Ultracold Bose gases under rotation, in latticepotentials, and bothbyVolker Schweikhard
Diplom Physik, University of Stuttgart, 2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2008

iii
Schweikhard, Volker (Ph.D., Physics)
Ultracold Bose gases under rotation, in lattice potentials, and both
Thesis directed by Prof. Eric A. Cornell

In the field of ultracold atoms, two major recent research directions have been
rapidly rotating Bose-Einstein condensates, and condensates loaded into optical latticepotentials. While past optical lattice work has concentrated largely on the regimeof few correlated particles per lattice site, in this thesis the opposite limit of tunnelcoupled arrays of macroscopic condensates – Josephson junction arrays – is studied.
Combining such Josephson junction arrays with rapidly rotating BECs yields highly
complex quantum fluid systems with many analogies to condensed-matter systems. In
particular, optical lattice potentials allow the creation of low-dimensional, or eﬀectively
low-dimensional systems.
• In this thesis, the first rotating two-dimensional optical lattice potential for
ultracold atoms was created, which served as a periodic pinning potential for
vortices, and induced a structural crossover in the vortex lattice.
• The Berezinskii-Kosterlitz-Thouless superfluid transition was studied in a nonrotating, finite-temperature, two-dimensional array of Josephson junctions.
• Rapidly rotating BECs were loaded into a one-dimensional optical lattice alignedwith the rotation axis. A crossover from a coherent array of Josephson-coupledrotating BECs, to an array of isolated, two-dimensional rotating BECs wasobserved, and the ensuing vortex lattice fluctuations were studied.
• Studies of vortex lattices in rotating BECs were extended to two-component
BECs, where a self-organization of vortices in the two components into twointerlaced square vortex lattices was observed.

DedicationFür Isabell , für meine Eltern und für meine Brüder.

v

Acknowledgements

It has been an extraordinary time for me since this thesis begun, and there aremany people I wish to acknowledge: First and foremost I would like to thank my advisor,
Eric Cornell. I have enjoyed a very pleasant working relation with him, and benefitted
enormously from his incredible intuition and enthusiasm for physics. I also thank him
for his thoughtful advice on all other matters.
I had the pleasure of working with several extremely talented and and enthusiastic colleagues during the years – they really deserve most thanks: Peter Engels, Ian
Coddington and Shihkuang Tung. Peter and Ian introduced me to the tricks and tradesof B224, and taught me an immense amount of physics during my first years on theexperiment. I also thank Peter for a thorough reading of this thesis and many helpfulcomments. Shihkuang worked tirelessly with me on almost all experiments describedin this thesis – I wish him success with all his future plans. Giacomo Lamporesi very
recently restored the European presence in the lab; I thank him for his very helpfulcomments on this thesis. I thank Debbie Jin and Carl Wieman for supervising our
experiment and the whole group in that crazy winter 2004/2005, and for countlesssuggestions on anything from the experiment to presentation skills. I benefitted enormously from discussions and equipment loans from all members of the Tri-Group – they
all helped make this thesis a very pleasant experience. A big thanks goes to Pam Leland
for keeping everyone organized, safe, happy, and environmentally sound. Thanks also to
Marco Cozzini, Veronique Mogendorﬀ and Melissa Friedman who made contributions

vi
as short-term visitors in the lab.
This work could not have been possible without the hard work of all earlier students and postdocs in B224, who did a terrific job in delivering an (almost) flawlessly
working BEC machine on which all the successes of this thesis were built. Just as important were the contributions from the excellent staﬀ of the JILA Electronics Workshop,
the Instrument Shop, the Supply Oﬃce, and all the other people who make JILA work.
We were extremely lucky to capture the interest of a large number of theorists
at various points. All of them contributed in one or the other way to an improved
understanding of our experiments: Very welcome local support on condensed-matter
related topics came from Leo Radzihovsky, Victor Gurarie and Dan Sheehy. Murray
Holland, Rajiv Bhat and Meret Krämer taught us a lot about rotating BECs, opticallattices, and other things. We had a very insightful collaboration on shock waves withMark Ablowitz and Mark Hoefer. An extraordinary amount of external input camefrom Jason Ho, Gordon Baym, Chris Pethick, Gentaro Watanabe, Alexander Fetter,
Sandro Stringari, Iacopo Carusotto, Rob Ballagh, Eugene Demler, Erich Mueller, James
Anglin, Nigel Cooper, Jasper Reijnders, Masahito Ueda, Kenichi Kasamatsu, Kazushige
Machida, Henk Stoof, Nick Bigelow, and their coworkers.
Lastly, I thank Tom Parker, Steve Jeﬀerts and Tom Heavner at NIST for introducing me to atomic fountain clocks, which made for a very good “first year experience”
in Boulder.

vii

Contents

Chapter
1 Introduction

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.1

Quantum statistics: Ideal gas Bose-Einstein condensation . . . .

5

1.2.2

Quantum fluids: Importance of interparticle-interactions . . . . .

8

1.2.3

Thomas-Fermi regime . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.4

Condensates at finite temperature . . . . . . . . . . . . . . . . .

10

1.3

1.4

1.5

1.6

Rotating Bose-Einstein condensates. . . . . . . . . . . . . . . . . . . .

11

1.3.1

Quantized vortices in a superfluid. . . . . . . . . . . . . . . . .

11

1.3.2

Vortices – from one to many . . . . . . . . . . . . . . . . . . . .

13

1.3.3

Manipulating vortices . . . . . . . . . . . . . . . . . . . . . . . .

16

The two-dimensional (2D) Bose gas . . . . . . . . . . . . . . . . . . . . .

19

1.4.1

The Berezinskii-Kosterlitz-Thouless transition . . . . . . . . . . .

20

Josephson Junction arrays of BECs in optical lattice potentials . . . . .

23

1.5.1

Optical dipole potentials - versatile tools for BEC manipulation .

23

1.5.2

Two Josephson-coupled BECs . . . . . . . . . . . . . . . . . . . .

25

1.5.3

Arrays of Josephson-coupled BECs . . . . . . . . . . . . . . . . .

29

1.5.4

Thomas-Fermi description of BECs in optical lattices. . . . . .

33

BEC imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

viii
1.7

History and organization of this thesis . . . . . . . . . . . . . . . . . . .

2 Vortex pinning in rotating Bose-Einstein condensates

37
39

2.1

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.2

Setup, alignment and diagnostics . . . . . . . . . . . . . . . . . . . . . .

41

2.3

Observation of vortex pinning in Bose-Einstein condensates . . . . . . .

52

2.3.1

Angular locking of the vortex lattice to a hexagonal optical lattice 52

2.3.2

Pinning-induced structural vortex lattice crossover . . . . . . . .

56

2.3.3

Defects in pinned vortex lattices . . . . . . . . . . . . . . . . . .

60

2.3.4

Pinning with two vortices per pinning site . . . . . . . . . . . . .

62

2.3.5

Pinning with one vortex per two pinning sites . . . . . . . . . . .

63

3 A 2D Josephson junction array in the Berezinskii-Kosterlitz-Thouless regime

64

3.1

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.2

The BKT transition in a Josephson junction array . . . . . . . . . . . .

65

3.3

Setup, calibrations and parameter determination . . . . . . . . . . . . .

67

3.3.1

Optical lattice intensity calibration . . . . . . . . . . . . . . . . .

69

3.3.2

Getting J/T right . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.3.3

The importance of the third (z) dimension . . . . . . . . . . . . .

74

Experiments and results . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.4.1

Thermally activated vortex proliferation . . . . . . . . . . . . . .

75

3.4.2

Studying vortex pair sizes and vortex pair unbinding . . . . . . .

81

3.4

4 Rapidly rotating 2D Bose gases in a 1D optical lattice
4.1

4.2

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87
87

4.1.1

The physics of a finite-temperature array of rotating 2D Bose gases 89

4.1.2

Setup and alignment . . . . . . . . . . . . . . . . . . . . . . . . .

95

Initial optical lattice characterization with a static BEC . . . . . . . . .

97

ix

4.3

4.2.1

Optical lattice depth determination from matter wave diﬀraction

97

4.2.2

Bloch oscillations as a probe of phase coherence in the lattice . .

99

Results: Rapidly rotating Bose-Einstein condensates in a 1D optical lattice102
4.3.1

Spin-down of the rotating BEC in the optical lattice . . . . . . . 103

4.3.2

Loss of vortex lattice visibility in the optical lattice . . . . . . . . 106

4.3.3

Technical considerations when loading a rotating BEC into a 1D
optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4

Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Vortex lattice dynamics in rotating spinor Bose-Einstein condensates

118

5.1

Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2

The magnetically trapped spinor BEC system in 87 Rb . . . . . . . . . . 120

5.3

Experiments and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Bibliography

130

Appendix
A Accurate optical dipole potential calculation

141

B Numerical determination of finite-T BEC parameters in a double well potential 143
C Calculation of the vortex pinning potential

149

D Rapidly rotating Bose gases: Vortex lattice physics and beyond

150

D.1 Lowest Landau level: Vortex core overlap and transition to normal state? 153
D.2 Vortex lattice melting in a quasi-2D BEC . . . . . . . . . . . . . . . . . 155
D.2.1 Quantum melting . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
D.2.2 Thermal melting . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

x
D.3 Fractional quantum Hall physics in a rotating Bose gas? . . . . . . . . . 159
E Image magnification calibration

162

E.1 Top view calibration from in-trap vortex lattice images . . . . . . . . . . 162
E.2 Side view calibration by releasing a BEC from a 1D optical lattice . . . 164
F The End

166

xi

TablesTable
1.1

Condition for BEC in 2D and 3D, homogeneous / trapped, interacting /noninteracting systems. . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1

Error budget for optical aberrations in a rotating 2D optical lattice. . .

45

3.1

Error budget on the Josephson coupling J. . . . . . . . . . . . . . . . . .

73

xii

FiguresFigure1.1

Vortices – from one to many . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2

Schematic phase diagram of a rotating 2D Bose gas. . . . . . . . . . . .

15

1.3

Principle of vortex pinning . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.4

Eﬀect of vortices and vortex-antivortex pairs on phase coherence . . . .

21

1.5

Berezinskii-Kosterlitz-Thouless transition in a continuous 2D system . .

22

1.6

Sketch of retro-reflected optical lattice (OL), acute-angle OL, and 2D OL

24

1.7

Parameters of a double well BEC . . . . . . . . . . . . . . . . . . . . . .

25

1.8

Dependence on optical lattice depth of parameters relevant to the BerezinskiiKosterlitz-Thouless transition . . . . . . . . . . . . . . . . . . . . . . . .

28

Vortices in a Josephson junction array. . . . . . . . . . . . . . . . . . . .

29

1.10 Schematic phase diagram of a Josephson junction array. . . . . . . . . .

32

1.9

1.11 Chemical potential in the eﬀective Thomas-Fermi description for a BEC
in a 2D optical lattice: Theory and experiment . . . . . . . . . . . . . .

35

1.12 Example of nondestructive BEC image and finite-T BEC fit results. . .

36

2.1

Pinning a vortex lattice . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.2

Schematic setup for a rotating 2D optical lattice . . . . . . . . . . . . .

42

2.3

Optical path length fluctuations lead to lattice displacements. . . . . . .

43

2.4

Analysis of aberrations in a rotating 2D optical lattice setup . . . . . . .

44

xiii
2.5

Light source and detailed setup for the rotating 2D optical lattice . . . .

47

2.6

Stability measurements of the rotating 2D optical lattice . . . . . . . . .

48

2.7

Long exposure images of rotating 2D optical lattices . . . . . . . . . . .

49

2.8

Experimental result: Spinning up a BEC using a rotating two-dimensional

2.9

optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Experimental result: Vortex lattice pinning in triangular optical lattice .

54

2.10 Experimental result: Pinning-induced structural vortex lattice crossoverfrom hexagonal to square vortex lattice . . . . . . . . . . . . . . . . . .

56

2.11 Theoretical predictions [128, 130] for pinning-induced structural crossover 57
2.12 Definition of k-vectors for analyzing pinning-induced structural crossover

58

2.13 Experimental result: Structure factors show the pinning-induced structural crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

2.14 Experimental result: Phase diagram for pinning in a square optical lattice 59
2.15 Pinning-induced defects in a vortex lattice . . . . . . . . . . . . . . . . .

60

2.16 Defects in pinned vortex lattices with varying rotation rate mismatch.

61

2.17 Vortex pinning with 2 vortices per pinning site . . . . . . . . . . . . . .

62

2.18 Theoretical predictions [128] for vortex pinning with 2 vortices per site .

63

3.1

Setup for static 2D optical lattice used to observe the BKT transition. .

67

3.2

Measured and calculated static 2D optical lattice intensity profile . . . .

70

3.3

Setup, system parameters and images showing vortex proliferation . . .

76

3.4

Converting phase winding defects in a Josephson junction array to vortices in a bulk BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

3.6

77

Experimental result: Quantitative study of thermally activated vortices
in the BKT crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Image showing vortex-antivortex annihilation . . . . . . . . . . . . . . .

81

xiv
3.7

Result: Varying the optical lattice ramp-down duration to distinguishfree vortices from tightly bound vortex-antivortex pairs . . . . . . . . .

82

3.8

Time-to-length scale mapping for vortex pair annihilation . . . . . . . .

83

3.9

Experimental result: Shift of vortex activation curve with ramp-down
duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.1

Motivation for loading a rotating BEC into a 1D optical lattice . . . . .

88

4.2

Idea: Study of single vortex dynamics in a 1D optical lattice . . . . . . .

89

4.3

Parameters of the 1D optical lattice . . . . . . . . . . . . . . . . . . . .

90

4.4

Model for vortex lattice melting and 3D-2D crossover in a 1D optical lattice 93

4.5

Light Source and setup for 1D optical lattice . . . . . . . . . . . . . . .

96

4.6

Characterization of 1D optical lattice depth . . . . . . . . . . . . . . . .

98

4.7

Bloch oscillations of a static phase coherent BEC in an optical lattice .

99

4.8

Bloch oscillations of a static BEC show the loss of phase coherence withincreasing lattice depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.9

Upon reduction of optical lattice depth, Bloch oscillations of a static BEC
show a revival of phase coherence. . . . . . . . . . . . . . . . . . . . . . 102

4.10 Rotating BEC loaded into a 1D optical lattice: Vortex number decay vs.
optical lattice hold time . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.11 Result: Vortex number decay vs. optical lattice depth . . . . . . . . . . 105
4.12 Result: BEC rotation rate decay in optical lattice vs. initial T /Tc . . . . 106
4.13 Result: Vortex lattice visibility vs. optical lattice depth . . . . . . . . . 107
4.14 Result: Vortex lattice visibility decay time vs. Josephson coupling J . . 109
4.15 Result: Revival of vortex lattice visibility in optical lattice . . . . . . . . 110
4.16 m = −2 quadrupole mode excitation when loading into azimuthally distorted optical lattice potential . . . . . . . . . . . . . . . . . . . . . . . . 112

xv
4.17 Tkachenko mode excitation due to radial anharmonicity when loading
into optical lattice potential . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.18 Scissors mode excitation due orientation mismatch between symmetry
axes of magnetic trap and optical lattice . . . . . . . . . . . . . . . . . . 114
5.1

Magnatically trapped states for two-component BEC in 87 Rb. . . . . . . 120

5.2

Experimental sequence for creating two-component BEC. . . . . . . . . 121

5.3

Hexagonal and square vortex lattice in real- and Fourier space. . . . . . 122

5.4

Experimental result: Evolution from hexagonal vortex lattice to interlaced square lattices in a rotating two-component BEC. . . . . . . . . . 123

5.5

Result: Tkachenko mode excitation in the square vortex lattice . . . . . 125

5.6

Experimental sequence for two-component vortex lattice interference. . . 126

5.7

Resultt: Vortex lattice interference in two-component BEC . . . . . . . 127

B.1 Numerical grid for simulation of the Gross-Pitaevskii equation (GPE) fora BEC in a double-well potential . . . . . . . . . . . . . . . . . . . . . . 144
B.2 Numerical GPE code schematic . . . . . . . . . . . . . . . . . . . . . . . 145
B.3 Double-well GPE simulations at zero temperature . . . . . . . . . . . . 146
B.4 Finite temperature corrections to GPE simulation results . . . . . . . . 147
D.1 Side view images of BECs with increasing rotation rate . . . . . . . . . 151
D.2 Physics of the lowest Landau level. . . . . . . . . . . . . . . . . . . . . . 152
D.3 Experimental result: Fractional condensate area occupied by vortex cores 155
D.4 Experimental result: Tkachenko modes of the vortex lattice . . . . . . . 156
E.1 Calibration of top view image magnification . . . . . . . . . . . . . . . . 163
E.2 Calibration of side view image magnification . . . . . . . . . . . . . . . . 165
F.1 Smiley BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Chapter 1
Introduction

The experiments described in this thesis revolve around two topics: rotating BoseEinstein condensates (BECs), and tunnel-coupled arrays of condensates in optical latticepotentials. It is shown that various combinations of both these themes yield highly
complex quantum fluid systems with many analogies to condensed-matter systems, andthus lead to new avenues in ultracold atom research. In the following, the relevant
aspects of superfluids, their rotation, their quantum and thermal fluctuations, and their
manipulation with tailored optical potentials will be introduced.

1.1

Background
During the course of this thesis, the field of ultracold atoms has undergone sig-

nificant developments. In the beginning, fundamental studies were in full bloom, whichhave in the meanwhile been described in two books [119, 122]. Nowadays, ultracold
atoms in many cases merely serve as a starting point for further adventures, see e.g. therecent reviews [29, 101]. What makes these dilute-gas ultracold atom samples so special
is that they represent a simple, to a large extent now well understood, macroscopic
quantum mechanical system, combined with the experimental capability to turn a variety of “knobs” that take the system into fundamentally new and poorly understood
regimes, often relevant to unsolved problems from other fields of physics. In a sense
this opens up a “bottom-up” approach to studying many-body quantum mechanics, as2
opposed to the “top-down” approach necessarily pursued e.g. in many other condensed
matter systems. Such “knobs” include but are not limited to• Optical lattices, i.e. spatially periodic potentials made from interfering laser
fields. The regime of few particles per lattice site is the realm of Hubbard
model physics [67, 85], relevant to high temperature superconductivity [80] andquantum magnetism [134]. In the opposite limit of many atoms per site, “classical” Josephson junction arrays [161] may be studied.
• Fast rotation of a quantum gas may allow the realization of a new system forstudying fractional quantum Hall physics [47, 165], and the creation of anyonicexcitations [118].
• Tunable interparticle interactions allow to access the full range from ideal (noninteracting) to strongly interacting quantum gases [29].
• Internal (spin) degrees of freedom add new complexity, see e.g. [137].
• Composite quantum systems, such as Rydberg-excited atoms in a BEC [75],trapped ions and BEC, as well as ultracold dipolar atoms [97] or molecules[113] will allow exploring the eﬀects of new interactions on quantum systems.
In addition, new probing capabilities are being developed, such as coupling many-bodyquantum systems to high-finesse cavities [31], or the study of noise correlations [8],complementing the traditional absorption and phase contrast imaging.
Relevant to this thesis, rotating Bose-Einstein condensates provide a conceptual
link between the physics of trapped quantum gases and the physics of condensed mattersystems such as superfluids [53], type-II superconductors [161] and quantum Hall eﬀect(QHE) systems [98]. In all these systems, striking counterintuitive eﬀects emerge whenan external flux penetrates the sample. For charged particles this flux can be provided

3
by a magnetic field, leading to the formation of Abrikosov flux line lattices in typeII superconductors, or in QHE systems to the formation of correlated electron-liquids
and composite quasiparticles made of electrons with attached flux quanta. For neutral
superfluids, the analog to a magnetic field is a rotation of the system, which similarly
spawns vortices. Recently, the creation and study of large ordered Abrikosov latticesof vortices in rotating atomic BECs [102, 5, 73, 78] has become possible. Experiments
have explored regimes of rotation far beyond the ones attainable in superfluid Helium[143, 44, 32]. In the limit of the highest possible rotation rates, analogies to integer
quantum Hall systems become apparent [143], and even a phase transition into a regimeof strongly correlated states and composite quasiparticle excitations, analogous to thefractional quantum Hall eﬀect, has been envisioned by theorists. These developments
have recently been excellently reviewed by Fetter [58].
In this thesis, several experiments were conducted that have expanded the rangeof studies of vortex lattices in rotating BECs: Periodic pinning potentials for vortices
were created, which allowed to induce a structural crossover from a hexagonal vortex
lattice to a pinned square lattice (Chapter 2). In Chapter 4, an array of two-dimensional
rapidly rotating BECs was created, which allowed the study of thermal vortex lattice
fluctuations, and realized a possible first step toward fractional quantum Hall states inrotating quantum gases with only few atoms per vortex. Finally, the first rapidly rotating two-component BECs were created (Chapter 5), and self-organization of vortices inthe two components into two interlaced square vortex lattices was observed.
Optical lattices, the second ingredient to this thesis, are periodic potential configurations created by interfering laser beams. The structures are reminiscent of crystalline
lattices in solid state physics, but are free of defects and phonons. Such lattices can forma very controlled environment for ultracold atoms: individual atoms can be confined
to single lattice sites, prevented from colliding with neighboring atoms, and cooled totheir motional ground state such that thermal motion is completely frozen out. In ad-4dition, optical lattices can turn a three-dimensional system into an array of two-, one-,or even zero-dimensional systems. This high level of control may be exploited for precision measurements, e.g. of the electron’s electric dipole moment [40], the fine structure
constant [43], and of absolute optical frequencies, e.g. in optical atomic clocks [30].
Optical lattices in the context of quantum degenerate gases have become a hot topic
since the prediction [85] and realization [67] of a transition from the BEC superfluid toa Mott-insulator state. Experiments typically operated in a regime of few particles per
site, where interactions and correlations between individual particles are important.
In this thesis we explored the opposite regime of large occupancy per site, where
Josephson junction arrays are realized, with many connections to condensed mattersystems. The first rotating two-dimensional optical lattice potential for ultracold atoms
was created, which served as a periodic pinning potential for vortices (Chapter 2). In
addition we made extensive use of optical lattice configurations to create arrays of lowdimensional systems (Chapter 4), or eﬀectively low-dimensional systems (Chapter 3).
A third important ingredient to this thesis is thus the influence of reduced dimensionality on an ordered state such as a BEC. In low-dimensional systems the establishment of order is often hampered by the increased importance of low-energy fluctuations[29]. Atomic systems are ideally suited to probe this regime, for one part due to theability to tailor very clean anisotropic optical potentials, based either on laser interference [146, 71] or beam shaping. Further, rather direct probes of order and correlations
exist, such as the interference of two or many atomic samples [71], or noise correlations[8]. Reduced-dimensionality systems remain a very open field, both theoretically andexperimentally, as both interparticle interactions and the trapping potential can lead to
nontrivial eﬀects.
In this thesis, the following low-dimensional systems were created: The BerezinskiiKosterlitz-Thouless superfluid-to-normal transition was studied in a finite-temperature,eﬀectively two-dimensional array of Josephson junctions (Chapter 3). A one-dimensional

5
optical lattice was employed to create a stack of two-dimensional rapidly rotating BECs,
and to observe the vortex lattice fluctuations resulting from this low-dimensional geometry ( Chapter 4).

1.2

Bose-Einstein condensation
In the following I will briefly review the topics of quantum degeneracy and con-densation, with an emphasis on the roles of dimensionality, finite temperature, andinterparticle-interactions as required for this thesis.
1.2.1

Quantum statistics: Ideal gas Bose-Einstein condensation

Bose-Einstein condensation (BEC) [54, 11] is defined as the occupation of a single
quantum mechanical state – typically the lowest-energy single-particle state – by amacroscopic number of particles. Its most powerful consequence is the development ofa macroscopic matter wave, in most instances with a high degree of phase coherence
across the whole condensed sample. This is in stark contrast with ordinary matter,
where each quantum mechanical state is negligibly occupied and ensembles of particles
behave incoherently.
Quantum Degeneracy. To understand when BEC will occur, two ingredients quantum mechanical uncertainty and the consequences of exchange symmetry of identical particles - are essential. The Heisenberg uncertainty relation states that position
and momentum of a particle may not simultaneously be measured sharply, but with uncertainties ∆x and ∆px that obey ∆x × ∆px ≈ �, where � is Planck’s constant divided
by 2π. For an ideal gas, the typical momentum spread associated with random thermal
motion at temperature T determines the width of the de Broglie wavepacket ∆x ≡ λth
�
2π�2
λth =
(1.1)
mkB T

6
where m is the mass of the particles and kB is Boltzmann’s constant. As de Broglie
wavepackets grow with decreasing T , neighboring wavepackets begin to overlap in a Ddimensional system when the phase space density reaches unity (nD is the D-dimensional
total particle density)nD × λDth = O(1)(1.2)

This is called the degeneracy condition 1 , under which identical particles are forced toreveal their quantum statistics: bosons will symmetrize their total wavefunction andhence allow unrestricted occupation of single-particle states, whereas fermions insist
on antisymmetrization, restricting single-particle level occupancy to below unity. The
quantum degeneracy condition (1.2) is thus an upper limit of when Bose-Einstein condensation might take place. But will it? The answer in many cases is yes, but in certain“marginal” situations (e.g. set by the dimensionality of the system, see below) it depends on additional factors, such as the presence and shape of a trapping potential, andimportantly the interactions between particles.
Condensation. To give a more definitive answer if condensation will occur,
one must examine how the quantum mechanical states of a system are occupied. The
occupation of a state with energy Ei in thermal equilibrium at temperature T is given
by Bose/Fermi statistics (+ for fermions, - for bosons)fi (Ei , µ, T ) =1
exp((Ei − µ)/kB T ) ± 1(1.3)

and we may set E0 ≡ 0. µ is the chemical potential, fixed by the total number of�particles, N = i fi . For an ideal gas of bosons (but not for fermions), µ ≤ 0 is enforced
in order to have physically meaningful positive occupation numbers fi > 0. It is now�∞�
tempting to approximate the sum i fi by an integral, N → N0 + ε>E0 f (ε, µ, T )g(ε)dε.

Here g(ε) is the density of states (DOS), i.e. g(ε)dε is the number of states in an intervaldε around an energy ε. The occupation N0 of the lowest state, in which BEC is expected
The air around you is non-degenerate, n3D λ3th ∼ 10−8 , despite a 6 orders of magnitude higher
density than typical BEC densities. This is because at room-T , λth is tiny, of order of atomic sizes.
1

7
to occur, is treated separately, as in some situations it is suppressed from the integral
by a vanishing DOS. At this point we distinguish between a homogeneous system anda (harmonic) trapping potential. For a homogeneous D-dimensional system of size Lg(ε) ∼

εD/2−1(�2 /mL2 )D/2

(1.4)

while in a D-dimensional harmonic trap potential with frequencies ωi [119]g(ε) ∼The integral Nex ≡

�∞

εD−1
�
(D − 1)! D
i=1 (�ωi )(1.5)

ε>E0 f (ε, µ, T )g(ε)dε gives the number of atoms in excited states.

For bosons this number reaches a maximum when µ = 0, and this enforces an upper
limit on the number of bosons that can be accomodated in excited states. If the total
number of particles exceeds this maximum,� ∞N > Nex,max ≡f (ε, µ = 0, T )g(ε)dε

(1.6)

ε>E0

then Bose-Einstein condensation of the excess atoms into the zero-energy state occurs.
Results of this analysis for two/three dimensions and homogeneous/trapped systems
are given in Table 1.1, as well as the eﬀects of interparticle interactions.
Table 1.1: Critical temperature Tc and phase-space density (PSD) for occurrence of
BEC in 2D/3D weakly interacting/noninteracting systems, both spatially homogeneousand in a harmonic trap [122, 119]. ntot,D : total density in D-dimensions, ω̄: geometricaverage of trap frequencies. TBKT , C, g̃: See Eq. (1.31). For interacting 2D systems(Section 1.4) and Josephson junction arrays (JJA, see Section 1.5), TBKT refers to thenormal-to-superfluid transition, not to the occurrence of BEC. J: Josephson energy.
homogeneous systemtransition temperature
3D nonint.
3D int.
2D nonint.
2D int.
2D JJA

PSD

trapped systemtransition temperaturekB Tc = 3.3 �m ntot,3D

2.61

kB Tc = 0.94�ω̄N 1/3

kB Tc = 0

≈ 2.61∞

kB Tc ≈ 0.94�ω̄N 1/3
kB Tc,2D = 0.78�ω̄N 1/2
��−1/2
TBKT /Tc,2D = 1 + π3g̃3 ln(C/g̃)2[81]
no theory

2

2/3

2

2/3
kB Tc ≈ 3.3 �m ntot,3D�2 n

tot,2D2π
TBKT = mk× ln(C/g̃)B
kB TBKT ≈ J

4Remarkably, in 3D the ideal Bose gas condenses at the maximum temperatureallowed by quantum statistics! This result turns out to be fairly robust also in the8presence of weak interparticle interactions and a trap potential. In the 2D systems on
the other hand, eﬀects of interactions, finite-particle number and of a trap are clearly
important [14]. The situation in 2D will be described in more detail below.
Macroscopic coherent matter wave. The many body wavefunction Ψ describing the condensed state is a product of identical single particle wavefunctions ψΨ(r�1 ,...,r�N ) =

N
�
i

�
�
ψ(�ri ) → Ψ(�r) ≡ �Ψ(�r) � eiφ(�r)(1.7)

This is the so-called mean field approximation: The BEC is described by a complex�
�2valued macroscopic wavefunction Ψ(�r) whose squared modulus �Ψ(�r) � ≡ n(�r) describes
the density of the ensemble of condensed particles, and whose phase gradient ∇φ(�r)defines the superfluid velocity�vs =�∇φ(�r)m(1.8)

In the 3D weakly interacting Bose gas, neither quantum nor thermal fluctuations arestrong enough to appreciably decohere the condensate phase. As a consequence, BoseEinstein condensation, phase coherence, and superfluidity occur simultaneously. In
lower dimensions the situation is more complicated, as discussed below.
1.2.2

Quantum fluids: Importance of interparticle-interactions

Neutral atoms interact through van der Waals interactions, and such interactions
modify many properties of the condensate, while leaving others almost unchanged. At
the low energies of ultracold atom experiments, an eﬀective description of this interaction
is valid, with the interaction strength parameterized by the s-wave scattering length asc .
The sign and magnitude of asc are determined by the precise shape of the two-bodymolecular potential, and are even tunable by external fields. In practice, unless special
steps are taken, interparticle interactions are “weak” in the sense that ntot a3sc � 1. Thisis the reason why interactions in general have little influence on the BEC transition as
well as the condensate fraction , see Table 1.1.

9
Despite being characterized as “weak”, interparticle interactions have several profound eﬀects. Interactions change the low-energy part of the BEC’s excitation spectrum
from quadratic to linear in momentum, and this linear excitation spectrum is fundamentally responsible for superfluidity. The macroscopic wavefunction Ψ of such a weaklyinteracting Bose condensate is described by the Gross-Pitaevskii equation:
�
�
�2
�
�2
�2 ∇
−
+ g �Ψ(�r) � + V(�r) Ψ(�r) = µΨ(�r)
(1.9)
2m
�
�2
In this nonlinear Schrödinger equation, the nonlinear term g �Ψ(�r) � Ψ(�r) describes the2

eﬀective contact interaction of strength g = 4π�masc . The external trapping potential
V(�r) is typically harmonic, and in all experiments in this thesis cylindrically symmetric,V(�r) = 12 m(ωr2 r2 + ωz2 z 2 ). The chemical potential µ gives the energy needed to add aparticle to the superfluid. It should be noted that, owing to the nonlinearity, (1.9) does
not have discrete eigenvalues µ, but has a solution for any µ > 0.
There are three length scales, found by balancing energy terms: the healing lengthξ is the only trap-independent length scale, determined by the balance of kinetic energy
and nonlinear interaction, with n ≡ |Ψ|2
� 2 �1/2�ξ=2mgn(1.10)

it determines the distance over which a density disturbance of the superfluid heals
back to the bulk value, and hence sets the size of intrinsic condensate defects, such as
solitons and vortices. Kinetic and potential energy determine the harmonic oscillator�
length, ar,z = mω�r,z . Balancing the potential energy with the nonlinear interaction
term gives the so-called Thomas-Fermi condensate radii, described in the next section.
1.2.3

Thomas-Fermi regime

The typical regime of condensate physics is the so-called Thomas-Fermi (TF)
regime where the interaction energy is much larger than the kinetic energy, and interactions push the BEC outward, typically increasing its size R to many times the

10noninteracting harmonic oscillator size, or in terms of length scales, R � ar,z � ξ. In
this regime, the chemical potential is given byµ=where ā =�ω̄
(15N asc /ā)2/5
2(1.11)

��/mω̄ and ω̄ = (ωx ωy ωz )1/3 is the geometric average of trap frequencies.
�The condensate density profile in the region (ri /Ri )2 ≤ 1 is an inverted parabola
nBEC (�r) =while it is zero where

�
�
µ�
1−
(ri /Ri )2g(i = x, y, z)(1.12)

�(ri /Ri )2 > 1. Interactions thus cause a fairly sharp drop in

the condensate density at its edge, making it visually easily distinguishable from thenoncondensed background atoms. Very importantly the chemical potential, and hence
using Eq. (1.11) the total BEC number, can be determined simply by measuring the
BEC radii and trap frequencies:
Ri =
1.2.4

�

2µ
mωi2

(1.13)

Condensates at finite temperature

At finite temperature T , only a fraction of atoms is condensed, with the remaining atoms occupying a lower-density “thermal cloud”. For the study of the finite-T
Berezinskii-Kosterlitz-Thouless transition in Chapter 3, the accurate determination of
BEC and thermal cloud parameters is essential. We thus give a detailed description of
the density profile of a finite-T BEC and its thermal cloud.
The Bose distribution describing noninteracting bosons in equilibrium at temperature T in an external potential V (�r) is given by [119]f (�r, p�) =1
exp((�
p2 /2m + V (�r) − µ)/kB T ) − 1(1.14)

where p� is the particle’s momentum and µ is the chemical potential. The spatial density
nth (�r) is obtained by integrating over the momentum coordinates, resulting in
nth (�r) =g3/2 (z(�r))
λ3th(1.15)

11
where g3/2 (z) ≡�∞zn
n=1 n3/2 is called the Bose function, λth is given by Eq. (1.1) andz(�r) = exp(−(V (�r) − µ)/kB T )(1.16)In order to realistically describe our experiments we need an approximate description ofinteracting bosons at finite T . We follow the description of Naraschewski and StamperKurn [112]. Repulsive interactions cause the high condensate density nBEC (�r) in thecenter of the trap to repel the thermal cloud. They also cause a self-repulsion of the thermal cloud (which is a weaker eﬀect due to the lower thermal cloud density). The resulting modification of the thermal cloud density profile is taken into account by treating the
BEC and thermal cloud densities as an eﬀective repulsive potential 2gnBEC (�r)+2gnth (�r), and including this eﬀective potential in Eq. (1.16):z(�r) = exp(−(V (�r) + 2gnBEC (�r) + 2gnth (�r) − µ)/kB T )(1.17)

Now, Eq. (1.15) with (1.17) forms an implicit equation for nth that can be solved
iteratively. The resulting “mean field modified thermal cloud profile” is used for fitting
the thermal cloud of finite-T BECs, an example of which is shown below in Figure 1.12
in Section 1.6 on BEC imaging. On the other hand, the thermal cloud density nth (�r)exerts a repulsive mean field pressure onto the BEC. This eﬀect is small however, sothat the BEC density nBEC (�r) is only slightly altered by the thermal cloud and still
follows the Thomas-Fermi profile, Eq. (1.12).

1.3

Rotating Bose-Einstein condensates

1.3.1

Quantized vortices in a superfluid

In this thesis there is no experiment that does not involve vortices in one form or
another. Here we explain their main features. At first sight, superfluids, being describedby a macroscopic wavefunction (1.7), are “irrotational”: Around any closed path inside
the superfluid, the single-valuedness of the wavefunction enforces a vanishing circulation

12
Γ≡

�

�
�vs · d�l = m(φ(end) − φ(start) ) = 0, with �vs given by (1.8) and φ(...) the condensate

phase at start and end point of the closed loop.
A vortex is nature’s way to allow a superfluid to rotate nonetheless. Picture avortex as a “quantum tornado”, with fluid circulating rapidly around a singular line (ina 2D system a point), where due to centrifugal pressure the superfluid density is forced
to zero. This zero density region marks a phase singularity, around which the phaseφ(�r) is allowed to undergo (multiples of) 2π winding. This leads to a nonvanishing,quantized circulation ΓΓ≡�

�vs · d�l = qκ

(1.18)hwhere κ = m
is the fundamental quantum of circulation, and q is the number of quantaenclosed by the loop (it is energetically favorable to distribute these q quanta onto q
singly quantized vortices). Vortices thus come with two possible signs of circulation (qin (1.18) can be positive or negative). The azimuthal superfluid flow around a singlyκ
quantized vortex is given by (1.18) to be vs = 2πr. The condensate wave function issuppressed as ψ ∝ r for r → 0. In the limit of isolated vortices this suppression isgoverned by a balance of interaction energy and kinetic energy, and is hence expected
to occur over a length scale comparable to the healing length ξ. From a numericalsimulation of the Gross-Pitaevskii equation we estimate the root-mean-square vortex
core radius
rv = 1.94 × ξ(1.19)The energy of a (straight) vortex line of length L in a condensate of radius R is dom�R
2 , with
inated by the kinetic energy of the superflow, Ev = ξ dr 2πr nSF (r) L 12 m vs(r)κ
vs = 2πr, the superfluid density nSF , and a short-distance cutoﬀ of order ξ, resulting in

Ev ≈

π�2
nSF L log(R/ξ)
m(1.20)

13

Increasing!

Figure 1.1: Images of vortex lattices in BECs rotating at diﬀerent rates Ω. With
increasing rotation, the vortex lattice’s unit cell shrinks, whereas the vortex core area
increases. At the same time, due to centrifugal forces the BEC expands outward.

1.3.2

Vortices – from one to many

If many vortices are present, a hexagonal vortex lattice (Abrikosov lattice [6])forms due to repulsive vortex-vortex interactions, with all vortex lines straight andaligned along rotation axis. For large enough numbers of vortices in a superfluid, thecorrespondence principle suggests that the rotation field, coarse grained over the superfluid, should go over to the classical limit of rigid-body rotation
� × �r�vf luid = Ω(1.21)

� the vortex lattice is stationary. Taking the contour ofIn the frame rotating at Ω,
integration in Eqn. (1.18) around a loop containing many vortices, and identifying the
integer q ≡ Nv with the number of vortices inside the loop, the areal density of vortices
nv is found to increase with rotation rate
nv ≡

Nv
mΩ
=
A
π�

(1.22)

14
The vortex density is often expressed in terms of two convenient length scales, either
the “magnetic length” �, i.e. the radius of the vortex lattice’s Wigner-Seitz unit cell
�=

�

b=�·

�

1
nv ≡ 2π��1/2(1.23)√
2π/ 3 ∼ 1.9 · �(1.24)

�mΩ

or the directly observable vortex spacing b
nv ≡

1
√
b2 3/2In Fig. 1.1, images of vortex lattices in BEC at diﬀerent rotation rates Ω are shown.
The increase in vortex density with rotation rate is clearly visible.
The shape of rotating BECs is influenced by centrifugal forces, which pull the
BEC radially outward. Chemical potential and BEC radii still follow the Thomas-Fermi�description, Eqs. (1.11) and (1.13), respectively, with the replacement ωρ → ωρ2 − Ω2(Appendix D). The rotation rate Ω is thus accurately determined from the aspect ratio
Rz (Ω)
AR(Ω) ≡ Rof the rotating BEC, compared to the aspect ratio of the static BEC
ρ (Ω)
�
ω
AR(0) ≡ ωρz , by the relation Ω/ωρ = 1 − (AR(Ω) /AR(0) )2 .

In a bigger picture, vortices have two, seemingly contradictory, roles in superflu-

idity:
Vortices are hallmarks of superfluidity. Although superfluidity is a multi-faceted
phenomenon, many of its most spectacular manifestations are associated with vortices
and its response to rotation. If a superfluid is placed into a slowly rotating container,
the quantization of circulation (1.18) leads to a minimum rotation rate required forthe first vortex to enter the system - this is the famous Hess-Fairbank eﬀect, one of
the hallmarks of superfluidity. An analogous eﬀect exists for Type-II superconductors,
where vortices are excluded from a sample below a critical applied magnetic field Hc1 .
A second phenomenon is the (meta-)stability of superflow. Once a vortex is nucleatedin an annular-shaped superfluid, the rotation of its environment may be stopped, but
the fluid will continue to rotate for astronomical timescales [99].

15

1

1
4" 3

staticBKTtransition

Normal state

BKT
vortex lattice
melting
Superfluid
]
,150
4
4
s
,1
ttice
[143
a
l
ex
vort

quantum
melting

T
TBKTT~
TBKTasc/az
FQH

Wigner
crystal~1

1/!

Figure 1.2: Schematic phase diagram of a rotating 2D Bose gas. 1/ν parametrizes
increasing rotation, see Eq. (1.25). The vertical axis is temperature in units of TBKT –the temperature at which a static superfluid undergoes a Berezinskii-Kosterlitz-Thouless(BKT) transition, given in Eq. (1.36). A BKT-type melting of the vortex lattice [64]in rotating BECs occurs at a diﬀerent, lower temperature, see Eq. (D.15) in Appendix
D [82]. Quantum fluctuations are expected to melt the vortex lattice at a filling factorν ∼ 10, see Eq. (D.11) [21]. At even faster rotation rates, strongly correlated quantum
Hall (FQH) states induced by rotation [165] are expected. For the thermal destructionof FQH states around T ∼ TBKT aascz see Eq. (D.17). At extremely large rotation rates,a Wigner crystal [117] may form. [143, 144, 150]: Experiments in this thesis.

Too many vortices destroy superfluidity. Vortices may be induced either by rotation, either of the superfluid’s environment or of the normal cloud or, in the absenceof rotation, by thermal agitation. Both situations lead to very diﬀerent physics, which
nonetheless share one common feature - too many vortices destroy superfluidity. Thermally activated vortices in a static BEC induce the Berezinskii-Kosterlitz-Thouless transition, described in Section 1.4. In the rapid rotation limit on the other hand, quantum
fluctuations of the vortex lattice destroy the macroscopic wavefunction responsible forsuperfluidity, and induce a transition to strongly correlated fractional quantum Hall
states. Such fluctuations are governed by the filling factor ν, i.e. the ratio of particles

16
Np to vortices Nv , or the ratio of areal densities nSF,2D and nv
ν ≡ Np /Nv ≡ nSF,2D /nv ;1/ν ∝ Ω

(1.25)

When ν � 10, quantum fluctuations are predicted to melt the vortex lattice and correlated fractional quantum Hall states are expected to appear, see Appendix D. The
prospect of enhancing quantum fluctuations of the vortex lattice has thus been a major
motivation for the general direction of experiments conducted during this thesis.
Vortex lattices thus continue to be a fascinating research topic. In this thesis, wehave extended vortex lattice studies to spinor Bose-Einstein-condensates (Chapter 5),
created periodic pinning potentials for vortices (Chapter 2), and loaded rapidly rotatingcondensates into a one-dimensional optical lattice (Chapter 4) aligned with the rotation
axis. Several further topics that have yet eluded experimental study are depicted inFig. 1.2, and are detailed in Appendix D.
1.3.3

Manipulating vortices

Having established the self-organized response of superfluids to rotation, we are
now in the position to think of ways to manipulate vortices, either on an individual
basis or acting on the vortex lattice collectively. In Chapter 2 of this thesis, we showthat vortex lattices can be collectively pinned to a rotating two-dimensional optical lattice. Beyond pure playfulness, such control may be desirable to establish defined initialconditions, e.g. in experiments studying vortex lattice fluctuations, or in experiments
studying persistent superfluid flow [136].
A vortex in a dissipation-free superfluid behaves much like a massless particle,
such that vortex motion is described by balancing all acting forces to zero. Two kinds offorces are at our disposal that act on a vortex. Spatial variations in the superfluid density
lead to a pinning force, illustrated in Fig. 1.3. The vortex energy, given by Equation(1.20), depends on the superfluid density surrounding the vortex core. Application of a17
repulsive potential V (x) to the superfluid leads to an energy gain when the vortex moves
to the lower-density location. In this way a repulsive potential for atoms is attractivefor vortices. As is shown in Appendix C, the pinning energy for a single vortex is
Epin (x) ≈ −L/asc V (x)(1.26)

where L is the vortex line length, asc is the scattering length and V (x) is the appliedexternal potential 2 . Eq. (1.26) is valid for V (x) < µ and for potentials V (x) smoothly
varying on the scale of the healing length ξ. The resulting pinning force is� Epin (x) ≈ L/asc ∇� V (x)F�pin (x) = −∇(1.27)The second force acting on a vortex is the Magnus force caused by relative motions
between the vortex (vvortex ) and the superfluid (vf luid ),F�mag (x) = m nSF (x)L (�vvortex − �vf luid ) × �κ

(1.28)nSF is the superfluid density and �κ = h/m ẑ. In the absence of pinning, a vortex willthus float in the velocity profile generated by all other vortices, while for a strongly
pinned vortex, �vvortex is determined by the motion of the pinning potential. There
are thus two sides to the Magnus force. In case the relative motion between pinned
vortices and the surrounding superfluid is too large, the Magnus force may overwhelm
the pinning force and lead to depinning. On the other hand, establishing controlled
superfluid flows may enable one to control vortex motion without acting directly on thevortex.
A rough dimensional argument gives insight in the conditions required for vortex
pinning: The pinning force exerted by an applied potential V varying on a spatial scale
w is Fpin ≈ Vw aLsc while the Magnus force on a pinned vortex is Fmag ≈ (µ − V ) aLsc m∆v
2�
2

One way to rationalize the prefactor L/asc in (1.26) is to note that L/asc = 2nSF L πrv2 , wherenSF is the superfluid density and rv is given by (1.19). The pinning energy is thus approximately thecollective potential energy of all the particles “missing” in the vortex core, Epin (x) ≈ −nSF L πrv2 ×V (x).

Repulsive
Potential Vn18

Kinetic
Energyreduction
V!

Vortex liner
Figure 1.3: Principle of vortex pinning. Application of a repulsive potential (of depthV ) to a superfluid (with chemical potential µ) allows a vortex to lower its kinetic energy
when moving to the lower-density region. The ratio V /µ gives the relative suppressionof superfluid density. Experimental demonstration using a single focused laser beam
(532 nm, V /µ ≈ 1). A vortex was pinned from an initially rapidly rotating BEC, andtrapped for up to 30 seconds. The bright spot surrounding the dark vortex core isa transient hydrodynamic eﬀect of BEC fluid filling in the void after removal of thepinning potential just prior to imaging.

for a relative velocity ∆v between superfluid and pinning potential. Using the speed of�sound cs = √2mξand healing length ξ, the ratio of pinning force to Magnus force is

Fpin /Fmag ≈Vcs ξ(µ − V ) ∆v w

(1.29)

with limits of validity wξ << 1 and Vµ < 1. Pinning occurs when Fpin /Fmag > 1: Either
the pinning potential V must be of order of the chemical potential µ, or the relative
velocity ∆v between superfluid and pinning potential has to be kept much smaller than
the speed of sound cs and the spatial scale w should not be too large compared to thehealing length ξ. In Chapter 2 we thus apply an optical lattice that roughly co-rotateswith the vortex lattice (∆v → 0) to facilitate pinning at small V /µ.
The concept of vortex pinning can be pushed to the quantum limit, where excitations are composite particles composed of atoms and flux quanta, so-called anyons withfractional (i.e. neither bosonic nor fermionic) statistics [98]. In this setting, an interferometric experiment[118] based on dragging pinned anyons around each other may one
day be used to prove the existence of anyons and their fractional statistics.

19

1.4

The two-dimensional (2D) Bose gas
Eﬀectively 2D systems form an important part of this thesis. In Chapter 4 westudy an array of rotating 2D Bose gases in a one-dimensional optical lattice. In Chapter 3 the Berezinskii-Kosterlitz-Thouless (BKT) transition is studied in an eﬀectively
2D system created in a 2D array of Josephson junctions. Here we summarize somedistinctive features of 2D Bose gases.
As Table 1.1 shows, condensation and superfluidity in 2D are much more subtle
issues than in 3D. The homogeneous ideal 2D Bose gas will not form a true condensate at any finite temperature, despite the overlapping of de Broglie waves. In general,in low-dimensional systems, the formation of phase-coherence is hampered by thermally induced fluctuations of the condensate phase. A simple argument shows why:
Equation (1.8) implies that spatial phase fluctuations cost an associated kinetic energy��2
E = dD x nSF 2m(∇φ)2 in D-dimensions (here, nSF is the D-dimensional superfluid�density). If we expand the phase in Fourier components φ(�r) = k Sk exp(i�k · �r), thenwe can obtain the Fourier amplitudes |Sk | using the result from equipartition that every�
Fourier mode of the phase contains an energy kB2T = 2m
nSF k 2 |Sk |2 . The phase variance2

in D-dimensions is then
∆φ =
2

�
k

2

|Sk | ∝

� k,maxk,mindk

k D−1 m kB T
× 2
k2
� nSF

(1.30)The lower momentum cutoﬀ kmin ∝ 1/R is given by the system size R, while kmax ∝1/λth gives the minimum physically reasonable length scale for a phase fluctuation. Eq.
(1.30) shows that in D ≤ 2 long wavelength fluctuations diverge with system size R.
The result is that in 2D, a degenerate Bose gas can only form a “quasicondensate”, i.e.
a condensate with spatially fluctuating phase. Local patches of quasicondensate “looklike” BEC, but there is no phase correlation between such local patches. This statement
holds irrespective of interparticle interactions.
The inclusion of interparticle interactions in 2D completely changes the nature of

20
the normal-to-superfluid transition: A homogeneous interacting Bose gas undergoes a
BKT transition, at a transition temperature TBKT determined by the total 2D density
n2D [127]kB TBKT =�2 n2D2π
×mln(C/g̃)(1.31)√
with C = 380±3 [127] and g̃ = aascz 8π (az is the axial oscillator length of the trap). The
BKT transition is clearly driven by interactions, as may be seen from the limit TBKT → 0
when g̃ → 0. Although this expression is therefore non-universal, it means that alreadyweakly repulsive interactions allow the 2D Bose gas to become superfluid close to the
maximum temperature allowed by quantum statistics! This fact is rather robust due
to its weak logarithmic dependence on g̃, with values of ln(C/g̃) ∼ 6..10 in practice.
The region below TBKT is characterized by the above-mentioned quasicondensate. The
existence of such local quasicondensates is suﬃcient for global superfluidity.
1.4.1

The Berezinskii-Kosterlitz-Thouless transition

It had been well known long before BKT theory was developed, that in 2D atrue BEC with long-range phase coherence exists only at T = 0, while at any finite
T > 0 in 2D true long range phase coherence is destroyed by thermally populated
long wavelength phase fluctuations, as described by Eq. (1.30). The new realizationof Berezinskii [23, 24], and Kosterlitz and Thouless [94], was that in addition to longwavelength excitations, another not-so-obvious type of low-energy excitation has tobe taken into account in order to understand superfluidity in 2D: thermally activatedvortex-antivortex pairs, i.e. pairs of vortices of opposite circulation.
To understand the influence of thermally activated vortices on superfluidity andphase coherence, it is essential to appreciate the diﬀerence between free vortices andbound vortex-antivortex pairs. In Fig. 1.4, the phase of an isolated vortex is shown it extends over the whole superfluid. A random collection of vortices therefore destroys
phase coherence over length scales larger than the mean vortex separation. A vortex-

21

Condensate Phase [radians]

3
2
1
0
-1
-2
-3

Figure 1.4: Eﬀect of vortices and vortex-antivortex pairs on phase coherence. Condensate phase is encoded in color. Left: a single free vortex, with superfluid flow indicatedby arrow. Center: phase decoherence caused by a random collection of vortices. Right:a vortex-antivortex pair on the other hand creates only a localized phase disturbance.

antivortex pair, in contrast, has its phase disturbance localized to the region in between
the vortices. This diﬀerence between free vortices and vortex-antivortex pairs has twoconsequences: First, the energy required to place an isolated “free” vortex into a 2D
superfluid of density nSF,2D is large, and increases with system size R as given by Eq.
(1.20), with the replacement nSF L → nSF,2D for a 2D system
Ev ≈

π�2
nSF,2D log(R/ξ)
m(1.32)(ξ is the healing length which sets the vortex core size), whereas the energy cost for
creating a bound vortex-antivortex pair can be very small, of the order of
Epair ≈π�2
nSF,2D log(r12 /ξ)m(1.33)

for a pair separated by r12 (Eq. (1.33) is meaningful only for r12 > ξ). Owing to thissmall energy cost, such tightly bound vortex-antivortex pairs occur at any finite T. On
the other hand, the very localized phase disturbance of small (tightly bound) vortexantivortex pairs means that such pairs have very little influence on phase coherence,and the system remains superfluid. With increasing temperature however, the densityof vortex-antivortex pairs increases, and the interaction between members of large pairs
becomes screened by smaller interpenetrating pairs. This causes the pairs to become
more and more loosely bound, and ultimately a plasma of free vortices forms. At this22
point, superfluidity and phase coherence are fully destroyed, and a finite T transition toa normal state occurs. This “thermally activated proliferation of vortices” is sketchedin Fig. 1.5.

In 2D:

superfluid

normal state

TBKT "!h 2 nSF
2mk B

T<TBKT

T>TBKTFew, tightly bound vortex antivortex pairs

Free vortices

T=0

Figure 1.5: Vortex-antivortex pair unbinding triggers the BKT transition from a 2D
superfluid to a normal state at the temperature TBKT . Adapted from [29].

A simple and fairly accurate estimate of the BKT transition temperature canthus be made by asking the question “under what conditions is it thermodynamicallyfavorable to have an isolated (free) vortex?” Placing a free vortex into a superfluid isthermodynamically favored (and hence will occur spontaneously) if its free energy
F = Ev − T Sv ≤ 0(1.34)The energy required to place a free vortex into a superfluid of density nSF,2D increases
with system size R as given by Eq. (1.32), but may be oﬀset by an entropy gainS ≈ 2kB log(R/ξ)(1.35)

due to the ≈ R2 /ξ 2 distinguishable locations for a vortex. Balancing energy and entropy
term results in the famous universal relation between TBKT and the 2D superfluid
density nSF at the transition temperature:
TBKT =π�2 nSF,2D
2mkB(1.36)

23
The diﬃculty, in contrast to Eq. (1.31), is that the superfluid density nSF,2D needs tobe determined independently, and in 2D is not necessarily equal to the BEC density.

1.5

Josephson Junction arrays of BECs in optical lattice potentials
In Chapter 3, we study the BKT transition in a 2D array of Josephson-coupled

BECs. In Chapter 4 we study vortex-lattice fluctuations in a one-dimensional Josephsoncoupled array of rapidly rotating 2D BECs. Here we give a basic description of sucharrays of Josephson-coupled condensates, and how we create them.
1.5.1

Optical dipole potentials - versatile tools for BEC manipulation

The optical dipole potential exerted on atoms by oﬀ-resonant light is the prime
tool for BEC manipulation exploited in this thesis. Multi-beam interference patterns
are used to create the periodic potentials from which Josephson junction arrays areconstructed. Here I only give approximate formulae useful to gain intuition and fororder-of-magnitude estimates, while the exact formulae used throughout this thesis tocalculate dipole potential strengths are given in Appendix A.
� induces a dipole moment d� = αE
� in a particle withClassically, an electric field E� ∝ αI(�r) proportional to thepolarizability α, which then experiences an energy −d� · E
local intensity I(�r). A useful approximation for the resulting dipole potential is
1 I(�r)Udip (�r) = �Γ
8
Isat

� �
Γ
δ

(1.37)

3 /(4πc2 ) = 1.64 mW/cm2 for the dominant 5S −→ 5P transition inwhere Isat = �Γωat87 Rb. c is the speed of light, δ = ω − ω

at is the detuning of the light field at frequency

ω from the atomic resonance at ωat , and Γ is the natural linewidth of the atomic
transition. A negative (“red”) detuning creates an attractive potential localizing theatoms to intensity maxima, while at positive (“blue”) detuning the potential is repulsive.
At the same time, decoherence and heating are caused by the spontaneous scattering of24photons by the atoms, which occurs at a rate of approximately
1 I(�r)Γscat (�r) = Γ
8 Isat� �2Γδ

(1.38)

From equations (1.37) and (1.38) it can be seen that by increasing the detuning δ andthe intensity I a fixed potential depth may be maintained while keeping a low enoughUscattering rate for the desired experiment: �Γdip= Γδ .
scat
Optical lattices have become a versatile tool owing to their variable geometry
and length scale, as well as time-dependent control over their strength. The latticegeometries relevant to this thesis are depicted in Figure 1.6. Depending on the number
a)

c)

b)

d!"#$2sin%)..

d!&"#$3sin%)
d!"#2

..

%

Figure 1.6: Diﬀerent optical lattice geometries employed in this thesis. a) Retro-reflected
optical lattice (OL) used in Chapter 4, b) acute-angle OL, and c) two-dimensional acuteangle OL used in Chapters 2 and 3.

of atoms per lattice site, two very diﬀerent regimes of optical lattices are possible.
For large numbers of atoms per lattice site, such that each site contains an individual
BEC with well-defined phase, a “classical” Josephson junction array is realized. Herethe individual superfluids communicate (i.e. exchange particles) by tunneling. In theother extreme limit of single- (or few-) particle occupation of lattice sites, a strongly
correlated system results, which undergoes a superfluid- to Mott-insulator transition
with increasing lattice depth, and holds promise for studying physics relevant to high-25
temperature superconductivity [80] and quantum magnetism [134].
Two Josephson-coupled BECs.

Energy. .
VOL.

#(Nwell+!N/2).
. .. .
....
. ..
...... ... .

J
!"
d

.
.T .
. .
. ..
. ... . .
.............
.
#(Nwell-!N/2)

1.5.2

r
Figure 1.7: A double well BEC. The barrier height VOL and the chemical potentialµ (fixed by the number of condensed atoms per well Nwell ), control the Josephson
coupling J, which acts to lock the relative phase ∆φ. A cloud of uncondensed atoms
at temperature T induces thermal fluctuations when J < T . The charging energy Ec
disfavors population imbalance ∆N between the wells, but is negligible in our system.

In order to build up the intuition for an array of Josephson junctions, we begin
with a brief look at a condensate loaded into a double-well potential, see Fig. 1.7. If we
restrict the dynamics of the system to two degrees of freedom, the relative number ∆Nof particles in the two wells and the relative phase ∆φ between the two condensates,
then the eﬀective Hamiltonian reads [99]
�
�
�
1
∆N 2
H = Ec ∆N 2 − J 1 −cos(∆φ)8
Nwell(1.39)

where Nwell is the average population per site. Ec , in analogy to superconducting
systems, is called the charging energy. In our neutral system, it is caused by the typically
repulsive contact interactions between condensate atoms. J is the Josephson (collective

26
tunneling) energy. We note here that the inclusion of additional degrees of freedom
beyond the relative number and phase, such as spatial phase variations inside the wells,
complicates the picture. (1.39) is therefore valid for small Josephson junctions, where
the dynamics of condensates inside respective wells is frozen out, as it would occur at
higher energies than the particle / phase dynamics.
Quantum fluctuations. It may be immediately inferred from (1.39) that thetwo energy terms lead to incompatible ground states: The charging energy alone disfavors population imbalances, and if dominant leads to fixed equal numbers of condensed
atoms on both sites. As a consequence, the quantum mechanical uncertainty between
relative number and relative phase, ∆N · ∆φ > 1/2 renders the relative phase between
the two sites completely uncertain. The Josephson coupling term on its own, on theother hand, favors phase coherence ∆φ = 0 between the two states. This in turn isincompatible with well-defined particle numbers on each site. The competition between
the two terms implies a crossover as a function of Ec /J. Quantitatively, one finds thefollowing quantum fluctuations of the relative number and phase:1∆φ2q =2�2 )/J(Ec + J/Nwell

1
∆N =
2
2

�

2 )
J/(Ec + J/Nwell(1.40)The transition between a phase coherent and a dephased system occurs roughly when
∆φ2q → 1/2, i.e. in the case of Nwell � 1 when Ec ≈ J.
Following Leggett [99], not only two but three diﬀerent regimes of zero-temperature
Josephson junction arrays have to be distinguished, based on the two energy scales,Ecand J, and the number of condensed particles per well Nwell :−2
Rabi regime. Here Ec /J << Nwell, i.e. the charging energy is unimportant, both forthe dynamics and the phase fluctuations (∆φ2q = 1/Nwell ).
−2
Josephson regime. Here Nwell<< Ec /J << 1, i.e. the charging energy speeds up

the dynamics of particle exchange (see “Dynamics” below), and begins to dominate the�quantum phase fluctuations (∆φ2q = Ec /J), which however remain small in absolute

27
magnitude.
Fock regime. Here 1 << Ec /J, i.e. finally Ec is strong enough to localize most atoms(∆N → 0) and quantum fluctuations dephase the system (∆φq > 1).
Thermal fluctuations. The experiments described in this thesis are perfomedin the Josephson regime, but with the additional ingredient of a finite temperature T .
The topic of study is a competition between J and T , with Ec unimportant. The eﬀects
of finite temperature lead to thermal phase fluctuations, which scale as∆φth �

�
T /J(1.41)

as long as T < J. The thermal fluctuations (1.41) of the double well system are shownin Fig. 1.8. J and Ec have been calculated numerically, see Appendix B, under theconditions relevant to our studies of the BKT transition in Chapter 3. It turns outthat the condition under which thermal fluctuations become important in a double well
BEC, i.e. J ≈ T , remains the same in an array of wells, allowing a rough estimateof the experimental conditions under which the BKT transition is expected in a 2D
Josephson junction array. Experiments on finite-T double well systems were pioneeredin Oberthaler’s group [60].
Dynamics. The dynamics of the double well system, for small displacementsfrom equilibrium ∆N = 0 and ∆φ = 0, is that of a harmonic oscillator, exchanging
energy between the phase and number degree of freedom, at the so-called Josephson
plasma frequency (with J and Ec measured in rad/s)
ωp =

�
�
2 )≈
J(Ec + J/NwellJEc(1.42)

with the approximation valid in the Josephson regime. The Josephson (particle) current
is given by J × sin(∆φ). Experiments have recently confirmed this oscillation [61]. For
larger displacements, the nonlinearities in the Hamiltonian (1.39) create more complex
behavior, such as population self-trapping in one well.

28

4000

Energy [Hz]

3500
3000
2500
VOL

2000
1500

!

1000
500 split
BKT

Energy [Hz]

104

60nK
30nK
100
MI

J
1
Ec

"#RMS [Rad]random
1.5
"#th
1

"#th
"#q

0.5
0

0

1000

2000

3000

4000

VOL [Hz]
Figure 1.8: Dependence of relevant parameters of the double well BEC on barrier height
VOL , under conditions typical for the experiments in Chapter 3. Upper panel: the chemical potential µ increases rather weakly with VOL , caused by increasing compression of
the BEC into the bottom of the well. As a result, already a weak ∼ 300 Hz barrier
physically splits the BEC into two. As soon as the BEC is split, the Josephson coupling
J is well defined and decreases approximately exponentially with Barrier height (central
panel). Thermal fluctuations ∆φth grow when J ≈ T , for realistic temperatures aroundVOL ≈ 1300 Hz – this is where the BKT transition is expected for a 2D Josephson junction array. Quantum fluctuations ∆φq become important only when J ≈ Ec , i.e. whenthermal fluctuations have long dephased the system. Hence the Mott-insulator (MI)transition is not observed. J and Ec have been calculated numerically, see Appendix B.

29
1.5.3

Arrays of Josephson-coupled BECs

In analogy to the double-well system, the Hamiltonian for a D-dimensional Josephson junction array (JJA) is given by�1 �
H = Ec∆Nj2 − J
8
j

j

�

1−

�

∆Nj
Nwell

�2

cos(∆φj )(1.43)

where j represents a D-dimensional sum over all Josephson junctions. Several newfeatures arise in an array compared to a single junction:(i) Vortices on a JJA. For a two-dimensional array, topological defects suchas vortices, can occur. A vortex in a JJA is defined by the phase diﬀerences it induces
along a closed loop of array sites, see Fig. 1.9. Vortex cores “live” in between lattice

Phase [radians]

sites – individual condensates are too small to sustain vortices inside them.

Figure 1.9: Vortices in a Josephson junction array. Condensate phases on array sites(circles) are color-coded. Left: single vortex, right: vortex-antivortex pair. The direction of Jospehson current flow is indicated by arrows.

(ii) Size scaling. In an extended 2D JJA one can expect sharper variations ofquantities such as phase fluctuations e.g. with J/T , compared to the behavior shownin Fig. 1.8 for a single Josephson junction.
(iii) Phonon modes of the JJA. The Josephson plasma frequency Eq. (1.42),
i.e. the single frequency at which particle exchange occurs in a double-well system, isreplaced by a whole spectrum of phonon modes. The phonon modes of a one-dimensional
JJA were described in Refs. [87, 35]. The spectrum resembles that of acoustic phonons

30(lattice vibrations) in a 1D crystal lattice of period d:
�
�
�
2J
2
2
ωq = 2J sin (qd/2) Ec + 2 sin (qd/2) .
Nwell(1.44)

Here, q = 2π/λphonon is the wavevector of the excitation, in a finite system of size 2R,πq ≈ i 2R
. The maximum frequency ωmax at the edge of the Brillouin zone (q = π/d)

is related to the Josephson plasma frequency, but with the replacement J → 2J, i.e.
�2 ). In the Josephson regime, the long-wavelength spectrum
ωmax = 2J(Ec + 2J/Nwell√is linear, ωq → 2JEc qd
2 . The lowest frequency mode in a finite system is ωmin =√πd
2JEc 2R, i.e. it can be estimated from the double-well plasma frequency (1.42) andthe number of lattice sites across the array R/d : ωmin ≈ ωp × d/R. The timescalerequired for adiabaticity with respect to these modes, e.g. when ramping up the opticallattice, can be very long, an aspect to be kept in mind in experiments.
(iv) 2D XY model. In the regime relevant to our experiments, the Hamiltonian(1.43) approximates the so-called XY Hamiltonian,
Hxy = −J�cos(∆φj )(1.45)

jwhich describes a system of spins constrained to rotate in the plane of the lattice (∆φjis the diﬀerence in spin orientation across junction j). This system was shown toundergo a BKT transition in the seminal paper by Kosterlitz and Thouless [94]. Insuch a “Josephson network”, it is not de Broglie wave overlap, but the strength of theJosephson coupling between sites, which decides on the development of (at least local)
phase coherence. In Chapter 3 I will describe our experiments on the BKT transition
in a 2D Josephson junction array, which occurs under condition
Tc ≈ J(1.46)(v) Frustrated 2D JJA. A highly fascinating topic is the application of an
external field, or a rotation of the array. This induces a vector-potential term Aj in the31relative phase, which renders the problem highly nontrivial [124, 25].
Hxy = −J�jcos(∆φj − Aj )Aj measures the integrated vector potential across a junction: Aj =(1.47)
��

junction j A(x) ·�dx/(2π).
We introduce the frustration index f , which measures the number of flux
quanta associated with the vector potential, per plaquette of the optical lattice 3���
f = A(x)· dx/(2π�)(1.48)
the loop integral is around one plaquette of the array. In the case of rotation-induced
� = mΩ
� × �r (see Appendix D), and f is identical to the number of vortices per
flux, Aplaquette, e.g. for a square JJA with the density of lattice sites nOL = 1/d2
f ≡ nv /nOL = 2mΩd2 /h(1.49)√
while for a triangular JJA with nOL = 1/(d2 3/4) 4√
f = mΩd2 3/(2h)(1.50)

with the areal density of vortices nv given by Eq. (1.22). Around rational values of f ,commensurate-incommensurate transitions occur between regular and irregular arrangements of vortices on the array [88], see Fig. 1.10. At finite temperature, these vortex
configurations undergo BKT transitions to a normal state, with an especially interesting
situation given for f = 1/2 [124] where the interplay between a BKT transition and an
Ising-type transition is neither theoretically nor experimentally fully understood. Withfurther increased rotation rate and at low temperature, fractional quantum Hall states
are expected to form, in analogy to the bulk rotating BEC. Experiments on a deep
rotating 2D optical lattice were only touched upon in this thesis, but may well form the
topic of another thesis to follow.
3
The frustration index is identical to the “filling factor” used in the literature relevant to vortex
pinning. We do not use this terminology here to avoid confusion with the filling factor used in thecontext of vortex lattice melting.
4
In a triangular array, the triangular “plaquette” on which f is calculated according to (1.48) is half
the conventional unit cell!

32

T/J
Normal state

rm
o
N

te
ta
s
al

~1
[146]

u
rf l
e
p
Su

id

static
BKT
FFJJA

[164]

f

C/IC

FQH
~1

1/!

~1
Ec/J

Mott insulator

Figure 1.10: Schematic phase diagram of a Josephson junction array. The vertical axis
is temperature, measured in units of the Josephson coupling J. Rotation rate increases
along the axis to the right: f = nv /nOL is the frustration index, Eq. (1.49), andν = nat /nv is the filling factor (1.25), i.e. the number of atoms per vortex. f and
1/ν are related by f = Nwell /ν where Nwell = nat /nOL is the lattice site occupation.
The Ec /J axis describes increasing quantum phase fluctuations and decreasing number
fluctuations, which drive the Mott insulator phase transition. C/IC: commensurateincommensurate phase transitions of the vortex lattice inside the JJA; FF-JJA: fully
frustrated JJA at f = 1/2, Ref. [124]; FQH: strongly correlated quantum Hall states
induced by rotation [25]; BKT: finite-T normal phase via Berezinskii-Kosterlitz-Thoulesstransition of the frustrated JJA, Ref. [160]. Mott-insulator – superfluid boundary for
rotating lattices: Ref. [114]. [164, 146]: Experiments in this thesis.

Josephson junction parameters. The important energy scale underlying mostparameters of optical lattice experiments, is the recoil energy
Er =�2 π 22md2(1.51)

where d is the lattice period. Both J and Ec are governed by it, and in addition it iscustomary to quote optical lattice depths in units of this energy scale.

33
In the limit where interparticle interactions are negligible compared to the energy
separation of Bloch bands at zero momentum, which is always larger than 4 Er , theJosephson energy is given by J = Nwell × t, with the single-particle tunneling energy t
given e.g. by Zwerger [170]�4t = √ Er (VOL /Er )3/4 exp(−2 VOL /Er )π(1.52)

As soon as interparticle interactions are not negligible compared to the recoil energy,
J must be found from a numerical solution of the GPE, Eq. (1.9). Our procedure isdetailed in Appendix B. The physical reason for the modification of J by interactions is
the interaction-induced increase of the BEC’s chemical potential, which in turn reduces
the tunneling barrier to neighboring lattice sites. An order-of-magnitude estimate for J
is possible by replacing VOL → (VOL − µlat ) in the exponent of Eq. (1.52), with µlat thechemical potential of a BEC loaded into an optical lattice, given by Eq. (1.54) below.
The interaction energy of Nwell particles on one lattice site is conventionally
written as U2 × Nwell (Nwell − 1), with U related to the charging energy Ec by Ec = 2U .
The charging energy depends on the strength of interparticle interactions, because arepulsively interacting BEC grows in size. Ec can be estimated as [122]Ec = 2∂µlat
4 µlat≈∂Nwell
5 Nwell(1.53)

with the chemical potential µlat given by Eq. (1.54) below.
1.5.4

Thomas-Fermi description of BECs in optical lattices

Stringari and coworkers [126] devised a useful concept for experimental determination of the parameters of a BEC loaded into an optical lattice. The central observation,from an experimental point of view, is that in-situ images of BEC in most cases do
not resolve individual lattice sites – instead a coarse-grained envelope density profile isobserved. Stringari and coworkers show that this coarse-grained profile in many cases

34
follows a renormalized Thomas-Fermi (TF) description. The renormalized chemical potential µlat depends only on the total number of condensed particles N in the array, andother known or measurable quantities 5 :���D �2/5
�ω̄ 15N ascd√µlat =
2ā
2πσ(1.54)

here, D is the dimensionality of the lattice and d is its period 6 , σ is the width of�an individual condensate on a lattice site, and ā = �/mω̄ and ω̄ = (ωx ωy ωz )1/3 are

the oscillator lengths and frequencies of the envelope potential formed by optical latticebeam envelope and magnetic trap. As long as interparticle interactions are insignificant,σ = σ0 ≡ π(Vis governed by the optical lattice depth and period alone, while�
2µlat
repulsive interactions further increase the width σ ≈ σ0 × �ω. Here ωlat is the
lat
d

1/4
OL /Er )lattice-site oscillator frequency. When including interparticle interactions in σ, Eq.
(1.54) becomes an implicit equation that can be solved for µlat .

Eq. (1.54) most importantly allows the determination of the overall BEC number�
2µlat
N by simply measuring the coarse-grained BEC radii Ri =, as shown in Fig.
mω 2
i

1.11. The important number of particles in the central lattice site, Nwell , then follows
�from N = i N(site i) where the number of particles N(site i) on each site follows froman integral of the coarse-grained TF density over the site volume Vsite : N(site i) /Nwell =
�
1
3
2
2
2
Vsite site i d x(1 − (x/Rx ) − (y/Ry ) − (z/Rz ) ). This results, e.g. in a 1D lattice inthe z-direction, in

Nwell =

15 d
×N
16 Rz

(1.55)or in a 2D hexagonal lattice in the xy plane,√
d2 3/2
Nwell = 2×N
πRρ2(1.56)

where Rρ is the envelope TF radius in the xy plane.
5Note that µlat in Eq. (1.54) is measured with respect to the zero-point energy of the trap potential,
as is Eq. (1.11) above. While usually this zero-point energy is negligible compared to the chemical
potential, this is no longer true in strong optical lattices.
6
VOL and d are assumed identical in all lattice dimensions but (1.54) can be generalized.

35" = VOLchemical potential
gn = 1/2 m!2zR2z [Hz]

600
500
400
300
200
100
0
0

200 400 600 800 1000120014001600

VOL [Hz]
Figure 1.11: Chemical potential µlat ≡ gn (symbols) measured from axial ThomasFermi radii Rz of BECs loaded into a hexagonal 2D optical lattice (period d = 4.7 µm)
in the xy plane, showing good agreement with eﬀective Thomas-Fermi formula (solidline) Eq. (1.54). The condensate number extracted from the data was N = 520000, i.e.
Nwell = 7000 using Eq. (1.56). This agrees with an independent analysis of the same
experimental data based on numerical simulations described in Appendix B. Dashed:
optical lattice depth VOL . For VOL = µ (at 200 − 300 Hz) the bulk BEC is split into a
Josephson junction array.

We note here that for our experiments on the BKT crossover in Chapter 3, where
precise knowledge of Nwell is crucial to determine J, we do not rely on this approximate
formalism, but employ numerical simulations of the Gross-Pitaevskii equation to provide
a relation between measured cloud radii and Nwell .

1.6

BEC imaging
For a detailed quantitative analysis of imaging, refer to the excellent chapters in

Haljan’s [74] and Matthews’ [105] theses. In this thesis we exploit two types of images,
both of which measure the density of atoms, integrated along the probe laser propagation
axis. Nondestructive sideview images (or sequences thereof) of BECs in trap provide allquantitative condensate parameters. An example is shown in Fig. 1.12.
Top view absorption images of the same condensate after expansion usually contain the “science” information. A note of caution is in order at this point: In most other
BEC experiments, expansion is used as a means to study the momentum distribution.

36

a)

d)

b)

c)

e)

f)

Figure 1.12: (a) Nondestructive image of a finite T BEC, (b) fit and (c) residuals. The
images show the density of atoms, integrated along the line of sight. The image area is
230×160 µm. The BEC actually resides in an optical lattice and is split into a Josephson
junction array. Individual lattice sites remain unresolved, and the image is analyzed
according to the eﬀective Thomas-Fermi description by Stringari and coworkers, seesection 1.5.4. The thermal cloud fit follows the description in Section 1.2.4. Horizontal
cut through (d) full density profile and fit, (e) extracted BEC profile, (f) extracted
thermal cloud profile - the dip in the center is caused by mean-field repulsion from the
BEC density.

In our system, on the other hand, expansion is employed as a means of magnifying
spatial information beyond the imaging resolution. This works for two reasons: (a) we
employ a “forced” (or “anti-trapped”) expansion on an inverted parabola potential thatmerely “stretches the fabric of the universe” [46], in contrast to the conventional “free”
expansion, which converts potential and interaction energy to kinetic energy. Our expansion is in addition designed to be almost exclusively in the two spatial dimensions of
the image plane. (b) we are often interested in topological defects such as vortices, andthese largely preserve their structure upon two-dimensional expansion. This techniqueallows a great detail of spatial information to survive the expansion – almost all BEC
images shown in this thesis are of this type. Possible systematic eﬀects of dynamics
during expansion nonetheless have to be kept in mind for quantitative analysis and37should be considered every time new quantities are sought to be retrieved from images.

1.7

History and organization of this thesis
Rotating BECs had been studied in our lab since 1999 [74]. Shortly after myarrival in the lab in 2002, we discovered how to reach the “rapidly rotating” regime[56, 44, 143] described in Ian Coddington’s thesis [46] . After finishing our fundamental
studies on this new system [45, 150, 57], we immediately started introducing various
optical lattice potentials [145, 164, 146] into the system, interrupted only at times by“side projects” [144, 79] when oﬄine developments were needed to make progress. This
thesis therefore could alternatively be entitled “bringing optical lattice potentials to theCornell labs”.
It consists of four essentially independent experiments, which are organized asfollows. Chapter 2 details results of experiments on vortex pinning in rotating BECs
loaded into a shallow (VOL /µ < 1) co-rotating two-dimensional optical lattice potential.
Chapter 3 presents our observation of the Berezinskii-Kosterlitz-Thouless crossover ina finite-T two-dimensional array of Josephson-coupled BECs. Chapter 4 summarizes
experiments on loading a rapidly rotating BEC into a one-dimensional optical latticewith its axis aligned along the rotation axis. In this way a stack of two-dimensional
rapidly rotating BECs with low filling factor is created, a possible first step toward
fractional quantum Hall states in rotating BECs. The extended Appendix D provides
additional background information relevant to Chapter 4. Chapter 5 reports on the firststudy of vortex lattices in a rotating two-component BEC, where self-organization of
two interlaced square vortex lattices in the two components is observed.
The underlying “BEC factory” has remained essentially unchanged since it has
been described in previous theses [74, 46]. Exceptions include the introduction of amore reliable repump laser system as well as slight changes to the setup for microwave
excitation. The major experimental additions to the machine all have to do with (a)38setting up laser sources for various optical lattice experiments and (b) the design andimplementation of various extremely stable optical lattice beam paths, each tailored forthe envisioned experiment. These setups are detailed in the respective chapters.

Chapter 2
Vortex pinning in rotating Bose-Einstein condensates

In this chapter we report the first observation of vortex pinning in rotating gaseous
Bose-Einstein condensates (BEC) [1]. A co-rotating two-dimensional optical lattice issuperimposed on the rotating BEC. The density of the superfluid is suppressed at theantinodes of the two-dimensional standing wave pattern of the optical lattice. These
antinodes therefore form an array of columnar pinning sites, regions of low potential
energy, for the superfluid vortices, as has been shown in Figure 1.3. We study the eﬀects
of two optical lattice structures, triangular and square. In both geometries we see anorientation locking between the vortex and the optical lattices. At suﬃcient intensity
the square optical lattice induces a structural cross-over in the vortex lattice.

2.1

Motivation
Some of the most appealing results from recent work in superfluid gases have had

to do with lattices, either optical lattices [67, 93, 159] or vortex lattices [102, 143, 5].
As distinct as these two periodic structures may be, there are reasons for trying tomarry them in the same experiment. For one thing, the extreme limits of rapid rotation(in the case of vortex lattices) [47] and deep potentials (in the case of optical lattices)[85] both lead to the same thing: correlated many-body states. For another, there
is considerable precedent, from various subdisciplines of physics, for interesting eﬀects
arising from the interplay between competing lattices [158, 15, 66]. Moreover, the40pinning of superconducting flux vortices to an array of pinning sites in solids is an areaof very active research [13, 69, 167, 129]. In superconductors, unpinned vortices are asource of dissipation, so pinning as a way of freezing vortex degrees of freedom is ofboth fundamental and technical interest. Here we are able to study this eﬀect in a clean
system with well-controlled pinning potential and vortex-vortex interaction, and with
time-dependent control over pinning potential strength and rotation rate.
With these considerations in mind, we undertook a preliminary experimental
study of the eﬀects of a rotating optical lattice on a vortex lattice in a Bose-condensed
sample of 87 Rb.
Pinning of vortex lattices. When a periodic pinning potential is superimposedon a rotating BEC, an ordered pinned structure, such as shown in Fig. 2.1, may be
expected for certain commensurate ratios r ≡ p/q of vortices p to pinning sites q 1 . For
r = 1, the vortex lattice can be pinned one-to-one to the pinning lattice. For rationalr < 1, vortices will form superlattice structures locked to the pinning lattice, whereas
for r > 1, some pinning sites contain more than one vortex. In the limit of strong
pinning, these may merge to form multiply quantized vortices.

Figure 2.1: A vortex lattice (left) rotating at a rate Ωf luid is exposed to a square opticallattice rotating at ΩOL . Once the pinning potential strength overcomes vortex-vortex
interactions (right), vortices are pinned to the antinodes (bright) of the optical lattice.

For any pinned vortex structure other than a hexagonal lattice of vortices, thepinning potential has to overcome the diﬀerence in vortex-vortex interaction between
1

I do not use the frustration index f here, which is defined as number of vortices per plaquette of
the optical lattice (Eq. (1.49)): For a triangular lattice there are two triangular plaquettes per latticeunit cell, but only one pinning site per unit cell – so r = 2f . For a square lattice, r = f .

41
the pinned structure and the unpinned hexagonal vortex lattice. Consider the pinninginduced transition to a square vortex lattice: In the absence of a pinning potential, theinteraction energy of a square vortex lattice is calculated to exceed that of a triangularlattice by less than 1 % [36], thus it is predicted [130, 131, 128] that the influence of evena relatively weak square optical lattice will be suﬃcient to induce a structural transitionin the vortex lattice. Prior to our experiments, numerical and analytical work existed forsome specific situations, regarding the phase transitions between pinned and unpinned
vortex phases [130, 131, 128]. Also [168, 25] discuss similar systems in the stronglyinteracting regime. Following the publication of our results, numerical simulations by
Kasamatsu and Tsubota [89] have qualitatively confirmed our results on the structural
crossover from hexagonal to square vortex lattice, and have added more informationon the dynamical behavior of vortices under the influence of a weak pinning potential.
Angular locking of the hexagonal vortex lattice to a hexagonal optical lattice was studied
numerically by Sato, Ishiyama and Nikuni [140]. The topic of rotating lattices has
become a very active field of ultracold atom theory. Several articles discussing related
issues have been published, e.g. [34], [114], [166].
Away from commensurate lattices: defects and vortex glass. Anotherquestion concerns the properties of vortices and vacancies at incommensurate frustrationindex, i.e. when the number of vortices is not a rational fraction of the number of pinningsites. Under certain conditions one expects that vortices could become frozen into adisordered but stationary phase known as a “vortex glass”. The statistical properties
and the finite-temperature behavior of this phase could be a very exciting subject ofstudy.

2.2

Setup, alignment and diagnostics
There are various possible ways of creating a rotating optical lattice potential.

Possibilities include a scheme based on crossed electro-optical modulators [37], holo-

Fiber output

(a)

42
(c)

(b)

!%

!"

90 O

Fghj

Lens

d1

Fghj

120

O

d2

O

90

120 O

Mask

$OL

(d)

(e)

Lens

$BEC

100#m

Figure 2.2: (a) Schematic diagram of our setup for the rotating 2D optical lattice.
Layouts of the masks for a triangular (b) and square (c) optical lattices. (d) and (e) are
pictures of triangular and square optical lattices, respectively.

graphic masks, or simply interference patterns created by a rotating shadow mask placed
placed in a large, cylindrically symmetric laser beam. We employed this last technique.
Making this experiment work was actually a quite laborious endeavor. For thepinning sites to appear static in the frame of a rotating BEC, the rotation of opticallattice and vortex lattice must be concentric, and such a quietly rotating optical latticepattern with few micron period is not exactly what you get when you try the firsttime. Mechanical instabilities and optical aberrations (which lead to epicyclic motionof the pinning sites) had to be canceled meticulously, as suggested in Fig. 2.3. Even
after optimization, residual undesired motion was such that the strength of the opticallattice had to be kept at less than 30 − 50 % of the condensate’s chemical potential or
unacceptable heating resulted over the experiment duration of tens of seconds. The
experiments in the following sections were thus performed in the weak pinning regime.
A conceptual view of the setup for creating a rotating optical lattice is shown in
Fig. 2.2(a). A mask with a set of holes is mounted onto a motor-driven rotary stage with

43
a 2” clear aperture 2 , and a laser beam (532 nm) emerging from a single-mode fiber
is expanded, collimated, and passed through the mask. After the mask the resulting
three beams are focused onto the BEC. The interference pattern at the focus constructs
a 2D optical lattice. The geometry and spatial extent of the triangular or the square
optical lattice are determined by the size and layout of the holes and the focal lengthof the second lens. We create a hexagonal optical lattice using three holes at 120◦ , ora square OL using three holes at 90◦ . The intuitively expected fourth hole is left out,
for the reason that three interfering laser beams create a stable interference structure,
while for four beams the topology of the lattice depends on the relative phases of thelaser beams [121]. The only consequence of omitting the fourth beam is that each site
is of elliptical shape, with the major axis at 45◦ with respect to the lattice axes.

OPD

!x"d
=OPD/#

Figure 2.3: Fluctuations in the optical path length diﬀerence (OPD) between lattice
beams lead to lattice displacements ∆x, which ultimately lead to heating. d: opticallattice period; λ: laser wavelength. Such fluctuations are easily introduced for a rotating
lattice by lens aberrations, optical imperfections, or mechanical vibrations.

Feasibility estimate. A first feasibility estimate for creating a smoothly rotating optical lattice is obtained by a ray tracing analysis of the experimental setup of
Fig. 2.2(a), taking into account expected imperfections in alignment.
Optical path length diﬀerence (OPD) curves provide a well suited measure of theeﬀect of aberrations on a rotating optical lattice: in a lattice of period d, a time-varying
optical path length diﬀerence between the beams leads to a time-varying displacement∆x of the lattice, given by OP D/λ = ∆x/d, as shown in Fig. 2.3. Only azimuthally
2

Aerotech ADRT 150 direct rotary stage.

44

A)

Fy
1

1

Fiber
outputOptical path length difference [!]B)Fx

1st 2ndlens lens

PBS

BEC

C)

D)

E)

Figure 2.4: Analysis of aberrations for the rotating 2D optical lattice used for thepinning experiments. A) Fans of rays in sagittal and paraxial plane. Coordinates Fx(in sagittal plane), Fy (in tangential plane) range from 0 to +1 from center to edge ofthis fan of rays. Optical path length diﬀerences are calculated in the image plane where
the optical lattice is formed. In the sagittal plane, by symmetry, only the range 0 to
+1 is plotted. B) Perfectly aligned system (the glass block (material: SF6) symbolizes
a 2” polarizing beamsplitter cube), showing only residual spherical aberrations, seen as
identical wavefront distortions in sagittal and paraxial plane. C) tilting the first lensby 5◦ induces astigmatism, seen as opposite curvatures in sagittal and paraxial plane.
Here the image plane is chosen between sagittal and paraxial waist - wavefronts still
converge in Fy, but already diverge in Fx. D) decentering the first lens by 5 mm. E)
tilting the polarizing beamsplitter cube by 5◦ .

45dependent aberrations such as astigmatism constitute a problem. Spherical aberration
is not a problem, as ideally the 3 beams trace spherically symmetric orbits. However,oﬀ center lenses convert spherical aberrations into azimuthally dependent aberrations.
In Fig. 2.4 examples of alignment imperfections relevant to the experimental setup andthe resulting aberrations (OPD curves) are shown 3 .
Table 2.1: Estimated error budget from ray-tracing analysis of aberrations in a rotating
2D optical lattice. In a lattice of period d, an optical path length diﬀerence (OPD)
leads to a displacement ∆x given by OP D/λ = ∆x/d. Such OPD may be introducedby the indicated misalignments of optical elements, either in the optical lattice beam
path between light source and BEC, or in the imaging beam path between BEC and
CCD camera where lattice stability is diagnosed. Note that the actual performance inthe experiment was found to be significantly better than 0.32λ.
Element
Optical Lattice Setup
first lens
first lens
shadow mask
second lenssecond lenscube
OPDRM S at BEC due to optical lattice setup
Imaging Setup
first imaging lens
first imaging lenssecond imaging lenssecond imaging lens
OPDRM S due to imaging setup
OPDRM S at camera, due to OL and imaging

Error

AmountOP D [λ]tiltdecenterdecentertilt
decenter
tilt

0.2◦
1 mm
2 mm
0.25◦
1 mm
1.5◦

0.1
0.07
0.05
0.15
0.03
0.1
0.225

tilt
decentertilt
decenter

3◦
2.5 mm0.2
0.1
negligible
negligible
0.225
0.32

Table 2.2 shows the expected error budget on vibrations of a 7 µm period rotating
2D optical lattice. The cumulative displacement ∆x due to aberrations is less than
one lattice period, showing the feasibility of such a setup. Of additional concern werepossible optical imperfections of the glass cell windows, as well as the possibility ofaberrations from imperfections of the fiber output facet, but experimentally we later
3

Simulations performed using the software Oslo LT.

46
found no indications of such eﬀects.
Detailed description of the actual setup. Fig. 2.5 (a) shows the light source
we set up for creating both blue-detuned (532 nm) and red-detuned (810 − 850 nm)
optical lattices. Output from a 10W Coherent Verdi is either directly coupled intoa single-mode fiber, or is used to pump a Tekhnoscan TIS-SF-077 Ti:Sapphire laser
4 .

A temperature-stabilized reference cavity serves as a frequency reference for theTi:Sapphire laser, while the Verdi is free-running 5 . In both cases, acousto-opticalmodulators that control the optical lattice intensity are positioned before the fiber.
This setup delivers a fiber output of 600 mW at 532 nm, limited by thermal eﬀects inthe fiber, or 550 mW at 810 nm, limited by the Ti:Sapphire laser output.
The rotating optical lattice setup is shown in Fig. 2.5 (b). The light exits froma fiber of N A = 0.12 and is collimated using a 2” achromatic doublet (f = 300 mm).
A shadow mask selects three beams on a cone of N A ≈ 0.04 6 . We use 2” coatedachromatic doublet lenses and protected silver mirrors everywhere in the beam path.
A f = 250 mm lens focuses the beams onto the BEC. A broadband 2” polarizing
beamsplitter cube (Karl Lambrecht) is used to combine the 532 nm lattice light with
780 nm MOT and imaging light. Following the glass cell, the light is collected on
a CCD camera through our ∼ 6× magnification vertical imaging setup (the dichroic
mirror before the CCD reflects > 99% of the optical lattice light; the remaining lightlevel is safe for the CCD and provides good signal/noise).
The optical lattice intensity is stabilized to the desired intensity waveform using
feedback from a photodiode monitoring a peripheral portion of light exiting the fiber.
A second photodiode monitors the actual optical lattice intensity.
Alignment procedure and diagnostics. To achieve an alignment of beams
4

The Ti:Sapph in addition allows for 720 − 810 nm operation using a second set of cavity mirrors.
The Verdi linewidth, measured on a timescale of 0.1 ms, is 0.02 M Hz. It drifts by 0.5 M Hz within
1 ms and on order of 10 M Hz over tens of seconds [142].
6
Triangular optical lattice, period d = 7.8 µm: φ1 = 2.5 mm, d1 = 11.5 mm (Fig. 2.2(b)). Squarelattice, period d = 7.0 µm: φ2 = 2.5 mm and d2 = 13.5 mm (Fig. 2.2(c)).
547a) Light source for red & blue detuned optical lattice

Coastal Connections
S-FAFU-3/125/20-5-H

550mW
power after fiber

large
aperture
AOM

NEOS
45100-5.-6.5DEG-5.1

1st order

shuttermagnifying telescope 1:2

600mW
Coastal Connections
S-FAFU-5/125/2-7-HAOM

1st order

shutter

Referencecavitytelescope f=250: f=200
!/2

Verdi
532 nm
10W to pump Ti:Sapph
2W for green optical lattice

removable
mirror30dB
isolator

Ti:Sapph 1W
810.1 nmb) Rotating optical lattice setup
MOT,
Imaging
780nm

Pinhole (looking at
single lattice site)

glass
cell

flip mirror

2” PBS

MOT
30x
objective

BEC

2” f=250

1.5” f=120

2” f=700

CCD

2” !/4
Shadow mask
on motor

removable
mirror2” f=300

Picomotors

Optical lattice
532nm

Dichroic
mirror
R532,T780

MonitorPD

Locking PD

Figure 2.5: a) Laser source for generating red- and blue-detuned optical lattices. Lightat 532 nm is either directly fiber-coupled to form a blue-detuned optical lattice, or isused to pump a Ti:Sapphire laser for a red-detuned optical lattice. b) Setup for rotating
2D optical lattice used for pinning experiments and 2D frustrated Josephson junction
arrays. A single-mode fiber supplies optical lattice light at 532 nm, which passes througha shadow mask mounted onto a motor-driven rotary stage, to define three rotating lattice
beams. These are combined with the MOT/imaging beam path and overlapped on the
BEC to form the rotating 2D lattice. The beams are imaged onto the CCD camera used
to optimize beam overlap, rotating lattice stability, and alignment with respect to the
BEC. Important alignment degrees of freedom are indicated. PD: photodiode.

48
and lenses of the accuracy given in Table 2.2, we begin by perfect centering of the lattice
beams onto the lenses, and orthogonal alignment of beams onto lenses and other opticalelements (waveplates, polarizing beam splitter cube). For this purpose both lenses in
the optical lattice beam path are mounted on 5-axis aligners. We use a narrow, central
beam created by inserting an aperture into the beam center (the beam center is foundby maximizing the transmitted light using a photodiode). We first align the reflections
from the first lens’ back and front surfaces back into this aperture. The central beamis then aligned with the center of the shadow mask, and the orientation of the mask
is made perpendicular to the beam using the reflection from an auxiliary mirror. The
second lens is aligned as the first one, using a second aperture.
The optical lattice light is then imaged onto a CCD camera using our ‘vertical’imaging system consisting of f = 120 mm and f = 700 mm achromatic doublet lenses.
Here we do not have as good control over the lens alignment, as these lenses do nothave tilt degrees of freedom. This is reflected in larger wavefront errors in Table 2.2.
lattice period = 7.2 !m6"x = 0.56 !m

2

5

0

4

-1

3

-2

"y = 0.47 !m

-3

2
1

-4

0

-5

-1

-6

0

20

40
60
80
time[seconds]

100

Y center [microns]

X center [microns]

1

-2
120

Figure 2.6: Rotating 2D optical lattice, 7.2 µm period. The central lattice site is marked
red. Position fluctuations of the central lattice site when rotating slowly (90 s rotation
period). Fluctuations are identical for a static lattice, possibly limited by fitting accuracy, camera pixelation, and vibrations of the camera mounting platform.

Using the CCD camera image of the optical lattice pattern we have two modesof diagnosis. A slowly rotating lattice allows real-time following of the optical lattice

49

a)

b)

c)

d)

e)

f)

Figure 2.7: (a)-(c) Long exposure images of rotating 2D optical lattice, 7.2 µm period.
(a) static, (b) rotation rate 1.3 Hz, (c) 8 Hz. In (b),(c) images are exposed for one lattice
revolution. The contrast of the central spot is 53%, for static and rotating lattices, forexposure times of 1 as well as 10 revolutions, indicating that neither aberrations nor
vibrations significantly deteriorate stability. (d)-(f) FFTs of (a)-(c).

motion while performing adjustments to lenses. An example of such analysis is shownin Fig. 2.6, where the two-dimensional position of the central interference maximum
is shown vs. time. This is the typical day-to-day mode for alignment. Alternatively,
long-time-exposure images of a fast-rotating lattice show the time-averaged contrast oflattice sites and thus allow assessment of the mean displacements during rotation, as
shown in Fig. 2.7. We consistently find the imperfections of optical lattice rotation
to be comparable at slow and fast rotation. We note that optical lattice imperfections
may be introduced either in the optical lattice beam path between the light source andthe BEC, or in the imaging beam path between BEC and CCD camera where latticestability is diagnosed, see Fig. 2.5. Thus a reliance solely on CCD camera diagnostics
may be dangerous, and we ultimately use the BEC itself as a diagnostic, as described
below.
It is worth noting that the initial step of perfect lens alignment never produces the50best smoothly rotating optical lattice pattern. We consistently have to tilt one lens by asignificant amount to achieve best performance. Such lens tilt introduces astigmatism,i.e. an optical path length diﬀerence varying as ∝ cos2 (φ), where φ is the azimuthal
angle on the lens, see Fig. 2.4.C). Transverse lens translation, i.e. “decentering” of thebeams, introduces path length diﬀerences ∝ cos(φ)), see Fig. 2.4.D). With these two
adjustments, and using the CCD camera as a precise monitor for optical lattice motions,
we are able to achieve a performance superior to the estimates in Table 2.2.
Alternatively, vibrations of the optical pattern may be quantified by imaging
the single central OL site onto a pinhole and monitoring the transmitted intensity
on a photodiode. This allows very quantitative measurements of position jitter, notlimited by CCD camera exposure time and repetition rate, but is not feasible for in-situ
monitoring of adjustments made to the beam path, as the position of central spot willchange and would have to be repositioned onto the pinhole after every adjustment.
As a last probe of lattice alignment we employ the atoms themselves. The consistently required significant tilt of one lens had lead us to doubt whether this was an
artifact, e.g. introduced on the beam path from BEC to CCD and thus not present at the
BEC position. This question was finally settled by performing both heating measurements on the atoms loaded into the optical lattice, and measurements of the minimum
optical lattice strength required to lock the vortex lattice to the optical lattice. The
results of both these measurements confirm that stability of the optical lattice patternon the CCD camera is the correct criterion, i.e. no significant lattice imperfections are
introduced in the beam path between the BEC and the CCD camera.
Having created a perfectly rotating optical lattice potential, as a last step it
remains to align the pattern perfectly centered onto the rotating BEC. A rough dailyalignment is performed using picomotors, taking CCD images of both the optical latticepattern and the BEC. A fine position alignment is then achieved using magnetic shimcoils to shift the position of the BEC. As a diagnostic, initially static BECs are spun up

51
using the rotating optical lattice pattern rotating at a constant frequency. We find that,when perfect alignment is achieved, the BEC can be spun up and come to rotational
equilibrium with the optical lattice at rotation rates as high as Ω/ωρ = 0.95. This iscomparable to what can be achieved by evaporative spin-up [73] starting from a rotating

1.0

250

0.9
0.8

200

0.7

150

!c

100

!BEC/"#

atoms remaining in BEC [a.u.]thermal cloud. The final condensate number is ≈ 2/3 of that of the initial static BEC.

0.6

!c

0.5
0.4
0.3
0.2

50

0.1

0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1!OL/"#

0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1!OL/"#

Figure 2.8: Spinning up a BEC using a rotating two-dimensional optical lattice. Ωc :commensurate rotation rate, given by Eq. (2.1) for this d = 8.4 µm period hexagonal
optical lattice. The BEC is spun up to rotation rates around Ω = 0.9ωρ and reaches
equilibrium with the optical lattice, without increase in number loss. Note that in otherruns we achieved Ω = 0.95ωρ , and that evaporatively pre-spun-up BECs can equilibratewith the rotating optical lattice up to Ω ≈ 0.97ωρ .
We tested the achievable precision of BEC – optical lattice center alignment by
sequences of phase contrast images of the BEC loaded into the optical lattice pattern.
Alignment was optimized by monitoring the amount of angular momentum imparted by
the rotating optical lattice onto an initially static BEC. From the images, the locationof the central optical lattice site as well as the BEC center were extracted, and found tocoincide within approximately 2 µm, i.e ≈ 1/4 of the optical lattice period, and ≈ 1/30of the condensate radius.
After the first-time alignment outlined above, the setup requires only a slightdaily tweak and is usually stable throughout the day.

52

2.3

Observation of vortex pinning in Bose-Einstein condensates
Here we describe the results of three experiments. We first study the angularlocking of the vortex lattice to a hexagonal optical lattice with one vortex per pinningsite. Secondly, the structural crossover of a vortex lattice pinned to a square opticallattice is examined. Finally we report preliminary eﬀorts of pinning to hexagonal opticallattices with two vortices per pinning site, and with one vortex per two pinning sites.
For all results presented here, the vortex lattice has had suﬃcient time to come torotational equilibrium with the optical lattice.
2.3.1

Angular locking of the vortex lattice to a hexagonal optical lattice

The experiments begin with condensates containing ∼ 3×106 87 Rb atoms, held in
the Zeeman state |F = 1, mf = −1� by an axial symmetric magnetic trap with trapping
frequencies {ωr , ωz } = 2π{8.5, 5.5} Hz. Before the optical lattice, rotating at angularfrequency ΩOL , is ramped on, the BEC is spun up [143] close to ΩOL . This leads, before
application of an optical lattice, to the formation of a near perfect triangular vortex
lattice with a random initial angular orientation in inertial space. Through dissipationa vortex lattice can come to equilibrium with an optical lattice, with their rotation
rates and angular orientations locked. In the absence of pinning sites, a vortex latticewith areal density of vortices nv will rotate at (approximately [45, 147]) Ω = ( �πm )nv , asdescribed in the introduction, see Eq. (1.22). This suggests that for an optical latticewith an areal density of pinning sites nOL 7 , locking between the two lattices will befacilitated if the optical lattice rotates at the commensurate rotation rate
Ωc ≡

�

�π
m

�

nOL

(2.1)

We measure the angular diﬀerence θOL − θV L between the orientation of the
optical and vortex lattice in reciprocal space (see Figs. 2.9(a),(b)). Fig. 2.9(c) shows
7

For a triangular optical lattice, nOL = d22√3 , while for a square lattice, nOL = d12 .

53θOL − θV L as a function of the pinning strength with an optical lattice rotation rate
ΩOL = 1.133 Ωc = 0.913 ωr . The strength of pinning is characterized by the ratio Upin /µ( µ is the chemical potential of the condensate 8 ), which gives the relative suppressionof the superfluid density at pinning sites. We can see the initially random angular
diﬀerence between the two lattices becomes smaller as the pinning strength Upin /µ
increases. For Upin /µ � 0.08 , the angular diﬀerences become very close to the locked
value. Figure 2.9(d) shows the phase diagram. The data points and error bars markthe minimum pinning strength (Upin /µ)min above which the lattices lock. We observe
two distinct regimes. First, for small rotation-rate mismatch ΩOL − Ωc , (Upin /µ)min
is rather independent of ΩOL − Ωc . Second, for rotation-rate mismatch beyond therange indicated by the dashed line in Fig. 2.9(d), angular orientation locking becomesvery diﬃcult for any Upin /µ in our experiment. Instead, an ordered vortex lattice withrandom overall angular orientation observed at low Upin /µ transforms into a disorderedvortex arrangement at high Upin /µ.
This box–like shape of the locked region in Upin − ΩOL space is worth considering.
Vortex motion in our system is governed by a balance of pinning force and Magnus force(Section 1.3.3). The pinning force arising from our sinusoidal optical lattice potential is−→F pin (x) ∝ ∇Epin (x) ∝ Upin /d(2.2)

where Upin and d are the pinning potential’s strength and period. The Magnus force,→−
acting on a vortex moving with velocity −v vortex in a superfluid with velocity →v f luid is
−
→
→
→
−
F mag (x) ∝ n(x) (−v vortex − −v f luid ) × →
κ

(2.3)

h
−
where →
κ = (m)ẑ, and n(x) is the superfluid density. With the pinning potential

applied, the superfluid density is modulated as n(x) ∝ µ − Upin (x). A pinned vortex
lattice is constrained to move at the angular velocity set by the pinning potential, giving
8

The chemical potential is determined from the axial Thomas-Fermi radius of the condensate obtained from in-trap images, and is proportional to the peak density of the condensate.

54

(a)

(b)
$OL

$VL

80
70

$OL-$VL (deg)

60
50

(c)

40
30
20

10
0.30
0
-10

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.90

0.95

Upin/!

0.25
-20
0.20

(d)

Upin/!

0.15

0.10

0.05

0.00
0.60

"c
0.65

0.70

0.75

0.80

0.85

"OL/#rFigure 2.9: (a) Triangular optical lattice (period 7.8 µm) and (b) vortex lattice in reciprocal space. Inset are the corresponding original real-space CCD-camera images. (c)The diﬀerence in orientation θOL − θV L versus the strength of pinning Upin /µ (the peakof the optical potential normalized by the condensate’s chemical potential) for the rotation rates ΩOL = 1.133 Ωc = 0.913 ωr . With increasing pinning strength, θOL − θV L
tends towards its locked value (the constant oﬀset to the values of θOL − θV L in (c) is
not physical, but is an artifact of our measurement procedure in this dataset – in independent experiments we have used in-trap images of vortex lattices to confirm that theabsolute value of θOL − θV L is zero for locked lattices). (d) Minimum pinning strength
needed for orientation locking between the two lattices as a function of the optical lattice rotation rate. Dashed: minimum and maximum rotation rates for which pinningwas observed. Solid red and dotted lines: see text.

55
→
−
→
−
→
−
v vortex (r) = −v OL (r) = Ω OL × →r . On the other hand the superfluid in solid-body−→→→
approximation rotates at a velocity −v f luid (r) = Ω f luid × −r , where Ωf luid = π�m nv θ̂ isset by the vortex density nv . When the optical lattice rotates at the commensurate
rotation rate the Magnus force vanishes, and pinning should result even for extremely
weak pinning strengths.
Comparing the magnitudes of both forces at r = R(Ωf luid )/2, where R(Ωf luid ) is
the centrifugal-force modified Thomas-Fermi radius, we obtain a minimum strength for
pinning(Upin /µ)min ≈��1√ R(Ωf luid )/d × (ΩOL − Ωf luid )/Ωc
2 3

(2.4)

What will be the fluid rotation rate Ωf luid in the presence of the pinning potential?
On the one hand, if vortices are tightly locked to the optical lattice sites, we have
Ωf luid = Ωc . The minimum strength (Upin /µ)min resulting from this assumption isplotted as solid red line in Fig. 2.9(d). The lack of decrease of measured (Upin /µ)minto zero around Ωc may be due to long equilibration times in a very shallow pinningpotential, as well as slight mismatches in alignment and initial rotation rate of the BEC
and the pinning potential. The ease of orientation locking with increasing rotation ratemismatch ΩOL − Ωc is less easy to explain in this model. On the other hand, in the
weak-pinning regime, the vortex lattice can accommodate a rotation rate mismatch by
stretching/compressing away from the pinning sites. This allows the fluid to co-rotatewith the optical lattice (Ωf luid ≈ ΩOL ) and reduce the Magnus force. This leads to avery low minimum pinning strength over a wide range of rotation rates, as suggested
by our data. However, the vortex lattice’s gain in pinning energy decreases rapidly inthe locked orientation when the mismatch between vortex spacing and optical lattice
constant increases to the point where the outermost vortices fall radially in between
two pinning sites. Then the preference for the locked angular orientation vanishes. Thispredicted limit is indicated by the vertical dotted lines in Fig. 2.9(d).

56
Our data thus reveal a major diﬀerence between our finite system and extended
systems: In extended systems, the pinned vortex density nv has to equal the densityof pinning sites, else defects will be induced - the physics, even for small rotation ratemismatch, lies in the competition between ordered states and defect generation. In
our finite system on the other hand, the pinned vortex lattice can stretch or compress
slightly away from the pinning sites to find a balance between Magnus and pinning force.
This induces a vortex density diﬀerent from the density of pinning sites, but allows thefluid to continue to remain pinned and to rotate with the lattice, even in the presenceof a modest rotation rate mismatch.
2.3.2

Pinning-induced structural vortex lattice crossover

Figure 2.10: Images of rotating condensates pinned to a square optical lattice (period
7.0 µm) at ΩOL = Ωc = 0.866ωr with increasing pinning strength, showing the structural
cross-over of the vortex lattice. Upper row: real-space absorption images of vortex
lattices after expansion. Bottom row: Fourier transforms of images in upper row.

In a second set of experiments we applied a square optical potential to the vortex
lattice. In the absence of a pinning potential, the interaction energy of a square vortex
lattice is calculated to exceed that of a triangular lattice by less than 1% [36], thus it

57

Numerical work:
H. Pu et al., PRL 94 190401 (2005)

Upin/!=0.01

Upin/!=0.044

Upin/!=0.11

Analytical theory:
Reijnders et al., PRL 93 060401 (2004)

Upin/!~0.03

Upin/!~0.01
Unpinned lattice

Half pinned state

Fully pinned lattice

Figure 2.11: Theoretical predictions for pinning-induced structural crossover (Refs.
[128], [130]). Numerically simulated vortex patterns coincide with our results in Fig.
2.10; pinning strengths for the structural crossover agree qualitatively.

is predicted [130, 131, 128] that the influence of even a relatively weak square opticallattice will be suﬃcient to induce a structural transition in the vortex lattice. This
structural vortex lattice crossover is observed in our experiment. Figure 2.10 shows how
the vortex lattice evolves from triangular to square as the pinning strength increases.
Over a wide range of pinning strengths, we observe that there is always at least one
lattice peak in reciprocal space that remains very strong. We define this peak to be
ko . Lattice peaks at 60◦ and 120◦ from ko are referred to as ktr , and, together with ko ,their strength is a measure of the continued presence of a triangular lattice. A peakat 90◦ , referred to as ksq , is instead a signal for the squareness of the vortex lattice.
With increasing pinning strength in Fig. 2.10 we see the triangle to square crossover
evolve. At intermediate strengths the vortex pattern shows ordering in one of the twolattice directions, while vortex planes in the other direction remain unlocked, as seen in
Fig. 2.10 at Upin /µ = 0.11; in reciprocal space we see the presence of structure at ktr
and ksq .
We quantify the crossover by means of an image-processing routine that locateseach vortex core, replaces it with a point with unit strength, Fourier transforms theresultant pattern, and calculates structure factors |S| [128] based on the strength of

St
ro
ng
ly

pi
nn
ed

ax

is

58

90o

60o

k0

ktr2
ksq
ktr1

Figure 2.12: Definition of k-vectors for pinning-induced structural crossover: ko is taken
to be the strongest peak; ktr1 , ksq , ktr2 are at 60◦ , 90◦ , 120◦ , respectively, from ko .
1.0

Structure Factors

0.8

0.6

0.4

0.2

0.0
0.00

0.05

0.10

0.15

0.20

Upin / !
Figure 2.13: Structure factors (a) |S(ksq )| (�), (b) |S(ktr )| (�) (average of |S(ktr1 )|and |S(ktr2 )|), and (c) |S(ko )| (�) versus the strength of the pinning lattice at thecommensurate rotation rate Ωc . |S(ksq )| is fitted to a smooth functional form: |S(ksq )| =2
A2 + A1 −A
where A1 , A2 , U0 , and δU are fit parameters. The fitting leads to a
U −U0
1+Exp[

δU]

maximum value 0.707 of |S(ksq )|. An ideal square vortex lattice would have |S(ksq )| = 1.

the images at lattice vectors ksq , ktr , and ko . In Fig. 2.13, we see with increasing
optical potential the turn-on of |S(ksq )| balanced by the turn-oﬀ of |S(ktr )|. We use
a fitting function to smooth the noisy data of |S(ksq )|. The structural crossover takes

59place around Upin /µ ≈ 8%, in rough agreement (see Fig. 2.11) with predictions ofUpin /µ ≈ 5% from numerical simulations [128] and Upin /µ ≈ 1% from analytic theory
for infinite lattices [130, 131]. The fact that one lattice peak remains strong for all
pinning strengths (the stars (�) in Fig. 2.13) suggests that as the pinning strength
is increased, one family of vortex rows represented by ko in Fourier space locks to thesquare pinning lattice and remains locked as the shape cross-over distorts the other two
families of vortex rows into a square geometry.
0.30(a)|S(k )|=0.7
sq

Upin / #

0.25

0.6

0.20

0.15

0.10

0.5
0.4
0.3
0.2
0.1

0.05

Maximum S(k sq)

0.00
0.00
0.80

(b)

0.60

0.40
0.78

0.82

0.86

0.90

0.94

Rotation rate (!/"r)
Figure 2.14: Eﬀect of a square pinning lattice. (a) Contours of |S(ksq )|, showing theeﬀect of the rotation rate and pinning strength on the squareness of the vortex lattice.
(b) Maximum observed squareness. In (a) and (b), for each rotation rate, the data
points are extracted from fits such as that shown in Fig. 2.13 for ΩOL = Ωc . Verticaldotted line plus arrow: possible range of Ωc consistent with the uncertainty in nOL .

60
The eﬀects of various rotation rates and optical potential strengths on the squareness of the vortex lattice is summarized in Fig. 2.14. We surmise that there are a numberof eﬀects at play. When ΩOL diﬀers from Ωc , pinning strength is required not only to
deform the shape of the vortex lattice from triangular to square, but also to compress or
expand it to match the density of the optical lattice sites. At higher optical intensities,
we know from separate observations that imperfections in the rotation of the opticallattice lead to heating of the condensate, which may limit the obtainable strength of
the square lattice.
2.3.3

Defects in pinned vortex lattices

Figure 2.15: Image of a dumbbell-shape defect consisting of two vortices locked to one
pinning site during the formation of the square vortex lattice. It is clearly visible how
adjacent vortices are pushed outward by the repulsion due to the additional vortex.
Above, a “sliding row” defect is visible too.

A dumbbell-shape lattice defect (Fig. 2.15), consisting of two vortices pinned tothe same site, is sometimes observed in the early stages of the square vortex lattice
formation when ΩOL > Ωc . In the weak-pinning regime, the defect will relax towards
the equilibrium configuration by pushing extra vortices at the edge of the condensate

61
outside the system. Defects of this nature, involving extra (or missing) vortices, arethe exception and not the rule in our observations, even for ΩOL �= Ωc . In an infinite
system, the physics of the lattice-lattice interaction would likely be dominated by thesepoint defects. In our finite system, would-be incommensurate lattices can accommodate
by stretching or compressing.
Defects,
Edges unpinned

Defects,
Defects,
zigzagDepinnedplanes arrangements

Maximum S(k sq)

0.80

0.60

0.40
0.78

0.82

0.86

0.90

0.94

Rotation rate (!/"r)
Figure 2.16: Defects in pinned vortex lattices with varying rotation rate mismatch.
Defects reduce the maximum attainable square structure factor of the pinned lattices.

Figure 2.16 shows typical images of vortex lattices pinned at increasingly noncommensurate rotation rates. Increasing numbers of defects can be seen towards theedge of the pinning range, indicating that the amount of compression/stretching thatallows vortex lattices to remain pinned over this large range of rotation rates, is limitedby the increasingly frequent appearance of defects.

62

Upin/!=0

Upin/!=0.16

Upin/!=0.2

Upin/!=0.35

Upin/!=0.4

Figure 2.17: Vortex pinning in a triangular optical lattice with 2 vortices per pinningsite. One-dimensional ordering is seen in real-space and Fourier images. This ordering
was not predicted theoretically.

2.3.4

Pinning with two vortices per pinning site

Here we describe experiments with, on average, two vortices per pinning site. The
expected evolution with increasing pinning potential, according to numerical simulations
by Pu and coworkers [128] on a square pinning lattice, is shown in Fig. 2.18 – atransition from two singly quantized vortices to doubly quantized vortices is seen. A
similar evolution is also expected in a triangular pinning potential.
In our experiments we used a triangular 9.54 µm period lattice, where two vortices per pinning site are obtained at a rotation rate Ω = 0.957ωr . To our surprise wefound signs of one-dimensional ordering, as shown in Fig. 2.17: Parallel rows of vortices
are seen, with roughly equidistant spacing to adjacent rows, but no obvious correlations
between vortex positions in neighboring rows, and also lacking perfect periodicity withineach row. We did however not observe fully ordered systems, and we never observed
the merging of multiple vortices pinned to one pinning site to one multiply quantizedvortex. A simple calculation shows that the vortex-vortex repulsion overcomes the pinning force arising from our sinusoidal potential unless the pinning strength VOL /µ ≈ 1.
As discussed, this regime is diﬃcult to work in due to strong heating eﬀects. One would

63

Figure 2.18: Numerical simulations of vortex pinning with 2 vortices per pinning site,from Ref. [128]. Full (hollow) circles represent singly (doubly) quantized vortices. A
transition to a pinned lattice of doubly quantized vortices is expected.

need steeper pinning sites to force vortices closer together. We have attempted to synthesize such potentials using the present setup, simply by adding more holes to theshadow mask. We found that such multi-beam potentials do not preserve their structure while rotating, as we do not actively control the relative phases between diﬀerent
beams against unwanted perturbations. I am optimistic that a future generation of
graduate students will be able to benefit from the development of fast enough spatial
light modulators that will do this kind of job sooner or later.
2.3.5

Pinning with one vortex per two pinning sites

We have also conducted experiments with one vortex per two pinning sites. Here,a superstructure of vortices is expected. We used a triangular 6.3 µm period lattice.
Due to various constraints in optical access, this period is close to the minimum period
achievable for a rotating lattice using the setup shown in Fig. 2.5. In this lattice and in
our “cigar” magnetic trap, one vortex per two pinning sites is obtained at Ω = 0.548ωr .
Experimentally, no ordered vortex structures were observed. The problem waslikely linked to the fact that due to the constraint on the lattice period we had towork at slow rotation rates, i.e. with a small BEC and a small number of vortices. In
this situation, strong radial density gradients and boundary eﬀects are expected to beimportant for vortex dynamics. In addition the BEC was not even close to quasi-2D, ie.
vortices could bend axially, making this experiment a more complicated 3D problem.

Chapter 3
A 2D Josephson junction array in the Berezinskii-Kosterlitz-Thoulessregime3.1

Motivation
In this chapter [2] I describe experiments on a finite-temperature two-dimensional

array of Josephson-coupled Bose-Einstein condensates undergoing a Berezinskii-KosterlitzThouless (BKT) transition from a superfluid to a normal state. Detailed data on thethermal activation of vortices are presented, providing a direct “microscopic” look at
the physical processes underlying the transition.
This study thus adds to a large body of work dealing with fluctuations in superfluids, which aﬀect long-range phase coherence [99]. Fluctuations may be quantum innature, as in the Mott-insulator transition [67, 85], or thermal fluctuations, as studied
e.g. in one-dimensional Bose gases [135, 76] and in a double-well system [60]. In two
dimensions (2D), Berezinskii [23, 24], Kosterlitz and Thouless [94] (BKT) in the 1970’s
developed an elegant description of thermal phase fluctuations based on the thermal
excitation and unbinding of vortex-antivortex pairs (i.e. pairs of vortices of opposite
circulation). The BKT picture applies to a wide variety of 2D systems, among them
Josephson junction arrays (JJA), i.e. arrays of superfluids in which phase coherence ismediated via a tunnel coupling J between adjacent sites.
Transport measurements, both in continuous superfluids [7, 138] and superconducting JJA [133] have confirmed the predictions of BKT, without however directly

65
observing its microscopic mechanism, vortex-antivortex unbinding. Ultracold atomic
gases, on the other hand, appear ideally suited to complement the traditional studies,because of the favorable length scales of vortex defects and direct “wavefunction imaging” capabilities. A recent beautiful experiment in Dalibard’s group at ENS in Paris[71, 96], performed in a continuous 2D Bose gas, opened up the field, and measured thephase-phase decay function through the BKT cross-over, and saw evidence for thermal
vortex formation. For related theoretical studies see e.g. [149, 65, 125]. In our work we
present detailed data on thermally activated vortex-formation, collected in a 2D arrayof BECs with experimentally controllable Josephson couplings. The system was studied
theoretically by Trombettoni et al. [163].
Since the completion of our studies, we have become aware of additional, yet
unpublished, work in Phillips’ group at NIST Gaithersburg [42], and Kasevich’s group in
Stanford [91]. The group of Rob Ballagh in Otago, New Zealand, have begun numericalstudies of our lattice system [17].

3.2

The BKT transition in a Josephson junction array
The BKT transition in a Josephson junction array proceeds very similarly to thecontinuous system described in the introduction. Vortex formation is governed by twocompeting energy scales: the Josephson (tunnel) coupling J between nearest-neighborcondensates acts to keep the relative phases of the condensates locked. A cloud of
uncondensed atoms, on the other hand, interacts with the condensates and induces
thermal phase fluctuations, which we observe as vortices. As long as the Josephsonenergy J exceeds the thermal energy T , the array is vortex-free. With decreasing J/T ,vortices appear in the system in ever greater numbers.
As for the BKT transition in a continuous superfluid, an estimate of the conditions
for the BKT transition is obtained by computing the free energy F of an isolated (free)vortex in a JJA, and demanding F = E − T S ≤ 0. In an array of period d the vortex

66
energy [161] diverges with array size R as 1
E ≈ J log(R/d)(3.1)

but may be oﬀset by an entropy gain 2
S ≈ log(R/d)(3.2)

due to the available ≈ R2 /d2 sites. This leads to a critical condition(J/T )crit ≈ 1(3.3)

independent of system size, below which free vortices will proliferate. In contrast,tightly bound vortex-antivortex pairs are less energetically costly and show up even above(J/T )crit . The overall vortex density is thus expected to grow smoothly with decreasing
J/T in the BKT crossover regime. Condition (3.3) for the BKT transition on a lattice isreally very similar to the one for the bulk 2D superfluid, Eq. (1.36). To see how similar� Nwell
the two conditions are, consider the well-established relation J ∼ 2m
. Here, m∗∗d2
2

is eﬀective mass of the bosons tunneling through the lattice, and Ndwell
is equivalent to2
the 2D superfluid density nSF,2D . Thus J ∼�2 nSF,2D
and it becomes obvious that Eqs.
2m∗(3.3) and (1.36) really represent the same condition.
There are however two important diﬀerences to a continuous 2D system: First,
analogously to superconductive Josephson junction arrays [161], there are two separated
transitions: Upon lowering the temperature of the system, first a BEC forms at eachindividual lattice site, with no coherent phase relation to neighboring condensates. Onlyat lower temperature, the array undergoes a BKT transition, upon which (at least local)
phase coherence is established between lattice sites. This separation of the transitions
has significant advantages in the experimental determination of system parameters: Dueto the separated transition to BEC on the individual lattice sites, the coherent coupling
1

Here I ignore factors of order unity related to the precise lattice geometry, as well as finite-systemsize corrections amounting to a finite “vortex core energy”
2Throughout this chapter I use energy units to express temperature, hence entropy is unitless.

67
J can be rather accurately defined experimentally as well as calculated numerically,
while the determination of the superfluid density nSF,2D is an experimental challenge[96]. The second large experimental advantage of an array over the continuous system isthe ability to tune the Josephson coupling J over orders of magnitude by changing thelattice depth. This is in contrast to the parameters of the continuous BKT transition,i.e. the temperature T and the 2D superfluid density nSF,2D , which are much harder toadjust accurately, especially while leaving other parameters equal.

Setup, calibrations and parameter determination
glass
cell

d=300mm2” f=150

2” f=150
2” PBS
780nm

1.5” f=120

Dichroic mirror
front R>99% 810nm
front&back AR 780nm
2” f=700

CCD

3.3

BEC
MOT,
Imaging!/4
MonitorPD

Shadow mask
Locking PD

Picomotors

2” f=300

Optical lattice
810nm

Figure 3.1: Optical lattice setup for creating a 2D Josephson junction array. 810 nm
light from a single-mode fiber passes through a shadow mask which defines three lattice
beams. These are combined with the MOT/imaging beam path using a polarizing
beam splitter cube (PBS), and overlapped on the BEC, where interference forms the 2D
lattice. Diagnostic photodiodes and a CCD camera are used to optimize beam overlap
and alignment with respect to the BEC. Important degrees of freedom for alignment
are indicated by arrows.

The setup, shown in Fig. 3.1, is similar in schematic to the one employed in thevortex pinning experiments, however the whole beam path was redesigned to provide

68
maximum stability to eliminate technical heating of the sample. The heating rate of
the BEC loaded into an optical lattice of VOL = 1.33 kHz depth was measured to be
1 nK/s, close to the value estimated from photon scattering. With a BEC transition
temperature around 50 − 100 nK this leaves plenty of time for experiments.
A lattice period of d = 4.7 µm was chosen, as a compromise between precise
control over J (facilitated by a small d), and minimization of phase fluctuations along thelattice axis (see below) requiring large d. To reach 4.7 µm lattice periods using 810 nm
light (as opposed to d = 7.2 µm using 532 nm light as in Chapter 2), a shorter focal
length lens (f = 150 mm) was used to focus the lattice beams onto the BEC. This lensneeded to be placed into the MOT/imaging beam path, with the associated insertion ofcompensating lenses to the MOT and imaging beam paths. An important improvement
was to change the position of the photodiode used for locking the lattice intensity tobehind the glass cell. The point is the following: thermal eﬀects, e.g. associated withvarying light level during lattice intensity ramps, in the single-mode fiber delivering thelattice light can cause significant polarization changes of the fiber output on few-second
timescales. The polarizing beam splitter cube converts such polarization fluctuations
into intensity fluctuations. The present setup compensates for the resulting intensity
fluctuations, in contrast to the setup for vortex pinning experiments (Fig. 2.5), where
the locking photodiode was placed before the polarizing beamsplitter cube.
One important issue with acute-angle optical lattices is a precise longitudinal focusing of the lattice onto the BEC, to achieve maximal beam overlap. This focusing wasachieved experimentally by monitoring the axial BEC radius, which grows when loaded
into a deep 2D lattice, and is thus maximized when the lattice beams are optimallyspatially overlapped on the BEC. The accuracy of this method was shown to be close
to the BEC size, within ±50 µm. This is very accurate compared to the longitudinalbeam overlap region of w/ sin(θ) = 580 µm, given by the beam waist w = 67 µm andthe inclination angle of the beams θ = 6.6◦ .

69
3.3.1

Optical lattice intensity calibration

The standard method of measuring an optical lattice depth, i.e. pulsing thelattice and analyzing the resulting matter-wave diﬀraction pattern [116, 51] does notwork for our long-period lattice, as the resulting momentum components do not cleanly
separate from each other. Therefore an accurate intensity measurement based on the
optical power, lattice beam waist and modulation contrast was required. Using themeasured wavelength λ = 810.1 nm, the optical lattice depth can then be calculatedusing Eq. (A.2) in Appendix A. The power P in the three beams was measured using a
power meter that was calibrated against several other power meters. Losses between themeasurement location and the BEC position were taken into account. The single-beam
intensity profile is given by an Airy pattern due to diﬀraction from the holes in the
shadow mask. The single-beam peak intensity is Ipk = 0.555 × wP2 for a 1/e2 waist w
2P(for a Gaussian it would be Ipk = πw
2 ).

The optical lattice contrast, i.e. intensity (max − min)/max, was measured on
the CCD camera, taking into account the finite camera pixel size 3 . The expected
contrast was calculated for comparison. The hexagonal geometry does not lead to fully
constructive and destructive interference along the axis connecting neighboring sites, butinstead gives (max − min)/max = 88.8 %. Known imperfections, such as slight (3%)
intensity imbalances, slightly imperfect beam overlap (3% of beam waist) and slightly
elliptical beam shapes (see Fig. 3.2) reduce the expected contrast to 86%. As themeasured (79%) and calculated (86%) contrast did not fully agree, we used the averageof measured and calculated contrast, 82.6(3.6)%. This is the largest contribution to theuncertainty in lattice depth. To calculate the intensity, the beam waist is measured on
3
In this measurement, “image resolution” is not limited in the conventional sense by the finite
f −number of the imaging system, nor deteriorated as usual by aberrations, as the image is formed
from three k-vectors only. We thus know that all k-vectors are collected by the imaging system, andin addition traverse the imaging system at identical radial displacements. This is in contrast to usual
images composed of a whole spectrum of k-vectors, of which only a finite cone is collected, and each ofwhich may be aﬀected diﬀerently by aberrations.

70
the CCD camera. The imaging magnification is known within 3%, see Appendix E.
a)

b)

c)

Figure 3.2: All images are shown using logarithmic intensity scale. (a) measured opticallattice intensity profile shows Airy pattern envelope. (b) individual beam profiles show
slight ellipticity with major axes orientation diﬀering by 120◦ - a direct consequence of
the acute-angle lattice geometry. (c) calculated intensity profile.

3.3.2

Getting J/T right

Vortex proliferation in a JJA is expected to depend on the ratio of Josephson
coupling to temperature, J/T – here I describe the accurate determination and error
analysis of J and T .
Determination of J and T. As I will explain, the only experimental parameters
required to obtain J are the axial condensate radius Rz , the lattice depth VOL and thesample temperature T 4 . These are measured independently for every shot. Rz and T
are determined from non-destructive side-view images of the finite-T BEC loaded into
the optical lattice. The images are analyzed using finite-T fits of the thermal cloud and
BEC density profile, as described in Section 1.2.4. The condensate density is analyzed
according to the coarse-grained Thomas-Fermi description of BECs in optical lattices(Section 1.5.4). VOL is determined according to the above Section 3.3.1.
As I describe in Appendix B, numerical simulations result in a prediction for the4
Note that the combined optical lattice and magnetic trap potential may be anharmonic in theradial direction (the optical lattice envelope Airy pattern is not infinitely large compared to the BECsize). This anharmonicity would complicate image analysis beyond the description of Sections 1.2.4 and1.5.4. The axial (z) potential however is purely harmonic even in the presence of the optical lattice. We
thus use only parameters extracted from the axial direction for our quantitative analysis.

71
Josephson coupling energy J as a function of the number of particles per well Nwell , theapplied optical lattice depth VOL , and temperature T , which is valid in the range ofNwell and VOL employed in our experiments
J(VOL , Nwell , T ) ≈ 0.315 nK × Nwell exp

�

Nwell
VOL
−
3950
244 Hz���0.59 T× 1+(3.4)
100 nKas well as a prediction for the axial condensate radius Rz
Rz (VOL , Nwell , T ) = Rz (VOL , Nwell , T = 0) − 0.55(10)µm × T /60nK

(3.5)

with
Rz (VOL , Nwell , T = 0) = 22.69 µm × (Nwell /7000)(1/3.8) × (VOL /1300 Hz)(1/4.5) (3.6)
The experimentally measured Rz , VOL and T can be converted into the required Nwell
using equation (3.6), which then allows determining J for every shot using (3.4) 5 . The
exponential dependence of J on Nwell and VOL therefore demands utmost care in theexperimental determination of Rz and VOL .
Systematic eﬀects and error budget. In the following we describe the mostcritical aspects in the determination of each parameter. There are in general two separate concerns, (i) systematic diﬀerences between “hot” and “cold” clouds which need tobe controlled in order to prove that vortex activation is thermally driven, and (ii) overallsystematics in the absolute value of J/T , which might prevent accurate estimates of the
BKT crossover position. The resulting error budget is summarized in Table 3.3.2.
Temperature T. We determine T from a fit to the BEC and thermal cloud,using a fit function including the BEC Thomas-Fermi profile and a mean-field modified
thermal cloud, as described in Section 1.2.4. We first fit the wings of the thermal
cloud outside a chosen ellipsoidal cut, then fix the thermal cloud parameters and fit
the whole distribution to the BEC Thomas-Fermi profile and the mean-field modified
5

For the most accurate parameter determination we do NOT employ the approximations (3.4) and(3.6), but interpolate for every shot between nearest numerical data points, see Appendix B.

72
thermal cloud. The cut position for fitting the thermal cloud is critical: If chosen
too close to BEC, one underestimates T by O(10%). This is a well known eﬀect, see[157]. The largest overall systematic in the temperature determination is the side image
magnification, which is uncertain within 3%. This leads to a temperature uncertainty
of 6%, very small in comparison to uncertainty in J, see below.
Josephson coupling J. As described, three primary parameters enter the extraction of J: Rz , VOL and T . The most crucial parameter for accurate comparison
between “hot” and “cold” samples is Rz . In the determination of Rz from the abovementioned fitting procedure, a subtle systematic eﬀect occurs: the fitted value of Rzvery slightly depends on the above mentioned fitting cut as well, and it does so differently for cold and hot samples. However an interval of cut positions exists where
fits to both hot and cold clouds are reliable. Nonetheless, we assign a ±0.75% relative systematic uncertainty in Rz between hot and cold clouds. Although this appears
tiny, it leads to a sizeable systematic uncertainty in J between hot and cold samplesof ∆J / J = 8 %. A second systematic eﬀect from Rz between hot and cold samples
arises from the modelling of the thermal cloud in the numerical simulations. We haveto include a smoothing function that prevents a kink of length scale less than λth at theboundary between BEC and thermal cloud, and the precise choice of this smoothing
function residually influences (a) J(Nwell , VOL ) for a fixed T and (b) the compression of
the BEC by the thermal cloud and hence Rz (Nwell , VOL ) at fixed T – see Appendix B.
The overall result is a further ≈ 10 % uncertainty in J between hot and cold samples.
The total uncertainty in J between hot and cold samples is thus ∆J / J = 13 %.
From here on we are dealing with overall uncertainties in J, as opposed to diﬀerences between “hot” and “cold” samples. The absolute uncertainty in Rz is dominatedby the side view magnification. It enters the correct determination of Nwell and leads
to ∆J / J ≈ 30 %.
The uncertainty in VOL is dominated by the optical lattice contrast of 82.6(3.6)%,

73
as described in Section 3.3.1. In addition the calibration of our optical power meter, as
well as the accuracy and reproducibility of our daily optical power calibration are taken
into account in Table 3.3.2.
The absolute value of J depends exponentially on the lattice constant d, which
is determined from top-view CCD camera images, and therefore depends on the uncertainty in the absolute top view imaging magnification calibration. However there are
two correlated eﬀects: First, if we overestimate d, we underestimate VOL by overestimating the beam waist. Second, an error in the lattice period d due to error in top
view magnification calibration influences the relation between Rz and Nwell in numericalsimulations. In Table 3.3.2 we give the combined result on J of these three correlated
eﬀects.
Table 3.1: Error budget on the Josephson coupling J. The diﬀerent contributions are
explained in the text. s.e.: standard error; stdev: standard deviation.
Technical Source
top view magnification
top view magnification
top view magnification
combined eﬀect
abs. Power meter calib.
daily OL cal. procedure
combined eﬀectside magnification
OL mod. contrast
syst. fit uncertainty
sim. finite-T corr. to J
sim. BEC compr. by THC
Total ∆J / J

Amount±3%±3%±3%

Couples todVOL
calc. Rz (Nwell )Amount
±3%
±6 %

∆J, +

∆J, −

+14 %

−12 %

+33 %
+32 %
+42 %
+8 %
+2 %
+7 %
+65 %

−24 %
−27 %
−29 %
−8 %
−2 %
−7 %
−50 %

3 % s.e.
2.2 % stdev±3 %(82.6 ± 3.6) %

Nwell
VOL
Rz
calc. Rz (Nwell )11 %±0.75 %As a last note, the statistical shot-to-shot fit uncertainty in Rz leads to an uncertainty in J, ∆J / J = 12 %. This uncertainty leads to a shot-to-shot scatter in the dataand is not taken into account here.

74
3.3.3

The importance of the third (z) dimension

Each lattice site in our setup is really an elongated tube. Axial phase fluctuations
inside each condensate may thus aﬀect the 2D nature of this experiment. The scale ofaxial phase fluctuations in each tube is set by equating the thermal energy kB T to theenergy cost of a phase fluctuation Φax . For long-wavelength fluctuations (length scale� → 2→
Rz ) which dominate at low temperatures, this cost is d3 −r � /(2m) n(−r )Φ2ax /Rz2 ∼→�2 /(mRz2 )Nwell Φ2ax , where n(−r ) is the condensate density. A more detailed, and experimentally verified ([135, 76, 83]), calculation by Walraven and coworkers [120] shows
the RMS fluctuations between positions −z, +z to be
�
32
kB T
2z
Φax,rms (z) =30 Nwell �2 /(mRz2 ) Rz

(3.7)

We consider the region z = Rz /3 where according to our simulations 85% of the tunnel
current between adjacent condensates is localized, and hence the relative phase between
the condensates is measured. Using Nwell = 7000, T = 62.5 nK and Rz = 22 µm, i.e.
typical parameters for the worst-case scenario of a deep lattice and the “hot” dataset (seebelow), we get Φax,rms ∼ 0.7 rad. Alternatively, considering z = 0.6Rz i.e. the region
occupied by 90% of atoms in a Thomas-Fermi profile, we get Φax,rms ∼ 1 rad. These
numbers show that while axial fluctuations cannot be completely ignored, they are notexpected to destroy the quantitative picture of a two-dimensional system undergoing a
BKT transition.

3.4

Experiments and results
The experiments reported below show vortex proliferation in a finite-T Josephson

junction array, confirm thermal activation as the vortex formation mechanism, andprovide information on the growth of bound vortex pairs, as J/T is lowered past thecritical value for the BKT transition.

75
3.4.1

Thermally activated vortex proliferation

Our experiment starts with production of a partially Bose-condensed sample of
87 Rb atoms in a harmonic, axially symmetric magnetic trap with oscillation frequencies{ωx,y , ωz } = 2π{6.95, 15.0} Hz. The number of condensed atoms is kept fixed around
6 × 105 as the temperature is varied. We then transform this system into a Josephson
junction array, as illustrated in Fig. 3.3. In a 10 s linear ramp, we raise the intensityof a 2D hexagonal optical lattice 6 of period d = 4.7µm in the x-y plane. The resulting potential barriers of height VOL between adjacent sites [Fig.3.3(b)] rise above thecondensate’s chemical potential around VOL ≈ 250 − 300 Hz, splitting it into an arrayof condensates which now communicate only through tunneling. This procedure is adiabatic even with respect to the longest-wavelength phonon modes of the array [87, 35]
over the full range of VOL in our experiments. Each of the ≈ 190 occupied sites (15sites across the BEC diameter 2 × RT F ≈ 68 µm 7 ) now contains a macroscopic BEC,
with Nwell ≈ 7000 condensed atoms in each of the central wells at a temperature T thatcan be adjusted between 30 − 70 nK. By varying VOL in a range between 500 Hz and
2 kHz we tune J between 1.5 µK and 5 nK, whereas the “charging” energy Ec , definedin [99], is on the order of a few pK, much smaller than both J and T . In this regime,�thermal fluctuations of the relative phases ∆φT h ≈ T /J are expected, while quantum
fluctuations ∆φQ ≈ (Ec /4J)1/4 are negligible [99].

The suppression of the Josephson coupling greatly suppresses the energy cost ofphase fluctuations in the x-y plane, between condensates, J[1 − cos(∆φ)], compared tothe cost of axial (z) phase fluctuations inside the condensates [120]. As a result, axial
phase fluctuations remain relatively small, and each condensate can be approximated6
Three circularly polarized laser beams (λ = 810.1nm) intersect in a tripodlike configuration, with
θ = 6.6◦ angles to the z-axis. Calculation of the optical dipole potential [70] includes counterrotating
terms and interaction with both the D1 and D2 lines, as well as the ‘fictitious magnetic field’ due tothe circular polarization. The tilted bias field of the TOP trap makes P in Ref. [70] ≈ 0.5.
7
To avoid radial flows during VOL ramp-up, RT F is kept constant by balancing the lattice-enhanced
mean field pressure with radial confinement due to the optical lattice envelope, by the choice of a 67µm
1/e2 intensity waist. Axial (z) confinement is due to the magnetic trap alone.

76

. .

b) . T .

. .
. .. .
....
. ..
...... ... .

RTF

J

.
.

.

. .
. .. .
. .. . .
..... .....

Nwell
Δφ

VOL

4.7µm
c)

Energy
VOL
10s

J

2s

image
tr

t

Figure 3.3: Experimental system. (a) 2D optical lattice intensity profile. A lattice ofJosephson-coupled BECs is created in the white-shaded area. The central box marks thebasic building block of our system, the double-well potential shown in (b). The barrierheight VOL and the number of condensed atoms per well, Nwell , control the Josephson
coupling J, which acts to lock the relative phase ∆φ. A cloud of uncondensed atoms
at temperature T induces thermal fluctuations and phase defects in the array when
J < T . (c) Experimental sequence: A BEC (i) is loaded into the optical lattice over
10 s, suppressing J to values around T. We allow 2 s for thermalization. To probe
the system, we ramp oﬀ the lattice on a faster timescale tr and take images of therecombined condensate. When J is reduced below T (ii)-(vii), vortices (dark spots)
appear as remnants of the thermal fluctuations in the array.

77

Phase [radians]

as a single-phase object 8 .

Figure 3.4: Ramping down the optical lattice depth converts phase winding defects in
the JJA (left) to vortices in a bulk BEC (right). The increasing circulating current isindicated by arrow grayscale.

After allowing 2 s for thermalization, we initiate our probe sequence. We firsttake a nondestructive thermometry image in the x-z plane, from which the temperature
T and, from the axial condensate size Rz , the number of condensed particles per well,
Nwell , is obtained (see below). To observe the phase fluctuations we then turn down
the optical lattice (Fig. 3.4) on a time-scale tr 9 , which is fast enough to trap phasewinding defects, but slow enough to allow neighboring condensates to merge, provided
their phase diﬀerence is small. Phase fluctuations are thus converted to vortices in thereconnected condensate, as has been observed in the experiments of Scherer et al. [141]and analyzed theoretically by Ghosh and Sols [62]. We then expand the condensate bya factor of 6 and take a destructive image in the x-y plane.
Figure 3.3(c) illustrates our observations: (ii)-(vii) is a sequence of images at
8

In the axial condensate region between z = −Rz /3, +Rz /3, where according to our 3D GPEsimulations 85% of the tunnel current is localized and hence the relative phase is measured, axial phasefluctuations [120] vary between ≈ 600 mrad (“cold” data in Fig. 3.5) and 800 mrad (“hot”) in theregime J/T ≈ 1.
9Within a dataset, the ramp-down rate is kept fixed, tr = τ × VOL /1.3 kHz, τ = 18 ms if nototherwise indicated.

78
successively smaller J/T (measured in the center of the array). Vortices, with theircores visible as dark “spots” in (iii)-(vii), occur in the BEC center around J/T = 1.
Vortices at the BEC edge appear earlier, as here the magnetic trap potential adds tothe tunnel barrier, suppressing the local J/T below the quoted value. That the observed“spots” are indeed circulation-carrying vortices and antivortices is inferred from theirslow ≈ 100 ms decay after the optical lattice ramp-down, presumably dominated by
vortex-antivortex annihilation. From extensive experiments on vortices in our system
we know that circulation-free “holes” fill so quickly due to positive mean field pressure,that they do not survive the pre-imaging expansion. Vortices with identical circulation
would decay by dissipative motion to the BEC edge, in our trap over � 10 s.
To investigate the thermal nature of phase fluctuations, we study vortex activation
while varying J at diﬀerent temperatures. For a quantitative study, accurate parameter
estimates are required. The Josephson-coupling energy J is obtained from 3D numericalsimulations of the Gross-Pitaevskii equation (GPE) for the central double-well system[9, 60] [Fig.3.3(b)], self-consistently including mean-field interactions of both condensed
and uncondensed atoms [112]. A useful approximation for J in our experiments is 10 :J(VOL , Nwell , T ) ≈ Nwell × 0.315 nK exp[Nwell /3950 − VOL /244Hz](1 + 0.59 T /100nK).
The finite-T correction to J arises from both the lifting-up of the BEC’s chemical potential and the axial compression by the thermal cloud’s repulsive mean field, but does nottake into account the eﬀects of phase fluctuations on J (in condensed-matter language,
we calculate the bare J). Nwell is determined by comparison of the experimentally measured Rz , to Rz (VOL , Nwell , T ) obtained from GPE simulations. Both experimental andsimulated Rz are obtained from a fit to the distribution of condensed and uncondensed
atoms, to a Thomas-Fermi profile plus mean-field-modified Bose function [112]. In determination of all J values, there is an overall systematic multiplicative uncertainty
10

J is averaged over junctions within the central 11% of the array area, from which all quantitative
experimental results are extracted.

79

(b)

0.06



0.10
vortex density (c)cold: 30<T<40nK
hot: 55<T<70nK

0.08

0.04
0.02

0.06

10

100
J [nK]

1000

0.04
0.02
0.0
10

100

J [nK](d)vortex density 



10

(e)

0.06

0.10

4

1000

0.04

0.08

0.02

0.06

0.1

1 J/T

10

0.04
cold: 30<T<40nK
hot: 55<T<70nK

0.02
0.0
0.1

1

J/T

10

100

Figure 3.5: Quantitative study of vortices. The areal density of vortices is quantified
by the plotted D defined in the text. D is extracted only from the central 11% of
the condensate region [circle in inset (a)] to minimize eﬀects of spatial inhomogeneity.
(b) D vs J for two datasets with distinct “cold” and “hot” temperatures. Each point
represents one experimental cycle. The increase in D with decreasing J � 100 nK
signals the spontaneous appearance of vortices, while the “background” D � 0.01 forJ � 200 nK is not associated with vortices. Vortices clearly proliferate at larger J for
the “hot” data, indicating thermal activation as the underlying mechanism. The large
scatter in D at low J is due to shot noise on the small average number of vortices in
the central condensate region. (c) same data as in (b), but averaged within bins ofsize ∆[log(J)] = 0.15. Error bars of D are standard errors. (d) same data as (b), butplotted vs J/T . “Cold” and “hot” datasets almost overlap on what appears to be auniversal vortex activation curve, as confirmed by averaging [inset (e)], clearly revealing
the underlying competition of J and T .

80∆J/J = ×
÷ 1.6, dominated by uncertainties in the optical lattice modulation contrast,
the absolute intensity calibration, and magnification in the image used to determine
Nwell . In comparing J for “hot” and “cold” clouds (see Fig. 3.5) there is a relative
systematic error of 15% associated with image fitting and theory uncertainties in thethermal-cloud mean-field correction to J.
The qualitative results of our work are consistent whether we use an automatedvortex-counting routine or count vortices by hand, but the former shows signs of saturation error at high vortex density, and the latter is vulnerable to subjective bias. As arobust vortex-density surrogate we therefore use the “roughness” D of the condensate
image caused by the vortex cores. Precisely, we define D as the normalized variance ofthe measured column density profile from a fit to a smooth finite-T Bose profile [112],
with a small constant oﬀset subtracted to account e.g. for imaging noise. To limit
spatial inhomogeneity in J, caused by spatially varying condensate density and opticallattice intensity, to < 10%, D is extracted only from the central 11% of the condensate
area which contains 20 lattice sites [Fig. 3.5(a)]. Comparison to automated vortexcounts shows that D is roughly linear in the observed number of vortices, irrespectiveof the sign of their circulation, with a sensitivity of ≈ 0.01/vortex.
Figure 3.5 shows results of our quantitative study. In Fig. 3.5(b), we plot D
vs J for two datasets with distinct temperatures. At large J � 200 nK a backgroundD � 0.01 is observed, that is not associated with vortices, but due to residual density
ripples remaining after the optical lattice ramp-down. Vortex proliferation, signaled bya rise of D above ≈ 0.01, occurs around J ≈ 100 nK for “hot” BECs and at a distinctly
lower J ≈ 50 nK for “cold” BECs [confirmed by the averaged data shown in Fig. 3.5(c)],indicating thermal activation as the vortex formation mechanism. Plotting the samedata vs J/T in Fig. 3.5(d) shows collapse onto a universal vortex activation curve,
providing strong evidence for thermal activation. A slight residual diﬀerence becomesvisible in the averaged “cold” vs “hot” data [Fig.3.5(e)], perhaps because of systematic

81
diﬀerences in our determination of J at diﬀerent temperatures.
3.4.2

Studying vortex pair sizes and vortex pair unbinding

Figure 3.6: Vortex-antivortex pairs, imaged just prior to their annihilation. Following
the optical lattice ramp-down, tightly bound pairs annihilate faster than loosely boundpairs, providing a time-to-length mapping that allows to extract information on vortex
pair sizes.

The vortex density D studied so far, by itself provides no distinction between
bound vortex-antivortex pairs and free vortices. In the following we exploit the flexibilityof optical potentials to distinguish free or loosely bound vortices from tightly boundvortex-antivortex pairs. Our technique for inferring vortex antivortex pair sizes makes
use of the fact that, once the optical lattice potential has been turned oﬀ, vortices
and antivortices annihilate in the bulk condensate over a ≈ 100 ms timescale. Figure
3.6 shows an example image of pairs of vortices just prior to their annihilation. It
is intuitively obvious that tightly bound vortex pairs will annihilate on a much fastertimescale than loosely bound pairs. A “slow” optical lattice ramp-down therefore allows
time for tightly bound pairs to annihilate before they can be imaged. By slowing down
the ramp-down duration τ [inset of Fig. 3.7], we can thus selectively probe vortex
pairs of increasing size. This represents an attempt to approach the “true” BKT vortex

82
unbinding crossover that is complementary to transport measurements employed so
successfully in superconductive and liquid Helium systems.

VOLτ

vortex density

0.03
time

0.02

ramp-down timescale ττ = 6 ms
τ = 36 ms

<

0.01
0.0
0.1

>
1

J/T

10

Figure 3.7: Vortex density D probed at diﬀerent optical lattice ramp-down timescalesτ . A slow ramp provides time for tightly bound vortex-antivortex pairs to annihilate,allowing selective counting of loosely bound or free vortices only, whereas a fast ramp
probes both free and tightly bound vortices. A fit to the vortex activation curve determines its midpoint (J/T )50% , its 27% − 73% width ∆(J/T )27−73 , and the limiting
values D< (D> ) well below (above) (J/T )50% .
Figure 3.7 shows vortex activation curves, probed with two diﬀerent ramp-downtimes. Two points are worth noticing: First, a slow ramp compared to a fast one showsa reduction of the vortex density D< in arrays with fully randomized phases at low
J/T . The diﬀerence directly shows the fraction of tightly bound pairs that annihilateon the long ramp. Second, a slower ramp shows vortex activation at lower (J/T )50% ,confirming that free or very loosely bound vortices occur only at higher T (lower J).
Specifically, the data clearly show a range around J/T ≈ 1.4 where only tightly boundpairs exist.
To map the experimental ramp-down time-scale to theoretically more accessible
vortex-antivortex pair sizes, we compare the observed number of vortices in fully randomized arrays at low J/T to simulations of vortex distributions in a hexagonal array

83

c) 6

a)

Vortices

5
4
3
2
1
0

+

III

+
e)

Io

II

1/ 3 10.14 I II
0.12

vortex density

+

2/ 3
III

I

II
III “>”
configuration

8
6

I

4

II
III
“>”
0
1/ 3 1
2/ 3 >1.5
smallest surviving pair size [d]pair size [d]
>1.5
“>”
2

smallest surviving pairs

10

0.10

8

0.08
0.06
0.04

6

< (J/T<<1)> (J/T>>1)

0

4
2

0.02
0.0

12

20
40
ramp-down timescaleτ [ms]inferred vortex numberb)Vortices surviving
ramp-down+ vortexo antivortex

d) 10

0
60

Figure 3.8: Time-to-length mapping based on vortex-antivortex annihilation.(a) simulation of vortices and antivortices in an array with random phases, (b) smallest√ possible
pair sizes in a √hexagonal array of period d: configuration I - pair size d/ 3, II: sized, III: size 2d/ 3. (c) simulated vortex pair size distribution (“>”: free vortices orpairs larger config. III), (d) cumulative distribution. (e) Mapping between ramp-downtimescale τ and estimated size of the smallest pairs surviving the ramp (upper axis).
The diﬀerence D< − D> measures the number of observed vortices surviving the ramp
(right axis). Comparison to the simulated vortex distribution yields a size estimate of
the smallest surviving pairs (upper axis).

84
with random phases. In these simulations, following Ref. [141], we count a vortex if
all three phase diﬀerences in an elemental triangle of junctions are ∈ (0, π), or if all are∈ (−π, 0). A snapshot of a simulated vortex distribution is shown in Fig. 3.8(a). Within
the central 20 lattice sites, comparable to the experimental region of interest we find,on average, a total of 10 vortices. 6 vortices occur in nearest-neighbor vortex-antivortex
pairs [configuration I in Fig. 3.8(b)], 1.7 (0.4) occur in configuration II (III) respectively, and 1.9 occur in larger pairs or as free vortices. To relate these time-independent
simulations to the experiment, we show in Fig. 3.8(d) the relevant cumulative vortex
distributions, i.e. all vortices occurring in pairs larger than a given lower cutoﬀ size. For
a given experimental ramp-down duration, we expect only those vortex configurations
to survive which are above a lower cutoﬀ pair size imposed by the ramp-down rate.
In Fig. 3.8(e) we compare the simulated cumulative vortex distributions to experimentally measured vortex numbers as a function of ramp down timescale, to obtain
the desired time-to-length mapping. Downward triangles show the decrease of the experimentally measured saturated (low-J/T ) vortex density D< with increasing ramp
timescale τ . The right axis shows the inferred number of vortices that survived theramp. Approximately 11 vortices are observed for the fastest ramps, in good agreementwith the total number of vortices expected from the simulations (indicated as grey bars).
For just somewhat slower ramps of τ ≈ 5 ms, only 3 vortices survive, consistent with
only vortices in configuration II & III or larger remaining (indicated in Fig. 3.8(e), topaxis). 11 . For τ � 30 ms ramps less than 2 vortices remain, according to our simula√tions spaced by more than 2d/ 3. Thus we infer that ramps of τ ≈ 30 ms or longer√
allow time for bound pairs of spacing � 2d/ 3 to decay before we observe them.
With this time-to-length mapping we now return to the observations in Fig.3.7.
For the slower ramp we observe vortex activation at lower (J/T )50% , confirming that11

The very short√ annihilation time of configuration I pairs [ τ < 5 ms in Fig. 3.8 ] is not unexpected:
their spacing d/ 3 = 2.8 µm is comparable to the diameter of a vortex core in the bulk condensate(after VOL rampdown).

85

midpoint (J/T)50%

1.6
1.4
1.2
1.0
0.8
0

20
40
ramp-down timescale ! [ms]All vortices(incl. smallest pairs)
contribute60

Only pairs > config. III
and free vortices
contribute

Figure 3.9: A glimpse at BKT vortex pair unbinding: A downshift in the midpoint(J/T )50% of vortex activation curves, such as shown in Fig. 3.7, is seen for slow rampdown times. This is consistent with the occurrence of tightly bound pairs around J/T ∼1.4, whereas loosely bound or free vortices occur at low J/T ∼ 1 only.

free or very loosely bound vortices occur only at higher T (lower J). In Fig. 3.9 weplot the midpoint (J/T )50% of vortex activation curves versus the applied ramp-downtime. The data quantitatively show a shift of (J/T )50% from 1.4 for fast ramp times
when all vortices are expected to contribute to the signal, to 1.0 for slow ramp times
when only loosely bound vortices survive. The data therefore reveal that loosely bound√pairs of size larger than 2d/ 3, or indeed free vortices, do not appear in quantity until
J/T ≤ 1.0, whereas more tightly bound vortex pairs appear in large number already for
J/T ≤ 1.4. This result clearly illustrates the mechanism of vortex-antivortex unbinding
with increasing temperature or decreasing superfluid coupling, which underlies BKT
theory.
A further interesting observation concerns the width of the vortex activation
curve. The relative width, determined from fits to data such as the ones shown in
Fig. 3.7, is ∆(J/T )27−73 /(J/T )50% ≈ 0.3, independent of ramp-down duration. This

86
width is neither as broad as in a double-well system [123, 60], where the coherence
factor rises over a range ∆(J/T )27−73 /(J/T )50% ≈ 1.4, nor as broad as expected fromour simulations 12 of an array of uncoupled phases, each fluctuating independently�with ∆φRM S = T /J, for which we find ∆(J/T )27−73 /(J/T )50% ≈ 0.85. Presumably

collective eﬀects in the highly multiply connected lattice narrow the curve. On theother hand, the width is 3 times larger than the limit due to spatial inhomogeneity in
J, suggesting contributions to the width due to finite-size eﬀects or perhaps revealing
the intrinsically smooth behavior of vortex activation in the BKT regime.
In conclusion, we have probed vortex proliferation in the BKT regime on a 2D
lattice of Josephson-coupled BECs. Allowing variable time for vortex-antivortex pairannihilation before probing the system provides a time-to-length mapping, which reveals
information on the size of pairs with varying J/T .

12

Following [141], in simulations we count a vortex if all three phase diﬀerences in an elemental
triangle of junctions are ∈ (0, π), or if all are ∈ (−π, 0).

Chapter 4
Rapidly rotating 2D Bose gases in a 1D optical lattice

4.1

Motivation
In this chapter I describe a set of experiments where rapidly rotating Bose-

Einstein condensates are loaded into a one-dimensional optical lattice aligned with therotation axis. In this way a stack of two-dimensional rotating Bose gases is created (Fig.
4.1). There are several conceptual motivations for these experiments.
The first one is the possibility of achieving low filling factors, i.e. low numbersof condensate atoms per vortex, as is required to reach fractional quantum Hall (FQH)
states in a rotating BEC (see Appendix D). This field has attracted a vast amount oftheoretical interest, e.g. [47, 118, 152, 165, 29], and spurred several previous experimental eﬀorts [143, 32]. The idea is sketched in Fig. 4.1: A rotating BEC with 3×106 atoms
and 150 vortices, i.e. a filling factor ν = 2 × 104 , is split by the optical potential intoa stack of ≈ 200 rotating two-dimensional “pancake” condensates. For each pancake
the filling factor is now ν ≈ 100, tantalizingly close to values around ν ≈ 10, where
vortex lattice melting due to quantum fluctuations, and the transition to FQH states, is
expected (see Appendix D). With a relatively modest further reduction of the numberof condensate atoms and careful attention to keeping temperatures low, this first step
toward fractional quantum Hall physics appeared to be within reach. Vortex lattice
melting in an optical lattice was analyzed theoretically e.g. in [154, 153].
The second motivation is that a layered stack of rotating superfluids bears many

88
analogies with the layered structures of high-temperature superconductors in a magnetic
field. By this analogy, a large range of finite-temperature physics of low-dimensional
systems might be accessible, the study of vortex lattice fluctuations being one example.
In addition, the ability to vary the interlayer tunnel coupling may allow studies of therich physics associated with the 2D-3D dimensional crossover [169].

(a)!"#$~8Hz(b)!"#$~60Hz%~100%~2 104gn~100Hzt~0.3Hz..
.
t~10kHzgn~1kHz..
.
~200 wells

Figure 4.1: (a) “Bulk” rotating BEC with ν ≈ 2 104 atoms per vortex. (b) Rotating
BEC loaded into a 1D optical lattice. The most dramatic changes are the reduction ofdimensionality of the individual “pancake” BECs, and the reduction of the filling factor
to ν ≈ 100 for each pancake. Also indicated are typical values for interaction energy gn
and single-particle tunneling frequency t.

As a third possible topic of study, even the physics of a single vortex in an opticallattice is rich and surprising, see the work of Stoof and coworkers [103], as well as [84].
Kelvin modes, the helical oscillations of the vortex line, in a continuous BEC of radius
R follow the dispersion relation [59] (k: Kelvin mode wavevector):RωK (k) = �k 2 /2m log( �)(1 − 2/(kR)2 )ξ 1 + (kR)2(4.1)

The negative frequencies of long-wavelength modes indicate an energetic (but not dynamical) instability of the vortex. These modes contribute to the vortex spiraling outof the BEC. In particular, the zero-energy mode is expected to be heavily thermally
excited. In an axial optical lattice of period d and single-particle tunneling frequency t,

89
the Kelvin mode dispersion relation is transformed toωK,OL (k)/(2π) ≈ 7.2 t × sin2 (kd/2) − �/(mR2 ) log(R/ξ)(4.2)

This expression shows that tuning the tunneling strength t allows control over thevortex dispersion relation; in particular it alters the wavevector at which the zeroenergy mode occurs, which should be observable in side-view images of the vortex. A
possible experiment studying such vortex-fluctuations is shown in Fig. 4.2.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.2: Study of single vortex dynamics in a 1D optical lattice. (a) A single vortex
in a bulk BEC is pinned (b) on a tightly focused single laser beam (dashed) to create
a controlled initial condition. (c) The optical lattice is applied and (d) the vortex isreleased from the pinning potential, allowing the study of its subsequent fluctuating
dynamics (e). (f) Expansion image of a single vortex line as in (a), demonstrating thefeasibility of imaging a single vortex in side-view.

4.1.1

The physics of a finite-temperature array of rotating 2D Bose gases

Rotating BECs in a one-dimensional lattice represent a complicated system, as
can be guessed from Fig. 4.1(b). Coherence along the lattice is established by the
Josephson coupling J = Nwell × t between lattice sites. However as the Josephson
coupling is weakened in a deep lattice, a dimensional crossover occurs from a bulk
3D BEC to a stack of 2D BECs on individual lattice sites. Even before the twodimensional rotating BECs in individual lattice sites become completely independent,
increasing thermal fluctuations of the 2D vortex lattices additionally weaken the axial

Energy [nK](a) 6000
5750
5500
5250
5000
4750
4500

(i) VOL=1Er(ii) VOL=4Er

VOL

"ax

1750
1500
1250
1000
750
500
250

VOL
(iii) VOL=40Erh"axh"ax/2gn
10(b) 105
Energy [nK]

90

20

30

J/Nv
103

40
J

T~!lat10

TM,BKT

0.1

loss ofvortexvisibility

Ec

10-3
0

10

20

30

40

VOL [Er]
Figure 4.3: (a) Optical lattice depth (red), lattice-site harmonic oscillator frequency
�ωax (black, solid) and zero point energy �ωax /2 (dashed), and interaction energy gn =µlat (Eq. (1.54), blue) vs. lattice depth measured in units of Er = 152 nK. A total of
2×106 atoms rotating at Ω/ωρ = 0.925 was assumed. Insets: (i) Optical potential (red),
approximate bound state energy (black), chemical potential (blue) at VOL = 1 Er – thelattice potential does not yet support a bound state, the BEC remains connected acrossthe lattice. (ii) At VOL = 4 Er a bound state exists at ≈ �ωax /2, and the BEC is split
into an array of condensates. Interactions and temperature are small compared to �ωax ,i.e. axial motion is frozen out and the condensates enter the 2D regime. (iii) Deep latticeat VOL = 40 Er . (b) Josephson coupling J (black), and coupling per vortex lattice unit
cell J/Nv (blue). Temperature T ≈ µlat (red) is always larger than the expected 2D
vortex lattice melting temperature (magenta, Eq. (D.15), Appendix D). Once J/Nv
is suﬃciently reduced (see text), 2D vortex lattice fluctuations develop on each latticesite. Experimentally, for VOL � (13.5 − 20) Er vortex visibility is lost (Section 4.3.2) inimages that are integrated through the whole optical lattice array.

91
phase coherence. In the following we will approach this problem piece by piece. The
important parameters of this system are shown in Fig. 4.3.
1.) Josephson coupling between lattice sites. At low lattice depth we expect
this to be the dominant energy scale. In all experiments in this chapter, interparticle
interactions are negligible compared to the energy separation of Bloch bands, thus the
single-particle tunnel coupling t (and hence the Josephson coupling J = Nwell × t) andthe charging energy Ec are quite accurately described by equations (1.52) and (1.53),
respectively. Similarly to the experiments on the BKT transition, the lattice systemhere is in the Josephson regime described in Section 1.5.2, where J ≡ Nwell × t >> Ec ,but Ec >> N 2J

(see Fig. 4.3). Upon reducing J by raising the lattice depth we thus

well

expect to see a competition between the coherence-preserving Josephson coupling andthermal phase fluctuations along the lattice direction, which will lead to decoherence
between lattice sites. Quantum fluctuations are not expected to play a significant role.
Based on the results from Chapter 3, the naive expectation for the condition when phasecoherence is lost between the the pancake lattice sites due to thermal fluctuations would
be J ≈ T (reached at VOL ≈ 45 Er ), similar to the BKT transition of a static BEC in a
2D optical lattice.
But in the case of a rotating BEC in a 1D lattice we need to think harder: Clearly,
a vortex lattice on each lattice site violates the assumption of a spatially uniform condensate phase on each site, which was the basis of the simple description of a Josephson
junction array only in terms of J, Ec and T . On intuitive grounds, if the 2D vortex
lattice undergoes significant fluctuations, each vortex lattice unit cell of size ∼ �2 may
act as an independent “piece” of the condensate, with a coupling strength to an adjacent
optical lattice site of order J × �2 /R2 ≡ J/Nv where R is the 2D condensate radius and
Nv = R2 /�2 is the number of vortices. This single-vortex-lattice-cell coupling strength
J/Nv is also shown shown in Fig. 4.3, and I will explicitly demonstrate its relevanceshortly.

922.) Rotating, finite-T, 2D BEC on each lattice site. For deep opticallattices the Josephson coupling is essentially zero, leaving us with a stack of individual
rotating pancake condensates. According to the discussion in Appendix D, under thepresent experimental conditions such 2D vortex lattices will undergo significant thermal
fluctuations, while quantum fluctuations of the vortex lattice are almost certainly notsignificant (the filling factor in all experiments is ν ∼ 100, still a factor 10 above thecritical filling factor for which quantum melting is predicted).
First, consider BKT-type vortex lattice melting due to the thermally activatedoccurrence and unbinding of vortex-antivortex pairs, which disturbs and ultimately
melts the surrounding ordered vortex lattice. The predicted melting temperature, Eq.
(D.15), is on the order of TM,BKT = (6 − 8) nK over the range of optical lattice depths
VOL = (10 − 40) Er . The actual temperature of even the coldest samples (Fig. 4.3(b),solid red line) is T ≈ µlat ≈ 50nK 1 , i.e. much higher than the melting temperature.
From this we conclude that, as soon as the 2D BECs experience a suﬃciently suppressed
axial Josephson coupling, the 2D vortex lattice should melt.
Secondly, also long-wavelength, phonon-like vortex lattice thermal fluctuations are
expected to be strong enough to induce vortex lattice melting. According to Eq. (D.14),
the respective melting temperature is TM,long ≈ (13 − 18) nK for VOL = (10 − 40) Er ,smaller than estimated sample temperatures.
3.) 3D - 2D crossover with increasing lattice depth. As shown in Fig.
4.3(a), as soon as the lattice depth reaches VOL � 4 Er the BECs on individual latticesites become two-dimensional, in the sense that only the lowest axial oscillator state oneach site is occupied and axial motion on-site is frozen out. However at this point thestack of 2D BECs is still strongly Josephson-coupled along the lattice direction and still
forms an eﬀectively 3D system.
1

Careful inspection of in-situ images reveals hardly any normal cloud, and if present its size hardly
extends beyond the BEC. In cases where a fit to this normal component is possible, T ≈ (1 − 1.5)µlat
is found.

3

(a)

(b)

2.5
J

!x2/l2!x93

C2=0 C2=0
T=100nK

2

1.5

"

0
!#
0

J/Nv

1

T=35nK

0.5

"
0

10

20

VOL [Er]30

40

Figure 4.4: (a) Simple model (see text) for thermally activated vortex displacements ∆x
in a stack of rotating BECs in an optical lattice (normalized to the vortex lattice unit
cell size �, Eq. (1.23)). Displacement of a single pancake vortex costs an in-plane energy∝ C2 ∆x2 due to vortex lattice deformation, and (zoom-in) a fraction ∝ ∆x2 /�2 of theJosephson coupling energy J/Nv due to the resulting phase diﬀerence ∆φ(�r) between
lattice sites. (b) Results: Blue (red) solid lines: sample T = 35 (100) nK, respectively.
Dashed lines: same, but assuming vanishing in-plane energy cost TM,BKT ∝ C2 = 0.
Solid black line: ∆x/� � 0.28 is the Lindemann criterion for melting of the vortex
lattice. Dashed black line: region of observed loss of vortex lattice visibility.

Here we develop a simple model for the crossover between this eﬀectively 3D
vortex lattice and a stack of truly independent 2D rotating BECs, that occurs uponfurther reduction of the Josephson coupling. We consider the thermally activated vortex
displacements under the influence of both the Josephson coupling and the in-plane
vortex lattice stiﬀness. We focus on a single vortex lattice unit cell and two adjacent
optical lattice sites. The situation is sketched in Fig. 4.4. The displacement of asingle pancake vortex by ∆x costs an in-plane energy 2 E2D ≈ 2C2 ∆x2 due to thevortex lattice deformation (C2 is the vortex lattice shear modulus). In addition, thisdisplacement results in a nonzero phase diﬀerence between condensates on adjacent
optical lattice sites, and thus costs a fraction of the Josephson coupling energy per unit
cell J/Nv : Eax = (J/Nv ) × �1 − cos(∆φ(�r) )�cell where ∆φ(�r) is the local phase diﬀerence2This expression comes from E2D =�

d 2 x C2�

� ∂εx �2
∂x

+

�

∂εx
∂y

�2 �

[21] where εx is the vortex dis-xx
placement, here chosen along x (Fig. 4.4). Both derivatives are of order ∂ε
≈ ∂ε
≈ ∆x
, and
∂x
∂y
�
2
integration is over one unit cell of area ∼ � .

94
between adjacent optical lattice sites, and � ... �cell is a spatial average over one vortex
lattice unit cell. Numerically we find Eax ≈ (J/Nv ) × 1.8∆x2 /�2 , where � is the sizeof the vortex lattice unit cell, Eq. (1.23). Equipartition tells us that there should be
an amount kB T of thermal energy (kB T /2 kinetic and potential energy) available forthis one degree of freedom of the vortex lattice, i.e. kB T = Eax + E2D , resulting in a
prediction for the average vortex displacements:
∆x2
kB T
=
�2
2C2 �2 + 1.8J/Nv(4.3)

with �2 = �/(mΩ), C2 = nSF,2D �Ω/8 for our parameter regime [50]. Using the 2D
vortex lattice melting temperature TM,BKT , Eq. (D.15), we can rewrite this in the
more meaningful form
∆x2
kB T
= √
2
�
2 3 TM,BKT + 1.8J/Nv(4.4)

This prediction is plotted in Fig. 4.4 for the expected range of condensate temperatures(T = (35 − 100) nK) in the optical lattice. An ordered lattice will melt as soon as themean-square displacements exceed a fraction of the lattice constant ∆x ∼ 0.28 � (theLindemann criterion, see e.g. [21]), which occurs in Fig. 4.4 between VOL = (15−20) Er
lattice depth 3 . The dashed lines in Fig. 4.4 are obtained by setting TM,BKT = 0, or
equivalently C2 = 0, showing that, as long as T > TM,BKT as in our experiments,
the precise value of the in-plane energy cost does not significantly influence the vortex
lattice melting condition.
Once the 2D vortex lattices have melted in our system, we also expect position
correlations between vortices in adjacent pancakes to get lost due to axial decoherence
across the lattice: Even if some degree of position correlation between vortices in diﬀerent sites remains in the vortex liquid, it is now very clearly the Josephson coupling per
vortex, J/Nv , that competes with kB T for axial coherence, and as Fig. 4.3 shows, J/Nv
3

2

To check our model we also apply it to an isolated 2D pancake (J = 0). With ∆x= 2√3TkB T
�2

M,BKT

and the Lindemann criterion we obtain a condition for vortex lattice melting, T /TM,BKT ≈ 0.3, close
to the expected value of unity.

95
approaches T for VOL = (15 − 20) Er , i.e. as soon as the 2D vortex lattices have melted.
Once axial coherence is lost, there is no incentive for vortices in diﬀerent optical latticesites to remain locked together.
We expect the prediction of this two-site model for the 3D-2D crossover to carryover to the full optical lattice system in a way similar to the double-well physics correctly predicting the approximate conditions for the BKT transition in a 2D Josephson
junction array.
In summary, raising an optical lattice in the axial dimension induces confinementof rotating BECs to the two-dimensional lattice sites. Upon further reduction of theJosephson coupling between lattice sites, phase fluctuations develop in the vortex lattices
on each site. On the other hand, the developing phase fluctuations in the 2D plane act
to enhance the decoherence between lattice sites. Once the 2D pancakes are suﬃciently
isolated from each other, in our system with temperatures T larger than the vortex
lattice melting temperature TM,BKT , the vortex lattice is certainly expected to melt.
Vortices on adjacent optical lattice sites act independently, and the 3D-2D crossover iscomplete. It is thus the competition between kB T and J/Nv that ultimately decides
on both the 2D vortex lattice melting and the 3D-2D crossover. Similar physics (butwithout rotation of the condensate, instead with 2D fluctuations arising from the BKTtransition of the static 2D superfluids) was investigated experimentally in Kasevich’s
group [91], and was studied theoretically by Giamarchi and coworkers [38].
4.1.2

Setup and alignment

The experimental setup is shown in Fig. 4.5. The light source for the opticallattice is a power amplifier seeded by an external cavity diode laser (ECDL) at 852 nm
4 .
4This system delivers 130 mW of light through a single-mode fiber. The optical

The amplifier chip was designed for 830 nm, but an available master laser diode and easy locking
to the 133 Cs line at 852 nm made us decide for 852 nm; the amplifier gain at 852 nm is reduced by 1/2,
but is suﬃcient for our purposes

-60dB

AOM
intensity
control

12mW Tapered
amplifier
500mW

ECDL
852nm

96

Monitor for light
returning from OLa) Light source for 1D optical lattice-30dB
frequency
stabilization
to Cs D2 lineb) 1D axial optical lattice setup

glass
cell

MOT,
Imaging !/4
780nm

Picomotors
2” f=700

CCD

1.5”
f=125

" BEC
Dichroic mirror
front R>99% 810nm
front&back AR 780nm

2” f=300

f=200

Optical lattice
852nm

Picomotors

-30dB

f=250

Angle
polished

Figure 4.5: (a) Light Source for the 1D optical lattice at 852 nm. An external cavity
diode laser (ECDL), frequency locked to the D2 transition of 133 Cs at 852 nm, seeds atapered amplifier chip. The light is intensity controlled by an acousto-optical modulator(AOM) and sent to the experiment via single-mode fiber. Optical isolators prevent
frequency noise caused by optical feedback to the ECDL. (b) Lattice setup. The opticallattice light is focused and noncollinearly combined with MOT and imaging beams. A
dichroic mirror retroreflects the lattice light onto the BEC where the optical lattice isformed, parallel to the BEC rotation axis ΩBEC .

lattice beam path, depicted in Fig. 4.5(b), is set up ≈ 1” above the optical tableto achieve maximum immunity to vibrations. A −30dB optical isolator is inserted toprevent a cavity eﬀect between the fiber output and the retroreflecting mirror. We
tried several options of combining optical lattice and MOT beam path using various
polarizing beamsplitter cubes (PBS), but found significantly reduced perturbation ofrotating BECs using a noncollinear geometry without PBS, as shown in Fig. 4.5(b).

97This setup in addition removes the quarter wave plate for the MOT from the latticebeam path. The incoming lattice beam is focused to a 1/e2 beam waist of 128 µm on
the BEC, and retroreflected by a dichroic mirror.
The initial lattice alignment is performed in the following steps: We first focus
the incoming lattice beam longitudinally (and of course transversely) onto the BEC,
by maximizing the aspect ratio Rz /Rρ of static BECs loaded into a single beam dipole
trap along the z-axis. The retroreflected beam is then overlapped with the incoming
beam by maximizing the power propagating back through the beam delivery fiber (wemonitor the light rejected by the optical isolator close to the fiber on the laser sourceside in Fig. 4.5(a) ). This step allows optimization of both longitudinal and transverse
beam overlap. A last fine adjustment of longitudinal overlap is done monitoring the
BEC aspect ratio when loaded into the lattice. Day-to-day alignment involves onlytransverse beam overlap.

4.2

Initial optical lattice characterization with a static BEC
Before attempting to load a rotating BEC into the optical lattice, we conducted acareful study of the optical lattice using static condensates. The goal of these studies was(a) to make sure that our understanding of the optical lattice depth based on Eq. (A.2)
was in agreement with experimental results, (b) to verify a procedure for adiabaticallyloading BECs into the lattice, and (c) to ensure that the optical lattice quality was
suﬃcient to avoid heating eﬀects beyond the unavoidable photon scattering.
4.2.1

Optical lattice depth determination from matter wave diﬀraction

As in the studies of the BKT transition, the precise knowledge of the opticallattice depth is essential for accurate knowledge of the site-to-site tunnel coupling. The
modulation contrast of the interference pattern creating the optical lattice is the largest
uncertainty. It depends crucially on longitudinal and transverse overlap of the the

98
+6 hk
+4 hk
BEC

+2 hk
0 hk
-2 hk
-4 hk
-6 hk

VOL = 41 Erecoil

time
Relative population

Pulse duration

1
0.8

3

5

7

9 11 13 15 17 19 21 23 25 [!s]

00 hk
hk

0.6
0.4

2 +2
hk hk
4 hk
+4
hk

0.2
0

5

10

15

20

Pulse duration [!s]25

30

Figure 4.6: Characterization of 1D optical lattice depth by matter wave diﬀraction.
Upper left: schematic of the experiment. A 1D optical lattice is pulsed on briefly, causing
the BEC to diﬀract into discrete momentum components corresponding to transfer of anumber Np of photons (of momentum �k) from one lattice beam to the other, imparting
momentum 2Np �k to the BEC. Population of the momentum components depends
on lattice depth. Upper right: resulting matter wave diﬀraction pattern vs. pulse
duration. Bottom: extracted relative population of momentum components (symbols)
vs. pulse duration, and calculation following Ref. [51] using the lattice depth VOL =41Er calculated from independent measurements of beam waist and laser power.

incoming and retroreflected beams. In addition, the retroreflected beam intensity isattenuated by ≈ 20% due to losses in the beam path. Fortunately, the optical lattice
depth can be measured precisely by pulsing on the lattice for very short amounts oftime, and recording the resulting matter-wave diﬀraction pattern as a function of pulse
duration for fixed intensity (or as a function of intensity at fixed pulse duration) [116, 51].
The result of such a measurement and a calculation following Ref. [51] is shown in Fig.
4.6. The theoretical curve is obtained without free parameters – we use the optical lattice

99
depth calculated from independently measured optical power and beam waist. The
agreement between theory and experiment is very good, confirming the good alignmentof lattice beams and our understanding of the optical dipole potential, Eq. (A.2).
4.2.2

Bloch oscillations as a probe of phase coherence in the lattice
Image

Magnetic trap

(a)

Thold
Optical lattice depth
Time
(b)

(i)

(c)

(ii)

(iii)

(iv)

(v)

(vi)

q-2!/dq
TholdFigure 4.7: Bloch oscillations of a static, phase coherent BEC in an optical lattice.
(a) time sequence: A BEC is loaded adiabatically (250 ms ramp time) into the opticallattice; the magnetic trap is turned oﬀ while the lattice remains on. The BEC accelerates
under the influence of gravity, causing its narrow momentum wavepacket to traverse theoptical lattice’s Brillouin zone (BZ) (b), following q(t) = mg
� t. Upon reaching the BZedge q = π/d, the wavepacket is briefly split into two by Bragg reflection (q = ±π/d),before completely reflecting to q = −π/d, and repeating the oscillation. (c) Momentum
images of BECs released from the optical lattice at various times Thold during the Blochoscillation (see associated labels in (b)). These momentum images are obtained bysuddenly switching oﬀ the lattice and imaging the cloud after 14 ms expansion. (iv) is
at the edge of the BZ – this timing is used in the following figures.

The previous matter wave diﬀraction experiment involves an extremely short100
pulse of optical lattice light and hence tells us nothing about long-term heating anddecoherence eﬀects in the lattice. On the other hand, the final goal here is to load
BECs into optical lattices adiabatically. In this adiabatic case, phase coherence may
be destroyed either by technical fluctuations in the lattice potential, or by fundamental
quantum or thermal fluctuations. In order to test for such eﬀects, we adiabatically(250 ms intensity ramp time) load static BECs into the optical lattice and employ Blochoscillations (see e.g. [108]) as a probe of phase coherence across the resulting array ofmini-condensates. Figure 4.7 explains the basic physics of Bloch oscillations. In thefollowing we concentrate on condensates released from the lattice after half a Blochoscillation, i.e. split into two momentum components at the edge of the optical lattice’s
Brillouin zone, as shown in Fig. 4.7(c)(iv). Phase decoherence across the optical lattice
is signalled by blurring, and eventually merging, of these two momentum components.
Figure 4.8 shows condensates released from the edge of the Brillouin zone, withincreasing optical lattice depth. In our case of a fairly large BEC (NBEC ≈ 1.1 × 106 forthese experiments) it is diﬃcult to disentangle several eﬀects occurring simultaneously:(a) s-wave scattering between the two momentum components released from the lattice,e.g. visible in Fig. 4.8 at VOL = 5.5 Er as a diﬀuse cloud centered between the two
momentum components, (b) additional mean field eﬀects in the lattice seen e.g. in Ref.
[108], increasing the complexity of the Bloch oscillation pattern, and (c) dephasing of
the array due to thermal [123] (or possibly quantum [115]) eﬀects. Our results are ingood qualitative agreement with earlier results obtained e.g. in Kasevich’s group [115].
Significant technical decoherence eﬀects can thus be ruled out. However due to theabove-mentioned diﬃculties we did not attempt to push these experiments towards aquantitative understanding. It must however be clarified that the blurring of the density
distributions in Bloch oscillation images in Fig. 4.8 is NOT due to strong heating: (a)the energy released in the horizontal direction for the cloud at VOL = 48.5 Er can beconverted to a temperature assuming the cloud is uncondensed. We obtain T = 76 nK.

101
Energy [nK]

105

J

103

TBKT

10

T~"lat48. 5 Erecoil

q

0.1

Ec
0

10

q-2!/d2030

4035 Erecoil50

VOL [Er]18. 5 Erecoil
opticallatticedepth

15 Erecoil

5. 5 Erecoil

q-2!/dq
TimeFigure 4.8: Bloch oscillations of a non-rotating BEC show loss of phase coherence. At
VOL = 5.5 Er two clearly separated momentum components are visible. With increasing
lattice depth, the components blur out and merge, partly due to phase decoherence,
partly due to axial spreading of the BECs due to lattice-enhanced mean field repulsion.
Blurring of the density distributions is NOT due to strong heating, see text. Inset: Fora static BEC, the temperature T ≈ µlat (red) remains smaller than the static BKTtemperature of 2D BECs on each lattice site TBKT (magenta) – in contrast to rotating
BECs, large in-plane fluctuations are not expected. Axial coherence is governed bycompetition between J (black) and T (red).

The ideal-gas BEC transition temperature for the 2D pancakes is much higher, Tc =218 nK, confirming the sample must be largely condensed. Also a hypothetical BKTtransition of the 2D BECs is not expected, as TBKT ≈ 200 nK (Eq. (1.31)). (b) an
in-situ image of the BEC in-lattice at VOL = 48.5 Er shows no bimodal distribution.
Fitting a zero-temperature Thomas-Fermi profile gives a condensate radius within 20 %of the expectation from Section 1.5.4. Fitting a thermal cloud profile gives a temperature
T = 85 nK, similar to the result from release energy, and much lower than Tc . (c) wehave loaded static BECs into VOL = 41 Er deep lattices and adiabatically released themback to the magnetic trap, with no significant heating.

102

48. 5 Erecoil

(a)

Thold
Magnetic trap
Optical lattice depth

7 Erecoil
Time

(b)

(i)

(ii)

(iii)

(iv)

(v)

q-2!/dq
TholdFigure 4.9: Revival of phase coherence of a non-rotating BEC after optical lattice ramp
to VOL = 48.5Er , 10 ms hold, and return to VOL = 7Er , followed by half a Bloch oscillation. This experiment proves the absence of strong technical sources of decoherence.

A final test for the adiabaticity of the loading process and the amount of irreversible decoherence mechanisms is the ability to recover phase coherent Bloch oscillations after ramping the optical lattice intensity slowly up (over 250 ms), and then backto a low lattice depth. Results are shown in Fig. 4.9. The optical lattice is ramped up
to VOL = 48.5 Er and then down to VOL = 7 Er , where a significant coherent fractionof atoms undergoing Bloch oscillations is recovered, further confirming the absence ofstrong technical sources of decoherence.

4.3

Results: Rapidly rotating Bose-Einstein condensates in a 1D
optical lattice
The following first results on rotating BECs loaded into an optical lattice fall into

three categories:• First, even under optimized conditions, we observe a spin-down of rotatingcondensates loaded adiabatically into an axial optical lattice. The reason for

103
this spin-down is suspected to be of technical nature.
• Second, we observe a loss of visibility of the vortices in expansion images, when
integrated over the full 1D lattice. This loss of visibility is indicative of the
breaking up of vortex lines of the bulk BEC into pancake vortices in individual
layers. The driving force for the loss of register between pancakes may wellbe the thermal fluctuations in the two-dimensional vortex lattices, as describedin Section 4.1.1, with the caveat that we cannot definitively exclude technicalreasons as well.
• Third, before even being able to observe the above eﬀects, we encountered andsolved various technical diﬃculties unique to the process of loading rotating
BECs into an axial optical lattice.
For most results, I quote the single-particle tunneling element t, as opposed to the full
Josephson coupling J = Nwell × t, or the Josephson coupling per vortex lattice unit cell
J/Nv . For almost all experiments however, Nwell = (1 − 2) × 104 and Nv ≈ 50 − 150.
4.3.1

Spin-down of the rotating BEC in the optical lattice

Although the spin-down of rotating condensates in an axial optical lattice is likelyof technical nature, we give details of our results here, as this is completely unexploredterritory. The optical lattice intensity ramp used in the experiments in this section isa roughly exponential 250 ms long ramp-up, followed by a specified “hold-time”, and amirror-symmetric ramp-down back to the magnetic trap.
When loading a rotating BEC into the optical lattice, a decay of vortices fromthe condensate is observed, roughly exponential with increasing optical lattice hold time
(Fig. 4.10). From experiments at diﬀerent optical lattice depths we find that the 1/evortex decay time scale τvortex decreases approximately inversely proportional to thelattice depth, τvortex ≈ 4 s/(VOL /Er ). From experiments with diﬀerent initial vortex

104

lattice
depth

image
hold timenumber of vortices remaining

01s,
no evaporation

0.25s

0.5s+hold time

200

400
600
800
hold time [ms]

time

120
100
80
60
40
20
0
0

1000

Figure 4.10: A rotating BEC is loaded into a VOL = 10 Er deep optical lattice, held fora variable time, and released back to the magnetic trap. The remaining vortex number
shows exponential decay with hold time. One radial trap period, which is the timescaleof vortex lattice rotation, is 30 ms.

number at fixed lattice depth we find the decay timescale to be roughly independent ofinitial vortex number.
For our typical optical lattice loading procedure, this vortex decay means that
we can load a rotating BEC, initially containing a vortex lattice with 150 vortices, intoa VOL = 41 Er deep lattice and back to the magnetic trap, with ≈ 40 − 50 vortices
remaining, as shown in Fig. 4.11. Under the conditions of VOL = 41 Er , J/Nv ≈ 2 nK,
i.e. at typical temperatures of (50 − 100) nK the system is way beyond the expectedvortex lattice melting and loss of axial phase coherence (Section 4.1.1). As a side note,
we also find that after a lattice hold time of 1.5 s at VOL = 20 Er , repeatedly one single
remaining vortex is observed, a good starting place for single-vortex experiments.

105

Number of vortices remaining

140
lattice
depth

120

image
10ms

100
0

80

0.25s
time

0.51s

60
40
t=60 Hz
t=8 Hzgn=400 Hz gn=600 Hz

20
0

0

5

t=0.3 Hz
gn=800 Hz

10 15 20 25 30 35 40 45
optical lattice depth [ERecoil]Figure 4.11: Number of vortices surviving adiabatic ramp to the given optical lattice
depth and back to the magnetic trap. Inset: time sequence – the optical lattice isramped up over 250 ms, held for 10 ms, ramped back down over 250 ms, and an insitu image is taken. Vortex number is calculated from BEC number and rotation rate.
Single-particle tunneling rates t and BEC chemical potential gn are indicated.

From experiments with varying initial condensate temperature T /Tc , shown in
Fig. 4.12, we find that the vortex number decay (i.e. rotation loss of the BEC) speeds
up with increasing T /Tc . For these experiments, the initial rotation rate is kept constant
while varying T /Tc (initial rotation rate and T /Tc are measured from a first in-situ image
in the magnetic trap). The finite-T rotating BEC is then loaded into an optical latticeof VOL = 9.5 Er depth, and adiabatically released back to the magnetic trap, where its
final rotation rate is measured from a second in-situ image. In Fig. 4.12(b), measured
initial and final rotation rates are shown, revealing a clearly enhanced loss of rotation
for T /Tc � 0.3, while rotation loss is negligible for cold BECs at this lattice depth.
These results indicate that interactions of the rotating condensate with its thermal
cloud contribute to the condensate’s spin-down, but do certainly not exclude additionaleﬀects unrelated to the thermal cloud. The results however point to the necessity of an106

latticedepth initial
image

(a)

final
image
0.2s

0.25s
evaporation off

time

1s

0.25s

0.9

BEC rotation rate !"#$

0.8

0.7

(b)

0.6

0.5

final T/Tc

0.7

0.6

0.5
0.4
0.3
0.2
0.1
0.0
0.0

(c)

0.4
0.3
0.2

initial rotation rate !initial
final rotation rate !final
0.1

0.2

0.3

0.4

initial T/Tc

0.1
0.5

0.6

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

initial T/Tc

Figure 4.12: BEC rotation rate decay vs. initial T /Tc at fixed optical lattice depth
VOL = 9.5Er . (a) time sequence. (b) initial and final BEC rotation rate vs. initial T /Tc
showing increased loss of rotation with increasing T /Tc . Lines are guides to the eye. (c)final T /Tc vs. initial T /Tc . The dashed line corresponds to no heating.

eﬃcient cooling scheme applicable to BECs inside the optical lattice.
4.3.2

Loss of vortex lattice visibility in the optical lattice

While the reason for condensate spin-down is likely technical, its timescale can bemuch longer than the timescale for the loss of vortex visibility discussed in this section,
allowing the separate study of this likely more fundamental process.
The optical lattice intensity ramp for experiments in this section is a roughly
exponential, 190 ms long ramp-up, followed by a specified “hold-time”. For most experiments, the optical lattice and magnetic trap are then turned oﬀ abruptly and simultaneously as shown in Fig. 4.13(a). This causes all 2D condensates from individual latticesites to recombine, followed by rapid radial expansion of the combined bulk condensate.

107

(a)

latticedepth(c) 0.6

image

V

(b)

0.5

time 0.19s

10ms

lattice
depth

image

time

0.4
0.3
0.2
0.1

V
0

vortex visibility

0

0.39s

2.39s

0.0

BEC without vortices

05

10 15 20 25 30 35 40
optical lattice depth Recoil[E] ][ERecoil(d)

V=9.5 Erecoil
t=80Hz

V=24 Erecoil
t=4Hz

V=28 Erecoil
t=2Hz

V=41 Erecoil
t=0.3Hz(e)

Figure 4.13: Vortex lattice visibility vs. optical lattice depth. (a) time sequence forresults shown in (d). Rotating BECs are loaded adiabatically into the optical lattice.
For imaging of vortex lattices inside the optical lattice, the optical lattice is turned oﬀ
abruptly. The released 2D pancake BECs recombine and expand before the image istaken. (b) time sequence for results in (e). Here the optical lattice is ramped down
adiabatically, to smoothly recombine vortices. A 2s wait time in the magnetic trap isinserted to allow vortices to straighten up after rampdown. (c) sample-averaged vortex
visibility extracted from data in (d). Results in (d), (e) show a complete loss of vortex
visibility in the optical lattice above VOL = 28Er , while BECs returned to the magnetic
trap show that large numbers of vortices are still present.

A destructive image is then taken, showing the full recombined condensate in the plane
perpendicular to the optical lattice axis. For control experiments, where indicated, a
190 ms exponential optical lattice ramp-down back to the magnetic trap is employed
and images are taken after an additional specified wait time in the magnetic trap.
Figure 4.13(d) shows images of vortex lattices loaded into optical lattices of in-108
creasing depth, and then abruptly released. At low lattice depths, e.g. VOL = 9.5 Er
in Fig. 4.13(d), a vortex lattice is clearly visible, albeit somewhat distorted, either by
excitations resulting from the optical lattice loading procedure, or possibly by longwavelength thermal fluctuations. With increasing optical lattice depth, vortex visibility
gradually degrades until it is fully lost. A sceptic might suspect that e.g. for the imageat VOL = 41 Er the condensate was heated above the condensation temperature, andwhat we observe is a normal cloud. The control experiment in Fig. 4.13(e) however
reveals that, when adiabatically released into the magnetic trap, a rotating condensate
with a significant remaining vortex lattice is obtained.
For the 190 ms lattice intensity ramps employed in both Fig. 4.13 and Fig.
4.14, the vortex lattice initially retains some visibility up to VOL ≈ 30 Er . For VOL �
15 Er however this visibility then decays with time, as shown for VOL = 20 Er in Fig.
4.14(a), lower panel. Figure 4.14(b) shows the time constant for vortex visibility loss
(black symbols), versus the single-particle tunneling rate t. The timescale for the vortex
visibility loss scales with t, indicating that the underlying reason is not an in-plane
process but associated with the weakening tunnel coupling. This appears consistent
with our model for vortex lattice melting in the 3D-2D crossover which is governed by
the competition between T and J/Nv , as described in Section 4.1.1. Also shown in Fig.
4.14(b) is the timescale for vortex number loss (red symbols). The dependencies of vortex
visibility loss and vortex number loss are markedly diﬀerent, so that it appears that these
two eﬀects have diﬀerent origins. Specifically, below t � 30 Hz (i.e. VOL � 15 Er ) vortex
visibility is lost much faster than vortex number, allowing vortex visibility to be studied
relatively undisturbed from condensate spin-down.
Loss of vortex visibility under equilibrium conditions occurs around VOL ≈ 15 Er :For lower lattice depth, some degree of vortex visibility remains at all times, indicatingthat some degree of phase coherence across the whole optical lattice is preserved, while
for deeper lattices vortex visibility is lost after some equilibration time. We comparea)Lattice
depth

Thold

V
0

b)

Thold

30ms

260ms

500ms

5ms

35ms

80 ms

time

t=8Hz (V=20 ERecoil)
Thold

1000
vortex visibility decay time [ms]

c)

109

t=60Hz (V=10 ERecoil)

Vortex visibility decay
Vortex number decay

100

10
1

10

100

t [Hz]
Figure 4.14: Vortex lattice visibility decay time vs. J. (a),(b) rotating BECs are loaded
adiabatically into optical lattices, and the hold time is varied. Vortex lattice images
after abrupt optical lattice turn-oﬀ show time-dependent loss of vortex visibility. Data
are shown for weak (a) VOL = 10 Er , and moderately strong (b) VOL = 20 Er lattice. (c)extracted timescale for loss of vortex visibility vs. tunneling strength t (black squares).
For t � 60 Hz vortices remain visible at all times (data point at t = 60 Hz is a lower
limit on vortex visibility decay time). Red symbols: timescale for vortex number decay
shows a markedly diﬀerent behavior than timescale for visibility decay. Importantly,
for low t (high VOL ) visibility is lost much faster than vortex number, allowing us toseparately study the two processes.

the conditions around VOL ≈ 15 Er in these experiments, to the expected conditions for110
vortex lattice melting and the 3D-2D crossover (see the discussion around Eq. (4.4)):
We have t ≈ 30 Hz, Nwell ≈ 16000 and Nv ≈ 100, resulting in J/Nv ≈ 230 nK.
The chemical potential is µlat ≈ 35 nK resulting in an estimate for the temperature ofT ≈ (35−50) nK. According to Eq. (4.4) this results in ∆x/� ≈ 0.14−0.2, very close to
the Lindemann condition for vortex lattice melting ∆x/�|melt ≈ 0.28 (melting according
to Eq. (4.4) would be expected around VOL = (17−19) Er ). We therefore speculate thatthe observed loss of vortex visibility is directly related to thermally activated in-plane
vortex lattice melting and the resulting 3D-2D crossover, in which position correlations
between vortices in diﬀerent lattice sites are lost.
V = 13.5 Erecoilt = 30 Hz

V

250 ms

image
Thold
time
Thold

0 ms

30ms

200ms 400ms

V = 8 Erecoilt = 100 Hz
Vi=28 recoils
ti=2 Hz

Thold

200ms

400ms

600ms

800ms

Figure 4.15: Vortex lattice visibility can be destroyed and restored in an optical lattice.Upper left: The lattice is initially ramped to (VOL = 28Er ), where vortex visibility is lost (lower left). Visibility is restored upon ramping the lattice depth down to(VOL = 8Er ) (lower right), whereas when ramping to (VOL = 13.5Er ) it takes much
longer to restore and reaches a lower final value.

As a last result, we show that vortex visibility, and even vortex lattice ordering can

111
be restored while the BEC remains in the optical lattice: In Fig. 4.15 the optical lattice
is initially ramped to a depth where vortex visibility is fully destroyed. The lattice isthen ramped back to a lower depth, and held for a time thold . When ramping back downto VOL = 8 Er , visibility of vortices is restored after a hold time thold ≈ 0.5 s while in
the optical lattice. After a longer hold time thold ≈ 1 s even an ordered vortex lattice isobserved. When instead ramping back down to VOL = 13.5 Er , just below the conditions
for loss of vortex lattice visibility, restoration of vortex visibility is incomplete, indicating
the vicinity of vortex lattice melting. This experiment shows that both the locking ofpancake vortices in diﬀerent layers of the optical lattice to a more or less straight vortex
line across the full lattice array, as well as positional order in the vortex lattice is not
lost irreversibly, but can be restored once the optical lattice is returned to conditions
where three-dimensional coherence across the array is expected. This appears to befurther proof that the loss of vortex visibility is not due to strong irreversible technicaldecoherence mechanisms, but is likely of a fundamental nature.
4.3.3

Technical considerations when loading a rotating BEC into a 1D
optical lattice

Before being able to observe the results detailed above, we have seen, and largely
removed, a host of excitations that are unique to the process of loading a rotating (asopposed to static) BEC into the optical lattice. A quadrupole (m = −2) surface mode[55] is excited by slight azimuthal ellipticity of the optical lattice potential, but canbe canceled by precise relative position alignment of incident and retroreflected opticallattice beams. Tkachenko modes [44] can be excited by radial anharmonicity of the
combined trapping potential of magnetic trap and optical lattice, but can be minimized
by choice of a large enough optical lattice beam waist. A scissors mode [39] of thevortex lattice may be excited by symmetry axis misalignment between magnetic trap
and optical lattice, but can be avoided by tilting the magnetic trap to align it with the112
optical lattice axis.

yxz

Figure 4.16: m = −2 quadrupole surface mode excitation [55] when loading into azimuthally elliptical optical lattice potential, e.g. created by imperfect overlap (leftsketch) of incident (solid) and reflected (dashed) lattice beam. This excitation can becancelled by slight adjustments to the lattice beams’ overlap in the xy-plane. Upper
left, imperfect overlap, strong excitation; lower right, good overlap, reduced excitation.

Surface waves. The first attempt to load a rapidly rotating lattice is almost
guaranteed to end up with the excitation of large-amplitude m = −2 quadrupole mode
excitation [55], as shown by the large amplitude elliptical deformation of the rotating
BEC in Fig. 4.16. Such modes in a rapidly rotating BEC are known to be nearresonantly driven by a static azimuthal ellipticity of the trap potential, and possibly
higher-order imperfections of the optical lattice intensity profile. To lowest order, thisexcitation can be removed by fine-tuning the transverse overlap between incoming andretroreflected lattice beam.
Tkachenko waves excited by anharmonicity. Stringari and coworkers [49]have shown that an anharmonic perturbation to the radial trap potential can excite
Tkachenko oscillations. While loading from magnetic trap to optical lattice potential,

113
V
r
(a)

(b)

0

195ms

(c)

(e)

(d)

225ms

!

255ms

top view
lattice
depth

17Erecoiltime

Figure 4.17: Tkachenko mode [44] excitation due to radial anharmonicity [49] whenloading a rotating BEC into the optical lattice. (a)-(d) time sequence of images during
optical lattice ramp-up showing the growing Tkachenko excitation, and sketches of the
BEC (grey) in the combined radial potential formed by optical lattice (narrow dimple)
and magnetic trap (sketches not to scale). The optical lattice intensity ramp is shown
below. (e) flow pattern of atoms in the rotating frame causing Tkachenko oscillation.

the BEC experiences some degree of radial anharmonicity (sketched in Fig. 4.17(a)-(d),not to scale), and the resulting Tkachenko excitation could be observed. This eﬀect wasremoved by increasing the optical lattice waist, such that the BEC is confined at all
times in the harmonic region of the trapping potential, and is also reduced by a slower
loading procedure.
Scissors mode due to optical lattice axis-tilt. The optical lattice axis initially was misaligned by a few degrees with respect to the magnetic trap symmetryaxis. When the BEC was loaded too rapidly from magnetic trap to optical lattice, asmall-amplitude scissors mode [39] was excited. The eﬀect is the same as the result oftipping a spinning top, i.e. a precessing motion of the rotation axis. To mitigate thiseﬀect, the axis of the TOP trap was tilted to match up with the optical lattice axis. As

(a)

(b)
!ini TOPlatticeaxis
depth TOP
OL
axis
axis

114

10ms
TOP
axis

!ini

!(t)
images

time

0.26s
(c)

!
!

!

!

BEC
time
(d) tilting the
TOP
axis

Figure 4.18: Scissors mode excitation due orientation mismatch (a) between magnetic
trap and optical lattice. (b) experimental time sequence for observing the scissors mode,
induced by transient application of the slightly tilted optical lattice. (c) time sequenceof in-situ side-view images showing the precessing motion of the BEC. Excitation of thismode was minimized by tilting the TOP magnetic trap axis which defines the initialrotation axis. (d) images of BECs in tilted TOP traps – the TOP trap axis can bealigned with the optical lattice.

a diagnostic, a large-amplitude scissors mode was excited by loading the rotating BEC
rather rapidly into the optical lattice (and releasing it back into the magnetic trap foreasier diagnosis), as shown in Fig. 4.18. When the magnetic trap and optical latticeaxes are aligned, no scissors mode is observed.
Multiple reflections. Multiple reflections of the optical lattice beam in ourcell, which is not antireflection-coated, may interfere (likely non-collinearly) with themain optical lattice beams, causing uncontrolled changes to the interference pattern,including modulations of the potential in the plane of the optical lattice sites. These
modulations are suspected to act as “speed bumps” to rotating BECs, while leaving

115
static BECs largely unaﬀected. This eﬀect may be suspected to be responsible forunexplainable heating and fast spin-down that we observed for certain optical latticealignments but not for others. This last technical diﬃculty was avoided by using a large
angle of incidence (≈ 5◦ ) with respect to the glass cell surfaces. However multi-passreflections are not fully controlled in this way.
An order of magnitude estimate of the amount of uncontrolled light that cancause trouble is as follows: A perturbation ∆ that reaches a significant fraction of the
BEC’s chemical potential gn would certainly aﬀect the vortex lattice. Interference ofthe full lattice potential with a perturbing potential Vpert leads to a perturbation ∆�
Vpert
given by the peak-peak interference modulation of V∆=
4
VOL . Assuming Vpert toOL
result from a nth order reflection from the glass cell windows (intensity In ) compared to�In VOLnthe single-lattice-beam intensity I0 ∝ VOL /4 we have ∆=4gn
4I0 gn . At VOL = 30 Er

the barrier height is VOL = 4 µK (see Fig. 4.3), whereas the chemical potential is
∆2
∆3
∆4
OL
1
gn = 50 nK, so Vgn= 80. This results in ∆gn = 20, gn = 3, gn = 0.5 and gn = 0.06

for first, second, third and fourth order reflections, respectively 5 . Geometrically, with(≈ 5◦ ) angle of incidence on the glass cell, we are sure up to third order no reflection
can hit the BEC, but higher order reflections are not controllable and still could present
a significant problem.

4.4

Conclusions and outlook
In this section, we have shown that it is possible to load rapidly rotating BECs

into a 1D optical lattice, and to create a stack of rotating 2D BECs. We have observeda loss of vortex visibility in images that are integrated along the rotation axis, throughthe whole array of 2D rotating condensates. This loss of vortex visibility occurs underconditions where we expect growing thermally activated in-plane vortex lattice fluctua5

I assume 4% reflection from glass cell windows and take into account beam spreading from the focus(Rayleigh range is 6 cm, roughly equal to the distance a reflected beam travels before returning to the
w2

w2

n

n

BEC). The intensity In is thus In /I0 = (4 10−2 )n × w20 with the ratio of beam waists w20 = 1/(1 + n2 ).

116tions and a resulting 3D-2D crossover, in which position correlations between vortices in
diﬀerent lattice sites are lost. Vortex visibility can in part be reversibly restored whenramping back to lower optical lattice depths, where coherence across the array again
creates an eﬀectively 3D system.
In our experiments, the filling factor is ν ≈ 100, to our knowledge the lowest
value reached to date. This is only a factor ≈ 10 away from the predicted melting of
the vortex lattice due to quantum fluctuations and a possible transition to fractional
quantum Hall states. To possibly reach this regime, however several technical diﬃculties
will have to be overcome: First, the observed fast spin-down of the vortex lattice has tobe slowed down significantly, as the timescales required to reach equilibrium in the fragile
vortex lattices of rapidly rotating BECs can be very long. Second, the temperature ofcondensates has to be reduced significantly, in order to suppress the presently dominant
thermal fluctuations, and to prevent thermal excitations of the very fragile fractional
quantum Hall states, as discussed in Appendix D. Third, a further fill factor reductionto ν ≈ 10 appears feasible.
With our experiments we certainly did not reach a full quantitative understandingof a stack of rotating pancake BECs in an optical lattice. As we suspected from the
beginning, the experiments described here had to overcome large technical diﬃculties,
and so the results presented in this chapter are at least in part technical in nature.
Nonetheless I think there is promise, especially for further studies of thermal eﬀects on
rotating 2D quantum gases.
For future experiments one would certainly like to develop more sophisticatedways of probing the properties of the rotating 2D BEC and vortex lattice on each lattice site. The vertical images in this thesis are integrated through the whole array oflattice sites, and can thus not distinguish between long-wavelength vortex excitations
and short-range phase fluctuations between sites. Can Bloch oscillations provide complementary information?

117
In addition, the 3D-2D dimensional crossover deserves further study: How dovortices influence the site-to-site phase coherence of a BEC in an axial optical lattice?
Are there connections to work on “large superconductive Josephson junctions” where
spatial variations of the phase are important? What is the role of dissipation when “reconnecting” vortices by ramping down the optical lattice? This question also concerns
the reconnection of vortices upon sudden optical lattice turn-oﬀ employed for imaging
of vortex lattices in this chapter.
I am confident that this work was not the last attempt at answering these questions about this interesting system.

Chapter 5
Vortex lattice dynamics in rotating spinor Bose-Einstein condensates

5.1

Background and motivation
In this chapter, a rotating two-component superfluid is studied, that is describedby a pseudo-spin-1/2 order parameter [3]. The work is an extension of our earlier workon rapidly rotating scalar BECs to the case of multicomponent, or spinor BECs. No
optical lattice potentials are involved.
As described in the introduction, a lot can be learned about a superfluid system when it is set rotating. Depending on the complexity of the order parameter asuperfluid can carry angular momentum in diﬀerent ways. A single-component superfluid exhibits quantized vortices that, in a dilute-gas single-component BEC, organize
into a regular hexagonal Abrikosov lattice [6][see Fig. 5.3(a)]. Such lattices are also
formed in Type II superconductors in a magnetic field [6] and in rotating superfluid 4 He
[53]. Multi-component superfluids, with their pseudo-spin order parameters, carry angular momentum in spin textures (Skyrmions) [92, 110], which may again form lattices[111, 90, 132], such as in the superfluid phases of liquid 3 He [139] and in certain regimesof two-dimensional electron systems [33].
The two-component, pseudo-spin-1/2, superfluid under study here consists of twomagnetically trapped internal states of 87 Rb [48, 104]. State |1� represents spin-down
and state |2� represents spin-up, and the azimuthal component of the spin is fixedby the relative phase Φ between the state |1� and |2� wavefunctions. Both the large-

119scale density profiles and the vortex lattice structure depend on the nature of interand intracomponent interactions. The interaction energy, determined by three scalar
coupling constants g11 , g22 and g12 and the population densities n1 and n2 , is given by
Eint =

�d3 x[g11 n21 + g22 n22 + g12 n1 n2 ](5.1)

For attractive intercomponent interaction (g12 < 0) it is energetically favorable forthe spatial profiles of both components to coincide, and a hexagonal vortex lattice
is expected, as in the one-component case. For repulsive intercomponent interaction,
realized in our 87 Rb system, it may be energetically favorable for the two components
to reduce spatial overlap. In a static BEC this may be achieved by macroscopic phaseseparation, which occurs when g12 >√g11 g22 . In a rotating two-component BEC on theother hand, the presence of vortex lattices in both components allows for more subtle
separation eﬀects, such as interlacing the two lattices. Vortices in either component may
be filled by fluid of the other component, making this phase separation favorable alreadyfor g12 > 0. The pseudo-spin-1/2 order parameter then forms a Skyrmion lattice.
Studies by Mueller and Ho [111] and Kasamatsu et al. [90] predict many diﬀerent
2 , which is approximated by ourstable vortex configurations in the vicinity of g11 g22 = g12
87 Rb system, and it appears conceivable that diﬀerent structures might be accessed e.g.

by population imbalances. Among the predicted structures, interlaced square lattices
occur over a wide range of parameters. Presumably this is because in a square lattice,unlike in a triangular lattice, each vortex in component |1� can have all its nearestneighbor vortices be in component |2�, and vice versa [100].
Under suitable conditions we indeed observe a spatial separation of vortices inthe two components, followed by the formation of ordered interlaced square lattice
structures, as shown in Fig. 5.3(b). Its formation and decay dynamics, as well as its
static and dynamic properties are examined in this work. In summary, we first preparea hexagonal vortex lattice in a one-component BEC in an internal atomic state |1�,RF ~1 MHz

120

|2!
mF = 1

F=2
microwave~6.8 GHz

Collisional loss:
Lifetime ~ few s

|1!

F=1

mF = "1Figure 5.1: Magnetically trapped states |1�, |2� used to create two-component BEC
in 87 Rb. A coherent coupling between the states is achieved by a resonance-enhanced
two-photon transition involving a microwave and an RF photon.

and then coherently transfer a fraction of the superfluid to a diﬀerent state |2�. The
subsequent evolution of this pseudo-spin-1/2 superfluid towards a state of oﬀset squarelattices involves an intriguing interplay of phase-separation and -mixing dynamics, both
macroscopically and on the length scale of the vortex cores, and a stage of vortex
turbulence. The stability of the square structure is proved by its response to applied
shear perturbations. An interference technique shows the spatial oﬀset between thetwo vortex lattices. Vortex cores in either component are filled by fluid of the othercomponent, such that the spin-1/2 order parameter forms a Skyrmion lattice.

5.2

The magnetically trapped spinor BEC system in 87 Rb
Our 87 Rb two-state system [48, 104], depicted in Fig. 5.1, consists of two mag-netically trappable hyperfine-Zeeman levels of the 87 Rb atom - |F = 1, mF = −1�,
henceforth called |1�, and |F = 2, mF = 1�, henceforth called |2�. Spatial overlap of thetwo states is realized in a harmonic, axially symmetric magnetic trap with oscillation
frequencies {ωρ , ωz } = 2π{7.7, 4.9} Hz. An electromagnetic coupling between the twostates can be achieved via a two-photon transition, involving a microwave photon at∼ 6.833 GHz and a radio frequency photon at ∼ 1 MHz. A short-pulse application ofthis coupling drive yields a desired, spatially uniform amount of population transfer

121

Figure 5.2: Experimental time sequence for creating a two-component BEC.

between the two states, instantaneous with respect to the external dynamics of the twostates. In the pseudo-spin-1/2 picture the eﬀect of the coupling drive is to cause spinrotations [48, 104]. Inelastic atomic collisions limit the lifetime of the |2� population toa few seconds. The decay does not cause |2� atoms to convert back to the |1� state but
rather to leave the trap altogether.

5.3

Experiments and results
To study the rotational properties of this two-superfluid system, we initially create

regular hexagonal vortex lattices in BECs in state |1�, as described in earlier work[56, 143, 45]. Here we start with near-pure condensates containing (3.5 − 4) × 106 atoms,
rotating at a rate Ω ≈ 0.75 × ωρ about the z-axis. As shown in Fig. 5.2 a “transfer
pulse” of the coupling drive then transfers a fraction of the population into state |2� , inthis work 80 − 85 % as discussed below. After a variable wait time we take two imagesof the system. A nondestructive phase-contrast image is taken either of one component
alone or of both components simultaneously [48, 104], along an axis perpendicular to therotation axis (“side view”) while the system is still trapped. In order to optically resolve
the vortex structures we expand either the |1� or |2� component of the condensate by afactor of 9, to a diameter of ∼ 600 µm, before a second, destructive image is taken along

122
the rotation axis (“top view”). The other component is removed at the beginning of
the expansion. The details of expansion and imaging have been described in Ref. [45].

b)

a)

1#
c)

1#
d)

$
%
|k|

k1
%!"
k2

Figure 5.3: (a) Hexagonal vortex lattice in a one-component BEC. (b) Square lattice,viewed in the |1� component of a two-component BEC. (c) Reciprocal (k−) space of thehexagonal lattice, obtained by 2D Fourier transform of (a), showing 6 peaks spaced by60◦ . (d) k− space of the square lattice. The reciprocal lattice vectors k1 and k2 enclosean angle ϕ12 = 90◦ .

In Fig. 5.4 we analyze the formation and decay dynamics of the square lattice
structure. Figs. 5.4(a) and (b) show time sequences of top-view images of the expanded |1� and |2� states, taken in diﬀerent experimental runs. The time evolution
after the transfer pulse involves several stages. For the first ∼ 0.1 − 0.25 s surprisingly
little dynamics is visible and certainly no structural transition in the vortex lattice isseen in either component. From ∼ 0.25 − 2 s a turbulent stage evolves in both components in which vortex visibility degrades significantly, shown in Fig. 5.4[a(ii)] and Fig.
5.4[b(ii)]. As we will show, this turbulence is directly linked with the transition fromoverlapping hexagonal vortex lattices to interlaced square lattices. From 2 − 3 s square

123

a)

i

ii

iii

iv

v

vi

|1! 0.05s
b)
i

1s

2.5s

4s

5.5s

9s

ii

iii

iv

v

vi

|2! 0.15s&c) "#$#%

1.5s

2.5s

4s

5s

9s

4

6

time [s]

9

300
240
180
120
60
0
0

d)
d)

2

i

ii

iii

iv

v

0.1s

0.6s

1s

2s

4s

Figure 5.4: (a) Time sequence of images of state |1�, after ∼ 80% population transfer
to |2�, showing evolution from a hexagonal lattice over a turbulent stage to a square
structure and back to a hexagonal lattice. (b) Also state |2� forms a square lattice (iv)
before its decay. (c) Detailed time evolution in reciprocal space. Intensity in an annulus
along the ϕ coordinate [defined in Fig. 5.3(c)] is shown on the ordinate. The initial
6-peak structure of the hexagonal lattice vanishes quickly because of turbulence. From
3 − 5.5 s square lattices give rise to a 4-peak structure. Around 6 − 9 s a transition backto hexagonal lattices occurs. (d) Two-color side view images of the initial turbulent
evolution. State |1� (|2�) appears bright (dark) on gray background. The fine verticalfilament structures in (ii-v) are due to mutual filling of vortex cores.

124
domains emerge from the turbulent state, and defects propagate out of the lattice {Fig.
5.4[a(iii)]}. From 3 − 5.5 s stable square lattices are observed in both components {Fig.
5.4[a(iv)] and Fig. 5.4[b(iv)]}. At this stage, around 4 s, despite the large (80 − 85 %)initial population transfer to state |2�, the number of |2� atoms has decreased to only
1.5 × 105 , while state |1� contains (5 − 7) × 105 atoms. As the |2� state population
continues to decay, the vortex lattice planes bend {Fig. 5.4[a(v)]} and a transition backto a hexagonal lattice in state |1� takes place {Fig. 5.4[a(vi)]}. During the transition
from square lattices to a hexagonal lattice no turbulence occurs.
A more quantitative analysis of these dynamics is possible in reciprocal space.
Figure 5.4(c) shows the time evolution of the intensity within an annulus in reciprocal
space, defined in Fig. 5.3(a), which contains the reciprocal lattice peaks. Initially six
peaks are visible, separated by 60◦ , forming the reciprocal lattice of a hexagonal vortex
lattice. Because of turbulence these peaks vanish between ∼ 200 ms and 2 s. After2 − 3 s a four-peak structure appears, that is stable for a period of ∼ 2.5 s. The observed
ratio of reciprocal lattice vector lengths [defined in Fig. 5.3(d)] k1 /k2 = 0.98(2) andthe angle ϕ12 = 95(3)◦ between k1 and k2 , clearly identify a square lattice. At ∼ 5.5 s
the appearance of two additional peaks signals the onset of a transition back to thehexagonal state, which is completed by ∼ 9 s.
To examine the origin of the initial turbulence, we show in Fig. 5.4(d) a time
sequence of in-trap side-view images, where the |1� (|2�) state appears bright (dark).
As visible in Fig. 5.4[d(i)], a macroscopic component separation from the initially homogeneous two-component superposition to a ball-shell structure takes place within
50 − 100 ms [48, 104]. During this period of dramatic axial separation, both the individual vortices and the overall vortex lattice remain remarkably quiet, as viewed along
the rotation axis {see Fig. 5.4[a(i)] and Fig. 5.4[b(i)]}. Around 600 ms, fine filament
structures appear at the intercomponent boundary {Fig. 5.4[d(ii)]}, coincident with thefull development of vortex turbulence seen in top-view images {Fig. 5.4[a(ii)]}. The

125
filament structures distort and fill out the whole BEC as the vortex turbulence peaks ataround 1 s {Fig. 5.4[d(iii)]}, and straighten up at around 2 s {Fig. 5.4[d(iv)]} coincident
with the restoration of vortex visibility in top view images. Subsequently the filaments
become less visible as state |2� decays {Fig. 5.4[d(v)]}. We interpret these structures asvortex cores in either component being filled by fluid of the other component, forminga Skyrmion lattice [110]. Filled vortex cores grow in size above the resolution limit ofour side view images and become observable, while empty vortices in single-component
BECs in equilibrium are well below this resolution limit and are not observed in-trap[10]. The initial macroscopic, predominantly axial phase separation has thus evolved
into a microscopic separation of two interlaced vortex lattices. It is this microscopic
separation which gives rise to the observed turbulence.

a)b)

Figure 5.5: (a) Tkachenko mode excitation in the square lattice, observed 250 ms after
beginning of excitation, and (b) after relaxation back to a square lattice structure (at850 ms).

To study the stability of the square lattice, we excite shear perturbations (Tkachenko
modes) [12, 20] in the square lattice by focusing a resonant laser beam onto the centerof the condensate [44]. As shown in Fig. 5.5, the perturbation relaxes to the equilibrium square configuration within 500 − 800 ms after excitation. However, in contrast to
single-component triangular lattices [44], no clear oscillation is observed. Two eﬀects
may contribute: The excitation may be overdamped because of ill-defined boundary

126

Figure 5.6: Experimental sequence for two-component vortex lattice interference.

conditions caused by imperfections in the outer region of the square lattice, where the|2� state population has already decayed. The oscillation may also be masked by random lattice excitations of comparable amplitude that cannot be completely removed in
the short time between formation and decay of the square lattice. However, the returnof the lattice to its square configuration is suﬃcient to demonstrate the stability of thesquare structure in the two-component system.
So far we have presented evidence that both components separately form regularand stable square lattices. Filament structures indirectly indicate a microscopic-scalespatial separation of vortices. In the following, we employ an interference techniquebetween the two superfluids to more precisely address two questions: Are the vortex
lattices really oﬀset from each other? Do the vortices really only separate from eachother during the turbulent stage, after a surprising delay of 200 ms? Figures 5.7[a(i)] and(iii) show results of a simple simulation of interference between two square vortex lattice
wavefunctions, spatially oﬀset by 1/2(�a + �b), where �a, �b are the two basis vectors of thesquare lattice. Two values (Φ = 0 and Φ = 45◦ ) of the relative quantum phase Φ between
the two wavefunctions 1 are considered. A population ratio of N|2> : N|1> = 20 : 80
1
The relative phase Φ is well-defined and measurable but is relevant only in the presence of coupling
between the states. Directly after population transfer it is zero and uniform across the sample, but witheach experimental run it quickly acquires an arbitrary value, e.g. because of slight irreproducibility inthe trapping potentials.

127

ii

a) i
! =0

o

4s

iv

iii
! =45

o

4s

b)

i

ii

iii

iv

0.2s

1s

3s

9s

Figure 5.7: (a) Vortex lattice interference. (i) and (iii): Simulations for diﬀerent valuesof the relative phase Φ between the |1� and |2� state wavefunctions. (ii) and (iv):
Experimental results. (b) Time evolution of the interference patterns shows no vortex
oﬀset before the turbulence (i) but patterns characteristic of oﬀset vortices afterwards[b(iii)],[a(ii)],[a(iv)].

128
was assumed, as in the experiment at 4 s. Evidently two very dissimilar patterns areseen, consisting either (Φ ∼ 0) of dark patches in between two vortices, surrounded by abright square structure, or (Φ ∼ 45◦ ) of staircase-like vortex arrangements separated bybright stripes. We checked that the qualitative appearance of such simulations changes
drastically when altering the spatial oﬀset from 1/2(�a + �b).
To observe such structures experimentally, we apply a π/2 “interference” pulse
to the two-component system just before expansion for the top view image, as shownin Fig. 5.6. This pulse results in a phase-coherent transfer of population between thetwo states, thus creating interference. If the π/2 pulse is applied under conditions
when regular square lattices are expected in both components, the observed images,
Fig. 5.7[a(ii)] and (iv), agree qualitatively with the simulations, clearly demonstratinga spatial oﬀset close to 1/2(�a + �b) between the vortex lattices.
The qualitative agreement between experimental results and simulation indicatesthat vortex lattice interference may be employed to examine the separation process ofthe vortex lattices. When varying the wait time between the transfer pulse and theπ/2 interference pulse, we observe the following qualitative features [Fig. 5.7(b)]: Until
turbulence occurs (after 200 − 300 ms) there are no indications for separation of thevortex lattices. Figure 5.7(b)(i) shows a hexagonal lattice with good vortex contrast,
very similar to images of each single component at this time. This confirms that during
the initial 200 − 300 ms of dramatic macroscopic component separation, vortices inthe two components continue to form identical and overlapped lattice structures, whenviewed along the rotation axis. Only after this surprisingly long delay, and after the
turbulent stage do interference patterns show a grouping characteristic of oﬀset vortices{Fig. 5.7[b(iii)]}. This observation is further proof of the direct link between the
turbulent period and the microscopic separation of the vortex lattices.
In conclusion, we have observed a new vortex lattice structure in rotating twocomponent BEC. Each component carries a square vortex lattice, and the lattices are

129
interlaced. Vortices in both components are filled by fluid of the other component, forming a Skyrmion lattice. This structure is stable, as evidenced by relaxation of applied
shear excitations back to the square structure. An intriguing turbulent stage accompanies the transition from overlapped hexagonal lattices to interlaced square lattices.
Vortex lattice interference has been used to study the oﬀset dynamics of the two lattices.

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Appendix A
Accurate optical dipole potential calculation

In this Appendix I describe the formulae used in this thesis to precisely calculate
the dipole potential exerted on atoms by oﬀ-resonant light. As shown in Section 1.5.1,
the potential energy of an atom in a light field of intensity I(�r) may be written as

Udip (�r) = −1Re(α)I(�r)
2�0 c

(A.1)

For multilevel atoms, the polarizability contains contributions from all excitedstates, which have to be summed up.
Depending on the laser detuning with respect to a given transition, fine structureand hyperfine structure components may have to be considered individually, with thetransition strength of the respective line distributed over all sublevels. In this thesis thelaser detuning is always much larger than any of the relevant hyperfine splittings, but
not necessarily much larger than fine structure splittings.
Following Grimm and coworkers [70], the polarizability then contains a scalarterm, independent of the ground state magnetic sublevel mF , as well as a vector term
which acts diﬀerently on the magnetic sublevels, and is therefore called a fictitious
magnetic field [52]. For the example of the dominant 5S1/2 −→ 5PJ transition of 87 Rb,
which is split into two fine structure components, the D1 line (J = 1/2) at a wavelength

142of λ = 794.98 nm and D2 line (J = 3/2, λ = 780.24 nm), one obtains
Udip (�r) =�

J=1/2,J=3/2πc2
32ωat,J

�

ΓJ
ΓJ
−
δJ
δ+,J

��

�
� · B̂ I(�r)c2J − iσJ gF mF (E�∗ × E)

(A.2)

Here, δ = ω − ωat is the detuning of the light field at frequency ω from the atomic
� r)resonance at ωat , and δ+ = ω + ωat represents the so-called counter-rotating term. E(�is the polarization unit vector, and ∗ denotes the complex conjugate. B̂ is a unit vector
along the magnetic field direction. Iˆ is the unity operator, cJ is a coeﬃcient for the� �transition 5S1/2 −→ 5PJ (c23/2 = 2, c21/2 = 1), and σJ = +1
−1 for the transition to 5PJ ,� �
3/2
with J = 1/2 .
Spontaneous scattering of laser photons occurs at a rate

Γscat (�r) =�

J=1/2,J=3/2πc2
32�ωat,J

�

ΓJ
ΓJ
−
δJ
δ+,J

�2 �

� · B̂
c2J − iσJ gF mF (E�∗ × E)�I(�r) (A.3)

Each scattering event imparts a recoil momentum to the atom, on average increasing
its energy by �2 k 2 /2m × 4�cos2 (θ)� = 2Erec . Here, θ is the half angle between incoming
and scattered photon, and Erec = �2 k 2 /2m is the laser recoil energy. The rate of
temperature increase is therefore of the order of Γscat (�r) × 2Erec .

Appendix B
Numerical determination of finite-T BEC parameters in a double wellpotential

Here, the numerical procedure is outlined for determining the Josephson coupling
J for the experiments on the BKT transition described in Chapter 3. To our knowledge, no accurate analytical formula exists that relates the Josephson coupling J to
the number of particles per well Nwell and the applied optical lattice depth VOL . In
the framework of the two-mode approximation, the Josephson coupling is found fromenergy diﬀerences between the two lowest states of the double well potential [9]:
1 AS
AS
SY M
SY M
J = Nwell ((Etot− Etot) − (Eint− Eint))2(B.1)

SY M (E AS ) is the total energy per particle of the lowest spatially symmetric
where Etot
tot(antisymmetric) state in the double well potential, given Nwell particles in each well,
SY M (E AS ) are the respective interaction energies per particle.
and Eint
int

To evaluate (B.1) we numerically find the three-dimensional ground and firstexcited state of the Gross-Pitaevskii equation, Eq. (1.9), for an interacting BEC in a
double well potential. This potential, a 2D cut of which is shown in in Fig. B.1, isconstructed from a hexagonal optical lattice of 4.7 µm period formed from three beams
intersecting in the xz plane 1

at 120◦ , and a harmonic confining potential with aharmonic oscillator frequency of 15 Hz along the y-direction. The potential is padded1
The convention for the y and z axes in this Appendix is diﬀerent than throughout the thesis!
Everywhere else, the 2D lattice is in the xy plane and z is the transverse axis.

144
with a constant value outside dashed lines in Fig. B.1 to confine the wavefunction to
two wells, with the section lines chosen such as to avoid discontinuities in the potential.
Initial, spatially symmetric (antisymmetric) trial wavefunctions, are constructed fromthe sum (diﬀerence) of two Gaussians in the xz plane, centered at the two potential
minima, while in the y-direction a wavefunction resulting in a parabolic Thomas-Fermi
density profile is employed.
Calculations are performed on a 3D grid of 210×180×70 points (directions x,y,z).
Here x is the axis connecting the two wells, z is the second dimension in the opticallattice plane, and y is the third dimension, along which each lattice site is elongatedand cylindrically symmetric. The real space size of the grid is 21 µm × 180 µm × 14 µm,
resulting in spatial grid steps of dx = 0.1 µm; dy = 1 µm; dz = 0.2 µm.
60Z [grid points]

50

40

30

20

10
40

60

80

100

X [grid points]

120

140

160

Figure B.1: 2D cut through the potential energy grid used for numerical solution of
GPE. Dark regions show the double well potential minima. Outside the dashed lines
the optical lattice potential was replaced by a constant potential to isolate one doublewell. Note that the vertical dimension of the image is compressed by 2x compared toreal space!

The iterative numerical procedure, based on a split-step-Fourier imaginary time
propagation, is sketched in Fig. B.2. As is well known, imaginary-time propagation
finds the ground state of a given system. What is less well known is that for an initial
trial wavefunction of odd spatial symmetry, that symmetry is preserved throughout the
propagation, and thus in our case the first excited state is found.
The zero-temperature part of the calculation is sketched in the upper loop of

145

For %R trial = symmetric and antisymmetric%R trialeF" iH K !

F-1

2

H K $ h 2 k 2 2m
H R $ H pot # H in
H pot $ V( r )

N

2

Every
10th
step

H in $ g (2 N well ) %R # 2 gnTH

! ( "i dtee
E pot $ %R H pot %R" iH K !

2%R %R

Ekin $ %K H K %K

%K %K

Updatethermal
cloud

F

%R %R

Ein $ %R H in %R

F : Fourier transformN : normalize" iH R!

nTH $1 max m 3 / 2&z m'T3 m $1
2

Etot $ E pot # Ein # Ekin

z$e

))

*

old
" (V( r ) # 2 g ( 2 N well ) %R # 2 gnTH
" Etot ) / kT)Sym
J $ N well EtotAS " Etot
" 1 / 2 EinAS " EinSym**

Figure B.2: Finite-temperature code for numerical solution of the Gross-Pitaevskii equation, to determine the Josephson coupling energy J. ΨR is a real-space wavefunction,that is propagated by a series of steps τ in imaginary time until convergence of the totalenergy Etot is achieved. The procedure is repeated twice, for a symmetric (antisymmetric) ΨR , to find the ground (first excited) state, respectively. J is then computed
according to Eq. (B.1). For finite-temperature calculations, a thermal cloud density
profile nT H is constructed and iteratively updated as well. Thermal cloud and condensate mutually influence each other through their mean-field repulsion.

Fig. B.2. To find the ground state, a spatially symmetric trial wavefunction is Fouriertransformed to k-space and propagated under the influence of the kinetic-energy term
Hk for 1/2 imaginary-time-step. After inverse Fourier-transform and normalization, it ispropagated for a full step under influence of potential (Hpot ) and nonlinear interactionenergy (Hin ). It is then again transformed to k-space, and propagated following the
kinetic-energy term Hk for the second 1/2 step. At this point, all contributions to thetotal energy are evaluated. The loop repeats until the total energy converges to a desired

146
accuracy. The calculation is then repeated to find the first excited state, starting froma spatially antisymmetric trial wavefunction. At this point J is calculated according to
Eq. (B.1).
303000 atoms/well4000 atoms/well5000 atoms/well6000 atoms/well7000 atoms/well
8000 atoms/well
10000 atoms/well

28
26

10000

22

J [Hz]

Ry [microns]

24

3000 atoms/well4000 atoms/well5000 atoms/well6000 atoms/well7000 atoms/well
8000 atoms/well
10000 atoms/well

20

1000

100

18
16

Ry=22.69!m*(Nwell/7000)(1/3.8)*(VOL/1300Hz)(1/4.5)

14
0

500

1000

1500

2000

10

J=6.57Hz*Nwell*exp(Nwell/3950-VOL/250Hz)
0

2500

500

1000

Ec=0.167Hz / Nwell(11/20) * (VOL/Hz)(1/2)

0.09

2500

3000 atoms/well6000 atoms/well
8000 atoms/well
10000 atoms/well

25

0.07

20

fPlasma [Hz]Ec [Hz]2000

30

3000 atoms/well6000 atoms/well
8000 atoms/well
10000 atoms/well

0.08

1500

VOL [Hz]VOL [Hz]

0.06
0.05

15

10
0.04

5
0.03

fPlasma=Sqrt(J*Ec)

0
0

500

1000

1500

VOL [Hz]

2000

2500

0

500

1000

1500

2000

2500

VOL [Hz]

Figure B.3: Results of double-well GPE simulations at zero T. Plotted are numerical
data points for axial condensate radius Ry , Josephson coupling J, charging energy Ec
and plasma frequency fp vs. optical lattice depth VOL for various number of condensed
particles per well Nwell .

Results of zero-temperature calculations, repeated for various optical lattice depths
VOL and condensate numbers per site, Nwell , are shown in Fig. B.3. In addition to
Ry (VOL , Nwell , T = 0) and J(VOL , Nwell , T = 0), the charging energy Ec and plasma
frequency fp are shown.
We have extended our procedure to take into account finite-T corrections, by
including the mutual mean-field repulsions between BEC and thermal cloud. This is0.07

Nwell=10000, VOL=1300Hz

24.5 Nwell=8000, VOL=1600Hz
24.0

0.06

(J(T)-J(0))/J(0)=0.30(4)*sqrt(T/(Nwell*"p))Nwell=7000, VOL=1600Hz

23.5

0.05
Nwell=8000, VOL=1300Hz

23.0
22.5

(J(T)-J(0))/J(0)Ry [microns],from TF+MFmodTHC fit to GPE result

147
25.0

Nwell=7000, VOL=1300Hz

22.0

Nwell=6000, VOL=1300Hz

21.5

Nwell=7000, VOL=1000Hz

21.0

19.5

0.03
0.02

Nwell=5000, VOL=1300Hz

20.5Nwell=6000, VOL=1000Hz
20.0

0.04

0.01

Ry(T)=Ry(0)-0.55(10)!m*(T/60nK)
for all data

0

10

20

30
T [nK]

40

0.00

50

60

T=0
0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

T/(Nwell*"p)Figure B.4: Finite T corrections to GPE simulation results. Left: slight compressionof the axial BEC radius with increasing T by the repulsive mean field of the thermal
cloud. Right: increase in J due to lifting-up of the condensate chemical potential by
the repulsive thermal cloud.

achieved by employing a Hartree-Fock-type density profile for the thermal cloud [112],
as given in the lower right box in Fig. B.2. This thermal cloud density nth enters the
BEC wavefunction through its repulsive mean field exerted onto the BEC, i.e. the 2gnth
term in Hin . Conversely, the BEC’s mean field repels the thermal cloud. Therefore thethermal cloud density is iteratively updated after every tenth step of the propagation,according to the evolution of the BEC density profile.
Finite-T corrections are shown in Fig. B.4. They consist of two separate eﬀects:
the left panel shows the slight compression of the axial BEC radius Ry with increasing
T due to mean field repulsion by the inhomogeneous thermal cloud density. The rightpanel shows the second eﬀect, the increase of J due to the lifting-up of the BEC chemical
potential by the thermal cloud’s mean field. The peculiar x-axis T /(Nwell ωp ) was chosenbecause for this parameter combination, numerical data from diﬀerent Nwell , T , VOL
showed best collapse onto one common curve.
Putting it all together, we obtain two results: First, accurate parameter determination for every experimental shot is achieved by interpolation between nearest
numerical data points, as described in Section 3.3. Second, a useful approximation for

148
J results, valid in the range of Nwell and VOL employed in our experiments
J(VOL , Nwell , T ) ≈ 0.315 nK × Nwell exp

�

Nwell
VOL
−
3950
244 Hz��0.59 T× 1+
100 nK�(B.2)

We note however that at the edges of the range of VOL in Fig. B.3, the limitations of
this approximation become visible. For the axial condensate radius we find
Ry (T ) = Ry (0) − 0.55(10)µm × T /60nK

(B.3)

with
Ry (0) ≡ Ry (VOL , Nwell , T = 0) = 22.69 µm × (Nwell /7000)(1/3.8) × (VOL /1300 Hz)(1/4.5)(B.4)
The charging energy is found to be(11/20)Ec = Ec = 0.167 Hz/Nwell× (VOL /Hz)(1/2)

which allows e.g. calculation of the plasma frequency according to fp =

(B.5)
√

J Ec .

Appendix C
Calculation of the vortex pinning potential

Here I calculate the pinning potential experienced by a vortex. Pinning is due tothe density modulation in the BEC created by the optical lattice. A vortex position (x)in a region of low density nSF (x) 1 is less costly, as can be seen by the total energy
E(x) of one vortex of length L given by Eq. (1.20) in the introduction
E(x) = π�2 /m nSF (x)L ln(R/ξ)(C.1)

Here R the Thomas-Fermi radius and ξ the healing length. The logarithmic term is∼ 4 − 5 for our BEC.
For the density I use the Thomas-Fermi approximation and assume VOL < µnSF (x) = 1/g (µ − VOL (x))(C.2)

with g = 4π�2 asc /m, the chemical potential µ and the optical lattice potential VOL (x) =V0 (sin2 (πx/d) with period d. Keeping only vortex-position dependent terms this givesa pinning energy
Epin (x) ≈ −L
VOL (x)asc(C.3)

which is up to a numerical factor consistent with the result of Reijnders et al. [130, 131]
(their results indicate a ∼ half as strong pinning potential).
1nSF (x) is the superfluid density imposed by the external potential, without the presence of thevortex. For the derivation to be valid, nSF (x) must vary smoothly on the scale of the vortex core.

Appendix D
Rapidly rotating Bose gases: Vortex lattice physics and beyond

This extended appendix details several interesting scenarios for rotating Bose
gases [4]. Many of the considerations described herein provided a major motivation for
the general direction of experiments conducted during this thesis.
We are interested here in superfluids containing large numbers of vortices, suchthat the coarse-grained superfluid velocity profile (and with it, the vortex lattice) under� r. One of the most fascinating aspects of such rotating
goes rigid-body rotation v�s = Ω×�superfluids is their fundamental analogy to superconductors, or two-dimensional electron gases, subjected to a strong magnetic field. This is perhaps easiest understood by� experienced by a particle with mass m
noting that the Coriolis force F�C = 2m�v × Ω� is fully analogous to the Lorentz force
moving with a velocity �v in the rotating frame Ω� on a charged particle (q ≡ 2e for Cooper pairs in a superconductor, q ≡ eF�L = q�v × B
� The fields Ω� and B� can be written in terms
for a 2D electron gas) in a magnetic field B.
� most conveniently expressed as A
�C = 1 Ω� r and A
�L = 1 B� r,of a vector potential A,
2 ×�
2 ×�
respectively.
Upon transforming to the coordinate frame co-rotating with the vortex lattice,� L� term. In order to clarify the physicalthe GPE, Eq. (1.9), is modified by the usual −Ω·
eﬀects occurring in the rotating frame, this equation may be rewritten to take the form
�

�
1
m 2 2
2
2
2
�
�
(−i�∇ − Π) + g |Ψ| + V(�r) − Ω (x + y ) Ψ = µΨ
2m
2

(D.1)

151
� ≡ 2mA� C describes the eﬀects of the Coriolis force, V(�r) = 1 m(ωρ2 r2 + ωz2 z 2 ) is
Here, Π2
the magnetic trapping potential, which is radially weakened by the anti-confining eﬀect
2 2
2−m
2 Ω (x + y ) of the centrifugal force, resulting in a reduced radial trap frequency�ωρ2 − Ω2 .

a)

b)

~
Ω=0
Ω
N = 3.8 106c)~Ω = 0.953
ΩN = 3.3 106~
Ω = 0.993
ΩN = 1.9 105

Figure D.1: Side view images of BECs in trap. The rotation axis is vertical. (a) Static
BEC. The aspect ratio Rz /Rρ = 1.57 follows the prolate trap shape. (b) Rotating BEC,Ω̃ = 0.953, (c) Ω̃ = 0.993. Centrifugal distortion causes the BEC to spin out into a
low-density quasi-2D cloud. From Ref. [143].
In this section we examine the high rotation limit, i.e. Ω̃ ≡ Ω/ωρ → 1 [143],
where the centrifugal force quite nearly cancels the radial confining force, and the BEC
spins out into a low-density cloud, as shown in Fig. D.1. For suﬃciently high rotation
rates, the BEC approaches the quasi-two-dimensional regime, where the strength ofgninteractions gn is reduced below the axial oscillator energy, Γ2D ≡ �ω
< 1, and axialz

motion is frozen out. We are then left with a quasi-2D system described by
�
�
1
2
2
� ⊥ − mΩ
� × �r) + g |Ψ| Ψ = µΨ
(−i�∇
2m

(D.2)

The first term describes the 2D kinetic energy of particles moving in the external
vector potential introduced by the rotation. It is identical to the integer quantum-Hall
Hamiltonian, whose eigenvalues are the Landau levels with an energy spacing 2�Ω. Theinteraction energy term provides the necessary ingredient to observe fractional quantum
Hall physics. The situation is illustrated in Fig. D.2.
The physics of such rapidly rotating 2D Bose gases is characterized by two parameters. The first one is the so-called Landau level parameterΓLLL =

gn
2�Ω

(D.3)

152
a)

b)

'$%$(

c)

'$)) !"'$* !"
2!"

!"
-2 -1

0

#$% !"&'
1

2

3 m

-2

-1

0

+%,

gn
1

2

3 m -2

-1

0

1

mmax
2

3 m

Figure D.2: a) single-particle energy levels of the static 2D harmonic oscillator, in acircular basis where m denotes angular momentum projection quantum number. b)
rotation Ω induces an energy splitting between positive and negative m sublevels. c) in
the limit of fast rotation the positive m states form a band of near-degenerate levels - thelowest Landau level (LLL). Higher Landau levels are separated by 2ωρ ∼ 2Ω. Dotted
line: the interaction energy per particle, gn, indicates which states are occupied by thegninteracting rotating Bose gas. If the LLL parameter ΓLLL = 2�Ωis smaller than unity,
only states in the LLL are occupied. The maximum index mmax is identical to thenumber of vortices (flux quanta) in the system. The filling factor ν gives the occupancyof m−sublevels by bosons, here ν = 3. For ν � 1 a mean field vortex lattice results,
while for ν ∼ 1 correlated states similar to fractional quantum Hall states appear.

which describes the ratio of interaction energy per particle to Landau level spacing.
Under the condition ΓLLL < 1, the condensate primarily occupies single-particle statesin the lowest Landau level (LLL). These form a ladder of near-degenerate states, witha frequency splitting of � = ωρ − Ω as shown in Fig. D.2 (the degeneracy is lifted by theresidual imbalance between centrifugal force and radial trapping force). The numberof occupied states in the LLL is mmax ≈ gn
�� , and is identical to the number of vortices(flux quanta) in the system. In order to observe eﬀects of the near-degeneracy in the
LLL, mmax � 1 is required.
The second parameter is the filling factor ν, given by the ratio of the number ofparticles Np to the number of vortices Nv (or occupied sublevels in the LLL) 1 :ν ≡ Np /Nv ≡ nSF,2D /nv(D.4)

which can interestingly be expressed in terms of the energy scale �Ω and the BKT1Note that for fermions ν ≤ ΓLLL (“=” applies at T = 0), enforced by the Pauli principle. For
bosons no such restriction applies, and ν may be arbitrarily big for any given ΓLLL .

153
temperature of the static superfluid kB TBKT
1/ν ≡

nv
nSF,2D

=

�Ω
2
2 π�
2m nSF,2D=

1 �Ω
2 kB TBKT(D.5)

For rotating bosons in the LLL, the filling factor distinguishes three regimes: For ν � 1
the condensate forms a classical superfluid in the mean-field LLL regime [77, 22, 21] andcontinues to form an ordered vortex lattice. With decreasing filling factor, the elastic
shear strength of the vortex lattice decreases. For filling factors below ν ≈ 10 the shear
strength is predicted to drop suﬃciently for quantum fluctuations to melt the vortex
lattice [47, 151]. For ν ∼ 1 a variety of correlated states similar to those in the Fermionicfractional QHE are predicted to appear [47], as well as quasiparticle excitations obeying
fractional statistics [118]. In the third regime, ν � 1, a Wigner-crystal like phase is
expected, where rotating bosons form an ordered lattice immersed in a huge numberof flux quanta (analogous to vortices forming a lattice when ν � 1). This regime isadmittedly esoteric given present experimental capabilities.
We note here that in a 2D system, Landau level parameter and filling factor areproportional to each otherΓLLL = ν ×�2
asc /lzπ(D.6)

where lz ≡�

D.1

Lowest Landau level: Vortex core overlap and transition to�mωz is the harmonic oscillator length of the axial trap potential.

normal state?
This section is a short note on the mean-field LLL regime, Refs. [143, 45]. For low
rotation rates, vortex cores are essentially line-like, well separated objects, whereas forlarge rotation rates, the vortex cores occupy a finite fraction of the condensate volume,
see Fig. D.3. This is analogous to the behavior of magnetic-field induced vortices in
Type-II superconductors, where the growth, and ultimately overlap, of vortex cores withincreasing magnetic field triggers the breakdown of superfluidity at the so-called upper

154
critical field Hc,2 . In this section we investigate whether this happens analogously in a
weakly interacting Bose gas.
We define the fractional surface area A occupied by the vortices to be A = nv πrv2 ,where nv is given by (1.22) and rv by (1.19). The resulting prediction for the fractionalvortex core area A may be cast in terms of the Landau level parameter ΓLLL
A = 1.34 × (ΓLLL )−1(D.7)

The prediction for A exceeds unity when ΓLLL < 1.34, which has led to the predictionthat vortices should merge in the LLL limit, as is observed in type-II superconductors(and as should be expected in a Cooper-paired rotating Fermi gas): Due to strong
interactions in these systems, the vortex cores are filled with material in the normal
phase, and vortex core overlap leads to the breakdown of type-II superconductivity.
Φ0This occurs at a critical field µ0 Hc,2 ≈ πξ
2 , where there is one magnetic flux quantum

Φ0 per vortex core area πξ 2 2 .
But careful, not so in a BEC! Vortex cores cease to grow according to (D.7) whenΓ−1
LLL reaches unity – they even get compressed with increasing rotation! Ultimately,
the reason for this diﬀerence in behavior is that the freedom of breaking a Cooper pair is
not available, and the system HAS to remain condensed until a diﬀerent kind of physics
destroys BEC. A more elaborate treatment by Baym and coworkers [22] consequently
predicts that A saturates at 0.225 in the LLL.
Our measured data for A are plotted in Fig. D.3 vs. Γ−1
LLL ≡ 2�Ω/(gn). For
Γ−1
LLL < 0.1 the data agree reasonably well with the numerical result (D.7). For larger
Γ−1
LLL the data clearly show saturation of A at a value close to the LLL limit A = 0.225[22], rather than a divergence of A as Ω̃ → 1.
2
For rotating He-II, this regime is unattainable: Due to the small healing length ξ ≈ a few Å, thecondition for vortex core overlap, ξ ≈ �, yields the unattainably large value Ω ≈ 1015 rad/s.

155

0.30

fractional core area

0.25
0.20
0.15
0.10
0.05
0.00
0.0

0.2

(!LLL)-1

0.4

0.6

Figure D.3: Fraction of the condensate surface area occupied by vortex cores, A, measured after condensate expansion, plotted vs. the inverse of the lowest Landau levelparameter, (ΓLLL )−1 = 2�Ω/µ. The data clearly show a saturation of A, as Ω̃ → 1.
Dashed line: prediction for isolated vortices, valid at low rotation rate and for superconductors. Dotted line: result of Ref.[8] for the saturated value of A in the LLL. Adaptedfrom Ref. [45].

D.2

Vortex lattice melting in a quasi-2D BEC
A second possibility of how the vortex lattice may play havoc with superfluidity

is the introduction of new low energy degrees of freedom to the superfluid, in the formof long-wavelength vortex lattice fluctuations.
Due to its ordered crystalline structure, the vortex lattice has an associated rigidity and supports transverse elastic modes. These modes were proposed for He-II by
Tkachenko in 1966 [162], but never conclusively observed [19]. It was suggested by
Anglin [12] that Tkachenko oscillations are also attainable in BEC. A description of
Tkachenko modes over the full range of rotation rates was outlined by Baym [20, 21, 63].
Further theoretical studies can be found in [49, 50, 106, 16, 41, 155, 27]. Fig. D.4 shows
our measurements of the frequency of the longest-wavelength Tkachenko oscillation in
BECs at various rotation rates. In the high rotation limit, the vortex lattice’s elastic
shear strength is reduced, as evidenced by the observed very low frequency of long wave-

a)

156

c)

0.8

b)

!(1,0) [Hz]

0.6
0.4
0.2

Expt. #LLL > 5
Expt. 5 > #LLL > 2
Expt. #LLL < 2

0.0

0.94

0.96 ~ 0.98
rotation rate "1

Figure D.4: Tkachenko oscillation in rotating BECs, (a) after the excitation drive isturned oﬀ. (b) after one half oscillation period. The low frequency is due to thevery small elastic shear strength of the vortex lattice. (c) Measured Tkachenko mode
frequency ω(1,0) vs. BEC rotation rate Ω̃. Adapted from Refs. [143, 44].

length Tkachenko oscillations. This eﬀect is a precursor to the predicted melting of thelattice under more extreme conditions of rotation or temperature.
The vortex lattice may melt due to quantum or thermal fluctuations [151, 152, 21],
which we describe separately below. The situation for a finite-size, trapped BEC has
been analyzed by Monte Carlo methods [95].
D.2.1

Quantum melting

To a certain degree it is valid to think of a vortex as an elementary quantum
mechanical object, and as such its coordinates (x, y) must undergo a zero point motion.
Motion along the two coordinates (x, y) is coupled by the Magnus force. For a twodimensional superfluid of density n2D , this force is F� = mn2D �v × κ̂, as seen from eq.
(1.28), with the replacement nSF L → n2D . Here, �v is the vortex velocity with respect tothe superfluid, and κ = h/m is the circulation quantum, Eq. (1.18). The corresponding
Hamilton’s equations reveal that (x, y) are canonically conjugate variables:∂H/∂x = −κmn2D ∂y/∂t∂H/∂y = κmn2D ∂x/∂t

(D.8)

157
Comparison to the conventional form of Hamilton’s equations∂H/∂q = −∂p/∂t
shows that q ≡√κmn2D x and p ≡√∂H/∂p = ∂q/∂t(D.9)

κmn2D y are canonically conjugate, and hencemust fulfill an uncertainty relation ∆q∆p > �, or ∆x∆y > �/(κmn2D ). For isotropic√
displacements ∆x = ∆y = ∆r/ 2 we find the zero point motion of the vortex, expressedin terms of the “magnetic length”, Eq. (1.23), which is related to the 2D vortex density
by nv = 1/(π�2 ):∆r2 /�2 = nv /n2D ≡1.
ν(D.10)

According to the Lindemann criterion (see e.g. [21]) a vortex lattice melts when
the zero point motion exceeds 10 − 20% of the vortex spacing, i.e. ∆r ∼ 0.15 b ∼ 0.28 �.
Inserting this criterion into Eq. (D.10) we obtain a result consistent with the one ofBaym’s elaborate calculation [21]nv /n2D ≡1∼ 0.07
ν(D.11)We thus predict a critical filling factor of νcrit ≈ 10 for vortex lattice quantum melting.
In Chapter 4 I describe experiments where the lowest filling factors to date, ν ≈ 100
were reached. Eﬀects of quantum fluctuations were however not observed, as they weresuperseded by thermal and possibly technical fluctuations.
D.2.2

Thermal melting

At finite temperature, the vortex lattice may melt due to thermal fluctuations.
We have to distinguish between two eﬀects. Low energy, long-wavelength, phononlike excitations can cause the loss of long-range positional order in the vortex lattice.
On the other hand, a BKT-type melting proceeds locally and suddenly, by formationand dissociation of thermally activated vortex-antivortex pairs, which will perturb andultimately destroy the ordering of the surrounding vortex lattice.

158
Long wavelength lattice fluctuations. At finite temperature, vortex position fluctuations become thermally activated. We describe the fluctuations by a 2D
displacement field ε(�r) ≡ (εx , εy ). In an ordered vortex lattice such fluctuations cost
��
�
�
� �
� 2
∂εy 2
∂εy 2
∂εx
∂εx
an energy E = d x C2
+ ∂y + ∂x[21], where C2 = n2D �Ω/8∂x − ∂y[50] is the vortex lattice shear modulus. Expansion of ε(�r) in Fourier components��ε(�r) = �k ε�k exp(i�k · �r) yields E ≈ �k C2 πR2 k 2 |ε�k |2 (R is the condensate radius, andk ≡ |�k|). The Fourier amplitudes |ε�k | are determined by equipartition, which states thatevery Fourier mode contains an energy kB2T ≈ C2 πR2 k 2 |ε�k |2 . The variance of thermally
activated vortex displacements (∆r)2 thus becomes(∆r) ≡2��k

1|ε�k | ≈(∆k)22

� k,maxk,min

2πk dkkB T
×2
k
2πC2 R2(D.12)The lower momentum cutoﬀ kmin ∝ 1/R is given by system size R, while kmax ∝2

1/� is set by the minimum lengthscale for vortex fluctuations. (∆k)2 ≈ πkNmax
= Rπ2
v
2

is the 2D k-space volume per Fourier mode, where we have used Nv = πR
. Withπ�2
� k,max
1
k,min dk/k = ln(R/�) = 2 ln(Nv ), the final result is, up to numerical factors of orderunity, in agreement with Baym’s elaborate calculation [21](∆r)2 /�2 ≈8 ln(Nv ).
n2D λ2th(D.13)√
λth = h/ 2πmkB T is the thermal de Broglie length. Again, the Lindemann criterion
for vortex lattice melting may be applied, ∆r ≈ 0.15 b ≈ 0.28 �, resulting in
TM,long ≈ TBKT

1.
2 ln(Nv )(D.14)

with the static BKT transition temperature TBKT given by Eq. (1.36). For typicalconditions ln(Nv ) ≈ 5, i.e. TM,long ≈ TBKT /10.
BKT type vortex lattice melting. The two-dimensional vortex lattice falls
into the class of “materials” undergoing a BKT transition to a normal state. Huberman and Doniach [82], as well as Giﬀord and Baym [64] give a condition relating the159
vortex lattice melting temperature TM,BKT to the BKT temperature TBKT of the static
underlying superfluid
TM,BKT ≈ TBKT ×

1
√
4π 3

(D.15)

again with Eq. (1.36) for TBKT . The relation (D.5) allows this to be rewritten as
kB TM,BKT ≈ �Ω ×νν√ ≈ �Ω ×
44
8π 3(D.16)For the rapidly rotating BECs obtained in our magnetic trap [143] we reached Ω �
0.99ωρ , i.e. Tc (Ω̃) = Tc,0 × (1 − Ω̃2 )1/3 roughly 5.5 nK. Filling factors were of orderν = 500, and with �Ω ∼ kB × 0.4 nK this resulted in a melting temperature TM,BKT ≈4.7nK, i.e. smaller than the condensation temperature Tc (Ω̃). An estimate of theactual temperature of the samples is as follows: The coldest temperatures achievable byevaporative cooling are of order of or larger than gn. For ΓLLL ≈ 1 in these condensates,
T ≈ gn = 2�Ω × ΓLLL ≈ 1 nK, i.e. lower than the melting temperature. This isconsistent with our observations, which showed that the vortex lattice remained ordered,
but became very fragile if temperatures were not kept to the very lowest values possible.
In the experiments on rotating BECs loaded into a 1D optical lattice, describedin Chapter 4, thermal fluctuations are much stronger, and thermally induced vortex
lattice melting is expected. Signatures consistent with vortex lattice melting are indeed
observed, as discussed in detail in Chapter 4.
In conclusion, even though quantum fluctuations are not yet expected to be observable in current experiments, thermal melting of the vortex lattice might well beobservable in current and possible near-future experiments rapidly rotating BECs.

D.3

Fractional quantum Hall physics in a rotating Bose gas?
From the considerations above we know that for a superfluid of very low two-

dimensional density, the vortex zero point motion is strong enough to melt the orderedvortex lattice. Once the vortex lattice has melted, theory has shown that – at low

160
enough temperatures, to be specified below – strongly correlated states of the rotatingboson system are expected, in analogy to the strongly correlated fractional quantum
Hall (FQHE) states of a two-dimensional electron gas in a strong magnetic field [98].
It is not the aim of this section to describe the physics of that regime, but rather toclarify the required experimental conditions and summarize our current knowledge ofexperimental limitations.
For a harmonic trapping potential, the upper limit on the number N of atoms for
obtaining the Laughlin state for a given rotation rate Ω and harmonic trap frequency ωρ√is on the order of (following e.g. [18]) N 2 (Ω − ωρ ) < g̃, where g̃ = 8πωρ asc /lz . Withg̃ = 0.5ωρ (realized for rotating BECs in optical lattices in Chapter 4) and Ω = 0.995ωρ(realized in our experiments of Ref [143]) 3 , the upper limit is of order N ∼ 10, a
number that would be extremely challenging to be studied in detail with present experimental methods. Experimental eﬀorts are thus blocked by the extreme requirements onachieving rotation frequencies extremely close to the external trapping frequency andworking with very small atomic samples, making the system extremely vulnerable toexperimental imperfections and instability, as well as placing high requirements on thedetection apparatus.
A further consideration in experiments is the achievement of the required low
temperatures. The Laughlin state, in essence, is a correlated state in which atoms avoid
being in the same place, thus avoiding their eﬀective contact interaction. Therefore theenergy gap ∆ that protects FQHE states is of the order of the interparticle interactionenergy, and the sample temperature must be lower than this gap for correlated states
to form:∆ ∼ g2D n2D ≡4π�2 asc
asc√
n2D ∼ 5 TBKT ×lz
m 2πlz(D.17)

where Eq. (1.31) was used. In absolute terms, it appears therefore to help if one works
in samples with high 2D density to boost TBKT , and in an axially tightly confined3

Note that Ω = 0.995ωρ and g̃ = 0.5ωρ were never realized in the same experimental setup.

161
situation, where alsc
is not a small number. Our magnetic trap is quite unsuitable inz
this respect, as alsc≈ 10−3 . In our deep 1D optical lattices in Chapter 4 on the otherz1
hand, axial confinement is very tight, alsc
≈ 10
. Another approach to boost alsc
would
z
z

be the use of a Feshbach resonance to increase asc [28]. Note however that large valuesof alsc
imply that for filling factors ν ≈ 1 the lowest Landau level approximation mayzbreak down, see Eq. (D.6), which might possibly alter the appearing correlated states.
On a more sobering note, evaporative cooling does not appear to be a safe strategy
to reach the fractional quantum Hall regime, as it is generally speaking limited to temperatures of the order of g2D n2D ∼ ∆. Other possible cooling schemes, such as coolingby superfluid immersion, proposed by Zoller and colleagues [68], may be desirable.
Having stated the problems, there are several possible routes that have not been
fully explored: The application of an axial optical lattice potential to reduce the filling
factor, and to boost axial confinement to make the system two-dimensional, which
we have begun to explore in Chapter 4, is certainly worth further study. Even if itturns out that fractional quantum Hall physics is not accessible in this system, thermal
fluctuations of 2D vortex lattices will allow further interesting studies.
A recent theoretical proposal [107] indicates that a modification of the radialtrapping potential can significantly relax the requirements on rotation rate and small
atom number for achieving fractional quantum Hall states.
A particularly attractive scenario would be the creation of FQHE states in rotating Josephson junction arrays. This situation has been studied in a series of articles by
R. Bhat, J. Cooper, M. Holland and coworkers [25, 26]. A closely related possibility is
the creation of artificial gauge fields in static optical lattices, as proposed by Sørensen
and coworkers [156, 72] and Mueller [86] and Jaksch and Zoller [109]. This proposal
would avoid all technical diﬃculties associated with mechanical rotation of a quantum
system.

Appendix E
Image magnification calibration

The precise calibration of the magnification factor of BEC images is essential
for accurate parameter estimation, as all quantitative information is retrieved frommeasured radii of BECs and thermal clouds. Here we present two – probably highly
unusual – ways of determining the magnification factor from in-situ BEC images.

E.1

Top view calibration from in-trap vortex lattice images
For the top view calibration we rely on the precise relation between vortex spacing

and BEC rotation rate, given by Eqs. (1.24) and (1.22). During this thesis, we were able
for the first time to take in-situ images of the vortex lattice, as shown in Fig. E.1 (a),allowing an accurate extraction of the vortex separation. Precisely, we perform a spatial
Fourier Transform of the images, and extract the vortex separation from the position of
the reciprocal lattice peaks in Fig. E.1 (a), right panel. The rotation rate of the samerotating cloud is accurately measured from the BEC aspect ratio in a non-destructiveside-view image. Results are shown in Fig. E.1 (b) for several BECs rotating at various
rotation rates.
There are two significant systematic eﬀects. The first one is a known slight
modification of the vortex spacing from the prediction of Eq. (1.24). This modification
has been calculated [147, 148] and measured [45] to be less than one percent for ourconditions. The second systematic eﬀect is a lensing eﬀect from the inhomogeneous

163
450

Fourier peak visibility [a.u.]

a)

c)

350
300
250
200
150
100

typ. noise background

50

4.0

2.12
magnification [!m / pixel]

vortex spacing [pixels]

b)

400

3.8

3.6

3.4

0.80

measured, 135 MHz detuning
measured, 160 MHz detuning
measured, 141-151 MHz detuning
theory, magnification 2.04 !m / pixel
theory, magnification 2.06 !m / pixel
theory, magnification 2.02 !m / pixel

0.85
0.90
rotation rate [trap units]

0.95

2.10
2.08
2.06

Model: dispersive
y=y0+A*G*(x-xc)/((x-xc)^2+G^2)
y0
xc
A
G

2.03854146.5 ±00.04251
18.51488±0.00315±0.00948±8.60933

d)

2.04
2.02
2.00
1.98

100 120 140 160 180
Probe AOM frequency [MHz]200

Figure E.1: Calibration of top view image magnification using in-trap vortex lattice
images. (a) in-trap vortex lattice image and its spatial Fourier transform (FFT). The
vortex separation is inferred from the position of the reciprocal lattice peaks occuring in
the FFT. (b) dependence of the measured vortex separation on rotation rate (symbols),
and theory (lines) assuming diﬀerent image magnifications. (c), (d) dependence ofFourier peak amplitude and inferred vortex separation on probe laser detuning from theD2 (F = 1 → F � = 0) optical transition (150 M Hz is on resonance). (c) the Fourier
peak amplitude tends to zero on-resonance due to the high optical density of the BEC
which overwhelms the density dips of the vortex cores. (d) dependence of the inferredvortex spacing on detuning, showing a slight dispersive variation, originating from alensing eﬀect originating from the inhomogeneous BEC density.

BEC density, which modifies the apparent magnification of the imaging system. Thislensing eﬀect follows a dispersive shape around the D2 (F = 1 → F � = 0) resonanceused for imaging. Figure E.1 (d) shows the inferred vortex spacing as a function ofprobe laser detuning from atomic resonance, exhibiting the expected slight dispersive
variation. The unperturbed vortex spacing is given by the constant value at large positive
and negative detuning.
The resulting image magnification, in lab units, is 2.04(2) µm per CCD pixel. The
CCD pixel separation is 13 µm, giving a magnification of 13µm/2.04µm = 6.37.

164

E.2

Side view calibration by releasing a BEC from a 1D opticallattice
For determination of the side view imaging magnification, we release the BEC

suddenly from an applied vertical optical lattice (Fig. E.2, left image) of known lattice
period d (d = λ/2 where λ = 852.2 nm is the wavelength of the lattice laser whose
frequency is locked to the D2 transition in 133 Cs). We employ a release procedure,
described below, which causes the BEC to split into two momentum components ±�k(k = π/d). These components separate spatially after a certain free time-of-flight t, seeFig. E.2, right image. This separation is given by (m is the mass of 87 Rb)∆xcalc =

2�k
t
m

(E.1)

From the accurately measured time-of-flight of t = 13.5 ms, the measured separationof the BEC momentum components ∆xmeas and the accurately known lattice period d∆xcalc [µm]
the image magnification is determined as ∆x. The result, an average over ≈ 20
meas [pixel]

individual shots, is 2.27(5) µm per camera pixel, i.e. a magnification of 5.73.
The precise experimental procedure is as follows: we adiabatically load a BEC
into the optical lattice and allow it to equilibrate. We then turn oﬀ the magnetic trap
but keep the optical lattice on. The hold time TB /2 = h/(2mgd), where h is Planck’s
constant and g = 9.81m/s, allows gravity to induce half a Bloch oscillation, as describedin Chapter 4 . The eﬀect is to transfer the BEC momentum state from the center tothe edge of the Brillouin zone, creating a superposition of two momentum components±�k. At this point we release the BEC abruptly from the optical lattice.
Again, there are possible systematic eﬀects. We worked with the tiniest possible
BEC to minimize a mean-field eﬀect arising when the two BECs separate, as well as
to minimize s-wave scattering during separation of the momentum components which
can distort their density profiles after separation. A second systematic is a possible tilt

165
MT

OL image

13.5msfree fall...
...

13.5ms
1/2TB

time

+hk
-hk

Figure E.2: Calibration of side view image magnification by releasing Bloch-oscillating
BEC from vertical optical lattice. Left: time sequence of the experiment. MT: magnetic
trap. OL: optical lattice. TB : Bloch period, see text. Center image: BEC in the
optical lattice. Right image: BEC after time-of-flight. From the separation of the two
momentum components the image magnification is determined.

of the optical lattice perpendicular to the image plane 1 . We estimate this tilt to be
not larger than ≈ 5◦ , and hence the error due to projection in the image plane to besmaller than 1%. A last possible systematic eﬀect has to do with the Guoy phase shiftacquired by a Gaussian beam upon traversing its focus. This phase shift causes theeﬀective optical lattice period at positions close to the focus to deviate slightly from2

λ/2. We find that the eﬀective period is λ2 (1 + 2πλ2 w2 ) where w ≈ 100 µm is the opticallattice waist. In our case, the relative change of the lattice period is negligible, ≈ 10−5 .
This eﬀect might however be important in precision measurements in optical lattices.
This method of determining the image magnification is preferable to switchingoﬀ the magnetic trap, releasing the BEC and measuring its fall distance vs. time. Inthat procedure, initial BEC slosh and possible accelerations in the process of magnetic
trap turn-oﬀ are could systematically influence the measurement. The measurement ofrelative motion performed here is largely immune to these eﬀects.
1

The visible tilt of the axis connecting the separating BECs is due to optical lattice tilt in the image
plane, and is taken into account in the measurement of the separation of the momentum components.

Appendix F
The EndFigure F.1: Thanks, everyone!