Ultracold Dipolar Erbium Atoms:
From Scattering Phenomena to
Quantum Simulations
DISSERTATION

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

advisors:
Univ.-Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences
Univ.-Prof. Dr. Gerhard Kirchmair,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences

Innsbruck, May 2018

Abstract
Remarkable progress in the field of experimental quantum physics has enabled the preparation of quantum systems across various experimental platforms. The ability to precisely
control and manipulate quantum states by experimental means, allows to study fundamental
laws of quantum mechanics and their impact at the many-body level.
Ultracold atomic gases are a powerful and flexible platform, providing precise control of key
parameters, such as temperature, density, internal and external degrees of freedom, dimensionality, or the trapping geometry. With the access of controllable interparticle interaction,a plethora of fascinating quantum phenomena has been observed. While the field was pioneered along studies with short-range contact interaction, nowadays dipolar interactions,featuring a long-range and anisotropic character, are attracting a large attention within thecommunity.
This thesis reports on the investigation of quantum phenomena emerging from dipolar interactions. As a workhorse for our studies, we use ultracold gases of strongly magnetic erbium
atoms. Erbium has first been Bose-Einstein condensed in 2012 in our laboratory. Shortly
after, we created the first degenerate Fermi gas of erbium. This thesis focuses on the use ofboth systems as a resource to investigate dipolar quantum phenomena from the few- to the
many-body level.
With dipolar fermions, we unveil the universal character of ultracold dipolar scattering,enabling a unique path towards quantum degenerate identical fermions. We further observea peculiar dependence of the total elastic scattering cross section on the dipole orientation.
The few-body collisional physics also impacts the behavior of the system at the many-bodylevel. Reporting on the first observation of a many-body effect in a dipolar Fermi gas, wedemonstrate the deformation of the Fermi surface. With bosonic particles, we investigate
the origin of a strong level repulsion in Feshbach spectra of magnetic lanthanides and trace
it back to the anisotropic van der Waals interaction among the atoms. Utilizing Feshbachresonances, we report on the first production of dipolar Feshbach molecules and reveal auniversal behavior of the stabilization of inelastic losses in reduced dimensions by dipolarinteractions.
As a major step towards strongly correlated dipolar systems, we investigate the system’s
behavior in a three-dimensional optical lattice. In particular, we report on the realization ofextended Hubbard models with dipolar bosonic and fermionic atoms. In bosonic systems, we
directly observe nearest-neighbor interactions activated by the long-range dipolar interaction.
We demonstrate the strengthening or weakening of the Mott insulator quantum many-bodyphase via solely changing the dipole orientation. For the fermionic counterpart, we add thespin-degree of freedom, giving rise to a large spin-19/2 system. A lattice protection techniqueallows to investigate in detail the elastic collisional properties of a two-state mixture. With
our method, we realize for the first time a strongly interacting dipolar Fermi gas.
The successful preparation of extended spinor Fermi Hubbard systems brings excitingprospects for future investigations at the interface with solid state physics. Offsite terms
emerging from dipolar interactions give rise to clustered states, exotic lattice spin models,
resonant demagnetization dynamics, or exotic quantum phases.
iContents
Abstracti

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

1
1
3
5

2. Ultracold erbium atoms
2.1. Erbium - a successful element . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Atomic and magnetic properties of erbium . . . . . . . . . . . . . . . . . . . .
2.2.1. Electron configuration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2. Atomic energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4. Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Laser cooling and trapping of erbium . . . . . . . . . . . . . . . . . . .
2.4. Degenerate gases of Erbium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1. Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2. Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
9
9
9
11
12
14
15
16
17
17
18

3. Few- and many-body scattering of dipolar atoms
3.1. Dipolar few-body scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2. Ultracold scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3. Universal dipolar scattering . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4. Anisotropic dipolar scattering . . . . . . . . . . . . . . . . . . . . . . .
3.2. Many-body effects in dipolar Fermi gases . . . . . . . . . . . . . . . . . . . .
3.2.1. Hartree-Fock theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2. Real and momentum space radii of dipolar atoms . . . . . . . . . . . .
3.3. Publication: Reaching Fermi Degeneracy via Universal Dipolar Scattering . .
3.4. Publication: Anisotropic Relaxation Dynamics in a Dipolar Fermi Gas Driven
Out of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Publication: Observation of Fermi surface deformation in a dipolar quantum
gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
21
21
22
23
24
27
28
29
30

4. Resonantly interacting lanthanide quantum gases
4.1. Quantum chaos in lanthanides . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1. Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50
51
51

ii

36
43

4.1.2. Feshbach spectrum of lanthanides . . . . . . . . . . . . . . . . . . . .
4.1.3. Chaotic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4. Interaction tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Dipolar Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Molecule formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2. Polar and dipolar molecules . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Publication: Emergence of Chaotic Scattering in Ultracold Er and Dy . . . .
4.4. Publication: Ultracold Dipolar Molecules Composed of Strongly Magnetic
Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54
56
57
58
58
59
61
75

5. A quantum simulator of extended Hubbard models
85
5.1. Periodic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.1. Bloch waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2. Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.3. Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2. Extended Hubbard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1. The Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.2. Extended Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . 92
5.2.3. Extended spinor Fermi-Hubbard model . . . . . . . . . . . . . . . . . 96
5.3. Quantum phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3.1. Superfluid-to-Mott insulator transition . . . . . . . . . . . . . . . . . . 99
5.3.2. Metal to band and Mott insulator transition . . . . . . . . . . . . . . 100
5.4. Extended Hubbard models in experiments with dipolar particles . . . . . . . 101
5.4.1. Lattice modulation spectroscopy . . . . . . . . . . . . . . . . . . . . . 101
5.4.2. A dipolar bosonic quantum simulator . . . . . . . . . . . . . . . . . . 102
5.4.3. A dipolar fermionic quantum simulator . . . . . . . . . . . . . . . . . 103
5.5. Publication: Extended Bose-Hubbard models with ultracold magnetic atoms 104
5.6. Publication: Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms 118
6. Conclusion and outlook
130
6.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2. Future investigations along the extended Fermi Hubbard model . . . . . . . . 131
A. Additional publications
134
A.1. Publication: Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein
Condensate to a Macrodroplet in a Dipolar Quantum Fluid . . . . . . . . . . 134
A.2. Publication: Anisotropic polarizability of erbium atoms . . . . . . . . . . . . 152
A.3. Publication: Observation of roton mode population in a dipolar quantum gas 160
B. Optical setup improvements

175

C. Magnetic field stabilization

180

Bibliography

184

Acknowledgments

203

iii

Chapter1

Introduction

1.1. Motivation
The advent of quantum theory at the beginning of the twentieth century has set the foundation for our current understanding of the microscopic world. Remarkably, already in theearly days the theory succeeded in capturing experimental observations such as black-bodyradiation [Pla06], or the photoelectric effect [Ein05]. However, the description of strongly
correlated quantum systems remains fairly difficult, already when they consist only of a
relatively small number of particles. Fascinating quantum phenomena emerging in such systems include low-temperature transport, quantum magnetism or most prominently high temperature superconductivity. While these phenomena have been studied extensively withincondensed matter physics, a comprehensive description is still lacking.
A promising approach to understand such phenomena can be undertaken via an experimental
bottom-up method. By forming a model system, where quantum particles are added oneby-one, a quantum simulator can be realized, allowing to reveal the properties of a complex
quantum system via the well-understood and well controlled model system. This approach, as
proposed by Richard Feynman [Fey82], requires the experimental control of single quantum
states, a scenario that was formerly thought to be inaccessible.
However, technological developments have made remarkable progress and nowadays coherent
control of single quantum states has been realized across many experimental platforms.
Indeed, experiments allow for the control and manipulation of single quantum particles suchas photons, ions in ion traps, atoms in dipole traps, electrons and nuclei in solid-state systems
such as quantum dots or vacancy centers, but also microwave photons in superconducting
circuits, see Refs. [Lad10, Tra12] for an overview.
Each experimental platform has different advantages and disadvantages and active research iscarried out along distinct lines. A striking advantage of trapped atomic systems, the experimental platform of choice within this thesis, is given by the fact that the used particles are intrinsically identical. This enables the preparation of large uniform quantum systems without
suffering from imperfections as often encountered in solid-state systems. An additional advantage arises from the accessibility of both fundamental classes of quantum particles, namely
11.1. MOTIVATION

2bosons and fermions. For reaching the quantum regime with this platform ultralow temperatures are required. In particular, the thermal de-Broglie wavelength of the atoms has toexceed the interparticle distance. In 1995 a major breakthrough within the field was achieved
with the first production of a Bose-Einstein condensate (BEC) [And95, Bra95, Dav95], anovel quantum state of matter predicted already 70 years before [Bos24, Ein24].
Since then, a leading research has focused on quantum effects that arise from the macroscopic
coherence of the interacting many-body system, such as matter-wave interference [And97],
the emergence of quantized vortices [Mat99, Mad00], or the creation of a continuous and wellcollimated atomic laser [Hag99, Blo99]. With the successful realization of a degenerate Fermi
gas (dFg) in 1999 [DeM99], the second fundamental class of quantum particles has become
available in experiments. In combination with the observation of Feshbach resonances, whichallow to change the strength and sign of the interparticle interaction, see Ref. [Chi10] for areview, the door was opened for experimental studies with tunable contact interactions. Fora a two-component Fermi gas the tuning from repulsive to attractive interactions acrossthe resonance has enabled studies of the well known BEC-to-BCS crossover, see Ref. [Ing07]
for an overview. From peculiar interest has been the unitary regime, i. e. the region in thevicinity of the Feshbach resonance, which has been shown to be closely related to the physics
of neutron matter in the crusts of neutron stars [Gez08]. Such relations can turn the ultracold
atomic system into a model system of complex quantum matter, as discussed above.
The ability of using ultracold atomic systems as quantum simulators, has further a strong
foundation in a pioneering theoretical proposal of the late 1990s. Here, the experimental
creation of a three-dimensional atomic crystal has been proposed via the use of a threedimensional optical lattice [Jak98]. Remarkably, the precise control of the optical potentials in combination with tunable atom-atom interaction allows for the preparation anddetailed study of condensed matter system Hamiltonians such as the Hubbard model, rendering the ultracold atomic system into an analog quantum simulator. The groundbreaking experimental demonstration of the superfluid-to-Mott insulator quantum many-bodyphase transition [Gre02], set the start to a large number of experimental studies along thislines. Those include the realization of Fermi-Hubbard models [Jör08], the study of orbital
physics [Mül07, Wir10], the implementation of superexchange couplings [Tro08], the exploration of sophisticated band structures [Tar12], or the realization of lattice spin models[Kra12]. With the experimental advance of single-site resolution for bosonic and fermionicsystems [She10, Bak10, Omr15, Gre16, Che16] via quantum gas microscopes, ultracold atom
experiments have reached a new era for quantum simulation applications, as nicely accounted
for by the spectacular recent experimental preparation of an antiferromagnetic ground statewith long-range order [Maz17b].
A natural requirement for the study of correlated states are interactions. Beside the availability of contact interactions, which have been the working horse for almost all the investigations
discussed above, a new type of interaction has become available in recent years, namely the
dipole-dipole interaction (DDI). With its long-range and anisotropic nature it allows to accessunique phenomena and quantum phases [Bar08, Lah09, Bar12]. This type of interaction canbe found in magnetic atoms, ground-state heteronuclear molecules or Rydberg atoms. An
important advantage of magnetic atoms lies in the experimental “simplicity”, as the dipolarinteraction is intrinsically carried by the atoms, without the need of molecule creation or1.2. THESIS OVERVIEW

3

Rydberg excitations.
The first magnetic atom brought to quantum degeneracy has been chromium [Gri05, Nay15].
It allowed for the observation of dipolar effects such as demagnetization cooling [Fat06],
a characteristic d-wave collapse [Lah08], or spontanous demagnetization at ultralow magnetic fields driven by the DDI [Pas11b]. The realization of even stronger atomic dipolesin the ultracold regime, as dysprosium [Lu11, Lu12] by the group of Benjamin Lev, anderbium [Aik12, Aik14] by our group in Innsbruck, has set the start to the strongly magnetic
era. The strong dipolar character of erbium and dysprosium has already led to a fair amountof observations of dipolar effects, among which several will be reported within the present
thesis.
Importantly, the long-range nature of the DDI gives also promising prospects in the context of quantum simulations of lattice models. Strikingly, nearest-neighbor interactions
emerge purely from offsite interactions and exotic quantum phases with long-range correlations become accessible, see e. g. Ref. [CS10]. Initial studies along this lines have already been performed with lattice confined samples of polar molecules [Yan13], Rydberg
atoms [Sch12a, Ber17], and dipolar atoms [dP13b, Bai16], as will also be reported within
this thesis. Those promising experimental investigations allow to envision a bright future forthe research direction of systems with DDI.

1.2. Thesis overview
This thesis focuses on the study of dipolar quantum effects by utilizing the experimental
platform of strongly magnetic ultracold erbium atoms. It reports on complementary studiesof the scattering properties of dipolar particles and resonantly interacting lanthanides. A
main focus lies on the experimental realization of an analog quantum simulator with dipolar
particles. The main scientific achievements of this thesis can be divided into three main
sections:(I) The production of the first degenerate Fermi gas of erbium, which enables clean observations of dipolar effects on the few- and many-body level.
(II) The study of the emergence of chaotic scattering in lanthanides with anisotropic interactions and the creation of the first strongly dipolar Feshbach molecules.
(III) The realization of extended Hubbard models by means of a three-dimensional opticallattice, further allowing for the production of the first strongly interacting dipolar
Fermi gas.
On a personal note, I have joined the Erbium team already in 2011 during my master studies.
I have contributed to the first realization of a BEC of erbium and my master thesis project,in which I have build an optical dipole trap with dynamically tunable trapping volume, has

1.2. THESIS OVERVIEW

4

served to improve the atom number of our BEC by a factor of three [Bai12a]. As a PhD
student, I have valuably contributed to the above mentioned scientific achievements. In thefirst half of my PhD work, the experiment was in the building and understanding phase.
The second part of my PhD work started with the implementation of a three-dimensional
optical lattice to the experiment. The ability to pin the atomic dipoles within a crystalline
structure has transformed our experiment into an analog quantum simulator of extended
Hubbard models. The study of strongly correlated dipolar particles in the optical lattice was
the major goal of my PhD work, and thus covers a main part within this thesis.
Additional scientific achievements, to which I have actively participated but that are not thecentral focus of this thesis, are covered in the Appendix.

Thesis outline
This thesis is based on seven scientific publications, which are grouped in thematic chapters.
The main chapters are arranged such that the basic concepts and background information
are given prior to the presentation of the publications.
Chapter 2 reviews the state-of-the-art knowledge of the properties of erbium and gives details
on our experimental setup. It further summarizes the production of degenerate erbium
quantum gases.
Chapter 3 is dedicated to degenerate dipolar Fermi gases. It describes the physics of universal dipolar interactions, discusses the angular dependence of elastic scattering processesof atomic dipoles and investigates on a many-body effects of dipolar Fermi gases. The publication on the production of deeply degenerate Fermi gases via universal dipolar scatteringand the observation of the Fermi surface deformation is also part of the PhD thesis of Albert
Frisch [Fri14a].
Chapter 4 focuses on the Feshbach spectrum of magnetic lanthanides and reports on theemergence of chaotic scattering. A detailed study of close to treshold molecular bound statesallows to identify the underlying molecular quantum numbers and enables the production ofstrongly magnetic Feshbach molecules.
Chapter 5 contains the study of strongly correlated dipolar quantum systems in threedimensional optical lattices. The DDI gives rise to extended Hubbard models that arestudied in detail. With the access of higher spin states the creation of a strongly interacting
dipolar Fermi gas is demonstrated.
Finally, Chapter 6 closes the thesis and briefly discusses a possible route for the Erbium
experiment in the future.
Important experimental upgrades, implemented during the course of this thesis, are describedin the Appendices B and C, including the optical setup of the three-dimensional optical lattice
and the setup for active magnetic field stabilization.

1.3. LIST OF PUBLICATIONS

5

1.3. List of publications
The publications discussed in this thesis are given in chronological order. The thematic
chapters for the publications are indicated. Additional publications are covered in Appendix
A.
(I) Few- and many-body scattering of dipolar fermions (Chapter 3)• Reaching Fermi Degeneracy via Universal Dipolar Scattering.
Kiyotaka Aikawa, Albert Frisch, Michael Mark, Simon Baier, Rudolf Grimm, andFrancesca Ferlaino,
Physical Review Letters 112, 010404 (2014).
• Observation of Fermi surface deformation in a dipolar quantum gas.
Kiyotaka Aikawa, Simon Baier, Albert Frisch, Michael Mark, Cornelis Ravensbergen,and Francesca Ferlaino,
Science 345, 6203 (2014).
• Anisotropic Relaxation Dynamics in a Dipolar Fermi Gas Driven Out of Equilibrium.
Kiyotaka Aikawa, Albert Frisch, Michael Mark, Simon Baier, Rudolf Grimm, John L.
Bohn, Deborah S. Jin, George M. Bruun and Francesca Ferlaino,
Physical Review Letters 113, 263201 (2014).

(II) Resonantly interacting lanthanide quantum gases (Chapter 4)• Ultracold Dipolar Molecules Composed of Strongly Magnetic Atoms.
Albert Frisch, Michael Mark, Kiyotaka Aikawa, Simon Baier, Rudolf Grimm, Alexander Petrov, Svetlana Kotochigova, Goulven Quéméner, Maxcence Lepers, Olivier Dulieu,
and Francesca Ferlaino,
Physical Review Letters 115, 203201 (2015).
• Emergence of Chaotic Scattering in Ultracold Er and Dy.
Thomas Maier, Holger Kadau, M. Schmitt, Matthias Wenzel, Igor Ferrier-Barbut,
Tilman Pfau, Albert Frisch, Simon Baier, Kiyotaka Aikawa, Lauriane Chomaz, Manfred J. Mark, Francesca Ferlaino, Constantinos Makrides Eite Tiesinga, Alexander
Petrov, and Svetlana Kotochigova,
Physical Review X 5, 041029 (2015).

(III) Dipolar interactions in optical lattices (Chapter 5)• Extended Bose-Hubbard models with ultracold magnetic atoms.
Simon Baier, Manfred J. Mark, Daniel Petter, Kiyotaka Aikawa, Lauriane Chomaz, Zi
Cai, Mikhail Baranov, Peter Zoller, and Francesca Ferlaino,
Science 352, 6282 (2016).

1.3. LIST OF PUBLICATIONS

6• Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms.
Simon Baier, Daniel Petter, Jan Hendrik Becher, Alexander Patscheider, Gabriele Natale, Lauriane Chomaz, Manfred J. Mark, and Francesca Ferlaino,
submitted for publication in Physical Review Letters,
online on arXiv [cond-mat.quant-gas] 1803.11445 (2018).

Additional publications• Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate to a
Macrodroplet in a Dipolar Quantum Fluid.
Lauriane Chomaz, Simon Baier, Daniel Petter, Manfred J. Mark, Falk Wächtler, Luis
Santos, and Francesca Ferlaino,
Physical Review X 6, 041039 (2016).
• Anisotropic polarizability of erbium atoms.
Jan Hendrik Becher⋆ , Simon Baier⋆ , Kiyotaka Aikawa, Maxence Lepers, Jean-François
Wyart, Olivier Dulieu, and Francesca Ferlaino
Phys. Rev. A 97, 012509 (2018).
⋆ equal contribution• Observation of roton mode population in a dipolar quantum gas.
Lauriane Chomaz, Rick M. W. van Bijnen, Daniel Petter, Giulia Faraoni, Simon Baier,Jan Hendrik Becher, Manfred J. Mark, Falk Wächtler, Luis Santos, and Francesca
Ferlaino,
Nature Physics 14, 442 (2018).

Chapter2

Ultracold erbium atoms

The realization of ultracold quantum gases demands a number of advanced experimental
techniques. For instance, ultracold experiments require ultra-high vacuum chambers to minimize scattering with the background gas, dedicated laser sources for laser cooling and trapping of the specific atomic species, high control on magnetic fields, specially designed laser
traps for the manipulation of the geometry of the confinement, and high-resolution cameras
to visualize the atomic sample. To address a particular atomic species, it is from prime
importance to know the atomic properties, such as the energy level spectrum or the detailed
physical properties of the atomic ground state.
When in 2009 our group started to plan an ultracold experiment with erbium, very little was
known about the properties of this atomic species. While proof-of-principle experiments in
NIST showed that erbium atoms can be cooled via laser light [McC06], it still was not clear if
the quantum degenerate regime can be reached. The successful realization of the first erbium
Bose-Einstein condensate (BEC) in 2012 within our group [Aik12] gave a striking answer tothis open question. This thesis connects to this worlds first and reports on the following
scientific journey undertaken at the Erbium experiment. As to set the stage, within thischapter we will summarize the up-to-date knowledge of the main properties of atomic erbium,
see Sec. 2.2, describe our experimental setup, see Sec. 2.3, and follow the route of erbium tothe ultracold regime, see Sec. 2.4.

2.1. Erbium - a successful element
Erbium, a rare-earth element, is part of the lanthanide series in the periodic table and has
the atomic number 68. In its pure solid form it is a soft, silvery metal and has nowadays
several applications. In an alloy it can enhance the workability of metals by lowering theirhardness, and as an oxide it allows to give a pink color to glass products [Gup05].
For the major technological application, erbium is used in the form of ions (Er3+ ). They
offer an optical transition around 1.55 µm, a wavelength that is most commonly used fortelecommunication as it exhibits minimum loss in optical fibers. When used as a dopant infused silica fibers, erbium ions enable amplification of this important wavelength regime via
7(a)6.98µB

2.1. ERBIUM - A SUCCESSFUL ELEMENT

La Ce Pr Nd Pm Sa Eu Gd Tb Dy Ho Er Tm Yb Lu

8
(b)

162

Er

0.14%

164

Er

1.61%

166

Er

167

Er

168

Er

170

Er

33.6%
23.0%
26.8%
15.0%

Figure 2.1.: Magnetic moments µ for atoms of the lanthanide series and isotope abundance of erbium.
(a) The strength of the magnetic character is indicated by the bar length. The magnitude of erbium’s
magnetic moment µ = 6.98 µB is highlighted in red. Laser cooling and trapping of lanthanide atoms
has been demonstrated for ytterbium [Hon99], erbium [McC06], dysprosium [Lu10], thulium [Suk10],
holmium [Mia14], and most recently for metastable europium [Ino18]. (b) Erbium offers six stable
isotopes - five bosonic (blue) and one fermionic (orange) isotope(s). Their natural abundances areindicated. Isotopes for which quantum degeneracy has been reached are highlighted.

optical pumping and have led to the development of erbium-doped fiber amplifiers (EDFA).
The large success of EDFA is nicely accounted for by their wide use within optical communication, where the low required maintenance even allows for in-line amplification in submarine(undersea) telecom cables [Auz99]. Beside the good photoluminescence properties, a very
recent experiment shows that also electroluminescence of erbium ions can be accessed via
quantum dot arrays [Zha18]. This technological advance would ease optical interconnection
application via chip-based near-infrared light generation and amplification.
Further, the key abilities of EDFA have reached research on quantum information and communication applications. Here, single photon protocols at telecom wavelengths are highly
recommended due to the low absorption loss, which makes Er3+ -based devices a natural
choice. Technical limitations have hindered such a fiber-based quantum device, but a recent experiment succeeded in demonstrating single-photon based light-matter interaction incryogenically cooled erbium-doped fibers [Sag15]. The demonstration of quantum storage
and recall of entangled quantum states of light gives exciting prospects for the realizationof a fully fiber-based future quantum network. The strength of erbium based devices fortelecom quantum applications is further highlighted by the achievement of an over a second
coherence time in 167 Er3+ doped Yttrium Orthosilicate crystals [Ran17].
Finally, erbium features remarkable properties in its pure atomic form, as used within thisthesis. Atoms of the lanthanide series host among the strongest magnetic moments foundin the periodic table. Figure 2.1(a) reports their strength and visualizes that the magnetic
moment1 of erbium ranges at a front position. To reveal this strongly magnetic character
it is necessary to enter into the ultracold regime. Here, the dipolar interaction between
the atoms becomes strong enough to unveil exotic and unobserved quantum phases. An
additional argument that marks erbium as an ideal choice for ultracold experiments is related
to its five bosonic and one fermionic isotope(s) with high natural abundance, as shown in
Fig. 2.1(b). This feature strongly increases the flexibility of accessible physics in experiments.
1

The magnetic moment of an element is usually given in units of the Bohr magneton µB , which wasintroduced as a physical constant to express the magnitude of the intrinsic magnetic moment of an electron.
The magnetic moment of alkali-metal atoms, which are widely used for ultracold experiments, is 1µB .

2.2. ATOMIC AND MAGNETIC PROPERTIES OF ERBIUM

9

The strong dipole-dipole interaction (DDI) of erbium and the ability to address different
quantum statistics by the choice of the isotope are at the heart of the results presented inthis thesis.

2.2. Atomic and magnetic properties of erbium
2.2.1. Electron configuration
Erbium atoms host in total 68 electrons that fill the electronic orbitals following the Madelungrule. As the 6s shell becomes filled before the 4f orbital, this inner shell is left partially
unfilled, resulting in a so called submerged-shell structure. The electron configuration of theground state is written as[Xe]4f 12 6s2 ,where [Xe] denotes the electron configuration of xenon. The two vacant electrons of the 4f
orbital give rise to large orbital momentum and spin quantum numbers of L = 5 and S = 1.
The ground state of erbium can be written in Russel-Saunders coupling scheme2 :ground state: 3 H6 ,with the standard notation of 2S+1 LJ . The unusually large spin-orbit coupling in erbium
gives rise to a large magnetic moment, see below. Further, the strongly anisotropic orbitalsof the 4f shell result in a highly anisotropic van der Waals interaction potential, leading to
intriguing scattering properties as discussed in Chapter 4.

2.2.2. Atomic energy spectrum
As a result of the submerged-shell electron configuration, erbium exhibits a complex energy
level structure. Reference [Kra18], which is based on the dataset of [Mar78], reports 672
atomic lines with angular momentum quantum numbers J ∈ [1, 12]. The ionization limit of
erbium is 49262 cm−1 , which corresponds to the energy of a photon with wavelength 203 nm.
More recent theoretical work predicts yet unobserved lines for the dipole allowed transitionsof the ground state, see Ref. [Lep14] and Appendix A.2, which underlines the developing
but still not complete knowledge on erbium spectroscopic data. In Fig. 2.2(a) we plot theatomic energy spectrum up to the strongest transition in erbium, which features an energyof 24943 cm−1 (≈ 401 nm).
For the purpose of laser cooling and trapping it is crucial to precisely know the transition probabilities of the excited levels. Extensive experimental data can be found in
Refs. [Law10, Har10]. In general, a large amount of excited states is unfavorable for laser
cooling applications, as the probability of pumping atoms to long-lived metastable states
2This coupling scheme is also known as LS-coupling and in the case of erbium is only applicable for the
ground state. For excited states the electronic spin-orbit interaction becomes more prominent then theindividual spin-spin and orbit-orbit interactions. Hence jj-coupling has to be used.

2.2. ATOMIC AND MAGNETIC PROPERTIES OF ERBIUM(a)6s 6p 1P1

25

401 nm light for cooling and imaging

10
F

(b)

∆HFS

x10
11/2 5511 MHz
9/2

5442 MHz

energy (103 cm-1)

20

13/2 5208 MHz

6s 6p 3P1

15532 nm light for trapping

7/2
5/2

583 nm light for MOT

5163 MHz

4809 MHz

15/2 4041 MHz

101064 nm light for trapping
17/2 2689 MHz

1570 nm light for trapping5
1 GHz
19/2 0 MHz

4f12 6s2 3H6 ground state

0
1

2

3

4

5

6

J

7

8

9

10

11

12

Figure 2.2.: Energy level spectrum of erbium and hyperfine structure of the ground state for thefermionic isotope. (a) Energy levels in the relevant range for laser cooling and trapping. Even andodd parity states are shown in red and black, respectively. All laser colors used in our experimentare indicated. For transversal cooling, Zeeman slowing and imaging 401 nm light (blue) is used. The
atoms are captured within a narrow-line magneto-optical trap (MOT) operated at 583 nm (yellow).
The optical potentials are formed by 1064 nm, 1570 nm, and 532 nm trapping light (red, dark red,and green, respectively). (b) While bosonic erbium shows no hyperfine structure, fermionic erbium
features eight hyperfine manifolds with total angular momentum quantum number F ∈ [19/2, 5/2].
The frequency splitting of the individual levels at zero magnetic field is given with respect to theground state manifold. Figure adapted from Ref. [Fri14a].

during cycling can be strongly enhanced. However, for the case of erbium, suitable electricdipole allowed laser transitions have been identified [Ban05], which led to the demonstrationof efficient laser cooling and trapping without repumping light via the (in principle) not
closed 401 nm line [McC06].
Encouraged by this result, within our experiment we have proposed and realized a simple
scheme for the preparation of cold atomic clouds of erbium with temperatures in the orderof 10’s of µK [Fri12]. Our approach for cooling and trapping is summarized in Fig. 2.2(a)and relies on two electric-dipole transitions with J → J + 1 where one of the 6s electrons
is excited to a 6p state. Initial cooling is performed on the 401 nm singlet 1 P1 line. The
broad natural linewidth of 29.7 MHz [Bai12a] allows for strong atom-light interaction, ideal
for transversal cooling, Zeeman slowing, and imaging applications. For achieving a cold
temperature of the atoms when captured in a magneto optical trap, it is from importanceto employ an excited state with a narrow linewidth, as the lowest reachable temperature,
the Doppler temperature, is directly proportional to this linewidth. Here, we decided forthe triplet 3 P1 state, an intercombination line that features a linewidth of 190 kHz, whichcorresponds to a Doppler temperature as low as 4.6 µK.

2.2. ATOMIC AND MAGNETIC PROPERTIES OF ERBIUM

11

For the production of ultracold atomic gases with temperatures well below the Doppler
temperature an additional approach is needed. The most common method in ultracold experiments relies on optical trapping with off-resonant laser light and subsequent evaporative
cooling [Gri00]. The depth of the optical potential for a given trapping wavelength is given by
the AC-stark shift on the ground state. This shift depends on the dynamical polarizability,
which accounts for the (off-resonant) contributions of all excited states via a sum-over-state
formula [Lep14]. In Fig. 2.2(a) all laser wavelengths used for optical trapping at our experiment are indicated. We have experimentally investigated the dynamical polarizability for1064 nm trapping light in Ref. [Bai12a] and over the years improved our understanding along
532 nm, 1064 nm, and 1570 nm light, as reported in Appendix A.2.
An additional important aspect of the energy spectrum is the energy distribution of theatomic ground state. In the case that the nucleus features a nuclear spin, I ̸= 0, theground state will be split by the interaction between the nuclear spin and the electron total
angular momentum, resulting in a hyperfine structure. While the bosonic erbium isotopes
lack hyperfine splitting, the fermionic isotope 167 Er features a nucler spin quantum numberof I = 7/2, giving rise to a total of eight hyperfine states F ∈ [J + I, J − I]. The states
and their corresponding energies are shown in Fig. 2.2(b). A detailed investigation of thehyperfine structure, including also the structures of the 6s6p 1 P1 and 6s6p 3 P1 excited states,
is given in Ref. [Fri13]. All experiments within this thesis are performed in the F = 19/2
hyperfine ground state manifold.

2.2.3. Magnetic properties
One of the landmark properties of several lanthanide elements is their exceptional large magnetic moment, see Fig. 2.1(a). These high values are a direct consequence of the submerged
shell structure, as the electron spin-orbit coupling leads to large total angular momentum
quantum numbers J. In particular, this results also in large magnetic quantum numbers
mJ , which are defined by the projection ot the total angular momentum quantum number
J on the quantization axis, given by an external magnetic field. The mJ values range from−J to +J, see Sec. 2.2.4, and are strongly related to the magnetic moment of an atom3 , as
it is defined byµ = mJ gJ µB .
2.1
To determine the magnetic moment, in addition the Landé g-factor gJ has to be known. In
the case of pure spin-orbit coupling it can be calculated via
gJ = 1 + (gS − 1)

J(J + 1) − L(L + 1) + S(S + 1)
with gS ≈ 2.00232.
2J(J + 1)

2.2

For the case of erbium additional correction have to be included, resulting in a slightly
reduced value. The experimentally reported value is gJ = 1.163801(1) [Con63] leading for3

We note that the scenario of magnetic dipoles is qualitatively different compared to the scenario of polar
molecules. The strength of the electric dipole moment of polar molecules scales with the applied electricfield magnitude, while the dipole moment of magnetic atoms is truly permant and persists also at zeromagnetic field.

2.2. ATOMIC AND MAGNETIC PROPERTIES OF ERBIUM

12

the bosonic isotopes to a magnetic moment of the lowest magnetic state mJ = −6 ofµ = −6.982806(6)µB .

2.3

For the fermionic isotope, the additional hyperfine structure requires to calculate the Landég-factor gF via
F (F + 1) − I(I + 1) + J(J + 1)
gF = gJ,
2.4
2F (F + 1)
resulting in gF = 0.735032. The magnetic moment of the energetically lowest state with
F = 19/2 and mF = −19/2 is found to beµ = mF gF µB = −6.982804µB ,2.5equaling the bosonic one. When different spin states are prepared the magnetic momentchanges accordingly.

2.2.4. Zeeman splitting
When a magnetic field B is applied, the Zeeman manifolds split into 2J + 1 (2F + 1) Zeeman
substates with magnetic quantum numbers mJ (mF ) for bosonic (the fermionic) isotope(s).
For the bosonic isotopes the energy of the substates scales linearly with the magnetic fieldstrength, while for the fermionic isotope, as a result of the hyperfine coupling, an additional
quadratic field dependence is encountered.
For bosonic erbium the energy of the ground state substates reads as
EZBo (B) = mJ gJ µB B = mJ q(B),2.6

with q(B) denoting the linear magnetic field dependence. This linear dependence holds forall experimental accessible magnetic field values, as a result of the large spin-orbit coupling
constant of erbium [Jud61]. In Fig. 2.3(a) the magnetic field dependence of the 13 mJ statesis plotted.
The energy splitting between two adjacent spin states is degenerate across the spin ensemble
and reads as∆EZBo (B) = (mJ − mJ+1 ) q(B) = hzB,|{z}

with z = 1.628879 MHz/G

2.7

1

being the linear bosonic Zeeman coefficient and h the Plank constant. The knowledge on
the energy splitting is from particular relevance as it allows to prepare different spin states
via e. g. radiofrequency coupling. In experiments, it is also used to calibrate the magnetic
field by driving the mJ = −6 to mJ = −5 transition and monitoring subsequent atomic loss
caused by dipolar relaxation.
For fermionic erbium, due to the Paschen-Back effect, which results in a decoupling of J
and I, the energy splitting starts to deviate from a pure linear magnetic field dependence2.2. ATOMIC AND MAGNETIC PROPERTIES OF ERBIUM
mJ
+6

500

(a)

400

+4
+3

energy (MHz)

200

+2

100

+1

0

0

-100

-1
-2

-200

-3

-300

∆2

-4

-400

∆1

-5
-6

10

20
30
magnetic field (G)4050
differential splitting (kHz)

0
0

-400
-800

+19/2

+11/2

+15/2

+7/2

mF

+3/2

-3/2

-7/2

-11/2

(c)
-15/2

-1200

-19/2

quadratic term (Hz/G2)

mF
+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)

+5

300

-500

13

0

10

20
30
magnetic field (G)

40

50

(d)

60
50
40
30
20
10
0

∆1-∆2
0

10

20
30
magnetic field (G)4050

Figure 2.3.: Zeeman energy for bosonic and fermionic erbium in the ground state manifold formagnetic field values up to 50 G. (a) Bosonic erbium shows a linear dependence of the different mJstates on the magnetic field. (b) For the fermionic isotope (F = 19/2 manifold) in addition to thelinear dependence a quadratic energy shift emerges due to the hyperfine coupling. (c) Extracted
quadratic term, see text, from a fit to the data of (b). With a parabolic fit (solid black line) zqu isdetermined. (d) Difference in the energy splittings between neighboring spin states, see inset of (b).

already at moderate B field values. An additional quadratic field dependence (EZFe ∝ B 2 )becomes apparent when calculating the hyperfine magnetic levels via exact diagonalizationof the atomic Hamiltonian [Smi65]. Figure 2.3(b) shows the 20 Zeeman substates for the
ground state manifold of the fermionic isotope. The resulting energy dependence of the mF
substates can be written in good approximation as
EZFe (B) = mF qli (B) + (F 2 − m2F )qqu (B 2 ) = h mF zli B + h (F 2 − m2F )zqu B 2 .
| {z }|{z}linear term

2.8

quadratic term

Here, qli (B) and qqu (B 2 ) account for the linear and quadratic magnetic field dependence, andzli and zqu are the fermionic linear and quadratic Zeeman coefficients. The mF dependent
linear and quadratic terms can be extracted from a polynomial fit of second order to thecalculated Zeeman energies4 . In Fig. 2.3(c), the quadratic fit term is plotted as a function of
the mF state. It depends on the magnetic sublevels mF in a quadratic way and results in adown shift in energy for all levels beside the edge states. From a parabolic fit, the quadratic
4

The fit is restricted up to 50 G and gives residua below 1 kHz. For larger magnetic field ranges a more
specialized fitting function should be used.

2.3. EXPERIMENTAL SETUP

14

Zeeman coefficient is determined to bezqu = −12.76(1) Hz/G2 .
A similar treatment gives for the linear Zeeman coefficient zli = 1.02874 MHz/G5 .
The major advance of the quadratic term arises for protocols of state selective addressing.
Here, it is necessary that the splitting between adjacent spin states in not degenerate, as toallow the coupling of two spin states without a coupling to all other spin states. Following
Eq. 2.8, the energy splitting between adjacent spin states reads as∆EZFe (B) = (mF − mF +1 )qli (B) + (−m2F + m2F +1 )qqu (B 2 )= hzli B + (2mF + 1)hzqu B

2.9

2The lift of the degenerate splitting between adjacent spin states becomes evident when the
differential splitting between three neighboring spin states (e. g. ∆EZFe between mF = −19/2
and m′F = 17/2 minus ∆EZFe between mF = 17/2 and m′F = 15/2, see inset Fig. 2.3(b))
is calculated, as plotted in Fig. 2.3(d). While the linear Zeeman contributions cancel, thequadratic coefficient zqu determines the differential splitting. Within our approximation the
differential splitting between three adjacent spin states is found to be∆(∆EZFe )(B) = −2qqu (B 2 ) = −2hzqu B 2 .

2.10

At a magnetic field value of 40 G this gives a differential splitting of 40.83 kHz, which matches
the actual value of 41.56 kHz within 2 %. The knowledge on the lift of the degenerate coupling
between adjacent spin states allows for a deterministic state preparation of the two lowest
spin states in fermionic erbium and enabled us to perform a clean study of the interspin
interaction properties, see Sec. 5.6.

2.3. Experimental setup
The building of the erbium apparatus has started in 2010, and led to the successful preparation of the first BEC of erbium already in early 2012. At the heart of the setup are ahigh-temperature oven, a transversal cooling (TC) section, a Zeeman slower (ZS) setup, andthe main chamber for trapping the atoms in a narrow-line magneto-optical trap (MOT),
evaporative cooling of the atoms in optical dipole traps, and absorption imaging of theatomic samples.
Within this section, we will summarize the most important experimental parts and report on
the performance of the apparatus. For a more detailed description of the experimental setup,
including technical details on the vacuum chamber, the blue and yellow laser light setups,
and the coil setup of the main chamber, the reader is referred to Ref. [Fri14a]. Updates on
the experimental setup, which are from particular importance for this thesis are reported in
the Appendices B and C.
5FeThis value matches nicely the expectation for a pure linear dependence with ∆EZ,li(B) = BgF µB =B × h × 1.0288 MHz/G.

2.3. EXPERIMENTAL SETUP

15

Ti-sub pump (hidden) 75 l/s ion getter pumpz
Ti-sub pump

20 l/s ion getter pumpsxfull-metal valve

port alignery
200 mmport aligner
bellowatomic beam shutterpressure gauge

UHV section
Zeeman pumping stage 2
slowermirrormain chamber

HV section
Zeeman slower

pumping stage 1

trans- high-temperature ovenversalcooling

Figure 2.4.: Drawing of the experimental chamber for the erbium experiment. Erbium pieces areheated in the effusion cell section (red) producing an atomic flux traveling from right to left. Aftera transversal cooling section (light blue), where the transversal velocity spread is reduced by nearresonant 401 nm laser light, the atoms enter the Zeeman slower section (green). The atoms are
slowed down by resonant 401 nm light, which enters via the Zeeman slower mirror section (blue). The
Zeeman slower also serves as a differential pumping section, which allows pressure differences between
the HV and the UHV sections of up to three orders of magnitude. Finally, the atoms are trappedat the main chamber (yellow) with 583 nm yellow laser light within a MOT. Ion getter pumps andTitanium sublimation pumps (gray) maintain the low pressure in the HV and UHV sections. Figure
adapted from Ref. [Fri14a].

2.3.1. Vacuum chamber
The erbium apparatus can be divided into two main sections. The high-vacuum (HV) section
includes the high-temperature oven and the TC chamber and enables a pressure of about
4×10−9 mbar. The ultra-high vacuum (UHV) section is connected via a differential pumping
section and includes the ZS and the main experimental chamber. In this section pressures
down to 1 × 10−11 mbar are reached, which renders collisions of the trapped atoms withbackground gas negligible. Figure 2.4 shows a drawing of the whole erbium apparatus,
including also to pumping stages for the HV and UHV sections.
The high-temperature oven6 consists out of two parts, the effusion cell and the hot lip,which are made out of tantalum and are separated by an aperture. Their temperatures arecontrolled independently to 1100 ◦ C and 1200 ◦ C, respectively. With this temperatures wereach an atomic flux right after the oven of 1014 s−1 sr−1 , which is sufficient for experimental
operations [Sch11]. For a filling of about 10 mG of solid erbium, the oven can be operated
for several years without refilling.

6

model DFC-40-10-WK-2B, from from CreaTec Fischer & Co. GmbH2.3. EXPERIMENTAL SETUP

16

2.3.2. Laser cooling and trapping of erbium
Right after the effusion cell the atomic beam enters the TC section. While apertures at theoven are implemented to collimate the beam, the remaining transversal velocity spread does
not allow all atoms to reach the main chamber through the differential pumping section.
As to reduce this spread, a slightly blue-detuned two-dimensional optical molasses in two
directions is implemented. This leads to a further collimation of the atomic beam andenhances the amount of atoms captured in the MOT by almost an order of magnitude, see
Appendix B.
Subsequentially, the atoms travel through the ZS section. A counter-propagating laser beam
reduces the longitudinal velocity via momentum transfer by scattering, which effectively
slows down the atoms. The ZS is designed such that the constantly slowing atoms are kept
at the atomic resonance, by compensating the varying Doppler shift via the Zeeman shift
induced by an external magnetic field. The initial longitudinal velocity of around 450 m/s isreduced to about 5 m/s, which enables to directly capture the atoms in the MOT.
Finally, the atoms enter the main chamber, where they are trapped within a MOT operated
at the narrow 583 nm transition. As discussed in Sec. 2.2.2, the narrow linewidth of 190 kHz
features a low Doppler temperature and allows for MOT temperatures around 10 µK. To
avoid residual light scattering of the ZS beam on the MOT, we use large light detunings of
up to 50 linewidths, which leads to an accumulation of the atoms below the ZS beam. With
our setup, we can realize MOTs of the five most abundant isotopes of erbium [Fri12]. For an
optimized experimental setup, we can achieve MOTs of up to 2×108 atoms at a temperatureof around 10 µK for the most abundant 166 Er isotope. This numbers are fairly sufficient forour experiment and do not limit us. Rather, the maximum atom numbers for further coolingare restricted by the amount of atoms that can be loaded to the optical dipole trap, seeSec. 2.4.1. Hence, we typically work with MOT numbers of 3 × 107 atoms, which are reached
for 3 s of MOT loading.
A major advantage of our narrow line MOT is that the atoms are naturally spin-polarized to
the lowest Zeeman substate. This is a result of the large detuning of the MOT, which leads toa so called ”gravitational sag”. As a consequence, atoms preferentially scatter light from thelower σ − -polarized MOT beam, which optically pumps the atoms to the lowest spin state.
Indeed, the influence of the top σ + -polarized MOT beam is negligible, which allowed us toremove it completely without influencing the MOT performance. A more detailed study
on such a five-beam MOT configuration has been performed recently [Ilz18]. The increased
optical access is particular valuable for our experiment, as it allows for the implementationof additional optical setups.

2.4. DEGENERATE GASES OF ERBIUM

17

2.4. Degenerate gases of Erbium
The demonstration of a cold MOT of erbium is the ideal starting point for loading theatoms into an optical dipole trap, and to further cool them by means of forced evaporation.
Utilizing this technique, we managed to achieve a BEC of 168 Er - a worlds first - in early2012 [Aik12]. This achievement was the turning point of our experiment and marked thestart for many exciting studies within the ultracold regime.
Together with the first BEC of dysprosium atoms in 2011 by the group of Benjamin Lev
at Stanford [Lu11], it also led to the beginning of a new era in ultracold experiments - theera of strongly magnetic dipoles. While pioneering work already has been performed with
dipolar chromium atoms [Gri05, Lah08, Pas11a], strongly magnetic lanthanides allow to access physical phenomena that are even more governed by the long-range and anisotropicinteraction, leading to unexpected quantum phases. The large impact on the communityof ultracold gases, is nicely accounted for by the recent building of several more strongly
magnetic lanthanide experiments. The groups of Tilman Pfau in Stuttgart as well the groupof Giovanni Modugno in Florence with dysprosium [Kad16, Luc18], and the group of Martin
Weitz in Bonn with erbium [Uli17] already succeeded in reaching the quantum degenerate
regime. Other groups are chasing behind these groups, as the group of Mark Saffman in
Wisconsin with holmium [Mia14], the group of Sylvain Nascimbene in Paris with dysprosium [Dre17], our group in Innsbruck with a mixture of erbium and dysprosium [Ilz18], thegroup of Rudolf Grimm in Innsbruck with a mixture of potassium and dysprosium [Rav18],
the group of Mikio Kozuma in Tokyo with europium [Ino18], the group of Patrick Windpassinger in Mainz with dysprosium7 , the group of Markus Greiner at Harvard in Boston
with erbium8 , as well as the groups of Zoran Hadzibabic in Cambridge with erbium andWolfgang Ketterle at MIT in Boston with dysprosium9 .

2.4.1. Optical trapping
Optical dipole traps are based on off-resonant laser light that induces an energy shift on theatomic state. This shift is known as the AC stark shift [Gri00] and is related to the dynamical polarizability. Appendix A.2 summarizes our up-to-date knowledge on the dynamical
polarizability of erbium for the three relevant wavelengths used in our experiment.
Directly after the MOT stage, we transfer the atoms to an optical dipole trap, operated
at 1064 nm. Good transfer efficiencies of up to 30% are achieved by means of a scanningsystem that allows to increase the trapping volume of the beam via a technique based ontime-averaged potentials [Bai12a]. The maximum transferable atom number is limited by
the total power of the dipole trap beam and reaches values of up to 1 × 107 atoms with atemperature of about 20 µK.

789Windpassinger dysprosium lab
Greiner erbium lab
private communication

2.4. DEGENERATE GASES OF ERBIUM

18

We have further demonstrated optical trapping with 1570 nm and 532 nm light. The use of1570 nm trapping light turned out to be necessary for the production of degenerate Fermi
gases of the 167 Er isotope, see Sec. 3.3, as with 1064 nm light significant heating of thermalgases was encountered. Laser light at 532 nm has proven to be particular valuable for the
production of optical lattices, as the short wavelength results in a small distance between
neighboring atoms in the lattice and enhances the effects from dipolar interactions, see
Chapter 5. The optical setups for all optical trapping beams are reported in Ref. [Fri14a]and Appendix B.

2.4.2. Evaporative cooling
To reach the quantum degenerate regime, the technique of forced evaporation is applied.
By adiabatically lowering the optical potential depth, the thermal Maxwell-Boltzmann distribution becomes truncated. As only the hottest atoms are removed from the trap, after
rethermalization due to elastic collisions, a lower temperature and most importantly a higher
phase-space density (PSD) is reached. The PSD of an atomic gas can be expressed as
PSD = N ω̄

3



~
kB T

3

,

2.11

with N being the atom number, ω̄ the mean trap frequency, and kB T the thermal energy.
The transition to a Bose-Einstein condensate, a state where a macroscopically large numberof bosons occupies the ground state, is related to a PSD of 2.6.
Efficient evaporative cooling relies on a favorable ratio between elastic and inelastic collisions,as the speed of rethermalization needs to exceed inelastic-loss timescales. For the case of
168 Er, ideal conditions are found at a magnetic field value of 0.4 G. At this field we find as-wave scattering length as of 137(1)a0 , see Sec. 4.1.1 for a definition of as , with a0 being
the Bohr radius. At this parameters we achieve an almost pure BEC of up to 2 × 105 atoms
within 7 s of forced evaporation [Bai12a]. In Fig. 2.5 the emergence of a BEC is visualized.
In 2015 we also succeeded in Bose-Einstein condensing 166 Er, the most abundant erbium isotope. The experimental procedure is very similar to the one for the 168 Er isotope. Here, wefind best evaporation efficiencies at a magnetic field value of 2.1 G, relating to a s-wave scattering length as of 83(2)a0 , see Appendix A.1. We typically create BECs of 1.1 × 105 atoms
with a BEC fraction of ≈ 70 %. This isotope features a conveniently broad Feshbach resonance centered at low magnetic field values of about 50 mG, allowing us to tune the scattering
length with a high accuracy and precision for arbitrary magnetic field orientations. The highlevel of control enabled us to investigate in detail the formation of a macro-droplet state, see
Appendix A.1, and the birth of Roton quasiparticles, see Appendix A.3.
The formation of a degenerate Fermi gas of 167 Er is investigated in detail in Sec. 3.3. Usually, identical fermions do not collide at ultralow temperatures. Hence, spin- or speciesmixtures have to be employed to enable an elastic cross section large enough for efficientcooling [DeM99, Tru01, Sch01, Had02, Roa02]. In contrast, our experimental procedure forevaporation is solely based on universal dipolar scattering between single-spin fermions. The2.4. DEGENERATE GASES OF ERBIUM

19

Figure 2.5.: Final evaporation steps to form an almost pure BEC of erbium - a worlds first realized with 168 Er. When the atoms are cooled down below a critical temperature, a sharp peakin the momentum distribution of the atomic cloud (in false color) arises. This resembles the major
fingerprint of a quantum degenerate bosonic gas. Temperatures of the thermal distributions areindicated.

large and anisotropic elastic cross section has given new prospects for the study of scattering
physics of identical fermions, see Chap. 3.

Chapter3

Few- and many-body scatteringof dipolar atoms

The dipole-dipole interaction (DDI) among particles gives rise to a universal scattering behavior in the ultracold regime [Boh09a]. To access elastic dipolar scattering in a cleanmanner, it is required to switch off all other types of interactions. With bosonic particles,
such a case can be realized when the isotropic contact interaction is lowered to zero bymeans of Feshbach tuning, see Chap. 4. However, in such a regime the dipolar interactions
start to dominate and can lead to a collapse of the atomic sample via the attractive partof the DDI [Lah08]. Nevertheless, for (small) contact interaction, particularly fascinating
many-body physics can occur, arising from the competition between the isotropic contact
interaction and the anisotropic DDI. Beautiful examples are given by the observation of selfconfining quantum droplets, see [Kad16, FB16] and Appendix A.1, or the emergence of roton
quasi-particles, see Appendix A.3. Importantly, these works represent scenarios in which,to explain the experimental results by theory, all the different scattering effects, including
dipolar interactions, have to be taken into account very carefully. While this is certainly
possible, it is not a straightforward method to access dipolar scattering.
A more elegant approach to experimentally treat elastic dipolar scattering does involve spinpolarized fermionic particles. In the fermionic case, contact interactions, due to the Wigner
treshold law governed by s-wave scattering, are absent at ultralow temperatures. In contrast, the scattering cross section, resulting from dipolar interactions, does not vanish, butconverges to a universal value in the ultracold regime, isolating dipolar effects. In addition,
fermionic systems deliver the advantage of being stable against a collapse, even for dominating attractive dipolar interactions. This is a result of quantum statistics where, for low
enough temperatures, the fermionic atoms occupy all lowest harmonic oscillator states (upto the Fermi energy) and a further compression (implosion) of the cloud is prohibited byFermi pressure. These two facts allow for a clean study of dipolar scattering properties withfermionic systems.
Within this chapter we will focus on the description of spin-polarized dipolar fermionic atoms
as a platform to investigate the effects of dipolar scattering. Section 3.1 reviews anisotropic
dipolar few-body scattering, and Sec. 3.2 focuses on the experimental observation of dipolar
many-body effects. The two Secs. 3.3 and 3.4 contain our publication on dipolar few-bodyeffects with fermionic erbium. Finally, Sec. 3.5 presents our publication on the observationof the Fermi surface deformation.
20

3.1. DIPOLAR FEW-BODY SCATTERING

21

3.1. Dipolar few-body scattering
3.1.1. Dipole-dipole interaction
The DDI has a long-range and anisotropic character. For the case of magnetic atoms, suchas erbium, the interaction of two dipoles µ1 and µ2 with permanent magnetic moments µ1
and µ2 , see Sec. 2.2.3, can be written in the general form asµ03
Udd (r) =(µ1 µ2 ) − 2 (µ1 r)(µ2 r)
3.1
4πr3
r
and depends on the interparticle distance r = r 1 − r 2 . Here, r = |r| and µ0 is the magnetic
constant. For the case of particles with an electric dipole moment the DDI is calculated
by substituting µi → di and µ0 → 1/ϵ0 . For electric dipoles the dipolar strength is set
by the strength of an external polarizing electric field. In stark contrast, magnetic dipoles
intrinsically carry a magnetic moment, where the strength does not depend on the external
field B. Nevertheless, the dipole orientation is given by the external magnetic field axis B̂,even for a weak field amplitude. Hence, for experimental conditions, the magnetic dipoles
become polarized and mutually align along the external field that sets the quantization axis.
As a result, µ1 ∥ µ2 and Eq. 3.1 simplifies to
Udd (r, θ) =µ0 µ1 µ2 1 − 3cos2 θ
4π
r3

3.2

with θ denoting the angle between the quantization axis and the interatomic axis of thetwo dipoles, see Fig. 3.1. Remarkably, the sign of the interaction can be changed when θis changed from 0◦ (head-to-tale configuration) to 90◦ (side-by-side configuration)1 . The
particles repel each other in a side-by-side configuration and experience an attractive force
in the head-to-tail configuration, see Fig. 3.1(b). As a consequence of the 1 − 3cos2 θ term,
the absolute strength for negative interaction is twice as large as for positive interaction at
the same interparticle separation r. The range of the DDI can be accessed via the so-called
dipolar length
mµ0 µ1 µ2
ad =
,
3.3
4π~2
which in addition to the magnetic moments depends on the mass m of the dipoles. As
erbium is a heavy element (m166 Er = 166 u)2 , in the lowest magnetic substate, i. e. mJ =−6 → µ1 = µ2 ≈ −7µB (Eq. 2.3), ad takes the value 196.2 a0 , which outperforms alkali atoms
as rubidium (ad,87 Rb = 2.1 a0 ) and even magnetic chromium (ad,52 Cr = 46.1 a0 ).

1

2

The strength of the interaction does also depend on the magnetic moments µ1 and µ2 , and the interaction
sign can be changed when the signs of the spins are changed from a parallel (sgn(µ1 ) = sgn(µ2 )) to an
antiparallel (sgn(µ1 ) = − sgn(µ2 )) configuration.
The atomic mass unit u equals 1.66 × 10−27 kg, which corresponds to 1/12 of the mass of carbon-12 atoms.

3.1. DIPOLAR FEW-BODY SCATTERING
0

(a)

22

(b)
θ

305.26

r

270

54.74

B

90

125.26234.74B
180

Figure 3.1.: DDI between two magnetic atoms (a) The interaction strength between two dipoles,aligned along B̂, relates to their relative distance r and the angle θ between the interatomic axis andthe quantization axis. Negative and positive potentials are shown in blue and red, respectively. The
dashed lines indicate the angles at which the repulsive and attractive part of the DDI cancel. (b)When a dipole (blue or red dipole) approaches another dipole, it depends crucially on the angle ofincidence if it feels a repulsive (red) or attractive (blue) potential. For visualization, the potential iscut off for small distances, at which it starts to diverge.

3.1.2. Ultracold scattering
In quantum mechanics the collision of two particles is described by the scattering theory, seee. g. Ref. [Lan77]. The problem of elastic scattering can be treated as a scattering event of asingle particle with reduced mass mr 3 within the field U (r). The scattered wave function atlarge distance is given by the sum of the incoming plane wave with wave vector k and theoutgoing spherical wave with wave vector k′ . The amplitude f (k′ , k) of the outgoing wave
depends on the scattering potential U (r) and the included angle with the incoming wave. It
is used to evaluate the differential cross sectiondσ(k′ , k) = |f (k′ , k)|2 .
dΩk′

3.4

The differential cross section is a measure of the scattering probability of incident direction
k into the outgoing direction k′ with the solid angle dΩk′ . The total scattering cross section
for a given incident direction k can thus be obtained via integrationZdσ(k′ , k).
3.5
σ(k) = dΩk′dΩk′
The collision energy of the scattering event reads as
E=

~2 k
2mr

3.6

with k = |k|.
3

m2
The reduced mass of two particles with mass m1 and m2 can be calculated via mr = mm11+m, which2
evaluates to mr = m/2 for m1 = m2 .

3.1. DIPOLAR FEW-BODY SCATTERING23

For short-range potentials and in the case of low-energy scattering, i. e. k → 0, the scattering amplitude is governed by scattering in the lowest partial wave l = 0, so-called s-wave
scattering, and is approximated tof (k′ , k) ≈ −as ,3.7

where as is the s-wave scattering length. In the case√ of identical bosons the scattering
amplitude has to be symmetrized, i. e. fB (k′ , k) = 1/ 2[f (k′ , k) + f (−k′ , k)], which leads tothe total cross section
σB = 8πa2s .
3.8This cross section is isotropic and thus does not depend on the incident scattering angle. The
case of identical fermions is in stark contrast to the bosonicscenario. As the scattering am√
plitude must be antisymmetrized, i. e. fF (k′ , k) = 1/ 2[f (k′ , k) − f (−k′ , k)], the scatteringcross section in the ultracold regime dissapears, i. e.
σF = 0.

3.9

As a result, identical fermions represent a perfectly non-interacting system. In the context of
ultracold experiments, this complicates the approach to cool identical fermions to quantum
degeneracy, as evaporative cooling, which is based on elastic scattering, is absent.

3.1.3. Universal dipolar scattering
The scattering behavior changes drastically in the case of DDI [Lan77, Bar08]. As a consequence of the 1/r3 long-range character, not only the s-wave but instead all partial waves
contribute to the scattering event. It has been shown that, in the strong dipolar limit where
DDI dominates, the elastic scattering cross section has a universal behavior that only depends
on the mass, the dipole moment and the collision energy of the scattered particles [Tic08].
Importantly, in the ultracold regime, the scattering cross section takes a constant value that
does not depend on the collision energy anymore [Boh09a]. For dipolar scattering the ultracold regime is reached when the collision energy of Eq. 3.6 is smaller then the natural energy
scale of the dipolar interactionµ0 µ1 µ2
ED =
3.10
4πa3D
where the nature length scale aD for two colliding dipoles is defined asaD =m r µ0 µ1 µ2= ad /2.
4π~2

3.11

For purely dipolar scattering within the first-order Born approximation4 the scattering amplitude reads as#"′ B̂)2
(
k̂
B̂
−
k̂
2
f (k′ , k) = aD
−
.
3.12
3
1 − k̂ k̂ ′4

The Born approximation can be applied when the scattering potential can be treated as a perturbation,i. e. when the scattered wave is not significantly changed from the asymptotic incident wave. It has beenshown that this approximation can be applied for long-range dipolar interaction potentials, where thescattering happens outside the centrifugal barrier, see e. g. Ref. [Yi01].

3.1. DIPOLAR FEW-BODY SCATTERING24

In stark contrast to the scattering at an isotropic potential, as it is the case for Eq. 3.7, thescattering amplitude within a dipolar scattering potential, see Eq. 3.2, strongly depends on
the angles between the incident, the outgoing, and the quantization axes, i. e. between k̂, k̂ ′ ,and B̂.
Following Eq. 3.5 and Eq. 3.12 one can evaluate the total cross section, averaged over all
incident directions k̂. Depending on the quantum statistics, again the antisymmetrization/symmetrization of the scattering amplitude has to be applied for identical particles,
see Sec. 3.1.2.
In the case of fermionic particles, this leads to the total cross section
σ̄F =

32π 2
a .
15 D

3.13

For bosonic particles, the s-wave short-range contribution has to be empirically added to thetotal cross section, which finally results inσ̄B = 8πa2s +

32π 2
a .
45 D

3.14

The obtained total cross sections only depend on the dipolar length scale aD (and the scattering length as for bosons). The found independence on energy reveals the universal behaviorof ultracold dipolar elastic collisions.
As a striking consequence of this unversal scattering behavior, dipolar interactions allow
for elastic collisions even in the case of spin-polarized fermions. Section 3.3 contains our
publication, in which we use universal dipolar scattering for evaporative cooling of fermionicerbium in the streched state. We observe, for the first time, that the deeply quantum degenerate regime can be reached with a single component Fermi gas. Initial work on dysprosium
indicated signatures of the very same effect [Lu12]. Our observed evaporation efficiencies
are remarkably high, underlining our simple and straightforward method to cool identicalfermions. We investigate the total elastic cross section via cross-dimensional thermalization
experiments and find a good agreement with the universal value, i. e. Eq. 3.13.

3.1.4. Anisotropic dipolar scattering
While in the case of isotropic interactions, σ(k), denoting for the scattering cross section ofincident wave vector k integrated over all outgoing directions k̂ ′ , does not depend on theincident scattering direction, the behavior is fundamentally different in the dipolar case. On
top of the universal behavior, dipolar scattering shows a peculiar angle dependence, as canbe seen from Eq. 3.12. The total cross section can be written as a function of the angle η,
which is spanned by the incident wave vector k̂ and the polarization axis B̂ [Boh14], seeFig. 3.2(a).
From Eq. 3.5 and Eq. 3.12 one finds for the scattering cross section of two identical fermionic
dipolesπ
σF (η) =3.15
3 + 18 cos2 η − 13 cos4 η a2D ,3

3.1. DIPOLAR FEW-BODY SCATTERING
(a)

25

10

(b)

8
σF = 32π/15k

σ /a 2D6η4
σB = 32π/45

2
B

0

0

30

60

90
120
η (degrees)

150

180

Figure 3.2.: Anisotropic dipolar scattering cross section. (a) Two dipoles collide via an angle ηthat is defined by the angle between the quantization axis given by B and the incident wave vectork. (b) Purely dipolar scattering cross section as a function of η for indistinguishable fermions (blue)and bosons (red) summed over all scattering angles k̂ ′ . σ(η) is cylindrically symmetric about thequantization axis. The angular averages σ̄B and σ̄F are indicated by the red and blue dashed line,
respectively.

while for indistinguishable dipolar bosons the total cross section reads asπ
72 a2s − 24 as 1 − 3 cos2 η aD + 11 a2D − 30 cos2 η a2D + 27 cos4 η a2D .
σB (η) =9

3.16

Figure 3.2(b) draws their functional form and reveals the strong anisotropy in the scattering
behavior. For the bosonic case, the scattering cross section is shown for vanishing contact
interactions, i. e. as = 0. Interestingly, fermionic and bosonic particles show a markedly
different behavior. While bosons scatter strongest when approaching each other in a side-byside configuration (η = 90◦ ), fermions preferentially scatter when they meet under an angleof η = 35◦ with respect to the quantization axis. When calculating the angular average ofthe total cross section
Z
1 +1
σ̄ =
d(cos η) σ(η),
3.17
2 −1
one finds back the values of σ̄B and σ̄F , see Eq. 3.13 and Eq. 3.14. For completeness, thesemean total cross sections are plotted in Fig. 3.2(b) as dashed lines. We emphasize that dipolar
fermions scatter in average three times more often then their bosonic counterparts.

The anisotropy of dipolar scattering can have a strong influence on the rethermalization
rate of an out-of-equilibrium atomic sample [Boh14]. Ultracold experiments are typically
performed in harmonic traps with an orthogonal coordinate system {x, y, z}. Collisions insuch a trap not only depend on the scattering angle η, but also on the angle β includedby the trap symmetry axis and the quantization axis, see Fig. 3.3(a). To access the angle
depenendence of the rethermalization rate, cross-dimensional thermalization experiments
can be applied. Here, one axis, e. g. the symmetry axis y, is brought out of equilibrium andthe time acquired for distributing the induced kinetic energy along the perpendicular axes
xz is monitored. This rethermalization time depends on the density and thermal velocity of
the atomic sample, but also on the elastic scattering cross section σ̄. Further, it is directly
proportional to the dimensionless constant α, which accounts for the number of collisions

3.1. DIPOLAR FEW-BODY SCATTERING
(a)

26

(b)
y

y

β

β
y

k

z

z

B

x

x

β

(c)
η
k

Figure 3.3.: Cross-dimensional thermalization experiments with identical fermionic dipoles. (a)Illustration of a scattering event within a cylindrically symmetric atomic cloud. For experiments, theangle β, included between the symmetry axis y and the quantization axis, is the relevant quantity.
The angle η is defined as previously discussed. (b) The number of collisions for rethermalization, α,
as a function of the angle β after the atomic cloud is brought out of thermal equilibrium along they-axis. For comparison, α is also shown for s- and p-wave collisions. (c) Differential cross section
dσ/dΩk′ as a function of the outgoing scattering direction k′ for the incident scattering direction kaligned along y for various scattering angles η. Figure adapted from Ref. [Boh14].

needed for rethermalization. α can be used as a measure of the efficiency of elastic collisions
towards thermal equilibrium.
Figure 3.3(b) shows the dependence of α on the angle β for the case of identical fermions. At
an angle β = 45◦ rethermalization happens fast, while for β = 90◦ more than a factor of two
more collisions are needed to distribute the energy along the perpendicular directions. To
get an intuition for this peculiar angle dependence, one can consider the following scenario:
As the trap is excited along the y-axis, it can be assumed that collisions happen mainly
along this principle axis, i. e. k ∥ y. For this scenario β ≡ η. As the scattering cross section isfound to be lowest for η = 90◦ , see Eq. 3.15 and Fig. 3.2, it is not a surprise that at β = 90◦
rethermalization requires the maximum amount of collisions. Further intuition is gained
via the angle dependence of the differential scattering amplitude. Thermalization will occur
fastest if the differential scattering amplitude is largest for perpendicular outgoing directions,i. e. if f (k′ , k) is maximal for k′ ⊥ k. As can be seen in Fig. 3.3(c) this scenario is found foran angle η of 45◦ . In contrast, for the cases η = 0◦ and 90◦ forward and backward scattering
is preferred, which hinders energy distribution from y to xz.
Section 3.4 presents our publication on the detailed experimental study of the anisotropic
dipolar scattering. Based on the method of cross-dimensional thermalization with an atomic
cloud of about 100 000 167 Er atoms, we beautifully reveal that the equilibration rate can varyby as much as a factor of four when we change the dipole orientation with respect to the
dynamic axis of the trap. We find remarkable quantitative agreement with the theoretical
expectation for our parameters. In addition, we observe that the rethermalization rate in
the ultracold regime undergoes a reduction due to Pauli blocking. Our experimental andtheoretical investigations reveal that this reduction does not depend on the dipolar angle,
i. e. on β.

3.2. MANY-BODY EFFECTS IN DIPOLAR FERMI GASES

27

A complementary approach to directly access the differential scattering cross section of colliding dipoles has more recently been applied with bosonic dysprosium. The method involves
the opposed acceleration of two ultracold clouds, which interfere and form a collisional halo.
The distribution of this halo depends on the differential cross section and can be observedin time-of-flight (TOF) experiments. While successfully implemented to visualize s- and dwave collisions of identical bosons in 87 Rb [Tho04, Bug04] and p-wave collisions of identicalfermions in 40 K [Tho16], the work with 162 Dy allowed to observe the angle dependence ofthe differential cross section for dipolar bosons [Bur16].

3.2. Many-body effects in dipolar Fermi gases
Ultracold atoms offer a wealth of possibilities to study the effects of the physics at play.
On the one hand, for few-body investigations, where the scattering between two or threeparticles is of interest, the large number of atoms available in experiments gives a boost to
the signal-to-noise ratio, as multiple scattering events are observed simultaneously. On theother hand, ultracold atoms allow to investigate many-body effects, which emerge due tothe collective coupling among all particles within the atomic sample. Remarkably, in this
scenario one does not need to have information on single scattering events, but rather can
describe the atomic system by a single quantum state.
In the case of dipolar particles, which only have been established in the recent years, manybody effects have mainly be explored with bosonic particles. Here, the competition between
the isotropic contact interaction and the anisotropic DDI can give rise to many intriguing
but also counterintuitive effects. With magnetic chromium atoms it has been observed that,when the dipolar interaction is dominating, the atomic cloud starts to collapse due to theattractive part of the DDI, forming a characteristic d-wave shape [Lah08]. Further, the DDI
can affect the collective oscillation frequency of a dipolar Bose-Einstein condensate [Bis10].
More recently, studies along the competing interactions observed the formation of multiple
quantum droplets [Kad16], and self-confining quantum liquids, see Ref. [Sch16] and Appendix
A.1. It has been shown that the leading role for the stabilization of such quantum dropletsis given by beyond mean-field effects, namely quantum fluctuations, see Ref. [FB16] andAppendix A.1, that only can manifest themselves since the other interactions almost cancel5 .
Another intriguing many-body effect, resulting from the competition of contact and dipolarinteractions, is given by the emergence of the so-called Roton spectrum that features a local
minimum in the energy dispersion relation at non-zero momentum, and is closely linked toelementary excitations in superfluid helium [Lan41]. For dipolar systems, the roton mode
has been predicted in 2003 [San03], which triggered a large amount of theoretical work along
this line [Ron07, Boh09b, Par09, Mar12, Bla12, JL13, Wil10, Nat14]. The first observationof this long-sought mode in dipolar quantum gases is reported in Appendix A.3.
For the case of fermionic dipolar atoms, many-body effects are hardly explored. In contrast
to bosonic systems, for spin-polarized fermions contact interactions are absent. For deeply5

It should be noted that such self-bound droplets can also be stabilized in purely contact interacting systems,
based on the competing attractive and repulsive interactions in e. g. alkali spin mixtures [Pet15, Cab17,
Sem17].

3.2. MANY-BODY EFFECTS IN DIPOLAR FERMI GASES

28

degenerate Fermi gases, the relevant energy scale is given by the Fermi energy EF . In orderto manifest themselves, dipolar interactions have to compete with this energy scale. The
strength of the expected many-body effect is given by the ratio
nµ0 µ2
,
4πEF

3.18

3/2

where n ∝ EF is the peak-number density, and µ is the magnetic moment of the spinpolarized gas. As for magnetic atoms the Fermi energy typically is much larger then theenergy of the DDI, dipolar many-body effects are subtle and difficult to observe.

3.2.1. Hartree-Fock theory
To describe many-body effects in a trapped single-component dipolar Fermi gas, HartreeFock mean-field theory can be used. Within the semiclassical Thomas-Fermi-Dirac approximiation [Gór01] the total energy of the system can be written as
E = Ekin + Etr + Ed + Eex

3.19

with Ekin and Etr accounting for the kinetic energy and for the external potential energy ofthe harmonic trap, respectively. The terms
Z
Z
Z
Z
1
3
3
3 ′
d k d3 k ′ Udd (r − r′ )g(r, k)g(r′ , k′ )
3.20
Ed =
d r d r2(2π)6and1
Eex = −
2(2π)6

Z

Z

Z

3 ′

3

Z

d k d3 k ′ Udd (r − r′ )d r d r 
r + r′
r + r′ ′i(k−k′ )(r−r′ )
×e
g
,k g
,k
2
2
3

3.21

are the Hartree direct energy and the Fock exchange energy, respectively, which arise fromthe DDI, see Eq. 3.1. Here, r and k denote for the coordinate and wave vector, respectively.
The Wigner distribution function g(r, k) can be written within the variational Ansatz
!
3
3
X
X
ri2
ki2
g(r, k) = Θ 1 −−3.22
Ri2 i=1 Ki2
i=1
where Θ is the Heaviside step function. Within this Ansatz, which has been confirmed
numerically in Ref. [Ron10], the variational paramters Ri and Ki describe the real space andmomentum space radii of the atomic cloud in the ith direction, respectively.
The Hartree-Fock theory can be used to calculate the ground state of a dipolar many-bodysystem. For purely contact interacting systems the Hartree and the Fock energy cancel eachother, as can be evaluated when Udd (r − r′ ) is replaced by ∝ δ(r − r′ ).

3.2. MANY-BODY EFFECTS IN DIPOLAR FERMI GASES

29

3.2.2. Real and momentum space radii of dipolar atoms
To calculate the ground state of a dipolar gas, the energy of Eq. 3.19 has to minimized. Interestingly, minimal energy is found if the cloud takes a deformed shape in real and momentum
space, see Refs. [Gór01, Miy08, Bai12b]. This deformation is a result of Ed and Eex and isabsent in purely contact interacting systems.
For dipolar bose gases, the deformation of the real space radius has been observed withmagnetic Cr [Stu05]. This effect is genuinly known as magnetostriction and results from theHartree direct interaction. Similar to the magnetostriction effect in real space, the surface
in momentum space can also be deformed. Here, the relevant term is given by the Fockexchange energy. In the case of dipolar Fermi gases the energy is minimized when the Fermi
sphere elongates along the dipole orientation, i. e. along the direction of maximum dipolar
attraction [Bai12b].
Section 3.5 presents our publication on the first experimental observation of an anisotropic
Fermi surface with a dipolar degenerate Fermi gas. Our investigations, based on TOF
experiments with 167 Er, reveal that the Fermi sphere indeed elongates along the dipole
orientation, and we demonstrate the control of the elongation axis via the external magnetic
field. We confirm that the observed deformation in momentum space mainly results fromthe Fock exchange interaction. To reach an accurate comparison between our experimental
observation and the theoretical expectation, we also consider the Hartree direct interaction.
During the intial expansion, where the atoms are still interacting, it leads to a non-negligible
deformation of the cloud, which we take into account. Remarkably, our observations disclosethat the Fermi surface deformation linearly depends on the ratio of the squared magnetic
moment to the Fermi energy, see Eq. 3.18, thus confirming the many-body nature of theobserved effect.
As we show, the strength of the Fermi surface deformation depends strongly on the competition between the dipolar interaction and the Fermi pressure. If a stronger deformation
has to be achieved, one could either increase the dipole moment via the use of polar or
dipolar fermionic molecules (e. g. 167 Er168 Er), or could access the hydrodynamic regime6 , see
Ref. [Vac17].

6

For the measurements in Sec. 3.5 the atomic sample is in the collisionless regime. Here, the mean-free pathis larger than the size of the atomic cloud. To enter the hydrodynamic regime the collisional rate has to
be increased. The hydrodynamic regime is reached when τ ω̄ ≪ 1, where τ is the rethermalization time,
see Sec. 3.1.4, and ω̄ the mean trap frequency.

3.3. Publication:
Reaching Fermi Degeneracy via Universal Dipolar Scattering†
Physical Review Letters 112, 010404 (2014)
submitted 21 Oct 2013; published 6 Jan 2014
DOI 10.1103/PhysRevLett.112.010404

Kiyotaka Aikawa,1 Albert Frisch,1 Michael Mark,1 Simon Baier,1 Rudolf Grimm,1,2 andFrancesca Ferlaino1
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria†

The author of the present thesis helped in taking the data, and contributed in writing the manuscript.
This publication is also part of the Phd thesis of Albert Frisch.

30

PRL 112, 010404 (2014)

week ending
10 JANUARY 2014

PHYSICAL REVIEW LETTERS

Reaching Fermi Degeneracy via Universal Dipolar Scattering
1

K. Aikawa,1 A. Frisch,1 M. Mark,1 S. Baier,1 R. Grimm,1,2 and F. Ferlaino1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria(Received 21 October 2013; published 6 January 2014)
We report on the creation of a degenerate dipolar Fermi gas of erbium atoms. We force evaporative
cooling in a fully spin-polarized sample down to temperatures as low as 0.2 times the Fermi temperature.
The strong magnetic dipole-dipole interaction enables elastic collisions between identical fermions even in
the zero-energy limit. The measured elastic scattering cross section agrees well with the predictions fromthe dipolar scattering theory, which follow a universal scaling law depending only on the dipole moment
and on the atomic mass. Our approach to quantum degeneracy proceeds with very high cooling efficiency
and provides large atomic densities, and it may be extended to various dipolar systems.
DOI: 10.1103/PhysRevLett.112.010404

PACS numbers: 03.75.Ss, 37.10.De, 51.60.+a, 67.85.Lm

Identical fermions with short-range interaction do notcollide at very low temperatures [1]. According to the rulesof quantum mechanics, the requirement of antisymmetry of
the fermionic wave function causes the scattering cross section to vanish in the ultracold regime. This makes ultracold
fermions special in many respects. For instance, they realize perfectly noninteracting quantum systems, which canserve for sensitive interferometers [2] and ultraprecise
atomic clocks [3]. From another point of view, the absenceof collisions means that direct evaporative coolingcannot work.
The inapplicability of direct evaporative cooling to fermions challenged scientists to develop alternative strategies. The common solution is to use mixtures of two
distinguishable atomic components [4]. In this scheme, fermions are sympathetically cooled through elastic s-wave
collisions with fermions in other spin states [4–8], withatoms belonging to a different isotope [9–13], or withatoms of a different chemical element [14–17].
The scenario is completely different in the presence ofthe long-range dipole-dipole interaction (DDI). Whilethe effect of the short-range van der Waals interaction still
freezes out at low temperatures, as it does for nondipolar
fermions, the DDI prevents the elastic cross section
between identical fermions from vanishing. The corresponding Wigner threshold law, governing the threshold
behavior of two-body scattering, gives a finite andenergy-independent elastic cross section [18–20]. As a
key consequence, identical dipolar fermions can collide
even in the zero-temperature limit.
Ultracold dipolar scattering is currently attracting arenewed interest in connection with recent experimentson polar molecules [21,22] and strongly magnetic atoms[13,23,24]. Early theoretical work on H atoms and atoms
in electric fields suggested that dipolar scattering could provide an elastic cross section that is large enough for direct
evaporative cooling of identical fermions [25–28]. Recent
0031-9007=14=112(1)=010404(5)theoretical work has elucidated the universal character ofthe dipolar scattering [29–31] and found that the elastic
dipolar cross section is determined only by the mass andthe dipole moment of the particles [30]. Recent experiments
on fermionic ground-state polar KRb molecules have tested
this prediction and have obtained evidence for the anisotropic character of the DDI [21]. Experiments on using
dipolar scattering for evaporative cooling have beenreported for fermionic Dy [13] and KRb molecules [32],both reaching temperatures on the order of the Fermi temperature T F .
In this Letter, we report on the creation of a quantum
degenerate dipolar Fermi gas of 167 Er atoms. We demonstrate a powerful approach in which the underlying coolingmechanism relies solely on dipolar scattering between spinpolarized fermions. We observe a remarkably high coolingefficiency, leading to very dense Fermi gases with typically
6.4 × 104 atoms at a temperature of T=T F ¼ 0.2 and a peakdensity of 4 × 1014 cm−3 . Finally, we confirm the prediction of the universal dipolar scattering theory [29,30] bymeasuring the Er elastic cross section in spin-polarized fermions via cross-dimensional thermalization [33]. Our work
opens up a conceptually novel pathway to quantum degeneracy in dipolar systems that can be generalized not only toother strongly magnetic atoms but also to ground-statepolar molecules, for which the implementation of sympathetic cooling might be difficult.
The strong dipolar character of Er originates from its
large magnetic moment μ of 7μB , where μB is the Bohrmagneton, and its large mass [20,34]. Among the six stable
isotopes, Er has one fermionic isotope, 167 Er, with a large
natural abundance of 23%. While the bosonic isotopes
have no hyperfine structure, 167 Er has a nuclear spin
I ¼ 7=2, giving rise to a manifold of eight hyperfine levels
and 104 magnetic sublevels in the electronic ground state[35]. In spite of the much more complex energy structure of
the fermionic isotope, our approach to quantum degeneracy

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is very similar to the one we have successfully used to condense the bosonic isotope 168 Er [24,36]. It consists of alaser cooling stage followed by direct evaporative coolingin an optical dipole trap (ODT). The fundamental difference with respect to the bosonic case is that the thermalization between spin-polarized fermions proceeds solely
through dipolar elastic collisions. In the present work,
we focus on spin-polarized fermions in the lowest hyperfine sublevel jF ¼ 19=2; mF ¼ −19=2i, where F is thetotal spin quantum number and mF is its projection along
the quantization axis.
Our laser cooling scheme relies on a Zeeman slower
operating at 401 nm and on a magneto-optical trap
(MOT) based on a narrow line at 583 nm [36]. Both lightfields act on transitions with quantum numbers
F ¼ 19=2 → F0 ¼ 21=2, which are sufficiently closed
for laser cooling. In our scheme, fermions in the MOT
are naturally spin-polarized into the lowest magnetic sublevel j19=2; −19=2i because of a combined effect of gravity and the MOT light [36]. We typically capture 1 × 107
atoms at T ¼ 7 μK in the MOT. All measurements in thepresent work are performed by absorption imaging on the
401-nm transition.
For evaporative cooling, we first transfer the atoms from
the MOT into a single-beam large-volume ODT at 1064 nm
and then into a tightly focused ODT at 1570 nm. The firsttrap is used as an intermediate step to increase the transfer
efficiency from the MOT. It consists of a single horizontal
beam with a power of 20 W and elliptical focus. The beam
waists are approximately 20 and 200 μm in the vertical andhorizontal direction, respectively. The corresponding trap
depth is roughly 100 μK. From the large-volume trap,
the atoms are loaded into a tightly focused ODT at
1570 nm. This second trap is made of a single horizontal
beam, which is collinear to the large-volume trapping beamand has a waist of 15 μm. The initial power of the 1570-nm
beam is 1.8 W, corresponding to trap frequencies ofðνx ; νy ; νz Þ ¼ ð2147; 51; 2316Þ Hz and a trap depth ofabout kB × 190 μK. Here, z is the direction of gravity.
At this stage, we have 1.5 × 106 atoms at T=T F ¼ 4.4 withT ¼ 28 μK and a peak density of about 1.2 × 1014 cm−3 .
The Fermi temperature is defined as T F ¼ hν̄ð6NÞ1=3 =kB ,where ν̄ is the geometric mean of the trap frequencies and his the Planck constant. We force evaporation by reducing
the power of the horizontal beam in a near-exponential
manner. When T F is reached, we introduce a vertical beamat 1570 nm to confine the fermions into the crossed region
created by the two beams and to preserve the atomic density. Its power is gradually increased and reaches 1.2 W at
the end of the evaporation. The vertical beam has a beam
waist of 33 μm. During evaporation, we apply a homogeneous guiding magnetic field to maintain the spin polarization in the system. At high temperature, the magnetic
field value is about 1.7 G, which is large enough to avoid
any thermal excitation into higher spin states. For

temperature below 3.2T F, we decrease the value of themagnetic field to 0.59 G, where we observe a slightly better
evaporation efficiency. After 10 s of forced evaporation, we
obtain a deeply degenerate Fermi gas.
Figure 1 shows a typical time-of-flight (TOF) absorption
image of a degenerate dipolar Fermi gas of N ¼ 6.4 × 104
and a peak density of n0 ¼ 4 × 1014 cm−3 at T=T F ¼
0.21ð1Þ with T F ¼ 1.33ð2Þ μK. At this point, our trap
frequencies are (470,346,345) Hz. Fermi degeneracy
reveals itself in a smooth change of the momentum distribution from a Maxwell-Boltzmann to a Fermi-Dirac distribution [37]. Correspondingly, the atomic density profile isexpected to change its Gaussian shape into a polylogarithmic one. A fit to TOF images reveals that at temperatures
above ≈0.5T F the Gaussian and polylogarithmic function
are hardly distinguishable from each other and both
describe the data well. By further decreasing the temperature, we observe a gradually increasing deviation from theGaussian shape. This deviation is evident in Fig. 1, whichshows a density profile at T=T F ¼ 0.21ð1Þ. A Gaussian fit
to the outer wings of the cloud, i.e., outside the disk withradius w, with w being the 1=e diameter of the Gaussian fit
to the entire cloud, clearly overestimates the population at
the center of the cloud. This is a fingerprint of Fermi degeneracy, meaning that the population of low-energy levels is
limited by the Pauli exclusion principle.
In all our measurements, we extract T=T F from fits to thedensity profiles by using either a polylogarithmic or a
Gaussian function. In the former case, the fit gives boththe fugacity ζ and the parameter σ characterizing thewidth of the distribution. The fugacity directly gives
T=T F ¼ ½−6 × Li3 ð−ζÞ−1=3 , with Lin being the nth-orderpolylogarithmic function [7,9]. The parameter σ is related
to the atomic temperature by T ¼ mσ 2 =ðkB t2TOF Þ, wheretTOF is the time of flight and m is the mass of 167 Er,(a)

0.5 mm(b) 6
linear density (108 m-1)

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z
x

0

0.10
0.20
optical density

x profile42
0
6

z profile

4
2
0
0

150
300
position (µm)

450

FIG. 1 (color online). Time-of-flight absorption image of a degenerate Fermi gas of Er atoms at T=T F ¼ 0.21ð1Þ after tTOF ¼
12 ms of expansion (a) and its density distribution integrated
along the z direction (upper panel) and x direction (lower panel)(b). The observed profiles (circles) are well described by fitting apolylogarithmic function to the data (solid lines), while they substantially deviate from a fit using a Gaussian distribution to theouter wings of the cloud, i.e., w (dashed lines). The absorption
image is averaged over six individual measurements.

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and together with T F , calculated from N and ν̄, gives amore indirect value for T=T F. We determine T=T F by using
both methods, which show well consistent results.
To get deeper insights into the evaporation process andthe underlying collisional properties, we study the evaporation trajectory. Figure 2 summarizes our results. We
observe that the evaporation first proceeds with high efficiency down to temperatures well below T F and then plateaus at about T=T F ¼ 0.2. The latter behavior suggeststhat further cooling is limited by Pauli blocking [4,6,7,9]
and that more thoroughly optimized evaporation ramps
might be needed to reach even lower temperatures. The
deepest degeneracy we attained is T=T F ¼ 0.19ð1Þ with
N ¼ 4.0 × 104 . From the slope of the evaporation trajectory, we obtain the efficiency parameter γ. This parameter
quantifies the gain in phase-space density (PSD) at theexpense of the atom number and can be written as
γ ¼ −dðln PSDÞ=dðln NÞ ¼ −3 × dðln T=T F Þ=dðln NÞ.
From a linear fit to the data down to T=T F ¼ 0.2, we findγ ¼ 3.5ð2Þ. This remarkably large number is in the leagueof the best evaporation efficiencies observed in experimentswith ultracold atoms based on s-wave scattering, including
our experiments with the bosonic 168 Er [24] and experiments on strongly interacting two-component Fermi gases[5,38,39].
Our interpretation of the cooling process in terms ofdipolar scattering relies on the full spin polarization of
the sample. Another spin state being present would leadto s-wave collisions in the sample. Therefore it is important
to make sure that we do not have any other spin state
present. For this reason, we carry out a dedicated set ofStern-Gerlach-type measurements at various stages of theevaporation. During the whole evaporation sequence, we
never observe any population in spin states different from
10

(a)

(b)

the mF ¼ −19=2 state. Figure 3 show the relevant portionof the TOF image, where atoms are observed. To identify
unambiguously the spatial positions of the different spincomponents, we intentionally prepare a spin mixture byradio-frequency (rf) transfer; see Fig. 3. It is worth mentioning that we observe fast spin relaxation when a multicomponent mixture is prepared [40].
The effectiveness of our evaporative cooling schemesuggests a very favorable ratio of the elastic scattering
rate to the inelastic one. We explore elastic scattering bymeasuring the elastic dipolar cross section σ el in our
spin-polarized fermionic sample via cross-dimensional
thermalization experiments [33]. We compress the system
in one spatial direction by increasing the power of the vertical beam by about a factor of 3. We then monitor the time
evolution of the temperature in the other direction, as
shown in the inset in Fig. 4. The time constant τ forcross-dimensional thermalization is directly connected toσ el through the relation τ ¼ α=ðn̄σ el vÞ, where α is the number of collisions
toﬃ thermalize, n̄ is the mean denﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
prequiredsity, and v ¼ 4 kB T=ðπmÞ is the mean relative velocity. A
delicate point of our analysis is the estimation of α, which
depends on the underlying scattering mechanism. We
employ α ¼ 4.1, which has been numerically calculated
for nondipolar p-wave collisions and has been applied to
KRb polar molecules [21]. Although p-wave collisions
are expected to be the leading term in dipolar scatteringof identical fermions, more detailed calculations of α might
be needed to fully account for the mixing of partial waves
resulting from the DDI [41].
In this way, we explore elastic scattering over a wide
range of atom numbers from 3 × 104 to 1.1 × 105 andfor various final temperatures ranging from 300 to
600 nK. Our findings at 0.59 G [42] are shown in
Fig. 4. In the nondegenerate regime (T ≳ T F ), we obtaina constant elastic cross section with a mean value of
2.0ð5Þ × 10−12 cm2 , corresponding to ½2.7ð3Þ × 102 a0 2 ,T/TF

T (µK)

10

(a)

(b)

(c)

(d)

(e)

(f)

1

mF
-15/2

1

-17/2
-19/2
0.1

0

5

10

Evaporation time (s)

15

0.1
10 4

10 5

10 6

T/TF = 0.90(6)N

FIG. 2 (color online). Evaporation trajectory to Fermi degeneracy. (a) Temperature evolution during the evaporation rampand (b) corresponding T=T F versus N. The ratio T=T F is obtained from the width σ of the distribution (triangles) and fromthe fugacity (circles); see the text. The error bars originate fromstatistical uncertainties in temperature, number of atoms, andtrap frequencies for the width measurements and the standard
deviations obtained from several independent measurements
for the fugacity. The solid line is a linear fit to the data for
0.2 < T=T F < 4.

0.71(1)0.47(1)0.33(1)0.27(1)

rf spin flip

FIG. 3 (color online). Absorption images of the atomic cloudwith a Stern-Gerlach separation of the spin components. A magnetic field gradient of about 40 G=cm is applied during the expansion for about 7 ms. (a)–(e) Along the entire evaporative
cooling sequence, atoms are always spin-polarized in the lowest
hyperfine sublevel jF ¼ 19=2; mF ¼ −19=2i. T=T F of theatomic samples are indicated in each panel. In (f), the image
is obtained right after rf mixing of the spin states for the sample
at T=T F ¼ 0.33ð1Þ. The three clouds correspond to the magnetic
sublevels mF ¼ −19=2, −17=2, and −15=2 from bottom to top.

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2

600
550

1

Tz (nK)

effective cross section (10-12 cm2)

PRL 112, 010404 (2014)

500
450
400
0

200
400
holding time (ms)

600

0
0.6

0.8

1.0

temperature after thermalization (TF)

FIG. 4 (color online). Effective elastic cross section as a function of T=T F after thermalization. In the nondegenerate regime,
the effective cross section is constant and gives a mean value of
2.0ð5Þ × 10−12 cm2 . The error bars for each point contain thestatistical uncertainties of the time constant for cross-dimensional
thermalization, of the trap frequencies, and of the temperature. A
typical cross-dimensional thermalization measurement with an
exponential fit to the data is shown in the inset. T z is the temperature along the z direction.

where a0 is the Bohr radius. The error bar is mainly due tosystematic uncertainties in trap frequencies, temperature,and number of atoms. Below T F, the effect of quantum
degeneracy becomes visible through a suppression of scattering events caused by Pauli blocking. In this regime, we
can interpret our measurements in terms of an effective
elastic cross section, which also includes the Pauli suppression factor. As expected, we observe a substantial decreaseof the effective σ el for decreasing T=T F , similarly to thecase of s-wave collisions between fermions in different spinstates [44].
Dipolar scattering theories predict an energy-independent
elastic cross section for identical fermions in the low-energy
regime [18–20]. The cross section is predicted to follow auniversal scaling law that is fully determined by a single
parameter—the dipolar length D [30]—and it reads as
σ el ¼ 6.702 × D2 ;(1)

where D ¼ 2π 2 d2 m=h2 with d2 ¼ μ0 μ2 =ð4πÞ and μ0 being
the vacuum permeability. This equation shows a clear analogy to the ordinary s-wave scattering, where D plays therole of the scattering length. For the Er parameters, the universal theory predicts σ el ¼ 1.8 × 10−12 cm2 , which is inreasonable agreement with the measured value. The small
deviation might be due to the chosen value for α, to systematic errors, or to a residual effect of the short-range physics,
which is not included in the theory.

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Our observations suggest that inelastic losses are very
weak. Since the atoms are fully polarized in the lowest spinstate, inelastic losses can be caused only by collisions withthe background gas and by three-body decay. To investigate
this more quantitatively, we carry out atom-decay measurements by recording the number of atoms as a function of thehold time in an ODT initially loaded with N ≃ 1 × 105
atoms at T=T F ≃ 0.47. In spite of the very high peakdensity of 3 × 1014 cm−3 , we find the atom number todecay in a purely exponential manner (time constant40 s) without showing any signature of three-body processes. From this observation we can derive an upper limit
for the three-body recombination rate constant as low
as L3 ≤ 3 × 10−30 cm6 =s.
The remarkable efficiency of evaporative cooling in asingle-component Fermi gas of Er and the exceptionallyhigh densities together with low inelastic collision rates
can be understood in terms of a very favorable combinationof the DDI with the p-wave barrier. While DDI is strong
enough to provide us with a sufficient cross section for elastic collisions, it is weak enough to preserve a substantial
repulsive barrier for any alignment of the colliding dipoles.
Even for the case of maximum dipolar attraction (head-totail configuration), the effective potential, given by theinterplay between the p-wave barrier and the DDI, featuresa repulsive barrier with a maximum height Vðrmax Þ ¼
2ℏ2 =ð27mD2 Þ at rmax ¼ 3D. For Er, the barrier height
still exceeds kB × 7 μK, which is much larger than all collision energies in the final evaporation stage. This prevents
atoms from getting close to each other, and three-bodydecay, which requires short-range interactions, is strongly
suppressed.
In conclusion, we produce a degenerate dipolar Fermi
gas of 167 Er atoms. We demonstrate direct evaporative cooling of identical fermions via universal dipolar scattering.
Our method provides two key advantages: feeble inelastic
losses and exceptionally high attainable densities. The former aspect is favorable for reaching low values of T=T F ,which are ultimately limited by the so-called hole-heating
mechanism caused by inelastic losses [45,46]. The latteraspect has important consequences for dipolar physics.
The relevant energy scale for dipolar phenomena at the
many-body level is given by n0 d2 [20,34]. Given the highdensities achieved here, our degenerate Fermi gas of Er currently is the most dipolar quantum gas available in experiments, with n0 d2 being 0.92% of the Fermi energy. We
speculate that even much higher densities than the ones
here attained may be achieved, since we do not see any limiting process. This may open a way for observing p-wave
pairing in dipolar gases and for the creation of an anisotropic Fermi superfluid [47,48].
We are grateful to J. Bohn, C. Salomon, M. Baranov, andM. Zwierlein for fruitful discussions. We thank M. Springer
for technical support and NKT Photonics for lending us the1570-nm fiber laser. This work is supported by the Austrian

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Ministry of Science and Research (BMWF) and theAustrian Science Fund (FWF) through a START grant
under Project No. Y479-N20 and by the European
Research Council under Project No. 259435. K. A. is supported within the Lise-Meitner program of the FWF.

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3.4. Publication:
Anisotropic Relaxation Dynamics in a Dipolar Fermi Gas
Driven Out of Equilibrium†
Physical Review Letters 113, 263201 (2014)
submitted 6 May 2014; published 23 Dec 2014
DOI 10.1103/PhysRevLett.113.263201

Kiyotaka Aikawa,1 Albert Frisch,1 Michael Mark,1 Simon Baier,1 Rudolf Grimm,1,2 John L.
Bohn,3 Deborah S. Jin,3 George M. Bruun4 and Francesca Ferlaino1,2
1Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3
JILA, NIST, and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
4
Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C,
Denmark†

The author of the present thesis helped performing the measurements and analyzing the data together
with A. F., M. M. and K. A., and contributed in writing the manuscript. The theoretical calculations of
this publication have been contributed by J. B., D. J., and G. B.

36

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week ending
31 DECEMBER 2014

Anisotropic Relaxation Dynamics in a Dipolar Fermi Gas Driven Out of Equilibrium
1

K. Aikawa,1 A. Frisch,1 M. Mark,1 S. Baier,1 R. Grimm,1,2 J. L. Bohn,3 D. S. Jin,3 G. M. Bruun,4 and F. Ferlaino1,2

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
JILA, NIST, and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
4
Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark(Received 6 May 2014; published 23 December 2014)
We report on the observation of a large anisotropy in the rethermalization dynamics of an ultracold
dipolar Fermi gas driven out of equilibrium. Our system consists of an ultracold sample of strongly
magnetic 167 Er fermions, spin polarized in the lowest Zeeman sublevel. In this system, elastic collisions
arise purely from universal dipolar scattering. Based on cross-dimensional rethermalization experiments,
we observe a strong anisotropy of the scattering, which manifests itself in a large angular dependence of thethermal relaxation dynamics. Our result is in good agreement with recent theoretical predictions.
Furthermore, we measure the rethermalization rate as a function of temperature for different angles
and find that the suppression of collisions by Pauli blocking is not influenced by the dipole orientation.
DOI: 10.1103/PhysRevLett.113.263201

PACS numbers: 34.50.Cx, 37.10.De, 47.70.Nd, 67.85.Lm

The behavior of any many-body system follows from theinteractions of its constituent particles. In some cases ofphysical interest, importantly at ultralow temperature,
where the de Broglie wavelength is the dominant lengthscale, these interactions can be simplified by appealing to
the Wigner threshold laws [1,2]. These laws, which havebeen extensively studied for particles interacting via vander Waals forces, both in experiment and theory [3],identify the interactions via simple isotropic parameterssuch as a scattering length. However, for dipolar particles,
the fundamental interaction is anisotropic and the systemproperties can depend on the orientation of the gas withrespect to a particular direction in space [4,5].
One of the major strengths of ultracold matter is its
susceptibility to being controlled by various means.
Striking examples include traversing the BEC-BES crossover [6,7], welding atoms together into molecules [8,9],
and inducing bosons to behave like fermions in one spatial
dimension [10,11]. Most often, such control exploits thequantum mechanical nature of a many-body gas at ultralow
temperature, and arises from the manipulation of isotropic
scattering between constituent particles. However, in thecase of dipolar particles the scattering is intrinsicallyanisotropic, affording novel opportunities to control the
behavior of the gas. For example, the anisotropic d-wave
collapse of a Bose-Einstein condensate of magnetic atoms[12,13] and the deformation of the Fermi sphere in a dipolar
Fermi gas [14] have been observed. These phenomena relyon the collective behavior of all the particles, occurring
according to their mean field energy.
Distinct from such many-body effects, dipoles can also
influence the properties of the gas via two-body scattering.
Since scattering of dipoles is highly anisotropic, properties
that require the collisional exchange of energy and0031-9007=14=113(26)=263201(5)momentum between the atoms, such as sound propagation,
viscosity, and virial coefficients [15], will be influenced by
the presence of dipoles. In particular, differential crosssections of dipolar particles are highly anisotropic, depending on both the initial, as well as scattered, relative
directions of the colliding particles, and it has recently
been predicted that dipolar anisotropy can exert a profoundinfluence on the nonequilibrium dynamics in such agas [16].
In this Letter, we demonstrate the control of the thermal
relaxation dynamics of a dipolar Fermi gas driven out of
equilibrium by adding excess momentum along one axis.
The control is achieved by changing the orientation of thedipoles relative to this direction, enabling us to substantially vary the rethermaliztion rate. As a striking consequence of the interaction anisotropy, we find that the rate ofequilibration can vary by as much as a factor of 4,
depending on the angle between the dipole orientationand the excitation axis. Furthermore, we observe that therethermalization rate decreases as the temperature is lowered. This effect is due to the lack of available final states
into which atoms can scatter and is known as Pauliblocking. Our results provide evidence that the Pauli
suppression of collisions does not contribute additional
anisotropic effects to the rethermalization.
To realize a dipolar Fermi gas, we use an ultracold spinpolarized sample of strongly magnetic erbium (Er) atoms,
which possess a magnetic dipole moment of 7 Bohrmagneton. This is an ideal system to study purely dipolar
scattering since short-range van der Waals forces give anegligible contribution to the scattering of identical fermions at ultralow temperatures [17]. Our experimental
procedure to create a degenerate Fermi gas of 167 Er atoms
follows the one described in Refs. [18,19]. In brief, it

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PRL 113, 263201 (2014)

comprises laser cooling in a narrow-line magneto-opticaltrap (MOT) [21] and direct evaporative cooling of a spinpolarized sample in an optical dipole trap (ODT) [18].
Evaporative cooling, which was successfully used to reach
Bose-Einstein condensation [22,23], relies on efficient
thermalization. In our case, this is achieved by elastic
dipolar collisions between spin-polarized fermions.
We optimize the evaporative cooling sequence andproduce a degenerate Fermi gas of about 3.0 × 104 atoms
at a temperature as low as T=T F ¼ 0.11ð1Þ. This gives apeak density of about 3 × 1014 cm−3 ; see Fig. 1. Here,T F ¼ 1.06ð5Þ μK. To achieve such a deeply degenerate
regime, we confine the atoms more tightly than in ourprevious work [18] by decreasing the beam waist of thevertical 1570 nm beam from 33 to 21 μm. The trap
frequencies in this configuration are ðνx ; νy ; νz Þ ¼
ð509; 447; 262Þ Hz. Our minimum temperature is comparable to the lowest ones achieved with sympathetic coolingschemes based on s-wave scattering [6,24,25].
While the crucial role of the long-range character ofthe dipole-dipole interaction (DDI) clearly emerges in theevaporation of spin-polarized fermions, the role of theanisotropy of the interaction is more subtle. We investigate
this aspect by studying for various dipole orientations how
the system rethermalizes when it is excited out of its
equilibrium state. The dipole orientation is controlled bychanging the direction of the polarizing magnetic field andis represented by the angle β between the magnetic field
and the weak axis of the ODT; see inset in Fig. 2.
The cross-dimensional rethermalization experiments
proceed as follows. We first prepare a nearly degenerate
Fermi gas of about 8 × 104 atoms at T=T F ≃ 0.6 withT F ≃ 600 nK in a cigar-shaped ODT with frequencies of
5

(b) 4
-1
8

0

0.04 0.08 0.12
Optical density(393,23,418) Hz. We then change the dipole orientationfrom β ¼ 90°, which is used for evaporative cooling, to thedesired value and excite the system by increasing thepower of the vertical beam by about a factor of 2.8 within
14 ms. After the excitation, the trap frequencies are(393,38,418) Hz. The excitation brings the system out of
equilibrium by transferring energy mainly in the directionof the weak axis (y), which we refer to as the excitation
axis. This process is nearly adiabatic with respect to the trap
period for the radial motion and introduces an initial
temperature imbalance of about T y =T z ¼ 2. We follow
the reequilibration dynamics by recording the time evolution of the temperature along the z direction T z and we
extract the rethermalization rate for the given dipole
orientation.
Figure 2 shows typical temperature evolutions, measured
for three different values of β. We observe that thereequilibration dynamics strongly depends on β withT z approaching the new equilibrium value in a nearexponential way. From an exponential fit to the data, we
extract the relaxation time constant τ. The slower rethermalization (i.e., largest τ) is found to occur for dipolesoriented perpendicular to the excitation axis, whereas bychanging β by about 45° we observe that the system
reequilibrates about 4 times faster. This strong angle
dependence clearly shows the anisotropic nature of therelaxation dynamics.
Since rethermalization relies on elastic collisions, its rate
1=τ should be directly proportional to the total elastic crosssection σ el [16]. The standard way to evaluate the latter isby integrating the differential cross sections dσ el =dΩðk0 ; kÞover all final directions of the atoms’ relative momentum
k0 , and averaging over incident directions k. For dipolar
550

X Profile

3
2

500

1
0
5
4
3
2
1
0

Tz(nK)

0.2 mmLinear density (10 m )(a)week ending
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PHYSICAL REVIEW LETTERS

Z Profile450

β = 90 deg.
β = 109 deg.
β = 138 deg.

β
0

100

200

300

400

400

500

Position (μm)

0FIG. 1 (color online). Time-of-flight absorption image of adegenerate Fermi gas of 3.0 × 104 Er atoms at T=T F ¼ 0.11ð1Þ
after 12 ms of expansion (a) and its density distribution integrated
along the x direction (upper panel) and z direction (lower panel)(b). The observed profiles (circles) are well described by fitting
the Fermi-Dirac distribution to the data (solid lines), while they
substantially deviate from a fit using a Gaussian distribution tothe outer wings of the cloud (dashed lines), i.e., outside the disk
with radius w, where w is the 1=e radius of the Gaussian fit to theentire cloud. The absorption image is averaged over 20 individual
measurements.

100

200

300

400

500

Holding time (ms)FIG. 2 (color online). Typical cross-dimensional thermalization
measurements for three dipole orientations: β ¼ 90° (squares),β ¼ 109° (circles), and β ¼ 138° (triangles). The time evolutionof the temperature in the z direction T z is plotted as a function ofholding time after the cloud is excited in the y direction. The
geometry of the cigar-shaped trap and the coordinates areindicated in the inset. The yellow arrow represents the dipole
orientation. After the equilibration, the Fermi gas is at T=T F ≃0.75 with T F ≃ 710 nK.

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PRL 113, 263201 (2014)α:n̄ησ el v

ð1Þ

The dimensionless proportionality constant α is commonlyreferred to as the number of collisions for thermalizationand can be computed from the known differential crosssections. For short-range interactions, Monte Carlocalculations yield α ¼ 2.7 and α ¼ 4.1 for s- and p-wave
collisions, respectively [28,29]. As predicted in Ref. [16],in the case of the DDI, which is long range and anisotropic,
α is a function of the angle β. In Eq. (1), we also include aPauli suppression factor η, which accounts for the reductionof the rethermalization rate in a degenerate Fermi gas
caused by Pauli blocking, where η ¼ 1 for nondegenerategases; see later discussion.
To explore the angle dependence of α, we perform crossdimensional rethermalization experiments under the same
conditions as in Fig. 2 for various values of β in a range
between 30° and 160°. We extract α by using Eq. (1), theexperimentally measured τ, and the elastic dipolar crosssection σ el ¼ 1.8 × 10−12 cm2 calculated for Er [26]. At
T=T F ≃ 0.75, which is our experimental condition after
excitation, η ¼ 0.93; see later discussion. Figure 3 shows
our experimental result together with the theoretical valueof α, which we calculate using Enskog’s equation similarly
to Ref. [16]. We observe a remarkably good quantitative
agreement between the experiment and the theory, which
does not have any free parameter. The theoretical curve iscalculated assuming a velocity distribution that is MaxwellBoltzmann in form that allows the width of the distribution
to be characterized by different “temperatures” T x , T y , andT z in the three directions of the trap. We then use thecollision integral in Enskog’s formulation to calculate therelaxation rate for T z for the following initial conditions:T z ¼ T x , and T y ¼ 2T z . From this we determine thetheoretical value of α along the z direction; see Fig. 3.
Owing to the anisotropy of the dipolar collision crosssection, we find that α is, in general, different in the three
directions i ¼ x; y; z.
Both the experimental and theoretical results show astrong angular dependence of α, which largely varies in a5
p waveNumber of collision α

fermions, the integration yields an energy-independent σ el ,which is universally related to the fermions’ dipole momentd by σ el ¼ ð32π=15ÞD2, where D ¼ 2π 2 d2 m=h2 , with m
the mass and h the Planck constant, is the characteristic
length scale of the dipolar interaction [26]. A collision rate
can then be definedpasn̄σ el v, where
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ n̄ is the mean number
0
density and v ¼ 16kB T =ðπmÞ is the mean relative
velocity with T 0 the effective temperature including amomentum spread by the Fermi energy [27].
The actual reequilibration occurs at a rate different fromthis collision rate since rethermalization emphasizes thosecollisions that significantly change the relative direction of
the atoms’ momenta. The characteristic time for therelaxation is inversely proportional to the collision rate
τ¼

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31 DECEMBER 2014

PHYSICAL REVIEW LETTERS

4

3s wave

2

1

0
0

30

60

90

120

150

180

Orientation angle β (deg.)

FIG. 3 (color online). Angle dependence of the number ofcollisions α required to rethermalize ultracold dipolar fermions.
The experimental data (circles) are compared with the parameterfree theoretical prediction calculated under our trapping andexcitation conditions (solid line).

sinelike manner from about 1 to 4. This behavior is unique
to dipolar scattering and occurs because dipolar particles
have a spatial orientation dictated by the magnetic field andthe collision processes depend on this orientation [16].
Given this angular dependence, dipoles can reequilibrate ona time scale that can be even faster than the one achievablewith short-range s-wave collisions (α ¼ 2.7). In particular,
the rethermalization is the most efficient (smallest α) when
β ¼ 45°. This can be seen qualitatively from the form of thecross sections in Ref. [16]. Consider a simplified scattering
event, where the magnetic field B, the relative incident
wave vector k, and the relative scattered wave vector k0
all lie in the same plane. Moreover, suppose that k makes
an angle θ with respect to B, while k0 makes an angle θ0 ,defining a scattering angle θs ¼ θ0 − θ. In this reduced
case, the differential cross section for scattering of dipolar
fermions has the simple angular dependence dσ el =dΩ ∝cos2 ð2θ − θs Þ; thus, the most likely scattering occurs whenθs ¼ 2θ. For the most efficient rethermalization one
requires scattering at right angles, θs ¼ 90°, and, therefore,
collisions in which the atoms approach one another at
θ ¼ 45°. At the same time, from the experimental geometry
one requires a high collision rate between the radial andaxial trap directions. From these considerations, the mostefficient rethermalization should occur for a field tilted at
an angle β ¼ 45° from the trap’s axis.
Another central aspect in the scattering of ultracold
fermions in the degenerate regime is related to Pauliblocking of collisions. This effect is caused by the lackof unoccupied final states in a scattering event and leads toa reduction of the elastic collision rate, which has been
observed in fermionic systems with short-range interaction[29,30]. In the case of dipolar scattering, it is an interestingquestion whether or not the Pauli blocking effect exhibits
anisotropy.
To address this question experimentally, we explore
the temperature dependences of τ for two different angles,

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PHYSICAL REVIEW LETTERS

β ¼ 90° and 110°, for which α differs by a factor of 2; see
Fig. 3. For each temperature, we measure τ, n, v, and we
derive η=α using Eq. (1). To extract the Pauli suppression
factor η, we normalize η=α to its value in the nondegenerate
regime where η ¼ 1. As shown in Fig. 4, for both angles weobserve the same pronounced reduction of η with decreasing temperature. Specifically, η decreases by about 70%
from 1 to 0.2T=T F . Our results indicate that the reductionof the rethermalization rate is independent from the dipole
orientation.
In order to quantitatively confirm that the observed
reduction of η is caused by Pauli blocking of collisions,
we use a theoretical model following the one developed fornondipolar fermions in Refs. [31,32]. Our model is basedon a variational approach for solving the Boltzmann
equation to calculate the thermal relaxation rate including
the Pauli blocking effect. This gives [33]1 hΓ½ΦT ΦT i¼;
τ2hΦ2T ið2Þwhere the trap and momentum average is defined as h…i ¼̬R 3 R 3̬d r d kfð1 − fÞ… with d3 k ¼ d3 k=ð2πÞ3 , and fðk;rÞ ¼
fexp½βðk2 =2mþVðrÞ−μÞþ1g−1 is the equilibrium Fermi
function with VðrÞ the trapping potential. We have definedΓ½ΦT  ¼ I½ΦT =fð1 − fÞ withZ
̬dσ jk − k2 j
I½Φ ¼ d3 k2 dΩ el
ΔΦff 2 ð1 − f 3 Þð1 − f 4 ÞmdΩ
ð3Þthe collision integral describing the rethermalization due tothe collision of two particles with incoming momenta k andPauli suppression factor η

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2

0.4

0.6

0.8

1.0

Temperature (TF)

FIG. 4 (color online). The Pauli suppression factor η as afunction of the temperature. The experimental data are taken for
two different orientations of the dipoles, i.e., for β ¼ 90°(squares) and for β ¼ 110° (circles). The experimental values
are compared with the theoretical predictions on Pauli blocking(solid line); see text.

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k2 and outgoing momenta k3 and k4 . Here, ΔΦ ¼ Φ1 þ
Φ2 − Φ3 − Φ4 and Ω is the solid angle of the outgoing
relative momentum ðk4 − k3 Þ=2. We use ΦT ¼ k2y − k2 =3
as a variational expression for the deviation function
corresponding to a thermal anisotropy in the y direction.
This is also the approach used to calculate α as a function of
β for a Maxwell-Boltzmann distribution in Ref. [16] andused to calculated the shear viscosity in a gas interactingwith s-wave interactions [31,32].
By calculating the thermal relaxation rate using Eq. (2)with either a pure s-wave or a pure p-wave interaction, we
find that the relative suppression of the rate compared to theclassical value is essentially the same in the two partial
wave channels for a given temperature. This indicates that
the Pauli blocking effect is largely insensitive to the angulardependence of the cross section of the atoms. Hence, we
can safely use the p-wave cross section as a stand-in for thetrue dipolar scattering in this calculation. In the hightemperature limit T ≫ T F , the integrals in Eqs. (2)–(3)can be solved analytically and we get Eq. (1) with α ¼25=6 for p-wave scattering and α ¼ 5=2 for s-wave
scattering. These variational values agree well with theMonte Carlo results stated above.
In Fig. 4, we plot the theoretically predicted temperaturedependence of η using Eqs. (2) and (3) and assuming ap-wave cross section, together with the experimental
values. As explained above, the calculation yields essentially the same result when we assume s-wave scattering.
By fitting the theoretical curve to the observed rethermalization rates with a single free scaling parameter for eachangle, we determine the values of σ el =α to be 0.51ð1Þð12Þ ×
10−12 cm2 at β ¼ 90° and 1.00ð2Þð24Þ × 10−12 cm2 at
β ¼ 110°, where the errors are statistical errors of fitting
and systematic uncertainties on temperature, trap frequencies, and atom number, respectively. The theory andexperimental results are in good agreement. Our findings
show that the reduction of the rethermalization rate isindeed due to Pauli blocking and that it is not significantly
influenced by the polarization angle.
In conclusion, we have studied the rethermalization
dynamics of indistinguishable dipolar fermions after thesystem is excited out of thermal equilibrium. We haveobserved that the rate of equilibration can strongly varydepending on the polarization direction of the atoms’magnetic dipole moments, which demonstrates a remarkable influence of anisotropic scattering. Further, we notethat the anisotropy has no significant influence on the Pauliblocking effect. Our results are fundamentally important forunderstanding the collisional behavior of dipolar particles,
such as strongly magnetic atoms and polar molecules.
We are grateful to M. Baranov for fruitful discussions.
This work is supported by the Austrian Ministry of Scienceand Research (BMWF) and the Austrian Science Fund(FWF) through a START Grant under Project No. Y479N20 and by the European Research Council under Project

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PHYSICAL REVIEW LETTERS

No. 259435. K. A. is supported within the Lise-Meitnerprogram of the FWF. J. L. B. and D. S. J. acknowledge
support from U.S. National Science Foundation Grant
No. 1125844.

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[17] According to the Wigner threshold law for the short-range
van der Waals interaction, the cross section for identicalfermions rapidly vanishes with temperature. As a consequence, both the isotropic and anisotropic part of the van der
Waals interaction plays a negligible role in the scattering.

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Ferlaino, Phys. Rev. Lett. 112, 010404 (2014).
[19] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.113.263201,
which
includes Ref. [20], for experimental details.
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Makrides, A. Petrov, and S. Kotochigova, Nature (London)
507, 475 (2014).
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1
SUPPLEMENTARY MATERIAL: ANISOTROPICRELAXATION DYNAMICS IN A DIPOLAR FERMI GASDRIVEN OUT OF EQUILIBRIUM
Creation of a spin-polarized sample of 167 Er

After the MOT stage, in which the sample is simultaneously cooled and spin-polarized into the lowest hyperfine sublevel, we typically transfer 1.5 × 106 atoms at a temperatureof T = 28 µ K into a crossed ODT. The latter consists of ahorizontal beam, which propagates along the y direction, anda vertical beam, propagating along the z direction (directionof gravity). The horizontal beam has a beam waist of 15 µ m
and a wavelength of 1570 nm. For the vertical beam, we use
light that operates either at 1570 nm or at 1064 nm and various
values of the beam waist, depending on the final temperaturesand densities required for the specific experiments.

avoid loss features associated to the recently observed FanoFeshbach resonances [1]. At 0.58 G, the Zeeman energy splitting between magnetic sublevels is 0.6 MHz × h. This value ismuch larger than the typical Fermi energy (about 20 kHz × h)and the typical energy of the DDI (about 100 Hz × h).

The angle β and the amplitude of the magnetic field arecontrolled by three independent sets of coils along the x, y, andz directions. Each coil set is independently calibrated by using
radio-frequency spectroscopy to within 5 mG, from which weestimate the error on angle to be within 1 degree.

Magnetic field control

We apply a magnetic bias field of 0.58 G, oriented along
the z direction. We select this value of the magnetic field to[1] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn,
C. Makrides, A. Petrov, and S. Kotochigova, Nature 507, 475(2014).

3.5. Publication:
Observation of Fermi surface deformation in a dipolar
quantum gas†‡
Science 345, 6203 (2014)
submitted 25 Apr 2014; published 15 Aug 2014
DOI 10.1126/science.1255259

Kiyotaka Aikawa,1 Simon Baier,1 Albert Frisch,1 Michael Mark,1 Cornelis Ravensbergen,1,2Francesca Ferlaino1,2
1Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria†‡

The author of the present thesis performed the measurements together with K. A. and the help of C. R.,
M. M., and A. F., and contributed in writing the manuscript. This publication is also part of the Phd thesis
of Albert Frisch.
The post-print version of the publication is attached for online access of this thesis.

43

Observation of Fermi surface deformation in a dipolar quantum gas
K. Aikawa,1 S. Baier,1 A. Frisch,1 M. Mark,1 C. Ravensbergen,1, 2 and F. Ferlaino1, 2

arXiv:1405.2154v2 [cond-mat.quant-gas] 17 Jul 2014

1 Institut für Experimentalphysik and Zentrum für Quantenphysik,
Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2 Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria(Dated: July 18, 2014)

The deformation of a Fermi surface is a fundamental phenomenon leading to a plethora of exotic quantum
phases. Understanding these phases, which play crucial roles in a wealth of systems, is a major challenge inatomic and condensed-matter physics. Here, we report on the observation of a Fermi surface deformation ina degenerate dipolar Fermi gas of erbium atoms. The deformation is caused by the interplay between strong
magnetic dipole-dipole interaction and the Pauli exclusion principle. We demonstrate the many-body natureof the effect and its tunability with the Fermi energy. Our observation provides basis for future studies on
anisotropic many-body phenomena in normal and superfluid phases.
PACS numbers: 03.75.Ss, 37.10.De, 51.60.+a, 67.85.Lm

The Fermi-liquid theory, formulated by Landau in the late50’s, is one of the most powerful tools in modern condensedmatter physics [1]. It captures the behavior of interacting
Fermi systems in the normal phase, such as electrons in metals and liquid 3 He [2]. Within this theory the interaction is
accounted by dressing the fermions as quasi-particles with
an effective mass and an effective interaction. The groundstate is the so-called Fermi sea, in which the quasi-particles
fill one-by-one all the states up to the Fermi momentum, kF .
The Fermi surface (FS), which separates occupied from empty
states in k-space, is a sphere of radius kF for isotropically interacting fermions in uniform space. The FS is crucial forunderstanding system excitations and Cooper pairing in superconductors. When complex interactions act, the FS can
get modified. For instance, strongly-correlated electron systems violates the Fermi-liquid picture, giving rise to a deformed FS, which spontaneously breaks the rotational invariance of the system [3]. Symmetry-breaking FSs have beenstudied in connection with electronic liquid crystal phases [4]and Pomeranchuk instability [5] in solid state systems. Particularly relevant is the nematic phase, in which anisotropic behaviors spontaneously emerge and the system acquires an orientational order, while preserving its translational invariance.
This exotic phase has recently been observed by transport andthermodynamics studies in ruthenates [6], in high-transitiontemperature superconductors such as cuprates [7], and in othersystems [3].
A completely distinct approach to study FSs is provided
by ultracold quantum gases. These systems are naturally freefrom impurities and crystal structures, realizing a situation
close to the ideal uniform case. Here, the shape of the FS candirectly reveal the fundamental interactions among particles.
Studies of FSs in strongly interacting Fermi gases have been
crucial in understanding the BEC-to-BCS crossover, where
the isotropic s-wave (contact) interaction causes a broadening of the always-spherical FS [8]. Recently, Fermi gaseswith anisotropic interactions have attracted remarkable attention in the context of p-wave superfluidity [9, 10] and dipolar physics [11]. Many theoretical studies have focused on

dipolar Fermi gases, predicting the existence of a deformed
FS [12–17]. These studies also include an extension of the
Landau Fermi-liquid theory to the case of anisotropic interactions [18]. Despite recent experimental advances in polar
molecules and magnetic atoms [19–22], the observation ofanisotropic FSs has so far been elusive.
In this letter, we present the direct observation of the deformed FS in dipolar Fermi gases of strongly magnetic erbium (Er) atoms. By virtue of the anisotropic dipole-dipole
interaction (DDI) among the particles, the FS is predicted
to be deformed into an ellipsoid, reflecting the underlying
symmetry of the interaction for polarized gases. To minimize the system’s energy, the FS elongates along the direction of the maximum attraction of the DDI, where the atomic
dipoles have a ’head-to-tail’ orientation. To understand theorigin of the Fermi surface deformation (FSD), one has to account for both the mechanical action of the DDI in k-space
and the Pauli exclusion principle, which imposes the manybody wave-function to be anti-symmetric. In the HartreeFock formalism, the FSD comes from the exchange interaction among fermions, known as the Fock term ( [12, 16] andSupplementary Materials). Our observations agree very wellwith parameter-free calculations based on the Hartree-Focktheory [12, 15, 17]. We demonstrate that the degree of deformation, related to the nematic susceptibility in the liquidcrystal vocabulary, can be controlled by varying the Fermi energy of the system and vanishes at high temperatures.
Our system is a single-component quantum degenerate
dipolar Fermi gas of Er atoms. Like other lanthanoids, a distinct feature of Er is its large permanent magnetic dipole moment µ (7 Bohr magneton), entailing the strong DDI among
the fermions. Similarly to our previous work [22], we takeadvantage of elastic dipole-dipole collisions to drive efficientevaporative cooling in spin-polarized fermions. The sample
is confined into a three-dimensional optical harmonic trap andtypically contains 7×104 atoms at a temperature of 0.18(1) TF
with TF = 1.12(4) µK (Supplementary Materials). We control
the alignment of the magnetic dipole moments by setting theorientation of an external polarizing magnetic field. The quan-

2
kz

kF

1.04
1.02

1.04

Aspect Ratiokx

Aspect Ratio

1.03
1.02

1.00
0.98
0.96

1.01

0.94

1.00

β

0.99

5

0.98
0.97
0

20

40

60

Dipole orientation β (deg.)

80

FIG. 1: (color online) AR of an expanding dipolar Fermi gas as afunction of the angle β . In this measurement, the trap frequencies are( fx , fy , fz ) = (579, 91, 611) Hz. The data are taken at tTOF = 12 ms.
Each individual point is obtained from about 39 independent measurements. The error bars indicate the standard errors of the mean.
For comparison, the calculated values are also shown for 0◦ and 90◦(crosses). The inset schematically illustrates the geometry of the system. Gravity is along the z direction. The atomic cloud is imaged
with an angle of 28◦ with respect to the y axis (Supplementary Materials). The magnetic field orientation is rotated on the plane withan angle of 14◦ with respect to the xz plane. Schematic illustrationsof the deformed FS are also shown above the panel. Here, the Fermi
momentum for an ideal Fermi gas is shown as kF .

tity β symbolizes the angle between the magnetic field and thez axis (inset Fig. 1).
To explore the impact of the DDI on the momentum distribution, we perform time-of-flight (TOF) experiments. Since
its first use as ”smoking-gun” evidence for Bose-Einsteincondensation [23, 24], this technique has proved its power
in revealing many-body quantum phenomena in momentum
space [8, 25]. TOF experiments are based on the study of the
expansion dynamics of the gas when released from a trap. For
sufficiently long expansion time, the size of the atomic cloudis dominated by the velocity dispersion and, in the case ofballistic (free) expansions, the TOF images purely reflect the
momentum distribution in the trap.
In our experiment, we first prepare the ultracold Fermi gas
with a given dipole orientation and then we let the sample expand by suddenly switching off the optical dipole trap (ODT).
From the TOF images, we derive the cloud aspect ratio (AR),which is defined as the ratio of the vertical to horizontal radiusof the cloud in the imaging plane (Supplementary Materials).
Figure 1 shows the AR for various values of β . For verticalorientation (β = 0◦ ), we observe a clear deviation of the AR

10Time of flight (ms)15

FIG. 2: (color online) Time evolution of the AR of the atomic cloud
during the expansion. Measurements are performed for two dipole
angles, β = 0◦ (squares) and β = 90◦ (circles) under the same conditions as in Fig. 1. The error bars are standard errors of the meanof about 17 independent measurements. The possible origin of thefluctuations in the AR is carefully discussed in the Supplementary
Materials. The theoretical curves show the full numerical calculations (solid lines), which include both the FSD and the NBE effects,
and the calculation in the case of ballistic expansions (dashed lines),i. e. in the absence of the NBE effect. For comparison, the calculation
for a non-interacting Fermi gas is also shown (dot-dashed line).

from unity with a cloud anisotropy of about 3 %. TOF images
show that the cloud has an ellipsoidal shape with elongation in
the direction of the dipole orientation. When changing β , weobserve that the cloud follows the rotation of the dipole orientation, keeping the major axis always parallel to the directionof the maximum attraction of the DDI. In a second set of experiments, we record the time evolution of the AR during the
expansion for β = 0◦ and β = 90◦ (Fig. 2). For both orientations, the AR differs from unity at long expansion times. Our
results are strikingly different from the ones of conventional
Fermi gases with isotropic contact interactions, in which the
FS is spherical (AR = 1) and the magnetic field orientation
has no influence on the cloud shape.
The one-to-one mapping of the original momentum distribution in the trap and the density distribution of the cloudafter long expansion time strictly holds only in the case ofpure ballistic expansions. In our experiments, the DDI is acting even during the expansion and could potentially mask theobservation of the FSD. We evaluate the effect of the nonballistic expansion (NBE) by performing numerical calculations based on the Hartree-Fock mean-field theory at zerotemperature and the Boltzmann-Vlasov equation for expansion dynamics [15, 17] (Supplementary Materials). In Fig. 2,
the theoretical curves do not have any free parameter and are
calculated both in presence (solid lines) and absence (dottedlines) of the NBE effect. We observe an excellent agreement
between experiment and theory, showing that our model accurately describes the behavior of the system. In addition,
the comparison between ballistic and non-ballistic expansion

3
A

B

4

2

2

1
0
0.1

1

2

Trap anisotropy
Trap anisotropy
4

10 0
6

400

600

f (Hz)

D

-2

4
0

E
3

-0.01

2

0.9

1.0

1.1

1.2
-2

η (10 )

1.3

1.4

0

0.01

Optical Depth

∆ (%)

5 C

200

∆(% )

∆ (%)

3

440 µmFIG. 3: (color online) ∆ for various trap geometries. We consider
a cigar-shaped trap with fx = fz in the calculations and show the
behaviors of the FSD (dashed lines) and thep
NBE (dotted lines) separately as a function of the trap anisotropy fx fy / fz at f¯ = 400 Hzp(A) and as a function of f¯ at
fx fy / fz = 5 (B). (C) Experimentally observed ∆ at tTOF = 12 ms are plotted as a function of η, together with the full calculation (solid line) and the calculation considering only FSD (dashed line). The shaded area shows the uncertainty originating from the uncertainty in determining η in our experiments. The sample contains 6 × 104 atoms at a typical temperatureof T /TF = 0.15(1). The error bars represent standard errors of themean of about 15 independent measurements. The variation of thetrap anisotropy in the experiment is indicated in the top axis. Visualization of the FSD at η = 0.009 from the experimental TOF image(D) and from the fitted image (E).

reveals that the latter plays a minor role in the final AR, showing that the observed anisotropy dominantly originates fromthe FSD.
Theoretical works have predicted that the degree of deformation depends on the Fermi energy and the dipole moment [12, 14–18]. In the limit of weak DDI, the magnitudeof the FSD in a trapped sample is expected to be linearly
proportional to the ratio of the DDI to the Fermi energy,
η = nd 2 /EF [16]. Here, n = 4π(2mEF /h2 )3/2 /3 is the peak
number density at zero temperature with h the Planck constant, m the mass, d 2 = µ0 µ 2 /(4π) the coupling constant for
the DDI, and µ0 the magnetic constant. For a harmonicallytrapped ideal Fermi gas, the Fermi energy EF depends on theatom number N and the mean trap frequency f¯ = ( fx fy fz )1/3 ,√EF = h f¯(6N)1/3 . Given that η ∝ EF , the FSD can be tuned
by varying EF .
To test the theoretical predictions, we first numericallystudy the degree of cloud deformation ∆, √defined as ∆ =AR − 1, as a function of the trap anisotropy, fx fz / fy , and/or
f¯. To distinguish the effect of the FSD and of the NBE,
we keep the two contributions separated in the calculations

0 .2

0 .4
T e m p e ra tu re (T F)

0 .6

0 .8

FIG. 4: (color online) ∆ as a function of the temperature of the cloud.
Measurements are performed for two dipole angles, β = 0◦ (squares)and β = 90◦ (circles) under the same conditions as in Fig. 1. The error bars are standard errors of the mean of about 26 independent measurements. The solid lines show the numerically calculated values atzero temperature for β = 0◦ and β = 90◦ .

(Fig. 3A and 3B). Our results clearly convey the following information: (i) the FSD gives the major contribution to ∆, (ii)the FSD is independent from the trap anisotropy, while it increases with f¯, (iii) the NBE effect is reminiscent of the trap
anisotropy and vanishes for a spherical trap [15].
In the experiment, we explore the dependence of ∆ on thetrap geometry for β = 0◦ by keeping the axial frequency ( fy )constant and varying the radial frequencies ( fx = fz within
5 %) (Fig. 3C). This leads to a simultaneous variation of boththe trap anisotropy and f¯. We observe an increase of ∆ withη, which is consistent with the theoretically predicted linear
dependence [16].
In analogy with studies in superconducting materials [26],we graphically emphasize the FSD in the measurements at
η = 0.009 by subtracting the TOF absorption image taken at
β = 90◦ from the one at β = 0◦ (Fig. 3D). The resulting image
exhibits a clover-leaf-like pattern, showing that the momentum spread along the orientation of the dipoles is larger thanin the other direction. For comparison, the same procedure
is applied for images obtained by a fit to the observed cloud(Fig. 3E). At η = 0.009, the trap anisotropy is so small that the
NBE effect is negligibly small and the deformation is caused
almost only by the FSD.
Finally, we investigate the temperature dependence of ∆(Fig. 4). We prepare samples at various temperatures by stopping the evaporative cooling procedure at various points. The
final trap geometry is kept constant. When reducing the temperature, we observe the emergence of the FSD, which becomes more and more pronounced at low temperatures andeventually approaches the zero-temperature limit. The qualitative behavior of the observed temperature dependence isconsistent with a theoretical result at finite temperatures [16],although further theoretical developments are needed for a4more quantitative comparison.
Our observation clearly shows the quantum many-body nature of the FSD and sets the basis for future investigations
on more complex dipolar phenomena, including collective
excitations [15, 17, 27, 28] and anisotropic superfluid pairing [29, 30]. Taking advantage of the wide tunability of cold
atom experiments, dipolar Fermi gases are ideally clean systems for exploring exotic and topological phases in a highly
controlled manner [11].
We are grateful to A. Pelster, M. Ueda, M. Baranov,R. Grimm, T. Pfau, B. L. Lev, and E. Fradkin for fruitful discussions. This work is supported by the Austrian Ministryof Science and Research (BMWF) and the Austrian Science
Fund (FWF) through a START grant under Project No. Y479N20 and by the European Research Council under Project No.
259435. K. A. is supported within the Lise-Meitner programof the FWF.
Supplementary Materials
Experimental setup

We obtain a quantum gas of fermionic 167 Er atoms via laser
cooling in a narrow-line magneto-optical trap [31] followedby evaporative cooling in an ODT [22]. The sample is trappedin a crossed ODT consisting of a horizontally (y axis) and a
vertically (z axis) propagating beam at 1570 nm. The beam
waist of the horizontal beam is 15 µm, while the one of thevertical beam is tuned in a range from 20 µm to 90 µm to vary
the trap geometry from a nearly spherical shape to a nearlycigar shape. During the entire experimental procedure, thefermions are fully polarized into 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 thequantization axis. For maintaining the spin polarization of thetrapped sample, we apply an external magnetic field of 0.58 G.
At this field value, we do not observe any influence of Feshbach resonances [32]. The magnetic field orientation is controlled with two sets of coils. During evaporative cooling, themagnetic field is vertically oriented (β = 0◦ ). For imaging,
we rotate the magnetic field orientation to the direction of theimaging axis to attain a maximum optical depth.
Measurement of the AR

We measure the deformation of the cloud shape in TOF absorption images by using a standard poly-logarithmic fit for
the integrated density distribution of an ideal Fermi gas [33–35]!
(h − h0 )2 (z − z0 )2−(1)n(X, Z) = BLi2 −ζ exp −
2σz2
2σh2
where Lin is the n-th order poly-logarithmic function, ζ is the
fugacity, h and z indicate horizontal and vertical coordinates

A

0

B

OD

C

0.27

FIG. 5: (color online) Time-of-flight absorption image (A) and its
integrated profiles in the horizontal (B) and vertical (C) directions.
The image corresponds to the measurement at β = 0◦ in Fig. 1 and isaveraged over 39 independent measurements. The integrated profilesof the fit with a poly-logarithmic function are shown by solid lines.
The AR of the cloud is about 1.03.

on the imaging plane, respectively, h0 and z0 are the positions,and σh and σz are the radii. We define the AR of the observed
cloud as σz /σh . The imaging axis has an angle of 28◦ with respect to the y axis (inset Fig. 1) and thus the horizontal radiusin the imaging plane σh is related to the radii in the x and y
directions, σx , σy , byq(2)
σh = σx2 cos2 (28◦ ) + σy2 sin2 (28◦ )The fugacity ζ is directly connected to T /TF through the relation T /TF = [−6 × Li3 (−ζ )]−1/3 . The optical depth is proportional to N. Although the optical depth is also related to ζ
through N and TF by TF = h f¯(6N)1/3 /kB , we leave both freein the fitting procedure and confirm that they are consistent
with each other. Here, we assume that the cloud has a constantfugacity over the entire cloud because the momentum deformation is small. Rigorously speaking, TF is anisotropic and T
is constant over the cloud, and thus ζ should be anisotropic.
Dealing with such a distribution is beyond the scope of thepresent work. Figure 5 shows a typical TOF absorption image
and its integrated profiles as well as the integrated profiles ofthe poly-logarithmic fit. The fit is in excellent agreement withthe observed distribution.
By taking the average of about 20 independent measurements, we are able to determine the AR with a typical precision of 0.1 %, corresponding to the standard error of themean for multiple measurements. In addition, we find six
possible sources of systematic errors in the measured AR. (a)Variation in pixel sizes. The variation in pixel sizes over thearea of the cloud can introduce a systematic error in the AR.
There is no measured data available for our CCD camera (Andor, iXon3). (b) Residual interference fringes. Interference
fringes, arising from dusts on the imaging optics, can produce
a fixed background pattern on the image. (c) Finite pixel number. The finite number of pixels can limit the resolution of themeasurement of the AR, in particular at short TOF. From the
TOF measurements shown in Fig. 2, where the position of theatomic cloud varies with TOF by a free fall, we estimate the
combined effect of (a), (b), and (c) to be within ±0.5 %. (d)Error in the fitting procedure. Although our fitting procedure5
assuming a constant fugacity may give rise to a systematic error in deriving the AR, it is difficult to quantitatively estimate
it owing to the lack of an appropriate model. Investigating thiseffect will be an important future work. (e) Fluctuations in themagnetic field. The influence of the fluctuation in magnetic
field, which results in a fluctuation in the dipole orientation,is negligibly small at β = 0◦ and β = 90◦ (< 0.05 % in deformation). (f) Tilt of the camera. Assuming that the camera isaligned perpendicular to the imaging beam path within 1◦ , weinfer that the influence of the tilt of the camera on the AR isnegligible (< 0.02 %).
Physical origin of the deformation of the FS

Within the Hartree-Fock theory for a many-body system,
the DDI contributes to the total energy of the system in two
distinct ways: the Hartree direct interaction and the Fock exchange interaction [12, 16]. As compared to the case with a
non-interacting gas, the Hartree term gives rise to a distortionin position space, whereas the exchange term gives rise to a
distortion in momentum space. Previously, magnetostrictionin position space was observed in a dipolar BEC of chromium
atoms [36]. In a BEC, the Fock term is zero because of thesymmetric character of the many-body wave function. In an
isotropically interacting Fermi gas, the Hartree and the Fockterms cancel out [12]. The existence of the exchange term in
dipolar Fermi gases arises from the combined effect of the
DDI and the Pauli exclusion principle. In our expansion measurements, both the Hartree and the Fock terms need to beconsidered. The first is responsible for the NBE, while thesecond gives the FSD.

The expansion dynamics is calculated using the BotzmannVlasov equation for the Wigner distribution function under thescaling ansatz [40–42]. The scaling parameters, representing
variations from the equilibrium condition, are described bya set of coupled time-dependent differential equations. The
NBE effect is naturally included in this framework and occurs
predominantly within 1 ms after the release from the trap. We
numerically solve the equations for the general triaxial geometry, where the trap frequencies in three directions are different and the dipoles are oriented in the direction of one of thetrap axes. This reflects our experimental situation at β = 0◦ .
Although at β = 90◦ the dipole orientation has an angle of 14◦
with respect to the x axis, we assume that the dipole orientation is parallel to the x axis in our calculation.
We calculate the radii of the cloud on the image plane, taking into account the angle of 28◦ between the imaging axis
and the y axis by using eq. (2). In all our measurements, weobserve an asymmetry between β = 0◦ and β = 90◦ , i. e. |∆|is larger at β = 0◦ than at β = 90◦ . We observe this asymmetry also in the subtracted images in Fig. 3D and Fig. 3E as ahigher contrast in the vertical direction than in the horizontal
direction. This asymmetry is well reproduced by our calculation and is understood as follows. At β = 0◦ , the major axis of
the ellipsoid is oriented to the z direction and is fully imaged.
By contrast, at β = 90◦ , the major axis is not perpendicular to
the imaging plane and we observe a combined radius between
the major and the minor axis of the ellipsoid. Therefore, theobserved deformation at β = 90◦ is always smaller than the
one at β = 0◦ .

Calculation of the deformation

In the present work, the collision rate associated with universal dipolar scattering [22, 37] is lower than the lowest
trap frequency. Therefore, our sample is in the collisionlessregime, where the mean free path is longer than the size ofthe cloud [38]. We describe the trapped dipolar Fermi gas in
the collisionless regime in the zero temperature limit with anansatz that the Wigner distribution function is given as an ellipsoid
!
3 k2
3 r2
j
j
(3)
g(r, k,t) = Θ 1 − ∑ 2 − ∑ 2
j=1 K j
j=1 R jwhere Θ denotes the Heaviside’s step function, and r, k, andt denote coordinate, wave vector, and time, respectively. The
parameters R j and K j represent the Thomas-Fermi radius andthe Fermi momentum in the jth direction, respectively. These
parameters are numerically determined by minimizing the total energy in the presence of the DDI. The validity of this approach was numerically confirmed [39]. At equilibrium, theparameters K j contain the information of the anisotropic FS.

Image subtraction for Fig. 3D,E

The image shown in Fig. 3D is obtained as follows. The
TOF absorption images from 18 independent measurements
are averaged and binned by 2 × 2 pixels to reduce backgroundnoise. This procedure is applied for the measurements at
β = 0◦ and β = 90◦ . We subtract the image at β = 90◦ fromthe one at β = 0◦ . This image subtraction is very sensitive tothe relative position of the clouds on the two images down toa sub-pixel level. We obtain accurate positions of the centerof the cloud from the fit and shift the coordinate of the imageat β = 90◦ such that the center positions of two images exactly agree. We then apply spline interpolation for the imageat β = 90◦ to estimate the optical depth of the cloud at eachpixel position in the image at β = 0◦ . Unlike the procedureused in Ref. [26], where the anisotropy is extracted by rotating a single image by 90◦ and subtracting it from the original
image, our procedure with two images at two dipole orientations allows us to extract only the anisotropy originating fromthe DDI.

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Chapter

4

Resonantly interacting lanthanide
quantum gases

Ultracold gases owe their success to the high degree of parameter control available in experiments. In particular, a major step forward has been the access of controllable contact interactions. Interaction tuning is nowadays a well established tool, which allows
to enter various physical regimes. Importantly, contact interaction can be changed frompositive to negative values. Repulsive interactions can be crucial to attain stable BoseEinstein condensates (BEC), and the access of tunable interactions has been necessary tocondense e. g. 85 Rb [Cor00], 133 Cs [Web03], or 39 K [Roa07]. Attractive interactions can result in an implosion of the BEC, a so-called Bosenova, as beautifully demonstrated with85 Rb [Don01], or allow for the spectacular observation of bright solitons in one-dimensional
geometries [Str02, Kha02, Mar13]. The tuning can further be used for periodic modulation of
the interaction strength, which enables to access highly correlated states such as matter-wave
jets [Cla17], or to mimic occupation-dependent tunneling processes via Floquet engineeringin an atomic Bose-Hubbard model [Mei16]. In the case of degenerate Fermi gases, interaction
tuning opens the way for studies along the BEC-to-BCS crossover [Ing07].
When dipolar interactions are added to the scenario, tuning of the contact interaction cangive rise to even more intriguing scenarios. It is the competition of the isotropic contact
interaction with the anisotropic dipolar interaction that can disclose new quantum phases.
For dominating attractive dipolar interactions, a d-wave collapse of a BEC of chromium has
been observed [Lah08], and with the advent of strongly magnetic lanthanides the formationof quantum droplets, see [Kad16, FB16] and Appendix A.1, or the observation of roton
excitations, see Appendix A.3, has been made possible.
The list of observations, which is far from being complete, relies on the experimental controlof the s-wave scattering length as , which is directly proportional to the isotropic contact
interaction. Its tunability is based on the existence of Feshbach resonances (FRs), whicharise when a molecular state is brought into resonance with the two-body scattering state.
The resonantly enhanced coupling can strongly change the scattering behavior [Chi10]. The
availability of FRs crucially depends on the employed atomic species and isotope. To access physical phenomena as discussed above, it is thus from prime importance to acquire
knowledge on the Feshbach spectrum of the specific atomic species. In the case of alkali
atoms, Feshbach spectra are rather dilute, i. e. the density of FRs is low. Further, theoretical50

4.1. QUANTUM CHAOS IN LANTHANIDES

51

calculation, based on coupled-channel models and predicted interaction potentials, provide
good expectations of resonance positions and enable to assign specific resonances to single
molecular states. The story changes drastically for the case of lanthanide atoms with anisotropic interactions where a very high density of resonances is encountered [Fri14b, Bau14].
This high density is explained by a large number of molecular states that contribute to theatomic scattering process. A strong coupling between many different molecular states greatly
challenges theoretical models and so far a direct assignment of the entire observed Feshbach
spectra is absent.
Beside the application of interaction tuning, FRs allow to access molecular bound states andto form molecules out of two free atoms. With the platform of cold and ultracold molecules
exciting physics becomes available, ranging from chemical reactions and collisional studies,over precision measurements of fundamental constants, up to the creation of new quantum
phases or quantum information and simulation applications; for a review see e. g. Ref. [Car09].
A routinely used method for molecule creation uses an adiabatic sweep of the molecularchannel across the two-atom scattering channel. The resonant channel-channel interaction
results in an avoided crossing, enabling association of so-called Feshbach molecules. Initially,molecules have been formed out of single atomic species [Reg03, Cub03, Joc03a, Str03,
Her03]. Molecule-creation methods have been further utilized for the successful creation ofheteronuclear molecules [Osp06], and can also be applied to magnetic atoms.
This chapter is dedicated to resonantly interacting dipolar quantum gases. In Sec. 4.1 we will
give the framework for Feshbach resonances and discuss the emergence of FRs in magnetic
lanthanides. Our publication on the emergence of quantum chaos in lanthanides is coveredin Sec. 4.3. Section 4.2 reviews the creation of ultracold molecules, and finally, Sec. 4.4
contains our publication on the creation of Feshbach molecules of bosonic erbium atoms at
low magnetic field values. Our measurements of the molecules’ magnetic moment prove to
be key for the assignment of specific FRs in lanthanides with anisotropic interactions.

4.1. Quantum chaos in lanthanides
4.1.1. Feshbach resonances
In Sec. 3.1 we have seen that in ultracold experiments the two-body scattering behavior canbe captured within a simple description. For isotropically interacting bosons, the interaction
is described by s-wave scattering and quantified via the s-wave scattering length as . This
simple description does not come without surprises, as in principal the interaction of two
particles is determined by their interaction potential, i. e. the Born-Oppenheimer potential,
which can have a complicated form [Chi10]. However, at ultralow temperatures T the thermalde Broglie wavelength λdB ∝ T −1/2 has a large extension, and hence the real shape ofthe short-range interaction potential becomes irrelevant. Indeed, it is only necessary toknow the impact of the underlying potential on the scattering wave at large interatomic
separation R. The real interatomic potential can then be replaced by a spherically symmetric4.1. QUANTUM CHAOS IN LANTHANIDES
(a)

E(B)

B0

(b)

2

as (abg)0.0

entrance channel

1

∆

abg

0

Bzero

-1
-2
-0.5

energy (arb. units)energy (arb. units)

3

closed channel

52

-1.0
2

4
6
8
atomic separation (arb. units)

10

1

(c)

δB
Bc

0
-1

-2

Ec
-1

Eb
0
B - B0 (arb. units)

1

2

Figure 4.1.: FR in a two-channel model (a) The energy E(B) of a molecular bound state can betuned by a magnetic field B such that a resonant coupling to the entrance channel of two scatteredatoms arises. At the atomic threshold a FR appears. (b) In the vicinity of such a FR the interparticle
scattering length as diverges. The width ∆ of the resonance is defined by the magnetic field separation
between the pole at B0 and the zero crossing of as at Bzero . Away from resonance the scattering
length approaches the background value abg . (c) At the pole position the molecular state (blue)
resonantly couples to the two-body entrance channel (black). In the universal regime, see text, due
to the coupling between the states, the bound states’ energy evolves quadratically, while it evolves
linearly further away from the resonance.

pseudopotential with the same effective interaction
Uc (R) =4π~2as δ(R)m

4.1

where m is the particle mass, δ(R) the Dirac delta function, and as the remaining relevant
parameter. A positive as results in a repulsive interaction while a negative as leads to an
attractive interaction.
To understand the concept of FRs, we have to add the internal structure of the collisionpartners to the description. Due to the e. g. atomic Zeeman structure, different internal
Zeeman states become available, resulting in additional molecular potentials as exemplified
in Fig. 4.1(a). For large separation R, these potential connect to two free atoms of the corresponding spin states. The scattering behavior of two atoms that approach each other in thelowest energy channel, the so-called entrance channel, dramatically depends on the coupling
to a molecular level within the other scattering potential, the so-called closed channel. In
particular, when the molecular bound state of the closed channel energetically approaches thetwo-body atomic threshold, the coupling becomes resonantly enhanced and the two channels
strongly mix. As a result of this strong mixing, a FR emerges and the interparticle interaction diverges. In order to tune the energy of the bound state, E(B), with respect to theatomic threshold, the states need to feature a differential magnetic moment δµ1 , as to allow
tuning via the bias magnetic field B.
1

The differential magnetic moment is defined via δµ = 2µa − µm . Here, µa and µm denote the magnetic
moment of the free atom and the bound molecule, respectively.

4.1. QUANTUM CHAOS IN LANTHANIDES

53

Surprisingly, despite the complex underlying process for the emergence of FRs, the dependence of the scattering length as on the magnetic field takes the simple expression [Chi10]∆
as (B) = abg 1 −,4.2
B − B0see Fig. 4.1(b), with abg denoting the background scattering length away from resonance,
B0 determining the resonance position, and ∆ defining the resonance width. The couplingstrength between the entrance and the closed channel affects the resonance width as wellas the resonance position B0 = Bc + δB, see Fig. 4.1(c). Close to the atomic threshold thecoupling results in an avoided crossing, leading to a bending of the energy of the molecularstate. In this universal regime the s-wave scattering length particularly proves its importanceas it solely determines the binding energy of the crossing molecular state
Eb =

~2
,
2mr a2s

4.3

with mr being the reduced mass, see Sec. 3.1.2. The binding energy is defined to be a positive
value [Chi10]. Away from resonance, the binding energy shows a linear dependence on B
and is given by
Ec = δµ(B0 − B).
4.4
As we have seen, FRs arise if a coupling between the entrance and the closed channel ispresent. This coupling is a result of the underlying interactions [Chi10]. For broad FRs,
typically electronic interactions, consisting of exchange interaction at short range and thevan der Waals interaction potential VvdW = −C6 /R6 at long range, are responsible forthe coupling. Here, C6 is the isotropic van der Waals coefficient. Due to the isotropy ofthis interaction, the angular momentum quantum number l is preserved and for an s-wave
entrance channel only s-wave closed channels can be coupled, i. e. molecular states without
rotation. To access molecular levels with ∆l ̸= 0, additional interactions such as spin-spin
dipole or second-order spin-orbit interactions are required. Those interactions can couple
partial waves with ∆l = 2 and give rise to e. g. d-wave and g-wave resonances, which are
typically narrow. In lanthanides that feature an anisotropic van der Waals potential more
exotic couplings arise as will be discussed below.
To distinguish between broad and narrow resonances, it is convenient to define the dimensionless resonance strength parameter2 [Chi10]
sres =abg δµ∆
.
āĒ

Here,
1
ā = 0.955978RvdW with RvdW =2
andĒ = 1.09422EvdW with EvdW =
2

4.5


2mr C6
~2

1/4

~2
1
2
2mr RvdW

4.6

4.7

We note that an alternative method to determine the strength of a FR is given by the length parameterR∗ = ~2 /(2mr abg δµ∆), see Ref. [Pet04a].

4.1. QUANTUM CHAOS IN LANTHANIDES

54

define the relevant length and energy scales of the underlying van der Waals potential. For
sres ≫ 1 the resonance is called entrance-channel dominated while for sres ≪ 1 it is called aclosed-channel dominated resonance. Typically, this is related to broad and narrow widths ∆,respectively. For closed-channel dominated resonances a universal bound state only extends
over a small fraction of ∆, imposing challenges on the experimental access of the universal
regime.

4.1.2. Feshbach spectrum of lanthanides
In stark contrast to alkali atoms, in lanthanide atoms orbital anisotropy can be found. This isa consequence of a submerged shell structure, see Sec. 2.2.1, and results in an anisotropic vander Waals potential, see Refs. [Kot11, Kot14]. This additional anisotropy gives qualitatively
new possibilities for the coupling to molecular bound states, and hence strongly influences
the Feshbach spectrum. Before we turn to the theoretical description of Feshbach spectrain lanthanides with anisotropic interactions, we will briefly review the experimental study of
FRs with magnetic atoms.
The first element used for investigations of the magnetic dipole-dipole interaction (DDI)in the ultracold regime has been chromium, which features a magnetic moment of 6 µB .
Measurements of the Feshbach spectrum of 52 Cr in the stretched state mj = −3 [Wer05]
have revealed a density of ≈ 0.02 resonances per gauss, see Fig. 4.2(a). Chromium has azero nuclear spin and hence hyperfine interaction is absent. As a result, isotropic electronicinteractions can only couple molecular states that have the same magnetic moment as thefree atom entrance channel. In such a scenario, a tuning of the bias magnetic field B does
not change the relative energy E(B), see Fig. 4.1, and no FRs would appear. The observed
resonances in 52 Cr are thus a result of first- and second-order spin-spin dipole interactions,which are 36 times stronger than in alkali-metal atoms.
An additional class of magnetic atoms has become available in experiments with the creationof ultracold atomic clouds of dysprosium and erbium. The first investigation of their Feshbach
spectra has been performed by our group with bosonic 168 Er in a magnetic field range of
0–3 G [Aik12]. Feshbach resonances have been observed via increased three-body loss in thevicinity of the resonance. Remarkably, already in this small field region six FRs have been
identified, which pointed towards a high density of resonances in magnetic lanthanides. A
more detailed study for two bosonic and one fermionic isotopes of erbium in a large magnetic
field range [Fri14b], see also Fig. 4.2(b), and three bosonic and one fermionic isotopes ofdysprosium in a smaller magnetic field range [Bau14] followed shortly after by our groupand the group of Benjamin Lev, respectively. Both experiments confirmed the initial workand revealed for the bosonic isotopes a density of resonances in the order of 3 per gauss, avalue previously unseen for elements in the ultracold regime. The fermionic isotopes exhibit
an even higher density with ≈ 27 and ≈ 11 resonances per gauss for 167 Er and 161 Dy,
respectively.
Remarkably, for erbium a two orders of magnitude higher density of FRs is observed whencompared to the spectrum of chromium, see Fig. 4.2. Both elements feature DDI that couples
partial waves with ∆l = 2. In addition, for erbium the anisotropic dispersion relation allows

4.1. QUANTUM CHAOS IN LANTHANIDES
115

55

(a)

52

a6,-6 (a0)

110

Cr

105
100
95
0

50

100

150

200

250

300

350
400 450magnetic field (G)

500

550

600

650

700 750

atom number

168

105

104

Er

(b)
0

10

20

30
40
magnetic field (G)

50

60

70

Figure 4.2.: Feshbach spectrum of chromium (a) and erbium (b). (a) Calculated scattering length
based on experimental Feshbach spectroscopy for 52 Cr in the lowest Zeeman state mj = −3 as afunction of the magnetic field. Figure adapted from Ref. [Wer05] (b) Feshbach spectrum for 168 Er in
the lowest Zeeman state mj = −6 within a magnetic field region as indicated by the red shading in(a). Every loss feature corresponds to a FR. Figure adapted from Ref. [Fri14b].

to couple rotational states with ∆l = 4. Moreover, as a consequence of the large angularmomentum quantum number j of bosonic erbium atoms, a large number of (j + 1)2 = 49
gerade molecular Born-Oppenheimer potential exist in the lowest Zeeman state. Thosepotentials provide a significant amount of molecular bound states that can be coupled, giving
an intuition for the observed high density of FRs.
To understand the experimental observations in more detail, theoretical models based oncoupled-channel calculation can be applied. Reference [Pet12] has studied the Feshbach
spectrum of dysprosium by taking into account the Zeeman, the magnetic dipole-dipoleVµµ ∝ 1/R3 , the electrostatic isotropic and anisotropic dispersion Vdisp ∝ 1/R6 , and theweak quadrupole-quadrupole Vqq ∝ 1/R5 interaction. The different anisotropies contributedifferently to the coupling of rotational states, as can be seen in Fig. 4.3(a). When all
interactions are included, the highest amount of resonances appears (upper panel), where alower number of resonances is present when either the anisotropic Vdisp (middle panel) orVµµ (lower panel) are excluded. A similar behavior is found for erbium, see Ref. [Fri14a].
Still, the experimentally observed density of FRs has not been captured by the theoretical
study of Ref. [Pet12].
Interestingly, the amount of resonances depends strongly on how many molecular states,
i. e. states with rotational quantum numbers l, are included into the theory, a scenario thatcannot be observed for collisions of alkali-metal atoms or chromium. The more higher partial
wave states are added, the more resonances appear, which suggests that the different partial
waves are strongly mixed. Indeed, it has been shown that the experimentally observed
density of resonances can only be reproduced if partial waves up to l = 40 are included,
see Fig. 4.3(b). The even higher density observed for fermionic erbium and dysprosium isa result of the additional hyperfine structure that leads to an increased number of possible

4.1. QUANTUM CHAOS IN LANTHANIDES
(a) 6

(b)

3
0

ρ(1/G)

as /100 (a0)

4
3

10
5
0
6

2
1

3
0
0

56

0
50

100
150
magnetic field (G)

200

0

10

20

30
Lmax

40

50

60

Figure 4.3.: Interaction and partial wave contributions to the Feshbach spectrum of lanthanides(a) Calculated scattering length for 164 Dy in the lowest Zeeman state mj = −8 as a function ofmagnetic field with all interactions included (top panel), when the anisotropic dispersion interaction
is set to zero (middle panel), and when the dipole-dipole contribution is excluded (lower panel). For
the simulation even partial waves up to Lmax = 10 are taken into account. Figure adapted from
Ref. [Pet12] (b) Calculation of the mean resonance density ρ for bosonic erbium when the numberof included partial waves Lmax is varied. Coupled-channel calculation (green circles) and analyticalestimate (black line) are compared to the experimentally measured ρ for 168 Er (dashed line) and166
Er (dashed-dotted line). Figure adapted from Ref. [Fri14b].

collisional channels.
Beside that many partial waves contribute to the Feshbach spectrum, the resonance positions
also show a significant dependence on the number of included partial waves [Pet12]. As aconsequence, the assignment of specific resonances by a single quantum number, as it istypically done for alkalis, is difficult. Hence, a different approach has to be applied toanalyze the Feshbach spectrum of lanthanides with anisotropic interactions.

4.1.3. Chaotic scattering
For the analysis of the Feshbach spectra of magnetic lanthanides a statistical method canbe utilized. This approach is based on the Random Matrix Theory (RMT), which has
been developed by Wigner and Dyson [Wig51, Meh63] and finds applications across various problems, ranging from nuclear physics [Lio72a, Lio72b], to atomic physics [Guh98],
towards neural networks [Raj06] or to the context of financial markets [Bou11]. In the caseof the scattering of lanthanide atoms with anisotropic interactions, the theory captures thecorrelation of the individual molecular levels among each other. In brief, the combinationof magnetic field and level coupling results in a certain distribution of FRs at the atomic
threshold. Hence, the statistical analysis of the nearest-neighbor-spacing (NNS) distributionof the Feshbach spectrum can give insights into the correlations of different channels.

4.1. QUANTUM CHAOS IN LANTHANIDES

57

An analysis of bosonic erbium [Fri14b] based on RMT has revealed that a strong level
repulsion has to be present3 . This is e. g. captured by the observed NNS distribution, whichshows that the magnetic field distance between neighboring resonances tends to maximize.
The distribution is well described by the Brody distribution
PB (s) = b(1 + η)sη exp (−bsη+1 ),4.8

which interpolates between the standard Poisson distribution PP (s) = exp (−s) and the
Wigner-Dyson distribution PWD (s) = π2 s exp (−πs2 /4). The dimensionless quantity s denotes the level spacing in units of the mean level spacing, η is the single fitting parameter,
and b = [Γ(η + 2)/(η + 1)]η+1 is a normalization constant where Γ depicts the Gamma
function. The observed distribution, matching better to a Wigner-Dyson distribution, is astrong signature of chaotic scattering [Fri14b]4 . The approach utilizing RMT has proven its
strength, as it allows to genuinely compare the experimental observation with theoretical
calculations without the need of an individual assignment of the resonances. While the workof Ref. [Fri14b] has given beautiful insights into the physical origin of resonant scattering
behavior of erbium atoms, it did not tackle the questions if lanthanides with anisotropicinteractions show a universal behavior and what is the dominating interaction for the level
correlation.
To address such questions we have formed a collaboration between the group of Tilman
Pfau in Stuttgart, the group of Svetlana Kotochigova in Philadelphia, and our group in
Innsbruck. In our joined effort we have investigated experimentally and theoretically in detail
the Feshbach spectrum of erbium and dysprosium atoms in a magnetic field range up to 70 G.
Our publication, reported in Sec. 4.3, shows that both, erbium and dysprosium, exhibit alarge amount of FRs. The observed densities can be nicely reproduced in theory when
including higher partial waves l up to a total Jmax = 39 and Jmax = 36, respectively. Further,
we theoretically show that a convergence of the resonance position requires Jmax > 22,
disclosing the strong mixing of such partial waves. We reveal that the emergence of chaotic
scattering is mainly determined by the anisotropic dispersion interaction and does not dependon the contribution of the anisotropic DDI. Finally, we report on a strong temperaturedependence of a significant number of resonances in 168 Er, in analogy to previous studies
with Dy [Bau14]. We find that this dependency, which cannot be explained by higherpartial wave two-body scattering, can be traced back to the temperature dependence of therecombination rate of three free particles into a resonant trimer state via a d-wave three-atom
entrance channel.

4.1.4. Interaction tuning
Beside the high complexity of the Feshbach spectra of lanthanides with anisotropic dispersion
interactions, interaction tuning via the s-wave scattering length does become possible if
isolated FRs are observed, see Eq. 4.2. In the context of atoms with DDI this gives thepossibility to change the relative strength between isotropic and anisotropic interactions.
3

4

Reference [Fri14a] gives further details on RMT and its application to the resonant scattering of erbium
atoms.
For an introduction to quantum chaos the reader is referred to e. g. Ref. [Jen92].

4.2. DIPOLAR FESHBACH MOLECULES

58

In recent years, several works have investigated the s-wave scattering length of dipolar lanthanides, both, away and in the vicinity of different FRs. The values of as for erbium anddysprosium have been studied via cross-dimensional thermalization experiments in thermalgases [Fri14a, Tan15], via the measurement of the anisotropic thermal expansion of a dipolar Bose gas [Tan16, Luc18], by comparing experimental measurements of the molecularbinding energy of dysprosium to a universal model [Mai15], or by comparing theoreticalsimulations with experimental measurements of the critical atom number of the liquid-togas phase transition [Sch16] or the frequency of collective excitations in a dipolar quantum
droplet [FB18].
A very precise method to derive s-wave interactions within a broad magnetic field range isbased on modulation-spectroscopy in a three-dimensional optical lattice, see Sec. 5.4.1. Thismethod is very successful as demonstrated within this thesis for 168 Er, 167 Er, and 166 Er,
see Sec. 5.5, Sec. 5.6, and Appendix A.1, respectively. Our obtained knowledge allows us
to precisely tune the atomic interactions in the vicinity of isolated FRs, as needed for the
study of dipolar phenomena that emerge from the competition between s-wave and contact
interactions, see Appendix A.1 and Appendix A.3, or for reaching the strongly interacting
regime of a two-component Fermi gas, see Sec. 5.6.
A further prospect emerging from the observation of isolated FRs is given by the possibility toaccess ultracold molecules. Details along this line will be discussed in the following section.

4.2. Dipolar Feshbach molecules
4.2.1. Molecule formation
In the vicinity of a FR the coupling between the two-body atomic threshold and the molecularstate is strongly enhanced, see Fig. 4.1. As a result, it becomes possible to associate Feshbach
molecules out of two free atoms [Köh06]. Further, the strength of the bond, see Eq. 4.4, canbe tuned by the magnetic field. For the creation of such molecules, different techniques canbe applied. Figure 4.4 gives two examples. The method of magnetic-field modulation allows
to directly couple to the bound state, even when the atomic sample is prepared away fromthe resonance [Tho05], see Fig. 4.4(a). Here, molecule formation is typically observed via aresonant atomic loss feature. By preparing the atoms at various magnetic field values, thismethod allows to directly map the molecular binding energy as a function of the magnetic
field. Hence, it is a convenient method to access the magnetic moment µm of a molecularstate, see Eq. 4.4.
Another method used for Feshbach molecule creation is based on adiabatic magnetic fieldsweeps, see Fig. 4.4(b). Two free atoms are prepared above the resonance and when adiabatically ramped across the resonance can follow into the molecular bound state along
an avoided crossing. Atom-to-molecule conversion efficiencies are strongly related to thephase-space density n of the atomic sample, the resonance strength, and the magnetic fieldsweep rate Ḃ via nabg ∆/Ḃ [Hod05]. Initially, such molecule creation has been achieved for4.2. DIPOLAR FESHBACH MOLECULES(b)binding energy / h (kHz)binding energy / h (kHz)

(a)

59

-100
0
100
200
0.85

0.90
0.95
magnetic field (G)

1.00

-100
0
100
200
0.85

0.90
0.95
magnetic field (G)1.00

Figure 4.4.: Creation of Feshbach molecules exemplified for 168 Er atoms via magnetic-field modulation (a) and magnetoassosciation (b) techniques. (a) The atoms are prepared below the FR. If theenergy of an oscillatory magnetic field matches the binding energy, Feshbach molecules are associated. The excess energy is taken away by the modulated field (green arrow). The resonant condition
typically is observed via increased loss, triggered by atom-dimer collisions, once the molecules areformed. (b) Adiabatic creation of Feshbach molecules via a magnetic field sweep across the avoided
crossing of the free-atom (black line) to the bound state (red line).
40 K [Reg03], 6 Li [Cub03, Joc03a, Str03], 133 Cs [Her03] 23 Na [Xu03], and also 87 Rb [Dür04],

all within one year. Collisional studies revealed a large loss rate of bosonic molecules causedby atom-dimer or dimer-dimer collisions, see e. g. Refs. [Muk04, Chi05, Sya06, Fer10], whilemolecules formed from two fermionic spin states can remain remarkably stable. This is a result of the Pauli exclusion principle, which suppresses few-body processes of three and moreparticles [Pet04b]. This collisional stability has paved the way for the creation of molecular
BECs out of weakly-bound Feshbach molecules [Joc03b, Gre03].

4.2.2. Polar and dipolar molecules
Molecule creation can also be used to access strongly dipolar systems. Such systems are
ideal candidates for e. g. quantum simulation of condensed matter [Bar12]. To realize thissystems, different experimental approaches are followed. One approach is based on the
production of ground state heteronuclear molecules that exhibit a strong electric dipole
moment. After Feshbach association the molecules can be transferred to the rovibrational
ground state via an optical stimulated Raman adiabatic passage (STIRAP) technique withhigh conversion efficiencies [Ber98, Vit17]. The access of lower-lying vibrational levels has
first been demonstrated with Rb2 [Win07]. Today, STIRAP is a well established method andhas led to the production of rovibrational ground state polar molecules such as KRb [Ni08],
RbCs [Tak14, Mol14], NaK [Par15], and NaRb [Guo16], which all posses a tunable electric
dipole moment.
A distinct approach towards the goal of ultracold polar molecules, involves the direct coolingof hot molecular samples. For certain molecular species remarkable progress has been made inrecent years and laser cooling and magneto-optical trapping have now been demonstrated forpolar SrF and CaF molecules [Bar14, Nor16, Tru17, And17, And18]. While temperatures
down to milikelvin and submilikelvin already have been reached, further cooling to theultracold regime remains an outstanding challenge.

4.2. DIPOLAR FESHBACH MOLECULES

60

A different method to access strongly dipolar interactions is given by elements that intrinsically carry a magnetic dipole moment such as chromium, dysprosium and erbium. Whenatoms of such elements are bond together to Feshbach molecules, a strong dipolar interactions can be accessed. Remarkably, the dipolar interaction comes for free without the needof bringing the weakly-bound molecules to the rovibrational ground state.
Our publication in Sec. 4.4, a joint effort between the group of Svetlana Kotochigova in
Philadelphia, the group of Olivier Dulieu in Orsay, and our group in Innsbruck, reports on
the first realization of magnetic dipolar molecules. We create them with strongly magnetic
168 Er atoms via the magnetoassociation technique, see Fig. 4.4(b), and explore the magnetic
moment of four molecular states by magnetic-field modulation spectroscopy5 , see Fig. 4.4(a).
For the strongest dipolar molecule we find ad = 1140a0 , see Eq. 3.3 for the definition of thedipolar length, which exceeds our typical s-wave scattering length by one order of magnitude.
Applying a new theoretical approach based on approximate adiabatic potentials allows us,
together with the experimental input, to assign the dominant quantum numbers of theinvestigated molecular states. We prove the dipolarity of the produced molecules via theobservation of a reduction of inelastic losses for a perpendicular dipole orientation withrespect to the weak confinement axes. Finally, we unveil that the observed ratios of theloss-rate between dominant repulsive and dominant attractive dipolar interactions follow auniversal behavior.
Our work sets the start for future investigations with magnetic dipolar molecules. A particularly interesting scenario is given by the creation of fermionic dipolar molecules. In sucha system inelastic losses are suppressed by Pauli pressure and the attainment of molecular
BECs is accessible, as demonstrated with alkalis [Joc03b, Gre03]. For the fermionic erbium
isotope 167 Er we already have achieved precise knowledge of the interaction between two
spin states along a comparatively broad FR, see Sec. 5.6, which brings the scenario of studies along the BEC-to-BCS crossover with dipolar interactions and the creation of a dipolar
molecular BEC at reach.

5

Magnetic-field modulation spectroscopy more recently has also been applied to detect the molecular binding
energy for broad resonances in 164 Dy [Mai15] and 162 Dy [Luc18].

4.3. Publication:
Emergence of Chaotic Scattering in Ultracold Er and Dy†
Physical Review X 5, 041029 (2015)
submitted 7 Jun 2015; published 21 Sep 2015
DOI 10.1103/PhysRevX.5.041029

Thomas Maier,1 Holger Kadau,1 M. Schmitt,1 Matthias Wenzel,1 Igor Ferrier-Barbut,1 Tilman
Pfau,1
1

5. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität
Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Albert Frisch,2,3 Simon Baier,2 Kiyotaka Aikawa,2 Lauriane Chomaz,2,3 Manfred J. Mark,2,3Francesca Ferlaino,2,32Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
3Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria

Constantinos Makrides,4 Eite Tiesinga,5 Alexander Petrov,4 and Svetlana Kotochigova44

Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
Joint Quantum Institute and Center for Quantum Information and Computer Science, National
Institute of Standards and Technology and the University of Maryland, 100 Bureau Drive, Stop 8423,
Gaithersburg, Maryland 20899, USA
5†

The author of the present thesis performed the measurements for erbium together with A. F., and K. A.,
and contributed in writing the manuscript. The theoretical calculations of this publication have been
contributed by C. M., E. T., A. P., and S. K.

61

PHYSICAL REVIEW X 5, 041029 (2015)

Emergence of Chaotic Scattering in Ultracold Er and Dy
T. Maier,1,* H. Kadau,1 M. Schmitt,1 M. Wenzel,1 I. Ferrier-Barbut,1 T. Pfau,1 A. Frisch,2,3 S. Baier,2 K. Aikawa,2,†
L. Chomaz,2,3 M. J. Mark,2,3 F. Ferlaino,2,3 C. Makrides,4 E. Tiesinga,5 A. Petrov,4,‡ and S. Kotochigova41

5. Physikalisches Institut and Center for Integrated Quantum Science and Technology,
Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
2
Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
3Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
4
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
5
Joint Quantum Institute and Center for Quantum Information and Computer Science,
National Institute of Standards and Technology and the University of Maryland,
100 Bureau Drive, Stop 8423, Gaithersburg, Maryland 20899, USA(Received 7 June 2015; revised manuscript received 21 September 2015; published 19 November 2015)
We show that for ultracold magnetic lanthanide atoms chaotic scattering emerges due to a combinationof anisotropic interaction potentials and Zeeman coupling under an external magnetic field. This scattering
is studied in a collaborative experimental and theoretical effort for both dysprosium and erbium. We present
extensive atom-loss measurements of their dense magnetic Feshbach-resonance spectra, analyze theirstatistical properties, and compare to predictions from a random-matrix-theory-inspired model. Furthermore, theoretical coupled-channels simulations of the anisotropic molecular Hamiltonian at zero magnetic
field show that weakly bound, near threshold diatomic levels form overlapping, uncoupled chaotic series
that when combined are randomly distributed. The Zeeman interaction shifts and couples these levels,
leading to a Feshbach spectrum of zero-energy bound states with nearest-neighbor spacings that changes
from randomly to chaotically distributed for increasing magnetic field. Finally, we show that the extreme
temperature sensitivity of a small, but sizable fraction of the resonances in the Dy and Er atom-loss spectrais due to resonant nonzero partial-wave collisions. Our threshold analysis for these resonances indicates alarge collision-energy dependence of the three-body recombination rate.
DOI: 10.1103/PhysRevX.5.041029

Subject Areas: Atomic and Molecular Physics,
Quantum Physics

I. INTRODUCTION
Anisotropic interactions are a central and modern toolfor engineering quantum few- and many-body processes[1]. A prominent example of such an interaction is thelong-range dipole-dipole interaction (DDI) acting, forinstance, between polar molecules [2], Rydberg atoms [3],or magnetic atoms [4]. Over the years, fascinating quantum
effects of the anisotropy have been observed, such as thed-wave collapse of a dipolar Bose-Einstein condensate [5],*Corresponding author.
t.maier@physik.uni‑stuttgart.de†Present address: Department of Physics, Graduate Schoolof Science and Engineering, Tokyo Institute of Technology,
Meguro-ku, Tokyo 152-8550, Japan.
‡
Also at: NRC “Kurchatov Institute” PNPI 188300, Divisionof Quantum Mechanics, St. Petersburg State University, 198904,
Russia.

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

2160-3308=15=5(4)=041029(13)the deformation of the Fermi surface [6], and the control
of stereodynamics in dipolar collisions [7]. Moreover, the
DDI is expected to give rise to a plethora of few- and manybody phenomena, which still await observation, such asuniversal few-body physics [8,9], rotonic features [10,11],
two-dimensional stable solitons [12], and the supersolid
phase [13].
Recently, atomic species in the lanthanide family became
available to the field of ultracold quantum gases. Theinteraction between magnetic lanthanide atoms, such as Er[14,15] and Dy [16,17], is highly anisotropic. This is not
only due to the long-range DDI, originating from their large
magnetic moment, but also to the shorter-ranged van der
Waals interaction [18], which exhibits anisotropic contributions arising from the large orbital angular momentum of
their valence electrons.
For magnetic lanthanides, which also include the successfully laser-cooled elements Ho [19] and Tm [20], theorbital anisotropy is a consequence of a partially filled
submerged 4f electron shell that underlies a closed outer 6sshell. This leads to an electronic ground state with a total
atomic angular momentum ~| with j ≫ 1. Consequently, in

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T. MAIER et al.

PHYS. REV. X 5, 041029 (2015)

collisions between such atoms there exist ðj þ 1Þ2 nondegenerate (gerade) molecular Born-Oppenheimer (BO)potentials and a correspondingly large manifold of collision
channels with associated molecular bound states. This is
in sharp contrast to the one or two BO potentials encountered in alkaline-earth and alkali-metal atom collisions.
In addition, the anisotropy or orientation dependence of
the BO potentials strongly mixes collision channels with~ between thelarge relative orbital angular momentum latoms even for our ultracold collisions with a l ¼ 0,
s-wave initial channel. The complexity of the molecularforces are reflected in a dense spectrum of Fano-Feshbachresonances as a function of magnetic field B, as recently
observed in Er [14,21] and Dy [22]. In Er a statisticalanalysis of the spacings between resonances has shown
correlations that revealed chaotic scattering. The data setof the initial Dy experiments was too small to extract
statistically significant correlations.
Chaotic behavior is manifest in a variety of complex
systems ranging from atomic to nuclear and solid-statephysics. In atomic physics, chaos was originally studied
with Rydberg states of H and He in a magnetic field [23].
Later on, a variety of more complex atoms and ions inhighly excited states showed signatures of chaotic spectral
distributions [24]. The origin of chaos in these systems wastraced back to a strong mixing of many-electron excitedstates by the Coulomb interaction [25]. A chaotic level
distribution is also common in a variety of solid-statesystems ranging from those with strong many-body interactions to the motion of particles in irregular potentials[26,27]. Experiments in nuclear physics [28,29] have also
produced substantial evidence for chaotic neutron resonance spectrum fluctuations, which agree with predictionsof random matrix theory (RMT). Similar agreement was
found from numerical simulations based on nuclear shellmodels [30,31]. Moreover, Refs. [32,33] suggested thatchaos is a generic property of nuclei with multiple degrees
of freedom (i.e., multiple active shells), which becomecompletely mixed.
This article describes a joint effort to understand ultracold scattering and Fano-Feshbach spectra of strongly
magnetic Er and Dy atoms. In particular, we report on
the measurement and statistical analysis of Fano-Feshbach
spectra for Dy and Er between B ¼ 0 and 70 G at gas temperatures T below and around 1 μK. Here, 1 G ¼ 0.1 mT.
We observe that both elements have similar chaotic
scattering. We present a RMT-inspired model to gain
insight into their statistical properties as well as theoreticalevidence based on coupled-channels calculations with amicroscopic Hamiltonian that chaotic scattering requires
both strong molecular anisotropy and Zeeman mixing tofully develop. Limitations of the RMT are also discussed.
Finally, we present experimental data and a comparison
to a resonant trimer model to show that our increase inresonance density with temperature is a consequence ofthe strong collision-energy dependence of transitions fromentrance d-wave channels of three free atoms to resonant
trimer states.
II. EXPERIMENT
A. Measurement
The experimental study of Fano-Feshbach resonances in
Er and Dy is based on high-resolution trap-loss spectroscopy on spin-polarized thermal samples. Ultracold bosonic
164 Dy samples are created by direct loading from a narrowline magneto-optical trap, operating on the 626 nm cyclingtransition, into a single-beam optical dipole trap (ODT)[34]. By moving the last focusing lens of the ODT, theatoms are transported from the magneto-optical trap chamber to the science cell. The ODT is created with a 100 W
fiber laser at a wavelength of 1070 nm. We achieve atransport efficiency close to unity. This fiber laser, however,causes atom loss due to its longitudinal multimode structure[35]. Therefore, we transfer the atoms into a second
single-beam ODT, created by a 55 W solid-state laser
at a wavelength of 1064 nm. Finally, forced evaporative
cooling in a crossed ODT leads to a sample of 105 atoms
in the energetically lowest Zeeman sublevel, mj ¼ −8
at T ¼ 600 nK.
High-resolution trap-loss spectroscopy is performed on aspin-polarized bosonic 168 Er sample at T ¼ 1400 nK andcompares this spectrum with that obtained at a 4 times
lower temperature measured in previous work for 168 Er as
well as for fermionic 167 Er [21]. The experimental procedures for creating bosonic and fermionic samples aredescribed in Refs. [14,15], respectively. Bosons (fermions)are prepared in the lowest Zeeman sublevel, mj ¼ −6
(mf ¼ −19=2). Erbium samples are trapped in a crossed
ODT and contain about 105 atoms.
B. Feshbach spectroscopy
Feshbach spectroscopy is performed in a similar manner
for the two species. The magnetic field is ramped up over a
few milliseconds to a magnetic-field value B, where theatoms are held in the ODT for 500 ms for Dy, 400 ms for
168
Er, and 100 ms for 167 Er. During this time, inelastic
three-body recombination causes atom loss from the ODT.
At resonance, the recombination process is enhanced
because of the coupling between the atomic-thresholdstate and a molecular state leading to a resonant increaseof the atom loss. We identify the field locations ofmaximum loss as the positions of Fano-Feshbach resonances [36]. The atom number is probed by standard timeof-flight absorption imaging at low magnetic field. We
record atom-loss features for magnetic-field values between
0 and 70 G in steps of a few mG. Figure 1(a) shows thenormalized loss spectrum for the 164 Dy isotope, where we
identify 309 resonances. For 168 Er at T ¼ 1.4 μK, there are041029-2

EMERGENCE OF CHAOTIC SCATTERING IN ULTRACOLD …

PHYS. REV. X 5, 041029 (2015)

normalized atom number(a) 100

-1

10

0

2

4

6

8

10

12

14

16

18

20

22

26

24

28

30

32

34

0

10

-1

10

36

38

40

42

44

46

48

50

52
54
magnetic field (G)

(c)
400
nK

300

164

D

n
400

2

Er 1
168

100

60

62

64

66

Er 35

0 nK

68

70

4
son
Pois

3

K

168

58

K
0n

y 60

Er

200

167

number of resonances

(b)

56

2
1
Wigner-Dyson

0

0

20

40
magnetic field (G)

0

60

0

1

2

N

3

4

FIG. 1. (a) Trap-loss spectroscopy mapping of the Fano-Feshbach spectrum of 164 Dy as a function of magnetic field between B ¼ 0
and 70 G with a data point every 14.5 mG and temperature T ¼ 600 nK. Each data point is an average of three measurements.
(b) Staircase function for the number of resonances as a function of B for 164 Dy, 168 Er at two temperatures, and fermionic 167 Er.
Dashed lines are linear fits forced to pass through the origin. Their slopes give a mean density of resonances of ρ̄ ¼ 4.3 G−1 for 164 Dy,
2.7 G−1 for 168 Er at T ¼ 350 nK, 3.4 G−1 for Er at T ¼ 1.4 μK, and 25.6 G−1 for 167 Er. (c) Number variance of the experimental data
as a function of scaled B-field interval N̄ ¼ ΔBρ̄. The experimental data for 164 Dy (blue line) and 168 Er at T ¼ 350 nK (orange line)
lie between the variances for an uncorrelated Poisson distribution (dashed line) and the correlated Wigner-Dyson distribution(dot-dashed line).

238 resonances. The Fano-Feshbach scan of fermionic167
Er is carried out from 0 to 4.4 G and yields 115
resonances.
The understanding of the richness of the scattering in
Er and Dy requires the development of sophisticatedmicroscopic coupled-channels scattering models. We defer
such analysis until later in this paper and first analyze
our data following the statistical approach based on the
RMT advocated by Ref. [21]. In particular, we study thecorrelations between resonance locations via the nearestneighbor spacing (NNS) distribution and set up a RMT-likemodel, which accounts for the structure of our B-dependent
microscopic Hamiltonian, to get intuition about these NNSs.
In our description of the coupled-channels calculations,limitations of such a RMT-like model are discussed.
C. Statistical analyses
Our statistical analysis starts with the construction of thestaircase function, which is a steplike function that counts
the number of resonances below magnetic-field value B[37]. Figure 1(b) shows the staircase function for Dy and
Er. For both species the function is well fit by a linear curve
forced to pass through the origin. Its slope ρ̄ corresponds

to the density of resonances. Deviations below and above
the fit occur for small and large B, respectively. The fitted
resonance densities are given in the caption. Remarkably,
the density of resonances of 168 Er at T ¼ 1.4 μK is 25%
higher than the one observed at 350 nK. The discussion of
the origin of this sensitivity is postponed until Sec. V. The
density ρ̄ for bosonic Dy is 50% larger than for bosonic Er.
This is caused by the larger ~| of Dy and, thus, its larger
number of allowed collision channels. The much larger
density ρ̄ of 25.6 G−1 for the fermionic 167 Er is due to its
additional hyperfine structure.
Fluctuations in the number of resonances within amagnetic-field interval ΔB is a second measure of thestatistical properties of the spacings between resonances.
Formally, it is defined as the dimensionless numberP
2variance Σ2 ¼ N 2 − N̄ 2 , where N̄ ¼ M−1
i¼0 N i =M, N ¼
PM−1 2i¼0 N i =M, and N i is the number of resonances in thefield interval [iΔB, ði þ 1ÞΔB], with i ¼ 0; …; M − 1,such that MΔB ¼ Bmax and Bmax ¼ 70 G for both species.
Consequently, N̄ ≡ ΔBρ̄. For shot noise or a Poissonian
distribution, we expect Σ2 ¼ N̄. Figure 1(c) compares Σ2
for our Dy and Er data as a function of N̄. The fluctuations
for both species monotonically increase with ΔB but are

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PHYS. REV. X 5, 041029 (2015)

substantially less than the shot-noise limit. While thisbehavior was previously demonstrated for Er [21], thepresent results provide the first evidence of correlation in
Dy and indicate similarity between the species.
These correlations between resonance locations arefurther studied using the nearest-neighbor spacings distribution PðsÞ, where s ¼ δBρ̄ and δB is the field spacing between two adjacent resonances in the spectrum. Figures 2(a)and 2(b) show the computed NNS distribution of ourexperimental data, derived from the number of NNS,
Si , with spacings s between iδs and ði þ 1Þδs, wherei ¼ 0; 1; … and δs ≈ 0.3. The NNS distributions have clear
deviations from both the Poisson PP ðsÞ ¼ expð−sÞ andWigner-Dyson PWD ðsÞ ¼ ðπ=2Þs exp½−ðπ=4Þs2  distribution, two well-known distributions within RMT [21].
A Poisson distribution corresponds to a random distributionof resonance locations, while a Wigner-Dyson distribution
corresponds to a situation where neighboring resonances“avoid” each other and PWD ðsÞ ∝ s for s → 0. Deviations
are also seen in Fig. 1(c), where for both atomic species thevariance Σ2 does not agree with the corresponding predictions for these distributions. The experimental NNS
distributions in Figs. 2(a) and 2(b) have also been fit to the
Brody distribution PB ðs; ηÞ ¼ bð1 þ ηÞsη exp½−bsηþ1 , an
1.0
164

168

164

168

Er theory
Er 350 nK
Er 1400 nK
167
Er

Dy theory
Dy

P(s)

168

0.5

(a)
0.0

0

(b)
1

s

(B)

1.0

2

3 0

1

s

2

3

Wigner-Dyson

(c)

(d)

0.5
0.0

Poisson

0

25

50
B (G)

0

25

50
B (G)

FIG. 2. (a) Nearest-neighbor spacing distribution PðsÞ determined from all observed Fano-Feshbach resonances for 164 Dy(blue markers). Dashed and dot-dashed curves are the Poisson
and Wigner-Dyson distribution, respectively. The solid line is a
Brody distribution with η ¼ 0.45ð7Þ fit to the experimental data.
(b) Distributions PðsÞ for 168 Er at T ¼ 350 nK and T ¼ 1.4 μK(orange and red markers, respectively), and 167 Er at T ¼ 0.4T F(green markers), where T F is the Fermi temperature of the gas.
The solid line is a Brody distribution with η ¼ 0.68ð9Þ fit to the
168 Er data at T ¼ 350 nK. Panels (c) and (d) show the magneticfield-resolved Brody parameter ηðBÞ as a function of magnetic
field for 164 Dy and 168 Er, respectively. The Brody parameters for
the Poisson and Wigner-Dyson distribution are 0 and 1, respectively. Gray markers and lines in all four panels are results fromour coupled-channels calculations. The 1σ error bars in (a) and(b) correspond to Poisson counting errors, while shaded bands in(c) and (d) are 1σ statistical uncertainties of the fits to the data.

empirical function that interpolates between PP ðsÞ and
PWD ðsÞ for η ¼ 0 and 1, respectively, and b is a normalization constant [38]. The values for η reported in thecaption indicate intermediate or mixed behavior of the data.
We present the magnetic-field resolved Brody parameterηðBÞ in Figs. 2(c) and 2(d) obtained from a fit to the NNS
distribution of resonances located in moving intervals½B − ΔB=2; B þ ΔB=2, with ΔB ¼ 20 G. It has a nonnegligible 1σ uncertainty equally limited by the quality of
the fit and the number of Feshbach resonances in an intervalor bin. The latter uncertainty is reflected in the bin-to-bin
variation of ηðBÞ. For Dy we observe that η increaselinearly with field for small B, which saturates at a value of≈0.5 for B > 30 G. For Er the Brody parameter fluctuatesaround 0.5. Interestingly, the Er data at our two temperatures have a similar behavior, indicating that the larger
density of resonances at higher T does not impact thedegree of correlation between their spacings.
III. RMT ENSEMBLE MODEL
Random matrix theory is based on the powerful
notion that the statistics of eigenvalues and eigenfunctionsof a complex system can be studied by replacing the
microscopic Hamiltonian by an ensemble of random
Hamiltonians. In this spirit, we construct a RMT-inspired
model for weakly bound molecular dimer states to test the
distribution of Fano-Feshbach resonances.
Our RMT model is based on the statistics of eigenvaluesof the N × N real, symmetric matrix HRMT ¼ H 0 þ HZ ,where matrices H0 and HZ represent the B ¼ 0
Hamiltonian and the Zeeman interaction of the two atoms,
respectively. Without loss of generality we can assume that
HZ is a diagonal matrix with matrix elements given by
mgμB B, where m is an integer between −2j and 2j,corresponding to the sum of the projection quantum
numbers of the atomic angular momenta, g is the atomic
Landé factor, and μB is the Bohr magneton. The Zeeman
interaction does not depend on the rotational state of themolecule and, thus, entries in HZ correspond to stateswith a definite value for l and its projection. H 0 is thenthe B ¼ 0 Hamiltonian expressed in this basis. It is also
convenient to define H 0 ¼ H d þ Hcpl , where diagonal
matrix Hd contains the diagonal matrix elements of H 0
and H cpl is the matrix of all its off-diagonal elements. The
eigenvalues of Hd can then be interpreted as the energiesof rovibrational levels of the isotropic contribution of
the molecular BO potentials, while Hcpl describes mixing
due to the anisotropic contributions of these potentials.
We generate members of our ensemble of HRMT bychoosing random matrix elements for HZ, Hd , and Hcpl
based on specific distributions. The values of m in HZ are
uniformly distributed integers between −2j and 2j. The
matrix elements of Hd are chosen according to a Brody
distribution with variable Brody parameter ηd ∈ ½0; 1 and041029-4

EMERGENCE OF CHAOTIC SCATTERING IN ULTRACOLD …
with a mean energy spacing between bound states ϵd .
Finally, matrix elements of Hcpl are chosen as Gaussiandistributed real numbers with zero mean and standard
deviation νcpl , thereby on average coupling all diagonal
elements equally. Notice that this construction deviatesfrom that for a true Gaussian orthogonal ensemble, where
all matrix elements of a symmetric Hamiltonian are
Gaussian distributed [39].
We apply the RMT model to the case of 168 Er. The
relevant species-specific quantities are j ¼ 6, g ¼ 1.16,
and ϵd is chosen to roughly reproduce the observed densityof Fano-Feshbach resonances of 168 Er and is set toϵd =h ¼ 6.4 MHz, where h is Planck’s constant. Figure 3(a)(b) 1

νcpl /h = 0 MHz

−20
−40

0.4

−80

0.2

0

2

4

6

8

νcpl /h = 10 MHz

0.6

−60

−100

νcpl /h = 2 MHz

0.8

P(s)

E/h (MHz)

(a) 0

0

10

0

1

2

3

s

B (G)

(d)

(c) 1
0.8

η

0.6
0.4

ηd = 0ηd = 0.25
ηd = 0.5

0.2

B=0

0
0

5

10

E=0
15

νcpl/h (MHz)

20

0

5

ηd = 1101520

νcpl/h (MHz)FIG. 3. Molecular spectrum and NNS distributions of FanoFeshbach resonances of 168 Er calculated from our RMT model.
(a) Example of a spectrum of molecular binding energies as afunction of B for νcpl =h ¼ 2 MHz, ηd ¼ 0. Here, h is Planck’s
constant. Squares at E=h ¼ 0 MHz indicate crossings of molecularlevels with the threshold of two mj ¼ −6 atoms and correspond to
the position of Feshbach resonances. For the sake of visibility,
we show only the spectrum between B ¼ 0 and 10 G. (b) NNS
distributions of simulated Feshbach resonances for ηd ¼ 0
and νcpl =h ¼ 0 MHz (circles), 2 MHz (squares), and 10 MHz(triangles). The dashed and dash-dotted lines are Poisson andWigner-Dyson distributions, respectively. Solid lines are
best-fit Brody distributions with η ¼ 0.03ð1Þ, η ¼ 0.41ð5Þ, andη ¼ 0.82ð1Þ for νcpl =h ¼ 0, 2, and 10 MHz, respectively. (c) Fitted
Brody parameters of the nearest-neighbor energy-spacing
distribution of the eigenvalues of H RMT at B ¼ 0 G as a functionof νcpl . Circles, squares, down triangles, and up triangles correspond to ηd ¼ 0, 0.25, 0.5, and 1, respectively. (d) Fitted Brody
parameters of the NNS distribution of the Feshbach resonances
as a function of νcpl and for four ηd using the same marker code
as in (c). In panels (b)–(d) distributions are obtained by averaging
over 15 realizations of H RMT , each of dimension 500 × 500 andusing Feshbach resonances computed up to B ¼ 85 G.

PHYS. REV. X 5, 041029 (2015)shows an example of a molecular spectrum, the eigenvaluesof HRMT obtained with our RMT model as a function of B
with ηd ¼ 0 and νcpl =h ¼ 2 MHz. We observe that as B
increases, weakly bound molecular states avoid each othermultiple times before reaching the two-atom thresholdcreating a Feshbach resonance. When we turn off Hcpl ,the levels cross. Similar B-field dependencies of theeigenvalues occur for ηd > 0.
We investigate the effect of the parameters νcpl and ηd on
the NNS distribution of the Fano-Feshbach resonances aswell as that of the B ¼ 0 molecular levels. Figure 3(b)shows the NNS distribution of Feshbach resonances,obtained by averaging over 15 realizations of HRMT , forfour values of νcpl and ηd ¼ 0. For negligible νcpl , thedistribution follows PP ðsÞ and approaches PWD ðsÞ when
the anisotropic coupling strength νcpl is large comparedto ϵd . In fact, we find that a larger ϵd requires a larger νcpl
to develop correlations.
Figures 3(c) and 3(d) show Brody parameters fit to NNS
distributions as functions of νcpl and ηd . Figure 3(c) shows ηfor the B ¼ 0 molecular binding energies. For νcpl ¼ 0,
the Brody parameter is simply ηd, as expected from the
distribution of the diagonal H d , while for larger interaction
anisotropy νcpl, the parameter η ≈ 0.9, close to a WignerDyson distribution, independent of ηd . Figure 3(d) shows η
extracted from the RMT Feshbach resonance locations as afunction of νcpl . It suggests that the correlation in the NNSof the resonances is caused by νcpl, whereas it appears fairly
independent of ηd . More precisely, the Brody parameter
fitted to these distributions rapidly increases from η ≈ 0
to η ≈ 0.8 for νcpl ≲ ϵd and tends to one for larger νcpl.
We conclude from the RMT model that the correlations
between the locations of the Fano-Feshbach resonance are
essentially due to the avoided crossings between weakly
bound molecular states at finite B and are only weakly
dependent on the energy distribution at B ¼ 0. In fact, these
correlations increase for increasing νcpl .
IV. MICROSCOPIC COUPLED-CHANNELS
MODEL
A. Realistic setup
A quantitative understanding of the origin of thechaotic resonance distribution requires coupled-channels
and bound-state calculations with physically realistic
angular-momentum couplings and interaction potentials.
We do so here based on the time-reversal symmetric
Hamiltonian for the relative motion of Dy and Er describedin Refs. [21,40]. It contains the Zeeman Hamiltonian, themolecular vibration and rotation, and the molecular interactions with isotropic (orientation-independent) and anisotropic (orientation-dependent) contributions, V̂ i ðRÞ and~ respectively, where R~ describes the separation RV̂ a ðRÞ,and orientation of the atom pair R̂. The potential has eight041029-5

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PHYS. REV. X 5, 041029 (2015)

tensor operators coupling the two atomic and relative~ For B ¼ 0, theorbital angular momenta, ~|1 , ~|2 , and l.
~ is conserved. For
total angular momentum J~ ¼ ~|1 þ ~|2 þ l~~ is conserved.
B > 0 G, only the projection M of J along B
The zero of energy of the Hamiltonian is the energy of an
atom pair in the absolute lowest Zeeman sublevel,mjα ¼ −jα .
~ contain short-ranged
The potentials V̂ i ðRÞ and V̂ a ðRÞ
exchange, medium-ranged van der Waals, as well as
long-range magnetic dipole-dipole interactions. We use theisotropic van der Waals coefficient C6 ¼ 1723Eh a60 andanisotropic coefficients spread over ΔC6 ¼ 174Eh a60 for Er[21]. For Dy we have improved the value of van der Waals
coefficients of Ref. [40] by including additional experimental and theoretical transition frequencies and oscillator
strengths [41–44] and now use C6 ¼ 2003Eh a60 and spread
ΔC6 ¼ 188Eh a60 . In particular, the anisotropic spread for Dy
has significantly increased. Here, Eh ¼ 4.360 × 10−18 J is
the Hartree energy and a0 ¼ 0.052 97 nm is the Bohr radius.
B. Bound-state calculations
In Ref. [21] we performed initial coupled-channels calculations of the scattering between ultracold Er atoms andpredicted that tens of partial waves l should have been
included as the strength of the anisotropic contribution islarge. We, however, were unable to reach numerical convergence with respect to the number of coupled equations.
Here, we circumvent this limitation by performing
multichannel bound-state calculations, in which we use
B ¼ 0 eigenstates as a basis for those at B > 0 G. For
B ¼ 0, where J is a good quantum number, at most 49 and
81 Bose-symmetrized and parity-conserving channels arecoupled for Er and Dy, respectively. The B ¼ 0 coupled
Schrödinger equations are discretized on the interval
R ∈ ½0; Rmax  assuming zero boundary conditions andsolved as a matrix eigenvalue problem [45–48]. For eachJ, only eigenstates with energies between ½E0 ; E1  surrounding the zero of energy are computed and stored. The
bound states for B > 0 G are solutions of the matrix
eigenvalue problem that includes all computed B ¼ 0
solutions with jMj ≤ J ≤ Jmax and their coupling due tothe Zeeman interaction. Selection rules of the Zeeman
interaction ensure that there only exists direct coupling
between J and J0 zero-field eigenstates with J − J0 ¼ 0,
1. For both species, Rmax ¼ 1000a0 , E0 =h ¼ −3 GHz,and E1 =h ¼ 0.9 GHz, ensuring that Feshbach resonancelocations below 70 G are converged.
In this section on the microscopic calculations we focus
on analyzing the spectra at our coldest temperatures, where
the initial collision channel has s-wave (l ¼ 0) character.
Hence, we need to consider only even-l channels with total
projection quantum number M ¼ −12 and −16 for 168 Er
and 164 Dy, respectively, and inclusion of zero-field solutions up to Jmax ¼ 36 for Dy and 39 for Er is sufficient toreproduce the experimental resonance densities. In Sec. V,
we discuss higher-temperature collisions between Er atoms,
where d-wave (l ¼ 2) entrance channels must be considered
and, hence, spectra at other M values (i.e., M between −14
and −10 for 168 Er) contribute.
C. Interaction anisotropies
We first look into the role of interaction anisotropies
on the level distribution of the most weakly boundmolecular energy levels at zero magnetic field. There are
two dominant components to the anisotropy, the dispersion~ and magnetic dipole-dipole V MDD ðRÞ~ contribuV ΔC6 ðRÞtion. To distinguish the contributions of these two terms,
we define~ ¼ λΔC V ΔC ðRÞ~ þ λMDD V MDD ðRÞ;~V̂ a ðRÞ
6
6

ð1Þ

with variable strength λΔC6 and λMDD . We systematicallyincrease the strengths λΔC6 and λMDD from zero, where werecover the full physical strength for λMDD ¼ λΔC6 ¼ 1.
For completeness, we note that the dominant tensor
operator for the anisotropic dispersion contribution is~ ¼
V ΔC6 ðRÞca X 1
pﬃﬃﬃ f3ðR̂ · ~|i ÞðR̂ · ~|i Þ − ~|i · ~|i g þ    ;R6 i¼1;2 6with strength ca < 0 found with the methodology discussed in Sec, IVA. Weaker contributions indicated by dots
are included in our calculations. Moreover,2~ ¼ − μ0 ðgμB Þ f3ðR̂ · ~|1 ÞðR̂ · ~|2 Þ − ~|1 · ~|2 g;
V MDD ðRÞ4π R3where μ0 is the magnetic constant.
Figures 4(b) and 4(d) show the most weakly boundB ¼ 0, J ¼ 16 levels of 164 Dy2 as a function of anisotropy
strength for purely dipolar (λΔC6 ¼ 0, varying λMDD ) anddispersive (λMDD ¼ 0, varying λΔC6 ) anisotropic interaction, respectively. For λMDD ¼ λΔC6 ¼ 0, the binding energies are regularly structured with many near degeneracies.
In fact, the corresponding states are rovibrational levels of
the isotropic centrifugal potentials V̂ i ðRÞ and labeled by l.
In our 3-GHz energy window an s-wave channel has at
most three bound states, while even l > 0 channels with
their centrifugal barriers have fewer [21,49]. For small λΔC6
and λMDD , the degeneracy is lifted and levels shift linearly.
The linear dependence for increasing strength of the dipoledipole is approximately valid up to the physical value of
λMDD ¼ 1. Hence, the dipole-dipole interaction does notlead to our chaotic level distributions. In fact, Fig. 4(a)shows that at λMDD ¼ 1 and λΔC6 ¼ 0 the NND distribution
is Poissonian.
On the other hand, for a relatively small anisotropic
dispersion strength λΔC6 ≈ 0.1, levels start to avoid each041029-6

EMERGENCE OF CHAOTIC SCATTERING IN ULTRACOLD …
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PHYS. REV. X 5, 041029 (2015)

0
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s

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3 0

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s

(d)

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1

1.2

P(s)

E/h (GHz)

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(f)

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-2

-3
0 0.2 0.4 0.6 0.8

λMDD

1 1.2 0 0.2 0.4 0.6 0.8

λΔC

1 1.2

1

0.5

0
0

1

s

2

3

6

FIG. 4. Interaction-anisotropy-induced chaos of B ¼ 0 near-threshold bound states. (b) Weakly bound J ¼ 16 bound-state energies of
164 Dy as a function of the anisotropy scale λMDD with λΔC6 ¼ 0. (a) NNS distribution (red circles) for the J ¼ 16 bound-state data in2(b) at λMDD ¼ 1 and λΔC6 ¼ 0. The solid red line is a Brody distribution fit to the data and agrees well with a Poisson distribution.
(d) Weakly bound J ¼ 16 bound-state energies of 164 Dy2 as a function of the anisotropy scale λΔC6 with λMDD ¼ 0. (c) NNS distribution(purple squares) for the J ¼ 16 bound-state data in (d) at λMDD ¼ 0 and λΔC6 ¼ 1. The solid purple line is a Brody distribution fit to thedata and is close to a Wigner-Dyson distribution. (e) Moving average of the Brody parameter η as a function of λΔC6 (purple squares) or
λMDD (red circles) with bins Δλ ¼ 0.2 obtained by fitting the NNS distribution for the J ¼ 16 bound-state data in (b) and (d) to Brody
distributions, respectively. The horizontal lines at η ¼ 0 and 1 correspond to the Brody parameter for a Poisson and Wigner-Dyson
distribution, respectively. The 1σ error bars combine statistical and fitting uncertainties. (f) The individual-J (blue squares) andcombined-J (red circles) NNS distributions PðsÞ at λMDD ¼ λΔC6 ¼ 1 as a function of the normalized energy spacing s. The distributions
are derived from B ¼ 0 bound-state data for J ¼ 16; …; 25. The gray shaded areas in (a), (b), and (f) indicate the Wigner-Dyson
distribution.

other. Starting from λΔC6 ≈ 0.5, most avoided crossingsare noticeable on the 3-GHz scale of the figure. At thenominal λΔC6 ¼ 1, where there are 56 levels with−3 < E=h < 0 GHz, a significant fraction of the levels
have undergone multiple avoided crossings and cannot be
described by a single dominant partial wave. The levelspacing is chaotic as confirmed by the NND distribution
for λΔC6 ¼ 1 and λMDD ¼ 0 in Fig. 4(c). We computethe weakly bound J ¼ 16 levels for λMDD ¼ λΔC6 ¼ 1.
Visually the level distribution is much the same as theone shown in Fig. 4(d). Similar results have been obtainedfor 168 Er2.
Figure 4(e) quantifies the intuition gained from
Figs. 4(a)–4(d) by showing the Brody parameter η of the
B ¼ 0 J ¼ 16 164 Dy2 levels as a function of λΔC6 or λMDD.
The Brody parameter is obtained by fitting a Brody
distribution to the NNS distribution of the bound statedata in Figs. 4(b) and 4(d). For increasing dipole-dipole
strength λMDD and no anisotropic dispersion (λΔC6 ¼ 0), theparameter is always zero, indicating the prevalence of small
level spacings. On the other hand, in the absence of the
DDI, increasing λΔC6 leads to an increasing η. It evolves
from η ¼ 0.2 for λΔC6 ≲ 0.5 to η ¼ 0.7 for λΔC6 ¼ 1,
indicating a depopulation of small energy spacings. Notethat our systems does not reach a Wigner-Dyson distribution, which corresponds to η ¼ 1.
In Fig. 4(f), we compare two NNS distributions of
B ¼ 0 weakly bound states of 164 Dy2 obtained for thefull anisotropic interaction (λMDD ¼ λΔC6 ¼ 1). Both
distributions are based on jE=hj < 3 GHz bound statescomputed for J ¼ 16 up to 25. The first so-called
individual-J distribution is constructed by averaging
the NNS distribution of levels for individual J’s assuming that individual distributions are the same. The
second, combined-J NNS distribution, is calculated
from a sorted list of all J ¼ 16; …; 25 levels. Data for
J > 25 are not included as the number of bound states
is too small for a reliable determination of the NNS
distribution.
The individual-J NNS distribution is non-Poissonian aslevels with the same J repel each other. The combined-J
distribution, however, follows a Poisson distribution indicating that energies of bound states with different J are
uncorrelated. In other words, even though the Hamiltonian,~ isi.e., the set of coupling operators between ~|1 , ~|2 and l,
the same, differences in the matrix elements and thus
coupling strengths between channels lead to uncorrelated
eigenenergies.

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Dysprosium

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Erbium(d)

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-50

-100

-150

-200

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10 50

55

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1050

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E/h (MHz)

800
400
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-400
-5
-10
-15
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52

54

56

58

60

magnetic field (G)

(b)

(c)
B (G)

3

ρ (1/G)The study of the B ¼ 0 G multichannel bound states has
shown that interaction anisotropies mix channels with thesame J, while states with different J are uncorrelated.
The Zeeman interaction mixes these molecular levels andleads to the Fano-Feshbach spectrum. Figures 5(a) and 5(b)show example 164 Dy2 M ¼ −16 bound-state spectra as afunction of B on two binding-energy and field regions.
Similarly, Figs. 5(c) and 5(d) show M ¼ −12 168 Er2 boundstates. In all cases, the full nominal anisotropy (λΔC6 ¼ 1
and λMDD ¼ 1) is used. For Dy and Er, channels with J up
to 36 and 39 are included, respectively. The figure showsthat the Dy level density is higher than that for Er. This
simply follows from the larger atomic angular momentumof Dy, leading to a larger number of channels with the same
J − jMj. We also observe that for both species the levelstructure in the 0–10 G, small field region is qualitatively
different from that in the larger field region. For small B,
the avoided crossings are substantially narrower than forlarger B. Moreover, at small field the levels cluster, whileat larger field they are more uniformly distributed. Thesechanges are a consequence of the linearly increasing
Zeeman coupling between vibrational levels with different
J’s as a function of B.
Figure 6(a) shows effective length as ðBÞ as a functionof B. It diverges at every resonance location and is closely
related to the scattering length of a zero-energy collision.
Our calculations cannot be directly used to define thescattering length as we use a hard-wall potential for
R ≥ Rmax . This wall leads to a discrete set of states withpositive energy, and using the lowest of these Es ðBÞ, we
can define the effective length as ðBÞ shown in the figure
by solving for Es ðBÞ ¼ ℏ2 π 2 =f2μr ½Rmax − as ðBÞ2 g, withμr ¼ m=2 and atomic mass m [50].

as /a 0

D. Atom scattering in a magnetic field

2
1
0

15

20

25

Jmax

30

35

40

55
54
53
52
51
50

15 20 25 30 35 40

Jmax

FIG. 6. Theoretical near-threshold bound states and Feshbachresonances in a magnetic field. (a) Near-threshold bound statesfor M ¼ −12 168 Er2 for fields between B ¼ 50 and 60 G (bottom
half of image). The top half of the image shows the effective
scattering length, defined in the text, as a function of B. It is
infinite at a resonance location where a bound state has zeroenergy. Calculations use the physical interaction anisotropies andchannel states with J ≤ J max ¼ 39. (b) Theoretical Feshbachresonance density ρ̄ as a function of J max (purple circles)
computed from resonance locations between 0 and 70 G. The
dashed horizontal line indicates the experimental density for
168
Er2 . (c) Convergence study of resonance locations (crosses)
between 50 and 55 G as a function of J max . Purple lines connect
resonances when their location has converged.

It is of interest to briefly discuss the convergenceproperties of our calculations. The data in Figs. 6(a), 5(c),and 5(d) are based on computations with channels with J
up to Jmax ¼ 39. Figure 6(b) shows the 168 Er2 Feshbachresonance density ρ̄ as a function of Jmax . The resonancedensity increases linearly from ≈0.5 1=G at Jmax ¼ 12 butthen is seen to “saturate” for larger Jmax. At Jmax ¼ 39 theexperimental density is reproduced. In addition, Fig. 6(c)shows the field location of resonances between 50 and 55 G
as a function of Jmax . The resonance locations change
significantly for Jmax < 22, but then rapidly converge. Thisimplies strong mixing among bound states with those J.
On the other hand, the location of resonances that appear
for J ≥ 22 is almost immediately converged indicating
weak mixing to smaller J states.

60

magnetic field (G)FIG. 5. Theoretical 164 Dy2 [(a),(b)] and 168 Er2 [(c),(d)] nearthreshold bound states as a function of magnetic field. Calculations have been performed with the full nominal anisotropy.
Panels (a) and (c) show the near-threshold region between B ¼ 0
and 10 G, while panels (b) and (d) show the region between
B ¼ 50 and 60 G. Red crosses indicate the location of Feshbachresonances.

E. Comparison of experiment andcoupled-channels model
In Figs. 2(a) and 2(b), we show the NNS distributionof converged Feshbach-resonance locations based on ourmultichannel data between B ¼ 0 and 70 G for 164 Dy2
with Jmax ¼ 36 and 168 Er2 with Jmax ¼ 39, respectively.
For both species the distribution clearly deviates from a041029-8

EMERGENCE OF CHAOTIC SCATTERING IN ULTRACOLD …
Poisson distribution, consistent with the experimental
distributions that are also shown. The fitted experimental
and coupled-channel Brody parameters agree within theirerror bars.
The anisotropy parameters λΔC6 and λMDD in the
coupled-channels calculations and the parameter νcpl in
the RMT play analogous roles in the Hamiltonian and in theemergence of chaotic level distributions, even though no
explicit quantitative connection exists. This role is mostmanifest in the Brody parameters of the B ¼ 0 G boundstates and that of the Feshbach-resonance spectra for thetwo models. For 168 Er the corresponding Brody parametersfrom the coupled-channels calculations are ≈0.01 and 0.68
at the physical λMDD ¼ λΔC6 ¼ 1, respectively. Within the
RMT model, the small η value for the B ¼ 0 G level
distribution requires weak coupling νcpl ≪ ϵd and ηd ≈ 0.
In contrast, the Brody parameter for the Feshbachresonance spectrum requires νcpl ≈ ϵd and points at limitations of the current RMT model. Similar conclusions holdfor bosonic Dy. Future advanced RMT models might
circumvent these limitations by incorporating overlapping,
uncoupled chaotic series as is found from our B ¼ 0 G
coupled-channels calculations.
We plot the B-field-resolved Brody parameter ηðBÞ ofthe theoretical coupled-channels data in Figs. 2(c) and 2(d).
A comparison with the experimental ηðBÞ shows excellent
agreement for 164 Dy, while the agreement for 168 Er is lesssatisfactory. A possible explanation for the discrepancies in
168
Er is the larger bin-to-bin fluctuations as bins containfewer resonances than for 164 Dy.
For 164 Dy the theoretical field-resolved Brody parameterin Fig. 2(c) linearly increases from zero for small B fields

PHYS. REV. X 5, 041029 (2015)

and saturates at ηðBÞ ≈ 0.5 for fields larger than 35 G,
where the size or width of the avoided crossings between
weakly bound states is larger. For 168 Er in Fig. 2(d) we finda much more rapid increase of ηðBÞ at small fields. This isfollowed by a plateau at ηðBÞ ≈ 0.5 between B ¼ 20 and
50 G, after which ηðBÞ → 0.9 with an uncertainty of 0.2
close to a Wigner distribution. The initial rise of ηðBÞfor both atomic species is a consequence of weakly
bound vibrational levels, uncoupled and randomly distributed when B ¼ 0 G, that start to repel each other as the
Zeeman interaction increases in strength for increasing B.
The plateau at ηðBÞ ≈ 0.5 and the sudden increase of ηðBÞto one for 168 Er have no simple explanation and aredetermined by the not-fully-explored complex interplay
between the Zeeman and anisotropic interatomic interactions. It does, however, indicate that Wigner’s assumptions on ensembles of Hamiltonians do not hold for fields
below 50 G.
V. TEMPERATURE DEPENDENCE
OF THE RESONANCE DENSITY
We now describe the origin of the strong temperaturedependence of some of the resonances in our atom-loss
spectra and thus explain the accompanying increase of theresonance density. Here, atom loss is solely due to threebody recombination, where three ultracold atoms collide toform a diatomic molecule and an atom that are both lostfrom the atom trap. Figure 7(a) shows atom-loss spectra forone such resonance for 168 Er at four temperatures below
2 μK. Atom loss, indeed, is larger for larger temperatures,
but we also observe a broadening of the B-field width anda shift of the maximum loss position to larger B fields.

1.1
T=230 nK

1

(c)

(b)

0.8

0.9

L3/Lscale

Normalized Atom Number

(a)
1

0.8
0.7
0.6

0.6
0.4

740 nK
2000 nK

0.2

0.5
1400 nK
0.4

1.5

1.55

1.6

0

1.48

1.5

B (G)

1.52

B (G)

1.54

1.48

1.5

1.52

1.54

B (G)

FIG. 7. Line shapes for a strongly temperature-dependent 168 Er Feshbach resonance near B ¼ 1.48 G. Panel (a) shows experimental
data (markers with error bars) as a function of B of the remaining atom number divided by the atom number away from resonancemeasured 400 ms after initial preparation. Black, red, green, and blue markers correspond to data for temperatures T ¼ 230, 740, 1400,
and 2000 nK, respectively. Dashed lines connecting the markers guide the eye. Solid lines are theoretical line shapes of the remaining
atom number based on the d-wave (N ¼ 2) recombination rates shown in (c). Panels (b) and (c) show simulated three-bodyrecombination rates for the same four temperatures assuming three-body entrance s- (N ¼ 0) and d-wave (N ¼ 2) scattering,
respectively. Curves are based on a thermally averaged line shape discussed in the text. Recombination rates are scaled such that thelargest value in each panel is one. For both panels, μ ¼ 3.1μB and Γbr =kB ¼ 250 nK, while ΓðEÞ=kB ¼ 0.2ðE=Eref Þ2 nK in (b) andΓðEÞ=kB ¼ 0.1ðE=Eref Þ4 nK in (c), where Eref =kB ¼ 1000 nK.

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PHYS. REV. X 5, 041029 (2015)

Resonances with a weak temperature dependence shownone of these behaviors.
We show with an intuitive resonant “trimer” model that astrongly temperature-dependent resonance is due to scattering processes with entrance d-wave channels even though
the two-body d-wave centrifugal barrier, V b =kB ¼ 250 μK,
is 100 times larger than our highest temperature, where kB is
the Boltzmann constant. The difference in the power-law
Wigner-threshold behavior of the recombination rate withcollision energy for s- and d-wave entrance-channel collisions can explain our observations.
Three-body recombination has been extensively studied
in the context of Efimov physics [51–54]. We follow
Refs. [55–57] and start from a coupled-channels description in the (mass-scaled) hyperradius ρ, which describes thesize of the three-atomic system, and basis functions in thefive other hyperspherical coordinates that are ρ-dependent
eigenstates of the squared “grand-angular-momentum operator.” Similar to the coupled-channels description for two
atoms, there are entrance, open, and closed channels. The
collision starts in one of the entrance channels with atoms
in the energetically lowest Zeeman state and relative threebody kinetic energy E3, the dimer plus atom are the open
channels, and bound states in closed channels can lead toresonances. These closed channels dissociate to three freeatom states with at least one atom in a Zeeman level withhigher internal energy. The bound states are resonant trimer
states giving us our name for the model. It should, however,be realized that their origin lies in bound states of pairs of
atoms and that the resonant state is better thought of as apair bound state that hops from pair to pair. We defineE3 ≡ ℏ3 k23 =ð2μ3 Þ ≡ μ3 v23 =2 with the three-body reduced
pﬃﬃﬃ
mass μ3 ¼ m= 3, where k3 and v3 are the relative wave
vector and velocity, respectively.
The potentials in the entrance channels have long-range
repulsive centrifugal potentials, governed by the asymptotic behavior of the grand-angular-momentum operator,~and depend on the relative orbital angular momentum Nof the three atoms. In fact, the centrifugal potentials areℏ2 ðλ þ 3=2Þðλ þ 5=2Þ=ð2μ3 ρ2 Þ with non-negative integer
quantum number λ [55]. For N ¼ 0, the least repulsive
potential has λ ¼ 0, while that for N ¼ 2 has λ ¼ 2.
For an isolated trimer resonance in a closed channel
coupled to both entrance and other open channels, we canapply the resonance theories by Fano and Feshbach andderive that the recombination rate coefficient at collisionenergy E3 and entrance channel with quantum number λ is
given by L3 ðE3 ; BÞ ¼ v3 σðE3 ; BÞ, where the cross sectionσðE3 ;BÞ¼ð2N þ1Þ192π 2 jSðE3 ;BÞj2 =k53 andjSðE3 ; BÞj2 ¼

ΓðE3 ÞΓbr½E3 − μðB − B0 Þ2 þ ½Γtot ðE3 Þ=22

is a resonant expression for the square of a dimensionlessS-matrix element, where B0 is the trimer resonancelocation and μ is the magnetic moment of the resonant
trimer relative to that of the entrance channel. The definition for jSðE3 ; BÞj2 also contains the entrance-channel
energy width ΓðE3 Þ ¼ Aλ Eλþ2
to the trimer resonance with3
a characteristic power-law energy dependence that reflects
the threshold behavior of the scattering solutions in thecentrifugal potentials. The energy width Γbr determines
the decay or breakup rate of the resonance into the fast
atom and dimer pair and is independent of E3 . Finally,Γtot ðE3 Þ ¼ ΓðE3 Þ þ Γbr . For simplicity, we assume that
nonresonant, direct recombination from the entrance toopen channels is weak. We also note that for N ¼ 0 and
λ ¼ 0, L3 ðE3 ; BÞ approaches a finite constant for E3 → 0
as expected.
In our experiments we have thermal samples of Er and
we require the thermally averaged rate coefficientZ
1 ∞ 2
L3 ðT; BÞ ¼
E dEL3 ðE; BÞe−E=kT
Z 0
R
and normalization Z ¼ 0∞ E2 dEe−E=kT ¼ 2ðkTÞ3 . In order
to increase the signal-to-noise ratio, we allow a significant
fraction of atoms to be lost [see Fig. 7(c)], which, assuming
a homogeneous sample, can be modeled by the rateequation dnðtÞ=dt ¼ −3L3 ðT; BÞn3 ðtÞ for atom densitynðtÞ [53] with solution
N0Nðth ; BÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ;
1 þ 6L3 ðT; BÞn20 thwhere Nðth ; BÞ is the remaining atom number after holdtime th , N 0 is the initial atom number, and n0 is the initial
density. This nonlinear time evolution adds additional
broadening to the lines.
Figures 7(b) and 7(c) show our model event rates
L3 ðT; BÞ as a function of B for N ¼ 0, λ ¼ 0 and
N ¼ 2, λ ¼ 2, respectively. Curves are for the same four
temperatures as in Fig. 7(a). A comparison of Figs. 7(b) and7(c) shows a striking difference. The strongest features inFig. 7(b) are for the smallest temperatures, while those in
Fig. 7(c) are for the largest temperatures. This behavior
naturally follows from an approximation of the integrant in
L3 ðT; BÞ under the conditions kT ≫ Γbr ≫ ΓðEÞ [58].
In this limit the Lorentzian is sharply peaked around E3 ¼
μðB − B0 Þ for B > B0, and after some algebra it follows
that L3 ðT; BÞ as a function of B has a maximum value
proportional to ðkTÞλ−1 located at B ¼ B0 þ ðλ þ 2ÞkT=μ.
Consequently, for λ ¼ 0 and 2 the maximum loss rate
decreases and increases with T, respectively. Even for lessrestrictive parameter values as used in Fig. 7 this trend
remains.
Our experimental data have a temperature trend as inFig. 7(c). In fact, Fig. 7(a) compares our experimental loss
data with model Nðth ; BÞ for N ¼ 2, λ ¼ 2 using thesame parameters as in Fig. 7(c) and requiring a ≈ 50%

041029-10

EMERGENCE OF CHAOTIC SCATTERING IN ULTRACOLD …
maximum atom loss as in the experiment. It is worth noting
that, from our theoretical calculations, the magnetic-field
width of L3 ðT; BÞ is noticeably smaller than that forNðth ; BÞ, indicating that the finite hold time does indeed
lead to broadening. The agreement of the experimental data
and the prediction of our model for the losses is satisfactory
for all four temperatures given the limitations and approximations within our modeling. We conclude that ourstrongly T-dependent resonances correspond to d-wave
or more precisely N ¼ 2 entrance-channel collisions. Notethat we have not observed any resonances with temperaturedependence similar to Fig. 7(b) in our spectra. In the caseof resonances with a three-body s-wave entrance channel,
which would correspond to such a dependence, we inferthat the loss spectra are saturated. This will be subject for
future investigations.
As a corollary, this implies that for two colliding atoms,
as described in Sec. IV, temperature-dependent resonances
are due to collisions with entrance d waves for which there
are multiple allowed values of the total angular projection
quantum number M. Here, M ¼ −14 to −10 for bosonic Er
and M ¼ −18 to −14 for bosonic Dy. Numerical computations, not presented here, show that their zero-field boundstates and thus resonance locations are again uncorrelated
and random.
VI. CONCLUSION
In summary, we experimentally and theoretically study
the resonant scattering of ultracold Er and Dy atoms
in a magnetic field. We show that chaotic scattering as
witnessed by chaotic nearest-neighbor spacings between
Feshbach-resonance locations emerges due to the anisotropy in the molecular dispersion.
Our study also reveals several unique features of colliding magnetic lanthanides that have not been observed inany other ultracold atomic system. These lanthanides arecharacterized by their exceptionally large electron orbital
angular momentum, which leads to large anisotropic
dispersion interactions between these atoms. Our theoretical estimate shows that in both Er and Dy collisions theratio of anisotropic to isotropic dispersion interaction
ΔC6 =C6 is about 10%. This anisotropy leads to significant
splittings among the 48 and 81 gerade short-range potentials that dissociate to the ground-state atomic limits of
Er and Dy, respectively. We show that each potential has
its own rovibrational structure, which by Coriolis forces
and the Zeeman interaction interacts with that of otherpotentials, creating a dense distribution of levels near thethreshold and initiating chaos. In fact, we find a very large
number of partial waves contributing to the creation of
Fano-Feshbach resonances.
On the other hand, if we just consider the anisotropy
from the magnetic dipole-dipole interaction alone, ourcoupled-channel calculations indicate that chaos in thelevel distribution does not appear. The strength of the

PHYS. REV. X 5, 041029 (2015)

dipole-dipole interaction is too small. In addition, weshow that the NNS distributions for Dy and Er are very
similar, as can be expected from their similar ΔC6 =C6 ratio.
The difference in their magnetic moment plays only asmall role. This further confirms that chaos is due to theanisotropic dispersion interaction.
The distribution of Feshbach resonances of ultracold
ground-state alkali-metal, alkaline-earth, Yb, and Cr atoms,
as experimental studies have shown, is not chaotic. This isbecause these atoms have a zero electron orbital angularmomentum and, hence, only an isotropic dispersion interaction. Even though alkali-metal and Cr atoms have anonzero magnetic moment of 1μB and 6μB , respectively,
these moments do not lead to chaos. We would expect that
other magnetic lanthanides and actinides with nonzero
orbital angular momentum will exhibit chaotic properties
in their collisions. In addition, collisions between mixed
species, such as magnetic lanthanides and alkali metals,
like K þ Dy or Na þ Er, might be susceptible to chaos.
A first theoretical analysis for Li þ Er [59], however,estimates a small 2% dispersion anisotropy and no chaos
is predicted.
Another interesting property of magnetic lanthanide
gases is the extreme sensitivity of the atom-loss spectra
and, in essence, three-body recombination to the temperature. This phenomenon was first observed in Ref. [22] forloss spectra of Dy. The number of Dy resonances increases
by 50% when the temperature is increased from 420 to
800 nK. Here, we observe a 25% increase in the Er
resonance density when the temperature rises from 250
to 1400 nK. We show by a comparison of resonanceprofiles taken at several temperatures and predictions of atheoretical model of three-body recombination via theformation of a trimer, or, more precisely, of a shared pairbound state, that the origin of the temperature-dependent
resonances lies in the “partial wave” of the three-atom
entrance channel. Entrance channels with zero and nonzero
total orbital angular momentum N lead to line shapeswith a different temperature behavior. Those with N ¼ 0
or “s-wave” entrance channels have sharply decreasing
recombination rates with temperature, whereas those with
N ¼ 2 or “d-wave” entrance channels have an increasing recombination rate. Temperature-sensitive resonances
can be explained only by “d-wave” collisions. It is worth
noting that for alkali-metal-atom collisions a number ofentrance-channel p-wave resonances have been observed(see, for example, Ref. [60] for cesium). Analysis of thetemperature-dependent rate coefficient, however, was not
performed.
ACKNOWLEDGMENTS
The Stuttgart group thanks Axel Griesmaier for the
support at the early stage of the experiment. The Dy work
is supported by the German Research Foundation (DFG)
within SFB/TRR21. H. K. acknowledges support by the041029-11

T. MAIER et al.

PHYS. REV. X 5, 041029 (2015)“Studienstiftung des deutschen Volkes.” The Er work is
supported by the Austrian Ministry of Science andResearch (BMWF) and the Austrian Science Fund(FWF) through a START grant under Project No. Y479N20 and by the European Research Council under ProjectNo. 259435. K. A. was supported within the Lise-Meitnerprogram of the FWF. L. C. acknowledges support by the
FWF through SFB FoQuS. Work at Temple University is
supported by the AFOSR Grant No. FA9550-14-1-0321
and the NSF Grant No. PHY-1308573. The work at JQI is
supported by the NSF Grant No. PHY-1506343.

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4.4. Publication:
Ultracold Dipolar Molecules Composed of Strongly Magnetic
Atoms†
Physical Review Letters 115, 203201 (2015)
submitted 17 April 2015; published 9 Nov 2015
DOI 10.1103/PhysRevLett.115.203201

Albert Frisch,1,2 Michael Mark,1 Kiyotaka Aikawa,1 Simon Baier,1 Rudolf Grimm,1,2 Alexander Petrov,3 Svetlana Kotochigova,3 Goulven Quéméner,4 Maxcence Lepers,4 Olivier Dulieu,4and Francesca Ferlaino1,2
1Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
4
Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, 91405 Orsay, France†

The author of the present thesis helped in performing the measurements, and contributed in writing themanuscript. The theoretical calculations of this publication have been contributed by A. P., S. K., G. Q.,
M. L., and O. D.

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Ultracold Dipolar Molecules Composed of Strongly Magnetic Atoms
A. Frisch,1,2 M. Mark,1 K. Aikawa,1,* S. Baier,1 R. Grimm,1,2 A. Petrov,3,† S. Kotochigova,3
G. Quéméner,4 M. Lepers,4 O. Dulieu,4 and F. Ferlaino1,2

1Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
4
Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, 91405 Orsay, France(Received 17 April 2015; published 9 November 2015)2

In a combined experimental and theoretical effort, we demonstrate a novel type of dipolar system madeof ultracold bosonic dipolar molecules with large magnetic dipole moments. Our dipolar molecules areformed in weakly bound Feshbach molecular states from a sample of strongly magnetic bosonic erbium
atoms. We show that the ultracold magnetic molecules can carry very large dipole moments and we
demonstrate how to create and characterize them, and how to change their orientation. Finally, we confirm
that the relaxation rates of molecules in a quasi-two-dimensional geometry can be reduced by using theanisotropy of the dipole-dipole interaction and that this reduction follows a universal dipolar behavior.
DOI: 10.1103/PhysRevLett.115.203201

PACS numbers: 34.50.Cx, 31.10.+z

Ultracold dipolar particles are at the heart of very intense
research activities that aim to study the effect of interactions
that are anisotropic and long range [1,2]. Dipolar quantum
phenomena require ultracold gases and a strong dipoledipole interaction (DDI). So far, strongly dipolar gases havebeen obtained using either atoms with a large magnetic
dipole moment or ground-state polar molecules with anelectric dipole moment [2]. With both systems, manyfascinating many-body quantum effects have been observed
and studied, such as the d-wave collapse of a dipolar BoseEinstein condensate [3,4], the deformation of the Fermi
sphere [5], and the spin-exchange phenomena [6,7].
Here, we introduce a novel kind of strongly dipolar
particles. These are weakly bound dipolar molecules
produced from a pair of atoms with large magnetic dipole
moments, such as erbium (Er). The central idea is that thesemolecules can possess a very large magnetic moment μ up
to twice that of atoms (e.g., 14 Bohr magneton, μB , for Er2 )and have twice the mass of the atoms. As a consequence,the degree of “dipolarity” of the magnetic molecules ismuch larger than the one of atoms. This can be quantified interms of the dipolar length ad ¼ mμ0 μ2 =ð4πℏ2 Þ [1], whichsolely depends on the molecular mass m and on μ; ℏ is the
Planck constant divided by 2π. To give an example, Er2
with μ ¼ 14μB has an ad of about 1600 a0 , which largely
exceeds the typical values of the s-wave scattering length.
Here, a0 is the Bohr radius. Moreover, in contrast to
ground-state heteronuclear molecules, the dipole momentof the magnetic molecules does not vanish at zero external(magnetic) field, opening the intriguing possibility ofinvestigating the physics of unpolarized dipoles.
In a joined experimental and theoretical effort, we study
the key aspects of ultracold dipolar Er2 molecules,
including the association process, the molecular energy
0031-9007=15=115(20)=203201(5)spectrum, the magnetic dipole moments, and the scattering properties in both three- (3D) and quasi-twodimensional (Q2D) geometries.
Erbium belongs to the class of strongly magnetic
lanthanides, which are currently attracting great attention
in the field of ultracold quantum gases [4,8–10]. Indeed,
these species exhibit unique interactions. Beside the longrange magnetic DDI, these species have both an isotropic
and an anisotropic contribution in the short range van der
Waals (vdW) potential. The latter results from the large
nonzero orbital momentum quantum number of the atoms[11,12]. This manifold leads to an extraordinary richmolecular spectrum, reflecting itself in a likewise dense
spectra of Feshbach resonances as demonstrated in recent
scattering experiments [4,13,14]. Each resonance position
marks an avoided crossing between the atomic scattering
threshold and a molecular bound state, which can be used
to associate molecules from atom pairs [15].
We create and probe Er2 dipolar molecules by using
standard magnetoassociation and imaging techniques [15].
Details of the production schemes are described in the
Supplemental Material [16]. In brief, we begin with anultracold sample of 168 Er atoms in an optical dipole trap
(ODT) in a crossed-beam configuration. The atoms are spinpolarized into the lowest Zeeman sublevel (j ¼ 6,
mj ¼ −6). Here, j is the atomic electronic angular momentum quantum number and mj is its projection on thequantization axis along the magnetic field. To associate
Er2 molecules, we ramp the magnetic field across one of thelow-field Feshbach resonances observed in Er [4,13]. We
experimentally optimize the ramping parameters, such as
the ramp speed and the magnetic-field sweep interval, by
maximizing the conversion efficiency. In our experiment
we typically achieve a conversion efficiency of 15%, which

203201-1© 2015 American Physical Societybinding energy / h (kHz)

0

0
μ1

μ2

μ3

μ4

atom loss

1

(a)
(b)

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TABLE I. Experimental (Expt.) and theoretical (Theo.) magnetic moments of four molecular states near the atomic threshold,Feshbach-resonance positions BFR , dipolar lengths, outer turning
points R , and dominant quantum numbers l, J, and M. For
convenience, the molecular states are labeled as μi withi ¼ 1; …; 4. The specified uncertainties correspond to the 1σstatistical errors.

400
0.0

0.5

1.0

1.5
2.0
magnetic field (G)

2.5

BFR
(G)

3.0

FIG. 1 (color online). Er2 weakly bound molecules. (a) Atomloss spectrum [4] from 0 to 3 G and (b) near-threshold binding
energy of the corresponding molecular states. The solid lines arefits to the experimental data and extrapolated to larger Eb up toh × 500 kHz. The error bars are smaller than the symbols.

is a common value for boson-composed Feshbach molecules [15]. To obtain a pure molecular sample, we remove
all the remaining atoms from the ODT by applying aresonant laser pulse. Our final molecular sample contains
about 2 × 104 Er2 Feshbach molecules at a temperature of
300 nK and at a density of about 8 × 1011 cm−3 [16].
A central question regards the magnitude of the dipole
moment owned by the magnetic molecules. We experimentally determine μ by using magnetic-field modulation
spectroscopy, a technique which was successfully applied
to alkali atoms [20–22]. With this method, we measure themolecular binding energy Eb near the atomic threshold as afunction of the magnetic field B. The binding energy isrelated to the differential magnetic moment of the molecules with respect to the atom-pair magnetic moment 2μa .
Here, μa ¼ −gmj μB ¼ 6.98μB in the case of Er, where g ¼
1.16 is the Er atomic Landé factor. We thus extract μ byusing the relation μ ¼ 2μa − jdEb ðBÞ=dBj. Our spectroscopic measurement begins with an ultracold atomic sample
near a Feshbach resonance. We then add a small sinusoidal
modulation to the bias magnetic field for 400 ms. The
modulation frequency is varied at each experimental run.
When it matches Eb =h, prominent atom losses appear
because of molecule formation. We trace the near-thresholdmolecular spectrum by repeating the measurement forvarious magnetic-field values. Figure 1 shows the Er2molecular spectrum in a magnetic-field range up to 3 G.
In our range of investigation, we identify four molecularenergy levels, which, near threshold, exhibit a linear dependence on B. For each state, we obtain a different μ value,
ranging from 8 to 12μB [23], as listed in Table I.
For alkali-metal atoms, which possess much simpler
interaction properties than lanthanides, theoretical
approaches based on coupled-channel calculations havebeen extremely successful in assigning the quantum numbers of the molecular energy levels and reproducing
molecular spectra [15]. However, a straightforward extension of these methods to the lanthanide case is out of reachbecause of their complex scattering physics involving

μ=μB
Expt. Theo.

ad
(a0 )

R
(a0 )

jl; J; Mi

μ1 0.91 11.30(7) 11.20 1041(13) 72.0 j4;12;−12=−10=−9i
μ2 2.16 11.51(4) 11.46 1080(8) 71.0
j4; 10; −10i
μ3 2.44 11.84(2) 11.75 1143(4) 86.0
j2; 12; −10i
μ4 2.47 7.96(3) 7.92 517(4) 57.0
j6; 10; −7= − 6ihighly anisotropic interactions and many partial waves[13]. Inspired by work on alkali-metal collisions [24–27],
we develop a new theoretical approach to identify themolecular quantum numbers, based on approximate adiabatic potentials and on the experimentally measured μ as
input parameters. Our scattering model is detailed in the
Supplemental Material [16], whereas we here summarize
the central ideas of our approach.
We first solve the eigenvalue problem of the full atomatom interaction potential operator [16], whose eigenvalues
are the adiabatic potentials Un ðR; BÞ.
PThe corresponding
eigenfunctions read as jn; Ri ¼ i cn;i ðRÞjii, wheren ¼ 1; 2; …, and cn;i ðRÞ are R-dependent coefficients.
The molecular state jii is uniquely determined by the setof angular momentum quantum numbers (l, J, M), wherel is the molecular orbital quantum number, J~ ¼ ~j1 þ ~j2 thetotal atomic angular momentum, and M its projection on
the internuclear axis.
To derive the corresponding “adiabatic” molecular magnetic moments, we calculate μcalc ≈ −dUn ðR; BÞ=dB at theposition of the outer classical turning point R ¼ R . This
choice is justified by the fact that most of the vibrational
wave function is localized around R .
From the Hellmann-Feynman
theorem it then follows
P
that μcalc ¼ −gμB i M i jci ðR Þj2 . Finally, we assume thatfor each Feshbach resonance a vibrational state is onresonance and we find the adiabatic potential that has amagnetic moment closest to the measured one within 1%.
Once the best match is identified, the corresponding jn; Ri
sets the molecular state jii, characterized by l, J, and M,
with the largest, dominant contribution. In our range ofinvestigation we observe d-, g-, and i-wave molecularstates; see Table I. These states show several dominant M
contributions. This fact is unusual and reflects the dominantrole of the DDI, which couples several adiabatic potentials
and M components. As shown in Fig. 2, this mixing effectis particularly dominant below 10 G, where the DDI at R islarger than the Zeeman interaction. Above 10 G, we predict
μ to be equal to integer multiples of gμB [16].

203201-2

16x103

13

12.80

12

11.64

11

molecule number

14

μ/μB = -M g = 13.97

10.47

10

9.31

9
8.15

8

6.98

7
2

4

6

8

10

12

12
8

(a) 3D

7x103

molecule number

μ/μB

15

6
0

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B

6

y

θ=0
5

x

z
θ = 90°

4

θ

3

y

x

2
0

As summarized in Table I, we find very good overall
agreement between the measured and the calculated
molecular magnetic moments. For the largest observed μ,
we calculate a corresponding dipolar length, ad ≈ 1150a0 .
This value exceeds the typical range of the vdW potentials,
setting the DDI as the dominant interaction in the system.
Remarkably, ad for Er2 is comparable to the one realizedwith ground-state KRb molecules [28], which are an
extensively investigated case serving as a benchmark dipolarsystem.
Following the methods introduced for KRb [29–31], wetest the dipolar character of Er2 by performing scatteringexperiments in a 3D and in a Q2D optical dipole trap. We
control the DDI between molecules by tuning the dipole
orientation, which is controlled by changing the directionof the magnetic field and is represented by the angle θ
between the magnetic-field axis and the gravity axis. Our
experiment begins with the atomic sample trapped either ina 3D or in a Q2D ODT. The Q2D trap is created bysuperimposing a vertically oriented, one-dimensional optical lattice [16]. After the magnetoassociation and theremoval of the remaining atoms, we probe the numberof molecules as a function of the holding time in the ODT.
We perform measurements for the molecular states μ1 , μ2 ,and μ4 [32]. For each of these states, we measure thecollisional stability of the sample for both in-plane(θ ¼ 90°) and out-of-plane (θ ¼ 0°) dipole orientation,and extract the corresponding relaxation rate coefficients,β⊥ and β∥ , using a standard two-body rate equation [33].
Figure 3 shows typical molecular decay curves in(a) 3D and in (b) Q2D. In 3D, we confirm that theθ = 0°

B

magnetic field (G)FIG. 2 (color online). Adiabatic magnetic moment as a functionof magnetic-field strength evaluated at the entrance channel
energy. Each curve corresponds to the adiabatic magnetic momentof one adiabatic potential U n ðR; BÞ. The magnetic moments in theasymptotic limit of large B are given. The dashed vertical lines
correspond to the field strength where we have observed Feshbachresonances. The red-filled circles represent the experimentallymeasured magnetic moments at these resonance locations.

z(b) q2D

5

10
holding time (ms)15

FIG. 3 (color online). Typical time evolution of the number ofmolecules for θ ¼ 90° (squares) and θ ¼ 0° (circles) in a 3D (a)and in a Q2D trap (b). The data refer to molecules in the state μ1
for the 3D case (a) and molecules in the state μ2 in Q2D (b). The
insets in (b) show an illustration of molecules in pancake-shaped
traps with out-of-plane (right) and in-plane (left) orientations.
The solid lines are two-body decay fits to the data. The error barsfor (a) and (b) are smaller than the data points and are not shown.
The data points in (a) are obtained by averaging five independentmeasurements and in (b) about 50 measurements have been
averaged.

inelastic decay does not depend on θ. We obtainβ3D ¼ 1.3ð2Þ × 10−10 cm3 =s. This is a typical value forboson-composed Feshbach molecules, which undergo a rapid
vibrational quenching into lower-lying molecular states, as
demonstrated with alkali atoms [33]. Contrary, in Q2D thedecay rates clearly depend on the dipole orientation. For each
investigated molecular state, β⊥ is larger than β∥ . We find a
reduction of losses of up to 30% for out-of-plane orientation,
for which the DDI is predominantly repulsive. The ratioðβ⊥ ðTÞ=β∥ ðTÞÞ increases with increasing μ; see Table II. We
note that stronger suppression of losses can be obtained using
a tighter two-dimensional confinement [29], which is presently not reachable with our experimental parameters.
TABLE II. Experimental and theoretical loss rate coefficients β
for T ¼ 400 nK and for various μ and θ at B ¼ 200 mG.
Uncertainties of β are statistical from fitting and systematicdue to number density uncertainty. For the slightly different
values of μ compared to Table I and the error discussion see theSupplemental Material [16].
μ=μB

β⊥ ð10−6 cm2 =sÞ
Expt.
Theo.

μ4 8.7(6) 12.5  0.3  3.3
μ1 10.9(5) 9.5  0.2  2.5
μ2 11.7(3) 11.3  0.2  2.9

203201-3

6.00
6.81
7.12

β∥ ð10−6 cm2 =sÞ
Expt.
Theo.
10.6  0.3  2.8
7.3  0.1  2.1
8.6  0.2  2.3

4.79
5.07
5.13

PHYSICAL REVIEW LETTERS

PRL 115, 203201 (2015)

The reduction of losses in Q2D draws a natural analogy
with the observations obtained with KRb molecules [31].
From a comparative analysis between Er2 and KRb, one
can unveil universal behavior attributed to the DDI, forsystems being different in nature, but sharing a similar
degree of dipolarity. We thus theoretically study thescattering behavior of Er2 using a theoretical approachsimilar to the one successfully applied to KRb. Ourformalism, which accounts for the DDI and the isotropic
vdW interaction, is described in Refs. [16,34].
We compute the Er2 þ Er2 loss rate coefficients βðTÞ in
3D and in Q2D for given values of μ, θ, and T. By
averaging over a 3D and a 2D Maxwell-Boltzmann distribution, we obtain the thermalized loss rate coefficientsβðTÞ in 3D and in Q2D, respectively. In 3D, we find a rate
coefficient of 1.0 × 10−10 cm3 =s at T ¼ 300 nK, which is
close to the experimental value [35]. In Q2D, our calculations show that the collision dynamics at long range, andthus the value of β, depends on the dipole orientation andmonotonically increases with μ. As in the experiments, ourcalculations show that collisions for in-plane orientation(β⊥ ) lead to larger molecular losses than for out-of-planeorientation (β∥ ). In Table II, we compare theory andexperiment. The absolute values of β agree within afactor of 2. This difference is well explained by the factthat our model does not include details of the short-range
physics, with the Er4 potential energy surfaces currently
unknown [16].
Remarkably, the experimental and calculated ratiosβ⊥ ðTÞ=β∥ ðTÞ agree very well with each other; see
Fig. 4. This suggests that β⊥ ðTÞ=β∥ ðTÞ for Er2 Feshbach
molecules is determined by the DDI and not by the shortrange physics, and that it can be correctly described using apoint-like-dipole formalism [16]. Figure 4 shows thecomparative analysis between bosonic 41 K87 Rb and
168 Er , and fermionic 40 K87 Rb and 167 Er168 Er based on2fermions

KRb theory
Er2 theory

2

β⊥/β

Er2 exp.

μ1

μ4

1
0

1

μ2
2

bosons3

ad /ã

FIG. 4 (color online). Universal loss rate ratio β⊥ =β∥ as afunction of ad =a~ for Er2 (circles) and KRb (squares) for a fixedvalue adB =aho ¼ 4.85 corresponding to T ¼ 400ð40Þ nK andνz ¼ 31.2ð1Þ kHz [16]. The gray shaded area is due to theuncertainty of T. Here, a~ ¼ aho for bosonic molecules (filled
symbols) and a~ ¼ avdW for fermionic molecules (open symbols).
The calculated loss rate ratios of Er2 are compared with theexperimental data for states μ1 , μ2 , and μ4 (triangles).

week ending
13 NOVEMBER 2015

our numerical calculations. Independent of the nature of themagnetic or electric dipolar system, we find universal~ one for bosons with a~ ¼
curves as a function of ad =a:
aho and one for fermions when a~ ¼ avdW . Here, aho is the
harmonic oscillator length and avdW ¼ ð2mC6 =ℏ2 Þ1=4 is the
vdW length with C6 the vdW coefficient. The fasterincrease of β⊥ =β∥ for fermions with respect to bosons isdue to the statistical fermionic suppression of β∥ in Q2D
that does not occur for bosons as explained in Ref. [36].
The universal behavior of ultracold dipolar scattering has
been previously pointed out in Ref. [37]. In the Wigner
regime, we derive simple universal scaling laws for dipolarbosonic and fermionic molecules [16,37]. For bosons with ad ,aho > avdW , which is the case of our Er2 molecules, we find½β⊥ ðTÞ=β∥ ðTÞ∼ðadB =aho Þ4 ðad =aho Þexp½2ðad =aho Þ2=5 . For
fermions with ad , avdW < aho , ðβ⊥ =β∥ Þ ∼ ðad =avdW Þ3 . Here,pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
adB ¼ h= 2πmkB T is the thermal de Broglie wavelength.
To conclude, our work reports on the study of strongly
dipolar molecules created by pairing ultracold atoms withlarge magnetic dipole moments. We anticipate that ourscheme can be generalized to other magnetic lanthanide
species and has the potential to open regimes of investigations, which have been unaccessible so far. First, theextraordinarily dense and rich molecular energy spectrumof Er opens the exciting prospect of cruising throughmolecular states of different magnetic moments or evencreating molecular-state mixtures with dipole imbalance[22,38,39]. Second, in contrast to electric polar molecules
where the electric dipole moment is zero in the absence of apolarizing electric field, magnetic dipolar molecules have apermanent dipole moment allowing us to study the physics
of unpolarized dipoles. In addition, strongly magnetic
Feshbach molecules offer a novel case of study for
scattering physics. These molecules are in fact diffuse inspace with a typical size on the order of the vdW length.
This novel situation can also have interesting consequences
and trigger the development of extended scattering models,
which account for multipolar effects and truly four-bodycontributions when the molecule size becomes comparable
to ad [40]. Finally, a very promising development will be tocreate fermionic Er2 dipolar molecules where vibrational
quenching processes are intrinsically suppressed because of
the Pauli exclusion principle [41,42].
The Innsbruck team acknowledges support by theAustrian Ministry of Science and Research (BMWF) andthe Austrian Science Fund (FWF) through a START Grant
under Project No. Y479-N20 and by the European ResearchCouncil under Project No. 259435. Research at Temple
University is supported by an AFOSR Grant No. FA955014-1-0321 and NSF Grant No. PHY-1308573. G. Q., M. L.,
and O. D. acknowledge the financial support of “Projet
Attractivité” from Université Paris-Sud and the
COPOMOL project (No. ANR-13-IS04-0004) from
Agence Nationale de la Recherche.

203201-4

PRL 115, 203201 (2015)*

PHYSICAL REVIEW LETTERS

Present address: Department of Physics, Graduate Schoolof Science and Engineering, Tokyo Institute of Technology,
2-12-1-S5-9, Ookayama, Meguro-ku, Tokyo 152-8551,
Japan.
†
Also at: NRC Kurchatov Institute PNPI 188300, Division of
Quantum Mechanics, St. Petersburg State University, St.
Petersburg 198904, Russia.
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[16] See Supplemental Material at http://link.aps.org/
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Er2 molecules, the theoretical description of the collision
formalism, and the adiabatic model.
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[23] For future experiments, it will be very interesting to accessmolecular levels parallel to the atomic threshold for which
μ ¼ 2μa [22].
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[32] The state μ3 is difficult to access because of its proximitywith μ4 .
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203201-5

Ultracold Dipolar Molecules Composed of Strongly Magnetic Atoms:
Supplemental Material
A. Frisch,1, 2 M. Mark,1 K. Aikawa,1, ∗ S. Baier,1 R. Grimm,1, 2 A. Petrov,3, †S. Kotochigova,3 G. Quéméner,4 M. Lepers,4 O. Dulieu,4 and F. Ferlaino1, 21Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
4
Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, 91405 Orsay, France

Creation of Er2 in 3D and Q2D

We create Feshbach molecules using standard techniques of magneto-association across a Feshbach resonance. As demonstrated in Refs. [1, 2], Er features anenormous number of Feshbach resonances. Here, we focus on the resonances observed below 3 G. In particular, we first create an ultracold atomic sample of about
3 × 105 168 Er atoms at a temperature of T ≈ 150 nK,
which is just above the onset of Bose condensation, see
Ref. [1]. The atoms are confined into a three-dimensional(3D) crossed optical dipole trap with frequencies νx =51.5(2) Hz, νy = 13.2(3) Hz, and νz = 207(1) Hz. We
choose magnetic fields of 1.4 G, 2.3 G, and 2.8 G for themolecular states µ1 , µ2 , and µ4 , respectively. We thenmagneto-associate molecules by ramping the magnetic
field 150 mG below the Feshbach resonance. The typicalramp speed is 90 mG/ms. After the molecule association,
we remove all the residual atoms from the optical dipole
trap by applying a short laser pulse. The pulse is on resonance with the strong atomic transition at 401 nm [3] andhas a duration of 1 µs with an intensity of ∼ 40 mW/cm2 .
To realize a Q2D geometry, we superimpose a one dimensional optical lattice beam to the system after finishing evaporation in the 3D trap. The lattice is realized
from a retro-reflected laser beam at 1064 nm, propagating along the vertical direction. The beam has a waist of
250 µm and a typical power of 8 W. As a result, the particles are confined into an array of Q2D pancakes withfrequencies νr = 33.0(3) Hz in the radial direction andνz = 31.2(1) kHz in the tightly confining axial direction. We first load the lattice from the atomic sample
and we then magneto-associate Er2 in the lattice. The
molecule conversion efficiency in the Q2D geometry is. 5 %, which is below the one observed in the 3D trap.
With this scheme, we produce about 1.1 × 104 molecules
at a temperature of 400 nK, corresponding to a density of
3.8 × 107 cm−2 . The molecules fill about 35 lattice layers.
We control the molecular dipole orientation by changing the orientation of the magnetic field. The orientation is quantified in term of the angle θ, which defines
the angle between the quantization axis, set by the magnetic field orientation, and the z-axis of the lattice trap.
We prepare the molecular samples at either θ = 0◦ or

90◦ , correspondingly side-by-side (repulsive) or head-totail (attractive) dipolar collisions. The magnetic field isrotated by using three pairs of independently-controlled
magnetic-field coils. We pay particular attention thatwhen changing the orientation of the magnetic field wekeep its magnitude constant. We check this by performing radio-frequency spectroscopy between Zeeman sublevels for different angles of rotation. We typically rotate
the field within ∼ 6 ms.
For all our loss-rate measurements, we jump to a magnetic field of about 200 mG after molecule association. At
this field, Eb is of the order of few h × 1 MHz. We choose
to perform our measurement at this magnetic-field value
because around 200 mG there are no Feshbach resonances
and the molecular spectrum might be less dense. Usinga Stern-Gerlach technique [4], we measure µ at 200 mG
for all the three target molecular states. We find a slightshift of µ in comparison with the values from Table II ofa few percent to 10.9(5) µB for µ1 , 11.7(3) µB for µ2 , and8.7(6) µB for µ4 .
The given uncertainties for the measured loss rates inTable II are composed of a statistical error with one standard deviation derived from fitting a two-body rate equation to the measured data, and a systematic uncertainty
coming from number density calibration. Due to the distribution of molecules across many lattice layers this is by
far the greatest uncertainty in the Q2D geometry. The
average 2D density and its uncertainty was calculatedusing a number-weighted average over occupied latticelayers similar to Ref. [5]. When calculating the loss rate
ratio β⊥ /βk , the systematic uncertainty in the density
can be neglected as it is highly correlated for the measurement of β⊥ and βk .
Collision Formalism

We briefly describe the theoretical formalism used inthis article to determine the collisional properties of Er2
molecules in free space (3D collisions) and in an onedimensional optical lattice (Q2D collisions), in an arbi~ More details can be found intrary magnetic field B.
Ref. [6, 7].
We use a time-independent quantum formalism based

2
on spherical coordinates ~r = (r, θr , φr ) describing the relative motion of two Er2 molecules. The quantization axisẑ is chosen to be the confinement axis of the optical lattice. A spherical harmonic basis set, summed over different partial waves ` with projections m` on the quantization axis, is used to expand the total colliding wave
function. The one dimensional optical lattice is supposed
to be deep enough to consider the collision taking placein an individual pancake. One pancake is represented as
an harmonic trap for the relative motion of reduced mass
mred
1
mred ω 2 z 2(1)
2
with ω = 2πν and ν = 31.2 kHz. The 3D collisions arerecovered by setting ν = 0. We consider molecules in the
ground state of the harmonic oscillator. A given state of
an Er2 Feshbach molecule is described by a rather complicated linear combination of atomic states which cannot
be precisely calculated as mentioned in the next sectionof this Supplemental Material. Therefore we consider
that the molecule has a magnetic moment of magnitudeµ aligned along the magnetic field which makes an angle
θ with the confinement axis. The interaction between
two molecules is provided by the magnetic dipole-dipole
interaction
Vho =

Vdd =

µ2 (1 − 3 cos2 (θr − θ)).
(4π/µ0 ) r3

(2)

We also used an isotropic Er2 + Er2 van der Waals interaction given by
C6
(3)
r6
with C6 = 4 × 1760 = 7040 a.u. which amounts to fourtimes the value of an isotropic atom-atom coefficient of1760 a.u. from the theoretical work of Ref. [8]. Note thatan alternative value of 1723 a.u. based on observed transitions was obtained in Ref. [2]. The Schrödinger equation
is solved for each radial intermolecular separations r using a log-derivative propagation method. Matching thecolliding wavefunction and its derivative with appropriate two-dimensional asymptotic boundary conditions at
long-range [6] provides the cross section and the rate coefficient as a function of the collision energy for any arbitrary configurations of magnetic fields and confinements.
Averaging the cross sections over a 3D and 2D dimensional Maxwell-Boltzmann distribution provides the corresponding thermalized rate coefficients β(T ) for a given
temperature.
At short range, we assume that the molecules undergoa full loss mechanism process with a unit probability (itcan be either an inelastic or a possible reactive process).
This assumption, which corresponds to the so-called universal regime in ultracold collisions, considers that thephysics is independent of the initial short-range scattering phase-shift [9] of the full potential energy surfaces of
VvdW = −Er4 . This is what it is usually assumed for theory as
nothing is known about this potential energy surface atshort range. Then, if the magnitude of the rates differs
between experiment and theory, one can learn that anexperimental system deviates from this universal regime
and short-range effects play a role.
To circumvent this, it is more convenient to computethe ratio of the theoretical rates of two different magnetic field orientations since we will start with the same
short-range physics condition for both orientations, andcompare it with the corresponding experimental ratio.
An analysis based on the universal behavior of dipolar
collisions in confinement of Ref. [10] using a Quantum
Threshold model leads to the following formula for theratio β⊥ (T )/βk (T ). For bosons, using Eq. 30 of Ref. [10]to describe β⊥ (dipole dominated) and Eq. 32 of the samereference for βk (confinement dominated) we find4adB
ad 2(ad /aho )2/5
β⊥ (T )∼e(4)
βk (T ) bos
ahoaho
when ad , aho > avdW for a fixed value of adB /aho where
adB is the thermal de Broglie wavelength. For fermions,using Eq. 16 of Ref. [10] to describe β⊥ (dipole dominated) and Eq. 14 of the same reference for βk (van der
Waals dominated), along with Eq. 27, we find

3
ad
β⊥
∼
(5)
βk fer
avdW
when ad , avdW < aho . These formulas suggest to plot theratio as a functionof ad /aho for bosons for a fixed ratiopadB /aho = 2π ν/kB T and as a function of ad /avdW for
fermions, as it has been done in Fig. 4 for the magnetic
dipolar molecules of Er2 and the electric polar moleculesof KRb.
Adiabatic Model

In Ref. [2] we presented the theoretical bosonic-erbium
Feshbach spectra derived from coupled-channels calculations. We concluded there that such first-principle evaluations can not quantitatively capture the complex scattering behavior of Er. In fact with the current computing
capabilities, the calculations can not be converged withrespect to the number of basis states required to explain
the experimental Feshbach-resonance density. For thisreason, we developed a novel approach based on adiabatic potentials (adiabats) Un (R; B).
Our adiabatic model starts from the Hamiltonian H =~ The first term is the radial−(~2 /2mr )d2 /dR2 + V (R).
~ describing the orientakinetic-energy operator with Rtion and the separation between the two atomic dipoles,
and mr is the reduced mass. The second term of the~ which deHamiltonian is the potential operator V (R),scribes the Zeeman and interatomic interactions. It reads

3
(b)
6

M=
i-wave-6-82g-wave

R*

-2

-9
-10

d-wave0-11

s-wave

60

80

100

R/a0

1.2

-7

4

-12

120

140

500

1000

R/a0

0.8
0.4

V/kB (mK)(a)

V/kB (mK)~ = ~2 ~`2 /(2mr R2 )+HZ +W elec (R)~ and incorporates
V (R)the rotational energy operator with molecular orbital angular momentum ~`, the Zeeman interaction of two atoms~ beHZ , and the electronic potential operator W elec (R)tween the particles. Our model assumes that the relative
vibrational motion of two Er atoms is slow compared tothe timescales of the rotational, Zeeman, and “electronic”
atom-atom interactions.
The Zeeman interaction is HZ = gµB (j1z + j2z )B.
Here, g = 1.16 is the Er g-factor, a magnetic field B
is aligned along the ẑ direction, and jiz is the z component of the angular momentum operator ~i of atom
~ dei = 1, 2. The electronic potential operator W elec (R),scribed in Refs. [2, 11, 12], is anisotropic, as it depends~ At large separation R, W elec (R)~on the orientation of R.
is given by the magnetic dipole-dipole interaction plus
both the isotropic and anisotropic contribution of thevan der Waals interaction. For R → ∞ the interaction~ → 0.
W elec (R)The Hamiltonian is evaluated in the basis |ii =|(j1 j2 )JM iY`m` (R̂), where J~ = ~1 + ~2 and Y`m` (R̂) is
a spherical harmonic. It conserves m` + M and parityp = (−1)` . In addition, for bosonic isotopes (−1)`+J = 1.
We focus on ultracold collisions between atomic states|j1 m1 i = |j2 m2 i = |6, −6i and, therefore, only include
basis functions satisfying m` + M = −12. We limit theincluded partial waves to even ` ≤ 6 and thus to stateswith even J, as the “adiabatic” magnetic moments of theresonances quickly converge with the included number ofpartial waves (In our calculation there is one s-wave channel, four d-wave channels, nine g-wave channels, and 16
i-wave channels.).
The adiabats Un (R; B) with n = 1, 2, . . . are eigen~ at a given field strength
values of the operator V (R)P
B. Their eigenfunctions are |n; Ri = i cn,i (R)|ii with
R-dependent coefficients cn,i (R). Note that we neglect
the coupling between Un (R; B) due to the radial part ofkinetic-energy operator.
Figure S1 shows the adiabats at B = 2.44 G. The scattering starts from the s-wave entrance channel correlating
to the energetically lowest adiabat. All other potentials
either have a centrifugal barrier and dissociate to two
atoms with M = −12, or dissociate to closed-channel
Zeeman sublevels with M > −12. We distinguish fourgroups of potentials, each associated with a dominantpartial wave `. Within a group, the potentials are splitby the Zeeman energy and the magnetic DDI and dissociate at different atomic thresholds. For each potential
Un (R; B) we can further assign the dominant J and M ,where J~ = ~1 + ~2 is the sum of electronic angular momenta of two atoms and M is the projection of J on
the internuclear axis. The figure also shows an example of predominantly d-wave Feshbach molecules with anouter classical turning point R∗ . Its “adiabatic” molecular magnetic moment is to good approximation given

0
1500

Figure S1. Adiabatic interaction potentials of two Er atoms
at medium- (a) and long- (b) interatomic separation R. The
calculation is performed at B = 2.44 G with m` + M = −12
and includes only states with even ` ≤ 6. The zero of energy is at the dissociation limit of two |j, mi = |6, −6i atoms.
Black, green, red, and blue curves indicate the dominant `wave character. The horizontal black line indicates a d-wave
Feshbach molecule with an outer turning point R = R∗ resonant with the s-wave entrance channel. Panel (b) also shows
the M projection for each of the Zeeman dissociation limit.

by µcalc ≈ −dUn (R∗ ; B)/dB, where we further use thatmost of the vibrational wavefunction is localized aroundR∗ . Interestingly, we observe that the µcalc value quickly
converges with the number of included ` (even ` ≤ 6 issufficient) and that it strongly depends on the DDI but
only weakly on the vdW dispersion potential. In fact, atR∗ the DDI dominates over the anisotropic part of thedispersion potential.
The adiabatic magnetic moment of a resonance isgiven by µadiab≡ −dEnv (B)/dB ≈ −dUn (R∗ ; B)/dB,
nv
where we realize that to good approximation mostof the adiabatic vibrational wavefunction is localizedaround the outer classical turning point. We further
note that dUn (R∗ ; B)/dB = hn; R∗ |dHZ /dB|n; R∗ i from=the Hellmann-Feyman
theorem and, hence, µadiab
nv
P−gµB i Mi |ci (R∗ )|2 , where Mi is the total atomic projection quantum number of state |ii. We assign a resonance by the quantum numbers of the basis state |ii
for which |ci (R∗ )|2 is largest and note that the absolutevalue of the magnetic moment of a resonance is always
smaller that 12gµB ≈ 14µB .
We further assume that the non-adiabatic coupling
between the adiabatic potentials is significantly smaller
than their spacings for R < 100a0 . Then a weakly boundlevel of adiabatic potential n can lead to a Feshbach resonance when its energy Env (B) coincides with the entrance channel energy. The outer turning point R∗ ofthis level satisfies Un (R; B) = 0. The resonance acquires
a width due to non-adiabatic coupling to the entrance
channel.
Finally, we determine the approximate quantum numbers of experimentally-observed resonances with Bres < 3
G, listed in Table I, based on a comparison of the experimental magnetic moment with those predicted by the
adiabatic model at the same resonant field. We find that4for these resonances there exist adiabats with a magnetic
moment that agrees within 1% uncertainty with the experimental values. A study of the largest coefficients
cn,i (R) at R = R∗ then enables us to assign the dominant quantum states shown in Table I.
Figure 2 (main text) shows the magnetic-field dependence of the adiabatic magnetic moment at the entrance channel energy for each of the adiabatic potentials Un (R; B). We see that for B > 10 G the magnetic moment values equal integer multiples of gµB corresponding to those of the atomic limits. For smaller fieldstrengths the adiabatic magnetic moments show mixingof the Zeeman sublevels. Here, the magnetic moment
value depends on the magnetic dipole-dipole interaction
but only weakly on the strength and anisotropy of thedispersion potential. The figure also shows our experimentally studied Feshbach resonance locations as well as
their magnetic moments µexp ; see Table I.

∗

Current address: Department of Physics, Graduate
School of Science and Engineering, Tokyo Institute ofTechnology, 2-12-1-S5-9, Ookayama, Meguro-ku, Tokyo
152-8551, Japan.
†
Alternative address: NRC “Kurchatov Institute” PNPI

188300, Division of Quantum Mechanics, St. Petersburg
State University, 198904, Russia.
[1] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler,
R. Grimm, and F. Ferlaino, Phys. Rev. Lett. 108, 210401(2012).
[2] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn,
C. Makrides, A. Petrov, and S. Kotochigova, Nature
507, 475 (2014).
[3] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler,
E. Zupanic, R. Grimm, and F. Ferlaino, Phys. Rev. A
85, 051401(R) (2012).
[4] M. Mark, F. Ferlaino, S. Knoop, J. G. Danzl, T. Kraemer,
C. Chin, H.-C. Nägerl, and R. Grimm, Phys. Rev. A 76,
042514 (2007).
[5] M. H. G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang,
G. Quéméner, S. Ospelkaus, J. L. Bohn, J. Ye, and D. S.
Jin, Nature Phys. 7, 502 (2011).
[6] G. Quéméner and J. L. Bohn, Phys. Rev. A 83, 012705(2011).
[7] G. Quéméner, M. Lepers, and O. Dulieu, Phys. Rev. A
92, 042706 (2015).
[8] M. Lepers, J.-F. Wyart, and O. Dulieu, Phys. Rev. A
89, 022505 (2014).
[9] Z. Idziaszek and P. S. Julienne, Phys. Rev. Lett. 104,
113202 (2010).
[10] G. Quéméner, J. L. Bohn, A. Petrov, and S. Kotochigova, Phys. Rev. A 84, 062703 (2011).
[11] S. Kotochigova and A. Petrov, Phys. Chem. Chem. Phys.
13, 19165 (2011).
[12] A. Petrov, E. Tiesinga, and S. Kotochigova, Phys. Rev.
Lett. 109, 103002 (2012).

Chapter5

A quantum simulator of extended
Hubbard models

A major advantage with ultracold atomic systems is the outstanding experimental flexibility
to tailor the confinement for the atoms via optical potentials. This unique control allows
to conveniently realize different trapping geometries, where the potentials are formed by
off-resonant laser light, exploiting the AC-stark effect [Gri00]. The first realization of opticaltraps for cold atom experiments featured harmonic potentials [Chu86], which are still widelyused in experiments today. In recent years also different potentials have become available
as e. g. box-like potentials [Gau13]. However, one of the cornerstones in experiments withultracold atoms is the ability to restrict the motion of the atoms along different directions
via strong confinements, allowing to access quantum effects in reduced dimensions.
A famous realization of reduced dimensions is given by the implementation of a threedimensional (3D) optical lattice. For very deep lattice potentials, this leads to a pinningof the atoms to individual lattice sites, effectively reducing the dimension to zero. Additionally, the precise and individual control of the depth of the periodic potentials along
the different geometrical axis also allows to relax the confinement along specific axis. This
enables the study of physical phenomena in tube- or pancake-traps, mimicking a one- or
two-dimensional world, respectively.
Three-dimensional optical lattices are nowadays one of the most important tools in ultracold
quantum-gas experiments, as they can access physical phenomena found in solid state systems via quantum simulations of the underlying Hamiltonians, see Ref. [Fey82] for the basic
idea. Several experimental milestones have been achieved in the past years, including theobservation of quantum many-body phase transitions [Gre02, Köh05, Jör08], the access oforbital physics [Mül07, Wir10], the observation of superexchange coupling [Tro08], the implementation of sophisticated band structures via specialized lattice geometries [Tar12], the realization of lattice spin models with short-range [Kra12] and long-range interactions [dP13b,
Yan13], and the creation of peculiar ground states, such as the antiferromagnet [Maz17b].
This list is far from being complete but strikingly shows how the remarkable experimental
progress in the engineering of ultracold atomic systems enables the access of fascinating
quantum phenomena in a controlled manner and allows to prepare peculiar quantum phases.

85

5.1. PERIODIC POTENTIALS

86

Within this thesis, we have upgraded our experimental apparatus by implementing a 3D
optical lattice, see Appendix B, transforming our system into an analog quantum simulatorof Hamiltonians with long-range correlations. The strongly magnetic character of erbium
atoms can allow to prepare exotic quantum phases resulting from the long-range and anisotropic dipole-dipole interaction (DDI) between the particles. In order to set the stage forthe results presented in this chapter, in Sec. 5.1 we will discuss the framework of an opticallattice in the single particle picture. Section 5.2 introduces the underlying extended Hubbard Hamiltonians of the engineered systems for bosonic and fermionic dipolar particles.
We will stress the influence of the DDI on the physical observables and its effect on thequantum many-body phase transitions, see Sec. 5.3. Finally, Sec. 5.4 sets the foundation forthe scientific results that are presented in Sec. 5.5 and Sec. 5.6. These sections contain our
publications on the implementation of extended Bose-Hubbard models and the realizationof strongly interacting dipolar Fermi gases, respectively.

5.1. Periodic potentials
5.1.1. Bloch waves
For a freely propagating particle in one dimension, the Schrödinger equation takes the form,
see e. g. Ref. [Hum85],p̂2
HΦ(x) =
Φ(x) = EΦ(x)5.12m
where the solution is found to consist of plane waves Φ(x) = eikx , and the particle is welldescribed by the energy-momentum relation E = ~2 k 2 /2m, where k is the wavevector of theparticle and m its mass. Here, the energy consists of a continuum of states. The situation
changes drastically when the particle is confined in a periodic potential, which leads to adiscretization of energy states, as will be shown below.
In nature, periodic potentials are typically found in solid state systems where nuclei arearranged in a crystalline structure. The coulomb attraction of the nuclei forms a periodical
potential V (x) that restricts the motion of the electrons along x, see e. g. Ref. [Ash76]. Inp̂2
this paradigmatic case, the solutions of the Schrödinger equation with H = 2m
+ V (x) are
given by the Bloch wave functionsiqx (n)Φ(n)uq (x)
q (x) = e

where

(n)
u(n)
q (x) = uq (x + d)

5.2

is a completely delocalized periodic function reflecting the periodicity d of the lattice, withq denoting the quasimomentum and n being the Bloch-band index.
In ultracold-atom experiments the role of the nuclei in solid state systems is taken by an
optical lattice, which in this case forms a periodic potential for the individual atoms. The
optical lattice can be formed by a retro-reflected gaussian laser beam, propagating along thex axis and being radially symmetric in r. The potential is described as


r2
2
2
V (r, x) = −V0 e−2r /w0 cos2 (kx) ≈ −V0 1 − 2 2 cos2 (kx),
5.3
w0

5.1. PERIODIC POTENTIALS

87

with V0 representing the lattice depth and k = 2π/λ denoting the lattice wave vector. The
finite focal waist of the lattice beam w0 results in an radial harmonic confinement of theatoms, which for the purpose of this introduction will be neglected, i. e. r is set to be zero.
In order to avoid inelastic atom-light scattering, the wavelength λ should be chosen such that
it is far detuned from atomic transitions. The depth of the optical lattice is conveniently
given in recoil energy ER as V0 = sER with ER = ~2 k 2 /2m linked to the absorption oremission of a lattice photon for an atom with mass m.
(n)Following from equation 5.2, the periodicity of the lattice allows to expand uq (x) and V (x)as discrete Fourier sums, leading to the expressions
X (n,q)
i2lkx
2
2ikx
−2ikx
u(n)
(x)
=
c
e
and
V
(x)
=
−V
cos
(kx)
=
−(V
/4)
e
+
e
+
2
. 5.4
0
0
q
l
l

The kinetic energy term reads asX (l × 2~k + q)2 X 
p̂2
q 2
=
=
2l +
ER .
2m
2m
~k
l

5.5

l

By inserting 5.4 and 5.5 into the stationary Schrödinger equation one can obtain its matrix
form expression: (2l + q/~k)2 ER − V0 /2 if l = l′X(n,q)(n) (n,q)−V0 /4
if |l − l′ | = 1 5.6
Hl,l′ cl= Eq clwith Hl,l′ =

l
0
else.

Periodicity allows to confine q to the first Brillouin zone [−~k, ~k] without the loss of generality. We can now solve the eigenvalue problem by numerically diagonalizing the Hamiltonian.
(n)
The extracted eigenvalues Eq give the energy at the quasimoment q for the n-th Blochband. The energy spread within q for a given lattice band is associated to the width of theband.

5.1.2. Band structure
In Fig. 5.1(a) we plot the energies of the five lowest bands as a function of the lattice depth.
The evolution of the different bands captures three different regimes. For zero lattice depth
the particle is well described by the free-particle dispersion relation, where no gaps between
the different bands are found, correspoding to a continuum of states. With increasing lattice
depth, band gaps open up, leading to a quantization of energy states. Finally, for an infinitely
deep lattice the width of the bands would reduce to zero and their energies would correspond√to the energies of an harmonic oscillator E (n) = ~ωlat (1/2 + n) with ωlat = 2 sER .
In contrast to solid state systems, ultracold atoms allow to conveniently explore the different
regimes, as the depth of the lattice potential can be tuned via V0 ∝ I α. This magic knob
is provided by the direct access to the intensity I = 2P/(πw02 ) of the lattice beam, where
P denotes the power of the laser light. In order to precisely know the resulting lattice
depth, it is crucial to determine the dynamical polarizability α of the specific element at

5.1. PERIODIC POTENTIALS
80

88
70

(a)

70

50

50

∆ E (n) (E R)

E (n) (E R)

60

40
30

40
30
20

20

10

10
0

(b)

60

0

20

40
60
lattice depth (ER)

80

100

0

0

20

40
60
lattice depth (ER)

80

100

Figure 5.1.: Energy of bands in a 1D optical lattice. (a) The energy of the five lowest Bloch bands(depicted by green, dark blue, orange, red, light blue from lowest to higher bands) evolve and theenergy spreads become narrower as the lattice depth is increased. In addition bandgaps emerge. Theenergy of the gaps between the ground band and the higher bands is shown in (b), respectively. Samecolor notation as in (a). The energies and the lattice depths are given in recoil energy ER .

the required laser wavelength. In the case of erbium, the dynamical polarizability prior
to our work was unknown. With our experiment, we have measured α for experimentally
relevant wavelengths as reported in Appendix A.2. For a calculation of the polarizability,
the contributions of all dipole-allowed transitions in erbium have to be taken into accountvia a sum-over-state formula [Lep14].
A majority of ultracold atom experiments focus on physics of the ground band in deep opticallattices, where effects arising from coupling to higher lattice bands are negligible [Lew12].
Still, the knowledge on the energy gap to higher bands is important as it can be used e. g. tocalibrate the lattice depth by measuring the excitation energy from the ground to higher
bands. Figure 5.1(b) depicts the band gap ∆E (n) between the ground and the excited bands
for a one-dimensional (1D) optical lattice.
Beside the physics in the ground band, an interesting direction is the investigation of orbital physics, accessed by transferring atoms to higher bands. Pioneering work have shownthat coherent states in higher lattice orbits indeed can be prepared, in the orbit of one
axis in an anisotropic 3D lattice [Mül07] and in the coupled two-dimensional orbit of a 2D
lattice [Wir10]. The access of higher orbits enables the studies of complex quantum manybody phases as e. g. p-wave superfluidity [Li16]. It should be noted that in the case of amore-dimensional lattice the wavevector q consists of components along all the orthogonallattice axes, leading both, to degeneracies of higher bands for certain quasimomenta q, andto different band gaps as compared to the 1D situation. As the investigation of higher orbits
is not the scope of the present thesis, the reader is referred to e. g. Ref. [Mül06] for moredetails.

5.1. PERIODIC POTENTIALS

89

5.1.3. Wannier functions
For experiments in deep optical lattices, it is convenient to switch from the description withdelocalized Bloch functions to a set of states that are localized at single lattice sites xi . Thisset of wavefunctions is given by the so-called Wannier functions that form a single-particle
basis for each band n and that are orthogonal to each other with respect to different latticesites and different bands. They are formed by a linear combination of all Bloch waves1 of agiven band as
1 X −iqxi (n)wn (x − xi ) = √eΦq (x)
5.7
M q
where q is again confined to the first Brillouin zone and M is a normalization factor [Wan37].
Within this thesis, we work with Wannier states in the lowest lattice band and hence theband index n is omitted from now on. Working with well localized Wannier functions gives
the advantage of easily placing an atomic wavefunction on an arbitrary lattice site. Indeed,
the Wannier function at lattice site j is constructed by a simple translation of the Wannierfunction at lattice site i, as wj (x) = w(x − xj ) = w(x − (xi + (j − i)d). For deep enough
optical lattices, i. e. in the tight binding regime, the ground state of a lattice site can also be
described by the harmonic oscillator ground state
who (x) =11/2
π 1/4 lho

2

e

− x2

2l
ho

with

lho =

d
πs1/4

5.8

being the single-site harmonic oscillator length. To visualize the validities of these two possible descriptions, we compare their functional form in Fig. 5.2 for various lattice depths(Fig. 5.2(a)). For an atom occupying the lattice site i = 0, the Wannier function wi (x)(Fig. 5.2(b)) and the harmonic oscillator ground state who (x) (Fig. 5.2(c)) as well as theirdifference w0 (x) − who (x) (Fig. 5.2(d)) are depicted. For small lattice depths the Wannierfunction extends over more lattice sites featuring side lobes, whereas it becomes more andmore localized for deeper lattices. While the harmonic oscillator state gives very similar
results for deep lattices, it particularly underestimates the occupation at neighboring lattice sites for shallow lattices, as nicely captured by the residual of the two descriptions(Fig. 5.2(d)). This can have a strong influence on the kinetic and potential energy termsof the system’s underlying Hamiltonian, i. e. the tunneling of a particle to a neighboring
lattice site and the interaction energy of neighboring particles, as this terms are directly
related to the wavefunction overlap between neighboring sites as will be discussed below.
The description with harmonic oscillator states has therefore to be used with care.

1(n)

The complex phases of Φq (x) have to be chosen such that the Wannier function of each band are real,
symmetric or antisymmetric around x = 0 or x = d/2 and fall of exponentially in x [Koh59].

(a)

(b)

V lat (a.u.)

0

0

x/d

-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2

V 0= 3 E R

-0.1

0

0.1

0

0.5

1

0

0.5

1

0

0.5

1

x/d

-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2

V 0= 5 E R

-0.1

0

0.1

0

0.5

1

0

0.5

1

0

0.5

1

x/d

-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2

V 0 = 10 E R

-0.1

0

0.1

0

0.5

1

0

0.5

1

0

0.5

1

x/d

-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2

V 0 = 20 E R

-0.1

0

0.1

0

0.5

1

0

0.5

1

0

0.5

1

x/d

-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2

V 0 = 50 E R

-0.1

0

0.1

0

0.5

1

0

0.5

1

0

0.5

1

x/d

-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2-2 -1 0 1 2

V 0 = 100 E R

Figure 5.2.: Wannier functions (b) and the harmonic oscillator ground state (c) for an atom at site i = 0 for different lattice dephts V0 (a) as afunction of the position x in the lattice in units of the distance d between neighboring lattice sites. The individual data of (a-c) is renormalized
by their respective maximum values for the case V0 = 100 ER . The residuum w − who (d) reveals the difference between the two functions used todescribe the population of a single lattice site.

-0.1

-0.1

x/d

0

0

-2 -1 0 1 2

0

0.1

0

0.5

(d) 0.1

0.5

1-2 -1 0 1 2

0.5

0.5

1

1

1

-2 -1 0 1 2

0

0

1

0.5

-2 -1 0 1 2

V 0= 1 E R

0.5

1

(c)

w (a.u.)who (a.u.)

w - w ho (a.u.)90

5.2. EXTENDED HUBBARD MODELS

91

5.2. Extended Hubbard models
5.2.1. The Hubbard model
The single-particle description in a periodic potential by itself leads to interesting phenomena
such as coherent Bloch oscillation in momentum space within the first Brillouin zone when theparticle is accelerated by an external force [Blo29, Dah96, Gus08, Fat08], or the tunnelingto higher bands at the edge of the Brillouin zone for small enough band gaps, known as
Landau-Zener tunneling [Lan32, Zen34, Tay10].
To capture real physical systems, in addition to the single particle description it is crucial
to take into account the correlations between the particles on the microscopic level, e. g. theelectron-electron interaction in solid state systems. In particular, the many-body dynamics
at low temperature depend critically on the interparticle interactions on single lattice sites
and are well described by the famous Hubbard model. The Hubbard Hamiltonian captures
the physical observables of a correlated many-body system in a lattice, i. e. the tunnelingof particles between neighboring sites, known as the term J, as well the onsite energy for
multiple particles occupying the same lattice site, denoted by the term U . Indeed this model,
being simple in its representation but still comprehensive with its description, is one of theprime examples of a many-body model in physics, and has since its developement [Hub63]
received significant attention. Originally, it was intended to explain the correlated behavior of fermionic particles, i. e. electrons, in a solid state system interacting via Coulomb
repulsion, resulting in more stringent conditions for the emergence of ferromagnetism. The
proposal to experimentally access the Hubbard model via ultracold atomic systems [Jak98]
led to a major popularization, resulting in the observation of the quantum phase transition from a superfluid state to a Mott insulator (SF-MI) within a bosonic system [Gre02],
as well as to the realization of the Hubbard model in the original context with fermionic
particles [Jör08]. These milestones have been followed by a vast amount of theoretical andexperimental studies, see Ref. [Lew12] for an overview.
More recently the Hubbard model has become accessible across different experimental platforms. Here, Fermi-Hubbard physics has been demonstrated with a 2D electron gas in an
artificial honeycomb lattice formed by a gallium arsenide heterostructure [Sin11], is modeledby digital quantum simulation via the use of a X-mon transmon qubit array in superconducting circuits [Bar15], and is realized for two sites by single-site resolution of subsurfaceboron dopants in silicon [Sal16]. Increasing experimental control nowadays allows to emulate
Fermi-Hubbard models in solid state systems without suffering from electrostatic disorder, as
beautifully demonstrated by single-electron control with gate-defined quantum dots [Hen17].
Despite that for superconducting circuits photonic excitations are used as ”quantum particles” [Hou12], theoretical works show that simulation of Bose-Hubbard physics including
the SF-MI phase transition is at reach, e. g. by coupling a charge qubit to a superconducting
transmission line resonator to form a lattice site [Den15]. An attractive 1D Bose-Hubbard
model already has been realized with a three-qubit transmon array [HG15]. This ongoing experimental and theoretical explorations of the Hubbard model throughout different quantum
system platforms underline its unbroken currentness.

5.2. EXTENDED HUBBARD MODELS

92

Quantum statistics
As discussed, the Hubbard model can be accessed with both classes of quantum particles,
namely bosons and fermions. The differences for this two classes arises from quantum statistics. While for identical bosons multiple particles can occupy the same lattice site, in thecase of identical fermions a multi-occupancy is forbidden by the Pauli exclusion principle.
As a result, the Fermi-Hubbard model of identical fermions includes only the tunneling termJ, while for a Bose-Hubbard system additionally the onsite interaction term U has to beconsidered. To realize the onsite term U also with a fermionic system, an additional spinstate has to be considered.
In the following, we will discuss Bose-Hubbard and Fermi-Hubbard models for the case ofdipolar particles, which open the opportunity to access new Hamiltonian terms that arise
from the long-range character of the DDI.

5.2.2. Extended Bose-Hubbard model
While the Bose-Hubbard model already gives a solid framework for the exploration of latticemodels, an enriched system can be achieved when correlations are extended beyond single
lattice sites as accessible with long-range interactions. Here, a possible route involves dipolar
particles, where the DDI gives significant nearest-neighbor interaction leading to exotic quantum phases [Gór02, Dut15]. The many-body dynamics of this system can be captured by theso-called extended Hubbard model, which is at the heart of the presented results within thischapter. Here, we will review the theoretical description of this model for the case of bosonic
dipolar atoms resulting in the extended Bose-Hubbard (eBH) model. A similar treatement
leads to the extended Fermi-Hubbard model, see Sec. 5.2.3.
Let us consider a dipolar Bose-Einstein condensate within a 3D periodic potential V (r). The
many-body Hamiltonian in second quantization reads as
 2 2

Z
Z
Z
~ ∇
1
†
Ĥ = drΨ (r) −+ V (r) Ψ̂(r) +dr dr′ Ψ̂† (r)Ψ̂† (r′ )U (r − r′ )Ψ̂(r′ )Ψ̂(r), 5.92m2
where Ψ̂† (r) and Ψ̂(r) are the bosonic creation and annihilation field operators, respectively.
The first integral of Eq. 5.9 describes the energy of a single particle in the periodic potential
V (r) and the second part considers the interaction U (r − r′ ) between two particles. Theyinteract with each other via the short-range contact interaction (∝ as ), see Eq. 4.1, and thelong-range DDI (∝ µ2 ), see Eq. 3.2,
U (r − r′ ) =µ0 µ2 1 − 3 cos2 θr−r′
4π~2 asδ(r − r′ ) +.
m
4π|r − r′ |3

5.10

PThe field operators can be expanded in the basis of Wannier functions as Ψ̂(r) = i b̂i wi (r)where bi is the bosonic annihilation operator for a particle on lattice site i. Here, the Wannierfunction is a three dimensional object and is given by the product of the individual orthogonal

5.2. EXTENDED HUBBARD MODELS
10 4
10 3

Us

(a)

(b1)

J

(b2)

U

(b3)

V

(b4)

∆J

Udd
Vdd

10 2

|energy| (Hz)

93

10 1
10 0

J
∆Js

10 -1

∆Jdd

10 -2

Vs

10 -3
10 -4
10 -5

Jpair
5

10

15
20
lattice depth sxyz

25

30

Figure 5.3.: Non-standard extended Bose-Hubbard terms. (a) Absolut energies of the individual
matrix elements for the case of erbium in a 3D lattice for our experimental lattice setup, see Appendix
B, as a function of the 3D lattice depth sxyz = sx = sy = sz . The terms are calculated along thehorizontal plane (xy) with a lattice spacing of d = 266 nm and a dipole orientation along the verticalz axis. The s-wave scattering length is set to as = 100 a0 . Beside the tunneling amplitude J, allterms have contribution from the contact interaction (solid lines) and the dipolar interactions (dashedlines), see text. (b1-b4) Illustrations of the relevant terms of the non-standard eBH model.

Wannier states w(r) = w(x)w(y)w(z). The two-body interaction of Eq. 5.9 then takes theform
1X
Uijkl b†i b†j bk bl
5.11
Ĥint =
2
ijkl

where the interaction strengthZ
Z
Uijkl = dr dr′ wi∗ (r)wj∗ (r′ )U (r − r′ )wk (r)wl (r′ )5.12

results from the wavefunction overlap of lattice sites i, j, k and l. Equation 5.12 contains thematrix elements of all possible processes involving two particles in a lattice. Taking only
onsite and nearest-neighbor scattering events into account, finally leads to the non-standard
extended Bose-Hubbard Hamiltonian in single-band approximation [Dut15]
H = Hsingle + Honsite + Hoffsite , withX †X
Hsingle = − J(bi bj + h.c.) +Vh,i ni ,
i

⟨ij⟩

UX
Honsite =ni (ni − 1), and
2
i
i
X
Xh †
X †2
Hoffsite = V
ni nj − ∆Jbi bj (ni + nj − 1) + h.c. + Jpairbi b2j .
⟨ij⟩

⟨ij⟩

5.13

⟨ij⟩

Here, ⟨ij⟩ denotes a pair of nearest-neighboring sites and ni = b†i bi is the onsite occupation
number. The Hamiltonian consists of three contributions: the single particle part Hsingle , theonsite part Honsite , and the offsite part Hoffsite that results from correlations between particles

5.2. EXTENDED HUBBARD MODELS

94

in neighboring lattice sites. The last part of the Hamiltonian represents new terms comparedto the standard Bose-Hubbard model. The two Hamiltonian parts Honsite and Hoffsite arise
from the interaction U (r − r′ ), where contact and dipolar interactions are at play. To accessthe terms of the Hamiltonian in experiments it is crucial to know their respective strengthfor experimental relevant parameters. In Fig. 5.3 we plot the individual contribution of theeBH terms as a function of equal lattice depths sxyz in the three orthogonal directions forour experimental conditions, see Appendix B. It should be stressed that, in contrast to solidstate systems, the individual terms and especially the relative strengths of the eBH terms canbe tuned conveniently via the control of the lattice depth, which allows to access different
regimes and quantum phases. The individual terms of the Hamiltonian will be discussed
below.

Single particle terms
Single particles can hop between neighboring lattice sites with a tunneling rate J/h(Fig. 5.3(b1)), which reads as

J =−

Z

drwi∗ (r)[−~2 ∇2+ V (r)]wj (r).
2m

5.14

Its strength depends on the overlap between the involved wavefunctions of neighboring latticesites i and j. As has been shown in Fig. 5.2, for deeper lattices the Wannier functions become
more and more localized, resulting in a diminishing overlap. Hence, while for shallow lattices
the tunneling J dominates the system dynamics it decreases for increasing lattice depths(dark blue line, Fig. 5.3(a)) and eventually vanishes for infinitely deep lattices.
Additional single particle contributions can arise from potential gradients that lead to anenergy offsets between different lattice sites [Sac02]. In particular, an important role can beplayed by the radial harmonic confinement of the lattice beam, see Eq. 5.3, which leads to a
position dependent potential energy Vh,i . Particles away from the center of the red-detuned
lattice beams experience a higher potential energy as compared to the central particles. Thiscan influence the hopping dynamics and can lead to different spatial regions of quantum
phases across the lattice for sufficient lattice dephts [Cam06, Föl06].

Onsite contributions
When a number of interacting atoms occupy the same lattice site, the particle correlations
lead to an energy change of the occupied state, i. e. the energy is increased (decreased)
for repulsive (attractive) interactions. In standard Bose-Hubbard models this onsite energy
U is solely given by the contact interaction between the particles, which is related to thescattering length as . In the case of dipolar particles the onsite energy reads as U = Us +Udd =Uiiii and captures both the contact and the dipolar interactions (Fig. 5.3(b2)). As depicted
in Fig. 5.3(a), the contact interaction (solid red line) dominates the onsite energy for theexperimental relevant scenario and in particular for deep lattices exceeds the energy of all5.2. EXTENDED HUBBARD MODELS

95

other eBH terms. This is a direct consequence of the localization of the 3D Wannier functions,effectively resulting in a compression of the particles. The dipolar contribution (dashed redline) scales similarly as the contact contribution but additionally depends on the shape of the3D Wannier function and the orientation of the atomic dipoles with respect to the symmetry
axis of the confinement potential [Wal13], i. e. Udd (w, θ).
As a result of the onsite energy, a gap opens up between e. g. the energies of a singlyoccupied and a doubly-occupied lattice site [Gre02]. The energy of this gap is also known
as the particle-hole excitation energy and in experiments can be conveniently accessed via
modulation spectroscopy, see Sec. 5.4.1.

Offsite contributions
The particle correlations can also give rise to offsite terms in the Hamiltonian. The probably
most exotic term is the nearest-neighbor interaction (NNI) term V = Vs + Vdd = Uijij(Fig. 5.3(b3)). We emphasize that the contribution of the DDI (Vdd ) to the NNI in a deep
lattice exceeds the contribution of the contact interaction (Vs ) by several orders of magnitude(dashed and solid green line in Fig. 5.3(a)), underlining the prominent role of the long-range
nature of the DDI. Especially for deep lattices Vs , can safely be neglected. Offsite interactions
are from particular interest as they can lead to special ordering of the particles within the
optical lattice, associated to exotic quantum phases. A promising theoretical study of such
quantum phases is performed in Ref. [CS10]. A system of hardcore dipolar bosons in a 2D
lattice studied via the path integral Monte Carlo method reveals the existence of various
Mott lobes depending on the filling fraction. In particular for half filling, the out-of-planeoriented dipoles arrange in a checkerboard order. Remarkably, the investigations, where no
cut-off on the dipolar interaction potential is used, identify parameter regions where theordered crystalline phase remains superfluid, resembling a supersolid phase. This phase isshown to remain stabilized even at finite temperature [CS10]. The experimental realizationof a supersolid phase, proposed for Helium already in the 1960s, is still of prime interest
within the quantum physics’ community2 .
It has been observed that offsite interactions can also arise from the so-called superexchangeinteraction, assosciated to two virtual tunneling processes of neighboring particles [Aue94].
The coupling strength J 2 /U of this second-order process can be large enough to study lattice spin models with two-component mixtures. While first observed locally for a pair ofbosonic atoms in a double well [Tro08], super-exchange coupling allowed to realize shortrange magnetic correlations in the fermionic Hubbard model in lower and later also in three
dimensions [Gre13, Har15], and led to the observation of the propagations of single spinexcitations [Fuk13a] and magnon bound states [Fuk13b] in a 1D-Bose-Hubbard spin chain.
Remarkably, when temperatures well below the superexchange scale are prepared, the correlations even can give rise to a long-range antiferromagnetic ordering, as recently observedin a 2D Fermi-Hubbard system [Maz17b]. However, superexchange interactions vanish for2

It should be noted that very recently experimentalists from ETH and MIT have managed to realize asupersolid phase by coupling a BEC to the modes of two optical cavities [Léo17] and by introducing a
spin-orbit coupling of BECs in an array of double-well potentials [Li17], respectively.

5.2. EXTENDED HUBBARD MODELS

96

very deep lattices. In stark contrast, nearest-neighbor correlations arising from the DDI are
only weakly affected by the lattice depth, see Fig. 5.3(a), and also enable direct coupling ofparticles beyond nearest-neighbors.
An additional term in the Hamiltonian of Eq. 5.13 resulting from offsite correlations is givenby the occupation dependent hopping ∆J, known as density-induced tunneling (DIT) [Lüh12,
Bis12] (Fig. 5.3(b4)). For the case of dipolar atoms, it has contributions both from the contact and the DDI and reads as ∆J = ∆Js +∆Jdd = −Uiiij . As can be seen in Fig. 5.3(a) (light
blue lines) the values of the DITs reach values of up to 10% of J and hence can significantly
influence the SF-to-MI quantum phase transition, see Ref. [Lüh12] and Sec. 5.3.1. While direct observations of DIT in a contact interacting tilted 1D Mott insulator nicely showed how
strong (isotropic) interaction can alter the tunneling dynamics [Jür14], the investigation of
DIT caused by DDI has been first performed with our system, see Sec. 5.5.
A final term involves the correlated pair tunneling Jpair = Jpair,s + Jpair,dd = Uiijj /2 of two
atoms originally occupying the same lattice site. As this term features very small energies(grey lines in Fig. 5.3(a)) it will not be further discussed.

5.2.3. Extended spinor Fermi-Hubbard model
Fermi-Hubbard models
Fermi-Hubbard models are very relevant as they can be directly related to the descriptionof electrons in solid state systems. Fermionic particles show completely distinct quantum
statistics with respect to their bosonic opponents as Pauli exclusion principle prohibits two
identical fermions to occupy the same quantum state. This has striking consequences forthe case of spin-polarized particles, as double occupancies (in the single-band picture) aresuppressed and the lattice Hamiltonian includes solely the tunneling term J.
The description changes once a fermionic spin-1/2 system is considered. Due to the distinguishability of the particles double occupancies become allowed, adding onsite interaction U
to the Hamiltonian. Additionally, nearest-neighbor exchange interactions can arise due tosuperexchange correlations. For the case of U ≫ J the Hamiltonian resembles the celebratedt − J model3 [Aue94]. Here, for the case of half-filling, i. e. one atom per lattice site andan in total equal spin number, the system is in a Mott insulator state and an antiferromagnetic ground state is expected [Man91]. A particular relevant scenario emerges when thesystem undergoes doping, i. e. when the filling fraction is lowered. Here, strong relations tohigh-temperature (high-Tc ) superconductors such as cuprates [And87] can be drawn. The
microscopic description of high-Tc superconductors still puzzles the community and it isbelieved that with the use of ultracold spin-1/2 fermions as a model system for electronic
spins in solids important insights can be be gained, see Ref. [Lee06] for a review. Indeed, thehigh control on parameters such as temperature, interaction, and doping, nowadays reached
3This model is well discussed in literature. Here, the notation typically differs from ours. t denotes forthe tunneling rate and J = 4t2 /U for a spin conserving nearest-neighbor correlation due to superexchange
coupling.

5.2. EXTENDED HUBBARD MODELS

97

the single-atom level via quantum gas microscopes techniques, see e. g. Ref. [Maz17b]. This
gives exciting prospects for future explorations with this already comprehensive quantum
simulators of regimes where theoretical simulations fail.
While the rather simple Hamiltonian of the t − J model already provides surprisingly rich
physics along the phase diagram, ultracold atoms give possibilities to further enrich the systems description. Adding long-range interactions yields Hubbard systems where the interparticle interaction ranges beyond nearest neighbors. Further, by upgrading the spin degreeof freedom, high spin Hubbard models can be accessed. Both upgrades can be implemented
via the use of fermionic erbium. First, erbium provides long-range dipolar interactions andsecond, the fermionic isotope features a large number of 20 spin states in the atomic groundstate, see Sec. 2.2.4. This gives promising prospects for the implementation of exotic latticemodels.

Fermi-Hubbard model with dipolar interaction
For single-spin dipolar particles, the extended Fermi Hubbard model includes in addition to
the tunneling term J the long-range NNI term V while the onsite term U remains absent.
To enrich the scenario, additional spin states can be added, resulting in the extended spinor
Fermi Hubbard (eFH) Hamiltonian. This Hamiltonian is derived via a very similar treatment
as for the bosonic case, see Sec. 5.2.2, and reads as [Aue94, Dut15]
 XX
X
H = −Jb†σi bσj + h.c. +Vh,i nσi + U
nσi nσi′⟨ij⟩,σ

+

X

i<j,σ



i,σ

i,σ̸=σ ′
1Vσi ,σj Fσzi Fσzj − (Fσ+i Fσ−j + Fσ−i Fσ+j ) .
4

5.15

Similarly to the bosonic case, ⟨ij⟩ denote pairs of adjacent sites, b†σi (bσi ) are the fermioniccreation (annihilation) operators of an atom at site i, and nσi = b†σi bσi is the associatednumber operator. The spin degree of freedom is encoded in σ, which for the case of fermionicerbium can be replaced by the state |F, mF ⟩. For the sake of simplicity the terms accounting
for density induced tunneling and pair hopping, see Sec. 5.2.2, are not included, while thepotential energy gradient term Vh,i is given for completeness. Figure 5.4 gives all relevantterms of the eFH model.
The single-particle tunneling J (Fig. 5.4(b1)) and the onsite energy Us that arises frominterspin contact interactions between two distinguishable fermions, show the same strengths
as for the bosonic case (compare blue and red solid lines in Fig. 5.4(a) and Fig. 5.3(a))4 . The
dipolar contribution to the onsite energy (red dashed line in Fig. 5.4(a)) gives similar results
as for the bosonic case. However, it should be noted that this energy crucially depends onthe composition of spin states.
In the case of a spinor dipolar Fermi gases in a 3D lattice the most prominent offsite terms
are given by the NNI Fσzi Fσzj (Fig. 5.4(b3)) and the flip-flop term Fσ+i Fσ−j (Fig. 5.4(b4)). The
4

We note that the tunneling term J is equal for all spin states.

5.2. EXTENDED HUBBARD MODELS
10 4

(a)

98

Us

(b1)

J

(b2)

U

(b3)

FzFz

(b4)

F+F -

10 3

|energy| (Hz)

Udd
10

2

FzFz
10 1

J

10 0
10 -1

F+F -

5

10

15
20
lattice depth s xyz

2530

Figure 5.4.: Extended Fermi-Hubbard terms. (a) Absolut energies of the individual matrix elements
for the case of the two lowest spin states of fermionic erbium in a 3D lattice for our experimental lattice
setup, see Appendix B, as a function of the 3D lattice depth sxyz = sx = sy = sz . The terms arecalculated along the horizontal plane (xy) with a lattice spacing of d = 266 nm and a dipole orientation
along the vertical z axis. The interspin s-wave scattering length is set to as = 100 a0 . Beside thetunneling amplitude J all terms have contributions from the interspin contact interaction (solid lines)
and the dipolar interactions (dashed lines), see text. (b1-b4) Illustrations of the relevant terms of
the eFH model for the case of two spin states | ↓⟩ and | ↑⟩ (blue and orange dipole, respectively).
Nearst-neighbor terms arising from the F z F z and the F + F − coupling are included.

spin operators obey the rules

andF ± |F, mF ⟩ =F z |F, mF ⟩ = mF |F, mF ⟩

5.16

p
F (F + 1) − mF (mF ± 1)|F, mF ± 1⟩.

5.17

Vσi ,σj , see Eq. 5.15, denotes the dipolar coupling strength and can be derived viaZZµ0 (gF µB )2 1 − 3 cos2 θr−r′′ ∗∗ ′wi (r)wj (r′ ).
Vσi ,σj = dr dr wi (r)wj (r )4π|r − r′ |3

5.18

While the NNI term features larger energies as the flip-flop term (dark and light green dashedlines in Fig. 5.4(a)), the flip-flop term can lead to resonant spin dynamics, see Sec. 6.2.
Additional (single particle) terms that have to be considered in experiments result fromspin-dependent potential energies. This energies arise from the Zeeman energyX
HZeeman =qli (B)σ + qqu (B 2 )(σ 2 − F 2 ) nσi ,
5.19
i,σ

see Sec. 2.2.4, and from spin-dependent light shifts
X
Hlight =qL (I)σi2 nσi .

5.20

i,σ

The light shift, being a consequence of the tensorial dynamical polarizability, see Appendix
A.2, is proportional to the intensity I of the confining laser traps. It should be noted that thepresented
can also feature terms that do not conserve the total magnetization
P Hamiltonianzz
Ftot = i Si as discussed in Sec. 6.2.

5.3. QUANTUM PHASE TRANSITIONS

99

5.3. Quantum phase transitions
The extended Hubbard models discussed in Sec. 5.2.2 and Sec. 5.2.3 do not only determine therelevant energy scales of the underlying system, but also give access to the dynamics of thestrongly correlated many-body wavefunction. In particular, quantum phase transitions canbe driven by changing the strength and the ratio of the different Hamiltonian parameters.
Within this section, we will briefly review the different phases expected for bosonic andfermionic systems in 3D lattices, and will set them in context with our experiment. In
the case of bosons, a superfluid-to-Mott insulator quantum phase transition can be driven,
where for a two-component Fermi gas the initial metal phase can undergo phase transitions
to either a band or a Mott insulating state.

5.3.1. Superfluid-to-Mott insulator transition
A seminal example where the many-body dynamics of a system are changed in a dramatic way
is the superfluid-to-Mott insulator (SF-MI) quantum phase transition. The superfluid phaseis characterized by a large (integer) number fluctuation on a single lattice site, resulting froma delocalized phase-coherent matter wave across the lattice. In contrast, in the insulating
regime, the ground state consists of exponentially localized bosonic wavefunctions leading
to a loss of global phase coherence. In the MI phase the excitations spectrum, associatedto the onsite energy U , is gapped and an incompressible state is formed. The quantum
phase transition can be driven by tuning the strength of the interaction term U with respect
to the tunneling term J. From a mean-field treatment for an average site occupation of⟨n⟩ = 1, the phase transition is predicted to occur at the critical value (U/J)c = z × 5.8
with z associated to the system’s dimensionality via the number of nearest neighbors. While
this value is found to be in reasonable agreement with Quantum Monte carlo simulations
for the 3D case, (U/J)c ≈ 29.3 [CS07], it differs significantly from Quantum Monte carlo
simulations for the 2D case, (U/J)c ≈ 16.7 [CS08], and from DMRG simulations in the 1D
case, (U/J)c ≈ 3.4 [Küh00].
In ultracold experiments, the respective Hamiltonian parameters can be conveniently tuned
via the lattice depth s. This technique led to the first experimental demonstration of the
SF-MI quantum phase transition in 2002 [Gre02]. Nowadays, the phase transition can evenbe studied on the single-atom level via quantum gas microscope techniques [She10, Bak10].
This methods allow to directly visualize the typical Mott structure, consisting of concentricshells with commensurate fillings of density ⟨n⟩ = 1, 2, 3 . . . (from outside to the lattice
center), resulting from a weak transversal confinement of the optical lattice beams.
As the SF-MI transition is driven by the competition of U and J, for the case of dipolar
particles the contributions of the DDI to these terms has to be taken into account. Precisely,
one has to consider the ratio between the total onsite energy Utotal = Us + Udd and the total
tunneling rate Jtotal = J + ∆Js + ∆Jdd along the three directions. In Sec. 5.5 we study thephase transition in such a dipolar system, proving a unique tunability of the critical value via
the dipole orientation. Further, for dipolar interactions, the phase diagram becomes enrichedby new and exotic quantum phases such as the checkerboard or the stripe phase [CS10]. The
preparation of such phases is at reach with our system.

5.3. QUANTUM PHASE TRANSITIONS

100

5.3.2. Metal to band and Mott insulator transition
A two-state fermionic mixture in an optical lattice shows a rich phase diagram. The relevantenergy scales that have to be compared are the onsite energy U , the width of the lowest Blochband W that is related to the tunneling rate5 , and the chemical potential µ6 . For the caseof weak interactions and a shallow trap (U ≪ µ ≪ W ), the system is in a metallic state andthe atomic wave functions are delocalized. The system is compressible and the central filling
per spin state ⟨nσ ⟩ remains below 1, resulting in the occurrence of holes but also doubly
occupied sites (doublons). When the interaction energy U is dominating (U ≫ µ ≫ W )doublons become suppressed and the system enters into the Mott insulating phase with⟨nσ ⟩ = 1/2. This phase shows a gapped excitation spectrum and can be described by two
Hubbard bands, where the upper band relates to the case of double occupancies. When thechemical potential exceeds the onsite energy (µ ≫ U, W ) filling of this upper band becomes
favorable and a band insulator forms with ⟨nσ ⟩ = 1 at zero temperature. Here, every site is
occupied by each spin state and a further compression of the sample is not possible due tothe Pauli exclusion principle. It should be noted that in the case of a single spin system aMott insulating phase is absent, and the system enters into a band insulator, i. e. ⟨nσ ⟩ = 1,
when µ exceeds W .
Pioneering experiments succeeded in observing band- [Köh05] and Mott-insulatingphases [Jör08, Sch08] by studying global observables. With the advent of fermionic quantum gas microscopes, the investigation of the phase transitions with single-site resolution
has become available. As a result of the variation of the (local) density across the lattice due to the harmonic confinement, the coexistance of all quantum phases within asingle sample is possible, which was nicely visualized with quantum gas microscope techniques [Omr15, Gre16, Che16].
All this experiments set the stage to further explore the fermionic phase diagram. Indeed,
for low enough temperature, the Mott insulator state can undergo a phase transition to an
antiferromagnetically ordered state owing to spin-spin interactions via superexchange coupling. While already in 2013 a promising experiment demonstrated short-range magnetic
ordering [Gre13], it remained an outstanding challenge to reach temperatures well below thesuperexchange energy scale. It was not before 2017 that the group of Markus Greiner inHarvard managed to engineer the entropy in such a way that a long-range ordered antiferromagnet was formed [Maz17b].
For the case of dipolar particles, additional quantum phases are expected due to the anisotropic and long-range nature of the DDI. While superexchange coupling remains isotropic,
dipolar interaction can show different strengths and even a different sign along perpendicularlattice directions. A recent theoretical work explores the possibility to adiabatically preparea striped ground state with dipolar atoms aligned within the plane of a 2D optical lattice.
The formation of the stripe state is driven by the competition of the long-range dipolar
correlations of neighboring spins that reside in adjacent Zeeman states [Maz17a].
5
6

For the case of an isotropic 3D lattice W = 12J.
The chemical potential scales with the strength of the radial harmonic confinement, see Vh,i in Eq. 5.15,
and with the total atom number.

5.4. EXTENDED HUBBARD MODELS IN EXPERIMENTS WITH DIPOLAR
PARTICLES

101

While the study of quantum phase transitions is one of the prime research directions withultracold experiments, the ability to prepare insulating phases is also a powerful tool toprotect the atomic sample from inelastic collisions. Pinning particles within a deep 3D
optical lattice strongly restricts their movement and their ability to interact with each other.
As a consequence, a large collisional protection can be realized [Tha06]. This can also nicely
facilitate the study of dipolar spin mixtures, as dipolar relaxations [Bur15] can be strongly
suppressed. In Sec. 5.6 we apply the method of lattice protection to prepare spin mixturesof the two lowest magnetic substates of fermionic erbium in a Mott-type state.

5.4. Extended Hubbard models in experiments with dipolar
particles
In order to experimentally access the extended Hubbard models that we have discussed inSec. 5.2.2 and Sec. 5.2.3 we have upgraded our experiment by implementing a 3D opticallattice. The lattice is formed by three retro-reflected orthogonal red-detuned laser beams7
operated at 532 nm along two horizontal axes and at 1064 nm along the vertical axis. The
resulting standing waves along the three orthogonal axes impose a 3D lattice structure on
the atomic cloud. The experimental setup of our optical lattice is described in detail in
Appendix B.
Importantly, the high natural abundance of bosonic and fermionic isotopes in erbium, seeSec. 2.1, allows us to prepare both eBH and eFH systems and gives us a large flexibility forthe study of different quantum models and quantum phases. Prior to the study of new andexotic quantum phases, it is necessary to properly understand and characterize our system.
This has been one of the major goals of this thesis. In particular, we have quantitatively
analyzed the eBH terms prepared with our dipolar bosonic system. Similar investigations
for our dipolar fermionic system are currently carried out, see Sec. 6.2. To characterize theindividual terms of the Hamiltonian, lattice spectroscopy can be applied. As a result of theinteraction terms, an energy gap can open up between neighboring lattice sites. This gap
can be measured via lattice modulation spectroscopy and carries information on the onsite
term U and the offsite term V of the Hamiltonian. As this method is one of the major toolsof our experiment, we will discuss it in the following section.

5.4.1. Lattice modulation spectroscopy
Depending on the number of particles occupying a given lattice site and on their interactions
the ground state energy of this lattice site can be modified. This is a direct consequence ofthe onsite energy U and results in an up- or downshift of the state for repulsive and attractive
interactions, respectively. This energy shift can be accessed by resonantly driving particlehole excitations, e. g. by forming a double occupancy (doublon) out of two single occupancies
on neighboring lattice sites. To allow for such a process, energy has to be deposited into the7

See Appendix A.2 for a definition of the red- and blue-detuned regions of the dynamical polarizability of
erbium.

5.4. EXTENDED HUBBARD MODELS IN EXPERIMENTS WITH DIPOLAR
PARTICLES

102

system. This can be conveniently utilized via amplitude modulation of the lattice potential
with a certain modulation frequency νmod , see Refs. [Stö04, Kol06, Cla06]. If the energy ofthe lattice modulation Emod = hνmod matches the onsite energy U a particle-hole excitationis resonantly driven.
As a probe of this resonant feature different methods are available. For the case of bosonicparticles, one can monitor the remaining fraction of atoms in the Bose-Einstein condensate(BEC) after adiabatically melting the lattice, see also Ref. [Mar11]. If particle-hole excitations have been driven, the system has gained energy and hence the final BEC fraction will
be reduced. In the case of fermionic particles, particle-hole excitations are only allowed if
additional spin states are available, as for spin-polarized fermions the Pauli exclusion principle prohibits doublons in a single lattice band. The resonant condition of the modulation
spectroscopy can be extracted via the increase in temperature after going back to a degenerate Fermi gas. For dipolar particles, the formation of two-spin state doublons can also bemonitored via particle loss if dipolar relaxations take place.
In the case of dipolar systems, the onsite energy U consists of contributions from the isotropic
contact interaction and the anisotropic DDI, as discussed in Sec. 5.2.2. Comparing the measured energy to numerical simulations of the system at hand enables to extract the individual
contributions. In particular, one can determine the s-wave scattering length as , which is directly proportional to Us . This is a powerful method to precisely measure the scattering
properties of the system. This information can in turn be used to investigate fascinating
physical phenomena, see also Sec. 4.1.4.
Further details on the modulation spectroscopy method and on how to extract as are givenin the publications presented in Sec. 5.5 and Sec. 5.6. Remarkably, the lattice modulation
spectroscopy also allows to extract the offsite interaction V , a method invented by our groupand reported in Sec. 5.5.

5.4.2. A dipolar bosonic quantum simulator
To prepare an analog quantum simulator of the eBH model, we adiabatically load a BEC
of erbium atoms into a deep 3D optical lattice. In our publication contained in Sec. 5.5 weinvestigate the effects of the individual terms of the eBH model and quantify their strength.
We show the first observation of the offsite dipolar interaction Vdd in eBH dynamics and set
the stage for future investigations with lattice models influenced or even governed by dipolar
long-range interactions [CS10]. In addition, we study the SF-to-MI phase transition, seeSec. 5.3.1, and reveal that the dipole orientation can significantly alter the phase transition
point. Here, the main influence can be traced back to the angle dependence of the onsite
dipolar interaction Udd , but in order to find a better agreement between our experimental
observations and theory, the dipolar DIT ∆Jdd has to be taken into account.

5.4. EXTENDED HUBBARD MODELS IN EXPERIMENTS WITH DIPOLAR
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103

5.4.3. A dipolar fermionic quantum simulator
To realize the Hamiltonian of Eq. 5.15 within our experiment, we first load a degenerate
Fermi gas of spin-polarized 167 Er atoms into the 3D optical lattice and subsequentially deterministically prepare a spin mixture8 . For our first investigation, we switch off the dipolar
spin exchange term Fσ+i Fσ−j by utilizing spin-dependent quadratic shifts via HZeeman + Hlight .
This allows us to investigate in detail the interaction properties of an effective spin-1/2 system consisting of the two lowest spin state mF = |–19/2⟩ and mF = |–17/2⟩. The interspin
scattering length is extracted by modulation spectroscopy, see Sec. 5.4.1, in combination with
our theoretical knowledge of the Hamiltonian terms in Eq. 5.15. Our publication that is contained in Sec. 5.6 reports on the finding of a relatively broad interspin Feshbach resonance,
on which we demonstrate interaction tuning. Our investigation realizes for the first time astrongly interacting dipolar Fermi gas. Importantly, our preparation method via the latticeprotection technique, see Sec. 5.3.2, allows to access deeply degenerate fermionic mixturesat any magnetic field, since the pinning of atoms to individual lattices sites in deep lattices
strongly suppresses dipolar relaxation losses.
Importantly, the performed study only sets the starting point for our experiment, as with
our dipolar fermionic quantum simulator at hand, a large amount of investigations become
accessible. A promising direction emerges from studies of spin dynamics along the Heisenberg
model, see Sec. 6.2. Further, the realization of a magnetically ordered ground state is atreach [Maz17a].

8

For the deterministic preparation of the spin states, we utilize the spin-dependent quadratic Zeeman shift,which leads to a lifting of the degenerate coupling between three adjacent spin states, see Eq. 2.10. The
differential splitting allows for selective spin-addressing already at moderate magnetic field values via
radio-frequency techniques.

5.5. Publication:
Extended Bose-Hubbard models with ultracold magnetic
atoms†‡
Science 352, 6282 (2016)
submitted 10 Jul 2015; published 8 Apr 2016
DOI 10.1126/science.aac9812

Simon Baier,1 Manfred J. Mark,1,2 Daniel Petter,1 Kiyotaka Aikawa,1 Lauriane Chomaz,1,2
Zi Cai,2 Mikhail Baranov,2 Peter Zoller,2,3 F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria†‡

The author of the present thesis developed the experimental procedures together with K. A., and M. J. M.,
performed the measurements with the help of D. P., K. A., and L. C., analyzed the data, and contributed
strongly in writing the manuscript. The theoretical calculations of this publication have been contributed
by Z. C., and M. B.
The post-print version of the publication is attached for online access of this thesis.

104Extended Bose-Hubbard Models with Ultracold Magnetic Atoms
S. Baier,1 M. J. Mark,1, 2 D. Petter,1 K. Aikawa,1, ∗ L. Chomaz,1, 2 Z. Cai,2 M. Baranov,2 P. Zoller,2, 3 and F. Ferlaino1, 2
1 Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

2 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3 Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria

arXiv:1507.03500v2 [cond-mat.quant-gas] 18 Apr 2016

(Dated: May 4, 2018)

The Hubbard model underlies our understanding of strongly correlated materials. Whereas its standard form
only comprises interactions between particles at the same lattice site, extending it to encompass long-range
interactions is predicted to profoundly alter the quantum behavior of the system. We realize the extended BoseHubbard model for an ultracold gas of strongly magnetic erbium atoms in a three-dimensional optical lattice.
Controlling the orientation of the atomic dipoles, we reveal the anisotropic character of the onsite interaction andhopping dynamics, and their influence on the superfluid-to-Mott insulator quantum phase transition. Moreover,
we observe nearest-neighbor interactions, a genuine consequence of the long-range nature of dipolar interactions. Our results lay the groundwork for future studies of exotic many-body quantum phases.
PACS numbers: 67.85.Hj, 37.10.De, 51.60.+a, 05.30.Rt

Dipolar interactions, reflecting the forces between a pairof magnetic or electric dipoles, account for many physicallyand biologically important phenomena. These range fromquantum many-body phases [1, 2], to liquid crystals and ferrofluids [3, 4], to the mechanisms underlying protein folding [5]. The distinguishing feature of the dipole-dipole interaction (DDI) is its long-range and anisotropic character [6]:two dipoles oriented in parallel repel each other, whereas theinteraction between two head-to-tail dipoles is attractive. Notable progress towards the ability to study DDI has been maderecently in systems containing electric dipoles, such as gasesof polar molecules [7] and Rydberg ensembles [8]; similarly,
the recent experimental advances in creating quantum degenerate gases of bosonic and fermionic magnetic atoms, including Cr [9–11] and the Lanthanides Er [12, 13] and Dy [14, 15],
have now opened the door to a study of magnetic dipolar interactions.
Ultracold Lanthanide atoms have an open electronic f shell, and anisotropic interactions; they are characterized byunconventional low-energy scattering properties, including
the proliferation of Feshbach resonances [16]. This complexity manifests itself in quantum many-body dynamics: by
preparing quantum degenerate Lanthanide gases in optical lattices we can realize extended Hubbard models for bosonic andfermionic atoms [2, 17]. In addition to the familiar singleparticle tunneling and isotropic onsite interactions (as forcontact interactions in Alkali), dipolar interactions give riseto anisotropic onsite interactions and density-induced tunneling (DIT), and activate nearest-neighbor (offsite) interactions(NNI). Such extended Hubbard models have been extensively
studied in theoretical condensed matter physics and quantum material science [18, 19], and it is the competition between these unconventional Hubbard interactions that underlies the prediction of exotic quantum phases such as supersolids, stripe and checkerboard phases [17, 20–26].
Here we report a first observation of the unique manifestations of magnetic dipolar interactions in extended Hubbard
dynamics. These observations are enabled by preparing an ul-

tracold sample of bosonic Er atoms in an three-dimensional(3D) optical lattice. It is the control of the optical lattice via
laser parameters in combination with a flexible alignment of
the magnetic dipoles in an external magnetic field, which allows us to reveal and explore the anisotropic onsite and offsite
interactions. Measurements of the excitation spectrum in the
Mott insulator state, and of the superfluid-to-Mott-insulator(SF-to-MI) quantum phase transition are employed as a tool
to detect these interactions and their competitions.

In our experiment an ultracold dipolar gas of 168 Er atoms
is prepared in a 3D optical lattice. The atoms are fully spinpolarized in their lowest Zeeman sublevel [12] and feature amagnetic moment µ of 7 Bohr magneton. The experimentstarts by adiabatically loading a Bose-Einstein condensate(BEC) of about 1.5 × 105 atoms from an optical dipole trap
(ODT) into a 3D optical lattice. The lattice is created by two
retroreflected 532-nm laser beams, defining the horizontal xyplane, and one 1064-nm beam, nearly collinear with the vertical (z) direction, defined by gravity (Fig. 1A) [27]. The lattice
has a cuboid unit cell with lattice constants dx,y = 266 nm anddz = 532 nm, which for 168 Er correspond to the recoil energies ER,x = ER,y = h × 4.2 kHz and ER,z = h × 1.05 kHz, hbeing Planck’s constant. The lattice can be controlled by independently changing the depths associated with the lattice
beams in each direction, (sx , sy , sz ), measured here in unitsof the corresponding recoil energies. The dipole orientation,
quantified by the polar angles θ and φ (Fig. 1A, inset), is varied by changing the direction of the polarizing magnetic field[27]. When increasing the lattice depths we can prepare theEr atoms in the Mott Insulator (MI) state, by driving the superfluid (SF)-to-MI phase transition, as described below.

The dynamics of Er atoms in the optical lattice are described by an extended Bose-Hubbard (eBH) model with

2

A

D

532 nmUdd /h (Hz)200

266 nm

n
66

2

x ϕ

m

B

-200

-400

y

lx

ly

∆ Jij,dd /h (Hz)lz

∆ Ji j

30
0
-30
-60

C
Jij

E

60

B

Vij /h (Hz)

zθ

0

F

1.0
0.5
0.0

-0.5

U

Vi j

0

30

60

Angle θ (°)

90

FIG. 1. Magnetic dipoles in a 3D optical lattice. (A) Schematic of our lattice geometry, where the lattice spacings are indicated. The dipole
orientation, given by the polarizing magnetic field B, is quantified by the polar angles θ and φ with respect to our coordinate system. (B)Illustration of the contributing terms in the eBH model: Tunneling matrix element Ji j , DIT matrix elements ∆Ji j , onsite interaction U, and
NNI Vi j . (C) Illustration of the single-site density distribution, where the harmonic oscillator lengths are indicated. (D to F) Calculated valuesof the DDI-dependent terms as a function of θ for φ = 0◦ in the MI phase, (sx , sy , sz ) = (15, 15, sz ) with sz set by the AR for the cases AR = 1(red) and AR = 2 (green). (D) shows Udd . (E) and (F) show Vi j and ∆Ji j,dd respectively, for bond and hopping directions i j both along x(solid lines) and y (dotted lines). The dashed lines indicate the case without DDI. Us and Ji j are independent of θ and their values for the two
configurations considered are Us = 3749 Hz (1775 Hz) for AR = 1 (2) and Ji j = 27 Hz for i j = x or y.

Hamiltonian [17, 28]ih
H = − ∑ (Ji j + ∆Ji j (ni + n j − 1)) b†i b j + h.c.
hi ji

+

U
ni (ni − 1) + ∑ Vi j ni n j .
2∑
i
hi ji

(1)

Here b†i (bi ) are the bosonic creation (annihilation) operators of an atom at site i, ni = b†i bi is the associated number operator, and hi ji denotes pairs of adjacent sites. All
terms of the eBH are illustrated in Fig. 1B. The interactions
manifest themselves in both the tunneling dynamics, the onsite (U) and the offsite (Vi j , approximated to NNI) interaction. In addition to the single-particle hopping, which here has
amplitudes Ji j reflecting the anisotropy of the optical lattice,
Eq. 1 includes DIT terms (∆Ji j ). The onsite interaction (U)and DIT terms (∆Ji j ) have contributions (Us and ∆Ji j,s ) fromthe short-range contact interaction, which is proportional to
the s-wave scattering length as ; they also have contributions(Udd and ∆Ji j,dd ) from the long-range DDI, which is proportional to µ 2 [17, 27]. The DIT have recently been observedin purely contact interacting systems [29, 30]. The NNI (Vi j )is a term that genuinely originates from the long-range DDI.
The NNI in spin-polarized dipolar systems is qualitatively different from Heisenberg spin-spin interaction between atoms
at neighboring sites hi ji [31, 32], which arises from superexchange processes in Hubbard dynamics in second-order vir-tual hopping processes (∼ Ji2j /U) in the limit of large onsite
interaction, and different from dipolar spin-exchange interactions [7] and full Heisenberg type interactions [10], driving
flip-flop dynamics.
The angle dependence of the DDI reveals itself more prominently in combination with anisotropic geometries [33]. Bychanging the lattice depths (sx , sy , sz ) in the three directions
independently, we can control the aspect ratio (AR) of thethree-dimensional density distribution at a given lattice site
(Fig. 1C), i.e. we can shape the anisotropy of the threedimensional Wannier function. For simplicity, we define theAR using the harmonic approximation, AR = lz /lx,y , where lz(lx = ly ) is the harmonic oscillator length along the z (xy) direction of the local atomic well. In our experiment sx = sy ,such that z is the anisotropy axis. The symmetric condition, defined as Udd = 0, is slightly shifted with respect to
AR = 1, when using Wannier functions (e.g. AR = 1.05 forsx = sy = 15 and AR = 1.07 for sx = sy = 10) [34]. For Udd ,the relative weight between the attractive and repulsive contributions can be tuned by changing the dipole orientation relative to the anisotropy axis of the onsite density distribution,and the AR (Fig. 1D). In contrast, the NNI Vi j is controlled
mainly through the orientation of the dipoles with respect tothe bond direction i j (Fig. 1E). Finally, ∆Ji j,dd depends bothon the orientation of the dipoles relative to the bond direction
and the anisotropy axis, and on the AR (Fig. 1F).
We first investigate the impact of the DDI on the onsite in-3
Excitation frequency νex (kHz)

A

2U

U

1.1
1.0

0.8
0.7
0.6

B

1.1
1.0

3

2

0.9
0.8
0.7
0.6
0

1

2

3

4

5

6

7

8

2U

6

1

4
2

0

∆ νex (Hz)normalized BEC fraction

0.9

C

4

1.0 1.2 1.4 1.6 1.8 2.0

600

D

400
200
0

-200

Modulation frequency νm (kHz)

1.0

1.2

1.4

1.6

Aspect ratio AR

1.8

2.0

FIG. 2. Measurement of the onsite interactions. (A and B) Excitation spectrum of the MI state for dipole orientations θ = 0◦ (A) and θ = 90◦(B) with φ = 0◦ . The modulation spectroscopy is performed along the x-axis at (sx , sy , sz ) = (15, 15, 52.5), corresponding to AR ≈ 1.46 andthe remaining BEC fraction is measured after adiabatically ramping down the lattice depths to zero. From a double Gaussian fit to the data(solid line) we extract the resonant excitation frequency νex for the U and 2U feature. (C) νex for the loss feature at U and 2U (inset) as afunction of AR for θ = 0◦ (squares) and θ = 90◦ (circles). (D) Difference in excitation frequencies νex relative to the two dipole orientations,∆νex , as a function of AR. The error bars for all figures are the sum of the SEM and systematic errors [27]. The theoretical model (solid lines
in C and D) also includes the effect of the NNI, which shifts the excitation frequency by up to 3 % (see Fig. 3). Dashed lines: calculations
accounting only for the isotropic (contact) interaction.

teraction (Udd ) by performing spectroscopic measurements.
We prepare our system deep in the MI phase and probe theenergy gap in the excitation spectrum for different dipole orientations. This energy gap, associated with particle-hole excitations, is U for atoms in singly- or doubly-occupied Mott
shells and 2U at the border between the two shells and at doublon defects within the first Mott-shell [35–38]. We excite theMI state by applying a sinusoidal modulation of frequency
νex on the intensity of the x-lattice beam. When hνex matches
U or 2U, we observe a resonant depletion of the remaining
condensate fraction after ramping down the lattice [27]. We
perform the measurement for θ = 0◦ and θ = 90◦ (Fig. 2, A
and B) with φ = 0◦ and observe that the resonance positions
clearly depend on the dipole orientation, consistent with our
expectation.
To further explore this effect, we repeat the measurementfor different values of the AR (Fig. 2C). At the symmetric condition (AR = 1.05), we observe that the excitation gap looses
its angle dependence showing that Udd averages to zero [39].
As the spatial distribution is deformed towards larger AR,
we find a clear deviation from the purely contact-interaction
case (dashed lines), with a smaller energy gap for dipolesat θ = 0◦ , and a larger one for dipoles at θ = 90◦ . For
large ARs, the dipole orientation substantially affects the incompressibility of the corresponding Mott state with the energy difference between the two dipole configurations of up
to h × ∆νex = h × 600 Hz (Fig. 2D). The observed angle dependence is well described within our eBH model (Fig. 2, C

and D, solid lines). The theory for Fig. 2D is parameter free,whereas for Fig. 2C as is the only fit parameter. From the fit
we find as = 137(1) a0 , with a0 being the Bohr radius, whichis consistent with previous measurements based on thermalization experiments [12]. Our measurement shows that Udd
plays a fundamental role in the stability of the MI phase: itcan either protect the MI phase for the dominantly repulsive
DDI (θ = 90◦ ) or make it more susceptible to excitations forthe dominantly attractive case (θ = 0◦ ).
The energy gap in the MI phase also depends on the NNI
between atoms occupying adjacent lattice sites. To isolate thecontribution of the NNI in the eBH, we design a dedicatedmeasurement scheme based on modulation spectroscopy inthe 2D short-spacing lattice plane (xy-plane), where the NNI
is stronger (Fig. 3A). For a system with only onsite interactions the energy gap associated with the particle-hole excitation does not depend on the direction of excitation, i.e. on
the direction of the modulated beam. In contrast, a system
including anisotropic NNI will exhibit a modification of theenergy gap according to the excitation direction as the energy
gap equals U − Vi j for excitations along the bond direction
i j. Hence the difference between the two resonance frequencies measured by modulating sx and sy , denoted as ∆VNNI /h,directly reveals the existence of the NNI as the onsite contribution cancels. Our scheme is illustrated in Fig. 3, A toC, for the case θ = 90◦ , φ = 90◦ . Here, one bond of attractive (repulsive) NNI with energy V att (V rep ) is destroyed during the excitation along (perpendicular to) the dipole orien-

4

B

Counts

7
5
3
1

rep

V

att

ϕ
ation
rient
o
le
Dipo

A
V

D

0°

z
x ϕ

θ y

90°
200 120B

θ = ϕ = 90°

C40-40 -120 -200

∆VNNI/h (Hz)FIG. 3. Nearest neighbor interactions. (A) Initial system in the MI regime with dipole orientation θ = 90◦ , φ = 90◦ . Driving excitations
along y (B) or x (C) leads to two different particle-hole energy gaps U −V att and U −V rep , respectively. Here, V att = Vi j with in-plane head-totail dipole orientations and V rep = Vi j for the in-plane side-by-side orientation. The difference between the two resonant energies ∆VNNI , equal
to −V att +V rep , reveals the NNI. (D) Histogram of our measurements of ∆VNNI for two dipole orientations φ = 0◦ and φ = 90◦ . The blacksolid curves are the normal distributions of the corresponding data. The two dashed lines show the theoretical expectation values, whereas theorange and green lines show the corresponding measured values with the shaded areas indicating the SEM.

tation such that ∆VNNI = −V att + V rep . Our measurement for
two dipole orientations in the plane with φ = 90◦ ( 0◦ ) give
∆VNNI /h = +74(10) Hz ( −87(14) Hz). Remarkably, |∆VNNI |is similar for both values of φ as expected from the symmetry
between these two configurations and is close to the theoretical expectation h × 91 Hz [27], as shown in Fig. 3D.
Finally, we use our understanding of the angle dependenceof the Hamiltonian, to modify the many-body phase transition from a SF to a MI state by changing θ and AR. Figure 4A shows the textbook signature of the phase transitionin the momentum distribution of a lattice-confined gas [40]:the interference pattern of the SF phase progressively disappears as the system is driven into the MI phase by increasing
the lattice depths. However, in stark contrast with the purely
contact-interaction case as in previous alkali experiments, weobserve that the dynamics of the phase transition has a clear
angle dependence. The full width at half maximum (FWHM)
of the central interference peak reveals a different evolution
with lattice depths for θ = 90◦ as compared to θ = 0◦ , particularly for the largest AR (Fig. 4, B to D). This behavior isqualitatively consistent with our previous observations on Udd
because the phase transition is governed by the competition
between the total onsite interaction, i. e. the sum of the dipolarand contact part, and the tunneling, U/J. For large ARs andθ = 90◦ , the DDI is mainly repulsive and thus strengthens
the pinning of particles in the MI phase with respect to the SF
phase, whereas the contrary happens for θ = 0◦ where the SF
phase is favored (Fig. 4B).
We systematically study the critical value of the lattice
depth, sc (θ ), corresponding to the onset of the phase transition, for two dipole orientations, θ = 0◦ and 90◦ , and various AR values. The determine sc (θ ) we use two independent methods. The first one, based on a double-line fit, identifies sc (θ ) as the point where the FWHM starts to grow,whereas the second method probes the visibility of the interference pattern [27, 41]. The visibility is more reliable forlarge ARs, where the system undergoes a three-dimensional
phase transition and the side interference peaks have a larger
contrast [27, 42]. Figure 4E shows ∆sc = sc (0◦ ) − sc (90◦ ) asa function of the AR. We find a substantial increase of ∆scfrom negative to positive values, providing the opportunity tomodify the phase diagram by tuning the strength and the signof the DDI. In addition, we observe that ∆sc crosses zero at
AR ≈ 1.2 (Fig. 4E). This value is surprising because, if the
DDI affected only the onsite interaction, one should record
the same symmetric condition as measured in the frozen gas
regime, i. e. AR = 1.07 (dotted line in Fig. 4E). However, at
AR = 1.07, we still observe an angle dependence of the phasetransition with the MI phase favored for θ = 0◦ (Fig. 4D).
This behavior could be explained if the DDI also affected thetunneling dynamics. In the eBH model, such a term appears
in the form of a dipolar-interaction-driven-tunneling mechanisms, i. e. DIT, on top of the standard single-particle tunneling; see Eq. 1 and [29]. For our typical experimental parameters, the DIT is predicted to be on the order of a few percent of
the single-particle tunneling and exhibits an angle dependence(Fig. 1F and inset in Fig. 4E), which leads for instance to a reduction of the overall tunneling for out-of-plane orientation(θ = 0◦ ) and thus to a shift of sc (0◦ ) to a smaller value. The
presence of the DIT is consistent with our observations, as
shown by the better agreement of our data with the mean-fieldcalculations including the DIT (solid line in Fig. 4E) with respect to the calculations without DIT (dotted line in Fig. 4E).
For completeness, we checked with Monte Carlo calculations
that the observed behavior can not be explained by the NNI,
as discussed in [27].
Quantum degenerate gases of magnetic Lanthanide atoms
in optical lattices offer a new avenue to access the physics of

5

A

1

2

3

4

5

FWHM (µm)

300
250
200
150
100
50

C

300
250
200
150
100
50

D

1.5

1.0

AR = 2

4

4

2

2

E

∆ Jx,dd /Jij (%)

B∆ Jx,dd /h (Hz)FWHM (µm)

300
250
200
150
100
50

FWHM (µm)

672µm

0

0

-2

-2
1.0 1.2 1.4 1.6 1.8 2.0
AR

∆s C

0.5

AR = 1.28
0.0

-0.5

AR = 1.07
6

8

10

12

14

Lattice depth xy-plane s

16

1.0

1.2

1.4

1.6

1.8

Aspect ratio AR

2.0

FIG. 4. Superfluid-to-Mott-insulator transition. (A) Time-of-flight absorption images of the atomic cloud at θ = φ = 0◦ taken 27 ms after asudden release from the 3D lattice with sx,y,z = s where s = 0, s = 10, s = 22 (A) 1 to 3, respectively and s = 4, s = 0 (A), 4 to 5, respectively.
The distribution of the interference peaks (A2) reflects the anisotropy of the lattice in combination with the orientation of the reciprocal latticewith respect to the imaging light [27]. (B to D) The FWHM of the central interference peak as a function of the lattice depth in the xy-plane
for AR = 2; 1.28; 1.07, with dipoles oriented along θ = 0◦ (blue squares) and θ = 90◦ (orange circles) with φ = 0◦ . (E) ∆sc as a functionof the AR. We extract sc (θ ) using both a double-line fit (triangles) and the visibility method (diamonds) [27]. The error bars for ∆sc represent
the SEM from the fits and the dot-dashed line is the weighted mean of the two methods. We compare our data with mean-field calculationsin presence (solid line) and absence of the DIT (dotted line). The grey shaded regions show the SEM, resulting from the resolution of thecalculation[27]. The dashed line represents the purely contact interacting case. The inset shows the calculated values of the DIT along thex-direction, ∆Jx,dd , for θ = 0◦ (blue) and θ = 90◦ (orange) with φ = 0◦ and sx,y = 10, which is close to the phase transition point, in units offrequency (left axis) and in units of the single-particle tunneling Ji j = 80.5 Hz in the xy-plane (right axis).

strongly correlated systems for both bosonic and fermionic
Hubbard dynamics in the presence of dipolar interactions,while building on the well-developed toolbox to prepare ultracold dense samples, and to manipulate and measure theseatomic gases. We have realized the extended Bose-Hubbard
Hamiltonian with anisotropic onsite and offsite interactions,which reveal themselves in the excitation spectrum and in the
many-body dynamics of the system. Our results show how tocontrol the Hamiltonian terms with the dipole orientation andaccomplish the long-awaited observation of NNI in Hubbard
dynamics. An outstanding challenge for future experiments is
the realization of many-body states in the lattice with spontaneously broken spatial symmetry due to NNI, such as thecheckerboard and stripe phase[17]. For our experimental parameters, the latter phase is expected to appear at temperaturesin the few nK regime [27]. Dipolar interactions can be further
increased by working with Feshbach molecules of magnetic
Lanthanides, essentially doubling the magnetic dipole moment [43], and sub-wavelength lattices [44, 45]. Lanthanides

offer unique opportunities to access the multitude of manybody phases predicted for dipolar quantum matter and arecomplemented by the remarkable experimental developmentswith heteronuclear molecules and Rydberg atoms [2].

We thank F. Meinert and H.-C.-Nägerl for fruitful discussions. The Innsbruck experimental group is supported by theAustrian Ministry of Science and Research (BMWF) and theAustrian Science Fund (FWF) through a START grant under project Y479-N20 and by the European Research Council(ERC) under project 259435. K.A. has been supported within
the Lise-Meitner program of the FWF. The Innsbruck theory
group is supported by the SFB FoQuS, by the ERC Synergy
Grant UQUAM, and by the EU FET Proactive Initiative SIQS.
The data leading to the histograms of Fig. 3D are presented in
[27].

6
SUPPLEMENTARY MATERIALSBEC production

We create a Bose-Einstein condensate (BEC) of about 1.5×
105 168 Er atoms by means of evaporative cooling in a crossed
optical dipole trap (ODT) [12]. The cloud has typically a BEC
fraction above 80 %, which is extracted by a two-dimensional
bimodal fit to an absorption image of the atomic cloud after atime-of-flight (TOF) of 27 ms [12]. The cloud temperature isestimated to be about 70 nK. The ODT is operated at 1064 nm
and is created by two beams, one propagating horizontallyand one vertically. The beams cross at their respective focal
points. The elliptic horizontal beam has a vertical (horizontal)
waist of about 18 µm (117 µm) and the elliptic vertical beam
has a waist of about 55 µm (110 µm) along (perpendicular to)
the axis of the horizontal beam. The measured trap frequencies are (ωx , ωy , ωz ) = 2π × (29.0(6), 22.2(4), 165.2(5)) Hz.
We observe a lifetime of the trapped cloud of about 10 s.
The atomic cloud is spin-polarized in the lowest Zeeman
sublevel (J = 6, mJ = −6), where J denotes the total angular momentum quantum number and mJ is its projection along
the quantization axis. The spin polarization already occurs in
the magneto-optical trap [46] and is maintained in the ODT by
applying a bias magnetic field with a fixed value of 0.40(1) G.
As discussed below, the magnitude of this field is kept constant for all the experiments, whereas its orientation is varied
to set the desired dipole orientation.
3D lattice setup

We describe the 3D lattice setup in the coordinate system given by the two horizontal lattice beams denoting thex and y-axis and the direction of gravity giving the z-axis(Fig. 1A, inset). The horizontal lattice beams are createdby two retroreflected beams with a waist of about 160 µm
and a wavelength λx = λy = 532 nm. The vertical lattice beam has a waist of about 300 µm and a wavelength
λz = 1064 nm. The resulting 3D optical lattice is given by
V (x, y, z) = Vx cos2 (kx x) + Vy cos2 (ky y) + Vz cos2 (kz z), where
Vi is the lattice depth in the i-direction and ki = 2π/λi thecorresponding lattice wavevector with i = (x, y, z). Becauseof the different wavelengths, the atoms experience different
recoil energies ER,i in the xy-plane with respect to the verticaldirection. The recoil energies given by ER,i = h2 /(2mλi2 ) are
ER,x = ER,y = h × 4.2 kHz and ER,z = h × 1.05 kHz. Here, his the Planck constant and m the mass of the Er atom. For
convenience, we give the lattice depth in units of the corresponding recoil energy si = Vi /ER,i . The maximum lattice depth we can achieve is (sx , sy , sz ) = (30, 30, 220). Because of the Gaussian profile of the lattice beams the atoms
experience an additional harmonic confinement. At a typical 3D lattice depth of (sx , sy , sz ) = (20, 20, 20) we measure(ωx , ωy , ωz ) = 2π × (34(1), 31(1), 43(1)) Hz.
We note that the vertical lattice beam is tilted from thevertical axis by θ = 10(2)◦ and has an azimuthal angleof φ = 5(5)◦ . This has two consequences: (a) The lattice spacings dx and dz are modified to dx = 270(2) nm anddz = 540(4) nm with respect to the λ /2 case and (b) thetilt of the wavefront of the vertical lattice beam gives riseto an additional potential difference between neighboring lattice sites along x of 200(40) Hz due to gravity. While (b)only leads to a broadening of the excitation resonances in themodulation spectroscopy measurements, (a) could in principle change the values of the extended Bose-Hubbard (eBH)
terms. Therefore, we recalculate them considering our effective lattice spacings for a typical experimental condition of(sx , sy , sz ) = (15, 15, 15). We find that the isotropic terms are
reduced by 3 % while the anisotropic terms can differ between
2-6 %, depending on the dipole orientation and the directionof the observed process (see Table S1). This gives rise to
a downshift of the phase transition point sc of about 1 % forboth θ = 0◦ and θ = 90◦ . However, all these shifts are notresolvable within our statistical errors and can therefore safely
be neglected.
TABLE S1. Difference of the eBH terms between the λ /2-spacing
and the actual spacing given in percentage of the λ /2-case for three
dipole orientations (θ = 90◦ , φ = 0◦ ), (θ = 90◦ , φ = 0◦ ), and θ =0◦ .
θ = 90◦ , φ = 0◦ θ = 90◦ , φ = 0◦ θ = 0◦
Us
3%
Ji j=x , Ji j=z
3%
Ji j=y
0%
Udd
6%
−2 %
2%
U
3%
−2 %
3%
∆VNNI
3%
2%
∆Ji j=x3%
2%
3%∆Ji j=y4%
2%
3%∆Ji j=z3%
2%
3%

Lattice depth calibration and onsite aspect ratio

To calibrate the depths of the horizontal lattice beams we
use the standard Kapitza-Dirac diffraction method [47]. For
the vertical lattice we use the technique of parametric heating,in which the atoms are excited from the first to the third lattice band [48, 49]. With these methods, we extract the lattice
depths with an uncertainty of up to 4 %.
The onsite aspect ratio (AR) is defined in terms of a Gaussian approximation to the corresponding Wannier function:
1/4
1/4
AR = lz /lx,y , where lx,y = dx,y /(πsx,y ) and lz = dz /(πsz )are the harmonic oscillator lengths associated with the lattice
beams along x, y and z, respectively (Note that we use sx = sy
in our measurements). The uncertainty of the AR results fromthe uncertainty of the lattice depths and is about 1 %.
Because of the non S-state character of Er atoms in theirelectronic ground state, the atomic polarizability of Er has atensorial contribution, which is about 3 % of the scalar one for7
an off-resonant trapping light [50]. In our system this effectgives rise to a different lattice depth depending on the dipole
orientation. For each lattice beam, we carefully studied thiseffect by comparing the measured lattice depth for the dipolesaligned parallel and orthogonal to the corresponding latticebeam. Our measurements reveal that the parallel orientation
gives a deeper confinement of up to 4 % as compared to theorthogonal orientation. We account for this effect as a systematical error, which gives rise to asymmetric error bars in
the AR in Fig. 2 (C and D) and Fig. 4E and in the xy-planelattice depth in Fig. 4 (B to D) and Fig. S4 (A to F).

tice, the dipole orientation is changed to the desired value in
38 ms, while the magnetic-field magnitude is kept constant.
Modulation spectroscopy in the MI

For our experiments the atoms are adiabatically loaded tothe 3D lattice by an exponential ramp to the final value within
150 ms, during which the vertical ODT is linearly lowered
to zero. To perform modulation spectroscopy, the horizontal ODT is switched off within 1 ms after the loading. For
the measurement of the BEC depletion, we exactly reverse
the described process. In the Mott-insulator (MI) phase weestimate a central density of two atoms per lattice site. The
external harmonic confinement leads to a density distribution
with a central doubly occupied Mott shell, consisting of up
to 40 % of the total atoms, surrounded by a singly occupied
shell. The external harmonic confinement is given by the sumof the ODT potentials and the Gaussian profiles of the lattice
beams during the lattice loading. For our typical lattice depth
condition (sx , sy , sz ) = (20, 20, 20), the lifetime of the atomic
sample in the lattice is 5(1) s. In addition, we observe a heating, which leads to a full depletion of the recovered BEC fora holding time in the lattice of about 1 s. The origin of thisheating is not fully understood and might be due to frequency
fluctuations of the 532 nm laser source. It should be noted thatbeside the case of AR = 2 the asymmetric lattice configuration (i.e. larger lattice spacing along the vertical direction) for
our experimental condition leads to a quasi-two-dimensional
system, since the tunneling rates along the vertical lattice aremuch smaller compared to those along the horizontal plane.

To probe the excitation gap in the MI we use a modulation
spectroscopy technique [35, 36]. We sinusoidally modulate
the power of one horizontal lattice beam with a typical total
amplitude between 30 % and 40 %, and a modulation time between 50 ms and 100 ms. With this method, we resonantly create particle-hole excitations in the system [37]. These excitations manifest themselves as a resonant depletion of the recovered BEC because of the extra energy stored in the system. We
record the remaining BEC fraction after ramping down the lattice as a function of the modulation frequency. The resulting
loss spectrum is then fitted with a double-Gaussian function,
whose centers give the excitation frequencies. The typical full
width at half maximum (FWHM) of the resonant loss features
is 1 kHz for excitations using the x-lattice beam and 0.8 kHz
for the y-lattice beam. The width is mainly determined by theexternal harmonic confinement. We note that the difference inwidth between the two excitation directions is due to the tiltof the vertical lattice beam as discussed above.
We also measure the onsite interaction by using an alternative method, known as the collapse-and-revival technique [51]. Here, we first prepare the system at the onset of
the SF-to-MI transition with a lattice depth of (sx , sy , sz ) =(10, 10, 10) and we then suddenly quench the system to(sx , sy , sz ) = (20, 20, 40) within 5 µs. As a result of the quench
the system oscillates between the MI and the superfluid (SF)phase. Figure S1 shows the evolution of the FWHM of thecentral interference peak as a function of the holding time after the quench for two different dipole orientations θ = 0◦and θ = 90◦ . We observe up to four collapses and revivals
and extract the onsite interaction from the oscillation frequency. For θ = 0◦ (θ = 90◦ ) we measure a frequencyof 2.07(16) kHz (2.98(5) kHz), which are consistent with thevalue of 2.15(3) kHz (2.77(3) kHz) obtained with the modulation spectroscopy technique.

Control of the dipole orientation

Analysis of the nearest-neighbor interaction (NNI)

The dipole orientation follows the direction of the magnetic
field, which we control using three pairs of independent coils
oriented perpendicular to each other. Each pair of coils is independently calibrated by performing radio-frequency spectroscopy, where resonant excitations to higher Zeeman sublevels can be used as a measure of the actual magnetic fieldat the position of the atoms. The dipole orientation can bechanged from θ = 0◦ to θ = 90◦ and for any value of φ .
Noise of the ambient magnetic field leads to fluctuations of
the absolute angles θ and φ by about 1◦ around their set values. During the evaporative cooling sequence the dipoles arealigned at θ = 0◦ . Before loading the atoms into the 3D lat-

To derive the NNI we perform a differential measurement
based on modulation spectroscopy, in which the orientationof dipoles is fixed but the direction of excitation is changed
between the horizontal lattice axes x and y. To explain theamount of energy needed to drive a particle-hole excitation
we consider the situation where the dipoles are aligned with
angles θ = 90◦ and φ = 90◦ , as also illustrated in Fig. 3(see main manuscript). Here we denote V att (V rep ) the attractive (repulsive) value of Vi j for the bond direction y (x). At
the starting configuration (Fig. 3A) the total energy is EA =12V att +12V rep . For an excitation along the y-axis the final energy of this configuration reads as EB = U + 11V att + 12V rep ,Loading of the 3D lattice

8
8 0 0

F W H M(µ m )

7 0 0
6 0 0
5 0 0
4 0 0
3 0 0
2 0 0
1 0 0
0

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

H o ld in g tim e ( m s )FIG. S1. Measurement of U by the collapse-and-revival technique. The FWHM of the central peak of the interference pattern
is monitored as a function of the holding time after a sudden quench
from the SF to the MI phase for an initial dipole orientation of θ = 0◦(squares) and θ = 90◦ (circles). The latter measurement is verticallyoffset by 300 µm for a better visualization of the two data sets. Eachpoint is obtained by two to four independent measurements and theshaded region indicates the SEM. The solid lines are the fits of adamped sine to the data, used to extract the oscillation frequency andhence U.

while for an x-excitation it is EC = U +12V att +11V rep . From
this consideration it becomes clear that the difference in energy EB −EC = −V att +V rep = ∆VNNI purely reveals the NNI.
Analogously the same consideration can be applied for an
initial dipole orientation of θ = 90◦ and φ = 0◦ leading to∆VNNI = −V rep + V att . From the theory we expect V rep /h =31.5 Hz, V att /h = −59.5 Hz and thus |∆VNNI |/h = 91 Hz. Including the corrections arising from the modification of thelattice spacings due to the tilt of the vertical lattice beam (seeabove) |∆VNNI |/h changes to 89 Hz, even closer to our measured values.
In Fig. S2A we show two excitation spectra obtained using
the method described above. The difference between the centers of the Gaussian fits to the data is found to be 72(30) Hz
and corresponds to one data point of Fig. S2B, where all taken
measurements are summarized. We believe that the fluctuation of ∆VNNI along the data sets is mainly caused by relative drifts of the lattice depths during a differential measurement. We carefully check for systematic errors on ∆VNNI using different initial lattice depths or atom numbers, but do notfind an effect within our measurement resolution. The usedlattice depths are (sx , sy , sz ) = (15, 15, 30), (14, 18, 30), and(20, 20, 40). The different depths can slightly modify ∆VNNI
by a maximum of 2 % which is not resolvable within our errorbar.

Analysis of the SF-to-MI transition point

To study the SF-to-MI transition we probe the momentum distribution of the atoms as a function of the lattice

depths [40]. In particular, we ramp simultaneously in 150 ms
the power of the three lattice beams up to the final desired
value while lowering the vertical ODT to zero. We then suddenly switch off all beams and let the atomic cloud expand for
27 ms. Here, we perform standard absorption imaging usinga resonant 401 nm laser beam [12]. The imaging light propagates horizontally with an angle of 20◦ with respect to thex-lattice beam, (θ = 90◦ , φ = −20◦ ). By increasing the lattice depth we observe a dramatic change in the momentum
distribution, which is the textbook signature for the SF-to-MItransition [40]. In the SF phase the distribution of the interference peaks (Fig.4A2) reflects the anisotropy of the latticein combination with the orientation of the reciprocal latticewith respect to the imaging light (see Fig. S3). In particular
we find in total six first-order interference peaks: two in thevertical direction and four in the horizontal one. Along thevertical direction, the two peaks have half the spacing fromthe zero-momentum central peak with respect to the peaks resulting from the y-axis lattice. In the horizontal direction, we
observe both the interference peaks from the y- and x-axis,
since the imaging axis is not completely collinear to the x direction. When the lattice depth is further increased the systementers into the MI as can be seen in the momentum distribution
by an increasing incoherent background (Fig. 4A3). An analysis of the observed momentum distribution for various final
lattice depths sx,y can thus reveal the SF-to-MI phase transition point. For a quantitative study of the dependence of thephase transition point we repeat the experiment for the twoangles θ = 0◦ and θ = 90◦ with φ = 0◦ and for various values of the AR. Beside the case of AR = 2 the physics of theSF-to-MI transition is effectively two-dimensional due to thenegligible tunneling rate along the vertical lattice.
The critical value for the SF-to-MI transition, sc , depends
on the ratio of the total onsite interaction to the total tunnelingrate. In presence of the dipole-dipole interaction (DDI), bothterms depend on the dipole orientation, imprinting an angle
dependence on the phase transition point sc (θ ), defined by
the value of the horizontal lattice depth sx,y = sx = sy . We
study the phase transition for θ = 0◦ and θ = 90◦ and extract
the difference in the critical point ∆sc = sc (0◦ ) − sc (90◦ ).
In general we observe a smooth transition from the SF
phase to the MI phase as expected for a trapped system with aspatially depending density. For this reason, the extraction of
sc is a delicate matter and we use two different methods [41].
The first method (a) is a double-line fit, which analyses the increase of the FWHM of the central interference peak along thehorizontal direction as a function of sx,y (Figure S4, A to C).
A weighted fitting function consisting of two smoothly connected lines is used to extract sc , which thus corresponds to
the knee of the experimental data. We apply the fit in a region
from sx,y = 5 to the values of sx,y for which the FWHM is
250 µm. In this region the experimental data are well approximated by the double-line fit.
The second method (b) is based on the visibility. Figure S4, D to E, shows the extracted visibility data for thesame data as in (a). The visibility is calculated from a two-

9
150 B

1.0

100

ΔVNN I / h (Hz)normalized BEC fraction

A

0.8

0.6

50
0
−50

−100
−150

0.4
1

2

3

4

Modulation frequency νm (kHz)

0

5

10

15

Dataset number

20

FIG. S2. Measurements for the NNI. (A) Excitation spectrum with modulation along x (diamonds) and y (triangles) with θ = 90◦ and
φ = 90◦ , forming one differential measurement. Each point is the average of about 5 independent measurements and the shaded region
indicates the SEM. The solid lines are weighted Gaussian fits to the data. The dashed lines indicate the obtained resonance frequencies. (B)Set of differential measurements as presented in (A) for dipoles aligned with θ = 90◦ and φ = 90◦ (squares), θ = 90◦ and φ = 0◦ (circles).
The solid lines are weighted fits with the shaded region being the SEM. The dashed lines are the theoretical expectations.

z
im

ag
in
g

y
x

For both methods, at large ARs, we observe a clear shiftof sc toward higher values for θ = 0◦ compared to θ = 90◦(Fig. S4, A and D). This shift vanishes around AR ≈ 1.2(Fig. S4, B and E), and changes sign for lower ARs (Fig. S4,
C and F). We note that method (b), is more reliable at large
ARs than at smaller ones, since in the latter case the phasetransition has a 2D nature (negligible tunneling rate along z).
Therefore, the interference peaks are broadened compared tothe 3D case resulting in a lower contrast, which might leads toa systematic shift of V . Method (a) can be applied to all ARs
but it can be more sensitive to experimental drifts and interaction broadening since it is based on the measurement of the
FWHM of the central peak.
Extended Bose-Hubbard model from microscopic Hamiltonian

FIG. S3. Observation of the interference peaks. Visualization of
the observed momentum distribution of the atoms in the SF phasereleased from the 3D lattice in TOF absorption imaging due to theorientation of the reciprocal lattice with respect to the imaging axis
(see also inset)

dimensional (2D) fit consisting of eight gaussians: one forthe central, six for the first-order interference peaks, and one
for the broad incoherent background (see Fig. S2G). The visibility is defined as V = A/(A + B), where A stands for themean amplitude of the first-order interference peaks and B for
the mean value of the incoherent background at the positionsof the interference peaks. We extract V as a function of sx,yand fit the whole dataset by the phenomenological function
V (sx,y ) = C/(1+exp(α(sx,y −sc )))−V0 (adapted from [41]).
Here C, α, sc , and V0 are fitting parameters, where sc corresponds to the phase transition point.

Here we present the details of derivation of the eBH model
Eq. 1 together with the expressions for all its coefficients interms of microscopic parameters of the system (see, e. g. [52]).
The microscopic Hamiltonian of the considered system ofpolarized (magnetic) dipolar atoms has the form:Ĥtot = Ĥ0 + Ĥint ,(S1)

where the first term
Ĥ0 =

Z

drΨ† (r)[−h̄2 ∇2+V (r)]Ψ(r)
2m

(S2)

is the single-particle Hamiltonian with Ψ(r) being a bosonfield operator, which describes the motion of an atom withmass m in the optical-lattice potential V (r) = Vx cos2 (kx x) +Vy cos2 (ky y) +Vz cos2 (kz z), and the second termĤint =

1
2

Z

dr

Z

dr0 Ψ† (r)Ψ† (r0 )U(r − r0 )Ψ(r0 )Ψ(r)Visibility

FWHM (µm)

10
350
300
250
200
150
100
50
1
0.8
0.6

G

C

B

A

AR = 1.07

AR = 1.28

AR = 2

0.4
0.2

D
4

E
6

8 10 12 14

4F
6

8 10 12 144

Lattice depth xy-plane sx,y6

8 10 12 14

FIG. S4. Derivation of the phase transition point sc . (A to F) The FWHM of the central interference peak and the visibility are displayed asa function of the final sx,y lattice depths for AR = 2 (A and D), 1.28 (B and E), and 1.07 (C and F), for dipole orientations θ = 0◦ (squares)and θ = 90◦ (circles). The solid lines show the corresponding fitting functions (see text). Each point is obtained from about 10 independentmeasurements. (G) Example of density distribution after TOF (top) and the corresponding 2D fit (bottom) used for the derivation of thevisibility.

corresponds to the interatomic interaction. In the consideredcase, the interaction contains a short-range part, which canbe modeled by a contact potential with the s-wave scattering
length as , and the dipole-dipole interaction (see, e.g. [53])U(r − r0 ) =µ0 µ 2 1 − 3 cos2 θr−r0
4π h̄asδ ( r − r0 ) +m
4π|r − r0 |3

with θr−r0 being the angle between the relative position of two
dipoles r − r0 and their polarization.
The Hamiltonian of Eq. S2 determines the single-particle
band structure,[−h̄2 ∇2+V (r)]uαp (r) = εα (p)uαp (r),2mwhere uαp (r) is the Bloch wavefunction corresponding to theband α and quasimomentum p from the Brillouin zone (BZ),
defined by −π/di < pi /h̄ ≤ π/di with di = π/ki being the lattice spacing along the i-direction, pi the corresponding component of p, and εα (p) is the corresponding energy. For our
purposes it is more convenient to work with Wannier functions φi,α (r) = ∑p∈BZ exp[−ip(r − Ri )]uαp (r), which are localized at different sites Ri of the lattice and orthogonal toeach other withR respect to both the lattice position i and the∗ (r)φ (r) = δ δ . Using these funcband index α, drφi,αi j αβj,βtions as a single-particle basis in the bosonic field operator,
Ψ(r) = ∑i,α bi,α φi,α (r), where bi,α are the bosonic annihilation operators for particles on the site i in the band α, we
can rewrite the initial Hamiltonian of Eq. S1 in terms of theoperators bi,α and b†i,α . To obtain the eBH model, we keep
terms with operators for the lowest energy band only. Notethat this approximation is legitimate because the interatomic
interaction in our case is order of magnitudes less than theband gap such that the admixture of the higher bands can beneglected. From the remaining terms we then neglect thosewhich contain square and higher power of exponentially small
spatial overlaps of the Wannier functions from different sites(see, [52] for details). Denoting the operators and the Wannierfunctions for the lowest band as bi , b†i and φi (r) , respectively,
we obtain
H = − ∑ Ji j (b†i b j + h.c) +hi jiUni (ni − 1) + ∑ Vi j ni n j
2∑
i
hi ji

− ∑ ∆Ji j [b†i b j (ni + n j − 1) + h.c]
hi ji

(S3)

where hi ji denotes a pair of nearest-neighboring sites. The
first two terms in this expression correspond to the standard
Hubbard model with the single-particle hopping amplitudeJi j = −Z

drφi∗ (r)[−h̄2 ∇2+V (r)]φ j (r)2mand the onsite interaction U = Us +Udd , where Us comes fromthe contact interaction,Us =4π h̄asmZ

dr |φi ( r)|4 ,and Udd from the dipole-dipole one,
Udd =µ0 µ 2
4π

Z

dr

Z

dr0 |φi (r)|2

1 − 3 cos2 θr−r02
φi (r0 ) .
|r − r0 |3

The third term in Eq. S3 corresponds to the NNI with
Vi j =µ0 µ 2
4π

Z

dr

Z

d r0 |φi (r)|2

1 − 3 cos2 θr−r02
φ j (r0 )|r − r0 |3

coming from the DDI (the contribution from the contact interaction is proportional to the square of the exponentially11small overlap and is therefore neglected). Note that the DDI
also generates interactions Vi j beyond the nearest-neighbors,−3
which decay as Ri − R j . The corresponding terms are neglected in the Hamiltonian of Eq. S3 because they are smaller
and play only minor role in the spatially homogeneous phases(SF and MI) relevant for our experiment. These terms however will be definitely relevant for inhomogeneous phases like,for example, the stripe and the checker-board phases. Finally,
the fourth term in Eq. S3 describes the density-induced tunneling (DIT) with the amplitude ∆Ji j = ∆Ji j,s + ∆Ji j,dd resulting
from the contribution from the contact interaction

n. In a spatially inhomogeneous system (e.g., in the presenceof a trapping potential Vh ), this value is site-independent, andthe system has typically a layered structure in which the Mott
states with different n (n = 1 and 2 in our case) are separated
by the SF phase.
It should be mentioned that, even though the MF approximation is known to overestimate the stability of the SF phase,
here we are interested not in the phase boundary of the SFto-MI transitions itself, but in the relative shift of this boundary when the dipolar polarization is changed from θ = 0◦ toθ = 90◦ .
In calculating this shift, the MF method turns out to be reZ
4π h̄asliable, as it is demonstrated on Fig. S5 where we present the
dr |φi (r)|2 φi∗ (r)φ j (r)∆Ji j,s = −mresults for the phase transition shift ∆sc in a spatially homogeneous system, obtained within quantum Monte Carlo (QMC)
and from the dipole-dipole one
and MF methods. One sees practically no effects of quanZZ2θ
tum fluctuations on ∆sc , and this is why the MF method can
0
1
−
3
cos
µ0 µ 2r−r ∗ 0dr dr0 |φi (r)|2φi (r )φ j (r0 ). be used for the quantitative description of our experimental∆Ji j,dd = −034π|r − r |results (Fig. 4E). To underline the role of the DIT in deterIt should be mentioned that in our experiments we also have amining ∆sc , we analyze the effect of the NNI on the SF-to-MI
shallow confining potential Vh (r). It can beR taken into accounttransition point by using QMC studies of the eBH model withby adding the term ∑i Vh,i ni with Vh,i = dr|φi (r)|2Vh (r) to
NNI, but without DIT. Fig. S5 shows that the inclusion of thethe Hamiltonian of Eq. S3.
NNI only slightly shifts ∆sc (compare dot-dashed and dot-dotThe above expressions, together with numerically comdashed line), and this shift turns out to be much smaller (5-6
puted Wannier functions, provide the theoretical values for thetimes in our case) as the shift due to DIT (Fig. S5, solid line),parameters Ji j ,U,Vi j , and ∆Ji j . During the calculations, thedespite the fact that the values of the NNI are much larger
singularity for |r − r0 | → 0 in the contributions from the DDI
than those of the DIT. An insight into this counterintuitive rewas resolved by performing the integration over θr−r0 before
sult can be gained by considering the SF-to-MI transition in
integrating over |r − r0 |. For our experimental conditions, thethe MF approximation. One can easily see that within this apcontributions from the DDI to the eBH parameters are typiproximation the contribution of the NNI to the energy of thecally smaller for U, in the same order of magnitude for ∆Ji j
system is the same for both phases. As a result, on the MF
and much larger for Vi j than those from the short-range conlevel the NNI does not affect the transition at all, and it is only
tact interaction. However, they strongly depend on the form
the fluctuation effects beyond the MF which introduces theof the Wannier function φi (r) , i.e. on the intensity and ratio
dependence of the transition point on the NNI. On the otherof sx , sy , and sz , and on the alignment of the dipoles relative tohand, the DIT affects the transition already on the MF level
the lattice axes (see Fig. 1, D to F).
and therefore, although being much smaller than the NNI, has
nevertheless a much more pronounced effect on the shift of
the transition.
SF-to-MI transition in the mean-field (MF) approximation

In the MF approximation (see, e.g. [54–56]), the groundstate wavefunction of the system is written as a product stateover sites:|ΨG i =O ∞i

( ∑ Ci |nii ),
(n)

n=0

where |nii denotes the Fock state with n bosons on site i.
(n)
The coefficients {Ci } are the variational parameters sub(n) 2jected to the constraint ∑∞= 1, which can be den=0 Ci
termined by minimizing the energy EG = hΨG |H|ΨG i. The
SF phase is characterized by the local order parameter hbi i =√ (n−1) (n)Ci 6= 0, which implies thathΨG |bi |ΨG i = ∑∞n=1 nCi(n)
Ci are non-zero for several adjacent values of n. In con(n)trast, in the MI phase Ci are non-zero for only one value of

Observability of the stripe phase

The stripe phase is an example of exotic quantum phases
induced by the NNI, which is characterized by spontaneoustranslational symmetry breaking along one direction. It canbe accessed in a deep optical lattice half-filled with atoms,
when the NNI overwhelms the effects of single-particle tunneling and temperature. In this case, the onsite interaction
is much larger than all the other parameters in the Hamiltonian, and prevents two atoms to be on the same lattice site.
We therefore can consider atoms as hardcore bosons, suchthat the number of atoms on a lattice site can be only zeroor one. We assume that the dipoles are polarized along thex-direction resulting in attractive V att (repulsive V rep ) NNI for
the bonds i j in the x(y)-direction, with −V att = 2V rep = 2V . To
calculate the critical temperature for the stripe phase, we con-

12
0.2
0.1

∆s C

0.0
-0.1
MF without DIT, without NNI
QMC without DIT, without NNI
QMC without DIT, with NNI
MF with DIT, without NNI

-0.2
-0.3
1.0

1.1

1.2

1.3

Aspect ratio AR

FIG. S5. Effects of fluctuations, the NNI and the DIT on the shiftof the phase transition ∆sc . Theoretical calculations of the shiftof the phase transition as a function of AR around the zero of ∆sc .
The results for ∆sc in the spatially homogeneous case without DIT
and NNI obtained within the MF (dotted line) and QMC (dot-dashedline) are similar. The effect of the NNI on ∆sc [compare QMC without (dot-dashed line) and with NNI (dot-dot-dashed line)] is muchsmaller than the effect of the DIT [compare MF without (dotted line)
and with DIT (solid line).

sider typical experimental conditions with sx = sy = 20 and
AR = 1. In such a lattice, we can ignore the DIT and thetunneling in the z-direction, and the single-particle tunneling
amplitudes Ji j do not depend on the direction of the hopping,Ji j = J = h × 20.5 Hz. We obtain V = h × 34 Hz and the NNI
value for the bonds in the z-direction is V /8 = h × 4.25 Hz.
After neglecting this small coupling in the z-direction, theHamiltonian for each xy-plane can now be written as
Hxy = −J ∑ (b†i b j + h.c) + ∑(−2V ni ni+êx +V ni ni+êy − µni ),
hi ji

i

(S4)
where b†i and bi are hard-core boson operators and i + êx(i + êy ) denotes the neighboring site of site i in the x (y) direction. We also add the chemical potential µ which is chosen
as µ = −V to satisfy the condition of half-filling hni i = 1/2.
To determine the critical temperature of the transition into thestripe phases, we perform Quantum Monte Carlo calculations
based on the worm algorithm for the Hamiltonian of Eq. S4,
which is free from the negative sign problem. For the above
parameters, the calculated value for the critical temperature is
Tc = 1.4J ' 1.5 nK.

∗ Current adress: Department of Physics, Graduate School of Sci-

ence and Engineering, Tokyo Institute of Technology, Meguroku, Tokyo, 152-8550 Japan[1] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau,
Rep. Prog. Phys. 72, 126401 (2009).
[2] M. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Chem.
Rev. 112, 5012 (2012).
[3] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals(Oxford Unversity Press, Oxford, 1995).

[4] R. E. Rosensweig, Ferrohydrodynamics (Cambridge University
Press, New York, 1985).
[5] K. A. Dill, Biochemistry 29, 7133 (1990).
2
2
0 µ 1−3 cos θ
where r[6] The magnetic DDI reads as VDDI (r, θ ) = µ4π
r3stands for the distance between the two particles, θ for the angle
between the dipole orientation and the interparticle axis, µ0 for
the vacuum permeability, and µ for the magnetic moment of theparticles.
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5.6. Publication:
Realization of a Strongly Interacting Fermi Gas of Dipolar
Atoms†
submitted for publication in Physical Review Letterssubmitted 30 Mar 2018
DOI arXiv:1803.11445

S. Baier1 , D. Petter1 , J. H. Becher1,⋆ , A. Patscheider1,2 , G. Natale1 , L. Chomaz1 , M. J.
Mark1,2 , F. Ferlaino1,2
1Institut für Experimentalphysik,Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria⋆
Current address: Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226, 69120
Heidelberg, Germany
2†

The author of the present thesis developed the experimental procedures together with J-H. B., and D. P.,
performed the measurements together with J-H. B., D. P., A. P., L. C. and G. N., analyzed the data, andcontributed strongly in writing the manuscript.

118

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

arXiv:1803.11445v1 [cond-mat.quant-gas] 30 Mar 2018

(Dated: April 2, 2018)
We report on the realization of a strongly interacting Fermi gas with a dipolar spin mixture offermionic erbium. Employing a lattice-protection technique, we prepare deeply degenerate Fermi
mixtures of the two lowest spin states and perform high-resolution Feshbach spectroscopy. We
identify a comparatively broad Feshbach resonance and precisely map the inter-spin scattering length
across the resonance. We show a remarkable collisional stability of the quantum mixture in thestrongly interacting regime, providing a first step towards BEC-to-BCS studies in presence of dipoledipole interaction.

The ability to prepare dipolar quantum gases of magnetic atoms [1–6] has led to the demonstration of fascinating, yet unexpected, phenomena, emerging from the peculiar traits of the dipole-dipole interaction (DDI) among
particles. In bosonic systems, the DDI can dominate thesystem’s behavior, leading to counter-intuitive effects,
such as a d-wave-patterned collapse [7], droplet stabilization [8–10], or the emergence of roton excitations [11].
With fermions, many-body dipolar phenomena have been
investigated only in spin-polarized systems. Here, theDDI competes with the Pauli pressure, 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 erbium (Er) has twenty available spinstates in the lowest hyperfine manifold. Bosonic dipolar spinor gases have been investigated in remarkable experiments with magnetic Cr atoms [13–16] (see also Rb
experiments [17, 18]), whereas the fermionic counterpart
remains rather unexplored in the quantum regime. Scattering experiments with fermionic Dy mixtures slightly
above quantum degeneracy, showed a large collisional
stability against inelastic dipolar relaxation [19], enablinge. g. the production of long-lived spin-orbit-coupled gases
via Raman excitations [20]. Near-degenerate gases offermionic ground-state molecules have been used to study
spin dynamics in lattices [21].
As yet, the realization of a two-component dipolar
Fermi mixture with tunable interactions has remained
elusive. Such a system is expected to give access to avariety of fascinating phenomena, from anisotropic quantum phases of matter, e. g. anisotropic Fermi liquids and
BEC-to-BCS pairing [22, 23], to dipolar magnetism [24],but also extended Fermi-Hubbard models with direct
off-site interactions [25]. A prime candidate for its realization is given by fermionic lanthanides that havea large magnetic moment. However, the astonishingly
large density of Feshbach resonances (FRs) even in spin-polarized gases [26–28] raises the question of whether stable fermionic quantum mixtures with tunable contact anddipolar interactions can be realized with lanthanides.
We here report on a powerful platform to produce
a two-component dipolar Fermi gas of pseudo-spin 1/2
and demonstrate tunability of the inter-spin interactions. By using highly magnetic 167 Er atoms and athree-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 wellisolated and comparatively broad inter-spin FR and precisely measure the scattering length between the two spinstates. By conducting scattering experiments, we reveal
a remarkable collisional stability of the Fermi mixture in
the strongly interacting regime.
Achieving a deterministic preparation of a spin-1/2
mixture and a precise control over the inter-spin interactions in highly-magnetic lanthanide atoms challenges
experimental schemes. Indeed, the enormous density of
FRs can cause collisional losses and severe heating, limiting the production and preparation of deeply degenerate mixtures at arbitrary magnetic fields (B), where
hundreds of FRs might need to be crossed; see e. g. [20].
Moreover, state-selective preparation of a spin-1/2 system
typically requires large B values for which the quadratic
Zeeman effect lifts the degeneracy on the Zeeman splitting among consecutive sublevels [19, 29].
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: We first produce a spin-polarized degenerate Fermi gas (dFg) in an
optical dipole trap (ODT) at low B (a1), we then load theatoms into the lowest band of a deep 3D optical lattice,which acts as a collisional shield (a2) [30, 31]. We thensweep to high B for spin preparation and perform radiofrequency (rf) transfer (a3), and sweep to the desired B
and eventually melt the lattice (a4).

2Experimentally, we prepare a spin-polarized dFg of
Er atoms in a crossed-beam ODT [5] (stage (a1);Fig. 1). All fermions occupy the lowest Zeeman state| ↓ i ≡ |F = 19/2, mF = –19/2i of the ground-state manifold. Here, F is the total spin quantum number and mF
its projection along the quantization axis. A homogeneous magnetic field of B = 0.6 G is applied along thevertical z direction to define the quantization axis andto maintain the spin polarization in the system. The
sample typically contains N = 2.4 × 104 atoms at about
T = 0.25 TF . Note that the ODT is shaped to optimize single-band loading of the optical lattice and yields
EF = kB × TF = kB × 170 nK = h × 3.6 kHz [32]. Here,TF is the Fermi temperature, h the Planck constant, andkB the Boltzmann constant.
In the next step, we transfer the spin-polarized dFg
into a 3D optical lattice (stage (a2); Fig. 1). We use alattice geometry and a loading scheme similar to the oneof our previous work [33]; for details see also Ref. [32]. In
order to pin the atoms in a one-fermion-per-lattice-site
configuration (unit filling), we use large lattice depthsof about (sx , sy , sz ) = (20, 20, 80), where si 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. Afterlattice loading, we obtain a single-component fermionicband-insulator (BI) of about 2.2 × 104 | ↓ i atoms. By
melting the lattice and re-loading the fermions into the
ODT, we measure a temperature of T ≈ 0.3 TF and extract a heating rate in the lattice as low as Ṫ = 0.03 TF /s.
Our system is well described by a single-band extended
Fermi-Hubbard model [25] with residual tunneling ratesof Jx,y = h × 10.5 Hz and Jz = h × 0.001 Hz, and nearestneighbor interactions in the order of h × 50 Hz [33]. We
confirm the single-band population by performing standard band-mapping measurements [34] [35]. In the horizontal (xy) plane, we observe that the fermions occupy
the lowest band and we do not record higher band occupation within our resolution; see Fig. 1(b) [36]. Along thez-axis, we detect a residual < 5 % population in the firstexcited band, resulting from the fact that EF > ER,z [37].
Because of the Pauli exclusion principle, doubly-occupied
sites (doublons) in a single band are strictly forbidden for
identical particles (| ↓ i).
In the BI regime, the lattice is expected to provide
a strong collisional protection to the particles, as previously observed with deeply-bound molecules [30, 31, 38].
As a first application, we use the lattice-protection technique to realize a spinor Fermi gas with pseudo-spin 1/2(stage (a3); Fig. 1). For such | ↓ i–| ↑ i mixtures, | ↑ i ≡|F = 19/2, mF = –17/2i. Experimentally, we start witha | ↓ i BI at B = 0.6 G and then ramp B in 40 ms toa value of about 40 G, for which the quadratic Zeeman
effect in 167 Er is large enough to lift the degenerate coupling of the individual spin levels [32]. After letting thefield stabilize for 120 ms, we use a standard rf-sweep technique to transfer part of the atoms into the | ↑ i state. By
167

2

(a)
1

4
3

rf
J

(b)

(c)

z

|mF〉
|-15/2〉|-17/2〉x|-19/2〉yFIG. 1. Spin-1/2 dipolar fermions in a 3D optical lattice.
(a) Sketch of the four key stages of our preparation scheme:
spin-polarized dFg in an ODT (a1), single-component BI (a2),rf-preparation of a spin mixture in the lattice (a3), and spinmixture in the ODT (a4). (b) Band population in the horizontal xy-plane, obtained by averaging 50 absorption images for a 12 ms time-of-flight (TOF). The red arrows indicate the first Brillouin zone of the lattice, i. e. 2h̄π/dx,y with
dx,y = 266 nm. (c) Spin-resolved band-mapping images after
9 ms of TOF in the vertical z x̃ plane, where x̃ accounts for theangle between the imaging beam and the y-axis of the lattice,for population imbalances δ = 1 (left panel), 0.02 (middle
panel), and −0.94 (right panel). The images are averages ofabout 20 absorption pictures. The spin states are separated
along the z-direction by a Stern-Gerlach technique and are
depicted by the blue and orange dipole, respectively.

tuning the rf-power, we can precisely control the population imbalance, δ = (N↓ − N↑ )/(N↓ + N↑ ), in the
mixture. Figure 1(c) shows exemplary spin-resolved absorption images of | ↓ i–| ↑ i mixtures for various δ after
B is swept back to low values [39]. At δ ≈ −1, we typically record N↑ = 1.6 × 104 | ↑ i atoms. By reloading
the atoms into the ODT, we extract a temperature ofT ≈ 0.3 TF (stage (a4); Fig. 1). For comparison, similar measurements in absence of the lattice clearly show amuch lower atom number of N↑ = 0.6 × 104 , proving thestrength of our lattice-protection scheme to circumvent
atomic loss [20] when cruising through the ultra-dense
Feshbach spectrum [26].
Figure 2 shows the high collisional stability of thelattice-confined spin mixture with lifetimes of several seconds. In particular, we probe the atom number of | ↓ i
and | ↑ i as a function of the holding time in the lattice;see Fig. 2(a). From an exponential fit to the data, we
extract long lifetimes of τ↓ = 31(3) s and τ↑ = 12.2(7) s.
The measurements are carried out at B = 3.99 G, where
no FRs occur [32]. We also observe that the lifetime ofeach spin state does not show a dependence on the population in the other state, i. e. on δ; see Fig. 2(b).
We note that, although very long for our purpose, we

3
(b)

2,0
1,5

lifetime (s)atom number (x104)

(a)

1,0
0,5
0,0

0

5
10
15
holding time (s)

20

40
30
20
10
0
-1,0 -0,5 0,0 0,5 1,0
pop. imbalance δ

FIG. 2.
Spin mixture of dipolar 167 Er in a 3D lattice.
(a) Lifetime measurements for spin-polarized samples of | ↓ i(squares) with δ = 1 and of | ↑ i (circles) with δ = −0.92 at
B = 3.99 G. Exponential decay functions (solid lines) are fitted to the respective data. (b) Lifetimes as a function of δ.
Constant fits extract mean lifetimes across δ of τ̄↓ = 29.9(3) sand τ̄↑ = 11.8(7) s. All error bars indicate the statistical uncertainty.

always record shorter lifetimes for a | ↑ i BI with respect
to the ones measured for a | ↓ i BI. Differently from the| ↓ i case, two-body relaxation processes for | ↑ i are allowed. At our magnetic fields, this process converts Zeeman energy into a large enough kinetic energy to let theatoms escape from the lattice [13, 40], and requires theparticles to collide at short distance (onsite) [19, 41]. In
the spin-polarized cases (e. g. δ = −1; | ↑ i), double occupancy necessarily involves population in higher bands
since the Pauli exclusion principle forbids doublons in thelowest band. In our system, a continuous transfer of asmall fraction of atoms into higher bands might be driven
by intensity and frequency noise of the lattice beams [31].
In the case of | ↑ i this would lead to subsequent fast relaxation and explain the observed difference in the lifetimes.
With the above described spin-preparation method, weare now able to conduct high-precision Feshbach spectroscopy in an ODT (stage (a4); Fig. 1) in search of interspecies loss features. For this, we first prepare thespin-1/2 mixture in a deep lattice and sweep B to the target value within 10 ms. After letting B stabilize for about
100 ms, we transfer the mixture back into the ODT, holdthe atoms for 500 ms, and finally measure the spin populations, as previously described. We then repeat themeasurement for different B values within the desired
range. Figure 3 exemplifies the high-precision Feshbach
spectroscopy for three values of δ within a narrow magnetic field range from B = 550 mG to 750 mG with aresolution of 1 mG. A lower-resolution and larger-range
scan is shown in Ref. [32].
As expected [26, 27], the atom-number trace as a function of B shows a high density of resonant loss features
on top of a constant background. By controlling δ, we areable to distinguish the spin nature of each of the observed
FRs. In the excerpt shown in Fig. 3, we identify three
narrow homo-spin FRs in a pure | ↓ i sample ((a), blue

shading) and four in a quasi-pure | ↑ i sample ((b), orange
shading). In the spin-polarized cases, all FRs are narrow
with measured widths of the order of our magnetic fieldstability of ≈ 1 mG. Thanks to our lattice-preparation
technique, the shape and the width of the FRs are notaffected by 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 = 0),
we observe five additional inter-spin FRs (Fig. 3(c), green
shading), where atoms in the two spin states are simultaneously lost. Because of the complicated scattering
behavior of Er, standard coupled-channel methods to assign the leading partial-wave character of the FRs are
currently not available [42]. However, the width of theFRs can give indications on the strength of the coupling
between open and closed channels [43].
Among the observed inter-spin FRs, the one at about
0.68 G stands out from the forest of narrow FRs, as it
is almost two orders of magnitude broader. This feature makes this FR a promising candidate for Fermi-gas
experiments in the strongly interacting regime. A thorough understanding of the collisional properties and interactions in its proximity becomes thus crucial to assert
the usefulness of this FR for such type of investigations.
To reach a precise knowledge of the inter-spin scattering
length, a↓↑ , we conduct modulation spectroscopy experiments in the 3D lattice to map the B-to-a↓↑ conversion. This technique, which has been very powerful to
precisely determine the scattering length in dipolar BoseEinstein condensates [9, 33], allows to measure the onsite
interaction energy U↓↑ . We prepare a spin-1/2 Fermi gas
in a one-atom-per-lattice-site configuration (stage (a3);Fig. 1). When the frequency of the lattice-depth modulation is resonant with U↓↑ /h, the sample undergoes increased losses; for details see Ref. [32]. In our system,U↓↑ = Uc + Udd is the sum of the inter-spin contact interaction, Uc , and the DDI, Udd . Thanks to the precise
knowledge of Udd for our lattice configuration [33] weare able to directly extract a↓↑ ∝ Uc = U↓↑ − Udd . Importantly, the angle dependence of the DDI makes our
method sensitive to the sign of a↓↑ : By changing the orientation of the dipoles, we can tune Udd from negative
to positive values, which increases (decreases) the totalenergy |U↓↑ | for positive (negative) a↓↑ [32, 33].

Figure 4(a) shows a↓↑ for various B values aroundthe comparatively broad inter-spin FR of Fig. 3(c). The
measured a↓↑ follows the typical behavior at an isolated FR, demonstrating the tunability of a↓↑ from positive to negative values. To obtain a precise B-toa↓↑ conversion, we fit 
the standard function a↓↑ (B) =
0

∆
∆
− B−B
to the data [43]. From theabg 1 − B−B000fit, we extract the background scattering length abg =91(8) a0 , the position of the comparatively broad FR
B0 = 687(1) mG and its width ∆ = 58(6) mG. Notethat our fitting function also accounts for a nearby inter-4atom 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 with a 1 mG step size for three different population imbalances in an ODT(cartoons): atoms in | ↓ i (squares) and | ↑ i (circles) for δ = 1 (a), −0.6 (b), and 0 (c) as a function of B. From the threedatasets one can determine the spin nature of the FRs as indicated by the blue (| ↓ i), orange (| ↑ i), and green (| ↓ i–| ↑ i) shaded
regions.

spin FR at B00 = 480 mG (out of range of Fig. 3 and
4) of width ∆0 = 29(4) mG, whereas narrower interspin FRs are neglected. Based on the extracted values, we can give an estimate of the length parameterR∗ = h̄2 /(mEr ∆abg δµ), which characterizes the strengthof a FR [43]. Here, mEr is the mass of 167 Er. For the considered FR the differential magnetic moment between the
open and closed channel, δµ, is not known. For bosonic
Er, we have previously measured a typical δµ ≈ 3 µB via
molecular spectroscopy of a few molecular states [42]. By
using this bosonic value we estimate R∗ ≈ 1000a0 . Universal scattering behavior is expected for a↓↑ > R∗ [44].
In experiments with strongly interacting alkali Fermi
gases of 40 K or 6 Li, the large collisional stability in
two-component mixtures has been essential for observing e. g. fermionic superfluidity and molecular BECs [45].
As direct consequence of the Pauli principle, three-bodyrecombination occurs primarily on the repulsive (BEC)side of a broad s-wave FR, where a weakly bound molecular level exists [46], whereas, on the attractive (BCS)side, large scattering lengths coexist with a remarkable
collisional stability [47–50]. Such an asymmetry in the
scattering behavior is identified as an essential attributeof BEC-to-BCS physics.
To investigate this aspect, we prepare an equally populated spin mixture (δ = 0) in an ODT, following thescheme of Fig.1(a), and probe the time evolution of thespin population as a function of the holding time in thetrap (stage (a4); Fig. 1) for various B across the FR.
Exemplary decay curves are shown in Fig. 4(b-c). On
the BEC side at a↓↑ = 880(140) a0 , we observe a fastatomic decay of both | ↑ i and | ↓ i atoms (Fig. 4(b)).
A simple exponential fit to the data gives lifetimes ofτ1/e ≈ 150 ms. In striking contrast, on the BCS side ata↓↑ = −1500(500) a0 (Fig. 4(d)), the spin mixture showsa remarkable collisional stability with lifetimes exceedingτ1/e = 1200 ms (Fig. 4(c)).

To get deeper insights, we systematically study the
initial decay rate, Ṅ /N0 , as a function of B. We determine the rates by using a linear fit to the data for
the initial time evolution. Figure 4(d) summarizes our
results, plotted in terms of the dimensionless coupling
constant 1/(kF a↓↑ ) with kF being the Fermi wave vector [32]. We clearly observe that the maximum in the loss
rate is shifted to the repulsive side of the FR pole andthat the Fermi mixture exhibits a remarkable stability inthe unitary and strongly attractive regime. The observed
shape of the loss rate reveals a remarkable similarity withmeasurements conducted with alkali fermions. It is interesting that the absolute values of the loss rates in 167 Er
are not only qualitatively similar but also quantitatively
very close to the ones measured in 40 K [49], with e. g. on
the BCS side lifetimes of about one second in both cases.
The existence of a comparatively broad FR, our acquired precise knowledge of the scattering length, andthe collisional stability in the strongly interacting regime
make fermionic Er atoms a new promising system foraccessing BEC-to-BCS crossover physics. Our mixture
adds DDI to the alkali scenario, paving the way forstudying Cooper pairs with large magnetic moments andstrongly dipolar molecular BECs [22].
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). LC
is supported within a Marie Curie Project (DipPhase,no. 706809) of the EU H2020.
†

Current address: Physikalisches Institut, Universität
Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg,
Germany* Correspondence and requests for materials should be
addressed to F.F. (email: francesca.ferlaino@uibk.ac.at).

5

scattering length a↓↑ (a0)

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)a0a↓↑ = 880(140)a0

6
4
2
0

(c)

(b)

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 1

.

10 0

10-1
15

10

5
0
coupling constant (1/(kFa↓↑))-5

FIG. 4. Inter-spin scattering length and collisional behaviorof the strongly interacting Fermi mixture (a) a↓↑ extracted
from modulation spectroscopy in the lattice; see text. Error
bars are smaller than the size of the data points. Verticaldashed lines indicate the position of narrow inter-spin FRs as
identified in Fig. 3(c). The solid line is a fit to the data (seetext) with the fit statistical uncertainty indicated as shaded
region. (b-c) Lifetimes of a spin mixture of | ↓ i (squares)
and | ↑ i (circles) with δ = 0 in an ODT for large positivea↓↑ at 680 mG (b) and for large negative a↓↑ at 690 mG. (d)Initial decay rate Ṅ /N0 of the normalized atom numbers as
a function of 1/(kF a↓↑ ) in the vicinity of the FR.

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SUPPLEMENTAL MATERIAL
Spin-polarized degenerate Fermi gases

Our experimental protocol for the preparation of
deeply degenerate Fermi gases (dFgs) of 167 Er follows theone described in Ref. [5]. The experiment starts with anarrow-line magneto-optical trap operated at 583 nm toprepare spin-polarized 167 Er atoms with N = 1.2 × 107
atoms in the lowest hyperfine sublevel |F = 19/2, mF =–19/2i, where F is the total spin quantum number andmF is its projection along the quantization axis. The
atoms are then transferred to a horizontal optical dipole
trap (ODT) operated at 1064 nm (ODT1064 ). The aspectratio AR = w⊥ /wz of this trap can be tuned from 1.6 to15 via a time-averaging potential technique [9] and allows to reach a good spatial mode overlap between theatomic cloud and the optical potential. Sub-sequentiallythe atomic cloud is compressed by reducing the AR andtransferred to a counter propagating tight ODT operated
at 1570 nm (ODT1570 ) with a beam waist of about 15 µm.
At this stage we typically have 1×106 atoms. During theevaporation procedure the atoms are further confined bya vertical ODT at 1570 nm with at waist of about 32 µm.
Following our previous work [5], we perform evaporative cooling based on elastic dipolar scattering among
identical fermions. Dipolar cooling has been proven to bevery efficient to produce samples in the deeply quantum
degenerate regime [5, 20]. At the end of the evaporation we typically obtain a spin-polarized dFg with up to
6 × 104 atoms and temperatures of T ≤ 0.15 TF , with TF
being the Fermi temperature. At this stage the Fermi
energy is EF = kB TF = kB × 630 nK = h × 13 kHz.
The final trap frequencies in ODT1570 are (ν⊥ , νk , νz ) =(286(3), 85(1), 255(3)) Hz with k (⊥) corresponding to theaxis along (perpendicular to) the horizontal beam and z
indicating the axis of gravity. The magnetic field has avalue of B = 0.6 G and is oriented along z, which sets
the quantization axis of the atomic dipoles.

Preparation for lattice loading

For deeply dFgs the atoms fill the Fermi sea up to the
Fermi energy EF = hν̄(6N )1/3 , where h is the Planck
constant, ν̄ the geometric mean of the trap frequencies and N the atom number. Hence, the Fermi energy gives the relevant energy scale of the system. The
number of populated bands, when the atoms are loaded
to an optical lattice, thus crucially depends on the ini-tial Fermi energy. Here, in first approximation, theFermi energy can be compared to the lattice recoil energy Erec = h2 /(2mλ2 ), with m being the mass of 167 Er
and λ the lattice wavelength. In particular, during theinitial increase of the lattice potential higher bands become populated if EF > Erec [37].
To minimize the occupation of higher bands due tothe loading procedure we reduce the Fermi energy of
our sample. Here, we transfer the atoms back to acrossed ODT1064 within 510 ms, which allows for a convenient control on ν̄ via the dynamically adjustable
AR. We optimize the ODT parameters by lowering ν̄and N while keeping a low temperature of the sample. Best conditions for subsequent lattice loading arereached for (ν⊥ , νk , νz ) = (63(1), 36(2), 137(1)) Hz andN = 2.4 × 104 atoms corresponding to a 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

After preparation, we adiabatically load the spinpolarized dFg into a three-dimensional (3D) optical lattice, created by two retro-reflected 532 nm laser beams
along the x- and y-axis and one retro-reflected 1064 nm
vertical laser beam along the z axis; see Fig. S1. The
lattice spacing are dx,y = 266 nm along the horizontal xy-plane and dz = 532 nm along the vertical zaxis [33]. We increase the lattice-beam intensities exponentially in 150 ms to the final value. With the available
power, we reach maximum lattice depths of (sx , sy , sz ) =(25, 25, 120), where si with i ∈ {x, y, z} is given in unitsof the respective recoil energies, ER;x,y = h × 4.2 kHz
and ER;z = h × 1.05 kHz. Typical lattice depths usedin the experiment are (sx , sy , sz ) = (20, 20, 80) corresponding to band gaps of h × 32.8 kHz along x and y andh × 17.7 kHz along z. Subsequentially, the ODT beams
are switched off in 10 ms and we additionally hold theatoms for 500 ms. This time is sufficient to remove residual atoms that have been pushed to higher bands of the
optical lattice by the Fermi pressure. We note that when
the atoms are loaded directly from ODT1570 we find up
to 25 % of higher bands population, which in this caseget strongly depopulated within 500 ms. A remaining
fraction of 5 % in the higher band of the vertical lattice
remains also for our most careful loading procedure; seemain manuscript.

Zeeman energy for fermionic Er

Fermionic Er exhibits a hyperfine structure with a nuclear spin quantum number I = 7/2 that couples withthe total electronic angular momentum J to the total
angular momentum F = J + I. In the lowest hyper-

dz

7

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.

fine manifold (F = 19/2) there are 2F + 1 = 20 magnetic substates mF resulting from the projection of F
on the quantization axis. The lowest magnetic state|F = 19/2; mF = –19/2i exhibits a magnetic momentof µ = mF gF µB = −6.982804µB where gF = 0.735032 is
the Landé g-factor and µB is the Bohr magneton [52].
The hyperfine energies can be calculated via exact diagonalization of the atomic Hamiltonian [53]. In Fig. S2
we plot the Zeeman substates of the lowest hyperfine
manifold as a function of the magnetic field. At firstglance it appears that the energies evolve linearly. However, at larger magnetic field due to the Paschen-Back effect J and I start to decouple. This leads to a quadraticterm of the Zeeman energy for states σ = mF with
Eσ = qqu (σ 2 − F 2 ) and qqu ∝ B 2 . The deviation fromthe linear Zeeman energy becomes evident in the differential splitting ∆EZ = (Eσ −Eσ+1 )−(Eσ+1 −Eσ+2 ) with
Eσ = qli σ + qqu (σ 2 − F 2 ) and σ ∈ [−19/2, . . . , 19/2] asshown in the inset of Fig. S2. The linear Zeeman effect
qli σ ∝ B cancels out when ∆EZ is evaluated.
Preparation of a spin mixture in the lattice

For deterministic spin preparation of the two lowest spin states we use typically magnetic field values of
B = 40.51 G where ∆EZ = 42.6 kHz, which is larger thanthe magnetic field noise at this field of ≈ 20 kHz. This
resembles an effective two level spin-1/2 system for spinspin couplings and enables a deterministic spin preparation of the lowest two spin states via radio-frequency(rf)-techniques. To couple the two hyperfine sub-states
we apply a rf-sweep by chirping the rf-frequency continuously from a value of (ν ? + 30 kHz) to (ν ? − 30 kHz)
within about 10 ms, where h × ν ? corresponds to the difference in Zeeman energy of the two lowest spin states(∆1; Fig. S2). We can prepare a well-reproducible mixture of | ↓ i and | ↑ i without populating the next higher

8of the magnetic field value. In our experiment this isgiven below 5 G where the noise is found to be ≈ 1 mG.
The second requirement is matched for a sufficiently
large magnetic field, where the quadratic Zeeman effectis strong enough to not be canceled by quadratic lightshifts [55]. Best conditions are found at a magnetic fieldvalue of 3.99 G, which is used for the lifetime measurements of Fig. 2. At this field the measured onsite interaction U = h × 2.43(2) kHz exceeds by far the relevant
tunneling rates Jx,y = h × 10.5 Hz and no spin dynamics
are observed.

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

State-resolved Feshbach spectroscopy

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) isof most relevance. Higher spin states are visualized by greylines. The linear Zeeman effect dominates the energy evolution while in the differential splitting ∆EZ = ∆1 − ∆2 (inset)
the quadratic Zeeman effect is evident.

spin state. The population imbalance δ between the two
spin states can be freely controlled by varying the powerof the rf signal. In particular, also almost all the atoms
can be transferred to | ↑ i reaching up to δ = −0.94 (seeFig. 1(c) in main manuscript). We note that, while our
preparation technique in the lattice initially leads to acoherent superposition of the two spin states, additional
measurements suggest a fast decoherence, leading to a
projection of pure states on the individual lattice sites
for experimental relevant time scales. In particular, weobserve that coherently driven Rabi oscillations between
the two spin states quickly damp within a few ms.

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 energy U to the tunneling rate J.
The magnetic field is also well away from Feshbach resonances (FRs). (ii) The Zeeman energies do not havean equidistant spacing. In such a regime magnetizationconserving spin exchange would lead to a change of thespin composition [15, 16, 21, 54].
Due to the high density of FRs the first requirement
can be more conveniently achieved with low technicalmagnetic field noise, which ensures a higher stability

To identify the magnetic field regions where promisinginter-spin 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 mere magnetic-fieldsweeps (e. g. losses during the sweeps).
For this measurement set, we first perform a Feshbachscan in a spin polarized gas in ODT1570 (Fig. S3, upper
panel). We jump to the final magnetic field and holdfor thold = 70 ms before time-of-flight 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. [26], we observe a high density of lossfeatures, which correspond to single-component (| ↓ i)
FRs of high partial-wave character. We then 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 | ↓ i homo-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, which mainly correspond to single-component | ↑ i FRs. In a last scan weobserve the loss features for a spin mixture prepared at
0.58 G in the same trap as for the pure | ↓ i measurement with thold = 50 ms (Fig. S3, lower panel). Here, theinitial temperature is slightly increased due to the spin
mixing to T = 0.24(1) TF . The individual homo-spin FRs
are still visible while we also find new inter-spin | ↓ 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 assign-932

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

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. The shading around the data points indicates statistical uncertainties, which are often smaller than thedata points. 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 inter-spin FR.

ment 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 inter-spin FR at about 700 mG,
which remains rather isolated from other homo-spin FRs;
see green shading in Fig. S3.
In a second set of measurements, we focus on themagnetic-field region around 700 mG and perform highresolution Feshbach spectroscopy, taking advantage ofa lattice-preparation scheme, as described in the maintext. The lattice-protection technique is very powerfulin removing technical broadening and artificial asymmetry of the loss peaks, as clearly appears from acomparison between the atom-number traces recordedwith ODT-preparation (Fig. S3) and lattice-preparation
schemes (Fig. 3). We note that the observed atomic losses
can be mainly attributed to resonant three-body recombination collisions in the short-range 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[43]. However, we do not expect this process to be enhanced at resonance.
We perform the measurements for Fig. 3 as follows.
We prepare a spin mixture in the lattice at high B
as described above and sub-sequentially ramp the field
to the desired value within 10 ms. After letting the B
stabilize for about 100 ms, the dipole trap beams areramped up within 10 ms and we unload the atoms fromthe lattice back into the ODT1064 within 150 ms. At
this stage, the sample contains N ≈ 1.6 × 104 atoms at
T ≈ 0.3 TF and the trap frequencies are (ν⊥ , νk , νz ) =(111.6(2), 35(1), 169.4(6)) Hz. We record the spin population after a holding time of 500 ms.

Modulation spectroscopy with a fermionic spin
mixture in the lattice

To measure the scattering length between two spinstates of 167 Er we rely on a method, which we havealready successfully implemented with 168 Er [33] and166
Er [9]. It is based on the measurement of the onsite
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 excitations of neighboring atoms
in different spin states 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. A convenient method to measure double occupancies uses thecoupling to a third spin state. Close to a molecular stateof the original doublon components, the third spin statefeatures a smaller inter-spin onsite energy and can thus
be used to detect an initial double occupancy [56]. As in
our system this method remains to be explored we detectdoublons via an increased atom loss once doublons are
created. We speculate that onsite dipolar driven relaxation is responsible for the observed loss [40]. Typically,
we modulate the lattice depth for 1 s with a peak-to-peakamplitude of 30%. Maximum loss occurs when the modulation frequency νmod reaches the resonance condition
νres (see Fig. S4). νres is directly related to the onsite
energy via U↓↑ = h × νres . Following our previous work
the onsite energy U↓↑ consists out of two contributions:the contact interaction Uc

Uc =4πh̄a↓↑
mEr

Z

4

dr |φi (r)| ,

6.0

1.0

5.5

0.9

ν res (kHz)U↓↑= hνresnormalized total atom number

10

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).

5.0

4.5

4.0
40

60

80

100

120

sz (ER,z)

and the DDI Udd
FIG. S5. Modulation resonance νres as a function of theZZ2µ0 µ↓ µ↑2 1 − 3 cos θr−r0
verticallattice power sz for sx,y = 20 at B = 650 mG. The00 2
dr dr |φi (r)||φi (r )| .
Udd =solid line shows a fit with our theory to extract the scattering
4π|r − r0 |3
Here, φi (r) denotes the Wannier function on site i,|r − r0 | is the interatomic distance and θr−r0 corresponds
to the angle between the polarization axis of the twodipoles with respect to their interparticle axis. The contact part depends on the inter-spin scattering length a↓↑ ,the reduced Plank constant h̄, and the mass mEr of Er,
while the DDI part is proportional to the vacuum permeability µ0 and the magnetic moments of the two spinstates µ↓ and µ↑ . The contribution of nearest-neighborinteractions are minor and therefore neglected.
Both, the strength but also the sign of Udd strongly
depend on the dipole orientation and the aspect ratio(AR) of the onsite Wannier function. As specified inour earlier work [33] the AR is defined by the ratio of
the onsite harmonic oscillator lengths AR = lz /lx,y with
1/4
li = di /(πsi ) for i ∈ {x, y, z}. For our typical lattice
parameters we find AR > 1 and hence Udd is negative(positive) for a dipole orientation out of (in) the xy-plane.
In the experiment we can use this fact to determine thesign of the scattering length a↓↑ . First, we perform a
modulation spectroscopy with a dipole orientation alongz. The extracted resonance frequency deduces the energy gap |U↓↑ | of the particle-hole excitation but does
not give any information on the sign of the total onsite
energy. We repeat the measurement at the same magnetic field with the dipoles oriented inside the xy-plane.
The rotation of the dipoles leads to a more positive total
onsite energy U↓↑ and thus to a shift to larger (smaller)
values of |U↓↑ | corresponds to a positive (negative) signof U↓↑ . With this method we determine the sign of thescattering length a↓↑ for 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 along

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.

z, to 650 mG and vary the depth of the z lattice. We repeat the modulation spectroscopy for different values ofsz and extract νres for each measurement. A comparison
to our theoretical model with a↓↑ being the only free parameter reveals a nice agreement, confirming the validityof our modulation spectroscopy technique. Here, the fit
gives a value for a↓↑ of 225(2) a0 .
The presented data in Fig. 4(a) for a given magnetic
field B shows the mean of all experimental data sets,taken with different lattice parameters. Table 1 summarizes all experimental lattice parameters and the expected contribution of Uc (for a↓↑ = 100 a0 ) and Uddfrom our theoretical model. From a given measurementof U↓↑ the inter-spin scattering length can be evaluatedby a↓↑ /a0 = (U↓↑ − Udd ) × 100/Uc .
Loss spectroscopy in the ODT at the inter-spin FR

For the measurements of the collisional properties ofthe fermionic spin mixture in the vicinity of the comparatively broad inter-spin FR (see Fig. 4(b-d)), we apply the following experimental procedure. We preparea spin mixture with δ = 0 in the deep 3D lattice andsweep to a magnetic field of B = 3.99 G. We thenjump with the magnetic field to the final value andlet it stabilize for 10 ms. We ramp up the ODT1064
beams in 10 ms and melt the lattice in 20 ms as to avoid
losses to happen already during the ramp-down procedure. The final trap frequencies are (ν⊥ , νk , νz ) =11(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

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

Udd /h (Hz)
−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 evaluate the onsite energy contributions Uc and Udd for an interspin scattering length of a↓↑ = 100 a0 . Here, the dipoles areoriented along z. This values are used to extract the interspin scattering length from the measured total onsite energyU↓↑ .

(111.6(2), 35(1), 169.4(6)) Hz. For this trap we typicallyrecord EF ≈ √kB × 150 nK corresponding to a Fermi wave
vector kF ≈ 2mEr EF /h̄ = 1 × 107 m−1 . We then store
the two-component mixture in the ODT1064 for a variable holding time and finally record the spin populations
with Stern-Gerlach imaging.
We record the atom decay for various magnetic fields
B across the FR. For each B, we extract an initial decayrate Ṅ /N0 by fitting a linear-decay function to the data,normalized to the initial atom number N0 . As fitting
range, we include all data for which the atom number
stays above a threshold of 75% of N0 . We checked thatthe extracted values of Ṅ /N0 do not change significantly
when varying the threshold between 65−85%. An analysis of the full data using exponential fits also yields similar
results.

Chapter6

Conclusion and outlook

6.1. Conclusion
This thesis is dedicated to the study of dipolar phenomena in the cold and ultracold regime.
It reports on effects of the long range and anisotropic dipole-dipole interaction (DDI) on the
few- and many-body behavior of an atomic sample. To this aim, we have utilized erbium
atoms, which combine the possibility for efficient laser cooling and trapping with a strongly
magnetic character. Our success of bringing not only bosonic but also fermionic erbium
atoms into the quantum degenerate regime has set the foundation for the observation of
dipolar phenomena with different governing quantum statistics. Our studies have also faced
the orbital anisotropy of erbium, an important feature of magnetic lanthanides. With theimplementation of a three-dimensional optical lattice, we have transformed the experimental
apparatus into a quantum simulator of condensed matter systems with long-range interactions. Being one among only a few dipolar experiments around the world has allowed us toconstantly break fresh experimental grounds. Our efforts have rewarded us with the attainment of fascinating physical systems, such as dipolar Feshbach molecules, extended Hubbard
models, or strongly interacting dipolar Fermi gases, and enabled observations of intriguing
physical phenomena, such as universal and anisotropic elastic few-body scattering, chaotic
inelastic scattering, or the altering of the quantum many-body phase transition from thesuperfluid to the insulating phase by the DDI.
In order to explore unknown territories, we have constantly upgraded the experimental apparatus as to enrich the possible research directions. A particular interesting prospect for our
experiment is given by our recent successful preparation of extended Fermi Hubbard models
with spin-mixtures of dipolar fermions, see Sec. 5.6. In the following section we will briefly
discuss possible experiments along this lines that are on the way or that can be undertakenwith our system.

1306.2. FUTURE INVESTIGATIONS ALONG THE EXTENDED FERMI HUBBARD
MODEL
131

6.2. Future investigations along the extended Fermi Hubbard
model
Within the course of this thesis, we have successfully prepared deeply degenerate Fermi
gases of 167 Er and have adiabatically transferred them into the lowest band of a threedimensional optical lattice. A major advantage of our fermionic system arises from quantum
statistics. For a spin-polarized gas, double occupancies on a single lattice site are forbidden,
resulting in a single-component band insulator with high filling fraction and long lifetime.
This clean preparation method together with the long-range dipolar character of the particles
provides a very clean test bed for the study of quantum phenomena that are solely driven
by offsite interactions. In addition, we can deterministically prepare individual spin statesof 167 Er, which features the unprecedented large number of 20 spin states in the lowest
hyperfine manifold, see Sec. 2.2.4. Our system gives exciting prospects for the studies oftransport dynamics, lattice spin models, or resonant demagnetization dynamics as we will
briefly discuss below.

Transport phenomena with spin-polarized dipolar fermions
In solid state systems, transport properties are closely linked to the underlying physicalphenomena, such as high-temperature superconductivity, topological insulators, or disorderphenomena. Well-controlled ultracold atomic systems in optical lattices can serve as quantum
simulators to investigate such phenomena via the study of transport dynamics. For the caseof long-range interactions, the transport properties of the many-body state can be markedly
influenced by the nearest-neighbor interaction (NNI) among particles. Such a system canbe conveniently prepared with spin-polarized dipolar fermions in a deep optical lattice. The
formed band insulator with high filling fraction gives ideal starting conditions for the studyof transport properties via expansion measurements, see e. g. Ref. [Sch12b]. In order toexpand, the atomic sample needs to locally break bonds of NNI between particles. This
process is suppressed by simple energy arguments and hence the system is expected to show
reduced expansion dynamics [San17]. As a result, a so-called clustered state is formed where
neighboring particles tend to stick together. Interestingly, the reduction of expansion is
expected to depend on the strength of the NNI, which in the case of dipolar interactions canbe conveniently controlled via the angle of the dipoles with respect to the interparticle axis,
see Eq. 3.2.

Extended spinor Fermi-Hubbard models
While already spin-polarized gases give exciting prospects for studies of peculiar quantum
effects, the story can become even more intriguing when the spin degree of freedom is added
to the system. Indeed, the availability of higher spin states allows to access lattice spinmodels, as represented by the extended spinor Fermi-Hubbard model.
For the Hamiltonian of Eq. 5.18 we have included all terms that conserve the total magneti-

6.2. FUTURE INVESTIGATIONS ALONG THE EXTENDED FERMI HUBBARD
MODEL
132
P z
z =
zation, i. e. where Ftot
i Si = const. In reality, the Hamiltonian can also feature terms
z , i. e. that lead to a demagnetization of the collective spin state.
that do not conserve Ftot
The offsite part of the Hamiltonian of this system reads as [Aue94, Zhu17]
Hoffsite = Hcons + Hnon−cons

6.1

where Hcons and Hnon−cons include the spin-conserving and spin-non-conserving terms, respectively. The effects of this terms will be discussed in the following. For the sake ofsimplicity, we will restrict the discussion to a one-dimensional systems, i. e. to a lattice chainof dipoles.

Spin conserving terms
The spin conserving offsite interaction terms read as


X ij
1 + −
2
z z
− +
′
Hcons =
V0 (1 − 3 cos θr−r ) Fσi Fσj − (Fσi Fσj + Fσi Fσj )
4

6.2

i<j,σ

with the general dipolar coupling strength


Z
Z
1
µ0 (gF µB )2
ij
′ ∗
∗ ′
wi (r)wj (r′ ).
V0 = dr dr wi (r)wj (r )4π|r − r′ |3

6.3

As already discussed in Sec. 5.2.3, Fσzi Fσzj represents the NNI while the term Fσ+i Fσ−j gives
rise to spin-conserving flip-flop dynamics among neighboring particles. This Hamiltonian has
been studied in pioneering works by the groups of Jun Ye in Boulder and Bruno LaburtheTolra in Paris by exploring off-site driven spin exchange with spin-1/2 KRb molecules [Yan13]
and with spin-3 bosonic chromium atoms [dP13b, Lep18]. With our experiment we have
now reached the point where we can investigate such exotic spin models in the context of an
unrivaled large spin-19/2 system.
The dynamics are expected to depend on various parameters. First of all, for a flip-flopprocess to take place, energy has to be conserved. Hence, the dynamics will strongly dependon the energy difference between adjacent spin states. In our system, this differential energy
can conveniently be controlled via spin-dependent quadratic Zeeman and quadratic lightshifts, see Sec. 5.2.3. With this tool at hand, we can switch on and off flip-flop dynamics atwill. Further, the deterministic preparation of pure spin states will allow us to investigate indetail the spin-dynamics time scales for different initial spin states. Depending on the spin
number, slower or faster time scales are expected, as follows from Eq. 5.17. Finally, the timescales are expected to be controllable via the dipole orientation θr−r′ . For dipoles aligned
along to the lattice chain, θr−r′ = 0◦ , the dynamics should be twice as fast compared toa perpendicular orientation, θr−r′ = 90◦ , and spin-exchange should be absent at the magic
angle θr−r′ = 54.74◦ .

6.2. FUTURE INVESTIGATIONS ALONG THE EXTENDED FERMI HUBBARD
MODEL
133
Spin non-conserving terms
The spin non-conserving offsite interaction terms read ashi
X ij  3
Hnon−cons = −V0
sin2 θr−r′ e−2iφ Fσ+i Fσ+j + h.c.
4
i<j,σ
h
i
3
−iφ
z +
+ z
+ sin 2θr−r′ e (Fσi Fσj + Fσi Fσj ) + h.c. .
4

6.4

Here, θr−r′ and φ denote the polar and azimuthal angles, respectively, and V0ij is the general dipolar coupling strength, see Eq. 6.3. The first term of the Hamiltonian changes themagnetization of two atoms by ±2 and the terms of the second line result in a change by±1. Magnetization changing terms are of particular interest as they allow to couple thespin and the orbital degree of freedom, assosciated to the famous Einstein-de-Haas effect,see e. g. Ref. [Lah09] for a review. In the case of lattice-confined particles, spin-orbit coupling can lead to excitations to higher lattice bands as beautifully demonstrated with onsite
interacting |S = 3, ms = 3⟩ bosonic chromium atoms [dP13a]. However, offsite-driven spin
demagnetization remains to be observed.
An offsite-driven demagnetization can occur when a resonant state is accessible. This scenario can emerge when the change of the NNI energy (Fσzi Fσzj ), resulting from the spin flip,
matches the respective Zeeman splitting. In our system this process requires very low magnetic field values well below 1 mG. The preparation of such a low magnetic field amplitude
demands a very precise magnetic field control. In our experiment we have already successfully implemented an active magnetic field stabilization, see Appendix C. However, to reachthe necessary magnetic field stability, further improvements have to be employed.
Interestingly, the strength of the terms in Eq. 6.4 show a peculiar angle dependence. In
particular, the sign of the terms can be changed by adjusting the dipole orientation and fora parallel dipole orientation, i. e. θr−r′ = 0◦ , the demagnetization terms would vanish.

Appendix

A

Additional publications

A.1. Publication: Quantum-Fluctuation-Driven Crossover from a
Dilute Bose-Einstein Condensate to a Macrodroplet in a
Dipolar Quantum Fluid†
Phys. Rev. X 6, 041039 (2016)
submitted 22 Jul 2016; published 22 Nov 2016
DOI 10.1103/PhysRevX.6.041039

Lauriane Chomaz,1,2 Simon Baier,1 Daniel Petter,1 Manfred J. Mark,1,2 Falk Wächtler,3 Luis
Santos,3 F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover,Germany†

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

134

PHYSICAL REVIEW X 6, 041039 (2016)

Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate
to a Macrodroplet in a Dipolar Quantum Fluid
L. Chomaz,1 S. Baier,1 D. Petter,1 M. J. Mark,1,2 F. Wächtler,3 L. Santos,3 and F. Ferlaino1,2,*
1

Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3Institut für Theoretische Physik, Leibniz Universität Hannover,Appelstrasse 2, 30167 Hannover, Germany(Received 22 July 2016; revised manuscript received 15 September 2016; published 22 November 2016)
In a joint experimental and theoretical effort, we report on the formation of a macrodroplet state in an
ultracold bosonic gas of erbium atoms with strong dipolar interactions. By precise tuning of the s-wave
scattering length below the so-called dipolar length, we observe a smooth crossover of the ground statefrom a dilute Bose-Einstein condensate to a dense macrodroplet state of more than 2 × 104 atoms. Basedon the study of collective excitations and loss features, we prove that quantum fluctuations stabilize theultracold gas far beyond the instability threshold imposed by mean-field interactions. Finally, we perform
expansion measurements, showing that although self-bound solutions are prevented by losses, the interplay
between quantum stabilization and losses results in a minimal time-of-flight expansion velocity at a finite
scattering length.
DOI: 10.1103/PhysRevX.6.041039

Subject Areas: Atomic and Molecular Physics,
Quantum Physics

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

Francesca.Ferlaino@uibk.ac.at

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

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

041039-1

Published by the American Physical Society

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)

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

160

2.0
εdd

140

1.5

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

0.4

0.8

0.6

80
⊥ -imaging

60

z

40

x
y

20

0

0.5

1.0

B

-

1.5
B (G)

2.0

2.5

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

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

041039-2

QUANTUM-FLUCTUATION-DRIVEN CROSSOVER FROM A …

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

0
30

40

80
-80 (b)ncol (µm 2 )

25

-40

z (µm)

(d)

35

-40

20

0
40

15

80
-80 (c)10-40

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

PHYS. REV. X 6, 041039 (2016)

0

5

40
80
-50

0

x (µm)

50

0
-100

-50

0

50

100

z (µm)

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

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

L. CHOMAZ et al.

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

1.32

dd

0.94 0.82 0.73 0.66 1.32

1.10
1.4

0.94 0.82 0.73 0.66

1.10

as= 80a0

1.2

2.5

R //

1.0
0.8
0

50

100

150

/

//

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

2.0

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

(b)
50

(c)
60

70

as (a0)80

90 100 50

60

70

80

1.5
90 100as (a0)

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

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

ð1Þ

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

PHYS. REV. X 6, 041039 (2016)

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

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)

εdd
1.65

1.10

0.94

t hold (ms)
0.82

0.73

0.66

10

5

10

100

1000

(a)

(c)

Ncore

Atom number

105

1.32

104

104

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

Atom number

10

5

(d)

20

10
8
6

104

4
th = 4 ms
th = 16 ms

40

50

60

70

as (a0)

80

90

100

2
40

50

60

70

80

as (a0)

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

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

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

QUANTUM-FLUCTUATION-DRIVEN CROSSOVER FROM A …

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

30

10

12

14

16

18

tTOF (ms)
20 22

(a)

24

26

28

30

32

σx (μm)

93a0
64a0

20

55.5a0
10

εdd
1.65

1.32

1.10

50

60

0.94

0.82

0.73

0.66

70

80

90

100

2
expansion velocities (μm/ms)

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

1

(b)
40

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

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

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

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

L. CHOMAZ et al.

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

work

by

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Gorceix, and B. Laburthe-Tolra, Collective Excitations of a
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in Dipolar Bose Gases, Phys. Rev. A 84, 041604(2011).
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[42] We note that the use of a generalized GPE, valid in the TF
regime, can be extended beyond this regime as long as the
LHY correction is dominated by short-wavelength excitations, whose momentum q fulfills qR ≫ 1, where R is thesize of the condensate along the direction of q. As discussed
in detail in Refs. [20,22], this is the case for large droplet
and εdd close to 1. Our experiments, performed in cigarshaped traps with relatively large BECs, offer privileged
conditions to be compared to generalized GPE predictions(in particular, enhanced compared to Refs. [15,16]). Indeed,
we quantitatively estimate the validity of this approximationin our study
by introducing ﬃ an anisotropic cutoff,pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qc ðθÞ ¼ qy cos2 ðθÞ þ K 2 sin2 ðθÞ, with K the aspect ratioof the BEC, assuming here an isotropic geometry around thedipole axis y, and θ denoting the angle ðy; qÞ [22]. For
typical experimental parameters, e.g., as ¼ 58a0 , N ¼
5 × 104 , we find an axial radius Ry ¼ 13 μm and
K ¼ 7.5. Introducing a typical cutoff value qy Ry ¼ 25,
the cutoff LHY correction [i.e., taking momentaq > qc ðθÞ] equals ∼0.8 of the value calculated in homogeneous space, demonstrating that short-wavelength excitations indeed have a dominant contribution and validating
our approach. We note that, when decreasing as , longwavelength excitations and unstable modes come to play amore significant role, and for this reason we consider our
simulation reliable down to as ≥ 50a0 .
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933 (1998).
[44] Yu Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Effect ofBose Condensation on Inelastic Processes in Gases, Pis’ma
Zh. Eksp. Teor. Fiz. 42, 169 (1985).
[45] We observe a 14(9)% increase of vth over the as range whendecreasing as that we attribute to fit artifacts. First, thedouble Gaussian fit does not properly account either for thecoherent or for the thermal parts of the gas which may resultin cross talk in the fit of these components. Second, flaws in
the partition in two components may occur, especially for

041039-9

L. CHOMAZ et al.

PHYS. REV. X 6, 041039 (2016)

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

041039-10

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

Appendix: Supplementary material
1.

Bose-Einstein condensation of 166 Er

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

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

2.

Experimental setup and axes

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

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

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

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

(A.1)

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

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

4.

Ramps in scattering length

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

(A.3)

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

3
5.

Determination of the three-body recombination
rate coefficient

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

(a)

65

as ( a 0 )

60
55
50

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

45

Lth
3 =

√

27 γ0



kB T
2πmν̄



(A.5)

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

(A.4)

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

Time-of-Flight measurements

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

m /s)

1,5
6

1,0

L (10

- 40

t r / a s d a s /dt

2,0

0,0

3

0,5

0, 00

0,25

0,50
t/t r

0,75

0,1

1, 00

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

0,01

0

10

20

30

40

50

60

70

80

a s (a 0)

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

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

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

8.

9.

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

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

Extracting the frequency of the collective modes

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

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

residu7.

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

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

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

0,984
0,982
40

50

60

70

80

90

100

as(a0)

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

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

11.

10.

Gaussian Ansatz including L3

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

ε dd

2000

1,32

1,10

0,94

0,82

50

60
as (a 0)7080

t slow (ms)

1500
1000
500
0

t fast (ms)

100

10

1
1
0,75
0,5

p

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

0.25

Describing the atom number decay

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

0,1
0,075
0,05

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

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

2

Y
p
ψ(r, t) = N (t)eiφ(t)

e

η
2
− 2w
2 +iη βη (t)η

1/2

η=x,y,zπ 1/4 wη,(A.6)

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

Z π

dθ sin θ

0

Z 2π

dφ

0

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

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

4

2

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

X v̇η00

η 00 6=η

vη00(A.11)
U=

 − ∂U , (A.12)∂vη

with P =q

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

12.

Self-bound droplets

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

13.

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

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

8

6

4

2

ER
ω

0

-2

-4

-6

-8
50

55

60

65

70

75

80

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

[1] K Aikawa, A Frisch, M Mark, S Baier, A Rietzler,
R Grimm, and F Ferlaino, “Bose-einstein condensationof erbium,” Phys. Rev. Lett. 108, 210401 (2012).
[2] A Frisch, K Aikawa, M Mark, A Rietzler, J Schindler,
E Zupanič, R Grimm, and F Ferlaino, “Narrow-line
magneto-optical trap for erbium,” Phys. Rev. A 85,
051401 (2012).
[3] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz,
Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, “Extended bose-hubbard models with ultracold magnetic
atoms,” Science 352, 201–205 (2016).
[4] Cheng Chin, Rudi Grimm, Paul S. Julienne, and Eite
Tiesinga, “Feshbach resonances in ultracold gases,” Rev.
Mod. Phys. 82, 1225–1286 (2010).
[5] Tino Weber, Jens Herbig, Michael Mark, HannsChristoph Nägerl, and Rudolf Grimm, “Three-body recombination at large scattering lengths in an ultracold
atomic gas,” Phys. Rev. Lett. 91, 123201 (2003).
[6] L. Pitaevskii and S. Stringari, Bose–Einstein condensation (Oxford University Press, Oxford, 2003).
[7] F. Wächtler and L. Santos, “Quantum filaments in dipolar bose-einstein condensates,” Phys. Rev. A 93, 061603(2016).

[8] F Wächtler and L Santos, “Ground-state properties andelementary excitations of quantum droplets in dipolarbose-einstein condensates,” arXiv:1605.08676 (2016).
[9] RN Bisset, RM Wilson, D Baillie, and PB Blakie,“Ground state phase diagram of a dipolar condensate
with quantum fluctuations,” arXiv:1605.04964 (2016).
[10] D Baillie, RM Wilson, RN Bisset, and PB Blakie, “Selfbound dipolar droplet: a localized matter-wave in freespace,” arXiv:1606.00824 (2016).
[11] Igor Ferrier-Barbut, Holger Kadau, Matthias Schmitt,Matthias Wenzel, and Tilman Pfau, “Observation ofquantum droplets in a strongly dipolar bose gas,” Phys.
Rev. Lett. 116, 215301 (2016).
[12] Holger Kadau, Matthias Schmitt, Matthias Wenzel,
Clarissa Wink, Thomas Maier, Igor Ferrier-Barbut, andTilman Pfau, “Observing the rosensweig instability of aquantum ferrofluid,” Nature 530, 194–197 (2016).
[13] Victo S. Filho, F. Kh. Abdullaev, A. Gammal, andLauro Tomio, “Autosolitons in trapped bose-einstein condensates with two- and three-body inelastic processes,”
Phys. Rev. A 63, 053603 (2001).

A.2. Publication: Anisotropic polarizability of erbium atoms†
Phys. Rev. A 97, 012509 (2018)
submitted 20 Oct 2017; published 19 Jan 2018
DOI 10.1103/PhysRevA.97.012509

Jan Hendrik Becher,1,⋆ Simon Baier,1,⋆ Kiyotaka Aikawa,1 Maxcence Lepers,3 Jean-François
Wyart,3 Olivier Dulieu,3 F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3
Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Paris-Saclay, Université Paris-Saclay,
91405 Orsay, France∗ These authors contribute equally to this work.

Supplemental Material:
The theoretical dataset containing information on all dipole allowed transitions from theground state, as used for the calculations within this publication, are available in Ref. [Bec18].

†

The author of the present thesis developed the experimental procedures together with K. A., performed
the measurements and analyzed the data together with J. H. B., and K. A., and contributed in writing themanuscript. The theoretical calculations of this publication have been contributed by M. L., J.-F. W. and
O. D.

152

PHYSICAL REVIEW A 97, 012509 (2018)

Anisotropic polarizability of erbium atoms
J. H. Becher,1,* S. Baier,1 K. Aikawa,1,† M. Lepers,2,‡ J.-F. Wyart,2 O. Dulieu,2 and F. Ferlaino1,3,§
1Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Paris-Saclay, Université Paris-Saclay, 91405 Orsay, France
3
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2(Received 20 October 2017; published 19 January 2018)
We report on the determination of the dynamical polarizability of ultracold erbium atoms in the ground and in
one excited state at three different wavelengths, which are particularly relevant for optical trapping. Our study
combines experimental measurements of the light shift and theoretical calculations. In particular, our experimental
approach allows us to isolate the different contributions to the polarizability, namely, the isotropic scalar andanisotropic tensor part. For the latter contribution, we observe a clear dependence of the atomic polarizabilityon the angle between the laser-field-polarization axis and the quantization axis, set by the external magnetic
field. Such an angle dependence is particularly pronounced in the excited-state polarizability. We compare ourexperimental findings with the theoretical values, based on semiempirical electronic structure calculations, andwe observe a very good overall agreement. Our results pave the way to exploit the anisotropy of the tensor
polarizability for spin-selective preparation and manipulation.
DOI: 10.1103/PhysRevA.97.012509

I. INTRODUCTION

Ultracold quantum gases provide many different degrees
of freedom, which can be controlled to a very high precision.
This makes them a reliable and versatile tool to study complex
many-body phenomena in the laboratory [1]. Some of thosedegrees of freedom rely on the interaction between atoms
and light. The strength of such an interaction depends on theatomic polarizability, which is a characterizing quantity of thespecific atomic species under examination. Over the courseof the last decades, tremendous progress has been made to
develop theoretical methods and experimental protocols todetermine the atomic polarizabilities, αtot , with an increasinglevel of accuracy [2,3]. With the gained control over quantum systems, the precise determination of αtot became evenmore fundamental with implications for quantum information
processing, precision measurements, collisional physics, andatom-trapping and optical cooling applications. Calculationsof αtot require a fine knowledge of the energy-level structure
and transition matrix elements, which is increasingly complex
to acquire with the increasing number of unpaired electrons in
the atomic species. For instance, alkali atoms with their single
valence electron allow a determination of the static atomic
polarizability with an accuracy below 1% [4,5] when the fullatomic spectrum is taken into account.
*
Present address: Physikalisches Institut, Universität Heidelberg, Im
Neuenheimer Feld 226, 69120 Heidelberg, Germany.
†
Present address: Department of Physics, Graduate School of
Science and Engineering, Tokyo Institute of Technology, Meguroku,
Tokyo 152-8550, Japan.
‡
Present address: Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne Franche-Comté, Dijon,
France.
§Corresponding author: Francesca.Ferlaino@uibk.ac.at2469-9926/2018/97(1)/012509(7)In the case of the multielectron lanthanide atoms (Ln),which have been recently brought to quantum degeneracy(ytterbium (Yb) [6,7], dysprosium (Dy) [8,9], erbium (Er)[10,11]), the atomic spectrum can be very dense with arich zoology of optical transitions, from being ultranarrow to
extremely broad. Beside Yb with its filled shell, the other Lnshow an electron vacancy in an inner and highly anisotropic
electronic shell (4f for all Ln besides lanthanum and lutetium),surrounded by a completely filled isotropic s shell. Because ofthis peculiar electronic configuration, such atomic species areoften referred to as submerged-shell atoms [12,13].
Capturing the complexity of Ln challenges spectroscopic
approaches and allows for stringent tests of ab initio calculations [14–20]. Besides being benchmark systems for theoretical models, Ln exhibit special optical properties, opening
novel possibilities for the control, manipulation, and detectionof Ln-based quantum gases [21,22]. One peculiar aspect ofmagnetic Ln is their sizable anisotropic contribution to thetotal atomic polarizability, originating from the unfilled 4f
shell. Particularly relevant is the anisotropy arising from thetensor polarizability. This term gives rise to a light shift, whichis quadratic in the angular-momentum projection quantum
number, mJ , and provides an additional tool for optical spin
manipulation, as recently studied in ultracold Dy experiments[23]. The anisotropy in the polarizability has been observed not
only in atoms with large orbital-momentum quantum number,but also in large-spin atomic systems, such as chromium (Cr),[24,25] and molecular systems [26–29].
This paper reports on the measurement of the dynamical
polarizability in ultracold Er atoms in both the ground stateand one excited state for trapping-relevant wavelengths. Our
approach allows us to isolate the spherically symmetric (scalar)and the anisotropic (tensor) contribution to the total polarizability. We observe that the latter contribution, although small
in the ground state, can be very large for the excited state. Our

012509-1©2018 American Physical Society

J. H. BECHER et al.

PHYSICAL REVIEW A 97, 012509 (2018)

results are in very good agreement with electronic structure
calculations of the atomic polarizability, showing a gained
control of the atom-light interaction in Er and its spectral
properties.
II. THEORY OF DYNAMICAL POLARIZABILITY

To understand the concept of anisotropic polarizability,
we first review the basic concepts of atom-light interaction[3,30]. When an isotropic medium is submitted to an external
electric field, e.g., a linearly polarized light field, it experiences a polarization parallel to the applied electric field.
However, in anisotropic media, an external electric field canalso induce a perpendicular polarization, which in the atomlight-interaction language corresponds to a polarizability witha tensorial character. As we will discuss in the following,
Er atoms can be viewed as an anisotropic medium becauseof their orbital anisotropy in the ground and excited states(nonzero orbital-momentum quantum number L = 0). The
atomic polarizability is then described by a 3×3 tensor, P.
The total light shift experienced by an atomic medium exposed
to an electric field E reads as
U = 21 E † P E.

(1)

Equation (1) can be decomposed into three parts. For this,we define the scalar polarizability tensor As (diagonal elements), the vectorial polarizability tensor Av (antisymmetric part of the off-diagonal elements), and the tensorial polarizability tensor At (symmetric part of the offdiagonal elements). Hence, a medium with polarizability
tensor P placed into an electric field E feels the total lightshift,
U = 21 E † [As + Av + At ]E.

(2)

We now consider the case of an atom in its electronic groundstate with nonzero angular-momentum quantum number J ,its projection on the quantization axis mJ , and a total polar 2,
izability αtot placed in a laser field of intensity I = 20 c |E|

polarization vector u, and frequency ω = 2π λc . Here, 0 is
the vacuum permittivity, c is the speed of light, and λ is
the wavelength of the laser field. For a given quantization
axis, which is typically set by an external magnetic field, we
furthermore define θk (θp ) as the angle between the propagation[31] (polarization) axis of the laser field and the quantization
axis (see inset in Fig. 1). As shown in Ref. [17], the tensor
product of Eq. (2) can be developed and the total light shift canbe expressed as the sum of the scalar (Us ), vector (Uv ), andtensor (Ut ) light shift as follows:1
I (r)αtot = Us + Uv + Ut
20 c

mJ
1
I (r) αs (ω) + |u∗ × u| cos θkαv (ω)
=−
20 c
2J

3m2J − J (J + 1) 3 cos2 θp − 1
αt (ω) .
(3)+
J (2J − 1)
2

U (ω) = −For convenience, we have explicitly separated the tensor
and vector term into two parts. The first part depends on theangles, J and mJ , and the second part on ω and J . We referto the latter as the polarizability coefficients {αs ,αv ,αt } for thescalar, vector, and tensor part, respectively.
Because of their J , u, and angle dependence, Uv and Ut
vanish for special configurations. In particular, Uv vanishes
for any linear polarization since u∗ ≡ u is a real vector, andthus |u∗ × u| = 0, and for ellipticalpolarization at θk = ±90 ◦ .
√1
Ut vanishes for cos θp0 = /3, i.e., for θp0 = 54.7 ◦ , or for
J = 1/2. The latter condition is always fulfilled by alkali atomic
species, which indeed have zero tensor light shift in the groundstate. As we will discuss later, this is an important difference
between alkali and magnetic Ln, such as Dy and Er, which have
J = 8 and J = 6 in the ground state, respectively. Finally, we
note that Ut shows a quadratic dependence on mJ , which paves
the way for a selective manipulation of individual Zeeman
substates.

FIG. 1. Calculated (solid line) and measured (filled circles) atomic polarizability αtot of Er in the ground state for θp = θk = 90 ◦ as afunction of the light-field wave number and wavelength in atomic units. A divergence of the polarizability indicates an optical dipole transition.
The finite amplitude of the peaks of the narrow transitions are an artifact caused by the finite number of calculated data points. The red andblue shadows indicate that there is a broad red-detuned region for long wavelengths without many resonances and also a mostly blue-detuned
region in the ultraviolet range. The inset illustrates the configuration of angles θk and θp for the shown data. B denotes the orientation of themagnetic field.

012509-2

ANISOTROPIC POLARIZABILITY OF ERBIUM ATOMSPHYSICAL REVIEW A 97, 012509 (2018)

The polarizability coefficients read as
1
αs (ω) = − √α (0) (ω),
3(2J + 1) J2Jα (1) (ω),αv (ω) =(J + 1)(2J + 1) J
2J (2J − 1)
α (2) (ω),αt (ω) =3(J + 1)(2J + 1)(2J + 3) J

(4)

where αJ(K) (ω), K ∈ {0,1,2}, is known as the coupled polarizability. To precisely calculate the value of the polarizability,
it is necessary to know the parameters of each dipole-allowedtransition, i.e., the energy of the transition h̄ωJ J and the naturallinewidth of the excited state γJ . In constant-sign convention[29], αJ(K) (ω) is indeed given by a sum-over-state formula overall dipole-allowed transitions ( J = 0,±1),√(−1)J +J
αJ(K) (ω) = 2K + 1 ×

×

1
J

1
× Re
h̄

J

K
J



1
| J ||d||J |2J
1+−
J J − iγJ /2


(−1)K
.
+
J J − iγJ /2(5)Here, | J ||d||J | is the reduced dipole transition element and±J J = ωJ J ± ω. The curly brackets denote the Wigner 6-j
symbol. Note that the imaginary part of the term in the squaredbrackets is connected to the off-resonant photon scattering rate.
As will be discussed in the next section, a precise knowledgeof the atomic spectrum is highly nontrivial for multielectron
atomic species with submerged-shell structure and requires
advanced spectroscopic calculations.
III. ATOMIC SPECTRUM OF ERBIUM

The submerged-shell electronic configurations of Er in its
ground state reads as [Xe]4f 12 6s 2 , accounting for a xenon
core, an open inner f shell with a two-electron vacancy, anda closed s shell. The corresponding total angular momentum
is J = 6, given by the sum of the orbital (L = 5) and the spin
(S = 1) quantum number.
The calculated static polarizability of ground-state Er is
149 a.u. [32]. To calculate the dynamical one, αtot (ω), we
use Eqs. (3) and (5), based on the semiempirical electronic
structure calculation from Ref. [18]. The result is shown in
Fig. 1 for the case of light propagating along the x axis
and linearly polarized along the y axis [θk = θp = 90 ◦ ; see
Fig. 1 (inset)]. Note that for this configuration, the vectorial
contribution vanishes and the tensor part is maximally negative.
The ground-state polarizability of Er is mainly determined by
the strong optical transitions around 400 nm (see Supplemental
Material [33]). The broadest transition is located at 401 nm
with a natural width of 2π ×29.7 MHz [34]. Apart fromthe broad transitions, Er also features a number of narrowtransitions. As indicated in Fig. 1 by the red-shaded region
to the left of the strong resonances, i.e., for wavelengths above
500 nm, there is a large red-detuned region. To the right,
i.e., for wavelengths below 380 nm, the atomic polarizability

FIG. 2. Ground-state polarizability of Er in the proximity of anarrow optical transition at 631.04 nm with a linewidth of 2π ×28 kHz.
(a) Polarizability coefficients αs (solid line), αv (dotted line), and αt(dashed line) vs the laser-field wavelength. The vertical dotted line
indicates the zero crossing of αs . (b) Total polarizability αtot as afunction of mJ , identifying the different Zeeman sublevels of theground-state manifold for θp = 90 ◦ (circles), θp0 = 54.7 ◦ (squares),and θp = 0 ◦ (stars) calculated with Eq. (3) for θk = 90 ◦ at 630.7 nm,
corresponding to the wavelength of the zero crossing of αs .

is mainly negative (blue-shaded region), which enables therealization of blue-detuned dipole traps for, e.g., boxlike
potentials [35].
As shown with Dy [23], narrow lines give prospects forstate-dependent manipulation of atomic samples. We find thata promising candidate for spin manipulation is the transition
coupling the ground state to the J = 7 excited state at
631.04 nm with a natural linewidth of 2π ×28 kHz [36], which
we investigate theoretically here. It is weak enough to allow
near-resonant operation with comparatively low scattering rate
and features large vector and tensor polarizabilities. Figure 2(a)shows the calculated values of αs , αv , and αt of the groundstate in the proximity of this optical transition, calculated with
Eqs. (4) and (5). Interestingly, αs has a sign opposite to αv
and αt and crosses zero around 630.7 nm, where still verylarge vector (680 a.u.) and tensor (175 a.u.) polarizabilities
persist. Such wavelengths are very interesting since they
allow one to freely tune the total light shift by changing
the polarization of the laser light. Figure 2(b) shows thetotal polarizability αtot as a function of mJ calculated with
Eq. (3) for the three angles θp ∈ {0 ◦ ,54.7 ◦ ,90 ◦ } at the zerocrossing of the scalar polarizability for θk = 90 ◦ . αtot depends
quadratically on mJ and can be tuned from positive to negativeby changing θp , while keeping θk constant. By changing θk ,the vertex of the parabola in Fig. 2 can be shifted towards
higher or lower values of mJ , such that αtot vanishes for aparticular mJ state. Such a feature can, in principle, be used
for a state-dependent manipulation or trapping of the atomic
sample [37].

012509-3

J. H. BECHER et al.

PHYSICAL REVIEW A 97, 012509 (2018)
IV. MEASUREMENTS

To extract the polarizability of Er, we measure the lightshift at three wavelengths: 532.26, 1064.5, and 1570.0 nm.
In addition, we study the polarizability of one excited state,located at 17 157 cm −1 ≡ 583 nm with respect to the groundstate for 1064.5 and 1570.0 nm. This optical line is particularly
relevant for ultracold Er experiments since it is used as thelaser-cooling transition in magneto-optical traps (MOT).
For the measurements, we initially cool down a sample of
168
Er in a MOT [38]. Here, the atoms are spin polarized to
the lowest level of the ground-state Zeeman manifold (J = 6,mJ = −6). We then transfer the sample into a crossed-beam
optical dipole trap at 1064 nm. We force evaporation by
decreasing the power of the trapping laser following theprocedure reported in [10] and cool the sample down totemperatures of several μK. All measurements of the lightshift are performed at a magnetic offset field of 0.4 G. At thisfield, the Zeeman shift is large enough to have the atoms in awell-defined magnetic sublevel mJ so that Eq. (3) is valid.
A. Measurement of the ground-state polarizability

For the measurement of the polarizability at ω = 2π c/λ,
we load the thermal sample from the crossed-beam dipole trap
into an optical dipole trap generated by a single focused beam,
operating at the desired wavelength λ. Typical beam waists
range from 18 to 46 μm. In this single-beam trap, the thermal
sample reaches typical peak densities ranging from 1013 to1014 cm−3 and temperatures of several μK. The propagation
direction of the beam is illustrated in the inset of Fig. 1, i.e., witha magnetic field oriented along the z axis and θk = θp = 90 ◦ .
We extract the corresponding light shift of the groundstate by employing the standard technique of trap-frequency
measurements. From the trapping frequencies, we infer thedepth of the optical potential U , which in turn is related to
αtot by Eq. (3). In harmonic approximation, for a Gaussian
beam of power P , which propagates along the x axis with22
− 2z), beamelliptical intensity profile I (y,z) = I0 exp (− 2ywywzwaists wy and wz , and I0 = πw2Py wz , the depth of the induced
dipole potential
U0 is related to the radial trapping frequencies√
by ωi = −4U0 /(wi2 m), where i ∈ {y,z}. m is the atomic
mass, and U0 = − 210 c αtot (ω)I0 . By combining the above
expressions, we find the relation

4αtot P.
(6)ωi =
0 cπ wy wz wi2 m
In Eq. (6), αtot is the only free parameter since we independently measure the wi and P , as discussed later.
We measure the radial trapping frequencies along the yand the z axis by exciting center-of-mass oscillations andmonitoring the time evolution of the position of the atomic
cloud in time-of-flight images. To excite the center-of-mass
oscillation, we instantly switch off the trapping beam forseveral hundreds of μs [39]. During this time, the atoms move
due to gravity and residual magnetic field gradients. When thetrapping beam is switched on again, the cloud starts to oscillate
in the trap and we probe the oscillation frequencies νz = ωz /2π
along the z axis and νy = ωy /2π along the y axis. In order

to extract αtot from Eq. (6), we precisely measure the beam
waists wy and wz . The most reliable measurements of the beam
waists are performed by using the knife-edge method [40]. We
measure the beam waists with an uncertainty of the order of 1%.
Aberrations and imperfections of the trapping beams, however,
introduce a systematic uncertainty in the measurement of thebeam waists. We estimate a conservative upper bound forsuch an effect of 2 μm, which provides the largest sourceof uncertainty in the measurement of the polarizability. The
corresponding systematic errors on αtot are up to about 35%.
We measure the trap frequencies as a function of the laser
powers P and we fit Eq. (6) to the measured frequencies,
leaving αtot the only free fitting parameter.
We apply the above-described procedure to three different
wavelengths of the trapping beam. The experimental andtheoretical values for αtot are summarized in Table I. For
completeness, we also give αstheor. . Comparatively speaking,at a wavelength of 1064.5 nm, we find that Er, as other Ln,
exhibits a weaker polarizability as compared, for instance, toalkali atoms (e.g., 687.3(5) a.u. (calculated) for rubidium [41]).
This is related to the submerged-shell electronic structure of Er
and the so-called lanthanide contraction, resulting in valence
electrons being more tightly bound to the atomic core, and somore difficult to polarize, than the single outermost electronof alkali atoms [18,42].
The comparison between the measured and calculated
values shows an overall very good agreement, especially at
λ = 1064.5 and 1570 nm. In this wavelength region, there
are very sparse and weak optical transitions and the polarizability approaches its static value; see Fig. 1. At λ = 532.26
nm, we observe a larger deviation between experiment andtheory. This can be due to the larger density of opticalresonances in this wavelength region. Here, the calculated
value of αs is thus much more sensitive to the precise parameters of the optical line (i.e., energy position and strength).
In addition, our theoretical model predicts a very narrowtransition at 18 774 cm−1 ≡ 532.7 nm with a linewidth ofγJ = 6.2×103 s−1 .
We point out that as a result of our improved methodology
to calculate transition probabilities, the theory value of αs =173 a.u. at λ = 1064.5 nm is slightly larger than the one previously reported in [18]. In particular, our present calculations
use a refined value of the scaling factor on monoelectronictransition dipole moments (Er + ) [43], which is now equal to0.807.
As previously discussed, Ln exhibit an anisotropic lightshift, arising from the sizable tensor contribution to the total
polarizability [see Eq. (3)]. This distinctive feature has been
experimentally observed in Dy in the proximity of a narrow
optical transition [23]. Here, we address this aspect with Er
atoms by measuring the light shift in the ground state andits angle dependence at 532.26 and 1064.5 nm. At thesewavelengths, our theory predicts that αt for the ground stateis of the order of a few percent of αs . To isolate this small
contribution and to clear the systematic uncertainties, which
could potentially mask the effect, we probe the tensor-to-scalarpolarizability ratio as follows. We first prepare the ultracold Er
sample in the lowest Zeeman sublevel (mj = −6) in the opticaltrap, operated at the desired wavelength. We then extract theangle-dependent light shift by repeating the measurements of012509-4

ANISOTROPIC POLARIZABILITY OF ERBIUM ATOMSPHYSICAL REVIEW A 97, 012509 (2018)

FIG. 3. Anisotropic polarizability of Er atoms in the ground state.
The plot shows the relative change of the light shift at 532.26 nm(squares) and 1064.5 nm (circles) for θk = 90 ◦ as a function of θp .
The variation of the total light shift unambiguously reveals the tensor
polarizability, which vanishes for an angle of θp ≈ 54.7 ◦ . The lines
are fits to the data with Eq. (3). The error bars indicate the statisticaluncertainties from the trapping-frequency measurements. The dottedlines represent the theory prediction.

the trap frequencies for different values of θp . This is done byeither rotating the magnetic field, while keeping an horizontal
polarization of the trapping light, or by rotating the polarization
axis of the trapping light at a constant magnetic field. In bothmeasurements, we choose θk = 90 ◦ such that the vector lightshift vanishes. Hence, the total light shift comes only fromαs and αt . Since the scalar light shift is independent of θp , adependence of the total light shift on θp is only caused by αt . We
quantify this variation by the relative change of the light shift,κ(θp ) =ω(θp )2 − ω(θp0 )2
U − UsUt==
Us
Us
ω(θp0 )2=3m2J − J (J + 1) 3 cos2 θp − 1 αt.
J (2J − 1)
2
αs

(7)

Note that the first factor in the second line of Eq. (7) is equal to
one for |J,mJ = |6,−6 , such that the peak-to-peak variationof κ(θp ) corresponds to κ0 = 1.5 × ααst . Figure 3 shows κ(θp )for 532.26 and 1064.5 nm. At both wavelengths, the data show
the expected sinusoidal dependence of κ on θp . We fit Eq. (7)to the data and extract κ0 and αt . Our results are summarizedin Table I. The systematic uncertainties of αt are obtained byerror propagating the systematical errors of αs . We observe that

αt for the ground state gives only a few-percent contribution
to the total atomic polarizability. However, the corresponding
tensor light shift for the typical power employed in opticaltrapping can already play an important role in spin-excitation
phenomena in Er quantum gases [44].
Given the complexity of the Er atomic spectrum and thesmall tensorial contribution, the good agreement between thetheoretical predictions of αt and the experimental value for bothinvestigated wavelengths is remarkable. The slightly smaller
values extracted from the experiments can be due to additional
systematic effects in the measurements. For comparison, wenote that at 1064 nm, κ0 for ground-state Er is slightly larger
theor.
than the one for Dy, which was predicted to be around κ0,Dy=
1.1% [15], and larger than the one of Cr atoms, which wastheor.
calculated to be κ0,Cr= 0.5% (at 1075 nm) [25] but was thenmeasured to be significantly lower [24]. In Cr experiments, thetensorial contribution to the total light shift was then enhanced
by using near-resonant light.
B. Measurement of the excited-state polarizability

Although small in the ground state, αt is expected to be
substantially larger in the excited state. Therefore, measuring
the 583 nm excited-state polarizability provides a further
test of the level calculations. To extract the excited-statepolarizability, we measure the shift of the atomic resonancein the dipole trap. As is depicted in Fig. 4(a), the dipole trap
induces a light shift not only to the ground state, but also to
the excited state. To measure the excited-state light shift, we
prepare the atomic sample as described above and apply a shortpulse of a circularly polarized probe light at 583 nm to thesample. This light couples the ground-state |J,mJ = |6,−6level to the |J ,mJ = |7,−7 sublevel of the excited-state
manifold of energy 17 157 cm−1 ([Xe]4f 12 6s6p(3 P1 )). We
find a resonant atom loss when the frequency of the probe
light matches the energy difference between the ground andthe excited state. By scanning the frequency of the probe light,
we extract the resonance frequency. This frequency is shiftedfrom that of the bare optical transition by the sum of theground-state polarizability and the excited-state polarizability.
Subtracting the ground-state shift reveals the light shift of theexcited state. For this, we use the experimental values of theground-state polarizability that are reported here and neglect
the angle dependence thereof since its anisotropy is two ordersof magnitude smaller than the anisotropy of the excited state.

TABLE I. Experimental and theoretical polarizabilities for Er of the ground state (0 cm−1 ) and of the 583 nm excited state (17 157 cm−1 )for three laser wavelengths λ. αtot for experiment and theory is given for the case θp = θk = 90◦ . The relative change of the light shift κ0 (seetext) and the tensor polarizability coefficient αt for the ground state and for the excited state are displayed. The polarizability is given in atomic
units. To convert atomic units into SI units, use a factor of α[Hz/(W mm−2 )] = α[a.u.] × 1.6488×10−35 /2h0 c. For αsexpt. , we give statisticaland systematic errors, respectively (see text).
E (cm−1 ) λ (nm)000532.261064.51570.0

17157
17157

1064.51570.0expt.

αtot (a.u.)(430 ± 8stat. ± 80syst. )(166 ± 3stat. ± 61syst. )(163 ± 9stat. ± 36syst. )αsexpt. (a.u.)(66.6 ± 0.5stat. ± 28syst. )(−203 ± 9stat. ± 50syst. )expt.

theor.
αtot(a.u.) αstheor. (a.u.)

317
176
162

κ0

(%)

κ0theor. (%)

expt.

αt

(a.u.)

αttheor. (a.u.)

308
173
159

(−5.3 ± 1)(−1.8 ± 0.8)

−9.2
−4.7
−4.1

(−15 ± 3stat. ± 6syst. )(−1.9 ± 0.8stat. ± 1.2syst. )

−19
−5.4
−4.3

91
−254

(−25.6 ± 1.6)
(104 ± 6)

−29.7
40.4

(−11.3 ± 0.5stat. ± 2.0syst. )(−141 ± 9stat. ± 19syst. )

−18
−68.5

012509-5

J. H. BECHER et al.

PHYSICAL REVIEW A 97, 012509 (2018)
V. CONCLUSION AND OUTLOOKFIG. 4. 583 nm excited-state polarizability. (a) Illustration of
the energy of atoms in an optical dipole trap with Gaussian shape.
The upper (lower) panel indicates the case with the excited-statepolarizability negative (positive). We measure the shift of the bareatomic resonance in the optical dipole trap (see text) for different
values of θp (dark-to-light red and light-to-dark blue). This shift
is given by the sum of the light shifts in the ground and in the
excited state (|J,mJ = |6,−6 → |J ,mJ = |7,−7 ). To extractthe excited-state light shift, we subtract the ground-state shift. (b)583 nm excited-state polarizability for 1064.5 nm (red squares) andfor 1570.0 nm (gray circles). The solid lines indicate fits to the data.

We repeat this measurement for various values of θp and finda large angle dependence, as we show in Fig. 4(b), for 1064.5
and 1570 nm. This is expected due to the highly anisotropic
wave function of the 6p electron in the 583 nm excited state.
From our data, similarly to the ground-state measurements, we
extract both the scalar and the tensor polarizability coefficients.
The results and the theoretical calculations are presented in thelower section of Table I. The scalar polarizability coefficientagrees within the error with the theoretical expectations, indicating a good understanding of the excited-state polarizability.
The tensor polarizability coefficients qualitatively match wellwith the theoretical values. The quantitative disagreement by
up to a factor of two is probably caused by uncertainties in theparameters of strong transitions close by.

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In this paper, we presented measurements of the scalar andtensor polarizability of Er atoms in the ground and the 583 nm
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For 1064.5 and 1570.0 nm, we find excellent agreementof the scalar polarizability. For 532.26 nm, we observe thatthe measured value of αs deviates from the calculated value,
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ACKNOWLEDGMENTS

We thank R. Grimm, M. Mark, E. Kirilov, G. Natale, andV. Kokoouline for fruitful discussions. M.L., J.-F.W., and O.D.
acknowledge support from “DIM Nano-K” under the project“InterDy,” and from “Agence Nationale de la Recherche”(ANR) under the project “COPOMOL” (Contract No. ANR13-IS04-0004-01). The Innsbruck group is supported throughan ERC Consolidator Grant (RARE, No. 681432) and a FET
Proactive project (RySQ, No. 640378) of the EU H2020.
The Innsbruck group thanks the Deutsche Forschungsgemeinschaft/Wissenschaftsfonds FWF (FOR2247).
J.H.B. and S.B. contributed equally to this work.

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012509-7

A.3. Publication: Observation of roton mode population in a
dipolar quantum gas†‡
Nature Physics 14, 442 (2018)
submitted 12 Dec 2017; published 05 Mar 2018
DOI 10.1038/s41567-018-0054-7

Lauriane Chomaz,1 Rick M. W. van Bijnen,2 Daniel Petter,1 Giulia Faraoni,1,3 Simon Baier,1
Jan Hendrik Becher,1 Manfred J. Mark,1,2 Falk Wächtler,4 Luis Santos,4 F. Ferlaino1,2
1Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria
3
Dipartimento di Fisica e Astronomia, Università di Firenze, Via Sansone 1, 50019 Sesto Fiorentino,
Italy
4
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover,Germany†‡

The author of the present thesis helped in performing the measurements, and contributed in writing themanuscript. The theoretical calculations of this publication have been contributed by R. v. B., F. W., andL. S.
The pre-print version of the publication is attached for online access of this thesis.

160

Observation of the Roton Mode in a Dipolar Quantum Gas

Institut für Experimentalphysik,Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
3
Dipartimento di Fisica e Astronomia, Università di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy
4
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany(Dated: May 22, 2017)
1

arXiv:1705.06914v1 [cond-mat.quant-gas] 19 May 2017

L. Chomaz1 , R. M. W. van Bijnen2 , D. Petter1 , G. Faraoni1,3 , S. Baier1 ,J. H. Becher1 , M. J. Mark1,2 , F. Wächtler4 , L. Santos4 , F. Ferlaino1,2,∗

The concept of a roton, a special kind of elementary excitation, forming a minimum of energy at finite momentum, has been essential tounderstand the properties of superfluid 4 He. In
quantum liquids, rotons arise from strong interparticle interactions, whose microscopic description remains debated. In the realm of highlycontrollable quantum gases, a roton mode has
been predicted to emerge due to dipolar interparticle interactions despite of their weaklyinteracting character. Yet it has remained elusive to observations. Here we report momentumdistribution measurements in dipolar quantum
gases of highly-magnetic erbium atoms, revealing the existence of the long-sought roton. We
observe the appearance of peculiar peaks at welldefined momentum matching the inverse of thetight confinement length as expected for dipolar
rotons. Our combined theoretical and experimental work demonstrates unambiguously the roton
softening of the excitation spectrum and provides
a further step in the quest towards supersolidity.

Introduction

Quantum properties of matter continuously challenge
our intuition, especially when many-body effects emerge
at a macroscopic scale. In this regard, the phenomenonof superfluidity is a paradigmatic case, which continues
to reveal fascinating facets since its discovery in the late
1930s [1, 2]. A major breakthrough in understanding
superfluidity thrived on the concept of quasiparticles, introduced by Landau in 1941 [3]. Quasiparticles are elementary excitations of momentum k, whose energies define the dispersion (energy-momentum) relation (k).
To explain the special thermodynamic properties ofsuperfluid 4 He, Landau postulated the existence of two
types of low-energy quasiparticles: phonons, referring
to low-k acoustic waves, and rotons, gapped excitations
at finite k initially interpreted as elementary vortices.
These two types of excitations were later unified in aunique dispersion relation [4], which continuously evolves
from linear at low k (phonons) to parabolic-like with aminimum (roton) at a finite k = krot . Neutron scattering experiments confirmed Landau‘s remarkable in-tuition [5]. In liquid 4 He, krot scales as the inverse of
the interatomic distance. This manifests a tendency ofthe system to establish a local order, which is driven by
the strong correlations among the atoms. Yet, the samestrongly-correlated nature of helium handicaps a microscopic understanding from first principles of the roton
properties [1, 6].
In the realm of low-temperature quantum physics,
ultra-cold quantum gases realise the other extreme limit
for which the interparticle interactions - and correlations- are weak, meaning that classically their range of action is much smaller than the mean interparticle distance [2, 7]. Because of this diluteness, roton excitations
are typically absent in ordinary quantum gases, i. e. in
Bose-Einstein condensates (BECs) with contact interactions [2]. However, the degree of tunability in BECs isremarkable and a roton-like softening has been induced
in hybrid systems via cavity-mediated interactions [8],and in spin-orbit-coupled BECs [9] and quantum gases
in shaken optical lattices [10] by engineering the singleparticle dispersion relation.
About 15 years ago, the existence of a roton minimum
was theoretically predicted in BECs with dipole-dipole
interactions (DDI) [11, 12]. The DDI is long-range andanisotropic; in particular it can change sign dependingon the dipole configuration, being attractive for head-totail dipoles and repulsive for side-by-side ones (Fig. 1a).
Despite the weakly-interacting character of the gas, theroton minimum in dipolar BECs (dBECs) is genuinely
interaction-induced as in superfluid 4 He. However, incontrast to helium, the dispersion minimum originatesfrom the peculiar k-dependence of the DDI rather thanfrom strong correlations (Fig. 1b).
The realisation of a roton minimum in dBECs would
allow for an unprecedented degree of control and microscopic understanding of the roton properties unavailable
in helium physics. This prospect has triggered remarkable theoretical works, devoted to unveil the special properties of the dipolar roton and to link them to experimental observables [13–27]. Despite the maturity achieved in
the theoretical understanding, the observation of dipolar
roton modes has remained so far an elusive goal. For along time, the only dBEC available in experiments consisted of chromium atoms [28, 29], for which the achievable dipolar character is hardly sufficient to support aroton mode. With the advent of the more magnetic lan-

2

A.

b

a

aS

Energy, ϵ (k)thanide atoms [30, 31], a broader range of dipolar parameters became available, providing new prospects for theobservation of rotons.

repulsion

Roton mode in dBECs and its signature

attraction

To elucidate the mechanism of rotonization in dBECs,
it is instructive to first review the case of cylindricallysymmetric pancake traps with the dipoles aligned along
the symmetry axis z (Fig. 1e1) [11, 13, 15–27]. In thisquasi-two-dimensional (q2D) geometry, we consider elementary excitations of planar momentum k, corresponding to in-plane density modulations of wavelength 2π/k(Fig. 1c-d). In a mean-field picture, the energy cost to
create such an excitation, (k), arises from both kinetic
and interparticle interaction contributions. The latter includes the contact interaction and the DDI, and the DDI
changes sign with k because of the much stronger confinement along z. At low k, the repulsive nature of thein-plane DDI prevails, stiffening the dispersion relation(Fig. 1c). In contrast, for k`z & 1, `z being the transverse confinement length, the three-dimensional (3D) nature of the excitation is reestablished and the attractive part of the DDI dominates, softening (k) (Fig. 1d).
This softening is counterbalanced by the contributionsof the repulsive interactions, namely the contact interaction at positive scattering length as , and of the kinetic
energy, increasing with k. For very large k, the kinetic
energy cost prevails and (k) eventually bends up intoa single-particle spectrum. For strong-enough DDI, thedipole-induced softening dominates at intermediate momenta, and (k) develops a minimum at krot ∼ 1/`z [11](Fig. 1b).
Ultracold atoms allow to regulate the impact of the
DDI in (k) by tuning the value of as through a Feshbachresonance (FR) [32]. This provides a powerful control
knob for the roton physics that is absent in the 4 He case.
The figure of merit is the parameter dd = add /as , defined as the ratio between the dipolar length, add =µ0 µ2 m/12π~2 , and as . Here m is the mass and µ the
magnetic moment of the atoms. Increasing dd (decreasing as ) mitigates the energy cost for large-k excitations
and the system can develop a roton minimum in (k).
By further increasing dd , the roton gap, ∆ = (krot ),decreases and eventually vanishes (Fig. 1b). In the latter case, the system undergoes a local instability [18, 19]
and develops a short-wavelength density modulation on
the scale of 1/krot , , resembling the case of the expectedsuperfluid-to-solid transition in overpressurized 4 He [33].
When the dBEC spectrum gets deeply rotonized (i. e.
small ∆), the momentum distribution can be profoundly
modified, providing a signature for the effect. In q2D
geometries (Fig. 1e1), the excitation softening develops
in plane - both in the x and y directions. This leads to aradial roton corresponding to a set of in-plane momenta
k of |k| = krot . The privileged population of these modes
translates into the emergence of a ring of radius krot in

Momentum, k

krot

d

c
z

dominantly repulsive DDI

dominantly attractive DDI

x,y plane

e1

f1
z

e2

x

f2

y

ñ(kx,ky)

1.0
0.8

kx

0.6

kx

0.4
0.2

ky

ky

0

Figure 1 | Roton mode in a dBEC. a, dipole orientations with repulsive (blue) and attractive (red) DDI.
b, illustration of the dispersion relation with real (solidlines) and imaginary (norm of the dotted line) parts of adBEC in constrained geometries (see e1-f1) and varyingas (dashed arrow) [11]. c-d, spatial density modulations
associated to small- (c) and large-k (d) excitations. The
shape along z and the color gradient illustrate the density
modulation. The color code, following that of a, pictures
the dominant DDI contribution to (k). In b, a roton
emerges at krot when the attractive DDI prevails. In q2D
geometries (e1), the roton in-plane momentum distribution shows a ring (e2). f1 illustrates the advantageous
q1D geometry, where the roton momentum distribution
is double-peaked (f2). e2 and f2 have the same total
population.
the momentum distribution (Fig. 1e2).
Here we extend the roton physics to a largely unexplored geometry: that of axially elongated dBECs with
dipoles oriented orthogonally to the elongation axis y(Fig. 1f1). Although the above energy arguments equally
apply to this quasi-one-dimensional (q1D) geometry, theaxial elongation provides an important difference. Here,the system develops a linear roton mode corresponding

3
0.9

a

0.8

-5
0
5

-10

0

y (µm)

1020

ñ (a.u.)1b1-5

n (a.u.)

0
-20

0.6
0.5
0.4
0.3

kx (µm-1)

0.5

Energy, ϵ (h z)

0.7

1 b

kx (µm-1)

n (a.u.)

1.5

0.2
0

ky (µm-1)

5

0

0.1
0

0

0.2

0.4

0.6

0.8

ky(eff) ℓz

1

1.2

1.4

1.6

1.5
1

c

0.5
0
-20

-5
0
5

-10

0

y (µm)

1020

ñ (a.u.)

1

c1
-5

0

ky (µm-1)

50

Figure 2 | BdG excitation spectrum. a, Excitation spectrum of the ground state of a BEC with N = 50.000
Er atoms in a trap with (νx , νy , νz ) = (267, 32, 456) Hz and scattering length as = 43.75 a0 , obtained by numericallysolving the BdG equations. Roton modes appear as isolated modes lying below the main branch of the spectrum,
forming a ’roton finger’. The lowest roton mode and an exemplary phonon mode are highlighted, with panels b, c
showing the corresponding excited state density modulation (with black dashed line the ground state, red solid line
the excited state) and, b1, c1, the momentum distribution from 30 ms ToF expansion (Methods). b, b1 phononmode, c, c1 roton mode.
166

to a narrow set of k with y component ky = ±krot . Asa consequence, the privileged population of this modetranslates into two marked peaks in the momentum distribution (Fig. 1f2). This geometrical focusing of the roton population greatly enhances the visibility of the effectcompared to the q2D case (Fig. 1e2-f2).
B.

Rotonized dispersion relation in q1D dBECs

The existence of a roton minimum in our q1D geometry is well explained by a simplified model inspired
from the uniform q2D calculations [11]. Here we consider that the condensate is trapped along x and z,
with harmonic frequencies νx and νz , but homogeneous,i.e. unconfined, along the axis y. The physics of thedBEC is well captured by a non-local Gross-Pitaevskii
equation (NLGPE) [2, 34], which contains the transverse confinement, the short-range interactions, and theDDI (Methods). Within the Thomas-Fermi (TF) approximation, the BEC density takes
the simple form
n(x, z) = n0 1 − (x/X)2 − (z/Z)2 , with n0 the homogeneous axial density [12] (Methods). The excitation
spectrum is obtained from the linearisation ofpthe NLGPE around the ground-state wavefunctionn(x, z).
Due to the homogeneity along y, the elementary excitations have a well-defined momentum ky . Proceeding as
in Ref. [11], we obtain an analytic form for (ky ) in therelevant case of 3D modes, i. e. for ky Z  1 (Methods).
For dominant DDI (dd ≥ 1), (ky ) indeed rotonizes. In
the vicinity of the roton minimum the dispersion acquires
a gapped quadratic form similar to that of helium rotons:
(ky )2 ' ∆2 +2
2~2 krot~2(ky − krot )2
m 2m

(1)

for
√ dd ∼ 1. Here the roton momentum is krot =2mEI /~, with EI = 2gn0 (dd − p1)/3 and g =4π~2 as /m, while its gap reads ∆ =E02 − EI2 , with2~
E02 = 2gdd n0 2m X −2 + Z −2 . Note that the TF radii
X and Z can be evaluated as functions of νx,z , dd andgn0 [12] and in particular X 2 , Z 2 ∝ gn0 . Then, close
to the instability, ∆ ' p
0, krot takes the simple form~/2πmνz and a geometricalkrot = κ/`z with `z =factor κ that depends only on νz /νx (Methods). In thecase νz /νx = 1, κ = 1.74.
To move beyond this simplified model and closer torealistic experimental conditions, we develop a numerical approach based on the 3D NLGPE for a genericanisotropic harmonic confinement of frequencies νx,y,z(Methods) [14]. For a quantitative understanding of theexperiments, our numerical treatment also includes quantum fluctuations (i. e. local Lee-Huang-Yang (LHY) corrections) [35–41], finite temperature effects [37, 42], andthree-body losses [43] (Methods). Our numerical platform offers rich possibilities to investigate the physics atplay: it allows for real-time evolution of the quantum gas
wavefunction, and provides access to the Bogoliubov-deGennes (BdG) excitation spectrum.
The excitation spectrum is obtained by linearizing the
NLGPE around a stationary state and numerically solving the resulting BdG equations (Methods) [44]. The
calculated spectrum is qualitatively modified comparedto that of an homogeneous system (Fig. 2). In orderto depict the spectrum as a quasi-dispersion relationeven in the presence of an axial confinement, we associate to each elementary excitation an effective momen(eff)tum ky= hky2 i1/2 [17]. The spectrum is discrete with(eff)

phonon-like collective modes at low ky. For higher

4

0.5

1

1.5

ñ (10 3 µm 2 )02

ñ (10 3 µm 2 )

0

c

b
5
kx (µm -1 )0.2 0.4 0.6 0.8

0.1 0.2 0.3 0.4

a z
y

d

x

0
-5

0.5
0.8

b1

1

0.4

0

0

c1

0.4

Contrast, C

2

ñ (10 3 µm 2 )Magnetic
field, Bñ (10 3 µm 2 )
0

d1

0.2

0.4
0.3
0.2
0.1

-8

-4

0

4

ky (µm -1 )

8

-8

-4

0

4

ky (µm -1 )

8

0

-8

-4

0

4

8

0

ky (µm -1 )

e
30

40

50

Scattering length, as (a 0)60

Figure 3 | Appearance of roton peaks in a q1D geometry. a, illustration of the in-trap Er BEC geometry,forming a cigar shape with y-elongation and dipoles aligned by an external uniform B-field along the z transverse
direction. b-d, observed momentum distributions ñ(kx , ky ) obtained by averaging 15-to-25 absorption images with(νz , λ) = (456(1) Hz, 14.4(4)), th = 3 ms and for different as : b, as = 54a0 , c, as = 44a0 and d, as = 37a0 .
b1-d1 show the corresponding central cuts with |kx | < 3.5µm−1 (red dots) and their fits with three-Gaussian distributions (blue lines), from which we extract krot and C (see text and Methods). e, evolution of C with as for(νz , λ) = (456(1) Hz, 14.4(4)) and th = 3 ms (dots). Error bars are deduced from the fit statistical uncertainties on
the amplitudes A∗,0 . We fit an empirically chosen linear step function to identify a∗s (line).
(eff)

the spectrum flattens, but eventually bends upky
wards again due to the dominant kinetic energy. In our
experimentally relevant q1D geometries, instead of developing a smooth minimum, roton excitations appear as
isolated low-lying modes at intermediate momenta thatdepart from the overall spectrum [17]. These so-called roton fingers are related to confinement of the roton modesin the inhomogeneous BEC of profile n0 (y) [16]. Using
the local-density approximation in Eq. (1), the spatiallydependent spectrum (k, n0 (y)) has a minimal roton gap
at the trap center, translating into an effective confinement of the mode.

GPE. This enables to fully account for the experimental
procedure, as described in the next Section, as well as theeffects due to finite temperature and atom losses (Methods). Also the real-time evolution shows a population of
the roton mode with the same signatures as in Fig. 2cc1, both in trap and in ToF. The extracted krot agrees
well, within 10%, with the BdG approach in its validity
regime, i. e. when a stationary dBEC state is found.

The confinement is evident from our BdG calculations,in which the lowest roton mode forms a short-wavelengthdensity modulation localized at the trap center (Fig. 2c).
This contrasts with phonon modes for which the modulation is delocalised over the entire condensate (Fig. 2b).
Whereas the density modulation (Fig. 2c) and the finitemomentum peaks (Fig. 1f2) are signatures of the same effect, from now on we will focus on the latter aspect since
it is the one we probe in the experiments via time-of-flight(ToF) expansion measurements [2]. Accordingly, we compute the momentum distribution from the ToF expanded
densities associated with the selected modes (Fig. 2b1-c1)
and observe that the roton mode indeed presents as two
symmetric peaks localised at positions corresponding to±krot .

Our experiment relies on the strong dipolar character of Er (add = 65.5a0 ) and on the ability to fine tune
as below add . The experimental sequence starts with a
166
Er dBEC in a trap elongated along the y axis. The
trap aspect ratio, λ = νz /νy , can span from about 4 to30, corresponding to νz ranging from 150 Hz to 800 Hz.
We note that νy and νz /νx are kept roughly constant to≈ 30 Hz and ≈ 1.6, respectively (Methods). An external
homogeneous magnetic field, B, defines the dipole orientation (magnetization) with respect to the trap axes
and sets the values of as through a magnetic FR, centered close to B = 0 G. The precise B-to-as conversion
has been measured in Ref. [39]. At the end of the preparation stage, we obtain a stable BEC at as = 61 a0(dd = 1.08, B = 0.4 G) with transverse (z) magnetization (Fig. 3a) (Methods).
To access the roton regime, we suddenly quench as to

To enrich the stationary-state picture, we additionallydevelop real-time evolution simulations of the 3D NL-

C.

Roton peaks in quench experiments

5First we investigate ñ(kx , ky ) as a function of as for(νz , λ) = (456 Hz, 14.4), and th = 3 ms. For large enoughas , ñ(kx , ky ) shows a single narrow peak with an inverted
aspect ratio compared to the trapped gas, typical of astable BEC [2] (Fig. 3b). We define the center of the
distribution as the origin of k. In contrast, when thesystem enters the dominantly dipolar regime by decreasing as , ñ(kx , ky ) changes fundamentally. We observe asudden appearance of two symmetric finite-momentum
peaks along the ky axis, located at ±ky∗ and of similar shape (Fig. 3c-d). Beside their remarkable symmetry,these finite-momentum peaks also exhibit a high shot-toshot reproducibility, as evidenced by their high contrast
in the averaged distributions (Fig. 3c-d). The observedside peaks show the privileged population of specific highmomentum modes (ky = ky∗ ) in the excitation spectrum
(ky ) of our trapped dBEC.
To quantitatively investigate the peak structures, we fit
a sum of three Gaussian distributions to the central cutsof the average ñ(kx , ky ) (Fig. 3(b1-d1)) (Methods). From
the fit we extract ky∗ , the amplitudes of the side and central peaks A∗,0 , and derive the contrast of the side peaks,
C = A∗ /A0 . Typical values of ky∗ are ∼ 4-5µm−1 . The
evolution of C with as reveals an interesting behaviour(Fig. 3e). For large as , we observe an essentially zeroand flat contrast. The residual C ∼ 3% arises from theusual thermal background and the imaging noise. Withdecreasing as , C first exhibits a sharp increase and thensaturates to an upper value of about 40%. We define theonset of the finite-momentum excitation, a∗s , as the valueof as at which C starts to rise. For the (νz , λ) parameter
here considered, we find a∗s = 49.0(2) a0 , corresponding
to dd = 1.34(1).
Our ToF observations directly reveal the presenceof the long-sought roton mode. The observed structures closely follow the predicted signature of this mode
(Fig. 2c1). In addition, the fitted ky∗ are in very good
agreement with the predicted krot from our theory(Sec. D). Finally, the high shot-to-shot repeatability of
the remarkable peak structure in ñ(kx , ky ) suggests thepersistence of a macroscopic phase coherence in the gas.

D.

Properties of the roton excitation.

6

Roton momentum, krot (µm-1)

the desired lower value and we shortly hold the atoms
in the trap for a time th , typically th = 3 ms. We notethat during th , B exponentially converges to its set value
with a 1/e-time of 0.98(5) ms (Methods). We then release the atoms from the trap, change as back to approximately its initial value, and let the cloud expand for
tf = 30 ms, after which we perform resonant absorption
imaging. The imaging beam propagates transversely, i. e.
nearly collinear with the z-axis. Hence the ToF images
probe the two-dimensional density distribution of the expanded cloud, nf (x, y, tf ). For tf long enough to ignore
the in-situ width of the cloud, nf (x, y, tf ) maps the momentum distribution of the gas in trap, ñ(kx , ky ), assuming negligible interactions during the expansion. Our
real-time simulations, accounting for the precise experimental sequence, confirm this assumption (Methods).

5

4

3

2
1.5

2.0

2.5

3.0

3.5

1/ℓz (µm-1)

Figure 4 | Geometrical scaling of the roton momentum. Measured krot at the onset of a roton population as = a∗s in the quench experiment (dots) as afunction of 1/`z . Error bars show the statistical uncertainty of the fit. Here short th ranging from 3 to 6 ms
are used (Methods). The dotted and dashed lines show
the theory predictions using the experimental parameters, from the analytic q1D model of Eq. (1) and fromour real-time simulations reproducing the experimental
procedure, respectively. Both theories use the as values
being just below the predicted roton softening (Methods). The blue solid line shows a linear fit passing by theorigin, to the data. The fitted slope is 1.61(4).
A major fingerprint of the roton mode in a dipolar
gas is its characteristic dependence on the trap geometry. Its geometrical nature has been proven in various contexts, from pancake- [11, 13, 15–17] to our cigarshaped traps (Sec. B), highlighting a universal scaling
with krot ∼ 1/`z . To experimentally investigate such geometrical properties in our elongated dBEC, we repeat
the quench measurements for different trap parameters(νz , λ) and we extract the corresponding value of krot
(Fig. 4) (Methods). Our data, plotted as a function of
1/`z , reveal a marked increase of krot , and a linear fit to
the data gives a slope of 1.61(4). This value is remarkably close to the mean krot `z = 1.59(15) calculated fromour simple analytic model using the experimental parameters. To firmly tie our observation to the roton excitation, we perform real-time simulations of the NLGPE, accounting for the specific experimental sequences (Methods). The extracted parameter-free curve (dashed line
in Fig. 4) is in excellent agreement with the data andalso captures the smooth deviation from a linear slope

6
Reduced roton momentum, krot ℓz
1.9

1.7

1.6

1.5

1.4

a

25

5

20

krot (µm-1)

Trap aspect ratio, λ

30

1.8

15
10

b

4.5
4
3.5

5

25
2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

Interaction parameter, εdd

30

35

40

45

50

Scattering length, as (a0)Figure 5 | Roton characteristic diagram. Summary of the measured roton properties as a function of dd andλ. Here short th ranging from 3 to 6 ms are used (Methods). The white-to-red colormap shows the reduced roton
momentum krot `z . In the blue region no roton excitation is observed (e. g. Fig. 3b), and in the grey region we do
not have a full set of data (Methods). The yellow triangles show the onset of the roton softening, a∗s , extracted fromcontrast measurements C at th = 3 ms (e. g. Fig. 3d) and the solid black line is a linear fit to a∗s (λ). b, krot as afunction of as for the reference trap (νz , λ) = (456(1) Hz, 14.4(4)), corresponding to the dashed line in a. Error bars
show the statistical uncertainty of the fit.
reflecting the experimental details. The comparison between the experimental observations and our two complementary theories unambiguously establishes the rotonic
nature of the finite-momentum peaks.
To gain further insights into the roton properties, we
additionally investigate the effects of λ and as . From
all our measurements, we construct a characteristic diagram, presented in terms of the dimensionless parameters
λ and dd (Fig. 5a). The former parameter governs thegeometric competition between the attractive and repulsive parts of the DDI, and dd embodies the interplay
between dipolar and short-range interactions (Methods).
The white-to-red color scale indicates the values of the
reduced roton momentum krot `z . Each column of the
diagram shows the variation of krot `z with λ, whereas
each row gives the evolution of krot with dd for a given
trap configuration. We observe krot `z varying at most by≈ 25% along the lines and rows, which is on the order ofour experimental precision. Close to the instability, both
dependencies are expected to be mild as krot remains
mainly set by its geometrical nature (see Secs. A, B ande. g. Refs.[11, 15]). The smooth dependence of krot withas is particularly relevant (Fig. 5b) as it contrasts the observed softening from a phonon instability in which theminimum in 2 (k) is expected to appear at k = 0 andthen move to larger values when lowering as . From ouranalytic q1D model for the roton softening, we extract a

residual increase of krot `z of 17% from as = 44a0 to 37a0
in the geometry (νz , λ) = (457 Hz, 14.3), similar to theexperiments.
Besides the geometric determination of krot , dd andλ control the roton gap ∆. In the experiment, we observe that the critical ∗dd = add /a∗s for the onset of theroton population moves to larger values for more elongated traps (blue-to-red border and C-measurements in
Fig. 5). This increase can be explained by the following
argument: a macroscopic roton population appears forsmall enough ∆ (Methods). This condition is realised
when the attractive DDI dominates over the total repulsive (contact and repulsive DDI) interaction. As the ratioof the attractive and repulsive contributions of the DDI
depends on λ, so will a∗s . A larger λ favors the repulsive part of the DDI, thus weaker contact interactions
are needed for a similar roton gap, explaining a lowera∗s . Predictions from our q1D model reproduce well theobserved a∗s , both its variation direction and the actual
values within 10% (Methods).

E.

Discussion

In conclusion, we report on the observation of the longsought roton mode in a dipolar BEC of Er atoms. Our investigations take advantage of an axially elongated geom-7etry, amplifying the roton signature in momentum space.
Our results, combining experimental studies, numericaland analytic theories, unambiguously establish the roton
softening of the excitation spectrum and the universal
geometrical scaling krot ∼ 1/`z of the excitation.
Our work opens fascinating new ground for the studyof roton physics in dipolar gases. In the future, it might
be interesting to explore out-of-equilibrium dynamics ofthe roton mode, to spectroscopically access the rotongap, and to directly probe in-trap density modulated

states. These density modulations, providing a signature of the roton softening in real space, are a precursor
of a supersolid phase, in which a phase-coherent densitymodulated ground-state would arise [45]. Although thedensity modulation is expected to be mean-field unstableagainst local collapses [46], quantum stabilization may
prevent collapse as for the case of recently explored quantum droplets [37–41]. Future investigations may be directed in exploring, both in theory and experiment, thepossibility of a stable supersolid state in dBECs.

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• Acknoledgements: We thank M. Baranov andE. Demler for fruitful discussions and G. Natale
for his support in the final stage of the experiment. This work is dedicated in memory of D. Jin
and her inspiring example. The Innsbruck groupis supported through a ERC Consolidator Grant(RARE, no. 681432) and a FET Proactive project(RySQ, no. 640378) of the EU H2020. LC is supported within a Marie Curie Project (DipPhase,no. 706809) of the EU H2020. FW and LS thankthe DFG (SFB 1227 DQ-mat). All the authors
thank the DFG/FWF (FOR 2247). Part of thecomputational results presented have been achieved
using the HPC infrastructure LEO of the University of Innsbruck.
• Competing Interests: The authors declare thatthey have no competing financial interests.
• Correspondence: Correspondence and requests
for materials should be addressed to F.F. (email:francesca.ferlaino@uibk.ac.at).
• Author contributions: FF, LC, DP, GF, MJM,
JHB, SB conceived and supervised the experimentand took the experimental data. LC analysed
them. RMWvB developed the BdG calculations.
FW and LS performed the real time simulations.
LS derived the q1D analytical model. LC, FF,
RMWvB and LS wrote the paper with the contributions of all the authors.
METHODSF.

Production of 166 Er BECs

We prepare a 166 Er BEC similarly to Refs. [31, 39].
From a narrow-line magneto-optical trap with 3 × 107
166
Er atoms, automatically spin-polarized in their absolute lowest Zeeman sub-level [47], at about 10µK, we
directly load the atomic gas in a crossed optical dipole
trap (ODT) with an efficiency of more than 30%. A
uniform magnetic field, B, is permanently applied along
the vertical z axis, fixing the dipole orientation, while
its value is varied during the experimental sequence, totune as (Method H). We achieve condensation by meansof evaporative cooling in the crossed ODT at B = 1.9 G(as = 80(2) a0 ). During the evaporation procedure, wefirst change the power and then the ellipticity of one of
the ODT beams (Method G). The final atom number,
typically 105 , and condensed fraction, typically 70%, areassessed by fitting the ToF absorption images of the gas
to a bimodal distribution, sum of a TF profile and abroad Gaussian background.
G.

Trapping geometries

The BEC is confined in a harmonic trapping potential V (r) = 2mπ 2 (νx2 x2 + νy2 y 2 + νz2 z 2 ), characterized
by the frequencies (νx , νy , νz ). The ODT results fromthe crossing of two red-detuned laser beams of 1064 nm

9wavelength at their respective focii. One beam propagates along the y-axis and is denoted hODTb; the other,called vODTb, propagates nearly collinear to the z-axis.
The vODTb has a maximum power of 7 W and an elliptical profile with waists of 110 and 55 µm along x andy respectively. The hODTb has a maximum power of
24 W, a vertical waist wz = 18 µm, and a controllable
horizontal waist, wx = λ0 wz . The ellipticity λ0 can betuned from 1.57 to 15 by time averaging the frequency of
the first-order deflection of an Acousto-Optic Modulator[48]. By adjusting independently λ0 and the powers ofthe vODTb and of the hODTb , we can widely control
the geometry of the trap in a dynamic manner. Precisely,νy is essentially set by the vODTb power, νz by that of
the hODTb. νx is controlled by both the power and ellipticity of the hODTb, with νz /νx ≈ λ0 . The independent
control of νy and νz yields an especially easy tuning of
the relevant trap aspect ratio, λ = νz /νy , for our q1D
geometry.
We use the tuning of the hODTb power and ellipticity to perform evaporative cooling to quantum degeneracy (Method F). After reaching condensation, we
again modify the beam parameters to shape the trap intoa favourable q1D configuration for observing the roton
physics (νy  νx , νz ). The different trapping geometries
probed in Figs. 3-5 are achieved by changing the hODTb
power with λ0 = 1.57 and the vODTb power set to its
maximum so that νy and νz /νx are kept roughly constant. The corresponding (λ, νx , νy , νz ) are calibrated via
the excitation and probing of the center-of-mass oscillation of thermal samples and reported in Extended Data
Table I. We note that the final atom number N , BEC
fraction f , and temperature T after the shaping procedure depend on the final configurations, as detailed in
Extended Data Table I. T is extracted from the evolution
with the ToF duration tf of the size of the background
Gaussian in the TF-plus-Gaussian bimodal fit to the corresponding ToF images of gas. The values of Nc = f N
and T are used for the initial states ψi of our real-time
simulations (Method N).

H.

Quench of the scattering length as

To control as we use a magnetic Feshbach resonance between 166 Er atoms in their absolute ground state, whichis centered around B = 0 G. The B-to-as conversion
has been previously precisely measured via lattice spectroscopy, as reported in Ref. [39]. Errors on as , taking
into account statistical uncertainties of the conversion
and effects of magnetic field fluctuations (e.g. from strayfields), are of 3-to-5 a0 for the relevant as range 27-67 a0
in this work. After the BEC preparation and in order toinvestigate the roton physics via an interaction quench,
we suddenly change the magnetic field set value, Bset ,twice. First we perform the quench itself and abruptly
change Bset from 0.4 G (as = 61 a0 ) to the desired lower
value at the beginning of the hold in trap (th = 0 ms).

Second we prepare the ToF expansion and imaging conditions (see Method I) and abruptly change Bset from thequenched value to 0.3 G (as = 57 a0 ) at the beginningof the ToF expansion (tf = 0 ms). Due to delays in theexperimental setup, e.g. coming from eddy currents in
our main chamber, the actual B value felt by the atoms
responds to a change of Bset via B(t) = Bset (t) + τ dB/dt[49]. By performing pulsed-radio-frequency spectroscopy
measurements (pulse duration 100 µs) on a BEC after
changing Bset (from 0.4 to 0.2 G), we verify this law andcalibrate τ = 0.98(5) ms. As a consequence as is evolving during th and tf . This effect is fully accounted in
the experiment and simulations and, we report the roton
properties as a function of the effective value of as at theend of th . We use th ranging from 3 to 6 ms. The lower
bound on th comes from the time needed for as to effectively reach the regime of interest. Here we then consider
the initial evolution for which th νy ; th /τcoll  1, 1/τcoll
being the characteristic collision rate. We estimate thatτcoll ranges typically from 40 to 90 ms in the initial BECs
of Extended Data Table I at as = 61 a0 . We note thatduring the considered th , the atom loss remains below
25% for our less confined geometries νz ≤ 600 Hz andcan go up 40% in the tightest traps of νz > 600 Hz. We
have checked that the roton, if it exists, has developed
within this range of th . In Figs. 4-5, the parameter th isoptimized for each as value to obtain the largest visibilityof the roton peaks (largest C).
I.

Imaging procedure

The in-trap density modulation associated with the roton excitation has a characteristic wavelength of d∗y =2π/krot ∼ 1 µm (e.g. Fig. 2c). This value is much smaller
than the axial width of the cloud (∼ 30 µm) and below
our imaging resolution (∼ 3.7 µm). In our experiments,
we employ ToF expansion measurements, accessing the
momentum distribution of the gas [2], to probe the roton mode population. We let the gas expand freely for
tf = 30 ms, which translates the imaging resolution in
space into a momentum resolution of ∼ 0.32µm−1 . Thismeans that we can resolve the population of modulation
modes with wavelength . 20 µm and the roton mode ofinterest should be well detectable.
In the experiments, we record 2D absorption picturesof the cloud after ToF expansion by means of standard
resonant absorption imaging on the atomic transition at
401 nm. The imaging beam propagates nearly vertically,
with a remaining angle of ∼ 15o compared to the z-axis
within the xz-plane. Thus the ToF images essentiallyprobe the spatial density distribution nf (x, y, tf ) in thexy plane. When releasing the cloud (tf = 0 ms), wechange back B to B = 0.3 G (Method H). This change
enables constant and optimal imaging conditions with afixed probing procedure, i. e. a maximal absorption crosssection. In addition, the associated increase of as to 57 a0
allows to minimize the time during which the evolution

10
happens in the small-as regime where roton physics develops. Thanks to this we are able to effectively probe
the short-time evolution of the gas. In this work, we use
the simple mapping:ñ(kx , ky ) =



~tf
m

2

nf



~kx tf ~ky tf
,
, tf
m
m



,

(2)

which neglects the initial size of the cloud in the trap
and the effect of interparticle interactions during the free
expansion. Using real-time simulations (Method N), we
simulate the experimental sequence and are able to compute both the real momentum density from the in-trap
wavefunction and the spatial ToF distribution 30 ms afterswitching off the trap. Using the mapping of Eq. (2) andour experimental parameters, the two calculated distributions are very similar, and, in particular, the two extracted momenta associated with the roton signal agrees
within 5%. This confirms that the interparticle interactions play little role during the expansion and justifies
the use of Eq. (2).
J.

In the ToF images, we qualitatively observe that the
momentum distribution broadens (i. e. larger hkx2 + ky2 i)when th increases and as decreases. This behaviour effectively limits the visibility of the roton peaks, and explainsthat the optimal th for the roton observation (see MethodH) gets shorter for lower as . In Fig. 5, the grey region
indicates the (as ,λ)-configurations where the whole momentum distribution was observed to be too spread out
even at the shortest th = 3 ms, so that the roton peaks
could not be clearly detected. This region was then excluded and no full set of measurements is available. We
note that, in this region of the diagram, 1/`z is the smallest, making the blurring effect due to the distribution
braodening more drastic, since the roton and BEC peaks
are closer together. The non-detectability of roton peaks
can also relate to the phase diagram characteristics. The
small-as -small-λ region, corresponding to the grey arealies deeper in the roton regime (as embodied by the increase of a∗s for decreasing λ, see main text). It is thenpossible that a larger number of modes are dynamicallydestabilized and populated, resulting in a blurring of theroton signal.

Fit procedure for the ToF images

For each data point of Figs. 3-5, we record between 12
and 25 ToF images. By fitting a two-dimensional Gaussian distribution to the individual images, we extract
their origin (kx , ky ) = (0, 0) and recenter each image.
From the recentered images, we compute the averagedñ(kx , ky ), from which we characterise the linear roton developing along ky . To do so, we extract a one-dimensional
profile ñ1 (ky ) by averaging the one-dimensional cut ofñ(kx , ky ) of fixed kx within |kx | ≤ km = 3.5µm−1 :RkRkñ1 (ky ) = −kmm ñ(kx , ky )dkx / −kmm dkx . To quantitatively
analyse the observed roton peaks, we fit a sum of threeGaussian distributions to ñ1 (ky ). One Gaussian accounts
for the central peak and its centre is imposed to k0 ∼ 0.
The two other Gaussians are symmetrically located at
k0 ± ky∗ , and we impose their size σ ∗ and amplitude A∗
to be identical. We focus on the roton side peaks by constraining ky∗ > 0.5 µm−1 and σ ∗ < 3 µm−1 . In all thefigures, the statistical error of the fit is characterised byits 95 % confidence interval.
In order to analyze the evolution of the contrast C of
the side peaks (see Fig. 3e and Fig. 5), we perform a second run of the fitting procedure, in which we constrain
more strictly the value of ky∗ . The interval of allowed
values is defined for each trapping geometry (Extended
Data Table I) and is set from the results of the first run of
the fitting procedure. We use the results of the (as , th )configurations where the peaks are clearly visible (seeFig.5) and we set the allowed ky∗ -range to that covered by
the 95% confidence intervals of the first-fitted ky∗ in these
configurations. This constraint enables that, for as > a∗s ,the fitting procedure estimates the residual backgroundpopulation on the relevant momentum range for the roton physics.

K.

Generalized Gross–Pitaevskii equation

Our theory is based on an extended version of the NLGPE
Z∂ψ(r, t)  ~2 ∇2i~= −+ V (r) + dr0 U (r − r0 )n(r0 )
∂t
2m
L3 
+ ∆µ[n] − i~ n2 ψ(r, t) (3)2L3 2 
≡ ĤGP [ψ] − i~ n ψ(r, t),(4)2

governing the evolution of a macroscopically occupied
wavefunction ψ(r, t), with corresponding atomic density
n = |ψ(r, t)|2 at position r and time t. The standard
dipolar NLGPE includes the kinetic energy, external trap
potential and the mean-field effect of the interactions.
These correspond to the three first terms of Eq. (3), where
the mean-field interaction potential takes the form of a
convolution of n with the binary interaction potential
3dd 1 − 3 cos2 θU (r) = g δ(r) +
,
(5)
4π
|r|3
for two particles separated by r [34]. The first term corresponds to contact interactions between the particles with2
strength g = 4π~m as . The DDI gives rise to the second
term, which depends on both the distance and orientation of the vector r compared to the polarisation axis (zaxis) of the dipoles (angle θ). Most properties of dBECs
are well captured by this standard NLGPE (mean-field)[2, 34].
Recent experimental and theoretical results, however,
have established the importance of accounting for quan-

11
tum fluctuations in dipolar condensates [37–41]. Their effect can be included in the NLGPE in a mean field treatment through a Lee-Huang-Yang correction to the chem√
ical potential, ∆µ[n] = 32g(nas )3/2 (1 + 32dd /2)/3 π,which is obtained under a local density approximation[35, 36]. The accuracy of this mean field treatment has
been established, e.g., in Refs. [37, 40, 41], and has
proven succesful in explaining recent experimental results[38, 39]. The final nonlinear term in the extended NLGPE accounts for three-body losses, with an experimentally determined loss parameter L3 , which is dependent
on as and typically of the order L3 ' 10−41 m6 s−1 , asreported in Ref. [39].
L.

BdG spectrum calculations

Collective excitations of the dBEC are obtained by linearising the NLGPE around a stationary state ψ0 , which
can be obtained by imaginary time propagation (Method
N). We write ψ = e−iµt/~ (ψ0 + η[ue−it/~ − v ∗ e+it/~ ]),where µ is the chemical potential associated with stateψ0 , and u, v are spatial modes oscillating in time withcharacteristic frequency /~ and η  1 [44]. Inserting
this ansatz in the NLGPE, and retaining only terms up
to linear order in η we obtain the BdG equations
 
 

ĤGP [ψ0 ] + A
−A
u
u
=
,
v
v
A
−ĤGP [ψ0 ] − A(6)where the operator A, acting on a function f and evaluated at point r, is defined as(Af )(r) =Z

dr0 ψ0 (r0 )U (r − r0 )f (r0 )ψ0 (r)


16
3 2
+ √ ga3/21+|ψ0 (r)|3 f (r). (7)
2 dd
π s

The above equations constitute an eigenvalue problem,
which we solve numerically using the Arnoldi method to
obtain eigenmodes (u, v) and corresponding excitationenergies . The equations presented here are a generalisation of the BdG equations for dipolar systems asderived in Ref. [44], to include the LHY correction for
quantum fluctuations. The LHY term generally serves tostabilise the excitation spectrum, causing the roton instability point to shift to lower scattering lengths. The excited states shown in Fig. 2b-c correspond to the density|ψ0 +η(u−v ∗ )|2 , for particular pairs of (u, v) corresponding to phonon and roton modes. Even while the amplitude η = 0.2 of both modes is taken to be equal in Figs. 2band 2c, the roton excitation (Fig. 2c) leads to markedly
larger local density modulations than the phonon excitation (Fig. 2b). The ToF signatures in Fig. 2 are computedby letting the wave function of the excitation, η(u − v ∗ ),expand ballistically for 30ms, i.e. neglecting interactions
during the expansion. The resulting density |η(u − v ∗ )|2
is then plotted (Fig. 2b1, c1).

M.

Analytical dispersion relation for an infinite
axially elongated geometry

Equation (1) in the main text results from a similar
procedure as that used in Ref. [11] for rotons in infinite q2D traps. We consider a dBEC homogeneous along
y but harmonically confined with frequencies νx and νz
along x and z. For sufficiently strong interactions the
BEC is in the TF regime on the xz plane, in which
p the
BEC wavefunction acquires the form ψ0 (ρ) = n(ρ),
with n(ρ) = n0 1 − (x/X)2 − (z/Z)2 , where X and Z
are the TF radii, and ρ = (x, z). The calculation of n0 ,X and Z is detailed at the end of this section.
Due to the axial homogeneity, the elementary excitations discussed in the previous section have a defined axial momentum ky , and take the form δψ(r, t) =u(ρ)eiky y−it/~ − v(ρ)e−iky y+it/~ . We consider the GPE
given by Eq. (3) without LHY correction and threebody losses, and insert the perturbed solution ψ(r, t) =(ψ0 (ρ) + ηδψ(r, t)) e−iµt/~ . After linearization we obtain
the BdG equations for f± (ρ) = u(ρ) ± v(ρ):
f− (ρ) = Hkin f+ (ρ),
f+ (ρ) = Hkin f− (ρ) + Hint [f− (ρ)],

(8)
(9)

where



~2
∇2 ψ 02
2
Hkin f± (ρ) =−∇ + ky +f± (ρ), (10)
2m
ψ0
Z
0
Hint [f− (ρ)] = 2 d3 r0 U (r − r 0 )e−iky (y−y )ψ0 (ρ)ψ0 (ρ0 )f− (ρ0 ).

(11)

Employing f+ (ρ) = W (ρ)ψ0 (ρ), and for ky  1/X, 1/Z,
we obtain the following equation for the function W (ρ):

 1 ∂2W
1 ∂2W+
0 = 2gn0 1 − x̃2 − z̃ 2
X 2 ∂ x̃2Z 2 ∂ z̃ 2


1 ∂W
1 ∂W
+ 2 z̃
− gn0 (1 + 2dd )x̃
X 2 ∂ x̃
Z
∂ z̃



2m 2
1
1
2
+
( − E(ky ) ) − 2gdd n0
+ 2
~2
X2
Z

4m
− 2 gn0 (1 − dd )E(ky ) 1 − x̃2 − z̃ 2 ,(12)~

where x̃ = x/X, z̃ = z/Z, E(ky ) = ~2 ky2 /2m. For dd = 1
the last term of Eq. (12) vanishes. In that case, thelowest-energy solution is given by W = 1, whose eigenenergy builds, as a function of ky , the dispersion 0 (ky )with0 (ky )2 = E(ky )2 + E02 ,

(13)


~2
1
1
with E02 = 2gdd n0 2m
In the vicinity of
X2 + Z2 .
dd = 1, the effect of the last term in Eq. (12) may beevaluated perturbatively, resulting in the dispersion
(ky )2 ' 0 (ky )2 − 2EI E(ky ),

(14)

12
with EI = 32 gn0 (dd − 1).
This expression for the dispersion√ presents a roton
minimum for dd > 1 at krot = ~1 2mEI . Expanding
2
Eq. (14) in theh vicinity
i of the roton minimum, (ky ) 'd2 2 (ky )dky2, we obtain Eq. (1) of theky =krotpmain text, with ∆ = (krot√) = E02 − EI2 . At the instability, ∆ = 0, and krot = ~1 2mE0 .
(krot )2 + 12

Employing a similar procedure as in Ref. [12] we obtainthat the BEC aspect ratio χ = Z/X fulfills:(1 − dd )(1 + χ)2 + 3dd
χ2
= λ2⊥ ,
(15)
(1 + 2dd )(1 + χ)2 − 3dd χ2
with λ⊥ = νx /νz and
Z2 =



3dd χ2
gn0
(1
+
2
)
−
dd
2π 2 mνz2(1 + χ)2(16)

These two equations fully determine the TF solution forgiven dd , gn0 , and the ratio νx /νz . By inserting theexpressions of X 2 and Z 2 in E0 , we find for dd ' 1:1h2 νz2(1 + χ)2 λ2⊥ +
E02 =
(17)
6
1 + 2χ
pwhereas χ simplifies into χ = λ⊥ (1 + 1 + 1/λ⊥ ). As
a result, at the instability krot `z depends only on thetransverse confinement aspect ratio λ⊥ , and we obtain
the geometrical factor κ introduced in the main text: 1/4
1/4
p2
1
κ = krot `z =1 + χ λ2⊥ +. (18)
3
1 + 2χ
In order to better compare quantitatively with ourquench experiments, we evaluate the 3D TF solution,n = n0 (1 − (x/X)2 − (y/Y )2 − (z/Z)2 ) for the axiallytrapped condensate prior to the quench. We then use
the central density n0 and the TF radii X and Z in theevaluation of the roton spectrum above, employing thevalue of dd after the quench. This procedure takes thequench of as into account by evaluating an initial (i. e.
at th = 0 ms) instantaneous roton spectrum, within theassumption of an immediate change of as . Further refinements might be required to fully describe the dynamical population of the roton mode, related to the roton
softening of the time-dependent instantaneous excitation
spectrum depicted in Method O. Yet, as being inspired
from the latter concept, our approach allows for a better description of our experiments than purely stationary
calculations.
The evaluation of the 3D TF solution is performed byfollowing the procedure of Ref. [50]. We introduce the
BEC aspect ratios: χx = Z/X, and χy = Z/Y . For agiven dd , these aspect ratios fulfil the equations: 2A(dd , χx , χy )
νx
χ2x
=
,
C(dd , χx , χy )νz 2B(dd , χx , χy )
νy
χ2y
,
=
C(dd , χx , χy )
νz

(19)
(20)

where A(dd , χx , χy ) = 1 + dd (F1 − F2 + F3 ),B(dd , χx , χy ) = 1 + dd (F1 − F2 − F3 ), andC(dd , χx , χy ) = 1 + dd (F1 + 2F2 ). Here we have introduced the functions:
#
Z 1 "
3u2
F1 (χx , χy ) = du p
−1 ,
(21)
α2 −β 20#
"Z 1  2
3u −1
3u2
p
−1 ,
F2 (χx , χy ) = du(22)2α2 −β 20#
" p 2Z 1  2
9u (1−u2 )α −β 2 − αp,(23)
F3 (χx , χy ) = du4
β α2 −β 20with α(χx , χy , u) = (χ2x + χ2y )(1 − u2 )/2 + u2 , andβ(χx , χy , u) = (χ2x − χ2y )(1 − u2 )/2. Once χx,y are determined one may then evaluate the TF radii:
gn0
A(dd , χx , χy ),2π 2 mνx2
gn0
Y2 =B(dd , χx , χy ),2π 2 mνy2
gn0
Z2 =C(dd , χx , χy ),2π 2 mνz2

X2 =

(24)
(25)
(26)

and the central density
n0 =1(15N )2/5,8π(ABC)1/5 (as ˜l4 )3/5

(27)

with ˜l2 = 2πm(νx~νy νz )1/3 .
Using this procedure and the experimental parameters(trap geometry and atom number), we calculate the values of krot shown in Fig. 4. These values are computed ata quenched as value taken to be just below the thresholdfor the roton softening. To estimate this threshold, we
use here our full numerical platform based on the generalized NLGPE (Method N), as it is expected to better
quantitatively capture the physical details of our experiments. Using our analytical model, we are also able toestimate a softening threshold in our quench experiment,
corresponding to ∆ = 0 within the above-described simplified approach. The predicted threshold values, a∞s ,in the experimentally relevant geometries are reported
in Extended Data Table II. Despite of the local density
approximation, and the limited validity of the approximate calculation of the roton spectrum performed above,
which is restricted to the vicinity of dd = 1, we note thatthe results for krot and a∗s are in very good agreementwith our experimental results (see Figs. 4-5, Extended
Data Table II)

13N.

Real-time evolution simulations

Our simulations of the NLGPE are performed using a
split operator technique. The evolution operator over atime ∆t (∆t → i∆t for imaginary time evolution) may beapproximately split as e−iĤ∆t/~ = e−iT̂ ∆t/~ e−iV̂ ∆t/~ +O(∆t2 ). In this expression, T̂ is the kinetic energy term,and V̂ the potential energy. The effective potential energy for the evolution is given by the sum of externalpotential, inter particle interactions, local LHY correction, and three-body losses.
We first evaluate using imaginary time evolution theinitial BEC wavefunction, ψ0 (r), prior to the quench ofas . The initial wavefunction for the subsequent real-time
evolution is then constructed via ψi = ψ0 + ∆ψ, where∆ψ accounts for thermal fluctuations, which we simulate by populating the eigenstates of the harmonic trap
as in Refs. [37, 42]. Starting with this initial wavefunction ψi , we mimic as close as possible the conditions of
our experiments, including ramping, holding, and ToF
times. In particular, we include an exponential ramp of
the magnetic field B(t) = Bf + (Bi − Bf )e−t/τ , with
τ = 0.98ms, and convert the magnetic field value to thescattering length as from the experimental measured relation. Moreover, for the value of the three-body loss
coefficient L3 we use a linear fit of the experimentally
determined values.
The simulation of the ToF expansion is performed in
two steps. First we use a multi-grid analysis in orderto rescale the size of the numerical box as the cloud expands during the ToF expansion. After some expansion
time the density drops significantly, and the subsequentevolution can be readily calculated via e−iT̂ t/~ . Our numerics show clearly that the effect of nonlinearity is small
during the first stages of the evolution, and hence that
the ToF expansion indeed may be employed to image themomentum distribution of the condensate at the time inwhich the trap is opened.
We obtain the Fourier Transform of the wavefunction ψ(r, t), ψ̃(k, t), at different times, evaluating the integrated momentum distribution ñ(ky , t) =R
dkx dkz |ψ̃(k, t)|2 . For a sufficiently large t, this momentum distribution shows clear roton side peaks. Thisis typically after a few ms, the exact t value depends onthe gas characteristics (in particular T ) and on as . We
evaluate the roton momentum as the mean value of the
momentum in the roton peak.
From the imaginary-time evolution simulations, we arealso able to predict the as threshold for the roton softening (i. e ∆ = 0), as it is equivalent to the threshold for the
instability of the BEC. To find the softening threshold,we then look for this instability boundary, which, in theimaginary-time, corresponds to the absence of mean-fieldstable solution [40, 41]. To do so, we proceed by steps.
We start by calculating the ground-state solution for agiven as that we know to be well within the stable regime.
We then reduce as in small steps and successively calcu-

late the corresponding ground-state solution using thesolution of the previous step as starting condition. We
do so until no mean-field stable solution can be found.
The predicted instability threshold ais is reported in Extended Data Table II. For the theory predictions shownin Fig. 4 (real-time evolution and analytical calculation(Method M)), we use quenched as values such that the
instability boundary is just slightly crossed, meaning as
is taken 1a0 smaller than the instability threshold ais .

O. Privileged population of the roton mode in aquench of as and instantaneous excitation spectrum

It is interesting to note that, even in the presence of afinite roton gap ∆ in the stationary excitation spectrum,
the roton mode can still be selectively dynamically populated in a quench of as [27]. Indeed the quench canbe viewed as performing a temporary destabilization of
the mode of lowest energies in the stationary spectrum,and thus in particular of the roton mode. This can be
grasped, based on our theoretical description, by decomposing the quench evolution in small time steps and linearising the NLGPE around the instantaneous state of
the gas. As the local state differs from the ground state,
the instantaneous spectrum is modified compared to thestationary one and the non stability of this state leads topushing the spectrum down in energy, destabilizing thelowest energy modes. The dynamically populated modes
are the ones that get instantly destabilized. Then the roton is favored in this respect as a local minimum in (k).
Moreover, the flattening of the dispersion relation aroundkrot makes the different modes in this region oscillate inphase. This might lead to a cooperative enhancement of
the population at short th , as suggested in Ref. [27]. Notethat our argument is based on a linear evolution assumption, which is expected to be valid at least for short th .
The dynamical population of the roton mode is an intriguing problem, which we plan to further investigate in
the future.
Extended Data Table I | dBEC parameters for theexperimental measurements (Fig. 5). The typical statistical uncertainties on νx and νz are below 1 %, and canbe up to 10 % for νy . The experimental repeatability results in 5-to-10 % shot-to-shot fluctuations of N , f andT.
λ νx (Hz) νy (Hz) νz (Hz) N (104 ) f (%) T (nK)
4.3 114
35
149
9
66
45
10.2 183
30
306
11
62
104
14.3 267
32
456
8.6
50
150
21.3 357
30
638
8.4
36
179
29.7 432
26
771
7
20
171

14
Extended Data Table II | Roton softening threshold for the experimental geometries (Table I), deduced
from the C fits for the experimental onset of the population (a∗s , also reported in Fig. 5), from the q1D analyticalimodel (a∞s ) and from imaginary time calculations (as ).
The theory and experiment values match well, yet witha noticeable stronger deviation for the (two) more elongated and tightest traps. We note that these latter cases
depart more from the dd = 1 regime.
i
λ νz (Hz) a∗s (a0 ) a∞s (a0 ) as (a0 )4.3 14951.3
54
50
10.2 306
50.2
52.9
46
14.3 456
4949.1
4321.3 638
47.7
44.8
39
29.7 771
45.2
34.2
30

Appendix

B

Optical setup improvements

Within this Appendix we discuss major experimental upgrades and improvements of the
optical setup that have been implemented during the course of this thesis. For a descriptionof the inital optical setup the reader is referred to Ref. [Fri14a].

Three-dimensional optical lattice
To perform experiments with erbium quantum gases in low dimensions, we implement a
three-dimensional (3D) optical lattice. The optical lattice is created in a standard way.
We retro-reflect the light of laser beams, which are focused on the position of the atoms.
The formed standing wave effectively slices the atomic cloud along the lattice beams. This
technique is applied along three (almost) orthogonal axes to form a crystal-like potential forthe atoms. Our lattice is build with a special geometry: Two orthogonal horizontal green
lattice beams at 532 nm1 and one vertical infrared lattice beam at 1064 nm2 create a cuboid
lattice with a twice as large lattice constant along the vertical direction. The cuboid unit
cell features lattice constants of dx = dy = 266 nm and dz = 532 nm, where x and y lie in
the horizontal plane and depict the axis of the two green lattice beams, and z denotes thevertical axis oriented along the infrared lattice beam. The optical setup of our off-resonant
trapping beams and resonant imaging beams is illustrated in Fig. B.1.
For the vertical lattice, we branch 1064 nm light from the horizontal optical dipole trap
(ODT) setup [Fri14a]. The optical lattice potentials are applied at the very end of the forced
evaporative cooling procedure, where the ODT has already reached low intensities and it
becomes possible to branch off the light. A polarization maintaining high-power fibre3 witha length of 3 m is used to clean the optical profile and delivers up to 6 W for the verticallattice. The angle of the lattice beam with respect to the axis of gravity is ≈ 11 ◦ and thewaist at the position of the atoms is ≈ 300 µm. At full power the harmonic confinement of theretro-reflected vertical lattice beam leads to radial trap frequencies of ωr1064 ≈ 2π × 25 Hz.
12
3

laser light from model Verdi V10, 10 W at 532 nm, from Coherent
laser light from model Mephisto MOPA, 42 W at 1064 nm, from Innolight (now Coherent)
model LMA-PM-15, from NKT Photonics GmbH175

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

100

λ/2

-50

to Dye laser

Green lattice 2

AOM-1
Green lattice 1

110MHz
λ/2

AOM

75

+1

-25 λ/2

λ/4110MHzMephisto

λ/2
300

CCD

λ/2
150

vertical imaging

λ/4

0

200

3

be
sa am
m
pl
er
30

1

2

horiz

PD

ontal

λ/2

Green lattice

1570
nm
ODT

ing

imag

-50

100

PD

ing
imag

vertical lattice 1064 nm

400

2

vert.

retro
GL1 tionreflec1

vertical ODT 1570 nm

PD

-25

3
dichroic
mirror

m
064 n

-25

0

30

ODT
1
horiz

100

250
ing
imag e
tiv
objec

ontal
horiz
ing
imag

CCD

r
am le
be mp
sa

ing

imag
300

ontal

vert.
GL2 retroreflection

50

150
150

400

300

NKT fibre laser

λ/4

250

vert. lattice
retro-reflection

400

vertical ODT 1064 nm

300

100

-50
PD

PD
cylindrical
telescope

Green lattice 1

IPG fiber laser

2

AOM

λ/2

Figure B.1.: Overview of the laser setup for optical trapping, absorption imaging and lattice experiments. The laser beams intersect at the central point of the experimental chamber, shown from aside and a top view. For the horizontal optical lattice beams a diode-pumped solid state laser (Verdi)
at 532 nm (green) is used. The vertical lattice utilizes 1064 nm (red) light from a single-mode master oscillator power amplifier (Mephisto MOPA). The lattice beams are created in a retro-reflecting
configuration. The optical dipole trap beams operated at 1064 nm (red) and 1570 nm (dark red) arealready introduced in Ref. [Fri14a] and shown here for completeness. Absorption imaging is performed
with 401 nm (blue) light that is created by a frequency doubled diode laser (TA-SHG-pro). Figure
adapted from [Fri14a].

176

177
The light for the horizontal lattice beams is prepared on a separate laser table, which islocated next to the experimental table. The 532 nm-light is split into two paths and passes
in each path through a large-aperture acusto-optic modulator4 (AOM), which fulfill two
purposes: First, they are used for an active stabilization of the lattice power and second,
they prohibit interference effects of the lattice beams via a detuning of their respective laser
frequencies to each other. Here, we use for the first green lattice beam the plus-first orderand for the second green lattice beam the minus-first order of the respective AOM, leading
to an absolute detuning of ≈ 220 MHz. Then, the light is delivered to the experimental table
via two 8 m polarization-maintaining fibres. The maximum deliverable power is limited due
to stimulated Brillouin scattering [Ipp72] and reaches values of up to 1.2 W, respectively. In
the future, shorter fibres could be implemented to increase the maximum available power.
Monitoring photodiodes (PD) on the experimental table allow for an active stabilization of
the laser power. The two green lattice beams are aligned such that they are orthogonal
to each other and their respective focal waists are ≈ 160 µm. At full power, the harmonic
confinement of the retro-reflected green lattice beams leads to radial trap frequencies ofωr532 ≈ 2π × 40 Hz, respectively.
Reference [Fri14a] gives more details on the additional optical dipole traps and imaging
beams.

Horizontal imaging
One of the most important technical requirements in ultracold atom experiments is a good
performance of the optical imaging. For absorption imaging, resonant laser light illuminates
the atoms and gets partially absorbed, which results in a reduction of the light intensity
when it passes through the atomic cloud. This shadow-like signature is compared to areference image and hence used to extract the density of the atoms, i. e. the optical depth.
An important parameter is the resolution of this imaging technique, as it gives the limits on
the size of observable structures within the atomic sample. Further, the quality can also be
limited by technical noise on the image.
In our experiment, we typically image the atomic cloud after a certain time-of-flight (TOF),
where the atoms have already expanded and reached a size of several tens of µm. Still,
to extract most information from the absorption images, it is important to have a good
performance in respect to the two relevant limitations: imaging resolution and imaging noise.
To give an example where a good performance has proven to be of relevance, noise-correlation
experiments can be mentioned. Here, pioneering works have shown that strongly correlated
quantum phases can be observed via density-density correlations of atoms in momentum
space in TOF experiments for high-resolution and low-noise images [Gre05, Föl05].

4model I-M110-2C10B6-3-GH26 AOM, from Gooch&Housego

178
Imaging resolution
The maximum possible imaging resolution for our horizontal imaging is given by our numerical aperture (NA), which is about 0.1. This value is set by the geometric size of the CF16
imaging viewport, see Fig. B.1. This NA results in a theoretical diffraction limit of 2.5 µm forour 401 nm imaging light. To reach this limit, we have implemented a self-designed imaging
objective, which has replaced the former final lense before the horizontal CCD-camera, see
Ref. [Fri14a]. The objective consists out of two achromatic doublets5 mounted close to eachother. The optimal combination of lenses has been found via ray-tracing simulations6 . With
the new objective the maximum imaging resolution is almost reached. The magnification of
the final setup is 2.9.

Imaging noise
One major contribution to imaging noise arises from residual interference fringes. Interference fringes appear from scattering of the imaging light at defects or dust on opticalcomponents. As the original absorption image is compared to a reference image, which only
differs by the absence of the atomic cloud, the fringes are expected to cancel after postprocessing. However, due to mechanical vibrations of the optical setup, it is possible thatfringes slightly change in position or amplitude for the two subsequent images and hence do
not vanish. The relevant parameter in this context is the timespan between the two images.
A to long time interval between the pictures facilitates the manifestation of vibrations, which
are typically governed by acoustic frequencies.
A reduction of this noise source for our imaging is tackled by exchanging the former CCD
sensor by a fast CMOS camera7 . The newly implemented CMOS model allows to improve
the sampling rate of our imaging system by three orders of magnitude, reducing the timespan
between the subsequent images from ≈ 100 ms to below 100 µs. This improvement results ina major suppression of residual fringes for the processed atomic picture, thus decreasing thenoise level.

Further improvements
Throughout the coarse of this thesis we have constantly maintained and renewed the experimental setup. This minor technical updates have had the aim to reduce the daily maintenancenecessities and to improve the performance and stability of the experiment.
Among these updates, the most important improvement, as compared to Ref.[Fri14a], relates
5
6
7

model AC508-150 and -250, from Thorlabs
software Zeemax Optics Studioscientific camera model Neo 5.5 sCMOS, front illuminated scientific CMOS, with 2560 × 2160 pixel resolution, 6.5 × 6.5 µm2 pixel size, 16 bit digitization, from Andor Technology

179
to the blue laser setup. The master laser was replaced by a commercial system8 that candeliver up to 1.2 W of 401 nm light. This allowed us to remove the injection-locked slavelaser for the Zeeman slower (ZS) setup, as the light can be transferred by an end-cap fiber9
to the experiment. For best operation of the ZS about 200 mW laser power after the fiber isneeded. With this setup, we find a nice long-term stability. In addition, we have replaced theinjection-locked slave laser for the transversal cooling section by a similar homemade slavelaser that delivers up to 300 mW, almost a factor of four more power than previously. With
this transversal cooling section, we can increase the atom number trapped in the MOT by up
to a factor of seven, as the perpendicular velocity spread of the atomic beam is reduced.

8
9

TA-SHG-pro system, from Toptica Photonics AG
model PMC-400Si, from Schäfter+Kirchhoff

Appendix

C

Magnetic field stabilization

One of the most relevant requirements for our experiment is a precise and accurate controlof the magnetic field. This control is from importance in different contexts:• Particle interaction: As a result of the dense Feshbach spectrum with many narrow
Feshbach resonances in lanthanides, see Chapter 4, tuning the scattering length as via
such resonances requires a high level of control on the magnetic field. This control isespecially needed in the strongly interacting regime, where small magnetic field changes
can strongly affect as .
• Low field regime: An interesting direction with dipolar atoms is the study of phenomena
emerging at very low magnetic field values, see Sec. 6.2. In this low-field region, astable magnetic field is from particular interest, as the observed phenomena can bedramatically influenced if the noise reaches similar amplitudes as the bias field. Hence,
the noise has to be reduced in a controlled manner.
• Dipole orientation: One of the major observables in our experiment is the dependence ofthe physical processes on the orientation of the dipoles with respect to the confinement
geometry and to each other. Here, it is from particular importance to have the angle ofthe dipole orientation, i. e. the magnetic field orientation, well under control to reveal
the underlying physics at play.
With this requirements in mind we have improved the magnetic field control along thethree orthogonal directions in our experiment. While small drifts during the day (typicallya few mG) can easily be calibrated and hence canceled, it is a major challenge to deal
with magnetic field frequency noise, mainly coming from the power lines surrounding theexperiment. This noise is related to the AC-frequency of the power line and shows frequency
components at 50 Hz and multiple higher harmonics, and can have different influences forvarious experimental conditions:• Long time effect: For experiments where we are interested in the properties of theatomic sample at timescales exceeding holding times of 20 ms, i. e. where also theoscillatory behavior of the lowest (50 Hz) noise component is covered, the magnetic
field noise leads to an effective broadening of the applied magnetic field. Hence, one
180

181
has to take into account the magnetic field spread and the resulting spread of e. g. thescattering length as for a specific experiment.
• Dynamical effect: When the effects we want to observe occur at short timescales, thesinusoidal oscillation of the magnetic field has to be taken into account. While for large
bias fields only oscillations of the noise along the bias field matters1 , for magnetic fields
in the order of the amplitude of the noise all three axes have to be accounted. This
proves to be particular striking when the noise oscillations of the different axis are notof the same amplitude and phase, leading to an oscillating rotation of the magnetic
field direction as a function of time.
As a consequence, for certain measurements it is necessary to cancel this magnetic field noisein order to observe physical effects that are sensitive to the absolute magnetic field and theorientation of the dipoles.

Experimental setup
Our aim is to actively cancel the magnetic field noise along the three orthogonal axes.
Therefore, we implement three sets of coils, which can be independently controlled via active
stabilization servo-loops. For each axis we implement a pair of coils with two windings eachon top of the already implemented compensation cage coils (see Ref. [Fri14a]). The coils areshaped rectangularly and have a geometric size of 1.2 × 1.2 × 0.85 m3 . While the center of thecoil pairs overlap with the center of the vacuum chamber in x and y, the center of the z-coils
is shifted by 0.25 m in positive z-direction. For our lab coordinate system see Fig. 2.4.
For active feedback, we implement a three-axis magnetic field sensor2 close to the experimental chamber. The sensor is about 10 cm away from the position of the atoms. The
sensor signal is fed back to three PID-controllers, which actively regulate the current sent
to the three cancellation coil pairs. In brief, the cancellation coils create a magnetic fieldthat has the opposite sign but similar amplitude as the respective noise field, thus cancelingthe magnetic field noise along the three orthogonal axes. The output stage consists of threehigh-current operational amplifiers3 , capable to deliver a maximum current of ±5 A each.
With this we can reach magnetic field values of about ±50 mG along x and y and about±100 mG along z. The PID-controllers are optimized for a frequency range below 1 kHz.
For our experiment this marks the relevant range of noise-cancellation operation as higher
frequency components are shielded by the steel chamber due to eddy currents and do notneed to be compensated.

1

2
3

The total magnetic field accounting
p for the field of three orthogonal axes is calculated by adding thequadratic components, i. e. |B| = Bx2 + By2 + Bz2 . Hence, for a large bias field along one of the axis,
e. g. Bx , the magnetic field noise along the other axes, which leads to small varying fields By and Bz , canbe neglected.
√
model Mag-03-MS-70, three-axis magnetic field sensor, noise < 10 pt/ Hz, from Bartington Instruments
model OPA549, from Texas Instruments

182
4

(a)

-20

(c)

-40

0

-120

-2

40

-60

-4
10

20
time (ms)

30

40

10
1

-100

0.1

-120

0

2

4
6
time (ms)

8

10

50

60

400

500

-80

(b)
2 |Vx| (mG)

0

0.01

-80

-40

Vx (dbVrms)

Vx (mG)

2

0

100

200
300
noise frequency (Hz)Figure C.1.: Performance of the magnetic-field-stabilization setup along the x-axis. The performance is measured with the magnetic field sensor, which is used for stabilization. Performance atthe position of the atomic cloud might differ. Magnetic field fluctuation without (black) and with(green) active stabilization in the time domain (a-b) and frequency domain (c). (a) Amplitude ofmagnetic field noise and (b) normalized total amplitude in logarithmic scale within a narrower time
window. (c) Frequency spectrum of the magnetic field fluctuations up to 500 Hz. On top of a smoothnoise floor, 50 Hz and higher harmonics are the dominant frequency components. The inset shows azoom-in into the 50 Hz region.

Performance
To investigate the performance of our magnetic-field-stabilization setup, we analyze the noisein time and frequency domain without and with active stabilization. The measurements are
performed directly with the magnetic field sensor that is used within the feedback loop4 . The
magnetic field noise close to the atomic cloud is found to have peak-to-peak amplitudes ofabout 8 mG along x and y and about 2 mG along z. We note that, the lower noise amplitude
along the z-axis results from a passive shielding by the massive optical table, which primarily
affects the z-noise.
The total performance of our magnetic-field-cancellation setup, exemplary shown for thex-axis, is summarized in Fig. C.1. Similar performance is found along the other two axes.
The initial peak-to-peak amplitude of ≈ 8 mG can be reduces below 1 mG when the stabilization is applied, see Fig. C.1(a-b). The most prominent frequency components are 50 Hz
and 150 Hz, while also other frequency components are contributing, see Fig. C.1(c). Our
magnetic field stabilization manages to reduce the noise amplitude by up to 45 db. At frequencies above 1 kHz the feedback loop acquires a π phase lag, resulting in an enhanced
noise at this frequency components. This high-frequency oscillations can be neglected, as
they are shielded by the steel chamber and do not reach the atomic sample.
4

The magnetic field sensor reports the noise amplitude in voltage, where 14.3 mV correspond to 1 mG.

183
relative magnetic field along x (mG)normalized atom number

-10

-8

-6

-4

-13.0

-9.8

-6.5

-2

0

2

4

6

8

10

-3.3
0.0
3.3
6.5
relative rf-frequency (kHz)

9.8

13.0

16.3

1.0

0.8

0.6
-16.3

Figure C.2.: Performance of the magnetic-field-stabilization setup measured with 168 Er via Zeeman
spectroscopy, see text. The measurements are performed for a bias field along the x-axis. The
normalized atom number without (squares) and with (circles) active stabilization is plotted as afunction of the frequency of the rf pulse, relative to the resonance frequency. The rf frequency can beconverted into magnetic field via the knowledge of the Zeeman splitting, see Eq. 2.7. To estimate thepeak-to-peak width of the noise, we use the ±2σ width of the Gaussian fitting functions (solid lines).
With this method we extract a noise amplitude without and with stabilization of 12.6(1.4) mG and
0.9(1) mG, respectively.

As an ultimate test of the performance, we measure the amplitude of the magnetic field
noise directly with the atomic cloud. For the measurement we resonantly excite the atoms
from the lowest to higher atomic Zeeman states by a radio-frequency (rf) pulse. The pulse
lasts for 20 ms as to cover also the lowest frequency noise component (50 Hz). When theenergy of the rf excitation matches the differential Zeeman energy, atom loss is encountered
due to dipolar relaxation of populated higher spin states. The width of the resonant lossfeature carries information on the magnetic field noise. In particular, it tells how muchthe magnetic field was drifting or oscillating during the rf spectroscopy. In turn, the width
can be used as a measure of the performance of our magnetic field stabilization. Figure
C.2 shows our measurements and reveals that the magnetic field noise across the atomic
sample can be reduced below 1 mG along the x-axis. Similar performance is found along thetwo orthogonal directions. Our observations confirm the results of Fig. C.1. The achievedmagnetic field stability is a promising starting point to investigate physical phenomena forwhich a precise magnetic field control is required.

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Acknowledgments
I have joined the Erbium experiment already during my master studies, and it turned out tobe one of the best decisions I could make. Beside the great team spirit, which has remained
curious and friendly during all the years, the experiment has been an exciting scientificworking place, which enabled me to learn and discover intriguing physics at a constantpace. Already back at my master thesis work, I had the luck to contribute to a worlds
first – the realization of a Bose-Einstein condensate of erbium. Greatly, this milestone only
set the start to the following scientific journey with many more worlds firsts. Working with
erbium, an element previously unstudied in the ultracold regime, imposed exciting challenges
on our team and every scientific project added another piece of knowledge to our growing
understanding. The access of physical phenomena in the quantum world requires a complex
experimental setup and the contribution of many people. Here, I want to thank all thepeople that not only have been important for the success of our scientific visions, but also
made the last five years a very enjoyable and exciting time.
First of all, I would like to thank my supervisor Francesca Ferlaino. She has introduced me
to the fascinating world of ultracold atoms and has given me the opportunity to perform mythesis at the Erbium experiment. Her constant scientific curiosity and her strong will formaking progress have enabled me to extract the most out of the last years. I have grown bothscientifically as well as a person by realizing also my own ideas in the lab and by presenting
our results on conferences around the world. I am grateful for her support during the various
phases of my thesis work, and her support along my future scientific career.
During the last years the people within the Erbium team have changed almost completely
and there are many people I want to thank for their assistance, patience and friendship.
In the first phase of my thesis I could learn a lot from our two postdocs Kiyotaka Aikawa
and Michael Mark, as well as from the first generation Phd student Albert Frisch. Kiyo
gave me deep insights into the scientific world and his constant energy and focus to reveal
the physics at play made a large impact on my scientific personality. Michael proved to bea master in solving every emerging problem and I enjoyed a lot his companion in the lab.
Albert made an excellent job in designing and building the Erbium machine from scratch
and his thorough work attitude and his way of documentation set the unquestioned base forthe today still fully functional Erbium machine. I have learned a lot from his technical skills
and his way of approaching physical problems.
The departure of the former Erbium members marked the start of the second phase of
my thesis. New team members filled in and together we smoothly carried on our scientificideas. Our senior scientist Manfred Mark constantly proves to be the important link for ourcoherently working Dipolar group. It is hard to find a question about electronics he is203not able to answer, and his great technical skills are complemented by his ability to have foralmost every question a Matlab script at hand. Our postdoc Lauriane Chomaz has a deep
understanding about our experiment also from the theoretical side and her work and mindful
ideas have been a great benefit for the whole team. Daniel Petter joined our team alreadyduring his master studies and has now become the person to ask for, when something in thelab decides to stop working. With the recent major upgrade by implementing a dynamical
controllable spatial light modulator into the experiment, he turned to the new lead Phd
student and I am confident that he will have a lot of fun working at the Erbium machine.
For one year our team obtained great support from the borrowed Phd students Giulia Faraoni
and Jan Hendrik Becher. Especially the time spent with Jan Hendrik as part of Team Spinin the lab – no matter if early in the morning or late in the night – is of fond memory. The
Hulk mode turned out to be indispensable for efficient lab progress. To the new generation of
Phd students Alexander Patscheider, and Gabriele Natale, who decided to stay in our teamafter their successful master thesis’ with us, I wish the best of luck for their future scientificprojects in the lab.
With the Rare experiment emerging, our team has grown larger and larger and I haveenjoyed the delightful discussions and the time shared together. I will never forget our
weekly Besprechung&Breakfast meetings and I have particularly enjoyed our yearly groupretreats to conferences all around Europe.
All people within the ultracold-atoms research group, which was initially set up by Rudi
Grimm, participate to the great environment in Innsbruck. Rudi took a special role during
the initial phase of my thesis as he played an important supervisory role for our first scientificresults. Despite that the Erbium team has relocated from the fourth to the second floor, we
still keep a lively and most valuable scientific exchange with the teams of Rudi Grimm andalso the teams of Hanns-Christoph Nägerl. In the lab area, behind almost every door you
can most probably find someone who is an expert on something you are struggling with. I
want to thank everyone for making work so enjoyable and instructive. Especially, I want tomention Michael Gröbner with whom I spent all my studies together and I am thankful forthe uncountable discussions about physics and other things. Outside the lab I am gratefulfor the contact, time and friendship with past and current ultracold-atoms members.
Further, I want to thank my second supervisor Gerhard Kirchmair who always has an open
ear to discuss about quantum physics outside the ultracold-atoms world. I am especiallygrateful for his encouragement towards my future scientific career.
A big thank you should also go to all our collaborators, which enriched our work withthoughtful discussions and striking ideas: George Bruun from Aarhus, John Bohn and Debbie
Jin from Boulder, Svetlana Kotochigova and her team from Philadelphia, Eite Tiesinga
from Maryland, Olivier Dulieu, Goulven Quéméner, Jean-François Wyart, and Maxcence
Lepers from Orsay, the group of Tilman Pfau from Stuttgart, Peter Zoller, Mikhail Baranov,Marcello Dalmonte, Andreas Läuchli and many more from Innsbruck, the team of Markus
Greiner from Harvard, Luis Santos and Falk Wächtler from Hannover, Antun Balaz and
Vladimir Veljic from Belgrade, Axel Pelster from Kaiserslautern, as well as Ana Maria Rey
and Bihui Zhu from Boulder. Certainly there are many more people who contributed in one
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way or the other and I want to thank everyone for the great discussions at conference-calls
or conferences and workshops around the world, from which I have learned a lot.
To be able to build, maintain and manage our experiments we also greatly rely on the nonscientific staff. The mechanical workshop with Armin Sailer, Anton Schönherr and Christoph
Wegscheider, and the electronic workshop with Wolfgang Kratz, Wolfgang Kuen and Gerhard
Hendl deliver almost everything needed from the technical side. The administrative supportof our secretaries Christine Götsch-Obmascher, Nicole Jorda, Michaela Palz, Carina-Theresa
Oberhöler, Carla Fantini, Gaby Holzer, Leonarda Garcia Lopez, and Silvia Bonazza eases
our life significantly and enables us to focus on the physics component of our work.
My biggest thanks goes to my friends and my whole family. In particular, I want to thankmy parents Reinhold and Julia, who have always supported my dreams and visions and also
my brother, who has always been there for chats, sports and beer. In addition, I want tothank my uncle Gottfried for his constant support along my scientific career. However, thisthesis truly belongs to my wife Katrin, who has always encouraged and pushed me to keep
going. Your love and support is invaluable and I truly feel lucky that we will spend the restof our lives together. Even though it seems that our newborn son Florian is now a bit taking
over – I could not be happier.

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