Vortices in a Bose-Einstein Condensate
byPaul C. Haljan
B.Sc.(Hons.), University of Alberta, 1997

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
2003

This thesis entitled:
Vortices in a Bose-Einstein Condensate
written by Paul C. Haljan
has been approved for the Department of Physics

Prof. Eric A. Cornell

Prof. Deborah S. Jin

Date

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁnd thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Haljan, Paul C. (Ph.D., Physics)
Vortices in a Bose-Einstein Condensate
Thesis directed by Prof. Eric A. Cornell

Since the advent of Bose-Einstein condensation in the dilute alkalis, there has been
considerable interest in observing eﬀects in atomic condensates akin to the hallmark
eﬀects associated with superﬂuidity and superconductivity. In particular, the study ofquantized vortices and vortex lattices represents an important connection between the
traditional ”super” systems such as liquid Helium and this new atomic system.
This thesis explores some of the ﬁrst vortex experiments in a condensate of magnetically trapped Rubidium-87. Single vortex lines and rings are created using a wavefunction engineering technique, which is an ideal starting point to study the dynamical
behavior of vortices within the condensate. An entirely diﬀerent approach of “intrinsic nucleation” has been developed to create rapidly rotating condensates with large
amounts of vorticity. A novel variation of forced evaporation is used to simultaneously
cool and spin up an ultracold gas. In this way, condensates can be formed that contain large, extraordinarily regular lattices of well over 100 vortices. Direct detectionof the vortex cores makes it possible to study the microscopic structure of the vortex
arrangements both at equilibrium and under dynamical conditions where severe appliedstresses distort the lattice far from its equilibrium conﬁguration.
In conclusion, the techniques developed in this thesis have opened up a new area ofrotating condensate physics and, in the future, may lead to regimes of extreme rotationand quantum Hall physics.

Dedication

To my parents and my two brothers, for a level of support that continues to baﬄe
and amaze me.

v

Acknowledgements

First, I would like to thank my supervisor, Eric Cornell, who more than anyone
has been responsible for making my graduate experience a positive one. Fortunate tohave been one of his students, I have learnt a tremendous amount about physics andbeneﬁted from his teaching, advice and enthusiasm.
Thanks are also due to Carl Wieman and Debbie Jin, who as surrogate supervisors
have provided helpful suggestions and advice on topics both scientiﬁc and career-related.
I have had the opportunity over the past 5 years to get to know and work withsome wonderful people. Jason Ensher, David Hall and Mike Matthews gave me mystart in the lab. As I was assuming a senior role in the lab, Brian Anderson arrived.
A fantastic postdoc, he taught me a considerable amount of experimental physics. In
the past couple of years I have had the privilege of working with another great postdoc
Peter Engels and a great graduate student Ian Coddington. Volker Schweikhard recently
joined the group and these three guys are already moving far ahead. I’m terribly jealousof the fun experiments they are pursuing! I have learnt a considerable amount from each
person and count each one as a colleague and friend; hopefully, I have oﬀered something
in return.
The electronics staﬀ, including Paul Beckingham, Mike Whitmore, James FungA-Fat and Terry Brown, contributed signiﬁcantly to this thesis as did the machine shop,
in particular Dave Alchenberger, Hans Green and Hans Rohner. Thanks to Pam Leland
for all her help. The support staﬀ is why JILA is a fantastic place to work.

vi
A number of theorists have contributed directly to this thesis work. I appreciatedthe chance to interact personally with Charles Clark, David Feder, Anthony Fetter,
Murray Holland, Anatoly Svidzinsky, Sandro Stringari, Jamie Williams, and Francesca
Zambelli. Their help and ideas have made the practice of science a pleasure.
Thanks to all the “degenerates” for making the Cornell-Jin-Wieman group suchan interesting atmosphere to work in.

Thanks especially to Kurt Miller, Heather

Lewandowski and Jeremy Richardson for the lunches and study sessions during comprehensives. Those laughs were therapeutic. Thanks to Neil Claussen for a great trip
and summer school in Corsica, and Tara Fortier for her friendship through the years.
Last of all, thanks to Emily Gibson and Quentin Diot for bolstering me through sometough times at the end.
Finally, I would like to recognize some great teachers who set me on my way
and encouraged me early on: John Beamish, John Macdonald and Leo Radzihovsky.
Misha Ivanov and Paul Corkum at the NRC in Canada gave me my ﬁrst gainful physics
employment, taught me a great deal of science in two summers and encouraged me topursue a graduate career. Finally, thanks are due to the NSERC program for helping
to fund my doctoral studies.

vii

Contents

Chapter
1 Introduction1

1.1

Bose-Einstein condensation and the “super” systems . . . . . . . . . . .

1

1.2

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

5

1.3

JILA Mark III: keeps on going, and going . . . . . . . . . . . . . . . . . .

82 Phase-contrast imaging

10

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Phase-contrast technique . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

3-D imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.4

Detailed calculation of imaging process . . . . . . . . . . . . . . . . . . .

15

2.4.1

Overview of calculation . . . . . . . . . . . . . . . . . . . . . . .

15

2.4.2

Atomic susceptibility . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.4.3

Object formation . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4.4

Phase-contrast image formation . . . . . . . . . . . . . . . . . . .

25

2.4.5

Phase-contrast signal inversion . . . . . . . . . . . . . . . . . . .

26

Nondestructive imaging . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.5.1

Spontaneous scattering and number loss . . . . . . . . . . . . . .

36

2.5.2

Stark eﬀects and measurement induced squeezing . . . . . . . . .

37

2.5

viii
3 Single Vortex Lines and Rings

45

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2

Coupled two-component condensates . . . . . . . . . . . . . . . . . . . .

48

3.3

Wavefunction engineering with two-component condensates . . . . . . .

52

3.3.1

The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.3.2

A closer look . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.3.3

Experimental realization: single vortex formation . . . . . . . . .

63

3.4

Dynamical stability of two-component vortices . . . . . . . . . . . . . .

69

3.5

Vortex Core precession [51] . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.5.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.5.2

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.5.3

Experimental technique . . . . . . . . . . . . . . . . . . . . . . .

75

3.5.4

Filled-core dynamics . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.5.5

Bare core dynamics. . . . . . . . . . . . . . . . . . . . . . . . .

80

Vortex tilting [98] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.6.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.6.2

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.6.3

Vortex formation . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.6.4

Disappearing vortices . . . . . . . . . . . . . . . . . . . . . . . .

87

3.6.5

Vortex tilt: three experiments . . . . . . . . . . . . . . . . . . . .

90

3.7

A position sensitive spectroscopy . . . . . . . . . . . . . . . . . . . . . .

94

3.8

Soliton engineering and decay to vortex rings [50] . . . . . . . . . . . . .

98

3.8.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

3.8.2

Introduction

3.8.3

Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . 102

3.8.4

Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . 105

3.8.5

Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

ix
3.9

Wavefunction engineering at DC . . . . . . . . . . . . . . . . . . . . . . 111

3.10 More angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Vortex Lattices

121

4.1

From quantum to classical ... and on to quantum . . . . . . . . . . . . . 121

4.2

Intrinsic vortex nucleation [97] . . . . . . . . . . . . . . . . . . . . . . . 123
4.2.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.2.2

Introduction

4.2.3

Evaporative spin-up . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.2.4

Highly rotating pure condensates . . . . . . . . . . . . . . . . . . 126

4.2.5

Mutually rotating condensate and normal cloud . . . . . . . . . . 131. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.3

Theory responds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.4

Direct observation of vortex lattices

4.5

Rotational suppression of the quantum degeneracy temperature . . . . . 144

4.6

Vortex lattice decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.7

Spinning up even faster: large lattices and rotation . . . . . . . . . . . . 153

4.8

Nonequilibrium dynamics of large vortex lattices [77] . . . . . . . . . . . 157

4.9. . . . . . . . . . . . . . . . . . . . 136

4.8.1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.8.2

Introduction

4.8.3

Experimental technique . . . . . . . . . . . . . . . . . . . . . . . 158

4.8.4

Lattice dynamics in the presence of an mz = −2 surface mode

. 161

4.8.5

Lattice dynamics in the presence of an mz = +2 surface mode

. 166

4.8.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Theory responds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Bibliography

171

x

FiguresFigure2.1

Phase-contrast imaging. . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2

Three-axis imaging setup . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3

Probe polarizations for three axes of imaging . . . . . . . . . . . . . . .

16

2.4

Relevant 87 Rb level structure . . . . . . . . . . . . . . . . . . . . . . . .

18

2.5

Magnetic bias ﬁeld orientation. . . . . . . . . . . . . . . . . . . . . . .

23

2.6

Optical phase shifts induced by a typical condensate . . . . . . . . . . .

30

2.7

Phase and phase-contrast signal variation due to rotating TOP bias ﬁeld. 31

2.8

Image distortion due to approximate signal inversion . . . . . . . . . . .

33

2.9

Phase wrap in the phase-contrast signal . . . . . . . . . . . . . . . . . .

34

2.10 Desloshing a condensate . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.11 Destructiveness of an oﬀ-resonant probe laser . . . . . . . . . . . . . . .

38

3.1

Singly quantized vortex . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.2

Coupled two-component condensates in 87 Rb . . . . . . . . . . . . . . .

49

3.3

Two-photon Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . .

50

3.4

Spin segregation in a two-component condensate . . . . . . . . . . . . .

51

3.5

ac Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.6

Local wavefunction engineering in a two-component condensate . . . . .

54

3.7

Point-by-point wavefunction engineering . . . . . . . . . . . . . . . . . .

55

xi
3.8

Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.9

Oﬀ-resonant Rabi oscillations on the Bloch sphere . . . . . . . . . . . .

60

3.10 Coherent control on the Bloch sphere . . . . . . . . . . . . . . . . . . . .

63

3.11 Engineering of vibrational wavepackets in a molecular system. . . . . . .

64

3.12 Sculpting beam for wavefunction engineering . . . . . . . . . . . . . . .

66

3.13 Two-component vortex growth . . . . . . . . . . . . . . . . . . . . . . .

67

3.14 Residual Rabi oscillations during vortex growth. . . . . . . . . . . . . .

68

3.15 Dynamical stability of two-component vortices . . . . . . . . . . . . . .

71

3.16 |2 vortices with a small |1 core. . . . . . . . . . . . . . . . . . . . . . .

73

3.17 Filled-core dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.18 Compiled data for ﬁlled vortex core precession . . . . . . . . . . . . . .

79

3.19 Bare-core dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

3.20 In situ vortex detection using quadrupolar surface-wave spectroscopy . .

89

3.21 Possible tilting behaviour of a vortex in a conﬁned BEC . . . . . . . . .

92

3.22 Visibility of a vortex undergoing tilt precession. . . . . . . . . . . . . . .

93

3.23 Trap deformation sequence to reverse the direction of a vortex. . . . .

95

3.24 Quadrupole precession frequency vs. radial vortex core position . . . . .

99

3.25 Quadrupole precession over a long time . . . . . . . . . . . . . . . . . . 100
3.26 Experimental setup for soliton formation . . . . . . . . . . . . . . . . . . 104
3.27 Numerical simulation of black-soliton decay in a BEC . . . . . . . . . . 107
3.28 Single-axis expansion images showing the decay products of solitons . . 107
3.29 Two-axis imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.30 Two-axis expansion images showing the decay products of solitons . . . 110
3.31 Vertical variation in two-photon intermediate state detuning across condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.32 Local pseudospin at each vertical point on the condensate . . . . . . . . 113
3.33 Soliton creation from a static gradient in Rabi frequency. . . . . . . . . 114

xii
3.34 Three-dimensional representation of a ﬁlled dark soliton . . . . . . . . . 115
3.35 Horizontal vortex ring derived from a winding soliton. . . . . . . . . . . 116
3.36 Stacks of solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.37 Simulation of  = 2 vortex engineering . . . . . . . . . . . . . . . . . . . 118
3.38 Multiple vorticity with multistep wavefunction engineering . . . . . . . . 119
4.1

Vortex lattices and rigid-body rotation . . . . . . . . . . . . . . . . . . . 122

4.2

Evaporation surface for prolate TOP conﬁnement. . . . . . . . . . . . . 126

4.3

Simultaneous evaporative cooling and spin-up. . . . . . . . . . . . . . . 127

4.4

Decay of a normal cloud’s rotation . . . . . . . . . . . . . . . . . . . . . 128

4.5

Highly rotating condensate with strong centrifugal distortion . . . . . . 129

4.6

Surface-wave spectroscopy of rapidly rotating condensates. . . . . . . . . 130

4.7

Aspect ratio comparison of condensate and normal components in a rotating sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.8

Vortex lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.9

Nucleation threshhold using direct vortex detection. . . . . . . . . . . . 139

4.10 Vortex number as a function of condensate rotation rate . . . . . . . . . 142
4.11 Discrepancy in observed vortex number with respect to rigid body prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.12 BEC transition in a rotating gas. . . . . . . . . . . . . . . . . . . . . . . 146
4.13 Rotational suppression of the BEC transition temperature. . . . . . . . 147
4.14 Lowest order surface roughness of the conﬁning potential. . . . . . . . . 149
4.15 Rotational relaxation in an asymmetric trap. . . . . . . . . . . . . . . . 151
4.16 Rotational relaxation as loss of vorticity . . . . . . . . . . . . . . . . . . 152
4.17 Rotational scissors mode of a pure normal gas . . . . . . . . . . . . . . . 154
4.18 Scissors spin-up technique. . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.19 Controllable condensate rotation. . . . . . . . . . . . . . . . . . . . . . . 156

xiii
4.20 Front and side perpectives of a vortex lattice in expansion . . . . . . . . 160
4.21 Evolution of the lattice and condensate density proﬁle in the presence ofan mz = −2 surface mode . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.22 Change of lattice structure in the presence of an mz = −2 surface mode

163

4.23 Sheet-like structure in the presence of an mz = −2 surface mode. . . . . 165
4.24 Lighthouse eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.25 Lateral vortex arrangement in the presence of a co-propagating quadrupolar surface mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.26 Macroscopic velocity ﬁeld of a rotating condensate in the presence of an
mz = −2 surface wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.27 Microscopic velocity ﬁeld of a rotating condensate in the presence of an
mz = −2 surface wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Chapter 1

Introduction

1.1

Bose-Einstein condensation and the “super” systems
The phenomenon of Bose-Einstein condensation(BEC) has a rich history over the

past century [155], beginning with the recognition of two classes of particles in nature,bosons and fermions, obeying Bose-Einstein and Fermi-Dirac statistics respectively: In asystem of indistiguishable particles, an arbitrary number of bosons may occupy a single
quantum state whereas fermions are restricted by the Pauli exclusion principle to a single
particle per state. Remarkably, according to the “spin statistics theorem,” particles innature with integer spin are bosons while those with half-integer spin are fermions. The
diﬀerent particle statistics have fundamental and far-reaching physical consequences;
however, eﬀects become evident only at low1 temperatures where particles gather inthe lowest energy states of a system and the average occupation of a state approaches
unity. Early in the 1900s, theory by Bose [56] and Einstein [76] predicted that if a gasof ideal(non-interacting) bosons is cooled down, a quantum phase transition will occur
at a critical temperature T = TC , below which a macroscopic population “condenses”
into the ground state of the system. At T = 0, the condensate fraction reaches one. It
certainly is not surprising that all the particles end up in the ground state at T = 0
since it is the state with the lowest energy. Signiﬁcantly, however, the macroscopic
1

A “low” temperature is deﬁned relative to the characteristic spacing of a system’s energy levels
and need not be low in any particular absolute terms.

2
occupation persists at ﬁnite temperatures up to the critical temperature.
Following the theory of Bose and Einstein, London [115] was the ﬁrst to suggestthat BEC could account for the experimentally observed superﬂuid transition in liquid
4 He, even though the strong interatomic interactions in the liquid clearly put the system

far from the concept of an ideal Bose gas. Implicit in this discussion is the fact that
4 He, although composed of fermions (electrons, protons, neutrons), is nevertheless acomposite boson, the constituent spins of which add up to give a total integer spin. The
original motivations of London are not delved into here; instead, some of the relevant
consequences are pointed out. In a liquid helium sample, the condensate is associatedwith the superﬂuid component. As a macroscopically occupied quantum state, it has anassociated wavefunction or order parameter Ψ = ρ(r)eiS(r) . As Tilley points out [155],
this fact in itself made London’s proposal a bold suggestion. After all, wavefunctions
were associated at the time with the microscopic quantum world of atoms but London
was suggesting that a continuous wavefunction exists over the scale of a bucket of liquid
helium!
Given the existence of an order parameter, currents of the superﬂuid or “superﬂows” are associated quantum mechanically with gradients in the phase S. As originallyemphasized by Onsager [131] and Feynman [85], this has a direct bearing on the meaning of a rotating quantum ﬂuid. Since the order parameter needs to be single valued, thephase around a closed loop must return on itself or, if there is a hole in the superﬂuid,
it can increment by an integer multiple of 2π. The latter case describes the multiplyconnected topology of a superﬂuid pierced by one or many quantized vortex cores. The
connection between quantum mechanical phase and velocity is a rather strong statement
permitting rotation to enter a superﬂuid only in the form of quantized vortices. In the“rotating bucket” experiments with liquid helium, a rotating normal ﬂuid is cooled below
the superﬂuid transition. Above a critical rotation rate ΩC , singly-quantized vortices
are nucleated at the edge of the superﬂuid and migrate inwards. Higher rotation rates

3
lead to the nucleation of more vortices, which come to form stationary arrays in therotating frame. In 1979, Yarmchuk and Packard [160] reported the direct observation ofindividual vortex cores in arrays of up to eleven vortices. The visual observation of vortex lattices, together with the quantization of vortex circulation as ﬁrst demonstrated
by Vinen [156], represents one of the hallmarks of superﬂuidity.
Following superﬂuid 4 He, the list of physical systems exhibiting a macroscopicallyoccupied quantum state grew to include superﬂuid 3 He, superconductors and lasers. The
latter two systems have resulted in countless technological advances. In the case of superconductors, conduction electrons pair up to form composite bosons which produce
persistent supercurrents, directly analogous to the superﬂows in liquid Helium II. Magnetic ﬁelds play the role of rotation in a superconductor. Thus, above a critical appliedmagnetic ﬁeld, Abrikosov lattices of magnetic ﬂux quanta are observed [1], having thesame triangular structure predicted for vortex lattices in superﬂuid 4 He.
In 1995, the newest member of the macroscopic quantum states was demonstrated
experimentally - BEC in a trapped dilute alkali gas. There are currently in excess of
30 gaseous BEC experiments around the world using several diﬀerent atoms (Cs, H,
He∗ , Li, K, Na, Rb ...). With a density 10−6 less than that of air at STP, a dilute-gas
condensate involves interatomic interactions that are relatively weak compared thosein helium liquid (although this is an evolving frontier [62]). The result is a theoreticaldescription that is more tractable. The dilute nature of the gas also means that the
ground state fraction is large, essentially 100% compared with ∼10% in Helium II,
where the interactions deplete the ground state [155]. A novel feature in the atomic
vapour experiments is the ability to directly see the density proﬁle of the condensate’s
wavefunction with optical imaging, central in fact to the results of this thesis. For thisreason in particular, dilute gas condensation as a visually accessible phenomenon has
become synonymous with the concept of ”BEC”.
One of the most interesting (and fun) aspects of atomic BEC is the interplay it

4
has with other areas of physics, be it quantum optics or condensed matter. For example,
a signiﬁcant amount of evidence has been amassed to connect superﬂuidity in 4 He to thephenomenon of Bose-Einstein condensation. One may ask though: is dilute-gas BEC asuperﬂuid? The question is more than just semantics. To answer it, one arguably should
demonstrate the canonical experimental properties that define superﬂuidity [114], suchas the rotating bucket experiments mentioned above, although the choice of experiments
ends up being somewhat subjective. It is useful to try to distill the basic features ofa condensate that contribute to its superﬂuid character. Consider rotation properties
speciﬁcally. The existence of a macroscopic wavefunction alone leads to the quantization of circulation characteristic of a superﬂuid; however, equally crucial is the role ofrepulsive interactions in the condensate since they are responsible for the “persistence”
of quantized vortices as stable topological objects in the ﬂuid. Garcia-Ripoll et al. [88]refer to this as the “structural stability” of a vortex, which may be understood as follows:
A quantized vortex cannot disappear from the condensate wavefunction by “spinning
down.” The multiply-connected topology of a wavefunction containing a vortex core,like a rubber sheet with a hole in it, can be destroyed only by cutting the density of
the condensate along a radial line from its edge to the vortex core. Since the repulsive
interatomic interactions give the condensate a ﬁnite compressibility, the suppressionof the condensate density along a radial line entails an energy cost. It is this energy
cost which makes a quantized vortex a structurally stable topological feature. In theabsence of suﬃciently strong repulsive interactions, an asymmetry introduced in thecondensate’s conﬁning potential would presumably destabilize the vortex [88]. An analogous situation has been experimentally demonstrated for an optical vortex containedin a laser beam propagating in free space [125]. Under an asymmetric perturbation froma cylindrical lens, the optical vortex dynamically inverts its topological charge, which isthus clearly not a stable quantity.

5

1.2

Rotating Bose-Einstein condensates
This thesis addresses the experimental creation of quantized vortices, from one tomany, in a dilute-gas condensate of 87 Rb. Concurrent with the results presented here,
a number of other experimental groups have also addressed the rotational behaviour ofa condensate, including studies of vortices [118, 48, 104, 103, 75] and the measurementof a condensate’s non-classical moment of inertia [119, 100]. The overall body of work,
both theoretical and experimental, is already sizeable, making rotating condensates its
own little sub-ﬁeld of gaseous BEC.
A general BEC workshop took place at JILA in 1999 near the outset of this thesis work. At the time, there was considerable discussion and uncertainty how a vortex
could be nucleated into a condensate. The discussion centered around the analog of
the “rotating bucket” experiment for trapped gases. The prime question related to
the nature of the energy barrier that a vortex would need to overcome to be nucleatedinto the condensate. This whole issue was initially circumvented when Williams andHolland proposed a novel way to directly create a condensate containing a vortex withwavefunction engineering [159]. Based on earlier experimental work in this group withcoupled two-component condensates, the wavefunction engineering technique led to theﬁrst creation of a (single) vortex in a dilute-gas condensate [122]. At about the sametime, the ENS group was successful at nucleating vortex lattices by stirring the condensate in a deformed conﬁning potential [118]. In very rapid succession, there were resultsfrom several groups providing diﬀerent methods of nucleation and elucidating vortex
behaviour. Vortices, it seemed, were “everywhere.” In retrospect, it seems strange that
there was any uncertainty at all in 1999 about vortex nucleation in a condensate! The
ﬁeld was and continues to be extremely active, which has made it a fascinating area towork in.
The wavefunction engineering technique, with its ability to introduce a vortex

6
into a condensate “artiﬁcially” with reproducible initial conditions, has proven to bea useful method to study a wide variety of vortex dynamics in a T = 0 condensate[Chapter 3]. Additionally, the technique allows the creation of two-component vortices
in which the vortex state is composed of one internal spin state of 87 Rb while thecore is ﬁlled with ﬂuid composed of another simultaneously trapped spin state. The
mutually repulsive interaction between the two components opens up the possibility to
study vortex dynamics in mixtures of interacting superﬂuids, where the vortex angularmomentum need not be conﬁned to one component.
As a general technique, wavefunction engineering has even been extended to makedark solitons in a condensate. The dark soliton, which is the matter wave equivalent of
the optical phenomenon, corresponds to a nodal plane cutting across the condensate andan associated π phase discontinuity. Unlike vortices, solitons are not topological featuresin the condensate wavefunction. Indeed, after their creation, the solitons are observed to
rapidly decay to vortex rings in our spherical trapping geometry. Although interesting
in its own right, this decay process can be considered an example of wavefunctionengineering whereby an unstable precursor state decays to a desired target state, thevortex ring. This may prove useful in the future for making more complicated topologicalstructures in the condensate.
The vortex work based on wavefunction engineering focused on the “quantum
limit” of single vortices in a condensate essentially at zero temperature. The variety
of experiments not only provided some interesting vortex physics but, perhaps mostimportantly, led to the acquisition of a number of experimental techniques for the manipulation and detection of vortices. These techniques have continued to prove usefulas the experimental direction has moved from the creation and study of condensateswith single vortices to the opposite limit of highly rotating condensates containing large
lattices of a hundred or more vortices.
The route to large vortex lattices in a condensate ultimately involves the mechan-7
ical addition of angular momentum to the trapped gas, just as for the original liquid
helium experiments. Coming full circle with the 1999 workshop, a point of controversy
has been the exact nature of the vortex nucleation mechanism, this time in the context
of experimental results. The group at ENS and later those at MIT and Oxford havenucleated vortex lattices by directly addressing the condensate with a stirring potential.
In this “extrinsic” nucleation technique, the stirring action of the potential has beenfound to drive surface waves on the condensate to instability. The crests of the waves
break, nucleating vortices in the resulting turbulence of the “surf”. The approach tovortex nucleation taken in this thesis [Chapter 4] has followed a somewhat diﬀerent
route, which at the time oﬀered the possibility of a diﬀerent physical mechanism fornucleation. In this new method, an ultracold gas above TC is ﬁrst brought into rotationand then evaporatively cooled to quantum degeneracy. Above a critical rotation rate,vortices are nucleated into the condensate. This “intrinsic” nucleation is due only tothe interaction of the normal cloud with the condensate since the trapping potential
is left static (and rotationally symmetric to conserve angular momentum) throughout
the evaporation. After a ﬂurry of discussion in the literature, the nucleation mechanism appears to be due again to surface-wave instabilities, driven not by a rotating trap
roughness but instead by thermal ﬂuctuations from the rotating normal cloud.
Nevertheless, a signiﬁcant beneﬁt of the intrinsic nucleation method is the generation of large amounts of vorticity in a condensate. A key feature of the evaporativetechnique is that it is designed to simultaneously cool and spin up a trapped gas tohigher rotation rates. As a result, highly rotating condensates of a million or more
atoms can be formed containing 130 or more singly quantized vortices. They are crystallized in remarkably ordered lattices, spanning ten or more lattice constants across thecondensate. In a particular sense, these vortex lattices are very much the opposite of the“quantum” limit of a single vortex. For the large numbers of vortices in the condensate,
the correspondence principle would suggest that the rotation ﬁeld, coarse grained over8
the vortex structure, should go over to the classical limit of rigid-body rotation. In
fact, the condensate does centrifugally distend like a rotating classical gas. From thedistortion to the condensate shape, an eﬀective rigid-body rotation rate as high as 95%of the centrifugal limit has be inferred (and this value is rising even higher in current
experiments). The large rotation rates and condensate number have been brought tothe point of reproducibility that the BEC machine is now truly a vortex machine.
The reproducible production of highly rotating condensates containing large, equilibrium vortex lattices is an ideal spring board for a number of diﬀerent experiments.
In the ﬁrst instance, a study has been made of the non-equilibrium lattice dynamics
induced when large amplitude shape deformations are applied to the condensate. The
anisotropic stress of the shape deformations induces a variety of strain responses in thevortex lattice including shifting of lattice planes, changes of lattice structure, and theformation of sheet-like structures in which individual vortices appear to have merged.
Finally, it was mentioned above how in presence of large amounts of vorticity
a condensate attains the classical limit of rigid-body rotation. However, in the limitof truly extreme rotation rates one approaches what may be called the “true quantum limit” where the amount of vorticity approaches the number of particles within
the condensate. Under these circumstances, recent theory [63, 133, 101, 146] predicts
the transformation of a condensate from a vortex lattice state to a strongly correlated
quantum system, analogous to the quantum Hall regime in condensed matter physics.
Given the versatile set of experimental tools that can be brought to bear on the condensate, it is a possibility more than a hope that quantum Hall physics will be accessedin dilute-gas BEC in the near future.

1.3

JILA Mark III: keeps on going, and going . . .
Underlying all the physics explored in this thesis is a BEC apparatus, JILA Mark

III, originally constructed by previous members of the group as a “next generation”

9machine [78, 120]. Diode laser technology and a vacuum system without moving parts
have made this experiment extraordinarily reliable as far as experiments go. Consider
that the glass cells have been under continuous high vacuum since their constructionin 1996. The original 87 Rb dispenser is still being used with several more available inchamber. Having a cycle time of ∼1min, the machine has produced about a half-million
condensates during the time of this thesis and averaged two successful experiments per
year. Admittedly, the apparatus has undergone a gradual overhaul with time. Lasers
have been changed and several more added. There are now at least as many lasers
devoted to manipulating and probing the condensate as there are ones devoted to its
production. Although practically every part of the setup has undergone modiﬁcation,
the same basic apparatus underlies it all (if you can still see it under the optics and coils!)
Only now is the experiment beginning to suﬀer from some limitations, particularly in
optical access, leaving room for plenty of ingenuity.

Chapter 2

Phase-contrast imaging

2.1

Introduction
A quick look through the JILA theses on atom trapping and cooling shows thatevery one covers the topic of optical imaging in one way or another; this thesis is no
exception. The reason is obvious: essentially all experimental data on an atom cloud
derives from images formed by scattering probe laser light oﬀ it. Every new imaging
technique opens up new experimental possibilities. Phase-contrast imaging [2], allowing
the non-destructive measurement of condensate samples in situ [52], has proven to bea particularly useful and general technique for our experimental apparatus, especiallyfor the vortex experiments detailed in this thesis. Because phase-contrast imaging has
become the workhorse for condensate assay in our experiment, more than the usual
detail is devoted here to outlining a proper quantitative model of the technique.
The imaging of an atom cloud can make use of the absorption or refractive index
properties of the atoms. For absorptive imaging, spontaneous photon scattering by
the atoms leaves a shadow imprint of the cloud on the probe beam: I = I0 e−D(r) . The
optical depth D(r) gives quantitative access to the density proﬁle of the cloud integrated
down the line of sight. There are a number of reasons why absorptive imaging is notparticularly useful in-trap. Optical depths for on-resonant probe light (D0 ∼ 102 − 103 )far exceed the limitations of imaging setups with a maximum observable optical depthof ∼ 3 or so [120]. Detuning the probe from resonance to reduce the optical depth

11
only leads to another problem, image distortions due to lens-like refraction by the atom
cloud [120]. From the perspective of non-destructiveness, the average recoil energy ofan atom following spontaneous emission (Er ∼ 45nK for 87 Rb) is comparable to thetemperature of Bose-condensed samples and thus results in the loss of the atom fromthe condensate. Since sample sizes are small (104 − 106 atoms), obtaining an image
with good signal-to-noise involves scattering a photon or more per atom. After a single
absorption image a condensate will be mostly melted if not blown away.
The alternative to absorption imaging is to use an oﬀ-resonant phase-contrast
technique sensitive to the phase shift induced by the cloud. A nondestructive image ispossible because the spontaneous scattering rate Γ dies out more quickly away from resonance than the phase shift ξ. (For a probe detuned δ from resonance, Γ ∝ 1/δ2 versusξ ∝ 1/δ). The high on-resonant optical depth D0 of a trapped condensate in particular allows for large probe detunings that signiﬁcantly suppress spontaneous scattering
while still maintaining a good imaging signal (ξ ∼ D0 /δ ∼ 1). In the last part of thischapter, the “nondestructive” nature of phase-contrast imaging will be characterized ina little more detail, including the ultimate limit of a quantum nondemolition (QND)
measurement for number.

2.2

Phase-contrast technique
To measure the phase shift induced by the atom cloud, an interference technique

is needed. One way to do this would be to have an interferometer with the atom cloudin one arm. Phase-contrast imaging, which uses Fourier optics techniques, works in avery similar way. As shown in Figure 2.1, a probe laser beam detuned from resonanceis incident on the atom cloud. The cloud acts as a phase object, diﬀracting (elasticallyscattering) some of the incident light. In analogy to the interferometer, the unscatteredcomponent forms the reference beam while the forward-scattered component carries theobject information. The components are collected by an imaging lens and spatially

12

BEC

L1

P1

L2

CCD

Figure 2.1: Phase-contrast imaging, using imaging lens L1, phase plate P1 in the backFourier plane of L1 and magniﬁcation lens L2. A CCD records the images.

separated in the back focal plane due to the Fourier transforming eﬀect of the lens[143]. This allows a small dielectric dot mounted on a glass substrate to selectively shift
the phase of the unscattered component by φ0 . A second magniﬁcation lens recombines
the two light components, which interfere at the image plane to form a phase-contrast
image of the atom cloud. A phase dot shift of φ0 = π/2 maximizes the contrast signal
for small phase shifts.
The phase plate is a 1/16” thick, 1” diameter glass substrate with a small MgFl2
dot at the center of one face. The dot, with a diameter of 100µm and a thicknessof λ/4(n − 1) = 513nm corresponding to a phase shift of π/2, was deposited onto
the substrate through a thin, stainless steel mask composed of a purchased opticalpinhole. The opposite surface of the phase plate was then anti-reﬂection coated for
780nm to reduce fringes. An issue which arose in the deposition process was the craterlike appearance of the pinhole (due to the laser ablation used in its manufacture).
In initial tests, the crater edges resulted in a non-uniform dot, presumably due toshadowing eﬀects. This was solved by carefully polishing the thin steel around the hole
to a ﬂat ﬁnish.
The diameter of the dot was chosen based on typical condensate and probe beam
sizes. The waist size for the probe beam incident on the atom cloud is 1mm compared to 30µm for the condensate radius; therefore, in the Fourier plane of the imaging

13
lens(f ∼100mm), the focused probe beam is 30µm compared with 1mm for the diﬀracted
light. A 100µm diameter dot is large enough to include most of the focused probe beam,
yet small enough to avoid signiﬁcantly aﬀecting the scattered light.

2.3

3-D imaging
There are a few points of interest relating to our imaging setup. The probe lasers

for phase contrast are typically detuned up to several GHz from resonance. Furthermore,
the detuning is changed often depending on the signal-to-noise and non-destructivenessdesired. External cavity diode lasers with a DAVLL frequency lock [65] provide theversatility for probing both near and far from resonance. A Fabry-Perot cavity is rather
useful to monitor the probe frequency relative to the well-deﬁned frequency of a laser
under saturation absorption lock [72].
To satisfy our insatiable desire for more and more information, three orthogonalaxes of (nondestructive) imaging are simultaneously available to probe an atom cloud(see Fig. 2.2). The horizontal axes share a single CCD array, which is more thanlarge enough to have side-by-side images of the atom cloud. The vertical imaging axis
is integrated into an optical path normally occupied by one of the beams used formagneto-optical trapping [78]. As a result, some beam switching is required during theexperimental cycle (see Fig. 2.2, side-view). All the complication of assembling a 3-D
puzzle of extraordinarily dense optics around a fragile glass vacuum cell is well worth
it, as each imaging direction has proven crucial to the vortex experiments discussed insubsequent chapters.

14

Top
view

Transfer tube

Sciencechamber

CCD #1side probe
front probe

pol
2f

Side
view

L1

MR2

L2

MOT
beam

L3

MOT
beam

MR1

λ/4

CCD #2

vertical probe
Figure 2.2: Experimental setup for 3-axis imaging. The top view shows the two horizontal imaging axes combined onto a single CCD camera. The more detailed side viewshows the vertical axis, which is integrated into a MOT beam path. New Focus ﬂipper
mounts MR1 and MR2 rotate into place to permit imaging. Lenses L1 (f = 120mm)
and L3 (f = 700mm) form the vertical imaging microscope. The matched pair of lenses
L1 and L2 form a one-to-one telescope for the MOT beam.

15

2.4

Detailed calculation of imaging process

2.4.1

Overview of calculation

The purpose of this section is to provide a quantitative analysis of phase-contrast
imaging, divided into the eﬀect of the atom cloud (“object formation”), the optics(“image formation”), and image processing (“signal inversion”). The only real eﬀort isto calculate the eﬀect of probe laser propagation through the atomic cloud; the rest of theimaging system involves basic Fourier optics and is straight forward. Condensates as wellas cold thermal clouds that are fairly dense can be imaged. For the purposes of imaging,a condensate is well approximated as a classical gas of independent quantum absorbers[126], in other words, a dielectric. Therefore, in order to understand object formation
we simply need to calculate the refractive index and absorption for a generic atom
cloud. The extra twist is that the magnetic conﬁning potential for the atoms provides
an ambient bias ﬁeld that breaks the symmetry of the system. As in anisotropic crystals,
the susceptibility of the atom cloud becomes a tensor quantity and the eigenmodes of
propagation are no longer the simple free-space ones. As an additional complication,
the TOP trap [78] used in this thesis work has a time-varying magnetic bias ﬁeld which
can be oriented in any direction in the horizontal plane. With three axes of imaging
having three diﬀerent polarizations constrained by optical access [Fig. 2.3], it is useful toinvestigate the eﬀect of the bias ﬁeld changing during a time-sequence of non-destructive
pictures. Finally, in pushing the limits of imaging, for example in the analysis of the
shape of an atom cloud, it is also useful to understand any nonlinear behaviour inphase-contrast imaging that will distort the image.
In calculating the optical response of an atom cloud, several references are useful[144, 143, 124] but the information is fairly scattered. Hopefully the expressions andtechniques collected here will prove useful for other trapping experiments, includingones using absorption imaging.

16
r

r

Eside

E frontrr

Bb (t )

Bq
r

Evert

Figure 2.3: Probe polarizations for the three orthogonal axes of imaging. Also shown
are the contributions to the magnetic bias ﬁeld seen by atoms trapped in the TOP trap:(i) the rotating bias ﬁeld Bb and (ii) a possible static component Bq depending on theposition of the atoms in the quadrupole trap.

2.4.2

Atomic susceptibility

The starting point for light propagation in an atom cloud (“the medium”) is theMaxwell wave equation in a nonpermeable (zero magnetization) dielectric medium2 − ∇(∇ · E)
 − µo ∂ D
 = 0
∇2 E
∂t2
 = 0
∇·D

(2.1)

 +P
 with P being the induced polarization of the medium, E the light’s
 = 0E
where D
electric ﬁeld, and 0 and µ0 the permittivity and permeability of free space respectively.
 r , t) of an ultracold trapped cloud of atoms (including a conThe polarization P(densate) is simply n(r)d , where n(r) is the number density and d  = −er is theinduced dipole moment of a stationary atom in the presence of the probe laser ﬁeld.
Velocity eﬀects are negligible for the trapped clouds in our experiment. Note that the
spatial dependence of the polarization arises only from the density distribution.
We want to consider the optical probing of a 87 Rb cloud in a particular ground
hyperﬁne state 5S1/2 |F, mF . Available transitions are the D1 and D2 lines reaching

17
the 5P1/2 and 5P3/2 electronic states respectively. We use the D2 line at λ = 780nm,
with hyperﬁne structure resolved, as shown in Fig. 2.4. Contributions from all allowedtransitions to the upper hyperﬁne manifold must be included when calculating the dipole
response.
We focus on a single pair of states, labeled {|g, |e} with associated resonant
frequency ωeg and linewidth γ. The state |g (|e) is in the ground(excited) hyperﬁnemanifold.
The incident probe laser with frequency ω has an electric ﬁeld, r , t) = E eiωt−ik0 z + c.c.
E(
2

(2.2)

The steady-state solution for probe durations τ  1/γ is discussed in many places[124, 144]. In the dipole approximation (eik0 z ∼ 1) and the rotating wave approximation(ω − ωeg  ω), the dipole moment of an atom can be written asˆd(t)=−i dge · E(t)deg + c.c.
 γ/2 + iδeg

(2.3)


where δeg = ω − ωeg is the detuning and deg is the dipole matrix element e|d|g.
Saturation eﬀects [124] are justiﬁably ignored in phase-contrast imaging since probe
detunings are many linewidths and probe intensities are low, typically 100−200µW/cm2
compared to the saturation intensity IS = 1.6mW/cm2 for Rubidium.
 d of the dipole moment can be reordered
The main numerator dependence (d · E)
as
†  
† 
† 
)E
E)deg = deg (degE) = (deg deg
(deg

(2.4)

†
† 
is a 3×3 (Hermitian) matrix (as opposed to deg
deg , which is a number).
where deg deg↔

↔

 where the susceptibility χ eg is: = χ eg E
This allows us to write the polarization as P†deg degn(r) γ/2 + iδeg↔
−iχ eg =

(2.5)

18

F=3
5P3/2

267MHz

F=2157MHz
72MHz

F=1
F=0

780nm “D2”

F=2

5S1/2
6834MHz

F=1
87Rb (I=3/2)

mF = -2

mF = -1

mF = 0mF = 1mF = 2F=26834MHzmF = -1

mF = 0F=1
mF = 1Figure 2.4: Relevant 87 Rb level structure showing the D2 optical transition with groundand excited hyperﬁne manifolds resolved. Also shown in the boxed region is the Zeeman sub-structure for the ground hyperﬁne states with ﬁlled circles identifying the twomagnetically trapped states used in the experiment.

19
To include all hyperﬁne states |e ∈ |F  , mF   in the upper manifold, we sum↔

up their contributions χ eg to the total susceptibility. The dipole decay rate from theupper hyperﬁne states is γ, independent of mF  sublevel; therefore, the excited state
↔
†
only.
dependence in χ eg is through the detuning and dipole matrix deg deg

It is very useful to deﬁne a dipole matrix normalized to the cycling transition
matrix element as follows:↔

C (e, g) =†deg deg|dcyc |2(2.6)

Thus the net expression for the susceptibility of an atom cloud in ground hyperﬁne state|g is:↔

χ = −i↔

e∈{F  ,mF  }nσ C (e, g)
k0 1 + i 2δeg

(2.7)

γ

where the sum is over all excited states {|e}. If more than one ground state is occupied,
the susceptibility should also be averaged over all of them. The value of σ is the standard
crosssection for a cycling transition
σ=

3λ2
2π

(2.8)

In obtaining Eqn. 2.7, the standard relation for the linewidth of a cycling transition was
also usedγ=ω 3 |dcyc |2
3π 0 c3(2.9)

We could express the susceptibility tensor in Cartesian co-ordinates; however, thenatural basis to use is the spherical tensor one deﬁned as:1
ˆ± = ∓ √ (x̂ ± iŷ), ˆ0 = ẑ2(2.10)

If we further choose to use “atom co-ordinates”, where the direction of the magnetic bias↔

ﬁeld lies along the ẑ-axis, then dipole selection rules cause the C (e, g) matrix [Eqn. 2.6]
to be diagonal for each of the terms in the susceptibility sum [Eqn. 2.7]. The total

20
susceptibility is, as a result, also diagonal in this basis with value:0  χ+ 0↔ χ=  0 χ
0 
0




0
0 χ−

(2.11)whereχq = −ie∈{F  ,mF  }nσ cq (e, g)
k0 1 + i 2δeg

(2.12)γ↔

The Clebsch-Gordan type terms cq (e, g) are the diagonal elements of C (e, g) in theatom basis [124]:
1 |ˆq · deg |2
N |dcyc |2
1(2J + 1)(2J  + 1)(2F + 1)(2F  + 1)=
N
J F  I
F
L J  S
×
mF
J L 1
F J 1

cq (F, mF ; F  , mF  ) =

1

F

q

−mF 

2

(2.13)

The curly-(round-)bracketed terms are 6-j (3-j) symbols, easily evaluated in Mathematica for example. The normalization N is chosen such that cq = 1 for the cyclingtransition, namely |2, 2 → |3, 3 for 87 Rb. The normalized values for the D2 line of
87 Rb (I=3/2) can be found listed in a previous thesis [120].

The contribution from each transition to the susceptibility in Eqn. 2.12 may be

written as the sum of a real and imaginary part, χq (e, g) + iχq (e, g) with values:
2δeg χ (e, g)γ q
nσ cq (e, g)χq (e, g) = −

2
k0
2δ
1 + γegχq (e, g) =(2.14)

where the imaginary part is responsible for absorption and the real part for the refractive
index. Clearly for large detunings (δeg  γ) the real part dominates.

21
2.4.3

Object formation

With the polarizability in hand, we can proceed with the eﬀect of light propagation through the atom cloud. Returning to the wave equation [Eqn. 2.1] and assuming r ) insidea probe laser frequency ω, we obtain an equation for the laser’s electric ﬁeld E(the medium:2 + k2∇ E − ∇(∇ · E)

↔
0

 =0
E

(2.15)

where↔

↔

= 0 (1+ χ )(2.16)First, diﬀraction eﬀects in the atom cloud are neglected, which amounts to dropping all transverse derivatives and keeping only those with respect to the propagation(ẑ-)direction. To estimate the magnitude of diﬀraction eﬀects we can use ray-optics
and treat the cloud as a sphere of radius R and uniform refractive index. Then thetransverse deviation of rays within the cloud is roughly ∆x ∼ ξλ where ξ is the peakphase shift of the probe through the cloud. Typically ξ  1, so diﬀraction eﬀects within
the cloud do not exceed the imaging resolution ( λ) and are unimportant.
Further, the density of the atom cloud is also assumed to change slowly compared r ) is
 = A( r )e−ik0 z where A(to λ. The laser ﬁeld can then be approximated as E
slowly varying. Called the paraxial or WKB approximation, this amounts to neglecting
reﬂection by the medium [143].
Incorporating these approximations into the wave equation Eqn. 2.15 and working↔

in the atom co-ordinates where χ (or↔

) is diagonal, we can readily solve for two

 b } in the medium, namely
 a, Eeigenmodes of propagation {E  (r)e−i
 i (r) = E
E
i

z

ki (
r  )dz , i ∈ {a, b}(2.17)

which satisfy the eigenvalue problem,
k02

↔

(r)
0

 =0
− k2 (1 − ẑ ẑ † ) E(2.18)

22
The spatially dependent eigenvalues ka,b (r) are determined from a generalized Fresnel
equation [143] for a spatially dependent anisotropic medium, |zq |2 (1 + χq (r))2q(1 + χq (r)) − kk2=0(2.19)

0where the components χq of the susceptibility in the atom frame have already been
determined [Eqn. 2.12]. The only diﬃculty to solving these equations is to calculate thevector components zq of the propagation direction in terms of the atom-frame sphericalco-ordinates. If the magnetic bias ﬁeld has polar co-ordinates {θ, φ} with respect tothe propagation direction [Figure 2.5], then the transformation from lab to atom coordinates is achieved with the rotation operator D(θ, φ) = Dy (−θ)Dz (−φ) in l = 1
representation






D(θ, φ) = 




1 iφ
2 e (1 + cos θ)√1 sin θ
21 −iφ(1 − cos θ) 2e− √12 eiφ sin θcos θ√1 e−iφ sin θ
21 iφ2 e (1 − cos θ)− √12 sin θ

1 −iφ(1 + cos θ)
2e







(2.20)

so that the z-direction in the atom co-ordinates used in equations (2.18) and (2.19) is1
 √2 sin θ (2.21)ẑ = 

cos
θ




− √12 sin θ
Since the ẑ components depend only on θ, the same is true of the wavevector andeigenmodes in the atom co-ordinates. The φ dependence arises when the eigenvectors
are transformed back to the lab co-ordinates with D † (θ, φ).
Equations (2.17)-(2.21) provide the total solution to propagation in the medium.
 in is some superposition of in , is incident on the atom cloud where E
A probe laser, E
the free-space circular polarization basis, E+ 
Ein =  0 




E−

(2.22)

23kˆ = zˆ
θ

r

B
atom
cloud

x̂

ŷ
φ
r

r

Ein ( r , t )Figure 2.5: Magnetic bias ﬁeld orientation relative to probe propagation direction k̂ = ẑ.

 out is obtained as follows:
The output laser ﬁeld E
 in onto the free-space limit of the medium’s eigenvectors, which are in lab1)Project E
co-ordinates: Ei+  i) = lim (D † (θ, φ)E 0  , i ∈ {a, b}
n→0




Ei−

(2.23)

 i |2 = 1 and D is given in Eqn. 2.20.
where |E
2)Apply phase shift and absorption to each of the eigenmodes.
3)Reproject (if necessary) onto the free-space circular basis.
In summary, long after the medium we have:


↔
†
 in
=
P
M
P
E
E out

(2.24)

where P is the projection onto the free-space limit of the eigenvectors [Eqn. 2.23] andcan be written generally in terms of angles {α, β} e−iα/2 cos β Ea+ 0 Ea−  
 

 

≡
P = 0
0
0 
0
 

 

Eb+ 0 Eb−e−iα/2 sin β0−eiα/2 sin β

0

0

0

eiα/2 cos β








(2.25)

24
M gives the eﬀect of propagation through the medium:


−iξ
a
0
0 
 e




(2.26)
M = 0
0
0 




0
0 e−iξb∞with ξa,b = −∞ (ka,b (r) − k0 ) dz. The real (imaginary) part of ξ describes the phaseshift(absorption) acquired by a probe passing through the cloud.
 i and ki are straightforward to solve, theAlthough the equations 2.17 & 2.18 for E
ﬁnal expressions in general involve all of the susceptibility matrix elements as well as
the angles θ and φ. The solutions are too complicated to be written out here; however,a relatively useful expression for the eigenvalues themselves is shown here for the casewhere the susceptibility is small:ka,b = kav ±δk2(2.27)

1 kav
3χ+ + 3χ− + 2χ0 + cos(2θ)(χ+ + χ− − 2χ0 )
≈1+
k0
16
1
δk
{ 9χ2+ + 4χ20 − 4χ0 χ− + 9χ2− − 2χ+ (2χ0 + 7χ− )≈k0
8+2 cos(2θ)(3χ+ − 2χ0 − χ− )(χ+ + 2χ0 − 3χ− )1+ cos2 (2θ)(χ+ − 2χ0 + χ− )2 } 2
To get a feeling for the size of the susceptibility under typical experimental conditions, we can consider the simple situation of the bias ﬁeld aligned with the propagation
axis of the probe laser. In this case the eigenmodes in the medium are the two circular√polarizations with wavevectors ka,b = k0 1 + χ± ∼ k0 (1 + χ± /2) for small χq . If wefocus on a single transition for one of the polarizations, say ˆ+ , then the susceptibility
χ+ can be written as
χ+ ∼ −

Dpk γ 
γ
1+i
, δγ
k0 R 2δ
2δ

(2.28)

where Dpk ∼ n0 σR is the peak on-resonant optical depth (absorption coeﬃcient), n0
the peak atom density, and R the radius of the cloud. Typical experimental parameters

25
for a condensate in-trap are 106 atoms, R ∼ 28µm and Dpk ∼ 300; therefore χ+ will
be small( 0.1) for δ  5γ ∼ 30MHz. For comparison, typical probe detunings in theexperiment are 100MHz or more from the nearest resonance.
Dpk γ
2 2δ for a single transition, theSince the peak phase shift of the probe is φpk =susceptibility can perhaps more usefully be written as
χ+ ∼ −

2φpk 
γ
1+i
,
k0 R
2δ

δγ

(2.29)

Suﬃcient signal-to-noise is achieved in images of atom clouds for peak phase shifts of∼ 0.5, a typical value for the experiment. Under these circumstances, for a condensate
with radius R ≥ 20µm, the magnitude of χ is less than 0.01, certainly small.

2.4.4

Phase-contrast image formation

After propagation through the atom cloud, the probe laser is the sum of scatteredand unscattered ﬁeld components, approximately identiﬁed as follows: out =
E

 in
E

unscattered

 −E
 )
+ (E
 out in(2.30)

scattered in since
The unscattered part is quite reasonably approximated by the incident ﬁeld E
the laser beam proﬁle (with waist w ∼1mm) is typically much larger than the cloud(with radius R ∼ 28µm), making the fraction of scattered power (∝ R2 /w2  1) small.
As mentioned above, the spatial Fourier transformation performed by the imaging lens [Fig. 2.1] spatially separates the light ﬁeld into its unscattered and scatteredcomponents with, crudely speaking, zero and non-zero spatial frequencies respectively.
This allows a small phase dot in the back Fourier plane of the imaging lens to selectively shift the phase of the unscattered component by φ0 . Thus, after the remainder
of propagation to the image plane, the total probe laser ﬁeld is: out − E in
 in + E image (x, y) = eiφ0 E
E


 in
= (eiφ0 − 1) + P † M P E(2.31)

26
 out from Eqn. 2.24. Any magniﬁcation m in the imaging systemusing the expression for E
has been omitted here but can easily be included in this and subsequent formulae by
adjusting the coordinates according to {x, y} → {mx, my}.
In the experiment, a CCD camera collects the image proportional to the incident image |2 ). For a given atom cloud, the standard protocol [78, 120]
intensity (I ∝ |E
actually involves a sequence of three images, a probe shot to obtain the intensity proﬁleI(r) of the probe laser modulated by the presence of the atom cloud, a normalization shot
to obtain the intensity proﬁle I0 (r) of the probe alone, and ﬁnally a background shot
without probe light, giving Ibackgr (r). The normalization shot can be realized either
by ﬁrst destroying the condensate or, more nondestructively, by detuning the probe
suﬃciently far from resonance. On a pixel-by-pixel basis, background subtraction andnormalization are performed to yield the raw phase-contrast signal:S(r) =

2.4.5

I(r) − Ibackgr (r)−1
I0 (r) − Ibacgr (r)(2.32)

Phase-contrast signal inversion

The ﬁnal step of imaging is to invert the raw signal S [Eqn. 2.32] for the column
density of the atom cloud on a pixel-by-pixel basis. This step requires knowledge of theexact form of S, which in general depends on the details of the imaging setup such as
the probe polarization, magnetic bias ﬁeld orientation and so on. It is most to the point
to consider the speciﬁc situations arising in the apparatus.
A variety of constraints on the optical table have resulted in the following arrangements for the three diﬀerent orthogonal probe directions [see Figs. 2.2 & 2.3]. The“front” horizontal axis uses a probe laser incident with horizontal, linear polarization.
Additionally, a horizontally polarizing beamsplitter cube is situated in front of the camera. The “side” horizontal axis uses a vertically polarized probe incident on the atoms.
Lastly, the vertical axis uses circular polarization. In all cases, the rotating magnetic

27
bias ﬁeld of the TOP trap may have an arbitrary orientation in the horizontal plane at
the time of imaging.
To calculate the phase-contrast signal, the detuning of the probe laser from allresonances is assumed to be large enough that the imaginary part of the susceptibility
can be disregarded [see Eqn. 2.14]. Under this approximation, the probe eigenmodesin the atom frame are real as are their wavevectors. A useful way to obtain the phase image and uses the general expression Eqn. 2.25
contrast signal starts with Eqn. 2.31 for E
for the projection matrix P in terms of angles α and β . Both angles depend on themagnetic bias ﬁeld orientation through α = 2φ and β = β(θ). The net signal for thefront axis, for example, is



φ0
δξ
φ0
φ0
Sf ront (ξav , δξ) = 4 sin
sin
+ cos
sin ξav −
2
2
2
2




δξ
φ0
φ0
cos α sin 2β cos ξav −sin+ 4 sin
2
2
2

δξ
1
3 − cos 2α + 2 cos2 α cos 4β sin2−42(2.33)

in which φ0 is the phase shift due to the phase dot, and ξav and δξ are the phase shifts
due to the atom cloud in terms of wavevectors kav and δk [Eqn. 2.27]:
 ∞ξav (x, y) =−∞∞δξ(x, y) =−∞(kav (r) − k0 ) dz
(δk(r) − k0 ) dz

(2.34)

Physically, ξav is the average phase shift of the probe passing through the cloud while δξ
is indicative of Faraday rotation of the probe’s polarization. Both quantities are directly
proportional to the column density of the atom cloud integrated along the probe lineof sight and have implicit dependencies on the bias ﬁeld orientation through the polar
angle θ (see Eqn. 2.27).
Phase-contrast signals such as Sf ront of Eqn. 2.33 cannot in general be algebraically inverted for the column density. However, a main point of this section is todemonstrate that, when phase shifts are not too large (ξav ∼ δξ  1), all the probe28
signals can be reasonably approximated by the simple result



φ0
φ0
φ0
sin
+ sin ξ −S0 (ξ) = 4 sin
2
2
2



f ront, side probe

 ξav
ξ =



 ξ + ηδξ
vertical probe
av(2.35)

where η = 0(±1) applies to a strong(weak) trapping potential (see below). The approximate signal is the same expression one would obtain for a scalar probe ﬁeld. For thecase of a linearly polarized probe laser such as the front axis, this amounts to neglecting
Faraday rotation eﬀects, which can be seen directly by setting δξ = 0 in the expression
for the front probe signal [Eqn. 2.33]. A nice feature of S0 (ξ) is that it can be readily
inverted for ξ

%
φ0
φ0
+
− sin
ξ = arcsin
φ0
2
2
4 sin
$

S0

(2.36)

2

Eqn. 2.36 is the signal inversion used in the experiment with S0 replaced by theactual acquired signal. In order to assess the validity of this approximation, as well asother eﬀects such as bias ﬁeld orientation, a representative example is now studied innumerical detail. The full phase-contrast signal Sf ront [Eqn. 2.33] for the front probe axis
is calculated for a condensate under typical experimental conditions and compared tothe approximate expression S0 [Eqn. 2.35]. The magnetically trapped ground hyperﬁnestate, |F = 1, mF = −1, is considered. In addition, the two typical conﬁgurationsof the TOP trap are assessed, (i) a strong trap and (ii) a weak trap where the atom
cloud sags downward by about 1mm under the pull of gravity. In the former case theatoms feel a bias ﬁeld Bb rotating in the horizontal plane. In the later sagged case,the atoms feel both a rotating ﬁeld of Bb ∼ 1.7G and an additional static downward
ﬁeld of Bq ∼ 3.2G. Relevant condensate parameters for the weak trap include numberN ∼ 106 and Thomas-Fermi radius R ∼ 28µm, giving a peak on-resonant optical depthof Dpk ∼ 300, assuming a cycling transition. The same optical depth is used for the29strong trap for comparison purposes.
First, a feeling for the size of the optical phase shifts induced by the condensate(in the weak trap) is given in Fig. 2.6 where ξav and δξ are plotted over the standard
range of probe detunings used. The probe is detuned below all resonances for the|1, −1 state such that the signal is positive or bright on the CCD screen relative to thebackground. The typical peak phase shift of |ξav | ∼ 0.3 − 0.5, which sets the standard
range of detunings, represents an operational compromise between nondestructivenessand a good signal-to-noise ratio [see Section 2.5].
Figure 2.7(a) shows the dependence of ξav and δξ on the orientation of the rotating bias ﬁeld Bb for the |F = 1, mF = −1 state and a ﬁxed detuning. The fractional
peak-to-peak variation of ξav is fairly small whereas the value of δξ varies considerably over bias orientation. The variation of the phase-contrast signal itself is shownin Figs. 2.7(b)&(c) for a variety of peak average phase shifts. The fractional variation
becomes worse when the phase shift is larger or, most importantly, when the probe frequency approaches a resonance, the nearest one corresponding to the F = 1 → F  = 0
transition at the right edge of the plots in Fig. 2.7. Additionally, the strong trap shows
more of an eﬀect than the weak one. By way of conclusion, in the weak trap where mostof the vortex work is pursued, reasonable phase shifts ξav ≤ 0.6 and probe frequencies
greater than 100MHz from resonance will ensure that variations in the peak signal will
not exceed 5% if the bias ﬁeld orientation is left uncontrolled. A more serious problem will arise for samples of smaller number, which force the probe detuning near toresonance to achieve a good signal.
Another important imaging eﬀect to consider is nonlinearity in the phase contrast
signal, which if left unaccounted for, will eﬀectively distort the image of an atom cloud,making it more “peaky” or ﬂat-topped depending on the trend of the nonlinearity. Since
we use only an approximate form of the true signal to invert for an image, there is thepotential for distortions due to nonlinearity. The maximum fractional diﬀerence between

30

2
100

Phase shift [rad]

50

F=1

→F '

1.5 0
-50

1-100

F‘=0 F‘=1 F‘=2-1000 -800

-600

-400

-200

0

200

0.5
0

-1400

-1200

-1000

-800

-600

ν – (ν2→3 +νhf) [MHz]′Figure 2.6: Average phase shift ξav (solid line) and diﬀerential phase shift δξ (dashedline) induced by a typical |F = 1, mF = −1 condensate versus probe laser frequency ν.
The frequency range plotted is the usual operating range in the experiment. The inset
shows the average phase shift over a larger frequency range including all resonances.

Phase per maximum

31

1
0.8
0.6
0.4
0.2

ξav
δξ

(a)

0
0

1

2

3

4

5

6

Bias field phase θ [rad]0.2

Fractional pk.-to-pk. signal var.

(b) weak trap
0.15

ξav = 1

0.1

0.6
0.2

0.05
0
0.2

-1400

-1200

-1000

(c) strong trap

-600

-800

-600

ξav = 1

0.15
0.1

0.2

0.05

-800

0.6

0
-1400

-1200

-1000

ν – (ν2→3 +νhf) [MHz]′ b (θ). Total fractional
Figure 2.7: (a)Phase variation due to rotating TOP bias ﬁeld B
variation of phase-contrast signal is shown in (b) and (c) as a function of probe frequency
ν for three diﬀerent magnitudes of the average optical phase shift ξav .

32
the approximate signal S0 (ξav ) [Eqn. 2.35] and the front probe signal Sf ront (ξav , δξ)[Eqn. 2.33] is shown in Fig. 2.8, again for the |1, −1 state over typical detunings. The
strong trap shows more of an eﬀect than the weak one. As a rule of thumb, an eﬀect
exceeding 10% should only arise for samples of small number where the probe needs tobe less than 100MHz detuned of the F = 1 → F  = 0 resonance to achieve a good signal(ξav ∼ 0.5).
There is one additional nonlinear eﬀect called “phase wrap,” which is intrinsic
to phase contrast. Since the signal is an interference eﬀect with 2π periodicity, thesignal must saturate for large enough phase shifts and begin to decrease [see Fig. 2.9].
The saturation point depends on the choice of phase dot retardance. For the choice of
φ0 = π/2 used in the experiment, the eﬀect is not much of an issue for positive phaseshifts as the saturation point occurs at a fairly high phase shift. However, it is an
issue for negative phase shifts since the phase-contrast signal is not symmetric. This is
the main reason why the probe detuning is chosen to give a positive phase shift whenlooking at a cloud composed of a single spin state.

2.5

Nondestructive imaging
The utility of phase-contrast imaging lies in its nondestructive character, allow-

ing time sequences of images or “movies” to be taken of a single atom cloud in-trap.
This is particularly valuable for measuring time-dependent excitations of a condensate
with a minimum amount of data-taking [e.g. Section 3.6]. As well, nondestructive
imaging can provide the only way to analyze condensate dynamics in situations when
the initial conditions ﬂuctuate from each realization of a condensate sample to thenext [e.g. Section 3.5]. A potentially interesting application of phase-contrast imaging
would be to use the information garnered from an image to feed back onto the condensate, perhaps for some type of coherent control. Admittedly, a servo-loop basedon the current implementation of the imaging would have a fairly low bandwidth since

33

0.2

S front − S0
S front(a) weak trap

ξav = 1

0.15

0.6

0.1

0.2
0.05
0
-1400

-1200

-1000

-800

-600

-800

-600

0

0.2
0.6

-0.05

S front − S0
S front-0.1

ξav = 1

-0.15
-0.2

(b) strong trap
-1400

-1200

-1000

ν – (ν2→3 +νhf) [MHz]′Figure 2.8: Fractional error between approximate and true phase-contrast signals, S0and Sf ront respectively, for the front probe direction as a function of probe laser frequency ν. Contours of ﬁxed optical phase shift ξav are labeled accordingly. Overlapping
contours would indicate a linear error independent of phase shift. The degree of image nonlinearity accrued by using the approximate signal for image processing can beinferred from the vertical spread of the contours at a ﬁxed probe frequency ν.

34

3π
4

6

S0(ξ)
[arb.]

π

4

2
2

φ0 =

0
-3

-2

-1

0

1

π
4
2

3

ξ [rad]Figure 2.9: Phase-contrast signal S0 as a function of induced optical phase shift ξ forthree diﬀerent phase dot retardances φ0 . The experiment uses φ0 = π/2

350µm

35

0ms

time

400ms

Figure 2.10: Nondestructive streak-camera image of a trapped condensate showing removal of its center-of-mass motion (up-down oscillation) by shifting the trap. The arrow
indicates the time where the trap center is suddenly shifted.

the readout and processing of a sequence of images from the CCD camera requires areasonable fraction of a second. As a simple example, phase contrast can be useful toassess and remove undesirable center-of-mass oscillations (“slosh”) of the condensate
in the conﬁning potential. Sloshing represents a hindrance in many experiments suchas wavefunction engineering [Chapter 3], where the condensate needs to be selectively
manipulated across its spatial extent. For the simple case of reproducible slosh fromshot to shot, a phase-contrast image can be made at the start of the day to measure
the oscillations. In subsequent shots, an “antislosh” step can be applied, where the trap
center is suddenly kicked or shifted with the correct phase and amplitude to bring thecondensate to a stop [Fig. 2.10].
The statement that phase-contrast imaging is “nondestructive” is not an exact
statement and requires some quantitative clariﬁcation. The two issues focused on hereare (i) condensate depletion and heating due to residual spontaneous scattering from theoﬀ-resonant probe laser and (ii) the eﬀect on condensate phase due to the laser-induced
ac Stark shift.

36
2.5.1

Spontaneous scattering and number loss

The nondestructive character of phase-contrast imaging is intimately related tothe signal-to-noise desired for the images. A discussion of signal-to-noise for phase
contrast can be found in a previous thesis [120]. A few points are added here as they
relate to non-destructive imaging in practice. For the sake of discussion, the atom cloudis assumed to induce a small phase shift ξ  1; then the phase contrast signal is S ∼ 2ξ.
The number of spontaneously scattered photons during the imaging is Nsc ∝ Iξ 2 where
I is the probe intensity incident on the atoms. Thus, the destructiveness, characterized
by Nsc , can be reduced by decreasing the intensity or increasing the probe detuning
δ since ξ ∝ 1/δ. However, signal-to-noise represents a diﬀerent issue. If the signal√
is assumed to be shot-noise limited then S/N ∼ Iξ. We can improve the signalto-noise ratio by either increasing the intensity or detuning closer to boost the phaseshift. Which is preferable to reduce destructiveness? The answer is that it does notmatter because the signal-to-noise depends only on the spontaneous scattering rate:
1/2

S/N ∝ Nsc . Therefore, in the shot-noise limit the signal-to-noise is ﬁxed by thescattering rate allowed for a stipulated destructiveness.
On the other hand, most of the time the image noise is dominated by backgroundinterference fringes proportional to the incident intensity. In this case, the signal-tonoise is S/N ∼ 2ξ/η where η characterizes the strength of the fringes. Under thesecircumstances, if the peak density of clouds being imaged is low, for example in thecase of thermal clouds, the only way to obtain better signal-to-noise is to tune theprobe closer to resonance. The intensity then is dropped to preserve nondestructivenessas much as possible. At some point the intensity is low enough that the shot noise
becomes of the same order as the fringe noise.
The destructive nature of the probe laser has been studied by shining it continuously on the condensate. A typical result is shown in Figure 2.11(a). The decay of total

37
condensate number matches fairly well with the calculated spontaneous decay rate (1.5times smaller), assuming a single frequency for the probe. The destructiveness can beconsiderably worse if, for example, the probe laser is running poorly single-mode.
When actually taking images, the probe does not shine continuously but is pulsed
on only for brief periods of exposure (100µs). To assess any delayed heating eﬀect
following a probe pulse, a sequence of eight images has been taken using three diﬀerent
time intervals and a deliberately destructive probe intensity [Fig. 2.11(b)]. Only arelatively small diﬀerence in condensate loss is seen between the image sequence lasting
0.14s and the one lasting 4s, indicating that most of the atom loss due to the probe
pulse occurs during the pulse time and not as a result of delayed heating.

2.5.2

Stark eﬀects and measurement induced squeezing

Nondestructive phase-contrast imaging of condensates oﬀers the intriguing possibility to produce measurement-induced squeezed states of atom number [57, 71], analogous to squeezed states for photons. A familiar concept from quantum optics is theexistence of an uncertainty relation between photon number and phase, δN δξ ≥ 1. At
the quantum limit where the equality sign is satisﬁed, a reduction of number ﬂuctuations
results in an increased variance in the phase. For a maximally squeezed state (δN ∼ 1),
the phase is undeﬁned. One can argue formally that the number-phase uncertainty relation is a feature of all systems of indistinguishable bosons and so applies to a condensate
as well. This would imply that a nondestructive measurement of condensate number
below the atom shot noise could create a squeezed condensate. The concomitant lossof a well-deﬁned condensate phase can be thought of as the back-action eﬀect of thenumber measurement. This formal discussion has not identiﬁed the physical mechanism
for the back-action. The object of this section is to show how shot noise on the probe
laser intensity together with the ac Stark eﬀect [3] produce the destructive eﬀect on theatoms’ phase. In the spirit of the “Heisenberg microscope,” a direct consideration of an

38

90

(a)

80

N [arb.]

70
60
50
40
30
20
10
0
0

100

200

300

400

500

time [ms]

(b)

1.0
0.8

N/N0 0.6
20ms/frame
100ms/frame
500ms/frame

0.4
0.2
0.0
1

2

3

4

5

6

7

8

frame
Figure 2.11: Destructive number reduction of a condensate in the presence of an oﬀresonant probe laser. (a)Condensate number decay due to a continuously shining probe
laser with an intensity of 150µW/cm2 and a detuning of 2.1GHz from the F = 1 →F  = 0 resonance. The ﬁt exponential decay rate is 2.9(2)s−1 compared to a calculated
spontaneous scattering rate of 1.9s−1 . (b)Time sequence of eight probe frames withthree diﬀerent time intervals between frames. The light pulses for each frame are 100µs
long with a large probe intensity of 2500µW/cm2 (a factor of 10 larger than typical)
and a detuning of 1.1GHz from the F = 1 → F  = 0 resonance.

39
experiment is used to arrive at the number-phase uncertainty relation for a condensate.
A side beneﬁt is a brief assessment of the feasibility to obtain number squeezing withphase-contrast imaging.
Number-squeezing due to interatomic interactions has already been observed in
multi-condensate systems [132, 93]. The loss of interference contrast between the individual condensates provides a signature for the large variance in relative phase. An
experiment to observe measurement-induced squeezing would use a similar technique.
For example, optical tweezers can be used to split a condensate spatially into two, oneof which is reserved as a phase reference. The other condensate is then subjected to anondestructive probe, after which the condensates are re-interfered to assess the probe’s
aﬀect on their relative phase. An interesting alternative that does not require any spatial separation makes use of a condensate composed of two internal spin states together
with a coherent coupling drive to create a beamsplitter in the time domain. A coupled two-component condensate composed of the simultaneously magnetically trapped
hyperﬁne states |1, −1 and |2, 1 [Fig. 2.4] has already been realized with this experimental apparatus [120] and is covered in more detail in Chapter 3. A key feature of thissystem is that phase contrast imaging can distinguish between the two states without
needing to spatially separate them. The probe laser frequency is detuned between thetwo states such that the |1, −1 state imparts a positive phase shift to the probe beam
while the |2, 1 state imparts a negative phase shift. As a result, in the presence of bothstates the net phase-contrast signal provides a direct readout of the number diﬀerence.
To simplify the discussion, the phase-contrast imaging of a single-component condensate is considered. The generalization to two components is straight forward andleads to similar results. The phase-contrast signal collected on the CCD camera provides a measure of the optical phase shift ξ imparted by the condensate, which in turn40
is proportional to its column density n̄(x, y),ξ = βpc n̄(x, y)(2.37)

where βpc is a constant which can be determined from the previous sections of thischapter. To obtain the total number of atoms in the condensate, the image needs to be
integrated over all Npix CCD pixels spanned by the condensate proﬁle. The uncertainty
in the total number of atoms for small phase shifts is thus

1/2
Npix
1 Apix   2 δNat =
δξi.
βpc m(2.38)

i=1

In the above equation, Apix is the area of a pixel, m accounts for the imaging magniﬁcation, and δξi is the phase uncertainty at the ith pixel. Neglecting technical noise, theoutput of the probe is very nearly a coherent state satisfying the uncertainty relation
δNph δξph ∼ 1 and having shot-noise ﬂuctuations about the average photon number,
1/2

δNph ∼ Nph . The uncertainty relation applies at each CCD pixel where Nph is thenumber of photons arriving at that pixel. For small phase shifts, the number of photons
will be approximately pixel-independent, allowing the uncertainty in atom number tobe reduced to the simple result
1 ApixδNat =
βpc m



Npix
ηQE

1/2
δξph

(2.39)

1/2

where δξph ≈ 1/Nph . The coeﬃcient ηQE has been added to account for the quantum
eﬃciency of the CCD detector. To connect with experimental parameters, the numberof photons at a CCD pixel can further be written in terms of the probe intensity I
incident on the atoms
Nph =τ ApixIω m(2.40)The various constants include the probe duration τ , the laser frequency ω, the pixel
area Apix and the magniﬁcation of the imaging system m.
Now that the nondestructive number measurement has been dealt with, it only
remains to consider the backaction eﬀect on the condensate phase. The oﬀ-resonant

41
probe induces an ac Stark shift in the internal energy state of the condensate atoms,
resulting in an intensity-dependent phase shift
ξat =

τ βst
I
ω

(2.41)

The constant βst can be determined from perturbation theory for weak intensities. The
average phase shift accumulated by the condensate can always be taken into accountfrom a measure of the average probe intensity; what remains, however, is the uncertainty introduced by ﬂuctuations in the intensity. Equation 2.40 can be used to relate
the intensity at the atoms to the photon number counted by the CCD. This allows
the uncertainty in condensate phase to be expressed in terms of the photon counting
statistics at the detector,
δξat = βst

m
δNph
Apix

(2.42)The uncertainties for atom number [Eqn. 2.39] and phase [Eqn. 2.42] can be combined
to yield the uncertainty relation&
δNat δξat ∼

βst
βpc

'

Npix
ηQE

1/2
(δNph δξph )  

(2.43)∼1

Thus the nondestructive measurement transfers the uncertainty relation for the probe
photons to one for the atoms. For simple probe arrangements, for example using asingle circular polarization and a magnetic bias ﬁeld aligned along the imaging axis,
the expression βst /βpc in Eqn. 2.43 is equal to one (and is expected to be larger than
one for more complicated situations). In the limit of a single CCD pixel and unity
quantum eﬃciency ηQE , the uncertainty relation for the atoms reaches the quantum
limit expected for a system of indistinguishable bosons.
The ﬁnal point addressed in this section is the feasibility of squeezing the ﬂuctuations in condensate number with a nondestructive probe. Measurement-induced squeezing is achieved by a determination of the number to better than the atom shot-noise.
The degree of squeezing can be assessed from the measurement uncertainty Eqn. 2.39

42
assuming an ideal detector with 100% eﬃciency. A practical limitation to this process
is the fraction of condensate atoms lost as a result of spontaneous scattering from theprobe. Clearly, the squeezing is not useful if the probe also destroys the condensate.
The number of scattered photons during the probe pulse is Nsc = Nat Γτ with Γ being the spontaneous scattering rate [124]. For a simple probe conﬁguration with large
detuning, the number uncertainty [Eqn. 2.39] for an ideal detector is easily written interms of NscδNat =

Acσc∗1/2 

Nat
Nsc

1/2
(2.44)The value of the coeﬃcient c∗ is of order unity or less and, in general, depends on
several Clebsch-Gordon coeﬃcients [Eqn. 2.13] as well as the probe detuning. If acycling transition is being used for probing, then c∗ = 1, independent of detuning. The
quantity σ ∼ λ2 /2 is the on-resonant light scattering crosssection [Eqn. 2.8] whereas
Ac =Npix Apix
is the crossectional area of the condensate in terms of detector variablesmwith the folowing qualiﬁcation: The crossectional area is ∼ πRc2 for a condensate withradius RC that is larger than the diﬀraction limit; otherwise, AC ∼ πλ2 and the ﬁrst
term in Eqn. 2.44 attains a minimum value (Ac /σ)1/2 ∼ 1. For typical experimental
parameters, a more appropriate value for this ratio is 10 − 100. Therefore, to achievea maximally squeezed state (δN ∼ 1), the number of scattered photons needs to be atleast as large as the atom number if not larger, assuredly killing the condensate [seeSection 2.5.1].
1/2

Even if the squeezing is not maximal, a reasonable eﬀect such that δNat = Nat /α
with α > 1 could nevertheless be useful for atom-interferometric applications. Whatconsiderations limit the maximum value of α? There are a number of possibilities
in connection with spontaneous scattering. First, the condensate number may be soheavily depleted following the probe measurement that δNat , albeit absolutely small,is not fractionally small anymore. This does not represent a signiﬁcant limitation and43
only plays a role if an overwhelming fraction of the condensate is destroyed. The second
limit is reached when the uncertainty δNsc in the number removed from the condensate
becomes comparable to the uncertainty δNat of the number measurement. This limit has
some ﬂexibility with regard to probe detuning and a discussion is left to a more detailed
treatment. Instead, a third issue is brieﬂy considered here, whereby the squeezing can
be strongly limited by the maximum particle density achievable in the condensate. The
maximum value of the density is determined by the rate of density-dependent inelastic
loss, which depends on the atomic species being used. The resulting limitation to
1/2

number squeezing can be seen to arise as follows: With δNat = Nat /α, Eqn. 2.44 canbe solved for α:


α=

σc∗
Nsc
Ac

1/2
(2.45)

It is evident that the level of squeezing is connected to the number Nsc of photons thatare spontaneously scattered. The ultimate limit to Nsc is on the order of the numberof atoms Nat . This leads to a maximum value of α = αmax , which can be expressed
explicitly in terms of the number density n ∼ Nat /(Ac L). Here, L is the size of thethird dimension of the condensate. The net expression for αmax is given by
αmax ∼

nλ

3 Lc∗

1/2

2λ

(2.46)For 87 Rb, the maximum density is about n ≈ 1014 cm−3 , yielding nλ3 ∼ 100. The size Ldepends on the number of atoms and the trapping arrangement. As a speciﬁc example,
in a lithographic microtrap where atom interferometry may be realized, approximateparameters include Nat ∼ 104 and L/λ ∼ 1 − 10 [4]. As a result, the limit to squeezing
imposed by a ﬁnite density is roughly αmax ∼ 10, which is a factor of 10 less thanmaximal squeezing. Of course, such a value is an upper limit based on a considerable
loss of atoms. The practical value of αmax will be lower depending on such issues as
how many atoms must be preserved to achieve good detection signal-to-noise.
A signiﬁcant improvement to this situation can be achieved by wrapping an optical

44
cavity with ﬁnesse F around the atoms. This has two eﬀects, (i) to increase the eﬀective
path length through the atom cloud (and optical phase shift) by F/π and (ii) to boost
the intensity in the cavity by F/π relative to the input [143]. If the intensity into thecavity is reduced to maintain a given amount of spontaneous scattering, the shot-noise
of the phase-contrast image will certainly increase by F/π; however, the image signal
itself increases much more - by F/π - due to the cavity-enhancement of the phase shift.
The net result is a signiﬁcant reduction in the number ﬂuctuations,(δN )cavity = (δN )f ree π 1/2
F(2.47)

with the cavity-free expression (δN )f ree given by Eqn. 2.44. At least a factor of ∼10
improvement in the squeezing could be achieved with a reasonable ﬁnesse of F ∼ 103 .

Chapter 3

Single Vortex Lines and Rings

3.1

Introduction
Quantized vortex lines, forming visible topological defects in a quantum ﬂuid, are

one of the most appealing demonstrations of superﬂuidity and superconductivity (Fora review, see [155, 74]). What sets quantum ﬂuids, including dilute-gas BEC, apartfrom their classical counterparts is that rotation can be manifested only in the formof quantized vortices. This is a direct consequence of the ﬂuid being associated with
a macroscopic order parameter(wavefunction) Ψ = ρ(r)eiS(r) . Because the ﬂuid’s
velocity ﬁeld v (r) is associated with the gradient of the order parameter’s phase [155],
any ﬂow is fundamentally restricted to be irrotational (curl-free)∇ × v (r) ∼ ∇ × (∇S(r)) = 0.

(3.1)

This immediately precludes the occurrence of solid-body rotation, so familiar from classical ﬂuids, as it is ostensibly non-irrotational. What is possible are vortices, characterized by an azimuthal phase gradient around a line defect that deﬁnes the vortex core.
The single valuedness of the order parameter requires that the phase accumulated in aclosed path around the core be an integer multiple of 2π or, in other words, that thecirculation be quantized in units of /m [Figure 3.1].
This chapter focuses on the creation and dynamical behaviour of singly quantizedvortices in a dilute-gas Bose-Einstein condensate. While the study of vortices in a

46

π/2
π

0
3π/2

vs

Figure 3.1: Singly quantized vortex in a quantum ﬂuid, characterized by a 2π phase wrap
around a ﬂuidless core. The azimuthal phase gradient is associated with a superﬂow
around the core, as indicate by the arrow.

condensate may be considered very much “condensed matter“ in ﬂavor, a truly atomic
physics oriented method - wavefunction engineering - is used to make the vortices. This
technique, based on a theoretical proposal by Williams and Holland [159], led to theﬁrst experiment to demonstrate vorticity in an atomic condensate [122]. A slightly
diﬀerent interpretation of the original proposal [159] is presented here to emphasize theexperimental route to generating condensates with arbitrary wavefunctions.
The idea for wavefunction engineering arose out of earlier experiments with twocomponent condensates, in which a microwave-frequency ﬁeld is used to transfer population between two internal spin states of 87 Rb [120]. The essence of creating an arbitrary
wavefunction is to control the amplitude and phase of the transferred population in a
spatially dependent way. A novel beneﬁt of this particular vortex creation techniqueis the ability to create two-component vortices, where ﬂuid composed of one spin state
contains vorticity while the ﬂuid in the other spin state is non-rotating and ﬁlls out
the vortex core. Because of the interatomic interactions, a condensate in a superposition of the diﬀerent spin states amounts to two interpenetrating, interacting superﬂuids[64, 120]. The relative density evolution of the two superﬂuids provides a fascinating
additional degree of freedom to the dynamics described in this chapter.

47
The deterministic introduction of a single vortex into a condensate via wavefunction engineering provides an excellent starting point to assess vortex dynamics. Studiesof two fundamental “bending modes” of a vortex have been made, namely precession of
the vortex core position about the condensate center and tilting of the vortex line’s orientation. In Section 3.5, the core precession will be shown to be the dynamical responseof the vortex to the radial forces pulling on it. The same forces more generally apply
a bending force to a vortex line oﬀset from the symmetrical position of the condensate
center [151, 89]. Under these circumstances, a curved shape can minimize the vortex
length. This is equivalent to energy minimization since for a condensate with repulsive
interactions there is an associated energy cost per unit length to suppress the density
along the vortex core. Vortex curvature is expected to be prominent in the case of vortices aligned with the long direction of a strongly prolate condensate, where signiﬁcantvortex shortening can be achieved with a bent shape. Indeed, increasing vortex curvature in the presence of thermal dissipation has recently been observed by the ENS groupin their cigar-shaped condensate [142]. On the other hand, for the roughly sphericallyshaped condensates of this experiment, the vortices are relatively short and stiﬀ and are
expected to be essentially straight except for very near to the cloud edge.
Nevertheless, we have been able to generate “maximally bent” vortices in theform of vortex rings fully situated within the bulk of the condensate. The formation
process of the vortex ring - through the catastrophic decay of a dark soliton - representsinteresting physics in and of itself, being a conﬂuence of ideas from nonlinear optics,
condensed matter and atomic BEC. Furthermore, the method demonstrates the use of
wavefunction engineering to create a precursor state, the soliton, that has the correct
symmetry to evolve through dynamical instabilities into a desired target state, thevortex ring. Perhaps this idea may prove useful to create objects in the condensate
with yet more complicated topologies.
As a ﬁnal comment, each of the experiments in this chapter has involved the48
implementation of new techniques to manipulate and probe condensates with vorticity.
These include the general technique of wavefunction engineering with two-component
condensates, tailored TOP conﬁning potentials for vortex manipulation [Section 3.6],
multiple axes of imaging [Section 3.8], and two complementary methods of vortex detection - direct imaging of vortex cores in ballistically expanded condensates [Section 3.5]
and angular momentum sensing using surface-wave spectroscopy [Section 3.6]. (Core detection in expansion was in fact ﬁrst demonstrated by the ENS group [118].) In general,
the study of single-vortex dynamics has provided a solid intuition and a proving groundfor techniques in the study of vortex behaviour in a condensate. The techniques havebeen broadly useful in the later experiments of Chapter 4 involving multiple vortices
and have spawned new ideas which might otherwise have been missed, for example, theinteraction of surface modes with highly rotating condensates [Section 4.8].

3.2

Coupled two-component condensates
Since two-component condensates form the basis for the wavefunction engineering

technique, a short overview of their main properties is presented here. A more detailed
account may be found in [120]. Rubidium has three magnetically trappable hyperﬁneground states, two of which have nearly the same magnetic moment and so can be held
simultaneously in overlapping TOP potentials [120]. These two states, |F = 1, mF =−1 and |F = 2, mF = 1 hereafter known as |1 and |2, can be coherently coupled witha ∼6.8GHz two-photon transition via the virtual intermediate state |F = 2, mF = 0[Figure 3.2]. As long as the radio frequency (rf) and microwave ﬁelds comprising thetwo-photon transition are suﬃciently far detuned from the intermediate state, it canbe adiabatically eliminated leaving an eﬀective two-level system {|1, |2}. The usual
coherent population transfer of Rabi oscillations may be observed between the states

49

|2〉
F= 2

rf

mF= 1

mF= 2

δ

∆

mF= 0

int

mF= -1

microwave
mF= -2~6.8GHz|1〉
mF= -1
mF= 0

F=1
mF= 1

δ

|2〉

~6.8GHz|1〉

Figure 3.2: Two-photon coupling between two hyperﬁne spin states of the 87 Rb groundstate 5S1/2 . The spin states, labeled |1 and |2 for short, may be simultaneously
conﬁned in overlapping magnetic potentials. The two-photon coupling is comprised ofa microwave photon (∼6.8GHz) and an rf photon (∼1MHz) as indicated. The intermediate state that mediates the coupling may be adiabatically eliminated, leaving a
two-level system with an eﬀective coupling drive (boxed inset).

140µm

50

0ms

time

60ms

Figure 3.3: Nondestructive imaging of a magnetically trapped condensate undergoing
spatially uniform two-photon Rabi oscillations between internal states |1 and |2. The
frequency of the probe laser is tuned such that it is primarily sensitive to the presenceof the |1 state, which appears as bright on a gray background.

|1 and |2 [Figure 3.3] with an eﬀective on-resonant (“bare”) Rabi frequency given byΩ0 =ΩµW Ωrf
4∆int

(3.2)

Approximate values for the intermediate state detuning ∆int , microwave and rf Rabi
frequencies ΩµW and Ωrf are ∼1MHz, ∼100KHz and ∼0-10KHz respectively, giving
an eﬀective Rabi frequency Ω0 in the range 0-1KHz. As an aside, the nondestructive
imaging of Rabi oscillations in Figure 3.3 shows nicely how phase contrast imaging is an
ideal probe for the two-component dynamics as it is state sensitive through the choiceof probe laser detuning.
The dynamics of the two-component system involve more than simply coupling ofinternal levels since the components interact through collisions. In the dilute-gas regime
characterizing the condensate samples, the weak interparticle interactions are well described by an elastic s-wave scattering length and an associated density-dependent meanﬁeld [69]. Three scattering lengths, a11 , a22 and a12 , refer to each of the intra- andinter-species two-body interactions possible for the |1 and |2 states. The values ofthe scattering lengths are all positive, indicating repulsive interactions, and are of very
nearly the same strength [5]. However, slight diﬀerences in the interactions make it suchthat it is energetically favorable for a mixture of the two ﬂuids to phase-separate intoa distribution dependent on particulars such as the conﬁning geometry [102, 99]. Figure 3.4 shows the phase separation behaviour typical of the spherical trap used in this180µm

51

0ms

30ms

60ms

90ms

120ms

150ms

180ms

210ms

Figure 3.4: Nondestructive imaging of mean-ﬁeld-induced spin segregation in a twocomponent condensate, comprised of the internal spin states |1 and |2. The frequencyof the probe laser is tuned such that it is primarily sensitive to the presence of the|1 state, which appears bright. At t = 0, the condensate is in |1, after which ashort coupling pulse transfers ∼1/2 the population to |2. The internal spin states
are simultaneously trapped in a spherical, harmonic magnetic trap (ω ∼ 2π8Hz). The
spherical symmetry causes a radial spin segregation into a |1 shell around a |2 ball by
120ms. The mutual density oscillations continue to later times.

chapter. After a short π/2 pulse of the two-photon drive creates an initially uniform
superposition of the two states, they separate radially into a ball-and-shell structurewith the |1 ﬂuid exterior to the |2’s. The |1 state in other words has a slightly greater
buoyancy than the |2 state. The mutual density oscillations continue about the equilibrium ball-and-shell conﬁguration with a characteristic time scale for the initial phaseseparation of roughly the harmonic trap period or 120ms.
In the hierarchy of time scales, the last eﬀect noted here, which ultimately limits
the kinds of behaviour that may be observed, is the magnetic trap loss of the twocomponents due to inelastic collisons. In the ﬁrst place, the situation is not as bad
as perhaps might initially be expected. The fortuitous suppression of spin-exchange
collisions in 87 Rb allows for the simultaneous coexistence of the |1 and |2 states without
rapid loss [128]. The individual lifetime of a |1-state condensate in the typical spherical
trap (ωtrap = 2π 8Hz) is ∼50s, most likely limited by collisions with hot background gas.1
A condensate in the |2 state has a considerably shorter density-dependent lifetime ofabout 1s. This is due to the availability of collisional channels to magnetically untrappedstates via spin exchange. The lifetime of the |2 state is a not-so-generous experimental
1

Loss in condensate fraction from sample heating can be suppressed by the continuous applicationof forced evaporative cooling. Empirically, this is found to somewhat improve the lifetime of a |1-state
condensate.

52
limit since only a few trap periods worth of dynamics may occur before the |2 atoms
are depleted. On the other hand, for single-component vortex experiments involving
the |1 state only, the time available is essentially “forever”.

3.3

Wavefunction engineering with two-component condensates

3.3.1

The idea

It has already been mentioned that the wavefunction engineering technique is associated with two-component condensates. However, some limited manipulation of thecondensate wavefunction is possible with single-component condensates alone. Dark
solitons in particular have been experimentally realized in this way [58, 73]. As amacroscopically occupied wavefunction, a single-component condensate has an associated scalar order parameter with a well-deﬁned amplitude and phase at each point inspace. One can spatially modify the phase using the intensity-dependent ac Stark shift
induced by an oﬀ-resonant laser [Figure 3.5]. The size of a typical trapped condensate(10-100µm) is large enough that it can be imprinted with intricate Stark-induced phasepatterns, limited in detail only by the wavelength of the illuminating laser (λ ∼ 1µm).
This technique provides access only to the phase, leaving the condensate density to rearrange itself via dissipative dynamics after the imprinting is complete. One hopes thatthe density evolution is suﬃciently well-bounded that the desired target state survives
the dynamics. This is by no means guaranteed since the target state has a higher energy
in general than the ground state.
Coupled two-component condensates provide a means to control both the amplitude and the phase of the wavefunction engineered. As was shown in Figure 3.3, acoupling drive coherently transfers population between spin states uniformly across acondensate. The aim of wavefunction engineering is to control the amplitude and phaseof the coherent transfer in a spatially dependent way. To develop a simple model, the

53
2P

3/2

λ∼780nm

|2〉

∆2

|1〉

∆1

Figure 3.5: The ac Stark shift of ground hyperﬁne states in 87 Rb due to a laser detuned
from resonance with the excited electronic state 5P3/2 .

two-photon Rabi frequency [Eqn. 3.2] is assumed to be suﬃciently large that population transfer between the spin states occurs quickly compared to the mean-ﬁeld driven
component separation in the trap. Under these circumstances, a simple “nailed down”
picture can be considered in which the two-component dynamics are described by alocal two-level system ﬁxed at each point in space.
The means to control the population transfer in the two-level system is again
the laser-induced ac Stark shift, which serves to alter the energy splitting between
the two levels [Figure 3.5] thereby aﬀecting the detuning with respect to the coupling(two-photon) drive. At least two control parameters are required to inﬂuence boththe amplitude and phase of the transfer. This results in a more sophisticated scheme
than phase imprinting whereby the incident intensity is modulated with characteristic
parameters of the dynamic control being the amplitude and phase of modulation. To
obtain spatially selective control over the population transfer, one can imagine using
a spatial light modulator array to put an arbitrary two-dimensional light pattern onto
the condensate. This amounts to focusing a laser beam at each point in the condensate

54

(a)
BEC
Off-resonant laser
I (t ) = I 0 + I m Sin(ωmt + φm )

(b)
|2〉

(c)
n=2
n=1

|2〉

n=0
Modulation~6.8GHz

n= -1

…
|1〉

ωm

~6.8GHz|1〉

Figure 3.6: Local wavefunction engineering in a two-component condensate. (a) An
oﬀ-resonant laser with modulated intensity is focused at a particular point in the condensate. The oscillating intensity modulates the local resonance between internal hyperﬁne states {|1, |2} via the ac Stark eﬀect. In the presence of a coupling drive, thisis equivalent to the modulation of the drive (b), which acquires sidebands at multiplesof the carrier frequency. A resonant sideband (c) allows coherent population transfer
between the spin states. Control of the sideband amplitude and phase allows control of
the amplitude and phase of the locally transferred population (see text).

|2〉

φ=π/2

φ=0

~6.8 GHz

φ=3π/2

Fluid
flow

55

φ=π

|1〉

Figure 3.7: Point-by-point wavefunction engineering in a two-component condensate toconstruct a singly quantized vortex.

[Figure 3.6]. Modulating the laser intensity at each point in the condensate dithers
the two-level resonance of the atoms. Equivalently the coupling drive itself can beconsidered to be locally modulated, putting FM sidebands onto the carrier frequency
at integer multiples of the modulation frequency. Thus it now becomes possible tocouple resonantly to the atoms using not only the carrier frequency but also one of the
sidebands [Figure 3.6]. If a sideband is tuned to the two-level resonance, the rate of theresulting population transfer may be controlled by the strength of the sideband, which
is directly related to the modulation depth of the laser intensity at that point. Whilethe control of transfer amplitude is straight forward, what is truly remarkable is that thephase of the transferred population can also be controlled in a completely deterministic
way - by the choice of modulation phase. This means that the modulation parametersof amplitude and phase may be dialed into the spatial light modulator pixel-by-pixel
to produce the desired (two-dimensional) condensate wavefunction in the ﬁnal internal
spin state [Figure 3.7]. Once the transfer to the target wavefunction is complete, the“scaﬀold state” formed by the remaining population in the initial spin state can (ifdesired) be selectively blown away with the light pressure from an on-resonant laser.

56
3.3.2

A closer look

Since the direct control of population amplitude and phase through associatedmodulation parameters is so fundamental a point, it is shown explicitly for the two-levelsystem at a particular point in the condensate. The model imagines a two-level system{|1,|2} with resonant frequency ω0 at about 6.8GHz and a coupling drive characterized
by frequency ω and bare Rabi frequency Ω0 . The resonance is Stark modulated as
ω0 (t) = ω0 + αm sin(ωm t + φm )(3.3)

with modulation frequency ωm , amplitude αm and phase φm . The value of ω0 includes
the time-averaged Stark shift of the states. There are four system parameters to accountfor, namely the amplitude and phase of both states |1 and |2. The total population in
the two-level system is conserved; thus only the population diﬀerence, p2 − p1 , matters.
Initially the state begins in a single spin state. The phases may be separated into thephase of the initial state and the phase of the target state referenced to the initial. As
long as the time-averaged Stark shift is the same across the condensate, there is only an
isotropic phase shift to the initial state, which can be considered a spatially ﬂat phasereference. As an additional caveat, the notion of a spatially ﬂat phase reference holds
only initially when population is being transferred from the initial state everywhere in
the condensate. Once the population begins to be transferred back to the initial state
at some points in the condensate, the phase evolution gets complicated (see below).
Except for Section 3.9, the wavefunction engineering focuses on the regime where population is still being transferred from the initial state; thus, we need care only about
the population diﬀerence and relative phase φ to understand the structure of the ﬁnal
state’s wavefunction. A particularly powerful pictorial way to understand the dynamics
of these two variables is the Bloch picture [49] where the system is characterized by apseudospin vector s in a three-dimensional space [Figure 3.8]. The vertical 3̂-axis represents the population diﬀerence and the azimuthal angle in the horizontal plane indicates

57

3

ΘB

s

φ

2

1

Figure 3.8: Bloch spherethe relative phase. In terms of the well-known one-to-one correspondence of a two-levelsystem with a spin-1/2 system, the Bloch vector axes represent the expectation valuesof the Pauli spin components, si = σi .
It is important to note that the Bloch-sphere representation suﬀers from a limitation in that the relative phase is undeﬁned for a pseudospin pointing along eitherof the poles. Only away from the poles where population exists for both states of the
two-level system is the relative phase a meaningful quantity. In fact, the loss of a welldeﬁned relative phase at the poles is necessary since there is a π phase change in φ as
the pseudospin traverses a pole. The phase change is associated with the state whose
population vanishes at the given pole. (See Section 3.9 for a physical implication of thiseﬀect).
The situation of a coupled two-level system in the absence of modulation (αm = 0)
is reviewed ﬁrst because it provides a paradigm for the solution with modulation. The
approach follows closely that of ref. [49]. The equations of motion for the pseudospin
in the presence of a coupling drive may be written as a torque problemd L (t) × s(t)s(t) = Ω
dt

(3.4)

58
where the torque vector is oscillating in time−iωt
+ c.c.  −Ω0 e
 L (t) = 
Ω


0




ω0

(3.5)

The bare Rabi frequency Ω0 is given by Eqn. 3.2. If the drive is not too far detuned fromresonance (|ω − ω0 |  ω0 ), the problem can be solved to a very good approximation bytransforming into a frame rotating about the 3̂-axis at the drive frequency ω and thenperforming a time-average to eliminate all rapidly oscillating torque components. Thisso-called “rotating-wave approximation” is justiﬁed since the fast oscillating components
will have no cumulative eﬀect on the pseudospin over time. What remains after thetime-averaging is again a torque equation for the pseudospin ρ(t) in the rotating framed ×ρρ(t) = Ω
(t)
dt

(3.6)

but now with a constant torque −Ω0 



 =
Ω
 0 




−δ

(3.7)

where δ = ω − ω0 is the detuning2 [Figure 3.2]. Since the rotating frame, called here
the “standard” Bloch frame, diﬀers from the lab frame by a trivial (time-dependent)
rotation about the 3̂-axis, the phase diﬀerence between the states can without loss ofgenerality be associated with the azimuthal angle in the standard frame. The solution
to Equation 3.6 is just a rotation of the pseudospin about the torque direction↔

ρ(t) =R (t) ρR(3.8)

with ρ
R being a constant vector characteristic of the initial conditions. The expression
2

The detuning diﬀers by a minus sign from the deﬁnition in [49].

59↔

for the rotation matrix R (t) is:R (t) = ↔



Ω20 +δ2 cos θ(t)Ω2δΩ sin θ(t)δΩ0
Ω2 (1 − cos θ(t)) −δΩ sin θ(t)cos θ(t)Ω0Ω sin θ(t)δΩ0
Ω2 (1 − cos θ(t))−Ω0Ω sin θ(t)

δ2 +Ω20 cos θ(t)
Ω2







(3.9)

The eﬀective oﬀ-resonant Rabi frequency Ω and the argument θ(t) are identiﬁed as =Ω = |Ω|(Ω20 + δ2 ,θ(t) = Ωt(3.10)The general dynamical behaviour, corresponding to Rabi oscillations, involves

precession of the pseudospin ρ, which traces out a cone centered on the torque vector Ω.
For example, at the start of the coherent control process, the population of the two-levelsystem is all in one state. For the case of the |2 state, this corresponds to a pseudospin
vector pointing along the north pole of the Bloch sphere. The coupling drive is far
detuned (|δ|  Ω0 ) such that the torque vector [Eqn. 3.7] is also nearly aligned with
the north pole. In the absence of any modulation, the pseudospin will precess in a tightcircle about the torque direction [Figure 3.9]. The deviation of the pseudospin from its
initial vertical orientation is small, as one would expect, since the large detuning of thedrive permits little population transfer to the ﬁnal state.
A time-dependent detuning is now added in accordance with Eqn. 3.3
δ(t) = δ0 + αm sin(ωm t + φm )(3.11)As a result, it is now necessary to solve for the pseudospin dynamics in the presenceof an oscillating torque. In similar fashion to the original Rabi problem, the torqueoscillations are expected to drive a slow tipping of the pseudospin at a new eﬀective
Rabi frequency ΩC , which characterizes the rate of population transfer for the coherent
control. Continuing with the analogy to the Rabi problem, a separation of time scales isevident between the instantaneous spin precession rate at Ω and the slow accumulated
spin tipping at ΩC . Using the standard technique to reduce the number of fast-oscillating

60

3

v

Ω

-δ

v

ρ
1’

-Ω0

2’

Figure 3.9: Rabi oscillations on the Bloch sphere in the standard rotating frame (seetext). The pseudospin ρ precesses in time, tracing out a cone about the torque vectorΩ.

variables, a transformation is made into the instantaneous moving frame of the torquevector, sometimes called the “quasi-static basis.” The moving frame is both rotating↔

and dithering in space. As a result the transformation matrix R (t) [Eqn. 3.9] acquires
extra time dependence throughΩ → Ω(t) =θ(t) =tΩ20 + δ(t)2Ω(t )dt

(3.12)and ρR of Eqn. 3.8, originally constant, becomes compensatorily time dependent. To
obtain the equation of motion in the co-moving frame, the torque equation 3.6, including
a time-dependent detuning, is ﬁrst written in matrix form and then expanded using↔

ρR (t). The resulting equations of motion for the pseudospin ρR in the coρ(t) =R (t)moving frame are↔

↔† ∂ R ∂Ω
d
ρR
=− R
dt
∂Ω ∂t

ρ
R

(3.13)

To obtain an analytical understanding of these equations we consider the limit where
the time-averaged detuning δ0 is large compared to both the bare Rabi frequency Ω0

61and the modulation amplitude αm . After an expansion to ﬁrst order in αm and Ω0 , thetime-average of the equations of motion [Eqn. 3.13] can (once again!) be written as aneﬀective torque equation:dC ×ρρR (t) = ΩR (t)
dt(3.14)

 C in the co-moving frame has components:
The time-averaged torque Ω
ΩC1
ΩC2
ΩC3

)

*
αm
ωm Ω0
sin(ωm t + φm ) sin Ωt += −sign(δ0 )αmsin(ωm t + φm )
Ω2
ωm
t

*
)
ωm Ω0
αm
= αm
sin(ωm t + φm )sin(ωm t + φm ) cos Ωt +
Ω2
ωm
t
 2
Ω0
= 0+O
(3.15)Ω2

The righthand side of the above expressions can be expanded in Bessel functions, which
are the familiar feature one obtains for all problems involving frequency modulation.
“Resonances,” deﬁned by a non-zero average torque, will occur when the modulation
frequency is an integer fraction of the eﬀective Rabi frequency
ωm =

Ω
,
n

n ∈ ±1, ±2 . . .

(3.16)The integer n refers to the index of the FM sideband being used for coupling [Figure 3.6].
The ﬁrst two resonances are evaluated as follows, corresponding to coupling via the ﬁrst
two sidebands:

sign(δ0 ) cos φm 
 C = − αm Ω0 J0
ωm = Ω → Ω− sin φm

2Ω

0


sign(δ0 ) cos 2φm 

 



Ω

 C = αm Ω0 J1 αm J0 2αm 
→ Ω
ωm =


− sin 2φm

2
2Ω
ωm
ωm 


0


 
αm 2 ωm 

(3.17)

Only the lowest order Bessel term has been retained for these expressions. The direction
 C is referenced to the moving frame; however, under the approximationof the torque Ω62already made that the average detuning is large and the modulation depth relatively
small, the moving frame is nearly equivalent to the standard Bloch frame within atrivial rotation about the 3-axis. (This can be seen by setting αm and Ω0 to zero in↔

the transformation matrix R [Eqn. 3.9].) As a result, the time-averaged torque canbe interpreted in the same way as for the familiar Rabi problem. This provides the C lies in the horizontal Blochremarkable conclusion that, when a sideband is resonant, Ωplane and has an azimuthal orientation controlled by the modulation phase. For theﬁrst sideband resonance the azimuthal angle is (within a sign) given by the modulation
phase; for the second sideband it is twice the modulation phase and so on. Clearly,
by controlling the orientation of the torque vector the pseudospin can be pointed in anarbitrary direction in the horizontal plane, or, in more physical terms, population canbe transferred in the two-level system with a deterministic arbitrary phase [Figure 3.10].
The amplitude of the time averaged torque, which corresponds to a new eﬀective
Rabi frequency ΩC , is directly proportional to the bare Rabi frequency Ω0 and, mostimportantly, the modulation amplitude αm . Thus, the amount of population transferred over a given time can be directly controlled by the modulation depth. As a side
note, from the Bessel dependence it is clear that the second FM sideband resonance isconsiderably suppressed relative to the ﬁrst in the low modulation regime (αm /ωm  1).
Concentrating on the ﬁrst sideband, the prescription for wavefunction engineeringof a two-component condensate is straightforward: apply oﬀ-resonant laser light at eachpoint in space to modulate the local two-component resonance. For each point, dial in
the amplitude and phase of modulation to obtain the desired amplitude and phase forthe wavefunction transfer. Direct point-by-point control is a novel feature of coherent
control with condensates and is due to the large size of the macroscopic wavefunction(10-100µm) relative to the wavelength (∼1µm) of the laser addressing it. It is interesting
to compare this situation to coherent control techniques in other atomic and molecularsystems, including the engineering of electronic wave-packets in Rydberg atoms [157]

63

v

ρ

v

ρ

v

ΩC

v

ΩC

v

ρ

v

ΩC

Figure 3.10: Coherent control of the pseudospin ρ on the Bloch sphere. The azimuthal
angle of the pseudospin can be controlled with the orientation of eﬀective torque vectorΩC (see text). Three diﬀerent orientations are illustrated. A particular polar angle ofthe pseudospin can also be achieved by applying the torque for the required amount oftime.

and vibrational wave-packets in molecules [92]. In both cases the ∼1µm wavelength of
the short-pulsed laser incident on the system is much larger than the typical size of thesystem, measured in Bohr radii. Therefore, coherent control is achieved in a diﬀerent
way. The large bandwidth of the laser pulses is used to couple to the superpositionof energy states required to construct the target wavefunction [Figure 3.11]. One canconsider wavefunction engineering in this fashion to be analogous to Fourier wave optics
whereas the point-by-point manipulation of the condensate describes the complementarycase analogous to ray optics.

3.3.3

Experimental realization: single vortex formation

The implementation of wavefunction engineering within the system {|1,|2} begins with a two-photon drive characterized by a bare Rabi frequency3 of Ω0 ∼33Hz
and a detuning of ∼100Hz from resonance, giving an eﬀective oﬀ-resonant Rabi frequency of Ω ∼100Hz. The coherent control makes use of the ﬁrst sideband resonance(ωm = Ω ∼100Hz) as it provides the strongest coupling. There exist a number of technological possibilities to generate the spatially and temporally modulated pattern oflaser intensity necessary to create arbitrary condensate wavefunctions. As mentioned3Ω0 is kept at a low value to suppress the magnitude of inhomogeneities in its value across thecondensate. The inhomegeneities arise from the magnetic trapping potential [Section 3.9].

64

R

λ

(a)

Ei

(b)

Ψ(R) =Σ Ψi(c)Figure 3.11: Wavefunction engineering in“Fourier space.” The cartoon illustrates thecreation of a vibrational wavepacket in a diatomic molecular system with a short pulsed
laser. (a) A laser pulse is incident on the molecule. (b) The frequency bandwidth of thepulse allows coupling to many vibrational states. (c) Shaping the frequency spectrumof the pulse allows the creation of the desired target state Ψ(R) as superposition ofvibrational modes Ψi .

earlier, a two-dimensional liquid crystal modulator provides the “ultimate” general solution. Simpler techniques, however, exist to create the topological features of immediate
interest, namely solitons and vortices. To make a singly quantized vortex, which is adonought shape with a 2π azimuthal phase winding, a spatially dependent intensity
modulation is required that increases from zero away from the cloud center and has a
2π phase dependence azimuthally. These speciﬁcations exactly describe a linear slopeof light intensity rotating about the center of the condensate. This is accomplishedin practice by rotating a laser beam that is oﬀset from the condensate center [Figure 3.12]. Since the rotation(i.e. modulation) frequency is a modest ∼100Hz, a pair ofpiezo-actuated mirrors provides a convenient and simple means to move the “sculpting”
beam in two-dimensions [Figure 3.12]. To minimize any spatially dependent phaseimprinting eﬀect from the time-averaged beam proﬁle, the radius of beam rotation ischosen to be ∼3/4 of its waist. This results in a uniform time-averaged intensity proﬁleacross the condensate extent. Typical beam parameters include a waist size of 95µm,
total power of ∼10nW, and a laser frequency ∼1100MHz below the F = 1 → F  = 0
transition [Figure 2.4]. These parameters yield an average ac Stark shift across the con-

65
densate of typically 60Hz (taken into account in the two-photon drive frequency) and apeak modulation amplitude at the condensate edge (R∼25µm) of 45Hz or ∼0.45Ω.
The condensate begins in a single initial spin state, chosen as either |1 or |2.
The coupling drive is suﬃciently far detuned (∼3Ω0 ) that negligible population transfer
occurs in the presence of the drive alone. However, when the sculpting beam is applied
and the two-photon drive is tuned to within a few Hz of the ﬁrst sideband, coupling tothe ﬁnal spin state occurs. As expected, population is transferred into the donought
shape of a vortex surrounding a core of the initial spin state [Figure 3.13]. The process
is continued for 50-70ms yielding a typical transfer eﬃciency of 50%. A vortex can be
successfully created starting with either internal state [Figure 3.13] although one would
expect that driving to the |1 state is preferable because it has a longer lifetime and its
situation outside a |2 core is the natural one in terms of the components’ buoyancies(Compare Figures 3.13(b)&3.4). In any case, vortices are habitually formed by driving
to the |1 state. If a |2 vortex is desired, a short (4ms) π pulse from a strong two-photon
drive can be subsequently applied to ﬂip the spin composition of vortex and core.
To obtain a reasonable transfer rate at the ﬁrst sideband, the bare Rabi frequency Ω0 and peak modulation depth αm are a reasonable fraction of the two-photon
detuning. This results in local residual Rabi oscillations during the engineering process[Figure 3.14]. Within one Rabi cycle (∼10ms), a partial transformation of the ﬁnal population occurs to a crescent shape that appears to rotate around the cloud. However,the non-symmetric crescent shapes are avoided by stopping the engineering process after
an integer multiple of the average eﬀective Rabi period (2π/Ω). At such points in time,neglecting technical imperfections, all azimuthal points must have experienced the same
evolution and the state must be symmetric, which it is in practice [Figure 3.14].
“By construction,” the wavefunction engineering should make a vortex with a 2π
winding as indicated by the simulation [Figure 3.13(d)]. The phase winding can be experimentally veriﬁed with an interferometric technique that couples the two components

66

(a)
fiber

AOM

PZT1
PZT2

servo
ECDL
780nm

BEC 50 cm

frequency
lock

(b)

Laser rotation, ωLaserprofileBEC

Figure 3.12: Sculpting beam setup for wavefunction engineering. (a) The output ofan external cavity diode laser (ECDL) is passed through an acousto-optic modulator(AOM) to provide control of the laser power. The light is transported to near thecondensate via optical ﬁber, providing some spatial mode cleaning as well. A small
amount of light is split oﬀ after the ﬁber to servo the laser power. A pair of piezomounted mirrors, PZT1 and PZT2, allow the beam to be moved in two dimensions withan arbitrary pattern. Three bounces between the mirrors provide ampliﬁcation of thebeam’s motion. The beam is then weakly focused onto the condensate to create, forexample, a rotating beam (b) for vortex engineering.

67

(a)

0ms

(b)

(c)
|2〉

10ms

20ms

30ms

π

(d)

40ms

50ms

40ms

-π

60ms

|1〉

70ms

|2〉
170 µmFigure 3.13: (a)&(b) Nondestructive imaging of vortex growth during the process ofwavefunction engineering in a trapped two-component condensate. Images are taken
at taken at 10 ms intervals corresponding to one period of the eﬀective oﬀ-resonant
Rabi oscillations present in the condensate. A vortex may be created in the internal|2 state (a) or |1 (b). In each case, the frequency of the probe laser is tuned to
be primarily sensitive to the state containing vorticity. (c) A numerical simulation of
the local Bloch equations [Eqn. 3.6] with detuning modulation across the condensate isshown for comparison. Parameters of the simulation, typical of those in the experiment,
are as given in the text. (d) The phase diﬀerence between the internal states is shownfor the simulation at t=40ms and may be interpreted as the phase of the ﬁnal spinstate. Although not perfectly uniform radially, the phase pattern clearly predicts the
2π azimuthal phase wrap of a singly quantized vortex.

68

0ms

2ms

4ms

6ms

8ms

10ms

Figure 3.14: Nondestructive imaging showing the residual local Rabi oscillations during
the growth of a vortex, in this case in the |2 state. The image series spans one cycleof the average oﬀ-resonant Rabi frequency (Ω=100Hz). At times (e.g. t=0ms) corresponding to an integer multiple of the Rabi period, the |2 ﬂuid forms a symmetric ringcorresponding to a centered vortex.

69
where the densities of the core and vortex overlap. This was done for the ﬁrst demonstration of vortex creation [122]. It is not discussed again here as two more heavily used
vortex diagnostics are already presented in later sections.
The handedness of the vortex created can be changed by reversing the rotationof the sculpting beam, but this is not done in practice. For a ﬁxed beam rotation
direction, the vortex handedness can also be chosen with the sign of the average twophoton detuning [Eqn. 3.17], which is demonstrated experimentally in Section 3.6.
The long term stability of the engineering process has been improved by a slow
intensity servo applied to the sculpting beam [Figure 3.12]. Residual diﬃculties include
the slow thermal drift of the piezoelectric elements, which could ultimately be stabilizedwith a beam-pointing servo. By far, however, reproducibility of the vortex formation
is most adversely aﬀected by undesirable center-of-mass motion (“slosh”) of the condensate. This causes both shot-to-shot variations in the relative position between thecondensate and sculpting beam as well as relative motion during the engineering process. Typically, the result is the creation of an oﬀ-center vortex. Careful attention isrequired on a day-to-day basis to minimize slosh as much as possible. Many shots (witha 1min cycle time) are still needed to obtain a collection of reasonably well-centeredvortices. For this reason, a good deal of eﬀort is made to assure that all aspects of theexperiment are stable over long periods of time, including condensate formation, beam
intensity, and so on.

3.4

Dynamical stability of two-component vortices
While it is essentially irrelevant which spin state is used initially to create a two-

component vortex [Figure 3.13], diﬀerences in the interparticle interactions of the twocomponents [Section 3.2] produce a stark dichotomy of dynamical behaviour dependingon the spin composition of the vortex and core. For convenience, the notation (1,0)
denotes the case of a |1 vortex and |2 core while (0,1) denotes the opposite conﬁg-

70
uration. On a simple level, a vortex in the |2 state will certainly be destroyed more
quickly due to the faster collisional loss of |2 atoms from the magnetic trap; however,a more interesting issue is the dynamical stability of a two-component vortex giventhat the two ﬂuids can exchange angular momentum through their mutual interactions.
A simpliﬁed analytical model by Pérez-Garcı́a and Garcı́a-Ripoll [90, 135] evokes thequalitative behaviour, which is shown to depend on two factors: the spin component
hosting the vortex, and, interestingly, the relative population N1 /N2 of the components.
A stability condition is derived for the (1,0) conﬁguration as
+

N1
−1
N2

2

>1−

a11
a22

(3.18)For 87 Rb, the values of the intra-species scattering lengths, aii [Section 3.2] are suchthat the stability condition is satisﬁed for any population distribution amongst the spincomponents. The opposite conﬁguration (0,1) is stable if
+

N2
−1
N1

2

>1−

a22
a21

(3.19)For 87 Rb this inequality fails over the range of populations N1 /N2 ∈ [0.73, 1.49].
The stability of the two diﬀerent vortex conﬁgurations has ﬁrst been compared
under the experimental condition of a roughly equal population distribution between
vortex and core [Figure 3.15]. The (1,0) conﬁguration remains stable over time whereas,in the (0,1) case, the |2 vortex crushes inward under the inﬂuence of its negative
bouyancy. The |2 vortex fragments into two pieces, never healing into an intact ring
shape in subsequent dynamics.
For a suﬃciently small |1 core, the stability condition for the (0,1) conﬁguration
will be satisﬁed. In order to approach this regime experimentally, a vortex has ﬁrst beencreated in the (1,0) state. The |2 core can then be reduced slowly over ∼1s through its
inherent decay or quickly (10ms) through the application of a short blast of selectively
resonant laser light. In either case, a 4ms two-photon π pulse is then administered to

71

|1 〉

(a)

(a)

|2〉

(b)

0 ms

200 ms

400 ms

600 msFigure 3.15: Dynamical stability of two-component vortices.(a) The free evolution of a|1 vortex (i.e. (1,0) state) in the magnetic trap. It is stable over a time long comparedto the trap oscillation period (128 ms). (b) The free evolution of a |2 vortex (i.e. (0,1)
state) is much more dynamic. It is seen shrinking quickly into the invisible |1 coreand rebounding into fragments. Each column is from a single run, where the times
indicated are referenced to the end of vortex creation. In these pictures the states
appear diﬀerently with respect to the background due to the diﬀerent signs of theimaging light detuning.

72
switch the spin-composition of the vortex from (1,0) to (0,1). This essentially instantaneous spin ﬂip provides a sharp initial time for the subsequent evolution of the (0,1)
conﬁguration.
Figure 3.16 presents typical results where one image sequence involves a moderately sized |1 core while the two remaining sequences are for a small |1 core. In thecase of the larger core, the two components separate into two pieces with the |1 ﬂuid
moving outward to edge of the condensate. The two components continue to rotate
around one another to preserve the total angular momentum of the system. The theoryof [90, 135] predicts that the instability of the initial (0,1) conﬁguration should induce
an oscillation between (0,1) and (1,0) in which the components fully exchange angularmomentum. The predicted intermediate state of two rotating halves sharing angularmomentum appears similar to that observed in Figure 3.16(a); however, the continued
evolution to the (1,0) conﬁguration has not been seen experimentally. Indeed, the factthat the |2 vortex fragments into a single rather than two pieces as in Figure 3.15(b)may be attributed to a radial bias in the initial core position introduced by the engineering process.
For the case of a small |1 core [Figure 3.16(b)and (c)], the mutual mean ﬁeld
energy clearly is suﬃciently reduced that the (0,1) conﬁguration is stable. The |1 atoms
remain conﬁned within the |2 ﬂuid and track the core as it precesses around the condensate center in the direction of the vortex ﬂow [see Section 3.5]. In the image sequenceof Figure 3.16(c) it may even be possible to distinguish a “Matryoshka nested doll” conﬁguration in which the |1 core itself is ﬁlled with a small amount of |2 ﬂuid trapped
inside due to its negative buoyancy.

73

(a)

0ms

(b)

(c)

50ms

100ms

150ms

200ms

250ms

150 µm

300ms

Figure 3.16: Three nondestructive image sequences showing the time evolution of twocomponent vortices in the (0,1) state. The image frames are spaced by the times indicated. (a) A |2 vortex (dark) with a moderately sized |1 core (bright) is unstable andbreaks up. The |1 and |2 components separate into two segments that rotate around
one another. (b)&(c) A |2 vortex with a small-sized |1 core is stable. The |1 ﬂuid
remains conﬁned in the |2 ﬂuid and tracks the precessional motion of the vortex core.

74

3.5

Vortex Core precession [51]3.5.1

Abstract

We have observed and characterized the dynamics of singly quantized vortices indilute-gas Bose-Einstein condensates. Our condensates are produced in a superpositionof two internal states of 87 Rb, with one state supporting a vortex and the other ﬁllingthe vortex core. Subsequently, the state ﬁlling the core can be partially or completely
removed, reducing the radius of the core by as much as a factor of 13, all the way down
to its bare value of the healing length. The corresponding superﬂuid rotation rates,
evaluated at the core radius, vary by a factor of 150, but the precession frequency ofthe vortex core about the condensate axis changes by only a factor of two.

3.5.2

Introduction

The dynamics of quantized vortices in superﬂuid helium and superconductors havebeen fascinating and important research areas in low-temperature physics [155, 74, 154].
Even at zero temperature, vortex motion within a superﬂuid is intricately related to thequantization of current around the vortex core. Besides these superﬂuid systems, studiesof the dynamics of optical vortices has also become an active area of research [87, 112].
More recently, demonstrations of the creation of quantized vortices in dilute-gas BoseEinstein condensates (BEC) [122, 118] have emphasized the similarities between thecondensed matter, optical, and dilute-gas quantum systems. Because of the observational capabilities and the techniques available to manipulate the quantum wavefunctionof the condensates, dilute-gas BEC experiments provide a unique approach to studies
of quantized vortices and their dynamics. This section reports direct observations andmeasurements of singly quantized vortex core precession in a BEC.
Numerous theoretical papers have explored the expected stability and behaviorof vortices in BEC [105, 141, 6, 55, 106, 82, 152, 153, 116, 147, 123, 90, 135]. One

75
interesting expected eﬀect is vortex core precession about the condensate axis [141, 55,
106, 82, 152, 153, 116, 147, 123]. Radial motion of the core within the condensate canalso occur, and may be understood as being due to energy dissipation and damping
processes. Core precession may be described in terms of a Magnus eﬀect – a familiar
concept in ﬂuid dynamics and superﬂuidity [155, 74]. An applied force on a rotating
cylinder in a ﬂuid leads to cylinder drift (due to pressure imbalances at the cylindersurface) that is orthogonal to the force. Analogously, a net force on a vortex core in asuperﬂuid results in core motion perpendicular to both the vortex quantization axis andthe force. In the condensate vortex case, these forces can be due to density gradients
within the condensate, for example, or the drag due to thermal atoms. The densitygradient force may be thought of as one component of an eﬀective buoyancy: just as a
bubble in a ﬂuid feels a force anti-parallel to the local pressure gradient, a vortex corein a condensate will feel a force towards lower condensate densities. The total eﬀective
buoyancy, however, is due less to displaced mass (the “bubble”) than it is to dynamical
eﬀects of the velocity-ﬁeld asymmetry, a consequence of a radially oﬀset core. Typically,
the total buoyancy force is away from the condensate center, and the net eﬀect is anazimuthal precession of the core via the Magnus eﬀect. Drag due to the motionless (onaverage) thermal atoms opposes core precession, causing the core to spiral outwards
towards the condensate surface. In the absence of this drag (for temperature ∼ 0),radial drift of the core may be negligible.

3.5.3

Experimental technique

Our techniques for creating and imaging a vortex in a coupled two-component condensate are described in refs. [122, 159]. The two components are the |F = 1, mF = −1
and |F = 2, mF = 1 internal ground states of 87 Rb, henceforth labeled as states |1 and|2 respectively. We start with a condensate of 106 |2 atoms, conﬁned in a sphericalpotential with oscillator frequency 7.8 Hz. A near-resonant microwave ﬁeld causes some76of the |2 atoms to convert to |1 atoms. The presence of a rotating, oﬀ-resonant laser
beam spatially modulates the amplitude and phase of the conversion. The net result
is a conversion of about half of the sample into an annular ring of |1 atoms with acontinuous quantum phase winding from 0 to 2π about the circumference – a singly
quantized vortex. The balance of the sample remains in the non-rotating |2 state andﬁlls the vortex core. With resonant light pressure we can selectively remove as muchof the core material as we desire. In the limit of complete removal, we are left with asingle-component, bare vortex state.
In this bare-core limit, the core radius is on the order of the condensate healing
length ξ = (8πn0 a)−1/2 , where n0 is the peak condensate density and a is the scattering
length. For our conditions, ξ = 0.65 µm, well under our imaging resolution limit. The
bare core can be observed after ballistic expansion [118] of the condensate, but this isa destructive measurement. On the other hand, if we leave some of the |2-state atoms
ﬁlling the core, the pressure of the ﬁlling material opens up the radius of the |1 vortex
core to the point where we can resolve the core in a time series of nondestructive phasecontrast images.

3.5.4

Filled-core dynamics

We ﬁrst discuss vortex dynamics in two-component condensates, where 10%−50%of the atoms were in the |2 ﬂuid ﬁlling the |1 vortex core. We took successive imagesof the |1 atoms in the magnetic trap, with up to 10 images of each vortex. The vortex
core is visible as a dark spot in a bright |1 distribution, as shown in Fig. 3.17(a).
Instabilities in our vortex creation process usually resulted in the creation of oﬀ-center
vortex cores, allowing us to observe precession of the cores. We observed precession out
to ∼ 2 s, after which the |2 ﬂuid had decayed to the point that the vortex core was too
small to be observed in the trapped condensate.
The recorded proﬁle of each trapped condensate was ﬁt with a smooth Thomas-

77

(a)
0 ms

50 ms

100 ms

150 ms200 ms

250 ms

300 ms

(b)

(b)

0
-45
-90

-135

15

(c)

r/ξ

θt (degrees)45

0

100 200 300

time (ms)

(d)

10
5
0

0

100 200 300

time (ms)Figure 3.17: (a) Seven successive images of a condensate with a vortex and (b) theircorresponding ﬁts. The 75-µm-square nondestructive images were taken at the times
listed, referenced to the ﬁrst image. The vortex core is visible as the dark region within
the bright condensate image. (c) The azimuthal angle of the core is determined for each
image, and is plotted vs. time held in the trap. A linear ﬁt to the data indicates a
precession frequency of 1.3(1) Hz for this data set. (d) Core radius r in units of healing
length ξ. The line shown is a linear ﬁt to the data.

78
Fermi distribution. Each vortex core proﬁle was ﬁt with a Gaussian distribution todetermine its radius and position within the condensate. From the ﬁts, we determined
the overall radius Rt of the trapped condensate (typically 22 µm), the HWHM radius r of
the ﬁlled vortex core, and the displacement dt and angle θt of the core center with respect
to the condensate center. Core angles and radii for the images in Fig. 3.17(a) are shownin Figs. 3.17(c) and (d). The vortex core is seen precessing in a clockwise direction,
which is the same direction as the vortex ﬂuid ﬂow around the core. The angularprecession frequency was determined from the time dependence of θt (Fig. 3.17(c)).
This and other similar data sets showed no reproducible radial motion of the core over
the times and parameters examined. However, consistent decrease in the size of the core
was observed, which we interpret as being due to known decay of the |2 ﬂuid throughinelastic atomic collisional processes.
For each data set, we determined a mean core radius and displacement. The data
cover a range of core radii (r = 7ξ to 13ξ), displacements (dt = 0.17Rt to 0.48Rt ),and percentage of atoms in the core (10% to 50%). Except for a few “rogue vortices”(discussed below), the measured precession frequencies are clustered around 1.4 Hz,
as shown in Fig. 3.18, precessing in the same direction as the ﬂuid rotation. The data
(Fig. 3.18(a)) suggest a slight increase in frequency for cores further from the condensate
center. We also see (Fig. 3.18(b)) a slight decrease in precession frequency for larger
cores. These measurements are in qualitative agreement with two-dimensional numericalsimulations for two-component condensates [123].
As indicated in Fig. 3.18, a few vortex cores exhibited precession opposite tothat of the ﬂuid ﬂow (negative frequencies). The quality of the corresponding vortex
images was routinely lower than for the positive-frequency precession points, with vortices looking more like crescents and “D” shaped objects rather than like the imagesof Fig. 3.17. We only found such occurances with our two-component (in-trap) measurements. We speculate that this “inverse precession” may be due to rare events in

2

(a)

1
0
-1
0.00

0.25

dt /Rt0.50

frequency (Hz)frequency (Hz)

79

2

(b)

1
0
-1
0

5

10

15

r/ξ

Figure 3.18: Compiled data for ﬁlled vortex core precession, with each data point extracted from a series (as in Fig. 3.17) of nondestructive images of a single vortex. Precession frequency is plotted vs. (a) core displacement dt in units of condensate radius
Rt , and (b) core radius r in units of healing length ξ. Circles correspond to positivefrequencies and ﬁlled squares to negative frequencies. (Positive frequency is deﬁned ascore precession having the same handedness as the vortex angular momentum.) The
triangle at r = ξ shows for reference the average measured precession frequency of manybare vortices (see text and Fig. 3.19(b)). A line is drawn as a guide to the trend in
frequency vs. core size.

80
which total angular momentum may be distributed in a complicated way among bothinternal states. Such cases might arise due to position instabilities of the rotating laser
beam during the vortex creation process. Recent theoretical attention has addressed thepossibly related situation of non-symmetric conﬁgurations of vortices in two-component
condensates [90, 135].

3.5.5

Bare core dynamics

To examine the dynamics of bare vortices, our procedure consisted of taking
a nondestructive phase-contrast picture of the partially ﬁlled |1 vortex distribution(Fig. 3.19(a) inset), as previously discussed, followed by complete removal of the coreﬁlling [7]. We then held the bare vortex in the trap for a variable hold time th , after
which the condensate was released from the trap. We took a ﬁnal near-resonance phasecontrast image [8] of the atomic distribution (Fig. 3.19(a)) after the condensate hadballistically expanded by a factor of ∼ 3.5 [9] and the empty core had expanded [70] toa ﬁt radius of ∼ 9µm.
Displacements dt and angular positions θt of the cores for the in-trap images wereextracted as described before. The images of the expanded clouds were ﬁt with identical
distributions, and the Thomas-Fermi radius Re of the expanded cloud and the vortex
core displacement de and angle θe were obtained for each image. For each pair of images,
we determined the angular diﬀerence ∆θet ≡ θe − θt between the cores in the expanded
and in-trap images. We also determined the core displacement ratio de /dt , an indicatorof the radial motion of the core during the hold time th .
From the measurements of ∆θet at diﬀerent hold times th (Fig. 3.19(b)), we ﬁnd
a bare core precession frequency of 1.8(1) Hz, slightly faster than the precession of ﬁlled
cores and consistent with the trend shown in Fig. 3.18(b) for ﬁlled cores. To emphasizethat our measurements of ﬁlled and empty cores are diﬀerent limits in a continuum ofﬁlling material, we indicate the measured bare core precession frequency in Fig. 3.18(b)200 µm(a)∆θet (degrees)

81

0

(b)

-360
-720

(c)

1.5
1.0
0.5
0.0

0.0

0.5
th (s)

1.0

core visibility(de /Re ) / (dt /Rt )

0.0

0.5
th (s)

1.0

1.0
(d)

0.5
0.0
0.0

0.5

1.0

th (s)

Figure 3.19: (a) Ballistic expansion image of a vortex after all |2 atoms have been
removed. The dark spot is the bare vortex core. Inset – the corresponding, preceding
in-trap nondestructive image of the partially-ﬁlled core. (b) Angular diﬀerences ∆θetbetween vortex cores from the in-trap and expansion images, plotted against hold time th
in the magnetic trap. The line is a ﬁt through the data, indicating a bare core precession
frequency of 1.8(1) Hz. (c) Radial core motion is determined by (de /Re )/(dt /Rt ), theratio of the fractional core displacements from the expansion and in-trap images of eachdata set. The data are shown as open circles, with the average of all data at each given
hold time plotted as a ﬁlled triangle. (d) Core visibility of an expanded vortex, deﬁned
as the conditional probability for observing a vortex in an expanded image given theobservation of a vortex in the corresponding, pre-expansion in-trap image. Visibility
drops dramatically for hold times th > 1 s.

82with a point at r = ξ.
From Fig. 3.18(b) it is apparent that the structure and content of the vortex core
have a relatively modest eﬀect on precession frequency. One can calculate, for instance,
the ﬂuid rotation rate νr at the inner core radius. The value of νr is given by thequantized azimuthal superﬂuid velocity evaluated at the radius of the core, divided by
the circumferential length at that radius. For bare vortices, νr is about 260 Hz, while
for the largest ﬁlled cores of Fig. 3.18(b) (for which nearly half of the sample mass iscomposed of core ﬁlling), νr is only about 1.7 Hz. Thus between vortices whose innerradius ﬂuid rotation rates vary by a factor of 150, we see only a factor of 2 diﬀerence inprecession frequency.
The slower precession of ﬁlled cores can be understood in terms of our buoyancy
picture. Due to its slightly smaller scattering length, |2 ﬂuid has negative buoyancy
with respect to |1 ﬂuid, and consequently tends to sink inward towards the center ofthe condensate [10]. With increasing amounts of |2 material in the core, the inward
force on the core begins to counteract the outward buoyancy of the vortex velocity ﬁeld,
resulting in a reduced precession velocity. It is predicted that with a ﬁlling material of
suﬃciently negative buoyancy in the core, the core precession may stop or even precess
in a direction opposite to the direction of the ﬂuid ﬂow [123], but our data do not reachthis regime.
Various theoretical techniques involving two- and three-dimensional numericaland analytical analyses have been explored to calculate the precession frequency of
a vortex core within a condensate. We brieﬂy compare those most readily applied
to our physical parameters, assuming a spherical, single-component condensate with
3 × 105 atoms in a non-rotating trap. Where relevant, we assume a core displacementof dt = 0.35Rt . A two-dimensional hydrodynamic image vortex analysis [11] has been
analytically explored in the homogeneous gas [82] and two-dimensional harmonic conﬁnement [116] limits. The latter of these predicts a bare core precession frequency

83of ∼ 0.8 Hz. Svidzinsky and Fetter’s three-dimensional [152] solution to the GrossPitaevskii equation predicts a precession frequency of ∼ 1.3 Hz. Jackson et. al. [106]have obtained results in close agreement with this analytical solution using a numericalsolution to the Gross-Pitaevskii equation. Finally, a path-integrals technique by Tempere and Devreese [153] predicts a 1.24-Hz precession, and a two-dimensional simulation
by McGee and Holland [123] using a steepest-descents technique predicts a precession
frequency of 1.2 Hz.
Measurements of de /dt for diﬀerent hold times th show the radial motion of
the bare cores and is a probe of energy dissipation of the vortex states. The plotof Fig. 3.19(c) displays no trend of the core towards the condensate surface during th ,indicating that thermal damping is negligible on the 1 s timescale [12]. However, we
notice a sharply decreasing visibility of expanded bare vortices for hold times greaterthan th ≥ 1 s, as indicated in Fig. 3.19(d). The absence of radial core motion suggests that decreased visibility is due to imaging limitations rather than true decay. One
hypothesis is that the vortex core may be tilting away from the imaging axis [151], suppressing contrast in optical depth below our signal-to-noise threshold. Such a situation
may arise if the trap is not perfectly spherical, and if the vortex is not aligned along a
principle axis of the trap. Evidence in support of this hypothesis [13] will be presented
in Section 3.6.
Through a combination of destructive and nondestructive imaging techniques wehave obtained measurements of vortex dynamics in bare- and ﬁlled-core vortices indilute-gas BEC. Vortex precession frequencies show only modest dependence on theradius and content of the vortex core. Further measurements of vortex dynamics incondensates may reveal the rate of loss of angular momentum at ﬁnite temperatures,
an indication of energy dissipation. Such measurements may suggest interpretations for“persistence of current” in condensates, further strengthening ties between BEC andsuperﬂuidity. In order to pursue these goals, we plan to extend the studies reported

84
here to investigate higher-order dynamical behavior and to characterize the dissipativeeﬀects of ﬁnite temperatures.

3.6

Vortex tilting [98]3.6.1

Abstract

A vortex in a condensate in a nonspherical trapping potential will in generalexperience a torque. The torque will induce tilting of the direction of the vortex axis.
We observe this behavior experimentally and show that by applying small distortions
to the trapping potential, we can control the tilting behaviour. By suppressing vortex
tilt, we have been able to hold the vortex axis along the line of sight for up to 15
seconds. Alternatively, we can induce a 180◦ tilt, eﬀectively reversing the charge on thevortex as observed in the lab frame. We characterize the vortex non-destructively witha surface-wave spectroscopic technique.

3.6.2

Introduction

The decay of “persistent” supercurrents, be they in superﬂuids or superconductors, is intimately connected to the dynamical behavior of vortices. A magneticallytrapped, gas-phase Bose-Einstein condensate (BEC) provides a useful laboratory forcharacterizing the microscopic behavior of individual vortices subject to various controlled perturbations [122, 118, 51, 61]. In one recent experiment, a vortex core in anear-spherical condensate “vanished” [51] from view apparently without moving out tothe edge of the sample and annihilating there. Another group [118] found that the empirical critical rotation velocity for the formation of a vortex in an elongated condensate
is much higher than can be accounted for by a simple model of the vortex as a rigid
line-defect without any dynamics along its length. Feder et al. [80] explain the latter
observation by showing that the higher rotation rate is necessary to suppress the growth85
of anomalous normal modes (“bending” modes) of the vortex. In this section we study
the lowest odd-order normal mode, which in our near-spherical geometry corresponds
not to a bend but to a tilting of the vortex orientation. We show that the “vanishing” vortex of ref. [51] was in fact due to tilting of the vortex away from the line ofsight. Such uncontrolled tilting was caused by residual asphericity in the condensate’s
conﬁning potential. By tailoring the asphericity, we have learned to control the tilting
dynamics.
The tilting behaviour of a vortex in a condensate, conﬁned in a slightly aspheric,parabolic potential, is discussed in detail by Svidzinsky and Fetter [151], starting fromthe Gross-Pitaevskii equation. Only a brief summary of their theoretical results isgiven here. The vortex direction satisﬁes a set of equations similar in form to theEuler equations for rigid body rotation, familiar from classical mechanics. As a result,
the tilting dynamics of a quantized vortex in a conﬁned BEC are reminiscent of the
behaviour of a freely spinning rigid body as seen in the body-ﬁxed frame [84]. In thecase of the spinning rigid body, there are two stable axes about which precession of
the spin direction will occur, namely the axes with the largest and smallest moments ofinertia. The intermediate axis is unstable and no precession occurs about that direction.
If the spin direction lies initially near the intermediate axis, it will evolve away fromits initial orientation. Similarly, for a quantized vortex in a BEC, tilt precession of
the vortex is predicted to occur about two stable axes, given in this case by the tightand weak directions of conﬁnement. The unstable axis corresponds to the direction ofintermediate trapping strength.
The tilting dynamics of a vortex are constrained by two integrals of motion, one
in particular corresponding to conservation of energy. Physically, this implies that, as
the vortex tilts, its direction follows an angular trajectory which is a contour of constantvortex length, or equivalently constant energy. As is the case for all the normal modes ofa vortex, the precession frequency for a vortex tilting about a stable axis is predicted to86
scale with the rate of vortex ﬂuid circulation evaluated at the condensate edge. For ourtypical conditions, this ﬂuid rotation is near 0.3 Hz. The theoretical frequency for tilt
precession depends additionally on the magnitude and character of the trap asymmetries
and the initial orientation of the vortex.

3.6.3

Vortex formation

In our experiment, we ﬁrst use a wavefunction engineering technique to makesingly quantized vortices in a two-component BEC [122, 159]. The two components,
which are two diﬀerent hyperﬁne levels of 87 Rb, are magnetically conﬁned together ina nominally spherical, harmonic TOP trap [78], parameterized by a trapping frequency
ωtrap /2π = 7.8(1) Hz. The vortex formation process leaves one component in the circulating state while the other component, which is non-rotating, ﬁlls out the core [14].
The vortex is formed initially aligned along the line of sight. We take a non-destructive
picture of the two-component vortex to record the initial displacement of the core withrespect to the center of the condensate cloud. We then create a bare vortex by selectively removing the ﬂuid ﬁlling the core [51], which shrinks down to a size below ourimaging resolution.
This section deals exclusively with the dynamics of bare vortices in a singlecomponent BEC. The bare vortex state is formed with about 2 · 105 atoms at a temperature of T /Tc = 0.8(1) where the critical temperature Tc is 20(6) nK. The number ofatoms in the condensate is determined (in the Thomas-Fermi limit) from the condensate radius R, equal to 20.8(4) µm on average. Following the creation process and core
removal, the condensate is held for a variable holding time in its conﬁning potential,
and then probed for the presence of the vortex.

87
3.6.4

Disappearing vortices

We have previously reported that the visibility of vortex cores was lost after aholding time of about 1 s in a nominally spherical trap [51]. In those experiments,
the conﬁning potential was suddenly removed, and the condensate allowed to expand
ballistically before imaging. The presence of a vortex was detected in the expansion
image as a dimple in the condensate’s density distribution. The topological nature of aquantized vortex means that the only way for a BEC to rid itself of a vortex is for thecore to make its way to the edge of the cloud and annihilate there; the vortex cannot
gradually spin down. Our vortices, however, seemed to disappear without any visible
radial motion of the cores outward. We did notice that the contrast of the cores as
observed in expansion decreased at longer holding times before being lost altogether.
These results could be explained by a tilting of the vortex away from the line of sight.
Because the vortex core is such a narrow feature, only small deviations from the lineof sight (∼ 20◦ ) are necessary for contrast to be lost below the noise threshold in the
expansion images.
In order to circumvent this imaging limitation, we have implemented an alternative method of vortex detection that is more robust against tilting of the vortex fromthe line of sight, and that has the additional advantage of being sensitive to the handedness of the vortex circulation. This detection technique uses the idea that the collective
excitation frequencies of a trapped condensate are sensitive to the presence of a vortex[15, 150]. We make use of quadrupolar surface-wave excitations, that is, surface-wave
excitations which carry angular momentum l = 2. (The angular momentum of these
perturbative excitations should not be confused with the angular momentum per particle of the bulk of the condensate, which in the presence of a well-centered vortex
approaches l = 1). The projection of angular momentum of an l = 2 surface-wave
excitation onto the axis of the vortex direction can have components m = 0, ±1, ±2.

88
In the absence of a vortex, the counter-propagating ±m modes are degenerate, but inthe presence of a vortex the handedness of the ﬂuid ﬂow breaks time-reversal symmetry
and lifts the degeneracy. The use of surface-wave spectroscopy for vortex detection was
ﬁrst demonstrated experimentally by Chevy et al. [61].
To detect a vortex, we excite an l = 2, mz = 0 excitation about the vertical ẑ
axis by modulating the trapping potential for a single cycle. Along the line of sight,
the x̂ axis, the mz = 0 mode projects onto a superposition of mx = +2, mx = −2, andmx = 0 modes. The mx = +2 and mx = −2 superposition may be thought of as astanding-wave of clockwise and counterclockwise surface waves. Along the line of sight
we observe the cloud alternately stretch along two principal axes, ﬁrst vertical, thenhorizontal. The excitation period is measured to be 11.0(2) Hz in agreement with the√predicted value 2ωtrap [148]. The presence of a vortex along the line of sight induces
a splitting of the clockwise and counterclockwise wave velocities, so that the nodes ofthe standing-wave are not completely ﬁxed. As a result we observe a precession of theprincipal axes of the quadrupolar excitation. We record the precession in a sequenceof seven non-destructive images, strobed at half the quadrupolar excitation period. A
typical data set is shown in Fig. 3.20. Opposite vortex circulation clearly leads toopposite precession of the principal axes, as is evident in Figs. 3.20(b) and (c).
For a vortex whose direction is tilted with respect to the line of sight, we still
expect precession of the quadrupolar principal axes to be induced about the vortex
direction. The main limitation to vortex detection in this case is that there must be
suﬃcient projection of the quadrupolar precession onto the imaging camera. In theextreme case where the vortex is oriented at 90◦ to the line of sight, any quadrupolar
precession will also be about an axis perpendicular to the line of sight, in other words,
not readily observable on the imaging camera.
Armed with our new detection method, we revisited the topic of longevity ofvortices in our nominally spherical trap. The precession of surface waves clearly revealed

89

0 ms

45.5 ms

(b)

50

(c)

∆θ (deg.)

(a)

(d)

25
0

-25

91 ms

-50

0

100

200

time (ms)136.5 ms

182 ms

(a)
(b)
(c)

227.5 ms

Figure 3.20: Using surface excitations for in situ detection of a vortex in a conﬁned
BEC. (a), (b) and (c) are each a series of non-destructive images of the quadrupolarmode, after excitation. The pictures are strobed at 45.5 ms, half the excitation period.
(a) is the case of a vortex free condensate; (b) and (c) show the excitations in thepresence of a vortex whose core is normal to the plane of the page. The vortices in (b)and (c) have opposite handedness. The principal axes of the ellipse-shaped quadrupolar
mode precess in the direction of the ﬂuid ﬂow. The images are each ﬁt to an elliptical
distribution with orientation ∆θ of the principal axes. The orientation, expressed as anangular deviation from the vertical and horizontal axes, is plotted versus time in (d) for
each of the cases (a), (b) and (c). A linear ﬁt has been applied to the data to determine
a precession frequency of the principal axes, -0.49(4) Hz for (b) and 0.45(5) Hz for (c).

90
the presence of vortices at 1.5 s and at 2 s holding time, well after the vortices ceased to
be visible in the expansion images [51]. Given that the surface-wave probe for vortices
should be relatively robust to small tilts of the vortices with respect to the line of sight,
we came to the tentative conclusion that our vortices were in fact tilting away from theline of sight during the ﬁrst few seconds after their creation. This behavior could be
accounted for by small (≤3%) residual asymmetries in our trapping potential.

3.6.5

Vortex tilt: three experiments

Further study of the tilting required better characterization of the trap potential,
and we achieved this by deliberately introducing tailored deformations to the conﬁning
potential, deformations that were certain to overwhelm the residual asymmetry. The
results of three such experiments are presented here: ﬁrst, the suppression of vortex tilt;
second, the generation and observation of vortex tilt about a stable axis; and, third, themanipulation of a vortex’s orientation through controlled deformations of the BEC’s
conﬁning potential.
In the ﬁrst experiment, the spherical conﬁning potential is squeezed by 10% along
the line of sight following the creation of a vortex in the BEC [16, 17]. This places thevortex along the tight, stable axis of the trap. As a result, tilting of the vortex awayfrom the line of sight should be suppressed. After a variable holding time in the squeezed
trap, the conﬁning potential is returned to spherical symmetry and the condensate isthen quickly probed for the presence of the vortex. Using both the surface-wave andexpansion techniques, we have detected vortices lasting for holding times up to 15 s.
This is a signiﬁcant improvement over our previously published limit of 1 s. Althoughvisibility has been restored at long times by holding the condensates in a squeezed trap,
it is important to note that only well-centered vortices have been found to survive tothe longest times. This is consistent with a model of initially oﬀset cores spiraling outof the condensate under the inﬂuence of thermal damping [18].

91
The second experiment proceeds in the same way as the ﬁrst except the spherical
trap is now squeezed along a horizontal axis at 45◦ to the line of sight. The squeezing
process produces a triaxial conﬁning potential, where the tight and weak directionsof conﬁnement lie in the horizontal plane while the intermediate axis is vertical. The
measured trap asymmetries together with the initial vortex direction indicate that thevortex should precess about the weak axis, although slightly diﬀerent conditions could
lead to rather diﬀerent behavior (see Fig. 3.21). In any case, as the vortex tilts away fromthe line of sight to a maximum excursion of 90◦ from the imaging axis, the visibilityof the vortex will gradually disappear. Eventually, if the vortex precesses about thestable axis and back into the line of sight, it should become visible again in a ‘revival ofvisibility’. This eﬀect is seen in Fig. 3.22, where vortex visibility is plotted as a functionof holding time in the squeezed trap. Visibility is deﬁned as the probability of detecting
a vortex at a given time and is obtained from a set of approximately ten shots for eachpoint plotted. Vortices whose initial oﬀset from trap center exceeds a prescribed valueof ∼0.15 R are omitted from the analysis. This is done to reduce blurring of the revival
due to possible position-dependent tilt rates, and, moreover, to prevent loss of contrast
at long times due to the drift of cores out of the condensate. The vortex detection isaccomplished with both surface-wave and expansion techniques. The revival measured
with expansion imaging is much more narrowly resolved in time, as is to be expectedfrom this method’s greater sensitivity to alignment of a vortex with the line of sight.
The frequency of the tilt mode, determined from the data in Fig. 3.22, is 0.25(2)Hz. The theory of Svidzinsky and Fetter [151] predicts the lowest-order odd-parity vortex mode should have a frequency below 0.3 Hz for our values of the Thomas-Fermi radius, conﬁnement asymmetry, and initial vortex angle. Uncertainties in the conﬁnement
asymmetry preclude a more precise determination of the theoretical tilting frequency.
The uncertainties are such that the vortex could lie on an initial trajectory arbitrarilyclose to a ‘saddle’ line, where the revival time should diverge (see Fig. 3.21). The high92Intermediate axis
(c)
(a)

(b)

Tight
axis

Weak
axis
45o

Line of Sight
Figure 3.21: Possible tilting behaviour of a vortex in a conﬁned BEC is illustrated forthe conditions of the second tilting experiment (see text). The axes of symmetry forthe triaxial conﬁning potential are indicated. Initially, the vortex is nearly anti-alignedwith the line of sight. Depending on the precise initial orientation of the vortex and therelative values of the conﬁnement strengths, the vortex direction could (a) precess about
the weak axis, (b) precess about the tight axis, or (c) drift towards the intermediateaxis along a ‘saddle’ line, as traced out on the surface of the condensate.

93

Visibility

1.0

0.5

0.0
0

2

4

6

8

holding time (s)Figure 3.22: Visibility of a vortex in a BEC is plotted versus the time the BEC is held ina trap squeezed along an axis at 45◦ to the line of sight. Visibility is the probability ofdetecting a vortex over several shots. Filled squares indicate vortex detection with thesurface-wave technique; open triangles indicate detection with the expansion technique.

94
contrast of the two revivals in Fig. 3.22 clearly indicates, however, that the shot-to-shot
ﬂuctuations of the conﬁnement asymmetry and of the initial position of the vortex aresmall.
The third and ﬁnal tilting experiment demonstrates the reversal of vortex handedness. The vortex, as usual, is formed along the line of sight. The direction of thevortex is then ﬂipped 180◦ by deforming the trapping potential in two steps, as illustrated in Fig. 3.23. From the perspective of the line of sight, the ‘lab frame,’ the vortex
has eﬀectively ﬂipped its handedness. Surface-wave detection is used to verify this, withpictures similar to Figs. 3.20(b) and (c) being obtained before and after the ﬂipping process respectively. Manipulation of vortex direction may prove to be a useful techniquein a TOP trap. For example, maneuvering a vortex to an arbitrary direction followed
by a second stage of wavefunction engineering could produce a condensate wavefunction
with more complicated, three-dimensional topological structure.
In conclusion, we have described a non-destructive method of vortex detection
and used it to characterize the tilt modes of a bare vortex in a trapped BEC. Control
over the tilt of a vortex has been demonstrated, including the suppression of tilting
altogether. The suppression of tilting, which maintains the visibility of vortices out tolong times, now permits the study of the lifetime of vortices at ﬁnite temperature.

3.7

A position sensitive spectroscopy
The vortex dynamics described to this point have been essentially at zero tem-

perature and well described by the Gross-Pitaevskii equation. The ﬁnite temperature
behaviour which has been alluded to in Sections 3.5 and 3.6 is the lifetime or “persistence” of a bare vortex in the condensate. As a topological feature, the vortex cannot“spin down.” To leave the condensate, it must spiral radially outward under the inﬂuence of the friction imparted by the normal cloud bathing the condensate (see also
Section 4.6). In order to assess this motion one would like to follow the radial position

95

Step 1

Step 2
Weakaxis 1
y

45o

Tight
axis 1

Line of Sight

x

45o

Tight
axis 2

Weakaxis 2

Figure 3.23: A sequence of trap deformations to reverse the direction of a vortex isshown. The ﬁgure shows a horizontal crosssection of the BEC as seen from above.
The vortex is initially formed along the line of sight and aligned with the -x̂ axis. In
step 1, the trap is squeezed in the horizontal plane along tight axis 1. The vortex isthen allowed to precess for 1.5 s, at which point it should be aligned with the ŷ axis.
The new direction of the vortex is indicated by a dotted arrow. In step 2, the trap issqueezed along tight axis 2 and the vortex allowed to precess another 1.5 s until it isaligned with the x̂ axis. To an observer looking down the line of sight, the vortex has
eﬀectively reversed its handedness. Changing the time allowed for vortex precession by
0.5 s inhibited the eﬀect. For the correct timing, the ﬂipping process was found to be
successful 80% of the time.

96of the vortex as it spirals outward to the condensate edge. However, a complicationarises if wavefunction engineering is used to create the vortex. As was mentioned in
Section 3.3.3, condensate slosh causes shot-to-shot variations in the initial core position.
This is an important consideration in comparing with theory since the lifetime of a vortex is predicted to depend logarithmically on its initial distance from the condensate
center (with a centered vortex being metastable) [82]. As a result, a nondestructive measurement scheme is required to determine both the initial and ﬁnal radial core positions
for each experimental run. Such a task is technically challenging for bare vortices since
the core size is below imaging resolution in-trap, where nondestructive phase-contrast
imaging can be used. The ﬁrst option available, as outlined in Section 3.5, is to measure the initial core position of a two-component vortex, remove the component thatﬁlls out the core to create a bare vortex, wait, then analyze the ﬁnal core position in
expansion. Besides being a relatively convoluted measurement, this technique suﬀers
from limitations to vortex visibility in the expansion images, where the “haze” of a sizable normal cloud surrounding the condensate obscures the vortex cores. Interestingly,
the surface-wave spectroscopy presented in the previous section provides an alternative
nondestructive technique to assess bare-vortex position in the condensate. Although
vortex-lifetime results obtained in this way ultimately proved inconclusive and are notdiscusssed here, the technique, which is interesting in its own right, is outlined here.
As alluded to previously, surface-wave spectroscopy is sensitive to the angularmomentum of a vortex in the condensate. For the quadrupolar (l = 2) surface waves
used in the experiment, the vortex ﬂow breaks the symmetry between co- and counterpropagating surface modes (m = +2 and m = −2), which acquire a frequency splitting[15]. A sum-rule argument [161] shows that the splitting depends on the angular momentum per particle z  in the condensate
ω+ − ω− =2 z 2 .
m r⊥

(3.20)

97
2 , the square of the co-ordinate perpendicular to the vortex
The expectation value of r⊥quantization axis, depends on the shape of the potential conﬁning the condensate. For
2  scales with the square of
the nominally spherical condensates in the experiment, r⊥

the size of the condensate given by the Thomas-Fermi radius R. To connect with theresults of Section 3.6, the standing-wave pattern formed by a superposition of m = +2
and m = −2 waves [Figure 3.20] precesses due to the splitting at a rateωp = (ω+ − ω− )/4.

(3.21)

While a singly-charged vortex at the center of a condensate contributes an angularmomentum per particle of , an oﬀset vortex will contribute progressively less as it moves
towards the edge of the condensate. (The core of course still has a 2π winding aroundit.) A calculation for a harmonic conﬁning potential [19] gives the angular momentum
per particle in terms of lateral vortex displacement r asr2
z  =  1 − 2
R

5/2
.

(3.22)

Thus, in so far as surface-wave spectroscopy is sensitive to the angular momentum perparticle in the condensate, it provides an indirect assay of vortex position. Recent
theory [83, 96] indicates that the sum-rule result for the frequency splitting [Eqn. 3.20]
may be inaccurate for a single oﬀset vortex; nevertheless, the qualitative behaviour iscertainly correct and provides an intuitive understanding of the eﬀect of vortex position
on surface-wave spectroscopy.
To verify this experimentally, multiple runs have been performed to create avortex in the standard way. To summarize brieﬂy, a two-component vortex with varying
radial position is engineered and the core ﬂuid is then removed to create a bare vortex.
An image of the two-component vortex is taken before the removal and later ﬁt to obtainthe vortex position. A standing wave of m = +2 and m = −2 waves is excited on thecondensate containing the bare vortex. The precession rate [Eqn. 3.21] of the standing

98
wave, determined from a series of non-destructive images [Figure 3.20], is plotted asa function of radial core displacement in Figure 3.24. Although the basic trend of
the data is as expected, the precession rate decreases faster with displacement thanthe predicted dependence according to Eqns. 3.20 and 3.22 as well as the prediction
according to ref. [83]. This discrepancy is likely due to systematic errors in the core
position obtained from ﬁts to the two-component images. Simulated images of twocomponent vortices indicate that the ﬁts are untrustworthy when the distance from thecloud center to core position is more than a few tens of percent of R.
As a ﬁnal note, the statement that the surface-wave technique is “nondestructive”
does require justiﬁcation as the presence of the surface-wave excitation has the potential
to perturb the vortex. At least in terms of eﬀects to the radial core position, thenondestructiveness can be readily assessed by monitoring any changes in the surfacewave precession frequency over time. As demonstrated qualitatively in Figure 3.25, thesurface-wave excitation apparently has no eﬀect on the radial position of the vortex forthe range of excitation amplitude used (20%-40%).

3.8

Soliton engineering and decay to vortex rings [50]3.8.1

Abstract

We have created spatial dark solitons in two-component Bose-Einstein condensates in which the soliton exists in one of the condensate components and the soliton
nodal plane is ﬁlled with the second component. The ﬁlled solitons are stable for hundreds of milliseconds. The ﬁlling can be selectively removed, making the soliton more
susceptible to dynamical instabilities. For a condensate in a spherically symmetric potential, these instabilities cause the dark soliton to decay into stable vortex rings. We
have imaged the resulting vortex rings.

99

Precession frequency [Hz]

0.6
r

R
0.4

0.2

0.0

0.0

0.2

0.4

0.6

0.8

1.0

r/R
Figure 3.24: Quadrupolar precession frequency vs. displacement r of a bare vortex
from the center of a condensate (see inset). The displacement is scaled by the ThomasFermi radius (R ≈ 22µm) of the nominally spherical condensate. Each data point
represents a single experimental run in which a singly-quantized vortex is created in acondensate, followed by quadrupolar surface-wave spectroscopy as per Figure 3.20. The
plot combines three data sets with diﬀerent amplitudes for the quadrupolar excitation,
characterized by an ellipticity of {0.2, 0.26, 0.4}. The solid line is the predictedbehaviour based on Eqns 3.20 & 3.22 while the dotted line is the prediction according
to ref. [83].

Data
(a)

100

Fits
(b)

0ms

182ms
100

Θ
Θ [deg.]

364ms
50

0

(c)
-50

0.0

0.5

1.0

time [s]Figure 3.25: Long-time precession of a quadrupolar standing-wave on the surface of acondensate containing an  = 1 vortex. (a) Sequence of nondestructive in-situ images,
which begins in the ﬁrst frame by showing a two-component vortex engineered into thecondensate. The ﬂuid ﬁlling the core is then removed (not shown) and a quadrupolar
surface wave excited. The second frame and following show the time evolution of thesurface wave where the indicated times are referenced to the time of surface-wave excitation. The images are spaced by two oscillation periods of the quadrupolar surface mode(2τ = 182ms) such that the major axis of the condensate deformation would remain
vertical in the absence of a vortex. The decrease in ellipticity of the cloud is primarily
due to thermal damping of the quadrupolar mode. (b) A two-dimensional ellipse is ﬁt
to each image. (c) The angular orientation of the ellipse’s major axis is extracted andplotted as a function of time. A linear ﬁt has been added to guide the eye.

101
3.8.2

Introduction

Topological structures such as vortices and vortex rings have fascinated scientists
and mathematicians for centuries. In a quantum ﬂuid, vortices have quantized ﬂow
around a one-dimensional core where the density vanishes. Vortex lines, which terminate at the boundaries of the quantum ﬂuid, have been observed in superﬂuid helium andsuperconductors [74, 155], and recently in Bose-Einstein condensates (BECs) [121, 118].
Similarly, vortex rings are vortices whose cores are closed loops; the poloidal quantized
ﬂow pattern resembles that of a smoke ring. Quantized vortex rings have been produced in superﬂuid helium, and were ﬁrst detected in pioneering work by Rayﬁeld andReif [139]. In this section we report direct experimental observations of vortex rings in
BECs.
Solitons are localized disturbances in a continuous nonlinear medium that preserve
their spatial proﬁle due to a balance between the eﬀects of dispersion and nonlinearity [20]. Dark solitons have been previously created in single-component BECs using
phase-imprinting methods [21]. These structures are characterized by a local decrease inﬂuid density (the depth). The macroscopic quantum phase of the BEC diﬀers on eitherside of the soliton; for a completely dark (black) soliton, the depth is 100% (a complete
absence of ﬂuid), the phase oﬀset is π, and the soliton velocity is zero. We have createdblack solitons (having a ﬂuid-free nodal plane) in nearly spherically symmetric 87 Rb
BECs, and have observed the subsequent decay of the solitons into vortex rings.
Unlike vortices and vortex rings, whose stability is ensured by Kelvin’s theorem [74], soliton stability depends on the nonlinearity and geometry of the medium [22].
Optical dark solitons in self-defocusing nonlinear media have been observed to decay into optical vortices via a ‘snake instability’ [23], conﬁrming predictions [110]. In
general, dark solitons in BECs are also expected to be inherently dynamically unstable [111, 24, 60, 81]. Because both the velocity and depth of a soliton are determined by

102
its phase oﬀset [140], small local perturbations of either the strength of the nonlinearity
or of the depth give the soliton a corresponding nonuniform transverse velocity proﬁle. Once the soliton begins to decay, it quickly breaks up into more stable structures.
Within our spherically symmetric BECs, the expected decay products are concentricvortex rings [81]. Because the theoretical background for soliton decay in condensateshas been previously described in detail in Ref. [81], we limit our paper to primarily aphenomenological presentation of data. Evidence for soliton decay is obtained throughdirect images of the BEC density distribution. We provide a visual comparison of datawith numerical simulations of soliton decay.

3.8.3

Experimental techniques

We create dark solitons using concepts that we have previously applied to making
singly quantized vortices in BECs [121, 159]. The magnetically trapped BEC can exist
as a superposition of two internal components, hyperﬁne states |1 ≡ |F = 1, mF = −1
and |2 ≡ |F = 2, mF = 1. Conversion between the two components is achieved witha microwave ﬁeld slightly detuned from the |1 to |2 internal conversion energy. A“modulation beam,” an oﬀ-resonant laser beam whose small spatial focus provides arequisite spatial selectivity, is dithered rapidly across the BEC. The laser beam inducesa small AC Stark shift in the |1 to |2 transition energy. The phase and amplitude ofthis eﬀective frequency modulation varies from point-to-point in the sample in a spatial
pattern determined by the time-varying position of the optical modulation beam; thus
the microwave-induced inter-component conversion varies in phase and amplitude acrossthe BEC.
In our experiment, we start with a uniform-phase (i.e., ground state) condensate
of component |2. To make a soliton, we choose a modulation beam pattern such thatinter-component conversion is suppressed in the middle of the sample, whereas theupper and lower parts of the BEC undergo conversion to component |1 with an initial

103
phase oﬀset of π between the two parts [Figure 3.26]. Uncontrolled variations in thetiming of the modulation cause the angular orientation of the soliton nodal plane tovary randomly from one soliton to the next. For further details, see our earlier vortex
work [121] and theory by Williams and Holland [159]. The soliton state contains about
3 × 105 |1 atoms, and the ﬁlling consists of about 7 × 105 |2 atoms. The axial (vertical)
trap frequency is 7.6(3) Hz, the radial (horizontal) frequencies are 7.8(3) Hz, and thecondensates have a Thomas-Fermi (TF) radius of 28 µm. The solitons are created attemperatures of T = 23(6) nK, or T /Tc = 0.8(1), where Tc is the BEC phase transition
temperature.
After removing the coupling ﬁeld, we observe the trapped excited-state condensates using nondestructive phase-contrast imaging. Our probe laser is tuned such that
only component |1 is visible, and these atoms appear bright on a dark background. As
described in [121], a ﬁlled-core vortex appears as a dark hole in a bright atom cloud.
Similarly, a soliton with a ﬁlled nodal plane appears as a dark band that divides the|1 BEC. This two-component BEC can be described as a bright soliton of component|2 trapped within the |1 dark soliton [25]. The ﬁlling material of |2 atoms stabilizes
the dark soliton against dynamical instabilities; the ﬁlled solitons are observed to last
for at least 800 ms.
To study the dynamical instabilities of the solitons, we ﬁrst use resonant lightpressure to selectively remove the |2 atoms that ﬁll the nodal plane. The |2 atoms are
adiabatically removed over 100 ms, and a bare dark soliton of component |1 remains.
The soliton node and any soliton decay products are then too small to be observed
while the BEC is held in the trap [26]. We thus remove the trapping potential andallow the BEC to ballistically expand, causing the variations, or ‘defects,’ in the density
distribution to also expand [70]. These density defects are then resolvable. We obtaina ﬁnal near-resonance phase-contrast image of the expanded atom cloud, using a probe
detuning of 20 MHz. Expansion imaging has previously been used to detect bare vortex

laser intensity104ωm =100Hz

δ
−δ

BEC

|2〉

~6.8 GHz|1〉

laser intensity

(a)

(b)

ωm

−δ

|2〉

BEC

|1〉

(c)

(d)

Figure 3.26: Experimental setup for soliton engineering is shown in (a,b). A linear
gradient in light intensity (a) from an oﬀset laser beam is rotated about the condensate
center at a frequency ωm . A pair of microwave coupling drives (b) is applied between
internal |1 and |2 with detunings of δ = ±ωm . Using the results of Section 3.3, theexperimental setup can be shown to be equivalent to the simpler wavefuction engineering
technique as described in (c,d). The light intensity undergoes a “see-saw” modulation(c) at frequency ωm and only one coupling drive (d) is used. The node of the soliton isformed along the pivot of the seesaw, where no intensity modulation occurs and, as aresult, no population transfer occurs between states |1 and |2 .

105
cores [118, 51].

3.8.4

Numerical calculations

Before discussing experimental data, we brieﬂy describe results of numericalcalculations.

The techniques employed have been described in detail in Ref. [81].

The equilibrium conﬁguration, dynamics, and Bogoliubov excitation spectrum for a
two-component condensate at low temperatures may all be obtained from the appropriate three-dimensional time-dependent Gross-Pitaevskii (GP) equation [137], with
N1 = 3 × 105 |1 atoms in a state initially constrained to have odd parity along one
axis, and 0 ≤ N2 ≤ 7 × 105 |2 atoms in an even-parity state. The spherical trap frequency is chosen to be ν0 = 7.8 Hz, and the intra- and inter-species scattering lengths
are (a11 , a22 , a12 ) = 5.5 × (1.03 , 0.97 , 1.0) nm, respectively [99]. The behavior of the
BEC in full two-component simulations, in which we approximate a slow removal of the|2 atoms, is found to be qualitatively similar to simulations where we set N2 = 0 at theoutset. In all cases, the soliton is found to undergo a snake instability, decaying into
vortex rings.
Indeed, the Bogoliubov spectrum contains modes with complex frequencies for all
N2 considered; such modes have been shown to drive the soliton instability [81]. For
N2 ≥ 4 × 105 , only one such mode remains, with imaginary frequency of magnitude ν ∼(4 × 104 /N2 )ν0 . Assuming a soliton decay time ∝ ν −1 , a ﬁlled soliton with N2 = 7 × 105
is expected to be stable for longer than 2 s, consistent with experimental observations.
The results of a one-component (N2 = 0) simulation are shown in Fig. 3.27. The
soliton decays into three nearly concentric vortex rings approximately 130 ms after initialformation. In general, the rings tend to migrate towards the condensate surface, andmay then grow, shrink, and reconnect with each other. The innermost ring is the moststable, remaining intact for at least 150 ms in the absence of a collision with anotherring. Along an imaging axis parallel to the plane of a given vortex ring, the integrated

106
column density reveals two dips in the density distribution, connected by a fainter line.
The ring, however, is not constrained to lie in any given plane, and may bend and tilt,
producing other types of images. In elongated traps with aspect ratios of two or higher,
simulations indicate that a soliton can decay directly into nearly parallel vortex lines
rather than rings; however, this behavior is not obtained for spherical traps.

3.8.5

Experimental results

In our experimental cycle, we ﬁrst create a ﬁlled dark soliton and obtain a nondestructive image of the initial trapped (ﬁlled) soliton. Over the next 100 ms, we removethe |2 atoms from the soliton nodal plane until only the |1 atoms remain. The trapped
single-component condensate is then held for a variable hold time before being allowed
to ballistically expand for 56 ms [27]. Finally, the expanded density distribution isrecorded.
Images of the decayed solitons typically reveal two dips in the condensate density
distribution, as shown in Fig. 3.28(b)-(e). While our images are consistent with the
expected signatures of vortex rings, they are inconclusive evidence for the decay ofsolitons into vortex rings; from a single image containing only a pair of density dips,
we can not diﬀerentiate between a single vortex ring and a pair of vortex lines.
For comparison, an expanded vortex line is shown in Fig. 3.28(a). We believe that oursignal-to-noise levels may limit our ability to see a dark line connecting the two clear
density defects that would indicate the sides of a ring. We also note that high-frequency
excitations visible in T = 0 simulations would be quickly damped for T > 0, and thus
not seen in the experiment.
Because detection of two-dimensional topological structures, such as dark solitonsand vortex rings, can be enhanced by probing along two orthogonal directions, we
added an additional probe beam. In our improved apparatus, the second probe beamis orthogonal to the original beam, and propagates horizontally (as does the original).

107

220 µmFigure 3.27: Results of numerical simulations showing the decay of a black soliton in a
BEC. The simulation corresponds to 3× 105 87 Rb atoms in a spherically symmetric trapwith frequency 7.8 Hz. Successive frames are shown at 50 ms intervals, with the ﬁrst
frame at 100 ms after the start of the simulation. The ﬁrst row shows the density proﬁleof the condensate, integrated down an axis parallel to the soliton plane. The low-density
regions within the cloud are also rendered (second row), with views perpendicular tothe soliton plane.

(a)

(b)

(c)

(d)

(e)

Figure 3.28: Typical images of expanded condensates, and their initial states (insets)
before the |2 atoms were removed. (a) A vortex, (b)-(e) the decay products of solitons.
Image (e) was taken with a hold time of 500 ms; all other images were taken with ahold time of 0 ms, in addition to the 100 ms |2 removal and the 56 ms expansion [22].

(a)Top view

CCD
array

vacuumchamber

front
probe

side
(b)

(d)

front

(c)

side

front

108

(e)

side probe
Figure 3.29: (a) An illustration of our probe beam paths. (b)-(e) Simultaneous 100µm-square images of trapped condensates from the front and side imaging directions.
Shown are (b) a ground state condensate of component |1, (c) a shell of |1 atoms
around a central ball of |2 atoms [21,22], (d) a |1 vortex ﬁlled with |2 atoms, and (e)a horizontally oriented dark soliton of component |1, ﬁlled with |2 atoms.

The beams intersect at the condensate. Phase-contrast imaging is used independently
with each beam path, and the probe beams propagate towards diﬀerent sides of asingle charge-coupled device (CCD) camera array. The apparatus allows us to takesimultaneous pictures of condensates from the ‘front’ (original) and the ‘side’ (added)
directions using a single camera, as illustrated in Fig. 3.29(a). While we can observe
the full depth of the ﬁlled soliton node along the front direction, regardless of the initialorientation, the nodal plane is usually at an oblique angle to the side probe, and soliton
contrast is thus reduced. Examples of simultaneous images of trapped condensatesfrom the front and side are shown in Figs. 3.29(b)-(e). In the pair of images shown in
Fig. 3.29(d), the ﬁlled core of a vortex is visible as the dark hole in the front image,and as the dark band in the side image. In Fig. 3.29(e), a ﬁlled horizontal dark soliton
appears as a band across the condensate in each probing direction, demonstrating thatthe BEC is split into two sections.
The use of two orthogonal probe beams conﬁrms that dark solitons indeed decay
into vortex rings. With the two beam paths, we observed pairs of density dips in simultaneous side and front expanded images of decayed solitons. In each image set shown109
in Figs. 3.30(b)-(d), pairs of density dips lie nearly horizontally and at corresponding
locations within the two BEC images. We have also obtained images in which a faintline corresponding to a decrease in integrated column density can be observed between
two defects in one of the expanded images, as shown in Figs. 3.30(e) and (f), providing
further striking demonstrations of soliton decay into vortex rings. Figures 3.30(e)-(g)emphasize that when the vortex ring is not close to horizontal, clear pairs of density dips
are not seen in the side images. However, a weak ring-shaped halo may be discernedin the side image residuals of Figs. 3.30(e) and (g), when the ring is in a near-verticalplane orthogonal to the side probe direction. Although the halo signal is only slightly
above the noise of the residuals, it can be noticed when compared to the residuals foran expanded ground state condensate, shown in Fig. 3.30(a). Images in Figs. 3.30(b)through (f) represent the best 10% of our data showing solitons decaying into rings;
Fig. 3.30(g) is typical.
Numerical as well as experimental data show great variability in the types ofimages that can be expected after a soliton decays. For example, Fig. 3.30(h) is arare example showing two clear density dips in the side image, but here a clear line
corresponding to the defect extends through the entire condensate in the front image
and residuals. It is possible that this image shows soliton decay into two vortex lines
rather than a ring, or breakup of a ring into vortex lines. We defer to a future paper a
discussion of the diversity of our images, which may result from interesting vortex ring
dynamics, including vortex ring bending, tilting, and reconnections.
Small perturbations unavoidably induce a dark soliton to decay. In a sphericallysymmetric BEC, a soliton decays into a relatively stable vortex ring. In addition todemonstrating this nonlinear decay, our experiment shows that condensates can indeed
support vortex rings, which may be detected via expansion imaging. Future experiments
may include studies of vortex ring stability, lifetime, and dynamics [108], as well as
investigations into vortex structures created by impurity motion above a superﬂuid

110

Figure 3.30: Images of expanded condensates and their initial states (insets), as viewed
along both imaging axes. Each subﬁgure consists of two expansion images (top row ofeach subﬁgure) and two images of residuals plots (bottom row), obtained after subtracting a TF ﬁt of each expansion image. The white regions in the residuals correspondto depletion of ﬂuid in the BEC. The residuals are superimposed with white ellipses
outlining the ﬁt TF proﬁles of the condensates, for position reference. (a) An expanded
ground state condensate. (b)-(h) The decay products of solitons. The hold times (between |2-atom removal and BEC expansion) were 0 ms for images (a)-(c),(f), and (g);50 ms for (d) and (e); and 150 ms for (h). For all images, we reduced background noise(fringes) using a digital fourier-transform ﬁlter.

111
critical velocity [130]. Possibilities to create vortex rings by other means may also be
explored, such as by passing objects through the BEC [107].

3.9

Wavefunction engineering at DC
The wavefunction engineering technique with two-component condensates uses

a space- and time-dependent modulation of the resonance between the two internal
components. In this way, arbitrary control over the population transfer between thestates is realized [Section 3.3]. It is, however, possible to exert some degree of spatial
control over the population transfer through the application of static gradients in theresonance across the condensate. This method of wavefunction engineering “at dc,”
which is an extension of earlier work in this group on two-component condensates [121],
provides an extremely simple way to generate dark solitons in a condensate and may be
extended to generate stacks of several solitons.
The method takes advantage of preexisting features of the 87 Rb two-photon transition used in the experiment [Figure 3.2] as well the magnetic potential conﬁning the condensate. In particular, the two-photon transition between the simultaneously trappedstates |F = 1, mF = −1 (“|1”) and |F = 2, mF = 1 (“|2”) occurs via a magneticallyuntrapped state, |F = 2, mF = 0. As a result, there arises a gradient in the transition’s intermediate state detuning ∆int across the vertical extent of the condensate[Figure 3.31]. Essentially this gradient is due to the force of gravity, which the magnetic
trapping force counterbalances for the trapped |1 and |2 states but cannot for theintermediate state because of its vanishing magnetic moment.
The gradient in intermediate state detuning has the peculiar eﬀect that it does
not change the two-photon resonance frequency across the condensate but instead adds
a vertical gradient to the on-resonant coupling strength (Rabi frequency Ω0 )[Eqn. 3.2].
In keeping with the local Bloch picture developed in Section 3.3, a “pseudospin” canbe associated with the internal two-level system {|1,|2} at each vertical point in the

112

|2〉
|F=2,mF=1 〉

∆
BEC

int

|F=2,mF=0 〉Energyz|1〉|F=1,mF=-1 〉zFigure 3.31: Vertical variation in two-photon intermediate state detuning ∆int across amagnetically trapped condensate. The magnetically untrapped intermediate state feels
a linear potential gradient (force) due to gravity.

113

3’

BEC
z

Ω0
Ω0
Ω0

e
d
c
b
a

Θ
φ

ρ
2’

1’

r
Figure 3.32: Local pseudospin ρ (inset) at each vertical point labeled “a...e” on the condensate. The on-resonant Rabi frequency Ω0 increases vertically, as indicated schematically.

condensate [Figure 3.32]. The pseudospin’s polar angle θ characterizes the population
diﬀerence between |2 and |1 while the azimuthal angle φ deﬁnes their relative phase.
At t = 0 before any coupling drive is applied, the condensate is in a single internal
state such that the pseudospin points down (for |1 ) or up (for |2 ) everywhere on
the condensate [Figure 3.33(i)]. The application of an on-resonant coupling drive tothe condensate is equivalent to introducing a torque vector along the 1̂-axis of the
Bloch sphere. This causes the pseudospin to precess in the 2-3 plane about the torquedirection, corresponding to on-resonant Rabi oscillations. Because of the gradient inΩ0 , the size of the torque changes for diﬀerent vertical positions across the condensate
and the precession of the pseudospin progresses at diﬀerent rates locally.
After some time, the Rabi oscillation at the top of the condensate will be 2π outof phase with the oscillation at the bottom. This corresponds to a full 2π spread in thepseudospin’s orientation across the condensate - one “winding” [Figure 3.33(iii)]. At thistime in the condensate, a central region of one spin component vertically separates twoouter regions of the other, forming a “hamburger-and-bun” appearance [Figures 3.33(iv)and 3.34]. From the Bloch sphere, it is clear that there must also a vertical step of π
in the relative phase φ across the middle of the condensate (Compare points b & d in

114

(i)

(iii)

3’

3’
e a

a,b,…e
d

2’

b

2’

1’
c

(ii)

(iv)

Ψ

Ψ

z

abcde

z

abcde

Figure 3.33: Filled dark soliton creation from a static vertical gradient in the Rabi
frequency. (i) The condensate begins in one internal state (|2) such that the local
pseudospin points up at all vertical positions on the condensate labeled a through e.
(ii) The initial condensate wavefunction Ψ is sketched as a function of vertical trap
co-ordinate z. (iii) Eventually the vertical gradient in Rabi frequency leads to a complete angular spread in the pseudospin across the condensate. (iv) The corresponding
wavefunction is shown for each of the two-components, |1 (dotted line) and |2 (solidline).

115

φ=π

φ=0

Figure 3.34: Three-dimensional density distribution of a ﬁlled dark soliton, inferred fromtwo orthogonal images shown in side panel projections. Only the bimodal distributionof the spin component containing the soliton is displayed. The other component ﬁlls
out the intervening space forming the horizontal soliton plane.

ﬁgure 3.33(iii)). This phase step is wholly associated with the spin component lying in
the outer regions since its density node permits the rapid phase change at the centerof the condensate. Very simply then, what has been created is a horizontally-oriented
ﬁlled dark soliton as for Section 3.8! To conﬁrm this indirectly, the central component
ﬁlling out the soliton plane has been removed to induce the decay of the soliton tovortex rings. After ballistic expansion of the condensate, two axes of images have beenused to identify the usual signature of a vortex ring [Figure 3.35], which shows up in
every shot. The reproducibility is due to the fact that the soliton (and consequently
the vortex ring) is always formed in a plane including the two imaging axes.
This soliton creation technique has a few additional features of potential interest.
First, if the drive is applied until two or more windings are present in the condensate,a stack of solitons may be created [Figure 3.36]. Unfortunately, corresponding stacks ofvortex rings were not observed in expansion. Second, if the coupling drive is not exactly
resonant, the phase steps will be smoothed out somewhat and be reduced from π, inother words creating ﬁlled gray solitons. Stacks of gray (moving) solitons suggest thepossibility of their collisions; however, since each member in the stack has a phase step

116

(a)

(b)

side

(c)

front

Side
Front
Figure 3.35: Typical vortex ring formed from a “winding soliton.” (a) The vortex ringis in a plane including the two orthogonal imaging axes, labeled “front” and “side.”(b & c) Expansion images showing the characteristic pairs of density dimples in both
imaging axes. The dimples correspond to the points on the vortex ring tangent to theimaging direction.

with the same orientation, the gray solitons will all move in the same direction probably
precluding any collisions.
As a ﬁnal note of general interest, soliton creation through winding demonstrates
the quantum weirdness of a spin-1/2 (SU(2)) system. At the time of a single winding in
the condensate, the Rabi oscillations at the top and bottom edges of the condensate are
2π out of phase, which for a classical oscillation would indicate rephasing. Both regions
are in the same spin state yet a π phase diﬀerence still exists between them, creating
the soliton. Only after a full 4π phase evolution has occurred will the local spin-1/2
systems at the top and bottom of the condensate be rephased.

3.10

More angular momentum
The wavefunction engineering implemented so far with frequency modulation has

used the ﬁrst sideband for coherent control [Section 3.3]. What do the higher-order
sidebands oﬀer? For the nth sideband, the phase of the engineered wavefunction is an
n-fold multiple of the modulation phase, as has been shown explicitly for the cases n = 1
and n = 2 [Eqn. 3.15]. This has an interesting consequence in the creation of quantizedvortices where a modulation pattern with a 2π azimuthal phase gradient is applied tothe condensate. As has been demonstrated in Section 3.3.3, the ﬁrst sideband can be117

t = 0 mst = 14 ms

t = 27 ms

t= 65 mst = 75 ms

Figure 3.36: Vertical stacks of solitons arising from a static vertical gradient in the Rabi
frequency. Successively larger number of solitons are obtained by applying the coupling
drive for a longer time t as indicated.

used to transfer the 2π phase winding from the modulation pattern onto the condensate
phase to form a singly quantized vortex. Tuning instead to the nth sideband should
transfer an n-fold winding of 2π, in other words an  = n vortex! A simulation of thesecond sideband case (ωm = Ω/2) is shown in Figure 3.37.
In practice, wavefunction engineering via the second sideband has proven to be
unsuccessful because the coupling is strongly reduced compared with the ﬁrst sideband[Eqn. 3.15]. More complicated modulation patterns might make it possible to create
multiply quantized vortices using only the ﬁrst sideband. Instead, a third method has
been attempted to create an  = 2 vortex. A multi-step engineering process is performed in which two  = 1 windings are sequentially engineered into the condensate[Figure 3.38]. An  = 2 vortex is expected to be unstable against decay to two  = 1
vortices, which is a lower energy conﬁguration [129]. Indeed, two  = 1 vortices areobserved in expansion following the engineering process [Figure 3.38(c)]. Unfortunately,
it is more likely that motion of the condensate in the intervening time between engineering stages has led to the direct creation of a pair of oﬀset  = 1 vortex cores rather
than a single  = 2 state. Recently, vortices with charge  = 2 have been created in acondensate using a diﬀerent technique that is nevertheless a form of coherent control[28].
The process of adding more vorticity to a condensate through sequential stagesof wavefunction engineering is limited to two iterations because of the 50% transfer

118

(a)
t=0

t=2

π

(b)

t=4

t=6

t=6

t=8

-π

t=10

(c)

t=5.0

t=5.1

t=5.2

t=5.3

t=5.4

t=5.5

t=5.6

t=5.7

t=5.8

t=5.9

t=6.0

Figure 3.37: Simulation of  = 2 vortex engineering using the second modulation sideband to control the transfer of population from state |2 to |1. (a) Growth of the vortex
population forming a ring of |1 ﬂuid at the condensate perimeter. Time is indicatedin units of the average eﬀective Rabi frequency Ω. (b) Relative phase at t=6 between|2 and |1 , which may be interpreted as the phase of |1 relative to the ﬂat phasereference of the initial state |2. (c) Transfer dynamics within one Rabi cycle (2π/Ω),showing two  = 1 cores separating, rotating around one another, and re-merging.

119

(a)
(d)

time

(b)

(c)

250 µmFigure 3.38: Multiple vorticity ( = 2 × 1) with multi-step wavefunction engineering.
(a) Four sequential nondestructive images show a |2 vortex being engineered into a
non-rotating trapped condensate. (b) The |1 ﬂuid ﬁlling the vortex core is selectively
removed and the remaining |2 vortex transferred into the |1 state with a fast coupling
pulse (not shown). (c) A second stage of wavefunction engineering, using the |1 vortex
as an initial state, adds another 2π winding to the condensate. (d) An image of theballistically expanded condensate shows two vortex cores, verifying the creation of acondensate with two units of circulation.

120
eﬃciency at each stage. In the next chapter a new approach is taken to introduce large
amounts of angular momentum into a condensate in the form of many singly-quantizedvortices.

Chapter 4

Vortex Lattices

4.1

From quantum to classical ... and on to quantum
Chapter 3 focused on the creation and behaviour of one quantum of circulation- in other words singly quantized vortices - in a condensate. This chapter deals withthe opposite limit of highly rotating condensates containing large amounts of vorticity.
The experimental technique has been honed to the point where condensates with large
numbers of both atoms and vortices can be made relatively reproducibly. The approachto creating vortex-full condensates is diﬀerent from that taken in other experimentalgroups and involves the evaporative cooling of a rotating cold gas to quantum degeneracy. In this way a condensate is created in equilibrium with the rotating environmentof the normal gas surrounding it. At high rotation rates, large, well-ordered triangularlattices of vortices are observed in the condensate. In the limit of large amounts of vorticity, the correspondence principle would suggest that the velocity ﬂow in the condensate
should mimic classical rigid-body rotation. This of course applies only on a macroscopic
scale coarse-grained over the vortex core structure [Figure 4.1]. Classical-like rotationaleﬀects, such as centrifugal distortion of the condensate’s density, are explored experimentally in this chapter. With direct detection of the vortex cores available, it has
also been possible to study the “microscopic” structure of the vortex arrangements,
both at equilibrium and under dynamical conditions where applied stress can distortthe equilibrium lattice conﬁguration.

122ΩLattice

ΩBEC

Top
View

(a)

(b)

Figure 4.1: (a) Vortex lattice and (b) classical rigid-body rotation.

As a ﬁnal note, in the rigid-body regime, the areal density nV of vortices grows
linearly with increasing rotation of the condensate. However, even though rotation
rates up to 95% of the centrifugal limit are demonstrated in this chapter, the vortex−1/2

distribution remains in the “dilute limit” where the spacing between vortices ∼ nV

is much larger than the typical core size given by the healing length. Since the core size
grows with decreasing atomic density, the limit of a very rapidly rotating, distended
condensate and reduced atom number allows vortices to begin to approach one another.
The “true” quantum limit is ultimately reached where a vortex occupies the same area as
an atom in the condensate. Under these circumstances recent theory [63, 133, 101, 146]
predicts the transformation of the condensate from a vortex lattice state to a strongly
correlated quantum system, associated with quantum Hall physics. The techniquesdeveloped in this chapter hopefully will provide the starting point for achieving thisnew regime of rotating condensate physics.

123

4.2

Intrinsic vortex nucleation [97]4.2.1

Abstract

We have developed an evaporative cooling technique that accelerates the rotationof an ultra-cold 87 Rb gas, conﬁned in a static harmonic potential. As a normal gas isevaporatively spun up and cooled below quantum degeneracy, it is found to nucleate
vorticity in a Bose-Einstein condensate. Measurements of the condensate’s aspect ratio
and surface-wave excitations are consistent with eﬀective rigid-body rotation. Rotation
rates of up to 94% of the centrifugal limit are inferred. A threshold in the normal cloud’s
rotation is observed for the intrinsic nucleation of the ﬁrst vortex. The threshold value
lies below the prediction for a nucleation mechanism involving the excitation of surfacewaves of the ground-state condensate.

4.2.2

Introduction

To paraphrase an ancient riddle, what happens when an irresistible torque meets
an irrotational ﬂuid? The answer has been known for more than 50 years: a quantizedvortex is nucleated. Vortices alone contribute to a superﬂuid’s rotation, so that thebulk of the ﬂuid may remain curl-free. The nucleation of vortices in bulk superﬂuid
Helium has been the topic of extensive study (for a review see [74]). In the archetypicalexperiment, a rotatable pot ﬁlled with a mixture of superﬂuid and normal liquid Helium
undergoes gradual angular acceleration. The normal ﬂuid and the walls of the pot
rotate together as a rigid body, deﬁning a rotating environment. At some threshold
angular velocity, a vortex line is nucleated at the circumference of the pot, and thenquickly migrates inward until it is collinear with the axis of rotation. Further angularacceleration results in the nucleation of more vortices; eventually the ﬂuid is ﬁlled withan array of vortex lines [29]. A central theme [30] of this research is the question: towhat extent is the nucleation process “extrinsic,” i.e. dependent on such details as the124roughness of the surface of the walls, and to what extent is it “intrinsic” [31, 32], i.e.
driven (in the limit of microscopically smooth walls) by the ﬂow of normal ﬂuid along
the boundary of the superﬂuid? In the analogous rotating-potential experiments with adilute-gas Bose-Einstein condensate (BEC), the conﬁning potential and the normal ﬂuid
typically rotate at diﬀerent rates [94]. In this context, the extrinsic-intrinsic question
can be restated as: is it the conﬁning potential or the normal ﬂuid that deﬁnes therotating environment?
Vortices in a BEC have been created with wavefunction engineering [33], throughthe decay of solitons [50, 75], and in the wake of moving objects [104, 34]. The ﬁrst
rotating-potential experiment to detect vortices in a BEC was performed by the Parisgroup [118]; results have also been obtained by the MIT [48] and Oxford [103] groups.
In these experiments the role of the normal ﬂuid was secondary to that of the rotating
potential; it is conceivable the normal ﬂuid was not rotating at all. This section presents
vortex nucleation experiments performed in the opposite limit, namely in the environment of a rotating normal gas in a static conﬁning potential. Such an environment
allows for the isolated study of the intrinsic mechanism for vortex nucleation.

4.2.3

Evaporative spin-up

Our experiments begin with a magnetically trapped cloud of about 6 · 106 87 Rb
atoms, in the |F = 1, mF = −1 hyperﬁne state, cooled close to the critical temperature Tc = 67nK. The atoms are initially conﬁned in an axially symmetric, oblate andharmonic potential [136] with axis of symmetry along the vertical (‘z’) axis. To induce
rotation of the cloud, we ﬁrst gradually apply an elliptical deformation to the potential
in its horizontal plane of symmetry, and then rotate the deformation [109] about thevertical axis at a ﬁxed angular frequency. The rotating potential is characterized byan axial frequency ωz = 2π(13.6)Hz, average radial frequency ω̄ρ = 2π(6.8)Hz and ahorizontal ellipticity of 25%. Such a large rotating trap asymmetry, accessible in the125
oblate conﬁguration of our apparatus, is found to be necessary not only to get the cloudrotating, but also to sustain ongoing rotation. Moreover, in steady state the cloud does
not reach the rotation rate of the applied asymmetry. We believe that the stirring
process is ﬁghting a small, static asymmetry that acts to despin the cloud [35].
In thermal equilibrium, a normal cloud rotates as a rigid body with the centrifugal
force causing the cloud to bulge outwards in the radial direction. In order to detect therotation, we use a nondestructive phase-contrast technique to image the cloud in situ
from the side. Four sequential pictures of a given cloud are taken to average over
oscillations in the widths and to improve the signal to noise. The cloud temperature is
extracted from its vertical width σz , which is unaﬀected by rotation about the verticalaxis. The rotation ΩN of a cylindrically symmetric cloud is determined from the aspectratio, λ = σz /σρ , using the relation:Ω/ωρ =



1 − (λ/λo )2(4.1)

where λo is the static aspect ratio. The technique of side-view imaging is crucial for
distinguishing between changes in radial size due to temperature and to rotation.
For stirring rates up to 2.5Hz, the rotation of the cloud reaches its steady-statevalue by 15s or less. After 15s, the rotating trap asymmetry is ramped oﬀ, leaving
non-condensed clouds a factor of 1.2 – 1.3 above Tc , for stirring frequencies 0 – 2.5Hz.
Radio frequency (rf) evaporation is then used to cool the normal cloud to BEC. In
the oblate conﬁguration of our TOP trap [136], we have found that rf evaporation
immediately quenches the rotation, presumably because the selection process, whichremoves atoms with large radial displacements, preferentially removes atoms with large
axial components of their angular momentum. By adiabatically distorting the trap
into a prolate geometry with {ωρ , ωz } = 2π{8.35, 5.45}Hz [78], we can instead cool
the cloud by removing atoms with large axial displacements and thereby reduce theeﬀect of the evaporation on the axial angular momentum [Figure 4.2]. As the normal

126

z

Trapped gas
y

Side
View

x

Evaporation
surfaceθFigure 4.2: Vertical crosssection of the evaporation surface in the prolate conﬁning
geometry of the TOP trap. The surface, which represents a contour of magnetic ﬁeld
at which the radio-frequency evaporation is resonant, forms a shallow cone in three
dimensions with an angle θ  10◦ from the horizontal. The prolate shape of the conﬁning
potential has the eﬀect of making the evaporation surface eﬀectively shallower (θef f 6◦ ).

cloud is evaporated, its aspect ratio is observed to decrease continuously, indicating amonotonically increasing rotation rate (Fig. 4.3a). During the evaporation, the angularmomentum per particle of the normal cloud remains roughly constant, even though thenumber of atoms is reduced by over a factor of ﬁve and temperature, by a factor of four(Fig. 4.3b). As the cloud cools and shrinks, it must spin up for the angular momentum
per particle to remain ﬁxed.
To reach signiﬁcant rotation rates by the end of evaporation requires the lifetimeof the normal cloud’s angular momentum to be comparable to the evaporation time.
The nearly one-dimensional nature of the evaporation together with the low average
trap frequencies make cooling to BEC in the prolate trap very slow (∼50s). We obtain
angular momentum lifetimes this long [Figure 4.4] by shimming the azimuthal trap
symmetry to better than 0.1%.

4.2.4

Highly rotating pure condensates

Towards the end of the evaporation, a condensate begins to appear at the centerof the rotating normal cloud [36]. We discuss ﬁrst the results of experiments in which

127

1.6

λN

1.4
1.2
1.0 (a)LZ / (N h)

60

200

40
100
0

TZ (nK)

80

300

20

(b)
0

10

20

30

40

50

0

evaporation time(s)Figure 4.3: (a)Aspect ratio λN of a rotating normal cloud during evaporation preferentially along the axis of rotation. Data for three initial cloud rotations are shown, obtained by ﬁrst stirring for 15s with an applied rotation of 0ω̄ρ (inverted triangles), 0.07ω̄ρ(open circles) and 0.37ω̄ρ (squares). (b)Angular momentum per particle (squares) andtemperature (triangles) of the normal cloud during evaporation for the 0.37ω̄ρ case.

128

1.7

1.5

λN
1.3

1.1
0

50

100

150

time (s)Figure 4.4: Aspect ratio of a normal cloud as a function of time after transfer to theprolate conﬁning geometry. Open (ﬁlled) squares indicate the case of a static (rotating) cloud. The lines correspond to exponential decay ﬁts with a single time constantτ1 =2.2(2)s for the static case and two time constants, τ1 =2.2s ﬁxed and τ2 =60(6)s, forthe rotating data. The initial rapid decay associated with τ1 is on the order of the collison rate in the gas and corresponds to rethermalization of the cloud shape to the new
conﬁning geometry. The longer decay of the rotating cloud’s aspect ratio is associatedwith the damping of its rotation as a result of residual trap asymmetries.

we continue the evaporation until little or no normal fraction remains. In this case,
we ﬁnd that the rotating normal component has given birth to a condensate distended
in its radial dimension, as one would expect for a classical rotating body under theinﬂuence of the centrifugal force [Figure 4.5]. This eﬀect is reminiscent of liquid Helium
experiments, in which the surface of a rotating bucket of superﬂuid exhibits the same
meniscus curvature as for an ordinary viscous ﬂuid [37]. For large enough numbers ofvortices in the condensate, the correspondence principle would suggest that the rotationﬁeld, coarse grained over the cloud, should go over to the classical limit of rigid-bodyrotation. In this limit, the classical Eq. (4.1) should connect condensate aspect ratio torotation rate.
Alternatively, we can study the angular momentum in the BEC directly by excit-

129

z

Rotating

Static

(b)

(a)
60 µm

Side
View

x

y

80 µm

Figure 4.5: Side view images of (a) a static and (b) a highly rotating condensate. The
imaged clouds are characterized by atom number {3.3·106 , 4.5·106 } and aspect ratio{1.586, 0.5}.

ing quadrupolar surface waves [61, 98] with a rotating weak deformity of the magnetic
trap. The quadrupolar surfaces waves are characterized by angular momentum quantum number mz = +(−)2 describing an excitation that is co-(counter-)propagating withthe rotation of the condensate. By varying the initial stir rate applied to the normal
cloud, condensates of diﬀerent aspect ratio can be accessed for study. In Fig. 4.6a, thefrequency of the mz = ±2 modes is shown as a function of condensate aspect ratio. The
mz = +2 mode is seen to speed up and the mz = −2 to slow down due to the presenceof vorticity in the condensate.
For small rotation rates, the splitting between the mz = ±2 modes is predicted tobe linearly proportional to the mean angular momentum of the condensate [39, 161] (see,though, Section 3.7). In the large-Ω limit of rigid-body rotation, Zambelli and Stringari[161, 38] have used a sum-rule argument to show that the splitting between the modes
is simply 2Ω, and, further, that the sum of the squared frequencies of the two modes is
independent of rotation rate. Cozzini and Stringari [66] have shown that hydrodynamic ∧ r reproduce the sum-rule result. The
equations including a rigid-body rotation ﬁeld Ωfrequencies of the m = ±2 modes are explicitly:(
ω±2 =2ωρ2 − Ω2 ± Ω.

(4.2)

130

( ν+ - ν- ) / (2νρ)

18

0.8

ν (Hz)

14
10

0.6

6
(a)
2
0.6 0.8 1.0 1.2 1.4 1.6
λBEC

0.4

(b)

0.4

0.6

ΩBEC / ωρ

0.8

Figure 4.6: Quadrupolar surface-wave spectroscopy of condensates formed in a rotating
normal cloud. (a)Quadrupolar frequency as a function of condensate aspect ratio forthe m = +2 (squares) and m = −2 (triangles) surface waves. Solid lines are a single ﬁt
to the combined data using the theory of Zambelli and Stringari [161, 38]. Dotted lines
indicate average frequency and aspect ratio for a static BEC. (b)The splitting between
m = ±2 frequencies scaled by twice the radial trap frequency, plotted explicitly as
a function of BEC rotation rate inferred from the aspect ratio. The solid line is theprediction from the same theory as in (a). Each plotted point is obtained from a single
m = −2 measurement in (a) combined with a spline interpolation to the relatively quiet
m = +2 data. Only aspect ratios smaller than 1.43 (corresponding to ΩBEC /ωρ ≥ 0.35)
are included to avoid obtaining imaginary rotation frequencies due to experimental noisein the aspect ratio.

131
A best ﬁt of this model to the combined m = ±2 data is shown in Fig. 4.6a, where
the rigid-body rotation rate has been inferred from the condensate aspect ratio and theclassical Eq. (4.1). Perhaps more intuitively, the frequency splitting is plotted explicitly
versus inferred rotation rate in Fig. 4.6b. The excellent agreement with the model ofZambelli and Stringari is compelling evidence in favor of the reasonableness of using
Eq. (4.1) to connect the condensate aspect ratio with its eﬀective rotation rate. This isfurther born out by extensive 3-D numerical simulations of the Gross-Pitaevskii equation
for the parameters of our experiment, by Feder and Clark. Their numerical simulations
conﬁrm that a condensate in an environment rotating at frequency Ω > 0.5 ωρ will
equilibrate close to the aspect ratio given by Eq. (1) [40].
In pure condensate samples we have observed aspect ratios as pronounced as
0.35 λo , corresponding to a rotation rate of 0.94 ωρ . The rapid rotation rate, combinedwith the increased condensate area arising from its radial bulge, mean that the condensate must be supporting a large number of vortices. Feder and Clark [40] calculate 56.
With these initial conditions, we have observed continued rotation for at least 140 s.

4.2.5

Mutually rotating condensate and normal cloudIf the evaporation is stopped before the normal cloud has been completely removed, a comparison can be made between the aspect ratios, and hence rotation rates,of the condensate and normal cloud. By adjusting the initial stir rate applied to thenormal cloud and the depth of the evaporation, we are able to reach diﬀerent rotation
rates for a given condensate fraction. After the evaporation is stopped, a time of 5 s
is allowed for the gas to rethermalize to the last evaporative cut. We then take four
nondestructive pictures and ﬁt the images to a two-component distribution. Figure 4.7
shows a plot of the condensate aspect ratio λBEC compared with the aspect ratio λNof the normal cloud where each point represents a single realization of the experiment.
For λN < 1.36 (ΩN > 0.5 ωρ ), the condensate aspect ratio closely tracks the

132
0.9

0.8

0.6

0.4

0

0

(a)

0.4

1.4

Ω BEC / ωρ1.6

Ω N / ωρ

0.6

λBEC

1.2

1.0

0.8
0.8

0.9
0.6
0.6

0.8

0.5

0.4

λN

1.2

0.3

1.4

1.6

Ω N / ωρ

0

(b)

λBEC

0

Ω BEC / ωρ

1.6

1.0

0.3

1.5

0.4
1.4

0.5
1.4

λN 1.5

1.6

Figure 4.7: (a)Aspect ratio of the BEC vs. that of the normal cloud after evaporation
halted. Right and top scales provide a conversion from aspect ratio to classical rigidbody rotation rate for BEC and normal cloud respectively. Data for evaporation ofboth a static (empty squares) and rotating cloud (ﬁlled squares) are shown. Dottedlines indicate average static aspect ratios for both BEC and normal cloud. A solid
1:1 line is superimposed on the data. Three representative integrated density proﬁlesof two-component clouds indicate the range of diﬀerent aspect ratios observed. (b)A
magniﬁed version of the region of high aspect ratio (low rotation rate) in (a). Added tothe plot are data (triangles) obtained with evaporative spin-up 3 times slower than forthe ﬁlled squares. A dashed line indicates the aspect ratio expected for a BEC with asingle, centered vortex as calculated by Feder and Clark [40].

133
normal aspect ratio, providing a further manifestation of the correspondence principle
for a highly rotating BEC. In the vicinity of λN = 1.48 (ΩN = 0.35 ωρ ), we observe
threshold behavior in the condensate rotation. At this low value of rotation we don’t
expect the rigid-body model to be valid for the condensate; but we make use of anumerical calculation by Feder and Clark [40] to indicate the change in condensate
aspect ratio associated with the presence of a single, centered vortex (Fig. 4.7b). Thereis considerable scatter in the data so one cannot make a strong statement about thenature of the threshold shape, but clearly, somewhere between 0.32 < ΩN /ωρ < 0.38
the ﬁrst vortex is nucleated.
A comparison of the observed threshold with two theoretical rotation rates provides some insight into the nature of the vortex nucleation. The ﬁrst theoretical valueis Ωc , the critical rotation rate for thermodynamic stability of a single vortex. For our
experiment (with 3 – 8·105 atoms in the condensate component) Ωc is 0.2 – 0.25 ωρ [41],distinctly lower than our observed threshold for nucleation. The second value is ωmin ,the frequency at which the slowest surface-wave mode propagates around the circumference of the condensate. The Paris group has shown that for their ”extrinsic” nucleation
process (vortices nucleated by a rotating asymmetric potential) the key mechanism isthe nonlinear excitation of surface waves [118, 117]. Results from MIT [34, 48] andfrom Oxford [103] are also consistent with such a mechanism. For the parameters of
our experiment, ωmin = 0.4 ωρ [68]. Our observed threshold for “intrinsic” nucleation
is clearly under this value; thus, interpreting our eﬀect in terms of a normal “wind”
exciting surface waves on the condensate is problematical. However, because the conﬁning potential is not rotating in our case, the vortex nucleation can only arise frominteraction with the rotating normal cloud.
The data presented in Fig. 4.7 include a range of condensate fraction from 0.1-0.45
for each rotation rate of the normal cloud, although no segregation relative to either
axis is evident for plots of diﬀerent BEC fractions. Moreover, by reducing the rate of

134
evaporation, we have decreased the rate of acceleration of the normal cloud rotation bya factor of 3, and still observe a threshold for vortex formation between 0.32 and 0.38
(Fig. 4.7b).
Threshold behaviour aside, the rotating normal cloud can create equilibrated
condensates with very large rotation rates. This may allow us in future work to approach
the regime for which the vortices are so close-packed that their separation becomescomparable to their core size, given by the healing length of the condensate [101].

4.3

Theory responds
The experimental results of the previous section have initiated a discussion in theliterature [158, 53, 134, 145] as to the mechanism for vortex nucleation in the presenceof a rotating normal cloud. The consensus of the theory to date is that the mechanism
is ultimately the same as in the “extrinsic” experiments where vortices are nucleatedwith a rotating asymmetric potential. In other words, the nucleation is driven in bothcases by surface-wave instabilities, which allow a vortex to overcome the energy barrierat the condensate surface and thereby enter into the bulk [68]. For a rotating potential,
the rotation threshhold for the onset of vortex nucleation is set by the minimum surfacewave frequencyΩS = min

ω 




(4.3)

where  deﬁnes the angular momentum of the surface mode with frequency ω . The
above equation is a generalization of the well-known Landau criterion [113] with thecritical velocity now given by vc = ΩS R on a condensate of radius R. For typical experimental parameters, min ≈ 10 [68, 53, 134]. In an actual experiment, the nucleationthreshhold may end up somewhat higher than ΩS if the surface roughness of the rotating potential does not contain a spherical harmonic order  high enough to couple tothe minimum surface wave [68].

135
In the intrinsic nucleation experiments, one can think heuristically of the “wind”
of the normal cloud, instead of the rotating potential, driving the surface-waves on acondensate. More technically, though, one can speak of the gain of a surface mode,
which depends on the mode’s velocity relative to the normal cloud [158, 134]:
γ=γ

(0)




ΩN
−1
ω

(4.4)

For a static normal cloud (ΩN = 0), all of the surface modes experience a negative gainor, in other words, are damped. However, if the normal cloud is brought into rotation,
the gain of one or several surface modes on the condensate may become positive leading
to an instability. The onset of the ﬁrst instability will occur when the normal cloudrotation is resonant with the slowest surface wave: ΩN = ΩS . More generally, for an
arbitrary rotation rate, Penckwitt et al. [134], using a condensate growth model, haveshown that the surface mode with the largest gain will dominate the vortex nucleation.
The  value corresponding to maximum gain can be found from ∂ ( ωΩ − 1) = 0 and

matches min from Eqn. 4.3 at threshhold.
The primary point of contact between theory and experiment is the location of
the threshhold for ﬁrst vortex nucleation. The experimentally observed threshhold fromthe previous section lies in the range 0.32 < ΩN /ωρ < 0.38, which is somewhat lower
than the slowest surface-wave ωmin /ωρ = 0.4 quoted for our experimental parameters[Section 4.2]. Anglin [53] provides an analytical expression for the rotation threshholdbased on the local critical velocity at the Thomas-Fermi surface of a condensate. Applied
to our experiment, this yields a threshhold value somewhat closer to the observed one Ω/ωρ = 0.41−0.36 for condensate number in the range 3−8·105 . The local surface theory
is valid in the limit of short surface-wavelength where the penetration depth δ of thesurface waves into the bulk of the condensate is small compared the Thomas-Fermi sizeof the condensate R. In terms of the radial harmonic oscillator length aρ = (/(mωρ )1/2 ,the penetration depth is δ = (a4ρ /R)1/3 . At threshhold, δ/R ≈ 0.1 and the theory is

136
expected to reasonably valid.

4.4

Direct observation of vortex lattices
In our initial vortex nucleation experiments [Section 4.2], the vorticity within acondensate was detected indirectly from the centrifugal distortion of the density proﬁle or from the presence of angular momentum sensed with surface-wave spectroscopy.
Complementary to these techniques assessing “macroscopic” rotational eﬀects is the
direct detection of vortex cores in the density proﬁle of a condensate when observed
along the rotation direction (see also chapter 3). There is considerable interest in the“microscopic” structure of large amounts of vorticity, where the term “microscopic” isused to refer to the size scale of the vortex core. In particular, the minimum energy
conﬁguration for a reasonably rotating condensate at equilibrium is expected to consist of a large number of singly-quantized vortex cores arranged in a triangular lattice[59, 79, 54].
Triangular vortex lattices were ﬁrst observed in a condensate at the ENS [118]and somewhat later at MIT [48] and Oxford [103]. Given that the intrinsic nucleation
technique can produce highly rotating condensates (ΩBEC = 0.95ωρ), the system oﬀers
the potential to study large amounts of densely packed vorticity. For this reason as
well as the added beneﬁts of direct vortex detection, we were led to add a verticallyoriented imaging axis to see down the rotation axis of the condensate. After the usual
evaporative spin-up sequence to produce near pure rotating condensates [Section 4.2],
the condensate’s conﬁning potential is discontinuously removed and the condensate
allowed to ballistically expand [27]. The expansion is continued until the vortex core sizeis large enough to be resolved with our imaging system at which point the density proﬁlealong the axis of rotation is imaged. Figure 4.8 shows typical images of condensates at
diﬀerent rotation rates, or equivalently, with diﬀerent amounts of vorticity. The highlyordered triangular lattice structure is apparent in the images out to the edge of the

137

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.8: (a). . .(f) Vortex lattices corresponding to condensate aspect ratio{1.53,1.54,1.42,0.96,0.5,0.54}. For (c). . .(f), the corresponding rigid-body rotation rate
ΩBEC /ωρ is {0.4,0.8,0.95,0.94}. The condensate in (f) has about the same rotation ascase (e) but is larger and contains more vortex cores because the number of atoms issigniﬁcantly larger.

138
condensate, even though the density proﬁle is non-uniform due to the inhomogeneous
conﬁning potential. It has recently been pointed out [79, 54] that the radially decreasing
density proﬁle in fact helps to preserve the triangular ordering by suppressing distortionsin the lattice structure at the surface of the condensate.
Two applications of vortex core detection are considered here, one at low vorticity
and one at high. In the ﬁrst application, the vortex nucleation process of Section 4.2 hasbeen revisited to verify the amount of vorticity at threshhold. In the original experiment,
numerical simulations [79] were relied upon to provide the aspect ratio for a condensate
in the presence of small numbers of vortices. This permitted the identiﬁcation of thenucleation threshhold with the creation of a single vortex. The new ability to detect
vortex cores in ballistic expansion allows this to be directly conﬁrmed in the experiment.
A set of data has been acquired similar to that in ﬁgure 4.7, where a normal
cloud is spun up to varying degrees and then evaporatively cooled until a condensate
component appears. The average condensate fraction for the data set is NBEC /N ∼0.34. An in-situ non-destructive image is taken from the side to permit a comparison of
the aspect ratios of the condensate and normal cloud. The sample is then ballistically
expanded and imaged from the top to allow for the visual identiﬁcation of vortex coresin the condensate. In ﬁgure 4.9, the aspect ratio of the condensate is plotted againstthat of the normal cloud near threshhold, where the data points are now identiﬁedaccording to vortex number. The vortex counting for the data set has been performedby eye. It is clear that the nucleation threshhold is indeed associated with the presenceof a single vortex in the condensate. In other words, the threshold is low enough that,after nucleation, single vortices are energetically favored. Only at a slightly higher
normal cloud rotation (0.4  ΩN /ωρ  0.5) do two and three vortices begin to appear.
This may help to solidify the connection between the experimental data and numericalsimulations of the nucleation [134] where single vortices should be found to occur at
threshhold.

139

0.5

0.4

0.3

Ω N / ωρ

0

λBEC

0

Ω BEC / ωρ

1.6

1.5

0.3
0.4
1.4

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

0.5
1.4

1.5

λN

1.6

Figure 4.9: The threshhold for vortex nucleation repeated as for ﬁgure 4.7, but withdirect vortex detection now used to identify the number of vortices for each data point(see legend). The open squares are obtained by evaporative cooling of a static normal
cloud. The axes and dashed and dotted lines are as deﬁned in Figure 4.7.

The aspect ratio for a condensate with a single centered vortex, as calculated
by Feder [79], is also corroborated by the data in ﬁgure 4.9. Notwithstanding themeasurement noise inherent in the data, it is not surprising that several single vortex
points appear in the plot region between static and single vortex aspect ratios: if a vortex
is oﬀset from the condensate center, it will have less of a centrifugal eﬀect on the radial
size. Indeed, by and large, the single vortices were not found to be perfectly centered on
the condensate. This eﬀect not only helps to explain the smoothness of the nucleationthreshhold when plotted in terms of aspect ratio but also substantiates the position of
the threshhold, 0.32 < ΩN /ωρ < 0.38, as originally identiﬁed. Further reﬁnement of themeasured critical rotation rate can be achieved only by reducing systematic errors andnoise in the determination of the normal cloud’s aspect ratio.

140
The data set has been continued to higher rotation rates of the normal cloud
where more and more vortices enter into the condensate. Under these circumstances it
is possible to study the number of vortices in the condensate as a function of its eﬀective
rigid-body rotation rate ΩBEC , which can be inferred from its aspect ratio λBEC in theusual way [Eqn. 4.1]. Since the condensates are formed in equilibrium with a rotating
normal cloud, the vortex number is expected to correspond to the equilibrium value.
The limit of eﬀective rigid-body rotation is characterized by a uniform distribution ofvorticity with areal density nV = 2Ω/κ [155]. Here κ = h/m is the quantized circulationof a single vortex and m is the mass of a condensate atom. For a condensate with aradial size given by Thomas-Fermi radius Rρ , the number of vortices in the rigid-body
approximation is πRρ2 nV , which can be expressed explicitly as:
NVrb =

2µ0Ω/ωρωρ (1 − Ω/ωρ )3/5

(4.5)

In the previous equation, µ0 = mωρ2 Rρ (0)2 is the chemical potential of the condensate in
the absence of rotation, expressed in terms the static Thomas-Fermi radius Rρ (0) in theradial direction. There is a weak dependence on the number of atoms in the condensate
through Rρ (0)2 ∼ N 2/5 . It is also worth noting that, in the rigid-body approximation,
the number of vortices included in a Thomas-Fermi condensate is related to the angularmomentum per particle [66] byNV =7 z 2 

(4.6)

In ﬁgure 4.10, the raw number of vortices for the current experimental data set
is plotted as a function of the aspect ratio of the condensate. Additionally, data froma later run using a more eﬃcient spin-up technique [Section 4.7] is included and involves experimental conditions that are signiﬁcantly diﬀerent in the following ways: The
evaporation has proceeded until little normal cloud is visible; higher rotation rates areincluded; and the condensates have larger number, corresponding to a static ThomasFermi radius Rρ ∼30µm compared with 21µm for the current set of data. The static

141
chemical potential µ0 has a correspondingly larger value of 33ωρ compared with 16ωρ .
As a result, it is not surprising that the later data set shows a larger number of vortices
for a given rotation rate [Figure 4.10(a)]. However, by plotting the reduced vorticity NV ωρ /2µ0 , the dependence of NV on condensate number can be removed [Figure 4.10(b)]. The ﬁt Thomas-Fermi radius for each plotted point is used to calculate
the chemical potential µ0 in order to remove intra-data-set ﬂuctuations as well. Expressed in terms of the reduced vorticity, the data sets are now seen to match over awide range of aspect ratio.
The rigid body prediction [Eqn. 4.5] has also been included in the plot for comparison. It is quite clear that the experimental data tracks but consistently lies below
the rigid-body value. Possible systematic errors will be discussed below; however, anumerical calculation by Feder and Clark [79] for a condensate with 2 · 105 atoms ina similar conﬁning potential matches the data extremely well over the entire range ofaspect ratio shown in ﬁgure 4.10(b). In their numerical simulations of rotating condensates, Feder and Clark have found that the aspect ratio and moment of inertia Lz /Ωapproach their classical rigid-body values with increasing rotation, but the number ofvortices lags considerably behind the expectation according to Eqn. 4.5. For example, inthe case of a condensate rotating at 95% of the centrifugal limit (ΩBEC /ωρ = 0.95), theaspect ratio and moment of inertia are found to be within a few percent of the rigid body
predictions, even though the vortex number is 65 compared to the rigid-body value of
89. The simulations further show that the vortex density does agree with the rigid-body
distribution in the bulk of the condensate; only at the edge of the condensate does adiscrepancy arise where at least one hexagonal ring of the triangular vortex lattice ismissing. The lattice is nevertheless stationary in the rotating frame which raises thequestion: what provides the extra rotation to make up for the missing vorticity? The
answer lies with the inhomogeneous conﬁning potential. The resulting non-uniform density proﬁle of the condensate contributes a radial force that, according to the Magnus

142

140

(a)

120
100

NV 80
60
40
20
0
0.6

0.8

1.0

λBEC

1.2

1.4

(b)

3

N

hωρ
V µ0

1.6

2

1

0
0.6

0.8

1.0

λBEC

1.2

1.4

1.6

Figure 4.10: (a)Vortex number as a function of BEC aspect ratio for two diﬀerent
experimental runs, in particular with an average condensate number of 0.4 · 106 (opensquares) and 2 · 106 (ﬁlled squares). A lower aspect ratio indicates a faster eﬀective rigidbody rotation rate. (b) Reduced vortex number plotted as a function of BEC aspectratio. The uniform rigid-body prediction (dotted line) and a numerical simulation byFeder and Clark [79](solid line) are added for comparison.

143
force [155], causes the vortices to rotate a little faster about the axis of the condensate.
This additional rotational motion is precisely the core precession observed for single
oﬀset vortices in Chapter 3. Recent analytical results [54] show the same “conspiracy”
of two microscopic rotational contributions to yield the overall macroscopic rigid bodyrotation ﬁeld.
The natural question arises whether the disagreement between the data and rigid
body prediction can be due to systematic errors in evaluating the reduced vorticity.
The main dependence on experimental numbers is NV /(ωρ Rρ (0)2 ). In ﬁgure 4.11, theratio of the observed vorticity to the rigid body expectation NV /NVrb is plotted as afunction of aspect ratio and is found to be roughly a constant around ∼ 0.75 − 0.8.
This could be accounted for by a systematic error of 25% in the radial trap frequencyωρ = 2π8.45Hz, which can be ruled out, or a 12% systematic error in the determinationof the condensate size Rρ , which cannot be immediately ruled out at this time. The
ﬁnal possibility for error is in the actual counting of vortices present in the condensate.
Because of the vanishing density at the edge of the cloud, the low signal-to-noise makes
it diﬃcult to identify vortices in this region of the image, particularly in the case of large
vortex number.(Compare ﬁgures 4.8(c)&(f)). It may simply not be possible to conclude
deﬁnitively whether or not there are vortices at the edge of the condensate contributing
to a rigid-body rotation ﬁeld. An automated vortex counting algorithm, not yet applied
to the data presented here, may help and will at least provide an unbiased estimate of
the counting error arising at the condensate surface.
In conclusion, direct vortex detection has been used to verify the single vortex
character of the intrinsic nucleation threshhold and to study ordered triangular lattices
composed of variable amounts of vorticity. The observed vortex number in a rotatingcondensate agrees with the rigid-body prediction within a constant factor of ∼25%.
Two possible systematics in the determination of condensate size and vortex number
preclude a deﬁnitive statement regarding the observed deviation from the rigid-body

144

1.2

0.8
NV
N Vrb

0.4

0.0
0.4

0.6

0.8

1.0

1.2

1.4

λBEC

Figure 4.11: Ratio between observed vortex number with respect to rigid body prediction. Two separate data sets as in ﬁgure 4.10 are indicated by open and ﬁlled squares.
The solid line represents the numerical simulation by Feder and Clark [79].

prediction. Nevertheless, the tantalizing agreement with theory justify further work in
the future, including more data at higher vortex number and a comparison of the vortex
lattice constant (vortex spacing) with theoretical predictions [79, 54]. It represents aninteresting application of the correspondence principle to understand how the microscopic quantum vortex structure reacts in the presence of an inhomogeneous conﬁning
potential to mimic the macroscopic classical limit of eﬀective rigid-body rotation.

4.5

Rotational suppression of the quantum degeneracy temperature
So far in this chapter, the phase boundary between vortex-free and vortex-fullcondensates has been considered as a function of the normal cloud rotation rate ΩN .
Another albeit simpler eﬀect of the rotation is to suppress the quantum degeneracy
temperature TC , which forms the boundary between normal and condensed phases [149].
This eﬀect may be accounted for by the centrifugal weakening of the radial harmonic(
conﬁnement, ω̃ρ = ωρ2 − Ω2 , which leads to a reduction in the normal gas density nth
relative to the non-rotating case. As a result, a lower temperature must be reached145
before the phase space density (∼ nth /T 3/2 ) is suﬃciently high to bring about the BEC
transition. The expression for TC as a function of the rigid-body rotation rate Ω of thegas sample is(0)
TC = TC(0)

In the above equation, TC



Ω2
1− 2
ωρ

1/3
(4.7)

is the static critical temperature. For a non-interacting(0)

gas in a harmonic potential, TC = 0.94ωho N 1/3 where N is the number of atoms inthe sample and ωho = (ωρ2 ωz )1/3 is the average trap frequency. In ﬁgure 4.12, the data
originally from ﬁgure 4.7 have been used to plot condensate fraction versus temperature
for three diﬀerent ranges of sample rotation rate. All quantities are obtained from ﬁts
to the non-destructive in-situ images of the trapped gas clouds. As usual, the rotation
rate has been assessed from the changing aspect ratio of the normal cloud according to
Eqn. 4.1. It is qualitatively clear from ﬁgure 4.12 that a lower temperature is required
at higher rotation rates to reach a given condensate fraction. For each data point, thetemperature and condensate fraction can be used together to infer a value for the criticaltemperature of the sample. To remove shot-to-shot variation in TC due to atom-number
ﬂuctuations, the value of the inferred transition temperature can be scaled by the static(0)value TC calculated from the measured atom number. The scaled value of TC is plottedagainst rotation rate in ﬁgure 4.13 for three ranges of temperature: “hot,” “medium”
and “cold,” corresponding to three diﬀerent ranges of condensate fraction.
Each group of data clearly show the suppression of TC towards higher rotation. The
overall suppression of TC is roughly consistent with an eﬀect associated with the interatomic interactions [69] although there may also exist systematic errors in the numberor temperature calibrations.
The centrifugal suppression of TC is perhaps less interesting than that due tomany-body interaction eﬀects; nevertheless, from a technical stand point, centrifugal
eﬀects are an important consideration in the evaporative cooling and spin-up process.

146

0.6

Ω N / ωρ
<0.5
0.5 - 0.67
>0.67

0.4

NC
N

0.2

0.0
0.4

0.6

0.8

1.0

( 0)
C

Tvert / T

Figure 4.12: Condensate fraction versus temperature for various sample rotation rates.
Condensate number NC and total number N are obtained from ﬁts to cloud images.
The temperature Tvert is extracted from the vertical width of the normal cloud while
the centrifugal distortion in the normal cloud’s aspect ratio yields its rotation ΩN . The(0)temperature is scaled by the static critical temperature TC for an ideal gas (see text).
The data has been grouped according to three diﬀerent ranges of ΩN (see legend). The
solid line is the theoretical dependence expected for a static ideal gas.

As the gas rotation is continually increased during evaporative spin-up, it aﬀects not
only the critical temperature but also the collision rate γcoll ∼ N/(T ω̃ρ2 ). The eﬀectof the weakening eﬀective trap frequency ω̃ρ is indeed a signiﬁcant eﬀect. Whereas
the temperature decreases by a factor of ∼ 10 during an evaporation proceeding to arotation of Ω/ωρ = 0.95, the value of ω̃ρ2 changes by a factor of ∼ 20 such that the density
and collision rate change little. Therefore the evaporative process lacks the hallmark“runaway” characteristic of an ever-increasing collision rate. In addition, the evaporative
process operates in a weak conﬁning potential (ωho ≈ 2π8Hz) and removes atoms along
only one dimension [Section 4.2]. In spite of violating the cherished rules of successful
evaporation, the current typical evapoartion can produce highly rotating condensates(Ω/ωρ ≈ 0.95) with sizeable atom number (N ≈ 2 · 106 )! Of course the evaporation,
starting not too far above TC , takes several minutes to make a pure condensate.

147

Inferred TC / TC(o)

1.0
0.8
0.6
0.4

NBEC / N<0.15
0.15 - 0.3
>0.3

0.2
0.0

0.0

0.2

0.4

0.6

0.8

1.0

ΩΝ / ωρ

Figure 4.13: Inferred critical temperature TC , scaled by the non-rotating expectation(0)
TC for an ideal gas, as a function of normal gas rotation ΩN /ωρ . For each data point,
TC is inferred from the measured condensate fraction and temperature of a sample. The(0)
total number of atoms is used to obtain TC . The data has been grouped according
to three diﬀerent condensate fractions as shown. A set of static points, where thenormal cloud is not stirred before evaporation, are deliberately plotted at zero rotation.
Otherwise, the rotation rate ΩN is obtained from the normal cloud’s aspect ratio. The
rotating data has been cropped at a minimum threshhold 0.2 < ΩN to avoid imaginary
rotation values arising from noise in near-static aspect ratios. The dotted line is the(0)theoretical expectation according to Eqn. 4.7 with TC as for an ideal gas. The solidline is a ﬁt of the data to Eqn. 4.7 with an arbitrary overall scaling of the vertical axis.
(0)
The ﬁt result is equivalent to assuming an eﬀective TC that is 87(1)% of the ideal gas
value.

148

4.6

Vortex lattice decay
Rounding out the “life cycle” of vortex lattices, a qualitative description of theirdecay from the condensate is presented here. The decay of vortex lattices [47] andsingle vortex lines [142] have also been studied in other experimental groups. As alreadymentioned in section 3.7, a rotating condensate spins down by passing vortices out toits edge where they decay in a shower of phonons. Since the vortex cores in a lattice arerotating about the condensate center, they reach the edge by spiraling outward. The
radial damping of the vortices is driven by the frictional drag from collisions with thenormal cloud [82]. If a vortex lattice is rotating in equilibrium with the normal cloud,the drag force on the vortices vanishes. If the normal cloud is faster, it pushes vortices
into the condensate to bring it into rotational equilibrium. If slower, the normal cloudpulls vortices out of the condensate. In the macroscopic limit of large vorticity where thecondensate and normal component may be characterized by rotation rates ΩBEC andΩN respectively, the eﬀect of the normal cloud can be summed up by a mutual friction
γ(ΩN − ΩBEC ), which is proportional to the relative velocity between the components[162] (see also Section 4.3). The phenomenological constant γ includes, among othereﬀects, the temperature dependence.
The evaporative spin-up process initially forms a condensate and normal cloudrotating together (about the vertical z-axis). In order to induce a decay in the rotationof the sample, a “surface roughness” must be present in the conﬁning potential tobreak its rotational symmetry. Angular momentum is then no longer conserved in thesystem. The walls of our magnetic trap are essentially inﬁnitely smooth, except for thelowest order ( = 2) roughness corresponding to an elliptical asymmetry [Figure 4.14].
This can be signiﬁcantly reduced with some care, though residual asymmetries in our
conﬁning potential still cause relatively cold condensates to lose their rotation over theNormalcloud

Staticpotential
contours

149

BEC
Top
View

Figure 4.14: An ellipticity (“surface roughness”) applied to the conﬁning potential
breaks rotational symmetry. This leads to the rotational relaxation of the initiallyco-rotating normal cloud and condensate.

span of minutes 1 . A theoretical model of angular momentum decay in the presenceof a trap asymmetry is detailed in [162]. In a loose sense, the rotating condensate
is a frictionless superﬂuid and is not slowed by the surface roughness of the potential(see, though, section 4.8). The rotation of the normal cloud will, however, damp dueto the friction imparted by the roughness. With the condensate and normal ﬂuid nolonger in rotational equilibrium, the mutual friction will in turn damp the condensate’s
rotation. Thus, there are two relevant times scales τV and τR characterizing the transfer
of angular momentum, ﬁrst, from the condensate to normal ﬂuid and, second, from thenormal ﬂuid to the trap walls, respectively. The value of τR in general depends on theatomic collision rate in the normal gas (typically ∼1Hz for our experimental parameters)
and the strength of the static elliptical deformation= (ωx2 − ωy2 )/(ωx2 + ωy2 ) present in

the conﬁning potential. (ωx and ωy are the harmonic trap frequencies in the horizontal
directions). By controlling the strength of the ellipticity, it is possible to explore two
regimes of rotational decay corresponding to a rapid (τR  τV ) or gentle (τV  τR )stopping of the normal cloud’s rotation.
In ﬁgure 4.15(a), the ﬁrst of these situations is shown where a “strong” horizontal
ellipticity ≈ 0.02 has been applied to mostly stop the normal cloud’s rotation after ∼11
The loss of rotation can be countered in steady state by the continuous application of rf evaporation,
which tends to spin up the cloud.

150
2s. Non-destructive, in-situ images are then taken of the trapped gas from the side andﬁt to extract the simultaneous aspect ratio of the condensate and normal component.
While the value for the normal cloud lies near the static value after the stopping, the low
value for the condensate indicates that it continues to rotate initially. Eventually therotation of the condensate itself damps, presumably under the inﬂuence of the mutual
friction from the static normal cloud. In ﬁgure 4.15(b) the rather diﬀerent scenario of a
weak applied asymmetry ( ∼ 10−3 ) is shown where the rotation of the condensate andnormal component decay together at the same rate. If the trap ellipticity is suﬃciently
small, the mutual friction between the components is expected to dominate the friction
between the normal component and trap walls. In this case, the condensate and normal
ﬂuid lock together as one system. The additional contribution of the condensate’s
moment of inertia to the total will cause the rotational relaxation of the system toproceed a little slower [162]. The quantitative analysis of the degree of “locking” forour experimental conditions as well as a detailed comparison with theoretical models isleft to a future publication.
The aspect ratio analysis of ﬁgure 4.15 provides a convenient method to assess
the macroscopic rotational decay of the condensate and normal cloud. A complimentary
viewpoint can obtained by directly imaging the vortex cores in a condensate over thecourse of the decay. Several experimental runs have been made in which the rotational
decay is allowed to proceed for a variable amount of time. The usual in-situ image
is ﬁrst made from the side to obtain aspect ratio information. The sample is thenballistically expanded and imaged vertically to reveal the vortex cores in the condensate.
In ﬁgure 4.16, the rotation rate of the condensate as obtained from its aspect ratio isshown as function of time in conjunction with the corresponding expansion images ofthe vortex distribution. An automated program is used to identify the vortex cores inthe density proﬁle and to obtain their average spacing, which is also plotted as functionof time. Figure 4.16 visually uniﬁes the diﬀerent perspectives of a condensate spinning

151

1.6

oit 1.4
artc 1.2ep
s 1.0
A(a) Strong static

0.8

trap asymmetry

0

5

10

15

20

time [s]1.6

BEC
Normaloit 1.4
artc 1.2
ep
s 1.0
A

cloud

(b) Weak static

0.8

trap asymmetry

0

40

80

120

time [s]Figure 4.15: Rotational relaxation of an initially co-rotating condensate and normalcomponent in an asymmetric conﬁning potential. The simultaneously obtained aspectratio is plotted for each of the components in the case of (a) a strong asymmetry and(b) a weak asymmetry applied at t = 0s. The aspect ratio allows a rigid-body rotation
rate to be inferred through Eqn. 4.1. An aspect ratio of ∼1.6 corresponds to a static
cloud, while a lower value indicates more rotation. The initial stopping of the normalcomponent is not included in graph (a); the data only begins after ∼1-2s when thenormal cloud’s aspect ratio has already reached the static value.

152

0s

(a)

(b)

4s

Ω BEC

ωρ

(c)

0.8
0.6
0.4

8s
50

Vortex 40
spacing[arb.] 30

(d)

20

10s

0

2

4

6

8

10

time [s]Figure 4.16: Rotational relaxation of a condensate from diﬀerent perspectives. (a)Expansion images of the condensate at successively longer times after an asymmetry
has been applied to the conﬁning potential. (b) Vortex cores identiﬁed by an automatedsearch algorithm. The large circle in both cases indicates the Thomas-Fermi radius of
the condensate. (c) The rotation rate of the condensate as inferred from its aspectratio. The aspect ratio is determined from a non-destructive image of the trappedsample before ballistic expansion. (d) The average vortex spacing as calculated fromthe expansion analysis in (b).

153
down: the loss of vortex number, the decrease in their density and the decrease inrotation rate as inferred from centrifugal distortion.

4.7

Spinning up even faster: large lattices and rotation
In the rotational decay experiments of the previous section, the application ofa strong, static ellipticity to the conﬁning potential was used to stop the rotation of
the normal cloud. A closer inspection of the initial behaviour in this case revealed
that the cloud had also been excited into damped, rotational scissors-mode oscillations[95, 119]. The behaviour of these oscillations is shown in ﬁgure 4.17 for a pure normal
cloud above TC . The torque from the applied static asymmetry brings the initiallyrotating gas to a stop with the density forming an elongated shape at ∼ 45◦ to thetrap deformation [Figure 4.17(c)]. At this point, the angular momentum from therotation has been stored as potential energy due to angular displacement. The motion
then rebounds leading to the reverse rotation. This behaviour is reminiscent of the
motion of a two-dimensional pendulum with slightly diﬀerent frequencies of oscillation
along two orthogonal directions. An equal superposition of pendulum motion in thetwo dimensions leads to a slow beating between circular and linear oscillation. By
analogy, the period τs = 2π/∆ωρ of the scissors mode depends on the asymmetry in the
conﬁning potential, ∆ωρ ≈ ω̄ρ for  1. Underdamped oscillations will only occur for
an ellipticity larger than a critical value

C = 1/4ωρ τ [94] such that the time between

atomic collisions τ is longer than the time τs for a single particle trajectory to feel thepotential asymmetry.
The behaviour of the scissors mode suggested the idea of using it in the ﬁrst
stage of vortex nucleation to induce rotation in the normal gas. Before the evaporative cooling to quantum degeneracy, a step-wise sequence is used to excite the scissors
mode in the normal cloud [Figure 4.18]. The excitation occurs in the same oblate conﬁguration of the TOP potential as was used for stirring in section 4.2. The cloud is

154

ε

Top view

(a)

0.5
0.4

1s

(c)

3s

(d)

4s

(e)

0.3
0.2
40
20

Θ [deg]

0
-20

(b)

-40
0

2

4

6

8

time [s]Static potential(f)time
Figure 4.17: Rotational scissors mode of an initially rotating normal gas subjected to asudden trap asymmetry. (a) Ellipticty of the cloud and (b) orientation θ of its major
axis as a function of time. The damped sinusoidal ﬁt to (a) and sinusoidal ﬁt to (b)yield a scissors oscillation period of τs =5.7(8)s and τs =5.9(2)s, respectively. (c...e) A
sampling of the in-trap images taken along the axis of scissors rotation and used toobtain graphs (b)&(c). Times indicated correspond to the horizontal axis of (b)&(c).
(f) Sketch of one complete scissors cycle including the direction of ﬂuid rotation. The
elliptical line represents one contour of the asymmetric conﬁning potential.

155

(a)

(c)

(b)

(d)

Figure 4.18: Stepwise excitation of the rotational scissors mode to impart angular momentum to a cold gas. (a) The gas is allowed to thermalize to an asymmetric TOP
potential. (b) The orientation of the trap asymmetry is suddenly jumped by 45◦ . (c)After a quarter period of the scissors mode, the gas is maximally rotating. (d) The trap
is quickly resymmetrized to preserve the angular momentum imparted to the gas.

ﬁrst adiabatically deformed to an elliptical shape by changing the shape of the potential. An extra 2s (of the order of the rethermalization time) is allowed for the cloudto thoroughly equilibrate to the new trapping geometry. Subsequently, the orientationof the potential asymmetry is suddenly changed by 45◦ to excite the scissors mode.
The strength of the trap asymmetry is also discontinuously changed at this point to avalue

mid ≈ 0.25 independent of the initial conditions.

After a short period of time,

the potential is made symmetric and the evaporative spin-up is implemented as before.
The excitation sequence generates the largest angular momentum, as assessed by theultimate rotation of the condensate formed after evaporation, around an excitation timeof τ ≈ τs /4 = 150ms, as expected for=

mid and ω̄ρ = 2π6.8Hz

2 . In a single parti-

cle model, the angular momentum imparted from the excitation depends on the initial
shape of the cloud through ∼ i x2 + y 2 . By controlling the initial trap deformity in theﬁrst stage of the excitation sequence, we can achieve good control over the angular momentum imparted to the normal cloud and, by extension, the vortex number generated
in the condensate [Figure 4.19]. The maximum available trap ellipticity ( i ≈ 0.41) isused to obtain the most highly rotating condensates (ΩBEC /ωρ ≈ 0.95) with the largest
2

The step-wise drive excites both a rotational and irrotational scissors mode at a low and highfrequency respectively. The irrotational component is ignored here since a simulation shows that thebulk of the angular momentum imparted to the gas is due the rotational component.

156

(a)

1.5
1.3

λBEC 1.1
0.9
0.7

100
80

NV 60
40
20

(b)

0
0.0

0.1

ε

0.2

0.3

initalFigure 4.19: Dependence of (a) condensate aspect ratio (in other words rotation) and(b) vortex number on the initial ellipticity used in the scissors excitation.

157
vorticity (NV ≈ 150).
Perhaps the most signiﬁcant feature of the scissors technique is that it allows theformation of highly rotating condensates with larger atom number than the original
stirring technique. In the case of stirring, the normal cloud needs to start quite closein temperature to TC , otherwise the evaporation cannot achieve condensation. The
main issue is the amount of heating and center of mass motion (“slosh”) induced by
the 15s of stirring required to reach steady state rotation. Both processes worsen due
to resonant eﬀects as the trap is rotated at higher frequencies. In the case of thescissors technique, the rotation is induced rapidly over 150ms. Slosh is straight forward
to minimize over the step-wise excitation sequence, leaving only the heating due tospurious shape excitations generated by the sudden trap changes. Therefore, it is not
surprising that the evaporation is successful for initial clouds further from TC . In fact,
it is found beneﬁcial to both ﬁnal condensate number and rotation to start at a higher
temperature initially, presumably because of the associated larger number of atoms
present(17 · 106 versus 6 · 106 for the stirring case). A successful evaporation in terms
of ﬁnal rotation rate has always proven to be intimately tied with the performance interms of condensate number.
The ability to reproducibly make highly rotating condensates with large numbersof vortices and large atom number makes an ideal starting point for a whole range ofexperiments, one of which is considered in the following section.

4.8

Nonequilibrium dynamics of large vortex lattices [77]4.8.1

Abstract

We have studied the dynamics of large vortex lattices in a dilute-gas BoseEinstein condensate. While undisturbed lattices have a regular triangular structure,large-amplitude quadrupolar shape oscillations of the condensate are shown to induce a

158
wealth of nonequilibrium lattice dynamics. When exciting an m = -2 mode, we observe
shifting of lattice planes, changes of lattice structure, and sheet-like structures in which
individual vortices appear to have merged. Excitation of an m = +2 mode dissolves theregular lattice, leading to randomly arranged but still strictly parallel vortex lines.

4.8.2

Introduction

The experimental study of vorticity in a dilute-gas Bose-Einstein condensate(BEC) provides an useful perspective on superﬂuidity. While the focus of early work
was on the study of single or few vortices [42, 118], recent advances in technique havemade it possible to create BECs containing large amounts of vorticity [117, 48, 97, 103].
In a rapidly rotating BEC, a lattice of vortices is the lowest energy state in the rotating
frame[59]. Experimental studies of lattices to date have concentrated on formation anddecay processes [118, 117, 48, 97, 103, 138, 47]. In this section we study the nonequilibrium behavior of vortex lattices under large-amplitude, anisotropic strain.

4.8.3

Experimental technique

For creating large amounts of vorticity in condensates, we use a modiﬁed versionof the spin-up technique described in our previous paper [97]. The starting point isa magnetically trapped cloud of 17 × 106 87 Rb atoms in the |F = 1, mF = −1 statewith a temperature approximately three times above the critical temperature for the
BEC phase transition, Tc = 67 nK. By manipulating the ellipticity of the conﬁning
potential, we can resonantly pump angular momentum into the as yet uncondensed
cloud [43]. We then tune the trapping potential to near perfect axial symmetry withtrap frequencies {ωρ , ωz } = 2π{8.35, 5.45}Hz. In this round trap the cloud continues
to rotate, and by performing a nearly one-dimensional evaporation along the axis ofrotation (z-axis) as described in Ref. [97], we cool down and further spin up the cloud
until a highly rotating condensate is born out of the rotating normal cloud. For the159experiments described in this paper, the evaporation is continued to a point where little
or no thermal fraction remains. As a result of this procedure, we obtain highly rotatingcondensates with typically 1 × 106 atoms and a Thomas-Fermi radius of Rρ = 51µm,
Rz = 26µm, containing 130 vortices or more.
To study these condensates we can image along the x, y, or z directions. Whenthe condensate is held in the trap, the core size of the vortices is close to the resolution
limit of our optical detection system so that individual vortices are not resolved by intrap imaging. However, by nondestructively imaging a trapped condensate from the side(along the x-direction) and measuring its aspect ratio, the rotation rate of the condensate
can be determined [97, 138]. For our highly rotating condensates we determine typicalaspect ratios of 0.5, which is markedly diﬀerent from the static aspect ratio of themagnetic trap, 1.53, because of centrifugal forces. Typical condensate rotation rates of0.95ωρ are inferred.
To resolve individual vortices, we can release a condensate from the trap andimage it 50 ms later after it has expanded by a factor of 5 [118, 51]. Such expansion
pictures taken along the z-direction reveal a triangular lattice of vortex cores in highlyrotating condensates [Fig. 4.20(a)]. The lattice structure is remarkably regular even at
the outer regions of the condensate, and is observed with very good contrast as shownby the cross section in Fig. 4.20(b).
When looking at such vortex lattices in expansion from the side (i.e., along the x
or y direction), good contrast is only obtained if the lattice is oriented in such a way thatvortices line up behind each other along the direction of view, or in other words when
the direction of view is parallel to lattice planes. The vortex lattices are not stationary
in the lab frame, but are rotating with the condensate, and furthermore the initialorientation of the lattice in the xy plane cannot be controlled in current experiments.
Therefore the times when vortices line up along the direction of sight are unpredictable,
and images revealing lattice planes are only obtained in a random subset of all trials.

160

a)

c)

310 µmb)
0450 µmFigure 4.20: (a) Expansion picture of a vortex lattice seen along the rotation axis. (b)One pixel wide cross section along the white line in (a). (c) Expansion picture of adiﬀerent condensate rotating at 0.75ωρ , i.e. more slowly than in (a), seen from the side.

One example is shown in Fig. 4.20(c), where the vortex planes are clearly visible asdark vertical lines. These images are direct veriﬁcations of the theoretical predictionthat vortex lines are only marginally bent for our experimental parameters [89, 79], as
is also conﬁrmed by the good contrast seen in the topview picture of Fig. 4.20(a).
Vortex lattices such as the one shown in Fig. 4.20 are the starting point forour studies of the eﬀects of dynamically generated strain. We apply the distortion byresonantly driving quadrupolar shape modes in the vortex-ﬁlled condensate [61, 98].
In the large-amplitude limit, the mz = +2 and mz = −2 modes distort the circularcross-section of the condensate into an ellipse and cause the major axis of the ellipse torotate along with, or against, the sense of the condensate rotation, respectively. The
generated strain is considerable – the ratio of major to minor axis of the condensate can

161
be larger than three to one. The frequencies of the mz = ±2 modes may be calculatedusing the sum rule argument given by Refs.[161, 44]. Our condensates are rotating
typically at 0.95ωρ , just 5% under the centrifugal limit. At the centrifugal limit, themz = +2 ellipse is ﬁxed in the rotating condensate frame, and the mz = −2 shape is
ﬁxed in the lab frame. At our rotation rate of 0.95ωρ , the distortion, as seen in theframe of the lattice, rotates at only 0.4 Hz for the mz = +2 mode, compared with 8.34
Hz for the mz = −2 mode. For this reason, we expect very diﬀerent resulting latticedynamics in the presence of either of the two modes. We emphasize as well that thetwo diﬀerent modes result in very diﬀerent ﬂuid-ﬂow patterns. In the centrifugal limit,
the mz = +2 mode very closely approximates rigid-body rotation, while much of the
mz = −2 velocity ﬁeld is provided by an irrotational, quadrupole ﬂow pattern. Thisdistinction may provide a parallel explanation for the correspondingly distinct observed
eﬀects [see Section 4.9].

4.8.4

Lattice dynamics in the presence of an mz = −2 surface mode

Because the mz = −2 mode is almost stationary in the lab frame for the parameters of our experiment, this mode can be excited conveniently and nearly resonantly
by a static trap deformation. Indeed we observe that by jumping from the round trap
described above to a trap that has the same mean horizontal frequency, but a horizontal
ellipticity of 3.6% [45], the ellipticity of the condensate increases over a time scale of 300
ms from approximately 0 to 40%, thereby exceeding the trap ellipticity by more thanan order of magnitude. Since the vortex lattice is rotating quickly with respect to thisalmost static deformation, it is an interesting and nontrivial question to ask whether or
not a lattice structure is maintained, and, if so, how it rearranges in the presence of theexcitation.
Experimentally, vortex lattices can indeed be observed in expansion images up
to 400 ms after the start of the continuously applied trap deformation (Fig. 4.21).

162

a)

b)

460 µmFigure 4.21: Evolution of the lattice and condensate density proﬁle after (a) 173 ms
and (b) 873 ms in the presence of an mz = −2 excitation. Pictures taken along axis ofrotation.

Subsequently, long range order is lost and vortex visibility becomes low, presumably
due to tilting [98] or bending of the vortex lines [48]. A closer look at the ﬁrst 400 ms
of this evolution reveals a wealth of intriguing vortex dynamics. As the vortex lattice
rotates in the deformed condensate, the lattice planes must continuously shift relative toeach other to accommodate to the elliptical shape of the condensate. One consequenceof this is transient changes of the lattice structure. In Fig. 4.22 the lattice has changed
from the triangular structure of an undisturbed lattice to a near rectangular structure.
The most striking observation, however, is of condensates containing sheet-likestructures rather than individual vortex cores, as shown in Fig. 4.23(a). We interpretthese sheets as rows of vorticity along which individual vortex cores have essentiallymerged. Given the near perfect contrast of these sheets, as shown by the cross sectionin Fig. 4.23(b), we can exclude the possibility that the sheets are merely formed by acollective tilting of vortices along a lattice plane, which would result in an intermediate
contrast. Similarly, we can rule out that this is an eﬀect merely of imaging resolution- had the vortices retained their original structure but come so close together that we

163

D

E

 µP

 µP

Figure 4.22: Change of lattice structure seen along rotation axis. (a) Triangular structure in an undisturbed lattice. (b) Near rectangular structure seen transiently during
lattice evolution after 273 ms in the presence of an mz = −2 quadrupolar surface mode.

164
could not spatially resolve them, we again would see white stripes set oﬀ by grey troughs,
rather than the near 100% alternating stripes of white and black. If there continue
to exist wispy ﬁngers of condensate that cross the stripes and diﬀerentiate individual
vortices, they must be very tenuous indeed to be consistent with the observed contrast.
The clouds have been continuously distorted from their original hexagonal symmetry,so the observation of 16 stripes along the major axis of the BEC in Fig. 4.23(a) tells
us that there must also be the equivalent of 16 units of vorticity along the length of
the stripes through the center of the cloud. By measuring the in-trap height of thecondensate, the ellipticity in the xy-plane and the number of atoms, we can infer the1,
ratio of unit vorticity spacing (along the minor axis) to healing length (given by √8πna

where n is the mean density and a the scattering length) to be on the order of 5.
In fact the contrast of this sheet-like phase is so deep that the structure can bedetected by looking at condensates in trap (along the x or z direction), even though
the structure is close to our optical resolution limit. By taking a time series of ten
nondestructive images of a single condensate along the x direction, we see that the
sheet structure appears and disappears periodically with a period of about 21.2 ms fora condensate rotating at a frequency of 0.95ωρ [Fig. 4.24].
This periodicity can be understood as follows: The sheet structure forms whena lattice vector lies along a minor axis of the cloud, presumably since this is whenthe vortices are closest together. As that vector rotates past the minor axis, structure
along the vector is reestablished, the two-dimensional (2D) pattern of individual vortices
reemerges, and the high-contrast stripes disappear from the in-trap images. A lattice
vector aligns with the minor axis six times per lattice rotation period due to the sixfold
symmetry of the unperturbed lattice, consistent with the observed frequency of periodic
stripe formation. This interpretation is bolstered by the expansion images, which showhigh contrast sheets when a lattice vector lies close to the minor axis, but 2D lattices(albeit distorted) for other orientations. We ﬁnd it remarkable that this lattice-to-sheet-

165

D

F

G

H

E
 µ2

 µ2

Figure 4.23: (a) Sheet-like structure seen after 173 ms in the presence of an mz = −2
quadrupolar surface mode. (b) Cross section integrated over the white box in (a). Even
though the box is wider than the calculated vortex core spacing, the observed contrast
is nearly perfect. (c-e) Same as (a), but observed within the ﬁrst tens of ms in a more
deformed trap with {ωx,y,z } = 2π{6.0, 8.6, 13.8}Hz. All pictures taken along axis ofrotation.

166

a)

b)

c)

d)

e)

f)
0

145 µmFigure 4.24: (a-e) Nondestructive in-trap images of the sheet-like structures seen along
the x-direction [conditions similar to Fig. 4.23(a)]. Spacing between images 10.6 ms.
Note the very diﬀerent spatial scale from expansion images (e.g., Fig. 4.20). (f) Cross
section of (a), integrated over the condensate.

to-lattice sequence persists through multiple cycles before long range order and visibility
are compromised.
Using a trap with a stronger distortion of 34% as in Fig. 4.23(c-e), we also observe
sheets breaking apart in the center. In Fig. 4.23(c), the density distribution along
the lattice vector that we see near-parallel with the minor axis is nearly featureless.
Figure 4.23(d,e) hint at the succeeding lattice evolution: Fig. 4.23(d) represents a small
rotation from Fig. 4.23(c), and we see that the sheets have begun to reconnect acrossthe cloud. Figure 4.23(e) appears to be a continuation of that process, with considerable
evidence of uniformity in the repeating structure.

4.8.5

Lattice dynamics in the presence of an mz = +2 surface mode

We excite the mz = +2 mode by elliptically deforming the magnetic trap androtating that ellipse around in the xy plane. When the mz = +2 mode is driven
approximately 1 Hz away from its resonance, the ellipticity of the condensate increases
from 0 to 50% over a timescale of 250 msec. Under the inﬂuence of an on-resonant
drive, the ellipticity increases continuously, even passing beyond 90% after 700 msec

167

a)

b) 3
counts620 µm

2
1
0

0

100

200

300

c) -1

400
500
distance [µm]Figure 4.25: (a) Irregular assembly of vortex cores as observed along rotation axis after
500 ms in the presence of an mz = +2 quadrupolar surface mode. (b) Histogramof distances between each pair of vortices in the regular triangular lattice shown in
Fig. 4.20a. The visibility of separate peaks reveals the high degree of long-range order.
(c) Same as (b) but calculated for the BEC of Fig. 4.25a. Peaks are barely visible here,
meaning that the lattice has almost reached a random disorder.

and approaching 100% [46]. As discussed above, the distortion rotates only very slowly
with respect to the lattice, and thus we expect it to be correspondingly “gentle” withrespect to its action on the lattice.
Indeed, as we show in Fig. 4.25(a), we do not see the lattices squeezed into sheets
for the mz = +2 mode. Instead we observe a gradual increase of disorder in the lattice.
400 msec after we start driving the excitation on resonance, we see a complete loss oflong-range order in the vortex structure. The imaging contrast of individual vortices
remains high, so we know they remain parallel to the line of sight (in contrast for instanceto the case of Fig. 4.21), but otherwise they appear to have liquiﬁed (see the correlation
histogram of Fig. 4.25). This behavior conﬁrms our expectation of the mz = +2 mode
being a gentler disturbance for the lattice than the mz = −2 mode.

4.8.6

Conclusions

In conclusion we have succeeded in observing a wealth of nonequilibrium vortex
lattice dynamics in a BEC under anisotropic compression. Key observations include the

168+
Rotational flow
due to vortex lattice=
Irrotational flow
due tom= - 2

Net flow

Figure 4.26: Macroscopic velocity ﬁeld of a rotating condensate in the presence of acounter-propagating m = −2 surface wave. The net ﬂow induces lattice shear along the
major axis of the quadrupolar deformation.

shifting of lattice planes relative to each other, leading to transient changes of the lattice
structure, the formation of an unexpected and very pronounced sheet structure, andﬁnally, a gradual transition from an ordered lattice to a rather liquid-like arrangement inwhich vortex cores remain visible, but their relative positions are irregular. The observed
sheet structure is of particular interest: at present the behavior of densely packed vortex
lattices is in the center of theoretical discussions about vortices [101, 63, 86, 133]. Whilemost of these theoretical discussions propose to enter this regime by having high rotation
rates almost inaccessible to experiment or inconveniently low atom numbers in thecondensate, this section demonstrates an alternative and unexpected way of bringing
vortices close together. We hope that the results presented here will stimulate further
theoretical discussions in the ﬁeld of vortex lattices.

4.9

Theory responds
Following the publication of the experimental results from the previous section,

several theory groups [127, 66, 91] have investigated the origin of the dynamical distortions of the lattice structure in the presence of the counter-propagating m = −2
mode. Although using diﬀerent approaches, both Mueller and Ho [127] and Cozzini

169
and Stringari [66] reproduce the transient orthorhombic lattice structure as well as theperiodic appearance of sheet structures aligned with the short axis of the quadrupolar
deformation. Both theoretical approaches conﬁrm that the irrotational ﬂow pattern ofthe quadrupolar mode applies a shear to the lattice and thus drives the rearrangementof the vortices [Figure 4.26]. Furthermore, it is interesting to note in this context thatthe velocity ﬁeld describing the sheet-like phase of the condensate really requires thepresence of a quadrupolar ﬂow ﬁeld [Figure 4.27].
Finally, on a macroscopic level, both refs. [127] and [66] predict the time evolutionof the amplitude and orientation of the m = −2 deformation in the presence of a constant
applied trap asymmetry. The predicted behaviour is qualitatively seen in the experimentand has already been brieﬂy described in section 4.8. A direct quantitative comparison
between experiment and theory will be forthcoming in the future [67].

170Line ofvortices
Net
circulation

(a)

(b)

Figure 4.27: (a)Expanded condensate density proﬁle at the time when stripes of vorticity
form aligned with the minor axis of the m = −2 deformation. (b) Inferred microscopic
velocity ﬁeld in the region of the two central stripes of vorticity surrounded by three
regions of condensate ﬂuid. Note the circulation about each stripe of vorticity composed of a row of > 10 vortex cores. The irrotational ﬂow pattern characteristic of aquadrupolar mode is required in the regions of condensate density to knit the vortex
stripes together with a continuous velocity ﬁeld.

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173
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