Creation of Erbium-Dysprosium
Dipolar Quantum Mixtures
and Their Interspecies Feshbach
Resonances

DISSERTATION
byGianmaria Durastante, M.Sc., B.Sc.

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

Advisors:

Univ.-Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of the Austrian Academy of
Sciences

Univ.-Prof. Mag. Dr. Paul Scheier,Institute of Ion Physics and Applied Physics, University of InnsbruckInnsbruck, 31 August 2020

Creation of Erbium-Dysprosium
Dipolar Quantum Mixtures
and Their Interspecies Feshbach
Resonances
Gianmaria Durastante

So remember to look up at the stars
and not down at your feet. Try to makesense of what you see and wonder about
what makes the universe exist. Be
curious. And however difficult life may
seem, there is always something you
can do and succeed at. It matters thatyou don’t just give up. Unleash yourimagination. Shape the future.
Stephen Hawking

Dedicated to my family.

Abstract
Ultracold quantum gases are a powerful platform for research in the field of quantum simulation as well as few- and many-body physics. They provide the possibility to precisely
control a wide range of parameters, ensuring the realization of simplified versions of more
complex systems typical of condensed matter, high-energy, and nuclear physics. Among thegreat experimental control, they allow for an accurate tuning of the interparticle contact
interactions by means of Feshbach resonances. Since the creation of the first Bose-Einsteincondensate, great progress has been made moving from single-component alkali experiments
to multi-component ones. More recently, aiming to further the complexity and extending
the possibilities of such systems, elements with permanent large magnetic dipole moments(chromium, dysprosium, and erbium overall) arouse interest due to the long-range and anisotropic character of the dipole-dipole interactions, paving the way towards the realizationof novel scenarios.
In this panorama, on the basis of the knowledge achieved on the topic, a further step led ourgroup to the world-first realization of a dipolar quantum gas mixture experiment combining
erbium and dysprosium atoms — two highly magnetic lanthanides. After a designing andbuilding phase with a duration of about three years, we are now able to compete at theforefront of the research in the field of ultracold atoms. The initial part of this manuscript
briefly summarizes the development of such state-of-the-art experiment from the review of
erbium and dysprosium properties, to the design of the experimental apparatus, to theachievement of quantum degeneracy in the two-components sample.
As the main topic of discussion, this thesis reports on the observation of heteronuclear magnetic Feshbach resonances in dipolar ultracold mixtures of erbium and dysprosium atoms.
We investigated several isotopic combinations including Bose-Bose, Bose-Fermi, and FermiFermi mixtures. Our research provides accessible experimental tools to control interactions
in dipolar quantum gases of erbium-dysprosium and it will help to understand fundamental
properties of the system such as the scattering dynamics. Furthermore, this work discloses
new possibilities for the investigation of dipolar polarons, miscibility/imiscibility phase diagram, vortex formation, and binary magnetic quantum droplets. Eventually, it will be possible to transfer this knowledge into the creation of erbium-dysprosium Feshbach molecules
via magnetoassociation with both electric and magnetic dipole moments.

iContents
Abstracti

1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Additional publications . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
3
4

2 Ultracold erbium-dysprosium atoms
2.1 Erbium and dysprosium properties . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Energy levels and atomic polarizabilities . . . . . . . . . . . . . . . . .
2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 General description of the apparatus . . . . . . . . . . . . . . . . . . .
2.2.2 Magnetic-field calibration . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Magnetic field system . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Publication: Two-species five-beam magneto-optical trap for erbium and dysprosium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
6
6
7
10
11
13
13
23

3 Quantum mixtures
3.1 Theory of quantum mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Contact and dipolar interactions . . . . . . . . . . . . . . . . . . . . .
3.1.2 Mean-field description and Gross-Pitaevskii equation . . . . . . . . . .
3.1.3 Solutions to the Gross-Pitaevskii equation . . . . . . . . . . . . . . . .
3.2 Mixture experiments overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Alkali-alkali mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Alkali-nonalkali mixtures . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Erbium-dysprosium mixture . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Publication: Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms .

31
31
32
33
34
36
37
38
39
41

4 Feshbach resonances
4.1 Theory of Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Ultracold collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Open- and closed-channel picture . . . . . . . . . . . . . . . . . . . . .
4.1.3 Resonance strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Inelastic collisions and trap-loss spectroscopy . . . . . . . . . . . . . .
4.2.2 Alkali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Mixtures and complex systems . . . . . . . . . . . . . . . . . . . . . .

49
50
50
51
52
53
54
55
55
56

iii

4.3

Feshbach resonances in lanthanides . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Role of anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Experimental observations . . . . . . . . . . . . . . . . . . . . . . . . .
Publication: Feshbach Resonances in an Erbium-Dysprosium Dipolar Mixture
Additional measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Feshbach resonance at 34 G in 166 Er-164 Dy . . . . . . . . . . . . . . . .
4.5.2 Additional measurements on the feature at 161 G in 161 Dy . . . . . . .

56
56
57
59
73
73
73

5 Conclusion and outlook
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Further studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77
77
78

4.4
4.5

A Additional publications
79
A.1 Publication: Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.2 Publication: Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B Extensive Feshbach scans

117

C Coil systems’ cooling

125

Bibliography

129

Acknowledgments

145

iv

List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3.1
4.1
4.2
4.3
4.4
4.5
B.1
B.2
B.3
B.4
B.5
B.6
B.7
C.1
C.2

Energy level diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8Atomic polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Coils setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Small-field coils numerical calculation . . . . . . . . . . . . . . . . . . . . . . 18
Small-field bias coils characterization . . . . . . . . . . . . . . . . . . . . . . . 19
Large-field coils numerical calculation . . . . . . . . . . . . . . . . . . . . . . 20
Compensation coils numerical calculation . . . . . . . . . . . . . . . . . . . . 22
Dipole-dipole interactions in polarized dipoles . . . . . . . . . . . . . . . . . . 32
Feshbach resonance two channels model . . . . . . . . . . . . . . . . . . . . . 52
Numerically calculated Feshbach spectrum for 164 Dy . . . . . . . . . . . . . . 57
Feshbach resonance at 34 G in 166 Er-164 Dy . . . . . . . . . . . . . . . . . . . . 74
Temperature dependence of the 161 G Feshbach resonance . . . . . . . . . . . 75
Er/Dy ratio dependence of the 161 G Feshbach resonance . . . . . . . . . . . 76
Extensive scan for 166 Er-164 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Extensive scan for 168 Er-164 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Extensive scan for 170 Er-164 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Extensive scan for 166 Er-161 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Extensive scan for 167 Er-161 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Extensive scan for 168 Er-161 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Extensive scan for 170 Er-161 Dy . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Large-field coils temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Small-field coils temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

List of Tables
2.1
2.2
2.3
3.1

Atomic properties of erbium and dysprosium . . . . . . . . . . . . . . . . . .
Employed optical transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coils technical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dipolar lengths comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

6
9
14
37

Chapter

1

Introduction

1.1 Motivation
From its very beginning [Pla06], quantum physics has transformed our view of the world andprovided us with a deeper understanding along with even more challenging questions. Asa matter of fact, natural phenomena might be too complicated to be described analyticallyor they happen on a scale too small or fast to be directly observable. High-temperature
superconductivity, quark-gluon plasma, or quantum behavior of electrons in semiconductors
are typical examples of such challenges. In extreme cases, some systems are not accessibleat all on Earth, like neutron stars and white dwarfs. More generally, quantum many-bodysystems are highly non-trivial and their numerical simulation gets exponentially demanding
with the increase in the number of particles involved [Gar14]. Beyond a few involved particles,
exact calculations including all correlations are challenging, getting eventually impossible
even with supercomputers for a few tens of particles. Complications may arise also in theother direction, when trying to validate a theory or a model. In strongly correlated orinteracting many-body systems, exact theoretical descriptions break down, and experiments
need to benchmark the limits and the approximations of such a model.
Scientists have searched for a solution to these issues for long time and one possibility is tofollow a bottom-up approach. That is, building a simpler system in which the fundamental
laws are identical to the one governing such complex systems. In the case of quantum physics,
these model systems are known as quantum simulators [Fey82]. In this panorama, ultracold
atoms have proven to be an excellent platform to investigate nature [Blo12, Tö14]. Their
strength lies in the extreme precision to which all the different external and internal degrees
of freedom can be controlled. Particularly important is the possibility to accurately tuneinterparticle interactions, e. g. by means of Feshbach resonances, exploiting non-interacting
as well as strongly interacting regimes [Chi10].
The field of ultracold gases began along with the investigation of simple systems featuringpurely short-range contact interactions, that is alkali-metal atoms. The major achievementof those years has been the attainment of quantum degeneracy in bosonic [And95, Dav95]
and fermionic [DeM99] samples, 70 years after the prediction of such a novel state of matter [Bos24, Ein24]. After the success set by this milestone, the field has seen remarkabletechnological progresses covering several fields of fundamental research and precision measurement as well [Cor02, Kuh16, Pez18]. Nevertheless, scientists have tried to push back the1

2

CHAPTER 1. INTRODUCTION

limits and delve further into the physics accessible with ultracold atoms. In this direction,
new intriguing scenarios — always dominated by contact interactions — have been introduced by using heteronuclear mixtures [Had02, Mod02] or more complex elements, e. g. ytterbium and strontium [Tak03, Ste09].
More recently, in the spirit of integrating new types of interactions into quantum simulators, the interest has moved to dipole-dipole interactions (DDI) displaying long-range andanisotropic characters. Systems characterized by such DDI are: magnetic atoms, groundstate heteronuclear molecules, and Rydberg atoms, in order of dipolar strength [Lah09,
Car09, Saf10]. In this thesis, we are interested in the peculiar case of magnetic atoms for theimmediate access to the dipolar regime they offer thanks to their permanent magnetic dipole
moment, avoiding complicated experimental procedures for molecule formations or Rydberg
excitation.
In the advent of dipolar quantum gases, bosonic chromium has been the first magnetic
species in which quantum degeneracy was attained [Gri05]. A number of different pioneeringexperiments demonstrated the importance of dipolar interactions in such a system, including
demagnetization cooling, its stability diagram, and d-wave collapse [Lah09]. Nevertheless,the turning point was reached with the realization of degenerate quantum gases of highly
magnetic species such as dysprosium and erbium [Lu11a, Lu12, Aik12, Aik14b]. The strong
dipolarity together with their intrinsic complexity have led to a vast amount of observations
proving the potentialities of such systems. Among these we mention the observation of Feshbach resonances and quantum chaos [Bau14, Mai15b], Fermi surface deformation [Aik14a]
and extended Bose-Hubbard models [Bai16], roton excitation [Cho18], the discovery of quantum droplets [FB16, Cho16] and supersolid states of matter [B1̈9, Tan19, Cho19].
To combine the manipulation schemes available with lanthanides together with the additional complexity of heteronuclear mixtures, our group worked towards the realizationof the world-first dipolar quantum gas mixture experiment combining erbium and dysprosium atoms. Only recently the interest in such binary dipolar systems aroused notableattention thanks to the striking prospects they offer. Phenomena arising from the complex
parameter landscape of heteronuclear dipolar mixture which have been already theoreticallyinvestigated include the miscibility of two-components BECs [Kum17a], vortices and vortex
lattices formation [Kum17b, Mar17], binary magnetic quantum droplets [Smi20], and dipolar
polarons [PA18, Wen18].

1.2 Thesis overview
This thesis reviews the first experimental realization of dipolar quantum gases of erbiumdysprosium isotopic mixtures with tunable interaction. The line followed by this work is
given by the chronological development of the experiment: from the realization of a dualspecies intercombination-line magneto-optical trap in a novel open-top configuration, to theachievement of quantum degeneracy in several isotopic combinations, to conclude with theobservation of interspecies Feshbach resonances in highly dipolar erbium-dysprosium mixtures. The work carried out until the achievement of quantum degeneracy is also includedin the thesis of Philipp Ilzhöfer [Ilz20].

1.2. THESIS OVERVIEW

3

Personally, I joined the group in the late 2015 when the design and planning of the experiment were coming to an end but at the very beginning of the building process (twoempty optical tables were the only components already set up). It took about three years
to move from an empty room to a fully operational quantum gas experiment competing at
the forefront of the research in the field.
This thesis resulted in three scientific publications setting the main discussion of themanuscript. Two additional publications on the topic of supersolidity in dipolar quantum
gases — in which I contributed but not part of the main discussion of this manuscript —are included at the end of the thesis. As a general overview, this manuscript is structured
as follows:
Chapter 2 summarizes the knowledge and the relevant properties of erbium and dysprosium
atoms and gives details on the experimental setup. Particular attention is drawn to
the magnetic field generation and control. It further describes the production of an
erbium-dysprosium cold thermal cloud as part of our first publication [Ilz18].
Chapter 3 reviews the state-of-the-art knowledge in the research topic of quantum mixturesfrom a theoretical as well as an experimental point of view. The chapter reports
the production of different heteronuclear dipolar BECs of several isotopic mixtures of
erbium and dysprosium as part of our second publication [Tra18].
Chapter 4 is dedicated to the topics of Feshbach resonance theory and experiments in ultracold atoms. The chapter develops from simple alkali system towards more complex
dipolar systems. The first experimental observation of Feshbach resonances in a heteronuclear lanthanide mixture is reported as part of the publication [Dur20].
Chapter 5 closes the thesis and discusses further steps to pursue as an outlook to future
investigations. The work performed during the development of this manuscript makes
the employment of erbium-dysprosium quantum mixtures with tunable interactions forthe purposes of quantum simulation and fundamental research possible.
Appendix A-C include the publications not covered by the main topic of this thesis [Cho19,
Ilz19], complete sets of measurement performed for the publication [Dur20] and additional information regarding the coils system.

1.2.1 List of publications
The publications discussed in the main text of this thesis are listed here in a chronologicalorder.
• Two-species five-beam magneto-optical trap for erbium and dysprosium
Philipp Ilzhöfer, Gianmaria Durastante, Alexander Patscheider, Arno Trautmann,
Manfred J. Mark, and Francesca Ferlaino
Physical Review A 97, 023633 (2018)

4

CHAPTER 1. INTRODUCTION• Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms
Arno Trautmann, Philipp Ilzhöfer, Gianmaria Durastante, Claudia Politi, Maximilian
Sohmen, Manfred J. Mark, and Francesca Ferlaino
Physical Review Letters 121, 213601 (2018)• Feshbach Resonances in an Erbium-Dysprosium Dipolar Mixture
Gianmaria Durastante, Claudia Politi, Maximilian Sohmen, Philipp Ilzhöfer, Manfred
J. Mark, Matthew A. Norcia, and Francesca Ferlaino
Physical Review A — accepted, in press (2020)

1.2.2 Additional publications
During my PhD work, I also contributed to study many-body dipolar physics, culminated in
the observation of supersolid phases in dysprosium quantum gases. The corresponding publications, which I co-authored, are listed below. Such publications are included in Appendix A
as “additional” since their subject is not part of the core topic of this thesis.
• Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases
Lauriane Chomaz, Daniel Petter, Philipp Ilzhöfer, Gabriele Natale, Arno Trautmann,
Claudia Politi, Gianmaria Durastante, Rick M. W. van Bijnen, Alexander Patscheider,
Maximilian Sohmen, Manfred J. Mark, and Francesca Ferlaino
Physical Review X 9, 021012 (2019)• Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms
Philipp Ilzhöfer, Maximilian Sohmen, Gianmaria Durastante, Claudia Politi, Arno
Trautmann, Giacomo Morpurgo, Thierry Giamarchi, Lauriane Chomaz, Manfred J.
Mark, and Francesca Ferlaino
arXiv:1912.10892

Chapter2

Ultracold erbium-dysprosium
atoms

In the periodic table, and more precisely in the sixth period from lanthanum to lutetium,
a series of elements has recently attracted great interest in the ultracold atom community.
Indeed, in the lanthanide series, several elements show peculiar properties which make themexcellent candidates to expand the physics accessible by quantum gas experiments. Typically, lanthanide elements offer several bosonic as well as fermionic isotopes with high natural abundance increasing the experiment flexibility. Due to the unpaired electrons in thef -orbital, most of the lanthanides possess a strong permanent magnetic dipole moment,
among the strongest on the periodic table, leading to new and fascinating few- and manybody phenomena in the ultracold regime. This dipole moment is particularly important tocreate dipolar quantum gases unaccompanied by the technical difficulties of working withmolecules (see Chap. 3) or Rydberg states. Furthermore, their complex electronic structureleads to a wealth of possible manipulation schemes. In this panorama, erbium and dysprosium are the two most widely used to date with large magnetic moments. After theirlaser cooling and trapping has been demonstrated [McC06, Lu10], several groups followed
the path towards quantum degeneracy [Lu11a, Lu12, Aik12, Aik14b, Tra18].
More generally, lanthanides are widely used in industrial applications as catalysts, luminescent devices, superconductors, magnets, and so on [McG00, Kri15]. Their ions are used in
optoelectronics such us lasers (e. g. Nd:YAG) and as a dopant in optical fibers (e. g. erbiumdoped fiber amplifiers in optical-fiber telecommunication) due to their favorable optical properties. Lanthanide atoms feature 58 core electrons in a xenon-like configuration and, the majority of them, a complicated valence electronic configuration with unpaired electrons in the
4f (even 5d) orbital. Peculiar is the submerged structure of the f -shell which is surroundedby the filled 6s-shell leading to the anisotropic electronic ground state (exceptions are Eu
and Yb) and the complex energy level distribution [Rum20].
In this chapter, the experimental apparatus needed to produce a cold thermal sample oferbium and dysprosium is presented. Sec. 2.1 reviews and compares the relevant atomic properties of erbium and dysprosium. The experimental setup is described in Sec. 2.2 with special
attention focused on the magnetic field generation and control. Sec. 2.3 includes our firstpublication Two-species five-beam magneto-optical trap for erbium and dysprosium [Ilz18].
56

CHAPTER 2. ULTRACOLD ERBIUM-DYSPROSIUM ATOMS

Table 2.1: List of erbium and dysprosium isotopes relevant for our experiment together with selected
properties. The sign on the nuclear spin I describes its relative orientation. B and F refer to bosonicand fermionic isotopes, respectively. Values taken from Ref. [Rum20].

Atomic mass (u)Abundance (%)
Nuclear spin
Statistics

166 Er

167 Er168 Er

170 Er161 Dy162 Dy163 Dy164 Dy

165.9
33.50
0
B

166.9
22.87
+ 27
F

167.9
26.98
0
B

169.9
14.91
0
B

160.9
18.89
+ 52
F

161.9
25.48
0
B

162.9
24.90
− 52
F

163.9
28.26
0
B

2.1 Erbium and dysprosium properties
Erbium and dysprosium are next-nearest neighbors on the periodic table with atomic numbers of 68 and 66 and atomic mass of 167.26 u and 162.50 u, respectively. They were discovered in the XVIII century and, like all the lanthanide elements, they can’t be found as
isolated elements in nature but always bound in mineral ores (dysprosium means indeed
hard to get from Greek). Their abundance on the Earth’s crust is about 3.5 ppm for erbium
and 5.2 ppm for dysprosium [Rum20]. The natural abundance and other properties for eachisotope are reported in Tab. 2.1.
In the following, Sec. 2.1.1 reviews the relevant properties of such elements in order to planand build an ultracold atoms experiment. Sec. 2.1.2 focuses on the energy level diagramsand the atomic polarizabilities of both species highlighting the most important aspects ofthem.

2.1.1 Physical properties
Erbium and dysprosium are solid under standard conditions with densities of 9.07 g/cm3 andEr = 1529 °C and T Dy = 1412 °C
8.55 g/cm3 , respectively. The melting temperatures are Tmmand this difference impacts their vapor pressures. At 1100 °C, temperature regime aroundwhich we evaporate the material, Dy has a vapor pressure ten times higher than Er.
As said in the introduction to this chapter, lanthanide elements have a peculiar electronic
configuration. In the specific case of erbium and dysprosium, it is
Er : [Xe]4f 12 6s2

andDy : [Xe]4f 10 6s2

2.1

which leads to the respective electronic ground state
Er : 3 H6

and

Dy : 5 I8 .

2.2

Here, the presence of a large total orbital momentum L — L Er = 5 and I Dy = 6 for
erbium and dysprosium respectively — modifies the scattering properties by introducing astrong anisotropy in the electronic ground state. L adds to the total spin S into the total
electronic angular momentum J leading to a fine structure with three states for Er and fivefor Dy (JEr ∈ {4, 5, 6} and JDy ∈ {4, 5, 6, 7, 8} respectively). Remarkably, all the bosonicisotopes have no nuclear spin (I = 0) leading to the absence of a hyperfine structure [Foo05].

2.1. ERBIUM AND DYSPROSIUM PROPERTIES

7

Whereas, in the case of fermionic isotopes, the situation is different with I Er = 7/2 andI Dy = 5/2. The hyperfine coupling of I with J to the total atomic angular momentum F
introduce a hyperfine structure for a total of eight states FEr ∈ {5/2, 7/2, . . . , 19/2} in erbium
and six FDy ∈ {11/2, 13/2, . . . , 21/2} in dysprosium with quantum number F .
In the case of erbium and dysprosium, the unpaired electrons in the valence f -shell give riseto their large magnetic moment. A complete derivation is possible by following Ref. [Foo05].
As reported in Tab. 3.1, the magnetic moment in the ground state manifold is µEr ≃ 7.0µB
and µDy ≃ 10µB (µB is the Bohr magneton). The strong dipolarity, hence the strong
long-range anisotropic dipole-dipole interaction, triggers the interest of using erbium anddysprosium in quantum regimes as it opens new ways for understanding nature (see Chap. 3
and Chap. 4).

2.1.2 Energy levels and atomic polarizabilities
Erbium and dysprosium show profound similarities in their energy level diagrams. In particular, the two transitions for both species used in our experiment to prepare and probe
the atomic sample promote a 6s valence electron to the 6p state (see colored arrows in
Fig. 2.1) [Mar78]. In both cases, a ∆J = +1 transitions to a singlet (triplet) 1 P1 - (3 P1 -)configuration of such 6s and 6p electrons is induced. For our purposes, we can distinguishthe transitions in two pairs: broad and narrow (see also Tab. 2.2).
Broad transitions This is the strongest transition to the singlet 1 P1 -configuration in both
species, particularly useful in cases where a large amount of photons need to be scatteredin a short time. It is employed in the initial transversal cooling of the atomic beam, theZeeman slower section, and for absorption imaging. The wavelength of this transition in
erbium (dysprosium) is λEr = 400.910 nm (λDy = 421.290 nm), with a natural linewidth of
29.4 MHz (32.2 MHz).
Narrow transitions Also referred as intercombination-line transition, it couples to thetriplet 3 P1 -configuration. It is employed in the magneto-optical trap (MOT) with wavelength λEr = 582.842 nm and λDy = 626.082 nm. Due to the dipole-forbidden nature, thenatural linewidth is 186 kHz and 135 kHZ, respectively. This comparative narrow linewidth
leads to a favorable low Doppler temperature of ∼4 µK, allowing to reach a final temperatureof the sample after the MOT stage of about 10 µK.
This MOT scheme has been efficiently employed in the single-species experiments describedin Refs. [Fri12, Mai14]. The decision of adopting for such approach simplifies the experiment
for the following reasons. Firstly, both transitions for both species are either nearly closed(the broad) or completely closed (the narrow) transitions, therefore there is no need forrepumpers reducing the complexity of the laser source. Secondly, the wavelengths for thetransitions are close enough to facilitate the design of the coatings of optics and viewports.
Additionally, the final temperature of the sample is low enough to allow for a direct loadingof the double-species MOT into an optical dipole trap beam. Nonetheless, in the pioneeringexperiments described in Refs. [McC06, Lu10], the authors obtained single-species MOTs on

(a)

(b)

Figure 2.1: Energy level diagram for erbium (a) and dysprosium (b) highlighting the optical transitions employed in the experiment described in
Tab. 2.2. J is the total angular momentum. (a) The violet (yellow) arrow indicate the 401 nm (583 nm) broad (narrow) transition. (b) The blue
(orange) arrow indicate the 421 nm (626 nm) broad (narrow) transition. The broad transitions are used for transversal cooling, Zeeman slower, andfor absorption imaging whereas the narrower transitions are used to generate the double-species MOT.

8

2.1. ERBIUM AND DYSPROSIUM PROPERTIES

9

Table 2.2: Optical transitions employed in our experiment. The broad ∼30 MHz transitions areused in the transversal cooling and Zeeman slowing stage and for absorption imaging. The narrower ∼100 kHz transitions are used to generate the double-species magneto-optical trap. Values are
extracted or calculated from Refs. [Law10, Lu11b, Kra19].

Transition property
Wavelength (nm)Lifetime
Natural linewidthSaturation intensity
Doppler temperature (µK)

Erbium
400.9105.41 ns
29.4 MHz59.7 mW
cm2
707

582.8420.85 µs
186 kHzµW
123 cm2
4.5

Dysprosium
421.290
4.94 ns
32.2 MHz
56.4 mW
cm2
774

626.082
1.2 µs135 kHz
µW
72 cm
2
3.2

the broad transition instead. In the dysprosium case, this method required an additional
MOT stage on a 8 kHz transition in order to load the atoms into a dipole trap for evaporative
cooling towards quantum degeneracy [Lu11a].
A relevant property directly connected to the level diagram is the atomic polarizability
α [Met99, Foo05]. Peculiar for every atomic species, α plays a fundamental role in techniques
involving light trapping such as optical-dipole traps and optical lattices [Gri00, Blo05]. Neutral atoms in an external electric field E exhibit an induced electric dipole moment d = α · E.
Worth to be noticed is that α depends on the field’s angular frequency and it is generally acomplex-numbered tensor of rank-two. In a laser field, the atomic polarizability dictates thelight potential seen by the atoms and the photon scattering rate through its real (R{α}) andimaginary part (I{α} ), respectively [Lep14, Bec18]. α itself can be decomposed in scalar
αs , vectorial αv , and tensorial αt polarizabilty where the scalar one is the dominant term in
our application [Bec18, Rav18a]. Despite its importance, the theoretical determination of α
for complex atomic species is a delicate and non-trivial task requiring deep understandingof the energy level diagrams and transition dipole moments. For erbium and dysprosium
such understanding is still limited compared to alkali species. In Fig. 2.2, the real parts ofthe scalar atomic polarizabilities for erbium and dysprosium are plotted. Here, the large
number of transitions in the range 300 nm to 700 nm gives rise to the interesting possibilityof implementing species-selective manipulation schemes [LeB07, Lud15]. Above this range,and especially at 1064 nm, there exists a relatively flat region in which the polarizabilities
converge becoming similar. This is of particular interest because, together with the similar
atomic masses, it allows for the implementation of a standard red-detuned far-off resonanceoptical dipole trap [Gri00] able to work efficiently on both atomic species (see also Sec. 3.2).
Experimentally determined values at 1064 nm are [Bec18, Rav18a]:R{α}Er = (166 ± 3stat ± 61syst ) a.u.
R{αs }Dy = 184.4(2.4) a.u.

and R{αt }Er = (−1.9 ± 0.8stat ± 1.2syst ) a.u.
and R{αt }Dy = 1.7(6) a.u.

2.3

where a.u. stands for atomic unit (1 a.u. = 4πϵ0 a30 with ϵ0 the vacuum permittivity and a0
the Bohr radius).

10

CHAPTER 2. ULTRACOLD ERBIUM-DYSPROSIUM ATOMS

Figure 2.2: Calculated real part of the scalar atomic polarizabilities R{αs } for erbium and dysprosium (red and blue, respectively) versus wavelength. At 1064 nm, value relevant for our opticaltrapping scheme, the calculated atomic polarizabilities are R{αs }Er = 161.8 a.u. and R{αs }Dy =198.6 a.u..

2.2 Experimental setup

The experimental apparatus to prepare a quantum-degenerate heteronuclear mixture consistsof an ultra-high vacuum (UHV) system in which hot erbium and dysprosium atoms comingfrom a high-temperature oven are trapped and cooled by means of laser lights and magnetic
fields (see Fig. 2.3). The general layout of the system resembles more a single-component
quantum gas experiment rather than a dual-species one. Such similarity is due to the similar
atomic properties of the two species allowing for a fairly simple design. It consists of aneffusion cell to evaporate at about 1400 K both species simultaneously, a transversal cooling(TC) section to collimate the atomic beam, a Zeeman slower (ZS) to decelerate longitudinallythe atoms, and a dual-species magneto-optical trap (MOT) to cool and trap the atoms down
to few µK. The last step towards the degeneracy implies an evaporative cooling scheme in a
red-detuned far-off resonance optical dipole trap (ODT) described in Chap. 3.
In this section we describe the key features and performances of our experimental apparatus. Sec. 2.2.1 treats the UHV system and the laser cooling and trapping of an erbiumdysprosium sample. Sec. 2.2.2 describes our procedure for magnetic-field calibration andSec. 2.2.3 focuses on the generation and control of the magnetic fields in the main chamber
section of the apparatus. More information including extensive technical descriptions anddetails are available in Ref. [Ilz20].

2.2. EXPERIMENTAL SETUP

11

Figure 2.3: Sketch of the vacuum system of the erbium-dysprosium experiment. The major highlights of the setup are indicated with arrows. The atomic beam generated in the effusion cell travels
from right to left through a transversal cooling and Zeeman slower section. The atoms are thentrapped and cooled to quantum degeneracy in the main chamber. Here, several coil pairs allow acomplete tuning of magnetic fields. Figure adapted from [Ilz20].

2.2.1 General description of the apparatus
Vacuum chamber
The whole apparatus is constructed with components made out of 1.4404 and/or 1.4429
stainless steel to ensure low magnetic permeability. This is particularly important to reduce
the risk of magnetizing the components themselves. The TC chamber and main chamberfeature an electrolytic-polished surface to reduce roughness and improve the vacuum quality [VV04]. Shown in Fig. 2.3, the vacuum system can be divided into three parts: theatomic beam source module, the atomic beam shutter module, and the main chamber module. These modules are separable via gate valves and they feature pressures on the low
10−10 mbar, few 10−11 mbar, and low 10−11 mbar level respectively due to two ion pumps
and an ion pump combined with a non-evaporative getter. A titanium sublimation pumpallows for refreshing the vacuum in case of oven refilling procedures. Additionally, two gate
valves on the main chamber’s sides allow for possible extensions of the experiment with two
more vacuum modules for future dedicated experiments.
The high-temperature oven has a dual-filament effusion cell design with a tantalum crucible
typically heated up to 1100 °C. An outer hot-lip section at about 1200 °C prevents evaporated
material to condense on the crucible’s aperture. The temperatures of both sections can becontrolled independently. The crucible is filled with granulated alloy composed from 33 %erbium and 67 % dysprosium. To compensate for the higher vapor pressure of dysprosium,
the hot-lip is filled with additional pure granulated erbium.

12

CHAPTER 2. ULTRACOLD ERBIUM-DYSPROSIUM ATOMSIn the atomic beam shutter module, an in-vacuum stainless steel plate can be moved bya servo driver and placed into the atomic beam to forbid atoms from traveling through the
ZS into the main chamber. This shutter is actuated after every MOT loading cycle. This
module also acts as a differential pumping stage between the first and the third module,
using the Zeeman slower tube and a short tube between the beam shutter and the first gate
valve as apertures maintaining the pressure difference.
The main chamber section is the heart of the experiment. Here, the double-species MOT
is produced and the sample is brought to quantum degeneracy. It consists of a custom-madevacuum chamber with full optical access thanks to eight viewports. Particularly important
are the two CF100 inverted viewports along the vertical direction. Such design allows for a
reduced working distance from the atoms and a large field-of-view supporting the installation of a high-resolution imaging along the z-axis. Moreover, it opens up the possibility of
implementing a series of coils for magnetic-field generation close to the atomic sample (seeSec. 2.2.3).

Laser cooling and imaging
The laser cooling employed in our experiment makes use of a robust three-stage scheme.
First, the TC section cools the transversal velocity components of the atomic beam, thereby
collimating it and increasing the atomic flux through the ZS to the main chamber. Its
operating principle is based on a red-detuned two-dimensional molasses technique[Met99] onthe ∼30 MHz broad transitions λEr = 401 nm and λDy = 421 nm of erbium and dysprosium,
respectively. The laser beams are elliptical to maximize the overlap with the atomic beam
and increase the amount of scattered light.
Secondly, the 350 mm spin-flip ZS stage simultaneously decelerate the two-components
atomic beam from velocity classes of up to 370 m/s down to 5 m/s, smaller than MOT
capture velocities of about 16 m/s. It operates on the same broad transitions employed in
the TC section.
Finally, the dual-species MOT traps and cools the atoms in the center of the main chamber down to a temperature of about 10 µK. The MOT operates at the same time on thetwo species on the narrower ∼100 kHz broad transitions λEr = 583 nm and λDy = 626 nm
of erbium and dysprosium, respectively. Because of the narrow-line character, the largedetuning operation, and the gravitational sag, the MOT spin-polarizes the atoms into thelowest Zeeman sublevel and it has the advantage of allowing for an open-top configuration
along the z-axis. In fact, the vertical down-propagating beam becomes redundant and it
can be removed without detrimental effects on the general performances ensuring an easy
full optical access from the top viewport. More details on the characterization of the MOT
operation are available in the publication Ref. [Ilz18] reported in Sec. 2.3.
The atomic sample is probed in time-of-flight (TOF) expansion by using a standard lowfield absorption imaging technique [Ket99]. The pictures are recorded under a ∼45° in thehorizontal x−y plane with respect to the transversal y-axis. The open-top MOT will support
the implementation of an additional µm-resolution vertical imaging suitable for in-situ as wellas TOF imaging.

2.2. EXPERIMENTAL SETUP

13

2.2.2 Magnetic-field calibration
To know precisely the value of each magnetic-field applied in the experiment at the mainchamber, we calibrate our bias coils by means of radio-frequency spectroscopy on a cold
thermal cloud of either erbium or dysprosium in trap. This is done by fixing the current
flowing in a specific coil pair and varying the frequency of the RF radiation1 . When the
Zeeman splitting matches the RF energy, atoms are transferred to higher mj states. In thesestates they experience dipolar relaxation processes heating up the atoms which induce losses
from the ODT. By recording such frequency-dependent atom-loss curves and because of the
known value of the Zeeman splitting (∆EZ,Er = 1.63 MHz/G and ∆EZ,Dy = 1.74 MHz/G forbosonic erbium and dysprosium, respectively), it is possible to connect the applied current
with the real magnetic field value at the atoms’ position. By repeating this procedure forseveral values of the current, we extract precise calibration curves for our coil pairs andwe get insights on the overall magnetic-field stability. In the following Sec. 2.2.3, values
extracted by applying such technique are reported. Note that not all the coils have beencharacterized, and in particular, measuring magnetic field gradients would require completely
different techniques [Xu17].

2.2.3 Magnetic field system
In a quantum gas experiment, magnetic fields are employed in ZS, MOT, and interactiontuning techniques, hence magnetic fields and the ability of control them are of fundamental
importance for a successful and reliable operation [Met99, Ket99]. In the following, we leaveaside the detailed description of the ZS to focus on the coil systems at the main chamber.
Our system is composed by eight coil pairs divided in three modules: inverted-viewportcoils, large-field coils, and compensation cage coils surrounding the main chamber itself(see Fig. 2.4). In Tab. 2.3, technical details about the qcoils system are reported. In thex,y,zfollowing, we define the total magnetic field Btot = Bx2 + By2 + Bz2 along x-, y-, and
x,y,z
x,y,z
z-axis, respectively. The gradient is then calculated as ∇Btot= ∂x,y,z (Btot). Moreover, the vertical and horizontal extension of a coil are indicated as windings and layers,respectively.
Several temperature sensors2 and resettable thermal fuses3 monitor the status of the system preventing overheating and switching off the power supplies in that case. In search
for the best heat-exchange performances, the copper wires are wound with high-thermal
conductivity glue4 .

1
2
3
4

RF source: in-house direct digital synthesizer (DDS), amplifier: Frankonia FFL-25A
10 kΩ NTC thermistor
Temperature thresholds ranging from 40 °C to 60 °C are used depending on their position
Fischer Elektronik WLK-10 and WLK-DK-50 for the large- and small-field coils and Electrolube
ERTCER75S for the compensation cage coils

Table 2.3: Details of the coil system employed at the main chamber. See Fig. 2.4 for the exact
location of the coils in the experimental setup. The vertical and horizontal extension of a coil are
indicated as windings × layers, respectively. Resistance and inductance are measured values. The
current-to-field conversion is reported as numerical calculated value (th) from the Biot–Savart law at
the center of the main chamber and as measured value (exp) from RF spectroscopy on a cold atomic
sample. The measured value is an average over independent repetitions on different days.

Coil purpose

Extension

Resistance(mΩ)

Inductance(µH)

Current-to-fieldthexp
4.80(1) G/A

Large bias

2 × 27150

1590

4.37 G/A

Large
gradient

2 × 27150

1590

0.48 (G/cm)/A

small bias

1×6

42

14

0.78 G/A

small
gradient

10 × 6330

796

1.55 (G/cm)/A

RF fields

1×1

30

0.50

0.14 G/A

Fast field

1×4

70

7

0.57 G/A

Compensation
x-axis2 × 11346

1006

1.06 G/A

1.06(2) G/A

Compensation
y-axis2 × 11326

898

1.16 G/A

1.13(2) G/A

14

0.822(5) G/A

(a)

Brass
cooling plates

Copper cylinder for
High-B-field
low-B-field coils 1 AHH-like coil, unit 1
High-B-field
HH-like coil, unit 1

Mounting
systemMounting
systemHigh-B-field
HH-like coil, unit 2
z
y

x

Brass
cooling plates(b)

Conicalcut

Copper cylinder for
High-B-field
low-B-field coils 2 AHH-like coil, unit 2
Water coolingconnections

Water coolingsystem

Low-B-field
AHH-like coil 1
Low-B-field
HH-like coil 1zxyCutout against
eddy currents

Fast response
coils 1Figure 2.4: Sectional view of the main chamber along the x-axis highlighting the coil system composed by six independent coil pairs. (a) The large-bias (orange) and large-gradient coils (violet) are inHelmholtz and anti-Helmholtz configuration, respectively. They are mounted on a water-cooled brass
plate featuring a small cutout against eddy current. Four additional coil pairs are meant for small
magnetic field generation and are wound on copper cylinders. Both mounting systems are rigidly
connected to the main chamber and symmetric with respect to its center. (b) Zoom of a water-cooled
copper block in the y − z plane. The cutout helps against eddy currents. The small-gradient coils(blue) are in anti-Helmholtz configuration to generate small gradients. The small-bias coils (yellow)
are in Helmholtz configuration to generate small bias. The fast coils (red) are meant for fast-field
generation. The single-winding single-layer RF coils (green) are meant to apply RF on the atoms.
Figure taken from [Ilz20].

15

16

CHAPTER 2. ULTRACOLD ERBIUM-DYSPROSIUM ATOMSInverted viewport coils

This module (see Fig. 2.4 (b)) is composed by four pairs of independent coils designed tocreate small magnetic-fields and wound around a copper cylinder of 81 mm of inner diameter.
This copper block is rigidly connected to the vacuum apparatus and it holds in place thecoils and cools them down thanks to an internal water cooling system. If the coil pairs areactivated at the maximum operating current of 10 A, the cooling system keeps the entire
module at a maximum steady-state temperature of ∼28 °C with 1 l/min of running waterat about 23 °C (see Fig. C.2). Due to the low resistivity of these four coil pairs, the power
dissipated is quite low and, even without water cooling, the temperature does not rise above∼50 °C. Worth to be noted is the presence of a cutout on the copper block and an additional
thin isolating layer between it and the viewport which prevent the buildup of large eddy
currents. The main characteristics of the four coil pairs in this module are summarized in
the following list:• Small-gradient coils Ten-windings six-layers coil built with a 2.7 mm × 1.0 mm copper
wire. The coils in the pair are at 100 mm distance and they are in anti-Helmholtz
configuration for quadrupole-field generation. They are employed in the creation ofgradients used in the MOT loading and/or for levitation of the atomic sample.
At the center of the main chamber, the calculated quadrupole field produced by thex,yz | = 1.55 (G/cm)/A (see Fig. 2.5 (a)).
coil pair is |∇Btot| = 0.78 (G/cm)/A and |∇Btot
Even though the actual value of the current-to-gradient conversion has not been measured, the performance of the double-species MOT was optimized by tuning the current
directly.
• Small-bias coils Single-winding six-layers coil built with a 2.7 mm × 1.0 mm copper wire.
The coils in the pair are at 70 mm distance and they are in Helmholtz configuration forhomogeneous-field generation to control the bias field along the z-axis. They are used
to generate the MOT-bias field and for precise intra-/inter-species interaction tuningin the ODT.
By employing RF spectroscopy on a cold sample (described in Sec. 2.2.2), we measure
x,y,z
a current-to-field conversion at the atoms’ position of Btot
= 0.822(5) G/A over independent repetitions on different days (see Fig. 2.6 (a) for a typical measurement).
x,y,z
As a comparison, the numerically calculated value is Btot
= 0.78 G/A with a relative−4
flatness of 10 within ±0.5 mm (see Fig. 2.5 (b)). By setting the magnetic field witha step function, we are able to follow the time-variation of the field at the atoms andextract a time constant τ from an exponential decay fit (see Fig. 2.6 (b)). We estimate
τ = 0.62(9) ms. The main limitation comes most likely from eddy currents in thesystems.
• RF coils Single-winding single-layer coil built with a 1 mm diameter copper wire. The
main purpose is radio-frequency generation. These coils are used e. g. to perform RF
spectroscopy for magnetic-field calibration.
• Fast coils Single-winding four-layers coil built with a 1 mm diameter copper wire. They
are meant for fast-field generation and/or precision tuning. Currently not employed.

2.2. EXPERIMENTAL SETUP

17

The small-bias and -gradiend coils are driven with two independent self-build bipolar current sources based on high-voltage/high-current operational amplifier5 capable of ±10 A of
constant current.

High-field coils
This module (see Fig. 2.4 (a)) is composed of two pairs of independent coils dedicated tolarge-gradient and large-bias generation. Each coil of each pair is composed of two singlewinding 27-layers parts of 165 mm of inner diameter separated by a 4 mm hollow brass plate.
The brass plates rigidly connect the coils to the vacuum apparatus and cool them down asa result of an internal water cooling system. In the inside of the cooling channel, the surfaceof the brass is machined with a high roughness in order to increase the surface and leading
to a turbulent water flow improving the heat exchange. If operated at the maximum steadystate current of 100 A, the temperature is kept at a maximal value of 40 °C (see Fig. C.1).
Cutouts on the brass plates work against eddy currents. Both pairs are manufactured byusing a 8 mm × 1 mm copper wire. The basic features of the two coil pairs are listed in thefollowing:• Large-gradient coils The coils in the pair are at 145 mm distance between each other andthey are in anti-Helmholtz configuration for quadrupole-field generation. Their purpose
is creating large-gradient fields useful for e. g. Stern-Gerlach separation.
The calculated quadrupole field generated at the center of the main chamber isx,yz | = 0.48 (G/cm)/A (see Fig. 2.7 (a)). If changed|∇Btot| = 0.24 (G/cm)/A and |∇Btot
to Helmholtz configuration, they can produce and homogeneous field of
x,y,z
Btot
= 3.24 G/A with a flatness of 10−4 within ±1 mm of the center. Alternatively,
using the pair in Helmholtz configuration, it can be used to extend the magnetic field
generated by the large-bias coils up to ∼800 G in steady-state operation. At the moment of the submission of this thesis, the large-gradient coils were not actively employedin the experiment.
• Large-bias coils The coils in the pair are 105 mm apart and they are in Helmholtz configuration for homogeneous-field generation. They are used to create large bias used forinteraction tuning out of the magnetic-field range achievable with the small-bias coils.
These large-bias coils are used to perform Feshbach spectroscopy (see Sec. 4.4).
At the position of the atoms we measure a current-to-field conversion of
x,y,z
Btot
= 4.80(1) G/A over independent repetitions on different days. As a comparx,y,z
ison, the numerically calculated value is Btot
= 4.37 G/A with a flatness of 10−4
within ±1.5 mm of the center (see Fig. 2.7 (b)). We checked that the response of thecurrent flowing in this coil pair can follow time ramps on the millisecond time-scale.
By setting the field with a step function of 10 G and following the time-variation of thegenerated field, we measure an effective change of the field at the sample position in
the order of 10 ms to settle at the part per thousand level.
5

OPA549T

8

2.0

6

1.5

|∇Btot | (G/cm)

Btot (G)

(a)

4
2

0.5
0

0
−40

0

40

80
|∇Btot | (G/cm)−80
0.8
Btot (G)

1.0

0.4
0
−4

−2

0

2

4

−80

−40

0

40

80

−4

−2

0

2

4

1.8
1.2
0.6

d (mm)d (mm)

0.20

0.75

0.15

|∇Btot | (G/cm)

1.00

0.50
x-axisy-axisz-axis

0.25
0
−80

−40

0

40

Btot (G)

0.779
−4

−2

0

0.05

80

0.785

0.773

0.10

0
|∇Btot | (G/cm)

Btot (G)

(b)

2

4

−80

−40

0

40

80

−4

−2

0

2

4

0.04
0.02
0
d (mm)d (mm)x,y,z
x,y,z| (right
Figure 2.5: Calculated magnetic-field values Btot
(left column) and gradient |∇Btot
column) as a function of the distance d from the main chamber center for a constant 1 A current
for the small-field coils. The orange curve shows the behavior along the z-axis. The violet and blue
curves show the behavior along the x- and y-axis, respectively. Each rectangular panel beneath therespective square one is a zoom-in of the field in the center. (a) Small-gradient coils in anti-Helmholtz
configuration. (b) Small-bias coils in Helmholtz configuration. Figure taken from [Ilz20].

18

2.2. EXPERIMENTAL SETUP

19

4

2.6

Magnetic field (G)Magnetic field (G)

(a)
3

2

1

(b)

2.4
2.2
2
1.8

0

1.6
0

1

2

3

Current (A)

4

5

0

2.5

5

7.5

10

Time (ms)Figure 2.6: Characterization of the magnetic field produced by the small-bias coil pair by employing
radio-frequency spectroscopy on either erbium or dysprosium (see Sec. 2.2.2). (a) Typical currentto-field measurement for magnetic-field calibration in which we fix the current flowing into a specificcoil pair and we measure the field by RF spectroscopy. Black points are experimental values whereas
the gray line is the fit over the datapoints. From independent repetitions of such measurement on
x,y,z
different days we extract a conversion factor of Btot
= 0.822(5) G/A. (b) Time variation of themagnetic-field measured at the atoms position when the field is set at 0 ms with a jump of 0.5 G (bluepoints) and 1 G (red points). The blue and red curves are exponential decay fits over the respective
dataset. We extract a time constant of 0.62(9) ms.

For each of the two pairs of coil, the field is controlled by stabilizing the current in the coils
by an active feedback loop. The current is provided by two independent DC power supplies6
and monitored by high-precision current transducers7 . The current is set via temperaturestabilized 18-bit digital-to-analog converters (DAC)8 resulting in an ideal shot-to-shot accuracy of ±1.7 mG.
Compensation cage coils
This module composed by two square coil pairs in Helmholtz configuration is meant forcanceling the Earth’s magnetic field and other external stray fields at the sample’s position
along the x- and y-axis. In addition, two independent bipolar power supplies (identical tothe one used for the small-bias and small-gradient coils) allow for rotations of the magnetic
field on the x − y plane e. g. towards the imaging axis to perform absorption imaging of
the sample. Each coil features a two-winding eleven-layers design and it is manufactured
with a 4 mm × 1 mm copper wire. The wire is wound inside an aluminum frame which isthen rigidly connected to the main chamber (see Fig. 2.3). The size of the two coil pairs
is slightly different to simplify the assembly in a configuration as close as possible to aperfect Helmholtz scheme and, at the same time, able to surround the entire main chamber.
Although the module is only passively cooled, the temperature at the maximum steady-state
current of 10 A does not raise above ∼30 °C. By optimizing the absorption imaging signal, it
678Delta Elektronica SM30-200
LEM IT200-S ULTRASTABDAC based on a LTC2757

0.5003|∇Btot | (G/cm)

Btot (G)

(a)

2

1

0.250
0.125
0

0
−75

0

75

150
|∇Btot | (G/cm)

−150
0.250
Btot (G)

0.375

0.125
0
−4

−2

0

2

4

−150

−75

0

75

150

−4

−2

0

2

4

0.60
0.40
0.20

d (mm)d (mm)(b)

0.75

5|∇Btot | (G/cm)

Btot (G)432x-axisy-axisz-axis

1

0

75

150
|∇Btot | (G/cm)−75

Btot (G)

4.374
4.372
4.370

−4

−2

0

0.25

0

0
−150

0.50

2

4

−150

−75

0

75

150

−4

−2

0

2

4

0.02
0.01
0
d (mm)d (mm)

x,y,z
x,y,z
Figure 2.7: Calculated magnetic-field values Btot
(left column) and gradient |∇Btot| (right
column) as a function of the distance d from the main chamber center for a constant 1 A current
for the large-field coils. The orange curve shows the behavior along the z-axis. The violet and blue
curves show the behavior along the x- and y-axis, respectively. Each rectangular panel beneath therespective square one is a zoom-in of the field in the center. (a) Large-gradient coils in anti-Helmholtz
configuration. (b) Large-bias coils in Helmholtz configuration. Figure taken from [Ilz20].

20

2.2. EXPERIMENTAL SETUP

21

has been possible to estimate the time for the field to settle after a change in its amplitudein the few-ms timescale.
• Compensation x-axis The dimension of each coil compensating for magnetic fields along
the x-axis is 380 mm × 380 mm with a distance between each other of 175 mm. The
field produced at the center of the main chamber by such configuration is calculated
x,y,z
to be Btot
= 1.04 G/A with a flatness of 2 × 10−6 within ±0.5 mm (see Fig. 2.8 (a)).
x,y,zWe measure Btot
= 1.06(2) G/A over different days.
• Compensation y-axis The dimension of each coil compensating for magnetic fields along
the y-axis is 350 mm × 350 mm with a distance between each other of 155 mm. The
x,y,zfield produced at the center of the main chamber is calculated to be Btot
= 1.12 G/A−6with a flatness of 2 × 10
within ±0.5 mm (see Fig. 2.8 (b)).
We measure
x,y,z
Btot
= 1.13(2) G/A over different days.

×10−2

×10−2

x,y,z
x,y,z
Figure 2.8: Calculated magnetic-field values Btot
(left column) and gradient |∇Btot| (right
column) as a function of the distance d from the main chamber center for a constant 1 A current
for the compensation coils. The orange curve shows the behavior along the z-axis. The violet
and blue curves show the behavior along the x- and y-axis, respectively. Each rectangular panel
beneath the respective square one is a zoom-in of the field in the center. (a) Compensation x-axis.
(b) Compensation y-axis.

222.3 Publication:Two-species five-beam magneto-optical trap for erbium anddysprosium†
Philipp Ilzhöfer1,2 , Gianmaria Durastante1,2 , Alexander Patscheider1,2 , Arno Trautmann2 ,Manfred J. Mark1,2 , and Francesca Ferlaino1,2
1Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria

Physical Review A 97, 023633 (2018)
submitted 20 Nov 2017; published 26 Feb 2018
DOI: 10.1103/PhysRevA.97.023633†

The author of the present thesis helped in developing the experimental procedure, taking the data, andcontributed in writing the manuscript. This publication is also part of the PhD thesis of Philipp Ilzhöfer.

23

PHYSICAL REVIEW A 97, 023633 (2018)

Two-species five-beam magneto-optical trap for erbium and dysprosium
P. Ilzhöfer,1,2 G. Durastante,1,2 A. Patscheider,1,2 A. Trautmann,2,* M. J. Mark,1,2 and F. Ferlaino1,2,†
1

Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria(Received 20 November 2017; published 26 February 2018)
We report on the first realization of a two-species magneto-optical trap (MOT) for the highly magnetic erbium
and dysprosium atoms. The MOT operates on an intercombination line for the respective species. Owing to thenarrow-line character of such a cooling transition and the action of gravity, we demonstrate a trap geometry
employing only five beams in the orthogonal configuration. We observe that the mixture is cooled and trappedvery efficiently, with up to 5 × 108 Er atoms and 109 Dy atoms at temperatures of about 10 μK. Our results offer
an ideal starting condition for the creation of a dipolar quantum mixture of highly magnetic atoms.
DOI: 10.1103/PhysRevA.97.023633

Within the very active research field of ultracold quantum
gases, heteronuclear mixtures of different atomic species offer
unique possibilities to study a broad range of quantum phenomena. In the past 15 years, various atomic species have beencombined to produce quantum degenerate mixtures. Each such
quantum mixture has its own characteristic traits. Among thewidely used alkali mixtures (e.g., [1,2]), the mass imbalance
and the selective tuning of the intra- and interspecies interaction have allowed to investigate fascinating phenomena, suchas heteronuclear Efimov states [3–6], polaron and impurity
physics in both bosonic and fermionic backgrounds [7–11], andheteronuclear molecules with large electric dipole moments[12–14].
The latter development is mainly driven by the interest in
studying phenomena arising from long-range and anisotropic
dipole-dipole interactions among the molecules [15]. As analternative approach, magnetic atoms have proven to be arobust system for study of few- and many-body dipolar physics.
The strength of magnetic atoms for the study of dipolar
physics was first shown using Bose-Einstein condensates ofchromium atoms [16,17]. More recently, both erbium (Er) anddysprosium (Dy), among the most magnetic and isotope-richatomic species, have been individually brought to quantum
degeneracy [18–21]. Using these species, remarkable manybody dipolar phenomena have been observed, including theobservation of deformed Fermi surfaces [22], quantum droplets[23–25], and roton excitations [26] and the recent study ofthermalization in many-body dipolar gases [27].
Adding the flexibility of mixtures to the richness of magnetic atoms, we here report on the first combination of thetwo highly magnetic atomic species Er and Dy in a single experimental apparatus. The Er-Dy system extends thecollection of available quantum mixtures by an unexploredcase, as the interplay between the interspecies contact anddipolar interactions and the dipolar imbalance among the two
species provides a new dimension in the parameter space of*†arno.trautmann@uibk.ac.at
Corresponding author: Francesca.Ferlaino@uibk.ac.at2469-9926/2018/97(2)/023633(6)accessible quantum phenomena. This impacts, e.g., the miscibility properties of the mixture [28]. Although imbalanced
dipolar mixture systems have not yet been considered in theory,
they are good candidates for observation of, e.g., long-range
dominated polarons, dipolar pairing, and the anisotropic BECBCS crossover with deformed Fermi surfaces.
While single-species magneto-optical traps (MOTs) of Er[29–31] and Dy [32–34] as well as other lanthanoid atoms[35–37] have already been attained, we simultaneously cooland trap Er and Dy in a two-species MOT operating on
intercombination lines. We observe a remarkably robust operation of the dual MOT, with atom numbers similar or evensurpassing the typical ones recorded in the single-species Er or
Dy experiments. Moreover, we demonstrate magneto-opticaltrapping using a unique beam configuration, allowing us toefficiently operate the MOT using only five beams (5B) in an
orthogonal open-top configuration; see Fig. 1(b). The working
principle of our orthogonal 5B MOT relies on the combined
effect of the narrow-line cooling and gravity [30,31,33–35,38]
and contrasts with the classical six-beam (6B) approach.
A beneficial factor for combining the multivalence-electron
atoms Er and Dy is their similarity in atomic properties [seetable in Fig. 1(e)]. They both have several stable isotopes
with a high natural abundance, >14%; in total, five bosonic(166 Er, 168 Er, 170 Er, 162 Dy, 164 Dy) and three fermionic (167 Er,
161
Dy, 163 Dy) isotopes. This isotope variety will allow us toprepare ultracold Bose-Bose, Bose-Fermi, and Fermi-Fermi
quantum mixtures of Er and Dy. Whereas all bosonic isotopes
have zero nuclear spin, the fermionic isotopes possess nuclear
spins of IEr = 7/2 and IDy = 5/2, leading to eight and six
hyperfine states in the electronic ground state, respectively.
Both elements exhibit a rich atomic energy spectrum, arisingfrom their submerged-shell electronic configuration, featuringa [Xe] core, a partially filled inner 4f shell, and a closed outer
6s shell. The electron vacancy in the 4f shell is responsible
for the large orbital quantum numbers and the respective highmagnetic moments, i.e., 7μB and 10μB for Er and Dy, with
μB being the Bohr magneton.
Figures 1(c) and 1(d) show the electronic levels of Er and
Dy for wave numbers up to 26 000 cm−1 [39]. While most of023633-1©2018 American Physical Society

P. ILZHÖFER et al.

PHYSICAL REVIEW A 97, 023633 (2018)

FIG. 1. Illustration of the vacuum apparatus, including the optical setup for the 5B MOT, and atomic properties of Er and Dy. (a) Er-Dy
vacuum apparatus, including the high-temperature oven, transversal cooling chamber, Zeeman slower (ZS), and main chamber. The atomic
beam propagates from right to left. The ZS beam is reflected by a metallic mirror in vacuum. (b) Sketch of the working principle of the open-top
MOT. Arrows depict MOT beams, and the blue region indicates the ZS beam. For clarity, we depict the atomic clouds displaced from eachother. (c, d) Energy level diagrams for Er and Dy up to 26 000 cm−1 at different total electronic angular momentum quantum numbers J . Stateswith odd (even) parity are indicated by black (orange) horizontal lines. Arrows show the broad and narrow laser-cooling transitions. (e) Table
listing the atomic properties of Er and Dy.

the possible transitions are dipole forbidden, both species offer
one particularly broad transition, suitable for laser cooling, tothe respective singlet 6s6p state [blue arrows in Figs. 1(c)and 1(d)], which we label broad in the following. The bluetransition light has a wavelength of 401 nm(421 nm), and thetransition line width is broad /2π = 27.5 MHz (32.2 MHz) forEr (Dy) [30,33]. We use this transition for transversal cooling,
Zeeman slowing, and absorption imaging. The laser light,
driving the broad transition, is derived from grating-stabilized
laser diodes, followed by tapered amplifiers and frequencydoubling cavities. The laser systems emit more than 1W ofpower each. Both systems are frequency-stabilized using signals from modulation-transfer spectroscopy in hollow-cathodelamps [30].
Following previous single-species experiments with Yb[35], Er [30], and Dy [33,34], we produce the MOT using anintercombination line driving the transition from the groundstate to the triplet 6s6p state [yellow (red) arrow in Figs. 1(c)and 1(d)] at a wavelength of 583 nm(626 nm) in Er (Dy)and a line width of 583 /2π = 190 kHz(626 /2π = 136 kHz).
The narrow-width character of these transitions leads to theconveniently low Doppler temperatures of TD,Er = 4.6 μK and
TD,Dy = 3.2 μK. The laser system for the Er MOT is based on a
Raman fiber-amplified diode laser at 1166 nm and a single-passfrequency-doubling stage, with an output power above 1.7 W.
The laser system for the Dy MOT is based on two fiber
lasers operating at 1050 and 1550 nm, which are amplified andfrequency-converted in a single-pass sum-generation stage,
resulting in more than 1.6 W of output power. Both MOT laser
systems are frequency-stabilized against long-term drifts on ahome-built ultralow expansion cavity via a Pound-Drever-Halllock [40] and have line widths below 100 kHz.
The experimental procedure generalizes our previously
demonstrated single-species MOT approach [30] to twospecies operation. The very similar strengths and wavelengthsof the laser-cooling transitions of Er and Dy and their similar
masses and melting points greatly simplify the design of thevacuum apparatus and the experimental procedure for the
mixture. Figure 1(a) shows the experimental apparatus. Er
and Dy atoms are emitted from a single high-temperatureoven, consisting of two sections: The first section (effusioncell) operates at a temperature of 1100 ◦ C, and the second
one (hot lip) operates at 1200 ◦ C, unless otherwise stated.
Three apertures of different diameters, placed inside the oven,geometrically collimate the Er-Dy atomic beam before it enters
the transversal-cooling chamber. We operate the transversal
cooling resonantly on the broad transitions with total powersof 300 mW(120 mW) for Er (Dy) and elliptic waists of023633-2

TWO-SPECIES FIVE-BEAM MAGNETO-OPTICAL TRAP …

PHYSICAL REVIEW A 97, 023633 (2018)
109

10

TEC (1000° C)
1.1

1.0

8

1.2

5
R (108 s−1 )atom number NDy , NErapproximately whoriz = 30 mm and wvert = 6 mm. We observethat the transversal cooling increases the MOT loading rate bya factor of 10 (6) for Er (Dy). The two-species atomic beam
is slowed down to about 5 m/s using a two-species Zeeman
slower (ZS) about 35 cm long in the spin-flip configuration.
The magnetic-field values along the ZS are experimentally
optimized for Dy and work equally well for Er. The two lightbeams for the ZS of Er and Dy are overlapped using dichroic
optics and guided through the ZS via a metallic in-vacuum
mirror [41]; see Fig. 1(a). The optimal performance of the
ZS has been found for laser powers of 57 mW(121 mW) withbeam waists of 4 mm at a detuning of −520 MHz(−530 MHz)
for Er (Dy).
The slow atoms are then captured into a two-species MOT,
operating on the respective intercombination line. Taking
advantage of the similar wavelengths, we combine the MOT
beams for both species into the same fibers. The MOT lightis spectrally broadened utilizing electro-optic modulators withresonance frequencies of 139 kHz(102 kHz) for Er (Dy), which
increases the capture range and thus the number of atoms in
the MOT by a factor of 5 (2). The recapture volume is further
enhanced by using large MOT beams, with diameters of about
36 mm. We observe an optimal loading for peak intensitiesof each laser beam of I583 = 50Isat,583 and I626 = 160Isat,626 .
Additionally, we endow our main chamber with inverted top
and bottom view ports with large clear apertures of 64 mm.
As discussed later, our special 5B MOT geometry allows us
to completely free the top view, where we will implement ahigh-resolution in situ imaging with a numerical aperture of
0.45, which can resolve structures down to 600 nm. A pair ofvertical coils creates a magnetic quadrupole field for the MOTof ∇B = 4.6 G/cm. A vertical bias field of B0 = 2.9 G shifts
the zero point of the quadrupole field downwards. Additional
coil pairs in the horizontal plane compensate for external
magnetic fields.
We produce and study the Er-Dy MOT using two beam
configurations. In the first one, we use a standard 6B geometry
with three pairs of orthogonal retroreflected beams. For thesecond configuration, we remove the top → bottom MOT
beam, demonstrating for the first time a 5B geometry with
an open top; see Fig. 1(b). Although this 5B MOT would
not work for alkali MOTs [42], we here demonstrate a very
robust operation for our lanthanoid mixture. In the first setof experiments, we study the loading performance of our
two-species MOT in both the 6B and the 5B configurationsfor 168 Er and 164 Dy. For all atom numbers we report in thispaper, we load the MOTs at the optimized detuning for eachsetting as discussed later (see Fig. 3), apply a compression
phase after MOT loading, and detect the atomic clouds using
absorption imaging, as described later. Figure 2 shows the
Er and Dy atom numbers as a function of the MOT loading
time. From a fit to the data using a standard loading function,
N(t) = Nss (1 − e−γ t ), we extract the loading rate R and decayrate γ with the steady-state atom number Nss = R/γ (seeTable I). In both the 6B and the 5B configurations, we observea very efficient loading of the two-species MOT. After about
10 s of loading, the atom numbers approach their steady-statevalue of some 108 atoms (see Table I). We do not observe any
mutual influence of one species on the other, as reported forsome alkaline mixtures, as, e.g., shown in Refs. [43] and [44].

6B 5B Er6B 5B Dy

2
1
0.5
0.2

7

10

0

5

15

10

20

25

30

loading time (s)FIG. 2. Loading curve of the two-species MOT for the 6B and
5B configurations. Filled and open squares (circles) show data forthe 6B and 5B Er (Dy) MOTs, respectively. The corresponding lines
are fits to the data, as detailed in the text. Fit parameters are listed
in Table I. Inset: Loading rate of the two species dependent on theeffusion cell temperature TEC . Note that the hot lip is always kept at
THL = TEC + 100 ◦ C.

Given the complex scattering properties and optical spectraof multielectron lanthanoids, this surprising result enables
the simultaneous MOT operation and greatly simplifies theexperimental sequence.
The difference in the loading curves between Er and Dy
is due to their different vapor pressures. The ratio of vapor
pressures of Er and Dy is typically higher than 10 at thesame temperature [45] and strongly temperature dependent,
which would prevent an efficient simultaneous MOT loading.
To mitigate this, we selectively heat up the atoms exploiting
the two-section design of the oven. We fill the effusion cellwith a 33%/67% alloy of Er/Dy and the hot-lip section withpure Er. We operate the oven with a differential temperature of
100 ◦ C between the two sections. In the temperature range from1000 ◦ C to 1200 ◦ C for the first section, we expect to reducethe vapor-pressure ratio among the two species to about 2.3.
We investigate this effect by repeating the loading experiments
for different temperatures of the effusion cell, while keeping
the hot lip always 100 ◦ C hotter; see Fig. 2 (inset). We observea roughly constant loading ratio of between 2.5 and 3.5, which
confirms the above expectations and shows that our concept ofdifferential heating works very efficiently.
We remarkably find that the performance of the 5B MOT is
only slightly below that of the 6B MOT. Moreover, even our 5B
double-species MOT shows atom numbers similar to or larger
than those previously reported for single-species Er or Dy MOT
experiments [29,30,32–34]. Judging from our experience withTABLE I. Loading rates R, decay rates γ , and steady-state atom
numbers Nss obtained from fits to the data shown in Fig. 2. Values forboth species in the 5B and 6B configurations are listed. Also listed
are the lifetimes obtained from the data in Fig. 4.
Er

R (108 s−1 )γ (s−1 )Nss (108 )Lifetime cMOT (ms)023633-3

Dy

5B

6B

5B

6B

0.35(1)0.100(2)3.5(1)
515(65)0.45(2)0.086(6)4.5(3)
475(50)1.21(4)0.166(7)7.3(4)
374(44)

1.79(3)0.187(5)9.6(3)
345(23)

P. ILZHÖFER et al.

PHYSICAL REVIEW A 97, 023633 (2018)

Er [30], we are thus confident that these numbers are sufficient
for reaching quantum degeneracy. The 5B MOT configuration
has the advantage of automatically spin-polarizing the atoms
and providing full optical access through the vertical top viewport, allowing implementation of optical setups which requirelarge numerical apertures, e.g., high-resolution imaging or
angle-resolved Bragg spectroscopy.
The simultaneous cooling and polarization of the intercombination MOT in lanthanoids has been studied for singlespecies Er [30] and Dy [34] 6B MOTs. In brief, the combined
effects of a narrow-line MOT and gravity yield a peculiar semishell-shaped MOT with its center lying below the zero of themagnetic quadrupole field; see Fig. 1(b). The center position of
the MOT can be adjusted by changing the MOT detuning. For
large enough vertical displacement of the MOT, i. e., detuning,
the atoms predominantly absorb σ − -polarized photons, which
are coming from the bottom-top beam. As a consequence, theatoms are spin-polarized into the lowest Zeeman sublevel. For
lower detuning, the atoms can absorb both σ + and σ − light andthe sample is unpolarized [34]. With our 5B MOT, we bringthis concept to the extreme: We completely remove the (σ + )top-bottom beam and force the atoms to sit only below thezero of the magnetic field. We verified the spin polarization
by performing spin-resolved absorption imaging using theStern-Gerlach technique. Within our experimental resolution,
we do not detect atoms in higher Zeeman sublevels [46].
Thanks to this spontaneous spin polarization, optical-pumping
schemes are not necessary. The spin purity is very beneficial
for future loading of the mixture into an optical dipole trap,
where the presence of additional spin states can lead to atomic
losses via inelastic dipolar relaxation processes.
In the second set of experiments, we systematically study
the effect of MOT-light detuning from the respective resonant
atomic frequency on the atom number and compare the results
for the 6B and 5B configurations after 5s of loading; see
Fig. 3. For both species, we see a clear rise of atom numberswith increasing detuning. After reaching a maximum value

×108
(a)

6
6B
5B

1.5

×108
(b)

5

Dysprosium
6B
5B

4

2 × 1083

1

atom numbers

atom number NDy , NEr

2

Erbiumthe numbers undergo a sharp decrease at large detunings. Thisdecrease can be simply explained by the spatial downshiftof the MOT position with increasing detuning, eventuallycausing the atoms to leave the recapture volume. Here, theequal behavior of the 6B and 5B MOTs indicates that the
top-bottom beam does not play a significant role. At intermediate detunings, however, the two configurations clearlyshow different behaviors. In particular, the 6B MOT has amuch broader range of operation than the 5B configuration.
We believe that this difference is due to the fact that thecentral cloud position approaches the magnetic-field zero pointwith decreasing detuning. In the absence of the top-bottom
beam, atoms above the magnetic-field zero do not experiencea restoring light force towards the trap center and may escapefrom the MOT. Contrariwise, in the 6B approach, these atoms
are retrapped, resulting in the broader operation range in termsof detuning.
As observed in previous experiments [33], the Dy atom
number shows a small dip at detunings around −70626 nm . We
attribute this feature to a partial overlap of the cloud with the
ZS light beam, which drives off-resonant pumping processes
resulting in atom losses [30]. For Er, the influence of the ZS
light is weaker due to its lower light intensity and larger relative
detuning, and this effect is not observed.
Finally, we study the lifetime of the mixture in the compressed MOT (cMOT). The compression phase is essential toefficiently load an optical dipole trap in future experiments,
as the compression reduces the temperature and increases thedensity of the mixture. After loading the MOT, we switch
off the spectral broadening electro-optic modulators, the ZS
beams, and block the atomic beam with a mechanical shutter.
The compression has a duration of 200 ms, during which we(i) reduce the detuning of the MOT light to 10 583 (18 626 ),(ii) decrease the MOT-beam power to I583 = 0.17Isat,583 andI626 = 0.6Isat,626 , (iii) ramp down the magnetic-field gradientto ∇B = 4.3 G cm−1 , and (iv) switch off the vertical bias
magnetic field. As shown in Fig. 4, we observe that the
double-species cMOT has a lifetime that is fully sufficient for
loading of atoms into an optical dipole trap, which typicallyrequires a holding time of about 100 ms. The data are taken indual operation, with both species present in all settings. Again,

2

0.5

1
0
-10 -20 -30 -40 -50 -60 -70
detuning (units of Γ583 nm )0

1 × 108

5 × 107-25 -50 -75 -100 -125
detuning (units of Γ626 nm )FIG. 3. Dependence of atom numbers in the cMOT on the initial
MOT detuning, in units of the respective linewidth of the narrowtransition of 583 /2π = 190 kHz(626 /2π = 136 kHz) for Er (Dy).
Both species show broad ranges of detuning for efficient MOT loadingin the 6B configuration, while 5B shows a narrower range. The optimal
detunings are nearly equal for the 5B and 6B configurations. Data weretaken for 5 s of MOT loading.

Er
Dy

0

200
400
600
800
compressed MOT hold time (ms)

1 000

FIG. 4. Lifetime of the cMOT for both Er (red) and Dy (blue) in
the 5B (open symbols) and 6B (filled symbols) configurations. We
adjust the MOT loading time to compare samples with equal atom
numbers. Solid lines show exponential fits to the respective data. From
the fits we extract lifetimes of 515(65) and 475(5) ms [374(44) and345(23) ms] for Er [Dy] in the 5B and 6B cMOTs, respectively.

023633-4

TWO-SPECIES FIVE-BEAM MAGNETO-OPTICAL TRAP …

PHYSICAL REVIEW A 97, 023633 (2018)

we do not see any interplay between the species. The resultsof fits to the lifetime data are listed in Table I. Additionally,
we extract temperatures of 11(1) μK [10(1) μK] for Er [Dy]in both the 5B and the 6B configurations from time-of-flightmeasurements. We observe that lower temperatures close to
the Doppler temperatures can be achieved with different setsof parameters, at the expense of lower atom numbers. From
our experience, we are certain that the observed lifetimes
and temperatures are fully sufficient for efficient loading
into an optical dipole trap as the next step towards quantum
degeneracy.
The data we present here refer to the mixture of 168 Er and164
Dy. Moreover, we are also able to trap and cool all otherabundant bosonic isotopes with equally good performance,
except for the 170 Er MOT, which has smaller numbers, as
expected from its low natural abundance. For future studiesof the fermionic isotopes, no changes in the experimental
apparatus are necessary.
In conclusion, we have demonstrated efficient cooling andtrapping of an Er-Dy mixture in a two-species MOT operatingon narrow-line transitions. In addition, we demonstrate a beamgeometry for our two-species MOT which consists of only fivelaser beams in an open-top orthogonal setting. This geometry
has the big advantage of providing free optical access fromthe top with a large numerical aperture, greatly simplifying theimplementation, e.g., of high-resolution imaging, as well as
optical lattices. Our recorded temperatures and atom numbers
provide ideal conditions for subsequent evaporative cooling,
towards the first production of a quantum-degenerate dipolar
mixture. Optimization of the optical-trap loading and evaporative cooling is under way in our laboratory.

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Phys. Rev. Lett. 94, 160401 (2005).
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Maréchal, L. Vernac, J.-C. Keller, and O. Gorceix, Phys. Rev. A
77, 061601 (2008).
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Lett. 107, 190401 (2011).
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[20] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler,
R. Grimm, and F. Ferlaino, Phys. Rev. Lett. 108, 210401(2012).
[21] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm, and F.
Ferlaino, Phys. Rev. Lett. 112, 010404 (2014).
[22] K. Aikawa, S. Baier, A. Frisch, M. Mark, C. Ravensbergen, and
F. Ferlaino, Science 345, 1484 (2014).
[23] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. FerrierBarbut, and T. Pfau, Nature 530, 194 (2016).
[24] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T.
Pfau, Nature 539, 259 (2016).
[25] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L.
Santos, and F. Ferlaino, Phys. Rev. X 6, 041039 (2016).
[26] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni, S. Baier,J. H. Becher, M. J. Mark, F. Waechtler, L. Santos, and F. Ferlaino,
arXiv:1705.06914.
[27] Y. Tang, W. Kao, K.-Y. Li, S. Seo, K. Mallayya, M. Rigol, S.
Gopalakrishnan, and B. L. Lev, arXiv:1707.07031.
[28] R. K. Kumar, P. Muruganandam, L. Tomio, and A. Gammal,
J. Phys. Commun. 1, 035012 (2017).
[29] A. J. Berglund, S. A. Lee, and J. J. McClelland, Phys. Rev. A
76, 053418 (2007).
[30] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler, E.
Zupanič, R. Grimm, and F. Ferlaino, Phys. Rev. A 85, 051401(2012).

We acknowledge A. Frisch, T. Pyragius, and C. Zhang for
support in the initial phase of the experiment. We thank the
ERBIUM and the Dy-K teams in Innsbruck, the Er team at
Harvard, and M. Sohmen and C. Politi for fruitful discussions.
This work was supported by an ERC Consolidator Grant(RARE; No. 681432) and a FET Proactive Project (RySQ; No.
640378) of EU H2020. G.D. acknowledges support from theAustrian Science Fund FWF within DK-ALM: W1259-N27.

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PHYSICAL REVIEW A 97, 023633 (2018)[31] J. Ulitzsch, D. Babik, R. Roell, and M. Weitz, Phys. Rev. A 95,
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J. Dalibard, and S. Nascimbene, J. Phys. B 50, 065005(2017).
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[41] The atomic beam of Er and Dy might deposit an additional
metallic layer on the mirror. However, we have tested this effectin our running Er machine and did not observe any sizable
reduction in the reflectivity of the mirror.
[42] For the sake of completeness, we note that early MOT experiments on alkali atoms have explored MOT geometries with fivenon–orthogonally intersecting beams (e.g., at an angle of 120°)[47,48], which constitutes a very different scenario compared toour open-top approach.
[43] U. Schlöder, H. Engler, U. Schünemann, R. Grimm, and M.
Weidemüller, Eur. Phys. J. D 7, 331 (1999).
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K. Dieckmann, Phys. Rev. A 73, 011402 (2006).
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(1964).
[46] Since the large spatial extension of the MOT might mask residual
atoms in higher spin states, we performed preliminary SternGerlach mapping after loading the atoms into an optical dipole
trap. Again, we did not detect atoms in higher Zeeman sublevels.
[47] J. Arlt, P. Bance, S. Hopkins, J. Martin, S. Webster, A. Wilson,
K. Zetie, and C. J. Foot, J. Phys. B 31, L321 (1998).
[48] A. di Stefano, D. Wilkowski, J. Müller, and E. Arimondo, Appl.
Phys. B 69, 263 (1999).

023633-6

Chapter3

Quantum mixtures

Ultracold atoms have proved to be a great system for fundamental research, precision measurements, and quantum simulations [Cor02, Chi15]. Their high degree of control allows
for an investigation of a multitude of phenomena beyond atomic physics: complex systems
typical of condensed matter, few-body and many-body physics, and cold chemistry — toname some of the possibilities. A natural step beyond the intriguing physics available inhomonuclear systems consists of studying more complex systems such as mixtures. They
include homonuclear systems in different spin states, isotopic combinations of the sameatomic species, and heteronuclear mixtures of multiple elements. More examples, not discussed within this thesis, are mixtures involving metastable noble gases [Vas12], trapped
cold molecules [Car09], and hybrid atom-ion systems [Tom19].
Mass and population imbalance, mixed statistic, species-selective manipulations, and intrinsic distinct properties are all tools made available by using mixtures. They become
suitable platforms to investigate new physical phenomena, e. g. impurity physics, polarons,
exotic phases and molecule formations. In most of these applications, an important role isplayed by Feshbach resonances, allowing the tuning of the interparticle interactions and themagnetoassociation of weakly bound molecules (see Chap. 4).
In this chapter, we present the first experimental realization of a quantum-degenerate
dipolar mixture of erbium-dysprosium atoms in several isotopic combinations. Up to ourknowledge, this is the first experiment which approached the further step of combining two
magnetic lanthanide species. The chapter begins with a theoretical introduction to the topic
of quantum degeneracy in heteronuclear mixtures described in Sec. 3.1. Thereafter, Sec. 3.2
reviews the advances in the field by comparing the results achieved in mixture experiments.
In the same section, a digression on the particular case of lanthanide mixture of erbiumdysprosium can be found. Sec. 3.3 includes the publication Quantum Mixtures of Erbium
and Dysprosium Atoms [Tra18].

3.1 Theory of quantum mixture
Predicted in the early 1920s by Satyendra Nath Bose and Albert Einstein [Bos24, Ein24,
Ein25], a Bose-Einstein condensate is a state of matter in which a large fraction of particles
of a system occupies the lowest quantum state. In an ideal non-interacting dilute gas in31

32

CHAPTER 3. QUANTUM MIXTURES

thermodynamic equilibrium at a temperature T , quantum-statistical effects become relevant
when the particles’ extension associated with de Broglie wavelength λdB is on the same orderof the interparticle spacing n−3 . In other words, when the phase space density (PSD)
P SD = nλ3dB = n



2πℏ
mkB T

3/2

3.1

is larger than one [Ket99]. Here ℏ is the reduced Planck constant, kB is the Boltzmann
constant, and m the particle’s mass. By further increasing the PSD, there is a criticalvalue for which the system undergoes a purely quantum-statistical phase transition. In athree-dimensional box, the condition is P SD ≳ 2.612.
To capture the behavior of a many-body system, the treatment must include interactions.
A very common mean-field description of BECs is based on the Gross-Pitaevskii equation(GPE) approach [Pit16]. In the following, we make use of this approach for a dilute gas ina harmonic trap.
This section introduces the reader to the physics of quantum gases and, in particular, tothe case of quantum mixtures. We first develop the mean-field description of an interacting
Bose gas in a harmonic trap in Sec. 3.1.2. Thereafter, we discuss the solutions of the GPE
in Sec. 3.1.3 in both cases of heteronuclear and dipolar systems. This section follows thetheoretical treatment proposed in Ref. [Pet08].

3.1.1 Contact and dipolar interactions
To understand a dipolar quantum gas in the mean-field description, we need to introduce theconcept of pseudo-potential, that is a simplified version of the complex interatomic potential.
In the case of dipolar atoms, for two particles at distance r, we can write such potential Vint (r)as a sum of the contributions coming from the contact Vc (r) and dipolar Vdd (r) interactions:Vint (r) = Vc (r) + Vdd (r).

(a)

(b)

µ
S

3.2

N
⃗r
S
µ

⃗
B

S

N

θ

S

N

Repulsive

N

(c)
S

N

S

N

Attractive

Figure 3.1: Dipole-dipole interactions of two polarized dipoles. (a) Sketch of the geometry. (b) Ina side-by-side configuration, the DDI are repulsive. (c) In a head-to-tail configuration, the DDI are
attractive.

3.1. THEORY OF QUANTUM MIXTURE

33

Contact interactions At low energy, we can replace the contact interaction term with aspherically symmetric zero-range contact potential in the form:Vc (r) = g1,2 δ(r) =2πℏ2
a1,2 δ(r)m12

3.3

where m12 is the particles’ reduced mass, δ(r) is the Dirac delta function, g accounts for the
coupling strength and a is the s-wave scattering length as discussed in Chap. 4. For the easeof reading, in the case of single-species interactions, we write gi,i = gi .
Dipole-dipole interactions The long-range and anisotropic DDI for two dipoles aligned by
an external magnetic field can be written as:
Vdd (r) =µ1 µ2 µ0 1 − 3 cos2 θ
4π
|r|3

3.4

where µ0 is the vacuum permeability, µ1 and µ2 are the particles’ magnetic moments, and θ
is the angle between r and the magnetic field (see Eq. 3.1). In analogy with Eq. 3.3, we candefine the characteristic dipolar length:add =µ1 µ2 µ0 m12
6πℏ2

3.5

in such a way to compare directly contact and dipolar strength via the ratio
ϵdd =

add
.
a

3.6

For ϵdd > 1, the system is DDI dominated. It is worth to note that, a — therefore ϵdd— can be tuned by means of Feshbach resonances (see Chap. 4), thus entering different
regimes of dipolarity. For the sake of completeness, a itself is generally dependent on the
DDI even though such dependency is weak and it is usually accounted in the experimental
measurements [Ron06].

3.1.2 Mean-field description and Gross-Pitaevskii equation
In a mean-field description, every particle interacts with a mean-field potential (see Sec. 3.1.1)
reproducing the complex interaction pattern coming from the rest of the system in a simplified way. Now, let’s consider an interacting gas of N identical bosonic particles trapped in
an external harmonic potential Vext . We can define the condensate wave-function ψ(r) and
its density distribution n(r) = |ψ(r)|2 . The time-dependent GPE acquires the form:
"

#

∂
ℏ2 2
iℏ ψ(r, t) = −∇ + Vext (r) + Vint (r) ψ(r, t)
∂t
2m

3.7

where Vint is described in Eq. 3.9. If we assume that ψ(r, t) = ψ(r) e−iµc t/ℏ with µc being thechemical potential, we can derive the time-independent GPE
"

#

ℏ2 2
µc ψ(r) = −∇ + Vext (r) + Vint (r) ψ(r).
2m

3.8

34

CHAPTER 3. QUANTUM MIXTURESIn the mean-field approach, Eq. 3.2 becomes
Vint (r) = gn(r) +Zµ2 µ0 1 − 3 cos2 θn(r) dr
4π
|r|3

3.9

and, given trap frequencies (ωx , ωy , ωz ), the harmonic trapping potential can be written as
Vext (r) =m 2 2
ωx x + ωz2 z 2 + ωz2 z 2 .
2

3.10

It is useful to define ω̄ = (ωx ωy ωz )1/3 denoting the geometric mean of the trap frequencies.
From here on, we work under the so called Thomas-Fermi approximation (TF) which neglects
the kinetic term in the GPE. Such approximation is true for a sufficiently large particle
number and leads toµc = Vext (r) + Vint (r).
3.11
Heteronuclear dipolar BECs
After preparing the formalism for a one-component BEC, the next step consists of extending
it to the case of two-components BECs. The problem results now in a system of coupledequations where the interspecies interactions (both contact and DD) have to be taken into
account. For two species 1 and 2, we can write:µc,1 = Vext,1 (r) + Vint,1 (r) + Vint,12 (r)µc,2 = Vext,2 (r) + Vint,2 (r) + Vint,21 (r)3.12

where Vext,1 and Vext,2 are written like in Eq. 3.9 and, for i ̸= j and i, j = 1, 2,
Vint,ij (r) =2πℏ2ai,j nj (r) +m12Zµi µj µ0 1 − 3 cos2 θnj (r) dr .
4π
|r|3

3.13

Note that, a12 = a21 indicates the interspecies scattering length which is unknown in erbiumdysprosium mixtures. Here, we did not consider the case of spinor condensate in which theparticle number is not conserved.

3.1.3 Solutions to the Gross-Pitaevskii equation
Although the study of systems described by Eq. 3.12 might be complicated, we review two
simpler cases: one-component dipolar BEC and two-component contact-interacting BECs.
The heteronuclear dipolar BEC case is only mentioned for completeness due to the intrinsic
difficulties of the methods required for achieving numerical solutions, going beyond the scopeof this thesis. For a more systematic treatment, the reader is encouraged to follow thereferences highlighted in the respective paragraph.
Homonuclear dipolar system
Let’s consider the time-independent GPE in the TF approximation leading to Eq. 3.11.
For simplicity, we assume the BEC to be confined in a cigar-shaped trap with radial (ρ)

3.1. THEORY OF QUANTUM MIXTURE

35

ωx = ωy = ω⊥ and axial (z) ω∥ confinement and dipoles oriented along the z-axis. The
condensate’s profile deviates from the typical Gaussian shape of the non-interacting gas toa quadratic shapen(r) = n0

ρ2z2
1− 2 − 2
Rρ Rz

!

15 N
=
8π Rρ2 Rzρ2z2
1− 2 − 2
Rρ Rz

!

3.14

where Rρ and Rz are the TF radii and n0 the peak density. The profile of the BEC ismodified due to the influence of the DDI term over the contact one and Rρ = κRz differsfrom ω∥ = λω⊥ where the cloud’s aspect ratio κ and the trap’s aspect ratio λ follow thecondition#!
"
fdip (κ)λ22+ (ϵdd − 1)(κ2 − λ2 ) = 0.
3.15
+13κ ϵdd2
1 − κ2
The function

√

2
2
1 + 2κ2 3κ arctan 1 − κ
fdip (κ) =−1 − κ2(1 − κ2 )3/2

3.16

is called dipolar anisotropic function and allows us to write the chemical potential asµc = gn0 [1 − ϵdd fdip (κ)]. The TF radii are then written as
15gN κ
2 1 + ϵdd
4πmω⊥
"

Rρ =

3 κ2 fdip (κ)−12 1 − κ2!#!1/51
Rz = Rρκ

3.17

and one of the typical effects arising from this is the so-called magnetostriction [Stu07]. In
this picture, a dipolar condensate is stable when the TF energy per particle is positive
E
1
λ2
2 2
= mω⊥
Rρ 2 + 2
N
14
κ

!

+

15ℏ2 aN(1 − ϵdd fdip (κ))
7m Rρ2 Rz

3.18

and one of the most known effects is the d-wave collapse reported for the first time in a BEC
of Cr atoms [Lah08].
Heteronuclear contact-interacting system
In the case of purely contact interacting BECs, we can neglect the DDI term in the GPE
leading to:µc,1 = Vext,1 (r) + g1 n1 (r) + g1,2 n2 (r)µc,2 = Vext,2 (r) + g2 n2 (r) + g1,2 n1 (r).

3.19

If the gas is homogeneous and the densities ni of the two components are constant, thestability is obtained when the energy increases under densities’ deviations. Such condition
can be put down in formula as
∂µc,1
> 0,
∂n1

∂µc,2
> 0,
∂n2

and

∂µc,1 ∂µc,2 ∂µc,1 ∂µc,2−> 0.
∂n1 ∂n2∂n2 ∂n1

3.20

36

CHAPTER 3. QUANTUM MIXTURES

If the gases are trapped in isotropic and harmonic potentials for which Vext,2 = λVext,1 andλ = m2 ω22/m1 ω12 , the density profiles can be derived to be as
µc,1
1
r2
g1,2
µc,2 g1,2
n1 (r) =− 2 1−λ1−2
gg1 1 − 1,2/g1 g2µc,1 g2
g2
R1
"



#

1
µc,1 g1,2
r2
1 g1,2
µc,2
1
−
−
1−
n2 (r) =2
2
g2 1 − g1,2/g1 g2µc,2 g1
λ g1
R2
"



#

3.21

where the length Ri is defined by µc,i = mi ωi2 Ri2/2. Depending on the interplay between
interspecies and intraspecies interactions, the two gases might merge in some region of spaceand might not in others. In such formalism, we recover the one-component case when oneof the species is absent. If one of the two components is a minority, it can be treated likean impurity. Such an impurity moves either to the center or to the surface of the majority
species depending on g1,2 .
Heteronuclear dipolar system
The coupled GPEs problem with magnetic species (Eq. 3.12) is rather non-trivial and it issolvable only numerically. The interest in such a system started to rise recently and the
exploration of a variety of phenomena just began. Among these, theoretical investigationon the miscibility of two-components BECs [Gac10, Wil12, Kum17a], on binary magnetic
quantum droplets [Smi20], and on vortices and vortex lattices [Zha15, Kum17b, Mar17] hasbeen done. In Refs. [Kai14, PA18, Wen18], the authors follow an impurity-physics approachto study properties of dipolar polarons.

3.2 Mixture experiments overview
In the introduction of this chapter we saw how single-component and multi-component ultracold gases are a valuable platform to approach physics on many different levels. Since
their prediction in the 1920’s, it took about 70 years of technical development to observe
the first BEC [And95, Dav95]. Subsequent to such milestone, many other elements werebrought to degeneracy from several groups around the world and many experiments withdifferent degrees of complexity are spreading new light on the topic every year. Up tonow, a total of 13 elements were brought to degeneracy, either BEC or DFG. Listed inorder of appearance on the periodic table, they are: hydrogen [Fri98], helium [Rob01,
PDS01, McN06], lithium [Bra95, Tru01, Sch01], sodium [Dav95], potassium [DeM99, Mod01],
rubidium [And95], cesium [Web03], calcium [Kra09], strontium [Ste09, dE09, DeS10, Tey10],chromium [Gri05, Nay15], dysprosium [Lu11a, Lu12], erbium [Aik12, Aik14b],
thulium [Dav20], and ytterbium [Tak03, Fuk07].
Over the years, ultracold atoms experiments have advanced so far that single-component
quantum gases can be controlled and probed down to individual atom precision (e. g. single
atom detection and quantum gas microscopy [Ott16, Kuh16]). From this point of view,experiments moved towards more complex objects and the focus has shifted to the creation ofmulti-component mixtures. Whether the final goal is the research performed on independent3.2. MIXTURE EXPERIMENTS OVERVIEW

37

Table 3.1: Dipole moments and dipolar lengths for different atomic and molecular species. Dipole
moments are in unit of µB for magnetic dipoles and D for electric dipoles. The dipolar length in unitsof a0 allows for a direct comparison across species. Values from Ref. [Gad16].

Species

Dipole momentadd (a0 )87 Rb

0.5 µB
6 µB
10 µB
7 µB
14 µB
0.57 D
1.2 D
2.7 D
3.3 D
1.9 D
3.5 D
4.1 D
5.5 D
1.5 D
0.21 D<0.1 D
1.7 D
10.4 D

0.18
15
130
67
533
3.9 × 103
3 × 104
4.4 × 104
1.1 × 105
6 × 104
5.3 × 104
2 × 105
4 × 105
3.9 × 104
1.1 × 103<200
4.5 × 103
1.2 × 10652 Cr

164 Dy168 Er168 Er

2

KRb
RbCs
NaK
NaRb
KCs
LiK
LiRb
LiCs
RbSr
RbYb
LiYb
∗ OH
KBr

components or associating them in ultracold molecules (eventually by magnetoassociation
over Feshbach resonances, see Chap. 4), the scientific interest and the techniques used arewidely shared. Often, these two directions of investigation move together and they can be
two parts of the same project. It has to be kept in mind that, in most of the cases, knowledge
about the mixture’s scattering properties is limited to theoretical prediction or completely
absent.
Despite single-component combinations of different spin states or different isotopes [Mya97,
Ste98, Tru01, Sch01], considerable importance has been given to heteronuclear mixtures forthe possibility of further enriching the properties of the system. In this section, we review the
progress achieved on alkali-alkali (Sec. 3.2.1) and alkali-nonalkali (Sec. 3.2.2) experiments.
In Sec. 3.2.3, our erbium-dysprosium system is introduced.

3.2.1 Alkali-alkali mixtures
Following the earliest advances and considering the extensive knowledge on single-species alkali systems, the most straightforward candidates to realize more complex ultracold objects
were heteronuclear mixtures of alkali. Such mixtures offer a wide spectrum of possibilities interms of masses (from 6 Li to 133 Cs), statistics (B-B, B-F, and F-F), and properties (e. g. electronegativity). Note that the common approach involves the combination of two species but apioneering work proposed in Ref. [Tag08] demonstrated the possibility to extend the method
to three species (Rb-K-Li).

38

CHAPTER 3. QUANTUM MIXTURES

Nowadays, nine out of the ten possible combinations of two alkali have been brought
to quantum degeneracy. Among these we enumerate: Li-Na [Had02], Li-K [Spi10, Wu17],
Li-Rb [Sil05], Li-Cs [DeS17], Na-K [Par12, Sch18], Na-Rb [Wan15], K-Rb [Mod02, Roa02,
Wac15], K-Cs [Grö16], and Rb-Cs [McC11, Ler11]. The remaining combination of Na andCs was never combined in a degenerate regime but it has been used recently to create coldpolar molecules in tweezers [Liu18].
Bialkali quantum gases offer a powerful tool to study phenomena arising from populationand mass-imbalance: impurity physics and polaron, pairing beyond the standard BCS theory
and superconductivity, BEC-BCS crossover, non-trivial Efimov physics, magnetic orderingand itinerant/localized ferromagnetism, and many more [Blo08, Blu12, Mas14].
The scientific interest is strongly driven by the possibility to create ground-state moleculeswith large electric dipole moments [Boh17]. For a comparison of the dipolar strengths
across such species, see Tab. 3.1. However, reaching quantum degeneracy in gases of diatomic molecules is particularly challenging due to low densities and high losses in the sample. So far deeply bound state molecules in the ultracold regime have been produced in
Na-K [Par15, See18, Vog20], Rb-K [Ni08], Rb-Cs [Tak14, Mol14], Na-Rb [Guo16], and
Li-Na [Rva17]. After many years of research, in Ref. [DM19], the authors observed a degenerate sample of fermionic polar molecules of 40 K-87 Rb for the first time.

3.2.2 Alkali-nonalkali mixtures
At the same time of the achievement of the first non-alkali quantum gas, a rising interest in
the field led to experimental and theoretical efforts to combine chemically different elements.
In this direction, the most promising mixtures include alkali together with alkali-earth (RbSr [Pas13] and Li-Sr [Ye20]), transitions metal (Li-Cr [Ner20]), or lanthanides (Li-Yb [Han11,
Har11], Rb-Yb [Vai15], Cs-Yb [Gut18], and K-Dy [Rav18b]). Two additional experiments
are being set up: Li-Er in Tübingen and Li-Dy in China.
These mixtures offer a wide variety of statistics: Sr, Cr, Dy, Er, and Yb have several bosonic and fermionic isotopes with a high natural abundance. The mass scalability is great as well, covering a broad range of values from a mass-ratio of about one for
Rb-Sr to the Li-Yb case where Efimov physics can be studied in the extreme regime ofmYb/mLi ≃ 29 [Bra06, End11].
Generally speaking, heteronuclear mixtures of atoms with diverse properties extend thepossibilities in designing Hamiltonians to simulate quantum many-body and few-body problems, testing of fundamental symmetries and creating scalable platforms for quantum information. In addition to the applications of heteronuclear systems explained in the previous
Sec. 3.2.1, going beyond bialkali mixtures allows other fields to be explored. They are goodcandidates for creating ground-state molecule exhibiting both electric and magnetic dipole
moments (see Tab. 3.1). Moreover, such molecules possess an unpaired electron, leading toa dense spin structure and a rich spectrum for manipulation. The scientific interest in polar
molecules is vast and reviews on the topic can be found in Ref. [Mos17, Gad16]. Among them
we find: novel superfluid phases, studies of lattice-spin models, geometrical phases, collective
spin excitations, and the investigation of fundamental questions e. g. the determination of3.2. MIXTURE EXPERIMENTS OVERVIEW

39

electron’s electric-dipole moment and proton-to-electron mass ratio.

3.2.3 Erbium-dysprosium mixture
Due to the dissimilar properties of the atomic species involved, mixture experiments are typically challenging from a scientific and technical point of view. Different melting temperaturesand vapor pressures might require a special design of the apparatus with individual lines foreach species. Different atomic transitions — thus laser wavelengths — might lead to difficulties in planning the optical setup and anti-reflection coatings. Moreover, Zeeman slower andthe magneto optical trap conditions could be inadequate for a simultaneous operation on
the two components requiring sequential loading of the species. Different polarizabilities andmasses strongly affect the confinement in optical dipole traps and the evaporative coolingmight be inefficient to reach quantum degeneracy. On top of these issues, the interspecies
scattering length is usually not accurately accessible a priori. For example, 6 Li-87 Rb has asmall background scattering length [Sil05, Deh08] leading to an inefficient sympathetic cooling, whereas in Rb-Cs or Rb-Sr, abg is large [And05, McC11, Pas13, Bar18], reducing themiscibility of the two condensates or leading to important atomic losses. Therefore, only adirect study of interspecies Feshbach resonances can give further information on the possibility of interaction tuning in the mixture to achieve good parameter regimes and experimental
conditions.
From this point of view, an Er-Dy mixture defines an exceptionally favorable starting
point for a two-species experiment. The technical knowledge on the two species required bya quantum gas experiment is well established [Lu11a, Lu12, Aik12]. Moreover, the physicalproperties of erbium and dysprosium are similar enough to allow a fairly simple experimental
apparatus with a dual-species oven, a ZS optimized for both components and full simultaneous operation (see Chap. 2). Worth to be noted is the similarity in atomic polarizabilities at1064 nm resulting in a comparable trapping potential seen by the two species into the ODT(see Eq. 2.3 in Sec. 2.1.2) [Gri00]. Only towards the end of the evaporation, the small difference (of about 6 %) plays an important role, leading to sympathetic cooling of Dy throughEr.
A more subtle issue regards the interspecies interaction properties of Er and Dy. Experimental values are not available and predictions are highly complicated (see Chap. 4).
In such species, even the intraspecies background scattering length is under debate for thedifficulties in identifying a real background value [FB18, Luc18]. Nevertheless, by tuning
the magnetic field to the mG precision, we reach double Bose-Einstein condensation withfive isotopic combinations and one Bose-Fermi degenerate mixture of 168 Er-161 Dy. Only the
Bose-Bose 166 Er-162 Dy mixture has revealed not to be adequate for the condensation in themagnetic field 0.1 G-5 G, probably due to a non optimal intraspecies vs. interspecies interactions regime. Further investigations might find a more favorable value of magnetic field toenhance evaporation efficiency. In the following Sec. 3.3, the experimental sequence from acold thermal sample down to degeneracy is presented in more details. Additional information
about the setup and an in-depth view on the evaporation scheme is part of Ref. [Ilz20].

3.3 Publication:
Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms†
Arno Trautmann1 , Philipp Ilzhöfer1,2 , Gianmaria Durastante1,2 , Claudia Politi1 ,Maximilian Sohmen1,2 , Manfred J. Mark1,2 , and Francesca Ferlaino1,2
1Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
2Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria

Physical Review Letters 121, 213601 (2018)
submitted 20 Jul 2018; published 21 Nov 2018
DOI: 10.1103/PhysRevLett.121.213601
Editors’ suggestion
Featured in Physics 11, s135. Synopsis: Making Mixtures of Magnetic Condensates†

The author of the present thesis helped in developing the experimental procedure, taking the data, andcontributed in writing the manuscript. This publication is also part of the PhD thesis of Philipp Ilzhöfer.

41

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

Featured in Physics

Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms
1

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

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

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

(b)

(c)

(d)

(e)

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

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

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

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

(a)

(b)

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

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

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

(b)

(c)

(d)

(b)

(e)
(d)
(e)

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

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

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

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

*
These authors contributed equally to this work.
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Coherent Interaction of a Single Fermion with a Small
Bosonic Field, Phys. Rev. Lett. 106, 115305 (2011).
[23] J. Heinze, S. Götze, J. S. Krauser, B. Hundt, N. Fläschner,
D.-S. Lühmann, C. Becker, and K. Sengstock, Multiband
Spectroscopy of Ultracold Fermions: Observation of Reduced Tunneling in Attractive Bose-Fermi Mixtures, Phys.
Rev. Lett. 107, 135303 (2011).
[24] N. Spethmann, F. Kindermann, S. John, C. Weber, D.
Meschede, and A. Widera, Dynamics of Single Neutral
Impurity Atoms Immersed in an Ultracold Gas, Phys. Rev.
Lett. 109, 235301 (2012).
[25] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M.
Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, andJ. J. Arlt, Observation of Attractive and Repulsive Polarons
in a Bose-Einstein Condensate, Phys. Rev. Lett. 117,055302 (2016).
[26] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A.
Cornell, and D. S. Jin, Bose Polarons in the Strongly
Interacting Regime, Phys. Rev. Lett. 117, 055301 (2016).
[27] T. Rentrop, A. Trautmann, F. A. Olivares, F. Jendrzejewski,
A. Komnik, and M. K. Oberthaler, Observation of the
Phononic Lamb Shift with a Synthetic Vacuum, Phys.
Rev. X 6, 041041 (2016).
[28] G. Gligorić, A. Maluckov, M. Stepić, L. Hadžievski, and
B. A. Malomed, Transition to miscibility in linearly coupledbinary dipolar Bose-Einstein condensates, Phys. Rev. A 82,
033624 (2010).
[29] R. K. Kumar, P. Muruganandam, L. Tomio, and A. Gammal,
Miscibility in coupled dipolar and non-dipolar Boseeinsteincondensates, J. Phys. Commun. 1, 035012 (2017).
[30] R. M. Wilson, C. Ticknor, J. L. Bohn, and E. Timmermans,
Roton immiscibility in a two-component dipolar Bose gas,
Phys. Rev. A 86, 033606 (2012).

213601-5

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

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

213601-6

Chapter4

Feshbach resonances

Since the first theoretical investigation in the 1930s [Ric33, Beu35, Fan35] and the experimental evidence in the 1970s [Spe75, Bry77], Feshbach resonances are considered part of afundamental step towards the understanding of the underlying properties of a specific system from both theoretical and experimental points of view [Fes58, Fan61, Fes62]. Firstly,
the study of Feshbach resonances gives insights in the collisional properties and it provides
very precise input for the calculation of interaction potentials [Tim99]. Secondly, and mostimportantly for the ultracold atoms community, Feshbach resonances are effective tools totune the scattering length, thus the interaction between atoms [Chi10]. It becomes possible
to investigate a wide range of interaction regimes, going from strongly attractive to noninteracting to strongly repulsive regimes. Few-body and many-body scenarios of particularinterest become also accessible.
Feshbach resonances (FR) are important in several research branches such as nuclear,atomic, molecular, and chemical physics [Chi10, K0̈6]. This thesis addresses the problem
from the ultracold atoms’ perspective. Here, particles in the dilute gas are close to the zeroenergy scattering regime. This allows a simpler description of the phenomenon of FRs. As amatter of fact, the two-body scattering in this regime can be described in terms of a single
parameter, a, called s-wave scattering length [Pet08]. While in single valence-electron species(alkali) methods to calculate as are well established, predictive scattering models for twoand multi-electron atoms are still under development and more approximations are typicallyneeded. In the case of complex atomic species such as lanthanides, the understanding is stillrather superficial because of the required treatment is highly non-trivial, involving manypotentials and complex interactions [Kot14]. The development of ultracold lanthanide-based
experiments calls for more theoretical input.
In this chapter, the first experimental observations of Feshbach resonances in an ultracold
atomic mixture of erbium and dysprosium are presented. The initial sections introduce thereader to the theory behind Feshbach resonances and the state-of-the-art knowledge in thecase of single-species and multi-species alkali systems (Sec. 4.1 and Sec. 4.2, respectively).
After that, in Sec. 4.3, the complex topic of FRs in lanthanides is discussed in more detail.
Sec. 4.4 includes the publication Feshbach Resonances in an Erbium-Dysprosium Dipolar
Mixture [Dur20]. The final Sec. 4.5, is dedicated to the presentation of additional informationand measurements collected during the development of the previously-mentioned publication.
In Appendix B, all the Feshbach scans performed in search of broad FRs are collected.
49

50

CHAPTER 4. FESHBACH RESONANCES

4.1 Theory of Feshbach resonances
The understanding of Feshbach resonances has its roots in the two- and three-body scattering
processes between atoms. In this section we develop the topic of FR with a treatment basedon the one presented in Refs. [K0̈6, Chi10, Kot14]. Generally speaking, collisions can be either
elastic or inelastic. In the first case, the relative kinetic energy of two particles undergoinga scattering process does not change, while the kinetic energy of the individual atoms canchange. In the second case, colliding particles can change their internal state during thescattering process. While elastic collisions are important for thermalization and evaporative
cooling, inelastic collisions typically release enough energy to cause a loss of atoms from thetrap. In the following description we neglect inelastic collisions.

4.1.1 Ultracold collision
Imagine two point-like particles of masses m1 and m2 colliding in an isotropic potential V (r)with r being the interatomic distance. The separated atoms can be initially described by
non-interacting plane waves. The plane waves can be re-written following a partial wave expansion by a sum over spherical harmonic Ylml (r̂), where l is the relative angular momentum
and ml is its projection along a fixed axis. In this picture, since there is no coupling among
them, each of these partial waves ψl (r) = ϕl (r)/r is a solution of the Schrödinger equation
Hϕl (r) = Eϕl (r).

4.1

Here the Hamiltonian is written as the sum of the relative kinetic energy and the interaction
potential asℏ2 d2ℏ2
H = Hk + V (r) = −+l(l + 1) + V (r)4.2
2m12 dr2 2m12 r2
where m12 is the reduced mass of the pair and the second term accounts for the centrifugal
potential. In the case of l ̸= 0, an energy barrier is added to the potential with the result ofpreventing atoms with low energy to reach the scattering center. In the following, we use thegeneral convention where the rotational states l = 0, 1, 2, . . . are called s, p, d, . . . waves. Ifr→∞we assume that V (r) −−−→ 0, every collisional channel can be expressed by its wave functionr→∞ϕl (R, E) −−−→ csin(kR − πl/2 + ηl (E)) iηl (E)
√
e
k

4.3

with k being the wavevector of the incoming particle, c = 2m12/(πℏ2 ) a normalization constant, and ηl a parameter describing the phase shift introduced by the scattering process
and strongly dependent on the potential itself. Such phase shift plays a fundamental roleenclosing the effect of the potential on the scattering event.
p

In ultracold atoms experiments, in which the relative momenta are very small, the leading scattering channel is typically the s-wave collision, and all higher partial waves can besafely neglected (this is not valid anymore in case of dipolar or anisotropic van der Waals
interactions where all partial waves can contribute). The phase shift parameter η0 assumes
the form
1 1
k cot η0 (E) = − + r0 k 2 ,4.4
a 2

4.1. THEORY OF FESHBACH RESONANCES

51

where r0 is the extension of the potential and a is the s-wave scattering length. In thezero-energy limit (k → 0), a further approximation leads toa = − lim

k→0tan η0 (E)
.
k

4.5

The parameter η0 can take any value from 0 to 2π depending on the potential, thus thescattering length a can vary from −∞ to +∞. a is typically given in units of the Bohrradius a0 ≃ 0.53 Å.

4.1.2 Open- and closed-channel picture
To describe the basis of Feshbach resonances, we assume the existence of at least two molecular potentials. The first one is associated with the asymptotically free atoms with colliding
energy close to the threshold and, from here on, such background potential Vbg (r) will becalled entrance channel, because it represents the energetically open channel. Nevertheless,this picture can change in the case of coupling from the free-body states to bound-states. The
second potential Vc (r), in which the energy of the colliding atoms is less than its asymptotic
value, is called closed channel and supports near-threshold molecular bound-states. If one ofthese bound-states in the closed channel becomes energetically resonant with the asymptotic
energy of the scattering state in the open channel, a small coupling can strongly impact
the scattering length mixing the two channels and leading to a divergence of a. We thenspeak about Feshbach resonance. A pictorial representation of the open- and closed-channel
picture is sketched in Fig. 4.1.
The separated atoms and the bound state might have different magnetic momentsδµ = µatoms − µc . If this is the case, the energy of the bound state in the closed channel Ec can be tuned relatively to the free-body open channel one by tuning the magnetic
field B such as Ec = δµ(B −Bc ). This case is know as magnetic tunable Feshbach resonances.
Here, we notice that, away from resonance, the bound-state energy is linear in B with theslope given by δµ.
Now, let’s consider the scattering phase shift η. Its background value ηbg is modified by
the presence of a coupling between the two channels leading to an additional resonant term
η(E) = ηbg (E) + ηres (E). Without deriving the full theory, in the limit of small k, one canwrite the scattering length asa(B) = abg + ares (B)
= abg

Γ0
1+
−E0 + i(γ/2)





4.6

where Γ0 and E0 are energy-dependent constants accounting for the width and the energythreshold resonance position of the resonance, respectively. γ takes into account the decayrate of the bound state into all possible loss channels. abg represent the background (offresonant) scattering length due to Vbg (r). In the case of magnetic Feshbach resonances,
keeping in mind the previous paragraph, we can rewrite Eq. 4.6 in the famous expressiona(B) = abg

∆1−
B − B0



4.7

52

CHAPTER 4. FESHBACH RESONANCESc

Figure 4.1: Basic description of Feshbach resonance in the two-channels model. (a) Pictorial representation of the entrance and closed channel (black and blue curve, respectively). In a magnetic FR,
the energy Ec (B) of a molecular bound state can be varied to resonantly couple the two channels bytuning the magnetic field. (b) Near-resonant behavior of the interparticle scattering length as (redcurve). The FR can be described by its position B0 and its width ∆. The width is defined as theseparation between the pole and the value of magnetic field Bzero for which as = 0 (also called zerocrossing). Far from the pole of the FR, as settle to the background scattering length abg . (c) At B0 ,where as is very large, the binding energy Eb follows a quadratic dependence typical of the universal
regime. Figure adapted from Ref. [Bai18b].

where ∆ = Γ0/δµ and B0 = Bc + δB are the width and the pole respectively and δB = −δE/δµ.
It is interesting to note that in the vicinity of the resonance’s pole where a ≫ 1, thebound-state energy bends with a quadratic dependence on the scattering length
Eb =

ℏ2
.
2m12 a2

4.8

This region of interaction is often known as universal regime and the state as halo state
which in turn is a very weak bound case of a molecule [Rii94, Jen04]. The wave function ofsuch state extends spatially on the order of a, beyond the turning point of the potential. In
Fig. 4.1, the typical near-resonant behavior of a and Eb is depicted.

4.1.3 Resonance strength
In the ultracold regime, for a large distance of the colliding particles, scattering events ofneutral atoms can be understood in good approximation by imagining the central potential
to be of the van der Waals (vdW) type V (r) = −C6/r6 with C6 being a coefficient. At shortdistances, the potential becomes repulsive and could be modeled as a hard-core repulsion
potential. We can define a length scale for the vdW potential
1
RvdW =
2

2m12 C6
ℏ2

1/4

.

4.9

4.1. THEORY OF FESHBACH RESONANCES

53

To simplify the physical interpretation, one can re-write this length scale as a mean scattering
length ā ≈ 0.956RvdW which defines a corresponding energy scale Ē = ℏ2/(2m12 ā2 ). In theseunits, it is possible to extract from Eq. 4.6 a dimensionless parameter associated with theresonance strengthabg δµ∆sres =.
4.10
ā Ē
If sres ≫ 1, the coupling between open and closed channel is strong and the resonance is said
to be “broad”. These types of resonances are also called open channel dominated because they
have the spin character of the entrance channel over a broad range of the width. Contrary tothat, when sres ≪ 1, the coupling is weak resulting in a “narrow” resonance which is labeled
as close channel dominated. Open-channel dominated resonances also feature an universal
regime which is much broader than the close-channel dominated case.

4.1.4 Hamiltonian
For the following development towards the complexity of lanthanides, it is necessary to do a
step forward and write down a more complete Hamiltonian describing the scattering event
and accounting for different interaction processes. In the following, s is the electronic spin, lthe electronic orbital angular momentum coupled to the total electronic angular momentum
j = s + l. Then, calling i the nuclear spin, the total angular momentum will be f = j + i.
mf and ml are the quantum numbers accounting for the projection of f and l (relative
angular momentum) on an external magnetic field. For two atoms (labeled as 1 and 2 ), thetwo-atoms Hamiltonian can be written as
H12 = Hk + Hhf + HZ + Hel + Hss

4.11

where Hk encloses the kinetic terms in the form of Eq. 4.2, Hhf is the hyperfine interaction,
HZ accounts for Zeeman splitting, Hel encloses the electronic interaction, and Hss is the
spin-spin interaction. Excluding the kinetic term, which was already discussed earlier in thechapter, the other terms can be summarized as follows.
The hyperfine interaction couples s to i by
Hhf =1
(ahf,1 i1 · s1 + ahf,2 i2 · s2 )
ℏ2

4.12

with ahf being the hyperfine constant.
The Zeeman term accounts for the energy-level shift of the atoms in a magnetic field B
by
µB
HZ =
B(gs s1 + gs s2 + gi,1 i1 + gi,2 i2 )4.13ℏ
where µB is the Bohr magneton and gs and gi are the Landé factors for electrons and nuclei.
The electronic interaction term includes the exchange term (Coulomb potential) and vdW
potential Hel = Hex + HvdW . The first one arises from the antisymmetry of the electronic
wavefunction and can be written as−r/rexHex = Cex e1+ 2s1 s2
2



4.14

54

CHAPTER 4. FESHBACH RESONANCES

with Cex and rex being constants. The vdW potential was already discussed in Sec. 4.1.3
and follows a dependence V (r) = −C6/r6 . Hel typically induces broad FR and it is diagonal
in l and ml , thus only even or odd partial waves are coupled together.
The last term in Eq. 4.11, encloses magnetic interactions and higher-orders spin-orbit
couplings. The magnetic interaction, here called dipole-dipole interaction, depends on therelative orientation of electronic spins and atomic axes, therefore it leads to anisotropy. The
higher orders spin-orbit coupling (here we just concentrate on the second order) can play animportant role on heavy atoms like Cs and it shows the same spin dependence as the DDI
with a different scaling law. Together they assume the shape11r
Hss = − α2 3 + Cso e /rso (s1 (3rr − 1)s2 )
2
r




4.15

where α is the fine structure constant, Cso and rso are constants. Hss instead causes narrow
FR to appear. Note that Hss is not diagonal in l, leading to coupling between different
partial waves (off-diagonal terms).
Another important aspect in ultracold scattering relates to symmetry arguments on the
wavefunctions of the atoms. Identical bosons (fermions) can collide only in even (odd)partial waves. In the case of collisions between nonidentical particles, all the values of l arein principle possible. Since all these models are highly sensitive on small variations of theinteraction potentials, ideally one would need to know the exact shape of all of them. Such
potentials are usually not directly accessible and a digression on their numerical calculation
would require a treatment going beyond this thesis. Good approximations are reached insimple systems like alkali and alkali-alkali mixtures where Hss is usually small and can beneglected, and the number of Born-Oppenheimer potentials is limited to a few. In the caseof complex elements supporting a large number of bound states and/or introducing a strong
Hss contribution (like strongly magnetic lanthanides), the calculations become involved andpredictions are hardly possible [Kot14]. As we will see later in Sec. 4.3, solving this scattering
problem is highly non-trivial for alkali and even more for lanthanides.

4.2 Experimental aspects
In quantum gas experiments, Feshbach resonances are of high importance to tune the interactions between particles and, in some cases, even to attain quantum degeneracy at all.
Pioneering observations of FRs in ultracold atoms were reported firstly in hydrogen [Rey86]
and, at a latter time, in Na and Rb [Ino98, Cou98]. A vast number of phenomena depend on
the existence of available FR: strongly and non-interacting systems, collapse and topologicaldefects, molecule formation via Feshbach association, BEC-BCS crossover, Efimov physics,
new phases of matter, and many more [Chi10]. In this section, we review some important
experimental evidence connected to the leading role of FR in the ultracold atoms community. This contextualizes the current state-of-the-art with future steps that can be done on
this topic. Firstly, we introduce the case of homonuclear alkali systems in, then we move to
mixtures of alkali, and finally we discuss the last advances on more complex atomic species.
A digression on the particular case of lanthanides is postponed to Sec. 4.3.

4.2. EXPERIMENTAL ASPECTS

55

4.2.1 Inelastic collisions and trap-loss spectroscopy
In the previous theoretical approach, we considered the ultracold scattering to be dominatedby elastic collisions. Nevertheless, an experimental tool used to investigate FR involves
trap-loss spectroscopy methods which are related to inelastic processes. Such methods werealready employed from the very first observations of FR in ultracold atom experiments [Ino98,
Rob00, Web03]. Inelastic losses can be due to one-, two-, and three-body losses (α, L2 , andL3 are the respective coefficients). In this picture, the time evolution of the trapped atom
number Ṅ is expressed by the following equation:Ṅ (t) = −αN (t) −Z

L2 n (r, t) d r −23

ZL3 n3 (r, t) d3 r

4.16

where n is the atomic density distribution. Note that the one-body decay is usually connectedto the lifetime of the sample due to background-gas collisions. L2 and L3 are enhanced near
FR and, in the particular case of bosons, L2 follows a Lorentzian profile while L3 ∝ a4 . In
the case of fermions, the treatment is more complicated.
Despite their shapes, FRs are generally identified by increased atom losses with the maximum loss at the location of the resonance pole. Such a technique is extensively used in our
experiment to identify and characterize FRs (see Sec. 4.4 and Appendix B).

4.2.2 Alkali
Nowadays, FRs are observed and employed in all the homonuclear alkali-metal systems(excluding francium) [Chi15]. Properties and appearance of FRs are strongly dependent
on the atomic species and, even within the same element, every isotope shows dramaticdifferences. The direct connection between theory and experiment is intriguing: starting fromthe vast knowledge on alkali elements and the vast amount of spectroscopy data, theoretical
calculations are possible and they are used to predict properties of FRs. In particular,
between different isotopes of the same species, corrections due to mass scaling are easily
implemented [Bur98, vK02]. Vice versa, their experimental observation and characterization
have a direct impact on the development of interatomic potential modeling [Set00, vK04,
Tie20].
Even though some alkali possess good collisional properties by themselves (23 Na, 41 K, and

87 Rb), interaction tuning via resonances is essential for Bose-Einstein condensation of 7 Li,

39 K, 85 Rb, and 133 Cs. Generally speaking, alkali have sparse Feshbach spectra if comparedto the more complex atomic species (in the lighter elements, FRs can be hundreds of Gauss
apart). The width of such resonances are also well distributed, varying from few mG tohundreds of G.
In Fermi gases and BECs of alkali-metal elements, FRs were used to successfully associate
atoms into weakly-bound dimers for the production of molecular BEC [K0̈6].

56

CHAPTER 4. FESHBACH RESONANCES

4.2.3 Mixtures and complex systems
After alkali, which were the first species to be investigated, a growing attention has been
focused on other species: alkaline-earth metals, transition metals, lanthanides, and various
combinations of all these atomic species. The case of lanthanides will be discussed in thenext Sec. 4.3.
A natural step towards an increased complexity of the system has been done by combining
different alkali species in mixture experiments. Starting from K-Rb [Ino04, Fer06], the first
mixture investigated to locate Feshbach resonances, observations have been reported for
Li-Na [Sta04], Li-K [Wil08], Li-Cs [Rep13], Li-Rb [Deh08], Na-K [Par12], Rb-Na [Wan13],
and Rb-Cs [Pil09]. There are no experimental observations for the case of Na-Cs. Isotopic
mixtures of homonuclear Rb, K, and Li were also investigated and FRs characterized [Zha05,
Pap06, Wu11, Tan18]. In the specific case of Li and K, both fermionic and bosonic isotopes
are available leading to the possibility of interaction tuning in Bose-Fermi isotopic mixtures.
Strontium (like Ytterbium) has no electronic magnetic dipole moment due to the antiparallel alignment of the two valence-electron spin. As a consequence, Sr presents no magnetic
resonances (optical FRs are possible [Bla11]). This problem does not arise in the case of
mixtures of strontium with alkali atoms in which Feshbach resonances have been observed(see Sr-Rb in Ref. [Bar18]).
Chromium has been the first atom with more than one valence electron in which FRs
were observed [Wer05]. In this element, the DDI term comes into play and calculations canbe performed only by including such a contribution into the Hamiltonian. Indeed, in 52 Cr
the resonances are due to the spin-spin interaction [Kot14]. Moreover, by using interaction
tuning to suppress the isotropic contact interaction, it was possible to enter regimes in which
DDI dominates the gas dynamics [Lah07, Koc08]. Quite recently, the creation of a cold gas of
Cr-Li was announced, but no investigation on the Feshbach properties was conduced [Ner20].

4.3 Feshbach resonances in lanthanides
This section deals with the current knowledge of Feshbach resonances in lanthanide elements.
In such systems, the vast amount of supported bound states combined with the complicatedcoupling schemes induced by the large spin-spin interaction term (see Eq. 4.11) and theanisotropy of the wave function, lead to a highly dense and intricate FR spectrum. Calculations and predictions are hardly possible, although, thanks to the great progress done on
Er and Dy during the last years, the interest in the field is increasing fast.

4.3.1 Role of anisotropy
If in the case of Cr the spin-spin interaction is roughly a factor 36 stronger than alkali atoms,
in lanthanides this factor can be even more substantial — in Er or Dy, it is about a hundred
times higher than the alkali one. In this condition, both the long- and short-range interaction
are influenced by this anisotropy and, to understand FRs at the first order, one has to consider

4.3. FESHBACH RESONANCES IN LANTHANIDES

57

Figure 4.2: Numerically calculated scattering length as a function of magnetic field for 164 Dy at
30 nK. In the top panel all anisotropic interactions are included, whereas in the middle and lower
panel the anisotropic dispersion and the dipole-dipole interactions are switched off, respectively. In
this model, only channels of even partial waves up to l = 10 are included. To observe the same density
as in the experiment, channels up to l > 40 have to be considered. Figure adapted from Ref. [Pet12]

at least the most prominent factors in the Hamiltonian (Eq. 4.11): dipole-dipole (∝ 1/r3 ),quadrupole-quadrupole (∝ 1/r5 ), and vdW dispersion (∝ 1/r6 ) interactions [Pet12, Kot14].
For the sake of clarity, it has been shown that the highest contribution in the vdW potential
comes from the standard isotropic term [Pet12]. However, in magnetic atoms the second
highest vdW contribution is due to the so called anisotropic dispersion interaction (ADI).
Without going into too much details in the calculations, by means of a coupled-channels
model, the authors in Ref. [Pet12] demonstrated that FRs in a sample of bosonic spinpolarized 164 Dy at 30 nK are purely caused by the anisotropic potentials. Fig. 4.2 shows
the result of such calculations. By excluding the anisotropic terms, the spectrum changes
drastically. However, the calculated density of resonances is much lower than the one experimentally reported. Here, the authors included channels with partial waves up to l = 10,
although an l > 40 is needed to reproduce the real density. Almost simultaneously, our group
has performed FR studies in erbium showing a number of resonances even higher than thepredicted one [Fri14]. Another limitation comes from the temperature taken into account.
In the experiments with lanthanides, a dramatic dependence of FR density on the sample
temperature has been observed [Bau14, Mai15b, Khl19]. Therefore, to achieve more realistic
results, calculations should include non-zero partial waves collisions and additional available
channels.
Despite the difficulty of carrying out calculations, qualitative studies of different lanthanide
systems were performed in Refs. [GM15, Gre16, ZK18, Fry19, Yan19, Kos20]. Nonetheless,to our knowledge, no work has been performed on heteronuclear lanthanide mixture systems.

4.3.2 Experimental observations
The first observations of FR in a lanthanides were reported by the Ferlaino’s group in gasesof bosonic Er atoms [Aik12]. Intensive Feshbach spectroscopy on both fermionic and bosonic

58

CHAPTER 4. FESHBACH RESONANCES

isotopes followed in Ref. [Fri14]. In Ref. [Bai18a], FR between different spin components offermionic erbium were observed. By using magneto-association, Feshbach molecules of Er
were created [Fri15]. Regarding Dy, preliminary observations of FR in gasses of bosonicand fermionic isotopes were performed by the Lev’s group [Bau14] followed by deeper investigations by other groups [Mai15a, Luc18]. Recently, FRs in bosonic 169 Tm atoms werereported [Khl19] together with a study on their statistic distribution. In all these systems, theexistence of several FRs located at low magnetic field is of particular experimental interest,
as it allows for precise interaction tuning and simplifying the experimental apparatus.
Ytterbium, has zero electronic angular momentum in the ground state due to the paired
electrons leading to absence of accessible and useful magnetic Feshbach resonances (beside
optical FR [Eno08]). A particular type of magnetic resonances, called orbital FR, wasobserved in a mixture of Yt atoms in different electronic states [Pag15, Cap19].
Moving outside the purely lanthanide systems, in Ref. [Rav20] the Grimm’s group reported
the first experimental observation of Feshbach resonances in a lanthanide-alkali mixture ofdysprosium-potassium. In Ref. [Gre20], FRs in a Yb-Li system has been reported.
As previously said, lanthanides possess a highly dense FR spectrum which in turn is areminiscence of the intrinsic complexity of the system. A direct effect of such complexity is the so called quantum chaos. It has been shown that FR spectra of lanthanides
follow a statistically chaotic distribution well described with the use of random-matrix theory [Fri14, Mai15b, Khl19]. In particular, the nearest-neighbor spacing between FRs resembles a Wigner-Dyson distribution (typical of chaotic systems), characterizing stronglyinteracting levels and level repulsion. It has also been proposed that chaos is more generally
the result of systems with multiple degrees of freedom which are mixed together [Wei09].
In Refs. [Mai15b, Acv18], the authors demonstrate that this chaotic behavior in the FRs
spectrum is generated by the substantial off-diagonal coupling arising from the DDI and
ADI.
Worth to be noticed is also the strong dependence of FR spectra on the sample temperature. The FR density in erbium increases of 25 % when the temperature grows from 350 nK
to 1.4 µK [Mai15b] and by ∼50 % in dysprosium in response to a doubling of the sample
temperature [Bau14]. A similar factor is reported in thulium to a six-times temperature increase [Khl19]. This behavior is due to the increasing contribution of different partial wavesof the entrance channels for increasing temperature [Mai15b], motivating the speculation
in Ref. [Kot11]. In particular, resonances with l > 0 entrance channels are suppressed bylowering the temperature.
Up to now, no experiments or theories covered the topic of FRs in mixture of different lanthanide species such as erbium-dysprosium mixtures. In Sec. 4.4 we present the observationof several FRs in different isotopic combinations of Er-Dy including Bose-Bose, Bose-Fermi,
and Fermi-Fermi mixtures. In Appendix B, the Feshbach spectra for such samples are collected and shown.

4.4 Publication:
Feshbach Resonances in an Erbium-Dysprosium Dipolar
Mixture†Gianmaria Durastante1,2 , Claudia Politi1 , Maximilian Sohmen1,2 , Philipp Ilzhöfer1 ,Manfred J. Mark1,2 , Matthew A. Norcia1 , and Francesca Ferlaino1,2
1Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
2Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria

Physical Review A — accepted, in press
submitted 11 Jun 2020
URL: arXiv:2006.06456†

The author of the present thesis developed the experimental procedure, performed the measurements,
analyzed the data, and wrote the manuscript with the help of the other authors. Particular support in thedata analysis and writing phase came from C.P. and M.N.

59

Feshbach Resonances in an Erbium-Dysprosium Dipolar Mixture
Gianmaria Durastante,1, 2 Claudia Politi,1 Maximilian Sohmen,1 Philipp
Ilzhfer,1 Manfred J. Mark,1, 2 Matthew A. Norcia,1 and Francesca Ferlaino1, 2
1

arXiv:2006.06456v1 [cond-mat.quant-gas] 11 Jun 2020

Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut für Experimentalphysik und Zentrum für Quantenoptik,
Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria(Dated: June 12, 2020)
We report on the observation of heteronuclear magnetic Feshbach resonances in several isotope
mixtures of the highly magnetic elements erbium and dysprosium. Among many narrow features,
we identify two resonances with a width greater than one Gauss. We characterize one of theseresonances, in a mixture of 168 Er and 164 Dy, in terms of loss rates and elastic cross section, andobserve a temperature dependence of the on-resonance loss rate suggestive of a universal scaling
associated with broad resonances. Our observations hold promise for the use of such a resonancefor tuning the interspecies scattering properties in a dipolar mixture. We further compare theprevalence of narrow resonances in an 166 Er-164 Dy mixture to the single-species case, and observe
an increased density of resonances in the mixture.

Ultracold quantum gases are a highly successful platform for physics research largely because it is possible tocreate simplified and controllable versions of condensed
matter systems [1]. As the field has advanced, great
progress has been made by reintroducing complexity in a
carefully controlled manner. This complexity can manifest in the form of interparticle interactions [2–4], thespecies and statistics of the particle under study [5–7], orin the form of the potential landscape, control protocols
and imaging techniques applied to the system [8, 9]. In
this work, we explore interspecies Feshbach resonances asa means of generating tunable interactions between two
different species of complex dipolar atoms.
Atoms with large magnetic dipole moments, suchas the lanthanide series elements erbium and dysprosium, interact in a manner that is both long-range andanisotropic. This is in contrast to more commonly usedatomic species, such as alkali and alkaline earth metals,
which primarily interact in a short-range and isotropic
way. The recent creation of degenerate Bose and Fermi
gases of such atoms [10–13] has enabled the observation ofa wealth of new phenomena including quantum-stabilized
droplet states [14–16], roton quasi-particles [17], supersolid states [18–20], and a non-isotropic Fermi surface [21].
In a separate direction, degenerate mixtures of multiple atomic species have also provided diverse opportunities for the study of new physical phenomena. Examples
include studies of polarons that arise when an impurity
species interacts with a background gas [22–27], and theformation of heteronuclear molecules with large electric
dipole moments [28–31].
We expect that combining dipolar interactions withheteronuclear mixtures will lead to a rich set of novel
physical phenomena, the exploration of which has only
recently begun. In particular, dipolar interactions are
expected to have dramatic consequences for the miscibil-

ity of binary condensates [32–34], and in turn on vortex
lattices that arise in such systems [35]. Further, novel
properties of polarons are predicted to emerge when either the background [36] or both background and impurity [37] particles experience dipolar interactions [38].
Dipolar heteronuclear mixtures have recently been
demonstrated [39], but so far the interspecies scattering properties have not been explored, either experimentally or theoretically. In these complex dipolar species,
scattering properties are dictated by both anisotropic
long-range dipolar interactions, which can be tuned
through a combination of system geometry and magnetic field angle, and by contact interactions, which
can be tuned through the use of interspecies Feshbachresonances. While scattering models and experimental
demonstrations exist for mixtures of single- and twovalence electron atoms (which lack strong dipolar interactions) [40, 41], the scenario of two multi-valence electron
atoms has yet to be considered, and represents a newfrontier for our understanding of ultracold scattering. In
many commonly used atomic systems, the strength, character, and location of magnetic Feshbach resonances canbe predicted with high precision through coupled-channel
calculations [3]. However, the complexity of the internal
level structure and coupling mechanisms present in lanthanide atoms lead to significant challenges for the development of a microscopic theory with predictive power,
and so necessitate an experimental survey to find resonances with favorable properties [42–46].
To this end, we searched for heteronuclear Feshbachresonances broad enough to provide a practical means
for tuning the interspecies interaction in Bose-Bose andBose-Fermi dipolar quantum mixtures. Using atomicloss spectroscopy to identify resonances, we perform surveys of fermionic 161 Dy and bosonic 164 Dy together with
166
Er, 168 Er, and 170 Er over a magnetic-field range fromzero to several hundred Gauss (the exact range varies

2
by isotope combination due to availability of favorable
evaporation conditions). We also explored a Fermi-Fermi
mixture of 167 Er and 161 Dy, but observed no broad resonances there. In Table I we summarize positions andwidths of these features observed in our surveys. As
an exemplary case, we present a more detailed characterization of the resonance near 13.5 G in the 168 Er164
Dy Bose-Bose mixture, through measurements of interspecies thermalization and the dependence of atomic
loss on temperature.
In addition, our dipolar mixtures host a large number of narrow interspecies resonances. In previous experiments with single species, the density and spacingof these narrow resonances has been studied to reveal
a pseudo-random distribution that can be modeled wellusing random matrices [43, 45, 46]. By performing highresolution scans over specific magnetic-field ranges, wefind that the average density of interspecies resonances
exceeds the combined density of intraspecies resonances,perhaps indicating the contribution of odd partial waves
or molecular states with antisymmetric electron configurations for the interspecies case, which are not present in
the scattering of identical bosons.
Finally, in each Fermi-Bose mixture involving 161 Dy
we observe a correlated loss feature between fermionic
Dy and bosonic Er atoms. Strangely, the loss feature
is present at the same magnetic-field value for all threebosonic erbium isotopes studied. Such behavior is inconsistent with a typical interspecies Feshbach resonance,
where the magnetic field at which the resonance occurs isstrongly dependent on the reduced mass of the atoms involved [47]. The mechanism behind this unusual feature
is as of yet unknown and calls for further experimental
and theoretical investigations.
Our experimental sequence is similar to the one introduced in our previous works [39, 48]. More details canbe found in the supplemental material [49]. In brief, after cooling the desired isotope combination of erbium anddysprosium atoms into dual-species magneto-optical trap
(MOT), we load the atoms into a crossed optical dipole
trap (ODT) created by 1064 nm laser light. Here we
perform evaporative cooling down to the desired sample
temperature. During the whole evaporation sequence, we
TABLE I. Comparatively broad resonances found in specific isotope mixtures together with estimated center positions
and widths (FWHM) from Gaussian fits to atom loss spectra.
Each value is an average between the fit values of Er and Dy.
Combination
168

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

Resonance magnetic field (G)Width (G)13.32(4)34.09(3)161.31(3)161.30(2)161.26(3)1.7(1)
1.5(1)
0.84(9)0.93(5)0.91(8)apply a constant and homogeneous magnetic field (Bev ),pointing along the z-direction opposite to gravity. Bev
preserves the spin-polarization into the lowest Zeeman
sublevel of both species. We use different values of Bev
to optimize the evaporation efficiency depending on theisotope combination and on the range of the target magnetic field (BFB ) to be investigated. The final ODT has
trap frequencies ωx,y,z = 2π × (222, 24, 194) s−1 . We typically obtain mixtures with atom numbers ranging from3 × 104 to 1 × 105 atoms for each species. The sample
is in thermal equilibrium at about 500 nK, which corresponds to about twice the critical temperature for condensation. Typical densities are up to a few 1012 cm−3for each species. After preparing the mixture, we linearly ramp the magnetic field from Bev to BFB in 5 ms,either in an increasing or decreasing manner. We hold the
mixture for a time ranging between 5 ms and 400 ms depending on the experiment. At the end of the hold time,
we release the atoms from the ODT in a 15 ms time-offlight (TOF) expansion after which we record an imageof the atoms using a standard low-field absorption imaging technique [12, 49]. Note that we adjust the relative
amount of erbium and dysprosium in the final thermal
mixture for the specific experiments by independently
tuning the MOT loading time for each species between
0.5 s to 5 s.
In the isotope combinations and range of magnetic
fields that we explore here, we observe two interspecies
resonances with widths greater than 1 G (see Table I). We
now turn to a more detailed characterization of a featurepresent in the 168 Er-164 Dy combination, for which atom
loss is shown in Figure 1(a). We chose to focus on thisfeature because it is relatively isolated from the many
narrow homonuclear and heteronuclear resonances typical of lanthanides. In this experiment, the starting mixture contains 6.2 × 104 erbium and 9.1 × 104 dysprosium
atoms and it is prepared by evaporation at Bev = 10.9 G.
In order to compensate for loss during magnetic-fieldramps and slow drifts of the atom number, we normalize
measurements performed with 200 ms hold times at BFB
to interleaved measurements at 10 ms hold time at thesame field. We further performed independent trap-loss
spectra in single-species operation to confirm the interspecies nature of the resonance. Moreover, such scansallow us to identify intraspecies resonances and exclude
them from the fit (see empty symbols in Figure 1(a)).
As shown in the inset for erbium, a high-resolution scanreveals a narrow region with less loss near the center ofour broad loss feature, probably due to the influence ofa second interspecies resonance. This structure is also
visible on the dysprosium loss feature but it is not shownin the inset for ease of reading.
A Gaussian fit to the loss profiles, with known narrow single-species resonance excluded, returns a center
value of 13.31(2) G and 13.33(4) G and a full width at half
maximum value of 1.95(5) G and 1.3(1) G for erbium and

3

0.6
0.4

3

Atom
numberNormalized
atom number

0.8

0.2

10

4

2
1
0

0

(10 -15 m 2 )ErDy

Cross-section

8

(b)

6
4
2
0

11

12

13

14

15

16

17

18

Magnetic field (G)FIG. 1. (a) Trap loss from the 13.5 G resonance in the BoseBose mixture 168 Er-164 Dy (red circles and blue squares points
respectively). Empty symbols correspond to narrow singlespecies resonances, which we exclude from fits. Each point isan average over four experimental repetitions. For each magnetic field, the atom number recorded after 200 ms of holdtime is normalized to that at a short hold time of 10 ms. Thelines are the Gaussian fits to the data. The inset shows erbium
loss measured in a different dataset with 5 mG resolution, andhighlights the structure present on the center of the feature.
The same structure is visible also for the dysprosium atoms
in the mixture. (b) Interspecies elastic cross-section σErDy
measured across the Feshbach resonance using cross-species
thermalization. Each value of σErDy is extracted from thermalization data using a numerical model for thermalizationthat includes temporal variation in atom number and temperature; see main text and supplemental material [49].

dysprosium, respectively. The observed difference in thefitted width of the two species can be explained by theimbalance in atom number: because this measurement
was performed with fewer erbium atoms than dysprosium, the fractional loss of erbium is higher than that ofdysprosium, leading to a greater saturation of loss andbroadening of the erbium loss feature.
To get insights on its effective strength and width,
we perform cross-species thermalization measurements
across the resonance (see Fig. 1(b)). Interspecies thermalization experiments are well established techniques
to extract effective thermalization cross sections, which
in turn depends on the scattering length [50–52]. Whileinferring on a precise value of the scattering length would
require the development of a detailed and rigorous modelthat accurately captures the temperature-dependence of
the interspecies and anisotropic dipolar scattering [53],and would go beyond the scope of this work, we areable to determine a thermally averaged scattering crosssection from which we can estimate the width of the resonance.
In this cross-thermalization experiment, we selectively
heat dysprosium by means of a near-resonant 421 nm
light pulse along the vertical direction. We confirmedthat the light pulse has no direct measurable effect on
erbium. The magnetic field is then jumped to the desired
value BFB and held for a variable amount of time, during which the temperature of erbium rises to equilibratewith dysprosium due to elastic collisions. We record thetemperature of the two species along a direction orthogonal to the heating pulse, as the effects of center of mass
motion are less prevalent here [54], and use a numericalmodel to extract a cross section from the rate of thermalization [49]. This simple model assumes an energy independent cross section, an assumption which may break
down near resonance where unitarity limits on scattering
may become significant.
From these thermalization measurements, we can see
a dramatic increase in the scattering cross section nearresonance, as one would expect for an interspecies Feshbach resonance. Further, we observe a significant modification of the cross section associated with the resonanceover a Gauss-scale range of magnetic fields, similar to thewidth we observe in loss measurements. For an isolated
resonance and pure contact interactions, a common way
to characterize the resonance width is the parameter ∆,given by the difference in magnetic field between the poleof the resonance, at which the thermalization rate is max-

3
2

Width (G)

(a)

2.5

L 3 (10 -24 cm6 /s)

1

2

1
0

1.5

0.5

1

1.5

Temperature (µK)
1
0.5
0
0.5

1

1.5

2

2.5

Temperature (µK)

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

4
imal, and the nearest zero-crossing in the thermalization
rate, which would correspond to a lack of scattering [3].
In lanthanides, the presence of anisotropic dipolar interactions leads to a scattering cross section that does
not completely vanish. In addition, multiple narrow andoverlapping resonances may be present, which may influence the interpretation of such a width measurement.
However, to get a rough estimate of the width of theresonance, we can consider the distance between the resonance pole and the apparent minimum in the thermalization rate at 17 G. This suggests a width of ∆ ' 3.5 G.
The dependence of the loss feature on the cloud temperature can provide additional information on the nature of the resonance. For broad resonances, a universal regime is expected to emerge near resonance where
the scattering cross section and loss are dictated primarily by the atomic momentum, rather than the scattering
length [55]. In this regime, the three-body loss parameterL3 follows a nearly universal form scaling as 1/T 2 , where
T is the temperature. Such scaling has been observed inbroad resonances of several atomic species [55–57].
We observe a temperature dependence of the loss rate
near resonance that is suggestive of such universal behavior. By varying the final depth of the ODT reached during evaporation, we tune the temperature of the atomic
mixture. For each temperature, we measure atom loss
on resonance at 13.4 G as a function of the hold time.
We then use a numerical model to extract the rate ofinterspecies three-body loss, and L3 [49].
These loss coefficients are plotted as a function of temperature in Fig. 2, along with a fit to a 1/T 2 dependence,
which provides a reasonable description of our data. The
universal temperature dependence arises from a maximum value of L3 possible at a given temperature, givenby:√h̄5 36 3π 2
λ3,max' 3.
(1)
L3,max =T2
m (kB T )2
Factors associated with Efimov physics [49] can leadto a lower value for L3 , but not higher [55, 58, 59].
From our fit to a 1/T 2 dependence for our data, we extract a value of λ3 = 1.0(2) × 10−24 µK2 cm6 s−1 , whichis compatible with the predicted bound of λ3,max =2.4 × 10−24 µK2 cm6 s−1 .
A reduction in the peak loss rate with increasing temperature can also result from thermal broadening of theresonance, especially for very narrow resonances [45].
This is unlikely to be the dominant effect here, as for typical differential magnetic moments between entrance andclosed channels in our lanthanide system [60], we would
expect broadening on the scale of a few times 10 mG for
temperatures near 1 µK, much narrower than the Gaussscale width of our feature. Further, suppression of peakloss is typically accompanied by a commensurate broadening and shift of the loss feature on the scale of its width,
which we do not observe (inset in Fig. 2).

In addition to the few relatively broad resonances,the lanthanides exhibit many narrow resonances, whose
statistical properties have been investigated for singlespecies gases [43, 45, 46]. In this section we compare
the abundance of interspecies resonances to single-species
resonances by performing high-resolution trap-loss spectroscopy on the isotope combination 166 Er-164 Dy (seeFig. 3). Here, we investigate four different magnetic-field
ranges, each 10 G wide, with a resolution 40 times higher
than the one used for the exploratory surveys. To enabledirect comparison with the previous works performed on
single species [43, 45], we use similar experimental conditions (isotope, atom number, temperature, and holdtime).
As expected, we observe many narrow homonuclear
resonances [43, 45]. In addition, we also identify many
narrow heteronuclear resonances. To distinguish these
two types of resonance, we first label features with a fractional loss above 30% as resonances. We then categorize
these resonances as interspecies if erbium and dysprosium loss features occur simultaneously within a rangeof ±10 mG and with a loss amplitude ratio in the range
0.5-2. Features that do not meet both of these criteria,
are labelled either as homonuclear or ambiguous, basedon comparison with separate scans performed with single
species, either within this work or from previously published data [43, 45]. The number of ambiguous features
define our confidence interval.
In order to visualize the number of resonances, we construct the staircase function N (B), which describes the
cumulative number of resonances from the start of a scan
range up to a given magnetic field BFB . Figure 4(ad) shows N (B) for the four investigated magnetic-field
ranges. The black lines represent heteronuclear Feshbachresonances, while the blue and the red lines represent thehomonuclear 166 Er and 164 Dy resonances, respectively.
The shaded regions represent our confidence interval defined by the total number of ambiguous Feshbach resonances.
Our analysis results in a total number of heteronuclear resonances of N ErDy (tot) = 339(16), counting all
magnetic-field ranges, and a number of homonuclear resonances of N Er (tot) = 116(16) and N Dy (tot) = 144(16).
Within our confidence intervals, we detect a total number of homonuclear resonances comparable with those ofprevious works [43, 45]. The corresponding total density of resonances ρ̄, given by the total number of resonances divided by the total range of magnetic fields
scanned are: ρ̄ErDy = 8.5(4) G−1 , ρ̄Er = 2.9(4) G−1 , andρ̄Dy = 3.6(4) G−1 .
For our combined dataset, we find that the total number of heteronuclear resonances exceeds the combined
number of homonuclear resonances for the two species:ρ̄ErDy = α(ρ̄Er + ρ̄Dy ), with α = 1.3(2). We would expect that the average density of heteronuclear resonances
should be greater than the sum of the two homonu-51Normalized atom number

0.5
0
0
1

1

2

3

4

5

6

7

8

9

10

23

24

25

26

27

28

29

30

31

32

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

0.5
0
22
1
0.5
0
50
1
0.5
0
60

Magnetic field (G)FIG. 3. High-resolution trap-loss spectroscopy for a balanced mixture of 166 Er and 164 Dy (red and blue curves respectively),with initial atom numbers of roughly 105 per species and a temperature of 500 nK and after 400 ms of interaction time. The
measurement is composed of four datasets [0,10] G, [22,32] G, [50,60] G, and [60,70] G with a stepsize of 5 mG Each point is anaverage over four experimental repetitions. Atom numbers are normalized to the maximum of each dataset for ease of reading.
The broad loss feature in Dy near 68.8 G was not observed in previous work [45], and may result from a technical source of loss
in our experiment.

clear resonance densities. This is because each species
contributes a set of internal states that can be coupled
to, and the heteronuclear resonances are not subject tothe same symmetrization requirements as the homonuclear resonances. In resonances involving distinguishable
particles, both gerade and ungerade Born-Oppenheimer
molecular potentials contribute, as well as both even andodd partial waves for the entrance channel. Our datais consistent with this expectation (α > 1). Note that
we do observe a lower number of interspecies resonancesin the range [50,60] G, perhaps as a result of the nonrandom distribution of resonances as observed in the
single-species case [43, 45], or to the presence of broadhomonuclear erbium resonances that could obscure theobservation of interspecies resonances.
Finally, we have also searched for broad (Gaussrange) resonances in Bose-Fermi mixtures consisting offermionic 161 Dy combined with different bosonic isotopesof erbium – 166 Er, 168 Er, and 170 Er, as well as FermiFermi mixtures of 161 Dy and 167 Er. For these combinations, we perform only coarse scans and thus only resolve broad features. In mixtures involving the bosonicisotopes of erbium we observe a correlated loss feature
between erbium and dysprosium near 161 G (see Fig. 5).
This loss feature is not present at our level of measurement sensitivity with either species alone, or in the mixture with the fermionic 167 Er. Surprisingly, the loss feature is centered at the same magnetic field (to within our
resolution of 0.1 G) for all bosonic isotopes of erbium.

This is quite unexpected as the magnetic-field value ofthe resonance position is typically highly sensitive to the
reduced mass of the atoms involved [47].
Several physical mechanisms could be consistent withsuch a feature. One possibility is that the resonance
we observe is associated with a bound state of a shallow molecular potential [61]. Mechanisms to create suchpotentials have been proposed for species with dipolarinteractions [62, 63]. However, none are obviously applicable to magnetic atoms in the lowest energy entrance
channel. Further, given the level of insensitivity to themass of erbium, we would expect to see additional resonances of a shallow potential in the magnetic-field range
over which we survey, which we do not. A second possibility is that the feature we observe is not a true interspecies resonance, but rather an intraspecies resonancein dysprosium whose loss rate is enhanced by the presence of bosonic erbium atoms. A similar effect was reported in a mixture of fermionic lithium and bosonic rubidium atoms [64]. Finally, it is possible that this feature is not a Feshbach resonance at all, but rather the
result of spin-changing processes resulting from unintentional radio-frequency tones in the laboratory, or of aninterspecies photoassociation resonance. We have ruledout the most likely culprits for the last effect by varying
the relative detuning between our horizontal and vertical
dipole traps and observing no change in the resonanceposition. We hope that our presentation of this mysterious feature may spur theoretical exploration of possible

6
1

100
(a)

0.5

(b)

60

Number of resonances

40
20
0
0
100

2

80

(c)

4

6

8

10

22

24

26

28

30

32

Normalized atom number

80

(a)

0
1
0.5

(b)

0
1
0.5

(c)

0
1

(d)
0.5

60

0
158

(d)
159

160

161

162

163

164

165

40

Magnetic field (G)20

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

0
50

52

54

56

58

60

60

62

64

66

68

70

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

physical mechanisms.
In conclusion, we have reported experimental observation of heteronuclear magnetic Feshbach resonancesin several isotope mixtures of erbium and dysprosium.
Among the Gauss-broad features identified in our surveys, we have characterized one in the combination 168 Er164
Dy by means of cross-species thermalization measurement and temperature dependence analysis. We performed high-resolution trap-loss spectroscopy in the combination 166 Er-164 Dy to compare the average resonancedensity of the mixture with respect to the single-species
case. In mixtures of fermionic 161 Dy and bosonic erbium
atoms, we observed a correlated loss feature which appears to be insensitive on the erbium isotope used but
absent in dysprosium alone. Our observations pave theway to realize tunable interactions in quantum degenerate mixtures of dipolar atoms, which will enable varied
opportunities including studies of the miscibility of binary condensates, of vortex lattices, and of dipolar polarons [32–37].
We thank Jeremy Hutson, Matthew Frye, John Bohn,
Arno Trautmann, and the Erbium and DyK teams
in Innsbruck for insightful discussions.
This work
is financially supported through an ERC Consolidator Grant (RARE, No. 681432), a NFRI Grant (MI-RARE, No. ÖAW0600) from the Austrian Academy of
Science, a QuantERA grant MAQS by the Austrian Science Fund FWF No I4391-N, and a DFG/FWF (FOR2247/PI2790). M. S. and G. D. acknowledge support by
the Austrian Science Fund FWF within the DK-ALM(No. W1259-N27). We also acknowledge the Innsbruck
Laser Core Facility, financed by the Austrian Federal
Ministry of Science, Research and Economy.
* Correspondence and requests for materials should be
addressed to Francesca.Ferlaino@uibk.ac.at.

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9
Supplemental Material

Cross-species thermalization

Preparation

At this point, we linearly ramp the magnetic field from
Bev to the target field BFB in 5 ms, either in an increasing or decreasing manner. We hold the mixture for a
specific time ranging between 5 ms and 400 ms depending on the experiment. At the end of the hold time,
we release the atoms from the ODT in a 15 ms time-offlight (TOF) expansion after which we record an imageof the atoms using a standard low-field absorption imaging technique [12]. We use pulses of resonant light in thehorizontal x − y plane at an angle of ∼ 45 ◦ with respect
to our weak trap axis y. 5 ms after the clouds being released from the ODT, we linearly ramp BFB to zero in
10 ms. At the same time, we activate a magnetic field of
about 4 G pointing along the imaging direction.
Note that we select the relative amount of erbium anddysprosium required in the final thermal cloud by the
specific experiments by independent tuning of the respective MOT loading times between 0.5 s to 5 s. Bev andBFB are generated by the same pair of coils in Helmholtz
configuration. We ramp them up linearly in the earlystage of the evaporation sequence 200 ms after loading
the atoms into the hODT. We checked that the responseof the current flowing in the coils can follow time ramps
on the millisecond time-scale. This translates in a effective change of the field at the sample position in the orderof 10 ms to settle at the part per thousand level.

3.5
Temperature K

The experimental sequence is similar to the one introduced in our previous works [39, 48]. After cooling the
erbium and dysprosium atoms into dual-species magnetooptical traps (MOTs) of the desired isotope combination,
we load about 1 − 5 × 106 atoms of both erbium and dysprosium into a single-beam optical dipole trap created by1064 nm laser light and horizontally propagating along
the y-direction (hODT). We perform an initial stage of
evaporative cooling of about 0.8 s. After that, a second
trap beam, coming from the same laser source but detuned by 220 MHz, is shone along the vertical directionz (vODT) onto the atoms forming a crossed ODT where
we continue the evaporation for an additional durationof about 4.3 s down to the desired sample temperature.
During the whole evaporation sequence, a constant andhomogeneous magnetic field (Bev ) pointing along the zdirection and opposite to gravity is applied. Different
values of Bev are used depending on the isotope combination and on the range of the target field (BFB ) to beinvestigated. We typically end up with 3−10×104 atoms
for each species, in thermal equilibrium at about 500 nK(about twice the critical temperature for condensation).
Final trap frequencies are ωx,y,z = 2π×(222, 24, 194) s−1 .

3.0
2.5
2.0
1.5

0

10

20
30
40
Hold time (ms)50

FIG. S1. Sample temperature traces for erbium (filled circles)
and dysprosium (hollow squares) after dysprosium is heated.
Purple, green, and orange correspond to magnetic fields of
12 G, 13.5 G, and 17 G, respectively. Fit lines represent theresults of the numerical integration of equation 3, which fits
the temperature profile of erbium based on its initial value
and the dysprosium temperatures. Different evaporation conditions cause the curves to have slightly different initial andfinal conditions (see main text).

As an exemplary case, we study in more detail theresonance found in the 168 Er-164 Dy Bose-Bose mixture,near 13.5 G. To quantify reliably the value of the interspecies cross-section, we developed the following scheme
for cross-species thermalization measurements [50–52].
To avoid heating of the sample by crossing Feshbach resonances, we evaporate the mixture at Bev close to resonance. Specifically, when measuring on the low(high)field side of the feature we evaporate at Bevap =10.8 G(16.4 G). Once the sample is prepared as previously described (here we use an unbalanced mixture withtwice as much Dy as Er), we compress the trap by linearly increasing the hODT power by a factor of five andthe vODT power by two in 500 ms to prevent any plain
evaporation. The final trap frequencies in the compressed
trap are ωx,y,z = 2π × (409, 26, 391) s−1 . Subsequently,
we ramp the magnetic field in 5 ms to either 10 G or 16 G.
Here, a pulse of near-resonant 421 nm light propagating
along the magnetic field direction (z) is used to selectively heat dysprosium. We fix the duration of the pulse
at 5.5 ms to roughly match the trap oscillation periodalong this direction and set the pulse intensity to give the10desired temperature increase of the dysprosium cloud (upto 4 µK). We confirmed that the light pulse has no direct
measurable effect on erbium. Finally with a quench fastcompared to the shortest thermalization rate, the magnetic field is set to the desired value BFB and held fora variable amount of time, during which the temperature of erbium rises to equilibrate with dysprosium dueto thermalizing collisions (Figure S1). We note that in
the temperature evolution of the clouds, the initial temperatures are slightly different. This behavior is mainly
due to different evaporation conditions on the two sidesof the resonance, the different strength in the quench tothe final BFB , and the heating caused by the resonanceitself. By comparing the two species’ temperature, we
ensure that these different conditions are consistent withgeneral offsets on the single measurement thus not affecting the final estimation of the cross-section.
To extract a scattering cross-section from our crossspecies thermalization data, we use a fit to a numericalmodel for the thermalization of two species. In principle,
a simple exponential fit to the temperature difference between the two species could also be used, but does notaccount for changes in the atom number or average temperature of the sample that may arise from residual evaporation during the thermalization time. Our numericalmodel follows that of Ref. [50]. We treat the scatteringcross section as independent of the energy of the colliding
particles, an assumption that greatly simplifies the analysis, but inevitably breaks down near enough to resonance
where unitarity considerations bound the scattering crosssection. This assumption leads to a collision rate for each
atom of species 1 with atoms of species 2 given by:
r
3/2
T1
T2
N2 m1 ω̄13
+
σ12
(2)
γ12 = 2
−2
3/2
m1
m2
π kB (T1 + β T2 )where m1 , m2 , T1 and T2 are the masses and temperatures of species 1 and 2, ω̄ = (ωx ωy ωz )1/3 characterizes the frequency of the trap, β 2 = m2 ω̄22 /m1 ω̄12 , andσ12 is the effective interspecies cross section. We assume that the energy exchanged per collision is given by∆E = ξkB (T2 − T1 ) where ξ = 4m1 m2 /(m1 + m2 )2 , andthat the heat capacity of each atom is 3kB . This leads toa differential equation for the temperature of erbium:
3/2

3ξkB (TDy − TEr )NDy mEr ω̄Er
dTEr
=
2
−2
3/2
dt
3π kB (TEr + β TDy )
s
TEr
TDy
×
+
σErDy
mEr
mDy

(3)

which we can numerically integrate using the instantaneous values for TDy and NDy , and from this extract the
scattering cross section σErDy that yields a thermalization profile that best matches our data, as determinedthrough a least-squares difference. Examples of threesuch fits, for 12 G, 13.5 G, and 17 G are shown in Fig. S1,
and generally describe our thermalization data well.

Temperature dependence of loss

We quantify the temperature dependence of threebody loss in terms of the interspecies three-body loss
coefficient. For a single species, the three-body loss coefficient L3 can be defined by: Ṅ /N = −L3Rhn2 i where N
is the total number of atoms, and hn2 i = d3 r n3 (r)/Nrepresents the average squared density of scattering partners for an atom in the gas. n(r) is the local density of
the gas.
We define analogous quantities for our two-species mixture, containing particles denoted i and j. In this case,Z−1Ṅin2i (r)nj (r)d3 r (2Li,i,j
=
3
Ni
3Ni
(4)
j,j,i
2
+L3 ni (r)nj (r)).
represents the loss rate due to collisions inHere, Li,i,j
3
volving two atoms of species i and one of j.
To arrive at simple expressions, we make several assumptions and approximations. First, we treat the mass,temperature, and polarizability of the two atomic species
as equal, which is a reasonable approximation for erbium and dysprosium isotopes in our 1064 nm wavelength
ODT [39]. This assumption implies equivalent spatial
distributions for the two species, which we assume tobe thermal in our three-dimensional harmonic trap. We
next set Li,i,j= Lj,j,i≡ Li3 near resonance, essentially33
assuming that the loss process is primarily determinedby the two pairwise interactions between the minority
participant and the two majority atoms. We find this assumption leads to a model consistent with our observed
relative loss between the two species. With these simplifications in place, we define Li3 using: Ṅi /Ni = −Li3 hn2 iieff ,where
hn2 iieff =(2Ni Nj + Nj2 )m3 ω̄ 6
5

3 2 8π 3 (kB T )3(5)

and ω̄ = (ωx ωy ωz )1/3 is the geometric mean of the trap
oscillation frequencies.
We extract the resonant value of L3 by measuring remaining atom number versus hold time in mixtures prepared at different temperatures, with the magnetic field
set near resonance at 13.4 G. We then fit the resulting
data by numerically integrating Eq. 4. Because we observe significant single-species loss of erbium (the majority species), we treat the erbium atom number measured
at each time-step as inputs to our fit, and extract thevalue of L3 that best predicts the loss of dysprosium.
≡ L3 . We boundHere, we assume that Li,i,j= Lj,j,i33
the effects of single-species loss in dysprosium by repeating the same measurement and analysis protocol off resonance at 11.5 G and 16.5 G. The error bars in Fig. 2of the main text include a contribution corresponding to
the extracted L3 in the off-resonant condition, which contain both the effects of single-species loss and the small

11effect of off-resonant interspecies loss. Also included areerrors associated with the observed change in temperature during the loss measurement, and relating to theapproximations made in estimating the density.
In a regime where the scattering√ length a exceeds thethermal wavelength λth = h/ 2πmkB T , and thermal
broadening is small compared to the width of the lossfeature, we expect roughly L3 ∝ 1/T 2 , as has been
observed in several experiments involving single atomic
species [55–57]. This picture becomes complicated somewhat in the case of a binary mixture due to stronger
Efimov effects, which lead to a temperature-dependent

modulation of loss relative to the simple 1/T 2 prediction.
In particular, the parameter s0 , which characterizes thestrength of the three-body Efimov potential, is equal toapproximately 1.006 for identical bosons, but approximately 0.41 for our binary mixture [58, 59]. The fractional importance of these temperature-dependent modifications scale as e−πs0 [55], making them a minor correction for identical bosons, but a potentially important
effect in mixtures. It is possible that such effects contribute to deviations of our data from a 1/T 2 form, but atrue calculation would require knowledge of short-range
inelastic processes in our system.

4.5. ADDITIONAL MEASUREMENTS

73

4.5 Additional measurements
In this Section, we present additional measurements performed during the preparation of
Ref. [Dur20] (see Sec. 4.4). If otherwise stated, the experimental procedure of these additional
measurements follows the one described in the supplemental material of Ref. [Dur20]. In
Sec. 4.5.1 we report the full observation of the mentioned feature observed in the mixture
166 Er-164 Dy at about 34 G. In Sec. 4.5.2, we present a deeper characterization of the feature
at about 161 G in mixtures of fermionic 161 Dy together with bosonic isotopes of Er. For theease of reading, all the surveys performed to identify eventual broad FRs are collected inAppendix B.

4.5.1 Feshbach resonance at 34 G in 166 Er-164 Dy
In the isotopic combination 166 Er-164 Dy, we identify by means of trap-loss spectroscopy a
Gauss-broad heteronuclear resonance as reported in Fig. 4.3 (a) together with Gaussian fits
to the loss feature. In this experiment, the starting sample contains a balanced mixtureof about 7 × 104 erbium and dysprosium atoms at about 500 nK. The evaporation is performed at 23.1 G and the hold time at BFB is 200 ms. The Gaussian fits return center valuesof 34.08(5) G and 34.10(7) G and FWHM values of 1.9(1) G and 1.6(2) G for erbium and dysprosium, respectively. Additionally, we performed single-species trap-loss spectra to confirm
the heteronuclear nature of the resonance.
With the same procedure of Ref. [Dur20], we perform cross-species thermalization measurements at different magnetic field over the resonance to estimate the elastic interspecies
cross section. More specifically for this experiment, the sample is prepared by evaporative
cooling at 23.1 G (36.2 G) when performing the measurement on the low-field (high-field) sideof the resonance. The mixture typically contains three times more dysprosium than erbium.
The resonant light pulse is shone at 31 G or 36.2 G, respectively. The analysis is performedby fitting a numerical model to the data as already discussed. Its results are reported inFig. 4.3 (b). Raising quantitative arguments on the strength of this resonance is hard dueto the complex pattern of narrower homo and heteronuclear resonances on top of the broadresonance. Nevertheless, we observe a dramatic increase of the scattering cross-section whenapproaching the center of the feature.

4.5.2 Additional measurements on the feature at 161 G in 161 Dy
As already discussed, we observe a correlated loss feature involving fermionic 161 Dy and thebosonic isotopes of Er. In the publication, we report the trap-loss spectra for the different
isotopic combinations. To shed more light on this puzzling feature, we conducted further
investigations on the exemplary case of 168 Er-161 Dy. In the first experiment, we vary thetemperatures of the sample and, in the second one, the unbalance of erbium with respect todysprosium in the final sample. In the following, more details are reported.

Atom number

8

104

6

4

2

(a)

(10 -15 m 2 )ErDy

Cross-section

0
10
8

(b)

6
4
2
0
31

32

33

34

35

36

Magnetic field (G)Figure 4.3: (a) Trap-loss profile from the 34 G resonance in the Bose-Bose mixture 166 Er-164 Dy(red circles and blue squares points respectively). Each point is an average over four experimental
repetitions. The lines are the Gaussian fits to the data. (b) Interspecies elastic cross-section σErDy
measured across the Feshbach resonance using cross-species thermalization. Each value of σErDy is
extracted from thermalization data using a numerical model for thermalization that includes temporalvariation in atom number and temperature (see Ref. [Dur20] for more details).

74

4.5. ADDITIONAL MEASUREMENTS(b) 1.5

1

Normalized loss coefficient

Normalized atom number

(a)

75

0.5
0
1
0.5
0
1
0.5
0
158

1.25

1

0.75

0.5
159

160

161

162

163

164

165

0.5

Magnetic field (G)

1

1.5

2

Temperature ( K)

Figure 4.4: Temperature dependence of the feature present in fermionic 161 Dy in combination withbosonic 168 Er. (a) Trap-loss spectra performed at a sample temperature of 600 nK, 1.3 µK, and 2 µK,
top to bottom panels, respectively. Red circles represent erbium, blues squares represent dysprosium.
For each magnetic field, the atom number recorded after 100 ms of hold time is normalized to the one
at a short hold time of 5 ms. For all panels, each point is an average over four experimental repetitions.
(b) Simple model describing the losses as either two-body Er-Dy loss (light blue) or three-body loss
with equal coefficients for Er-Er-Dy and Er-Dy-Dy (green). The traces are normalized to their meansin order to better visualize the temperature dependence.

Temperature dependence By varying the final depth of the ODT, we probe three different
temperature regimes of the mixture: 600 nK, 1.3 µK, and 2 µK. The hold time is 100 ms andthe result is analyzed by normalizing the atom number at this value to the one at a shorthold time of 5 ms to compensate for slow drifts in the experiment. The trap loss spectra
are plot in Fig. 4.4 (a). To extract qualitative information, we model the losses as either
two-body Er-Dy loss or three-body loss with equal coefficients for Er-Er-Dy and Er-Dy-Dy.
Given the densities of the two species, we define the two-body Γ2B and three-body Γ3B loss
coefficients as
Γ2B =

ṅ
2nEr nDy

Γ3B =

ṅ
2
3(nEr nDy + nEr n2Dy )4.17

where ṅ is the total loss rate of atoms assumed to be linear. The result of the analysis is plot
in Fig. 4.4 (b) where the traces are normalized to their means in order to better visualize thetemperature dependence. Although there might be a slight increase of the loss coefficientswith temperature, it does not appear to be dramatic.
Er/Dy ratio dependence By varying the loading time of the Er MOT — therefore thenumber of atoms loaded into the ODT — we explore three different ratios of unbalance inthe atom number of Er and Dy in the final sample. We performed loss measurements with
erbium numbers being 1.3 × 104 , 2.6 × 104 , and 3 × 104 (with the number of dysprosium
fixed at about 3.3 × 104 ). The hold time is 100 ms and the result is analyzed by normalizing

76

CHAPTER 4. FESHBACH RESONANCES

(a)

(b)

Figure 4.5: Er/Dy atoms ratio dependence of the feature present in fermionic 161 Dy in combination
with bosonic 168 Er. (a) Trap-loss spectra performed with erbium numbers being 1.3 × 104 , 2.6 × 104 ,and 3 × 104 , top to bottom panels, respectively. Dysprosium atoms are fixed at about 3.3 × 104 .
Red circles represent erbium, blues squares represent dysprosium. For each magnetic field, the atom
number recorded after 100 ms of hold time is normalized to the one at a short hold time of 5 ms. For
all panels, each point is an average over four experimental repetitions. (b) Simple model describing
the losses as either two-body Er-Dy loss (light blue) or three-body loss with equal coefficients for ErEr-Dy and Er-Dy-Dy (green). The traces are normalized to their means in order to better visualize
the dependence.

the atom number at this value to the one at a short hold time of 5 ms to compensate for slow
drifts in the experiment. The resulting trap loss spectra are plot in Fig. 4.5 (a). In each case
we found that the number of erbium lost was roughly equal to the number of dysprosium lost,
and that the number of total atoms lost was roughly proportional to the number of erbium
present. Following the same approach described in the previous paragraph, we analyze theloss feature in terms of two- and three-body loss coefficients as in Eq. 4.17. The result issummarized in Fig. 4.5 (b). The traces are normalized to their means in order to better
visualize the dependence on the different Er/Dy ratio.

Chapter5

Conclusion and outlook

5.1 Conclusion

This thesis reviews the experimental realization of quantum degeneracy in dipolar mixturesof erbium and dysprosium, focusing the discussion on the first observation of interspecies
Feshbach resonances essential for interaction tuning.
In order to study dipolar systems in quantum regimes, we make use of mixtures of highly
magnetic erbium and dysprosium atoms for the first time. These species possess among thestrongest magnetic dipole moments on the periodic table — 7µB and 10µB for Er and Dy
respectively — leading to a dipolar strength of about a hundred times the one in alkali-metal
atoms. By employing their similar physical and atomic properties and the recent advancesin the field [Lu11a, Aik12], we were able to build an ultracold atom experiment, competingat the forefront of the research in about three years. Remarkably, the experiment led to
five publications, three of which set the main discussion in this thesis: the realization of adual-species intercombination-line magneto-optical trap in open-top configuration [Ilz18], theachievement of quantum degeneracy in several erbium-dysprosium isotopic mixtures [Tra18],
and the experimental observation of interspecies magnetic Feshbach resonances [Dur20].
Our new apparatus combines the fascinating physics of dipolar systems with the flexibility
and opportunity offered by mixture experiments in which mass and population imbalance,
mixed statistic, species-selective manipulations, and intrinsic distinct properties are available. This new type of experiment enters a regime of complexity in which the theoreticalunderstanding is only at the beginning and predictive models are hardly possible, e. g. for
the Er-Dy interspecies scattering properties. With this thesis, in which interspecies Feshbachresonances are investigated, we want to make a step forward into the knowledge required bymore advanced experiments. In particular, the work performed will allow for the creationof dipolar heteronuclear quantum gases with tunable interaction unveiling new intriguing
possibilities such as dipolar polarons, miscibility/immiscibility diagram, vortex formation,and binary magnetic quantum droplets [Dur20].
77

78

CHAPTER 5. CONCLUSION AND OUTLOOK

5.2 Further studies
The apparatus is designed to receive several major updates to expand the possible research
topics of the experiment. Among these, the first in chronological order will be the implementation of a high-resolution imaging along the vertical z-axis in the main chamber1 . Sucha setup will allow for in-situ absorption or phase-contrast imaging with a sub-micrometer
diffraction-limited resolution at the broad blue transitions. The high-resolution verticalimaging will boost the performance of the experiment in all directions, e. g. in the direct
characterization of quantum droplets and supersolid states [Ilz19]. The utilization of a digital micromirror device will be of central importance to study vortexes. On top of this,another open question regards the accurate characterization of interactions in the Er-Dy
quantum mixtures, fundamental to understand their miscibility/immiscibility and stability
diagrams. The identification of several Feshbach resonances made the first step in this direction, whereas more investigation is needed to extract precise values of intra- and inter-species
scattering lengths. For this purpose, the technique developed in Ref. [Tra18] makes use of thecenter-of-mass shift of the two species originating from the in-trap displacement and observable from TOF pictures. Such in-trap displacement is directly connected to the interparticle
interactions and, by monitoring it for different magnetic-field values and orientations, will
be possible to infer on a reliable value of a12 (which is currently unknown)2 .
The apparatus features the possibility to directly connect two additional vacuum modules
in which the atoms can be optically transported from the main chamber. The second majorchange on the apparatus will regard the implementation of one of these modules dedicatedto quantum-gas microscopy (QGM). Our design is based on a glass-cell chamber containing a high-NA in-vacuum objective with an estimated diffraction-limited resolution of a few
hundred nanometer and designed to be achromatic for both erbium and dysprosium. Here,single-atom identification and manipulation will possibly open the path to a deeper understanding of many-body quantum systems with the additional — and so far unexplored — caseof dipolar interactions. Quantum magnetism, topological quantum matter, lattice physics
with fermions and bosons in periodic potentials, out-of-equilibrium systems, and atomtronic
circuits are topics made available by QGM [Ott16, Gro17] on top of which the DDI will playa fundamental role.
The last planned update would have been the implementation of the third science module
dedicated to Rydberg physics [Ši18]. In erbium and dysprosium, the complex electronic
configuration with an open f -shell and an optically active core allow for a wealth of possible
experimental techniques such as new excitation and manipulation schemes including trappingand cooling of Rydberg atoms [Ilz20]. The custom-made stainless steel chamber features afull-3D electric field control, microchannel plate detectors, and a sub-micrometer imaging
scheme. However, due to the intrinsic complexity of the project and the fact that it can be
optimized for a faster and more robust operation independently from the standard quantumgas scheme proposed here, a whole new experiment will be set up3 .
1

2
3

The employed objective is a customized commercial solution with a numerical aperture of 0.45 and aworking distance of 55 mm. It is provided by Special Optics, a Navitar company
A separate publication on this project is under preparation
Tweezer array for Rydberg atoms of Erbium for Quantum Simulation (T-REQS). More information areavailable at the web page: http://www.erbium.at/FF/?page_id=37462

Appendix

A

Additional publications

In the following appendix, two publications achieved during my PhD and which I co-authored
are reported. Their subject is not part of the core of this thesis, therefore they are included
as “additional” material. Such publications are:• Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases [Cho19]
reported in Sec. A.1,
• Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms [Ilz19]
reported in Sec. A.2,

79

A.1 Publication:
Long-Lived and Transient Supersolid Behaviors in Dipolar
Quantum Gases†
Lauriane Chomaz1 , Daniel Petter1 , Philipp Ilzhöfer2 , Gabriele Natale1 , Arno Trautmann2 ,Claudia Politi2 , Gianmaria Durastante1,2 , Rick M. W. van Bijnen2 , Alexander
Patscheider1 , Maximilian Sohmen1,2 , Manfred J. Mark1,2 , and Francesca Ferlaino1,2
1Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria

Physical Review X 9, 021012 (2019)submitted 11 Mar 2019; published 19 Apr 2019
DOI: 10.1103/PhysRevX.9.021012
Featured in Physics 12, 38. Viewpoint: Dipolar Quantum Gases go Supersolid†

The author of the present thesis helped in taking the dysprosium data, and contributed in writing themanuscript. This publication is also part of the PhD thesis of Philipp Ilzhöfer.

81

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

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

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

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

DOI: 10.1103/PhysRevX.9.021012

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

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

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

021012-1

Published by the American Physical Society

L. CHOMAZ et al.

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

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

021012-2

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
z

0

y

B

4

0

4

5
0
46

48

50

52

54

-1

2 (c)

4

n (10 µm )aS (a0)

1
0

0.8

(d)

0.4
-10 0 10y (µm)

0

(g)

8

2
N( 10 )y (µm)(b)44

Sn (10 µm )10 (f)
4x-5

0.1 0.2 0.3 0.4 0.5

-1

ID

SSP

ID

0

*

0.6
0.4
0.2

6

0

-10

0
10
y (µm)

SSP
BEC

2

166

0.2
-10 0 10y (µm)BEC

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

164

Er
55

90

95

Dy
100

aS (a0)

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

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

021012-3

L. CHOMAZ et al.

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

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

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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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L. CHOMAZ et al.

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

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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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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
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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

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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.

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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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Correction: The inadvertent omission of a marker indicating“Featured in Physics” has been fixed.

021012-12

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

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

GROUND STATE CALCULATIONS

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

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

Z

s

∗
ψi−1
(y)ψi (y)dy

(1)
(0)

(0)

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

!

.

(2)

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

LINK STRENGTH AND ESTIMATE OFTUNNELING RATE

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

2

TOY MODEL FOR THE INTERFERENCEPATTERN

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

4
3
2
1
0

c

n(µm)

F[n]

n(µm)

F[n]

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

x104

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

F[n]

x104

x104

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

n(µm)

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

F[n]

(3)

F[n]

+

n(µm)

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

(0)

(yi

n(µm)

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

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

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

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

3

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

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

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

F[n]

n(µm)

n(µm)

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

4c

F[n]

5b

F[n]

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

x104

x104

5a

F[n]

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

x104

n(µm)

F[n]

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

F[n]

n(µm)

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

x104

F[n]

n(µm)

x104
6c
5
4
3
2
1
0

n(µm)

n(µm)

n(µm)

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

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

F[n]

n(µm)

5
4
3
2
1
0

x104

F[n]

6a

F[n]

x104

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

n(µm)

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

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

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

IMAGING ANALYSIS: 164 Dy AND 166 Er

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

DETAILS ON THE FOURIER ANALYSIS

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

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

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

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

3

5

4

AФ

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

3

2

1

0

0

10

20

30
tr (ms)

40

50

60

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

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

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

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

e

b

d

f

2

120
AФ(103)

Scattering length (a0)

c

4

140

100
80

0
5
4
3
2
1
0

60
40
1.1

a

1.3

1.5

1.7

1.9
B (G)

2.1

2.3

2.5

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

ATOM LOSSES IN 166 Er AND 164 Dy

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

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

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

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

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

anism limiting the lifetime of the supersolid properties inDy.

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

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

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

a

c

e

b

d

f

2
1
0
12

AФ(102)tN (ms)32(5)51(9)
46(12)

N (104)3

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

8
4
0

0

50

100

th (ms)

0

50

100

th (ms)

0

50

100

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

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

tN (ms)300(12)728(34)926(36)tΦ (ms)12(5)152(13)133(25)[1] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
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[3] A. R. P. Lima and A. Pelster, “Quantum fluctuations in
dipolar Bose gases,” Phys. Rev. A 84, 041604 (2011).
[4] A. R. P. Lima and A. Pelster, “Beyond mean-field lowlying excitations of dipolar Bose gases,” Phys. Rev. A 86,
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Bose-Einstein condensates,” Phys. Rev. A 94, 043618(2016).
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[10] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen,
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[19] We note that we have also checked our analysis without
performing the recentering step and the same features
remain. For instance, for our test data of Fig. 2, the effectbeing mainly that the side peaks in (e) are more washed
out and a slight difference occurs between nM and nΦ ,both showing still side peaks.
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A.2 Publication:
Phase coherence in out-of-equilibrium supersolid states ofultracold dipolar atoms†
Philipp Ilzhöfer1 , Maximilian Sohmen1,2 , Gianmaria Durastante1,2 , Claudia Politi1 , Arno
Trautmann1 , Giacomo Morpurgo3 , Thierry Giamarchi3 , Lauriane Chomaz2 , Manfred J.
Mark1,2 , and Francesca Ferlaino1,2
1Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
2Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
3
DQMP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

arXiv:1912.10892
submitted 23 Dec 2019
URL: arXiv:1912.10892†

The author of the present thesis helped in taking the data, and contributed in writing the manuscript.
This publication is also part of the PhD thesis of Philipp Ilzhöfer.

103

Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms
P. Ilzhöfer,1 M. Sohmen,1, 2 G. Durastante,1, 2 C. Politi,1 A. Trautmann,1
G. Morpurgo,3 T. Giamarchi,3 L. Chomaz,2 M. J. Mark,1, 2 and F. Ferlaino 1, 2 ,*1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik und Zentrum für Quantenoptik,

Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

3 DQMP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

arXiv:1912.10892v1 [cond-mat.quant-gas] 23 Dec 2019

(Dated: December 24, 2019)

A supersolid is a fascinating phase of matter, combining
the global phase coherence of a superfluid with hallmarksof solids, e. g. a spontaneous breaking of the translational
symmetry. Recently, states with such counter-intuitive
properties have been realized in experiments using ultracold quantum gases with strong dipolar interactions.
Here, we investigate the response of a supersolid state tophase excitations which shatter the global phase coherence. After the creation of those excitations, we observea rapid re-establishment of a global phase coherence, suggesting the presence of a superfluid flow across the wholesample and an efficient dissipation mechanism. We areable to identify a well-defined region where rephasing occurs, indicating the phase boundary between the solid-likeand the supersolid phase. Our observations call for the development of theoretical descriptions able to capture thenon-equilibrium dynamics in the recently discovered supersolid states of quantum matter.
The notion of phase coherence lies at the foundation ofquantum physics. It is considered a master property in understanding many-body quantum phenomena [1, 2], ranging
from superfluidity and the Josephson effect to the more applied examples of matter-wave interference, atom lasing processes, and quantum transport in mesoscopic and macroscopic
systems. A coherent state can be described in terms of single amplitude and phase fields. However, the phase itself is
not a physical observable and the study of coherence relies on
measurements of phase differences between a set of coherent
matter waves. In the context of atomic Bose–Einstein condensates (BECs), sets of spatially separated clouds have beencreated, for instance, by splitting a BEC into two or more parts
or by loading an ultracold gas into an optical lattice or into adouble-well potential [3].
The reverse process, i. e. in-trap merging of BECs, and therelated study of the phase evolution, is a much less exploredand more subtle problem, invoking, for instance, the growthof thermal correlations in isolated systems [4], the complex
interaction-mediated collapse and revival of many-body coherence [5], dissipative dynamics [6], or even the exponential
growth of unstable modes and topological defects in connection with the Kibble-Zurek mechanism [7–9]. Despite important theoretical and experimental progress, no generic framework exists yet to understand the quantum-phase evolution
and relaxation dynamics in quantum many-body systems outof equilibrium [10].
The physical understanding of the dynamical reestablishment of coherence remains even more elusive

for many-body quantum states that feature a spontaneous
breaking of the translational symmetry. Prime examples are
the supersolid states. For a long time mainly considered atheoretical notion [11–14], such states have been recently
observed in quantum gases [15–19]. These supersolids canbe seen as coherent matter waves with short-wavelength
modulations – shorter than the system size. Remarkably,in dipolar quantum gases the density modulation is not
imprinted by external fields but truly emerges from the
many-body interactions between atoms. Here, the symmetry
breaking is driven by the interplay between short- (contact)
and long-range (dipolar) interactions [20–25] and is connected to the softening of the roton mode in the excitation
spectrum [26, 27].
The density modulation is predicted to robustly surviveboth in the limit of infinite system size [21] and trapped quantum gases [17, 19]. In the latter case, for a given trap geometry, the modulation contrast can be controlled by tuning the scattering length as – parametrizing the contact interaction – or changing the atom number, N, in the system. The different quantum phases of a cigar-shaped dipolar quantum gas with vertical dipole orientation are shownin Fig. 1a. The phase diagram is constructed by numericallysolving the extended Gross-Pitaevskii equation (eGPE; Methods) [19], describing our trapped system and including therecently discovered quantum-fluctuation-driven stabilization
mechanism [28–31]. The color map encodes the strengthof the density modulation via the number Ce = 1 − C, with
C = (nmax − nmin )/(nmax + nmin ) the dimensionless modulation contrast. Here, nmax (nmin ) is the density maximum (minimum) in the central region of the calculated in-situ density
distribution. Ce equals unity for a non-modulated and zero fora fully modulated state.
The phase diagram shows three distinct regions; see Fig. 1a.
For large enough as , the system is a non-modulated dipolar
BEC with Ce = 1 (grey region). By lowering as , the influence of the dipolar interaction increases. When reaching a
critical value of as , the system undergoes a phase transition
to a supersolid phase (SSP). Here, a density modulation with
Ce < 1 appears in the ground-state density profile (red region).
By further lowering as , the system evolves into an array of independent droplets (ID) with an exponentially vanishing density link between individual droplets and Ce approaching zero
(blue region).
Recent experiments have shown a connection between thestrength of the density modulation and the coherence proper-

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0

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0
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th (ms)Phase scramblingchange aS

scramble timeb3

Rephasingchange aS

hold timetS
θ0th = 100 ms

θ0thθ1 θ 2 θ 3 θ 4

θ 1 θ 2 θ 3 θ4θ‘1 θ‘2 θ‘3 θ‘4Fig. 1 | Phase diagram, experimental sequence, and starting conditions. a, Ground-state phase diagram for our cigar-shaped
e The grey color indicates a non-modulated BEC, while the red andtrapped 164 Dy gas. The color map shows the values of C.
blue regions correspond to a SSP and ID phase, respectively. The insets show illustrations of the density profiles along the weakaxis for the different phases. b, Illustration of the phase scrambling sequence: Starting from a SSP (b1), we reduce as to enter
the ID regime (b2). During tS , the phases of the droplets can evolve differently, leading to a phase scrambling between theindividual droplets. Eventually, we jump as back to its initial value, re-entering the supersolid regime (b3), where we study thetime evolution of the global phase coherence. c, Amplitudes AM and AΦ , and d, ∆Φ for our evaporatively cooled SSP plotted
over th . Each data point is derived from q = 80–90 individual experimental realizations. The error bars (almost covered by plot
symbols) are the one-σ confidence intervals calculated using a bias-corrected accelerated bootstrapping analysis (Methods) [32].
e, Polar scatter plot for Pi and f, histogram of the probability density function (PDF) for Φi at th = 100 ms.
ties of the system, revealing a clear difference between the
SSP and ID phase [17–19]. In the SSP, a global phase ispresent along the whole system, whereas, in the ID case, phasecoherence is absent. The latter behavior can be understoodby considering that any fluctuation or excitation within a single isolated droplet will drive an independent evolution of thephases, which cannot lock to each other since particle flow isabsent [3]. This type of dephasing has been studied in split
BECs and atomic Josephson-junction arrays [3].
While the phase evolution when moving from a SSP to an
ID can be understood intuitively, highly fundamental and nontrivial questions arise when considering the opposite route,
i. e. when phase-incoherent isolated droplets are linked back
together. First, will the out-of-equilibrium system spontaneously re-establish phase coherence? And, if yes, will
it relax into its supersolid ground state or reach a quasistationary state? Second, which mechanism sets the rephasing
timescale? Finally, whereto will the excitation energy be dis-

sipated? Many-body quantum descriptions, as e. g. a standard
eGPE approach, are often inherently phase coherent and thus
cannot capture such types of non-equilibrium dynamics.
Here, we take first steps to experimentally answer thosequestions by studying the out-of-equilibrium phase dynamics of a supersolid state after a phase-scrambling excitation.
Our excitation scheme relies on an interaction quench and exploits the different coherence characters of the SSP and ID
phase, as illustrated in Fig. 1b. In particular, after preparing a dipolar quantum gas in the SSP via direct evaporative
cooling (b1), we drive the system into the ID regime by lowering as (b2). Here, we observe that the phase coherence
gets quickly lost while the system remains density modulated.
When going back to the parameter regime where the SSP is
again the ground state (b3), we observe efficient rephasing
dynamics, re-establishing the global phase coherence of thesupersolid. Our measurements indicate the presence of superfluid flow with particles delocalizing across the density mod-

3

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ulated gas as well as a dynamical mechanism dissipating thecreated phase excitations.
As starting point for the experiments, we produce the initial supersolid state by direct evaporative cooling from a thermal sample. As demonstrated in Ref. [19], this is a powerfulapproach to create a long-lived supersolid state with a highdegree of phase coherence. For the present work, our supersolid state contains about N = 1.4 × 104 164 Dy atoms and isconfined in an axially elongated optical-dipole trap of harmonic frequencies ωx,y,z = 2π × (225, 37, 165) s−1 . During
the whole evaporation sequence, we apply a vertical magnetic
field of B = 2.430(4) G to set the dipole orientation and thedesired as -value in the SSP region.
Our investigation relies on the ability to probe the system’sphase coherence and density modulation, whose co-existence
is a hallmark of supersolidity. To this aim, we developed
an analysis based on matter-wave-interference [17, 19, 33–36], which is capable of capturing the degree of phase coherence and the density-modulation strength (Methods). In brief,for each individual experimental realization i, we take an absorption image after a time-of-flight (TOF) expansion, which
exhibits an interference pattern in case of an in-situ density
modulation. Via Fourier transform, we extract the phasor
Pi = ρi · e−i·Φi , revealing the amplitude ρi and phase Φi at the
spatial frequency of the interference pattern. Whereas a single
Pi characterises the degree of density modulation, the statistical average over an ensemble q of many realizations reveals
information about the global phase coherence. We calculate
the phase amplitude, AΦ = |hPi i|, and the density-modulation
amplitude, Aq
M = h|Pi |i, as well as the circular phase variance
2
2
q
q
1
∆Φ = 1 − q∑i=1 cos (Φi ) + ∑i=1 sin (Φi ) [37]. We
note that for a perfect supersolid (resp. ID) state and in thelimit q → ∞, AΦ = AM > 0 (resp. AΦ = 0, AM > 0) and ∆Φ = 0(resp. 1).
To demonstrate the power of this analysis, we apply it to our
initial state, whose supersolid properties have been previously
investigated [19]. As shown in Fig. 1c–d, AΦ , AM and ∆Φ areroughly constant during holding times th up to 100 ms. We
observe almost equal values for AΦ and AM and a mean valueh∆Φi = 0.142(8), confirming a high degree of global phasecoherence for our density-modulated initial state. For th =100 ms, we also show a polar plot of Pi (Fig. 1e) as well as
the corresponding histogram for Φi (Fig. 1f), both displaying
a narrow distribution.
After preparing the initial supersolid state, we apply ourphase-scrambling protocol; see Fig. 1b2. We ramp the Bfield within 20 ms from 2.43 G (SSP) to 1.65 G (ID phase)and let the system evolve for a variable time tS . Exploiting the magnetic-field tunability of as via Feshbach resonances, the B-field ramp corresponds to a change from about
as,SSP = 88 a0 to as,ID = 77 a0 (Methods). As shown in Fig. 2a,
we observe a rapid initial increase of ∆Φ on a time scale oftS ' 20 ms [38], after which ∆Φ saturates close at a meanvalue of h∆ΦitS ≥30 ms = 0.92(2). We note that the saturation
value is not expected to reach unity because of our finite sample size (q ' 95). Indeed, it is comparable to the one calculated from a toy model, which considers a sample with the

0.5
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100
80

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ts = 100 ms
Fig. 2 | Phase scrambling. a, ∆Φ as a function of tS for
the ID phase at 1.65 G (as,ID = 77 a0 ). Each point is derived
from q = 90–100 independent experimental realizations. The
error bars are the one-σ confidence intervals calculated using a bias-corrected accelerated bootstrapping analysis (Methods) [32]. The grey shaded area indicates the theoretical oneσ confidence interval for ∆Φ using the same sample size anda uniformly random phase. The inset shows the according AΦ(cyan) as well as AM (blue). b, Polar scatter plot for Pi and c,histogram of the PDF for Φi at tS = 100 ms.
same q and fully random (i. e. uniformly distributed) phases(Methods).
Simultaneous to the increase of ∆Φ, we observe that AΦ
decreases quickly towards zero, while AM slightly increases.
This behavior shows that the density modulation is maintained
while losing global phase coherence; see inset. As expected,
in the ID phase, quantum and thermal fluctuations as well as
atom losses can give rise to a different time evolution for thephases of the individual droplets. Apparently the vanishingsmall density overlap between droplets (Ce ' 0) prevents anefficient phase locking, resulting in the observed loss of global
phase coherence.
We now move to the core of our experiment and investigate the phase-relocking after the phase scrambling. We set
as back to its initial value, i. e. where the system’s groundstate is a supersolid, by a B-field jump, and study the system’s evolution; see Fig. 1b3. As shown in Fig. 3a, we observe first a rapid reduction of ∆Φ, occurring in the first
20 ms, and then a much slower dynamics with ∆Φ saturating

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Fig. 3 | Rephasing dynamics. a, ∆Φ as a function of th after
a jump from the ID phase back to the SSP regime at 2.43 G(as,SSP = 88 a0 ). For each point q = 66–74. The error bars
are the one-σ confidence intervals calculated using a biascorrected accelerated bootstrapping analysis (Methods) [32].
The solid black line is an exponential fit to guide the eye. The
inset shows the according AΦ (light red) as well as AM (red).
b, Polar scatter plot for Pi and c, histogram of the PDF for Φiat th = 0 ms and at th = 100 ms (d–e).
to h∆Φith ≥30 ms = 0.20(2). Accordingly, AΦ approaches AM
on the same time scale, whereas AM remains nearly constant.
This re-establishment of global phase coherence is further illustrated with individual polar scatter plots and histogramsin Fig. 3, confirming a reduction of the phase distribution’s
width with increasing th .
Our system of multiple superfluid parts with different
phases interconnected via weak links is reminiscent of a
Josephson-Junction array (JJA) [39], opening the question
whether a JJA framework can capture the main ingredientsof our system’s dynamics. Although our array of droplets
is soft, meaning that the droplets’ shape and their distancechange with as , we construct a simple model in terms of a onedimensional array of coupled grains (Methods). This is justi-

fied as the strongest effect of the change of the system’s statewith as is the change of the wavefunction overlap between thedroplets, i. e. the tunneling rate. Using this model, we simulate
quenches of the tunneling rate and look at the time evolutionof the correlation function of the phases in the array, whichcorresponds to the experimental observable AΦ /AM .
The model gives dephasing and rephasing dynamics, similar to the observations of Fig. 2-3. A more quantitative description goes beyond the scope of this paper. It would require (i) to find proper relations between the parameters of
the JJA model and the real system, (ii) to achieve a macroscopic modelling of the dissipation mechanisms by including coupling with a thermal bath and/or with the excited
droplet modes [40], or even (iii) to go beyond the hard-grainmodel. Even in experiments with non-dipolar coupled quasicondensates, realizing a case closer to an ideal JJA, the observed phase dynamics and full phase-locking have no theoretical explanation up to now [41]. Another important ingredient in the phase relaxation dynamics is the phase defects
formed at the boundaries between the distinct grains when
they merge [6, 8, 9, 42]. These defects, forming e. g. solitons, are expected to propagate and interact with each otherand with excitations from the thermal bath, and thus eventually decay.
To further investigate the role of the density links amongdroplets – i. e. the Josephson coupling –, we study the rephaseing dynamics as a function of the theoretically calculated C;
see Fig. 1a. Although our system is out of equilibrium, we use
the ground-state quantity Ce as an estimate for the strength ofthe density link [17–19]. For each value of as , we assign Ce
and record the time trace of ∆Φ for different th . As shownin Fig. 4a, we see different rephasing dynamics depending one In the case of small C,e associated with the ID regime, noC.
rephasing occurs with ∆Φ remaining large (> 0.5) for all th(blue region). As Ce and thus the link strength increases, thesystem starts to rephase with ∆Φ approaching a small saturation value (≈ 0.15) at long evolution times (red region).
The time traces clearly show the existence of two regimes
for ∆Φ, one in which phase re-locking occurs and one inwhich the system remains incoherent. To further investigate
these regimes and their interface, we study the long-time dynamics of AΦ and AM and record their asymptotic values. As
shown in Fig. 4b, AM remains large and shows only slight varie This indicates theations over the full investigated range of C.
persistence of density modulation in the system. In contrast, a
striking change is found in the evolution of the ratio betweenAΦ and AM . At large Ce > 0.01, AΦ and AM are nearly equal.
This shows the re-establishement of a global phase coherence, and the relaxation towards a SSP. Differently, at small
Ce < 0.001, AΦ nearly vanishes while AM remains large andalmost constant, evidencing a final phase-incoherent state (IDe AΦ and AM show an in-between
regime). At intermediate C,
behavior with AΦ smaller than AM but non-vanishing, showing a partial recoherence of the state. These three distinct
behaviors are also reflected in the asymptotic values of ∆Φ(Fig. 4c), showing full recoherence (∆Φ ≈ 0.15) in the supersolid regime, persistence of a full incoherence (∆Φ ≈ 0.9) in

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Fig. 4 | Time traces of the phase dynamics and their characterization. a, Temporal evolution of ∆Φ (color map) after thee For each th we record q ≥ 35 individual experimental
complete phase scrambling sequence plotted as a function of th and C.
realizations. In the red region (∆Φ ' 0) the system has recovered its global phase coherence, while for the blue one (∆Φ ' 1)
no global phase coherence is present. b, AΦ (light red) and AM (red), c, saturation value (∆Φ)sat and d, rephasing rate |R| ase AΦ , AM , and (∆Φ) are the mean values at long th . The grey shaded area indicates the theoretical one-σ
a function of C.
satconfidence interval for ∆Φ using the same sample size as the experiment and a uniformly random phase.

e i. e. the differA further question is whether the value of C,
ent regimes, also dictates the speed of the rephasing dynamics. To explore this aspect, we study the early time dynamicsof ∆Φ by performing a linear fit to the data for th ≤ 20 ms. The
extracted slope characterizes the initial rephasing rate |R|. As
shown in Fig. 4d, in the supersolid regime, we always record
a large rephasing rate, which remarkably is roughly indepene with |R| ≈ 30s−1 . This value is comparable todent of C,
the weak-axis trap frequency, ωy /2π, and compatible with thetime needed for a sound wave or soliton to propagate along thesystem [8, 43]. In contrast, when crossing from the supersolid
to the intermediate regime, we observe a sudden decrease of|R| by almost a factor of two. Evolving from this intermediate
regime to the ID, |R| continuously decreases with decreasing
Ce until it vanishes. While such a decrease of |R| is consistent
with a JJA picture in which the tunneling between the droplets
dictates the rephasing dynamics, the underlying reason for aconstant rephasing rate in the supersolid regime remains an
open question. It might indicate the action of other mechanisms, related for instance to the soft nature of our JJA, or theformation and slow decay of phase defects in the array.
In conclusion, we have reported the first study of theout-of-equilibrium dynamics of a dipolar supersolid after aninteraction-driven phase excitation that fully randomizes thephases. In the SSP regime, we have demonstrated that thesystem re-establishes a high-degree of phase coherence on
the timescale of one trap period by almost perfect rephasing. When tunneling is suppressed by a too weak density
link across our spontaneously-modulated quantum state, therephasing substantially slows down at a rate depending on thetunneling and eventually ceases in the deep ID regime. Our
observations might shed new light on the properties of the particle flow in a SSP and its superfluid properties, whose general understanding is still elusive. Future experimental works,
combined with advanced out-of-equilibrium theoretical models, will be crucial to understand the relaxation dynamics anddissipation mechanisms in isolated and open supersolid statesof quantum matter.

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• We are grateful to S. Erne, J. Schmiedmayer and the
ERBIUM team for insightful discussions. We acknowledge R. M. W. van Bijnen for developing the codefor our eGPE ground-state simulations. This work
is financially supported through an ERC Consolidator
Grant (RARE, No. 681432), a NFRI Grant (MIRARE,
No. ÖAW0600) from the Austrian Academy of Science and a DFG/FWF (FOR 2247/PI2790), and by the
Swiss National Science Fundation under Division II.
M. S. and G. D. acknowledge support by the Austrian
Science Fund FWF within the DK-ALM (No. W1259N27). A. T. acknowledges support by the Austrian
Science Fund FWF within the Lise-Meitner-Programm(No. M2683-N36). We also acknowledge the Innsbruck
Laser Core Facility, financed by the Austrian Federal
Ministry of Science, Research and Economy. Part ofthe computational results presented have been achieved
using the HPC infrastructure LEO of the University of
Innsbruck.
• Competing Interests: The authors declare no competinginterests.
• Author information: Correspondence to F. F. (e-mail:Francesca.Ferlaino@uibk.ac.at).
• Author contributions: P. I. , G. D. and A T. conducted the experiment and collected the experimental
data. M. S. and C. P. analyzed the data. M. J. M. performed the eGPE ground-state simulations. G. M. and
T. G. contributed to the theoretical analysis. All authors
contributed to the writing of the paper. F. F. supervised
the project.

8
I. METHODS
A.

Phase diagram and contrast.

Our numerical calculations of the ground-state phase diagram of a cigar-shaped 164 Dy dipolar quantum gas follow
the procedure described in our earlier works [19, 27]. In
brief, the calculations are based on minimizing the energyfunctional of the extended Gross-Pitaevskii equation (eGPE)
using the conjugate-gradients technique [44]. The eGPE
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 [29, 30]
and [45]. From the derived three-dimensional wavefunction
ψ(r) we calculate
the one-dimensional in-situ density proR
file n(y) = |ψ(r)|2 dxdz. We evaluate the in-situ density
contrast C = (nmax − nmin )/(nmax + nmin ) for profiles whichfeature density modulations by searching central extrema ofn(y) and determining the overall maximum (nmax ) and minimum (nmin ) value. For profiles without density modulation(ordinary BEC), we set C = 0. We use the quantity Ce = 1 −C
to estimate the density link between the droplets, which isconnected to the tunneling strength [19].
B. Experimental sequence.

We apply our phase scrambling protocol to the evaporatively cooled SSP of 164 Dy atoms [19]. For this, we initiallyload our 5-beam open-top magneto-optical trap (MOT) for 3 s
and apply a MOT compression phase, which lasts 400 ms [46].
We then load about 8 × 106 atoms into a single-beam horizontal optical dipole trap (hODT), propagating along our y-axis.
The hODT is derived from a 1064 nm focused laser beam. After loading, we apply forced evaporative cooling by exponentially reducing the optical power in the hODT for 0.9 s. Subsequently, we switch on a second ODT beam along the verticalz-axis to form a crossed ODT and continue with the last stageof evaporative cooling for 2 s [47], until the SSP is reached.
During the entire evaporation sequence, a magnetic field of
B = 2.43 G, pointing opposite to gravity along our z-direction,
is maintained. The final trap geometry is cigar-shaped withharmonic frequencies ωx,y,z = 2π × (225, 37, 165) s−1 . After the initial-state preparation (SSP), we apply our phasescrambling protocol. For that, without any additional waitingtime after the evaporative cooling, we change the B-field to
1.65 G deep in the ID regime. Here, we allow the system’s
global phase to freely evolve for tS = 20 ms. We have explored two types of protocols: jumping, which results in an
effective ≈ 1 ms change of the B-field due to the finite time
response of the system, and ramping within 20 ms. We observe a similar scrambling behavior in ∆Φ for both the jump
and the ramp protocol. We complete our phase-scrambling sequence by jumping the B-field back to its initial value and byletting the system evolve for a variable hold time th . Finally,
we perform a matter-wave interference-type experiment during time-of-flight (TOF) expansion and record the resulting

interference pattern by absorption imaging. A TOF durationof 26.5 ms ensures a mapping onto momentum space. The
imaging beam propagates along x̃ in the horizontal x–y-planeat an angle of ∼ 45 ◦ with respect to the weak trap axis alongy.
C.

Tuning the scattering length.

To connect the experimental B-field values with thecontact scattering length as , we use the well establishedformula for overlapping Feshbach resonances as (B) =abg ∏i (1 − ∆Bi / (B − B0,i )) [48], with B0,i the poles, ∆Bi thecorresponding distance from the pole to the zero-crossing andabg the (local) background scattering length. We determine
the poles and zero-crossings in our B-field region of interestby performing loss spectroscopy and thermalization measurements. Starting from a thermal cloud prepared at 2.55 G wefirst ramp the magnetic field to the final value within 5 ms,
then lower the trap depth to its final value within 50 ms, andwait an additional hold time of about 400 ms. In absence ofFeshbach resonances, we typically end up with a thermal gasof 5 × 105 atoms with a temperature of about 500 nK. Whenscanning the magnetic field in our region of interest, we observe several atom loss features together with peaks in theatom cloud temperature, which we fit with gaussian functions
to extract the positions of the poles B0,i and the widths ∆Bi .
The value of the background scattering length of 164 Dy
is a more subtle topic, as several measurements give varying values in the range between 60–100 a0 [49]. These
measurements were using different methods (e. g. crossthermalization, theory-experiment comparisons of oscillation
frequencies), different initial states (thermal gases and quantum droplets) and were performed at different magnetic fields.
Especially the existence of very broad resonances at higher
magnetic fields [50] will affect the measured local backgroundscattering lengths. Therefore, we set the value of abg in sucha way that the B-to-as conversion reproduces the calculated
critical scattering length as = 91 a0 at the experimentally estimated phase transition point between BEC and SSP around
2.5 G. This gives a value of abg = 73 a0 which lies within theerror bars of the latest published value of abg = 69(4) a0 [49].
Extended Data Figure 1 shows the resulting calculated B-to-as
conversion from which we estimate as,SSP = 88 a0 at 2.43 G in
the SSP and as,ID = 77 a0 at 1.65 G in the ID as used in theexperiment.
D.

Interference pattern analysis.

Our analysis is similar to the one described in Ref. [19]. We
record q = 30–100 experimental repetitions for each parameter set P. Each recorded picture i (i = 1 . . . q) is processedby first subtracting the thermal background via a symmetric
2D-Gaussian fit to the wings of the density distribution. Next,
we recenter the image of the degenerate cloud and integrate
its central region, where the matter-wave interference signal
is concentrated, along the z-direction within ±2 µm−1 . We

100
80
60
1.5

1.7

1.9

2.1

B (G)

2.3

b 10

0.2

2.5

Extended Data Figure 1 | Estimated scattering length. Calculated B-to-as conversion for 164 Dy. Red and blue shaded
areas indicate the SSP and the ID region, respectively. The
grey area indicates the BEC region, while the yellow areas
indicate regions around the two narrow Feshbach resonances
located at 2.174 G and 2.336 G where we observe increased
atom loss. We estimate as,SSP = 88 a0 in the SSP at 2.43 G
and as,ID = 77 a0 in the ID at 1.65 G.
obtain a momentum density profile which we normalize by its
sum. From such a momentum profile, a fast Fourier transformation (FFT) yields the 1D density profile ni (ỹ). An in-situ
density modulation in an atomic cloud will lead to side peaks
in ni (ỹ), symmetrically centred around the zero-momentum
peak. To isolate the centre of this specific modulation, we
calculate the incoherent and coherent means of ni (ỹ), which
we denote nM (ỹ) = h|ni (ỹ) |iP and nΦ (ỹ) = |hni (ỹ)iP |, respectively. The incoherent mean nM reflects the mean modulation amplitude of the cloud at the respective wavelengthỹ. The coherent mean nΦ . nM if the phases of the interference pattern among the q repetitions at the respective ỹ areroughly constant, and nΦ → 0 (and hence nΦ  nM ) if thephases are random. Therefore, the most pronounced difference nM − nΦ is observed for the ID regime (see Extended
Data Fig. 2a). From the maximum of this difference we read
off the modulation wavelength (or ‘droplet distance’) ỹ ≡ d.
The FFT phasors at d we call Pi = ni (d) = ρi · e−i·Φi , yielding sets {P1 , . . . , Pq }P . To characterise the distribution ofphases Φi withincalculate the circular varianceq our sets, we
2
2
q
q
1
∆Φ = 1 − q∑i=1 cos (Φi ) + ∑i=1 sin (Φi ) [37]. For aphase-coherent sample, and hence interference fringes stablewithin the envelope, ∆Φ is small, whereas for an incoherent
sample it approaches unity. To estimate the confidence intervals of our circular variance data we apply a bias-corrected accelerated bootstrapping scheme [32] for each P, resampling
106 times from the respective q experimental values.
E.

0.4

PDF

a

120

nM – n (a.u.)

Scattering length (as )9Effect of finite sample size

Even the circular variance ∆Φ of a sample of q angles
Φ1 , . . . , Φq drawn from a completely random distribution willapproach unity only in the limit q → ∞. To estimate the fullyincoherent limit of ∆Φ for our finite q, we calculate 106 values for ∆q Φ, each for q independent draws from a theoretical,
uniform distribution in [0, 2π). The histograms of ∆q Φ areshown in Extended Data Fig. 2b. The indicated one-σ confi-

0-10 -5

0

m

5 10

5
0

0.6

0.8

1

q

Extended Data Figure 2 | Wavelength of the modulationand finite-sampling effect. a, Difference between incoherent and coherent mean of the density profiles in the ID regime(1.65 G), peaking at the modulation wavelength d ' ±2 µm(dashed lines). b, Histograms of 106 realisations (each) for
calculations of ∆q Φ from uniformly random phases Φi , forq = 35 (green) and q = 100 (yellow) draws, respectively. The
dashed vertical lines reflect the confidence interval enclosing
68.3 % (‘one σ ’) of the calculated values. The solid lines reflect a Beta distribution with same mean and variance as thedrawn distributions of ∆q Φ (no free fit parameters).
dence intervals are [0.77, 0.93] for q = 35 and [0.86, 0.96] for
q = 100 draws. We note that the histograms of ∆q Φ follow a
Beta distribution [51], even if one generalizes the underlying
distribution of phases Φi to a von Mises distribution, of which
the uniform distribution is just a degenerate case.
F.

Interference pattern analysis and simple model of a droplet
array

For simplicity let us consider here that the state is made of
ND identical droplets. In that case the total wavefunction of
the system would be
NDψ (x, y, z) = ∑ f (x, y − R j , z) eiθ j

(1)

j=1

where R j is the spatial coordinate of the jth droplet, θ j is
its phase, taken to be uniform over the droplet, and f is the
wavefunction of a single droplet localized around y = 0. With
such a wavefunction, the phasor extracted from one realization would be
Pi =

Z

dky

ND

∑ eiky (R j1 −R j2 −d) ei(θ j1 −θ j2 ) | f˜ (ky ) |2(2)

j1 , j2 =1

where f˜ is the Fourier transform of the function f and d isthe distance between neighboring droplets d = hR j+1 − R j i. It
simplifies in
Pi =

ND

∑ g (R j1 − R j2 − d) ei(θ j1 −θ j2 )(3)

j1 , j2 =1≈ g (0) ∑ ei(θ j+1 −θ j )
j

(4)

10
with g (y) the Fourier transform of | f˜ (ky ) |2 , which is thus apeak function with a width of the order of the droplet size.
Formula (4) yields
AM = h|Pi |iP ≈ (ND − 1) g (0) ,(5)which is essentially independent on the phase relation between the droplets and shows only a weak dependence on thedroplets’ shape. Here h.iP denote the average over an ensemble of realizations P.
On the contrary the function AΦ = |hPi iP | contains the average of the phases with
|hPi iP |AΦ=' |hhei(θ j+1 −θ j ) i j iP |.
AM
h|Pi |iP(6)

h.i j denotes the average over the droplet array. The ratio AΦ /AM thus measures essentially the mean difference ofphases between two neighboring droplets in the array. We
also read from Eq. (4) that the phase of the phasor is Φ ≈hθ j+1 − θ j i j .
The circular variance ∆Φ for q realizations can be expressed
as
v
u q
q
1u
(7)
∆Φ = 1 − t ∑ eiΦi1 ∑ e−iΦi2
q i1 =1i2 =1For a totally phase coherent state, Φ = 0 for all realizations
leading to ∆Φ = 0 while for a totally phase incoherent sample
only the diagonal terms in (7) survive, leading to ∆Φ = 1 − √1q
for q independent measurements.
G.

Modelization by a Josephson junction array

Let us modelize the system by a set of independent droplets,
each one having a number of particles N j (whose average will
be denoted N j ) and a phase θ j . The Hamiltonian of such asystem is
H =∑
j

"

Nj − Nj
2C j

2− J j cos θ j+1 − θ j

#


(8)

The first term is the “charging” energy of the droplet (corresponding to its mean interaction energy) with the “capacitance” C j while the second term describes the Josephson tunnelling of particles between droplets with the Josephson amplitude J j . This is the well known JJA description [39]. Such a
description is well adapted if the droplets are reasonably well
separated in space, and thus should work adequately when the
SSP is established. For simplicity we assume in the subsequent calculations that all parameters C j and J j are independent of the droplet and their values are later denoted C and J
respectively. In addition, C is taken as constant.

H.

Time evolution for the JJA model

To follow the experimental protocol, we used the equilibrium state of our Hamiltonian as initial state and looked at its
evolution when we applied quenches to the Hamiltonian. We
took a set of 4 droplets with, for simplicity, periodic boundary
conditions. First, we quenched our initial state with a Hamiltonian having a JS much smaller than the original J for a given
time tS . This corresponds to a decrease of the tunneling between droplets and therefore lets them evolve independently
from each other. Then, we quenched our state again by letting
it evolve a time th with the original Hamiltonian (J). Thismeans that we reinstate the original tunneling between thedroplets which is what happens in the experimental protocol.
We then looked at the correlation function| hψ (t)| heiθ j+1 e−iθ j i j |ψ (t)i | which corresponds to AΦ /AM .
Given the perfect coherence that exists in the JJA model
described above, this correlation function would show undamped oscillations corresponding to the time periodic natureof a system with a finite number of frequencies. In order tomake the plateaus apparent in the correlation function we havedamped these oscillations by an artificial damping term e−ωk t
for the mode with a frequency ωk . The choice of such damping rather than the usual constant exponential one, is to get
rid efficiently of the high frequency oscillations without having to recourse to the coupling to a bath for example. It isclear that a more precise and microscopically correct way of
including the damping should be considered, but as discussed
in the main text, what mechanism leads to damping is a wholequestion in itself in this system.
From time 0 to ts , this correlation function shows the decrease in AΦ /AM and corresponds qualitatively to the phasescrambling as can be seen in Extended Data Figure 3a. The
first minimum in the figure and the corresponding time scale
would correspond to the dephasing discussed in the main text.
In total absence of residual coupling between the droplets
one would have lost completely the phase coherence on thistimescale. Because we have put a small but finite coupling JS
remaining between the droplets one can also see at later times
a partial recovery of the phase coherence whose value is ofcourse controlled by the value of JS .
Furthermore, if looked up to time th , this correlation function shows the increase in AΦ /AM and therefore corresponds
to the rephasing of the system as shown in Extended Data Figure 3b. In the calculation, the correlation does not go backto a value of 1, which means that the rephasing is not perfect, in contrast with the experiment. This happens since thissimplified model has no energy dissipation mechanism. The
initial state not being an eigenstate of the final Hamiltonian
thus leads to a final state which is a thermal-like state (withpossibly more complicated distributions than a simple thermalone), where the extra energy has been converted to a distribution over the eigenstates. An energy dissipation mechanism,for example via the normal part of the fluid, will thus be necessary to converge back to the initial state. Such extra effects
can be potentially incorporated in subsequent studies.

11

a

b
1

1

0.5

0.5

0
0

50

100

0
0

5

10

15

Extended Data Figure 3 | Theoretical predictions for the
JJA model with parameters J = 100, JS = 1, C = 1, h̄ = 1 and
j1 , j2 are neighbours. a, Evolution of the correlation when thedroplets evolve independently from each other as a function ofscrambling time tS . b, Evolution of the correlation when thedroplets are p
re-coupled to each other as a function of th withtS = 1000 [h̄ C/J ].

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Appendix

B

Extensive Feshbach scans

The following appendix collects the extensive Feshbach spectroscopy required for Ref. [Dur20].
These exploratory surveys were performed with a resolution of 200 mG in order to identify
possible FRs of width greater than ∼1 G. All the experimental conditions are kept constant beside the laser locking parameters necessary to change the isotopic combination. Ourinvestigation covered the following mixtures:• 166 Er-164 Dy in Fig. B.1,
• 168 Er-164 Dy in Fig. B.2,
• 170 Er-164 Dy in Fig. B.3,
• 166 Er-161 Dy in Fig. B.4,
• 167 Er-161 Dy in Fig. B.5,
• 168 Er-161 Dy in Fig. B.6,
• 170 Er-161 Dy in Fig. B.7.

117

30

40

190

140

90

50

200

150

100

0

0.5

0
300
1

0.5

0
250
1

0.5

0
200
1

1

20

180

130

80

1

10

170

120

70

0.5
0

160

110

60

0.5
0
1
0.5
0
50
1
0.5
0
100
1
0.5
0
150

360

Magnetic field (G)

260

210

270

220

280

230

340

290

240

350

300

250

440

330

420

320

400

310

380

Figure B.1: Exploratory trap-loss spectroscopy in the mixture 166 Er-164 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

118

Normalized atom number

119

0
150

0.5

0
100
1

0.5

0
50
1

0.5

1

0

160

110

60

10

170

120

70

20

180

130

80

30

190

140

90

40

0
350

0.5

0
300
1

0.5

0
250
1

0.5

0
200
1

Magnetic field (G)

200

150

100

50

0.5

0.5

0

1

1

375

310

260

210

320

270

220

400

330

280

230

425

340

290

240

450

350

300

250

Figure B.2: Exploratory trap-loss spectroscopy in the mixture 168 Er-164 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

Normalized atom number

30

40

190

140

90

50

200

150

100

0
350

0.5

0
300
1

0.5

0
250
1

0.5

0
200
1

1

20

180

130

80

1

10

170

120

70

0.5
0

160

110

60

0.5
0
1
0.5
0
50
1
0.5
0
100
1
0.5
0
150

Magnetic field (G)

360

310

260

210

370

320

270

220

380

330

280

230

390

340

290

240

400

350

300

250

Figure B.3: Exploratory trap-loss spectroscopy in the mixture 170 Er-164 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

120

Normalized atom number

121

Figure B.4: Exploratory trap-loss spectroscopy in the mixture 166 Er-161 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

Figure B.5: Exploratory trap-loss spectroscopy in the mixture 167 Er-161 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

122

123

Figure B.6: Exploratory trap-loss spectroscopy in the mixture 168 Er-161 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

Figure B.7: Exploratory trap-loss spectroscopy in the mixture 170 Er-161 Dy (red and blue curves respectively) performed with a stepsize of 200 mG.
For the ease of reading, the atom numbers are normalized to the maximum of each panel. Each point consists of an average over two to four
experimental repetitions.

124

Appendix

C

Coil systems’ coolingIn this appendix, additional information on the actual performance of the coils’ water coolingsystem are reported together with a comparison to the simulated ones. Specifically, Fig. C.1
and Fig. C.2 summarize details about the large-field and the small-field coils respectively.
The simulations are performed in SolidWorks© on the 3D-model and they assume perfectcontact between the copper of the coils and the cooling surface. This simple model does nottake in consideration either small gaps filled with thermal conductive glue or the isolation
layer of the wire itself. The large-field coil is approximated with a solid copper block, heatedby a homogeneous power density equivalent to the total power dissipated at the steady-state
current of 100 A. Instead, for the small-field coils, the heating is modeled with an equivalent
power density at the steady-state current of 10 A acting at the lower part of the copper
cylinder.

125

(a)

(b)

30 °C

40 °C

Figure C.1: Steady-state temperature for one of the large-bias/gradient coils at 100 A of constantoperating current. The water runs clockwise from the inlet on the bottom-left of the figure to theoutlet on the bottom right. (a) Temperature measured with an infrared thermal camera after a
waiting time of a few minutes to ensure steady-state conditions. With ∼1.5 l/min of running waterat about 23 °C, the coil temperature rises up to 30 °C at the inlet and 40 °C at the outlet of the water
circuit. (b) Top view of the coil showing the simulated temperature with 1 l/min of running water at
20 °C. The increase in temperature of ∼11 °C at the warmest spot is consistent with the measured
value.

126

(a)

(b)

28 °C

51 °C

(c)
Figure C.2: Steady-state temperature of the inverted viewport coil module at 10 A of operating
current. The measurement has been performed with an infrared thermal camera after a waiting timeof a few minutes to ensure stable conditions. (a) Deactivating the water cooling, the coils temperaturerises to about 50 °C. (b) With ∼1 l/min of running water at about 23 °C, the coils temperature rises
up to 28 °C. (c) Sectional view of the copper mounting system showing the simulated temperature
with 1 l/min of running water at 20 °C. The increase in the temperature of 4 °C is consistent withthe measured value.

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Acknowledgments
Here I am at the end of this long process of personal growth that lasted about five years
called PhD. In these five years I got to know many people and I had the chance to meet evenmore of them. I had to face uplifting and upsetting moments, exciting hours, and frustrating
situations which are intrinsic in experimental physics research done at its top level. Finding
the right words to describe all this is quite challenging for me, nevertheless I will try my
best.
The first thanks go to my supervisor Francesca Ferlaino. She gave me the possibility ofjoining a group competing at the leading edge of the investigation of nature and she trusted
me for the development of an impressive experiment. In less than five years we saw thisexperiment growing with us, building something from scratch. I really pushed myself into
this project and I am personally proud of what we achieved. I hope this will be the case forall my present and future colleagues working in this laboratory. Even brighter prospects willcome, I am sure.
Another big thanks go to the people who assisted me with such hard work, which the PhD
is. My team mates (M.J.M., A.T., P.I., M.S., C.P., M.N.) and my old and new colleagues
from the Ferlaino’s group firstly, but also the whole ultracold group of the University of
Innsbruck and the people participating in the DK-ALM. We shared a big part of our life,
supporting and learning from each other. A special thanks goes to all the administrationand mechanical and electronics workshop personnel of the University and IQOQI because
all these big machines keep running just because of them. I wish the best to all of you.
There are tens of people I would like to acknowledge here, from my grandparents to myaunts and uncles, from university friends to schoolmates to my tireless holiday buddies. I
tend to forget things but, believe me, I bring all of you in my heart every day. These words
are for everyone of you who gave me a single smile over all these years.
During my PhD I had the chance to travel a lot, spending a significant amount of timeaway from the people I love. I am really grateful for this opportunity. Nonetheless, in aworld which is more connected every day and in which people are constantly aware of eachother, I understood the meaning of loneliness. During one of these journeys I lost a person
without having the possibility to say goodbye. This achievement is also for you Grandma.
Love is what makes humanity alive. It can throw us from the brightest stars in the sky
down to the darkest voids of our soul. Sometimes love paints the world with the most vivid
colors and sometimes it drains all the energy we have, harming us and hurting our hearts.
There is a person that I want to thank for all the good things she did in my life. I hope we
can forgive each other for our mistakes.
T.K. entered my life in a way that I will never be able to forget. She put together the lost145

pieces of myself, giving me renewed faith for what I believe in. Love is what makes us alive
and, after so many years, I finally am. She taught me that our dreams are more important
than anything and to follow them till they come true. Loving someone every single day poses
its challenges but there is nobody I would rather face them with. Wherever this will bringus, being together is the only thing which matters. With you I am genuinely happy cause I
understood the meaning of true love. You deserve the best that this life can donate. I love
you so. Thank you!
Generally, I would not be the person I actually am without my friends from Novaglie. We
are together since we came to life about 25 to 30 years ago. Ours is more than a simple
friendship, we are the family we chose to have. We shared great moments and fights, andeven though life is bringing us a bit further away from each other, I wish our friendship could
last forever. Without you I am simply lost. I want to sit with all of you in 50 years at thebench of our hometown and laugh about every little thing which happened in our lives. To
you I owe one of the greatest and most sincere thanks.
Lastly, all that I am goes back to the people closest to my heart. My family. My dad S.D.,
my mum P.P., and my sister E.D.. I think there exist no words able to describe what I haveinside. Maybe for this reason I find sometimes hard to express my feelings and sometimes I
forget to tell you how much I love you. Since I came to Innsbruck, I realized what it actuallymeans living. I understood how you put myself and my happiness — and my sister’s — infront of everything you might have wanted from your life. You pushed me to follow my own
path without imposing anything on me. You believed in me in the same moment I opened
my eyes for the first time. You raised and protected us despite all the efforts it required.
Sometimes I forget to tell you that I love you all. You are the best family I could havedreamed for. You are the best father, the best mother, and the best sister. I am proud of
you and I wish I could make you proud of me as well.
All this is for you family. You gave me more love than I could ever give you back.

Innsbruck, 31 August 2020

146

G. D.

Ringraziamenti
Eccomi alla fine di questo lungo processo di crescita personale durato cinque anni e chiamato PhD. In questi cinque anni ho conosciuto molte persone ed ho avuto la possibilità diincontrarne ancora di più. Ho dovuto affrontare sia momenti edificanti che di sconforto, ore
eccitanti ed anche situazioni frustranti intrinseche nella ricerca in fisica sperimentale compiuta al suo massimo livello. Trovare le giuste parole per descrivere tutto ciò è compito arduo
per me; ciononostante farò del mio meglio.
Il primo grazie va alla mia supervisor Francesca Ferlaino. Mi ha dato la possibilità diunirmi ad un gruppo all’avanguardia nello studio della natura, dandomi fiducia nello sviluppo
di un esperimento così ambizioso. In meno di cinque anni abbiamo visto questo esperimento
crescere con noi, costruendo qualcosa dal niente. Personalmente ho dato molto di me inquesto progetto e sono fiero di ciò che abbiamo raggiunto. Spero che lo stesso possa valereanche per i miei colleghi presenti e futuri in questo laboratorio. Un avvenire ancora migliore
è alle porte, ne sono certo.
Un altro grande grazie va a tutte le persone che mi hanno assistito in questo duro lavoro
che è il PhD. I miei compagni di team (M.J.M., A.T., P.I., M.S., C.P., M.N.) ed i miei vecchi
e nuovi colleghi del gruppo della Francesca, così come l’intero ultracold group dell’università
di Innsbruck e le persone partecipanti al DK-ALM. Abbiamo condiviso una parte importantedella nostra vita supportandoci a vicenda ed imparando gli uni dagli altri. Un grazie specialeè dedicato a tutto il personale dell’amministrazione e dell’officina meccanica ed elettronica
dell’università e dell’IQOQI poiché questa grande macchina continua a girare solo per merito
loro. Auguro il meglio a tutti voi.
Ci sono decine di persone che vorrei ringraziare in queste righe, dai miei nonni alle mie zie
e zii, dai soci dell’università ai compagni di scuola agli instancabili amici delle mie vacanze.
Tendo ad essere smemorato ma fidatevi, vi porto tutti nel mio cuore ogni giorno. Queste
parole sono per tutti voi che mi avete concesso anche solo un sorriso durante tutti questi
anni.
Durante il mio dottorato ho avuto la possibilità di viaggiare molto, trascorrendo un notevole lasso di tempo lontano dalle persone che amo. Sono veramente grato per questa
opportunità. Tuttavia, in un mondo ogni giorno sempre più connesso nel quale le persone
sono costantemente al corrente di tutto, ho capito il significato del termine solitudine. Durante uno di questi viaggi ho perduto una persona senza la possibilità di dirle addio. Questa
conquista è anche per te nonna.
L’amore è ciò che rende vivo l’essere umano. Ci può sbattere dalle stelle più luminosedel cielo giù fino al nulla più buio della nostra anima. Talvolta l’amore dipinge il mondo
con i colori più intensi e talvolta ci prosciuga di tutte le nostre energie facendoci del male e147ferendo i nostri cuori. C’è una persona a cui vorrei dire grazie per tutte le cose belle che haportato nella mia vita. Spero di poterci perdonare per i nostri errori.
T.K. è entrata nella mia vita in un modo che non sarò mai capace di dimenticare, mettendo
insieme i pezzi mancanti di me stesso e dandomi rinnovata fede per ciò in cui credo. L’amore
ci rende vivi e, dopo tanti anni, finalmente lo sono anche io. Mi ha insegnato che i nostri
sogni sono più importanti di ogni altra cosa e d’inseguirli finché non si realizzano. Amare
qualcuno tutti i giorni pone costantemente delle sfide ma non c’è nessun altro con cui leaffronterei. Ovunque questo ci porterà, essere insieme è l’unica cosa che conta. Con te sono
sinceramente felice giacché comprendo finalmente il significato di amare. Ti meriti il meglio
che questa vita ha da offrire. Ti amo così. Grazie!
In generale non sarei la persona che sono senza i miei amici di Novaglie. Ci conosciamo daquando siamo nati circa 25-30 anni fa. La nostra e più che una semplice amicizia. Noi siamo
la famiglia che abbiamo scelto di avere. Abbiamo condiviso bei momenti e tante liti e, anche
se la vita ci sta allontanando un poco gli uni dagli altri, spero che tutto questo possa durare
per sempre. Senza di voi sono semplicemente perso. Voglio potermi sedere con tutti voi tra
50 anni alle panchine del nostro paese a ridere insieme per ogni piccola cosa accaduta nelle
nostre vite. A voi devo uno tra i più grandi e autentici grazie.
Infine, tutto quello che sono lo devo alle persone più care del mio cuore. La mia famiglia.
Mio papà S.D., mia mamma P.P. e mia sorella E.D.. Non credo esistano parole capaci didescrivere ciò che provo dentro di me. Forse è per questa ragione che certe volte mi viene
difficile esprimere i miei sentimenti e spesso mi dimentichi di dirvi quanto vi voglia bene.
Da quando sono ad Innsbruck ho capito cosa significhi vivere. Ho capito come voi abbiate
messo me e la mia felicità (e quella di mia sorella) prima di ogni cosa che possiate mai avervoluto dalla vostra vita. Mi avere spinto a seguire il mio percorso senza mai impormi niente.
Avete creduto in me nel momento stesso in cui aprii i miei occhi per la prima volta. Ci
avete cresciuto e protetto nonostante tutto l’impegno e la dedizione richiesti. Certe volte mi
dimentico di dirvi che vi amo. Siete la famiglia migliore che potessi mai sognare di avere.
Siete il miglior papà, la migliore mamma e la miglior sorella. Sono fiero di voi e spero dipotervi rendere fieri di me.
Tutto questo è per voi famiglia. Mi avete donato più amore di quanto io potrò mai
ricambiarvi.

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G. D.