Creation of
Dipolar Quantum Mixturesof Erbium and Dysprosium– DISSERTATION –byPhilipp Ilzhöfer, MSc., BSc.

submitted to the Faculty of Mathematics, Computer Science, andPhysics of the University of Innsbruckin partial fulfillment of the requirements for the degree ofDoctor of Philosophy (PhD)
advisors:
Univ. - Prof. Dr. Francesca Ferlaino,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences
Univ. - Prof. Dr. Rudolf Grimm,
Institute for Experimental Physics, University of InnsbruckInstitute for Quantum Optics and Quantum Information of theAustrian Academy of Sciences
Innsbruck, August, 2020

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

Statutory declaration
I hereby declare by my personal signature that I have authored this thesis independently
and that I have not used other than the declared sources and resources. All material
which has been quoted either literally or by content from the used sources has been
explicitly marked.
The present thesis has not yet been submitted in the same or a similar form as a
Magister/master’s/diploma thesis/dissertation.

Philipp Ilzhöfer

Innsbruck, den

3

Synopsis
Over the past two decades, ultracold atom systems and especially ultracold quantum
gas experiments underwent a steep progression curve and emerged as highly versatile
experimental platforms. Rapid technological advances opened new paths to a large variety
of different experimental techniques allowing nowadays not only for a high degree ofcontrol at the quantum level but also for the investigation of the most complex elements
within the periodic table of elements. Recently, the highly magnetic lanthanoids erbium
and dysprosium, the most magnetic element of all, moved into focus, as both introduce
the magnetic dipole-dipole interaction, a long-range anisotropic interaction, to thosehighly controllable experimental platforms via their intrinsically large magnetic moments.
This PhD thesis presents the first experimental apparatus for heteronuclear quantumdegenerate dipolar mixtures of erbium and dysprosium. Its structure follows the developments over approximately five years from this endeavor’s very beginning in early2015 to the achievement of a state-of-the-art system in the field of dipolar quantum
gases today. Following a comparison of both elements’ experimentally relevant physicaland atomic properties as well as the theoretical description of a (heteronuclear) dipolar
Bose-Einstein condensate, it reviews the complete layout of the experimental apparatus
and the obtained results in three chapters.
First, it covers a dual-species intercombination-line magneto-optical trap in a novel opentop configuration as starting point for all further experimental steps. This encompasses
a detailed description of the ultra-high-vacuum apparatus, the magnetic field systems aswell as the laser systems and their integration to the experimental apparatus.
Second, it addresses the production of five different heteronuclear dipolar Bose-Einsteincondensates and of one quantum-degenerate Bose-Fermi mixture via evaporative cooling inan optical dipole trap. The experimental results revealed complex evaporation dynamics
due to sympathetic cooling as well as first evidence for in-trap interspecies interactions
between the Bose-Einstein condensates of erbium and dysprosium.
Third, it presents its contributions to the first experimental realization of supersolid
phases in dipolar quantum gases of erbium and dysprosium in early 2019. Among the
ground-breaking work in three different research groups, it contributed the achievementof a supersolid phase in a dipolar quantum gas of dysprosium with long lifetimes andthe direct evaporative cooling into it, and, in a second study, insights into the phasecoherence dynamics of an out-of-equilibrium dysprosium supersolid.
This PhD thesis concludes with a summary of the achieved results and provides an
outlook at upgrades to the experimental apparatus, next characterization measurements
and first experiments with quantum-degenerate dipolar mixtures of erbium and dysprosium. Further, it discusses prospects for multivalent Rydberg atoms, as the complex
valence electron configurations of lanthanoids might facilitate novel Rydberg manipulation
methods. For this purpose, this PhD work developed a dedicated science module.
5Zusammenfassung
Im Verlauf der letzten zwei Jahrzehnte durchliefen ultrakalte Atomsysteme und insbesondere ultrakalte Quantengas-Experimente eine steile Progressionskurve und gingen
hierbei als äußerst vielseitige experimentelle Plattformen hervor. Rapide technologische
Fortschritte eröffneten neue Pfade zu einer großen Vielfalt an verschiedenen experimentellen Techniken, die heutzutage nicht nur ein hohes Maß an Kontrolle auf der Quantenebene, sondern auch die Erforschung der komplexesten Elemente im Periodensystem
der Elemente ermöglichen. Vor Kurzem rückten die hochmagnetischen Lanthanoide Erbium and Dysprosium, das magnetischste aller Elemente, in den Fokus, da beide durch ihre
intrinsisch großen magnetischen Momente die magnetische Dipol-Dipol-Wechselwirkung,eine langreichweitige, anisotrope Wechselwirkung, in diese hochgradig kontrollierbaren
experimentellen Plattformen einbringen.
Diese Dissertation präsentiert die erste Versuchsapparatur für heteronukleare, quantenentartete, dipolare Mischungen aus Erbium und Dysprosium. Ihre Struktur folgt hierbei
den Entwicklungen über einen Zeitraum von etwa fünf Jahren, von den Anfängen dieses
Unterfangens zu Beginn des Jahres 2015, hin zum Erreichen eines hochmodernen Systems
auf dem Gebiet der dipolaren Quantengase heute. Anschließend an einen Vergleich derexperimentell relevanten physikalischen und atomaren Eigenschaften beider Elemente,sowie der theoretischen Beschreibung eines (heteronuklearen) dipolaren Bose-EinsteinKondensats, schildert sie in drei Kapiteln das gesamte Layout der Versuchsapparatursowie die erzielten Forschungsergebnisse.
Zunächst behandelt sie eine magneto-optische Falle für beide Spezien, die auf einerInterkombinationslinie in einer neuartigen open-top Konfiguration operiert und als Startpunkt aller weiterer experimenteller Schritte dient. Dies umfasst eine detaillierte Beschreibung der Ultrahochvakuum-Apparatur, der Magnetfeldsysteme, sowie der Lasersysteme
und deren Integration in die Versuchsapparatur.
Anschließend befasst sie sich mit der Erzeugung von fünf verschiedenen heteronuklearen,
dipolaren Bose-Einstein-Kondensaten und einer quanten-entarteten Bose-Fermi-Mischungdurch Verdampfungskühlung in einer optischen Dipolfalle. Die experimentellen Resultate
zeigten hierbei eine komplexe Verdampfungsdynamik aufgrund von sympathetischem
Kühlens, sowie erste Hinweise auf Interspezies-Wechselwirkungen zwischen den gefangenen
Bose-Einstein-Kondensaten von Erbium und Dysprosium.
Drittens stellt sie ihre Beiträge zur ersten experimentellen Realisierung von suprasoliden
Phasen in dipolaren Quantengasen aus Erbium und Dysprosium zu Beginn des Jahres
2019 vor. Unter den bahnbrechenden Arbeiten in drei verschiedenen Forschungsgruppen
trug sie das Erreichen einer suprasoliden Phase in einem dipolaren Quantengas aus
Dysprosium mit langer Lebensdauer und das direkte Verdampfungskühlen in diese, und,in einer zweiten Studie, Einblicke in die Phasenkohärenzdynamik eines DysprosiumSuprafestkörpers außerhalb seines Gleichgewichts bei.
7

Diese Dissertation schließt mit einer Zusammenfassung der erreichten Ergebnisse
und gibt einen Ausblick auf Erweiterungen der Versuchsapparatur, nächste Charakterisierungsmessungen und erste Experimente mit quanten-entarteten, dipolaren Mischungen
aus Erbium und Dysprosium. Außerdem diskutiert sie Perspektiven für multivalente
Rydbergatome, da die komplexen Valenzelektronenkonfigurationen der Lanthanoide
möglicherweise neue Manipulationsmethoden für Rydbergatome ermöglichen könnten.
Für diesen Zweck hat diese Doktorarbeit ein dediziertes Wissenschaftsmodul entwickelt.

8

Contents
Synopsis

5

Zusammenfassung

7

1 Introduction
11
1.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Thesis research topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Erbium and Dysprosium
2.1 Physical properties and isotopes . . . . . . . . . . . . . . . . . . . . . . .
2.2 Atomic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Energy level diagrams and laser cooling transitions . . . . . . . . .
2.2.2 Atomic polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Dipolar Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Interactions within the mean-field description . . . . . . . . . . . .
2.3.2 Heteronuclear dipolar Bose-Einstein condensates . . . . . . . . . .

17
17
18
21
24
26
27
30

3 Dual-species intercombination-line magneto-optical trap
3.1 Ultra-high vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Magnetic field systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Spin-flip Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Layout of the main chamber coils . . . . . . . . . . . . . . . . . . .
3.3 Laser systems and optical setups . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Broad transition: 401 nm and 421 nm . . . . . . . . . . . . . . . . .
3.3.2 Intercombination-line transition: 583 nm and 626 nm . . . . . . . .
3.3.3 Integration into the experimental apparatus . . . . . . . . . . . . .
3.4 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Publication I:
Two-species five-beam magneto-optical trap for erbium and dysprosium .

31
32
36
36
39
44
44
46
50
55
56

4 Heteronuclear dipolar quantum mixtures
4.1 Optical dipole trap: 1064 nm . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Evaporative cooling to quantum degeneracy . . . . . . . . . . . . . . . . .
4.2.1 Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Evaporation dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Publication II:
Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms . . . . . .

65
66
67
69
70

57

75
9

Contents
5 Supersolidity in dipolar quantum gases of dysprosium
83
5.1 Publication III:
Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases 89
5.2 Publication IV:
Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar
atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 Conclusion and Outlook
125
6.1 Dipolar quantum few-/many-body physics . . . . . . . . . . . . . . . . . . 126
6.2 Multivalent Rydberg atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7 Bibliography131

Acknowledgments

145

10

Chapter1

Introduction
Already in the early days of quantum mechanics, fundamentally important theoreticalwork devoted to the quantum statistical character of particles. In 1924, Albert Einstein
extended Satyendra Nath Bose’s approach, to reconcile the photon hypothesis with
Planck’s law [1], to an ideal gas [2], with a direct proceeding in 1925 [3]. Shortly after,in 1926, Enrico Fermi [4] and Paul Dirac [5] independently described the influence ofPauli’s exclusion principle. These initial findings gave rise to the understanding that
all integer-spin particles (bosons) follow Bose-Einstein statistics and can undergo aquantum statistical phase transition, the Bose-Einstein condensation, to a macroscopic
occupation of a system’s lowest energy or ground state [6, 7], whereas half integer-spin
particles (fermions) follow the Fermi-Dirac statistics, governing a quantum-degenerate
Fermi gas (DFG) [8]. However, for quantum statistics to generally take over, and thus todetermine the character of a system, the particles’ thermal de Broglie wavelength must
be comparable to or larger than the mean interparticle distance [6, 7].
After decades of extensive research, ultracold dilute atomic gases emerged as prime
experimental platform to study a regime of dominating quantum statistics in a highly
controllable environment. The groundbreaking work on techniques to laser-cool and trap
atomic gases, for which Steven Chu, Claude Cohen-Tannoudji and William D. Phillips
received the Nobel prize award for development of methods to cool and trap atoms withlaser light in 1997 [9], eventually led to the first realization of Bose-Einstein condensates(BECs) of rubidium [10] and of sodium [11] in 1995. In the following, a series ofexperiments confirmed for instance the coherence of a BEC by interfering two BECs infree expansion [12] or the superfluid character of interacting BECs by quantized vortex
excitations [13, 14, 15]. The major importance of the first observation of Bose-Einsteincondensation led to the prompt Nobel prize award to Eric A. Cornell, Carl E. Wieman
and Wolfgang Ketterle for the achievement of Bose-Einstein condensation in dilute gasesof alkali atoms, and for early fundamental studies of the properties of the condensatessoon after in 2001 [16].
Meanwhile, in 1999, Deborah S. Jin’s group reported the first realization of a two spincomponent DFG of potassium at temperatures as low as half the Fermi temperature [17].
In the following, for instance [18] demonstrated strong interactions in a DFG of lithiumor [19] and [20, 21] utilized the BEC-BCS crossover to produce molecular BECs offermionic potassium and of fermionic lithium, respectively.
Nowadays, there exist BECs of atomic hydrogen [22], all alkali metals except francium [10, 11, 23, 24, 25], the alkaline-earth metals calcium [26] and strontium [27], the11Chapter 1 Introduction
transition metal chromium [28], the lanthanoids dysprosium [29], erbium [30], thulium [31]and ytterbium [32], and the noble gas helium [33], as well as DFGs of potassium [17],lithium [34], strontium [35], chromium [36], dysprosium [37], erbium [38], ytterbium [39]and helium [40]. All these elements differ in their properties starting from their isotopes’
quantum statistics ranging over their laser cooling properties to their interactions. The
research field of ultracold quantum gases reflects this diversity not only in the vastvariety of raised scientific questions to be answered and of pursued research directions,for instance the field of quantum simulation [41] or the field of quantum metrology [42]with exemplarily three-dimensional optical lattice clocks utilizing DFGs [43], but also in
the available experimental techniques. A particular strength of ultracold quantum gas
experiments lies within the multitude of external control and manipulation parameters.
In general, magnetic traps [7] and single- or crossed-beam optical dipole traps [44]provide ideal conditions for the initial production steps, but their respective working
principle can even further extend to tailored trapping potential landscapes specific toan addressed scientific question. Here, optical lattices emerged as a powerful tool toinvestigate quantum many-body systems and their quantum phase transitions suchas the superfluid-Mott insulator transition [45]. In combination with high-resolution
imaging, those are even observable at the single-atom level, as for instance [46] and [47]demonstrated with rubidium for the transition to a Mott insulator phase or [48] withfermionic lithium for a transition to an antiferromagnet.
Moreover, ultracold quantum gases offer a precise control over their interparticle
interactions. Magnetically tunable Feshbach resonances enable the adjustment of thecontact interaction [49], which opens a wide spectrum of research directions alreadyfor purely contact-interacting systems. One particular research focus are heteronuclear
mixtures of purely contact-interacting elements, as the additional interspecies interactions
increase the complexity and open new routes of manipulation. Experiments achieved
quantum degeneracy for instance for Bose-Bose mixtures of rubidium and potassium [50],for Bose-Fermi mixtures of sodium and lithium [51] or for Fermi-Fermi mixtures of lithium
and potassium [52]. One among many research directions leads along the productionof polar ground-state molecules by initially associating weakly-bound heteronuclear
Feshbach molecules, such as potassium-rubidium [53], rubidium-cesium [54, 55] or sodiumpotassium [56] molecules, and the achievement of quantum degeneracy [57]. As theirconstituents differ in electronegativity, these molecules have tunable electric dipole
moments, and thus give rise to the long-range anisotropic electric dipole-dipole interaction.
This aspect is of interest, among others, in the field of ultracold chemistry or for quantum
information or quantum simulation applications [58]. Because of the complexity of thisapproach, however, the direct laser cooling and trapping of polar molecules such as
strontium monofluoride [59, 60, 61], yttrium oxide [62], calcium monofluoride [63, 64] orcalcium monohydroxide [65] is another viable option.
Alternatively, magnetic elements such as chromium, erbium or dysprosium, the mostmagnetic one in the periodic table of elements, facilitate direct access to the longrange anisotropic magnetic dipole-dipole interaction via their intrinsically large magnetic
moments [66], and thus awakened great interest in the community. The aim of this PhD
thesis was the first production of heteronuclear quantum-degenerate dipolar mixtures of
erbium and dysprosium. The following Section 1.1 gives a brief overview over the major
experimental developments in the field of magnetically dipolar quantum gases up to thisPhD work’s starting point in early 2015 and Section 1.2 highlights the research topics.
12

1.1 State of the art

1.1 State of the art
In 2005, [28] reported the first observation of Bose-Einstein condensation of chromium,and thus opened a path to investigate dipolar quantum gases. Consequently, while theinfluence of the magnetic dipole-dipole interaction already became evident with thedemonstration of demagnetization cooling of chromium [67], a series of experiments
studied a chromium BEC, for instance its behavior in the strongly dipolar regime [68], its
stability diagram depending on the contact interaction strength and the trap geometry [69]or its so-called d-wave collapse [70]. Although these first results represent only a glimpse of
the magnetic dipole-dipole interaction’s true potential in the context of dipolar quantum
gases, they reveal at the same time a major experimental drawback of a chromium BEC,
as it generally requires the adjustment of the contact interaction strength to be in thestrongly dipolar regime.
Meanwhile, pioneering work on laser cooling and trapping concepts for erbium wasunderway and, in 2006, led to the first erbium magneto-optical trap (MOT) [71] and soonafter, in 2008, to a further extension by a blue-detuned narrow-line MOT [72]. Becauseof the elements’ great similarities, the same techniques are applicable to dysprosium andformed the basis for the first dysprosium MOT in 2010 [73] as well as the first dysprosium
BEC, and thus the first intrinsically strongly dipolar Bose-Einstein condensate (dBEC) in2011 [29]. In the following year 2012, a number of rapid developments occurred: first, [74]presented an intercombination-line MOT for erbium, immediately followed by the firstobservation of an erbium dBEC [30] and of a dysprosium DFG [37], respectively. For thelatter, bosonic dysprosium served as sympathetic coolant but did not reach quantum
degeneracy. In 2014, [38] realized the first erbium DFG via pure universal dipolar
scattering and [75] adopted the intercombination-line MOT approach to dysprosium.
Subsequent to these fundamental results, which established first production routes
for dipolar quantum gases of erbium or dysprosium, first investigations devoted to theirrespective Feshbach resonance spectra [76, 77]. Here, [77] revealed a chaotic scattering
behavior in erbium, which a comparative study confirmed also for dysprosium [78].

1.2 Thesis research topics
This PhD thesis presents the first experimental apparatus for combining erbium anddysprosium in the field of dipolar quantum gases. The endeavor started in early 2015
with an empty lab and the aim to develop and to construct an experimental apparatus
for a rapid but highly robust production of heteronuclear quantum-degenerate dipolar
mixtures of both elements with a similar or even superior performance in comparison torespective single-species experiments. A second key aspect in the development process
was the achievement of a high versatility to allow for more advanced experiments inspecialized experimental environments in the future.
The process from initial conceptual ideas to the start of construction took approximately
one year. Over the course of the following three years, the experimental apparatus became
progressively operational. It first demonstrated a dual-species intercombination-line MOT
in a novel open-top configuration and subsequently various heteronuclear dBECs, thebinary Bose-Einstein condensation in atomic mixtures of two highly magnetic elements,
and a quantum-degenerate Bose-Fermi mixture. In early 2019, it joined in at the forefrontof the search for supersolidity in dipolar quantum gases.
13

Chapter 1 Introduction
This PhD thesis led to four publications in total and highlights the complete PhD
work over the next five chapters in the following structure:
Chapter 2 summarizes and compares the most important properties of erbium anddysprosium, encompassing their vapor pressures, electronic ground state structures, magnetic moments as well as laser cooling and trapping properties. Moreover, it examines
dBECs and their interactions in the single-species and the heteronuclear case.
Chapter 3 describes the experimental apparatus in the context of this PhD thesis’first publication on a dual-species intercombination-line MOT for erbium and dysprosiumin a novel open-top configuration. It details the ultra-high vacuum apparatus, the magnetic field systems, the various laser systems and their integration into the experimental
apparatus as well as the imaging and the control system.
Publication I:
Two-species five-beam magneto-optical trap for erbium and dysprosium
Philipp Ilzhöfer, Gianmaria Durastante, Alexander Patscheider, Arno Trautmann,
Manfred J. Mark and Francesca Ferlaino
Physical Review A, 97, 023633, (2018), DOI: 10.1103/PhysRevA.97.023633
Chapter 4 reviews the production of five different heteronuclear dBECs and of aquantum-degenerate Bose-Fermi mixture in the context of this PhD thesis’ second publication. It addresses the optical dipole trap, the experimental sequence as well as theevaporation dynamics for all studied isotope mixtures of erbium and dysprosium.
Publication II:
Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms
Arno Trautmann, Philipp Ilzhöfer, Gianmaria Durastante, Claudia Politi, Maximilian
Sohmen, Manfred J. Mark and Francesca Ferlaino
Physical Review Letters, 121, 213601, (2018), DOI: 10.1103/PhysRevLett.121.213601,
Editors’ suggestion,
Featured in Physics 11, s135, Synopsis: Making Mixtures of Magnetic Condensates
Chapter 5 highlights supersolidity in dipolar quantum gases in the context of thisPhD thesis’ third publication, where it contributed the experimental realization of along-lived dysprosium supersolid, even via direct evaporative cooling, and of its fourth
publication on the rigidity of global phase coherence and the rephasing dynamics of an
out-of-equilibrium dysprosium supersolid. It briefly reviews past research efforts anddiscusses the special aspects supporting a supersolid phase in a dipolar quantum gas.
Publication III:
Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases
Lauriane Chomaz, Daniel Petter, Philipp Ilzhöfer, Gabriele Natale, Arno Trautmann,
Claudia Politi, Gianmaria Durastante, Rick M. W. van Bijnen, Alexander Patscheider,
Maximilian Sohmen, Manfred J. Mark and Francesca Ferlaino
Physical Review X, 9, 021012 (2019), DOI: 10.1103/PhysRevX.9.021012,
Featured in Physics 12, 38, Viewpoint: Dipolar Quantum Gases go Supersolid
141.2 Thesis research topics
Publication IV:
Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms
Philipp Ilzhöfer, Maximilian Sohmen, Gianmaria Durastante, Claudia Politi, Arno
Trautmann, Giacomo Morpurgo, Thierry Giamarchi, Lauriane Chomaz, Manfred J. Markand Francesca Ferlaino
arXiv-preprint, arXiv:1912.10892
Chapter 6 concludes this PhD thesis and provides an outlook on the next experimental steps employing these first quantum-degenerate dipolar mixtures of erbium anddysprosium. Further, it explores multivalent Rydberg atoms of lanthanoids and describes
a dedicated science module, which comprises an in-vacuum three-dimensional electricfield control and microchannel plate detectors.
Note that, as this PhD thesis focuses on the two highly magnetic elements erbium anddysprosium, the following discussion and the introduced concepts are exclusively in viewof the magnetic dipole-dipole interaction, even if concepts are analogous to an electricallydipolar system.

15

2

Chapter

Erbium and Dysprosium

Erbium (Er) and dysprosium (Dy) are, as Figure 2.1 shows, both part of the lanthanoid
series in the sixth period of the periodic table of elements [79]. As next-nearest neighbors,
uniquely characterized by proton numbers ZEr = 68 and ZDy = 66 and only separated
by holmium [79], they manifest qualitatively similar but distinct properties.
This second chapter provides a careful comparison of both elements with a special focus
on the experimentally relevant aspects. Section 2.1 starts with their physical properties
and isotopes. Then, Section 2.2 studies their atomic properties incorporating energy level
diagrams, laser cooling transitions and atomic polarizabilities. Additionally, Section 2.3
explores single-species and heteronuclear dBECs and their interactions.

2.1 Physical properties and isotopes
Assuming room temperature and standard conditions for pressure, erbium and dysprosium
exist in solid phases exhibiting a hexagonally close-packed lattice [79] with paramagnetic ordering [81, 82]. Moreover, both have similar densities ρEr,25 °C = 9.07 g/cm3 andρDy,25 °C = 8.55 g/cm3 , melting points Tm,Er = 1529 °C and Tm,Dy = 1412 °C as well as
boiling points Tb,Er = 2868 °C and Tb,Dy = 2567 °C [79].
From an experimental point of view, the great similarity in Tm,Er and Tm,Dy is ofparticular interest, which, besides the similar laser cooling properties (see Section 2.2.1),
facilitates the realization of a dual-species atomic beam source. However, despite thesebeneficial aspects, the vapor pressures pvap,Er and pvap,Dy differ significantly and scale in57

La

138.915d6s2
2D3/2
1.2 µB

58

Ce

59

Pr

60

Nd

61

Pm

62

Sm

63

Eu

140.12140.91144.24(145)

150.36151.964f5d6s2
1
G4

4f3 6s24I9/2
3.3 µB

4f4 6s2
5I4

4f5 6s26H5/2
0.8 µB

4f6 6s2
7
F0

4f7 6s28S7/2
7.0 µB

3.8 µB

2.4 µB

0 µB

64

Gd

65

Tb

66

Dy

158.93162.504f7 5d6s2 4f9 6s2
6
9
H15/2
D2

4f10 6s2
5I8

157.255.3 µB

9.9 µB

9.9 µB67

Ho

164.93
4f11 6s2
4I15/2
9.0 µB

68

Er

69

Tm

167.26168.93

4f12 6s2
3
H6

4f13 6s22
F7/2
4.0 µB

7.0 µB70

Yb173.0571Lu

174.97

4f14 6s2 4f14 5d6s2
1
2
S0
D3/2

0 µB

1.2 µB

Figure 2.1: The lanthanoid series in the periodic table of elements. The elements’ symbols,
proton numbers, standard atomic weights, valence electron configurations and electronic
ground states are from [79]. Moreover, equation (2.11) provides with the Landé g-factors
gJ in [80] the magnetic moments µ for the stretched states of maximal |mJ | in the groundstate manifolds. Erbium and dysprosium are in violet and orange.
17Chapter 2 Erbium and Dysprosium
102
10

1

ErbiumDysprosium

pvap (Pa)

100
10−1
10−2
10−3
10−4

900

1 000

1 100
T (°C)

1 200

1 300

Figure 2.2: Calculated vapor pressures pvap,Er (violet) and pvap,Dy (orange) as a functionof temperature T in °C by replacing T with (T + 273.15 K) in equation (2.1). The verticaldashed line indicates T = 1100 °C with pvap,Er = 0.098 Pa and pvap,Dy = 0.850 Pa.
units of Pa as a function of temperature T in units of K as [79]pvap (T ) = 105.006+A+ /T +C log10 T + /T .
B

D

3

(2.1)

The quadruple (A, B, C, D) denotes parameters characteristic to each element with
(9.916, −16642, −1.2154, −)Er or (9.579, −15336, −1.1114, −)Dy [79]. Figure 2.2 shows
pvap,Er and pvap,Dy in the temperature range 850 °C to 1350 °C, where pvap,Er = 0.098 Pa
and pvap,Dy = 0.850 Pa at a typical atomic beam source temperature of T = 1100 °C. A
possible layout for a dual-species atomic beam source must pay special attention to thisfactor of close to nine in pvap,Er and pvap,Dy , and must balance it. Chapter 3 describes a
simple modification to a commercial dual-filament effusion cell as one possible solution.
Both elements have multiple naturally occurring stable isotopes varying widely in theirabundance [79]. Four isotopes each exhibit an abundance larger than 14 %, large enoughfor an efficient use in the experiment. Table 2.1 compares their basic characteristics.
Considering Table 2.1, it is worth highlighting that for both elements bosonic as well as
fermionic isotopes, and thus almost mass-balanced heteronuclear Bose-Bose, Bose-Fermi
or Fermi-Fermi mixtures are experimentally accessible.

2.2 Atomic properties
All lanthanoids combine 58 core electrons in a xenon-like configuration ([Xe]) withcomplicated valence electron configurations [79] (see Figure 2.1). As part of the f-blockin the periodic table of elements, their majority exhibits unpaired electrons in the 4f (inspecific cases also in the 5d) valence shell, which lies within a surrounding entirely filled
6s valence shell (see Figure 2.3). This submerged structure has far-reaching consequences
for the lanthanoids. It is for instance the cause for their anisotropic electronic groundstates except europium and ytterbium [79] (see Figure 2.1), or their highly complex
18

2.2 Atomic properties

Table 2.1: Characteristics of all erbium and dysprosium isotopes with an abundancelarger than 14 % [79]. The atomic masses m, the abundances as well as the nuclear spinquantum numbers I are from [79]. The + and the − sign describe the relative orientationof the nuclear spin I. The neutron numbers N result from subtraction of the proton
number ZEr = 68 or ZDy = 66 from the respective isotope’s mass number and determine
the quantum statistics (B=bosonic or F=fermionic).
ErbiumAtomic mass (u) mAbundance (%)
Neutron number N
Statistics
Nuclear spin
I

Dysprosium

166 Er

167 Er168 Er

170 Er161 Dy162 Dy163 Dy164 Dy

165.9
33.50
98
B
0

166.9
22.87
99
F
+7/2

167.9
26.98
100
B
0

169.9
14.91
102
B
0

160.9
18.89
95
F
+5/2

161.9
25.48
96
B
0

162.9
24.90
97
F
−5/2

163.9
28.26
98
B
0

energy level diagrams featuring valence electron excitations from either the 6s or the 4f
valence shell (see Section 2.2.1).
As Figure 2.3 illustrates, erbium and dysprosium in their respective electronic groundstate distribute 14 or 12 valence electrons among the partially filled 4f and the full 6svalence shell, leaving two or four electrons unpaired. Hence, their characteristic electron
configurations are [79]
Er: [Xe] 4f12 6s2

Dy: [Xe] 4f10 6s2 .

(2.2)

Commonly, the quantum number triplet (L, S, J) further characterizes their electronic
ground states, analogous to all lanthanoids, in accordance with Russel-Saunders coupling.
Here, the valence electrons’ individual angular momenta l and spins s couple to the total
orbital momentum L and the total spin S, which combine to the total electronic angularmomentum J [83]. The resulting fine structure has three states JEr ∈ {4, 5, 6} or fivestates JDy ∈ {4, 5, . . . , 8} with the electronic ground states [79]
Er: 3 H6

Dy: 5 I8 .

(2.3)

These states characterize all bosons (166 Er, 168 Er, 170 Er, 162 Dy and 164 Dy), which show
no nuclear spin I (quantum numbers IEr = IDy = 0 [79], see Table 2.1), and have asa direct consequence only a fine structure. The fermions (167 Er, 161 Dy and 163 Dy) are
more complex due to the hyperfine coupling of their I with J to the total atomic angularmomentum F [83]. Considering IEr = 7/2 and IDy = 5/2 [79] (see Table 2.1), the emerging
hyperfine structure with quantum numbers F has eight states FEr ∈ {5/2, 7/2, . . . , 19/2} orsix states FDy ∈ {11/2, 13/2, . . . , 21/2} with the electronic ground states167

Er: 3 H6 , F = 19/2

161

Dy: 5 I8 , F = 21/2

163

Dy: 5 I8 , F = 11/2. (2.4)

The inversion in F for 163 Dy in comparison to 161 Dy is due to the relative orientationof I [84] (see Table 2.1). The large L introduce an anisotropy in the electronic groundstates (2.3) and (2.4). This strongly modifies the atomic polarizabilities as well as thescattering properties, as Sections 2.2.2 and 2.3 discuss in greater detail.
19Chapter 2 Erbium and Dysprosium

Energy4f valence shell:

ml : -3

-2

-1

0

1

6s valence shell:

2

3
Submerged shell:

ml : 0[Xe]-like core

Figure 2.3: Illustration of the electronic ground state structures of erbium and dysprosium.
In total 58 electrons form a xenon-like core ([Xe]), while 14 and 12 valence electrons(arrows) distribute over the 4f and the 6s valence shell, respectively. The black arrows
indicate the arrangement for dysprosium, whereas the two additional red arrows complete
the picture for erbium. Further, the absolute squared of spherical harmonics depict in al, ml -basis the anisotropy and the isotropy in the probability densities of the 4f and the6s valence electrons, respectively. The inset sketches the submerged shell structure forthe 4f (orange) and the surrounding 6s (violet) valence shell.
The lanthanoids’ unpaired 4f valence electrons can give rise to large magnetic momentsµ & 7 µB (see Figure 2.1), with the Bohr magneton µB . However, it is important to notethat an anisotropic electronic ground state not necessarily coincides with a large µ andvice versa. Out of the 15 lanthanoids only europium, terbium, dysprosium, holmiumand erbium have µ & 7 µB . Contrary to the others, europium has the isotropic electronic
ground state 8 S7/2 , but its µ ≈ 7.0 µB purely arises from S.
This paragraph follows [83] and derives µ as the projection of the atomic magnetic
moment µ onto the system’s quantization axis. Depending on the basis, µ calculates asµJ = −gLµBµB
L − gS
S
h̄
h̄

(2.5)

or

µB
µN
µB
J + gII ' −gJ
J,
(2.6)
h̄
h̄
h̄
with the Landé g-factors gL = 1, gS ≈ 2.0023, gJ and gI for L, S, J and I, respectively,
the reduced Planck constant h̄ and the nuclear magneton µN  µB . This µ couples to a
weak external magnetic field B of value B as
µF = −gJ

H = −µJ,F · B.
20

(2.7)

2.2 Atomic properties
The projection in equation (2.7) of the single contributions of µJ,F along the respective
basis vector J or F leads to the Zeeman shift
EZS = gJ,F mJ,F µB B.

(2.8)

The magnetic quantum numbers mJ,F describe the projections of J and F onto the quantization axis with values {−J, −(J − 1), . . . , J} or {−F, −(F − 1), . . . , F }. Furthermore,
3 S (S + 1) − L (L + 1)+
2
2J (J + 1)

(2.9)

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

(2.10)gJ =andgF = gJ
From equation (2.8)µ = −gJ,F mJ,F µB .

(2.11)

Equation (2.9) leads with (L, S, J) from (2.3) to gJ,Er = 1.17 and gJ,Dy = 1.25.
However, this simple picture does not fully hold for erbium and dysprosium, and various
corrections are necessary to reproduce the experimental gJ [85, 86]. For gJ,Er = 1.16381
and gJ,Dy = 1.24159 in [80] and the ground state manifolds’ stretched states of maximal|mJ,Er | = 6 and |mJ,Dy | = 8, equation (2.11) results in |µEr | = 6.982 86 µB ≈ 7.0 µB and|µDy | = 9.932 72 µB ≈ 9.9 µB . This places both among the most magnetic elements in the
periodic table of elements. A large µ gives rise to the long-range anisotropic magnetic
dipole-dipole interaction, which opens new paths for dipolar quantum gas experiments,
as Section 2.3 examines more closely in the context of dBECs.

2.2.1 Energy level diagrams and laser cooling transitions
The first ionization energies for reaching the ionic ground states [Xe] 4f12 6s and [Xe] 4f10 6sof erbium and dysprosium, respectively, are EI,Er = 6.1077 eV and EI,Dy = 5.9389 eV [79],which convert to λI,Er = 202.997 nm and λI,Dy = 208.766 nm. Figure 2.4 depicts excerptsof the energy level diagrams in the experimentally relevant regime from the electronic
ground states up to wavenumbers ν̃ of 26 000 cm−1 or wavelengths λ of 384.615 nm [87],which is far below the first ionization thresholds at EI,Er or EI,Dy . Despite the high complexity due to the submerged shell structure, the level diagrams reveal similar excitation
paths. The arrows in Figure 2.4 indicate two transition types, each a transition of a 6s
valence electron to the 6p state. Contrary to the electronic ground states, a J1 J2 -coupling
scheme characterizes the corresponding excited states via the notation (J1 , J2 )J [80].
Here, on the one hand the 4f valence electrons, and on the other hand the 6s and the
6p electron, couple l and s according to the Russell-Saunders scheme to J1 and J2 that
result in J [89]. Following this scheme, both types introduce a ∆J = +1-transition toa singlet 1 P1 - or a triplet 3 P1 -configuration of the remaining 6s and the excited 6p electron.
The broad transition
to thesinglet 1 P1 -configuration addresses
 the respective excited
12 3state [Xe] 4f
H6 6s6p 1 P1 (6, 1)7 or [Xe] 4f10 5 I8 6s6p 1 P1 (8, 1)9 (violet and blue
arrow in Figure 2.4). It is the strongest transition for both elements at a wavelength
λEr = 400.910 nm or λDy = 421.290 nm with a natural linewidth ∆νEr,401 nm = 29.4 MHzor ∆νDy,421 nm = 32.2 MHz [90] (see Table 2.2).
21

Chapter 2 Erbium and Dysprosium

400

(a)
25

1000

401 nm

4f12 (3 H6 )6s6p(3 P1 °)(6,1)7 °583 nmλ (nm)

15

10

2000

ν̃ (103 cm-1 )

20

667

500

4f12 (3 H6 )6s6p(1 P1 °)(6,1)7 °5
4f12 (3 H6 )6s2 ground state

0

2

3

4

5

6

7
J

8

9

10

11

12

400

(b)

15

4f10 (5 I8 )6s6p(3 P1 °)(8,1)9 °626 nm1000

421 nm10

λ (nm)

4f10 (5 I8 )6s6p(1 P1 °)(8,1)9 °

2000ν̃ (103 cm-1 )

20

667

500

25

5
4f10 (5 I8 )6s2 ground state

0

2

3

4

5

6

7
J

8

9

10

11

12

Figure 2.4: Energy level diagrams of (a) erbium and (b) dysprosium up to wavenumbers ν̃
of 26 000 cm−1 or wavelengths λ of 384.615 nm over the total angular momentum quantum
1 P (6, 1)
number J. The
violet
and
the
blue
arrow
indicatethebroadtransition
to
the
1
7

and the 1 P1 (8, 1)9 state at λEr = 400.910 nm and λDy = 421.290 nm, whereas
theyellow
and theorange arrow point along the intercombination-line to the 3 P1 (6, 1)7 and the
3 P (8, 1) state at λ1
Er = 582.842 nm and λDy = 626.082 nm, respectively. Table 2.2
9summarizes and compares the transition properties. The broad transition is ideal fortransverse cooling, Zeeman slowing and absorption imaging. The intercombination-line
transition serves for a dual-species intercombination-line MOT (see Chapter 3). The
figure originates from [88] with data from [87].
222.2 Atomic properties

Table 2.2: Optical transitions for dual-species laser cooling and trapping of erbium
and dysprosium. The wavenumbers ν̃ from [87] provide the wavelengths λ = 1/ν̃ andfrequencies ν = cν̃, with the speed of light c. For erbium, the decay rates Γ in [91]give the lifetimes τ = 1/Γ, the natural linewidths ∆ν = Γ/2π, the saturation intensities
Isat = πhcΓ/3λ3 and the Doppler temperatures TD = hΓ/4πkB . Furthermore, the recoil
temperatures Tr = h2/mλ2 kB . Here, the Planck constant h, the Boltzmann constant kB
and the mass m. All formulas are either directly in or straightforwardly extractablefrom [83]. The dysprosium values are from [90]. The excited state Landé g-factors gJ0
in [80] and the electronic ground state Landé g-factors gJ (see Section 2.2) determine thetransitions’ differential magnetic moments ∆µ = (gJ0 m0J − gJ mJ ) µB [83], with the Bohrmagneton µB , and values exemplarily for the stretched states of maximal |m0J | and |mJ |.
ErbiumTransition
Wavenumber (1/cm) ν̃
Wavelength(nm) λFrequency(THz) νDecay rate(1/s) ΓLifetimeτ
Natural linewidth∆νSaturation intensity
Isat
Doppler temperature TD
Recoil temperature
Tr
Landé g-factor
gJ0
Differential µ(µB ) ∆µ1

P1 (6, 1)7


24943.272
400.910747.7811.85 × 108
5.41 ns
29.4 MHz59.7 mW
cm2
707 µK712 nK
1.160
1.137

3

Dysprosium

P1 (6, 1)7


17157.307
582.842514.363
1.17 × 106
0.85 µs
186 kHzµW
123 cm2
4.5 µK337 nK
1.195
1.382

1

P1 (8, 1)9


23736.610
421.290711.606
2.02 × 108
4.94 ns
32.2 MHz
56.4 mW
cm2774 µK660 nK
1.22
1.047

3


P1 (8, 1)915972.35
626.082478.839
8.5 × 105
1.2 µs135 kHzµW
72 cm
2
3.2 µK298 nK
1.29
1.677

The intercombination-line
transition
to the triplet 3 P1 -configuration couples to the ex12 3
10 5
33
cited state [Xe] 4f
H6 6s6p P1 (6, 1)7 or [Xe] 4f
I8 6s6p P1 (8, 1)9 (yellow andorange arrow in Figure 2.4). It has a wavelength of λEr = 582.842 nm or λDy = 626.082 nm
and being dipole-forbidden due to the change of the spin multiplet manifests itself in anatural linewidth ∆νEr,583 nm = 186 kHz or ∆νDy,626 nm = 135 kHz [90] (see Table 2.2).
As first single-species experiments, [74] and [75] employed these two transitions torealize a robust intercombination-line MOT for erbium and dysprosium, respectively, that
directly loads from a slowed atomic beam without the need for any optical repumping. The
transition comparison in Table 2.2 shows, besides sufficiently large differential magnetic
moments ∆µ, which is generally relevant for laser cooling and trapping techniques,significant differences, which determine the application in the experiment [74, 75, 92].
The broad transition’s large decay rate ΓEr,401 nm or ΓDy,421 nm is optimal for transverse
cooling or Zeeman slowing of an atomic beam as well as the absorption imaging of cold
atomic samples (see Chapter 3). The intercombination-line transition’s small ΓEr,583 nm or
ΓDy,626 nm , on the contrary, results in a more than 100 times smaller Doppler temperature
TD,Er,583 nm = 4.5 µK or TD,Dy,626 nm = 3.2 µK [90] (see Table 2.2). This represents an
ideal working regime for a MOT on the path to quantum degeneracy. Moreover, becauseof the great similarities between the two elements, the single-species intercombination23

Chapter 2 Erbium and Dysprosiumline MOT approach also extends to their dual-species laser cooling and trapping. The
following Chapter 3 focuses on the first experimental realization [88].
For completeness, the first MOTs for erbium [71] and dysprosium [73] operated on
the broad transition to the 1 P1 states. Here, the magnetic trapping of metastable statescompensates for loss channels due to the broad transition’s open character. In the courseof creating the first dysprosium dBEC, [29] extended this approach by an additional
blue-detuned narrow-line MOT stage at a wavelength of 741 nm, whose concept [72]demonstrated first for erbium at 841 nm. Such multiple MOT stages are not necessary
for the intercombination-line MOT approach and a significantly simpler overall layout of
the experimental apparatus is possible.
Although the broad and the intercombination-line transition exist in all lanthanoids,
the intercombination-line MOT approach is not simply applicable for all series elements.
Exemplarily, the laser cooling of europium on its broad transition might require several
repumping lasers, and thus an alternative route utilizing a metastable state instead ismore favorable [93].

2.2.2 Atomic polarizabilities
In the presence of an external electric field E, a neutral atom typically develops an
induced electric dipole moment [83, 94]
d = α · E,

(2.12)

through which it interacts with the E-field. Here, the polarizability α, in a general
setting a complex-numbered second-rank tensor and dependent on the E-field’s angularfrequency ω [95, 96], characterizes the initial response of the atom to the E-field. Its
diagonal and off-diagonal elements determine the response collinear or in the orthogonalplane to the E-field. The latter is especially important for elements such as the lanthanoids
owing to the L-introduced anisotropy in their electronic ground states [97].
As [98] and [99] studied for erbium and dysprosium atoms, respectively, in a laser
field, α has a scalar representation with real and imaginary part < {α} and = {α}. In
particular, < {α} determines the potential of an atom in the laser field [98]1
I (r) < {α (ω)}
20 c
h
1
mJ
=−
I (r) < {αsca (ω)} + A cos θk< {αvec (ω)}
20 c
2J
i
3m2 − J (J + 1) 3 cos2 θp − 1+ J< {αten (ω)}
J (2J − 1)2

V (r, ω) = −(2.13)

and = {α} the photon scattering rate [98]1
I (r) = {α (ω)}
h̄0 ch1mJ=
I (r) = {αsca (ω)} + A cos θk= {αvec (ω)}
h̄0 c2Ji3m2 − J (J + 1) 3 cos2 θp − 1+ J= {αten (ω)} ,J (2J − 1)
2

Rscat (r, ω) =24

(2.14)

2.2 Atomic properties
800

ErbiumDysprosium< {αsca } (a. u.)

600
400
200
0
−200
−400
300

400

500

600

700
λ (nm)

800

900

1 000

1 100

Figure 2.5: Real parts of the theoretical scalar polarizabilities < {αsca }Er (violet) and< {αsca }Dy (orange) as a function of wavelength λ. In the low-wavelength regime, varioustransitions, in particular the broad transition at λEr = 401 nm or λDy = 421 nm, dominate
the individual behavior, whereas above λ = 700 nm essentially no transitions occur. The
vertical dashed line indicates λ = 1064 nm, where < {αsca,1064 nm }Er = 161.8 a. u. and< {αsca,1064 nm }Dy = 198.6 a. u., as a typical ODT regime.
with the vacuum permittivity 0 , the speed of light c and the laser field intensity I at
position r. Further, < {α} and = {α} each split into three contributions, the real and theimaginary parts of the scalar (αsca ), the vector (αvec ) and the tensor (αten ) polarizability.
Here, θk and θp characterize the inclination angles of the light’s propagation or polarization
axis with respect to the system’s quantization axis, and A ∈ {−1, 0, 1} describes the stateof polarization. The detailed expressions for the individual contributions are in [98].
Equations (2.13) and (2.14) emphasize the necessity for a deep understanding of α,
as it determines the choice of laser wavelength for experimental techniques such as
optical dipole traps (ODTs) [44], one- and two-dimensional ODT arrays [100, 101] or
optical lattices [102], with the latter for instance at a magic wavelength [103, 104], with
underlying double wells [105] or with a species-selective character [106].
In general, however, the precise characterization of αEr and αDy , the polarizabilities of
erbium and dysprosium, is highly demanding. The theoretical model requires detailed
knowledge about the energy level diagrams and the transition dipole moments [98, 99],
whereas the experimental determination is strongly susceptible to the laser’s 1/e2 -beamwaist and power due to the intensity-dependence, as well as to its polarization due to theA- and θp -dependence in equations (2.13) and (2.14), respectively.
Figure 2.5 shows the real parts of the theoretical scalar polarizabilities < {αsca }Er and< {αsca }Dy as a function of wavelength λ ranging from 300 nm to 1100 nm [107]. Both areentirely independent of either the atom’s, or the light’s state (see equation (2.13)), andrepresent the main contribution to < {α}Er and < {α}Dy for the electronic ground states
and linearly polarized laser light (see below). In the shown wavelength regime, a large
amount of transitions, especially the broad transition at λEr = 401 nm or λDy = 421 nm,
govern the behavior of < {αsca }Er and < {αsca }Dy up to around λ = 700 nm. Above
25

Chapter 2 Erbium and Dysprosiumthat point, the individual behavior settles down and < {αsca }Er and < {αsca }Dy approacheach other while remaining positive. In the region around λ = 1064 nm no transitions
which could possibly introduce off-resonant scattering exist. This is especially beneficial
for the realization of an ODT owing to the availability of commercial high-power laser
systems. At λ = 1064 nm the theoretical values are < {αsca,1064 nm }Er = 161.8 a. u. and< {αsca,1064 nm }Dy = 198.6 a. u., respectively.
Two independent experiments investigated < {α} at λ = 1064 nm with linearly polarized laser light (A = 0, and thus no vector contribution) and measured in good agreementwith theory for erbium < {α1064 nm }Er = (166 ± 3stat. ± 61syst. ) a. u. with the tensorial
contribution < {αten,1064 nm }Er = (−1.9 ± 0.8stat. ± 1.2syst. ) a. u. [97], as well as for dysprosium < {αsca,1064 nm }Dy = 184.4(2.4) a. u. and < {αten,1064 nm }Dy = 1.7(6) a. u. [108].
The great similarity not only in < {αsca }Er and < {αsca }Dy but also in masses mEr andmDy (see Table 2.1) enables a combined optical trapping approach for both elements.
However, as Chapter 4 addresses more thoroughly, the small differences in both properties
are already sufficient to play a crucial role for the dynamics during evaporative coolingto quantum degeneracy.

2.3 Dipolar Bose-Einstein condensates
Bose-Einstein statistics predict that an ideal Bose gas in thermodynamic equilibrium
at temperature T occupies each state η of an external trap with a mean occupation
number [6]1Nη = Eη −µ /k T,(2.15)cheB − 1ewith the energy Eη of a considered η, the chemical potential µche and the Boltzmann
constant kB . As T lowers below a critical temperature TC , equation (2.15) reveals
quantum degeneracy with a macroscopic occupation of the trap ground state (η = 0) by apurely quantum statistical phase transition [7], well known as Bose-Einstein condensation.
Here, TC depends on the dimensionality of the system as well as on the number density
n [6]. Hence, not the Bose gas’ T or n alone, but the phase space density (PSD), theproduct of n and the inverse number of occupied states per unit volume approximately
scaling as the thermal de Broglie wavelength λdB cubed [6]

3/2
2πh̄2
3
PSD = nλdB = n,(2.16)
mkB T
with the mass m, determines whether a BEC forms. In a three-dimensional box potential
the PSD must be equal to or larger than ζ (3/2) ≈ 2.612 [6], with the Riemann zetafunction ζ. A BEC for an interacting Bose gas within a three-dimensional harmonic trap
requires a PSD on the same order of magnitude [109]. This represents together with theevaporation efficiency the figure of merit for evaporative cooling (see Chapter 4).
However, while the quantum statistics of a Bose gas may be at the heart of Bose-Einsteincondensation, it is two-body interactions that shape and control a BEC (neglecting threebody processes) [6, 110]. For magnetic elements, and thus for erbium and dysprosium, twointeraction types are relevant, the contact interaction as well as the long-range anisotropic
magnetic dipole-dipole interaction [66], whose interplay gives rise to a dBEC [29, 30, 66].
The following Section 2.3.1 treats a dBEC in a mean-field description and Section 2.3.2
briefly extends the discussion to the heteronuclear case.
26

2.3 Dipolar Bose-Einstein condensates

2.3.1 Interactions within the mean-field description
Considering an interacting Bose gas in an external harmonic trap Vtrap , the time-dependent
Gross-Pitaevskii equation (GPE) describes a BEC in a mean-field description as [110]
 2 2

∂
h̄ ∇
ih̄ ψBEC (r, t) = −+ Vtrap (r, t) + Vint (r, t) ψBEC (r, t) ,(2.17)∂t2mwith the position r, the time t, the kinetic energy term −h̄2 ∇2/2m and the mean-fieldinteractions Vint . The BEC wave function ψBEC fulfills the normalization to the BEC
atom number [110]Z|ψBEC (r) |2 dr(2.18)

nBEC (r) = |ψBEC (r) |2 .

(2.19)NBEC =and gives the BEC number density [110]For a dBEC, the GPE approach introduces the mean-field contact interaction Vcon as
well as the mean-field magnetic dipole-dipole interaction Vdd as [66]Vint (r, t) = Vcon (r, t) + Vdd (r, t) .

(2.20)

Contact interaction
At low energies, scattering theory commonly replaces the van der Waals-type interaction
between two atoms at positions r and r0 by the Fermi pseudopotential [6] 2πh̄2 as
Vcon r, r0 =
δ r − r0 ,mr(2.21)

with the s-wave scattering length as and the reduced mass mr = (m1 m2 )/(m1 +m2 ) of theatoms’ individual masses m1 and m2 , respectively. The Dirac delta function δ (r − r0 )describes a contact-like interaction. The approximation by the isotropic as holds, as
s-wave scattering is the main contribution to low-energy scattering [6]. In the mean-field
description, equation (2.21) leads for a single species (m1 = m2 = m) to [6, 110]
Vcon (r, t) =4πh̄2 as|ψBEC (r, t) |2 .
m

(2.22)

Magnetic dipole-dipole interaction
In the experimentally relevant setting of an external B-field, two atoms with atomic
magnetic moments µ1 and µ2 and at positions r and r0 interact via the magnetic
dipole-dipole interaction as [66] µ1 µ2 µ0 1 − 3 cos2 θ
Vdd r, r0 =,4π
|r − r0 |3(2.23)

with the vacuum permeability µ0 and the magnetic moments µ1 and µ2 of the two dipoles,
respectively. The scaling by the inverse distance cubed, 1/|r−r0 |3 , results in a long-range
27

Chapter 2 Erbium and Dysprosium(a)Bθ = 90 °

(b)

B

µ2

θ = 0°µ2µ1|r − r0 ||r − r0 |µ1

Figure 2.6: Illustration of the magnetic dipole-dipole interaction Vdd between two atoms
with atomic magnetic moments µ1 and µ2 at a distance |r − r0 | in the presence of an
external magnetic field B. The two extremes of Vdd are (a) a repulsive side-by-side and(b) an attractive head-to-tail arrangement with θ = 90 ° or θ = 0 ° between the twodipoles of magnetic moments µ1 and µ2 , and the interatomic axis r − r0 .
non-local effect, whereas θ captures the orientation of the two dipoles with respect tothe interatomic axis r − r0 , and thus the anisotropic behavior of Vdd controllable via
the B-field direction. Figure 2.6 depicts the two extremes, a repulsive side-by-side andan attractive head-to-tail arrangement for θ = 90 ° and θ = 0 °, respectively. In themean-field description, equation (2.23) leads for a single species (µ1 = µ2 = µ) to [6, 66]
Z 2

µ µ0 1 − 3 cos2 θ
Vdd (r, t) =|ψBEC r0 , t |2 dr0 .
(2.24)
0
3
4π |r − r |Equation (2.20) extends with equations (2.22) and (2.24) to
Z 2
 2 0
4πh̄2 asµ µ0 1 − 3 cos2 θ20Vint (r, t) =|ψBEC (r, t) | +|ψr,t| dr .
BECm
4π |r − r0 |3
Hence, moving from equation (2.17) to the time-independent GPE [6, 110]
 2 2

h̄ ∇
µche ψBEC (r) = −+ Vtrap (r) + Vint (r) ψBEC (r)2m(2.25)(2.26)and comparing both sides, results under the Thomas-Fermi approximation [6, 110], which
neglects the kinetic energy term −h̄2 ∇2/2m, and with equations (2.19) and (2.25) in
Z 2
4πh̄2 asµ µ0 1 − 3 cos2 θµche = Vtrap (r) +nBEC (r) +nBEC r0 dr0 .
(2.27)
03m
4π |r − r |In the Thomas-Fermi regime, a dBEC in a Vtrap of trapping frequencies ωr and ωz
in the radial r = (x, y)- and the axial z-direction, respectively, obtains a quadratic
profile [110]




r2
z2
15 NBEC
r2
z2nBEC (r) = n̂BEC 1 − 2 − 2=
1− 2 − 2,(2.28)2 Z
8π RTFRTF ZTF
RTF ZTF
TF
with the peak number density n̂BEC , the radial and axial Thomas-Fermi radius RTF and
ZTF as well as the dipoles along the z-axis. Contrary to a purely contact-interacting BEC,28

2.3 Dipolar Bose-Einstein condensates
(a)

(b)

B

B

Figure 2.7: Illustration of the mean-field magnetic dipole-dipole interaction Vdd in a
dBEC within a cigar-shaped harmonic trap Vtrap . An external magnetic field B aligns
the dipoles, where the two extremes of (a) a parallel and (b) an orthogonal orientation
with respect to the symmetry axis of Vtrap result in an attractive or a repulsive Vdd .
however, a dBEC deforms under the influence of Vdd and its aspect ratio κ = RTF/ZTF
differs from the trap aspect ratio χ = ωz/ωr with [110] 21/515h̄2 κas NBEC
3 κ f (κ)
RTF =
1
+

−
1
(2.29)ddm2
ωr2
2 1 − κ2
 2
1/5
1 15h̄2 κas NBEC
3 κ f (κ)
ZTF =1 + dd−1.
(2.30)
κ m2
ωr2
2 1 − κ2
At this [110]f (κ) =p
1 + 2κ2
3κ2
−1
−
tanh
1 − κ2
3/2
2
1 − κ2(1 − κ )and κ and χ follow the condition [110]
 2



χ
f (κ)
2
2
2
3κ dd
+1
−
1
+
(
−
1)
κ
−
χ
= 0.
dd
2
1 − κ2(2.31)(2.32)

Equations (2.29) and (2.30) capture the interplay between Vdd and Vcon via the ratioof the dipolar length add and as [66]
dd =

add
mµ0 µ2=,as
12πh̄2 as(2.33)

with dd > 1 in the dipolar regime. It reveals two possibilities to adjust the impact
of Vdd , either by µ via mJ,F (see equation (2.11)) or by as via Feshbach resonances [49].
While the first method is limited by dipolar relaxation processes [111], the second one isof particular experimental interest, but has its own complexity. For lanthanoids, eventhe van der Waals interaction is anisotropic due to the anisotropic electronic groundstates (for instance JEr = 6 or JDy = 8, see Section 2.2), and thus facilitates couplings torotating bound states [112, 113]. This results for erbium and dysprosium in particularlycomplex Feshbach spectra [76, 77, 78]. For completeness, [114] investigated recently the
Feshbach spectrum in thulium and observed a temperature-dependent transition fromrandom to chaotic statistics. Alternatively, as a second method, a rotating B-field allows
for the control over Vdd [66], as [115] demonstrated in a dysprosium dBEC.
For a dBEC to be stable, the Thomas-Fermi energy [110]


E
1
χ2
15h̄2 as NBEC
2 2
= mωr RTF 2 + 2 +(2.34)(1 − dd f (κ)) ,
2 Z
NBEC
14
κ
7m RTF
TF
29

Chapter 2 Erbium and Dysprosium
the sum of trap and interaction energy, must remain positive. As a repulsive Vdd canequate an attractive Vcon and vice versa, and due to the anisotropy of Vdd , the geometryof Vtrap influences the stability of a dBEC, where the critical scattering length [69, 110]
as,crit = add f (κ (χ)) .

(2.35)The unstable regime for as < as,crit gives rise to a mean-field instability, the so-calledd-wave collapse [70]. In view of Chapter 5 on supersolidity in dipolar quantum gases, it
should not be confused with the roton-instability [66, 116].
Considering the asymptotic behavior of equation (2.35), as,crit = add for χ → 0 andas,crit = −2add for χ → ∞ [66, 69], respectively, for the initially defined alignment of thedipoles along the z-axis and χ = ωz/ωr . The difference in as,crit arises from the change
in Vdd for the different geometries of Vtrap , since a cigar-shaped Vtrap (χ → 0) leads toa predominant head-to-tail arrangement of the dipoles, and thus to an attractive Vdd ,and a pancake-shaped Vtrap (χ → ∞) to a predominant side-by-side arrangement of thedipoles, and thus to a repulsive Vdd .
In general, this stability consideration depends on the orientation of the dipoles withrespect to the symmetry axis of Vtrap . Figure 2.7 exemplarily illustrates the two extremes
for a dBEC in a cigar-shaped Vtrap . If a B-field aligns the dipoles parallel to the trap
symmetry axis, Vdd turns attractive and acts destabilizing (see Figure 2.7 (a)). On thecontrary, if a B-field, and thus the dipoles point orthogonal to the trap symmetry axis,
Vdd is repulsive and stabilizes the dBEC (see Figure 2.7 (b)). In an analog discussion, adBEC in a pancake-shaped Vtrap shows an inverted behavior.

2.3.2 Heteronuclear dipolar Bose-Einstein condensates
The previous Section 2.3.1 laid the foundation to describe a heteronuclear dBEC, thebinary Bose-Einstein condensation in an atomic mixture of two highly magnetic elements.
A coupled GPE approach extends equation (2.27) for two highly magnetic elements 1
and 2, respectively, with the interspecies mean-field contact interaction Vinter-con and theinterspecies mean-field magnetic dipole-dipole interaction Vinter-dd toµ0che,1 = µche,1 + Vinter-con,12 (r) + Vinter-dd,12 (r)Z
2πh̄2 as,12µ1 µ2 µ0 1 − 3 cos2 θ= µche,1 +n2 (r) +n2 r0 dr003mr
4π|r − r |0µche,2 = µche,2 + Vinter-con,21 (r) + Vinter-dd,21 (r)Z2πh̄2 as,12µ1 µ2 µ0 1 − 3 cos2 θ= µche,2 +n1 (r) +n1 r0 dr0 ,
0
3
mr
4π
|r − r |(2.36)(2.37)

with the interspecies scattering length as,12 . Here, Vinter-con and Vinter-dd deduce fromthe general settings (different masses, m1 6= m2 , and magnetic moments, µ1 6= µ2 ) inequations (2.22) and (2.24) as well as by extending the basic theoretical concepts for apurely contact-interacting mixture in [6].
This PhD thesis’ second publication [117] reported on the first production of such
heteronuclear dBECs by combining erbium and dysprosium (see Chapter 4). However,their experimental investigation and understanding are still at the beginning, where for
instance even the interspecies scattering length of erbium and dysprosium as,Er-Dy is still
unknown. Accordingly, Chapter 6 briefly presents the next steps for a first thoroughcharacterization of these heteronuclear quantum-degenerate dipolar mixtures.
30

Chapter3

Dual-species
intercombination-line
magneto-optical trap
The first milestone of this PhD work was the development and the construction of thefirst experimental apparatus for heteronuclear quantum-degenerate dipolar mixturesof erbium and dysprosium. This milestone proved successful with the realization of adual-species MOT operating in an open-top configuration on the intercombination-line
transition to the 3 P1 state at λEr = 583 nm and λDy = 626 nm [88], respectively.
The experimental apparatus’ core element is an ultra-high vacuum (UHV) apparatus,
as Figure 3.1 illustrates, very much reminiscent of a single-species experiment. Its layout
combines three experiment stages for the loading of the dual-species MOT. First, aspart of a dual-species atomic beam source, a dual-filament effusion cell evaporates bothelements and a transverse cooling (TC) unit collimates the emitted atomic beam. Next,
a 350 mm long spin-flip Zeeman slower (ZS) decelerates the atomic beam starting frominitial velocity classes of up to 370 m/s, and finally, the dual-species MOT itself traps andcools slowed atoms arriving with around 5 m/s at the center of the main chamber.
Typically, a MOT loading time tMOT = 3 s−5 s provides sufficiently high atom numbers
to apply a compression phase to a compressed MOT (cMOT) (see Section 3.3.3) and totransfer both elements into the same ODT for evaporative cooling to quantum degeneracy.
Chapter 4 discusses this step in the experimental sequence more thoroughly.
This third chapter describes the complete experimental apparatus. The initial Section 3.1 covers the UHV apparatus, followed by Section 3.2 about the magnetic fieldsystems featuring the spin-flip ZS. Section 3.3 reviews the laser systems operating on the
broad and the intercombination-line transition, respectively, as well as their implementation into the UHV apparatus. The next two Sections 3.4 and 3.5 give more informationon the imaging and the control system. Section 3.6 concludes with presenting the firstpublication Two-species five-beam magneto-optical trap for erbium and dysprosium [88].
Some information provided throughout this chapter might coincide with [88] and is not
explicitly referenced. Further, experimental parameters might differ from the publishedones [88], as the production of quantum-degenerate dipolar mixtures requires a slightly
different parameter regime than the sole MOT operation.
31

Chapter 3 Dual-species intercombination-line magneto-optical trap
Ion pump,
40 l/sIon pump,
20 l/sAll-metal
angled valve

Titanium
sublimation pump
Ionization
gauge

Gate valves

Ionizationgauge

All-metal
angled valve
Zeeman slower

Gate valve 2

Non-evaporative getterand ion pumpMainchamberModule III:
Mainchamber

Compensation
cage

Gate valve 1

Atomic beam
shutter

Port aligner

TC unit’soctagon chamber

Dual-filament
effusion cellzxyModule II:
Atomic beam
shutterModule I:
Dual-species
atomic beam source

Figure 3.1: Ultra-high vacuum apparatus for heteronuclear quantum-degenerate dipolar
mixtures of erbium and dysprosium. It consists of three individual modules, the dualspecies atomic beam source (I), the atomic beam shutter (II) and the main chamber (III).
Module I emits a collimated atomic beam of both elements, which propagates along thesystem’s x-axis through Module II into Module III. Here, a spin-flip ZS slows the atomic
beam (see Section 3.2.1) and a dual-species MOT traps and cools slow atoms at thecenter of the main chamber (see Section 3.3.3). Additionally, the main chamber provides
various B-field coil systems for homogeneous or quadrupole B-fields (see Section 3.2.2).

3.1 Ultra-high vacuum apparatus
From the project’s very beginning, the development process focused on the UHV apparatus(see Figure 3.1) and especially on the main chamber. The primary objective was therealization of a highly specialized but at the same time highly versatile platform for stateof-the-art experiments with heteronuclear quantum-degenerate dipolar mixtures. This
process strongly benefited from the first-generation erbium experiment in Innsbruck [89].
The UHV apparatus comprises three individual modules separable via gate valves1(Gate valve 1 and Gate valve 2 in Figure 3.1), namely the dual-species atomic beam source,
the atomic beam shutter and the main chamber. Each of these three sub-assemblies
has an ionization gauge2 and an all-metal angled valve3 for an independent pressure
measurement and the use of an external pumping station, respectively. Additionally, two
all-metal gate valves4 at the main chamber provide the possibility to attach two science
modules for more advanced experiments.
1

VAT Vakuumventile AG, 48132-CE01-0002
Agilent Technologies, Bayard-Alpert gauge tube with tungsten filaments, UHV-24P
3
VAT Vakuumventile AG, 54132-GE02-0001
4
VAT Vakuumventile AG, 48132-CE01-AHL1 and 48132-CE01-X
2

32

3.1 Ultra-high vacuum apparatus
Water coolingsystemzyElectrical
feedthroughs

Bulk zone

Port aligner
Hot lipTo
TC unit

Tantalumcruciblex

Nozzles

Disk aperture

Figure 3.2: Vertical sectional view of the dual-filament effusion cell along the atomic
beam propagation axis (x-axis). Two heating circuits (not shown) operate the crucible’s
bulk zone at a temperature TBZ = 1100 °C and its hot-lip at THL = 1200 °C. The internal
water cooling system is necessary due to these high operating temperatures. The nozzles
within the crucible have a length of 30 mm and a 3 mm aperture, whereas the disk
aperture close to the transverse cooling (TC) unit is 0.5 mm thick and provides an 8 mm
aperture. The crucible and all apertures are from tantalum.
Throughout the entire UHV apparatus, all stainless steel components are from 1.4404
and/or 1.4429 stainless steel. Both types inherently do not develop a large magnetic
permeability during machining. In this regard, 1.4429 stainless steel is slightly superior.
This material property is particularly important since magnetized steel parts would
compromise the ability to achieve the highly precise B-field control needed. The TC
unit’s octagon chamber as well as the main chamber are from 1.4429 stainless steel andhave a surface finish via electrolytic polishing [118], a common technique to improve thevacuum quality by reducing the surface roughness of UHV components.
Module I: Dual-species atomic beam source
The first module is an atomic beam source for both elements, whose construction benefited
from the great similarities in melting points Tm,Er and Tm,Dy as well as in laser coolingproperties (see Chapter 2). Its two central components are a dual-filament effusion cell5
and a TC unit, which together produce a collimated atomic beam. The atomic beam
propagation axis, starting from the atomic beam source, points along the system’s x-axis
through the atomic beam shutter as well as the spin-flip ZS into the main chamber, andis fine-adjustable via a port aligner (see Figure 3.1).
Because of Tm,Er = 1529 °C and Tm,Dy = 1412 °C [79], the decision was to implement acommercial dual-filament effusion cell. Its working principle relies on heating the materialof interest in a crucible to a given target temperature. As the vapor pressure increases,
material evaporates and passes as vapor through one or more apertures geometricallyshaping the atomic beam. The chosen type of evaporator is commonly in use in epitaxy
applications but also operates successful in the first-generation erbium experiment [89].
5Createc Fischer & Co. GmbH, DFC-40-10-WK-2B

33

Chapter 3 Dual-species intercombination-line magneto-optical trap
Figure 3.2 shows a vertical sectional view of the installed dual-filament effusion cell. It
has two heating circuits that establish two temperature regions in a tantalum crucible.
In standard operation, the bulk zone, which serves as reservoir for a granulated alloy of
erbium and dysprosium in a ratio of 33 % to 67 %, is at a temperature TBZ = 1100 °C
and the hot-lip at a temperature THL = 1200 °C. The temperature gradient prevents
condensation of evaporated material on the crucible’s internal apertures, as they areat the same temperature as the hot-lip. However, as Chapter 2 discussed, the vapor
pressures pvap,Er and pvap,Dy differ by a factor of close to nine in the given temperature
region. By exploiting this temperature gradient, enriching the hot-lip with additional
pure granulated erbium helps to compensate the difference in pvap,Er and pvap,Dy , as it
individually increases the atomic flux, and thus the MOT loading rate of erbium.
In total three tantalum apertures provide an initial geometric collimation of the emitted
atomic beam, namely two 30 mm long nozzles with an aperture of 3 mm diameter at thebulk zone and the hot-lip, respectively, as well as an external 0.5 mm thick disk aperture
with an aperture of 8 mm diameter right in front of the TC unit’s octagon chamber. The
implementation of the latter results in a more favorable overall performance [119]. In
comparison to the aperture setup in the first-generation erbium experiment [89, 119],a nozzle at the hot-lip, instead of another disk aperture, achieves a stronger geometriccollimation of the here additionally evaporated pure erbium.
The TC unit includes an octagon chamber featuring two pairs of anti-reflection coated
CF63 viewports6 under 45 ° to the horizontal x-y plane and two uncoated CF40 viewports
for additional optical access within the plane (see Figure 3.12). A single 20 l/s ion pump7
at the top connection port of the octagon chamber maintains the pressure of the entire
module at a low 10−10 mbar level.
Module II: Atomic beam shutter
The next module is the atomic beam shutter (ABS). After the MOT loading, an external
servo drives a simple in-vacuum stainless steel plate, which intrudes into the atomic beampath, from its open to closed position. This prevents the atomic beam from entering themain chamber, and thus any possible influence on the experiments.
Furthermore, this sub-assembly is at the center of the UHV apparatus’ differential
pumping system. Here, a 50 mm long connector tube with an inner diameter of 8 mm
acts as an aperture to the atomic beam source, whereas the spin-flip ZS accomplishes thisfor the opposite side with respect to the main chamber. A 40 l/s ion pump8 completes
the differential pumping scheme and establishes a vacuum pressure of a few 10−11 mbar
in the entire module. The additional titanium sublimation pump9 refreshes only after an
exchange of the dual-filament effusion cell.
Module III: Main chamber
The final module is the main chamber, the core element for all experiments during
this PhD work. Its adjunct pumping assembly includes an additional viewport and an
6

Anti-reflection coating for 401 nm, 421 nm, 532 nm, 583 nm, 626 nm and 1550 nm
Agilent Technologies, VacIon Plus 20 StarCell
8
Agilent Technologies, VacIon Plus 40 StarCell
9
Agilent Technologies, titanium sublimation pump
7

34

3.1 Ultra-high vacuum apparatus(a)CF63 Viewport,
AR2, MOT axis 3-4

CF40 connection
tube

CF63 Viewport,
AR1, MOT axis 1-2

CF100 Viewport,
AR1, vertical MOT axis

CF40 connection
port to ZS

CF40 connection port
to pumping assembly

Cutout for
high-B-field coils
CF63 Viewport,
AR1, MOT axis 1-2(b)

CF40 connection
tube

CF63 Viewport,
AR2, MOT axis 3-4

(c)

z
x-y

z
y

x

Figure 3.3: Custom-made main chamber from 1.4429 stainless steel in (a) a top view andvertical sectional views along (b) the MOT axis 1-2 and (c) the atomic beam propagation
axis (x-axis). Along the three orthogonal MOT axes, the main chamber features four
CF63 viewports (x-y plane) as well as two CF100 inverted viewports (z-axis) with antireflection coatings AR1 or AR2 (see text). The inverted viewports minimize the working
distance along the vertical z-axis to 35 mm. The CF40 connection ports for the ZS andthe pumping assembly as well as the CF40 connection tubes for the gate valves reducethe distance to the chamber center to 75 mm and 124 mm, respectively. The cutouts atthe top and the bottom allow for high-B-field coils (see Section 3.2.2 and Figure 3.5).
in-vacuum aluminum mirror10 for incoupling of the ZS light (see Section 3.3.3) as well asa single pump11 that combines a non-evaporative getter material with an ion pump andkeeps the vacuum pressure of the entire sub-assembly on a low 10−11 mbar level. The
spin-flip ZS (see Section 3.2.1) interfaces the main chamber with the previous Module II.
The main chamber itself is entirely custom-made. Its layout adapts to the stringent
experimental requirements (see below), whereas providing a high versatility at the sametime. Figure 3.3 depicts the main chamber in various views.
10
11

Kugler GmbH, optical-grade aluminum mirror (AlSi1MgMn)
SAES Getters S.p.A., NEXTorr D500-5 with mu-metal shielding

35

Chapter 3 Dual-species intercombination-line magneto-optical trap
Along the vertical z-axis, two CF100 inverted viewports offer a clear aperture of 64 mm
and reduce the distance from chamber center to the viewports’ outside to 35 mm. The
inverting serves two purposes. First, the reduced working distance allows in combination
with the large field of view for the implementation of a vertical high-resolution imaging
system (see Chapter 6). Second, it facilitates the integration of various low-B-field coils
close to the atoms (see Section 3.2.2).
In the horizontal x-y plane, two orthogonal pairs of CF63 viewports provide sufficient
optical access for MOT beams with 1/e2 -waists wMOT = 19 mm (see Section 3.3.3). These
enable a large MOT capture volume, which is necessary for the direct loading of adual-species intercombination-line MOT for erbium and dysprosium due to not only
the small MOT capture velocities (see Section 3.2.1) but also the substantial spatial
downshift of the MOT position (see Section 3.3.3) [74, 75, 88, 92].
The majority of viewports has an anti-reflection coating12 (AR1) for the most importantlaser cooling and trapping wavelengths, while two CF63 viewports along one of thehorizontal MOT axes have a broad anti-reflection coating (AR2) for the wavelength
regime ranging from 400 nm to 800 nm. This combination ensures a high flexibility ofusable laser wavelengths. Figure 3.3, 3.13, and 3.14 illustrate the coating arrangements.
Besides optimized optical access, the main chamber also features a special shape. The
recessed CF40 connection ports for the spin-flip ZS and the pumping assembly leavea distance of 75 mm to the chamber center. This increases the amount of slow atoms
entering the MOT capture volume and improves the pump performance within thechamber. Further, the welded CF40 connection tubes permit a close installation of thetwo gate valves at a distance of 124 mm to the chamber center. This minimizes thetransport distance to either of the two (possible) future science modules. Finally, cutouts
around the CF100 inverted viewports create space for two additional high-B-field coils in
a Helmholtz (HH) and an anti-Helmholtz (AHH)-like configuration (see Section 3.2.2).

3.2 Magnetic field systems
Many experimental techniques in ultracold atom experiments, such as Zeeman slowing
or magneto-optical trapping, rely on atom-light interactions under the influence of an
external magnetic field B [83, 120]. Equally important, as Chapter 2 briefly addressed,
the contact and the magnetic dipole-dipole intra- and interspecies interactions are tunableby varying the strength or the orientation of a B-field. This is a crucial aspect not only forevaporative cooling but also for more advanced experiments. The different experimental
techniques and application areas require specific B-field systems. Section 3.2.1 covers
the spin-flip ZS, including a short review of its basic working principle, and Section 3.2.2
describes the layout of the various B-field coils at the main chamber.

3.2.1 Spin-flip Zeeman slower
The dual-filament effusion cell operates at TBZ = 1100 °C and THL = 1200 °C in orderto achieve sufficiently high pvap,Er and pvap,Dy for a high atomic flux towards the mainchamber. Thus, the emitted atomic beam has longitudinal velocity distributions peaking
at calculated values of v̂Er = 451 m/s and v̂Dy = 457 m/s. This makes it necessary to
12Anti-reflection coating for 401 nm, 421 nm, 532 nm, 583 nm, 626 nm, 1064 nm and 1550 nm

36

3.2 Magnetic field systems
decelerate the atomic beam down to the MOT capture velocities vcap,Er and vcap,Dy , themaximum velocities at which the dual-species MOT can still capture erbium or dysprosium
atoms. An efficient technique is Zeeman slowing, as [121] first reported in 1982 for sodium.
Its working principle relies on slowing atoms with velocity v via the scattering force Fscatof a counterpropagating red-detuned laser beam with wavenumber k, while simultaneouslycompensating the change in the Doppler shift kv via a Zeeman shift EZS of the atoms’energy levels (see equation (2.8)). Many textbooks, for instance [83] or [120], offer adetailed description.
The following discussion considers the steps in [83] and derives the optimal B-field
profile Bopt for Zeeman slowing. In a ZS along the x-axis, an atom is subject to thescattering force of a counterpropagating red-detuned laser beam as
I/Isat1
Fscat (2πδ + kv) = h̄kΓ,2
2
1 + I/Isat + 4(2πδ+kv) /Γ2(3.1)

with the photon momentum h̄k and k = 2π/λ = 2πν/c, the decay rate Γ, the laser beam
intensity I, and the light detuning δ = ν − ν0 from resonance to an atomic transition
at frequency ν0 and with saturation intensity Isat . If Fscat saturates at h̄kΓ/2 in thehigh-intensity limit, integrating for the resulting acceleration of the atom with mass m
dv
dx dv
dv
Fscat
h̄kΓ
=
=v
=−
=−
dt
dt dx
dx
m
2m

(3.2)

from the initial velocity vini to the final velocity vfin and from x0 = 0 to x0 = x, with thedeceleration length x, leads to2
2
vini− vfin
=s

h̄kΓ
x.
m

(3.3)

The safety factor s ∈ (0, 1) controls the actual acceleration below the maximal value inequation (3.2). Equation (3.3) yields the position-dependent final velocitysh̄kΓvfin (x) = vini 1 − s
(3.4)
2 x.
mvini
This illustrates the superiority of the broad transition to the 1 P1 state for Zeeman slowing,as the large ΓEr,401 nm or ΓDy,421 nm (see Table 2.2) and the large h̄kEr,401 nm or h̄kDy,421 nm
reduce x for slowing to a target vfin . Further, equation (3.4) gives an estimate for vcap,Er
1 2
and vcap,Dy . The MOT beams with√ /e -waists wMOT = 19 mm under 45 ° to the atomic
beam propagation axis set x = 2 × 38 mm. For vfin = 0 and s = 1, vcap,Er = 16.0 m/s
and vcap,Dy = 13.4 m/s. Tables 2.1 and 2.2 provide the necessary values for m, λ and Γ.
By comparing kvfin to 2πδ and the differential EZS of the transition, the resonancecondition is
s
h̄kΓ
∆µ
kvini 1 − s(3.5)
2 x = −2πδ + h̄ Bopt (x) ,mvini
with the transition’s differential magnetic moment ∆µ (see Chapter 2), and as a resultsh̄kvinih̄kΓh̄2πδBopt (x) =
1−s
x+
.
(3.6)
2
∆µ
∆µ
mvini
37

Chapter 3 Dual-species intercombination-line magneto-optical trap(a)Start coil

Profile coilsyOuter tube

Inner tube

Water cooling(b)

End coilzx

Conical part
Bias coilCompensation coil

400
Bopt
BZS

B (G)

200

0

−200
−100

0

100

200

300

400

500

600

x (mm)

Figure 3.4: Spin-flip ZS for erbium and dysprosium. (a) Vertical sectional view along
the atomic beam propagation axis (x-axis). The ZS tube consists of two 435 mm longstainless steel tubes with 2 mm wall thickness and outer diameters of 26 mm and 10 mm,
respectively. The inner tube conically expands over the last 100 mm. The ZS’s start, end
and profile coils are in violet, whereas its bias coil and its first compensation coil are ingreen and blue. (b) Calculated spin-flip ZS B-field profile BZS as a function of position x.
In total 13 individual coils, the start, the end and the profile coils (violet), model BZS(orange) to reproduce the optimal B-field profile Bopt (black) for a slowing distance of
350 mm. The ZS’s compensation coils (blue) compensate the leakage field at the centerof the main chamber (vertical dashed line). The start of Bopt defines the zero position.
For the present endeavor, the realization of a dual-species atomic beam source and thesimilar laser cooling properties (see Table 2.2), again becoming evident in the similarvcap,Er and vcap,Dy , allow for a single ZS for both elements. Conveniently, such an
approach requires only fine adjustments due to the elements’ differences in order toobtain an optimal ZS performance.
Figure 3.4 (a) displays a vertical sectional view through the ZS. A 435 mm long stainlesssteel tube with an outer diameter of 26 mm and 2 mm wall thickness holds a 370 mm longstack of 14 B-field coils13 . The atomic beam travels through a second 435 mm long innerstainless steel tube with an outer diameter of 10 mm and 2 mm wall thickness, which
13

Wire material: copper, geometry: 2.7 mm × 1.0 mm

38

3.2 Magnetic field systems
conically increases its diameter over the last 100 mm where it connects to the outer tube.
This reduces the probability of slow atoms colliding with the tube wall, and thus increases
the fraction of atoms arriving within the MOT capture volume. Moreover, the innertube completes the differential pumping scheme between Modules II and III of the UHV
apparatus (see Section 3.1), whereas the hollow volume in between both tubes serves aswater circulation path as part of the water cooling system for the coil stack.
The ZS operates on the broad transition at λEr = 401 nm or λDy = 421 nm withan increasing B-field in spin-flip configuration. It decelerates initial velocity classes of
up to vini,Er = 370 m/s and vini,Dy = 358 m/s over a total distance of 350 mm down tovfin,Er = vfin,Dy = 5 m/s, which corresponds to sEr = 0.35 and sDy = 0.31. Here, as aprecaution, vfin,Er < vcap,Er and vfin,Dy < vcap,Dy . Figure 3.4 (b) compares the calculated
spin-flip ZS B-field profile BZS to the optimal B-field profile Bopt (see equation (3.6)). In
total 13 out of the 14 coils, the start, the end and 11 profile coils, differing in layers andwindings (values range between 1-11 and 4-17), shape BZS such that it deviates from Bopt
by only 1.5 % from x = 25 mm to x = 345 mm, and thus almost over the entire slowing
distance. Further, a bias coil, with two layers of 128 windings, can tune the zero-crossingof BZS , and thus vini . Two coils, each with four layers of four windings, at a distance of
75 mm around the center of the main chamber compensate the leakage B-field.

3.2.2 Layout of the main chamber coils
The main chamber holds eight sets of coil pairs symmetrically around its center for
homogeneous and/or quadrupole B-fields. Every coil pair serves a specific purpose andis individually controllable. Figure 3.1 illustrates the complete coil assembly at the
UHV apparatus, whereas Figure 3.5 shows the coil pairs along the main chamber’s
vertical z-axis in sectional views. The following description14 characterizes the generated
B-fields aroundq the center of the main chamber by the calculated total B-field value(Btot )x,y,z = Bx2 + By2 + Bz2 along the x-, y- and z-axis, respectively, as well as by therespective gradient (∇Btot )x,y,z = ∂x,y,z (Btot )x,y,z .
Low-magnetic-field coils
The inverted viewports along the vertical z-axis of the main chamber support each acopper mounting cylinder with an inner diameter of 81 mm (see Figure 3.5 (a)). It has
a cutout preventing eddy currents and a conical cut to fit the future high-resolution
imaging system along this axis (see Figure 3.5 (b)). Each cylinder stacks the individual
coils for in total four different coil pairs at an inner diameter of 85 mm as close as possible
to the center of the main chamber. The internal water cooling system maintains the coils
at a steady-state temperature of 28 °C at a maximum coil current of 10 A.
The first coil pair combines two 100 mm distant coils15 , each with six layers of ten
windings, in an AHH-like connection scheme. Around the center, it produces a quadrupole
B-field BAH with | (∇Btot )x,y | = 0.775 (G/cm)/A and | (∇Btot )z | = 1.550 (G/cm)/A (seeFigure 3.6 (a)). For the MOT, this AHH-like
BAH,MOT coil provides the quadrupole B-fieldwith typical gradient values ∇BAH,MOT x,y = 2.25 G/cm and ∇BAH,MOT z = 4.50 G/cm.
14

The upcoming PhD thesis of Gianmaria Durastante will provide a more detailed discussion of theentire coil system including characterization measurements.
15
Wire material: copper, geometry: 2.7 mm × 1.0 mm, same for the HH-like coil

39

Chapter 3 Dual-species intercombination-line magneto-optical trap
(a)

Brass
cooling plates

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

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

x

Brass
cooling plates(b)

Conicalcut

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

Water coolingsystem

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

Fast response
coils 1Figure 3.5: Coil systems at the main chamber. (a) Vertical sectional view of the mainchamber along the atomic beam propagation axis (x-axis). The main chamber holds six
B-field coil pairs in close proximity symmetrically around its center. The high-B-fieldanti-Helmholtz (AHH)-like (violet) and Helmholtz (HH)-like (orange) coil pair operate in
the designated cutouts. The respective coil units include a brass water cooling plate witha slit against eddy currents (out of sectional plane). The remaining four coil pairs are for
low B-fields and within copper cylinders in the CF100 inverted viewports. (b) Verticalsectional view of a single copper cylinder in the y-z plane. The four coils are part of an
AHH-like (blue) and a HH-like (yellow) coil pair as well as fast response coil pairs (green
and red). The cylinder has a water cooling system and a cutout against eddy currents.
40

3.2 Magnetic field systems
The second one pairs two coils, each with six layers of a single winding, at a distanceof 70 mm in a HH-like connection scheme. It produces a homogeneous B-field BHH of(Btot )x,y,z = 0.779 G/A with a (relative) flatness of 10−4 within ±0.5 mm of the center
pointing along the z-axis (see Figure 3.6 (b)). For the MOT loading, this HH-like coilshifts the origin of the quadrupole B-field BAH,MOT via a bias B-field BHH,MOT oftypically BHH,MOT = −3.245 G below the center of the main chamber. Moreover, it
maintains a homogeneous B-field BHH,evap of constant value during evaporative coolingand can provide precise B-fields for low-field Feshbach spectroscopy.
The two remaining coil pairs have one and four layers of a single winding per coil16 ,respectively. This makes both ideal for homogeneous and time-varying B-fields with ashort time response.
High-magnetic-field coils
The designated cutouts in the main chamber allow for two additional coil pairs along
the z-axis, one in an AHH- and the other one in a HH-like connection scheme (seeFigure 3.5 (a)). The individual coil units are 145 mm and 105 mm apart, respectively.
Each of the four coil units consists of two coils17 , each with an inner diameter of 165 mm
and 27 layers of a single winding, and a 4 mm thick brass water cooling plate inbetween.
The cooling plate has a slit to prevent eddy currents and its internal water cooling systemallows for coil currents of up to 100 A at a steady-state temperature of 40 °C.
Figure 3.7 summarizes the performance of both coil pairs. The AHH-like coil generates
a quadrupole B-field with | (∇Btot )x,y | = 0.240 (G/cm)/A and | (∇Btot )z | = 0.480 (G/cm)/A
around the center (see Figure 3.7 (a)), whereas the HH-like coil provides a homogeneousB-field of (Btot )x,y,z = 4.372 G/A with a flatness of 10−4 within ±1.5 mm of the center
pointing along the z-axis (see Figure 3.7 (b)). Furthermore, the AHH-like coil can switch
to a HH-like operation, where it creates a homogeneous B-field of (Btot )x,y,z = 3.244 G/Awith a flatness of 10−4 within ±1 mm of the center. This is particularly interesting for
Feshbach spectroscopy in order to extend the B-field range up to 800 G in steady state.
Compensation cage
The experimental apparatus also controls the B-field in the horizontal x-y plane. For that
purpose, two square coil pairs of 380 mm × 380 mm and 350 mm × 350 mm, respectively,
surround the main chamber along the system’s x- and y-axis and form a cage (seeFigure 3.1). Each coil18 has eleven layers of two windings within an aluminum U-channel.
The larger coil pair produces a homogeneous B-field of (Btot )x,y,z = 1.04 G/A pointing
along the x-axis, whereas the smaller one achieves (Btot )x,y,z = 1.12 G/A pointing along
the y-axis, both with a flatness of 2 × 10−6 within ±0.5 mm of the center. The maximum
current rating for both is 10 A.
During the experimental sequence, this cage compensates external B-fields, for instancethe earth’s B-field in the horizontal x-y plane, and rotates the B-field for absorption
imaging (see Section 3.4). Moreover, its combined operation with the HH-like coil within
the inverted viewports generates B-fields of up to 5 G in any direction on a sphere.
16

Wire material: copper, geometry: 1.0 mm diameter
Wire material: copper, geometry: 8.0 mm × 1.0 mm
18
Wire material: copper, geometry: 4.0 mm × 1.0 mm
1741

Chapter 3 Dual-species intercombination-line magneto-optical trap

8

2.0

6

1.5

|∇Btot | (G/cm)

Btot (G)

(a)

4
2

0.5
0

0
−40

0

40

80

0.8
0.4
0
−4

−2

0

2

4

|∇Btot | (G/cm)−80
Btot (G)

1.0

−80

−40

0

40

80

−4

−2

0

2

4

1.8
1.2
0.6

d (mm)d (mm)

0.20

0.75

0.15

|∇Btot | (G/cm)

1.00

0.50
x-axisy-axisz-axis

0.25
0
−80

−40

0

40

Btot (G)

0.779
−4

−2

0
d (mm)

0.05

80

0.785

0.773

0.10

0

2

4

|∇Btot | (G/cm)

Btot (G)

(b)

−80

−40

0

40

80

−4

−2

0

2

4

0.04
0.02
0
d (mm)

Figure 3.6: Calculated total B-field value (Btot )x,y,z (left column) and the corresponding
gradient (∇Btot )x,y,z (right column) as a function of distance d from the center of themain chamber for the low-B-field (a) anti-Helmholtz (AHH)-like and (b) Helmholtz(HH)-like coil within the inverted viewports (see Figure 3.5). The orange data shows the
behavior along the z-axis, whereas the violet and blue data corresponds to the orthogonalplane along the x- and y-axis, respectively. The current for the calculation is 1 A. The
AHH-like coil produces a quadrupole B-field BAH with | (∇Btot )x,y | = 0.775 (G/cm)/A
and | (∇Btot )z | = 1.550 (G/cm)/A around the center, whereas the HH-like coil generates
a homogeneous B-field BHH of (Btot )x,y,z = 0.779 G/A with a flatness of 10−4 within±0.5 mm of the center pointing along the z-axis.
423.2 Magnetic field systems

0.500
3|∇Btot | (G/cm)

Btot (G)

(a)

2

1

0.250
0.125
0

0
−75

0

75

150

0.250
0.125
0
−4

−2

0

2

4

|∇Btot | (G/cm)−150
Btot (G)

0.375

−150

−75

0

75

150

−4

−2

0

2

4

0.60
0.40
0.20

d (mm)d (mm)(b)

0.75

5|∇Btot | (G/cm)

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

1

0

75

150

Btot (G)

4.374
4.372
4.370

−4

−2

0
d (mm)

2

4|∇Btot | (G/cm)

−75

0.25

0

0
−150

0.50

−150

−75

0

75

150

−4

−2

0

2

4

0.02
0.01
0
d (mm)

Figure 3.7: Calculated total B-field value (Btot )x,y,z (left column) and the corresponding
gradient (∇Btot )x,y,z (right column) as a function of distance d from the center of the mainchamber for the high-B-field (a) anti-Helmholtz (AHH)-like and (b) Helmholtz (HH)-likecoil within the main chamber’s cutouts (see Figure 3.5). The orange data shows the
behavior along the z-axis, whereas the violet and blue data corresponds to the orthogonalplane along the x- and y-axis, respectively. The current for the calculation is 1 A. The
AHH-like coil produces a quadrupole B-field BAH with | (∇Btot )x,y | = 0.240 (G/cm)/A
and | (∇Btot )z | = 0.480 (G/cm)/A around the center, whereas the HH-like coil generates ahomogeneous B-field of (Btot )x,y,z = 4.372 G/A with a flatness of 10−4 within ±1.5 mmof the center pointing along the z-axis.
43

Chapter 3 Dual-species intercombination-line magneto-optical trap

3.3 Laser systems and optical setups
The efficient and robust operation of the dual-species MOT relies on the broad and theintercombination-line transition to the 1 P1 and the 3 P1 states, respectively, as Chapter 2
discussed. This section reviews the laser systems, their individual optical setups as wellas their combined implementation and operation at the UHV apparatus. Section 3.3.1
starts with the laser systems for λEr = 401 nm and λDy = 421 nm, driving the broadtransition for the TC, the spin-flip ZS, an absorption imaging as well as a blow beam.
Next, Section 3.3.2 introduces the λEr = 583 nm and the λDy = 626 nm laser system forthe dual-species MOT itself. Finally, Section 3.3.3 addresses the various optical setups at
the UHV apparatus for the TC unit, the spin-flip ZS as well as the dual-species MOT.

3.3.1 Broad transition: 401 nm and 421 nm
The laser setups at λEr = 401 nm (see Figure 3.8) and at λDy = 421 nm (see Figure 3.9)
are essentially identical. Hence, the following discussion considers the basic structure ofa single laser setup only for simplicity, but emphasizes their differences.
The optical setup arranges around a commercial multi-staged laser system19 employing
second harmonic generation (SHG). A grating-stabilized diode (master) laser, whichemits at λEr = 802 nm or λDy = 842 nm, seeds a tapered amplifier (TA), while someof the master laser’s light is available for monitoring on a fiber-coupled wavemeter20 .
The high output power from the TA optically pumps a resonant SHG unit combining anonlinear crystal with a ring resonator in bow-tie configuration. This concept delivers amaximum second-harmonic power P401 nm,SHG = 2 W or P421 nm,SHG = 1.3 W. After an
initial beam shaping, the frequency-doubled laser light splits among five beam paths.
The spectroscopy path provides the possibility to frequency stabilize the laser system toan atomic reference. A first single-pass acusto-optic modulator21 (AOM) frequency-shifts
the laser light by ν401 nm,spec = ν421 nm,spec = 110 MHz, which then arrives polarizationmaintaining (PM) fiber-coupled at the spectroscopy setup. Here, a modulation transfer
spectroscopy generates a dispersive signal for frequency stabilization utilizing a hollowcathode lamp22 (HCL) and a home-built electro-optic modulator (EOM) [122, 123].
Thus, considering the initial ν401 nm,spec or ν421 nm,spec , the stabilized
laser systemhas aneffective detuning δ401 nm = δ421 nm = −110 MHz to the 1 P1 (6, 1)7 and the 1 P1 (8, 1)9
state, respectively. This approach reduces the total amount of required AOMs in theother beam paths and makes a single-pass operation for their majority possible. The
experiences from [89] suggested that the modulation transfer signals of the fermionicisotopes are too small to be suited for frequency stabilization. Hence, this beam pathcontains one additional double-pass AOM23 for the λEr = 401 nm and two additional
double-pass AOMs24 for the λDy = 421 nm laser setup. This allows for adjusting the laser
system’s overall δ401 nm or δ421 nm such that the experiment addresses a fermionic isotope,
while the master laser remains stabilized to a bosonic isotope. Contrary to [89], however,
19

TOPTICA Photonics AG, DLC TA-SHG PRO
HighFinesse GmbH/Ångstrom, WS/6-200
21Gooch & Housego PLC, 3100-125, center frequency 100 MHz, bandwidth 25 MHz
22
Heraeus Holding GmbH, 3QQAY/Er for erbium and 3QQAY/Dy for dysprosium, each with 4 mbar ofargon, high-voltage power supply FuG Elektronik GmbH, MCP 350-650
23Gooch & Housego PLC, 3080-125 center, frequency 80 MHz, bandwidth 25 MHz
24
Brimrose Corporation, TEF-270-100-401, center frequency 270 MHz, bandwidth 100 MHz
20

44

Hollow cathode lamp
Spectroscopy

Master50

19 MHz200

Wavemeter

PD

TA

100

EOM
-50

SHG

f

f

Iris
Lens withfocal length f
in mm
MirrorPolarizing beam
splitter cube
λ/4-plate

ZS-50

Optical isolator

TA-SHG PRO 401 nm

3.3 Laser systems and optical setups

λ/2-plate

AOM 110 MHz

250

100
150

100

-75
AOM 200 MHz

150

-50

-100 125

Retroreflector

270 MHz AOM
125

75 -50
AOM 100 MHz

Spectroscopy

-50

100

Blow

TC

-50 80 Cylindrical

AOM 100 MHz

150
AOM 80 MHz

100 MHz AOM
Imaging

Figure 3.8: Optical setup for the λEr = 401 nm laser system driving the transition to the
1 P (6, 1) state in erbium. The laser light that a SHG unit generates from an amplified
1
7
802 nm master laser distributes between five different beam paths. Each individuallyfrequency-shifts and subsequently PM fiber-couples the light to its designated location
at the UHV apparatus (see Section 3.3.3). The modulation transfer spectroscopy setup
in the top left serves for the frequency stabilization of the master laser. The various
frequency values specify the AOM center frequencies, whereas the actual frequencies arein the main text.
a direct frequency stabilization to the fermionic isotopes is possible for the present laser
systems without any notable influence for instance on the long-term stability during the
production of quantum-degenerate Bose-Fermi mixtures (see Chapter 4).
The beam paths for the TC unit and the spin-flip ZS are the two high-power paths of thelaser system. Each utilizes a quartz AOM to reduce thermal lensing in the AOM crystal
owing to the wavelength regime. A single double-pass AOM25 in the ZS path detunes
the laser light by additional ν401 nm,ZS = −410 MHz or ν421 nm,ZS = −420 MHz further tothe red. Here, a retroreflector from two square mirrors under 90 ° helps overcome thestrong polarization dependence of the diffraction efficiency, typical for a quartz AOM,and thus ensures a high double-pass efficiency. In the TC path, a single-pass AOM26
frequency-shifts the laser light back towards resonance by an experimentally optimized
value ν401 nm,TC = 106 MHz and ν421 nm,TC = 108 MHz, respectively.
The remaining two beam paths are for an imaging and a blow beam, respectively. In
each beam path, a single-pass AOM27 compensates the initial ν401 nm,spec or ν421 nm,spec .
Moreover, the imaging light is further detunable via a double-pass AOM28 . This will benecessary when switching the future vertical high-resolution imaging from absorption
imaging to an insitu technique such as phase-contrast imaging [7] (see Chapter 6).
25

Gooch & Housego PLC, 3200-1210, center frequency 200 MHz, bandwidth 100 MHz
Gooch & Housego PLC, I-M110-2C10B6-3-GH26, center frequency 110 MHz
27
Gooch & Housego PLC, 3100-125, center frequency 100 MHz, bandwidth 25 MHz
28
Brimrose Corporation, TEF-270-100-401, center frequency 270 MHz, bandwidth 100 MHz
26

45

Hollow cathode lamp

MasterSpectroscopy

50
21 MHz200

Optical isolator

Wavemeter

PD

TA

EOM
-50

SHG

TA-SHG PRO 421 nm

Chapter 3 Dual-species intercombination-line magneto-optical trapf

Iris
Lens withfocal length f
in mm

f

MirrorPolarizing beam
splitter cube
λ/4-plate
λ/2-plate

-75

-75

250
AOM 200 MHz

200

100 MHz AOM

AOM 110 MHz

Retroreflector

AOM 100 MHz
Spectroscopy

100

ZS

125

250

TC

-70 100 Cylindrical

Blow
100

270 MHz AOM
125

Imaging

100
270 MHz AOM

AOM 100 MHz

75
200
AOM 270 MHz

75

Figure 3.9:
 Optical setup for the λDy = 421 nm laser system driving the transition to
the 1 P1 (8, 1)9 state in dysprosium. The laser light that a SHG unit generates froman amplified 842 nm master laser distributes between five different beam paths. Eachindividually frequency-shifts and subsequently PM fiber-couples the light to its designatedlocation at the UHV apparatus (see Section 3.3.3). The modulation transfer spectroscopy
setup in the top left serves for the frequency stabilization of the master laser. The various
frequency values specify the AOM center frequencies, whereas the actual frequencies arein the main text.
Finally, PM single mode fibers connect the different beam paths to their designatedoptical setups at the UHV apparatus (see Section 3.3.3). While for the low-power paths
standard PM single mode fibers are sufficient, for the ZS as well as the TC path highpower PM single mode fibers29 with end caps and a pure silica core are in operation
without any obvious signs of degradation since their implementation at the very beginningof construction.

3.3.2 Intercombination-line transition: 583 nm and 626 nm
The λEr = 583 nm laser system relies on the combination of a grating-stabilized diodelaser30 at λEr = 1166 nm and a Raman-fiber amplifier31 (see Figure 3.10). An integrated
fiber-coupled SHG unit in single-pass configuration operates at 65.0 °C to achieve phasematching [96], and produces a maximum second-harmonic power P583 nm,SHG = 1.7 W.
Similar to single-species experiments with dysprosium [124], the λDy = 626 nm laser
system is a sum-frequency generation (SFG) setup in single-pass configuration (seeFigure 3.10). The optical pumping stage utilizes a 40 mm long magnesium oxide doped
periodically poled lithium niobate (MgO:PPLN) crystal32 and its entire layout is in29

Schäfter + Kirchhoff GmbH, PMC-E-400Si-2.8-NA011-3-APC.EC-750-P
TOPTICA Photonics AG, DLC DL PRO
31
MPB Communications Inc., VRFA-P-1800-583-SF
32
Covesion Ltd, MSFG626-0.5-40
30

46

3.3 Laser systems and optical setups
DL PRO 1166 nm

100

175

100
SFG

100
-50

Dichroic

Dichroic

Master150 100f

Mirror1550 nmPolarizing beam
splitter cube

AOM 80 MHz

100 501050 nm

200 50λ/2-plateRaman
fiber amplifier
SHG
EOM

CavityCavityWavemeter

104 kHz

Distribution

Distribution

λ/4-plate
200

EOM

f

Iris
Lens withfocal length f
in mm

139 kHz-50 100

Wavemeter

AOM
80 MHz 100

100 75Figure 3.10: Optical setups for the λEr = 583 nm (right) and the λDy= 626 nm (left)
lasersystem driving the intercombination-line transition to the 3 P1 (6, 1)7 and the
3 P (8, 1) state in erbium and dysprosium, respectively. The λ
1
Er = 583 nm laser
9system comprises a grating-stabilized diode laser at λEr = 1166 nm and a Raman fiber
amplifier with a fiber-coupled single-pass SHG unit, whereas for the λDy = 626 nm laser
system two amplified DFB fiber lasers at λDy = 1050 nm and λDy = 1550 nm pump a
SFG process in a MgO:PPLN crystal. In both setups, a fiber-coupled EOM guides laser
light to a reference cavity setup for a PDH frequency stabilization (see Figure 3.11 (a)),
while a spectral-broadening EOM enhances the MOT light’s spectral width right in frontof the PM fiber coupling to the MOT distribution (see Figure 3.11 (b)).
accordance to the Boyd-Kleinmann model [96, 125], which generally predicts a pumpbeam’s ideal 1/e2 -waist at the center of a non-linear crystal as [96]slλwSFG,λ =,(3.7)
2.84 × 2πnλwith the crystal length l, the pump light wavelength λ and the corresponding refractive
index nλ 33 . Two distributed feedback (DFB) fiber lasers34 at λDy = 1050 nm and
λDy = 1550 nm, respectively, seed amplifier units with 5 W output power each. The
emitted pump beams pass through initial beam shaping optics to achieve the required
1/e2 -waists wSFG,1050 nm = 33 µm and wSFG,1550 nm = 40 µm. Translation stages for thelast focusing lenses and the crystal facilitate a precise centering of both foci in the centerof the crystal. A dichroic mirror35 overlaps both pump beams right in front of the crystal.
Further, a commercial oven36 stabilizes the crystal at 168.70 °C in order to maintain thephase matching condition of the SFG process. At peak performance, a second dichroic
mirror37 separates a sum-frequency power P626 nm,SFG = 1.6 W from the pump light.
33

MgO:PPLN: n1050 nm = 2.203 and n1550 nm = 2.222
NKT Photonics A/S, Koheras Boostik Y10 at 1050 nm and Koheras Boostik E15 at 1550 nm
35
Thorlabs Inc., DMSP1180
36
Covesion Ltd, PV40 in combination with the temperature controller OC1
37
Thorlabs Inc., DMLP650R
34

47

Chapter 3 Dual-species intercombination-line magneto-optical trap
In a first step, lenses reshape the visible laser light, which then distributes among three
beam paths leading to a reference cavity for frequency stabilization (see Figure 3.11 (a)),
a MOT distribution setup (see Figure 3.11 (b)), as well as the wavemeter for monitoring.
The beam path leading via a PM fiber coupling to the MOT distribution guides mostof the power P583 nm or P626 nm . An initial double-pass AOM38 switches the laser lighton and off during the experimental sequence, and stabilizes P583 nm or P626 nm in the
MOT distribution. Additionally, a spectral-broadening EOM39 in front of the PM fiber
coupling enhances the spectral width of the MOT light. This technique enables a fiveand twofold MOT loading improvement for erbium and dysprosium [88], respectively.
Originally, the AOM operated in a single-pass configuration. However, thermal drifts in
the AOM crystal led to fluctuations in the fiber-coupling efficiency, and thus in P583 nm
or P626 nm within the MOT distribution, which resulted in an unstable MOT operation.
Reference cavity
Because of the intercombination-line transition’s small natural linewidth ∆νEr,583 nm or∆νDy,626 nm (see Table 2.2), a home-built plano-concave cavity under vacuum serves as
frequency reference via the Pound-Drever-Hall (PDH) technique [126, 127]. For bothlaser systems, a fiber-coupled EOM40 delivers low-power laser light to the cavity setup,
as Figure 3.11 (a) sketches. Here, a dichroic mirror41 combines the λEr = 583 nm andthe λDy = 626 nm laser beam, and a single lens focuses the bichromatic beam throughthe cavity’s concave mirror onto its plane mirror. In each beam path, a fine-adjustable
collimation lens in the fiber coupler and two mirrors facilitate an individual modematching to the TEM00 -mode of the resonator. A 50/50-beam splitter cube separates therespective back-reflected light, which a photodiode detects for the derivation of the PDH
signal. The cavity transmission signal is for coupling and monitoring purposes.
For the cavity and its vacuum setup, [128] provides a detailed description and various
characterization measurements. The entire cavity is from ultra-low expansion (ULE)
glass and has an experimentally determined zero-crossing temperature of the coefficientof thermal expansion T0 = 21.78 °C [128]. Further measured characteristics are a free
spectral range FSR = 989(8) MHz and a finesse F = 1636(4) at λDy = 626 nm [128].
All three master lasers exhibit an excellent short-term stability42 . Thus, the PDH
frequency stabilization mainly acts against long-term frequency drifts by feeding backto the λEr = 1166 nm diode laser and to the λDy = 1050 nm DFB fiber laser. The
λDy = 1550 nm DFB fiber laser remains free-running, as it is intrinsically stable enough.
The fiber-coupled EOM has a large bandwidth, and thus serves two purposes, modulating sidebands for the PDH signal as well as tunable sidebands for a variable stabilization
point. Hence, the laser frequency is dynamically modifiable during the experimental
sequence without the need for a complex frequency shifting setup involving multiple
double-pass AOMs. This flexibility is advantageous for both, adjusting the detuningδ583 nm or δ626 nm for an efficient MOT loading to a value δ583 nm,MOT or δ626 nm,MOT and reducing it to δ583 nm,cMOT or δ626 nm,cMOT during the compression phase (see Section 3.3.3).
38

Gooch & Housego PLC, 3080-125, center frequency 80 MHz, bandwidth 25 MHz
QUBIG GmbH, EO-0.1M3-VIS, resonance frequency 139 kHz for erbium and 104 kHz for dysprosium
40
JENOPTIK Optical Systems GmbH, PM594 for erbium and PM635 for dysprosium
41
Thorlabs Inc., DMLP605
42
Manufacturer information: 1166 nm diode laser < 20 kHz (over 5 µs), 1050 nm and 1550 nm DFB fiber
laser < 20 kHz and < 1 kHz (timescales not specified)39

48

3.3 Laser systems and optical setups

583 nm(a)

PD

Lens withfocal length fin mmShielding

Vacuum chamber

f

50/50
50

f

MirrorPD

Dichroic mirror

50

Polarizing beam
splitter cube

400

λ/4-plate
λ/2-plate

PD 50
50/50

Level 1:
1583 nm

To 2

50/50

3a

To 3a
2

Level 5:

Level 4:

70/30

3b

3c

To 3c

To 3b

3a

3c

3b

Pick off
To 4
PD

50/50

MOT

To 3a

4

5

Level 3:

Pick off

PD
3

To 3b

MOT

2

Level 2:MOT(b)reference cavity

626 nmTo 3c

626 nm4

70/30

Figure 3.11: Reference cavity and MOT distribution. (a) Optical setup for the PDH
frequency stabilization to the home-built reference cavity. A vacuum chamber and twosurrounding shieldings against thermal radiation isolate the reference cavity from theenvironment. Photodiodes detect the back-reflected light for the PDH signals as wellas the transmission for coupling and monitoring, respectively. Additionally, the ULE
spacer hosts a second reference cavity ranging from 380 nm to 450 nm [128]. (b) MOT
distribution and the arrangement of its five individual levels according to the schemeon the left. The arrival levels 1 and 5 transfer the λEr = 583 nm and the λDy = 626 nm
laser light to the subsequent levels 2 and 4, which divide the respective laser light with afixed power ratio among three beam paths. The center level 3 combines the laser light inthree PM single mode fibers guiding it to the MOT setup (see Section 3.3.3).

Magneto-optical trap distribution
The dual-species MOT employs retroreflected bichromatic MOT beams for a robust andlow-maintenance operation (see Section 3.3.3). For this purpose, a home-built MOT
distribution overlaps the λEr = 583 nm and the λDy = 626 nm laser light in three standard
PM single mode fibers, one for each MOT axis, as Figure 3.11 (b) depicts. It consists offive levels on top of each other for a compact overall layout. Two mirrors in each beam
path interconnect the different levels, and thus ensure complete alignment autonomy.
First, at the incoming level 1 and level 5, respectively, some part of the polarizationcleaned λEr = 583 nm and λDy = 626 nm laser light splits off for a power stabilization
49

Chapter 3 Dual-species intercombination-line magneto-optical trap
feedback loop via the AOM in the respective laser setup. This allows for an exponential
reduction of P583 nm or P626 nm in the MOT distribution during the compression phase,and thus of all MOT beams powers simultaneously (see Section 3.3.3).
In the following level, 2 for the λEr = 583 nm and 4 for the λDy = 626 nm laser light, a70/30- and a 50/50-beam splitter cube divide the laser beam with a fixed power ratio in
three. Each is individually adjustable via a motorized λ/2-plate and a polarizing beam
splitter cube to control the power ratio of the individual MOT beams (see Table 3.1).
Finally in level 3, a dichroic mirror43 in front of each of the three outgoing PM single
mode fibers overlaps both wavelengths. Two mirrors, one λ/4-plate as well as one λ/2-plateallow for an individual PM fiber coupling of each laser beam.

3.3.3 Integration into the experimental apparatus
For a rapid and robust MOT loading, the three experiment stages, the TC unit, thespin-flip ZS and the dual-species MOT, must seamlessly integrate and adjust to eachother. The following paragraphs describe their optical setups at the UHV apparatus.
Transverse cooling unit
The TC unit, as part of the dual-species atomic beam source, is a crucial aspect for the
MOT loading. Its collimation effect on the emitted atomic beam along two orthogonalaxes in transverse direction strongly improves the MOT loading rates (see below). The
collimation along each of the two axes relies on an optical molasses technique [120], wherean atom with velocity v along a molasses axis experiences the scattering forces Fscat (seeequation (3.1)) from two counterpropagating red-detuned laser beams as [83]Fmol = Fscat (2πδ − kv) − Fscat (2πδ + kv) ,(3.8)

with the linearization [83]Fmol = −αv = −4h̄k 2−2(2πδ+kv)/ΓI

 v
Isat 1 + 4(2πδ+kv)2/Γ2 2


(3.9)

for small v along the molasses axis. As Fmol points opposite to v, it damps the transversal
atomic motion, and thus leads to an effective collimation of the atomic beam [120].
Figure 3.12 sketches the layout of the TC unit. First, PM single mode fibers guidethe λEr = 401 nm and the λDy = 421 nm laser light from their respective laser setup (seeFigures 3.8 and 3.9) to the TC unit’s octagon chamber (see Figure 3.12 (a)). In each beampath, a telescope expands the polarization-cleaned laser light to an 1/e2 -waist wTC = 3 mm.
Subsequently, a dichroic mirror44 overlaps both laser beams. The emerging bichromatic
beam splits into two beams, which then travel up into the TC axes (see Figure 3.12 (b)).
A 50/50-beam splitter cube ensures an equal power ratio between the two TC axes, andthus a balanced transverse cooling of the atomic beam. Figure 3.12 (c) depicts the opticalsetup along each of the two orthogonal TC axes. Initially, a custom-made dichroic
λ/4-plate45 changes the light’s polarization to circular. A cylindrical telescope reshapes
43

Thorlabs Inc., DMLP605R
Chroma Technology Corp., T412lpxt-UF3
45
LENS-Optics GmbH, multi order λ/4-plate for 401 nm and 421 nm
4450

3.3 Laser systems and optical setups(b)y421 nm

(c)

z
y

x

Ion pump

(a)

z

x

Retroreflector-50
401 nm500
Dichroic-50

Atomic beam
1

500
50/50

f

f

Lens withfocal length f
in mm
MirrorPolarizing beam
splitter cubeam

ic b eAtom12
BeneathTC chamber

λ/2-plate

2

Cylindricallenses200-20

Dichroic
λ/4-plate
Transverse
beam profile

Figure 3.12: Overview over the TC unit. (a) Optical setup for a bichromatic beam of
λEr = 401 nm and λDy = 421 nm laser light. Telescopes collimate the two laser beams
individually to 1/e2 -waists wTC = 3 mm. A dichroic mirror overlaps both wavelengths andthe emerging bichromatic beam splits subsequently with an equal power ratio into two.
(b) Front view of the TC unit’s octagon chamber in the y-z plane. The two incoming
bichromatic TC beams travel diagonally upwards along the TC axes. After the octagon
chamber, retroreflectors send both TC beams back, balancing the TC for the atomic
beam, which propagates into the plane. (c) Optical setup along each of the TC axes.
The setup includes a dichroic λ/4-plate, a cylindrical telescope and a retroreflector.
the TC beam and increases its major axis parallel to the atomic beam propagation axis
0(x-axis) from wx,TC = 3 mm to wx,TC= 30 mm. A retroreflector46 completes the TC
concept, while maintaining the circular polarization of the laser light.
For dysprosium, the TC saturates at P421 nm,TC = 60 mW along each TC axis, corresponding to a peak intensity Iˆ421 nm,TC = 0.8Isat,421 nm , with a sixfold improvement of
the MOT loading rate [88]. For erbium, on the contrary, despite P401 nm,TC = 150 mW ineach TC beam (Iˆ401 nm,TC = 1.8Isat,401 nm ) and a tenfold increase of the MOT loading
rate [88], the TC effect does not show any saturation. A possible explanation might bethat the dual-filament effusion cell additionally evaporates pure erbium from its hot-lip tocompensate the difference in pvap,Er and pvap,Dy (see Section 3.1). As this contribution to
the atomic beam only passes through two of the in total three apertures (see Figure 3.2),
this might result in a broader transverse velocity distribution requiring higher P401 nm,TC .
Spin-flip Zeeman slower
Figure 3.13 includes the optical setup for the spin-flip ZS. Identical to the TC unit, theoverlapping of polarization-cleaned λEr = 401 nm and λDy = 421 nm laser light via adichroic mirror leads to a bichromatic ZS beam. A telescope collimates the ZS beam46

Two 2 " × 2 " UV-enhanced aluminum mirrors (Thorlabs, PFSQ20-03-F01) under 90 °51

Chapter 3 Dual-species intercombination-line magneto-optical trap
to an 1/e2 -waist wZS = 4 mm and a vertical periscope guides it onto a dichroic λ/4-plate,identical to the TC unit, which rotates the light’s polarization to circular. A dichroic
mirror47 (Img-DM) reflects the ZS beam upwards through a CF63 viewport into the UHV
apparatus, where an in-vacuum aluminum mirror subsequently deflects the ZS beam into
the spin-flip ZS (see sectional view in the inset of Figure 3.13).
An efficient ZS operation requires P401 nm,ZS = 57 mW (Iˆ401 nm,ZS = 3.8Isat,401 nm ) andP421 nm,ZS = 120 mW (Iˆ421 nm,ZS = 8.5Isat,421 nm ).
Dual-species intercombination-line magneto-optical trap
A textbook-like MOT traps atoms in all three dimensions by combining three orthogonal pairs of counterpropagating red-detuned circularly-polarized laser beams with aquadrupole B-field [83], as [129] first realized in 1987 with sodium. Neglecting gravity, it
exerts a force along each of the three orthogonal MOT beam axes l as [83]
−

+

σ
σ
FMOT,l = Fscat(2πδ − kv − βl) − Fscat(2πδ + kv + βl)α' −αv − βlkα ∆µ= −αv −(∇B)l l,k h̄(3.10)

with α in equation (3.9), the wavenumber k and the circular polarization’s handedness σ ±with respect to a fixed quantization axis. The position-dependent term βl describes the
Zeeman shift EZS due to the quadrupole B-field, and depends on the MOT transition’s
differential magnetic moment ∆µ, the gradient (∇B)l and the position l along the MOT
beam axis. It gives rise to the restoring force character of FMOT that allows a MOT to
act not only in momentum space, such as an optical molasses, but also as a trap [83].
The dual-species intercombination-line MOT at λEr = 583 nm and λDy = 626 nm
operates with orthogonal bichromatic MOT beams in the horizontal x-y plane andalong the vertical z-axis. Its narrow-line laser cooling character results in very low
final MOT temperatures close to the Doppler temperatures TD,Er,583 nm = 4.5 µK and
TD,Dy,626 nm = 3.2 µK [88] (see Table 2.2), however, in comparison to a textbook-like
MOT, its working regime is entirely different. For an efficient MOT loading, the dualspecies MOT requires MOT light detunings δ583 nm and δ626 nm of typically several tensof ∆νEr,583 nm and ∆νDy,626 nm (see Table 2.2). As [130] investigated for strontium, sucha trapping regime provides hard walls rather than a linear restoring force over the entire
MOT (see equation (3.10)). In particular, the interplay between gravity, the scattering
forces Fscat and the MOT temperature TMOT determines the MOT trapping behavior
along the vertical z-axis and results in atoms accumulating at the MOT bottom [130].
For the dual-species MOT, this not only leads to an intrinsic spin-polarization in thelowest Zeeman sublevel [74, 92] but also makes the top-down MOT beam redundant,
allowing for a novel open-top MOT configuration along the vertical z-axis with [130]−σFMOT,z = Fscat(2πδ − kv − βz) − mg,(3.11)

with the gravitational acceleration g. This PhD thesis’ first publication [88] provided
corresponding characterization measurements (see Section 3.6).
47

LENS-Optics GmbH, highly reflective for 401 nm and 421 nm and highly transmittive for 532 nm,
583 nm, 626 nm, 1064 nm and 1550 nm

52

3.3 Laser systems and optical setups

421 nmImaging

MirrorPolarizing beam
splitter cubexz

401 nmFirst breadboard layer

λ/2-plate
AR2

Gate valve

AR1

Dichroic

1
Periscopealong z-axis

Main chamber
Effusion celland TCy

401 nmf

Dichroic
421 nmf

Iris
Lens withfocal length f
in mm

250
Aluminum mirror

Dichroic
λ/4-plate

Zeeman slower

15.29

Neo sCMOSz
Pumping section

Periscopealong z-axis

x
y

2

AR1

Gate valve

AR1

AR2

Figure 3.13: Top view on the UHV apparatus around the main chamber with nearby
optical setups. At the main chamber, arrows indicate the arrangement for the horizontal
MOT beams (yellow and red arrows) and for the horizontal imaging beam (violet andblue arrows, see Section 3.4). Figure 3.14 illustrates the optical setup along the horizontal
MOT axis 1-2 in greater detail. A preparation setup next to the main chamber overlaps
the imaging light at λEr = 401 nm and λDy = 421 nm, and PM fiber-couples it into
the imaging path. A periscope after the main chamber guides the imaging light onto
a sCMOS camera. The ZS setup in the top right prepares a bichromatic ZS beam of
λEr = 401 nm and λDy = 421 nm laser light. A telescope expands the ZS beam to a
1/e2 -waist wZS = 4 mm and an in-vacuum aluminum mirror reflects it into the spin-flip
ZS, as the inset depicts in a vertical sectional view along the atomic beam propagation
axis (x-axis). The corresponding dichroic mirror is out of the sectional plane. AR1 andAR2 label the viewports’ antireflection coatings (see Section 3.1).
Figure 3.13 illustrates the arrangement of the two retroreflected horizontal MOT beams
in a top view and Figure 3.14 shows a vertical sectional view along one of the horizontal
MOT axis. The inset of Figure 4.1 includes the optical setup for the vertical MOT beam.
For each of the three orthogonal MOT axes, the MOT distribution provides an individual
PM single mode fiber guiding the λEr = 583 nm and the λDy = 626 nm laser light to themain chamber (see Section 3.3.2). A fiber-coupled achromatic beam expander48 extends
the bichromatic MOT beam to a 1/e2 -waist wMOT = 19 mm , then a dichroic mirror49(MOT-DM) reflects the MOT beam into the main chamber (see Figures 3.14 and 4.1).
The achromatic beam expander includes a dichroic λ/4-plate, and thus fixes the light’s
polarization to the required circular polarization. The MOT-DMs’ dielectric coating has
an optimized PM performance especially preserving the polarization’s ellipticity.
48
49

Schäfter + Kirchhoff GmbH, 60FC-Q-4-M200-04-amag
LENS-Optics GmbH, highly reflective for 583 nm and 626 nm and highly transmittive for 401 nm,
421 nm, 1064 nm and 1550 nm

53Dichroic
λ/4-plate

Objectivex-y
AR1

Imagingz

MOT distribution

Chapter 3 Dual-species intercombination-line magneto-optical trap

2

1
MOT-DM

Img-DM

MOT-DM

Img-DM

Figure 3.14: Vertical sectional view along the horizontal MOT axis 1-2 (see Figure 3.13).
Starting from the achromatic beam expander, a MOT-DM deflects the incoming bichromatic MOT beam of λEr = 583 nm and λDy = 626 nm laser light into the main chamber.
On the opposite side, a retroreflection setup, which consists of a second MOT-DM, adichroic λ/4-plate and a broadband dielectric mirror under 0 °, reflects the MOT beam
back. Additionally, this particular MOT axis includes the horizontal imaging setup. It
comprises an achromatic beam expander for the imaging light at λEr = 401 nm and
λDy = 421 nm, two Img-DMs as well as a home-built objective (see Section 3.4). The
influence of the Img-DM on the retroreflected MOT beam, and thus on the dual-species
MOT is negligible. AR1 refers to the viewports’ antireflection coating (see Section 3.1).
For each horizontal MOT axis, a second MOT-DM after the main chamber reflects thehorizontal MOT beam out of the horizontal optical axis through a dichroic λ/4-plate50
on a broadband dielectric mirror under 0 ° serving as retroreflector (see Figure 3.14).
This arrangement moves the required λ/4-plate out of the horizontal optical axis, but still
ensures the correct polarization for the retroreflected MOT beam. The two incoming
horizontal MOT beams arrive opposite to the spin-flip ZS port to support the initial
MOT trapping by providing their higher powers counter-propagating to the arriving
atoms (in comparison to the retroreflected MOT beams).
Despite the large MOT beams on each axis, the combination of achromatic beam
expander, dichroic mirrors and retroreflection setup is extremely compact and provides
sufficient optical access for laser beams of different wavelengths for future experiments.
For an efficient MOT loading, a bias B-field BHH,MOT additionally shifts the origin of
the quadrupole B-field BAH,MOT below the center of the main chamber (see Section 3.2.2).
Hence, considering the initial working regime of the dual-species MOT at large MOT lightdetunings δ583 nm and δ626 nm , the atoms accumulate at the MOT bottom well below thecenter of the main chamber. Thus, subsequent to the MOT loading, a compression phaseof tcomp = 400 ms moves the MOT up towards the center of the main chamber, while
reducing its size as well as its temperatures from TMOT,Er and TMOT,Dy to TcMOT,Er and
TcMOT,Dy in the cMOT, respectively. Table 3.1 summarizes typical operating parameters
for the MOT loading and the cMOT, while Figure 4.2 shows their time traces in thecomplete experimental sequence. During the compression phase, δ583 nm and δ626 nm changeby ∆δ583 nm = δ583 nm,MOT − δ583 nm,cMOT and ∆δ626 nm = δ626 nm,MOT − δ626 nm,cMOT oftens of ∆νEr,583 nm and ∆νDy,626 nm (see Table 2.2), respectively. Their precise values
depend on the chosen isotopes as well as the operating state of the system correlatingwith all other parameters, and thus Table 3.1 does not provide reference values.
50

LENS-Optics GmbH, multi-order λ/4-plate for 583 nm and 626 nm

54

3.4 Imaging system

Table 3.1: Typical operating parameters for the dual-species intercombination-line MOT
at λEr = 583 nm and λDy = 626 nm in an open-top configuration. The compression phaseto the cMOT, changes the B-field bias value BHH and the gradient value ∇BAH linearly,
and reduces the horizontal and vertical MOT beam powers Phor and Pver exponentially.
Parameters
Bias
Gradient
Horizontal beams(values per beam)

Vertical beam

MOT loadingcMOT

BHH (G)x,y,z−3.2451.020∇BAH (G/cm)

x,y
z

2.25
4.50

1.94
3.88

Phor (mW)

583 nm626 nm

36
69

0.12
0.25

Iˆhor (Isat )583 nm626 nm

52
168

0.17
0.61

Pver (mW)

583 nm626 nm

32
64

0.10
0.24

Iˆver (Isat )583 nm626 nm

46
157

0.15
0.59

3.4 Imaging system
An absorption imaging on the broad transition at λEr = 401 nm and λDy = 421 nm images
the cMOT or the ODT (see Chapter 4). The laser light first arrives PM fiber-coupledfrom the laser setups (see Figures 3.8 and 3.9) at a preparation setup next to the UHV
apparatus (see Figure 3.13). A dichroic mirror51 overlaps both polarization-cleaned
beams to a bichromatic beam and a PM single mode fiber leads to the imaging setup.
Figure 3.14 sketches the imaging path at the main chamber. Initially, a fiber-coupled
achromatic beam expander52 with an integrated dichroic λ/4-plate provides a circularly
polarized bichromatic imaging beam with a 1/e2 -waist wimg = 5 mm. An Img-DM deflects
the imaging beam through the MOT-DM into the main chamber. On the outgoingside, in order to avoid aberrations, a second Img-DM reflects the imaging beam directlyin front of the MOT beam’s retroreflection setup out of the horizontal optical axis
upwards to the objective. Subsequently, a periscope guides the imaging beam onto ascientific-CMOS-based camera53 (see Figure 3.13).
The objective is home-built and consists of a 2 "-lens triplet54 . The imaging system
operates at a working distance of 257 mm, a magnification of 3.2 and provides a diffraction
limited resolution of 3.5 µm at λEr = 401 nm and λDy = 421 nm, respectively.
At the beginning of the absorption imaging sequence, the camera takes a cleaning image,the trap as well as the inverted-viewport HH-like coil switch off and the compensation
cage rotates the B-field within 9 ms into the imaging axis. The camera’s field of view
51

Chroma Technology Corp., T412lpxt-UF3
Schäfter + Kirchhoff GmbH, 60FC-Q-4-M50L-01-AM
53
Andor Neo 5.5 sCMOS, 16.6 mm × 14.0 mm chip size, 2560 × 2160 pixel, 6.5 µm × 6.5 µm pixel size,
AR-coated window for 401 nm,421 nm,583 nm and 626 nm
54
Thorlabs, two achromatic doublets, AR coating 400 nm-700 nm, focal length f = 500 mm, one planoconvex lens, AR coating 350 nm-700 nm, f = 1000 mm
5255

Chapter 3 Dual-species intercombination-line magneto-optical trap
allows for a variable time-of-flight (TOF) expansion of up to 27 ms, after which thecamera records the required pictures in the following order. First, it captures the erbium
and dysprosium samples on two individual pictures 10 µs apart. Following a processing
time of 40 ms, it then takes the reference pictures for the imaging light with the sametime delay of 10 µs, and finally, with another wait time of 40 ms, it records a dark picture.
A compact CMOS camera55 images the cMOT along the second horizontal MOT
axis by collecting its fluorescence for 70 ms directly at the beginning of either a cMOT
holding time or a ODT loading time (see Chapter 4). A reference picture at the very
end of the experimental sequence completes the fluorescence imaging. This imaging is
non-destructive and especially beneficial for experiments in the ODT, where otherwise
no information about the MOT or the cMOT performance is available.

3.5 Control system
A network of four computers controls and monitors the entire experimental apparatus. A
control computer provides the possibility to define the consecutive sequence commands in
script format and to set the individual sequence parameters in a graphical user interface.
For every experimental repetition, this unit compiles the experimental sequence andprograms a digital I/O PCI card56 . It transfers its output via a BUS system to racks
holding individually programmable direct digital synthesizers and analog or digital output
channels, which for instance generate the AOMs’ or fiber-coupled EOMs’ RF signals,
set RF power levels or provide the camera triggers. The time resolution between single
sequence commands is 5 µs. The second computer interfaces the cameras for absorption
and fluorescence imaging (see Section 3.4) and stores the obtained images. At the endof every experimental repetition, a third computer accesses, analyzes and stores theseimages fully automatically. This provides essentially a real-time analysis of the obtained
data. The last computer monitors all laser systems and the controller of the dual-filament
effusion cell as well as records data from a temperature and a cooling water monitoring.

55
56

Allied Vision, Guppy PRO F503B mono, 2588x1940 pixel, 2.2 µm × 2.2 µm pixel size
National Instruments PCI-6534

56

3.6 Publication I:

Two-species five-beam magneto-optical trap for
erbium and dysprosium
P. Ilzhöfer1,2,† , G. Durastante1,2 , A. Patscheider1,2 , A. Trautmann1 , M. J. Mark1,2 and
F. Ferlaino1,2
1 Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck,
Technikerstrasse 25, 6020 Innsbruck, Austria
2 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, 6020 Innsbruck, Austria

Physical Review A, 97, 023633, (2018), DOI: 10.1103/PhysRevA.97.023633,
received 20 November 2017, published 26 February 2018

Note: All atom numbers within this publication are a factor of 1.4 too large due to a
miscalibration of the imaging system.
†

The author of this PhD thesis participated in the development and the construction of the experimental
apparatus, the data recording and analysis as well as in the writing process of this publication.

57

PHYSICAL REVIEW A 97, 023633 (2018)

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

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

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

P. ILZHÖFER et al.

PHYSICAL REVIEW A 97, 023633 (2018)

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

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

TWO-SPECIES FIVE-BEAM MAGNETO-OPTICAL TRAP …

PHYSICAL REVIEW A 97, 023633 (2018)
109

10

TEC (1000° C)
1.1

1.0

8

1.2

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

6B 5B Er6B 5B Dy

2
1
0.5
0.2

7

10

0

5

15

10

20

25

30

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

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

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

Dy

5B

6B

5B

6B

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

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

P. ILZHÖFER et al.

PHYSICAL REVIEW A 97, 023633 (2018)

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

×108
(a)

6
6B
5B

1.5

×108
(b)

5

Dysprosium
6B
5B

4

2 × 1083

1

atom numbers

atom number NDy , NEr

2

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

2

0.5

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

1 × 108

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

Er
Dy

0

200
400
600
800
compressed MOT hold time (ms)

1 000

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

023633-4

TWO-SPECIES FIVE-BEAM MAGNETO-OPTICAL TRAP …

PHYSICAL REVIEW A 97, 023633 (2018)

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

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We acknowledge A. Frisch, T. Pyragius, and C. Zhang for
support in the initial phase of the experiment. We thank the
ERBIUM and the Dy-K teams in Innsbruck, the Er team at
Harvard, and M. Sohmen and C. Politi for fruitful discussions.
This work was supported by an ERC Consolidator Grant(RARE; No. 681432) and a FET Proactive Project (RySQ; No.
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[46] Since the large spatial extension of the MOT might mask residual
atoms in higher spin states, we performed preliminary SternGerlach mapping after loading the atoms into an optical dipole
trap. Again, we did not detect atoms in higher Zeeman sublevels.
[47] J. Arlt, P. Bance, S. Hopkins, J. Martin, S. Webster, A. Wilson,
K. Zetie, and C. J. Foot, J. Phys. B 31, L321 (1998).
[48] A. di Stefano, D. Wilkowski, J. Müller, and E. Arimondo, Appl.
Phys. B 69, 263 (1999).

023633-6

Chapter4

Heteronuclear dipolar
quantum mixtures

Upon accomplishing the first milestone, the dual-species intercombination-line MOT [88],the experimental efforts aimed for the second milestone, the first production of heteronuclear dBECs for various bosonic isotope mixtures of erbium and dysprosium [117].
The figure of merit for Bose-Einstein condensation to occur is the phase space density [6]

3/2
2πh̄2
PSD = n
,
(4.1)
mkB T
as Chapter 2 described in equation (2.16). The PSD scaling with n and T −3/2 implies alarge number density n at a low temperature T in order to achieve the critical point of
the phase transition. Hence, ultracold quantum gas experiments typically switch fromthe initially applied laser cooling to evaporative cooling [6, 131] either in an ODT [44,132] or in a magnetic trap [7, 133]. Only recently experiments achieved Bose-Einsteincondensation in strontium [134] and rubidium [135, 136] via laser cooling.
The dual-species MOT, after a loading time of tMOT = 3 s and after the compression
phase, provides atom numbers of up to NEr = 6.7(1) × 107 and NDy = 2.1(1) × 108 at temperatures TcMOT,Er and TcMOT,Dy close to the Doppler temperatures TD,Er,583 nm = 4.5 µK
and TD,Dy,626 nm = 3.2 µK [88] (see Table 2.2). These are ideal conditions for a directtransfer from the cMOT into an ODT to increase the PSD via evaporative cooling. This isnecessary, since the intercombination-line MOT approach results in PSDs several orders ofmagnitudes below the phase transition’s critical point for erbium [74] and dysprosium [75].
Following single-species experiments [29, 30], the experimental apparatus operates withan ODT at λ = 1064 nm. As it is red-detuned from the broad transition at λEr = 401 nm
or λDy = 421 nm and far off-resonant from any other optical transitions (see Chapter 2),
the ODT provides for both elements a conservative trap with low off-resonant scattering.
This fourth chapter highlights the production of heteronuclear quantum-degenerate
dipolar mixtures of erbium and dysprosium. First, Section 4.1 covers the ODT setup
at 1064 nm, then, Section 4.2 moves to the evaporative cooling to quantum degeneracy,
providing details on the experimental sequence and on the evaporation dynamics. Finally,
Section 4.3 presents the second publication Dipolar Quantum Mixtures of Erbium andDysprosium Atoms [117]. Some information provided throughout this chapter might
coincide with [117] and is not explicitly referenced.
65

Chapter 4 Heteronuclear dipolar quantum mixtures

4.1 Optical dipole trap: 1064 nm
A high evaporative cooling efficiency requires a high elastic collision rate, which scales linearly with the number density n. [7, 131]. This imposes two rather contrary requirementson the ODT setup in order to achieve high n throughout evaporative cooling. First, amaximal mode matching between cMOT and ODT with a sufficiently deep ODT potential
VODT for a high transfer efficiency during the initial ODT loading, and second, a tightly
confining ODT for a small trapping volume during the later stages of evaporative cooling.
A combined approach of a large reservoir ODT and of a tightly focused ODT can fulfill
both requirements. However, the present ODT setup utilizes a time-averaged potential
technique for a geometrically tunable ODT, as [137] successfully demonstrated for erbium.
This approach additionally provides control over the final trapping geometry, which isadvantageous due to the geometrical stability requirements of a dBEC (see Chapter 2).
In general, the VODT an atom experiences in a laser field is
VODT = V (r, ω) = −1
I (r) < {α (ω)} ,
20 c

(4.2)

as Chapter 2 covered in equation (2.13). Because of the linear dependence on theintensity I, VODT benefits from a high power at the cMOT position. Even more, since
the applied time-averaged potential technique increases the mode matching to the cMOT
during the ODT loading at the expense of the VODT depth.
The entire ODT setup operates directly next to the UHV apparatus, as Figure 4.1
depicts. Avoiding any fiber couplings prevents loss in power, and thus increases the power
reserves during the experimental sequence. However, the resulting long beam paths makethe ODT more susceptible to mechanical instabilities within the experimental apparatus.
A commercial master oscillator power amplifier (MOPA) system57 at λ = 1064 nm
produces a total output power P1064 nm = 55 W with a specified linewidth as low as 1 kHz
over a timescale of 100 ms. It combines a master laser, which employs a non-planarring resonator in a neodymium-doped yttrium aluminum garnet (Nd:YAG) crystal, withmultiple amplifier modules. Directly behind the laser output, an optical isolator58 protects
the laser system from back reflections and a set of lenses optimizes the beam shape.
The laser light splits among two beam paths, one for a horizontal and another one for a
vertical ODT (h/vODT) forming together a crossed ODT (xODT) at the cMOT position.
A motorized rotation mount controls a λ/2-plate and allows for a dynamic distributionof P1064 nm between hODT (PhODT ) and vODT (PvODT ) during the evaporative cooling.
This is necessary, since the initial ODT loading requires for the hODT the maximallyavailable PhODT = 32 W at the cMOT position (see Section 4.2.1).
Each beam path has an AOM59 for switching as well as for power stabilization.
Accordingly, photodiodes monitor PhODT in the periscope setup and PvODT above themain chamber for a power stabilization feedback loop, respectively. The AOM in thehODT path enables the geometrically tunable ODT [137], where the modulation of the
AOM’s carrier frequency leads effectively to an increase of the horizontal beam waist,and thus to a time-averaged potential. A gain voltage GhODT dynamically adjusts themodulation amplitude, and thus the hODT aspect ratio (AR), the ratio of horizontal tovertical beam waist, during the experimental sequence.
57

Coherent Inc., Mephisto MOPA 55 W
Electro-Optics Technology, Inc., 110-23052-0002-ISO, 5 mm, 1045 nm − 1080 nm, WPO, ULTRA
59
Gooch & Housego PLC, 3110-197, center frequency 110 MHz, bandwidth 15 MHz
58

66

4.2 Evaporative cooling to quantum degeneracy
z

y
PD

x

x
y

z

1

MOT-DM
MOT distribution

First breadboard
layer
300

f

f

Lens withfocal length f
in mm
MirrorPolarizing beam
splitter cube
λ/2-plate

1000

1

150 -50

500

-50

150

Periscopealong z-axis

150
AOM

50

AOM

Mephisto 1064 nm 55 W

110 MHz
Optical isolator

110 MHzPD-50 100

1000

Figure 4.1: Optical setup for the ODT at λ = 1064 nm directly next to the main chamber.
The MOPA laser light distributes between the horizontal and the vertical ODT (h/vODT)
with a dynamically adjustable power ratio by a motorized λ/2-plate. Both AOMs are forswitching and for power stabilization, while the one in the hODT additionally produces atime-averaged potential [137]. In the main chamber, the ODT beams have foci with 1/e2 waists whODT = 22 µm and wvODT = 130 µm for the hODT and the vODT, respectively.
In the inset, a vertical sectional view along the atomic beam propagation axis (x-axis)
illustrates the optical setup for the vertical MOT beam and for the vODT.
The remaining optical setup subsequently guides the laser beams to the main chamberand focuses the hODT and the vODT to 1/e2 -waists whODT = 22 µm and wvODT = 130 µm.
As an additional degree of freedom, translation stages enable the position adjustment of
the last focusing lenses, and thus of the focus positions for a precisely overlapping xODT
at the cMOT position. Furthermore a λ/2-plate rotates the polarization of the hODT tominimize off-resonant scattering (Rscat , see equation (2.14)).

4.2 Evaporative cooling to quantum degeneracy
This PhD thesis’ second publication [117] reported on the production of five different heteronuclear dBECs as well as of one quantum-degenerate Bose-Fermi mixture (168 Er-161 Dy)
at T/TF ≈ 0.5, with the Fermi temperature TF . Since the production steps are very similar
for the different isotope mixtures, [117] almost exclusively focused on experimental datafor 166 Er-164 Dy. This section reviews the remaining isotope mixtures in greater detailhighlighting the experimental sequence in Section 4.2.1 and their PSD evolution during
evaporative cooling in Section 4.2.2. The discussion also includes the isotope mixture
166 Er-162 Dy, which so far has not reached a heteronuclear dBEC.
67

Chapter 4 Heteronuclear dipolar quantum mixtures
ODT and evaporative cooling ramps
401 nm421 nm150
75
0
1

PvODT (W) GhODT (V) PhODT (W) BHH (G) ∇BAH (G/cm) ∆δ (∆ν)

P (mW)PZS (mW)

ABS

PTC (mW)

MOT loading cMOT0120401 nm421 nm60
040

583 nm,ver
626 nm,ver

20

583 nm,hor
626 nm,hor

00583 nm626 nm−25−50
5x-/y-axis
z-axis

2.5
0
4
0
−4
35

1.5

17.5

0.75

0
3.5

0

1.75
0
15

10

7.5

5

0

0
0 0.15

2.5

3

3.5

4

6.5
4.5
t (s)

7

7.5

8

8.5

9

Figure 4.2: Experimental sequence with time t exemplarily for 170 Er-162 Dy. The time
traces detail the temporal development of all relevant experimental parameters during
the MOT loading, the compression phase, and the ODT and the evaporative coolingramps. Note that, the time axis interrupts at ramp four to provide a higher resolution
and P626 ,MOT deviates from its typical values in Table 3.1 (see text).
684.2 Evaporative cooling to quantum degeneracy

4.2.1 Experimental Sequence
The following paragraphs discuss the generic steps for the production of quantumdegenerate dipolar mixtures. The experimental sequence consists of three steps, the MOT
loading, the compression phase as well as the ODT and the evaporative cooling ramps.
Figure 4.2 depicts the individual time traces of the various experimental parameters withvalues exemplarily for the production of a heteronuclear dBEC of 170 Er-162 Dy.
Initially, the TC and the ZS are active, the ABS is in open position and the dualspecies MOT loads at its typical operating parameters, in particular at MOT beam
powers P583 nm,MOT and P626 nm,MOT , MOT light detunings δ583 nm,MOT and δ626 nm,MOT ,a quadrupole B-field BAH with ∇BAH,MOT and a bias B-field BHH with BHH,MOT (seeTable 3.1). All investigated isotope mixtures, however, require dysprosium as minority
since erbium acts as sympathetic coolant [117] (see Section 4.2.2). For all achieved
heteronuclear dBECs, this imbalance is achievable by adjusting the individual MOT
loading times to tMOT,Dy < tMOT,Er (see Table 4.1), and by reducing P626 nm,MOT ,depending on the system’s operating state, below the values in Table 3.1, but keeping theratio. On the contrary, the Bose-Fermi mixture 168 Er-161 Dy requires tMOT,Dy > tMOT,Erat the typical MOT loading parameters due to the weaker MOT loading performance of
161 Dy arising from the hyperfine structure of the fermionic isotopes (see Chapter 2).
In the next step, the TC and the ZS deactivate and the ABS blocks the atomic beam.
Simultaneously, the hODT switches on at its maximal PhODT = 32 W with AR = 4(GhODT = 3 V) to prevent thermal drifts during the actual ODT loading. Within thenext tcomp = 400 ms, the MOT compresses and moves upwards as the compression
phase reduces the operating parameters to their final values P583 nm,cMOT , P626 nm,cMOT ,δ583 nm,cMOT , δ626 nm,cMOT and ∇BAH,cMOT , respectively, while BHH inverts its direction
to BHH,cMOT (see Table 3.1). The cMOT’s small size and low temperatures TcMOT,Erand TcMOT,Dy together with the increased AR of the hODT optimize the ODT transfer
efficiency, facilitating an ODT loading time of tload = 100 ms. Then, the MOT systems
turn off and the homogeneous BHH jumps to a constant evaporation field value BHH,evap .
At this stage, the first evaporative cooling ramp starts by lowering PhODT and GhODT
in linear ramps. During the second ramp, the motorized λ/2-plate redistributes PhODT
and the vODT increases to PvODT = 15 W to form a xODT. At the end of the thirdramp, the aspect ratio of the hODT reaches AR = 1 (GhODT = 0 V). The evaporative
cooling continues as PhODT and PvODT decrease further. In ramp seven and eight, thelast two evaporation steps, the hODT reshapes to AR = 5 (GhODT = 3.5 V) and, in order
to compensate for the accompanying reduction in VODT , increases to PhODT = 0.41 W.
The vODT remains constant at PvODT = 3 W. After a hold time in the xODT of up
to thold = 100 ms, the imaging system records absorption images with a variable TOF
expansion of up to 27 ms (see Chapter 3).
The switching between different isotope mixtures requires the adjustment, besides oftMOT,Dy , tMOT,Er and P626 nm,MOT , additionally of the MOT light detuning, δ583 nm or
δ626 nm , and of the evaporation field value BHH,evap . The precise detuning values for the
MOT loading and the cMOT depend on the isotopes, whereas the change in BHH,evap
adjusts the intra- and interspecies contact interactions that drive the thermalization
during evaporative cooling. All other experimental parameters stay unchanged (except the
final ODT parameters for 168 Er-161 Dy [117]). Table 4.1 summarizes tMOT and BHH,evap
for the various isotope mixtures as well as all for single-species dBECs. This includes
values for a dBEC of 170 Er, which this PhD work achieved for the first time.
69

Chapter 4 Heteronuclear dipolar quantum mixtures

Table 4.1: MOT loading times tMOT and evaporation field values BHH,evap for all achieved(heteronuclear) dBECs and for the quantum-degenerate Bose-Fermi mixture (168 Er-161 Dy)
in the investigated B-field range BHH,evap = 0.1 G − 5 G. Additionally, γini and γsym
characterize the efficiency of the evaporative cooling at the underlined values of BHH,evap(see Section 4.2.2). Here, [117] published all values with (*). The values for γini do notprovide an error (see text). Note that, 166 Er-162 Dy did not reach a heteronuclear dBEC
within the investigated B-field range (see text).
Isotopes
166 Er168 Er
170 Er

162 Dy164 DytMOT (s)BHH,evap (G)

γini

γsym

3–5
3–5
3–5
3–5
3–5

2.075
4.795
1.500
4.795
2.500

-

-

168 Er-162 Dy

Er
Dy

3
2

3.300*, 3.700, 4.400, 4.795

1.1
1.6

2.6 (0.3)
6.0 (2.0)*

170 Er-162 Dy

Er
Dy

3
2

1.540*, 4.060

0.9
1.0

2.2 (0.6)
11.0 (7.7)*

166 Er-164 Dy

Er
Dy

3
1

2.075*, 5.900-6.100

1.2
1.1

2.2 (0.4)
7.2 (1.8)*

168 Er-164 Dy

Er
Dy

3
1

3.300*

0.8
1.0

2.1 (1.0)
5.9 (1.6)*

170 Er-164 Dy

Er
Dy

3
1

3.210*

0.6
1.5

1.3 (0.7)
3.3 (0.6)*

168 Er-161 Dy

Er
Dy

3.5
5

3.455*

1.6
0.7

1.8 (1.4)
4.1 (1.1)*

166 Er-162 Dy

Er
Dy

3
1

1.600

1.0
1.3

1.4 (0.5)
2.3 (0.5)

4.2.2 Evaporation dynamics
The difference in the real part of the scalar polarizabilities, < {αsca,1064 nm }Er and< {αsca,1064 nm }Dy , at λ = 1064 nm (see Chapter 2) manifests itself in differing trapping
potentials VODT,Er and VODT,Dy , and thus plays a crucial role in the evaporation dynamics.
Figure 4.3 compares VODT,Er and VODT,Dy for 170 Er-162 Dy along the system’s verticalz-axis including gravity for three settings, namely (a) the ODT loading, (b) the xODT at
the beginning of ramp four and (c) the final ODT during thold . While gravity only weakly
influences the settings (a) and (b), it becomes noticeable as VODT,Er and VODT,Dy getshallower, until it has a major influence in (c). As < {αsca,1064 nm }Er < < {αsca,1064 nm }Dy ,VODT,Er is shallower than VODT,Dy , and thus, as for the masses mEr > mDy , the relative
influence of gravity is larger for VODT,Er than for VODT,Dy . These differences lead in ashallow trapping regime to sympathetic cooling of dysprosium by erbium [117].
This behavior occurs for all investigated isotope mixtures in their PSD evolution during
the evaporative cooling. Because of the non-uniform density distribution in a harmonic704.2 Evaporative cooling to quantum degeneracy
(a)

(b)

0

VODT (µK)

VODT (µK)

−100
−200
−300

0
−100
−200
−300

−400
−40

−20

0

20

−40

40

−20

z (µm)(c)

20

40

z (µm)

0ErbiumDysprosium−2
VODT (µK)

0

−4
−6
−8
−10
−12

−20 −10

0

10

20

30

z (µm)

Figure 4.3: Calculated ODT potentials VODT,Er (violet) and VODT,Dy (orange) for
170 Er-162 Dy along the system’s vertical z-axis under the influence of gravity. The

calculation parameters are (a) ODT loading, PhODT = 32 W, PvODT = 0 W and hODT
AR = 4 (GhODT = 3 V), (b) xODT at the beginning of ramp four, PhODT = 5.8 W,
PvODT = 15 W and hODT AR = 1 (GhODT = 0 V), (c) final xODT, PhODT = 0.41 W,
PvODT = 3 W and hODT AR = 5 (GhODT = 3.5 V). The calculation employs the
ODT beam parameters in Section 4.1, for dysprosium the real part of the scalarpolarizability < {αsca,1064 nm }Dy = 184.4 a. u. from [108] and for simplicity the ratio< {αsca,1064 nm }Er = <{αsca,1064 nm }Dy/1.06 [117].
trap, equation (4.1) replaces the number density n by the peak number density n̂. The
harmonic oscillator density distribution [6]n (r) =

π3/2

N−x2 2 −y 2 2 −z 2 2
e /Rx e /Ry e /Rz ,Rx Ry Rz

(4.3)

p
2kB T/mω 2
with the position r, the atom number N , Rx,y,z =
x,y,z and the trapping
frequency ωx,y,z along the corresponding x-, y- or z-axis, gives

n̂ =

m
2πkB T3/2
N (ωx ωy ωz ) .

(4.4)
71

Chapter 4 Heteronuclear dipolar quantum mixtures
Hence, absorption images after each ramp provide via N and T together with calculated
ωx,y,z sufficient information to extract the PSDs via equation (4.1). Starting from thesecond ramp, Figure 4.4 shows for each ramp ln (PSD/PSD0 ) over ln (N/N0 ) at the underlined
values of BHH,evap in Table 4.1. The second ramp’s PSD0 and N0 serve as normalization
to the individual PSD and N . Identical to 166 Er-164 Dy [117] (see Section 4.3), all otherisotope mixtures display a change in the evaporation efficiency [131]γ=−d ln (PSD/PSD0 )d ln (N/N0 )(4.5)

upon evaporative cooling from the xODT at ramp four. Linear fits (dashed and solid lines)
access γ up to and starting from ramp four, γini and γsym , respectively, to characterize
the change in the evaporation behavior. The gain in γ is significantly larger for γsym,Dythan for γsym,Er indicating sympathetic cooling of dysprosium by erbium [117]. Table 4.1
summarizes the results, which [117] already partially published. The limited amount ofdata points prevents a thoroughly fitting of γini . Hence, Table 4.1 does not provide thecorresponding errors and the dashed lines in Figure 4.4 are a guide to the eye.
For the investigated B-field range BHH,evap = 0.1 G − 5 G, 166 Er-162 Dy did not reacha heteronuclear dBEC. A possible explanation might involve the low gain in γsym,Dy incomparison to the other isotope mixtures, even for the best evaporation performanceat BHH,evap = 1.600 G (see Table 4.1). However, to conclude for instance on either
increased losses from the isotope mixture or an unfavorable thermalization behavior in
the investigated B-field range, further experimental studies are necessary.

724.2 Evaporative cooling to quantum degeneracy
(a)

162

4
2

−3

−2

−1

0

5.0
2.5

−3

ln (N/N0 )
(c)

10.0

−2

−1

(d)

Er
162
Dy

10.0
168

ln (PSD/PSD0 )ln (PSD/PSD0 )

5.0
2.5

164

7.5

−4

5.0
2.5

−3

−2

−1

−4

0

−3

8

−2

Er
164
Dy

(f)

8

170

168

ln (PSD/PSD0 )

6
4
2
0
−5

−4

−3

−2

ln (N/N0 )−10ln (N/N0 )ln (N/N0 )ln (PSD/PSD0 )

Er
Dy

0

0

(e)

0

ln (N/N0 )

170

7.5

Er
Dy

7.5

0
−4

0
−4

168

10.0
ln (PSD/PSD0 )6
ln (PSD/PSD0 )

(b)

Er
162
Dy
166

−1

0

161

6

Er
Dy

4
2
0
−5

−4

−3

−2

−1

0

ln (N/N0 )Figure 4.4: Evolution of the phase space density PSD over the atom number N from the second ramp onwards for (a) 166 Er-162 Dy, (b) 168 Er-162 Dy, (c) 170 Er-162 Dy, (d) 168 Er-164 Dy,(e) 170 Er-164 Dy and (f) 168 Er-161 Dy. Each plot shows ln (PSD/PSD0 ) over ln (N/N0 ) withthe second ramp’s PSD0 and N0 serving as normalization. The violet and orange lines
are linear fits to the experimental data (violet and orange diamonds) to extract theevaporation efficiencies γini up to (dashed lines) and γsym starting from (solid lines) ramp
four for erbium and dysprosium, respectively. Table 4.1 provides the MOT loading times
tMOT and the evaporation field values BHH,evap for the individual data sets and lists thefit results. Note, 166 Er-162 Dy did not reach a heteronuclear dBEC.
73

4.3 Publication II:

Dipolar Quantum Mixtures of Erbium andDysprosium Atoms
A. Trautmann1 , P. Ilzhöfer1,2,† , G. Durastante1,2 , C. Politi1 , M. Sohmen1,2 ,M. J. Mark1,2 , F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck,
Technikerstrasse 25, 6020 Innsbruck, Austria

Physical Review Letters, 121, 213601, (2018), DOI: 10.1103/PhysRevLett.121.213601,
received 20 July 2018, revised 8 October 2018, published 21 November 2018,
Editors’ suggestion
Featured in Physics 11, s135, Synopsis: Making Mixtures of Magnetic Condensates†

The author of this PhD thesis participated in the data recording and analysis as well as in the writing
process of this publication.

75

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

Featured in Physics

Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms
1

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

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

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

(b)

(c)

(d)

(e)

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

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

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

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

(a)

(b)

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

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

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

(b)

(c)

(d)

(b)

(e)
(d)
(e)

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

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

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

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

*
These authors contributed equally to this work.
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[52] Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and B. L. Lev,s-wave scattering lengths of the strongly dipolar bosons
162
Dy and 164Dy, Phys. Rev. A 92, 022703 (2015).
[53] C.-H. Wu, I. Santiago, J. W. Park, P. Ahmadi, and M. W.
Zwierlein, Strongly interacting isotopic bose-fermi mixture
immersed in a fermi sea, Phys. Rev. A 84, 011601 (2011).
[54] The trap frequencies along the y and z axes are extracted
from center-of-mass-oscillation measurement; the frequency along x is calculated based on the beam waist,
calibrated via the z frequency, and our knowledge of thepolarizability [14,49].
[55] W. Ketterle and N. V. Druten, in Evaporative Cooling ofTrapped Atoms, edited by B. Bederson and H. Walther,
Advances in Atomic, Molecular, and Optical Physics Vol. 37(Academic Press, New York, 1996), pp. 181–236.
[56] A. Frisch, Dipolar quantum gases of erbium, Ph.D. thesis,
University of Innsbruck, 2014.
[57] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C.
Makrides, A. Petrov, and S. Kotochigova, Nature (London)
507, 475 (2014).
[58] K. Baumann, N. Q. Burdick, M. Lu, and B. L. Lev,
Observation of low-field fano-feshbach resonances in ultracold gases of dysprosium, Phys. Rev. A 89, 020701 (2014).
[59] T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. FerrierBarbut, T. Pfau, A. Frisch, S. Baier, K. Aikawa, L. Chomaz,
M. J. Mark, F. Ferlaino, C. Makrides, E. Tiesinga, A. Petrov,and S. Kotochigova, Emergence of Chaotic Scattering in
Ultracold Er and Dy, Phys. Rev. X 5, 041029 (2015).
[60] For the Bose-Fermi mixture, slightly different final trap
parameters are required, with typically νEr ¼ (95ð5Þ;
33ð1Þ; 120ð1Þ), νDy ¼ (106ð5Þ; 34ð1Þ; 142ð1Þ).
[61] The light pulse at the respective imaging transition has aduration of 1 ms. We have checked that the resonant light of
one species does not affect the other.
[62] S. Kotochigova, Controlling interactions between highly
magnetic atoms with feshbach resonances, Rep. Prog. Phys.
77, 093901 (2014).

213601-6

Chapter

5

Supersolidity in dipolar
quantum gases of dysprosium

In 1969, A. F. Andreev and I. M. Lifshitz [138] and, in 1970, G. V. Chester [139] andA. J. Leggett [140] suggested a new state of matter, the so-called supersolid, which at
the same time exhibits the crystalline order of a crystal and the flow of a superfluid. A
more rigorous description connects to the spontaneous formation of a density long-rangeorder as well as of an off-diagonal long-range order in a homogeneous system [141]. The
first implies a broken translational symmetry intrinsic to crystalline solidity, whereas thelatter attributes for delocalization characteristic to superfluidity [141]. Because of thesecontradicting properties, supersolidity attracted a wide interest across different research
fields in physics.
In a first proposal, [140] considered the possibility to observe in solid helium a supersolidstate of matter via its non-classical rotational inertia, also present in the superfluid phaseof helium. Eventually, torsional oscillator experiments with solid 4 He, first within aporous Vycor glass disk [142] and then in bulk form [143], observed a non-classicalrotational inertia and interpreted their results as signature of supersolidity. A refinedapparatus, however, disproved the initial claims, putting also the results in the bulk casein question [144]. Although over the years various alternative techniques arose [145], forinstance the search for mass flow [141, 145] or fourth sound propagation [146] through asupersolid, none achieved a positive outcome. Hence, all efforts to confirm supersolidity
in solid 4 He, even with more recent studies [147], miss definite evidence so far [141, 145].
Ultracold quantum gas systems joined the endeavor as an alternative platform dueto their high degree of control. Experiments with purely contact-interacting quantum
gases reported for instance the softening of a roton-type mode at the phase transition
to a supersolid utilizing an optical cavity [148], a supersolid in vicinity of two opticalcavities [149] and the characterization of its Higgs and Goldstone mode [150], or a spinorbit-coupled BEC with supersolid properties [151]. Despite the indisputable significanceof these results, all these system impose a stiff crystal wavelength by external laser lightand miss a purely interparticle-interaction-driven formation of density long-range order.
Scientific attention increasingly directed towards dipolar quantum gases, as theircomplex interparticle interactions spontaneously support a supersolid phase [141, 152].
Recently, four experiments, one in the group of Tilman Pfau in Stuttgart [153], one in
the group of Giovanni Modugno in Florence [154] and two experiments in the groupof Francesca Ferlaino in Innsbruck [155], achieved erbium and dysprosium supersolids
83Chapter 5 Supersolidity in dipolar quantum gases of dysprosiumin absence of any optical cavity or optical lattice. The following paragraphs review
the developments that led to this first observation of supersolidity in three-dimensional
magnetically dipolar quantum gases.
In a pure mean-field description, the behavior of a trapped dipolar quantum gas follows
the time-dependent GPE [110]
 2 2

∂
h̄ ∇
ih̄ ψBEC (r, t) = −+ Vtrap (r, t) + Vint (r, t) ψBEC (r, t) ,
(5.1)
∂t
2m
as Chapter 2 discussed in equation (2.17), predicting a non-modulated dBEC as the
only stable phase. In general, an important characteristic of a BEC is its Bogoliubovde Gennes excitation spectrum E [110], which depends on the momentum q. A purely
contact-interacting BEC shows a phononic (linear in q) and a free-particle (quadraticin q) dispersion for small and large momentum values q, respectively, where [6, 110]sh̄4 4 4πh̄4 as nBEC 2
E (q) =
q +
q ,
(5.2)
4m2
m2
with the mass m, the s-wave scattering length as and the BEC number density nBEC (seeequation (2.19)). In comparison, an uniform dBEC exhibits a more complex excitation
spectrum [110]sh̄4 4 4πh̄4 as nBEC 2
E (q) =q +q (1 + dd (3 cos2 α − 1)).
(5.3)
4m2
m2
The additional term in equation (5.3) arises from the magnetic dipole-dipole interaction Vdd (see equation (2.23), µ1 = µ2 = µ), and involves dd (see equation (2.33)) as
well as the Fourier transform of Vdd [66, 110]Z 2
 µ2 µ0µ µ0 1 − 3 cos2 θ −iq·(r−r0 )0(FVdd ) (q) =
e
d
r
−
r
=
3 cos2 α − 1 .
(5.4)
0
3
4π |r − r |3Here, α denotes the angle between q and the orientation of the dipoles, and thus introduces
an anisotropy in equation (5.3). As a result, equation (5.3) combines, similarly to a purely
contact-interacting BEC, a phononic and a free-particle dispersion, but additionallyreveals via its α-dependence an anisotropic dispersion behavior for an uniform dBEC
with anisotropic phonons differing in their sound velocities for q → 0 [110].
A non-uniform dBEC, however, for instance due to a pancake- or a cigar-shaped external
harmonic trap Vtrap , requires a more thorough analysis [116, 156, 157]. It shows, similarly
to superfluid helium, a roton-maxon excitation spectrum [116, 157], which manifests itself
in a characteristic maximum-minimum energy spectrum at intermediate q. Figure 5.1 (a)compares the corresponding basic dependence on q to the one of equation (5.2). The
roton-maxon behavior arises from the q-dependence of the non-local Vdd (or the electric
dipole-dipole interaction for instance in a dBEC of polar molecules) [116], as evident fromequation (5.4) via its α-dependence, and the interplay between Vdd , the contact interaction
and the kinetic energy [157]. The rotonized E appears, depending on the geometry ofVtrap , only along distinct spatial directions of the system [116, 157]. Moreover, it isadjustable for instance via Vtrap (see below) or as , and thus dd (see equation (2.33)) [116],
as [158] demonstrated by Bragg spectroscopy in an erbium dBEC.
84

(a)

E

BEC
dBEC

∆rot = E (qrot )
q
(b)

B

(c)

z
y
x

Figure 5.1: Bogoliubov-de Gennes excitation spectrum E and roton instability of a
non-uniform dBEC. (a) Basic functional behavior of E with momentum value q for apurely contact-interacting BEC (violet) according to equation (5.2) and a non-uniform
dBEC (orange) [116]. The comparison illustrates the phononic and the free-particle
dispersion in a purely contact-interacting BEC, and the roton-maxon exciation spectrum
in a non-uniform dBEC. Here, the roton gap ∆rot characterizes the local minimum at
the roton momentum qrot . (b) Illustration of a non-modulated dBEC at q = 0 in acigar-shaped harmonic trap Vtrap . The symmetry axis of Vtrap is along the y-axis and an
external magnetic field B, and thus the dipoles point along the z-axis. The rotonizedE occurs along the y-axis. (c) Illustration of a macroscopic density modulation in a
dBEC at qrot along the symmetry axis of the cigar-shaped Vtrap , arising due to the roton
excitation. In the density maxima, the mean-field magnetic dipole-dipole interaction Vdd
is attractive. This forms the basis for the roton instability.
At particular interest is the local minimum in the excitation energy, the so-called
roton minimum60 at the roton momentum qrot and the roton gap ∆rot = E (qrot ) (seeFigure 5.1 (a)). The following discussion considers the situation of a dBEC in a cigarshaped Vtrap with the trap symmetry axis along the y-axis (see Figure 5.1 (b)). An
external magnetic field B aligns the dipoles along the z-axis orthogonal to the trap
symmetry axis.
Here, the rotonized E occurs along the y-axis with its minimum atpqy = qrot ∼ mωz/h̄ [157], with the q-component qy and the trapping frequency ωz along60

The name ’roton’ originates from superfluid helium, where in a vivid picture this excitation is a particle
moving as the closely packed neighbors rotate out of its path [159].

85

Chapter 5 Supersolidity in dipolar quantum gases of dysprosium
the y- and z-axis, respectively. Remarkably, thermal and quantum fluctuations canprovide an initial seed population to the roton excitation with far-reaching consequences.
As ∆rot decreases with increasing dd softening the rotonized E, the population of theroton excitation experiences an exponential growth upon ∆rot turning imaginary [157].
In this regime, the dBEC in its initially non-modulated state at q = 0 (see Figure 5.1 (b))
spontaneously develops a macroscopic density modulation at qrot along the y-axis (seeFigure 5.1 (c)). As a result, the mean-field magnetic dipole-dipole interaction Vdd(see equation (2.24)) turns in the density maxima from repulsive to locally attractive,
surpassing the stabilizing characters of the mean-field contact interaction Vcon (seeequation (2.22)) as well as of the kinetic energy. This leads, in an analogous discussion
also for a dBEC in a pancake-shaped Vtrap , to the roton instability [66, 116, 157], acollapse of a non-uniform dBEC in the applied mean-field description.
This was the physical picture prior to 2016. Independent theoretical and experimental work identified a stabilization mechanism beyond the mean-field description, which
supports in a non-uniform dipolar quantum gas besides the dBEC phase two additional
phases, an isolated droplet and a supersolid phase. Initially, [160] reported the fragmentation of a dysprosium dBEC in a pancake-shaped Vtrap into individual droplets uponentering the roton instability. For a possible explanation, the results of [161] moved into
focus, which considered quantum fluctuations in form of a Lee-Huang-Yang correction to
the mean-field description as stabilization mechanism for an otherwise unstable Bose-Bose
mixture. While a conservative three-body interaction was under debate as well [162, 163],
further theoretical investigations on the Lee-Huang-Yang correction [164, 165, 166] aswell as experimental work on droplets in a waveguide [167], the self-bound characterof an individual droplet [168] or the transition from a dBEC to a macrodroplet [169]eventually confirmed quantum fluctuations as stabilization mechanism.
Hence, the theoretical description of a three-dimensional dipolar quantum gas, neglecting three-body processes, involves not only the mean-field terms Vdd and Vcon but also abeyond-mean-field Lee-Huang-Yang term extending equation (5.1) to [164, 169]"∂h̄2 ∇24πh̄2 asih̄ ψgs (r, t) = −+ Vtrap (r, t) +ngs (r, t)
∂t
2m
m
Z 2

µ µ0 1 − 3 cos2 θ+
ngs r0 , t dr0
(5.5)
0
3
4π |r − r |#√128 πh̄2
3/2 5/2+
ngs (r, t) as F (dd ) ψgs (r, t) ,
| 3m
{z
}
Lee-Huang-Yang termwith the ground state wave function ψgs and the number density ngs (see equation (2.19)),
respectively, and [164, 169]
Z

5/2
1
F (dd ) =sin α 1 + dd 3 cos2 α − 1
dα.
(5.6)
2
Because of its scaling as ngs in comparison to the linear scaling with ngs of Vcon and
Vdd , the additional beyond-mean-field Lee-Huang-Yang term plays a crucial role owing
to its repulsive, and thus stabilizing character. It facilitates a radically different phase
diagram to the one arising from equation (5.1) and supports a supersolid phase [153,3/286154, 155]. The broken continuous translational and gauge symmetry in this state ofmatter, corresponding to the initially described density long-range order and off-diagonal
long-range order, respectively, give rise to a complex E with a crystal and a superfluid
excitation branch, as recent work reported for erbium [170] or for dysprosium [171, 172],
also including the confirmation of phase rigidity via the observation of the system’s
low-energy Goldstone mode [173]. For completeness, [174] and [175] provided first studies
in dysprosium on the non-classical rotational inertia of the the supersolid phase, andfurther, [176] theoretically and experimentally investigated the response behavior of aerbium supersolid utilizing Bragg spectroscopy.
The extended GPE (eGPE, see equation (5.5)) offers a fully coherent description of athree-dimensional dipolar quantum gas, and thus does not capture any change in thephase coherence properties across the phase diagram. A stochastic eGPE approach would
be an appropriate choice for a more thorough theoretical analysis, since the dBEC andthe supersolid phase have a fundamentally different phase coherence behavior than theisolated droplet phase, but is so far not available.
The experimental realization of a long-lived dysprosium supersolid via direct evaporative
cooling and well-established analysis tools capable to distinguish the isolated droplet andthe supersolid phase [155], enabled a series of matter-wave-type experiments to study therigidity of global phase coherence and the rephasing dynamics of an out-of-equilibrium
dysprosium supersolid [177]. The remainder of this chapter presents in Sections 5.1
and 5.2 these results as this PhD thesis’ third and fourth publication Long-Lived andTransient Supersolid Behaviors in Dipolar Quantum Gases [155] and Phase coherence inout-of-equilibrium supersolid states of ultracold dipolar atoms [177].

87

5.1 Publication III:

Long-Lived and Transient Supersolid Behaviors
in Dipolar Quantum Gases
L. Chomaz1 , D. Petter1 , P. Ilzhöfer2,† , G. Natale1 , A. Trautmann2 , C. Politi2 ,G. Durastante1,2 , R. M. W. van Bijnen2 , A. Patscheider1 , M. Sohmen1,2 , M. J. Mark1,2
and F. Ferlaino1,2
1 Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck,
Technikerstrasse 25, 6020 Innsbruck, Austria
2 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, 6020 Innsbruck, Austria

Physical Review X, 9, 021012 (2019), DOI: 10.1103/PhysRevX.9.021012,
received 11 March 2019, published 19 April 2019, corrected 25 April 2019
Featured in Physics 12, 38, Viewpoint: Dipolar Quantum Gases go Supersolid†

The author of this PhD thesis participated in the recording of the dysprosium data as well as in thewriting process of this publication.

89

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

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

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

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

DOI: 10.1103/PhysRevX.9.021012

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

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

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

021012-1

Published by the American Physical Society

L. CHOMAZ et al.

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

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

021012-2

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
z

0

y

B

4

0

4

5
0
46

48

50

52

54

-1

2 (c)

4

n (10 µm )aS (a0)

1
0

0.8

(d)

0.4
-10 0 10y (µm)

0

(g)

8

2
N( 10 )y (µm)(b)44

Sn (10 µm )10 (f)
4x-5

0.1 0.2 0.3 0.4 0.5

-1

ID

SSP

ID

0

*

0.6
0.4
0.2

6

0

-10

0
10
y (µm)

SSP
BEC

2

166

0.2
-10 0 10y (µm)BEC

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

164

Er
55

90

95

Dy
100

aS (a0)

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

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

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L. CHOMAZ et al.

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

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

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

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

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L. CHOMAZ et al.

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

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

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

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

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PHYS. REV. X 9, 021012 (2019)

also causes an increase of the three-body-loss rate. In
contrast, 164Dy realizes the opposite case with the background scattering length smaller than add . This featurebrings the important advantage of requiring tuning B awayfrom the FR pole to reach the supersolid regime. As we
describe below, this important difference in scattering
properties leads to a strikingly longer lifetime of the
164Dy supersolid properties with respect to 166Er and to
the recently observed behavior in 162Dy [35,36].
The measurements proceed as follows. For both 166Er
and 164Dy, we first prepare the quantum gas in the stable
BEC regime and then ramp as to a fixed value in the
supersolid regime for which the system exhibits a state ofcoherent droplets (i.e., AΦ =AM ≈ 1); see previous discussion. Finally, we record the TOF images after a variable th
and we extract the time evolution of both AΦ and AM .
The study of these two amplitudes will allow us to answer
the question of whether the droplet structure—i.e., thedensity modulation in space—persists in time whereas
the coherence among droplets is lost (AM > AΦ → 0) or
if the density structures themselves vanish in time
(AM ≈ AΦ → 0).
As shown in Fig. 4, for both species, we observe that AΦ
and AM decay almost synchronously with a remarkably
longer lifetime for 164Dy [Fig. 4(b)] than 166Er [Fig. 4(a)].
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Interestingly, AΦ and AM remain approximately equal
during the whole time dynamics; see insets of Figs. 4(a)and 4(b). This behavior indicates that it is the strength of thedensity modulation itself and not the phase coherence
among droplets that decays over time. Similar results havebeen found theoretically in Ref. [67]. We connect thisdecay mainly to three-body losses, especially detrimental
for 166Er, and possible excitations created while crossing
the BEC-to-supersolid phase transition [40]. For comparison, the inset of Fig. 4(a) shows also the behavior in the ID
regime for 166Er, where AΦ =AM < 1 already at short th andremains so during the time evolution [40].
To get a quantitative estimate of the survival time ofthe phase-coherent and density-modulated state, we fit asimple exponential function to AΦ and extract tΦ , definedas the 1=10 lifetime; see Fig. 4. For 166Er, we extracttΦ ¼ 38ð6Þ ms. For th > tΦ, the interference patterns
become undetectable in our experiment and we recovera signal similar to the one of a nonmodulated BEC state [asin Figs. 2(a) and 2(d)]. These results are consistent withrecent observations of transient supersolid properties in162
Dy [35]. For 164Dy, we observe that the coherent densitymodulated state is remarkably long-lived. Here, we findtΦ ¼ 152ð13Þ ms.
The striking difference in the lifetime and robustness ofthe supersolid properties between 166Er and 164Dy becomeseven more visible when studying tΦ as a function of as(B for Dy). As shown in Fig. 5, tΦ for Er remains
comparatively low in the investigated supersolid regime
and slightly varies between 20 and 40 ms. Similarly to therecent studies with 162Dy, this finding reveals the transientcharacter of the state and opens the question of whether astationary supersolid state can be reached with these
species. On the contrary, for 164Dy we observe that tΦ
first increases with B in the range from 1.8 G to about
1.98 G. Then, for B > 1.98 G, tΦ acquires a remarkably
large and almost constant value of about 150 ms over awide B range. This shows the long-lived character of thesupersolid properties in our 164Dy quantum gas. We notethat over the investigated range, as is expected to monotonously increase with B [40]. Such a large value of tΦ
exceeds not only the estimated tunneling time acrossneighboring droplets but also the weak-axis trap period,
which together set the typical timescale to achieve global
equilibrium and to study collective excitations.
VI. CREATION OF STATES WITH SUPERSOLID
PROPERTIES BY EVAPORATIVE COOLING

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

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

LONG-LIVED AND TRANSIENT SUPERSOLID BEHAVIORS …
(b)

(a)

20

(e)

10 ms

(i)

(f)

50 ms

(j)

(g)

100 ms(k)(h)

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

15

150

10

0

100

(b)

0
53.5

164

Er

54 54.5 1.8
aS (a0)

1.9

2

Dy
2.1

B (G)

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

n (102 µm)

15
10
5
0

(c)

20
n (102 µm)

166

20

15
10
5
0

(d)

20
n (102 µm)

50

Evaporation time (ms)t (ms)

515
10

10 µm-1

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

02

4 6 8 10n (102 µm2)

50-15 -10 -5 0 5 10 15
kY (µm-1)

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

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

021012-9

L. CHOMAZ et al.
300

PHYS. REV. X 9, 021012 (2019)

(a)

(b)

250

10
150
100
50
0
0

1

(c)

A / AM

A , AM

200
8
6
4

0
0
50

200
100

400
150

200

5 µm-1
250

300

350

400

th (ms)20n (102 µm2)

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

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

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

021012-12

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

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

GROUND STATE CALCULATIONS

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

s
=

∗
ψi−1
(y)ψi (y)dy

(1)
(0)

(0)

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

!
.

(2)

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

LINK STRENGTH AND ESTIMATE OFTUNNELING RATE

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

2

TOY MODEL FOR THE INTERFERENCEPATTERN

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

4
3
2
1
0

c

n(µm)

F[n]

n(µm)

F[n]

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

x104

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

F[n]

x104

x104

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

n(µm)

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

F[n]

(3)

F[n]

+

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

n(µm)

(0)

(yi

n(µm)

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

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

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

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

3

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

3
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5g
4
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-4 -2 0 2 4ky(µm-1)

x104
5 d
4
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5 h
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-6-4-2 0 2 4 6
y (µm)

F[n]

n(µm)

n(µm)

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

4c

F[n]

5b

F[n]

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

x104

x104

5a

F[n]

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

x104

n(µm)

F[n]

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

F[n]

n(µm)

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

x104

F[n]

n(µm)

x104
6c
5
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1
0

n(µm)

n(µm)

n(µm)

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

5 b
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5 f
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5 j
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y (µm)

F[n]

n(µm)

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F[n]

6a

F[n]

x104

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

n(µm)

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

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

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

IMAGING ANALYSIS: 164 Dy AND 166 Er

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

DETAILS ON THE FOURIER ANALYSIS

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

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

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

To be able to condense directly into the supersolid,
we modify the dipole trap to condense at a stronger
confinement of ωx,y,z = 2π × (225, 37, 134) Hz. After a
total evaporative cooling duration of 3.1 s, we achieve
Bose-Einstein condensation at 2.55 G and reach a statewith supersolid properties at 2.43 G, keeping the magnetic field constant throughout the entire evaporation
sequence for both cases.
Time of flight and imaging for 166 Er and 164 Dy - Inorder to probe the momentum distribution of the Dy (Er)gases, we switch off the confining laser beams and let theatoms expand freely for 18 ms (15 ms), while keeping themagnetic field constant. Consecutively the amplitude of
B is increased to a fixed amplitude of 5.4 G (0.6 G). In thecase of 164 Dy, the magnetic field orientation is rotatedin order to point along the imaging axis. This ensuresconstant imaging conditions for different as . After an
additional 9 ms (15 ms) we perform a standard absorption
imaging.

TUNING THE SCATTERING LENGTH IN 166 Er
AND 164 Dy
166Erbium - All measurements start with a BEC at
64.5 a0 . In order to probe the BEC-supersolid-ID region,
we linearly ramp as to its target value in tr = 20 ms
by performing a corresponding ramp in B. Due to a
finite time delay of the magnetic field in our experimental
setup and the highly precise values of as needed for theexperiment, we let the magnetic field stabilize for another15 ms before th = 0 starts. By this, we ensure that theresidual lowering of as during the entire hold time is <∼0.3 a0 . In the main text, we always give the as at th = 0 .
Furthermore, we estimate our magnetic field uncertainty
to be ±2.5 mG, leading to a ±0.2 a0 uncertainty of as in
our experiments.
To choose the best ramping scheme, we have performedexperiments varying tr from 0.5 ms to 60 ms, ramping toa fixed as lying in the supersolid regime, and holding forth = 5 ms after a fixed 15 ms waiting time. We record theevolution of AΦ as a function of tr ; see Fig. S4. Whenincreasing tr , we first observe that AΦ increases, up totr = 20 ms, and then AΦ gradually decreases. The initial
increase can be due to diabatic effects and larger excitation when fast-crossing the phase transition. On theother hand, the slow decrease at longer tr can be explained by larger atom loss during the ramp. We thenchoose tr = 20 ms as an optimum value where a supersolid behavior develops and maintains itself over a significant time while the losses are minimal.
x10

3

5

4

AФ

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

3

2

1

0

0

10

20

30
tr (ms)

40

50

60

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

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

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

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

e

b

d

f

2

120
AФ(103)

Scattering length (a0)

c

4

140

100
80

0
5
4
3
2
1
0

60
40
1.1

a

1.3

1.5

1.7

1.9
B (G)

2.1

2.3

2.5

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

ATOM LOSSES IN 166 Er AND 164 Dy

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

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

0 20 40 60
t h (ms)

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

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

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

anism limiting the lifetime of the supersolid properties inDy.

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

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

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

a

c

e

b

d

f

2
1
0
12

AФ(102)tN (ms)32(5)51(9)
46(12)

N (104)3

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

8
4
0

0

50

100

th (ms)

0

50

100

th (ms)

0

50

100

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

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

tN (ms)300(12)728(34)926(36)tΦ (ms)12(5)152(13)133(25)[1] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
S. Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, “Observation of roton mode population in a dipolar quantum gas,” Nat. Phys. 14, 442(2018).
[2] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, “Bogoliubov modes of a dipolar condensate in a cylindricaltrap,” Phys. Rev. A 74, 013623 (2006).
[3] A. R. P. Lima and A. Pelster, “Quantum fluctuations in
dipolar Bose gases,” Phys. Rev. A 84, 041604 (2011).
[4] A. R. P. Lima and A. Pelster, “Beyond mean-field lowlying excitations of dipolar Bose gases,” Phys. Rev. A 86,
063609 (2012).
[5] F. Wächtler and L. Santos, “Quantum filaments in
dipolar Bose-Einstein condensates,” Phys. Rev. A 93,
061603(R) (2016).
[6] F. Wächtler and L. Santos, “Ground-state properties andelementary excitations of quantum droplets in dipolar
Bose-Einstein condensates,” Phys. Rev. A 94, 043618(2016).
[7] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,“Ground-state phase diagram of a dipolar condensate

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[8] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,
L. Santos, and F. Ferlaino, “Quantum-fluctuation-driven
crossover from a dilute Bose-Einstein condensate to a
macrodroplet in a dipolar quantum fluid,” Phys. Rev.
X 6, 041039 (2016).
[9] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut,
and T. Pfau, “Self-bound droplets of a dilute magnetic
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[10] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen,
M. Isoard, S. Stringari, and T. Pfau, “Scissors Mode of
Dipolar Quantum Droplets of Dysprosium Atoms,” Phys.
Rev. Lett. 120, 160402 (2018).
[11] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas,
P. Cheiney, and L. Tarruell, “Quantum liquid droplets
in a mixture of Bose-Einstein condensates,” Science 359,
301–304 (2018).
[12] V. Cikojevi, L. Vranje Marki, G. E. Astrakharchik, andJ. Boronat, “Universality in ultradilute liquid Bose-Bose
mixtures,” arXiv:1811.04436 (2018).
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tunnelling,” Phys. Lett. 1, 251 – 253 (1962).
[14] A. Barone and G. Patern, Physics and applications of theJosephson effect (Wiley, New York, NY, 1982).
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traps containing Bose condensates,” Phys. Rev. Lett. 57,
3164–3166 (1986).
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[17] M. Wenzel, F. Böttcher, J.-N. Schmidt, M. Eisenmann,
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Superfluid Behavior of a Dipolar Bose-Einstein Condensate,” Phys. Rev. Lett. 121, 030401 (2018).
[18] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and
J. Dalibard, “Interference of an Array of Independent
Bose-Einstein Condensates,” Phys. Rev. Lett. 93, 180403(2004).
[19] We note that we have also checked our analysis without
performing the recentering step and the same features
remain. For instance, for our test data of Fig. 2, the effectbeing mainly that the side peaks in (e) are more washed
out and a slight difference occurs between nM and nΦ ,both showing still side peaks.
[20] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm,
and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature (London)
449, 324 (2007).
[21] 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).
[22] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz, and F. Ferlaino, “Probing
the roton excitation spectrum of a stable dipolar Bosegas,” arXiv:1811.12115 (2018).
[23] A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi,
M. Sohmen, M. J. Mark, and F. Ferlaino, “Dipolar
Quantum Mixtures of Erbium and Dysprosium Atoms,”
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5.2 Publication IV:

Phase coherence in out-of-equilibrium supersolidstates of ultracold dipolar atoms
P. Ilzhöfer1,† , M. Sohmen1,2 , G. Durastante1,2 , C. Politi1 , A. Trautmann1 ,G. Morpurgo3 , T. Giamarchi3 , L. Chomaz2 , M. J. Mark1,2 and F. Ferlaino1,2
1 Institut für Quantenoptik und Quanteninformation, Österreichische Akademie derWissenschaften, 6020 Innsbruck, Austria
2 Institut für Experimentalphysik und Zentrum für Quantenoptik, Universität Innsbruck,
Technikerstrasse 25, 6020 Innsbruck, Austria
3 DQMP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

arXiv-preprint, arXiv:1912.10892,
submitted 23 December 2019†

The author of this PhD thesis participated in the data recording as well as in the writing process of
this publication.

111

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

Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
3 DQMP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

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

(Dated: December 24, 2019)

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

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

2

10−3

10−2

10−1

1

c

~
C

30
25

N (103)

20
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10 ID
5
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b1 Preparation

90aS (a0)
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95

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d

BEC

SSP

e

PDF

−4

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a 10

0

100

0

50

0
-

100

0

th (ms)Phase scramblingchange aS

scramble timeb3

Rephasingchange aS

hold timetS
θ0th = 100 ms

θ0thθ1 θ 2 θ 3 θ 4

θ 1 θ 2 θ 3 θ4θ‘1 θ‘2 θ‘3 θ‘4Fig. 1 | Phase diagram, experimental sequence, and starting conditions. a, Ground-state phase diagram for our cigar-shaped
e The grey color indicates a non-modulated BEC, while the red andtrapped 164 Dy gas. The color map shows the values of C.
blue regions correspond to a SSP and ID phase, respectively. The insets show illustrations of the density profiles along the weakaxis for the different phases. b, Illustration of the phase scrambling sequence: Starting from a SSP (b1), we reduce as to enter
the ID regime (b2). During tS , the phases of the droplets can evolve differently, leading to a phase scrambling between theindividual droplets. Eventually, we jump as back to its initial value, re-entering the supersolid regime (b3), where we study thetime evolution of the global phase coherence. c, Amplitudes AM and AΦ , and d, ∆Φ for our evaporatively cooled SSP plotted
over th . Each data point is derived from q = 80–90 individual experimental realizations. The error bars (almost covered by plot
symbols) are the one-σ confidence intervals calculated using a bias-corrected accelerated bootstrapping analysis (Methods) [32].
e, Polar scatter plot for Pi and f, histogram of the probability density function (PDF) for Φi at th = 100 ms.

ties of the system, revealing a clear difference between the
SSP and ID phase [17–19]. In the SSP, a global phase ispresent along the whole system, whereas, in the ID case, phasecoherence is absent. The latter behavior can be understoodby considering that any fluctuation or excitation within a single isolated droplet will drive an independent evolution of thephases, which cannot lock to each other since particle flow isabsent [3]. This type of dephasing has been studied in split
BECs and atomic Josephson-junction arrays [3].
While the phase evolution when moving from a SSP to an
ID can be understood intuitively, highly fundamental and nontrivial questions arise when considering the opposite route,
i. e. when phase-incoherent isolated droplets are linked back
together. First, will the out-of-equilibrium system spontaneously re-establish phase coherence? And, if yes, will
it relax into its supersolid ground state or reach a quasistationary state? Second, which mechanism sets the rephasing
timescale? Finally, whereto will the excitation energy be dis-

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

3

a

1
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A , A M (a.u.)

0.4
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0

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ts (ms)

b
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1

PDF

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

0.5
0
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100
80

100

0

ts = 100 ms
Fig. 2 | Phase scrambling. a, ∆Φ as a function of tS for
the ID phase at 1.65 G (as,ID = 77 a0 ). Each point is derived
from q = 90–100 independent experimental realizations. The
error bars are the one-σ confidence intervals calculated using a bias-corrected accelerated bootstrapping analysis (Methods) [32]. The grey shaded area indicates the theoretical oneσ confidence interval for ∆Φ using the same sample size anda uniformly random phase. The inset shows the according AΦ(cyan) as well as AM (blue). b, Polar scatter plot for Pi and c,histogram of the PDF for Φi at tS = 100 ms.

same q and fully random (i. e. uniformly distributed) phases(Methods).
Simultaneous to the increase of ∆Φ, we observe that AΦ
decreases quickly towards zero, while AM slightly increases.
This behavior shows that the density modulation is maintained
while losing global phase coherence; see inset. As expected,
in the ID phase, quantum and thermal fluctuations as well as
atom losses can give rise to a different time evolution for thephases of the individual droplets. Apparently the vanishingsmall density overlap between droplets (Ce ' 0) prevents anefficient phase locking, resulting in the observed loss of global
phase coherence.
We now move to the core of our experiment and investigate the phase-relocking after the phase scrambling. We set
as back to its initial value, i. e. where the system’s groundstate is a supersolid, by a B-field jump, and study the system’s evolution; see Fig. 1b3. As shown in Fig. 3a, we observe first a rapid reduction of ∆Φ, occurring in the first
20 ms, and then a much slower dynamics with ∆Φ saturating

4

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0

Fig. 3 | Rephasing dynamics. a, ∆Φ as a function of th after
a jump from the ID phase back to the SSP regime at 2.43 G(as,SSP = 88 a0 ). For each point q = 66–74. The error bars
are the one-σ confidence intervals calculated using a biascorrected accelerated bootstrapping analysis (Methods) [32].
The solid black line is an exponential fit to guide the eye. The
inset shows the according AΦ (light red) as well as AM (red).
b, Polar scatter plot for Pi and c, histogram of the PDF for Φiat th = 0 ms and at th = 100 ms (d–e).

to h∆Φith ≥30 ms = 0.20(2). Accordingly, AΦ approaches AM
on the same time scale, whereas AM remains nearly constant.
This re-establishment of global phase coherence is further illustrated with individual polar scatter plots and histogramsin Fig. 3, confirming a reduction of the phase distribution’s
width with increasing th .
Our system of multiple superfluid parts with different
phases interconnected via weak links is reminiscent of a
Josephson-Junction array (JJA) [39], opening the question
whether a JJA framework can capture the main ingredientsof our system’s dynamics. Although our array of droplets
is soft, meaning that the droplets’ shape and their distancechange with as , we construct a simple model in terms of a onedimensional array of coupled grains (Methods). This is justi-

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

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10–3

10–1

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

the ID regime, and partial recoherence (∆Φ ≈ 0.5) in the intermediate regime.
e i. e. the differA further question is whether the value of C,
ent regimes, also dictates the speed of the rephasing dynamics. To explore this aspect, we study the early time dynamicsof ∆Φ by performing a linear fit to the data for th ≤ 20 ms. The
extracted slope characterizes the initial rephasing rate |R|. As
shown in Fig. 4d, in the supersolid regime, we always record
a large rephasing rate, which remarkably is roughly indepene with |R| ≈ 30s−1 . This value is comparable todent of C,
the weak-axis trap frequency, ωy /2π, and compatible with thetime needed for a sound wave or soliton to propagate along thesystem [8, 43]. In contrast, when crossing from the supersolid
to the intermediate regime, we observe a sudden decrease of|R| by almost a factor of two. Evolving from this intermediate
regime to the ID, |R| continuously decreases with decreasing
Ce until it vanishes. While such a decrease of |R| is consistent
with a JJA picture in which the tunneling between the droplets
dictates the rephasing dynamics, the underlying reason for aconstant rephasing rate in the supersolid regime remains an
open question. It might indicate the action of other mechanisms, related for instance to the soft nature of our JJA, or theformation and slow decay of phase defects in the array.
In conclusion, we have reported the first study of theout-of-equilibrium dynamics of a dipolar supersolid after aninteraction-driven phase excitation that fully randomizes thephases. In the SSP regime, we have demonstrated that thesystem re-establishes a high-degree of phase coherence on
the timescale of one trap period by almost perfect rephasing. When tunneling is suppressed by a too weak density
link across our spontaneously-modulated quantum state, therephasing substantially slows down at a rate depending on thetunneling and eventually ceases in the deep ID regime. Our
observations might shed new light on the properties of the particle flow in a SSP and its superfluid properties, whose general understanding is still elusive. Future experimental works,
combined with advanced out-of-equilibrium theoretical models, will be crucial to understand the relaxation dynamics anddissipation mechanisms in isolated and open supersolid statesof quantum matter.

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

8
I. METHODS
A.

Phase diagram and contrast.

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

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

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

C.

Tuning the scattering length.

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

D.

Interference pattern analysis.

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

100
80
60
1.5

1.7

1.9

2.1

B (G)

2.3

0.4

b 10

0.2

PDF

a

120

nM – n (a.u.)

Scattering length (as )9

2.5

Extended Data Figure 1 | Estimated scattering length. Calculated B-to-as conversion for 164 Dy. Red and blue shaded
areas indicate the SSP and the ID region, respectively. The
grey area indicates the BEC region, while the yellow areas
indicate regions around the two narrow Feshbach resonances
located at 2.174 G and 2.336 G where we observe increased
atom loss. We estimate as,SSP = 88 a0 in the SSP at 2.43 G
and as,ID = 77 a0 in the ID at 1.65 G.

obtain a momentum density profile which we normalize by its
sum. From such a momentum profile, a fast Fourier transformation (FFT) yields the 1D density profile ni (ỹ). An in-situ
density modulation in an atomic cloud will lead to side peaks
in ni (ỹ), symmetrically centred around the zero-momentum
peak. To isolate the centre of this specific modulation, we
calculate the incoherent and coherent means of ni (ỹ), which
we denote nM (ỹ) = h|ni (ỹ) |iP and nΦ (ỹ) = |hni (ỹ)iP |, respectively. The incoherent mean nM reflects the mean modulation amplitude of the cloud at the respective wavelengthỹ. The coherent mean nΦ . nM if the phases of the interference pattern among the q repetitions at the respective ỹ areroughly constant, and nΦ → 0 (and hence nΦ  nM ) if thephases are random. Therefore, the most pronounced difference nM − nΦ is observed for the ID regime (see Extended
Data Fig. 2a). From the maximum of this difference we read
off the modulation wavelength (or ‘droplet distance’) ỹ ≡ d.
The FFT phasors at d we call Pi = ni (d) = ρi · e−i·Φi , yielding sets {P1 , . . . , Pq }P . To characterise the distribution ofphases Φi withincalculate the circular varianceq our sets, we
2
2
q
q
1
∆Φ = 1 − q∑i=1 cos (Φi ) + ∑i=1 sin (Φi ) [37]. For aphase-coherent sample, and hence interference fringes stablewithin the envelope, ∆Φ is small, whereas for an incoherent
sample it approaches unity. To estimate the confidence intervals of our circular variance data we apply a bias-corrected accelerated bootstrapping scheme [32] for each P, resampling
106 times from the respective q experimental values.

0-10 -5

0

m

5 10

5
0

0.6

0.8

1

q

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

Interference pattern analysis and simple model of a droplet
array

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

(1)

j=1

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

Z

Pi =

dky

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

j1 , j2 =1

E.

Effect of finite sample size

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

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

Pi =

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

j1 , j2 =1≈ g (0) ∑ ei(θ j+1 −θ j )
j

(4)

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

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

G.

Modelization by a Josephson junction array

Let us modelize the system by a set of independent droplets,
each one having a number of particles N j (whose average will
be denoted N j ) and a phase θ j . The Hamiltonian of such asystem is"H =∑jNj − Nj
2C j

2− J j cos θ j+1 − θ j

#


(8)

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

H.

Time evolution for the JJA model

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

11

a

b
1

1

0.5

0.5

0
0

50

100

0
0

5

10

15

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

[44] Ronen, S., Bortolotti, D. C. E. & Bohn, J. L. Bogoliubov
modes of a dipolar condensate in a cylindrical trap. Phys. Rev.
A 74, 013623 (2006). URL https://doi.org/10.1103/
PhysRevA.74.013623.
[45] Wächtler , F. & Santos, L. Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einsteincondensates. Phys. Rev. A 94, 043618 (2016). URL https://doi.org/10.1103/PhysRevA.94.043618.
[46] Ilzhöfer, P. et al. Two-species five-beam magneto-optical trap
for erbium and dysprosium. Phys. Rev. A 97, 023633 (2018).
URL https://doi.org/10.1103/PhysRevA.97.023633.
[47] Trautmann, A. et al. Dipolar Quantum Mixtures of Erbium andDysprosium Atoms. Phys. Rev. Lett. 121, 213601 (2018). URL
https://doi.org/10.1103/PhysRevLett.121.213601.
[48] Chin, C., Grimm, R., Julienne, P. S. & Tiesinga, E. Feshbachresonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286(2010). URL https://doi.org/10.1103/RevModPhys.
82.1225.
[49] Ferrier-Barbut, I. et al. Scissors Mode of Dipolar Quantum
Droplets of Dysprosium Atoms. Phys. Rev. Lett. 120, 160402(2018). URL https://doi.org/10.1103/PhysRevLett.
120.160402.
[50] Maier, T. et al. Broad universal Feshbach resonances in the
chaotic spectrum of dysprosium atoms. Phys. Rev. A 92, 060702(2015). URL https://doi.org/10.1103/PhysRevA.92.
060702.
[51] Evans, M., Hastings, N. & Peacock, B. Beta distribution. In Statistical Distributions, chap. 5, 34–42 (Wiley, New York, 2000),
3 edn. URL https://doi.org/10.1002/9780470627242.

Chapter6

Conclusion and Outlook
This PhD thesis covered the evolution of the first experimental apparatus for heteronuclear
quantum-degenerate dipolar mixtures of erbium and dysprosium over the first five years
from its initial development process and construction phase to first experiments on dipolar
quantum many-body physics.
Exploiting the great similarities between both elements not only in physical but
also in atomic and in laser cooling properties, as well as the pioneering work fromrespective single-species experiments [74, 75] allowed for a straightforward and overallrobust layout of the experimental apparatus. The initial experiment stages involve adual-species atomic beam source, a single spin-flip Zeeman slower and a dual-species
intercombination-line MOT. After a development process and a construction phase of
approximately two and a half years, the dual-species MOT began to operate reliably ina novel open-top configuration, possible due to its narrow-line laser cooling character.
This first milestone led to the publication Two-species five-beam magneto-optical trap for
erbium and dysprosium [88], which presented a thorough characterization of the system.
This achievement paved the way to further extend the experimental apparatus by
an ODT for evaporative cooling to quantum degeneracy. While initial progress led tosingle-species dBECs soon after, the actual production of heteronuclear dBECs required
deep insight into both elements’ specific differences and their influence on the evaporation
dynamics. Eventually, after another year, the experimental apparatus achieved singlespecies dBECs for five bosonic isotopes, including for the first time also 170 Er, as wellas the first quantum-degenerate dipolar mixtures of erbium and dysprosium with fivedifferent heteronuclear dBECs and one quantum-degnerate Bose-Fermi mixture. The
second publication Dipolar Quantum Mixtures of Erbium and Dysprosium Atoms presented
this second milestone and additionally reported first evidence for in-trap interspecies
interactions between the dBECs of the isotope mixtures. [117].
At this point, the experimental apparatus started to prove its capabilities as it joined
the search for a supersolid phase in dipolar quantum gases. In single-species operation, it
achieved with 164 Dy a supersolid phase with long lifetimes and the direct evaporative
cooling into it, and thus made an important contribution to the third publication LongLived and Transient Supersolid Behaviors in Dipolar Quantum Gases [155]. Utilizing
this unique setting, it then focused, as this PhD work’s last project, on the rigidity of
global phase coherence and the rephasing dynamics of an out-of-equilibrium dysprosium
supersolid by exploiting the isolated droplet phase. This led to the fourth publication
Phase coherence in out-of-equilibrium supersolid states of ultracold dipolar atoms [177].
125

Chapter 6 Conclusion and Outlook
The experimental apparatus adds with quantum-degenerate dipolar mixtures of erbium
and dysprosium a new type to the existing selection of heteronuclear mixtures, namely
almost mass-balanced ones with not only contact but also magnetic dipole-dipole intraand interspecies interactions. Many properties of such dipolar mixtures are still unknown
and demand careful determination, before more advanced experiments are feasible.
Furthermore, the experimental apparatus allows for two additional science modules. By
optically transporting single- or dual-species samples in either of the two, more advanced
experiments in an optimized environment and with specific experimental techniques will
be possible. During this PhD work two ideas took shape, the realization of a quantum-gas
microscope and the exploration of multivalent Rydberg atoms, for which this PhD work
developed a complete Rydberg science module.
The following Section 6.1 presents the next steps in the direction of dipolar quantum
few-/many-body physics and Section 6.2 summarizes basic conceptual ideas on multivalent
Rydberg atoms as well as the main aspects of the Rydberg science module.

6.1 Dipolar quantum few-/many-body physics
An important first upgrade to the experimental apparatus, and thus to its capabilities will
be a high-resolution imaging system along the vertical z-axis above the upper inverted
viewport of the main chamber. It comprises a commercial infinity-corrected objective61
with a working distance of 55 mm, an effective focal length of 65 mm and a numericalaperture of 0.45. The objective is achromatic for λEr = 401 nm and λDy = 421 nm, andprovides a diffraction-limited resolution of 0.54 µm and 0.57 µm, respectively. Moreover,with an adjustment of the focusing plane and a reduction of the numerical aperture
to 0.38, it operates non-achromatically at λEr = 583 nm and λDy = 626 nm with adiffraction-limited resolution of 0.94 µm or 1 µm. The entire imaging setup is easily
switchable from absorption imaging to non-destructive insitu phase-contrast imaging
and vice versa. The superior overall performance compared to the horizontal absorption
imaging will be of major importance for all future experiments.
Near future studies will characterize the quantum-degenerate dipolar mixtures, inparticular in view of their interactions, their miscibility or immiscibility as well as theirstability diagrams. Currently, there is ongoing work on high-resolution Feshbach spectroscopy with the aim of identifying broad heteronuclear Feshbach resonances applicablefor interaction control [49]. As a first result, [178] investigated various isotope mixturesand confirmed for instance a multitude of heteronuclear resonances exceeding the homonuclear ones [76, 77, 78]. However, as the homonuclear Feshbach spectra are already toocomplex to extract a global single-species background scattering length for the respective
isotope, also for the isotope mixtures a careful determination of the intra- and interspecies
scattering lengths as,Er-Er , as,Dy-Dy and as,Er-Dy for a given magnetic field range of interestis necessary. Here, either a well established technique such as a cross-species thermalization measurement [178, 179, 180], or the recorded TOF center-of-mass oscillations [117],
which arise from an interspecies-interaction-induced in-trap shift, are viable possibilities.
These results will directly connect to the miscibility behavior [181, 182, 183, 184], whichis the basis for all future experiments. Here, the vertical high-resolution imaging will be
crucial, as it will provide direct insight to the in-trap behavior of the isotope mixtures.
61

Special Optics, A Navitar Company

126

6.2 Multivalent Rydberg atoms
Once the experimental apparatus establishes full control over the produced quantumdegenerate dipolar mixtures, an interesting question concerns quantized vortices, which
served as probe to demonstrate superfluidity in purely contact-interacting BECs [13,14, 15] or in strongly interacting DFGs within the BEC-BCS crossover [185]. Despitethat their experimental realization in dipolar quantum gases is still missing, extensive
theoretical work studied their properties in dBECs [66]. More recently, [186] and [187]investigated the formation of vortex lattices and patterns specifically in heteronuclear
dBECs of erbium and dysprosium, and [188] the creation of a vortex line in a dBEC ofdysprosium. The high-resolution imaging system in combination with a digital mirror
device will offer the possibility to study quantized vortices in single-species or in dualspecies operation. Following the recent observation of a supersolid phase in dipolar
quantum gases [153, 154, 155], this probe will also be applicable to confirm the superfluid
character of such a state of matter, and thus its phase coherence as well as to answer
whether and how supersolidity is inducible in one species via the presence of the other.

6.2 Multivalent Rydberg atoms
Rydberg atoms are a diverse research field, with the conceptual idea to explore andexploit the exaggerated characteristics of atoms with valence electrons in highly excitedstates [189]. There is a great interest in their large polarizability, and thus their high sensitivity to an external electric field E [189] or their strong and long-range interactions [190]and associated effects such as the Rydberg blockade [191, 192, 193].
So far, atomic physics platforms focused mainly on the alkali metals rubidium andcesium due to their favorable laser excitation schemes. Hot vapor Rydberg systems
demonstrated for instance the measurement of microwave E-fields [194], terahertz imaging [195] or a single-photon source [196]. Ultracold Rydberg systems for instance proved
the formation of ultralong-range Rydberg molecules [197, 198], determined the van der
Waals interaction utilizing optical tweezers [199], or reported a CNOT quantum gate [200],
single-photon sources [201, 202] and single-photon transistors [203, 204].
More recently, strontium and ytterbium with their two 5s and 6s valence electrons,
respectively, moved into the focus, as their core remains optically active upon a single
Rydberg excitation, and thus facilitate additional possibilities for Rydberg manipulation.
Corresponding experiments investigated with strontium the autoionization process andthe arising electron-Rydberg atom collisions [205] or the direct imaging of Rydberg atoms
and the impact of an ultracold seed plasma [206], and demonstrated with ytterbium thetrapping of Rydberg atoms in optical tweezers [207].
At this point, lanthanoids with their more complicated valence electron configurationsof 4f and 6s valence electrons (see Chapter 2) allow not only for the same concepts but
possibly also for an extended range of experimental techniques. Their electronic groundstates’ large total electronic angular momentum J leads with a non-zero nuclear spin I
to a large set of hyperfine sublevels in the ground state manifold. For instance, the 128
hyperfine sublevels of holmium could serve for collective register encoding in a Rydberg
gate quantum computer [208].
Furthermore, the lanthanoids’ highly complex energy level diagrams open new paths
for very different Rydberg excitation schemes. Figure 6.1 shows exemplarily the energy
level diagrams of neutral erbium (Er I) and of singly ionized erbium (Er II) up to127Chapter 6 Conclusion and Outlook
J, Er II
3/2

5/2

7/2

9/2

11/2 13/2 15/2 17/2 19/2 21/2 23/2 25/2

ν̃ (103 cm-1 )80

4f12 (3 H6 )13/2 6pns/nd

70
60

391 nm
4f12 (3 H6 )13/2 6sns/nd

50
4f

2

( F°)5d6s �

25

Ry states

411 nm11 4

4f12 (3 H6 )6s6p(1 P1 °)(6,1)7 °ν̃ (103 cm-1 )20
15

401 nm

4f12 (3 H6 )6s6p(3 P1 °)(6,1)7 °583 nm

4f11 (4 I13/2 °)5d3/2 6s2 (13/2,3/2)7 °631 nm389 nm10841 nm1299 nm504f11 (4 I15/2 °)5d5/2 6s2 (15/2,5/2)7 °4f11 (4 I15/2 °)5d3/2 6s2 (15/2,3/2)7 °4f12 (3 H6 )6s2 ground state

2

3

4

5

6

7
J, Er I

8

9

10

11

12

Figure 6.1: Extended energy level diagram of erbium up to wavenumbers ν̃ of 26 000 cm−1for neutral erbium (Er I, bottom) and of 83 000 cm−1 for singly ionized erbium (Er II, top)
over the total angular momentum quantum number J. The violet and the yellow arrow
indicate, identicallyto Figure 2.4,the broad and the intercombinatione-line transition
1
to the P1 (6, 1)7 and the 3 P1 (6, 1)7 state at λEr = 401 nm and λEr = 583 nm,
respectively. The remaining four arrows from the electronic ground state in Er I indicate
transitions of a 4f valence electron to a 5d3/2 or a 5d5/2 state in a wavelength range
1
λEr = 389 nm − 1299 nm. Starting from the P1 (6, 1)7 state, the ns or the nd Rydberg
series connected to the lowest ionization threshold are addressable with 411 nm laser
light. Accordingly, singly ionized erbium Er II in its electronic ground state offers for
instance a transition of the remaining 6s valence electron to a 6p state at 391 nm. The
figure originates from [209] with data from [87].

wavenumbers ν̃ of 26 000 cm−1 and of 83 000 cm−1 with respect to the electronic groundstate in Er I over the total angular momentum quantumnumber J [87]. Besides
 the broad
1
3
and the intercombination-line transition to the P1 (6, 1)7 or the P1 (6, 1)7 state(see Chapter 2), it highlights from the electronic ground state in Er I four transitions
1286.2 Multivalent Rydberg atoms
MCP CF40
mounting flange

MCP unitwith wire gridand electrodes

Mounting plate

Notch mounting
adapterInsulatingbushings

Field plates

Wire guiding
systemMCP connections

Figure 6.2: In-vacuum E-field control system. In total eight field plates in a cloverleafoctupole arrangement provide a complete three-dimensional E-field control to either
compensate external stray E-fields, to apply precise E-fields during the experiment or forfield ionization of the excited Rydberg atoms. Two microchannel plate detector (MCP)
units, each consisting of an MCP, a housing, a wire grid as well as four electrodes, candetect ions or electrons after field ionization. All these parts are from 1.4429 stainlesssteel. A wire-guiding system and insulating bushings, all from macor (white), lead theconnection wires (not shown) from the field plates and MCP units to their respective
electrical feedthroughs (not shown for the field plates). The entire E-field control system
connects via four notch mounting adapters to a notch system within the surrounding
1.4429 stainless steel octagon chamber (not shown).
of a 4f valence electron to a 5d3/2 or a 5d5/2 state differing in natural linewidths ∆ν andtransition wavelengths λ within the range λEr = 389 nm − 1299 nm. Hence, corresponding
two-photon excitation schemes can couple to Rydberg series of relatively high angularmomentum quantum numbers l in comparison to the two 6s valence electrons. This might
suppress undesirable autoionization processes. Moreover,the corresponding Rydbergstates might offer real parts of their polarizabilities < αRyd sufficiently large for trapping,as the 6s valence electrons remain unchanged and provide a significant contribution
to < {α}. As a direct consequence, this might enable the trapping and cooling of Rydberg
atoms. Additionally, Figure 6.1 illustrates the 401 nm-411 nm two-photon Rydberg
excitation scheme of a 6s valence electron via the broad transition to the ns or the nd
Rydberg series connected to the lowest ionization threshold, with the principal quantum
number n. Here, the ionic core excitation of the second 6s valence electron to the 6p state
at 391 nm could serve for instance for Rydberg imaging or further Rydberg manipulation.
This PhD work developed a complete layout for a Rydberg science module. It comprises
a custom-made stainless steel octagon chamber from 1.4429 stainless steel with four
pairs of CF40 ports in-plane and two opposing CF100 ports orthogonal to the plane,
and provides optical access along three orthogonal axes via six anti-reflection coated129Chapter 6 Conclusion and Outlook
viewports62 . Further, it features a high-resolution imaging objective identical to the one
at the main chamber (see above). Its core feature, however, is an in-vacuum E-field
control system from 1.4429 stainless steel, as Figure 6.2 illustrates. Here, two opposing
sets of four field plates in a cloverleaf arrangement, at a distance of 36 mm and with25.5 mm inner diameter, form an octupole. Each field plate is individually addressablewith up to 5 kV via a separate electrical feedthrough. High-voltage solid-state switches63
for four out of the eight field plates allow for a dynamic switching between low but highly
precise, and high voltage64 sources. During the experimental sequence, this enables not
only the compensation of stray E-fields in all three dimensions but also the creation ofhigh E-fields of up to E = 830 V/cm. Thus, the combined operation with two microchannel
plate detector (MCP) units allows for Rydberg detection via field-ionization [189]. EachMCP unit encloses the MCP65 to prevent any additional stray E-fields. A wire grid
serves as shielding on the entrance of the MCP unit and is together with four additional
electrodes individually addressable to redirect the ions or the electrons emerging fromfield ionization onto the MCP.
However, due to the overall complexity of this project, the results from Rydberg tweezer
experiments in one [210] and two [211] dimensions as well as the recent developments withstrontium [212, 213] and ytterbium [207], this Rydberg project split off. An entirely newexperimental apparatus aiming to study erbium Rydberg atoms in optical tweezer arrays
will mostly replicate the presented UHV apparatus (see Chapter 3) and will integrate
the developed science module as its core element in a slightly modified form.

62

Four CF40 viewports, MgF2 anti-reflection coating, and two CF100 viewports, 68 mm diameter viewport,anti-reflection coating for 401 nm, 421 nm, 1064 nm and 1570 nm
63
Behlke Power Electronics GmbH, HTS 31-02-HB-LC-C
64
iseg Spezialelektronik GmbH, ECH 124-INH rack with four slide-in modules, DPr 30 405 24 5, voltages
up to 3 kV
65
Hamamatsu Photonics K.K., F4655-10S184

130Chapter

7

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Acknowledgments
Finally, I would like to take the opportunity and acknowledge and thank all the people
without whose support and contributions over the past five years this PhD thesis would
not have been possible.
First of all, I would like to thank my supervisor Francesca Ferlaino for the opportunity
to pursue a PhD in her research group. Thank you for your trust in me to lay and shape
the foundation of this ambitious project as the very first PhD student. I truly hope that
this new experimental apparatus will push the frontiers to unexplored areas and will
provide the right answers for many more years to come.
It is hardly possible to build and operate such a complex experimental apparatus alone,
therefore a big thank you to the RARE team, our senior scientist Manfred J. Mark, ourfirst Postdoc Arno Trautmann, and to my fellow PhD students Gianmaria Durastante,
Maximilian Sohmen and Claudia Politi. Looking at the work over the past five years, allthis was only possible thanks to the hard work and commitment of all of us. I wish you
all, including the new team members Matthew Norcia and Lauritz Klaus, an absolutely
successful time and many exciting scientific questions to explore.
I would especially like to thank Arno. You absolutely made sure that I will remember
your strong opinions on random topics and the many new insights into LATEX. Our
countless long nights and weekends in the lab were an absolute pleasure, I just want torecall our successful run when we were able to condense various mixtures during a single
night simply in the spirit of condensing heavy metals with heavy metal.
I also would like to thank Max. Your tireless dedication to guide us to Innsbruck’s
famous karaoke bar resulted in many memorable nights out. Many thanks also for
proofreading this PhD thesis.
A big thank you also goes to the Erbium team, the postdoc Lauriane Chomaz and
the PhD students Daniel Petter, Alexander Patscheider and Gabriele Natale, for many
interesting discussions and the big fun during our joint group activities.
I would especially like to thank Alex. It was a great help having you as the first Masterstudent during the construction phase of this experimental apparatus. Without you, itwould not have been possible to assemble the entire UHV apparatus in a few days time.
I also would like to express my gratitude to Bernhard Öttl, Andreas Strasser, Stefan
Haslwanter and Gerhard Hendl from the mechanical and electronics workshop of the
IQOQI Innsbruck. Only your phenomenal work made it possible that many ideas for theexperimental apparatus not only remained ideas, but actually became reality.
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A special thank you goes to Christian Veit, Matthias Wenzel and Tobias Bäuerle.
Looking back to autumn 2009, when the four of us began our studies in Stuttgart, andnow in 2020, when eventually all of us will finish this journey with a PhD, makes me realize how far we have come and how many shared memories we have collected over the years.
The very last words of this PhD thesis I would like to devote to my parents and mybrother. Without your unconditional support over the years, I would not have been able
to pursue my studies in all the freedom I had. From all of my heart - thank you!

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