsite energyU↓↑ .

Scattering-length tunability and magnetic-fieldstability.

A precise control of the magnetic-field value is crucial
for tuning the interaction strength in the spin mixture.
For the FR of Fig. 4(a), a width of ∆ = 58(6) mG and relative strength abg /R∗ = 0.1 have been estimated. Hence,reaching a↓↑ (B) = R∗ requires to sit ≈ 6 mG away fromthe resonance pole. Based on an rf-spectroscopy calibration scheme, an accuracy of the order of 100 µG and astability of ≈ 1mG on the magnetic-field value are esti-

7
mated, corresponding to a relative stability of 10−3 atthe resonance position B0 = 687 mG. Note that in alkali
Fermi experiments, for which B0 is typically 3 orders ofmagnitude larger, a much larger relative stability of 10−5
is usually required even if the FR is effectively broader.

Loss spectroscopy in the ODT at the interspin FR

For the measurements of the collisional properties ofthe fermionic spin mixture in the vicinity of the comparatively broad interspin FR (see Fig. 4(b-d)), we apply the following experimental procedure. We preparea spin mixture with δ = 0 in the deep 3D lattice following the scheme detailed above, that is applying a
RF-sweep at large B. After the application of the RFsweep, we ramp the magnetic-field value to an intermediate lower value B = 3.99 G in 100 ms. We thenjump with the magnetic field from the later intermediate value to the desired final value and let it stabilize
for 10 ms. Finally, we ramp up the ODT1064 beams
in 10 ms, melt the lattice down in 20 ms. This shorter
timescale for the lattice rampdown (compared, e. g. , tothe Feshbach spectroscopy measurements, see above) is
chosen to avoid significant losses to happen already atthis stage. The final trap frequencies in the ODT1064
are (ν⊥ , νk , νz ) = (111.6(2), 35(1), 169.4(6)) Hz. For thistrap, the typical atom numbers recorded (see Fig. 4(bc)) correspond to the Fermi energy EF ≈ kB × 150 nK
for each spin component,
which in turn gives a Fermi√
wave vector kF ≈ 2mEr EF /h̄ = 1 × 107 m−1 . We thenhold the two-component mixture in the ODT1064 for avariable holding time, t, at the selected B-field and ultimately record the spin populations via Stern-Gerlach
imaging.
We record the atom number decay with t for various
magnetic fields B across the FR. For each B and eachspin component, we extract an initial decay rate Ṅ /N0
by fitting a linear-decay function to the recorded atom
number N , normalized to its initial value N0 , as a function of t. We fit all data for which the atom number stays
above a threshold of 75% of N0 . We checked that the extracted values of Ṅ /N0 do not change significantly whenvarying this threshold between 65 − 85%. An analysisof the full data using exponential fits also yields similar
decay rate values.

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F. Ferlaino, Phys. Rev. Lett. 112, 010404 (2014).
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[4] S. Will, PhD Thesis, Johannes Gutenberg-Universität,
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Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Science
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6.2 Publication: Bragg scattering of an ultracold dipolar gas
across the phase transition from Bose-Einstein condensate tosupersolid in the free-particle regime.
Physical Review A 104, L011302 (2021)a
submitted 5 May 2020; published 22 July 2021
DOI: 10.1103/PhysRevA.104.L011302

D. Petter,1 A. Patscheider,1 G. Natale,1 M. J. Mark,1,2 M. A. Baranov,2
R. M. W. van Bijnen,2,3 S. M. Roccuzzo,4,5 A. Recati,4,5 B. Blakie,6,7 D. Baillie,6,7
L. Chomaz,1 and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
Center for Quantum Physics, University of Innsbruck, Austria
4
INO-CNR BEC Center and Dipartimento di Fisica, Universitá degli Studi di Trento, 38123
Povo, Italy
5
Trento Institute for Fundamental Physics and Applications, INFN, 38123, Trento, Italy
6
The Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin9054, New Zealand
7
Department of Physics, University of Otago, Dunedin 9016, New Zealand2a

The author of the present thesis performed the measurements together with D. P., A. P., and L. C.,
and contributed in writing the manuscript.

151

PHYSICAL REVIEW A 104, L011302 (2021)
LetterBragg scattering of an ultracold dipolar gas across the phase transition from Bose-Einsteincondensate to supersolid in the free-particle regime
D. Petter ,1 A. Patscheider ,1 G. Natale,1 M. J. Mark ,1,2 M. A. Baranov,2,3 R. van Bijnen,2,3 S. M. Roccuzzo,4,5
A. Recati,4,5 B. Blakie,6,7 D. Baillie ,6,7 L. Chomaz,1 and F. Ferlaino1,2,*
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria
3
Center for Quantum Physics, University of Innsbruck, 6020 Innsbruck, Austria
4
INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy
5
Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy
6
The Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin 9054, New Zealand
7
Department of Physics, University of Otago, Dunedin 9016, New Zealand
2(Received 5 May 2020; revised 30 March 2021; accepted 9 June 2021; published 22 July 2021)
We present an experimental and theoretical study of the response of a dipolar supersolid to a Bragg excitation
at high-energy defined by the impulse approximation regime. We experimentally observe a continuous reductionof the response when tuning the contact interaction from an ordinary Bose-Einstein condensate to a supersolidstate and ultimately to an incoherent array of droplets. Already in the supersolid regime, the observed reduction
is faster than the one theoretically predicted by the Bogoliubov–de Gennes theory. By comparing experiments
and theories, we are able to attribute this discrepancy to the presence of coherent phase dynamics induced by thecrossing of the phase transition. The phase variations are found to change character along the phase diagram andbecome predominantly incoherent only when reaching the incoherent-droplet regime.
DOI: 10.1103/PhysRevA.104.L011302

Recently, supersolid states have been realized in laboratories using ultracold quantum gases of dipolar atoms [1–3].
Predicted more than half a century ago [4–7] and longsearched for in helium [8], this intriguing phase of matter
spontaneously breaks two symmetries, namely the translational and the gauge symmetry. The breaking of the former
one gives rise to a periodic order in space with the system
ground state developing a density modulation, recalling acrystalline structure, whereas the breaking of the gauge symmetry introduces a superfluid flow of particles.
The supersolid phase (SSP) transition is typically controlled by the interaction. By varying its strength, a quantum
system may pass from an unmodulated superfluid to a fully
localized crystal state of insulating droplets (ID). Between
these two extremes, the system is supersolid, showing a coexistence of these two apparently antithetical orders. Theinterplay between localization and superfluidity has raised
lively debates [4–7]. In a seminal work, Leggett derived an
upper-bound relation for the superfluid fraction in a supersolid [9], which directly connects the loss of superfluidity
with the increase of localization, the latter being quantified interms of a modulation contrast. Importantly, Leggett’s famous
formula is valid at equilibrium and requires the macroscopic
phase of the quantum system to be stationary.

*
Author to whom correspondence and requests for materials should
be addressed: Francesca.ferlaino@uibk.ac.at2469-9926/2021/104(1)/L011302(6)However, in experiments, a common path to create dipolar supersolids relies on interaction tuning [1–3,10–13]. The
corresponding dynamical crossing of the phase transition may
introduce excitations, which, on the one hand, naturally questions the applicability of equilibrium theories. On the other
hand, excitations typically entail phase variations, raising interest in understanding their role and impact on the systembehavior, calling for the development of theoretical models
accounting for out-of-equilibrium effects.
Interestingly, phase variations across the system display
different natures. They can be coherent and deterministic, orincoherent and random depending whether they arise fromcollective dynamics or, e.g., from quantum and thermal fluctuations. While experiments pointed to the existence of phasevariations in dipolar supersolids [11–13], a comprehensive
study of their characteristics and origins is lacking. Providing
access to local properties of the system, high-energy scattering
measurements may help by bridging this gap. Such scattering
protocols have been successfully used across a vast range of
disciplines, from high-energy [14–17] to condensed-matterphysics [18,19]. They allowed measurements, e.g., of thecondensate fraction in superfluid liquid helium [20] and of
beyond-mean-field effects in ultracold gases [21–27].
In this Letter, we study the response of a dipolar supersolid to a high-energy two-photon Bragg scattering probe. As
the system crosses from the Bose-Einstein condensate (BEC)
into the SSP, we observe a strong reduction of the scattering response, that eventually vanishes in the ID regime. A
comparison with theory reveals that the response is reduced

L011302-1©2021 American Physical Society

PHYSICAL REVIEW A 104, L011302 (2021)j× δ( h̄ω − h̄ω j ),(1)

where we neglect the creation of multiple excitations. Here,ψ0 is the system’s macroscopic ground-state wave function.
Equation (1) gives different information depending on
the momentum and energy ranges [28]: For low-k transfer, S(k, ω) is sensitive to the system’s collective response,
whereas, in the high-k and high-energy regime, the DSF is
proportional to the momentum distribution of the system,ñ(k). Here, we focus on the latter regime to probe the impact of density modulation in a superfluid state. We study
the response along the density-modulated direction, y, withk = (0, ky , 0). In the regime of free-particle excitations (u j →eik j y , v j → 0, ω j → h̄k 2j /2m with m the atomic mass), theimpulse approximation becomes valid and we find [28–32]
S(ky , ω) =ñ(0, ky − k j , 0) δ( h̄ω − h̄ω j ).
(2)
jOn resonance, ω = ω j and ky = k j , the DSF is uniquelydetermined by the system’s momentum distribution
at k = 0, independent on the probed momentum k j ,S(k j ) ≡ S(k j , ω j ) ∝ ñ(k = 0).
To identify the free-particle regime, we calculate theexcitation spectrum. Following the Bogoliubov–de Gennes(BdG) theory, a free-particle excitation is an elementary
excitation of plane wave character. This occurs for excitations of high enough energy and single-particle character(u j  = |u j (r)|2 dr = 1 and v j  = 0) [28,33].
In order to gain an intuition, we begin with calculating
the Bogoliubov amplitudes and S(k, ω) in the thermodynamic
limit. We consider the BdG theory for an infinitely elongatederbium quantum gas. As shown in Figs. 1(a) and 1(b), thesupersolid spectrum exhibits a periodic structure in momentum space with a period given by the reciprocal lattice vector
kc . The state develops a density modulation along the axial
direction with wavelength 2π /kc [see inset in Fig. 1(b)]. The
two lowest branches correspond to the superfluid and crystal
branches, respectively [35]. At higher energies, excitations
follow a gapped parabolic dispersion branch and a flat bandat ω ≈ 1.25 ωz (corresponding to transverse breathing modesof single-droplets). Importantly, the excitation modes of theparabolic branch have a free-particle character, when u = 1.

3

4

>5

||u||

1

(b)

104

)

1.5 (a)z(units of

2 >2.60(k, )(units of S*)

)

2

0

1.0

-2

(

0

)

2

0.5
0
0

1
k (units of k )2 0

1
k (units of k )(c)2

SSP BEC

1

0.8

1

53.2

as (units of a0)
54
54.8

C

2
 

 ∗
 ikr
∗


S(k, ω) = dr u j (r) + v j (r) e ψ0 (r)1

S(k) (units of S )stronger than expected from the emergence of a density modulation in the SSP. Going beyond equilibrium expectations, wefind that coherent phase variations, emerging due to the crossing of the BEC-SSP transition, are the cause of the anomalous
response suppression.
We start by reviewing the description of the dynamical
response of an interacting many-body system to a weak
scattering probe within the linear-response theory [28]. The
dynamic structure factor (DSF), S(k, ω), quantifies the density
response of a system to a probe of momentum, h̄k, and energy,h̄ω. For weak interactions at equilibrium, the DSF is related tothe excitation spectrum via the Bogoliubov amplitudes, u j andv j , describing the excitation mode j of energy h̄ω j . It reads-1

D. PETTER et al.

00.6 53.2

53.6

54
54.4
as (units of a0)

54.8

FIG. 1. (a) Axial excitation spectrum of the transversely symmetric modes and (b) corresponding DSF of an infinitely elongateddipolar supersolid at as = 51 a0 in a harmonic trap with ωx,y,z =2π × (250, 0, 160) Hz. The color maps correspond to u andS(k, ω), respectively. The inset shows the integrated axial density profile n(y) of the ground state with mean density 4.7 ×
103 μm−1 . (c) S(k) for the 3D-trapped system with ωx,y,z = 2π ×(250, 31, 160) Hz. S(k) is calculated at k = 4.2 μm−1 ≈ 1.8 kc (greyline) and normalized by its value at the BEC-SSP phase transition,S ∗ . The atom number is varied with as to match the experimental
conditions [34]. The red (blue) line shows the result from the SIA(DAA). (upper inset) Integrated density profile of the ground state at
as = 54.49 a0 and N = 5 × 104 atoms. (lower inset) Evolution of theground state’s central contrast C. For the infinite (3D-trapped) case,kc = 2.3 μm−1 (2.4 μm−1 ).

We now move on to the three-dimensional (3D) trappedcase for the experimentally relevant parameters. Previous
works have shown that the main spectral features qualitatively
persist when changing from the infinite to the finite sized system [10–12,36,37]. We calculate the spectrum of excitations
as in Refs. [10,38] and extract the Bogoliubov amplitudes.
Similar to the infinite system, we find a free-particle characterfor excitations with h̄ω  0.6 h̄ωz [34]. This enables the impulse approximation for the later experiments, which are doneat an exemplary momentum of k ≈ 1.8kc . Figure 1(c) showsS(k ≈ 1.8kc ), which decreases when entering the SSP fromthe BEC and further reduces when lowering as . Simultaneously, the ground-state density develops a spatial modulation(upper inset), whose contrast C rapidly increases (lower inset).
Note that C evolves faster with as than S(k). For instance,
at as = 53 a0 , C ≈ 1, whereas S(k) reduces only by about
35 %. Here, C = (nmax − nmin )/(nmax + nmin ) with nmax (nmin )L011302-2

BRAGG SCATTERING OF AN ULTRACOLD DIPOLAR GAS …

FIG. 2. Fraction of Bragg-excited atoms as a function of ω forvarious as across the BEC-SSP-ID regimes (see labels). The spectra
are vertically offset for visibility. Here and throughout the Letter,the error bars correspond to one standard error. Solid lines show theGaussian fits to the data.

For the Bragg excitation, we project on the atoms an opticallattice potential of constant depth V for a duration τ = 7 ms.
The lattice has a constant wave vector k = 4.2(3)μm−1 alongy and moves with a variable frequency ω. After the Bragg
excitation, we measure the integrated momentum distribution,ñ(kx , ky ), using a time-of-flight expansion of 30 ms. The number of excited atoms Nexc is extracted in a narrow region ofinterest around k [34]. For a fixed as , we find a clear resonancein Nexc /N as we vary ω. From a Gaussian fit we extract theresonance peak’s amplitude, F. From linear response theory,
we expect F ∝ V 2 τ S(k) [43]. For the relevant as range, we

5

ID

SSP

BEC

54.5

55 55.5 56
as (units of a0)

4
F (%)

being the central maximum (minimum) of the integrated density [34].
To gain an intuitive understanding of the density-response
reduction, we develop a 1D model [32]. Using two different wave-function ansatzes, we evaluate S(k) in the weak
and strong density-modulation regimes. As discussed inRefs. [39–41], for weakly modulated supersolids, with C  1,
the ground-state wave function can be approximated by afully coherent sine-modulated function on top of a √uniformbackground. At leading order in C, it reads ψ (y) = n(1 +C sin(kc y)/2), with n being the mean density. Applying thissine ansatz (SIA) in Eq. (2), we find S(k) ∝ n(1 − C 2 /8).
This result shows that an increasing contrast directly causes
a suppression of the DSF. We find a similar C dependence
for the superfluid fraction derived from Leggett’s formula [9],fSF = 1 − C 2 /2. Therefore, in the weakly modulated regime,
the reduction of the high-energy scattering response connects to the reduction of the superfluid fraction [32]. We
benchmark our SIA results with the BdG calculations for an
equilibrium supersolid state, by evaluating C from the full
Gross-Pitaevskii equation (GPE) solution [34]. As shown in
Fig. 1(c), despite its simplicity, the SIA scaling reproducesvery well the full numerics up to C  40 %. For larger C, as
expected, the model breaks down.
For large C, we employ a droplet-array ansatz (DAA),
describing
the system as an array of ND droplets, ψ (y) =NDiθ j[3,42]. Each droplet is described by aj=1 χ (y − jd )eGaussian function, χ (y), of size σ , separated by a distanced > σ from its neighbours. Each droplet is allowed to havean independent, yet uniform, phase θ j . Within the DAA, the D iθ j 2
e | σ /d.
DSF shows the proportionality S(k) ∝ n| N1D Nj=1
It decreases with both the density overlap between droplets,
set by σ /d, and the phase variance along the array. The phasevariance can not be accounted for in the ground-state GPE
theory, which describes a state possessing a uniform phase.
To benchmark the DAA results with the BdG calculations, we
thus set θ j = 0 for all j [34]. We find a very good agreement
for C > 80 %. The effect of phase variations on the scattering
response will be later studied using dynamical simulations;
see Figs. 3 and 4.
To summarize, at equilibrium, the high-energy response
decreases when the contrast increases. For a small density modulation, the response can be directly connected toLeggett’s estimate for the superfluid fraction. Moreover, thepresence of phase variations further decreases the response,
as we explicitly show using the DAA. We now compare ourtheory expectation with the experiment.
In the experiments, we access the density response of asupersolid by performing high-energy Bragg scattering on a
166
Er dipolar quantum gas, confined in an axially elongatedharmonic trap. A transverse homogeneous magnetic field orients the atomic dipoles and sets as [3]. We initially prepare
the system in the ordinary BEC phase, and enter the SSP
via interaction tuning by linearly lowering as below a criticalvalue, as∗ , for which the BEC-SSP phase transition occurs.
Similar to previous experiments [3,10], as∗ is extracted withan interferometric technique. For the present trap and atom
+28
numbers, N, we measure as∗ = 54.94−13
a0 . Furthermore, we
observe the ID regime below as ≈ 53.9 a0 , see Ref. [34].

PHYSICAL REVIEW A 104, L011302 (2021)

3
2
1
0
53.5

54

56.5

57

FIG. 3. Experimental F (circles) versus as across the BEC-SSPID phases. For the lowest three as , we do not observe a resonanceand plot the standard deviation of the data as an error estimate.
Horizontal error bars correspond to uncertainties of the magnetic
field [34]. Theoretical F (lines) from the BdG calculations on the
ground state (grey), from the RTE simulations (black), and from therescaled BdG calculations that include
obtained from the RTE(blue). The gray shading corresponds to the uncertainty in as of theexperimental phase transition (vertical line).

L011302-3

D. PETTER et al.

ny (arb. u.)

1

PHYSICAL REVIEW A 104, L011302 (2021)

(a)

0

7

(b)

1

0
(rad)

0 (c)

-8

-4

y

0

4

8

53.6
54.2
54.8
as (units of a0)0

FIG. 4. RTE simulations without Bragg excitation. (a) Time evolution of the integrated in situ density of the wave function for(squares) versus as .
as = 54.04 a0 . (b) C τ (triangles) and
The grey line corresponds to the central contrast obtained from theground-state theory. The solid blue line is a smooth interpolationof, fixed to unity at the phase transition point. The shadings
give the standard deviation obtained from five simulation runs. The
vertical line corresponds to the phase transition point. (c) Phase cuts
corresponding to the simulation shown in (a).

have checked the scaling with τ and V [34]. Figure 2 shows
examples of the Bragg-excitation spectrum for various as . In
the BEC regime until the onset of the SSP, we observe a downward shift of the resonance frequency without a significant
change in F [34]. In contrast, as we enter into the SSP regime,
F undergoes a stark reduction. In the ID regime, the resonancepeak completely vanishes.
Figure 3 shows the evolution of F across the BEC-SSP-ID
phase diagram. The as extension of the three phases (see background colors), has been determined from independent measurements of the phase coherence and density modulation of
the states [3,34]. When reducing as , F first slightly increases
in the BEC phase, continuously crosses at the BEC-SSP transition, and then drastically reduces to  1%, close to ourdetection level, when lowering as further by ∼0.5 a0 . Finally,
for as < 54 a0 , we do not observe any resonant response.
We compare the experimental results with our BdG theory
for the stationary, trapped gas. While in the BEC regime, experiment and theory show a good agreement, in the SSP they
start to substantially deviate from each other. The data shows amuch faster reduction of F than the one predicted by the BdG
theory. This suggests that an important ingredient is missingin the ground-state theory. Our DAA model provides a firstintuitive explanation, showing that, not only the increasing
crystalline modulation but also phase variations can lead toa reduction of the system response. We envision two sourcesof phase variations. First, quantum and thermal fluctuations,which are expected to dominate in the ID regime, yield phasepatterns varying from shot to shot. Second, coherent dynamics, as, e. g., induced by the crossing of the BEC-SSP phasetransition, leading to reproducible phase patterns. Neither
phenomena are accounted for in the BdG calculations.
To investigate these effects, we simulate the system realtime evolution (RTE) [44]. Our calculations reproduce the fullexperimental sequence [34]. Random shot-to-shot variations
are included by adding an initial population of BdG modes
from quantum and thermal noises [34]. From the simulated
momentum distributions, we extract the excited fractions, as
done for the experimental data. Contrary to the BdG results,
the RTE simulations describe remarkably well the data bothin the BEC and SSP phase; see Fig. 3.
The impact of the changing contrast and phase variations
across SSP-ID phase can be seen from RTE without a Bragg
excitation for different holding times. As shown in Fig. 4(a),the density profiles n(y) exhibit only a slight reduction of
the contrast with time due to atom loss. As expected, thecalculated C τ , time averaged over the Bragg scattering duration, increase with decreasing as . However, for each as , weobserve a 10–30 % lower contrast than the one extracted fromthe ground-state calculations. Since a reduced contrast would
mean an increase in F, the varying contrast cannot explain themismatch between the BdG theory and both the experimental
and RTE results; see Fig. 3.
The RTE calculations also reveal that the phase of the wave
function, θ (y), develops a nonuniform profile. For instanceat as = 54.04 a0 , θ (y) exhibits a stairlike profile with fairly
constant values within the density peaks and discrete phasesteps in between them; see Fig. 4(c). This behavior suggeststhat each density peak acquires an independent phase, despite
their density links. We also observe that the phase pattern
is fairly reproducible between simulation runs and mainly
affected by the coherent dynamics arising by the crossing of
the phase transition [34].
Following the DAA model, phase variations are expected D iθ j 2
e τ | [28,34].
to reduce S(k) by a factor≈ | N1D Nj=1
As shown in Fig. 4(b),is almost unity close to the BECSSP phase transition and significantly drops when lowering as
towards the ID regime, where it starts to flatten. The standard
deviation ofrelates to the shot-to-shot reproducibility ofthe phase pattern. In the SSP, the deviation remains small,confirming that the phase variations originate from coherent
dynamics. In contrast, the deviation increases when reaching
the ID regime. This highlights the increasing effects of fluctuations, showing that the phase variations ultimately become
incoherent. We empirically account for the effect of phasevariations in the BdG theory by scaling the DSF withover
the whole SSP-ID regimes. As shown in Fig. 3, this simple
inclusion ofshows the pronounced impact of the coherent
phase variations for the experimentally observed response.
In conclusion, we demonstrate high-energy two-photon
Bragg scattering spectroscopy as a sensitive probe of density
modulations, and coherent and incoherent phase variations
in a quantum system. Accounting for the phase variations
is crucial to fully capture the behavior of supersolid statescreated in experiments via a dynamical tuning of the interactions. Our work provides important steps to a more complete
vision of the dipolar supersolid, including out-of-equilibrium
phenomena, and opens the door for future exploration ofcritical phenomena induced by the dynamical crossing of the
BEC-SSP phase transition [45–47].
We thank S. Stringari for insightful discussions and B.
Yang for the careful reading of the manuscript. Part of thecomputational results presented have been achieved using

L011302-4

BRAGG SCATTERING OF AN ULTRACOLD DIPOLAR GAS …

PHYSICAL REVIEW A 104, L011302 (2021)

the HPC infrastructure LEO of the University of Innsbruck. This work is financially supported through an ERC
Consolidator Grant (RARE, No. 681432), a DFG/FWF(FOR 2247/PI2790) and a joint-project grant from the FWF(No. I 4426, RSF/Russland 2019). L. C. acknowledges the

support of the FWF via the Elise Richter Fellowship number
V792. A. R. and S. M. R. acknowledge support from Provincia
Autonoma di Trento and the Q@TN initiative. We also acknowledge the Innsbruck Laser Core Facility, financed by theAustrian Federal Ministry of Science, Research and Economy.

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L011302-6

Supplemental Material: Bragg scattering of an ultracold dipolar gas across the phasetransition from Bose-Einstein condensate to supersolid in the free-particle regime
D. Petter,1 A. Patscheider,1 G. Natale,1 M. J. Mark,1, 2 M. A. Baranov,2, 3 R. van Bijnen,2, 3
S. M. Roccuzzo,4, 5 A. Recati,4, 5 B. Blakie,6, 7 D. Baillie,6, 7 L. Chomaz,1 and F. Ferlaino1, 21Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
3
Center for Quantum Physics, University of Innsbruck, Austria
4
INO-CNR BEC Center and Dipartimento di Fisica,
Università degli Studi di Trento, 38123 Povo, Italy
5
Trento Institute for Fundamental Physics and Applications, INFN, 38123, Trento, Italy
6
The Dodd-Walls Centre for Photonic and Quantum Technologies,
University of Otago, Dunedin 9054, New Zealand
7
Department of Physics, University of Otago, Dunedin 9016, New Zealand(Dated: June 16, 2021)

A. Preparation of the BECs

3

SSP

BEC

We prepare a BEC of 166 Er by loading about 3 × 106
thermal atoms into a crossed optical dipole trap (ODT)
and subsequent evaporative cooling, see Ref. [1, 2]. During the evaporative cooling, a homogeneous magnetic
field of 1.9 G is present to ensure high enough rethermalization rates to obtain ultracold temperatures. After theevaporation, we adiabatically modify the corresponding
ODTs laser powers and beam waists, to shape the confining potential Vtrap (r) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )/2 to acigar-shaped geometry with harmonic trapping frequencies ωx,y,z = 2π × [250(1), 31.7(13), 156(2)] Hz. Consecutively, the magnetic field is lowered to a value corresponding to 64.9 a0 . After this preparation procedure,
we obtain a BEC with a total atom number of 1.2 × 105
atoms and a condensed fraction of 70 %. The temperature of 95(5) nK is obtained from time-of-flight (ToF)
expansion measurements.

To enter the BEC-SSP-ID regimes, we lower down as
linearly in 20 ms to the corresponding values given in themain manuscript. We then let the system equilibrate for
10 ms and consecutively apply a Bragg pulse of 7 ms duration. In order to access the momentum distributionof our atomic cloud, we perform a ToF expansion, by
abruptly switching off all trapping potentials directly after the Bragg excitation. After 30 ms of free expansion,
we take an absorption image of the cloud along the dipole
direction. We note that, due to residual magnetic field
drifts in the experiment (estimated to be ±2 mG), the uncertainty on as , during the Bragg pulse, ranges between±0.1 a0 and ±0.2 a0 , increasing for lower as . This uncertainty is represented by the corresponding error bars on
our data in the main manuscript.

fside (%)

2.5
2

1.5
1

0.5
0

*
Δas,P
*
Δas,B
Δas*

53.8 5454.2 54.4 54.6 54.8 55
as (units of a0)

55.2 55.4

FIG. S1. Fraction of atoms in one side peak of the atomic
cloud’s interference pattern across the BEC-SSP phase transition. Error bars denote one standard error obtained fromabout 30 measurements. The vertical line shows the obtainedphase transition point. The different grey shadings correspond to the different uncertainties that are taken into account to obtain the total uncertainty, ∆a∗s , of a∗s (see text).

B. Determination of experimental BEC-SSP phasetransition

In order to determine a∗s for our experimental parameters, we perform a time-of-flight expansion of the system in the BEC or SSP regime after the equilibration
time. Here, no Bragg pulse is applied. We find either
an expanded ordinary BEC or an interference patternof the expanded supersolid, where a part of the atoms
appear in two side peaks around ky ≈ ±kc . The atom
number in these two side peaks is directly related to themodulation contrast of the in-situ cloud [2–4]. We measure the fraction of atoms in a single side peak, fside ,and monitor its evolution versus as ; see Fig. S1. In the
BEC regime, where no density modulation is present, we2find fside = 0 down to as,1 = 55.00 a0 . After crossing
the BEC-SSP phase transition, we observe fside > 0 for
as ≤ 54.88 a0 = as,2 , which increases with lower as . Taking the mean, (as,1 −as,2 )/2, we find a∗s = 54.94 a0 with anuncertainty of ∆a∗s,P = 0.05 a0 , coming from our resolution in as ; see shadings in Fig. S1. We include a magnetic
field uncertainty corresponding to 2 mG (±0.12 a0 at a∗s ),which increases the uncertainty to ∆a∗s,B = ±0.13 a0 .
Furthermore, we take a finite resolution of fside ≈ 0.2 %into account and obtain the final estimate of the criticalpoint of the BEC-SSP phase transition a∗s = 54.94+28−13 a0 .

dynamics.

D. Calibration of atom loss and atom number for
BdG theory

x104
7

SSP

BEC

6

C. Transition from SSP to ID

x102
ID

BEC

SSP

AФ, AM

2

N

5
4
3
0
54

54.5
as (units of a0)

55

FIG. S2. Amplitudes Aφ (red circles) AM (blue squares)
versus as . Error bars denote the standard error from about 30
experimental realizations [2]. Non-zero values of AM enable
us to identify modulated states and confirms the BEC-SSP
transition point (vertical line, gray shaded area). The SSP is
identified by AM ≈ Aφ > 0 and extends down to as = 53.9 a0 .
For lower as an ID state is observed (Aφ < AM > 0).

We use the same analysis of Aφ and AM as in Ref. [2]to distinguish in the experiment the SSP and the ID
regime. In brief, Aφ relates to a reproducible interference pattern in time of flight and thus reveals a coherent
and modulated state. AM relates only to the presenceof an in-situ density modulation (structures in the ToF
images). By combining both observables, one can distinguish the SSP (Aφ ≈ AM > 0), the ID (Aφ < AM )and the ordinary BEC regimes (Aφ = AM = 0) in theexperiment, see Fig. S2. We find that for as < 53.9 a0
the system is in the ID regime. The measurements are
performed directly after the equilibration time and without a Bragg excitation (same timings as in Sec. B). We
note that the ratio Aφ /AM is mostly sensitive to phasefluctuations, which lead to different interference patterns
in different experimental runs and is insensitive to reproducible phase variations in the system. The latter could
affect the structure of the interference patterns, yet ina reproducible way. Therefore, Aφ /AM is an observablethat is adapted to describe the coherence of the systembut does not measure the phase variations, investigatedin the main manuscript, which are induced by diabatic

2
53

54

55
56
as (units of a0)

57

FIG. S3. Atom number in the BEC versus as for different
times in the experiment, corresponding to the beginning (circles), middle (triangles) and end (squares) of the Bragg pulse.
Error bars denote the standard error from 5 measurements.
The lines are spline interpolations to the corresponding data.
The measurements were obtained without applying a Bragg
pulse. The vertical line indicates the measured phase transition point.

Due to three-body recombination losses, the atom
number, N , in the condensed part is decreasing during
the 7 ms of the Bragg pulse by typically 10-30 %. Therefore, the atom number in the experiment varies withas , which we include in our BdG theory of the threedimensionally trapped system. We note that we do not
observe additional atom loss due to the presence of the
Bragg excitation, as the wavelength of the used laser lightis far enough detuned from any atomic resonance (seeSec. E).
To extract N we perform an additional set of measurements in which we do not apply a Bragg pulse and, after agiven hold time in trap, take absorption images after a 30
ms ToF expansion. From these images, the thermal component is fitted by an isotropic 2D Gaussian function andsubtracted. A final numerical integration over the image
yields N without the need of an additional fitting of thecondensed part itself. In Fig. S3, we show N across thephase diagram for different timings in the experiment,
corresponding to the beginning, the middle and the end

3

E. Bragg spectroscopy

The Bragg excitation beams are realized holographically with a digital-micromirror device (DMD), as detailed in Ref. [5]. In short, the setup uses a near-resonant
laser light, red-detuned by 71(1) GHz from the 401 nm
transition of 166 Er. These two Bragg beams interfere under an angle on the atoms’ position, giving rise to an interference pattern. In our setup this angle can be tuned,but for this current work we keep it fixed to obtain aninterference pattern with a wave vector k = 4.2(3) µm−1
along y. The value and uncertainty on k is deduced
from offline measurements of the angle between the two
Bragg beams. To excite the system, the Bragg scatteringneeds to supply energy, h̄ω, which is introduced with afrequency difference, ω, between the two Bragg beams.
Here, we use a sequence of holographic gratings that is
uploaded on the DMD and continuously shifts the phaseof one beam in 9 steps from 0 to 2π. Depending on theframe-rate of the uploaded sequence, we can vary ω from
0 Hz to 1000 Hz.
To calibrate the depth V of our Bragg potential,
we perform Kapitza-Dirac-diffraction measurements [6].
For these measurements, we tune the laser lightcloser to the atomic transition (20.6 GHz) and use
the maximally available power for our Bragg beams.
By doing so, we achieve a maximum optical depthof Vmax /h = 430(50) Hz, corresponding to 3.2(4) Erec ,where the recoil energy Erec = h̄2 (k/2)2 /(2m) = 135 Hz.
In order to extract the potential depth V of our Bragg
scattering probe, we rescale this calibrated value according to the corresponding laser light detuning andpower used for the Bragg scattering [7]. We obtain
V = 42(5) Hz, which is well inside the linear scattering regime (see Sec. F and Fig. S6). The exact calibration of the potential depth does not include systematiceffects, as for example inhomogeneities of V cross theatomic cloud or in-trap dynamics of the atoms during
the Kapitza-Dirac-diffraction measurement. Nevertheless, we note that the estimation of the linear response
regime is insensitive to the exact calibration of V . We
furthermore note, that a direct comparison of F from theexperiment with the one from the RTE suggests that V
needs to be rescaled by about 1.7.
Figure S4 gives examples of our images for a resonant
and an off-resonant Bragg scattering frequency. For the
resonant case (Fig. S4, left panel) we find scattered atoms
at a high momentum around ky ≈ k. As they appear outside of the interference patterns, observed from the unperturbed system, they constitute a clean signal for the

6

x102
3

ky (μm-1 )

4
2
0

2

-2

1

-4

-5

0
kx (μm-1 )

5

-5

0
kx (μm-1 )

50n (kx, ky) (μm2 )of the Bragg pulse. Each timing is interpolated with aspline fit. The fitted values of the intermediate timing(orange line in Fig. S3) is used as the atom number in
our BdG theory.

FIG. S4. Examples of momentum distributions of a supersolid at as = 54.59(13) a0 after an applied Bragg excitation(a) on resonance at 1.7 h̄ωz and (b) off resonance at 3.7 h̄ωz .
The two side peaks appearing around ky ≈ ±2.4µm−1 constitute, together with the central peak at k ≈ 0µm−1 , theinterference pattern obtained when expanding a supersolidstate. The black box indicates the region of interest fromwhich Nexc is extracted. Each image is an average of 15 experimental realizations.

analysis of the excited fraction. We count the numberof atoms, Nexc , in a region of interest (ROI), indicatedby the black boxes in Fig. S4. We note that we carefully
checked that neither F , nor ωk , changes within the uncertainties when increasing the ROI size by 30%.. We
measure the total atom number, N , for each measurement individually, by performing a similar count on arectangle of 12 µm−1 by 14 µm−1 , covering all condensed
and scattered atoms. By measuring Nexc /N for different
excitation frequencies, we obtain a spectroscopy of the
Bragg scattering. We note that due to thermal atoms,present in the analyzed region of interest, all Bragg resonances show a small offset, which is extracted from theoffset of the Gaussian fit to the resonance and then subtracted.
From the Bragg excitation spectra, we extract theresonance peak’s amplitude, as discussed in the main
manuscript, and a resonance frequency, ωk . The latteris shown in Fig. S5 as a function of as . We observe that
ωk decreases monotonically from high to low as , acrossthe BEC-SSP phase transition. This behaviour is consistent with the extracted ωk from the RTE theory. Furthermore, we calculate the expected resonance frequency
from the BdG calculations. In the BEC regime, the BdG
theory predicts a softening of the measured excitation
modes that relates to the roton softening (see Section
G). This equilibrium BEC expectation agrees with theexperimental results and the dynamical simulations. In
contrast, a disagreement arises after crossing the BECSSP phase transition with lowering as . Here, the BdG
theory predicts a hardening of the measured excitation
modes in the supersolid regime, which is not observed
neither in the experiment, nor in the RTE simulations.
This qualitative difference can stem from the increased
phase variations that develop in the system. Indeed, the4phase variations intrinsically connect to finite velocity
fields in the system that effectively make the excitation
modes softer.

SSPωk (units of ωz )

2.5

BEC
G. Evolution of the excitation spectrum with as forthe infinite cigar-shaped gas

2

1.5

54

55

56
as (units of a0)

57

FIG. S5. Extracted resonance frequencies, ωk , versus as (circles) and their comparison with the expected resonance position from the BdG theory (gray line) and ωk from the RTE(connected dots). The vertical line indicates the BEC-SSP
phase transition point and its shading the uncertainty on a∗s .

F. Variations of the excited fraction with the Bragg
pulse duration

6

F (%)

10

F (%)

of F with τ , which is consistent with the expected scalingfrom BdG theory, F ∝ V 2 τ S(k) [8]. Furthermore, weprobe the quadratic scaling of F with V in the SSP andfind also here an agreement up to V = 80 Hz (see inset).

4

0

5

2

0

40
80
V/h (Hz)

120

0
0

2

τ (ms)

4

6

FIG. S6. Evolution of F during the Bragg pulse for three exemplary as = [55.5(1), 54.0(1), 53.3(1)] a0 (black, red, blue),corresponding to the BEC, SSP, ID regimes, respectively.
Each data point corresponds to an average of 5 to 15 measurements and its uncertainty to the standard error. The
solid lines correspond to a linear fit and its shading to the
fit’s 68 %-confidence bound. The inset shows the evolutionof F with the applied potential depth, V , of the Bragg pulsefor τ = 7 ms for a supersolid state at as = 54.28(14) a0 . The
solid line corresponds to a quadratic fit up to V = 80 Hz, thedashed line is the extension of the fitting.

In Figure S6, we present the measured evolution of F
with the Bragg pulse duration from 0 to 7 ms for a fixed ω.
Across the BEC-SSP-ID regimes, we find a linear scaling

We calculate the excitation spectrum and the dynamic
structure factor (DSF) of an infinitely elongated, cigarshaped dipolar supersolid in Fig. 1(a, b) of the main
manuscript. In Figure S7 we present the evolution of
the excitation spectrum across the BEC-SSP-ID regimes.
Figure S7 (a1-a5) shows the integrated density profiles of
the ground state along the unconfined direction for different values of as and a fixed mean axial density of
4.7 × 103 µm−1 . Figure S7 (b1-b5) shows the corresponding excitation spectrum. At large enough as , the groundstate has a uniform density along the unconfined direction (a1, a2 - BEC phase) and its excitation spectrum
shows the typical phonon-maxon-roton spectrum, firstpredicted in [9, 10]. When decreasing as below a critical value, the ground state becomes density modulated(a3, b3 - SSP phase) with a modulation wave number
kc = 2.3 µm−1 close to the BEC’s roton momentum (b2),underlying the connection between roton softening andcrystallization. The density modulation has a finite contrast and its value increases when lowering as further
down (a4, a5).
When crossing the BEC-SSP phase transition, the excitation spectrum changes dramatically, becoming periodic, with the appearance of two gapless Goldstone
branches associated with phase (lower energy branch)and density (higher energy branch) excitations, respectively [11–13]. In addition to these gapless branches, one
observes gapped parabolic branches of excitations withenergetic minima at integer multiples of kc . The one
branch at ky = kc is the one investigated in the main
manuscript. For decreasing as , the energy minimum of
this parabolic branch increases towards the ID regime[Fig. S7 (b3-b5)].
As described in the main paper, we use the
R norm ofthe calculated Bogoliubov amplitude kuj k = |uj (r)|2 dr
to distinguish whether a mode j is a collective excitationor has a single-particle character [14, 15]. Collective excitations feature kuj k  1 whereas single-particle excitations have kuj k ' 1. In Figure S7 (b1-b5), we color each
excitation mode according to kuk. We find that lower
energy modes, such as the roton mode in the BEC andthe Goldstone modes in the SSP have a clear collective
nature. The energetically higher modes (h̄ω >∼ 0.5 h̄ωz ),of the parabolic branch in the SSP and the ky > kc branch in the BEC, have kuj k ' 1 across the BEC-SSP
phase transition. We note that the condition kuj k ' 1
does not directly identify an excitation mode as a free-ω (units of ωz)n (x104 μm-1)

5

1.5

(a1)

0
-4

-2

(a2)

0
2
y (μm)1(b1) ||u||

1.5

3

>5

-2
(b2)

-2

0
2
y (μm)(a4)

0
2
y (μm)

(b3)

-2
(b4)

(a5)

0
2
y (μm)-20
2
y (μm)

(b5)

1.0
0.5
0

0

0.5 1 1.5 2 0
ky (units of kc)
0ω (units of ωz)

4

(a3)

1.5

(c1) S (k, ω)

12 >2.6

momentum (kc)

0.5 1 1.5 2 0
ky (units of kc)(c2)

0.5 1 1.5 2 0
ky (units of kc)(c3)

0.5 1 1.5 2 0
ky (units of kc)(c4)

0.5 1 1.5ky (units of kc)

2

(c5)

1.0
0.5
0

0

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5ky (units of kc)

2

FIG. S7. Axial excitation spectra of the infinitely, extended gas across the BEC-SSP-ID regimes in a ωx,y,z = 2π(250, 0, 160) Hz
trap with fixed axial density n0 = 4.7 × 103 µm−1 . (a1-a5) Integrated density profiles along the unconfined direction, for
as = (52.00, 51.40, 51.25, 51.00, 49.75) a0 . (b1-b5) Transverse symmetric modes of the corresponding excitation spectrum
colored according to kuk. (c1-c5) The corresponding S(k, ω). For visibility, the DSF is broadened with a Gaussian function.

particle. To obtain a free-particle excitation, the modeneeds to be of single particle character and additionallyits energy needs to be mainly given by the kinetic energy.
Therefore, free-particle excitations have a wave functionthat is a plane wave [15]. We note that for our experimentally relevant energy regime, the probed excitation
modes are described well by a plane wave, as shown inSec. I. and Fig. S10 (b).

From our simulations, we also calculate the DSF. In
the BEC phase [Fig. S7 (c1, c2)] the DSF is dominatedby the roton mode at ky = kc . Moreover, the deeper
the roton minimum, the stronger is its response to small
density perturbations. We note that, in the BEC phase,
this affects also the density response and the energy evenfor momenta higher then the roton momentum. Aftercrossing the phase transition into the SSP, we find thatthe DSF of the parabolic branch [Fig. S7 (c3)] smoothly
connects to the free-particle branch of the BEC phase.
For decreasing as , the DSF of the free particle branch
becomes smaller [Fig. S7 (c4, c5)].

H. BdG theory for the three-dimensional trappedgas

For the current manuscript, we employ similar BdG
and ground state calculations as already described inRefs. [5, 13, 16]. In this theory the gas is trapped in
all three dimensions. For calculating the ground states,
we use the experimentally extracted atom number at
the intermediate timing of the Bragg pulse (presented inFig. S3, triangles). The radially integrated density profiles of the ground states in the SSP regime are presented
in Fig. S8. We note that in this theory, the BEC-SSP
phase transition lies at 3.79 a0 below the experimentally
determined one. This shift in as between theory andexperiment has also been found in Refs. [2, 5, 17]. Therefore, throughout the manuscript, the BdG theory and theground state calculations are presented with an up-shiftof 3.79 a0 for as .
The presence of an axial trapping potential, leads todiscrete excitation modes in the spectrum, see Fig. S9 (a).
Furthermore, each mode is broadened in ky . The finite
duration of the Bragg pulse gives an energy broadeningof each excitation mode (Fourier broadening ∼ h×130 Hz
full width at half maximum) which is much bigger than

6n (x103 μm-1)

0

1

2

3

4

5

10

(a)

0

1

>2

(b)

ω (unis of ωz)

3
2

5
y (μm)S (k, ω) (units of S*)

4

1

0

0

-5
-10
53.5

54
as (units of a0)

54.5

FIG. S8. Radially integrated in-situ density profiles versus
as of the supersolid ground states in a trap with harmonicfrequencies ωx,y,z = 2π × (250, 31, 160) Hz. For each as , weuse the atom number measured in the experiment (see Fig. S3,
triangles).

the energy spacing between the modes in the spectrum.
Therefore, in the Bragg spectroscopy only a single resonance is visible, which is constituted of multiple excitation modes. To account for this, we calculate S(k, ω)
while broadening each mode in energy according to the
Fourier-broadening, expected from a 7 ms Bragg pulse,
see Fig. S9 (b). After calculating the broadened S(k, ω)and evaluating it at the experimental k, we also find in
the BdG theory a single resonance in energy. To comparewith the experiment, we extract S(k) from a Gaussian fit
to this resonance; see also [5].

I. Free particle regime in the BdG theory for atrapped gas

To transfer the insights from the BdG calculations ofan infinitely extended system (see Sec. G) to the experimentally trapped case, we also analyze kuk and kvk of theexcitation modes obtained from the BdG calculations ofa three-dimensional trapped gas. Similar to the infinitely
extended system, we find that modes with h̄ω >∼ 0.5 h̄ωz
have kuk ≈ 1 and therefore a single-particle character
across the BEC-SSP-ID regimes. This is exemplified inFig. S10 (a) where we show, for an exemplary state in the
SSP, the norm of the obtained Bogoliubov amplitudes
versus the energy of the corresponding mode.
As mentioned already in Sec. G, to further identify asingle particle excitation as a free particle one, the ex-

0

1
2
3
ky (units of lz)

40

1
2
3
ky (units of lz)

4

FIG. S9. Dynamic structure factor calculations for the 3Dtrapped system in the SSP at 54.4 a0 . (a) DSF with a small
broadening of each excitation mode,p showing the discretenessof the spectrum. Here, lz =h̄/mωz ≈ 0.6µm denotes
the harmonic oscillator length along z. (b) DSF after applying a broadening, corresponding to a 7ms long Bragg excitation pulse (square-shaped pulse, sinc-squared broadening
function). Note that the lowest energy mode at ky lz ≈ 1.5
has a very strong contribution to the DSF, giving rise to visible harmonic components at higher energies. The vertical,
dashed lines indicate the experimentally probed momentum.

citation’s wave function needs to be a plane wave. To
investigate this aspect, we study the excitation modes’
density profiles and find for modes in the experimentally
relevant energy regime a clear plane wave character. Figure S10 (b) shows the radially integrated density profile ofan exemplary excitation mode of a supersolid and compares it to the integrated density of the ground state.
The plane wave character is clearly visible as a modulation with k ≈ 4.1 µm−1 across the whole system. We
only find a mild reduction of the plane wave’s amplitude
towards the outer region of the system. As a comparison,
we show in Figure S10 (c) the integrated density profile
of an excitation mode at lower energy, 0.55 h̄ωz , which
also has kuk ≈ 1, but is clearly not a plane wave.
We note that, from the theory of infinite systems,
the low-energy low-momentum scattering response is expected to directly relate to the superfluid density, via thestiffness of the lowest-energy branch [15, 18]. However,such a relation is difficult to establish for the finite-sized
experimental systems and measurements in this regime
are thus expected to be hard to interpret. Therefore,
our measurements in the high-energy high-momentum
regime circumvent this issue and offer an alternative
probe scheme, less affected by finite-size limitations.

J. Comparison of the finite-size BdG theory to theself-consistent SIA and DAA model

From the ground state profiles of the three-dimensional
trapped BdG theory in Fig. S8, we numerically evaluate

7

(a)

||u||, ||v||

10

with the ground state properties and not its absolutevalue, we renormalize the SIA and the DAA. For thecomparison in Fig. 1 of the main manuscript we show thefinite BdG and the SIA rescaled to unity for the point at
the phase transition. The DAA is rescaled directly on the
BdG theory to match its values in the lower as regime.
Over the investigated as range, we find that both, the
SIA and the DAA, describe the BdG theory well for low
and large C, respectively (see main text), and for momenta k ≥ 4.0 µm−1 . This gives an estimate for which
momenta the impulse approximation becomes valid.

freeparticles

collective1

0.1

nexc , nGS (arb. u.)

nexc , nGS (arb. u.)0

0.5
energy (units of ħωz )

1

1 (b)
K. Real time evolution of the Bragg scattering

0
1 (c)

0

-10

-5

0
y (μm)

5

10

FIG. S10. (a) Bogoluibov amplitudes, kvk (crosses) and kuk(circles), for the excitation modes of a trapped supersolid at
as = 54.49 a0 . (b) Integrated density profile, nexc , of an exemplary excitation mode with an energy of 1.7 h̄ωz (red line) andk ≈ 4.1 µm−1 and a comparison with the integrated groundstate density, nGS , (grey line). (c) The same as in (b) but foran excitation mode at 0.55 h̄ωz .

our SIA and DAA model on the corresponding groundstates. To self-consistently evaluate the SIA result fromthe ground state, we need to estimate the contrast of thedensity modulation, C = (nmax − nmin )/(nmax + nmin ).
We determine nmin from the minimum density at y = 0
and nmax from the density of one of the two most-central
density peaks. To evaluate the DAA model, we numerically extract the 1/e-size, σ, of the two central droplets
and the distance d between them. To estimate the density, we calculate the mean density in the central region
between the two central droplets. Therefore, our modelcomparison takes only the central part of the system into
account and neglects the outer density regions. Furthermore, as our models extract only the scalings of the DSF

Our theory for the RTE simulations was already presented in Ref. [16]. For the current manuscript, the simulations start with the ground state wave function with
N = 8.5 × 104 atoms at as = 60.9 a0 . We add a thermal population, corresponding to a randomly drawn occupation of the system’s excited states (incl. a random phase) with a Poisson distribution, whose mean isgiven by the Bose distribution (+1/2 to simulate quantum fluctuations) for the mode’s energy at a temperatureof 100 nK [19]. This increases the total atom number toabout 1 × 105 (similar to the experimental situation) andsimulates thermal and quantum fluctuations in the system.
In the time evolution, we reproduce the experimental
sequence, including a 20 ms long linear as ramp, followed
by a 10 ms holding at the final as and a consecutive 7 ms
Bragg excitation along y. In order to obtain the system’s
momentum distribution, we perform a Fourier-transformof its wave function. On this momentum distribution, we
perform the same analysis as on the experimental data,
i. e. we analyse the fraction of excited atoms in a regionof interest around k ≈ 4.2 µm−1 for various ω. The resonances are fitted with a Gaussian function to obtain F
from the RTE simulations. For the same reasons as mentioned for the BdG calculations (Sec. H.), the crossingof the BEC-SSP phase transition in the RTE happens
3.34 a0 below the experimental phase transition. Therefore, throughout the manuscript, the RTE theory is upshifted in as by 3.34 a0 . We note that, when performing the RTE directly on the corresponding ground statesfrom the BdG theory, i. e. we do not include thermal
noise, the as -ramp and atom loss, we recover an excited
fraction that is well described by the BdG theory. This
indicates that the chosen analysis in ToF gives a reliablemeasurement of S(k) and in particular a consistent resultwith the 3D-trapped BdG calculations.

8
L. Real-time evolution and characteristics of thestate’s wave function

modulation in the system and the timing of the as ramp
(which occurs during [−30, −10] ms).

To study the time evolution of the contrast and thephase of the dynamically created supersolid states, we
perform RTE simulations without applying a Bragg excitation and monitor the axial density and the phase profiles from the calculated wave functions [see Fig. 4 (a, c)
in main manuscript]. Typically, in the RTE we observe
ND = 4 − 6 droplets, containing a variable atom number across the system. We evaluate the time-dependent
central contrast, C, between the two central density
peaks
numerically. The phase-variation factor ∆Θ =PN
Dheiθj iτ |2 is calculated by extracting the mean
| N1D j=1
phase, θj , of each single density peak over its full-widthat half maximum.

To give another insight into the time evolution of thesystem’s phaseR profile, we extract the global phase variation, α = L1 L |φ(0, y, 0) − hφiL |dy, of the wave function, which is extracted along a cut of φ along y. Here,hφiL denotes the averaged phase over the central region
L = [−7, 7] µm, see also [3]. In Figure S11 (b), we show
the time evolution of α for the whole simulation time.
For all as one sees a first local maximum in α (around−20 ms), coming from an axial breathing mode which is
excited due to the as ramp. At longer times, we find for∗54.5 a0 <∼ as ≤ as , that α remains small while the density
contrast is finite. For as < 54.5 a0 , we find that α seems
to approach a constant value, which increases with lower
as .

(a)

C

1

ramp of as

holdtiming of
Bragg pulse

53.94 a054.24 a054.54 a054.74 a054.84 a054.94 a055.14 a00

α (rad)2π (b)

0
-30

-20

-10
time (ms)07FIG. S11. RTE simulations with V = 0 (no Bragg excitation applied). (a) The time evolution of the extracted central
contrast of the integrated in-situ density distributions for different as (see legend). (b) The extracted phase incoherenceof a central cut through the wave function (see text). The
shadings represent the standard deviation from 5 simulation
runs with different statistical draws of the thermal population. The data presented in Fig. ?? (a, b) corresponds to thetime window of [0, 7] ms.

Figure S11 (a) shows C(t) over the whole simulation
time. For early times, we observe a small, but finite C
due to density noise in the simulations, which is coming from the included thermal fluctuations. During theholding time ([−10, 7] ms), for as < a∗s , we observe that
C first increases and consecutively slightly decreases due
to atom loss. Only for as < 54.2 a0 we find an oscillating
behaviour of the contrast in time. We note that there
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[16] 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).

9[17] 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, Phys. Rev. Research 1, 033088 (2019).

[18] A. Griffin, Excitations in a Bose-condensed Liquid (Cambridge University Press, New York, 1993).
[19] P. Blakie, A. Bradley, M. Davis, R. Ballagh, and C. Gardiner, Adv. Phys. 57, 363 (2008).

6.3 Publication: Observation of a narrow inner-shell orbitaltransition in atomic erbium at 1299 nm.
Physical Review Research 3, 033256 (2021) a
submitted 04 May 2021; published 17 September 2021
DOI: 10.1103/PhysRevResearch.3.033256

A. Patscheider,1 B. Yang,1 G. Natale,1 D. Petter,1 L. Chomaz,1 M. J. Mark,1,2
G. Hovannesyan,3 M. Lepers,3 and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne FrancheComté, 21078 Dijon, France

2a

The author of the present thesis developed the experimental procedures together with B. Y., A. P.,
D. P., and L. C., performed the measurements together with B. Y., A. P., and D. P., and contributed
in writing the manuscript.

169

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

Observation of a narrow inner-shell orbital transition in atomic erbium at 1299 nm
A. Patscheider ,1,* B. Yang ,1,* G. Natale ,1 D. Petter ,1,† L. Chomaz ,1,‡ M. J. Mark ,1,2
G. Hovhannesyan ,3 M. Lepers ,3 and F. Ferlaino 1,2,§
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria
3
Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne Franche-Comté, 21078 Dijon, France
2(Received 4 May 2021; accepted 30 July 2021; published 17 September 2021)
We report on the observation and coherent excitation of atoms on the narrow inner-shell orbital transition, connecting the erbium ground state [Xe]4 f 12 (3 H6 )6s2 to the excited state [Xe]4 f 11 ((4 I15/2 )0 )5d (5 D3/2 )6s2 (15/2, 3/2 )07 .
This transition corresponds to a wavelength of 1299 nm and is optically closed. We perform high-resolution
spectroscopy to extract the gJ factor of the 1299-nm state and to determine the frequency shift for four bosonicisotopes. We further demonstrate coherent control of the atomic state and extract a lifetime of 178(19) ms,
which corresponds to a linewidth of 0.9(1) Hz. The experimental findings are in good agreement with oursemi-empirical model. In addition, we present theoretical calculations of the atomic polarizability, revealing
several different magic-wavelength conditions. Finally, we make use of the vectorial polarizability and confirm
a possible magic wavelength at 532 nm.
DOI: 10.1103/PhysRevResearch.3.033256

I. INTRODUCTION

Ultranarrow atomic transitions are an extremely powerful
resource for high-precision measurements and for controllingand manipulating atoms on a quantum level [1]. Prominent
examples are clock transitions in alkaline-earth-like atoms[2–4]. The small spectral linewidth of these transitions enables the high-resolution detection of energy shifts on veryfine scales. This unique property made it possible, e.g., toobserve SU(N)-symmetric interactions in both, ytterbium andstrontium [5,6]. An additional important avenue paved bynarrow transitions is the optical manipulation and coherent
control of ultracold atoms. The tuning of the interparticle interactions using optical Feshbach resonances has been
demonstrated and benefits from the narrow linewidth due tothe suppressed photon scattering rate [7–11]. Coherent control
enabled the creation of ultracold molecules via Raman statetransfer [12–15], the preparation of the atoms in different
nuclear spin configurations [5,6], and the creation of spinorbit coupled quantum gases [16–19]. Finally, the coherent*

These authors contributed equally to this work.
Present address: Optical Materials Engineering Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092
Zurich, Switzerland.
‡
Present address: Physikalisches Institut, University of Heidelberg,
69120 Heidelberg, Germany.
§
Corresponding author: Francesca.Ferlaino@uibk.ac.at†

Published by the American Physical Society under the terms of theCreative 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.
2643-1564/2021/3(3)/033256(12)033256-1

excitation allows for the realization of quantum computationand quantum simulations, e.g., with neutral atoms loaded into
optical lattices [20–22].
Atomic species of the lanthanide family are multivalence
electron atoms and possess a special electron configuration, aso-called submerged shell, in which the 6s subshell is filled,
while the lower-lying 4 f or 5d subshells are open, being
partially unoccupied. This leads to a large variety of opticaltransitions in these elements, whose linewidths range fromtens of μHz to tens of MHz [23–25]. In contrast to alkalineearth-like atoms, which do not carry a magnetic moment intheir ground state, a selection of lanthanides allow for the
combination of a narrow transition with a large magnetic
moment. While narrow and ultranarrow transitions have been
extensively studied in alkaline-earth and ytterbium atoms,
only little is known for the other elements of the lanthanide
series. Some spectroscopic studies have been carried out fordysprosium [26] and thulium [27,28].
For the specific case of erbium, there is a prediction of anarrow inner-shell orbital transition, which has a change inthe total angular moment of J = +1 (|J = 6 → |J  = 7)and a change in the total spin of S = 1 [23]. The transition
involves the excitation of a 4 f ground-state electron to a 5d
state; see Fig. 1. Theoretical calculations predict a linewidth ofabout 2 Hz [23], which fills a gap between ultranarrow transitions in the mHz regime and transitions having linewidths
on the order of kHz, available in alkaline-earth atoms andpreviously explored in lanthanide atoms. Moreover, in contrast to most narrow transitions in other atomic species, thewavelength of 1299 nm lies within the telecom-wavelength
window, which, e.g., is advantageous for the application inquantum communication systems [29–32]. Here, we report on
the experimental observation of this transition. We perform a
Published by the American Physical Society

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)[Xe]4f 12 6s6p

4f 11

mJ’ = 75d
6s2

energy11[Xe]4f 5d 6s

401 nm2
mJ’ = -7

4f 12
6s2

mJ = 61299 nm[Xe]4f 12 6s 2
mJ = -6

6

7

8

J

FIG. 1. Schematic level scheme illustrating the [Xe]4 f 12
( H6 )6s2 → [Xe]4 f 11 ((4 I15/2 )0 )5d (5 D3/2 )6s2 (15/2, 3/2 )07 inner-shell orbital transition at 1299 nm (|g → |e) and the [Xe]4 f 12 (3 H6 )6s2 →[Xe]4 f 12 (3 H6 )6s6p(1 P01 )(6, 1)07 transition at 401 nm, used for absorption imaging. The horizontal lines indicate the energy levels
for |g (blue, even parity), |e (red, odd parity), and the state at
401 nm (black, odd parity). [Xe] stands for the electron configurationof xenon. The insets illustrate the electron configurations. The grey
shaded boxes represent the Zeeman manifold for |g and |e. Energies
are not to scale.
3

degeneracy. The BEC typically contains N = 1–3×104 atoms
with BEC fractions ranging from 30–80%, depending on theisotope choice. For the fermionic 167 Er isotope, we obtain adegenerate quantum gas of N = 2×104 atoms at a temperature of T ≈ 0.5TF , where TF is the Fermi temperature. During
the evaporation, a homogeneous magnetic field B is applied
to ensure that the atomic cloud remains spin-polarized in thelowest Zeeman level mJ = −6 (mF = −19/2) for the bosonic(fermionic) isotopes.
The light for driving the narrow inner-shell transition is
generated from an external-cavity diode laser (ECDL) operating at 1299 nm. We determine the absolute frequency ofthe laser by measuring the frequency-doubled light with acalibrated wavemeter [34], which has an accuracy of 60 MHz.
For our coarse spectroscopy, we use the wide tunability of
the ECDL via the control of a piezoelectric element, whichallows us to change the laser frequency. Furthermore, we cannarrow the laser linewidth and stabilize the frequency using
a high-finesse reference cavity made of ultralow expansion
glass. The reference cavity has a free spectral range (FSR)
of 1.4972462(3) GHz and finesse of about 175000. The stabilized laser system has an Allan deviation of 3.1×10−15 overan observation time of 1 s. The coherence time is extracted
from the phase noise power spectral density and corresponds
to 96 ms [35]. We measure a linear frequency drift of the highfinesse cavity of 4.34(7) kHz/day. In the experiment, we use
the frequency stabilized configuration for the high-resolution
spectroscopy.

careful experimental survey and characterization of the 1299nm transition, realizing the first crucial step towards extended
applications, e.g., to explore novel few and many-body phenomena in dipolar or large spin systems.
We experimentally observe the transition at 1299 nm forthe bosonic isotopes 164 Er, 166 Er, 168 Er, and 170 Er and forthe fermionic isotope 167 Er. We perform high-resolution spectroscopy to determine the gJ  factor of the excited atomic
state (|e) and to measure the frequency shift for the fourbosonic isotopes. We further demonstrate coherent controlof the atomic state and measure an excited-state lifetime of178(19) ms. We carry out trap frequency measurements todetermine the atomic polarizability of the excited state relative
to the ground state with the trapping light at 532.2 nm. As
we vary the polarization of the light we take advantage of thevectorial term of the atomic polarizability and we are able toget close to a magic-wavelength condition, where the groundstate (|g) and |e feature the same polarizability. We finallypresent theoretical calculations of the atomic polarizability
based on a sum-over-states formula and report on several
alternative options for magic wavelengths.

In the 1960s, the atomic spectra of lanthanides began toattract interest. Absorption lines were observed using King’s
furnace or oxyacetylene flames in the range between 650 to
250 nm [36,37]. In early spectroscopic works, the configuration of the low-lying energy levels, and particularly thatof the odd parity states, was not known. The first work to
identify the odd-parity level 4 f 11 5d6s2 —i.e., ortho-erbium
ground state—as the lowest-lying configuration above |g was
Ref. [38]. In this configuration, the angular momentum J1 =7/2 of the 4 f 11 core couples via J1 − j coupling to the j of the
5d electron, which leads to a total of 10 fine structure levels
with J ranging from 5 to 10. The assignment of the fine structure levels has been deduced by analyzing energy differences
between absorption lines using a spectroscopic-level searching algorithm. The NIST database reports the energies of thecorresponding fine-structure levels, referring to unpublishedmeasurements from mid 70 s [39]. For our level of interest|e = 4 f 11 ((4 I15/2 )0 )5d (5 D3/2 )6s2 (15/2, 3/2 )07 , the wavenumber
given by NIST [40], not accounting for the isotope shift, is

II. EXPERIMENTAL SETUPν̄NIST = 7696.956 cm−1 .

We search for the narrow inner-shell transition by performing spectroscopic measurements on a trapped quantumdegenerate erbium gas. Our experimental procedure to create
an erbium Bose-Einstein condensate (BEC) follows Ref. [33].
In brief, after laser cooling in a magneto-optical trap, we load
the atoms into a crossed optical dipole trap (ODT) operating
at 1064 nm and perform evaporative cooling down to quantum

To the best of our knowledge, prior to this paper, there has
been no direct measurement of the 1299-nm transition. Our
ultracold quantum gas provides a new opportunity to observeand characterize this transition.
We start our search of the line by performing a coarse
spectroscopy over a broad frequency range around ν̄NIST(corresponding to νNIST = 230.738 THz). After preparing an

III. COARSE SPECTROSCOPY

033256-2(1)

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)
IV. HIGH-RESOLUTION SPECTROSCOPY

1
0.5
0

164

Er

166

Er

167

Er

168

Er

170

Er

normalized atom number in g

1
0.5
0
1
0.5
0
1
0.5
0
1
0.5
0

230.746230.748230.75230.752

frequency (THz)
FIG. 2. Coarse spectroscopy results for the four bosonic isotopes
Er, 166 Er, 168 Er, and 170 Er at B = 1.355(5) G and for the fermionicisotope 167 Er at B = 0.52(5) G. The normalized atom number in|g is plotted versus the laser frequency, which is controlled by thepiezoelectric element of the ECDL. The atom number is normalized
to the moving median formed by 21 data points.
164

optically-trapped ultracold erbium gas in |g, we shine the1299-nm spectroscopy light on the sample. The spectroscopy
beam has a peak intensity of Ipeak ≈ 0.8 W/cm2 and a 1/e2
waist of about 110 μm. The irradiation time is 100 ms,
during which we sweep the laser frequency with an amplitude of about ±40 MHz. After irradiation, we release theatoms from the trap for a free expansion of 30 ms. We
record the number of remaining |g atoms by performing
standard absorption imaging using resonant light at 401 nm(see Fig. 1).
We record the absorption spectrum by repeating the measurement over a wide frequency range with a step sizeof 40 MHz. Figure 2 summarizes our results for the fourmost abundant bosonic erbium isotopes (164 Er, 166 Er, 168 Er,
and 170 Er) and for the fermionic 167 Er isotope. As expected from their zero nuclear spin (I = 0), each bosonicisotope exhibit just one resonant absorption line, detected
as a sharp dip in the number of |g atoms when varying the 1299-nm laser frequency. For the fermionic 167 Er
isotope, possessing a hyperfine structure (I = 7/2), we identify three resonances. We attribute the three resonances tothe transitions |F = 19/2 → |F  = 21/2, |F = 19/2 →|F  = 19/2, and |F = 19/2 → |F  = 17/2, respectively.
Notably, at the resonance positions, the population of |greaches values below 0.5, indicating an underlying loss mechanism, such as heating from a reduced trapping potential for|e or an interaction-based loss processes.

Thanks to the coarse spectroscopy measurements, we can
now restrict the frequency region of interest and perform aspectroscopy survey with a much higher spectral resolution
and lower laser intensity, allowing also to resolve the magnetic
Zeeman sublevels.
For this, we stabilize the laser frequency to the high-finesse
reference cavity and then precisely tune the laser frequency
using an acousto-optical modulator. The recorded absorption
spectra have a step size ranging from 1 to 8 kHz, depending on the measurement. Moreover, the spectroscopy is now
performed with a free-falling gas to eliminate possible ac
Stark shifts, eventually caused by the ODT light at 1064 nm.
Therefore, after sample preparation, we switch off all trapping lights, and then irradiate the sample with a 1299 nm
spectroscopy pulse of 1 ms, corresponding to a Fourier limited linewidth of 800 Hz. The pulse has a peak intensityof Ipeak ≈ 25 mW/cm2 . To minimize the possible frequency
shifts caused by the Doppler effect in a free-falling sample,
the 1299-nm laser beam propagates in a plane orthogonal tothe vertical direction, defined by gravity. The quantization
axis, defined by our bias magnetic field, is oriented along thevertical direction. The light contains contributions from alllight polarizations, such that the 1299-nm beam can induce
σ + , σ − , and π transitions; see inset in Fig. 3(a).
Figure 3(a) shows the ground-state population for the 168 Er
isotope as a function of the laser detuning, plotted with respect
to the central frequency of the π transition. We clearly observe
three resonant dips in the ground-state population, corresponding to the transitions from the ground-state level mJ =−6 to the excited Zeeman sublevels mJ  = −7, −6 and − 5.
We extract the center frequency and the transition linewidthby fitting the spectroscopy signals with a Lorentzian function.
The extracted linewidths are 2.4(1) kHz and 2.5(1) kHz forthe π and σ + transition and 20(1) kHz for the σ − transition.
The different linewidths of the spectroscopy resonances canbe explained by a power broadening effect, due to the different
Clebsch-Gordan coefficients of the magnetic sublevels, andthe composition of the light polarization.
We use the wavemeter to determine the absolute wavenumber asν̄168 = 7696.955(2) cm−1 .

(2)

Our measurement is consistent with the value reported in
the NIST database [40] [see Eq. (1)]. The accuracy of theabsolute wavenumber is limited by the wavemeter. However,
our spectroscopy measurement has a precision of about 2 kHz,
which provides the opportunity to improve the accuracy of
the absolute frequency by several orders of magnitude using
advanced measurement techniques, such as frequency combs[41].
V. LANDÉ FACTOR FOR BOSONIC ISOTOPES

From the observed Zeeman structure in the bosonic isotopes, we can extract the Landé gJ factor. This is an important
quantity, e.g., to describe the atomic interaction with an external magnetic field, to describe the interaction between
different atoms via their magnetic dipoles, and to benchmark

033256-3

A. PATSCHEIDER et al.

norm. population in g(a)

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)
TABLE I. Isotope shifts for three bosonic isotopes in dependenceof the 168 Er isotope. The error bars denote the statistical error, mainly
given by uncertainties of B. Systematic errors are not taken into
account.

1

0.5

isotope pairν0 − ν0168 (MHz)0164−168
166−168
170−168–2732.290(3)–1371.710(3)1414.920(5)-2.45 -2.4 -2.35-0.01 0 0.01

2.39 2.4 2.41(b)represents the dominant limitation on the precision of ourmeasurements.
Figure 3(b) shows the experimentally extracted values of
the gJ  factor as a function of the isotope mass number for thefour bosonic isotopes. We find that, as expected, the values forthe gJ  factor are the same within one standard deviation for
all four isotopes. We combine the results by calculating theweighted mean and determine the gJ  factor of |e tomass number

170
168
166
164
1.258

1.26

1.262

1.264

1.266

gJ  = 1.2599(5).

FIG. 3. (a) High-resolution spectroscopy of 168 Er at B =1.358(2) G, unveiling the σ − , π , and σ + transition (|mJ = −6 →|mJ  = −7, |mJ = −6 → |mJ  = −6, and |mJ = −6 →|mJ  = −5). The normalized population in |g is measured against
the laser frequency relative to the frequency position of the πtransition. (b) Measured gJ  factor for the four bosonic isotopes
for the different atomic mass numbers. The error bars denote the1σ -standard deviation. The red solid line represents the weighted
mean of the experimental data and the grey shaded area corresponds
to the combined standard deviation. The black solid lines represent
the gJ  factor calculated using our semi-empirical method and thevalue given in the NIST database [40].

atomic spectrum calculations. Here, we use the relative frequencies of the π and σ + transitions to determine the valueof the gJ  factor for the excited state. In small magnetic fields,
the Zeeman splitting is linear and the transition frequencies
can be written asν π = ν0 − mJ (gJ − gJ  )μB B/h,(3)

for the π transition andν σ + = ν0 − [mJ gJ − (mJ + 1)gJ  ]μB B/h,(4)+

for the σ transition, where ν0 is the absolute transition frequency at B = 0 G, μB is the Bohr magneton and h is the
Planck constant. By taking the difference of Eq. (4) from
Eq. (3) one obtains
gJ  =(ν σ + − ν π )h
,
μB B

(5)

which allows us to extract the value of gJ  , where the uncertainties are arising from the measured frequencies ν π , ν σ + ,and the applied B.
We calibrate B, before and after each spectroscopic measurement, by driving the atomic radio-frequency transition of
the atoms in |g from mJ = −6 → mJ = −5. We evaluate
possible drifts of B from these spectroscopy measurements
and estimate them to be ≈1 mG. This uncertainty on B

(6)

The individual gJ  factors are weighted by their standard deviation and the final error corresponds to the combined 1σ
standard deviation. We compare our experimentally determined gJ  factor to the value specified in the NIST database[40], gNIST= 1.266, and find agreement at the 1% level. AJ
careful study of systematic effects such as calibration errors
on the magnetic field or collisional shifts (not included in thepresented uncertainty) could refine this comparison further,
providing a useful benchmark for atomic structure calculations.
VI. ISOTOPE SHIFTIn addition to the gJ  factor, the high-resolution spectroscopy allows us to extract the isotope shift between thefour bosonic isotopes with high precision. Because the πtransition is less sensitive to magnetic field fluctuations, we fix
B and, with the knowledge of the FSR, determine the relative
frequency difference directly from the individual transitions.
Table I gives the isotope shifts relative to the transition frequency of the 168 Er isotope.
At leading order, isotope shifts are caused by two effects,
the field shift and the mass shift, which arise from the changeof the nuclear size and the mass, respectively. Here, for the
involved 4 f → 5d transition, both of the contributions arecomparably large [42,43]. Isotope shifts of two different transitions, plotted against each other, follow at leading order alinear dependence, which is referred to as King’s linearity[44]. Violations of the linearity can provide an exceptional
insight into intranuclear interactions and can help to shed lighton processes that are not described by the current theory of
the standard model [45–50]. The availability of four isotopes
that have zero nuclear spin and a large number of different
narrow transitions make erbium a potential candidate for investigations along this route. In particular, erbium features
further narrow transitions that involve the excitation of anelectron from the 4 f orbital to the 5d orbital. Consequently, in
combination with our transition at 1299 nm, the nonlinearity
might be less sensitive to field shifts induced by different

033256-4

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

An important opportunity that comes along with narrow
line transitions and plays a fundamental role, e.g., in quantum
information and communication protocols, is the possibility
to coherently control the atomic state. To demonstrate theability to drive coherent excitations, we measure Rabi oscillations on the closed σ − transition for a thermal cloud of the168
Er isotope. We use a thermal cloud to reduce the effect ofinteractions, by allowing for lower densities. From theoretical calculations (see description in Sec. VIII), it is expectedthat the atomic polarizability of atoms in |e is very low,or even negative at 1064 nm, depending on the polarizationof the trapping light. Therefore, we transfer the atoms after
evaporation into a crossed ODT that is created by two intersecting laser beams at 532.2 nm and 1570 nm, resulting in trap
frequencies of (ωx , ωy , ωz ) = 2π [232(6), 117(7), 209(3)] Hz
for the ground-state atoms. At this stage, we measure 2×104
atoms at a temperature of T ≈ 700 nK, corresponding to apeak density of about 9×12 cm−3 . After the preparation of
the atomic cloud, we shine a resonant narrow-line laser witha peak intensity of about 4.9 W/cm2 for a pulse duration of
tpulse onto the atomic sample and measure the atom number in|g after a time of flight of 10 ms.
Figure 4(a) shows the population in |g, normalized to the
maximum atom number as a function of tpulse . We observe a
damped oscillation of the population in |g, which is well described via pg (t ) = 0.5e−t/τosc cos(Rt ) + 0.5, where R is the
Rabi frequency and τosc is the 1/e decay time of the contrast of
the oscillation. We find that R = 2π ×50.02(8) kHz, which=corresponds to a normalized Rabi frequency of norm.
R2
2π ×0.76(6) kHz/ mW/cm . For the decay time of thecontrast we find τosc = 192(20) μs, indicating a strong decoherence mechanism. Possible mechanisms that might lead tothe decoherence are, e.g. atomic interactions, intensity noiseon the trapping light, intensity inhomogeneities of the probe
light over the atomic cloud, or fluctuations of the magnetic
field. Nonetheless, the coherent control allows us transfer
atoms from |g to |e with an efficiency >97%.
The ability to transfer atoms from |g to |e with high efficiency enables us to measure the lifetime of atoms in |e. At
high densities we observe a short lifetime of the sample in |e,suggesting a density-dependent loss mechanism, similarly toRef. [51]. Therefore, we reduce the atom number to N ≈ 300
atoms by using a shorter loading time of the magneto-opticaltrap and we stop the evaporative cooling process at an earlier stage, leading to a temperature of T ≈ 1 μK. Here, we
obtain a peak number density of about 7.5×1010 cm−3 . In
this regime of low density, radiative losses dominate overfew-body collisional losses.
To measure the lifetime of |e, we carry out two complementary measurements. First, we perform a π pulse to transfer|g atoms to |e with high efficiency. To obtain a pure sample
of |e atoms, we remove the small remaining fraction of |g
atoms with a resonant pulse at 401 nm. Note that, the light at
401 nm is not resonant for atoms in |e. We then measure them J’ = -7

1
norm. population in gVII. COHERENT CONTROL AND LIFETIMEMEASUREMENTS(a)-m J = -6

0.8
0.6
0.4
0.2
0
0

(b)

20

40

tpulse (µs)

60

80

(i)

blow

hold time

imaging

(ii)

blow

hold time

imaging

100

time

(c)

time
400

atom number

electron configurations and therefore interesting for future
investigations [45,46,48].

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

300
200
100
0
0

200

400

600

800

1000

thold (ms)FIG. 4. (a) Coherent Rabi oscillations for 168 Er on the|mJ = −6 → |mJ  = −7 transition. Shown is the normalized atom
number in |g against the duration of the laser pulse. The dashedline represents a fit of a damped sinusoidal oscillation to the experimental data points. (b) Schematic illustration of the two applied
measurement sequences to extract state populations (see main text).
(c) blue circles [red squares] represent the atom number in |g[|e] independence of the hold time for the measurements sequence (i) [(ii)].
The grey diamonds represent the total atom number. The solid lines
are exponential fits to the experimental data. Error bars denote thestandard error of 4 (a) and 10 (c) repetitions.

lifetime of the excited sample in two independent measurements, (i) the number of atoms in |g, which decayed from |edue to spontaneous emission, and (ii) the atomic population in|e by applying a second π pulse to invert the populations in|e and |g in order to directly measure the excited-state atoms;
see Fig. 4(b).
Figure 4(c) shows the measured atom number for bothmeasurement sequences at different holding times thold . We
observe a decay of the atom number in |e, which is consistent
with the simultaneous growth of the atom number in |g. Note
that, the sum of the atom number in both states remains constant over the observed timescale, indicating that atoms in |e,
indeed, decay dominantly to |g. We extract the lifetime, forboth measurement protocols, by fitting an exponential function N (t ) = ae−t/τe + d to the non-averaged atom numbers.
Here, a denotes the amplitude and d the offset of the growth(decay) of the atom number in |g (|e). The characteristic
time τe represents the lifetime of |e. We extract a lifetime of033256-5

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

162(23) ms [212(33) ms] through the measurement sequence(i) [(ii)]. We combine both results by calculating the weighted
mean and obtain a mean lifetime ofτe = 178(19) ms.

(7)

This lifetime corresponds to a natural linewidth of 0.9(1) Hz,
which is in agreement within error bars with the theoretical predicted value of 2(1) Hz in Ref. [23]. Note that, themeasured lifetime is consistent with the natural linewidth
predicted from R determined above.
VIII. THEORETICAL PREDICTIONS

We compare our experimental findings with the resultsof a semi-empirical model, which has previously been very
successful in predicting the properties of broader optical transitions in erbium and dysprosium [52,53]. Our calculations
are based on the semi-empirical method provided by the
COWAN suite of codes [54,55], and extended by us [52].
In a first step, ab initio radial wave functions Pn for all thesubshells n of the considered configurations, with n andbeing the principal and orbital quantum numbers, are computed with the relativistic Hartree-Fock (HFR) method. Thosewave functions are then used to calculate energy parametersthat are the building blocks of the atomic Hamiltonian. In
a second step, the energy parameters are adjusted so thatthe eigenvalues of the Hamiltonian best fit the experimental
energies of the NIST database [40], using Kramida’s versionof the least-square fitting COWAN code RCE [55]. The Pn
wave functions also serve to calculate
the mono-electronictransition integrals n |r|n   = drPn (r)rPn  (r), that are
the building blocks of Einstein coefficients for spontaneousemission Aik . In a third step, the n |r|n   integrals are adjusted to minimize the difference between experimental andtheoretical Aik coefficients [52].
For the even-parity levels of erbium, we use the sameenergy parameters as Ref. [56]. Briefly, the even electronic
configurations are separated into three groups:
4 f 12 6s2 + 4 f 12 5d6s + 4 f 11 6s2 6p,
4 f 12 6s2 + 4 f 11 5d6s6p,
and
4 f 12 6s2 + 4 f 12 6s7s + 4 f 12 6s6d + 4 f 12 6p2 .
Each group is associated with a different least-square fitting
calculation with experimental levels belonging to the corresponding configurations.
Compared to Refs. [56,57], the odd-parity level calculations have been improved by adding some high-lying
experimental energy levels that were previously excluded
from the fitting procedure, as well as incorporating a larger
number of free configuration-interaction parameters into thefitting procedure. The following configurations are included in
the calculation: 4 f 11 5d6s2 , 4 f 11 5d 2 6s, 4 f 12 6s6p, 4 f 12 5d6p,
and 4 f 13 6s. The latter is included for technical purpose, butdoes not play a physical role. The fitting procedure is performed using a total of 30 free groups of parameters and219 levels. The standard deviation between experimental andcalculated energies is equal to 53 cm−1 , which is satisfactory

for a semi-empirical calculation. Details on the parameters
for the first four odd parity configurations are given in the
Appendix.
The n |r|n   transition integrals were adjusted using
the set of experimental Aik coefficients of Ref. [58], especially the transitions involving levels of the ground-state
configuration [Xe]4 f 12 6s2 . Following Ref. [52], we seek tominimize the standard deviation σA on Einstein coefficientsAik [52]. Because the latter is poorly sensitive to 4 f |r|5d,
we could not find a value of that integral minimizing σA ,we have taken a scaling factor with respect to the HFR integral equal to f4 f ,5d = 0.95, following previous works on
dysprosium [53] and holmium [59]. We applied the fitting
procedure on 6s|r|6p, and found f6s,6p = 0.786. We have
fitted 77 experimental lines and found a standard deviation
σA = 8.085×106 s−1 .
With this optimized set of energies and transition integrals, we have calculated the polarizabilities of the groundand excited states using the sum-over-state formula comingfrom second-order perturbation theory. The polarizability ofthe excited level also depends on 5d|r|6p, for which we tooka scaling factor of 0.8.
From our theory we obtain a wavenumber of ν̄ th =7729.3 cm−1 , a gJ  factor of gthJ  = 1.2622, and an excited-state
lifetime of τ th = 602 ms. For ν̄ th and gthJ  we find satisfactory agreement with the values reported from the current
experimental work. Note that, by included least-square fitted
energy parameters in the theoretical calculations (comparedwith Ref. [60]), we obtain better agreement with the experimental data. The extracted lifetime is about a factor of 3
longer compared to the experimentally measured value. This
discrepancy comes from the fact that the underlying transition
dipole moment involves small components in the eigenvector
associated with level |e. Those small components are moredifficult to optimize, as they are less affected by the leastsquare fitting procedure on energies.
Figure 5(a) shows the calculated atomic polarizability α(ω)
for |e in a broad wavelength range from 350 nm to 1500 nm.
The polarizability spectrum becomes very dense at lower
wavelengths. The background value of α(ω) is dominantly
positive; however, a strong transition at around 1140 nm
causes a negative value of α(ω) around 1064 nm, commonlyused for optical dipole traps. Further, this strong transition
creates two interesting situations appearing at 1010 nm and1070 nm. Here, while α(ω) is finite for |g, α(ω) of |e is either
0 (1010 nm) or has the same absolute value with opposite
sign (1070 nm). These circumstances are beneficial for therealization of spin-dependent lattice configurations [61]. Dueto the weak coupling to |g, the effect of the transition at
1299 nm is not visible in this plotting range.
IX. MAGIC-WAVELENGTH CONDITIONS

A very important ingredient for the coherent control of our
two-level system is the atomic polarizability of each state,α(ω), and their ratio α|e/α|g . Figures 5(b)–5(d) show examples of interesting wavelength regions in which, for a given
light polarization (π ), the |g and |e state polarizabilities
cross, meaning that atoms in both states will experience thesame optical trapping potential. These specific values of the033256-6

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

FIG. 5. (a) Theoretically calculated α(ω) for |e as a function of the wavelength λ for π -polarized light. [(b)–(e)] Zoom-in into specificwavelength regions showing α(ω) for |e (red) and |g (blue). The green stars indicate possible magic wavelengths. The dashed black lines in(e) denote the wavelength of 1010 nm, 1064 nm, and 1070 nm.

wavelength realize the so-called magic condition. Our calculation shows wavelength values for which both red-detuned[Figs. 5(c) and 5(d)] or blue detuned magic [Fig. 5(b)] trapping is possible. Moreover, Fig. 5(e) shows a case in which thepolarizability of |g and |e has opposite sign. This situation
might be interesting to create an effectively subwavelength
lattice in which |g and |e atoms are spaced by λ/4.
The existence of such type of crossings is a rather generalfeature also in more simpler atomic species. Additionally,
α|e/α|g , and thus the exact crossing position, can be tuned
by changing the light polarization or the magnetic-fieldorientation [56,62–64]. These features, named in analogy“magic”-polarization condition [64,65], originate from thevectorial and tensorial part of the atomic polarizability, following the equation:[u∗ × u] · Jαv (ω)2J
J (J + 1) − 3m2j 1 − 3 cos2 θ pα(ω) = αs (ω) + i+

J (2J − 1)2

αt (ω).

(8)

Here, αs (ω), αv (ω), and αt (ω) are the scalar, the vectorial, andthe tensorial polarizabilities and u is the polarization vector of
the laser field. The angle θ p defines the orientation of u withrespect to B.
We aim at exploring the impact of the vectorial polarizability on α|e/α|g for a commonly used trapping wavelengthof 532.2 nm. As for Sec. VII, we prepare our |g atoms in acrossed optical dipole trap, with a horizontal (vertical) beam
operating at 532.2 nm (1570 nm). For reference, we first excite
the center-of-mass (COM) motion and extract the vertical trap
frequency (mainly determined by the light at 532.2 nm) of
the atoms in |g; see Fig. 6(a). We then repeat the same trapfrequency measurements for atoms in |e and use the relationof α|e/α|g = (ω|e/ω|g )2 [63].

To study the impact of the vectorial polarizability, we measure α|e/α|g for different polarization of the light at 532.2 nm.
In each measurement, θ p = 90◦ , while the angle φ defining u
is varied [66]; see inset in Fig. 6(b). Figure 6(b) summarizes
our results. We observe a periodic behavior of α|e/α|g as afunction of φ, which reaches its maximum for σ − -polarized
light (φ = 45◦ ). At this angle, we find α|e/α|g = 0.98(3), whichis consistent within the experimental uncertainty to the magictrapping condition. The figure also shows our theoretical
calculation, which qualitatively reproduces the experimental values, although with a smaller amplitude, which might
indicate a shift of the close by transition as well as an underestimation of the transition strength. At φ = 45◦ , the theory
gives (α|e/α|g )theo = 0.97, and predicts crossings at 528.9 nm
and 532.5 nm, which are interesting options to explore withtunable laser systems. The polarizabilities for the special casesof σ − - and σ + -polarized light are shown in Fig. 6 for an
extended wavelength region.
We estimate from our theoretical calculations, that the relative change of α|e − α|g for φ = 45◦ , in linear approximation,
follows a slope of about 4 a. u./nm at 528.9 nm and a slope ofabout 80 a. u./nm at 532.5 nm. This implies that a sufficientlaser stability to resolve our observed linewidth can be reachedwith standard laser locking techniques. Further, the measured
BEC lifetime at 532.2 nm is about 2 s, which suggests a low
photon scattering rate. Furthermore, as the detuning to the
nearby ground-state transition is increased, for the crossing
at 528.9 nm a potentially lower scattering rate can be found[see Fig. 6(c)].
X. CONCLUSION

In conclusion, we have observed the narrow inner-shellorbital transition at 1299 nm for the four bosonic, as well
as for the fermionic isotope. We characterized the transition

033256-7

A. PATSCHEIDER et al.

x (µm)(a)

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

20

x (µm)

-2
4
0
-4
0

5

10

15

20

tCOM (ms)

(b)

0.8

uy

0.6

p

u

e

g

1

B

ux

0.4
-40

0

40

528

529

530

80

120

160

531

532

533

tot

(a.u.)

(c)
350
300
250

(nm)
FIG. 6. (a) Center position of the thermal cloud in dependenceof the hold time for |g (blue) and |e (red) for the case of linearlypolarized light with u pointing along y and B aligned opposite to
the propagation axis of the 532.2 nm trapping light. Atoms aretransferred to |e following the protocol (ii) described in Sec. VII.
(b) Polarizability ratio α|e/α|g for different polarizations of the trapping light defined by the angle φ between u and the optical axis of
the λ/4-waveplate. The black solid line is a guide to the eye based ona sinusoidal fit to the experimental data and the grey solid represents
theoretical results obtained from the sum-over-states method. The
inset illustrates the experimental configuration for the shown data.
(c) Theoretically calculated α(ω) for |e (red) and |g (blue) in theirlowest magnetic sublevels as a function of the wavelength λ for σ − (solid line) and σ + -polarized (dashed line) light. The green solid line
indicates the wavelength for the measurement in (b) and the green
stars indicate possible magic wavelengths for σ − -polarized light.

by measuring the gJ  factor and the atomic polarizability at
532.2 nm, which we compare to a semi-empirical model.
Further, we demonstrated the ability to coherently control theatomic state. The narrow transition, at a wavelength within
the telecom window, with a linewidth of 0.9(1) Hz and a
related long lifetime of 178(19) ms, represents a very versatile
tool with outstanding potential for a broad range of possible
applications.
For the realization of efficient long-distance quantum
communication, atom-photon interfaces at wavelengths thatare compatible with telecom wavelengths are highly desired[29–31]. To date, most of the existing approaches rely either

on frequency conversion of the photons [67,68], leading toundesired noise and reduced efficiency, or suffer from various other dephasing mechanisms [69,70]. Here, our transitionat 1299 nm, lying within the O-band of the telecom wavelength architecture, is a potential candidate to circumventthese issues. Finally, the photon storage time can be drastically
improved by collective scattering in ordered atomic arrays,
exceeding the natural lifetime [71,72].
The relatively long wavelength of this transition may also
provide a favorable platform for studies of collective scattering from ordered atomic samples. Such effects, which include
geometry-dependent enhancement or suppression of emission[73,74], can be more significant when the spacing between
atoms is small relative to the transition wavelength. For our
266 nm spacing, typical of lattice confinement with 532 nm
light [75], this condition is well met in our system, in contrast to the more common situation present in alkaline atoms
where the wavelength of trapping light typically exceeds the
wavelength of the scattering transition.
Additionally, the advantages of encoding qubits in the
ground and the metastable state and the possibility for independent control of atomic motions by lattice light is promisingfor quantum computational tasks [20–22]. Moreover, thecontact interaction can be tuned using the technique of anarrow-line optical Feshbach resonance, where the systemsuffers only weak atom loss [7–9].
Finally, this transition enables the coherent control of magnetic Zeeman levels for dipolar bosonic atoms, which has been
elusive so far, due to the absent hyperfine interactions [76].
The 13 magnetic Zeeman levels in the ground state of erbium

TABLE II. Parameter names, constraints (see text), fitted values
and scaling factors fX = Xfit /XHFR , for the 4 f 11 5d6s2 and 4 f 11 5d 2 6s
configurations of neutral Er. All parameters are in cm−1 .
Parameter X

Constraint

Xfit
11

fX

4f 5d 6s
Eav
F 2 (4f 4f)
F 4 (4f 4f)
F 6 (4f 4f)αβγ
F 2 (5d 5d)
F 4 (5d 5d)
ζ4 f
ζ5d
F 1 (4f 5d)
F 2 (4f 5d)
F 4 (4f 5d)
G1 (4f 5d)
G2 (4f 5d)
G3 (4f 5d)
G4 (4f 5d)
G5 (4f 5d)
G3 (4f 6s)
G2 (5d 6s)

033256-8

r1
r2
r3
r4
fix
fix

r5
r6
r7
r8
r9
r10
r11
r12
r13
r14
r15
r17

464120981776692640500680
200
–6500
20000

23898
7882
7417
157117105582
50541
17174
64003
16306
38097

2

0.761
0.856
0.861

0.984
0.831
0.775
1.149
0.580
0.928
0.732

Xfit
11

fX
2

4f 5d 6s
65531.6
98004.7
69134.049972.2
20.0–650.02000.021668.3
17208.2
2387.8652.6
741.7
13597.8
8970.94325.81717.45422.61630.63216.71254.311696.8

0.761
0.856
0.861

0.663
0.831
0.984
0.831
0.775
1.149
0.580
0.928
0.732
0.844
0.609

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

can be employed as spin states and allow for the realizationof large bosonic spin systems with dipolar interactions. In
combination with optical lattices, this enables the possibility to study many-body dynamics in extended Bose-Hubbard
models [77,78]. Generally, in such large spin systems, the long
lifetime of the excited state is helpful for the realization ofadvanced, spin-resolved imaging-shelving techniques [79,80].

nancial support of “Région Bourgogne Franche Comté” under
the Project 2018Y.07063 “ThéCUP”. Calculations have been
performed using HPC resources from DNUM CCUB (Centre
de Calcul de l’Université de Bourgogne).

ACKNOWLEDGMENTS

Here we present the parameters of the first four odd-parity
configurations of neutral erbium. The ones of the 4 f 13 6s
configuration are not shown, as the latter does not play any
physical role in the level interpretation.
Tables II and III show the one-configuration parameters,
like the direct F k , exchange Gk , or spin-orbit ζn integrals.
Table IV shows the configuration-interaction ones. We also
give the scaling factor fX = Xfit /XHFR between the fitted andab initio value of the parameter X . During the fitting procedure, some groups of parameters are constrained to vary
within the same scaling factors; such groups are characterized
by the same fX = rn value in the second column of Tables II–
IV. The word “fix” means that the corresponding parameters
are not adjusted. Finally, we use so-called “effective” parameters, like α, β, γ , or F1 (4 f 5d ), which cannot be calculated
ab initio, but which are there to compensate the absence ofelectronic configurations not included in the model. Theirinitial values are known from previous studies.

APPENDIX
Odd-parity energy parameters

We thank H. Ritsch and M. A. Norcia for insightful discussions and M. A. Norcia for careful reading of the manuscript.
This work is financially supported through an ERC Consolidator Grant (RARE, No. 681432), a DFG/FWF (FOR2247/I4317-N36) and a joint-project Grant from the FWF(I4426, RSF/Russland 2019). L.C. acknowledges the supportof the FWF via the Elise Richter Fellowship No. V792.
We also acknowledge the Innsbruck Laser Core Facility, financed by the Austrian Federal Ministry of Science, Researchand Economy. G.H. and M.L. acknowledge support fromthe NeoDip Project (ANR-19-CE30-0018-01 from “Agence
Nationale de la Recherche”). M.L. also acknowledges the fiTABLE III. Parameter names, constraints (see text), fitted values
and scaling factors fX = Xfit /XHFR , for the 4 f 12 6s6p and 4 f 12 5d6p
configurations of neutral Er. All parameters are in cm−1 .
Parameter X

Constraint

Xfit

fX
12

4f 6s 6p
Eav
F 2 (4f 4f)
F 4 (4f 4f)
F 6 (4f 4f)
α
β
γ
ζ4 f
ζ5d
ζ6p
F 1 (4f 5d)
F 2 (4f 5d)
F 4 (4f 5d)
F 1 (4f 6p)
F 2 (4f 6p)
F 2 (6p 5d)
G1 (4f 5d)
G2 (4f 5d)
G3 (4f 5d)
G4 (4f 5d)
G5 (4f 5d)
G3 (4f 6s)
G2 (4f 6p)
G4 (4f 6p)
G1 (6s 6p)
G1 (6p 5d)
G3 (6p 5d)

r1
r2
r3
r4
fix
fix
r5
r6
r18
r7
r8
r9
fix
r19
fix
fix
fix
fix
fix
fix
r15
r19
r19
fix
fix

36278.6
92716.3
65123.3
46996.3
20.0–650.02000.02250.30.984

1513.0

1.461

100.0
3776.91405.1864.4
751.9
12209.7

0.761
0.856
0.861

1.156

0.843
1.155
1.156
0.522

Xfit

fX
12

4f 5d 6p
60020.5
92496.864956.1
46876.2
20.0–650.02000.02250.3454.7
1119.6741.7
11092.6
7222.8150.0
3006.111470.1
3897.81091.64397.41028.32761.2

0.761
0.856
0.861

TABLE IV. Configuration-interaction parameters: parameternames, constraints (see text), fitted values and scaling factors fX =Xfit /XHFR , for odd-parity configuration pairs of neutral Er. All parameters are in cm−1 .
Parameter X

Constraint

Xfit
11

fX
2

4f 5d 6s - 4 f
0.985
0.831
1.461

R2 (4f 6s, 4f 5d)
R3 (4f 6s, 4f 5d)
R2 (5d 6s, 5d 5d)

r21
r21
r17

–799.81128.5–13663.3

11

5d2 6s
0.852
1.465
0.621

4f11 5d 6s2 - 4f12 6s 6p
0.775
1.149
1.151
0.794
0.585

R1 (5d 6s, 4f 6p)
R3 (5d 6s, 6p 4f)

r20
r20

656.3
568.2

1.155
1.157

7880.05052.0

0.600
0.600

0.461
0.461

4f11 5d2 6s - 4f12 6s 6p
R1 (5d 5d, 4f 6p)
R3 (5d 5d, 4f 6p)

r22
r22

2405.6
643.7

0.647
0.647

4f11 5d2 6s - 4f12 5d 6p

0.895
0.761

–3173.6–679.9R1 (5d 6s, 4f 6p)
R3 (5d 6s, 4f 6p)

r22
r22

–2536.8–564.5

0.427
0.427

4f12 6s 6p - 4f12 5d 6p
R2 (4f 6s, 4f 5d)
R3 (4f 6s, 5d 4f)
R2 (6s 6p, 5d 6p)
R1 (6s 6p, 6p 5d)033256-9

r23
r23
r22
r22

–5359.92758.3–6833.7–7463.2

A. PATSCHEIDER et al.

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033256-12

6.4 Publication: Determination of the scattering length of erbium
atoms.
Physical Review A 105, 063307 (2022)a
submitted 22 Dec 2021; published 8 June 2022
DOI: 10.1103/PhysRevA.105.063307

A. Patscheider,1 L. Chomaz,1 G. Natale,1 D. Petter,1 M. J. Mark,1,2 S. Baier,1
B. Yang,1 R. R. W. Wang,3 J. L. Bohn 3 and F. Ferlaino 1,21Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
JILA and Department of Physics, University of Colorado, Boulder Colorado 80309, USA2a

The author of the present thesis performed the measurements together with A. P., L. C., D. P., S. B.,
B. Y. and contributed in writing the manuscript.

183

PHYSICAL REVIEW A 105, 063307 (2022)

Determination of the scattering length of erbium atoms
A. Patscheider ,1 L. Chomaz ,1,* G. Natale ,1 D. Petter ,1,† M. J. Mark ,1,2 S. Baier ,1 B. Yang ,1,‡
R. R. W. Wang ,3 J. L. Bohn ,3 and F. Ferlaino 1,2,§
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria
3
JILA and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

2(Received 22 December 2021; accepted 2 May 2022; published 8 June 2022)
An accurate knowledge of the scattering length is fundamental in ultracold quantum gas experiments andessential for the characterization of the system as well as for a meaningful comparison to theoretical models.
Here, we perform a careful characterization of the s-wave scattering length as for the four highest-abundanceisotopes of erbium, in the magnetic field range from 0 to 5 G. We report on cross-dimensional thermalization
measurements and apply the Enskog equations of change to numerically simulate the thermalization process andto analytically extract an expression for the so-called number of collisions per rethermalization (NCPR) to obtainas from our experimental data. We benchmark the applied cross-dimensional thermalization technique with theexperimentally more demanding lattice modulation spectroscopy and find good agreement for our parameter
regime. Our experiments are compatible with a dependence of the NCPR with as , as theoretically expected in
the case of strongly dipolar gases. Surprisingly, we experimentally observe a dependency of the NCPR on thedensity, which might arise due to deviations from an ideal harmonic trapping configuration. Finally, we apply amodel for the dependency of the background scattering length with the isotope mass, allowing us to estimate thenumber of bound states of erbium.
DOI: 10.1103/PhysRevA.105.063307

I. INTRODUCTION

The high degree of environmental isolation and the highcontrol over the large parameter-space of ultracold quantum
gases are key for their success [1]. One of the most decisive
properties in determining the many-body phases of a quantum
gas is the interaction force between atoms. Among neutral
particles, it can be isotropic and short range, as in alkalimetal atoms, and/or anisotropic and long range. Open-shelllanthanides, such as erbium (Er) and dysprosium (Dy), haveboth interactions in place [2]. Their strong magnetic characteris reflected in a large dipole-dipole interaction (DDI), while
the contact potential is governed by the well-known scattering
length, whose value as , as in alkali-metal atoms, can be largely
controlled by so-called Fano-Feshbach resonances [3–5].
Although the concept of the scattering length itself iswell known by now, theoretical challenges to calculate as
depend on the atomic species of interest. For lanthanides,
predicting as remains a major challenge of quantum chemistry and microscopic scattering theories [6]. The complexity*

Present address: Physikalisches Institut, University of Heidelberg,
69120 Heidelberg, Germany.
†
Present address: Optical Materials Engineering Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092
Zurich, Switzerland.
‡
Present address: Southern University of Science and Technology,
Shenzhen 518055, China.
§
Corresponding author: Francesca.Ferlaino@uibk.ac.at2469-9926/2022/105(6)/063307(12)of describing such atoms has several reasons: the multiple valence electrons, the strongly anisotropic orbital shells,
the strong coupling between core and valence electrons,
and the relativistic contributions, also made important by
the large atomic mass. To date, there are still no ab initio
models with the capacity for quantitative predictions, although many general properties of the interaction potentials(e.g., Born-Oppenheimer potentials) have been studied andunderstood [7].
Yet, knowledge of the scattering length remains of prime
importance since it is an essential regulator of few- andmany-body quantum phenomena. For instance, the fascinatingsupersolid state, recently discovered in Dy [8–10] and Er [9],lives in a narrow range of only a few a0 (a0 is the Bohr radius),
and the functional forms of beyond-mean-field corrections,which are still under discussion [11–14], depend on as ina subtle way. In the absence of complete microscopic andab initio potential models, the study of as in lanthanides,
therefore, relies on experimental investigations and empiricalmodels.
Several different experimental methods have been appliedin previous works to extract as for Er and Dy. These include spectroscopy of the molecular binding energy close
to a broad Fano-Feshbach resonance [15,16], the anisotropic
expansion of a thermal gas [17], and the cross-dimensional
thermalization technique [18–21]. Furthermore, for the 166 Er
isotope, as has been determined with high accuracy based ona measurement of the particle-hole excitation gap in the Mott
insulator regime via lattice modulation spectroscopy [22,23].

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A. PATSCHEIDER et al.

PHYSICAL REVIEW A 105, 063307 (2022)

These techniques did not always provide consistent values,
opening up a number of fundamental questions, e.g., from thevalidity of the additivity of the interaction pseudopotentials[24–27] to the appropriateness of the Lee-Huang-Yang form
for beyond-mean-field effects [12–14,28].
In this work, we extensively study the scattering length of
the four most abundant bosonic isotopes of erbium (164 Er,
166
Er, 168 Er, and 170 Er) and its magnetic-field dependence.
For each isotope, we perform high-resolution Fano-Feshbach
spectroscopy in the low-magnetic-field region (0 to 5 G)
and identify previously unreported scattering resonances. In
this range, we then accurately determine the erbium scattering length as by developing a model based on the Enskog
equations to extract as from cross-dimensional thermalization
experiments. We benchmark our results with the ones obtained from high-precision lattice-modulation spectroscopy,
which has been previously developed for 166 Er [23,29] andhere expanded to 168 Er. Finally, from the magnetic-field mapping of as , we extract for each isotope an effective backgroundscattering length asbg at zero B field and we discuss the resultsin the context of the isotope-mass scaling.
II. CROSS-DIMENSIONAL THERMALIZATIONThe cross-dimensional thermalization technique is a very
powerful method to experimentally determine the scattering length. First successfully applied to alkali-metal atoms[30–33], this technique has proved to be very general and,more recently, has been used for more complex atomic
species, such as chromium [34], specific isotopes of erbium[19] and dysprosium [21], and molecular systems [35,36].
Starting from a cold thermal cloud, the basic idea ofthe cross-dimensional thermalization method is to excite thesystem by increasing the potential energy along one spatial dimension of the atomic cloud and to measure the characteristic
time τ that the system needs to re-thermalize in the orthogonal
directions [30]. In the regime of small excitations, for an
atomic cloud at a temperature T and a total atom number N,
the characteristic time is related to the total scattering crosssection σ̄ by
α
τ=
,
(1)
n̄σ̄ vr
where n̄ is the mean number density

3/2
m
N ω̄3
n̄ = √
8 2π kB T(2)and vr is the mean relative velocity for two colliding atoms

16kB T
.
(3)
vr =
πm
Here, ω̄ is the geometric mean of the harmonic trapping
frequencies, m is the atomic mass, and kB is the Boltzmann
constant. Because multiple collisions, not all contributing
equally to rethermalization, are occurring during the thermalization process, the parameter α can be interpreted as a
rescaling of σ̄ and, therefore, as a number of collisions per
rethermalization (NCPR). Experimentally, the knowledge of
α is fundamental for the extraction of the total scattering crosssection.

Equation (1) has two unknown parameters: as and α.
In contrast to alkali-metal atoms, where the scattering isisotropic, the situation is more complex for dipolar atoms suchas Er and Dy [18,20]. Here, the total cross section for bosons is
not only given by the contact scattering length as , but also by
an additional contribution from the nonisotropic DDI, which
for two atoms at a distance r and polarized by an external
magnetic field B, reads as
Vdd (r, θ ) =μ0 μ2 1 − 3 cos2 θ
.
4π
|r|3

(4)

Here, μ0 is the magnetic permeability, μ is the magnetic
dipole moment, and θ is the angle between B and r. Taking
an angular average of the total cross section leads toσ̄ = 8π as2 +

32π 2
a ,
45 d

(5)

0μ
where ad = mμis the dipolar length (ad = 98.2a0 for 166 Er),8π h̄2
with h̄ being the reduced Planck constant. Finally, we canrewrite Eq. (1) asαα.
= 4Nmω̄  2
(6)
τ=
4 2
n̄σ̄ vras + 45 ad
πkB T02

The interplay between the isotropic scattering length and theanisotropic dipolar cross section leads to a dependence of α onboth the dipole orientation θ and as [37]. In the limit of weakexcitation, an analytic form of α(as , θ ) can be found based on
the Enskog equations; see later discussion.
III. EXPERIMENTAL PROCEDURE

In our experiment, we produce a spin-polarized thermal
cloud of Er atoms in the lowest Zeeman sublevel, similarly
to Ref. [38]. In brief, after cooling and trapping the Er atomic
ensemble in a narrow-line magneto-optical trap [39], we transfer the atoms into a crossed optical dipole trap. Here, wefirst further cool the atoms via standard evaporative cooling,
and then we tighten the trapping confinement to avoid atom
loss due to residual evaporation. Simultaneously, we ramp
B to the desired value. At this stage we typically reach atemperature of T = 250–300 nK with N ≈ 1×105 . The exact
numbers depend on the isotope choice and the individual
set of measurements. The typical final trap frequencies are(ωx , ωy , ωz ) = 2π ×[65(1), 19(1), 300(2)] Hz. For all sets ofmeasurements the critical temperature for the onset of BoseEinstein condensation, Tc , lies between 150 and 200 nK, suchthat T  1.5×Tc . The orientation of the magnetic dipoles iscontrolled by the direction of the polarizing B and is represented by the angle θ between B and the vertical directionz, defined by gravity (see inset Fig. 1). We calibrate thevalue of B with an accuracy of about 1 mG by driving theradio-frequency transition between the two lowest Zeeman
sublevels mJ = −6 to mJ = −5.
After preparing the thermal sample, we perform crossdimensional thermalization experiments [19]. In particular,
we excite the cloud along the y direction and probe the thermalization dynamics in the z direction. Our excitation scheme
relies on a rapid increase in the power of one trapping beam,leading to a 60% increase of the trapping frequency, whileleaving the other two directions mostly unaffected. We extract

063307-2

DETERMINATION OF THE SCATTERING LENGTH OF …

PHYSICAL REVIEW A 105, 063307 (2022)
3.5

6

600

3.5
2.5

3

500

4.5

1.5
0

40

p-wave

400

2.5

300

20

s-wave3s-wave
0

50

100

150

200

250

300

FIG. 1. Effective temperatures Tz (blue circles) and Ty (red diamonds) after the increase of the trapping potential along the weakest
trapping direction y. The measurement was performed at 1 G and◦θ = 0 for the 166 Er isotope. The red dashed line represents a guideto the eye. The black solid line denotes the results of the Enskog simulations for this specific data set. The error bars denote the standard
error for three repetitions. The inset shows a schematic representationof our experimental system.

the effective temperature Tz (Ty ) for a variable in-trap holdtime th from the width of the momentum distribution σz (th )[σy (th )] after a time of flight of tToF = 25 ms (20 ms). This
scheme, illustrated in the inset of Fig. 1, leads to an outof-equilibrium cloud with an effective temperature increase
along y from about 300 to 600 nK.
Figure 1 shows Tz and Ty as a function of th at B = 1 G. As
we excite the system along y, we observe the expected rapid
increase of Ty . After reaching a maximum effective temperature, Ty starts to decay, and simultaneously Tz increases, bothreaching the same equilibrium temperature and thus showing
thermalization dynamics. We observe oscillations in Ty , which
we attribute to a breathing mode that gets induced by the
excitation. For Tz we observe an exponential-type growth of
the formTz (t ) = Tf (1 − Te−t/τ ).

2(7)Here, Tf denotes the final temperature and T denotes thetemperature increase due to the added energy. However, using
this simple fit we cannot directly extract as as additional
knowledge on α(as, θ ) is needed [see Eq. (6)].
IV. THEORETICAL ESTIMATE OF α(as , θ)To compute α(as , θ ), we utilize the Enskog equations ofchange [40]: a coupled set of differential equations derived inclosed-form for dipolar gases, by linearization of the Boltzmann equation, and the assertion of a Gaussian phase-space
distribution [41]. These equations permit an analytic derivation of α(as , θ ) in the limit of short times and small excitations[37]. For the current experiment with excitation along y andthermalization measured along z, the NCPR is described by asimple analytic formula, which reads
14 45as2 + 4ad2.
(8)α(as , θ ) =252as2 + 96as ad + (3 cos(4θ ) + 13)ad2

1.5
0

30

60

90

1.5
0

100

200

300

FIG. 2. (a) Dependency of α on θ and as for as = 0a0 , 5a0 , 10a0 ,36.5a0 , and 68.3a0 . These values are chosen such that the angle
dependence at small as becomes visible. Note that, at 68.3a0 (as at
1 G, see later measurements) the variation of α with θ is strongly◦
suppressed. (b) α vs as for θ = 0 . The inset shows an enlargementof the region for as between 0a0 and 400 . The gray dashed lines show
the values of α for s-wave and p-wave scattering, respectively.

The quantity α(as , θ ) exhibits an anisotropic character via its
angle dependence, as already observed for dipolar fermionic
atoms [19] and molecules [36].
Figure 2 shows α(as , θ ) as a function of θ [panel (a)]and as [panel (b)], for our experimental configuration of apancake-shaped trap. Figure 2(a) shows that the anisotropic
character of α(as , θ ) competes with the contact one. Indeed,
while for small as ( 10a0 ), α(as , θ ) exhibits a pronounced
angle-dependence with a maximum at 45◦ , for increasing as
such behavior progressively washes out. For as ≈ 70a0 , thethermalization behavior becomes basically independent of θ ;however, α(as , θ ) acquires a number below the one expected
for purely contact-interacting s-wave collisions. This suggests
faster thermalization for dipolar particles, arising from a more
efficient diversion of velocities of the scattering constituents.
In the experiment, we only measure rethermalization for relatively large values of as  30a0 , and therefore, we are notsensitive to the angle dependence of α(as , θ ). In the courseof this work, we thus focus on the case θ = 0◦ , simplifying
Eq. (8) to
14 45as2 + 4ad2◦.
(9)α(as , θ = 0 ) =252as2 + 96as ad + 16ad2◦

As shown in Fig. 2(b), after an initial decrease, α(as , 0 )increases for as  36.7a0 —and thus the thermalization loses
efficiency—moving to the regime of contact-dominated in◦teraction, eventually reaching the α(as , 0 ) = 2.5 limit ofnonmagnetic atoms [18,42,43]. We note that, by setting◦
θ = 0 and ad /as ≈ 2.7, the NCPR is minimized with thevalue α ≈ 1.65, indicating highly efficient collisional thermalization. This is directly attributed to the innate anisotropic
differential cross section in dipolar bosons [18].

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0.4

30
0

0.2
0

0.5

1

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2

2.5

3

3.5

4

4.5

5

FIG. 3. Atom-loss spectroscopy (orange circles) as a function of B for a fixed holding time of 250 ms. For each B value, the data pointis an average of three repetitions and it is normalized to the maximum averaged atom number recorded in the explored magnetic-field range.
Further, as extracted from cross-dimensional thermalization measurements using both the Enskog equations (red squares) and the analytic
formula of Eq. (8) (blue diamonds) is shown for 166 Er. Additionally asLMS (black triangles) values obtained from lattice modulation spectroscopy
measurements are given. The solid black lines represent a fit of Eq. (10) to asLMS . Error bars and the shaded area of the fitting results denote thestandard error.
V. MAPPING OF as AS A FUNCTION OF B FOR 166 Er

Before taking cross-dimensional thermalization measurements for 166 Er, we perform a high-resolution scan of the atom
number as a function of the magnetic field in order to record
the spectrum of Fano-Feshbach resonances, which we know
to be exceptionally dense [4,5]. We record the Fano-Feshbach
spectra in a magnetic field region from 0 to 5 G (see Fig. 3
and Appendix D). In all the measurements the magnetic fieldis oriented along z.
We then perform thermalization measurements at valuesof the magnetic field, where the system is not dominated byresonant atom loss. For each thermalization curve, we extract
as using two different approaches, one numerical and one
semianalytical. The first, constitutes a direct fit of the full
Enskog solutions to the experimental data, leaving as as a float
parameter of the theory (see Appendix B for more details).
The second method is based on the exponential growth rateτ , from Eq. (1), using the analytic expression for α(as , 0 ◦ )in Eq. (9). For the latter, since as is unknown a priori, we
use an iterative approach to determine α(as , 0 ◦ ) starting fromα(as , 0 ◦ ) = 1.7. We use the calculated as and the analytic
formula [see Eq. (9)] to obtain a new value for α(as , 0 ◦ ).
We stop the iteration once the relative change of α(as , 0 ◦ ) is
 1×10−7 .
Figure 3 summarizes as for 166 Er in the region from 0 to
5 G. In the studied B-field regime, the scattering behavior
is essentially dominated by a broad resonance at 3 G and asecond one around B = 0 G. The as extracted from the Enskog
model and the semianalytic one are in very good agreementwith each other, reflecting the strength of the analytic formulaof Eq. (9).

VI. BENCHMARKING WITH LATTICE SPECTROSCOPY

To evaluate the robustness of our approach to extract as , webenchmark our cross-dimensional thermalization results withthe one obtained using an alternative technique based on lattice modulation spectroscopy (LMS). Such a technique, which
we have developed in the past for 166 Er [23,29] and 167 Er[44], is based on the measurement of the on-site interaction—
related to as —of a lattice-confined dipolar gas in a Mott

insulator state. The LMS is able to provide accurate values ofasLMS , but at the price of being experimentally more involved
due to its requirements of an optical lattice together with ahighly degenerate sample. Here we compare the values ofas obtained with cross-dimensional thermalization on a lowdensity thermal sample, with asLMS values obtained from thelattice modulation spectroscopy obtained in Ref. [29]. In brief
we extract asLMS as follows. We prepare an ultracold sample
of 166 Er atoms in a three-dimensional optical lattice, createdby two retro reflected laser beams at 532 nm in the horizontal
plane and by one retroreflected laser beam at 1064 nm along
the vertical z direction, defined by gravity. The final lattice
depth along the three directions is (sx , sy , sz ) = (20, 20, 100),in units of Erec = 4.2 kHz (1.05 kHz) for 532 nm (1064 nm).
The uncertainty on sx , sy , and sz is about 5%. In such a deep
lattice, the atoms are in the Mott insulator phase [23].
We then create particle-hole excitations by sinusoidally
modulating the power of the horizontal lattice beams for 90 ms
with a peak-to-peak amplitude of about 30% and measure
the recovered Bose-Einstein condensation (BEC) fraction after melting of the lattice. At the resonance condition, where
the modulation frequency matches the particle-hole excitation
gap, we observe a resonant reduction in the BEC fraction[45]. The particle-hole excitation gap is directly given by
the on-site interaction U = Uc + Udd . Here, Uc is the contact
interaction—and thus depends on the unknown asLMS —while
the on-site dipolar interaction Udd can be accurately calculated. We repeat the measurements at various magnetic-fieldvalues and, for each, we extract asLMS .
In Fig. 3, we compare asLMS with as extracted from thethermalization measurements. We see an overall very good
agreement between the value of as extracted using the twotechniques. This shows that the cross-dimensional thermalization approach combined with the Enskog equations is a very
reliable method to extract as , even in the case of complex
atoms for which the knowledge of α(as , θ ) is not a priori
given.
VII. DENSITY DEPENDENCE

Our measurements for the 166 Er isotope were performed
in a regime of relatively low density (n̄  0.5×1013 cm−3 ).

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4
3
2
1
0.5

1

1.5

2

2.5

3
10 13

FIG. 4. Measurements of α(as , θ ) as a function of n̄. The blue
circles correspond to the data sets at 1 G, shown in Fig. 3. The blacksolid line marks the value given by the analytic formula in Eq. (9).
◦All measurements were performed with θ = 0 . Error bars denotethe standard error.

Interestingly, when applying the same method in a regime ofhigh density, we observe a dependence of the thermalization
rate on the density which goes beyond the Enskog approach.
For instance, we repeat the cross-dimensional thermalization
measurements for 166 Er at B = 1 G and the variable clouddensity n̄. We control the density by either increasing N or applying a tighter trapping configuration of (ωxcyl , ωycyl , ωzcyl ) ≈2π ×(300, 19, 300) Hz before compression, or both. From thelattice modulation spectroscopy, we have extracted the value
as = 68.3(7)a0 at B = 1 G. By fixing this value—meaning toimpose that the scattering length does not depend on density—◦and using Eq. (1), we can determine α(as , 0 ) as a function ofn̄. Note that for all measurements we find a Knudsen number,given by the ratio of the mean free path and the size ofthe atomic cloud, >10. This implies that we are far awayfrom the hydrodynamic regime, which could otherwise leadto modifications in the thermalization behavior [46,47].
◦
Figure 4 shows α(as , 0 ) for different values of n̄. We finda pronounced dependency on n̄, with a rapid increase andan eventual saturation at high densities. Such a behavior is
not captured by our theoretical model, which, as reflected◦in the definition of α(as , 0 ) in Eq. (1), predicts no density
dependence. To the best of our knowledge, such a dependence
has not been reported in previous works on cross-dimensional
thermalization. Possible explanations are rooted in various
causes, either physical or technical nature. Although being above Tc , precursors of quantum many-body phenomena
might influence the scattering behavior. Exemplary, we tried
to explicitly include effects coming from Bose enhancement
into our theoretical framework. This did not have significant
influence on the thermalization behavior. Note that, in theexperiment, we varied the initial temperature of the atomic
cloud and the excitation strength, which did not show any
influence on the observations.
Another possible explanation, based on unavoidable experimental imperfections, is rooted in deviations from an ideal
harmonic trapping condition, leading to a modification of the
kinetic energy and the mean density. Such a variation would◦
manifest in an apparent change of α(as , 0 ) [see Eq. (1)].

Indeed, Eqs. (2) and (3) are only valid for an ideal harmonic
trapping confinement. Furthermore, trap anharmonicities leadto a larger kinetic temperature after thermalization comparedto the case of purely harmonic traps as seen when formulating
the dynamics using the scaling ansatz method [48]. As thiseffect is stronger for larger densities, this further suggests thatthe range of low densities is the appropriate one to consider.
First Monte Carlo simulations performed by using a realistic
Gaussian trapping potential seem to support this assumption(see Appendix C).
We emphasize that, due to the agreement with the lattice
modulation spectroscopy results and the above discussion onanharmonicities, and since our measurements to extract ashave been performed at low densities, we are confident that
our method remains valid.
VIII. SCATTERING LENGTH FOR 164 Er and 170 Er

After the detailed study on 166 Er and the benchmarking of the results with high-precision lattice modulation
spectroscopy, we confidently apply our cross-dimensional
thermalization approach to two other isotopes, 164 Er and
170
Er. Again we start with a Fano-Feshbach spectroscopy
between 0 and 5 G to identify the position of the scatteringresonances as shown in Fig. 5. We note that these FanoFeshbach spectra have not been reported previously. For the
cross-dimensional thermalization measurements we follow asimilar experimental procedure as described above. From thethermalization curve, we again use both the full fit of the◦
Enskog equations and the iterative approach on α(as , 0 ) todetermine as from the exponential growth rate τ .
Figure 5 shows as for the isotopes 164 Er [panel (a)] and170
Er [panel (b)]. While the scattering behavior for 164 Er
is, similarly to 166 Er, dominated by two broad resonances
at 1.5 and 3.3 G, 170 Er features several narrow overlapping
resonances, providing different test scenarios for our crossdimensional thermalization. Although minor deviations canbe observed in the vicinity of Fano-Feshbach resonances, forboth isotopes the extracted as values using the two approaches
are once more in good agreement.
IX. SCALING OF BACKGROUND SCATTERINGLENGTH WITH MASS

The knowledge on as as a function of the magnetic field
allows us to extract an effective background scattering length
asbg for each isotope. The general behavior of as with B can be
described by generalizing the well-known formula [49]Nres 
Bi1−
as (B) = asbg + sB
B − Bii=1(10)

to the case of Nres overlapping resonances of position Bi andwidth Bi and allowing for a smooth off-resonant variationof as with B. We observe that a linear variation of slope s
already well reproduces the data with as (0) defined as theeffective asbg . We note that different mechanisms could leadto an off-resonant variation of as . For instance, the influenceof broad Fano-Feshbach resonances, which are not within our
measurement range, could lead to a smooth variation of the063307-5

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0.4

100

0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
5

FIG. 5. Atom-loss spectroscopy (orange circles) as a function of B for a fixed holding time of (a) 250 ms for 164 Er and (b) 500 ms for 170 Er.
For each B value, the data point is an average of four to five repetitions and it is normalized to the maximum averaged atom number recordedin the explored magnetic-field range. Further, as values extracted from cross-dimensional thermalization measurements using both the Enskog
equations (red squares) and the analytic formula of Eq. (8) (blue diamonds) are shown. The solid black lines represent a fit of Eq. (10) to as
obtained using the Enskog equations. Error bars and the shaded area of the fitting results denote the standard error.

background behavior, similar to that observed for cesium [50].
Alternatively, the effect could be due to the coupling inducedby DDI between the incident scattering channel and Zeeman
states that lie higher in energy. As a consequence this results
in a perturbation of the molecular potential, whose strength
depends on the magnetic field, leading to an increasing valueof the van der Waals C6 coefficient [51].
To parametrize as as a function of B, we fit Eq. (10) to themeasured as for 164 Er, 166 Er, 168 Er, and 170 Er. For 166 Er and
168
Er, we use the scattering lengths obtained from the lattice
modulation spectroscopy, corresponding to our most accurate
determination (see solid lines in Figs. 3 and 8. For 164 Er and
170
Er, we fit Eq. (10) to the as data obtained by applying
the Enskog equations to the cross-dimensional thermalization
measurements (see solid lines in Fig. 5). More details on
the fitting procedure as well as the complete list of the fit
parameters are given in Appendix F. In general, we observethat the fitting function reproduces very well the behavior ofas for every isotope.
Figure 6 shows the value of asbg from the fit as a functionof the isotope mass. We observe a monotonic rising of asbg
with increasing m, which might be compatible with different
functional forms, including a simple linear increase. Under theassumption that erbium has a similar behavior to ytterbium
and cesium, we can use the model for the mass scaling as
developed in Refs. [52–54]. Such a model assumes that as isonly given by the Van der Waals potential U (r) = −C6 /r 6 ,with C6 being the Van der Waals coefficient. This might bea rather severe approximation for magnetic atoms, but, in theabsence of alternative models, it is interesting to compare thesimple mass-scaling approach to erbium.
As introduced in Ref. [52], as can be written as



π
,
(11)
as = ā 1 − tan φ −8(3/4) mC6 1/4( h̄2 ) being the characteristic lengthwith ā = 2−3/2 (5/4)
and
√
m ∞
−U (r)dr.
(12)
φ=
h̄ R0

Here, (x) is the Gamma function and R0 is the classicalturning point of U (r). Although the exact shape of U (r) is unknown, Eq. (11) can be employed
to extract a mass scaling due√
to the dependence of φ ∝ m [53]. Such a scaling is valid, aslong as the mass-dependent modification of U (r) is negligible.
Furthermore, φ allows for the calculation of the number ofbound states NB via the relation NB = φ/π − 5/8, where  denotes the floor integer function.

140
120
100
80
60
40
163

164

165

166

167

168

169

170

171

FIG. 6. Background scattering length asbg for four bosonic isotopes (red circles). The solid line represents the best fit with φ/π =144(1) (see text). The shaded area, enclosed by the dotted lines,
represents the fitting function for φ = 143 and φ = 145. The error
bars denote the standard error of the fit of Eq. (10) to the experimental
data.

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We now apply this model to our Er case. Figure 6 shows
the fit of Eq. (11) to the experimental data (see Appendix G
for details). We obtain the best agreement for φ/π = 144(1),leading to NB = 143(1) for 168 Er. Despite the similar C6 coefficient, NB is approximately a factor of 2 larger than for
ytterbium [53]. Note that NB is in agreement with the result
obtained when using the same approach but assuming a hardcore potential (see Appendix H). We would like to emphasize
once more that this model does not consider any contribution
arising from the DDI. An improved description calls for thedevelopment of advanced theoretical models.
X. CONCLUSION

In conclusion, we report on an accurate study of the scattering length of four different isotopes of erbium. Our work
focuses on the low-magnetic-field region, which is the rangeof most interest in current experiments. Our experimental survey combines two different techniques: a high-precision, yetdemanding, approach based on the measurement of the on-site
interaction in a Mott insulator phase, and another one basedon measuring the re-equilibration time in cross-dimensional
thermalization experiments. From the latter, we extract thethe value of as by both numerically applying the full Enskog
equations and using the analytic formulation for α(as , θ ). All
these different approaches, benchmarked one with respect tothe others, provide a very consistent measure of the scattering
length in the region of interest. These results will be relevant
for current experiments and moreover point to a practicalmanner to extract as with reduced experimental effort, which
can be readily generalized to other magnetic lanthanides.

follows from Ref. [37], but we present a brief outline here forcompleteness. The gas is assumed to be close to equilibrium,
allowing us to treat r j and p j as Gaussian distributed. Thermalization trajectories are then tracked using the Gaussian
widths along each axis to compute the energy differentialχ j ≡ E j − kB T f ,(A2)

where T f = (Tx + Ty + Tz )/3 is the final equilibration temperature (obtained from the equipartition theorem), . . .
denotes an ensemble average assuming a Gaussian phasespace distribution whose widths are allowed to vary, andE j = p2j /(2m) + mω2j r 2j /2 is the sum of kinetic andpotential energies in the jth direction. The Enskog equations dictate that the relaxation of χ j follows the differential
equationd χj= C[χ j ].
dt

(A3)

For small deviations from equilibrium and at short times,
rethermalization can be approximated with a single decay rate
γ , such that C[χ ] ≈ −γ χ . This results in the relationdE j= −γy j (E j − kB T f ) = C[E j ],
dt

(A4)

where the subscript on γy j indicates that the gas was excitedalong y, and rethermalization was measured along j. This thenpermits us to computeE j − kB T fn̄σ̄ vr
αy j ==(A5)n̄σ̄ vr ,γy j
C[E j ]

which for j = z has the form in Eq. (8).
ACKNOWLEDGMENTS

This work is financially supported through an ERC Consolidator Grant (RARE, Grant No. 681432), a DFG/FWF grant(Grant No. FOR 2247/I4317-N36), a joint-project grant fromthe FWF (Grant No. I4426-N), and a FWF grant (Grant No.
I4391-N). We also acknowledge the Innsbruck Laser Core
Facility, financed by the Austrian Federal Ministry of Science,
Research and Economy. R.R.W.W. and J.L.B. acknowledge
that this material is based on work supported by the National
Science Foundation under Grants No. PHY-1734006 and No.
PHY-2110327.
APPENDIX A: ANALYTIC NUMBER OF COLLISIONSPER RETHERMALIZATION

Analytic expressions for α(as , θ ) can be derived under ashort-time approximation, with the Enskog equations
d q2j2
q j p j = 0,
m

(A1a)

 
+ 2mω2j q j p j = C p2j ,

(A1b)

dt
d p2j
dt

−

APPENDIX B: FITTING ENSKOG EQUATIONSTO EXPERIMENTAL DATA

The extraction of the scattering lengths as from crossdimensional thermalization data was done here by means offull numerical solutions to the Enskog equations. To do so,
as was left as a float parameter in the theory and then varied
until a best fit between the theoretical and experimental data
was obtained. A feature we noticed during fitting was the highsensitivity of thermalization rates to variations in the trapping
frequencies ω, over the finite-time quench. Measurement uncertainties, therefore, motivate us to also leave ω as a float
parameter, with allowed values within its 1-σ error bars. Thisis applied to the trapping frequencies both before and after thequench.
We performed fits using a χ 2 optimization criterion,
tend T (t ) − TE [T (0); ω, as ](t ) 2
min
,
(B1)
ω,as
δT (t )
t=t
0

d qj pj
1 2
p + mω2j q2j = 0,
(A1c)
−
dt
m j
where r j and p j are positions and momenta, respectively( j = x, y, z), and C is the collision integral. The derivation

where the sum runs over measurement time instances t, T (t )is the temperature data from the experiment, δT (t ) is thetemperature measurement uncertainty, and TE [T (0); ω, as ] isthe solution to the Enskog equations with initial condition
T (0) and fit parameters as and ω.
To reduce biasing of the fits, we run an iterative algorithm
that recursively fits ω and as in succession until they converge

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340

potential


300

2



2

2





−2 2x + 2z
2Ũ2 P2−
e w2,x (y) w2,z (y) + mgz,
π w2,x (y)w2,y (y)
(C1)

280
260

where P is the laser power, Ũ is an atomic polarizability
parameter, andz2w(z) = w0 1 + 2 ,
(C2)
zR

240
220

2

−2 2x + 2y2Ũ1 P1
e w1,x (z) w1,y (z)VODT (r) = −π w1,x (z)w1,y (z)

320

0

50

100

150

200

250

300

FIG. 7. Benchmarking of the Enskog simulation results for Tz(red solid line) with Monte Carlo simulations (black dashed line).
The data set is the same as that in Fig. 1.

with zR and w0 denoting Rayleigh lengths and beam widths,
respectively.
Such a potential limits the applicability of the aforementioned Enskog equations as formulated in Ref. [41]. Instead,more robust molecular dynamics (MD) methods are required
to accurately predict thermalization trajectories. We implement an MD simulation similar to that in Ref. [20], which
evolves simulation particles under the action of VODT via theVerlet symplectic integrator:

to stable values. Such a procedure would take exceedingly
long times (approximately weeks) with full Monte Carlo(MC) simulations, but can be done in minutes with the Enskog
equations on a current-day computing device.
Solutions to the Enskog equations have shown themselves
accurate when compared to MC simulations [37,41]. Weshow their accuracy here yet again, using the parametersfrom the current experimental setup. An illustrative example
is provided in the plot of Fig. 7, comparing an instance of
the Enskog solutions (red solid line), MC simulations (black
dashed line), and the experimental data (blue circles).

t
(C3a)
p (t ),2m k
pk (t + t ) = pk (t ) + F k t,(C3b)t
rk (t + t ) = qk +(C3c)
p (t + t ),2m k
where subscripts k denote the kth simulation particle, t is
the simulation time-step, t is the time, and
qk = rk (t ) +F k = −∇VODT (rk ).

APPENDIX C: MONTE CARLO SIMULATIONSINCLUDING TRAP ANHARMONICITIES

Dipolar collisions are then computed with the directsimulation Monte Carlo method [56], which determines
postcollision momenta via stochastic sampling of the differential cross section.
In a preliminary study of the density dependence, ideal
Gaussian beam profiles are assumed, along with perfectly
accurate beam widths and Rayleigh lengths. Following a
trap quench, thermalization of the out-of-equilibrium gas in

Optical dipole traps are, in many studies, assumed to
be well modeled by purely harmonic potentials. This may,
however, be inadequate in regimes with significant trap anharmonicity effects, which we currently attribute the density
dependence of α to. In such cases, the potential is better modeled as two cross-propagating Gaussian-profile beams along
the y and z axes (with gravity). This produces the confinement
1

400

0.8

300

0.6

200

0.4

100

0.2
0
0

(C4)

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
5

FIG. 8. Atom-loss spectroscopy (orange circles) as a function of B for a fixed holding time of 500 ms. For each B value, the data pointis an average of six to seven repetitions and it is normalized to the maximum averaged atom number recorded in the explored magnetic-field
range. Further, the measured scattering lengths as obtained for 168 Er from lattice modulation spectroscopy measurements are shown. The solid
black line represents a fit to asLMS . The shaded area and the error bars denote the standard error.
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TABLE I. Values for asbg and s obtained from the fit of Eq. (10) to
as for the four bosonic isotopes. The error denotes the fit error of one
standard deviation.
Isotopea0 bg (a0 )s (a0 /G)

164
166
168
170

52(6)
61(3)
110(2)
129(9)9(3)5.4(9)
11(2)20(10)VODT indeed shows an apparent increase of α with density, qualitatively similar to that observed in the experiment.
This effect is absent in simulations with an ideal harmonic
trap. Furthermore, in higher density regimes, the simulations
with VODT predict the experimentally observed equilibration
temperatures more accurately compared to the harmonic trapcase. These early findings on density dependence from trap
anharmonicities are intriguing, and a cautionary tale for future
experiments. However, we do not develop this idea further
here and leave such analysis for future works.
APPENDIX D: FANO-FESHBACH SPECTROSCOPY

To identify the positions of the Fano-Feshbach resonances
we perform high-resolution loss spectroscopy in a cylindrically symmetric trap. We evaporatively cool the atoms
until they reach a temperature between T = 300 and 400
nK. At this stage, the atom number is between 6×104
and 1.2×104 with typical trap frequencies of (ωx , ωy , ωz ) =2π ×(300, 30, 300) Hz. The exact values depend on the isotope choice. After reaching thermal equilibrium, we change B,
oriented along the z axis, in 1 ms to the desired value and wait
for a holding time between 250 and 500 ms. We use different
holding times for different data sets to avoid saturation effectsof the resonances for higher densities. After the holding time,
we measure the atom number using absorption imaging after
a time-of-flight expansion of 25 ms. The results of the lossspectroscopy measurements are shown in Figs. 3, 5, 8.
APPENDIX E: SCATTERING LENGTH FOR 168 Er

To obtain as for the 168 Er isotope, we follow a similar
approach as for 166 Er. First, we perform loss spectroscopy
to identify the position of Fano-Feshbach resonances. We
then transfer the atoms into an optical lattice with a depth of(sx , sy , sz ) = (20, 20, 40)Erec and apply the lattice modulation
TABLE II. Parameters for the Fano-Feshbach resonances included in the fit of Eq. (10) to as for 164 Er. The error denotes thefit error of one standard deviation. Values without error are fixed in
the fitting procedure.
Position Bi (G)
1.52
2.67
2.83
3.26

Width Bi (G)0.22(3)0.005
0.005
0.10(3)

TABLE III. Parameters for the Fano-Feshbach resonances included in the fit of Eq. (10) to as for 166 Er. The error denotes thefit error of one standard deviation. Values without error are fixed in
the fitting procedure.
Position Bi (G)
0.02(5)3.04(5)
4.208
4.96

Width Bi (G)0.05(2)0.15(2)0.01
0.005

spectroscopy technique to extract as . The lattice modulation
spectroscopy follows the same lines as for the 166 Er isotope(see main text). Figure 8 summarizes the results for 168 Er andshows the Fano-Feshbach spectroscopy result as well as as asa function of B in the magnetic field range from 0 to 5 G.
APPENDIX F: EXTRACTING BACKGROUNDSCATTERING LENGTH

To obtain a value for asbg , we fit Eq. (10) either to as
obtained from the full Enksog equations (164 Er and 170 Er) or
to asLMS (166 Er and 168 Er). Due to the different numbers ofFano-Feshbach resonances compared to the number of available data points for as , we slightly vary the fitting approachfor the individual isotopes. Depending on the position andthe width of the resonance, for some resonances, we fix theposition Bi to the minimum of the loss feature and keep only
the width Bi as a floating parameter. For the very narrow
resonances, which have a negligible influence on the overall
scattering behavior, we fix both Bi and Bi .
Table I gives the results for the background scattering
lengths asbg and the slopes s for all four isotopes. Moreover,Tables II–V contain a detailed listing of all Fano-Feshbachresonances and how they are included in the fitting procedure. Note that for 170 Er we are aware of the existence ofa particularly broad resonance at 6.91 G [57], which we include with variable width. When looking closely, the onsetof this resonance can actually be seen as a reduction of N
towards higher magnetic-field values in the loss spectroscopy[see Fig. 5(b)].
TABLE IV. Parameters for the Fano-Feshbach resonances included in the fit of Eq. (10) to as for 168 Er. The error denotes thefit error of one standard deviation. Values without error are fixed in
the fitting procedure.
Position Bi (G)0.49
0.911(6)1.51
2.174(4)2.471(9)
2.86
3.79
4.23
4.5

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0.032(2)0.01
0.038(2)0.19(1)0.005
0.006(5)
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PHYSICAL REVIEW A 105, 063307 (2022)

TABLE V. Parameters for the Fano-Feshbach resonances included in the fit of Eq. (10) to as for 170 Er. The error denotes thefit error of one standard deviation. Values without error are fixed in
the fitting procedure.

φ we fit a quadratic function to the local minima. We extract
the limits of the confidence interval by considering the region
where χ 2  χ 2 + 1.
APPENDIX H: HARD-CORE POTENTIAL
FOR MASS SCALING

Width Bi (G)Position Bi (G)
0.35
0.86
1.12
1.62
2.17
2.74
3.3
3.57
4.38
4.49
6.91

0.005
0.028(12)
0.005
0.01
0.067(7)0.134(9)0.01(1)
0.01
0.005
0.01
0.8(7)

APPENDIX G: χ2 ANALYSIS FOR MASS SCALING

In this section, we describe our analysis of the backgroundas of the four Er isotopes (Fig. 6) with Eq. (10). To findthe best-fitting parameter φ, we analyze the agreement of thetheoretical model in Eq. (11) with our experimental data. For
each value of φ, we calculate the χ 2 via
χ =
2

2
4  mod

a
− ai
s

i=1

s

σsi

.

(G1)

Here, asmod is the scattering length given by the model for thecorresponding φ, and asi and σsi are the measured as with thecorresponding standard error.
The behavior of χ 2 is nonmonotonic with the appearanceof several minima. We identify the absolute minimum of χ 2
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The model contains the assumption that the s-wave scattering length is given at large distances by the van-der-Waals
potential scaling with UvdW (r) ∝ −C6 /r 6 , with C6 being the
Van der Waals coefficient, and at short distances r < rc by ahard-core potential [52]. In this specific case, the scaling ofasbg can be described byasbg = ā tan(),(H1)a , with ac = ( 2mh̄r2C6 )1/4 being the charwhere ā = 2√(2)(3/4)(5/4) cacteristic scattering length scale of the potential, and  =ac2− 3π
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2rc2
8
From theoretical calculations in Ref. [6] we use C6 =1723 a.u. and we estimate from the theoretical interaction
potential given in Ref. [6] that rc ≈ 4a0 –8a0 . We fit Eq. (H2)
to asbg of the four bosonic isotopes. Due to a large number ofpossible local minima, we combine the fitting with a minimization of the χ 2 value while varying the start parameter forrc . We obtain the best agreement for rc = 5.05(5)a0 .
In addition, the Levinson theorem [58] allows us to estimate the number of bound states NB , which can be calculated
from the semiclassical phase  using

 3
−
+ 1,
(H2)
NB =
π
8
where the square brackets mean the integer part. For thecurrent fitting we obtain NB ranging from 141 to 144, inagreement with the approach in the main text. We want toemphasize, that this modeling of asbg is a simple approach anda more thorough analysis could add deeper valuable insights.

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Acknowledgments
I would like to start by thanking Francesca for giving me the possibility to do my PhD in hergroup and for her ability to drive the research in the right direction thanks to her knowledge,
experience, and intuition. I am also grateful to her for organizing events and excursions to
discuss physics, brainstorm on future directions or simply build up team spirit.
Another big thank goes to Manfred and his incredibly deep scientific and technical knowledge.
Every time something was not clear, I knocked on his office door, and he was always available(apart from when he is smoking outside...) to answer my questions. I remember one pieceof advice he gave me when I arrived in Innsbruck: “when you don’t know something, ask”.
I took advantage of his patience, and I hope he does not regret this now.
I would also like to thanks Lauriane Chomaz and Bing Yang, the postdocs I was lucky towork with. In particular I thank Lauriane Chomaz, who was a senior postdoc in our teamfor many years. She was one of the driving forces of our team. Her knowledge helped
us to understand many of the dipolar surprises. I learned from her the importance to be
meticulous in doing scientific research, but I will also always remember the lessons she gave
me, when we were in a group retreat in Grenoble, on how to cut and serve cheese.
I am extremely grateful to Daniel Petter and Alexander Patscheider. Their always positive
attitude made it nice and easy to work in the ERBIUM team. With them inside the lab,
I learn a lot from their scientific and technical knowledge and their skill in problem-solving
made the foundation of our group very solid. I am extremely grateful to them for introducing
me to the mountains, sunrise hikes, snowboarding, and eating pizza during the weekend in
the labs when pushing for getting nice measurements.
I would also like to thank the previous generation of PhD students that built the vacuum
apparatus and in particular, Simon Baier and Jan Hendrik Becher with whom I overlapped
during my Master’s thesis. I own to Simon the accurate knowledge of the conversion B-field
to scattering length that allowed a better comparison with the theoretical predictions.
A big thanks goes also to Rick van Bijnen. He theoretically guided us in the first years of
my PhD and patiently found time to teach me how to run, analyze and improve his code. I
drop by his office many times and he always found time to discuss with me.
I would like to thank the new generation of PhD students in my lab, Arfor Houwman, andLouis Lafforgue (that will start as PhD student in January) for being able to grasp quickly
all the knowledge they needed to run the lab and allowing me to concentrate on writing mythesis. They are extremely motivated and fearless in changing the system to improve it. I211am confident the lab is in good hands with them.
I would also like to thank the Master’s students and interns that worked during my PhD. In
particular, Simon Gschwendtner and Daniel Grün, with whom I really enjoyed working on
the last project of this PhD thesis. Let me also thank Théo Mezieres, who helped us during
his internship for three months.
A big thanks goes also to the Ultracold group. In particular, Cornee Ravensbergen, Vincent
Corre, Gregor Anich, and Milena Horvath. They were extremely helpful inside and outside
the labs. I would also like to thank them for the chat in the corridors apart from the lunch,dinners, excursions, and activities together. I would also like to thank Emil Kirilov for being
always available to discuss technical and physical details as well as for raising our spirits
with nice anecdotes.
Another big thank goes to the italian PhD community in Innsbruck. In particular, to Lorenzo
Dania, Elisa Soave, Deborah Capecchi, Cosetta Baroni, and Elvia Colella. The many dinners
we organized were fundamental to support each other and in particular to ”survive” theTiroler küche.
I would also like to thank the new generations of PhD students, in particular: Eva Casotti,
Lauritz Klaus, Samuel White, Daniel Grün, Clemens Ulm, Alberto Canali. I wish them asuccessful PhD with the same great support I received.
I would also like to thank the theory group (Thomas Bland and Elena Poli), first for letting
me enter their office every time I needed a break, and second for the help to cruise ourexperimental research toward the goal.
We also benefit a lot from the discussion with Russell Bisset & Wyatt Kirkby. I particularly
like to thank Wyatt for asking me every day if my thesis was submitted. Adding in this way,
a little bit of positive pressure to my days.
I thank everybody working in the mechanical and electronic workshop that keep much simpler
our student life.
I would also like to thank the administrative stuff, and in particular Silvia Bonazza, Leonarda
Garcia Lopez, Laura Uiberreiter, Elisabeth Huck e Nikolaus G. Falschlunger. I need toespecially thank Silvia for helping with everything Deutsch related.
A big thanks go to my family for their unconditional love and support. In the future, I need
to to visit them more often. In fact, I also need to transmit my passion for physics to myniece and nephew (Sofia & Luca).
Finally, I want to thank Claudia, my girlfriend. I need to thank her from a scientific andpersonal perspective. Since our PhDs were simultaneous, we studied and discussed papers
together, learnt from each other’s mistakes, and supported in the more difficult times. From amore personal side, our activities and hobbies ranging from coffee roasting to mountain biking
helped me stay mentally and physically healthy and her love and support were fundamental212

to keeping my motivation high.
That’s All Folks!1

1

Bosko (Looney Tunes, 1929)213

Eidesstattliche Erklärung
Ich erkläre hiermit an Eides statt durch meine eigenhändige Unterschrift, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Alle Stellen, die wörtlich oder inhaltlich den angegebenen Quellen
entnommen wurden, sind als solche kenntlich gemacht.
Die vorliegende Arbeit wurde bisher in gleicher oder ähnlicher Form noch nicht als Magister/Master-/Diplomarbeit/Dissertation eingereicht.

Datum

Unterschrift