Coherences and correlations in an ultracold Bosegas
by
Heather Jean Lewandowski
B.S. Physics, Michigan Technological University, 1997
M.S. Physics, University of Colorado, 2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2002This thesis entitled:
Coherences and correlations in an ultracold Bose gas
written by Heather Jean Lewandowski
has been approved for the Department of Physics

Eric A. Cornell

Deborah S. Jin

Date

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

iii
Lewandowski, Heather Jean (Ph. D. Physics)
Coherences and correlations in an ultracold Bose gas
Thesis directed by Dr. Eric A. Cornell

The development of a new and simplified system to create a Bose-Einstein condensate (BEC) is presented, as well as experimental studies of two spin-state condensed
and non-condensed samples. The first part of this thesis describes in detail our apparatus and experiential procedure for creating a BEC. We designed a system to createa BEC that is simple and robust enough to be constructed by someone outside of the
field of atom cooling and trapping. Our system includes several novel features that reduce the complexity of the apparatus, which include mechanical transfer of atoms anda hybrid magnetic trap.
The second part of this thesis describes studies of two spin-state clouds. We studied the correlations and coherences of non-condensed clouds using Ramsey spectroscopy
to gain an understanding of interatomic interactions. We were able measure precisely
the mean-field interactions between coherent particles, as well as determine mechanisms that preserve and destroy coherence. These experiments led us to the study ofspin waves, in which scattering of indistinguishable particles gives rise to coherent spin
oscillations. Using sensitive Ramsey spectroscopy, we were able to fully examine thespatio-temporal spin-state oscillations. Following the studies of normal cloud coherence,
we went on to explore coherence effects in condensates.

iv

Acknowledgements

First I would like to acknowledge my advisor Eric Cornell. He always amazes mewith his physical insight and ability to be entranced by “shiny bits of physics,” which
led us to a lot of great science. He has the right balance of knowing when to give advice
and when to let us direct the experiment. I am grateful to have worked with suchan inspiring physicist. I would also like to acknowledge Carl Wieman who has offeredguidance throughout my graduate career.
I have had the privilege to work with many talented people. During his postdoc,
Dwight Whitaker helped to construct the apparatus and always kept us going evenduring the two months of vacuum hell. I could not have asked for a better graduate
student to have in the lab than Dave Harber. His unquenchable enthusiasm combinedwith his laid back attitude made working long hours in the lab truly fun. Anotherpostdoc I worked with, Jeff McGuirk, brought his training in precision measurement toour lab. He really made me think about the experiment in a new way. He was also
a great friend who kept me at least partially sane during the writing of this thesis. It
was great to have so many other graduate students and postdocs to talk to and learn
from over the years including Paul Haljan, Kurt Miller, Brian DeMarco, Neil Claussen,
Ian Coddington, Peter Schwindt, Ying-Ju Wang, John Goldwin, David Hall, Brian
Anderson, Stephan Duerr, Peter Engles, Shin Inouye and Tetsuo Kisimoto.
The construction of the apparatus would not have gone so quickly and smoothly
if not for the JILA instrument and electronics staff. Hans Rohner, Hans Green, Blainv
Horner, James Fung-a-Fat, and Terry Brown were always willing to help figure out asolution to the many problems I presented them.
Lastly I would like to thank my family for all of their support. My husband Don
put up with a lot of late nights and was always supportive. My parents Kathleen and
Dennis have always encouraged me through the years. I would not be writing this thesis
if not for them.

Contents

Chapter
1 Introduction

1

1.1

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

3

1.2

Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . .

32 Experimental design and procedure5

2.1

Experimental resources . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Frequency control and stabilization . . . . . . . . . . . . . . . . .

9

2.2.2

Shutters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2.3

Spatial filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.1

Chamber design . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.2

Chamber construction . . . . . . . . . . . . . . . . . . . . . . . .

18

2.3.3

Chamber bake out . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.4.1

MOT alignment . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.4.2

MOT characterization . . . . . . . . . . . . . . . . . . . . . . . .

26

From MOT to Ioffe-Pritchard trap . . . . . . . . . . . . . . . . . . . . .

27

2.5.1

27

2.3

2.4

2.5

Transfer into the quadrupole magnetic trap . . . . . . . . . . . .

vii

2.6

2.7

2.8

2.9

2.5.2

Fluorescence imaging . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.5.3

Transfer from vapor cell to UHV region . . . . . . . . . . . . . .

33

2.5.4

Ioffe-Pritchard Magnetic Trap . . . . . . . . . . . . . . . . . . . .

35

2.5.5

Transfer between magnetic traps . . . . . . . . . . . . . . . . . .

40

Rf evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.6.1

Evaporation optimization . . . . . . . . . . . . . . . . . . . . . .

43

2.6.2

rf coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Absorption imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.7.1

Optical set up. . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.7.2

Imaging in a non-uniform magnetic field . . . . . . . . . . . . . .

48

2.7.3

Producing and controlling microwaves . . . . . . . . . . . . . . .

50

2.7.4

Imaging procedure . . . . . . . . . . . . . . . . . . . . . . . . . .

52

2.7.5

Focusing the image . . . . . . . . . . . . . . . . . . . . . . . . . .

56

2.7.6

Measuring the image magnification . . . . . . . . . . . . . . . . .

57

Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.8.1

Image processing . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.8.2

Image fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

2.8.3

Calculating cloud parameters . . . . . . . . . . . . . . . . . . . .

59

Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3 Driven two-level atom

65

3.1

Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.2

Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.3

Two-photon transition in 87 Rb . . . . . . . . . . . . . . . . . . . . . . .

68

4 Spectroscopy

70

4.1

Spectroscopy in a magnetic trap . . . . . . . . . . . . . . . . . . . . . .

71

4.2

Cold-collision energy shift . . . . . . . . . . . . . . . . . . . . . . . . . .

75

viii
4.3

Missing frequency chirp . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.4

Coherence of normal clouds . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.5

Coherence of condensates . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.6

Speculation on precision metrology . . . . . . . . . . . . . . . . . . . . .

90

5 Spin waves

92

5.1

Theoretical description of spin waves . . . . . . . . . . . . . . . . . . . .

93

5.2

Description of system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.3

Bloch-sphere representation . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.4

Experimental method for studying spin waves . . . . . . . . . . . . . . .

97

5.4.1

Extracting longitudinal phase . . . . . . . . . . . . . . . . . . . .

97

5.4.2

Extracting transverse phase . . . . . . . . . . . . . . . . . . . . .

98

5.5

Frequency of oscillation of spin waves. . . . . . . . . . . . . . . . . . . 101

6 Coherence effects in a finite temperature condensate

104

6.1

Dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2

Symmetries of the dressed-state system . . . . . . . . . . . . . . . . . . 107

6.3

6.2.1

Stability of dressed states . . . . . . . . . . . . . . . . . . . . . . 108

6.2.2

Separation of dressed states . . . . . . . . . . . . . . . . . . . . . 111

6.2.3

Evaporation of dressed states . . . . . . . . . . . . . . . . . . . . 113

Decoherence induced melting of a finite temperature condensate

Bibliography. . . . 115

118

Appendix
A Everything you ever wanted to know about Rubidium

123TablesTable
2.1

Vacuum component temperature limits . . . . . . . . . . . . . . . . . . .

21

2.2

Parameters for trapping and loading into a magnetic trap . . . . . . . .

29

2.3

Image timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.1

Breit-Rabi equation constants for 87 Rb . . . . . . . . . . . . . . . . . . .

69

A.1 Properties of 87 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Figures

Figure2.1

Optical set-up for saturated absorption spectroscopy. . . . . . . . . . . .

2.2

Saturated absorption spectra of the hyperfine structure on the Rubidium

10

5S1/2 →5P3/2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Laser frequency control for the three lasers in our experiment. . . . . . .

12

2.4

Vacuum system layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.5

Diagram of the MOT cell. . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.6

Getter assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.7

Schematic of the MOT optical layout. . . . . . . . . . . . . . . . . . . .

23

2.8

Picture of the vertical-beam optics for the MOT. . . . . . . . . . . . . .

24

2.9

Schematic of the MOT/quadrupole trap servo circuit. . . . . . . . . . .

25

2.10 Optical layout of fluorescence imaging. . . . . . . . . . . . . . . . . . . .

31

2.11 Illustration of vignetting. . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.12 Illustration of the motion of the quadrupole trap coils. . . . . . . . . . .

34

2.13 Science cell region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.14 Ioffe-Pritchard magnetic trap (end on view). . . . . . . . . . . . . . . . .

38

2.15 Ioffe-Pritchard magnetic trap (side view). . . . . . . . . . . . . . . . . .

38

2.16 Machine drawing of the Boron-nitride HIP-trap form. . . . . . . . . . .

39

2.17 Sample evaporation trajectory. . . . . . . . . . . . . . . . . . . . . . . .

45

2.18 Optical system for absorption imaging. . . . . . . . . . . . . . . . . . . .

48

xi
2.19 Possible transitions with different probe beam polarizations . . . . . . .

49

2.20 Microwave frequency chain. . . . . . . . . . . . . . . . . . . . . . . . . .

51

2.21 Energy level diagram showing imaging transitions. . . . . . . . . . . . .

56

2.22 Computer control diagram. . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.1

Ramsey’s method of separated oscillatory fields. . . . . . . . . . . . . . .

67

3.2

Energy level diagram for the ground state of 87 Rb. . . . . . . . . . . . .

68

4.1

Spatial dependence of the transition frequency. . . . . . . . . . . . . . .

73

4.2

Differential Zeeman shift at low magnetic fields for the |1i → |2i transition. 74

4.3

Measurement of the cold collision shift.

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

77

4.4

Testing the exchange symmetry between the |1i and |2i states. . . . . .

79

4.5

A data set of Ramsey fringes probing for frequency shifts as a functionof coherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.6

Coherence times in a normal cloud. . . . . . . . . . . . . . . . . . . . . .

82

4.7

The frequency shift across the axial direction of the cloud. . . . . . . . .

84

4.8

The transition frequencies at different positions along the cloud are plotted as a function of Bbias . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Ramsey fringes of the center bin of a condensate. . . . . . . . . . . . . .

88

4.10 Alternate method of taking coherence data. . . . . . . . . . . . . . . . .

89

5.1

Intuitive picture of spin rotation during collisions.

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

95

5.2

Creation of spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.3

False color plots of the longitudinal and transverse components of the

4.9

Bloch vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.4

Bloch vector trajectories across the cloud. . . . . . . . . . . . . . . . . . 100

5.5Frequency and damping of spin waves as a function of density. . . . . . 102

6.1

Energy level diagram of the dressed states showing the avoided crossing. 106

xii
6.2

Transfer rate is shown as a function of Rabi frequency. . . . . . . . . . . 110

6.3

Component separation in a dressed state system. . . . . . . . . . . . . . 112

6.4

Condensate Rabi flopping after evaporation in the dressed states. . . . . 114

6.5

Measured phase space density as a function of time. . . . . . . . . . . . 116

A.1 Energy level diagram for 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . 124
A.2 Branching ratios for 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 1

Introduction

The creation of the field of ultracold atomic physics has opened up many new
possibilities for studying the effects of quantum mechanics in theoretically accessible
systems. Ultracold temperatures reduce the complexity of the system by reducing thepossible states a system can occupy, thus allowing a more complete understanding of thesystem. An extreme case of a sample of ultracold atoms is a Bose-Einstein condensate(BEC), in which all of the atoms occupy a single quantum state.
The theory of Bose-Einstein condensation was first suggested by Bose [1] andexpanded upon by Einstein [2] more than 75 years ago. However it was not until 1995
that a BEC was first produced in a dilute vapor. The time lag in the experimental
realization of the predicted phase transition is not so incredible, once one realizes theextraordinary conditions that must occur in order to Bose condense a gas.
In everyday life, quantum statistics does not play a major role in our experiences
because the thermal wavelength of particles is small compared to the interparticle spacing. However if a particle’s wavelength is made comparable to the interparticle spacing,quantum statistics will become important. The relevant parameter that determines
whether particles obey classical or quantum statistics is the phase-space density, whichis the number density times the thermal de Broglie wavelength cubed. As this parameter approaches one, quantum statistics becomes relevant. For Rubidium gas at room
temperature and typical vacuum density of 10−9 torr, the phase space density of the2sample is ∼ 10−20 , which is twenty orders of magnitude away from the Bose-Einsteincondensation transition. It was predicted that one would have to cool Rubidium vapor
below 1 millionth of a degree above absolute zero to observe a BEC. Remarkably, withthe advent of laser and evaporative cooling, this goal turned out to be obtainable. Now
over 35 groups in the world have condensed eight different atomic species. This accomplishment was recently recognized by the awarding of the 2001 Nobel prize in physics“for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, andfor early fundamental studies of the properties of the condensates” [3].
It is somewhat ironic that, after finally achieving the goal of producing a condensate, the studies in our group concentrated on the most basic system in quantum
mechanics: the two-level system. We wanted to study the macroscopic quantum phaseof a condensate, and how this phase is affected by both internal and external couplings.
One cannot measure the absolute phase of a particle, but rather only the relative phasewith respect to another particle. Therefore if we want to study the phase of a condensate, we must have another condensate as a phase reference. We meet this requirement
by producing a condensate in two distinct spin states, which we can manipulate andmeasure by standard atomic-physics techniques. The ultimate goal was to study how anormal cloud interacting with a condensate affects the phase of the condensate. Before
we began our condensate coherence studies, we felt we should understand fully normal
cloud coherences and correlations. This initial study led to many significant results
including the unexpected observation of spin waves and precision measurements of interatomic collisions and correlations. These studies are directly relevant to the field of
atomic clocks, where interatomic collisions currently limit the accuracy. After an extensive study of normal clouds in two spin states, we began studies of pure-condensate
coherence and the effects of the normal cloud on condensate coherence. The studies
of condensate coherence are still ongoing with the hope that we may be able to understand the coupling between the normal cloud and the condensate. It is a pleasure

3
to note that a two-level system can still yield surprises, 75 years after the advent of
quantum mechanics.

1.1

Overview
We originally set out with two goals in mind. First, we wanted to design andbuild a new type of BEC apparatus, which would be simpler and more robust than was
currently in use in dozens of labs around the world. Second, we wanted to use our stableapparatus to study phase coherence of condensates.
In the seven years since their first observation, dilute vapor condensates havebeen studied extensively. In most cases the condensate properties themselves are thefocus of the investigations. Relatively little work has been done using a condensate asa tool to explore questions in other fields. We feel that physicists in other fields such ascondensed matter have a different and unique perspective on possible experiments that
could make use of condensates. Therefore, developing a system that could be used forthese purposes is worthwhile. The current experimental systems [4, 5, 6] were designed
by people with a tremendous amount of knowledge and experience in experimental
atomic physics, and until now producing a BEC without expertise in ultracold atom
trapping has been a daunting task. We felt, however, that with some modifications
to the current experimental design, any experimental physicist, regardless of discipline,
could produce a BEC in their lab.

1.2

Organization of this thesis
This thesis is organized as follows. The first part of this thesis (Chapter 2) will

describe most of the basic steps in detail on how to build a BEC apparatus. Many of thetechniques described have been developed by others [7, 8, 9] over the last 20 years but
are included so that this thesis may serve as a ‘recipe’ for creating a BEC. Besides the
traditional methods we also describe several new features of our design, which include

4
a new method of atom transfer and a hybrid magnetic trap.
The second part of the thesis (Chapters 3-6) is devoted to the scientific studies of two spin-state ultracold and condensed atom samples. We started our study of
two-component samples by spectroscopically measuring cold-collision properties, which
include mean-field and Zeeman energy shifts, as well as correlations between the spinstates. The basic theory behind the spectroscopy is described in Chapter 3, and theresults of these studies are described in Chapter 4. During our spectroscopic measurements, we observed by accident an effect we initially called anomalous spin-state segregation. We observed the spin states separating spatially in the trap in an unexpected
way. Eventually, with the help of several theorists, we understood the spin dynamics
to be spin waves. Chapter 5 discusses the origin of and our extensive measurements ofspin waves produced in our two-component cloud. Armed with the knowledge of coherences and correlations gained from our experiments of cold collisions and spin waves, westudied how loss of coherence affected two-component condensates. Chapter 6 describes
our results on coherence effects on finite temperature condensates.

Chapter 2

Experimental design and procedure

A large fraction of the effort represented in this thesis was the development of a
new and improved system for creating a BEC. This system was developed from scratch,with little more than a bare optical table as a starting point. Several of the components
developed for this apparatus have proven so useful that several groups at JILA, andat universities throughout the United States, are now incorporating them into theirexperimental systems.
We recommend several sources that contain background information on a variety
of subjects not covered in depth in this text. For a textbook description of atom coolingand trapping we recommend Laser cooling and trapping by Metcalf and van der Straten[10]. Two useful papers that describe diode lasers and saturated absorption spectroscopy
are “Using diode lasers for atomic physics” [11] and “A narrow-band tunable diode laser
system with grating feedback and a saturated absorption spectrometer for Cs and Rb”[12]. A third paper by Wieman and coworkers describes the components and the process
for constructing a magneto-optical trap (MOT) for use in undergraduate laboratories[13]. A review of many ideas in evaporative cooling may be found in van Druten and
Ketterle [14]. There is a good section on imaging cold clouds in “Making, probing, andunderstanding Bose-Einstein condensates” by Ketterle and colleagues [15].
Our design has many features that allow a condensate to be made even if thesystem is not particularly optimized. We use a MOT with large beams (5 cm diameter)6with a considerable amount of laser power (400mW) to collect a large number of atoms.
The large number of atoms in our MOT is efficiently and reproducibly translated 45 cm
to a final magnetic trap via a mechanical transfer mechanism. We are able to place
the magnetic trapping magnets and coils very close to our atoms thus producing verystrong confinement and high collision rate. All of these features insure that evaporation
will work well and produce a condensate consistently.
One of the main concerns when designing any BEC apparatus is the need to
optically collect many atoms and yet to have a long lifetime for the atoms in the magnetic
trap. These two constraints require orders of magnitude different vapor pressures. It is
not easy to change the vapor pressure in a vacuum system by two orders of magnitude ina reasonable time of seconds. There are two conventional ways of solving this problem: adouble MOT apparatus [16] and an atomic beam [17]. The double MOT system collects
atoms in a MOT in one region of the vacuum system, which has a high alkali vapor
pressure, and transfers the pre-cooled atoms to a second MOT in an UHV chamber, witha pressure in the low 10−11 torr, for further cooling. The transfer is done essentially bypushing the atoms between MOTs with a laser beam. There are several disadvantages
to this method. Two MOTs are necessary, which requires the system to use more laser
power than a single MOT system. Also the optics and optical access needed for thesecond MOT restrict the space for the magnetic trap coils, so that to create a specified
magnetic gradient requires considerable electrical power. Finally, the push beam and
MOTs are very sensitive to optical alignment making optimum performance difficult tosustain. The other main method is to use a laser-cooled atomic beam from a Zeeman
slower. Zeeman slowers are large (1-2 m) and require a high temperature oven. Systems
with Zeeman slowers have only one MOT, but they suffer from the same optical accessproblems around the magnetic trap as does the double MOT design.
A design similar to ours, developed by the Hänsch group in Garching [18], uses a
series of electromagnetic coils to move the atoms from one chamber to the other. The

7
atoms are initially transferred from the MOT into a magnetic quadrupole trap; then by
ramping the current in successive sets of quadrupole coils lined along the transfer tube,
the atoms are transferred between chambers. This design has the benefit of being able
to move the atoms easily around a corner to reduce line-of-sight between the chambers,
which reduces trap loss from background collisions. However, the ten overlapping setsof quadrupole coils require a large amount of power to run, take up considerable spacearound the apparatus, and require effort to design, construct, and optomize.
Our system uses a moving magnetic trap to transfer atoms between regions of
the vacuum system. Magnetic coils are mounted on a linear track and translated fromone section of the system to the other; this technique has many advantages. It is very
easy to use. The track, motor, and controller are commercially available as a turnkey
system. The transfer worked the first time and works essentially every time without
any maintenance. As with the Garching group’s design, our system does not need asecond MOT in the UHV region, so we can place our magnetic trap close to the atoms
and produce very strong confining fields.
The other new feature in our system is a hybrid magnetic trap. It uses strong
permanent magnets to produce radial confining fields and low power electromagnetic
coils to produce axial confinement and a bias field. The permanent magnets do notconsume power and thus do not need to be actively cooled, as would electromagnetic coils
producing the same field. The tight confinement from the permanent magnets insuresthat we have the necessary collision rate to evaporatively cool the atoms. Permanent
magnets are however plagued by long-term stability problems. The longitudinal bias
field, the only parameter for which stability is critical, is produced by servo-controlled
electromagnetic coils, which are air cooled. Our design allows tight confinement fromthe permanent magnets and bias-field stability from the coils. The magnetic trap isdesigned to have tight enough confinement so that we can produce a condensate even
if the system is not well optimized.

8
We will outline in detail the steps required to make a Bose-Einstein condensate
using our particular experimental system. This Chapter begins by giving directions forsetting up the necessary vacuum and optical systems. Next we describe the experimental
procedure for pre-cooling atoms in a MOT, transferring the atoms to a magnetic trap,
and evaporatively cooling them to create a condensate. There is also an extensive
Section dedicated to imaging the condensate and extracting useful parameters from theimages.

2.1

Experimental resources
Our experimental apparatus does not require a large amount of space comparedto other BEC systems, particularly those with Zeeman slowers. Our lab contains threecomponents: an optical table, electronics, and control computers. We have the vacuum
system, lasers, and all of the optics on one 4’×8’ optical table. The system is notcongested, and we still have several square feet of free space on the table. In addition tothe optical table, we have a canopy the size of the table, which is half full of electronics,
and one full electronics rack. We also have two computers in the lab, which control theexperiment.

2.2

Lasers
The experiment requires three different wavelengths of laser light: two to makethe magneto-optical trap and one to image the condensate. We use a different laser foreach of these tasks. We use a commercial external cavity diode laser (ECDL) [12], whichis amplified by a single pass through a tapered amplifier chip for the MOT trapping
beams, in a master-oscillator power-amplifier (MOPA) configuration. This system willnominally produce 500 mW of power at 780 nm. The MOT hyperfine repump and probe
beams are supplied by two separate ECDLs, which each produce ∼ 8 mW.

9
2.2.1

Frequency control and stabilization

All three lasers are locked to atomic transitions in 87 Rb using saturated absorption
spectroscopy [12]. This type of frequency stabilization is standard practice and thus will
not be described here in detail. The basic idea is to produce sub-Doppler spectral lines,
which can be used as feedback to stabilize the laser. The optical layout for saturated
absorption spectroscopy is shown in Fig. 2.1.
We lock each laser to the peak of an atomic transition. The frequency locationof the peak of the transition is relatively insensitive to intensity and broadening effects,
which would change the locking set point if the laser were locked instead to the side ofthe line. Unfortunately a servo can only lock to a region where there is a slope of the line
to use as feedback. We solve this problem by generating a derivative of the saturated
absorption signal. We modulate the frequency of the laser, by either modulating thelaser current or the radio frequency (rf) driving an acoustic optic modulator (AOM).
The modulation has a depth of 5 MHz at a rate of 50 kHz, which is slow enough for the
AOMs to respond and fast enough to be above the bandwidth of the servo. The signal
from the saturated absorption spectrometer is routed to a homemade lock-in detector
which gives the derivative of the original transition lines. The derivative changes signat the peak, and thus is a convenient error signal for our servo.
We use AOMs to shift the frequency of the light that is used in the experiment.
The AOMs are driven by amplified voltage-controlled oscillators. A diagram of the
optical set-up for the AOMs is shown in Fig. 2.3. A portion of the light from the
MOPA is sent through a 120 MHz AOM, which is frequency modulated. The negative
first order diffracted beam is used in the saturated absorption spectrometer. This scheme
allows us to lock the laser to the peak of the (2 → 20 , 2 → 30 ) crossover line (peak A
in Fig. 2.2) and have the trapping light red detuned by several natural linewidths fromthe 2 → 30 cooling transition. The unprimed states refer to the 5S1/2 manifold and thephoto diodes
focusing lens

10
eam
ing b
t
a
r
Satu
lucitebeamsplitter
Rb vapor cell

Figure 2.1: Optical set-up for saturated absorption spectroscopy. A small amount oflight (0.1-1 mW) is split off from the laser and sent through a half inch thick piece ofLucite to produce two weak beams. The two reflected beams, about 1-2 mm wide, passthrough the Rb vapor cell and are focused onto two photodiodes. The saturating beamthat passes through the lucite is sent into the vapor cell counter-propagating with oneof the weak beams.

primed states refer to the 5P3/2 manifold. The repump laser is locked directly to the
1 → 20 transition. We modulate the current of the repump laser instead of the frequencyof an AOM. The probe beam is sent through two AOMs in our design. One AOM isessentially a fast shutter, while the other allows us to modulate the frequency sent tothe saturated absorption and not impart the frequency modulation onto the probe beamitself. We need to shift the frequency of the probe beam only a few MHz from the lockfrequency. Therefore the two AOMs are set to almost the same frequency, and theircenter frequency of 260 MHz is arbitrary. We lock the probe laser to the peak of the(2 → 20 , 2 → 30 ) crossover transition.
Vibration isolation is important when operating ECDLs, in our case commercial
New Focus Vortex lasers. Vibration can cause undesirable frequency noise at a levelthat the feedback may not be able to fully correct. We mount the repump laser on apiece of 1/4” thick sorbathane sheeting to reduce the effect of vibrations from the table.
The probe beam laser is mounted directly on the table. Mounting it on sorbathane
could cause the beam pointing to drift and thus decrease the coupling into the fiber.
We also mount our shutters on the same material, so that vibrations induced by the

1187

Rb (F=1 F’)
1

2

F’ =0

85

Rb (F=2 F’)
2
3

1

85

Rb (F=3 F’)

3287

Rb (F=2 F’)

4

A

2

3

1
0

600 MHz

Figure 2.2: Saturated absorption spectra of the hyperfine structure on the Rubidium
5S1/2 →5P3/2 transition. Widths and relative heights of the peaks are affected by beamalignment, intensity, and polarization. These are only representative signals.

solenoids when they open or close is not transmitted to the optical table.

2.2.2

Shutters

We use both mechanical shutters and AOMs to control the timing of the laser
light. The mechanical shutters provide an excellent extinction ratio but are slow and canhave jitter in their timing. We can also use an AOM as a shutter by using a diffracted
order. AOMs are fast, with response times on the order of a 1 µs, but their extinction
ratio is less impressive. For critical light pulses such as the probe pulse, we use both

Feedback120 MHz AOM

12shutter

MOPA

MOT-1 ordermodulation

Sat. spec.

Lock-inServo

main beam
Hyperfine repump

MOT

modulation

FeedbackServobypass repump
Lock-inFeedback

260 MHz AOM

Sat. spec.

260 MHz AOM
Dump

Probe Laser
modulation

Servo

Lock-in-1 orderSat. spec.

-1 orderProbe beam into
single mode fiber

Figure 2.3: Laser frequency control for the three lasers in our experiment. For the lesscritical repump laser, the frequency modulation for the lock-in detection is applied to
the laser frequency itself through current modulation. For the more critical trapping
and probing beams, the frequency modulation is applied to the AOMs, and thus thefrequency modulation is not on the light used on the atoms.

types of shutters. We create a pulse by first opening the mechanical shutter several ms
before the start of the pulse, turning on the AOM for the desired pulse length, thenclosing the mechanical shutter several ms later. Mechanical shutters can cause noiseon the lasers when they open or close through acoustical vibrations. We are careful toopen the shutter early enough for the laser servo to recover from the acoustical noise
before triggering the AOM for the probe pulse.

13
2.2.3

Spatial filtering

The spatial beam profile from diode lasers is not Gaussian and may contain highcontrast stripes, which must be reduced before the light is used in the experiment.
The output of a MOPA system can be nonuniform depending on the amplifier chip. A
MOT does not require perfectly uniform beams but high contrast features will lower
the trapping efficiency. We focus the MOPA beam, f# 16, through a large pinhole(50 - 100 µm), eliminating any high-frequency spatial modes while retaining most of thelaser power (∼75 %). The MOT is largely insensitive to the spatial profile of the repump
beam, so we do not spatially filter this light. The probe beam, on the other hand, musthave a very uniform and flat intensity profile. We use a single-mode angle-polishedpolarization-preserving fiber to filter spatially the probe beam. It is critical to use
both an angle-polished and polarization-preserving fiber to reduce temporal intensity
fluctuations of the beam. The facets of a flat-polished fiber can form an etalon, which
will produce high-frequency intensity fluctuations on the output. A non-polarizationpreserving fiber will scramble the input polarization depending on the stress (thermalor mechanical) applied to the length of the fiber. These polarization fluctuations will
be converted into intensity fluctuations when the light pases through a polarizer. One
good method to align the input light polarization with the axis of the fiber is to firstadjust the angle of the initial linear polarization until pure linear light is emitted fromthe fiber. This procedure may be too coarse to accurately align the axis, so the next
step is to tap on or warm up with your hand the center of the fiber, so as to not change
the coupling at the ends, and to watch the intensity fluctuations on the output after apolarizer. One can now more finely adjust the input polarization until a minimum ofpolarization fluctuations is observed on the output.

14

2.3

Vacuum system

2.3.1

Chamber design

The vacuum chamber is a simple single ion pump system that has a high vacuum
vapor cell (10−8 − 10−9 torr) and an ultra-high vacuum (UHV)(10−11 torr) science cell.
Three vacuum pumps are used in the system, but only one of them is used on a continual
basis. A diagram of the chamber is shown in Fig. 2.4. The turbo pump, connectedto the system by an all-metal valve, is used only during initial pump down and bakeout. The Ti-sublimation pump is turned on only every couple of years to remove extra
Rb or H from the system. The workhorse pump is the 40 l/s ion pump, which pumps
continuously on the chamber during normal operations. The ion pump is conduction
limited for some atomic species and thus it may be larger than needed. However wewould recommend the 40 l/s pump to ensure the system has adequate pumping speed.
The valve on the ion pump was used only as a diagnostic tool during original testingof the system and would be removed if the system was reconstructed. The valve used
to close off the turbo pump is necessary to remove the pump from the system after thebake out. The sealing surface of the valves and not the bellows should always face thevacuum side; this configuration reduces the surface area in the UHV chamber.
One thing to consider when deciding on Ti-sublimation pump placement is where
the Titanium will deposit. The Ti-sublimation pump’s filaments should be placed sothat there is not a direct line of sight to any valve or pump. The Ti will coat all surfaces
in direct line with the filaments and can cause a valve sealing or pumping problem. See
Fig. 2.4 for position of the filaments in our system.
The differential pumping between the two chambers is maintained by placing asmall aperture on each side of the bellows to reduce the conductance. The apertures are
5 mm diameter holes in the solid copper gaskets connecting the bellow to the system.
With our design the pressure differential between the two chambers is about a factorRb getters

15

Vapor cellTi-sublimation pump

Ti-filaments

Bellows
3/4” tubing

Turbo pump
mounted vertically2” tubing

Science cell

All metal valves

Ion Pump
40 l/sFigure 2.4: Vacuum system layout. The system is suspended with a series of clamps(not shown), so the center line of all the horizontal tubing is 18 cm above the optical
table.

of 17 for Rb. A second ion pump could be added between the two cells to increase thepressure differential and allow a shorter MOT loading time without sacrificing sciencecell lifetime.
The conductance of gas through a tube and an aperture in the molecular-flow
regime is
Ctube =

3.81D 3 q
T /M liters sec−1
Lq

Caperture = 3.7A T /M liters sec−1 ,

(2.1)
(2.2)

16
where D and L are the diameter and length of the tube in cm [19]. A is the crosssectional area of the aperture in cm2 , T is the temperature in Kelvin, and M is themolecular weight of the gas. The conductance of a system can be found by adding
the conductance of each individual part like capacitors in an electrical circuit as in thefollowing: (Cparallel = C1 + C2 + · · ·, 1/Cseries = 1/C1 + 1/C2 + · · ·).
The science and MOT cells are cylindrical glass cells attached to glass-to-metal
seals. The science cell is a 10 cm length of 1.4 cm outside diameter pyrex tubing witha window on one end and a glass-to-metal seal on the other. The MOT cell is a 20 cm
length of 5 cm outside diameter pyrex tubing necked down on each end. One end is
attached to a glass-to-metal seal, while a getter assembly is fused to the other end(Fig. 2.5).

1.9 cm
1.9 cm

4 cm 4 cm

5 cm

25 cm

8 cm

Figure 2.5: Diagram of the MOT cell showing the getter assembly and glass-to-metal
seal welded onto a knife-edge seal flange.

The getter assembly consists of a current feedthrough and two Rb dispensers(getters). The feedthrough, called a pin press, is a glass section with Tungsten pins
inserted. The Rb getters are spot welded to the pins of the pin press. The Rb getters
are a controllable source of Rb vapor. A getter is a small foil container of a Rb salt,
which releases Rb when a moderate current of 2-6 A is run through the device. It is
important to use a current limited power supply for the getters, because the resistanceof the getters changes drastically as the temperature increases. The getter assembly canbe seen in Fig. 2.6.
Special care must be taken with the getters to insure they will produce clean

17getters

pinpress

Tungsten pins

Figure 2.6: Getter assembly showing two getters and reentrant glass with pins inserted.

Rb vapor. First, the getter material can easily absorb water, so we keep them undervacuum with desiccant during storage and flow only dry gas during the glass fusing
process. Second, they release Rb as a double exponential function of temperature.
Thus, we make sure that while the pin press is being fused to the cell, the getters are
not heated significantly by the fusing torch. Avoiding moisture and heat, we are usuallyable to have getters that produce clean Rb vapor. However occasionally the getterswill produce other products as well as Rb. In our current system when we turn on thegetters the number of atoms in the MOT decreases, presumably due to contaminants
being released from the getter. Our mode of operation is to turn on the getters at 3.5 A
for 10 minutes to supply the MOT cell with a day’s worth of Rb, and then to allow
10-20 min for the contaminants to pump out of the system before taking data.
A good question to ask is why we chose to use cylindrical glass cells insteadof square cross-section cells. The main reason is it is easier to make cylindrical cells.
Anyone with a small amount of glassblowing training can fabricate the cylindrical cells,
whereas constructing square cells usually requires significantly more equipment andexpertise. Another advantage to using a cylindrical MOT cell is the interference fringes

18
on the trapping beams have higher spatial frequency and are spaced less regularly thanwith a square cell. One can shake the glass MOT cell around by a 1 cm and see little
movement of the MOT cloud. MOT alignment with a square cell can be more difficult
because it is important to place the minimum of the fringes away from the center of
the trap. This type of alignment requires more frequent adjustments and is more atrial-and-error process than simply overlapping the beams at the correct angles, as isrequired for a cylindrical cell. A cylindrical cell distorts the trapping beam, but thisdoes not seem to be a large problem for us.
The cylindrical science cell, on the other hand, is less desirable. The probe beam is
focused by the cell, which acts as two cylindrical lenses. This is generally not a problem
for absorption imaging, but can be for phase contrast imaging. Phase contrast imaging
requires placing a material in the fourier plane to shift the phase of the light. The large
astigmatism induced by the cell requires placing a phase shifting line rather than a dot,
thus making alignment more difficult. If we were to reconstruct our experiment, we
would replace the cylindrical science cell with a commercial square cell, which is easy
to obtain from companies who specialize in producing spectroscopic cells.

2.3.2

Chamber construction

Obtaining UHV pressures requires careful assembly of the vacuum components.
The most important thing is to make sure all of the components are clean. We start
the cleaning process by placing the submersible parts (no valves, pumps, or cells) in an
ultrasonic cleaner with strong soap for 1 hour. If the valves are cleaned in the ultrasoniccleaner they must be re-greased before they are used. The long cleaning time allows
the strong soap to remove residual oil from the factory. Typically when a vacuum partis baked in air it will become a golden color, which we assume is residual burnt factoryoil. A one hour bath in a strong basic soap will remove this coating, and it will not
return with subsequent air bakes. The long bath is not absolutely necessary and may

19
be reduced to a few minutes to just remove any particulates from the parts. After theultrasonic bath the parts are rinsed with deionized water, acetone, and then methanol.
Next the parts are baked in air for 4 hours at 400◦ C to drive off any residual material
still on the parts. Once the parts have cooled they are wrapped in oil-free aluminum
foil until assembly.
It is important to avoid contamination of the vacuum system during assembly.
We always wear powder-free latex gloves and change them often. All knife edges andcopper gaskets are wiped with ultra-pure methanol before installation to remove any
factory residue.
We use silvered bolts on the knife edge flanges to reduce the possibility of bolts
seizing in the flanges during the bake out. If silvered bolts are not available we placesome molybdenum disulfide powder suspended in methanol on the threads of the bolts
for lubrication. Suspending the molybdenum in methanol reduces the chance that it
will fall into the vacuum system during assembly and become a contaminant.
After the entire vacuum system has been assembled it is pumped out and checked
for leaks. We use a small turbo pump backed by a dry, four-stage diaphragm pump to
initially pump out the chamber. We use a diaphragm pump rather than a standard
oil-filled roughing pump, because the diaphragm pump does does not contain any oil,which could leak into the system. Once the turbo pump has spun up to full speed wespray a small amount of ultra-pure methanol on all the flanges and cells. If there isa large leak, the pressure in the tubing connecting the turbo pump to the diaphragm
pump, read by a thermocouple gauge, will change when methanol is applied. The system
is pumped on overnight before the bake out is started. A carefully cleaned system, which
does not contain any large leaks, should pump out overnight with the pressure reachingaround several 10−8 torr, read from the ion pump.

20
2.3.3

Chamber bake out

The vacuum system must be baked at high temperatures under vacuum to remove
contaminants to obtain UHV pressures. We bake most of the chamber at 300 ◦ C for
several days. Before the bake out, we run about 6 A through the getters for 30 seconds
to verify the presence of Rb, which can be seen by either laser absorption or fluorescence.
The high current also degasses the getters.
The system can now be prepared for the bake. The first step in the bake out
process is to wrap the glass cells with clean fiberglass cloth. The cloth will protect
the cells from anything that may melt onto the cells during the bake. Next we place
Teflon-coated thermocouples on the chamber at critical places such as the cells, glassto-metal seals, and pumps. We then wrap the system with high-temperature resistive
heater tape. It is not important to wrap the entire surface of the chamber with tape,but rather to have a constanttape-to-chamber surface area ratio. Heater tape is applied to different objects proportional to their surface area and not the mass of the object. The mass only defines
the time constant for thermal equilibration, whereas the ultimate temperature is determined by the heat flow in and out of the region, which is proportional to the surface
area. The tape should never overlap itself, or the intense heat will cause the tape toburn. Several short tapes are used to wrap the system so each section may be controlled
independently. The turbo pump is not baked because it is not part of the final chamberand can not handle high temperatures. We do however bake the entire ion pump withthe magnets in place. Typical maximum baking temperatures for different components
are listed in Table 2.3.3. After the tapes are in place the chamber is wrapped with stripsof fiberglass and then several loose layers of aluminum foil to insulate the system fromthe environment.
The system is slowly brought up to the final temperature over 6-8 hours. The ion

Vacuum element
turbo pump inlet flange
ion pump magnets
ion pump body
ion pump cablebakeable valve, open
bakeable valve, closed
Ti-sublimation pumpglass/metal seals

Maximum baking temperature (◦ C)
120
250
400
250
450
300
350
300

21

Table 2.1: Vacuum component temperature limits

pump is still off at this point. There is a large amount of material driven off the wallsof the chamber during the initial several hours of the bake. We prefer to have the turbo
pump, which has an almost infinite pumping lifetime, pump out initially rather than
the finite lifetime ion pump. Each heater tape is powered by a variable AC transformer(variac) which controls the temperature of each section of the chamber. Generally weincrease the temperature by at most 50◦ C per hour. Temperature gradients can apply
significant stress to the system. We prefer to keep the temperature gradients to under30◦ C across the glass cells and glass-to-metal seals, which are the most susceptible
components to failure. Caution must be taken when the system is near 300◦ C becausesome parts of the system could overshoot in temperature due to long thermal time
constants. During the bake 3 A are run continuously through the getters to clean them.
If the getters are left off they will be the coldest part of the system, and contaminants
will accumulate on them. We also run 25 A through the Ti-sublimation filaments during
most of the bake. Throughout the warming up process, thermocouple readings, variac
settings, and pressure readings are recorded to facilitate future bakes. Once the system
is at the desired temperature we bake with just the turbo pump on for 12 hours. At
this point we degas the getters and Ti-sublimation filaments. To degas the getters,
we increase the current in each getter for 30 seconds to 5 A to drive off any surface
contaminants. After the first 12 hours we turn on the ion pump and valve off the still-

22
running turbo pump. When the valve is above room temperature we close it only finger
tight. Therefore, we do not allow the turbo pump to spin down until the system isback to room temperature, and the valve can be properly closed with a torque wrench.
We allow the system to bake with the ion pump on for ∼ 2 days, or until the pressure
on the ion pump reads in the middle 10−9 torr. We cool the system down slowly over
4-6 hours. At this point the ion pump should read the lowest possible pressure, which
is 10−10 torr for most pumps, if there is no leakage current. The ultimate test of thevacuum pressure is the lifetime of the atoms in a magnetic trap.

2.4

Magneto-optical trap
We have a large six-beam MOT. We are forced to use six beams and not retroreflect the beams because the cloud of atoms casts a shadow and would cause an intensity
imbalance between the beams. Also the cylindrical cell distorts the beams as they passthrough both sides of the cell. The diameter D, of the beams is approximately 5 cm.
Large beams greatly increase the number of atoms in a MOT because the loading rate
scales theoretically as D∼4 for intensities even as low as half saturation [20].
The optical layout is shown in Fig. 2.7 with the vertical beam paths shown in
Fig. 2.8. We use 5 cm diameter polarizing beamsplitting cubes and waveplates and7.5 cm mirrors to accommodate the large beams. The repump beam enters the system
via the backside of a cube; therefore the polarization of the repump beam will not be
optimum when it enters the trapping region. This is not a problem because a Rb MOT
requires very little repump power.
The MOT coils, which also serve as the quadrupole trap coils, are made of 24 turnsof square hollow copper tubing coated with Kapton. The wire has a square crosssection of 4.15 mm on a side with a round 2.5 mm diameter hole in the center. The
coils are cooled by running water through the center region of the wire. The wireis wound onto a phenolic spool and secured with epoxy. Phenolic was chosen as thePolarizing
beam splitting
cube
l
2

Bottom vertical
beam
l
2

Mirror23l
2 Trapping beaml2

Repump beam

l
4

l
4

l
4
l
4

l
2

Top verticalbeam

Figure 2.7: Schematic of the MOT optical layout. Some of the mirrors and waveplates
for the vertical beams are not shown.

spool material because it will not support eddy currents when the current is abruptly
changed in the coils. The inner diameter of the coils is 5 cm, and their centers are
separated by 10 cm. The current in the coils, run in series, is controlled by a simple
servo circuit (Fig. 2.9), which controls three power MOSFETs. We use a 580 Amp, 8 V
switching power supply, run in voltage-controlled mode, to supply current to the MOT
coils. The coil configuration produces a magnetic field gradient of 1 Gauss/cm/Amp
along the axis of the coils.

24

Figure 2.8: Picture of the vertical-beam optics for the MOT omitted from Fig. 2.7.
Figure shows the location of the track (blue arrow) and the vertical MOT beams (redarrows). The coils of the MOT/quadrupole trap are translated left, towards the science
cell, out of the field of view of this photograph.

2.4.1

MOT alignment

The alignment of our MOT is not as sensitive as it would be for a MOT with
1” beams. We start aligning the MOT by placing an iris in the beam from the MOPA

1N414B

Pos terminalof power supply

25

12 pF
Control
voltageCoils
10 kOhm
1 kOhm_122 pF

Power MOSFET
water cooled
APT10M07JVR

10 kOhm+

200 Ohm

OPA627
Hall sensor
Differential
amplifierCurrent source output
475 Ohm

Ref
SENS
+IN
OUT

Neg terminalof power supply-IN

Figure 2.9: Schematic of the MOT/Quadrupole trap servo circuit. The circuit has
standard PI loop gain, where the proportional gain comes from the feedback from theclosed-loop Hall current sensor. We use 2/0 gauge welding cable to carry 300 A fromthe power supply to the coils and MOSFETs. The three MOSFETs are mounted on awater cooled copper plate to remove the heat generated by the MOSFETs.

before it is split into six. Closing the aperture allows us to align the centers of thebeams. Once we get all the beams roughly aligned with respect to the magnetic coils
and each other, we balance the power in the beams. We have λ/2 plates mounted on
rotating mounts before every polarizing beamsplitting cube to adjust the power in eachbeam. We measure the power just before the beams enter the MOT cell and adjust the
waveplates until the power is equal in each of the six beams to better than 10%. Next we
open the iris and attempt to see a trapped cloud. A simple security camera can image
the fluorescence from the cloud, which can be viewed on a monitor. It may take several
hours with the getters running for the cell to be coated with Rb and a significant Rbvapor pressure to be established. Once a cloud is visible, an easy way to adjust finely
the alignment is to reduce the beam size, optimize the cloud for number and roundness,

26
and repeat, all the time making sure the beam pairs are kept counterpropagating.
The position of the trapped cloud should be centered with the quadrupole magnetic trap to minimize energy gained by the cloud when it is transferred into the magnetic trap. To test this, we increase the current in the MOT coils until the cloud size is
reduced greatly by the large magnetic field-induced detuning. The position of the small
cloud, in a very large magnetic field gradient, is reliably at the center of the magnetic
field, and thus the center of the magnetic trap. We then decrease the field, and adjust
the the beam alignment and power balance until the cloud center is in the same location
at high and low magnetic fields.
Another test for quality of the alignment is to look at optical molasses [21]. We
quickly turn off the magnetic field of the MOT and look at the expanding cloud. If it
moves rapidly in one direction this could be a sign of beam imbalance (from incorrect
polarization or intensity splitting between the beam), poor alignment, or stray magnetic
fields. The goal is have slow, spatially-uniform expansion during optical molasses. In
traditional systems, it is necessary to use shim coils to cancel ambient magnetic fields
and to have the atoms in the optical molasses expand uniformly. A feature of oursystem is the lack of need for shim coils. We transfer atoms to our quadrupole trap at
a relatively high temperature, where the small reduction in energy from the shim coils
would not make a great improvement in the phase space density of the magneticallytrapped cloud.

2.4.2

MOT characterization

We calculate the number of atoms in the MOT by measuring fluorescence fromthe trapped cloud on a photodiode. We focus the fluorescence onto the photodiode witha single lens. The number of atoms in the MOT is
N=

4π(photodiode current)(solid angle)(responsivity)(energy of a photon)(R)(0.96) k

,

(2.3)

27
where solid angle refers to the solid angle subtended by the collection lens, the responsivity refers to the current produced for a given power incident on the photodiode, andk is the number of uncoated glass surfaces between the atoms and the detector. R thephoton scattering rate in photons/sec/atom, is
R=

I
Is πΓ
,
2
1 + IIs + 4( ∆
Γ)

(2.4)

where I is the total intensity of the six beams impinging on the atoms, Is is the saturation intensity, which is 4.1 mW/cm2 for random polarization for Rb. Γ is the naturallinewidth of 6 MHz for Rb, and ∆ is the detuning from resonance. In our experience,using the Is appropriate for random polarization gives the most accurate number ofatoms in a MOT.
We servo the MOT load to increase reproducibility in condensate number. We
could load the MOT for a fixed time each shot. However, if the pressure of Rb changes
throughout the day, the loading rate, and thus the total number loaded into the MOT
will drift with time. We instead measure the voltage output from a photodiode collecting
light from the MOT; when a particular value is reached we stop the loading and proceed
to transfer atoms to the quadrupole trap. This also allows us to vary the resulting
number of atoms in the final cloud by adjusting the MOT load level.

2.5

From MOT to Ioffe-Pritchard trap

2.5.1

Transfer into the quadrupole magnetic trap

There are three main steps to transferring atoms from a MOT into a magnetic
trap: compressed MOT (CMOT) [22], optical pumping, and turn on of the magnetic
trap. Our goal is to transfer the atoms into the quadrupole trap with the highest phasespace density. When the atoms are caught in the magnetic trap, most of the resultant
energy of the atoms comes from the added potential energy due to the Zeeman energy
from magnetic field. The larger the cloud is when the magnetic trap is turned on the28greater the potential energy gained by the atoms. We can not adiabatically ramp on ourmagnetic trap from zero gradient, because at low magnetic gradients the trap centeris significantly offset from the cloud center due to gravity. The center offset inducesslosh in the trap, which turns into thermal energy. Therefore reducing the initial spatial
extent of our CMOT cloud is more important than obtaining the coldest temperature.
Our CMOT stage consists of a MOT with greatly reduced repump laser power
and large red detuning of the trapping laser. MOTs with large numbers of atoms havea maximum density of around 1011 atoms/cm3 , which is limited by radiation pressure.
The CMOT has the effect of reducing radiation pressure in the trap and thus creatinga denser cloud of atoms. Reducing the repump power reduces the time the atoms
spend in the state (F= 2) resonant with the trapping light. Increasing the detuning of
the trapping laser decreases the scattering rate and thus the absorption of reradiatedphotons.
We use a short CMOT stage in preparation for transfer to the magnetic trap.
The CMOT has a much smaller loading rate than a regular MOT. Therefore we want tominimize the time spent in the CMOT stage and just go briefly to a CMOT configuration
before the magnetic trap is turned on. The repump power is reduced from several mW
to 50 µW for the CMOT stage. We have two separate repump beams entering thevapor cell. One beam is the main repump beam with several mW of power, and theother, which we call the bypass beam, has only 50 µW of power. Using two shutters weare able to have either full repump power or quickly reduce the power for the CMOT
stage. We also jump the detuning of the trapping laser 50 MHz red of resonance. Thisfrequency jump is accomplished by unlocking the laser, applying a voltage to the laser
piezo electric tuner (PZT) for the CMOT and optical pumping stages, and then after
the shutters are closed turning off the applied voltage and relocking the laser. We keep
the magnetic field gradient constant at the MOT value during the CMOT stage. The
entire CMOT stage lasts about 10 ms and is not very sensitive to changes in trapping

29
laser detuning on the order of 10 MHz.
We can characterize the atom cloud in the CMOT using fluorescence imaging.
The position of the cloud in the CMOT may be very different from the position of thecloud in the MOT or magnetic trap because of beam imbalances or misalignment. We
adjust the alignment and half-wave plates controlling the power in the beams to overlap
the position CMOT with that of the magnetic trap, using the same high magnetic field
gradient alignment technique used for the MOT/magnetic trap alignment. A large offsetof the centers will increase the temperature of the magnetically trapped cloud.
After the CMOT stage we optically pump the atoms into the lower hyperfine
ground state with arbitrary population in the magnetic sublevels. An atom excited to
the F0 = 3 state has about a 1 in 1000 chance of decaying to the F = 1; therefore theatoms will be pumped into the F = 1 state in less than one ms if the repump light isturned off. We trap only the mf = −1 Zeeman state in the magnetic trap. Naively one
might think that we trap only 1/3 of the atoms, surprisingly this is usually not the case.
We can, in certain circumstances, get over 50% of the atoms in the right Zeeman state,
depending on the MOT beam polarization. The population distribution in a MOT is
not an easily controlled parameter but can be adjusted with small random tweaks of
the MOT beams. We check to see that we are effectively pumping the atoms into the
F= 1 state by imaging the cloud with F= 2 → F’= 3 light. If the atoms fluoresce, they
have not been fully pumped into the F= 1 state, and the optical pumping time must
be increased.
Stage
MOT
CMOT
Optical pumping
Magnetic catch
Magnetic trap ramp

Trapping/Repump
On/3 mW
On/50 µW
On/Off
Off/Off
Off/Off

Detuning-2.5 Γ
-10Γ
-10Γ
—
—

Magnetic gradient
8 G/cm
8 G/cm
8 G/cm
100 G/cm
100→250 G/cm

Time
∼10 sec
20 ms
1 ms
200 µs
500 msTable 2.2: Parameters for trapping and loading into a magnetic trap

30
The atoms are now ready to be caught in the magnetic trap. As stated before
we do not slowly ramp the magnetic field from zero due to the effect of gravity; wealso do not want to turn on the magnetic trap at the highest gradient possible because
this will add an excess amount of energy to the cloud. Our procedure consists of
diabatically turning on the magnetic trap to a point where gravity has a minimal effectand yet the magnetic trap adds as little potential energy as possible. This catch point
is determined empirically to be around 100 G/cm in the axial (vertical) direction. Afterthe initial catch we adiabatically ramp the magnetic field gradient to 250 G/cm in 0.5 s.
We optimize the transfer by imaging the atoms after they have been loaded into themagnetic trap, since we ultimately care about the temperature and number of atoms inthe magnetic trap. We get 1-2×109 atoms at 250-400 µK in the fully compressed trap.

2.5.2

Fluorescence imaging

We need to image the atom cloud in the MOT cell for a variety of diagnostics,
such as loading efficiency into and temperature in the magnetic trap, transfer efficiency
to the science cell, and magnetic trap lifetime in different regions of the vacuum system.
For these diagnostics it is not important to measure the absolute temperature andnumber of atoms in the trap but rather relative quantities. We use fluorescence imaging
because, although it is far inferior to absorption imaging, it is easy to setup and gives
us the information we require.
We optimize the transfer to the magnetic trap by imaging the cloud using fluorescence. We turn off the quadrupole trap, open the camera shutter, turn on the repump
beam, flash the MOT beams for less than 1 ms, and image the fluorescence from thecloud onto a CCD camera. We image the atoms directly out of the magnetic trap without any expansion. We can not take an image too soon after the MOT stage because thefluorescence from the MOT will saturate the camera if the shutter is open. We therefore
must wait several ms for the large camera shutter to open after the MOT stage before

31
we can take an image. The optical layout is shown in Fig. 2.10.
CameraTop View

Side View
Shutter

Objective lensf= 100 mm
MOT
cell

Camera5 cm

f= 50mm
mirror13 cm

Figure 2.10: Top and side views of the optical layout for fluorescence imaging. The
objective lens is apertured with an iris to 10 mm in diameter, and the second lens is
30 mm in diameter.

We calculate temperature of the cloud using the fitting parameters from a Gaussian fit. The cloud is not Gaussian in shape in a quadrupole trap. However, the cloud’s
profile is not far from Gaussian, and all we really want is a measure of the cloud thatis monotonic with respect to size and fluorescence intensity. The temperature T andpeak density n0 are
T =2 µB gh 0
Bx σF W HM
5 kb

n0 = 1.27

N,
σF3 W HM

(2.5)

(2.6)

where g is the Landé g factor, and σF W HM is the full width at half max size of thecloud using the correct functional form of atoms in a spherical quadrupole trap. We
calibrate the number N using the photodiode. Unfortunately our cylindrical glass cellcauses some problems with imaging. The cell lenses the scattered light so that the sizeof the cloud is distorted by about 25% in the vertical direction. We get the temperaturefrom the horizontal direction, on which the cylindrical cell has no effect.
Vignetting is such a common imaging systematic for fluorescence imaging that it
deserves to be elaborated on here [23]. Vignetting occurs in a multiple lens system when
imaging an extended object. It happens when there is more than one effective aperture

32
in the system. A common example is when a ray of light that passes through theobjective lens does not make it through the second lens. The light that does not makeit to the second lens comes primarily from the edge of the object as seen in Fig. 2.11.
The decrease in imaged light from the edge of the cloud decreases the apparent size ofthe cloud.

Light lost from system

Figure 2.11: Illustration of vignetting. Vignetting occurs when rays of light from theedge of an extended object are removed from the imaging system by a second aperture,in this case the second lens.

There are a few easy ways to check if an imaging system has vignetting issues.
One way to check for a problem is to reduce the size of the first lens by a factor of two indiameter with an iris. This should decrease the total intensity of the image by a factor
of four without changing the apparent width if vignetting is not a problem. One canalso measure how close the system is to being affected by vignetting by reducing thesize of the second lens with an iris. The size of the image will remain the same until
the second lens begins to become an aperture in the system. There are several ways toeliminate vignetting: replace the second lens with a larger diameter lens, aperture theobjective lens, or move the lens closer together.

33
2.5.3

Transfer from vapor cell to UHV region

We use moving magnetic coils to transfer the atoms from the relatively high pressure vapor cell to the UHV region, where we evaporativly cool to BEC. The quadrupole
coils are mounted on a linear stage that is driven by a servo motor and controlled by acomputer (Fig. 2.12). The maximum possible acceleration of the coils is about 1/3 g,
which is much less than the trapping force from the magnetic trap at 4 g; the atoms
are therefore not heated any detectable amount. Also we do not see any atom loss frommoving the atoms. We want to get the atoms out of the vapor cell as quickly as possible
because collisions with the background gas limit the lifetime to 5-15 s depending on
the Rb vapor pressure. However, we must slow down the coils as the trapped cloudenters the fringing fields of the Ioffe-Pritchard trap in order to adiabatically compress
the cloud. We move the atoms out of the vapor cell, into the UHV region, and towithin 4 cm from the center of Ioffe-Pritchard trap in about 1 s. We decelerate to a
speed of 1 cm/s as the atoms enter the permanent magnetic region and are adiabaticallycompressed.
We purchased a commercial servo-linear track to move the coils from one end of
the vacuum system to the other. The track itself is a table, mounted on a ground ballscrew, which can accommodate higher speeds and has less backlash than a traditional
lead screw. The servo motor in our system has reproducibility of 5 µm, which is measured by a rotary encoder in the motor housing. The encoder signal, although sent
through a shielded cable, is radiated from the encoder cable; typically we see 20 kHz
spikes of 3 µs duration coming from the cable. So far this radiation has not caused any
problems with other equipment or our ability to make a condensate.
The ability to move the atoms in the magnetic trap to different regions of thevacuum system allows us to measure the background pressure and also to localize possible places where stray light impinges on the the system. We need long, background-34

Figure 2.12: Illustration of the motion of quadrupole trap coils.

gas-limited lifetimes in the UHV region to be able to efficiently evaporate and form acondensate. We do not know the lower limit on the necessary lifetime, but we do knowthat our 170 s lifetime is much more than sufficient. We measure the lifetime in themagnetic trap by loading atoms into the quadrupole magnetic trap, moving the cloudto the desired position in the vacuum system, waiting some time, moving the atoms
back to the MOT region, and imaging the cloud. We fit an exponential to the numberof atoms remaining as a function of time. The exponential time constant gives us the
lifetime in the trap.
There are two additional loss mechanisms that could reduce the lifetime in themagnetic trap. One is resonant light impinging on the atoms; the other is Majorana
spin flips [24]. An atom absorbing a single photon has a large probability of falling backto an untrapped state and thus being ejected from the trap. There are two schools ofthought on shielding laser light from the atoms in a magnetic trap. One is to place a box
around the lasers and frequency stabilization optics; the other is to enclose the vacuum
chamber and MOT optics. We choose to place a large (137 cm × 124 cm × 53 cm) box
made from an aluminum frame with opaque lucite panels around the large MOT optics
and vacuum system. This choice has the added benefit of also blocking room lights from35
the experiment and thus allowing the experiment to be run with the room lights on.
The other loss mechanism is due to Majorana spin flips. Majorana or diabatic
spin flips happen in a magnetic trap only when the trap has a zero of the magnetic
field. Atoms can undergo a spin flip if the time rate of change of the magnetic field isgreater than the Larmor frequency. In a quadrupole trap atoms which pass through anellipsoid near the center of the trap can be lost due to spin flips to a non-magneticallytrapped state. The lifetime associated with this loss rate is proportional to the squareof the size of the cloud and is given byτ = αl2 ,(2.7)

where α is determined experimentally to be
α = 3.7(7) × 104

s
,
cm2

(2.8)

and l is the radial half width half maximum of the cloud [24]. The loss rate due to spin
flips is much smaller than the loss rate from background gas collisions for the typicalcloud temperatures (200-400 µK) we have in the quadrupole trap.

2.5.4

Ioffe-Pritchard Magnetic Trap

We have a hybrid Ioffe-Pritchard trap (HIP trap) which contains both permanent
magnets and electromagnetic coils. Permanent magnets are useful because they produce
large magnetic field gradients with no power consumption. However permanent magnets
are sensitive to temperature fluctuations and thus can lead to instabilities if used to
produce a bias field for a magnetic trap. The bias field is the most important parameterto stabilize because it determines the final evaporative cut and thus the temperature. In
our trap the two permanent magnets produce a quadrupole field in the radial direction
and no field along the axial or bias field direction.
The bias field and axial confinement are created by four electromagnetic coils.
The outer two coils produce essentially all of the axial curvature, and the inner two

36
coils control the value of the bias field. Each pair of coils is run in series and controlled independently by a bipolar power supply. The power supplies internally servo
the current to better than 1 part in 104 using an analog voltage set point supplied by
a computer-controlled digital-to-analog converter. We increase the long term stabilityof our trap by continually running the operating current through our coils except for a
4 s period when the atoms are entering the HIP trapping region. The trap is therefore
always at the same temperature even if our experimental timing changes. If we run
operating current through the coils continually the coils reach a steady state temperature of 75◦ C. Although this temperature does not effect the operation of the magnetic
trap it does increase the temperature of the glass cell, which it surrounds. Raising thetemperature of the glass cell causes an undesirable increase in background pressure. We
use forced air cooling to reduce the temperature of the coils from 75 ◦ C to 35◦ C. We
have two air cooling ports fed by filtered compressed air as seen in Fig. 2.13. We do
not use a fan to cool the trap because it adds noise to our trapping fields.
Our trap has the advantages of tight radial confinement from permanent magnets
and also a stable bias field from well-servoed axial coils. Our trapping frequencies are(230, 230, 7) Hz with a 3 G bias field. The radial frequencies can be adjusted bychanging the bias field. The radial frequency is
1
ν=
2π

r

µB mf gf B0
√
m
B0

(2.9)

where µb is the Bohr magneton, gf is the Landé gfactor, m is the mass of Rb, B’ is thefield gradient, and B0 is the bias field.
The trap configuration is shown in Figs. 2.14 and 2.15. The permanent magnets
are 2” X 3/4” X 1/4” grade 35 Nd/Fe/B, which produce a gradient of 450 G/cm. Initially we used permanent magnets that produced a quadrupole gradient of 1200 G/cm.
This gradient gave us 700 Hz radial trap frequencies at a bias field of 3 G. We foundthat having such tight confinement caused density dependent losses, which were so large

37

Air cooling ports
Imaging port

Centering coil

Axial coils

Microwave waveguide
MagnetSecond lensCameraAir cooling ports

Figure 2.13: Science cell region showing the magnetic trap holder. Two stainless-steel
endcaps on the ends of the Boron nitride form (white) are attached to a support structure
behind the coil form. The microwave waveguide is shown on the left side of the picture
directed towards the trapping region. Not shown is the objective lens on the back of
the coil form.

that the final evaporation stage was not efficient, presumably due to inelastic collisions,and thus we produced smaller condensates with shorter lifetimes. With a sufficiently
large number of atoms trapped initially, a larger transverse quadrupole gradient is notalways better.
The outer(inner) coils are each 20(10) turns of 18 gauge magnet wire held in placewith thermally conductive epoxy. In the normal configuration we run 13 A through theouter coils and 6.5 A through the inner coils. All of the trap components are mounted ona form made from Boron nitride. This ceramic has a high thermal conductivity, similar

38

Imaging axis
N
S
Magnetic field
lines

S
N
Permanent magnets I-P coils

Figure 2.14: Ioffe-Pritchard magnetic trap (end on view). The permanent magnets areat a 450 angle with respect to the horizontal axis so as to provide a magnetic field inthe same direction as the magnetic field from the quadrupole trap used to transport theatoms. Four permanent magnets, magnetized through the thin dimension, will also work
to provide a two dimensional quadrupole field with no field along the axial direction of
the trap. However, using just two magnets magnetized through the thin direction will
create a significant gradient along the axial direction of the trap.

50mm
Permanent magnet

38mm

Axial coils

1cm 1cm 1cm
Figure 2.15: Ioffe-Pritchard magnetic trap (side view) showing the permanent magnet
and axial coil positions. Outer(inner) coils are each 20(10) turns of 18 gauge magnet
wire and provide the axial confinment/bias.

to Aluminum, so that the heat generated from the coils can be removed. It also has a low
coefficient of thermal expansion, smaller than stainless steel, which allows the position,and thus the trapping field, to remain constant as the trap holder changes temperature.

39
Boron nitride also allows microwaves through without attenuation for frequencies below
10 GHz. Being transparent to microwaves is important for our imaging procedure, inwhich we make transitions between hyperfine ground states, and for assorted scientificgoals. Boron nitride has the consistency of a hard chalk but can be machined into simple
shapes (Fig. 2.16).
80.00
65.40
55.00
49.00
44.00
36.00
31.00
25.00
14.60

19.05 to fit magnet

ø48.00ø38.00ø33.00bolt circle

9.00

ø24.00ø19.05
ø5.00

4 X 1/4-20 X 9.525 deep

Figure 2.16: Machine drawing of the Boron-nitride HIP-trap form.(all units in mm)

We choose to use a HIP trap in the experiment because of its great stability, but
it is obviously not the only solution. A fully electromagnetic trap, such as the TOP
trap would also work [24]. The main requirement for a trap is for it to have around
450 G/cm quadrupole gradient, which is not hard to achieve with electromagnetic coils
close to the 1 cm diameter cell. The quadrupole gradient must be large to have an
acceptable initial collision rate of at least a few Hz to begin evaporation. Initially thecloud is not in the harmonic region of the trap and is mostly confined by the quadrupole
gradient. Therefore the quadrupole gradient defines the initial collision rate. A cloud40
is in the harmonic region of the trap when the mean energy is less than one “bias fieldin energy”, µB gf Bbias , above the bottom of the trap; for a 3 G bias field a cloud is in
the harmonic region when its temperature is below 30µK.

2.5.5Transfer between magnetic traps

We optimize the transfer between traps by transferring the atoms from thequadrupole trap to the HIP trap and back again. We then move the atoms back down
to the MOT region and image them using fluorescence. We do this procedure instead ofimaging in the HIP trap because we can not obtain accurate values for the temperatureand number of hot clouds in the HIP trap. One reason it is nearly impossible to image
in the HIP trap is that the cloud is too large in the axial direction for the image to fit
on the CCD. Secondly there is a large magnetic-field-induced detuning across the radial
direction of the cloud from the permanent magnetic field. Lastly the cloud’s opticaldepth is large in the magnetic trap, which leads to systematics in determining thenumber. The cloud’s temperature and number measured in the MOT cell after beingbrought back from the HIP trap are not a completely accurate representation of theparameters that existed in the HIP trap, but the comparisons are at least monotonic.
Transferring atoms between a quadrupole trap and an Ioffe-Pritchard trap can
not be done completely adiabatically due to the relative directions of the magnetic fields
in each trap. If the transfer is done correctly however, one can limit loss in phase spacedensity to a factor of two to four during transfer. We start by slowly ramping down
the quadrupole gradient (∼ 500 ms) to a point where the cloud is approximately modematched in the axial direction; for our trap this corresponds to a gradient of 100 G/cm.
Next we turn on the HIP axial coils discontinuously and turn off the quadrupole coils.
We find that the timing of the traps turning on and off is not critical to within 10 ms.
After the quadrupole trap is off, we quickly move the coils down to the MOT region in
preparation for the next run.

41

2.6

Rf evaporation
Now that we have atoms in the HIP trap we can evaporatively cool them todegeneracy. The basic idea of evaporation is to remove atoms with more than the
average energy of the cloud and allow the ensemble to equilibrate to a lower temperature
through collisions [10]. We need an adequate elastic collision rate to have the sample
reequilibrate before there is a large loss of atom number or a large increase in energy ofthe sample from inelastic collisions. The elastic collision rate isγ = n(8πa2 )vrel ,(2.10)

where the peak density for atoms in a quadrupole trap isn=0.633 N,σx σy σz(2.11)and the peak density in a harmonic trap is
n=

1
N
.
3/2
(2π) σx σy σz(2.12)The relative velocity isvrel =

s

16kb T,
3πmRb(2.13)

where N is the number of atoms, a is the s-wave scattering length, σi is the full widthat half max of the cloud in the i dimension, and T is the temperature of the cloud.
There are three types of inelastic collisions we have to be concerned with during
evaporation: one-, two- and three-body processes. One-body loss from collisions withbackground gas atoms will cause essentially only number loss and does not induce
heating, because all atoms in the trap have about the same probability of removal.
During the initial stages of evaporation one-body loss is the dominant factor because
the density is low, inhibiting density-dependent collisions. As the density increases
two- and three-body process become important. Two-body processes are significantly
suppressed with a spin-polarized gas in the maximum angular momentum state of a42ground hyperfine state. Due to angular momentum conservation two atoms can notcollide and spin flip to a different Zeeman level in the lower hyperfine state. The only
way for atoms to change their spin is for one of the atoms to be excited to the upper
hyperfine state, which requires 6.8 GHz of energy. An upper bound on the rate constant
has been determined experimentally to be 1.6 × 10−16 cm3 /s [25]. Two-body loss isseldom an issue for 87 Rb in the lower hyperfine state. Three-body loss happens whenthree atoms collide, two forming a molecule, and the other taking away the residual
angular momentum. Three-body processes are proportional to density squared; thethree-body rate constant has been measured to be 4.31(1.8) × 10 −29 cm6 /s for 87 Rb
atoms [25] in the |F = 1, mf = −1i state, and becomes a concern when the density
approaches 1013 cm−3 . The three-body process not only causes atom loss but also heating
because atoms are preferentially lost from the highest density region of the cloud, whichcorresponds to the atoms with the least energy in a magnetic trap. We call this processes
anti-evaporation. A good rule of thumb for efficient evaporation is to have the ratio ofelastic to inelastic collisions be ∼ 100. We define runaway evaporation as the situation
where the collision rate increases as the cloud cools. Inelastic collisions always increaseas the density increases, so without runaway evaporation one will reach a point where
the elastic to inelastic collision ratio will drop below 100.
Our evaporation consists of applying rf radiation to the atoms to spin flip themto an anti-trapped state. The spin flipping is done in an energy dependent manner bytaking advantage of the Zeeman shift from the magnetic trap. High energy atoms spend
more of their time in high magnetic fields near the edge of the cloud compared to thelow energy atoms, and thus the high energy atoms have a larger Zeeman energy splitting
between magnetic sublevels. We can tune the rf to be resonant with the Zeeman energy
splitting for atoms with the most energy, removing them from the trap. We continue
to lower the rf, removing lower and lower energy atoms, all the time approximately
maintaining thermal equilibrium. We want to remove the same fraction of energy from43
the cloud for each evaporation frequency step if the collision rate is constant. Thiscondition corresponds to an exponentially decreasing frequency ramp. As the collision
rate changes so will the optimum exponential time constant. The functional form we
use isν(t) = (νstart − νo )e−t/τ + νo ,(2.14)

where νstart is the frequency where we begin evaporating, νo is the frequency corresponding to the bottom of the trap, and τ is the exponential time constant of the ramp.
The time constant is set by the collision rate and loss rate.

2.6.1

Evaporation optimization

We need several stages of evaporation, each with different parameters. Throughout evaporation both the elastic and inelastic collision rates change as well as the rf
coupling to the atoms, thus we must adjust the evaporation time constant and the rf
power for each stage. We decrease both the time constant, due to the increased collisionrate, and the rf power delivered to the atoms, to avoid power broadening effects, as theatoms cool.
Power broadening of the rf knife will cause the evaporation to lose energy selectivity as the width of the knife becomes comparable to the temperature of the cloud. For
the initial stages of evaporation we need more power to have the same large probability
that an atom will spin flip as when the cloud is cold, because the atoms initially havea larger velocity. The probability of being ejected from the trap scales as[1-exp(-rf power/velocity)]. As the cloud approaches zero temperature one must bevery careful about not applying too much rf power. Another concern with setting the rf
power is coil or amplifier resonances. The rf coil may have a self-resonant frequency inthe frequency range spanned by the evaporation. An easy way to check for resonancesis measure the rf power delivered to the atoms using a small pick-up coil placed near
the the evaporation coil. The rf power from the synthesizer may have to be drastically44
reduced near a resonance to avoid power broadening.
We break up the evaporation into enough stages so that we decrease the temperature by a factor of two or three with each stage; this criterion sets the start and stopfrequency for each stage, points A-E in Fig. 2.17. We start with a time constant of 5 s.
In general the time constant of the evaporation ramp should be about a factor
10-20 greater than 1/collision rate. We do not actually ramp continuously the frequencyof the rf for the initial stages but instead have discrete steps sent to the synthesizer
through GPIB. Typically a single GPIB command will take between 30-50 ms to bereceived and executed; therefore we send a new frequency command every 50 ms. The
discrete nature of the frequency ramp is not a problem for the initial stages when eachstep is small compared to the temperature of the cloud, but is a problem in the laststage of evaporation. For the last stage we sometimes use a programmable frequency
synthesizer, which can ramp the evaporation frequency phase continuously. The extra
synthesizer is not necessary but will produce larger condensates.
We want to optimize collision rate for each stage of the evaporation. If we image
the cloud right after the stage we are optimizing the cloud may not be in equilibrium
due to a too-rapid cut and appear to have an anomalously large collision rate. Imaging
a cloud out of equilibrium can systematically misrepresent the collision rate. However if
we add an additional evaporation stage before imaging, we can circumvent this problem.
The second stage will not be as efficient if the cloud is out of equilibrium, and thus thecloud will have less number after the second stage.
Optimizing the initial stages of evaporation in our trap is difficult because of
our inability to image hot clouds. As stated before, we can not obtain an accurate
temperature or number of atoms in our cloud when the temperature is above 1µK
because of the magnetic field gradients. However, we have created an optimization
procedure for the first stages of evaporation that is acceptable. We start with twostages as seen in Fig. 2.17. The parameters of the first stage, segment A-B, are varied,45
and the second stage, segment B-C, parameters are kept constant. We image the cloudat point C and maximize the peak OD by changing the parameters for segment A-B.
Even with imperfect imaging, the peak OD measured after ramp B-C is monotonic in
the true equilibrated collision rate produced by ramp A-B. Next we add a stage C-D
and optimize segment B-C and so on. It is important to iteratively adjust the time
constant and rf power because they are coupled. Luckily the initial evaporation is notvery sensitive to the parameters of the cut so this procedure works well.

log (n-n0)

A
B
C
D
E
Time
Figure 2.17: Sample evaporation trajectory with four segments shown. We typically use
8 segments, each providing a factor of 2-3 decrease in (ν − ν0 )The final stages of evaporation are more critical than the first stages. However
for the last stages we can accurately determine the temperature and density of thecloud. The procedure for extracting these numbers from the images will be discussed in
Section 2.8. Except when optimizing the very last stage, we characterize a given stage
by optimizing number in the cloud after an additional stage. It is easy to walk theparameters in the wrong direction, especially with the final stage; one tends to have too
short a time constant and too much rf power. We reduce the rf power 15 db from thefirst to the last stage. Generally we can change the rf power by 5-10 db in the upperstages and 3 db in the later stages without observing a significant change in evaporation

46
efficiency. When optimizing the last few stages it is also important to remeasure thetrap bottom ν0 , which can be more accurately determined now that one has a cold
cloud.

2.6.2

rf coil

We use a simple single-loop coil to deliver rf to the atoms for evaporation. We
typically evaporate from 40→2 MHz. The large range of frequencies we use restricts us
from impedance matching the coil to gain better coupling. Because we are in the nearfield limit of the radiation for all evaporation frequencies, one can think of the rf as justan oscillating magnetic field. Only the component of the magnetic field produced by thecoil perpendicular to the atoms’ quantization field will cause transitions between the
Zeeman states. Therefore we place the coil directly outside the glass cell with the axisof the loop perpendicular to the bias field to maximize the coupling to the atoms when
they are cold. Hot clouds will have atoms in the quadrupole field with quantization axesin every direction, and there will be small regions in the cloud that are not be affectedby the rf. This does not appear to pose a serious problem, however. The coil must not
be placed closer than one radius to any electrically conductive object; the conductive
object will reduce the flux return path and thus the magnetic field produced. The sizeof our loop is about 1 cm in diameter, made from 18 gauge magnet wire. We want tominimize the enclosed area of the loop that is not near the atoms to maximize the fieldat the atoms, so the loop is soldered directly to a RG 175 cable, which is powered by a
5 Watt rf amplifier.

2.7

Absorption imaging
We image clouds in the HIP trap using laser absorption. We illuminate the cloudwith resonant light, atoms scatter photons out of the beam, and we focus the shadow
cast by the atoms onto a charge coupled device (CCD) array. The amount of light47
absorbed gives the column optical density (OD) at a particular position in the cloud,which can be used to calculate the number and temperature of the sample. Opticaldensity is defined by Beer’s law and is given by
I = Io e−OD ,(2.15)

where I is the intensity impinging on the camera and I 0 is the initial intensity of the
beam.

2.7.1

Optical set up

The imaging optics are shown in Fig. 2.18. We use a probe beam that has been
filtered spatially by a single mode fiber. The probe beam is expanded to a diameter of
1 cm so that the intensity across a 100 µm condensate is nearly constant. The light firstpasses through a polarizer and then through a λ/2 plate so that we can adjust the angleof the linear polarization. We place a 4 mm2 aperture made from electrical tape on themagnetic trap form to eliminate excess light that could scatter onto the camera fromdefects in the glass cell. The shadow of the atoms is focused onto the camera with twolenses. We use, as the objective, a 1 cm diameter gradient index singlet lens, which isa diffraction limited optic. The objective lens is mounted directly on the trap coil form
to collect the largest possible solid angle. We use a 30 mm diameter achromat doublet
as the second lens. We use an achromat, not for its reduction in chromatic aberrations,but for its low spherical abberations when oriented correctly. Our CCD camera, which
is no longer commercially available, is a front illuminated CCD array with pixels 9 µm
on a side. The entire array is 720 X 560 pixels and has 16 bit resolution. The quantum
efficiency at 780 nm is around 35%.

PBS

F=25.2mm
gradium singlet

CCD Camera

Atoms

~40mm

~150mm½ wave plate~25mm

light from a single
mode fiber

48

F=40 mmachromatic doublet

Figure 2.18: Optical system for absorption imaging. Light from a single mode fiber is
put through a polarizing beamsplitting cube (PBS) and a half waveplate to adjust thepolarization. After the light passes through the cloud, the image is focused onto a CCD
camera by two lenses.

2.7.2

Imaging in a non-uniform magnetic field

Imaging atoms in a strong quadrupole magnetic field presents many problems.
Our goals are to i) expand the cloud above the resolution limit of our imaging system, ii)
image through the radial direction so we can observe dynamics along the axial direction,and iii) extract useful parameters from the image without systematic effects. These goals
are difficult to obtain, because first we can not turn off the quadrupole field createdby the permanent magnets and allow the cloud to ballistically expand as is usually
done. Second imaging in the radial direction causes the bias (quantization) axis to beperpendicular to the propagation direction of the probe beam, and thus does not allow
us to drive purely σ+ or σ− transitions, which is desirable because it would give us a
cycling transition and thus a large signal-to-noise ratio. The last obstacle to overcomeis the magnetic field gradient, which causes a spatially varying energy shift due to the
Zeeman effect. Therefore, we can not apply light which is resonant with the entire cloud.
The spatially varying detuning causes the image to have systematically the wrong widthand intensity.
We have found ways to reduce or eliminate all of our imaging problems. We
expand the cloud by transferring the atoms to an anti-trapped state and allowing themPropagation direction
wrt quantization axis

Polarization

Transitions

ParallelRight circulars+

ParallelLeft circulars-

Parallel

Linears+, s-

Perpendicular

Circulars+, s-, p

Perpendicular

Linear along
quantization fieldp

Perpendicular

Linearperpendicular toquantiazaiton fields+, s-

49

Figure 2.19: Possible transitions with different probe beam polarizations

to fall off of the potential created by the magnetic trap. We use a microwave adiabatic
rapid passage (ARP) to transfer coherently the atoms from the |1, −1i to the |2, −2istate [26]. ARP is best described in the dressed state picture, which is discussed in
Chapter 6.
To ARP the atoms from one bare state to another we turn on a coupling field
far off resonance, ramp the frequency slowly, compared to the Rabi frequency, throughresonance, and then turn off the field. This will coherently transfer the atoms between
the two bare states.
The other problem to solve is the incorrect imaging polarization. We would liketo drive a cycling transition from the |2, −2i to the |30 , −3i state. If we choose our probe
beam polarization linear and perpendicular to the bias field we can drive not only thedesired transition but also the |2, −2i → |30 , −1i transition, which is obviously not acycling transition. However we image in 100 G bias field which breaks the degeneracyof these two transitions by 31 linewidths and allows us to have a cycling transition. Table 2.19 lists the possible transitions for the different probe beam propagation directions

50
and polarizations.
Naively one might expect that at maximum only half of the light could be absorbed, because we think of linear light as an equal amount of σ+ and σ− light. Howeverin the atoms’ frame these two polarizations are not the correct basis and are actuallycoupled. Absorption of this type is typically know as the Voight effect. In fact allof the light can be absorbed by the atoms, and the only effect of the direction of thepolarization is a reduction in the line strength by a factor of two.
Lastly, we reduce the Zeeman detuning across the cloud by increasing our bias
field to 100 G; this is easily accomplished by reversing the current in the inner coils.
The gradient adds in quadrature with the bias field thus reducing the spatial variationof the magnetic field from 2.4 G to 0.1 G for a typical expanded radial cloud radius of
100 µm. The residual variation in magnetic field across the cloud changes the imaging
detuning by 140 kHz, which is much less than a natural linewidth.

2.7.3

Producing and controlling microwaves

We use microwaves for manipulating the state of the atoms during our imaging
procedure as well as during our spectroscopic measurements. We use two different
systems to produce microwaves. One system is a commercial microwave synthesizer,
which, although flexible and easy to use, has a frequency stability of only 5 ×10−12 ina second. This stability is acceptable for ARP during our imaging procedure, but notquite for long interrogation times during Ramsey spectroscopy. The other system we use
to produce microwave frequencies is a home-assembled frequency chain as diagramed inFig. 2.20. This frequency chain has frequency stability better than 10−12 in a second.
Our frequency chain begins with a reference signal from a global positioning
system (GPS) disciplined oscillator. This device provides only long-term frequency
accuracy and stability, where long term refers to a time scale on the order of hours. The
GPS-referenced oscillator removes long-term drift of a Wenzel 10 MHz phase-locked

51

GPS 10 MHz
referenceslow PLL

feedback

2 X 5 MHz
PLO

100 MHz
PLO

two-way
splitter

6.80 GHz
PLO

amplifierthree-way
splittersingle side-band
mixer

transformer

transformer

10 MHznotch filter

10 MHznotch filter

rf synthesizer

device6.4 watt
microwave
amplifiercirculator
square-law
detector

directional
couplermicrowave
wavguideFigure 2.20: Microwave frequency chain showing all of the components we use to produces variable frequency microwaves at 6.8 GHz. The solid arrows indicate transmissionof 10 MHz signals. See text for further description.

oscillator (PLO), via an external, slow-response phase-locked loop (PLL), which has
a bandwidth of about 1 s. The Wenzel 10 MHz signal is split, with one portion sent
to a Wenzel 100 MHz PLO and the other portion sent to another splitter. We use thesignals from the second splitter to phase lock other rf devices in the experiment. We gain
short term frequency stability from the 10 MHz PLO and mid-range frequency stability
from the 100 MHz PLO. The cross-over region between the two PLOs is around 5 kHz.
The 100 MHz signal is multiplied up to 6.80 GHz and phase-locked to a 6.8 GHz crystal
oscillator by a Microlambda device. We require the ability to adjust the frequency of the

52
microwaves, so we mix adjustable-frequency rf and the 6.80 GHz signal with a single
sideband mixer (image reject mixer). We use a single- instead of a double-sideband
mixer because it does not produce significant power in the unwanted sideband; thus itallows greater power output of the desired sideband by the amplifier. The microwave
signal is then amplified by a 6.4 Watt solid state amplifier. We place a circulator on
the output of the amplifier to reduce reflected power coming back to the amplifier. Also
for diagnostic purposes, we place a directional coupler after the circulator, such thatthe directional coupler splits off a small amount of the reflected power and sends it to asquare-law detector. The square-law detector produces a dc signal proportional to themicrowave power.
The microwaves are transmitted to the atoms by a sawed-off waveguide, whichis placed near the HIP trap, directed along the axis of the trap. We use low-loss semiflexible microwave cables to transfer the microwaves from the amplifier to the waveguideto maximize the power delivered to the atoms.

2.7.4

Imaging procedure

We start the imaging procedure with the atoms in the |1, −1i state. We ARP
the atoms to the |2, −2i using microwaves. The microwave field must be swept phase
continuously for atoms to remain in one dressed state. The easiest way to have aphase-continuous sweep is to hold the microwave frequency constant and ramp the bias
magnetic field. We typically start about 1 MHz away from resonance and sweep throughin 0.3 ms; our Rabi frequency is around 100 kHz. Next we jump the bias field to 48 G,
and wait for the anti-trapped atoms to expand. If we expanded in our normal 3 G
bias field trap the atoms would expand too rapidly into the anharmonic the region of
the trap, thus making it difficult to calculate the effect of this expansion. If we jump
directly to a 100 G bias field, the atoms would expand too slowly and fall under gravity,
once again into the anharmonic region of the trap. The intermediate field keeps theTime (ms)
0
9
10
10.3
10.3
10.3 + Expansion time
10.4 + Expansion time
10.402 + Expansion time
20

Event
Camera triggered, probe beam shutter open
Centering coil on
Microwave on, bias field ramp for ARP started
bias ramp stopped, microwave off
Bias field jumped to 48 G for expansion
Bias field jumped to 100 G for imaging
Probe beam AOM on
Probe beam AOM off
Probe beam shutter closed

53

Table 2.3: Image timing

atoms in the harmonic region of the trap during the entire expansion. The atoms arealso slightly sagged in the trap due to gravity, so when they begin to expand they are
sitting on the side of the potential, which induces some asymmetry to the expansion.
We correct for this sag by applying a small magnetic field to shift the center of the trap
to the center of the cloud just before the expansion. We find the correct magnitude of
the centering field by imaging the cloud after long expansion times and adjusting theadded field until the cloud remains in the same position in the vertical direction.
After the cloud has expanded the desired amount, we jump the bias field to 100 G
and flash the probe beam for 20 µs. We use a short 20 µs pulse for two reasons. First we
do not want the atoms to be excited to the to the |30 , −1i state and fall back to a darkstate. We are only 200 MHz detuned from the |2, −2i → |30 , −1i transition and therefore
will drive transitions to this state, although with a very low probability. Second if theatoms in the cloud scatter many photons they will pick up enough momentum to movealong the direction of the probe beam; this motion could blur the image or cause theatoms’ transition frequency to change as they experience a different magnitude of themagnetic field.
We take two additional pictures for normalization purposes. One picture is takenwith the probe beam on but with no atoms present Ilight ; this gives our light image
which we use to calculate percent absorption. We also take a third picture with the54probe beam off and the camera shutter open Idark . This image will give a calibration of
any stray light that does not come from the probe beam as well as camera dark current.
We calculate the optical density of each pixel, which is given by

Ilight − Idark
ODmeas = ln
Iatoms − Idark.

(2.16)

There are two common systematics that should be addressed with any absorption
imaging system. One is the effect of a maximum observable optical density. Any lightcollected by the camera that can not be absorbed by the atoms will reduce the observed
OD. Two usual culprits are off-resonant light and probe-beam light that does not passthough the cloud. A good way to check how much of the probe beam is far off resonant
is to send the probe beam through a heated Rb vapor cell and measure the percent
absorbed. Some diodes have a broad pedestal of light that is not in the main frequency
mode of the laser and thus can cause a reduction in the observed OD. The second reason
for a low maximum observable OD is probe light which is reflected off of objects andis indirectly scattered onto the CCD. We place a small aperture in front of the cell to
reduce scattering light from the cell onto the CCD. We observe a maximum OD (OD sat )around 2.8, even for clouds for which the actual OD is much greater. This OD sat value
is acceptable, but we must correct for it in our analysis of the image. The modified OD,
taking out the effect of OD saturation, is
ODmod = ln

1 − e−ODsat.
e−ODmeas − e−ODsat(2.17)We measure the maximum observable OD by creating a large cloud and expanding
it for 1 ms. The center of the cloud will have a flat top where the OD is saturated
at the maximum value. If the correction factor between ODmeas and ODmod is toolarge, the potential for error increases. We increase the expansion of the cloud until
ODmeas <ODsat /2.
The other systematic with absorption imaging is the effect of probe beam intensity

55
saturation. The actual OD isODactual = ODmod + (1 − e−ODmod )

I
Isat

,

(2.18)

where I is the intensity of the probe at the position of the cloud and Isat is 1.6 mW/cm2
for Rb on a cycling transition. Isat is increased by a factor of two for our case because theline strength is reduced for our imaging polarization. We like to minimize the correction
factor, so we work at less than 1/10 of saturation intensity.
The resonant frequency for the imaging transition can be calculated easily because
both the initial and final states are maximum angular momentum states. F and mf are
therefore always good quantum numbers, and the frequency splitting between the twostate is ∆ν = µB B. We confirm we are on resonance by taking a transition lineshape,
which involves producing identical clouds, and probing them with different frequencies.
We change the frequency of the first probe AOM and measure the peak optical depth.
The resulting curve should be a Lorentzian with the natural linewidth, Γ. Measuring thenatural linewidth with the expected center implies that many parameters are correct in
the imaging system, including narrow laser linewidth, accurate calibration of magnetic
fields, probe beam well below saturation, and correct control of probe frequency.
The lineshape can also be a useful diagnostic for probe laser frequency noise.
Frequently the probe laser frequency may be affected by shutter-induced vibrations orcurrent transients right before imaging. Therefore it is important to measure the noiseon the laser during the imaging pulse. One can find the shot-to-shot standard deviationof the number of atoms imaged while the probe is tuned on resonance and contrast
with that of a half linewidth off resonance. Comparing the two measurements rejects
uncorrelated atom number fluctuations. A significant increase in shot-to-shot noise
when the laser is tuned a half linewidth off resonance indicates probe laser frequency ormagnetic field noise.

56

F=3’
~93 MHz

mf=-3

Energy~780 nm
Imaging
Transition

F=2
70 MHzmf=-2~6.8 Ghz
ARP

mf=-1

70 MHz

F=1

Figure 2.21: Energy level diagram showing imaging transitions in a 100 G magnetic
field. The microwave ARP transition and optical imaging transition are shown withsolid arrows.

2.7.5

Focusing the image

We focus the image onto the CCD camera by imaging a small cloud that has
not expanded much. Before we focus the image we first take a lineshape to ensure
we are on resonance. Imaging on resonance will minimize the cloud lensing the light,
which leads to incorrectly focusing of the image. We also use a cloud with a peak OD
of less than one to reduce effects of lensing. Once we have tuned the probe laser tothe resonant frequency of the atomic transition, we adjust the position of the camera
along imaging axis. The focus of the image will be at the minimum cloud width. We
focus the image by measuring the width in the radial direction, which is not necessarily
the focus in the axial direction of the cloud because of the astigmatism induced by the

57cylindrical cell. If the cloud is exactly in the center of the glass cell the image would not
be astigmatic because all the rays of light hit perpendicular to the glass and therefore
are not refracted.

2.7.6

Measuring the image magnification

We measure the magnification of our imaging system by watching a cloud fallunder the influence of gravity. We start with atoms in the |1, −1i state and perform an
ARP to place them in the |2, 0i state, which is affected only slightly by the magnetic
field gradient. We allow the atoms to fall for a varying time and measure their resulting
position. A cloud’s position as a function of time including the small acceleration due
to the second order Zeeman shift isz(t) = −√a(cos ct − 1) + z0 ,c 

(2.19)

where a is the acceleration due to gravity in pixels/ms2 , z0 is the cloud’s initial position,and c is
c=



4πh̄
f B 02 ,mRb(2.20)

where f is the second order Zeeman shift of 287.57 Hz/G2 , and B ’ is the magnetic field
gradient. Fitting the position vs. drop time data will give a value for a, which can beused to find the magnification M, where M is given byM=
The factor of√

9.81µm/ms√.
2a(2.21)

2 is included because our imaging axis is at 45o with respect to gravity.

2.8

Image analysis

2.8.1

Image processing

After the three images have been downloaded to the computer we apply some
image processing before fitting the images. We first calculate a measured OD for each

58
pixel using Eqn. 2.16. Occasionally we will get pixels that have anomalously high or low
values due to noise or readout error. We remove these pixels by systematically going
through the image array comparing nearest neighbor pixels. If there is a difference of 6,
in units of OD, or greater the pixel is replaced by the average of the eight adjacent pixels.
We perform the same procedure a second time, this time using a difference thresholdof 0.8. After the spikes are removed, we apply corrections for saturation effects using
Eqns. 2.17 and 2.18. Finally, depending on the size of the cloud, we bin the array. For
most condensate images we use 2 X 2 binning, making a modest sacrifice in resolution
in order to reduce calculation time during the fitting routine.

2.8.2

Image fitting

We use three different fitting routines depending on the degeneracy of the cloud[15]. For clouds above the condensation temperature we fit the image to a 2-D Gaussian.
Clouds at finite temperature but with a condensate present we fit to a modified Gaussianplus a Thomas-Fermi distribution. In some cases, where the cloud has no detectable
thermal fraction, we just use a Thomas-Fermi distribution.
Thermal clouds are fit using a standard 2-D Gaussian function which is
1 (x − cx )2 1 (y − cy )2zGauss = |P eakOD| exp −
−
2
s2x
2
s2y

!

,

(2.22)

where the adjustable parameters are cx , cy , the cloud centers in both directions, sx , sy ,the Gaussian widths in both directions, and the peak OD. The fitting is done using a
Matlab script called from inside LabVIEW.
We use a slightly more complicated procedure to fit clouds that have both acondensate and thermal cloud present. The condensate portion of the image can be fit
to a Thomas-Fermi profile, which is parabola integrated in one dimension. The thermal
cloud is no longer an ordinary Gaussian when it is degenerate but is modified by Bosestatistics. We first use only the pixels that have an OD from ODpeak to 70% ODpeak for59
a fit to a Thomas-fermi profile given by

fT F = |T F peakOD|MAX 1 −

x − cxsT F x2y − cysT F y−!2 3/2


where sT F x , sT F y are the radii of the profile in each direction.

, 0 ,
(2.23)The Thomas-Fermi fit curve is subtracted from the image and the resulting image
is fit to the modified Gaussian given by
fmodGauss =g 3 (fGauss )2

g 3 (1),(2.24)

2

where g 3 is2

g 3 (z) = z +
2

z3
z4
z5
z6
z7
z8
z9
z 10
z2
+
+
+
+
+
+
+
+
... (2.25)
23/2 33/2 43/2 53/2 63/2 73/2 83/2 93/2 103/2

We use just the first 11 terms of g 3 in the fitting procedure. We also impose a constraint
2

on the Gaussian fit to have the same center as the Thomas-Fermi fit.

2.8.3

Calculating cloud parameters

Once we fit the image and extract the fitting parameters we can calculate theproperties of the cloud. The first step is to calculate the size of the cloud in themagnetic trap based on our anti-trapped expansion. The Boltzmann equation for theequilibrium distribution function F, which describes the anti-trapped expansion, is

LF (x, v, t) = 0,(2.26)

where F and L are given by

F (x, v, t) =N m3Rb λ6dB |ω|mRb →−x 2 ))
exp(−(−
v 2 + ω2→
6
h
2kB T

L=

δ
−
→
− 2→
−
+→
v · ∇−→→v,
x − ( ω x ) · ∇−δt

(2.27)(2.28)

60
and the matrix of the trap frequencies is


 ωx


w=
 0



0

0



ωy

0 

0

ωz



0 
.



(2.29)The solution to the Boltzmann equation, for each dimension of the cloud, is givenbyσ(t = 0) = q

σmeas ωω 2 + (ω 2 + ω02 ) sinh2 (ωt),(2.30)

where σmeas is the cloud size after expansion, ω is the harmonic trap frequency during
the expansion, ω0 is the original trapping frequency, and t is the expansion time. This
treatment assumes that the initial position and velocity are uncorrelated and that themean-field does not contribute significantly to the expansion. The mean-field contribution to the expansion is negligible in the axial direction, from which we calculate all size
dependent parameters.
The expansion is dominated by the anti-trapping potential and not the mean field,so the expansion is essentially the same for the condensate and the normal component.
We calculate the temperature of the cloud from the axial width because there is very
little expansion in this direction, and the frequency of the potential does not changemuch during the expansion.
The following equations allow one to calculate the useful properties of normal
clouds and condensates. The absorption cross-section A, is
A = (branching ratio)

3λ2
2π

!

×

1
2

1 + 4∆
Γ2(2.31)

where λ is the wavelength of the transition, ∆ is the detuning from resonance, and Γ isthe natural linewidth. The number, temperature, peak number density, collision rate,
relative velocity and phase space density of a normal cloud are respectively
Nnormal = 4.88512 ODpeak sradial saxial A(2.32)

61

T =

mRb
(ωaxial saxial )2kB(2.33)

N a.r.21(2 π)3/2 s3axial(2.34)

nnormal =C.R.peak = n (8πa2 ) v

s

16 kB T3 π mRbh
2πmRb kB Tv=

P SD = n √(2.35)(2.36)

3,(2.37)

where sradial/axial is the Gaussian size as defined in Eqn. 2.22, ωi is the trap frequency
axial), and a is thein the i dimension, a.r. is the aspect ratio of the trap (a.r.= ωωradials-wave scattering length. The number, peak number density, and chemical potential ofa condensate are given respectively by the following

NBEC =

2π
ODpeak sT F axial sT F radial A
5nBEC =



15
8π



NBEC a.r.2
s3axial

15NBEC a
1µBEC = h̄ωho
2
aho


2/5

(2.38)(2.39).

(2.40)The harmonic oscillator length is
aho =

s

h̄
,
mRb ωho(2.41)and the harmonic oscillator frequency is2ωho = (ωaxial ωradial)1/3 ,

where sT F radial/axial is defined in Eqn. 2.23.

(2.42)

62

2.9

Computer control
We use two computers to run our experiment; one computer controls all of thetiming, digital, analog, and GPIB commands, while the other is dedicated to runningthe camera. To initiate an image acquisition, the control computer externally triggers
the camera, which in turn triggers the camera computer. It is useful to allocate thesetasks to different computers so that the camera computer can analyze the data from theprevious shot while the control computer moves on to the next shot. The preliminary
analysis of each shot, which includes calculating cloud parameters, is completed in realtime, greatly increasing the amount of data that can be compiled and digested in a day.

Control Computer
MOT
PhotodiodeShutters,
Switches,etc.

analog input
DIO-128
digital outputs(timing clock)

MOT coil
servo

analog output
board
trigger

MOPA laser
servo

GPIB

rf sysnthesizer
Inner coil
power supply(Kepco)Outer coil
power supply(Kepco)

DAC 1
DAC 2

CameraCamera Computer
Cameraboard

Fitting andanalysis

Figure 2.22: Computer control diagram showing the different computer boards andwhat they control.

A Bose-Einstein condensation experiment requires precise temporal control of63
a variety of components. Most functions require timing resolution on the ms scale,but for certain key tasks, such as imaging, expansion, and microwave spectroscopy, weneed timing on the microsecond scale. There are several basic types of outputs andinputs our control system needs to handle. We need TTL, analog, serial, and GPIB
outputs. Most of the experiment is controlled by digital outputs, which control items
such as shutters and switches. Our magnetic coils, both quadrupole and HIP trap,
have servos that require analog voltage set points. The servo motor driving the trackis controlled via a serial connection. We also have several instruments including rf andmicrowave synthesizers that use GPIB as the main mode of communication. Our only
input port, excluding the camera, is an analog voltage from a photodiode that monitors
the fluorescence from the atoms in the MOT. This input is fed into a multipurpose
analog input board produced by National Instruments.
We have come up with a complete timing system that includes all the different
I/Os. The programming software we chose is LabVIEW; it is a well known and easy
to use software package. LabVIEW does have some limitations because it runs in a
Windows environment. LabVIEW timing can vary by up to 10 ms shot-to-shot, because
the operating system can interrupt the program at any time. Therefore we need anothersource to handle our precise timing. LabVIEW can still handle our imprecise events,
such as GPIB commands, quadrupole coils current ramps, and track motion. Fourcomputer boards and two external digital-to-analog converters (DACs) make up thecontrol hardware.
We use a DIO-128 board produced by Viewpoint USA as the main clock in oursystem. It has 64 digital inputs and outputs and an internal oscillator that has 500 ns
resolution [27]. We load an array of time stamps and port levels, which is the state(hi/low) of each digital port at each time stamp, into the first-in-first-out buffer on
the board via LabVIEW. The buffer can hold up to 16000 words. When we want thetiming sequence to start we send a trigger to the DIO-128 from within LabVIEW. From

64
that point on until the buffer is cleared, the board no longer communicates with thecomputer and thus is not susceptible to operating system interrupts.
We have two different methods to produce the analog output voltages required in
the experiment. One way we produce an analog voltage is via a National Instruments
analog output board, which resides inside the noisy environment of the computer. We
use this board to control items that can handle the amount of noise produced, suchas the quadrupole coil servos. The analog output board can store an array of voltage
values and output them when triggered by the DIO-128. We create voltages with alower level of noise using external 16 bit DACs digitally controlled by the DIO-128. The
power supplies which drive the HIP trap coils are controlled with the low noise DACs.

Chapter 3

Driven two-level atom

The goal of this chapter is to describe how a two-level atom is affected by acoupling radiation field and a specific case of two hyperfine levels in 87 Rb coupled bya two-photon field. This chapter includes a basic description of Rabi flopping andRamsey’s method of separated oscillatory fields, which will be used as the methodof spectroscopy for the following experiments. The goal of both of these methods ofspectroscopy is to measure the energy difference between two atomic states. The choiceof the two specific hyperfine levels and method of coupling them will also be discussed.

3.1

Rabi flopping
A two-level atom in an oscillating field can be described by a time dependent

Hamiltonian, which is given by



H=


h̄ω0
2

ΩR

ΩR

eiωt/2−h̄ω02e−iωt/2




,


(3.1)

where ω0 is the the frequency difference between the two states, ω is the frequency of thedriving field, ΩR is the Rabi frequency, and t is time. The time-dependent Schrödinger
equation with this Hamiltonian can be solved using the rotating wave approximationand the initial condition that the atom is in the ground state. The probability of being
found in the excited state as a function of time is
P (t) =

ΩR
Ω0R

!2

sin

2



Ω0R
t ,
2


(3.2)

66
where the effective Rabi frequency is
Ω0R =
=

q

q

Ω2R + (ω0 − ω)2(3.3)Ω2R + δ 2 ,(3.4)

and δ is the detuning from resonance. An atom in the field oscillates between the twostates at the effective Rabi frequency, where the contrast of the oscillation is determinedby the detuning. This sinusoidal population transfer is referred to as Rabi flopping.
Rabi’s resonance method can be used to measure the energy splitting between two
states. However, it has some limitations. Rabi’s method probes the transition frequency
while an atom is interacting with the coupling field, which can lead to frequency shifts
from the AC Stark effect or additional phase noise. The frequency of the Rabi fringes
depends not only on the frequency between the two states but also on the power ofthe driving field. Originally, in molecular beam experiments, the molecules would passthrough a single region where they were illuminated by radiation and then be detected.
It is reasonable to think that increasing the length of the interaction region l, would
decrease the width of the spectral lines. However if the field is not uniform throughout
the interaction region, increasing l will, at some point, lead to a broadening of thespectral lines. This broadening limits the resolution of Rabi spectroscopy. Ramsey’s
method of separated oscillatory fields was developed to overcome these limitations.

3.2

Ramsey spectroscopy
Ramsey’s method uses two interaction regions separated by free propagation as

seen in Fig. 3.1. Following a two-pulse Ramsey sequence, the probability of being
in the excited state as a function of time, in the short intense pulse limit, with theapproximations that δ  ΩR and τ T is
Pt (T, δ) =1 1
+ cos (δ T ),
2 2

(3.5)

Magnetic Field

67

T

t

t

Figure 3.1: Ramsey’s method of separated oscillatory fields, where two pulses length τ
are separated by free propagation length T.

where T is the free propagation time. A π pulse corresponds to 100% transfer between
states, and a π/2 pulse creates an equal superposition of the two states. The two pulses
do not need to be π/2 pulses. However this condition gives the highest signal-to-noise
ratio and full contrast fringes. Note that the Ramsey fringe frequency only depends
on the field detuning from resonance and thus on the energy difference of the atomic
states and not on the power of the field, as long as τ ΩR is constant. The driving fieldmust still be stable during the pulses; this is a shorter time and the constraint on field
amplitude stability does not increase with increasing free propagation time.
Ramsey’s method can be thought of as an interferometer in the following way. If
an atom is initially in the ground state, the first π/2 pulse places the atom in an equal
coherent superposition of the ground and excited states. The splitting is identical to thesplitting by a beamsplitter in a optical interferometer. Next, during the propagation
time, the relative phase of the two states evolves at a rate proportional to the energy
difference between the states, and the coupling drive accumulates a phase ωT. The
second π/2 pulse recombines the two states, interferometrically comparing the relative
phase accumulated.

68

3.3

Two-photon transition in 87 Rb
We want to measure the phase properties of an ensemble of atoms to gain un-derstanding of phase coherence and mechanisms that cause decoherence; these measurements will be described in depth in chapter 4. The absolute phase of an atom can
not be measured but rather only the phase relative to another system. We study thephase properties of a system that is made up of two hyperfine ground states in 87 Rb.
These states must have the same magnetic moment to first order to be trapped, overlapped spatially within the trap, and have nearly identical responses to magnetic field
for spectroscopy.

F=2|2, 1>|2, 0>
DIM=0.7 MHz

5S1/ 2~ 3 MHz~ 6.8 GHz|1, -1>F=1Figure 3.2: Energy level diagram for the ground state of 87 Rb in a field of 3.2 G showing
the relevant two-photon transition.

The two states we use are the |1i ≡ |F = 1, mf = −1i and |2i ≡ |F = 2, mf = 1i
states in the 5S1/2 manifold. They satisfy the requirement of having nearly identicalmagnetic moments. We can coherently transfer population between the two states via

µb
h
ni

gi
gj
νhf

1.399624624 (56) MHz/G
3/2
0.9951414(10) ×10−3 [28]2.00233113(20)6834.68261090434(3) MHz [29]69

Table 3.1: Breit-Rabi equation constants for 87 Rb

a two-photon transition. We need two photons because the states differ by two units ofangular momentum. We use a microwave photon (∼ 6.8 GHz) and a rf photon(∼ 3 MHz) to couple the two states as shown in Fig. 3.2. When the detuning fromthe intermediate state |2, 0i is large compared to the Rabi frequency for the one photontransition, the probability of making a transition to the intermediate state is small. In
this limit the system can be thought of as just a two-level system with effective coupling
given by the two-photon Rabi frequency, which isΩ=Ωµwave Ωrf
,
2 ∆IM

(3.6)

where Ωµwave is the Rabi frequency for the |1, −1i → |2, 0i transition, Ωrf is the Rabi
frequency for the |2, 1i → |2, 0i transition, and ∆IM is the intermediate state detuning.
The magnetic field dependence of this transition can be calculated using the Breit-Rabi
formula, which is

νhf
νhf− g i µB m f B ±ν(B) = −2(2ni + 1)
2

s

1+

4 mf
x(B) + x2 (B),
2 ni + 1

(3.7)

where
x(B) =(gj + gi )µB B
.
νhf

(3.8)

B is the magnitude of the magnetic field, and the constants for the equation are given
in Table 3.1.

Chapter 4

Spectroscopy

This chapter, which was adapted from “Effect of cold collisions on spin coherence
and resonance shifts in a magnetically trapped ultracold gas” published in PhysicalReview A [30], describes precision microwave spectroscopy of magnetically trapped,
ultra-cold 87 Rb. We use this spectroscopy to characterize intra- and inter-state density
correlations. The cold collision shifts for both normal and condensed clouds are measured. The results verify the presence of the sometimes controversial “factors of two,”in normal-cloud mean-field energies, both within a particular state and between two
distinct spin species. One might expect that as two spin species decohere, the interstate factor of two would revert to unity, but the associated frequency chirp one naively
expects from such a trend is not observed in our data. We also measure coherence times
of normal clouds as a function of potential inhomogeneity. Our simple model of decoherence, which ignores collisions, systematically underestimates the coherence times
leading to the conclusion that collisions help to preserve coherence across a trappedsample.
In the cold collision regime, quantum statistical effects due to particle indistinguishability dominate scattering processes. The symmetrization requirement for identical particles in an ultra-cold Bose gas enhances the probability of finding two particles
nearby, causing enhanced density fluctuations. At lower temperatures the statisticalnature of the Bose gas causes the phenomena of Bose-Einstein condensation, where all71
atoms in the condensate share the same wavefunction, suppressing the density fluctuations found in a noncondensed sample.
In other work, suppression of second-order density fluctuations in a condensate
has been measured through analysis of the expansion energy of condensates [31, 32]. Ina separate experiment, the suppression of third-order density fluctuations was probed
by comparing the three-body loss rate of a condensate to that of a normal cloud [33].
The effect of cold collisions has also been measured as a density-dependent energy shift
in atomic fountain clocks [34, 35, 36]. These shifts are quite small (∼0.1-10 mHz) due tothe low densities at which the clocks operate, but are measurable because of their highprecision. The uncertainty associated with these collisional shifts can be problematic; infact the next generation of atomic fountain clocks are based on 87 Rb rather than 133 Cs
because the collisional shift of 87 Rb is ∼ 30 times smaller. In recent ultra-cold hydrogenexperiments the cold collision shift provided the signature of Bose-condensation; below
the BEC transition a large frequency shift of the 1S-2S transition was seen, reflecting
the high density of the condensate [37].

4.1

Spectroscopy in a magnetic trap
We performed precision microwave spectroscopy on ultra-cold and condensed 87 Rb

atoms confined in a magnetic trap. Due to the high densities achievable in a magnetic
trap, the collisional energy shifts are 105 greater than those in 87 Rb atomic clocks, allowing a high-precision measurement of the shifts of the magnetically trappable states
to be made with relative ease. The collisional shifts for both a normal and condensed
sample were measured, providing a useful probe of the quantum statistics of the system. Additionally, magnetic confinement permits long interrogation times, allowing
us to characterize temporal coherence of the normal cloud under various experimental conditions. Comparison of measured coherence times with a collisionless numericalsimulation suggests that collisions preserve coherence in normal clouds.

72
Precision spectroscopy of trapped samples is difficult because atom trapping relieson spatial inhomogeneity of the atomic energy levels. Spatial inhomogeneity of theenergy levels broadens the transition frequency, thus limiting the precision attainablethrough spectroscopy. In other work, this difficulty has been avoided by confining atoms
in a blue-detuned optical dipole trap, where atoms spend little time interacting with thetrapping fields [38]. In our work spatial inhomogeneity of the transition frequency was
minimized through the use of a pair of energy levels which experience the same trapping
potential. At a magnetic field of ∼ 3.23 G, the |1i and |2i hyperfine levels experience
the same first-order Zeeman shift. For a normal cloud at 500 nK, each energy level
is Zeeman shifted by ∼10 kHz across the extent of the cloud; however, at 3.23 G the
differential shift of the two levels across the cloud is ∼1 Hz. Compared to the differential
Zeeman shift, the energy shift due to cold collisions is then a relatively large effect at
high densities, making measurements of collisional shifts in this system possible. The
small inhomogeneity allows for long coherence times, ∼2 seconds and longer for lowdensity clouds, making this system attractive for precision measurements as well as forthe study of condensate coherence in the presence of a thermal cloud.
A two-photon microwave-rf transition is used to transfer atoms between the |1i
and |2i states. A detuning of 0.7 MHz from the |2, 0i intermediate state provides
a two-photon Rabi frequency of ∼ 2.5 kHz. Ramsey spectroscopy of the |1i → |2itransition is performed by measuring the total number of atoms remaining in state |1i
after two π2 pulses separated by a variable time delay are applied [39]. The frequencyof the resulting Ramsey fringes is the difference between the transition frequency ν 12
and the two-photon drive frequency. We can measure local variations of ν 12 (e.g. frommagnetic or mean-field energy shifts) by detecting the number of atoms remaining instate |1i at specific spatial locations along the axis of the normal cloud Fig. 4.1a. We
bin the cloud only in the axial direction because the collision rate is less than the radial
trap frequency; therefore the atoms thermally average over the radial direction during

73
an experiment. For a particular peak density we measure the axial spatial variationin frequency by binning the cloud into several sections and determining a frequency
for each bin Fig. 4.1b. This measurement is done at Bbias = 3.23 G to minimize themagnetic contribution to the inhomogeneity. The transition frequency vs. axial position
along the cloud is shown in Fig. 4.1c.
62.35 ms

65.05 ms

67.75 ms(b)

Center bin optical density

(a)

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0

10

20

30

40

50

60

Time between p/2 pulses (ms)Offset frequency (Hz)

(c)
-4494
-4496
-4498
-4500
-4502
-600

-400
-200
0
200
400
Axial distance from center of cloud (mm)600

Figure 4.1: (a) Images of |1i atoms (upper) and |2i atoms (lower) after two π2 pulses.
The delay between the pulses is listed at the top. The edges of the cloud have becomeout of phase with the center. The white lines show how the cloud is divided up along
the axial direction to measure the spatial dependence of the transition frequency. Each
image is 1 mm on the horizontal (axial) dimension. (b) The center-bin |1i state opticaldensity as a function of time between the two π2 pulses. (c) The plot shows the spatial
dependence of the transition frequency in the axial direction. Offset frequency is definedas the transition frequency - 6834682610.9 Hz. The solid line is a gaussian fit to thedata. These data were taken at Bbias = 3.23 G.

74
By analyzing the spatio-temporal variations of ν12 , combined with the measured
evolution of the |1i state after a single π2 pulse, we were able to resolve spatially theevolution of spin waves [40], which will be discussed in depth in chapter 5. In orderto perform measurements of ν12 insensitive to spin waves, one of the following two
techniques was used. With one technique the entire cloud, rather than specific spatial
locations, was monitored to average out the effects of spin waves. Alternatively, Ramsey
spectroscopy was restricted to interrogation times short compared to the spin wave
frequency [41].

Figure 4.2: Differential Zeeman shift at low magnetic fields for the |1i → |2i transition.
The solid line is the predicted splitting from the Breit-Rabi formula. The inset plot
expands the bias field region where most studies are performed.

One effect which shifts the transition frequency ν12 is the differential Zeeman
shift. The Breit-Rabi formula predicts a minimum in ν12 at B0 = 3.228917(3) Gauss,thus the differential Zeeman shift between the |1i and |2i energy levels is first-order

75
magnetic field independent at B = B0 . The differential Zeeman shift about B0 can beapproximated as ν12 = νmin +β(B−B0 )2 [42]. Measuring ν12 for different magnetic fields
allows us to calibrate our magnetic field from the expected dependence, see Fig. 4.2. By
working in the vicinity of B = B0 we greatly reduce spatial inhomogeneity of ν12 andalso become first-order insensitive to temporal magnetic field fluctuations.

4.2

Cold-collision energy shift
A second effect which shifts ν12 arises from atom-atom interactions. In thes-wave regime, where the thermal de Broglie wavelength of the atoms is greater than2

their scattering length, atoms experience an energy shift equal to α 4πh̄m an, where α isthe two-particle correlation at zero separation, n is the atom number density, a is thescattering length, and m is the atom mass. Therefore for a two-component sample theexpected energy shift of each state is
4πh̄2(α11 a11 n1 + α12 a12 n2 )m4πh̄2(α12 a12 n1 + α22 a22 n2 ),
δµ2 =
m
δµ1 =

(4.1)
(4.2)

where n1 and n2 are the |1i and |2i state density andαij =hΨ†i Ψ†j Ψi Ψj ihΨ†i Ψi ihΨ†j Ψj i

.

(4.3)

The shift of the transition frequency in Hz can then be written as∆ν12 ==2h̄(α12 a12 n1 + α22 a22 n2 − α11 a11 n1 − α12 a12 n2 )mh̄n(α22 a22 − α11 a11 + (2α12 a12 − α11 a11 − α22 a22 )f ),
m

(4.4)

2
where f = n1 −n
and n = n1 + n2 .
n

For noncondensed, indistinguishable bosons, α = 2 due to exchange symmetry;nc = αnc = 2 in a cold normal cloud (where the superscript c or nc refers
therefore α1122

to condensed of noncondensed atoms respectively). Distinguishable particles do not76nc = 1 for an incoherent two-component mixmaintain exchange symmetry, making α12ture. However if a two-component sample is prepared by coherently transferring atoms
from a single component, such as in Ramsey spectroscopy, then the excitation process
nc = 2 [43]. In this scenario themaintains exchange symmetry, and we might expect α12
nc = αnc = αnc = 2, leading to a predictedcollisional shift should be calculated using α112212frequency shift of∆ν12 =2h̄n(a22 − a11 + (2a12 − a11 − a22 )f ).
m

(4.5)

This result can also be obtained by solving the transport equation [44, 45]. From spectroscopic studies [46] the three 87 Rb scattering lengths of interest have been determined
to be a22 = 95.47a0 , a12 = 98.09a0 , and a11 = 100.44a0 , where a0 is the Bohr radius.
The frequency shift can then be written as∆ν12 =2h̄
a0 n(−4.97 + 0.27f ).
m

(4.6)

If on the other hand the |1i and |2i states do not maintain exchange symmetry, such
nc = 1, then the frequency shift would instead bethat α12∆ν12 =2h̄
a0 n(−4.97 − 97.82f ).
m

(4.7)

These two models are clearly distinguished by the dependence of ν12 on f .
When we perform Ramsey spectroscopy with a pair of π2 pulses, the populationsof the |1i and |2i states are equal, and thus f = 0 during the interrogation time. Fromnc and a
Eq. 4.4 it is apparent that with f = 0 the collisional shift is sensitive only to αiiii

terms. For these measurements the bias field was set to B0 , and the transition frequency
was measured for a range of densities. To adjust density of the sample, the number ofatoms in the initial MOT load was varied. All normal cloud data was taken at thesame temperature of 480 nK, and all condensate data was taken with high condensate
fractions in order to minimize effects due to the normal cloud. The density for thenormal cloud was found by fitting Gaussian profiles to absorption images of the clouds

77
and extracting the number, temperature, and density. To measure condensate density,
Thomas-Fermi profiles were fit to absorption images of the condensates and the total
number, N0 , in the condensates and the Thomas-Fermi radius along the long axis, Z,
were extracted.

Figure 4.3: Measurement of the cold collision shift. Solid and open circles represent
measurements of the normal cloud and condensate respectively. The solid line is a fit
to the normal cloud data ∆ν12 = 0.1(0.4) − 3.9(0.3)10−13 n; the dashed line is a fit to
the condensate data ∆ν12 = −0.1(1.4) − 1.9(0.2)10−13 n, where ∆ν12 is in Hz and n isin cm−3 .

The results of this measurement are shown in Fig. 4.3. Comparing the collisional shift measured for the normal cloud to that measured for a condensate gives
nc /αc = 2.1(2). If instead we assume αnc = 2 and αc = 1, then the data for both
αii
ii
ii
ii

the condensate and normal cloud can be used to obtain a value for the difference inscattering lengths of a22 − a11 = −4.92(28)a0 , in agreement with values determinedfrom molecular spectroscopy [46].
Many systematics can plague density measurements made through absorption
imaging.

In order to test independently our density calibration for both the nor-

78
mal and condensed samples, each of which can suffer from different errors, we used
the Bose-condensation phenomenon. The density of normal clouds was tested throughmeasurement of the critical temperature, and condensate density was tested with theThomas-Fermi approximation. Assuming that disagreements are due only to errors inestimation of atom number, the worst case scenario, leads us to reduce normal clouddensity by 11(4)% and increase condensate density by 11(3)%. Adjusting the cold collinc /αc = 1.7(2) [47].
sion shifts accordingly would yield a worst-case corrected value of αii
ii

The adjusted normal and condensate density shifts can be combined as above to give avalue for the difference in scattering lengths of a22 −a11 = −4.85(31)a0 ; not significantly
different from our unadjusted measurement.
The remaining significant systematic is atom loss due to |2i-|2i collisional dipolar
relaxation. In order to minimize effects of this loss, interrogation times were kept as short
as possible. Nevertheless for the highest density condensate measurements the |2i-|2iloss causes the total density to drop by 3% in 20 ms, the maximum interrogation time.
For all other densities the loss was no larger than this, and in most cases much smaller.
Finally, the |1i and |2i states begin undergoing spatial separation in the condensate
after the first π2 pulse [48]; however, the timescale for the separation is much longer
than our 20 ms interrogation time.
Exchange symmetry between the |1i and |2i states can be tested by working at
a fixed density and varying the relative |1i to |2i population by varying the lengthof the first Ramsey pulse [49]. In this case the first term in Eq. 4.4 will be connc and ancstant and the measurement will test α1212 as well as the αii and aii terms(see Eq. 4.6 and 4.7). To minimize systematics the interrogation times were kept
short, making precise frequency determination difficult. Nevertheless, our measurenc /αnc
ment (Fig. 4.4) indicates α1211,22 = 1.01(2), where we have used the spectroscopi-cally determined scattering lengths. This clearly indicates that exchange symmetry is
maintained between the |1i and |2i states. A similar measurement was made on the

79

Figure 4.4: Testing the exchange symmetry between the |1i and |2i states. The tran2
is varied for a normal cloud at fixed peaksition frequency is measured as f = n1 −nn12−3density of 7×10 cm and temperature of 510 nK. The solid line is the fit, which yields
nc /αnc
α1211,22 = 1.01(2), which is to say, inter- and intra-state density correlations are quite
nc /αnc
accurately the same. The dotted line indicates the expected slope for α12
11,22 = 1/2.

|F = 1, mf = 0i → |F = 2, mf = 0i transition by Fertig and Gibble [35].

4.3

Missing frequency chirp
As a thought experiment, imagine distinct thermal populations of |1i and |2i

atoms, separately prepared, then mixed together, with the energy of interaction (pronc ) measured for instance calorimetrically. Surely in this case the density
portional to α12
nc would be deterfluctuations in state |1i and in state |2i would be uncorrelated, and α12mined to be 1, not 2. We lack the experimental sensitivity to make such a calorimetricmeasurement, and our Ramsey-fringe method of measuring energy differences obviously
nc = 2would not work for incoherent mixtures. We speculated, however, that if α 12
nc = 1 for incoherent mixtures, then for partiallyfor coherent superpositions, and if α12
nc would take on some intermediate value. So by performing a
decohered samples, α1280Figure 4.5: A data set of Ramsey fringes probing for frequency shifts as a function ofcoherence. For this measurement normal clouds at a temperature of 480 nK and a peakdensity of 3.2 × 1013 cm−3 were used.

measurement similar to that in Fig. 4.4 we might expect to see a more negative slope
for a partially decohered sample; alternatively a frequency chirp in the Ramsey fringes
may be seen as the sample decoheres.
nc in a way similar to Fig. 4.4; however rather
We probed the time evolution of α12than varying f we set f ' 0.8 then measured ν12 with long interrogation times, looking
for a frequency chirp as the fringe contrast decreased. This method has the advantagethat there is a relatively small |2i state population, so effects arising from |2i loss areminimized. Seven data sets were taken for this measurement; an example is shown in
Fig. 4.5. By allowing a linear frequency chirp in the fit of the Ramsey fringes, thefrequency shift can be constrained to −0.2(3) Hz by the time the fringe contrast has

81
nc goes from 2 to 1 linearly as
reduced to 1/e [50]. However if we hypothesize that α12fringe contrast goes from 100% to 0% we would expect a frequency shift of −20(2) Hz
as the fringe decayed, while the experimental limit is 40 times smaller. The lack ofchirp disagrees with the traditional notion that the frequency of the transverse spin
precession is a direct measurement of the energy difference between spin up and spin
down. Clearly this appealing but unrigorous model is far too naive.

4.4

Coherence of normal clouds
Ramsey spectroscopy not only allows us to probe the energy difference between

the two states but also permits the measurement of the coherence between the twostates. Coherence measurements were performed using the same time domain method
used to measure ν12 ; however, interrogation times were extended until the Ramsey fringe2

contrast was lost. The resulting data was fit to a e−(t/τ ) decay, where τ is the coherence
time. Additionally, fitting allowed for both a loss in total atom number and a linear
frequency chirp of ν12 . To ensure that the 1/e atom loss times were much longer thanthe coherence times, fractional transfers (f ' 0.8) to the |2i state were used for thehigh density data points. The results of coherence measurements for different magnetic
fields and at three different densities are shown in Fig. 4.6.

82

Figure 4.6: Coherence times in a normal cloud. The temperature was ∼ 480 nK andthe normal cloud peak density for each plot is: (a) 4 × 1012 cm−3 ; (b) 1.3 × 1013 cm−3 ;(c) 3.2 × 1013 cm−3 . The solid line in each plot corresponds to the 1/e times obtained
from a numerical simulation (see text). The noise in the simulation data is due to thefinite number of particles used and the random initialization.

83
We expect that the primary source of decoherence is inhomogeneity in ν12 acrossthe cloud, to which both the differential Zeeman and collisional shifts contribute. The
collisional shift scales directly with density, and thus provides a Gaussian-shaped ν12
profile across the cloud. Inhomogeneity arising from the differential Zeeman effect depends on the bias field. The magnitude of the magnetic field near the bottom of the00

trap can be written as B(z) = B2 z 2 + Bbias , thus the inhomogeneity due to the Zeeman
002

shift is ν12 ∝ B4 z 4 + (Bbias − B0 )B 00 z 2 . By setting Bbias (>, <, =)B0 the curvature ofν12 can be adjusted to be positive, negative, or nearly zero, respectively. This allows inhomogeneity due to the collisional shift to be roughly cancelled by an opposing Zeeman
inhomogeneity Figs. 4.7 and 4.8 [51].
This effect of this cancellation can be seen in Fig. 4.6; as the density is increased
from (a) to (c), the inhomogeneity induced by the collisional shift increases, so that alarger opposing Zeeman inhomogeneity is necessary for cancellation. Therefore the bias
field for peak coherence time decreases as cloud density increases.
In an attempt to compare the measured decoherence times to the known spatial
inhomogeneity of the transition frequency, we performed the following numerical simulation: consistent with a Maxwell-Boltzmann distribution, we randomly assign initial
positions and velocities to ten thousand simulated atoms. Ignoring the effects of collisions, we calculate the three-dimensional trajectory of each atom for several simulated
seconds, keeping track of the time integral of ν12 along the trajectory. At each point
in time, we calculate the spatially integrated transverse magnetization and, as inhomogeneities cause this magnetization to wash out, find the time it takes the integrated
transverse magnetization to reduce to 1/e of its original value [52]. This model should
correctly account for the effect of motional averaging except that all collisional effects
are explicitly excluded. The resulting modelled damping times are plotted as a solid line
along with the experimentally measured damping times in Fig. 4.6. While the model
does a reasonable job predicting the value of the bias field for which the coherenceBC - 0.15 Gauss

Cancellation field (BC)84
BC + 0.15 Gauss

25

Frequency Shift (Hz)

20
15
10
5
0
-5

-400 -200 0

200 400

Axial Position (mm)Cold Collision Shift
Differential Zeeman Shift
Total Energy Shift

Figure 4.7: The frequency shift across the axial direction of the cloud is plotted forthree different magnetic bias fields. The dotted curves indicate the the cold collisionalenergy shift, the dashed curves indicate the Zeeman energy shift, and the solid line is asum of the two energy shifts. The cancellation field is shown in the center where there
is a minimal variation of the frequency across the cloud. Increasing the bias field causes
the energy shifts to have the same curvature and produces an overall larger variationof frequency across the cloud; reducing the bias field below BC causes the Zeeman shift
contribution to overwhelm the cold collision shift and thus dominate the energy vs.
position profile.

time peaks, it consistently underestimates (in some cases by a factor of eight) the actual value of the coherence time. Our model neglects both velocity-changing collisions
and the exchange-type collisions that lead to spin waves; it appears that these effects
contribute significantly to preserving coherence across the trapped atom cloud.
The extreme aspect ratio of our trapped cloud complicates a proper quantitative
analysis of the effects of collisions on coherence. Along the axial direction, other work
has shown that the effects of spin-waves are to keep local magnetization across the cloudfrom straying too far from its spatially averaged value [see in particular Fig. 5.3(a) in
chapter 5]. In the radial directions, the motional oscillation frequency exceeds the mean-

Offset frequency (Hz)

3.37
3.23
Bbias3.10
(G)
3.37
2.97
3.23
2.90
3.10
2.83
2.97
2.90
2.83

-4440

-4460

85

-4480

-4500
-600

-400

-200

0

200

400

600

Axial distance from center of cloud (µm)
Figure 4.8: The transition frequencies at different positions along the cloud are plotted
as a function of Bbias . The cloud has a peak density of 4 ×1013 cm−3 . Offset frequency is defined as the transition frequency - 6834682610.9 Hz. The frequency shift at
3.23 G (negligible magnetic contribution to the inhomogeneity) has a gaussian shape.
At Bbias = 3.37 G the magnetic shift and the mean-field shift are in the same direction
and therefore the inhomogeneity is across the cloud is increased. When the cloud is in afield of 2.83 G inhomogeneity due to the mean-field shift is almost completely cancelledby the magnetic shift. Data points are connected to guide the eye.

field exchange frequency in the cloud, and thus the effects of spin-waves are probably
not relevant. On the other hand, velocity-changing collisions are likely important – intheir absence, atoms with small transverse energies would stay near the axis of thetrap, while atoms with large transverse energies would preferentially sample the larger
magnetic fields and lower densities further from the axis. Velocity-changing collisions
which re-randomize the transverse trajectories of these different classes of atoms (before
they have a time to accumulate a radian or more of relative phase-difference) will serve
to extend the coherence time of the sample. In the highest density data set presented[Fig. 4.6(c)] the mean elastic collision rate was 74 Hz, which should be compared to the86measured coherence times of around 0.5 to 1.2 seconds.

4.5

Coherence of condensates
We were also able to explore the coherence of nearly pure condensates. There are

two standard data taking methods to measure coherence using Ramsey spectroscopy.
One is to take equally spaced points and fit the data to an exponentially decaying sine
wave (Fig. 4.9). This measurement technique is sensitive to phase noise because phasenoise will artificially decrease the fitted coherence time. The functional form of thefringes is derived as follows. The same result can be more rigourously derived using adensity matrix approach. The initial state after the first π/2 pulse is1|ψi = √ [|1i + |2i].
2

(4.8)

After some time t the state is modified by loss in both states and phase evolution due
to the energy difference between the states and can be written as1|ψi = √ [e−t/2τ1 |1i + e−t/2τ2 eiφ |2i],
2

(4.9)

where τi is the lifetime of state i. The approximation of an exponential loss rate is notentirely correct. The main loss rate is 2-body for the |2i state and 3-body for the |1i.
However the condensate density distribution is changing in time due to the mean-field
separation effect and therefore the loss rate is just modelled as exponential. After sometime t a second π/2 pulse is applied, producing the state:|ψi =1 −t/2τ1[e(|1i + |2i) + e−t/2τ2 eiφ (−|1i + |2i)].
2(4.10)

From this state we can calculate the probability of being in the |1i state as afunction of time, which is1
P|1i (t) = [e−t/τ1 + e−t/τ2 − 2 cos φe−(t/2τ1 +t/2τ2 ) ].
4

(4.11)

87
The |1i state has a much longer lifetime than the |2i state so the term e −t/2τ1 in
Eqn. 4.11 can be set equal to unity, and an exponential decoherence term can be addedwith time constant τcoh . The probability of being in the |1i state then is1
P|1i (t) = [1 + e−t/τ2 − 2e−t/τcoh cos φe−t/2τ2 ].
4(4.12)

The function we using in our fitting routine is1y(t) = A [1 + e−t/τ2 − 2e−t/τcoh cos (2π(ν + c t)t + θ)e−t/2τ2 ],4(4.13)

where the free parameters are an amplitude A, a |2i state lifetime τ2 , a decoherencetime τcoh , a fringe frequency ν, a linear frequency chirp c, and a phase offset θ.
The other method for measuring coherence time, which is insensitive to phasenoise, is to take data (∼10 points) over a time interval equal to a single fringe at several
different propagation times (Fig. 4.10a). Then the standard deviation Σ, is calculated
for each fringe and fit to a decaying exponential in quadrature with a background noiselevel (Fig. 4.10b). Σ is

Σ(t) =

r

1 −2t/τ2e.
8(4.14)

The functional form we use in the fitting routine isy(t) =s

A2 −2t/τ2e+ y02 ,
8

(4.15)

where y0 is the background noise level.
Measuring condensate coherence time is difficult because of the high loss rate in
the |2i state and phase noise due to shot-to-shot number fluctuations. The high density
in the condensate causes large dipolar relaxation rates and reduced lifetime of the |2i
state. Therefore we lose signal rapidly as the |2i atoms are lost from the trap. We canreduce this loss rate by not applying a π/2 pulse but rather transfer only 20% of theatoms to the |2i state, which decreases the density and thus the loss in this state.

88

Number in |1ñ state (arb.)

16
14
12
10
8
6
4
2
0
0

100

200

300

400

500

600

Time (ms)Figure 4.9: Ramsey fringes of the center bin of a condensate with peak density of
8.9(3)×1013 with a condensate fraction of 0.70(3). 24% of the condensate is initiallyplaced in the |2i state. The fit gives a value for τ = 268(28)ms. (see text for discussion)

The other complication with measuring condensate coherence is phase noise induced by shot-to-shot number fluctuations. The cold-collision frequency shift for typicallarge condensates with densities of 1014 cm−3 is ∼ 20 Hz. A 5% variation of the density
will then cause a phase shift of 0.5 rad at 0.5 seconds. Therefore we must make condensates with small and repeatable density to measure coherence times with minimal
phase noise. We create small stable condensates by first evaporating normally to produce a medium-sized condensate (105 atoms); then leave the rf on close to the edge ofthe condensate for several seconds. Any noise in the bias field will cause the rf knife to
cut into the condensate. However the noise will average out over several seconds thus
producing a small, consistently-sized condensates.
One example of a coherence measurement of a condensate is shown in Fig. 4.9.
The fringe contrast and the mean decay at the same rate, which is not predicted by oura)Number in |1ñ state (arb.)

89

b)

16
12
8
4
0

Standard deviation

6
5
4
3
2
1
0
0

200

400

600

800

1000

Time (ms)Figure 4.10: Alternate method of taking coherence data.(a) Ramsey fringes taken over
one fringe at six different propagation times. (b) Standard deviation of each of the six
fringe sets normalized by the mean of the data points within that set.

simple model. Our model predicts that, in absence of additional decoherence, the fringe
contrast should decrease at half the rate of the mean. One might assume initially that
there is just additional decoherence, but there are two reasons why we think this might
not be the case. First the decay rate for both the contrast and the mean coincide withan independent measurement of the |2i state decay rate. Second the contrast and themean decay at the same rate over a wide parameter space, which includes a variety ofcondensate fractions and number densities. One would think the additional decoherence
would change as these parameters are varied.

90

4.6

Speculation on precision metrology
It is interesting to consider the usefulness of magnetically trapped atoms for pre-cision metrology. Peak coherence times of approximately 2.5 s were realized with cold,
low density samples. An interrogation time of 2.5 s provides a 0.2 Hz linewidth, which
naturally leads one to consider using such a system for precision measurements. By
working at Bbias = B0 , coherence times are slightly reduced. However, perturbationsof ν12 due to magnetic field fluctuations become very small, on the order of 4 mHz for
current typical experimental conditions. With careful design of the confinement coils,
the current supply, and magnetic shielding of external fields, it should be possible tosuppress fluctuations of the bias field below 1 mG, reducing Zeeman-induced frequency
shifts to below 0.1 mHz. Perturbations of ν12 due to the cold collision shift are moresignificant; shot-to-shot density fluctuations will introduce frequency noise, so it is advantageous to work with the minimum possible density. However as atom number isreduced the maximum signal-to-noise ratio will decrease due to shot-noise [53]; therefore
the optimum strategy is to work with an atom number such that the frequency uncertainty due to shot-noise is on the order of the uncertainty due to density fluctuations.
For example a normal cloud of 400 nK and a peak density 1.5 × 1012 cm−3 has
6 × 104 atoms. The shot-noise-limited signal-to-noise ratio is then 245:1. With aninterrogation time of 1 second the single-shot statistical uncertainty is then 0.65 mHz;
including the effects of decoherence and atom loss will increase this to approximately
0.9 mHz. Assuming that shot-to-shot number fluctuations are also shot-noise limited,
then statistical uncertainty from the density shift is 0.84 mHz. Combining these gives a
total single-shot uncertainty of 1.24 mHz; with our current 30 second evaporation time,√the duty-cycle is such that an absolute precision of 6.8 mHz/ Hz can be realized. This√
corresponds to a relative precision of 1 × 10−12 / Hz, which in terms of measurement
precision does not reach the level of atomic fountain clocks. This system however has

91
the advantage that small energy shifts can be measured in a compact, stationary spatial
position; a 400 nK cloud occupies approximately only a 1040 × 32 × 32 µm region of
space. It is certainly feasible to perform spectroscopy at this level within 100 µm of asurface, which might allow the measurement of short range atom-surface interactions.
We have demonstrated precise spectroscopy in an ultra-cold magnetically trapped
gas. This permitted measurement of the cold collision shift in both a condensate anda normal cloud, allowing a probe of the quantum statistics of the system. Working at
low densities minimizes the effect of the collisional shift, allowing long coherence times
and precise determination of ν12 ; however the measurement of any quantity not related
to atom-atom interactions will at some level be limited by this shift. An intriguing
alternative would be to instead use a fermionic atom, which should have no collisional
shift, and thus density induced frequency noise will not be an issue. On the other hand,
the lack of collisions may also lead to more rapid decoherence: collisions appear topreserve our bosonic system from the decohering effects of spatial inhomogeneity.

Chapter 5

Spin waves

This Chapter is adapted from “Observation of anomalous spin-state segregation
in a trapped ultra-cold vapor” and “Spatial resolution of spin waves in an ultra-coldgas” published in Physical Review Letters [51, 40].
Macroscopic collective behavior can arise in a quantum gas even above the onset
of degeneracy. When indistinguishable atoms collide, the scattering can be significantly
altered by the need to symmetrize the scattered waves. Examples of quantum scattering
effects are seen in the strong polarization dependence of heat conduction and viscosity coefficients in spin polarized 3 He [54] and of thermalization rates in nondegenerate
Fermi gases [55]. Quantum collisional effects can also produce spatio-temporal spin
oscillations known as spin waves. In this Chapter we present spatially resolved imagesof coupled transverse and longitudinal spin perturbances propagating through a magnetically trapped atomic cloud. Frequencies and damping rates for the standing-wave
excitations are studied as a function of density and temperature.
Although the concept of spin waves in ferromagnets dates back to Bloch’s predictions in 1930 [56], the theory of spin waves in liquids and in dilute gases was notformulated until much later [57]. The first evidence for spin waves (in the form of extra
resonances in nuclear magnetic resonance spectra) occurred in spin-polarized hydrogenand soon after in polarized 3 He and dilute mixtures of 3 He in 4 He [58]. On a microscopic
level, spin waves arise from interference effects in lowest partial-wave collisions. When93
identical particles collide, the 180◦ backscattering event causes a scattered atom to propagate along a trajectory indistinguishable from that of either a forward scattered atom
or an unscattered atom. When the two atoms are spin aligned, the backward scattering
contribution is indistinguishable from the forward scattering contribution, which adds afactor of two to the mean-field collisional energy shift. When the spins are antiparallel,
the backward scattering event is distinguishable, and the mean-field shift arises fromthe forward scattering event only. In the case of intermediate spin alignment, the backward scattering event is only partially distinguishable from the unscattered event, andone needs to add coherently two unparallel spin amplitudes, leading to a rotation of thespins of the scattered atoms. The cumulative effect of many such spin-rotating collisions
is such that inhomogeneities of spin propagate like waves rather than diffusively. The
characteristic frequency scale for the spin rotation effect is the exchange collision rate,ωexch = 4πh̄an/m, where m, n, and a are the mass of the atoms, the number density,and the s-wave scattering length respectively.

5.1

Theoretical description of spin waves
The proceeding theoretical description follows a similar formalism to that of

J. Williams and colleagues [45]. Ultimately we want to calculate the spin density as afunction of position in the cloud and time. We start with the Hartree-Fock mean-field
theory of a Bose gas, which leads to the collisionless Boltzmann equation. The solution
to the Boltzmann equation predicts spin waves, but because it does not include collisions, it can not predict damping of the spin waves. The collision integrals will not be
dealt with in this text but can be found in reference [59]. The Hamiltonian for two-levelatoms in a trap isH=h̄ →−h̄2 2−−∇ + U0 (r) Iˆ + Ω (r) · →
σ,
2m
2
!

(5.1)

94
where the first two terms are just the center of mass contribution including the kinetic
energy and the potential, which is
U0 (r) = Uext (r) +

4πh̄2
a12
[a11 n1 + a22 n2 +(n1 + n2 )].
m
2

(5.2)

Uext is the magnetic trapping potential, and the other terms in Eqn. 5.2 are the mean
field potential terms. For our experiments 4πh̄2 an/m  kB T so the mean field terms
can be ignored. The last term in the Hamiltonian is the contribution from the spin,−
where →σ are the Pauli matrices. The coupling field, a combination of the drive field,energy difference from the mean field and Zeeman energy shifts as well as from the spin
density, is→−
2πh̄2 a12 →−→−S (r, t),Ω = Ω0 +
m

(5.3)

→
−0
Ω = [Ωx , Ωy , ∆BR (r, t) + ∆MF (r, t)],

(5.4)

where

where the components of Ω0 are in Bloch space. The hyperfine splitting has been
removed by taking the rotating wave approximation. ∆BR is the differential Zeeman
shift calculated from the Breit-Rabi formula, and the differential mean-field energy is
∆MF =

4πh̄2
[2a11 n1 − 2a22 n2 + 2a12 (n1 − n2 )],
m

(5.5)

where ni is the density in state i. The factor of two in front of the a12 comes fromthe addition of the forward and backward scattering events as described previously in
Chapter 4. Ωi is the resonant Rabi frequency about the axis i and zero when the coupling
field is off, such as after the π/2 pulse. The second term in Eqn. 5.3 is responsible forspin waves and comes from the extra factor of two in the mean-field interaction for
identical particles. The spin density as a function of position in the cloud and time is→−
S (r, t) =R−
dp × →σ (r, p, t)
,
2πh̄3

(5.6)

−
where →σ (r, p, t), the spin distribution function, can be obtained from the spin Boltzmann equation. The longitudinal component of the spin density Sz is the difference in

95
the population of the two states [n1 (r, t)−n2 (r, t)]. Sx and Sy are the real and imaginaryparts of the internal coherence.
Now that the problem is defined, we can numerically integrate the kinetic equation
to find the spin distribution function and subsequently the spin density, which was doneby J. Williams and collegues[45]. The kinetic equation, ignoring collisions, is
−
p
∂→
σ
→
− −
−
−
+
· ∇→
σ − ∇Uext · ∇p →σ − Ω ×→σ = 0,∂tm(5.7)

where the second and third terms are the free streaming evolution, and the last term
depicts the spin precessing about the local field, which is a combination of the driving
field and the local spin.
Another slightly more intuitive way to understand the origin of spin waves is tothink about the spins of each atom precessing about the total spin during a collision.
When two atoms with different spins collide, their total spin must be conserved after thecollision. However they can each precess about the total spin and thus have a different
spin after the collision. This is depicted graphically in Fig. 5.1. The initial direction of

Before collision

Spins precess
about total spin

After collision
Figure 5.1: Intuitive picture of spin rotation during collisions. Atoms initially withdifferent relative phases precess about the total spin during a collision.

96
the spin is dependent on the “history” or previous trajectory of each atom. Imagine atest atom coming from the center of the cloud with a certain spin; as it moves throughthe cloud its spin rotates because of its interaction with the average local spin of thecloud. A number of such interactions, because they are coherent, build up a coherent
oscillation.

5.2

Description of system
The experimental apparatus and spectroscopy method used to prepare and probe

the ultra-cold gas are described in Chapters 2-4. Typical atom cloud parameters forthese spin wave studies are n ∼ 1013 cm−3 and T ∼ 600 nK, several times the condensation temperature. The radial frequency is high enough that for the effects consideredhere it is sufficient to average the cloud over the radial dimensions, effectively reducing the cloud to a one dimensional density distribution. The atomic hyperfine statesare |1i and |2i, which together make up a pseudo-spin doublet [60]. The spin state is
manipulated using the two-photon coupling transition described in Chapter 3.
The transition frequency ∆(z) is in general a function of the axial position z,
due to the spatial inhomogeneity of the Zeeman shift and of the density-dependentmean-field shift. As stated earlier in Chapter 4, we can tune the bias magnetic fieldso as to cause the Zeeman shifts to cancel in whole or in part the mean-field shifts.
As a consequence, the spatial curvature of the frequency, ∂ 2∆(z)/∂z 2 , is a controllable
parameter of the experiment which may be tuned in real time, even as a spin wave is
propagating.

5.3

Bloch-sphere representation
To describe the effective two-level system, we use the language of the Bloch sphere(see [61]) and take the axis w as the “longitudinal” population inversion, and the axes u
and v as the “transverse” coherences [62]. In a typical experiment, the atom-preparation

97
cycle concludes with the atoms initially all in the |1i state, along the w -axis (Fig. 5.2a).
Applying a π/2 pulse starts the evolution of the wave by rotating all the spins to lie along
the v -axis (Fig. 5.2b). When ∆(z) is set to have nonzero curvature, the spins along
the long axis of the cloud begin to fan out in the u-v plane as the relative phase of the(|1i + |2i) coherent superposition develops a spatial dependence (Fig. 5.2c). Optionallyat this point in the evolution, we can tune ∂ 2∆(z)/∂z 2 back to zero or leave it unchanged
and watch the spins evolve in an inhomogeneous potential. In either case, the atoms’
thermal motion now carries spin information back and forth along the length of thecloud, and spin-rotating collisions can tilt the spins up out of the u-v plane (Fig. 5.2d)[63]. Eventually, spin-currents develop that drive oscillations about an equilibrium spin
distribution. Our driving potential, determined by the curvature of ∆(z), is even inz, and we predominantly drive the lowest order symmetric mode: a two-node standing
wave.

5.4

Experimental method for studying spin waves

5.4.1

Extracting longitudinal phase

The longitudinal and transverse components of the Bloch vectors are measured asa function of time and space. The method for projecting the longitudinal component issimilar to that described in [51] and is depicted graphically in Fig. 5.2. After an initialresonant π/2 pulse, the Bloch vectors are allowed to evolve for times up to t = 800 ms,
and then the atoms are imaged, thereby projecting the Bloch vectors along the cloud
onto the two spin states |1i and |2i. The cloud is radially averaged and broken up into 25
equal bins along the axial direction for spatial resolution of the spin dynamics. The |1i
and |2i populations are measured on separate experimental shots, and the longitudinalangle θ of the Bloch vector is extracted. (θ = π/2 corresponds to equal populations,and θ = 0 or π correspond to all the atoms in |1i or |2i respectively.) The evolution

98a) Prepare ~ 600 nK normal cloud of atoms in state |1ñ.

Absorption Images:

Axis of Cloud

|2ñ

|1ñ

b) Apply a two-photon pulse to rotate the Bloch vectors into the u-v plane.

Time = 1 msc) The n12 inhomogeneity causes the Bloch vectors to precess at
different rates in the u-v plane.
Time = 40 msd) Exchange collisions rotate the Bloch vectors out of the u-v plane.

Time = 110 ms
Figure 5.2: Creation of spin waves (see text).

is traced out by repeating the experiment for many values of evolution time. Fig. 5.3ashows the time evolution of the longitudinal component of the spin across the cloud.

5.4.2

Extracting transverse phase

The transverse spin measurement consists of a two-pulse π/2 − π/2 Ramsey pulse
sequence separated by an evolution time t [39]. For each time evolution time t andfor each spatial bin along the trap axis, we measure the final density of the atoms
in |1i, resulting in a set of roughly sinusoidal (but with slightly time-varying phaseand amplitude) Ramsey fringes for each location in space. The transverse phase andamplitude of the Bloch vector are extracted from the fringes at each time t for eachradially-averaged section of the cloud by performing a sinusoidal fit over a small window

99

600 a)

b)
0.74

0.62

Time (ms)

0.50

0.38

200

Phase angle (π rad)

400

0.24

0
-500

500 -500
Position across cloud (µm)

500

0.12

Figure 5.3: False color plots of the a) longitudinal and b) transverse components of the
Bloch vectors across the cloud as a function of time for a density of 2 × 10 13 cm−3 anda temperature of 800 nK (corresponding to a 380 µm full width, half maximum cloudsize). The angles have been interpolated between bins and time steps of a) 10 ms andb) 3 ms.

centered about t in each spatial bin [64]. Fig. 5.3b shows typical phases extracted withthis method. The spatial and temporal oscillations around the mean phase are thetransverse projections of the spin waves.
The complete trajectories of the Bloch vector can be determined for each spatial
bin by combining the transverse and longitudinal components. Normalizing the lengthof each vector to the length at time t = 0 removes any effects of decoherence (e.g. frominelastic loss and magnetic field fluctuations) and allows for a study of the equilibriumphases of the Bloch vectors as a function of the position across the cloud for various

100∆(z). Fig. 5.4 shows the trajectories of the reconstructed Bloch vectors in each spatial
bin. Each bin has been offset by 1 rad in the transverse direction from the adjacent bins
for clarity and to depict schematically the position across the cloud. The oscillations
can be fit to extract spin wave frequency, amplitude, and damping rate.

Longitudinal phase angle (π rad)Cloud edge

Cloud center

Cloud edge

0.8
0.5
0.2 a)
0.8
0.5
0.2 b)

-8

-6

-4

-2
0
2
Transverse phase angle (rad)

4

6

8

Figure 5.4: Bloch vector trajectories across the cloud for T = 600 nK andn = 2 ×1013 cm−3 . Each spatial bin (∼ 40 µm/bin) is shifted by 1 rad from theadjacent bins (see text). In each bin, the path traced by the Bloch vectors is shown
starting with the red points near time t = 0 and spiralling towards the blue points at
t = 0.3 s as the Bloch vectors precess about the equilibrium position. a) Spin waves at∂ 2 ν/∂z 2 ≈ 90 Hz/mm2 . The equilibrium position (center of precession) has a curvature
to match the curved potential. b) Spin waves are excited for 70 ms at the same curvature, and then at t = 0 the bias magnetic field is ramped quickly so that ∂ 2 ν/∂z 2 ≈ 0,
giving an approximately flat equilibrium configuration near the v -axis for all spatialbins.

The steady-state Bloch vector arrangement is determined by the curvature of
the potential. In order to equilibrate, the Bloch vectors across the cloud must arrange
themselves so that the torque from the driving potential ∆(z) (which tends to spreadthe vectors in the u-v plane) is balanced by a torque from the local curvature in the spin
field, which is spread out in the v-w plane. In the presence of a nonuniform potential,
the equilibrium configuration of the Bloch vectors is curved longitudinally across thecloud to match the curvature of ∆(z), i.e. the Bloch vectors lie in the v-w plane but are
not all collinear with the v -axis. However, if the bias magnetic field is chosen to cancel
the density shift, then ∆(z) is predominantly flat across the cloud, and the equilibrium

101
corresponds to the Bloch vectors everywhere in the cloud collinear with the v -axis.
These equilibrium configurations are illustrated in Fig. 5.4a and Fig. 5.4b, which show
spin-wave evolution in curved and flat potentials respectively.

5.5

Frequency of oscillation of spin waves
Next, the density dependence of the spin wave frequency was studied. The spinwave frequency exhibits a strong density dependence, scaling roughly as 1/n, shownin Fig. 5.5a. The linear regime for these excitations occurs only at extremely small
amplitude, where spin waves are difficult to observe due to signal-to-noise limitations.
In the nonlinear regime there is a dependence of frequency on amplitude. In order toremove this dependence, we excite spin waves over a range of amplitudes by controllingthe curvature of the potential, ∂ 2 ∆(z)/∂z 2 , and we extrapolate the frequency to the lowamplitude limit for each density. The solid line is a numerical calculation predicting thefrequency obtained from solving the one-dimensional Boltzmann transport equations[59]. The two dotted lines represent the weak and strong coupling limits, ωexch  ω◦and ωexch  ω◦ respectively:ωexch
22 (k T )keffb.
m ωexchωw = 2ω◦ −

(5.8)

ωs =

(5.9)

where ω◦ is the axial trap frequency and keff is an effective wave vector. The spatial wave
vector k = 2π/λ of the spin waves is studied by finding the distance λ/2 between thespatial nodes, i.e. the bins where the orientation of the Bloch vector remains constant.
The wave vectors k and keff differ by a numerical factor of order unity due to the finite
size of the sample. For all the work discussed here, the lowest order symmetric mode,η = 2, is excited by the symmetric inhomogeneous potential [65]. The wavelength ofthe η = 2 mode is observed to be directly proportional to the size of the trap. As
predicted by Eqs. 5.8 and 5.9, there is no temperature dependence for the excitation

102
frequency, since we observe k to scale inversely as the size of the cloud, i.e. as T −1/2 .
The numerical calculation agrees well with experiment (see Fig. 5.5a).

a)14Frequency (Hz)12

Numerical calc.
Coupling limits

10
8
6
4
2
0

Damping Rate (Hz)

b)
8
6
4
2
0
0

1

2
3
19
Density (10 m-3)

4

5

Figure 5.5: a) Frequency of the transverse spin wave component as a function of thepeak density. Each frequency is an extrapolation to the limit of small amplitude spinwaves to remove the dependence on driving amplitude. The solid line is the predicteddispersion relation, and the dotted lines show the weak and strong coupling limits. Alldata is for T = 600 nK except for the three tightly clustered points at n = 2 × 10 19 m−3which correspond to T = 350, 600, and 1000 nK. b) Damping rate as a function ofdensity, including a numerical calculation (solid line) which includes elastic collisionsand Landau damping.

The Bloch vectors initially precess about the equilibrium value with some initial
amplitude. As the Bloch vectors precess, their trajectories spiral towards the equilibrium
configuration due to damping of the spin wave excitation. The damping is predicted to
be a combination of two effects: spin diffusion and Landau damping [66]. Spin diffusion
dominates in the high density, or hydrodynamic, regime. For intermediate densities,

103
there is a maximum in the damping rate due to the combination of the two effects.
It near this intermediate regime that these experiments are performed. We observea relationship between spin wave lifetime and density which is good agreement withtheory (see Fig. 5.5b).
We have observed collective spin excitations in a nondegenerate ultra-cold gas.
Particle indistinguishability and the need to symmetrize collisions between like particles
in coherent superpositions give rise to spatial and temporal oscillations of the Blochvectors across a cloud of trapped atoms. We have separately imaged both transverseand longitudinal components of these spin waves and studied the effects of temperatureand density on the spin wave frequency, paving the way for studying spin dynamics ina finite-temperature condensate system.
Our original motivation to explore spin dynamics in uncondensed clouds arosefrom our plan to extend to finite temperatures our earlier studies of spin coherence ina condensate as described in Chapter 4 and in particular to understand the extent towhich an unpolarized normal cloud acts as a decohering thermal reservoir. The existenceof spin waves complicates these studies. A pure condensate cannot support spin waves,
but it will trivially separate due to small differences in the scattering lengths [48]. Spinoscillations in the thermal cloud can have a considerable effect on the spin dynamics
of a finite temperature condensate, and conversely, having a small stationary clusterof condensed atoms will significantly perturb spin waves in a thermal cloud. Such a
two component system is a rich system for studying spin dynamics in ultra-cold gases[66]. The next chapter will present our efforts to study a finite temperature condensate
system.

Chapter 6

Coherence effects in a finite temperature condensate

Multiple spin-component Bose-Einstein condensates have permitted many studies
of global symmetries and quantum statistical correlation effects in macroscopic quantum
systems[67, 68, 25]. Interparticle correlations, or coherences, are fundamental properties
in ultra-cold atomic gases as well as in the field of quantum optics. First-order coherences, referring to the deterministic nature of the phase of a system, can be seen in bothfields. First-order coherence has been seen by overlapping two Bose-Einstein condensates and observing spatial density fringes [69], which is a counterpart to interference
between two laser beams. We investigate further first-order correlations by watching
finite-temperature condensate melting due to decoherence of the normal cloud. Using
the Bose-Einstein phase transition as calibration, we examine the thermodynamics ofthe system to gain understanding of the correlations between atoms.
In this chapter we study how correlations affect the condensing and melting process in a condensate. We start by discussing the properties of the states in which wewill study loss of coherence. We use two dressed states in 87 Rb, because these statesenhance the natural symmetries of the system and allow us to more easily measure thecorrelations. Studies of the dressed states include measurements of stability of the statesagainst transitions and experiments to determine the equilibrium spatial density distribution. Next we discuss the situation where we cool through the transition temperature
with a partially coherent cloud to examine the symmetry of the condensed state, and105
finally how loss of coherence leads to melting of the condensate.

6.1

Dressed states
The dressed-state formalism is useful to describe a two-state system coupled bya radiation field [26]. The energies of the bare states
1
Ee/g = ± h̄ω0
2

(6.1)

are shifted by the presence of the coupling field to1
Eg = − h̄ω0 + N h̄ω
2

(6.2)

1
Ee = + h̄ω0 + (N − 1)h̄ω,
2

(6.3)

where ω0 is the frequency difference between the two bare states, ω is the frequency of the coupling field, and N is the number of photons in the field. Solving for the
eigenstates of the new Hamiltonian including the coupling field produces two eigenstatesthat are linear combinations of the bare states. These dressed states|+i = e|−i = e−i(Ω0R −δ−ωo ) 2ti(Ω0R −δ−ωo ) 2t

"

"s

−e

1
δ
(1 + 0 )|1i + e−iωt
2
ΩR

−iωt

s

1
δ
(1 − 0 )|1i +
2
ΩR

s

s

1
δ
(1 − 0 )|2i
2
ΩR

#

(6.4)

#

(6.5)

1
δ
(1 + 0 ), |2i2ΩRare separated in energy by the effective Rabi frequency
Ω0R =

q

(ω0 − ω)2 + Ω2R ,(6.6)

where ΩR is the resonant Rabi frequency. A diagram of the dressed states as a functionof detuning from resonance is shown in Fig. 6.1. When the coupling field is detuned
far red of resonance the ground state is essentially the |−i state, and the excited stateis essentially the |+i state. As the detuning approaches zero the dressed states become

106

|+

2

wo

|+
|

Engery||+AC Stark shift

WR
4d

|+

2

wo

WR

|
-d

|
0
Detuning+dFigure 6.1: Energy level diagram of the dressed states showing the avoided crossing.

equal superpositions of the bare states. Finally, if the detuning increases in the blue
direction the |−i state is now the excited state.
An atom initially in one dressed state will remain in that dressed state if thedetuning is changed adiabatically. Adiabaticity is maintained if Υ is much greater thanone, where Υ, the adiabaticity parameter, is
Υ=

Ω2R
dδ
dt

,

(6.7)

where dδdt is the time rate of change of the detuning from resonance. The probability
that an atom will make a transition to another dressed state is [70]P = e−2πΥ .

(6.8)

107

6.2

Symmetries of the dressed-state system
Before looking at correlation effects, we must first understand the properties andsymmetries of the dressed state system. The dressed states on resonance, being equalsuperpositions of the bare states, have convenient properties that are not present in thebare state system. These properties include identical collisional and loss rates as well
as nearly identical trapping potentials.
A study of the symmetries includes the stability of atoms in a single dressed state.
A collection of ultra-cold atoms in a binary mixture of two spin states has two global
U(1) symmetries, which correspond to conservation of particle number in each species.
However, when a coupling radiation field is tuned to the transition frequency between
the spin states, one of the U(1) symmetries is broken. The symmetry is broken by theconstant phase relation between the two species imposed by the field. The symmetrythat remains corresponds to a conservation of total atom number. If the energy between
the two states could be reduced to zero the system would be fully SU(2) symmetric.
Such as system would be ideal to produce proposed Schrödinger cat states [71], but isexcessively difficult with our system.
The experimental system is similar to the one described in Chapter 2. The system
consists of a sample of atoms in the |1i hyperfine ground state of 87 Rb cooled to below
500 nK in the HIP trap. The dressing field consists of a microwave photon (∼ 6.8 GHz)and a rf photon (3 MHz) that couples atoms to the |2i state through the intermediate
state |2, 0i [72]. Typical resonant Rabi frequencies are between 0.1-8 kHz and canbe varied dynamically during an experimental run. The atoms can be prepared in asingle dressed state by adiabatically ramping the dressing-field detuning from a large
positive or negative value to zero. The atoms can also be prepared in an equal coherent
superposition of the two dressed states by diabatically turning on a resonant dressing
field.

108
We can selectively image atoms in either the dressed- or bare-state basis. To
image atoms in the dressed basis we adiabatically ramp the dressing-field frequency far
from resonance in either direction before turning off the field. For example ramping thefrequency to a larger frequency transforms the |+i state into the |1i state and the |−istate into the |2i state. We can then use absorption imaging to probe the atoms onlyin the |1i state. We can also quickly turn off the dressing field while still on resonanceto project the atoms into the bare-state basis, and therefore probe the coherence of the
dressed states, which manifests itself as Rabi flopping in the bare states.

6.2.1

Stability of dressed states

We are unaware of any symmetry that implies conservation of atom number separately in each dressed state, and thus transitions between dressed states are not explicitly
forbidden. We set out to observe transitions between dressed states because we thoughttransitions might be possible when the energy splitting of the dressed states is less than
the thermal energy kB T .
There are several mechanisms that can transfer population between dressed states.
The most interesting type of transfer arises from collisions. During a collision the relative
phase between the |1i and |2i bare parts of the wavefunction may acquire a phase shift,which results in population transfer in the dressed basis. Another mechanism of transfer
is driven resonantly by the magnetic trapping fields. As atoms oscillate in the harmonic
trap there appears, in the atoms’ frame, to be an oscillating magnetic field. If theenergy splitting between the dressed states, i.e. Rabi frequency, is tuned such that it is
twice the trap oscillation frequency, the atoms have a probability of making a transition
between dressed states. The radial frequency is large compared to the collision rate
which allows an atom to oscillate freely in the trap for several cycles before a collision
takes place and changes the effective oscillating magnetic field seen by the atom. This
resonant transfer is an incoherent process, because the phase of the oscillating field for109each atoms depends on its position and velocity, which are distributed thermally. The
populations in each state should therefore not oscillate coherently but relax instead toan equal mixture.
Besides collisionally and resonantly induced transfer, atoms can make a transition
between dressed state via the Landau-Zener process [73, 74]. If an atom cannot adiabatically follow the torque applied by the dressing field it may transfer between dressedstates. If dδ/dt > Ω02
R , where δ is the detuning of the dressing field, there is a large
probability an atom will undergo a Landau-Zener transition, as described in section 6.1.
The detuning from the intermediate state varies spatially across the cloud due to the
non-uniform density profile, leading to a spatially dependent mean-field energy shift,
and also due to the Zeeman energy shift from the magnetic trapping fields. As theatoms oscillate in the trap they experience a time rate of change of detuning as theysample different densities and magnetic fields, which induces transfer.
We measure the rate of transfer between dressed states by adiabatically dressing
the atoms in purely one dressed state on resonance and detecting atoms appearing in
the other state. We start with the atoms in the |−i state and wait up to two seconds.
We then ramp the drive detuning 10×ΩR in either direction to adiabatically convert oneof the dressed states into the |1i state, which we image using absorption. We determine
the transfer rate at different Rabi frequencies by extracting a linear slope of the fraction
transferred as a function of time (Fig.6.2).
There are two features to note with the data in Fig. 6.2. First, there is a resonant
increase in transfer where the Rabi frequency is twice the radial frequency for the reason
described above. Second there is a background rate, which decreases as Rabi frequency
increases, which is most likely due to Landau-Zener transitions. We can estimate thetransfer probability per collision far from the resonance, above 3 kHz, to be no more than
4(2) × 10−5 . Although collisional transfer from the |−i to the |+i state is endothermic,
it is energetically allowed because the atoms’ thermal energy is larger than the splitting

110

0.4

Transfer rate (Hz)

0.3

0.3
0.2

0.2

0.1

0.0
300

400

500

600

ΩRabi (Hz)

0.1

0.0
1

10

ΩRabi (kHz)
Figure 6.2: Transfer rate is shown as a function of Rabi frequency. The inset graphhighlights the resonant transfer at twice the radial trap frequency (ν radial = 230.5Hz);
the solid line is a fit to a plausible functional form of a decaying exponential plus a
Lorentzian. The error bars reflect the statistical uncertainty in the fitting. Initial cloudparameters are T = 300 nK and density = 2 × 1013 cm−3 .

between the dressed states. Collisional transitions maybe suppressed in 87 Rb because
the triplet and singlet scattering lengths are nearly identical, which reduces the rate
for atoms to spin-flip between hyperfine ground states during collisions. The small
collisionally induced transfer rate implies an approximate symmetry. The system is notfree to explore fully the entire SU(2) space, which would be useful for verifying several
theoretical predictions such as Mermin-Ho vortices [75]. For our purposes, the stabilityof atoms in the dressed states is fortunate. We want to compare the thermodynamics
of atoms in an equal superposition of states to that of atoms in one dressed state. This
would not be possible if the atoms, initially prepared in one state, would over short111
times equilibrate to an equal populations in both states.

6.2.2

Separation of dressed states

Working with dressed states instead of bare states can also increase the spatial
symmetry of a two-component condensate. A condensate in a superposition of two barestates will spatially separate in the trap to reduce the total energy of the system [48, 76].
The two spin states we work with in 87 Rb have slightly different mean field interactions,which drives the separation. In other work it has been shown that the |1i state, thestate with a larger s-wave scattering length, will move to the lower density region of
the condensate, and the |2i state will concentrate in the center of the cloud where thedensity is the highest [48].
The dressed states on resonance, on the other hand, are composed of equal superpositions of bare states. The mean-field interaction energy between the dressed states
is equal and therefore does not drive spatial separation. However any significant spatial
gradient in the differential potential between the dressed states, such as produced by aspatially inhomogeneous Rabi frequency, will cause separation. Atoms at different positions in the harmonic trap are detuned differently from the intermediate state, |2, 0i,
due to the Zeeman effect. The spatial variation of the intermediate state detuning results in a variation of the Rabi frequency across the cloud. Typically the Rabi frequency
varies by at most a few percent across the entire condensate. However at large Ω R , a few
percent can mean 100 Hz difference, more than enough to drive spatial separation. The
Rabi frequency inhomogeneity has the functional form of the magnetic trap: parabolic.
When we operate with large Rabi frequencies (>1kHz) and negative intermediate state
detuning, the dressed states separate with the |−i occupying the outer region of thecondensate and the |+i in the center (Fig. 6.3). Changing the sign of the detuningcauses the states to reverse positions in the trap. The goal for spectroscopy is to havethe two states overlapped as much as possible in the trap. The dressed states in the112condensate remain completely overlapped, within experimental resolution, when the intermediate state detuning is increased to 1.4 MHz and the Rabi frequency is decreased
300 Hz as shown in Fig. 6.3 .

Ω

Rabi

= 1900 HzΩRabi= 300 Hz0s

0.5 s
Figure 6.3: Normalized cross sections through the clouds comprised of |+i atoms in red
and |−i in blue as a function of time; the difference between the cross sections is inblack (second and fourth columns). There is significant separation of the two dressedstates when the Rabi frequency is 1.9 kHz and ∆IM = 0.7 kHz. Decreasing the Rabi
frequency to 300 Hz and increasing ∆IM = 1.4 MHz results in the clouds remaining
overlapped for at least 0.5 seconds.

113
6.2.3

Evaporation of dressed states

We investigate the symmetries of the dress-state system also by condensing withthe dressing field present. An interesting question to ask is how the condensation process
is affected by the dressing field. Does the condensate form in the lowest energy statein the system or in an excited state such as a coherent superposition of dressed states?
We examine these questions by starting with a cloud of atoms above the transition
temperature, applying the resonant dressing field, and evaporatively cooling the cloud
until a condensate forms.
We begin the experiment with a 50/50 fully-coherent superposition of |+i and|−i states, above the condensation temperature. During the evaporation the atoms
dephase due to the spatial inhomogeneity of the Rabi frequency. The dephasing time is
proportional to the magnitude of the Rabi frequency and is on the order of 100 ms fora Rabi frequency of 370 Hz. We cool the cloud in 120 ms using simultaneous microwave
evaporation on both the |1i and |2i states. By the time the condensate forms the normal
cloud has dephased significantly but not entirely. In this manner we can create a cloudwith a condensate fraction up to 10%. After the condensate forms we investigate thestate of the condensate by waiting a variable time before diabatically turning off the
dressing field and imaging the atoms in the |1i state. We integrate the number througha column in the center of the cloud which includes most of the condensate and some of
the normal cloud. The data in Fig. 6.4 show the number in the center of the cloud,which is 52(4)% condensate, Rabi flopping with 86(4)% contrast.
The high contrast fringes demonstrate that the condensate, which is condensed
out of a normal cloud with residual coherence, forms in a superposition of dressed states.
One might expect this result because the normal cloud retains some local coherence when
the condensate forms. However, it is still surprising that the condensate is created ina state that cycles between the bare states, despite the powerful statistical mechanical114
preference for groundstate condensation.
We are unable to fully answer the question of how correlations are lost, or whatstate a condensate is created in when the normal cloud has lost all correlations, becauseof the limited parameter space we can explore. By the time the condensate forms, thenormal cloud has significantly decohered because of dephasing during the relatively longevaporation time. This limits the measurement to samples with already small correlations. On the other side, we can not form a condensate where the correlations haveessentially disappeared. This inability to condense suggests that the correlations themselves could promote condensation. We can barely create a condensate when correlations
are present because we lose a significant number of atoms due to dipolar relaxation in
the |2i state. Therefore it is not surprising that we can not produce a condensate when
the correlations are lost, because we would need a factor of 2 more atoms. Becauseof these limitations we also looked at the loss of correlations from the perspective ofTotal number (arb.)

melting instead of condensing.
1.0
0.8
0.6
0.4
0.2
0.0
0

1

2

3

4

5

6

7

Time (ms)Figure 6.4: The density of atoms in the center of the cloud in the |1i state Rabi flopping
as a function of hold time after evaporation. 52% of the central density consists ofcondensate atoms.

115

6.3

Decoherence induced melting of a finite temperature condensate
We use the statistics of the Bose phase transition to look at the correlations. A

partially condensed system made up of fully coherent atoms in an equal superpositionof two spin states is, in effect, composed of a single species of atom. If the noncondensed
atoms dephase, due to an inhomogeneous potential, they become distinguishable particles and must be thought of atoms in two distinct populations. A condensate at finite
temperature initially in equilibrium with a fully coherent normal cloud will presumably
melt as the coherence is lost in the normal cloud. This can be understood by realizing the loss of coherence essentially reduces the effective number in the state by two(Nef f = N/2). The condensate must then melt to restore thermal equilibrium.
We observe condensate melting induced by decoherence in the normal cloud byinitially creating a cloud in the |1, −1i state with about 45% condensate fraction. Next
we dress the cloud in an equal coherent superposition of dressed states. We measure thetemperature and density of the cloud over a time of 200 ms and calculate the apparent
phase space density [77] of the normal cloud as the condensate melts. For a system
in a single dressed state or a fully coherent superposition of states, the phase spacedensity for the normal cloud should be equal to one, normalized by 2.61, as long as acondensate is present. If the system is an uncorrelated binary mixture of two states,
the total normal cloud will appear to be at twice the phase space density. We also
performed the same experiment with the sample in only one dressed state.
The one state data allows us to remove imaging and fitting systematics. Thereis a systematic error in calculating phase space density of the normal cloud when acondensate is present due to our fitting routine. We find the calculated phase spacedensity increases linearly with increasing condensate fraction for the one component
cloud, which should remain at a constant 2.61 when a condensate is present. We remove

116
this systematic by fitting a line to the phase space density vs. condensate fraction data,
and then we remove this slope from the both the single and superposition state data.
Also, when we took these data we did not have enough microwave power to transfer
100% of the atoms via ARP for imaging. Therefore, we could not detect all of theatoms, which led to an underestimate of the phase space density. We eliminated thisproblem by normalizing each phase space density data point by the average of the single
state phase space density.

Scaled phase space density

2.5Superposition of dressed states
Single dressed state

2.0

1.5

1.0

0.5
0

50

100

150

200

Time (ms)Figure 6.5: The measured phase space density (PSD), normalized by 2.61, as a functionof decohering time. The solid circles are the PSD of the normal cloud in a single state,
whereas the open circles represent the PSD of the normal cloud in an initially fully
coherent superposition of dressed states. The superposition state PSD approaches two
in a time scale of 40 ms, which is consistent with the normal cloud coherence time of30(5) ms. Each point is a weighted average of nine independent measurements.

We are able to observe the apparent phase space density of the superposition casego from 1→∼2 [Fig. 6.5]. The time scale for the correlations to go away is on the order117
of 40 ms, as extracted from an exponential fit to the data in Fig. 6.5. One would expect
the atoms to become uncorrelated on the same time scale as the decoherence, ignoring
the thermal equilibration time of the system. We measure the coherence time of thethermal atoms at the edge of the cloud to give an estimate of the coherence time at thecenter of the cloud. We find the coherence time of the thermal atoms to be 30(5) ms
which is consistent with the time for the correlations to disappear.
In the special case where the normal cloud has lost all correlations and can bethought of as a binary mixture of two states, the question arises what state describes thecondensate? One possibility is that collisions into and out of the condensate from thenormal cloud could create two condensates in a binary mixture. The two condensates
would still have spatial coherence but would have lost internal coherence. There would
be a distinct relative phase between the condensates in each dressed state but would
vary from experimental shot-to-shot. In the end, any product of the decohering process
would be a novel quantum state. Unfortunately we are currently unable to determine
the state of the condensate after the decohering process, because by the time all of thecorrelations are lost there is very little to no condensate present due to the large loss
rate in the |2i state.
In conclusion we have studied the properties of a two-state dressed system. Atoms
in a dressed state show remarkable stability against relaxing to a different dressed state.
This stability allows us to use these states to examine the effects of correlations on condensing and melting Bose-Einstein condensates. We were able to watch the correlations
disappear and observe decoherence-induced melting in a finite-temperature condensate.
The ultimate state of the condensate after all normal cloud correlations are gone would
be interesting to study in the future.

Bibliography[1] S. N. Bose, Z. Phys. 26, 178 (1924).
[2] A. Einstein, Sitzungsber Kgl. Preuss. Akad. Wiss. 1924, 261 (1924); ibid. 1925, 3(1925).
[3] 2001 Nobel prize in physics citation.
[4] M. Anderson, J. Ensher, M. Matthews, C. Wieman, and E. Cornell, Science 269,
198 (1995).
[5] K. Davis, M.-O. Mewes, M. Andrews, N. van Druten, D. Durfee, D. Kurn, and W.
Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
[6] C. Bradley, C. Sackett, J. Tollett, and R. Hulet, Phys. Rev. Lett. 75, 1687 (1995),
ibid. 79, 1170 (1997).
[7] E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Phys. Rev. Lett. 59, 2631
(1987).
[8] A. L. Migdall, J. V. Prodan, W. D. Phillips, T. H. Bergeman, and H. J. Metcalf,
Phys. Rev. Lett. 54, 2596 (1985).
[9] N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. Hess,and G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988).
[10] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag,
1999).
[11] C. Wieman and L. Hollberg, Rev. Sci. Instrum. 62, 1 (1991). 113, 151 (1998).
[12] K. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys. 60, 1098 (1992).
[13] C. Wieman, G. Flowers, and S. Gilbert, Am. J. Phys. 63, 317 (1995).
[14] W. Ketterle and N. van Druten, Adv. At. Mol. Opt. Phys. 37, 181 (1996).
[15] W. Ketterle, D. Durfee, and D. Stamper-Kurn, in Proceedings of the International
School of Physics - Enrico Fermi, (IOS Press, 1999), p. 67.
[16] C. Myatt, N. Newbury, R. Ghrist, S. Loutzenhiser, and C. Wieman, Opt. Lett. 21,
290 (1996).

119[17] W. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).
[18] M. Greiner, I. Bloch and T. Esslinger, Phys. Rev. A 63, 031401(R) (2001).
[19] J. H. Moore and M. A. Coplan, Building Scientific Apparatus, Second Edition(Perseus Books, 1991).
[20] S. K. Kurt E. Gibble and S. Chu, Opt. Lett. 17, 526 (1992).
[21] P. D. Lett, R. N. Watts, C. E. Tanner, S. L. Rolston, W. D. Philllips, and C. I.
Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).
[22] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, J. Opt. Soc. Am. B
11, 1332 (1994).
[23] E. Hecht, Optics, Second Edition (Addison-Wesley, 1990).
[24] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74,
3352 (1995).
[25] E. Burt, R. Ghrist, C. Myatt, M. Holland, E. Cornell, and C. Wieman, Phys. Rev.
Lett. 79, 337 (1997).
[26] C. Cohen-Tannoudju, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions(Wiley-Interscience, 1992).
[27] Viewpoint USA has recently begun to produce a similar board (DIO-64) with 64
outputs and an improved resolution of 50 ns. This new board is PCI rather than
ISA and should be compatible with the exsisting software drivers written for the
DIO-128.
[28] E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977).
[29] C. Salomon, Y. Sortais, S. Bize, M. Abgrall, S. Zhang, C. Nicolas, C. Mandache,
P. Lemonde, P. Laurent, G. Santarelli, A. Clairon, N. Dimarcq, P. Petit, A. Mann,
A. Luiten, and S. Chang, in Proceedings of the 17th International Conference on
Atomic Physics (ICAP 2000), (American Institute of Physics, 2000), p. 23.
[30] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, Phys. Rev.
A 66, 053616 (2002).
[31] M. J. Holland, D. S. Jin, M. L. Chiofalo, and J. Cooper, Phys. Rev. Lett. 78, 3801
(1997).
[32] W. Ketterle and H.-J. Miesner, Phys. Rev. A 56, 3291 (1997).
[33] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E.
Wieman, Phys. Rev. Lett. 79, 337 (1997).
[34] K. Gibble and S. Chu, 70, 1771 (1993).
[35] C. Fertig and K. Gibble, Phys. Rev. Lett. 85, 1622 (2000).

120[36] Y. Sortais, S. Bize, C. Nicolas, A. Clairon, C. Salomon, and C. Williams, Phys.
Rev. Lett. 85, 3117 (2000).
[37] T. C. Killian, D. G. Fried, L. Willmann, D. Landhuis, S. C. Moss, T. J. Greytak,
and D. Kleppner, Phys. Rev. Lett. 81, 3807 (1998); D. G. Fried, T. C. Killian, L.
Willmann, D. Landhuis, S. C. Moss, D. Kleppner, and T. J. Greytak, Phys. Rev.
Lett. 81, 3811 (1998).
[38] N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, Phys. Rev. Lett.
74, 1311 (1995).
[39] N. F. Ramsey, Molecular Beams (Clarendon Press, Oxford, 1956).
[40] J. M. McGuirk, H. J. Lewandowski, D. M. Harber, T. Nikuni, J. E. Williams, andE. A. Cornell, Phys. Rev. Lett. 89, 090402 (2002);[41] At n = 1013 cm −3 νspin wave = 5 Hz, so interrogation times of 20 ms are acceptable.
[42] Where β = 431.35957(9) Hz/G2 and νmin = 6834678113.59(2) Hz [51].
[43] M. Ö. Oktel, T. C. Killian, D. Kleppner and L. S. Levitov, Phys. Rev. A 65, 033617(2002).
[44] M. Ö. Oktel and L. S. Levitov, Phys. Rev. A 65, 063604 (2002).
[45] J. E. Williams, T. Nikuni, and C. W. Clark, Phys. Rev. Lett. 88, 230405 (2002).
[46] E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, and B. J. Verhaar,Phys. Rev. Lett. 88, 093201 (2002).
[47] The Thomas-Fermi radius (and to a lesser extent the critical temperature) are ofcourse interaction dependent, and therefore using measurements of these quantities
to calibrate densities in order to characterize interaction strengths is somewhat cirnc /αc = 2.1(2), on the other hand, is based
cular. Our directly measured value of αii
ii
predominantly on measured spatial extent of the cloud and on an absolute determination of the absorption depth, and therefore it is only weakly model dependent.
nc /αc = 1.7(2) in order to put a
We quote the “worst-case corrected value” of αiiii
reasonable limit on the size of possible systematic errors, and not because we feel itis the preferred value.
[48] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, Phys.
Rev. Lett. 81, 1539 (1998).
[49] When we changed the length of the first Ramsey pulse, the second pulse length
was also changed such that the two pulses combined to form a π pulse, which gives
us the maximum signal-to-noise.
[50] The quoted chirp limit of -0.2(3) Hz includes a correction to account for a small frequency change associated with decay of total atom number during the measurement
time. For each data set the number decay was found by allowing an exponential
decay in the fit of the Ramsey fringes (not to be confused with the decay in Ramsey

121
fringe contrast associated with decoherence), resulting in a correction amounting to-0.74 Hz on average.
[51] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and E. A. Cornell, Phys. Rev.
Lett. 88, 070403-1 (2002).
2[52] Fitting the decay of the integrated transverse magnetization to e −(t/τ ) , the samefunctional form as we fit to the actual experimental data, gives values typicallywithin 5%, and at worst 12%, of the 1/e time.
[53] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L.
Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A 47, 3554 (1993).
[54] See for instance Spin polarized quatum systems, Proc. of the Aussois Conference,
J. Physique Colloq. 41, C-7 (1980).
[55] B. DeMarco, J. L. Bohn, J. P. Burke, Jr., M. Holland, and D. S. Jin, Phys. Rev.
Lett. 82, 4208 (1999).
[56] F. Bloch, Z. Physik 61, 201 (1930).
[57] E. P. Bashkin, JETP Lett. 33, 8 (1981). C. Lhuillier and F. Laloë, J. Physique 43,
225 (1982); C. Lhuillier and F. Laloë, J. Physique 43, 197 (1982); A. Leggett and
M. Rice, Phys. Rev. Lett. 20, 586 (1968), errata Phys. Rev. Lett. 21, 506 (1968);
K. Miyake, W. Mullin, and P. C. E. Stamp, J. Physique 46, 663 (1985); L. P. Lévy
and A. E. Ruckenstein, Phys. Rev. Lett. 52, 1512 (1984), G. Tastevin, P. J. Nacher,
M. Luduc, F. Laloë, J. Phys. Lett. 46, 249 (1985).
[58] B. R. Johnson et al., Phys. Rev. Lett. 52, 1508 (1984); P. J. Nacher, et al., J. Phys.
(Paris) Lett. 9, L441 (1984); W. J. Gully and W. J. Mullin, Phys. Rev. Lett. 52,
1810 (1984).
[59] T. Nikuni, J. E. Williams, C. W. Clark, Phys. Rev. A 66, 043411 (2002).
[60] The two states in question are |1i ≡ |F = 1, mF = −1i and |2i ≡ |F = 2, mF = 1i,
which are pseudo-spin states since the system is not a true spin-1/2 system.
[61] See for instance L. C. Allen and J. H. Eberly, Optical Resonance and Two Level
Atoms (Dover, New York, 1956).
[62] Alternately to the Bloch sphere picture, one can think of the u, v, and w compo→−
nents as the expectation values of the pseudo-spin state S : hSx i, hSy i, and hSz i
respectively.
[63] It is important to realize that the difference in mean-field interactions between thetwo states, (a22 − a11 ) ≈ 2.6 Å, is extremely small in 87 Rb and is not the mechanism
for spin wave propagation. For the purposes of these experiments (a22 −a11 ) provides
merely a contribution to the inhomogeneity in ∆(z). The total mean-field effect ω exch
is proportional to a = (a11 + a22 )/2 ≈ 50 Å.

122[64] The mean phase of the Bloch vector evolves at the average hyperfine splitting
frequency of ∆ = 6.8 GHz; the Bloch vector only moves slowly in the frame rotatingat ∆. It is in this reference frame the the effects of spin waves are considered. The
local oscillator is set so that the Ramsey fringe frequency is much larger than the spinwave frequency, simplifying analysis of the “instantaneous” phase and amplitude of
the Bloch vectors.
[65] The quadrupole mode is excited with a frequency splitting ∆(z) that varies quadratically with distance from the trap center. The dipole mode could be driven by applying a linear gradient to the trapped atoms.
[66] M. Ö. Oktel and L. S. Levitov, Phys. Rev. Lett. 83, 6 (1999).
[67] M. Naraschewski and R. J. Glauber, Phys. Rev. A, 59, 4595 (1999).
[68] H. J. Miesner et al., Phys. Rev. Lett. 82, 2228 (1999).
[69] M. R. Andrews et al., Science 275, 627 (1997).
[70] J. R. Rubbmark, M. M. Kash, M. G. Littman, and D. Kleppner, Phys. Rev. A 23,
3108 (1981).
[71] A. B. Kuklov and B. .V. Svistunov, cond-mat/0202538.
[72] M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998).
[73] L. D. Landau, Z. Sowjetunion 2,46 (1932).
[74] G. Zener, Proc. R. Soc. London A, 137, 696 (1932).
[75] Tin-Lun Ho and Lan Yin, Phys. Rev. Lett. 84, 2302 (2000).
[76] B. D. Esry, C. H. Greene, J. P. Burke, and J. L. Bohn, Phys. Rev. Lett. 78, 3594
(1997).
[77] The apparent phase space density is not correct in the sense that we are counting
all of the atoms in the normal cloud regardless of which state they are in. Thisis similar to a situation with two condensates of different atomic species, say Rb
and Na condensates in the same trap. If the total number was measured, the phasespace density would be calculated to be factor of two larger than if only one state
was correctly considered.

Appendix A

Everything you ever wanted to know about Rubidium

Property
cooling transition
nuclear spin
wavelength in vacuum
mass
lifetime of upper state
natural linewidth
absorption cross section
saturation intensity(stretched transition)
recoil temperature
recoil velocity
ground hyperfine splitting|1, −1i s-wave scattering length|1, −1i/|2, 1i s-wave scattering length|2, 1i s-wave scattering length

symbol
I
λ
mRb
τn
Γ/2π
σabs
Isat
Trec
vrec
ωhf
a11
a12
a22

Value
5S1/2 , F=2→5P3/2 , F=3
3/2
780.23 nm
1.44×10−25 kg
27 ns
5.9 MHz290.7×10−15 m2
1.6 mW/cm2
180 nK
0.59 cm/s
6834.68261090434(3) MHz
100.44 a0
98.09 a095.47 a0

Table A.1: Properties of 87 Rb
.

124

gF
F=3

2/3

F=2

2/3

F=1
F=0

2/3
0

F=2

1/6

F=1

-1/6

F=2

1/2

F=1

-1/2

267 MHz
P 3/2
157 MHz72 MHzl~780nm
P 1/2

812 MHzl~795nm

S 1/26835 MHz

Figure A.1: Energy level diagram for 87 Rb showing Landé g factors for each state.

125

D2-line (780 nm)

87Rb

D1-line (795 nm)

F’=3
1
X
15

15

10
5

6
8

3
9

1
8

5

1

3

6

10

2

3

3

2

15

F=2
F’=2
1
X
12

4

1
2

0
3

1

4
2

3

1
X
12

F=2
F’=1
1

1
X
60

3
6

3
4

3

6
3

1
X
12

3

1
X
12

1

F=2
F’=2
1
X
12

3

6
3

1
4

1

6

3

F=1
F’=1
1

5
X
12

1

1
0

1

1
1

1
X
12

F=1
F’=0
1

1
X
3

1
1
F=1

Figure A.2: Branching ratios for 87 Rb. Multiply by the circled number in the left(right)
column to get the branching ration for the D2(D1) line.