Controlling and Understanding ofDipolar Quantum Gases of
Erbium Atoms– DISSERTATION –byAlexander Patscheider, MSc

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

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

Abstract
In recent decades, great progress has been made in the realization of various atomic, molecular, and optical physics (AMO) platforms to study quantum mechanical many-body phenomena, e. g. atomic and molecular quantum gases [Blo12, Gro17], trapped ions [Bla12, Hem18],
and Rydberg atoms in optical tweezers [Ber17, Bro20]. Atomic quantum gases offer a powerful collection of desirable properties: tunability of interactions, geometry, and statistics(bosonic/fermionic), the manipulation of the internal degree of freedom (spin), and an intrinsic scalability of the system size. Additionally, their high environmental isolation makes
them attractive for fundamental research as well as promising candidates for various technological applications.
As progress has advanced, new ideas have emerged about the possibility of quantum simulating a variety of phases of matter, or even creating some that have no counterpart in othersystems. Some of these exciting proposals required a radical change in the nature of theinteraction that governs the many-body behavior. From the traditional short-range“ inter”
action, whose strength is appealing because of its tunability, the proposals started to focus onplatforms in which the atoms interact via a long-range potential that activates a connection
between atoms over a long distance. Platforms such as ultracold molecules, Rydberg atoms,
cavity-mediated systems, and magnetic atoms started to appear on the experiment map.
This thesis focuses on a new class of magnetic quantum gases consisting of magnetic lanthanides, which are gaining a remarkable momentum in the scientific community due totheir extraordinarily large magnetic dipole moment. In erbium and dysprosium, the magnetic interaction is a factor > 100 larger than in alkali atoms and several times larger than intransition metals. Our group has pioneered this field with the first Bose-Einstein condensateof erbium [Aik12], now celebrating its 10th anniversary, and with the observation of novel
phenomena such as roton modes [Cho18, Pet19] and supersolidity [Böt19b, Tan19, Cho19].
Within this thesis, we present our progress in increasing the level of control over these new
systems, based on the special properties of lanthanides. We realize a quantum simulator for
the XXZ-Heisenberg model with large spins in a three-dimensional optical lattice and study
magnetization conserving spin-exchange dynamics. The large spin manifold is encoded in
the 20 Zeeman levels associated with the atomic ground state of fermionic erbium. We
demonstrate experimental control over the rate at which the dynamics occur by changing
the initial spin state, by tuning the dipole orientation, or by exploiting ac-Stark shifts toalter the resonance condition.
To obtain enhanced control over the internal atomic states, we extend our experimental
toolbox with a narrow optical transition at 1299 nm, with a spectral linewidth on the order of
only 1 Hz. We perform a detailed characterization of the fundamental transition parameters,such as lifetime, isotope shifts, and Landé gJ -factor. Studies of the dynamic polarizabilityof the excited state with respect to the ground state open up interesting opportunities formagic-wavelength lattices and other exciting applications.
Finally, we expand our understanding on the collision properties of dipolar atoms and study
thermalization dynamics of cold, non-degenerate quantum gases. Within this work, we applyithe cross-dimensional thermalization technique to gain knowledge on the contact scattering
length for four bosonic isotopes over a broad magnetic field range. This also allows us to
extract a background scattering length for each isotope and analyze its scaling with theisotope mass. Since theoretical calculations of the scattering length for complex magnetic
atoms are very challenging, our results represent a valuable contribution for the developmentof advanced theoretical models.

ii

Zusammenfassung
In den letzten Jahrzehnten wurden große Fortschritte bei der Realisierung verschiedener
Plattformen der atomaren, molekularen und optischen Physik (AMO) gemacht, um quantenmechanische Vielteilchenphänomene zu untersuchen, z. B. atomare und molekulare Quantengase [Blo12, Gro17], gefangene Ionen [Bla12, Hem18] und Rydbergatome in optischen
Pinzetten [Ber17, Bro20]. Atomare Quantengase bieten eine leistungsstarke Sammlungwünschenswerter Eigenschaften: Abstimmbarkeit von Wechselwirkungen, Geometrie undStatistik (bosonisch/fermionisch), die Manipulation des internen Freiheitsgrades (Spin) und
eine intrinsische Skalierbarkeit der Systemgröße. Darüber hinaus sind sie aufgrund ihrer hohen Isolation von der Umgebung attraktiv für die Grundlagenforschung und vielversprechende
Kandidaten für verschiedene technologische Anwendungen.
Im Zuge des Fortschritts sind neue Ideen über die Möglichkeit der Quantensimulation einer
Vielzahl von Materiephasen oder sogar der Erzeugung von Phasen, die in anderen Systemen kein Pendant haben, entstanden. Einige dieser aufregenden Vorschläge erforderten eineradikale Änderung der Art der Wechselwirkung, die das Vielteilchenverhalten bestimmt. Von
der traditionellen kurzreichweitigen Wechselwirkung, deren Stärke aufgrund ihrer Abstimmbarkeit so verlockend war, konzentrierten sich die Vorschläge auf Plattformen, bei denen die
Atome über ein langreichweitiges Potenzial wechselwirken, das eine Verbindung zwischen
den Atomen über eine große Entfernung aktiviert. Plattformen wie ultrakalte Moleküle,Rydberg-Atome, resonatorbasierte Systeme und magnetische Atome tauchten auf der Experimentkarte auf.
Diese Arbeit konzentriert sich auf eine neue Klasse magnetischer Quantengase, die aus magnetischen Lanthaniden bestehen, die aufgrund ihres außerordentlich großen magnetischen
Dipolmoments in der wissenschaftlichen Gemeinschaft eine bemerkenswerte Dynamik entwickeln. Bei Erbium und Dysprosium ist die magnetische Wechselwirkung um einen Faktor> 100 größer als bei Alkaliatomen und um ein Vielfaches größer als bei Übergangsmetallen.
Unsere Gruppe hat auf diesem Gebiet mit dem ersten Bose-Einstein-Kondensat von Erbium [Aik12], das jetzt sein 10-jähriges Jubiläum feiert, und mit der Beobachtung neuartigerPhänomene wie Roton-Moden [Cho18, Pet19] und Supersolidität [Böt19b, Tan19, Cho19] Pionierarbeit geleistet.
Im Rahmen dieser Arbeit stellen wir unsere Fortschritte bei der Verbesserung der Kontrolle über diese neuen Systeme vor, die auf den besonderen Eigenschaften der Lanthaniden
basieren. Wir realisieren einen Quantensimulator für ein XXZ-Heisenberg-Modell mit großen
Spins in einem dreidimensionalen optischen Gitter und untersuchen die magnetisierungserhaltende Spin-Austausch-Dynamik. Die große Spin-Mannigfaltigkeit ist in den 20 ZeemanNiveaus kodiert, die mit dem atomaren Grundzustand von fermionischem Erbium verbundensind. Wir demonstrieren die experimentelle Kontrolle über die Geschwindigkeit, mit der die
Dynamik auftritt, indem wir den anfänglichen Spin-Zustand, die Orientierung des magnetischen Dipolmoments oder die ac-Stark-Verschiebungen ändern, um die Resonanzbedingung
zu verschieben.
Um eine fortgeschrittene Kontrolle über die internen Atomzustände zu erhalten, erweitern wir unser experimentelles Instrumentarium um einen schmalen optischen Übergangiii

bei 1299 nm, mit einer spektralen Linienbreite in der Größenordnung von nur 1 Hz. Wir
führen eine detaillierte Charakterisierung der grundlegenden Übergangsparameter durch,
wie Lebensdauer, Isotopenverschiebungen und Landé gJ -Faktor. Untersuchungen der dynamischen Polarisierbarkeit des angeregten Zustands in Bezug auf den Grundzustand zeigen
interessante Möglichkeiten für magische Gitterwellenlängen und andere Anwendungen auf.
Schließlich erweitern wir unser Verständnis der Kollisionseigenschaften von dipolaren Atomen
und untersuchen die Thermalisierungsdynamik von kalten, nicht entarteten Quantengasen.
Im Rahmen dieser Arbeit wenden wir die Technik der kreuzdimensionalen Thermalisierungan, um die Kontaktstreulänge für vier bosonische Isotope über einen breiten Magnetfeldbereich zu ermitteln. Dies erlaubt uns auch, eine Hintergrundstreulänge für jedes Isotop zu
extrahieren und ihre Skalierung mit der Isotopenmasse zu analysieren. Da Berechnungen derStreulänge für komplexe magnetische Atome sehr anspruchsvoll sind, stellen unsere Ergebnisse einen wertvollen Beitrag für die Entwicklung fortgeschrittener theoretischer Modelledar.
Zusammenfassend lässt sich sagen, dass unsere Ergebnisse einen wichtigen Baustein für das
Verständnis dipolarer Quantengase bilden, wobei der schmale Übergang bei 1299 nm einenbedeutenden Fortschritt für die Kontrolle und Manipulation von Erbiumatomen darstellt.

iv

Contents
Abstracti

Zusammenfassungiii

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

1
1
4
5

2. Erbium: From the basic properties to atoms in an optical lattice
2.1. Basic properties of erbium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Historical overview on spectroscopy measurements and ground-stateelectron configuration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Large permanent magnetic dipole moment . . . . . . . . . . . . . . . .
2.1.3. Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. The experimental apparatus and cooling sequence . . . . . . . . . . . . . . .
2.2.1. Vacuum chamber and first cooling and trapping stage . . . . . . . . .
2.2.2. Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. The optical lattice: A platform for quantum simulation . . . . . . . . . . . .
2.3.1. A brief introduction to optical lattices . . . . . . . . . . . . . . . . . .
2.3.2. The lattice setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
8
8
11
12
14
14
16
17
17
18

3. Dipolar spin-exchange dynamics in a three-dimensional optical lattice
3.1. Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. General description of two interacting dipoles . . . . . . . . . . . . . .
3.1.2. Spin-exchange and dipolar relaxation due to DDI . . . . . . . . . . . .
3.2. Dipolar Fermions in an optical lattice . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Hamiltonian for spin lattice models . . . . . . . . . . . . . . . . . . . .
3.2.2. The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3. Our experimental system . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Publication: Controlling dipolar exchange interactions in a dense
three-dimensional array of large-spin fermions. . . . . . . . . . . . . . . . . .

25
27
27
28
30
30
31
33

4. Narrow inner-shell orbital transition at 1299 nm
4.1. Electron configuration and transition parameters . . . . . . . . . . . . . . . .
4.1.1. The J1 j-coupling and energy level diagram of the 4f 11 5d6s2 electron
configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2. Lifetime of the excited state and other transition parameters . . . . .

53
55

35

55
57
v

4.2. Publication: Observation of a narrow inner-shell orbital transition in atomic
erbium at 1299 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1. Hyperfine constants of the fermionic isotope . . . . . . . . . . . . . . .
4.3.2. Dependence of the excited state lifetime on the atomic density . . . .
4.3.3. Preparation of bosonic erbium atoms in higher Zeeman states . . . . .
5. Determination of the contact scattering length via cross-dimensional thermalization
5.1. Dipolar scattering in the ultracold regime . . . . . . . . . . . . . . . . . . . .
5.2. Determining the scattering length via coss-dimensional thermalization . . . .
5.3. Publication: Determination of the scattering length of magnetic erbium via
cross-thermalization measurements. . . . . . . . . . . . . . . . . . . . . . . . .

61
75
75
77
79

81
82
87
91

6. Conclusion and outlook
107
6.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2. The transition at 1299 nm: A versatile experimental tool . . . . . . . . . . . . 108
6.2.1. Prospects for combining the optical lattice and the narrow inner-shellorbital transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2.2. Further potential applications . . . . . . . . . . . . . . . . . . . . . . . 111
7. Additional publications
113
7.1. Publication: A two-species five-beam magneto-optical trap for highly magnetic Er and Dy atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2. Publication: Realization of a Strongly Interacting Fermi Gas of Dipolar
Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3. Publication: Probing the roton excitation spectrum of a stable dipolar
Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4. Publication: Long-lived and transient supersolid behaviors in dipolar quantum
gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.5. Publication: Excitation spectrum of a trapped dipolar supersolid and its experimental evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.6. Publication: Bragg scattering of an ultracold dipolar gas across the phasetransition from Bose-Einstein condensate to supersolid in the free-particle
regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
A. Upgraded distribution for the light at 532 nm

205

B. Laser system at 1299 nm

209

C. Magnetic field sensitivity of the atomic state at 1299 nm

213

Bibliography

217

Acknowledgments

235

vi

Chapter1

Introduction

1.1. Motivation
Throughout history, mankind has witnessed a number of technological developments thathave had a fundamental impact on daily life. Many of these are directly related to theunderstanding and application of new physical processes. Examples are the development of
the steam engine by J. Watt, based on the laws of thermodynamics, that led to the industrial
revolution [Dea79], or the development of microprocessors enabled by solid-state physics,
which resulted in the so-called digital revolution in which we currently find ourselves [Arn10].
A very exciting research area that promises another major step in technological advancement
is represented by quantum mechanics [Jae19].
First steps to explore the quantum world have been taken at the beginning of the 20th century, e. g. by M. Planck, which observed black-body radiation and found the relation of thelights energy E to its frequency ν and the Planck constant h, given by E = hν [Pla48], or
A. Einstein, which applied the quantization of light to explain the photoelectric effect [Ein05].
Such type of discoveries gave rise to the concept of wave-particle duality, postulated by
L. de Broglie in 1924 [Kub70]. Noteworthy, the quantum mechanical description of particles
leads to two distinct classes of statistical behavior, represented by bosons and fermions. As
a consequence, atoms at ultra low temperatures can form two fundamentally different quantum phases: the Bose-Einstein condensate, proposed by A. Einstein in 1925 [Ein25], andthe degenerate Fermi gas, whose theoretical framework was pioneered by E. Fermi [Fer26]
and P. Dirac [Dir26]. After their theoretical predictions, it took about 70 years, until thefirst realization of a Bose-Einstein condensate in rubidium [And95], lithium [Bra95], andsodium [Dav95] as well as the first degenerate Fermi gas in potassium [DeM99b] were demonstrated.
The creation of degenerate quantum gases constitutes a milestone in the field of ultracold
atoms. In the spirit of R. Feynman, who proposed the use of quantum simulators to solve
complex quantum mechanical questions that are too demanding to be answered with classical computers1 [Fey82], ultracold atoms combine several requirements that make them a
1

For a complete description of a system with N spin-1/2 particles, 2N coefficients are necessary, whichbecomes impossible for N > 50.

1

2

1. Introduction

promising platform: (i) they are environmentally well isolated, (ii) they have a high degreeof controllability, (iii) they provide access to a large physical parameter space, and (iv) theycome along with efficient probing techniques [Blo12]. A prominent example that goes towards a quantum simulator is the investigation of superfluid behavior in Fermi gases, which
is based on the formation of Cooper pairs [Bar57]. Over the last two decades, numerous
experiments have investigated the BEC-BCS crossover regime by tuning the inter-particle
interaction with the help of a Feshbach resonances [Bar04, Bou04, Alt07, Ing07].
While the above mentioned example of quantum simulation involves a spatially continuous
system, an alternative approach goes back to a proposal from 1998 [Jak98] and is basedon ultracold atoms loaded into a so-called optical lattice that is created by a laser-lightfield in standing-wave configuration. Atoms in the periodic potential formed by an opticallattice can mimic the physics of a broad class of condensed-matter systems [Gro17, Sch20a].
The hopping of particles between lattice sites is characterized by the tunneling rate, which
depends on the potential depth and thus can be controlled by the light intensity of thelaser beam. In combination with the tunability of interactions, optical lattices therefore
provide for example a natural setting for the simulation of Hubbard models, which describe
correlated many-body lattice systems [Hub63, Jak05, Aro21]. One of the first demonstrationsof the outstanding capabilities was the observation of the transition from a superfluid to a
Mott insulator state [Gre02], which was predicted in 1989 [Fis89]. In recent years, a further
step in the development has been taken, which includes the implementation of quantum gas
microscopes [Bak09, She10, Che16] enabling manipulation and detection of individual atoms
and giving access to local observables.
An important role in such quantum systems is taken on by interactions. In optical latticesystems, which typically contain hundreds to thousands of atoms, they are for example essential for the establishment of correlations and entanglement among the atoms [Blo08a].
While, most experiments involving optical lattices so far have been performed with dominantly contact-interacting particles, exciting prospects arise from dipole-dipole interactions,which occur for example in magnetic atoms [Cho22], hetero-nuclear molecules [Car09], orRydberg atoms [Lim13]. The long-range and anisotropic character of dipolar interactions canlead to entirely new quantum phases [Dut15], such as checkerboard and stripe Mott crystals
which feature a spontaneously broken spatial symmetry [CS10]. An extended Bose-Hubbard
model has been experimentally realized for the first time within our group in 2016 [Bai16].
The main focus for Hubbard models lies on the competition between tunneling and interparticle interactions. On the other hand, atoms in deep optical lattices featuring a spin degreeof freedom enable also the realization of spin lattice models and the study of quantum magnetism [Aue94]. A prominent example is the Heisenberg model [Hei28], which has proven to
successfully describe inter-spin interactions. For atoms interacting only at short range, suchas alkali atoms, spin-exchange interactions are limited to neighboring lattice sites and rely ona second-order tunneling process called super-exchange. The strength of the super-exchangeinteraction depends on both, the on-site interaction and the tunneling rate. Due to the deep
lattice depth in spin lattice models, typically the super-exchange rates are relatively small
compared to other energy scales in the system, thus making the experimental observation
challenging [Tro08, Gre13]. In contrast, thanks to the long-range character, dipole-dipole
interactions induce a direct coupling of spins which goes beyond nearest neighbors and ex-1.1. Motivation3

ceeds super-exchange interaction strengths significantly [Lep19, Cho22]. This enables thestudy of spin lattice models even in the regime of very deep optical lattices, where atoms
are pinned to individual lattice sites and tunneling is absent. To date, dipolar spin latticesystems have been realized based on fermionic KRb molecules [Yan13] or bosonic chromium
atoms [dP13b, Lep19]. While in case of molecules the spin is encode in two rotational statesof the ground state, in the case of magnetic atoms the spin can be included via the populationof different Zeeman sublevels.
Fermionic erbium atoms are advantageous for the preparation of atoms in different Zeeman states, because they possess a quadratic Zeeman shift which arises from the additional
hyperfine interaction. As the Zeeman shift depends quadratically on the specific magnetic
quantum number, this intrinsically introduces a difference in the energy splitting between
subsequent spin states and enables therefore deterministic preparation using radio-frequency
pulses [Bai18a]. Bosonic erbium isotopes do not have a nuclear spin and therefore a quadratic
Zeeman shift is absent, making deterministic preparation schemes challenging. A promising
tool to circumvent this issue are narrow and ultranarrow optical transitions that were originally used for the implementation of atomic clocks [Lud15]. On the one hand, atomic clocks
have paved the way for an incredible improvement of time keeping, that has reached an accuracy good enough to realize advanced global navigation satellite systems [Guo21]. On theother hand, the high precision and high accuracy of atomic clocks allows them to function as
an accurate measurement and preparation tool in ultracold quantum gas systems [Zha16].
To drive the progress of fundamental research, it is promising to combine technical advancesthat have been successfully implemented in different systems. Magnetic atoms have a very
dense energy spectrum and it is predicted that they also contain narrow and ultranarrowtransitions [Ban05]. Transferring the extended knowledge on these types of transitions tomagnetic atoms brings them a number of new experimental possibilities, such as efficient
preparation of spin states for bosonic isotopes, the possibility to coherently control theatomic population, or the potential application of spin-resolved imaging techniques [Fra21].
Prior to this thesis, a first survey on a narrow transition in magnetic atoms has been doneusing a thermal cloud of dysprosium atoms [Pet20a].
In addition to the precise control on the atomic spin state, accurate knowledge on the interparticle contact scattering length is of fundamental importance as it contributes to thegoverning the many-body state. Thanks to the tunability of the contact interaction bymeans of a Feshbach resonance [Chi10], its strength relative to the dipole-dipole interaction
can be controlled carefully. Remarkably, this control has led to the recent realization ofquantum droplets [Kad16, Cho16a, Sch16] and the related demonstration of the appearanceof a roton minimum in the bosonic excitation spectrum [Cho18, Pet19, Sch21]. Building up
on these results, recently, dipolar supersolids were realized in one- [Cho19, Böt19b, Tan19]
and two-dimensions [Nor21]. The latter are particularly fascinating, because they feature aperiodic density modulation while simultaneously the atoms maintain phase coherence.

4

1. Introduction

1.2. Thesis overview
This thesis covers experimental work of fundamental research based on dipolar erbium atoms.
The results obtained represent an important contribution to the understanding, the control,
and the manipulation of dipolar atoms and bring us closer to the goal of realizing a quantum
simulator. The main achievements of this work are organized along three sections:(I) The control of dipolar spin-exchange dynamics of fermionic erbium atoms loaded intoa three-dimensional optical lattice.
(II) The experimental observation and extensive characterization of the narrow inner-shellorbital transition at 1299 nm.
(III) The determination of the contact scattering length for four bosonic erbium isotopes
based on cross-dimensional thermalization measurements.
Chapter 2 provides a general overview of atomic erbium and specifically defines the properties that will be of importance throughout this thesis. In addition, the chapter contains abrief description of the experimental apparatus that was used for the realization of bosonicand fermionic quantum gases. This covers in particular the cooling steps starting from hot
erbium vapor to the attainment of a degenerate quantum state. This chapter concludes witha short introduction to optical lattices and a description of our lattice setup including asummary of the sources of inhomogeneities in the experimental realization.
Chapter 3 describes the scenario of fermionic erbium atoms loaded into a three-dimensionallattice. First, the dipole-dipole interaction is introduced and its special role in spin-exchange
dynamics is described. Second, the spin lattice Hamiltonion is discussed and it becomes clearthat our system represents a quantum simulator for the large spin XXZ-Heisenberg model.
Chapter 4 presents the observation of a narrow inner-shell orbital transition at 1299 nm.
First a general overview on the transition parameters is given. Then, following the publication in which the transition has been identified and carefully characterized, further results
are presented, including: the determination of the A and B hyperfine coefficients and thedensity dependence of the excited-state lifetime. Finally, we demonstrated first experimental
results in which we use the transition for the preparation of bosonic atoms in higher magnetic
sublevels.
Chapter 5 is devoted to the scattering properties of dipolar quantum gases. The first focus
lies on the impact of dipolar interactions on the scattering amplitude, which depends on both,the orientation of the atomic dipole moment and on the relative momenta of the scattering
constituents. Furthermore, the influence of this anisotropy on the thermalization process ofa thermal gas brought out of equilibrium is shown.
To conclude, Chapter 6 provides a summary of this thesis and gives a short outlook on
possible future realizations.

1.3. List of publications

5

Chapter 7 contains a list of all additional publications that have emerged over the courseof this thesis and in which the author has been involved.

1.3. List of publications
The following list contains the publications that form the main part of this thesis. They are
given in chronological order, which also reflects the thematic framework of the chapters. In
addition to the main publications, the author has contributed extensively to a number ofother projects within the course of this work. However, a detailed discussion of all results
would go beyond the scope of this thesis. These further publications are each presented witha brief summary in Chapter 7.
 Controlling dipolar exchange interactions in a dense three-dimensional array of large-spin fermions.
A. Patscheider, B. Zhu, L. Chomaz, D. Petter, S. Baier, A.-M. Rey, F. Ferlaino, M. J. Mark,
Physical Review Research, 2, 023050 (2020).
 Observation of a narrow inner-shell orbital transition in atomic erbium at1299 nm.
A. Patscheider, B. Yang, G. Natale, D. Petter, L. Chomaz, M. J. Mark, G. Hovannesyan,
M. Lepers, and F. Ferlaino
Physical Review Research 3, 033256 (2021).
 Accurate determination of the scattering length of erbium atoms.
A. Patscheider, L. Chomaz, G. Natale, D. Petter, M. J. Mark, S. Baier, B. Yang,
R. R. W. Wang, J. L. Bohn, F. Ferlaino
Submitted to Physical Review A (2021).

Additional publications
 A two-species five-beam magneto-optical trap for highly magnetic Er and
Dy atoms.
P. Ilzhöfer, G. Durastante, A. Patscheider, A. Trautmann, M. J. Mark, F. Ferlaino,
Physical Review A 97, 023633 (2018).
 Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms.
S. Baier, D. Petter, J. H. Becher, A. Patscheider, G. Natale, L. Chomaz, M. J. Mark,F. Ferlaino,
Physical Review Letters 121, 093602 (2018).

6

1. Introduction
 Probing the roton excitation spectrum of a stable dipolar Bose gas.
D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz,
F. Ferlaino,
Physical Review Letters, 122, 183401 (2019).
 Long-lived and transient supersolid behaviors in dipolar quantum gases.
L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante,
R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, F. Ferlaino,
Physical Review X, 9, 021012 (2019).
 Excitation spectrum of a trapped dipolar supersolid and its experimental
evidence.
G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter, M. J. Mark, L. Chomaz,
F. Ferlaino,
Physical Review Letters 123, 050402 (2019).
 Bragg scattering of an ultracold dipolar gas across the phase transition from
Bose-Einstein condensate to supersolid in the free-particle regime.
D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. Baranov, R. M. W. van Bijnen,
S. M. Roccuzzo, A. Recati, B. Blakie, D. Baillie, L. Chomaz, F. Ferlaino,
Physical Review A 104, L011302 (2021).

Chapter2

Erbium: From the basic
properties to atoms in an opticallattice

Atomic erbium belongs to the lanthanide series of the periodic table, which includes elementswith atomic numbers from 57-71. The successful story of the lanthanides started back in 1789,
when the mineral gadolinite was discovered by Carl Axel Arrhenius in a quarry in Ytterby,
near Stockholm [Din16]. It took more than 50 years, until Carl Gustaf Mosander, in 1843,
identified three constituents of gadolinite: yttrium, erbium, and terbium [Mos43]. Overthe years further elements have been identified, and today we know that the mineral foundby Arrhenius, contains additional six elements, all of which are assigned to the lanthanide
series.
The application of lanthanides in modern world covers a wide spectrum, ranging from theproduction of permanent magnets [Rol16, Mü17], which form a key component, e. g. in windturbines and hybrid cars, to applications in medical diagnostics [Cot12]. In laser technology, triply ionized erbium is of key importance for the production of erbium doped fiber
amplifiers (EDFA). The emission of EDFA lies within the telecommunication C-band andL-band1 , which exhibit the lowest optical loss in current optical fibers of the telecommunication wavelength range. Therefore, this makes erbium an advantageous dopant [Bou11]. In
the near future, EDFA might play a key role in the realization of photon-based quantum
networks [Sag15].
In addition to its various industrial applications, atomic erbium (and magnetic lanthandidesin general) has many attractive properties that make it appealing for fundamental research.
This is especially true in the field of ultracold quantum gases. It was in 2012, when thefirst Bose-Einstein condensate of erbium was realized by our group [Aik14b] in Innsbruck.
Since then, the interest in the use of erbium for ultracold experiments has been growing
steadily. This can be recognized in the rising number of erbium experiments that have either
been built or are in the planning and construction phase. Examples of new experimentsthat are currently realized are in Bonn [Uli17], Cambridge (UK) [Hof21], Cambridge (U. S.,
1

The telecommunication C-band and L-band correspond to the wavelength ranges of 1530 nm - 1565 nm
and 1565 nm - 1625 nm.

78

2. Erbium: From the basic properties to atoms in an optical lattice

Table 2.1.: Atomic mass number of all stable erbium isotopes, their relative abundance, and theirassociated quantum statistics [Ber11].

Atomic mass number

natural abundance

quantum statistics

162
164
166
167
168
170

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

bosonic
bosonic
bosonic
fermionic
bosonic
bosonic

Harvard) [Phe20], and Hong Kong [Seo20].
In this Chapter, Sec. 2.1 introduces the key properties of atomic erbium that are relevant in
the context of this thesis, such as the special electron configuration and the large magnetic
dipole moment. Furthermore, Sec. 2.2 contains a description of the experimental apparatus
and the experimental sequence to create degenerate quantum gases of erbium for both,
fermionic and bosonic isotopes. Finally, Sec. 2.3 gives a brief introduction to optical lattices
and presents our setup including a discussion on experimental inhomogeneities.

2.1. Basic properties of erbium
Erbium has an atomic mass number of 68 and can be found in the periodic table between
holmium and thulium. Naturally, erbium contains six stable isotopes whose natural abundances range from only 0.14 % for 162 Er to 33.5 % for 166 Er [Ber11, Hol18]. The relative
atomic mass, calculated from the atomic mass of the individual isotopes and their relative
isotopic abundances, corresponds to 167.259(3) u2 . Table 2.1 gives an overview on the natural abundance of the various isotopes and the corresponding quantum statistic they follow.
While five isotopes follow bosonic statistics, only 167 Er possesses fermionic character. Besidethe 162 Er isotope, whose natural abundance is too low to realize efficient cooling and trapping to obtain sufficiently high atom numbers, all isotopes have been brought to quantum
degeneracy [Aik12, Aik14b, Cho16b, Tra18]. Note that the first Bose-Einstein condensate(BEC) of 164 Er has been realized within the work of this thesis.

2.1.1. Historical overview on spectroscopy measurements and ground-stateelectron configuration
The 68 electrons of erbium arrange in a special configuration around the atomic core. The
largest number of electrons, i. e. 54, form a closed electronic core, which resembles the electron
configuration of xenon and is denoted as [Xe]. The remaining 14 valence electrons form aso-called submerged-shell electron configuration, where the 4f -orbital is only partially filled
2

1 unified atomic mass unit = 1 u = 1 Da = 1.660 539 066 60(50) × 10−27 kg

2.1. Basic properties of erbium

9

104
2.5

2

1.5

1

0.5

0

Figure 2.1.: Energy level spectrum of erbium containing all energy levels up to 2.6 × 104 cm−1 andshowing all relevant cooling transitions as well as the narrow transition at 1299 nm. In addition laser
wavelengths used for optical trapping are shown. Even (odd) parity states are shown in red (blue).
Energy levels are taken from Ref. [Kra20].

while the 6s-orbital is complete. Thus, out of 14 possible electrons, only 12 effectively locate
in the 4f -orbital. The ground state electron configuration is written as[Xe]4f 12 (3 H6 )6s2 .
As we will see in the following, the electron structure, containing 14 valence electrons, leads
to a rich and somehow complex energy level structure.
In addition to the obvious interest in atomic transitions for applications in the field of
ultracold quantum gases, spectroscopic measurements represent an important contribution,e. g. , to the understanding of complex electronic structures of atoms or the determinationof material compositions due to their unique spectral signatures [Lag04]. Originally, in the
1960’s, the latter was the driving force behind an increased interest in atomic spectra. For the
specific case of erbium, the first reported energy levels originate from the analysis of emission
and absorption spectra that have been recorded by L. C. Marquet and S. P. Davies [Mar65b]
as well as V. G. Mossotti and V. A. Fassel [Mos64].
The experiments of L. C. Marquet and S. P. Davies were based on emission spectra of Er2 I3(erbium iodide) placed in a discharge tube and excited by microwave radiation. The atomic
source was heated up to 750 °C in an electric furnace to enhance the spectroscopic signal. The
wavelength selective recording was realized with a Czerny-Turner type spectrometer [Neu14],

10

2. Erbium: From the basic properties to atoms in an optical lattice

Table 2.2.: Summary of the important atomic transitions used with this thesis. Data for the 401 nm
and 583 nm transitions is taken from Ref. [Har10]. The wavelength as well as the characteristic parameters for the transition at 1299 nm were determined within the work of this thesis (see Chapter 4).

wavelength (nm)linewidth (Hz)

lifetime (ms)

401
583
1299

29.7(6) × 106
186(10) × 103
0.9(1)

5.4(1) × 10−60.86(5) × 10−3
178(19)which contains a diffraction grating that acts as a dispersive element. In this experimental
configuration, the diffracted light and thus the emission spectra of the source material was
photographed and recorded by a spectrograph camera [Rea63]. Due to technical limitations,
the wavelength range accessible in their work was limited from 280 nm to 630 nm [Mar65b].
While the work of L. C. Marquet and S. P. Davies was based on the optical emission spectra,
V. G. Mossotti and V. A. Fassel [Mos64] investigated the absorption spectra of erbium. To
record the latter, sample material was fed into fuel-rich oxyacetylene flames and illuminatedwith a light source extending over a broad wavelength range. In their case, the backgroundlight was created by a xenon discharge arc and a tungsten filament, resulting in an observable wavelength range from 250 nm to 650 nm [Mos64]. A specific arrangement of mirrors
in the spectroscopy setup ensured that the light passed multiple times through the oxyacetylene flames before being photographed in a spectrometer, amplifying the spectroscopic
signal [Fas63].
However, although a large number of self-reversed lines3 was observed, in their early works
only a low number of lines, including the ground state, could be classified. The large number of unclassified self-reversed lines suggested the existence of another low lying electron
configuration, which, however, could not be directly resolved due to the limited measurement ranges. In various additional works, the interpretation of the observed energy levels
was advanced [Mar65a, Spe66, Rac66, Spe67]. In particular, N. Spector extended the list ofobserved absorption lines in the range from 270 nm to 470 nm [Spe65]. By combining theavailable sets of data and by calculating the energy difference from a selection of strong lines
with the whole energy spectra, N. Spector was able to identify a low-lying manifold of energy
levels with odd parity corresponding to the electron configuration 4f 11 5d6s2 . In contrast to
the ground state, one of the 12 electrons in the 4f -orbital is excited to the 5d-orbital. A
detailed discussion of this electron configuration and of one specific level of this manifold isgiven in Chapter 4.
Today, the NIST database consists of 674 energy levels and is based on the collection ofenergy levels in Ref. [Mar78]. The majority of lines originates from unpublished data of vanKleef and Koot from 1975, which performed extensive measurements on the atomic spectra of
erbium. Figure 2.1 shows all energy levels for a total angular momentum quantum number
J ranging from 2 to 12 and up to an energy of 2.6 × 104 cm−1 , which corresponds to awavelength of about 384 nm. Since the lowest energy level can be found at a wavelength of3

Self-reversed lines result from a reduction of the emission signal due to re-absorption of the emitted lightby surrounding atoms.

2.1. Basic properties of erbium

11

1986 nm, in principle, all shown energy levels are within reach with standard laser techniques.
The linewidths of the transitions range from µHz to MHz and a selection of energy levels
have transition parameters that make them appealing for the various cooling techniques (seeSec. 2.2.2) as well as for the optical manipulation of erbium atoms (see Chapter 4). Table 2.2
summarizes the important atomic transitions used within this thesis and gives the respective
linewidths and corresponding excited state lifetime.

2.1.2. Large permanent magnetic dipole moment
A property of the erbium ground state that results from the special electron configurationis the large angular momentum quantum number, which again leads to a large permanent
magnetic dipole moment µ. The 12 electrons located in the 4f -orbital arrange following
Hund’s rule and tend to maximize both, the multiplicity 2S + 1 and the angular momentum
quantum number L [Ste20]. This gives the quantum numbers S = 1, L = 5, and, following
the LS-coupling scheme, the angular momentum quantum number J = L + S = 5 + 1 = 6.
For the bosonic erbium isotopes the nuclear spin I = 0, and therefore the magnetic momentµJ is given byS
LµJ = −gS µB − gL µB ,2.1ℏℏwith gS ≈ 2.0023 and gL = 1 being the electrons spin and angular momentum g-factor andµB being the Bohr magneton4 .
In contrast to the bosonic counterpart, the fermionic 167 Er isotope possesses an additional
nuclear spin of I = 7/2. The coupling of J with I leads to the hyperfine interaction and
F = I + J. As a consequence, for the magnetic moment the additional nuclear spin andg-factor gI need to be considered and [Foo05]µF = −gJ µB

JI+ gI µN ,ℏℏ

2.2with gJ being the Landé g-factor and µN being the nuclear magneton5 . Since µB >> µN(me/mp = 1/1836), the magnetic moment can be approximated for most experimental situations
byJ
2.3µF ≈ −gJ µB .
ℏ
The projection of µJ (µF ), first, along the corresponding basis vectors J and F and second,
along the quantization axis given by an external magnetic field leads to the magnetic momentsofµ = −mJ gJ µB(bosons)2.4
andµ = −mF gF µB
45(fermions).

2.5eℏµB = 2m, with e the elementary charge and me the electron mass. For our experimental regime it iseconvenient to use µB/h = 1.3996 MHz/GeℏµN = 2m, with e the elementary charge and mp the proton mass.
p12

2. Erbium: From the basic properties to atoms in an optical lattice

Here, the magnetic quantum number mJ (mF ) defines the projection of J (F) along thequantization axis and ranges from mJ = −J to mJ = +J (mF = −F to mF = +F ). For
the respective ground states with mJ = −6 (mF = −19/2) and gJ = 1.163 801(1) [Con63](gF = 0.735 032 0(7)), the magnetic moments areµ = 6.982 806(6) µB(bosons)2.6µ = 6.982 80(1) µB(fermions).

2.7

and

The large µ, which is one of the largest of all lanthanides in the periodic table of chemicalelements6 , leads to the fascinating dipole-dipole interaction. Note also that the strengthof the dipole moment depends on the specific magnetic sublevel and can be tuned by the
preparation of atoms in different states. Both the dipole-dipole interaction as well as the
preparation in different Zeeman states will be of relevance in Chapter 3.

2.1.3. Zeeman effectPlaced in an external magnetic field, the coupling of µ to B is described by
HB = −µ · B,

2.8

and in the bosonic (fermionic) case the 2J + 1 (2F + 1) magnetic sublevels, labeled by mJ(mF ) are no longer degenerate. In the following, the energy shift experienced by an atom
placed in a magnetic field is discussed. For simplicity, B is assumed to be pointing alongz.

Fine-structure Zeeman splitting
To derive a description for the fine-structure, we consider an atom in the internal state |JmJ ⟩.
Using Eq. 2.1, the interaction of the electron spin and the orbital angular momentum with
B is described by the Hamiltonian [Ste20]
HB = −µS · B − µL · B =µB(gS Sz + gL Lz ) · B.
ℏ2.9

If the magnetic field is weak compared to the fine-structure energy, B can be considered
as a small perturbation and the energy shift can be calculated using Jz = Sz + Lz and
Jz |JmJ ⟩ = ℏmJ |JmJ ⟩ via∆EZ (J, mJ ) = ⟨JmJ | HB |JmJ ⟩µB B=⟨JmJ | (gS Sz + gL Lz ) |JmJ ⟩ℏµB B(gS − gL )= µB BgL mJ +⟨JmJ | Sz |JmJ ⟩ .
ℏ
6

2.10

Only the elements dysprosium (µ = 9.93 µB ), terbium (µ = 9.94 µB ), holmium (µ = 8.96 µB ), and europium(µ = 6.98 µB ) have a comparably large magnetic moment [Fri14a].

2.1. Basic properties of erbium

13

By applying the projection theorem and the relation
1
S · J = (J2 + S2 − L2 )2

2.11

to the second term of Eq. 2.10, leads to⟨JmJ | Sz |JmJ ⟩ =J(J + 1) + S(S + 1) − L(L + 1)
mJ ℏ.
2J(J + 1)

2.12

Inserting Eq. 2.12 into Eq. 2.10 results in the familiar Zeeman energy shift∆EZ (J, mJ ) = µB gJ mJ B,

2.13

where the Landé g-factor is identified as
gJ = gL + (gS − gL )J(J + 1) + S(S + 1) − L(L + 1).
2J(J + 1)

2.14

Hyperfine-structure Zeeman splitting
We now consider the case of hyperfine interaction, where I ̸= 0. As long as the magnetic
field is small compared to the fine-structure splitting, J is a good quantum number and themagnetic dipole interaction of the nucleus simply adds to the interaction Hamiltonian [Ste20].
Therefore, for a weak magnetic field the solution of the Hamiltonian follows the previously
discussed fine-structure case and the substitutions J → F , S → I, and L → J can be
performed. The result is consistent with Eq. 2.13, however, the Landé g-factor needs to be
replace by the g-factor gF and mJ with mF . The g-factor is commonly written as
gF ≈ gJ

F (F + 1) − I(I + 1) + J(J + 1),
2F (F + 1)

2.15

which includes again the approximation, that µB is much larger than µN (see Eq. 2.2).

Paschen-Back effect
When the fine-structure or hyperfine splitting become comparable to the interaction with B,
the angular momentum quantum number J or total angular momentum quantum number
F are no longer good quantum numbers. This is called the Paschen-Back effect and asa consequence the Zeeman splitting starts to deviate from a linear behavior. For bosonicerbium the fine-structure splitting is O(1 THz) and thus much larger than the interaction withB (∆EZ = O(10 MHz/G)) for reasonable magnetic fields. In contrast, this effect becomes
more important for the fermionic 167 Er isotope. The hyperfine splitting in the ground stateis O(GHz) and therefore comparable to the Zeeman energy at experimentally realistic B
fields. On magnetic field scales that are relevant for our experiment, i. e. in the range from
0 G to about 50 G, the deviation from a purely linear behavior can be corrected by adding aquadratic term such that Eq. 2.13 turns into
∆EZ (F, mF ) = µB gF mF B + ∆Eq (F 2 − m2F )B 2 .

2.16

14

2. Erbium: From the basic properties to atoms in an optical lattice

In Ref. [Bai18b], the correction factor ∆Eq was extracted from a fit to results obtained via
exact diagonalization and determined to∆Eq = −h × 12.76(1) Hz/G2 .

2.17

We will exploit the quadratic energy shift in Chapter 3 for the preparation of fermionic atoms
in higher magnetic sublevels. At B = 40 G, the energy difference between the transitions|mF = −19/2⟩ → |mF = −17/2⟩ and |mF = −17/2⟩ → |mF = −15/2⟩ is about 41 kHz. Thisdifference is large enough to prevent populating of the |mF = −15/2⟩ state when driving the|mF = −19/2⟩ → |mF = −17/2⟩ transition using radio-frequency. This enables a deterministic
preparation of atoms in higher magnetic sublevels.

2.2. The experimental apparatus and cooling sequence
Many individual steps and an advanced technical scheme are necessary to convert a solid
piece of erbium into an atomic cloud that forms a degenerate quantum gas. More than
10 years ago from today, in 2009, the planning, designing, and building of the experimental
apparatus has started and was followed by the first degenerate quantum gas of bosonic erbium
in early 2012 [Aik12]. Since then, the apparatus and the technical components have been
continuously developed and upgraded. A detailed description of the experimental system canbe found in Ref. [Fri14a] and various upgrades are discussed in Ref. [Bai18b] and [Pet20b].
In this Section, a brief introduction to the vacuum setup and the most important steps forcooling and trapping erbium atoms is given.

2.2.1. Vacuum chamber and first cooling and trapping stage
Figure 2.2 shows a drawing of the complete vacuum chamber. The erbium granules are heatedin a high-temperature oven consisting of two segments at slightly different temperatures. The
so-called effusion cell, in which the atomic sample is placed, runs at a temperature of 1100 °C
and generates an atomic vapor. The second segment, the hot lip, runs at 1200 °C and createsa temperature gradient to ensure that the atoms move towards the oven aperture. Atoms thatleave the oven have an average atomic velocity distribution at about 430 m/s. Naturally, theatomic beam that leaves the high temperature oven has a certain divergence, which reduces
the number of atoms reaching the main chamber. To increase the atomic flux, the atoms
are cooled radially by a 2D-optical molasses, also referred to as transversal cooling, which
effectively collimates the atomic beam.
Following the transveral cooling stage, the atoms reach the Zeeman slower, whose entrance
marks the end of the first vaccum section. Here, the pressure is on the order of 1 × 10−9 mbar
and this part is therefore referred to as high-vacuum (HV) section. The Zeeman slower itself
represents a differential pumping stage and connects the HV section to the main chamber inthe ultra-high-vacuum region (UHV) with a pressure on the order of 1 × 10−11 mbar. The
atoms that leave the Zeeman slower are slowed down to a mean velocity of 5-10 m/s. For

2.2. The experimental apparatus and cooling sequence

Ti-sub pump (hidden)

15z

75 l/s ion getter pumpx

20 l/s ion getter pumps

Ti-sub pumpy
200 mmfull-metal valve
port aligner

port aligner

bellowpressure gauge

atomic beam shutter

UHV section
Zeeman slower
mirrorpumping stage 2

main chamber

HV section
Zeeman slower

pumping stage 1

transversal
coolinghigh-temperature oven

Figure 2.2.: Drawing of the vacuum chamber used for the creation of degenerate quantum gases.
Solid pieces of erbium are heated up in a high temperature oven (red). The atomic beam is collimatedin the transversal cooling section (blue) and the atoms are longitudinally slowed down by the Zeeman
slower (green). After leaving the Zeeman slower the atoms enter the main chamber where they are
first trapped and cooled in the MOT (yellow) and finally also evaporatively cooled. Figure adaptedfrom Ref. [Fri14a].

both, the transversal cooling as well as for the Zeeman slower we use the atomic transition
at 401 nm, because of its broad linewidth. This has the advantage of a large capture range
and allows for a high photon scattering rate due to the relatively small lifetime of the excitedstate (see Tab. 2.2).
In the main chamber, the atoms are cooled and trapped in a three-dimensional magnetooptical trap (MOT) that operates on the narrow transition at 583 nm. The transition has
a linewidth of about 190 kHz resulting in a low Doppler limited temperature of several µK.
This low temperature is advantageous for subsequent loading of atoms into an optical dipole
trap (ODT). An additional aspect associated with the narrow line MOT and the large massof erbium is that it can be realized with only five laser beams, omitting the vertical backreflection [Ilz18]. This is in contrast to the textbook six-beam configuration and brings theadvantage of additional space for optical access, e. g. the implementation of high-resolution
imaging.
Originally, the light at 583 nm for the MOT was generated with a dye laser system. In thecourse of this thesis, the dye laser was replaced by a combined laser system consisting of adiode laser7 at 1166 nm and a single-frequency Raman-Fiber Amplifier (RFA) including asecond-harmonic generator8 to convert the laser wavelength to 583 nm. Further, the looking
scheme for the laser frequency has been upgraded to a fiber-coupled electro-optical modulator, similar to Ref. [Pat17]. This brings the advantage, that switching between isotopes is7
8

Tuneable Diode Laser DLPro, TOPTICA Photonics AG, www.toptica.com.
Single Frequency Raman Fiber Amplifier, MPB Communications Inc, www.mpbcommunications.com.

16

2. Erbium: From the basic properties to atoms in an optical lattice

possible without any realigning of laser beams since no additional frequency shifting setup
is necessary. Compared to the previous laser setup based on the dye laser, the new system
improved the overall experimental stability during daily operation. For the bosonic 166 Er
isotope, with a MOT loading time of 3 s, we are able to trap about 2.5 × 107 atoms at atemperature of about 10 µK.

2.2.2. Evaporative cooling
The figure of merit to realize a quantum degenerate gas is the phase space denstiy (PSD),
which in an harmonic trap is given by [Pit16]ℏ 3
PSD = N ω̄ 3
.
2.18
kB T

Here, N is the atom number, ω̄ is the geometric mean trap frequency, T is the atomic temperature, and kB is the Boltzmann constant. The transition to Bose-Einstein condensation
occurs when PSD ≥ 2.6.

The PSD of the atoms in the MOT at about 10 µK is on the order of 4 × 10−6 [Fri12].
To reach the value of 2.6, an additional cooling step, called evaporative cooling, must be
performed. For this purpose, the atoms are loaded into a crossed ODT generated by two
orthogonally intersecting laser beams at a wavelength of 1064 nm and with a power of about
7 W. An additional scanning technique implemented for one of the trapping beams (seeRef. [Bai12] for details) allows us to transfer about 10 % of the atoms from the MOT into
the ODT.
To increase the PSD, the optical potential is then gradually reduced by decreasing the powerof the trapping beams. Due to the lower potential depth, the hottest atoms can no longer
be held in the trap and are lost. Consequently, this leads to a truncation of the Boltzmann
velocity distribution. During and after each step of laser power reduction, the atoms rethermalize via elastic collisions, which finally results in an effectively lower temperature (thescattering of dipolar atoms is discussed in more detail in Chapter 5). For example, for thebosonic 166 Er isotope, we obtain a BEC of about 1.2 × 105 atoms and a condensate fractionof 80 % after the evaporative cooling sequence.
The overall similarity of the bosonic isotopes allows for an easy switching between them.
The only two parameters that need to be considered for the evaporative cooling are themagnetic field to set the desired scattering length and the duration of the ramps for the laser
powers. The latter is necessary due to a different value of the scattering length and therefore
different thermalization rates. Within this thesis, BECs of the four bosonic isotopes 164 Er,
166 Er, 168 Er, and 170 Er were used.
Thanks to the dipole-dipole interaction, elastic collisions between identical fermions are notsuppressed. Therefore, direct evaporative cooling can also be used for the fermionic 167 Er
isotope [Aik14b] to approach the Fermi temperature
TF =ℏω̄ 3 (6N )1/3
.
kB

2.19

2.3. The optical lattice: A platform for quantum simulation

17

The major difference compared to the bosonic counterpart is a reduced atomic lifetime inan optical trap at 1064 nm. Therefore, an additional step in which the atoms are transferred
into a crossed ODT operating at 1570 nm is introduced. At the end of the evaporation, we
end up with a degenerate Fermi gas of about 4 × 104 atoms with T/TF ≈ 0.1. The fermionicisotope is used in Chapter 3.

2.3. The optical lattice: A platform for quantum simulation
2.3.1. A brief introduction to optical lattices
Optical lattices represent a versatile platform which allows to study various quantum manybody phenomena. These cover for example the realization of Hubbard models [Jak98, Jak05,
Dut15] or the investigation of spin lattice models, e. g. the Heisenberg model [Sac08, Blo08b,
Tro08, Gre13]. In contrast to the study of complex crystal configurations in condensed
matter systems, the experimental implementation of optical lattices is relatively simple.
As already for moderate system sizes, theoretical simulations of the underlying models arevery challenging, following the famous approach by R. Feynman, ultracold atoms loaded into
optical lattices can serve as platform for quantum simulation [Fey82].
The easiest configuration for an optical lattice consists of three orthogonal pairs of counterpropagating Gaussian laser beams leading to a cubic lattice geometry. The potential experienced by a particle of mass m can be described via [Gre08, Gro17]m
Vlatt (x, y, z) ≈ Vx sin2 (kx x) + Vy sin2 (ky y) + Vz sin2 (kz z) + (ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) . 2.20
|
{z
} |2
{z
}
periodic potential

harmonic confinement

Here, the first part describes the periodic potential landscape with lattice depths of Vx ,Vy , and Vz along the corresponding lattice directions with wave vectors kx , ky , and kz .
This periodic term arises from the interference of counter-propagating laser beams forminga standing wave. The second part describes the residual harmonic confinement given by
the Gaussian intensity distribution of the laser beams and characterized by the harmonic
trapping frequencies ωx , ωy , and ωz .
A particle placed in such a periodic potential can be described by Bloch wave functions, which(n)have the form of a plane wave eiqx/ℏ that is modulated by a periodic function uq (x) [Jak98].
As an example, in one-dimension, the Bloch wave function can be written as [Blo29, Ash76]
iqx/ℏ (n)ϕ(n)
uq (x),
q (x) = e

2.21

with q being the quasi-momentum and n indicating the nth energy band. Note that the(n)periodicity of uq (x) is congruent with the potential period.
From Eq. 2.21 is becomes clearly visible, that the Bloch wave functions are completely delocalized over the whole crystalline structure. An alternative, localized description for a18

2. Erbium: From the basic properties to atoms in an optical lattice

-

-

-

-

-

+

+

+

+

+

Figure 2.3.: Schematic illustration of (a) atoms (red circles) in an optical lattice and (b) electrons(green circles) in a periodic potential created by the ionic cores (blue circles). The red solid line
represents the atom (electron) wavefunction, which is similar for both systems.

particle in band n is provided by the so-called Wannier functions [Koh59], which are givenby the Fourier transform of the Bloch wave functions [Tre11]
1 X −iqxi /ℏ (n)eϕq (x).
wni (x − xi ) = √
M q

2.22

The index i of the position xi describes the ith lattice site and M is a normalization constant.
For an orthogonal optical lattice, the three-dimensional Wannier function w(r) is formedby the product of the one-dimensional functions w(r) = w(x)w(y)w(z). Note that theWannier function in Eq. 2.22 is not uniquely defined. To obtain the maximally localized(n)Wannier function the complex phase of ϕq (x) needs to be chosen such that wni (x − xi )is real, falls off exponentially, and is symmetric or anti-symmetric about either x = 0 or
x = q/2 [Koh59, Dal05]. The usage of Wannier functions is particularly useful in the tightbinding approximation for deep optical lattices, where populations in higher bands are highly
suppressed, and for the description of local interactions.
Figure 2.3 shows a schematic comparison of a one-dimensional optical lattice and a basic
crystalline structure. Electrons that move in the periodic potential created by the positively
charged ions forming the crystal are represented by atoms trapped in the optical potential.
Importantly, the atoms in the optical lattice are described by a similar wavefunction as theelectrons in the crystal array, underlying once more the potential to utilize ultracold atoms
in an optical lattice as a quantum simulator [Geo14, Sch20a, Alt21].

2.3.2. The lattice setup
In our experiment, we use a three-dimensional optical lattice setup created by three retroreflected laser beams that are perpendicular to each other. Along the vertical direction, defined by gravity, we use laser light at 1064 nm, leading to a lattice spacing of dz = 1064 nm/2 =532 nm. In the orthogonal, horizontal plane, the lattice is created by two retro-reflected laser

532 nm
10 W

λ/2

5.6W

λ/2

4.4W

Verdi-V10-50 λ/2

150

λ/4

λ/2

50
:5
0

100

λ/2

-50

λ/2

-50

to Dye laser

100Green lattice 2

AOM-1
Green lattice 1

110MHz
λ/2

75

+1

AOM

-25 λ/2

λ/4110MHz

PD

ing

imag

Green lattice 2

troGL1 re ontireflec

300

CCD

λ/2

vertical imaging

be
sa am
m
pl
er

0

30

λ/4

200

vertical lattice 1064 nm

vert. lattice
retro-reflection

400

λ/4

dichroic
mirror150

vert. im

CCD

r

λ/2
Mephisto

PD-50

Green lattice 1

PD

0

30

vertical lattice 1064 nm

400
100

ontal
horiz
ing
imag

250
ing
imag e
tiv
objec

am le
be mp
sa

horiz
o

aging

GL2 retroreflection

ntal O
D

400

T 106
4 nm

150

Figure 2.4.: Overview over the optical lattice setup. The alignment of the laser beams on the mainchamber is shown in a side-view (left) and a top-view (right). The box on top shows the distributionof the light at 532 nm based on a diode-pumped solid-state laser reaching an output power of 10 W.
The light at 532 nm is transported to the experiment using 8 m long optical fibers. The lower box
shows the last stage of the optical setup before reflection onto the atoms for the vertical lattice beamat 1064 nm with the light generated from a Mephisto MOPA. For reference, also the beams of thehorizontal optical dipole trap at 1064 nm and the 401-nm light for absorption imaging are shown.
Figure adapted from Ref. [Bai18b]

19

20

2. Erbium: From the basic properties to atoms in an optical lattice

beams at 532 nm, resulting in half the vertical lattice spacing, i. e. dx,y = dz/2 = 266 nm. The
lattice potential formed at these wavelengths is attractive for erbium atoms, which meansthat the atoms will locate at the intensity maxima. Note that, due to alignment purposes,
our vertical lattice beam has a small angle of about 11(2)° with respect to the vertical axis.
This translates into a slight deviation from a perfect cuboid lattice geometry leading tolattice spacing of (dx , dy , dz ) = (271(2), 266(1), 542(2)) nm.
We use two different laser systems to generate the light for three lattice beams. For thelight at 1064 nm for the vertical lattice, we use part of the power from our horizontal ODT.
This special configuration is possible, because both trapping beams are never switched on
simultaneously at maximum power. The lattice beams are ramped-up only at the end of theevaporative cooling stage, when the power in the horizontal trap is at low level. The laser
system used here is based on a Mephisto MOPA. The beam waist is 300 µm which results ina radial harmonic trapping frequency of about 2π × 25 Hz at 6 W. For the light at 532 nm,
we utilize a diode-pumped solid-state laser system9 . The beam waist for the lattice beams
is 160 µm, leading to a radial trapping frequency of about 2π × 40 Hz at a power of 1.2 W.
Figure 2.4 shows a schematic illustration of the optical lattice setup as implemented in theexperiment.
In addition to the optical lattice, the light at 532 nm is used at the same time as for opticalpumping of the dye laser for the generation of light at 583 nm. In combination with the 8 mlong optical fibers our maximal available power at the experiment table is limited to about
1.2 W. Since this also limits the maximum achievable lattice depth and because the dye laser
was replaced which provides more power, an upgraded light distribution was implemented
during the time of this thesis; see Appendix A.

Inhomogeneities over the atomic cloud
For the experimental realization of (spin) lattice models, it is important to have a homogeneous potential and energy landscape over the entire atomic sample. Any deviation from an
ideal configuration can have implications on the observed quantum phases or dynamics in thesystem. Inhomogeneities can be caused by various technical sources, such as the Gaussian
shape of the laser beams or external magnetic field gradients. In the following, we briefly
review the different sources of inhomogeneities with respect to our lattice system.
Let us look at the potential generated by two orthogonal lattice beams along the propagation
direction of one beam; see Fig. 2.5(a). For simplicity, we consider two sections, one throughthe maximum intensity and a second one slightly displaced from the center of one beam.
Figure 2.5(b) shows an amplified schematic illustration of the lattice inhomogeneity along
these lines. Note that the lattice potential is created by one of the two considered beams,
while the strong Gaussian confinement is generated by the second lattice beam in orthogonal
direction.
One consequence of the non-uniform intensity distribution is a spatial variation of the lattice9

Coherent Verdi V10, 10W at 532 nm.

2.3. The optical lattice: A platform for quantum simulation

21

Figure 2.5.: Schematic illustration of the optical lattice potential along one dimension. (a) Illustration of the two intersecting Gaussian beams (black solid lines). (b) The solid blue (red) line represents
the lattice potential V (x) along the dashed lines in (a). The black dashed line indicates the Gaussianenvelope of the intensity distribution of the laser beam. The horizontal green lines illustrate the lowest
energy level on the corresponding lattice site and ∆E ij indicates the energy difference between two
neighboring lattice sites. Note that this represents the energy shift in a strongly amplified mannersince in experiments atoms occupy dominantly the high lattice depth region.

depth, which follows the Gaussian envelope of the laser beam intensity. This results forexample in a variation of the tunneling rate t, defined by the overlap of the Wannier functionof two lattice sites i and j as

 2 2
Z
ℏ ∇
+ Vlatt (r) wj (r).
2.23
t = − d3 r wi∗ (r) −2mHere, wi∗ (r) corresponds to the Wannier function in the first lattice band as defined in
Eq. 2.22. Importantly, the change in potential depth leads to a modification of the local
Wannier functions and thus to a spatially non-uniform t. Figure 2.6(a) shows the spatial
variation of t along one horizontal dimension of our experimental system, considering amaximum lattice depth of V0 = 20 Erec 10 and a lattice size of about 40-80 sites. Assumingthat the atoms are located exactly in the center of the laser beams, the tunneling rateincreases by about 4 % towards the edge of the lattice. Note that in most experimental
realizations, the atoms dominantly occupy the lattice sites at the center of the intensity
distribution and therefore the variation of t is strongly reduced for larger beam waists.
Depending on the underlying physics, the energy offset ∆E ij between two adjacent latticesites (see Fig. 2.5(b)) takes on an important role. The origin of ∆E ij can be manifold. In oursystem, on the one hand, energy differences arise from the Gaussian intensity distribution
which leads to spatial dependent light shifts and a residual harmonic trapping confinement.
On the other hand, uncontrolled magnetic field gradients lead to a spatially varying Zeeman
shift. Therefore, for our experimental configuration, ∆E ij can be decomposed in threedominant contributions as
ij
ij
ij∆E ij = ∆Eharm+ ∆Equad+ ∆Elin
.

2.24

ij
ij
Here, ∆Eharmand ∆Equadaccount for energy differences that arise due to the residualij
harmonic trapping confinement and tensorial light shifts, and ∆Elin
accounts for the linear
102Erec = ℏ2 klat
/2m is the photon recoil energy with klat being the lattice wavevector.

22

2. Erbium: From the basic properties to atoms in an optical lattice

11

50

10.9

40
30

10.8

20

10.7

10

10.6

0
-40

-20

0

20

40

300

-40

-20

0

20

40

-3/2 3/2

11/2

19/2

10

150
0
0
-10
-150
-20

-300
-40

-20

0

20

40

-19/2 -11/2

ij
Figure 2.6.: Overview over the spatial variation of (a) t, (b) ∆Eharm, and (c) ∆Elin as a functionof the lattice site. Lattice site i = 0 corresponds to the center of the Gaussian beam-intensity
mF
distribution. (d) ∆Equad
as a function the mF states at lattice site i = 0.

Zeeman shift due to a magnetic field gradient. Despite all energy shifts are contributing to∆E ij , their impact depends on the underlying physical system. In the following, these threecontributions are discussed individually.
ij
between neighboring lattice sites i and i + 1 can be calculated
The energy difference ∆Eharm
considering the second part of Eq. 2.20 asij=∆Eharmmωx2 d2x 2(i − (i + 1)2 ).
2

2.25

ij
Figure 2.6(b) shows the variation of ∆Eharm
as function of the distance to the central latticesite for our experimental configuration. The energy difference increases linearly towards theedge of the lattice. For deep optical lattices, where tunneling is strongly reduced, the role ofij
ij∆Eharmis negligible for many experimental aspects. The role of ∆Eharm
becomes importantfor processes that rely on the contribution of t. A prominent example are super-exchangeinteractions, where strength of the exchange process is ∝ t2 [Tro08, Bro15]. Moreover, whenij
∆Eharm> 4t, particle tunneling is suppressed and thus it can lead to a spatially reduced
tunneling rate.
ij
ij
In contrast, ∆Elinand ∆Equad
can become important at large lattice depths, when atoms
are pinned to local sites. As we will see in Chapter 3, these contributions are particularly
relevant to the observation of spin-exchange dynamics, where atoms change their internal

2.3. The optical lattice: A platform for quantum simulation

23

magnetic sublevel. If the energy shift between lattice sites of involved atoms is too large,
these dynamics are strongly affected.
A magnetic field gradient translates into an energy shift at lattice site i with respect to thecenter of∆Elin = gF mF µB idx ∆B.
2.26
Figure 2.6(c) shows the resulting energy shift across the lattice array for atoms in mF = −19/2
due to a magnetic field gradient of ∆B = 0.023 mG/µm11 . Moreover, between adjacentij
lattice sites this leads to ∆Elin≈ 6 Hz.
What makes the quadratic light shift special is that, in addition to the intensity of thetrapping light, it depends quadratically on the magnetic sublevel. The light shift has its
origin in the tensorial polarizability and is given by [LK13, Lep14]
mF∆Equad= IαtF (F + 1) − 3m2F ) 1 − 3 cos2 θp.
F (2F − 1)2

2.27

Here, θp denotes the angle between the light polarization and the quantization axis and αt isthe tensorial polarizability coefficient. In Ref. [Bec18], αt was experimentally determined toij
mF
is estimated to be < 0.2 Hz across our atomic sample, ∆Equadbe −15 a.u.12 . While ∆Equadis changing significantly on a single lattice site for different mF states. Figure 2.6(d) showsmF
mF
isfor an atom sitting at the center at lattice site i = 0. The difference of ∆Equad∆Equadthe largest for the edge mF -states being about 10 Hz. Note that due to the dependence onθp , the magnitude of the light shift also depends on the orientation of the external magnetic
field and can therefore potentially be tuned by the light polarization.

11∆B was estimated for our system using radio-frequency spectroscopy measurements of the mJ = −6 tomJ = −5 transition in bosonic erbium.
12
1 a.u. = 4πϵ0 a30 = 1.6488 × 10−35 /2hϵ0 c Hz/Wmm−2 .

Chapter3

Dipolar spin-exchange dynamics
in a three-dimensional opticallattice

One of the most ambitious goals for experimental physicists is the realization of a quantum
simulator that allows for the understanding of complex many body quantum systems aswell as physical processes occurring in chemistry and biology [Geo14, Blo12, Gro17]. The
classical computational resources that are necessary for an efficient prediction of quantum
phenomena scale exponentially with the system size and reach therefore quickly fundamental
limitations. To date, a large number of different approaches to realize such a simulator are
based on various platforms and subject to ongoing investigations [Alt21]. Some of the mostpromising ideas include, e. g. trapped ions [Bla12, Hem18], superconducting circuits [Aru19,
Kja20], Rydberg atoms in optical tweezers [Ber17, Bro20], and neutral atoms in opticallattices [Jak05, Str11, Geo14].
Platforms based on neutral atoms bring along a number of advantages. On the one hand,
they are well isolated from the environment, for instance, the atoms do not couple to any
external thermal bath [Gro17]. On the other hand, these platforms have various tuning knobs
for both, internal degrees of freedom, for example the preparation of atoms in spin states, andexternal degrees of freedom, such as shaping of the corresponding trapping potential. Thanks
to the great technical development over the decades various possibilities have emerged to
exploit these degrees of freedom, including lasers featuring a narrow linewidth to address theatomic spin [Zha14, Sca14], spatial light modulators (SLM) [Gau13, Nog14] and digital micro
mirror devices (DMD) [Maz17, Nic19] for the creation of almost arbitrary optical potentials,
or quantum gas microscopes which enable single atom detection [Bak09, She10, Che16].
Putting all together, it has been proposed and successfully demonstrated that this represents powerful platforms to simulate relevant condensed matter phenomena [Alt21]. For
example, an approach to describe the electronic behavior in solid state systems is basedon the Hubbard model, whose central idea is the competition of inter particle interactions
and hopping between lattice sites [Aue94]. Over the last two decades, great progress on therealization of predictions made by the Hubbard model has been made using alkali atoms
loaded into optical lattices [Gro17], which includes, e. g. , the observation of the superfluid
to Mott insulator transition [Gre02], the confirmation of the existence of the Higgs mode25

26

3. Dipolar spin-exchange dynamics in a three-dimensional optical latticein two dimensional systems [End12], and the engineering of topological phases and artificial
gauge fields [Aid13, Aid15].
In common Hubbard models, including the experiments based on alkali atoms mentioned
above, the inter-particle interaction is dominantly short-ranged. As a consequence onlyinteractions that occur between particles sitting on the same lattice site, the so-called onsite interactions, are considered. Although the electrostatic interaction scales as ∝ 1/r andis therefore long-ranged, due to the electric field screening effect it represents is a good
approximation for many experimental situations. However, there exists a large class ofmodels, which include long-range interactions and are therefore hardly accessible with alkali
atoms [Dut15]. An important example are spin-spin interactions between atoms located indifferent lattice sites and the related realization of spin lattice models such as the Heisenberg
Hamiltonian [Hei28]. Partially, this problem has been circumvented by the implementationof super-exchange interactions, which involves a virtual tunneling process. As this processscales with the tunneling rate, these interactions are weak in deep optical lattices. They
are further limited to nearest neighbors and therefore a large class of Hubbard models are
hardly accessible with alkali atoms. This makes the observation of quantum magnetism andlong-range ordering in these systems challenging due to the small energy scales related andthe required low temperatures. Nevertheless, they enable the observation of new orderedground states such as the anti-ferromagnetic ordering [Maz17].
First experimental realizations of optical lattice systems with long-range interactions havebeen demonstrated with polar molecules featuring an electric dipole moment [Yan13, Haz14]
and with magnetic atoms such as chromium [dP13b, Lep19] and erbium [Bai16] featuringa magnetic dipole moment. While the magnetic dipole moment is permanent, the electric
dipole moment needs to be induced by an external electric field. Typically, the strength ofthe electric dipole moment is about one to two orders of magnitude larger than for magnetic
dipoles. However, molecules suffer from a more complex and less controlled preparation
protocols blocking these systems to very low filling factor. Recently, the field of ultracold
molecules has experienced a substantial improvement by the creation of the first moleculardegenerate quantum gases [Mar19, Dud21].
In this work, we have realized the first simulator for quantum magnetism based on fermionic
dipolar erbium atoms. The basis for the simulator is the dipole-dipole interaction, which
is described in Sec. 3.1. In the next step, in Sec. 3.2 we consider atoms which are loaded
into an optical lattice. We introduce the Hamiltonian describing our system and discuss
the XXZ-Heisenberg model. Finally, Sec. 3.3 presents our publication on the observation of
dipolar exchange interactions for large-spin fermions.

3.1. Dipole-dipole interaction

27

3.1. Dipole-dipole interaction
3.1.1. General description of two interacting dipoles
The dipole-dipole interaction (DDI) has led to the discovery of many fascinating phenomena [Cho22], such as the Fermi-surface deformation [Aik14a] or the recent realization ofsupersolid states in erbium [Cho19] and dysprosium [Böt19b, Tan19, Cho19, Nor21]. The
potential that describes the interaction between two non-polarized magnetic dipoles, µ1 andµ2 , can be written in a general form as [Lah09]µ0
UDDI (r) =
4πr3



3
µ1 µ2 − 2 (µ1 r) (µ2 r) .
r

3.1

Here, µ0 is the vacuum permeability1 and r = r1 − r2 denotes the distance between the twodipoles. Note that for electric dipoles, the equivalent formula is valid with the substitutionsµ0 → 1/ϵ0 and µi → di , with ϵ0 being the vacuum permittivity and di being the electric dipole
moment. However, while for magnetic atoms the dipole moment is permanent, the strengthof the electric dipole moment, for example of polar molecules, depends on the amplitude ofan external polarizing electric field.
In most experimental situations, the atoms are polarized by an external magnetic field B
and Eq. 3.1 can be simplified to
UDDI (r, θ) =µ0 µ2 1 − 3 cos2 θ
.
4π
r3

3.2

The strength of the DDI depends on both, the orientation of B with respect to the interparticle axis r, given by θ, and on the distance r = |r| between the interacting particles; see
Fig. 3.1(a).
Equation 3.2 reflects two important characteristics of the DDI:
 From a thermodynamic point of view, the DDI has long-range character in threedimensions, meaning that the energy is intensive in the thermodynamic limit anddepends on the total atom number and the atomic density [Ast08]. This manifests
by the 1/r3 dependence of the interaction energy, that leads to the divergence of theintegral of the interaction potential [Lah09, Cho22].
 The DDI is anisotropic and exhibits d-wave symmetry (characterized by four lobes withalternating phase), which is reflected in the dependence on 1 − 3 cos2 θ. Depending on
θ, the corresponding factor can vary from −2 to +1 and thus UDDI (r, θ) is attractive for
dipoles in head-to-tail configuration and repulsive for dipoles in side-by-side configuration; see Fig. 3.1(b)-(c). For the so-called magic-angle of θ = 54.7° the term 1 − 3 cos2 θand in consequence the DDI vanishes.
1µ0 = 1.256 637 062 12 × 10−6 N/A2

28

3. Dipolar spin-exchange dynamics in a three-dimensional optical lattice

40
20
0
-20
-40
-60
0

30

60

90

Figure 3.1.: (a) The interaction between two dipoles µ1 and µ2 polarized by an external magnetic
field depends on the orientation of B with respect to the inter-particle axis r. (b) Three-dimensional
representation of Eq. 3.2 for two interacting magnetic dipoles. Red (blue) color represent repulsive(attractive) interaction. (c) UDDI for two magnetic dipoles separated by 266 nm (see our latticeconfiguration in Sec. 2.3.2 and 3.2.3) as a function of the dipole orientation θ.

A further important consequence that arises directly from the DDI and has not been considered in Eq. 3.2 are spin-exchange interactions and spin-relaxation dynamics [Hen03, Pas10],which will be introduced in the upcoming section.

3.1.2. Spin-exchange and dipolar relaxation due to DDI
For simplicity, in the following the interaction potential between two atoms, labeled by the
numbers 1 and 2, where the magnetic interaction arises only from the electronic spin isconsidered (L = 0 and I = 0). The spin changing property becomes clear by rewriting thetensorial part (µ1 µ2 − 3 (µ1 er ) (µ2 er )) of Eq. 3.1, with er = r/|r| and µi = gs µB Si , in termsof the quantum mechanical operators as
1(S1 · S2 ) − 3(S1 · er )(S2 · er ) = S1z S2z + (S1+ S2− + S1− S2+ )2
3− (2rz S1z + r− S1+ + r+ S1− ) × (2rz S2z + r− S2+ + r+ S2− )
4


1 + −
− +
2
z z
= (1 − 3rz ) S1 S2 − (S1 S2 + S1 S2 )4
3 2 + + 3 2 − −− r− S1 S2 − r+ S1 S2
4
4
3
3
z +
− rz r− (S1 S2 + S1+ S2z ) − rz r+ (S1z S2− + S1− S2z )
2
2

3.3

with r± = rx ± iry and S ± = S x ± iS y [Hen03, Dut15]. The coefficients rx,y,z correspond tothe vector elements of er . A more intuitive picture can be obtained by including the transformation from the Cartesian into a spherical coordinate system, leading to rx = sin θ cos ϕ,
ry = sin θ sin ϕ, and rz = cos θ. Here, θ and ϕ are the polar and azimuth angle of r with respect to the Cartesian coordinate system. Inserting the coordinate transformation in Eq. 3.3

3.1. Dipole-dipole interaction

29

results in1(S1 · S2 ) − 3(S1 · er )(S2 · er ) = (1 − 3 cos2 θ) S1z S2z − (S1+ S2− + S1− S2+ )4
2 
3 sin θ −2iϕ + +−e
S1 S2 + h.c.
4

3 sin 2θ  −iϕ z +−e
S1 S2 + S1+ S2z + h.c. ,4

3.4

where h.c. denotes the Hermitian conjugate. Equation 3.4 reveals once more the special
tunability of dipolar interactions, since each individual term has a significant dependence on
θ.
To understand the impact of Eq. 3.4 we need to consider the eigenvalues of the individual
operators acting on the respective eigenstate. Acting on the state |S, ms ⟩, the operators Siz
and Si± result in
and
Si± |S, ms ⟩ =Siz |S, ms ⟩ = ms |S, ms ⟩3.5p
S(S + 1) − ms (ms ± 1) |S, ms ± 1⟩ .

3.6

While Siz leaves the magnetic sublevel unchanged, the ladder operators increment or decrement the magnetic quantum number ms .
Knowing the results of Siz and Si± acting on a quantum state |S, mS ⟩, the terms in Eq. 3.4
allow for an intuitive physical interpretation. In the first line, the interaction terms aremagnetization conserving, meaning that the total magnetization M 2 remains unchanged.
While, as shown in Eq. 3.5, S1z S2z does not change the magnetic quantum number for anyof the two atoms, the term S1+ S2− (S1− S2+ ) increases and decreases the magnetic quantum
numbers symmetrically. While ms increments for particle 1, it decrements for particle 2 andvice versa. This term is of particular interest for the aim of this thesis, because it allows forthe investigation of magnetization conserving spin-exchange dynamics in an optical lattice.
We will encounter this interaction term again in the following Secs. 3.2.1 and 3.2.2. Notethat the first term, (1 − 3 cos2 θ)S1z S2z corresponds to Eq. 3.2 and describes elastic dipolarinteractions.
In contrast to the first line, the terms in the second and third line of Eq. 3.4 do not preserve
the total magnetization. The term S1+ S2+ (S1− S2− ) increments (decrements) ms for both
atoms simultaneously and leads therefore to a change of M by ∆M = ±2. Finally, in thelast line of Eq. 3.4, ms of one atom remains unchanged, while the second atom changes its
projection by ±1.
Although all spin exchange and spin relaxation terms are present at any time, they are often
suppressed due to arguments based on energy conservation. In the following, we will consider
the case of fermionic erbium atoms confined in an optical lattice.
2

The total magnetization is defined as the sum of the magnetic quantum numbers as M =the superscript i denotes the particle index.

Pii ms , where

30

3. Dipolar spin-exchange dynamics in a three-dimensional optical lattice

3.2. Dipolar Fermions in an optical lattice
For dipolar atoms confined in an optical lattice, based on the decomposition of the DDI
given in Eq. 3.4, the Hamiltonian governing the dipolar interactions can be written as


X
1 + −
− +
2
z z
HDDI =
Vi,j {(1 − 3 cos θ) Fi Fj − (Fi Fj + Fi Fj )4i,j̸=i3.7 3 sin 2θ 
3 sin2 θ  −2iϕ + +e
Fi Fj + h.c. −e−iϕ Fiz Fj+ + Fi+ Fjz + h.c. }.
−44This contains also the substitution of S → F to account for the total angular moment
quantum number. Moreover, Vi,j denotes the coupling strength between two atoms sitting
on lattice sites i and j. It can be calculated using the definition of Wannier functions in
Eq. 2.22 byZµ0 µ2B gF2
1 − 3 cos2 θ
d3 rd3 r′|wi (r)|2 |wj (r′ )|2 .
3.8
Vi,j =
4π|r − r′ |3Note that the Hamiltonian in Eq. 3.7 accounts for the fact that each lattice site is occupied
by maximally one atom.
It is further important to mention, that for atoms in the lowest energy band, the nonmagnetization conserving terms in Eq. 3.7 are allowed only if the released Zeeman energy
is compensated by the change in the Fiz Fjz energy contribution. As this is only the casefor very low magnetic fields (B << 1 mG), for our experimental configuration only themagnetization conserving term needs to be considered. Note that demagnetization processes
involving higher energy bands have been nicely demonstrated in Ref. [dP13a] using bosonicchromium atoms.

3.2.1. Hamiltonian for spin lattice models
Depending on the underlying system properties, e. g. the fermionic or bosonic nature of theatoms or spin degree of freedom, different lattice models fit best to describe the system’s
behavior. As mentioned above, the system discussed withing this thesis consists of dipolar
fermionic erbium atoms loaded into a three-dimensional optical lattice. Considering both,spin dynamics (internal degree of freedom) and particle hopping (external degree of freedom),
the system is well described by a general spin-lattice model given by [Dut15]
X †H = −t(fi,σ fj,σ + h.c.)⟨i,j⟩,σ+UXi,σ̸=σ ′+ni,σ ni,σ′


3.9


1X
1
Vi,j Fiz Fjz − (Fi+ Fj− + Fi− Fj+ ) .
2
4
i̸=j

The first line in Eq. 3.9 describes the tunneling between neighboring lattice sites i and j†denoted by ⟨i, j⟩. The operators fi,σand fi,σ are the creation and annihilation operator for3.2. Dipolar Fermions in an optical lattice31

an atom on lattice site i with spin state σ. The tunneling rate t depends on the overlap
between two neighboring Wannier functions and was defined previously in Eq. 2.23.
The second line includes the on-site interaction U = Uc + UDDI , i. e. the interaction between
atoms sitting on the same lattice site, accounting for both [Bai16], contactZ

Uc =4πℏ2 asmZ

d3 rd3 r′

and dipolarµ0 µ2B gF2
UDDI =
4π

d3 r|w(r)|2 ,1 − 3 cos2 θ|w(r)|2 |w(r′ )|2 .
|r − r′ |3

3.10

3.11

interactions. The parameter as denotes the s-wave scattering length.

Ultimately, the last line of Eq. 3.9, describes the off-site contribution due to the DDI. While
the symmetric term Fiz Fjz accounts for the diagonal part of the interactions, the off-diagonalpart is given by Fi+ Fj− + Fi− Fj+ , describing the dipolar-exchange interactions. The nonmagnetization conserving terms in Eq. 3.7 have been omitted here since they are highly
suppressed in our experimental parameter regime.
Importantly, the full Hamiltonian of Eq. 3.9 has never been realized under the aspect ofcompeting t and U . Up to today, only limited situations with very low tunneling have been
explored experimentally [Yan13, Lep19] (see also Sec. 3.2.3) and the experiments focused on
spin-spin interactions.

3.2.2. The Heisenberg Model
From a classical point of view, the Bohr -Van Leuween theorem, demonstrated by N. Bohrand J. H. Van Leuween in their respective doctoral theses [Boh11, Van21], says that according to classical mechanics the net magnetization of a particle ensemble vanishes in thermal
equilibrium. This argumentation roots in the fact that following statistical mechanics, theoccupation probability for any state of motion depends on the kinetic energy. An external
magnetic field exhibits a force orthogonal to the particle velocity and has therefore no influence on the kinetic energy. This means that the motion of the particle is the same whether amagnetic field is applied or not, and therefore no effects such as para- or diamagnetism canbe explained. Moreover, this underlines, that magnetism is a purely quantum mechanicalphenomenon [Fey11].
A model, which in the past has proven to successfully describe magnetic interactions, is the
Heisenberg model, developed by W. Heisenberg in 1928 [Hei28]. Among the most famous
magnetic orders are the ferro- and antiferromagnetic ground states, in which the microscopic
spins are parallel or anti-parallel aligned [Aue94, Maz17]. For two fermionic erbium atoms,
the Hamiltonian can be written as [Bru04]
X
HH = −Jij Fi · Fj .
3.12i,j̸=i32

3. Dipolar spin-exchange dynamics in a three-dimensional optical lattice

Here, Jij describes the spin interaction between particles sitting at site i and at site j and Fi
is the spin vector. From Eq. 3.12 it follows directly that the spins energetically prefer parallel(anti-parallel) ordering for Jij > 0 (Jij < 0) [Aue94]. Note again that in our case the spinis encoded in the Zeeman sublevels of the hyperfine manifold and we therefore substitute S
with F.
By carrying out the vector multiplication, Eq. 3.12 becomes
HH = −X

Jx Fix Fjx + Jy Fiy Fjy + Jz Fiz Fjz .

3.13i,j̸=iFour special cases result from this description:
 In one dimension, where Jx = Jy = 0, the Heisenberg Hamiltonian simplifies to the
Ising model in which the number of spin components is reduced to one being parallelor anti-parallel to the quantization axis.
 The XYZ-Heisenberg model for which Jx ̸= Jy ̸= Jz . For dipolar interactions the realization of the XYZ-Heisenberg model is directly associated with the non-magnetizationconserving terms in Eq. 3.7, as [Cho22]

αSix Sjx + βSiy Sjy = α−β α+β  + −Si Sj + Si− Sj+ +Si+ Sj+ + Si− Sj− .
2
4 The isotropic XXX-Heisenberg model for equal coupling strengths Jx = Jy = Jz .
 The anisotropic XXZ-Heisenberg model for Jx = Jy ̸= Jz .

Introducing once more the raising and lowering operators Fi± = Fix ± iFiy , the XXZHeisenberg model, which is of central interest within this work, can be re-written as
HXXZ =XJx  + −Jz Fiz Fjz +Fi Fj + Fi− Fj+ .
2

3.14

i,j̸=i

When comparing Eq. 3.14 with the last line of Eq. 3.9 a strong similarity becomes evident.
Importantly, this underlines the fact that our system represents a quantum simulator for
the XXZ-Heisenberg Hamiltonian, arising directly from the dipole-dipole interaction. To
avoid possible confusion it should be emphasized that in literature J is often used as anabbreviation for the tunneling rate, while here Jx,y,z denotes the coupling strength between
the spin orientations.
The results of the operators Fiz and Fi+ acting on the atomic state |F, mF ⟩ is analogous
to Eq. 3.5 and Eq. 3.6, where S is replaced by F . By considering Eq. 3.14, Eq. 3.9, andapplying the operator rules given in Eq. 3.6, it becomes clear, that the early timescale forthe spin-exchange dynamics driven by Fi+ Fj− (and Fi− Fj+ ), for a fixed lattice configuration,
is depending on two factors:

3.2. Dipolar Fermions in an optical lattice33(i) on the initial mF state, aspp
F + F − |F, mF ⟩ = F (F + 1) − mF (mF + 1) F (F + 1) − mF (mF − 1) |F, mF ⟩= γ(mF ) |F, mF ⟩and(ii) on the strength of the DDI given by Vi,j and therefore on the dipole orientation θ; see
Eq. 3.8.
In our experimental work presented in Sec. 3.3, we demonstrate control over both the initial
spin state and the dipole orientation.

3.2.3. Our experimental system
In this section we give a brief reminder about our experimental lattice setup and describein particular how it relates to the Hamiltonian given in Eq. 3.9. Note that a more detailed
description of the setup is given in Sec. 2.3.2.
Our lattice is created by three orthogonal retro-reflected laser beams, whereby two beams
that are at a wavelength of 532 nm form a horizontal plane and a third beam at 1064 nm
propagates along gravity. This leads to lattice spacing of 266 nm between two sites within
the horizontal plane and a spacing of 532 nm along the vertical direction. We perform ourmeasurements at lattice depths of (sx , sy , sz ) = (20, 20, 80), given in units of the respective
x,y,z. The related tunneling rates are of about 10 Hz along x and y, andrecoil energy Erec
about 1 mHz along z. As we will see below, these rates are comparatively small with respect
to timescales of the spin-exchange dynamics and the contribution of tunneling to Eq. 3.9 istherefore strongly suppressed.
To prepare atoms in different initial Zeeman states, we apply a technique that was developed
within our group in Ref. [Bai18a]. In brief, we exploit the quadratic Zeeman shift that arises
for fermionic erbium due to the additional nuclear spin to induce a differential energy shift
between neighboring Zeeman levels; see also Sec. 2.1.3. We ramp the magnetic field to about
40 G and transfer the atoms, initially polarized in the lowest magnetic sublevel mF = −19/2,
by means of a rapid-adiabatic passage. Importantly, prior ramping the magnetic field, weload the atoms into the optical lattice and use the latter as a protection shield against
three-body losses when crossing the dense Feshbach spectrum [Fri14b, Mai15]. The fact that
we are using fermionic erbium and that we are loading a spin polarized sample into thelattice leads, in addition to the suppressed tunneling, to a second important consequence tothe Hamiltonian in Eq. 3.9: The Pauli exclusion principle prevents double occupancies andtherefore also onsite interactions are negligible. Considering the Hamiltonian in Eq. 3.9, we
can therefore consider only the last line, which describes the spin-exchange interactions.
However, as experimental realizations are never perfect, inhomogeneities over our atomic
sample that derive either from ac-Stark shifts or Zeeman shifts need to be considered. A

34

3. Dipolar spin-exchange dynamics in a three-dimensional optical lattice

80

40

60

30

40

20

20

10

0
-19/2 -11/2

0-3/2 3/2

11/2

19/2

0

30

60

90

Figure 3.2.: Interaction strength governing the early spin dynamics for the simplified configurationof 2 atoms in a double well potential for different (a) initial mF states with θ = 0 ◦ and (b) dipole
orientations for mF = −17/2. The interaction strength Vdd is calculated using Eq. 3.8. The dashedline in (a) is a quadratic function as a guide for the eye to indicate the quadratic dependence on mF .
The dotted line in (b) indicates the magic angle at θ = 54.7 ◦ where the DDI vanishes.

quantitative description of inhomogeneities in our lattice system is given in Sec. 2.3.2. Theenergy contribution regarding these additional terms can be written as
X
X
Hinh =
δi (Fiz )2 +Bi Fiz3.15iiHere, δi accounts for all quadratic shifts and Bi considers a lattice site dependent magnetic
field gradient. It is important to mention that in Sec. 3.3 we use these inhomogeneities forour advantage to temporally control the spin-exchange dynamics.
Finally, the Hamiltonian describing our system can be written as

 X
X
1 + −
1X
− +
z z
Vi,j Fi Fj − (Fi Fj + Fi Fj ) +δi (Fiz )2 +Bi Fiz .
H=
2
4
i̸=j

i

3.16

i

For our particular system, Fig. 3.2 shows the contributions of γ(mF ) and the angle-dependent
Vi,j for the simplified situation of two dipoles in a double well potential, separated by 266 nm.
This extraordinary tunability leads to two different control knobs, which allow for a carefuland controlled tuning of the spin-exchange dynamics.

3.3. Publication: Controlling dipolar exchange interactions in adense three-dimensional array of large-spin fermions.
Physical Review Research 2, 023050 (2020)3
submitted 27 Sep 2019; published 17 Apr 2020
DOI: 10.1103/PhysRevResearch.2.023050

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

Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
3
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
4
JILA, NIST, and Department of Physics, University of Colorado, 440 UCB, Boulder, Colorado
80309, USA
5
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
2

3

The author of the present thesis developed the experimental procedures together with L. C., D. P., andS. B., performed the measurements together with L. C. and D. P., analyzed the data, and contributed inwriting the manuscript. The theoretical calculations of this publication have been contributed by B. Z. andA.-M. R..

35

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

1,5,*

1

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

I. INTRODUCTION

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

Corresponding author: manfred.mark@uibk.ac.at

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.
2643-1564/2020/2(2)/023050(16)023050-1

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

A. PATSCHEIDER et al.

(a)

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

(b)

z
B

Θ
y

Φ
x

Activation(c)
Initialization

Time evolution

Δ1
Δ2

early timelong time

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

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

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

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

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

δ / h (Hz)
0 20 40 60 80

(b)

-12/2

1

(c) -19/2

1

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

0.9

(d) -19/2

0.5

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

0.4

Spin state (mF)

1 -40 -20

M(t)

(a)

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

-13/2

0.8
Δ2

0.6
Δ1
Δ2

0.4

50

100

0.6

Spin state (mF)

Δ1Spin population

Spin population

0.8

0.4
0.2

0.2
0

-14/2
0

0
1

1.5
2
2.5
Magnetic field (G)

25

0

50
75
Time (ms)

100

0.8
0.7
0.6

0.3
0.2
0.1
25

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

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

023050-3

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

IV. COMPARISON TO GDTWA

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

V. CONTROL OF THE INTERACTION STRENGTH

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

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

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

0.8
0.6
0.4
0.2
0
1

Spin population

(a)
a)

(d)

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

42
28
14

0.6
0.4
0.2
0
25

50 75 100 0Time (ms)25

50 75 100 0Time (ms)

25

-17/2

-9/2

(h) 1

0.8

0

56

0

Spin population

Spin population

1

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

17/2

0.8

0.6

0.4
0

0.5

1
Veff / ħ . t

1.5

2

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

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

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

VII. CONCLUSION AND OUTLOOKVI. OPTICAL CONTROL

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

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

023050-5

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

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

Spin population

15
0.8

10

0.6
5
ACKNOWLEDGMENTS

0.4
0

25 50
75 100Time (ms)1.2

Spin population

Spin population

(c)

1
0.8

0

0

20 40 60
Θ (deg)

80

50

75
100
δiZ / h (Hz)

125

75

100

1

0.7

0.4
25

0.6
0.4
0.2

0

25

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

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

APPENDIX A: EXPERIMENTAL SETUP
AND LATTICE LOADING

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

023050-6

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

(a)
0.5

0

E/h (MHz)

-0.5

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

-1

-2

avoided crossing
-380

100

(b)

95
90
85
80

-370

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

Counting efficiency (%)

Preparation efficiency (%)

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

(c)

80
60
40
20

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

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

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

APPENDIX C: QUENCH PROTOCOL AND
DETECTION SEQUENCE

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

023050-7

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

tfit (ms)

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

− 17
2

34.2

1.8

0.7

− 13
2

15.7

7.2

0.69

0.68

5.3

0.66

0.64

19.6

− 29
− 25
− 21
1
2
3
2
9
2
13
2

0.55

11.3

8.7

0.67

0.62

25.1

0.51

9.6
9.0

13.7

0.66

0.58

27.7

0.48

12.2

0.65

0.57

36.1

0.42

9.0

13.5

0.65

0.56

36.7

0.41

9.2

9.2

0.64

0.58

34.0

0.43

11.3

6.7

0.63

0.59

23.1

0.49

15.7

4.9

0.62

0.59

21.9

0.48

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

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

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

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

0
10
20
30
35
40
50
60
80

26.8
30.1
36.7
47.0
52.0
53.2
46.1
37.0
30.0

13.8
11.9
6.9
8.6
6.7
7.2
12
8.9
10.4

0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7

0.60
0.61
0.65
0.64
0.65
0.65
0.61
0.63
0.62

30.1
25.6
18.2
17.8
11.4
13.2
17.2
19.2
19.3

0.49
0.52
0.57
0.57
0.62
0.60
0.58
0.56
0.56

023050-8

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

|5/2>

Bz

|3/2>

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

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

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

APPENDIX D: LIFETIME AND LOSSES IN THE LATTICE

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

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

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

i

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

(F1)

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

023050-9

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

Vdd =μ0 (μB gF )4π

d 3 rd 3 r

(F2)

1
I.
N

iD =

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

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

APPENDIX G: THE GDTWA METHOD

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

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

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

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


μ

cμi iμ ,

with



cμi = Tr iμ Ôi ,

(G5)
(G6)

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

diμ
dt

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

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

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


{aμ }

p({aμ })λiμ,{aμ } (t ).

(G8)

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

1

8

0.6

Average neighbors

Spin population

0.8

Spin population= |-9/2>

1
0.8
0.6
0.4
0.2

0

25

50
75
Time (ms)100

Random removal
NN-dependent removal

6
4
2
0
0

0.4

0.2
0

20

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

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

0.5

1
1.5
2
2.5
Relative distance (dy)3

3.5

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

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

023050-11

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

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

Spin population

A. PATSCHEIDER et al.

1

0.6
0.4
0

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

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

5

10
15
Time (ms)20

25

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

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

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

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

= |-9/2>

0.8

0.6

0.4
0

0.5

1
V /ħ.t1.5

2

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

023050-12

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

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

(J1)
(J2)


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

m,n

i

(J3)

m,n

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

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

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

1

2

3

4

1

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

0.8|+3/2>

0.6
0.4
0.2

= |-17/2>

0.8
Spin population

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

Fermi-Hubbard Hamiltonian:

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

0

0.6

0

0.4
0.2
0
0

20

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

0.5

1
V /ħ.t1.5

2

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

023050-13

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

be generally rewritten in terms of
Umnkl =

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

mS =−S

S=0,2,...

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

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

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

0
−a
0
0
a
0
0
0
0
0
0
0
0
0
0
0

0
0
b
0
0
0
0
0
−b
0
0
0
0
0
0
0

0
0
0
c
0
0
d
0
0
−d
0
0
−c
0
0
0

0
a
0
0
−a
0
0
0
0
0
0
0
0
0
0
0

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

0
0
0
d
0
0
e
0
0
−e
0
0
−d
0
0
0

i,j

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

0
0
0
0
0
0
0
b
0
0
0
0
0
−b
0
0

0
0
−b
0
0
0
0
0
b
0
0
0
0
0
0
0

0
0
0
−d
0
0
−e
0
0
e
0
0
d
0
0
0

where

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

0
0
0
0
0
0
0
0
0
0
0
−a
0
0
a
0

0
0
0
−c
0
0
−d
0
0
d
0
0
c
0
0
0

0
0
0
0
0
0
0
−b
0
0
0
0
0
b
0
0

0
0
0
0
0
0
0
0
0
0
0
a
0
0
−a
0

(J13)

0

d=

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

(J17)

e=

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

(J18)

a=

4J 2
,
3Udd + 2Us

(J14)

b=

8J 2
,
3Udd − 4Us(J15)

J is the nearest-neighbor tunneling rate,c=

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

(J16)

Us =

023050-14

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

4π h̄2 a2Mdr|φ(r)|4 ,(J19)

CONTROLLING DIPOLAR EXCHANGE INTERACTIONS IN …

PHYSICAL REVIEW RESEARCH 2, 023050 (2020)

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

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

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n−1/2 = 1 −3t 2  2
V + O(t 4 ),4 i, j=i i, j

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023050-16

Chapter

4

Narrow inner-shell orbitaltransition at 1299 nm

One of the strengths of precision atomic physics is the possibility to resolve the ladderstructure of the energy levels of an atomic spectrum. In particular, a tremendous advancehas been the identification, manipulation, and control of two isolated energy levels in an atomthat form a two-level system. Such levels can be used as an atomic oscillator with powerfulapplication in metrology. An atomic oscillator can indeed be used as a high precision clock,being an ideal frequency reference. The reason for this are the relatively small linewidths ofthe excited states, which be intuitively interpreted as measurement uncertainties.
Time occupies not only a special role in the organization of our everyday humans life, butalso represents a very important physical quantity. Its accurate knowledge is fundamental fora large variety of applications, e. g. to increase the accuracy of Global Navigation Satellite
Systems [Lud15] or to determine the earth’s geoid [Bon12]. Historically, the first clocks
were built by the ancient Sumerians and Egyptians around 3000 B. C. based on the positionof the sun. Although many new technologies to measure time have been developed overtime, such as candle clocks, hourglasses, pendulum clocks, and quartz clocks, the concept
has remained the same for all technical approaches, i. e. counting periods of a periodicallyrepeating observable [Bru00]. Modern definition of time use an atomic oscillator as reference.
Since 1967, the second is defined via the microwave transition between the two hyperfine
states of the ground state in atomic caesium-133 [Aud02]. Today’s best clocks are basedon optical atomic transitions and reach an accuracy on the order of 10−18 , leading to aclock uncertainty of 1 s over more than 300 billion years [Gao18, Bot19a, Bel21]. Indeed, theInternational Bureau of Weights and Measures is pursuing the goal of redefining the secondon the basis of ultranarrow optical transitions around the year 2030 [Pet21].
In addition to the application in atomic clocks, narrow and ultranarrow atomic transitions
have demonstrated to be a very versatile tool in ultracold quantum gas experiments. Thanks
to the large Q-factor1 they are attractive for the application in high-precision measurements
as well as for the manipulation and the control of atoms. An outstanding example is the
1 S → 3 P transition in fermionic ytterbium and strontium featuring a linewidth in the00mHz regime. The reason for the existence of this transition, even though it is spin for1

The ratio between the transition frequency and the linewidth is referred to as quality-factor (Q-factor).

53

54

4. Narrow inner-shell orbital transition at 1299 nm
104
1
0.8
0.6
0.4
0.2
0
Er

Yb

Dy Tm Ho

Tb

Gd

Eu

Sm Pm

Nd

Pr

Ce

La

Lu

Figure 4.1.: Energy levels with ∆J = 0, ±1 with respect to the ground state for all atomic species ofthe lanthanide series up to an energy of 10.5 × 104 cm−1 . The horizontal red (blue) lines correspondto even (odd) parity. The shaded are denotes the telecommunication-wavelength window ranging
from 1260 nm until 1625 nm, being the lower (upper) limit of the O-band (L-band). Until today, laser
cooling has been realized for Er, Yb, Dy, Tm, Ho, and Eu.

bidden, is the hyperfine interaction, which causes a mixing between states of the same
total spin F , including the 3 P1 , 3 P2 , and 1 P1 levels [Boy07]. The high spectral resolution allowed to accurately determine the s- and p-wave interaction strengths by performing
lattice- or Ramsey-spectroscopy [Zha14, Sca14] and enabled the observation of many-bodyeffects including the building up of correlations in optical lattice clocks [Mar13, Rey14].
Recent experiments showed, that ultranarrow transitions can also be employed to coherently create molecules [Ni08, Cap19] and realize novel synthetic phases via spin-orbit coupling [Kol17, Liv16]. Ultimately, the coherent control over individual atomic states is afundamental ingredient for quantum simulation and quantum computation platforms basedon neutral atoms [Dal08, Gor09]. Very recently, in Ref. [Hei20] it was demonstrated that inan optical lattice, which operates at a tune-out wavelength for the ground state2 , losses due
to excited state interactions can be highly suppressed, paving the path for the applicationof quantum simulation and quantum computation schemes.
Although the study of ultranarrow atomic transitions has been mostly focused on atomic
species having two addressable valence electrons, the existence of narrow transitions can begenerally expected in multi-electron atoms. In this thesis, we have performed a first experimental study in search of a clock-type transition in ultracold atomic erbium. Prior tothis work, to the best of our knowledge, the transition at 1299 nm has not been directly
observed. The NIST database [Kra20] refers to unpublished spectroscopic data of van Kleef
and Koot [vK75] from the 70’s. A more detailed historical overview on spectroscopic measurements of erbium in given in Sec. 2.1.1. Complementary, H. Y. Ban and co-workers in
Ref. [Ban05] performed theoretical calculations of the atomic structure and qualitatively reproduced the reported energy levels. For atoms of the lanthanide series, very recently, firstexperimental studies have been performed using dysprosium [Pet20a] and thulium [Tre20],
which possess narrow atomic transitions at 1001 nm and 1140 nm, respectively. While dysprosium, similar to erbium, has a large permanent magnetic dipole moment, the interest
2

At the tune-out wavelength the atomic polarizability is zero and therefore the trapping potential vanishes.

4.1. Electron configuration and transition parameters55

in thulium arises from the fact that the transition corresponds to an inner-shell f -f excitation. This makes the transition frequency less sensitive to energy shifts induced byblack-body radiation, which constitutes one of the biggest limitations in current opticalclock systems [Gol19, Fed20].
It is interesting to note, that although almost all atomic species of the lanthanide series exhibit a large number of optical transitions, erbium is so far the only atomic species that has
an atomic transition in the telecommunication-wavelength window and has been successfullylaser cooled; see Fig. 4.1. This could be an important aspect in the field of quantum information in the future, making erbium an interesting candidate for this type of application.
Within the work of this thesis, we extend our experimental toolbox with the narrow innershell orbital transition at 1299 nm. We observe the atomic transition and characterize important parameters such as the excited state lifetime and related linewidth, the gJ -factor,
and the dynamic polarizability of the excited state relative to the ground state at 532.2 nm.
The first part of this chapter, Sec. 4.1, will introduce the electron configuration of the stateof interest and the most important transition parameters. The publication that derived fromthe work on this transition is presented in Sec. 4.2. Finally, in Sec. 4.3 we present further
work on the atomic transition, i. e. the determination of the A and B hyperfine constants,
the dependency of the excited state lifetime on the atomic density, and the preparation of
atoms in higher Zeeman sublevels. A description of the laser system that is used to generateand stabilize the light at 1299 nm is given in Appendix B.
On a personal note, as part of this project, I had the privilege to visit Jun Ye’s group in
Boulder (Co) for two months. Being one of the world’s leading groups in the field of threedimensional lattice clocks and high-precision metrology, I was able to gain a deep insight
into the world of ultra-stable laser systems during this time. Furthermore, I visited Stable
Laser Systems Inc. during my stay in Boulder, which was an important aspect in the further
planning and realization of this project in terms of the laser system.

4.1. Electron configuration and transition parameters4.1.1. The J1 j-coupling and energy level diagram of the 4f 11 5d6s2 electron
configuration
The valence electron configuration relevant in this chapter is the one of 4f 11 5d6s2 , which,in contrast to the ground state, involves the excitation of an electron from the 4f -shell to
the 5d-shell. Interestingly, both orbitals have a lower principle quantum number than the
6s orbital occupied by two electrons. This circumstance is the motivation for naming thetransition an inner-shell orbital transition. Fig. 4.2 shows a schematic illustration of theinvolved orbitals.
The coupling of valence electrons is an important effect that must be considered to describe the energy levels of an atomic system. Due to the large mass of erbium, contribution

56

4. Narrow inner-shell orbital transition at 1299 nm

Figure 4.2.: Schematic illustration of the inner-shell orbital transition corresponding to the excitationof an electron in the 4f to the 5d orbital. The red shell illustrates the 6s orbital, while the blue shellcorresponds to the (a) 4f (ml = 1) and (b) 5d (ml = 2) orbital. Note that the inner shells are scaled
up for better visibility.

of the spin-orbit interaction becomes significant with respect to the Coulomb interactions
among the electrons and makes therefore the application of coupling schemes other thanLS-coupling3 necessary. A coupling scheme that is suitable for this electronic configurationis the J1 J2 -coupling, also referred to as J1 j-coupling [Dra96, Wyb07]. This scheme is particularly relevant in configurations, where a single electron of an orbital l′ is weakly coupled
to a core containing N electrons. Applied to our configuration, this means that the electronin the 5d orbital couples to the 11 remaining electrons in the 4f orbital.
In the following we briefly outline the application of the J1 j-coupling scheme to the 4f 11 5d6s2
electron configuration. First, the 11 remaining valence electrons in the 4f -core are considered, which couple via LS-coupling. This leads to angular momentum and total spin quantum
numbers of L = 6 and S = 3/2. Following the LS-coupling scheme, this results in a possible
total angular momentum ranging from J1 = L−S = 9/2 to J1 = L+S = 15/2. In the manifoldof energy levels that we are interested in, the inner-valence electrons couple to J1 = 15/2, also
written as 4 I15/2 4 .
Subsequently, the spin s = 1/2 of the electron in the 5d-orbital adds vectorially to the orbital
angular momentum l = 2 leading to j = s + l and j = 3/2 or j = 5/2. Note that the quantum
numbers are written in lowercase letters to indicate that they refer to a single electron.
Finally, following the J1 j-coupling, j couples to J1 (J1 = 15/2) resulting in J = J1 + j withangular momentum quantum numbers ranging from J = J1 − j to J = J1 + j. The final
atomic state with total angular momentum J is usually labeled as (J1 , j)J . Accounting for
all coupling possibilities for the combinations of J1 and j leads to six and four different
energy levels, respectively, with each group representing the fine-structure. Interestingly,
this manifold of energy levels represents an odd parity ground state configuration, as there
is no lower lying energy level with the same parity.
Figure 4.3 shows the manifold of energy levels with inner-valence electrons coupled to 4 I15/2 .
3

4

P
Also known as Russel-Saunders coupling scheme. For N electrons, the total spin S = N
i si and the total
PN
orbital angular momentum L = i li couple to a total angular momentum J = S + L.
The electronic state is denoted as S LJ .

4.1. Electron configuration and transition parameters

104
2.5

57

103
12

2
1.5
1

10

8

0.5
0

6

Figure 4.3.: (a) Full energy level spectrum (see also Fig. 2.1) and (b) energy levels of the electron
configuration where the electron in 5d couples to the f 11 (4 I15/2 ) core via J1 j-coupling. The levels
correspond to (J1 , j) = (15/2, 3/2) and (J1 , j) = (15/2, 5/2), respectively. The dashed line connects theenergy levels that belong to the same (J1 , j) coupling manifold. All the energy levels shown in (b)have odd parity.

As mentioned in Sec. 2.1.1, the levels were assigned first by N. Spector [Spe65] in 1965.
All 10 levels are energetically between 7177 cm−1 and 11 888 cm−1 and their respective J
values range from 5 to 10. Remember, that the ground state of erbium has a total angularmomentum of J = 6 and therefore only half of the levels fullfill the selection rule ∆J =0, ±1. As discussed above, the energy levels can be separated into two manifolds, which
can be distinguished by the coupling of the electronic spin and the angular momentum of
the electron in the 5d orbital. The manifolds are denoted by (J1 , j)J = (15/2, 3/2)J and(J1 , j)J = (15/2, 5/2)J , containing four and six energy levels, respectively.
The state at an energy of 7969 cm−1 with respect to the ground state, which corresponds toa wavelength of 1299 nm, has quantum numbers of j = 3/2 and J = 7. Considering also theatomic core, the atomic state is denoted as[Xe]f 11 (4 I15/2 )5d(2 D3/2 )6s2 (15/2, 3/2)07 .

4.1

Note that 0 indicates the odd parity of the atomic state.

4.1.2. Lifetime of the excited state and other transition parameters
For application in ultracold quantum gas experiments, a number of parameters characterizing
the atomic transition play an important role. One of them is the lifetime, which describes
the time an electron remains in the excited state before decaying back to the ground state byspontaneous emission of a photon. To derive an expression for the decay rate Γ the couplingof the atom to all modes of the vacuum field needs to be considered [Wei30], which, however,goes beyond the scope of this section. At this point we refer to further literature, for example
Ref. [Ste20, CT98].

58

4. Narrow inner-shell orbital transition at 1299 nm

Table 4.1.: Theoretical and experimental parameters for the atomic state at 1299 nm. The theoreticalparameters are deduced from calculations in Ref. [Ban05] while the experimental ones are determinedwithin the work of this thesis.

Property

Theoretical values

Experimental value

13(7) s−1
2.1(1.1) Hz
75(40) ms
1.3(7) µW/m2

5.6(6) s−1
0.9(1) Hz
178(19) ms
0.56(6) µW/m2transition rate Γ
linewidth ∆ν
lifetime τsaturation intensity Isat
recoil temperature Trec
recoil velocity vrec

33.8 nK
1.83 mm/s

For a two-level system formed by the two states |g⟩ and |e⟩5 , Γ is given by [Ste20]Γ=ω03 × |⟨g| d |e⟩|2,
3πϵ0 ℏc3

4.2

with ω0 being the transition frequency arising from the energy difference between |g⟩ and |e⟩.
Moreover, the transition rate depends on the dipole matrix element ⟨g| d |e⟩ of the involved
states, which corresponds to the electric dipole moment associated with the transition, andc denotes the speed of light.
In systems involving more than two atomic levels, all possible transition rates to lower-energy
states must be considered for the lifetime of a single state. In addition, for experimental applications it is important that the transition is optically closed, meaning that the decay toother states than the ground state is highly suppressed. We can make a few simple considerations that indicate there are no significant optical leaks [Ban05]. First, in the entire
energy spectrum of erbium (see Fig. 2.1 and Fig. 4.3(a)), no energy level exists below thestate at 1299 nm, which has even parity and satisfies the dipole selection rule ∆J = 0, ±1.
This leads to the conclusion that no spontaneous emission can occur via an electric dipole
transition (E1) to states other than the ground state. Second, possible higher order transitions via magnetic dipole radiation (M1) and electric quadrupole radiation (M2) are highly
suppressed due to a low energy difference between the involved states and the dependence ofΓ ∝ ω03 . Combining these arguments and considerations, there will be no significant opticalleakage to different states and the transition at 1299 nm can therefore be considered to beclosed for our experimental purposes.
Prior to the work within this thesis, a first theoretical study of Γ, which relates to the excitedstate lifetime τ = 1/Γ and the natural transition linewidth ∆ν = Γ/2π, has been performedby H. Y. Ban and co-workers in Ref. [Ban05]. The calculations of the dipole matrix elementin Eq. 4.2 are based on the relativistic Hartree-Fock code of Cowan [Cow81] and yield atransition rate of Γ = 13(7) s−1 , which converts into a lifetime of τ = 75(40) ms and alinewidth of ∆ν = 2(1) Hz. The relatively long lifetime arises as a direct consequence of the
major quintet character of the excited state (5 H7 ) in contrast to the triplet character of theground state (3 H6 ).
5Here and in the following |g⟩ refers to the ground state and |e⟩ to the excited state of the system.

4.1. Electron configuration and transition parameters

59

Table 4.1 summarizes some important transition parameters that arise from the lifetime ofthe excited state and gives, where it is appropriate, a comparison to the experimental values
determined within this thesis; see Sec. 4.2. In addition to Γ, ∆ν, and τ , the saturation
intensity6 , the recoil temperature7 , and recoil velocity8 are given. Note that, the theoretical
calculations agree within two standard deviations with the experimentally determined values.
A description of the magnetic field sensitivity of the atomic state is given in Appendix C.

6
7
8

πhc
Saturation intensity Isat = 3λ3τ .
ℏ2 k 2
Recoil temperature Trec = 2mk.
BℏkRecoil velocity vrec = 2 .

4.2. Publication: Observation of a narrow inner-shell orbitaltransition in atomic erbium at 1299 nm.
Physical Review Research 3, 033256 (2021) 9
submitted 04 May 2021; published 17 September 2021
DOI: 10.1103/PhysRevResearch.3.033256

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

9

The author of the present thesis developed the experimental procedures together with B. Y., G. N., D. P.,
and L. C., performed the measurements together with B. Y., G. N., and D. P., analyzed the data with
B. Y. and contributed in writing the manuscript. The theoretical calculations of this publication have been
contributed by G. H. and M. L..

61

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

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

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

I. INTRODUCTION

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

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

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.
2643-1564/2021/3(3)/033256(12)033256-1

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

A. PATSCHEIDER et al.

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

4f 11

mJ’ = 75d
6s2

energy11[Xe]4f 5d 6s

401 nm2
mJ’ = -7

4f 12
6s2

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

6

7

8

J

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

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

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

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

II. EXPERIMENTAL SETUPν̄NIST = 7696.956 cm−1 .

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

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

III. COARSE SPECTROSCOPY

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OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

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

1
0.5
0

164

Er

166

Er

167

Er

168

Er

170

Er

normalized atom number in g

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

230.746230.748230.75230.752

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

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

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

(2)

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

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

033256-3

A. PATSCHEIDER et al.

norm. population in g(a)

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

1

0.5

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

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

170
168
166
164
1.258

1.26

1.262

1.264

1.266

gJ  = 1.2599(5).

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

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

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

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

(5)

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

(6)

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

033256-4

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

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

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

0.8
0.6
0.4
0.2
0
0

(b)

20

40

tpulse (µs)

60

80

(i)

blow

hold time

imaging

(ii)

blow

hold time

imaging

100

time

(c)

time
400

atom number

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

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

300
200
100
0
0

200

400

600

800

1000

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

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

A. PATSCHEIDER et al.

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

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

(7)

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

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

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

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

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

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

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

J (2J − 1)2

αt (ω).

(8)

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

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

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

033256-7

A. PATSCHEIDER et al.

x (µm)(a)

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

20

x (µm)

-2
4
0
-4
0

5

10

15

20

tCOM (ms)

(b)

0.8

uy

0.6

p

u

e

g

1

B

ux

0.4
-40

0

40

528

529

530

80

120

160

531

532

533

tot

(a.u.)

(c)
350
300
250

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

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

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

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

Constraint

Xfit
11

fX

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

033256-8

r1
r2
r3
r4
fix
fix

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

464120981776692640500680
200
–6500
20000

23898
7882
7417
157117105582
50541
17174
64003
16306
38097

2

0.761
0.856
0.861

0.984
0.831
0.775
1.149
0.580
0.928
0.732

Xfit
11

fX
2

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

0.761
0.856
0.861

0.663
0.831
0.984
0.831
0.775
1.149
0.580
0.928
0.732
0.844
0.609

OBSERVATION OF A NARROW INNER-SHELL ORBITAL …

PHYSICAL REVIEW RESEARCH 3, 033256 (2021)

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

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

ACKNOWLEDGMENTS

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

APPENDIX
Odd-parity energy parameters

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

Constraint

Xfit

fX
12

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

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

36278.6
92716.3
65123.3
46996.3
20.0–650.02000.02250.30.984

1513.0

1.461

100.0
3776.91405.1864.4
751.9
12209.7

0.761
0.856
0.861

1.156

0.843
1.155
1.156
0.522

Xfit

fX
12

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

0.761
0.856
0.861

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

Constraint

Xfit
11

fX
2

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

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

r21
r21
r17

–799.81128.5–13663.3

11

5d2 6s
0.852
1.465
0.621

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

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

r20
r20

656.3
568.2

1.155
1.157

7880.05052.0

0.600
0.600

0.461
0.461

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

r22
r22

2405.6
643.7

0.647
0.647

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

0.895
0.761

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

r22
r22

–2536.8–564.5

0.427
0.427

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

r23
r23
r22
r22

–5359.92758.3–6833.7–7463.2

A. PATSCHEIDER et al.

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033256-12

4.3. Further work

75

1
0.8
0.6
-160

-159

-158

-157

-156

-155

-154

-153

-152

-131

-130

-129

-128

-127

-126

-125

-124

-123

-170

-169.5-169-168.5-168-167.5-167-166.5

-166

1
0.8
0.6

1
0.8
0.6

Figure 4.4.: Spectroscopic measurements for three different hyperfine states. The normalized atom
number in the ground state is measured as a function of the relative frequency of the spectroscopy
light, given as the radio-frequency applied to the AOM relative to the locking point of the laser. The
solid lines are a triple-, double-, and single-Gaussian fit to the experimental data points, respectively.

4.3. Further work
4.3.1. Hyperfine constants of the fermionic isotope
The interaction between the total angular momentum J and the nuclear spin I of the atomic
core leads to a hyperfine splitting of the energy levels. While for the bosonic isotopes I = 0,
the fermionic 167 Er isotope carries a nuclear spin of I = 7/2. The basic concept of thehyperfine structure roots on the interaction between the magnetic moment of the nucleus
and the magnetic field generated by the negatively charged electrons surrounding the core
and thus creating an effective current. By considering the magnetic dipole and the electric
quadrupole interaction, the hyperfine energy shift is given by [Sch55, Ste20]
3
C(C + 1) − 2I(I + 1)J(J + 1)
1
∆Ehfs (F, J, I) = Ahfs C + Bhfs 2,
2
4I(2I − 1)J(2J − 1)

4.3

C = F (F + 1) − I(I + 1) − J(J + 1).

4.4

with
Here, Ahfs and Bhfs denote the magnetic dipole and the electric quadrupole hyperfine constants, respectively. For an accurate description of the energy levels for the fermionic isotopes, knowledge of the hyperfine constants is of course a prerequisite.

76

4. Narrow inner-shell orbital transition at 1299 nm

Figure 4.5.: Schematic illustration of the allowed transitions in fermionic erbium 167 Er. Startingfrom the ground state |F = 19/2, mF = −19/2⟩, a total of six excited states that fullfill the angularmomentum conservation ∆mF ′ = 0, ±1 with respect to the ground state can be reached using σ + (red dotted line), π-(blue dashed line), and σ − - (green solid line) transitions.

We can use spectroscopic measurements on 167 Er to extract the values for Ahfs and Bhfs withhigh accuracy. The measurement sequence follows the lines given in Sec. 4.2. After coolingand trapping in the magneto optical trap, the atoms are loaded into a crossed optical dipole
trap, formed by two orthogonal intersecting laser beams at 1064 nm and 1570 nm. We further
evaporatively cool the atomic sample until the atoms reach a temperature of about 1 µK.
Subsequently, we irradiate the atoms with light at 1299 nm and record the remaining atom
number via absorption imaging.
Figure 4.4 shows the normalized atom number measured as a function of the laser frequency
for three different frequency ranges. In total, we observe six transitions, which we identify
by a dip in the atom number. Starting from the ground state |F = 19/2, mF = −19/2⟩, theobserved resonances can be grouped as transitions
 |F = 19/2, mF = −19/2⟩ → |F ′ = 21/2, mF ′ ⟩,
 |F = 19/2, mF = −19/2⟩ → |F ′ = 19/2, mF ′ ⟩, and |F = 19/2, mF = −19/2⟩ → |F ′ = 17/2, mF ′ ⟩.

To understand the exact number of transitions observed, we must consider that starting fromthe ground state, only σ + -, π-, and σ − -transitions are allowed, leading to ∆mF = 0, ±1
with respect to mF = −19/2. While for |F = 19/2, mF = −19/2⟩ → |F ′ = 21/2, mF ′ ⟩ excitations to three magnetic sublevels (mF ′ = −21/2, −19/2, and −17/2) are possible, for|F = 19/2, mF = −19/2⟩ → |F ′ = 19/2, mF ′ ⟩ and |F = 19/2, mF = −19/2⟩ → |F ′ = 17/2, mF ′ ⟩only two (mF ′ = −19/2, −17/2) and one (mF ′ = −17/2) magnetic sublevel can be reached, respectively. Figure 4.5 shows a schematic illustration of the involved energy levels and theallowed transitions.
Importantly, the σ + -transition (mF ′ = mF + 1) is the only excitation that can be observed

4.3. Further work77

in all three spectra. We can therefore use this transition to derive the values for Ahfs andBhfs from the spectroscopic measurements. Using Eq. 4.3, the frequency of the transition|F, mF ⟩ → |F ′ , mF ′ ⟩ can be written asν σ+ (F ′ , mF ′ ) = ℏω − (mF gF + (mF ′ + 1)gF ′ )µB B − ∆Ehfs (F, J, I) + ∆Ehfs (F ′ , J ′ , I), 4.5

where ω is the bare transition frequency. By taking the difference between ν σ+ (F ′ , mF ′ ) for
two different hyperfine states F ′ , all terms beside the hyperfine energy shift of the excitedstate cancel each other. We combine the three spectroscopy measurements and determine
the hyperfine constants to
Ahfs = −125.62(1) MHz
4.6

and

Bhfs = −3050.7(5) MHz.

4.7

4.3.2. Dependence of the excited state lifetime on the atomic density
Collisions between atoms are an important constituent in ultracold quantum gas systems.
For experiments involving narrow and ultranarrow transitions, both elastic and inelastic
collisions must be considered. For example, while elastic collisions result in a shift of theresonance frequency [Cam09, Lem11], inelastic collisions between atoms in the excited statelead to two-body losses and a suppression of excitation [Bis11]. Furthermore, the latter cause
automatically a reduced lifetime for the excited state.
To estimate the magnitude of the two-body loss in our experiment, we measure the excitedstate lifetime for a dense atomic cloud. In brief, after evaporative cooling, we load theatoms into a crossed optical dipole trap created by to intersecting laser beams at 532 nm and1570 nm, respectively. Note that we use light at 532 nm because of a negative polarizabilityof the excited state for the usual 1064-nm light. At this stage, we obtain about N = 3 × 104
atoms at a density of n ≈ 1 × 1014 cm−3 and temperature of T ≈ 1 µK. We efficiently transfer
the ground-state atoms to the excited state by applying a π-laser pulse at 1299 nm. Aftera certain hold time, we apply a second π-pulse to re-transfer the atoms to the ground stateand quantify the atom number by absorption imaging.
Figure 4.6(a) shows the measured atom number in the excited state as a function of hold time.
Although the natural lifetime of atoms in the excited state is about 180 ms (see Sec. 4.2), we
observe a much faster decay that occurs on the timescale of several ms. Importantly, we do
not observe atoms decaying to the ground state, suggesting that the atoms are lost from thetrapping potential.
To describe the behavior of the atom number, we follow the approach of Ref. [Bis11] andadopt a model that accounts for two-body loss occurring for atoms in the excited state. The
atomic density in the excited state ne can be described by [Tra09, Lis09]dne= −Γob ne − Kee n2e ,4.8dt
with Γob being the one-body loss rate and Kee being the coefficient describing the two-bodyloss rate arising from collision between atoms in the excited state. Note that this approach

78

4. Narrow inner-shell orbital transition at 1299 nm
104
3
10 4

2.5

3

2

2

102

1.5
1

1
0

0.5

1

1.5

0.5
0
0

2

4

6

8

10

1011

1012

1013

hold time (ms)Figure 4.6.: (a) Atom number as a function of hold time for the excited state. The red solid line
represents a two-body loss fit (Eq. 4.9) to the experimental data. The green sold line in the inset
indicates an exponential fit to the data. (b) Lifetime of the excited state in dependence of the peak
number density of the atomic cloud. The red solid line is an empirical exponential fit. Error bars
denote one standard deviation.

does not take into account collisions between atoms in the ground and excited state, since
their contribution is expected to be negligible. For our experimental parameter regime, Γob
is dominated by spontaneous emission and therefore depending on the natural lifetime of theexcited state. An expression for the time dependent atom number N (T ) can be found byintegration of Eq. 4.8 [Bis11], which leads to
N (t) =

1+

N0 exp(−Γob t)
.
N0 Kee/(π 3/2 Γbg wx wy wz ) [1 − exp(−Γob t)]

4.9

Here, N0 is the atom number at t = 0 and wx , wy , and wz describe the 1/e2 -radius of theatomic cloud along the corresponding direction.
We fit Eq. 4.9 to the atom number shown in Fig. 4.6(a). For the fitting procedure we fix
Γob to 1/180 ms. The decay of the atom number is well described by Eq. 4.9 and we extract
a two-body loss rate coefficient of Kee ≈ 5 × 10−21 cm3 /s for our experimental condition.
In contrast, a pure exponential decay of the form N (t) = A + Be−t/τ fails to reproduce theatom decay, in particular at early times; see inset Fig. 4.6(a).
Since knowledge of the atomic lifetime at different densities is important for various experimental realizations, we repeat the above measurement for different atomic numbers, keeping
the trap and temperature the same. We control the atom number by changing the loading
time of the magnet-optical trap. Although an exponential fit does not describe the experimental data perfectly in all density regimes, we can use it to obtain a quantitative estimation
for the excited-state lifetime as a function of density.
Figure 4.6(b) summarizes the excited-state lifetime as a function of the atomic density.
For low densities, where collisions are strongly reduced, the lifetime approaches the natural
lifetime of about 180 ms; see Sec. 4.2. In contrast, an increasing density goes hand in handwith strong reduction of the lifetime. Finally, we find empirically that the dependence of the4.3. Further work

79

lifetime on the density follows an exponential behavior.

4.3.3. Preparation of bosonic erbium atoms in higher Zeeman states
Finally, we now discuss the application of our transition at 1299 nm for spin manipulation.
The preparation of atoms in different magnetic sublevels that can be utilized as spin states isvery attractive, as discussed in Chapter 3. In contrast to fermions, bosonic isotopes lack thequadratic Zeeman shift, making deterministic preparation impossible with this technique.
Here, the inter-orbital atomic transition opens up new opportunities for transferring atoms
to energetically higher states.
The narrow transition is a flexible tool and allows for the implementation of different preparation protocols. Due to the large energy difference of about 1.62 MHz/G10 between subsequent
magnetic sublevels, a straightforward preparation follows the stimulated Raman adiabatic
passage [He90]. Starting from the mJ = −6 ground state, the subsequent application of
σ + and π transitions allows to move within the spin-manifold to any desired spin state.
For example, we were able to transfer atoms from mJ = −6 to mJ = 0 with an efficiency> 80 % using two consecutive laser pulses, even without careful optimization of the laserpulse times.
A second, alternative method relies on the ac-Stark shift induced by the light at 1299 nm.
For an atom with total angular momentum J (as it is true for the bosonic erbium isotopes
for which I = 0), following the derivation of Ref. [Ros09, LK13], the energy shift induced bya light field oscillating at a certain frequency ω is given by
1α(ω)I(r, ω)cϵ0"[u∗ × u] · J
1αs (ω) + i
αv (ω)+=−cϵ02J#
J(J + 1) − 3m2j 1 − 3 cos2 θp+αt (ω) I(r, ω).
J(2J − 1)
2

U (r, ω) = −4.10

Here, u and I(r, ω) are the polarization vector and the intensity11 of the laser field, respectively. The angle θp defines the orientation of u with respect to B. Further, αs (ω), αv (ω),
and αt (ω) are the scalar, the vectorial, and the tensorial polarizabilities.
The contributions of αs (ω), αv (ω), and αt (ω) are on the same order of magnitude, which inconsequence allows for a large tunability of α(ω). Looking at Eq. 4.10 and on the dependenceof α(ω) on u, B, and mJ , various possibilities to control the light shift for the atomic state
become apparent. Particularly appealing for the preparation of atoms in spin states is thedependence of the contribution of αt (ω) on m2J . This results in a light shift that dependents
quadratically on the magnetic sublevel and allows to control the energy difference between
two subsequent mJ levels.
1011

About 1 MHz/G for fermionic Er.
I(r, ω) = cϵ0 |E(r, ω)|2

80

4. Narrow inner-shell orbital transition at 1299 nm

1
0.8
0.6
0.4
0.2
0
0

50

100

150

t (µs)

200

250

300

Figure 4.7.: Influence of light at 1299 nm for preparation of atoms in different magnetic Zeeman
levels for the bosonic 168 Er isotope. (a) Absorption image after Stern-Gerlach separation and timeof flight expansion of 30 ms. Starting with an atomic sample polarized in the lowest Zeeman level,a resonant radio-frequency pulse without light at 1299 nm causes a spreading of the population overmany spin states. (b) Rabi oscillations between the two lowest magnetic sublevels mJ = −6 (blue)
and mJ = −5 (red). Almost no population can be detected in the mJ = −4 (green) state. To lift thedegeneracy of the energy difference between subsequent mJ states, 0.5 mW of light at 1299 nm, on
resonance with the mJ = −4 → mJ ′ = −4 transition, is shined onto the atoms. The upper images
show the absorption image for the two spin states again after Stern-Gerlach separation and time offlight expansion of 30 ms.

This quadratic dependence on the laser intensity corresponds to the same effect as thequadratic Zeeman shift for fermionic isotopes and enables the coherent preparation of atoms
in different mJ states by driving a radio-frequency transition. To test this technique in theexperiment, we shine light at 1299 nm, resonant with the transition mJ = −4 → mJ ′ = −4,on the atoms and apply subsequently a radio-frequency pulse at the transition frequency
of mJ = −6 → mJ = −5 for variable hold time. While without light at 1299 nm a radiofrequency pulse causes a board distribution of population among the Zeeman levels (seeFig. 4.7(a)), the light at 1299 nm enables the isolation of only two levels. Figure 4.7(b)shows the observed Rabi oscillations between the two lowest magnetic sublevels mJ = −6
and mJ = −5 for a Bose-Einstein condensate of the 168 Er isotope. Importantly, during
the Rabi oscillations, no higher Zeeman sublevels are populated enabling deterministic spin
preparation for future experimental realizations.

Chapter5

Determination of the contact
scattering length via
cross-dimensional thermalization

Atomic interactions constitute a key ingredient in ultracold quantum gases as they havean important impact on the physical system. In dipolar quantum gases, a very interesting
aspect is the interplay between the dipole-dipole interaction (DDI) and the so-called contact
interaction, whose strength is given by s-wave scattering length as . In the case of magnetic
atoms, the maximum strength of the DDI cannot be tuned in the experiment because its is
intrinsically given by atomic magnetic moment. On the other hand, as can be controlled, forexample with an external magnetic field by means of a Feshbach resonance [Chi10]. Therefore, to carefully control the competition between the interactions an accurate knowledge
on as and its dependence on the magnetic field is required. A fascinating example is therecently observed one-dimensional supersolid state, which exists only in a small parameter
range of about 3 a0 [Böt19b, Tan19, Cho19].
Away from any Feshbach resonance [Chi10], for alkali atoms the scattering length, also referred to as background scattering length, is typically constant (an exception is cesium [Kra06]).
Due to the dense Feshbach spectra in lanthanide atoms, which for example in the case ofbosonic (fermionic) erbium contains about three (20) resonances per gauss [Fri14b], thedefinition of a unique background scattering length is a general problem. Moreover, thetheoretical prediction of as for lanthanides is very challenging, although knowledge on theinteractions potentials has been obtained [Kot14]. A special theoretical challenge arises fromthe coupling of different angular momentum states making the allocation of resonances difficult [Pet12]. Altogether, experimental techniques are required to contribute to the accurate
determination of as .
Over the last years, a number of different approaches have been realized to experimentally
determine the scattering length of erbium and dysprosium. For the latter, examples are
based on the anisotropic expansion of a thermal cloud [Tan16, Luc18], the measurement of
the molecular binding energy [Mai15, Luc18], cross-dimensional thermalization [Tan15], or bydirect comparison between experimental observables and theory in the droplet regime [FB18,
Böt19c]. Table 5.1 summarizes the various results for the two isotopes 162 Dy and 164 Dy
that have been obtained via the different approaches. Interestingly, these results do not81

82

5. Determination of the contact scattering length via cross-dimensional thermalization

Table 5.1.: Summary of experimentally determined as for the dysprosium isotopes 162 Dy and 164 Dy.
Additionally, the respective magnetic field at which the measurement was performed and the corresponding reference are given.

Isotopeas (a0 )B (G)Reference

162 Dy180(50)220(50)122(10)154(22)157(4)

22, 2722, 27
1.6
1.6
5.2[Luc18][Luc18][Tan15][Tan16][Tan16]96(22)
92(8)
91(16)
62.5
69(4)

1.6
1.6
80, 180
6.6
6.6[Tan16][Tan15][Mai15][Sch16][FB18, Böt19c]

162 Dy162 Dy162 Dy162 Dy164 Dy164 Dy164 Dy164 Dy164 Dyappear to be entirely consistent at first glance. Although a potential explanation for thisinconsistency could ly in the different magnetic field values at which the measurements havebeen performed and nearby Feshbach resonances could affect the results, this opens thechallenge for an accurate determination of as over a broader magnetic field range.
For erbium on the other hand, a different experimental approach has been applied, which isbased on high-precision spectroscopy of the on-site interaction energy in a three-dimensional
optical lattice. This technique, often referred to as lattice modulation spectroscopy, is known
for its accuracy, however, it also is technically more demanding because it requires theimplementation of an optical lattice. Thanks to its exceptional accuracy, lattice modulation
spectroscopy can be used as a benchmark for other methods that are experimentally easierto realize; see Sec. 5.3.
This Chapter covers the work on determination of the scattering length over a magnetic
field range from 0 G to 5 G for four bosonic erbium isotopes, using the cross-dimensional
thermalization technique. In Section 5.1 a brief introduction into ultracold scattering is given,
with the aim, to show the influence of the DDI on the scattering cross section and to highlightthe competition between the two types of interactions. The following Sec. 5.2 provides an
overview over the cross-dimensional thermalization technique and gives in particular an
explanation for the geometrical dependence of the thermalization process.

5.1. Dipolar scattering in the ultracold regime
To derive a description for the scattering of two particles 1 and 2 with reduced mass mr =m1 m2
m1 +m2 , one needs to solve the Hamiltonian describing the relative motion of the scattering
constituents [Dal98]
Ĥ =

p2
p2
+ V (r) =+ Vc (r) + VDDI (r),
2mr
2mr

5.1

5.1. Dipolar scattering in the ultracold regime

83

where p = (p1 −p2 )/2 and r = r1 − r2 are the relative momentum and position coordinates,
Vc (r) is the short range potential accounting for the contact interaction, and VDDI (r) is the
dipolar interaction potential. This Hamiltonian can be interpreted as the scattering eventof a single particle with reduced mass mr from the potential V (r).
At large distances from the scattering potential, the general solution for the wave function
describing the scattering process requires the following asymptotic form [Lan81]
ψk (r) ∼ eikr + f (k, k′ )

eikr
.
r

5.2

Physically, this solution corresponds to a superposition of an incoming plane wave with
momentum k and a scattered spherical wave function. The amplitude of the latter, f (k, k′ ),is called the scattering amplitude and is generally unknown as it depends on the form of theunderlying potential. Moreover, it depends on the energy of the particle given though |k| andthe relative angle between incoming momentum k and outgoing momentum k′ [Boh14].
The scattering amplitude can be used to determine the differential scattering cross section [Lan81]
dσ(k, k′ ) = |f (k, k′ )|2 ,5.3
dΩk′which can be interpreted as a probability measure that an incident particle at momentum k
is scattered into k′ . Finally, integration of the differential scattering cross section over thefull three-dimensional solid angle leads to the total scattering cross sectionZdσ
σ(k) = dΩk′(k, k′ ).
5.4dΩk′
For simplicity, we first consider the particular case of colliding particles with negligible DDI,e. g. as it is the case for alkali atoms. Here, the only interaction potential that needs to beconsidered is the short range van der Waals potential, scaling as Vc (r) = −C6/r−6 , with C6
being the van der Waals C6 coefficient. For low enough energies1 , f (k, k′ ) is isotropic andtherefore independent on the specific scattering direction [Dal98]. The scattering amplitude
can be calculated by making use of the partial wave expansion, leading to the total scatteringcross section of
8π X
σB (k) = 2(2l + 1) sin2 δl (k)(bosons)5.5kl evenandσF (k) =8π X(2l + 1) sin2 δl (k)k2(fermions)

5.6

l odd

for bosons and fermions, respectively. Furthermore, for low scattering energies, the contribution of all partial waves with l ̸= 0 vanishes and Eq. 5.5 and 5.6 become [Lan81]
l=0
lim σB(k) = 8πa2sk→0(bosons)

5.7(fermions).

5.8

and
lim σFl=0 (k) = 0

k→0
1

2

This comprises the regime, where Ecoll ≪ ℏ /2mr b2 , with b describing the range of the potential.

84

5. Determination of the contact scattering length via cross-dimensional thermalization

Equation 5.8 has strong consequences for fermionic atoms, as this implies that no scattering occurs between individual particles at ultracold temperatures. Therefore, for example, alternative approaches need to be applied to realize evaporative cooling of fermionic
atoms [DeM99b].
We now turn our attention to the DDI (see also Sec. 3.1), which strongly modifies the atomic
scattering behavior [Lan81, Dal98, Cho22]. Note that the following discussion will focus onmagnetic dipolar interaction, although the same description holds for electric dipoles. An
important consequence of the long-range DDI2 , scaling as VDDI (r) ∝ r−3 , is that even in theregime of low collision energy all partial waves contribute to the scattering process, leading,as we will see below, to an anisotropic scattering cross section.
For purely dipolar interacting systems at low collision energies, theoretical works suggested
the existence of a universal scattering behavior, i. e. different dipolar systems show the same
scattering behavior and the dynamics are independent on the collision energy [Tic08, Boh09].
Moreover, the scattering cross section takes on a constant value in this low energy (ultracold)
regime. The threshold energy below which the ultracold regime is reached is defined via theratio of the collision energy Ecoll = ℏ2 k2/2mr , where k = |k|, and the natural energy scale ofthe dipolar interaction [Boh09] asED =mr µ0 µ2
16π 2 ℏ6.
=
4πa3D
m3r µ20 µ4

5.9

with

m r µ 0 µ2
,
5.10
4πℏ2
being the typical length scale characterizing the DDI and referred to as dipolar length. For
two colliding erbium atoms of the 166 Er isotope, aD = 98.2 a0 and ED/kB ≈ 200 µK. As wewill see later in Sec. 5.3, the measurements in this Chapter are performed at temperatures< 1 µK and thus in the ultracold regime.
aD =In the case of a weak scattering potential, i. e. when the incident plane wave is only weakly
modified compared to the asymptotically outgoing wave, the potential can be assumed as aperturbation, which allows for the application of the Born-approximation [Yi01]. Within thefirst order Born-approximation, the scattering amplitude for dipolar particles can be written
as [Boh14]
f (k, k′ ) = −as + f (1) (k, k′ )aD ,
5.11
with
f (1) (k, k′ ) = −

1
2π

Z

′

d3 re−ik r VDDI (r)eikr .

5.12Here, as account for the s-wave scattering length and f (1) (k, k′ ) denotes the first order Bornapproximation for the dipolar scattering contribution.
The integral in Eq. 5.12 can be solved by applying the spherical wave expansion to the plane′wave ei(k−k )r . Details on the calculation can be found for example in Ref. [Boh14]. Finally,
2

The DDI is long-range in three dimensions; see 3.1.

5.1. Dipolar scattering in the ultracold regime

85

25
20
15
10
5
0

0

30

60

90

120 150 180

Figure 5.1.: (a) Scattering cross sections σB (aD = 98.2 a0 , solid lines) and σF (aD = 98.8 a0 , reddashed line) as a function of the angle η between B and k. For the bosonic case σB (η) is shown for
as = 0 a0 , 15 a0 , 30 a0 , 45 a0 , 60 a0 , and 500 a0 . For better visibility σB (η) for as = 500 a0 is rescaled
by a factor 30. (b) Three-dimensional illustration of σB (η) for as = 0 a0 . The color code indicates theabsolute value of σB (η). (c) Same as (b) for σF (η). Note that the size of (b) and (c) is not to scale.

f (k, k′ ) can be written in terms of the incoming and outgoing momenta as well as an external
polarizing magnetic field as [Boh14](k · B − k′ · B)2 3
′
+
aD .
5.13
f (k, k ) = −as −1 − kk′2
Note that, this expression assumes elastic scattering with |k| = |k′ |.
Depending on the considered quantum statistics, i. e. fermionic or bosonic, the scattering
amplitude needs to be symmetrized accordingly, as1fB,F (k, k′ ) = √ f (k, k′ ) ± f (k, −k′ ) .
2

5.14

Inserting the symmetrized scattering amplitudes into Eqs. 5.3 and 5.4 results in the total
scattering cross section ofσB (η) =π
72a2s − 24as 1 − 3 cos2 (η) + 11 − 30 cos2 (η) + 27 cos4 (η) a2D
95.15

for the case of indistinguishable bosons, andσF (η) =π
3 + 18 cos2 (η) − 13 cos4 (η) a2D3

5.16

for the case of indistinguishable fermions. Here, η denotes the angle enclosed by the incoming
relative momentum k and the orientation of the dipoles polarized by an external field B.
Figure 5.1(a) shows σB (η) and σF (η) as a function of η as well as σB (η) for different as .
Let us first consider the case of pure dipolar interaction, i. e. as = 0 a0 for σB (η). In
this case, a remarkable difference between σB (η) and σF (η) can be observed, which roots inthe symmetrization of the scattering amplitudes in Eq. 5.14. For purely dipolar interacting

86

5. Determination of the contact scattering length via cross-dimensional thermalization

Figure 5.2.: Scattering cross section for dipolar bosons σB with aD = 98.2 a0 in dependence of therelative momentum k and the orientation of an external, polarizing magnetic field B at 0 a0 , 15 a0 ,30 a0 , and 500 a0 . Note that the size of the individual geometric shapes is not to scale.

bosons (as = 0) the scattering cross section is the largest if the atoms collide in a sideby-side configuration (η = 90°) and minimal if they collide at η ≈ 45°. This is in strong
contrast to the fermionic case, where σF (η) is the largest for atoms approaching under anangle of about η ≈ 45° and is reduced for the side-by-side configuration (η = 90°). For a
more intuitive picture, Fig. 5.1(b)-(c) show three-dimensional surface plots that represent (b)σB (η) (as = 0 a0 ) and (c) σF (η), once again indicating the distinct difference in geometricshape. Note that σB,F (η) is cylindrically symmetric about the magnetic field axis.
While, obviously, the total scattering cross section has a purely dipolar nature for fermions,σB (η) changes significantly for as > 0; see Fig. 5.1(a). For larger as , the probability to scatterin head-to-tail configuration (η = 0°) increases with respect to the side-by-side configuration(η = 90°). Finally, for large as , the geometrical shape of σB (η) approaches spherical symmetry, reflecting the dominant contact interaction. Similar to Fig. 5.1(b)-(c), Fig. 5.2 illustrates
the geometrical shape of σB (η) for different as , reflecting the transition from the spinning
top like (as = 0 a0 ) to almost spherical geometry (as = 500 a0 ).
Finally, for experimental purposes it is convenient to define an angular averaged scatteringcross section. Calculating the angular average via [Boh14]
σ̄B,F =

1
2

Z 1

d cos(η) σB,F (η)

5.17

−1

leads toσ̄B = 8πa2s +

32π 2
a
45 D

(bosons)

5.18

and

32π 2
a
(fermions).
5.19
15 D
The averaged cross sections depend only on the dipolar length aD (and as for bosons), which
reveals once more the universal scattering behavior in the ultracold regime.
σ̄F =The scattering anisotropy of dipolar atoms has important implications on the thermalization behavior of an atomic cloud out of thermal equilibrium, as it has been beautifully5.2. Determining the scattering length via coss-dimensional thermalization

87

demonstrated in experiments using fermionic erbium [Aik14b] and KRb molecules [Li21].
In the following, we will focus on cross-dimensional thermalization measurements to extract
knowledge on the contact scattering length of bosonic erbium atoms.

5.2. Determining the scattering length via coss-dimensional
thermalization
A cold atomic cloud brought out of thermal equilibrium re-thermalizes by elastic collisions
and the rate at which thermalization occurs depends on the elastic scattering cross section.
For atomic systems where the scattering length and thus the scattering cross section is
unknown a priori, measurements of the thermalization rate offer the possibility to gain insight
into the scattering properties. The cross-dimensional thermalization method represents anefficient technique to determine the thermalization rate for an atomic cloud confined in an
harmonic trapping potential [Mon93].
Starting from an atomic cloud that is in thermal equilibrium, the concept of the crossdimensional thermalization technique is based on a change of the systems energy along one
spatial dimension and to record the quasi-temperature along the orthogonal dimension. Notethat in general, excitation along two spatial dimensions is also possible, but for simplicity
in the following we will focus on the excitation along a single direction. The thermalization
rate determines the time the atomic cloud need to reach thermal equilibrium again after
excitation.
One approach to increase the energy along one spatial direction is based on a compressionof the trapping potential, which leads to an increase of the potential energy. The mean totalenergy per particle in a harmonic trap can be written as [Pit16]
E0 = Ekin + Epot
X 1
1=
mvq2 + mωq2 q 2 = 3kB T0 ,22
q=x,y,z

5.20
5.21

with ωq being the harmonic trapping frequency along the spatial dimension q. After anincrease of the trapping frequency along one dimension, here exemplary along y, the energy
increases to
5
1
E∞ = kB T0 + m(ωy + ∆ωy )2 y 2 = 3kB T∞ .
2
2

5.22

Here, T∞ denotes the temperature of the atoms after thermalization and can be calculated
by inserting kB T0 = mωy2 y 2 into Eq. 5.22 as
T∞ = T0
5 1 (ωy + ∆ωy )2
+
.
6 6
ωy2

5.23

This means, an increase of the harmonic trapping frequency by 50 % results in a total temperature increase of about 20 %.

88

5. Determination of the contact scattering length via cross-dimensional thermalization

After being brought out of thermal equilibrium, the atoms will thermalize by redistributing
the gained energy along the orthogonal spatial dimensions through elastic collisions. The
characteristic rate γth at which an atomic cloud re-thermalizes is given by [Rei65]
γth = nvr σrth ,5.24

with n being the average number density3 , vr being the relative velocity4 , and σrth being thescattering cross section relevant for re-thermalization. Equation 5.24 can be interpreted as
the combination of the probability of two atoms to meet, given by 1/nvr , and the probabilityof two atoms to scatter, represented by σrth .
Due to the anisotropy of the scattering cross section and the related dependence on thetrapping geometry and the magnetic field orientation, σrth cannot simply be replaced by theaveraged total cross section of Eq. 5.18. However, they can be related via a proportionality
constant leading to [Boh14, Syk15]σ̄B
σrth =.
5.25α(η)
For an intuitive interpretation, α(η) is often referred to as the number of collisions per
re-thermalization. This describes the fact that not all collision contribute equally to thethermalization process, as for example forward and backward scattering does not contributeto the redistribution of energy. The dependence of α(η) on η will be discussed below. Note
that, although this specific designation, α(η) does not take on only integer values.
For the specific case of alkali atoms, which have an isotopic scattering behavior, α(η) ≡ α
is independent on external parameters such as the trap geometry or the magnetic fieldorientation. In Ref. [DeM99a], the value of α was determined to be α = 2.5 for s-wave
collisions and α = 4.17 for p-wave collisions. The higher value for p-wave collisions originatesin the fact that here forward and backward scattering is favored compared to sideward
scattering, making thermalization less efficient.
The description of α(η) is becoming more complicated in the case of strongly dipolar atoms.
The reason for this is two fold: On the one hand the DDI leads to an anisotropic scatteringcross section that needs to be included in α(η). As a consequence, α(η) depends on both,the orientation of the magnetic dipole and the geometry of the trapping potential. On theother hand, due to the competition between contact and dipolar interactions α(η) depends
on the scattering length as .
The dependence of α(η) on the underlying geometry and dipole orientation can be understoodby considering the total and the differential scattering cross section. The efficiency of thethermalization process is strongly based on two arguments:
 a large scattering cross section σB (η)
 an efficient transfer of momentum into the dimensions orthogonal to the excitation,dσ
described by dΩ(k, k′ ).
′k34

n = 1/q
N
vr =

R

3

2

d r n(⃗r) .

8kB T, with µ being the reduced mass.
πµ5.2. Determining the scattering length via coss-dimensional thermalization

89

dσ
Figure 5.3.: Geometric representation of dΩ(k, k′ ) = |f (k, k′ )|2 giving the probability for a momen′kdσtum transfer from k to k′ for different as and η. The color indicates the absolute vale of dΩ(k, k′ ).
k′Note that the size of the individual geometric shapes is not to scale.

While σB (η) is shown in Fig. 5.2 and has been discussed earlier, various differential scatteringcross sections describing the probability of momentum transfer into k′ are shown in Fig. 5.3
for different η and as . For two atoms scattering in a head-to-tail configuration (η = 0°) thedσgeometric shape of dΩ(k, k′ ) shows spherical symmetry, independent on as . In contrast,k′for η > 0° both, as and the orientation of B have an influence on the probability to scatter
into different dimensions. A significant influence of as can be observed for η = 90° (atoms inside-by-side configuration), where the dimension for the most efficient momentum transfer
becomes inverted from as = 0 a0 to as = 60 a0 .
dσThe exact form of α(η) is a complex interplay between σB (η) and dΩ(k, k′ ), and thereforek′strongly depending on the underlying system parameters such as trap geometry, direction of
excitation, and orientation of B. In Sec. 5.3 α(η) was calculated by our theory collaborators
R. R. W. Wang and J. L. Bohn (JILA, Boulder Co) for our specific experimental configuration. Moreover, R. R. W. Wang and J. L. Bohn applied a theoretical framework based on the
Enskog’s equations of change to extract as from cross-dimensional thermalization measurements.

5.3. Publication: Determination of the scattering length ofmagnetic erbium via cross-thermalization measurements.
Submitted to Physical Review A (2021)5
submitted 22 Dec 2021;
DOI: arXiv:2112.11883
A. Patscheider,1 L. Chomaz,1 G. Natale,1 D. Petter,1 M. J. Mark,1,2 S. Baier,1 B. Yang,1 R. R. W. Wang,3
J. L. Bohn,3 and F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3
JILA and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
2

5

The author of the present thesis developed the experimental procedures together with L. C., D. P., and
G. N., performed the measurements together with D. P., G. N., and B. Y., analyzed the data with L. C. andcontributed in writing the manuscript. The theoretical calculations of this publication have been contributed by R. R. W. W. and J. L. B..

91

Accurate determination of the scattering length of erbium atoms
A. Patscheider,1 L. Chomaz,1, ∗ G. Natale,1 D. Petter,1, † M. J. Mark,1, 2
S. Baier,1 B. Yang,1, ‡ R. R. W. Wang,3 J. L. Bohn,3 and F. Ferlaino1, 2

arXiv:2112.11883v1 [cond-mat.quant-gas] 22 Dec 2021

1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
3
JILA and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA(Dated: December 23, 2021)
An accurate knowledge of the scattering length is fundamental in ultracold quantum gas experiments and essential for the characterisation of the system as well as for a meaningful comparison totheoretical models. Here, we perform a careful characterisation of the s-wave scattering length asfor the four highest-abundance isotopes of erbium, in the magnetic field range from 0 G to 5 G. We
report on cross-dimensional thermalization measurements and apply the Enskog equations of change
to numerically simulate the thermalization process and to analytically extract an expression for theso-called number of collisions per re-thermalization (NCPR) to obtain as from our experimental
data. We benchmark the applied cross-dimensional thermalization technique with the experimentally more demanding lattice modulation spectroscopy and find good agreement for our parameter
regime. Our experiments are compatible with a dependence of the NCPR with as , as theoretically
expected in the case of strongly dipolar gases. Surprisingly, we experimentally observe a dependencyof the NCPR on the density, which might arise due to deviations from an ideal harmonic trapping
configuration. Finally, we apply a model for the dependency of the background scattering lengthwith the isotope mass, allowing to estimate the number of bound states of erbium.
I.

INTRODUCTION

The high degree of environmental isolation and thehigh control over the large parameter-space of ultracold
quantum gases are key for their success [1]. One of themost decisive properties in determining the many-bodyphases of a quantum gas is the interaction force between
atoms. Among neutral particles, it can be isotropic andshort-range, as in alkali atoms, and/or anisotropic andlong-range. Open-shell lanthanides, such as erbium (Er)and dysprosium (Dy), have both interactions in place [2].
Their strong magnetic character is reflected in a large
dipole-dipole interaction (DDI), while the contact potential is governed by the well-known scattering length,
whose value as , as in alkali atoms, can be largely controlled by so-called Fano-Feshbach resonances [3–5].
Although the concept of the scattering length itself iswell known by now, theoretical challenges to calculate as
depend on the atomic species of interest. For lanthanides,
predicting as remains a major challenge of quantum
chemistry and microscopic scattering theories [6]. The
complexity of describing such atoms has several reasons:the multiple valence electrons, the strongly anisotropic
orbital shells, the strong coupling between core and valence electrons, and the relativistic contributions, also

∗ Present address:

Physikalisches Institut, University of Heidelberg, 69120 Heidelberg, Germany.
† Present address: Optical Materials Engineering Laboratory, Department of Mechanical and Process Engineering, ETH Zurich,
8092 Zurich, Switzerland.
‡ Present address: Southern University of Science and Technology,
Shenzhen 518055, China

made important by the large atomic mass. To date,there are still no ab-initio models with the capacity forquantitative predictions, although many general properties of the interaction potentials (e. g. Born-Oppenheimer
potentials) have been studied and understood [7].
Yet, knowledge of the scattering length remains ofprime importance since it is an essential regulator of fewand many-body quantum phenomena. For instance, thefascinating supersolid state, recently discovered in Dy [8–10] and Er [9], lives in a narrow range of only a few
a0 (a0 is the Bohr radius), or the functional forms of
beyond-mean-field corrections, which are still under discussion [11–14], depend on as in a subtle way. In theabsence of complete microscopic and ab-initio potential
models, the study of as in lanthanides therefore relies on
experimental investigations and empirical models.
Several different experimental methods have been applied in previous works to extract as for Er and Dy. These
include spectroscopy of the molecular binding energy
close to a broad Fano-Feshbach resonance [15, 16], theanisotropic expansion of a thermal gas [17], and the crossdimensional thermalization technique [18–21]. Furthermore, for the 166 Er isotope, as has been determined withhigh accuracy based on a measurement of the particlehole excitation gap in the Mott insulator regime via lattice modulation spectroscopy [22, 23]. These techniques
did not always provide consistent values, opening up a
number of fundamental questions, e. g. from the validityof the additivity of the interaction pseudo-potentials [24–27] to the appropriateness of the Lee-Huang-Yang form
for beyond-mean field effects [12–14, 28].
In this work, we extensively study the scattering lengthof the four most abundant bosonic isotopes of erbium(164 Er, 166 Er, 168 Er, and 170 Er) and its magnetic-field de-2pendence. For each isotope, we perform high-resolution
Fano-Feshbach spectroscopy in the low magnetic-field region (0 to 5 G) and identify previously unreported scattering resonances. In this range, we then accurately determine the erbium scattering length, as , by developing a model based on the Enskog equations to extract
as from cross-dimensional-thermalization experiments.
We benchmark our results with the ones obtained fromhigh-precision lattice-modulation spectroscopy, which
has been previously developed for 166 Er [23, 29] and here
expanded to 168 Er. Finally, from the magnetic-field mapping of as , we extract for each isotope an effective background scattering length abg
s at zero B field and we discuss the results in the context of the isotope-mass scaling.
II.

CROSS-DIMENSIONAL THERMALIZATIONThe cross-dimensional-thermalization technique is avery powerful method to experimentally determine thescattering length. First successfully applied to alkali
atoms [30–33], this technique has proved to be very general and, more recently, has been used for more complex
atomic species, such as chromium [34], specific isotopesof erbium [19] and dysprosium [21], and molecular systems [35, 36].
Starting from a cold thermal cloud, the basic idea ofthe cross-dimensional thermalization method is to excite
the system by increasing the potential energy along one
spatial dimension of the atomic cloud and to measure
the characteristic time τ that the system needs to rethermalize in an orthogonal directions [30]. In the regimeof small excitations, for an atomic cloud at a temperature
T and a total atom number N , the characteristic time isrelated to the total scattering cross section σ̄ byτ=α,n̄σ̄vrwhere n̄ is the mean number density3/2
N ω̄ 3mn̄ = √
8 2πkB T(1)(2)

and vr the mean relative velocity for two colliding atomsr
16kB Tvr =(3)πmHere, ω̄ is the geometric mean of the harmonic trapping frequencies, m is the atomic mass, and kB is theBoltzmann constant. Because multiple collisions, not allcontributing equally to re-thermalization, are occurring
during the thermalization process, the parameter α canbe interpreted as a re-scaling of σ̄ and therefore as anumber of collisions per re-thermalization (NCPR). Experimentally, the knowledge of α is fundamental for the
extraction of the total scattering cross section.
Equation 1 has two unknown parameters: as and α. In
contrast to alkali atoms, where the scattering is isotropic,

the situation is more complex for dipolar atoms such as
Er and Dy [18, 20]. Here, the total cross section forbosons is not only given by the contact scattering length
as , but an additional contribution from the non-isotropic
DDI, which for two atoms at a distance r and polarized
by an external magnetic field B, reads as
Vdd (r, θ) =µ0 µ2 1 − 3 cos2 θ
.
4π
|r|3

(4)

Here, µ0 is the magnetic permeability, µ is the magnetic
dipole moment, and θ the angle between B and r. Taking
an angular average of the total cross section leads toσ̄ = 8πa2s +

32π 2
a ,
45 d

(5)

2

0µ
where ad = mµ8π~2 is the dipolar length (ad = 98.2 a0
166forEr), with ~ being the reduced Planck constant.
Finally, we can rewrite Eq. 1 as

τ=αα
= 4N mω̄ 2.
4 2n̄σ̄vr(as + 45 ad )πkB T0(6)

The interplay between the isotropic scattering lengthand the anisotropic dipolar cross section leads to a dependence of α on both, the dipole orientation θ andas [37]. In the limit of weak excitation, an analytic formof α(as , θ) can be found based on the Enskog equations;
see later discussion.
III.

EXPERIMENTAL PROCEDURE

In our experiment, we produce a spin-polarized thermal cloud of Er atoms in the lowest Zeeman sublevel,similarly to Ref. [38]. In brief, after cooling and trapping
the Er atomic ensemble in a narrow-line magneto-opticaltrap [39], we transfer the atoms into a crossed optical
dipole trap (cODT). Here, we first further cool the atoms
via standard evaporative cooling, and then tighten thetrapping confinement to avoid atom loss due to residual
evaporation. Simultaneously, we ramp B to the desired
value. At this stage we typically reach a temperature of
T = 250 nK − 300 nK with N ≈ 1 × 105 . The exact numbers depend on the isotope choice and the individual setof measurement. The typical final trap frequencies are(ωx , ωy , ωz ) = 2π × (65(1), 19(1), 300(2)) Hz. For all setsof measurements the critical temperature for the onset of
Bose-Einstein condensation Tc lies between 150 nK and
200 nK, such that T & 1.5 × Tc . The orientation of themagnetic dipoles is controlled by the direction of the polarizing B and is represented by the angle θ between B
and the vertical direction z, defined by gravity; see inset
Fig. 1.
After preparing the thermal sample, we perform crossdimensional thermalization experiments [19]. In particular, we excite the cloud along the y direction and probe
the thermalization dynamics in the z direction. Our excitation scheme relies on a rapid increase in power of one

3
3.5

6

3.5

600

2.5

3

500

4.5

20

40

p-wave400

2.5s-wave3300

s-wave

0

50

100

150

200

250

2

300
1.5

FIG. 1. Effective temperatures Tz (blue circles) and Ty (reddiamonds) after the increase of the trapping potential along
the weakest trapping direction y. The measurement was performed at 1 G and θ = 0° for the 166 Er isotope. The reddashed line represents a guide to the eye. The black solid line
denotes the results of the Enskog simulations for this specificdataset. The errorbars denote the standard error for 3 repetitions. The inset shows a schematic representation of ourexperimental system.

trapping beam, leading to a 60 % increase of the trapping
frequency, while leaving the other two directions mostly
unaffected. We extract the effective temperature Tz (Ty )for a variable in-trap hold time th from the width of the
momentum distribution σz (th ) (σy (th )) after a time offlight of tToF = 25 ms (20 ms). This scheme, illustrated
in the inset of Fig. 1, leads to an out-of-equilibrium cloudwith an effective temperature increase along y from about
300 nK to 600 nK.
Figure 1 shows Tz and Ty as a function of th at
B = 1 G. As we excite the system along y, we observe theexpected rapid increase of Ty . After reaching a maximum
effective temperature, Ty starts to decay, and simultaneously Tz increases, both reaching the same equilibrium
temperature, thus showing thermalization dynamics. We
observe oscillations in Ty , which we attribute to a breathing mode that gets induced by the excitation. We observe
an exponential-type growth of the formTz (t) = Tf (1 − ∆T e−t/τ ).

(7)Here, Tf denotes the final temperature and ∆T denotes
the temperature increase due to the added energy. However, using this simple fit we can not directly extract as
as additional knowledge on α(as , θ) is needed (see Eq. 6).
IV.

1.5
0

THEORETICAL ESTIMATE OF α(as , θ)To compute α(as , θ), we utilize the Enskog equationsof change [40]: a coupled set of differential equations derived in closed-form for dipolar gases, by linearization of
the Boltzmann equation, and the assertion of a Gaussian

0

30

60

90

1.5
0

100

200

300

FIG. 2. (a) Dependency of α on θ and as for as = 0 a0 , 5 a0 ,10 a0 , 36.5 a0 , and 68.3 a0 . These values are chosen such thatthe angle dependence at small as becomes visible. Note that,at 68.3 a0 (as at 1 G, see later measurements) the variationof α with θ is strongly suppressed. (b) α vs. as for θ = 0°.
The inset shows an enlargement of the region for as between
0 a0 and 40 a0 . The grey dashed lines show the values of α fors-wave and p-wave scattering, respectively.

phase-space distribution [41]. These equations permit
an analytic derivation of α(as , θ) in the limit of shorttimes and small excitations [37]. For the current experiment with excitation along y and thermalization measured along z, the NCPR is described by a simple analytic formula, which reads
14 45a2s + 4a2dα(as , θ) =.
252a2s + 96as ad + (3 cos(4θ) + 13)a2d(8)

The quantity α(as , θ) exhibits an anisotropic character
via its angle dependence, as already observed for dipolar
fermionic atoms [19] and molecules [36].
Figure 2 shows α(as , θ) as a function of θ (a) and as (b),for our experimental configuration of a pancake shaped
trap. Figure 2(a) shows that the anisotropic character ofα(as , θ) competes with the contact one. Indeed, while forsmall as (. 10 a0 ), α(as , θ) exhibits a pronounced angledependence with a maximum at 45°, for increasing as
such behavior progressively washes out. For as ≈ 70 a0 ,the thermalization behavior becomes basically independent of θ, however α(as , θ) acquires a number below the
one expected for purely contact interacting s-wave collisions. This suggests faster thermalization for dipolar
particles, arising from a more efficient diversion of velocities of the scattering constituents. In the experiment, weonly measure re-thermalization for relatively large valuesof as & 30 a0 , and therefore we are not sensitive to theangle dependence of α(as , θ). In the course of this work,
we will thus focus on the case θ = 0°, simplifying Eq. 8

4
1

150
120

0.8

90
0.6
60
0.4

30
0

0.2
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

FIG. 3. Normalized atom number (orange circles) and as extracted from cross-dimensional thermalization measurements using
both, the Enskog equations (red squares) and the analytic formula of Eq. 8 (blue diamonds) are shown for 166 Er. AdditionallyaLMS(black triangles) obtained from lattice modulation spectroscopy measurements are given. The solid black lines representsa fit of Eq. 10 to aLMS
. Error bars and the shaded area of the fitting results denote the standard error.
sto
14 45a2s + 4a2d.
α(as , θ = 0°) =252a2s + 96as ad + 16a2d(9)

As shown in Figure 2(b), after an initial decrease,α(as , 0°) increases for as & 36.7 a0 – and thus the thermalization loses efficiency – moving to the regime ofcontact dominated interaction, eventually reaching theα(as , 0°) = 2.5 limit of non-magnetic atoms [18, 42]. We
note that by setting θ = 0° and ad /as ≈ 2.7, the NCPR is
minimized with value α ≈ 1.65, indicating highly efficientcollisional thermalization. This is directly attributed tothe innate anisotropic differential cross-section in dipolarbosons [18].
V.

MAPPING OF as AS A FUNCTION OF B
FOR 166 Er

Before taking cross-dimensional thermalization measurements for 166 Er, we perform a high resolution scanof the atom number as a function of the magnetic fieldin order to record the spectrum of Fano-Feshbach resonances, which we know to be exceptionally dense [4, 5].
We record the Fano-Feshbach spectra in a magnetic fieldregion from 0 G to 5 G; see Fig. 3 and Ref. [43]. In all themeasurements the magnetic field is oriented along z.
We then perform thermalization measurements at values of the magnetic field, where the system is not dominated by resonant atom loss. For each thermalization
curve, we extract as using two different approaches, one
numerical and one semi-analytical. The first, constitutes
a direct fit of the full Enskog solutions to the experimental data, leaving as as a float parameter of the theory;see [43] for more details. The second method, is basedon the exponential growth rate τ , from Eq. 1 using theanalytic expressin for α(as , 0°) in Eq. 9. For the latter,
since as is unknown a priori, we use an iterative approachto determine α(as , 0°) starting from α(as , 0°) = 1.7. We

use the calculated as and the analytic formula (see Eq. 9)
to obtain a new value for α(as , 0°). We stop the iteration
once the relative change of α(as , 0°) is ≤ 1 × 10−7 .
Figure 3 summarizes as for 166 Er in the region from
0 G to 5 G. In the studied B-field regime, the scattering
behavior is essentially dominated by a broad resonance at
3 G and a second one around B = 0 G. The as extracted
from the Enskog model and the semi-analytic one arein very good agreement with each other, reflecting thestrength of the analytic formula of Eq. 9.

VI.

BENCHMARKING WITH LATTICE
SPECTROSCOPY

To evaluate the robustness of our approach to extract
as , we benchmark our cross-dimensional thermalization
results with the one obtained using an alternative technique based on lattice modulation spectroscopy (LMS).
Such a technique, which we have developed in the past
for 166 Er [23, 29] and 167 Er [44], is based on the measurement of the on-site interaction - related to as - ofa lattice-confined dipolar gas in a Mott insulator state.
The LMS is able to provide accurate values of aLMS
, buts
at the price of being experimentally more involved due
to its requirements of an optical lattice together with aclean degenerate sample. Here we compare the valuesof as obtained with cross-dimensional thermalization ona low-density thermal sample, with aLMS
obtained fromsthe lattice modulation spectroscopy obtained in Ref. [29].
In brief we extract aLMS
as follows. We prepare an ultras
cold sample of 166 Er atoms in a three-dimensional optical lattice, created by two retro-reflected laser beams at
532 nm in the horizontal plane and by one retro-reflected
laser beam at 1064 nm along the vertical z direction,
defined by gravity. The final lattice depth along thethree directions is (sx , sy , sz ) = (20, 20, 100), in units of
Erec = 4.2 kHz (1.05 kHz) for 532 nm (1064 nm). The uncertainty on sx , sy , and sz is about 5 %. In such a deep

5
lattice, the atoms are in the Mott insulator phase [23].
We then create particle-hole-excitations by sinusoidally modulating the power of the horizontal lattice
beams for 90 ms with a peak-to-peak amplitude of about
30 % and measure the recovered BEC fraction after melting of the lattice. At the resonance condition, where themodulation frequency matches the particle-hole excitation gap, we observe a resonant reduction in the BEC
fraction [45]. The particle-hole excitation gap is directly
given by the on-site interaction U = Uc + Udd . Here, Uc
is the contact interaction – and thus depends on the unknown aLMS
– while the on-site dipolar interaction, Udd ,scan be accurately calculated. We repeat the measurements at various magnetic-field values and, for each, we
extract aLMS.
s
In Fig. 3, we compare aLMS
with as extracted froms
the thermalization measurements. We see an overallvery good agreement between the value of as extractedusing the two techniques. This shows that the crossdimensional thermalization approach combined with the
Enskog equations is a very reliable method to extract as ,even in the case of complex atoms for which the knowledge of α(as , θ) is not a priori given.

VII.

DENSITY DEPENDENCE

Our measurements for the 166 Er isotope were performed in a regime of relatively low density (n̄ ≤0.5 × 1013 cm−3 ). Interestingly, when applying the same
method in a regime of high density, we observe a dependence of the thermalization rate on the density whichgoes beyond the Enskog approach. For instance, we repeat the cross-dimensional thermalization measurements
for 166 Er at B = 1 G and variable cloud density, n̄. We
control the density by either increasing N or by applying a tighter trapping configuration of (ωxcyl , ωycyl , ωzcyl ) ≈2π × (300, 19, 300) Hz before compression, or both. Fromthe lattice modulation spectroscopy, we have extracted avalue as = 68.3(7) a0 at B = 1 G. By fixing this value– meaning to impose that the scattering length does notdepend on density – and using Eq. 1, we can determineα(as , 0°) as a function of n̄.
Figure 4 shows α(as , 0°) for different values of n̄. We
find a pronounced dependency on n̄, with a rapid increase
and an eventual saturation at high densities. Such a behavior is not captured by our theoretical model, which, as
reflected in the definition of α(as , 0°) in Eq. 1, predicts no
density dependence. To the best of our knowledge, sucha dependence has not been reported in previous works oncross-dimensional thermalization. Possible explanations
root in various causes, either physical or technical nature.
Although being above Tc , precursors of quantum manybody phenomena might influence the scattering behavior.
Exemplary, we tried to explicitly include effects comingfrom Bose-enhancement into our theoretical framework.
This did not have significant influence on the thermalization behavior.

5
4
3
2
1
0.5

1

1.5

2

2.5

3
10 13

FIG. 4. Measurements of α(as , θ) as a function of n̄. The
blue circles correspond to the datasets at 1 G, shown in Fig. 3.
The black solid line marks the value given by the analytic
formula in Eq. 9. All measurements are performed with θ =0°. Errorbars denote the standard error.

Another possible explanation, based on unavoidable
experimental imperfections, roots in deviations from an
ideal harmonic trapping condition, leading to a modification of the kinetic energy and the mean density. Sucha variation would manifest in an apparent change ofα(as , 0°); see Eq. 1. Indeed, Eq. 2 and 3 are only valid
for an ideal harmonic trapping confinement. First MonteCarlo simulations performed by using a realistic gaussian
trapping potential seem to support this assumption [43].
Note that, varying the initial temperature of the atomic
cloud and the excitation strength did not show any influence on the observations. We emphasize that, since ourmeasurements to extract as have been performed at low
densities, our method should remain valid.

VIII.

SCATTERING LENGTH FOR 164 Er AND
170
ErAfter the detailed study on 166 Er and the benchmarking of the results with high-precision lattice modulation
spectroscopy, we confidently apply our cross-dimensional
thermalization approach to other two isotopes, 164 Er and
170
Er. Again we start with a Fano-Feshbach spectroscopy
between 0 G and 5G to identify the position of the scattering resonances as shown in Fig. 5. We note that thisFano-Feshbach spectra have not been reported previously. For the cross-dimensional thermalization measurements we follow a similar experimental procedure as described above. From the thermalization curve, we again
use both, the full fit of the Enskog equations as well as
the iterative approach on α(as , 0°) to determine as fromthe exponential growth rate τ .
Figure 5 shows as for the isotopes (a) 164 Er and (b)170
Er. While the scattering behavior for 164 Er is, similarly to 166 Er, dominated by two broad resonances at
1.5 G and 3.3 G, 170 Er features several narrow overlapping resonances, providing different test scenarios for our

6

1

150

0.8

120

0.6

90

0.4

60

0.2

30

0

0

1

400

0.8

300

0.6
200
0.4
100

0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
5

FIG. 5. Normalized atom number (orange circles) and as extracted from cross-dimensional thermalization measurements using
both, the Enskog equations (red squares) and the analytic formula of Eq. 8 (blue diamonds) are shown for (a) 164 Er and (b)170
Er. The solid black lines represent a fit of Eq. 10 to as obtained using the Enskog equations. Error bars and the shaded areaof the fitting results denote the standard error.

cross-dimensional thermalization. Although minor deviations can be observed in the vicinity of Fano-Feshbachresonances, for both isotopes, the extracted as using thetwo approaches are once more in good agreement.

DDI between the incident scattering channel and Zeeman
states that lie higher in energy. As a consequence this results in a perturbation of the molecular potential, whosestrength depends on the magnetic field, leading to anincreasing value of the van der Waals C6 coefficient [48].

(10)

To parametrize as as a function of B, we fit Eq. 10 tothe measured as for 164 Er, 166 Er, 168 Er, and 170 Er. For
166
Er and 168 Er, we use the scattering lengths obtained
from the lattice modulation spectroscopy, corresponding
to our most accurate determination; see solid lines inFig. 3 and [43]. For 164 Er and 170 Er, we fit Eq. 10 to theas data obtained by applying the Enskog equations tothe cross-dimensional thermalization measurements; seesolid lines in Fig. 5. More details on the fitting procedure
as well as the complete list of the fit parameters is given in
Ref. [43]. In general, we observe that the fitting function
reproduces very well the behavior of as for every isotope.

to the case of Nres overlapping resonances of position Bi
and width ∆Bi , and allowing for a smooth off-resonantvariation of as with B. We observe that a linear variation of slope s already well reproduces the data withas (0) defined as the effective abg
s . We note that different
mechanisms could lead to an off-resonant variation of as .
For instance, the influence of broad Fano-Feshbach resonances, which are not within our measurement range,
could lead to a smooth variation of the background behavior, similar to that observed for cesium [47]. Alternatively, the effect could be due to the coupling induced by

Figure 6 shows the value of abg
s from the fit as a function of the isotope mass. We observe a monotonic rising of abg
s with increasing m, which might be compatible
with different functional forms, including a simple linear
increase. Under the assumption that erbium has a similar behavior to ytterbium and cesium, we can use themodel for the mass scaling as developed in Ref. [49–51].
Such a model assumes that as is only given by the Van
der Waals potential U (r) = −C6 /r6 , with C6 being the
Van der Waals coefficient. This might be a rather severe
approximation for magnetic atoms but, in absence of alternative models, it is interesting to compare the simple

IX. SCALING OF BACKGROUNDSCATTERING LENGTH WITH MASS

The knowledge on as as a function of the magneticfield allows us to extract an effective background scattering length abg
s for each isotope. The general behaviorof as with B can be described by generalizing the wellknown formula [46]as (B) = (abgs + sB) ×

N
res 
Y
i=1

1−

∆Bi
B − Bi,

7
the similar C6 coefficient, NB is approximately a factorof 2 larger than for ytterbium [50]. Note that, NB is
in agreement with the result obtained when using thesame approach but assuming a hard core potential [43].
We would like to emphasize once more that this model
does not consider any contribution arising from the DDI.
An improved description calls for the development of advanced theoretical models.

140
120
100
80
60
40
163

164

165

166

167

168

169

170

171

FIG. 6. Background scattering length abg
for four bosonicsisotopes (red circles). The solid line represents the best fit
with φ = 144(1); see text. The shaded area, enclosed by thedotted lines, represents the fitting function for φ = 143 and
φ = 145. The errorbars denote the standard error of the fitof Eq. 10 to the experimental data.

mass-scaling approach to erbium.
As introduced in Ref. [49], as can be written ashπ i
as = ā 1 − tan φ −,8
mC6 1/4
being the characteristic~2

with ā = 2−3/2 Γ(3/4)Γ(5/4)
length and
φ=

(11)

√

m
~

Z ∞p
R0

−U (r)dr.

X.

CONCLUSION

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

(12)
ACKNOWLEDGMENTS

Here, Γ(x) is the gamma-function and R0 is the classicalturning point of U (r). Although the exact shape of U (r)is unknown, Eq. 11 can be employed to√extract a massscaling due to the dependence of φ ∝ m [50]. Such a
scaling is valid, as long as the mass dependent modification of U (r) is negligible. Furthermore, φ allows for the
calculation of the number of bound states NB via relation
NB = bφ/π − 5/8c, where b c denotes the floor integerfunction.
We now apply this model to our Er case. Figure 6
shows the fit of Eq. 11 to the experimental data; see
Ref. [43] for details. We obtain the best agreement for
φ = 144(1), leading to NB = 143(1) for 168 Er. Despite

This work is financially supported through an ERC
Consolidator Grant (RARE, no. 681432), a DFG/FWF(FOR 2247/I4317-N36) and a joint-project grant fromthe FWF (I4426, RSF/Russland 2019). We also acknowledge the Innsbruck Laser Core Facility, financed by theAustrian Federal Ministry of Science, Research and Economy. R. R. W. W. and J. L. B. acknowledge that this material is based on work supported by the National Science Foundation under grants numbered PHY-1734006
and PHY-2110327.
* Correspondence and requests for materials should be
addressed to Francesca.Ferlaino@uibk.ac.at.

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[52] G. A. Bird, The DSMC method (CreateSpace Independent Publishing Platform, 2013).
[53] G. Durastante, C. Politi, M. Sohmen, P. Ilzhöfer, M. J.
Mark, M. A. Norcia, and F. Ferlaino, Feshbach resonances in an erbium-dysprosium dipolar mixture, Phys.
Rev. A 102, 033330 (2020).
[54] G.-j. Ni, The Levinson theorem and its generalization in
relativistic quantum mechanics, Chinese Physics C 3, 432(1979).

10
SUPPLEMENTARY MATERIAL
A.

340
320

Analytic number of collisions per
re-thermalization

300

Analytic expressions for α(as , θ) can be derived undera short-time approximation, with the Enskog equations
dhqj2 i−2
hqj pj i = 0,m

dt
dhp2j i
+ 2mωj2 hqj pj i = C [p2j ],dtdhqj pj i
1 2−
p + mωj2 hqj2 i = 0,
dt
m j

(13a)

(13c)

(14)

(15)

For small deviations from equilibrium and at shorttimes, re-thermalization can be approximated with a single decay rate γ, such that C [χ] ≈ −γhχi. This results in
the relationdEj= −γyj (Ej − kB Tf ) = C [Ej ],
dt

240
220

where Tf = (Tx +Ty +Tz )/3 is the final equilibration temperature (obtained from the equipartition theorem), h. . .idenotes an ensemble average assuming a Gaussian phasespace distribution whose widths are allowed to vary, andEj = hp2j i/(2m)+mωj2 hrj2 i/2 is the sum of kinetic and potential energies in the j-th direction. The Enskog equations dictate that the relaxation of hχj i follows the differential equationdhχj i= C [χj ].
dt

260

(13b)

where rj and pj are positions and momenta respectively(j = x, y, z), and C is the collision integral. The derivation follows from Ref. [37], but we present a brief outline here for completeness. The gas is assumed close-toequilibrium, allowing us to treat rj and pj as Gaussian
distributed. Thermalization trajectories are then tracked
using the Gaussian widths along each axis, to computethe energy differentialhχj i ≡ Ej − kB Tf ,

280

(16)

where the subscript on γyj indicates that the gas was
excited along y, and re-thermalization measured along j.
This then permits us to computeEj − kB Tfn̄σ̄vrαyj ==n̄σ̄vr ,(17)
γyj
C [Ej ]

0

50

100

150

200

250

300

FIG. S1. Benchmarking of the Enskog simulation resultsfor Tz (red solid line) with Monte-Carlo simulations (black
dashed line). The dataset is the same as in Fig. 1.

of full numerical solutions to the Enskog equations. To
do so, as was left as a float parameter in the theory, thenvaried until a best fit between the theory and experimental data was obtained. A feature we noticed during fitting
was the high sensitivity of thermalization rates to variations in the trapping frequencies ω, over the finite-time
quench. Measurement uncertainties therefore motivate
us to also leave ω a float parameter, with allowed values
within its 1-sigma errorbars. This is applied both to thetrapping frequencies before and after the quench.
We performed fits using a χ2 optimization criterion
2tend 
X
T (t) − T E [T (0); ω, as ] (t)
min
,
(18)
ω,as
δT (t)
t=t0

where the sum runs over measurement time instancest, T (t) is the temperature data from the experiment,δT (t) is the temperature measurement uncertainty, andT E [T (0); ω, as ] is the solution to the Enskog equations
with initial condition T (0), and fit parameters as and ω.
To reduce biasing of the fits, we run an iterative algorithm that recursively fits ω and as in succession until
they converge to stable values. Such a procedure would
take exceedingly long times (∼ weeks) with full Monte
Carlo (MC) simulations, but can be done in minutes with
the Enskog equations on a current-day computing device.
Solutions to the Enskog equations have shown themselves accurate when compared to MC simulations [37,41]. We show their accuracy here yet again, using theparameters from the current experimental set-up. An
illustrative example is provided in the plot of Fig. S1,
comparing an instance of the Enskog solutions (red solidline), MC simulations (black dashed line) and the experimental data (blue circles).

which for j = z, has the form in Eq. (8).
C.
B.

Fitting Enskog equations to experimental data

The extraction of the scattering lengths as , from crossdimensional thermalization data was done here by means

Monte-Carlo simulations including trap
anharmonicities

Optical dipole traps are, in many studies, assumed to
be well modeled by purely harmonic potentials. This

11
may however be inadequate in regimes with significant
trap anharmonicity effects, which we currently attributethe density dependence of α to. In such cases, the potential is better modeled as two cross-propagating Gaussianprofile beams along the y and z axes (with gravity). This
produces the confinement potential−2
2Ũ1 P1e
VODT (r) = −πw1,x (z)w1,y (z)−22Ũ2 P2−eπw2,x (y)w2,y (y)+ mgz,y2x2
2 (z) + w2 (z)
w1,x
1,y





z2
x2
2 (y) + w2 (y)
w2,x
2,z



(19)

where P is the laser power Ũ is an atomic polarizability
parameter andsz2
w(z) = w0 1 + 2 ,(20)zRwith zR and w0 denoting Rayleigh lengths and beamwidths respectively.
Such a potential limits the applicability of the aforementioned Enskog equations as formulated in Ref. [41].
Instead, more robust MD methods are required to accurately predict thermalization trajectories. We implement a MD simulation similar to that in Ref. [20], which
evolves simulation particles under the action of VODT via
the Verlet symplectic integrator∆tp (t),2m k
pk (t + ∆t) = pk (t) + F k ∆t,∆t
p (t + ∆t),r k (t + ∆t) = q k +2m k
q k = r k (t) +

(21a)
(21b)
(21c)

where subscripts k denote the k-th simulation particle,∆t is the simulation time-step, t is the time andF k = −∇VODT (r k ).

(22)

Dipolar collisions are then computed with the direct simulation Monte Carlo method [52], that determines postcollision momenta via stochastic sampling of the differential cross section.
In a preliminary study of the density dependence, ideal
Gaussian beam profiles are assumed, along with perfectly
accurate beam widths and Rayleigh lengths. Following a
trap quench, thermalization of the out-of-equilibrium gas
in VODT indeed shows an apparent increase of α with density, qualitatively similar to that observed in the experiment. This effect is absent in simulations with an ideal
harmonic trap. Furthermore, in higher density regimes,
the simulations with VODT predict the experimentally observed equilibration temperatures more accurately compared to the harmonic trap case. These early findings
on density dependence from trap anharmonicities are intriguing, and a cautionary tale for future experiments.
However, we do not develop this idea further here andleave such analysis for future works.

D.

Fano-Feshbach spectroscopy

To identify the positions of the Fano-Feshbach resonances we perform high-resolution loss spectroscopy ina cylindrically symmetric trap. We evaporatively cool
the atoms until they reach a temperature between T =300 nK and 400 nK. At this stage, the atom number is between 6 × 104 and 1.2 × 104 with typical trap frequenciesof (ωx , ωy , ωz ) = 2π × (300, 30, 300) Hz. The exact values depend on the isotope choice. After reaching thermal
equilibrium, we change B, oriented along the z axis, in
1 ms to the desired value and wait for a holding time between 250 ms and 500 ms. We use different holding times
for different datasets to avoid saturation effects of theresonances for higher densities. After the holding time,
we measure the atom number using absorption imaging
after a time of flight expansion of 25 ms. The results of
the loss-spectroscopy measurements are shown in Fig. 3,
Fig. 5 and Fig. S2.
E.

Scattering length for 168 Er

To obtain as for the 168 Er isotope, we follow a similar approach as for 166 Er. First, we perform loss spectroscopy to identify the position of Fano-Feshbach resonances. We then transfer the atoms into an optical lattice with a depth of (sx , sy , sz ) = (20, 20, 40)Erec andapply the lattice modulation spectroscopy technique toextract as . The lattice modulation spectroscopy follows
the same lines as for the 166 Er isotope; see main text.
Fig. S2 summarizes the results for 168 Er and shows theFano-Feshbach spectroscopy result as well as as as a function of B in the magnetic field range from 0 G to 5 G.
F.

Extracting background scattering length

To obtain a value for abg
s , we fit Eq. 10 either to as obtained from the full Enksog equations (164 Er and 170 Er)
or aLMS(166 Er and 168 Er). Due to the different numberssof Fano-Feshbach resonances compared to the number ofavailable data points for as , we slightly vary the fitting
approach for the individual isotopes. Depending on theposition and the width of the resonance, for some resonances, we fix the position Bi to the minimum of theloss feature and keep only the width ∆Bi as a floating
parameter. For the very narrow resonances, which havea negligible influence on the overall scattering behavior,
we fix both Bi and ∆Bi .
Table I gives the results for the background scattering
lengths abg
s and the slopes s for all four isotopes. Moreover, Tables II–V contain a detailed listing of all FanoFeshbach resonances and how they are included in the fitting procedure. Note that for 170 Er, we are aware of theexistence of a particularly broad resonance at 6.91 G [53],which we include with variable width. When looking
closely, the onset of this resonance can actually be seen

12
1

400

0.8

300

0.6
200
0.4
100

0.2
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
5

FIG. S2. Normalized atom number (yellow circles) and measured scattering lengths as obtained for 168 Er from lattice
modulation spectroscopy measurements. The solid black line represents a fit to aLMS
. The shaded area as well as the errorsbars denote the standard error.

TABLE I. Values for abg
s and s obtained from the fit of Eq. 10
to as for the four bosonic isotopes. The error denotes the fit
error of one standard deviation.
abgs (a0 )52(6)61(3)
110(2)
129(9)

isotope
164
166
168
170

s (a0 /G)9(3)5.4(9)
11(2)20(10)

Position Bi (G)
0.02(5)3.04(5)4.208
4.96

TABLE II. Parameters for the Fano-Feshbach resonances included into the fit of Eq. 10 to as for 164 Er. The error denotes
the fit error of one standard deviation. Values without error
are fixed in the fitting procedure.
Position Bi (G)
1.52
2.67
2.83
3.26

Width ∆Bi (G)0.22(3)0.005
0.005
0.10(3)as a reduction of N towards higher magnetic field values
in the loss spectroscopy (see Fig. 5(b)).

G.

TABLE III. Parameters for the Fano-Feshbach resonances
included into the fit of Eq. 10 to as for 166 Er. The error denotes the fit error of one standard deviation. Values without
error are fixed in the fitting procedure.
Width ∆Bi (G)0.05(2)0.15(2)0.01
0.005

TABLE IV. Parameters for the Fano-Feshbach resonances included into the fit of Eq. 10 to as for 168 Er. The error denotes
the fit error of one standard deviation. Values without error
are fixed in the fitting procedure.
Position Bi (G)0.49
0.911(6)1.51
2.174(4)2.471(9)
2.86
3.79
4.23
4.5

Width ∆Bi (G)0.005
0.032(2)0.01
0.038(2)0.19(1)0.005
0.006(5)
0.005
0.005

χ2 analysis for mass scaling

In this section, we describe our analysis of the background as of the 4 Er isotopes (Fig. 6) with Eq. 10. To
find the best fitting parameter φ, we analyze the agreement of the theoretical model in Eq. 11 with our experimental data. For each value of φ, we calculate the χ2
via
2

χ =

2
4  mod
X
a
− ai
s

i=1

s

σsi

.

(23)

Here, amod
is the scattering length given by the models
for the corresponding φ and ais and σsi are the measured
as with the corresponding standard error.
The behavior of χ2 is non-monotonic with the appearance of several minima. We identify the absolute minimum of χ2 for φ = 144.03. To further obtain an estimate
for the error of φ we fit a quadratic function to the local
minima. We extract the limits of the confidence interval
by considering the region where χ2 ≤ χ2 + 1.

13
TABLE V. Parameters for the Fano-Feshbach resonances included into the fit of Eq. 10 to as for 170 Er. The error denotes
the fit error of one standard deviation. Values without error
are fixed in the fitting procedure.

where ā = √Γ(3/4)2(2)Γ(5/4)

ac with ac =

2mr C6 1/4
being the~2characteristic scattering length scale of the potential, anda2
Φ = 2rc2 − 3π
8 is the semi-classical phase [49].
cPosition Bi (G)
0.35
0.86
1.12
1.62
2.17
2.74
3.3
3.57
4.38
4.49
6.91

H.

From theoretical calculations in Ref. [6] we use C6 =1723 a.u. and we estimate from the theoretical interaction potential given in Ref.[6] that rc ≈ 4 − 8 a0 . We
fit Eq. 24 to abg
s of the four bosonic isotopes. Due to alarge number of possible local minima, we combine thefitting with a minimization of the χ2 -value while varying
the start parameter for rc . We obtain the best agreement
for rc = 5.05(5) a0 .

Width ∆Bi (G)0.005
0.028(12)
0.005
0.01
0.067(7)0.134(9)0.01(1)
0.01
0.005
0.01
0.8(7)In addition, the Levinson theorem [54] allows us toestimate the number of bound states NB , which can becalculated from the semi-classical phase Φ using

NB =Hard-core potential for mass scaling

The model contains the assumption, that the s-wave
scattering length is given at large distances by the vander-Waals potential scaling with UvdW (r) ∝ −C6 /r6 ,with C6 being the Van der Waals coefficient, and at shortdistances r < rc by a hard core potential [49]. In thisspecific case, the scaling of abg
s can be described byabg
s = ā tan(Φ),

(24)




Φ 5
−
+ 1,
π
8

(25)

where the square brackets mean the integer part. For
the current fitting we obtain NB ranging from 141 to
144, in agreement with the approach in the main text.
We want to emphasize, that this modelling of abg
s is asimple approach and a more thorough analysis could add
deeper valuable insights.

Chapter6

Conclusion and outlook

6.1. Conclusion
This work describes experiments with quantum gases composed of ultracold fermionic andbosonic erbium atoms, which possess a large permanent magnetic dipole moment. The
overarching objective of my PhD thesis was to push the frontier of accurate knowledge
about fundamental aspects of quantum physics with dipolar gases. This include the mappingof spin-exchange interactions, the search, observation, and characterization of a clock-type
optical transition, and an accurate determination of the erbium scattering length.
The first part of the thesis describes our experimental efforts to study lattice spin physics withfermionic erbium. In particular, we have developed a precise control over the internal degreeof freedom (spin) of our system in order to realize and study a XXZ-Heisenberg model basedon dipolar interactions. The spin states are encoded in the magnetic Zeeman sub-levels,
and we find that under the condition that the energy difference between subsequent levels
is equal, resonant spin-exchange dynamics occur. We have measured the out-of equilibrium
dynamics of spin populations, which represent a global quantity, and studied the spreading
and redistribution driven by the exchange interaction. We have shown that the rate at which
the dynamics occur depends strongly on the initial spin state and on the magnetic dipole
orientation. Moreover, we have demonstrated a new protocol for switching the dynamics
fast on and off by exploiting the tensorial ac-Stark shift induced by an off-resonant lightfield. This work was carried out in collaboration with A.-M. Rey and B. Zhu, who developed
numerical calculations that show excellent quantitative agreement with our experimental
findings.
The second part of the thesis focuses on endowing our erbium quantum gas platform withclock-type laser capabilities. The existence of a narrow optical transition at 1299 nm witha linewidth in the Hz range has been predicted theoretically, although with low precision.
We performed high-resolution spectroscopic measurements for various isotopes and were able
to accurately determine the resonance position. We have characterized the atomic transition in detail and determined important parameters such as the lifetime of the atomic state
and the Landé gJ -factor. Moreover, we were able to demonstrate the ability to coherently
control the atomic state. As an important input for the design of future experiments, we
107

108

6. Conclusion and outlook

measured the ratio of the polarizability between the ground state and the excited state.
Comparison to calculations based on a semi-empirical method, performed by our collaborators G. Hovhannesyan and M. Lepers, showed good agreement. Finally, we have performed
initial experiments on spin manipulation that show efficient preparation of spin states andspin mixtures for bosonic isotopes, which would otherwise not be possible because of thelack of hyperfine structure.
Finally, the third part of the thesis describes our investigation aiming at expanding theknowledge of the scattering length in various erbium isotopes and at obtaining an accurate
magnetic-field to scattering length conversion. This work has been conducted in collaboration with theorists from JILA, i. e. J. Bohn and R. Wang. This project has been developed
following two paths. On the one hand, we have performed high-precision lattice spectroscopy
and extracted the scattering length with low uncertainty. On the other hand, we have used
these results to benchmark the scattering length obtained with the much easier approach
using cross-dimensional thermalization measurements. For the latter we applied the theoretical Enskog equations and an analytic formula based on the number of collisions per
re-thermalization. We carried out cross-dimensional thermalization measurements for fourbosonic isotopes in a magnetic field range from 0 G to 5 G, which allowed us to determine abackground scattering length for this magnetic field range. Finally, we applied a simplified
theoretical approach to model the background scattering length scaling with isotope mass.
Overall, the knowledge gained about erbium atoms and the characterization of the narrow atomic transition represent a promising starting point for future experiments. In thefollowing, a number of possible applications, especially for the transition at 1299 nm, areoutlined.

6.2. The transition at 1299 nm: A versatile experimental tool
6.2.1. Prospects for combining the optical lattice and the narrow inner-shellorbital transition
The successful observation of dipolar spin-exchange dynamics with fermionic erbium in athree-dimensional optical lattice, opens the door to a number of interesting questions about
many-body dynamics that remain to be answered. To address many of these questions, theimplementation of new tools is required. For our system, the narrow optical transition at
1299 nm is of particular importance because it represents a unique tool the preparation, detection, and identification of quantum states. In this section, we will discuss two particularlyinteresting aspects.

Thermal equilibrium for spin-exchange dynamics
One direction for further investigation of spin-exchange dynamics is to observe the attainment of an eventual thermal equilibrium [Lan15]. From theoretical calculations based on the6.2. The transition at 1299 nm: A versatile experimental tool109

generalized discrete truncated Wigner approximation [Zhu19], it is expected that the atomic
distribution among the magnetic sublevels reaches a thermal equilibrium state, which depends crucially on the spin quantum number. While for an integer spin, as it is the case forbosonic erbium (J = 6), a homogeneous spin distribution is expected, for fermionic erbium(F = 19/2) an imbalanced population of Zeeman levels should occur due to the half-integer
spin. Moreover, for fermionic erbium, the equilibrium state depends significantly on the sizeof the spin manifold.
To be able to study this behavior, two major improvements need to be implemented: First,
it is important to decouple the spin-exchange dynamics from other processes that may occur
due to tunneling effects. This includes, e. g. losses due to on-site dipolar relaxation in cases
where the gain in energy exceeds the lattice depth [dP13a]. In our current experimental
system, the tunneling rate is about 10 Hz, which limits the usable timescale of those experiments to several 100 ms. Thanks to a technical upgrade of our laser-light distribution forthe optical lattice (see Appendix A), a reduction of the tunneling rate by a factor of about
250 is within reach. This will allow to observe pure spin-exchange dynamics on much longer
timescales.
Second, a deterministic preparation of Zeeman-states for bosons as well as a procedure for theisolation of sub-systems within the spin manifold is necessary. Our narrow-line transition at
1299 nm can be employed for both. It enables the preparation of bosonic atoms in higher spinstates (see Sec. 4.3.3), which has not been realized so far due to the lack of a quadratic Zeeman
shift. On the other hand it allows for the isolation of sub-systems within the spin manifoldby exploiting the quadratic ac-Stark shift and shifting Zeeman levels out of resonance for the
spin dynamics. The latter enables therefore the possibility to study thermalization behavior
for smaller system sizes.

Spin-resolved imaging technique
Up to now, experiments studying dipolar spin-exchange dynamics [Yan13, Lep19, Ala22] arelacking information about the spatial distribution of spin populations. Similar to our case,in previous experiments that studied dipolar-spin exchange dynamics with large spin, theatomic distribution within the spin manifold was measured via Stern-Gerlach separation.
For this purpose, the atoms were released from the lattice confinement and a magnetic field
gradient is applied during time of flight expansion, which causes a spatial separation of thedifferent spin states. In this way, detailed information about the spin populations is obtained,
but the spatial atomic distribution in the lattice remains inaccessible.
The realization of quantum gas microscopes, so far implemented with alkali and alkaline-earth
like atoms, provides detailed spatial information due to the detection of individual atoms on
single lattice sites [Bak09, She10]. In addition to single atom resolution, recent experiments
demonstrated spin resolved imaging for spin-1/2 systems using 6 Li [Bol16, Ber18]. However,in optical lattices these techniques rely on a super-lattice and a magnetic field gradient,
limiting the detection scheme to one-dimensional spin chains. For dipolar atoms, so far noquantum gas microscope has been realized, however, different groups around the world areworking towards its implementation [Phe20, Soh21, Uer21].

110

6. Conclusion and outlook

Figure 6.1.: Schematic illustration of the shelving technique to image different spin states (indicatedby blue, red, and green circles) in a single experimental cycle, with the energy level at 1299 nm serving
as a storage state. (a) By applying a π-pulse, all atoms, except those in the spin state which is imaged
first, are transferred to the excited state. (b) Subsequently, an imaging pulse with light at 401 nm
is applied to image the ground-state atoms. (c) Step by step, all atoms in different spin states are
pumped to the ground state and imaged.

The narrow line at 1299 nm opens here new opportunities to realize a spin resolved imaging
technique in a quantum gas microscope. Here, the excited state takes on the role of a storage
state for atoms in different spin states. Figure 6.1 shows a schematic illustration of spinresolved imaging using this shelving technique. First, all atoms except those in one spin state
are transferred to the excited state by application of a π-pulse using the light at 1299 nm.
Subsequently, the remaining atoms in the ground state are detected, e. g. via fluorescenceimaging as considered in Ref. [Pic19], using the broad transition at 401 nm. In the following,atoms in other spin states are successively transferred to the ground state and detected.
Note that a similar scheme might be realized with the corresponding transition at 1001 nm
in dysprosium [Pet20a].
An important aspect for this imaging technique is taken on by the different time scales, suchas the duration of the π-pulse and the imaging pulse length. Since the excited state lifetime
is finite, an extended imaging time leads to a reduced detection fidelity. Considering the
Rabi oscillation as measured in Chap. 4, the duration for the π-pulse corresponds to about
10 µs. Further, in Ref. [Pic19], an optimal imaging-pulse length was determined to 1.5 µs. In
the case of fermionic erbium, which has 21 spin states, an estimation for the imaging time ofall spin states leads to about 250 µs. Considering the excited-state lifetime of about 180 ms,
this leads to detection efficiency > 99.9 %. Importantly, during our measurements performed
in Chapter 4 we did not observe any influence of the light at 401 nm on the excited state.
In addition to spin-resolved imaging, the shelving technique can be used to enable imaging of
single atoms beyond the diffraction limit, thereby achieving an effectively higher imaging resolution [Fra21]. This allows the implementation of very short lattice spacing, which is relevant
for dipolar systems since the interaction strength (∝ 1/r3 ) can be significantly increased. The
conceptual idea follows the stochastic optical reconstruction microscopy (STORM)1 , which1

In 2014 the Nobel Prize in chemistry was awarded for the development of super-resolved fluorescence6.2. The transition at 1299 nm: A versatile experimental tool111

is widely applied in biology to distinguish overlapping fluorophores [Hua08]. In a controlledmanner, only a fraction of the fluorophores are activated simultaneously, effectively reducing the density and thus the number of overlapping molecular complexes. Using multiple
images, the overall image can thus be reconstructed with higher resolution. Analogously,
by promoting the majority of atoms to the excited state, the number of atoms that are
imaged simultaneously is reduced and neighboring atoms whose separation is smaller than
the imaging resolution can be identified.

6.2.2. Further potential applications
The transition at 1299 nm has further potential applications, which may not be immediately
obvious, since they are not directly related to our experimental apparatus. This shows once
again the flexibility the transition and the variety of new possibilities that the transition
brings. Here we will describe two fields for potential future applications.

Quantum Communication

For the realization of quantum networks and the associated transmission of quantum information over long-distances, it is of special interest to take advantage of already existing
technologies. One aspect is the use of optical fibers in the telecommunication-wavelength
window, since these exhibit a particularly low photon absorption. Most of the existing platforms, based for example on trapped ions or nitrogen-vacancy centers, use transitions which
are outside this particular wavelength range. Therefore, they rely on frequency conversionof light to the telecommunication-wavelength range [Dra18, Kru19], which limits current
performance. Today, best conversion schemes reach an efficiency of about 25 % [Kru19],
however they suffer from various dephasing mechanisms [Bau20]. Being directly in thetelecommunication-wavelength O-band, the transition at 1299 nm is an interesting candidate for future realizations of quantum information systems.
Generally, for the successful realization of quantum networks it is fundamental to have large
entanglement generation rate. More specifically, it is important that the entanglement generation rate exceeds the decoherence rate of the entangled state [Hum18]. Current stateof-the-art systems achieve entanglement rate on the order of tens of Hz [Hum18, Lan21].
At first glance, this seems unattainable due to the relatively long lifetime of the state at1299 nm for erbium. However, a possible workaround could be obtained by exploiting the
Purcell effect in optical cavities [Pur46], which leads to a reduced lifetime due to the atomcavity coupling. Similar concepts have been successfully realized using trapped ions [Cas15],
nitrogen-vacancy centers [Ruf21], and rubidium atoms [Wel21]. Although a more detailed
characterization of the potential system performance is needed, erbium might be a promising
candidate for future quantum networks.
microscopy.

112

6. Conclusion and outlook

Rydberg atoms
Rydberg states in neutral atoms have become a very promising platform for quantum simulation and quantum computation. While many exciting results have been obtained usingalkali [Ber17, Sch20b] and alkaline-earth like [Nor18, Coo18] atomic species in configurable
tweezer experiments, lanthanide atoms have not been exploit in this context up to know.
Only recently, the first high-resolution spectroscopic measurement of Rydberg states in erbium atoms, performed here in Innsbruck, was reported [Tra21].
The transition at 1299 nm potentially provides additional control knobs when combinedwith Rydberg atoms. Thanks to the multi-valence electrons of erbium, the electronic coreremains optically active after excitation of an electron to a Rydberg state. A first interesting
aspect is the inner-shell character of the 1299-nm transition, which describes the fact thatboth orbitals involved in the transition have a lower principle quantum number (4f and 5d)than the 6s orbital. In strontium it was shown that the excitation of an electron in the 5s
orbital to the 5p orbital leads to strongly enhanced auto-ionization [Mad20]. In the caseof erbium, if an electron of the 6s orbital is promoted to a Rydberg state, the inner-shellelectrons might still be shielded by the remaining electron in 6s, which eventually reduces
the effect of auto-ionization. Within the same work of Ref. [Mad20], an alternative single
photon excitation protocol was demonstrated, where the long-lived clock state in strontium
is used as a meta-stable ground state and an improved transfer fidelity was obtained. Sincealso the lifetime of the state at 1299 nm is much longer than the typical one of Rydberg
states, a similar approach can be applied. Additionally, starting from the state at 1299 nm,
larger angular momentum Rydberg states can be reached in a simplified approach, leading
to longer lifetimes and modified interactions.

Chapter7

Additional publications

7.1. Publication: A two-species five-beam magneto-optical trap
for highly magnetic Er and Dy atoms
Physical Review A 97, 023633 (2018)a
submitted 20 Nov 2017; published 26 Feb 2018
DOI: 10.1103/PhysRevA.97.023633

P. Ilzhöfer,1,2 G. Durastante,1,2 A. Patscheider,1,2 A. Trautmann,1,2 M. J. Mark,1,2 and
F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2a

The author of the present thesis contributed to the building of the experimental apparatus together
with P. I. and G. D., performed the measurements together with P. I., A. T., and G. D., and contributed
in writing the manuscript.

Summary
In this publication, we demonstrate the realization of the first magneto-optical trap thatworks simultaneously for the two highly magnetic atoms erbium and dysprosium. For both
species, we use the narrow intercombination transitions that lead to final temperatures ofabout 10 µK, which is beneficial for the subsequent creation of dipolar quantum mixtures.
A special aspect of our magneto-optical trap is that we achieved a trapping configurationconsisting of only 5-beams, showing similar atom numbers and final temperatures comparedto the standard 6-beam approach.

113

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.

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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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J. Phys. Commun. 1, 035012 (2017).
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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.

023633-5

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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.
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trap. Again, we did not detect atoms in higher Zeeman sublevels.
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K. Zetie, and C. J. Foot, J. Phys. B 31, L321 (1998).
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023633-6

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

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

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

SummaryThis publication covers the realization of a strongly interacting dipolar Fermi gas of erbium
atoms. The atoms are prepared in a higher Zeeman level with high efficiency using a latticeprotection technique. We perform high-resolution Feshbach spectroscopy over a magnetic
field range of 200 mG for atoms initialized in the two lowest hyperfine sublevels. In addition
to several narrow intra- and inter-spin resonances, we identify a relatively broad inter-spin
resonance and characterize the inter-spin scattering length in its vicinity using lattice modulation spectroscopy. Our experiment represents the starting point for future experimentswith strongly interacting Fermi gases.

121

PHYSICAL REVIEW LETTERS 121, 093602 (2018)

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

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

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

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

PHYSICAL REVIEW LETTERS 121, 093602 (2018)
4
rf

3
J
(b)

(c)

z

(b)

2,0
1,5

lifetime (s)

(a)

2

1

atom number (x104)

(a)

1,0
0,5

0

y

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

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

30
20
10

0,0

x

40

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

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

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

093602-2

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

2
1

(a)

0
1.0
0.5
0.0
1.0

(b)

0.5

(c)
0.0

560

580

600

620

640
660
magnetic field (mG)

680

700

720

740

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

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

093602-3

scattering length a (a0)

PHYSICAL REVIEW LETTERS 121, 093602 (2018)
300

(a)

200
100
0

-100
-200

initial decay rate N/N0 (s -1)

atom number (x103)

560

.

8

580

600

620 640 660 680
magnetic field (mG)

700

720

740

a = -1500(500)a0

a = 880(140)a0

6
4
2

(c)

(b)
0

10 1

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

1.0

1.5

2.0

(d)

10 0

10-1
15

10

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

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

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

093602-4*

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

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093602-6

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

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

Spin-polarized degenerate Fermi gases

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

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

Preparation for lattice loading

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

Three-dimensional optical lattice and its loading

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

dz

2

z

x

dy

y

dx

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

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

Higher-band populations

Zeeman energy for fermionic Er

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

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

3
0

-200

differential splitting (kHz)Zeeman energy (MHz)

-100

-300

-400

0

40

∆1-∆2

30
20

∆2

10

∆1

0
0

10

5

10

20

30

40

15
20
25
magnetic field (G)

30

35

40

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

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

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

Lifetime of the spin mixture in a deep lattice

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

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

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

5
3
2

atom number (x104)

1
0
0.4
0.2
0.0
3
2
1
0
0.0

0.2

0.4

0.6

0.8

1.0
magnetic field (G)

1.2

1.4

1.6

1.8

2.0

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

1.0

0.9

U↓↑= hνres

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

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

0.8
3

4
5
6
modulation frequency ν Mod (kHz)78

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

contact interaction Uc
Uc =4πh̄a↓↑
mEr

Z

4

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

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

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

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

6.0

5.0

4.5

4.0
40

60

80

100

120

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

AR
1.68
1.52
1.41
1.34
1.28
1.32
1.45

(100)

Uc

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

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

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

Scattering-length tunability and magnetic-fieldstability.

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

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

Loss spectroscopy in the ODT at the interspin FR

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

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F. Ferlaino, Phys. Rev. Lett. 112, 010404 (2014).
[2] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Phys. Rev. X 6, 041039(2016).
[3] N. Q. Burdick, Y. Tang, and B. L. Lev, Phys. Rev. X 6,031022 (2016).
[4] S. Will, PhD Thesis, Johannes Gutenberg-Universität,
Mainz (2011).
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Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Science
352, 201 (2016).
[6] M. Köhl, H. Moritz, T. Stöferle, K. Günter, andT. Esslinger, Phys. Rev. Lett. 94, 080403 (2005).
[7] J. G. Conway and B. G. Wybourne, Phys. Rev. 130, 2325
(1963).
[8] K. F. Smith and P. J. Unsworth, Proceedings of the Physical Society 86, 1249 (1965).
[9] A. de Paz, A. Sharma, A. Chotia, E. Maréchal, J. H.
Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, andB. Laburthe-Tolra, Phys. Rev. Lett. 111, 185305 (2013).
[10] S. Lepoutre, J. Schachenmayer, L. Gabardos, B. Zhu,
B. Naylor, E. Marechal, O. Gorceix, A. M. Rey,
L. Vernac, and B. Laburthe-Tolra, ArXiv e-prints(2018), arXiv:1803.02628 [cond-mat.quant-gas].
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[12] S. Baier et al., in preparation (2018).
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C. Makrides, A. Petrov, and S. Kotochigova, Nature(London) 507, 475 (2014).
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7.3. Publication: Probing the roton excitation spectrum of astable dipolar Bose gas
Physical Review Letters 122, 183401 (2019)a
submitted 29 Nov 2018; published 8 May 2019
DOI: 10.1103/PhysRevLett.122.183401

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

2a

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

Summary
In this work, we directly probe the excitation spectrum of a dipolar Bose-Einstein condensate by means of Bragg spectroscopy, where our flexible spectroscopy setup based on adigital micro-mirror device allows us to span a wide momentum range. In the experiment,
we precisely control the contact interaction and determine the spectrum for various relative
strengths ϵdd of the dipolar interaction. As ϵdd increases, we observe an increasing deviationof the excitation spectrum from a linear phononic behavior. At the transition to a dominant dipole-dipole interaction, we observe a softening of the excitation spectrum and theemergence of a roton minimum. We compare our findings to theoretical calculations basedon mean-field Bogoliubov theory and find good agreement with our results. Interestingly,
the inclusion of beyond-mean-field corrections leads to a deviation of the theory from theexperiment, calling into question the validity of the applied description.

137

PHYSICAL REVIEW LETTERS 122, 183401 (2019)

Probing the Roton Excitation Spectrum of a Stable Dipolar Bose Gas
D. Petter,1 G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 M. J. Mark,1,2 L. Chomaz,1 and F. Ferlaino1,2,*
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria(Received 29 November 2018; published 8 May 2019)

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

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

183401-1© 2019 American Physical Society

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

Bragg beam 2

Bragg beam 1

B

l
l

y
x

2π/q
(c) 0

(b)

1

(d)

qx

n(qy )

1energy

l

0
-3.2

qy

(e)

0

3.2

qy (um-1)

6.4

‹qy2›

1

q

0

momentum

0

0.5

1

1.5

2

2.5

( z)

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

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

π 2 V 20 τS̃0 ðq; ωÞ;h2ð1Þ

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

183401-2

PHYSICAL REVIEW LETTERS 122, 183401 (2019)
dd

<1

dd

(c)

(e)

‹qy2 ›

(g)

1.5

1.5
1

/

z

2.0 (a)

>1

1.0
0.5

0.5
0
2.0 (b)

0

(d)

(f)

norm.

(h)

~
S0(q,

)

1.5

/

z

3

2

1.0

1

0.5

0

0

0.4

0.8 1.2
qlz

1.6

0

0.4

0.8 1.2
qlz

1.6 0

0.4

0.8 1.2
qlz

1.6 0

0.4

0.8 1.2
qlz

1.6

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

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

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

183401-3

PHYSICAL REVIEW LETTERS 122, 183401 (2019)
dd

>1

dd

<1

dd

2.5
25

0.7

10
0

15

0.6

0.5

20

40 60
V0 /h (Hz)

45

80

1.5

1

0.5
0.4
0.3

(arb. units)

0.8

2

rot )

1

30

20

~

res (%)zres (%)50

S0(qrot ,

1.5

0

>1

10

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

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

52

54
56
58
scattering length as (a0 )60

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

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

183401-4

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

*

Corresponding author.
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183401-6

Supplemental Material: Probing the roton excitation spectrum of a stable dipolar
Bose gas
D. Petter,1 G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 M. J. Mark,1,2 L. Chomaz,1 and F. Ferlaino1,2,∗
1

Institut für Experimentalphysik, Universität Innsbruck,
Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria(Dated: April 8, 2019)

Bragg
pulse

th

14
σy,ToF (um)

PREPARATION AND CALIBRATION OF THE
dBECs

12

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

108vat (100 um/s)

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

2
1
0
-1
-2
0

10

20

30
th(ms)

40

50

60

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

ACCOUNTING FOR THE BREATHING AND
SLOSHING MODES: TIMING AND DOPPLERSHIFTS

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

2

BRAGG SETUP

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

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

0.6
0.5
0.4
0.3

- qrotvat= 133 µm/s

- qrotvat= -9.9 µm/s

+ qrotvat= 133 µm/s

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

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

3incoming beamx

DMD
mirror area(a) Top view~25°

ydumped(e)filterdown to1 : 4 telescope

vacuumchamber(b) Side view
DMD
mirror area

z

25 um

+1st orders

qlz=1.65d
0th and -1st orders

y
(c)

whor

(d)
f = 200 mmwz

ω=dφ/dtΘφ
qlz=1.01

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

φ/2�=6/9...
qlz=0.27

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

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

4
1
0.8

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

0.6
0.4
0.2
0.8

1

1.2

qlZ

1.4

1.6

1.8

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

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

5(a)SCATTERING-LENGTH VALUE AND ITSUNCERTAINTIES

(b)

(c)

0

CONFIDENCE LEVEL OF THE EXISTENCE OF
A ROTON MINIMUM

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

-0.1

0.5

∆ (ωz )

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

ω (ωz )

0.6

-0.2

0.4
0.74 1.27
qlz

0.74 1.27
qlz

52.2 52.5
aS (a0)

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

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

THEORY

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

(4)l

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

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

Fl =

(5)
(6)

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

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

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

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

Summary
In this publication, we demonstrate, for the first time, the existence of states with supersolid properties in dipolar systems of erbium and dysprosium. In this particular regime, thequantum gas spontaneously develops a density modulation while maintaining phase coherence and therefore superfluidity. This counterintuitive quantum phase is in strong relation
to the roton softening of the excitation spectrum and a stabilization mechanism due to quantum fluctuations. In our work, we identify three parameters regimes, corresponding to thephases of a regular Bose-Einstein condensate, of the supersolid, and of an insulating droplet
array. Moreover, we find that the supersolid formed by 164 Dy isotopes has a much longer
lifetime compared to the 166 Er isotopes, which roots in the different atom loss rates. Our
work opens the door to experimental studies of the counter-intuitive supersolid phase.

153

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].

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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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LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …

(b)

-6
-4
-2
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2
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6

(c)

6
5
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3
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(e)

4
3
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(f)

2.5
2
1.5
1
0.5
0

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

kx (µm-1)

kx (µm-1)

kx (µm-1)

kx (µm-1)

25 x10

3

x10 4

(g)
20

5

x10 4

(j)

5

4

4

3

3

2

2

5

1

1

0

0

n (µm)

-6
-4
-2
0
2
4
6

(d)

15
10

14
12
10
8
6
4
2
0

(h)

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

5n (µm)

(a)

(k)

5

4

4

3
2

3
2

1

1

0
5
n (µm)ky (µm-1)

ky (µm-1)

ky (µm-1)

n (103 µm2)
-6
-4
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PHYS. REV. X 9, 021012 (2019)

(n)

0

(l)

5 (o)

4

4

3

3

2

2

1

1

0

(m)

-4 -2 0 2
ky (µm-1)

4

0-6 -4 -2 0 2 4 6ky (µm-1)

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

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

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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

021012-6

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
x103
1

A /AM

(a)
A AM

4

0
-2

-1

0

1

as - as*(a0)

2

0
0.8

(b)

S

0.6
0.4
0.2
0
-2

-1

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

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

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

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L. CHOMAZ et al.

PHYS. REV. X 9, 021012 (2019)

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

(a)
A /AM

1

A AM

4
3

0

2

0

4

8
12 16
th (ms)

1
0
12

166

x10

(b)

1
A /AM

A AM

Er

2

8

0

0

60

4

th (ms)

120

164

0
0

40

80

Dy

120

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

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

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

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
(b)

(a)

20

(e)

10 ms

(i)

(f)

50 ms

(j)

(g)

100 ms(k)(h)

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

15

150

10

0

100

(b)

0
53.5

164

Er

54 54.5 1.8
aS (a0)

1.9

2

Dy
2.1

B (G)

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

n (102 µm)

15
10
5
0

(c)

20
n (102 µm)

166

20

15
10
5
0

(d)

20
n (102 µm)

50

Evaporation time (ms)t (ms)

515
10

10 µm-1

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

02

4 6 8 10n (102 µm2)

50-15 -10 -5 0 5 10 15
kY (µm-1)

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

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

021012-9

L. CHOMAZ et al.
300

PHYS. REV. X 9, 021012 (2019)

(a)

(b)

250

10
150
100
50
0
0

1

(c)

A / AM

A , AM

200
8
6
4

0
0
50

200
100

400
150

200

5 µm-1
250

300

350

400

th (ms)20n (102 µm2)

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

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

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Filaments in the Ground State of Trapped Dipolar Bose
Gases, Phys. Rev. A 96, 013627 (2017).
[71] Y. Kora and M. Boninsegni, Patterned Supersolids in
Dipolar Bose Systems, arXiv:1902.08256.
[72] Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidity arounda Critical Point in Dipolar Bose Einstein Condensates,
arXiv:1903.06161.
Correction: The inadvertent omission of a marker indicating“Featured in Physics” has been fixed.

021012-12

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

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

GROUND STATE CALCULATIONS

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

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

Z

s

∗
ψi−1
(y)ψi (y)dy

(1)
(0)

(0)

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

!

.

(2)

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

LINK STRENGTH AND ESTIMATE OFTUNNELING RATE

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

2

TOY MODEL FOR THE INTERFERENCEPATTERN

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

4
3
2
1
0

c

n(µm)

F[n]

n(µm)

F[n]

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

x104

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

F[n]

x104

x104

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

n(µm)

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

F[n]

(3)

F[n]

+

n(µm)

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

(0)

(yi

n(µm)

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

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

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

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

3

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

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

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

F[n]

n(µm)

n(µm)

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

4c

F[n]

5b

F[n]

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

x104

x104

5a

F[n]

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

x104

n(µm)

F[n]

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

F[n]

n(µm)

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

x104

F[n]

n(µm)

x104
6c
5
4
3
2
1
0

n(µm)

n(µm)

n(µm)

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

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

F[n]

n(µm)

5
4
3
2
1
0

x104

F[n]

6a

F[n]

x104

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

n(µm)

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

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

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

IMAGING ANALYSIS: 164 Dy AND 166 Er

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

DETAILS ON THE FOURIER ANALYSIS

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

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

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

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

3

5

4

AФ

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

3

2

1

0

0

10

20

30
tr (ms)

40

50

60

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

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

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

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

e

b

d

f

2

120
AФ(103)

Scattering length (a0)

c

4

140

100
80

0
5
4
3
2
1
0

60
40
1.1

a

1.3

1.5

1.7

1.9
B (G)

2.1

2.3

2.5

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

ATOM LOSSES IN 166 Er AND 164 Dy

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

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

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

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

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

anism limiting the lifetime of the supersolid properties inDy.

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

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

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

a

c

e

b

d

f

2
1
0
12

AФ(102)tN (ms)32(5)51(9)
46(12)

N (104)3

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

8
4
0

0

50

100

th (ms)

0

50

100

th (ms)

0

50

100

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

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

tN (ms)300(12)728(34)926(36)tΦ (ms)12(5)152(13)133(25)[1] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
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7.5. Publication: Excitation spectrum of a trapped dipolar
supersolid and its experimental evidence
Physical Review Letters 123, 050402 (2019)a
submitted 3 Jul 2019; published 1 Aug 2019
DOI: 10.1103/PhysRevLett.123.050402

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

2a

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

Summary
The experimental realization of supersolid states in dipolar quantum gases has triggered
numerous questions. This work covers the study of the excitation spectrum of a dipolar
quantum gas across the phase transition from a regular Bose-Einstein condensate to a supersolid phase. Theoretical calculations reveal the appearance of two distinct excitation
branches when crossing the phase transition, which are associated with crystalline and superfluid excitations. In the experiment we probe the collective compression modes using theprinciple component analysis. While we measure a quadrupole oscillation in the regime of the
Bose-Einstein condensate, we observe two distinct oscillation frequencies in the supersolid
regime which are related to the spontaneous symmetry breaking.

175

PHYSICAL REVIEW LETTERS 123, 050402 (2019)
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Excitation Spectrum of a Trapped Dipolar Supersolid and Its Experimental Evidence
G. Natale,1 R. M. W. van Bijnen,2 A. Patscheider,1 D. Petter,1 M. J. Mark,1,2 L. Chomaz,1 and F. Ferlaino1,2,*
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
Technikerstraße 21a, 6020 Innsbruck, Austria(Received 3 July 2019; published 1 August 2019)

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

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

050402-1© 2019 American Physical Society

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

(c)

(e)

(g)

(b)

(d)

(f)

(h)

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

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

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

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

(b)

(c)

(d)

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

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

050402-3

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

(b)

(c)

(d)

(e)

(f)

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

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

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

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

050402-4

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

*

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050402-6

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

Calculation of the Bogoliubov Spectrum

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

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

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

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

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

Nin(out)

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

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

y∈Rj

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

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

0

1.6

≥3.2
C x1034

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

2

0

0

1

ky lz

2

3

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

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

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

Ns
X
p=1

wp,iC p (kx , ky ),

(3)

3

-4

ñ(kx ,ky) (arb. u.)

a

-0.1

0

0

2

0.1

b

ky (μm-1)

-2
0
2
4
-4

-2

0

2

kx (μm-1)

4

-4

-2

kx (μm-1)

4

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

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

Identifying the elementary modes of a quantum gas via the
PCA

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

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

Partial recomposition

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

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

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7.6. Publication: Bragg scattering of an ultracold dipolar gas
across the phase transition from Bose-Einstein condensate tosupersolid in the free-particle regime
Physical Review A 104, L011302 (2021)a
submitted 5 May 2020; published 22 July 2021
DOI: 10.1103/PhysRevA.104.L011302

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

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

Summary
In this publication, we present Bragg spectroscopy measurements in the high-energy regimeof a dipolar quantum gas across the transition from a ordinary Bose-Einstein condensate toa supersolid phase and finally to an incoherent droplet array. Our measurements reveal a
continuous reduction of the systems response, occurring faster than theoretically predicted.
We are able to identify coherent phase dynamics, which arise from the crossing of the phasetransition, as cause for the faster decay. In conclusion, our work confirms that Bragg spectroscopy is a sensitive probe for density modulations as well as coherent and incoherent phasevariations.

187

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

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

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

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

L011302-1©2021 American Physical Society

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

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

3

4

>5

||u||

1

(b)

104

)

1.5 (a)z(units of

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

)

2

0

1.0

-2

(

0

)

2

0.5
0
0

1
k (units of k )2 0

1
k (units of k )(c)2

SSP BEC

1

0.8

1

53.2

as (units of a0)
54
54.8

C

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

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

D. PETTER et al.

00.6 53.2

53.6

54
54.4
as (units of a0)

54.8

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

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

BRAGG SCATTERING OF AN ULTRACOLD DIPOLAR GAS …

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

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

5

ID

SSP

BEC

54.5

55 55.5 56
as (units of a0)

4
F (%)

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

PHYSICAL REVIEW A 104, L011302 (2021)

3
2
1
0
53.5

54

56.5

57

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

L011302-3

D. PETTER et al.

ny (arb. u.)

1

PHYSICAL REVIEW A 104, L011302 (2021)

(a)

0

7

(b)

1

0
(rad)

0 (c)

-8

-4

y

0

4

8

53.6
54.2
54.8
as (units of a0)0

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

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

L011302-4

BRAGG SCATTERING OF AN ULTRACOLD DIPOLAR GAS …

PHYSICAL REVIEW A 104, L011302 (2021)

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

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

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L011302-6

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

A. Preparation of the BECs

3

SSP

BEC

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

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

fside (%)

2.5
2

1.5
1

0.5
0

*
Δas,P
*
Δas,B
Δas*

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

55.2 55.4

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

B. Determination of experimental BEC-SSP phasetransition

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

dynamics.

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

x104
7

SSP

BEC

6

C. Transition from SSP to ID

x102
ID

BEC

SSP

AФ, AM

2

N

5
4
3
0
54

54.5
as (units of a0)

55

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

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

2
53

54

55
56
as (units of a0)

57

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

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

3

E. Bragg spectroscopy

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

6

x102
3

ky (μm-1 )

4
2
0

2

-2

1

-4

-5

0
kx (μm-1 )

5

-5

0
kx (μm-1 )

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

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

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

SSPωk (units of ωz )

2.5

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

2

1.5

54

55

56
as (units of a0)

57

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

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

6

F (%)

10

F (%)

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

4

0

5

2

0

40
80
V/h (Hz)

120

0
0

2

τ (ms)

4

6

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

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

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

5

1.5

(a1)

0
-4

-2

(a2)

0
2
y (μm)1(b1) ||u||

1.5

3

>5

-2
(b2)

-2

0
2
y (μm)(a4)

0
2
y (μm)

(b3)

-2
(b4)

(a5)

0
2
y (μm)-20
2
y (μm)

(b5)

1.0
0.5
0

0

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

4

(a3)

1.5

(c1) S (k, ω)

12 >2.6

momentum (kc)

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

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

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

0.5 1 1.5ky (units of kc)

2

(c5)

1.0
0.5
0

0

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5 2 0
ky (units of kc)

0.5 1 1.5ky (units of kc)

2

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

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

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

H. BdG theory for the three-dimensional trappedgas

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

6n (x103 μm-1)

0

1

2

3

4

5

10

(a)

0

1

>2

(b)

ω (unis of ωz)

3
2

5
y (μm)S (k, ω) (units of S*)

4

1

0

0

-5
-10
53.5

54
as (units of a0)

54.5

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

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

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

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

0

1
2
3
ky (units of lz)

40

1
2
3
ky (units of lz)

4

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

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

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

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

7

(a)

||u||, ||v||

10

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

freeparticles

collective1

0.1

nexc , nGS (arb. u.)

nexc , nGS (arb. u.)0

0.5
energy (units of ħωz )

1

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

0
1 (c)

0

-10

-5

0
y (μm)

5

10

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

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

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

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

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

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

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

(a)

C

1

ramp of as

holdtiming of
Bragg pulse

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

α (rad)2π (b)

0
-30

-20

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

Figure S11 (a) shows C(t) over the whole simulation
time. For early times, we observe a small, but finite C
due to density noise in the simulations, which is coming from the included thermal fluctuations. During theholding time ([−10, 7] ms), for as < a∗s , we observe that
C first increases and consecutively slightly decreases due
to atom loss. Only for as < 54.2 a0 we find an oscillating
behaviour of the contrast in time. We note that there
is a time delay between the development of the density[1] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, Phys. Rev. X 6, 041039(2016).
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A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Phys. Rev. X 9, 021012 (2019).
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Lett. 122, 183401 (2019).
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[10] D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, Phys.
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A 87, 061602(R) (2013).
[12] S. Saccani, S. Moroni, and M. Boninsegni, Phys. Rev.
Lett. 108, 175301 (2012).
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Lett. 123, 050402 (2019).
[14] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Phys.
Rev. A 74, 013623 (2006).
[15] L. Pitaevskii and S. Stringari, Bose-Einstein condensation and superfluidity, Vol. 164 (Oxford University Press,
Oxford, 2016).
[16] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
S. Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Nat. Phys. 14, 442 (2018).

9[17] F. Böttcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Langen, I. Ferrier-Barbut, T. Pfau, R. Bombı́n, J. SánchezBaena, J. Boronat, and F. Mazzanti, Phys. Rev. Research 1, 033088 (2019).

[18] A. Griffin, Excitations in a Bose-condensed Liquid (Cambridge University Press, New York, 1993).
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Appendix

A

Upgraded distribution for thelight at 532 nm

At the time, when the experiments in Chapter 3 have been performed, our green latticelaser was simultaneously used to pump the dye laser for generation of the 583-nm light. In
practice, this lead to two disadvantages: As the laser system provides maximum 10 W andhalf of the power was necessary for the dye laser, the power available for our lattice setup
was limited. Second, to cover the distance from the laser to the experiment relatively longfibers of 8 m were required. For such long fibers, Brillouin scattering plays a significant role
because it limits the maximally transmittable power. Moreover, Brillouin scattering leads
to undesired intensity noise.
Thanks to the switching to a diode laser system at 583 nm (see Chapter 2.2), we havemore power at 532 nm available and we have been able to upgrade the light distribution.
To reach higher powers for each lattice beam, within this thesis, we rebuilt the opticalsetup closer to the experimental table. The implementation of shorter high power fibers1
increased the available power for each lattice beam from maximally 1.2 W per beam to about
2.5 W per beam, limited by the maximum laser power and not by Brillouin scattering. This
improvement will allow us to realize larger lattice beams and deeper lattice depths in future
experiments; see also Chapter 6.
Figure A.1 shows a drawing of the new lattice setup. The full power of 10 W is divided into
two paths, one for each lattice beam. Both lattice beams are controlled individually using
an acousto-optic modulator (AOM) made of crystalline quartz2 to minimize thermal lensing
effects as they occur for example with TeO2 [Sim19]. The AOMs are used simultaneously asswitches as well as to stabilize the beam intensity.
The laser power gained on the experimental table allows us to make a few design consideration
for an upgrade of the optical lattice. Generally, two aspects are of major interest:(i) To observe thermalization behavior due to spin-exchange dynamics, it is important to
reduce the tunneling rate in the system. The observed exchange dynamics occur on the
12

Schaefter-Kirchhoff, PMC-E-460Si-4.0-APC.EC-300-P, 3 m length.
Gooch & Housego, I-M110-2C10B6-3-GH26.

205

206

A. Upgraded distribution for the light at 532 nm

λ/2
λ/2
AOM

f = 7.5 mm-1th order
λ/4
λ/2

Green Lattice 1

f = -70 mm
f = 100 mm

AOM+1th order
λ/4
λ/2

λ/2

λ/2

f = 6.4 mm

532 nm
10 W
Verdi-V10

Green Lattice 2

Figure A.1.: Schematic drawing of the upgraded distribution for the light at 532 nm used for the
optical lattice. The light is obtained from a 10 W-Verdi laser system and split in two paths. The
light is transmitted to the experimental table with two optical fibers with a length of 3 m. For eachlattice beam the intensity is controlled with an AOM.

timescale of several 100 ms. With a lattice depth of 60 Erec the tunnel rate corresponds
to about 0.036 Hz, which means that only after 25 s every atom has tunneled approximately once. With this lattice depth, we can ensure that only a negligible number ofatoms are tunneled during thermalization up to a few seconds.
(ii) The energy difference between neighboring lattice sites due to the harmonic confinement of the laser beams should be as small as possible to reduce inhomogeneities; see
Chapter 2.3.2. This can be reached by using large lattice beams.
It quickly becomes clear that the two requirements are strongly correlated. A deeper lattice
depth can be achieved either by a smaller laser beam or an increased power. However, bothlead to a stronger harmonic confinement. Since in our system, due to the larger lattice spacingin the third dimension (see Chapter 2.3), the dipolar exchange interactions are dominantly
occurring in the horizontal planes of our three-dimensional lattice, we can reach an larger
lattice depth combined with a lower confinement by changing the aspect ratio (AR) of thelattice beams.
Figure A.2 shows calculations of the lattice depth s and the harmonic radial confinement
νr as a function of the beam waist for AR = 1, 2, 3. Due to expected losses caused by anadditional optical isolator, we have assumed a power of 2 W. For the polarizability, we used
the theoretical value of 317 a. u. as determined in Ref. [Bec18].
As expected, a larger AR is advantageous to reach a deeper lattice depth while reducing
simultaneously the harmonic confinement along the weaker dimension of the trapping beam.
Keeping a lattice depth of 60 Erec as a minimum requirement, the harmonic confinement

207

120
100
80
60
40
20
150

175

200

225

250

275

300

250

275

300

waist (µm)
120
100
80
60
40
20
150

175

200

225

waist (µm)

Figure A.2.: Lattice depth (a) and harmonic confinement (b) as a function of the beam waist for
laser beams with AR = 1 (blue dashed line), AR = 2 (red dotted line), and AR = 3 (green solidline). For AR > 1, the beam waist corresponds to the larger dimension. The calculations are done fora laser power of 2 W and a dynamical polarizability of 317 a. u. [Bec18]. The horizontal dashed linein (a) corresponds to 60 Ered and the vertical dashed lines denote our current beam waist of about
160 µm.

could be reduced by about 25 % for AR = 2. Interestingly, the condition of AR = 3 brings
only a slight improvement compared to the AR = 2 case (≈ 5 % − 10 %).
In conclusion, for our experimental system this means that with an increased lattice beam
power of 2 W, a larger lattice depth and at the same time a reduced harmonic confinement
can be achieved for a trapping beam with AR = 2.

Appendix

B

Laser system at 1299 nm

The light at 1299 nm is generated by a diode laser from Toptica1 , based on an external cavity
diode laser and a tampered amplifier. The laser system reaches a maximum output powerof 1.5 W. This relatively high output power will allow us to provide all labs of the Dipolar
Quantum Gas-group with ultra-stable light at 1299 nm in the near future.
For frequency stabilization of the laser light, we use a commercial system from Stable Laser
Systems 2 . The non-amplified laser light of the external cavity diode laser is locked to a highfinesse cavity following the Pound-Drever-Hall technique [Bla01]. We use a fiber-coupledelectro-optical modulator3 to introduce frequency side-bands on the laser light. In thefrequency-locked configuration, the laser systems possesses a measured coherence time of
96 ms and an Allan deviation of 3.1 × 10−15 over 1 s. Note that, the coherence time as wellas the Allan deviation have been characterized by Stable Laser Systems Inc..

Cavity characteristics
To accurately determine the free spectral range (FSR) of the cavity, which describes thefrequency difference between two fundamental cavity modes, we perform two high resolution
spectroscopy measurements. For each measurement, the laser is frequency stabilized to one
of two adjacent fundamental modes of the cavity. We use the fiber-coupled EOM to overcome
the frequency offset resulting from the difference between the atomic transition to the cavity
mode. We record the frequency difference between the fundamental cavity modes and thecenter position of the transition and by summing these frequency shifts we directly obtainthe FSR as
FSR = 1.497 246 2(3) GHz.
B.1
Obviously, as the cavity contracts over time, a minimal increase of the effective FSR can be
expected.
A characteristic quantity that describes the quality of a cavity is given by the finesse F,
1
2
3

DLC TA Pro, TOPTICA Photonics AG, www.toptica.com.
Stable Laser Systems Inc., www.stablelasers.com.
Integrated Optical Phase Modulator, Jenoptik AG, www.jenoptik.com.

209

210

B. Laser system at 1299 nm

0.12
0.1
0.08
0.06
0.04
0.02
-0.1

-0.05

0

0.05

0.1

0.15

Figure B.1.: Cavity ring-down measurement to determine the photon-storage time of the high-finesse
cavity. The blue line represents the light intensity measured by a photodiode in transmission of thetcavity. We extract the photon lifetime τph from an exponential fit of the form U (t) = A + Be /τph .
The red line denotes the fit result.

which corresponds to the ratio between the Lorentzian linewidth of the cavity transmission
and the free spectral range (FSR). The Lorentzian linewidth is directly related to the photon
lifetime in the cavity, which we extract from a ring-down measurement. While the laser isfrequency stabilized, we suddenly shift the laser frequency with respect to the cavity modeand monitor the cavity transmission UPD using a photodiode. Figure B.1 shows the recorded
signal. We fit UPD with an exponential decay function to extract the photon lifetime anddetermine the finesse toF ≈ 173 000.
B.2
A special feature of this cavity is that the coating of the resonator mirrors is in addition to1299 nm also highly reflective at 1001 nm. The reason for this is that dysprosium shares asimilar transition to erbium at this wavelength [Pet20a]. Since the Er-Dy experiment within
our group works in dual species operation, it will be advantageous if a second laser systemrunning at 1001 nm can be added in the future to enable the tool of a narrow transition also
for dysprosium.

Frequency conversion for wavelength meter monitoring
To monitor the laser frequency over a wide range, for example to find the correct cavity
mode, we need to overcome the disadvantage that our wavelength meter works only in arange up to 1100 nm. This makes it impossible to directly measure the laser wavelength at
1299 nm. We resolve this problem by doubling the frequency of the laser light to 649.5 nm
using second harmonic generation using a periodically poled lithium niobate (PPLN) crystal4
with a periodicity of 12.45 µm.
To hold the crystal, we use a self-designed mounting system that directly includes a temperature stabilization based on two Peltier devices to ensure that the condition for phasematching is satisfied; see Fig. B.2. The lower of the two Peltier devices5 runs at a constant
4
5

SHRED-MB, HC Photonics Corp., www.hcphotonics.com.
Peltier Cooler Module MPADV-127-140170-S, 40 mm × 40 mm × 3.9 mm,15.4 V DC, 42.8 W.

211
SHG crystal
Crystal holder

Peltier device
Cooling plate
Peltier device
Cooling plate

Figure B.2.: Schematic image of the holder for the PPLN crystal including temperature stabilization.
The cooling plates as well as the crystal mount are done out of copper. The crystal can be clamped
down using copper sheets and M4-screws.

power of 2 W to improve the heat dissipation. The upper, smaller Peltier device6 is used
to actively stabilize the temperature of the crystal holder. The temperature of the crystal
holder is monitored via a negative temperature coefficient thermistor, that is glued into ahole of the crystal holder. Finally, operating at a temperature of about 48 °C, the frequencydoubling setup allows us to generate several 100 µW with a few mW of pump power which
is sufficiently large for the sensitivity of the wavelength meter.

6

Peltier Cooler Module MCPE1-12704AC-S, 30 mm × 30 mm × 3.65 mm,15.7 V DC, 35.2 W.

Appendix

C

Magnetic field sensitivity of theatomic state at 1299 nm

The influences of external magnetic fields on the atomic energy level and thus on the transition frequency between atomic states are two-fold. Constant magnetic fields lead to anenergy shift of the involved Zeeman levels and consequently to a variation of the absolutetransition frequency. In contrast, magnetic field noise at high frequencies, which occurs ontimescales faster than the probing time, leads to a broadening of the effectively observed
linewidth of the transition. In addition, magnetic field noise translates to decoherence inexcited state atoms. Therefore, it is important to know the behavior of atoms in magnetic
fields.
The response of the atomic state to an external magnetic field follows the same behavior
as it has it been discussed for the ground state in Chapter 2. For the bosonic isotopes,
the excited state has a total angular momentum of J ′ = 7 and therefore the energy level
splits up in 15 magnetic sublevels that range from mJ ′ = −7 to mJ ′ = 7. As described by
Eq. 2.13, the response of an energy level to an external magnetic field is proportional to themJ ′ quantum number and the gJ ′ -factor. Within this thesis, the gJ ′ -factor of the excitedstate was determined to be gJ ′ = 1.2599(2); see Sec. 4.2. A similar behavior occurs for thefermionic isotope, however, here the nuclear spin I = 7/2 needs to be accounted for, leading
to a total angular momentum quantum number F ′ = 21/2 and 21 magnetic sublevels, ranging
from mF ′ = −21/2 to mF ′ = 21/2. Figure C.1 shows the splitting of the energy levels forbosons and fermions at low magnetic fields. Note that, for the fermionic case the quadratic
contribution that arises from the coupling of the nuclear spin to the angular momentum is
not included.
For various experimental conditions and to exploit the full capacity of the narrow transition it is important that the frequency broadening of the latter, induced by magnetic field
fluctuations, is smaller than any other relevant energy scale of the system. As an example,
for an efficient addressing of the mJ = −6 → mJ ′ = −7 transition, the broadening of thetransition frequency needs to be smaller than the energy difference to neighboring sublevels.
For example, as shown in Fig. C.2, the energy differences ∆E(mJ , m′J ) and ∆E(mJ ′ , m′J ′ )need to be larger than the transition linewidth ∆ν. The sensitivity of the observed transition
linewidth to external magnetic field noise depends on the starting and final mJ (mF )-levelsof the involved states and therefore on the corresponding transition being used. To fullfill
213

214

C. Magnetic field sensitivity of the atomic state at 1299 nm(a)(b) 150

150
-7

100

7

100

50

50

0

0

-50

-50

-100

-100

-150

-21/2

21/2

-150
0

2

4

6

8

10

0

2

4

6

8

10

Figure C.1.: Zeeman splitting for the magnetic sublevels of the excited state at 1299 nm for (a)bosons and (b) fermions in the range from 0 G to 10 G. The energy shift corresponds to (a) EZB =mJ ′ × 1.7634(3) MHz/G and (b) EZF = mF ′ × 1.1756(2) MHz/G.

the conservation of angular momentum, only three transitions are allowed, i. e. , σ − -, π-, or
σ + -transitions that corresponds to an angular momentum difference of mJ − mJ ′ = 0, ±1(see Fig. C.3(a)).
Figure C.3(b) shows the transition broadening for the case of bosonic erbium as a functionof the starting mJ state for a magnetic field noise of 1 mG, which is the current magnetic
field stability in our experiment without active field stabilization. The largest broadening ofabout 2.5 kHz can be observed for the σ − - and σ + -transitions at the edge-levels mJ = −6 andmJ = 6, respectively. In contrast, the transition mJ = 0 → mJ ′ = 0 is insensitive to magnetic
field noise and therefore ideal for high-precision applications. Note that, the insensitivity tomagnetic field noise goes along with a sacrifice of the magnetic dipole moment. Therefore,
depending on the specific application, a compromise must be found between the magnetica)-3

mJ = -6

0

b)

-5
-6
-7

-5

-4

-3

-2

-1

0

∆E (mJ’’ , mJ’’’ )∆E (mJ’ , mJ’’ )Energy-6-4

EnergymJ’ = -7

-5

-2

-1

∆E (mJ , mJ’ )-5-6

Figure C.2.: (a) Zeeman sublevels with mJ ≤ 0 for the ground state and the excited state. (b)Relevant energy differences for the efficient addressing of magnetic sublevels.

215
a)

b)

c)

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0
-6 -4 -2

0

2

4

6

-19/2 -9/2

0

9/2

19/2

Figure C.3.: (a) Schematic illustration of the σ − -, π-, and σ + -transitions. (b)-(c) Broadening of
the transition linewidth assuming a magnetic field noise of 1 mG in dependence of the starting statefor (b) bosons and (c) fermions. The broadening is shown for the π- (red circles), the σ − - (green
squares), and the σ + -transition (blue diamonds), respectively.

moment and the sensitivity to magnetic field noise. In the fermionic case (see Fig. C.3(c)),
interesting conditions appear for σ − - ( σ + -)transitions at mF = −17/2 and mF = −15/2(mF = 17/2 and mF = 15/2) and for π-transitions at mF = ±1/2. This has the advantage thata large magnetic moment can be combined with a high spectroscopic precision.

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Acknowledgments
It has been a great pleasure for me to be part of the ERBIUM team for the last 5 years.
Of course, all the work that led to this thesis was only possible with a broad and strong
support from all sides, both from a scientific and a social background. At this point, I would
like to thank everyone I met over the years and everyone who has contributed in one way or
another to the success of this thesis.
From a scientific point of view, I would first like to thank Francesca Ferlaino for giving me
the opportunity to do my PhD in her group. Thanks for all the support in so many ways,
from preparing talks to writing publications. I am also grateful that she enabled me to visit
Boulder for two months, which was a special experience for me.
Next, I would like to thank the more senior“ members that have been (and partially are”still) part of our group. First, Manfred Mark, who has always been a big help in answeringscientific and technical questions. Second, I would like to thank our long-time Post-Doc
Lauriane Chomaz. Beside being a very inspiring scientist, she has always supported us inevery project. I will always remember the days when we took measurements together andlistened to the French Fip-Radio. Third, it was a great pleasure to work with Bing Yang,especially on the project about the transition at 1299 nm, and to discuss about Chinese food.
All three of them, Manfred, Lauriane, and Bing have always been able to help us build abigger scientific picture.
Of course, a very special thanks goes to my lab-colleagues Daniel Petter and Gabriele Natale.
I think we can really call ourselves a team and not just a group. We worked well together
at all times and kept a positive spirit, even in more challenging and stressful times. It wasa particular pleasure to enjoy and share the same taste of great music with Daniel and tohave the right songs for every situation. Also, I will always remember the many (in-depth)
discussions with Gabriele about food, bikes, and most recently Covid-19. The sunrise hikeof the three of us to the Nockspitze has gone down in the history books.
Although our overlap in the lab was only short, I would like to thank Simon Baier for his
contributions to the projects. Many of my projects have benefited from the great work Simonhas done during the years. A thanks for a great time also goes to all the interns and students
we have had in our lab over the years.
Obviously, a big thanks goes to all our collaborators who contributed to the theoreticalunderstanding of our experimental results. Without your input, our work would have beenmuch more challenging/impossible. I would like to thank here Ana-Maria Rey, Bihui Zhu,
Maxence Lepers, Gohar Hovannesyan, John Bohn, and Reuben Wang. Thank you for always
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being very patient and for providing us with a large number of simulations.
I would also like to thank the members of the ErDy and the T-Reqs experiments, as wellas our theory subdivision for their great collaborations and entertaining lunches. A special
thank goes to Arno Trautmann and Philipp Ilzhöfer. In particular, Philipp played a very
important role as an excellent mentor during my master thesis and provided me with hisgreat technical expertise.
Next, I want to thank all the surrounding labs of the Ultracold -collaboration consisting thegroups of Hanns-Christoph Nägerl and Rudi Grimm. As a representative for the Ultracold collaboration I would like to mention Emil Kirilov, who was always very helpful with his
profound physical and technical knowledge.
As I mentioned earlier, I had the great opportunity to spent two months at JILA, in Jun
Ye’s group. Besides Francesca, who made this possible for me, I would of course like tothank Jun Ye and his group for welcoming me with open arms. It was a great time, thankyou Jun, Christian, Will, Ross, John, Aki, . . .
An important role in our group is taken by our administrative staff Silvia and Leonarda.
They were always a big help when it came to organize traveling or to order equipment.
Recently, Silvia also regularly demonstrated her cake baking skills to us. Further, I would
like to thank our mechanical and electronic workshops at university and in particular at
IQOQI for their support. Thanks also to Andreas for the occasional bike rides. You are still
responsible for my best time to Gries im Sellrain.
A special thank you goes to my long-time study colleagues Roman and Paul. We had many
entertaining evenings together and a lot of fun on our regular bike tours.
Finally and most importantly, I would like to thank my family: my parents, my grandparents
and my girlfriend. They have all supported me to the best of their efforts at all time andwithout them I would not have been able to complete this thesis successfully. Above all, mygirlfriend Barbara made sure that I never lost sight of other important things despite all the
excitement for the lab work. I am looking forward to our next adventure(s) together!
Thanks a lot to all of you!

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

Datum

Unterschrift