Dipolar Quantum Gases of Erbium
DISSERTATION

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

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

Innsbruck, October 2014

Attention: This is not the original version of the thesis! Changes have been made to
the original version to correct a number of spelling mistakes. The contents, however, hasremained entirely untouched. The original version can be obtained directly from the author,
or through the University of Innsbruck. The publications presented in Chapter 10 and in
Chapter 11 are replaced by their respective preprint versions for online access of this thesis.

Summary
Since the preparation of the first Bose-Einstein condensate about two decades ago and thefirst degenerate Fermi gas following four years later a plethora of fascinating quantum phenomena have been explored. The vast majority of experiments focused on quantum degenerate atomic gases with short-range contact interaction between particles.
Atomic species with large magnetic dipole moments, such as chromium, dysprosium, anderbium, offer unique possibilities to investigate phenomena arising from dipolar interaction.
This kind of interaction is not only long-range but also anisotropic in character and imprints
qualitatively novel features on the system. Prominent examples are the d-wave collapse ofa dipolar Bose-Einstein condensate of chromium atoms realized by the group in Stuttgart,
the spin magnetization and demagnetization dynamics observed by groups in Stuttgart,
Paris, and Stanford, and the deformation of the Fermi surface observed by our group in
Innsbruck.
This thesis reports on the creation and study of the first Bose-Einstein condensate anddegenerate Fermi gas of erbium atoms. Erbium belongs to the lanthanide group of elements
and has a large magnetic moment of seven Bohr magneton. In particular, this thesis describes
the experimental apparatus and the sequence for producing a dipolar quantum gas. Thereis an emphasis on the production of the narrow-line magneto-optical trap of erbium since
this represents a very efficient and robust laser-cooling scheme that greatly simplifies theexperimental procedure.
After describing the experimental setup this thesis focuses on several fundamental questions
related to the dipolar character of erbium and to its lanthanide nature. A first set of studies
centers on the scattering properties of ultracold erbium atoms, including the elastic and theinelastic cross section and the spectrum of Feshbach resonances. Specifically, we observethat identical dipolar fermions do collide and rethermalize even at low temperatures. The
corresponding elastic cross section predicted by the theory of universal dipolar scattering
scales only with the particle’s mass and its magnetic moment and is temperature independent. This represents a dramatic difference compared to non-dipolar fermions, which exhibit
a rapidly vanishing elastic cross section at lower temperatures.
Another distinctive feature of erbium is its spectrum of Feshbach resonances. Using bosonicerbium we observed an enormous density of resonances exceeding that of alkali metals by atleast a factor of ten. We statistically analyzed the spectrum in terms of the random matrix
theory and demonstrated that the resonances bare a strong correlation to each other. We
identify the origin of this correlation in the highly anisotropic van der Waals interaction
potential of erbium. This is a phenomenon not previously encountered in ultracold quantum
gases.
At the many-body level we observed the d-wave collapse of the Bose-Einstein condensate aspreviously observed in the chromium experiment in Stuttgart. With the dipolar Fermi gas
we demonstrated that the Fermi surface deforms into an ellipsoid induced by the action of
the dipole-dipole interaction in momentum space.

iZusammenfassung
Die Erzeugung des ersten Bose-Einstein-Kondensats vor ungefähr zwei Jahrzehnten sowie
des ersten entarteten Fermigases vier Jahre später, öffnete die Tür zur Erkundung einer
Vielzahl faszinierender Quantenphänomene. Der Großteil dieser Experimente konzentrierte
sich dabei auf quantenentartete atomare Gase mit kurzreichweitiger Kontaktwechselwirkungzwischen den Teilchen.
Atome mit großem magnetischen Dipolmoment, wie zum Beispiel Chrom, Dysprosium und
Erbium, bieten einmalige Möglichkeiten, durch die Dipolwechselwirkung hervorgerufene Phänomene zu erforschen. Der Charakter dieser Wechselwirkung ist nicht nur langreichweitig
sondern auch richtungsabhängig und prägt dem System qualitativ neue Eigenschaften auf.
Herausragende Beispiele hierfür sind der d-Wellen-Kollaps eines dipolaren Bose-EinsteinKondensats aus Chromatomen, realisiert von der Gruppe in Stuttgart, die Spin-Magnetisierungs- und Demagnetisierungsdynamik, beobachtet von Gruppen in Stuttgart, Paris und
Stanford sowie eine Deformation der Fermifläche, welche von unserer Gruppe beobachtet
werden konnte.
Die vorliegende Dissertation behandelt die Erzeugung und Untersuchung des ersten BoseEinstein-Kondensats und des ersten entarteten Fermigases von Erbiumatomen. Erbium
gehört zur Gruppe der Lanthanoide und besitzt ein großes magnetisches Moment von sieben
Bohr Magnetonen. Diese Arbeit beschreibt im Besonderen die experimentelle Apparatur und
den Ablauf, welcher zur Erzeugung eines dipolaren Quantengases notwendig ist. Das Hauptaugenmerk liegt dabei auf der Produktion der schmalbandigen magneto-optischen Falle fürErbium, die eine sehr effiziente und robuste Art der Laserkühlung darstellt und das experimentelle Herstellungsverfahren maßgeblich vereinfacht.
Nach der Beschreibung des experimentellen Aufbaus werden in der Arbeit einige fundamentale Fragen bezüglich des dipolaren Charakters von Erbium und seiner Natur als Element der Lanthanoiden behandelt. Erste Untersuchungen konzentrieren sich dabei auf die
Streueigenschaften bei ultrakalten Temperaturen, welche den elastischen und inelastischen
Streuquerschnitt sowie das Spektrum von Feshbach-Resonanzen einschließen. Im Speziellen
beobachten wir, dass identische dipolare Fermionen kollidieren und rethermalisieren können,
und dies sogar im tiefen Temperaturbereich. Der damit verbundene elastische Streuquerschnitt, welcher im Rahmen der Theorie der universellen dipolaren Streuung berechnet werden kann, hängt nur von der Masse des Teilchens und seinem magnetischen Moment ab
und ist temperaturunabhängig. Dies stellt einen entscheidenden Unterschied im Vergleich
zu nicht dipolaren Fermionen dar, welche einen mit der Temperatur schnell abklingenden
elastischen Streuquerschnitt aufweisen.
Eine weitere markante Eigenschaft von Erbium ist dessen Spektrum von Feshbach-Resonanzen. Unter der Verwendung von bosonischem Erbium beobachteten wir eine enorme Dichte
von Resonanzen, welche die bei Alkalimetallen vorhandene Resonanzdichte um mehr als das
Zehnfache übersteigt. Wir führten eine statistische Analyse der Verteilung von FeshbachResonanzen im Sinne der Theorie der Zufallsmatrizen durch und zeigten, dass die Resonanzeneine starke Korrelation aufweisen. Wir identifizierten den Urprung dieser Korrelation in demiihochgradig richtungsabhängigen van der Waals Potential von Erbium, wie es erstmals inultrakalten Quantengasen beobachtet werden konnte.
In Bezug auf Mehrkörperwechselwirkungen konnten wir den d-Wellen-Kollaps des BoseEinstein-Kondensates beobachten, ähnlich wie im Chrom-Experiment in Stuttgart. Mithilfe
des dipolaren Fermigases demonstrierten wir darüberhinaus die Deformation der Fermiflächein ein Ellipsoid, welche durch Auswirkungen der Dipolwechselwirkung im Impulsraum zu
erklären ist.

iii

Contents
Summaryi

Zusammenfassungii

1. Introduction
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
5

2. Erbium properties
2.1. The discovery of erbium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Electron configuration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Atomic energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Laser-cooling transitions . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3. Isotope shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1. Magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2. Landé g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3. Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4. Hyperfine Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . .

8
8
9
9
12
13
16
16
17
17
19
20
21

3. Experimental setup
3.1. Experimental requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Vacuum chamber design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. High vacuum section . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2. Ultra-high vacuum section . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Slow atom beam with high flux . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Transversal cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2. Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3. Velocity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Coil setup at main chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1. Main coil setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2. Compensation cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Laser light setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1. Blue laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2. Yellow laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3. Optical dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
24
25
29
32
32
32
37
38
39
42
42
42
45
49

iv

4. Ultracold dipolar scattering
4.1. Anisotropic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1. Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2. Anisotropic dispersion interaction . . . . . . . . . . . . . . . . . . . . .
4.2. Dipolar scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Universal dipolar scattering . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Anisotropy-induced Feshbach resonances . . . . . . . . . . . . . . . . . . . . .

53
54
54
56
59
61
62

5. Random matrix theory
5.1. Introduction to RMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1. General idea of RMT . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2. Applications of RMT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3. Symmetry groups and Gaussian ensembles . . . . . . . . . . . . . . . .
5.1.4. Statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Quantum chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1. Bohigas-Giannoni-Schmit conjecture . . . . . . . . . . . . . . . . . . .
5.2.2. Sinai billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3. Quantum chaos in atomic physics . . . . . . . . . . . . . . . . . . . . .
5.3. Ultracold resonant scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .

68
69
69
70
70
71
75
76
77
78
78

6. Publication: Narrow-line laser cooling of erbium
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1. Narrow-line cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Publication: Narrow-line magneto-optical trap for erbium . . . . . . . . . . .
6.3. Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1. Spin polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
83
84
90
96
96

7. Publication: Bose-Einstein condensation of erbium
98
7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.1.1. Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.1.2. Dipolar Bose-Einstein condensate . . . . . . . . . . . . . . . . . . . . . 102
7.2. Publication: Bose-Einstein Condensation of Erbium . . . . . . . . . . . . . . 104
7.3. Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3.1. Cross-dimensional rethermalization . . . . . . . . . . . . . . . . . . . . 110
7.3.2. Scattering length mass scaling . . . . . . . . . . . . . . . . . . . . . . 111
8. Publication: Hyperfine structure of fermionic erbium
115
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.1.1. Hollow-cathode lamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.1.2. Modulation-transfer spectroscopy . . . . . . . . . . . . . . . . . . . . . 117
8.2. Publication: Hyperfine structure of laser-cooling transitions
in fermionic erbium-167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9. Publication: Degenerate Fermi gas of erbium
127
9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.1.1. Degenerate dipolar Fermi gas . . . . . . . . . . . . . . . . . . . . . . . 127
9.1.2. Collisions between identical fermions . . . . . . . . . . . . . . . . . . . 128
9.1.3. Evaporative cooling of fermions . . . . . . . . . . . . . . . . . . . . . . 128v

9.2. Publication: Reaching Fermi degeneracy via universal dipolar scattering . . . 131
10.Publication: Quantum chaos in ultracold erbium
137
10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.1.1. High-resolution Feshbach spectroscopy . . . . . . . . . . . . . . . . . . 138
10.1.2. Statistical analysis of Feshbach resonances . . . . . . . . . . . . . . . . 138
10.2. Publication: Quantum chaos in ultracold collisions of gas-phase erbium atoms 140
10.3. Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.3.1. RMT and Feshbach spectroscopy . . . . . . . . . . . . . . . . . . . . . 151
10.3.2. Feshbach resonance width distribution . . . . . . . . . . . . . . . . . . 153
11.Publication: Fermi surface deformation
155
11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
11.2. Publication: Observation of Fermi surface deformation
in a dipolar quantum gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.Conclusion and outlook
163
12.1. Further investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
12.2. Our vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A. Measurement of the natural linewidth of the 401-nm transition

167

B. Erbium high-temperature oven169
B.2. Erbium vapor pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.3. Oven refilling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
C. Notes on the blue master laser system

177

D. Feshbach resonances data tables of bosonic erbium

179

E. Full level diagram of erbium

182

Bibliography

184

Acknowledgments

200

vi

Chapter1

Introduction

1.1. Motivation
About 90 years ago Satyendranath Bose and Albert Einstein predicted a novel quantum
state of matter, called Bose-Einstein condensate (BEC). This state of matter is formedby the macroscopic occupation of a single quantum state with bosonic particles. It took
70 years and major technological advances to produce the first BEC in experiment, see
Refs. [And95, Dav95]. A key development was the advent of laser cooling and atom trapping,see Ref. [Chu85]. For atomic systems, cooling to ultralow temperatures is necessary to achieve
the BEC, since it requires the thermal de Broglie wavelength to be larger than the mean
distance between particles.
From then on the field of atomic physics experienced an enormous boost in the number of
research groups working on ultracold gases. Some of the early ground-breaking studies on
ultracold atomic systems include the verification of matter-wave interferences, the creationof vortices in the superfluid, and the realization of an atomic laser, see Ref. [Blo08] for a summary. Nowadays, the condensation of particles into a BEC is demonstrated not to be limited
to ultracold atomic gases only but is also possible with other bosonic systems like photons or
solid-state quasi-particles as magnons or exciton-polaritons, for which condensation happens
even at room temperature, see Ref. [Den10].
Parallel to the research with bosons, experimental efforts focused on the production of a degenerate Fermi gas (DFG) to access the other fundamental class of particles. The interest in
fermions is great because they are often found in nature, e. g. within the electron gas of metals, as neutrons of heavy stars, or as quarks in plasmas as constituent elements of composite
particles. The production of a DFG has been achieved in the late 1990s, see Refs. [DeM99b],
and it challenged experimentalists for a simple reason: identical alkali fermions do not collide at ultralow temperatures. This stringent property is derived from the Pauli exclusion
principle and the Wigner threshold law. However, experimentalists have found an elegant
way to overcome this fundamental limitation by introducing a second spin state, anotherspecies, or a different isotope, making particle collisions and cooling possible. Beautiful
experiments with fermions include the study of the famous BEC-BCS crossover, the production of fermionic mixtures with unequal densities, and the measurement of fundamental
11.1. Motivation2

properties of the superfluid in a strongly interacting Fermi gas, see Ref. [Gio08, Ket08] foran overview.
Several tools exist for ultracold gases to manipulate the interaction between particles andconfine them in various geometries. Tuning the interactions between particles via a Feshbachresonance (FR) is one of the most prominent example. Dependent on an external magnetic
field, the scattering length can be changed from zero (non interacting) to very large values(strongly interacting) across a resonance. FRs are further used to associate two ultracold
particles to weakly-bound molecules, which in the next step can be optically transferred into
their rovibrational ground state. A second commonly-used tool are optical lattices, periodic
structures of light mimicking solid-state crystal-like structures. They offer full control overlattice parameters, e. g. the tunneling rate, and are free of lattice dynamics induced by theenvironment.
A trend has developed in atomic physics within the last ten years from exploring degeneratematter towards the realization of interesting theoretical models like the well-known BoseHubbard model in lattices [Jak98], the superfluid to Mott-insulator transition [Gre02], orAnderson localization using disorder [Bil08]. It is precisely this trend which marked a turning
point from the mere awareness and observation of quantum effects in ultracold gases to thepossibility of engineering quantum mechanical systems with an unprecedented large degreeof control and flexibility.
Dramatic advances in laser technology and more sophisticated cooling techniques made it
possible to condense elements other than alkali metals. Figure 1.1 shows a periodic table ofelements with those elements, which have been brought to degeneracy to this day, highlighted.
Today, a total of 13 elements are available in ultracold quantum gas experiments, three of
which allow for the study of dipolar systems.
In contrast to conventional atomic gases, dipolar systems add an interaction between particles, the so-called dipole-dipole interaction (DDI), which is anisotropic and long-range. As
reviewed in Ref. [Bar02a] it is possible to study fundamentally different phenomena in dipolarsystems compared to experiments with alkali metals. Dipolar systems had been proposed foraccessing exotic quantum phases already in the late 1990s. Nowadays, at least three dipolarsystems are experimentally accessible, namely atoms with a large magnetic dipole momentin the ground state, heteronuclear molecules with an electric dipole moment, and atoms inhighly excited Rydberg states.
Chromium was the first dipolar element to be Bose-Einstein condensed by the group ofT. Pfau in Stuttgart (Germany) about a decade ago, see Ref. [Gri05]. It has a magnetic
moment of 6 µB and a total quantum number of three. Several interesting phenomena,
which are purely caused by the DDI, had been realized with chromium, such as the principleof demagnetization cooling [Fat06], the d-wave collapse of the BEC [Lah08], and the spin
relaxation in an optical lattice [Pas11].
In the last several years, a new class of dipolar atoms from the lanthanide series have comeinto play, which posses an even larger magnetic moment than that of chromium. Elements
like dysprosium or erbium have a magnetic moment of up to 10 µB and a large total spin

3

1.1. Motivation

year of first BEC

1998

1995

2000

2001

2005

2010

2014

1995

1995

2001

2009

1995

2009

2004

2002

lanthanide series
2011

2012

2003

actinide series

Figure 1.1.: Periodic table showing Bose-Einstein-condensed elements up to 2014. Initially, only
alkali atoms were brought to degeneracy, but in the last years the interest in alkaline-earth elements
and elements with a large magnetic dipole moment (blue frame) increased significantly. List ofreferences: Na [Dav95], Rb [And95], Li [Bra95], H [Fri98], He [Rob01], K [Mod01], Cs [Web03], Yb[Tak03], Cr [Gri05], Ca [Kra09], Sr [Ste09], Dy [Lu11], and Er [Aik12].

quantum number in the ground state, which makes them particularly interesting for ultracold
gas experiments.
Early experimental work on trapping and laser cooling erbium was done by the group of
J.J. McClelland at NIST (USA), see Refs. [McC06b, Ber08]. Different laser-cooling schemes
and magnetic-trapping methods were successfully demonstrated but the atomic sample wasnever brought to degeneracy in these experiments. Laser cooling of dysprosium atoms was
soon to follow by the group of B. Lev now in Stanford (USA), see Ref. [Lu10]. The same groupproduced the first dysprosium Bose-Einstein condensate [Lu11] and first degenerate Fermi gas[Lu12]. In our group in Innsbruck we have been able to laser cool erbium using a narrow-line
transition and to create the first Bose-Einstein condensate and the first degenerate Fermi gasof erbium atoms. This forms a solid basis for future experiments on dipolar quantum gases.
Moreover, thulium [Suk10] and holmium [Mia14] are two more elements from the lanthanide
series, which have been successfully laser cooled. Nowadays, bosonic as well as fermionicsystems (or even mixtures thereof) in combination with a large magnetic dipole moment arereadily available for experimental studies. The success of lanthanides in ultracold atomic
physics can be already observed, as several research groups around the globe are setting up
new experiments on lanthanide atoms at the moment.
Alternatively to magnetic atoms, one can use ultracold diatomic, heteronuclear groundstate molecules to study dipolar effects. Such molecules can be associated from ultracold
alkali-metal atoms and have a strong electric dipole moment. Commonly used molecules incurrent experiments are KRb [Ni08], LiCs [Dei08], or RbCs [Tak14]. Up to date, only gasesof diatomic molecules composed of alkali-metal atoms are available at ultracold temperatures4

1.1. Motivation
10

7

LiCs
10

dipole length (a0)

10

10

10

6

5

heteronuclear groundstate molecules

SrRb

KCs

RbCs
MgH
H2O
KRb
OH

4

Er2*

3

Dy
10

NaCl

NaCs

magnetic atoms/
Feshbach molecules

Er

2

Cr
10

10

1

0

Rb
K
0.01

alkali atoms
0.1

1
dipole moment (D)10Figure 1.2.: Comparison of alkali atoms with magnetic atoms and heternuclear molecules. The
dipole interaction length is plotted in units of the Bohr radius against the dipole moment given in
Debye. Weakly-bound Feshbach molecules are marked with a star. Rydberg atoms have an extremely
high dipole moment of several thousand Debye, which is out of scale for this figure.

but developments in slowing and trapping polyatomic molecules look very promising. Veryrecently a magneto-optical trap of SrF molecules has been shown in Ref. [Bar14], whichmight have future applications for testing the standard model of particle physics [Hun12] orfor searching for variations of fundamental constants [Chi09].
A relevant quantity to compare the dipolar character of particles with, is the so-called dipole
length, ad . It is defined for magnetic atoms byad =mµ0 µ2,4π~21.1

and depends on the magnetic moment µ and the mass of a single dipole m. Here, µ0 is themagnetic constant and ~ the Planck constant divided by 2π. Figure 1.2 shows a plot of thedipole length as function of the dipole moment for several dipolar systems in comparision
with alkali atoms. The plot includes magnetic atoms, Feshbach molecules from magnetic
atoms, and heteronuclear molecules. Alkali atoms have the smallest dipole moment, whereas
heteronuclear molecules have a very high dipole moment. Magnetic atoms lie in between
these two cases and Feshbach molecules of magnetic atoms, such as weakly bound Er2 , havea dipole length, which is only one order of magnitude smaller than that of polar molecules.
Note that Rydberg atoms have a much higher dipole moment than heteronuclear molecules
and are not shown in Fig. 1.2.
The large magnetic moment of lanthanide atoms is caused by an unfilled 4f electron shell.
The lanthanide electron configuration forms a so-called submerged-shell structure as the 4f
shells are immersed in the outer-lying 6s shell, which is completely filled. This structure
provides a rich atomic energy spectrum with transition linewidths ranging from broad to1.2. Thesis overview

5

ultra-narrow. Several atomic transitions turn out to be very convenient for implementing
efficient laser-cooling techniques, which makes it surprisingly simple to prepare lanthanide
atoms in ultracold gas experiments.
All these reasons make lanthanide systems to be very suitable candidates for exploring dipolar
quantum phenomena.

1.2. Thesis overview
This thesis describes the production and study of a dipolar quantum gas of ultracold bosonicand fermionic erbium atoms. In particular, the thesis covers the following achievements:(I) production of the first Bose-Einstein condensate and (II) the first degenerate Fermi gas of
erbium, (III) observation of Feshbach resonances and chaotic scattering, and (IV) observationof anisotropic dipolar scattering and the deformation of the Fermi surface.
On a personal note, I joined the erbium team in 2009, when the experiment was still in
its conceptual phase. In the beginning, the design and setup of the vacuum apparatus was
solely undertaken by me under the guidance of Francesca Ferlaino. Once the experimentapproached the point of first measurements, the team quickly enlarged. I actively participated in the above mentioned achievements of the experiment. The scope of this thesis
covers many of the technical challenges, so that it might serve as a reference for future PhD
students.
The thesis is structured as follows:
Chapter 2 presents the relevant properties of atomic erbium for ultracold gas experiments.
It contains detailed information on the electron configuration, the magnetic properties, andthe atomic energy spectrum, including the laser-cooling transitions of erbium.
Chapter 3 describes the erbium experimental setup. It is structured in four sections including
the vacuum apparatus, the preparation of a slow atom beam, the main coil system, and thelaser light setup.
Chapter 4 offers an introduction to ultracold dipolar scattering. The aim is to highlightthe difference arising in scattering physics between short- and long-range interactions andintroduce the anisotropic van der Waals interaction. The chapter starts with a discussion
on anisotropic interactions between atoms. Then the theory of universal dipolar scattering
is presented. The last part of this chapter offers an overview of the emergence of Feshbachresonances in a dipolar quantum gas.
Chapter 5 is devoted to random matrix theory and quantum chaos. A very general introduction is given as many readers of this thesis might be unfamiliar with that topic. The
second part of this chapter draws a connection between quantum chaos and atomic physics
and presents an application of random matrix theory for calculating scattering properties ofcomplex systems by means of the theory of ultracold resonant scattering.

1.2. Thesis overview

6

The next five chapters present scientific results, which have been published within the scopeof this thesis. Each of these chapters contains one publication. The publications are given inchronological order and every chapter follows the same structure, with experimental background information at the beginning, the actual publication in the middle, and further
unpublished work at the end.
Chapter 6 contains our publication on the magneto-optical trap for erbium.
Narrow-line magneto-optical trap for erbium
Albert Frisch, Kiyotaka Aikawa, Michael Mark, Alexander Rietzler, Johannes Schindler, Eric
Zupanič, Rudolf Grimm, and Francesca Ferlaino
Physical Review A 85, 051401(R) (2012). DOI 10.1103/PhysRevA.85.051401
This publication shows the experimental realization of the narrow-line magneto-optical trap
for erbium. Specifically, it addresses the Zeeman slowing process, the physical limitations of
the narrow-line magneto-optical trap and the spin polarization of the atom cloud.
Chapter 7 presents our publication on the first Bose-Einstein condensate of erbium.
Bose-Einstein Condensation of Erbium
Kiyotaka Aikawa, Albert Frisch, Michael Mark, Simon Baier, Alexander Rietzler, Rudolf
Grimm, and Francesca Ferlaino
Physical Review Letters 108, 210401 (2012). DOI 10.1103/PhysRevLett.108.210401
Editor’s suggestion and Viewpoint in physics.
It presents detailed information on evaporative cooling towards a degenerate Bose gas. Furthermore, this chapter contains a measurement of the scattering length for different isotopesof erbium.
Chapter 8 contains our publication on the hyperfine structure of erbium.
Hyperfine structure of laser-cooling transitions in fermionic erbium-167
Albert Frisch, Kiyotaka Aikawa, Michael Mark, Francesca Ferlaino, Ekaterina Berseneva,
and Svetlana Kotochigova
Physical Review A 88, 032508 (2013). DOI 10.1103/PhysRevA.88.032508
It shows a spectroscopic measurement of the hyperfine structure of fermionic erbium using
a hollow cathode lamp.
Chapter 9 contains our publication on the first degenerate Fermi gas of erbium.
Reaching Fermi degeneracy via universal dipolar scattering
Kiyotaka Aikawa, Albert Frisch, Michael Mark, Simon Baier, Rudolf Grimm, and Francesca
Ferlaino
Physical Review Letters 112, 010404 (2014). DOI 10.1103/PhysRevLett.112.010404
Editor’s suggestion.
The publication sets the path for producing a degenerate Fermi gas using a similar approachas for bosons. It turns out that evaporative cooling of a single, dipolar fermionic species canbe employed with equal efficiency as in the bosonic case.

1.2. Thesis overview

7

Chapter 10 presents our publication on signatures of quantum chaos in erbium.
Quantum chaos in ultracold collisions of gas-phase erbium atoms
Albert Frisch, Michael Mark, Kiyotaka Aikawa, Francesca Ferlaino, John L. Bohn, Constantinos Makrides, Alexander Petrov, and Svetlana Kotochigova
Nature 507, 475–479 (2014). DOI 10.1038/nature13137
Nature News & Views associated article.
It provides a high-resolution Feshbach spectroscopy measurement revealing up to 200 Feshbach resonances at low magnetic fields. A statistical analysis of these resonances indicatesquantum chaos in the collision of erbium atoms.
Chapter 11 contains our publication on the observation of the Fermi surface deformation.
Observation of Fermi surface deformation in a dipolar quantum gas
Kiyotaka Aikawa, Simon Baier, Albert Frisch, Michael Mark, Cornelis Ravensbergen, andFrancesca Ferlaino
Science 345, 1484 (2014). DOI 10.1126/science.1255259
It contains the first experimental realization and observation of the deformation of the Fermi
surface induced by anisotropic interactions. The deformation is a quantum many-body effectand depends on the temperature of the gas.
Finally, I conclude the thesis in Chapter 12 with an outlook on possible future directions in
the erbium experiment.

Chapter

The rare earth elements perplex us in our researches, baffle us in our speculations, and haunt us
in our very dreams. They stretch like an unknown
sea before us mocking, mystifying and murmuring
strange revelations and possibilities.

2(Sir William Crookes 1887)

Erbium properties

Erbium (Er) is a rare-earth metal belonging to the lanthanide series of the periodic tableof elements. It has an atomic number of Z = 68 and an atomic mass of 167.26 amu, with
1 amu = 1.6605402 × 10−27 kg. Pure solid erbium is a silvery-white metal and needs to be
artificially isolated as it naturally occurs only in chemical compounds on Earth mostly inoxides with trivalent bonds. As a solid it is a malleable and soft metal.
Erbium can be found in the Earth crust with a concentration of about 3.5 ppm and thus
belongs to the rare-earth metals, see Ref. [Lid10]. It is a very important element for manytechnological applications. For example, it is commonly used as dopant in the form of
Er3+ -ions in silica-glass fibers for erbium-doped fiber amplifiers. Such laser light amplifiers
are usually operated at wavelengths in ranges from 1525 nm to 1565 nm and 1570 nm to1610 nm and are nowadays the core elements in optical telecommunication networks. The
same erbium ion has a prominent transition at 2940 nm, which is highly absorbed by waterand thus perfectly suitable for laser surgery and other medical applications. In a very recent
publication the electrical detection of an optical excitation of Er3+ has been demonstrated inRef. [Yin13]. This technique might lead to a novel interface between optical-based quantum
computing and common silicon-based semiconductor technologies.
In the 1990s the price for erbium oxide, as well as for other rare-earth elements, dropped
significantly due to the construction of large production facilities in China. Recently, prices
were rising again as up to about 90 % of the worldwide production of rare-earth elements isnow carried out in China. This lead to a global discussion about China’s monopoly in the
production of rare-earth metals and its strong influence on the high-tech industry.

2.1. The discovery of erbium
The erbium element was first mentioned by Carl Gustaf Mosander in 1843, see Ref. [Mos43].
From the mineral gadolinite he was able to separate three fractions of oxides that he called
yttria, erbia, and terbia. Gadolinite was found in large amounts close to the village Ytterby
near Stockholm, Sweden. Eventually also the elements ytterbium, terbium and yttrium weredirectly named after this place. Other elements, which were discovered at the same place892.2. Basic properties
isotope

162 Er

164 Er166 Er

167 Er168 Er

170 Erabundance
statistics

0.14 %
boson

1.61 %
boson

33.6 %
boson

23.0 %
fermion26.8 %
boson

15.0 %
boson

Table 2.1.: Relative abundances and type of quantum mechanical statistics for all stable erbium
isotopes. Isotopes commonly used in the work of this thesis are highlighted.

but later, were named gadolinium (after gadolinite), holmium (after the Latin name for
Stockholm), and thulium (with thule an ancient word for Scandinavia). All together, nearly
half of all elements from the lanthanide series were discovered in Ytterby for the first time.
Many years later it turned out that the originally separated parts of gadolinite were actuallya mixture of several oxides and pure erbium as a solid metal could be isolated by Wilhelm
Klemm and Heinrich Bommer as late as in 1934.

2.2. Basic properties
As many other lanthanides, erbium has a high melting point at 1529 ◦ C and boiling pointat 2900 ◦ C. It oxidizes and tarnishes slowly under air atmosphere but burns readily to form
erbium(III)-oxide at high temperatures. Thin structures made out of erbium, like foils or
powder, are easily flammable when brought from an inert atmosphere into air. It reacts withwater and produces hydrogen gas. Erbium can be dissolved in dilute sulfuric acid, formingrose red hydration complexes containing Er3+ ions. Vacuum gettering properties of erbium
in the form of evaporated films are studied in Ref. [Mül72].
Erbium has six stable isotopes, of which five (162 Er, 164 Er, 166 Er, 168 Er, and 170 Er) arebosons and one (167 Er) is a fermion, see Table 2.1 for a list of all stable erbium isotopes
and their abundances. This thesis focuses mostly on the bosonic isotope 168 Er and on thefermionic isotope 167 Er, which both have natural abundances above 20 % as well as convenient
scattering properties for efficient evaporative cooling.
Another basic property of erbium is its large magnetic moment of about 7 µB , with µB the
Bohr magneton. Although a large magnetic moment is not common to all elements within
the lanthanide series, some of them have the highest magnetic moment of all elements. This
includes dysprosium and terbium, which both have the largest magnetic moment of 10 µB ,and holmium with 9 µB . See Table 2.6 for a full list of all lanthanide elements.

2.2.1. Electron configuration
According to the Aufbau principle ground-state erbium has an electron configuration, which
is often referred to as a submerged-shell configuration. Electrons are distributed among10

2.2. Basic properties
electron orbitals1 simply as(1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 ) 4f 12 6s2 ,2.1

Following the Madelung rule with increasing number of electrons, the 6s orbital is filled up
before the 4f orbitals. Because of this, all orbitals except the 4f orbitals are completely filledin the electron configuration of erbium. The part of above electron configuration written inbrackets equals the electron configuration of xenon and is often abbreviated as [Xe].
There are seven 4f orbitals available, which can in principle take up to 14 electrons in total.
In case of erbium two electrons are missing in the 4f shells, having an angular momentum
projection quantum number of mℓ = +2 and +3. In both cases the spin projection quantum
number of the missing electrons is ms = −1/2, see Table 2.2. Furthermore, the 4f electron
orbitals are highly anisotropic due to their large angular momentum quantum number ofℓ = 3, whereas the 6s orbital is spherically symmetric with ℓ = 0. See corresponding figures
in Table 2.2. It has to be noted here, that these anisotropic orbitals result in a highly
anisotropic van der Waals interaction potential between atoms, not yet observed in ultracold
gas experiments so far, see Sec. 4.1.2.
In case of heavy elements the coupling of electrons is more complicated than for light elements. The reason for this is that the spin-orbit interaction gets comparable or even stronger
than the Coulomb interaction for electrons in higher orbitals. The well-known and simple
spin-orbit (LS) coupling scheme of electrons2 is not applicable any more and the jj-coupling
scheme has to be used. A special case of this is the J1 J2 -coupling, which is important formost excited states of lanthanide elements, see Ref. [Wyb07]. In this scheme, all inner electrons, i. e. electrons in the [Xe] configuration including the 4f shells, as well as all outerelectrons in the 6s shell couple independently in a LS-coupling scheme to states with angularmomentum quantum numbers J1 and J2 . They then sum up to the overall total angularmomentum J = J1 + J2 with quantum number J, which is denoted as (J1 , J2 )J .
Erbium in its ground state (g. s.) is in an H-state with an orbital angular momentum
quantum number of L = 5 due to LS-coupling of all electrons3 . This angular momentum is
unusually large compared to alkali and alkaline-earth atoms, which have L = 0 in the groundstate. Using S and J as the spin and total electronic angular momentum quantum numbers,
respectively, the ground state of erbium is written as
g. s. :
L = 5,3H
6

S = 1,

J =6.

We take the prominent 401-nm transition in erbium as an example for the electron couplingof an excited state. Its excited state (e. s.) results from the excitation of an electron fromthe 6s orbital into the 6p orbital. Following jj-coupling, first the 6s and 6p electrons couple
1

2
3

Using the standard spectroscopic notation, nℓe , with n the principal quantum number, followed by theorbital quantum number, ℓ (denoted by s = 0, p = 1, d = 2, f = 3, . . . ), and the number of electrons, e,in each orbital as superscript.
The LS-coupling is also known as Russel-Saunders coupling scheme.
As shown in Sec. 2.4.2 small corrections on the order of 0.3 % have to be made compared to a pure LScoupling scheme.

11

2.2. Basic properties

4f
mℓ
ms

−3
↑↓

−2
↑↓

−1
↑↓

0
↑↓

+1
↑↓

+2
↑

+3
↑

6s
0
↑↓

mℓ
ms

Table 2.2.: Electron configuration of erbium in the ground state for the 4f and 6s orbitals4 . Electrons
are filled into the 6s orbital before the 4f orbitals according to the Aufbau principle. Arrows denoteoccupied states with a spin quantum number of ms = +1/2 and −1/2 by |↑⟩ and |↓⟩, respectively.
The two vacancies in the 4f shells with ml = +2, +3 give rise to the large magnetic moment and thestrongly anisotropic van der Waals potential of erbium.

to a 1 P1 state and the rest of the inner electrons couple to 3 H6 by LS-coupling. Then, these
two states couple to a (J1 , J2 )J state. The 401-nm excited state is thus written as
401 nm e. s. :[Xe]4f 12 (3 H6 )6s6p(1 P1 )(6, 1)7

J1 = 6,J2 = 1,J =7.

All bosonic isotopes of erbium have a zero nuclear spin quantum number I. On the contrary,
the fermionic isotope has I = 7/2, which leads to hyperfine structure as J couples with thenuclear spin to give the total angular momentum quantum number F .
164

Er, 166 Er, 168 Er, 170 Er :
167

Er :

I=0
I = 7/2

g. s. : F = 19/2 .

It is interesting to point out that nuclei with an even number of protons and neutrons will
always have zero nuclear spin, whereas odd-even or even-odd nuclei will have half-integer andodd-odd nuclei an integer nuclear spin, see Ref. [Wyb07]. More details about the hyperfine
structure of fermionic erbium in the ground state is found in Sec. 2.3.2 and Chapter 8.
On the one hand, erbium’s exceptional electron configuration gives rise to its large magnetic
moment, as shown in Sec. 2.4 and on the other hand, it causes a strongly anisotropic vander Waals potential, see Sec. 4.1.2. In conclusion, the submerged-shell structure of erbiumwith two vacancies in the anisotropic 4f orbitals is at the foundations of its exotic scattering
properties, which will be discussed in detail in Chapter 4.

4The size of the 6s orbital is shown out of scale with respect to the 4f orbitals.

122.3. Atomic energy spectrum
(a)

(b)
J=7

401 nm25

rel. transition probability

1

20

10-2
10-4
10-6

6s6p 3P1
10-8

15631 nm1299 nm

5

3

0
1

2

H6 ground state

3

4

5

6

7

8

9

10

11

12

J=6

583 nm - M

10

OT

ing

841 nm401 nm - ZS
, TC & imag

energy (103 cm-1)

583 nm6s6p 1P1

4f12 6s2 3H6 g.s.

J

Figure 2.1.: Energy level scheme of erbium. (a) Levels up to an energy of a wavenumber of
25,000 cm−1 . The full level scheme is presented in appendix E. States with even (odd) parity are
shown in red (black). (b) Energy levels of transitions from the ground state to an excited state withJ → J + 1. The two important transitions for the erbium experiment are at a wavelength of 401 nm(blue) and 583 nm (yellow). The blue transition is used for Zeeman slowing (ZS), transversal cooling (TC), and imaging, whereas the yellow transition is used for creating the magneto-optical trap
(MOT).

2.3. Atomic energy spectrum
The submerged-shell electron configuration of erbium is the reason for its complicated energy
level structure. Figure 2.1(a) shows part of the erbium level scheme up to a wavenumberof 25,000 cm−1 . The full level scheme is presented in appendix E. A comprehensive set ofspectroscopic data of all lanthanide elements can be found in Refs. [Ral11, Mar78]. For
erbium a total of 312 states with odd parity5 and 358 states with even parity are known
up-to-date. Its ground state has even parity.
The natural linewidth of several excited states of erbium was initially determined by measuring the radiative lifetime after excitation by a pulsed laser. These measurements had
been carried out in Ref. [Mar80]. More recent measurements on a slow atom beam of erbiuminclude many more transition probabilities of excited states, see Refs. [Law10, Har10].

5

The parity refers to the symmetry property of the electronic wave function. Let the parity operator P̂
invert the wave function through the origin. If P̂ Ψ(⃗r) = Ψ(−⃗r) the wave function has even parity, ifP̂ Ψ(⃗r) = −Ψ(−⃗r) it has odd parity.

13

2.3. Atomic energy spectrum
energy(cm−1 )
24943.272

17157.307

15846.549

11887.503

7696.956

wavelength
in vacuum (nm)400.91582.84631.04841.221299.21naturallinewidth

excited stateel. conf.
jj-coupling

29.7(6) MHz[Xe]4f 12 (3 H6 )6s6p (1 Po1 )(6, 1)o7

186(10) kHz[Xe]4f 12 (3 H6 )6s6p (3 Po1 )(6, 1)o7

28(7) kHz[Xe]4f 11 (4 Io13/2 )
5d3/2
6s2

(13/2, 3/2)o7

8.0(2) kHz[Xe]4f 11 (4 Io15/2 )
5d5/2
6s2

(15/2, 5/2)o7

2.1(1.1) Hz[Xe]4f 11 (4 Io15/2 )
5d3/2
6s2

(15/2, 3/2)o7

Table 2.3.: Laser-cooling transitions with J → J + 1 of erbium in the ground state. Note thedecreasing natural linewidth of these transitions. The electron configuration (el. conf.) of the excitedstate is given. Electrons in the inner shells (up to the 4f shell) first couple via LS-coupling (shownin brackets) and then couple via jj-coupling with electrons in outer shells. Spectroscopic data istaken from Refs. [Ban05, Har10, Law10]. The natural linewidth of the 401-nm transition has beenremeasured within the work of this thesis, see Appendix A.

Due to the large number of energy levels it is possible that an excited state decays with alarge probability into a lower-lying state under the constriction of changing the parity ofthe electronic wave function. Dependent on the lifetime of the target state it decays further
down and eventually reaches the ground state again. Intermediate states with much longer
lifetimes compared to the lifetime of the initially excited state are also called metastable statesand have to be considered in laser-cooling applications, see Ref. [McC06b]. In principle, afew metastable states can be actively depleted by using a repumper laser. To reduce large
technical costs, a simple cooling strategy should avoid repumping many metastable states.

2.3.1. Laser-cooling transitions
Electric-dipole transitions are commonly used for laser cooling. In case of erbium, this restricts possible transitions from the ground state to an excited state with J = 5 or 7. A
broad transition allows an efficient operation of the Zeeman slower (ZS), whereas a narrowtransition is better suited for operating a magneto-optical trap (MOT) due to a lower achievable temperature of atoms, see Chapter 3 for the ZS and Chapter 6 for the MOT. To reduce
the number of possible decay channels into metastable states, the excited state should nothave a too high energy as the number of possible decay channels increases with increasingenergy.

14

2.3. Atomic energy spectrum

laser cooling parametertransition
401 nm583 nmtransition rateΓ (s−1 )1.87 × 108

1.17 × 106

lifetime

τ (ns)

5.35

857

natural linewidth∆ν (MHz)

29.7

0.19

saturation intensity(mW/cm2 )

60.3

0.13

Doppler temperature

TD (µK)714

4.6

Doppler velocityvD (mm/s)

267

21

recoil temperature

Tr (nK)

717

339

recoil velocity

vr (mm/s)

6.0[Har10]Appendix A

4.1[Har10][Law10]IS

Refs.

Table 2.4.: Relevant laser-cooling parameters of the 401-nm and 583-nm transition in erbium usedin the present work. See also Appendix A for a recent measurement of the natural linewidth of the
401-nm transition using a sample of cold atoms.

Early work on laser-cooling of erbium atoms have identified several suitable electric-dipole
transitions from the ground state, with J = 6 and even parity, to an excited state, with
J = 7 and odd parity, see Ref. [Ban05]. The linewidths of these transitions range from alarge linewidth, of a few ten MHz, to a very small one, of only a few Hz. In Fig. 2.1(b) theselaser-cooling transitions are highlighted and Table 2.3 gives a summary of their properties.
For laser cooling in the current work only two transitions are relevant, which will be discussed
in more detail below.
The first transition is at a wavelength of 401 nm (blue), where a 6s electron is excited into a
6p state and couples with the second 6s electron to the singlet state 1 P1 . This transition has
a natural linewidth of 29.7 MHz and is comparable to the broad |S1/2 , F = 2⟩ → |P3/2 , F = 3⟩transition of the D2 line in alkali-metal atoms, which are commonly used in ultracold gas
experiments. The large linewidth makes this transition very suitable for Zeeman slowing,transversal cooling, and imaging purposes. The second transition is at a wavelength of
583 nm (yellow), where the 6s and the excited 6p electron couple to the triplet state 3 P1 . It
is an intercombination line and has a narrow linewidth of only 190 kHz, which implies several
advantages for operating the MOT on this transition, see Chapter 6.
Table 2.4 gives many laser-cooling parameters of these two transitions. It consists of theΓtransition rate Γ, the lifetime τ = Γ1 , the natural linewidth ∆ν = 2π
, the saturation intensity
q
kB TD
~Γ
IS = πhcΓ,theDopplertemperature
T
=
,
the
Dopplervelocity
v
=
D
D
2kB
m , the recoil
3λ3~ ktemperature Tr = 2mk, and the recoil velocity vr = ~km . Here, h is Planck’s constant withBh
~ = 2π , kB the Boltzmann constant, m the atomic mass, and k = 2π
λ the wavenumber with
λ the wavelength of the transition.
2 2

x10

9/2 1758 MHz
11/2 1752 MHz
7/2

1654 MHz

13/2 1557 MHz
15/2 1077 MHz
1
P1
6s6p

17/2 205 MHz

19/2 -1182 MHz

21/2 -3223 MHz

x10
9/2
7/2

3196 MHz
3170 MHz

11/2 3035 MHzF

13/2 2572 MHz

∆HFS

x10

15/2 1670 MHz
11/2 1843 MHz
9/2

3
P1
6s6p

1774 MHz

17/2 173 MHz

13/2 1540 MHz
7/2
5/2

19/2 -2099 MHz

1495 MHz

1141 MHz

15/2 673 MHz
0 MHz
3

H6
g.s.

17/2 -979 MHz
21/2 -5345 MHz

1 GHz
19/2 -3668 MHz

J=6

J=7

Figure 2.2.: Hyperfine structure of the ground state and two excited states. The frequency splitting,∆HFS , for each hyperfine state with total angular momentum quantum number F is given with respect
to an assumed state with I = 0 depicted by a dashed line. Gray boxes indicate a magnification in
frequency by a factor of ten.

15

16

2.3. Atomic energy spectrumlevel

A (MHz)B (MHz)Ref.

ground state-120.487(1)-4552.984(10)[Chi83]401 nm e.s.

-172.70(7)-4457.2(29)[Jin90]583 nm e.s.

-100.1(3)-3079(30)

Chap. 8

Table 2.5.: Magnetic dipole, A, and electric quadrupole, B, hyperfine constants in units of frequency
for the ground state and two excited states (e.s.) used for laser cooling of the fermionic isotope 167 Er.

2.3.2. Hyperfine structure
Fermionic 167 Er has a rich hyperfine structure due to its nonzero nuclear spin of I = 7/2.
This gives a total of eight hyperfine states with quantum numbers ranging from F = J − 7/2
to J + 7/2. The relevant A and B constants for the ground state and for the 401-nm and583-nm excited states are presented in Table 2.5. Using these two coefficients the transitionenergy from the ground state to an excited state with hyperfine energies Eg (Fg , Jg , I) andEe (Fe , Je , I), respectively, can be calculated by~ωFe Fg = ∆167 + ~ω166 + Ee (Fe , Je , I) − Eg (Fg , Jg , I) ,2.2

with ~ω166 the transition energy of the most abundant isotope 166 Er. For 167 Er an isotope
shift ∆167 with respect to 166 Er is specified by evaluating the ‘center of gravity’ of the full
hyperfine structure, details are presented in Chapter 8. The hyperfine energies, Ei with
i = g, e, are calculated via
1
1 3Ci (Ci + 1) − 4I(I + 1)Ji (Ji + 1)
Ai C i + B i222I(2I − 1)2Ji (2Ji − 1)
= Fi (Fi + 1) − Ji (Ji + 1) − I(I + 1) .

Ei (Fi , Ji , I) =
Ci

2.3
2.4

The resulting hyperfine structure of the ground state and the two excited states are shownin Fig. 2.2 for zero magnetic field.

2.3.3. Isotope shift
Due to a change in the charge distribution of the nucleus for different isotopes the transitionenergy changes linearly dependent on the atomic mass. This is known as isotope shift. The
isotope shift between the two nearest bosonic erbium isotopes is about 820 MHz for the 401nm transition and 980 MHz for the 583-nm transition, corresponding to a shift per atomic
mass unit of 410 MHz and 490 MHz, respectively.
In Fig. 2.3 the isotope shift is shown for both transitions by plotting data taken from literature, see Refs. [Jin90, Con10]. The expected linear dependence of the isotope shift on theatomic mass can be observed for both transitions within the uncertainty of the reference
values. Within the present work, the isotope shift was experimentally verified by using ahollow-cathode lamp as spectroscopy cell, see Chapter 8.

172.4. Magnetic properties

shift (GHz)2

401 nm583 nm

1

0

-1
164

166

168

170

isotope (amu)

Figure 2.3.: Isotope shift for bosonic isotopes 164 Er, 168 Er, and 170 Er with respect to 166 Er. The
isotope shift is plotted for the 401-nm transition in circles and for the 583-nm transition in squares.
Lines are a linear fit to the data points. Data is taken from Refs. [Jin90, Con10].

2.4. Magnetic properties
Lanthanides are ferromagnetic metals but they have a Curie temperature, which is below
room temperature. However, they easily form compounds with transition metals like iron,nickel or cobalt and are frequently used in alloys to create some of the strongest permanent
magnets, known as rare-earth magnets6 . Using these magnets it is possible to create permanent magnetic fields of up to 1.2 T. Rare-earth magnets can maintain such large magnetic
fields because electrons in the 4f shells are strongly localized close to the atomic nucleus.
This means that the 4f electrons tend to have no overlap with orbitals of neighboring atoms
within the compound and lanthanide atoms form so-called paramagnetic centers. These paramagnetic centers prevent a reduction of the magnetic moment of the lanthanide atom, which
would happen in case of overlapping orbitals7 . This process explains the large magnetic fieldstrength of rare-earth magnets, see Ref. [Jen91].
Several elements of the lanthanide series have a magnetic moment, which is amongst thelargest magnetic moments of all elements. The coupling of electrons results in exceptionallylarge quantum numbers. Specifically, large values of the total orbital angular momentum L
and total angular momentum J can be found, see Ref. [Mar78].

2.4.1. Magnetic moment
A full comparison of the atomic magnetic moment, µ, of all elements belonging to thelanthanide series is shown in Table 2.6. The magnetic moment is given in units of the Bohrmagneton8 , µB . Elements with the largest magnetic moment are terbium and dysprosiumwith µ = 10 µB , followed by holmium with 8 µB , as well as europium and erbium with 7 µB .
67

8

Neodymium magnets and samarium-cobalt magnets are most commonly available.
A single hydrogen atom is paramagnetic, but in the form of molecular hydrogen as H2 it has diamagnetic
properties, because opposite electron spins pair up and the magnetic moment is lost.
e~
The Bohr magneton is defined as µB = 2m
with e the elementary charge and me the electron mass. It iseconvenient to use µB ≈ 1.39962 MHz/G.

18

2.4. Magnetic properties
element
symbol
La
Ce
Pr
Nd
Pm
Sa
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu

isotopesgroundbos. ferm. state

4f n
n

5dm
m

gJ

µ (µB )

2D
3/2
1G
4
4I
9/2
5I
4
6H
5/2
7F
0
8S
7/2
9D
2
6H
15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
2D
3/2

0
1
3
4
5
6
7
7
9
10
11
12
13
14
14

1
1
0
0
0
0
0
1
0
0
0
0
0
0
1

0.80
0.95
0.73
0.60
0.31
0
1.99
2.65
1.33
1.24
1.20
1.16
1.14
0
0.80

1.20
3.78
3.29
2.41
0.76
0
6.98
5.30
9.94
9.93
8.96
6.98
3.99
0
1.20

1
3
1
3
0
4
1
3
1
5
1
5
1
5
1

0
0
0
2
0
0
0
2
0
2
0
1
0
2
0

has manyisotopes

is highly
magnetic

X

X
X
X

X
X
X
X
X

X

Table 2.6.: Number of different isotopes, ground-state electron configuration, and magnetic properties of all elements belonging to the group of lanthanides in comparison. The number of stable bosonic(bos.) and fermionic (ferm.) isotopes of each element is given. The electron configuration is written
as the number of electrons in the 4f and 5d shells using the notation [Xe]4f n 5dm 6s2 . Two selection
criteria are included as a guide for the reader. Having many isotopes requires that an element has
more than three stable isotopes, of which at least one is fermionic. Being highly magnetic means thatµ > 6 µB . Elements of interest for novel laser-cooling experiments are written in bold. Data is takenfrom Ref. [Mar78].

This is in stark contrast to many other atomic species. Alkaline-earth metals and ytterbium
are in a 1 S0 state and do not have any magnetic moment at all, whereas alkali-metal atoms
have µ = 1 µB in the ground state due to their single electron in the s shell. Chromium, on
the other hand, has a large magnetic moment of 6 µB .
The component of the atomic magnetic moment along the direction of an external magnetic
field9 in z-direction can be calculated byµ = mJ gJ µB ,2.5

with mJ the magnetic quantum number, i. e. the projection of the total angular momentum
J along the direction of the magnetic field B. Values for mJ range from −J to +J. Furthermore, gJ is the Landé g-factor, which is also known as the gyromagnetic ratio. It canbe calculated and measured in the experiment by observing the Zeeman splitting of energy
levels in presence of a magnetic field. The precise knowledge of gJ is very important for theexperiment and its exact value for erbium will be derived in the following.

9

The homogeneous magnetic field vector points in the vertical z-direction, unless otherwise noted, and gives
the quantization axis of the system.

19

2.4. Magnetic properties

groundstate

Landé
value

LScorrection

relativisticand diamagn.
correction

calc.
gJ

exp.
gJ

3H

1.1670533-0.0013345-0.00192

1.1637985

1.163801(1)

Eq. 2.6[Con63][Jud61][Con63][Con63]

6

Refs.

Table 2.7.: Landé value gJ for the ground state of bosonic erbium including corrections. The relative
deviation between theoretical and experimental values is only 2 × 10−6 .

2.4.2. Landé g-factor
For the calculation of the Landé g-factor, gJ , two different cases have to be taken into account,which correspond to either LS-coupling or jj-coupling. In the case of simple LS-coupling,gJ can be calculated according to
gJ = 1 + (gS − 1)

J(J + 1) − L(L + 1) + S(S + 1).
2J(J + 1)

with gS the gyromagnetic ratio of the electron spin. It is given byα+ . . . ≈ 2.00232 ,gS = 2 1 +
2π

2.6

2.7

with α ≈ 1/137 the fine-structure constant. In the case of lanthanides, several corrections
have to be taken into account for an exact calculation of gJ . A theoretical derivation of all
corrections is given in Ref. [Jud61]. The first correction is due to deviations from perfectLS-coupling. Second, a relativistic correction addresses the fact that high-orbit electrons oflanthanides posses a relativistic kinetic energy. The third correction is a diamagnetic correction and depends on the electron density at the core. All corrections and the experimental
value for gJ of bosonic erbium in the ground state are summed up in Table 2.7, see also
Refs. [Cab61, Con63]. The most precise experimental value for gJ of erbium is 1.163801(1)and thus the magnetic moment of bosonic erbium in the energetically lowest state with
mJ = −6 isµ = mJ gJ µB = −6.982806(6) µB ≈ −9.77 MHz/G .
2.8
In the case of jj-coupling, with a total electronic angular momentum quantum number J1and J2 of the two coupled parts, the gyromagnetic ratio can be calculated by
J(J + 1) + J1 (J1 + 1) − J2 (J2 + 1)
2J(J + 1)
J(J + 1) + J2 (J2 + 1) − J1 (J1 + 1)
+ gJ2.
2J(J + 1)

g J = g J1

2.9

Here, gJ1 and gJ2 are the Landé g-factors for each part according to Eq. 2.6. The excited states of the 401-nm and 583-nm transitions are [Xe]4f 12 (3 H6 )6s6p(1 P1 )(6, 1)7 and[Xe]4f 12 (3 H6 )6s6p(3 P1 )(6, 1)7 , respectively. Including above mentioned corrections for LScoupling for gJ1 and gJ2 , Eq. 2.9 gives a value of gJ = 1.160 for the excited state of the 401-nm
transition and gJ = 1.195 for the excited state of the 583-nm transition, see Ref. [Mar78].

20

2.4. Magnetic properties
mJ
+6
+5
+4
+3
+2
+1
0
-1
-2
-3
-4
-5
-6

energy (MHz)

100
50
0
-50

-100
0

2

4
6
magnetic field (G)810Figure 2.4.: Zeeman energy splitting of bosonic erbium in the ground state with J = 6 for all
Zeeman substates. For this magnetic field range, the Zeeman splitting is linear (see text).

In case of fermionic 167 Er the total electronic angular momentum J couples with the nonzero
nuclear spin I = 7/2 to form the total angular momentum F = J + I. This splits energy
levels into a hyperfine structure, see Sec. 2.3.2, and the Landé g-factor has to be calculated
according to
F (F + 1) − I(I + 1) + J(J + 1)
gF = gJ,
2.10
2F (F + 1)
with gJ as given in Table 2.7 for the ground state. The magnetic moment is then given
by µ = mF gF µB . Here, mF is the projection of F on the quantization axis and ranges
from −F to +F . In the energetically lowest state, with F = 19/2 and mF = −19/2,
this results in gF = 0.735032 and the magnetic moment of the fermionic ground state isµ = −6.982804 µB , which equals the magnetic moment of bosons in the ground state, with
J = 6 and mJ = −6.

2.4.3. Zeeman splitting
In presence of an external magnetic field with field strength B each energy level with total
angular momentum quantum number J splits in a total of 2J + 1 Zeeman substates. Eachstate has a magnetic quantum number mJ . The Zeeman energy shift, ∆EZ , with respect tothe level at zero magnetic field is given by∆EZ (B) = mJ gJ µB B .

2.11

Figure 2.4 shows the Zeeman splitting of bosonic erbium from 0 to 10 G. The energeticallylowest state has mJ = −6. For low magnetic fields the energy splitting between the twonearest mJ states is linear10 and has a slope of 1.628879 MHz/G. In fact, the range of linear
Zeeman splitting extends to magnetic fields higher than technically achievable in ultracold gas
experiments, due to the unusually large spin-orbit coupling constant of erbium of 2242 cm−1 ,see Ref. [Jud61]. Using radio-frequency fields one can drive transitions from mJ → m′J =mJ ± 1, see Sec. 6.3. In the experiment measurements of the transition frequency frommJ = −6 to m′J = −5 are used to calibrate magnetic fields.
10

We have experimentally verified the linearity of the Zeeman splitting up to a magnetic field of 50 G within
an uncertainty of 0.02 % using RF spectroscopy of the transition from mJ = −6 to m′J = −5.

212.4. Magnetic properties
(a)
11/2
9/2
13/2
7/2

2

energy (GHz)

5/2
15/2

0
17/2

-2

-4

F = 19/2
mF 19/2-19/2-6∆E (MHz)(b) 200
F=19/2, mF = -19/2 → -17/2

150
100

∆E (MHz)

(c)

exact

50

linear00residual

-3
-6
-9
0

20

40

60

80
100
120
magnetic field (G)

140

160

180

200

Figure 2.5.: Zeeman splitting of the hyperfine states of fermionic 167 Er in the ground state within an
external magnetic field. (a) Hyperfine states with a total quantum number F split up in 2F + 1 magnetic substates with quantum numbers mF . (b) Energy difference, ∆E, between the two energeticallylowest substates with mF = −19/2 and −17/2 derived from exact calculation (red) compared to thelinear splitting according to Eq. 2.12 (black). The residual between the two curves is shown in (c).

2.4.4. Hyperfine Zeeman splitting
In case of fermionic erbium each hyperfine state F splits in the presence of a magnetic field
B into a manifold of 2F + 1 substates with quantum numbers mF from −F to +F , seeFig. 2.5. Consequently, the Zeeman energy of the hyperfine substates can be approximatedby∆EZ (B) = mF gF µB B ,2.12
for magnetic fields below 20 G. Here, gF is the Landé g-factor from Eq. 2.10. For larger magnetic fields, a decoupling of J and I becomes apparent, which is also known as Paschen-Backeffect, and the energy splitting deviates form the linear case. In this regime, the hyperfine
energy levels have to be calculated via exact diagonalization of the atomic Hamiltonian, see
Ref. [Smi65].
Figure 2.5(b) shows the deviation of the exact calculation from the linear splitting following
Eq. 2.12. The residual between these two curves is separately shown in Fig. 2.5(c). The
frequency difference of the transitions from mF = −19/2 to m′F = −17/2 and mF = −17/2

2.4. Magnetic properties

22

to m′F = −15/2 is 65.5 kHz at a magnetic field of 50 G. This separation would be large
enough to prevent a population of the mF = −15/2 state when driving a transition frommF = −19/2 to m′F = −17/2 in an ultracold atomic sample using a radio-frequency. Thismethod allows for the preparation of a spin mixture of fermionic erbium atoms.

Chapter3

Experimental setup

This chapter focuses on the experimental setup used to produce ultracold fermionic andbosonic quantum gases of erbium. The design phase of the experiment started in the second
half of 2009. The vacuum chamber and laser system have been installed in 2010 and 2011.
We obtained the first erbium magneto-optical trap in late 2011, the first Bose-Einsteincondensate in early 2012, and the first degenerate Fermi gas in 2013.
In the following, the erbium apparatus will be described in detail. The chapter is structured
in four sections: Section 3.1 reviews the key experimental requirements of the apparatus,
Sec. 3.2 offers an overview of the vacuum chamber including a high-temperature effusion cell,Sec. 3.4 introduces the coil system for creating magnetic fields and Sec. 3.5 shows the laser
system for laser cooling and trapping.

3.1. Experimental requirements
The erbium machine was designed from scratch with the aim of creating a quantum gas
experiment. Several parts of the vacuum chamber were standard to ultracold atom experiments, while other parts required a more careful design. In designing the apparatus thefollowing list of guidelines was taken into account:• The pressure in the vacuum chamber should allow for a long lifetime of the atomic
sample, which corresponds to a collision rate with the background gas below 0.01 s−1 .
This requires a pressure below 10−10 mbar, i. e. within the ultra-high vacuum regime.
• Because of the high melting point of erbium a high-temperature oven has to be used
for evaporating the source material at temperatures of up to 1500 ◦ C.
• The amount of source material in the oven should allow for an oven lifetime of morethan a year.
• The magneto-optical trap (MOT) should operate on the narrow transition (with wavelength λ = 583 nm and linewidth ∆ν = 190 kHz) to guarantee temperatures low enoughfor direct loading into an optical dipole trap.
23

3.2. Vacuum chamber design24• Hot atoms from the oven have to be slowed to velocities below the capture velocityof the narrow-line MOT. The usage of a Zeeman slower (ZS), operating on a strongtransition (with λ = 401 nm and ∆ν = 30 MHz), ensures a slow atom beam with ahigh flux at the center of the main chamber.
• For future experiments with a Er-Li mixture, a lithium oven should be attachable
to the vacuum chamber, allowing for evaporating lithium at a temperature of up to500 ◦ C. The two atom beams should be combined in order to require only one ZS.
• The ZS should be usable for both species simultaneously. In case of erbium the ZS
should capture atoms with velocities as high as possible to maximize the atomic fluxand increase the lifetime of the source material in the oven. This is less critical in caseof lithium.
• It should be possible to create homogeneous magnetic fields up to 400 G for Feshbach
spectroscopy1 and gradient fields above 50 G/cm for MOT operation and Stern-Gerlach
experiments. Additionally, coils should be installed to allow for compensation of external magnetic fields, active reduction of the magnetic field noise, and creation of asmall homogeneous polarizing magnetic field in any spatial direction.
• The stability of the experiment should be as high as possible in order to be able
to carry out measurements over several days/weeks without significant noise. This
includes short-term stability, e. g. by reducing vibrations as well as long-term stability
by stable environmental conditions, like temperature, humidity, and external magnetic
fields.
During the construction of the erbium machine, we could benefit from various technical developments in other experimental setups in Innsbruck, like the Sr or the FeLiKx experiment,
see Refs. [Ste13] and [Wil09], respectively.

3.2. Vacuum chamber design
Figure 3.1 gives an overview of the main parts of the vacuum apparatus. The apparatus
can be divided into a high vacuum (HV) section and an ultra-high vacuum (UHV) section.
Each individual part of the vacuum chamber will be discussed in the following sections, seeSec. 3.2.1 for the HV and Sec. 3.2.2 for the UHV section.
• the high vacuum section includes:– the high-temperature effusion cell for erbium– the transversal cooling section– the low-temperature oven for lithium
1

Nothing was known about the Feshbach spectrum of erbium or other lanthanides at the beginning of theexperiment.

3.2. Vacuum chamber design25– the atom beam shutter– a first pumping stage• the ultra-high vacuum section includes:– the Zeeman slower– the main chamber– a second pumping stage– the Zeeman slower mirror
A pressure of about 4 × 10−9 mbar in the HV section and 1 × 10−11 mbar in the UHV section
has to be maintained for the experiment. A differential pumping tube connects the two
sections and provides a pressure gradient of up to three orders of magnitude. The sections
can be disconnected from each other by an all-metal gate valve. This allows for refilling theoven without breaking the vacuum in the UHV section.
The center of the vacuum chamber is situated 200 mm above the surface of an optical table,
making the use of two layers of optical setup with a beam height of 75 mm each possible.
Solid aluminum bread boards with a thickness of 25 mm support the upper optical layer.
Stainless-steel supporting structures are placed on various critical points of the vacuum
chamber to suppress mechanical vibrations to a minimum. A welded bellow is introduced
at the gate valve to reduce mechanical stress while baking out the main chamber, whichleads to a thermal expansion of the vacuum chamber. Furthermore, it allows for a roughalignment of the two sections with respect to each other. This is crucial to point the atom
beam into the center of the main chamber.

3.2.1. High vacuum section
Figure 3.2 shows a section view of the main part of the HV section. It consists of the hightemperature erbium oven and a transversal cooling stage, connected by a port aligner. The
latter allows for aligning the angle between the axis of the effusion cell and the main axis of
the vacuum chamber. It is primarily used to compensate a slight downward bending of theinner parts of the high-temperature oven, which was observed within a few years of operation
at high temperatures. For this, the port aligner is oriented such that a single screw changes
the angle of the oven axis with respect to the horizontal plane. Each subsection of the highvacuum section will be discussed in more detail in the following.

High-temperature oven For heating erbium to the required temperatures, a commerciallyavailable high-temperature oven is used2 . The oven consists of two separately controllable
sections, the so-called effusion cell and the hot lip that are heated by two independent heating
filaments. In the former, solid pieces of erbium (about 10 g) are evaporated and in the latter,a collimated atom beam is formed using a set of apertures. The two heating filaments make2

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

pumping stage 2pressure gauge

main chamber

UHV section

75 l/s ion getter pump

Zeeman slower

bellowfull-metal valve

transversal
cooling

HV sectionx

200 mmyhigh-temperature ovenport aligner

20 l/s ion getter pumps

atomic beam shutter

pumping stage 1

Ti-sub pumpz

Figure 3.1.: Full model of the vacuum chamber. Individual sections are highlighted in different colors. Laser beams for transversal cooling andZeeman slowing as well as MOT operation are shown by blue and yellow arrows, respectively. The oven chamber is situated in the high vacuum(HV) section and the main chamber in the ultra-high vacuum (UHV) section. In case of refilling the high-temperature oven, the two sections can beseparated from each other by a full-metal gate valve. A differential pumping tube in the center of the Zeeman slower (green) maintains a pressure
difference of up to three orders of magnitude. Sufficient pumping is achieved by two 20 l/s (HV section) and one 75 l/s (UHV section) ion getterpumps as well as by one titanium sublimation pump in each section.

Zeeman slower
mirrorport aligner

Ti-sub pump (hidden)

27

3.2. Vacuum chamber designtransversal cooling beamelectrical and water
connections
port aligner

AP3 - disk

water coolingzhigh temperature crucible

40 mm
x

heating filaments

effusion cell1100 °CØ12

AP1 - tube

Ø3

hot lip
T=1200 °C

Ø3

Ø25

AP2 - diskyerbium source material
11

50

30

61

Figure 3.2.: High-temperature oven and transversal cooling section. For a better view the model isshown in a section view. The oven crucible (red) contains pieces of erbium and is shown in the inset
with dimensions in mm. Two apertures (AP1 and AP2), the first in shape of a tube (blue) and thesecond like a disk (green), create a collimated atom beam directed at the center of the main chamber.
To prevent clogging of the apertures they are heated to a higher temperature than the effusion cell.
A third aperture (AP3) with an inner diameter of 8 mm basically blocks excessive material exiting
the oven and keeps the transversal cooling section clean.

it possible to set the temperature of the aperture setup to a higher value compared to thatof the effusion cell. Typical operating temperatures are as high as 1100 ◦ C for the effusion
cell and 1200 ◦ C for the hot lip, which results in an atom flux density of 1014 s−1 sr−1 at theposition of the transversal cooling section, see Ref. [Sch11].
Effusion cell The effusion cell consists of a tantalum crucible, see inset of Fig. 3.2. Tantalum is used due to its high temperature resistivity, inertness to chemical reactions, andsuppression of forming low-melting alloys with erbium3 . It is filled with small pieces of naturally occurring erbium4 . A filling volume of 7 cm3 must not be exceeded to prevent damageof the effusion cell due to thermal expansion of the source material. More details about theoven filling procedure can be found in Appendix B.
Aperture setup A set of three apertures creates a collimated atom beam, two of which are
shown in the inset of Fig. 3.2 (AP1-AP2) and the third aperture (AP3) is located between
the high-temperature oven and the transversal cooling section. Like the oven crucible all34

Any low-melting alloys will destroy the high-temperature oven as they will easily leak out of the crucible
in horizontal orientation and short-circuit the heating filaments.
10 g of erbium distilled dentritic, purchased from Alfa Aesar

3.2. Vacuum chamber design

28

apertures are made out of tantalum. The first two apertures are mounted as an inset directly
into the crucible. Tantalum cylinders are used as spacers, which are held in place by twisted
tantalum wires at the front of the crucible setup. More details about the full characterizationof the atom beam and a test of different aperture geometries for optimum collimation of theatom beam can be found in Ref. [Sch11].

2D transversal cooling section This section allows for laser cooling the atomic beam inboth transversal directions, see left-hand side of Fig. 3.2 and Sec. 3.3 for more details. The
homemade stainless-steel chamber has an octagonal shape and provides optical access in two
orthogonal directions transversally to the atom beam. On top of the transversal coolingchamber an all-metal aluminum mirror is mounted on a linear translation stage, which canbe lowered to check the alignment of the Zeeman slower laser beam.
Intermediate pumping stage An intermediate pumping stage is connected to the transversal
cooling section to pump off the volume between the high-temperature oven and the optionallithium oven. Installing the lithium oven setup would strongly reduce the pumping efficiency,
therefor this intermediate pumping stage is necessary. It consists of a 20 l/s ion getter pump5 ,a hot-filament ionization gauge6 for pressure readings, and an angled valve for connecting a
pumping station.

Lithium oven After the transversal cooling section for erbium, it is possible to attach asecond oven. It is intended to evaporate lithium, which has a low melting point of 181 ◦ C. To
achieve an optimized overlap with the atom beam coming from the high-temperature erbium
oven, an oven with a design similar to Ref. [Wil09] is intended to be used. Here, small micro
tubes with an inner diameter of 200 µm create a collimated atom beam. The volume of theoven is divided into three sections, separated by thin walls. One section contains the micro
tubes, whereas the other two sections are empty and erbium atoms can pass through. This
will create a mixed erbium and lithium atom beam. Currently this oven is not installed but
replaced by a dummy tube.

Pumping stage 1 A 20 l/s ion getter pump5 and a titanium sublimation pump7 are mounted
to the vacuum chamber to provide pressures below 1 × 10−9 mbar. Furthermore, a hotfilament pressure gauge6 is connected for pressure monitoring and an angled valve for initial
pumping. Due to the large gettering properties of erbium, see Ref. [Mül72], it is not necessary
to bake the high vacuum section in order to reach suitable pressures. This makes a refillingof the high-temperature oven very convenient and can be achieved within a single week ofpumping, see Appendix B.

5
6
7

VacIon Plus 20 Ion StarCell, from Agilent Technologies
UHV-24 dual filament gauge, from Agilent Technologies
TSP Filament Cartridge CF40, from Agilent Technologies

3.2. Vacuum chamber design

29

Atom beam shutter An atom beam shutter is mounted on the lower flange of the pumping
stage. After loading the MOT is finished the shutter blocks the atom beam and prevents
erbium atoms from entering the UHV section. It consists of a stainless-steel disk with
20 mm diameter and 2 mm thickness, mounted on a homemade wobble stick. The disk canbe operated by a servo motor from outside the vacuum utilizing a welded bellow.
Full-metal gate valve A full-metal gate valve8 separates the HV section from the UHV
section in case of having to break the vacuum in one of the sections.

3.2.2. Ultra-high vacuum section
As is known, collisions with the background gas limit the lifetime of the trapped atomicgas. Only at an ultra-low pressure the lifetime is long enough for ultracold gas experiments.
At a typical pressure of 1 × 10−11 mbar in the UHV section the measured lifetime of thetrapped atomic gas is longer than 40 s. This lifetime is much longer than the duration of
the experimental cycle. Thus collisions with the background gas can be neglected for theexperiment. The UHV section consists of four subsections including the main experimental
chamber.

Zeeman slower tube The ZS vacuum tube has a total length of 460 mm and an innerdiameter of 24 mm. It contains a single stainless-steel differential pumping tube with alength of 300 mm and a diameter of 8 mm. The differential pumping tube is welded on a
CF40 flange, mounted between the gate valve and the ZS tube. Any heating of the vacuum
chamber due to an excessive power dissipation of the ZS magnetic field coils would lead toan increase of the pressure in the main chamber. To prevent this, a second tube with 40 mm
outer diameter encloses the ZS tube, which creates a 5 mm thick cylindrical spacing along
the whole ZS, allowing for water-cooling. The ZS coils are directly mounted onto the outertube for most efficient cooling. The principle of operation and the design of the ZS will be
discussed in Sec. 3.3 in more detail.

Main chamber Atoms are trapped, cooled, and manipulated in the main chamber. Figure
3.3 shows the main vacuum chamber in a section view. It provides optical access for loading
atoms into the MOT, employing various optical dipole traps and optical lattices, and imaging
the atoms using CCD cameras. It is an in-house-manufactured stainless-steel9 chamber with
six different optical axes. As the slow atom beam is strongly divergent when it leaves the ZS,
the distance from the end of the ZS to the center of the main chamber has to be minimized
for a high MOT loading efficiency. This distance is about 120 mm in the current chamber andit was one of the major geometrical restrictions for the design. Two sets of coils are mounted
on top and below the main chamber for creating magnetic fields, which are symmetric aroundthe z-axis, see Sec. 3.4. The two sets of coils are mounted in water-cooled copper casings.
The height of each casing is 35 mm with an outer diameter of 136 mm.
89All-metal gate valve, size CF40, Series 481, from VAT Vacuumvalves AG
stainless steel code 316LN (after AISI) or 1.4429 (after ESU)clampscopper casing

CF40 windowc

low-field coilsagradient coilsbatomic beam

Feshbach coilsz

MOT center position
x

y

20 mm
cooling water inlet

cooling water pocket

Figure 3.3.: Main vacuum chamber and coil setup shown in a section view. The stainless-steel
vacuum chamber (yellow) has a total of ten anti-reflection coated windows for optical access. Additionally, two CF40 flanges are used to connect the Zeeman slower (right side) and the UHV pumping
stage (left side), not shown here. The vertical dimension of the chamber between the upper and lower
coil casing is only 38 mm. The bores for the atom beam and MOT beams have a diameter of 34 mm.
Both, upper and lower coil sets are identically built and consist of three coils each. (a) For operating
the MOT, two coils (green) produce a magnetic field gradient in all three spatial directions by aquadrupole field. (b) Two coils in Helmholtz configuration (blue) give a homogeneous magnetic field
for Feshbach spectroscopy. (c) A smaller set of coils (purple) is used to create stable homogeneous
magnetic fields below 4 G. The coils are held in place by a copper casing (brown), which enables
efficient cooling by water flowing through a pocket on the outer side of the casing. For sufficient heattransport the coils are moulded into the casing with a thermally conductive resin.

30

31

3.2. Vacuum chamber design

port aligner
aluminum mirror

40 mm
CF63 window
Zeeman slower
laser beam

Figure 3.4.: Zeeman slower mirror setup in a section view. The ZS light (blue) enters the vacuum
chamber through a CF63 window. It is then reflected by a solid aluminum mirror (orange) and counter
propagates to the atom beam through the whole vacuum apparatus. The usage of an aluminum mirror
prevents coating of the vacuum window by atoms not being slowed by the ZS and not captured in the
MOT. The mirror is mounted on a port aligner that can be moved and tilted for optical alignment.

Pumping stage 2 The UHV pumping stage consists of a 75 l/s ion getter pump10 , a pressuregauge11 , an angled valve, and a titanium sublimation pump7 . For an optimized pumping
speed the titanium sublimation pump is contained within a large CF100 vacuum tube.

Zeeman slower mirror In many experimental setups the laser light for the Zeeman slower
is brought collinear to the atom beam into the vacuum chamber via a vacuum window. In
the erbium experiment this would result in an irreversible coating of the window with erbium
atoms coming from the atom beam. Compared to alkali and alkali-earth metals, erbium can
not be heated off the window due to its high melting point. The transmittance of such a
coated window would decrease significantly within a short period of time. A solution forthis problem is to use a mirror inside the vacuum and let the laser beam enter the chamber
perpendicular to the atom beam. The mirror will get coated by a thin layer of erbium overtime but its reflectivity will not suffer much from the coating. Therefore, a solid aluminum
mirror12 is mounted on a port aligner at the end of the vacuum chamber, see Fig. 3.4. We
estimated the reduction of the mirror’s reflectivity by coating with a thin layer of erbium
onto a test mirror to be less than 1.3 %.

10

VacIon Plus 75 Ion StarCell, from Agilent Technologies
UHV-24p dual filament, from Agilent Technologies
12
diamond fly-cut, optical-grade aluminum mirror, AlMgSi1, manufactured by Kugler GmbH11

3.3. Slow atom beam with high flux

32

3.3. Slow atom beam with high flux
The preparation of a slow atom beam with high flux in the main chamber is important for
efficient MOT loading. This will result in a larger number of atoms in the MOT, whichis advantageous for further experiments. In the erbium experiment an estimated flux ofslow atoms13 of about 4 × 108 s−1 at the MOT position is achieved by a combination of a
two-dimensional transversal cooling (TC) stage and a Zeeman slower (ZS).
The maximum longitudinal velocity of the erbium atom beam exiting the high-temperatureoven is about 500 m/s, see Sec. B.2 in the appendix. The ZS has to be well designed andoptimized to slow down atoms close to the MOT capture velocity. The capture velocity is
only 5.6 m/s because of the narrow-line character of the 583-nm transition, see Sec. 6.1.1.
The TC reduces the transversal velocity of the atom beam immediately after leaving thehigh-temperature oven. Both, ZS and TC, work best on a broad atomic transition, which
has a short lifetime of the excited state. This enables a sufficiently large number of scatteredphotons within the time it takes for the atom to fly through the TC section or the ZS tube.
A suitable broad transition in erbium is the blue transition at a wavelength of 401 nm witha linewidth of 29.7 MHz corresponding to a lifetime of 5.4 ns, see Sec. 2.3.

3.3.1. Transversal cooling
After atoms exit the oven they are cooled in the transversal directions by two elliptical laser
beams. The ellipticity increases the overlap between the laser beam and the atom beam andmaximizes the amount of scattered light. Both laser beams are retro-reflected to form a twodimensional optical molasses. Each beam has a total power of about 40 mW and the lightis generated by a slave diode laser, seeded by the blue master laser, see Sec. 3.5.1. The light
has a detuning of −10 MHz to the atomic resonance, which was determined by maximizing
the MOT loading rate. Under optimal conditions, the TC increases the MOT loading rateby a factor of up to three to four.

3.3.2. Zeeman slower
The ZS reduces the longitudinal velocity of single atoms by exerting a strong light force
in opposite direction of the atom’s direction of propagation. This is achieved by shining alaser beam in collinear and counter-propagating direction to the atom beam. To keep atoms
resonant with the light during the full slowing process one exploits the Zeeman shift of atomic
energy levels in a magnetic field. The ZS efficiently reduces the atom’s longitudinal velocity
from a maximum velocity, so-called capture velocity, vcZS , down to a final velocity vf . The
capture velocity is given by√
vcZS = 2aZS x0 ,3.1

13Here, slow means a velocity close to the capture velocity of the MOT as depicted in Sec. 3.3.3.

33

3.3. Slow atom beam with high flux(a)
400

comp 1comp 2bias
profilecoil start
coil end
comp 1comp 2
total
ideal

profile300

coil start

MOT

200

Bx (G)

bias

100

coil end

x

0

MOT

-100
-200

(b)
0.6

η

0.4
0.2
0.0

0

100

200

300
x (mm)

400

500

600

Figure 3.5.: Increasing-field spin-flip Zeeman slower for erbium and lithium. (a) Six separately
controllable coils (inset) are used to create the necessary magnetic field. Utilizing a bias coil (blue)
for creating a homogeneous magnetic field the capture velocity of the ZS can be tuned (see text). The
resulting total magnetic field (red) is calculated and approximates the ideal field (light-red dashed)well except at the beginning and the end of the ZS. Two compensation coils before and after the mainchamber (comp 1 and comp 2) are used to reduce the residual magnetic field and its gradient to zeroat the position of the MOT (gray dashed). (b) The resulting value for η calculated for erbium (solidline) using the total magnetic field from (a). Its mean value (dashed line) is as large as 0.34.

where aZS is the actual deceleration acting on the atoms within the ZS and x0 is the total
length of the ZS. The capture velocity predominantly determines the flux of slow atoms
leaving the ZS as it gives the upper velocity bound in the Maxwell-Boltzmann distributionof the atom beam coming from the high-temperature oven. The thermal velocity distribution
is shown in Fig. B.3 in Appendix B.
The ZS in the erbium experiment is an increasing-field spin-flip ZS with a length of x0 =360 mm, see Fig. 3.5. It is designed for a vcZS for erbium of about 400 m/s, which is close to the
maximum of the thermal velocity distribution. By using a spin-flip ZS it is possible to have alarge capture velocity with moderately low absolute values of the magnetic field. Currently,
the ZS is operated with vcZS = 325 m/s, which is limited by the available light intensity fromthe injection-locked laser, see Sec. 3.5.1. Furthermore, the ZS can be simultaneously used toslow lithium as a second species. The general principle of operation is similar to existing ZSs
for many other elements, see Refs. [Lis99, Kro02, Har03, Gün04, Wil09]. A major differenceof the ZS in the erbium experiment to other setups is the ability of changing its capture
velocity in a wide range by the utilization of an additional bias field coil. With such a
coil setup the slowing efficiency can be easily optimized to its maximum, see Sec. 3.3.3. A
consequence of changing the bias field is that the position of the zero-crossing of the magnetic
field moves along the axis of the ZS.

34

3.3. Slow atom beam with high flux

In the following, we will summarize the general working principle of the ZS before we discuss
the coil setup and the operational parameters of the erbium ZS. For a maximum slowing
efficiency atoms with transition frequency ω0 have to be in resonance with the laser light offrequency ω over the full extension of the ZS. During the slowing process the Doppler shiftof the atom changes as the atom’s velocity, v(x), changes as a function of position x. The
Doppler shift is compensated by a Zeeman shift induced by the external magnetic field B(x).
Thus the resonance condition equals
ω0 − kv(x) +µ′ B(x)
=ω .
~

3.2

Here, µ′ is the difference of the magnetic moment between the ground and excited state given
by µ′ = (ge me − gg mg )µB with magnetic quantum numbers mg , me and Landé g-factors gg ,ge , see Sec. 2.4.3. For the blue transition µ′ = 1.13719 µB and k = 2π
λ is the wavenumber of
the transition with wavelength λ. The maximum possible deceleration, amax , which is givenby the light force acting on the atom, depends on the transition parameters as
amax =

~kΓ
,
2m

3.3

with Γ the natural linewidth of the transition and m the atomic mass. For erbium this gives
amax = 6.6 × 105 m/s2 . A constant deceleration of atoms in the ZS leads to a square-root
dependence of the velocity on the position along the ZS by
r
x
ZS
1−
v(x) = vc.
3.4x0
The necessary ZS magnetic field can therefore be written asrx.
B(x) = Bb + B0 1 −x0

3.5Here, Bb is a magnetic bias field, which can be understood as a shift of the square-root-likemagnetic field profile, and B0 is the overall height of the profile. Utilizing Eq. 3.2 these twoparameters can be calculated by
Bb =

~δ
µ′

and B0 =~kvcZS,µ′3.6

where δ = ω − ω0 is the detuning of the ZS laser light. The required total length of the ZS
to slow atoms from vcZS to a final velocity, vf , is simply given by
x0 =(vcZS − vf )2
.
2aZS

3.7

The number of scattering events needed for the slowing process can be calculated by
Nsc =m(vi − vf )
,
~k

3.8

with vi the initial velocity, which can be in general smaller than vcZS . This results for the ZS
of erbium in Nsc = 5.4 × 104 for an assumed vf = 5 m/s and vi = vcZS = 325 m/s. Atoms
which enter the slower with vi < vcZS need a smaller number of scattering events.

3.3. Slow atom beam with high flux

35

According to Eq. 3.3 one wants to have large values of Γ and k, i. e. small values of λ, fora large maximum deceleration. The broad 4f 12 6s2 1 S1 - 4f 12 6s6p 1 P1 transition fulfills both
requirements with λ = 401 nm and Γ = 2π × 29.7 MHz, see Sec. 2.3. At a first glance, thelarge number of metastable states in which the excited 4f 12 6s6p 1 P1 state can decay into,
see Fig. 2.1, might suggest that this transition is not suitable for slowing at all. However, acareful analysis of the branching ratio of the excited state shows different. The value for the
branching probability into metastable states is 7.7 × 10−6 , see Ref. [McC06b]. Comparing
this with the required number of scattering events from Eq. 3.8 shows that on average 1/3of the atoms in this velocity class decays into metastable states and can not be loaded into
the MOT. For lower initial velocities the loss fraction decreases, e. g. for vi = 150 m/s theloss fraction is only 18 %. Although this loss fraction seems large it can be easily overcome
by slightly increasing the oven temperature to have a sufficiently large flux of slow atoms forloading the MOT14 .
From the right-hand-side expression in Eq. 3.6 one can easily see that a larger B0 increases
the capture velocity of the ZS and by this also the atom flux. As mentioned earlier, the main
advantage of a spin-flip ZS is the ability to achieve large values of B0 with a moderate set ofcoils as the absolute magnetic fields on both ends of the ZS can be smaller than compared toa non spin-flip ZS. This is achieved by shifting the magnetic field by Bb , see ideal magnetic
field in Fig. 3.5(a). To minimize the light force of the ZS laser beam on atoms which are
already loaded into the narrow-line MOT the detuning of the ZS light should be as large as
possible. Following Eq. 3.6, the detuning δ also fixes the value of Bb . In the experiment, adetuning of about −530 MHz corresponding to δ = −17.8Γ is used that results in Bb = 333 G
and B0 is tunable by the bias field coil.
If there are too strong deviations from the ideal magnetic field given in Eq. 3.5, it can easily
happen that atoms will not be resonant with the ZS light and are not contributing to theslowing process any more. It is therefore advisable for the design of the ZS to assume adeceleration, aZS , not larger than about 2/3 of the maximum possible deceleration15 . Thisratio is known as deceleration parameter and is written as
aZS
η=
,
3.9
amax
where 0 ≤ η ≤ 1. If the atomic transition is not fully saturated η has to be chosen evensmaller due to the limited light force acting on the atoms. Its upper limit is then set bys0
η < 1+s
with the saturation parameter s0 = IIS . In the erbium experiment the current
0
maximum power of the ZS beam is limited to about 90 mW, which gives s0 = 5 for a beam
waist of the ZS laser of 4 mm. This sets the maximum value of η to 0.83. To ensure reliableslowing at the beginning of the ZS where atoms are fastest the laser beam is slightly focused
to increase the intensity at this point and make a larger η possible. Figure 3.5(b) shows thedependence of η over the full length of the ZS calculated for the total magnetic field.
The coil setup of the erbium ZS is as follows. In total six independently supplied coils create
the required magnetic field, see Fig. 3.5(a). Most important is the profile coil, which roughly
An estimation of the flux of atoms, which can be captured by the MOT, gives a value of 4 × 108 s−1 ata temperature of the oven effusion cell of 1100 ◦ C, see Ref. [Sch11]. Increasing the temperature by 150 ◦ Cwill increase the flux by one order of magntidue.
15
The factor of 2/3 is chosen from the experience of previous experiments.

14

36

3.3. Slow atom beam with high flux

creates a square-root-like magnetic field according to Eq. 3.5. This is achieved by a variable
number of windings dependent on the position along the ZS tube. Secondly, a cylindricalbias coil provides the homogeneous bias field for shifting the profile magnetic field. Two
additional coils are located at the beginning and at the end of the ZS. They are constructed
such that the magnetic field drops off as fast as possible when moving away from the ZS16 .
Another two coils are used to compensate any residual magnetic field and magnetic field
gradient to zero at the MOT position. Figure 3.5(a) shows the magnetic field created byeach of the coils and the total magnetic field for η = 0.35. To tune η the electric current
through each of the six coils has to be changed accordingly, which results in a shift of theposition of the zero crossing. The value of η can thus be changed from 0.1 to 0.5, whichmakes the ZS highly versatile.
We wind the ZS coils using standard enameled copper wires with a special coating for hightemperature resistivity and long-term stability17 . This is important as the ZS coils havebeen mounted before performing the bake-out of the UHV chamber. This high-grade copper
wire easily withstands the bake-out temperature of 200 ◦ C for a time much longer than two
weeks. The electrical isolation of low-grade copper wire would degrade at this temperature
within a few hours and electrical isolation would not be guaranteed anymore.
Next we will estimate the usability of the ZS for two different atomic species and give thefinal ZS operating parameters. Equations 3.1, 3.3, and 3.6 can be combined for an expressionof the deceleration parameter, which can be written asη=mµ′2 B02.
x0 Γ~3 k 3

3.10

An efficient multi-species ZS for two atomic species A and B can be constructed only if η issimilar for both species, see Ref. [Wil09]. This can be estimated by the ratio
3mA µ′2
ηA
A kB ΓB=,
3
ηB
mB µ′2
B kA ΓA

3.11

In case of a dual-species ZS for erbium and lithium ηEr /ηLi = 1.091. Thus it is possible touse the same ZS for both elements and slowing should work efficiently. The final parametersof the currently operating ZS for erbium are
vcZS = 325 m/s vf = 5 m/s

Bb = 333 G

B0 = −627 G η = 0.34 δ = −17.8 Γ .

Assuming the same magnetic field parameters, Bb and B0 , this would result in the following
parameters for a ZS for lithium with
vcZS = 600 m/s

vf = 100 m/s η = 0.31

δ = −83 Γ ,which is very similar to other slowers used for lithium, see Refs. [Joc04, Wil09].

16
17This is achieved by a short coil geometry and utilizing a larger current than that for the other coils.
wire class W200, manufactured by Synflex Elektro GmbH37

3.3. Slow atom beam with high flux
purely transversal velocity
2

mirrorZS off

atom
beam
45°

ZS beam

1

90°

ZS on

Gaussian fit-0.2 m/s
centerwidth
1.6 m/sCCD

signal (arb. unit)

(a)

probe

z
x

y

0

(b)

transversal and longitudinal velocity
ZS off

atom
beam

x
y
in plane

45°

4

ZS beam
45°

2

ZS on

Gaussian fit
8.4 m/scenterwidth
5.8 m/sCCD

signal (arb. unit)

6

probe

0
-10

0
10
20
velocity (m/s)

30

-10

0
10
20
velocity (m/s)

30

Figure 3.6.: Velocity distribution along probing direction oriented (a) purely transversal and (b)under an angle of 45◦ with respect to the atom beam (see insets). The measurement was done in thecenter of the main chamber using a CCD camera for imaging the fluorescent light. The effect of the
Zeeman slower (ZS) is clearly seen when turned off (left panels) and on (right panels). The centerof the transversal velocity distribution with the ZS switched off is used as a marker for zero-velocity
calibration. With turned-off ZS the velocity distribution in (b) is outside the shown range at muchlarger values because the thermal longitudinal velocity of atoms exiting the oven is about 500 m/s.
When the ZS is turned on a distribution of atoms with a maximum at 8.4 m/s can be observed, seeGaussian fit (dashed line). A second peak is appears at negative velocities, which is caused by atoms
being stopped by the ZS light and being accelerated back in direction of the oven. Note that also
the transversal velocity distribution gets slightly changed and deviates from the Gaussian fit. This isdue to transversal heating when the ZS is turned on.

3.3.3. Velocity distribution
For optimizing the operational parameters of the ZS we measured the velocity distribution of
erbium atoms after the slowing process in the center of the main chamber. Narrow-linewidth
light at a wavelength of 583-nm was shined on the slow atom beam and the fluorescence signal
was recorded on a CCD camera. The signal was achieved by integrating pixel values withina small region of interest of about 1 × 2 mm2 around the atom beam. Figure 3.6 shows asketch of the experimental setup and the measured velocity distributions. We oriented theprobing laser beam either vertically, i. e. under an angle of 90◦ to the atom beam, or within
the horizontal plane under 45◦ with respect to the atom beam, see insets in Fig. 3.6. The
former setup allowed to probe only the velocity component in transversal direction whereas
the latter gave information on both transversal and longitudinal components. The beam
in vertical direction was retro-reflected and used as a zero-velocity marker for calibration
purposes.

38

3.4. Coil setup at main chamber

The 401-nm transition in erbium has a linewidth of 29.7 MHz. This is too large to probe
the velocity distribution with a desired small resolution of maximal 1 m/s. The narrow-line
transition at 583 nm provides a better velocity resolution of 0.1 m/s. For this transition theexperienced velocity-dependent Doppler-shift of the atoms, δν, is given byδν = v

ν0
c

⇒

m/s
v
c
= λ = 0.58
=
,
δν
ν0
MHz

3.12

where ν0 is the transition frequency and c is the speed of light. As the frequency of the
probing laser could only be detuned within ±60 MHz using a non-specialized AOM setup
the velocity distribution could be recorded between −35 m/s and +35 m/s. The velocity
distribution in Fig. 3.6(b) shows a clear effect of the ZS as a fluorescence signal appears only
when the ZS light is turned on. A fitted Gaussian curve gives a center velocity of 8.4 m/s
with a full-width-half-maximum of 13.4 m/s. This nicely covers the capture velocity of thenarrow-line MOT of 5.6 m/s, refer to Sec. 6.1.1 and Eq. 6.6. With the ZS turned off thevelocity of the atoms is too large and the Doppler shift is larger than the tuning-range of
the AOM and thus no signal could be recorded.
The purpose of this measurement was to verify whether atoms with a velocity comparable tothe capture velocity of the MOT are indeed available in the main chamber. An optimizationof the detuning of the slowing light and the magnetic field was carried out. Final fine-tuningof the ZS was performed by maximizing the atom number and the loading rate of the MOT.
Only a small change of the initial values was necessary. These results were promising forsetting up the first erbium MOT on the narrow-line transition. The setup of the narrow-line
MOT will be described in Chapter 6.

3.4. Coil setup at main chamber
Magnetic fields play a very important role in ultracold gas experiments. Due to the Zeeman
shift of atomic energy levels, see Sec. 2.4.3 and 2.4.4, and a spin-dependent force acting
on atoms in a magnetic field gradient (so-called Stern-Gerlach experiment) atoms can beconveniently manipulated from outside the vacuum chamber using magnetic fields. At thesame time this implies that atoms with a large magnetic moment are very sensitive tounwanted but persistent magnetic field noise.
In particular, three major coil setups are used in the erbium experiment:• Zeeman slower coils for operating the ZS, see previous description in Sec. 3.3.2.
Regarding the main chamber there are two coils utilized to compensate the magnetic
field and its gradient generated by the ZS at the center position of the main chamber.
• Main coil setup consisting of gradient coils for creating a magnetic field gradient, e. g. during MOT operation and Stern-Gerlach experiments, as well as low- andhigh-field coils for a homogeneous magnetic field, e. g. for Feshbach spectroscopy, seeSec. 3.4.1.

3.4. Coil setup at main chamber

39• Compensation cage for compensating the Earth’s magnetic field as well as otherexternal magnetic fields, e. g. from the static magnets of ion getter pumps. Furthermore, it can be used to rotate the magnetic field with the possibility of leaving its
absolute value constant, see Sec. 3.4.2.
The main chamber and attached parts are made of stainless steel with extremely low magnetic
permeability18 . This prevents any residual magnetic stray fields generated by the vacuum
chamber after ramping between high and low magnetic fields. The largest amount of periodic
magnetic field noise is introduced by the mains electricity with a characteristic frequencyof 50 Hz and higher harmonics. Other strong non-periodic noise sources come mostly fromneighboring experiments and a mechanical workshop close-by. This magnetic field noise
easily turns into a major technical problem especially if the magnetic field should be reduced
to very small values close to zero. In this case, an active feedback loop with an ultra-lownoise magnetic field sensor19 has to be used. Furthermore, a high relative magnetic fieldstability on the order of 10−4 even at large absolute values of the magnetic field is desired for
Feshbach spectroscopy. This is achieved by using an actively stabilized current source, which
relies on a homemade analog feedback loop with an ultra-stable current transducer20 .

3.4.1. Main coil setup
The main coil setup is used to create homogeneous magnetic fields as well as gradient fields bya set of three coil pairs. Each pair of coils consists of an upper and lower coil. Figure 3.3 showsa schematic overview of the coil geometries and Fig. 3.7 gives a plot of the corresponding
magnetic fields, i. e. the total magnetic field Btot (ri ) and its gradient ∂i Btot = ∂Btot /∂ri
with i = x, y, z, calculated for a unit current of 1 A. All three sets of coils are made up ofenameled copper wire21 placed in a copper casing. For thermal conductivity, the coils arevacuum moulded into the casing using thermally conductive resin22 .
The main coil system is cooled by a water pocket around the perimeter of the copper casing,see Fig. 3.3. Temperature sensors are placed inside and outside the coils as well as inside
the copper casing for monitoring purposes. An automatic safety system, the so-called interlock system, switches off the coil power supplies when the coil temperature exceeds a given
threshold.
The magnetic fields created by the gradient and the high-field coils can be switched off
in a short time by high-current IGBT switches23 . We measured the switch-off-time of themagnetic field utilizing a pick-up coil. The time is limited to about 5 − 6 ms due to eddy
currents in the steel chamber. Every pair of wires supplying a single coil is twisted as tightas possible and positioned symmetrically with the other pair of wires of the opposite coil18

µr < 1.005, see Ref. [Lee05], for stainless steel code 316LN (after AISI)√or 1.4429 (after ESU)
model Mag-03-MS-70, three-axis magnetic field sensor, noise < 10 pT/ Hz, from Bartington Instruments
20
ITN 12-P, 12.5 A measuring range, 0.0509 % accuracy, from LEM
21
2.50 × 1.00 mm2 W200 copper wire, outer dimension 2.96 × 1.22 mm2 , from Synflex Elektro GmbH
22
type RAKU-POX 22-G110/7-7, two-component electro casting resin, from RAMPF Holding GmbH & Co
KG
23
dual IGBT module 2MBI100U4A-120-50, Ic = 100 A and Uce = 1200 V, from Fuji Electric

19

10

6

grad

grad

|Btot | (G)

0.3

x, y
z

8

|∂iBtot | (G/mm)

(a)

4
2
0

(b)

0.2

0.1

0.0

16

|∂iBtot | (G/mm)

10
8
6

FB

14.29114.290414.289

2

-8 -4

0

0

4

8

-100 -75 -50 -25 0
25
position (mm)
(c)
478.4

50

75

100

477.6

0.2
0.1

-100 -75 -50 -25 0
25
position (mm)
1.6

lf

477.2
476.8
476.4
-1.5

0.3

1.4

x, y
z

478.0

x, y
z

0.4

0.0

|∂iBtot| (mG/mm)

FB|Btot | (G)12|Btot| (mG)

0.5

x, y
z

14

lf

x, y
z

50

75

100

x, y
z

1.2
1.0
0.8
0.6
0.4
0.2
0.0

-1.0

-0.5
0.0
0.5
position (mm)

1.0

1.5

-1.5

-1.0

-0.5
0.0
0.5
position (mm)

1.0

1.5

Figure 3.7.: Magnetic field strengths generated by the main coil system. The absolute values of thetotal magnetic field Btot (ri ) (left) and its gradient ∂i Btot = ∂Btot /∂ri with i = x, y, z (right) along
the x-, y-, and z-direction (colors) for a unit current of 1 A are shown. (a) Magnetic field generated
by the gradient coils (grad) for MOT operation and Stern-Gerlach experiments. (b) Homogeneous
magnetic field induced by a set of coils in Helmholtz configuration for high-field Feshbach (FB)
spectroscopy. (c) Magnetic field of low-field coils (lf) usable up to a maximum field of 4.5 G. Zeroposition corresponds to the center of the main chamber.

40

3.4. Coil setup at main chamber

41

in respect to the center of the main chamber. Doing so, the magnetic field created by thecurrent running into the coil cancels with that of the current returning from the coil. This
is intended to reduce any residual magnetic fields created by the power supply wires.

Gradient coils Two coils in anti-Helmholtz configuration create a magnetic field gradientin every spatial direction, i. e. by creating a quadrupole magnetic field. In this configuration
the gradient in x- and y-direction is half as large as the one in z-direction. Each coil is madeup of six layers with 14 windings each, which makes it possible to use a power supply24 at amoderate output current, i. e. below 50 A reducing thermal heating. The conversion factor
for the magnetic gradient field is about 2.698 G/cm
in z-direction. The maximum gradient isAabout 80 G/cm, which is limited by thermal heating at a maximum current of 30 A. The total
resistance of the gradient coils is 440 mΩ. At maximum current the gradient coils dissipate
a total power of 400 W in form of heat, which is removed by the water-cooling system.

Feshbach coils The high-field coils are capable of creating a homogeneous magnetic field of
up to 500 G. Due to its Helmholtz configuration the spatial homogeneity of the magnetic field
is better than 0.2 mG/mm at the center of the main chamber, see inset of Fig. 3.7(b). The
calculated conversion factor for the Feshbach coil is 14.290 G/A. The maximum current islimited to 35 A and the total resistance is 430 mΩ. This corresponds to a dissipated power ofabout 530 W. At maximum current the temperature at the center of the coil stack increases
to about 40 ◦ C but due to the water-cooling system the temperature of the copper casing does
not exceed 25 ◦ C. We measured the frequency response of the high-field coils revealing a cutoff frequency of about 500 Hz. High-current relays25 make it possible to switch between two
different power supplies, one for a high and one for a low current operation. The first power
supply26 is capable of driving the maximum current whereas the second power supply27 islimited to about 5 A giving a maximum magnetic field of 70 G. In both cases the current is
actively stabilized using a current transducer28 and a PID control loop to achieve a relative
short-term stability of the magnetic field of better than 2 × 10−4 for the low-current power
supply. The magnetic field of the Feshbach coils was calibrated using atomic radio-frequency(RF) transitions from mJ = −6 to −5. The average magnetic field noise is estimated to be
40 mG for the high-current power supply and 5 mG for the low-current power supply. In bothcases the main contribution comes from the power supply itself, which could be replaced bylead-acid batteries for further noise reduction.

Low-field coils A pair of two small coils consisting of two layers with four windings eachcreates a magnetic field of up to 5 G. The conversion factor is 477.22 mG
A and the maximum
current should not exceed 10 A. Although they are not in perfect Helmholtz configuration
the coils have a calculated homogeneity of about 0.9 mG/mm. For a typical cloud size of
20 µm this would in principle give a spatial deviation of less than 18 µG, which is much less24

switching power supply, model SM 18-50, from Delta Elektronika
model G9EA-1-B-24VDC, for 60 A and 400 VDC, from Omron Corporation
26
switching power supply, model SM 15-200 D, from Delta Elektronika
27
linear power supply, model EA-PS 1032-10 B, from Elektro-Automatik
28
IT 60-1000 S (high current) and ITN 12-P (low current), from LEM
25

3.5. Laser light setup

42

than the magnetic field noise. The low-field coils are supplied by a linear power supply27 with
low noise but an active stabilization of the current is not implemented yet. RF-calibration
yields a total magnetic field noise of about 600 µG. A high precision feedback loop would benecessary for performing low-field experiments below 100 µG.

3.4.2. Compensation cage
Three sets of rectangular-shaped coils forming a rectangular cage are used to compensate any
residual magnetic field in x-, y-, and z-direction. An external field is for instance generatedby Earth and the permanent magnets of the ion pumps. The geometric size of the cage is1200 × 1200 × 850 mm3 . The center of the cage overlaps with the center of the main chamberin x- and y-direction but is shifted by 250 mm in positive z-direction. The coils are composedof 40 windings in x- and y-direction and 30 windings in z-direction. This results in a total
inductance of about 15 mH for each set of coils.
Additionally, the compensation cage can be used to change the direction of the magnetic
field even while keeping the value of the absolute magnetic field constant. Such a magnetic
field rotation is limited to absolute magnetic fields smaller than 1.2 G. In the experiment
this allows for changing the polarization direction of the magnetic dipoles with respect tothe trapping geometry by rotating the orientation of the magnetic field. Due to the large
inductance of the coils the frequency response function shows a high-frequency cut-off at
about 1 kHz. This limits the maximum speed, at which the magnetic field can be rotated,
to about 15 ◦ /ms.

3.5. Laser light setup
For the production of quantum gases of erbium we use laser light at a wavelength of 401 nm(blue light, see Sec. 3.5.1) for ZS, TC, and imaging and light at 583 nm (yellow light, seeSec. 3.5.2) for magneto-optical trapping. Additionally, for optical dipole traps we employ
far-off-resonance laser light at a wavelength of 1064 nm and 1570 nm, see Sec. 3.5.3.

3.5.1. Blue laser system
For the 401-nm light we use a master laser and two slave lasers. Figure 3.8 illustrates the
optical setup for the master laser whereas Fig. 3.10 shows the optical setup for the slave lasers.
The master laser is utilized for the locking setup, seeding the slave lasers, and imaging atoms
in the experiment. It is a commercial laser29 . The light of a diode laser operating at 802 nm
is first tapered-amplified and then frequency-doubled in a bow-tie ring cavity. The maximum
output power of the master laser is 210 mW from which 80 mW are needed for operating theexperiment, see Appendix C. For the ZS- and TC-light injection-locked slave lasers provide
29

TA-SHG-pro system, from Toptica Photonics AG

3.5. Laser light setup

43

a high optical power on the experiment table. They are seeded by light coming from themaster laser through optical fibers. The master laser is locked by the modulation-transfer
technique to a spectroscopy cell in form of a hollow-cathode lamp (HCL). We observe about
40 % absorption in the HCL for typical operating parameters of a voltage of 110 V at adischarge current of 6 mA. The spectroscopy signal contains a clear signal of the transitionsof all bosonic isotopes as well as features coming from the hyperfine structure of the fermionicisotope. The modulation-transfer technique creates a signal, which can be conveniently used
for laser locking. It is described in Sec. 8.1.2.
In total, five acousto-optic modulators30 (AOMs) are used to shift the laser light frequency tothe desired values, see Fig. 3.8. The master laser is locked on the laser frequency necessary for
the ZS. This is achieved by shifting the light used for the spectroscopy cell with one AOM in
double-pass (DP) configuration by an amount equal to the ZS-light detuning of −530 MHz.
For fermions the spectroscopy signal of the hyperfine transition mF = −19/2 → −21/2
is weak and sits on top of the broad background signal of the bosonic isotope 166 Er, see
Chapter 8. Therefore we lock the laser on the bosonic transition and use an additional DP
AOM31 to shift the light by another +150 MHz for experiments with fermionic 167 Er.
Because the laser is locked away from the atomic resonance another DP AOM can be used
for shifting the light back on or close to resonance with a variable detuning. This is done
to prepare light for TC, which needs to have a detuning of about −10 MHz with respect
to the atomic transition. The final value of the detuning was determined empirically by
maximizing the improvement of the MOT loading rate when TC is switched on. The lightfor absorption imaging is shifted close to resonance in a similar way with a detuning suitable
for the current experiment32 . Finally, for selectively removing atoms from an atom-molecule
mixture a weak blowing (or pushing) beam is prepared. It is tuned exactly on resonancewith the atomic transition to have a largest possible light-force acting on the atoms during
the blowing sequence. Usually a pulse with a duration of 1 µs and a beam power of 2 mW
giving an intensity on the order of the saturation intensity is sufficient to remove all atoms
from the sample.

Injection-locked lasers
Two injection-locked diode lasers are used as slave lasers on the experiment table, seeFig. 3.10. Both slave lasers are homemade and share a similar setup. Each one of themdelivers 80 mW of laser light after seeding them with about 2 mW from the master laser.
The first slave laser is used for the ZS and the second one for TC. In this setup only a low
power has to be carried via optical fibers from the laser table to the experiment table. This
makes high-power fibers unnecessary resulting in a larger long-term stability of the laser
system.

30

For the blue laser system mostly AOMs, model BRI-TEF-270-100-401, from Brimrose Corporation areused, which have a center frequency of fc = 270 MHz and a 3 dB-bandwidth of B = 100 MHz.
31
model BRI-TEF-80-20-583, with fc = 80 MHz and B = 20 MHz, from Brimrose Corporation
32
Off-resonant light is used for imaging optically thick atomic clouds to prevent saturation effects.

14MHz

EOM

HCL

λ/2

-50
100

DP 270MHz

AOM

spectr. 2

λ/2

300

ZS seed

200

λ/4

AOM
DP 270MHzTC

DP 270MHz

AOM

blow

λ/2

λ/2

300

300

75

-50

λ/2

λ/2

λ/2

λ/2

300

300

100

100

λ/2

λ/2

λ/2

λ/2

DP 270MHz

AOM

spectr. 1

-50

-50

λ/4

200

λ/4

802 nm
10 mW

TA-SHG-pro

DP 270MHz

AOM

imaging

300

TC seedblue blow

wavemeter

EOM

AOM

λ/2 λ/4

100 -50

fiber-coupler

electro-optic
modulator

acousto-optic
modulatoriris

polarizing
beam-splitter

waveplatelensmirrorFigure 3.8.: Blue laser setup for the 401-nm transition. A frequency-doubled, tapered-amplified diode laser operating at 802 nm (TA-SHG-pro) isused as a master laser (blue lines). Light for seeding the Zeeman slower (ZS) and transversal cooling (TC) slave lasers, absorption imaging, andresonantly blowing atoms is prepared by frequency-shifting using acousto-optical modulators (AOMs). The master laser is locked to a spectroscopy
cell using modulation transfer spectroscopy. It operates off-resonance with a detuning given by the ZS light. Two AOMs in double-pass (DP)configuration are utilized for shifting the laser frequency to resonance for spectroscopy (see text).

PD

λ/2

300

spectroscopy setup

λ/4

200

imaging

λ/4

401 nm
80 mW

3.5. Laser light setup

45

For the slave laser a wavelength-selected single-mode laser diode33 is stabilized in temperature34 and is powered by a constant current source up to 120 mA. For suppressing unwanted
back-reflections into the laser diode the light passes a polarization-dependent optical isolator35 , which is simultaneously used to couple in the light coming from the master laser.
To ensure single-mode operation of the slave laser some light is picked-off and sent to ahomemade wavemeter.
In the case of the ZS the laser beam from the slave laser is first expanded using two telescopes
to a diameter of more than 30 mm before a single lens with a focal length of 1000 mm focuses
the light into the vacuum chamber. Here, the focus is situated at the high-temperature oven.
A standard quarter-wave plate sets the polarization of the beam to circularly polarized.
For TC the laser beam is first elongated to an aspect ratio of about 10 : 1 using cylindricallenses before it is split into two beams. The large aspect ratio maximizes photon scatteringby simultaneously having a high intensity and a long interaction time with the atom beam.
Each laser beam is retro-reflected to form the two-dimensional optical molasses.

Imaging setup
Absorption imaging of the atom cloud can be done either in the horizontal or the verticaldirection, see Fig. 3.11. The imaging laser light from the blue laser setup is either sent into thevacuum chamber through a CF16 flange for horizontal imaging or is recoupled into anotherfiber for vertical imaging. The second fiber carries the light below the vacuum chamber
before it enters the vacuum chamber through the lower CF40 window. In both cases thelight is circular polarized by quarter-wave plates in front of the windows.
For horizontal imaging an objective with a magnification of 1.9 creates an image on the
CCD sensor36 . The numerical aperture is limited by the geometric size of the CF16 flange toabout 0.1 giving a theoretical resolution limit of 2.5 µm. The objective for vertical imaging
has a magnification of 2.5 and creates an image on a second CCD sensor37 . In this case thenumerical aperture is determined by the objective and has a value of about 0.08.

3.5.2. Yellow laser system
We use a continuous-wave ring-cavity dye laser38 with rhodamine 6G as an active medium
for a laser light source for the MOT operating at a wavelength of 583 nm. The dye laser is
33model NDV4313 from Nichia Corporationby the temperature controller WTC3243, from Wavelength Electronics
35
model IO-5-405-LP, from Thorlabs
36
scientific camera model iXon3 897, back-illuminated AR-coated EMCCD, with 512 × 512 pixel resolution,
16 × 16 µm2 pixel size, 14 bit digitization, from Andor Technology
37
scientific camera model Luca R 604, AR-coated EMCCD in OEM housing, with 1004×1002 pixel resolution,
8 × 8 µm2 pixel size, 14 bit digitization, from Andor Technology
38
CW ring dye laser, from Radiant Dyes
34

3.5. Laser light setup

46

pumped by a 10 W solid-state laser39 at 532 nm. The bare dye laser has a maximum output
power of 1 W, a linewidth of about 1 MHz, and a frequency drift of 100 MHz/h.
For MOT operation the linewidth has to be decreased below 200 kHz, which is given by thenatural linewidth of the yellow transition, and a frequency-drift lower than a few kHz/h has
to be achieved. The reduction in linewidth is accomplished by introducing an electro-opticalmodulator (EOM) into the cavity of the dye laser for a fast feedback-loop with a response
frequency of up to a few MHz. The error signal is generated from a low-finesse cavity
using the Pound-Drever-Hall (PDH) method. We improve the frequency-drift and long-term
stability by locking the low-finesse cavity to an ultra-low expansion (ULE) cavity. The ULE
cavity consists of a ULE spacer40 and two fused silica mirrors, which are optically contacted
to the spacer. The cavity has a finesse of 1865(6) and a free spectral-range of 998.69(2) MHz,
which is similar to the isotope frequency shift of erbium. This turns out to be convenient forlocking to different isotopes of erbium. To achieve sufficiently high stability the ULE cavity
is put under vacuum and shielded from outside temperature fluctuations by two aluminum
shields. The temperature of the ULE spacer is stabilized to its zero-expansion temperatureof −5.65(9) ◦ C using four Peltier elements. Utilizing this locking setup the linewidth of thedye laser could be decreased to 45.0(5) kHz within 10 ms and the drift was measured to be580(70) Hz/h over a timescale of 40 days. A full description of the locking procedure andULE cavity setup can be found in Ref. [Rie12].
Figure 3.9 shows the optical setup of the yellow laser-system. It consists of the dye laser,
the locking setup including the low-finesse and ULE cavities, two DP AOMs for shifting thefrequency of the MOT light and yellow blow light, and the MOT distribution setup. The
output of the dye laser can be monitored on a wavelength-meter and a power-meter. The
dye laser has to be realigned whenever the dye needs to be exchanged. This realignment
would lead to a misalignment of the full optical setup. For convenient realignment a shortfiber is placed between the dye laser and the rest of the optical setup.
Error signals from the low-finesse and ULE cavities are generated by a standard PDH method
using EOMs for sideband creation. The error signal of the ULE cavity is fed to the piezo
crystal of the low-finesse cavity controlling the length of the low-finesse cavity. The error
signal of the low-finesse cavity is then split up in three signals of different frequency ranges
feeding three optical elements inside the dye laser cavity. The low-frequency part, whichgoes up to a few 100 Hz, drives a Brewster plate, the mid-frequency part, up to several kHz,
controls a cavity mirror mounted on a Piezo crystal, and higher frequencies up to a few MHz
are applied to the intra-cavity EOM. As the length of the ULE cavity is fixed the frequencyof the laser light used for locking has to be shifted by two DP AOMs41 such that the light
generated by the dye laser is on resonance with the atomic transition. The lock-shift setup
was designed for a frequency tuning range covering at least the free spectral-range of the
ULE cavity of about 1 GHz. One of the two AOMs is used for stabilizing the intensity toabout 100 µW. This ensures a constant and small heat input to the ULE cavity. The shifted
light is then carried to the cavity lock setup using an optical fiber and is modulated byanother EOM for the PDH technique.
39

Verdi V10, from Coherent
ULE premiumgrade, length 150 mm and diameter 60 mm, from Hellma Optics
41
model AOMO 3200-125 with fc = 200 MHz from Crystal Technology, Inc. (now Gooch & Housego)40

Verdi-V10

yellow blow

λ/2532 nm
5.6 W

λ/4
λ/4
MOT

AOM
80MHz

AOM

DP

80MHz

λ/2

DP

λ/2

blow

low-finesse
cavityMOT distr.

λ/4lock shift
λ/2

300

rhodamine 6G

λ/2

EOM

dye laserPDlock shift
λ/2

λ/2

λ/2

AOM

DP 200MHz

lock

AOM

λ/2
λ/2

583 nm> 600 mW

λ/4

DP 200MHz

lock

lock shift

powermeterwavemeter

λ/2

λ/2

λ/4

1000

λ/2
cavity lockcavity lockPD

MOT vert.

MOT hor. 1

MOT hor. 2

MOT distribution

λ/2

cavity lock

ULE cavity

λ/2

λ/2

λ/2
λ/2

31MHz

λ/2

PD
MOT distr.

λ/2

EOM

λ/2

λ/4

λ/2

Figure 3.9.: Yellow laser setup for the 583-nm transition. Here, the light source is a dye laser
operating with rhodamine 6G (yellow), which is pumped by 532-nm light (green). The light for
the three MOT beams can be detuned using an AOM. Furthermore, resonant light is prepared forblowing atoms out of the optical dipole trap. For narrowing the laser linewidth lower than the naturallinewidth of the atomic transition an intra-cavity electro-optical modulator (EOM) is utilized (notshown). The laser is locked to a low-finesse cavity, which length is locked to an ultra-low-expansion(ULE) cavity using the Pound-Drever-Hall technique.

47

401 nm
80 mW
pumping

1000

λ/2

λ/2
200
-100

λ/2

50

ZS seed

-50
λ/4

200

100

wavemeter

0

30

4

λ/

T

O

M

r.
1

ho

300

λ/4

λ/4

4

λ/

-50

150

wavemeter

vertical MOT beam(out of plane)

λ/2

cyl.

10

MOT vert.

Zeeman slower

cyl.

100

λ/2

λ/2

TC seed

4
λ/

λ/

4

4

TC light

λ/transversal cooling(out of plane)

4

slave
laser 2

λ/oven

Figure 3.10.: Overview of the laser setup for transversal cooling (TC), Zeeman slowing (ZS), and MOT beams. The vacuum chamber is schematicallyshown in a top view. The retro-reflection of the MOT beams (yellow) is done in the way such that highest intensities are achieved in the direction
against the atom beam as well as against gravity. This ensures maximum loading efficiency into the MOT. The injection-locked slave lasers generate
about 80 mW each (blue). The TC beams are highly elliptical and are displaced from each other in direction of the atom beam. This maximizes thescattering force on the atoms.

periscope

p

pe

co

is
er

e

op

0

is
c
4

λ/

30

pe
r

MOT light

4

1
r.

ho

slave
laser 1
λ/

T

O

M

ZS light

pumping

401 nm
80 mW

3.5. Laser light setup

49

The frequency of the MOT light can be detuned by an AOM42 in double-pass configuration.
Afterwards, the light is carried to the MOT distribution board in a single fiber. After thefiber a photo-diode monitors the light intensity for intensity stabilization utilizing the AOM.
The light is then split up in three beams, which are carried to the experiment table inthree separate fibers. Two beams are used for the horizontal MOT direction and the thirdfor the vertical direction. On the experiment table each beam is increased to a diameterof more than 20 mm using a telescope. This increases the capture velocity of the MOT
according to Eq. 6.6. The enlarged beams are then brought close to the vacuum chamberusing periscopes, see Fig. 3.10. All beams are circularly polarized by achromatic true zeroorder quarter-wave plates43 before entering the vacuum chamber and are retroreflected on
the opposite side of the chamber. The minimum total output power of the dye laser has to
be at least 600 mW otherwise the intensity of the MOT beams is not high enough for MOT
operation. Typically intensity values at the vacuum chamber are as high as 12 IS for eachbeam with IS = 0.13 mW/cm2 the saturation intensity of the transition.

3.5.3. Optical dipole traps
After laser cooling atoms in the MOT they are usually loaded into a red-detuned optical
dipole trap (ODT) comprising of various trap geometries and laser wavelengths. A so-calledscanning setup was implemented to achieve a highly efficient loading from the MOT into
the ODT. In this setup the laser beam is deflected in the horizontal, transversal direction
by an AOM with a frequency much larger than any trapping frequencies. This creates atime-averaged optical potential, which can be changed by tuning the amplitude and speed ofdeflection. Using the scanning system the ODT can be increased in size to have a good spatial
overlap between atoms in the MOT and the ODT. This optimizes the loading efficiency toabout 35 %. Furthermore, the scanning system can be used to adiabatically change theaspect ratio of the ODT, i. e. the ratio of the trapping frequencies in two directions, from 1.5
to 15. A comprehensive description of the scanning system is presented in Ref. [Bai12a].
As soon as atoms are loaded into the ODT further cooling is done via evaporative cooling in acrossed beam setup, which consists of a horizontal and a vertical trapping beam. A detailed
description of the evaporation procedure can be found in Sec. 7.1.1 for cooling bosons and in
Sec. 9.1.3 for cooling fermions.
In case of fermions the lifetime in the ODT operating at a wavelength of 1064 nm turned out
to be too short for evaporative cooling and further measurements. Due to that restriction
fermions are loaded from the 1064 nm trap into an ODT at 1570 nm. In this trap the lifetime
is about 40 s, which is sufficiently long. The reason for this reduced lifetime of the fermionsin the 1064-nm ODT is not understood so far.
The current ODT setup in the erbium experiment consists of five different beams; two beams
in horizontal and three in vertical direction. The setup is shown schematically in Fig. 3.11.
Each ODT beam is briefly discussed in the following.
42
43

model BRI-TEF-80-20-583, from Brimrose Corporation
model APAW, usable from 500 − 750 nm, 40 mm diameter, from Astropribor

1

1D lattice 1064 nm
400
λ/2

PD

lattice

distribution 1570 nm

dichroic
mirror1570 vert.

50
25
300

100

1570 hor.

-50

PD

-25

λ/2

1570

IPG fiber laser

hor.

ing
imag

ing
imag

AOM

AOM

vert.

300
PD

200

2

+1

AOM

vertical ODT 1064 nm
λ/2

0
-50

λ/2

150

CCD

λ/2
NKT fiber laser

λ/4vertical imaging

3

150

200

2
vertical ODT 1570 nm

1

PD

-25

250

3

imag

AOM

ing

vert.

imag

150

ing
0

1570 vert.

100

latticedichroic
mirror

-100

200

50
-25

ontal
horiz

λ/4

latticeretro-reflection

400

300

CCD

+1

100

λ/2
100

-50
λ/2

λ/2

250

AOM

Mephisto
MOPA

1000

1064 nm
42 W

PD

Figure 3.11.: Overview of the laser setup for absorption imaging (blue) and optical dipole-trap(ODT). The main chamber is shown from the front (left side) and from the top (right side). Imaging
the atomic cloud can be done in horizontal or vertical direction. The laser light for the horizontal
dipole trap and the one-dimensional optical lattice is derived from a single-mode master oscillator
power amplifier (MOPA) at 1064 nm (light red). For the vertical dipole trap a Yb fiber laser is utilized
operating at 1064 nm. Additionally, a second horizontal and vertical dipole trap from an Er-doped
fiber laser at 1570 nm is used for trapping fermions (dark red).

50

3.5. Laser light setup

51

Horizontal 1064 nm ODT The laser light for this ODT has a wavelength of 1064 nm andis generated by a high-power, single-mode, ultra-narrow linewidth laser44 . Light with a total
power of 42 W is sent through a high-power AOM45 . The zeroth-order beam is used for thevertical lattice, see later. For the above mentioned scanning system the first-order beam
is deflected by the AOM. This deflection can be translated into a parallel displacement by
putting the AOM in the focal point of a lens, see Ref. [Bai12a]. The beam is then collimatedby a second lens with a focal length of 250 mm before it is focused into the main chamberby a third 150 mm lens. This results in a beam waist of wx = 16.1 µm and wy = 24.2 µm
without scanning. The beam waist in the x-direction can be increased with the scanningsystem by up to a factor of ten. Using this ODT with a maximum power of up to 20 W
traps with potential depths of more than kB × 1 mK can be created without scanning andkB ×180 µK with maximum scanning. The largest scanning is very suitable for loading atoms
from the MOT into the ODT because of two reasons. First, the large trap provides a good
spatial overlap with the MOT and second, atoms in the MOT have an initial temperature of
15 µK, which is one order of magnitude smaller than the trap depth. A minimum trap depthof a factor of ten higher than the initial MOT temperature is required for a high loading
efficiency.

Vertical 1064 nm ODT This trap is created by light coming from a broadband Yb fiber
laser46 operating at 1064 nm with a maximum output power of 10 W. The vertical beam is
focused by a 400 mm lens and has waists of wx = 224 µm and wy = 140 µm.

Horizontal 1570 nm ODT A second horizontal trapping beam is created by light from asingle-mode, Er-doped fiber laser47 at 1570 nm. The light is split up using an AOM andcarried to the vacuum chamber by high power, polarization-maintaining fibers48 . The firstdiffraction order from the AOM is used for the horizontal beam, which is overlapped withthe horizontal 1064 nm trapping beam using dichroic mirrors.

Vertical 1570 nm ODT The light for this beam is taken from the zeroth diffraction orderfrom the AOM in the horizontal 1570 nm setup. The light is frequency-shifted by a second
AOM to prevent interference with the horizontal beam. The same AOM is used to control
and stabilize the laser power. The beam is then focused into the chamber by a 300 mm
lens.

Vertical 1D lattice with 1064 nm light For confining atoms in a quasi-2D geometry a onedimensional optical lattice with a lattice constant of 532 nm can be created in the erbium
experiment. An array of pancake-shaped traps is formed by retro-reflecting the lattice beam.
Light coming from the zeroth order of the AOM of the horizontal 1064 nm trap is sent through44

model Mephisto MOPA, 42 W at 1064 nm, from Innolight (now Coherent)
model 3110-197, numerical aperture of 3 mm, from Crystal Technology, Inc. (now Gooch & Housego)46
model PYL-10-1064-LP, 10 W at 1064 nm, from IPG Photonics
47
model Koheras Boostik HPA C15, System PM Pzt, 10 W at 1570 nm, from NKT Photonics GmbH
48
LMA-PM-15, from NKT Photonics GmbH45

3.5. Laser light setup

52

a second AOM for detuning the frequency with respect to the horizontal beam preventing
unwanted interference effects. Furthermore, an intensity stabilization is implemented using
the AOM. The light is then coupled into a high-power fiber for mode-cleaning. It is brought
above the vacuum chamber before it is focused into the chamber by a 400 mm lens. On theother side of the chamber the beam is collimated again and retro-reflected forming an opticalstanding wave. Monitoring the amount of laser light traveling in backward direction throughthe fiber is a convenient method for optimizing the quality of retro-reflection. In the final
setup the lattice beam has a waist of 250 µm and gives a trap frequency of up to 31 kHz in
the direction of strong confinement for a power of 8 W. The radial trap frequencies in thelattice are as low as 33 Hz.

Chapter

4

Ultracold dipolar scattering

Since the very first experiments on ultracold quantum gases have been carried out the scattering properties of particles were of fundamental importance. The reason for this is the factthat scattering properties reveal explicit information on the interaction between particles.
At small collision energies, which are only achievable with ultracold gases, one can explore
the regime of lowest partial waves. In this regime, the contact interaction between two particles has been extensively studied during the last years. It is an isotropic and short-range
interaction and can be described by a collision of two hard spheres.
The dipole-dipole interaction (DDI) fundamentally changes the scattering properties of particles. Its long-range and anisotropic character has strong influences on the scattering crosssection at ultracold temperatures. A new length scale, which is set by the dipole moment of
the particle and its mass, comes into play, the so-called dipole length. The stronger the DDI,
the larger its contribution to the particle interaction is, up to a point, where it becomes thedominant interaction in the system. A convenient method exists in ultracold gas experiments
to tune the relative strength between contact interaction and DDI. This tunability is inducedby the existence of Feshbach resonances.
One of the most striking features of the DDI is that the anisotropic nature of the dipole
potential causes the elastic scattering cross section not to vanish for decreasing collisionenergies. This is the case for contact interaction, except for the well-known s-wave scatteringof bosons. But with DDI, the contribution of higher partial waves can not be neglected,
even in case of identical fermions. Here, the scattering length will rather converge for lower
collision energies to a universal value, which is independent of temperature. These novel
scattering properties are summarized under the term universal dipolar scattering.
In this chapter, we will discuss the scattering physics of dipoles at low collision energies. The
first section is dedicated to anisotropic interactions, i. e. the dipole-dipole interaction and theanisotropic dispersion interaction between atoms. Then the theory of dipolar scattering ispresented in Sec. 4.2 with the aim of developing a framework for universal dipolar scattering.
Finally, anisotropy induced Feshbach resonances are subject for discussion in Sec. 4.3.

53

54

4.1. Anisotropic interactions

4.1. Anisotropic interactions
The interaction potential U of two dipolar atoms separated by a distance r has two main
contributions. On the one hand, there is the dipole-dipole interaction Udd and on the other
hand, the van der Waals interaction UvdW .
U (r) = Udd (r) + UvdW (r)4.1

Here, higher order interactions, like quadrupole-quadrupole or dipole-quadrupole, have been
neglected. The DDI is intrinsically anisotropic as the sign and the strength of interaction
depends on the orientation of the two dipoles. The van der Waals interaction is isotropic for
simple atoms, like alkali metals, but it can also be anisotropic for atoms with complex electronic structure, like for lanthanides. Both interactions will be discussed in the following.

4.1.1. Dipole-dipole interaction
The dipole-dipole interaction is an anisotropic and long-range interaction. Its anisotropic
nature makes the DDI dependent on the geometry the system. Due to these reasons, it
gained large interest in the field of atomic physics within the last few years. The interaction
potential Udd between two dipoles of identical dipole moment pointing in directions e1 ande2 , see Fig. 4.1(a), can be described in a general form by
Udd (r) =Cdd (e1 · e2 )r2 − 3(e1 · r)(e2 · r)
.
4π
r5

4.2

Here, r = |r| is the distance between the two dipoles and Cdd the dipolar coupling constant.
The coupling constant is given by µ0 µ2 for particles with a permanent magnetic dipole
moment µ, as in case of magnetic atoms, and Cdd = d2 /ϵ0 for particles with an electric
dipole moment d, e. g. for heteronuclear molecules. The magnetic constant µ0 and the electric
constant ϵ0 are used here. The magnetic dipole moment is usually expressed in units of µB ,whereas an electric dipole moment is given in units of Debye1 , D, due to convenience. The
conversion factor2 between magnetic and electric dipole moments is 1 µB ≈ 9.274×10−3 D.
The dipolar coupling constant for electric dipoles with a dipole moment of 1 D is muchlarger than that for magnetic dipoles with a dipole moment of 1 µB . This becomes obvious
by expressing the magnetic and electric dipole moments by the fundamental definitions ofµB and the Bohr radius3 , a0 . The ratio of magnetic to electric coupling constant is givenbyµ 0 µ2α2≈≈ 3 × 10−5 ,4.3
d2 /ϵ01.6
1
2

3

A Debye is defined in SI units as 1 D = 1c 10−21 C/m, with c the speed of light.
√
The conversion factor can be calculated by setting d2 /ϵ0 = µ0 µ2 and using ϵ0 µ0 = 1/c, which gives for1e~≈ 9.274 × 10−3 .
d = 1 D and µ = 1µB the following relation cDµB = 1021 2me
20~
The Bohr radius is defined as a0 = 4πϵ≈ 0.529 × 10−10 m ≈ 0.05 nm, with me the electron rest massme e2and e the elementary charge.

55

4.1. Anisotropic interactions
(a)

(b)

(c)

θ

µ

µ

+

+
-

Figure 4.1.: Dipole-dipole interaction (DDI) between two dipoles. Schematic view of the geometryof two interacting dipoles (a) in the unpolarized case and (b) in the polarized case. (c) Interaction
strength of the DDI, showing its anisotropic character. Regions where two dipoles repel each otherare shown in red (positive potential) and where they attract one another are shown in blue (negative
potential). The central white region is an artifact coming from a cutoff in the interaction strength
used for plotting. It directly shows the d-wave angular symmetry. Dashed lines depict the angle atwhich the DDI is zero.

with α the fine-structure constant4 . From now on, we will focus on atomic magnetic dipoles
unless otherwise stated.
In presence of an external magnetic field, the dipoles will align along the direction of the
field such that the atomic sample gets polarized, see Fig. 4.1(b). For two aligned dipoles,
Eq. 4.2 simplifies toµ0 µ2 1 − 3 cos2 θ.
4.4
Udd (r, θ) =4π
r3Here, θ is the angle between the orientation of the dipoles with respect to the interatomic
axis and r denotes the distance between two dipoles. At θm ≈ 54.7 ◦ the DDI vanishes.
When changing θ from 0 to π/2 the interaction changes its sign. For angles smaller than θmthe potential is negative and dipoles attract each other, whereas above θm the potential is
positive, which leads to a repulsion between dipoles. The term 1−3 cos2 θ ranges from −2 to 1,
i. e. maximum attraction is twice as large as maximum repulsion. A configuration with θ = 0,
is called a head-to-tail configuration and with θ = π/2 a side-by-side configuration of dipoles.
It is interesting to note that the angular symmetry of the DDI equals the Legendre polynomial
P2 (cos θ) of second order and the interaction is thus d-wave symmetric, see Fig. 4.1(c).
The classification of the DDI as a long-range interaction is subtle as it depends on thedimensionality of the system. Following an approach in Ref. [Ast08], a classification can bedone by checking whether the interaction energy of a system is extensive or intensive. An
extensive energy is given when the integral of the interaction potential U (r) converges,
Z∞

U (r)dD r ,

4.5

r0

with D the dimensionality of the system and r0 some short-distance cutoff parameter. If theintegral does not converge, one speaks of an intensive energy. For a system with extensive
energy the chemical potential µ depends only on the particle density n with µ = µ(n). In
41
The fine-structure constant is given by α = me~ca0 ≈ 137.

56

4.1. Anisotropic interactions

contrast, for a system with intensive energy the chemical potential also depends on the total
number of particles N and such µ = µ(N, n). This difference is used in the following toclassify short-range and long-range potentials. For interaction potentials decaying as 1/ri
at large distances, the integral in Eq. 4.5 converges if D < i. This means that the DDI
with i = 3 has long-range character in three dimensions and the chemical potential is anextensive property of the system. In one and two dimensions the DDI has to be consideredas short-range. As shown later, the van der Waals interaction has i = 6 and such it can be
fully considered as a short-range interaction, even in three dimensions.

4.1.2. Anisotropic dispersion interaction
In the simplest case the attractive electronic interaction of two atoms is given by the dispersion potential5 , also known as van der Waals potential,
Udisp (r) = −

C6
,
r6

4.6

where r is the internuclear distance and C6 is a constant called the van der Waals coefficient.
The potential is attractive, drops off with r−6 and is usually spherically symmetric. It
is a short-range interaction and is caused by the interaction of the electrons of the two
atoms. Its exact shape can be calculated from spectroscopic data, especially in case ofalkali, alkali-earth, and alkali-earth-like atoms. For this, the well-known Born-Oppenheimer(BO) approximation is applied, in which the electronic part is decoupled from the nuclear
part of the molecular wave function. Solving the full interaction Hamiltonian with the BO
approximation gives the full electronic interaction potential, known as Born-Oppenheimer
potential.
For lanthanide atoms, the dispersion interaction is anisotropic and the calculation of the BO
potentials is more difficult than for alkali atoms as the electronic structure of the groundstate is very complex and has many energy levels. Nevertheless, the group of S. Kotochigova
at Temple University (USA) pioneered in calculating the dispersion potential for dysprosium atoms, see Ref. [Kot11]. The same approach was also applied to erbium atoms, see
Ref. [Kot14].
Due to the strong spin-orbit coupling of lanthanide atoms in the ground state, their electron
spin S is coupled to their electron orbital angular momentum L, such that the total atomic
electron angular momentum j is formed. In the following, two colliding atoms with a total
angular momentum j1 and j2 are investigated. Due to the strong magnetic field created byeach of the atoms during a collision, j1 and j2 precess around the internuclear axis and giverise to the magnetic quantum numbers m1 and m2 , see Fig. 4.2(a).
The collisional state is defined by the total angular moment J = j1 + j2 , its projection M
along the magnetic field, and its projection Ω along the internuclear axis, see Fig. 4.2(b).
Values of Ω range from 0, 1, . . . , J. Each projection of J onto the internuclear axis gives rise5

The dispersion interaction, also known as London dispersion, describes only the attractive part of the
Lennard-Jones potential. Its repulsive part becomes important at very small values of r and is not considered in Eq. 4.6

57

4.1. Anisotropic interactions
(a)

(b)
j2

B

(c)

J even → gerade BO potential
J odd → ungerade BO potential

j2

B
m2

J = 12M
j1...

JΩ = 12Ω = m1 + m2

11

10

9

...

0

j1

BO potentials in total:
m1

J = 10Ω = 1049 gerade
42 ungerade

...
9

8 ...

0

Figure 4.2.: Total angular moments j1 and j2 of two colliding atoms (shown as black dots) in an
external magnetic field B. The internuclear axis is shown as gray dashed line. (a) The projection of
j1 and j2 onto the nuclear axis is labeled with m1 and m2 . (b) Both angular moments are coupled andform J, which precesses around the internuclear axis and results in the projection Ω. Its projectionin the direction of the magnetic field is called M . (c) Different values of Ω for J = 12 and 10. Evenvalues of J give rise to 49 gerade Born-Oppenheimer (BO) potentials and odd values to 42 ungerade
BO potentials (see text).

to a separate BO potential, see Fig. 4.2(c). Only the projections on the internuclear axis aregood quantum numbers during a collision process, see Ref. [Kot11]. The quantum numbers
m1 and m2 can change in a collision to m′1 and m′2 as long as Ω = m1 + m2 = m′1 + m′2 stays
conserved.
For two erbium atoms, having j = j1 = j2 = 6 and J = 12, there are (2j + 1) − Ω different
potentials with gerade and ungerade parity for each value of Ω. This results in a total of 91
BO potentials for Er2 . There are 49 potentials with gerade and 42 potentials with ungerade
symmetry. Gerade BO potentials correspond to even values of J and ungerade potentials to
odd values.
In case of colliding bosons with the same value of ji , only gerade states have to be considered.
Using atomic spectroscopy data, the BO potentials have been calculated within an ab initio
relativistic, multi-reference configuration-interaction method developed in Ref. [Kot97]. This
results in a matrix of van der Waals coefficients, C6 , dependent on the quantum numbers
m1 , m2 , m′1 , and m′2 . Using this matrix, the dispersion potential has be written as
Udisp (r) = −C6 (m1 m2 , m′1 m′2 )
,
r6

4.7

with r the separation between the atoms. The dispersion coefficients are calculated byevaluating the coupling from the initial state |j1 m1 , j2 m2 ⟩ to the final state |j1 m′1 , j2 m′2 ⟩by coupling through all possible electronic states |na ja ma , nb jb mb ⟩ of intermediate electron
configurations a and b. Here, n denotes the main quantum number of the electronic states.
This calculation can be formally written as, see Ref. [Kot11],⟨j1 m1 , j2 m2 | Udisp |j1 m′1 , j2 m′2 ⟩ =Xna ja manb jb mb

1
(E1 + E2 ) − (Ena ja + Enb jb )× ⟨j1 m1 , j2 m2 | Udd |na ja ma , nb jb mb ⟩× ⟨na ja ma , nb jb mb | Udd |j1 m′1 , j2 m′2 ⟩ ,

4.8

58

4.1. Anisotropic interactions
(a)

(b)

-600

2000

Udisp (cm-1)

C6 (a.u.)Ω = 0 to 12
1900

1800

-700

1700
0

2

4

6
Ω

8

10

12

-800

8

9
r (a0)

10

Figure 4.3.: Van der Waals C6 coefficients of erbium. (a) Different values of the C6 coefficients of
the 49 gerade BO potentials of Er2 dependent on Ω. The red arrows show the orientation of J, when
the internuclear axis is taken to be along the horizontal direction. (b) Born-Oppenheimer potential
minima dependent on the internuclear distance r. In this graph, the different potential depth of all
49 gerade BO potentials is shown. Both graphs are taken from Ref. [Kot14].

with Ena ja and Enb jb the energies of the electron configuration a and b, and E1 and E2 theground state energies of the colliding atoms. The coupling between two electronic states is
induced by the electric dipole-dipole interaction as
Udd (r) =1 d1 · d2 − 3d1 d2
.
4πϵ0
r3

4.9

Here, di are the electric dipole operators and di their projection along the internuclear axis.
Figure 4.3 shows the values of all calculated C6 coefficients of the 49 gerade BO potentials of
erbium dependent on the quantum number Ω. Note, that for different values of Ω a different
number of C6 coefficients exist, as shown in Fig. 4.2(c). The spread of the coefficients iscaused by the anisotropic electric dipole interaction between electrons according to Eq. 4.9,
which is in case of lanthanide atoms due to electrons in the 4f shell. It is often referred toas anisotropic dispersion interaction (ADI). The amount of anisotropy in the C6 coefficientscan be characterized by introducing ∆C6 . Results of the calculation for erbium atoms are
C6 = 1723 a.u. and ∆C6 ≈ 350 a.u., see Ref. [Kot14].
Figure 4.4 gives an overview of relevant interaction terms for the collision of two erbium
atoms. For comparison, the energy of the Zeeman splitting for a typical magnetic fieldof B = 1 G and 100 G is shown additionally. Spin-changing collisions can only occur atinternuclear distances where the anisotropic interaction terms, i. e. magnetic dipole-dipole
interaction and anisotropic dispersion interaction, become larger than the Zeeman splitting,which is the case for distances of r < 170 a0 at B = 1 G but r < 45 a0 at B = 100 G.
One of the most important consequences of anisotropic interactions, namely both DDI and
ADI, is the fact, that they provide a coupling between electronic degrees of freedom andmolecular rotation. The DDI couples rotational states with ∆ℓ = 2 and the ADI with∆ℓ = 4, see Ref. [San03b]. This means, that during a collision, the m quantum numbers canchange as long as the total projection quantum number Mtot = m1 +m2 +mℓ stays conserved.

59

4.2. Dipolar scatteringinteraction potential U (a.u.)

10-6

Zeeman splitting

C6/r6

at 100 G
10-8
C3/r3

at 1 G

10-10∆C6/r6
5

10-12

10-14

C5/r

50

100
150
interparticle distance r (a0)200

Figure 4.4.: Comparison of interaction potentials U in atomic units (a. u.) of two colliding erbium
atoms dependent on the interparticle distances r. Here, we compare the magnetic dipole-dipole interaction Cr33 (solid red), the electrostatic quadrupole-quadrupole interaction Cr55 (dash-dotted green),6
the isotropic dispersion interaction Cr66 (dashed blue), and the anisotropic dispersion interaction ∆C
r6(solid purple). The quadrupole-quadrupole interaction can be neglected due to its small value. The
long-range character of the dipole-dipole interaction can be observed, whereas at short-distances thedispersion interaction dominates. Taken from Ref. [Kot14].

This will be of importance, when we discuss the emergence of Feshbach resonances induced
by anisotropic interactions, see Sec. 4.3.
In conclusion, the peculiar electron configuration of lanthanides leads to a highly anisotropic dispersion interaction potential. In combination with the DDI, these two interactions
effectively couple atomic electronic states with molecular rotational states.

4.2. Dipolar scattering
The scattering process of two particles can be in the simplest case described by writing thetotal wave function as the sum of an incoming plane wave and an outgoing scattered wave.
From the knowledge of the scattering amplitude, f (θ), one can calculate the differential
cross section, i. e. the cross section per unit solid angle. The amplitude of the outgoing wave
depends on the angle θ with respect to the incoming wave. The differential cross section isdefined bydσ= |f (θ)|2 .
4.10dΩThe scattering cross section can then be calculated by integration, giving
∞

σ=

4π X
(2ℓ + 1) sin2 δℓ ,
k2
ℓ=0

4.11

60

4.2. Dipolar scatteringwith k the wave number of the incoming wave and δl the phase shift at r → ∞ between theincoming and outgoing wave with an angular momentum quantum number6 ℓ. The collisionenergy is given by~2 k 2
E=
.
4.12
2mr
m2
Here, mr is the reduced mass with mr = mm11+m
for two particles of mass m1 and m2 , which2
simplifies to mr = m
forparticleswithidentical
mass m.
2

In case of an anisotropic interaction potential, a multichannel description of the scattering
process has to be used, because the scattering problem can not be reduced to only one single
channel, see Ref. [Mar98]. This would usually be the s-wave channel with ℓ = 0 for colliding
bosons. A calculation of the total elastic cross section between identical particles can thenbe carried out byX
X ℓ′ m′σ B = 8π|tℓmℓℓ |2 ,4.13
el(F )′
evenℓ,ℓ′ =(odd) mℓ ,mℓfor bosons (B) and fermions (F ), respectively. Here, mℓ is the projection of the angularℓ′ m ′ℓ′ m′quantum number and tℓmℓℓ = Tℓmℓℓ /k are the reduced T -matrix elements, which contains
the asymptotic phase shift, δℓ , of each partial wave. Note, that in the limit of sphericallysymmetric collisions, i. e. without any dipole-dipole interaction potential, the matrix reducesℓ′ m′
to tℓmℓℓ = −as δℓ0 δℓ′ 0 δmℓ 0 δm′ℓ 0 , with δij the Kronecker delta.
In the case of a short-range potential, the phase shift changes as δℓ ∝ k 2ℓ+1 for small kaccording to the Wigner threshold law. Assuming an interaction potential varying like r−iat large distances this is only true for partial waves with ℓ < i−3
2 . For higher partial wavesi−2δℓ ∝ k , see Refs. [Lan77, Pet02]. The scattering phase shift for partial waves can be
written astan δℓ ∝ Ak 2ℓ+1 + Bk i−2 ,4.14
with constants A and B, which are determined by the interaction potential. The first term
describes the action of the short-range potential, whereas the second is due to long-range
scattering taking place outside the centrifugal barrier, see Ref. [Sad00].
For a van der Waals interaction potential with i = 6, this argument implies that all phaseshifts of partial waves other than the s-wave become negligible small as k → 0. In this case,the s-wave scattering cross section σs is only determined by the s-wave scattering length,
as , and the phase shift can be written as δ0 = −kas . From Eq. 4.11 and also from Eq. 4.13
follows by approximation the well-known result for identical bosons7 of
σel B = σs =8π 2
δ = 8πa2s .
k2 0

4.15

The fact that the s-wave scattering length is one of the most important parameters forcollisions in the ultracold regime can be described by √the following. Due to the large size ofthe thermal de Broglie wavelength, given by λdB = h/ 2πmkB T , at low temperatures T the
6

7

The notation of partial waves follows the usual rule for labeling an angular momentum quantum number
using s, p, d, f, g, . . . for ℓ = 0, 1, 2, 3, 4, . . ..
For bosons, Eq. 4.11 has to be multiplied by two due to the symmetrization of the wave function.

61

4.2. Dipolar scattering

inner-lying small-scale structure of the interaction potential can not be resolved anymore.
Therefore, the shape of the potential does not matter as long as it gives the same value forthe scattering length and such it can be replaced by a simpler model with the same effective
interaction. This is known as the so-called hard-core model describing the contact interaction
between particles.
In presence of DDI the picture of the scattering process changes fundamentally, see Ref.
[Mar98]. Its long-range character dropping off as r−3 implies, that for all partial waves, thephase shift δℓ ∼ k for small k and thus the phase shift does not vanish. It is explicitly shown
in Ref. [Deb01], that the reduced T -matrix elements from Eq. 4.13 stay non-zero for k → 0.
This means that all partial waves contribute to the scattering amplitude and the scatteringcross section contains a sum over phase shifts of all higher partial waves.
A direct consequence arising from dipolar scattering can be nicely observed using a Fermi
gas in a single spin state. Namely, the scattering cross section of identical, dipolar fermionsdoes not vanish at low temperatures. It rather approaches a constant, nonzero value whengoing to zero temperature as will be shown in the following. This is in stark contrast to
the ordinary situation without DDI, for which σel F = 0 for T → 0. The consequence of thenonzero value is that scattering can be used to accomplish evaporative cooling of a singlestate dipolar Fermi gas, which had been predicted by theory and was recently experimentallyshown, see Ref. [Mar98] for theory and Chapter 9 for the experimental results.

4.2.1. Universal dipolar scattering
Dipolar scattering theories predict an energy-independent elastic scattering cross sectionin the regime of low energies, see Refs. [Lan77, Sad00, Bar08]. In this regime, σel takes aconstant value, which is for fermions purely determined by the strength of the dipole-dipole
interaction and the particle mass, whereas for bosons the s-wave contribution from contact
interaction has to be added. When the DDI becomes the dominant interaction, i. e. when the
scattering process is insensitive to short-range interactions, the emergence of an universal
behavior can be observed, see Ref. [Tic08]. This causes different dipolar systems, bosonic
as well as fermionic ones, to show basically the same scattering behavior. According toRef. [Boh09], a universal scaling law can be found in two distinct regimes. The first regime
is at low collision energies with a constant scattering cross section, whereas the second one isfound at large collision energies. To estimate the transition point between these two regimes,
it is important to compare the collision energy from Eq. 4.12 with the energy scale set by
the DDI as
Cdd
16π 2 ~6
ED =
=
.
4.16
4πa3D
m3r µ20 µ4Here, aD is a typical length scale for characterizing two colliding dipoles8 . This gives
aD =8

mr Cdd
,
4π~2

4.17
2

It can be obtained by equating the centrifugal energy EZ = m~a2 with the dipolar energy ED . Its definitionr D
is similar to the dipole length, ad , with the only difference that for two colliding dipoles the reduced mass
has to be considered.

62

4.3. Anisotropy-induced Feshbach resonances

which results for two colliding 168 Er atoms in aD = 99.4 a0 . Taken this into account, thedipolar energy of 168 Er is ED = kB ×209 µK and for fully polarized KRb as low as kB ×83 nK.
Temperatures below this energy belong to the ultracold and above to the cold regime. Onlyin the ultracold regime, purely elastic threshold scattering can be observed, whereas in thecold regime inelastic collision have to be taken into account.
The second universal behavior can be observed for collisional energies larger than the dipolarenergy, i. e. outside the threshold region. In this cold regime, the cross section scales asσ ∼ aD /k. Here k is√the wave number of relative motion of the two collision partners andthus σ drops off as 1/ E. At even higher temperatures, the collision becomes semiclassical.
By applying the Born approximation to the scattering of dipoles one can calculate expressions
for the elastic cross section in the ultracold limit with a collision energy E < ED . The Born
approximation assumes that the incoming scattering wave is not substantially changed by theinteraction potential, which is true for long-range scattering outside the centrifugal barrier,see Refs. [Sha72, Gao99, Gao08]. The result for the total elastic cross section, σel , depends
on whether scattering occurs in even or odd partial waves withee
= σBorn+ σs =σel

16π 2
a + 4πa2s ,
45 D

16π 2
o
o
aD ,
σel
= σBorn
=

even

4.18
odd

15

(i. e. for bosons)(i. e. for fermions) .

For even partial waves the second term gives the contribution coming from the contact
interaction characterized by the s-wave scattering length as . For indistinguishable particles
both cross sections in Eq. 4.18 have to be multiplied by two. For distinguishable particles indifferent partial waves both cross sections of even and odd partial waves have to be added.
In case of s-wave scattering of identical bosons the elastic cross section reads
σel B =

32π 2
a + 8πa2s ,
45 D

4.19

whereas for identical fermions colliding in p-wave the cross section reads
σel F =

32π 2
a .
15 D

4.20

It is remarkable that σel only depends on the dipole length aD and on as for bosons, but is
independent of energy and temperature. The universal behavior of the scattering length wasstudied in detail with a deeply degenerate Fermi gas of 167 Er for the first time, see Chapter 9.
Signatures of this effect have been observed in an experiment using 161 Dy, see Ref. [Lu12].

4.3. Anisotropy-induced Feshbach resonances
Feshbach resonances (FR) are well-studied in collisions of alkali-metal atoms, see Ref. [Chi10]
for a review. They are used in many applications in ultracold-gas experiments and have developed to a standard tool within the last decade. The existence of FR allows for example
to tune the particle’s scattering length and convert a weakly interacting gas into a strongly

4.3. Anisotropy-induced Feshbach resonances

63

interacting gas of atoms. Alternatively, the scattering length can be tuned to zero, which
essentially turns off interactions and an ideal Bose or Fermi gas can be prepared. FRs had
also been the key element in observing the crossover from a molecular BEC to a degenerate
Fermi gas, the so-called BEC-BCS crossover [Bar04]. Furthermore, two particles can be associated to a weakly-bound molecule using a FR [Köh06], which is the first step in creating
ultracold polar molecules in its rovibronic ground state by transferring them with the helpof the stimulated Raman adiabatic passage (STIRAP) technique [Dan08, Lan08, Tak14]. Asa last example, FRs make it possible to study three-body Efimov physics [Kra06, Fer11].
In most applications, FRs are either induced magnetically/electrically by an external magnetic/electric field or optically by a light field. The method of using a magnetic field iscommonly implemented because of its ease of use in the experiment and will be the focus of
this thesis.
The exotic interaction properties of erbium give rise to a very rich Feshbach spectrum, see
Ref. [Kot14]. It is instructive to compare FRs in erbium with those in ordinary alkali metals.
In the latter case, FRs are induced by a different magnetic moment of different molecular
hyperfine states. For bosonic erbium, FRs occur due to the coupling of rotational boundstates because hyperfine structure is missing here. The coupling is achieved by the anisotropic
dispersion potential as well as the anisotropic dipole-dipole interaction, see Ref. [San03b].
Calculations have shown that both interactions give significant contributions to the Feshbach
spectrum, see Ref. [Pet12] and Fig. 4.6. The very same argument also applies to the collisionof dysprosium atoms, but does not apply to chromium atoms, as in this case anisotropic
dispersion interaction is absent.
FRs can be studied only at ultracold temperatures, where losses due to inelastic three-bodycollisions are suppressed. In case of bosonic erbium, all atoms are prepared in the lowest
Zeeman sublevel with a total atomic angular momentum of j1 = j2 = 6 and its projection
along the magnetic field mj1 = mj2 = −6. There is no hyperfine structure for bosons due tozero nuclear spin quantum number. For collisions of atoms in this stretched state exists only
one collision channel with zero relative nuclear orbital angular momentum quantum numberℓ, resembling a s-wave collision. FRs only appear due to the coupling to bound states withℓ ̸= 0, see Ref. [Pet12]. The Hamilton operator describing the collision can be written as
2

Ĥ = −~2 d2ℓ̂++ gj µB (mj1 + mj2 )B + V̂ (r, τ ) ,2mr dr2 2mr r2

4.21

with mr the reduced mass and B a homogeneous magnetic field along the z-direction. Herethe first two terms give the relative kinetic and rotational energies of the collisional state andthe third term denotes the Zeeman interaction energy, with gj the g-factor. The electronic
Hamiltonian, V̂ (r, τ ), is anisotropic and depends on the orientation and separation between
the two atoms, with an electronic configuration labeled by τ . It consists of the magnetic
dipole-dipole and quadrupole-quadrupole interaction as well as the anisotropic dispersion
interaction. As shown in Sec. 4.1.2, the anisotropic dispersion interaction is caused for lanthanides by electrons distributed in the 4f shells. These shells lie deep within the atom’s
electron cloud and do not overlap with the electron cloud of the nearby atom. Thus, thebonding of lanthanide molecules is mainly due to an overlap of the outer-lying isotropic 6sshells.

64

4.3. Anisotropy-induced Feshbach resonances

200

40
30
20

0

10

4

-9

-200
-1

energy (cm )energy (cm-1)

l=0

-400

-10
2

B
0

-600

6

-11

10

8

EZ(B)

4
2

∆E

mJ = -12

l=0

5

10

15

20

25
30
r (a0)

35

40

45

50

-800
5

10

15

20

r (a0)

Figure 4.5.: Anisotropy-induced Feshbach resonances of bosonic erbium atoms. The molecularpotential, which depends on the internuclear distance r, is shown only for selected values of ℓ. In
principle, 49 different BO potentials are present for every value of ℓ, as in Fig. 4.3(b), but for simplicity
only the molecular potential for Ω = 12 is plotted. The red lines show rotational states up to ℓ = 40 insteps of ten for the energetically lowest magnetic state with mJ = −12. The Zeeman splitting EZ (B)for a magnetic field of 50 G is only visible after scaling the gray shaded area of the plot, see inset.
Here, the open channel is shown as a solid black line, and closed channels by dotted lines. States withdifferent magnetic quantum numbers mJ are coded in color. As the Zeeman splitting depends on B,
the energy of each bound state is tunable with B (indicated by arrows). A FR appears, whenever
the energy of a bound state crosses the threshold of the open channel making ∆E = 0. Due to thelarge number of magnetic states and the coupling of rotational states up to ℓ ≈ 50, the density ofaccessible closed channels is extremely high. This large density can’t be observed with alkali metals.

As shown in Ref. [Pet12], coupled-channel (CC) calculations can be performed in the basis|(j1 j2 )jmj , ℓmℓ ⟩ = Yℓm (θ, φ) |(j1 j2 )jmj ⟩, with Yℓm (θ, φ) the spherical harmonic functions andthe angles θ and φ give the orientation of the internuclear axis with respect to the magnetic
field. The Hamiltonian of Eq. 4.21 conserves Mtot = mJ + mℓ and is invariant under parity
operation, so that only even or odd ℓ states are being coupled with each other. In case ofhomonuclear collisions only basis states with even J + ℓ exist and for s-wave collisions oferbium atoms it gives Mtot = −12.
Figure 4.5 shows the magnetic tunability of molecular potentials, which is a necessary premisefor the existence of magnetic FRs. States with a rotational quantum number up to ℓ = 40
are shown in the figure. A change of ℓ is possible due to strong anisotropic interactions
between the two atoms, which couples rotational states with ∆ℓ = ±2. Atoms collide in an
incoming or open channel and can be resonantly transferred into a bound molecular state orclosed channel, when the energies of both channels are degenerate. The separation between
the two channels, ∆E, is tunable by an external magnetic field B via the Zeeman shift.
Early CC calculations of the scattering length have revealed many FRs at magnetic fields
below 200 G already when taking only up to ℓ = 10 partial waves into account, see Fig. 4.6.

65

4.3. Anisotropy-induced Feshbach resonances
3000

scattering length (a0)

2000

1000

0

-1000
no ∆C6all interactions

-2000

0

50

100
magnetic field (G)

150

200

Figure 4.6.: Preliminary calculated scattering length of 168 Er dependent on the magnetic field toshow the important role of the anisotropic dispersion interaction. When neglecting the anisotropic
dispersion interaction (red), the number of FRs is strongly decreased compared to the case, in which
all interactions are taken into account (blue). Only rotational states up to ℓ = 10 have been considered
for coupled-channel calculations. This calculation was carried out for a temperature of 30 nK andzero collision energy. Taken from Ref. [Kot].

A qualitative analysis showed that more resonances appear as more and more partial waves
were included in the calculations. Recent measurements verified a number of resonances
corresponding to a density of about three per Gauss, see Refs. [Bau14, Pfa] for dysprosium
and Chapter 10 for erbium. An estimation using a random quantum defect theory approachsuggests that up to ℓ ≈ 50 partial waves need to be included for describing the experimental
results.
Within the coupled-channel calculations, not only the density of resonances changes withincreasing values of ℓ but also the position of a resonance shifts significantly. The position
converges within a few Gauss for a partial wave number being four larger than the partial
wave number at which the resonance has initially appeared. This behavior with increasing ℓand the strong coupling of rotational states can not be observed in ultracold gas experimentswith alkali-metal atoms or chromium. Moreover, it even requires a new way of labelingFRs. It was suggested in Ref. [Pet12] to use the specific ℓ value at which a FR appears
in calculations for labeling purposes. Recent work has shown that measuring the magnetic
moment of the bound state might be utilized to assign rotational quantum numbers for
specific FRs, see Refs. [Aik14a, Kot].
As mentioned earlier, a FR occurs when the energy of the closed state, Ec = ∆E, is equal
to the energy of the open channel, such that Ec = δµ(B − Bc ) = 0. Here, δµ is the magnetic
moment of the molecular state with respect to the magnetic moment of the atomic threshold
and Bc is the magnetic field, at which the uncoupled closed channel crosses the threshold.
Then the bound state is strongly coupled to the open channel and the scattering length

66
2

500

1

a (a0)

1000

abg

δB

Bc

0

∆
0

Bzero

energy (arb. units)

4.3. Anisotropy-induced Feshbach resonances

Eb
Ec
-500
0.7

-1

B0
0.8

0.9
B (G)

1.0

1.1

Figure 4.7.: Scattering length (red) and binding energy (blue) close to a Feshbach resonance of168
Er. The scattering length diverges at the position of the resonance B0 and has a zero crossing at
Bzero , which is characterized by the resonance width ∆. The resonance position is shifted by δB due
to coupling between open and closed channels. The bound state energy, Eb , varies quadratically close
to the resonance, whereas the energy of the bare, uncoupled closed channel, Ec , depends linearly andcrosses the threshold at Bc .

depends on the magnetic field like, see Ref. [Chi10],∆a(B) = abg 1 −,B − B04.22

where ∆ = |Bzero − B0 | is the width and B0 is the position of the resonance, see Fig. 4.7.
Here Bzero is the magnetic field at which a = 0. The background scattering length abg
characterizes the scattering length away from the resonance. The position of the FR, B0 ,at which a diverges, is shifted away from the position without coupling, which is denotedby Bc , due to the coupling between the closed and open channel. The shift is given byδB = −δE/δµ, which results in B0 = Bc + δB.
Note that in case of alkali atoms only the van der Waals interaction determines abg , whereas
for dipolar systems an additional dipolar contribution to the scattering length has to beconsidered. See Sec. 7.3.1 for a measurement of the scattering length of different erbium
isotopes via the cross-dimensional rethermalization method.
Close to the FR the energy of the bound state depends quadratically on the magnetic fieldwith~2
Eb = −.
4.23
2mr a2Further away from the FR, the energy of the closed channel Ec varies linearly with themagnetic field as
Ec = −δµB .
4.24
The binding energy of the closed channel can be measured relative to the two-atom thresholdenergy by magnetic field modulation spectroscopy. In the erbium experiment, this technique4.3. Anisotropy-induced Feshbach resonances67

was recently used to measure the magnetic moment of five different molecular states at
FRs below 2.5 G. The values of the measured molecular magnetic moments range fromµ = 7.96 µB to 11.8 µB , see Ref. [Aik14a].
In principle, for higher magnetic fields the magnetic moment of the molecular state can be
any combination of the magnetic moment of two atoms in some mJ states. The maximum
magnetic moment would be twice the magnetic moment of atoms in mJ = −6, which results
in a maximum magnetic moment of 13.96 µB for the Er2 molecule. This molecular state lies
parallel to the atomic threshold and is difficult to prepare by ramping the magnetic field
because it has to be entered utilizing another molecular state with lower magnetic moment.
But preparing such a molecular state will be interesting in future, as this gives a maximallylarge dipole length four times as large as that for atoms making Er2 molecules competitive
to current experiments using polar heteronuclear molecules.

Chapter

What is here required is a new kind of statisticalmechanics, in which we renounce exact knowledge
not of the state of a system but of the nature of thesystem itself.

5(Freeman J. Dyson 1961)

Random matrix theory

Random matrix theory (RMT) was introduced by Wigner in the 1950s, see Refs. [Wig51a,
Wig51b]. RMT was developed to describe statistical properties of neutron scattering spectra. This approach was necessary as the recorded nuclear spectra revealed unprecedented
complexity and an exact theoretical description was not possible. Early experiments in the
1930s revealed numerous narrow resonances in the neutron scattering on light nuclei, see
Refs. [Fer34, Fri36]. It was the insight of Bohr that these spectra could only be explained bya nucleus of great complexity with strongly interacting constituents, see Ref. [Boh36]. Later
this idea led to the discovery of the nuclear shell model in 1949.
The main advantage of RMT is that one can describe generic spectral properties without theknowledge of the Hamiltonian governing the system’s behavior. This is done by substituting
the real Hamiltonian by an ensemble of randomly generated Hamiltonians, whose matrix elements obey a certain probability distribution, hence the name random matrices. According
to the symmetry of the underlying system these Hamiltonians belong to a specific ensemble,
e. g. real and symmetric matrices in the case of time-reversal symmetry. A comprehensive
theory of the statistical properties of RMT was developed in the 1960s by Dyson and Mehta,
see Refs. [Dys62a, Dys62b, Dys62c, Dys63, Meh63]. Such statistical properties include thecharacter of spectral fluctuations, which are revealed by spacing distributions and number
variances.
It was speculated in Ref. [Bro81] that the study of spectral fluctuations in complicated systems can be applied to other many-body systems outside nuclear physics. In 1984 a behavior
according to RMT was found in a classical scattering experiment as well as in its quantummechanical analog, see Ref. [Boh84b]. This led to a strong connection between quantum
physics and classical chaos theory by the formulation of the so-called Bohigas-GiannoniSchmit (BGS) conjecture. The field of studying spectral fluctuations and correlations inquantum few- and many-body systems is today known as quantum chaos and has since the
BGS conjecture acquired much attention.
In the following section we will give a short introduction to the theoretical formalism of
RMT. Secction 5.2 presents the connection between classical chaos and quantum chaos andSec. 5.3 provides an application of RMT to ultracold scattering processes.

685.1. Introduction to RMT

69

Figure 5.1.: Six different level spectra for various types of level correlations but with the same meanlevel density. In (a) the levels are fully uncorrelated (Poisson), whereas in (c) (neutron scattering on
166
Er atoms) and (d) (Sinai billiard, see Sec. 5.2) the level distribution follows the GOE prediction.
In (f) all levels are uniformly distributed. Taken from Ref. [Boh84a].

5.1. Introduction to RMT
5.1.1. General idea of RMTTrying to tackle a complex system either from the experimental or the theoretical point ofview can be very difficult, especially if the exact Hamiltonian of the system is unknown in
the first place. Applying RMT to these problems can certainly not provide an exact solution,e. g. predicting the position of individual resonances, but what it can do is offering a first
insight into ‘global’ properties of the system. There exists a strong similarity between RMTand thermodynamics in the sense that only a few global properties define a rather complex
system. This implies some sort of universality in the application of the random matrix
approach, which allows it to be used in many different fields, see Refs. [Ros60, Guh98]. The
most intriguing consequence of RMT is that levels or eigenstates of the system are correlated.
This correlation is revealed by a characteristic fingerprint in several of the system’s statisticalproperties.
Providing an example for this correlation Fig. 5.1 shows different sets of level spectra fromthe fully uncorrelated case, to level distributions according to RMT, to uniformly distributed
levels. These sets have all the same level density but they are distinguishable by analyzing
the degree of correlation between levels.

70

5.1. Introduction to RMT

5.1.2. Applications of RMTAbove mentioned universality makes the RMT applicable to many different fields beyond
its origin in nuclear physics. The general predictions of RMT were found in many complex
systems ranging from analyzing financial correlation matrices [Lal99, Bou09], to wirelessmultiuser communication techniques [Tul04, Mül13], electric grid topology planning [Mar10],
neural networks [Raj06], to RNA folding theories [Orl02], just to name a few.

5.1.3. Symmetry groups and Gaussian ensembles
RMT classifies systems by their symmetry properties, see Ref. [Wei09]. A system with N
correlated energy levels is described by a Hamiltonian H of size N ×N with entries Hnm . All
entries are randomly chosen according to a Gaussian distribution. As defined in Ref. [Dys62a]
there are three possible Gaussian symmetry classes available within RMT. These symmetry
classes are labeled by the Dyson index β. In the case of time reversal and rotational symmetry(e. g. when the system has integer spin) the matrix H is real symmetric (β = 1). For halfinteger spin rotational symmetry is broken and H is self-dual with quaternionic entries(β = 4). For a non time-reversal symmetric system all Hnm are complex numbers (β = 2).
Diagonalizing the random matrix H = U −1 xU gives N eigenvalues x = diag(x1 , . . . , xN ).
The diagonalizing matrix U belongs to the orthogonal group O(N ) for β = 1, to the unitarygroup U(N ) for β = 2, and to the unitary-symplectic group USp(2N ) for β = 4. Due tothe use of a Gaussian distribution for constructing the random Hamiltonian the ensembles
are therefore called Gaussian orthogonal (GOE), unitary (GUE), and symplectic ensemble(GSE) for β = 1, 2, 4, respectively.
Each entry of H is chosen randomly from a Gaussian distribution with a probability density(β)
function PN (H). It has a zero mean value and a variance fixed by λ. The probability
density is given as


Nβ
(β)
(β)
2
PN (H) = CN exp − 2 trH.
5.1
4λHere, the constant λ sets the energy scale, which is defined by the level density, see Eq. 5.3,
(β)
and the constant CN ensures normalization. Note that no basis set had to be chosen for(β)(β)
constructing PN (H) and thus H only depends on its eigenvalues, for which PN (H) =(β)
PN (x) holds.
The mean level density of a spectrum, ρ(x), is defined as
ρ(x) =Xi

δ(x − xi ) ,5.2

with i an index over individual levels at position xi , δ the delta function, and the bar denoting
an average value. It is equivalent to the density of states of the system and is for Gaussian

71

5.1. Introduction to RMT

1.0
0.8
0.6
0.4
0.2
0.0

-1.0

-0.5

0.0

0.5

1.0

Figure 5.2.: Histogram of the calculated mean level density ρ(x), for 10,000 individual randommatrices of size 10,000 × 10,000 (blue). Spectrum according to Eq. 5.3 (red line). The semicircularshape of the spectrum is clearly visible.

ensembles given by
ρ(x) =

(

N
2πλ2

0

√

4λ2 − x2

if |x| ≤ 2λif |x| > 2λ .

5.3This equation is known as the semicircle law, which was found by Wigner, see Refs. [Wig57a,
Wig58]. The spectrum of ρ(x) has the shape of a semicircle with radius 2λ and a mean levelNdensity at the center of the semicircle of ρ(0) = πλ, see Fig. 5.2. Usually the limit N → ∞has to be taken to suppress cutoff effects in the spectrum. In reality almost no physicalsystem has a bound spectrum in shape of a semicircle as one can always observe resonancesx
at larger values of x and thus ρ(x) is not zero even for 2λ> 1, see Ref. [Guh98]. But forx
small values of 2λ the semicircle is a good approximation.
The mean level spacing between consecutive levels is defined byd(x) =

1
.
ρ(x)

5.4

It tends to zero as N → ∞ for constant λ because N eigenvalues have to fit into a finite
energy interval from −2λ to +2λ. For x = 0 the mean level spacing equals d(0) = πλN .

5.1.4. Statistical properties
RMT was developed to describe the statistical properties of energy spectra recorded in nuclear scattering experiments, see Ref. [Wei09]. There exists a great number of tools foranalyzing these statistical properties. A comprehensive description of most relevant observables can be found in Refs. [Bro81, Wei09]. More recently developed methods include forexample analyzing the 1/f noise of quantum chaotic spectra, see Ref. [Rel02], or the usage ofmaximum likelihood methods for correcting possibly missed levels, see Ref. [Mul11]. Only asmall selection of the most important measures of RMT is discussed in the following. Somequantities show the short-range correlations, like the nearest-neighbor spacing probability,

72

5.1. Introduction to RMT
1000

410

800
400
390

600

390

400

410

400

200
Poisson
GOE
0

0

200

400
600
B (arb. unit)

800

1000

Figure 5.3.: Simulated staircase function of uncorrelated levels (Poisson) and levels according toa Gaussian orthogonal ensemble (GOE), both assuming a density of resonances of unity. The inset
emphasizes the integer nature of the staircase function. Note the larger fluctuations of levels in
the case of a Poisson distribution compared to levels following a GOE, which are visible as larger
deviations from a line with unity slope.

whereas others probe the long-range correlation of the spectrum, e. g. the number varianceand the two-gap correlation function. For applying these methods to Feshbach spectra wewill redefine all necessary statistical properties in terms of a dependence on the magnetic
field B.
It is common in the RMT data analyzing process to first plot the position of resonances inthe so-called staircase function, see Ref. [Wei09]. This step-like function counts the numberof resonances up to some magnetic field value. It is defined as
N (B) =

ZB
0

dB ′

X
i

δ(B ′ − Bi ) ,5.5

with δ being the delta function and Bi the position of the i-th resonance. Figure 5.3 shows
the staircase functions for two randomly generated sets of data of uncorrelated and correlated
resonances with a density of one per unit. It can easily be seen that the staircase function ofcorrelated resonances deviates less from a line with unity slope than the staircase functionof uncorrelated resonances.
One of the central observables to discriminate uncorrelated from correlated resonances in a
spectrum is to calculate the nearest-neighbor spacing (NNS) distribution P (s), see Fig. 5.4(a).
¯ As shown
Here, s is the actual spacing between levels in units of the mean level spacing d.
in Refs. [Ber77, Bro81] the probability distribution of a system with uncorrelated classical73

5.1. Introduction to RMT(a)1,0Poisson
GOE

0,8

P

0,6
0,4
0,2
0,0

0

1

2

s

3

4

5

(b)
2,0

Poisson
GOE

Σ2

1,5
1,0
0,5
0,0
0,0

0,5

1,0

∆B

1,5

2,0

Figure 5.4.: (a) Nearest-neighbor spacing distribution for uncorrelated levels (Poisson) and levels
according to a Gaussian orthogonal ensemble (GOE). Lines show the distributions given by Eq. 5.6
and 5.7. Points are calculated from a randomly generated dataset of 100,000 levels assuming a Poissonor Wigner-Dyson distribution, respectively. (b) Number variance of levels according to a Poisson andWigner-Dyson distribution. Lines show the theoretical values from Eq. 5.12 and 5.13 and points show
calculated values of the generated dataset.

dynamics is simply given by the Poisson distribution
PP (s) = exp(−s) .

5.6

An exact expression for the spacing distribution of a Gaussian orthogonal ensemble can not
be written down in a simple form but according to the Wigner surmise, see Ref. [Wig57b], a
good approximation is provided by the Wigner-Dyson distribution
PWD (s) =πs exp(−πs2 /4) .
2

5.7

R
Note that both distributions Rare not only normalized with ds P (s) = 1 but they havealso the same mean value as ds s P (s) = 1. What makes
R them differ strongly from eachother is their variance. For the Poisson distribution it is ds (s − 1)2 PP (s) = 1 whereas for
the Wigner-Dyson distribution the variance is 0.5224, see Ref. [Gut90]. This difference invariances can be easily probed by analyzing the local fluctuations of levels, i. e. analyzing
the level correlations.
Regarding the NNS distribution the properties of the other two Gaussian ensembles, i. e. GUE
and GSE, differ from the GOE as P (s) shows a dependence on s, which is proportional to74

5.1. Introduction to RMT

sβ for small values of s. This means that for a GOE P (s) shows a linear increase, seeFig. 5.4(a), whereas for a GUE it increases quadratically, and for a GSE it increases withthe fourth power. This implies that the eigenvalues of a GUE and a GSE are even stronger
correlated and repel each other more than in the GOE case. Uncorrelated levels show no
repulsion at all and are often referred to by β = 0.
In Ref. [Bro73] an empirical function was introduced to interpolate between PP and PWD .
It is called the Brody distribution and has a single fitting parameter η. It quantifies thetendency (but not the degree of chaoticity) of the observed distribution to be more Poissonlike (η = 0) or more Wigner-Dyson-like (η = 1). The Brody distribution is defined by
PB (s) = Asη exp(−αsη+1 )

5.8

A = (η + 1)α η + 2 η+1
α = Γ
,
η+1

5.9
5.10

where Γ denotes the Gamma function.
To calculate the NNS probability distribution P (s) from a measured spectrum a histogram oflevel spacings has to be constructed using an appropriate bin size or number of bins Nbin for
distinguishing between the two distributions. There is no fixed rule for choosing an optimum√
bin size for an arbitrary histogram, see Ref. [Tay97]. Using a number of bins of Nbin = Nres
where Nres is the number of observed resonances usually provides reasonably good results.
A too small bin size increases the error on each value of the histogram and a too large bin
size makes it difficult to differentiate between different distributions.
For a reliable statistical analysis following RMT one has to make sure that the staircase
function increases linearly over a large range. If this is not the case a so-called unfolding
procedure has to be carried out before further analysis to artificially ‘linearize’ the staircase
function, see Ref. [Bro81]. A reliable unfolding procedure requires knowledge on the averagedensity of resonances over a large range of the spectrum, see Refs. [Wei09, Mor11]. In the caseof a Feshbach spectrum of ultracold erbium atoms there is no physical model available up
to now, describing the density of resonances. Due to this limitation an unfolding procedure
can not be carried out in the present case. Luckily, the staircase function of erbium is indeed
linear above a magnetic field of 30 G and the problem of unfolding can be circumvented byonly considering resonances above that value for further analysis.
A second statistical quantity of interest is the number variance Σ2 , representing the varianceof number of resonances within a certain range of the spectrum. The number variancedepends on long-range correlations between resonance spacings within an interval ∆B andis defined byΣ2 (∆B) = n2 (B0 , ∆B) − (n(B0 , ∆B))2 ,5.11
with n(B0 , ∆B) = N (B0 + ∆B) − N (B0 ) giving the number of resonances in the interval[B0 , B0 + ∆B] and the bar denotes the mean value over all possible B0 . It follows for a
Poisson distribution that
Σ2P = ∆B .
5.12

75

5.2. Quantum chaos
In contrast, for a spectrum according to the GOE one expects for large ∆B2π2Σ2WD =ln(2π∆B)
+
γ
+
1
−
π2
8
2
≈
ln(∆B) + 0.442 ,
π2

5.13
5.14

where γ = 0.5772... is Euler’s constant, see Ref. [Dys63]. Figure 5.4(b) shows a plot of Σ2
for the Poisson and GOE case. The fact that Σ2WD ∝ ln(∆B) shows that there are only verysmall fluctuations around an average number of resonances within a given interval of size
∆B. This suppression of fluctuations is often referred to as spectral rigidity. Compared to
the NNS distribution the number variance is more suitable to probe long distances in the
spectrum including many resonances. A clear signature of level repulsion on the one handand a large spectral rigidity on the other hand are central properties of strong correlations
between levels.
In contrast to analyzing the NNS distribution and number variance an unfolding independent
method was developed in Refs. [Oga07, Kol10]. This method consists of calculating the ratioof two consecutive gaps between resonances. The so-called two-gap correlation function isdefined bymin(di , di−1 )0 ≤ ri =≤1,
5.15
max(di , di−1 )with di the spacing between two neighboring resonances. As long as the density of states
does not vary on the scale of the mean level spacing an unfolding procedure is not needed for
calculating this quantity. A Poisson distribution of resonances shows a two-gap correlation
distribution following P (r) = 2/(1 + r)2 with a mean value of ⟨r⟩P = 2 ln 2 − 1 ≈ 0.386. In
contrast, the mean value of a Wigner-Dyson distribution can be calculated to be ⟨r⟩WD ≈0.53(1), see Ref. [Kol10]. Utilizing the mean values of both distributions we can introduce aquantity ηr which is proportional to the proximity to either a Poisson distribution (ηr = 0),a Wigner-Dyson distribution (ηr = 1), or any intermediate value byηr =⟨r⟩ − ⟨r⟩P.
⟨r⟩WD − ⟨r⟩P

5.16

In general, a larger sample size increases the significance of statistical observables. Since theearly years of RMT the number of nuclear resonances in the ‘Nuclear Data Ensemble’ has
increased to several thousands, see Refs. [Haq82, Guh98]. Also theoretical simulations offer
a much greater number of resonances, e. g. the Sinai billiard, as shown in Sec. 5.2.2, or thezeros of the Rieman zeta function, see Ref. [Tim06].

5.2. Quantum chaos
Following Bohr’s correspondence principle a quantum mechanical system in its macroscopic
limit shows coinciding behaviors with a classical system. This might easily suggest thatsignatures of classical chaos can be found in quantum mechanical systems as well. But5.2. Quantum chaos

76

Figure 5.5.: The nearest-neighbor spacing distribution of eigenmodes of the Sinai billiard (histogram)follows a Wigner-Dyson distribution (solid line) instead of a Poisson distribution (dashed line). The
inset shows the actual form of the billiard walls; it has a triangular shape with one rounded corner.
Note that the normalized level spacing is denoted by x instead of s. Taken from Ref. [Boh84b].

one of the central properties of classical chaotic systems namely the strong dependenceof the system’s state on initial parameters is not present in quantum mechanical systems.
Despite its probabilistic nature and the fundamental limit of the uncertainty principle inquantum mechanics an analog to the classically extremely sensitive evolution in time caused
by uncertainties adding up exponentially can not be found in quantum mechanical systems.
But it turns out that there exist quantum mechanical systems, for which the correspondingclassical Hamiltionian indeed exhibits chaotic behavior. The field of studying and describing
such systems is referred to as the field of quantum chaos. Since its origin in the middleof the 1980s various experiments confirmed quantum chaotic systems with high precision.
One of the most intriguing examples is the study of resonances in two-dimensional microwave
resonators, commonly known as Sinai billiard. These experiments are described in Sec. 5.2.2.
A nice introduction into the subject of quantum chaos can be found in Refs. [Pec84, Jen92].

5.2.1. Bohigas-Giannoni-Schmit conjecture
The origin of quantum chaos can be traced back to the publication of the Bohigas-GiannoniSchmit (BGS) conjecture, see Ref. [Boh84b]. The eigenmodes of a classical particle in a flat
triangular-shaped billiard, depicted as Sinai billiard system, had been calculated. From thisspectrum the NNS probability distribution and the number variance had been derived andboth were found to comply with the predictions of RMT for the case of a GOE. Figure 5.5
shows the calculated NNS distribution and the billiard system. Within the quantum mechanical viewpoint the classical particle can be described by a wavefunction with the Schrödinger
equation giving the evolution in time.
This result has lead to the formulation of the BGS conjecture, conveying two central messages. First, the level fluctuations of a quantum mechanical system with a corresponding5.2. Quantum chaos

77

classical Hamiltonian showing chaotic behavior can be described using RMT. Second, thesefluctuation laws are universal for complex single-, few-, and many-body quantum systems.
Recently the BGS conjecture was formally proven for the special case of time-reversal invariant graphs in Refs. [Plu13, Plu14]. Although a general proof is not yet found the conjecture
is by now numerically well established. There are also strong theoretical efforts to prove theconjecture following a semiclassical approach in Refs. [Sie01, Mül04] based on the Gutzwiller’s
periodic orbit theory, see Ref. [Gut90].

5.2.2. Sinai billiard
An extensively studied example of a single-body quantum system, showing chaotic behavior
in the corresponding classical limit, is the case of Sinai billiard (SB). After the theoreticalinvestigation, which built the foundation of the BGS conjecture, experimentalists realized
such a system and measured the absorption spectrum of a microwave field confined in aflat superconducting resonator of irregular shape, see Refs. [Grä92, Alt95]. The recorded
spectrum shows several thousand absorption dips and its statistical properties fully comply
with the description by RMT. It is possible to access two regimes with SB-type experiments. The first one is the regime of non-overlapping resonances and the second one is theregime of weakly overlapping resonances where Ericson fluctuations start to be prominent,
see Refs. [Die08, Mit10]. In a recent study a hexagonal structure inside the superconducting
resonator, mimicking the structure of graphene, makes it possible to study quantum chaoswith a relativistic time evolution given by the Dirac equation, see Ref. [Bit12].
After injecting an electromagnetic field, which is tunable in the GHz range, at one site of
the resonator, the amplitude and phase at another site is measured using a vector network
analyzer. By varying the frequency of the input microwave and recording the spectrumwith high resolution one can reconstruct the full scattering matrix of the superconducting
resonator SB system including the phase shift between incoming and outgoing waves. The
position of input and output sites are kept fixed during measurements. Such a resonator,
having an entrance as well as an exit port for the microwave, models a scattering system of
two particles with an incoming and outgoing channel, e. g. a system of a neutron colliding
with a compound nucleus, see Ref. [Mit10], or colliding ultracold particles as it is the case in
the present thesis.
In the following we will outline the analogy of the classical SB-type experiment and a quantum mechanical particle with boundary conditions. As described in Ref. [Ber81] theories andmeasurements of SB-type experiments determine the energy eigenvalues En of the stationary
Schrödinger equation
2mEn
∆Ψn +
Ψn = 0 ,5.17~2
with Ψn the wave function. For convenience the wave number kn and the rescaled energy En
is defined by
2mEn
kn2 =
= 4π 2 En .
5.18
~2

78

5.3. Ultracold resonant scattering

Dirichlet boundary conditions are assumed as the wave function has to vanish at the wallsof the resonator. The classical limit of the Schrödinger equation corresponds to the limit~ → 0 and this leads to kn → ∞ and En → ∞. By this, Eq. 5.17 can be rewritten to∆Ψn + kn2 Ψn = 0 ,5.19

which resembles the Helmholtz equation of a static microwave field in the resonator. It was
shown that this classical system reveals chaotic properties and thus its quantum mechanicalanalog can be called quantum chaotic.

5.2.3. Quantum chaos in atomic physics
First hints to quantum chaos in the field of atomic physics were given in Ref. [Ros60]. Here,the repulsion of energy levels for the complex atomic spectra of atoms with strong spindependent forces, such as Hf, Ta, W, Re, Os, and Ir, were investigated and the fingerprintsof Wigner-Dyson distributions were found. The study of quantum chaos in many-electron
atoms is described in Ref. [Con97] and a more recent measurement of energy levels in singly
ionized thorium can be found in Ref. [HS13].
Another system showing quantum chaos was found in Ref. [Has89] by studying the diamagnetic Kepler problem, which can be mimicked by highly-excited Rydberg atoms in a
homogeneous magnetic field as realized in Ref. [Jia96]. In particular, quantum chaos wasobserved measuring the Stark spectrum of atomic lithium in Ref. [Cou95]. A transition tochaos can be driven by increasing the electric field, responsible for the Stark splitting. This
is in contrast to the regular Stark system of hydrogen. A possible explanation for this discrepancy is offered by the scattering of electrons into many trajectories when passing thenucleus and thus is able to explore the full phase space. These multiple scattering processes
are known as so-called closed-orbit recurrences.

5.3. Ultracold resonant scattering
Complex systems such as lanthanide atoms and atom-dimer or dimer-dimer mixtures support a large number of Feshbach resonances (FR). In particular, for lanthanide atoms thisoriginates from the strong anisotropic interaction between two atoms, mixing higher angular momentum states, see Sec. 4.3 and Refs. [Kot11, Pet12]. For atom-dimer and dimerdimer mixtures the large number of FRs is due to the additional ro-vibrational structureof molecules. Applying RMT to ultracold collision processes allows for deriving statisticalproperties of these complex Feshbach spectra, which can be compared with experimental values. Specifically, theoretical calculations were carried out for atom-dimer and dimer-dimer
mixtures in Refs. [May12, May13] and first experimental signatures of RMT were found for
erbium atom-atom collisions within the present work of this thesis, see Chapter 10.
Due to the strong coupling between rotational molecular states many of these states can be
explored by the collision constituents within a single collision process. During this time of79

5.3. Ultracold resonant scattering‘exploration’ the constituents are quasi-bound and form the so-called collision complex. The
dissociation of this complex is described by the Rice-Ramsberger-Kassel-Marcus (RRKM)
theory, see Ref. [Lev05]. The mean rate of dissociation, kRRKM , of a particular configurationof the complex is expressed as
1 Na
kRRKM =.
5.20
2π ~ρE
Here, ρE denotes the density of states with respect to energy and Na is the number of allavailable quantum states, which can be explored during the collision. Following Ref. [May12]
the mean lifetime of the collision complex, τ , can be calculated by
τ=

2π~ρE
.
No

5.21

Here, No is the number of open channels, available for outgoing collision products after thecollision process. In the ultracold regime No can be as small as one, resulting in a lifetime of
the complex on the order of a few µs and in the case of two colliding dimers on the order of
1–10 ms. During this long lifetime collisions with other particles become probable, ultimately
leading to a loss of complexes. This loss process has to be considered in experiments withultracold molecules.
For low-energy atom-dimer collisions it was just recently shown in Ref. [Cro14] that a fractallike structure can be observed in the lifetime of the collision complex dependent on the angle
between the interatomic axis of the dimer and the collision direction of the two collisionpartners. An extensive study of the fractal dimension versus collision energy suggests atransition from a classical to a chaotic collision process when energies are smaller than about
kB × 400 K. Thus, ultracold gases are supposed to be deep in the regime of chaotic collisions.
For these calculations only the van der Waals coefficient C6 and the well depth De of the
Lennard-Jones potential need to be taken into account.
In Ref. [May12] a multichannel quantum defect theory (MQDT) approach is utilized to derive
the scattering matrix S phys for alkali atom-dimer collisions. A set of parameters is requiredfor MQDT. Once this set of parameters is acquired for a given class of molecular potentials it
gives a direct connection between the short-range reactance matrix K sr , valid for distances
R < Rm , and S phys , valid for R → ∞, see Fig. 5.6. Here, Rm is a characteristic distance,
splitting the interaction range into a short-range and long-range part. In the short-range
region forms the collision complex at R < Rm and incomming channels a are strongly
mixed with each other. In the long-range region the scattering channels are assumed to be
independent. The long-range interaction is of the form
Va (R) = −C6~2 La (La + 1)+
+ Ea (B) ,R6
2mr R2

5.22

where C6 denotes the isotropic van der Waals coefficient and mr is the reduced mass of thescattering partners. Here, Ea (B) is the energy threshold, which depends on some external
magnetic field B, and La is the angular momentum of channel a.
The wavefunction of a scattering process from the incoming channel a into the outgoingchannel a′ can be written as(a)+Ψa′ (R) = δa′ a fa− (R) − S physa′ a fa′ (R) ,

5.23

80

5.3. Ultracold resonant scattering

Figure 5.6.: MQDT approach to resonant scattering of alkali atom-dimer collisions. The MQDT
treatment connects the short-rage K matrix K sr , containing information about the ro-vibrational
resonances, to the physical scattering matrix S phys at long ranges, where only hyperfine channels
are taken into account. From S phys elastic and inelastic cross sections can be deduced, which areobservable in the experiment. For R → ∞ the number of energetically closed and open channels is
given by Nc and No , respectively. The collision complex forms at distances R < Rm . Taken from
Ref. [May12].

with
fa± =

r

2mr ±i(ka R−La π/2)e,
π~2 ka5.24

the incoming (+) and outgoing (-) spherical waves.
For a given number of hyperfine channels Na at short ranges there are only No open channels
at long ranges available for scattering products to exit to infinity as the energy is above
threshold with Ea > E. The remaining Nc = Na − No channels are energetically closed.
Closed channels have to be removed within the MQDT formalism by eliminating them. For
accomplishing this K sr is partitioned in a first step into
 sr
K oo K sr
sr
oc
.
5.25
K =
sr
K sr
co K cc
Then a closed-channel MQDT parameter β is introduced and the modified reactance matrix
f is defined byK
f = K sr − K sr (K sr + tan β)−1 K sr .
K
5.26
oo
oc
cc
co

The modified reactance matrix can be transformed into an energy-normalized long-range
representation by
1
f + GKf−1 )A 12 .
K = A 2 K(15.27
A, G, η, and β are diagonal matrices containing the relevant MQDT parameters. The
physical scattering matrix S phys is finally achieved by
S phys = eiη (1 + iK)(1 − iK)−1 eiη .

5.28

The MQDT has been extensively used for describing resonant scattering processes at ultracold temperatures in Refs. [Bur98, Mie00, Gao05, Gao08]. The reason for the success of81

5.3. Ultracold resonant scattering

MQDT lies within the fact that at a matching radius of R = Rm classically closed states at
R → ∞ become open. Thus the reactance matrix K exhibits only a weak dependence on
the energy and complex spectra can be described. The connection to the scattering matrix
S phys can then be established by the use of MQDT.
In the case of atom-atom collisions of lanthanides or atom-dimer collisions of alkali-metal
atoms the exact details about the interaction potential as in Eq. 5.22 are not known andthus the short-range reactance matrix K sr can not be determined. A possible solution tothis problem is to use a fully statistical K matrix according to RMT.
As shown in Refs. [May12, Mit10] in great detail such a statistical K matrix can be received
by writing
NX
W aµ W µb
sr
K a,b = −π,
5.29
E − Eµµ=1with W an energy-independent coupling-matrix, representing the coupling between shortrange resonances. Here, a and b are the asymptotic channels and µ denotes intermediate
states. A mean coupling strength of channel a to short-range resonances can be written as
Ra(0) =πρE Γa ,
2

5.30

where ρE is the density of states at incident energy E and
N

Γa =

2π X
|W µa |2
N

5.31

µ=1

the partial width of channel a.
As K sr is expressed in the frame, where the corresponding Hamiltonian is diagonal, thecoupling W becomes a random process and can be constructed as a random matrix, where
its elements have a zero mean value and a variance of νa2 , see Ref. [Mit10] and Sec. 5.1.3. WithΓa = 2πνa2 the variance is only defined by the density of states ρE and the mean coupling(0)
strength Ra by
(0)
Ra
νa2 =
.
5.32
ρE π 2(0)

According to Ref. [May12] it can be assumed that Ra = 1 as this gives a transmission
coefficient of Ta = 1 between short-range and long-range channels by
(0)

(0)

Ta = 

4Ra

 .
(0) 2

5.33

1 + RaIn fact, Ra can be left as a fitting parameter to some real spectra, giving more information
on short-range physics of complex collision processes. The expression of W by a randommatrix directly leads to a distribution of energy levels with a NNS distribution, resemblinga Wigner-Dyson distribution as shown in Eq. 5.7.

5.3. Ultracold resonant scattering

82

An estimation of the density of states for atom-dimer collisions for various combinations ofalkali-metal atoms can be found in Ref. [May12]. Much higher density of states, like on theorder of 103 per Gauss, are predicted for dimer-dimer collisions, e. g. 87 Rb133 Cs + 87 Rb133 Cs
or 40 K87 Rb + 40 K87 Rb, see Ref. [May13]. In this regime resonances start to overlap and can
not be individually resolved any more. This leads to strong and random fluctuations of thescattering cross section, which are known as Ericson fluctuations, see Ref. [Eri60].

Chapter

6

Publication: Narrow-line laser
cooling of erbium

6.1. Introduction
Narrow-line magneto-optical traps (MOTs) are a very powerful tool to reach low temperatures. The Doppler temperature, TD , directly depends on the natural linewidth, Γ, of the~Γ. For comparison, an ordinary MOT of alkali atoms,
laser-cooling transition with TD = 2k
B
operating on the D2 transition with a linewidth, ∆ν = Γ/(2π), of a few MHz, provides typicaltemperatures of 150 µK. Multi-valence-electron species have a richer energy structure thanalkali atoms and offer a number of different opportunities for laser cooling. For instance, a
MOT of ytterbium atoms is realized with a narrow-line transition, i. e. an intercombinationline with ∆ν = 200 kHz resulting in TD = 4.8 µK, and an narrow-line MOT, with ∆ν of a few
kHz and TD as low as a few hundred nK, has been realized with strontium, see Ref. [Kat99].
In case of ytterbium, atoms from a Zeeman slowed atom beam are directly captured by the
MOT, while in the case of strontium the narrow-line MOT acts as a second cooling stage
within a two-stage MOT setup.
Although the electron structure of lanthanides is much more complicated than compared
to ytterbium it is possible to implement a narrow-line MOT with erbium. To provide an
overview, typically achievable temperatures in the presented erbium MOT are as low as 10 µK
with a maximum atom number of a few 108 atoms. This narrow-line cooling of erbium ispresented in the following.
Magnetic lanthanides have several broad and narrow transitions usable for laser cooling as
discussed in Chapter 2. One difference to simple systems, like alkali, alkaline-earth, andalkaline-earth-like elements, is the existence of many metastable states, in which an excitedstate can decay into, see Fig. 2.1. In general, the number of possible decay channels increases
the shorter the transition wavelength is. However, pioneering work with erbium by the groupof J.J. McClelland at NIST (USA) demonstrated a MOT operating on the open transition
at a wavelength of 401 nm. Similar results have been obtained in later experiments withdysprosium, see Ref. [Lu10]. Atoms participating in the cooling cycle are trapped and cooledin the MOT. Part of these atoms decay into some metastable state, become dark, and do not83

6.1. Introduction

84

scatter the MOT light anymore until they return into the ground state again via relaxation.
But due to the large atomic magnetic moment atoms in metastable states stay trapped in
the magnetic gradient field of the MOT. The dynamics of such a MOT with a reservoir ofmagnetically trapped atoms was studied in Ref. [McC06b]. In a later experiment erbium
atoms were then loaded from this broad-line MOT into a narrow-line MOT operating at awavelength of 841 nm. Typical final temperatures after this two-stage erbium MOT were as
low as 2 µK, see Ref. [Ber08].
We use a different approach with erbium, inspired by the successful demonstration of the ytterbium MOT, see Ref. [Kuw99]. The transition 4f 12 6s2 1 S1 - 4f 12 6s6p 3 P1 has a wavelengthof 583 nm and a linewidth of 190 kHz, which is very similar to the transition in ytterbium.
It was not fully clear from the beginning, whether this cooling approach would work out
because of two issues. First, there are two metastable states below the excited 4f 12 6s6p 3 P1state of erbium, into which the excited state can decay. However, the calculated branchingratio into these two states is negligible small. Second, the narrow linewidth of this transition
results in a small capture velocity of the MOT of 5.6 m/s, which makes loading the MOTwith an atom beam from a high-temperature oven a delicate issue. As shown in Sec. 3.3.2
one can indeed prepare a slow atom beam with a velocity smaller than the capture velocityof this narrow-line transition. This makes it possible to load the narrow-line erbium MOT
directly from the ZS. The erbium narrow-line cooling strategy is very simple and robust.
The final temperature of the atoms in the MOT is as low as 10 µK and the peak density is
1.5×1011 cm−3 , which corresponds to a phase-space density of 1.2×10−5 . Another advantage
of our narrow-line MOT approach is an automatic spin-polarization of the atoms without
the need of optically repumping them into their lowest spin state. We successfully realized
the directly-loaded narrow-line MOT for erbium atoms in our group for the first time in late2011 and the group in Stuttgart just recently used a similar idea for dysprosium atoms, see
Ref. [Mai14].
In the following, we describe the creation of a narrow-line MOT operating on the 583-nm
transition for different erbium isotopes. In Sec. 6.1.1 we discuss its relevant experimental
parameters and then we show a measurement of the spin purity of atoms in the MOT
measured by a Stern-Gerlach experiment in Sec. 6.3.1.

6.1.1. Narrow-line cooling
The narrow-line MOT for the bosonic isotopes 164 Er, 166 Er, 168 Er, and 170 Er and thefermionic isotope 167 Er operates on the yellow 583-nm transition. The natural linewidthΓof this transition is ∆ν = 2π
= 190 kHz, which results in a Doppler temperature of 4.6 µK,
see Sec. 2.3.1. Figure 6.1 shows two real-color photographs of the fluorescence of trapped
atoms. Figure 6.2 summarizes the experimental sequence for the MOT, e. g. the light detuning and intensity, the magnetic gradient field, and the homogeneous magnetic field. In
the following we will first describe the erbium MOT loading procedure and then discuss thetheory of narrow-line cooling.
The ZS laser beam travels in collinear direction with the atom beam through the vacuum
chamber and overlaps with the center of the MOT. Schemes that combine a broad transition

85

6.1. Introduction

(a)

(b)

far-detuned MOT

compressed MOT

Figure 6.1.: Real-color photographs of the narrow-line erbium MOT. (a) The typical bowl-shaped
MOT is due to the narrow-line laser-cooling transition in combination with a large detuning and thegravitational sag. In this picture the cloud contains approximately 107 atoms. (b) After finishingloading the MOT the atomic cloud is compressed into the center of the trap and gets sphericallysymmetric. The much smaller volume of the MOT enables one to load atoms efficiently into an
optical dipole trap for further experiments.

for ZS and a narrow transition for MOT potentially suffer from an undesired light force,
acting on atoms from the ZS light during the MOT loading1 . To avoid this effect we spatiallyseparate the MOT from the ZS laser beam by the use of a large detuning of the MOT laser
light in combination with the proper magnetic field gradient. This shifts the atoms outof the trap center and effectively prevents any light force of the ZS laser beam acting on
the atoms. For erbium a detuning as large as δ = −7.6 MHz ≈ −40 ∆ν and a gradient of∂z B = ∂B∂z = 2.6 G/cm is needed, see Fig. 6.2. At such a large detuning compared to thenatural linewidth atoms in the MOT will not feel a homogeneous light force over the fullvolume of the MOT but will rather be resonant with the light on an ellipsoid of constantmagnetic field. Adding the gravitational potential the trapped atoms will accumulate at
the lower part of this ellipsoid. Figure 6.3 gives a schematic view of this MOT working
principle. The so-called gravitational sag will create a characteristic ‘bowl-shaped MOT’,
see Fig. 6.1(a). Dependent on the laser detuning, ∆ = 2πδ, the position of the cloud inz-direction, z0 , will adjust itself such that the effective detuning ∆ − µ∂z Bz0 stays constant,see Ref. [Lof04]. Here, µ is the atomic magnetic moment. An important consequence of thenarrow-line MOT is that the damping and diffusion coefficients of atoms in the MOT are
independent of the laser light detuning. This is in contrast to broad-line MOTs and strongly
affects the equilibrium temperature in the MOT as it will be shown below.
After loading sufficiently many atoms into the MOT the ZS light and ZS magnetic field
are switched of and the atom beam is blocked by a shutter. Atoms are then being shifted
towards the center position, which creates a nearly-symmetric compressed MOT (cMOT), seeFig. 6.1(b). The cMOT is achieved by ramping the laser-light detuning closer to resonance,
decreasing the light intensity, and reducing the magnetic field gradient2 . This is achievedin a sequence of two linear ramps. During the first ramp, which has a duration of only
1

2

The ZS light gives a scattering force on the atoms, which is two orders of magnitude larger than for the
MOT light. This estimation is based on the actual experimental parameters for light intensity and detuning
for the blue and yellow transition, respectively.
It has to be noted here that this is in contrast with most other schemes for MOT compression, in which
the gradient is usually increased.

MOT loadingZS(a)cMOT - 600ms

ODT

ON

detuning (MHz)

OFF
(b)

0
-2
-4
-6
-8

(d)

gradient (G/cm)

intensity (IS)

(c)

(e)

12
8
4

0.02 IS

0
3
2
1
0

B (G)

1.0
0.5
0.0
0.0

0.2

4.8

5.0

5.2

5.4

5.6

time (s)Figure 6.2.: Relevant parameters for MOT loading and compressing. The MOT loading is shown fora loading time of 5 s, which is varied dependent on the specific isotope or experimental needs whereas
the cMOT ramping procedure is kept unchanged. (a) After switching off the Zeeman slower (ZS)
a two-stage ramp is used to compress the MOT (cMOT). (b) This involves changing the detuning,(c) the intensity (in units of the saturation intensity, IS ), and (d) the gradient field (in z-direction).
After an additional holding time of 200 ms a final temperature of as low as 10 µK can be achieved in
the cMOT. The atoms are then loaded into an optical dipole trap (ODT) for further experiments.
(e) A homogeneous magnetic field is ramped up within 10 ms at the beginning of the cMOT stage,to keep the atomic sample fully spin-polarized.

86

87

6.1. Introduction(a)energymJ = +1

(b) z

σ+
x

small δlarge δsmall δg

mJ = -1large δ
σ

+

σ

-

z

σ-

Figure 6.3.: Working principle of the narrow-line MOT. (a) Position-dependent detuning in the
MOT, shown in one direction for a simplified MOT with a J = 0 → J ′ = 1 transition. Due to thesmall linewidth trapped atoms fulfill the resonance condition only in a small region far outside thecenter of the gradient field (yellow regions). The distance from the center is set by the detuning (seetext). (b) In three dimensions atoms are resonant on an ellipsoid. The gravitational force pulls theatoms to the lower part of the ellipsoid, where they preferably scatter σ − -polarized light from thelower MOT beam and the sample automatically gets spin polarized.

10 ms, the laser beam peak intensity is ramped to 4 IS with IS the saturation intensity and ahomogeneous magnetic field of 0.9 G is introduced. Within the second ramp with a durationof 390 ms the magnetic field gradient is changed to 1.4 G/cm, the light intensity to 0.02 IS
and the detuning to −1.8 MHz. In the following holding time of 200 ms the atoms are further
cooled in the cMOT until they reach their final temperature. To achieve low temperatures
the peak intensity of the cooling laser beams has to be chosen small, see Fig. 6.4. Figure 6.2
shows a full list of necessary parameters for loading the MOT and ramping to the cMOT.
A very convenient consequence of a narrow-line MOT, operated with a large detuning, is its
ability to intrinsically spin-polarize the atoms. In such a narrow-line MOT the lower verticallaser beam3 supports the atoms against gravity and atoms predominantly scatter light fromthis beam as demonstrated in Fig. 6.3(b). This beam happens to be σ − -polarized, which
successively transfers all atoms into the energetically lowest Zeeman sublevel mJ = −6.
As long as they do not scatter light from the upper vertical beam they stay polarized inthis stretched spin state. A homogeneous magnetic field during compressing the MOT andloading it into the optical dipole trap (ODT) preserves the spin polarization, see Fig. 6.2(e).
The degree of spin polarization in the ODT is verified by a Stern-Gerlach measurement,
which is presented in Sec. 6.3.1.
Now we will discuss the theory of narrow-line MOT cooling. The total force along thez-direction, acting on atoms with velocity vz at position z in the MOT, is given by, see
Ref. [Lof04],
"
#
s
~kΓ
− mg .
6.1
FMOT (vz , z) = Fsc + Fg =
2
z Bz)
2
1 + s′ + 4(∆−kvz −µ∂2ΓThe first term is due to light scattering whereas the second term gives the gravitational
force with m the atomic mass and g the gravitational acceleration. The scattering force, Fsc ,3

For perfect spin polarization it is very important that the vertical MOT axis is oriented along the directionof gravity.

6.1. Introduction

88

depends on the transition linewidth Γ, the laser light detuning ∆ = 2πδ, and ∂z B. Here,I′k = 2πλ , s = IS is the saturation parameter, s ≥ s takes care of saturation effects induced
by beams in other directions, and I is the single-beam peak intensity. We will now derive
an expression for the damping coefficient in the narrow-line MOT from which the capture
velocity and the equilibrium temperature of the MOT can be calculated.
Dependent on the detuning of the MOT light one has to distinguish between two regimes
in which the MOT dynamics turn out to be completely different, see Ref. [Lof04].
For√
a detuning, being larger than the power-broadened natural linewidth ΓE = Γ 1 + s and|∆| > ΓE , atoms can be treated in a free flight picture with hard walls as boundaries
induced by the MOT. This regime is called regime (I). The hard walls are located far outside
the MOT center where atoms are resonant with the cooling light and experience a strong
light force, see Fig. 6.3. Including the gravitational potential atoms will sag to the lower
part of the trap and scatter mostly the lower beam. This applies during the erbium MOT
loading procedure as well as in the cMOT, as for both stages ∆ ≈ −10 ΓE . The regime (II)is reached for small detunings when |∆| < ΓE . In this case atoms experience a radiative
force over the full volume of the trap. The resulting restoring force changes linearly with theposition in the trap, which is described by damped harmonic motion, see Ref. [Xu02]. Thisis usually the case, e. g. for a MOT of alkali-metal atoms and lanthanides, where the MOT
is operated on a broad transition.
Atoms in the MOT experience a cooling power, Pcool , which is proportional to the square of
their velocity, v, see Refs.[Let89, Sch03]. The power can be written as
dE
Pcool == Fsc v = −αv 2 ,6.2
dt cool
where Fsc = −αv is the friction force in the MOT given by Eq. 6.1 with (∂Fsc /∂v) = αevaluated at vz = 0, z = z0 . Here, α is the damping coefficient. Accordingly, the kinetic
energy of the atoms, E = 21 mv 2 , will decrease exponentially with a time constant τ , theso-called thermal equilibration time. Utilizing Eq. 6.2 the cooling rate in the MOT can becalculated by
1
Pcool
2α
=−
=
6.3
τ
E
m
The thermal equilibration time has to be considered when optimizing the MOT ramping
parameters. As shown in Ref. [Lof04] α is given in regime (I) by√
2~k 2 Rs − s′ − 1
α=
,
6.4
R2 s
~kΓ
with R = 2mg. Note that in this regime α depends on the atomic mass, m. Using this result
the equilibration time for the erbium MOT is about 78 ms and for the cMOT τ = 3.6 ms.
For completeness, in regime (II) α is according to ordinary Doppler theory given by, see
Ref. [Xu02],

−2
It
∆2
4 2 It |∆|α = ~k
1+
+4 2
,
6.5
3
IS Γ
IS
Γ

where It is the total intensity of all six MOT beams.

89

6.1. Introduction

Teq (µK)

100

10

0.01

0.1

1

10

s

Figure 6.4.: Equilibration temperature of 168 Er in the MOT (solid line). Note that the equilibration
temperature is independent of the detuning and only depends on the saturation parameter, s, in asquare-root-like behavior for large values of s, according to Eq. 6.8. The lower temperature limit isset by the Doppler temperature (dashed line).

Utilizing the result for α one can estimate the capture velocity of the MOT, vc , which is
the maximum velocity that atoms can have when entering the MOT and they will still be
captured by the MOT. The capture velocity is given by
vc =

2dα
,
m

6.6

and depends on the size of the capture volume of the MOT, which is simply set by the beam
diameter d. A Value for vc for the narrow-line transition of erbium is 5.6 m/s for d = 25 mm.
This result is important for the design of the Zeeman slower as depicted in Sec. 3.3.2.
From Eq. 6.1 an expression for the diffusion coefficient in regime (I) can be derived as, see
Ref. [Lof04],~2 k 2 Γ
Dp =
.
6.7
2R
Using this expression one can calculate the equilibrium temperature Teq as the ratio between
diffusion and damping coefficient. The former results in an effective heating process whereas
the latter cools the atomic sample. Equilibrium is reached when both heating and coolingbalance each other. Taking the damping coefficient from Eq. 6.4 the equilibrium temperaturein the narrow-line MOT is thus,
√
Dp
~Γ s
R
p
Teq =
=
.
6.8
kB |α|
2kB 2 R − s′ /s − 1/sNote, that Teq is independent on the detuning of the MOT light put is a function of s.
Figure 6.4 shows the dependence of Teq on the laser-beam intensity. During the erbium
MOT loading stage this gives Teq = 126 µK for s = s′ = 12 and in the cMOT the theoreticalequilibrium temperature is as small as 5.7 µK for s = s′ = 0.02, which is close to the Doppler~Γtemperature of TD = 2k
= 4.6 µK.
B
Utilizing this simple and efficient approach to a narrow-line MOT we could successfully
demonstrate a directly loaded MOT with up to 2 × 108 erbium atoms at a temperature of
10 µK and a peak density of 1.5×1011 cm−3 , corresponding to a phase-space density of about
1.2 × 10−5 .

6.2. Publication:
Narrow-line magneto-optical trap for erbium†
Physical Review A 85, 051401(R) (2012)
submitted 7 March 2012; published 7 May 2012
DOI 10.1103/PhysRevA.85.051401
Albert Frisch,1 Kiyotaka Aikawa,1 Michael Mark,1 Alexander Rietzler,1 Johannes Schindler,1
Eric Zupanič,1,2 Rudolf Grimm,1,3 and Francesca Ferlaino1
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria2
Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
3
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria†

The author of the present thesis constructed the experimental apparatus together with J. S. and A. R.,
performed the measurements with K. A. and M. M., analyzed the data, and contributed in writing themanuscript.

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RAPID COMMUNICATIONSPHYSICAL REVIEW A 85, 051401(R) (2012)

Narrow-line magneto-optical trap for erbium
A. Frisch,1 K. Aikawa,1 M. Mark,1 A. Rietzler,1 J. Schindler,1 E. Zupanič,1,2 R. Grimm,1,3 and F. Ferlaino1
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria2
Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
3
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, A-6020 Innsbruck, Austria(Received 7 March 2012; published 7 May 2012)
We report on the experimental realization of a robust and efficient magneto-optical trap for erbium atoms,
based on a narrow cooling transition at 583 nm. We observe up to N = 2 × 108 atoms at a temperature of about
T = 15 μK. This simple scheme provides better starting conditions for direct loading of dipole traps as compared
to approaches based on the strong cooling transition alone, or on a combination of a strong and a narrow kHz
transition. Our results on Er point to a general, simple, and efficient approach to laser cool samples of otherlanthanide atoms (Ho, Dy, and Tm) for the production of quantum-degenerate samples.
DOI: 10.1103/PhysRevA.85.051401

PACS number(s): 37.10.De, 67.85.−d, 37.10.Vz

Laser cooling of non-alkali-metal atoms has become avery active and challenging field of research. The great
appeal of unconventional atomic systems for experiments on
ultracold atomic quantum gases stems from the possibilityof engineering complex interactions and of accessing richatomic energy spectra. Both features are at the foundation of anumber of novel and fascinating phenomena. For instance,
the energy spectra of two-valence-electron species, suchas alkaline-earth-metal and alkaline-earth-metal-like atoms,
feature narrow and ultranarrow optical transitions, which are
key ingredients for ultraprecise atomic clocks [1], efficientquantum computation schemes [2], and novel laser coolingapproaches, as demonstrated in experiments with Sr, Yb, andCa [3–5].
As a next step in complexity, multivalence-electron atoms
with a non-S electronic ground state such as lanthanides are
currently attracting increasing experimental and theoreticalinterest. Among many, one of the special features of lanthanides is the exceptionally large magnetic dipole momentof atoms in the electronic ground state (e.g., 7μB for Er and10μB for both Dy and Tb), which provides a unique chance
to study strongly dipolar phenomena with atoms. Highly
magnetic atoms interact with each other not only via theusual contact interaction but also via an anisotropic and longrange interaction, known as the dipole-dipole interaction [6].
Chromium was the first atomic species used for experiments
on atomic dipolar quantum gases [7,8], and the even moremagnetic lanthanides are nowadays in the limelight thanks tolaser cooling experiments on Er and Tm [9,10] and to therecent realization of quantum-degenerate Dy gases [11,12].
Similarly to Yb and the alkaline-earth-metal atoms, theatomic energy spectra of magnetic lanthanides include broad,narrow, and ultranarrow optical transitions. This collectionof lines is reflected in a wide choice of possible schemes
for laser cooling experiments. However, all experiments on
Zeeman slowing and cooling in a magneto-optical trap (MOT)with magnetic lanthanides so far have relied on an approachthat is essentially based on the strongest cycling transition[9,10,13]. This broad transition typically lies in the blue
between 400 and 430 nm and has a linewidth on the order of
few tens of MHz. As a consequence, the Doppler temperature
is close to 1 mK. Such a high temperature makes direct loading
1050-2947/2012/85(5)/051401(5)from a MOT into a dipole trap difficult and inefficient, evenwhen sub-Doppler mechanisms take place [14,15]. To further
decrease the temperature of atoms prior to the dipole trap
loading, an additional MOT stage based on an ultranarrow
kHz-linewidth transition was applied in Refs. [11,13], making
the whole experimental procedure more involved.
Taking advantage of the rich atomic spectrum of lanthanides, we identify a different transition to be the mostsuitable one for MOT operation toward production of aquantum-degenerate gas. This transition, which for Er liesat 583 nm and has a linewidth of 190 kHz, represents anintermediate case between the broad (blue) and the ultranarrow(red) transitions available in the lanthanide spectra and hasvery similar properties to the Yb intercombination line [16,17].
Based on this narrow yellow line, we demonstrate a MOT of
2 × 108 Er atoms at a temperature as low as 15 μK. Our
approach, inspired by Yb experiments [4], provides better
starting conditions for direct loading into a dipole trap thanthe conditions achieved in other experiments with magnetic
lanthanides [11].
Erbium is a heavy rare-earth element of the lanthanide
series. It has six stable isotopes, among which the three bosonicisotopes 166 Er (34%), 168 Er (27%), and 170 Er (15%) and thefermionic isotope 167 Er (23%) have a high natural abundance.
The Er electronic configuration is characterized by a xenonlike
core, an inner open 4f shell, and an outer closed 6s shell,[Xe]4f 12 6s 2 . The electron vacancies in the inner f shell are acommon feature of all the lanthanides (with the exception ofYb) and are at the origin of the strong magnetism as well as
various interesting collisional effects [18–20].
The atomic level spectrum of Er is shown in Fig. 1.
In the ground state, Er has a highly anisotropic electronic
density distribution with a large orbital angular momentum
L = 5 (H state) and a total electronic angular momentum
J = 6. The bosonic isotopes have zero nuclear spin (I = 0)
and consequently do not exhibit a hyperfine structure. On
the contrary, the fermionic isotope 167 Er has a nuclear spin
I = 7/2, leading to eight hyperfine levels, from F = 5/2 to
F = 19/2, in the electronic ground state.
In Ref. [22], five different J → J + 1 laser coolingtransitions were identified with linewidths ranging from tensof MHz to a few Hz. Here we focus on the blue and051401-1©2012 American Physical Society

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PHYSICAL REVIEW A 85, 051401(R) (2012)energy (1000 cm -1 )25

20

4f12(3H6) 6s6p(1P1)

15

4f12(3H6) 6s6p(3P1)

10

ZS light(401 nm)5

MOT light(583 nm)
4f126s2 3H6

0
2

3

4

5

6

7

8

9

10

11

12

J
FIG. 1. (Color online) Energy levels of atomic erbium up to
25 000 cm−1 for different total electronic angular momentum quantum numbers J [21]. States with odd (even) parity are indicatedby black (red/gray) horizontal lines. The two relevant laser coolingtransitions at 401 and 583 nm are indicated by arrows [22].

the yellow transition at 401 and 583 nm, respectively; see
the arrows in Fig. 1. The corresponding excited levels are
the singlet 1P1 and triplet 3P1 states coming from the transitionof an s electron into a p shell. The strong blue transition
at 401 nm has a linewidth of Ŵ401 /2π = 27.5 MHz [23],corresponding to a Doppler temperature of h̄Ŵ401 /(2kB ) =660 μK. We use the blue light for both Zeeman slowing
and transversal cooling of the Er atomic beam. The MOT
operates on the narrow yellow transition at 583 nm, which has anatural linewidth of Ŵ583 /2π = 190 kHz and a corresponding
Doppler temperature of 4.6 μK. The line may exhibit weakleaks from the excited level into two intermediate levels
with very low calculated leakage rates (0.017 and 0.0049
s−1 ) [22]. Note that we find these losses irrelevant for allpractical purposes. For 167 Er the hyperfine structure in the401- and 583-nm excited states gives rise to eight levels
ranging from F ′ = 7/2 to F ′ = 21/2. For the 583-nm line,
the hyperfine constants are known [26]. On the contrary
the hyperfine constants for the 401-nm line are currently
unknown, making the operation of the fermionic MOT morechallenging.
We generate the blue light from two independent blue diodelasers injection locked to a master laser. The master laser
light is produced by frequency doubling methods based ona tapered amplified diode laser at 802 nm. The blue light fromthe master laser is locked to a hollow-cathode discharge lamp[4] via modulation transfer spectroscopy. With this setup wespectroscopically resolve the lines of the four most abundantbosonic isotopes as well as the hyperfine structure of thefermionic isotope 167 Er. Since the hyperfine constants of theexcited level at 401 nm are unknown, we could not assign
the absorption features to specific hyperfine transitions. We
derive the yellow light from a dye laser operating with

Rhodamin 6G. By using an intracavity electro-optic modulator(EOM) and an external reference cavity, we stabilize the laser
to a linewidth of about 50 kHz. By additionally locking thelaser to an ultralow-expansion cavity, we achieve a long-term
stability better than 30 kHz within a day [27].
Our experimental procedure is as follows. We load the Zeeman slowed atomic beam from an effusive, high-temperatureoven directly into the narrow-line MOT. Our commercial oventypically operates at a temperature of 1300 ◦ C, which is about
200 ◦ C below the Er melting point. Two 3-mm apertures,
separated by 50 mm, provide a geometrical collimation of theatomic beam. In addition, the atomic beam is further collimatedand transversally cooled by a two-dimensional (2D) opticalmolasses, working on the broad 401-nm transition with a total
power of about 100 mW. The beams are elliptically shaped
to increase the interaction time between the atoms and thelight. Thanks to the transversal cooling stage, we increase theloading flux by almost an order of magnitude.
The atomic beam then enters the Zeeman slower (ZS).
Because of the limited capture velocity (a few m/s) imposedby the narrow cooling transition used for the MOT, it is
crucial to design a ZS that provides enough atomic flux at
low velocities [4]. We build a 360-mm-long spin-flip Zeeman
slower, which can slow the atoms from 500 to about 5 m/s. The
ZS light is focused at the oven position and has a total powerof about 60 mW. At the MOT position, we estimate a beam
diameter of about 11 mm, corresponding to an intensity of
about one Is,401 , where Is,401 = 56 mW/cm2 is the saturation
intensity. Our ZS operates with light detuned by about −20Ŵ401(−540 MHz) from the unshifted resonance.
The narrow-line MOT is operated in a standard sixbeam configuration with retroreflected beams. To increase
the capture velocity of the MOT, we use large MOT beam
diameters of about 30 mm. Typical MOT loading parameters
include a magnetic field gradient B ′ (along the symmetryaxis) of 4 G/cm, a laser intensity of 12Is,583 per beam with
Is,583 = 0.13 mW/cm2 , and a detuning δ583 from the atomic
transition of −50Ŵ583 (−9.5 MHz). To measure the number ofatoms in the MOT after loading, we optically compress the
MOT by reducing δ583 to −0.5Ŵ583 and we apply standard
absorption imaging on the blue transition.
A special feature of our narrow-line MOT is the large
detuning of the MOT light (typically −50Ŵ583 ) needed for
optimal loading. At this detuning, we observe a very strong
effect of gravity on the position and shape of the atom cloud [3].
The atoms are located well below the center of the magnetic
quadrupole field, and the cloud takes the form of a large
semishell. To elucidate the reason for the large detuning, wemonitor the loading dynamics and the lifetime of the MOT;
see Fig. 2. Our measurements focus on the 166 Er isotope, but
we have observed the same qualitative behavior also for theother isotopes.
Figure 2(a) shows the atom number in the MOT as afunction of the loading time for different values of δ583 . We
fit our data by using a standard loading rate equation [28],which includes a capture rate R and a decay rate γ ; thelatter accounts for both collisions between trapped atoms
and collisions with the background gas. For a detuning of−50Ŵ583 the atom number approaches its steady state in about
10 s with Nss ≈ 2 × 108 . For a lower value of the detuning

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PHYSICAL REVIEW A 85, 051401(R) (2012)

FIG. 2. (Color online) Loading (a) and decay (b) of the narrowline MOT with 166 Er atoms for δ583 = −50Ŵ583 (circles) and δ583 =−32Ŵ583 (triangles). (a) Atom numbers are plotted as a function of
the MOT loading time. The solid lines are fits to the data using
N (t) = Nss (1 − e−γ t ) with Nss = R/γ . From the fits we obtain
γ = 0.137(2) s−1 , R = 2.6(4) × 107 s−1 , and γ = 0.200(4) s−1 ,R = 1.1(1) × 107 s−1 for −50Ŵ583 and −32Ŵ583 , respectively. The
inset shows the atom number after 10-s loading time as a function of
the MOT detuning. (b) Atom numbers are plotted as a function of theholding time in the MOT in presence (open symbols) and absence of
the ZS light (solid symbols). The solid lines are fits to the data using
a double-exponential function; see text.

to −32Ŵ583 we observe a substantial decrease of the atom
number to Nss ≈ 5 × 107 . This behavior is clearly shown in
the inset, where we monitor the number of atoms Nss for a fixedloading time of 10 s as a function of the MOT light detuning.
For detunings exceeding δ583 = −30Ŵ583 we observe a rapid
increase in the atom number. When further increasing thedetuning, Nss first stays constant and then rapidly decreases.
We believe that the large detuning of the MOT light serves
to minimize the detrimental effects of off-resonant pumping
processes driven by the ZS light. According to the Zeeman
shift in the quadrupole field of the MOT, we observe a
spatial displacement of the atomic cloud with δ583 of about
1.4 mm/MHz [29]. For large detunings, this shift becomes solarge that the atoms can no longer be kept in the MOT. Forintermediate detunings in the range from −40 to −55Ŵ583 ,the cloud displacement is advantageous for MOT operation
since the atoms become spatially separated from the region ofinteraction with the ZS light. For small detunings, the ZS lightleads to substantial losses of atoms.
The effect of the ZS light also shows up in lifetime
measurements, where we monitor the number of atoms in the
MOT as a function of the holding time with the ZS light being
present or absent; see Fig. 2(b). For these measurements weswitch off the ZS magnetic field and the atomic beam after
10 s of MOT loading. For large detuning (δ583 = −50Ŵ583 ) the
evolution of the atom number is not affected by the ZS light. In
both cases, i.e., with the ZS light on and off, we observe fasterlosses in the earlier stage of the decay, which we attribute toinelastic two-body collisional processes, and a slower decay at
a later time, which is finally limited by background collisions.
For simplicity we use a double-exponential fit function toestimate the time constants [30]. We extract time constants of
19(3) and 80(20) s for the fast and slow dynamics, respectively.
At lower MOT detuning (−32Ŵ583 ), the ZS light strongly
affects the decay. When the blue light is turned off, ourobservations are qualitatively similar to the ones at −50Ŵ583
while with the blue light turned on the atomic loss dramaticallyincreases. In this case the decay curve is well described by asingle exponential function with a time constant τ = 9.0(1)s. This decay time is consistent with a simple estimate ofpumping losses. By considering the absorption rate Ŵa of the
401-nm light at the actual intensity and detuning [31] andthe branching ratio b for decay from the excited state to all
the possible metastable states [9], we estimate a decay rate
1/τ = bŴa of the order of 0.1 s−1 .
We could also demonstrate trapping of all the other Er
isotopes, with the exception of the rare 162 Er (0.1% naturalabundance). For all the bosonic isotopes we used about thesame values for the detuning of the ZS and the MOT light.
Figure 3(a) shows the atom number in the MOT for thedifferent isotopes as a function of their natural abundance.
For a long loading time of 20 s we observe similar atom
numbers exceeding 108 for the three most abundant bosonicisotopes, indicating that saturation effects might apply. For ashort loading time of 5 s, the atom number increases with thenatural abundance. However, we observe a more complicated

FIG. 3. (Color online) Behavior of different Er isotopes. (a) MOT
atom number for the four most abundant bosonic isotopes as well
as for the 167 Er fermionic isotope is plotted as a function of thenatural abundance for 20-s (squares) and 5-s (circles) loading times.
(b) Spectroscopy signal of the blue transition shows the isotope shifts
for the bosonic isotopes and the HF structure of 167 Er. The arrow
indicates the locking point used for the ZS light to produce the 167 Er
MOT.

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PHYSICAL REVIEW A 85, 051401(R) (2012)

behavior than the expected linear growth. This might be due
to slight differences in the optimal MOT parameters, but may
also point to differences in scattering and collisional properties
among the different isotopes.
For the fermionic isotope 167 Er we observe a MOT withatom number of about 3 × 107 , which is substantially lower
than the numbers measured for the bosonic isotopes. A simple
explanation of this behavior can be that we decelerate andcool only atoms in the F = 19/2 hyperfine state, which has astatistical weight of about 20%. Note that a similar behavior
has been observed with the fermionic 161 Dy MOT, which also
shows lower atom numbers than the ones of bosonic MOTs[13]. An additional complication stems from the unknown
hyperfine splitting of the 401-nm line. From the spectroscopic
signal we could not identify a priori where to lock the Zeeman
slower light to be resonant with the desired F = 19/2 → F ′ =21/2 hyperfine transition. To produce the 167 Er MOT, we had toproceed blindly by first locking the 583-nm MOT light on thecooling transition and then by trying different locking points
for the ZS light until the MOT was visible. We finally succeedin creating a MOT by locking the ZS light to the spectroscopic
line located 150 MHz below the blue transition frequency of
the 166 Er isotope; see the arrow in Fig. 3(b).
After loading the MOT we apply a stage of compression to
reduce the temperature of the atomic cloud and to achieve
good starting conditions for direct loading of an optical
dipole trap. With the optimized parameters δ583 = −0.5Ŵ583 ,I = 0.16Is,583 , and B ′ = 0.8 G/cm, we measure a temperatureof 15 μK via time-of-flight experiments. In addition, SternGerlach experiments indicate that atoms in the compressed
MOT phase are naturally pumped in the lowest Zeeman
sublevel mJ = −6 by the MOT light. This can be explained byconsidering a combined effect of the narrow cooling transition
used for the MOT and of gravity. The latter pushes the atoms

downward, creating an effective imbalance in the trap andleading to a preferential absorbtion of the σ − polarized lightfrom the lower vertical beam. For 2 × 108 atoms at T = 15 μK
we estimate a peak number density of the polarized sample
of 1.5 × 1011 cm−3 , corresponding to a phase-space densityof about 4 × 10−6 . These values are similar to the ones
observed in Yb experiments using the intercombination light[4]. Compared to other experiments on magnetic lanthanides,
such as Dy [11], our much simpler approach based on a single
cooling light for the MOT provides higher atom numbers andsimilar final temperatures. We suggest that this scheme canbe successfully implemented with Dy, Ho, and Tm, using
the 626-nm (Ŵ/2π = 135 kHz) [32], the 598-nm (Ŵ/2π =146 kHz) [33], and the 531-nm (Ŵ/2π = 370 kHz) [34]transitions, respectively.
In conclusion, we have demonstrated an efficient and simple
approach for an Er MOT based on a single narrow-line
transition. Our scheme works with all abundant Er isotopes
and allows for direct loading of an optical dipole trap. In
first experiments we were able to load up to 107 Er atoms
into the dipole trap in a single focused-beam configuration.
Optimization of the dipole trap loading and evaporative
cooling experiments are under way in our laboratory.

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We are grateful to J. J. McClelland and A. J. Berglund forsharing with us their knowledge on Er and to S. Kotochigova,
O. Dulieu, M. Lepers, and J. F. Wyart for fruitful discussions.
We also thank the Sr team in Innsbruck and the Yb team inTokyo for their support. This work is supported by the Austrian
Ministry of Science and Research (BMWF) and the Austrian
Science Fund (FWF) through a START grant under Project
No. Y479-N20 and by the European Research Council under
Project No. 259435.

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PHYSICAL REVIEW A 85, 051401(R) (2012)[21] Y. Ralchenko, A. Kramida, J. Reader, and N. A. Team, NIST
Atomic Spectra Database, 2011.
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and J. J. McClelland, Opt. Express 13, 3185
(2005).
[23] In Ref. [24] a linewidth of Ŵ401 /2π = 36 MHz has beenmeasured via fluorescence spectroscopy on atomic beams
while in Ref. [25] a value of Ŵ401 /2π = 29.5 MHz has beenobtained from lifetime measurements. We measure a linewidthof 27.5(4) MHz by using an absorption imaging technique onmagneto-optically trapped atoms in the regime of low saturation
parameters.
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[27] Although we could clearly observe the spectroscopic signals forthe different Er isotopes using a hollow-cathode lamp, we prefer

to lock the 583-nm light to an ultralow-expansion cavity in orderto reach higher-frequency stability.
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Phys. 71, 1 (1999).
[29] The atomic cloud displacement x from the trap center is givenby μB ′ x = h̄583 , where μ is the difference in magnetic
moments between the ground and excited state.
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the distorted (semishell) shape of the MOT makes it difficult to
extract reliable values for the density and because the MOT may
be in the high-density limit regime [28].
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6.3. Further work96

6.3. Further work
6.3.1. Spin polarization
As pointed out earlier an advantage of the narrow-line MOT with large detuning is thatthe atomic sample gets spin-polarized without the need for an extra repumping laser1 . Thiseffect is related to the gravitational sag in the MOT, which causes the atoms to preferably
scatter the lower MOT beam, see Fig. 6.3(b). Dependent on the polarization of the lower
beam atoms get pumped into a stretched Zeeman state, which is in case of σ − -polarized lightfor the lower beam the energetically lowest mJ = −6 state of erbium.
The polarization of the spin state happens already during MOT loading. By applying anhomogeneous magnetic field of 855 mG at the beginning of the cMOT stage, see Fig. 6.2(e),the spin-polarization can be easily maintained also when one turns off the magnetic field
gradient for loading into the ODT. The degree of spin-polarization was checked by performinga Stern-Gerlach-type experiment. Atoms are first loaded from the cMOT into the ODT whereforced evaporation was performed. This results in a final temperature of 140 nK. To separate
different mJ states a magnetic field gradient of about 5 G/cm was applied for 4 ms afterswitching off the trap. The magnetic field gradient leads to a spin-dependent force, which
spatially separates different spin states. Further separation happens during a subsequenttime-of-flight of another 16 ms such that different spin components can be separately imagedon the CCD.
Figure 6.5(a) shows the result of this measurement. The spin-purity is determined by theratio between the area of a Gaussian fit to mJ = −6 and the overall area enclosed by thedata points. This gives a spin purity of 99(1)%. By driving a radio-frequency transition
in resonance with the energy of the Zeeman splitting energetically higher mJ -states canbe intentionally populated, see Fig. 6.5(b-d). This was done in presence of a homogeneous
magnetic field of 855 mG.

1

Due to a finite branching ratio into unwanted spin states in case of alkali metals and unwanted electricstates in case of alkali-earth metals during the MOT loading, repumping lasers have to be used to bringatoms into the desired spin state at the end of MOT loading.

(a)

(b)

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210

2005 tRF (µs)

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0.5
1.0
norm. atom number

Figure 6.5.: Population of Zeeman sub-states in an ultracold thermal sample measured in a SternGerlach-type experiment. (a) When no radio-frequency (RF) is applied only the lowest spin state ispopulated and thus spin polarization of the atomic sample is verified. (b) When applying a RF for
90 µs at a power of 23 dBm energetically higher-lying Zeeman states are populated. (c) After 210 µs
mostly states with mJ ≥ 0 are populated. (d) For even longer RF durations the atoms are distributed
equally in all Zeeman sub states. (e) Integrated atom number from (d) is fitted by Gaussian peaks
for each mJ state (line). Small deviations in the spacing between different mJ -peaks are due to a
spatial inhomogeneity of the magnetic gradient field.

97

Chapter

7

Publication: Bose-Einsteincondensation of erbium

7.1. Introduction
The successful demonstration of narrow-line laser cooling using the transition at a wavelengthof 583 nm was important to further proceed with achieving a Bose-Einstein condensate (BEC)of erbium atoms for the first time. After preparing the compressed MOT (cMOT), atoms
are loaded into an optical dipole trap (ODT) where they undergo forced evaporative cooling.
Regarding dipolar lanthanide atoms, the approach of forced evaporation has been also applied
to prepare a dipolar BEC of dysprosium atoms, published in Ref. [Lu11].
Since the atomic polarizability of erbium was not known in the beginning a first crossed
ODT was set up using two fiber-amplified lasers operating at a wavelength of 1064 nm and1070 nm. The first laser was used to create a trapping beam in horizontal direction whereas
the second one was used in vertical direction. Atoms had been loaded with an efficiency of
about 10 % from the cMOT into the ODT within the first 150 ms with only a small heatingof up to a temperature of 42 µK. After a total loading time of 600 ms forced evaporative
cooling was performed. Initial experiments on 166 Er showed insufficient thermalization rates.
Later it was found out that this isotope has only a small background scattering length.
Changing the isotope to 168 Er lead directly to the observation of a bimodal distribution
at low temperatures1 , which is the smoking gun of the appearance of a condensed phase.
An onset of condensation was seen at a temperature of about 400 nK with 2 × 105 atoms.
Finally, a pure BEC was achieved with 6.8 × 104 atoms by further evaporation, showing
its characteristic Thomas-Fermi density distribution and an inversion of the aspect ratio
after releasing it from the trap. A Feshbach scan at magnetic fields below 3 G revealed up
to six Feshbach resonances and gave important information about the scattering properties
for optimizing the evaporation procedure. The BEC was mutually brought to collapse byramping from above closer and closer to a prominent Feshbach resonance at 914 mG. Thisrevealed the tunability of the s-wave scattering length as well as the presence of dipole-dipole
interaction by the observation of an anisotropic d-wave pattern in the absorption image.
1

The first observation of the erbium BEC was made on the 8th of March 2012, about two years after starting
to design and set up the experiment.

98

99

7.1. Introduction

In the following section an improved evaporation procedure will be described. Section 7.2
contains the publication on the erbium BEC. In Sec. 7.3 further additional work is presented.
In the first half of that section the cross-dimensional rethermalization method used for measuring the scattering length for various isotopes of erbium is discussed. In the second half
the mass dependence of the s-wave scattering length for various isotopes is shown and fitted
by a theoretical model.

7.1.1. Evaporative cooling
Due to the small phase-space density in the MOT atoms need to be evaporatively cooled in
the ODT to produce a degenerate quantum gas, see Ref. [Ket99]. The phase-space density2of a quantum gas is defined as
PSD = n0 λ3dB ,7.1
where n0 is the peak number√density of the gas and λdB the thermal de Broglie wavelength,
which is proportional to 1/ T . The transition into the condensed phase happens whena macroscopically large number of bosons occupy the ground state and the phase-spacedensity3 approaches a value of 2.6. A large value of the phase-space density requires both
low temperature and high density of the gas. In a harmonic trapping potential with trap
frequencies ωi with i = x, y, z in the three spatial directions and geometrical mean trap
frequency ω = (ωx ωy ωz )1/3 the peak number density can be calculated asn0 = N ω

3m
2πkB T

3
2

,

7.2

with N being the total atom number. The phase-space density can be expressed by using
this equation as


~ 3
3
.
7.3
PSD = N ω
kB T

Increasing the PSD by evaporative cooling is based on actively truncating the thermal
Maxwell-Boltzmann distribution by removing only hot particles from the system and letting the residual atoms rethermalize via elastic collisions. In dipolar systems the elastic
cross section arises from the long-range dipole-dipole interaction (DDI). Refer to Sec. 4.2.1
and Eqs. 4.19 and 4.20 for expressions of the elastic cross section of bosons and fermions,
respectively. Note that for bosons the DDI gives an additional contribution to the contactinteraction cross section whereas for fermions it causes the cross section to be nonzero incontrast to non-dipolar fermions. This opens the possibility to evaporatively cool of indistinguishable fermions, see Chapter 9. In both cases the truncation of hot particles can beeasily achieved by adiabatically lowering4 the potential depth of the ODT. Increasing thephase-space density by evaporative cooling relies on a favorable ratio of elastic to inelastic

2

3

4

The phase-space density
relates the mean particle distance, expressed by n0 , to the extension of the particle,√
given by λdB = h/ 2πmkB T .
For a 3D gas in a harmonic trapping potential, one would expect the onset of condensation at a criticalphase-space density of PSD = 1.2, see Ref. [Pet02], but it can be shown that for kB T ≪ ~ω the criticalphase-space density is 2.6, see Ref. [Cas01].
It is crucial to lower the potential depth adiabatically because of two reasons. First, the gas has to haveenough time to rethermalize and second, excitations in the trap have to be avoided.

ramp 1 2

3

4

5

6

7

8

9

10

duration 50 350
(ms)

900

700

900

900

700

700

800

900

2
3
4
5
evaporation time

6

(a)

20
y

power (W)15

vertical beam

wx = 112 µm
wz = 70 µm

ARw = w *AR µm

yhorizontal beam wx = 16.1
µm

10
8
6
4
2
0

10

1

7

5
0
trap frequency (kHz)

(b)
fx

1

fy
fz

0.1

magn. field (G)0.01(c) 2.0

atom number

(d)

1.5

(2)

1.0

FR at 0.914 G

0.5

(1)

0.0
107

106

105

(e)

BEC

psd

1
10-1
10-2
10-3
0

1

2

3
4
evaporation time (s)

5

6

7

Figure 7.1.: Evaporation trajectory after loading 168 Er into the ODT. Parameters of each ramp
have been optimized to achieve the largest possible increase of the phase-space density (e) with thesmallest loss of atom number (d). (b) Trap frequencies are calculated from the laser power (a) andmeasured waists of the horizontal and vertical beams. (c) The magnetic field is shown with theposition of a Feshbach resonance (FR) at 914 mG (gray dashed line). Evaporative cooling is mostly
carried out at 0.4 G (1) but for special applications, e. g. where the cloud has to be prepared above
the FR, we jump to 1.2 G at the beginning of the ninth ramp. This is indicated by the dashed line
(2).

100

101

7.1. Introduction

collision rates. The former is needed to provide efficient rethermalization between atoms
within a reasonable time whereas the latter leads to losses especially at high densities andshould therefore be small.
A major improvement of the loading from the cMOT into the ODT was made by replacing
the initially used multi-mode fiber laser in the horizontal direction by a single-mode diodepumped solid state laser5 and implementing the scanning system for the horizontal trappingbeam, see Ref. [Bai12a] and Sec. 3.5.3. By deflecting the trapping beam in a fast way using
an accusto-optical modulator a time-averaged potential is created. This lets one optimize
the spatial overlap between the cMOT and the ODT and improves the loading efficiency
from 10% to 35%. Furthermore, it allows for a dynamical change of the aspect ratio of the
ODT during the experiment.
For efficient evaporation it is important to find an optimum set of parameters for an adiabatic
ramp of the potential depth. Lowering the trapping potential depth is done by decreasing
the trapping laser power. As described in Sec. 3.5.3 atoms are initially loaded from thecMOT into a single horizontal trapping laser beam. Typical conditions at this stage are
1.2 × 106 atoms at a temperature of 22 µK, which corresponds to a PSD of 3.1 × 10−4 . The
power of the horizontal laser beam is decreased in an exponential-like piecewise linear ramp
consisting of a total of ten steps. Figure 7.1 presents the important ramp parameters forevery evaporation step, including the laser power, trap frequencies, homogeneous magnetic
field, atom number, and phase-space density. During the first two ramps the aspect ratio of
the horizontal beam is lowered from a value of ten, used for loading from the MOT, to two,see inset in Fig. 7.1. This spatially compresses the sample before evaporation is performed.
At the fourth step the power of the vertical beam6 is ramped up, which gives a much stronger
confinement in the axial direction of the horizontal beam and thus a strong increase of thedensity as well as the phase-space density is observed, see Fig. 7.1(e). The magnetic field is
usually changed to 0.4 G after the third ramp, which gives a more favorable scattering length
during evaporation of about 240 a0 . As shown in Fig. 7.1(c) there is a Feshbach resonance(FR) located at 914 mG. For special applications, which require to be at a magnetic field
above the FR, like for driving the dipolar collapse or creating Er2 molecules via magnetoassociation, we jump back above the FR to 1.2 G for the last two evaporation stages, seeFig. 7.1(c). After completing the full evaporation ramp up to 2 × 105 atoms are prepared in
an almost pure BEC within 7 s. Final laser powers are 0.3 W for the horizontal and 0.7 W
for the vertical trapping beam, giving trap frequencies of (ωx , ωy , ωz ) = 2π × (35, 98, 31) Hz.
An in-detail description of the evaporation process and its optimization can be found in
Ref. [Bai12a].
The evaporation efficiency equals the order of magnitude increase of the phase-space density
at the expense of an order of magnitude of atom number, see Ref. [Ket96]. It is given byγ=−d(ln PSD).
d(ln N )7.4

The evaporation efficiency can be easily determined by plotting the phase-space density
against the atom number in a log-log plot, as seen in Fig. 7.2. The slope of a linear fit gives
5
6

model Mephisto MOPA, 42 W at 1064 nm, from Innolight (now Coherent)
created by a fiber laser, model PYL-10-1064-LP, 10 W at 1064 nm, from IPG Photonics

102

7.1. Introduction

1

psd

10-1
10-2
10-3
10-4

106

107
atom number

evaporation ramp

Figure 7.2.: Evaporation efficiency of cooling 168 Er. The increase of the phase-space density isplotted against the loss of atom number. Evaporative cooling proceeds along the direction following
an increasing phase-space density, indicated by the arrow. The slope of the linear fit following Eq. 7.4(solid line) gives an overall efficiency of the evaporation process of γ = 3.4. For the final evaporation at
small atom numbers the efficiency is about two (upper dashed line), which is similar to the efficiency
at the beginning of evaporation (lower dashed line).

the overall evaporation efficiency, which is in case of 168 Er evaluated to be γ = 3.4. The
sudden increase of the phase-space density is caused by the ramping of the vertical trapping
beam during the fourth evaporation ramp. This evaporation efficiency is among the highest
efficiencies available in modern quantum gas experiments and is usually achievable by using
a so-called dimple technique or other more elaborate techniques. For non-dipolar systems ina simple crossed dipole trap, like in the presented case, typical values for γ range from about
one up to two.

7.1.2. Dipolar Bose-Einstein condensate
At the many-body level the s-wave contact interaction is given by the contact potential
Ucont (r) =4π~2 asδ(r) = gδ(r) ,
m

7.5

2

with δ the delta function and g = 4π~m as the coupling constant. The contact interaction
can be changed by tuning the s-wave scattering length as . The s-wave scattering length canbe compared to the dipolar scattering length, aD , see Eq. 4.17, by calculating the dipolar
constant ϵdd as
2 aD
.
7.6
ϵdd =
3 as
It is important that ϵdd is non negligible if dipolar effects should be observable in a systemof ultracold particles. Some numerical values of aD and ϵdd for magnetic and electric dipolarsystems are summarized in Table 7.1.
A Bose-Einstein condensate with contact and dipole-dipole interaction is described by the

103

7.1. Introduction
species

dipole momentaD (a0 )

ϵdd

87 Rb

1 µB
6 µB
7 µB
10 µB
0.6 D
3.0 D

1.1
24
99
196
3.0 × 103
3.6 × 104

0.007
0.2
0.3
0.7
20
240

52 Cr
168 Er
164 DyKRb
HCN

Table 7.1.: Dipolar scattering length aD and dipolar constant ϵdd for different systems in comparison.
To observe dipolar effects at ultracold temperatures ϵdd should be non-negligible. Most of the data
was taken from [Lah09].

following Gross-Pitaevskii equation,
i~


∂ψ
~2
=−
∆ψ + Uext + g|ψ|2 + Φdd ψ ,
∂t
2m

7.7

with ψ the many-body wave function, Uext an external trapping potential, and Φdd thedipolar contribution. The dipolar contribution can further be written asZΦdd (r, t) = |ψ(r′ , t)|2 Udd (r − r′ )d3 r′ ,7.8with Udd the dipole-dipole interaction potential, see Eq. 4.2. In case of a spherically symmetric cloud of radius R = Ri with i = x, y, z trapped in an ODT with trapping frequencyω = ωi the dipolar contribution Φdd can be written to first order in ϵdd as, see Ref. [Gio02],(
r2 if r < Rmω 2
2(1 − 3 cos θ) R5
7.9
Φdd (r) = ϵdd
5
if r > R .
r3
Thus the dipolar mean-field potential has the shape of a saddle with two minima along thedirection of the dipole orientation. This means that the cloud will be elongated in that
direction because it is energetically more favorable.
The elongation is the most obvious example for an effect of the DDI in a harmonicallyconfined BEC. The effect is often called magnetostriction and was predicted in Ref. [San00].
The first observation of this effect was made using a BEC of 7 Li atoms in spite of its small
magnetic moment of 1 µB , see Ref. [Pol09]. Magnetostriction stems from magnetic forces
within the cloud, which change the cloud’s shape and volume.
A second effect of the DDI is the emergence of a d-wave pattern when the BEC collapses.
The numeric factor in Eq. 7.6 is chosen such that a BEC will become unstable with ϵdd ≥ 1.
The contact interaction can be tuned in a broad range by an external magnetic field usinga Feshbach resonance and thus also ϵdd can be changed. Utilizing this tunability a stablecondensate can be mutually brought to collapse. This demonstration of a collapse is used
as a smoking gun for a tunable and dipolar quantum gas and was first demonstrated by
the group of T. Pfau in Stuttgart (Germany) in Ref. [Lah08] for a gas of chromium atoms.
A confirmation of the erbium BEC collapse is given within the publication in the following
section.

7.2. Publication:
Bose-Einstein Condensation of Erbium†
Physical Review Letters 108, 210401 (2012)
submitted 6 April 2012; published 25 May 2012
DOI 10.1103/PhysRevLett.108.210401
Editor’s suggestion and selected for a Viewpoint in Physics
Kiyotaka Aikawa,1 Albert Frisch,1 Michael Mark,1 Simon Baier,1 Alexander Rietzler,1 Rudolf
Grimm,1,2 and Francesca Ferlaino1
1

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

The author of the present thesis made strong contributions in constructing the experimental apparatus
together with A. R., S. B., K. A. and M. M., analyzing the data, and writing the manuscript.

104

PRL 108, 210401 (2012)

Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS

week ending
25 MAY 2012

Bose-Einstein Condensation of Erbium
K. Aikawa,1 A. Frisch,1 M. Mark,1 S. Baier,1 A. Rietzler,1 R. Grimm,1,2 and F. Ferlaino1
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria(Received 6 April 2012; published 21 May 2012)
We report on the achievement of Bose-Einstein condensation of erbium atoms and on the observation ofmagnetic Feshbach resonances at low magnetic fields. By means of evaporative cooling in an optical dipole
trap, we produce pure condensates of 168 Er, containing up to 7  104 atoms. Feshbach spectroscopy reveals
an extraordinary rich loss spectrum with six loss resonances already in a narrow magnetic-field range up to
3 G. Finally, we demonstrate the application of a low-field Feshbach resonance to produce a tunable dipolar
Bose-Einstein condensate and we observe its characteristic d-wave collapse.
DOI: 10.1103/PhysRevLett.108.210401

PACS numbers: 03.75.Nt, 37.10.De, 51.60.+a, 67.85.Hj

Ultracold quantum gases have proven to be ideal
systems for observing spectacular many- and few-bodyquantum effects. The large majority of these phenomena
rely on the high degree of control over the interparticle
interaction achieved with ultracold atoms. In the widelyused alkalis, ultracold atoms interact isotropically via ashort-range contact potential. A novel exciting frontier inquantum gas experiments is to access unexplored physicalscenarios based on the anisotropic and long-range dipoledipole interaction (DDI) [1,2]. A dipolar quantum gas is
expected to exhibit fascinating phenomena, including
novel many-body quantum phases [3–6]. The DDI actsin systems having sizable electric or magnetic dipole
moments [1].
In the context of ultracold atomic quantum gases,
pioneering experimental work on strong DDI has been
carried out with chromium atoms [7–9]. Magnetic lanthanides offer new possibilities for dipolar physics. In such
systems, the combination of a large magnetic moment anda large atomic mass leads to a particularly strong dipolar
character. The demonstration of the first magneto-opticaltrap of erbium atoms [10] stimulated growing interest insuch species for quantum gas experiments. Very recently, a
Bose-Einstein condensate (BEC) and a degenerate Fermi
gas of dysprosium have been produced [11,12]. We choose
erbium as a promising candidate for experiments on dipolar quantum gases. This species has a number of very
appealing features, including a large magnetic moment of 7 times the Bohr magneton, several stable isotopes, arich energy level scheme [13] with a non-S electronic
ground state [14], and interesting cold collisional
phenomena [15,16].
In strongly magnetic atoms, the competition between the
DDI and the contact interaction is very important and gives
rise to many intriguing phenomena. The contact interaction
is determined by the s-wave scattering length a and can beoften tuned with external magnetic fields via Feshbachresonances [17]. Tuning of a also controls the balance of
these two interactions. In the case of a novel species in
0031-9007=12=108(21)=210401(5)quantum gas experiments, Feshbach resonances and scattering lengths are a priori unknown. Magnetic lanthanides
such as erbium with their large magnetic moments andtheir non-S electronic ground states present a completely
unexplored terrain in ultracold scattering physics. Here theanisotropic interaction is expected to give rise to novel
scattering scenarios, which are not accessible with alkali
atoms [18,19].
In this Letter, we report on the attainment of
Bose-Einstein condensation of erbium atoms and on theobservation of Feshbach resonances in the region of low
magnetic fields. We obtain pure optically trapped BECs of
168
Er containing 7  104 atoms. The remarkably highefficiency of evaporative cooling in a standard optical
dipole trap indicates favorable scattering properties of the
168 Er isotope. In addition, the magnetic Feshbach spectroscopy provides first valuable information on the scattering
behavior of submerged-shell atoms at ultralow temperatures. Moreover, we demonstrate low-field Feshbach tuning of the contact interaction in our strongly dipolar BEC.
Our experimental procedure to create a BEC of Er
follows a simple and straightforward scheme, inspired bywork on Yb atoms [20,21]. Our starting point is thenarrow-line yellow magneto-optical trap (MOT) describedin our very recent work [22]; it operates on the 583 nm line
(natural linewidth 190 kHz). We choose this approach
because narrow-line MOTs permit us to obtain samples
with a large number of atoms at temperatures in the lower
microkelvin region. This allows a direct and efficienttransfer of atoms into optical dipole traps without theneed for additional cooling stages [20,21]. Our MOT gives
about 108 atoms at a temperature of 15 K [23].
An additional very advantageous feature of our approachis that the MOT light automatically pumps the atoms into
the lowest Zeeman sublevel mJ ¼ 6, where mJ is the
projection quantum number of the total electronic angularmomentum J ¼ 6. This effect results from the interplay
between gravity and weak radiation pressure, which leads
to a spatial down shift with respect to the zero of the210401-1

Ó 2012 American Physical Society

week ending
25 MAY 2012PHYSICAL REVIEW LETTERS

magnetic quadrupole field [24] and thus to a preferential
absorption of the vertical MOT beam with  polarization[22]. The polarization of the sample is confirmed by
Stern-Gerlach-type measurements.
Our optical dipole trap geometry follows the concepts
originally applied in experiments with Yb BEC [20]. The
trap is created by crossing a tightly confining horizontal
beam (y axis) and a less focused vertical beam (z axis). The
basic idea is that initially the atoms are predominantly
trapped by the horizontal beam, whereas the verticalbeam provides confinement relevant in the final stage of
evaporation. The horizontal beam is derived from a 100 W
broadband Yb fiber laser operating at 1075 nm and has an
initial power of 10 W. The beam has an elliptic crosssection with a waist of 30ð40Þ m along the vertical(horizontal) direction. The vertical beam is produced by
a 10 W Yb fiber laser source at 1064 nm and has an initial
power of 8 W. The beam profile is elliptic with a waist of55ð110Þ m along (perpendicular to) the axis of the horizontal beam.
We load the dipole trap during the MOT compression
phase. We observe that the time period in which the compressed MOT and dipole trap coexist is crucial for efficientloading. The number of atoms in the optical dipole trap
exponentially approaches its maximum value with a time
constant of about 150 ms. After 600 ms of loading, we turnoff the MOT beams and the quadrupole magnetic field, andwe switch on a weak homogeneous magnetic field of about
300 mG along the vertical direction to preserve the spinpolarization of the sample. We obtain 2:6  106 atoms at atemperature of 42 K in the optical dipole trap. At thispoint, the atoms are mainly trapped by the horizontal
beam. We measure oscillation frequencies ðx ; y ; z Þ ¼
ð1:3; 0:016; 1:95Þ kHz; the potential depth is estimated to
be 560 K. The peak density and the peak phase-spacedensity are 1:7  1013 cm 3 and 1:6  10 4 , respectively.
These are our starting conditions for the evaporative
cooling process.
Forced evaporative cooling is performed by reducing thepower of the trapping beams in a nearly exponential ramp.
The overall evaporation sequence has a duration of 5.5 s[25]. We then turn off the trapping beams and let the atomic
cloud expand before applying standard absorption imaging. For imaging, we illuminate the atomic cloud with a
50-s probe beam pulse [26]. The probe beam propagateshorizontally at an angle of 14 with respect to the propagation axis (y axis) of the horizontal trapping beam.
The phase transition from a thermal cloud to BEC manifests itself in a textbooklike bimodal distribution in the timeof-flight absorption images. Figure 1 shows the absorption
images and the corresponding linear density profiles fordifferent final temperatures, i.e., for different stages of theevaporation. At higher temperatures the atomic distribution
is thermal with the expected Gaussian profile resulting
from the Maxwell-Boltzmann distribution; see Fig. 1(a).

(a)

T=1100 nK(b)

408 nK

linear density (108 m-1)

PRL 108, 210401 (2012)

0.4

(c)

222 nK

(d)

0.3
0.2
0.1
0

1mm

position (µm)

FIG. 1 (color online). Absorption images and integrated
density profiles showing the BEC phase transition for different
evaporation times. The absorption images are an average of four
images taken after 24 ms of expansion. The color bar shows the
optical density. The solid lines are fits to the data using
Gaussian (a), bimodal (b) and (c), and Thomas-Fermi (d)distribution. The dotted lines represent the Gaussian part ofthe bimodal fit, describing the thermal atoms. From the fit
we extract: N ¼ 3:9  105 , T ¼ 1100 nK (a), N ¼ 2:1  105 ,T ¼ 408 nK (b), N ¼ 1:6  105 , T ¼ 222 nK (c), N ¼
6:8  104 (d), where N is the total atom number. For (b) and(c), we extract a condensate fraction of 5% and 20%, respectively.

By cooling the atomic sample below the critical temperature, we clearly observe that the atomic density distribution
has a bimodal profile with a narrower and denser peak at thecenter, which represents the BEC (b). By further evaporating
the sample, the BEC fraction continuously increases(c) until the thermal component is not anymore discernible
and an essentially pure BEC is formed with 7  104
atoms (d).
To analyze our data, we fit a bimodal distribution to theintegrated time-of-flight absorption images. This distribution consists of a Gaussian function, which accounts for thethermal atoms, and an inverted integrated parabolic function for the BEC component in the Thomas-Fermi limit.

210401-2

PRL 108, 210401 (2012)PHYSICAL REVIEW LETTERS

Just after the onset of quantum degeneracy (BEC fraction5%), we measure trap frequencies of ðx ; y ; z Þ ¼
ð208; 70; 299Þ Hz, atom number of N ¼ 2:1  105 , and atemperature of T ¼ 408 nK. The critical temperature of
417 nK as calculated from standard BEC theory (without
interaction shift) is consistent with this observation.
The evaporation efficiency is found to be remarkably
high as 3.5 orders of magnitude in phase-space density are
gained by losing a factor of 10 in atom number. Thisobservation already points to favorable scattering parameters of the 168 Er isotope. First evaporative cooling experiments on the most abundant 168 Er isotope reveal a lower
efficiency in the final stage of evaporation, suggesting thata different strategy might be needed to reach BEC.
To gain insight into the ultracold collisional properties
of erbium we perform Feshbach spectroscopy [17] at low
magnetic fields. This measurement is done in a way thatallows us to identify both the poles and zero crossings ofFeshbach resonances [27,28]. The basic idea here is toprepare the system at a variable target value of the magnetic field and then to rapidly (50 ms) decrease the depth ofthe optical dipole trap by almost a factor of 2. The sample
stays near thermal equilibrium with an effective temperature of 2:2 K but features a truncated energy distribution.
We then let the system evolve at a constant trap depth
for 250 ms, during which plain evaporative cooling andinelastic losses can occur depending on the scattering
length. We finally switch off the trap and take time-offlight images to determine the temperature and number of
the atoms. The measurement is then repeated for variablemagnetic-field values. Such a Feshbach scan shows resonance poles as loss features and zero crossings as temperature maxima.
Figure 2 shows the loss spectrum and the corresponding
temperatures in the low magnetic-field range up to 3.2 G[29]. Already in this narrow magnetic-field range, the loss
spectrum is very rich. We identify six pronounced resonant
minima in the atom number that we interpret as beingcaused by Feshbach resonances. For convenience, we determine the resonance positions with Gaussian fits, yielding 0.72, 0.91, 1.51, 2.16, 2.48, and 2.85 G. The lossfeatures show different strengths and widths. For the threebroader resonances at 0.91, 2.16, and 2.48 G, we also
observe the appearance of temperature maxima to the rightof the loss minima (arrows in Fig. 2). These temperature
maxima mark the zero crossings of the scattering length.
The other loss features are too narrow to provide clear
signatures of the zero crossing. From the difference inpositions between the minima in the atom number andthe maxima in temperature we estimate the widths  ofthe resonances. We find  ¼ 65, 60, and 180 mG for the
resonances at 0.91, 2.16, and 2.48 G, respectively.
In a strongly dipolar atomic gas, universal dipolar scattering is present [30–32], so that the total cross section forelastic scattering does not vanish at the zero crossings ofweek ending
25 MAY 2012

FIG. 2 (color online). Observation of Feshbach resonances in
Er-Er collisions. The measured temperature (a) and atom
number (b) are plotted as a function of the magnetic field. The
minima in the atom number indicate the Feshbach resonancepoles, marked by the thin vertical lines. The maxima in thetemperatures to the right of the three stronger loss features(arrows) are attributed to the respective zero crossings of thescattering length. The varying background in the atom number ispresumably due to ramping effects caused by the sweep of themagnetic field across the resonances.

the scattering length. For Er, a minimum cross section
dip ¼ 8ð30a0 Þ2 results from universal dipolar scattering, where a0 is the Bohr radius. The fact that we observe
temperature maxima near the zero crossings suggests a
dominant role of s-wave scattering and not of dipolar
scattering. Preliminary cross-dimensional thermalization
measurements indeed point to a scattering length between
150 and 200a0 .
The existence of Feshbach resonances at low magnetic
fields makes the manipulation of the contact interaction in
the Er BEC very convenient and straightforward. As aproof-of-principle experiment, we explore the controlled
d-wave collapse of the BEC, following the procedure
successfully applied by the Stuttgart group [7]. We first
produce a pure BEC by evaporative cooling at 1.2 G, which
is above the position of the first broad Feshbach resonance(0.91 G). Here we obtain 3  104 atoms in the BEC,
indicating that forced evaporation at this magnetic field isslightly less efficient. We then ramp down the magnetic
field within 2 ms to a variable target value and let the
sample evolve for 2 ms before switching off the trap. The
magnetic field is kept constant at its target value during thefirst stage of the expansion (15 ms), where the main
dynamics happens. We then set the magnetic field along
the y axis and we image the atomic cloud after an additional 11 ms of expansion. Our results are summarized inFig. 3, where we show time-of-flight absorption images fordifferent values of the target magnetic field. We observe a
dramatic change in the shape of the condensate when the210401-3

PHYSICAL REVIEW LETTERS

PRL 108, 210401 (2012)
(a)

(b)

(c)

0.4

0.3

(d)

(e)

(f)

0.2

0.1
0

FIG. 3 (color online). Absorption images showing the d-wave
collapse of the BEC. The field of view is 290 m  290 m.
The images are an average of eight pictures. The color bar shows
the optical density. The images are taken for different target
values of the magnetic field: 1.208 G (a), 0.963 G (b), 0.947 (c),0.942 G (d), 0.939 G (e), and 0.934 (f). We note that the actual
magnetic-field value might be slightly higher (10 mG) becauseof ramping issues and eddy currents [7].

magnetic field is reduced towards the zero crossing of
the scattering length. At the magnetic field of evaporation,
the aspect-ratio of the cloud is close to the one observed atzero magnetic field; see Fig. 3(a). By changing the magnetic field to lower target values, the BEC shows a more
and more anisotropic shape (b) and (c). Below a criticalmagnetic-field value, the BEC develops a complicatedcloverleaf pattern (d)–(f) which is the striking signatureof the d-wave collapse in a dipolar BEC [7].
In conclusion, we have demonstrated the first BEC of
Er atoms and the tunability of its interparticle interaction
via Feshbach resonances. Our scattering data provide firstsensitive input to understand the complex collisional
behavior of submerged-shell atoms. The observation of
the d-wave collapse is very encouraging in view of future
experiments dedicated to the rich phenomena expected instrongly dipolar quantum gases.
We are grateful to F. Schreck, S. Stellmer, and
B. Pasquiou (Innsbruck Sr team) for their continuous support. We thank the Yb teams in Kyoto and Tokyo for theiradvice, M. Springer for technical support, and E. Zupanič
for contributions in an earlier stage of the experiments.
This work is supported by the Austrian Ministry of Scienceand Research (BMWF) and the Austrian Science Fund(FWF) through a START grant under Project No. Y479N20 and by the European Research Council under ProjectNo. 259435.

[1] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T.
Pfau, Rep. Prog. Phys. 72, 126401 (2009).
[2] M. A. Baranov, Phys. Rep. 464, 71 (2008).
[3] K. Goral, K. Rzazewski, and T. Pfau, Phys. Rev. A 61,
051601 (2000).

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25 MAY 2012[4] B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S.
Kotochigova, P. Julienne, and P. Zoller, Phys. Rev. Lett.
90, 110401 (2003).
[5] G. Pupillo, A. Micheli, H. P. Büchler, and P. Zoller, in
Cold Molecules: Theory, Experiment, Applications, editedby R. V. Krems, W. C. Stwalley, and B. Friedrich (CRC
Press, Boca Raton, 2009), Chap. 12.
[6] D. S. Jin and J. Ye, Phys. Today 64, 27 (2011).
[7] T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A.
Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda,
Phys. Rev. Lett. 101, 080401 (2008).
[8] T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A.
Griesmaier, S. Giovanazzi, and T. Pfau, Nature (London)
448, 672 (2007).
[9] B. Pasquiou, G. Bismut, E. Maréchal, P. Pedri, L. Vernac,
O. Gorceix, and B. Laburthe-Tolra, Phys. Rev. Lett. 106,015301 (2011).
[10] J. J. McClelland and J. L. Hanssen, Phys. Rev. Lett. 96,
143005 (2006).
[11] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys.
Rev. Lett. 107, 190401 (2011).
[12] M. Lu, N. Q. Burdick, and B. L. Lev, arXiv:1202.4444[Phys. Rev. Lett. (to be published)].
[13] H. Y. Ban, M. Jacka, J. L. Hanssen, J. Readerand, and J. J.
McClelland, Opt. Express 13, 3185 (2005).
[14] The Er electronic configuration is [Xe] 4f12 6s2 . The
electronic ground state is a 3 H6 state, meaning L ¼ 5(H state), S ¼ 1, and J ¼ 6. L, S, J denote the quantum
numbers for the orbital angular momentum, for the spin,and for the total electronic angular momentum, respectively. 168 Er has zero nuclear spin (I ¼ 0) and consequently does not exhibit a hyperfine structure.
[15] R. V. Krems, G. C. Groenenboom, and A. Dalgarno, J.
Phys. Chem. A 108, 8941 (2004).
[16] C. B. Connolly, Y. S. Au, S. C. Doret, W. Ketterle, andJ. M. Doyle, Phys. Rev. A 81, 010702 (2010).
[17] C. Chin, R. Grimm, P S. Julienne, and E. Tiesinga, Rev.
Mod. Phys. 82, 1225 (2010).
[18] S. Kotochigova and A. Petrov, Phys. Chem. Chem. Phys.
13, 19165 (2011).
[19] A. Petrov, E. Tiesinga, and S. Kotochigova, arXiv:1203.4172.
[20] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M.
Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev.
Lett. 91, 040404 (2003).
[21] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A
76, 051604(R) (2007).
[22] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler,
E. Zupanic, R. Grimm, and F. Ferlaino, arXiv:1203.1460[Phys. Rev. A (to be published)].
[23] The MOT stage includes a 15 s loading phase, in which thelight is red detuned by 9:5 MHz, and a compression
phase, in which we reduce the detuning to 0:8 MHz and
the MOT beam intensity by about a factor of 400. The
compression phase is used to obtain colder and denser
samples and it is optimized for loading efficiency into thedipole trap.
[24] H. Katori, T. Ido, Y. Isoya, and M. Kuwata-Gonokami,
Phys. Rev. Lett. 82, 1116 (1999).
[25] Initially only the power of the horizontal beam is reduced
to its final value of 230 mW, while the vertical beam power
is reduced during the last 600 ms of evaporation to its final

210401-4

PRL 108, 210401 (2012)PHYSICAL REVIEW LETTERS

value of 1.5 W.
[26] We use the strong blue transition at 401 nm for imaging.
[27] S. Jochim, M. Bartenstein, G. Hendl, J. H. Denschlag, R.
Grimm, A. Mosk, and W. Weidemüller, Phys. Rev. Lett.
89, 273202 (2002).
[28] M. Zaccanti, C. D’Errico, F. Ferlaino, G. Roati, M.
Inguscio, and G. Modugno, Phys. Rev. A 74, 041605(2006).
[29] The magnetic field is calibrated via rf-spectroscopy
between Zeeman sublevels. From the calibration weweek ending
25 MAY 2012

estimate a short-term stability of 1 mG and a long-term
stability of about 10 mG.
[30] V. Roudnev and M. Cavagnero, Phys. Rev. A 79, 014701(2009).
[31] J. L. Bohn, M. Cavagnero, and C. Ticknor, New J. Phys.
11, 055039 (2009).
[32] K. K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B.
Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and
D. S. Jin, Nature (London) 464, 1324 (2010).

210401-5

110

7.3. Further work

7.3. Further work
7.3.1. Cross-dimensional rethermalization
The cross-dimensional rethermalization method is a powerful method for determining thescattering length of an atomic sample at ultracold temperatures, see Ref. [Mon93]. For thisthe atomic cloud has to be in thermal equilibrium before it is brought out of equilibrium in
one spatial direction. This can be achieved by increasing the temperature in one direction via
a sudden compression of the cloud. When the cloud is placed in a crossed ODT this compression can be easily realized by increasing the power of the vertical trapping beam. To reduce
systematic errors care has to be taken that only one direction is affected and no collective
oscillations are excited by this compression. The temperature increase in the direction ofcompression will eventually be distributed in the other directions as the kinetic energy of theparticles will be ‘distributed’ by elastic collisions and the gas will reach thermal equilibrium
again, i. e. a rethermalization process happens across different spatial dimensions.
The timescale of the rethermalization can be conveniently measured in the experiment byobserving the change of the size of the atomic cloud in the direction orthogonal to the direction of excitation. In the experiment the cloud is usually excited in the horizontal direction(y-direction) and the rethermalization is observed in the vertical direction (z-direction).
Figure 7.3 shows the increase of the vertical size after compression. It is described by an exponential function with time constant τ , which equals the thermalization time of the cloud.
It is inversely proportional to the elastic scattering cross section, σel , with
τ=α.
nσel v

7.10

Here, the term nσel v gives a collision rate with the mean number densityn0
n= √ ,
8

7.11

where n0 is the peak density in a harmonic trapping potential, see Eq. 7.2 and Ref. [Sch03],
and the mean relative thermal velocity, v, at temperature T with reduced mass mr is given
by
r
8kB T
v=
.
7.12
πmr

In Eq. 7.10 the dimensionless constant α has been introduced, which is often referred toas the number of collisions necessary for rethermalization. It can be computed from the
known differential cross section and depends on the probability of two horizontally colliding
particles to exit the collision in the vertical direction. It has been shown theoretically byMonte-Carlo simulations and also experimentally that for alkali-metal atoms α = 2.7 andα = 4.17 for s- and p-wave collisions, respectively, see Refs. [Mon93, DeM99a]. For particles
with a large dipole moment the scattering amplitude is anisotropic and thus α depends
on the angle θ between the excitation direction and the orientation of the dipoles set by apolarizing magnetic or electric field, see Ref. [Boh14]. This angular dependence of α has been
demonstrated in a cross-dimensional rethermalization measurement using a dipolar Fermi
gas of 167 Er atoms, see Ref. [Aik14b]. Surprisingly, α can be as small as one for θ = 45◦ for

111

7.3. Further work

vertical size (µm)

105
100
95
90

0

25

50

75

waiting time (ms)Figure 7.3.: A typical cross-dimensional rethermalization measurement for isotopes 168 Er (bluecircles) and 170 Er (red triangles). The time constant for thermalization is extracted by an exponential
fit (solid lines).

fermions, which makes rethermalization extremely efficient. The angle dependence is quite
different for bosons. In this case the minimum of α can be found at an angle of 90◦ with avalue of about two whereas in the fermionic case it is largest at exactly this angle with α = 4.
In both fermionic and bosonic cases α further depends on the initial degree of temperature
anisotropy Ty /Tz , which is being induced by the trap compression and is usually as large as
two, see Ref. [Boh14].
The cross-dimensional rethermalization method was used to determine the total elastic scattering cross section, σel , away from Feshbach resonances for various bosonic isotopes of
erbium. Atoms were trapped in the crossed ODT and a holding time of 500 ms ensured
thermal equilibrium. Then the power of the vertical beam was ramped from 1.6 W to 8.0 W
within 5 ms in a parabolic way to prevent collective oscillations. Typical final trap frequencies were (ωx , ωy , ωz ) = 2π ×(61, 35, 260) Hz. The angle was set to θ = 90◦ which corresponds
to a value for α = 2. Two examples of a measurement of the cloud size in z-direction versus
the waiting time after compression in the y-direction can be found in Fig. 7.3.
By using Eqs. 4.17 and 4.19 one can estimate a value for the s-wave scattering length, as , bysubtracting the contribution of the dipole-dipole scattering length, aD , from the measured
scattering length, a. Using the expression for the elastic cross section σel = 8πa2 we getr
2.234 2
as = a2 −
a .
7.13
8π D
Experimental values of the scattering length for the bosonic isotopes 164 Er, 166 Er, 168 Er, and170 Er are measured via the cross-dimensional rethermalization method and are summarizedin Table 7.2.

7.3.2. Scattering length mass scaling
The scattering length depends on the atomic mass of different isotopes in a tangent-likemanner. This was derived and supported by measurements using ytterbium in Ref. [Kit08]

112

7.3. Further work
isotope(amu)

magn. field
(G)

164
166
168
170

0.8, 1.0
0.4, 0.7
0.4
0.4, 0.5

scattering lengtha (a0 ) aD (a0 ) as (a0 )86(10)
78(12)
202(23)
223(22)

97.0
98.2
99.4
100.5

81(10)
72(13)
200(23)-221(22)

Table 7.2.: Homonuclear scattering lengths for four bosonic erbium isotopes determined by thecross-dimensional rethermalization method. The measured scattering length is given by a, the dipole
length is aD , and the s-wave scattering length by as . The values for the scattering length are averaged
over a few measurements at different magnetic fields far away from Feshbach resonances. The valuesof the magnetic field at which measurements were taken are given. The sign of a for 170 Er wasdetermined by locating the zero crossing of the scattering length on the left-hand side of the nearest
Feshbach resonance, which was found at 0.858 G.

and strontium in Ref. [Ste13]. The mass scaling is given by a simple expression ashπ ia = ā 1 − tan Φ −
,
8

7.14

with the characteristic length ā, which is associated to the van der Waals potential byā = 2−3/2Γ(3/4)
Γ(5/4)

2mr C6
~2

1/4

.

7.15

Here, Γ is the gamma function and mr = m/2 the reduced mass. The semiclassical phase Φin Eq. 7.14 is defined by
√
Z∞ p
2mr
Φ=
−U (r)dr ,
7.16
~
r0

where U (r) is the adiabatic Born-Oppenheimer interaction potential between two atoms andr0 is the inner classical turning point at zero energy. By knowing Φ one can calculate thenumber of bound states N by, see Ref. [Fla99],


Φ 5
N=
−
+1 .
7.17
π
8
The brackets denote taking only the integer part. Given these relations the scattering lengthfor different isotopes can be derived by the knowledge of a and N of one single isotope.
Figure 7.4 shows a plot of the values for as for four bosonic isotopes of erbium together witha fit of the mass scaling following Eq. 7.14. For fitting the value for C6 was assumed to be1723 a.u., see Ref. [Kot14]. The value for Φ is extracted from the fit, which yields for thenumber of bound channels N = 142(1).
In initial experiments, the thermalization of 166 Er was insufficiently working, which can beunderstood by its small scattering length of 78(12) a0 . A BEC of erbium atoms was finallysuccessfully produced using the isotope 168 Er. Its scattering length of a = 202(23) a0 madethe evaporative cooling very efficient. After knowing the scattering lengths and Feshbachresonance spectra of isotopes 166 Er and 170 Er it was possible to evaporatively cool these two
isotopes to a BEC using slightly modified evaporation procedures.

113

7.3. Further work

300
200

as (a0)

100
0

-100
-200
-300
163

164

165

166

167
168
isotope (amu)

169

170

171

Figure 7.4.: Mass scaling of the scattering length. Measurements of the scattering length for fourbosonic isotopes have been carried out by cross-dimensional rethermalization measurements awayfrom Feshbach resonances (blue circles). A tangent function according to Eq. 7.14 with a fixed value
for C6 = 1723 a.u. (solid red line) is fit to the data.

Identifying the s-wave scattering length, as , with the background scattering length, abg , andcombining this with the Feshbach resonance properties, like positions and widths, we canestimate the scattering length dependent on the magnetic field. Figure 7.5 shows a plot of
this dependence from 0 to 5 G for the isotopes 166 Er, 168 Er, and from 0 to 1.6 G for 170 Er.

(a)

200

scattering length (a0)

150
100
50
0
166

Er

168

Er

170

Er

-50

scattering length (a0)

(b)

500
400
300
200
100
0
-100

(c)

scattering length (a0)

50
0
-50
-100
-150
-200
-250
-300
0

1

2
3
magnetic field (G)45Figure 7.5.: The s-wave scattering length, as , dependent on the magnetic field for the three bosonicisotopes 166 Er (a), 168 Er (b), and 170 Er (c). These plots are intended as a rough estimation of the scattering length due to the large error of the background scattering length in Table 7.2. The plots weregenerated using Eq. 4.22 for multiple resonances assuming a constant background scattering length
over the full plotting range. Positions and widths of Feshbach resonances are given in Appendix D.
For 170 Er the Feshbach scan was only carried out between 0 and 1.6 G.

114

Chapter

8

Publication: Hyperfine structureof fermionic erbium

8.1. Introduction
In contrast to bosonic erbium the fermionic isotope has a nonzero nuclear spin of I = 7/2.
This gives rise to the hyperfine structure of the ground and excited states. Knowledge of thehyperfine energy structure is fundamental for laser cooling experiments. For erbium only
poor knowledge exists of its hyperfine structure. The hyperfine coefficients of the excitedstate of the 401-nm transition have not been known at all, which made a determination with
spectroscopic measurements necessary. In the presented publication the hyperfine structure
and isotope shifts for the 401-nm and 583-nm transitions were recorded using modulationtransfer spectroscopy. We carried out a fitting of the measured spectra using a theoreticalmodel in order to assign quantum numbers to the spectral features and extract values forthe hyperfine A and B coefficients.
In the following section the spectroscopic setup utilizing a hollow-cathode lamp is described
and the technique of modulation-transfer spectroscopy is briefly discussed. In Sec. 8.2 thepublication is presented.

8.1.1. Hollow-cathode lamp
Hollow-cathode lamps (HCLs) have been used extensively in spectroscopic setups as a spectral line source since the 1960s. In such setups the plasma, which is created in the HCL, isused as an emitter of incoherent light that can be used as a frequency reference. Nowadays,
hollow-cathode lamps are commonly utilized as spectroscopy cells in ultracold gas experiments also for elements other than erbium, like calcium [Han05], chromium [Chr09], andytterbium [WL11].
Figure 8.1 shows a photograph and a schematic view of a HCL. The working principle of a
HCL is the following: the hollow cathode is formed by a metallic cylinder with a center borein axial direction. It can be either made of a solid piece of the desired metal or its surface
115

116

8.1. Introduction

(a)

(b)

1
2

anode

2

cathodeanode

optical axis
3

34

isolation
Ar-gas at 4 mbar

fused silica glass

5
feed-throughs

Figure 8.1.: (a) The photograph of the hollow-cathode lamp shows the cathode (1), the two anodes(2), the electrical connections via feedthroughs (3), the glass body (4), and the lamp holder (5). (b)The schematic view highlights the mechanical construction and shows the optical axis (dotted line).
The lamp is filled with an Ar-gas at a pressure of 4 mbar. Electric isolation between the cathode andanodes is achieved by thin mica insulator disks (orange) and ceramic tubes (gray).

can be coated with the desired metal. In the latter case, the lifetime of the HCL will bemuch shorter because of the small layer thickness. Two anodes are mounted on each side
next to the cathode. The full setup is inside a glass cell, which is filled with a buffer gas at apressure of a few millibar. Nobel gases are best suited as a buffer gas. In our case the lamp
is filled with argon at a pressure of 4 mbar. A high voltage of 110 V is applied between thecathode and the anodes. This creates a plasma discharge by electric-field ionization of atoms
in the buffer gas. Positively charged ions are then accelerated into the center of the hollow
cathode. When an argon ion with sufficient kinetic energy hits the surface of the cathodeneutral atoms are emitted. This is known as a so-called sputtering effect, which was studied
in Ref. [Mus62]. Electrical isolation between the two electrodes is realized by two thin mica
insulator disks. This is important to direct the plasma into the center of the cathode andmaximize the number of target atoms along the optical axis.
Hollow-cathode lamps can be acquired from a manufacturer with almost any metallic element
forming the cathode. This broad availability was the reason to use such lamps as spectroscopy
cells instead of light sources. In the beginning spectroscopy was done in a setup taking
advantage of the optogalvanic effect, see Ref. [Bar90]. Later HCLs were slightly modified as
see-through HCLs such that they can be used in Doppler-free absorption spectroscopy setups,
see Ref. [Law81]. A major drawback of the HCL is given by the large pressure of the buffergas, which broadens the natural linewidth by several MHz. The usage of HCLs as frequency
references has been nowadays surpassed by modern optical frequency standards, like highfinesse cavities and frequency combs. But its simple operation makes it advantageous forproviding a source of elements for spectroscopic setups. This saves the construction of a hightemperature oven in case of elements with high melting point. Lasers for broad transitions
can be reliably stabilized on the spectroscopy signal coming from a HCL whereas narrowtransitions suffer from line broadening effects.
The electrical current, at which the plasma discharge is operated, should not exceed 20 mA
to prevent arc discharges inside the lamp. Typical values of the discharge current for its
purpose as a spectroscopy cell are below 10 mA. A too high current will reduce the lifetimeof the lamp significantly. The current can be controlled by changing the applied voltage.
When turning on the lamp it requires a higher voltage of about 400 V for ignition of the8.1. Introduction

117

plasma discharge. Typical guaranteed lifetimes are specified to be around 1000 hrs but it
was observed that even after 3 years of daily use in the lab, which corresponds to about
15.000 operating hours, spectroscopy still works well. However, strong signs of degradation
are already visible, like a deposition of black metallic vapor on the inside of the lamp as wellas an increase of the plasma discharge volume.

8.1.2. Modulation-transfer spectroscopy
Compared to other spectroscopic methods, e. g. saturated absorption spectroscopy, modulation-transfer spectroscopy (MTS) produces a Doppler-free signal with a flat background,which can be directly used for laser-locking purposes. One of the main advantages of MTS
is that it directly gives dispersive-shaped curves with zero crossings at the frequencies ofatomic transitions.
The optical setup consists of a high-power pump beam, which is aligned through the center of
the atomic vapor in the center of the HCL1 , and a weaker probe beam in counter-propagating
direction. Both laser beams are derived from the same laser source. The pump beam is phasemodulated via an electro-optical modulator (EOM) and thus contains additional sidebands
in the frequency spectrum. These sidebands are transfered onto the probe beam when thelaser frequency is close to resonance with an atomic cycling-transition. The modulationtransfer happens due to an optical four-wave mixing process in a sufficiently nonlinear gas,see Ref. [Shi82]. The probe beam is recorded using a photodiode (PD) and its sidebands aredetected by mixing the signal of the PD with a local oscillator (LO) from which also theinitial modulation frequency is derived. Dependent on the phase delay between signal andLO a signal proportional to either the dispersion or the absorption of the atomic gas can berecorded, i. e. it is sensitive to either the in-phase or quadrature component.
For laser-locking the dispersion signal is used. This signal can be optimized to largest
steepness around a spectroscopic feature and a maximum signal-to-noise ratio (SNR) by
choosing the right modulation frequency, fmod , see Refs. [Ebl07, McC08]. The optimum isfound when fmod equals half of the natural linewidth of the atomic transition to be probed.
In case of the 401-nm transition of erbium this sets fmod ≈ 15 MHz. Strongest contributionsin the spectroscopy signal come from the bosonic isotopes of erbium, which makes locking
to these features convenient and reliable. But also the hyperfine structure of the fermionicisotope is observed as presented in the publication in the next section. The shape of featuresin the dispersion signal equals the expected shape of a differentiated Lorentzian curve.
The spectroscopy signal from MTS is Doppler-free but can be affected by other broadening
effects, like pressure and power broadening. This has to be considered when using MTS in
combination with a HCL as spectroscopy cell. Due to the large background pressure thecollisional broadening can be as large as 8.2 MHz. Because of the small erbium atom number
density inside the HCL it is difficult to achieve a sufficiently large SNR for stable locking.
For a large signal the intensity of the pump laser has to be chosen quite large,
p leading to asignificant power broadening. Typical broadening factors are as large as 1 + I/IS = 2.3

1

see-through HCL 3QQAYEr, from Heraeus Noblelight8.1. Introduction

118

with I the total intensity of pump and probe beam and IS the saturation intensity of thetransition. These two effects broaden the natural linewidth from 30 MHz to about 81 MHz.
Nevertheless, the steepness of the spectroscopy signal is estimated to be 2 MHz/V, whichallows for reliable laser-locking with a sufficiently high stability.
In summary, modulation transfer spectroscopy on a hollow-cathode lamp is a very reliable
and easy-to-use spectroscopic technique, which can be used for direct laser-locking on atomic
transitions with a natural linewidth of several MHz. For narrower transitions, like the 583nm transition with a linewidth of 190 kHz, a signal can only be observed if the intensity of
the laser beam is strongly increased. This causes a significant broadening of the transitionof up to a factor of 100, making laser locking impossible.
As shown, locking a laser on a broad transition using a HCL can be done even for elements,
which would otherwise require a high-temperature oven for spectroscopy or different spectroscopic techniques, like locking to a reference cavity. A setup, which is very similar to theone described above is presented in Ref. [Bra12] but the spectroscopy signal suffers from an
unusually high discharge current of 70 mA at an applied voltage of 1.3 kV. Here, the HCL is
operated at a lower pressure of 1 mbar intended to increase the signal amplitude and SNR
to a usable magnitude.

8.2. Publication:
Hyperfine structure of laser-cooling transitions in fermionicerbium-167†
Physical Review A 88, 032508 (2013)
submitted 11 April 2013; published 13 September 2013
DOI 10.1103/PhysRevA.88.032508
Albert Frisch,1 Kiyotaka Aikawa,1 Michael Mark,1 Francesca Ferlaino,1 Ekaterina Berseneva,2and Svetlana Kotochigova2
1

Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA†

The author of the present thesis developed the experimental procedures, performed the measurements
together with K. A., analyzed the data, and contributed in writing the manuscript. The theoretical calculations of this publication were contributed by E. B. and S. K.

119

PHYSICAL REVIEW A 88, 032508 (2013)

Hyperfine structure of laser-cooling transitions in fermionic erbium-167
Albert Frisch, Kiyotaka Aikawa, Michael Mark, and Francesca Ferlaino
Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

Ekaterina Berseneva* and Svetlana Kotochigova
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA(Received 11 April 2013; published 13 September 2013)
We have measured and analyzed the hyperfine structure of two lines, one at 583 nm and one at 401 nm,of the only stable fermionic isotope of atomic erbium as well as determined its isotope shift relative to thefour most-abundant bosonic isotopes. Our work focuses on the J → J + 1 laser cooling transitions from the[Xe]4f 12 6s 2 (3 H6 ) ground state to two levels of the excited [Xe]4f 12 6s6p configuration, which are of majorinterest for experiments on quantum degenerate dipolar Fermi gases. From a fit to the observed spectra of thestrong optical transition at 401 nm we find that the magnetic dipole and electric quadrupole hyperfine constantsfor the excited state are Ae / h = −100.1(3) MHz and Be / h = −3079(30) MHz, respectively. The hyperfine
spectrum of the narrow transition at 583 nm, was previously observed and accurate Ae and Be coefficientsare available. A simulated spectrum based on these coefficients agrees well with our measurements. We havealso determined the hyperfine constants using relativistic configuration-interaction ab initio calculations. The
agreement between the ab initio and fitted data for the ground state is better than 0.1%, while for the two excitedstates the agreement is 1% and 11% for the Ae and Be constants, respectively.
DOI: 10.1103/PhysRevA.88.032508

PACS number(s): 32.10.Fn, 32.30.Jc, 37.10.De

I. INTRODUCTION

The field of ultracold quantum gases has historicallyheavily relied on alkali-metal atoms. Only recently, the useof non-alkali-metal atoms has gained attention as a means
to explore fascinating quantum phases of matter that are notaccessible with alkali-metal species. Species with multiple
unpaired valence electrons have rich atomic energy spectra
and exhibit various types of coupling between the electronic
angular momentum J and the nuclear spin I of the atom.
For instance, fermionic alkaline-earth-metal atoms have J = 0
and I = 0 and the electronic and nuclear angular momenta
decouple. This decoupling is at the center of proposals for
efficient quantum simulation [1–3] and quantum magnetism[4–7]. Recently, degenerate Bose and Fermi gases of Ca [8],Sr [9–11], and the alkaline-earth-metal-like Yb atoms [12,13]
have been realized.
Lanthanides with submerged 4f -shell electrons are a novel
class of atoms that attract attention in the field of ultracold
quantum physics. Lanthanide atoms can have an exceptionallylarge electronic angular momentum J resulting from thealignment of the angular momenta of the submerged electrons. Consequently, these species can have strong magnetic
moments μ as large as 10μB , where μB is the Bohr magneton.
The mutual interaction is dominated by long-range magnetic
dipole-dipole forces. Their dipolar character can be one
hundred times larger than that for alkali-metal atoms. This key
property makes lanthanides prime candidates for the study ofatomic dipolar physics [14–16]. Dy [17,18] and Er [19], with
μ = 10μB and 7μB , respectively, have been recently brought
to quantum degeneracy, while others are under investigation[20–22].

*Present address: Division of Quantum Mechanics, St. Petersburg
State University 198904, Russia.

1050-2947/2013/88(3)/032508(7)The success of quantum-degenerate-gas experiments relies
on a precise understanding of the atomic properties, such asenergy levels, hyperfine structures, and atomic polarizabilities. However, for unconventional atomic species, such as
lanthanides, the available knowledge is in many instances
insufficient for laser-cooling and trapping purposes. Therefore,
dedicated experiments need to be conducted en route toquantum degeneracy [21,23–25].
In this paper, we present a combined experimental andtheoretical investigation of the hyperfine structure of the
only stable fermionic erbium isotope, 167 Er. In particular,
we obtain the magnetic dipole, A, and electric quadrupole,B, hyperfine structure constants for the ground and twoelectronically excited states of 167 Er, which are relevant for
laser-cooling experiments [23,26]. The two electronic excitedstates investigated are the one at a wavelength of 582.67 nm[corresponding to photon energy E/(hc) = 17 157.307 cm−1 ]and one at 400.796 nm [E/(hc) = 24 943.272 cm−1 ] from theground state [27]. Here h is Planck’s constant and c is the speedof light. In addition to the study of the hyperfine constants,
we also obtained the isotope shift of 167 Er relative to themost-abundant bosonic isotopes. Our work provides important
information for future experiments on quantum-degenerate
Fermi gases of strongly dipolar Er atoms.
In a previous work, we used the optical transitions at about
401 nm and 583 nm for Zeeman slowing (ZS) and magnetooptical trapping (MOT) applications [26]. We demonstrated
efficient laser cooling for five Er isotopes, including thefermionic one. However, the realization of a MOT of fermionic
Er isotope was challenging, since only the hyperfine structureof the ground and the 583-nm-excited state were known[28,29], while the one of the state at 401 nm was unknown
prior to this work. To operate the Zeeman slower and thetransversal cooling stage we had in fact to proceed empiricallyand try different locking points for the light at 401 nm before
being able to produce a MOT of fermions.

032508-1©2013 American Physical Society

ALBERT FRISCH et al.

PHYSICAL REVIEW A 88, 032508 (2013)energy (1000 cm -1 )25

20

4f12(3H6) 6s6p(1P1)

15

4f12(3H6) 6s6p(3P1)

10

ZS light(401 nm)5

MOT light(583 nm)
4f126s2 3H6

We use a commercially available HCL, which is filled withan argon gas at a fixed pressure of 4 mbar [32]. By applying
a high voltage on the electrodes, the argon gas is ionized andaccelerated into the center of the Er-coated cathode. Whenhitting the surface the kinetic energy of the Ar ions is highenough to free neutral erbium atoms by sputtering processes[33]. We typically operate the HCL with a voltage of 110 V,
giving a discharge current of 9.2 mA.
We perform a Doppler-free modulation-transfer spectroscopy in the HCL [34,35]. The laser beam is split into apump and a probe beam, as shown in Fig. 2(a). The pumplight is modulated with an electro-optical modulator (EOM)
driven by a local oscillator (LO) at a frequency of 14.2 MHz
and with a power of 23 dBm [36]. A four-wave mixing processtransfers the sidebands from the modulated pump beam onto
the counterpropagating probe beam [37]. We acquire the

0
2

3

4

5

6

7

8

9

10

11

12

J

LO 14.2 MHz(a)spectroscopy signal

FIG. 1. (Color online) Energy levels of atomic Er up to E/(hc) =25000 cm−1 for different electronic angular momentum quantum
numbers J [27,30]. States with odd (even) parity are indicated byblack (red) horizontal lines. The two relevant laser-cooling transitions
at 401 nm and 583 nm are indicated by arrows.

from
laser

EOM
pump

Figure 1 shows the atomic level scheme of Er. The electronic ground state belongs to the [Xe]4f 12 6s 2 configurationand has a large orbital angular momentum quantum number
L = 5 (H state) and a total electronic angular momentum
quantum number J = 6. The excited states at 401 nm and583 nm belong to the [Xe]4f 12 6s6p configuration and havesinglet 1 P1 and triplet 3 P1 character for the outer two valence
electrons, respectively. Both excited states have a total electron
angular momentum J = 7. Erbium has six stable isotopes
with natural abundance being 33.6% for 166 Er, 26.8% for168
Er, 23.0% for 167 Er, 14.9% for 170 Er, 1.61% for 164 Er, and
0.14% for 162 Er. 167 Er is the only stable fermionic isotope.
The bosonic isotopes have zero nuclear spin (I = 0), while
the fermionic one has I = 7/2 and shows hyperfine structure.
All three electronic states of 167 Er have eight hyperfine
levels ranging from F = J − 7/2 to F = J + 7/2, whereF = J + I.
This paper is structured as follows. Section II describes our
experimental methods to investigate the hyperfine structureof the relevant states. Section III reports on our least-squares
fitting procedure to obtain the hyperfine constants and isotope
shifts from the measured spectra. Section IV describes ourab initio relativistic configuration-interaction calculations andcompares the ab initio hyperfine constants with the fitted
values. We present our conclusions in Sec. V.
II. ATOMIC SPECTROSCOPY

We measure the hyperfine structure of the 167 Er isotope using modulation-transfer spectroscopy [31]. The spectroscopy
is performed on an atomic Er vapor created with a hollow cathode discharge lamp (HCL). The HCL, based on a sputtering
process, has the advantage of providing atomic vapors without
the need of a high-temperature atomic source.

λ/2

HCL

PBS

PD

PBS

probe

HV 9.2 mA 110 V

(b)

.

.

.

.

.

.

.

(c)
.
.
.
.

.

.

.

.

.

.

FIG. 2. (Color online) Modulation transfer spectroscopy of Er for
the 401 nm and 583 nm transitions. (a) Laser setup for spectroscopy ona hollow cathode discharge lamp (HCL); see text. The pump (probe)
light has a power of 3.3 mW (0.6 mW) for the 401 nm transition
and 20 mW (1 mW) for the 583 nm transition. (b), (c) Obtained
spectroscopy signals for the 401 nm and the 583 nm transitions ofdifferent isotopes. Signals related to the hyperfine structure of 167 Er
are indicated by arrows. The relative amplitudes of the observed
signals reflect the natural isotope abundances.

032508-2

HYPERFINE STRUCTURE OF LASER-COOLING . . .

PHYSICAL REVIEW A 88, 032508 (2013)

spectroscopy signal by mixing the LO signal with the probe
beam signal detected by a photodiode (PD). By setting the LO
and the signal either in phase or shifted by π , one can obtain asignal proportional to the dispersion or to the absorption of theatomic sample, respectively. In our setup we use the dispersion
signal.
In a first set of experiments, we measure the hyperfine
structure and the isotope shift of the excited state at 401 nm,with a natural linewidth of 2π × 29.7(5) MHz [38–40]. A
frequency-doubled diode laser is used for the spectroscopy.
Figure 2(b) shows the dispersive spectroscopy signal for thistransition. The signal is averaged over 16 scans with a scanningspeed of 2.4 GHz/s [41].
Our measurement reveals the full hyperfine structure forthe fermionic 167 Er. The discussion and the assignment of theobserved spectral features are given in Sec. III. In addition,
we determine the isotope shifts for the bosonic isotopes
relative to 166 Er. We measure a shift of −1681(14) MHz for
170
Er, −840(14) MHz for 168 Er, and +849(17) MHz for
164
Er, which is in good agreement with Ref. [42]. The
linewidths are extracted by fitting the derivative of a Lorentzian
curve to the data. This gives an averaged value of 2π ×88(8) MHz, corresponding to about three times the naturallinewidth. This broadening of the transition can be explained
as a combined effect of collisional and power broadening.
For a number density of about 1017 cm−3 and an argon
background pressure of 4 mbar, we calculate a collisional
broadening of 2π × 8.2 MHz. Considering a total intensityof the pump and probe beams of I =250 mW/cm2 , weestimate a power broadening of a factor of 1 + I /I0 = 2.3
with I0 = 60.3 mW/cm2 being the saturation intensity.
Combining the two contributions, we estimate a broadened
linewidth of 2π × 81 MHz, which is in agreement with theobserved value.
In a second set of measurements, we focus on the hyperfine
structure of the excited state at 583 nm, with a linewidth of
2π × 186 kHz [43]. The spectroscopy is performed with a dyelaser, which is frequency stabilized to an ultralow expansion
cavity within 30 kHz [26]. We use a spectroscopy setup
similar to the one described above for the 401 nm transition.
Figure 2(c) shows the corresponding spectroscopy signal.
Despite the narrow-line nature of the transition, we could
observe five features related to the hyperfine structure of thefermionic isotope and three features for the bosonic ones.
The discussion of the hyperfine structure is given in Sec. III.
We measure an isotope shift of −975(15) MHz for 168 Er and−1966(14) MHz for 170 Er relative to 166 Er, respectively. These
values are in good agreement with Ref. [29].
For this transition, we extract an averaged value for the
linewidth of 2π × 23(5) MHz, corresponding to about 120
times the natural linewidth. This large broadening can again
be explained in terms of collisional and power broadening.
Considering the saturation intensity of I0 = 0.13 mW/cm2
and our total intensity of I = 1.3 × 103 mW/cm2 , we calculate a power broadening of a factor of 100. Adding the effectof collisional broadening, we obtain an overall linewidth of
2π × 19.3 MHz, which is in agreement with the measured
value. Because of this large broadening, we could operate themodulation-transfer spectroscopy at the same LO frequency
as the one used for the 401 nm transition.

III. ANALYSIS OF HYPERFINE STRUCTURE

In this section we describe our fitting procedure to theobserved spectra of the five most abundant Er isotopes and we
present the resulting hyperfine-structure constants Ae and Be
for 167 Er. The bosonic features are easily assigned as shownin Fig. 2. The remaining weaker features, which sometimes
overlap with those from the bosonic isotopes, are due to 167 Er.
We start with the definition of the transition energies
between the ground and an excited state of 167 Er including
hyperfine interactions [44]h̄ωFe Fg = 167 + h̄ω166 + Ee (Fe ,Je ,I ) − Eg (Fg ,Jg ,I ), (1)where 167 is the 167 Er isotope shift relative to the transitionenergy h̄ω166 of the bosonic 166 Er atom, the most abundantisotope, and Ee (Fe ,Je ,I ) and Eg (Fg ,Jg ,I ) are hyperfine
energies of the excited and ground state, respectively. The
quantum numbers Fi and Ji with i = e or g are the total
atomic and electronic angular momentum of the excited andground state, respectively, andEi (Fi ,Ji ,I ) =

1
Ai Ci
2
1 3Ci (Ci + 1) − 4I (I + 1)Ji (Ji + 1),
+ Bi
2
2I (2I − 1)2Ji (2Ji − 1)(2)

where Ci = Fi (Fi + 1) − Ji (Ji + 1) − I (I + 1). Finally, thetransition energies A + h̄ω166 define the isotope shift forbosonic Er isotopes with atomic number A.
In addition to the resonance positions, we can calculate theline shape of the fermionic spectral features S(ω) by notingthat the signal is well approximated bydL(ω − ωFe Fg ,γ )S(ω) ∝ −(3)
QFe ,Fg
dω
F F
e

g

as a function of laser frequency ω, where the sum is over all(Fg ,Fe ) hyperfine lines, and L(ω,γ ) is a Lorentzian centeredaround zero with linewidth γ [45]. Consequently, for an
isolated line the resonance occurs when the signal is zero.
The fluorescence line strength QFe ,Fg is
QFe ,Fg =|(Jg I )Fg Mg |d1q |(Je I )Fe Me |2
Me Mg q= F̂g F̂e Jˆg



Fg
Je

Fe
Jg

1
I

2

|Jg ||d||Je |2 ,(4)

where the Mi are magnetic quantum numbers, d1q is theelectric dipole-moment operator, and we have assumed equal
population for all hyperfine states Fe Me of the electronic···
excited state. Finally, F̂ = 2F + 1, ( ···) is a six-j symbol, andJg ||d||Je  is a reduced dipole matrix element independent of
Fg and Fe .
We use a nonlinear least-squares fit to the experimental
spectra to determine the hyperfine constants and isotope
shift 167 of the excited states. The fit is based on six
resolved hyperfine features for the 401 nm line and five
resolved features for the narrow 583 nm line. In our analysis, we hold the hyperfine constants for the ground state
to the literature values of Ag / h = −120.487(1) MHz and032508-3

0

-1
168

166

Er

Er

-2
-1.2

-0.9

-0.6

-0.3

detuning with respect to

0
166

0.3

(5/2-7/2)-1
Q-branch

-2

168

170

166

Er

Er

-2

-1

Er

0

detuning with respect to

Bg / h = −4552.984(10) MHz [28], which have significantly
lower uncertainties than those for the excited states.
Figures 3 and 4 are the results of our fit for the 401 nm
and 583 nm line, respectively. We observe remarkable agreements between the simulated and experimental spectra. For
the excited 401 nm level, we extract the best value for thehyperfine coefficients to be Ae / h = −100.1(3) MHz and
Be / h = −3079(30) MHz. Using these coefficients and thosefor the ground state, we obtain resonance positions thatagree to better than 11 MHz with the experimental values.
For the excited 583 nm level, we fit the line shape of thespectral features while the resonance positions are calculated
by using the hyperfine constants of the excited states, Ae / h =−172.7 MHz and Be / h = −4457.2 MHz, from Ref. [29].
We note that the additional structure in the experimental
data, which is not fitting to the theoretical curve, originatesfrom a slightly misadjusted phase of the local oscillator
in the spectroscopy setup. Table I compares the theoreticaland experimental hyperfine energies 167 + Ee (Fe ,Je ,I ) +Ee (Fg ,Jg ,I ) for the 583 nm and 401 nm transitions in 167 Er
and lists the corresponding quantum numbers of Fg and Fe .
Table II and Fig. 5 show the resulting isotope shifts A as afunction of the mass number A relative to the energy of the166
Er isotope.

(7/2-9/2)0Er line (GHz)

FIG. 3. (Color online) Spectroscopy signal and hyperfine assignment of the 401 nm transition of the fermionic 167 Er. The blacksolid line is the recorded line. The red line is a simulated line shapeobtained using a nonlinear fit to the line positions and a linewidthof γ /(2π ) = 90 MHz. The simulated line shape is a sum of the first
derivative of several Lorentzians, one for each hyperfine transition,
whose relative strength is given by a theoretical estimate of the line
strength. We scaled the overall size of the simulated line shape to
fit to the experiment. The assignment of the P -branch transitions(Fg → Fe = Fg + 1) is shown by vertical lines and pairs (Fg ,Fe ). The
hyperfine coefficients of the excited state are Ae / h = −100.1 MHz
and Be / h = −3079 MHz.

(9/2-11/2)(11/2-13/2)

1

-3

0.5

(13/2-15/2)

2(17/2, 19/2)

relative intensity (arb. units)(19/2, 21/2)(5/2, 7/2)

(7/2, 9/2)

(9/2, 11/2)(17/2, 19/2)

relative intensity (arb. units)1(11/2, 13/2)(15/2, 17/2)(13/2, 15/2)

3
2(15/2, 17/2)

PHYSICAL REVIEW A 88, 032508 (2013)(19/2, 21/2)

ALBERT FRISCH et al.

1
166

2

Er line (GHz)

FIG. 4. (Color online) Spectroscopy signal and hyperfine assignment of the 583 nm transition of the fermionic 167 Er. The solid blackline is the experimental spectrum, while the red line is a simulatedline shape using a linewidth of γ /(2π ) = 20 MHz. The P -branchtransitions (Fg → Fe = Fg + 1) are assigned by pairs (Fg ,Fe ).
Three P -branch resonances and several Q-branch (Fg → Fe = Fg )resonances are predicted to lie outside of the measurement range.
The simulated line shape is a sum of the first derivative of several
Lorentzians, one for each hyperfine transition, whose relative strength
is given by a theoretical estimate of the line strength. We scaled theoverall size of the simulated line shape to fit to the experiment. The
hyperfine coefficients of the excited state are Ae / h = −172.7 MHz
and Be / h = −4457.2 MHz.

calculations of the magnetic dipole A and electric quadrupole
B hyperfine constants. They describe the coupling of thenuclear spin I to the total electron angular momentum J, due
to the magnetic dipole and electric quadrupole interaction,
respectively. The latter originates from the electric-field
gradient created by the electrons at the nuclear location. We

TABLE I. Observed and calculated hyperfine energies 167 +Ee (Fe ,Je ,I ) + Ee (Fg ,Jg ,I ) for the 583 nm and 401 nm lines in 167 Er.
The theoretical values are based on the hyperfine coefficients Ae / h =−172.7 MHz and Be / h = −4457.2 MHz [29] for the 583 nm line
and our values Ae / h = −100.1 MHz and Be / h = −3079 MHz forthe 401 nm line.
Obs.
energy(MHz)

Calc.
(Fg ,Fe )energy(MHz)
583 nmObs.
energy(MHz)

Calc.
(Fg ,Fe )energy(MHz)
401 nm

−2011
−1449
−820
−200
393

−2011
−1454
−834
−203
396
941
1369
1709

−761

−762
−757
−580
−498
−315
−31
150
220

IV. ab initio HYPERFINE CONSTANTS

In conjunction with the experimental measurements andfits, we have performed extensive ab initio electronic structure
032508-4(19/2, 21/2)(17/2, 19/2)(15/2, 17/2)(13/2, 15/2)(11/2, 13/2)(9/2, 11/2)(7/2, 9/2)
(5/2, 7/2)

−589
−498
−325
150
220

(15/2, 17/2)(13/2, 15/2)(11/2, 13/2)(17/2, 19/2)(9/2, 11/2)(7/2, 9/2)
(19/2, 21/2)(5/2, 7/2)

HYPERFINE STRUCTURE OF LASER-COOLING . . .

PHYSICAL REVIEW A 88, 032508 (2013)

TABLE II. Observed isotope shift of the Er isotopes for the
583 nm and 401 nm lines. The transition energy for the bosonicisotope 166 Er is taken as energy reference. The isotope shift of thecenter of gravity of fermionic 167 Er was obtained from fitting.
IsotopeObs.
energy(MHz)

Calc.
energy
(MHz)

Obs.
Calc.
energyenergy(MHz)(MHz)
401 nm583 nm−1966(14)−975(15)

170
168
167
166
164

−1681(14)−840(14)−337(11)−297(6)00+849(17)

shift with respect to166

Er line (GHz)were interested to reproduce the known constants for the
Er ground state as well as those of the excited state at the
583 nm line obtained by Refs. [28,29]. We can then confirm
our measurement of the unknown constants of the excited levelat the 401 nm line.
The ab initio calculations of the hyperfine structure constants have been performed using a relativistic multiconfiguration Dirac-Fock (MCDF) method [46]. In this method
we perform an all-electron calculation of the wave function
leading to an accurate description of the electron-spin density
near the nucleus. The eigenfunctions are superpositions ofnonorthogonal many-electron determinants of one-electron
Dirac-Fock functions for the core and valence orbitals andSturm functions for virtual orbitals. Both types of one-electron
orbitals are optimized for either the 4f 12 6s 2 ground or
4f 12 6s6p excited-state reference configurations.
The hyperfine splittings of atomic levels are due to interactions between electrons and nuclear multipole moments. In theconfiguration-interaction picture and using atomic units the A1
0.50λ=401 nm-0.5

λ=583 nm

-1
-1.5
-2
164

166

168

170

isotope number
FIG. 5. (Color online) Isotope shifts for the 583 nm and 401 nm
lines of the isotopes 164 Er up to 170 Er as a function of mass number
where the transition energy for the bosonic isotope 166 Er is taken as
energy reference. The isotope shift of the bosonic isotopes falls on asingle straight line, with the isotope shift of the center of gravity of
the fermionic 167 Er isotope, green cross and green square, is slightly
displaced from this linear dependence.

and B constants are given by [ri × α i ]00
gI μN
,J MJ ||,J MJ , (5)
MJ
ri3
i

2Q 2J (2J − 1)(2J + 1)
B(J ) =MJ
(2J + 2)(2J + 3)
 Y20 (r̂i )|,J MJ ,(6)× ,J MJ |ri3iwhere the sum i is over all electrons with positions ri withrespect to the nucleus, Yℓm (r̂) are spherical harmonics, and α iis the Dirac matrix for electron i. Furthermore, gI is the nuclear
g factor, μN is the nuclear magneton in atomic units, and Q
is the nuclear quadrupole moment.
The relativistic electroniceigenfunctions |,J MJ  = β cβ |φβ ,J MJ , obtained fromthe configuration-interaction calculations with relativistic determinants |φβ ,J MJ  and CI coefficients cβ , have total angularmomentum J and projection MJ .
When an atom has open or unfilled electron shells, it leads
to an unbalanced electron-spin density near the location of
the nucleus. As hyperfine constants are proportional to thedifference in electron-spin densities this leads to nonzero A
and B coefficients. To account for this effect we use a model,
where the single-electron orbitals differ for each spin direction
or more precisely for each spinor of the Dirac-Fock equation.
Alternatively, this implies different exchange potentials forelectrons with spin up or down.
We use three restricted active spaces (RAS) to classify theelectron Dirac-Fock and Sturm orbitals, ensuring an efficientand compact CI expansion that, nevertheless, remains accurate.
The first group of orbitals, RAS1, contains the occupied
spinors of the relevant reference configuration. We havestudied convergence of the hyperfine structure constants as
the active set of orbitals was systematically increased. For our
most-precise Er atom calculation RAS1 includes the occupied
6
2
4
1s 2 ,2s 2 ,2p1/2,2p3/2, . . ., and 4f5/2
shell electrons. We allow
up to one electron to be excited out of RAS1 into the twoother active spaces. The second group, RAS2, contains the62, and 6p1/2 ,6p3/2 spinors, while the, 6s1/2
open-shell 4f7/2
third group, RAS3, contains spinors that are unoccupied in
the reference configuration. These latter virtual Sturm orbitals
are the high-lying s-wave spinors from 7s up to 13s, p-wave
spinors from 7p up to 11p, and the 5d spinor. For both RAS2
and RAS3 we allow up to two electrons to enter or leave.
With this basis our finite-nuclear-size and finite-nuclear
mass corrected ab initio values of the A and B hyperfine
constants are −120.42 MHz and −4554 MHz for the groundstate level, −174 MHz and −4057 MHz for the excited level at
583 nm, and −100 MHz and −3424 MHz for the excited levelat 401 nm, respectively. Consequently, the ab initio A constants
agree with experimentally determined values to better than 1%,
whereas the B constants differ by up to 11% for the two excitedstates. For the ground state the agreement for the B constantis also better than 1%.
A(J ) =

V. CONCLUSIONS

We have used laser modulation-transfer spectroscopy on
atomic Er as well as performed ab initio electronic structure

032508-5

ALBERT FRISCH et al.

PHYSICAL REVIEW A 88, 032508 (2013)

TABLE III. Summary of the relevant hyperfine A and B constantsfor ground and excited states (e.s.) usable for laser cooling of 167 Er.
State

J

A/ h (MHz)B/ h (MHz)

Ref.

Ground state
Ab initio
583 nm e.s.
Ab initio
401 nm e.s.
Ab initio

6−120.487(1)−120.42−172.70(7)−174−100.1(3)−100−4552.984(10)−4554−4457.2(29)−4057−3079(30)
−3424

[28]
This work[29]
This work
This work
This work

7
7

calculations of Er to obtain the magnetic dipole and electric
quadrupole constants for the only stable fermionic isotope,
167
Er. We focused on transitions from the 4f 12 6s 2 (J = 6)
ground state to two J = 7 levels within the excited 4f 12 6s6p
configuration. A least-squares algorithm applied to the experimentally measured hyperfine-structure energies gives accurate
values for the two constants as well as values for the isotope
shift of five isotopes. The ab initio calculation is based ona multiconfiguration Dirac-Fock method where we allow no
more than two electrons to be excited from and between
the active spaces. The method has no further adjustableparameters.
Our results are summarized in Table III and Fig. 6. We
find that the ab initio A coefficients for all three states
and the B coefficient for the ground state agree to better
than 1% with the experimental values, which is in a surprisingly good agreement considering the complex electronshell structure of the Er atom. We note that the ab initio
electric quadrupole constants B for the two excited states
exhibit a larger deviation from the experimental values. Thismight be a consequence of missing key configurations: theexcited states have three open shells, 4f 12 , 6s, and 6p, fromwhich more than two electrons might need to be excited.
In addition, Sternheimer shielding (e.g., distortions in theelectron shells by the nuclear quadrupole moment), which is
not considered in our MCDF theory, might cause significant
corrections.

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101, 170504 (2008).
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J. Ye, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 102, 110503(2009).
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135301 (2009).
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P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nat. Phys. 6,289 (2010).
[6] M. Foss-Feig, M. Hermele, V. Gurarie, and A. M. Rey,
Phys. Rev. A 82, 053624 (2010).

F’
9/2
11/2
7/2
13/2
15/2
17/2
19/2
21/2

4f126s6p (1P1)

167

5.84 MHz
98.2 MHz
97.3 MHz479 MHz872 MHz1.39 GHz
2.04 GHz

401 nm
F
11/2
9/2
13/2
7/2
5/2
15/2
17/2
19/2

F’’

Er

4f126s6p (3P1)

9/2
7/2
11/2
13/2
15/2
17/2
19/2
21/2

25.8 MHz
135 MHz463 MHz902 MHz1.50 GHz
2.27 GHz
3.25 GHz583 nm69.1 MHz234 MHz44.5 MHz354 MHz468 MHz1.65 GHz
2.69 GHz
4f126s2 3H6 g.s.

FIG. 6. (Color online) Hyperfine levels of the ground (g.s.) andtwo excited states of 167 Er with particular interest for laser cooling.
The level splitting was calculated using A and B constants given
in Table III for the respective transitions. The arrows depict twolaser-cooling transitions. The transition at 401 nm used for Zeeman
slowing is shown in blue and the transition used for magneto-opticaltrapping at 583 nm is shown in yellow.
ACKNOWLEDGMENTS

We would like to acknowledge R. Grimm for fruitful
discussions and A. Rietzler, S. Baier, and M. Springer fortechnical support. The Innsbruck team acknowledge support
by the Austrian Ministry of Science and Research (BMWF)
and the Austrian Science Fund (FWF) through a START grant
under Project No. Y479-N20 and by the European ResearchCouncil under Project No. 259435. C.B. and S.K. acknowledge
support by the AFOSR Grant No. FA 9550-11-1-0243 andARO MURI Grant No. W911NF-12-1-0476. They also thankDr. I. I. Tupitsyn for useful discussions.

[7] H.-H. Hung, Y. Wang, and C. Wu, Phys. Rev. B 84, 054406(2011).
[8] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Phys. Rev.
Lett. 103, 130401 (2009).
[9] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck,
Phys. Rev. Lett. 103, 200401 (2009).
[10] Y. N. Martinez de Escobar, P. G. Mickelson, M. Yan, B. J.
DeSalvo, S. B. Nagel, and T. C. Killian, Phys. Rev. Lett. 103,
200402 (2009).
[11] S. Stellmer, R. Grimm, and F. Schreck, Phys. Rev. A 84, 043611(2011).
[12] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda,
M. Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett.
91, 040404 (2003).

032508-6

HYPERFINE STRUCTURE OF LASER-COOLING . . .

PHYSICAL REVIEW A 88, 032508 (2013)[13] T. Fukuhara, Y. Takasu, M. Kumakura, and Y. Takahashi,
Phys. Rev. Lett. 98, 030401 (2007).
[14] M. A. Baranov, Phys. Rep. 464, 71 (2008).
[15] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau,
Rep. Prog. Phys. 72, 126401 (2009).
[16] M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller,
Chem. Rev. 112, 5012 (2012).
[17] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys. Rev.
Lett. 107, 190401 (2011).
[18] M. Lu, N. Q. Burdick, and B. L. Lev, Phys. Rev. Lett. 108,215301 (2012).
[19] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm,
and F. Ferlaino, Phys. Rev. Lett. 108, 210401 (2012).
[20] M. Saffman and K. Mølmer, Phys. Rev. A 78, 012336 (2008).
[21] D. Sukachev, A. Sokolov, K. Chebakov, A. Akimov,
S. Kanorsky, N. Kolachevsky, and V. Sorokin, Phys. Rev. A
82, 011405 (2010).
[22] D. Sukachev, K. Chebakov, A. Sokolov, A. Akimov,
N. Kolachevsky, and V. Sorokin, Opt. Spectrosc. 111, 633(2011).
[23] J. J. McClelland and J. L. Hanssen, Phys. Rev. Lett. 96, 143005(2006).
[24] S. H. Youn, M. Lu, U. Ray, and B. L. Lev, Phys. Rev. A 82,
043425 (2010).
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032502 (2011).
[26] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler,
E. Zupanic, R. Grimm, and F. Ferlaino, Phys. Rev. A 85, 051401(2012).
[27] Y. Ralchenko, A. Kramida, J. Reader, and N. A. Team, NIST
Atomic Spectra Database (2011).
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3402 (1983).
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McClelland, Opt. Express 13, 3185 (2005).

[31] J. H. Shirley, Opt. Lett. 7, 537 (1982).
[32] Manufacturer Heraeus Noblelight GmbH, model number
3QQAYEr.
[33] T. Musha, J. Phys. Soc. Jpn. 17, 1440 (1962).
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J. Appl. Phys. 52, 4375 (1981).
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B 106, 405 (2012).
[36] The modulation frequency is chosen such that the steepness ofthe spectroscopy signal is maximized, according to [37].
[37] D. J. McCarron, S. A. King, and S. L. Cornish, Meas. Sci.
Technol. 19, 105601 (2008).
[38] Prior to our work on ultracold atoms, two values of the naturallinewidth were available in literature, 2π × 35.6(1.2) MHz [39]and 2π × 29.5(1.6) MHz [40]. Our measurement is based on
absorption imaging experiments on magneto-optically trapped
atoms of three bosonic isotopes. Our value of 2π × 29.7(5) MHz
confirms the findings of Ref. [40]. We note that our previous
value of 2π × 27.5(4) MHz [26] was affected by calibration
issues.
[39] J. J. McClelland, Phys. Rev. A 73, 064502 (2006).
[40] J. E. Lawler, J.-F. Wyart, and E. A. D. Hartog, J. Phys. B: At.
Mol. Opt. Phys. 43, 235001 (2010).
[41] The frequency is monitored during recording the spectroscopy
signal and the systematic uncertainty due to nonlinear scanningis estimated to be 12 MHz.
[42] C. B. Connolly, Y. S. Au, S. C. Doret, W. Ketterle, and J. M.
Doyle, Phys. Rev. A 81, 010702 (2010).
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155004 (2010).
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37 (1977).
[45] Pressure broadening, giving a Gaussian contribution to the full
profile, is small and can thus be neglected.
[46] P. Villemoes, R. van Leeuwen, A. Arnesen, F. Heijkenskjöld,
A. Kastberg, M. O. Larsson, and S. A. Kotochigova, Phys. Rev.
A 45, 6241 (1992).

032508-7

Chapter

9

Publication: Degenerate Fermi
gas of erbium

9.1. Introduction
In 1999 the group of D. Jin at JILA (USA) produced a degenerate Fermi gas (DFG) of
neutral atoms for the very first time, see Ref. [DeM99b]. The atomic species chosen in thispioneering experiment was potassium, one of the two fermionic species available in the seriesof alkali-metal atoms. Since then many experiments on DFGs have been built all around theworld and many spectacular effects have been observed.
Prior to our work the only fermionic species, which had been brought to quantum degeneracy
were potassium and lithium from the alkali metals, ytterbium and dysprosium from thelanthanide series, strontium from the alkali-earth metals and helium. The latter opened
the fascinating possibility to study dipolar effects with fermions, which could not be donewith chromium before. Above mentioned experiments use sympathetic cooling of fermions
with either another spin state or a second species. In case of fermions with a large dipole
moment elastic collisions between identical fermions driven by dipolar interactions are notsuppressed at low temperatures, see Sec. 4.2. This offers the new approach in preparing adegenerate Fermi gas with high densities and low temperatures by direct evaporative cooling.
Additionally to erbium this idea has been recently applied for Dy [Lu12] and RbK [Ye]. With
erbium we demonstrated very efficient direct evaporative cooling down to deep degeneracy
for the first time.

9.1.1. Degenerate dipolar Fermi gas
Quantum degeneracy in Fermi gases arises as a smooth crossover when the thermal energy,
kB T , reaches the Fermi energy, EF . The associated Fermi temperature reads
TF =EF~ω(6N )1/3
=
,
kB
kB

127

9.1

9.1. Introduction

128

for harmonically trapped atoms with ω the mean trap frequency and N the number ofparticles. At T = 0 the Fermi energy equals the energy of the highest occupied level. For
temperatures T < TF the gas is said to be quantum degenerate. In the current erbium
experiment temperatures of down to 0.11(1) TF are usually achieved with an atomic sample
consisting of 3 × 104 particles with TF = 1.06(5) µK, see Ref. [Aik14b]. From EF the Fermi
wave number can be derived via
rr
2mEF
2mω(6N )1/3
kF =
=
.
9.2
2
~
~
The Fermi wave number resembles the radius of the Fermi surface in reciprocal space andrelates to the Fermi momentum by pF = ~kF . In a recent experiment we observe a deformation of the Fermi surface caused by the strong magnetic DDI, see Chapter 11. This is apurely many-body quantum effect and vanishes for high temperatures around TF .

9.1.2. Collisions between identical fermions
As shown in Eq. 4.20 the elastic scattering cross section for dipolar fermions is energy independent. Its non-zero value causes identical fermions to collide even at low temperatures.
At temperatures below TF collisions get more and more suppressed by the Pauli exclusion
principle. Although the DDI reduces the p-wave barrier the effective barrier height for erbium, which is about kB × 7 µK, is still much larger than the collision energy. This prevents
atoms from getting close and efficiently suppresses losses induced by three-body collisions,requiring short-range interactions.
These two features, namely the non-vanishing elastic scattering and a suppression of losses
by three-body collisions, make a fermionic gas of dipolar atoms to make an ideal candidate
for preparing a strongly-correlated degenerate Fermi gas.

9.1.3. Evaporative cooling of fermions
Observing elastic collisions between identical fermions opens the possibility of employing
evaporative cooling of a single spin-component Fermi gas. This strongly simplifies the preparation of a degenerate Fermi gas. We cool the fermionic sample like in the bosonic case byloading atoms from the magneto-optical trap (MOT) into an optical dipole trap (ODT) andapplying evaporative cooling in a straightforward way.
We drive direct evaporative cooling in a crossed ODT, consisting of a horizontal and a verticalbeam. Both beams are operated at a wavelength of 1570 nm. See Sec. 3.5.3 for the opticalsetup. The setup is similar to the one used for bosons but the trapping beams have smaller
waists to increase the particle density for a more efficient evaporation. The horizontal beam
has a waist of 15 µm and the vertical beam of 33 µm.
Figure 9.1 summarizes the parameters of our evaporation procedure starting from the MOT.
As we are using a tightly focused ODT for evaporation we have to introduce an intermediateramp 1 2 3 4

5

6

7

8

9

10

11

duration
110 1000(ms) 300 110

1200

1500

1400

1600

1600

1800

200

power (W)2.0

power (W)

1.5
1.0

20
15
10
5
0

horizontal 1064nm
AR

0.0

0.1
0.2
evap. time (s)

20
15
10
5
0

AR

(a)
1570 nm ODThorizontal beamvertical beam

0.3

0.5
0.0

magn. field (G)

(b)

2.0
1.5
1.0
0.5
0.0

(c)
atom number

106

T/TF

(d)

105

104

1

0.1

0

1

2

3

4

5
6
7
evaporation time (s)

8

9

10

11

Figure 9.1.: Evaporative cooling of a 167 Er Fermi gas in an ODT operated at 1570 nm. (a) Atoms
are initially loaded from the MOT into a single-beam ODT at 1064 nm and after initial compression
within 300 ms into the horizontal beam at 1570 nm (see inset). This intermediate step makes loadingfrom the MOT more efficient. Forced evaporation is accomplished by ramping down the horizontal
trapping beam (orange) in an exponential manner and ramping up the vertical trapping beam tocompress the trap in x- and y-direction and preserve a large density. Typical trapping frequencies atthe end of evaporation are (νx , νy , νz ) = (398, 282, 304) Hz. (b) At the beginning the magnetic field
is changed to 0.59 G. The positions of Feshbach resonances are highlighted by gray lines. (c) Atom
number and (d) temperature of the cloud are measured at the end of every evaporation step.

129

130

9.1. Introduction(a)

T (µK)

101

167

Er fermionic
Er bosonic

0.1

168

104

105

106
atom number

107

(b)
167

Er fermionic
Er bosonic

1

168

psd

10-1
10-2
10-3
10-4

105

106
atom number

107

evaporation ramp

Figure 9.2.: Comparison between evaporation trajectories of fermionic 167 Er (red) and bosonic168
Er (blue). (a) Temperature and (b) phase-space density are shown in dependence on the atom
number. In both panels the path of evaporation is from right to left (arrow). The overall evaporation
efficiencies, given by a linear fit in the log-log plot (lines), are roughly equal for both isotopes. We
determine the efficiency for the fermionic isotope to be 3.5.

step by loading the atoms first from the MOT into the horizontal ODT operating at 1064 nm.
Then we compress the cloud by decreasing the aspect ratio of the ODT with the scanningsystem before we can load the sample into the 1570-nm ODT. This intermediate loading
step is done within the first 300 ms. In the 1570-nm ODT the power of the horizontal beamis then reduced in a piecewise exponential-like manner and the power of the vertical beamis stepwise ramped up. Note that due to the high density of Feshbach resonances it is noteasy to find a suitable spot for evaporation. We change the magnetic field after 520 ms to
0.59 G for a better evaporation efficiency. The full evaporation procedure usually takes up
to 11 s to achieve temperatures of 0.11 TF with 3 × 104 atoms.
Atom number and temperatures are measured at each evaporation step, see Fig. 9.1(c-d).
These values are used to compare the evaporation trajectory of fermionic 167 Er with theone of bosonic 168 Er, see Fig. 9.2. A linear fit to the log-log plot of the phase-space density
versus atom number gives the evaporation efficiency, γ, similar as in Sec. 7.1.1. Remarkably,
we observe that fermions can be cooled just like bosons with an efficiency of γ = 3.5.

9.2. Publication:
Reaching Fermi degeneracy via universal dipolar scattering†
Physical Review Letters 112, 010404 (2014)
submitted 21 October 2013; published 6 January 2014
DOI 10.1103/PhysRevLett.112.010404
Editor’s suggestion
Kiyotaka Aikawa,1 Albert Frisch,1 Michael Mark,1 Simon Baier,1 Rudolf Grimm,1,2 andFrancesca Ferlaino1
1

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

The author of the present thesis helped in taking the data, analyzed the data together with K. A., andcontributed in writing the manuscript.

131

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PHYSICAL REVIEW LETTERS

Reaching Fermi Degeneracy via Universal Dipolar Scattering
1

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

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

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

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

010404-1© 2014 American Physical Society

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PHYSICAL REVIEW LETTERS

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

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

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

PRL 112, 010404 (2014)

z
x

0

0.10
0.20
optical density

x profile42
0
6

z profile

4
2
0
0

150
300
position (µm)

450

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

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

(a)

(b)

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

T (µK)

10

(a)

(b)

(c)

(d)

(e)

(f)

1

mF
-15/2

1

-17/2
-19/2
0.1

0

5

10

Evaporation time (s)

15

0.1
10 4

10 5

10 6

T/TF = 0.90(6)N

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

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

rf spin flip

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

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PHYSICAL REVIEW LETTERS

2

600
550

1

Tz (nK)

effective cross section (10-12 cm2)

PRL 112, 010404 (2014)

500
450
400
0

200
400
holding time (ms)

600

0
0.6

0.8

1.0

temperature after thermalization (TF)

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

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

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

week ending
10 JANUARY 2014

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

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PRL 112, 010404 (2014)

PHYSICAL REVIEW LETTERS

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

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Chapter

10

Publication: Quantum chaos
in ultracold erbium

10.1. Introduction
In the presented publication the correlation and distribution of Feshbach resonances (FR)in the collision of atomic erbium were studied by performing high-resolution Feshbach spectroscopy. A total of 190 FRs were found for each of the two bosonic isotopes 168 Er and 166 Er
in the magnetic field range from 0 to 70 G. This corresponds to a density of about threeresonances per Gauss. For the fermionic isotope 167 Er 115 resonances have been observed
up to 4.5 G, resulting in a density of about 25 resonances per Gauss. Compared to alkali
atoms, which typically have one resonance each ten Gauss, the density of FRs in erbium istremendously high.
Feshbach spectroscopy is a commonly used technique to study collisional properties of various
atomic species at ultracold temperatures, as outlined in Sec. 4.3. The level of control over theparticle interaction and the energy resolution of the collision energy are unprecedentedly highcompared to other spectroscopic techniques, e. g. neutron scattering experiments. Ultracold
physics is currently about to enter a new regime where far more complex atomic species likemagnetic lanthanide atoms or even molecules are being studied.
Instead of following a microscopic approach based on coupled-channel calculations, as isroutinely done with alkali atoms, we analyzed the Feshbach spectrum using statistical tools.
Coupled-channel calculations are not easy to carry out in case of erbium because of thelarge number of scattering parameters of the 49 gerade Born-Oppenheimer potentials andthe anisotropic interaction potential, see Sec. 4.1.2. The statistical analysis of FRs is a
novel approach in ultracold atom experiments. The characteristic measures provided by the
random matrix theory could be confirmed within this analysis of the Feshbach spectrumof erbium. This suggests that quantum chaotic scattering controls the supposedly ‘simple’
collision process of two erbium atoms. Consequently, a strong coupling between individual
molecular states is expected as outlined in Sec. 10.3 and Ref. [Kot].

13710.1. Introduction

138

10.1.1. High-resolution Feshbach spectroscopy
Feshbach resonances are an important tool for understanding the scattering properties of asystem. The first observation of FRs in a sample of lanthanide atoms had been made by ourgroup, see Chapter 7. Already in these early experiments we observed an unusually large
density of resonances. In the present publication we extended the magnetic field scanning
range from 0 to 70 G. Compared to the density of FRs of other atomic samples, like alkali
atoms as 133 Cs [Chi04], dipolar atoms as 52 Cr [Wer05] or mixtures of alkali atoms as K+Rb
[Ino04, Fer06] and Rb+Cs [Pil09], the density of resonances in the case of erbium is one totwo orders of magnitude larger.
For the high-resolution Feshbach scan we have chosen a magnetic-field resolution of 20 mG.
The experimental procedure consisted of preparing an ultracold sample of 168 Er or 166 Er in
the energetically lowest Zeeman sublevel mJ = −6 at a temperature of 330 nK, which wasjust before the onset of condensation. We then changed the magnetic field to a target value
and kept the atoms at this field for 400 ms. After switching of the trap and recording theatom number the procedure was repeated for the next value off the magnetic field. For a highreproducibility the electric current for creating the magnetic field was actively stabilized bya feedback loop at the same time ensuring a fast but smooth change of the current to thetarget value within a few ms. The maximum ramp speed of the magnetic field was limited
by eddy currents in the vacuum chamber. The trap-loss spectroscopy scan was performed
successively with a step size of 20 mG for ranges of 1 G. If a trap-loss feature was found in
this range it was rescanned with a smaller step size down to 5 mG. Before continuing thescan for the next region the stability of the atom number was confirmed by repeating themeasurement for several previously measured magnetic field values. Scanning the magnetic
field from 0 to 70 G for both isotopes required 13,200 individual measurements lasting forabout 20 s each.
Recently a Feshbach scan of four different isotopes of dysprosium was reported in Ref. [Bau14].
A similar density of roughly 3.5 resonances per Gauss was observed in the atom-loss spectraof bosons in the range from 0 to 6 G at similar temperatures as in the erbium case. For
this set of measurements the total number of resonances was not large enough to perform astatistical analysis of the resonance spacings and correlations for dysprosium.

10.1.2. Statistical analysis of Feshbach resonances
We analyzed the distribution of FRs using the statistical methods introduced by randommatrix theory, see Chapter 5. First, the position of maximum atom loss was determined.
For simplicity this was done by fitting a Gaussian to the loss maxima. Then the staircase
function of Feshbach resonances was created, see Sec. 5.1.4. The staircase function has asimilar dependence on the magnetic field for 168 Er, 166 Er, and 167 Er. After a non-linear
increase at low magnetic field values up to about 20 G follows a linear increase up to the
maximum measured field. For the fermionic isotope 167 Er the linear dependence already sets
in at about 1.5 G and the density of resonances is about a factor of five higher than for thebosons.

10.1. Introduction

139

The density of resonances, ρ(B), is derived by calculating the slope of the staircase function.
The mean density of resonances, ρ, and the mean width, Γ, are important statistical values
for comparisons with theoretical coupled-channel calculations. In the presented trap-loss
spectroscopy the width of the resonances can be derived either by locating the zero crossingof the scattering length or by determining the width of a Gaussian fit to the loss feature. The
former method is very reliable, but is only possible if a local maximum of the atomic cloudsize can be observed, whereas the second method is only to be taken as a rough estimate of
the resonance width. A complete set of extracted resonance positions and widths for 168 Er
and 166 Er is given in Appendix D.

10.2. Publication:
Quantum chaos in ultracold collisions of gas-phase erbium
atoms†‡Nature 507, 475–479 (27 March 2014)
submitted 6 December 2013, published online 12 March 2014
DOI 10.1038/nature13137
Nature News & Views associated article
Albert Frisch,1 Michael Mark,1 Kiyotaka Aikawa,1 Francesca Ferlaino,1 John L. Bohn,2 Constantinos Makrides,3 Alexander Petrov,3,4,5 Svetlana Kotochigova3
1

Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria
JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440, USA
3
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
4
St. Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
5
Division of Quantum Mechanics, St. Petersburg State University, 198904, Russia
2†‡

The author of the present thesis developed the experimental procedures, performed the measurements,
analyzed the data, and strongly contributed in writing the manuscript. The theoretical calculations of thispublication were contributed by C. M., A. P., and S. K.
The preprint version of the publication is attached for online access of this thesis.

140

Quantum Chaos in Ultracold Collisions of Erbium
Albert Frisch, Michael Mark, Kiyotaka Aikawa, and Francesca Ferlaino∗
Institut für Experimentalphysik, Universität Innsbruck, Technikerstraÿe 25, 6020 Innsbruck, Austria

John L. Bohn
JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440, USA

Constantinos Makrides, Alexander Petrov,† and Svetlana Kotochigova
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA(Dated: December 6, 2013)

Atomic and molecular samples reduced to temperatures below 1 microkelvin, yet still in the gas
phase, aord unprecedented energy resolution inprobing and manipulating how their constituent
particles interact with one another. As a resultof this resolution, atoms can be made to scatter
resonantly at the experimenter's whim, by precisely controlling the value of a magnetic eld[1]. For simple atoms, such as alkalis, scattering resonances are extremely well-characterized[2]. However, ultracold physics is now poised
to enter a new regime, where far more complex
species can be cooled and studied, including magnetic lanthanide atoms and even molecules. For
molecules, it has been speculated [3, 4] that adense forest of resonances in ultracold collisioncross sections will likely express essentially random uctuations, much as the observed energy
spectra of nuclear scattering do [5]. According
to the Bohigas-Giannoni-Schmit conjecture, theseuctuations would imply chaotic dynamics of theunderlying classical motion driving the collision[6, 7]. This would provide a paradigm shift in ultracold atomic and molecular physics, necessitating new ways of looking at the fundamental interactions of atoms in this regime, as well as perhaps
new chaos-driven states of ultracold matter.
In this report we provide the rst experimental demonstration that random spectra are indeed
found at ultralow temperatures. In the experiment, an ultracold gas of erbium atoms is shown
to exhibit many Fano-Feshbach resonances, forbosons on the order of 3 per gauss. Analysis of
their statistics veries that their distribution ofnearest-neighbor spacings is what one would expect from random matrix theory [8]. The density
and statistics of these resonances are explainedby fully-quantum mechanical scattering calculations that locate their origin in the anisotropy of
the atoms' potential energy surface. Our results
therefore reveal for the rst time chaotic behavior
in the native interaction between ultracold atoms.
In the common perception, atoms are regarded as sim-ple systems in sharp contrast to complex molecules, whose

behavior is dictated by many (rotational and vibrational)

degrees of freedom. The recent realization of dipolar
Bose-Einstein condensates and Fermi gases of magnetic
lanthanides [912] made available a novel class of atoms
in the ultracold regime. These exotic species, such as erbium (Er), allow to bridge the enormous conceptual gap
between simple atoms and molecules, potentially providing a natural testbed to explore complex scattering dynamics in a controlled environment. The rich scattering
behavior of lanthanides has been pointed out in pioneering experiments at millikelvin temperatures [13, 14] andtheoretical work on cold collisions of atoms with non-zero
angular momenta [15, 16].
A wealth of intriguing properties in Er, which is thefocus of this paper, originates from its exotic electronicconguration. Er is a submerged-shell atom with electron vacancies in the inner anisotropic 4f 12 shell, whichlies beneath a lled 6s2 shell. As a consequence, it not
only has a large magnetic moment of 7 Bohr magnetons(µB ) but also has a large electronic orbital (total) angularmomentum quantum number of L = 5 (J = 6); note thatfor bosonic (fermionic) isotope the nuclear angular quantum number is I = 0 (I = 7/2). Large values for L and
J are sources of anisotropy in the interatomic interaction. Moreover, the two-body scattering is controlled byas many as 91 electronic Born-Oppenheimer (BO) interaction potentials, each potential accounting for a specicorientation of J with respect to the internuclear axis.
All BO potentials are anisotropic and include at large internuclear separations a strong dipole-dipole interaction(DDI) and anisotropic van der Waals dispersion potentials. This situation is in contrast to that of conventional ultracold atoms, such as alkali-metal atoms, where
the scattering is determined mainly by the isotropic singlet and triplet BO potentials [2]. Recent theoreticalwork predicted the existence of anisotropy-induced FanoFeshbach resonances in magnetic lanthanides [17]. This
greater complexity brings signicant new challenges inunderstanding and exploiting scattering processes.
Our experimental study is based on high-resolution
trap-loss spectroscopy of Fano-Feshbach resonances in an
optically-trapped ultracold sample of Er atoms in theenergetically lowest magnetic Zeeman sublevel. We prepare the ultracold sample by following a similar coolingand trapping approach to that described in Ref. [11] forbosons and Ref. [12] for fermions (Method Summary).

2
a
105

Atom number

104

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

105

104

36

38

40

42

44

46

48

50

52
54
Magnetic field (G)

56

58

60

62

64

66

68

70

b
105

Atom number

104

0

2

4

6

8

10

12

14

16

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22

24

26

28

30

32

34

105

104

36

38

40

42

44

46

48

50

52
54
Magnetic field (G)

56

58

60

62

64

66

68

Figure 1 | Fano-Feshbach spectrum of 168 Er and 166 Er from 0 to 70 G. The trap-loss spectroscopy is performed
in an optically trapped sample of Er atoms in the energetically lowest Zeeman sublevel mJ = −6 at a temperature of
330 nK. The atom number is measured after a holding time of 400 ms. a, We observe 190 Fano-Feshbach resonancesfor 168 Er and b, 189 resonances for 166 Er. Resonance positions are extracted by tting a Gaussian shape to individualloss features; a full list is given in the Supplementary Information.
After the preparation procedure, the ultracold sample
typically contains about 105 atoms at a temperaturearound 400 nK. High-resolution trap-loss spectroscopy
consists of many experimental cycles. In each cycle, we
ramp the magnetic eld to a target value B and hold theatoms for 400 ms in the optical dipole trap, during which
they undergo elastic and inelastic collisions. To probe theloss of atoms from the trap, we record the atom number
by applying standard time-of-ight absorption imaging
at zero magnetic eld. In the next experimental cycle,
we vary the magnetic eld value from 0 to 70 G with step
sizes of a few mG and repeat the measurement. Figure 1 shows the loss spectra for 168 Er and 166 Er. For
both isotopes, we observe an enormous number of resonant loss features, which we interpret as being caused
by Fano-Feshbach resonances [2]. We identify 190 resonances for 168 Er and 189 resonances for 166 Er, meaning that we observe about 3 resonances per gauss. We
performed similar spectroscopic measurements with thefermionic isotope 167 Er, revealing a much higher densityof resonances that exceeds 20 resonances per gauss (Extended Data Fig. 1). The fermionic case is complicatedby its additional hyperne structure and detailed studies
will be subject of future work.
The immense density of resonances in Er is without
precedent in ultracold quantum gases. For comparison,
the density of resonances observed in experiments withultracold alkali-metal atoms or even mixtures is about
two orders of magnitude lower than Er (c. f. Ref. [18, 19]).
In Er, it is unclear whether a quantitative mapping of
the observed resonances is possible at all. In principle
there are at least 91 unknown parameters, corresponding to the phase shifts introduced by the BO potentials[17]. Instead, we focus our theoretical analysis on fundamental questions, such as: Can the observed densityof resonances be reproduced by microscopic calculations?
Do our results imply the presence of highly anisotropicinteractions, which call into play resonant states of high

70

3
4

3

(1/G)

orbital momentum? We answer these questions in thearmative using full coupled-channel (CC) calculations,
supported by an analytical model.
We construct a rst-principle CC model for Er+Er
scattering to calculate the spectrum of Fano-Feshbachresonances for the experimental conditions. Following
Ref. [17], our model uses the atomic basis set and Hamiltonian (Methods) that includes the radial kinetic and rotational energy operators, the Zeeman interaction, andthe 91 anisotropic BO potentials. For small interatomic
separations R, the BO potentials are calculated using
the ab initio relativistic multi-reference congurationinteraction method [20]. At intermediate to large R, theBO potentials are expressed as a sum of multipolar interaction terms. The van der Waals dispersion interaction potentials (∝ 1/R6 ) are determined from experimental data on atomic transition frequencies and oscillator
strengths [21, 22]. An important point is that the dispersion potentials have both isotropic and anisotropic
contributions. The latter comes from the non-S statecharacter of the Er electronic ground state. The BO potentials induce thus either isotropic (` and m` conserving) or anisotropic (` or m` changing) couplings. Here,` and m` are the partial wave quantum number and its
projection on the magnetic-eld quantization axis.
We perform CC calculations for bosonic 168 Er, considering s-wave (` = 0) collisions and couplings to molecularstates with even ` up to Lmax = 20. We calculate the elastic collisional rate coecient as a function of magnetic
eld to obtain the Fano-Feshbach resonance spectrum.
For Lmax = 20, we observe a very dense resonance spectrum with about 1.5 resonances per gauss, which qualitatively reproduces our experimental observation (Extended Data Fig. 2). To get deeper insight into the roleof the anisotropy of the potentials, we calculate the meandensity of resonances ρ from our CC calculations for different values of the maximum partial wave Lmax (Fig. 2).
For Lmax up to 20, we observe that ρ increases with Lmax
in a quadratic manner. This dependence stands in stark
contrast to alkali-metal atoms, where high-partial-wave
resonances tend to be too narrow to be observed.
Since our limited computational resources do not allow us to perform calculations for Lmax > 20, it is worth
estimating the density of resonances in a simpler way,
based on the separated atom quantum numbers [3]. The
key ideas of our model are the following. For each channel |j1 mJ,1 , j2 mJ,2 , `m` i we dene the long-range potential −C6 /R6 + ~2 `(` + 1)/(2µR2 ) + gµB (mJ,1 + mJ,2 )B ,with the isotropic van der Waals C6 coecient of the
BO potentials. Here µ is the reduced mass, g is theatomic g-factor, and for ground state Er C6 = 1723
a.u.. Fano-Feshbach resonances in our open (mJ,1 =−6) + (mJ,2 = −6) channel are due to couplings to themost-weakly bound rovibrational level of closed channels. For a van der Waals potential [2, 23] this boundstate has a binding energy that must fall within the `dependent energy window [−∆` , 0] with ∆` > 0. The
short range potentials are not accurately known and,

2

1

0

0

10

20

30

40

50

60

70

Lmax

Figure 2 | Mean resonance density for bosonic
Er as a function of largest included partial wave
Lmax . CC calculations for Lmax up to 20 (circles) and

RQDT calculation (solid line) for a magnetic-eld region
from 0 G to 70 G. For calculations a collision energy of
E/kB = 360 nK is assumed. The mean densities of resonances measured in the experiment are shown for 168 Er(dashed line) and for 166 Er (dash-dotted line) with onesigma condence bands (shaded areas).

for each closed channel, there is a probability dEb /∆`of nding a bound state with a binding energy between
Eb and Eb + dEb . From Ref. [23] and numerical simulations we nd ∆` /EvdW ≈ 38.7 + 25.5`+ 3.17`2 , wherep422
EvdW = ~ /(2µxvdW ) and xvdW = 2µC6 /~2 /2. Eachclosed channel contributes gµB δm/∆` to the mean resonance density, where gµB δm > 0 is the magnetic-momentdierence of the closed and open channels and δm istheir dierence in molecular projection quantum numbers. Adding the contributions for the closed channels
gives ρ. This counting technique, which we here namerandom quantum defect theory (RQDT), yields the meandensity of states shown in Fig. 2. For Lmax 6 20, the results of our analytic RQDT agrees very well with theexact CC calculations. For larger Lmax , the density ofresonances keeps growing and eventually saturates to avalue comparable to the one observed in the experiment.
RQDT shows that at least 40 partial waves need to beconsidered to reproduce the experimental observations.
Our microscopic models reproduce well the qualitative
behavior of the system. However, given the complexityof the scattering, the analysis of ultracold collision data
can not and should not aim anymore at the assignmentof individual resonances and the fundamental question of
how to tackle complex scattering naturally arises. Historically, spectra of great complexity have been understoodwithin the framework of random matrix theory (RMT),
as originally developed by Wigner to describe heavy nuclei containing a very large number of degrees of freedom[24]. This is an alternative view of the quantum mechanics of complex systems, where individual energy levels
and resonances are not theoretically reproduced one-byone, yet their statistics can be described [25]. RMT characterizes spectra by uctuations of their energy levels and4classies their statistical behavior in terms of symmetry
classes, e. g. the Gaussian-orthogonal ensemble (GOE) in
the case of a system with time-reversal symmetry, suchas neutral atoms.
Following RMT, the distribution of spacings between
neighboring levels (or resonances) characterizes the spectral uctuations of the system and reects the absenceor the presence of level correlations in terms of a dimensionless parameter, s, i. e. the actual spacing between neighboring levels in units of the mean spacing,d = 1/ρ. Whereas the nearest-neighbor spacing (NNS)
distribution P (s) of non-interacting levels is Poissonian,
PP = exp(−s), strongly interacting levels obey a totallydierent distribution which, in the case of GOE statistics, is known as the Wigner-Dyson (WD) distribution or
Wigner surmise [25]

PWD =

π
s exp(−πs2 /4),
2

(1)which shows a strong level repulsion for small s,
PWD (0) = 0. The eld of application of the WD distribution is so vast as to make it a universal feature of very
complex systems, such as heavy nuclei, disordered conductors, zeros of the Riemann function in number theory,and even risk management models in nance [5]. Remarkably, the Bohigas-Giannoni-Schmit conjecture further enriched the eld of applications of GOE statistics[6], showing that it applies generally to chaotic systems,
such as Rydberg atoms in strong magnetic elds or Sinai
billiards, where only few degrees of freedom are relevant,but where motion in these degrees of freedom occurs on ahighly anisotropic potential energy surface [7]. Recently,
it has been speculated that even cold and ultracold atommolecule collisions will show essential features of GOE
statistics [3, 4].
Inspired by these works, we statistically analyze boththe experimental and calculated Fano-Feshbach spectrum
according to RMT. To obtain the NNS distribution ofresonances, we rst derive ρ and the mean spacing between resonances, d, by constructing the so-called staircase function [7]. This step-like function counts the number of resonances below a magnetic eld value B and isRBPdened as N (B) = dB 0
δ(B 0 − Bi ), with δ being the0idelta function and Bi the position of the i-th resonance.
For our experimental data (Fig. 3a) the staircase function
shows an increase of the number of resonances with B ,which proceeds linearly at large B and attens out towards lower magnetic-eld values (Fig. 3b). The densityof resonances is given by the derivative of the staircase
function. We evaluate ρ in the region above 30 G, where
the staircase function shows a linear progression (Supplementary Information) and we obtain ρ = 3.0(3) G−1
and d = 0.33(3) G. We perform a similar analysis with
166
Er and nd ρ = 3.3(3) G−1 and d = 0.31(3) G (Extended Data Fig. 3). For CC-calculation data, we ndρ = 3.3(3) G−1 for Lmax = 20 (Fig. 2). We nally derive the NNS distribution for the experimental and CC-

a
168

Er

b
200
110

150

100
90

100

36

38

40
B (G)

42

44

50

0
0

10

20

30

40

50

60

70

B (G)

Figure 3 | Loss-maxima position and staircase
function for 168 Er. a, Positions of the measured loss
maxima of Fig. 1 are shown as vertical lines. b, The stair-case function shows a linear dependence on the magnetic
eld at large values. A linear t to the data above 30 G
is plotted in light colors. The inset shows a magnication of the data to emphasize the step-like nature of thestaircase function.
calculated data by constructing a histogram of resonancespacings.
We choose a number of bins on the order of√
N , with N being the number of Fano-Feshbach resonances used for analysis [26]. We then rescale the histogram in terms of the dimensionless quantity s = B/dand normalize the distribution in order to obtain P (s).
Figure 4 is the main result of our statistical analysis for 168 Er. The plot shows the NNS distribution of
the experimental and the CC-calculated Fano-Feshbachresonances together with the parameter-free Poisson andWigner-Dyson distributions (Eq. 1). We see an impressive agreement between the experimental result and theCC calculations. Remarkably, both follow a distribution
much closer to the WD than to the Poissonian one. To
quantify the agreement with the GOE statistics, we evaluate the reduced chi squared, χ̃2 , between our data andthe Poisson and WD distribution. We nd χ̃2WD = 0.9
and χ̃2P = 2.3 for our experimental data and χ̃2WD = 0.8
and χ̃2P = 3.0 for the data of the CC calculations. The
fact that χ̃2WD 6 1 conrms that our data are well described by a WD distribution. Similar results are foundfor 166 Er (Extended Data Fig. 4).
To further investigate the spectral correlations, we analyze our data in terms of other statistical quantities,
such as the number variance and the two-gap distributionfunction (Supplementary Information) [27]. The number
variance Σ2 (∆B) measures the uctuations of the number of resonances in a magnetic-eld interval ∆B (Methods) [7] . For non-correlated (Poissonian-distributed)
levels, Σ2 = ∆B , indicating large uctuations arounda mean value. For quantum chaotic systems, the correlations are strong and the uctuations are thus less spreadout. In this case, Σ2 ∝ ln(∆B). This slower increase5aFigure 4 | NNS distribution and number variance. a, 168 Er NNS distribution above 30 G with a bin

features predicted by GOE statistics for chaotic systems,
the level repulsion and the spectral rigidity. This implies
a degree of complexity in Er+Er cold collisions unprecedented in any previous ultracold scattering system. Our
results bring the powerful analytical tools of quantum
chaos to bear [7]. In particular, these approaches connect
the large-scale structure of the spectra to simple features
such as the shortest closed classical orbits in the potential energy surface, where these connections are made by
the Gutzwiller trace formula [28]. Identifying the mostimportant closed orbits will then shed light on the potential energy surface itself, providing a route to describing
ultracold collisions that is complementary to the elaborate close-coupling calculations that will be dicult toconnect in detail with the data.
Erbium represents the rst occasion where statisticalanalyses and chaotic behavior are important to ultracold collisions, but they will not be the last. Specically, much experimental eort is being exerted toward
producing ultracold molecular samples, which also enjoy
highly anisotropic potential energy surfaces. Learning toread complex spectra, by acknowledging their essentiallychaotic nature, represents a turning point in how the eldwill consider ultracold collisions in the future and provide
new inroads into ultracold chemistry.

of the number variance is regarded as a strong spectral
rigidity of the system [7]. Our observations clearly deviate from the Poissonian behavior showing that Σ2 tends
to the WD case (Fig. 4b) and conrm the presence ofcorrelations in our system.
To conclude, our observations reproduce the salient

ration we follow the approach of Ref. [11]. We obtainabout 3 × 105 optically-trapped atoms at a density of
3×1013 cm−3 . The trap-loss spectroscopy is performed ina trap with frequencies of (νx , νy , νz ) = (65, 26, 270) Hz.
The temperature of the cloud is measured by time-ofight imaging at 0.4 G and gives T168 = 326(4) nK and
T166 = 415(4) nK, respectively. We ramp the magnetic
eld within 10 ms to a probe value between 0 and 70 G,
and hold the atomic cloud for 400 ms in the optical dipole
trap. We observe an increase of the temperature up to560 nK at a magnetic eld of about 50 G due to the ramping over many Fano-Feshbach resonances. For fermionicsample preparation we follow the approach of Ref. [12].
We obtain about 1.2 × 105 fermionic atoms at a densityof 2 × 1014 cm−3 and at a temperature of 0.4 TF , where
TF = 1.0(1) µK is the Fermi temperature. The trap frequencies are (427, 66, 457) Hz.

1.0
0.8

P

0.6
0.4
0.2
0.0
0

1

2

3

4

s

b

4

3

2

1

0
0

1

2
∆B (G)

3

4

size of 160 mG. The plot shows the experimental data(circles) with the corresponding Brody distribution (solidline), the Brody distribution for the CC calculation with
Lmax = 20 (dotted line), and the parameter free distributions PP (dashed line) and PWD (short-dashed line).
The Brody distribution is given in the Methods section.
For the error bars in the experimental data, we assume
a Poisson counting error. b, Number variance for theexperimental data (solid line) with a two-sigma condence band (shaded area), the CC-calculation data (dotted line), PP (dashed line), and PWD (short-dashed line).

∗†

francesca.ferlaino@uibk.ac.at; corresponding author

METHODS SUMMARY
Sample preparation. For bosonic sample prepa-[3] M. Mayle, B. P. Ruzic,

and J. L. Bohn,  Statistical as-

Alternative address: St. Petersburg Nuclear Physics In-

pects of ultracold resonant scattering. Phys. Rev. A

stitute, Gatchina, 188300; Division of Quantum Mechan-

062712 (2012).

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82, 1225 (2010).

and J. L. Bohn,

 Scattering of ultracold molecules in the highly resonant
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87, 012709 (2013).

[5] T. Guhr, A. Müller-Groeling,and H. A. Weidenmüller, Random-matrix theories in quantum physics: common
and E. Tiesinga, Feshbach resonances in ultracold gases.
Phys.

[4] M. Mayle, G. Quéméner, B. P. Ruzic,

85,

Rev. Mod.

concepts. Physics Reports

299, 189 (1998).

[6] O. Bohigas, M. J. Giannoni,

and C. Schmit,  Charac-terization of chaotic quantum spectra and universality of6level uctuation laws. Phys. Rev. Lett.

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J. Flores,

J. B. French,

P. A. Mello,

and B. L. Lev,

A. Pandey, and S. S. M. Wong,  Random-matrix physics:

 Strongly dipolar Bose-Einstein condensate of dyspro-

Spectrum and strength uctuations. Rev. Mod. Phys.

sium. Phys. Rev. Lett.

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[10] M. Lu, N. Q. Burdick, and B. L. Lev,  Quantum degenerate dipolar Fermi gas. Phys. Rev. Lett.

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[28] M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, 1990).

[29] A. Frisch, K. Aikawa, M. Mark, A. Rietzler, J. Schindler,

R. Grimm, and F. Ferlaino,  Bose-Einstein condensation

E. Zupani£, R. Grimm,of erbium. Phys. Rev. Lett.

magneto-optical trap for erbium.

108, 210401 (2012).

[12] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm,
and F. Ferlaino,  Reaching Fermi degeneracy via universal

dipolarscattering.

ArXiv

e-prints(2013),

arXiv:1310.5676.
J. M. Doyle,  Magnetic trapping of rare-earth atoms at
millikelvin temperatures. Nature

431, 281 (2004).

[14] C. B. Connolly, Y. S. Au, S. C. Doret, W. Ketterle,
and J. M. Doyle,  Large spin relaxation rates in trapped
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[15] V. Kokoouline, R. Santra, and C. H. Greene,  Multichannel cold collisions between metastable Sr atoms. Phys.

90, 253201 (2003).

[16] R. V. Krems, G. C. Groenenboom,and A. Dalgarno, Electronic interaction anisotropy between atoms in arbitrary angular momentum states. J. Phys. Chem. A

108, 8941 (2004).

[17] A. Petrov, E. Tiesinga, and S. Kotochigova,  Anisotropyinduced Feshbach resonances in a quantum dipolar gasof highly magnetic atoms. Phys. Rev. Lett.

109, 103002(2012).
[18] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C.
Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M.
Hutson,  Feshbach resonances, weakly bound molecularstates, and coupled-channel potentials for cesium at highmagnetic elds. Phys. Rev. A

87, 032517 (2013).

[19] T. Takekoshi, M. Debatin, R. Rameshan, F. Ferlaino,
R. Grimm, H.-C. Nägerl, C. R. Le Sueur, J. M. Hutson, P. S. Julienne, S. Kotochigova,

and E. Tiemann, Towards the production of ultracold ground-state RbCs
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Author contributions

A.F., M.M, K.A., and F.F. did the experimental workand statistical analysis of the data, C.M., A.P., and S.K.
did the theoretical work on CC calculations and RQDT,
J.L.B. did the theoretical work on RMT. The manuscript
was written with substantial contributions from all authors.

Acknowledgements

The Innsbruck group thanks R. Grimm for inspiring discussions and S. Baier, C. Ravensbergen, and M. Brownutt
for careful reading of the manuscript.
S. K. and
A. P. thank E. Tiesinga for useful discussions. J. L. B. is
supported by an ARO MURI. The Innsbruck team is
supported by the Austrian Science Fund (FWF) througha START grant under Project No. Y479-N20 and by theEuropean Research Council under Project No. 259435.
K. A. is supported within the Lise-Meitner program of
the FWF. Research at Temple University is supported
by AFOSR and NSF PHY-1308573.

7

METHODS

Experimental procedures. For bosonic sample
preparation we follow the approach of Ref. [11]. In brief,
after the magneto-optical trap [29], we load the atoms
in an optical dipole trap composed of two laser beams
in horizontal (1,064 nm, 0.4 W, single-mode) and vertical direction (1,064 nm, 4.0 W, broadband Yb berlaser). In the trap, we force evaporation by ramping
down the power of both trapping laser beams within
6.2 s, in the presence of a homogeneous magnetic eldof 0.4 G to prevent spin-ips to other Zeeman states.
We stop evaporative cooling before the onset of BoseEinstein condensation. Our nal trap has frequencies of(νx , νy , νz ) = (65, 26, 270) Hz and contains about 3 × 105
atoms at a density of 3 × 1013 cm−3 . The temperature of
the atomic cloud is measured by time-of-ight imaging
for both isotopes at 0.4 G and gives T168 = 326(4) nK
and T166 = 415(4) nK. We ramp the homogeneous magnetic probe eld up to 70 G within 10 ms and hold theatomic cloud for 400 ms in the optical dipole trap. The
magnetic eld is suddenly (< 5 ms, limited by eddy currents) switched o and the atom number and size of thecloud is measured via absorption imaging after a time of
ight of 15 ms. We observe an increase of the temperature up to 560 nK at a magnetic eld of about 50 G due
to ramping over many Fano-Feshbach resonances. For
fermionic sample preparation we follow the approach ofRef. [12]. We obtain about 1.2 × 105 fermionic atoms at adensity of 2 × 1014 cm−3 and at a temperature of 0.4 TF ,where TF = 1.0(1) µK is the Fermi temperature. The
trap frequencies are (427, 66, 457) Hz.
Magnetic-eld control. An analog feedback loop
stabilizes the current for the homogeneous magnetic-eldcoils with a relative short-term stability of better than
2×10−4 . Calibration of the magnetic eld is done by driving a radio-frequency transition between Zeeman statesmJ = −6 and mJ = −5. Trap-loss spectroscopy is carried out in steps of 20 mG (out of resonance) and 5 mG
(on resonance). The long-term oset stability of the magnetic eld was observed during the data recording period
to be better than 4 mG within one week.
Coupled-channel calculations. We perform exact CC calculations for Er+Er scattering in the basis|j1 mJ,1 , j2 mJ,2 , `m` i ≡ Y`m` (θ, φ)|j1 mJ,1 i|j2 mJ,2 i, where~ja=1,2 are the atomic angular momenta with space-xed
projection mJ,a=1,2 along the magnetic-eld direction,
the spherical harmonics Y`m` (θ, φ) describe molecular rotation with partial wave ~`, and where the angles θ and φorient the internuclear axis relative to the magnetic eld.
For a closed-coupling calculation of the rovibrationalmotion and of the scattering of the atoms we need all electronic potentials dissociating to two ground-state atoms.
There are 91 BO potentials for Er2 , of which 49 are
gerade and 42 are ungerade potentials. For collisionsof bosons in the same Zeeman state only gerade states
matter. These potential surfaces have been obtained using an ab initio relativistic multi-reference conguration-interaction method (RMRCI) [20], and converted into atensor operator form with R-dependent coecients. Examples of tensor operators are the exchange interaction
Vex (R)~j1 · ~j2 and the anisotropic quadrupole-rotation operator VQ (R)Y2 (R̂) · [~j1 ⊗ ~j1 ]2 coupling the quadrupole
operator [~j1 ⊗ ~j1 ]2 of one atom with angular momentum
j1 to the rotation of the molecule. See [17] for otheroperators.
Collisions of submerged 4f-shell atoms at low temperatures also depend on the intermediate to long-range
isotropic and anisotropic dispersion, magnetic dipoledipole and quadrupole-quadrupole interatomic interactions. The van der Waals dispersion potentials fortwo ground-state atoms are obtained using the transition frequencies and oscillator strengths [21, 22]. The
quadrupole moment of Er is calculated using an unrestricted atomic coupled-cluster method with single, double, and perturbative triple excitations uccsd(t) [30] andshown to be small at Q = 0.029 a.u..
We use a rst-principle coupled-channel model tocalculate anisotropy-induced magnetic Fano-Feshbachresonance spectra of bosonic Erbium. The model treats
the Zeeman, magnetic dipole-dipole, and isotropic andanisotropic dispersion interactions on equal footing. The
Hamiltonian includes

H=−

~` 2
~2 d2
~ τ) ,++ HZ + V (R,
2
2µ dR
2µR2

~ describes the orientation of and separation bewhere R
tween the two atoms. The rst two terms are the radial kinetic and rotational energy operators, respectively.
The Zeeman interaction is HZ = gµB (j1z + j2z )B , where
g is an atomic g-factor and jiz is the z component of theangular momentum operator ~i of atom i = 1, 2. The in~ τ ), includes the Born-Oppenheimer andteraction, V (R,the magnetic dipole-dipole interaction potentials, which
are anisotropic, and τ labels the electronic variables. Finally, µ is the reduced mass and for R → ∞ the inter~ τ ) → 0. Coupling between the basis states
action V (R,~ τ ), inducing either isotropic (` and m`
is due to V (R,conserving) or anisotropic (` or m` changing) couplings.
The Hamiltonian conserves Mtot = mJ,1 + mJ,2 + m` andis invariant under the parity operation so that only even(odd) ` are coupled. In the atomic basis set, the Zeeman
and rotational interaction are diagonal.
NNS probability distribution. As the density ofresonances is not constant below 30 G we restrict ouranalysis to resonances appearing from 30 to 70 G. We
plot a histogram of spacings between adjacent resonances
given by di = Bi+1 − Bi . For this an appropriate
num√
ber of bins is chosen on the order of N , with N being the total number of Fano-Feshbach resonances observed up to 70 G. This ensures a bin size at least anorder of magnitude larger than the mean resolution of
the trap-loss spectroscopy scan. For every bin a statistical counting error according to a Poisson distribu-

8tion is assigned. Next, the magnetic-eld axis of the histogram is divided by the mean spacing of resonances to
get the dimensionless quantity s = B/d. To calculate
the NNS probability distribution P (s) the histogram has
R∞
to be normalized such that ds P (s) = 1. As shown
0

in Ref. [27], the probability distribution of uncorrelated
random numbers is simply given by the Poisson distribution PP (s) = exp(−s). A theoretical spacing distributionof random matrices can not be written in a simple form
but, according to the Wigner surmise, an excellent approximation is given by the Wigner-Dyson distribution
PWD (s) = π2 s exp(−πs2 /4). A way of discriminating between these two distributions is to t the so-called Brody
distribution to the NNS distribution [8]. It is an empiricalfunction with a single tting parameter η , which interpolates between PWD and PP and quanties the tendency(and not the degree of chaoticty) of the observed distribution to be more Poisson-like (η = 0) or more WignerDyson-like (η = 1). It is dened by

PB (s) = Asη exp(−αsη+1 )
A = (η + 1)α
 
η+1
η+2
α= Γ
,
η+1
where Γ denotes the Gamma function. From a leastsquares t to the experimental data, we obtain η168 =0.66(10) for 168 Er and η166 = 0.73(18) for 166 Er, and at to the CC-calculation data gives ηCC = 0.72(18).
Number variance. The number variance Σ2 is aquantity that depends on long-range correlations between
resonance spacings within an interval ∆B . It is denedbyΣ2 (∆B) = n2 (B0 , ∆B) − (n(B0 , ∆B))2 ,with n(B0 , ∆B) = N (B0 + ∆B) − N (B0 ) giving thenumber of resonances in the interval [B0 , B0 + ∆B]
and the bar denotes the mean value over all B0 .
For a Poisson distribution, Σ2 = ∆B .
By contrast, for a spectrum according to RMT one expects
Σ2 = 1/π 2 ln(2π∆B) + γ + 1 − π 2 /8 , for large ∆B andwhere γ = 0.5772... is Euler's constant [25]. This behavior reects that there are only very small uctuations
around an average number of resonances within a given
interval of size ∆B (spectral rigidity). Compared to the
NNS distribution the number variance is more suitable toprobe long distances in the spectrum. A clear signatureof level repulsion on the one hand and a large spectral
rigidity on the other are central properties of strong correlations between levels according to RMT [27].

9Atom number

105

104

0.0

0.5

1.0

1.5

2.0
2.5
Magnetic field (G)

3.0

3.5

4.0

4.5

Extended Data Figure 1 | Fano-Feshbach spectrum of fermionic 167 Er from 0 to 4.5 G. The trap-loss

spectroscopy is performed in an optically trapped sample of fermionic Er atoms at a temperature of 0.4 TF , where
TF = 1.0(1) µK is the Fermi temperature. The atoms are spin-polarized in the lowest Zeeman sublevel, mF = −19/2.
We keep the atomic sample at the magnetic probing eld for a holding time of 100 ms. We observe 115 resonances
up to 4.5 G, which we attribute to be Fano-Feshbach resonances between identical fermions. The corresponding meandensity is about 26 resonances per gauss.
-12

10

-11

3

Elastic rate (cm /s)

10

10

-10

10

-9

0

10

20
30
Magnetic field (G)40

50

Extended Data Figure 2 | Elastic rate coecient of mJ = −6 168 Er collisions. The s-wave elastic ratecoecient as a function of magnetic eld assuming a collision energy of E/kB = 360 nK. Partial waves ` up to 20 are
included. A divergence of the elastic rate coecient, i.e. the position of a Fano-Feshbach resonance, is marked withsquares.
a
166

Er

b
200

110
100

150
90
80
36

100

38

40
B (G)

42

44

50

0
0

10

20

30

40

50

60

70

B (G)

Extended Data Figure 3 | Statistical analysis of high-density Fano-Feshbach resonances of isotope
Er. a, Position of the resonances are marked with vertical lines. b, The staircase function shows a similar behavior

166

to 168 Er (Fig. 3). A linear t to the data above 30 G is plotted in light colors. From the staircase function we
calculate a mean density of resonances of ρ = 3.3(3) G−1 , which corresponds to a mean distance between resonancesof d = 0.31(3) G.

10
a 1.0
0.8

P

0.6
0.4
0.2
0.0
0

1

2

3

4

s

b

4

3

2

1

0
0

1

2
∆B (G)

3

4

Extended Data Figure 4 | NNS distribution and number variance. a, 168 Er NNS distribution above 30 Gwith a bin size of 140 mG. For the error bars we assume a Poisson counting error. The plot shows the experimental
data (circles) with the corresponding Brody distribution (solid line). The parameter free distributions PP (shortdashed line) and PWD are shown and reduced chi-squared values are χ̃2P = 2.32 for the Poisson and χ̃2WD = 1.85 for
the Wigner-Dyson distribution. b, Number variance Σ2 for the same experimental data (solid line) with a two-sigma
condence band (shaded area). The number variance from experimental data shows a clear deviation from the number
variance of a Poisson distribution.

151

10.3. Further work

10.3. Further work
10.3.1. RMT and Feshbach spectroscopy
To gain a deeper insight into the coupling between molecular states we developed a toy model,
which allows to calculate the binding energy of a random set of coupled molecular states
dependent on the magnetic field. Utilizing this model we can then determine the position,
at which the states cross the threshold at zero energy and derive a ‘simulated Feshbach
spectrum’. We analyze the NNS distribution of these artificial Feshbach resonances fordifferent coupling strengths and observe a Wigner-Dyson distribution only for strong couplingstrengths. The energy dependence was derived by diagonalizing the model’s Hamiltonian
H = H0 + Hcpl − µmol B ,10.1

where H0 gives the energy of the states at zero magnetic field B, Hcpl contains the couplingstrengths between different states in the off-diagonal elements, and the last term gives the
Zeeman shift of the states. For simplicity, the magnetic moment of the simulated molecularstates, µmol , is chosen randomly from all possible combinations of mJ states of two erbium
atoms. Furthermore, a representative density of resonances of about three per Gauss, likeobserved in the experiment, was assumed. The coupling strength between two molecularlevels are randomly chosen from a Gaussian distribution with a given variance of νa2 , seeFig. 10.1.
Figure 10.2 shows the magnetic-field dependence of the simulated binding energy of coupledmolecular states for different values of νa2 . For larger values of the coupling avoided crossingsof states become visible. As the slope of the state changes with the magnetic field whenmoving over an avoided crossing, its magnetic moment changes accordingly. For stronger
coupling strengths this effect leads to a more homogeneous distribution of magnetic momentsof states, as seen in Fig. 10.2(c-d).
(a)

(b)

ν2a=10-4

2x104
occurrence

occurrence

2x104

1x104

0
-0.04

ν2a=10-2

-0.02
0.00
0.02
coupling strength

0.04

1x104

0
-0.4

-0.2
0.0
0.2
coupling strength

0.4

Figure 10.1.: Random Gaussian-distributed coupling strength for the molecular state coupling simulation shown in Fig. 10.2. The histogram (blue) plots the occurrence of coupling matrix elementswith zero mean and a variance of (a) νa2 = 10−4 and (b) νa2 = 10−2 , respectively. Each histogram
contains 124,750 matrix elements, which are needed for a coupling matrix Hcpl of size 500 × 500. The
solid line (gray) shows a Gaussian fit to the histogram.

152

10.3. Further work

energy (MHz)

(a)
0.0

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1.0

0

1

3

4

5

6

ν =10
2
a

(c)

energy (MHz)

2

-3

-1.0

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8
0

1

2
3
4
magnetic field (G)

0

1

5

6

-1.0

2

3

4

5

6

2
3
4
magnetic field (G)

5

6

ν =10
2
a

(d)

0.0

-1.0

ν2a=10-4(b)

no coupling

0

1

-2

Figure 10.2.: Magnetic field dependence of the energy of simulated molecular states (blue) withrandom magnetic moments assuming (a) no coupling between molecular states and randomly chosen
Gaussian distributed coupling strengths with (b) νa2 = 10−4 , (c) νa2 = 10−3 , and (d) νa2 = 10−2 .
The slope of the states equals their magnetic moment. Additionally, the crossing of a state with thethreshold, i. e. the position of a Feshbach resonance, (E = 0) is marked with red circles and the energyof the state at zero magnetic field (B = 0) is shown by green squares. Avoided crossings appear whena coupling between states is present.

The strong coupling between molecular states leads to a significant change of the distributionof FR positions, as shown in Fig. 10.3. The molecular states are assumed to be distributed
according to a Wigner-Dyson distribution with respect to energy. The distribution of FRschanges from uncorrelated resonances (without coupling) to strongly correlated resonances(for coupled molecular states) following a Wigner-Dyson distribution.
Contrary to a simple system of, e. g. alkali-metal atoms, where coupled-channel calculations
can be easily performed and the Feshbach spectrum fully assigned [Chi10], lanthanide atoms
are challenging for theorists. Due to the unprecedented high density of FRs and strongly
coupled molecular states, the way of describing a Feshbach spectrum by employing randommatrix theory seems to be the most fruitful at the moment.

153

10.3. Further work
(a)

(b)

no coupling(c)ν2a=10-4

1.0

ν2a=10-2

P(s)
P(s)

0.8
0.6
0.4
0.2
0.0

0

1

2

3

4

0

1

2
s

s
1.0

0.0

0.8

-0.2
energy (MHz)

P(s)

4

0

1

2
s

3

4

2
3
4
magnetic field (G)

5

6

(e)

(d)

0.6
0.4
0.2
0.0

3

-0.4
-0.6
-0.8

0

1

2

3

4

s

-1.0

0

1

Figure 10.3.: NNS distribution P (s) of molecular states (a-c) along the magnetic field axis and (d)along the energy axis. P (s) dependent on the magnetic field is shown for non-coupled states (a) and
Gaussian-distributed coupling strengths with (b) a variance of νa2 = 10−4 and (c) with νa2 = 10−2 . (d)In all three cases the energy of the states at zero magnetic field are assumed to be randomly distributed
according to a Wigner-Dyson distribution. The solid line shows a Wigner-Dyson distribution and thedashed line a Poisson distribution. A total of 500 states was used to derive the NNS distributions.
Panel (e) shows only a small region of the magnetic field dependence of states as in Fig. 10.2 for acoupling strength of νa2 = 10−2 .

10.3.2. Feshbach resonance width distribution
It was shown in Ref. [Por56] that widths from nuclear reactions can be described by a chisquared (χ2 ) distribution. This distribution is defined asPχ2 (s) =1(ns)n−1 e−ns n ,
Γ(n)

10.2

where Γ is the Gamma function and ν = 2n is the so-called degree of freedom. For ν = 2
this gives Pχ2 (s) = exp (−s), being an exponential distribution.
Widths of resonances in nuclear spectra were best fitted by a distribution with one degreeof freedom. Figure 10.4 shows the distribution, P (s), of erbium Feshbach resonance widths.
The histogram was created by binning the width data from trap-loss spectroscopy into ahistogram with a bin size of 25 mG. Normalizing the width is done by s = Γred /Γred ,with Γ being the arithmetic mean of widths. This results in P (s), which is compared withchi-squared distributions of different degrees of freedom. We observe that it fits best bya χ2 -distribution with one degree of freedom. Following Ref. [Por56] this corresponds to a

154

10.3. Further work

2.0
168

Er
Er
DOF=1
DOF=2
DOF=3

1.6

166

P(s)

1.2
0.8
0.4
0.0

0

1

2

3

4

5

s

Figure 10.4.: Normalized width distribution of Feshbach resonances in 168 Er (red circles) and 166 Er(blue squares) above 30 G. The χ2 -distributions with degrees of freedom (DOF) of one, two, andthree are plotted in gray colors for comparison. The data points agree best with the χ2 -distribution
with DOF = 1. Error√ bars result from statistical uncertainties from creating the histogram. The
error is taken to be N with N the number of counts per bin.

single outgoing channel No , which is as expected by the preparation of erbium atoms in theirenergetically lowest Zeeman state.

Chapter

11

Publication: Fermi surface
deformation

11.1. Introduction
Dipolar many-body effects in atomic Fermi gases are a rather unexplored field. This is due tothe fact that only recently suitable fermionic systems have become available. The magnitude
2of dipolar effects is given by the ratio nd
EF , which is typically much smaller than unity. Here,
EF is the Fermi energy, n the peak number density, and d2 the coupling constant of the20µdipole-dipole interaction (DDI), being for magnetic dipoles µ4π.
A first dipolar many-body effect should be noticeable in a deformation of the Fermi surface. The Fermi surface deformation (FSD) is predicted in many theoretical works, see
Refs. [Miy08, Fre09, Bai12b], but has not been experimentally observed before. In the presented publication we confirm the deformation of the Fermi surface in a degenerate Fermi
gas of erbium atoms. Furthermore, we demonstrate that it is indeed a many-body effect.
The FSD in the erbium system is caused by the anisotropy of the magnetic DDI. The DDI
induces two additional contributions in the total energy of the system. The first one is theHartree direct term and the second one is the Fock exchange term, see Ref. [Miy08]. In caseof isotropic contact interaction these two terms cancel each other and the Fermi surface isalways isotropic.
The amount of deformation for the parameters of our experiment is expected to be very small,just on the order of a few percent, which makes the detection of the FSD very difficult. We
have shown that the amount of deformation depends on the Fermi energy of the system as
it increases for higher Fermi energies, clearly showing the quantum many-body nature of
the FSD. Furthermore, the direction of the FSD follows the orientation of the polarizing
magnetic field. In time-of-flight images, revealing the momentum distribution of the cloud,it was possible to visualize the FSD by an anisotropic shape of the atom cloud.
This first experimental verification of the FSD might lead to further investigations on collective excitations and anisotropic superfluid pairing, see Ref. [Bar02b].

155

11.2. Publication:
Observation of Fermi surface deformation in a dipolar
quantum gas†‡
Science 345, 1484 (2014)
submitted 25 April 2014; published 15 August 2014
DOI 10.1126/science.1255259
Kiyotaka Aikawa,1 Simon Baier,1 Albert Frisch,1 Michael Mark,1 Cornelis Ravensbergen,1,2
and Francesca Ferlaino1,2
1Institut für Experimentalphysik and Zentrum für Quantenphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
2Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften,
6020 Innsbruck, Austria†‡

The author of the present thesis helped in taking the data and contributed in writing the manuscript.
The preprint version of the publication is attached for online access of this thesis.

156

Observation of Fermi surface deformation in a dipolar gas: a quantum nematic phase in theultracold regime
K. Aikawa,1 S. Baier,1 A. Frisch,1 M. Mark,1 C. Ravensbergen,1, 2 and F. Ferlaino1
1 Institut für Experimentalphysik and Zentrum für Quantenphysik,

Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

2 Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria(Dated: May 8, 2014)

The deformation of a Fermi surface is a fundamental phenomenon leading to a plethora of exotic quantum
phases with broken symmetries. Understanding these phases, which play crucial roles in a wealth of systems, is amajor challenge in condensed matter physics. Here, we report on the observation of a Fermi surface deformation
in a quantum gas of erbium atoms. The deformation is triggered by the strong magnetic dipole-dipole interaction
among erbium atoms. We demonstrate that the Fermi surface deformation is tunable via an external trapping
potential. Our work shows that the ground state of a dipolar Fermi gas is a nematic phase and that a dipolar
Fermi gas provides an ideal clean system for exploring a nematic phase in a highly controlled manner.
PACS numbers: 03.75.Ss, 37.10.De, 51.60.+a, 67.85.Lm

The Fermi-liquid theory, formulated by Landau in the late50’s, is one of the most powerful tools in modern condensed
matter physics [1]. It captures the behavior of interacting
Fermi systems in the normal phase, such as electrons in metals
and liquid 3 He [2], and provides the basis for understanding
Cooper pairing in conventional superconductors. Within thistheory, the interaction is accounted by dressing the fermionsas quasi-particles with effective mass and effective interaction, and with an isotropic Fermi surface (FS) in momentum space. In strongly correlated electron systems, however,
the electron-electron (Coulomb) interaction can violate theFermi-liquid picture and give rise to a deformed FS, which
breaks the symmetry of the system. The deformation of a FS
is often referred to as a Pomeranchuk instability of a normal
Fermi liquid [3]. The simplest realization of such a situation,
where the rotational symmetry of the FS is broken, is a quantum nematic phase, which is named in analogy with liquid
crystal phases [4]. Symmetry-breaking FSs in solids have attracted enormous attention for their potential impact on pseudogap phases in high-transition-temperature superconductors
and quantum criticality [5–7]. Recent experimental work reported on signatures of such a phenomenon in transport, mechanical, and surface properties of various materials, including superconductors such as cuprates [8–11] and iron pnictides [12–14]. However, the direct observation of spontaneously deformed Fermi surfaces in solids remains an elusive
goal since a FS is strongly affected by disorders, impurities,
and underlying crystalline structures [15, 16].
A completely distinct approach to quantum nematic phases
is to explore ultracold Fermi gases with strong dipole-dipole
interaction (DDI) [17–20], as realized with ultracold polar
molecules and magnetic atoms [21–23]. Differently from swave interaction that can entail an isotropic broadening of thealways-spherical FS [24, 25], the anisotropic DDI is predicted
to impact the momentum distribution of fermions in a profound and subtle way, giving rise to a permanent deformationof the FS [19, 20, 26–28]. The realization of such a quantum
nematic phase in experiments would strengthen the role of ul-βFIG. 1: (color online) AR of an expanding dipolar Fermi gas as afunction of the angle β . The sample contains 7 × 104 atoms at a typical temperature of T /TF = 0.18. In this measurement, the trap frequencies in three directions are ( fx , fy , fz ) = (579, 91, 611) Hz. The
data are taken at tTOF = 12 ms. The error bars are standard errorsof more than 20 independent measurements. For comparison, thecalculated values are also shown for 0◦ and 90◦ (squares). The inset
schematically illustrates the geometry of the system. Gravity is along
the z direction. The atomic cloud is imaged with an angle of 28◦ withrespect to the y axis. The magnetic field orientation is rotated on theplane with an angle of 14◦ with respect to the xz plane. Schematic
illustrations of the imaged atomic cloud emphasizing the anisotropy
are also shown above the panel.

tracold matters as an excellent quantum simulator for exotic
condensed matter phenomena.
In this paper, we report on the direct observation of a
Fermi surface deformation (FSD) in a dipolar quantum gasof fermionic 167 Er atoms. The large DDI among the fermions
results from the large Er magnetic moment µ of 7 Bohr magneton. We demonstrate that the magnitude and the orientationof the FSD can be controlled with external fields. The deformation vanishes at high temperatures of around TF , where TF

2
A

B

Trap anisotropy
Trap anisotropy

C

D
0.01

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

is the Fermi temperature, pointing to the quantum-mechanicalnature of the observed effect. Our observations are in very
good agreement with calculations based on Refs. [26–28].
This work constitutes the first observation of a many-bodyquantum effect in a dipolar Fermi gas in three-dimensions
and provides direct evidence that the ground state of a dipolar
Fermi gas is a nematic phase.
Our starting point is a single-component quantum degenerate dipolar Fermi gas of 167 Er atoms [23] (Supplementary
Materials). The ultracold sample is confined in a threedimensional harmonic trap consisting of two laser beams ina crossed-beam configuration. We control the orientation of
the atoms’ dipole moment with an external polarizing magnetic field; see inset Fig. 1.
To explore the impact of the DDI on the momentum distribution, we study the expansion dynamics of the gas whenreleased from the trap. Such a time-of-flight (TOF) experiment is extremely powerful to probe properties of trapped
gases and many-body phenomena [24, 25], providing, for instance, the early ”smoking-gun” evidence for Bose-Einsteincondensation [29, 30]. The crucial point of this method is thatfor a long expansion time, the size of the atomic cloud is dominated by the velocity dispersion and, in the case of ballistic(free) expansions, the TOF images purely reflect the momentum distribution in the trap. In our experiment, we release thedegenerate Fermi gas from an optical dipole trap (ODT) andimage the sample after a long expansion time tTOF . We thenanalyze the aspect ratio (AR) of the cloud, i. e. the vertical-tohorizontal ratio of the observed radii (Supplementary Materials).
Figure 1 shows the AR of the Fermi gas for various dipole
orientations β , where β is defined as the angle of the magnetic

Optical Depth

0

E

-0.01
440 µmFIG. 3: (color online) Deformation ∆ for various trap geometries.
The calculations consider a cigar-shaped trap with fx = fz and show
separately the behavior of the FSD (dashed lines)
p and the NBE (dotted lines) as a function of the trap anisotropy fx fy / fz at f¯ = 400 Hzp(A) and as a function of f¯ at fx fy / fz = 5 (B). (C) Experimentallyobserved deformations at tTOF = 12 ms are plotted as a function of η,
together with the full calculation (solid line) and the calculation considering only FSD (dashed line). The sample contains 6 × 104 atoms
at a typical temperature of T /TF = 0.15. The error bars represents
standard errors of about 15 independent measurements. The variation of the trap anisotropy in the experiment is indicated in the top
axis. (D) Visualized FSD for the observed TOF image at η = 0.14.
(E) Visualized FSD for the fitted image at η = 0.14.

field orientation with respect to the z axis (see inset Fig. 1). We
observe a clear deviation of the AR from unity, corresponding
to an ellipsoidal distribution with cloud elongation along theorientation of the polarizing magnetic field. For vertical polarization, we estimate a cloud anisotropy of about 3 %. Remarkably, we observe that the ellipticity of the Fermi gas rotateswith the magnetic field orientation, keeping the major axis of
the ellipse always aligned with the dipole orientation. This behavior clearly shows that the DDI is the origin of the observed
anisotropy and stretches the FS along the orientation of thedipoles. Subsequently, we investigate the expansion dynamics for two dipole orientations by following the time evolutionof the AR (Fig. 2). At short tTOF , AR rapidly changes as aconsequence of the radial expansion (acceleration). Eventually, for tTOF > 10 ms, it plateaus for both angles, revealing
the momentum distribution of the sample.
The one-to-one mapping between the initial momentum
distribution of the trapped Fermi gas and the position distribution of the expanding cloud after long tTOF strictly holds
only in the case of pure ballistic (i. e. free) expansions. In our
experiments, the DDI is acting even during the expansion andcould potentially mask the observation of the FSD. To evaluate the effect of the non-ballistic expansion (NBE), we perform numerical calculations based on a Hartree-Fock mean-

3√
η = ACdd EF(1)Here, A = c0 m3/2 /h3 is a numerical factor with c0 =8π 3 6−1/6 ' 184, m the mass of 167 Er, and h the Planck con-stant, Cdd = µ0 µ 2 /(4π) denotes the dipolar coupling constantwith µ0 vacuum permeability, and EF = kB TF = h f¯(6N)1/3
is the Fermi energy for a non-interacting Fermi gas in a harmonic trap with kB the Boltzmann constant, f¯ = ( fx fy fz )1/3
the mean trap frequency, and N the number of atoms. Intuitively, this scaling is interpreted as follows: a larger EF ,resulting in a higher number density and thus in a smaller interparticle spacing, gives rise to larger DDI in a Fermi gas.
These arguments open up the possibility of tuning the FSD
by changing EF , i. e. the external trapping potential and thenumber density.
We explore the dependence on the external trapping potential in experiment and substantiate the experimental results by
calculations. For convenience, we define the deformation factor, ∆, as ∆ = 1 − AR.
√ We first simulate how ∆ varies when
the trap anisotropy, fx fz / fy , (Fig. 3A) and/or f¯ (Fig. 3B) arevaried. In the calculations, we keep the FSD and the NBE contributions separated. Our theoretical results clearly convey thefollowing information: (i) the FSD gives the major contribution to ∆ and is truly independent from the trap anisotropy,
while it increases with f¯, (ii) the NBE effect is reminiscent of
the trap anisotropy and vanishes for a spherical trap [27].
In the experiment, we explore the dependence of ∆ on thetrap geometry for β = 0◦ by keeping the axial frequency ( fy )constant and vary the radial frequencies ( fx = fz within 5 %)(Fig. 3C). This leads to a variation in the trap anisotropy, inf¯, and thus in η. We observe a linear dependence of ∆ on
η in agreement with Refs. [26, 27] and our numerical calculations. The excellent agreement between experiments andtheory shows that our system is a clean sample allowing us toinvestigate a nematic phase under a highly controlled environment.

4

2

∆(% )

field theory in the zero temperature limit [26–28] (Supplementary Materials). In Fig. 2, we compare the experimental datawith the parameter-free theoretical curves, which include both
the FSD and the NBE effects. We observe an excellent overall
agreement between the experiment and the theory, showingthat our model accurately describes the behavior of our system. In addition, we plot the numerical simulations in thecase of pure ballistic expansions. The comparison between
the ballistic and non-ballistic expansion reveals that the latter
plays a minor role in the final AR, showing that the observed
anisotropy dominantly originates from the FSD. For the sakeof completeness, Fig. 2 also shows the calculation for a noninteracting Fermi gas, whose FS is spheric.
The FSD is a quantum phenomena that comes from the antisymmetrization of many-body fermion wavefunctions. It has
been predicted that its magnitude increases with the Fermi energy and the dipole moment and that. In the limit of weak
DDI, the magnitude of the FSD is linearly proportional to adimensionless parameter [26, 27]

0

-2

0 .2

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

0 .6

0 .8

FIG. 4: (color online) Deformation as a function of the temperatureof the cloud. Measurements are performed for two dipole angles,
β = 0◦ (squares) and β = 90◦ (circles) under the same conditions
as in Fig. 2. The experimental data are taken at tTOF = 12 ms. The
plot shows the numerically calculated values at zero temperature for
β = 0◦ (solid line) and β = 90◦ (dashed line).

In analogy with a study on nematicity in a superconducting material [31], we graphically emphasize the FSD in themeasurements at η = 0.14 by subtracting the TOF absorption
image taken at β = 90◦ from the one at β = 0◦ (Fig. 3D). The
resulting image shows a clover-leaf-like pattern, visualizingthat the momentum spread in the orientation of the dipoles islarger than in the other direction. For comparison, the sameprocedure is applied for images obtained by a fit to the observed cloud (Fig. 3E). At η = 0.14, the trap anisotropy is sosmall that the NBE effect is negligibly small. Therefore, thedeformation visualized here is almost purely the FSD.
Finally, we investigate the temperature dependence of ∆(Fig. 4). We prepare samples at various temperatures by stopping the evaporative cooling procedure at arbitrary points [32].
The final trap geometry is kept constant. By reducing thetemperature of the Fermi gas, we observe the emergence of
the FSD which becomes more and more pronounced at low
temperature and eventually approaches the zero-temperature
limit. The observed temperature dependence qualitatively
agrees with a theoretical result at finite temperatures [33]. Our
observation clearly shows the quantum many-body nature of
the FSD.
The observation of a deformed Fermi surface sets the basis for future investigations on quantum many-body dipolar phenomena, including dipolar collective excitations [27,28, 34, 35], and exotic phases in strongly correlated dipolargases [18, 19]. Taking advantage of the wide tunability of cold
atom experiments, dipolar Fermi gases are ideally clean system for exploring an exotic and topological phases in a highly
controlled manner.
We are grateful to A. Pelster, M. Ueda, M. Baranov,R. Grimm, and T. Pfau for fruitful discussions. This work is
supported by the Austrian Ministry of Science and Research(BMWF) and the Austrian Science Fund (FWF) through a
START grant under Project No. Y479-N20 and by the Eu-

4ropean Research Council under Project No. 259435. K. A. is
supported within the Lise-Meitner program of the FWF.

Supplementary Materials
Experimental setup

We obtain a quantum gas of fermionic 167 Er atoms via laser
cooling in a narrow-line magneto-optical trap [36] followedby evaporative cooling in an ODT [23]. During the entire experimental procedure, the fermions are fully polarized into thelowest hyperfine sublevel |F = 19/2, mF = −19/2i, where F
is the total angular momentum quantum number and mF is its
projection along the quantization axis. The sample is trappedin a crossed ODT consisting of a horizontally (y axis) and a
vertically (z axis) propagating beam at 1570 nm. The beam
waist of the horizontal beam is kept constant to be 15 µm,
while that of the vertical beam can be tuned from 20 µm to
90 µm. By controlling the power of the two beams and thebeam waist of the vertical beam, we are able to vary thetrap geometry from a nearly spherical shape to a cigar shape.
The magnetic field amplitude is always kept constant to be
0.58 G where we do not see any influence of Feshbach resonances [37]. The magnetic field orientation is controlled with
two sets of coils. During evaporative cooling, the magnetic
field is vertically oriented (β = 0◦ ).

Imaging and fitting procedure

We measure the deformation of the cloud shape in the
TOF absorption image by applying the conventional polylogarithmic fit for an ideal Fermi gas [38–40]. The fit has
six free parameters: the positions and sizes in two directions, the fugacity, and the optical depth. The fugacity ζ
is directly connected to T /TF through the relation T /TF =[−6 × Li3 (−ζ )]−1/3 , with Lin being the n-th order polylogarithmic function. The optical depth is proportional to N.
Although the optical depth is related to ζ through N and TF
by TF = h f¯(6N)1/3 /kB , we leave both free in the fitting procedure and confirm that they are consistent with each other.
Here, we assume that the cloud has a constant fugacity overthe entire cloud because the momentum deformation is small.
Rigorously speaking, TF is anisotropic and T is constant overthe cloud, and thus ζ should be anisotropic. Dealing with sucha distribution is beyond the scope of the present work.

Errors in AR

By taking the average of more than 20 independent measurements, we are able to determine the AR with a typicalprecision of 0.1 %, corresponding to the standard error of
multiple measurements. In addition, there are five possible

sources of systematic errors in the measured AR. (a) Variation in the pixel sizes. The variation in pixel sizes over thearea of the cloud can introduce a systematic error in deformation. There is no measured data available for our CCD camera(Andor, iXon3). (b) Residual interference fringes. Interference fringes, arising from dusts on the imaging optics, canproduce a fixed background pattern on the image. From the
TOF measurements shown in Fig. 2, where the position of theatomic cloud varies with TOF by a free fall, we estimate the
combined effect of (a) and (b) to be within ±0.5 %. (c) Errorin the fitting procedure. Although our fitting procedure assuming a constant fugacity may give rise to a systematic error
in deriving the AR, it is difficult to quantitatively estimate it
owing to the lack of an appropriate model. Investigating thiseffect will be an important future work. (d) Fluctuations in
the magnetic field. The effect of the fluctuation in magnetic
field, which results in a fluctuation in the dipole orientation,is negligibly small at β = 0◦ and β = 90◦ (< 0.05 % in deformation). (e) Tilt of the camera. Assuming that the camera isaligned perpendicular to the imaging beam path within 1◦ , weinfer that the influence of the tilt of the camera on the AR isnegligible (< 0.02 %).
Calculation of the deformation

In the present work, the collision rate associated with universal dipolar scattering [23, 41] is lower than the lowest
trap frequency. Therefore, our sample is in the collisionlessregime, where the mean free path is longer than the size ofthe cloud [42]. We describe the trapped dipolar Fermi gas in
the collisionless regime in the zero temperature limit with anansatz that the Wigner distribution function is given as an ellipse
!
3 k2
3 r2
j
j
(2)
g(r, k,t) = Θ 1 − ∑ 2 − ∑ 2
j=1 K j
j=1 R jwhere Θ denotes the Heaviside’s step function, and r, k, andt denote coordinate, wave vector, and time, respectively. The
parameters R j and K j represent the Thomas-Fermi radius andthe Fermi momentum in the jth direction. These parameters
are numerically determined by minimizing the total energy inthe presence of the DDI. The validity of this approach wasnumerically confirmed [43]. At equilibrium, the parameters
K j include the information of the anisotropic FS.
The expansion dynamics is calculated using the BotzmannVlasov equation for the Wigner distribution function under thescaling ansatz [44–46]. The scaling parameters, representing
variations from the equilibrium condition, are described bya set of coupled time-dependent differential equations. The
NBE effect is naturally included in this framework. We numerically solve the equations for the general triaxial geometry, where the trap frequencies in three directions are different
and the dipoles are oriented in the direction of one of the trap
axes. This reflects our experimental situation at β = 0◦ and
β = 90◦ .

5
We calculate the sizes of the cloud on the image plane, taking into account the angle of 28◦ between the imaging axis
and the y axis (see inset Fig. 1). In all our measurements, weobserve an asymmetry between β = 0◦ and β = 90◦ , i. e. |∆| islarger at β = 0◦ than at β = 90◦ . We observe this asymmetry
also in the subtracted images in Fig. 3D and Fig. 3E as a higher
contrast in the vertical direction than in the horizontal direction. This asymmetry is well reproduced by our calculationand is understood as follows. At β = 0◦ , the major axis (x)of the ellipse is oriented to the z direction and is fully imaged.
By contrast, at β = 90◦ , the major axis is not perpendicular to
the imaging plane and we observe a combined size between
the major and minor (y) axis of the ellipse. Therefore, the observed deformation at β = 90◦ is always smaller than that at
β = 0◦ .

Image subtraction for Fig. 3D,E

The image shown in Fig. 3D is obtained in the following
procedure. The TOF absorption images from 18 independentmeasurements are averaged and binned by 2 × 2 pixels to reduce background noise. This procedure is applied for the measurements at β = 0◦ and β = 90◦ , yielding two images. We
subtract the image at β = 90◦ from the one at β = 0◦ . This
image subtraction is very sensitive to the relative position of
the clouds on the two images down to a sub-pixel level. We
obtain accurate positions of the center of the cloud from thefit and shift the coordinate of the image at β = 90◦ such thatthe center positions of two images exactly agree. We then apply spline interpolation for the image at β = 90◦ to estimate
the optical depth of the cloud at each pixel position in the image β = 0◦ . Unlike the procedure used in Ref. [31], where theanisotropy is extracted by rotating a single image by 90◦ andsubtracting it from the original image, our procedure with two
images at two dipole orientations allows us to extract only theanisotropy originating from the DDI.

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Any sufficiently advanced technology is indistinguishable from magic.

Chapter(Arthur C. Clarke)

12

Conclusion and outlook

This thesis describes the creation and study of bosonic and fermionic degenerate quantum
gases of erbium atoms. It documents setting up the experimental apparatus from a completely empty lab to a fully working experiment. We had to face the challenge of designing
an apparatus for an atomic species, which was unknown to quantum gas experiments and not
well characterized in its relevant atomic properties. In the planning phase, our greatest concerns included, whether the chosen narrow-line laser-cooling approach using erbium would
turn out to work as good as expected, where Feshbach resonances would be located, andwhether the lifetime of the atomic sample in the optical dipole trap would be long enough,i. e. not limited by light-assisted collisions or off-resonant absorption. The reported results
after the successful demonstration of the narrow-line magneto-optical trap of erbium werebeyond our expectations. Not only that this experiment allowed for the first production ofdegenerate gases of erbium, but also the number of further investigations is vast.
The slow transformation from an idea to a room-filling experiment took place in many small
steps, each of which brought us closer to exploring interesting physics. Many more steps
will be made using the erbium machine to continue studying ultracold dipolar gases. Someof these steps are under preparation and undertaken right now or will be done in the near
future. These further developments will be briefly listed in the following section. Other
steps lie in the more distant future and are difficult to predict as they strongly depend on
the direction in which the experiment will go. Nevertheless, some of the long term goals forthe erbium experiment are presented in Sec. 12.2.

12.1. Further investigations
Weakly-bound molecules Weakly-bound molecules can be created using the magneto-association technique across by ramping over a Feshbach resonance, see Ref. [Chi10]. This
standard technique can also be applied to erbium associating two atoms to a Er2 molecule.
Dependent on how the individual total angular moments of both atoms add up, the moleculecan have a magnetic moment of up to about 14 µB . The combination of such a large magnetic
moment and the large mass of Er2 makes this system regarding its dipolarity competitive tocurrent experiments using polar heteronuclear molecules, like KRb [Ni08] or RbCs [Tak14].
163

12.1. Further investigations

164

We have conducted preliminary measurements with 168 Er, in which we utilized a Feshbachresonance at 912(1) mG with a width of 39(2) mG to prepare up to 2 × 104 Er2 molecules at
a temperature of 300 nK and a molecule density of 8 × 1011 cm−3 . The simple preparation of
weakly-bound molecules is the first step towards experiments with deeply-bound molecules
possessing an exceptionally large magnetic dipole moment.

Molecular binding energy The binding energy of a weakly-bound molecule depends on themagnetic field. A sinusoidal modulation of the constant magnetic field in the vicinity of a
Feshbach resonance drives a resonant conversion of atoms into a weakly-bound molecule, only
if the modulation frequency equals the binding energy of the molecule, see Ref. [Tho05]. Thisso-called wiggling technique is often used to measure the binding energy at different valuesof the magnetic field. Very close to the Feshbach resonance the binding energy changes
quadratically with the magnetic field, whereas further away from the resonance a linear
dependence is given, see Sec. 4.3. From the linear dependency the magnetic moment of themolecular state with respect to the magnetic moment of two free atoms can be derived. In
the erbium experiment this technique was recently used to measure the magnetic momentof five different molecular states at Feshbach resonances below 2.5 G. The values of themagnetic moments range from µ = 7.96 µB to 11.8 µB , see Ref. [Aik14a]. In the near future
wiggling measurements at Feshbach resonances above 3 G are important to characterize thespectrum of molecular states in more detail. First theoretical efforts show that in contrast to
alkali atoms one single molecular state of lanthanide atoms turns out to posses a magnetic
moment, which is a linear combination of magnetic moments of different Zeeman sublevels.

Observation of the roton minimum It has been theoretically shown that a so-called roton
minimum should emerge in the excitation spectrum of an ultracold Bose gas with shortrange two-body interaction. This roton minimum can be observed as a local minimum ina non-trivial energy dispersion relation and is known from condensed matter systems, likesuperfluid helium, where the roton resembles an elementary excitation of rotation. Morerecently, the roton spectrum of a quasi-two-dimensional dipolar Bose-Einstein condensate
has been theoretically studied in Ref. [San03a]. Dipolar atoms like 52 Cr, 164 Dy, or 168 Er in
current experiments are expected to reveal the roton spectrum when the atom cloud is tightly
confined in the polarization direction of the dipoles, see Ref. [Bla12]. The roton minimum
can then be measured in the ultracold gas using Bragg spectroscopy. We plan to pursue
this route towards the roton minimum with erbium atoms in an array of two-dimensional
trapping potentials. We will carry out Bragg spectroscopy with both 1064 nm and 532 nm
light, which enables us to measure the excitation spectrum and observe signatures of theroton minimum.

Exotic optical potentials By carefully engineering optical potentials of larger complexity,
like honeycomb, triangular, or dimer potentials, it is possible to explore condensed-matterphenomena in a highly flexible way using ultracold atoms. For example, it has been experimentally demonstrated in Ref. [Tar12] that so-called Dirac points can be created using a
Fermi gas in a honeycomb lattice. As in this experiment the massless electrons of graphene12.2. Our vision165

are mimicked, it is often referred to as artificial graphene. What makes an important difference to experiments in condensed-matter physics is the high degree of tunability in ultracold
atomic systems. Adding long-range interactions between atoms in such a honeycomb potential would open the opportunity to study strong correlation effects, which are not accessible
in the field of condensed-matter physics. As a first approach we are working on creatinga three-dimensional optical lattice using standing waves of 532-nm light in two directions
and 1064-nm light in the third direction. This gives a twice as large lattice constant in one
direction than in the other two. It is speculated, that in a dipolar system the superfluid
to Mott-insulator transition will be modified dependent on the orientation direction of thedipoles with respect to the optical lattice. Theoretical calculations also predict a modification of the Mott lobes by the dipole-dipole interaction, see Ref. [CS10]. We are in close
collaboration with the group of P. Zoller from Innsbruck to investigate the parameter range,
which will be accessible with this lattice setup in the current erbium experiment.

12.2. Our vision
LiEr molecules From the beginning the erbium experiment has been designed such thatlithium can be added as a second species in the distant future. A lithium oven can be mounted
between the erbium transversal cooling stage and the atom beam shutter, which will form a
combined Er/Li atom beam. The Zeeman slower was designed to operate with both species
simultaneously and should allow an efficient loading of an erbium and a lithium magnetooptical trap during a single experimental cycle. After cooling both elements to ultracold
temperatures via forced evaporation one can think of many interesting experiments andmeasurements with such a heteronuclear mixture. For example, three-body bound states,
so-called Efimov states could be investigated, or a two-species superfluid mixture of fermions
could be created. In both cases the extreme mass imbalance between erbium and lithium andthe magnetic moment of erbium makes this combination favorable compared to mixtures ofalkali atoms. One could further think of associating a lithium and erbium atom to a weaklybound LiEr molecule via the magneto-association technique. Then, in principle, a STIRAP(Stimulated Raman adiabatic passage) process could transfer the molecule into its groundstate, where the two atoms will be strongly bound. This process creates a stable polar
molecule with both large magnetic and electric dipole moments. The electric dipole moment
is tunable via an external electric field and allows for the full control of the electric dipoledipole interaction strength. This might open the field of so-called ultracold chemistry, which
focuses on studying reaction dynamics and was mentioned in Ref. [Hut10]. With the erbium
experiment the reaction dynamics between molecules and atoms or molecules and moleculesin three or lower dimensions can then be explored. An early experimental study was doneby letting KRb molecules collide with K or Rb atoms, respectively, see Ref. [Osp10]. Veryrecently, it was theoretically found that quantum entanglement plays an important role foractivating and driving an ultracold chemical reaction towards its dynamical equilibrium, see
Ref. [Ric14].

Spin-orbit coupling Atoms with large magnetic moments are promising to study the physics
of spin-orbit (SO) coupling in ultracold gases, which was first demonstrated in a BEC of ru-

12.2. Our vision166

bidium atoms, see Ref. [Lin11]. SO coupling, i. e. an interaction between a particle’s spin
and its angular momentum, is important in condensed-matter systems for elucidating thespin-Hall effect in semi conductors [Kat04], topological insulators [Has10], and Majorana
fermions [Sau10]. Furthermore, SO coupling can be accomplished with non-dipolar gases
using artificial gauge fields, which have been realized within optical lattices, see Ref. [Aid11].
Dipolar atoms with a large magnetic dipole moment are favorable for these kind of experiments due to their intrinsic SO coupling induced by anisotropic interaction. This manifests
itself in expected large dipolar relaxation rates, see Ref. [Nas13].

Topological superfluids Utilizing strong spin-orbit coupling in a dipolar system one might
be able to prepare topological superfluids. They are considered to be exotic states of matterand are of major interest since the discovery of topological insulators and superconductors,
see Refs. [Has10, Qi11]. The fascinating thing about topological superfluids is the fact that
their fundamental properties can only change if the system passes through a quantum phasetransition, i. e. the properties are ‘topologically protected’. In Ref. [Nas13] the implementation of a topological superfluid using lanthanide atoms is proposed. Here, the experimental
setup consists of a one-dimensional Fermi gas, which is placed in the proximity of a twodimensional superfluid made of condensed Feshbach molecules. For creating a topologicalsuperfluid the Fermi gas has to be trapped in a spin-dependent optical dipole potential,
which is difficult to create for fermionic alkali atoms due to large heating rates induced byspontaneous emission. The reason for this heating is because of the small fine-splitting of
the excited P-state of alkali atoms in comparison to the large linewidth of the laser-coolingtransition. This problem should be overcome by using optical lattices with wavelengths
closely tuned to the narrow-line transitions in 161 Dy or 168 Er, which makes it possible tocreate strong spin-dependent potentials with small heating rates. According to the proposal,
lanthanide atoms should enable the study of topological superfluids. We do not know yet,whether these will be realistic experiments in our setup. However, topological superfluids
represent an interesting research direction, which deserve further investigation.

Appendix

A

Measurement of the naturallinewidth of the 401-nm
transition

The natural linewidth of the 401-nm transition was measured by absorption imaging of an
atomic sample in a magneto-optical trap (MOT) for three different isotopes of erbium. Figure
A.1 shows the results of this measurement. A comparison between values of the linewidth
found in literature and the result of the present work is found in Table A.1. The temperatureof the atomic sample was determined by time-of-flight absorption imaging to be 11.1 µK for
166 Er, 13.9 µK for 168 Er, and 14.6 µK for 170 Er. Typical atom numbers ranged between 1 and
2.5 × 107 . Due to the low temperature of the atomic sample the thermal Doppler broadening
is as small as 60 kHz. Collisional broadening of the natural linewidth is estimated to beabout 52 kHz for a peak number density of 8 × 109 cm−3 . Both broadening effects are oneorder of magnitude smaller than the systematic and statistic error and are thus neglected.

2.5

166

Er 29.2(2) MHz
Er 30.3(4) MHz
170
Er 29.7(2) MHz
168

atom number (107)

2.0
1.5
1.0
0.5
0.0

-40

-20

0
20
detuning (MHz)40

60

Figure A.1.: Linewidth measurement of the 401-nm transition for three different bosonic isotopes
Er, 168 Er, and 170 Er. A Lorentzian fit is applied to the data points and the natural linewidth
is given for each isotope with a statistical error. Averaging these three values and considering thesystematic error (see text) the value for the natural linewidth is ∆ν = 29.7(6) MHz, which is in very
good agreement with a lifetime measurement of the excited state in Ref. [Har10].

166

167

168
linewidth (MHz)measurement methodRef.

27.5(2.8)

lifetime of excited state[Mar80, Ral11]35.6(1.2)

laser-induced fluorescence[McC06a]29.5(1.5)

lifetime of excited state[Har10, Law10]29.7(6)

absorption measurement

this work

Table A.1.: Comparison of natural linewidth measurements of the erbium 401-nm transition using
various methods found in literature and the one presented in this work.

After preparing the atomic sample in the MOT absorption imaging was applied using 401-nm
laser light with variable detuning from −60 to +60 MHz with respect to the atomic transition frequency. Absorption imaging gives a two-dimensional image of the optical density of
the atomic cloud. The total atom number is then found by separately integrating the optical density along both image axes, fitting a Gaussian-shaped curve to the one-dimensional
density distribution, and integrating this Gaussian fit. The assumption of a Gaussian density distribution is valid for a purely thermal atom cloud, which is the case in the present
measurement. This imaging procedure is commonly used as a standard absorption imaging
technique.
Deriving the optical density from an absorption image needs knowledge of the detuning of
the imaging light and of the natural linewidth of the transition. If the linewidth is unknown,
it can be measured by setting the detuning to zero in the calculation of the optical density.
Measuring the total atom number dependent on the detuning reveals the natural linewidth,see Fig. A.1. Note, that this method can only be applied if at least a small amount of atoms
can be detected, otherwise determining the atom number from the absorption image will notwork properly. In spite of this restriction, the measurable range is sufficiently large to apply
a Lorentzian fit to the data points and determine the linewidth.
The imaging laser light was detuned using an accusto-optical modulator (AOM) in doublepass configuration. It is driven by a radio frequency produced by a voltage controlled oscillator (VCO). The nonlinearity of the VCO gives a systematic error of the detuning of about
0.6 MHz over the full range of detuning.
Considering this systematic error and measuring the linewidth for three bosonic isotopes,
see Fig. A.1, we determine the natural linewidth of the 401-nm transition of erbium to beΓ∆ν = 2π= 29.7(6) MHz. This agrees with some of the most recent measurements found inliterature, see Table A.1.

Appendix

B

Erbium high-temperature oven

The erbium high-temperature oven1 creates a roughly collimated atom beam by evaporating
solid pieces of erbium. It consists of two separate heating filaments. The first filament
heats the erbium source material contained in a tantalum crucible, in the so-called effusion
cell, and the second filament heats the aperture setup used for collimation in the hot lip
section. The temperature of both filaments can be independently controlled, but due to heatradiation and heat conduction by the crucible itself, the temperature difference can not belarger than about 250 ◦ C. Usually the temperature of the aperture setup is 100 ◦ C higher
than the temperature of the effusion cell to prevent a blockage of the apertures by excessive
source material. Typical operating temperatures are 1100 ◦ C for the effusion cell and 1200 ◦ C
for the aperture setup.
The heating filaments are made of tantalum wires. The wires are held in place by boron
nitride (PBN) ceramic insulator disks. The insulator disks have a thickness of about 2 mm
and are very fragile. Furthermore, they support the crucible setup in the center of theoven. Close to each filament a thermoelectric temperature sensor2 is placed for temperature
measurements and stabilization via a PID controller3 . The controller allows for a temperaturestability4 of about ±0.1 ◦ C. The filaments are connected to two independent power supplies
delivering up to a power of 700 W each with a maximum output current of 13 A. According
to the manufacturer the maximum achievable temperature of the crucible is 1600 ◦ C. For
erbium a temperature of not more than 1500 ◦ C is advisable to prevent melting the source
material. The current of the heating filaments should not exceed 3.5 A for the effusion celland 6.1 A for the hot lip.
To prevent excessive heating of the vacuum chamber, several temperature shields are implemented within the high-temperature oven. First, three layers of tantalum foil provide
a passive shielding of heat radiation. Second, an outer tube enclosing the oven provides
water cooling. A flow rate of at least 2 l/min must be guaranteed during oven operation. If
this is not the case, the power supplies of the oven will be switched off automatically by a
1

2
3
4

dual-filament effusion cell, model DFC-40-10-WK-2B with control unit CU-3504-S2-DC
from CreaTec Fischer & Co. GmbH
Effusion cells are commonly used for molecular beam epitaxy systems.
thermocouple type C, usable up to 2300 ◦ Cmodel 3504, from Invensys Eurotherm Ltd
at temperatures below 500 ◦ C the PID regulator tends to oscillate within ±20 ◦ C

169heating power (W)

170
200
180
160
140
120
100
80
60
40
20
0

May 2014

effusion-cell
aperture setup
April 2013

July 2012

0

100

200

300
400
500
operating time (days)

600

700

Figure B.1.: Heating power needed for operating the high-temperature oven over the timespan of
about two years. The temperature of the effusion cell and aperture setup is 1100 ◦ C and 1200 ◦ C,
respectively. Lines are shown as a guide to the eye.

water watch dog. Furthermore, the cooling water circuit must not be tightly switched-off
to prevent any high pressure from building up, in case the residual cooling water starts toboil and steam needs to escape. This is achieved by a one-way gate valve in the back-flowof the cooling water circuit. Due to the multi-stage temperature shielding the vacuum tube
containing the oven usually has a temperature of 40 ◦ C on the outer side. Strong heat radiation in front of the oven, which is not effectively shielded, heats those parts of the vacuum
chamber to about 120 ◦ C.
The oven must not be operated at pressures larger than 10−5 mbar, otherwise the heating
filaments can be damaged easily. Temperature ramps for heating and cooling from 5 to30 ◦ C/min are standard. If the temperature is ramped up too fast, thermal stress can breakthe ceramic insulators. When the experiment is out of operation, the oven is ramped down
to a standby temperature of typically 700 ◦ C for the effusion cell and 800 ◦ C for the hot lip.
This increases the lifetime of the source material significantly. Lower standby-temperatures
are not advisable, as the crucible might slightly deform over many ramping cycles due tothermal stress. In case of strong deformation, the crucible can’t be extracted from the oveneasily any more, which might cause some of the ceramic insulator disks to break5 .
The crucible has a total length of 150 mm and an inner diameter of 12 mm, see Fig. 3.2 forfurther dimensions. This corresponds to a volume capacity of 17 cm3 . After the aperture
inset is mounted inside the crucible, a volume of about 7 cm3 remains for the source material.
The crucible is usually filled with up to 10 g of erbium, which turns out to be enough for morethan 5200 hours of operation6 and continuing, at an effusion-cell temperature of 1100 ◦ C.

5
6

In the case of oven damage, the manufacturer offers a full refurbishment of the oven.
The source material in the crucible was not yet emptied at the time this thesis was written.

171

B.2. Erbium vapor pressure

vapor pressure (mbar)

104

Cs

Li
Yb

102
Er
Dy
Cr

100
10-2
10-4
10-6

0

500

1000

1500
T (°C)

2000

2500

3000

Figure B.2.: Vapor pressure for various elements. Chromium, dysprosium and erbium have a similarvapor pressure and require a high-temperature oven. Cesium is shown for comparison as an elementwith a low melting point of only 28 ◦ C, whereas lithium and ytterbium have an intermediate vapor
pressure. The region of a suitable vapor pressure for ultracold gas experiments is indicated by thegray shaded area.

B.2. Erbium vapor pressure
The high melting point of erbium at a temperature of 1529 ◦ C makes the construction of anoven for evaporating atoms technically difficult. In general, the saturated vapor pressure ofan element, pvap , is the figure of merit for choosing a suitable type of oven for the experimental
setup. By the precise knowledge of its temperature dependence one can estimate the fluxof an atom beam and thus the loading time of the magneto-optical trap in the further
experiment, see Chapter 6. The temperature dependence of pvap is accurately described by
the Antoine equationB
pvap (T ) = 10A− C+T ,B.1
with A, B, and C being empirical material constants without any units. A full set of vapor
pressure data for various elements and chemical compounds can be found in Ref. [Gra72]. A
nonlinear least-squares fit to the vapor pressure data of erbium using Eq. B.1 gives AEr =7.103(4), BEr = 12170(20), and CEr = 100(2) for T in units of ◦ C and p in units of mbar.
Figure B.2 shows the vapor pressure pvap for a broad selection of elements commonly usedin ultracold gas experiments.
Having an expression for pvap we can roughly estimate the atom flux in dependence on thetemperature of the oven. For this the longitudinal velocity distribution, fem , of atoms withmass m is calculated from the Maxwell-Boltzmann distribution byfem (v, T ) =



πDem
2

2

pvap (T )
kB T



m
2πkB T

3
2



mv 2
v exp −
2kB T
3sin2 (θem ) ,B.2

see also Refs. [Ram05, Chi07]. Here, atoms are assumed to escape the oven through a diskshaped aperture of diameter Dem under a solid angle θem and kB is the Boltzmann constant.

172

B.3. Oven refilling procedure(b)
71200°C
1150°C
1100°C
1050°C

6
5
4
3
2

1013
1012

atom flux (atoms/s)

lin. distribution (106 atoms/m)

(a)

1011
1010
109
108
107

1
0

200

400

600 800
velocity (m/s)

1000 1200

106
800

1000
1200
temperature (°C)

1400

Figure B.3.: (a) Velocity distribution for four different oven temperatures typically used in theexperiment. The position of the maximum shifts only a small amount from 443 m/s for a temperatureof 1050 ◦ C to 468 m/s for 1200 ◦ C. The maximum Zeeman slower capture velocity of 400 m/s is shown
as a dotted line. (b) Estimated atom flux into the main chamber for an oven aperture with Dem =3 mm, θem = 0.03 , and a Zeeman slower capture velocity of 400 m/s. A typical oven temperature of1100 ◦ C is highlighted by the dashed line.

q
The velocity distribution has its maximum located at vmax = 3kmB T ≈ 450 m/s for a typicaltemperature of 1100 ◦ C in the case of erbium, see Fig. B.3(a). By integrating fem up to
the Zeeman slower capture velocity, vc , the atom flux, Φ, which will be available after the
Zeeman slower, can be estimated as
Φ(T ) =

Zvc

fem (v, T )dv ∝0

pvap (T )
√
.
T

B.3

Figure B.3(b) shows the atom flux into the main chamber, which increases more than fiveorders of magnitude when the oven temperature is raised from 800 ◦ C to 1400 ◦ C.

B.3. Oven refilling procedure
A depletion of the source material in the erbium oven can be detected easily by a rapid
decrease of the atom number in the magneto-optical trap (MOT) within the timespan ofa few days. During this time, the MOT loading rate shows the following characteristic
signature. After reaching operational temperature of the oven, the atom number in the
MOT first seems to be as usual but after one to two hours of operation it drops significantly
to less than half of its original value. Furthermore, the fluorescence of erbium atoms in thetransversal cooling section gets weaker. In such a case the oven crucible has to be refilled.
The refilling procedure is described in the following and takes at least ten days until theexperiment can be continued.
Erbium source material can be purchased in various shapes from different suppliers7 . We
have made good experiences with erbium pieces, which are distilled dendritic, although the7usually 25 g of 99.9 % pure erbium, distilled dendritic, purchased from Alfa Aesar

173

B.3. Oven refilling procedure

(a)

(b)

Figure B.4.: Cleaning the high-temperature oven from old source material. (a) A thin erbium foil
is deposited on the aperture before the transversal cooling section. Peeling off foils under air canlead to a spontaneous ignition of the erbium foil. (b) Heating filaments are kept in place by ceramic
spacers (arrow). After a long time of operation the degraded heat shield of the oven has to be cleaned
to avoid direct contact with the crucible.

price per gram is twice as large as compared to normal pieces. Dendritic pieces have theadvantage that they can be easily cut into smaller pieces fitting better into the crucible.
Normal, solid pieces of erbium would need much rougher handling.
Refilling the high-temperature oven consists of two major tasks. First, the oven has to bedismounted from the vacuum chamber and old source material has to be removed. In caseof a damage of the oven, it has to be repaired or even replaced. Because of excessive erbium
material deposited in the aperture setup, the crucible can usually not be dismounted. A
new crucible setup has to be ordered and prepared beforehand. Then follows the second
task, which consists of filling the crucible with new source material, mounting the oven, andrestoring the vacuum again.

Removing old source material
This task includes flooding the oven chamber with argon gas, dismounting the oven from thevacuum chamber, removing the crucible, and cleaning the crucible and oven. The task will
be presented in the following steps.
1. Tightly close the gate valve between the UHV section and the oven section.
2. Switch off the oven power supply and close the cooling water cycle.
3. Prepare a needle valve with a pressure monitor mounted on a CF40 T-piece for flooding
the oven chamber with argon gas. A simple latex glove acting as a balloon suits
perfectly as a pressure monitor.
4. Mount the T-piece to the angled valve of the oven chamber.
5. Connect a scroll pump to the third flange of the T-piece.

B.3. Oven refilling procedure174

6. Clean the T-piece by repeatedly pumping off and slowly flooding with argon throughthe needle valve.
7. Disconnect the pumping station and close the T-piece with a blind flange.
8. Switch off the ion getter pump and pressure gauge mounted on the high vacuum partof the vacuum chamber.
9. Carefully open the angled valve, while keeping the T-piece filled with argon at a pressure slightly above ambient pressure. The glove should always be filled like a small
balloon.
10. As the angled valve is very slowly opened, adjust the needle valve to keep the T-pieceand the balloon slightly over-pressurized at all times.
11. When the oven chamber is fully flooded with argon, the high-temperature oven canbe unmounted between the CF40 port aligner and the oven vacuum tube. Remove
also the port aligner and the additional aperture disk from the transversal coolingvacuum chamber. Close the transversal cooling section with a CF40 blind flange again
to prevent argon gas from leaking out.
12. During operation of the oven thin foils of erbium are deposited inside the port aligner,see Fig. B.4. Take care as they can spontaneously ignite and burn when brought fromthe inert argon atmosphere into air. DO NOT put burning pieces of erbium into water!
Erbium produces hydrogen gas when put in water! Erbium foil and other small pieces
can be safely contained in paraffin oil, which reduces the risk of ignition.
13. Clean the aperture and the port aligner from deposited erbium material.
14. Carefully take out the crucible from the oven tube. Due to slight expansion of the
crucible it might get stuck easily. Try to carefully rotate the crucible setup before
pulling it out slowly.
15. Inspect the ceramic isolators inside the oven. They might easily break during handlingof the crucible. If the heating filaments can not be kept safely in place anymore, theoven has to be refurbished by the manufacturer.
16. Try to disassemble the crucible. Erbium deposited in the aperture setup makes thisstep difficult or even impossible. If it can not be disassembled a new crucible setup has
to be used for refilling.
17. Make sure the oven is clean. Remove any loose pieces of the heat shield as it might get
dissolved at high temperatures over longer times, see Fig. B.4(b).

Filling with new source material
Filling the oven is done in the following steps including preparing erbium pieces, assembling
the crucible setup, mounting the oven on the vacuum chamber, and pumping off the ovenchamber.
1. An accurate scale is needed to measure the amount of erbium filled into the crucible.
Properly clean all tools needed with isopropyl alcohol before continuing.
2. Prepare all necessary parts for the crucible setup. Clean the parts in an ultrasonic175

B.3. Oven refilling procedure

(a)

(b)

Figure B.5.: Refilling the crucible with source material. (a) Preparing small pieces of erbium.
Distilled dendritic pieces of erbium can be easily cut using cutting pliers. (b) The crucible setup isassembled. Tantalum wires (arrow) fix the aperture setup on the crucible tube.

bath, first using acetone, then isopropyl alcohol, about 15 min each is sufficient.
3. Dendritic erbium is usually shipped in 5 mm thick disks. Remove small pieces from thedisk using cutting pliers, see Fig. B.5(a). The pieces should not be larger than 5 mm
to easily fit into the crucible.
4. Fill about 10 g of erbium pieces into the crucible. Slightly compress the erbium pieces
to the back part of the crucible by tapping the crucible carefully on the table.
5. Assemble the crucible setup according to the oven’s manual, see also Fig. B.5(b).
6. Slide the crucible setup into the oven and ensure a snug fit.
7. First mount the additional aperture and the port aligner on the transversal coolingvacuum chamber.
8. Mount the oven onto the port aligner on the vacuum chamber.
9. Start pumping off the vacuum chamber.
10. After reaching less than 10−7 mbar switch on the ion getter pumps of the oven chamber.
11. Switch on the oven power supply and cooling water.
12. When the pressure is below 10−8 mbar the oven can be heated slowly up to 200 ◦ Cusing a ramp of 10 ◦ C/min.
13. Observe the vacuum pressure during the heat up. When the pressure again drops below
10−7 mbar, continue heating up the oven to standby temperature in steps of 100 ◦ C.
14. Heat the oven to operating temperature and try to observe a fluorescence signal in thetransversal cooling stage using the TC laser light.
15. Ramp the oven back to room temperature and switch it off.
16. Flash the titanium sublimation pump in the oven chamber for 180 s with 42 A.
17. If the total pressure reaches less than 10−8 mbar, close the angled valve, switch off thepumping station. Disconnect the pumping station.
18. Open the gate valve between the UHV section and the oven section.

B.3. Oven refilling procedure

176

19. Continue with the experiment.
Note, that no extra baking of the oven vacuum chamber is necessary to achieve a pressure
suitable for the experiment due to the getter properties of erbium, see Ref. [Mül72]. Aftera few heat-up cycles of the oven to operating temperatures the pressure should be low
enough for continuing with the experiment. Typical values for the total pressure are 6 ×
10−9 mbar when the oven is at room temperature and 2×10−8 mbar when it is at an operating
temperature of 1100 ◦ C one day after the refilling has been completed. The pressure at hightemperature of the oven should drop to a value of 1 × 10−9 mbar after some days of operationand the experiment can be continued.

Appendix

C

Notes on the blue master laser
systemThe blue laser system, which is presented in Sec. 3.5.1, ultimately relies on the stable operation of the blue master laser1 . Its working principle and the most important operating
parameters will be described in the following. For further details, please refer to the guidelines and datasheets provided by the manufacturer.
The master laser works as follows. It amplifies the light coming from a diode laser by
a tapered amplifier (TA) and then doubles its frequency by second harmonic generation(SHG). The diode laser comprises of an external-cavity diode laser (ECDL), which operates
at 802 nm and delivers about 44 mW output power. The laser-diode current is actively
stabilized by a PID control loop to 92 mA and the temperature is set to 22.3 ◦ C. About
10 mW of the laser power exits the system at this point and is used for monitoring on a
wavelength meter2 as reference. The remaining power is then amplified by the TA to about
635 mW. The TA-chip is usually operated with a current of 1780 mA at a temperature of
20.3 ◦ C. After passing some mode-matching lenses the light enters the SHG cavity, which islocked to the wavelength of the master laser using the Pound-Drever-Hall (PDH) technique.
The required sidebands for this locking technique are generated by modulating the laserdiode current with a radio-frequency signal at a frequency of about 20 MHz.
The laser light frequency is doubled in the SHG cavity by a non-linear crystal. The temperature of the crystal has to be precisely controlled to 39.4 ◦ C for optimal phase matching
between the fundamental light and the frequency-doubled light. The temperature is chosenhigh to prevent dust from sticking to the surfaces of the crystal. The maximum doubling
efficiency is about 33 %. In this setup a lithium triborate (LBO) crystal is used for frequencydoubling. An early version of this SHG cavity suffered from strong degradation of the crystal
due to large light intensities over a timespan of a few weeks. The output power dropped byabout 25 % within one week. Shifting the crystal by a few 10 µm recovered the doubling efficiency. This optimization had to be done every few weeks. Finally, the doubling cavity was
refurbished by the manufacturer and the crystal was replaced. The new cavity shows a lossof output power of less than 1 % over the timespan of a week. After the frequency-doubling
1
2

TA-SHG-pro system, from Toptica Photonics AG
model WS7-R MC, from HighFinesse GmbH

177

178
stage a total of 209 mW of light at 401 nm is available at optimum conditions in the newsetup. An output power of at least 80 mW is necessary for the experiment.

Appendix

DFeshbach resonances data tablesof bosonic erbium

The Feshbach resonance positions, Bres , and widths, ∆, are determined from high-resolution
Feshbach spectroscopy as described in Chapter 10. For convenience, the loss features arereported in ascending order with respect to their positions and labeled with index i. The
error in brackets gives one standard deviation. ∆ is determined either from a Gaussian fit
as the half-width at 1/e2 (method G) or by measuring the zero crossing of the resonance,
Bzero , being ∆ = |Bres − Bzero | (method Z). Feshbach resonance data for isotope 168 Er canbe found in Table D.1 and for isotope 166 Er in Table D.2.

179

Table D.1.: Feshbach resonance data for isotope 168 Er.
i

Bres
(G)

∆
(mG)

meth.

i

Bres
(G)

∆
(mG)

meth.

i

Bres
(G)

∆
(mG)

meth.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

0.912(1)2.167(1)2.436(3)2.476(1)2.838(9)3.778(1)4.481(1)6.558(1)6.594(1)6.674(5)7.048(5)7.725(1)8.549(7)8.909(1)9.311(1)9.946(1)
11.004(1)11.274(1)11.329(1)11.807(1)11.913(1)12.250(1)12.873(1)13.029(1)13.968(1)14.614(1)14.711(1)15.126(1)15.336(1)15.851(1)17.180(1)17.450(1)17.808(3)18.229(1)18.566(1)18.948(1)19.258(1)19.469(1)19.66(1)20.138(1)20.627(1)20.884(1)20.965(1)21.588(1)21.934(1)22.146(1)22.34(1)22.960(1)23.224(1)23.413(1)23.953(1)24.40(2)24.549(1)24.649(1)24.908(2)25.117(1)25.265(1)25.925(1)26.202(1)26.534(1)26.877(1)27.098(1)27.447(1)28.070(1)

39(2)
36(1)
14(9)
153(6)
16(2)
22(2)
23(2)
29(5)
22(9)
217(10)343(8)
15(2)
900(16)
18(2)
13(2)
18(2)
17(1)
17(1)
11(1)
22(1)
18(1)
18(1)
11(1)
15(1)
16(1)
20(1)
14(1)
14(1)
19(1)
11(1)
23(1)
21(2)
207(4)
18(1)
19(2)
20(1)
21(2)
145(1)
171(13)
16(1)
18(2)
20(2)
25(2)
47(2)
43(2)
16(1)
190(13)159(2)
21(1)
21(2)
67(3)
48(15)73(3)14(3)
100(5)
42(3)
17(2)
26(3)
12(1)
19(1)
99(4)
18(1)
20(1)
23(1)

Z
Z
G
G
G
G
G
G
G
Z
Z
G
Z
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
Z
G
G
G
G
Z
Z
G
G
G
Z
Z
Z
G
Z
Z
G
G
Z
Z
Z
G
Z
Z
G
G
G
G
Z
G
G
G

65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128

28.643(1)28.737(1)29.233(1)29.448(1)29.797(1)30.057(1)30.541(1)31.369(1)31.602(1)31.903(1)32.243(1)32.556(1)32.753(2)33.245(1)33.703(1)33.903(1)34.206(1)34.340(1)34.647(1)34.939(1)35.524(1)35.713(1)35.836(1)36.148(1)36.399(1)37.015(1)37.105(1)37.137(1)37.198(1)37.856(1)37.944(1)38.362(1)38.416(1)38.728(1)38.895(1)39.075(1)39.458(1)39.739(8)39.919(1)40.252(6)40.772(2)40.901(1)42.429(1)43.019(1)43.078(1)43.969(1)44.404(1)44.731(1)44.823(1)45.165(1)45.571(1)45.772(1)45.851(1)46.343(1)46.815(1)46.900(1)47.284(3)47.485(7)48.035(1)48.082(1)48.207(1)48.473(1)48.842(2)49.251(1)

21(3)
19(1)
17(1)
32(2)
25(2)
26(2)
15(1)
19(2)
18(1)
16(2)
25(2)
25(2)
283(5)
17(1)
21(1)
61(2)
13(1)
21(1)
16(1)
53(1)
17(2)
45(1)
15(1)
132(4)
13(3)
83(1)
4(1)
17(2)
54(2)
178(24)152(1)
17(2)
18(2)
19(2)
23(2)
16(1)
16(3)
1722
16(3)
202(7)
106(4)
17(2)
20(1)
20(2)
13(1)
23(1)
44(4)
109(7)
18(2)27(4)
156(8)
19(3)
88(4)
89(3)
20(2)
12(3)
58(10)
1193
141(7)
7(2)
21(2)
98(3)
138(5)
14(2)

G
G
G
Z
Z
G
Z
G
G
G
Z
G
Z
G
G
Z
G
G
G
Z
Z
Z
G
Z
G
Z
G
G
Z
G
Z
G
G
G
G
G
G
Z
G
Z
Z
G
G
G
G
Z
G
Z
G
G
Z
G
Z
Z
G
G
G
Z
Z
G
G
Z
Z
G

129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190

49.735(1)50.019(1)50.113(1)50.325(1)50.788(1)51.624(1)51.651(1)52.110(1)52.552(2)52.835(1)53.082(1)53.195(1)53.245(1)53.299(1)53.738(1)53.925(1)54.272(2)54.404(1)54.763(4)55.035(1)55.239(1)55.692(1)56.314(1)56.436(1)57.06(2)57.61258.167(2)58.426(1)58.528(1)58.795(2)59.078(1)59.609(1)59.875(3)60.585(1)60.646(1)61.241(1)61.759(1)62.271(1)62.510(1)63.174(1)63.367(1)63.398(1)63.727(1)63.985(3)64.374(4)64.522(1)64.889(1)64.985(1)65.302(1)65.671(1)65.989(1)66.401(1)66.475(1)66.790(1)67.103(1)67.354(1)67.767(1)68.240(1)68.810(1)68.895(1)69.046(1)69.727(1)

31(1)
15(3)
104(4)
18(2)
15(2)
15(1)
12(2)
18(1)
66(3)
14(1)
13(2)
33(5)
14(3)
33(4)
31(3)
19(1)
20(5)
12(1)
57(4)
18(2)
18(3)
300
109(3)
15(3)
3531
89(1)
13(2)
14(2)
7(1)
27(4)
55(2)
16(2)
85(10)
51(3)
8(2)
22(1)
13(1)
15(1)
52(3)
9(2)
6(2)
18(3)
17(3)
348(7)
76(5)
16(1)
17(1)
20(1)
37(1)
33(3)
46(2)
58(2)
51(1)
118(3)
497
18(2)
12(1)
113(3)26(1)16(4)
347(3)
19(2)

Z
G
Z
G
G
G
G
G
Z
G
G
G
G
Z
G
G
Z
G
Z
G
G
G
Z
G
Z
Z
Z
G
Z
Z
Z
G
Z
G
G
G
G
G
Z
Z
G
Z
G
Z
Z
G
G
G
Z
Z
Z
Z
Z
Z
Z
G
G
Z
Z
G
Z
G

180

Table D.2.: Feshbach resonance data for isotope 166 Er. Four loss features, at positions 121 mG,
244 mG, 366 mG, and 490 mG, which have been reported in the extended data of Ref. [Fri14], havebeen caused by a technical problem and are not due to Feshbach resonances.
i

Bres
(G)

∆
(mG)

meth.

i

Bres
(G)

∆
(mG)

meth.

i

Bres
(G)

∆
(mG)

meth.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62

3.056(2)4.012(5)4.955(1)6.546(1)6.768(1)6.884(1)8.053(1)8.553(1)9.910(1)10.228(3)11.108(1)11.623(1)12.978(1)13.278(1)13.426(2)13.686(1)14.177(1)14.801(1)14.835(1)15.077(1)15.371(1)15.726(1)16.107(1)16.187(1)17.053(1)17.977(1)18.961(1)19.152(1)19.23919.543(1)19.82(1)20.539(4)21.047(2)21.803(1)21.922(1)22.068(1)22.350(1)23.055(1)23.345(1)23.585(1)23.725(1)23.898(1)24.272(1)24.924(1)25.272(1)26.036(1)26.206(1)26.621(1)26.915(1)27.324(1)27.681(1)28.488(1)28.726(1)29.024(1)29.357(1)29.515(1)30.023(1)30.319(1)30.508(1)30.703(1)30.810(2)31.220(2)124(4)
23(1)
9(2)
38(2)
23(2)
26(2)
29(2)
20(2)
47(5)
116(6)
52(6)
27(1)
14(2)
34(1)
92(7)
16(2)
19(2)
17(2)
13(1)
18(2)
70(2)
68(2)
19(2)
38(2)
25(2)
17(3)
20(3)
22(1)
20(1)
21(3)
158(12)417(18)
63(8)
21(2)
22(2)
50(5)
27(2)
17(2)
19(2)
72(3)
15(4)
15(2)
12(1)
14(2)
19(1)
17(1)
13(2)
10(2)
19(1)
30(1)
23(1)
31(1)
25(1)
14(1)
25(1)
16(1)
44(2)
21(2)
31(1)
16(4)
154(4)
95(10)

Z
G
G
Z
G
G
G
G
Z
Z
Z
Z
G
Z
Z
G
G
G
G
G
Z
Z
G
Z
G
G
G
G
G
Z
Z
G
Z
G
G
Z
G
G
Z
G
G
G
G
G
G
G
G
G
G
Z
G
Z
Z
G
Z
G
Z
G
Z
G
Z
G

63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124

31.683(1)32.106(1)32.145(1)32.656(1)32.909(1)33.166(1)34.309(1)34.408(1)34.734(1)34.824(1)35.345(1)35.463(1)35.784(1)36.006(1)36.876(1)37.077(1)37.253(1)37.662(1)38.048(1)38.145(1)38.573(1)38.788(1)38.888(1)39.354(1)39.563(2)39.652(1)40.099(1)40.459(1)41.649(1)41.756(1)41.926(1)42.003(1)42.040(1)42.229(7)42.817(1)42.906(1)43.146(1)43.273(1)43.716(1)43.844(1)43.922(1)44.436(1)44.589(1)44.737(1)44.961(1)45.111(1)45.168(1)45.405(1)45.850(1)45.989(1)46.359(2)46.664(4)46.803(1)47.282(1)47.478(8)47.851(1)47.962(6)48.873(1)49.384(1)49.676(2)49.793(1)50.072(1)

33(2)
16(2)
13(2)
17(2)
14(2)
13(2)
15(1)
77(4)
19(1)
13(1)
15(1)
18(1)
14(1)
18(1)
14(2)
49(5)
17(2)
45(3)
23(1)
190(5)
39(3)
12(2)
84(5)
11(2)
52(2)
110(5)
13(1)
87(4)
27(1)
36(2)
17(2)
18(3)
14(1)
473(29)
53(4)
19
14(1)
14(1)
26(1)
47(1)
23(1)
16(1)
19(6)
12(1)
15(1)
20(2)
40(2)
16(1)
29(1)
22(1)
63(7)
311(5)10(4)16(2)
366(36)27
238(23)17(1)90(3)
105(4)
15(3)
90(4)

G
G
G
G
G
G
Z
Z
G
G
G
G
G
G
G
Z
Z
Z
G
Z
Z
G
Z
Z
Z
Z
G
Z
Z
Z
G
G
G
G
G
G
G
G
Z
Z
G
G
G
G
G
G
Z
G
Z
G
G
Z
G
G
G
G
G
G
Z
Z
G
G

125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185

50.119(1)50.249(1)50.32(1)51.321(1)51.684(1)52.04(1)52.395(1)52.435(1)52.795(1)53.114(2)53.267(1)53.863(4)54.207(2)54.488(1)54.932(1)55.263(1)55.431(1)55.974(1)56.062(1)56.456(1)56.668(1)56.740(3)56.768(1)57.556(1)57.753(1)57.978(1)58.135(1)58.239(1)58.987(1)59.179(1)59.718(2)59.890(1)60.111(2)60.637(1)61.104(1)61.522(1)61.667(1)61.829(1)62.348(3)62.600(2)62.804(1)63.126(1)63.500(1)63.524(1)64.078(1)64.535(1)65.089(1)65.334(1)66.287(1)66.375(1)66.898(1)67.186(1)67.333(1)67.519(2)67.776(1)68.243(1)68.426(1)68.642(1)68.875(1)69.200(1)69.728(2)56(10)
24(4)483(30)65(3)48(3)
161(11)27(4)76(2)
235(15)55(5)19(1)
88(10)
101(5)
19(2)
17(1)
17(2)
35(2)
131(13)144(33)149(4)
121(3)
4(4)
29(3)
15(1)
21(1)
33(2)
19(2)
31(2)
13(1)
18(1)
77(6)
21(1)
97(7)
45(2)
24(2)
8(1)
33(2)
53(5)
156(12)
97(33)
16(2)
16(1)
5(1)
21(1)
18(1)
17(2)
5(2)
16(1)
14(2)
18(1)
12(1)
15(1)
20(2)
72(6)
6(3)
38(4)
17(1)
13(1)
16(1)
35(2)
97(5)

G
G
G
G
Z
Z
G
Z
G
Z
G
G
Z
G
G
G
Z
G
G
Z
Z
Z
G
G
G
G
G
G
G
G
G
G
G
G
G
G
Z
Z
G
G
Z
G
Z
G
G
G
G
G
G
G
G
G
Z
G
Z
G
G
G
G
G
G

181

Appendix

E

Full level diagram of erbium

Figure E.1 shows all known energy levels of neutral erbium (ErI) according to Ref. [Ral11],
which uses [Mar78] as a primary data source. In total, 312 states with odd and 358 stateswith even parity are known up-to-date. Transition energies range from 5035.1 cm−1 to
46,970.8 cm−1 from the ground state. The ionization energy of erbium is, according toRefs. [Wor78, Mar78],
E = 49,262(5) cm−1 = 6.1078(6) eV ≈ 203 nm .

182

E.1

40 000

energy (cm-1)

30 000

20 000

10 000

0
1

2

3

4

5

6

7

8

9

10

11

12

J

Figure E.1.: Full energy spectrum of erbium. Levels have a total angular momentum quantum
number J ranging from one to twelve. States with even (odd) parity are shown in red (black). The
data is taken from Ref. [Ral11].

183

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Unix was not designed to stop its users fromdoing stupid things, as that would also stop themfrom doing clever things.
(Doug Gwyn)

Acknowledgments
The design, construction, and operation of the erbium experiment has been a vast undertaking. From the beginning it required hard work but we were soon rewarded with beautiful
results. The experiment held several surprises, luckily most of them were positive in nature,
which made the last five years an enjoyable adventure. Setting up such a huge experimentmakes one dependent on the help of a tremendous amount of people. Only if all of themwork together flawlessly a project of this scale can succeed. And succeed it did. Therefor,
I would like to utilize the last pages in concluding my thesis for thanking everybody who
contributed in one way or the other to my work on the erbium experiment.
First and most of all, I would like to thank my supervisor Francesca Ferlaino for givingme the great opportunity and trust to set up the erbium machine. Her understanding andintuition for some of the most interesting topics in ultracold atomic physics and her ability
to put together a perfectly working team is unsurpassed. Specifically, I would like to thankFrancesca for her endless drive in exploring new physics, for making time for discussions,as well as for carefully reading and correcting my thesis. Many little things, like organizing
pizza when we achieved the first BEC in the middle of the night and our somewhat regular‘B&B-meetings’, which included a full-sized breakfast, made the daily work in the lab very
pleasant and became fond memories for me. Because of this, I wish her all the best for her
scientific future as well as for her charming family with Pello and their two kids Michele andCora.
I’d also like to thank Rudi Grimm who founded the ultracold-atoms research group in Innsbruck many years ago. He generously offered his laboratory space after my former master thesis advisor Johannes Hecker-Denschlag went to Ulm and left behind an empty lab.
Rudi’s exceptional experience and deep understanding of nearly everything relating to atomic
physics has been a key element in the development of the erbium experiment. Having erbium as the third element to be Bose-Einstein-condensed in Innsbruck for the first time, after
cesium and strontium, made him especially proud as a ‘grandfather’ to the experiment.
The rest of the erbium team deserves huge thanks as they are the ones who made the timespent in the lab worthwhile. This includes Kiyotaka Aikawa and Michael Mark, two postdocs, Simon Baier, another PhD student, Johannes Schindler, Alexander Rietzler, Daniel
Petter, and Michael Springer, all of whom master’s students, and Eric Zupanič, a long-term
visitor. Having two experienced post-docs in the team was tremendously helpful. Working
with Kiyo in the lab was a once in a lifetime experience. I could learn so many things from
him and enjoyed his subtle sense of humor a lot. Good luck with you future scientific career!
Especially to Simon I give my best wishes, since he is going to continue with the experimentand will lead it to new frontiers. May working with the erbium machine be even more fun

200

for him than it was for me. Recently, Tadas Pyragius started to work on the new erbium
experiment and I wish him all the best with that.
Some people in our research group played a very special and important role during the design and setup of the erbium experiment. These include Florian Schreck, with his invaluable
efforts in setting up a versatile experimental control system, Simon Stellmer, who discussed
the narrow-line cooling approach and the experimental setup with us, Manfred Mark together with Gerhard Hendl, with their comprehensive understanding of electronics and thedevelopment of most of our homebuilt electronic devices, and Anton Schönherr and Matthias
Szabados, who helped with designing and manufacturing the main chamber and many othermechanical parts. Each one of you has been of great help and was a source of inspiration forme!
The following list contains all coworkers of the many other experiments which were conducted simultaneously during my PhD studies. I would like to thank each one of them fortheir individual contributions, which added up to a unique working environment. This includes the people from the LevT experiment: Alessandro Zenesini, Martin Berninger, andBo Huang; from the RbCs: Tetsu Takekoshi, with whom I shared my office for many years,
Lukas Reichsöllner, Andreas Schindewolf, Silva Mezinska, Markus Debatin, Michael Kugler,
Bastian Schuster, and Almar Lercher; from the CsIII: Manfred Mark, Katharina Lauber,
Florian Meinert, Johann Danzl, Elmar Haller, and Russell Hart; from the KCs: Emil Kirilov, Philipp Weinmann, and Michael Gröbner; from the DyK: Slava Tzanova and Cornelis
Ravensbergen; from the FeLiKx: Marco Cetina, Rianne Lous, Michael Jag, Matteo Zaccanti, Devang Naik, Christoph Kohstall, Andreas Trenkwalder, and Frederik Spiegelhalder;from the Li: Meng Khoon Tey, Leonid Sidorenkov, and Edmundo Sanchez; and from the Sr:
Benjamin Pasquiou, Simon Stellmer, and Alex Bayerle.
The mechanical workshop consisting of Armin Sailer, Anton Schönherr, Helmut Jordan, andChristoph Wegscheider as well as the electronic workshop with Gerhard Hendl, Wolfgang
Kratz, Michael Hofer, and Wolfgang Kuen were most important for setting up an experiment.
I really enjoyed the open atmosphere when working with these people and their offer to use
most of their available equipment. The secretaries Nicole Jorda, Carla Fantini, ChristineGötsch-Obmascher, Karin Köhle, Michaela Palz, Patricia Moser, and Renate Rupprechterare real masters of administrative work. Thanks to all of you for doing a great job!
I would also like to thank some master’s students from the ultracold group, including David
Baier, Walter Harm, Andreas Klinger, and Stefan Besler. I enjoyed the company of every
single one of you. At the beginning of my PhD work I had some overlap with the guys fromthe BaRbI experiment, who later moved to the university of Ulm, namely Stefan Schmid andArne Härter. I knew them since writing my master’s thesis, which dealt with that experiment.
Working with them in the lab was always great fun. I am grateful to Christoph Kohstall,
Michael Niedermayer, and Birgit Brandstätter for always being around for interesting side
projects involving more applied physics.
Some of my best friends Richard Schwarz, Markus Piff, Robert Spindler, Richard Spindler,
and Christoph Sengthaler as well as some people from the ‘Innsbrucker hackerspace’ were
there for me whenever I needed a different point of view on scientific problems or justsome timeout from physics. I’ve always very much appreciated their willingness for lengthy
discussions about any topic whatsoever.
My biggest thanks is to my family, especially to my parents Pia and Günter, for their strong
initial support and for offering me the possibility to study physics in the first place. My uncle
Walter introduced me to the fascinating field of electronics when I was just five years old,which I still remember vividly. I am very lucky to have my two sisters Birgit and Veronika
standing by my side. They will always have a special place in my heart. But my heart truly
belongs to my beloved Sabine: I have the utmost gratitude for your love and support inevery condition of life.