Study of Quantum Droplets, Rotonic
Quasiparticles and Supersolid States in
Ultracold Quantum Gases of Dipolar
Erbium Atoms
DISSERTATION

byDaniel Petter, 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)

advisors:
Univ.-Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences
Univ.-Prof. Mag. Dr. Paul Scheier,Institute for Ion Physics and Applied Physics, University of InnsbruckInnsbruck, August 2020

Gewidmet meiner Familie und meiner Freundin,für ihre Unterstützung und Liebe.

iEidesstattliche Erklärung
Ich erkläre hiermit an Eides statt durch meine eigenständige Unterschrift, dass ich die vorliegende Arbeit selbstständig 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 Dissertation eingereicht.

Datum, UnterschriftiiiAbstract
The current era of dilute, ultracold quantum gas experiments, allows experimentalists tostudy a vast number of new physical topics, ranging from Hubbard-models to strongly correlated quantum matter to liquid-like, gaseous droplets. At the heart of all these experiments
lies the exquisite control over system parameters and over the interactions between atomic
constituents. So far, the majority of experiments worked with contact-interacting atoms,
and only recently the magnetic atoms chromium, dysprosium, and erbium could be brought
to quantum degeneracy. The magnetic interaction between these atoms gives rise to numerous new physical phenomena and effects. One prime example is the formation of a roton
minimum in the excitation spectrum of a dipolar Bose-Einstein condensate (BEC), similar
to the one observed in superfluid helium. Other important features arise from the fact thatdipolar interactions can be used to carefully cancel contact interactions in a BEC. This yields
small interaction energies at the mean-field level in the system, where higher order effects
from quantum fluctuations become important.
The work in the present thesis is carried out with bosonic erbium, an element with oneof the highest magnetic interactions among current atomic quantum gas experiments. The
aim of our studies lies in further understanding dipolar quantum gases in a regime, where
dipolar and contact interactions cancel and the system is close to a mean-field instability.
Our experiments focus on two regimes: (i) in the first one, a global collapse of the BEC
is expected from standard mean-field theory, whereas (ii) in the second one, the BEC is
expected to collapse under a modulational instability. Starting with the former case, insteadof globally collapsing, our experiments showed a smooth crossover from a BEC to a highly
dense macrodroplet state. A careful comparison of the excitation frequency of a low-lying
collective mode with an extended mean-field theory – including a Lee-Huang-Yang (LHY)
energy correction – could show that the macrodroplet state is stabilised by quantum fluctuations. In the second case, the BEC is expected to exhibit a roton minimum in its excitation
spectrum. Here, a fully softened roton minimum is expected to trigger a collapse of the
BEC under a modulational instability. We could prove the existence of the roton minimum
in a dipolar BEC for the first time and confirmed the tunability of the roton momentum
with the trapping geometry of the BEC. Furthermore, by performing Bragg spectroscopy
measurements of the full excitation spectrum, we could observed the formation of the roton
minimum in the excitation spectrum and its energy softening when tuning the interatomic
interactions.
Due to the stabilising role of quantum fluctuations, also the modulational instability of a
BEC is arrested, resulting typically in an incoherent assembly of small quantum droplets,
first observed in the group of T. Pfau in Stuttgart with dysprosium atoms. However, for highenough densities, our two groups and the group from G. Modugno in Pisa, found a parameter
regime, where the dipolar BEC features a globally linked and phase coherent state, that isdensity modulated and stabilised by quantum fluctuations. This state shows the spontaneous
breaking of the U (1) gauge symmetry and the translational invariance simultaneously and,hence, has properties of a supersolid. The present thesis investigates the lifetime of thesesupersolid states with erbium and examines their coherence properties, as well as dynamics
that appear due to the dynamical formation process.
v

Our experimental results, together with the groups from Stuttgart and Pisa, expand thelist of possible phases in dipolar quantum gases to states, that are stabilised by quantum
fluctuations. They provide new grounds for theorists and experiments to study the behaviour
and the properties of these exciting new states and open a bridge to superfluid droplets ofliquid helium.

vi

Contents
Abstractv

1. Introduction
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3. List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
4

2. Degenerate Bose gases of erbium atoms
2.1. Erbium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Atomic shell structure . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Vacuum setup and magneto optical trapping . . . . . . . . . . . . . .
2.2.2. Evaporative cooling to quantum degeneracy . . . . . . . . . . . . . . .
2.3. Scattering properties of 166 Er . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Magnetic dipole-dipole interaction . . . . . . . . . . . . . . . . . . . .
2.3.2. Contact interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
7
8
9
9
10
12
12
13

3. Bogoliubov theory and quantum fluctuations
17
3.1. Gross-Pitaevskii equation approach: an overview . . . . . . . . . . . . . . . . 17
3.1.1. GPE and mean field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.2. Spectrum of excitations within mean field theory . . . . . . . . . . . . 21
3.1.3. Excitation spectrum of a homogeneous BEC . . . . . . . . . . . . . . 23
3.2. Beyond mean field effects: quantum fluctuations . . . . . . . . . . . . . . . . 24
3.2.1. Quantum depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2. Lee-Huang-Yang energy correction . . . . . . . . . . . . . . . . . . . . 26
3.2.3. Experimental observations of the LHY correction in contact interacting
BECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3. Mean-field theory including quantum fluctuations . . . . . . . . . . . . . . . . 28
3.4. Publication: Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein
Condensate to a Macrodroplet . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4. The rotonic excitation spectrum of a dipolar Bose-Einstein condensate
4.1. Excitation spectrum of a BEC confined along the dipole direction . . . . . . .
4.2. The dynamic structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. General description and Bogoliubov approach . . . . . . . . . . . . . .
4.2.2. Cigar-shaped dipolar BECs . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1. Basic theoretical description . . . . . . . . . . . . . . . . . . . . . . . .
vii

51
52
55
56
57
58
58

4.3.2. Experimental implementation . . . . . . . . . . . . . . . . . . . . . . .
4.4. Publication: Observation of Roton Mode Population in a Dipolar Quantum
Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5. Publication: Probing the Roton Excitation Spectrum of a Stable Dipolar Bose
Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59
62
77

5. Supersolid states in dipolar Bose–Einstein condensates
93
5.1. Historical idea of a superfluid solid . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2. Supersolidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3. A new type of quantum stabilised ground state in dipolar gases . . . . . . . . 95
5.4. Publication: Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5. Publication: High-energy Bragg Scattering Measurements of a Dipolar Supersolid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6. Conclusion

137

A. Additional publications
139
A.1. Publication: Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms 139
A.2. Publication: Controlling Dipolar Exchange Interactions in a Dense 3D Arrayof Large Spin Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
A.3. Publication: Excitation spectrum of a trapped dipolar supersolid and its experimental evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B. Optical setup for Bragg spectroscopy

189

Bibliography

191

Acknowledgments

207

viii

Chapter1

Introduction

1.1. Motivation
Nature surrounds us in our everyday life and reveals itself in an incredible number of facets:
From our sun to the seas and mountains on our earth. From forests to concrete cities. But,
even though its complexity goes far beyond what an individual mind could possibly capture,
its sophistication is rooted in small, similar building blocks - single atoms. The shape andall the properties of every matter, is given by the interactions between its constituent atoms
and their kinetic energy. Matter may appear therefore in different forms, called phases. If
the interaction between the atoms dominates over their kinetic energy, the atoms start toarrange in an ordered, solid structure, which possesses a fixed volume and shape. As the
kinetic energy increases and eventually becomes comparable with the interaction energy,
the solid structure might disappear and matter turns into an unordered liquid. It loses its
ability to keep a fixed shape but still possesses an essentially fixed volume. Once the kinetic
energy dominates over the interactions between the atoms, matter also loses its property
to occupy a fixed volume and turns into a gas.1 Throughout modern history, humanity
devoted a lot of time and resources to understand the different phases of matter. Researchers,
early on, characterised the chemical properties of different matter and ordered its different
constituent atoms in a periodic table of elements. It was found that different elements canhave very diverse properties: The atoms may vary in their chemical reactivity with others,
their stability, and also their optical and magnetic properties might differ. Therefore, they
constitute a rich toolbox that nature greatly knows how to apply.
In the 20th century, quantum theory emerged and scientists realised that matter and its
phases need to be described by energetically discrete quantum states, each represented bya wave function. Furthermore, it was discovered that all atoms can be divided into two
distinct classes - fermions and bosons. Fermions were found to obey the Pauli exclusion
principle and, as a consequence, two identical fermions are prohibited to occupy the same
quantum state. Bosons, however, do not obey the aforementioned principle and therefore
multiple bosons may exist in the same quantum state. In 1924, Albert Einstein [Ein24] andSatyendranath Bose [Bos24] predicted a novel phase of matter for non-interacting bosonic1

Note that for even higher kinetic energies, eventually the constituting atoms may break apart, which results
in a plasma phase.

1

2

CHAPTER 1. INTRODUCTION

particles, called a Bose-Einstein condensate (BEC). This new phase was predicted to occur
only below a critical temperature, Tc , and reveals itself in a macroscopic occupation of asingle quantum state. At first, the BEC phase of matter seems hard to picture next to theclassical phases such as a solid, liquid, or gas. As we will see, it should not be viewed as asimple expansion of the list of possible phases, but the BEC rather establishes an extensionof the classical phases to the quantum world. Indeed, the first experimental observation of
Bose-Einstein condensation was in a liquid.
At the beginning of the 20th century, 4 He was liquefied and allowed experimentalists for the
first time to reach cold temperatures of a few Kelvin [Del10]. It was found that, as theliquid 4 He was cooled below a temperature of about Tc ≈ 2.2 K, the fluid suddenly exhibiteda phase transition to a state with an extraordinarily high heat conductivity [All37] and an
essentially frictionless flow [Kap38, All38]. This new phase was in stark contrast to what
was known from ordinary fluids, and therefore it was called superfluid. It was speculated
early on that the superfluid phase emerged in connection with a Bose-Einstein condensate
that formed in 4 He, as the temperature of the liquid crossed Tc [Lon38]. Only decades later
it was experimentally proven, that up to ∼ 10 % of the helium atoms are indeed occupying
the same quantum mechanical ground state [Sno92] and thus form a BEC.
In Einstein’s early proposal [Ein24], he was considering the BEC phase for non-interacting
particles, where at zero temperature all atoms occupy the same quantum mechanical state.
The reason why for superfluid helium, even at zero temperature, the fraction of condensed
atoms does not reach unity is the strong interactions between the helium atoms. As superfluid
helium is a liquid, its high density means that the average distance, d, between two atoms2
is in the same range as their interaction potential, aHe . Therefore, helium is considered astrongly interacting system, in which ∼ 90 % of the atoms are depleted out of the BEC. Thisstrongly interacting character of the superfluid also leads to a peculiarity in the system’s
dispersion relation (i. e. excitation spectrum), which exhibits a so-called roton minimum.
The roton minimum describes density wave excitations, which have a wavelength in the orderof aHe and form an energetic minimum in the excitation spectrum of superfluid helium. It
was first predicted by Landau [Lan41] to describe essential properties of helium, such as thecritical velocity of the superfluid flow or the fraction of the system that is superfluid. The
direct observation of the roton minimum in the excitation spectrum was only later achievedin scattering experiments [Pal58, Gri93].
Around 1970, as more and more about superfluid helium was understood, theoretical proposals were made about a possible observation of Bose-Einstein condensation also in solid
helium [And69, Che70, Leg70]. As superfluid helium stays liquid at room pressure even atzero temperature[Sim34],3 one needs to pressurize the liquid to obtain solidify helium [Kee26].
It was speculated, that also solid helium might allow for a superfluid flow within solid matter [Bon12b]. This counter-intuitive property was coined supersolidity. It is still an active
research topic, where not only the connection of the roton minimum to the solidification
point but also the observation of supersolidity itself is under ongoing discussions [Bal10].
In 1995, a BEC was also observed for the first time in gaseous matter. With the advent
2
3

The typical interatomic distance is in the order of a few Å. With 1Å = 1 · 10−10 m.
Due to its high zero-point energy, it is energetically favourable for helium to stay liquid rather than freeze.

1.2. THESIS OVERVIEW

3of laser cooling techniques, researchers were able to cool atomic gases of rubidium [And95],
lithium [Bra95] and sodium [Dav95] to low enough temperatures that the gas undergoes aphase transition to a BEC and, similar to helium, becomes superfluid4 [Mat99, Mad00]. In
contrast to superfluid helium, gaseous BECs feature a density that is about 8 orders of magnitudes lower. Therefore, the interaction range between two atoms is much smaller than theinter-particle distance, d, and gaseous BECs are considered weakly interacting. In this regimeof low density and low temperature, the interatomic interaction is well described by a single
parameter, the s-wave scattering length, as . The interactions in this regime are called contact
interactions, which turned out to be highly tunable via Feshbach resonances [Chi10].
The connection between gaseous BECs and superfluid helium is ever since a flourishingresearch direction. The high interaction tunability of gaseous BECs allowed to performmeasurements in a regime where as ∼ d [Pap08] and observe effects in the regime of strong
interactions, which are known to be important in helium. However, essential properties
of superfluid helium, are typically absent in gaseous BECs with contact interactions. For
example, the excitation spectrum does not feature a roton minimum [Ste02]. Therefore,
in 2003 theoretical proposals [San03, O’D03] tried to establish a roton minimum in the
excitation spectrum of a gaseous BEC and suggested to use dipolar interactions between theatoms. In this case, the roton minimum is expected to be highly tunable in experiments andits energy can be tuned to zero, leading to a potential modulational instability, where thegas collapses in a density wave.
Over the last 25 years, more and more elements were brought to quantum degeneracy. Researchers realised BECs with alkali atoms [Dav95, And95, Bra95, Fri98, Mod01, Web03,
Zwi03], alkaline earth atoms [Kra09, Ste09], gaseous He [Rob01], Yb [Tak03], Cr [Gri05] and,most recently, Dy [Lu11] and Er [Aik12]. Nowadays, each atomic element has its own quality
and application for research, where the latter three ones - Cr, Dy, and Er - feature a highmagnetic dipole moment that is inherent to the atoms. Therefore, these elements exhibit astrong and permanent magnetic dipole-dipole interaction.
As shown in this thesis, Er proved to be a good choice to observe a roton minimum inthe excitation spectrum of a gaseous BEC. Furthermore, recent works, including the onespresented in this thesis, could tune the roton’s energy to zero and enter the mean-field
unstable regime. They revealed the pronounced impact of quantum fluctuations, which canstabilise the quantum gas against collapse, where a new type of quantum state was shown
to exist. A new phase of dipolar quantum matter – the dipolar supersolid – that exhibits
the spontaneous breaking of both, the system’s gauge and translational symmetry.

1.2. Thesis overview
The present thesis focusses on the study of dipolar Bose-Einstein condensates of highly
magnetic erbium atoms. Its main achievements cover:4

In gaseous BECs superfluidity was proven via the observation of quantizes vortices. See also [Yar79, Pit03]
for the case of superfluid helium.

4

CHAPTER 1. INTRODUCTION(I) The discovery of a crossover from a Bose-Einstein condensate to a dense macrodroplet
state, which is stabilised by quantum fluctuations against collapse.
(II) The experimental verification of a roton minimum in the excitation spectrum of dipolar
Bose-Einstein condensates.

(III) The realisation of density modulated states in dipolar Bose-Einstein condensatesbreaking both, the translational and the gauge symmetry simultaneously - so-called
dipolar supersolid states.
Chapter 2 reviews the basic properties of an erbium atom, which are important for understanding the fundamental ingredients of the physics investigated in this thesis. It recapitulates briefly the experimental apparatus that our group in Innsbruck is using to produce
degenerate quantum gases of erbium, before focussing on one specific isotope – 166 Er. Thisbosonic isotope will accompany us throughout this thesis.
Chapter 3 introduces the excitation spectrum of a dipolar Bose-Einstein condensate, obtained with Bogoliubov theory. After recapping the analytic case of a homogeneous gas, we
discuss two effects beyond this standard mean-field theory – the quantum depletion and the
Lee-Huang-Yang energy correction. The latter one is of prime importance for the results
shown in this thesis as the Lee-Huang-Yang correction has an important influence on the
ground state stability of a dipolar BEC. We will see that it allows for a stabilisation of an
expected collapse of the BEC and gives rise to a highly dense macrodroplet state, which we
confirm experimentally.
In Chapter 4, we focus again on the excitation spectrum but in a more particular case, where
the BEC is confined along the direction of the atomic dipoles. We will see that this peculiar
geometry allows the BEC to develop an excitation spectrum with a roton minimum, which
was predicted for dipolar BECs already in 2003 [San03], but took until now to be observed
experimentally.
Chapter 5 presents a new state of matter, newly found in dipolar quantum gases, thatshows the simultaneous presence of a density modulation with global phase coherence. It
was coined dipolar supersolid, referring to supersolidity, which originated in the context ofhelium. As we will find out, the dipolar supersolid combines both, the quantum mechanicalstabilisation from the Lee-Huang-Yang correction and the roton mode, discussed in theprevious chapters.
Finally, Chapter 6 concludes the present thesis.

1.3. List of publications
The following list contains all publications discussed in this thesis. They are given in chronological order that also defines the thematic structure of the chapters. The additional publications that emerged during the time of this thesis are covered in Appendix A.

1.3. LIST OF PUBLICATIONS

5• Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a
macro-droplet in a dipolar quantum fluid.
L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, F. Ferlaino,
Physical Review X, 6, 041039 (2016)• Observation of roton mode population in a dipolar quantum gas.
L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S. Baier, J. H. Becher, M. J.
Mark, F. Wächtler, L. Santos, F. Ferlaino,
Nature Physics, 14, 442-446 (2018)• 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)• High-energy Bragg scattering measurements of a dipolar supersolid.
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,
Submitted to Physical Review Letters, (2020)

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)• 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)• Controlling Dipolar Exchange Interactions in a Dense 3D Array of Large Spin Fermions.
A. Patscheider, B. Zhu, L. Chomaz, D. Petter, S. Baier, A.-M. Rey, F. Ferlaino, M. J.
Mark,Physical Review Research 2, 023050 (2020)

Chapter2

Degenerate Bose gases of erbium
atoms

Erbium is a chemical element from the lanthanide series of the periodic system. It was
first found in the middle of the 19th century [Mos43]. In nature, it occurs mainly in chemical compounds with phosphor and oxygen, known as monazit, and therefore needs to beleached with acids [Kum15]. Over the second half of the previous century, together withother lanthanides, erbium became an important element for highly advanced technologies ofmodern society. In its oxidized form, erbium is used for example in LEDs and liquid crystal
displays [Ken05, Sin16]. Due to its optical properties, it is used as a dopant in fiber-based
telecommunication systems or for lasers [Bec99]. Therefore, the need and supply of erbium
almost constantly increased over the last decades, where nowadays, most of erbium-oxide is
produced from China [Gor02]. Next to the flourishing technological advances in our modern
society, erbium also offers interesting properties to study fundamental research.
In this chapter, I will introduce in Sec. 2.1 the properties of an erbium atom that are relevant
for this thesis and compare them to other elements, nowadays used in degenerate quantum
gas experiments. In Sec. 2.2, I discuss the experimental apparatus in our laboratory in
Innsbruck and sketch the underlying scheme with which our group produces Bose-Einsteincondensates for our scientific research. Afterwards, the dipolar and the contact interaction
between two atoms is discussed in Sec. 2.3.

2.1. Erbium properties
2.1.1. Atomic shell structure
Atomic erbium has an average atomic mass of 167.26 u1 and has six stable isotopes, withnatural abundances ranging between ∼0.1% and ∼34% [Mic11]. Five isotopes are bosonicwhereas one is fermionic, see Table 2.1. Up to now, 166 Er, 167 Er, 168 Er, and 170 Er have been
1

1 unified atomic mass unit = 1 u = 1.66053906660(50) · 10−27 kg.

78

CHAPTER 2. DEGENERATE BOSE GASES OF ERBIUM ATOMS

brought to quantum degeneracy [Aik12, Aik14, Cho16, Tra18].2 The 166 Er isotope was first
Bose-Einstein condensed within the framework of this thesis. It is the working horse for all
studies presented here.
Table 2.1.: All stable isotopes of erbium together with their natural abundance and their quantum
statistics. Data taken from [Mic11].

Isotopenat. abundance

162 Er

0.14%
1.6%
33.5%
22.9%
27.0%
14.9%

164 Er166 Er
167 Er168 Er
170 Erbosonic
bosonic
bosonic
fermionic
bosonic
bosonic

A single erbium atom consists of 68 electrons that surround the atomic nuclei. The majorityof these electrons fill up a xenon core, while 14 are distributed on the two outermost shells.
These 14 electrons fill up the outermost 6s-shell completely and only partially fill up a
submerged 4f -shell. Therefore, the total electronic configuration of the atom is written[Xe]4f 12 6s2 .
It is this electronic configuration that gives rise to a multitude of important properties of
the atom. It leads to a plethora of optical transitions on the one hand and a large magnetic moment on the other hand. Individual electrons from either the 4f - or the 6s-shell,
can be easily excited with visible or infrared laser light. Among all these optical transitions, one finds electronically excited states that feature lifetimes from a few nanoseconds
up to tens of milliseconds [Ral11, Mar78]. Therefore, different transitions can be used inexperiments for different tasks, such as cooling, imaging or transferring atoms into different
internal states [McC06, Aik12]. Transitions with particularly long lifetimes might be used
to implement new coupling-schemes between atoms or even optical clocks [Liv16, Wil02].

2.1.2. Magnetic properties
The configuration of the electronic shell has also another important consequence on theatomic properties. As the 4f -shell contains only 12 out of 14 possible electrons, the atom
experiences a high orbital angular momentum of L = 5 together with a spin angular momentum of S = 1. Following the LS-coupling scheme, one obtains a ground state with a large
total angular momentum of J = 6, which gives the atom a high magnetic moment ofµ = mJ gJ µB = 6.982806(6) µB .
2

Recently, we managed to create a BEC of 164 Er by evaporating the atoms in a magnetic field of 3.1 G.
However, due to the low natural abundance of this isotope, the BEC contained only 5 × 103 atoms.

2.2. EXPERIMENTAL APPARATUS

9

Here, mJ = −6 describes the projection of the total angular momentum on the quantisation
axis for an atom in the lowest Zeeman state. Furthermore, the g-factor gJ = 1.163801(1)was determined experimentally in Ref. [Con63] and µB denotes Bohr’s magneton.
Table 2.2.: A few representative elements with their respective magnetic moment (rounded to theclosest integer). The data is taken from Refs. [Bra97, Dav95, Gri05, And95, Ste09, Web03, Ino18,
Aik12, Mia14, Lu11, Suk10, Tak03].

Isotopeµ (µB )

Isotopeµ (µB )

1
1
6
0
1
1

153 Eu

7
10
9
7
4
0

7 Li
23 Na
52 Cr
84 Sr
87 Rb
133 Cs164 Dy165 Ho166 Er
169 Tm174 Yb

To put the magnetic moment of erbium in comparison with other elements, used in (ultra-)cold
experiments nowadays, Table 2.2 lists a few representative examples. Alkali elements have intheir ground state a magnetic moment of ∼ 1 µB . Due to a single electron in the outermost sshell, they have mJ = 1/2 and gJ ≈ 2. Other specific elements, like 84 Sr or 174 Yb, have completely filled electronic shells and therefore feature an essentially vanishing magnetic moment.
These elements are interesting for example in precision-measurements [Hu17, Jam14, Har15],
where insensitivity to external magnetic fields is necessary. In contrast to all of these elements, 52 Cr and most of the lanthanides possess a more complex electronic shell structure.
For them, multiple electrons contribute to the atom’s total angular momentum and givethese elements much larger magnetic moments than alkalis. As one can see, Er is among
the elements with the highest magnetic moment, together with Dy, Ho, and Eu. It is thishigh magnetic moment that leads to a multitude of interesting quantum phenomena, someof which are presented in the publications throughout this thesis.

2.2. Experimental apparatus
The design and the building of the experimental apparatus started in 2009 in our group.
Over the years, the machine was constantly improved and upgraded. Its main technicalparts and working principles are described in more detail in the PhD thesis of Ref. [Fri14a].
Here, only the apparatus’ main parts are briefly reviewed and the recent upgrades during
the last few years are discussed for completeness.

2.2.1. Vacuum setup and magneto optical trapping
At the heart of the apparatus lies a vacuum system that is seperated in an ulta-high vacuum(UHV) and a high vacuum (HV) section via a differential pumping stage, see Fig. 2.1. At the
HV section, solid pieces of erbium are heated up to 1100 ◦ C with a high tempearture oven. A

10

CHAPTER 2. DEGENERATE BOSE GASES OF ERBIUM ATOMS

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 2.1.: Drawing of the experimental apparatus. A high-temperature oven (red) creates anatomic vapour beam in which atoms are flying through a transversal cooling stage (light blue) anda ZS (green), which slows the atoms down (right to left in image). After the ZS, the atoms are slow
enough to be captured with a MOT in the main chamber (yellow). The grey-shaded elements depict
essential parts of the vacuum chamber such as ion pumps, gate valves, and an atomic beam shutter
to block the atomic beam. For further details see text. Image courtesy of Albert Frisch [Fri14a].

small exit-aperture on the oven creates a hot atomic vapour beam that is travelling towards
the UHV section. Directly out of the oven the mean velocity of the atoms is ⟨v⟩ ≈ 420 m/s. In
a first stage, the atoms are radially cooled by a two-dimensional optical molasses (transversal
cooling), that is used to increase the flux of the atomic beam. Consecutively, the atoms are
slowed down by a Zeeman-Slower (ZS) to a velocity of a few m/s and enter the UHV section.
Here, a narrow-line magneto optical trap (MOT) is used to trap and cool the atoms. During
the course of this thesis the laser setup for the MOT was upgraded. We changed from a
dye-based laser system to a fiber-based system,3 which is seeded with a diode-laser.4 The
upgrade of this laser system improved the overall stability of the atom number in the magneto
optical trap and allows us to load 2.3 × 107 atoms in 3 s from the atomic beam. Due to the
MOT’s narrow transition, the atoms are spin-polarized in the lowest Zeeman sublevel andhave a temperature of around 8 µK [Fri12].

2.2.2. Evaporative cooling to quantum degeneracy
For further cooling, the atoms need to be loaded from the MOT to a far-detuned optical
dipole trap (ODT) operated at 1064 nm [Gri00]. The initial laser for the ODT was a single3

4

Raman Fiber Amplifier from MPB incl. frequency doubling stage. It converts 28 mW of seed light intoaround 1 W of 583 nm light.
DLpro from Toptica with a wavelength of 1166 nm.

2.2. EXPERIMENTAL APPARATUS

11mode MOPA Mephisto,5 which degraded in power and had to be replaced. Here, we changed
to a fiber-based laser system,6 whose laser light is separated by an additional high-power
fiber7 from the experimental apparatus, see Fig. 2.2. The laser power after this additional
fiber is currently limited by Brioullin scattering to about 15 W but is sufficient for loading
enough atoms from the MOT into the ODT. As described in Ref. [Bai12], the ODT has atunable beam waist (scanning system), which allows us to load around 5.3 × 106 atoms witha final temperature of typically 23 µK. The atoms are cooled further by forced evaporative
cooling. Here, the trap-depth of the ODT is stepwise linearly reduced in 10 s to obtain a
Bose-Einstein condensate (BEC) of 1.2 × 105 166 Er atoms, surrounded by additional 2 × 104
thermal atoms, corresponding to a thermal fraction of about 15%.
horizontal
ODT

main chamber(top view)

λ/2

laser head
1064 nm
45 W

isolator-75
scanning
system
AOM

-100
200

150
λ/2

λ/2 6.24

100

100

-50

rotatable
λ/2150

horizontal ODTwater-cooled high power fiber
200

-150

250

periscopehorizontal ODT

Figure 2.2.: Laser setup for horizontal ODT. The laser light is supplied by a 45 W single-frequency
laser, operating at 1064 nm. From the available laser power, only 20 W are needed and sent througha water-cooled high-power fiber to the experiment table. Before the fiber, a rotatable waveplate incombination with a polarising beam splitter cube allows us to cycle the amount of laser power that issent through (inset). In an experimental run, typically 20 W on the fiber are only needed to load theatoms from the MOT into the ODT. For the consecutive evaporation in the experiment, the power
is reduced to reduce thermal stress on the fiber. The outcoupler of the high power fiber replaced thedegraded MOPA laser, where a few beam-shaping optics were included. Furthermore, we changed a-50:100 mm cylindrical telescope in front of the chamber to a spherical one, to obtain an ODT beamwith a minimum aspect ratio of 1, that allowed us to obtain higher atomic densities for the BECs in
the experiments.

5
6
7

MOPA Mephisto from Innolight/Coherent with 42W output power.
High-power fiber laser from Azurlight with 45 W output power at 1064 nm.
Photonic crystal fiber from NKT photonics with 5 m length.

12

CHAPTER 2. DEGENERATE BOSE GASES OF ERBIUM ATOMS

2.3. Scattering properties of 166 Er
For evaporatively cooling the erbium atoms, they need to rethermalise and maintain a thermal equilibrium as the system transitions into a Bose-Einstein condensate. This thermalisation is ensured by interactions between the atoms, which are in the case of erbium twofold.
The atoms possess on the one hand the dipolar interaction, due to their magnetic moment,
and on the other hand the contact interaction, originating from s-wave scattering between
the atoms. As we will see, these interactions are not only important for cooling the sample,
but also dictate the physics in the obtained Bose-Einstein condensate.

2.3.1. Magnetic dipole-dipole interaction
Due to their large magnetic moments, discussed in Sec. 2.1.2, two atoms in the Bose-Einsteincondensate may interact via a long-range and anisotropic dipole-dipole interaction (DDI).
This interaction depends on the atoms respective magnetic moments, µ1,2 , and their distance, r. Its potential is described by
VDDI (r) =µ0 (µ1 · µ2 )r2 − 3(µ1 · r)(µ2 · r)
,
4π
r5

2.1

with µ0 being the vacuum permeability constant. In usual experiments, a magnetic field
is applied that polarizes the atoms along the same direction, eB . Furthermore, if both
atoms are in the same internal Zeeman state, they possess the same magnetic moment andµ1 = µ2 = µeB . Under these conditions the DDI becomes
VDDI (r) =µ0 µ2 1 − 3cos2 θ
4π
r3

2.2

and depends on the relative angle, θ, between the atomic dipoles with respect to theirpolarization axis, see Fig. 2.3. From Eq. 2.2 one sees that the DDI is proportional to r3 .

θ
rFigure 2.3.: Schematic illustration of the magnetic interaction between two atomic dipoles, polarizedalong the same direction (arrows).

This long-range character means that the total energy of a three-dimensional system of
dipolar atoms is an extensive quantity and thus not only depends on the system’s density,
but on the size of the total system. The other peculiar property of the DDI is its anisotropy,
as the strength and the sign of the interaction depends on θ. When two magnetic atoms

2.3. SCATTERING PROPERTIES OF 166 ER

13

are orientated head-to-tail, i. e. θ = 0◦ , they experience an attractive force, whereas for aside-by-side orientation, θ = 90◦ , they repel each other. It is interesting to note that for thecondition θ ≈ 54.7◦ the DDI vanishes completely.
As will become clear when discussing the excitation spectrum of a BEC, in Sec. 3.1.3, an
important quantity is the Fourier-transform of the DDI potential [Lah09]

1
2
ṼDDI (k) = µ0 µ cos α −
.
3
2

2.3

Here, α is the angle between the momentum vector, k, and the orientation of the magnetic
dipole moment. It is important to realise that, eventhough the DDI is longrange, its Fouriertransform in Eq. 2.3 only depends on α and not on the absolute value of k.
To quantify the length scale of the DDI, one can define a so-called dipolar scattering lengthadd =µ0 µ2 m
,
12π~2

2.4

with ~ being the reduced Planck’s constant. The dipolar scattering length is particularlyhigh for magnetic lanthanides, due to both their high magnetic moment and the high atomic
mass m, see Table 2.3. It defines the relevant length scale of the DDI and can be directly used
to compare the strength of the dipolar interaction to the contact interaction. The numericalfactors in Eq. 2.4 are chosen such that a homogeneous, dipolar BEC is stable as long as thelength scale associated with the contact interaction exceeds add .
Table 2.3.: Dipolar scattering length of a few representative elements used in experiments with(ultra-)cold gases. The values are given in units of Bohr’s radius, a0 . The data is obtained with thereferences in Table 2.2.

Isotope
7 Li
23 Na
52 Cr
84 Sr
87 Rb
133 Csadd (a0 )Isotopeadd (a0 )

0.06
0.19
15.1
0
0.70
1.08

153 Eu

60.7
133
108
65.5
21.9
0

164 Dy165 Ho166 Er
169 Tm174 Yb

2.3.2. Contact interaction
Next to the magnetic dipole-dipole interaction, the atoms exhibit also a contact interaction, which is typically well approximated by the so-called s-wave scattering length, as ,for low enough temperatures [Tay06]. In contrast to the dipolar interaction, the contact
one is isotropic and short-ranged. It can be shown that the interatomic interaction poten-

14

CHAPTER 2. DEGENERATE BOSE GASES OF ERBIUM ATOMS

tial, usually given by Lennard-Jones type potentials, is well approximated by the pseudopotential [Jon24, Pit03, Hua57, Chi10]
Vcon (r) =4π~2 as∂∂δ(r) r = gδ(r) r.
m
∂r
∂r

2.5

Its strength can be tuned in experiments with a homogeneous magnetic field, B, by meansof a Feshbach resonance (FR) which directly allows to change as (see Ref. [Chi10] for an
extensive review).
To explore physical phenomena with dipolar atoms, it is important to have a precise tunability of the relative strength between the dipolar and the contact interaction. Here, the, allows for a classification ofratios between the two interactions’ length scales, ϵdd = aadds
two regimes. One is the dipolar dominated regime, for ϵdd > 1. The other one is the contact
dominated regime, for ϵdd < 1. In experiments, as the magnetic interaction is mainly fixed by
the magnetic moment of the atoms, one therefore usually tunes as with a FR. It is therefore
important to identify suitable magnetic fields that allow for a precisely tunable as to coverboth regimes.
In our experiment, the first isotope that was brought to Bose-Einstein condensation was
168 Er [Aik12]. It features multiple resonances at a low magnetic field of a few Gauss.8
However, to obtain BECs of this isotope with high atom numbers (∼ 1 × 105 ), the atoms
need to be evaporatively cooled at a magnetic field of 300 mG, which is below the magnetic
field of the first Feshbach resonance. Furthermore, at this magnetic field ϵdd > 1. Therefore,
in order to enter ϵdd < 1, one would have to magnetically jump across the first Feshbachresonance and utilise its high-field side, where as can be tuned below add . This additional
preparation step limits the production of 168 Er BECs with high atom numbers in a regime
where ϵdd < 1.
Therefore, our group investigated a new isotope of erbium, 166 Er, and achieved first Bosecondensates in September 2015. The evaporation for this isotope yields atom numbers of
1.2 × 105 at a magnetic field of 2.1 G with a condensed fraction of around 80 %. Early
investigations to look for Feshbach resonances were performed by loss spectroscopy. Here,after the preparation of the BEC, the magnetic field is switched within a few milliseconds to a
final value ranging between 0 G and 5.5 G. Consecutively the atomic cloud is hold for 240 ms
before counting the atom number, NBEC , after a time of flight expansion. Figure 2.4 shows
the obtained loss spectrum which indicates four pronounced loss features in the magnetic field
range investigated. These loss features are a direct consequence of the presence of Feshbachresonances at the corresponding magnetic fields [Ino98, Ste99a, Esr99, Wer05, Fri14b]. Closeto the pole of a FR as diverges and increased three-body recombination is present, leading tothe observed atom loss. Importantly, this spectroscopic measurement shows a pronounced
loss feature close to 0 G, which is a strong indication that this isotope features a Feshbachresonance below its magnetic field for evaporation.
In a next step, we performed a careful mapping of the relation between as and B for thisisotope using lattice spectroscopy - a technique that has been used successfully for example inRef. [Bai16a]. Here, the dipolar BEC was loaded adiabatically into a deep, three-dimensional
8

1 Gauss = 1 G = 10−4 T.

2.3. SCATTERING PROPERTIES OF 166 ER

15

·105

NBEC

1

0.5

0
0

0.5

1

1.5

2

2.5
3
B (G)

3.5

4

4.5

5

5.5

Figure 2.4.: Atomic loss spectrum of an 166 Er BEC in an optical dipole trap. The arrows indicate
pronounced loss features, which indicate the presence of Feshbach resonances. As the BEC is preparedat 2.1 G, the overall atom number gets reduced for B & 3 G, as a pronounced resonance is crossed in
the preparation.

optical lattice. Consecutively, particle-hole excitations are created by modulating the intensity of one lattice beam with a given frequency ω. This process results in a modulation of
the tunnelling rate between neighbouring atoms in the optical lattice and supplies energy,~ω, for particles to hop on neighbouring lattice sites. The needed energy for this process has
multiple contributions coming from dipolar and contact interactions, but if the lattice depth
is carefully calibrated, the only unknown parameter is as . After modulating, the latticeis adiabatically ramped back and the recovered atom number in the BEC is measured. If
the in-lattice modulation creates particle-hole excitations, energy is pumped into the systemand, hence, the recovered BEC fraction is reduced. By performing measurements at various
ω, one obtains a resonant feature, whos center defines the excitation energy of particle-holes
and allows to extract as with rather high precision.
The measurements for as at various B are presented in Fig. 2.5. The data is modelled witha product of five FR [Chi10]
as (B) = abg (B)

5 
Y
i=1

∆i
1−
B − Bi.

2.6

It includes the four FR, shown in Fig. 2.4, and another one at 6.55 G, which is also seen withloss spectroscopy. To improve the fit performance we assume a linearly varying backgroundscattering lengthabg (B) = A + k · B.
2.7
The necessity to include in the model a linearly varying background points to the difficulties
to define a clear background scattering length for erbium. Due to the high density of Feshbach

16

CHAPTER 2. DEGENERATE BOSE GASES OF ERBIUM ATOMS
140
ϵdd

120

as (a0 )

100

1.6
1.4
1.2
1
0.8

0.2

80

0.4

0.6 0.8
B (G)

1

1.2

60
40
20

0

0.5

1

1.5
B (G)

2

2.5

3

Figure 2.5.: Conversion of as -to-B for 166 Er obtained from lattice spectroscopy, see text andRef. [Bai16a]. The solid line is a model fit. Two resonances are shown as solid vertical lines. The fit
also includes three more Feshbach resonances at magnetic fields, which are seen in loss spectroscopy,
but their positions and widths are kept constant. Their inclusion in the fitting slightly improves theoverall matching with the experimental data. The shading shows the fits statistical uncertainty corresponding to one standard deviation. For all fit values see Table 2.4. The inset shows ϵdd , extracted
from the fit, in the region of low B.

resonances and the presence of a few very broad ones [Fri14b], the assignment of a uniquevalue of the background scattering length is difficult. Therefore, including a linear variationin abg yields a good description of as in the B-field range investigated here. The fit results
and their standard error are summarized in Table 2.4. The corresponding values of ϵdd areshown in the inset of Fig. 2.5. As one can see, the FR close to 0 G allows one to smoothly vary
ϵdd from the contact to the dipole dominated regime. The latter is entered for B . 0.7 G. It
is the regime that is explored in this thesis.

Table 2.4.: Fit results for conversion between as and B for the 166 Er isotope at low magnetic field.

A = 61(4)a0
B1 = 0.04(3) G
B2 = 3.06(9) G

k = 5(1) a0 /G∆1 = 0.04(2) G∆2 = 0.15(4) G

B3 = 4.01 G
B4 = 4.95 G
B5 = 6.55 G∆3 = 0.02 G∆4 = 0.009 G∆5 = 0.04 G

Chapter3

Bogoliubov theory and quantum
fluctuations

This Chapter discusses the important role of quantum fluctuations in dipolar Bose-Einsteincondensates. We will see that the inclusion of the first order correction – called Lee-HuangYang energy correction – to the mean field description tremendously enriches the phase
diagram of a dipolar BEC. New quantum phases appear, ranging from macrodroplets tocrystal-like droplet arrays and supersolids. The existence of such phases crucially depends
on the stabilisation mechanism arising from quantum fluctuations. The publication at theend of this chapter discusses our experimental investigations of an unexplored region in thisnew phase diagram, where the dipolar BEC smoothly evolves into a highly dense macrodroplet
state. Our findings, together with the work from the group of Tilman Pfau [Kad16, FB16],
prove the role of the LHY correction for experiments with dipolar BECs.
Once we introduced the Gross-Pitaevskii equation (GPE) and derived the Bogoliubov-deGennes (BdG) equations in Sec. 3.1, we will discuss the excitation spectrum of a homogeneous BEC with contact and dipolar interactions. Moving on to Sec. 3.2, we will introduce
the concept of quantum fluctuations and see that they can be quantitatively deduced fromthe excitation spectrum. Section 3.3 then presents the extended Gross-Pitaevskii equation(eGPE) theory that incorporates the LHY correction at the mean-field level and discusses
the resulting phase diagram for dipolar BECs. The Chapter ends with our publication in
Sec. 3.4.

3.1. Gross-Pitaevskii equation approach: an overview
Before I present the theoretical description of a gaseous Bose-Einstein condensate at themean-field level, it is important to recall a few of its fundamental properties, see also
Refs. [Pit03, Leg06]. First of all, due to the very low temperature achieved in experiments(typically few tens or hundreds of nanokelvin), one obtains BECs with very low thermal fractions. Therefore, most of the atoms are occupying the system’s energetically lowest quantum
state, described by a wave function Ψ0 (r), and only a small fraction of atoms is distributed
on excited states, described by ψi (r). Secondly, for our experimental parameters, we have17

18

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

around 105 atoms in a trap volume† of the order 10 µm3 . Therefore, the BEC has a density
n ≈ 1020 m−3 and a mean particle distance of around 50 nm. As we have seen in Sec. 2.3.2,
the contact interaction is in the order of as = 100 a0 ≈ 5 nm and thus much smaller thanthe mean distance between the particles. Therefore, we are in the limit of weak interactions,n|as |3 ≪ 1, and we can assume to a good approximation that only two atoms are interactingwith each other at a given time.
Before being able to describe a gaseous BEC, we have to briefly review the formalism of
second quantisation, which can be used to define the Hamiltonian for a quantum many-bodysystem. The idea is to describe the system’s state with field operators [Pit03],XΨ̂† (r) = Ψ0 (r)â†0 +ψi (r)â†i ,
3.1
i̸=0

Ψ̂(r) = Ψ0 (r)â0 +X

ψi (r)âi ,

3.2

i̸=0

where â†i (âi ) describes the creation (annihilation) of a single particle in state i.

As we aredealing with bosonic atoms in this thesis, it is assumed that the field operators obey thebosonic commutation relations

and[âi , â†j ] = âi â†j − â†j âi = δij

3.3[âi , âj ] = [â†i , â†j ] = 0.

3.4

The field operators can be then used to construct the Hamiltonian [Pit03, Gro61, Pit61,
Bar12] of the system, which reads
Ĥ =

Z 


Z
~2
†
∇Ψ̂ (r)∇Ψ̂(r) dr + Vtrap (r)Ψ̂† (r)Ψ̂(r)dr+2m
Z1+Ψ̂† (r′ )Ψ̂† (r)Vint (r′ − r)Ψ̂(r′ )Ψ̂(r)dr′ dr. 3.5
2

The first term describes the kinetic energy of the atoms, the second one the energy contribution from the trapping confinement, Vtrap (r), and the last term describes the contact anddipolar interactions, Vint (r′ − r), between two particles.
As the Hamiltonian contains all properties of the system, it unambiguously defines its evolution in time. One can calculate the time-dependence of the field operator via the Heisenberg
equation, i~∂t Ψ̂(r, t) = [Ψ̂(r, t), Ĥ] and obtains [Pit03]
 2 2

Z
1
~ ∇
† ′
′
′
′
i~∂t Ψ̂(r, t) = −+ Vtrap (r) +Ψ̂ (r , t)Vint (r − r)Ψ̂(r , t)dr Ψ̂(r, t),
3.62m2
where we used the commutation relations (Eq. 3.3 and Eq. 3.4). Equation 3.6 describes the
temporal evolution of the system and the particles in all quantum states under the influence†

A volume can be defined by means of the harmonic oscillator lengths, lx,y,z =beeing the harmonic trapping frequency along direction i.

p~/mωx,y,z , with 2π × ωi

3.1. GROSS-PITAEVSKII EQUATION APPROACH: AN OVERVIEW

19of the trapping potential and the internal interactions between the atoms, but unfortunately
it still consists of coupled equations between all the different quantum states, as can be see
from Eq. 3.1. For this reason, it is in general very hard to solve Eq. 3.6 directly and further
approximations are needed.
Looking at Eq. 3.1, one may easily see the separation of the field operator into a BEC part,proportional to Ψ0 (r), and a remaining, non-condensed part, given by the sum over theexcited states. This separation is useful, as on the one hand, it allows us in the following
to easily take the limit where the non-condensed part of the system is negligible. Under
this approximation, Eq. 3.6 turns into an equation, that contains only a single field (i. e.
wave function), as we will see. On the other hand, the non-condensed part can be used
to investigate corrections to the system’s wave function. It is intimately connected to thepresence of quantum fluctuations, which will be discussed in Sec. 3.2.

3.1.1. GPE and mean field
For most of the experiments in gaseous BECs, the majority of all atoms occupy the system’senergetically lowest state. Therefore, one can neglect occupations of atoms in higher excitedstates and one may approximate
3.7
Ψ̂† (r) ≈ Ψ0 (r)â†0 .
Furthermore, the atom number in the BEC is given by the expectation value ⟨â†0 â0 ⟩ = N .
In the limit of high enough atom numbers,
N ≫ 1, one can approximate the operators â0√
and â†0 with their respective c-number N and obtains [Pit03]√Ψ̂† (r) ≈ N Ψ0 (r).
3.8This means, in the limit where most of the atoms sit in the condensate and we are dealing
with high particle numbers, the field operators of the system’s Hamiltonian can be replacedwith the classical field Ψ0 (r) which is typically called the system’s wave function. Thisapproximation has the advantage that it describes the BEC by a mean field, that a single
atom in the condensate sees. It, therefore, simplifies Eq. 3.6 to a single equation – theGross-Pitaevskii equation (GPE) [Gro61, Pit61] – that describes the evolution of the mean
field
 2 2

Z
~ ∇
′
′
′
′
2
i~∂t Ψ(r, t) = −+ Vtrap (r) + g|Ψ(r, t)| + Ψ(r , t)VDDI (r − r)Ψ(r , t)dr Ψ(r, t).
2m
3.9
′
′
′
Here, the interaction potential Vint (r − r) = Vcon (r ) + VDDI (r − r) is already inserted (seeEq. 2.2 and Eq. 2.5).
The GPE can be readily used to describe both, the stationary ground state and the timedependent dynamics of a BEC. In general, its solutions have to be calculated numericallyand only under certain approximations one might obtain analytic results. For example,
analytic solutions of the stationary ground states can be obtained with a Gaussian ansatz
for the wave function or within the Thomas-Fermi approximation, where the kinetic energy
is neglected [Pit03, Bar12, Lah09].

20

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

Let us investigate the energetic stability of a dipolar BEC in different trap geometries andfor interaction parameter regions with Eq. 3.9. For this, it is useful to look at the different
contributions to the system’s energy. Here, we find that the total energy of the BEC is givenby
E = Ekin + Etrap + Econ + Edip
3.10
with
Ekin =
Etrap =
Econ =
Edip =Z

Z~2|∇Ψ(r)|2 dr,
2m

Vtrap |Ψ(r)|2 dr,Zg|Ψ(r)|4 dr,2
ZZ1|Ψ(r)|2 VDDI (r′ − r)|Ψ(r′ )|2 dr′ dr.
2

3.11
3.12
3.13
3.14

The regimes where a stable BEC can exist can be calculated for example with a variational
Gaussian ansatz for the BEC wave function. If, for a given given as and trap geometry, one
finds a finite-sized solution which posses an energetic minimum in E, it is a (meta)stablestate [Bis16]. If no such solution exists, the GPE does not support a stable BEC. Figure 3.1 (a) shows the stability diagram for an 166 Er BEC for various trapping geometries,
which differ by their harmonic trapping frequencies ω∥ (ω⊥ ) along (perpendicular) to thedirection of the atomic dipoles. The trapping geometries are classified by their aspect ratio λ = ω∥ /ω⊥ and are varied from a cigar, λ < 1, to a pancake trap, λ > 1. For cigar
shaped traps, the BEC is elongated along the axial direction and the DDI is on average
attractive. Therefore, the critical s-wave scattering length for a collapse, a∗s , lies close to
add . As λ increases, the BEC becomes more pancake shaped and the average DDI becomesless attractive. Therefore, the GPE supports stable ground states at lower as .
To gain insight into the collapse, Figs. 3.1 (b-d) show the E-landscape of the variational
calculations for different BEC sizes at three different as . In the stable regime at as > add(b), one finds a state with a global minimum in E. This state is elongated along the weakerconfining axis. As as is reduced close to a∗s , the total-energy landscape exhibits a second
minimum (c) for a solution that is infinitely small along the radial direction. The global
minimum from (b) turns into a local minimum, indicating that the system is only metastable
at a finite size and close to a collapse. As as is reduced further, (d), this local minimum
vanishes and the GPE does not support any finite-sized ground state solution any more.
Here, it is energetically favourable for the BEC to collapse radially.
The stability diagram of a dipolar BEC was first investigated with chromium atoms [Koc08].
There, the authors prepared a BEC in a harmonic trapping potential and consecutively
reduced as until they observed a collapse of the BEC. Repeating this measurement for different λ, changing from a cigar to a pancake trap, they could confirm the stability regimes
expected from the GPE. Furthermore, the authors observed slight discrepancies for λ & 5,
where a∗s was higher than expected from a Gaussian ansatz for the GPE. It was later shown
in Ref. [Boh09] that this Gaussian ansatz may fail for pancake shaped traps, as the collapse

3.1. GROSS-PITAEVSKII EQUATION APPROACH: AN OVERVIEW

21

may happen not globally, but locally due to density fluctuations. This inherently connects
to the presence of roton excitations that develop in a BEC, as we will see in Chapter 4.

3.1.2. Spectrum of excitations within mean field theory
The GPE can be not only used to study the ground-state properties of a dipolar BEC but
also its dynamical response to external perturbations, e. g. a time-dependent Vtrap (r, t). This
dynamics can be described within Bogoliubov theory [Bog47, Pit03, Gri93], which allows
one to decompose the response of the BEC into elementary excitations. These excitations
describe an orthogonal set of new quantum states, which characterise the time evolutionof the system. To name two intuitive examples, a simple elementary excitation can be acenter-of-mass excitation of a trapped BEC or a breathing of the BEC’s size, but also more
intriguing elementary excitations, like phonons or rotons, exist (see Chapter 4). Together,
all elementary excitations describe the spectrum of excitations of the BEC – a concept that
we will introduce now.
To obtain the governing equations that describe the wave function and the time-dependency
of an elementary excitation, one typically considers a small perturbing wave function, ψ(r, t),which is added to the ground-state wave function Ψ0 (r) of the BEC. The total wave functionof the system is then assumed to beΨ(r, t) = Ψ0 (r) + ηψ(r, t),

3.15

with η ≪ 1. Inserting Eq. 3.15 in the GPE (Eq. 3.9) and neglecting terms of O(η 2 ) yields


 

Z
′
′
′
′
i~ ∂t Ψ0 (r) + ηψ(r) ≈ H0 + Ψ0 (r )Vint (r − r )Ψ0 (r )dr Ψ0 (r)+
Z

Z
′
′
∗ ′
′
′
′
′
′
+η
Ψ0 (r )Vint (r − r )ψ (r )dr + ψ(r )Vint (r − r )Ψ0 (r )dr Ψ0 (r)+


Z
′
′
′
′
+ η H0 + Ψ0 (r )Vint (r − r )Ψ0 (r )dr ψ(r), 3.16~
∇2 + Vtrap (r). The t-dependence was omitted for readability. It was firstwith H0 = − 2m
realised by Nikolay Bogoliobov [Bog47] that Eq. 3.16 may be directly solved via
2

ψ(r, t) = u(r)e−iωt + v ∗ (r)eiωt ,with u(r) and v(r) fulfillingZ|u(r)|2 − |v(r)|2 dr = 1.

3.17

3.18

As we will see below, this definition of ψ(r, t) directly defines an excitation mode of thesystem with an eigenenergy ϵ = ~ω. When inserting Eq. 3.17 into Eq. 3.16, one obtains two
coupled equations, which can be written in a matrix representation [Ron06]





HGP [Ψ0 ] + B
B
u(r)
u(r)
=ϵ
.
3.19
−B
−HGP [Ψ0 ] − B
v(r)
v(r)

< -100 -50 0
E (Nħωz)70

σ║ (l║)

60
stable

101

100

50

aS = 67 a0(c)σ║ (l║)

unstableas (a0)

>100

(b)

(a)

~ add

40

101

100
30

aS = 64.5 a0

σ║ (l║)

(d)

20

10

50

101
unstable

100
0.5

1

1.5

2
λ

2.5

3

3.5

4

10-1

aS = 40 a0100σ┴(l┴)

101

Figure 3.1.: (a) Stability diagram of an 166 Er BEC with 1 × 105 atoms, for different values of as andtrap geometries λ (see insets). For all calculations the geometric mean of the trapping frequencies
2 1/3
is kept constant at ω̄ = (ω∥ ω⊥)= 2π × 100 Hz. The results are obtained from the GPE with aGaussian ansatz for the BEC’s wave function. (b-d) Examples of the E-landscape versus the radial
and axial size (σ⊥ and σ∥ , respectively) of the BEC for three exemplary as in a cigar shaped trap
[indicated in (a)]. (b) At high as > add = 65.5 a0 an energetic global minimum (black cross) exists
at a finite size, indicating a stable ground state. (c) As as is decreased, the global minimum turns
into a local minimum, where the system is still metastable. (d) For even lower as the local minimum
disappears and the system becomes unstable, as it can minimise its energy by becoming infinitely
small in σ⊥ . Note that the color scales are truncated for visibility. The calculations are based on thetheory in Ref. [Bis16].

22

3.1. GROSS-PITAEVSKII EQUATION APPROACH: AN OVERVIEWHere,
HGP [Ψ0 ] = H0 +ZΨ0 (r′ )Vint (r − r′ )Ψ∗0 (r′ )dr′

23

3.20

describes the direct interaction term, i. e. the interaction between the particles inside thecondensate. Therefore, this term relates directly to the energy of the system’s ground state.
Furthermore, B is an operator defined viaZ
Bh (r) = Ψ0 (r′ )Vint (r − r′ )h(r′ )dr′ Ψ∗0 (r),3.21
acts on a function h = u(r), v(r) and is evaluated at position r. Its integral describes theinteraction between particles in the excited state and the ground state and is called exchangeinteraction.

It can be shown that the matrix in Eq. 3.19 can be diagonalised and that the chosen ansatz
for ψ(r, t) are the corresponding eigenstates. To see this, I refer the reader to Ref. [Ron06],
where a transformation of u(r) = 1/2(f (r) − g(r)) and v(r) = 1/2(f (r) + g(r)) simplifies
Eq. 3.19 to





f (r)
f (r)
0
HGP [Ψ0 ]
.
3.22
=ϵ
g(r)
g(r)
HGP [Ψ0 ] + 2B
0
After taking the square on both sides, one arrives at two independent equations

HGP [Ψ0 ] HGP [Ψ0 ] + 2B f (r) = ϵ2 f (r),
HGP [Ψ0 ] + 2B HGP [Ψ0 ] g(r) = ϵ2 g(r),

3.23
3.24

from which only one has to be solved to obtain ϵ, u(r), and v(r). These solutions have to
be calculated numerically in general. Only under further assumptions, like a homogeneousdensity, or an infinitely extended Bose-Einstein condensate, an analytic solution can beobtained.

3.1.3. Excitation spectrum of a homogeneous BEC
To give a first insight into the excitation spectrum [Pit03, Lah09, Bar12, Ron06, Bla17], let
us consider the simplest example of an unconfined, homogeneous BEC and look for solutions√of Eq. 3.23. The system’s ground state may be written as Ψ0 (x, y, z) = n0 . As the BEC
is unconfined, the atoms are completely delocalised and momentum, k, is a good quantum
number to describe the elementary excitations. Therefore, we can choose plane waves as acomplete basis set for u(r) = ueikr and v(r) = veikr . The amplitudes u and v are chosensuch that the normalisation in Eq. 3.18 is fulfilled. With this ansatz, we get for the directinteraction term [Eq. 3.20]
Z~2~2∆ + n0 Vint (r)dr = −∆ + E0 ,3.25
HGP [Ψ0 ] = −2m2m
which contains a constant energy offset E0 , which is the interaction energy that a single
particle has with the remaining atoms in the BEC’s ground state, i. e. the chemical potential.
As we are working in the limit of small perturbations, we can assume that any excitation

24

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

does not alter the chemical potential in the BEC. Then, E0 is a constant energy offset for
both, the particles in the ground state and the ones in the excitation mode, and we canneglect it. For the exchange interaction term (Eq. 3.21), we find [similar for v(r)]ZBu (r) = un0 Vint (r)e−ikr dr = un0 Ṽint (k),3.26

where Ṽint (k) represents the Fourier transform of the interaction potential at momentumk.

This plane-wave ansatz allows us to easily calculate the eigenenergies ϵ of the excitation
modes. As we solve Eq. 3.23 with f (r) = v(r) + u(r), we obtain
~2 k 2 ~2 k 2+ 2n0 Ṽint (k) v + u = ϵ2 v + u .
3.27
2m
2m
k
, and
In a last step, one typically introduces the excitation energy of a free particle, ϵ0k = ~2m
2
can insert the Fourier transform of the interaction potential Ṽint (k) = g 1 + ϵdd (cos α − 1) ,with α being the angle between the momentum vector k and the magnetic dipole orientation,
as derived in Sec. 2.3. This gives the Bogoliubov description of a homogeneous, interacting
BEC with an excitation spectrums ϵ(k) = ϵ0k ϵ0k + 2n0 g 1 + ϵdd (cos2 α − 1)
3.28
2 2

and the Bogoliubov amplitudess
ϵ0k + n0 g 1 + ϵdd (cos2 α − 1)
1
uk =
+ ,
2ϵk
2
s

 ϵ0k + n0 g 1 + ϵdd (cos2 α − 1)1
vk = sign Ṽint (k)
− .
2ϵk
2

3.29
3.30

As the Hamiltonian in the Bogoliubov theory is diagonal with the Bogoliubov Ansatz, ψ(r, t)
can now be interpreted as elementary excitations of the BEC with eigenenergy ϵ(k).
On the one hand, the concept of elementary excitations allows us to describe the dynamics ofthe BEC. On the other hand, it also directly contains important information on the systemproperties. For example, it defines the sound velocity of the gas cα = limk→0 ϵ(k)/k, whichbecomes anisotropic for dipolar interactions [Bis12], or describes fluctuations that are present
in the BEC. These fluctuations can be viewed as particles constantly fluctuating between
different elementary excitation modes and the BEC and can have surprising consequences
for dipolar BECs. As we will see in the publications of Sec. 3.4 and Chapter 5, they give riseto new types of states, which were unexplored prior to this thesis.

3.2. Beyond mean field effects: quantum fluctuations
When deriving the GPE equation in Sec. 3.1.1, we neglected particles occupying excitedstates ψ̂(r) in order to approximate the system’s field operator Ψ̂(r) with its wave function

3.2. BEYOND MEAN FIELD EFFECTS: QUANTUM FLUCTUATIONS

25

Ψ0 (r). We discussed, that this is a good approximation for systems like gaseous Bose-Einsteincondensates. Here, the interaction between the particles can be typically considered as weak,as the density n is small enough to fulfil n|as |3 ≪ 1. Nevertheless, real physical systems
always exhibit finite fluctuations of particles, which occupy excited states. These fluctuations
may stem for example from thermal atoms, but also at zero temperature, it is known thatthe interaction between the particles depletes them from the condensate into excited states.
This depletion can have a back-action on the ground-state energy of the system itself, which
is known as the Lee-Huang-Yang (LHY) correction.
The pronounced impact that quantum fluctuations is best seen for the case of superfluid
helium. In this liquid BEC, the system’s high density leads to an inter-particle distancethat is in the same order as the range of interactions, aHe . This has the consequence thatsuperfluid helium is considered a strongly interacting system, na3He ∼ 1, where it was foundthat only ∼ 10% of the particles actually are in the BEC and the remaining ∼ 90% aredepleted [Sos90, Sno92, Sno95]. This situation makes it challenging to theoretically describe
superfluid helium, as pertubative methods fail and more modern theoretical approaches,
like variational methods [Usm82, Vit88] and Monte Carlo calculations [Kü84, Whi87] areneeded.
Over the last century, the scientific community focussed a lot on the study of superfluid
helium and, therefore, corrections to mean-field theory and finite particle occupations in
excited states are rather well understood [Pen56, Hua57, Lee57, McM65]. They can bereadily applied to the case of gaseous BECs, where for most of the experiments the LHY
correction and the depletion of the mean-field ground state is typically small. In this case,both effects can be calculated from Bogoliubov theory, as we will see now. For simplicity, we
still assume a homogeneous, unconfined BEC. Therefore, the following calculations might beconsidered within a local density approximation (LDA).

3.2.1. Quantum depletion
In order to calculate the density of depleted atoms, ndep , in a dipolar BEC, we may use theansatz from Bogoliubov theory in Eq. 3.17. The atom number in an individual excited state,k, is given by
ndep,k = |ψk (r)|2 = |uk eikr + vk∗ e−ikr |2 = |vk |2 +

1
2

3.31

where in the last step the normalisation (Eq. 3.18) is used. One sees, that the number ofdepleted atoms is only determined by the vk amplitude in the Bogoliubov theory and an
additional 1/2, which is called vacuum occupation. By integrating over all excited modes,
excluding the BEC at k = 0, we obtain the density of depleted atoms [Bog47, Lim11]
ndep =

Zk̸=0

ndep,k =

Z

k̸=0



1
|vk | +
2
2



.

3.32

26

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

Inserting the Bogoliubov amplitude for an interacting BEC, obtained in Eq. 3.30, yields [Lim11]Zp
ϵ0k + n0 g 1 + ϵdd (cos2 α − 1)
8n0
ndep =
dk = √ Q3 (ϵdd ) n0 a3s
3.33
2ϵk
3 π
k̸=0
which depends on the BEC’s density n0 as and a function
Q3 (ϵdd ) =Z 10

1 − ϵdd + 3ϵdd u2

3/2

du.

3.34

Figure 3.2 (a) shows the functional dependence of Q3 with the relative strength of the dipolarinteractions ϵdd . For the purely contact-interacting BEC we find Q3 (0) = 1, whereas for a
BEC with dominant dipolar interactions Q3 (1) ≈ 1.3, showing that the presence of dipolarinteractions is increasing the quantum depletion. It is important to realise, that Q3 (ϵdd )becomes imaginary for ϵdd > 1, as we assumed an unconfined BEC, which becomes energetically unstable in the GPE mean-field theory and hence would collapse. In experiments witha BEC of 166 Er, one obtains typical densities of n0 ≈ 5 × 1020 m−3 , which results in a small
depletion of ∼ 1% at ϵdd = 1.

3.2.2. Lee-Huang-Yang energy correction
Besides the depletion of atoms from of the BEC, the interactions are also having an effect on the total energy of the system. From Bogoliubov theory one obtains the following
correction [Lim11, Lim12]Z

1∆ELHY =
ϵk − ϵ0k − n0 g 1 + ϵdd (cos2 α − 1) dk =3.35
2 k̸=0p
64= √ gn20 Q5 (ϵdd ) n0 a3s
3.36
15 π
with

Q5 (ϵdd ) =Z 10

1 − ϵdd + 3ϵdd u2

5/2

du.

3.37

∆ELHY is known as the LHY energy correction and increases the ground-state energy of thesystem. The functional dependence of Q5 is shown in Figure 3.2 (b). Again, one finds a
monotonic increase from Q5 (0) = 1 to Q5 (1) ≈ 2.6 revealing also the increase of this energy
correction with the dipolar interaction strength. Also here, Q5 acquires an imaginary valuefor ϵdd > 1.

3.2.3. Experimental observations of the LHY correction in contact interacting
BECs
Usually, in BEC experiments, effects from quantum fluctuations are small corrections compared to the atom number and energy of the ground state. Therefore, early experimentswith contact-interacting BECs tried to enhanced quantum fluctuations. One experiment

1.8

5
(a)
4





real Q3

real Q5

(b)

1.6
1.4

1

1

imag Q5 · 102

2


imag Q3 · 102

1.2

3

0

2

−5

−10

0

0

0.5

1

1.5

ϵdd

0

0.5

1

1.5

ϵdd

Figure 3.2.: Real and imaginary part of the functions (a) Q3 (ϵdd ) and (b) Q5 (ϵdd ), that describe
the scaling of the quantum depletion and the LHY energy correction, respectively, in a dipolar BEC.
For 0 < ϵdd < 1, both functions are real and monotonously increasing with ϵdd , which shows theincreasing effect of quantum fluctuations as the DDI becomes dominant in the BEC. For ϵdd > 1 thereal parts still monotoneously increase, but the functions exhibit an imaginary part, owed to the factthat the results were derived from a homogeneous dipolar BEC, which becomes unstable for ϵdd > 1.

27

28

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

with 85 Rb observed quantum fluctuations
via increasing as with a Feshbach resonance topenter the strongly interacting regime na3s ∼ 1 [Pap08]. Another interesting approach wasaltering the excitation spectrum of a 23 Na BEC with an optical lattice [Xu06], which leadto a strong increase of the quantum depletion. Other early studies focussed on precise measurements of collective oscillations in a molecular BEC, obtained from a two-component
fermionic mixture of 6 Li. Here, the LHY correction was observed in an upshift of the oscillation frequency compared to mean-field theory [Alt07]. Beyond mean-field effects werealso observed by carefully studying in-situ density profiles of 6 Li molecular BECs to directly
obtain the system’s equation of state [Shi08, Nav10] or from studies of the critical temperature of Bose-Einstein condensation in a trapped 39 K gas [Smi11]. More recent experiments
realised a measurement of the single-atom momentum distribution in a weakly-interacting
gaseous BEC of metastable 4 He∗ [Cha16] and studied the quantum depletion in a homogeneous experimental setup with 39 K [Lop17]. Even though, these experiments proved theimportance of quantum fluctuations in gaseous Bose-Einstein condensates, so far no experiment had been realised with additional dipolar interactions.

3.3. Mean-field theory including quantum fluctuations
In 2016 an experiment [Kad16] with strongly dipolar BECs of 164 Dy found a surprising result.
The authors tuned the interactions of the BEC into a region where a collapse was expectedfrom GPE theory. However, instead of collapsing, the BEC formed a crystal of multiple,
small droplets with typically ∼ 1000 atoms each and a few 100 ms lifetime. This result cameas a surprise, as the stability of a dipolar BEC was expected to be understood from themeasurements with 52 Cr [Koc08]. The main difference between these two elements was thestrength of their magnetic interactions: while 52 Cr possesses a dipolar scattering length ofadd ≈ 15 a0 , the one from 164 Dy is about 130 a0 and therefore almost an order of magnitude
higher. In the early days in which this experiment was published, there were two ideas to explain this unexpected stabilisation mechanism. One was the LHY correction [Wä16b, Bis16]
and another idea was suggesting repulsive three-body forces to explain the experimentallyfound droplets [Lu15, Bis15, Bla16].
To understand the role, that the LHY correction can play for a dipolar BEC, Refs. [Lim11,
Wä16b, Bis16] proposed an extended Gross-Pitaevskii equation (eGPE)

Z
~2 ∇2
2
i~ ∂t Ψ(r) = −+ Vtrap (r) + g|Ψ(r)| + Ψ(r′ )VDDI (r − r′ )Ψ(r′ )dr′ +
2m

64
3/2
+ √ gQ5 (ϵdd ) (n(r)as )
Ψ(r),
3.38
15 π
|
{z
}
LHY correction

which is based on the GPE (Eq. 3.9) but contains an additional energy term, coming fromthe LHY correction (see Eq. 3.36). Note that, even though the LHY is a beyond-mean-field
effect, it is here incorporated again at the mean field level. Furthermore, Eq. 3.38 relies on
a local density approximation, because the LHY correction is obtained from a homogeneous
BEC (see Sec. 3.1.3 and Sec. 3.2.2). That means for a given density, n(r), in space, we assume
its LHY correction is given by the homogeneous case of the same density.

3.3. MEAN-FIELD THEORY INCLUDING QUANTUM FLUCTUATIONS

29

To understand how the LHY correction can stabilise a collapsing BEC, one may notice two
important properties: (i) the LHY energy correction is always positive, meaning that it will
act similar to a repulsive force between the atoms, as the density in the BEC increases.
(ii) Its density dependence is n5/2 and, therefore, stronger than the n2 dependence of theinteraction energies (see Eq. 3.14). Consequently, even for attractive interactions in the BEC,
at high enough density, the repulsion coming from the LHY correction will stabilise the BEC.
Figure 3.3 shows two exemplary variational calculations for a BEC in a cigar shaped trap
and for interaction parameters, where the system is expected to collapse radially in GPE
theory. The presence of the LHY correction results in an energy barrier, which prevents
the system from reaching too small radial sizes, i. e. too high densities. Hence, the BEC
can form a stable state – called droplet. For high enough atom numbers, the density of thedroplet saturates and the state becomes incompressible [Wä16a, Bai17].
< -100 -50 0
E (Nħωz)

50

>100

101

(b)
σ║ (l║)σ║ (l║)(a)unstable

100
10-1

101

100
100
σ┴(l┴)

101

10-1

100
σ┴(l┴)

101

Figure 3.3.: Comparison of variational calculations of (a) the GPE with (b) the eGPE. The color
maps show the total-energy landscape of a 166 Er BEC with 1 × 105 atoms in a cigar-shaped trapwith λ = 0.5, ω̄ = 2π × 100 Hz and as = 40 a0 (see also Fig. 3.1). In the GPE theory, the system
is expected to collapse radially, while for the eGPE theory the LHY correction acts as a repulsive
barrier for small σ⊥ and creates a stable ground state solution at finite size (cross). Note that thecolor scales are truncated for visibility. The calculations are based on the theory in Ref. [Bis16].

Within our group, Rick van Bijnen and Gabriele Natale developed a code to numericallycalculate ground states, excitation spectra and real-time evolutions of a dipolar BEC within
the eGPE theory. These calculations were tremendously helpful for us to get a deeper insight
into our physical system and also to provide quantitative comparisons between experimentand theory. Figure 3.4 shows the ground-state phase diagram obtained from numericallysolving the eGPE. For high as & 51 a0 , similar to the GPE theory, one finds a regular BEC;
see (b). However, at lower as , the eGPE exhibits a much richer phase diagram than theGPE (see Fig. 3.1). For λ . 1.5, one finds a smooth crossover from the regular BEC solution
to a single macrodroplet; see (c) [Wä16a]. This droplet state has a much smaller radial
size than a regular BEC. Therefore, it is very elongated along the dipole direction and hasa density than can be up to an order of magnitude higher. Interestingly, when changing
the trap geometry from a cigar to a pancake trap, i. e. increasing λ, the axial confinement
can squeeze the single droplet axially and prevent it from elongating. As a result, it canbe energetically more favourable for a macrodroplet to split into multiple, smaller ones andform a droplet crystal; see (d) [Wen17, Bai18]. These types of states were initially observedin Ref. [Kad16].

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

50

crossoveras (a0)

52

48
46
44

regular BEC

dropletcrystalmacrodroplet
1

1.5

(d)

λ

2

2.5

(c)

(d)

4

4

4

0

-4

(c)

0.5

(b)

3

2
x ( 0 -2µm)

z (µm)(b)z (µm)(a) 54z (µm)

30

0-4

2 x 20-2 0 m) (µm -2
)
y (µ

2
-2 0 m)
y (µ

0
-4

2
x ( 0 -2µm)

2-2 0 m)µy(Figure 3.4.: Ground-state phase diagram of 1 × 105 166 Er atoms calculated numerically from theeGPE in different trapping geometries with ω̄ = 2π × 100 Hz. (a) Show the three regimes of a regular
BEC, which also exists in the GPE theory, a macrodroplet regime and a droplet crystal regime.
The two latter ones only exists due to the presence of quantum fluctuations in the system. The
regular BEC is connected via a crossover to the macrodroplet regime. To access the droplet crystal
regime a phase transition has to be crossed. (b-d) gives exemplary isodensity profiles, at 90% of thecorresponding maximum density, in the three regimes depicted in (a). Here, the atomic dipoles areoriented along the z direction.

The LHY correction in the eGPE Hamiltonian has been initially studied theoretically inthe context of non-dipolar mixtures of bosons in Ref. [Pet15a]. There, the author predicted
the existence of a self-bound mixture BEC in free space, which was later confirmed by two
independent groups with spin-mixture BECs of 39 K [Cab18, Sem18]. To form a self-bounddroplet, a carefully tuned balance between repulsive inter -species and attractive intra-species
interactions are needed, which renders the energy contribution from the LHY term dominantand stabilises the mixture to a self-bound object. For the case of dipolar atoms, the DDI
replaces the role of the inter-species interactions and the droplet exists due to a carefully
tuned interplay between contact and dipolar interactions. Also for the dipolar case, thedroplets can be self-bound, as was theoretically proposed in Ref. [Bai16b] and later confirmed
experimentally in Ref. [Sch16].
For both, non-dipolar and dipolar systems, the droplets show great similarities to liquid
droplets of superfluid helium, which is a highly active field of research. On the one hand,
helium droplets are seen as a bridge between Bose gases and bulk superfluid helium, to study
the transition from the weakly to the strongly interacting regime [Dal01]. On the other hand,
helium droplets provide an ultracold surrounding to study chemical reactions [Nor01, Toe01,
Toe04].
It was, therefore, for our team very interesting to experimentally study the nature of thisstabilization mechanism, which led to the publication that is presented at the end of this
Chapter. Here, we investigated for the first time the crossover region from an BEC to a
macrodroplet in a cigar shaped trap. When reducing the contact interactions, we foundindeed a smooth crossover to a high-density macrodroplet state, that was stable beyond the
GPE mean-field stability regime. Compared to the pancake trap geometry, the crossover3.3. MEAN-FIELD THEORY INCLUDING QUANTUM FLUCTUATIONS

31

provides a smooth transition to a droplet state and one does not have to cross a phase transition to a droplet crystal. The latter typically creates excitations in the system during the
preparation. Therefore, in the crossover region, the system stays close to the ground state.
Furthermore, we performed measurements of an energetically low-lying collective mode, as
initially proposed to test quantum fluctuations [Lim12]. A comparison of the obtained oscillation frequencies with numerical studies of the eGPE theory yielded a very good agreement.
Also the group from Tilman Pfau in Stuttgart obtained evidence for the presence of quantum
fluctuations in 164 Dy droplets, by estimating the density of the droplets [FB16]. Therefore,
our two groups could establish the important role of the LHY correction as being the stabilisation mechanism that leads to the formation of dipolar droplets.
After the first observations of dilute droplets in dipolar BECs and the confirmation of thestabilising role played by quantum fluctuations, a wealth of theoretical papers were published.
To name just a few, the authors from Ref. [Bai17] studied the full spectrum of collective
excitations of a single self-bound dipolar droplet in free space and also in a trap. They
found that the droplet is expected to support axial low-angular-momentum excitations andinvestigated the excitation spectrum for increasing atom numbers, as the droplet transitions
from a compressible to an incompressible liquid-like state. Another interesting directionis the study of vortices in droplets, where the authors in Ref. [Lee18] found that vortices
in a BEC may be brought over to a dipolar droplet via interaction tuning. However, theauthors also found that a vortex-line along the axial direction of the droplet is expected
to be unstable against a quadrupole mode (see also [Cid18]). Other works focussed on theextension of the dipolar droplets to lower spatial dimensions [Edl17]. Furthermore, there
has been intensive theoretical efforts to quantify quantum fluctuations in the dimensionalcrossover regimes for dipolar and non-dipolar quantum gases [Ilg18] and also for the presenceof optical lattices [Kum19].

32

CHAPTER 3. BOGOLIUBOV THEORY AND QUANTUM FLUCTUATIONS

3.4. Publication: Quantum-Fluctuation-Driven Crossover from a
Dilute Bose-Einstein Condensate to a Macrodroplet
Physical Review X, 6, 041039 (2016)‡
submitted 22 Jul 2016; published 22 Nov 2016
DOI: 10.1103/PhysRevX.6.041039

L. Chomaz,1 S. Baier,1 D. Petter,1 M. J. Mark,1,2 F. Wächtler,3 L. Santos,3 and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover,Germany‡

The author of the present thesis developed the experimental procedures together with L. C. and S.B.,
performed the measurements and analyzed the data together with L. C. and S.B., and contributed inwriting the manuscript. The theoretical calculations of this publication have been contributed by F. W.
and L. S.

PHYSICAL REVIEW X 6, 041039 (2016)

Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate
to a Macrodroplet in a Dipolar Quantum Fluid
L. Chomaz,1 S. Baier,1 D. Petter,1 M. J. Mark,1,2 F. Wächtler,3 L. Santos,3 and 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
3Institut für Theoretische Physik, Leibniz Universität Hannover,Appelstrasse 2, 30167 Hannover, Germany(Received 22 July 2016; revised manuscript received 15 September 2016; published 22 November 2016)
In a joint experimental and theoretical effort, we report on the formation of a macrodroplet state in an
ultracold bosonic gas of erbium atoms with strong dipolar interactions. By precise tuning of the s-wave
scattering length below the so-called dipolar length, we observe a smooth crossover of the ground statefrom a dilute Bose-Einstein condensate to a dense macrodroplet state of more than 2 × 104 atoms. Basedon the study of collective excitations and loss features, we prove that quantum fluctuations stabilize theultracold gas far beyond the instability threshold imposed by mean-field interactions. Finally, we perform
expansion measurements, showing that although self-bound solutions are prevented by losses, the interplay
between quantum stabilization and losses results in a minimal time-of-flight expansion velocity at a finite
scattering length.
DOI: 10.1103/PhysRevX.6.041039

Subject Areas: Atomic and Molecular Physics,
Quantum Physics

I. INTRODUCTION
The extraordinary success of ultracold quantum gases
largely stems from the simplicity with which the physics at
the many-body level can be controlled and described,
allowing access to a wide range of theoretical models ofgeneral interest [1]. Notably, the actual many-body interactions are often very well captured via simple mean-field(MF) potentials, proportional to the local particle density n
and accounting for the average mutual effect of allneighboring particles [1]. Moreover, short-ranged interactions, even if complex or unknown, can be simply
accounted for via a contact potential and parametrized
by the sole s-wave scattering length as , which in turn canbe widely tuned by means of Feshbach resonances (FRs)[2]. The MF treatment of a Bose gas leads to the celebrated
Gross-Pitaevskii equation (GPE) and Bogoliubov–deGennes (BdG) spectrum of collective modes, which arevery powerful in describing the physics of an ultracoldbosonic gas: its ground-state properties as a Bose-Einsteincondensate (BEC), as well as its dynamics [1].
Beyond the great achievements of dilute gases as a test
bed for MF theories, the quest for beyond-MF effects has
triggered great interest in the ultracold community. The*

Francesca.Ferlaino@uibk.ac.at

Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

2160-3308=16=6(4)=041039(10)general question of how the many-body ground state ofbosons is modified by quantum fluctuations (QFs) ofelementary excitations was first addressed by Lee,
Huang, and Yang (LHY) in the 1950s [3]. The so-called
LHY term, which accounts for the first-order correction to
the condensatepﬃﬃﬃﬃﬃﬃﬃﬃ energy, scales for a contact-interacting gas
as as n na3s . While in the weakly interacting regime theeffect of QFs is negligible and difficult to isolate from MF
contributions, it can be sufficiently amplified by increasing
as via a FR. Based on this concept, recent experiments withalkali have observed clear shifts of the BdG spectrum andequation of state caused by the LHY term in stronglyinteracting Fermi [4–6] and Bose gases [7,8].
While in these measurements the LHY correction does
not modify the qualitative behavior of the gas, it has been
recently pointed out [9] that, in systems with competinginteractions of different origin, the MF interaction can bemade small and the LHY term dominant, so that the latter
dictates the physics of the system, even in weakly interacting
gases. In this regime, a novel phase of matter is expected toappear, namely, a liquidlike droplet state. For purely contactinteracting gases, this situation is hard to realize since itwould require, for instance, Bose-Bose mixtures withcoincidental overlapping FRs [9]. In contrast, dipole-dipole
interaction (DDI) genuinely offers this possibility in asingle-component atomic gas by competing with the isotropic MF contact interaction [10,11]. In the pure MF
picture, a paradigm of the competition between DDI andcontact interaction is embodied by the ability of quenching a
dipolar BEC to collapse by varying εdd ¼ add =as , where

041039-1

Published by the American Physical Society

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)

II. EXPERIMENTAL PROCEDURES
The atomic properties of Er offer a privileged platform to
explore a variety of interaction scenarios. Besides its
strongly magnetic character and its many FRs [25], Er
has several stable isotopes. This feature adds an important
flexibility in terms of the choice of the background as [26].
In our early work on Er BECs, we employed the 168 Er
isotope, which has a background as about twice as large as
the dipolar length, add ¼ 65a0 [27,28].

160

2.0
εdd

140

1.5

120
1.0
0.2
a s (a 0 )add ¼ μ0 μ2 m=12πℏ2 is a characteristic length set by the
DDI, with m the mass and μ the magnetic moment of theatoms [12,13]. Here, ℏ stands for the reduced Planck
constant and μ0 for the vacuum permeability. In general,
because of the special geometrical tunability of DDI with theexternal trapping potential and dipole orientation, thestability and phase diagram remarkably depend on
λ ¼ ν∥ =ν⊥ , where ν∥ (ν⊥ ) is the trapping frequency along(perpendicular to) the dipole orientation [11,12,14].
In parallel, recent breakthrough experiments with an
oblate dysprosium (Dy) dipolar BEC (λ > 1) have shownthat when quenching up εdd , the system, instead of collapsing, forms a metastable state of several small droplets[15,16]. This observation has triggered an intense debate
on the nature of such a state and its underlying stabilization
mechanism [17–23]. Eventually, Dy experiments indicatedQFs as the origin of the stabilization [16], which werequickly confirmed by theoretical works [20–22].
Furthermore, these theoretical studies highlight the richnessof the dipolar-gas phase diagram, in which a dilute-BEC, amultidroplet, and a single-droplet phase are found for
distinct as , add , atom number N, and λ. Up to now, droplet
physics has only been investigated in a single setup, using
Dy BEC and exploring a specific region of the phase
diagram: In the considered pancake geometry (λ > 1),
multiple stable solutions—single droplet or multidroplet—coexist, resulting in the formation of variable mesoscopic
assemblies of a small droplet in the experiments.
In the present work, we (i) demonstrate the generality ofdroplet physics, by using a dipolar BEC of erbium (Er) atoms[24], (ii) quantitatively investigate the specific role played by
QFs in dipolar systems, and (iii) explore a pristine region of
the phase diagram, studying a cigar-shaped geometry(λ ≪ 1), and observe the crossover from a dilute BEC to asingle macrodroplet state when increasing εdd , as predicted inRefs. [21,22]. Given the complexity of the physics at play, we
combine distinct measurements, based on the observations of
the density distributions, collectives excitations, expansion
dynamics, and lifetime of the dipolar quantum gas, which
together offer a comprehensive picture of droplet physics.
The exquisite control of the scattering length gained in our
experiment, together with a direct comparison to parameterfree simulations including QF effects, ultimately enable us todepict in which way QFs dictate the physics at play, beyond
proving their crucial stabilizing role.

0.4

0.8

0.6

80
⊥ -imaging

60

z

40

x
y

20

0

0.5

1.0

B

-

1.5
B (G)

2.0

2.5

FIG. 1. Scattering length in 166 Er. as as a function of B. The data
points (circles) are extracted from spectroscopic measurements in a
lattice-confined gas and the solid line is a fit to the data with its
statistical uncertainty (gray shaded region [30]). Upper inset:
Zoom-in of εdd as a function of B. The gray dashed line marks
εdd ¼ 1; see also the other figures. The lower inset illustrates thegeometry of our experimental setup, the relevant axes (x, y, z), theoptical-dipole-trap beam (shaded region), the magnetic fieldorientation (green arrow) along which the dipoles are aligned,
and the ∥- and ⊥-imaging view axes (blue arrows). The dashedlines picture the small angles of these axes to y and z [30].

In the work reported here, we produce and use a BEC of
Er in the lowest internal state. This isotope provides us
with two major advantages. First, its background as iscomparable to its dipolar length, add ¼ 65.5a0 , realizing
εdd ¼ add =as ≈ 1 without the need of Feshbach tuning.
Second, 166 Er features a very convenient FR at ultralow
magnetic-field values B. To precisely map as as a function ofB, we use a spectroscopic technique based on the measurement of the energy gap of the Mott insulator state in a deep
three-dimensional optical lattice [28,29]. A detailed description is given in the Supplemental Material [30]. Between 0
and 3 G, we observe a smooth variation of as , which resultsfrom two low-lying FRs whose centers are fitted to 0.05(5)and 3.0(1) G, respectively; see Fig. 1. This feature gives easy
access into the εdd > 1 regime, allowing variation of εdd
from 0.70(2) to 1.58(18) by changing B from 2.5 to 0.15 G;
see Fig. 1 upper inset. By fitting our data [2], we extractas ðBÞ valid for B in the [0.15, 2.5]-G range, which we use
throughout this paper [30].
We achieve Bose-Einstein condensation of 166 Er using
an all-optical scheme very similar to Ref. [27] with coolingparameters optimized for 166 Er [30]. In short, we drive
forced evaporative cooling at a magnetic field B ¼ 1.9 G,
corresponding to as ¼ 81ð2Þa0 [εdd ¼ 0.81ð2Þ]. In thisphase, B is oriented along the vertical z axis. At the endof the evaporation, we obtain a BEC of N ¼ 1.2 × 105
atoms with a condensed fraction above 80%.
166

041039-2

QUANTUM-FLUCTUATION-DRIVEN CROSSOVER FROM A …

III. DENSITY DISTRIBUTION
In a first set of experiments, we study the stability ofour dipolar Er BEC by probing the evolution of the
TOF density distribution for different as . Figures 2(a)–2(c)show the absorption images acquired with ∥ imaging[Figs. 2(a)–2(c)] and the corresponding central cuts
(x ¼ 0) of the 2D column density profiles [Fig. 2(d)]. In
striking contrast to the MF predictions, we observe that thesystem remains stable for as well below add, with a central
coherent core surviving for times much longer than 1=ν⊥(from several tens to hundreds of ms). The density distribution does not exhibit any special patterns, which is typical of acollapsing cloud [12,13,32].
For as > add [Fig. 2(a)], the density distribution of thegas shows good agreement with the MF Thomas-Fermi(TF) profile on top of a broad Gaussian distribution,2n col (µm )35 30 25 20 15 10 5 040-80 (a)z (µm)(a) a s = 93a 0(b) a s = 57a 0(c) a s = 50a0

0
30

40

80
-80 (b)ncol (µm 2 )

25

-40

z (µm)

(d)

35

-40

20

0
40

15

80
-80 (c)10-40

z (µm)To reach the λ ≪ 1 regime, we slowly modify, in thelast step of the evaporation, the confining potential to thefinal cigar shape, with typical frequencies ðνx ; νy ; νz Þ ¼½156ð1Þ; 17.2ð4Þ; 198ð2Þ Hz. Simultaneously, we decrease
B to 0.8 G [as ¼ 67ð2Þa0 ] and then change the magneticfield orientation to the weak trapping axis (y) while keeping
its amplitude constant [30]. Finally, we ramp B to thedesired target value (and equivalently as ) in tr [30], hold fora time th , and perform absorption imaging of our gas after atime-of-flight (TOF) of tTOF . Two imaging setups are usedin order to measure the density distribution integrated either
along the dipoles (∥ imaging) or perpendicular to them(⊥ imaging) [30]. Figure 1 (lower inset) illustrates
the final
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
geometry of our system with ν∥ ¼ νy , ν⊥ ¼ ðνx þ ν2z Þ=2,
giving λ ¼ 0.097ð3Þ, and defines the relevant axes.
Here, we explore the properties of the system when therepulsive MF contact interaction is weakened enough to be
overcome by the DDI (λ ≪ 1, εdd > 1), after adiabaticallychanging (tr ≥ 45 ms) or quenching (tr ¼ 10 ms) as to its
target value [30]. For tr ≥ 45 ms, the system evolves
following its ground state and gives access to the slow
dynamics, whereas for the tr ¼ 10 ms case, we can probe
the fast dynamics and study the relaxation towards anequilibrium. The key question is whether QFs protect thesystem from collapsing. Indeed, in this regime, the MF
treatment would imply that the attractive BEC becomes
unstable, leading to a twofold dramatic consequence [1].
First, some modes of the BdG spectrum acquire complex
frequencies. Second, in a trap, the density distribution of
the cloud undergoes a marked change on short time scales(≤1=ν⊥ ), described as a “collapse”, which can develop intoa rapid loss of coherence [12,31], and pattern formations,such as anisotropic atom bursts (“bosenova”) and speciald-wave-type structures, as observed in rubidium [32] anddipolar gases of chromium [12,13], respectively. This fast
dynamics has been proved to be well encompassed by GPE
simulation [13,14,33].

PHYS. REV. X 6, 041039 (2016)

0

5

40
80
-50

0

x (µm)

50

0
-100

-50

0

50

100

z (µm)

FIG. 2. Density profiles in the BEC-to-droplet crossover.
(a)–(c) 2D column density distributions probed with ∥ imaging
and (d) corresponding central cuts along the x ¼ 0 line (dots) fortr ¼ 10 ms, th ¼ 6 ms (>1=ν⊥ ), and different as (see legend).
Each distribution is obtained by averaging four absorption images
taken after tTOF ¼ 27 ms. In (d), the lines show the central cuts ofthe 2D bimodal fit results, the solid (dashed) lines showing thetwo-Gaussian (MF-TF plus Gaussian) distributions and thedotted lines the corresponding broad thermal Gaussian part.

accounting for the thermal atoms; see Fig. 2(d), dashedlines. When lowering as below add [Fig. 2(b)], we observe a
sharpening of the central core, whose profile starts todeviate from the MF-TF shape (see Ref. [30] for aquantitative description). When decreasing as even further[Fig. 2(c)], a similar bimodal structure holds on although
the dense core loses atoms. Because of the high density
reached, three-body (3B) collisions regulate the lifetime of
the central core; see discussion below and Ref. [23]. We
note that we observe a similar qualitative behavior of thedensity distribution when using an adiabatic ramp of as .
However, the importance of the central peak is reduced as,in this case, losses already set in during the ramp.
In contrast with the behavior of the central core, thedistribution of the thermal atoms, encompassed by thebroad Gaussian function of the bimodal fits [see Fig. 2(d),dotted lines], remains mainly unaffected by the change ofas , highlighting an absence of significant heating andpopulation transfer, and thus an apparent decoupling of
the evolution of the coherent and thermal parts.
For further analysis, we fit the data to a bimodal
distribution made of the sum of two Gaussian functions,
as it offers a smaller residue than the fit to the MF-TF
distribution for as ≲ 70a0 ; see Fig. 2(d). We note that the
beyond-MF effects on the density profile are expected to be
more sophisticated than a Gaussian shaping. However,041039-3

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)
(a)
B
dd

1.32

dd

0.94 0.82 0.73 0.66 1.32

1.10
1.4

0.94 0.82 0.73 0.66

1.10

as= 80a0

1.2

2.5

R //

1.0
0.8
0

50

100

150

/

//

thold (ms)axialtheoretical studies show good agreement between theGaussian ansatz and the full numerical solution for ourparameter range [22,23].
Being a smoking gun for long-range phase coherence,the survival of a bimodal profile in the TOF distribution far
beyond the MF instability threshold points to a persistentcoherent behavior. This absence of a collapse advocates theoutbreak of an additional stabilization mechanism, which
we now further investigate by probing global properties of
the gas.

2.0

IV. COLLECTIVE OSCILLATION
In a second set of experiments, we unveil the origin of
the stabilization mechanism by studying the elementary
excitations of the coherent cloud. This is a very powerfulprobe of the fundamental properties in quantum degenerategases [1,34]. In particular, collapse is intimately related
to the softening of some collective modes at the MFinstability threshold. We focus here on the axial mode,
which is the lowest-lying excitation in the system above thedipole mode. It corresponds to a collective oscillation of thecondensate length along y (R∥ ) with frequency νaxial.
The axial oscillation comes along with a smaller-amplitude
oscillation of the radial sizes in phase opposition; see
Fig. 3(a). As a result, this mode has a mixed character
between a compression and a surface mode [1]. The
compression character is particularly relevant since it
involves a change in the density and it is therefore sensitive
to the LHY corrections [35].
We excite the axial mode either by ramping B during thefinal preparation stage or by transiently increasing the powerof the vertical optical dipole trap beam, after ramping B to
Bf . Here, ν∥ is abruptly changed from 17 Hz to typically
21 Hz, kept at this higher value for 8 ms, and finally set backto 17 Hz. Following the excitation, we let the cloud evolvefor a variable th and image its TOF density distribution with⊥ imaging. To extract νaxial , we probe the axial width R∥ ofthe central coherent component of the gas [30] with th and fit
it to a damped sine; see inset of Fig. 3(b).
Figure 3 shows the observed νaxial normalized to thetrapping frequency ν∥ [36] as a function of as for adiabatic[Fig. 3(b)] and nonadiabatic [Fig. 3(c)] ramps. Both cases
exhibit a similar qualitative behavior. For as > add, theoscillations show a smooth dependence on εdd , with νaxial
increasing by about 5% with an average value of 1.70ν∥[37]. When lowering as , the oscillation of the coherent part
remains visible well below the εdd ¼ 1 threshold and νaxial
exhibits a marked increase. νaxial =ν∥ grows up to 2.6(1) at
as ¼ 54a0 for tr ¼ 100 ms [Fig. 3(b)]. For tr ¼ 10 ms[Fig. 3(c)], νaxial =ν∥ first increases similarly to the adiabaticcase [Fig. 3(b)], reaches a maximum of ∼2.13ð7Þ at 57a0
(εdd ¼ 1.15), and finally decreases for even smaller as(open squares). The latter behavior can be explained by thefact that the larger quenches in the interaction excites

(b)
50

(c)
60

70

as (a0)80

90 100 50

60

70

80

1.5
90 100as (a0)

FIG. 3. Axial mode. (a) Illustration of the axial mode in our
experimental setup. The black arrows sketch the oscillations of
the widths of the coherent gas along the characteristic axes of thetrap, with weights indicating their relative amplitudes. (b),(c) Measured νaxial =ν∥ (squares) as a function of as together
with the theoretical predictions, including (solid line) or not(dashed lines) the LHY term for tr ¼ 100 ms (b) and tr ¼ 10 ms(c). Theoretical predictions are obtained from RTE (see text) for
as varied from 50a0 to 95a0 . In the MF case, predictions fail foras ≤ ac (orange area) due to the occurrence of the collapsing
dynamics which rules out the collective excitation picture.
ac ¼ 57a0 [ac ¼ 64a0 ] in (b) [(c)]. In (c), νaxial cannot be reliably
extracted for quenches to as ≤ 56a0 , nor from the experiment(open squares) or from the LHY theory (open circles, thin line).
The inset in (b) exemplifies a measurement of R∥ (triangles) and
its fit to a damped sine (solid line) for as ¼ 80a0. We typically fit
4–5 oscillations for all our as.

additional high-energy modes while it drives the system
away from the linear response regime [38]. A similar
behavior is found from our theory predictions including the
LHY term (see below), thus highlighting a qualitative
agreement even in this small-as range.
V. THEORY
To account for our observation and discern between the
MF instability picture and QF mechanisms, we develop abeyond-MF treatment of our system at T ¼ 0. The coherent
gas is described here by means of the generalized nonlocal
nonlinear-Schrödinger equation (gNLNLSE), which
includes the first-order correction from QF effects, i.e.,the LHY term, and 3B loss processes. The gNLNLSE
reads as [20,23]


∂ψ
L
iℏ
¼ Ĥ0 þ μMF ðn; ϵdd Þ þ Δμðn; ϵdd Þ − iℏ 3 n2 ψ;∂t2041039-4

ð1Þ

QUANTUM-FLUCTUATION-DRIVEN CROSSOVER FROM A …
where Ĥ 0 ¼ ½ð−ℏ2 ΔÞ=2m þ VðrÞP is the noninteracting
Hamiltonian and VðrÞ ¼ 2π 2 m η¼x;y;z ν2η η2 the harmonic
confinement.
MF chemical potential, μMF ½nðrÞ; ϵdd  ¼
R 3 The0gnðrÞ þ d r V dd ðr − r0 Þnðr0 Þ, results from the competition between short-range interactions, controlled by the
coupling constant g ¼ 4πℏ2 as =m, and the DDI term with
V dd ðrÞ ¼ ½ðμ0 μ2 Þ=4πr3 ð1 − 3cos2 θÞ and θ the angle sustained by r and the dipole moment μ. Here, nðrÞ ¼ jψðrÞj2 .
The beyond-MF physics is encoded in the LHY term,
leading to an additional repulsive termin ﬃ the chemical
pﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
potential, Δμðn; ϵddRÞ ¼ ½32=ð3 π Þgn na3 Fðϵdd Þ. The
function Fðϵdd Þ ¼ 12 dθk sin θk ½1 þ ϵdd ð3 cos2 θk − 1Þ5=2
is obtained from the LHY correction in homogeneous 3D
dipolar BECs [39–41] using local-density approximation[42]. The last non-Hermitian term in Eq. (1) accounts for
3B loss processes [43]. In our calculations, we use theexperimentally determined values of the 3B recombination
rate of the condensate L3 ðas Þ [30].
As discussed in Refs. [22,23], due to the repulsive LHY
term, Eq. (1) sustains stable ground-state solutions for any
as and λ. For pancake traps (λ > 1), the solution of Eq. (1)is not unique. The phase diagram reveals three types ofsolutions: the one of a dilute BEC, a single droplet solution,
and a third one, which separates the previous two phases,that corresponds to a metastable region of multidroplet
states. The latter has been observed in Dy experiments [15].
However, the single-droplet solution appears difficult toaccess because of the overhead multidroplet state and thestringent 3B loss mechanisms. Remarkably, in cigar-shaped
traps (λ < 1), Eq. (1) has only one possible solution. In theεdd parameter space, the corresponding wave function
exhibits a smooth crossover from a dilute BEC to a single,
high-density, macrodroplet solution for increasing εdd . It is
worth noting that the crossover physics, e.g., the formationand lifetime of the droplet state, is expected to cruciallydepend on the 3B collisional processes. In the following,
we concentrate on the λ < 1 case, which corresponds to ourexperimental setting.
The continuous and smooth change of the static properties of the system with increasing εdd is consistent with
our observations on the evolution from a dilute into anhigh-density state; see Fig. 2.
Based on Eq. (1), we theoretically investigate thedynamics of the coherent gas. In order to compare as close
as possible the theory to our experimental results, we
precisely account for the experimental sequence by performing real-time evolution (RTE) starting from the ground stateof Eq. (1) at as ¼ 67a0 with N ¼ 1.2 × 105 atoms. We
simulate a linear ramp in as from 67a0 to a variable final
value of as in tr , followed by a compression of the axial
trap from ν∥ ¼ 17.3 to 21 Hz during 8 ms. We then record
the axial
width from the standard deviation of nðrÞ,pﬃﬃﬃﬃﬃﬃﬃﬃ
σ y ¼ hy2 i, as a function of the subsequent holding time.
The evolution of σ y is well fitted by a sinusoidal function,

PHYS. REV. X 6, 041039 (2016)

whose frequency constitutes our theoretical predictionof νaxial .
In Fig. 3, we present our calculations with and without
the LHY term. The MF simulations reveal a criticalscattering length ac < add below which the system collapses, thus ruling out the collective mode excitation picture
for as < ac. This is in qualitative disagreement withthe experimental observations. Moreover, for decreasingas ≥ ac , the MF predictions of νaxial are sizably shiftedcompared to our measurements. In contrast, the experimentshows an excellent match with the theory including the
LHY term, thus ruling out the MF scenario and demonstrating the crucial role played by QFs in stabilizing thesystem. Then, QFs qualitative modify the phase diagramand drive the formation of a special coherent state, namely,a single macrodroplet [20–23]. The lowering of ac ¼ 57a0
found in Fig. 3(b) compared to Fig. 3(c) (ac ¼ 64a0 ) arises
from the more stringent interplay between QFs and 3B
losses within this longer ramp, both mechanisms being able
to drive the system out of the instability region.
VI. LOSS DYNAMICS
To further investigate the respective roles of 3B losses
and QFs, we study the time evolution of the atom numberof both the central core (N core ) and thermal (N th ) components along the BEC-to-droplet crossover. Since in thedroplet regime the core density ncore ðrÞ dramaticallyincreases, 3B losses are expected to play an importantrole even for moderate and low values of L3 [23].
Notwithstanding, 3B losses and QFs exhibit different power
dependencies on nðrÞ [see Eq. (1)] and, thus, the atom-loss
dynamics should disclose their competition: while QFs tend
to stabilize a high-density state, namely the droplet, 3B
losses favor lower densities.
Figures 4(a) and 4(b) show N core and N th , extracted fromthe double-Gaussian fit as a function of as after a nonadiabatic [Fig. 4(a)] and adiabatic [Fig. 4(b)] change of as .
Both cases show a similar evolution. When lowering as ,N core is first constant for as > add, then shows a sharp drop
starting around as ∼ add , and finally curves up for lower as.
We note that in the adiabatic case, N core decreases faster ascompared to the nonadiabatic one and finally saturates
around 7 × 103 at lower as. We attribute these to the longer
timing involved, and we observe a similar trend as well as asimilar saturation value for longer th [see, e.g., Fig. 4(c)].
Remarkably, N th remains mainly unaltered over thewhole range of as and the whole system does not show
any appreciable heating. This suggests that the condensed
atoms, which are ejected from in the core, leave the trap
instead of being transferred to the thermal component,
confirming a picture in which the thermal and the condensed component have uncoupled dynamics.
We now compare the experiment with the theory, which,as previously, precisely accounts for the experimental
sequence and its timing by performing RTE along041039-5

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)

εdd
1.65

1.10

0.94

t hold (ms)
0.82

0.73

0.66

10

5

10

100

1000

(a)

(c)

Ncore

Atom number

105

1.32

104

104

as = 50a0
as = 57a0
as = 65a040(b)n (1020 m-3)

Atom number

10

5

(d)

20

10
8
6

104

4
th = 4 ms
th = 16 ms

40

50

60

70

as (a0)

80

90

100

2
40

50

60

70

80

as (a0)

FIG. 4. Lifetime and in situ density of the high-density core. (a),(b) Measured N core (squares) and N th (circles) versus as after (a) a
nonadiabatic (tr ¼ 10 ms, th ¼ 8 ms) and (b) an adiabatic(tr ¼ 45 ms, th ¼ 0 ms) ramp. The data show a better agreementwith the theory with the LHY term (solid line) as compared to theMF theory (dashed line). (c) Time decay of N core for as ¼ 65a0
(triangles), 57a0 (circles), and 50a0 (squares) after quenching as(tr ¼ 10 ms). We fit a double exponential function to the data(solid lines) [30]. (d) From the fit, we deduct the mean in situ
density of the core n̄ (see text) for th ¼ 4 ms (triangles) and16 ms (squares) and as a function of as . The error bars include thestatistical errors on the fit and on L3 . The solid lines show resultsof the RTE including the LHY correction for th ¼ 0 ms (red),th ¼ 25 ms (blue).

Eq.
R (1).3 Here, we compute the final atom number N ¼
nðrÞd r as a function of as with and without the LHY
term. Remarkably, the observed evolution of N core is very
well reproduced by our beyond-MF calculations (solidlines), whereas in the absence of the LHY stabilization, thecalculations predict losses in the condensed core to occur atvalues of as too large compared to the measured ones; see
Figs. 4(a) and 4(b).
The observed evolution of N core is well reproduced byour beyond-MF calculations (solid lines). The agreement isparticularly remarkable for the quench [Fig. 4(a)] while it isslightly degraded in the adiabatic ramp [Fig. 4(b)], with anoverestimation of the remaining N core at small as . This canbe explained by noting that, due to the longer time during
which the losses set in, a more acute importance is given to
L3 , and by considering the effects of QFs on its value.
Indeed, it is of interest to note that many-body effects
modify the 3B correlation function g3 [44], leading to an
enhanced loss rate. This then justifies the larger predicted
N core in our simulation based on the simple noninteracting

value g3 ¼ 1 [30] compared to the experiment (we estimate
g3 ∼ 1.3 for our typical parameters), and the increased
discrepancy with decreasing as , where QFs are doomed toprevail. In contrast, the MF calculations deviate from theexperiment with enhanced losses in the as ∼ add region. We
note that the abrupt and high saturation of N core at as < add ,distinct from the experimental observations, is a signatureof the collapse, reestablishing lower density in the gas via
fast “explosive” dynamics.
Finally, we investigate the in-trap time evolution of N core
after quenching as in the droplet regime; see Fig. 4(c). Our
measurements reveal three different time scales for thelosses. At very short th (≈0–3.5 ms), N core is roughly
constant, which we attribute to the time needed for thehigh-density state to develop. It follows a fast decay(≈3.5–25 ms), in which the atoms are ejected from thehigh-density core via 3B losses, and witnesses the formation of a high-density coherent state. The steepness of
this fast decay appears to critically depend on as , with amarked acceleration below the MF instability threshold.
Then, the loss dynamics substantially slows down(≈25–1000 ms) while a coherent central core is still visible
in the density profile (with N core ∼ 104 atoms).
From the loss curves [Fig. 4(c)] and using the general 3B
2
loss relation ð1=N core ÞðdN core =dth Þ ¼ −L
3 n̄ , we are able
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
to extract the mean in situ density n̄ ¼ hncore ðrÞ2 i of thehigh-density component in the BEC-to-droplet crossover.
Here, we estimate N core ðth Þ and ðdN core =dth Þ from anempirical fit to our data and compute n̄ using an independent measurement of the 3B loss coefficient L3 [30].
Figure 4(d) shows our results for th ¼ 4, 16 ms. We
observe a prominent increase of n̄ across the thresholdas ∼ add , and a surviving high-density state deep into the
MF instability regime.
The formation of the droplet state is particularly visible
for the th ¼4ms case. Here, n̄ grows from 6.2ð9Þ ×1020 m−3 at as ¼67a0 to a maximum of 35ð7Þ×1020 m−3
at as ¼ 57a0 , while it is slightly reduced to ∼24 × 1020 m−3
at as ∼ 46a0 . This direct estimate of n̄ advocates theactivation of the LHY term when lowering as ; additionally,
its magnitude as well as its evolution are in good agreementwith our simulations including the LHY correction.
Our results together with the good agreement between
theory and experiments provide an alternative confirmationof the central role of beyond-mean-field physics. The
lifetime of the high-density core reveals, on the one hand,the activation of the LHY term and the crossover toward a
dense droplet state, and on the other hand, the counteractingrole of 3B losses in regulating the maximum density in thedroplet regime.
VII. EXPANSION DYNAMICS
Besides their dissimilar stability diagram, collective
excitations, and density distribution, a dilute BEC in the041039-6

QUANTUM-FLUCTUATION-DRIVEN CROSSOVER FROM A …

shows vx in an εdd range from 0.7 to 1.5. When the systemapproaches the droplet regime with decreasing scattering
length (as < add ), vx undergoes a strong reduction anddrops to a minimum equal to vx ¼ 0.40ð2Þ μm=ms at about
56a0 (εdd ∼ 1.17). For further lowering of as , vx starts toincrease again. A similar behavior is observed for vz. We
note that only the high-density component reveals thisintriguing dependency on as , whereas the thermal partshows an almost constant expansion velocity [45].
Considering the fit-free character of our simulations aswell as the experimental challenge of accurately estimating
the expansion velocities [46], we conclude that our observations agree well with the theory predictions including the
LHY term; see solid line in Fig. 5(b). The TOF evolution is
calculated using a multigrid numerical scheme [30]. We
record the evolution of σ η with tTOF and extract thecorresponding expansion velocities from the asymptotic
behavior of dσ η =dtTOF . Our simulations show a slowingdown with a minimum of vx ¼ 0.32 μm=ms at as ∼ 56a0
(εdd ∼ 1.17), followed by an increase at lower as. In
contrast, calculations in the absence of beyond-MF corrections fail to reproduce the experimental data. Here, thevelocity is first slightly more reduced above the MF
instability threshold εdd ∼ 1 than is expected with LHY

30

10

12

14

16

18

tTOF (ms)
20 22

(a)

24

26

28

30

32

σx (μm)

93a0
64a0

20

55.5a0
10

εdd
1.65

1.32

1.10

50

60

0.94

0.82

0.73

0.66

70

80

90

100

2
expansion velocities (μm/ms)

MF regime and a quantum droplet are also expected to
exhibit a markedly different expansion dynamics. While thefirst is confined by an external trapping potential and thus
freely expands in its absence, a droplet state is self-bound(SB) by its underlying interaction in analogy with theHe-droplet case [20–23]. As in our previous discussions,the evolution from a trap-bound to a self-bound solution is
expected to be regulated by the interplay between QFs and
3B loss processes.
We investigate the expansion dynamics of our system forvarious as . To preserve the high density of the coherent
component, our measurements focus on short time scaleswith tr ¼ 10 ms and th ¼ 5 ms. After preparing the systemat the desired as , we abruptly switch off the optical dipole
trap, let the gas expand for a variable tTOF , and probe thecloud width using the ∥ imaging. We fit the observed
density distribution to a double-Gaussian function, as
previously described. To extract the width σ η of thehigh-density
core (ncore ), we compute the second momentsRσ 2η ¼ η2 ncore ðrÞdr along η ¼ x, z, where ncore is extracted
from the double-Gaussian fit. Figure 5(a) exemplifies the
TOF evolution of σ η¼x at as ¼ 93a0 , 64a0 , and 55.5a0 .
When entering the εdd > 1 regime, atoms in the highdensity core exhibit a marked slowing-down of the expansion dynamics, which cannot be explained within the MF
approach.
To systematically study this effect, we repeat the abovemeasurements for different values of as (i.e., εdd ). Fromσ η ðtTOF Þ, we extract the value of the expansion velocity vη
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
by fitting the data to σ η ðtTOF Þ¼ σ 2η;0 þv2η t2TOF . Figure 5(b)PHYS. REV. X 6, 041039 (2016)

1

(b)
40

a s (a 0 )FIG. 5. Expansion dynamics across the BEC-to-droplet crossover. (a) TOF evolution of the width σ x of the high-density
component for as ¼ 93a0 (squares), 64a0 (circles), 55.5a0(triangles). qThelines ﬃ are fit to the data using
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
σ x ðtTOF Þ ¼

σ 2x;0 þ v2x t2TOF , from which we extract vx . (b) vx

as a function of as (squares). For comparison, the as -independent
expansion velocities of the thermal component are also shown(circles). The experimental data are in very good agreement with
our parameter-free theory from RTE simulations including the
LHY term (solid line) and rule out the MF scenario (dotted line).
For clarity, we show only vx ; similar results are found for vz.

corrections, it then already increases at this threshold. The
first point relies on the trivial slowing-down of a BEC
whose mean repulsion energy is weakened (by reducing as
or decreasing its population N core ). The second point
reveals a collapsing behavior that gives rise to an explosive
evolution of the density profile. The minimal velocity isfound here to be vx ¼ 0.56 μm=ms at as ¼ 68a0 , which isa much higher value than both our experimental results andour theory predictions including the LHY correction.
The expansion behavior can be qualitatively well understood considering the so-called released, or internal, energy
ER . This is the energy of the system when subtracting theenergy related to the confinement [1]. In the MF scenario,
ER > 0, as long as the ground state is stable. The BEC
expands ballistically and vη decreases for decreasing as and
N. In the unstable regime, the expansion velocity depends
crucially on the value of th at which the trap is switched off
due to the occurrence of an in situ collapse dynamic. On thecontrary, in the presence of QF, a stable ground state always
exists. The sharp variability in th is expected to be041039-7

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)suppressed. Assuming a fixed N core (i.e., no 3B losses), one
can show that ER decreases with decreasing as and eventually reaches ER < 0 for as < aSB, marking the onset of the
SB solution (e.g., aSB ¼ 56a0 for N ¼ 1.2 × 105 ) [30].
However, in stark contrast to the MF case, ER increases
with decreasing N core in the droplet regime. We note that aSB
is then shifted to lower values when N core gets reduced by 3B
losses, thus affecting the lifetime of the self-bound solution.
The existence of a minimal expansion velocity is thus a
direct consequence of the competition between the decreaseof ER for decreasing as at a fixed N core and the increase of ER
for decreasing N core in the droplet regime. In the crossover
regime, the system smoothly evolves towards a fully selfbound state (vη ¼ 0) until 3B losses, occurring in the trap orin the initial phase of the expansion, set in to unbind thesystem and to reduce the lifetime of the droplet state.
VIII. CONCLUSION
In summary, we demonstrate the existence of the crossover from a dilute BEC to a quantum droplet state driven by
QFs. Our experiments not only demonstrate that LHY
stabilization is a general feature of strongly dipolar gases,
but also thoroughly investigate the driving role of QFs in
dictating the system properties, in particular, its collectivemode, its atom losses, and expansion dynamics. This clear
and quantitative demonstration of the impact of QFs in
dipolar gases intrinsically relies on our unique and precise
knowledge of as that alone enables a direct comparison to aparameter-free theory, which is based on a generalized GPEwith LHY correction. Our combined experimental andtheoretical results ultimately offer an experimental validation of the modeling proposed in Ref. [20] and thus of thelatter results of Refs. [21–23].
ACKNOWLEDGEMENT
We thank M. Baranov, R. van Bijnen, R. N. Bisset, B.
Blakie, E. Demler, R. Grimm, T. Pfau, and the Stuttgart Dy
team for multiple useful discussions. We thank G. Faraoni
for her support in the final stage of the experiment. The
Innsbruck group thanks the European Research Council
within the ERC Consolidator Grant RARE No. 681432.
L. C. is supported within the Marie Curie Project
DIPPHASE No. 706809 of the European Commission.
The Hannover group acknowledges the support of the DFG(RTG 1729). Both groups acknowledge the support of the
DFG/FWF (FOR 2247).
Note added in the proof.—A related
M. Schmitt et al. recently appeared [47].

work

by

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[42] We note that the use of a generalized GPE, valid in the TF
regime, can be extended beyond this regime as long as the
LHY correction is dominated by short-wavelength excitations, whose momentum q fulfills qR ≫ 1, where R is thesize of the condensate along the direction of q. As discussed
in detail in Refs. [20,22], this is the case for large droplet
and εdd close to 1. Our experiments, performed in cigarshaped traps with relatively large BECs, offer privileged
conditions to be compared to generalized GPE predictions(in particular, enhanced compared to Refs. [15,16]). Indeed,
we quantitatively estimate the validity of this approximationin our study
by introducing ﬃ an anisotropic cutoff,pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qc ðθÞ ¼ qy cos2 ðθÞ þ K 2 sin2 ðθÞ, with K the aspect ratioof the BEC, assuming here an isotropic geometry around thedipole axis y, and θ denoting the angle ðy; qÞ [22]. For
typical experimental parameters, e.g., as ¼ 58a0 , N ¼
5 × 104 , we find an axial radius Ry ¼ 13 μm and
K ¼ 7.5. Introducing a typical cutoff value qy Ry ¼ 25,
the cutoff LHY correction [i.e., taking momentaq > qc ðθÞ] equals ∼0.8 of the value calculated in homogeneous space, demonstrating that short-wavelength excitations indeed have a dominant contribution and validating
our approach. We note that, when decreasing as , longwavelength excitations and unstable modes come to play amore significant role, and for this reason we consider our
simulation reliable down to as ≥ 50a0 .
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[45] We observe a 14(9)% increase of vth over the as range whendecreasing as that we attribute to fit artifacts. First, thedouble Gaussian fit does not properly account either for thecoherent or for the thermal parts of the gas which may resultin cross talk in the fit of these components. Second, flaws in
the partition in two components may occur, especially for

041039-9

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)

the configuration where both components have similar sizes(short TOF). We note that at long tTOF the thermal sizes arenearly constant, while a trend in their evolution with asappears at short tTOF . Because they are less subject to
artifacts, the long tTOF are expected to more reliably estimate
v and we hence conclude that the observed slight heating is,
at least mainly, an artifact.
[46] The observed slight overall shift of the experimental datacompared to the theoretical predictions may be attributed toexperimental artifacts coming from (i) the restricted range oftTOF experimentally accessible, (ii) artifacts of the expansion
fit arising, in particular, from the interplay of the two fit
parameters, the empirical expression of the fit, or the finite
range of expansion time [cf. (i)], or (iii) artifacts comingfrom the empirical double Gaussian fit used here to extract
σ. This may lead to misassessment of the size of thecoherent part and of its evolution with tTOF .
[47] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, T.
Pfau, Self-Bound Droplets of a Dilute Magnetic Quantum
Liquid, Nature (London) 539, 259 (2016).

041039-10

Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a
macro-droplet in a dipolar quantum fluid: Supplementary Material
L. Chomaz1 , S. Baier1 , D. Petter1 , M. J. Mark1,2 , F. Wächtler3 , L. Santos3 , F. Ferlaino1,2∗1Institut für Experimentalphysik,Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany(Dated: October 18, 2016)
PACS numbers: 30.0, 67.85.-d, 03.75.Kk

Appendix: Supplementary material
1.

Bose-Einstein condensation of 166 Er

We prepare an ultracold gas of the 166 Er isotope following a similar trapping and cooling scheme as the one
employed for 168 Er [1, 2]. We load a crossed-ODT from anarrow-line MOT of 3 × 107 166 Er atoms at about 10µK.
At the end of the MOT sequence, the atoms are automatically spin-polarized in their lowest Zeeman sub-level [2].
The dipole orientation follows the one of the external
applied magnetic field, B. In our experiment, the latter is controlled by independent tuning of the components Bx , By , Bz along the experimental coordinate system (x, y, z), as defined in Fig. 1 (lower inset).
The ODT results from the crossing at their foci of two
red-detuned laser beams at a wavelength of 1064 nm. One
beam propagates horizontally along the y-axis, and theother propagates vertically and nearly collinear to the zaxis. The z-beam has a maximum power of 10 W and an(z)(z)
elliptical profile defined by its waists of (wx , wy ) =(110, 55) µm. The y-beam has a maximum power of(y)
27 W, a vertical waist wz = 18 µm, and a tunable hori(y)zontal waist, wx . The latter can be conveniently tuned(y)(y)
from 1.57 wz to 15 wz by time averaging the frequencyof the first-order deflection of an Acousto-Optic Modulator (AOM). This scanning scheme enables both an efficient loading of the MOT into the ODT (> 30% of
the atoms are loaded) and an adiabatic and controlled
tuning of the trap aspect ratio λ over a broad range.
We achieve Bose-Einstein condensation of 166 Er atoms
by means of evaporative cooling in the crossed ODT
at |B| = Bz = 1.9 G (as = 80(2) a0 ). Typically, we
first rapidly (in 600 ms) reduce the power and aspect ra(y)(y)
tio wx /wz of the y-beam from 24 W to 4 W and 10
to 1.6, respectively. We further decrease the power ofthe y-beam from 4 W to 0.3 W in 3 s in an exponential
manner and then exponentially increase the aspect ratio(y)(y)
wx /wz from 1.6 to 8 in 2.5 s. The final trap frequencies are typically of (νx ; νy ; νz ) ∼ (40; 40; 180) Hz. We finally obtain BECs of typically N = 1.2 × 105 atoms with
∗ Francesca.Ferlaino@uibk.ac.at

more than 80% condensed fraction and a temperature
T ∼ 70 nK. We typically measure N and the condensed
fraction from a bimodal fit of the 2D column density distribution measured along //-imaging with tToF = 27 ms.
T is extracted from the evolution of the thermal size ofthe bimodal fit with tToF varying from 14 to 28 ms.

2.

Experimental setup and axes

In our setup we define the orthonormal (x, y, z)coordinate system in the following way: the vertical axis
z is aligned with gravity and the y axis with the horizontal ODT beam; see Fig. 1. The polarizing magnetic
field is created by three orthogonal pairs of coils. Thesepairs of coils define an orthonormal (X, Y, Z)-coordinate
system with Z = z and (X, Y ) rotated by a small angle
θ as compared to (x, y). The magnetic field components
BX , BY , BZ , each created by each pair of coils, can becontrolled independently. We estimate θ to be about 15oby sensing directly the atomic cloud, as its dipolar character makes it sensitive both to the trap geometry andto the magnetic field direction.
The small tilt θ between the dipoles and y causes asmall reduction of the mean DDI energy and corresponding small shifts of the expected characteristics comparedto the one predicted for θ = 0: the MF collapse thresholdshould appear at a lower as and, for a given as , νaxial /ν//and vη are shifted respectively down and up. We haveexperimentally evaluated thep shift of νaxial deep in the
2 + B 2 = 2 G) to be ofstable BEC regime (|B| = BXY
the order of 2% and in the droplet regime to be about 10
to 15%, as confirmed also by our theoretical predictions.
Finally, we also note a tilt between the//-imaging beam
and our reference frame, corresponding to an angle of
o
im
o
θ/im
/,0 ∼ 28 compared to y and θ// = 13 compared to Y
in the xy-plane. The ⊥-imaging axis is tilted by ∼ 15o ,mainly in the xz-plane. Such tilts shift the observed size
compared to the ideal case of imaging along and perpendicular to the dipoles. Such an effect is not expected
to impact the measurement of the collective frequencies,
whereas it might bring a systematic shift of vx becauseof a mixed projection of vX and vY , the two first velocity components in the (X, Y, Z)-coordinate system, which
are respectively perpendicular and along the dipoles.

2
In the theoretical calculations presented in the main
text (Eq. (1), RTE and Gaussian ansatz), for simplicity,
we do not account for these angles, whose effects are estimated to be smaller than our systematics.

3. Precision measurements of the as -to-B
conversion in a three-dimensional optical lattice

Precise determination of the as -to-B conversion is adelicate issue, especially in the case of complex species,
such as Er, for which comprehensive multi-channel calculations are still out of reach and the knowledge ofas should thus rely on experimental investigations. We
perform lattice modulation spectroscopy in a threedimensional optical lattice. From the measurements ofthe energy gap in the Mott insulator state we extractas (B). Our lattice experiments focuses in the region of
low B-field [0, 2.5 G] and are based on a lattice setup
and procedure similar to the one described in Ref. [3].
In brief, after producing the BEC, we load the atoms ina three-dimensional optical lattice by exponentially increasing the lattice-beam power in 150 ms. The typical
final depths are (sx , sy , sz ) = (20, 20, 100), given in unitsof the respective recoil energies h × (4.2, 4.2, 1.05) kHz.
At these lattice depths, the gas is in the Mott insulator
state. We then vary B = (0, 0, Bz ) to the desired value
by rapidly changing Bz in 2 ms, either just before or justafter loading the lattice. We use the latter option for thesmallest B values at which L3 is enhanced because of its
proximity to the near-zero-field resonance. In this case,
we further hold 20 ms to make sure the magnetic field isfully established before performing the modulation.
To perform spectroscopy measurements, we sinusoidally modulate sy at a variable frequency νm for 90 ms
with a peak-to-peak amplitude of about 30%. Finally, we
ramp down the lattice depths to zero in 150 ms, and measure the recovered condensed fraction as a function of νmfrom //-imaging ToF picture. For the smallest B values
considered, we also ramp B back to 2 G in 2 ms before
switching off the lattice-beams in order to again minimize
3B loss effects.
When varying νm , we observe a resonant depletion of
the condensate due to particle-hole excitations. The resonance condition in the Mott-insulator regime is givenbyhνex = Us + Udd − Vdd .

(A.1)

Here Us , Udd and Vdd are respectively the on-site contact interaction, the on-site dipolar interaction and thenearest-neighbor dipolar interaction along y in the corresponding extended Bose-Hubbard model. Udd and Vdd
can be accurately predicted from the knowledge of thelattice depths and dipole orientation and in our typicalexperimental condition, they are equal to h × −344.8 Hz
and h × 31.5 Hz respectively. By subtracting the theoretical dipolar contributions to the measured frequency, we

extract Us , which is directly proportional to the scattering length as . A precise mapping of as in the ultralow
B-field region is then obtained by repeating the abovemeasurements at various B values; see Fig. 1.
In the low B-field region shown in Fig.1, our lattice
spectroscopy reveals the presence of two FRs, one locatedat about zero B field and the other one at about 3 G.
The existence and position of these two FRs agree with
our Feshbach spectroscopy measurements performed inan harmonically trapped thermal cloud, where the maxima in 3B losses approximately pinpoint the resonancepositions. In this measurement, further FRs are identified at 4.1 G and 5 G.
The scattering length of 166 Er can be parametrized by
the following simple expression [4]
"
#
4
X
∆Bi
as (B) = abg (B) 1 +(A.2)
B − Bii=1
in which the specific positions (Bi ) and widths (∆Bi )of the two first resonances as well as the backgroundscattering length are obtained from a fit to our lattice spectroscopy measurement. From the fit, we obtain B1 = 48(45) mG, ∆B1 = 39(20) mG, B2 = 3.0(1) G,∆B2 = 110(35) mG. The B-dependent background scattering length abg (B) accounts for the overlapping resonances and reads as abg (B) = 62(4) + kB with k =5.8(1.2) a0 /G. We also account for the small effect of
the two next resonances, whose positions B3 , B4 andwidths ∆B3 , ∆B4 are fixed to their estimates from theloss-spectroscopy measurements to 10 mG. We check thatthe precise values of this parameters has little effect on
our empirical description along Eq. A.2 of as in the Brange of interest here, namely [0, 2.5] G.

4.

Ramps in scattering length

Our measurements rely on controlled variations of thescattering length as (B). In our experiments, we either
adiabatically change as using tr = 45 ms or we quench itusing tr = 10 ms. The adiabatic condition for as reads as
1 das≤ min (νx , νy , νz ) = ν// for λ  1
as dt

(A.3)

As shown in Fig. S1, we use two different types of time
variations of B and thus of as : (i) a simple linear ramp
in B and (ii) we design a specific B(t) variation in ordersto minimize the adiabaticity parameter a1s da
dt /ν//. The
resulting as shows an exponential-type variation with t.
The adiabaticity condition of Eq. (A.3) is more stringent
for lower as . For ramping down to as = 48 a0 , we findthat (i) verifies Eq. (A.3) for tr & 100 ms and (ii) fortr & 20 ms. Data from Figs. 1, 3 and 4 (a, c-d) (resp.
Figs. 2, 4 (b) and 5) use ramp (i) (resp. (ii)).
For our theoretical description, we use a linear changeof as (t), similar to case (ii).

3
5.

Determination of the three-body recombination
rate coefficient

Since three-body inelastic losses play a crucial rolein the many-body dynamics and lifetime of the droplet
state, we run a dedicated set of measurements to determine L3 . We first prepare a non-degenerate thermal
sample of Er atoms at T = 490(10) nK in an harmonic
trap. We then record the decay of the atom number,
Nth , as a function of th in a range from 0 to 1 s. Nth
is obtained from a Gaussian fit to the measured density
distribution.
To fit the time evolution of Nth , we use the integratedth 2th
3B rate equation, which reads as N1th dN
dth = −L3 hn i.
Here, hn2 i is the mean square density on the cloud. To
describe the scaling of hn2 i with Nth , we use its prediction for an ideal gas at thermal equilibrium at T whosestate occupancies follow Boltzmann law and take into
account the anti-evaporation effect [5]. Then Lth
3 is extracted from a fit of Nth (th ) along:
Nth (th ) =N0(1 + 3γ0 N02 th )1/3

(a)

65

as ( a 0 )

60
55
50

linear ramp in B (i)polynomial ramp in B (ii)
RTE

45

Lth
3 =

√

27 γ0



kB T
2πmν̄



(A.5)

with kB the Boltzmann constant and ν̄ = (νx νy νz )1/3 .
We account for the as -dependence of Lth
3 near a FR byrepeating the measurement at different B. We check thatthe measured Lth
3 does not depend on the B orientationand measure its B-dependency using |B| = By and for B
varying in 0.1 to 1.9 G. Note that, due to the bosonization effect, the L3 in a quantum degenerate bosonic gas
is equal to Lth
3 /3!. Figure S2 shows L3 in a quantum
degenerate bosonic gas of 166 Er as a function of as using
the measured as -to-B conversion. Despite the existenceof many coupled molecular potential in Er, we measure a
low L3 (as ), comparable to the typical values reported inalkali atoms. L3 (as ) varies between 1.7(3)×10−40 m6 /s at
as = 18(17) a0 and 3.2(3) × 10−42 m6 /s at as = 80(2) a0 .
6.

(A.4)

where N0 is the atom number at th = 0 ms and γ0 is
70defined via the relation

Time-of-Flight measurements

For our ToF measurements, we abruptly extinguish the
ODT in about 100 µs. In order to accurately image ourgas while minimally influencing its dynamics during the
expansion, we split the ToF in two parts. During a firstpart, lasting tToF − tB , B remains unchanged and the
dynamics occur at the original as and dipole orientation.
At time tB before the image is taken, B is modified bothin amplitude and in direction, in order to correctly set thequantization axis for the imaging light to be σ− polarized.
The amplitude |B| of the imaging field is chosen constant2, 5 (b)
1

m /s)

1,5
6

1,0

L (10

- 40

t r / a s d a s /dt

2,0

0,0

3

0,5

0, 00

0,25

0,50
t/t r

0,75

0,1

1, 00

FIG. S1.
Predicted evolution of the scattering length ass(upper panel) and the adiabaticity parameter a1s da/ν// (lowerdtpanel) over the ramp from B = 0.8 G to the extreme Bf =0.17 G (as = 45 a0 ). The parameters are shown as a functionof the normalized time t/tr and the universal variation of theadiabaticity parameter is obtained from normalizing to ν// tr .
We show the cases of three different ramps: a linear ramp in
B (dashed red line) which is used in a first set of experiments(i), a polynomial ramp in B (solid blue line) which is used ina second set of experiments (ii), a linear ramp in as (dottedgreen line) which is used in the simulation (RTE).

0,01

0

10

20

30

40

50

60

70

80

a s (a 0)

FIG. S2.
Measured 3B recombination rate coefficient L3of a quantum degenerate gas of 166 Er as a function of as for
B varying from 0.1 to 1.9 G. We extract L3 from the measurement of Lth
3 on a thermal gas at T = 490(10) nK using
Boltzmann law and taking into account anti-evaporation effect; see text.

4and equal to 0.31 G for all as considered. tB is set tobe 12 ms for //-imaging and 15 ms for ⊥-imaging where
the change in B is more drastic for the typical dipole
orientation (Y ) used in this experiment.
We note that our resolution limit for both //- and ⊥imaging is estimated to be & 3 µm. Moreover the effective pixel sizes are set to 8.4 µm and 3.1 µm in our setup.
These limit the size of the structure we are able to observe as well as the minimal tToF we can use, which istypically tToF ≥ 16 ms.
Averaging experimental density distribution

In order to obtain a better image quality and resolution, we typically average four experimental absorption
images taken in the same condition and with the same
experimental series (i.e. within less than a few hours interval). In order to average the images we first definea region of interest (ROI) of typically 300 µm × 300 µm
containing the cloud shadow and translate the ROI to superimpose the cloud centers. To estimate the translation
amplitudes for each individual image, we use the centerfrom a simple Gaussian fit to the 2D density distribution
ROIs. In this averaging process, we use a sub-grid resolution of 1/10 of a pixel to more accurately superimpose
the centers. We fit the averaged density distribution after binning back to the original resolution. We note thatfits on the sub-pixel resolved images give similar results.

8.

9.

Bimodal fits of the density distribution in//-imaging.

To quantitatively analyse the experimental column
density distribution imaged along //-imaging, we perform
bimodal fits on the 2D averaged profiles n//(x, z). The
bimodal fits are made of the sum of two peak distributions, describing respectively a high-density, coherent part and a thermal incoherent background. To account for the change of the profile of the density distribution across the BEC-to-droplet crossover, we use
two types of fitting functions ffit (x, z): (i) A sum of a
MF-TF and a Gaussian distribution, which account respectively for the coherent and the thermal part. For

Extracting the frequency of the collective modes

As stated in the main text, we focus on the axial mode,
which reveals itself by a collective oscillation of the axial
size R// of the BEC, along with smaller amplitude oscillations of the radial size in phase opposition. We extract its
frequency νaxial by studying the larger amplitude oscillation of R//. For this, we probe the ToF density distribution of the gas with ⊥-imaging after a ToF of 30 ms. We
focus on the sizes of the central, high-density componentof the cloud, which we study as a function of th for different as . We note that the precise shape of the column
density profile is expected to change as a function of as
and this in a different way for the two axis x and y underobservation in ⊥-imaging. This complicates the analysis, in particular compared to the //-imaging where bothaxes are nearly equivalent. Here, we extract the sizes of
the central component, using various methods and selectthe most reliable method according to as . Typically, we
use a bimodal MF TF plus Gauss fit for as ≥ 57 a0 . For
as ≤ 57 a0 , we select a central region in the cloud and perform a simple Gaussian fit on it. Such a determination
should give access to the variations, if not to its absolute value, of R// with th at fixed as and thus enables todetermine νaxial . In particular, we have checked that forlarge as , the two fits (Gaussian on a central region and
MF-TF plus gaussian bimodal fits) give very similar and

1,000
0,998
0,996
0,994

residu7.

compatible values of νaxial . For example for as = 93 a0
we find νaxial = 30.65(4) Hz and 30.55(6) Hz for the MFTF and Gaussian fit respectively. For as = 60 a0 , we findrespectively νaxial = 36.1(3) Hz and 36.7(3) Hz.
To fit νaxial , we use a damped-sine function of th . Typically we fit the evolution of R// for th varying from 0 to
few hundreds of ms, depending on the damping rate observed. The upper value of th used is never less than
150 ms such that at least 4 to 5 oscillations are observed
and fitted. We also note that for our shortest tr = 10 ms
we typically discard the first 4 ms of the evolution in order to ensure that the magnetic field is safely stabilized
at its target value.

0,992
0,990
0,988
0,986

d o u b le g a u s s ia n fit
M F - T F p lu s g a u s s ia n fit

0,984
0,982
40

50

60

70

80

90

100

as(a0)

FIG. S3. Residues of the two bimodal fits used in our analysis of the //-images as a function of as for tr = 10 ms andth = 6 ms: MF-TF plus Gauss fit (blue triangle) and double
Gaussian fit (red square).

5
We note that Ncore (th ) is well accounted for by a double exponential decay evolution of respective amplitude
N0 (1 − p) and N0 p. N0 is the initial atom number. Eachdecay corresponds to a different time constant, respectively tfast and tslow , and starts after a different delaytime, respectively td and tD . We fix tD = td + 2tslow .
For tr = 10 ms td is approximately constant and equal to3.65 ms. The absence of evolution for th < td indicatesthat the magnetic field may not have reached the target
value in the first ms. For tr = 45 ms, td can be set to
0. tfast , tslow and p (and N0 ) depend on as , typically decreasing with it. The evolution is illustrated in Fig. S4
for tr = 10 ms.

11.

10.

Gaussian Ansatz including L3

A good qualitative (and to a large extent quantitative)
insight in the physics of dipolar condensates in the presence of LHY stabilization may be gained from a simplified

ε dd

2000

1,32

1,10

0,94

0,82

50

60
as (a 0)7080

t slow (ms)

1500
1000
500
0

t fast (ms)

100

10

1
1
0,75
0,5

p

the integrated column density, the MF TF distribution
3/2

2
2
0)
0)
− (z−z[6].(ii) The
writes fTF (x, z) = 1 − (x−x2
RxRz2
sum of two Gaussian distributions. Typically the central
Gaussian can be anisotropic with any orientation axis.
As expected, the thermal background is broader thanthe central coherent component and for the tToF considered its width is typically at least 1.6 times larger. It is
first fitted by taking out the central part of the density
distribution at radius typically smaller than 1.3 times its
width.
The quality of the bimodal
fit is evaluated by the normR2dx dz (n//(x,z)−ffit (x,z))Rof the residue ρ = 1 −. For both2dx dz n//(x,z)
distributions, it satisfies ρ > 0.98.
The evolution of ρ with as informs us further on thephysical properties our gas, as stressed in the main text inSec. III. Fig. S3 shows the extensive variations of ρ, thus
completing the data of Fig. 2 and the physical picture described there. First, as stated in Sec. III, by considering
the agreement of MF-TF plus Gauss distribution, one isable to identify a deviation from the MF-TF behavior.
Indeed ρ shows here a marked decrease when lowering
as below add from 0.997 to 0.985 and a further saturation around this value. On the contrary, the residue ofthe double Gaussian fit increases when lowering as below add before decreasing again for as below 57 a0 , andremains above 0.99 over the whole range of as .
However, we point out that we are not able to distinguish, from the mere density profiles, between a MF
or beyond-MF character of the deviation from MF-TF
regime. Even though theoretical studies point out thedifference in density profile expected by the arising of
beyond-MF effects compared to the simple reduction of
the MF interactions [7–10], bimodal fit agreement does
not enable a reliable distinction. Indeed, the difference isminute and surely blurred by the presence of a thermal
fraction. Hence, even though a more complex profile isexpected, the double Gaussian fit agrees particularly wellto our data and is here used to extract global propertiesof the system for further studies. We note that such a
procedure has been also adopted in [11, 12] for the arrayof micro droplets, and the Gaussian ansatz has also beenused in theoretical works [8, 9], in which the Gaussian
solution shows a good approximation as compared to themuch heavier numerical solution. To further prove therole of QFs in our paper, we extensively account for emblematic global properties that characterize the droplet
physics (see Figs. 3, 4and 5).

0.25

Describing the atom number decay

In Figure 4 (c) and in the main text we have briefly
described the evolution of the atom number in the coherent part Ncore with th . There our aim was to extract
the mean density and we did not expand on the mere
description of the Ncore (th ).

0,1
0,075
0,05

FIG. S4. Fit parameters for the evolution of Ncore (th ).

6
L3
with R = π3 35/2
, and
ω̃ l̃6Gaussian ansatz of the form

1 X −2  ωη 2 2
2 P QN 3/2vη +vη + 
 32
2 η
ω̃
3 Q
η vη



PN
vy vy
Q
+
,
,
(A.13)
1 + dd F
vx vz
η vη

2

Y
p
ψ(r, t) = N (t)eiφ(t)

e

η
2
− 2w
2 +iη βη (t)η

1/2

η=x,y,zπ 1/4 wη,(A.6)

where the variational parameters are the number ofatoms N (t), the global phase φ(t), the widths wη , andthe phase curvatures βη . The Lagrangian density reads:i~∂ψ ∗ (r, t)~2∗ ∂ψ(r, t)
L=
ψ
−ψ
+
|∇ψ(r, t)|2
2
∂t
∂t
2m
2
g
+ V (r)|ψ(r, t)|2 + |ψ(r, t)|4 + gLHY |ψ(r, t)|5
2
5
Z
1
+
d3 r0 Vdd (r − r0 )|ψ(r, t)|2 |ψ(r0 , t)|2 .
(A.7)
2We insert ansatz
R (A.6) into Eq. (A.7), obtaining the Lagrangian L = d3 rL:(mX 2 2~X
β̇wη2 +ω w
L = N ~φ̇ +2 η
4 η η η

)
~2 X
1
2 2
+2
βη w η +
2m η
4wη2
)
(
g(1 + dd F (wy /wx , wy /wz ))2Q+N2(2π)3/2 η wη
( 
)
5/2
2
gLHY
5/2
+N
(A.8)
Q 3(2 ,
5
π 9/4 η wηwith1
F (κx , κz ) =
4π
"

Z π

dθ sin θ

0

Z 2π

dφ

0

#
3 cos2 θ− 1 . (A.9)κ2x cos2 φ + κ2z sin2 φ sin2 θ + cos2 θ

The variational parameters are determined from the corresponding Euler-Lagrange equations [13]:




Z
∂L
∂ψ ∗
∂ψ
d ∂L
−
= d3 r Γ+ Γ∗, (A.10)dt ∂ λ̇∂λ∂λ∂λwhere λ = N, φ, wη , βη , and Γ(r) = −i~ L23 |ψ(r)|4 ψ(r).
Introducing
the dimensionless units τ = ω̃t, wη = ˜lvη , ˜l =pQ~/mω̃, with ω̃ = ( ωη )1/3 , and after some algebra, we
obtain a close set of equations for the Gaussian widths
and the number of atoms:3RṄ = − Q 2 N 3 ,
η vη
2

4

2

7R N
2RN
v̈η = −vη  Q 4 + Q 2
v
0
0
η η
η 0 vη 0

X v̇η00

η 00 6=η

vη00(A.11)
U=

 − ∂U , (A.12)∂vη

with P =q

512 F (dd )2 a
˜ 3/2 .
π l̃ and Q = 25√5π 45 (a/l )Due to their simplicity, Eqs. (A.11) and (A.12) permit a much more flexible simulation of the exact experimental conditions and sequences compared to the obviously more exact but numerically much more cumbersome simulation of the gNLNLSE. We have checked thatthe results of the Gaussian ansatz approach are in excellent agreement both qualitative and to a large extent
also quantitative to full simulations of the gNLNLSE, inwhat concerns lowest-lying excitations, atom losses, andexpansion velocities.

12.

Self-bound droplets

The Gaussian ansatz approach allows for an intuitive
understanding of the degree of self-bound (SB) character of the system. As mentioned in the main text, a SB
solution is characterized by a negative released energy,
ER < 0. In absence of losses, we may evaluate ER bymeans of the Gaussian Ansatz for the ground-state of atrapped BEC with scattering length as and fixed N . Figure S5 shows the results for ER for different N values.
Whereas ER increases with growing N for large as , forsmall as in the droplet regime ER increases with decreasing N . For each N there is a finite scattering length aSB
such that if as ≤ aSB the droplet will be fully self-bound(vη = 0). Given its N -dependence, aSB shifts to lower
values with decreasing N . 3B losses add a time dependence on N and thus on aSB that governs the lifetimeof the droplet state. We note that for small (tr , th ) ER
may change its sign during the ToF due to atom losses,
i.e. a SB solution may unbind during the ToF. In our experiments, the interplay of losses and LHY stabilization
leads to a minimal released energy, that translates into aminimal expansion velocity as shown in the main text.

13.

Simulation of the ToF expansion using the
gNLNLSEToF expansion is simulated using the gNLNLSE bymeans of a multi-grid method, i.e. dynamically enlarging the numerical grid following the expansion. This isnecessary due to the obvious difference in length scales atthe beginning and at the end of the ToF. We note that the
precise description of the ToF dynamics is very relevant,since in contrast to standard cases, nonlinear dynamics

7
and losses here may play an important role during the
expansion, especially within the LHY stabilized regime.

8

6

4

2

ER
ω

0

-2

-4

-6

-8
50

55

60

65

70

75

80

aS / a0
FIG. S5. Released energy ER as a function of as for N =1.2 × 105 (solid), 5 × 104 (dashed), and 104 (dotted). Notethat ER < 0 indicates a SB solution.

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50

Chapter

4

The rotonic excitation spectrumof a dipolar Bose-Einsteincondensate

This Chapter focusses on one of the central results of this thesis, namely the study of theexcitation spectrum of a trapped dipolar BEC. Key concepts of the excitation spectrum
have been already introduced for the homogeneous system in Chapter 3. Here, we extend
the discussion to the case of trapped dipolar BECs. Off-resonant laser lights are commonlyused to create optical traps for neutral atoms. The shape of the trap is typically harmonicwith frequencies (ωx , ωy , ωz ). By adjusting the optical power and laser beam sizes, it ispossible to modify the trap from a spherical one (ωx = ωy = ωz ) to a pancake (ωx , ωy ≪ ωz )or a cigar (ωy ≪ ωz , ωx ,). As we will see, for the latter two cases the orientation of theatomic dipoles in the BEC plays an important role. If the dipoles are oriented along atightly confining axis, the BEC develops a so-called roton-maxon excitation spectrum along
the weaker trap axes. This excitation spectrum has striking similarities to the excitation
spectrum found in superfluid helium [Gri93].

In Sec. 4.1, I will discuss what is a roton and how a dipolar BEC may develop it in its
excitation spectrum. To understand the basic properties of the roton and its necessary
ingredients it is useful to study the analytic case of an infinitely pancake shaped BEC.
Section 4.2 introduces the dynamic structure factor (DSF), which is a useful way to visualise
the excitation spectrum. Additionally, the DSF describes the strength of the density responseof the system when it is perturbed externally with momentum, ~k, and energy, ~ω. How
this external perturbation can be described theoretically and is realised in experiments is the
topic of Bragg spectroscopy in Sec. 4.3. Here, I will also briefly review the setup for Bragg
spectroscopy in our experiment, which has been one of the major experimental upgrades
done during this thesis. Finally, Sec. 4.4 and 4.5 are two publications in which our groupproved the existence and characterized the rotonic excitation spectrum in a dipolar BEC of
166 Er.
51

CHAPTER 4. THE ROTONIC EXCITATION SPECTRUM OF A DIPOLAR
BOSE-EINSTEIN CONDENSATE

52

4.1. Excitation spectrum of a BEC confined along the dipole
direction
In a seminal work from Santos et al. [San03]† , the authors showed that the trap geometry of adipolar Bose-Einstein condensate has not only important consequences on the stability of thesystem, but also strongly modifies its excitation spectrum. They consider a pancake-shaped
dipolar BEC, harmonically trapped, with a frequency ωz , along the confining direction (z)and infinite in the other two directions. The atomic dipoles point parallel to z, out of theconfining plane. This additional confinement provides anotherlength scale for the system,pwhich is given by the harmonic oscillator length, lz = ~/mωz . As we will see below, lz
allows one to separate the in-plane excitations into quasi-two-dimensional ones and threedimensional ones. The former excitations are associated with wavelengths much bigger thanlz , where the transverse degree of freedom is not influencing the excitation mode and thedipolar interactions between the atoms in the excited mode are essentially repulsive. The
latter excitations, with wavelengths in the order (or smaller) than lz have a three dimensional character, meaning that the dipolar repulsion between atoms reduces with shorter
wavelengths and eventually becomes attractive.
To study the excitation spectrum of a dipolar pancake-shaped BEC, one can use an analytic
approach given in Refs. [San03, Ped05, Fis06, Lah09, Bla12]. Here, the BEC wave functionin the transverse direction is considered to be a Gaussian function. Under this assumption,
one can integrate out the z-dimension of the GPE (see Eq. 3.9 and Sec. 3.1.1) and reduce theproblem to the remaining radial dimension in plane, ρ. Importantly, this ansatz allows to
describe the DDI (see Sec. 2.3) by an effective two-dimensional interaction potential, whose
Fourier transform becomes

with

√
µ0 µ2
2D
ṼDDI
(kρ ) = √F⊥ (kρ lz / 2)3 2πlz

4.1√
2
F⊥ (x) = 2 − 3 πxex erfc(x),4.2where kρ describes the momentum in the radial direction and erfc(x) is the complementary
error function. In contrast to the unconfined system, where the DDI interaction is independent on the modulus of k (see Eq. 2.3), the confined case features a momentum-dependence
2D (k ) with higher momenta, which eventuallyin kρ , see Fig 4.1. One finds a reduction of ṼDDIρchanges sign and becomes negative for kρ lz & 1. Thus, for small momenta the DDI is mainly
repulsive, but for higher momenta it turns attractive. This momentum dependence has important consequences on the excitation spectrum, which can be calculated from Bogoliubov
theory (Sec. 3.1). One finds an analytical expression for the excitation spectrum
s 


√ 
g 
ϵ2D (kρ ) = ϵ0k ϵ0k + 2n2D √1 + ϵdd F⊥ kρ lz / 2
,
4.3
2πlz
with n2D being the areal density of the atomic cloud. This excitation spectrum is shownin Fig. 4.2 for various ϵdd . For low ϵdd (i. e. as ≫ add ) the contact interactions dominate in the system and its excitation spectrum shows an almost linear dispersion relation.
†

See also the work from O’Dell et al. [O’D03] which appeared at a similar time.

4.1. EXCITATION SPECTRUM OF A BEC CONFINED ALONG THE DIPOLE
DIRECTION

53

2D (~ω /n )
ṼDDI
z
2D

4

2

0

−2

0

0.5

1

1.5
kρ lz

2

2.5

3

2D
Figure 4.1.: Effective two-dimensional dipolar interaction potential ṼDDI
for an infinitely pancake166
14 −2
shapedEr BEC with n2D = 1.85 × 10 m and ωz = 2π × 160 Hz.

However, as ϵdd increases, the dipolar interaction, with its momentum-dependence, become
more dominant and starts to visibly ’bend down’ the excitation spectrum in a momentum
region around kρ lz ≈ 1.5. Eventually, for high enough ϵdd ( & 6 for the parameters in
Fig. 4.2) the spectrum shows an energetic maximum at kmax lz ≈ 0.8, followed by an energetic minimum at krot lz ≈ 1.7. This maximum and minimum is called maxon and roton,
respectively, and their names come from the excitation spectrum observed in superfluid helium [Lan41, Gri93, Pal58, Yar59]. In contrast to the excitation spectrum in superfluid
helium, the excitation spectrum in a dipolar quantum gas is easily tunable with the transverse confinement length scale lz and with ϵdd . The former allows to modify the momentumof the roton minimum, krot , and the latter allows to change its energy, ∆ = ϵ2D (krot ).
The early work from Santos et al. on the possibility of a rotonic excitation spectrum leadto a wealth of theoretical works, which continue up to date. Early questions regarded thestability diagram and ground states of dipolar BECs in finite-sized pancake traps, whichwere investigated for example in Refs. [Ron07, Mar12]. Here, it was found that in certain
parameter regions, the system might acquire unusual density profiles due to the DDI, forexample a blood-cell shape, close to instability. In the same parameter region roton modes,
carrying an angular momentum, were found, see also Ref. [Bis13a]. These angular rotons
were theoretically shown to drive an angular collapse of the BEC, breaking spontaneously
the radial symmetry in a perfectly cylindrical system [Wil09]. Other works focussed on
the new length scale that the roton mode establishes in dipolar BECs. Reference [Wil08]
showed that, when the BEC is subjected to a small, local density perturbation, the roton
minimum reveals itself as density ripples in the condensate with a wavelength ∼ 1/krot .
Reference [Nat10] reported that the roton mode may also dominate the wavelength of faraday
patterns, that emerge when a BEC is periodically non-linearly modulated. For radiallyanisotropic systems, like slightly deformed pancakes or tilted atomic dipoles, the rotonic
excitation spectrum can become anisotropic in the plane of the pancake. It was shown

54

CHAPTER 4. THE ROTONIC EXCITATION SPECTRUM OF A DIPOLAR
BOSE-EINSTEIN CONDENSATE
4z

ϵ2D (~ωz )

3

y
x

2

1
roton

maxon
0

unstable
0

0.5

1

1.5
kρ lz

2

2.5

3

Figure 4.2.: Radial excitation spectra of an infinitely pancake-shaped 166 Er BEC (see inset), withparameters corresponding to the ones in Fig. 4.1. Solid lines show the spectrum for different ϵdd =(0.22, 0.42, 1.00 2.62, 6.55, 16.38), top to bottom, respectively. The dotted line corresponds to thefree particle energy, ϵ0k . The dashed line exemplifies an excitation spectrum at ϵdd = 21.8, where themodes around the roton minimum are unstable with an imaginary eigenenergy, see text. Below the
ϵ2D = 0−line, the y-scale shows thus the norm of the excitation spectrum multiplied by −1.

first theoretically in Ref. [Tic11] and also experimentally proven, though not with a rotonic
spectrum, that dipolar BECs feature an anisotropic speed of sound and critical superfluid
velocity [Bis12, Wen18] .
As the experimental observation of a rotonic excitation spectrum remained elusive, further
theoretical efforts investigated different experimental signatures of the roton minimum. For
example, in Refs. [JL13a, Bis13b, Bla13] the authors investigated the spatial dependence ofthe roton excitations and found that the roton mode is concentrated in high density regions,which is typically the trap center for an inhomogeneously trapped BEC. In this central
region, the local atom number fluctuations of the BEC were found to be amplified in thepresence of a roton mode in the spectrum. As the experimental observation of in-situ atom
number fluctuations, relies on rather good imaging resolution, other approaches proposed touse weak optical lattices [Cor13, JL13b, Bis19]. Here, it was found that the whole BEC canbe easily density modulated or even destabilized when the optical lattice wavelength matches∼ 1/krot .
Despite all the theoretical efforts that took place, the experimental observation of a rotonmode in dipolar BECs remained elusive up to the works presented at the end of this chapter.
There might be two reasons, why experiments lagged behind theory. The first one is simply
the limited number of laboratories working with dipolar quantum gases, which concentrateon chromium, erbium and dysprosium. Furthermore, from these three elements, chromium
has a too small magnetic dipole moment to exhibit a roton in its excitation spectrum forcurrent experimental atom numbers. The second one relies on the fact that the pancake-4.2. THE DYNAMIC STRUCTURE FACTOR

55

shaped geometry is not working as fruitfully for typical experimental parameters as expected.
Looking on the parameters in Fig. 4.2, to observe a rotonic excitation spectrum in a pancake
shaped dipolar BEC, one finds that for typical densities in experiments one needs rather large
values of ϵdd . For 166 Er a value of ϵdd ≈ 6 would correspond to as ≈ 11 a0 , which is close to the
0-Gauss-Feshbach resonance used in our experiment (see Sec. 2.3). Usually, when interactions
are tuned closer to a Feshbach resonance, three-body loss limits the BECs lifetime (at leastin the case of the 0-Gauss-Feshbach resonance of 166 Er) below any reasonable timescale toperform measurements on the excitation spectrum. In our experiments we therefore choseanother geometry, a cigar-shape, to observe the roton minimum in the excitation spectrum.
As will be discussed in Sec. 4.3 and Sec. 4.4, this geometry is advantageous as it allows forhigher densities in experiments and also geometrically ’focusses’ the roton mode along one
predefined axis in the experiment.
For our publication in Sec. 4.4, we did choose to look on the dynamics in the BEC, when its
interactions are quenched into the instability regime of the GPE theory. Here, it is important to understand that, as the system comes closer to instability, ∆ approaches zero in the
excitation spectrum. Within Bogoliubov theory, the unstable regime is characterised by an
excitation spectrum with an imaginary ∆ (see Fig. 4.2), where the excitation modes close to
the roton minimum become imaginary. As a consequence (see Eq. 3.17) these modes will exponentially grow in time and the system dynamically transforms away from the ground-statesolution, closely before the instability point. It undergoes thus a modulational instability
in which a density modulation with a wavelength ∼ 1/krot exponentially grows in time.
In Sec. 4.4, we utilised this instability to prove the existence of the roton mode in dipolar
quantum gases for the first time. We studied the momentum distribution in time and foundan exponentially increasing number of atoms, appearing around krot as the roton minimum
gets populated. By carefully measuring the momentum of these atoms for different trapping
potentials, we could furthermore confirm the scaling of krot with 1/lz . These first measurements proved the existence of the roton mode in the excitation spectrum, but were limited
to the unstable regime.‡ We therefore pursued another technique, called Bragg spectroscopy,
which allowed us to study the rotonic excitation spectrum also in the stable regime. This
technique is presented in Sec. 4.3 but it first requires the introduction of a new quantity ofthe excitation spectrum - the dynamic structure factor.

4.2. The dynamic structure factor
A well established way to describe the response of a many-body system to an external
perturbation is given by the framework of linear response theory. A general discussion ofthis theory is found in Refs. [Pit03, Kub56, Kub57]. Here, I will only briefly review thetheoretical parts, that are necessary for the further discussion of the experimental results in
Sec. 4.4 and Sec. 4.5.
‡

Note that when we speak here of an unstable regime, it referes to the GPE theory without LHY correction.
In reality, the BEC does not collapse completely under a density modulation, but is also stabilised by the
LHY, as will be discussed in Chapter 5.

56

CHAPTER 4. THE ROTONIC EXCITATION SPECTRUM OF A DIPOLAR
BOSE-EINSTEIN CONDENSATE

4.2.1. General description and Bogoliubov approach
Within linear response theory, one of the most studied observables is the density operator,n̂(r), and its fluctuation, δn̂(r) = n̂(r) − ⟨n̂(r)⟩, around the system’s equilibrium density,⟨n̂(r)⟩. To derive general expressions for the density response, one typically considers theFourier transformation of the density fluctuationsZδ ρ̂k = dre−ikr δn̂(r).
4.4
To obtain the density response of the system to an external perturbation with momentum,~k, and energy, ~ω, one needs to consider the coupling that the perturbation introduces
between a state |m⟩, of energy Em , and another state |l⟩, of energy El . As discussed inRef. [Pin66, Pit03], the density response of the system can be calculated from perturbation
theory. One obtains the so-called dynamic structure factor
2
1 X −Em /kB T
ST (k, ω) =⟨l| δ ρ̂†k |m⟩ δ(~ω − ~ωm,l ),
4.5
e
Q
m,l

P

where Q = m e−Em /kB T is the partition function, δ(~ω − ~ωm,l ) is the Dirac delta functionand ~ωm,l = Em − El is the energy difference between the corresponding states. Here, kdefines the momentum at which the density of the system responds. Note that the DSF isgenerally defined for a finite temperature, T , and contains therefore the Boltzmann constant,
kB . To gain more insights in the DSF from Eq. 4.5 for Bose-Einstein condensates, the matrix
elements ⟨l| δ ρ̂†k |m⟩ should be evaluated. In order to do this, one needs to know the system’s
ground state, |m = 0⟩ (resp. Ψ0 (r) in mean-field theory), and its excited states |m ̸= 0⟩. As
shown for example in Ref. [Cso96, Bla02] a useful approach to calculate the excited states is
the Bogoliubov theory, which is discussed in Sec. 3.1 and Sec. 3.1.3 of this thesis. One finds
the linear response of a Bose condensed gas at temperature T
ST (k, ω) =

X

m̸=0

2

∗
u∗m (r) + vm(r) eikr Ψ0 (r) ×

×

h

i

⟨m̂⟩ + 1 δ(~ω − ~ωm ) + ⟨m̂⟩δ(~ω + ~ωm ) , 4.6∗ (r), of the excited states are given via
where the Bogoliubov amplitudes, u∗m (r) and vm
Eq. 3.17. At non-zero tempearture each excitation mode has a mean occupation of ⟨m̂⟩. The
two last terms in Eq. 4.6 describe two processes that may happen in the system. The firstterm, proportional to δ(~ω − ~ωm ), describes particles that are excited out of the condensate
into the excited state |m⟩. The second term, proportional to δ(~ω + ~ωm ), describes excited
particles that are de-excited into the condensate by the perturbing field. An important limitof Eq. 4.6 can be done by assuming T = 0, where ⟨m̂⟩ = 0 for all excited states. Then the
DSF simplifies to
X
X
2

∗
S(k, ω) =u∗m (r) + vm(r) eikr Ψ0 (r) × δ(~ω − ~ωm ) ≡Sm (k, ω).
4.7
m̸=0

m̸=0

As one can see, S(k, ω) contains not only all excited states from the excitation spectrum,
but includes also the coupling strengths between the ground-state and the excited states. It

4.2. THE DYNAMIC STRUCTURE FACTOR

57

is therefore an ideal way to analyse and visualise the excitation spectrum of a BEC, whenits density is perturbed with a momentum ~k at an energy ~ω.

4.2.2. Cigar-shaped dipolar BECs
In our experiments, the dipolar BEC is always confined in all three dimensions in a harmonic
trap with trapping frequencies ωx,y,z . We employ a cigar-shape geometry that has only one
weaker confining direction, which is in our case ωy ≪ ωx , ωz . One main difference comparedto the pancake shape from Sec. 4.1 is that one achieves higher densities in experiments thanfor the case where ωy , ωx ≪ ωz and thus, the spectrum shows a roton minimum alreadyat smaller ϵdd (or higher as , further away from the pole of the used Feshbach resonance,
see Sec. 2.3). Furthermore, the roton mode develops only along the weak axis, y, instead ofradially symmetric in x and y.
In order to be able to thoroughly compare the theoretical excitation spectrum with theexperiment, we rely on numerical calculations of the ground state and the corresponding
excitation modes with Bogoliubov theory (see Sec. 3.1). This is necessary, as the rotonized
spectrum was shown to be sensitive to different ansatzes, yielding slightly different spectra [Bai15]. Figure 4.3 shows two examples of S(k, ω) along the axial momentum, ky , of an
166 Er BEC. In Fig. 4.3 (a) the contact interactions dominate and ϵ
dd < 1. When comparing
2
(a)

(b)

0

ϵ(~ωz )

1.5
1

x
0.5

∗ )
S(k, ω) (Srot

1

z

y
roton

0

0

0.5

1
ky lz

1.5

2

0

0.5

1
ky lz

1.5

2

Figure 4.3.: Axial excitation spectra for a cigar-shaped 166 Er BEC with 2.45 × 104 atoms (inset) and(a) ϵdd = 0.82 and (b) ϵdd = 1.43. In both cases the trap has harmonic frequencies (ωx , ωy , ωz ) =2π × (260.8, 26.7, 256.4) Hz and the atomic dipoles are oriented along z. Each excitation mode isslightly broadened in energy with a Gaussian function for visibility. The color-scale in (a) and (b) is∗
normalised to the maximum DSF, Srot
= S(krot , ∆), of the roton mode (indicated by arrow).

to the infinite pancake case from Fig. 4.2, one realizes two important differences. On the one
hand, the excitation spectrum contains distinct excitation modes that are discrete in energy,
and does not show a continuum of modes. This comes from the fact that the present calculations describe a finite sized system, which can contain only a discrete number of modes.

58

CHAPTER 4. THE ROTONIC EXCITATION SPECTRUM OF A DIPOLAR
BOSE-EINSTEIN CONDENSATEOn the other hand, each individual mode shows a broadening in ky . Also this is the effect of
the finite system size, as an individual excitation mode cannot be any more well describedby a plane-wave and contains contributions from a broader momentum range. The color
code indicates the strength of S(k, ω) of each mode, where one sees that modes at low ky(and low energy) show less response than higher-ky modes. This suppression is known asphononic suppression and directly connects to the finite compressibility of the interactinggas [Pit03].
Figure 4.3 (b) shows the case of dominat dipolar interactions with ϵdd > 1. For the given
system parameters, one finds a well isolated roton mode. Similar to the roton minimumin Fig. 4.2, this mode can be easily tuned in ∆ with ϵdd and its mean momentum, krot ,with the z-confinement. Importantly, the DSF reveals a high value of S(k, ω) of the rotonmode, which can exceed the values of the other modes in Fig. 4.3 (a) and (b) easily by anorder of magnitude. This high value of S(k, ω) shows a high susceptibility of the system’s
response when it is perturbed at the roton momentum and leads for example to the postulated
instability, mentioned in Sec. 4.1 and Refs. [Cor13, JL13b, Bis19], when the BEC is exposed toa weak lattice potential. It furthermore reveals a pronounced increase of density fluctuations
at wavelengths around 2π/krot present in the BEC.

4.3. Bragg spectroscopy
In our second experimental work, presented in Sec. 4.5, we mapped out the dipolar BEC’s
response over a similar momentum and energy range as presented in Fig. 4.3. This wasachieved with Bragg spectroscopy, which is a widely used tool in physics and was established about two decades ago also for ultracold quantum gas experiments. It was used forexample to show the existence of the phononic branch in the excitation spectrum of contactinteracting BECs [SK99, Ste02] and to characterize the system’s intrinsic momentum uncertainty [Ste99b]. Bragg spectroscopy was also employed to measure phase fluctuationsof one-dimensional BECs [Ric03] and to characterize the transition from a one-dimensional
superfluid to a Mott insulating state [Sto04]. Besides these early works, Bragg spectroscopy
could be used to measure the quantum depletion of a homogeneous BEC [Lop17] and was
also implemented in various quantum gas experiments with fermionic atoms [Vee08, Kuh10,
Hoi13].

4.3.1. Basic theoretical description
A carefully written discussion of the Bragg spectroscopy technique and its theoretical approaches are given in Ref. [Bla02]. I therefore remind here only on the essential ingredientsto understand Bragg spectroscopy within the (e)GPE approach. An ideal way to experimentally probe the DSF of a quantum system is to look on its response when it is perturbed
with a plane-wave potential applied for a given time τ(V0 cos(k · r − ωt). if 0 < t < τV (t) =
4.8
0.
otherwise4.3. BRAGG SPECTROSCOPY59

The potential has a depth of V0 and a wave vector k. It furthermore contains a frequency ω,that makes its wave fronts move with a velocity ω/|k| in the direction of k. The perturbing
potential in Eq. 4.8 can be included in the (e)GPE. As shown in Ref. [Mor98, Bla02], one may
use Bogoliubov theory to calculate the occupation of each excitation mode with time. Thisapproach relies on the assumption of a negligible perturbation of the ground state itself. In
the limiting case of zero temperature, one obtains for the occupation of a single excitation
mode after τ
Z
Z τ
2

′ iωm t′′∗∗dt ecos(k · r − ωt ) um (r) + vm (r) |Ψ0 (r)|4.9⟨m⟩ = N0
drV00

where N0 describes the total atom number in the initially unperturbed condensate. One canrewrite the time-dependent integral in Eq. 4.9 using Euler’s formula to obtain
Z τ
0

′

dt′ eiωm t cos(k · r − ωt′ ) =

1
2

Z τ



′
′
dt′ eik·r−i(ω−ωm )t + e−ik·r+i(ω+ωm )t0
Z
1 τ ′ ik·r−i(ω−ωm )t′sin ((ω − ωm )τ /2) ik·r
≈
dt e
=τ
e
4.10
2 0
(ω − ωm )τ /2

where the term, proportional to ω + ωm can be neglected within the rotating wave approximation. Under this approximation Eq. 4.9 simplifies to
πN0 V02 τ
⟨m⟩ =
2

Z



2

∗
dr u∗m (r) + vm(r) eik·r |Ψ0 (r)|The function
F (ω − ωm , τ ) =F (ω − ωm , τ ).

2 sin2 ((ω − ωm )τ /2)πτ (ω − ωm )2

4.11

4.12

takes the finite time of the perturbation into account, which can be seen as a frequency
broadening of the perturbing potential in ω. In the limit of long pulses,
lim F (ω − ωm , τ ) = δ(ω − ωm )

τ →∞

4.13

becomes a delta function and Eq. 4.11 can be written with Eq. 4.7 to⟨m⟩ =πN0 V02 τ
Sm (k, ω).
2

4.14

Therefore, the occupation of a single mode |m⟩ is directly determined by its DSF. Thatmeans for a well characterised Bragg potential, one can measure the DSF directly via the
population in |m⟩ after the Bragg excitation.

4.3.2. Experimental implementation
To experimentally probe the DSF of a quantum system, one essentially relies on a clean
implementation of the periodic potential, given in Eq. 4.8. Most of the quantum gas experiments in the last decade followed the early ideas of the Ketterle group [Ste99b, SK99] andthe Davidson group [Ste02], which are based on Ref. [Ber97]. This scheme is illustrated in

60

CHAPTER 4. THE ROTONIC EXCITATION SPECTRUM OF A DIPOLAR
BOSE-EINSTEIN CONDENSATE

k 1, ω L

k2, ωL-ωV (t)yFigure 4.4.: Bragg excitation scheme with two laser beams (solid arrows). In experiments, thebeams interfere at the atom’s position and create an optical potential with wave vector k. A frequency
difference ω between the two beams makes the interference pattern move with a velocity ω/|k| (dashedarrow). This scheme excites the system with a rate proportional to S(k, ω).

Fig. 4.4. Two coherent laser beams of frequency ωL , coming from different spatial directions,with wave vectors k1 and k2 , are directed on the atomic cloud, where they create an interference pattern with a wave vector k = k1 − k2 . An additional frequency difference, ω ≪ ωL ,between the two beams introduces a constant movement of the interference pattern.
In experiments, ω can be easily tuned for example via acusto-optical modulators. A change
in k needs a change of k1 or k2 , which typically requires a realignment of the laser beams. As
this realignment usually involves direct work on the experimental machine and inherently is
not very reproducible, we developed a Bragg spectroscopy setup, that allows for an easy andreproducible way to change both k and ω. It is implemented by using holographic gratings,
that are uploaded on a digital micro-mirror device. The underlying concept is discussed in
the supplemental material of the publication in Sec. 4.5 and in Ref. [Pet15b].
To give the reader an intuition how a Bragg spectroscopy pulse is perturbing a BEC, Fig. 4.5
shows a numerical real-time simulation of a Bose Einstein condensate in a cigar-shaped trap
(see also Sec. 4.4). In the example the perturbing potential (Eq. 4.8) is applied along thepositive y direction over the whole timing from t ∈ [0, 7] ms. The frequency is chosen suchthat the Bragg excitation is resonantly exciting the BEC. Fig. 4.5 (a) shows the integrated insitu density along y versus t. At the beginning of the simulation, at time t = 0 ms, the density
has an inverted parabolic shape, similar to a BEC in the Thomas-Fermi regime [Pit03]. As
the time progresses, the Bragg excitation excites atoms from the BEC, which start to movealong the positive y direction. The superposition of the wave function of the excited atoms
and the wave function of the ground-state atoms leads to a density wave with a wave vectork, which travels through the BEC. The contrast of this density wave increases with time, as
more and more atoms are occupying the excited mode. Fig. 4.5 (b) shows the corresponding
momentum distribution of the whole system, where the excited particles are visible as apronounced side peak appearing at ky = 4.3 µm−1 , next to the peak of the ground state
atoms at ky ≈ 0 µm−1 . This insight is relevant for experiments, as it shows that the Bragg
excitation can be either probed directly in-situ (given a good enough imaging resolution) or
via the momentum-space, which can be accessed by standard time-of-flight imaging.

4.3. BRAGG SPECTROSCOPY

ny (103 μm-1)

0

0

61
1

2

3

nk (μm)101 102 103

104

time t (ms)

1
2
3
4
5
6
7

-15

-10

-5

0
5
y (μm)

10

15 -6

-4

-2

0
2
ky (μm-1)

4

6

Figure 4.5.: Simulation of a Bragg spectroscopy pulse on a dipolar BEC with 5 × 104 atoms in a(ωx , ωy , ωz ) = (250, 31, 160) Hz trap. The Bragg excitation has a duration τ = 7 ms, ω = 2π ×375 Hz,−1
V0 /h =
R 14 Hz and k = 4.3 µm along the positive y direction. (a) Integrated in-situ density profile
ny = n(x, y, z)dxdz versus time. (b) Corresponding integrated momentum distribution, nk , along
y of the atomic cloud.

In our publication in Sec. 4.5 we measured the excited fraction of atoms after the Bragg
excitation in time-of-flight. By performing measurements over a broad momentum andenergy range, we were able to show the developement of the roton minimum in the excitation
spectrum for increasing ϵdd . We furthermore probed the excited fraction at krot for a fixed
V0 and for various ϵdd , that directly reveals the increase of the DSF at the roton momentum
in the system. Interestingly, when comparing our measurements to theory, we found aquantitative deviation from the established theory in Sec. 3.4, which includes the Lee-HuangYang correction. This deviation questioned for the first time if the inclusion of the LHY asa mean-field energy shift is justified or not. Also a further experimental work with dipolar
droplets [Bö19b] found similar discrepancies and showed the importance of correlations inthe system. Up to date, it is still an open point how these discrepancies can be best includedin current theories.

62

CHAPTER 4. THE ROTONIC EXCITATION SPECTRUM OF A DIPOLAR
BOSE-EINSTEIN CONDENSATE

4.4. Publication: Observation of Roton Mode Population in a
Dipolar Quantum Gas
Nature Physics, 14, 442–446 (2018)§
submitted 12 Dec 2017; published 5 Mar 2018
DOI: 10.1038/s41567-018-0054-7

L. Chomaz,1 R. M. W. van Bijnen,2 D. Petter,1 G. Faraoni,1,3 S. Baier,1 J. H. Becher,1 M.
J. Mark,1,2 F. Wächtler,4 L. Santos,4 and F. Ferlaino 1,21Institut für Experimentalphysik, Universität Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Austria3
Dipartimento di Fisica e Astronomia, Università di Firenze, Sesto Fiorentino, Italy
4
Institut für Theoretische Physik, Leibniz Universität Hannover, Germany
2§

The author of the present thesis developed the experimental procedures together with L. C. and S.B.,
performed the measurements together with L. C., S. B., G. F., and J. H. B., and contributed in writing
the manuscript. The theoretical calculations of this publication have been contributed by R. M. W. v. B.,
F. W., and L. S.

Letters
https://doi.org/10.1038/s41567-018-0054-7

Observation of roton mode population in a dipolar
quantum gas
L. Chomaz1, R. M. W. van Bijnen2, D. Petter1, G. Faraoni1,3, S. Baier1, J. H. Becher1, M. J. Mark1,2,
F. Wächtler4, L. Santos4 and F. Ferlaino 1,2*
The concept of a roton, a special kind of elementary excitation
forming a minimum of energy at finite momentum, has been
essential for the understanding of the properties of superfluid
4
He (ref. 1). In quantum liquids, rotons arise from the strong
interparticle interactions, whose microscopic description
remains debated2. In the realm of highly controllable quantum gases, a roton mode has been predicted to emerge due
to magnetic dipole–dipole interactions despite their weaklyinteracting character3. This prospect has raised considerable
interest4–12; yet roton modes in dipolar quantum gases haveremained elusive to observations. Here we report experimental and theoretical studies of the momentum distribution in Bose–Einstein condensates of highly magnetic erbium
atoms, revealing the existence of the long-sought roton mode.
Following an interaction quench, the roton mode manifests
itself with the appearance of symmetric peaks at well-defined
finite momentum. The roton momentum follows the predicted
geometrical scaling with the inverse of the confinement
length along the magnetization axis. From the growth of theroton population, we probe the roton softening of the excitation spectrum in time and extract the corresponding imaginary roton gap. Our results provide a further step in the quest
towards supersolidity in dipolar quantum gases13.
Quantum properties of matter continuously challenge our intuition, especially when many-body effects emerge at a macroscopic
scale. In this regard, the phenomenon of superfluidity is a paradigmatic case, which continues to reveal fascinating facets following its
discovery in the late 1930s2,12. A major breakthrough in understanding superfluidity thrived on the concept of quasiparticles, introduced by Landau in 19411. Quasiparticles are elementary excitationsof momentum k, whose energies ε define the dispersion (energy–
momentum) relation ε(k).
To explain the special thermodynamic properties of superfluid
4
He, Landau postulated the existence of two types of low-energy
quasiparticle: phonons, referring to low-k acoustic waves, androtons, gapped excitations at finite k initially interpreted as elementary vortices. The dispersion relation continuously evolves from linear at low k (phonons) to parabolic-like with a minimum (roton) ata finite k =​  krot. Neutron scattering experiments confirmed Landau’s
remarkable intuition14. In liquid 4He, krot scales as the inverse of theinteratomic distance. This manifests a tendency of the system toestablish a local order, which is driven by the strong correlations
among the atoms2.
In the realm of low-temperature quantum physics, ultracold quantum gases realize the other extreme limit for which theinterparticle interactions—and correlations—are typically weak,meaning that classically their range of action is much smaller thanthe mean interparticle distance. As a result of this diluteness, roton
excitations are absent in ordinary quantum gases, that is, in Bose–
Einstein condensates (BECs) with contact (short-range) interactions12. However, about 15 years ago, seminal theoretical works
predicted the existence of a roton minimum both in BECs withmagnetic dipole–dipole interactions (DDIs)3 and in BECs irradiated by off-resonant laser light15. Following the lines of the latter
proposal, a roton softening has recently been observed in BECs
coupled to an optical cavity16. Here, the excitation arises from theinfinite-range photon-mediated interactions and the inverse of thelaser wavelength sets the value of krot. In addition, roton-like softening has been created in spin–orbit-coupled BECs17,18 and quantum
gases in shaken optical lattices19 by engineering the single-particle
dispersion relation.
Our work focuses on dipolar BECs (dBECs). As in superfluid
4
He, the roton spectrum in such systems is a genuine consequenceof the underlying interactions among the particles. However, incontrast to helium, the emergence of a minimum at finite momentum does not require strong interparticle interactions. It instead
exists in the weakly interacting regime and originates from thepeculiar anisotropic and long-range character of the DDI in real
and momentum space3–12. Despite the maturity achieved in the theoretical understanding, the observation of dipolar roton modes hasremained so far an elusive goal. For a long time, the only dBEC available in experiments consisted of chromium atoms20, for which theachievable dipolar character is hardly sufficient to support a rotonmode. With the advent of the more magnetic lanthanide atoms21,22,
a broader range of dipolar parameters became available, opening
the way to access the regime of dominant DDI. In this regime, novel
exciting many-body phenomena have been recently observed, as theformation of droplet states stabilized by quantum fluctuations23–25,
which may become self-bound26. Lanthanide dBECs hence open
new roads toward the long-sought observation of roton modes.
Prior to this work, dipolar rotons have been mostly connected topancake-like geometries3–6,8–12. Here, we extend the study of roton
physics to the case of a cigar-like geometry with trap elongation
along only one direction (y) transverse to the magnetization axis(z) (Fig. 1a). The anisotropic character of the DDI (Fig. 1b, inset)
together with the tighter confinement along z is responsible for therotonization of the excitation spectrum along y (Fig. 1b). To illustrate this phenomenon, we consider an infinite cigar-shaped dBEC
and focus on its axial elementary excitations, of momentum ky.
These excitations correspond in real space to a density modulation

Institut für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria. 2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie
der Wissenschaften, Innsbruck, Austria. 3Dipartimento di Fisica e Astronomia, Università di Firenze, Sesto Fiorentino, Italy. 4Institut für Theoretische
Physik, Leibniz Universität Hannover, Hannover, Germany. *e-mail: francesca.ferlaino@uibk.ac.at
1442

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Letters

Nature Physicsaczñ (a.u.)1.0añ (103 µm2)
0

0.5

1

1.5

2

b

ñ (103 µm2)c

ñ (103 µm2)0

0.2 0.4 0.6 0.8

0

0.1 0.2 0.3 0.4

0.8
y

0.6

kx

b

0.4
0.2

Attraction
Repulsion

0ky

5ky (µm–1)

x

Energy, ε(k)d–52ñ (103 µm2)

as

kx

0.8
0.4

1

0

krot

Momentum, ky

0

0.4

–8 –4

0

4

0

8

0.2

–8 –4

ky (µm–1)

ky

0

4

8

0

ky (µm–1)

d

–8 –4

0

4

8

ky (µm–1)ez
Dominantly repulsive DDIyDominantly attractive DDI

Fig. 1 | Roton mode in a dBEC. a, An axially elongated geometry with
dipoles oriented transversely. b, Real (solid lines) and imaginary (norm of
the dotted line) parts of the dispersion relation of a dBEC in the geometry
in a, showing the emergence of a roton minimum for decreasing as(dashed arrow). The DDI changes from repulsive (blue) to attractive (red)depending on the dipole alignment (inset). The bottom panels show the
dipole alignment (colour code as in the inset) associated with small-ky(left panel) and large-ky (right panel) density modulations, respectively.
c,d, Distributions on the kxky plane associated with the roton population incigar (a) or pancake (d) geometries with an identical roton population andcolour scale.

along y of wavelength 2π​/ky. For low ky, the atoms sit mainly sideby-side and the repulsive nature of the DDI prevails, stiffening thephononic part of the dispersion relation (Fig. 1b, bottom left panel).
In contrast, for k y ℓz ≳​  1, where ℓz is the characteristic z confinement
length, the excitation favours head-to-tail alignments and the DDI
contribution to ε(ky) eventually changes sign3 (Fig. 1b, bottom rightpanel). The resulting softening of ε(ky) is counterbalanced by thecontributions of the repulsive contact interaction, and of the kinetic
energy, which ultimately dominates at very large ky. For strong
enough DDI, this competition gives rise to a roton minimum inε(ky), occurring at a momentum ky =​  krot set by the geometrical scaling k rot ~ 1 ∕ ℓz (see below and, for example, refs 3,8,9).
Similar to the helium case, the roton energy gap, Δ​  =​  ε(krot),
depends on the density and on the strength of the interactions. In
ultracold gases, both quantities can be controlled. In particular,
the scattering length as, setting the strength of the contact interaction, can be tuned using Feshbach resonances12. As as is reduced,
Δ​decreases, vanishes and eventually becomes imaginary (Fig. 1b).
In this last case, the system undergoes a roton instability and the population at ky =​ 0 is transferred to ±​krot at an exponential rate5,6. The
population of the roton mode is then readily visible in the momentum distribution of the gas (Fig. 1c,d). In the extensively studied
pancake geometries, the roton population in k-space spreads over aring of radius k =​  krot because of the radial symmetry of the confinement (Fig. 1d). Such a spread can be avoided using a cigar geometry. Here, the roton population focuses in two prominent peaks at
ky =​  ±​krot, enhancing the visibility of the effect (Fig. 1c).
We explore the above-described physics using strongly magnetic 166Er atoms. The experiment starts with a stable dBEC in a

Roton momentum, krot (µm–1)

6

5

4

3

2
1.5

2

2.5

3

3.5

z-confinement scale, 1/ℓz (µm–1)

35

40

45

50

Scattering length, as (a0)Fig. 2 | Observed roton peaks and characteristic scalings. a–c, The top
panels show n∼(kx , ky ) obtained by averaging 15–25 absorption imagesfor (νz, λ) =​ (456 Hz,14.4), th =​ 3 ms and as =​ 54a0 (a), as =​ 44a0 (b) andas =​ 37a0 (c). The bottom panels show the corresponding cuts at kx ≈​ 0
(dots) and their fits to three-Gauss distributions (lines), from which
we extract krot and A*. d, Measured krot depending on 1 ∕ ℓz at as ≲​ as* forνy ≈​35 Hz (circles) and νy =​ 17(1) Hz (triangle). e, krot depending on as for(νz, λ) =​ (149 Hz, 4.3) (circles). In d,e, the error bars show the 95%
confidence interval of the three-Gauss fits (see the bottom panels in a–c).
The squares (diamonds) show predictions from the SSM (NS) in theirrange of validity (Methods). The lines are guides to the eye.

cigar-shaped harmonic trap of frequencies νx,y,z, elongated along
the y axis. The trap aspect ratio, λ =​  νz/νy, can be tuned from about
4 to 30, corresponding to νz ranging from 150 to 800 Hz, whereas
νy and νz/νx are kept constant at about 35 Hz and 1.6, respectively(Methods). An external homogeneous magnetic field, B, fixes the
dipole orientation (magnetization) with respect to the trap axes
and sets the values of as through a magnetic Feshbach resonance,
centred close to B =​ 0 G. In previous experiments, we precisely
calibrated the B-to-as conversion for this resonance25. The BEC isprepared at asi = 61a 0 (B =​ 0.4 G) with transverse (z) magnetization.
The characteristic dipolar length, defined as add = μ 0 μ m2 m ∕ 12π ℏ2,
is 65.5a0, where m is the mass and μm is the magnetic moment of
the atoms, ħ =​  h/2π​is the reduced Planck constant, μ0 is the vacuum
permeability and a0 is the Bohr radius.
To excite the roton mode, we quench as to a desired lower value,asf , and shortly hold the atoms in the trap for a time th. We measurethat as converges to its set value with a characteristic time constantof 1 ms during th. We then release the atoms from the trap, change
as back close to its initial value and let the cloud expand for 30 ms.
We probe the momentum distribution n∼ (kx , k y ) by performing

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443

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d

1

–1
–2

Renormalized roton population

b

–6

0
Momentum, ky (µm–1)

Δ2 (h2ν2)

a

8

100
10

6

1
0.1

4

0.01
0.001

–4

ñ1 (μm)>6,000

–2

5,000
4,000

0

3,000
2

2,000

4
0

1

2

2 3 4
th (ms)

5

6

1,000
0

6

e –10
n1 (μm–1)

0

10,000–5
Position, y (µm)

Roton momentum, krot (µm–1)

c

3
2

8,000

0

6,000
4,000

5

1

2,000

0
0

1

2
3
4
Hold time, th (ms)

5

6

10

0

2

4
Hold time, th (ms)

6

8

0

Fig. 3 | Dynamics of the roton mode. Results depending on th after quenching to asf = 50a0 in (νz, λ) =​ (149 Hz, 4.3). a–c, Δ2 at y =​0, amplitude andmomentum of the roton peak, from the SSM (solid lines), the NS (dashed lines) and the experiments (circles) with their 95% confidence interval from thethree-Gauss fit (error bars). The shaded areas identify the dynamical destabilization. For comparison, the amplitudes are renormalized to their th =​ 3 msvalues and the NS results are translated by −​3.7 ms. krot is reliably extracted from the experiments (NS) for th ≥​3 ms (2.5 ms). Inset: same plot in log scale.
d,e, Integrated 1D density profile, in momentum n∼1(ky ) and real space n1(y), from one run of the NS.

standard resonant absorption imaging on the expanded cloud(Methods). The measurement is then repeated at various values ofas <​  add in a fixed trap geometry. The momentum distribution showsa striking behaviour (Fig. 2). For large enough as, ñ (k x , k y ) shows asingle narrow peak with an inverted aspect ratio compared to thetrapped gas, typical of a stable BEC12 (Fig. 2a, top). We define thecentre of the distribution as the origin of k. In contrast, when further decreasing as below a critical value as*, we observe a sudden
appearance of two symmetric finite-momentum peaks, of similar
shape and located at ky =​  ±​krot (Fig. 2b,c, top panels). By repeating
the experiment several times, we observe that the peaks consistently
appear at the same locations, and they are visible in the averaged
distributions. To quantitatively investigate the peak structures, we
fit a sum of three Gaussian distributions to the central cuts of theaverage n∼ (kx , k y ) (Fig. 2a–c, bottom panels). From the fit, we extract
the central momentum, krot, and the amplitude, A*, of the side peaks.
A major fingerprint of the roton mode in dBECs is its geometricalnature, leading to a universal scaling k rot ~ 1 ∕ ℓz (see, for example,
refs 3,4,8,9). In addition, the dependency of krot on as close to the instability is expected to be mild as krot remains mainly set by its geometrical nature3,8. We investigate both dependencies in the experiment.
In a first set of experiments, we repeat the quench measurementsfor various trap parameters and extract krot. We clearly observe theexpected geometrical scaling. krot shows a marked increase with1 ∕ ℓz = 2π mνz ∕ h , matching well with a linear progression witha slope of 1.61(4) (Fig. 2d). Note that no dependence of krot on νy isobserved. In a second set of experiments, we fix the trap geometry
and explore the dependence of krot on as. We observe that, within
our experimental uncertainty, krot stays constant when decreasing
as (Fig. 2e). This behaviour contrasts with the one expected fora phonon-driven modulation instability that exhibits a strong
as-dependence27.
444

To gain a deeper understanding of the roton excitations in oursystem and its dynamical population, we develop both an analytical model and full numerical simulations (NSs) and compare
the findings with our experimental data. Our analytical model
starts by calculating the roton spectrum of the stationary BEC,
generalizing the results of ref. 3 to a non-radially symmetric configuration. Since the roton wavelength is much smaller than theextension of our three-dimensional (3D) BEC along y, this modecan be evaluated using a local density approximation in y. Hence,for our model, we consider a dBEC homogeneous along y, of axial
density n0, harmonically confined along x and z. To analyticallyevaluate the roton spectrum, we approximate the BEC wavefunction using the Thomas–Fermi (TF) approximation. For dominant
DDI, εdd =​  add/as ≥​ 1, we find that ε(ky) indeed rotonizes (Methods).
In the vicinity of the roton minimum and for εdd ~ 1, the computed
dispersion acquires a gapped quadratic form similar to that ofhelium rotons:
ε (k y ) 2 ≃ Δ2 +2
2ℏ2k rotℏ2m2m(k y −k rot ) 2

(1)
1∕4

The roton momentum reads as k rot = 2m (E 02−Δ2 ) ∕ℏ,
where Δ = E 02−E I2 , EI =​  2gn0(εdd −​  1)/3, g =​  4π​ħ2as/m andℏ2
E 02 = 2gεdd n 0 2m (X −2 + Z −2 ) . The TF radii X,Z satisfy X2,Z2 ∝​  gn0, sothat E0 ∝​ hνz, with a scale set by εdd and νz/νx but independent of gn0(ref. 15). Close to the instability ( Δ ≃ 0), the roton momentum thus
follows a simple geometrical scaling, k rot = κ ∕ ℓz with the geometrical factor κ depending on νz/νx alone. For completeness, we have also
performed full 3D numerical calculations of the static Bogoliubov
spectrum of a finite trapped dBEC, confirming the existence andscaling of the roton mode (Supplementary Information).
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a

b

0.4

Amplitude, A* (103 µm2)

af = 39a0

Imaginary gap, Im[Δ]/h (Hz)af = 47a0
af = 50a0
af = 52a0

0.2

Exp. fit

0.1

0.0

c

af = 44a0

0.3

2

4
6
Hold time, th (ms)

0.1

1
10
Rescaled time, Th (ms)

0

100
0

200

50
100

300

150
200

400
40

45

0

1

50
Scattering length, as (a0)2
3
45Hold time, th (ms)

6

55

60

Fig. 4 | Population growth and roton gap. Results for (νz, λ) =​ (149 Hz,
4.3). a,b, A* depending on th and on the rescaled time Th, after quenchingto asf = 39a0, 44a0, 47a0, 50a0 and 52a0. The shaded areas show the
95% confidence interval from the three-Gauss fit. The black line shows
an exponential fit to the full data set with Th <​ 3 ms. c, Extracted Im[Δ ]depending on as and on th (inset) from the experiments (dashed line withpentagons) with the propagated errors (shaded area) from the timerescaling analysis and exponential fit (see b), and from the SSM (solid line).
The dotted line and the corresponding shaded area show the experimental
as* and the confidence interval of its fit. The inset shows the same
configuration as Fig. 3 (asf = 50a0).

̄The above stationary description accounts for the existence ofthe roton mode in the cigar geometry used in experiments, and predicts the scaling of krot and Δ with the system parameters. However,the quench of as introduces dynamics, which is crucial to quantitatively reproduce the experimental observations. The reduction of as
decreases the contact interaction and additionally induces a compression of the cloud. This yields a dynamical modification of the local
roton dispersion relation, and the roton may be destabilized during
the evolution. This dynamical destabilization is well accounted forby a self-similar model (SSM) describing the evolution of the cloudshape after the quench12. In particular, we consider a 3D harmonic
confinement, and, starting from the stationary TF profile at asi, we
evaluate the evolution of the cloud along the change of as assuming that the TF shape is maintained but with time-dependent TF
radii (Methods). We then estimate the local (along y) instantaneous
roton spectrum ε(ky, y,th) using a local density approximation, thatis, evaluating equation (1) with the experimentally calibrated as(th),and the n0(y, th), X(y,th) and Z(y,th) estimated from the 3D profile.
We indeed find that the roton gap decreases and eventually turns
imaginary (Δ(th)2 <​  0) (Fig. 3a). When this occurs the population atky ≈​  krot exponentially grows with an instantaneous local rate 2Im[ε(ky,y,th)]/ħ, with Im[.] indicating the imaginary part, giving riseto two symmetric side peaks in the axial momentum distributionthn∼1 (k y , t h ) ∝ ∫ exp 2 ∫ Im[ε (k y , y, t )]dt dy (Fig. 3b) (Methods).
0
The centre of the peaks, ±​krot(th), also evolves with th but quickly()

converges after a few milliseconds to its final value, krot (Fig. 3c).
The measured krot and the calculated values from our parameterfree theory are in remarkable agreement (Figs. 2d,e and 3c).
Our SSM quantitatively explains the experimental observations
and provides us with a physical understanding. For completeness,
we perform NSs of the system dynamics. We calculate the time
evolution of the generalized non-local Gross–Pitaevskii equation(NLGPE), which accounts also for quantum fluctuations, threebody loss processes and finite temperature, not included in the SSM(Supplementary Information). The first two contributions limit thepeak density of the atomic cloud and stabilize the dBEC against collapse4–6,24,25, whereas the last term thermally seeds the initial rotonmode population. The NSs confirm both the SSM results and theexperimental observations. Indeed, the calculations show that, a few
milliseconds after the quench, the system develops roton peaks inmomentum space (Fig. 3d) and short-wavelength density modulations at the centre of the BEC (Fig. 3e), showing the predicted roton
confinement9. The extracted value of krot and its geometrical scaling
are in very good quantitative agreement with both the experimental
data and the SSM calculations (Figs. 2d,e and 3c). Interestingly, thetime scales for the emergence of roton peaks in the NS are a few milliseconds longer than those observed in the experiment. The originof this time shift remains an open question, whose answer could, forexample, require a refinement of current models of quantum fluctuations. However, despite this delay, the growth rate of the roton
population in the NS matches both the experimental observations
and the SSM predictions in the early dynamics (Fig. 3b).
In the experiment, we study the time evolution of the roton modein a fixed geometry (νz, λ). In a first set of measurements, we fix asf
and follow the dynamics by recording the momentum distribution
at various th. We observe that krot does not change significantly while
the roton population initially grows, in excellent agreement with thetheories (Fig. 3b,c). The growth rate is a particularly relevant quantity as it is directly connected to the imaginary excitation energy inthe Bogoliubov description (for the roton, its gap), as revealed byour theory. In a second set of experiments, we thus systematicallystudy the growth rate of the roton population for various asf (Fig. 4).
Our data show that the roton mode begins to become populated
from a critical value of the scattering length, as*. For asf > 52a 0, we do
not observe the roton peaks at any time. For asf ≲​  52a0, after a timedelay, A* undergoes an abrupt increase. At longer time, A* then saturates and eventually slowly decreases while atoms are coincidentally
lost (Fig. 4a). By further lowering asf , the roton population exhibits
a faster growth rate and shorter time delay.
From the growth rate of the roton population, we now extract anoverall roton gap Δ (as ) (Methods). In brief, the dynamical variation of the gap is determined by the leading time dependence oftas and A (asf , t h ) ∝ exp 2 ∫ h dt Im[Δ (as(t ))] ∕ℏ . To investigate0*
the scaling Δ (as ), we first consider the analytical expression of theelementary local roton gap, Δ as a function of as (see equation (1)).
Developing Δ in the vicinity of as = as* yields Δ(as ) ∝ (as*−as ) 1 ∕ 2.
Here we confirm this scaling by considering the generic power-law

̄

̄

(

)

̄

̄

dependence Im[Δ (as )]∕ℏ = Γδ (as ) with δ (as ) =

(

th

(

a s* − a sa0)) (a ≤ a *) .
βssConsequently, A* grows as exp 2Γ ∫ δ (as(t ))dt = exp (2ΓTh ) . By0trescaling the time variable as t h → Th = ∫ h δ (as(t ))dt , we determine0
the parameters as* and β such that all of the curves of Fig. 4a fallon top of each other (Fig. 4b). The best overlap of the experimental curves is found for as* = 53.0(4)a 0 and β =​ 0.55(8). We then perform an exponential fit to all of the rescaled data for Th <​ 3 ms andextract Γ =​ 465(83) s−1. The same analysis, applied to the calculated
population growth from the SSM for different asf , gives as* ≃ 52.8a 0,
β ≃ 0.55 and Γ ≃ 472s −1, which are very close to the experimental
values. This time-resolved study allows us to readily extract theimaginary roton gap, Im[Δ ], from both the experiment and thēNature Physics | VOL 14 | MAY 2018 | 442–446 | www.nature.com/naturephysics© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

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SSM, as a function of as and th (Fig. 4c). Our observations show thesoftening of the roton mode at as = as* and the expected increase ofIm[Δ ] for as < as*, nicely grasping the dynamics during the growthof the roton population (inset of Fig. 4c). The observed behaviour
shows again a remarkable agreement with the theory predictions.
To conclude, our work demonstrates the power of weakly interacting dipolar quantum gases to access the regime of large-momentum, yet low-energy, excitations dressed by interactions. This newly
accessible regime, which is largely unexplored in ultracold gases,
raises fundamental questions and opens novel directions. Future
key developments are to study the impact of such low-lying excitations on the superfluid behaviour of a dBEC8–10,12, the interplay
between phononic and rotonic modes in the stability diagram of
the quantum gas4,7 and the possible role of roton excitations as thetriggering mechanism of the recently observed instability leading
to the formation of metastable droplet arrays in elongated traps24,28.
Of particular interest is the prospect of creating a supersolid andstriped ground states in dBECs13,28. Indeed, the short-wavelengthdensity modulation tied to the roton softening together with thequantum stabilization mechanism can favour the formation of suchan exotic phase of matter, in which crystalline order coexists withphase coherence. In contrast to recent experiments29,30, where thedensity modulation is imposed by external fields yielding a supersolid-like arrangement of infinite stiffness, a dipolar supersolid
would be compressible. Hence, experiments on dBECs provide theexciting opportunity to unveil similarities and differences among
complementary approaches to supersolidity.

̄Methods

Methods, including statements of data availability and any associated accession codes and references, are available at https://doi.
org/10.1038/s41567-018-0054-7.
Received: 12 December 2017; Accepted: 22 January 2018;
Published online: 5 March 2018

References

1. Landau, L. D. The theory of superfluidity of helium II. J. Phys. (Mosc.) 5,
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Acknowledgements

We are particularly grateful to B. Blakie for many inspiring exchanges. We thank
D. O’Dell, M. Baranov, E. Demler, A. Sykes, T. Pfau, I. Ferrier-Barbut and H. P. Büchler
for fruitful discussions, and G. Natale for his support in the final stage of the experiment.
This work is dedicated to the memory of D. Jin and her inspiring example. The
Innsbruck group is supported through an ERC Consolidator Grant (RARE, no. 681432)
and a FET Proactive project (RySQ, no. 640378) of the EU H2020. L.C. is supported
within a Marie Curie Project (DipPhase, no. 706809) of the EU H2020. F.W. and L.S.
thank the DFG (SFB 1227 DQ-mat). All authors thank the DFG/FWF (FOR 2247). Partof the computational results presented have been achieved using the HPC infrastructure
LEO of the University of Innsbruck.

Author contributions

F.F., L.C., D.P., G.F., M.J.M., J.H.B. and S.B. conceived and supervised the experimentand collected the experimental data. L.C. analysed the data. R.M.W.v.B. developed the
Bogoliubov–de Gennes calculations. F.W., R.M.W.v.B. and L.S. performed the real-time
simulations. L.S. derived the analytical model and the SSM. L.C., F.F., R.M.W.v.B. and L.S.
wrote the paper with contributions from all of the authors.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information is available for this paper at https://doi.org/10.1038/
s41567-018-0054-7.
Reprints and permissions information is available at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to F.F.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

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Letters

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Methods

Trapping geometries. The BEC is confined in a harmonic trapping potential
V (r) = 2mπ 2 (νx2x 2 + νy2y 2 + νz2z 2) , characterized by the frequencies (νx,νy,νz).
The trap results from the crossing of two red-detuned laser beams of 1,064 nm
wavelength at their respective foci. One beam, called vODTb, propagates nearlycollinear to the z axis and the other, denoted hODTb, propagates along the y axis.
By adjusting independently the parameters of the vODTb and hODTb, we canwidely and dynamically control the geometry of the trap (see Supplementary
Information). In particular, νy is essentially set by the vODTb power while νz (andνx) is set independently by that of the hODTb. This yields an easy tuning of the trap
aspect ratio, λ =​  νz/νy, relevant for our cigar-like geometry.
After reaching condensation (see Supplementary Information), we modify thebeam parameters to shape the trap into an axially elongated configuration, favourable
for observing the roton physics (νy ≪ νx , νz ) . The trapping geometries probed in theexperiments, whose (λ,νx,νy,νz) are reported in Supplementary Table 1, are achievedby changing the hODTb power with the vODTb power set to 7 W, so that νy and νz/νx
are kept roughly constant. Only the green triangle in Fig. 2d is obtained in a distinct
configuration, with a vODTb power of 2 W, leading to (νx,νy,νz) =​ (156, 17, 198)Hz
and λ =​ 11.6. The (λ,νx,νy,νz) are experimentally calibrated via exciting and probing thecentre-of-mass oscillation of thermal samples. We note that the final atom number
N, BEC fraction f and temperature T after the shaping procedure depend on the final
configurations, as detailed in Supplementary Table 1.
Quench of the scattering length as. To control as, we use a magnetic Feshbachresonance between 166Er atoms in their absolute ground state that is centred around
B =​ 0 G (ref. 31). The B-to-as conversion has previously been precisely measured
via lattice spectroscopy, as reported in ref. 25. Errors on as, taking into accountstatistical uncertainties of the conversion and effects of magnetic field fluctuations(for example, from stray fields), are of 3–5a0 for the relevant as range 27–67a0 inthis work. After the BEC preparation and to investigate the roton physics via aninteraction quench, we suddenly change the magnetic field set value, Bset, twice.
First, we perform the quench itself and abruptly change Bset from 0.4 G (asi = 61a 0)
to the desired lower value (corresponding to asf ) at the beginning of the hold in
the trap (th =​ 0 ms). Second, we prepare the time-of-flight (TOF) expansion andimaging conditions (see the section entitled Imaging procedure) and abruptly
change Bset from the quenched value back to 0.3 G (as =​  57a0) at the beginningof the TOF expansion (tf =​ 0 ms). Due to delays in the experimental set-up (forexample, coming from eddy currents in our main chamber), the actual B value
felt by the atoms responds to a change of Bset via B(t) =​  Bset(t) +​  τdB/dt (ref. 32). By
performing pulsed-radiofrequency spectroscopy measurements (pulse duration
100 μ​s) on a BEC after changing Bset (from 0.4 to 0.2 G), we verify this law andextract a time constant τ =​ 0.98(5) ms. Hence, as is also evolving during th and tf
on a similar timescale. This effect is fully accounted for in the experiments andsimulations, and we report the roton properties as a function of the effective valueof as at th. We use th ranging from 2 to 7 ms. The lower bound on th comes from thetime needed for as to effectively reach the regime of interest. We then consider the
initial evolution for which t h ∕ νy , t h ∕ τcoll ≪ 1, with 1/τcoll being the characteristiccollision rate. We estimate that τcoll ranges typically from 40 to 90 ms in the initial
BECs of Supplementary Table 1 at asi = 61a 0. Experimentally, we observe that theroton, if it ever develops, has developed within the considered range of th.
Imaging procedure. In our experiments, we employ TOF expansion measurements,
accessing the momentum distribution of the gas12, to probe the roton mode
population. We let the gas expand freely for tf =​ 30 ms, which translates the spatial
imaging resolution (~ 3.7 μ​m) into a momentum resolution of ~0.32 μ​m−1 while ourtypical roton momentum verifies krot ≳​  2 μ​m−1. After TOF expansion, we record 2D
absorption pictures of the cloud by means of standard resonant absorption imaging
on the atomic transition at 401 nm. The imaging beam propagates nearly vertically,
with a remaining angle of ~15° compared to the z axis within the xz plane. Thus, the
TOF images essentially probe the spatial density distribution nTOF(x,y,tf ) in the xy
plane. When releasing the cloud (tf =​ 0 ms), we change B back to B =​ 0.3 G (see thesection entitled Quench of the scattering length as). This change enables constantand optimal imaging conditions with a fixed probing procedure (that is, a maximal
absorption cross-section). In addition, the associated increase of as to 57a0 allows
us to minimize the time during which the evolution happens in the small-as regime
where the roton physics develops, such that we effectively probe only the short-time
evolution in this respect. In this work, we use the simple mapping:2
 ℏk t ℏkyt f 
 ℏt 
n∼ (k x , k y ) =  f  n TOF  x f ,, t f  m 
m
 m

(2)

which neglects the initial size of the cloud in the trap and the effect of interparticle
interactions during the free expansion. Using real-time simulations (seeSupplementary Information), we simulate the experimental sequence and are able tocompute both the real momentum distribution from the in-trap wavefunction andthe spatial TOF distribution 30 ms after switching off the trap. Using the mapping ofequation (2) and our experimental parameters, the two calculated distributions arevery similar, and, in particular, the two extracted momenta associated with the roton

signal agree within 5%. This confirms that the interparticle interactions play little
role during the expansion and justifies the use of equation (2).
Fit procedure for the TOF images. For each data point in Figs. 2–4, we record
between 12 and 25 TOF images. By fitting a 2D Gaussian distribution to theindividual images, we extract their origin (kx, ky) =​ (0, 0) and recentre each
image. From the recentred images, we compute the averaged n∼(kx , ky ), fromwhich we characterize the linear roton developing along ky. To do so, we extracta 1D profile n∼1(k y ) by averaging n∼(kx , ky ) over kx within ∣k x∣ ≤ k m = 3.5 μ​m−1:
km
km
n∼1(k y ) = ∫ n∼(k x , k y )dkx ∕ ∫ dk x . To quantitatively analyse the observed roton
−k m
−k m
∼
peaks, we fit a sum of three Gaussian distributions to n1(k y ). One Gaussian accounts
for the central peak and its centre is constrained to k0 ~ 0. Its amplitude is denoted
A0. The two other Gaussians are symmetrically located at k 0 ± k y*, and we constrain
their sizes and amplitudes to be identical, respectively equal to σ* and A*. We focus
on the roton side peaks by constraining ky* > 0.5 μ​m−1 and σ* <​  3 μ​m−1 (peak at finite
momentum of moderate extension compared to the overall distribution).
To analyse the onset and evolution of the roton population (see Fig. 4), we
perform a second run of the fitting procedure, in which we constrain the valueof ky* more strictly. The interval of allowed values is defined for each trapping
geometry on the basis of the results of the first run of the fitting procedure. We
use the results of the (asf , th) configurations where the peaks are clearly visible andwe set the allowed ky* range to that covered by the 95% confidence intervals of thefirst-fitted ky* in these configurations. This constraint enables the fitting procedure
to estimate the residual background population on the relevant momentum range
for the roton physics, even for as >​ a*s (see, for example, Fig. 4a).
Time-rescaling analysis and roton gap estimate. In Fig. 4a,b, we systematicallyanalyse the time evolution of the roton population for various asf and link it to theroton spectrum in a quench picture. The roton population is embodied by the
amplitude A* of the three-Gaussian fit (see the section entitled Fit procedure for
the TOF images), which measures the density n∼1(k rot). A* is observed to initially
increase if asf < as*, its growth rate increases for decreasing asf and as* depends on
the trap geometry.
If dynamically unstable, the krot component is expected to grow exponentiallywith an instantaneous rate 2Im[ε(krot, th)]/ħ. We hence expect an initial growth of
A* of the form:A (asf , t h) ∝ exp 2*∫0 dt Im[Δ ̄(asf , t )] ∕ℏ
th

(3)

̄

where Δ (asf , t h) is the instantaneous value of the overall roton gap,
corresponding to an average over the cloud, after quenching as to asf . Our results
show that the most relevant effect of the quench on the roton spectrum is givenby the reduction of as itself. Hence, the time dependence of Δ is determined byas(t) and, by monitoring A (asf , t h) , one can readily extract the scattering-length*
dependent gap Δ (as), Δ (asf , t ) = Δ (as(t )). To investigate the scaling Δ (as), we
first consider the simpler case of the static and local roton gap Δ. From equation(1) of the main text characterizing this elementary gap, assuming n0, X and Z
fixed, and using the fact that Δ (as = as*) = 0, one can easily obtain that for as in
the vicinity of as*, Δ(as) ∝ (as*−as)1 ∕2. Here we verify this scaling for the global
and dynamical quantity Δ (as), considering the generic power-law dependence

̄

̄

̄

̄

̄

( )
*

̄

β

Im[Δ(as)]/ħ =​  Γδ(as) with δ(as) = a s − a s (as(t ) ≤ as*) , in which the parameters Γ​,a0
as* and β mainly depend on the trap geometry.
f
Our full set of data A (as , t h) for the geometry (νz, λ) =​ (149 Hz, 4.3) enables us*
to assess this scaling. Indeed, equation (3) then reads:A (asf , t h) ∝ exp 2Γ
*


th



∫0 δ(as(t ))dt t

(4)This defines a time rescaling t h → Th = ∫ h δ(as(t ))dt along which all our0experimental data A*(as,th) should collapse in a unique curve, marked by an initial
exponential growth of rate 2Γ. The relevant values of as*, β and Γ are the ones that
result in the best overlap of the data for the initial growth of A* (minimal spread in Th).
To determine as* and β, we then plot our full data set as a function of Th forvarious trial values of these parameters and evaluate the relevance of the trial
couple (as*, β ). Precisely, we assess the dispersion in Th of the full data set fora few fixed A = A i . We use a panel of 10 values of A i within [80, 200] μ​m2,***
corresponding to the range of the initial growth of A* for the geometry of Fig. 4a,b.
We interpolate the experimental data A*(Th) for each asf using piecewise cubic
polynomial interpolation, and extract the corresponding set of Thi at which
A (Thi ) = A i . For each i, we evaluate the spread of the Thi (asf ) by two complementary**
quantities: the square root of their variance and the discrepancy between the
maximal and minimal value of the set. We finally estimate the accuracy of a unified
dependence A*(Th) for a given (as*, β ) via the geometrical average of these twocomplementary quantities for all of the ten A i values. We fit the relevant as* and*β by minimizing this averaged spread. For the experimental data of Fig. 4a,b, thisresults in as* = 53.0(4)a 0 and β =​  0.55(8).

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Letters

Nature Physics

Using these values of as* and β, we observe that the initial growth of A* extends
for Th ≤​ 3 ms (before saturating and decreasing for longer Th). We estimate Γ byperforming an exponential fit on the full set of data A*(Th) with Th ≤​ 3 ms. This gives
Γ =​ 465(83) s−1. Note that Fig. 4a,b show only 5 values of asf while the reported analysis
considers all available experimental data (that is, 11 values between 31a0 and 52a0).
From the formula with the estimated as*, β and Γ, we can then compute the2
global roton gap Δ . The extracted value is meaningful only for Δ < 0. For theexperiments, we also restrict its relevance to Th ≤​ 3 ms so that, for example, in theinset of Fig. 4c, Im[Δ ] is shown up to th =​ 4 ms, after which the roton population isobserved to deviate from the exponential growth picture (see Figs. 3b and 4a). Notethat the Im[Δ ] estimates at the different th are not independent.
In Fig. 2, we report on the roton momentum as a function of the systemcharacteristics (as and ℓz ). Here we estimate krot from a th value that is individually
optimized for each as and ℓz investigated (largest visibility). We point out that the
selected th corresponds, in the A* dynamics, to the late stage of the exponential
growth, close to the maximum (that is, Th ~ 3 ms). Here, the atom loss remains at
the level of a few per cent.
In Fig. 2d, we show the value of krot at the onset of the population of the rotonmode (that is, at as = as*). We estimate as* for each trap geometry by analysing theevolution of the roton population with as. We employ a simplified approach withrespect to the one in Fig. 4 (see Supplementary Information), which we estimate
to lead to a maximum underestimate for as* of about 1.5a0, lying within our
experimental uncertainty on as (see Supplementary Table 2).

̄

̄

̄

̄

Analytical dispersion relation for an infinite axially elongated geometry.
Equation (1) in the main text results from a similar procedure to that used in ref. 3
for rotons in infinite pancake traps. We consider a dBEC homogeneous along y but
harmonically confined with frequencies νx and νz along x and z. For sufficiently
strong interactions, the BEC is in the TF regime on the xz plane, in which the BEC
wavefunction acquires the form ψ0(ρ) = n(ρ) , with n(ρ) =​  n0(1 −​  (x/X)2 −​  (z/Z)2),
where X and Z are the TF radii, and ρ =​  (x,z). The calculation of n0, X and Z isdetailed at the end of this section.
Due to the axial homogeneity, the elementary excitations of the Bogoliubov–
de Gennes spectrum have a defined axial momentum ky, and take the formδψ (r , t ) = u(ρ)eikyy −iεt ∕ℏ−v(ρ)e−ikyy +iεt ∕ℏ, where u and v denote the amplitudesof the spatial modes oscillating in time with characteristic frequency ε/ħ (seeSupplementary Information for more details). We consider the standard NLGPE
without beyond mean-field correction and three-body losses, and insert the
perturbed solution ψ(r,t) =​  (ψ0(ρ) +​  ηδψ(r,t))e−iμt/ħ, where μ is the chemical potential
associated with ψ0 and η ≪ 1. After linearization, we obtain the Bogoliubov–deGennes equations for f±(ρ) =​  u(ρ) ±​  v(ρ):εf− (ρ) = Hkin f+ (ρ)(5)εf+ (ρ) = Hkin f− (ρ) + Hint[f− (ρ)]

(6)

where
Hkin f± (ρ) =∇2 ψ0 
ℏ2  2
 f (ρ)
−∇ + k y2 +2m 
ψ0  ±Hint [f− (ρ)] = 2∫ d3r ′U (r −r ′)e−iky(y−y′)ψ0(ρ)ψ0(ρ′)f− (ρ′)

(7)

(8)

2 
where U (r) = g δ(r) + 3εdd 1 − 3cos3 θ  is the binary interaction potential for two

4π

∣r∣
particles separated by r including both contact and dipolar interactions and4π ℏ2 ag = m s (see Supplementary Information). θ is the angle between r and themagnetization axis (z).
Employing f+ (ρ) =​  W(ρ)ψ0(ρ), and for k y ≫ 1 ∕ X , 1 ∕ Z , we obtain thefollowing equation for the function W(ρ):ε0(ky )2 = E(k y )2 + E 02
ℏ2

(

(10)

)

1
1
+ 2 . In the vicinity of εdd =​ 1, the effect of the last term in
2m X 2Zequation (9) may be evaluated perturbatively, resulting in the dispersion

with E 02 = 2gεddn 0ε(k y )2 ≃ ε0(ky )2 −2E I E(ky )

(11)

with
2
E I = gn 0(εdd −1)3This expression for the dispersion presents a roton minimum for εdd >​  1 at
1
k rot = ℏ 2mE I . Expanding equation (11) in the vicinity of the roton minimum,
2 2
1  d ε (ky) ε(k y )2 ≃ ε(k rot)2 + 2  2 , we obtain equation (1) of the main text, with dky  ky=krot

1

Δ = ε(k rot) = E 02−E I2 . At the instability, Δ =​  0, and k rot = ℏ 2mE 0 .

Employing a similar procedure as in ref. , we obtain that the BEC aspect ratio
χ =​  Z/X fulfils:15 (1−ε )(1 + χ )2 + 3ε

dd
dd 
χ 2 
= λ⊥2
2
2
(1
2
)(1
)
3
+
ε
+
χ
−
ε
χ


dd
dd

(12)

2 

(1 + 2ε )− 3εddχ 
dd(1 + χ )2 (13)with λ⊥ =​  νx/νz and
Z2=

gn 0

2π mνz2 
2These two equations fully determine the TF solution for the given εdd, gn0 and λ⊥.
By inserting the expressions of X2 and Z2 into E0, we find for εdd ≃ 1:
E 02 =


h2νz2
1 

(1 + χ )2 λ⊥2 +6
1 + 2χ 

(9)

(14)

whereas χ simplifies to χ = λ⊥ (1 + 1 + 1 ∕ λ⊥ ) . As a result, at the instability, k rotℓz
depends only on the transverse confinement aspect ratio λ⊥, giving the geometricalfactor κ:
2
κ = k rotℓz =   3 

1 ∕4

1 ∕4

1 

1 + χ λ⊥2 +
1 + 2χ (15)

Self-similar model and instantaneous roton spectrum. The dynamics of thecondensate during and after the quench is crucial for the understanding of theresulting momentum peaks and their direct relation to the growth of roton
excitations. Due to its large characteristic momentum, the roton spectrum may
be evaluated at any time using local density approximation. On the other hand,the evolution of the local density is determined by the global dynamics of thecondensate induced by the quench. Hence, interestingly, the analysis of the effect of
the quench on the roton spectrum may be performed by combining a self-similar
theory of the global dynamics33, describing the evolution at low ky, and the modelof the roton spectrum developed in the section entitled Analytical dispersion
relation for an infinite axially elongated geometry, accounting for the high-ky
region of interest. We note that this analysis approximates the real-time evolution
provided by the standard NLGPE12,34,35. This treatment is valid for moderateenough quenches so that each of the descriptions applies (see below for an in-depth
discussion) and as long as the high-k mode population only minimally impact the
BEC dynamics, that is, for short-enough timescales.
Here, we assume that the condensate preserves its TF shape during theevolution:  x  2  y  2  z  2 − 
 −  n(r, t ) = n 0(t ) 1− 
 
 
 
  X (t )   Y (t )   Z(t )  2
1 ∂ 2W 
2  1 ∂ W
0 = 2gn 0 (1−x∼2−∼z )  2 ∼2 + 2 2 Z ∂∼z  X ∂x 1 ∂W
1 ∂W 
− gn 0(1 + 2εdd)  2 x∼ ∼ + 2 z∼ ∼  X ∂x
Z ∂z 
 2m
 1
1 
+  2 (ε2−E(ky )2 ) −2gεddn 0  2 + 2  
 X
 ℏ
Z  
4m
2
gn 0(1−εdd)E(ky )(1−x∼2−∼z )−ℏ2where x∼ = x ∕ X , ∼z = z ∕ Z and E(ky ) = ℏ2k y2 ∕ 2m. For εdd =​ 1, the last term ofequation (9) vanishes. In that case, the lowest-energy solution is given by W =​  1,
whose eigen energy builds, as a function of ky, the dispersion ε0(ky) with

(16)

where X(t), Y(t) and Z(t) are the rescaled TF radii. Their evolution after the quench
can be deduced from solving the hydrodynamics equations (see Supplementary
Information). Prior to the quench of as, the stationary TF solution is obtained
from the self-consistent equations deduced from the hydrodynamics equations bycancelling all time derivatives. Solving these equations and using normalization
provides X(0), Y(0), Z(0) and n(0) for known atom number N and trap frequencies
νx,y,z. We use this stationary solution as the initial condition at the start of thequench (t =​ 0), and solve the system of differential equations resulting from the

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Letters

Nature Physics
hydrodynamics evolution to obtain the rescaled radii. An example for the relevant
parameters of Fig. 3 is shown in Supplementary Fig. 3.
For a given position y, we then evaluate the local TF profile:n(r,t) =​  n0(y,t)[1 −​  (x/X(y,t))2 −​  (z/Z(y,t))2], where n0(y,t) =​  n0(t)[1 −​  (y/Y(t))2],
X(y,t) =​  X(t)[1 −​  (y/Y(t))2]1/2 and Z(y,t) =​  Z(t)[1 −​  (y/Y(t))2]1/2, withn0(t) =​  n0(0)X(0)Y(0)Z(0)/X(t)Y(t)Z(t). We may then employ a local density
approximation and evaluate the local (in y) instantaneous roton spectrum using
the results of the section entitled Analytical dispersion relation for an infinite
axially elongated geometry:ε 2 (k y , y , t )(hνz )2 lz4≃

k y4 8π− n (y , t )(add−as(t ))k y2
4 3 0

 1
1 
+ 4πn 0(y , t )add +
2Z(y , t )2  X (y , t )(17)

The roton population grows exponentially when the spectrum becomes imaginary,leading to the appearance of peaks at large momenta. From the local instantaneous
roton spectrum, we may estimate the associated peak in the momentum
distribution asn∼1(k y , t ) ~



t



∫ dy e2 ∫0 dt Im[ε (ky,y,t )] −1
′

′





(18)

where we assume equal seeding for all unstable modes. We then evaluate the total
population of the peak N (t ) = ∫ dk yn∼1 (k y , t ) , as well as the roton momentum*n∼1 (ky, t).
⟨k⟩ = ∫ dk yk y
N (t)

*

The previously discussed SSM does not properly describe the evolution when
the shrinking of the condensate, and hence the increase of its peak density, is toolarge. As the gas gets dense, quantum fluctuations, which are not considered in the
SSM, introduce in our experiments an effective repulsion that crucially prevents large
densities or condensate collapse (see Supplementary Information). This limits the useof the SSM to the vicinity of as*. Moreover, for our tightest trap, the SSM results arealso unreliable, since the theory predicts condensate collapse before the momentum
peak develops. That said, as shown in the main text, the SSM provides not only aclear intuitive understanding of our measurements, but to a very large extent alsoa very good quantitative agreement with both our experimental results and our NS
based on a generalized NLGPE, which is detailed in the Supplementary Information.
Data availability. The data that support the plots within this paper and otherfindings of this study are available from the corresponding author upon
reasonable request.

References

31. Chin, C., Grimm, R., Julienne, P. S. & Tiesinga, E. Feshbach resonances inultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010).
32. Lahaye, T. et al. d-wave collapse and explosion of a dipolar Bose–Einsteincondensate. Phys. Rev. Lett. 101, 080401 (2008).
33. Giovanazzi, S. et al. Expansion dynamics of a dipolar Bose–Einsteincondensate. Phys. Rev. A 74, 013621 (2006).
34. Baranov, M. Theoretical progress in many-body physics with ultracold dipolargases. Phys. Rep. 464, 71–111 (2008).
35. Blakie, P., Bradley, A., Davis, M., Ballagh, R. & Gardiner, C. Dynamics andstatistical mechanics of ultra-cold Bose gases using c-field techniques.
Adv. Phys. 57, 363–455 (2008).

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Letters
SUPPLEMENTARY INFORMATION
https://doi.org/10.1038/s41567-018-0054-7

In the format provided by the authors and unedited.

Observation of roton mode population in a dipolar
quantum gas
L. Chomaz1, R. M. W. van Bijnen2, D. Petter1, G. Faraoni1,3, S. Baier1, J. H. Becher1, M. J. Mark1,2,
F. Wächtler4, L. Santos4 and F. Ferlaino1,2*Institut für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria. 2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie
der Wissenschaften, Innsbruck, Austria. 3Dipartimento di Fisica e Astronomia, Università di Firenze, Sesto Fiorentino, Italy. 4Institut für Theoretische
Physik, Leibniz Universität Hannover, Hannover, Germany. *e-mail: francesca.ferlaino@uibk.ac.at
1

Nature Physics | www.nature.com/naturephysics© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

Supplementary Information for ’Observation of Roton Mode Population in a Dipolar Quantum
Gas’
L. Chomaz1 , R. M. W. van Bijnen2 , D. Petter1 , G. Faraoni1,3 , S. Baier1 ,J. H. Becher1 , M. J. Mark1,2 , F. Wächtler4 , L. Santos4 , F. Ferlaino1,2,∗

1 Institut für Experimentalphysik,Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2 Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3 Dipartimento di Fisica e Astronomia, Università di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy
4 Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany(Dated: January 18, 2018)

A.

Production of 166 Er BECs.

We prepare a 166 Er BEC similarly to Refs. [1, 2]. From anarrow-line magneto-optical trap with 3 × 107 166 Er atoms,
automatically spin-polarized in their absolute lowest Zeeman
sub-level [3], at about 10µK, we directly load the atomic gas
in a crossed optical dipole trap (ODT) with an efficiency ofmore than 30%. A uniform magnetic field, B, is permanently
applied along the vertical z axis, fixing the dipole orientation,
while its value is varied during the experimental sequence, totune as (see Method Quench of the scattering length as ).
We achieve condensation by means of evaporative cooling in
the crossed ODT at B = 1.9 G (as = 80(2) a0 ). During theevaporation procedure, we first change the power and then theellipticity of one of the ODT beams (see Section B). The final
atom number, typically 105 , and condensed fraction, typically
70%, are assessed by fitting the time-of-flight (ToF) absorption images of the gas to a bimodal distribution, sum of a TF
profile and a broad Gaussian background.
B.

Details on the trapping geometries.

The harmonic potential V (r) = 2mπ 2 (νx2 x2 + νy2 y2 + νz2 z2 )trapping the 166 Er atoms results from the crossing of two
red-detuned laser beams of 1064 nm wavelength at their respective focii; the hODTb propagating along the y-axis andthe vODTb, propagating nearly collinear to the z-axis. The
vODTb has a maximum power of 7 W and an elliptical profile with waists of 110 and 55 µm along x and y respectively.
The hODTb has a maximum power of 24 W, a vertical waist
wz = 18 µm, and a controllable horizontal waist, wx = λ 0 wz .
The ellipticity λ 0 can be tuned from 1.57 to 15 by time averaging the frequency of the first-order deflection of an AcoustoOptic Modulator [4]. By adjusting independently λ 0 and thepowers of the vODTb and of the hODTb, we can control
the geometry of the trap. Precisely, νy is essentially set by
the vODTb power, νz by that of the hODTb, and νx is controlled by both the power and ellipticity of the hODTb, withνz /νx ≈ λ 0 .
We use the tuning of the hODTb power and ellipticity toperform evaporative cooling to quantum degeneracy (see Section A). After reaching condensation, we again modify thebeam parameters to shape the trap into an axially elongatedconfiguration (νy  νx , νz ). The trapping geometries probedin the experiments, (λ , νx , νy , νz ), are reported in Supplemen-tary Table 1. They are achieved by changing the hODTb
power with λ 0 = 1.57 and the vODTb power set to its maximum so that νy and νz /νx are kept roughly constant. Only
the green triangle in Fig. 2d is obtained in a distinct configuration, with a vODTb power of 2W, leading to (νx , νy , νz ) =(156, 17, 198) Hz and λ = 11.6. The (λ , νx , νy , νz ) are experimentally calibrated via exciting and probing the center-ofmass oscillation of thermal samples. We note that the final
atom number N, BEC fraction f , and temperature T after theshaping procedure depend on the final configurations, as detailed in Supplementary Table 1. T is extracted from the evolution with the ToF duration tf of the size of the background
Gaussian in the TF-plus-Gaussian bimodal fit to the corresponding ToF images of the gas. The values of Nc = f N and
T are used for the initial states ψi of our real-time simulations(see Section G).
Supplementary Table I — dBEC parameters for the experimental measurements (Figs. 2-4). The typical statistical uncertainties on νx and νz are below 1 %, and can be up to 10 %for νy . The experimental repeatability results in 5-to-10 %shot-to-shot fluctuations of N, f and T .
λ
4.3
4.9
10.2
14.3
21.3
29.7

νx (Hz ) νy (Hz ) νz (Hz ) N (104 ) f (%) T (nK)
114
154
183
267
357
432

35
40
30
32
30
26

149
195
306
456
638
771

9
10
11
8.6
8.4
7

66
65
62
50
36
20

45
50
104
150
179
171

C. a∗s for various trap geometries.

We estimate a∗s for each trap geometry of Fig. 2d by studying the roton population as a function of as . When decreasing as , the roton population and in particular its contrast
C = A∗ /A0 (see Method Fit procedure for the ToF images)
exhibits a sharp increase from an essentially zero and flat contrast. We extract as∗,c as the value of the scattering length
corresponding to the onset of the increase of the contrast forth = 3 ms; see Supplementary Fig. 1 and Supplementary Table 2. This is a simplified approach with respect to the one in
Fig. 4 (see Method Time-rescaling analysis and roton gap

2estimate.), which we estimate to lead to a maximum underestimate for a∗s of about 1.5 a0 , which lies within our experimental uncertainty on as .
Supplementary Table II — Roton softening threshold forthe geometries realized in the experiment (Table I), deduced
from the C fits for the experimental onset of the population
∗,st
(a∗,c
s ) and from imaginary time NS (as ). The fit uncertainties
on the experimental values are typically of 0.2a0 , except for
the last trap it is of 0.8 a0 .
λ νz (Hz ) as∗,c (a0 ) a∗,sts (a0 )
4.3
149
51.4
50
10.2 30650.2
4614.3 45649
4321.3 63847.7
3929.7 771
45.2
30

Contrast, C

0,4
0,3
0,2
0,1
40
45
50
Scattering length, as(a0)55Supplementary Figure 1 — Onset of the roton population. Evolution of C with as at th = 3 ms, in the geometries (νz , λ ) = (149 Hz, 4.3) (green squares) and (νz , λ ) =(456 Hz, 14.4) (blue circles). The error bars correspond to the
propagated errors from the 95% confidence interval on A∗,0
from the three-Gauss fit. We empirically fit a linear step function to identify a∗,cs (line).

D.

Generalized Non-local Gross-Pitaevskii equation.

Our theory is based on an extended version of the non-linear
Gross-Pitaevskii Equation (NLGPE)
ih̄



2

h̄ ∇2

∂ ψ(r,t)= −+V (r) +
∂t
2m

Z



3εdd 1 − 3 cos2 θU(r) = g δ (r) +
,
4π
|r|3

(3)

for two particles separated by r [6]. The first term corresponds
to contact interactions between the particles with strength2
g = 4π h̄m as . The DDI gives rise to the second term, which
depends on both the distance and orientation (angle θ ) of
the vector r compared to the polarisation axis (z axis) of thedipoles. Most properties of dBECs are well captured by thisstandard NLGPE (mean-field) [5, 6].
Recent experimental and theoretical results, however, haveestablished the importance of accounting for quantum fluctuations in dBECs [1, 7–10]. Their effect can be included in the
NLGPE in a mean field treatment through a Lee-Huang-Yang
correction √to the chemical potential, ∆µ[n] = 32g(nas )3/2 (1 +23εdd /2)/3 π, which is obtained under a local density approximation [11, 12]. The accuracy of this mean field treatment
has been established, e.g., in Refs. [8–10], and has proven succesful in explaining recent experimental results [1, 7]. The final nonlinear term in the extended NLGPE accounts for threebody losses [13], with an experimentally determined loss parameter L3 , which is dependent on as and typically of the order
L3 ' 10−41 m6 s−1 , as reported in Ref. [1].

0,5

0,0
35

governing the evolution of a macroscopically occupied wavefunction ψ(r,t), with corresponding atomic density n(r,t) =|ψ(r,t)|2 at position r and time t. The standard dipolar NLGPE includes the kinetic energy, external trap potential andthe mean-field effect of the interactions [5, 6]. These correspond to the three first terms of Eq. (1), where the mean-fieldinteraction potential takes the form of a convolution of n withthe binary interaction potential

dr0U(r − r0 )n(r0 )L3 + ∆µ[n] − ih̄ n2 ψ(r,t) (1)2L3 2 
≡ ĤGP [ψ] − ih̄ n ψ(r,t),
(2)
2

E.

Bogoliubov-de Gennes spectrum.

Collective excitations of the dBEC are obtained by linearising the NLGPE (see Section D) around a stationary stateψ0 , which can be obtained by imaginary time propagation(see Section G). We write ψ = e−iµt/h̄ (ψ0 + η[ue−iεt/h̄ −v∗ e+iεt/h̄ ]), where µ is the chemical potential associated withstate ψ0 , and u, v are spatial modes oscillating in time withcharacteristic frequency ε/h̄ and η  1 [14]. Inserting thisansatz in the NLGPE, and retaining only terms up to linear
order in η we obtain the Bogoliubov-de Gennes (BdG) equations!
!
!
ĤGP [ψ0 ] + A
−A
u
u
=ε
, (4)
A
−ĤGP [ψ0 ] − Avv
where the operator A, acting on a function f and evaluated atpoint r, is defined as(A f )(r) =Z

dr0 ψ0 (r0 )U(r − r0 ) f (r0 )ψ0 (r)
16 3/23 2+ √ gas1 + εdd |ψ0 (r)|3 f (r). (5)2π3The above equations constitute an eigenvalue problem, which
we solve numerically using the Arnoldi method to obtaineigenmodes (u, v) and corresponding excitation energies ε.
The equations presented here are a generalization of the BdG
equations for dipolar systems as derived in Ref. [14], to include the LHY correction accounting for quantum fluctuations.
In order to depict the spectrum as a quasi-dispersion relation even in the presence of an axial confinement, we associate to each elementary excitation an effective momentum(eff)ky = hky2 i1/2 [15]. The spectrum is discrete with phonon(eff)(eff)

like collective modes at low ky . For higher ky , the spectrum flattens, but eventually bends upwards again due to thedominant kinetic energy. Instead of developing a smooth minimum, roton excitations appear as isolated low-lying modes
at intermediate momenta that depart from the overall spectrum [15]. These so-called roton fingers are related to confinement of the roton modes in the inhomogeneous BEC ofprofile n0 (y) [16], see also discussion in the main text.
The confinement is evident from the BdG calculations, inwhich the lowest roton mode forms a short-wavelength density modulation localized at the trap center (Supplementary
Fig. 2e). This contrasts with phonon modes for which themodulation is delocalised over the entire condensate (Supplementary Fig. 2c). The excited states shown in Supplementary Fig. 2b-e correspond to the density |ψ0 + η(u − v∗ )|2 , forparticular pairs of (u, v) corresponding to phonon and rotonmodes (Supplementary Fig. 2c and e), and exemplary modes
at higher energies (Supplementary Fig. 2b and d). Even while
the amplitude η = 0.2 of the excited modes is taken to beequal in Figs. 2b - 2e, the roton excitation (Supplementary
Fig. 2e) leads to markedly larger local density modulations
than the phonon excitation (Supplementary Fig. 2c). The ToF
signatures in Supplementary Fig. 2 are computed by letting
the wave function of the excitation, η(u − v∗ ), expand ballistically for 30ms, i.e. neglecting interactions during the expansion. The resulting density |η(u − v∗ )|2 is then plotted (Supplementary Fig. 2b1 - e1).
Carrying out the numerical BdG spectrum calculation forvarious trap parameters, scattering lengths and atom numbers
confirms the geometrical scaling krot ∼ 1/`z , as well as its
weak dependence on as close to the instability as expectedfrom the analytical model. We note in particular a good quantitative agreement of krot `z in Supplementary Fig. 2 with the(stationary) analytic factor κ = 1.3 for this same trap geometry (see Method Analytical dispersion relation for an infinite axially elongated geometry). A qualitative agreementof the BdG stationary spectrum with the experimental observations is however not fully reached because of the role of the
dynamics. A treatment of this effect is provided by the SSM
and NS as discussed in the main text (see also the corresponding Method and Section G).
Finally, an alternative way of visualizing the excitation
spectrum is achieved by computing the dynamical structure

factor at T = 0 for each mode (u, v) [17, 18],
S(k, ω 0 ) = ∑jZ2dr[u∗ (r) + v∗ (r)]eik·r ψ0 (r) δ (ω 0 − ω),(6)where ω = ε/h̄. The structure factor of the axial modes of our
finite trapped system (Supplementary Fig. 2a) presents features resembling a roton-maxon spectrum, its continuum limit
being a single-valued spectrum as depicted in Fig. 1b. To enhance the visibility, the delta function in Eq. (6) is replaced
by a Gaussian with a small finite width in ω. Since the dynamical structure factor determines the response of the system
when probed at specific energies and momenta, such as e.g.
in Bragg spectroscopy experiments [18, 19], it is interesting
to note the difference in amplitude between the roton modes
and other parts of the spectrum. In particular, in our quench
experiments where the system is effectively driven at a range
of energies and momenta, one would expect the strongest response from the roton modes.
F.

Self-similar dynamics of the BEC.

In our self-similar model we account for the dynamics ofthe BEC profile and its effect on the roton spectrum (see details in the corresponding Method). To do so, we assume thatthe condensate preserves its TF shape during the evolution:
"

2 
 
2 #
x
y 2
z
n(r,t) = n0 (t) 1 −
,
−
−
X(t)
Y (t)
Z(t)
(7)
where X(t) = bx (t)X0 , Y (t) = by (t)Y0 , and Z(t) = bz (t)Z0
are the re-scaled TF radii, with bx,y,z (t) the scaling coefficients (bx,y,z (0) = 1). The corresponding hydrodynamic equations reduce to:mẌ 2 Ÿ 2 Z̈ 2∇
x + y + z = −∇µ(r,t),
(8)
2
X
y
Z
R

where µ(r,t) = V (r) + gn(r,t) + d 3 r0Vdd (r − r0 )n(r0 ,t) is
the local chemical potential. The latter acquires the form: y 2
 z 2  x 2µ(r,t) = V (r) + gn0 D − A
−B
−C
,
X
Y
Z
(9)
where D(t) = 1 + εdd F1 , A(t) = 1 + εdd (F1 − F2 + F3 ), B(t) =1 + εdd (F1 − F2 − F3 ), and C(t) = 1 + εdd (F1 + 2F2 ). Here wehave introduced the functions:
"
#
Z 1
3u2
F1 (χx , χy ) = du p
−1 ,
(10)
0
α 2 −β 2#"Z 1  2
3u2
3u −1
p
−1 ,
(11)
F2 (χx , χy ) = du20
α 2 −β 2#
" p 2Z 1  2
9u (1−u2 )α −β 2 − αpF3 (χx , χy ) = du, (12)
40
β α 2 −β 2

0
5

0

-5

1.5
1 c
0.5
0
-20

0
5

0
m -1 )

5n (a.u.)
1

0

0.8

0
y ( m)

1020
n~ (a.u.)

0

-5

0.2

1

c1
0
k y ( m -1 )

5

0
5

0

-10

0
y ( m)

0

0

0.5

1

1.5

2

1.5
1 e
0.5
0
-20

0
5

(eff)
ky
z

20
n~ (a.u.)
1

-5

-5

10

d1
ky (

0.4
-10

-5

0.6

n (a.u.)n (a.u.)20
n (a.u.)
~

1

ky (

k x ( m -1 )10

k x ( m -1 )0y ( m)-5-5

S (arb.u.)

1.0
-10

1.5
1 d
0.5
0
-20

k x ( m -1 )

1.5
1
0.5
0
-20

E (h z )k x ( m -1 )n (a.u.)4-100
m -1 )0y ( m)

5

10

20
n~ (a.u.)
1

e1

0

-5

0
k y ( m -1 )5

Supplementary Figure 2 — BdG excitation spectrum. a, Excitation spectrum of the ground state of a BEC with N = 50.000

166 Er atoms in a trap with (ν , ν , ν ) = (267, 32, 456) Hz and scattering length a = 43.75 a , obtained by numerically solving
x y z
s
0

1 b̈x
1 A(t)= −bx + 2,(2π)2 νx2
bx by bz A(0)

(13)

1 b̈y
1 B(t)= −by +,
(2π)2 νy2
bx b2y bz B(0)(14)

1 C(t)
1 b̈z= −bz +.
(2π)2 νz2
bx by b2z C(0)(15)

Prior to the quench of as , the stationary TF solution
pis obtained
from the self-consistent equations χx (0) = ννxz C(0)/A(0)ν pand χy (0) = νyz C(0)/B(0). Solving these equations andusing normalisation provides X(0), Y (0), Z(0), and n(0) for
known numbers of atoms N and trap frequencies νx,y,z . We use
this stationary solution as the initial condition at the start of
the quench (t = 0), and solve the system of differential equations (13), (14) and (15) to obtain the scaling coefficients. An
example for the relevant parameters of Fig. 3 is shown in Supplementary Fig. 3.
G.

1.2

60

1.1

58

1

56

0.9
X
Y

54
52

0.8
0.7

Z,
n0 ,

50
0

1

0.6

as

2

3

th(ms)

4

5

6

Rescaled parameterswith χx (t) = Z(t)/X(t), χy (t) = Z(t)/Y (t), α(χx , χy , u) =(χx2 + χy2 )(1 − u2 )/2 + u2 , and β (χx , χy , u) = (χx2 − χy2 )(1 −u2 )/2. Substituting Eq. (9) into Eq. (8) we obtain a closed setof equations for the scaling parameters:

Scattering length, as(a0)the BdG equations. Roton modes appear as isolated modes lying below the main branch of the spectrum, forming a ’roton
finger’. The dynamic structure factor S corresponding to each of the modes is indicated with colored shading. Highlighted ingreen are several exemplary modes, with panels b - e showing the corresponding excited state density modulation (black dashedline indicating the ground state, red solid line the excited state), and panels b1 - e1, the corresponding momentum distribution
from 30 ms ToF expansion (Methods). b, b1 intermediate mode, c, c1 phonon mode, d, d1 single particle mode, e, e1 roton
mode.

0.5

Supplementary Figure 3 — Self-similar dynamics inquench. SSM results after a quench from ais = 61 a0 to
afs = 50 a0 in the trap (νz , λ ) = (149 Hz, 4.3) and using the experimental cloud characteristics (configuration of Fig. 3). For
reference, we plot the experimentally known as (th ) (circles,
left axis). We show the 3D TF radii X (down triangles), Y(diamonds), Z (up triangles) and n0 (squares) renormalized bytheir th = 0-values (right axis). The compression mainly occurs along X. The subsequent increase of n0 is less than 2 %when ∆(y = 0) touches zero in this configuration (Fig. 3a).
This demonstrates that the dominant effect on the roton spectrum at the instability comes from the reduction of as itself,even in the case of afs relatively close to a∗s .

Numerical simulations of the evolution.

Our simulations of the NLGPE are performed using a
split operator technique. The evolution operator over a time∆t (∆t → i∆t for imaginary time evolution) may be approximately split as e−iĤ∆t/h̄ = e−iT̂ ∆t/h̄ e−iV̂ ∆t/h̄ + O(∆t 2 ). In thisexpression, T̂ is the kinetic energy term, and V̂ the potential energy. The effective potential energy for the evolution
is given by the sum of external potential, interparticle interactions, local LHY correction, and three-body losses (see Section D).

5
We first evaluate using imaginary time evolution the initial BEC wavefunction, ψ0 (r), prior to the quench of as . The
initial wavefunction for the subsequent real-time evolution isthen constructed via ψi = ψ0 + ∆ψ, where ∆ψ accounts forthermal fluctuations, which we simulate by populating the excited states of the system as described in Ref. [20]. The excited states used in this procedure are obtained from the full
BdG calculation detailed in Section E. Starting with this initial wavefunction ψi , we mimic as close as possible the conditions of our experiments, including ramping, holding, and ToF
times. In particular, we include the experimentally calibrated
as (t) (Method Quench of the scattering length as ). Moreover, for the value of the three-body loss coefficient L3 we use
a linear fit of the experimentally determined values [1]. Fromthe simulated evolution, we obtain the 3D wavefunction of thegas as a function of th , ψ(r, th ), from which we can extract the
spatial and momentum distributions.
The simulation of the ToF expansion is performed in two
steps. First we use a multi-grid analysis in order to rescale
the size of the numerical box as the cloud expands during the
ToF expansion. After some expansion time the density drops
significantly, and the subsequent evolution can be readily calculated via e−iT̂ t/h̄ . Our NS show clearly that the effect of
nonlinearity is small during the first stages of the evolution,and hence that the ToF expansion indeed may be employed

to image the momentum distribution of the condensate at thetime in which the trap is opened.
We evaluate
the integrated momentum distributionRñ(ky ,th ) = dkx dkz |ψ̃(k,th )|2 with ψ̃(k,th ) the Fouriertransform of ψ(r, th ). After th of a few ms, ñ(ky ,th ) shows
clear roton peaks. The exact th value for the peak emergence
depends on the gas characteristics (in particular T ) and on as .
We evaluate the roton momentum as the mean value of the
momentum in the roton peak.
From the imaginary-time evolution simulations, we are alsoable to predict the as = a∗,st
threshold for the mean-field ins
stability of the BEC, which corresponds to the absence ofa mean-field stable solution [9, 10]. To find this instability
boundary, we proceed by steps. We start by calculating the
ground-state solution for a given as that we know to be wellwithin the stable regime. We then reduce as in small steps
and successively calculate the corresponding ground-state solution using the solution of the previous step as starting condition. We do so until no mean-field stable solution can befound. The predicted as∗,st are reported in Supplementary Table II. For the theory predictions shown in Fig. 2d (NS andSSM; see Method Analytical dispersion relation for an infinite axially elongated geometry), we use quenched as values such that the instability boundary is just slightly crossed,
as = as∗,st − 1a0 .

[1] Chomaz, L. et al. Quantum-fluctuation-driven crossover from adilute Bose-Einstein condensate to a macrodroplet in a dipolar
quantum fluid. Phys. Rev. X 6, 041039 (2016).
[2] Aikawa, K. et al. Bose-Einstein condensation of erbium. Phys.
Rev. Lett. 108, 210401 (2012).
[3] Frisch, A. et al. Narrow-line magneto-optical trap for erbium.
Phys. Rev. A 85, 051401 (2012).
[4] Ahmadi, P., Timmons, B. P. & Summy, G. S. Geometrical effects in the loading of an optical atom trap. Phys. Rev. A 72,
023411 (2005).
[5] Pitaevskii, L. & Stringari, S. Bose-Einstein condensation andsuperfluidity, vol. 164 (Oxford University Press, 2016).
[6] Baranov, M. Theoretical progress in many-body physics withultracold dipolar gases. Physics Reports 464, 71 – 111 (2008).
[7] Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau,
T. Observation of quantum droplets in a strongly dipolar Bosegas. Phys. Rev. Lett. 116, 215301 (2016).
[8] Wächtler, F. & Santos, L. Quantum filaments in dipolar BoseEinstein condensates. Phys. Rev. A 93, 061603 (2016).
[9] Wächtler, F. & Santos, L. Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einsteincondensates. Phys. Rev. A 94, 043618 (2016).
[10] Bisset, R. N., Wilson, R. M., Baillie, D. & Blakie, P. B. Groundstate phase diagram of a dipolar condensate with quantum fluctuations. Phys. Rev. A 94, 033619 (2016).
[11] Lima, A. R. P. & Pelster, A. Quantum fluctuations in dipolar
Bose gases. Phys. Rev. A 84, 041604 (2011).

[12] Lima, A. R. P. & Pelster, A. Beyond mean-field low-lying excitations of dipolar Bose gases. Phys. Rev. A 86, 063609 (2012).
[13] Kagan, Y., Muryshev, A. E. & Shlyapnikov, G. V. Collapse andBose-Einstein condensation in a trapped Bose gas with negative
scattering length. Phys. Rev. Lett. 81, 933–937 (1998).
[14] Ronen, S., Bortolotti, D. C. E. & Bohn, J. L. Bogoliubov modesof a dipolar condensate in a cylindrical trap. Phys. Rev. A 74,
013623 (2006).
[15] Bisset, R. N., Baillie, D. & Blakie, P. B. Roton excitations in
a trapped dipolar Bose-Einstein condensate. Phys. Rev. A 88,
043606 (2013).
[16] Jona-Lasinio, M., Łakomy, K. & Santos, L. Roton confinementin trapped dipolar Bose-Einstein condensates. Phys. Rev. A 88,
013619 (2013).
[17] Zambelli, F., Pitaevskii, L., Stamper-Kurn, D. M. & Stringari,
S. Dynamic structure factor and momentum distribution of atrapped Bose gas. Phys. Rev. A 61, 063608 (2000).
[18] Blakie, P. B., Baillie, D. & Bisset, R. N. Roton spectroscopy in aharmonically trapped dipolar Bose-Einstein condensate. Phys.
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[20] Blakie, P., Bradley, A., Davis, M., Ballagh, R. & Gardiner, C.
Dynamics and statistical mechanics of ultra-cold Bose gases
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4.5. PUBLICATION: PROBING THE ROTON EXCITATION SPECTRUM OF A
STABLE DIPOLAR BOSE GAS

77

4.5. Publication: Probing the Roton Excitation Spectrum of a
Stable Dipolar Bose Gas
Physical Review Letters, 122, 183401 (2019)∥
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. Ferlaino 1,21Institut für Experimentalphysik, Universität Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Austria2∥

The author of the present thesis developed the setup for Bragg spectroscopy, developed the experimental
procedures with G. N., A. P., and L. C., performed the measurements together with G. N., A. P., and L.
C., analysed the data, performed and analysed the theoretical calculations with G. N. and R. M. W. v.
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

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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).

92

Chapter

5

Supersolid states in dipolar
Bose–Einstein condensates

In this chapter, we discuss recent experiments on supersolidity, which opened a new avenue
with dipolar Bose-Einstein condensates. It is a particularly interesting research topic for ourgroup, and me personally, as it connects to our previous works on the Lee-Huang-Yang stabilization and the roton mode, which are presented in Chapter 3 and Chapter 4, respectively.
Simultaneously with two other experimental groups, the one lead by G. Modugno [Tan19a]
and the one from T. Pfau [Bö19a], we discovered the existence of a new, exotic state in dipolar
quantum gases, that exhibits global phase coherence together with a spontaneously formed
density modulation. These two properties relate directly to the spontaneous breaking of two
continuous symmetries in our system. On the one hand, the state breaks the U (1) gauge
symmetry, leading to a global phase coherence [Pit03]. On the other hand, it possesses aninherent density modulation, that breaks the translational symmetry. Both properties arehallmarks of supersolidity, and, hence, these type of states are called dipolar supersolids.
In Sec. 5.1, I will present the historical idea of supersolidity. A more thorough theoreticaldefinition is then given in Sec. 5.2 together with the state-of-the-art of current experiments.
Section 5.3 is discussing supersolidity in the context of dipolar BECs and sets the stage forour group’s two publications, presented in Sec. 5.4 and Sec. 5.5.

5.1. Historical idea of a superfluid solid
In the late 1930s, experiments by Kapitza [Kap38] and Allen and Misener [All38] found ananomalously low viscosity of liquid 4 He, when cooled below a temperature of about 2.2 K.
The authors observed for the first time superfluidity - an essentially frictionless flow of thehelium fluid. It took about two decades for scientists to understand the main features ofthis intriguing new phase of matter and explain its accompanying properties, such as theextra-ordinarily high heat conductivity [All37] or the phenomenon of second sound [Pes44,
Lan47]. Nowadays, it is known that the characteristics of superfluid helium are intimately
connected to two important properties of the superfluid: (i) The occurrence of Bose-Einsteincondensation that happens in this system below the critical temperature of 2.2 K and (ii)93

94

CHAPTER 5. SUPERSOLID STATES IN DIPOLAR BOSE–EINSTEIN
CONDENSATES

the strong interactions between the particles, which give rise to a maxon-roton excitation
spectrum [Lon38, Leg99, Pit03].
Even at zero temperature, superfluid helium does not freeze at room pressure. Therefore, toobtain solid helium, it needs to be pressurised [Kee26, Sim34, Kee38]. It was known early onthat there is a phase transition between superfluid and solid helium that can be crossed byapplying pressure. Based on early theoretical publications that extended the definition of a
BEC to an interacting system [Pen56] and described the possibility of a density modulated,superfluid state [Gro57], consecutive works [And69, Che70, Leg70, Mat70, Mul71] introduced
the idea of Bose-Einstein condensation, and the accompanied superfluidity, also in the context
of solid helium. This possible property of superfluidity in solid helium is called supersolidity
and seems at first sight counter-intuitive - or even paradoxical.
To get an intuitive picture of a supersolid object Ref. [Bon12b] gives an insight into someof its peculiar properties: A supersolid can be viewed as a rigid object, where a part ofits constituents is expected to be localised on a lattice structure, while the remaining partis delocalised over the system. This delocalised part can be for example delocalised defects [And69, Che70], which are expected to undergo Bose-Einstein condensation and becomesuperfluid. As the delocalised part still experiences the lattice structure of the solid, its wave
function may feature a periodic modulation. This delocalised part might therefore showsuperfluid flow within solid helium. Along with its first theoretical description, theorists
also looked for experimental signatures of supersolidity in helium. An early proposal [Leg70]
suggested a torsional oscillator, to measure the moment or inertia (MoI) of solid helium. It
was expected that, as solid helium is cooled into a possible supersolid phase, its MoI should
reduce below the expected MoI of a classical, rigid object. This non-classical moment ofinertia should arise in supersolid helium, as the superfluid part of the solid would remain atrest and hence would not contribute to the MoI.
Another, mind-boggling, property of supersolid helium concerns mass transport within thesolid [Bon12b]. If one imagines an experiment, where the supersolid is brought in directcontact with a superfluid of the same kind of constituents, the superfluid outside of thesupersolid may couple to the superfluid part inside the supersolid. As a consequence, thesuperfluid outside, might flow through the supersolid. Therefore, mass transport of a superfluid through a supersolid may happen. However, this property would vanish, when asuperfluid of another constituent would be used.

5.2. Supersolidity
For a more theoretically thorough description, a supersolid implies two main properties[Bon12b]. Due to its solid character, it exhibits density long-range ordering, while the superfluid character requires off-diagonal long-range ordering. The density long-range ordering
breaks the translational invariance in the system, as exhibited in crystalline structures suchas solids, where the density
ρ(r) = ρ(r + Rc )5.1

5.3. A NEW TYPE OF QUANTUM STABILISED GROUND STATE IN DIPOLAR
GASES

95

is periodic with a vector Rc . Importantly, for a supersolid phase the breaking of the translational invariance needs to happen spontaneously, e. g. due to a change in the internalinteractions between the particles in the system. It is, for example, not sufficient to externally impose a periodic structure on the system. Superfluidity is intimately connected toBose-Einstein condensation [Leg06] and a supersolid is expected to feature a macroscopicallyoccupied quantum mechanical state, in which the condensed fraction of particles occupies
the k = 0 momentum and the reciprocal lattice momentum, kc = 2π/|Rc |, simultaneously [Bon12b].
The broken translational symmetry has interesting consequences on the properties of thesupersolid. For example, Leggett showed in Ref. [Leg70] that the presence of a density
modulation limits the superfluid fraction of the system to values smaller than one, even atzero temperature. This is in stark contrast to the superfluid phase, where the superfluid
fraction approaches unity at zero temperature even in presence of strong quantum depletion(i.e. a small condensed fraction). Another interesting feature is the presence of two gaplessexcitation branches in the excitation spectrum of a supersolid, as has been shown for example
in Ref. [Sac12] for soft-core bosons. The two branches reflect the two spontaneously broken
symmetries in the system, where one is connected to excitations of the crystal structure andthe other one is connected to a superfluid flow of particles.
Up to date, multiple platforms have been investigated to realise and study states that possess
the above discussed properties of a supersolid. The earliest candidate for supersolidity washelium [Tho69, And69, Che70]. A non-classical moment of inertia [Kim04b, Kim04a, Bal10]
and superfluid mass transport in solid helium [Ray08] showed first indications of a supersolid state, but were later found to be explainable with temperature-dependent dislocations,
impurities or a changing shear modulus in the solid [Kim08, Cho10, Cho18a]. The advent of ultracold quantum gas experiments provided additional systems, with high purity
and great tunability of inter-particle interactions, to look for supersolidity [Bon12a]. Recent
theoretical proposals for Rydberg states [Hen10, Cin10] or dipolar bosons with additional
three-body repulsion [Lu15] were identified to exhibit supersolid properties. Experimentally,
recently the group from T. Esslinger realised a supersolid-like state with a BEC coupled toa crossed optical cavity [Léo17] and the group of W. Ketterle found supersolid properties
in a spin-orbit coupled BEC [Li17]. In both experiments the translational invariance breaks
spontaneously. However, the wavelength of the density modulation is set externally by the
wavelength of lasers. This has an important consequence, as the created supersolid states
exhibit a rigid density modulation. Therefore, no low-momentum phonons may be excitedin these systems and the crystal structure shows no possible excitation.

5.3. A new type of quantum stabilised ground state in dipolargases
All the mentioned platforms at the end of Sec. 5.2 have one common ingredient, that seems
necessary for a phase transition from a superfluid to a (potentially) supersolid phase. It is
the presence of a roton mode in the excitation spectrum that becomes soft at the transition
point [Mot12, Ji15, San03]. As the roton mode lowers its energy towards the ground state

96

CHAPTER 5. SUPERSOLID STATES IN DIPOLAR BOSE–EINSTEIN
CONDENSATES

energy, the density fluctuations in the system at the roton wavelength strongly increase (seealso Sec. 4.2). For superfluid helium, once the roton mode has a low enough energy thesystem crystallises and transitions into a solid [Sch71]. In the case of gaseous BECs, it
was long believed, that the system should collapse at the roton instability. Only recently,
the discovery of the stabilisation mechanism by quantum fluctuations raised hopes again toobserve supersolidity in dipolar quantum gases, as was speculated early on from the groupof Tilman Pfau [Kad16] and also our group (see Sec. 4.4).
In Sec. 3.1.1 the GPE equation (Eq. 3.9) for a dipolar BEC was introduced and its stabilityof the ground state was discussed. There, we argued that in this mean-field theory a collapseof the BEC was inevitable once the total interaction energies in the system become negative,
which typically coincides with a fully softened excitation mode. But this picture changed withthe observation of the Lee-Huang-Yang stabilisation mechanism, which prevents the systemfrom reaching too high densities (see Sec. 3.3) and therefore can establish a stable state,beyond the modulational instability triggered by a roton mode – similar to the solidification insuperfluid helium. In fact, the very first experimental observation of dipolar droplets [Kad16]
was in a crystal-shaped assembly, that had a great similarity to a solid phase. However,in these early measurements, the observed droplet crystals did not show any global phasecoherence [FB16]. Consecutive other works [Wen17, Bai18] showed that the eGPE (Eq. 3.38),
which includes the LHY correction, supports stable crystals of droplets and, even new groundstates were found where the droplets showed a non-negligible density overlap[Bai18, Roc19].
These later works raised hopes for the possibility to establish coherence across the droplet
crystal and, thus, eventually a supersolid phase for dipolar BECs.
In our previous observations of roton mode population (see Sec. 4.5), we performed an interaction quench of the BEC it into a regime where it becomes favourable to spontaneously andcoherently populate the roton mode. In these measurements, we were able to create density
modulated dipolar BECs, which preserved on short timescales the phase coherence acrossthe system. However, the created states were quickly decohering due to non-equilibrium
dynamics and the fact that we were working in a regime, where no ground-state with significant density overlap between droplets is expected. Nevertheless, this work provided
important insights for more recent experiments, of our group (see Sec. 5.4), the group of
G. Modugno [Tan19a] and the one from T. Pfau [Bö19a]. Especially Ref. [Tan19a] did afirst, fine tuning of the contact interactions and could show that a global phase coherence
can persist in a density modulated quantum gas of dysprosium atoms. Only shortly after,
also our group found good parameter regions for 166 Er and could observe density modulated
states, that maintain a global phase coherence across the system.
It turned out that the important ingredients contain: (i) a roton mode in the system, (ii) a
rather shallow, cigar-shaped trapping potential that the kinetic energy of the atoms remains
weak while keeping the density moderately high, (iii) high enough atom numbers and (iv)a very precise tuning of as , better than the level of 1 a0 . Fig. 5.1 (a) shows ground-statecalculations of the density profile of a BEC. The interaction parameters are chosen suchthat the system exhibits a roton mode (see also Fig. 4.3). As discussed in Sec. 4.2, when thecontact interactions between the atoms are slightly reduced, the roton mode may lower its
energy towards zero. At this point, the ground state is found to exhibit a phase transition
to a density modulated state, exemplified in Fig. 5.1 (b). More importantly, this new ground5.3. A NEW TYPE OF QUANTUM STABILISED GROUND STATE IN DIPOLAR
GASES

97

state exhibits a wave function, that shows a clear density link between the peaks, whichallows for superfluid flow across the entire system. It is therefore a good candidate to lookfor density modulated states that might be able to preserve a global phase coherence inexperiments.
·1014
8

(b)
n2D (m-2)

n2D (m-2)

(a)

4

0
4
x (μm) 0
-4

-10

-5

0
y (μm)

5

10

·1014
12
6

0
4
x (μm) 0
-4

-10

-5

0
y (μm)

5

10

Figure 5.1.: Ground state density profile, n2D (integrated along the atomic dipole direction, z), ofa dipolar BEC in a (ωx , ωy , ωz ) = 2π × (250, 31, 160) Hz trap with 5 × 104 atoms. (a) Shows a regular
BEC for as = 51.0 a0 while in (b) the ground state develops a spontaneous density modulation for
as = 50.8 a0 , while preserving a density link across the whole system. This type of state is referred
to as a dipolar supersolid.

Fig. 5.2 shows axial cuts of the wave function (along y) of the roton mode on the BEC side of
the phase transition and the supersolid ground state on the other side of the phase transition.
As the regular BEC approaches the transition point, one observes a small reduction of theroton mode’s wavelength, combined with an increase of its wave function amplitude. The
latter is directly connected to the increase of density fluctuations and the dynamic structure
factor of the BEC, as discussed in Sec. 4.2. Once the system crosses the phase transition, adensity modulation appears in the ground state whose wavelength matches well the rotonmode. This underlines the important role that the roton mode plays in the transition froma regular BEC to a supersolid phase.
In order to study these supersolid states, our group performed experiments in both laboratories, the ERBIUM and the Dysprosium lab. The former one investigated 166 Er, while theother one focussed on 164 Dy. The resulting publication is presented in Sec. 5.4. We found forboth elements states that showed global phase coherence in combination with a spontaneousdensity modulation in a small interaction parameter region, as expected from the groundstate calculations. We furthermore observed that the system decoheres, as one reduces theinteractions further, towards a regime where the density link in the ground state vanishes.
Here, one enters the regime of an incoherent droplet array. This observation suggests theidea of a solid-like phase, in which the density modulation of the system persists, but it has
lost its global phase coherence, which is consistent with the earlier observations of dipolar
droplet assemblies [Kad16, Wen17]. Interestingly, our study found a pronounced differencein the lifetime of the created states. In the case of 166 Er the state survives a few tens of
milliseconds, which is limited by three-body recombination processes that cause atom loss
in this highly dense state. Contrary, for 164 Dy the lifetime exceeded hundreds of milliseconds, as three-body losses are less prominent. Ultimately, this allowed for the creation ofsupersolid states via direct evaporative cooling of 164 Dy, which was also demonstrated in the

98

CHAPTER 5. SUPERSOLID STATES IN DIPOLAR BOSE–EINSTEIN
CONDENSATESψGS (y) (ψm, BEC)
0
1
2
10

supersolid roton mode ofground state ordinary BEC5
y (μm)

ψrot (y) (arb. u.)
-1
0
1

0
-5
-10

50.2

50.6

51
aS (a0)

51.4

51.8

Figure 5.2.: Axial wave function cuts of the roton mode (ψrot , red) and the ground state (ψGS , blue)
across the BEC-to-supersolid phase transition (vertical, black line at 51 a0 ). The figure shows how
the axial modulation of the roton mode is evolving across the phase transition into the supersolid
ground state. The atom number is fixed to 5 × 104 and the trap has harmonic trapping frequencies
are (ωx , ωy , ωz ) = (250, 31, 160) Hz. ψGS is normalised to the maximum value, ψm, BEC on the BEC
side of the phase transition.

publication in Sec. 5.4.
The observation of global phase coherence in this density modulated state is, in itself, nota proof of supersolidity, as it does not directly inform about superfluidity in this system.
Therefore, additional measurements that revealed signatures of superfluid behaviour quickly
followed in experiments. As mentioned in Sec. 5.2, the presence of two broken symmetries
reveals itself as two gapless excitation branches in the excitation spectrum of a supersolid,
where the energetically lower lying branch is expected to be connected to superfluid flow
between different density peaks in the system. The spectrum of a dipolar supersolid wasexperimentally investigated in Refs. [Tan19c, Guo19, Nat19]. Another experiment [Tan19b]
could also show the reduction of the moment of inertia of the supersolid states compared toits classical value, which is an observable that was also used to search for supersolidity inhelium. Reference [Ilz19] studied the response of the supersolid states after creating phase
excitations in the system. This experiment found a rather fast re-establishment of phasecoherence in the supersolid and was furthermore able to identify a parameter region wherere-phasing did not any more occur.
In our most-recent work with 166 Er, we studied the created supersolid states with high-energy
Bragg spectroscopy. The corresponding publication is given in Sec. 5.5 and shows that the
probing of free-particle excitations in the supersolid state gives a response of the system thatis determined by both, the density modulation contrast and by phase variations in the wave
function. The established technique, therefore, allowed us to identify the important role ofvariations in the phase of the wave function in the experimentally created supersolid states.
Furthermore, we could show the occurrence of coherent phase variations mainly due to thecrossing of the phase transition in the preparation scheme, which emphasises the importanceof dynamics in the currently investigated supersolid states.

5.4. PUBLICATION: LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS INDIPOLAR QUANTUM GASES
99

5.4. Publication: Long-Lived and Transient Supersolid Behaviors
in Dipolar Quantum Gases
Physical Review X, 9, 021012 (2019)†
submitted 11 Mar 2019; published 19 April 2019
DOI: 10.1103/PhysRevX.9.021012

L. Chomaz,1 D. Petter,1 P. Ilzhöfer,2 G. Natale,1 A. Trautmann,2 C. Politi,2 G. Durastante,2R. M. W. van Bijnen,2 A. Patscheider,1 M. Sohmen,2 M. J. Mark,1,2 and F. Ferlaino 1,21Institut für Experimentalphysik, Universität Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Austria2†

The author of the present thesis developed the experimental procedures and performed the measurements
for erbium together with L. C., G. N., and A. P., analysed the data with L. C., and contributed in writing
the manuscript. The theoretical calculations for the publication were performed and analysed by G. N.,
R. M. W. v. B., and L. C.

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

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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].

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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

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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

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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

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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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[out of the scale of Fig. 1(g)].
[51] We note that a similar behavior is also found in our 166Er
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the dynamics and the atom losses [40], can affect theexperimental observations.
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Correction: The inadvertent omission of a marker indicating“Featured in Physics” has been fixed.

021012-12

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

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

GROUND STATE CALCULATIONS

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

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

Z

s

∗
ψi−1
(y)ψi (y)dy

(1)
(0)

(0)

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

!

.

(2)

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

LINK STRENGTH AND ESTIMATE OFTUNNELING RATE

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

2

TOY MODEL FOR THE INTERFERENCEPATTERN

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

4
3
2
1
0

c

n(µm)

F[n]

n(µm)

F[n]

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

x104

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

F[n]

x104

x104

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

n(µm)

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

F[n]

(3)

F[n]

+

n(µm)

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

(0)

(yi

n(µm)

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

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

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

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

3

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

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

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

F[n]

n(µm)

n(µm)

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

4c

F[n]

5b

F[n]

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

x104

x104

5a

F[n]

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

x104

n(µm)

F[n]

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

F[n]

n(µm)

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

x104

F[n]

n(µm)

x104
6c
5
4
3
2
1
0

n(µm)

n(µm)

n(µm)

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

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

F[n]

n(µm)

5
4
3
2
1
0

x104

F[n]

6a

F[n]

x104

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

n(µm)

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

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

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

IMAGING ANALYSIS: 164 Dy AND 166 Er

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

DETAILS ON THE FOURIER ANALYSIS

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

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

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

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

3

5

4

AФ

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

3

2

1

0

0

10

20

30
tr (ms)

40

50

60

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

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

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

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

e

b

d

f

2

120
AФ(103)

Scattering length (a0)

c

4

140

100
80

0
5
4
3
2
1
0

60
40
1.1

a

1.3

1.5

1.7

1.9
B (G)

2.1

2.3

2.5

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

ATOM LOSSES IN 166 Er AND 164 Dy

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

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

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

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

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

anism limiting the lifetime of the supersolid properties inDy.

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

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

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

a

c

e

b

d

f

2
1
0
12

AФ(102)tN (ms)32(5)51(9)
46(12)

N (104)3

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

8
4
0

0

50

100

th (ms)

0

50

100

th (ms)

0

50

100

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

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

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5.5. PUBLICATION: HIGH-ENERGY BRAGG SCATTERING MEASUREMENTS OFA DIPOLAR SUPERSOLID
121

5.5. Publication: High-energy Bragg Scattering Measurements ofa Dipolar Supersolid
Submitted to Physical Review Letters, (2020)‡
submitted 05 Mar 2020

D. Petter,1 A. Patscheider,1 G. Natale,1 M. J. Mark,1,2 M. 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. Ferlaino 1,21Institut für Experimentalphysik, Universität Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
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, Dunedin9054, New Zealand
7
Department of Physics, University of Otago, Dunedin 9016, New Zealand2‡

The author of the present thesis developed the experimental procedures and performed the measurements
together with A. P., G. N., and L. C., analysed the data, performed and analysed theoretical calculations
with G. N., R. M. W. v. B., and L. C. and contributed in writing the manuscript. The theoretical
calculations for the inifite cigar trap were performed S. M. R. and A. R.

High-energy Bragg scattering measurements of a dipolar supersolid
D. Petter,1 A. Patscheider,1 G. Natale,1 M. J. Mark,1, 2 M. A. Baranov,2 R. v. 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

arXiv:submit/3161535 [quant-ph] 4 May 2020

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
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: May 4, 2020)
We present an experimental and theoretical study of the high-energy excitation spectra of adipolar supersolid. Using Bragg spectroscopy, we study the scattering response of the system toa high-energy probe, enabling measurements of the dynamic structure factor. We experimentallyobserve a continuous reduction of the response when tuning the contact interaction from an ordinary
Bose-Einstein condensate to a supersolid state. Yet the observed reduction is faster than the one
theoretically predicted by the Bogoliubov-de-Gennes theory. Based on an intuitive semi-analytic
model and real-time simulations, we primarily attribute such a discrepancy to the out-of-equilibriumphase dynamics, which although not affecting the system global coherence, reduces its response.

The field of quantum gases has moved towards thestudy and realization of novel quantum states of matter,
often showing exotic properties [1]. A recent example,
still challenging scientist’s intuition, is the long-soughtsupersolid phase, recently observed in atom-light-coupledsystems [2, 3] and dipolar quantum gases [4–6]. A supersolid state spontaneously develops a density modulationin space, breaking the translation symmetry, and a global
phase coherence, breaking the gauge symmetry [7–10]. Ina dipolar gas, the supersolid phase (SSP) lives in a narrow interaction-parameter range, sandwiched between an
ordinary Bose-Einstein condensate (BEC) and an incoherent array of droplets (ID), showing extreme densitymodulation [5, 6, 11–16].
In a dipolar supersolid, fundamental properties related
to its quantum-fluid nature remain to be understood.
This includes the relation between condensation and superfluidity, as well as their connection to density modulation and phase fluctuations across the phase diagramof a dipolar gas. In a seminal work [17], Leggett asks
how solid is a supersolid, deriving an upper bound forthe superfluid fraction of a stationary supersolid state,
which connects to the degree of localization, i. e. the density modulation. In the recently observed dipolar supersolid [4–6], the situation might be more complex becauseof many-body out-of equilibrium phenomena. Indeed,
the macroscopic phase of a supersolid might dynamicallydevelop variations in space, caused e. g. by the crossingof the BEC-SSP phase transition, or by thermal andquantum fluctuations [4, 5, 12, 18]. Such effects could
impact the superfluid properties of the system, going beyond Leggett’s original prediction.
Local phase variations are typically not readily acces-

sible in experiments. However, the study of the dynamical response of a physical system to a high-energy scattering probe has proven to contain key information on
the state properties. Such powerful scattering protocols have been widely used across different physical disciplines, ranging from high-energy [19–22] to condensedmatter physics [23, 24]. For instance, scattering of fastneutrons from superfluid liquid helium has enabled thefirst measurement of the condensate fraction in a stronglyinteracting many-body system [25]. In the realm of ultracold quantum gases, a similar concept has been employed
to reveal e. g. beyond-mean-field effects, to measure quantum depletion and coherent fractions, or Tan’s universal
contact parameter [26–32].
In this Letter, we experimentally study the dynamical
response of a dipolar supersolid to a high-energy scattering probe by performing two-photon Bragg excitationin the free-particle-excitation regime (high energy andhigh momentum). We observe that the system response
strongly reduces in the supersolid regime before vanishing in the ID phase. By benchmarking our data withtheoretical models, we identify the role of the densitymodulation contrast and the phase variations in the observed response. Our study reveals the importance of thecoherent phase dynamics induced by the crossing of the
BEC-to-supersolid phase transition.
The dynamical response of an interacting many-bodysystem to a weak scattering probe can be describedwithin the linear-response theory. An essential quantity
is the dynamic structure factor (DSF), S(k, ω), whichcharacterizes the density response of a system to a scattering probe of momentum, h̄k, and energy, h̄ω [33]. For
weak interatomic interactions, the DSF can be directly

21

related to the excitation spectrum via the Bogoliubov
amplitudes, uj and vj , describing the excitation mode jof energy h̄ωj . It readsjS(ky , ω) =X
jñ(0, ky − kj , 0) δ(h̄ω − h̄ωj ).

(2)

On resonance, ω = ωj and ky = kj , the DSF becomes S(kj ) ≡ S(kj , ωj ) ∝ ñ(k = 0) and is uniquely determined by the system’s momentum distribution at
k = 0.
To identify the free-particle regime in our system, we
briefly review the basic properties of the excitation spectrum of a supersolid, considering the simpler case of
an infinitely elongated cigar-shaped trap. A more extensive description, including calculations for the full
three-dimensional (3D) confined case, which we use hereafter for comparison with the experiments, can be foundin Refs. [18, 38–42]. As shown in Fig. 1 (a), the supersolid 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 [43]. They have been recently investigated in experiments [18, 38, 39]. The upper branch, appearing athigh energy and showing a gapped parabolic dispersion,is the one of interest here for its free-particle character.
In addition, the flat band at ω ≈ 1.25 ωz corresponds toa single-droplet excitation (mainly transverse breathingmodes), which couples to the parabolic branch, opening
small energy gaps. Figure 1 (b) shows the corresponding
DSF values. Interestingly, we observe that the DSF does
not reflect the periodicity of the energy spectrum, ande. g. for the upper branch it shows large values at the

>5

||u||

2 >2.6
S (k, ω) (S*)

0

1

(b)

104

n (μm-1)

1.0

0

-2

0 2
y (μm)

0.5

× δ(h̄ω − h̄ωj ), (1)

0

1

0

1
ky (kc)(c)

2 0

1ky (kc)SSP BEC

B || z

n2D
y
x

0.8

2

1

53.2

as (a0)
54

54.8

C

where we neglect the creation of multiple excitations.
Here, ψ0 is the system’s macroscopic ground-state wave
function and δ(·) is the Dirac delta function.
Equation (1) gives different information depending on
the momentum and energy ranges [33]: 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 informs about the momentum distribution of thesystem, ñ(k). We explore the latter regime for our supersolid state, focusing on the response along the densitymodulated direction, y, with k = (0, ky , 0). In theregime of free-particle excitations (uj → eikj y , vj → 0,
ωj → h̄kj2 /2m with m the atomic mass), the impulse
approximation becomes valid and we find [33–37]

4

1.5 (a)ω (ωz)

2
dr u∗j (r) + vj∗ (r) eikr ψ0 (r) ×3

S(k) (S *)S(k, ω) =X Z

2

0.6 53.2

0

53.6

54
as (a0)

54.4

54.8

FIG. 1. (a) Axial excitation spectrum of the transversely
symmetric modes and (b) corresponding DSF of an infinitely
elongated dipolar supersolid at as = 51 a0 in a harmonic trapwith ωx,y,z = 2π × (250, 0, 160) Hz. The color maps correspond to kuk and S(k, ω), respectively. The inset shows theintegrated axial density profile n(y) of the ground state withmean density 4.7 × 103 µm−1 . (c) S(k) for the 3D-trappedsystem with ωx,y,z = 2π × (250, 31, 160) Hz. S(k) is calculated at k = 4.2 µm−1 ≈ 1.8 kc (grey line) and normalized
by its value at the BEC-SSP phase transition, S ∗ . The atom
number is varied with as to match the experimental conditions [42]. The red (blue) line shows the result from the SIA(DAA). (upper inset) Integrated density profile of the groundstate at as = 54.49 a0 and N = 5 × 104 atoms. (lower inset)
Evolution of the ground state’s central contrast C. For theinfinite (3D-trapped) case, kc = 2.3(2.4) µm−1 .

momenta that continuously connect to the free-particle
excitations in the ordinary BEC [42].
In a contact-interacting BEC, the inverse healing
length provides the scale of the crossover from the collective to the free-particle character of the excitations [33].
This notion can not be simply exported to the case ofdipolar gases because of the momentum dependence ofthe dipole-dipole interaction. We thus follow the definition based on the Bogoliubov-de-Gennes (BdG) theory. A free-particle excitation is an elementary excitation whose wave function is well approximated bya plane wave. This is typically justified for excitations of highenough energy and single-particle characterR(kuj k = |uj (r)|2 dr = 1 and kvj k = 0, see color map
Fig. 1 (a)) [33, 44]. For our parameters, we find that the

3the density overlap between droplets, set by σ/d, and thephase variance along the array. The latter effect is notincluded in the BdG theory, which describes a state possessing a uniform phase. To benchmark the DAA resultswith the BdG calculations, we thus set θj = 0 for allj [42]. We find a very good agreement for C > 80 %.
In the experiments, we access the density responseof a supersolid by performing high-energy Bragg scat-5
58.19(8) a00
BEC56.0(1) a0
55.3(1) a0
54.7(1) a0

SSP

Nex /N (%)

54.4(1) a0
54.0(1) a0
53.8(1) a0

IDdensity profile of the mode, |uj (r)|2 , shows a plane-wave
character for h̄ω >∼ 0.6 h̄ωz .
To quantitatively compare theory with experiments,
we perform similar ground-state and BdG-spectrum calculations for the 3D-trapped case [38], and extract
S(k) [42]. As shown in Fig. 1 (c), in the free particle
regime (k ≈ 1.8kc ), S(k) starts to decrease when entering the SSP and further reduces by 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 withas 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 ) being the central maximum (minimum) of the integrated
density [42].
To gain an intuitive understanding of the densityresponse reduction, we develop a 1D model [37]. Using two different wavefunction ansatzes, we evaluate S(k)in the weak and strong density-modulation regimes. As
discussed in Refs. [11, 16, 45], for weakly modulated supersolids, with C  1, the ground-state wave function
can be approximated by a fully coherent sine-modulated
function on top of a uniform√background. At leading order in C, it reads ψ(y) = n (1 + C sin(kc y)/2), withn the mean density. Applying this sine ansatz (SIA)in Eq. (2), we find S(k) ∝ n(1 − C 2 /8). This resultshows 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 [17], 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. We benchmark our SIA results with the
BdG calculations by evaluating C from the full GPE solution [42]. As shown in Fig. 1 (c), despite its simplicity,
the SIA scaling reproduces very 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) =PNDiθj[6, 12]. Each droplet is described
j=1 χ(y − jd)e
by a Gaussian function, χ(y), of size σ, separated bya distance d > σ from its neighbours. Each droplet
is allowed to have an independent, yet uniform, phaseθj . Within the DAA, the DSF shows the proportionalPND iθj 2
ity S(k) ∝ n N1D j=1e
σ/d. It decreases with both012ħω (ħωz)

3

4

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 the Gaussian fits to the data.

tering on a 166 Er dipolar quantum gas, confined in anaxially elongated harmonic trap. A transverse homogeneous magnetic field orientates the atomic dipoles andsets as [6]. We initially prepare the system in the ordinary BEC phase, and enter the SSP via interaction
tuning by linearly lowering as below a critical value, a∗s ,for which the BEC-SSP phase transition occurs. Similar
to previous experiments [6, 38], a∗s is extracted with aninterferometric technique. For the present trap and atom
numbers, N , we measure a∗s = 54.94+28−13 a0 ; see [42].
For the Bragg excitation, we project on the atoms an
optical lattice potential of constant depth V for a duration τ = 7 ms. The lattice has a constant wave vector k = 4.2(3)µm−1 along y and moves with a variablefrequency ω. After the Bragg excitation, we measure
the integrated momentum distribution, ñ(kx , ky ), using atime-of-flight expansion of 30 ms. The number of excitedatoms Nexc is extracted in a narrow region of interest
around k [42]. For a fixed as , we find a clear resonancein Nexc /N as we vary ω. From a Gaussian fit we extract
the resonance peak’s amplitude, F . From linear response
theory, we expect F ∝ V 2 τ S(k) [46]. For the relevant as
range, we have checked the scaling with τ and V [42].
Figure 2 shows examples of the Bragg-excitation spectrum for various as . In the BEC regime until the onsetof the SSP, we observe a downward shift of the resonancefrequency without a significant change in F [42]. In contrast, as we enter into the SSP regime, F undergoes astark reduction. In the ID regime, the resonance peakcompletely vanishes.
Figure 3 shows the evolution of F across the BEC-SSPID phase diagram. The as -extension of the three phases(see background colors), has been determined from independent measurements of the phase coherence and den-

4

5

ID

SSP

ramp in as , the Bragg excitation, the three-body losses,
and an initial population of BdG modes from quantum
and thermal noises [42]. From the simulated momentum
distributions, we extract the excited fractions, following
the same procedure as for the experimental ones. As
shown in Fig. 3, contrary to the BdG results, the RTE
simulations describe remarkably well the data both in
the BEC and SSP phase.

BEC

F (%)

4
3
2
1
54

54.5

55 55.5
as (a0)

56

56.5

57

FIG. 3. Experimental F (circles) versus as across the BECSSP-ID phases (white, red, blue shadings). For the lowest
three as , we do not observe a resonance and plot the standard
deviation of the data as an error estimate. Horizontal error
bars correspond to uncertainties of the magnetic field [42].
We show the excited fraction expected from the BdG calculations on the corresponding ground states (gray line) and fromthe RTE simulations (connected dots, one RTE simulationrun). In the RTE we cannot extract clear resonances for thelowest as . We furthermore show the rescaled BdG calculations, which include ∆Θ obtained from the RTE (blue line).
The gray shading corresponds to the uncertainty in as of theexperimental phase transition (vertical line).

sity modulation of the states [6, 42]. When reducing as
in the BEC phase, we find a slight increase of F . On
both sides of the BEC-SSP phase transition, we observe
similar F values, indicating a continuous behavior acrossthe transition. Remarkably, as soon as we lower as further by ∼0.5 a0 , F drastically reduces to <∼ 1 %, which is
close to our detection level. Finally, for as < 54 a0 , wedo not observe any resonant response.
We compare the experimental results with our BdG
theory for the 3D-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 a much faster reduction of the system’s
response than the one predicted from the BdG theory.
This result suggests that an important ingredient is missing in the theory. Our DAA model provides a first intuitive explanation: It suggests that the presence of phasevariations across the system can yield a reduction of the
DSF. We envision two sources of phase variations. First,
quantum and thermal fluctuations, which are expected todominate in the ID regime, yield phase patterns varying
from shot to shot. Second, coherent dynamics, as e. g. induced by the crossing of the BEC-SSP phase transition,
leading to reproducible phase patterns. Neither phenomena are accounted for in the BdG calculations.
To investigate these effects, we simulate the system
real-time evolution (RTE) [47]. Our calculations reproduce the experimental sequence and include the linear

ny (arb. u.)

53.5

(a)

time (ms)

0

7

1

(b)

0
0 (c)
Θ (rad)

0

1

-π
-8

-4

0
y (μm)

4

8

53.6

54.2
as (a0)54.8

0FIG. 4. RTE simulations without Bragg excitation. (a) Timeevolution of the integrated in-situ density of the wave function for as = 54.04 a0 . (b) hCiτ (triangles) and ∆Θ (squares)
versus as . The grey line corresponds to the central contrast
obtained from the ground-state theory. The solid blue line
is a smooth interpolation of ∆Θ, fixed to unity at the phasetransition point. The shadings give the standard deviation obtained from 5 simulation runs. The vertical line corresponds
to the phase transition point. (c) Phase-cuts corresponding
to the simulation shown in (a).

To highlight the role of the contrast and phase variations, we perform RTE simulations without the 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, the calculated hCiτ , time-averaged over the
Bragg scattering duration, increase with decreasing as .
However, for each as , we observe a 10-30 % lower contrast than the one extracted from the ground-state calculations. Since a reduced contrast would mean an increase in F , we deduce that the varying contrast can notexplain the mismatch between the BdG theory and boththe experimental and RTE results; see Fig. 3.
We now study the phase variations and its dynamics.
The RTE calculations reveals that the phase of the wave
function, θ(y), develops a non-uniform profile. For instance at as = 54.04 a0 , θ(y) exhibits a stair-like profile
with fairly constant values within the density peaks anddiscrete phase steps in between them; see Fig. 4 (c). Thisbehaviour suggests that each density peak acquires an
independent phase, despite their density links. We also
observe that the phase patterns is fairly reproducible be-5tween simulation runs and mainly induced by the coherent dynamics arising by the crossing of the phase transition [42].
Following the DAA model, phase variations are
expected to reduce S(k) by a factor ∆Θ ≈PND iθj 2
he iτ | [33, 42]. As shown in Fig. 4 (b), ∆Θ| N1D j=1
is almost unity close to the BEC-SSP phase transitionand significantly drops when lowering as towards the ID
regime, where starts to flatten. The standard deviationof ∆Θ relates to the shot-to-shot reproducibility of thephase 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, highlighting the increasing
effects of fluctuations. We empirically account for the effect of phase variations in the BdG theory by scaling the
DSF with ∆Θ over the whole SSP-ID regimes. As shownin Fig. 3, this simple inclusion of ∆Θ demonstrates the
pronounced impact of the coherent phase variations forthe experimentally observed response.
In conclusion, we demonstrate that the supersolidstates, when created via a dynamical tuning of the interactions, develop important phase variations across thesystem, which have to be taken into account to understand the system behavior. Those phase variations occur
even in presence of sufficiently strong density links between the droplets. Our work provides first steps to amore complete vision of the dipolar supersolid, including
out-of-equilibrium phenomena, and opens the door for
future exploration of critical phenomena induced by the
dynamical crossing of the BEC-SSP phase transition [48–50].
We thank S. Stringari for insightful discussions andB. Yang for the careful reading of the manuscript.
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), a DFG/FWF (FOR 2247/PI2790) and a
joint-project grant from the FWF (I 4426, RSF/Russland2019). 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.
* Correspondence and requests for materials should be
addressed to Francesca.Ferlaino@uibk.ac.at.

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7

A. Preparation of the BECs

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. [6, 51]. 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.

ing 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 .

3

BEC

2

1.5
1
*
Δas,P
*
Δas,B
Δas*

0.5
0

53.8 5454.2 54.4 54.6 54.8 55
as (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).

C. Transition from SSP to ID

B. Determination of experimental BEC-SSP phasetransition

2

x102
ID

BEC

SSP

AФ, AM

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 [4–6]. 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, wefind 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 . Tak-

SSP

2.5
fside (%)

Supplemental Material

0
54

54.5
as (a0)55

FIG. S2. Amplitudes Aφ (red circles) AM (blue squares)
versus as . Error bars denote the standard error from about 30
experimental realizations [6]. 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. [6]

8

x104
7

SSP

BEC

6
5
N

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
dynamics.

4
3
2
53

54

55
as (a0)56

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.

D. Calibration of atom loss and atom number for
BdG theory
E. Bragg spectroscopy

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 endof 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.

The Bragg excitation beams are realized holographically with a digital-micromirror device (DMD), as detailed in Ref. [56]. In short, the setup uses a nearresonant 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 an interference 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, theBragg scattering needs to supply energy, h̄ω, which is introduced with a frequency 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 phase of one beam in 9 steps from 0 to 2π.
Depending on the frame-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 [52].
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.

9

6

x102
3

ky (μm-1 )

4
2
0

2

-2

1

-4

-5

0
kx (μm-1 )

5

-5

0
kx (μm-1 )0

5n (kx, ky) (μm2 )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.

SSP

2.5
ωk (ωz )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 [1]. 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
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 beheviour is consistent with the extracted ωk from the RTE theory. Furthermore, we calculate the expected resonance frequency
from the BdG calculations, in which the resonance frequency is increasing again after crossing the BEC-SSP
phase transition with lowering as . Therefore, the BdG
theory predicts a hardening of the measured excitation
modes, which is not observed neither in the experiment,
nor in the RTE simulations. This qualitative difference
might stem from the increased phase variations that develop in the system, but further studies are needed toelucidate this point.

BEC

2

1.5

54

55

as (a0)56

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

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 scalingof F with τ , which is consistent with the expected scalingfrom BdG theory, F ∝ V 2 τ S(k) [46]. Furthermore, weprobe the quadratic scaling of F with V in the SSP andfind also here an agreement up to V = 80 Hz (see inset).

10

6

F (%)

F (%)

10

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.

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

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 [53, 54]. 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 [38, 43, 55]. In addition to these gapless branches,
one observes gapped parabolic branches of excitations

with energetic 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 [33, 44]. 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 freeparticle. 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 [33]. 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. S9 (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 even for momenta
higher then the roton momentum. After crossing thephase transition into the SSP, we find that the DSF of
the parabolic branch [Fig. S7 (c3)] smoothly connects tothe 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. [38, 47, 56]. 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 and experiment has also been found in Refs. [6, 56, 57]. There-n (x104 μm-1)

11

1.5

a1

0
-4

ω (ωz)

-2
b1

1.5

0
2
y (μm)

4||u||1

>5

3

a3

-2

-2

0
2
y (μm)b2

a4

02
y (μm)

a5

-2

b3

0
2
y (μm)b4-2

0
2
y (μm)

0.5

1 1.5
ky (kc)

2

0.5

1 1.5
ky (kc)

2

b5

1.0
0.5
0

0

0.5

20

1 1.5
ky (kc)c1 S (k, ω)

0

1

2 >2.6

momentum (kc)

1.5
ω (ωz)

a2

0.5

1 1.5
ky (kc)

c2

2 0

0.5

1 1.5
ky (kc)

20

c3

0.5

1 1.5
ky (kc)

c4

2 0
c5

1.0
0.5
0

0

0.5

1 1.5
ky (kc)

20

0.5

1 1.5
ky (kc)

2 0

0.5

1 1.5
ky (kc)

20

0.5

1 1.5
ky (kc)2 0

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.

fore, 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 (typical energy
spacing ∼ h × 20 Hz). Furthermore, each mode is broadened in ky . The finite duration of the Bragg pulse gives
an energy broadening of each excitation mode (Fourier
broadening ∼ h × 130 Hz full width at half maximum)
which is much bigger than the energy spacing between
the modes in the spectrum. Therefore, in the Bragg
spectroscopy only a single resonance is visible, which isconstituted of multiple excitation modes. To account for
this, we calculate S(k, ω) while broadening each modein energy according to the Fourier-broadening, expectedfrom a 7 ms Bragg pulse. 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 compare with the experiment, we extract S(k) froma Gaussian fit to this resonance; see also [56].

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. S9 (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 excitation’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 S9 (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 modu-

12

n (x103 μm-1)

0

1

2

3

4

5

||u||, ||v||

10
5
y (μm)

(a)

10

free
particles

collective

1

0
0.1

-5

53.5

54
as (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).

lation 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 S9 (c) the integrated density profile ofan excitation mode at lower energy, 0.55 h̄ωz , which also
has kuk ≈ 1, but is clearly not a plane wave.
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
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
with the ground state properties and not its absolutevalue, we renormalize the SIA and the DAA. For the

nexc , nGS (arb. u.)-10nexc , nGS (arb. u.)0

0.5
energy (ħωz )

1

1 (b)

0
1 (c)

0

-10

-5

0
y (μm)

5

10

FIG. S9. (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 .

comparison 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.

13

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

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-width1

ramp of as(a)C

Our theory for the RTE simulations was already presented in Ref. [47]. 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 [58]. 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.

at half maximum.
holdtiming of
Bragg pulse

53.94 a054.24 a054.54 a054.74 a054.84 a054.94 a055.14 a00
2π (b)α (rad)K. Real time evolution of the Bragg scattering

0
-30

-20

-10
time (ms)07FIG. S10. 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. 4 (a, b) corresponds to thetime window of [0, 7] ms.

Figure S10 (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
is a time delay between the development of the density
modulation in the system and the timing of the as ramp
(which occurs during [−30, −10] ms).
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 [4]. In Figure S10 (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 .

Chapter6

Conclusion

This thesis reports on our experimental works with gaseous dipolar Bose-Einstein condensates, made of highly magnetic 166 Er atoms. The main emphasis lies on a parameter regime,
where standard mean-field theory predicts a collapse of the quantum gas. Our experiments
reveal the importance of quantum fluctuations, which lead to a surprising stabilisation of
the quantum gas and enrich tremendously the phase diagram of the dipolar BEC. This stabilisation mechanism introduces new types of ground states such as a macrodroplet, droplet
crystals and even a supersolid phase.
In our first experiment, we identified a crossover region from a dipolar Bose-Einstein condensate to a dense macrodroplet. With this investigations, we could prove that, indeed, quantum
fluctuations stabilise the dipolar BECs against collapse. Furthermore, our findings showedthat an extended Gross-Pitaevskii theory, including the Lee-Huang-Yang energy correction,
can accurately describe the macrodroplet state.
A second set of experiments focussed on the theoretically predicted roton minimum in the
excitation spectrum of a dipolar BEC. Here, standard mean-field theory predicts that afully softened roton minimum leads to a modulational instability under which the system
collapses. We were in the fortunate position to prove the existence of this roton minimum
for the first time and could probe some of its peculiar properties, such as the increase ofdensity fluctuations that appear in the quantum gas, as the roton minimum forms in the
spectrum.
These two observations – the stabilising role of quantum fluctuations and the existence ofa roton minimum – constitute the main building blocks for our third series of experiments.
We could show that quantum fluctuations also stabilise the roton’s modulational instability
and establish a phase transition to a density modulated state, that can exhibit global phasecoherence. This new state shows properties of supersolidity in dipolar Bose-Einstein condensates, where the translational and the gauge symmetry are spontaneously broken at thesame time.
Our studies helped paving the way for an interesting, new research direction of dipolar
quantum gases that are stabilised by quantum fluctuations. They contributed to, but also
benefited strongly from the works of others. From the experimental side, the group of
137

138

CHAPTER 6. CONCLUSION

Tilman Pfau in Stuttgart made the first observation of dipolar droplets, which set our stage
to investigate quantum fluctuations in dipolar BECs. The Stuttgart group and also thegroup of Giovanni Modugno in Pisa observed, at a similar time as us, the aforementioned
supersolid states [Tan19a, Bö19a, Cho18b] and our three groups continue to investigate theproperties of this exciting new state of matter up to today. From the theoretical side, we
received ideas and strong support from the groups of Luis Santos in Hannover, Blair Blakie
in Otago, Alessio Recati in Trento and also from Rick van Bijnen and Misha Baranov in
Innsbruck. The numerous collaborations I was involved in and the investigated topics, madethe last years an exciting time. I feel privileged to provide a humble contribution to theflourishing research direction of dipolar quantum gases and hope it will be beneficial also
for other research groups and the next generation of experiments of highly magnetic atoms,
which are being currently build up all over the world.

Appendix

A

Additional publications

A.1. Publication: Realization of a Strongly Interacting Fermi Gas
of Dipolar Atoms
Physical Review Letters 121, 093602 (2018)†
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,1,2 G. Natale,1 L. Chomaz,1 M. J. Mark,1,2
and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Austria2†

The author of the present thesis developed the experimental procedures together with J. H. B. and S. B.,
performed the measurements together with J. H. B., S. B., A. P., L. C., and G. N., and contributed inwriting the manuscript.

139

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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093602-6

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.

[1] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm, and
F. Ferlaino, Phys. Rev. Lett. 112, 010404 (2014).
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L. Santos, and F. Ferlaino, Phys. Rev. X 6, 041039(2016).
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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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[10] S. Lepoutre, J. Schachenmayer, L. Gabardos, B. Zhu,
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A.2. Publication: Controlling Dipolar Exchange Interactions in a
Dense 3D Array of Large Spin FermionsPhysical Review Research 2, 023050 (2020)‡
submitted 27 Sep 2019; published 17 Apr 2020
DOI:

A. Patscheider,1 B. Zhu,2,3,4 L. Chomaz,1 D. Petter,1 S. Baier,1 A.-M. Rey,2 F. Ferlaino,1,5and M. J. Mark1,5
1

Institut für Experimentalphysik, Universität Innsbruck, Austria
Department of Physics, University of Colorado, 440 UCB, Boulder, USA
3
Harvard-Smithsonian Center for Astrophysics, Cambridge, USA
4
Department of Physics, Harvard University, USA
5
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Austria2‡

The author of the present thesis developed the experimental procedures and performed the measurements
together with S. B., A. P, and L. C., and contributed in writing the manuscript. The theoretical calculations
for the publication were performed by B. Z. and A. M. R.

154

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

Controlling dipolar exchange interactions in a dense three-dimensional array of large-spin fermions
A. Patscheider ,1 B. Zhu ,2,3,4 L. Chomaz ,1 D. Petter ,1 S. Baier ,1 A.-M. Rey ,4 F. Ferlaino ,1,5 and M. J. Mark

1,5,*

1

Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
3
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
4
JILA, NIST, and Department of Physics, University of Colorado, 440 UCB, Boulder, Colorado 80309, USA
5
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2(Received 27 September 2019; revised manuscript received 19 December 2019; accepted 13 March 2020;
published 17 April 2020)
Dipolar interactions are ubiquitous in nature and rule the behavior of a broad range of systems spanning fromenergy transfer in biological systems to quantum magnetism. Here we study magnetization-conserving dipolar
induced spin-exchange dynamics in dense arrays of fermionic erbium atoms confined in a deep three-dimensionallattice. Harnessing the special atomic properties of erbium, we demonstrate control over the spin dynamics bytuning the dipole orientation and changing the initial spin state within the large 20-spin hyperfine manifold.
Furthermore, we demonstrate the capability to quickly turn on and off the dipolar exchange dynamics via opticalcontrol. The experimental observations are in excellent quantitative agreement with numerical calculations basedon discrete phase-space methods, which capture entanglement and beyond-mean-field effects. Our experimentsets the stage for future explorations of rich magnetic behaviors in long-range interacting dipoles, including
exotic phases of matter and applications for quantum information processing.
DOI: 10.1103/PhysRevResearch.2.023050

I. INTRODUCTION

Spin lattice models of localized magnetic moments (spins),
which interact with one another via exchange interactions,are paradigmatic examples of strongly correlated many-bodyquantum systems. Their implementation in clean, isolated,
and fully controllable lattice confined ultracold atoms opens
a path for a new generation of synthetic quantum magnets,
featuring highly entangled states, especially when driven outof equilibrium, with broad applications ranging from precision sensing and navigation to quantum simulation and quantum information processing [1,2]. However, the extremely
small energy scales associated with the nearest-neighbor spin
interactions in lattice-confined atoms with dominant contact
interactions have made the observation of quantum magnetic
behaviors extremely challenging [3,4]. On the contrary, evenunder frozen motional conditions, dipolar gases, featuringlong-range and anisotropic interactions, offer the opportunity
to bring ultracold systems several steps ahead toward theambitious attempt of modeling and understanding quantum
magnetism. Great advances in studying quantum magnetism
have been achieved using arrays of Rydberg atoms [5–8],
trapped ions [9–11], polar molecules [12,13], and spin-3*

Corresponding author: manfred.mark@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.
2643-1564/2020/2(2)/023050(16)023050-1

bosonic chromium atoms [14,15]. Most of these studies so
far have been limited to spin-1/2 mesoscopic arrays of at
most a few hundred particles or to macroscopic but dilute(< 0.1 filling fractions) samples of molecules in lattices orwere contaminated by the effects of double occupancies inthe system. Only very recently, experiments using bosonicchromium gas have revealed the power of magnetic atoms
toward the understanding of quantum magnetism [16].
Here we report on a substantial leap forward in the control
and understanding of atomic dipolar platforms for simulating
quantum magnetism. Relying on the atomic properties of
erbium, we realize a quantum simulator of the long-range
X X Z Heisenberg model in a dense array of fermionic magnetic atoms with unique control knobs. Our platform rootson the special atomic properties of 167 Er. The erbium groundstate bears large angular momentum quantum numbers with
I = 7/2 for the nuclear spin and J = 6 for the electronic angular momentum, resulting in a F = 19/2 hyperfine manifold.
These large numbers bring in several important consequences
and provide novel control capabilities. First, they give accessto a fully addressable landscape of 2F + 1 = 20 internal
levels, as depicted in Fig. 1(a). Second, they are responsible
for the large magnetic moment in erbium, leading to a strong
dipolar coupling between atoms in neighboring lattice sites.
Such coupling can be up to 49 times larger than the ones
felt by F = 1/2 alkali atoms and about 40% larger than the
one in chromium [17]. Finally, the non-S character of theground state leads to comparatively large tensorial light shifts[18], which enables spin-dependent optical manipulation in
addition to the quadratic Zeeman effect [19].
In our work, any of the 20 internal spin states available inerbium can be selected to initialize our simulator in a quantum
Published by the American Physical Society

A. PATSCHEIDER et al.

(a)

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

B|+19/2>|+17/2>|+15/2>|+13/2>|+11/2>|+9/2>|+7/2>|+5/2>|+3/2>|+1/2>|-1/2>|-3/2>|-5/2>|-7/2>|-9/2>|-11/2>|-13/2>|-15/2>|-17/2>|-19/2>

(b)

z
B

Θ
y

Φ
x

Activation(c)
Initialization

Time evolution

Δ1
Δ2

early timelong time

FIG. 1. (a) Illustration of the total spin space of a single 167 Er√atom in the lowest hyperfine level |F = 19/2 with all 20 mF states. The angleof the symbols indicates the orientation of the total spin |F| = F (F + 1) in relation to the quantization axis. (b) Sketch of the experimental
system, an anisotropic three-dimensional lattice structure filled with fermionic 167 Er with a quantization axis tunable by the angles (, φ)of the external magnetic field B. (c) Illustration of the experimental sequence (from left to right): The system is initialized by preparing all
atoms in one starting state, here |–17/2. We activate the spin dynamics by changing the magnetic field to set δ̄ = 0. Early-time dynamics are
happening mainly among nearest-neighbor atoms. Subsequently, interactions between atoms at larger distances are involved in the dynamics.

spin Fock state. After activating spin-exchange processes,
arising from the large magnetic dipole interaction of erbium,
we study the spreading of the spin population, characterize
the effective strength of the dipolar coupling, and observe evidence of beyond-nearest-neighbor couplings. We benchmark
our simulator with an advanced theoretical approach, whichtakes quantum entanglement and spatial inhomogeneities into
account [20]; see Appendix H. Moreover, we show that the
spin-exchange dynamics can be fully controlled via opticallight fields on timescales much faster than typical interaction
times. The reported demonstration of these new control knobs,
some without equivalence in alkali and chromium atoms,
constitutes an important step toward a fully controllable quantum simulator, e.g., for the realization of synthetic dimension[21–23] or as qudits for quantum computation [24–26].
II. THEORETICAL DESCRIPTIONThe X X Z Heisenberg model that rules the magnetizationconserving spin dynamics of our system can be conveniently

written using spin-19/2 dimensionless angular momentum
operators F̂i = {F̂ix , F̂iy , F̂iz }, acting on site i and satisfying
the commutation relation [F̂ix , F̂iy ] = iF̂iz . We use the eigenbasis of F̂ z denoted as |mF  with 0  |mF |  F [27,28](see Appendix F):


1
1 + −
z z
− +
Ĥ =
Vi, j F̂i F̂j − (F̂i F̂j + F̂i F̂j )2 i, j=i4  2
δi F̂iz .
+(1)

i1−3 cos2 (θ )i, jThe coupling constants Vi, j = Vdd
describe theri3j
direct dipole-dipole interactions (DDI), which have longrange character and thus couple beyond nearest neighbors.
The dipolar coupling strength between two dipoles locatedat ri and rj depends on their relative distance ri j = |ri − rj |and on their orientation, described by the angle θi, j between
the dipolar axis, set by the external magnetic field, and the023050-2

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

δ / h (Hz)
0 20 40 60 80

(b)

-12/2

1

(c) -19/2

1

-17/2
-15/2
-13/2
-11/2
-9/2
-7/2

0.9

(d) -19/2

0.5

-17/2
-15/2
-13/2
-11/2
-9/2
-7/2
0

0.4

Spin state (mF)

1 -40 -20

M(t)

(a)

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

-13/2

0.8
Δ2

0.6
Δ1
Δ2

0.4

50

100

0.6

Spin state (mF)

Δ1Spin population

Spin population

0.8

0.4
0.2

0.2
0

-14/2
0

0
1

1.5
2
2.5
Magnetic field (G)

25

0

50
75
Time (ms)

100

0.8
0.7
0.6

0.3
0.2
0.1
25

50
75
Time (ms)1000FIG. 2. (a) Measured spin populations in states mF = |–17/2 (circles) and mF ± 1 (diamonds and squares) after 50 ms hold time as afunction of the magnetic field with  = 0◦ . A Gaussian fit (not shown) to the data provides a resonant magnetic field value of ≈1.67 G. The
top axis shows the mean total detuning δ̄ from the resonance condition. (b) Measured spin population in states mF = |–13/2 (circles) andmF ± 1 (diamonds and squares) as a function of the hold time after quenching onto the spin-exchange resonance with  = 0◦ . The dashed line
shows the simple mean-field result, the dashed-dotted line gives the NNI-GDTWA result, the solid lines represent the full-GDTWA result, andthe dotted line shows the full-GDTWA for the ideal case of a lattice with unit filling. The inset shows the total magnetization M(t ). [(c) and(d)] Spin diffusion with initial state mF0 = |–13/2 plotting the population of states from mF − 3 to mF + 3 as a function of the hold time, forthe experiment (c) and the full-GDTWA model (d), with the same initial conditions as (b). Data points consist of a minimum of four individual
realizations and error bars denote the standard error of the mean.
μ g2 μ2interparticle axis; see Fig. 1(b). Here Vdd ≈ 0 4πF B denotes the
dipolar coupling strength, with gF ≈ 0.735 for 167 Er, μ0 themagnetic permeability of vacuum, and μB the Bohr magneton.
The F̂iz F̂jz terms in the Hamiltonian account for the diagonalpart of the interactions while the F̂i+ F̂j− + F̂i− F̂j+ terms describe dipolar exchange processes. The second sum denotes
the single-particle quadratic term δi (F̂iz )2 with δi = δiZ + δiT ,accounting for the quadratic Zeeman effect ∝ δiZ and tensorial
light shifts ∝ δiT . These two contributions can be independently controlled in our experiment.
The quadratic Zeeman shift allows us to selectively prepare all atoms in one target state of the spin manifold; see
Appendix B. The tensorial light shift can compete or cooperate with the quadratic Zeeman shift and can be used as anadditional control knob to activate/deactivate the exchange
processes. Note that, for all measurements, a large linear
Zeeman shift is always present, but since it does not influence
the spin-conserving dynamics, it is omitted in Eq. (1).
III. EXPERIMENTAL SEQUENCE

In the experiment, we first load a spin-polarized quantum
degenerate Fermi gas of ≈104 Er atoms into a deep threedimensional (3D) optical lattice, following the scheme ofRef. [29]. The cuboid lattice geometry with lattice constants(dx , dy , dz ) = (271(2), 266(1), 542(4)) nm results in weakly
coupled 2D planes, with typical tunneling rates of ∼10 Hz
inside the planes and ∼mHz between them; see Appendix E.
The external magnetic field orientation, setting the quantization axis as well as the dipolar coupling strengths, is definedby the polar angles  and φ in the laboratory frame; see
Fig. 1(b). The fermionic statistics of the atoms enables us
to prepare a dense band insulator with at most one atom perlattice site. Additionally, Pauli exclusion blocks tunneling aswell as superexchange processes on the initial timescales,
helping in realizing a frozen condition, as required for aclean implementation of the X X Z Heisenberg model; see
Appendix J. This is an advantage of fermionic atoms ascompared to bosonic systems, which typically require filtering
protocols to remove doublons and higher lattice depths to
avoid tunneling [16].
Our experimental sequence to study the spin dynamics is illustrated in Fig. 1(c). In particular, we prepare the system into
the targeted spin Fock state, with a near-unity population of
the mF0 level, by using the lattice-protection protocol demonstrated in Ref. [29]. At the end of the preparation, the majority
of atoms are in the desired mF0 (>80%) at B ≈ 4 G. We notethat atom losses during the spin preparation stage reduces thefilling factor to about 60% of the initial one; see Appendix D.
We then activate the spin dynamics by quenching the magnetic field to a value for which δ̄ = i δi = 0, providing aresonance condition for the magnetization-conserving spinexchange processes; see Fig. 2(a) with mF0 = |–17/2. Aftera desired time of evolution, we stop the dynamics by rapidly
increasing the magnetic field, leaving the resonance condition.
We finally extract the atom number in each spin state via aspin-resolved band-mapping technique and derive the relative
state populations by normalization to the initial total atom
number. Note that, throughout all our experiments, within
our detection resolution, we do not observe any populationin higher bands.
We now probe the evolution of the spin-state population
as a function of the hold time on resonance. We observea redistribution of the population from the initial state tomultiple neighboring states in mF space, as for exemple shown
for an initial state of |–13/2 in Figs. 2(b) and 2(c). The

023050-3

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

dynamics preserves the total magnetization; see the inset of
Fig. 2(b). We observe similar behavior independently of theinitialized mF0 states. The spin transfer happens sequentially.
At short times it is dominated by the transfer to states directly
coupled by the dipolar exchange Hamiltonian, i.e., those ones
which differ by plus/minus one unit of angular momentum(mF = ±1). At longer times, subsequent processes transfer
atoms to states with |mF |  2; see Figs. 2(c) and 2(d).

IV. COMPARISON TO GDTWA

To benchmark our quantum simulator, we use a semiclassical phase-space sampling method, the so-called generalized discrete truncated Wigner approximation (GDTWA)[16,20,30–32]; see Appendix G. The method accounts forquantum correlation in the many-body dynamics and is
adapted to tackle the complex dynamics of a large-spin system
in a regime where exact diagonalization techniques are impossible to implement with current computers. The GDTWA
calculations take into account actual experimental parameterssuch as spatial inhomogeneites, density distribution after thelattice loading, initial spin distribution, and effective latticefilling, including the loss during the spin preparation protocol;
see Appendices D and H. Figure 2(b) shows the experimental
dynamics together with the GDTWA simulations. Although
the model does not include corrections due to losses andtunneling during the dynamics, it successfully captures the
behavior of our dense system not only at short time but also
at long time, where the population dynamics slows down andstarts to reach an equilibrium. A similar level of agreement
between experiment and theory is shown in Figs. 2(c) and2(d) where we directly compare the spreading of the spin
population as a function of time.
Our choice of using a quantum Fock state instead of aspin coherent state [16] as initial state combined with ourpossibility to cancel the quadratic energy shift (i.e., set δ̄ = 0)
allows us to directly reveal the important role of quantum
effects in the observed spin dynamics. This can be shown
both at the quantitative and qualitative levels by contrasting
the GDTWA simulation with a mean-field calculation. Indeed,
the mean-field calculation fails in capturing the system behavior. It predicts a too-slow population dynamics for nonperfect spin-state initialization, as in the experiments shown in
Fig. 2(b), and no dynamics for the ideal case where all atoms
are prepared in the same internal state. To emphasize thebeyond-nearest-neighbor effects, we also compare the experiment with a numerical simulation that only includes nearestneighbor interactions (NNI-GDTWA). Here we again observea very slow spin evolution, which largely deviates from themeasurements. The agreement of the full GDTWA predictions
with our experimental observations points to the long-range
many-body nature of the underlying time evolution. Our
theory calculations also support the build-up of entanglement
during the observed time evolution. To illustrate the quantum
spin dynamics under ideal conditions, in Fig. 2(b), we also
show the results for a system with unity filling and perfectinitialization in state mF0 = |–13/2. In this case a speedupof the dynamics due to the increased effective interaction
strength is visible, quickly approaching an equilibrium state.

V. CONTROL OF THE INTERACTION STRENGTH

Different spin configurations feature distinct effective interaction strengths, which also depend on the orientation of
the dipoles with respect to the lattice. We demonstrate ourability to control this interaction, which governs the rateof population exchange, by the choice of the initial mF0
state and the orientation of the external magnetic field. This
capability provides us with two tuning knobs to manipulate dipolar exchange interactions in our quantum simulator.
Figures 3(a)–3(f) plot the dynamics of the populations forthree neighboring spin states after the quench, starting fromdifferent initial spin states. Solid lines show the results ofthe full-GDTWA calculations. For each initial mF0 , we finda remarkable agreement between theory and experiment. We
observe a strong speedup for states with large spin projections
perpendicular to the quantization axis, as is expected from theexpectation value of F̂i+ F̂j− , which gives a prefactor γ (mF0 ) =√√F (F + 1) − mF0 (mF0 + 1) F (F + 1) − mF0 (mF0 − 1). The
initial dynamics can be well described by a perturbative
expansion up to the second order (see Appendix I), resulting in
the analytic expression for the normalized population nmF (t )of the initial state:


V2 2
.
(2)
nmF0 (t ) = nmF0 (0) 1 − nmF0 (0) effth̄2
γ 2 (m0 ) 
22Here Veff≡ 8N Fi, j=i Vi j is the overall effective interaction strength summed over N atoms and nmF0 (0) denotes
the purity of the initial state preparation. For a quantitative
analysis of the early-time spin evolution, we compare thetheoretically calculated Veff from the initial atomic distribution
used in the GDTWA model with the one extracted from a fit ofEq. (2) to the experimental data. Here we consider the data up
to t < 0.5 Vh̄eff estimated using the theoretically calculated Veff[33]. Figure 3(g) plots both the theoretical and experimental
Veff as a function of the initial mF0 and highlights once moretheir quantitative agreement. The interaction parameter Veff
can also be used to rescale the time axis. As shown in
Fig. 3(h), all data sets now collapse onto each other fortVeff< 0.5, revealing the invariant character of the short-timeh̄
dynamics under the internal state initialization. At longer
timescales, the theory shows that the timetraces start to deviate
from each others and saturate to different values, indicatingthat thermal-like states are on reach. In the experiment, weobserve a similar behavior but here the saturation value mightalso be affected by losses and residual tunneling.
Because of the anisotropic character of the DDI, thestrength of the dipolar exchange interaction can be controlled
by changing the angle ; see Fig. 1(b). As shown in Fig. 4(a)for |–17/2, the observed evolution speed of the spin populations strongly depends on , changing by about a factorof 2 between  = 40◦ and 80◦ . The GDTWA results show
a very good quantitative agreement with the experiment. We
repeat the above measurements for different values of  andwe extract Veff ; Fig. 4(b). It is worthwhile to mention that,while the dipolar interactions can be completely switched off
at a given angle in a 1D chain, in a 3D system the situation
is more complicated. However, as expected by geometricalarguments, we observe that the overall exchange strength023050-4

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

= |-17/2>(b)= |-9/2>(c)= |-5/2>(g) 70Veff / ħ (sec-1)

0.8
0.6
0.4
0.2
0
1

Spin population

(a)
a)

(d)

= |1/2>(e)= |9/2>(f )= |13/2>

42
28
14

0.6
0.4
0.2
0
25

50 75 100 0Time (ms)25

50 75 100 0Time (ms)

25

-17/2

-9/2

(h) 1

0.8

0

56

0

Spin population

Spin population

1

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

50 75 100
Time (ms)-3/2 3/2
9/2
Initial state (m0F)

17/2

0.8

0.6

0.4
0

0.5

1
Veff / ħ . t

1.5

2

FIG. 3. [(a)–(f)] Dynamic evolution of the initial states |–17/2 (a), |–9/2 (b), |–5/2 (c), |1/2 (d), |9/2 (e), and |13/2 (f) and of thecorresponding neighboring states mF ± 1 together with the full-GDTWA results (solid lines) for  = 0◦ . (g) Extracted Veff as a function of theinitial state mF0 from a fit to the experimental data (cyan triangles) and numerically computed from the initial spin distribution (black circles).
Error bars denote the 68% confidence interval of the fits. (h) All datasets with mF0 < 0 used in (g) together with the corresponding full-GDTWA
results (solid lines) plotted in units of the rescaled time τ = Veff /h̄ · t. Note that all experiment and theory data are plotted for times t  100 msof (a)–(f). To account for the different preparation fidelity, the populations of the initial states are shifted to 1 by adding a constant offset. For
clarity error bars are omitted here.

becomes minimal for a specific dipole orientation (c ≈ 35◦ ,φc = 45◦ ). We compare our measured Veff with the ones
calculated from the initial spin distribution, which is a goodquantity to describe the early-time dynamics. Theory andexperiment show a similar trend, in particular reaching aminimum at about c . Note that the simple analytic formula[Eq. (2)], used for fitting the data, deviates from the actual
evolution at longer times. This leads to a small down-shift ofthe experimental values; see Appendix I. Our study ultimately
demonstrates the ability to tune the lattice spin model throughthe magnetic field orientation. This paves the way to thesimulation of various spin-lattice models [34].

By exploiting the rich energy spectrum of lanthanides like
erbium, narrow optical transitions [38] allow an even morerefined manipulation of the spin manifold like, e.g., blocking
specific spin-exchange channels or additional Floquet engineering. This capabilities can be an excellent resource forquantum information processing, e.g., we could use dipolar
exchange processes to efficiently prepare highly entangledstates between different parts of a quantum system and thenstore the quantum information at longer times by turning theinteractions off.

VII. CONCLUSION AND OUTLOOKVI. OPTICAL CONTROL

Finally, we demonstrate fast optical control of the spin
dynamics relying on the remarkably large tensorial light shiftof erbium compared to alkali atoms. As shown in Fig. 4(c),we can almost fully suppress the spin-exchange dynamics
by suddenly switching on a homogeneous light field after an
initial evolution time on resonance. Therefore, the tensorial
light shift, inducing a detuning from the resonance condition(see inset), allows a full spatial and temporal control overthe exchange processes as light fields can be easily shapedin space and time, in stark contrast to magnetic fields. For
example, the light power can be changed orders of magnitude faster than the typical interaction times and can address
even single lattice sites in quantum gas microscopes [35–37].

In our work, we explore the dipolar exchange dynamics
and benchmark our simulator with an advanced theoreticalmodel, which takes quantum entanglement and spatial inhomogeneities into account. In particular, we initialize thesystem into a desired spin Fock state and activate the spin
dynamics, while the motional degree of freedom mainly remains frozen. We study the spreading of the spin population
with different choices of the macroscopically populated initial
Zeeman level as a function of both the specific initial spinstate and the dipole orientation. We demonstrate that the
spin dynamics at short evolution time follows a scaling thatis invariant under internal state initialization and that is set
by the effective strength of the dipolar coupling. On thecontrary, at longer times, the many-body dynamics is affectedby the accessible spin space and the long-range character

023050-5

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

tunneling is only allowed in one direction. Moreover, thedemonstrated control over the initial state population of arbitrary hyperfine levels, our capability to tune the strengthof the direct dipolar exchange coupling via the magnetic
field angle, and the possibility of the dynamical and spatial
control of the hyperfine manifold via tensorial light shifts
make our system a potential resource for quantum information
processing with a qudit-type multilevel encoding using the 20
different interconnected hyperfine levels [24–26].

(b) 20(a) 1
Θ = 80°Θ = 40°Veff / ħ (sec-1)

Spin population

15
0.8

10

0.6
5
ACKNOWLEDGMENTS

0.4
0

25 50
75 100Time (ms)1.2

Spin population

Spin population

(c)

1
0.8

0

0

20 40 60
Θ (deg)

80

50

75
100
δiZ / h (Hz)

125

75

100

1

0.7

0.4
25

0.6
0.4
0.2

0

25

50
Time (ms)FIG. 4. (a) Exemplary measurements of the time evolution forthe starting spin state |–17/2 for  = 40◦ , 80◦ . Solid lines show the
full-GDTWA results. (b) Extracted Veff as a function of  from a fit
to the experimental data (orange circles) and numerically computedfrom the initial spin distribution (black circles). Error bars denote the
68% confidence interval of the fits. (c) Time evolution of the initialstate |–9/2 at δ̄ = 0 and  = 0◦ without (filled circles) and with
(open circles) switching on an additional light field after 20 ms ofevolution. Solid (dashed) lines are the corresponding full-GDTWA
calculations. The inset shows the population of the initial spin state
after 50 ms evolution time as a function of the quadratic Zeeman shift
without (filled circles) and with (open circles) the additional lightfield. Determining the centers of the resonances via a fit yields anabsolute shift of the resonance condition by h × 27(1) Hz between
both conditions.

of dipolar interactions beyond nearest neighbors. We further
demonstrate temporal control of the exchange dynamics using
off resonant laser light.
The excellent agreement between the experiment and thetheory not only verifies our quantum simulator but also sets
the stage toward its use for the study of new regimes intractable to theory. For example, by operating in a shallow
lattice where motion is involved, the dynamics is no longer
described by a spin model but by a high spin Fermi-Hubbard
model with long-range interactions. Alternatively by treating
the internal hyperfine levels as a synthetic dimension [21,22]
while adding Raman transitions to couple them, one could
engineer nontrivial synthetic gauge field models even whenWe thank J. Schachenmayer for fruitful discussions andArghavan Safavi-Naini for helping us understanding the loading into the lattice. We thank J. H. Becher and G. Natale for
their help in the experimental measurements and for fruitful
discussions. We also thank Rahul Nandkishore and Itamar
Kimchi for reviewing the manuscript. The Innsbruck group is
supported through an ERC Consolidator Grant (RARE, Grant
No. 681432), an EC Horizon 2020 FET Proactive project(RySQ, Grant No. 640378), and a DFG/FWF Forschergruppe (FOR 2247/PI2790). L.C. is supported though an
EC Horizon 2020 Marie Curie Project (DipPhase, Grant No.
706809). A.M.R is supported by the ARO single investigatorAward No. W911NF-19-1-0210, DARPA-DRINQs, the NSF
PHY1820885, JILA-NSF PFC-173400 grants, and by NIST.
B.Z. is supported by the NSF through a grant to ITAMP.

APPENDIX A: EXPERIMENTAL SETUP
AND LATTICE LOADING

In our experiment we start with a degenerate Fermi gasof about 2.4 × 104 167 Er atoms in the lowest spin state|F = 19/2, mF = –19/2 = |–19/2 and a temperature ofT ≈ 0.3 TF [29,39]. The atoms are confined in a crossed
optical dipole trap (ODT) and the trap frequencies are(ν⊥ , ν , νz ) = (63(1), 36(2), 137(1)) Hz, where ν⊥ (ν ) are
the trap frequencies perpendicular to (along) the horizontal
ODT and νz is measured along the vertical direction definedby gravity. We load the atomic sample adiabatically into a
3D lattice that is created by two retroreflected laser beams at
532 nm in the x-y plane and one retroreflected laser beam at
1064 nm nearly along the z direction, defined by gravity andorthogonal to the x-y plane. Note that due to a small angle ofabout 11(2)◦ between the vertical lattice beam and the z axis
we obtain a 3D lattice, slightly deviating from an ideal situation of a rectangular unit cell and our parallelepipedic cell has
the unit lattice distances of dx = 271(2) nm, dy = 266(1) nm,
and dz = 542(2) nm. The lattice geometry is similar to theone used in our previous works [29,40]. We ramp up thelattice beams in 150 ms to their final power and switch off
the ODT subsequently in 10 ms and wait for 500 ms toeliminate residual atoms in higher bands [29]. For our typicallattice depths used in the measurements reported here of(sx , sy , sz ) = (20, 20, 80), where si with i ∈ x, y, z is given in
the respective recoil energy ER,i with ER;x,y = h × 4.2 kHz andER;z = h × 1.05 kHz, the atoms can be considered pinned on
single lattice sites with low tunneling rates Jx,y = h × 10.5 Hz
and Jz = h × 1 mHz.

023050-6

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …
APPENDIX B: STATE PREPARATION
AND DETECTION EFFICIENCYPHYSICAL REVIEW RESEARCH 2, 023050 (2020)

(a)
0.5

0

E/h (MHz)

-0.5

-540
-550
-560
-1.5 -570
-580
-590
-390

-1

-2

avoided crossing
-380

100

(b)

95
90
85
80

-370

mF = +19/2
mF = +17/2
mF = +15/2
mF = +13/2
mF = +11/2
mF = +9/2
mF = +7/2
mF = +5/2
mF = +3/2
mF = +1/2
mF = -1/2
mF = -3/2
mF = -5/2
mF = -7/2
mF = -9/2
mF = -11/2
mF = -13/2
mF = -15/2
mF = -17/2
mF = -19/2-360-400 -300 -200 -100 0 100 200 300 400
RF detuning (kHz)100

Counting efficiency (%)

Preparation efficiency (%)

To prepare the atoms in the desired spin state, after loading them into the lattice, we use a technique based on a
rapid-adiabatic passage (RAP). During the full preparation
procedure, the optical lattice operates as a protection shield
to avoid atom loss and heating due to the large density ofFeshbach resonances [29]. To reach a reliable preparation
with high fidelity it is necessary that the change in the energy difference between subsequent neighboring spin states issufficiently large. Therefore, we ramp the magnetic field in
40 ms to 40.6 G to get a large-enough differential quadratic
Zeeman shift, which is on the order of ≈h × 40 kHz. Afterthe magnetic field ramp we wait for 80 ms to allow the latterto stabilize before performing the RAP procedure. The follow
up RAP protocol depends on the target state. For example, totransfer the atoms from |–19/2 into the |–7/2 state, we apply
a radio-frequency (RF) pulse at 41.31 MHz and reduce themagnetic field with a linear ramp, e.g., by 500 mG in 42 ms.
The variation of the magnetic field is analogous to the moreconventional scheme where the frequency of the RF is varied[see Fig. 5(a)]. For the preparation of higher (lower) spinstates we perform a larger (smaller) reduction of the magnetic
field on a longer (shorter) timescale. After the RAP ramp weswitch off the RF field and ramp the magnetic field in 10 ms
to B = 3.99 G. Here we wait again for 100 ms to allow themagnetic field to stabilize. During the ramp up and down to
40 G of the magnetic field we loose 25(2)% of the atoms. We
attribute this loss mainly to the dense Feshbach spectra that weare crossing when ramping the magnetic field. The exact loss
mechanism has not been yet identified, constituting a topic ofinterest for latter investigation. At 3.99 G, before switchingon the spin dynamics, about 1.7 × 104 atoms remain in thelattice. The losses affect the lattice filling at which the spin
dynamics occur. Our simulations account for this initiallyreduced filling; see Appendix H.
Additionally to the losses due to the magnetic field ramps,
we also observe losses caused by the RAP itself. To quantify
the preparation efficiency, i.e., the loss of atoms due to thespin preparation via RAP as a function of the target mF
state, we perform additional measurements where we either
do not perform a RAP or where we add an inverse RAP totransfer all atoms back into the |–19/2 state. By comparing
the atom numbers from measurements without RAP and withdouble-RAP and assuming that the loss process is symmetric,
we derive the preparation efficiency as plotted in Fig. 5(b).
We also account for the difference in the counting efficiencyof the individual spin states, which arises from different
effective scattering cross sections for the imaging light. Herewe compare the measured atom number in a target mF state to
the expected atom number taking into account the previously
determined preparation efficiency as discussed above and theatom number without RAP; see Fig. 5(c).
The counting and preparation efficiencies are determined
for the |–17/2, |–15/2, |–9/2, and |9/2 states and interpolated assuming a linear dependency of these efficiencies on mF [see Figs. 5(b) and 5(c)]. We estimate thepreparation efficiency of the respective mF state to be
0.86(1) − 0.008(1) × mF . We attribute the lower preparation-17-11 -5 1 7 13
2 2 2 2 2 2Spin state (mF)

(c)

80
60
40
20

-17-11 -5 1 7 13
2 2 2 2 2 2Spin state (mF)

FIG. 5. (a) Energy levels of the ground-state hyperfine manifoldin the dressed-state picture in dependence of the detuning between
the applied just RF and the atomic resonance condition for the |–1/2
to the |1/2 hyperfine levels. The solid red arrow exemplary shows
the RAP for preparation of atoms into the mF0 = |–7/2 state. The
insets show a zoom of one avoided crossing. [(b) and (c)] Spinpreparation and atom-counting efficiency measured for |–17/2,|–15/2, |–9/2, and |9/2. The obtained values are interpolated
linearly assuming a linear dependence on the mF0 state.

efficiency for higher spin states to the larger number of
avoided crossing between spin states that come into playduring the RAP procedure. Overall, we expect that the latticefilling over the whole sample, taking into account the losses
due to magnetic field ramping and spin state preparation,
reduces from close to unity down to a value between 0.6 and0.7; see also Tables I and II.

APPENDIX C: QUENCH PROTOCOL AND
DETECTION SEQUENCE

In our experiment we exploit both the light and themagnetic shifts of the energies of each spin state to reacha resonant condition where the energy difference between
neighboring spin states is canceled and therefore spin changing dynamics preserving the total magnetization become energetically allowed. In particular, we exploit the tensorial light

023050-7

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

TABLE I. Percentage of lost atoms and extracted, effective filling fraction ν for different |mF  states at t = tfit and t = 100 ms.
mF0

tfit (ms)

Nloss (tfit ) (%)ν (0)ν (tfit )Nloss (100 ms) (%)ν (100 ms)

− 17
2

34.2

1.8

0.7

− 13
2

15.7

7.2

0.69

0.68

5.3

0.66

0.64

19.6

− 29
− 25
− 21
1
2
3
2
9
2
13
2

0.55

11.3

8.7

0.67

0.62

25.1

0.51

9.6
9.0

13.7

0.66

0.58

27.7

0.48

12.2

0.65

0.57

36.1

0.42

9.0

13.5

0.65

0.56

36.7

0.41

9.2

9.2

0.64

0.58

34.0

0.43

11.3

6.7

0.63

0.59

23.1

0.49

15.7

4.9

0.62

0.59

21.9

0.48

shift of the spin states energies [18]Ut =3mF2 − F (F + 1) 3 cos2 θ p − 1
αt (ω),F (2F − 1)
2

present in atomic erbium to initialize the dynamic evolutionof the spin population. The tensorial light shift depends
quadratically on the mF state as well as on the angle θ p
between the magnetic field axis and the axis of polarizationof the light. Here αt refers to the tensorial polarizability
coefficient and ω to the light frequency. After the preparationof the respective spin state we start all our measurements at
B = 3.99 G, pointing in the z direction. However, to reachthe resonance condition we use two slightly different quench
protocols for the measurement sets with fixed  = 0◦ for thedifferent initial spin state and for the sets of measurements
where |mF  = |–17/2 and  ∈ (0◦ , 80◦ ). The measurement
sequences differ on the one hand by the way we jump on
resonance to initialize the spin dynamics and on the other
hand by shining in an additional laser beam of wavelength1064 nm and power of 7 W. This additional light is necessary
because changing  reduces simultaneously θ p , resulting ina smaller tensorial light shift and therefore in a shift of theresonance position to lower magnetic field values. For large
 the light shift of the lattice beams is smaller and therefore
the resonance is very close to 0 G which we want to avoid
to prevent spin relaxation processes. For the sets of measurements where we keep  = 0◦ but vary the initial mF0 state wequench the magnetic field directly after the preparation, from3.99 G to resonance. In contrast we use a different approachfor the measurements where  is varied. After the preparation
we ramp in 10 ms the additional laser beam to 7 W. Due tothe reduced speed of our magnetic field coils in the x andy directions we first rotate the magnetic field such that thetransverse components Bx and By are already at their target
values while keeping an additional offset of 2 G in the z
direction. The quench to resonance is then done using only thecoils for the magnetic field in the z direction. The additional
offset field of 2 G is large enough to suppress dynamics. We
measure the evolution of the magnetic field by performing
RF spectroscopy and find that for both quench procedures
the magnetic field evolves exponentially toward its quench
value with 1/e decay times of 1.4 and 1.2 ms, respectively.
After holding on resonance for a certain time we quench
the magnetic field back to 3.99 G and we rotate the latterback to  = 0◦ . After a waiting time of 50 ms we perform aband-mapping measurement combined with a Stern-Gerlach
technique, i.e., we ramp the lattice down in 1 ms and apply
a magnetic field gradient that is large enough to separate
the individual spin states after a time of flight (TOF) of
15 ms. This allows us to image the first Brillouin zone for thedifferent spin states. During TOF the magnetic field is rotatedtoward the imaging axis. We typically record the populationof the initially prepared |mF , of its four neighbors, and of|–19/2 by summing the 2D atomic density over a region ofinterest. Figure 6 shows examples of the imaging of different
spin states for the cases of a nonadjusted RAP as well as forthe preparation of the atoms in |–9/2, |3/2, and |5/2. In thecase of |3/2 residual atoms in |–19/2, |–17/2, and |5/2
are visible due to a nonperfect preparation.

TABLE II. Percentage of lost atoms and extracted effective filling fraction ν for different  at t = tfit and t = 100 ms.
 (deg)tfit (ms)

Nloss (tfit ) (%)ν (0)ν (tfit )Nloss (100 ms) (%)ν (100 ms)

0
10
20
30
35
40
50
60
80

26.8
30.1
36.7
47.0
52.0
53.2
46.1
37.0
30.0

13.8
11.9
6.9
8.6
6.7
7.2
12
8.9
10.4

0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7

0.60
0.61
0.65
0.64
0.65
0.65
0.61
0.63
0.62

30.1
25.6
18.2
17.8
11.4
13.2
17.2
19.2
19.3

0.49
0.52
0.57
0.57
0.62
0.60
0.58
0.56
0.56

023050-8

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

|5/2>

Bz

|3/2>

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

cal prediction without losses; see Incorporating Experimental
Conditions in Numerical Simulation. A proper description of
the long-time dynamics will certainly require to account andunderstand such effects.

|1/2>
APPENDIX E: EXPERIMENTAL UNCERTAINTIESAND INHOMOGENEITIES|-1/2>

Spin states (mF )|-3/2>|-5/2>|-7/2>|-9/2>|-11/2>|-13/2>|-15/2>|-17/2>|-19/2>
FIG. 6. Absorption images for a nonadjusted RAP and for the
preparation of |–9/2, |3/2, and |5/2. Whereas for the |–9/2 and|5/2 case no residual atoms in other spin states are visible, for the|3/2 case we observe a small amount of residual atoms in other spinstates due to a nonperfect preparation.

APPENDIX D: LIFETIME AND LOSSES IN THE LATTICE

Off-resonance, i.e., at a magnetic field of B = 3.99 G,
we measure the lifetime of the prepared spin state to be on
the order of a few seconds, being slightly shorter for higher
spin states. Note that here we do not observe any population
growing in the neighboring spin states. Differently, for themeasurements on resonance, we observe a faster loss happening on the timescale of the first 20–30 ms followed by loss atlower speed for the remaining atoms. We fit an exponential
decay to extract the atoms loss and change in filling over thetimescale that we use to extract Veff , tfit , (see Appendix I) as
well as over the full 100 ms of the dynamics reported in themain text (Figs. 2 and 3). Table I gives the corresponding
numbers for the sets of data for the different initial mF0 states.
During the fitting timescale we observe atoms loss on theorder of 5–10%. This atom loss can be converted into a changeof the effective filling of the lattice compared to the stateobtained after the lattice loading giving a minimum fillingof ν = 0.56 for the mF0 = |1/2 case. For longer timescales
larger losses in the range between 10 and 35% are observed.
In general, we note that the amount of loss depends on theinitial mF0 state, resulting larger for the central |mF 0 states.
Similar numbers are obtained for the sets of measurement
where we vary  (see Table II). The exact mechanism leading
to these losses is not yet understood and will be the topic offuture studies. Thanks to their limited importance over the
early-time dynamics, we here compare our results to theoreti-

Ideally, all atoms in the sample experience the same linear
and quadratic Zeeman shift and the same quadratic light shift.
However, in the experiment inhomogeneities from the magnetic field and light intensities lead to a spatial dependence of
those shifts. An upper bound of the variation of Zeeman shifts
can be deduced from RF-spectroscopy measurements donewith bosonic erbium. From the width of the RF resonance(≈500 Hz) and the size of the cloud (≈15 μm) we estimate a maximum magnetic field gradient of 230 mG/cm,
assuming the gradient as main broadening mechanism forthe resonance width, neglecting magnetic field noise andFourier broadening. This translates into a differential linear
Zeeman shift of h × 6 Hz between adjacent lattice sites in
the horizontal x-y plane and h × 12 Hz between adjacentplanes along the z direction. Together with the magnetic fieldvalues used in the spin dynamic experiments, the variationof the quadratic Zeeman shift is negligible compared toother inhomogeneities (h × 0.1 Hz). The inhomogeneity ofthe quadratic light shifts can be estimated by considering
the shape of the lattice light beams [Gaussian beams withwaists of about (wx , wy , wz ) = (160, 160, 300) μm] and theresonance condition of the magnetic field, translated into aquadratic Zeeman shift of h × 71(1) Hz. These considerations
can be used to obtain an estimation for a site-dependent lightshift compared to the center of the atomic sample. If wetake a possible displacement of the atoms by 10 μm in all
directions, from the center of the lattice to the center of thebeams, into account, then we can estimate an upper bound forthe light shift of δiT  h × 2 Hz at 20 lattice sites away fromthe center along the y direction.
APPENDIX F: SPIN HAMILTONIAN

The experiment operates in a deep lattice regime, where
tunneling is suppressed. At the achieved initial conditions,the 167 Er atoms are restricted to occupy the lowest lattice
band, and Fermi statistics prevents more than one atom perlattice site. In the presence of a magnetic field strong enoughto generate Zeeman splittings larger than nearest-neighbor
dipolar interactions, only those processes that conserve the total magnetization are energetically allowed [15]. Under these
considerations, the dynamics is described by the following
secular Hamiltonian:  2 Ĥ =δi F̂iz +Bi F̂iz
i

i



1 + −
z z
+
Vi, j F̂i F̂j − (F̂i F̂j + H.c.) .
2 i, j=i
4
1

(F1)

Here the operators Fiz,± are spin 19/2 angular momentum
operators acting on lattice site i. The first two terms accountfor the site-dependent quadratic and linear shifts, respectively,

023050-9

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

where δi includes both Zeeman terms and tensorial lightshifts as discussed in the main text. Bi = B + Bi denotes
the linear Zeeman shift at site i. While the constant anduniform contribution, B, commutes with all other terms, thus
can be rotated out, the spatially varying contribution, Bi , isrelatively small in the experiment but still is accounted for in
the theory calculations. The last term is the long-range dipolarinteraction between atoms in different sites, with
1 − 3 cos2 θi, j
Vi,j ≡ Vdd dy3,|ri − r j |3

Vdd =μ0 (μB gF )4π

d 3 rd 3 r

(F2)

1
I.
N

iD =

1 − 3 cos θrr |φi (r)|2 |φ j (r )|2 ,|r − r |3(F3)
2

where φi (r) denotes the lowest band Wannier function centered at lattice site i. For the experimental lattice depths(sx , sy , sz ) = (20, 20, 80) in units of the corresponding recoil
energies, Vdd is estimated to be h × 0.336 Hz.

APPENDIX G: THE GDTWA METHOD

To account for quantum many-body effects during the dynamics generated by long-range dipolar interactions in thesecomplex macroscopic spin F = 19/2 3D lattice array, weapply the so-called GDTWA, first introduced in Ref. [16].
The underlying idea of the method is to supplement themean-field dynamics of a spin F system with appropriate
sampling over the initial conditions in order to quantitatively
account for the build-up of quantum correlations. For a spin-F
atom i with N = 2F + 1 spin states, its density matrix ρ̂iconsists of D = N × N elements. Correspondingly, we candefine D Hermitian operators, iμ , with μ = 1, . . . , D, using
the generalized Gell-Mann matrices (GGM) and the identity
matrix [41]:1iμ=1,...,N (N −1)/2 = √ (|β α| + H.c.),2iμ=N (N −1)+1,...,N 2 −1
⎞
⎛
α

1
⎝
=√
|β β| − α |α + 1 α + 1|⎠, (G3)α(α + 1) β=1
for 1  α < N ,where θi j is the angle between the dipolar orientation set by an
external magnetic field and the interparticle spacing ri − r j .
μ0 g2F μ2B
corresponds to the interaction strength between
Vdd ≈ 4πd3y
two atoms, i and j, separated by the smallest lattice constant|ri − r j | = dy = 266 nm and forming an angle θi, j = π /2
with the quantization axis. Here gF ≈ 0.735 is the Landeg factor for Er atoms, μ0 is the magnetic permeability of
vacuum, and μB is the Bohr magneton. We compute Vdd from2 

for α > β, 1  α, β  N ,(G1)for α > β, 1  α, β  N ,(G4)

With these operators, the local density matrix ρ̂i , as well as
any operator Ôi of local observables can be represented as
Ôi =


μ

cμi iμ ,

with



cμi = Tr iμ Ôi ,

(G5)
(G6)

and μ = 1, 2, . . . , D. This allows expressing both one-bodyand two-body Hamiltonians in the form Ĥi = μ cμi iμ and
ij
Ĥi j = μ,ν cμν
iμ νj . The Heisenberg equations of motionfor iμ can be written as
i h̄

diμ
dt



= iμ , Ĥ 
  ij  i=cνi iμ , iν +cσ,ν μ , iσ νj .
μ(G7)σ, j,ν

In the experiment, the initial state is
a product state of singleatom density matrices ρ̂(t = 0) = ρ̂ i (t = 0). If we adopt
a factorization iμ νj · · · kσ  = iμ νj  · · · kσ  for any
nonequal i, j, . . . , k (i.e., each operator acts on a different atom) and arbitrary μ, ν, σ , then Eq. (G7) becomes aclosed set of nonlinear equations for λiμ = iμ . Withina mean-field treatment, the initial condition is fixed byλiμ (t = 0) = Tr[iμ ρ̂(t = 0)], which determines the ensuingdynamics from Eq. (G7). This treatment neglects any correlations between atoms. In the GDTWA method, the initial valueof λiμ is instead sampled from a probability distribution inphase space, with statistical average λiμ (0) = Tr[iμ ρ̂(t = 0)].
Specifically, each iμ can be decomposed via its eigenval
ues and eigenvectors as iμ = aμi aμi |aμi  aμi |. We takeaμi as the allowed values of iμ in phase space, andthen for an initial state ρ̂ i (t = 0), the probability distribution is p(aμi ) = Tr[ρ̂ i (t = 0) |aμi  aμi |]. From Eq. (G7),each sampled initial configuration for the N atom array,{aμ } = {aμi11 , aμi22 , . . . , aμiNN }, leads to a trajectory of iμ , which
we denote as λiμ,{aμ } (t ). The quantum dynamics can be obtained by averaging over sufficient number of trajectories

1iμ=N (N −1)/2+1,...,N (N −1) = √ (|β α| − H.c.), (G2)2i023050-10λiμ (t ) ≈ λiμ (t ) =


{aμ }

p({aμ })λiμ,{aμ } (t ).

(G8)

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

1

8

0.6

Average neighbors

Spin population

0.8

Spin population= |-9/2>

1
0.8
0.6
0.4
0.2

0

25

50
75
Time (ms)100

Random removal
NN-dependent removal

6
4
2
0
0

0.4

0.2
0

20

40
60
Time (ms)80100FIG. 7. Spin dynamics for mF0 = |–9/2 as shown in Fig. 3 ofthe main text. The solid line represents the spin population and theshaded area denotes the standard deviation of individual trajectories
which shows that the spread of the GDTWA trajectories does not
grow large with time. The inset visualizes 10 typical trajectories
obtained from the GDTWA simulation. Note that the different trajectories include both the quantum noise, which is essential to accountfor beyond-mean-field effects, and statistical noise, which averages
out when sampling over enough trajectories.

This approach has been shown capable of capturing the buildup of quantum correlations [16,31]. In Fig. 7, we illustrate
the typical trajectories and their spread in the spin dynamics
obtained with GDWTA for mF0 = |–9/2.
APPENDIX H: INCORPORATING EXPERIMENTAL
CONDITIONS IN NUMERICAL SIMULATIONIn our experiment, the lattice filling fraction is not unity
when the spin dynamics takes place. The reduced filling fraction is due to two effects: the finite temperature and atom loss
during the initial state preparation. To account for the effectof a finite temperature, we first obtain the density distribution
before ramping up the lattice from a Fermi-Dirac distribution1
n0 (ri ) = 1+exp{β[(r, with parameters β = 1/kB T and μi )−μ]}
matching the inferred experiment temperature T and the total
atom number N0 = 2.4 × 104 . The function (ri ) accounts
for the weak external harmonic confinement. We computethe density distribution function after loading the atoms in
the lattice, nF (ri ), by simulating the lattice ramp which ispossible since to an excellent approximation we can treat
the system as noninteracting. Indeed, we neglect the dipolarinteraction in the loading given that their magnitude is much
lower than the Fermi energy of the gas. In the numericalsimulation, we then sample the position of atoms ri in thelattice according to a distribution p(ri ) = nF (ri )/N0 . In practice, to reduce computation cost we need to reduce the total
atom number in our calculations and use a smaller latticewith fewer populated lattice sites. In this case, we reduce
the number of lattice sites by a factor ξ = (Nsim /Nexp )1/3 ,where Nsim(exp) are the number of atoms in the simulation(experiment), while keeping the lattice spacings the sameas in experiment, (dx , dy , dz ) = (272, 266, 544) nm. That is,for an initial lattice with Lx sites along x direction, in our

0.5

1
1.5
2
2.5
Relative distance (dy)3

3.5

FIG. 8. Histogram showing the average number of atoms in
distances normalized to the lattice direction along y for random
removal of atoms and for removal depending on the number ofnearest neighbors (NN-dependent removal).

simulations there are ξ Lx sites while the separation between
two adjacent lattice sites is still dx . We then sample the initial
distribution of atoms in the lattice with p̃(r̃i ) = ξ 3 p(ξ r̃i ),which preserves the local density and is similar to sampling
in a coarse-grained lattice. In our simulations, we chose
Nsim  350 and checked that the convergence in Nsim has beenreached.
As discussed in Appendix B, a fraction of atoms is lost
during the ramp up and down of the magnetic field before
initializing the spin dynamics over the sample. While a rigorous treatment on how these losses modify the distribution is
not currently accessible with our current experimental setup,
we try to account for it in the simulation by preferentiallyremoving those atoms with a probablity ∝ p(ri )Nnn , where
Nnn is the number of nearest neighbors (separation dy ),until N = ν(0)N0 atoms are left. According to experimentestimates, the filling fractions before the initialization of
the spin dynamics are ν(0) = 0.6–0.7 (see Tables I and II).
Figure 8 shows the histogram of neighbors in the resulting
atom distribution. Such distribution effectively reduces thenearest-neighbor interactions and is found to give a better
agreement with experiment.
Both the quadratic and linear shifts in the experimentare inhomogeneous across the lattice as discussed in Appendix E, and we include them in our numerical simulation as site-dependent terms δi (F̂iz )2 and Bi F̂iz , with
δi = a|ri |2 and Bi = b(xi + yi + zi ). Based on experimental
estimation, we have chosen the values of a and b such that
δi = h × 1.6 Hz (h × 0.7 Hz) at 20 sites along y away fromthe lattice center, and Bi differs by h × 6 Hz (h × 1.8 Hz)
between adjacent sites, for Fig. 2 and 3 (Fig. 4) in thesimulation.
To illustrate the quantum spin dynamics under ideal conditions, in Fig. 2(b), we also show the results for atoms frozen in
the lattice with unity filling fraction and zero temperature andall initialized in a single mF0 state. In the numerical simulation
for this ideal case, a lattice size of 9 × 9 × 3 is used, which
accounts for the fact that the lattice spacing along z is larger
and thus the dipolar coupling decreases significantly within a
few lattice sites. Even though numerical simulations cannot be
performed for a lattice as large as that one in experiment, as

023050-11

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

shown in Ref. [16], for such a lattice configuration finite-size
effects are negligible.
APPENDIX I: SHORT-TIME POPULATION DYNAMICS

Considering a fixed initial atomic distribution over thelattice, the population dynamics at early times can be derived
via a perturbative short-time expansion

Spin population

A. PATSCHEIDER et al.

1

0.6
0.4
0

nmF (t ) ≡ n̂mF (t ) = n̂mF  + i[Ĥ, n̂mF ]t/h̄−i[Ĥ , [Ĥ , [Ĥ , n̂mF ]]]t 3 /3!h̄3(I1)

Here the average · is over the initial state, which isassumed to be a pure state, n̂mF = ( i PimF )/N, where
PimF = |mF ii mF | is the onsite projector for an atom at sitei in state |mF  and N denotes the total number of atoms. Notethat here the sums are always carried out over the populated
lattice sites in the initial lattice configuration. We obtain 
γ 2 mF0  2
2
Veff
=
V ,
(I2)
8N i, j=i i, j

  

γ mF0 = F (F + 1) − mF0 mF0 + 1

× F (F + 1) − mF0 mF0 − 1 ,(I3)where nmF0 denotes the population on the selected target state.
To obtain Eq. (I2), we have assumed that initially most of the
population is in this target state, i.e., nmF0 (0) ∼ 1. In the experiment, this assumption is always satisfied and therefore Eq. (I4)is expected to reproduce well the short-time dynamics.
The dependence of γ (mF0 ) on the initial state
mF0 is a consequence of the dependence of dipolar
exchange processes on the spin coherences, i.e.,|i : mF0 +1, j : mF0 −1|F̂i+ F̂j− |i : mF0 , j : mF0 |. Therefore thesmaller the value |mF0 | of the initial populated states, thefaster the early-time dynamics. Notably, to order t 2 theinitial dynamics is independent of quadratic shifts andexternal magnetic field gradients. This is because both oftheir corresponding Hamiltonians commute with the spin
population operator n̂mF . From this simple perturbative
treatment one learns that by preparing different initialstates with different mF0 , the decay rates of the short-time
population dynamics provide information of Veff and thus of
the underlying dipolar couplings. As discussed in Appendices
B and H, the lattice filling fraction is not unity and the initialatomic density distribution in the lattice may vary from shot
to shot. To account for this effect, we perform a statistical
average of Eq. (I3) calculated for each lattice configuration
generated with the procedure in Appendix H to obtain thetheoretical values in Fig. 3(g) and Fig. 4(b).
It is important here to compare the predictions obtained
from a simple mean-field analysis. In contrast to Eq. (I4),
neglecting quantum correlations yields
Mean−Field
nm
(t )
0
F


2

 Veff2
4= nmF0 (0) 1 − nmF0 (0) 1 − nmF0 (0) 2 t + O(t ) . (I4)
h̄

5

10
15
Time (ms)20

25

FIG. 9. The dotted-dashed line exemplary shows the fit ofEq. (I2) to the experimental data to extract Veff for mF0 = |–9/2. The
solid green line indicates the time tfit up to which the fit is performed.

At the mean-field level, therefore, if initially the atoms areprepared such that nmF0 (0) = 1, then there is no population dynamics. This is in stark contrast to the quantum systems where
dynamics is enabled by quantum fluctuations. To extract Veff
from our experimental data and to compare it to the theoreticalsimulations we fit the initial dynamics with Eq. (I4). We
define the timescale for the fitting via tfit < 0.5 Vh̄eff , whichcorresponds to the timescale on which each atom did on
average half a spin flip. We note that on this timescale thetime evolution starts already to deviate from the short-time
expansion [Eq. (2)], leading to a systematic downshift of theexperimentally fitted Veff ; see Fig. 4(b). However, a minimum
timescale has to be chosen to ensure that the fit is performed
using a large-enough number of data points. Figure 9 shows
exemplary the fit to the experimental data for mF0 = |–9/2.
In Fig. 10 we show for completeness the experimental data
as well as the theory calculations for all spin states with therescaled time axis [see Fig. 3(h)].
APPENDIX J: EFFECT FROM TUNNELING PROCESSES

In our experiment, the tunneling rate is small but finite.
To understand its possible effect on the measured spin dynamics, we recall that our system can be described by a1

0.8
Spin population−[Ĥ , [Ĥ , n̂mF ]]t 2 /2h̄2+[Ĥ , [Ĥ , [Ĥ , [Ĥ , n̂mF ]]]t 4 /4!h̄4 + O(t 5 ).

= |-9/2>

0.8

0.6

0.4
0

0.5

1
V /ħ.t1.5

2

FIG. 10. Expansion of Fig. 3(h) showing all measured spin states
and the corresponding theory predictions.

023050-12

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

ĤFH = Ĥtun + Ĥinho + ĤVdd + ĤUdd + ĤswithĤtun = −† ˆ(Ji,j fˆim
fjm + H.c.),i,j,mĤinho =iĤVdd =

(J1)
(J2)


†  z 2 ˆ† z ˆF̂mn fin ,Fmn fin +δi fˆimBi fˆim

m,n

i

(J3)

m,n



1 
1 + −
z
− +
Vi,j Fmn
Fklz − (Fmn
Fkl + Fmn
Fkl )2 i=j m,n,k,l4† ˆ† ˆ ˆ(J4)
× fˆimfik fjl fjn ,


 
1
1 + −
z
− +
ĤUdd = Udd
Fmn
Fklz − (Fmn
Fkl + Fmn
Fkl )2
4
i m,n,k,lĤs =† ˆ† ˆ ˆ
fik fil fin ,× fˆim(J5)

1 † ˆ† ˆ ˆ
Umnkl fˆim
fik fil fin ,2 i m,n,k,l(J6)†
where fˆim ( fˆim
) annihilates (creates) a fermion of spin statez,+,−
m on site i = {ix , iy , iz }, and Fmn
are the matrix elementsof the corresponding spin-19/2 angular momentum operators.
The first term describes the single-particle tunneling between
adjacent sites, with the tunneling rate J calculated from theintegral over the lowest band Wannier functions:
 2 2


h̄ ∇
3
∗
Ji,j = − d rφi (r) −+ Vlatt (r) φj (r), (J7)2M

1with the integral over Wannier functions at the same lattice
site, φ(r). For dipoles oriented along z, (i.e.,  = 0, seeFig. 1), the experimental lattice geometry leads to an attractive
net Udd . For s-wave scattering, the interaction kernel in Ĥs can
J /ħ.t
0

1

2

3

4

1

5
|-3/2>|-1/2>|+1/2>

0.8|+3/2>

0.6
0.4
0.2

= |-17/2>

0.8
Spin population

where M denotes the atomic mass and Vlatt is the externallattice potential. The second term Ĥinho includes the sitedependent quadratic and linear shifts, respectively. In theexperiment, initially all atoms are prepared in the same spinstate. At the achieved initial temperature at which atoms are
restricted to occupy the lowest lattice band, Fermi statistics
prevents more than one atom per lattice site. Dipolar exchange
processes can change the atomic internal spin states, allowing
tunneling to happen. When two Er atoms occupy the samelattice site they interact with each other via both a contact
interaction governed by the Hamiltonian Ĥs and an on-site
dipolar interaction governed by ĤUdd , which are the last two
terms in Eq. (J1). The strength of on-site dipolar interaction
can be obtained from1 − 3 cos2 θ
μ0 (μB gF )2
Udd =|φ(r)|2 |φ(r )|2 ,d 3 rd 3 r
4π|r − r |3(J8)Spin population

Fermi-Hubbard Hamiltonian:

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

0

0.6

0

0.4
0.2
0
0

20

40
60
Time (ms)80100FIG. 11. Comparison between spin dynamics obtained with the
reduced spin model Eq. (F1) (solid lines) and the Hamiltonian
Eq. (J12) accounting for superexchange interactions (circles) for initial state mF0 = |−17/2 and typical experimental conditions. Results
are shown for the three most populated spin states, mF0 , mF0 ± 1. For
the on-site interaction strengths, we adopt for a18 the value measured
in Ref. [29] and assume the rest aS ’s are randomly distributed within±40% with respect to a18 . For identical aS and absent of Udd thesuperexchange Hamiltonian has a SU(2F + 1) symmetry that does
not change spin population. We expect that the reasonable variation
between aS ’s of 167 Er atoms does not exceed ±40 %.

0.5

1
V /ħ.t1.5

2

FIG. 12. Spin dynamics from the Fermi-Hubbard model with
F = 3/2 (symbols). A one-dimensional system with eight sites isused, with uniform nearest-neighbor tunneling rate Ji,j = J. To reduce finite-size effect, a periodic boundary condition is adopted. We
assume an on-site contact interaction Us with SU(2F + 1) symmetry
and strength Us /J = 200, approximately corresponding to the value
measured in Ref. [29] and J/h̄ = 10 Hz. Since Us is much stronger
than the on-site dipolar interaction for the erbium experiment [29],we do not include Udd in these calculations. We have also neglected
external inhomogeneous fields [Eq. (J3)] in the model. Initially fivesites are occupied with atoms all prepared in the mF0 = 1/2 state. The
spin dynamics is calculated using exact diagonalization and averaged
over all possible initial distribution of empty sites. To provide areference for the F = 19/2 erbium case, we have focused on atimescale similar to the one probed in experiment, with Veff defined
as in the main text and setting F = 3/2 in the expression for γ (mF0 ).
As a comparison, the spin dynamics obtained with the frozen spinmodel [Eq. (F1)] with the same lattice configuration is also plottedwith solid lines.

023050-13

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

be generally rewritten in terms of
Umnkl =

2F
−1
cS (aS )Sforbidden and they contribute only as second-order virtual
processes also known as superexchange interactions. They
take the form α| Ĥtun |γ  γ | Ĥtun |βi,j|α β|,
(J11)
=−
Ĥex
γ | Ĥs |γ  + γ | ĤUdd |γ γ(F, m; F, k|S, MS 

mS =−S

S=0,2,...

× S, MS |F, n; F, l),(J9)where α, β denote a basis set spanned by states where two
adjacent sites i, j are each occupied by one atom and γ denotes
the set of states where two atoms occupy the same site andare diagonal in both the s-wave and on-site dipolar interaction
Hamiltonians, Ĥs , ĤUdd . The superscript i, j emphasizes thatthe superexchange interaction only occurs between nearestneighbor sites.
After accounting for all these interactions, we can obtainan effective spin Hamiltonian by projecting to the physicalsubspace with at most one atom per site and also neglecting
tunneling processes to empty sites, which anyway do not
modify the magnetic character of the system:i,jĤ  = Ĥ +
Ĥex
,
(J12)

where S is the total angular momentum of the colliding
particles and MS its projection along the quantization axis.
The subscripts m, n, k, l run from −19/2 to 19/2. Odd valuesof F are forbidden for s-wave collision. Here
4π h̄2 aS(J10)cS (aS ) =d 3 r|φ(r)|4 ,Mcharacterizing the scattering in the total spin S channel and aS
is the corresponding background scattering length. For 167 Er,with F = 19/2 there are in principle 10 different scattering
lengths. Most of the aS remain unknown, except the one
between m = −19/2 and n = −17/2, which was measured tobe ∼91(8) a0 with a0 the Bohr radius [29]. Given the complex
molecular potential of Er atoms we expect not significant
variations between them.
In the deep lattice regime, the on-site interactions are of
the order of kHz [29] and therefore much stronger than thetunneling rate ∼10 Hz. As a consequence, tunneling processes
between two adjacent occupied lattice sites are energetically

⎛
0
⎜0
⎜
⎜0
⎜0
⎜
⎜0
⎜
⎜0
⎜
⎜0
⎜
⎜0
i,j
Ĥex
=⎜
⎜0
⎜
⎜0
⎜
⎜0
⎜
⎜0
⎜
⎜0
⎜
⎜0
⎝0
0

0
−a
0
0
a
0
0
0
0
0
0
0
0
0
0
0

0
0
b
0
0
0
0
0
−b
0
0
0
0
0
0
0

0
0
0
c
0
0
d
0
0
−d
0
0
−c
0
0
0

0
a
0
0
−a
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
d
0
0
e
0
0
−e
0
0
−d
0
0
0

i,j

where Ĥ is the spin Hamiltonian given in Appendix F.
To get a basic idea of the resulting superexchange interactions, we first provide as an example the case of F = 3/2, forwhich the Hamiltonian obtained using the above procedure for
two adjacent sites is

0
0
0
0
0
0
0
b
0
0
0
0
0
−b
0
0

0
0
−b
0
0
0
0
0
b
0
0
0
0
0
0
0

0
0
0
−d
0
0
−e
0
0
e
0
0
d
0
0
0

where

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
−a
0
0
a
0

0
0
0
−c
0
0
−d
0
0
d
0
0
c
0
0
0

0
0
0
0
0
0
0
−b
0
0
0
0
0
b
0
0

0
0
0
0
0
0
0
0
0
0
0
a
0
0
−a
0

(J13)

0

d=

2J 2 (3Udd + 2Us x0 ),2
9Udd + 6Udd Us (1 + x0 ) − 4Us2 (1 + x0 )

(J17)

e=

2J 2 [−9Udd + 2Us (2 + x0 )],2
9Udd + 6Udd Us (1 + x0 ) − 4Us2 (1 + x0 )

(J18)

a=

4J 2
,
3Udd + 2Us

(J14)

b=

8J 2
,
3Udd − 4Us(J15)

J is the nearest-neighbor tunneling rate,c=

2J 2 [3Udd + 2Us (2 + x0 )],2
9Udd + 6Udd Us (1 + x0 ) − 4Us2 (1 + x0 )

(J16)

Us =

023050-14

⎞
0
0⎟
⎟
0⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟,
0⎟
⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟
0⎟
⎟
0⎟
0⎠

4π h̄2 a2Mdr|φ(r)|4 ,(J19)

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

and x0 = a0 /a2 − 1 denotes the fractional difference between
the scattering lengths of the two channels with total spin 0 and
2. With this explicit form of the superexchange Hamiltonian,
we can find the short-time dynamics for atoms initialized in
the same state mF0 = |–1/2:which indicates that superexchange interactions do notaffect the initial dynamics, as one would expect from Fermi
statistics.
Although for the case of Er atoms with F = 19/2 we do
not have a simple analytical expression for the final Hamiltonian Eq. (J12) we can obtain it numerically. Moreover, sincefor Er most aS are unknown we assume random values for aS
and solve the ensuing dynamics with the GDTWA approach.

In Fig. 11, we plot the result for a typical initial state mF0 =|−17/2, which shows that the addition of superexchangeinteractions hardly affects the measured spin dynamics during
the relevant timescale.
To further analyze the role of tunneling during the dynamics and account for the presence of initially unoccupied sites, we calculate the spin dynamics from the FermiHubbard model Eq. (J1). In Fig. 12, we plot the results
obtained for a small-size system using exact diagonalization and assuming ∼40% holes at t = 0. At short times
J/h̄ · t ∼ 1, tunneling effect is negligible. At longer times,
tunneling interplays with interaction and Fermi statistics andmodifies the quantum dynamics. Nevertheless, these results
suggest that such modification is not significant for the most
relevant timescales probed in this work. Therefore, for thecomparison with experimental observations in this work, we
use the reduced spin model Eq. (F1) and neglect tunneling
altogether.

[1] I. Bloch, Quantum coherence and entanglement with ultracold
atoms in optical lattices, Nature 453, 1016 (2008).
[2] C. Gross and I. Bloch, Quantum simulations with ultracold
atoms in optical lattices, Science 357, 995 (2017).
[3] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics withultracold gases, Rev. Mod. Phys. 80, 885 (2008).
[4] D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger,
Short-range quantum magnetism of ultracold fermions in an
optical lattice, Science 340, 1307 (2013).
[5] J. Zeiher, J.-Y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen,
I. Bloch, and C. Gross, Coherent Many-Body Spin Dynamics ina Long-Range Interacting Ising Chain, Phys. Rev. X 7, 041063(2017).
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023050-16

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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A.3. Publication: Excitation spectrum of a trapped dipolar
supersolid and its experimental evidence
Physical Review Letters 123, 050402 (2019)§
submitted 3 Jul 2019; published 1 Aug 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, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Austria2§

The author of the present thesis developed the experimental procedures and performed the measurements
together with G. N., A. P., and L. C., and contributed in writing the manuscript. The theoretical calculations for the publication were performed by G. N., R. M. W. v. B., and L. C.

178

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
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Featured in Physics

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)].

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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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Optical setup for Bragg
spectroscopy

One major upgrade of the experimental apparatus was the implementation of a setup to
perform Bragg spectroscopy with a digital micro mirror device (DMD). Our group used thistechnique to perform measurements of the excitation spectrum of dipolar BECs which led
to the two publications, discussed in Sec. 4.5 and Sec. 5.5, where the working principle andmore details of the setup are discussed.

802 nm
10 mW
λ/2

100

GHz-PD forbeat-note lock

50
:5

0

401 nm
220 mW

λ/2

AOM

DMD

110 MHzλ/2

ZS reference light

Toptica TA-SHG-50100cylindricalwavemeter

Appendix

B

Figure B.1.: The laser setup that provides stabilised light at 401 nm for the Bragg spectroscopy.
The light is transferred via an optical fiber (labelled ’DMD’) to the Bragg spectroscopy setup; see
Fig. B.2. The setup includes mirrors, lenses, polarising beam splitter cubes, half-wave plates (λ/2),an acousto-optical modulator (AOM) and a 50:50 beam splitter to combine the laser light with apossible reference laser beam, obtained from the Zeeman slower from the experiment (ZS reference
light).

Figure B.1 shows the setup which provides the laser light for the Bragg spectroscopy. As alaser source we use a diode laser which provides 802 nm, which is amplified and frequency
doubled. The obtained 401 nm light is typically 71(1) GHz red-detuned from the strongest
atomic resonance in erbium and allows us to achieve suitable potential depths for the Bragg
spectroscopy, while having negligible photon scattering of the atoms in the BEC. Due tothe passive stability of the laser system, for this detuning, the diode laser does not need to
189

190

APPENDIX B. OPTICAL SETUP FOR BRAGG SPECTROSCOPY

be frequency locked to a reference. However, the laser can be also locked on the ZS lightof the experimental setup with an offset beat-note lock. Here, a fast gigahertz-photodiode(GHz-PD) allows a lock detuning of up to 2.6 GHz from the ZS frequency. This beat-note
lock is currently not used in our experiment, but it allows to also lock this laser to a variablefrequency around the atomic line of erbium.

9
1:
9
9

532 nm

1:
9

-30

DMD

PD

PD

cyl.

r.
ch r
di irrom

Top view

reflecti
on down to main chamber

Figure B.2 shows the Bragg spectroscopy setup, which is situated on top of the main chamber.
The setup was initially designed and calibrated for two wavelengths, 401 nm and also 532 nm.
However, for the experiment, only 401 nm is used currently. A photodiode (PD) after thefiber allows to stabilise the laser light via a feedback-loop on the AOM of the laser setup
in Fig. B.1. A cylindrical telescope expands the beam in the vertical axis (out of plane in
Fig. B.2) to an aspect ratio of 1:10 mm, which is used to illuminate the DMD. By meansof different uploaded holograms, the DMD creates the two Bragg beams for the experiment(both beams are on top of each other in the top view in Fig. B.2). These two beams pass
another telescope which includes a filtering iris in its focal point. Afterwards the Bragg laser
beams are directed down to the main chamber of the experiment. For further details on the
setup after the DMD, see also supplemental material of the publication in Sec. 4.5.

DMD mirror board

DMD controller

cyl.
300
3"

achrm.
1"
60

250

main chamber
at lower leveliris

Figure B.2.: Laser schematics for the Bragg spectroscopy setup. The laser light is transferred via an
optical fiber (labelled ’DMD’) from the laser setup in Fig. B.1. For details see text and Sec. 4.5. The
solid outline of the image represents the used breadboard, which is cutted out in the region where
the Bragg beams are reflected down to the main chamber.

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Acknowledgments
I would like to thank my supervisor Francesca Ferlaino for giving me the possibility to workand perform this thesis on the ERBIUM experiment. Her scientific curiosity, as well as her
constant drive for progress helped me to get the best out of my last years, where I developedscientifically and also personally a lot. She gave me the freedom to realise my own ideas in
the experiment and the opportunity to present our results on numerous conferences. I am
thankful for her support throughout the various phases of my PhD studies – from the firstpublic talks at the beginning to the final reading of this thesis.
Over the last years, the lab members were gradually changing and, therefore, there are a lotof people I would like to thank for their advice, support, patience, and friendship. Startingwith the ’older generation’ PhD student, I want to thank Simon Baier for introducing me
into the details of the experimental apparatus and for providing a role model how scientificlab work is carried out. A big thanks goes also to my Post-Doc, Lauriane Chomaz, forher constant scientific advice, whether about questions on the bigger physical pictures, theway to model simulations or help with the experiment. Another thanks goes to our seniorscientist, Manfred Mark, whos technical and physical expertise was always helpful.
A special thanks goes to the other two PhD students in our lab, Alex Patscheider andGabriele Natale. They made the daily lab work truly enjoyable, where time was flying bythanks to Alex’ great taste of music and Gabriele’s discussions about the best way to makepizza and coffee. Outside of the lab, I also enjoyed our evening dinners and the hiking we
did together. For one year, two more PhD students, Jan-Hendrik Becher and Giulia Faraoni,
visited our group and I would like to also thank them also for their great support in the lab
and an enjoyable time.
Over the years, our dipolar quantum gases–group was getting bigger and bigger. I also wantto thank the people in the other two experiments of our group – in particular Claudia, Max,
Arno, Philipp, Gianmaria – for their experimental advise, for lending optics or electronics and also for our Karaoke evenings and boulder sessions. One big thanks also goes toour administrative office, where Sivlia, Leonarda and Nicole helped me all the time withorganisational work from ordering to bookings.
Our group is also closely connected to the groups of Rudi Grimm and Hanns-ChristophNägerl, where I want to thank especially Cornee, Deborah, Milena, Elisa, Erich, Andreas
and Gregor for dinner and cocktail invitations, the climbing together and simply chatting in
the lab-hallway.
I would also like to thank all our theory collaborators, especially Blair Blakie, Rick van
207Bijnen, Alessio Recati, Santo Roccuzzo and Misha Baranov, for their deep insights andpatient nature when discussing.
I want to especially thank my family. My parents, Hugo and Doris, for their constant support
throughout my whole life. Without them, I would not be writing these lines now. I want tothank Katha and Niki (together with Emma and Greta) for being a part of my life and forreminding me of what is really important. Finally, the biggest thanks goes to my girlfriend,
Magda, for her support, for all our activities and conversations, for her love and for being astrong basis in my life. I feel lucky to have and love her.

208