Two-Component Bose-Einstein Condensationby
Michael Robin Matthews
B.S., University of Texas, 1994

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
1999

This thesis entitled:
Two-Component Bose-Einstein Condensation
written by Michael Robin Matthews
has been approved for the Department of Physics

Prof. Eric Cornell

Prof. Deborah Jin

Date

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

iii
Matthews, Michael Robin (Ph.D., Physics)
Two-Component Bose-Einstein Condensation
Thesis directed by Prof. Eric Cornell

Bose-Einstein condensates in dilute atomic gases have been around for almost 5 years now. The ﬁrst experiments dealt with density related properties of
the condensate – things like internal energy, collective excitations, and density proﬁles. All of these aspects are well described by a non-linear Schrodinger equation,in which the non-linearity arises from repulsive potentials between 87 Rb atoms
that make up the condensate. While interesting in their own right, these experiments had little to do with the “strangeness” of a condensate as a macroscopic
quantum object.
In this work the new era of condensate physics is described. These are experiments mainly concerned with the quantum phase and coherence properties. We
have constructed a robust system for creating internal-state superpositions, withthe ability to image the interference between them. Our ﬁrst observations show
how condensates, initially created with a well-deﬁned phase, remain coherent, unlike a sample of thermal atoms. We have devised schemes to spatially manipulate
the relative phase between two condensates in unique ways which brings about
very unexpected behavior. By choosing a very speciﬁc scheme we were able tocreate a vortex state and measure the unique 2π phase winding, making this avery simple system for utilizing quantum control.

Dedication

I dedicate this work to my wife Melanie, and my parents Ann and Robin.
Melanie has really given up a lot so that I could follow physics. To her it meant
living apart for many years, and always having to cut short our time together. She
has been incredibly patient and supportive, and understanding when sometimes
physics came ﬁrst. I hope I can make it up to her.
Sometimes people ask my parents where I got my talent at physics. For somereason people assume that it could not have been from a kindergarten teacher anda highschool geography teacher. Well, it was. From as early as I can remember,they motivated in me an interest in everything. They indulged me when ever they
could aﬀord to – things from archery to cadaver frogs. One of the ﬁrst times I
remember thinking about the world was when I asked my dad why the Battlestar
Gallactica ships seemed to have so many parts that weren’t aerodynamic. He
answered “That’s smart thinking, but remember, there is no air is space”. To
this I replied, “Then why do all the movies have ships that look so much like
airplanes?”. Eventually my scientiﬁc knowledge outgrew theirs, but their loveand support continued. Both my sister and I owe them everything for our talentand conﬁdence to pursue the most diﬃcult paths.

Acknowledgements

Given the stereotype of physicists as introverted and antisocial, it seems
amazing that I have not had to work with a single one. The Wieman, Cornell andJin groups have known some truly dedicated, bright, and fun people.
First, I need to acknowledge Eric Cornell for the opportunity and guidance.
He knew when to lend advice, and when to let me go oﬀ on my own. His graspof physics inspired me to understand things from new perspectives. It’s been aprivilege to work with him. Carl Wieman is another great physicist who led me
through my graduate career. His technical intuition always motivated me to seekout new methods and ﬁnd simple but useful techniques. Even though their statusis among the elite, and the Nobel Prize may well come to JILA, they always
treated the grad students as respectable, intelligent scientists in their own right.
It’s part of what made JILA a stimulating environment.
There have been other great scientists who shared equally in this work. Mike
Anderson who pushed me into the experiment head ﬁrst, and was responsible formost of the work leading to the original BEC. Jason Ensher was a co-grad student
for most of my time here. He and I worked very well together with complimentarystyles. The second post-doc I had the privilege of working with was Deborah
Jin. Coming from a condensed matter background, she had little atomic physics
training, and yet always seems to stay a step ahead of everything. She, Jason andI made a great team, and did some of the best early condensate studies in theviworld. After feeling a personal responsibility for the success of our experiment,
David Hall came to our group as a post-doc. He was very well trained in the art ofprecision measurement, which led to numerous arguments between us concerning
nearly every detail of the apparatus. Because of this, the experiment was taken
to another level. We ferreted out some of the last bugs and made the experimentextremely robust, making the best compromises between making things work andmaking things right. It was a great experience working with David (except when
he suggested to Carl that we ﬁnd some cots for the lab so we could sleep there
too). Paul Haljan recently entered the group, and has contributed a lot to thiswork. He and the newest post-doc Brian Anderson have been instrumental in the
recent vortex studies. The experiment is in good hands.
On another front, much of this work was based on and/or motivated by thework of theorists at JILA. Jinx Cooper, Chris Greene, Murray Holland, and John
“Weird” Bohn have given a lot of support and ideas. Jim Burke, Brett Esry, andJamie Williams have devoted much of their eﬀorts to understanding and providing
new directions for our experiment.
I would like to acknowledge the older post-docs and grad-students who not
only made JILA a great place to work, but taught me a lot of physics: post-docs
Eric Abraham, Eric Burt, Marla Dowell, Tim Dinneen, Zheng-Tian Lu, Nate
Newbury, and Mike Renn, and grad students Steve Bennett, Kristan Corwin,
Elizabeth Donley, Chris Myatt, Brad Paul, Jake Roberts, Kurt Vogel, and Chris
Wood. No one could ask for a better group of people to work with over the years.
The Wieman/Cornell and now Jin groups are blessed with a new generation of post-docs and grad student. Post-docs Simon Cornish and Simon
Kuppens, along with graduate students Neil Claussen, Brian DeMarco, Heather
Lewandowski, Kurt Miller, and Dirk Mueller have been very helpful.
This work relied heavily on the expertise of the instrument and electonics

vii
shop people: David Alchenberger, Hans Green, Seth Wieman, and Hans Rohner,from the instruments shop and Paul Beckingham, Terry Brown, James Fung-AFat, and Michael Whitmore from the electronics shop .
Fran Haas, Pam Leland, and Karen Melcher helped make JILA a fun place
to be, not to mention the daunting task of keeping Eric and Carl organized andon time, which made my work that much easier.

viiiContents

Chapter
1 Introduction

1

1.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Who cares? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 Finite Temperature excitations [1]11

3 Two-Photon transition

22

3.1

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

22

3.2

Transition for a free atom . . . . . . . . . . . . . . . . . . . . . .

22

3.3

Transition for a trapped condensate . . . . . . . . . . . . . . . . .

31

3.3.1

Quantum mechanical derivation . . . . . . . . . . . . . . .

32

3.3.2

Classical derivation . . . . . . . . . . . . . . . . . . . . . .

33

3.3.3

Observable eﬀects . . . . . . . . . . . . . . . . . . . . . . .

37

Rabi frequency gradient across the condensate . . . . . . . . . . .

43

3.4.1

Trap oﬀset . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.4.2

Intermediate-state detuning . . . . . . . . . . . . . . . . .

44

3.4.3

Drive polarization . . . . . . . . . . . . . . . . . . . . . . .

45

3.5

Density shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.6

Tilted TOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.4

ix
4 Imaging51

4.1

Near resonance - absorption imaging . . . . . . . . . . . . . . . .

51

4.1.1

Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.1.2

Signal-to-noise . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.1.3

Magniﬁcation measurement . . . . . . . . . . . . . . . . .

58

4.1.4

Double condensate imaging . . . . . . . . . . . . . . . . .

60

Polarization Imaging . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.2.1

Signal-to-noise . . . . . . . . . . . . . . . . . . . . . . . . .

69

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

71

4.3.1

Non-destructive imaging techniques . . . . . . . . . . . . .

74

4.3.2

Signal to noise . . . . . . . . . . . . . . . . . . . . . . . . .

75

Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.2

4.3

4.4

5 Double Condensate experiments – density related [2]

81

5.1

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

81

5.2

Dynamical response-instantaneous change in scattering length . .

83

5.3

Mixtures of condensates . . . . . . . . . . . . . . . . . . . . . . .

87

6 Double condensate experiments – phase related [3]98

7 Unique phase patterns [4]

106

7.1

Twisting the phase . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2

Dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8 Vortex creation [5]122xBibliography

134

Appendix
A Phase-contrast imaging

145

xi

FiguresFigure2.1

Experimental quantities . . . . . . . . . . . . . . . . . . . . . . .

14

2.2

Finite temperature oscillations . . . . . . . . . . . . . . . . . . . .

15

2.3

Excitation spectrum . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4

Excitation model . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.5

Nonlinear response . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.6

Frequency response and phase shift . . . . . . . . . . . . . . . . .

21

3.1

Energy structure of 87 Rb and the two photon drive . . . . . . . .

24

3.2

Energy level splittings of 87 Rb. . . . . . . . . . . . . . . . . . .

26

3.3

Two-photon drive setup . . . . . . . . . . . . . . . . . . . . . . .

27

3.4

Two-photon timing example . . . . . . . . . . . . . . . . . . . . .

30

3.5

Two-photon resonance shape . . . . . . . . . . . . . . . . . . . . .

30

3.6

Ramsey oscillations . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.7

Eﬀect of the rotating ﬁeld . . . . . . . . . . . . . . . . . . . . . .

35

3.8

TOP trap ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.9

Separation between |1, −1 and |2, 1 . . . . . . . . . . . . . . . .

39

3.10 Two-photon transition frequency II . . . . . . . . . . . . . . . . .

41

3.11 Slope of the two-photon transition . . . . . . . . . . . . . . . . . .

42

3.12 Eﬀect of two-photon polarization . . . . . . . . . . . . . . . . . .

47

xii
4.1

Optical depth correction . . . . . . . . . . . . . . . . . . . . . . .

53

4.2

Lensing due to a condensate . . . . . . . . . . . . . . . . . . . . .

56

4.3

Optimal signal to noise . . . . . . . . . . . . . . . . . . . . . . . .

59

4.4

Magniﬁcation measurement setup . . . . . . . . . . . . . . . . . .

61

4.5

Destructive double condensate image . . . . . . . . . . . . . . . .

63

4.6

Clebsh-Gordon coeﬃcients for 87 Rb . . . . . . . . . . . . . . . . .

64

4.7

Polarization imaging setup . . . . . . . . . . . . . . . . . . . . . .

67

4.8

Polarization signal and data . . . . . . . . . . . . . . . . . . . . .

68

4.9

Polarization signal to noise . . . . . . . . . . . . . . . . . . . . . .

70

4.10 Phase-contrast imaging setup . . . . . . . . . . . . . . . . . . . .

73

4.11 Discrete, non-destructive images . . . . . . . . . . . . . . . . . . .

76

4.12 Continuous, non-destructive images . . . . . . . . . . . . . . . . .

77

4.13 Phase contrast signal to noise . . . . . . . . . . . . . . . . . . . .

79

5.1

Oscillations caused by change in a . . . . . . . . . . . . . . . . . .

86

5.2

RF systematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.3

Density distributions . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.4

Radial separation of the double condensate . . . . . . . . . . . . .

92

5.5

Time-evolution of densities . . . . . . . . . . . . . . . . . . . . . .

93

5.6

Center-of-mass motion . . . . . . . . . . . . . . . . . . . . . . . .

95

5.7

Cross section for diﬀerent mixtures . . . . . . . . . . . . . . . . .

97

6.1

Condensate interferometer schematic . . . . . . . . . . . . . . . . 101

6.2

Density proﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3

Interference as a function of time . . . . . . . . . . . . . . . . . . 105

7.1

Continuous Rabi oscillation . . . . . . . . . . . . . . . . . . . . . 109

7.2

Schematic of the condensate phase . . . . . . . . . . . . . . . . . 111

xiii
7.3

“Nailed down” theory . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.4

Model and actual experiment . . . . . . . . . . . . . . . . . . . . 113

7.5

Density distribution for phase winding . . . . . . . . . . . . . . . 115

7.6

Theory and experiment comparison . . . . . . . . . . . . . . . . . 116

7.7

Recurrence of Rabi oscillations vs two-photon detuning . . . . . . 117

7.8

Fraction in the |2, 1 state . . . . . . . . . . . . . . . . . . . . . . 120

7.9

Density proﬁle of dressed condensates . . . . . . . . . . . . . . . . 121

8.1

Vortex creation schematic . . . . . . . . . . . . . . . . . . . . . . 125

8.2

Build-up of |1, −1 atoms from vortex drive . . . . . . . . . . . . 128

8.3

Vortex density and phase images . . . . . . . . . . . . . . . . . . 130

8.4

Ball-and-shell density and phase image . . . . . . . . . . . . . . . 131

8.5

Vortex evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

TablesTable
3.1

Useful fundamental constants . . . . . . . . . . . . . . . . . . . .

25

4.1

Far oﬀ resonance CB coeﬃcients . . . . . . . . . . . . . . . . . . .

65

Chapter 1
Introduction

1.1

Overview
Bose-Condensation is formed when a macroscopic population of bosons oc-

cupy the same quantum state at non-zero temperature. Our experiment can bethought of as a series of steps that make all degrees of freedom for a large number
Rubidium atoms exactly the same. Luckily, most of the steps are simply selectingout atoms which are already in the same state. The methods we go through follow
a progression from higher to lower energy scales, becoming more diﬃcult as theenergy lowers. This of course begins by selecting Rb to put in our vacuum chamber (this part is simple, because our living at room temperature means we ﬁnd
Rb is already non-degenerate with other elements – only chemically combined).
Next we select out the isotope of 87 Rb from 85 Rb by tuning our magneto-opticaltrapping (MOT) lasers to the proper wavelength so that only 87 Rb is captured in
our trap. The only remaining degrees of freedom are the internal energy levelsof 87 Rb (the hyperﬁne and Zeeman levels) and the external, motional degree of
freedom (the harmonic oscillator energy level). With optical pumping techniques
we can force most of the atoms into the same internal level. Those that we don’t
get are eﬀectively removed from the sample by our magnetic trap’s selectivity –only speciﬁc internal levels are conﬁned. The last step is the most diﬃcult. At
this point in the experiment the atoms are conﬁned in a harmonic, magnetic trap,

2
but their energy is much larger than the spacing of the oscillator levels, so theprobability of even a single atom existing in a given state is small. Forced evaporation [61] is used to cool the sample until nearly all atoms exist in the sameground state; the zero-point energy of our trap.
In principle BEC can occur in states other than the ground state, but thelarge state space available to particles makes arbitrary-state BEC creation unlikely. The ground state posses this one advantage; it is the end of the road fora system being cooled. Atoms cooled in a harmonic trap must eventually run
into the ground state, piling up as they run out of energy levels to occupy. Of
course, BEC is much more than a simple state-space argument. The symmetryof bosons brings about a stimulated process whereby atoms have a preference toscatter into a state already occupied. There are, however, so many states available for an atom to occupy when at higher temperatures, that the enhancement
is generally immaterial. The number of accessible states can be limited by colder
temperatures (i.e. narrowing of the distribution and truncation of the state spaceby the zero point). Only when the atomic wavefunctions overlap and when there
is signiﬁcant probability to exist in the same state does the Bosonic symmetry become important. This brings about the temperature and density requirements forcondensation. The small enhancement factor also grows as the number of atoms
in the ground state becomes larger. The wavefunction for N atoms in the groundstate is the product wavefunction
ψBEC = ψ0 (1)ψ0 (2)...ψ0 (N).

(1.1)

Eq. 1.1 based on a harmonic oscillator potential is not quite a complete
description of the multiparticle ground state. Two colliding atoms experience repulsive (in the case of 87 Rb ) potentials which can be well modeled by a hard
sphere interaction. The size of this sphere is given by the s-wave scattering length3a, and leads to a nonlinear term in the wave equation ψBEC for a condensate,
proportional to aN|ψBEC |2 . For 87 Rb the positive scattering length imposes a resistance to compression on the condensate. Condensates with many atoms (likecurrent experiments with ∼ 106 ) are many times larger in extent, and correspondingly less dense than if a = 0. Most of the original BEC experiments over theﬁrst two years after 1995 centered on validating the non-linear wave equation,and studying the added richness of the system from the standpoint of the density
distribution |ψ|2.
Another interesting trait of BEC is its phase. A phase factor may be written
in front of Eq. 1.1, but of course, it takes another condensate in order for thephase to be meaningful. Double condensates were ﬁrst produced by Myatt et al.
[108] through poor optical pumping which led to creation of condensates in the
two long-lived trappable states of 87 Rb (the |2, 2 and |1, −1 ). These two states
are not ideal for most double condensate studies since they feel a factor of two
diﬀerence in trapping potential, and sit in quite distinct positions in the trap due
to gravity. 87 Rb has another trapped state, the |2, 1 , which is conﬁned with thesame strength as the |1, −1 . Thus started our studies on double condensates. It
is not trivial to cool these two states simultaneously into condensation since the|2, 1 state has a relatively large spin-exchange loss rate. This means it is mucheasier to create the |2, 1 state from an existing |1, −1 condensate, which has theadded advantage of starting the system with a well deﬁned relative phase between
the two condensates.
This is accomplished through a two photon-transition. The energy splitting
between the |1, −1 and |2, 1 is second order in magnetic ﬁeld, centered about the“magic” ﬁeld value of 3.24 G. We try to operate as close to this ﬁeld as possible
so that the eﬀect of noise from various sources is small. It turns out there is aninteresting (and useful) side eﬀect of the TOP trap [114] (our implementation of

4a harmonic, magnetic trap) on the two-photon transition. The time-dependent
rotating ﬁeld of the TOP trap is equivalent to a ﬁctitious, static magnetic ﬁeld
which acts oppositely on the two states. This translates into linear oﬀsets in therelative position of the states in the trap, and eﬀects the two-photon transitionenergy. As it turns out, we can use this eﬀect to cancel another oﬀset imposedby the diﬀering magnetic moments (from the higher-order B dependence). This
allows for unique control over the separation of the condensates in the TOP trap.
The addition of the new condensate state requires modiﬁcation of the wavefunction describing each condensate. An extra nonlinear term in each state’s wave
equation represents the pressure of one condensate on the other. Interestingly, all
the scattering lengths involved in the double condensate system are very similar
in magnitude. This provides for subtle behavior: each component in a mixture of
the two states is able to rearrange its density distribution without much energy
cost, as long as it does not aﬀect the overall density distribution.
When the two-photon drive is on, there can be coherent transfer of population between the two states. This can make the dynamics very complicatedgiven the nonlinear interaction between the states. However, when the time scale
to make coherent transfer is short compared to the trap oscillation period andthe time needed for the interactions kick in, the system can be modeled simply.
The atoms can then be treated as discrete, non-interacting two-level systems, andthe equations become the two-level Bloch equations applied locally to each partof the condensate. We can then think in terms of familiar concepts to atomic
physicists: things like the Bloch sphere, Rabi oscillations, and π-pulses. As theﬁrst demonstration of these concepts we placed a condensate in a superpositionof the two spin states, and at a later time, read out the relative phase, much likea standard atomic or optical interferometer.
There is of course more to a condensate than just single atom spectroscopy.

5
In fact, we have an interesting new piece of physics here that is not so accessible
in atomic physics. This is a wavefunction which is macroscopic in size, so that
we can aﬀect the relative phase and population in a spatially selective manner. It
becomes possible to think of multiply connected phase structures such as vortices.
Interestingly these experiments can also occur quickly compared to trap oscillations or density-changing behaviour, which means we can apply our two-level
model locally to create phase and population structures. Experimentally we canadjust the local detuning and Rabi frequency of the two-photon drive to control
the evolution of each discrete point. Of course our ability to implement arbitrary
control is ﬁnite, and we have only succeeded in creating large scale (order of thecondensate) structures.
The ﬁrst of these was a simple linear phase gradient across the condensate.
When the two-photon drive is instantly turned on the condensate oscillates between the two spin states. A continuous spatial variation in the detuning and/or
Rabi frequency of the coupling drive means the local phase accumulates at diﬀerent rates for diﬀerent positions.
Experimentally what we see corresponds exactly to our local view of discrete
two-level systems (at least for short times, before mean-ﬁeld has much eﬀect). For
a vertical gradient in the Rabi frequency, the population oscillations eventually
get out of phase between the top and bottom of the condensate. As more timegoes by, other parts of the condensate get further out of phase and the condensates
appear to be striped, alternating between population of |1, −1 and |2, 1 along thevertical axis. Some of these structures actually have quite good overlap with theharmonic oscillator states, which makes this the ﬁrst quantum-state engineering
done with a condensate. For longer times with the two-photon drive on, the phasewinds up tighter and tighter.
There turns out to be a subtle condensed matter analogy in this experiment.

6
If we don’t ignore the mean-ﬁeld and potential terms in the wave equation (orwe run the experiment for longer time), then we ﬁnd a peculiar behavior. The
phase which has wound up begins to unwind, even with the two-photon drive stillpresent. In fact it unwinds back to its initial condition of all atoms in the |1, −1state. An analogy may be drawn to spin-echo [74], except that this recurrence
occurs in an inhomogeneous system without the application of additional uniquepulses to reverse the spin evolution. In this case there is an energy cost associatedwith the spatial change in phase, and it is the hydrodynamics which drive theatoms to change their superposition of the two spin states. The two-dimensional
vector that deﬁnes the geometric phase actually has access to a higher dimension –the relative phase between the two states (or equivalently, the relative populationsof the dressed states). The fact that it is able to use its extra spin degree of freedom
to minimize the energy is a unique feature of our two-component condensate.
One might ask why the phase, instead of unwinding, doesn’t just break atsome point in order to remove windings. The answer is again the energy cost.
Since the condensate is not simply the sum of the parts from our local model,
there is a large amount of energy required to support a large gradient in phase. If
the phase were to become discontinuous at a point, then density must go to zero atthat point. This is energetically unfavorable since the pressure from the repulsive
interactions opposes that. There is a kind of threshold predicted in this qualitativemodel. If the winding of the phase is slow enough, then the condensate shouldhave enough energy to continuously remove the phase gradient. We have seen thiskind of behavior; for very small gradients in Rabi frequency the condensate neverdevelops stripes, even though it should if a simple scaling applied from the caseswith larger gradients. This has only been qualitatively explored.
Given this kind of interpretation of our technique, Williams and Holland [6]proposed another scheme. Their motivation was simple – if a linear phase gradient7can be applied in the vertical direction, is it possible to apply the gradient aroundthe condensate, in such a way to create a vortex state? The lowest angularmomentum state is one with a single quanta of rotation, in which the phase winds
by 2π around condensates center. A simple angular dependent phase describes
the new wavefunction:
Φ(r) = eiφ Φ(r).

(1.2)

One intriguing aspect of superﬂuid vortices is their metastability. This comesfrom the same energy argument discussed earlier. Since the phase in a vortex
state winds continuously by 2π around the core, it can only unwind by creatinga discontinuity – “breaking” the phase. This discontinuity must force the density
to zero at that point, creating a large slope in the wavefunction, equivalent to alarge kinetic energy. If this energy is not available, the state is metastable.
Numerically solving the nonlinear wave equation, and including an arbitrary
mechanism for oﬀsetting the conﬁning potentials for the two spin states (one way
to make the phase gradient), Williams and Holland found that such a state could
be generated. They proposed applying the two-photon drive to the initial |1, −1state, while applying a rotating oﬀset. The oﬀset instantaneously produces a
gradient in detuning across the cloud, but it is rotated about the cloud’s centerat a rate comparable to the Rabi frequency. The result is a vortex state in |2, 1
with the core region occupied by the |1, −1 state.
This method gets around one of the concepts stated earlier in a unique way.
The problem with creating a vortex in an existing, single-state condensate is thatthe uniform phase of the condensate must be broken, then twisted, then put back
together in order to achieve the 2π winding. We found earlier that a condensate
resists this breaking (or even tightly twisting) of the phase since it requires alarge amount of energy. Using a two-state condensate, we write on the phase8
as the population is transferred into the |2, 1 state. Thus we are not bound by
the condensates stiﬀness since there is not yet a condensate in the target state.
There is yet another attractive feature of this technique. Since the central regionof the condensate is in the |1, −1 state when the vortex is created in the |2, 1
state, the states partially overlap. The presence of the |1, −1 enables us to use
standard atomic interferometry to read out the relative phase between the twostates. What we ﬁnd is exactly the 2π winding in the phase of the ring shaped|2, 1 state.
1.2

Who cares?
These new experiments dealing with and manipulating the condensate’s

phase, and related work going on in other groups, will be the main focus of BEC
physics for many years. So why is this interesting and who cares? (“who” refers
to Joe Scientist, not Joe Six-pack). Before BEC was observed in a dilute atomic
system, every well-rounded physicist certainly had some interest in the creationof an ideal Bose-Einstein condensate. Not because the theory was thought lacking, but because it was a regime in phase-space which had never been accessible
experimentally. Of course physicists in condensed matter and liquid helium havelong been able to produce super-conductors and ﬂuids. Unfortunately these arefar from an ideal realization of BEC, and required heroic eﬀorts in probing thesystem on the microscopic level. The thing that draws atomic physicists to theidea of a dilute system is twofold – the ability to have nearly 100% of the particles
in the condensate, and the ability to do atomic spectroscopy on them.
The ﬁrst reason is obvious. Nearly any measurement in atomic physics musttake into account broadening from some source, almost always including thermal.
With a million or so atoms in the exact same state, the prospects for precision
measurements are phenomenal (well, almost... see below), since the broadening

9is erased, and the signal to noise that much better. Schooled in the atomic clock
techniques of magnetic resonance and precision spectroscopy, I ﬁnd this a strong
motivation for creating long lived, unique superpositions and entangled states withthe ability to test fundamental physics. Most physicists surely ﬁnd it extremely
appealing to measure quantities with far reaching implications such as parity nonconservation, or the electric dipole moment of the electron, on the top of a small
table, in a small room, with only a handful of money and scientists. It is somehow
more elegant, or at least, complimentary to high energy experiments.
Unfortunately, some of the experiments in this thesis show precision measurement with condensates still has some hurdles to clear. In the most straightforward form, a precision measurement utilizes the relative phase accumulation
between two states which exist in superposition. This phase is easily read using
many well developed techniques. It is the amount of relative phase gained by
the states while in superposition that is proportional to some energy diﬀerence
between the states. The real trick then, is to arrange the superposition so thatthe energy involves the quantity to be measured, and aﬀects the two states diﬀerently. Unfortunately, the interactions can produce very unpleasant results. Whenwe create the two-state superposition, the interactions drive some very complex
spatial rearrangement of the states. This is unfortunate not only because the state
separation makes a relative phase measurement diﬃcult, but also since there isanother source of energy which aﬀects the two states diﬀerently. There are somepossible solutions to this, like optically (or magnetically if the energy of the states
can be insensitive to ﬁelds) turning oﬀ the interactions, or by going to a beam type
experiment. The later is one very real application for coherently removing atoms
from the condensate (the so-called “atom-laser” [97]). Things like continuous BEC
production will be a critical step in this and many other applications.
There is another area I have learned to appreciate and take an interest in,10mostly because of Eric’s motivation. These are the experiments to make connections with condensed matter physics. Coming into this mostly ignorant ofsuperﬂuid and superconductor physics, partly because of their theoretical complexity, my real motivation was exploring the quantum nature of BEC, similar toold atomic physics work. And some of our original work with collective modes andother “ﬂuid dynamics” type behavior I found somewhat dry and unenlightening.
However there are some great experiments on liquid helium that we have beenable to draw strong analogies to, which we can also talk about in atomic physics
terms. Plus we have the luxury of superior imaging and probing to get at theimportant quantities like phase and state distribution. I think atomic physicists
will be able to add a new perspective on many helium experiments in the near
future.
This thesis focuses mainly on double condensate experiments, including
mean-ﬁeld interactions and relative phase measurements. Chapter 2 begins withﬁnite temperature excitations, the only remnant of the ﬁrst BEC apparatus notcovered by Ensher [61]. Chapters 3 and 4 focus on two-photon and imaging
technology respectively, with emphasis on double condensate experiments. The
remaining chapters cover mean-ﬁeld interactions (Chapter 5), the measurementof relative phase (Chapter 6), creation and evolution due to a gradient in phaseacross the condensate (Chapter 7), and creation of a vortex state (Chapter 8).

Chapter 2
Finite Temperature excitations [1]Bose-Einstein condensation in a dilute atomic vapor [36, 54, 43] has been
the subject of several experimental studies characterizing this weakly-interacting
quantum ﬂuid. In particular, measurements of interparticle interactions and the
low-lying collective excitations of the condensate [91, 92, 79, 107] show excellent
agreement with theoretical predictions based on a mean-ﬁeld description of thecondensate. This ability to make quantitative comparison between experimentand theory is one of the primary advantages of BEC in a dilute atomic gas.
Previous excitation measurements were performed in a regime for which there isno detectable non-condensate fraction. This chapter expands the study of lowlying collective excitations of condensates to include higher temperatures, a regime
where theoretical predictions do not yet exist. At these temperatures the interplay
between condensate and non-condensate components, a potentially dissipative
process not included in the usual mean-ﬁeld theoretical treatment [81, 128, 59, 112,
51, 62, 66, 127, 135, 104, 67, 85, 109], strongly aﬀects the physics. We ﬁnd large
energy shifts and unexpected structure in the condensate excitation spectrum,
as well as a damping rate which plummets with decreasing temperature. These
dramatic temperature eﬀects have been met with interest from the theoreticalcommunity [65, 82, 103, 115, 122, 125, 126].
The apparatus and techniques used for creating a Bose-condensed sample in12a dilute atomic gas of 87 Rb [36, 114, 113], accurately determining its temperature[60], and probing the spectrum of low-lying collective excitations [91] are described
elsewhere. We ﬁrst optically trap and cool the atoms, then load them into a purely
magnetic TOP (Time-averaged Orbiting Potential) trap [114]. This trap provides
a cylindrically symmetric, harmonic conﬁnement potential. Cooling the atoms in
the TOP trap proceeds by forced evaporation, whereby an applied weak radiofrequency (rf) magnetic ﬁeld induces Zeeman transitions to untrapped spin states[77]. The ﬁnal temperature is controlled by the ﬁnal frequency of the rf ﬁeld. The
atom cloud is observed by absorption imaging, in which the shadow of the atomic
cloud is imaged onto a charge-coupled device array. To circumvent limitations of
optical imaging resolution we allow the atom cloud to expand freely for 10 ms
before imaging [36, 91].
The condensates are produced in a trap with radial frequency νr =129 Hz(365 Hz axial), with the same evaporation parameters as used in our recent quantitative study of the BEC phase transition [60]. We thus perform ﬁnite-temperaturestudies of condensate excitations in a well-characterized system, complementing
our previous low-temperature study of collective excitations which examined thedependence on relative interaction strength. We report our results as a functionof reduced temperature T  ≡ T /To , where To is the predicted BEC transition
temperature for a harmonically conﬁned ideal gas [7]. The estimated systematic
uncertainty in T  is 5% for T  > 0.6, and 10% for T  < 0.6. At T  < 0.6 thenon-condensate component becomes unobservably small and the temperature isinferred from the ﬁnal frequency of the rf ﬁeld. The lowest attainable T  is limitedby the evaporation parameters and the extended size of the condensate; deeper
evaporative cuts quickly reduce the number of atoms in the condensate. Whileplotting our results as a function of T  , we emphasize that the evaporative coolingprocess changes the total number of atoms as well as the temperature. Fig. 1

13shows various quantities relevant for our particular evaporation parameters.
The basic spectroscopic approach for studying collective excitations follows
Ref. [91]. First, a small applied sinusoidal, time-dependent perturbation to thetransverse trap potential distorts the cloud. We then turn oﬀ the perturbation andthe cloud freely oscillates in the unperturbed trap. Finally, the cloud is suddenly
allowed to expand and the resulting cloud shape imaged. The symmetry of thedrive perturbation can be varied in order to match the symmetry of a particularcondensate mode. The modes are labeled by their angular momentum projection
m on the trap axis. In this work, we examine the previously observed m = 0 and
m = 2 modes [91, 107]. Frequency, amplitude, and damping rate of the excitations
are determined as shown in Fig. 2.1 . The main results of this work are shownin Fig. 2.3. For these data, the perturbative drive pulse duration was typically
14 ms, with the center frequency set to match the frequency of the excitation
being studied. The response amplitudes (deﬁned in caption for Fig. 2.2) for
these data were kept small, between 5 and 15%. The radial trap frequency νr iscalibrated by driving a m = 1, or “sloshing”, excitation of a cloud at T  = 1.3.
Interatomic interactions should not aﬀect this excitation which consists solely of
rigid-body center-of-mass translation. We have also checked that anharmonicities
in the conﬁning potential are negligible in our measurements.
The normalized response frequencies as a function of T  are shown in Fig.
2.3a. Collective excitations of a non-interacting cloud should occur at twice thetrap frequency (dashed line). Indeed, above the BEC phase transition, the m = 0
and m = 2 excitations are basically degenerate at ν/νr ≈ 2. The non-condensate
frequencies are consistently about 1% higher than 2νr , presumably because interactions are not completely negligible even at the relatively low densities of theseclouds [8]. The non-condensate excitation frequencies do not vary signiﬁcantlywith temperature, even for T  < 1.

14

Figure 2.1: We plot temperature T (a) and total number of atoms N (b) as
a function of normalized temperature T  . The relationship between N and T is
not fundamental but rather a consequence of our evaporative trajectory. In (c) weplot the number of condensate atoms NBEC and in (d) the inferred non-condensate
number Nnc . Because of their diﬀerent spatial extents, the peak density of thecondensate component is an order of magnitude larger than that of the noncondensate component below T  ≈ 0.9.

15

Figure 2.2: In this example, we plot the observed widths, obtained from 2Gaussian surface ﬁts to absorption images, of a T  = 0.79 cloud with an m = 0
excitation. The widths of each component, condensate and non-condensate,of the freely oscillating cloud are ﬁt by an exponentially damped sine wave:
Aexp(−γt)sin(2πνt + φ) + B, from which we obtain the response frequency ν,
initial fractional amplitude A/B, and decay rate γ. For each pair of points (condensate and non-condensate widths) a fresh cloud of atoms is cooled, excited, andallowed to evolve a time t before a single destructive measurement.

16
In contrast, the excitation spectrum of the condensate exhibits strong temperature dependence. While some frequency shift might be expected because of
the temperature dependence of the number of condensate atoms (see Fig. 2.4),
and therefore the relative interaction strength, the magnitude of the observed
temperature-dependent frequency shifts is several times larger than that observed
due to interaction eﬀects [91]. In addition, the frequencies of these two modes,
which are degenerate in the limit of zero interactions, actually shift in opposite
directions, with only the m = 0 heading to the non-interacting limit. A sharpchange is apparent in the slope of the temperature dependence of the m = 0 excitation frequency. Repeated measurements conﬁrm that this distinct feature atT  ≈ 0.62 reproduces. We speculate this might arise from coupling with anothermode, perhaps with a strongly temperature-dependent second-sound excitation
or with the excitation of the non-condensate component. No corresponding distinct feature is evident in the temperature dependence of the m = 2 condensate
excitation frequency.
Fig. 2.3b presents the decay rate γ as a function of temperature. While
the frequencies of the two condensate modes behave very diﬀerently, their decay
rates appear to fall on a single curve, with γ quickly decreasing with decreasing
T  [9]. Another remarkable feature is that for temperatures where two-component
clouds are observed, the condensate excitations damp much faster than their noncondensate counterparts (see also Fig. 2.2). The strong temperature dependence
suggests that ﬁnite decay times reported in Refs. [91] and [107] were due to theﬁnite temperature of the samples and that condensate excitations may persist forvery long times at lower T  .
In the limit of low amplitude response, the measured spectrum corresponds
to the elementary excitations of BEC in a dilute gas. At our lowest temperature,T  ≈ 0.48, we examine the condensate excitations as a function of increasing

17

Figure 2.3: Temperature-dependent excitation spectrum: (a) Frequencies (normalized by the radial trap frequency) for m = 0 (triangles) and m = 2 (circles)
collective excitation symmetries are shown as a function of normalized temperature T  . Oscillations of both the condensate (solid symbols) and non-condensate(open symbols) clouds are observed. Short lines extending from the left side of
the plot mark the mean-ﬁeld theoretical predictions in the T = 0 limit (for 6000
atoms in our trap) [8]. (b) For both the m = 0 and m = 2 condensate excitations
the damping rate γ quickly decreases with decreasing temperature.

18

Figure 2.4: A simple model based on the theory by Edwards [59], which only takes
into account the change in mean ﬁeld due to the changing number of atoms in thecondensate as a function of temperature.

19
response amplitude. The m = 0 mode exhibits a signiﬁcantly smaller anharmonicfrequency shift than the mode with m = 2 symmetry (Fig. 2.5a). The decay
rates γ for the m = 0 and m = 2 condensate excitations show some dependence
on response amplitude (Fig. 2.5b), but in neither case does γ approach zeroin the limit of small response. These data demonstrate that ν and γ measured
for a typical response amplitude of 10% are fairly close to the zero-amplitude
perturbation limit.
For comparison with the results of our free-evolution spectroscopy technique,Fig. 2.6 presents a more conventional resonance measurement. While the central
frequency is in good agreement with that measured from observing the freely oscillating widths as discussed above, the resonance technique is ineﬃcient in termsof data collection time and moreover vulnerable to systematic errors, particularly
when one tries to extract a damping rate from the linewidth.
This experiment extends previous studies of condensate excitations to temperatures for which both condensate and non-condensate atoms are present insigniﬁcant numbers. This regime is not yet well understood theoretically, as thestandard mean-ﬁeld theoretical treatment of collective excitations of a harmonically conﬁned dilute BEC is done for the T = 0 limit and does not include
dissipation. Both m = 0 and m = 2 condensate modes exhibit large temperaturedependent frequency shifts. The decay rate of the condensate collective excitations
decreases rapidly with temperature, and shows no signs of leveling oﬀ. More recently, there have been further theoretical investigations into ﬁnite temperaturemodes [56, 73, 110].

20

Figure 2.5: For large response amplitudes at our lowest temperature, T  ≈ 0.48,
we measure the shift in the response frequency ν and damping rate γ. In (a)the frequencies, normalized by the small drive limit, show a larger anharmonic
shift for the m = 2 condensate excitation than for the m = 0 case. (Some of them = 2 data is the same as presented in Ref. 4 and is reproduced here to facilitatecomparison.) Solid lines are guides to the eye. In (b) the decay rates γ for them = 0 and m = 2 condensate modes shift in opposite directions with increasing
response amplitude.

21

Figure 2.6: The response of the m = 0 condensate excitation at T  = 0.79 as afunction of the center drive frequency. Solid lines show ﬁts to the expected form
for the amplitude (a) and phase (b) of a driven harmonic oscillator in the presenceof damping. For these measurements a constant amplitude drive is applied witha duration of 150 ms (about 7 Hz FWHM in frequency), after which the resulting
excitation is observed for 10 ms to determine amplitude and phase.

Chapter 3
Two-Photon transition

3.1

Introduction
One of the unique qualities of degenerate quantum systems is that all par-ticles share the same phase. Although this phase is random between successively
created condensates, the phase remains coherent over time for each production ofa condensate [41, 58, 89, 90, 99, 102]. This is only true in the absence of external eﬀects, such as thermal atoms and magnetic ﬁeld variations. Investigation ofphase properties requires the presence of another object which can interfere withthe condensate in order to make the phase observable. The internal states of 87 Rb
prove very useful, in that standard atomic physics magnetic-resonance techniques
can be employed for interferometry. For experiments on trapped condensates theinternal states used must have the same sign and magnitude of ∂E/∂B (the magnetic moment) in order that the states overlap spatially. Only the |1, −1 and|2, 1 states satisfy this. This chapter describes the two-photon transition ﬁrst
for a free atom in a uniform magnetic ﬁeld, then for the complicated case of a
conﬁned atom in a time dependent magnetic ﬁeld.
3.2

Transition for a free atom
The levels |1, −1 and |2, 1 (hereafter called |1, −1 and |2, 1 respectively)

have the same magnetic moment to ﬁrst order, and are separated by the hyperﬁne

23
splitting νhs ∼6.8 GHz. A two-photon drive is required since there are two unitsof angular momentum between them. Fig. 3.1 shows the 5S1/2 F = 1 and F = 2
ground states of 87 Rb in a small magnetic ﬁeld which produces splittings of ∼ 0.7
MHz/G between adjacent Zeeman levels. Also shown are the radio-frequency (∼ 1
MHz) and microwave (∼ 6.8 GHz) photons and their relative detunings. Whenthe intermediate detuning is larger than the Rabi frequencies of the two level
microwave and rf transitions (Ω2−1→0 and Ω20→2  ∆2 ) then the probability fora transition to the intermediate state |2, 0 is small. The three-level system canthen be treated as a two-level system [68] with a Rabi frequencyΩ0 ≡Ω−1→0 Ω0→2
.
2∆IM

(3.1)

where all Ω are bare Rabi frequencies and ∆IM is the detuning from the intermediate state. For arbitrary two-photon polarizations and ∆ multiple paths must be
considered to avoid interference or transitions to untrapped states. As an example, another possible arrangement is a microwave detuning below the |2, 0 stateand increasing the rf energy. This opens up path to the |2, −2 state throughthe |2, −1 intermediate state if the required polarizations are present. Likewise,
increasing the microwave frequency so that the intermediate state is higher in energy than the |2, 1 , and changing the rf correspondingly, opens another path tothe same ﬁnal state |2, 1 , but through the |1, 0 , causing interference. The states|2, 1 and |1, −1 are suﬃciently long lived that the linewidth is interaction time
limited, meaning it is possible to measure the transition frequency to ∼ 10Hz. At
this level the transitions have magnetic ﬁeld dependence given by the Breit-Rabi
formulaν(B) = −

µn
νhs 
νhs
− gI BmF ±1 + mF x(B) + x2 (B)
8
h
2
gI µb
B
x(B) = (gJ +)Mr νhs h

(3.2)
(3.3)

24

|2,2ñ
rf photon
Intermediate state D =1.24 MHz {

|2,1ñ

|2,0ñ

F=2

|1,-1ñ

6834.6826128 MHz|2,-2ñ

micr
owa
ve p
hoto
n

|2,-1ñ

rf evaporation
|1,0ñ

F=1
|1,1ñ

Figure 3.1: The ground states of 87 Rb in a small magnetic ﬁeld. The states marked
with squares have nearly the same magnetic moment. The state marked with thetriangle is the only other trapped, but with twice the conﬁnement of the othertwo. Microwave and rf photons couple the two states through an intermediatelevel.

µb
h
µn
h

p
Mr = m
me
gI
gJ
h
νhs

25

1.399624 MHz/G
762.2591 Hz/G
1836.152701
2.75124 × 23
2.002319 × 1.0000059
6.6260755 × 10−27 erg·s
6834.6826128 MHz [39]Table 3.1: Useful constants

where the plus (minus) sign is used for F = 2 (F = 1) and the fundamental values
in table 3.1 are useful [35]. This equation is exact for purposes here; the sphericityof the electronic orbit means that the ﬁeld gradient at the nucleus is zero, giving
no interaction with the nuclear charge distribution. All other interaction terms
vanish by symmetry arguments, up to the small magnetic octupole moment [121].
Expansion of the function of x(B) in Eq. 3.2 about small x(B) is valid when
the Zeeman splitting is small compared to the hyperﬁne splitting (µb B  hνhs ),yielding the useful approximation for the frequency of any state




µn 1
gI µb
1 1
± (gJ +)mF B
ν(B) = (− ± )νhs + −gI
8 2
h
4
Mr h

2
mF
gI µb
1
(1 −
) B .
±
) (gJ +
4νhs
4
Mr h

(3.4)

This evaluates to the simple expression for the transition frequency between|1, −1 and |2, 1 , good to < 1Hz up to B =20G
∆ν(B) = −2796.21B + 431.361B 2 + νhs .

(3.5)

Eq. 3.5 is plotted in Fig. 3.2. The setup for driving the two-photon transitionis shown in Fig. 3.3. Rf is applied through the same path as the evaporation rf;
capacitively coupled onto the top quadrupole coil. Microwave is applied througha sawed-oﬀ waveguide of cross-section 2.2 cm by 1.0 cm, ∼15 cm long, and ∼3 cmfrom the center of the cell to the end of the guide. The cutoﬀ frequency for the26energyF=287

Rb (I=3/2)

F=1

mf

−2
−1
0
+1

ν|2,1Ô − ν|1,-1Ô − νhfs (kHz)

10
+2
+1
0
−1

(a)

(b)

8
6
4
2
0

3.24 G

-2
-4
-6

0

5000

10000

0

2

magnetic field (G)

4

6

8

10

magnetic field (G)Figure 3.2: (a) The hyperﬁne and Zeeman levels as a function of bias ﬁeld. The
frequency splitting at B=0 is νhs . Figure (b) shows the energy diﬀerence between|1, −1 and |2, 1 , where the energy diﬀerence at B=0 has been subtracted. At
the special ﬁeld value of B = 3.24G the transition has no ﬁeld dependence.

T E10 mode is 6.82 GHz (λ = 4.4 cm) [86]. Magnetic ﬁeld polarization is along
the long direction of the waveguide, or out of the page in Fig. 3.3, in the same
direction as the rf. Since the quantization axis of a trapped atom is parallel tothe page (rotating with the TOP ﬁeld), this conﬁguration provides equal amountsof σ+ and σ− microwave and rf for all angles of the TOP ﬁeld. Unfortunately,
the presence of other conduction objects near the trap (coil supports, copper
coils) distort the microwave ﬁeld in an unknown way.

Power is applied to the

waveguide through rigid coax from a Hughes traveling-wave tube ampliﬁer (20W),
which is driven by an HP 8672A frequency source. Since our expected precision
was ∼ 1 part in 1010 , and we wanted the option of making absolute frequency
measurements, the HP source was locked to an HP58503A GPS receiver with root
Allen variance of ∼ 1 × 10−12 over 1 second (10MHz signal). Before the ampliﬁer
input is a microwave switch for fast turn on. The ampliﬁer output passes ﬁrst
through a high power directional coupler to prevent damaging reﬂections, thenthrough a power splitter in reverse. The splitter takes power from the reﬂectedwave to a square law detector for measuring the power on a scope.

27

Figure 3.3: The position of the microwave waveguide is shown relative to the
trap. Rf is capacitavly coupled to the top quadrupole coil (it is the same as theevaporation rf).

28
The rf is generated either by the rf synthesizer used for evaporation, ora separate synthesizer if the frequency switching time is not fast enough to go
between evaporation and the two-photon drive. An rf switch is used to toggle
between the two synthesizers. The remainder of the rf path is the same as forevaporation. Stability requirements for the rf are easily met by the devices without
externally locking to the 10MHz GPS signal (although we do anyway).
For general superposition experiments, the rf and microwave pulses must be
short compared to the trap frequency so that no density evolution occurs during
the population transfer. Other experiments require arbitrary pulse sequences andtimings. We use a programmable HP8672A for this task. Following is a descriptionof the general timing for two-photon pulses. See [61] for details on condensate
creation.
Programming of the HP8672A can easily require a few seconds of time given
the 100 commands that must be uploaded for complex pulse sequences. This isdone in two pieces; one after a command is changed from the experiment-computer
keyboard, and the other once the experiment run has begun (after the atoms aretransferred into the TOP trap). At this time the amplitude and frequency ofvarious synthesizers is set, and the HP8176A is awaiting a trigger to begin its
pulse sequence . Evaporation proceeds and a nearly pure condensate is created in
the F = 1, mF = −1 state. The trap can then be ramped to achieve the desired
conﬁnement. After things have settled (quadrupole coil temperature for example)
a trigger from the computer is sent to the AF synchronization box, which outputs
a trigger to the HP8176A only when the TOP ﬁeld has a speciﬁc phase. This way
all experiments are locked to the rotating TOP ﬁeld, so that speciﬁc events will
occur at the same ﬁeld direction. This is critical for destructive, near-resonanceimaging, obtaining a repeatable position of a dropped condensate, and keeping
strange sideband behavior constant (see the Tilted TOP section in this chapter).

29
Fig. 3.4 shows a timing diagram for a simple separated oscillatory ﬁeld experimentcontaining two two-photon pulses.
For initial experiments we drove single-photon microwave transitions froma dropped (Bq turned to zero) condensate in the |1, −1 state to the F = 2
states. Initial linewidths were ∼ .3MHz, making this a fair probe of magnetic ﬁeld
magnitude. This is interesting for calibration issues, especially one of rounding
the rotating TOP ﬁeld. Single-photon rf or microwave transitions means they are
ﬁeld sensitive at 0.7 MHz/G for rf drives, and multiples (up to 3) of .7 MHz/G
for microwave. Given that the linewidth is interaction time limited, and that thedrive must be on for, at most, 1/4 of a TOP rotation to sample only a given
angle, means that the transition width is 4νT OP . This eﬀectively detects a ﬁeld
4
νT OP /.7 × 106 Hz if the line can be split to 1 in 10. For a 7.2kHz
diﬀerence of 10

rotation frequency this means an eccentricity of 4mG can be observed in the bestcase scenario.
The two-photon resonance from |1, −1 to |2, 1 was easily observed by applying both rf and microwave drives for a few milliseconds and observing theatoms in the F = 2 levels. Figure 3.5 shows typical data. Sidebands are a common feature in these lineshapes and are separated from the main peak by the
TOP rotation frequency. An asymmetry is very common, with the right sideband
larger than the left. This will be covered in more detail later.
Ramsey’s method of separated oscillatory ﬁelds [121] was also employed fora host of frequency measurements in characterizing the system. Two common
implementations are shown in Fig. 3.6 for dropped condensates. Graph 3.6(a)shows the signal as the time between two π/2 pulses of length 278 µs is changed,
and the two-photon drive is detuned by 850 Hz. The observed oscillation frequency
is the diﬀerence between the atomic splitting and the drive. Essentially, this is
a comparison of the rate of phase accumulation (energy) between the synthesizer

30µw-enable
rf-enableµw-switch
rf-switchprobe AOM
probe shutter

Time
Figure 3.4: A basic timing diagram for a two-pulse experiment. The microwave
and rf switch pulses can be any desired length, but are ∼1ms for most π-pulses.

1200

Number in |2,1> (arb.)

1000

800

600

400

200

0
1.000 1.002 1.004 1.006 1.008 1.010 1.012 1.014 1.016 1.018

Rf (MHz)

Figure 3.5: A typical Rabi lineshape for the two-photon resonance in a dropped
condensate. The microwave is 6833.668 MHz, and the pulse is 690µs long. The
sideband is separated by ∼7.2 kHz, equal to the TOP rotation frequency.

31

Figure 3.6: Ramsey’s method of separated oscillatory ﬁelds by changing the time
between (a) or the frequency of (b) the two pulses. (a) The ﬁt frequency gives
the detuning between the drive and the atomic transition. (b) The solid line is aﬁt with adjustable amplitude and central frequency.

and the condensate, during the time between the pulses. The mean-ﬁeld energy is
the same for each point in this case; the system is in the same superposition of thetwo states for each data point. A more common case is shown in Fig. 3.6(b), in
which the time between the pulses is constant at 1.02 ms, but the drive frequencychanges. Since the superposition changes as a function of detuning, a systematic
broadening could occur if the mean-ﬁeld energy of the two states were unequal.
The ﬁrst case is used extensively in the following chapters as calibration of ﬁeld
amplitudes and in measuring phase properties of a condensate.
3.3

Transition for a trapped condensate
The preceding analysis is suﬃcient for a static magnetic trap. Additional

complications arise, however, when a time-dependent magnetic trap (such as the
TOP trap) is used to provide the conﬁnement. Referred to as the “Weird Bohn
Eﬀect”, the time dependence leads to an additional eﬀective static magnetic ﬁeld
which yields energy level shifts and vertical oﬀsets between |1, −1 and |2, 1 . It

32
has become a useful tool in double-condensate systems.

3.3.1

Quantum mechanical derivation

The TOP trap involves a magnetic ﬁeld of magnitude B0 rotating counterclockwise (as viewed from the positive z-axis) at angular frequency ωt in thexy-plane,B0 (t) = B0 (cos ωt tx̂ + sin ωt tŷ).

(3.6)

The eﬀective Hamiltonian in a frame co-rotating with the magnetic ﬁeld transforms asĤeﬀ = R(−ωt t)ĤBR R† (−ωt t) − ih̄R† (−ωt t)∂
R(−ωt t)∂t

(3.7)

with the time-dependent rotation operator R deﬁned byiR(−ωt t) = exp( Fz ωt t).
h̄

(3.8)

The operator Fz is the z-component of the total (nuclear plus electronic) spin
vector, and the sign of the argument of R is chosen to rotate the coordinate axesin the same sense as the ﬁeld. The Breit-Rabi Hamiltonian ĤBR (with eigenvalues
Eq.3.2) is invariant under the transformation. We therefore obtainĤeﬀ = ĤBR − Fz ωt .

(3.9)

The expectation value of this operator on a state |F, mF  isF, mF | Ĥeﬀ |F, mF  = EBR − h̄mF ωt = gF mF µB B − h̄mF ωt1
51
gF =1 = − gJ −
gI −
4
4Mr
2
1
3
1
gI
gF =2 = gJ −4
4Mr
2
where EBR is written as only the largest part of Eq. 3.2, i.e. the ﬁrst order
Zeeman shift. Interpreting the expectation value of Ĥeﬀ as an eﬀective magnetic

33
ﬁeld gF mF µB Beﬀ , we see that the rotating magnetic ﬁeld becomes simply anotherstatic ﬁeld contribution:
Beﬀ = B0 −

3.3.2

ωth̄
µB gF

(3.10)Classical derivation

A classical derivation is more intuitive and yields the same result [121]. The
equation of motion for a spin vector F in a constant magnetic ﬁeld B ish̄dF
= µB gF F × B0 .
dt

(3.11)For magnetic ﬁeld rotating at ωt , the time derivative in the rotating frame becomes
dF −
∂F
=→
ωt × F +.
dt∂t(3.12)

Combining these two and solving for the motion of the vector in the rotating frame
gives
h̄

∂F
= µB gF F × Beﬀ∂t
Beﬀ = B0 + Bω
→
h̄−
ωt
Bω ≡
µB gF

(3.13)

which is the same as Eq. 3.10. This extra static ﬁeld can be interpreted as an
imaginary ﬁeld which causes the spin vector to precess at a Larmor frequency ωt .
The magnitude of Bω is only 2.6 mG for ωt = 1.8kHz, but switches sign for thetwo diﬀerent hyperﬁne states. However since Bω is perpendicular to B0 , the totalﬁeld magnitude remains constant in this picture. Only when another ﬁeld withsome component along Bω is present does this modify the actual magnitude, andhence two-photon transition energy. This is the case for an untrapped condensate,

34
as long as there is an external ﬁeld along Bω . The total ﬁeld is|Btotal | =B02 + (Bz ± Bω )2(3.14)

where Bz is an external uniform ﬁeld in the same direction as Bω and the plus
(minus) sign is used for the |2, 1 (|1, −1 ). The ﬁeld for each state from Eq. 3.14
must be used in Eq. 3.4, and the diﬀerence taken to ﬁnd the transition frequency.
A useful approximation for the transition frequency is∆ν = −2796.21B0 + 431.361B02 + 699579

Bω Bz
B0

(3.15)

valid for B0  Bω , Bz . The ﬁrst data investigating this eﬀect was taken using
condensates dropped from the trap, so that the magnetic gradient was oﬀ andonly the rotating ﬁeld B0 and background ﬁeld Bz present. Figure 3.7 shows thisdata and plots of the exact version of 3.15 for diﬀerent directions of B0 rotation(or equivalently, diﬀerent signs of Bz . The extra term in Eq. 3.15 applies an oﬀset
whose magnitude drops oﬀ as B0 dominates the total ﬁeld.
The situation is similar for a trapped condensate, except that the conﬁning
gradient provides Bz . In the TOP trap gravity displaces the equilibrium positionof the condensate from z = 0. The total instantaneous ﬁeld at any point in thetrap is|Btotal (z)| =(B0 cos(ωt t) +

Bq 2
B
x) x̂ + (B0 sin(ωt ) + q y)2 ŷ + (−Bq z ± Bω )2 ẑ.
22(3.16)

Figure 3.8 shows graphically the orientation of the ﬁelds in the TOP trap. The→
ω . Since z = 0 when gravity is included the eﬀect
imaginary ﬁeld Bω points along −of Bw is to modify the vertical quadrupole gradient with an oﬀset. This can be
rewritten as an oﬀset in z, i.e. ,(−Bq z ± Bω )2 = (−Bq (z ± zω ))2
zω ≡

Bω
Bq

35

Transition frequency offset (Hz)4000

Bz = 0.12 G
ω = 7202 kHz ; Bω = .0103 G

2000

+ω

0

-2000

ω=0

-ω

-4000

-6000

0

1

2

3

4

5

6

7

Rotating field amplitude Bω (G)Figure 3.7: The data points are for a dropped condensate. The frequency ofrotation of B0 is 7.2 kHz and the vertical component of the background ﬁeld Bz isfound from a ﬁt of Eq. 3.15. Also shown are the curves for the opposite rotationof B0 and when there is no rotation.

36

z

ω
B’q

Bω

quadrupole
coils
Figure 3.8: Diagram of the magnetic ﬁelds in a TOP trap. The curved arrows
represent the ﬁeld line from the quadrupole ﬁeld Bq produced by the coils. A
uniform ﬁeld Bω rotates about the central axis, providing an imaginary ﬁeld Bω
→along −
ω.

37
It is the total ﬁeld in Eq. 3.16, along with the position z ± zω of the atoms in
the trap, which must be substituted into Eq. 3.2 to ﬁnd two-photon transition
frequency.
3.3.3

Observable eﬀects

These equations can all be put together to yield useful quantities, such as therelative separation between the states and the transition frequency as a functionof trap parameters. For the following quantities, the condensate is assumed to beat its equilibrium position, as deﬁned by the trap parameters B0 , Bq and Bω . The
following are small eﬀects which are only observable for speciﬁc trap geometries,
or when looking at diﬀerences in large quantities.
In general the position of an atom in a magnetic potential is found by setting
the spatial derivative of the total energy to zero. From Eq. 3.2 and Eq. 3.16, thefollowing must be solved for z;∂(hν(|Btotal (z)|) + mgz) = 0.
∂zThis is a very complicated equation in z and not easily solved exactly. Using the
expansion in Eq. 3.4 gives a quartic equation in the vertical position z whose
solution is approximated by the expansion in +
η
η
B0
B02
√
−
2
+
Bq 1 − η 2
Bq (1 − η 2 )23
B 3 3η 2 + 2 2
4 3η + 2η 3
−2 0
+
+
16B
+
7
0
Bq (1 − η 2 ) 2(1 − η 2 )4z = ∓zω +(3.17)

mg
in which η ≡ KB
 , + ≡ L/K, and K and L are the coeﬃcients of the linear andqquadratic Zeeman terms in Eq. 3.4 respectively. This reduces to Ensher’s result(Eq. 6.9 [61]) when the nuclear and higher order Zeeman structure vanish ( + → 0
and K → 14 gJ µhB ). K, L and z0 depend on the atomic state mF . The size of thehigher order terms is actually quite small compared to zero order. It is only when

38
looking at the diﬀerence in vertical position of the two states that these terms
play signiﬁcant role. As will be shown later these diﬀerences have quite a large
eﬀect in double condensate behavior.
Fig. 3.9 shows the eﬀect of the trap rotation and the higher order magnetic
ﬁeld dependence on the relative vertical position of the two states. The eﬀect oftrap rotation is only to apply a vertical oﬀset between the states in Fig. 3.9a. For
many double-condensate experiments we are interested in traps where the relative
oﬀset is zero so that the condensates completely overlap. Fig. 3.9b is a plot of therotating ﬁeld magnitude B0 necessary for overlap as a function of the quadrupole
gradient Bq . Only a very small range is accessible when the rotating ﬁeld has apositive direction. Qualitatively the oﬀset from rotation is opposing (−Bω case)or enhancing (+Bω case) the relative sag caused by the diﬀerent higher order
Zeeman shifts. All of the double-condensate experiments are done with the −Bωrotation.
Using Eq. 3.17 for the position of atoms in the trap the absolute magnetic
ﬁeld, and hence the transition frequency, can be found as a function trap parameters. The total ﬁeld at the equilibrium position of a state is given by theapproximation|Btotal | =







η+
z0


B02 + Bq2 z02 − 2Bq B0222
2(1 − η )B0 + Bq2 z02(3.18)

B0
1 − η2B0
η√≡Bq 1 − η 2√z0where z0 is the lowest order term in the vertical position from Eq. 3.17. This isindependent of the rotating ﬁeld frequency since its eﬀect is to apply an oﬀset in
the vertical ﬁeld which only moves the equilibrium position, but does not change
the ﬁeld the atoms experience. To calculate the transition frequency between

39

5
2

-Bω

0

B0 for overlapping states (G)Separation |2>-|1> (µm)

4
-2

+Bω

-4

-Bω
-6

Bq'=69 G/cm
Bq'=43 G/cm

-8

+Bω

-Bω

3

Bω =0

2

-10

+Bω
-12
0

1

2

3

B0 (Gauss)

4

5

1
30

35

40

45

50

55

60

65

70

Vertical Gradient, Bq' (G/cm)

Figure 3.9: (a) The relative positions of the two states in the trap for two diﬀerent
values of the vertical gradient, with ωt = 1800 kHz. In each pair of lines are theplots for diﬀerent rotation directions of the TOP ﬁeld. As B0 → 0 the lines fora particular gradient are separated by 2Bw /Bq = 2z0 . (b) The magnitude of B0
necessary to have overlapping condensates as a function of Bq , for the diﬀerent
rotation directions and without including the rotating eﬀect.

40|1, −1 and |2, 1 , Eq. 3.18 must be used to calculate the ﬁeld seen by each state.
Since the atoms will be at the equilibrium position of the initial state (|1, −1 forinstance), Eq. 3.18 is used with the appropriate η, z0 and + for |1, −1 . The ﬁeld
B2 experienced by |2, 1 is not only diﬀerent through the terms η, z0 and +, butalso because it is oﬀset from its equilibrium position by an amount 2Bω /Bq , whichmust be added to z0 in Eq. 3.18. The transition frequency of Eq. 3.5, minus thehyperﬁne splitting νhs , becomes


µn
µb
gI
3
gI µb
∆ν = −gI (B1 +B2 )+ (gJ +)(B2 −B1 )+
(gJ +
)
h
4h
Mr
16νhs
Mr h

2

(B12 +B22 )(3.19)

This is plotted in Fig. 3.10 as a function of the rotating ﬁeld B0 for two diﬀerent
values of Bq . The eﬀect of changing the direction of the rotating ﬁeld is to oﬀset
the transition frequency. For many experiments we wish to operate in the region
where the transition is insensitive to ﬁeld noise (or rotating ﬁeld asymmetries).
This is the zero-slope point in Fig. 3.10, where the total ﬁeld experienced by theatoms is 3.241 G. The rotating ﬁeld magnitude required to reach this point isgiven by the approximation


B0 =

1
Bω2
1 − η 2 (Bq ) ηBω + 3.241 1 − (1 − η 2 )23.2412

3.241 1 − η 2 (Bq ).

(3.20)The second approximation is very good; it is nearly independent of Bω (oﬀ by 10
mG for a factor of 10 change in the rotation frequency).
Unfortunately, the requirements for overlapping condensates is not the sameas the requirement in Eq. 3.20. A contour plot of the slope of the transition(∂∆ν/∂B0 ) is shown in Fig. 3.11 along with the trajectory for overlapping condensates from Fig.3.9b.

41

3.0

4

2x10

+Bω

Bq' =172 G/c
Bq' =43 G/c

Rotating field B0 (G)Transition frequency (Hz)

3.5

-Bω

4

1x10

+Bω

0

-Bω

4

-1x10

0

2

4

6

Rotating field B0 (Gauss)

8

2.5

2.0

1.5

1.0
20

40

60

80

100 120 140 160 180

Quadrapole gradient Bq' (G/cm)

Figure 3.10: (a) The transition frequency as a function of the rotating ﬁeld B0 ,similar to Fig. 3.2(b), but here the complete, time-dependent model is included.
Each pair is for a diﬀerent vertical gradient. Within each pair the eﬀect of therotating ﬁeld is shown for the two directions of rotation ±Bω . (b) The value of
B0 where the transition is insensitive to changes in ﬁeld, plotted against Bq . The
dashed line is 3.241 G.

42

10

9

5e+0038

Rotating field (G)762e+0035
1e+003
4

500
0

3

-500
-1e+003

2

1

40

60

80

100
120
Quadrapole gradient (G/cm)

140

160

Figure 3.11: The contour plot shows the slope of the two-photon transition withrespect to B0 . Lines are labeled in Hz. The dotted line is the B0 required forcompletely overlapping traps, for the frequency of ﬁeld rotation of 1800 Hz.

43
3.4

Rabi frequency gradient across the condensate
The previous section dealt with the basic eﬀects of the TOP trap and thehigher order Zeeman structure on the two-photon transition. This section will
specialize to a remarkably useful consequence of these eﬀects; a gradient in the
two-photon eﬀective Rabi frequency Ωeﬀ across the condensate. As will be shownin Chapter 7, the gradient provides a way to manipulate the phase of a condensate spatially and temporally. There are essentially three diﬀerent (although not
independent) eﬀects: trap oﬀset, intermediate state detuning, and polarization of
the two-photon drive.
3.4.1

Trap oﬀset

The vertical oﬀset between the equilibrium positions of the two states wasgiven in Eq. 3.17 and Fig.3.9. This comes into Ωeﬀ through the detuning of thetwo-photon drive from resonance:


Ωeﬀ =

Ω20 (z) + ∆2 (z)(3.21)

where Ω0 , the bare Rabi frequency from Eq. 3.1 and has no z dependence in thiscase, and1m 2 2 1m 2ω z −ω (z + zoﬀset )2
2h
2h
m 2
→ δ + ω zoﬀset z.
h∆(z) = δ +(3.22)

This equation for ∆(z) includes an overall detuning δ (a simple detuning of thedrive from the resonance) and the detuning due to oﬀset harmonic oscillator potentials. For strong ﬁeld gradients the oﬀset is simply 2Bω /Bq . The trap oscillation
frequency ω is nearly the same for both states. In the limit that Ω0  ∆(z) orδ  the detuning across the condensate, the eﬀective Rabi frequency becomes
simply ∆(z) and there is a linear gradient in Ωeﬀ along the vertical dimension.

44This gradient can be from 0 (overlapping traps) to a few kHz over the extent of
the condensate.

3.4.2

Intermediate-state detuning

The bare Rabi frequency Ω0 from Eq. 3.1 may also be modiﬁed by changing
the intermediate-state detuning ∆IM as a function of vertical position (refer toFig. 3.1). Since the |2, 0 state has no magnetic ﬁeld dependence, and the |2, 1
and |1, −1 shift essentially together at ∼ µb /(2h)0.7 MHz/G, the virtualintermediate-state will shift, locked with the |2, 1 and |1, −1 at 0.7 MHz/G.
The gradient B  in which the condensate sits is where gravity is canceled, i.e.
B  µb /2 = mg (higher order Zeeman terms are ignored). B  is approximately 30.5
G/cm for both |2, 1 and |1, −1 . The total intermediate state detuning is then
µb 
Bz
2h
mg
z
= ∆IM 0 +hkHz= ∆IM 0 + 2140zµm∆IM = ∆IM 0 +(3.23)

where z = 0 is at the equilibrium position of the condensate, and ∆IM 0 is thedetuning of the microwave and rf from the |2, 0 where z = 0. The magnitude ofthis eﬀect can be controlled by changing the size of the condensate (by changing
ω), changing the relative strength of Ω0 and the detuning, or by changing ∆IH .
The size of this eﬀect is variable from a few Hz to kHz. A gradient in Ωeﬀ due toa gradient in Ω0 is advantageous since the entire condensate is still in resonance,
as opposed to a gradient in the detuning ∆ in which the transfer between the twostates also varies along z.

45
3.4.3

Drive polarization

The polarization of the two-photon drive can also be made to vary across thevertical direction. This occurs by changing the direction of an atom’s quantization
axis by varying the relative magnitude of the ﬁelds B0 and Bq . As the conﬁning
potential is lowered, the atoms sag further down due to gravity, experiencing a
diﬀerence in the vertical magnetic ﬁeld component. This alters the projection of
the microwave and rf drives onto the quantization axis, changing the bare Rabi
frequency for each drive in Eq. 3.1. Since the rf and microwave have the samepolarization, this eﬀect gets squared for the two-photon transition. Taking theﬁrst order term for the position of the atoms in the trap (Eq. 3.17), the bare Rabi
frequency and its gradient for the full two-photon transition become respectivelyΩ0 → Ω0 (1 − η 2 )∂Ω0∂z= −2Ω0BqB0η(1 − η 2 )3/2 .

(3.24)(3.25)

These are plotted in Fig. 3.12. Fig. 3.12a is a plot of the fractional change inΩ0 at the center of the condensate as the quadrupole gradient is changed. It isinteresting that this is independent of the rotating ﬁeld B0 . Fig. 3.12b is a plotof the fractional gradient in Ω0 . In Fig. 3.12c, this gradient is multiplied by theThomas-Fermi vertical width (full-width at zero) [53] of the condensate to show
the fraction change in Ω0 across the condensate. In general this eﬀect accounts
for only a few Hz across the cloud, but could be made ∼ 100 Hz.
The previous analysis assumed that the polarization of the microwave andrf were along the rotating ﬁeld’s axis. This is nominally true, but interaction
with the conducting coil forms and supports likely modiﬁes these ﬁelds in an
unknown way. As an estimate, I assume the ﬁeld polarization angle to maximize
the gradient of polarization across the cloud, which gives an increase by a factor of
2.4. This is the maximum increase, and only applies for a speciﬁc Bq in Fig. 3.12,

46
which depends on the real angle of ﬁeld polarization. For the case of the Rabi
frequency gradient in Chapter 7, this maximum corresponds to ∼ 18 Hz acrossthe condensate.
3.5

Density shift
This shift is not a property of the magnetic trap as all the previous eﬀects

have been, but still provides a systematic in the two-photon transition frequency.
Repulsive interactions between 87 Rb atoms in a BEC lead to an energy oﬀset fromthe ground state of the harmonic oscillator. Since the strength of the interactions
is diﬀerent for diﬀerent internal states, there can be a systematic shift in thetransition frequency between the states which will depend on relative state populations. This is a small eﬀect by the standards here, but well known in atomic
clocks as the clock-shift, or pressure shift [98]. The diﬀerence in interaction energy
between the states is [53]E = 5πh̄2[(a11 n1  − a22 n2 ) + a12 ( n2  − n1 )] .
m(3.26)

where n is the density-weighted average density in the Thomas-Fermi limit,
and aij is the s-wave scattering length between states i and j. The ﬁrst groupof terms comes from the self-interaction energy and the second group from themutual interaction between the two states. We are concerned with only the following values; a11 = 107, a22 = 101, and a12 = 104 in units of the Bohr radius
a0 = 0.529 × 10−8 cm [46]. In a usual Ramsey-type clock, the two states involved
have equal superpositions during the time between the π/2-pulses. For typicalcondensate densities of 5 × 1013 cm−3 , the shift is 30 Hz.

47

1.0
0

0.8

-1

Fractional gradient of Ω0 (cm )Fraction of bare Rabi frequency

0.9

0.7
0.6
0.5
0.4
0.3

(a)

0.2
0.1

40

60
80 100 120 140 160
Quadrapole gradient Bq' (G/cm)

-10
-20

B0= 3 G-30
-40
B0= 1 G

-50
-60

180

B0= 10 G

(b)
40

60
80 100 120 140 160
Quadrapole gradient Bq' (G/cm)

180-0.005Fractional difference in Ω0 across BEC

B0= 10 G-0.010B0= 3 G

-0.015-0.020B0= 1 G

-0.025-0.030(c)-0.035-0.040

40

60

80

100

120

140

160

180

Quadrapole gradient Bq' (G/cm)

Figure 3.12: (a) The fractional change in bare Rabi frequency as a function ofBq , due to a spatial gradient in polarization direction of the microwave and rf.
(b) The fractional gradient in Ω0 for three diﬀerent values of the rotating ﬁeld
B0 . (c) The fractional diﬀerence in bare Rabi frequency across the cloud due toa gradient in the two-photon drive polarization. Multiply this by Ω0 to get theabsolute diﬀerence in bare Rabi frequency across the condensate’s diameter.

48
3.6

Tilted TOP
One last eﬀect is introduced as a possible explanation for two eﬀects common

in two-state experiments; sidebands and radial symmetry breaking. A systematic
study of sidebands on the transition proﬁle has not been done, and the only realknowledge we have is that they are separated from the central carrier by the TOP
rotation frequency ωt . Radial symmetry breaking is observed as a separationof two-component condensates in a speciﬁc direction perpendicular to the trap’s
symmetry axis (see Fig. 3.8) as will be discussed later. A possible explanation isa tilt of the axis of ωt in Fig. 3.8 relative to the axis of the quadrupole coils. The
resulting ﬁeld from a tilt by angle θ in the x̂ direction is|Btotal (z)| =  B0 cos θ cos(ωt t) +
+

2
Bq
B 2
x ± Bω sin θ + B0 sin(ωt ) + q y22

2−Bq z ± Bω cos θ + B0 cos θ cos(ωt t)

1
2

.

(3.27)

Two eﬀects are present here. First, since the condensate experiences a ﬁeld component in the ẑ direction due to gravitational sag, the magnitude of the ẑ-component
oscillates at the TOP frequency ωt (third term). It causes a modulation of thetransition frequency and thus sidebands. This can be thought of as a modulated
drive; ω(t) = ωc + ∆ω cos ωm t where ωc and ωm are the carrier and modulation
frequencies, and ∆ω is the amount of frequency modulation. To estimate thesize of this eﬀect, I start by estimating the on resonance Rabi frequency of the
sideband Ωsb , as ∼ 0.4 of the Rabi frequency of the carrier Ωc (determined fromﬁg. 3.5). In the region of wide-band phase modulation the electric ﬁeld can be
written as
E(t) = Ac J0 (β) cos ωc t − Ac J1 (β)(cos(ωc + ωm )t − cos(ωc − ωm )t) + ...

(3.28)

where Ac is the amplitude of the carrier for zero modulation, J0 and J1 are Bessel
functions, and β ≡ ∆ω/ωm is the modulation index. The ratio of Rabi frequencies

49
is then just that of the amplitudes squared (J1 (β)/J0 (β))2 which implies β1.

It is now necessary to estimate how big the TOP tilt must be to produce
this size of modulation of the two-photon transition. To produce a modulationindex of 1, ∆ωωt = 1800 kHz. This amount of modulation gives an amount ofmagnetic ﬁeld modulation through the derivative of Eq. 3.5, which is 100 Hz/G
for Btotal = 3.5 G. I take instead the very conservative limit of 500 Hz/G, implying
a ﬁeld modulation of 3.6 G which is larger than the total ﬁeld. Obviously thiscannot be the cause of the sidebands.
For completeness, I estimate the eﬀect of a more reasonable tilt. The peak
to peak amplitude of oscillation of the total magnetic ﬁeld for small tilts θ, from
Eq. 3.27 is∆Bpp
A tilt of 5o gives ∆B

2η(Bq )B0 θ.

0.3 G out of(3.29)

3.5 G total ﬁeld. The slope is again

taken to be 500 Hz/G (ﬁve times larger than the actual slope at Btotal = 3.5 G)
which means a frequency modulation of ∆ω = 150 Hz and a modulation index
β = ∆ω/ωtop = 150/1800.
Unfortunately a detailed study of sidebands has not been done, even though
they are frequently an annoying presence. There are many features which have yet
to be explained: Sidebands are present even for a dropped atom cloud (Bq = 0,Bω = 0), implying the trap is not the cause (although an external ﬁeld could
take the place of the vertical ﬁeld from the quadrupole); sidebands move and evenswitch sides about the carrier for diﬀerent turn on times of the two-photon drivewith respect to the rotating TOP ﬁeld. This was true even when the two-photon
pulse length is comparable to the TOP rotation period (It should be stressedthat this latter observation was made with a diﬀerent placement of the microwave
waveguide, and has not been investigated for the current placement shown in Fig.

50
3.3).
A second eﬀect of the tilt is a component of the imaginary ﬁeld Bω along
the x̂ direction (ﬁrst term), thereby giving an oﬀset similar to that discussed
previously in the ẑ direction. In this case however, gravity is perpendicular tothe oﬀset and so the higher-order Zeeman shifts cannot be used to cancel theoﬀset as in Fig. 3.9. The magnitude of the oﬀset is 4Bω sin θ/Bq . For θ =5
degrees, a vertical gradient of Bq = 122 G/cm (17 Hz radial frequency), and a
TOP rotation of 1800 kHz, this comes out to ∼ 0.08µm . See Fig. 5.4 for thedata corresponding to this trap, showing the radial oﬀset. To put this oﬀset inperspective, it is compared with the vertical separation in a trap with verticalgradient of 129 G/cm, vertical frequency of 62 Hz, and an oﬀset of 0.4 µm, seenin Fig. 5.5. In that case the time for appreciable separation was 10 ms. In thecase of radial separation in Fig. 5.4 the radial separation time was about 100 ms.
A rough scaling of the separation time in each case by the oﬀset and the relevant
trap frequency gives (17 × .08 × 100) = 136 and (62 × .4 × 10) = 248 shows afactor of two diﬀerence for a 5 degree tilt. A 10 degree tilt is somewhat unlikely,
but the estimate shows the eﬀect is in the right neighborhood.

Chapter 4
Imaging

4.1

Near resonance - absorption imaging
For a majority of the experiments described in this thesis, absorption imag-

ing has been employed. This has been covered in detail in Ensher’s thesis [61],but a few additional aspects are addressed here. Essentially, absorption imaging
relies on the resonant scattering of light out of a probe beam, and the decreasein probe beam intensity is measured. Intensity decays exponentially through thecondensate; Im = I0 e−OD . The optical depth represents the column density ofatoms along the probe direction; OD(r) =n(r, z)σ0 dz for the scattering cross-section σ0 . It is extracted by imaging the spatially dependent probe intensity ontoa CCD through the following three pictures; Ishadow – the intensity proﬁle with thecondensate present, Ilight – the intensity proﬁle of the probe beam only, and Idark– an image with the probe beam oﬀ, so that only external sources of oﬀsets aremeasured (for example, dark current on the CCD, or room lights). The measured
optical depth is then calculated throughODmeas (r) = ln

Ilight (r) − Idark (r).
Ishadow (r) − Idark (r)(4.1)

This equation is applied for each pixel on the CCD array. For probe intensities
comparable to the saturation intensity (1.6 mW/cm2 ), the absorption no longer
follows this equation, and Ensher [61] has derived a correction to obtain the real

52
optical depth from the measured ODmeas (r) and the probe intensity.
There are two more eﬀects worth mentioning. The ﬁrst has to do with ageneral correction which can be applied to account for many eﬀects. Since theprevious formulae rely on the assumption that all of the probe light interacts
with the atoms the same way, the presence of oﬀ-resonant or incorrectly polarized
light will lead to systematic eﬀects. Since the probe laser linewidth is much lessthan the atomic linewidth, the light can be broken up into two pieces; on-resonantand correctly polarized light, oﬀ-resonant, incorrectly polarized, or scattered light.
The ﬁrst piece is the probe light which provides a signal proportional to the column
density through the condensate. The remaining pieces are part of the probe beam
but cannot be absorbed by the atoms. These can be lumped together into α, thefraction of the intensity which is not absorbed, and treated the same way as theother oﬀsets Idark . The measured optical depth using Eq. 4.1 becomes
ODmeas (r) = ln

Ilight (r) − Idark (r).
Ishadow (r) + αIlight (r) − Idark (r)(4.2)

It is not practical to directly measure αIlight (r) simply through another image on
the CCD, so its value must be found through other means. Once it is measured,
the following rearrangement of Eq. 4.2 may be applied on a pixel by pixel basis
to get the actual optical depth:

ODreal (r) = ODmeas (r) − ln 1 − α(r)eODmeas ($r) + ln(1 − α(r)).

(4.3)

This is plotted in Fig. 4.1 for three values of α.
An example of the above correction is the “polarization correction”, which
accounts for the variation in the direction of the atoms’ quantization axis as theTOP ﬁeld rotates during the probing. Ideally, the probing would only occur while
the TOP ﬁeld points along (or toward) the circularly polarized probe beam sothat the cycling transition F = 2, mF = 2 → F  = 3, mF = 3 is driven. Since the

53

16

14

α=0

12

ODmeas108

α = 0.001

64

α = 0.05

2

0

0

2

4

6

8

10

12

14

ODactual
Figure 4.1: For a real optical depth of ODactual the observed value (ODmeas ) isplotted for diﬀerent values of the fraction of light which cannot absorbed by theatoms α.

54
probe has ﬁnite duration tp , the ﬁeld rotating at angular frequency ωt sweeps out
an angle θ = tp ωt . This can either be thought of as a time-varying polarization
applied to the mF = 2 state, or as a constant circular polarization applied toa time-varying state superposition. In the second picture the population in the
mF = 2 state varies as cos4 (θ/2). Integrating this from −θ/2 → θ/2 (assumes
that the TOP ﬁeld points along the probe beam half-way through the probe pulse)
gives the fraction of correctly polarized light (i.e. = 1 − α) :
f =

θ
θ
θ
3 1
3
+ cos3
cos
sin
+
8 θ
4
4
2θ
4
1
1 4
1 − θ2 +
θ
24
768

(4.4)

for small θ. The resulting incorrectly polarized fraction can be substituted into Eq.
4.3 for alpha. In early experiments where ωt = 2π × 7200 Hz and tp = 25.7µs, thisaccounted for a 10% decrease in observed optical depth for a condensate with real
optical depth of 1. This does not take into account other possible transitions.

4.1.1

Lensing

Part of the simple model of absorption imaging is that the light travelingthrough the condensate is only attenuated by resonant scattering. For suﬃciently
dense atom clouds, there can also be a large spatial change in the index of refraction. The rays traveling through the cloud will suﬀer some refraction due to thecurvature of the condensate, as well as its density proﬁle. In this way it behaves
as both a geometric lens and a gradient-index (GRIN) lens. The type of lens(converging or diverging) depends on the detuning of the probe from resonanceas the index is greater or less than one respectively. This behavior can lead tosystematics in measurements of the temperature and density. The focal length of

55
a ball of radius R with uniform density is given byf=R
2(n − 1)

(4.5)

where n is the index of refraction (see Eq. 4.10). Lensing is a problem when the
focal length is shorter than the focal depth of the imaging system, or when theimaging system is improperly focused. For our imaging setup with an f-numberof ∼ 5, the focal depth is 200 µm. A condensate of 20 µm radius implies that(n − 1)  .05 for minimal lensing, which occurs for a probe detuning ∆ > 600
MHz. At this detuning the observed optical depth is down to 0.05, which is visible
but with signal to noise near unity.
A schematic of lensing is shown in Fig. 4.2. The condensate is represented
by the dark grey ball and the probe beam is incident from the left. For theblue detuned case, the index is less than 1 so the condensate is a diverging lens.
The solid lines represent the path of the probe light and the dashed lines show
the boundary outside which the light does not travel through the condensate. At
position B, light from the inside of the condensate is refracted outside the cylinder
deﬁned by the dashed boundary. Where this occurs (represented by the light grey
areas) there is more light than the non-refracting model would suggest. If the
focal plane of the imaging system were at B, the resulting optical depth would
look basically normal near the center of the condensate (except for an enhanced
OD), but would actually go negative at the edges due to the increase in light in
the presence of a condensate. If the focal plane where at position A, then thevirtual image must be taken into account, represented by the dotted lines. Here
there is a cone of excess light, and the resulting optical depth has a corresponding
negative dip in the center. Similar but reversed pictures occur in the case of red
detuning.
Lensing is a serious complication when probing a condensate in the magnetic

56

Blue detuned
A

B
Position x

Incident beamvirtual ray

Red detunedCD
virtual ray

Position x

Incident beam

Optical Depth

Actual profile0

Focal plane at position B,C0

Position x

Focal plane at position A,D

0

Position x

Position x

Figure 4.2: Schematic of a uniform ball with index = 1 in the path of a probe
beam incident from the left. Grey regions enclosed by beam paths indicate anincrease in the light level over the non-refracting model. Regions outside thedashed lines (a cylinder around the condensate) are where the probe light does
not intercept the condensate. Dotted lines represent virtual rays. At the bottom
are the actual image in the absence of lensing, and the observed optical depth isshown for various focal plane locations.

57
trap. On resonance, the index is 1 so lensing is not a problem. However the opticaldepth is much too large (> 100), so that small contributions to α in Eq. 4.3 canseverely eﬀect the optical depth, leading to a saturation. In fact, from Eq. 4.3
the maximum observable optical depth is 

ODmax = ln

1
α

(4.6)

which turns out to be ∼ 3 for our experiment, implying α = .05. Detuning further
from resonance does make lensing better, but the index of refraction only drops oﬀ
as 1/∆ and the signal drops as 1/∆2 so the “signal-to-lensing” ratio gets worse. For
this reason destructive imaging is most useful when looking at dropped, expanded
condensates when the density is low and the radius of curvature big.

4.1.2

Signal-to-noise

The signal-to-noise is given in terms of the real optical depth by;
OD∆ (1 − α) exp−OD∆
S/N = σS2 + σL2 ((1 − α) exp(−OD∆ ) + α)

(4.7)

where OD∆ = ODres / (1 + 4(∆/γ)2 ) for the on-resonance, real optical depth
ODres , ∆ is the detuning from resonance, γ is the natural linewidth of 87 Rb (5.9
MHz), and σL and σS is the fractional intensity noise in the light and shadow
frames respectively. Noise on the dark frame is ignored in this analysis. This isjustiﬁed since dark noise is about 1 count versus 7 counts of noise from the othersources listed here (dark noise can vary from chip to chip even for the same type,
depending on CCD quality, the quality of the vacuum, and the temperature ofthe CCD). In the case of shot noise, σS = 1/ (1 − α)I0 exp(−OD∆ ) + αI0 and√
σL = 1/ I0 and the signal-to-noise becomes√

I0 OD∆ (1 − α) exp−OD∆

S/N shot = (1 − α) exp(−OD∆ ) + α + 1 (1 − α) exp(−OD∆ ) + α

(4.8)

58
where I0 represents the number of photons in the probe (actually, it must also
include the quantum eﬃciency of the detector). Since the signal for near-resonanceimaging is a reduction in light level, the S/N shot actually gets worse for large
signals. The optical depth with maximum S/N shot is 2.2 when α = 0, implyingthat the detuning ∆ should be set to achieve that measured OD. Figure 4.3 shows
the optimal optical depth for diﬀerent values of α.
A major source of noise above shot noise for usual intensities is spatial
ﬂuctuations of the probe intensity between the shadow and light frames. This isdue to small scale fringes on the probe beam that change position between the
Ishadow and Ilight . In this case the noise is linear in the light intensity, and σ isthe standard deviation for a sinusoidal function with fractional amplitude A. The
signal-to-noise is√
S/Nfringe =2 OD∆ (1 − α) exp−OD∆
A ((1 − α) exp(−OD∆ ) + α)

(4.9)

assuming maximum variation in the fringe spacing. The optimal optical depth
in this case is also plotted in Fig. 4.3. These are compared to other imaging
techniques at the end of this chapter.

4.1.3

Magniﬁcation measurement

For many quantitative measurements knowing the imaging magniﬁcation isextremely important. If an atom cloud is to be used for measuring the magniﬁcation, then the camera must be set at the focal point of the imaging system.
Ensher [61] described a method for doing this by destructively imaging small atom
clouds in the magnetic trap. This technique was plagued by lensing eﬀects andgave only modest results. Non-destructive imaging brought the ability to tolerate large optical depths so that very small clouds could be imaged in the trap.
The position of the camera was then adjusted to give the smallest observed cloud

59

3.5

Optimal optical depth

3.0

2.5

2.0

1.5

1.0
0.0

0.2

0.4

0.6

0.8

α

Figure 4.3: α is the fraction of the probe intensity which cannot be absorbed by
the atoms. This plot shows the optical depth required for the best signal to noisefor a given value of α in the shot noise limit (solid line) and for the intensity noise
limit (dashed line).

60
diameter.
The ﬁrst method used for a magniﬁcation measurement was to obtain a value
for g by turning oﬀ the trap and watching the atom cloud fall. A new picture wastaken for each diﬀerent time after the trap was turned oﬀ. The resulting vertical
position was ﬁt to a gravitational acceleration with a size scaling, and an initial
velocity. The size scaling directly gives the magniﬁcation.
A separate method for measuring the magniﬁcation was to use the interference of two beams which cross at the same location of the atom cloud, as
illustrated in Fig. 4.4. Essentially, the input angle α was measured, and thefringe spacing from the interference on the CCD was measured to yield the angle
β. The magniﬁcation is then the ratio α/β. It is crucial that the two beams
cross at the location of the atom cloud. This was done by ﬁrst centering theprobe beam on the atoms. While looking at the CCD image, the split-oﬀ beam
(dotted line) was overlapped with the probe beam on the CCD, guaranteeing thatthey crossed at the atom cloud since the imaging system was previously set atthe focal point for the cloud. The current magniﬁcation from this method is 9.4,
or 1 camera pixel = 2.5µm in the trap. This method and the dropping method
yield very similar results, so one is not clearly better than the other. They arecomplimentary though since they rely on very diﬀerent measurements. A possible
systematic aﬀecting the optical measurement is curvature in the glass windowsthat changes the angle between the two beams before the lens L1. This eﬀect
could be manifested in diﬀerent ways between the optical method and dropping
method, since they use diﬀerent axes and regions of the window(s).

4.1.4

Double condensate imaging

Early double condensate experiments used destructive imaging of ballistically expanded condensates. Ensher [61] provides an explanation of the sequence

tr a

ns

fe

rt

ub

e

61

α

condensate
position

L1

BS
M1
probe
beam

CCD

L2
β

Figure 4.4: A schematic for measurement of the imaging system magniﬁcation
using interference. The probe beam is split into two pieces which cross with angle
α where the condensate would be. They overlap again on the CCD array, but at
a diﬀerent angle β, measured by observing the fringe spacing of the interference
pattern on the CCD . The ratio of α/β determines the magniﬁcation.

62of laser pulses necessary to see either of the |1, −1 or |2, 1 states. However, toimage |1, −1 in the presence of |2, 1 it was necessary to “blow away” the |2, 1
atoms with a ∼ 2 ms pulse of MOT light about 10 ms before the |1, −1 atoms
were imaged (the expansion time was typically 22 ms, so the MOT pulse occurred
after ∼ 12 ms of expansion). For signiﬁcant attenuation (8 × 10−3 ) of the MOT
beams this was suﬃcient to not disturb the |1, −1 atoms.
In order to image both states in a single shot a similar technique was used.
The |2, 1 atoms were ﬁrst imaged using a F = 2 → F  = 3 probe pulse. About 1
ms later the attenuated MOT light was ﬂashed on for 2.8 ms to discard the |2, 1
atoms as described above. One millisecond after that, the |1, −1 atoms wereimaged by applying a short repump pulse and then the probe pulse [61]. Since
there is signiﬁcant time between the imaging of the states, the |1, −1 condensate
expands and drops further as seen in Fig. 4.5.
4.2

Polarization Imaging
Polarization imaging was used as a ﬁrst attempt at non-destructive imaging

of a condensate. When a condensate is in a single state, the Clebsh-Gordon
coeﬃcients K (Fig. 4.6 and Tab. 4.1) are in general diﬀerent for left (σ− ) andright (σ+ ) circularly polarized light. If the input probe beam is linearly polarized,then the output beam will have its polarization axis rotated slightly, since theσ + and σ − components experience diﬀerent phase shifts through the condensate.
This is the same birefringent process exploited in polarization spectroscopy [55].
For light with frequency ω and detuning ∆ from the atomic transition, thephase shift [10] through a media of length d isdω
Re (n(∆))
c
∆i
Ne2 
Ki
Re (n(∆)) = 1 +28π+0 mω i ∆i + (γ/2)2
φ =(4.10)

63

Figure 4.5: Image of both the |2, 1 (top) and |1, −1 (bottom) condensates after
dropped from the magnetic trap. In the trap the |2, 1 sits below the |1, −1 .

64Figure 4.6: Branching ratios for 87 Rb .

65

state
σ+
σ−
π
|2, 2
1/2 1/6 1/3|2, 1 5/12 1/4 1/3|1, −1 5/12 1/4 1/3

Table 4.1: The eﬀective Clebsh-Gordon coeﬃcients for light far detuned from the
5P3/2 states.

where Nd is the integrated through density, e and m are the charge and mass ofthe electron and the sum is over all possible transitions. It must also be true that
the Clebsh-Gordon coeﬃcients are normalized



i Ki

= 1. Specializing to only

two levels (for σ + and σ − ) which are assumed to be at nearly the same detuningfrom the initial state (as compared to ∆), the diﬀerence in phase shift isδφ = σ0∆
Nd(K + − K − )γ 2
28π ∆ + (γ/2)(4.11)

in which σ0 is the on resonant cross-section for the atom to scatter a photon. Since
the on-resonance optical depth is simply σ0 Nd, this can be rewritten in order tocompare with destructive imaging;
ODres =

8π∆

(K + − K − ) γ

δφ(4.12)

when ∆  γ.
The setup to measure the rotation of polarization due to δφ is shown in Fig.
4.7. A Glan-laser input polarizer (P1) deﬁnes the polarization of the incomingprobe beam. The condensate is imaged when the rotating magnetic ﬁeld has thecorrect phase to point along the probe beam so that the linear polarization iscomposed of equal parts σ + and σ − . A second Glan-laser polarizer (P2) at angle
θ to the ﬁrst allows a measurement of the amount of rotation of the polarization.
For an input intensity (after the ﬁrst polarizer) I0 and a relative phase shift of δφ,the polarization is rotated by δφ/2. The intensity at the CCD is
Im = I0 cos2 (θ − δφ/2) .

(4.13)

66
As in the destructive measurements, three images are taken in order to measure
the beam intensity distribution and any oﬀsets. The optical depth can be written
in terms of the measured quantities and applied on a pixel by pixel basis (similar
to Eq. 4.1):
∆
16π
cos−1
ODres =+−(K − K ) γIshadow − Idark−θ .
Ilight − Idark(4.14)The quantity δφ/(σ0 Nd) from Eq. 4.12 (using the full sum Eq. 4.10) is
plotted in ﬁgure 4.8 for the three trapped states of 87 Rb . In (c) data is shown fora non-condensed cloud of |2, 2 atoms over regions of minimal scattering rate. For(c) the probe laser frequency was determined by measuring the beat frequency
between it and the MOT laser with an avalanche photodiode.
This signal may be used for nondestructive measurements of double condensate systems. For large detunings (∆  γ) from both the |2, 1 and |1, −1 states,
the signal for atoms in either hyperﬁne state and the trappable mF Zeeman stateisδφ =ODres γ |mF |8π ∆ 6(4.15)

where the presence of |mF | is only a useful coincidence; it has no physical meaning.
When the probe is tuned between the resonances for |1, −1 and |2, 1 , the signal
is positive for |2, 1 atoms and negative for |1, −1 atoms since the detuning ∆switches signs. This makes polarization imaging useful for double condensate
measurements since the states are distinguishable. In this case the range of phaseshift is the diﬀerence between all the atoms in |1, −1 and all the atoms in |2, 1 ;
δφ =ODres γ.
24π ∆(4.16)

67

P1

θ

B
P2
CCD
Figure 4.7: Schematic for polarization imaging. The probe beam incident fromthe upper-left travels through a polarizer (P1) to purify the polarization. It travels
through the condensate whose birefringence rotates the polarization of the probe.
The beam exits through the second polarizer (P2) which is at an angle θ to theﬁrst. The transmission through P2 depends on the amount of polarization rotation
caused by the condensate which is proportional to the column density.

68

difference in phase shift per OD (rad)

0.10

0.05

0.10

(a)
Initial state F=2, mF=2(b)

Initial state F=1, mF=-1

0.05

0.00

0.00

F'=1
F'=2
-0.05

F'=0

-0.05

F'=3

-0.10

F'=1

F'=2

-0.10
-600

-400

-200

0

200

400

-600

Detuning from F=2 to F'=3 (MHz)

-400

-200

0

200

400

Detuning from F=1 to F'=2 (MHz)
15

0.10

(c)

(d)

Initial state F=2, mF=1
10

phase shift (deg.)

0.05

0.00

F'=1

F'=2

-0.05

5

0

F'=3
-5

-0.10
-600

-400

-200

0

200

400

Detuning from F=2 to F'=3 (MHz)-2000 -1500 -1000 -5000500 1000 1500

Detuning from F=2 to F'=3 (MHz)Figure 4.8: Plots of the signal from polarization imaging as a function of detuning.
(a) For atoms in the F = 2, mF = +2 state. (b) For atoms in the F = 1, mF = −1
state. (c) For atoms in the F = 2, mF = +1 state. (d) Rotation of the probe’s
polarization as a function of detuning for atoms in the F = 2, mF = +2 state.
The data points are for a cloud with optical depth ∼ 70.

69
4.2.1

Signal-to-noise

The signal to noise for polarization imaging is
S/N = δφ(∆, ODres )1sin (θ − δφ(∆, ODres)/2)cos (θ − δφ(∆, ODres )/2) σ 2 + σ 2
L(4.17)

S

where δφ(∆, ODres) is given in Eq. 4.11. For the shot noise limit on the probe
intensity, σS2 = 1/I0 cos2 (θ −δφ(∆, ODres )) and σL2 = 1/I0 cos2 θ the signal to noiseis

S/Nshot =sin(θ − δφ(∆, ODres)/2) cos θI0 δφ(∆, ODres ) cos2 (θ − δφ(∆, ODres)) + cos2 θ(4.18)

When the noise is dominated by fringes with fractional amplitude A on the probe
intensity, the signal to noise is simply√
sin(θ − δφ(∆, ODres )/2) 2
S/Nfringe = δφ(∆, ODres )cos(θ − δφ(∆, ODres)/2) A(4.19)

Both of these have a maximum near θ = π/2 when the phase shift δφ is small.
Data was taken looking at a condensate expanded 10 ms from a 80 Hz axial
frequency trap. Figure 4.9 shows this and a ﬁt of the shot noise limiting form of
Eq. 4.18.
Birefringence of the vacuum windows contributes an oﬀset which can add
to the noise. The birefringence was measured by rotating the input polarization
with a waveplate, and correspondingly rotating the input polarizer. The output
polarizer was then rotated to ﬁnd the extinction ratio. It was found that the best
extinction ratio was for vertically polarized light, at which point the extinctionwas ∼ 100 times worse than the polarizers themselves (Glan laser polarizers were
measured to be 5 × 10−6 ). The worst extinction was 2 × 10−2 for horizontal
polarization.
Eqs. 4.18 and 4.19 indicate that the signal to noise can be shot noise limitedat θ = π/2. However the small amount of light which makes it though both

70

Figure 4.9: Plot of experimental signal to noise as a function of crossing angle θ
between the two polarizers. For this data δφ = 3.6 degrees. The solid line is a ﬁtof the shot-noise limiting form.

71
polarizers (due to window birefringence and ﬁnite polarizer extinction) meansthat the minimum noise is the shot noise (or fractional noise) on this light. These
comments are somewhat unnecessary though, since we must allow some lightthrough in order to do a pixel by pixel calibration of the light intensity; the signal
at each pixel is determined not only by the phase shift from the condensate,
but also by spatial variations in probe intensity. These variations cannot bemeasured if θ = π/2, without removing the second polarizer (which would no
doubt result in a change in the spatial variations for a real-world polarizer). Thisis a general problem with dark-ground imaging schemes, even though they yieldsuperior signal to noise in other applications (like polarization spectroscopy). It
should be mentioned that schlieren [11] imaging does not suﬀer from this sincehigh spatial frequency components do make it to the imaging plane.
4.3

Phase-contrast imaging
Phase-contrast imaging [12, 37] is another nondestructive technique cur-rently in use. It relies on a comparison between the phase shifts of the lighttraveling through the condensate versus that which does not travel through. The
eﬀect may be derived using a straight forward argument (see Appendix A for acomplete derivation).
At the plane perpendicular to the imaging axis, and just after the condensate, the electric ﬁeld can be written as
E(x, t) = E0 cos(ωt + φ(x))(4.20)

where φ(x) is a spatially dependent phase shift on the probe light from its interaction with the condensate. Assuming the phase shift is small this isE(x, t) = E0 (cos(ωt) + φ(x) sin(ωt)).

(4.21)

72
For small phase shifts this of course does not yield a signal since the time averageof |E|2 does not depend on φ(x). We require an additional phase shift in only oneof the terms in Eq. 4.21. If a π/2 shift is applied to only the second term, thenthe intensity isI(x) = I0 (1 + 2φ(x))(4.22)

which reproduces the same phase pattern as that produced by the condensate.
Since the intensity is linearly dependent on the phase shift, the signal reverses signs depending on which side of resonance the probe beam is located. Fora mixture of condensates in the |1, −1 and |2, 1 states, a beam tuned in the
middle of the transitions gives a positive signal for |1, −1 atoms and a negative
signal for |2, 1 atoms. When the states are spatially overlapping, the signal is
proportional to the population diﬀerence.
The implementation is shown in ﬁgure 4.10. Light which is not aﬀected by
the condensate (solid lines) is focused at P1 by the ﬁrst imaging lens. The part ofthe light which is diﬀracted by the condensate (dashed lines) is collimated at P1
since the condensate is at the focal point of the ﬁrst lens. These two components
are the two terms in Eq. 4.21. Since they are spatially separated by the lens, a π/2phase shift can be applied by inserting a window with a “phase dot” – a layer of
MgF with the proper thickness. The diameter of the dot must be large comparedto the waist size of the probe beam, but small compared to the diﬀracted beam.
For our experiment with beam size ∼ 2 mm and condensate size ∼ 20µm, thefocused probe is 50 µm and the diﬀracted beam is 5000 µm, providing a large
possible range for the phase dot size. We chose 100 µm diameter in order to beable to image non-condensed atom clouds (at least the colder ones) that are muchlarger in size than the condensate.
In general it is not trivial to align the phase dot with the probe beam. The

73

P1

BEC

CCD

6 cm

Phase
plate

50 cm

Figure 4.10: Schematic for phase contrast imaging. A probe beam is incident fromthe left, and obtains a spatially dependent phase shift from the condensate. It
travels through the ﬁrst imaging lens which focuses the probe light onto the phase
dot at P1. The phase dot imparts a π/2 phase shift to only the probe beam, but
not to the diﬀracted light from the condensate. The second lens completes the
microscope and images the condensate onto the CCD array.

74
most successful technique is to ﬁrst place the dot so that it is not at the focal
point of the ﬁrst lens, so the dot does not encompass the focusing probe. A bull’seye pattern is then observed between the second lens and the CCD. As the phase
dot is moved toward the beam focus the bull’s eye pattern grows until, at thefocus, the proﬁle is ﬂat.

4.3.1

Non-destructive imaging techniques

Both destructive and polarization imaging depend on atom orientation todeﬁne a symmetry axis. This is deﬁned by the direction of the rotating TOP
ﬁeld. In the extreme case where the axis switches direction (1/2 of a TOP period
later), near-resonance imaging sees a large change in the detuning from resonance,and polarization imaging sees reversed Clebsh-Gordon coeﬃcients, resulting in an
opposite polarization rotation. Linearly polarized and far detuned, phase-contrast
imaging gives nearly equal signals no matter which direction the TOP ﬁeld points(see Table 4.1). Where the other methods require strobing of the probe light inorder to wait for proper ﬁeld alignment in the rotating TOP trap, phase-contrast
images may be taken at any time, or even continuously.
Discrete, non-destructive pictures of a condensate are taken using kinetics
mode of the Princeton Instruments camera (with ST138 controller). In this mode,the camera waits for either an internal or external trigger, at which time it advances the charge on each row of pixels by a set number of rows. A region of
N rows may be deﬁned as the exposed area by physically masking the remaining
512 − N rows directly in front of the CCD. Once a trigger is received, the charge
accumulated on the N rows is advanced into the unexposed area. This advancement occurs at a user-speciﬁed speed of 19.2 × (integer multiples up to 16) µs per
row. Once the shift is complete the camera waits for a new trigger and repeats theprocess until the array is ﬁlled. The internal trigger mode has a ﬁxed repetition

75
rate, but it is simple to provide arbitrary triggers in the external trigger mode.
This latter method can be used to obtain any desired probe timing. Once the
CCD is ﬁlled, the charge is read from the entire array, digitized and sent to thecomputer for analysis. In general, we are able to get ∼ 12 to 30 pictures of anindividual condensate, depending on the condensate size (i.e. trap frequencies)and center-of-mass slosh. Figure 4.11 shows a non-destructive series of picturesof a condensate undergoing Rabi oscillations between the |2, 1 and |1, −1 states,with a Rabi period of about 5 images.
A semi-continuous movie may also be taken by scrolling the charge on the
CCD at a constant rate (or, one can think of scrolling each row discretely as anindividual frame). The scroll speed is still given by integer multiples of 19.2 µs
per row for internal triggering, which means that the total time ranges from 10 ms
to 157 ms. External triggers again enable arbitrarily long frames. Since the condensate image occupies many pixel rows, this method smears out vertical (along
the direction of scrolling) density information. However, it is extremely usefulfor quickly measuring center-of-mass slosh, and quantitatively useful in measuring Rabi frequencies. Figure 4.12 shows a continuous movie of (a) a condensate
undergoing Rabi oscillations between the |2, 1 and |1, −1 states, and (b) radial
and axial center-of-mass slosh.

4.3.2

Signal to noise

Assuming the phase dot thickness is π/2 and φ is small, the signal to noisein phase-contrast is√

I0 φ
S/Nshot = (φ + 1)(φ + 2)

(4.23)

for shot noise, and for “fringe noise” it is√
φ 2
S/Nfringe =A(φ + 1)σ(4.24)

76

Figure 4.11: Discrete, non-destructive imaging of a condensate undergoing Rabi
oscillations. For each image, the probe beam is ﬂashed on and oﬀ quickly, and theresulting CCD charge scrolled away from the exposed area. The phase contrast
signal is positive (white) for the |1, −1 state and negative (black) for the |2, 1
state. The two-photon drive is on during for the duration of the images.

77

Figure 4.12: Continuous, non-destructive images. (a) A condensate undergoing
Rabi oscillations (with collapse and revival). (b) A single condensate in the |1, −1state oscillating radially (horizontally in the picture) and axially (vertically in the
picture). The axial oscillation is manifested as an increase (decrease) in the signal
as the condensate moves with (against) the CCD scrolling.

78
The shot noise limit was experimentally observed in ﬁgure 4.13 for a trapped
condensate in a 62 Hz axial trap.
4.4

Comparisons
The next section compares the three imaging techniques based on their use-

fulness for a speciﬁc measurement. The criteria are measurement of density distributions, signal to noise, double condensate signal, and non-destructive movies.
Single and two-state condensate density distributions are best imaged after
ballistic expansion with destructive imaging. The main reason is that the imaging
resolution is ∼ 5µm, too small for a common 10 µm cloud (although reasonable for
50µm condensates in very weak traps). With destructive imaging we are sensitive
to < 5% deviations in overall condensate size when looking at collective excitations. The major drawback here is that time dependence cannot be studied using
a single condensate; each time step point must be taken with a fresh condensate,requiring more experiment stability. It is also necessary to mathematically reconstruct the condensate’s image as it was in the trap before expansion using thenonlinear Schrödinger equation. However, qualitative reconstruction is generally
possible using scaling and scattering length arguments.
Signal to noise in overall condensate number is generally the best for polarization imaging in the trap. This is the case when the noise is dominated by spatialintensity ﬂuctuations on the probe laser since the polarizers may be crossed toeliminate background light. In order to get spatial information, the proﬁle of theprobe must be ﬂat on the scale of the condensate in order to forgo the backgroundsubtraction which is not possible for crossed polarizers. If the imaging resolution
is larger than the condensate, then it is better to destructively image a dropped
cloud near resonance.
Phase-contrast is the best technique for double condensate imaging and for

79

90
80
70

signal/noise

60
50
40
30
20

room lights onlights off

1050

100 150 200 250 300 350 400
light-dark (bin 1x1)

Figure 4.13: Plot of the experimental signal to noise of a condensate in the trap,
as a function of probe light level I for phase contrast
 imaging. The solid line is aﬁt to the shot noise limit which is proportional to (I).

80
multiple images of the same condensate. Although polarization imaging has better
signal to noise for crossed polarizers, no state speciﬁc information is possible (both+ and - polarization rotations give the same signal) in that conﬁguration. Phasecontrast is still better than polarization (with uncrossed polarizers) due to its
smaller dependence on magnetic ﬁeld direction.
The biggest noise contribution is usually spatial variations in the beam intensity that change position between the condensate picture and the backgroundpicture. These seem to be due to interference implanted by multiple reﬂections
from optical surfaces, or diﬀraction from aperaturing of the beam or from small
imperfections on the optics (dust for example). The size scale is often near the
scale of whatever is being imaged, which seems to indicate that multiple patterns
and sizes are present. There are a few techniques which have helped to diminish
the eﬀect of the stripes. Changing the alignment of optics (beamsplitters, polarizers) before and after the trap has helped. Occasionally, changing the magnetic
trap parameters from the condensate image to the background image has an effect. Since the rotating magnetic ﬁeld produces audible noise, it causes vibrationsin the optics and windows, which modulates the fringe pattern. It has helped toﬁnd speciﬁc phases and amplitudes of the rotating ﬁeld to minimize the diﬀerence
between the two frames.

Chapter 5
Double Condensate experiments – density related [2]5.1

Introduction
An alluring aspect of atomic Bose-Einstein condensates has been their ideal

representation of the fundamental physics of particle statistics. Other systems
such as 3 He and superconductors mask the basic phenomenon through overwhelming inter-particle interactions, due to their relatively high densities. Atomic condensates also experience interatomic potentials, but the separation between atoms
is much larger than the scale of these interactions, making the condensate dilute inthis sense. Despite complicated interatomic potentials, the mean-ﬁeld interaction
in condensates is well-characterized by a single parameter, the s-wave scattering length a. Qualitatively, a system is dilute when the scattering “volume” ismuch smaller than the inverse density, i.e. a3 n  1. These interactions have an
important eﬀect on most condensate physics, including density and momentum
distributions [106, 80], collective excitations [91, 107], speciﬁc heat [60, 69], speedof sound [37], and limited condensate number[42]. In the present context of double
condensates, there are three important values of the scattering; the self-repulsion
for each state and the mutual repulsion between the states.
The ﬁrst demonstration of a binary mixture of condensates by Myatt etal.[108] produced overlapping condensates of the 5S1/2 |F = 1, mf = −1 and|F = 2, mf = 2 states of 87 Rb. The ratio of the magnetic moments of these states

82
is 1:2, so the condensates experience diﬀerent potentials in a magnetic trap andare displaced unequally from the trap center by gravity. Due to an accidental coincidence between the singlet and triplet scattering lengths of 87 Rb, collisional loss
is reduced and any mixture of spin states will be relatively long-lived [108, 94, 45].
Here, we use mixtures of |1, −1 and |2, 1 states, which possess several advantages. First, these two states have essentially identical magnetic moments, andhence feel identical conﬁning potentials. Second, one can conveniently and quickly
change atoms from the |1, −1 state to the |2, 1 state by a two-photon transition(microwave plus rf). Finally, we can selectively image the diﬀerent components
using appropriately tuned lasers.
Using the two-photon drive the |1, −1 and |2, 1 states can be treated asa two level system. When the two-photon drive is oﬀ the energy diﬀerence ismuch larger than the energy available from kinetic or potential contributions (afew kHz compared to 6.8 GHz), so that two condensates in each of these states
are distinguishable, scalar condensates. In the presence of the drive, atoms interconvert between the hyperﬁne states coherently. This chapter is not concernedwith the phase or coherence between the two states, but only with their density
proﬁle and dynamics as governed by the scattering lengths.
The evolution of the double condensate system, including release from thetrap and subsequent expansion [79], is governed by a pair of coupled GrossPitaevskii equations for condensate amplitudes Φi :∂Φih̄2 ∇2= −+ Vi + Ui + Uij Φi
ih̄
∂t
2m

(5.1)

where i, j = 1, 2 (i = j), Vi is the magnetic trapping potential for state i, themean-ﬁeld potentials are Ui = 4πh̄2 ai |Φi |2 /m and Uij = 4πh̄2 aij |Φj |2 /m, m is themass of the Rb atom, and the intraspecies and interspecies scattering lengths are
ai and aij . This equation is numerically integrated by various authors to compare

83with experiment.
The following experiments are all done on a time scale faster than the decayof atoms from inelastic processes. In Eq. 5.1 it is the real part of a which governs
the elastic, and the imaginary part which determines the inelastic collisions. For
our system of condensates, it is the inelastic spin-exchange process which occurs
the fastest, whereby two |2, 1 atoms collide to produce one |2, 0 and one |2, 2
atom (or possibly |1, 0 and |2, 2 ). These atoms are eﬀectively lost from the trap.
We have measured an upper limit to this loss rate as 3 × 10−13 cm3 /s.
5.2

Dynamical response-instantaneous change in scattering length
As an initial experiment we apply a short π-pulse to convert the initial|1, −1 condensate instantaneously into a |2, 1 condensate. Our analysis of theensuing behaviour does not require the interatomic term in Eq. 5.1 since thetwo states do not exist at the same time. Because these two states have slightly
diﬀerent values of a, the sudden change in self-interaction gives rise to oscillatory
spatial behavior of the condensate wave function [96]. The scattering length ratio can be extracted from a model using analytical equations of motion for thecondensate widths [96, 95, 52] based on Gross-Pitaevskii theory.
We have investigated the dynamical behavior of the condensate after morethan 99.5% of the atoms are transferred from the |1, −1 state to the |2, 1 state.
After the transfer, the condensate is allowed to evolve in the trap for a time interval T before the cloud is released from the trap and probed. The experiment isrepeated for diﬀerent values of T , and the axial and radial widths of the condensate are measured using a ﬁt to the two-dimensional condensate image, using the
integrated-through Thomas-Fermi form. Fig. 5.1 illustrates the time dependenceof both the axial and radial size of the atom cloud. Qualitatively, the data forboth dimensions are consistent with a “compression” oscillation. The condensate

84
shrinks for very early times, indicating a weaker mean-ﬁeld repulsion in the |2, 1
state.
In order to describe the response to the change in scattering length, we
use the Gross-Pitaevskii (GP) equation for the condensate wave function, with atime-dependent interaction term:h̄2 ∇2∂Φ(r, t)= −+ V (r) + U(t)|Φ(r, t)|2 Φ(r, t) ,
ih̄
∂t
2m

(5.2)

where U(t) = 4πh̄2 a(t)/m. For t < 0, a(t) = a1 , the scattering length for |1, −1
atoms on |1, −1 atoms. For t > 0, a(t) = a2 , the |2, 1 on |2, 1 scattering
length. The condensate is formed in its ground-state in which the density proﬁlen(r) = |Φ(r)|2 is constant in time, and with the axial and radial widths (wz , wr )determined by V (r), a1 , and the number of atoms N. After a sudden change ina(t), Φ(r, t) is projected onto a coherent superposition of its new ground-state andcollective vibrational modes. Instead of solving the complete GP equation, one
can use the Thomas-Fermi (TF) approximation, which corresponds to neglecting
the quantum pressure term in the kinetic energy of the condensate. In this case,
one can replace the GP equation with a pair of scalar equations of motion for thecondensate widths [96, 95, 52]:15h̄2 Na(t)ẅi + (2πνi ) wi −75/2 m22Here the widths wr =r 2  and wz =

1
wr2 wz wi

=0.

(5.3)

z 2  are the radial and axial rms radii,

and the subscript i is to be replaced with either r or z for the respective widths.
The TF approximation holds when N is large [13] and, in this limit, the GP theory
coincides with the hydrodynamic theory of superﬂuids [130].
Time-dependent behavior is predicted by numerically integrating Eqs. 5.3.
The initial conditions (t < 0) are that the widths are at the values determined by
the ground-state solution, ẇr = ẇz = 0, and a(t) = a1 . After t = 0, a(t) = a2 and85
the integration proceeds for time T . We model the removal of the trap by setting
νr = νz = 0 and continue the numerical integration for the 22 ms free expansion.
The axial and radial widths subsequently expand with diﬀerent speeds. In the
TF approximation, the ratio between the two expanded widths, averaged over T ,can be shown to depend only on νr , νz and the expansion time [52]. In our case,the theory predicts wz /wr = 1.29, in good agreement with the measured value
wz /wr = 1.31. The oscillations of the widths correspond to a superposition of
two m = 0 modes. Since no angular momentum is imparted to the condensate by
the change of scattering length, these are the only modes excited. The calculated
mode frequencies are ν/νr = 1.80 and ν/νr = 4.99, and turn out to be independentof the amplitude in the range considered here. The two modes contribute to theoscillations in z and r with a diﬀerent phase and amplitude, the axial motion
behaving mainly as the fast mode and the radial motion as the slow one.
In each direction, the ratio of the amplitudes of the two modes is predicted
to be constant over the range of possible scattering lengths relevant to the experiment. Thus, only the initial size and an overall amplitude are used as adjustableparameters in comparison of theory with experiment. The solid lines in Fig. 5.1
show the theoretical prediction using the best ﬁt value of the oscillation amplitude,
which is related to the scattering length ratio a1 /a2 . The predicted oscillations
agree remarkably well with the shape of the data, for both the frequency andthe phase of the two modes. From the ﬁt amplitude we obtain the ratio of thescattering lengths a1 /a2 = 1.062 ±12, which is consistent with the ratio of 1.059+4−6
obtained in a theoretical calculation of binary collision parameters [46, 131].
Analysis of the data is complicated slightly by an observed systematic dependence of the oscillation amplitude on the magnitude of the rf drive during thetwo-photon pulse. We believe that this eﬀect is partly due to a small dressingof the atoms to the |1, 0 or |2, 0 state by the rf, and partly due to a coupling

86

56

axial

72

radialsize (µm)

54
70
52
68
50
66
0

1

2

3
-1

time (units of ω r )

4

0

1

2

3

4

-1

time (units of ω r )Figure 5.1: Oscillation in the width of the cloud in both the axial and radial
direction due to the instantaneous change in scattering length. The widths arefor condensates as a function of free evolution time in units of the radial trap
period (ωr−1 = 9.4 ms), followed by 22 ms of ballistic expansion. Each point is
the average of approximately 10 measurements. Note that the fractional change
in width is quite small. The solid line is a ﬁt of the model to the data, with only
the amplitude of the oscillation and the initial size as free parameters.

87of the rf onto electronics controlling the trap potential. Both result in a change
in strength of the conﬁning potential during the two-photon pulse which returns
to normal when the pulse is complete. After this impulsive perturbation, theBEC can be thought of as freely evolving with an initial “velocity” in the width(ẇr , ẇz = 0). This is in contrast to the discrete change in scattering length, inwhich the BEC width has an initial oﬀset from the equilibrium width, but no initial velocity. The systematic is manifested as an initial oﬀset in the phase of theresulting oscillation, which is indeed observed for large rf amplitudes. Our result
for the ratio was obtained by extrapolating to zero rf amplitude with a quadraticﬁt as shown in Fig. 5.2.
5.3

Mixtures of condensates
In this experiment, we report results from initial studies of simultaneously

trapped BECs in the |2 and |1 states. The condensates begin with a welldeﬁned relative phase, spatial extent, and “sag” — the position at which themagnetic trapping forces balance gravity for each state. The ﬁne experimental
control of this double condensate system permits us to study its subsequent timeevolution under a variety of interesting conditions, most notably those in which
there remains substantial spatial overlap between the two states.
The double condensate system is prepared from the single |1 condensate
by driving the two-photon transition. We are able to transfer quickly any desired
fraction of the atoms to the |2 state by selecting the length and amplitude of
the two-photon pulse. The two condensates are created with identical density
distributions, after which they evolve and redistribute themselves for some time
T . We then turn oﬀ the magnetic trap and allow the atoms to expand for 22 ms
for imaging.
We selectively image the densities of either of the two states (n1 and n2 ) or

88

0.03

fractional amplitude

0.020
0.015

0.02
0.010
0.01
0.005

radial

axial
0.000
0.0

0.1

0.2

rf amplitude (arb.)

0.3

0.00
0.0

0.1

0.2

0.3

rf amplitude (arb.)

Figure 5.2: Dependence of the oscillation amplitude on the rf strength in the twophoton transition. The vertical axis represents the ﬁtted value of the fractional
excitation amplitude, with error bars from uncertainty in the ﬁt. The solid line
shows extrapolation to zero rf amplitude.

89
the combined density distribution (nT ) by changing the sequence of laser beams
applied to the condensates for probing (see Chapter 4). Since the expansion andimaging are destructive processes, each image is taken with a diﬀerent condensate;
the excellent reproducibility of the condensates permits us to study the timeevolution of the system by changing the time T . The images of the condensates
do not always appear in the same absolute locations on the CCD-array detector,
however, and we compensate for this shot-to-shot jitter by reconstructing therelative positions of the condensates from the images of n1 , n2 , and nT at eachtime T .
The evolution of the double condensate system, including the release fromthe trap and subsequent expansion [36, 79], is governed by a pair of coupled
Gross-Pitaevskii equations in Eq. 5.1. In the Thomas-Fermi limit, the condensate
density distributions are dominated by the potential energy terms of Eq. (5.1).
Consequently, the expanded density distributions retain their spatial informationand emerge with their gross features (such as the relative position of the condensates) intact.
The similarity in scattering lengths a1 , a2 , and a12 implies that the total
density nT will not change signiﬁcantly from its initial conﬁguration even though
the two components may redistribute themselves dramatically during the evolution time T . In 87 Rb, the scattering lengths are known at the 1% level to bein the proportion a1 : a12 : a2 :: 1.03 : 1 : 0.97, with the average of the three
being 55(3) Å [46, 105]. The near-preservation of the total density nT can be approached theoretically by deriving from Eq. (5.1) the hydrodynamic equations of
motion [136] for nT and evaluating them in the limit that the fractional diﬀerences
between the scattering lengths are small. The pressures that tend to redistributenT must also be small. A similar argument pertains if the minima of the trapping
potentials V1 and V2 are displaced from one another (see below) by a distance that90
is small compared to the size of the total condensate; once again, the eﬀects on
the equilibrium distribution of the individual components may be profound but
the total density should remain largely unperturbed [63].
The rotating magnetic ﬁeld of the TOP trap gives rise to a subtle behavior
that permits us to displace the minima of the trapping potentials V1 and V2 withrespect to one another (Chapter 3). By adjusting trap parameters, we can change
the sign of the relative sag or cause it to vanish while preserving (to ﬁrst order)√
the same radial (νr ) and axial (νz = 8νr ) trap oscillation frequencies.
In a ﬁrst experiment, we choose a trap that has zero relative sag (νz = 47 Hz)
and transfer 50% of the atoms to the |2 state with a ∼ 400 µs pulse. When
T = 30 ms, we observe a “crater” in the image of the |1 atoms (Fig. 5.3a). The“crater” corresponds to a region occupied by the |2 atoms (Fig. 5.3b), indicatingthat the |1 atoms have formed a shell about the |2 atoms. This is consistent
with the theoretical prediction that it is energetically favorable for the atoms withthe larger scattering length (|1) to form a lower-density shell about the atoms
with the smaller scattering length (|2) [119]. At later times the |2 atoms breakradial symmetry and drift transversely away from the center of the cloud, possibly
due to a tilt of the rotating TOP ﬁeld with respect to the quadrupole ﬁeld axis
shown in Fig. 5.4.
In order to explore the boundary between the two condensates, we perform
a series of experiments in a trap in which we displace the trapping potentials suchthat the minimum of V2 is 0.4 µm lower than that of V1 , or approximately 3%of the (total) extent of the combined density distribution in the vertical direction. The subsequent time-evolution of the system is shown in Figs. 5.5 and 5.6.
The two states almost completely separate (Fig. 5.5a-c) after 10 ms; they then“bounce” back until at T = 25 ms the centers-of-mass are once more almost exactly superimposed (Fig. 5.6), although a distinctive (and reproducible) vertical

91

(a)

(b)
|1〉

(c)

|1〉

|2〉

Figure 5.3: (a) The image of the |1 condensate exhibits a “crater,” corresponding
to a shell in which the |2 atoms (b) reside. For this trap, νz = 47 Hz with zero
relative sag. By changing the strength of the magnetic quadrupole ﬁeld we canintroduce a nonzero relative sag, which shifts the location of the “crater” (c).
(Each square in this post-expansion image is 136 µm on a side.)92

Figure 5.4: A streak camera image of a 50-50 mixture of |1, −1 and |2, 1 atoms
in a condensate. The trap is 47 Hz axial. Time increases from the top to thebottom (total time is 154 ms), and left-right is the “radial” direction. At the
top of the frame, a |1, −1 condensate (white) is put into a 50-50 mixture by aπ/2 pulse. At the end of the pulse the condensate becomes grey; invisible to thephase-contrast imaging. As time progresses we begin to see separation of the twostates, with the |1, −1 atoms favoring a shell outside the |2, 1 atoms. For evenlonger times the cylindrical symmetry is broken as the |2, 1 atoms move to theright.

93

Figure 5.5: Time-evolution of the double-condensate system with a relative sagof 0.4 µm (3% of the width of the combined distribution prior to expansion) anda trap frequency νz = 59 Hz.

94
structure has formed (Fig. 5.5d-e-f). By T = 65 ms, the system has apparently reached a steady state (Fig. 5.5g-h, Fig. 5.6) in which the separation of thecenters-of-mass is 20% of the extent of the entire condensate. From these images
we observe: (i) the fractional steady-state separation of the expanded image islarge compared to the fractional amount of applied symmetry breaking, as we
expect for a repulsive interspecies potential; (ii) the placid total density proﬁle(rightmost column of Fig. 5.5) betrays little hint of the underlying violent rearrangement of the component species; and (iii) the component separation is highly
damped, although it is not yet certain what mechanism [124, 117, 96, 116, 103, 65]
is responsible. With respect to the damping, the excitation is not small and may
therefore be poorly modeled by theories that treat the low-lying, small-amplitude
excitations [70, 47, 72, 62, 118] of double condensates.
It should be noted that after many months since this data was taken, thelong time behavior is not reproduced. The qualitative behavior is the same but
no longer do we see the center of mass of the two states damp as signiﬁcantly
as in Fig. 5.5. A few experimental details have changed. The rotating TOP
ﬁeld is controlled by electronics with much less noise. We have also adjusted theroundness of the TOP ﬁeld. It is possible that our evaporation has changed sothat a smaller thermal fraction remains now than at the time of this experiment.

Finally, we show the optical density as a function of relative number and position on the condensate vertical axis in order to better appreciate the amount ofoverlap between the two states at T = 65 ms (Fig. 5.7), which remains substantial
despite the underlying separation. Each plot is averaged across a ∼ 14 µm widevertical cut through the centers of the two condensates. From the overlap shown,
one could determine the magnitude of the interspecies scattering length a12 bycomparison to theoretical calculations conducted within the Thomas-Fermi ap-

95

20

|2Ô
|1Ô

vertical position (µm)

15
10
5
0
-5
-10
-15
-20

0

20

40
60
time (ms)80

100Figure 5.6: The relative motion of the centers-of-mass of the two condensates
under the same conditions as those in Fig. 5.5.

96
proximation [78] and numerical solutions of the Gross-Pitaevskii [118] or HartreeFock [64] equations.

97

optical density

2.0 (a) N|2Ô / N|1Ô = 2

|2Ô
|1Ô
sum
T-F fit

1.5
1.0

T = 65 ms

0.5
0.0
-150

-100

-50

0

50

100

150

0

50

100

150

0

50

100

150

optical density

2.0 (b) N|2Ô / N|1Ô = 1
1.5
1.0
0.5
0.0
-150

-100

-50

optical density

2.0 (c) N|2Ô / N|1Ô = 0.5
1.5
1.0
0.5
0.0
-150

-100

-50

vertical position (µm)

Figure 5.7: Vertical cross-sections of the density proﬁles at T = 65 ms for diﬀerent
relative numbers of atoms in the two states. The combined density distribution(solid line) is shown for comparison to the Thomas-Fermi parabolic ﬁt (dashedline). The trap parameters are the same as those in Fig. 5.5.

Chapter 6
Double condensate experiments – phase related [3]The relative quantum phase between two Bose-Einstein condensates is expected to give rise to a variety of interesting behaviors, most notably those analogous to the Josephson eﬀects in superconductors and superﬂuid 3 He [40]. Experiments with condensates realized in the dilute alkali gases [36, 54, 43] have recently
drawn considerable theoretical attention, with a number of papers addressing
schemes [88, 83, 123] by which to measure the relative phase. Two independent
condensates are expected to possess [41] (or develop upon measurement [87, 49])a relative phase which is essentially random in each realization of the experiment.
The experimental observation at MIT of a spatially uniform interference patternformed by condensates released from two independent traps conﬁrms the existenceof a single relative phase [14]. In the next three chapters the phase is measured
and manipulated in ways which have strong analogies to liquid helium physics. In
this ﬁrst experiment, we use an interferometric technique to measure the relative
phase (and its subsequent time-evolution) between two trapped condensates [15]that are created with a particular relative phase. This system permits us to characterize the eﬀects of couplings to the environment on the coherence [16] between
the condensates.
As in our previous work [105, 76], we create a condensate of approximately
5 × 105 Rb-87 atoms, conﬁned in the |F = 1, mf = −1 (|1) state in a TOP mag-99
netic trap. The rotating magnetic ﬁeld (νAF = 1800 Hz) is ramped to 3.4 G and
the quadrupole gradient to 130 G/cm, resulting in a trap with an axial frequency
νz = 59 Hz. The ﬁelds are chosen to make the hyperﬁne transition frequency
nearly ﬁeld-independent [75]. We create the second condensate by applying ashort (∼ 400 µs) two-photon pulse that transfers 50% of the atoms ( π2 -pulse) fromthe |1 spin state to the |F = 2, mF = 1 (|2) spin state. The coupling drive has
an eﬀective frequency of 6834.6774 MHz and is detuned slightly (∼ 100 Hz) fromthe expected transition frequency in our trap [17]. After an evolution time T
and an optional second π2 -pulse, we release the condensates from the trap, allow
them to expand, and image either of the two density distributions [105]. The
post-expansion images preserve the relative positions and gross spatial features of
the condensates as they were in the trap [76, 18].
The evolution of the double condensate system, including the coupling drive,is governed by a pair of coupled Gross-Pitaevskii equations for condensate amplitudes Φ1 and Φ2 :ih̄Φ̇1 = (T + V1 + U1 + U12 ) Φ1 +h̄Ω(t) iωrf t
e Φ2
2

(6.1)

and
ih̄Φ̇2 = (T + V2 + Vhf + U2 + U21 ) Φ2 +h̄Ω(t) −iωrf t
e
Φ1
2

(6.2)

where T = −(h̄2 /2m)∇2 is the kinetic energy, m is the mass of the Rb atom, Vhf
is the magnetic ﬁeld-dependent hyperﬁne splitting between the two states in theabsence of interactions, condensate mean-ﬁeld potentials are Ui = 4πh̄2 ai ni /mand Uij = 4πh̄2 aij nj /m, ni = |Φi |2 is the condensate density, and the intraspecies
and interspecies scattering lengths [105, 76] are ai and aij = aji . For the trap
parameters given above, the harmonic magnetic trapping potentials V1 and V2 aredisplaced from one another by 0.4 µm along the axis of the trap [75]. The coupling
drive is represented here in the rotating wave approximation and is characterized

100by the sum of the microwave and rf frequencies, ωrf , and by an eﬀective Rabi
frequency Ω(t), whereΩ(t) =

 2π · 625 Hz,

coupling drive on;coupling drive oﬀ.

0,

(6.3)

Phase-sensitive population transfer between the |1 and |2 states occurs with thedrive on, but the two condensates become completely distinguishable once thedrive is switched oﬀ [76]. The ﬁrst π2 -pulse [Fig. 6.1(b)] creates the |2 condensate
with a repeatable and well-deﬁned relative phase with respect to the |1 condensate
at t = 0. The relative phase between the two condensates subsequently evolves
at a rate proportional to the local diﬀerence in chemical potentials between thetwo condensates ω21 (r, t), which in general is a function of both time and space.
Couplings to the environment [20] can induce an additional (and uncharacterized)
diﬀusive precession of the relative phase, leading to an rms uncertainty in its
value ∆ϕdiﬀ [101, 21]. After an evolution time T , therefore, the condensates haveaccumulated a relative phaseT0ω21 (r, t) dt + ∆ϕdiﬀ (T ). During the same time,

the coupling drive accumulates a phase ωrf T . A second π2 -pulse [Fig. 6.1(e)] thenrecombines the |1 and |2 condensates, comparing the relative phase accumulatedby the condensates to the phase accumulated by the coupling drive. The resulting
phase-dependent beat note is manifested in a diﬀerence in the condensate density
between the two states. Immediately after the second pulse the density in the |2state (n2f ) is11n2f (r) = n1 (r)+ n2 (r)+ n1 (r)n2 (r) cos
2
2

 
T
0



ω21 (r, t) dt − ωrf T + ∆ϕdiﬀ (T ) .
(6.4)

In this equation, ni denote the densities prior to the application of the second π2 pulse. The interference term in Eq. 6.4 shows that measurement of n2f (r) in theoverlap region is sensitive to the relative phase. Each realization of the experiment(with a freshly-prepared condensate) yields a measurement of the relative phase

101

n2
(a)

t<0
z

n1
z

 -pulse

(b)

t=0

(c)

0 < t d 45 ms(d)

45 ms d t < T -pulse(e)t=T
expansion

(f)

t>T

Figure 6.1: A schematic of the condensate interferometer [19]. (a) The experiment
begins with all of the atoms in condensate |1 at steady-state. (b) After the ﬁrst π2 pulse, the condensate has been split into two components with a well-deﬁned initial
relative phase. (c) The components begin to separate in a complicated fashion due
to mutual repulsion as well as a 0.4 µm vertical oﬀset in the conﬁning potentials(see also Fig. 3 of Ref. [76]). (d) The relative motion between the components
eventually damps with the clouds mutually oﬀset but with some residual overlap.
Relative phase continues to accumulate between the condensates until (e) at timeT a second π2 -pulse remixes the components; the two possible paths by which thecondensate can arrive in one of the two states in the hatched regions interfere. (f)The cloud is released immediately after the second pulse and allowed to expand
for imaging. In the case shown, the relative phase between the two states at thetime of the second pulse was such as to lead to destructive interference in the |1state and a corresponding constructive interference in the |2 state.

102
for a particular T ; by varying T , we can measure the evolution of the relative
phase.
At short times T , for which the overlap between the condensates remains
high, varying the moment at which the second π2 -pulse is applied causes an oscillation of the total resulting number of atoms in the |2 state. The oscillation occurs
at the detuning frequency δ = ω21 − ωrf and is completely analogous to that observed in separated-oscillatory-ﬁeld measurements in thermal atomic beams [121]or in cold (but noncondensed) atoms in a magnetic trap [50]. The fringe contrast,
initially 100%, decreases as the condensates separate. After ∼ 45 ms the relative
center-of-mass motion damps and comes to equilibrium, leaving the components
with a well-deﬁned overlap region at their boundary, as shown in Figs. 6.1(d)and 6.2(a); see also Fig. 5(b) of Ref. [76]. Application of a second π2 -pulse at
T > 45 ms results in a density proﬁle in which the interference occurs only in theoverlap region; see Figs. 6.1(f) and 6.2(b). We look at the density of atoms in the|2 state at the center of the overlap region [22] in order to examine the intriguing
issue of the reproducibility of the relative phase accumulated by the condensates
during the complicated approach to equilibrium. If the phase diﬀusion term in
Eq. 6.4 is so large that the uncertainty is greater than π, then repeated measurements for the same values of T will yield an incoherent (i.e., random) ensemble ofinterference patterns. In the opposite extreme, (i.e., very little phase diﬀusion),
repeated measurements will give essentially the same interference pattern at T ineach experimental run. We plot the optical density in the center of the overlap
region as a function of T in Fig. 6.3, and observe an oscillation at the detuning
frequency with a visibility of approximately 50%, corresponding to an rms phase
diﬀusion ∆ϕdiﬀ (T ) < π3 . At longer times the maximum contrast observed in a single realization of the experiment decreases slightly, possibly due to the increasingpresence of thermal atoms as the condensates decay.

103

2.5

(a) before second π /2 - pulse

|2Ô
|1Ô

2.0
1.5

optical density

1.0
0.5
0.0
2.5
(b) after second π /2 - pulse

|2Ô
|1Ô

2.0
1.5
1.0
0.5
0.0
-100

-50

0

50

100

vertical position (µm)
Figure 6.2: (a) The post-expansion density proﬁles of the condensates in thesteady-state attained after a single π2 -pulse. These density proﬁles vary little fromshot-to-shot (and day-to-day). (b) The density proﬁles after the second π2 -pulse.
The density in the overlap region depends on the relative phase between the twocondensates at the time of the pulse; in the case shown, we observe constructive
interference in the |2 state and destructive interference in |1. The patterns in (b)are much less stable than those in (a), possibly as a result of unresolved higherorder condensate excitations, issues associated with the expansion, or technicalinstabilities of the apparatus.

104
The stable interference patterns show that the condensates retain a clear
memory of their initial relative phase despite the complicated rearrangement dynamics of the two states following the ﬁrst π2 -pulse. This is rather surprising,since the center-of-mass motion of the double condensate system is strongly (andcompletely) damped, and, in general, decoherence times in entangled states tend
to be much shorter than damping times [48, 132, 44]. The intuition one develops
in understanding few-particle quantum mechanics may not apply to experiments
involving condensates. The phase between the two condensates seems to possess arobustness which preserves coherence in the face of the “phase-diﬀusing” couplings
to the environment.
The oscillation pattern of Fig. 6.3 seems to wash out by 100 ms. The peaks
and valleys still retain nearly the full contrast, but any given location in time
will show a random phase relationship between the two condensates. This can be
explained as a π variation in phase at 100 ms, implying a 2π × 5 Hz ×h̄ variationin the relative energy of the two condensates. This is likely due to magnetic ﬁeld
ﬂuctuations in instead of phase diﬀusion or decoherence from thermal atoms [71].
For the trap of 130 G/cm and a rotating ﬁeld of 3.4 G, the ﬁeld sensitivity of thetwo-photon transition is ∼ 250 Hz/G, implying ﬁeld ﬂuctuations of 20 mG. A 10%
variation in the quadrupole, and a 0.6% variation in the rotating ﬁeld magnitude
can explain this if the time scale is between ∼ 100 ms and ∼ 3 minutes . If it were
faster than 100 ms it would average out, and if it were longer than 3 minutes,
we would notice that successive shots where reproducible. Measurement indicatethat a 10% variation of the quadrupole ﬁeld is unreasonable, but the rotating ﬁeld
changes by 1 to 2 % on a time scale of ∼ 200 ms. It is therefore very reasonable
that our system has irreproducible phase from technical issues.

105

1.00
0.75

optical density (arbitrary units)

0.50
0.25

(a)
0.00
1.00
0.75
0.50
0.25

(b)
0.00

64

66

68

70

72

pulse separation (ms)Figure 6.3: The value of the condensate density in the |2 state is extracted at
the center of the overlap region (inset) and plotted (a) as a function of T . Eachpoint represents the average of 6 separate realizations and the thin bars denote therms scatter in the measured interference for an individual realization. The thicklines are sinusoidal ﬁts to the data, from which we extract the angular frequency
ω21 − ωrf . In (b), the frequency of the coupling drive ωrf has been increased by
2π · 150 Hz, leading to the expected reduction in fringe spacing.

Chapter 7
Unique phase patterns [4]Quantization and persistence of current in superconductors and superﬂuids
can be understood in terms of the topology of the order parameter. Current arises
from a gradient in the phase of the order parameter. Quantization of ﬂow arounda closed path is a consequence of the requirement that the order parameter besingle-valued; metastability, or “persistence”, arises from the fact the number ofphase windings (the multiple of 2π by which the phase changes) around the pathcan be changed only by forcing the amplitude of the order parameter to zeroat some point. If the energy this requires exceeds that available from thermal
excitations, then the current will be immune to viscous damping. This familiar
argument relies, however, on the order parameter’s belonging to a very simple
rotation group. The order parameter in superﬂuid 4 He, for example, is a single
complex number. Its phase can be thought of as a point lying somewhere on acircle, and is subject to the topological constraints mentioned above. The orderparameter of a more complicated superﬂuid, on the other hand, will in general
be capable of ridding itself of unwanted kinetic energy by moving continuously
through a higher-dimensional order-parameter space in such a way as to reduce,even to zero, its winding number. Presumably this ability will reduce a superﬂuid’s
critical velocity; in the limit that the order-parameter space is fully symmetric,
the critical velocity may even vanish [23].

107
7.1

Twisting the phase
Our experiments utilize a gas-phase Bose-Einstein condensate with two in-

ternal levels [24]. This is equivalent to a spin-1/2 ﬂuid: the order parameter has
SU(2) rotation properties. A diﬀerential torque across the sample is applied tothe order parameter so that with time it becomes increasingly twisted. Eventually the sample distorts through SU(2) space so that the steadily applied torque
now has the eﬀect of untwisting the order parameter, which returns nearly to its
unperturbed condition. The pattern of twisting and then untwisting is manifested
experimentally as a washing-out followed by a recurrence of an extended series ofoscillations in the population between the spin states. Related behavior has beenpreviously observed in A-phase 3 He [111]; a major diﬀerence in this work is that
we can directly observe components of the order parameter with temporal andspatial resolution.
Magnetically conﬁned 87 Rb can exist in a superposition of the two internal
states |1 and |2. The two internal states are separated by the relatively large
87

Rb hyperﬁne energy, but in the presence of a near-resonant coupling ﬁeld thestates appear, in the rotating frame, to be nearly degenerate. The condensate canthen dynamically convert between internal states. The order parameter for thecondensate is the pair of complex ﬁeld amplitudes Φ1 and Φ2 of states |1 and |2.
Evolution of these ﬁelds is governed by a pair of coupled Gross-Pitaevskii equations
which model the coupling drive, the external conﬁning potential, kinetic energy
eﬀects and mean-ﬁeld interactions [50, 76, 57]. The SU(2) nature of the orderparameter (Φ1 ,Φ2 ) is more evident if we write
Φ1 = cos(θ/2)e−iφ/2 nt eiα(7.1)

Φ2 = sin(θ/2)eiφ/2 nt eiα

(7.2)

1/2

and
1/2

108
where θ, φ, nt , and α are purely real functions of space and time. θ and φ give therelative amplitude and phase of the two internal components, and may be thoughtof respectively as the polar and azimuthal angles of a vector whose tip lies on asphere in SU(2) space. The total density and mean phase, nt and α respectively,
remain relatively constant [75] during the condensate evolution described in thispaper.
The starting point for the measurements is a magnetically conﬁned cloudof ∼ 8 × 105 evaporatively cooled, Bose-Einstein-condensed 87 Rb atoms near zerotemperature. The combined gravitational and magnetic potentials [114] yield anaxially symmetric, harmonic conﬁning potential V1 (V2 ) for particles in the |1(|2) state, in which the aspect ratio of the axial oscillation frequency in the trap
to the radial frequency ωz /ωr can be varied from 2.8 to 0.95 [61]. V1 and V2 arenearly identical but can optionally be spatially oﬀset a distance z0 in the axial
direction (see Chapter 3). The coupling ﬁeld has a detuning δ from the local|1 to |2 resonance. If z0 is nonzero, δ depends on the axial position z, with
δ(z) − δ(z = 0) linear in z and in z0 . The strength, characterized by the Rabi
frequency Ω, of the coupling ﬁeld also varies with an axial gradient.
We are able to measure the population of both spin states nondestructively
using phase-contrast microscopy [12, 38]. We tune the probe laser between theresonant optical frequencies for the |1 and |2 states. Since the probe detuning
has opposite sign for the two states, the resulting phase shift imposed on theprobe light has opposite sign, such that the |1 atoms appear white and the |2
atoms appear black against a gray background on the CCD array. We can acquire
multiple, nondestructive images of the spatial distribution of the |1 and |2 atoms
at various discrete moments in time, or we can acquire a quasi-continuous time
record (streak image) of the diﬀerence of the populations in the |1 and |2 states,
integrated across the spatial extent of the cloud.

109

Figure 7.1: (a) With the trap parameters adjusted for high spatial uniformity inΩeﬀ , we drive the coupling transition and record a streak-camera image of 60 Rabi
oscillations between the |1 (white) and |2 (black) states. The vertical dimensionof the ﬁgure is 80 µm. (b) The value of δN, the total number of atoms in |2
minus the total in |1, is extracted from the image in part (a). The contrast ratio
remains near unity — observed loss of signal is due to overall shrinkage of thecondensate through collisional decay.

The eﬀect of the coupling drive is to induce a precession of the order parameter at the local eﬀective Rabi frequency Ωeﬀ (z) ≡ (Ω(z)2 + δ(z)2 )1/2 . In apreliminary experiment, we chose parameters so as to make Ωeﬀ nearly uniform,
with ωz = 2π × 63Hz, ωr = 2π × 23 Hz, Ω

2π × 340 Hz and δ(z)0. A conden-sate at near-zero temperature was prepared in the pure |1 state. The coupling
drive was then turned on suddenly, inducing an extended series of oscillations of
the total population from the |1 to the |2 state (“Rabi oscillations”) [Fig. 7.1].
The robustness of the Rabi oscillations is proof that our imaging does not signiﬁcantly perturb the quantum phase of the sample [25](population transfer via Rabi
oscillations is phase-sensitive).
If there is an axial gradient to Ωeﬀ , then a relative torque is applied to theorder parameter across the condensate, which can cause a twist to develop along
the axial direction. If we naively model the sample as a collection of individual
atoms, each held ﬁxed at its respective location, then the order parameter at eachpoint in space rotates independently at the local eﬀective Rabi frequency, Ωeﬀ .

110
In Fig. 7.2(a) and Fig. 7.3we see the implications of the “ﬁxed-atom” model forΩ = 2π×700 Hz and δ(z) = 2π×(100+14z)Hz with z in microns: a twist develops
in the order parameter which leads to a washing out of the Rabi oscillations [Fig.
7.2(c)]. In contrast, the kinetic energy provides stiﬀness for a true condensate.
Simulation of the condensate [26] shows that, for early times, the order parameter
begins twisting as in the “ﬁxed-atom” model but the twisting process self-limits
about 40 ms into the simulation. At this point there is nearly a full winding acrossthe condensate. Thereafter, though the two ends of the order parameter continue
to twist with respect to one another, the order parameter has been suﬃciently
wrapped around the SU(2) sphere that the eﬀect of further torque is to return thecondensate close to its unperturbed condition [Fig. 7.2(d)]. The Rabi oscillations
exhibit a corresponding revival [Fig. 7.2(e)]. The factor driving the untwistingprocess is the increasing kinetic energy cost associated with an increasing twist in
the order parameter. For a simple U(1) order parameter, continuously increasing
the winding ultimately results in a “snap”, in which the order parameter is driven
to zero and a discontinuous (and presumably dissipative [27]) process releases theexcess windings. The revival in the present case is made possible by the larger
rotation space available to a two-component cloud.
Under experimental conditions similar to those of the simulations in Fig.
7.2, we have observed as many as three complete cycles of Rabi-oscillation decay
and recurrence. These data appear in Ref. [133]. In this paper we present datathat correspond to the case of a more vigorous twisting. We increase the axial
dimension of the condensate cloud by a factor of four; the kinetic energy costof twisting the condensate is correspondingly lower, so that at the point in time
when the condensate is maximally distorted there are four windings across thecloud. The parameters of the experiment were as follows: ωr = ωz = 2π × 7.8 Hz
and mean Ωeﬀ = 2π × 225Hz. There was a gradient in both δ and Ω across the

111

Figure 7.2: (a) We represent the polar-vector order parameter as an arrow in thesesimulations. The angle θ from the vertical axis determines the relative population,and the azimuthal angle φ is the relative phase of states |1 and |2 (see Eqs. 1).
Each column in the arrow-array is at ﬁxed time, and each row at ﬁxed axial
location. Ω̂ is perpendicular to the plane of the page, so that a uniform, onresonance Rabi oscillation would correspond to all the arrows rotating in unison,in the plane of the image. The tips of all the arrows are (on the relatively fast timescale of Ωeﬀ ) tracing out circles nearly parallel to the plane of the page (in ourrotating-frame representation, small excursions out of the page are a consequenceof ﬁnite detuning). In the ﬁgures, we “strobe” the motion just as the central arrowapproaches vertical, to emphasize the more slowly evolving “textural” behavior.
(b) The total density of the condensate nt maintains a Thomas-Fermi distribution(integrated through one dimension, as imaged) and changes only slightly during
the evolution of the cloud. (c) In a simple model of individual, ﬁxed atoms, acontinuous inhomogeneity in Ωeﬀ will cause the Rabi oscillations in δN to wash
out. (d) When a condensate is simulated [19], the kinetic energy causes the orderparameter to precess through the full SU(2) space, coming out of the page to castoﬀ the winding and thus reduce its kinetic energy. (e) The corresponding plotof δN shows that when the arrows are once more aligned, the Rabi oscillations
recur.

112

Figure 7.3: Note this schematic is rotated with respect to the previous. Thismodel is similar to Fig. 7.2a, with density plots included. Solid lines represent
the |1, −1 density cross section and dashed lines are the |2, 1 density (of course,
one Rabi period later they switch). Individual atoms are treated as discrete twolevel systems each with a diﬀerent along the axial position. The detuning causes
a gradient in eﬀective Rabi frequency which means the population transfer slowly
accumulates diﬀerences across the cloud. This is manifested as increasingly ﬁne
structure on the modeled density proﬁle.

113

Figure 7.4: The actual density behaves quite diﬀerently than the model in Fig.
7.3 suggests. Initially the phase winds up similar to the no-kinetic energy model.
After time, the phase begins to unwind on its own, appearing to retrace its steps
to ﬁnd the original density distribution. The arrows are only a qualitative model
suggested by the density evolution, but are very similar to the simulation in Fig.
7.2d.

114
54µm axial extent of the cloud, resulting in a ∼ 2π × 60Hz diﬀerence in Ωeﬀ from
top to bottom of the condensate. The result of the experiment is seen in Fig. 7.5.
The observed recurrence of the Rabi oscillations at 180 ms [Fig. 7.5(a)], whencorrected for overall decay of the cloud, corresponds to 60 percent contrast. We
ﬁnd it remarkable that the distorted order-parameter ﬁeld seen in Fig. 7.5(b) attimes 65 and 75 ms should ﬁnd its own way back to a nearly uniform conﬁguration.

The simulations qualitatively reproduce the integrated number and statespeciﬁc density distributions observed in the experiments. For large inhomogeneity in Ωeﬀ , however, the simulations predict the development of small-scale spatial
structure not observed in the experiment. The simulations contain no dissipation,
whereas ﬁnite-temperature damping may occur in the experiment [28].
Heuristically, what value do we expect for the recurrence time trecur forthe data in Fig. 7.5? The diﬀerence in Ωeﬀ from the top to the bottom of thecondensate is about 60 Hz. From the data in Fig. 7.5(b) we see that the recurrence
occurs only after four windings have one-by-one been twisted in and then twisted
out of the condensate. A rough estimate then would be trecur = (4+4)/60Hz= 133
ms, shorter than the observed value of 180 ms, but reasonable given that edge
eﬀects have been neglected. This scaling is born out in Fig. 7.7, aside from an
oﬀset in the rf detuning, which has not yet been understood.
An interesting theoretical challenge would be to develop simple argumentsthat would allow an a priori prediction of the spacing of the windings at the instantof maximum twist. For particularly strong torques, one might expect the total
density to be suppressed to zero along a plane transverse through the cloud, so as
to allow for discontinuous relaxation of the order parameter. Indeed we have seensuch behavior in numerical simulations. Under what conditions should suppressionof total density, rather than continuous evolution through SU(2) space, be the115

Figure 7.5: A condensate with large axial extent undergoes twisting. (a) The
streak camera data shows a rapid decay in the Rabi oscillation in the integrated
population diﬀerence, from full contrast at t=0 to near zero by t = 20ms. The
oscillations recur at 180 ms. (b) Individual phase-contrast images (at distinct
moments in time) of the spatial distribution of |1-state atoms show that the
spatially inhomogeneous Rabi frequency is twisting the order parameter, cranking
successively more windings into the condensate until by ∼ 75 ms four distinct
windings are visible. Further evolution results not in more but in fewer windings
until, at time 180 ms, the order parameter is once more uniform across the cloud.
Each image block is 100 µm on a side, and the probe laser is tuned much closer tothe |1 state than to the |2 state. (c) The numerical simulation reproduces thequalitative features of the corresponding experimental plot (a). The simulation
used δ(z) = 0, Ω(z = 0) = 2π × 225Hz and a 2π × 60Hz spread in Ω across theextent of the condensate.

116

1.0
0.5

Population

0.0
-0.5
-1.0
20

40

60

80

100

120

140

20

40

60

80

100

120

140

1.0
0.5
0.0
-0.5
-1.0
0

Time (ms)Figure 7.6: Experiment (top) and theory (bottom) for a condensate undergoing
Rabi oscillations between the two spin states, with a gradient in the two-photon
drive detuning. The overall detuning of the two-photon drive from resonance is
80 Hz red. The gradient is determined by the 0.4 µm oﬀset between the traps of
frequency 62 Hz in the axial direction.

117

100

Recurrence time (ms)

80

60

40

20

0
-400

-200

0

200

400

radio frequency detuning (Hz)Figure 7.7: The time for the ﬁrst recurrence of the Rabi oscillations as a functionof detuning. This data was for a 62 Hz (axial) trap with a bare Rabi frequencyof 370 Hz.

118
preferred mode of relieving accumulated stress?
7.2

Dressed states
We have observed that the presence of the coupling drive need not result in

population oscillations. For any given frequency and coupling strength, there are
two steady-state solutions which are completely analogous to the dressed-states
solutions of the single-atom problem [29]. The lower and upper dressed states fora single atom (|a and |b respectively) are
|a =






1 
∆(z)
∆(z)
1 
√ 1+
|1, −1 − √ 1 −|2, 1 ,

|b =






1
1 
∆(z)
∆(z)

√ 1−
|1, −1 + √ 1 +|2, 1 ,2Ωe (z)2Ωe (z)2Ωe (z)2Ωe (z)(7.3)



where ∆(z) is the detuning from resonance, and Ωe (z) =∆(z)2 + Ω(z)2 is theeﬀective Rabi frequency. The population in these states is independent of time.
These are the true, local dressed states for our double condensate system if meanﬁeld is taken into account in the terms ∆(z) and Ωe (z) (for times short comparedto the time for density evolution to occur).
We have been able experimentally to put the condensate in such states via
an adiabatic process: the strength of the drive is increased gradually from zero,and the frequency (initially far detuned) is gradually ramped onto resonance. The
fraction transferred to the |2, 1 is determined by
1 − e−2πΓ

(7.4)

2

where Γ =1πΩ
2 ∂ν/∂t

for the rate of frequency ramp ∂ν/∂t. Our frequency ramp speed is such thatthe transfer is > 95% for a ramp completely through resonance. This ramp
corresponds to adiabatic rapid passage transfer. If the ramp is halted on resonance119(∆ = 0), the condensate is in the lower dressed state |a. This resulting “dressed
condensate” is extremely stable— after the ramp is complete, the cloud remains
motionless in a near-equal superposition of the two bare states. Figure 7.8 shows
the population in the |2, 1 state as a function of time.
After a pure dressed state is created, the drive amplitude may be turned
down slowly so that the condensate moves adiabatically from the dressed state to
the spin states. Since the two spins sit in slightly oﬀset traps, the spin states begin
to separate. In the limit that the drive strength goes to zero, the ground state ofthe double condensate system is produced. This ground-state mixture of the two
spin states no longer has a ﬁxed phase relationship since the drive is turned oﬀ– it is equivalent to the condensate mixture that would be formed if both states
were simultaneously evaporated into condensation. The density distributions of
the spin states for two diﬀerent drives are shown in Fig. 7.9.

Fraction of initial (t=40ms) atoms in |2>

120

1.0

Full transfer
Half transfer

0.5

0.0
30

40

50 60 70
Time (ms)80

Figure 7.8: The drive frequency is ramped from 10kHz below to near resonancein 40 ms. The two curves show data for a ramp to resonance resulting in 100%transfer, and a ramp to on resonance resulting in 50% transfer. The decay after
40 ms is the loss due to spin exchange of |2, 1 atoms.

121

|2ñ

|1ñ

|1ñ

|1ñ
|2ñ

optical depth
-75

|2ñ

reduce
drive

-50

-25

0

25

position (µm)

50

75 -75

-50

-25

0

25

50

75

position (µm)

Figure 7.9: Density proﬁles of dressed condensates, for two values of the drive
power. As the drive is reduced, the dressed state becomes an equal mixture of thespin states.

Chapter 8
Vortex creation [5]The ability to fundamentally alter the topological properties of a condensate
has already proven useful. When the coupling drive is turned oﬀ, the SU(2)properties vanish, and states |1 and |2 become distinct species, each forced tolive in its separate U(1) space. Figure 7.5 illustrates how this can be used tochange Φ1 (z) in a controlled manner. With a variation on this technique we havecreated a vortex-state condensate.
The concept of a vortex is at the center of our understanding of superﬂuidity. A vortex is a topological feature of a superﬂuid – in a closed path around thevortex, the phase undergoes a 2π winding and the superﬂuid ﬂow is quantized.
Following the experimental realization of a dilute atomic Bose-Einstein condensate (BEC) [30], much theoretical eﬀort has been directed towards the formation
and behavior of vortices in atomic BEC [31, 32, 120]. This chapter presents theexperimental realization and imaging of a vortex in BEC. We use the method proposed by Williams and Holland [6] to create vortices in a two-component BEC.
An interference technique is used to obtain phase images of the vortex state andconﬁrm the 2π phase winding required by the quantization condition. We havealso carried out preliminary studies of the stability of the vortices.
Vortices are created in superﬂuid helium by cooling a rotating bucket of
helium through the superﬂuid transition, and a vortex forms for each unit of an-123
gular momentum. This does not work for BEC because it is formed in a harmonicmagnetic trap. When the condensate ﬁrst forms it occupies a tiny cross-sectional
area at the center of the trap and is too small to support a vortex. Eventually, thecondensate grows to suﬃcient size that it can support vortices, but the time scale
for vortices to be generated in the vortex-free condensate due to coupling withthe rotating environment is unknown, and may well be longer than the lifetimeof the condensate. This is the potential diﬃculty with using an optical “stirringbeam” or magnetic ﬁeld distortion to rotate the cloud during condensation, as
has been frequently proposed. Another proposal has been to use optical beams
with appropriate topologies to “imprint” a phase on an existing condensate. This
technique must drive the local density to zero at some point and then rely on
uncertain dissipative processes for the condensate to relax into a vortex state.
We have avoided these uncertainties by creating vortices using a coherent
process that directly forms the desired vortex wave function via transitions between two internal spin states of 87 Rb. The two spin states, henceforth |1 and |2,are separated by the ground-state hyperﬁne splitting and can be simultaneously
conﬁned in identical and fully overlapping magnetic trap potentials. A two-photon
microwave ﬁeld induces transitions between the states. As we have seen in previous experiments, this coupled two-component condensate is exempt from the
topological rules governing single-component superﬂuids - rules that make it difﬁcult to implant a vortex within an existing condensate in a controlled manner.
In the coupled system, we can directly create a |2 (or |1) state wave function
having a wide variety of shapes [6] out of a |1 (or |2) ground-state wave function by controlling the spatial and temporal dependence of the microwave-induced
conversion of |1 into |2.
We control the conversion by shifting the transition frequency using the AC
Stark eﬀect. A spatially inhomogeneous and moveable optical ﬁeld (a focused laser

124beam) provides the desired spatial and temporal control of the AC Stark shift.
The vortex state is an axially symmetric ring with a 2π phase winding aroundthe vortex core where the local density is zero. To create a wave function withthis spatial symmetry, the laser beam is rotated around the initial condensate
as in Fig. 8.1a. The desired spatial phase dependence is obtained by detuning
the microwave frequency from the transition, and rotating the laser beam at theappropriate frequency ω to make the coupling resonant. For large microwave detunings δ, the necessary rotation frequency is simply δ. For smaller detunings, therotation frequency must be the eﬀective Rabi frequency of the microwave transition [33]. As shown in Fig. 8.1b for large detunings, the energy resonance condition
now means that atoms can only change internal state through the coupling of thetime-varying perturbation, and are therefore obliged to obey any selection rulethat the spatial symmetry of that perturbation might impose. The center of thecondensate (at the axis of the beam rotation) feels no time-varying change, while
regions near the circumference of the condensate feel a near-sinusoidal variation,with a phase delay equal to the azimuthal angle θ around the circumference ofthe cloud. Williams and Holland show that this is precisely the geometry best
suited to couple the condensate into a vortex state. It should be emphasized that
it is not simply the mechanical forces of the optical ﬁeld that excite the vortex:
a laser beam rotating clockwise can produce clockwise or counterclockwise vortex
circulation, depending on the sign of the microwave detuning.
In the absence of the microwave coupling ﬁeld, the two states can be thoughtof as two distinguishable, interpenetrating superﬂuids that interact with eachother and with themselves via a mean-ﬁeld repulsion proportional to the local
densities. The interaction coeﬃcients diﬀer slightly [105, 46], so the |1 ﬂuid hasslight positive buoyancy with respect to the |2 ﬂuid [75]. When the |1 ﬂuid
has a net angular momentum, it forms an equatorial ring around the central |2

125

|

2>

Laser rotation, ω

l =1
l =0

ω

Laser
profileBEC

Two-photon
microwave

(a)

|

1>

(b)

Figure 8.1: (a) A basic schematic of the technique used to create a vortex. An
oﬀ-resonant laser provides a rotating gradient in the AC Stark shift across thecondensate as a microwave drive of detuning δ is applied. (b) A level diagram
showing the microwave transition to very near the |2 state, and the modulation
due to the laser rotation frequency that couples only to the angular momentum
l = 1 state when ω = δ. In the ﬁgure, the energy splitting (< 1 Hz) between thel = 1 and l = 0 states is exaggerated.

126
ﬂuid. The |1 ﬂuid partially penetrates the constant-phase |2 ﬂuid, which creates a central potential barrier. Conversely, a |2 vortex forms a ring that tends
to contract down into the |1 ﬂuid. We use the overlap of the |1 and |2 ﬂuids
to image the phase proﬁle of the vortex state via the interconversion interferencetechnique that we introduced in [76]. In the presence of a near-resonant microwave
ﬁeld (and no perturbing optical ﬁeld), the two states interconvert at a rate sensitive to the local diﬀerence in the quantum phases of the two states. Thus theapplication of a resonant π/2 microwave pulse transforms the original two-ﬂuid
density distribution into a distribution that reﬂects the local phase diﬀerence, a“phase interferogram.” Looking at the condensate both before and after the interconversion pulse provides images of both the amplitude and phase of the vortex
ring.
The basic experimental setup for forming condensates and driving thembetween diﬀerent spin states is the same as in [105]. Using laser cooling andtrapping, followed by trapping in a TOP magnetic trap and evaporative cooling,
we produce a condensate of typically ∼ 8 × 105 atoms in the |1 state (F = 1,
mF = −1). We then adiabaticaly convert the trap to a spherically symmetricpotential by reducing the quadrupole magnetic ﬁeld gradient [61]. This leaves us
with a condensate 54 microns in diameter in a trap with oscillation frequencies of
7.8 ±0.1 Hz in the radial and axial directions for both spin states. In this trap,a |1 state condensate has a lifetime of 75 s and the |2 state (F = 2, mF = +1)
about 1 s. Oscillating magnetic ﬁelds can then be pulsed on to drive the microwave
transition between the |1 and |2 states. The power and oscillation frequency of
the ﬁelds are adjusted to obtain the desired eﬀective Rabi frequency for the |1 to|2 transition. To create a vortex in |2 we add a 10 nW, 780 nm laser beam thathas a waist of 180 µm and is detuned 0.8 GHz blue of resonant excitation of the|2 state. Using piezoelectric actuators we rotate the beam in a ∼75 µm radius

127
circle around the condensate at 100 Hz. The following procedure is used to obtain
the precise location of the laser beam that is required to create a vortex. We
ﬁrst set the eﬀective Rabi frequency to 100 Hz by adjusting the detuning of the
microwave frequency away from resonance in the presence of the light. Typicallythe resonant Rabi frequency is 35 Hz and the total detuning about 94 Hz from the|1 to |2 transition. By making ﬁne adjustments of the detuning we then optimize
the amount of transfer (typically 50%) to the |2 state, as shown in Fig. 8.2. We
then adjust the center of rotation of the beam to obtain the most symmetric rings.
After ∼ 70 ms the vortex has been formed, and we turn oﬀ the laser beam andthe microwave drive.
We can take multiple images of a vortex both during and after formation
using nondestructive state-selective phase contrast imaging [38]. Rapid control ofthe microwave power and frequency allows us to apply various pulse sequences to
explore many diﬀerent options for the creation, manipulation, and observation ofa single vortex. For example, we can put the initial condensate into either the |1
or |2 state and then make a vortex in the |2 or |1 state respectively, and we canobtain phase interferograms or quickly switch the internal state of the vortex at
any time after the rotating laser beam is oﬀ. We can also watch the evolution of
the vortex over time scales from milliseconds to seconds. All of these techniques
are nondestructive to both the density and phase.
In Fig. 8.3 we show a detailed picture of the phase proﬁle of a |2vortex.
To obtain this we ﬁrst take a picture of the vortex (Fig. 8.3a), then we apply aresonant microwave π-pulse. Half-way through the 4 ms long π-pulse, we take asecond image (Fig. 8.3b), and at the completion of the pulse we take a third image(Fig. 8.3c) that shows the original density distribution of the interior |1 state.
Normalizing by the density distributions of the vortex and interior states we obtain
the phase image in Fig. 8.3d [34]. The ﬁgure dramatically shows the variation

128

Figure 8.2: Atoms transferred into the |1, −1 state due to the vortex drive. Thisis slightly diﬀerent than described in the text. We start with a |1, −1 condensate.
Just before t = 10 ms we apply a π-pulse to convert the condensate into |2, 1 .
Then at 10 ms the vortex drive is turned on to create a vortex in the |1, −1 state.
The gradual increase in |1, −1 population (dark line) is a signature of successful
coupling to this unique resonance. The grey line is the background.

129
and continuity of the phase around the ring (Fig. 8.3e) that are required by thequantization of angular momentum. As a check, we have taken the same typeof phase pictures for our “ball and shell” density arrangement which occurs for a
simple mixture of the two states. This is shown in Fig. 8.4. The density pictures(top and bottom) show similar structures to our vortex pictures. However theinterference picture shows only symmetric structure, indicated there is no relative
phase accumulation around the condensate.
In Fig. 8.5 are pictures of the time evolution of vortices to show their dynamics and stability. As expected, the dynamics of the |1 state vortices (Fig.
8.5a and Fig. 8.5b) are diﬀerent from the |2 state vortices (Fig. 8.5c) due to thediﬀerent scattering lengths. The |2 vortex ring sinks in toward the trap center(Fig. 8.5c), and then rebounds and apparently breaks up. This pattern is repeatedly observed in measurements of the |2 vortices. Conversely, the equilibrium
position of a |1 state ﬂuid is obtained by “ﬂoating” outside the |2 state ﬂuid.
The inner radius of the |1 state ring shrinks slowly as the interior |2 ﬂuid decaysaway with a ∼ 1 s lifetime. A variety of additional behaviors has been seen for|1 vortices. Initial asymmetry is very sensitive to beam position and condensate
slosh. For small diﬀerences in the initial vortex state density distribution, asymmetric density distributions sometimes develop and/or “heal” between 0.5 and 1s.
The nonrotating |2 ﬂuid within a |1 vortex is analogous to the defects
that pin vortices in superconductors. This |2 “defect” can be removed quickly,
by a properly tuned laser pulse, or allowed to decay slowly (as in Figs. 8.5a and8.5b). Thus this system can be varied between two relevant physical limits. In
the limit of a large repulsive central potential (produced by a large amount of |2ﬂuid) the system most closely resembles quantized ﬂow in a ﬁxed, 3-D toroidal
potential. The vortex core is pinned and its size is determined by the central

130Figure 8.3: Condensate images (100 µm on a side) in which the imaging laser
is tuned such that only |2 is visible. (a) The intial image of the vortex, (b)after π/2 of the interconversion pulse, and (c) after completion of the π-pulse.
The vortex is now invisible (in |1) and the interior ﬂuid is imaged (in |2). (d)The normalized diﬀerence in densities between the local average ring and interior
densities ((a) and (c) respectively) and the phase interferogram density (b) for
each corresponding point in the images. This is approximately the cosine of thelocal phase diﬀerence φ between the vortex state and the interior state [34]. The
values are shown only for those regions where the densities of each state werehigh enough to give adequate signal to noise for the phase reconstruction. (e)The radial average at each angle θ around the ring is shown in (d). (The data isrepeated after the azimuthal angle 2π to better show the continuity around thering.)

131

Figure 8.4: Condensate images for a 50-50 mixture of the |1, −1 and |2, 1 states,
produced by a simple π/2 pulse. Each vertical strip is the same data, but scaled
diﬀerently to show speciﬁc features. Time is in the vertical direction. The ﬁrst
image is that of the |2, 1 “ball”, just before a π-pulse is applied. The second
image is after π/2 of the π-pulse, which shows the interference between the twostates. The third image is after the π-pulse is complete and shows the |1, −1“shell”.

132

|1ñ

|2ñ

t = 600 ms (b)

(c)

|1ñ

t = 0 mst = 200 ms

t = 400 ms(a)

Figure 8.5: (a,b) Two separate instances of the free evolution of a |1 state vortex
in the magnetic trap. It is stable over a time long compared to the trap oscillation
period (128 ms). (c) The free evolution of a |2 vortex is much more dynamic. It
is seen shrinking quickly into the invisible |1 ﬂuid and rebounding into fragments.
Each column is from a single run, where time t is referenced to the end of vortex
creation (t is the same for each row). The |1 and |2 state images appear diﬀerent
due to diﬀerent signs of the probe detuning.

133
potential. Figs. 8.5a and 8.5b are near this limit for t= 0 and 200 ms. In this casethe density distribution of the vortex state is vulnerable to instabilities due tothe fact that the relative densities of |1 and |2 may evolve with relatively little
energy cost so long as the total density remains constant [75]. In the opposite
limit (a small amount of |2 ﬂuid), the central potential is negligible and the sizeof the vortex core (in equilibrium) is determined entirely by its own centrifugal
barrier. Figs. 8.5a and 8.5b at t = 600 ms are evolving towards this limit. In
Fig. 8.5a, we see that the interior |2 ﬂuid is no longer pinning the vortex at 600
ms, but rather is being dragged around by the precessing vortex. The |2 vortex
provides an interesting mixture of the above limits. The |1 ﬂuid “ﬂoats” to theoutside so there is no pinning of the vortex core, but the tendency towards density
instabilities remains.
Expanded studies of stability issues are underway. We also expect to beable to observe interesting transitional behavior between these limiting cases. For
example, it is a straightforward extension of our method to create an l = 2 vortex.
In the presence of a strong pinning potential, l = 2 vortices should be stable, but
in the weak potential limit, l = 2 vortices are predicted to spontaneously bifurcate[120].

Bibliography[1] This is a revised version of [93], with the addition of a comparison to theexicitation spectrum when only the changing number in the condensate istaken into account.
[2] This chapter is a revised version of [105] and [75], with the addition of an
introduction and data showing radial separation of condensates in real time.
[3] This is a revised version of [76], including an analysis of the randomizationof the relative phase due to experimental noise.
[4] This is a revised version of Matthews(/cond-mat/9906288). Additions include two plots further describing our “phase-twisting” model, and anotherplot comparing theory to experiment in another regime.
[5] This chapter is a revised version of Matthews(/cond-mat/9908209), with theaddition of data showing an interferogram of our “ball and shell”, non-vortex
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4 to be too low by 1%. We thank W. Ketterle for pointing out this possible
source of systematic error.
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136
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Appendix A
Phase-contrast imaging

Following is a more detailed derivation of phase contrast imaging, specializing to our speciﬁc case. The only assumptions made previously are qualiﬁed here;that the phase shift from a condensate is small, and that the probe beam size
is much larger than the condensate. Due to the cylindrical symmetry along the
probing direction (aside from a constant scaling factor) the analysis can be donein only one dimension, perpendicular to the direction of beam propagation.
I begin by writing down the electric ﬁeld for a gaussian beam (of size σ =√1/ 2a) propagating along the ẑ direction with angular frequency ω and spatial
phase proﬁle φ(x) = φ0 exp(−bx2 );$2E(x, z) = E0 e−ax eiωt+iφ(x) e−ik·$z
2E0 e−ax (1 + iφ(x) − φ2 (x)/2)(A.1)

where the approximation is valid for small φ0 , and the time and z terms can be
dropped. This is the ﬁeld just after the atom cloud. The ﬁeld at the focal point(P1) of the ﬁrst imaging lens (Fig. 4.10) is the Fourier transform of the ﬁeld at
the condensate, evaluated at k = x/λf1 ;E(x, P 1) =

 ∞
−∞
−

= e

E0 e−ax

π2x2
aλ2 f 2
1

2



1 + iφ(x) − φ2 (x)/2 e−2πikx dx22− π2 2 x2− π2 2 x2iφ1
φ2√ + √ 0 e bλ f1 − √ 0(A.2).
e 2bλ f1aa+b
2 a + 2b

146
In this equation the coeﬃcient to x2 in the exponent is the size of the beam/phaseproﬁle in the focal plane of the ﬁrst lens. The factors in front are also dropped
for convenience, but can be easily replaced at the end by requiring the intensity
to be normalized.
This is the point where the phase mask is inserted. The phase mask applies
a spatially dependent shift of the following form



 θ

if −c < x < c 0otherwise

θ(x) 

(A.3)

where 2c is the diameter of the magnesium ﬂuoride dot responsible for the phaseshift.
Again the Fourier transform is taken and evaluated at k = x/λf2 for thesecond imaging lens which focuses onto the CCD array. The transform is broken
up into two parts due to the discontinuity in the phase dot;
 cE(x, P 2) = 2

eiθ E(x, P 1) + 2

0f2− 12 ax2

= e

f

2

iθ

 ∞
c



E(x, P 1)
f2− 12 (a+b)x2(e − 1)A + 1 + iφ0 e f2

(eiθ − 1)B + 1



f2


φ20 − f12 (a+2b)x2e 2−(eiθ − 1)C + 1 .
2

(A.4)

The coeﬃcients in the exponents are now the size of the beam/phase proﬁle in theimage plane, properly scaled by the magniﬁcation. A, B and C are error functions
from the integrals in Eq. A.4;
πc
√
λf1 a
πc
√
B ≡ erf
λf1 a + b
πc√
C ≡ erf
λf1 a + 2b
A ≡ erf(A.5)

where the argument of the error functions is the ratio of the size of the phase dot
to the beam (in A) or phase proﬁle (in B) in the plane P1.

147
To ﬁnd the intensity at the CCD array, the ﬁeld from Eq. A.4 is squared
and time-averaged to givef2

− 12 ax2

I(x) = 2I0 e f2

(1 − cos θ)A2 + (cos θ − 1)A + 1/2

f2− 12 bx2+ 2I0 φ0 e f2

(A − B) sin θ.

(A.6)

This shows that the signal is proportional to the phase shift, and that the spatial
distribution is recreated on the CCD. The best signal is obtained when θ = π/2
and A = 1, B = 0. The ﬁrst condition, A = 1, requires that the probe beam
waist at the plane P1 is much smaller than the phase dot, and B = 0 requiresthat the diﬀracted wave from the condensate be much larger than the phase dot
at P1. With these conditions the signal isI(x) = I0 (1 + 2φ(x)).

(A.7)