Department of Physics
Cavendish Laboratory

Degenerate Bose Gases in a
Uniform Potential

Igor Gotlibovych
Churchill College
April 2014

This dissertation is submitted for the degree ofDoctor of Philosophy

Igor Gotlibovych, 2014
Cavendish Laboratory
University of Cambridge
J J Thomson Avenue
Cambridge CB3 0HEUK

Declaration
I declare that this thesis is my own work and is not substantially the same as any that I have submitted or am
currently submitting for a degree, diploma or any otherqualification at any other university. No part of thisthesis has already been or is being concurrently submitted for any such degree, diploma or any other qualification. This thesis does not exceed the word limit of sixty
thousand words, including tables, footnotes, bibliography and appendices, set out by the Faculty of Physics
and Chemistry.

3

Abstract
This thesis describes the first ever implementation of an atomic Bose-Einstein condensate (BEC) in a uniform three-dimensional system. We prepare a degeneratesample of 87 Rb in a customizable uniform box trap and perform a range of experiments demonstrating fundamental quantum-mechanical and statistical effects.
A dilute vapour of 87 Rb atoms is first cooled to degeneracy in a conventional
harmonic trap. The single-chamber apparatus described in this thesis utilizes evaporative cooling in an optimized hybrid trap to produce large degenerate samples ofover 3 × 105 atoms with a repetition rate of 25 s. The partially condensed gas isthen loaded into a uniform optical box formed by intersecting three specially shaped
blue-detuned laser beams: two parallel sheet-like beams cross a tube beam at rightangles, creating a dark cylinder-shaped trap. The gravitational force is cancelledusing an external magnetic field gradient, creating a region of space with a nearly
uniform potential. The gas is than evaporatively cooled to condensation in theuniform trap.
We than study partially condensed samples using absorption imaging in time offlight. We develop the methods necessary to extract thermodynamic properties ofthe sample from these images. We characterize the critical point for condensationand observe saturation of the thermal component in a partially condensed cloud, inagreement with Einsteins textbook picture of a purely statistical phase transition.
Further, we observe the quantum Joule-Thomson effect, namely isoenthalpic coolingof an (essentially) ideal gas. In our experiments this cooling occurs spontaneously,
due to energy-independent collisions with the background gas in the vacuum chamber. We extract a Joule-Thomson coefcient µJT > 109 K/bar, about ten orders ofmagnitude larger than observed in classical gases. We show that quantum-statisticaleffects dominate over the interaction effects.
Lastly, we study the ground-state properties of the homogeneous atomic Bosegas using high-resolution Bragg spectroscopy. For a range of box sizes, up to 70 µm,
we directly observe Heisenberg-limited momentum uncertainty of Bose-condensed
5atoms. We measure the condensate interaction energy with a precision of kB ×100 pK
and study, both experimentally and numerically, the dynamics of free expansion of
the interacting condensate released from the box potential. All our measurements
are in good agreement with theoretical expectations for a perfectly homogeneousgas, which also establishes the uniformity of our optical-box system on a sub-nK
energy scale.
While uniform systems have always provided a basis for the theoretical description of ultra-cold Bose gases, all experimental studies to date where conducted in
non-uniform external potentials, most commonly in harmonic traps. Our experiments allowed the first direct experimental validation of many theoretical predictions, and paved the way for future novel studies.

6

Acknowledgements
This thesis is the culmination of several years of hard work. It was a dominant
and formative part of my life throughout these years, and it would not have been
the same without the people who guided and supported me, both in and outside
the lab. First and foremost, my thanks goes to my supervisor Zoran Hadzibabic
for allowing me to join his group. While our world views might differ, I could nothave wished for a better mentor in the field of physics. It is due to his vision andoptimism that our research has advanced to the stage presented here. I would further
like to thank the original members of our group who laid the foundations for every
experiment I have been involved with: during their time here, Naaman Tammuz andRobbie Campbell always offered helpful advice, and their publications were essential
for every newcomer to the group. Continuity was upheld by Stuart Moulder and
Scott Beattie, who were at hand for most of my PhD. Alex Gaunt’s experimental
and computational contributions to this work are hard to over-estimate. Further,both Alex and Richard Fletcher were a great help with the preparation of every
manuscript, including this thesis. I would like to extend my gratitude to Robert
Smith, who has been (and continues being) a pillar of our group, and possesses theuncanny ability to bring any experiment back to life when all other attempts havefailed. With Nir Navon and Martin Robert-de-Saint-Vincent, the latest postdoctoral
researchers to join our group, providing advice and supervision, I feel that current
as well as future students are in safe hands.
Of all the members of our research group, it is undeniably Tobias Schmidutz,
whom I have worked with closely for three years, who deserves my deepest thanks. I
have spent more hours with him than any other person over the course of my PhD,
and I could not have wished for a better friend and colleague. Without his help, thecompletion of this work would have hardly been possible.
Last but not least, I would like to thank Matt, Giulia, my fiends and my family
for believing in me unconditionally.

7

Contents
1 Introduction

13

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3

Publications arising from this work . . . . . . . . . . . . . . . . . . . 17

2 Theory of the Ideal Bose Gas in an External Potential

19

3 The Testbed: a Single-Chamber Ultracold Atom Experiment

23

3.1

The vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1

General considerations . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2

Sources of 87 Rb atoms . . . . . . . . . . . . . . . . . . . . . . 25

3.1.3

Assembly and performance . . . . . . . . . . . . . . . . . . . . 27

3.2

Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3

Laser system for BEC experiments . . . . . . . . . . . . . . . . . . . 29

3.4

3.5

3.3.1

Atomic level structure of 87 Rb . . . . . . . . . . . . . . . . . . 29

3.3.2

Laser system for cooling and trapping . . . . . . . . . . . . . . 31

3.3.3

Fiber optics and beam delivery . . . . . . . . . . . . . . . . . 34

3.3.4

Dipole trap beam . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.5

Computer control . . . . . . . . . . . . . . . . . . . . . . . . . 36

Magneto-optical trapping and laser cooling . . . . . . . . . . . . . . . 36
3.4.1

Theoretical foundations. . . . . . . . . . . . . . . . . . . . . 36

3.4.2

Doppler limit and sub-Doppler cooling . . . . . . . . . . . . . 38

3.4.3

Experimental implementation . . . . . . . . . . . . . . . . . . 38

Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1

Theoretical basics . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.2

Magnetic trap with radio-frequecncy (rf) evaporation . . . . . 40

3.5.3

Optical dipole trap (ODT) . . . . . . . . . . . . . . . . . . . . 42

3.5.4

Magneto-optical hybrid trap . . . . . . . . . . . . . . . . . . . 43
9

CONTENTS

3.5.5
3.6

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . 44

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6.1

Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6.2

High-intensity imaging . . . . . . . . . . . . . . . . . . . . . . 49

3.6.3

Condensate Density . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Towards a Uniform BEC
4.1

4.2

4.3

4.4

53

Adding the box potential . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1

Dark optical trap . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.2

Gravity compensation . . . . . . . . . . . . . . . . . . . . . . 56

Loading the BEC into the box . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . 57

4.2.2

Effect of transfer on phase space density and condensate fraction 58

4.2.3

Evaporative cooling in the uniform trap

Thermometry on a uniform BEC . . . . . . . . . . . . . . . . . . . . 60
4.3.1

ToF expansion from an ideal box . . . . . . . . . . . . . . . . 61

4.3.2

ToF expansion from a quasi-uniform power-law trap . . . . . . 65

4.3.3

Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

How good is our trap? . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Thermodynamics of a Uniform BEC
5.1

5.2

5.3

5.4

. . . . . . . . . . . . 59

71

Transition temperature of a uniform BEC . . . . . . . . . . . . . . . 71
5.1.1

Theoretical description . . . . . . . . . . . . . . . . . . . . . . 71

5.1.2

Measuring the critical point . . . . . . . . . . . . . . . . . . . 75

Saturation of the thermal gas in a box . . . . . . . . . . . . . . . . . 76
5.2.1

The Popov approximation . . . . . . . . . . . . . . . . . . . . 76

5.2.2

Experimental results . . . . . . . . . . . . . . . . . . . . . . . 79

Energy and heat capacity . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.1

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.2

Extracting energy from ToF images . . . . . . . . . . . . . . . 81

5.3.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Quantum Joule-Thomson effect in a Bose gas . . . . . . . . . . . . . 84
5.4.1

Joule-Thomson effect in partially condensed clouds . . . . . . 85

5.4.2

Experimental results . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.3

Joule-Thomson cooling in a thermal Bose gas . . . . . . . . . 86

5.4.4

Effect of heating and evaporation . . . . . . . . . . . . . . . . 87
10

CONTENTS

5.4.5

The Joule-Thomson coefficient . . . . . . . . . . . . . . . . . . 88

6 Spectroscopic Studies of a Uniform BEC
6.1

91

Theory of Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . 92
6.1.1

Raman picture . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1.2

Moving lattice picture . . . . . . . . . . . . . . . . . . . . . . 95

6.1.3

Bloch picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1.4

Condensate wavefunction in momentum space . . . . . . . . . 98

6.1.5

Effect of Rabi frequency and pulse length . . . . . . . . . . . . 100

6.1.6

The mean-field shift . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.7

Time evolution and the GPE . . . . . . . . . . . . . . . . . . 102

6.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3

Ground-state wavefunction in momentum space . . . . . . . . . . . . 104

6.4

6.3.1

Spectrum width . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.2

Mean-field shifts . . . . . . . . . . . . . . . . . . . . . . . . . 105

Expansion of a quasi-pure BEC from a uniform trap . . . . . . . . . . 107

7 Conclusions and Outlook

111

Appendices
A Performance of Grating-Stabilized Diode Lasers

115

B Saturated Absorption Spectroscopy and Locking

121

C IGBT driver electronics

127

D Analysis Software

131

E The polylogarithm function

135

F Efficient numerical evaluation of the polylogarithm functions

137

11

Chapter 1
Introduction
1.1

Background

Quantum mechanics has proven to be one of the most successful modern-day physical theories, its fundamentals remaining unchallenged for nearly a century. Fromthe early 20th century onwards, the description of many-particle systems has been
central to the development, understanding and applications of the quantum theory:
while a single particle in an external potential is, in many cases, a useful model, ourworld is inherently many-particle. Indeed, most phenomena in areas as varied as
atomic physics, condensed matter physics, and stellar physics, can only be understood in terms of a fully quantum theory. Historically significant examples include
superfluidity, superconductivity, and neutron stars.
Interestingly, Planck’s quantized description of black-body radiation - one of
the most significant steps towards a consistent quantum theory, implicitly introduced several of the key concepts that were to prove instrumental in extending
single-particle quantum mechanics (which was, in fact, not formulated carefully until decades after Planck) to many-body systems. These concepts are: first, theidentification of microstates in statistical mechanics with different occupations ofquantized energy levels; second, the indistinguishability of identical particles; andthird, the classification of all particles (elementary or composite) into two classes,
namely bosons and fermions, depending on their spin. We briefly review each in
turn.
First, we recall that for any particle confined in an external potential, quantum
theory predicts the existence of discrete energy levels, each with their associatedeigenstate, that the particle can occupy. For example, a quantum-mechanical harmonic oscillator has the energy levels En = (n + 1/2)~ω for n = 0, 1, 2, . . ., where ω13

1.1. BACKGROUND

is the corresponding “classical” oscillation frequency. If the oscillator is in thermal
equilibrium with a bath of temperature T , the probability to find state N occupied
is proportional to the Boltzmann factor exp[−En /(kB T )]. The probalility to find theparticle in it’s ground state (n = 0) increases rapidly as the temperature is reduced
below kB T ∼ ~ω. We would say that at this temperature the system transitions
from a predominantly incoherent thermal state, described well by classical statisticalmechanics, into one that is predominantly coherent and thus fully quantum.
If the same harmonic trap contains many non-interacting particles, each will
occupy one of the energy levels En . We would have to associate a microstate, with
its respective energy, with each possible arrangement of the particles between thelevels. For example, 3 particles could be distributed as follows: particle 1 in level 0,particles 2 and 3 in level 1, or particle 2 in level 0, particles 1 and 3 in level 1, etc.
[Fig. 1.1 (a)]. However, as long as the particles are distinguishable, each will behave
essentially independently from the others.
2

2

1

2

0

1

3

2

1

1

0

2

3

2

1

1

0

3

2

1
0

(a)

(b)

Figure 1.1: Three different microstates of distinguishable particles (a), and thecorresponding single microstate of indistinguishable particles (b).

This changes if the particles are indistinguishable - since we can only tell thenumber of particles in each state, all three microstates in the above example are
identical, and there is only one microstate corresponding to 1 particle in level 0 and
2 particles in level 1 [Fig. 1.1 (b)]. In addition, quantum mechanics places limitations
on the allowed occupation number for each level: for particles with a half-integer
spin - fermions - it can only be 0 or 1, while for particles with integer spin - bosons- it is unrestricted1 .
The occupations of different levels in an ensemble of identical particles will thus
be described by a different statistical distribution depending on the type of particles:
Fermi-Dirac statistics [1, 2] for fermions, and Bose-Einstein statistics [3] for bosons.
Further, both distributions will deviate significantly from their classical counterpart
1

Strictly speaking, the constraint is on the exchange symmetry of the many-particle wavefunc-tion (odd for fermions, even for bosons). We must thus consider a basis of suitable many-particle
eigenstates. However, it can be shown that the description given in the text is identical for the
purposes of statistical mechanics.

14

1.1. BACKGROUND

for temperatures much higher than ~ω: in fact, quantum-mechanical nature will
manifest itself for kB T ∼ N 1/3 × ~ω, where N is the number of particles - a large
factor for most macroscopic systems2 . Such a gas is called degenerate. A degenerate Fermi gas is characterised by nearly constant occupation over low-lying levels
up to a given threshold, with a rapid decrease in occupation for higher energies[Fig. 1.2 (a)]. A degenerate Bose gas exhibits macroscopic occupation of the groundstate [Fig. 1.2 (b)] - this is because at any given T , Bose-Einstein statistics only allow
finitely many particles in the excited states (n > 0), the excess forcibly accumulating in the ground state. This behaviour of bosons was first studied theoretically by

(a)

(b)

Figure 1.2: Typical occupation of low-lying energy levels in a degenerate Fermi (a)and Bose (b) gas.

Einstein [4, 5], and bears the name of Bose-Einstein condensation, hinting at thefact that, thermodynamically, it represents a (second-order) phase transition - onedriven by quantum statistics rather than inter-particle interactions. Both superfluidity and superconductivity have since been linked to the behaviour of degeneratebosons. While these systems are generally dense and strongly interacting, the production of weakly interacting “textbook” condensates eluded experimentalists for afurther 70 years. Most gases undergo an interaction-driven transition to a liquid (orsolid) phase long before they reach quantum degeneracy. This necessitates the useof extremely dilute clouds, pushing the transition temperature into the nK range.
Development of techniques that would allow cooling to these low temperatures thus
presented the main experimental challenge, that was eventually overcome by thegroups of Wieman and Cornell [6] and Ketterle [7] in 1995.
The Bose-Einstein condensates (BECs) of ultra-cold atoms have since proven to
be a highly prolific system for fundamental studies, not least because they allow
tuning of a wide range of parameters - from trapping geometry to inter-particle interaction strength. Suggested applications include quantum computing, simulationsof superfluid and superconducting behaviour, and precision metrology [8].
2NThe exact scaling with N will depend on the external trapping potential, e.g. taking the form

2/3

in a uniform system (see Chapter 2)15

1.2. THESIS OUTLINE

However, most experiments on ultra-cold atomic gases to date have been conducted in harmonically trapped atomic clouds, while conventional many-body systems are uniform and transitionally invariant. Various methods have been developed
to overcome this problem and extract uniform-system properties from a harmonically trapped sample [9–18] - most rely on the local density approximation (LDA, seelater). Sometimes harmonic trapping can even be advantageous, allowing simultaneous mapping of uniform-system properties at different (local) particle densities. On
the other hand, in many important situations local approaches are inherently limiting, for example, for studies of critical behavior with diverging correlation lengths.
The possibility to directly study a spatially uniform quantum-degenerate gas has
thus remained an important experimental challenge. So far, atomic Bose-Einsteincondensates (BECs) have been loaded into elongated [9] or toroidal [10] traps thatare uniform along only one direction while still harmonic along the other two directions.
This Thesis describes the first ever implementation of a truly uniform degenerate Bose gas, as well as a variety of experimental studies performed on this system.
While many of the techniques we use to study uniform Bose-Einstein condensates
derive from those used previously on harmonically trapped clouds, there are alsosignificant differences which we seek to highlight throughout this work. Our measurements allow the most direct tests of the theory of the uniform Bose gas, butalso serve as a benchmark for the quality of our system and its suitability for futurestudies.

1.2

Thesis outline

This thesis is structured as follows: Chapter 2 provides a review the theoretical framework used to describe Bose-

Einstein condensates in arbitrary power-low potentials, including the uniformcase.
 Chapter 3 contains technical details of our BEC experiment, and describes

production of harmonically trapped condensates.
 Chapter 4 explains how we succeeded in creating a uniform box trap, andexplains how the atoms can be loaded from the harmonic into the box trap
and cooled to degeneracy.
16

1.3. PUBLICATIONS ARISING FROM THIS WORK
 Chapter 5 presents an extensive characterization of the thermodynamic prop-

erties of partially degenerate uniform clouds, including transition temperatureand heat capacity.
 Chapter 6 focuses on the ground state properties of the uniform BEC, andpresents results of two-photon Bragg spectroscopy performed on quasi-puresamples.
 Finally, in Chapter 7 we summarize our findings and point out potential im-

plications of our work and future research directions.

1.3

Publications arising from this work

Many of the results presented in this Thesis were published over the course of thepast few years.
1. Methods described in Chapter 3 are summarized in
I. Gotlibovych, T. F. Schmidutz, S. Moulder, R. L. D. Campbell, N. Tammuz,
R. J. Fletcher, A. L. Gaunt, S. Beattie, R. P. Smith, and Z. Hadzibabic, “Acompact single-chamber apparatus for Bose-Einstein condensation of 87 Rb,”
arXiv:1212.4108, Dec. 2012 [19]2. Key experimental results of Chapter 4 first appeared in
A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. Smith, and Z. Hadzibabic,“Bose-Einstein Condensation of Atoms in a Uniform Potential,” Physical Review Letters, vol. 110, p. 200406, May 2013 [20]3. Experimental studies presented in Chapter 5 formed the basis forT. F. Schmidutz, I. Gotlibovych, A. L. Gaunt, R. P. Smith, N. Navon, and
Z. Hadzibabic, “Quantum Joule-Thomson Effect in a Saturated Homogeneous
Bose Gas,” Physical Review Letters, vol. 112, p. 040403, Jan. 2014 [21]4. Measurements presented in Chapter 6 were published in
I. Gotlibovych, T. F. Schmidutz, A. L. Gaunt, N. Navon, R. O. Smith, and Z.
Hadzibabic “Observing the Ground-State Properties of an Interacting Homogeneous Bose Gas” in preparation, 2014 [22]17Chapter 2Theory of the Ideal Bose Gas in
an External Potential
In this Chapter, we briefly review the important theoretical results on the thermodynamics of an ideal Bose gas confined in a power-law potential, to be referred torepeatedly in subsequent chapters. We start by considering a power-law potentialof the form
U (r) =X

Ai |xi |si

(2.1)

i

in three dimensions (i = 1, 2, 3). A potential with s1 = s2 = s3 = s (or, more
−1
−1
generally, s−1 = (s−1
1 + s2 + s3 )/3) will have the same thermodynamic properties

as the corresponding spherically-symmetric potential U (r) = Ars . Thus, s = 2 corresponds to a harmonic trap, while a uniform trap is obtained in the limit s → ∞1 .
Finding the discrete single-particle energy levels is generally non-trivial2 . However,most of the time we are dealing with a large number of excited states, and it isconvenient to replace all thermodynamic sums over these states by integrals of the
form
n
X
i=1

Z n
f (i )i ≈f ()g()d,

(2.2)

0

where we have defined the density of states g(). Approximating the number ofaccessible states with energies in the interval [,  + d] by g()d is known as thesemi-classical approximation [23]. In degenerate Bose gases, this approximation
obviously breaks down for the ground state, which must be treated separately.
At the same time, we must set Ai ∝ r0−si if we are to maintain a constant volume.
2
Well-known textbook cases which have analytic solutions include the harmonic trap and the1

rectangular uniform box.

19

The functional form of g() depends on the trapping potential. We define theparameter α according to
α=

3 X1
+
2
si
i

or α =

3 3
+ .
2 s

(2.3)

Our definition of α is consistent with [23], while [24] uses η ≡ α − 1. Using thisdefinition, the in-trap density of states can be written as [23, 24]α−1g() = Cα where Γ(α) =

R∞
0

1
≡
Γ(α)Eα




Eα

α−1
,

(2.4)

xα−1 e−x dx is the gamma function. On dimensional grounds, Eα

must be proportional to the ground state energy E0 of the respective trap. For aharmonic trap with frequencies (ωx , ωy , ωz ) and a uniform trap with volume V , wefind the following results:
harmonicα=ω̄ = (ωx ωy ωz )1/3

uniform¯l = (lx ly lz )1/3 = V 1/3

3

3/2

Ē0 = ~ω̄

π 2 ~2 /(2m¯l2 )Eα = Ē0(4/π)Ē0

The thermodynamics of a Bose gas in the potential U (r) can be described uniquelyin terms of g(), temperature T and the chemical potential µ. In the case of a uniform system with constant volume, these reduce to the known variables (V, T, µ),suggesting the grand potential Ω(V, T, µ) as a suitable starting point [25] for all thermodynamic calculations3 . Within the semi-classical approximation, we can computethe total atom number N , energy E and grand potential Ω in terms of the chemical potential µ. Noticing that positive occupation numbers in Bose statistics implyµ ≤ 0, we find that the integral over the thermal occupancies of all excited states
Nth → Nc as µ → 0− . Using the above notation, we find

α
kB T
Nc = ζ(α)
,
(2.5)
Eα
P
−α
where ζ(α) = ∞is the Riemann zeta function. For N > Nc the ground statei=1 i
occupation N0 becomes macroscopic, and must be treated as an additive term to all
our integrals - the particles in the ground state form a Bose-Einstein condensate.
3(V, T, µ) are known as the natural variables of the thermodynamic potential Ω. The system isfully described by Ω(V, T, µ), for example the energy can be expressed as E = dΩ
dβ .

20

Inverting the above equation, we can define the critical temperature Tc at fixed atom
number as
kB Tc =

N
ζ(α)

1/α
Eα .

(2.6)

We define the dimensionless temperature τ by rescaling to the critical temperature(2.6):
T
.
Tc

(2.7)

Z = eβµ ,

(2.8)

τ=
Another useful quantity is the fugacitywith β = kB1T .
The entropy S can be calculated from Ω using S = − dΩ
. It is easy to showdT V,µthat the contribution of the ground state atoms to Ω, E and S is negligible forall but the lowest temperatures (T  Tc ). We can therefore set the ground stateenergy to 0. In Table 2.1, we summarize the results in the cases T ≤ Tc , T ≥ Tc andP−α iz is the polylogarithm function,
T  Tc . Throughout this thesis, gα (z) = ∞i=1 iwith gα (0) = ζ(α). For T ≤ Tc , we have µ = 0 and thus Z = 1. For T  Tc wehave Z  1 and we can approximate gα with its truncated series.
Table 2.1: Important thermodynamic properties of a BEC in a general power-law
trapping potential.

τ ≤1

τ ≥1

τ 1

fugacity Z = 1

Z<1

Z1

N = Nc + N0gα (Z)Nc
ζ(α)Z
Nζ(α) c

Nth
=
N

1

1

temperature

τα

Ω
kB T

= − ζ(α+1)
Nth
ζ(α)

(Z)
− gα+1Ngα (Z)−NE
kB T= α ζ(α+1)
Nth
ζ(α)

(Z)
α gα+1Ngα (Z)ihgα+1 (Z)(α + 1) gα (Z) − ln(Z) NαNh α iτ(α + 1) + ln ζ(α)
N

S
=
kB

(α + 1) ζ(α+1)
Nth
ζ(α)

It is worth noting that within the local density approximation (LDA), the peakthermal phase space density in the trap ρ = nλ3T , defined in terms of peak atom
21

1/2

density n and the thermal wavelength λT = (2π~2 /mkB T ), approaches its peakvalue of ζ(3/2) = 2.612 for τ → 1+ . Since ρ ∝ n, we can writeρ = ζ(3/2)τ −α .

(2.9)

In the limit τ  1, this allows us to write the entropy S in terms of ρ only - weshall exploit this fact subsequently.

22

Chapter 3
The Testbed: a Single-Chamber
Ultracold Atom Experiment
The aim of this Chapter is to provide an overview of the design and constructionof the apparatus [19] that formed the backbone of all experiments described in this
Thesis. The system was initially built for training and was used in the early coolingexperiments in our group [26–28]. This was to form the seed for a new, independent BEC experiment. While re-using an existing design promised to shorten theconstruction time, we were hoping to create a machine that would offer the performance and flexibility sufficient for a variety of proof-of-principle studies. Specifically,building on the existing design, we were hoping to optimize it to achieve most of
the following: robust performance high atom numbers in a harmonic BEC
 high repitition rates
 easy optical access for future experiments space for additional coils, optics etc.

As proven by the various experiments described in subsequent Chapters, the systemfulfilled all of these criteria. During my PhD, the original “donor” vacuum system
had to be disassembled, cleaned, reassembled and baked. The new laser systemwent through several iterations before becoming fully operational, and the magnetic
coil arrangement evolved in accordance with our experimental needs. We further
chose to implement a hybrid magnetic-optical trap introduced by Lin et al. [29], in23

3.1. THE VACUUM SYSTEM
which we were able to reliably produce quasi-pure BECs of > 3 × 105 atoms witha repetition of over 1/30 s−1 . This is comparable to or exceeds most multi-chamber87

Rb setups. Efficient production of harmonic BECs provides an indispensable tool

for further studies described in this Thesis. Nonetheless, considering the low cost,
simplicity and excellent optical access of the system described in this Chapter, we
feel that it warrants some interest in its own right. Our implementation will beof interest to groups wanting to design simple and cost-efficient BEC machines forvarious applications
Below, we describe the design considerations that guided the construction of
the system, explain the BEC production sequence and quantify its performance atvarious stages.

3.1

The vacuum system

3.1.1

General considerations

Ultra-cold atom experiments typically require an ultra-high vacuum (UHV) environment. The atomic sample of relatively high density n is confined to a certain regionof the vacuum chamber using a variety of techniques. Outside this region, UHV
conditions prevail, and the background gas has a much lower density nbg  n. Thisis necessary because scattering and heating due to (two-body) collisions with thebackground gas limit any experiment to a timescale of the order τbg = Γ−1bg , whereΓbg is the corresponding collision rate [27]. Besides limiting the time available for
conducting any useful experiments, τbg first and foremost determines the feasibilityof creating a degenerate sample in thermal equilibrium. In fact, as we shall see below(see 3.5), the timescale for thermal equilibration, τeq = Γ−1
eq , is the lower bound on
the time needed to cool any sample evaporatively. The simple timescale argument
suggests that the collision rates in our experiment must satisfy
τbg & τeq ,(3.1)

or, equivalently, Γbg . Γeq .

(3.2)

Note that for simplicity we consider two-body collisions only. The collision rates
for two- and three-body processes scale as n and n2 , respectively, meaning that thetwo-body rate dominates for all but the highest densities in our experiments (∼101 4 cm−3 ; see e.g. [30] and references therein). Then, assuming the mean velocitiesof the trapped and background atoms are v̄ and v̄bg , respectively, and with σ being243.1. THE VACUUM SYSTEM

the typical collision cross-section, we find:Γbg ∼ σnbg v̄bg ,(3.3)

Γeq ∼ σnv̄.

(3.4)

Strictly speaking, the applicable value for σ depends on the relative velocity, but weshall neglect the fact for an order-of-magnitude estimate. Typical values will range
from the high-temperature hard-ball limit, 4πr2 (r being the atomic radius), tothe low-temperature scattering cross-section 8πa2 (a being the effective scattering
length), but the difference is irrelevant for the argument presented here. Now,
assuming our sample is at a temperature T , while the background gas has Tbg ≈√300 K, and remembering that v̄ ∝ T ,we can re-write (3.2) asq
nbg . n Tbg /T .
(3.5)
Consider we want to crate a uniform BEC of N = 105 atoms at T = 30 nK in a box
with volume V = 30 × 30 × 30 µm3 . The atom density in the trap is n = N/V =4×1012 cm−3 , the temperature ratio is Tbg /T = 1010 , and (3.5) requires nbg . 10−5 n,corresponding to a background pressure
pbg = nbg kB T . 1.5 × 10−9 mbar.
It has to be remembered that this is an absolute upper bound on pbg , while typicalpressures will be in the region of 10−11 mbar. During the early cooling stages, n is
typically much smaller than the value assumed above.
τbg can be easily identified as the 1/e lifetime of a trapped sample: assuming
atoms are predominantly lost via two-body collisions with the background gas, thein-trap atom number will evolve according toṄ = −

1
N = −c1 nbg N.
τbg

(3.6)

Measuring τbg gives us a simple way to quantify the background pressure in theimmediate vicinity of the cold atomic cloud, without resorting to expensive gauges.

3.1.2

Sources of 87 Rb atoms

At this point, it might seem that our aim is to reduce the background pressure
as far as possible. However, loading atoms into the trap imposes contradicting
requirements. The starting point of most cooling sequences is loading the atoms
into a magneto-optical trap (MOT, see 3.4). This can be achieved in one of several
ways:
25

3.1. THE VACUUM SYSTEM
 using a chirped [31] or Zeeman [32] slower
 from a 2D MOT [33, 34] directly from the background gas, as described here.

The last option was chosen in our setup because of its proven simplicity and reliability. It relies on a the background pressure of 87 Rb in the chamber to load the
MOT. The loading rate depends only on the MOT geometry and is proportional tothe background density of 87 Rb, nRb . Releasing 87 Rb into the chamber increases the
background gas density, which we now write as nbg = nbg,0 + nRb . We can now write
the rate equation similar to (3.7) for the atom number in the MOT:Ṅ = −c1 (nbg,0 + nRb )N + c2 nRb .

(3.7)

Setting N (0) = 0, we find the solution has the form
N (t) = Nmax (1 − exp(−t/τMOT )) ,

(3.8)

−1
−1
τMOT
= c1 (nbg,0 + nRb ) = c1 nbg = τbg,c2
nRb
Nmax =.
c1 nbg,0 + nRb(3.9)(3.10)The two noteworthy features of this solution are:
 first, the 1/e MOT loading time τMOT is identical to the lifetime of a trappedsample τbg , and both decrease with nRb
 second, the maximum MOT loading Nmax increases with nRb until nRb  nbg,0We see that, while reducing the density of contaminants nbg,0 remains beneficial, therequired background 87 Rb density is determined by balancing the benefits of increasing the atom number in the MOT against the decrease in lifetime. The resulting
limitations on trapping time can be circumvented in a two-chamber experiment,
where the MOT chamber is kept at a higher background pressure than the “science chamber” by differential pumping, and the atoms are transferred between thetwo spatially separated regions by using suitable moving trapping potentials. The
improved lifetime generally comes at the cost of increased complexity and reduced
repetition rate. In our experiments, we followed a different approach: building on
the group’s previous experience with a single-chamber setup, we were hoping to optimize the background pressure (as well as the later stages of the BEC production
sequence) in order to achieve excellent performance within the limitations of our26

3.1. THE VACUUM SYSTEM

vacuum topology. Further, our system provides sufficient optical access for complex
experiments.
The choice of atomic sources affects the performance of a single-chamber 87 Rb
system markedly. This is because the natural abundance of 87 Rb is only 28%. Whenusing an atom source with the natural isotope ratio, the remaining 72% of 85 Rb willcontribute to nbg,0 , but not to nRb in (3.9) and (3.10). A simple estimate using theabove model shows that, the background pressure and thus the lifetime τbg being
equal, the achievable MOT loading scales directly as the relative abundance of thetrappable isotope. By using a source of isotopically pure 87 Rb, we can theoreticallyachieve a 3.6-fold enhancement in the MOT loading. We shall see that this translates
into even greater gains later in the evaporation sequence.

3.1.3

Assembly and performance

The original single-chamber vacuum system, developed for the early proof-of-principle
experiments in our group [26–28], allowed us to cool atoms in a magneto-opticaltrap [35]. However, the system used vapour sources of Rb in its natural isotope
mixture, and the measured lifetimes in a magnetic trap did not exceed 3.5 s. It was
thus decided to rebuild the system, with particular care given to cleaning, baking
and eliminating potential sources of leaks and contaminations.

Figure 3.1: Single-chamber vacuum system: 1 - turbo pump port, 2 - valve, 3 - ion
pump, 4 - 87 Rb dispensers, 5 - viewport, 6 - glass cell. We also show the magnetic
coils for creating: 7 - quadrupole field, 8 - bias field, 9 - guide field, 10 - RF field. All
coils except 9 and 10 are paired (only one coil per pair is shown).

The layout of the vacuum system is shown in Fig. 3.1. The system is constructed27

3.1. THE VACUUM SYSTEM

from standard vacuum components, with the exception of a pyrex cell by Triad
Technology [36] which allows optical access via two 50 mm (top and bottom), seven
25 mm anti-reflection-coated windows and the 25 mm vacuum tube. The system
uses standard CF flanges with copper gaskets. In day-to-day operation, the vacuum
is maintained by a single 45 l/s ion pump (Oerlikon IZ50 [37]). Additional highvolume pumps (a turbo pump backed by a scroll pump), used during the bake-out
only, are separated by an UHV all-metal angle valve (Series 540, DN40 by VAT
Vacuum [38]). For the atomic sources, we installed 98% enriched 87 Rb atomic vapour
sources (AS-3-Rb87(98%)-50-V from Alvatec [39]).
All components were cleaned in several stages:
1. mechanical cleaning with a scouring pad (steel parts only)
2. ultrasonic cleaning in an acetone bath (steel parts only)
3. rinsing with isopropanol
4. rinsing with water
5. rinsing with methanol
After assembly, the system was baked at 200 ◦ C for one week. The temperature
gradients across the system, as well as the heating and cooling rates, were monitored
using multiple thermocouples and held within the manufacturer specified limits forthe respective components.
Pure 87 Rb is released into the system by applying a heating current to the atomic
vapour source. Initially, the walls of the vacuum system adsorb most of the atoms,
making it impossible to maintain a high enough pressure in the MOT region. Oncethe surfaces saturate, the atoms are predominantly removed by the ion pump, anda near-constant pressure can be maintained by activating the 87 Rb sources daily1
to match the average loss rate. The released quantity is controlled by the durationand magnitude of the applied current.
We achieved magnetic trap lifetimes of over 20 s, corresponding to a backgroundpressure of ∼ 2 × 10−11 mbar. Under typical experimental conditions, we increase
the background pressure to improve the MOT loading, reducing τbg to about 10 s.
1

The flow rates in our system are on the order of 0.1 h−1 , making more frequent adjustments

unnecessary28

3.2. MAGNETIC FIELD COILS

3.2

Magnetic field coils

The magnetic quadrupole field for both the MOT and the magnetic trap is createdby an anti-Helmholtz pair of 16-turn water-cooled coils wound from 4 mm copper
tubing (“7” in Fig. 3.1). These create an axial gradient of B 0 = 400 G/cm whencarrying 200 A.
The quantisation axis for optical pumping is provided by a magnetic field along
the imaging axis. We use a low-inductance planar 10 turn coil wound from 5 mmcopper wire (“9” in Fig. 3.1), providing a field of 5 G at the atoms.
Gravity compensation during long-time-of-flight (ToF) measurements is achievedby combining a quadrupole field (created by the MOT coils) with a homogeneousvertical bias field. Residual potential curvature is minimized by a high bias fieldstrength. We use two 50-turn coils in a Helmholtz configuration (“8” in Fig. 3.1),
providing a bias field of 70 G when run at 10 A.
A radio-frequency (RF) field for forced evaporative cooling is created by a threeturn coil mounted against the top viewport of the cell. For transitions between
neighbouring mF states separated by up to 15 MHz, we achieve Rabi frequencies
above 20 kHz using 600 mW of RF power.

3.3

Laser system for BEC experiments

3.3.1

Atomic level structure of 87 Rb

Alkali atoms have long been favoured for atomic physics experiments due to theirsimple and well-understood electronic level structure. Since they only possess one
electron in the outermost shell, their excitations at low energies are described extremely well by the single active electron approximation. Due to LS-coupling [41],87

Rb exhibits a fine-structure doublet, comprising the D1 line (52 S1/2 → 52 P1/2 )at 795 nm and the D2 line (52 S1/2 → 52 P3/2 ) at 780 nm [40]. These are well separated, and in our experiments we only use the D2 line. The total electron angularmomentum J = L + S further couples to the nuclear angular momentum I = 3/2,
leading to the hyperfine splitting of the ground state into the F = 1 and F = 2
states. The D2 excited states is split into the F 0 = 0, 1, 2, 3 hyperfine states. The
resulting level structure is shown in Fig. 3.2. The transition between stretchedstates F = 2 → F 0 = 3 provides an essentially closed two-level system, which we
can utilize for laser cooling and trapping. Still, spontaneous decay into the F = 1
29

3.3. LASER SYSTEM FOR BEC EXPERIMENTS

F=3

193.7408(46) MHz

double-pass AOM
2x -105 MHz

variable detuningwith double-pass AOM
cooling AOM-91.5 Mhz
D= -15 Mhz

imaging AOM-77 Mhzpump AOM-230 Mhz(2,3) crossover

72.9113(32) MHz2

5 P3/2F=2229.8518(56) MHz

repump AOM+78.5 Mhz

(1,2) crossover

F=2

repump laser

780.241209686(13) nm
384.2304844685(62) THz
12816.54938993(21) cm-1
1.589049439(58) eV

spectroscopy

F=0cooling laser

F=1

spectroscopy

302.0738(88) MHz

2.56300597908911(4) GHz2

5 S1/24.27167663181519(6) GHz

F=1

Figure 3.2: 87 Rb D2 line structure (reproduced from [40], showing laser frequencies,
transitions used for locking and AOM-induced shifts (see text for details).

ground state is possible via higher-order transitions, and any atom in this state ispermanently lost from the cooling process. To counteract population growth in thisdark state, a second laser beam (the repump beam) is used to pump atoms from theF = 1 to the F 0 = 2 (or any other excited) state. These atoms can then return tothe cooling cycle via spontaneous decay.
Besides laser cooling, we also use resonant light to pump the atoms into themagnetically trappable Zeeman states via the F = 2, mF → F 0 = 2, mF 0 = mF + 1
transition, as well as to implement absorption imaging on the F = 2 → F 0 = 3
line. This means our laser system must produce four different laser frequencies,
each stable to a few 100s kHz while tunable over many MHz, as summarized inTable 3.1.
30

3.3. LASER SYSTEM FOR BEC EXPERIMENTS
Table 3.1: Laser frequencies for the 87 Rb BEC experiment. Note that pumping andimaging detunings refer to Zeeman-shifted lines.

beamtransition

detuning ∆ (MHz )power (mW)cooling

F = 2 → F0 = 3-15..-70∼ 100

imaging

F = 2 → F0 = 3

0

∼ 100

pumping F = 2 → F = 2

0

∼ 10

F = 1 → F0 = 3

0

∼ 10

0

repump3.3.2

Laser system for cooling and trapping

All the required frequencies are derived from external-cavity diode lasers (ECDLs)[42–47]. These offer a good balance of stability, cost, linewidth, and tunability.
Many designs are available, and we have experimented with several home-built aswell as commercial products. The operating principles, design and performance of
ECDLs are described in Appendix A. We eventually settled for a commercial model(DLpro from Toptica [48]). The necessary stability is achieved by locking each
ECDL to an atomic line of 87 Rb using saturation absorption spectroscopy [49]. We
describe the locking technique in detail in Appendix B; here, it suffices to say thatthe technique allows us to stabilize the laser frequency to that of any of the dipoleallowed transitions within the D2 line, but also to the so-called cross-over frequencies.
The cross-over frequency is the mean of the frequencies of two transitions sharing
the same ground state. Stabilizing an ECDL to a cross-over peak is often easierdue to a stronger spectroscopic feature (see Appendix B for details). Acousto-optic
modulators (AOMs) can be used to change a laser frequency by up to a few 100 MHz,
allowing us to derive frequencies within this range of the locking frequency. However,using AOMs to bridge the hyperfine splitting of the 52 S1/2 ground state (6.8 GHz)requires complicated multi-pass setups that suffer from power losses. We therefore
use two separate laser sources: one for repump and and one for cooling, imaging
and optical pumping.
The locking and AOM configuration we use is shown in Fig. 3.2. The red arrows
indicate ECDL frequencies, the black arrows indicate AOM-induced shifts, and theblue arrows indicate the reference frequency used for locking. The dashed bluelines indicate cross-over frequencies. The optical layout implementing this scheme
is shown in Fig. 3.3.
The part of cooling light used for spectroscopy is frequency-shifted using a31

3.3. LASER SYSTEM FOR BEC EXPERIMENTS

double-pass AOM. The AOM is placed between two focusing lenses, such thatthe minus-first order is always retroreflected back through the AOM, experiencing the same negative frequency shift (∼ 105 MHz) twice. The laser is locked to the
F = 2 → F 0 = (2, 3) cross-over peak, meaning that the laser frequency in the main
output is ca. 210 MHz higher than the cross-over frequency. The double-pass AOM
remains aligned for any frequency shift and can thus be used for easy frequency tuning. Up to 50 mW of the available output power are used to seed a tapered amplifier(BoosTA from Toptica [48]), providing up to 1.5 W of output power. Imaging andpump beams are derived from the amplifier’s output using a beam splitter and two
AOMs (see Fig. 3.2) and recombined using a polarizing beam splitter. A final AOM
after the tapered amplifier is used to reach the cooling frequency.
Each of the output beams passes through a shutter. This serves several purposes:
it enables us to keep the AOM at operating temperature while the beam is shut,thus improving power stability, and eliminates any light diffracted due to stray radiofrequency signals from the AOM driver. AOMs can still be used for precise timing
and allow a much faster switching time compared to mechanical shutters.

32

3.3. LASER SYSTEM FOR BEC EXPERIMENTS

PD

repumpRb-8781→P1)2R

λ/4

repump8light

Toptica8BoosTA
tapered8amplifier

Rb-8782→P2)3R

λ/2

Toptica8DL8pro
780nm8diode8laser
-77

PD
λ/4

,78.5

pump

-105

λ/2

-230

λ/2

imaging

imaging8D8pumpcooling8lightλ/2

beam81

beam82-91.5

PD

cooling

beam83
beam86
beam84
beam85anamorphicyprismypair

opticalyisolatoryw/35dBD

mirror/9125

polarizingybeamysplitterywPBSD

λ/2

whalf/DwaveyplatePDdouble
pass

λ/2

λ/4

Toptica8DL8pro
780nm8diode8laser

Bragg8light

AOM7yfrequencyyinyHz
photodiodeshutterpolarizer

Rb/87

fiberycoupler

free/spaceybeam

vapourycell

singleymodeyopticalyfiber

Figure 3.3: Layout of the laser system

33

3.3. LASER SYSTEM FOR BEC EXPERIMENTS

3.3.3

Fiber optics and beam delivery

Polarization-maintaining single-mode optical fibers (PM630-HP with APC couplers
from Thorlabs [50]) are used to guide beams to the BEC setup. The cooling andrepump beams are delivered to the eight-way fiber port cluster shown in Fig. 3.3
and Fig. 3.4. This is used to combine cooling and repump light and split it 6-way for
the MOT beams. The fiber coupling efficiency is determined by the overlap between
the free-space laser mode and the eigenmode of the fiber. Assuming the free-space
beam is collimated and the fiber tip is exactly in focus of the collimation lens, themaximum efficiency achievable when coupling an elliptic Gaussian beam with waists
wx and wy into a fiber with a mode field diameter (MFD) of 2wf isC=with a maximum at wf =√4wf2 wx wy,(wx2 + wf2 )(wy2 + wf2 )(3.11)

wx wy . One can see that the maximum coupling efficiency

Cmax = 4wx wy /(wx + wy )2 depends weakly on the beam ellipticity, only dropping to
89% for a 2:1 beam. Beam shaping is therefore secondary, as long as we choose theappropriate lens and collimate the beams well. We achieved good results using an
aspheric lens mounted on a micrometre translation stage for precise collimation, andtwo steering mirrors providing decoupled degrees of freedom for spatial alignment.
In practice, we achieve a free-space-to-fiber coupling efficiency of at least 45% at
every stage. About 20% of the available cooling light power is therefore delivered
to the MOT.
in 1
coolingout 6
in 2
repump
out 5
out 4

out 1
out 3

out 2

Figure 3.4: Fiber port cluster used for beam splitting

Similarly, imaging and pump beams are coupled into polarization-maintaining34

3.3. LASER SYSTEM FOR BEC EXPERIMENTS

fibers and delivered to the MOT setup. Fig. 3.5 shows the arrangement of MOT,
imaging and pumping beams relative to the vacuum system.
-1051064nmtYbtfibertlaser

λ/4PD
glasstcell
CCD
NPBS

PD

λ/4

λ/4

vacuumtsystem

λ/4beamt1

λ/4

beamt4

beamt3

beamt2

Figure 3.5: Optical layout of the single chamber system, showing four of the six
MOT beams (1-4), the optical path for the imaging and pump beams (left to right)
and the ODT beam (top to bottom).

The MOT beams, as well as the pump and imaging beams, are collimated to the
maximum size permitted by the science cell’s 1” windows, simplifying alignment of
the MOT. All beams are circularly polarized using quarter-wave plates. The choiceof polarizations is explained below.

3.3.4

Dipole trap beam

Figure 3.5 also shows the optical-dipole-trap (ODT) beam used for the hybrid trap.
It is derived from a 20 W ytterbium fiber laser (YLR-20-LP from IPG Photonics).
After passing through an AOM, the beam is focused by a concave-convex pair oflenses, allowing the adjustment of both waist size and position. We typically use abeam waist of 65 µm and a maximum power of 5 W at the atoms. Higher powers
35

3.4. MAGNETO-OPTICAL TRAPPING AND LASER COOLING

lead to excessive atom losses, which can be attributed to two-photon transitions [51];these transitions are caused by the large linewidth of the trapping laser (1.8 nm in
our case). The focal point is positioned between one and two beam waists below thezero of the magnetic quadrupole field. The beam power is controlled in the range
0.01 − 5 W using a feedback loop consisting of a photodiode, a proportional-integralderivative (PID) controller, and the AOM. Throughout this range, we achieve an
root-mean-square (rms) power noise below 1 mW over a bandwidth of 1 kHz.

3.3.5

Computer control

The experiment is controlled using a system of PXI-compatible output boards from
National Instruments [52]. The current system consists of a PXIe-1082 chassis holding two PXI-6536 digital output cards, and a PXI-6713 and PXI-6733 analog output
cards. Each digital card provides 32 TTL-compatible channels, and each analog
card provides 8 outputs. The digital channels are individually isolated using optocouplers. The high coil current can be switched using an IGBT driver, and its
strength is controlled via an analog input of the power supply. Laser beams can beswitched and detuned using digital and analogue signals, respectively. The outputs
are programmed using Cicero software by Aviv Keshet using NI DAQmx drivers.
All outputs are synchronized by distributing a common clock signal between the
PXI cards. One of the digital outputs is used to provide trigger signals for theimaging camera. Images recorded by the CCD camera are read out and processedusing a LabView program written by Tobias Schmidutz, and further image analysis
is explained subsequently.

3.4

Magneto-optical trapping and laser cooling

3.4.1

Theoretical foundations

The principle of magneto-optical trapping is to combine laser cooling by six pairwise
counter-propagating red-detuned beams with a quadrupole field to provide confinement in both momentum space and real space [41, 53]. Consider the beam and field
configuration shown in Fig. 3.6 (a) below.
We will restrict our discussion to a model atom having a ground state with total
angular momentum F = 0 and an excited state with F = 1. In an external field,
the upper level will thus be Zeeman-split into the mF = −1, 0, +1 states, as shown36

3.4. MAGNETO-OPTICAL TRAPPING AND LASER COOLING
E

RHCP

B

mF
x

-1

I

B

LHCP

LHCP

mF
+1

F=1

0

0

+1

-1
s+

z

s-

LHCP

y

LHCP
F=0

I
RHCP

z

(a)

(b)

Figure 3.6: Magneto-optical trapping: (a) field orientation and corresponding beampolarizations (y beams not shown); (b) operating principle

in Fig. 3.6 (b). The combination of the fields and beam polarizations chosen ensuresthat the left-hand circularly polarized (LHCP) beam propagating in the positive z
direction drives the |0, 0i → |1, 1i (σ + ) transition, which is closer to resonance forz < 0, where B points in the negative z direction, and vice versa. The cooling andconfinement is easily seen by considering the time-averaged force acting on the atom
due to each of the two counter-propagating beams along z [53]:F± = ±~kΓS,
2 1 + S + (2δ± /Γ)2

(3.12)

where k is the wavenumber, Γ is the natural linewidth of the transition, S = I/Isat2

is the ratio of beam intensity I to saturation intensity Isat = 2π3 ~Γc, and δ is theλ3
effective detuning as seen by the atom. The latter is affected by three factors: thedetuning of the laser δ0 , the Doppler shift due to the atom’s velocity ∓kv, and the
Zeeman shift due to the quadrupole field ∓gF µB B 0 z/~ = ∓αz. The resultant force
isF = F+ + F−
"
#
~kΓS
1
1
=
2 −
2
2
1 + S + 2 δ0 −kv−αz
1 + S + 2 δ0 +kv+αz
Γ

(3.14)

Γ

8~kS(−δ0 )≈ −(kv + αz) h
γ 1+S+(3.13)

 i2 .
2δ0 2
Γ(3.15)

In 3.15 we assumed kv  |δ0 |, αz  |δ0 |. This shows that the force is approximately
linear in v and z, and will provide both confinement and cooling for red (negative)
detuning. Cooling is achieved by damping the motion of individual atoms. Moregenerally, the cooling region (MOT volume) will be limited to αz . |δ0 |, and the37

3.4. MAGNETO-OPTICAL TRAPPING AND LASER COOLING

maximum initial velocity (and thus temperature) of atoms that can be cooled will
be limited to kv . |δ0 |.

3.4.2

Doppler limit and sub-Doppler cooling

The temperature achievable in a magneto-optical trap is limited to what is known
as the Doppler limit, TD . The limit arises due to the stochastic nature of photon
re-emission by the atoms, resulting in heating [54]. The cooling effect is balanced
by stochastic heating at a temperaturekB TD =

~Γ.
2

(3.16)For 87 Rb, Γ = 2π × 6.1 MHz [40] and TD ≈ 150 µK. However, it has been shownthat multi-level atoms moving in a light field with spatially varying polarization canbe cooled to temperatures below the Doppler limit [53, 55–57]. In the counterpropagating beam configuration described above (σ + − σ − ), the atomic cloud will reach
lower temperatures if the magnetic field gradient is turned off and the laser detuning
is increased further. This cooling stage at the end of MOT loading is referred to as
optical molasses [58]. The ultimate cooling limit withing the semiclassical theory of
atom-light interaction (that is, treating the external degrees of freedom of the atom,
as well as light, classically) is the recoil temperature TR given by [56]kB TR =

~2 k 2.
2m(3.17)For 87 Rb, the recoil limit is TR ≈ 0.2 µK - 3 orders of magnitude below the Doppler
limit, leaving significant scope for optimizing the starting conditions before transferring atoms from the MOT into a purely magnetic trap. It is worth noting thattemperatures below TR have also been achieved [59], and can be explained within
the quantum atom-light model.

3.4.3

Experimental implementation

For the MOT we use an axial field gradient of B 0 = 13 G/cm and a cooling-lightdetuning of −25 MHz. Typically, we load a MOT of 109 atoms in 15 s, but 5 s is often
sufficient to produce BECs with > 105 atoms. MOT atom numbers can be deduced
using either absorption imaging or the intensity of fluorescence light collected ona photodiode. The latter is less accurate, with the signal deviating from a simple
linear scaling for optically dense clouds. However, it is our preferred method as it can
38

3.5. EVAPORATIVE COOLING

be used for non-destructive in-sequence diagnostics, following the initial calibration
using absorption imaging.
Following the initial loading stage, we compress the MOT by increasing B 0 to
64 G/cm for 20 ms, and then detune the cooling light to −68 MHz for 1 ms (CMOT,[60]). We found that these steps reduce the temperature and improve transfer into
the magnetic trap. These steps are followed by 2 ms of optical molasses, during
which the magnetic field is switched off and the cooling-light detuning of −68 MHz
is maintained. This cools the cloud to ≈ 15 µK - well below the Doppler limit.
Lower temperatures are possible, albeit at the expense of atom number.

3.5

Evaporative cooling

3.5.1

Theoretical basics

To overcome the fundamental limitations of optical cooling, a different mechanism
is required. A commonly used method that leads to further reduction in T while
increasing the cloud’s phase-space density is evaporative cooling. The idea behindevaporative cooling is to allow the atoms carrying energy well above the thermal
average to escape - which after rethermalization leads to a decrease in the temperature. This process can be repeated by gradually decreasing the trap depth. An
analytical model proposed in [61] allows us to describe the evolution of all relevant
quantities during evaporation by simple power laws. If we assume that atoms withenergies above ηkB T are allowed to escape from a trap with characteristic power α2 ,η
we can show that these carry away γ = α+3−1 times the average thermal energy per

atom. We then find that the relevant quantities scale as N q during the evaporative
cooling, with characteristic exponents q shown in Table 3.2 (note that [61] definesparameter ξ ≡ α + 3/2).
Successful evaporation towards BEC requires an increase in the phase-space density ρ as N decreases, i.e. qρ < 0. Additionally, continuing evaporation requires asufficiently fast thermalization rate, which is proportional to the collision rate κ.
Runaway evaporation is possible if κ increases, requiring qκ < 0. The range of η
where efficient evaporation is possible depends on the trap shape. The conditions
for runaway evaporation in different trapping potentials are shown in Table 3.3.
We compare linear, harmonic and uniform trapping potentials (s = 1, 2 and ∞,respectively).
2

As defined previously, α = 3/2 + 3/s for an rs potential39

3.5. EVAPORATIVE COOLING

Table 3.2: Scaling exponents during evaporative cooling; reproduced from [61]variablesymbolexponent q

atom number N

1temperatureTγ

volumeVγ(α − 3/2)

densityn

1 − γ(α − 3/2)

p.s.d.

ρ = nλ3T

1 − γαcollision rate

κ = nσv̄

1 − γ(α − 2)

It can be seen that traps with higher s (and thus steeper walls) can generally achieve gains in phase space density at lower η, meaning faster experimental
sequences. Large η will require slow evaporation, which must be balanced withatomic losses due to background collisions and thus finite in-trap lifetime. However,runaway evaporation is only possible for s < 6, excluding especially the case of theuniform trap3 . Further, in Table 3.3 we show the characteristic exponents in different traps at fixed η = 10. It can be seen that, while ρ increases fastest in traps
with larger s, κ (and n) shows the opposite trend. This in part justifies the use of alinear trap, i.e. a magnetic quadrupole trap, in the early stages of the evaporation
sequence, where density is low.
Having discussed the basics, we now consider three possible implementations oftraps suitable for evaporative cooling, before presenting the experimental details.

3.5.2

Magnetic trap with radio-frequecncy (rf ) evaporation

A magnetic quadrupole field B = B 0 (x, −y/2, −z/2)T provides a convenient linear
trap for atoms with a permanent magnetic moment µB . Consider the Zeeman energy
V = −µB · B = gF µB mF B, where F is the total angular momentum quantum
number, mF - its projection along the direction of B, and gF is the corresponding
Landé g-factor. This potential will confine states with gF mF > 0. Considering
hyperfine sub-levels of the 2S1/2 state of 87 Rb, we find that the trappable states are
F = 2, mF = +1, +2 and F = 1, mF = −1. Removal of the most energetic atoms
from the magnetic trap is achieved by selectively transferring these to untrapped
3

Evaporation in a uniform trap will be discussed in Chapter 4.

40

3.5. EVAPORATIVE COOLING

Table 3.3: Comparison of evaporation conditions in common trap types.

Trap type:linearharmonic

uniform

α

9/2

3

3/2

p.s.d. gain (qρ < 0) forη > 5.5

η>4

η > 2.5

runaway evaporation (qκ < 0) forη > 6.3

η>6

never

Typical exponents for η = 10:
N

1

1

1

T

1.2

2.3

5.7

V

3.7

3.5

0

n

-2.7

-2.5

1

ρ

-4.5

-6

-7.5

κ

-2.1

-1.3

3.8

states. The process is described in more detail in [62]. The underlying principle is
as follows: the Zeeman splitting of the hyperfine states increases with B, and thus
with distance from the centre of the trap, x. Choosing a radio frequency (rf) suchthat hfrf = µB B(x) for some x will cause transitions to untrapped states in atoms
energetic enough to reach the corresponding region of the trap, i.e. with energies
E > E0 = gF µB mF B(x). An atom starting in the F = 2, mF = 2 state will be
successively transferred to the states with the same F and mF = 1, mF = 0, etc. It
can be seen that E0 /(kB T ) will define the evaporation parameter γ. The value ofγ, which in turn determines the evaporation rate, can thus be adjusted by changing
fRF .

The field direction is not constant inside the trap, and perfect trapping requiresthat µB follows B adiabatically. The Larmor precession frequency must therefore
be much higher than the trapping frequency. For sufficiently strong fields, this is thecase everywhere except at the trap centre. The associated losses (Majorana spin-flip
transitions) limit the cooling in magnetic quadrupole traps to a few µK. Magnetic
traps without a zero have been used to avoid these Majorana losses [6, 7, 63]. Analternative is to use an optical trap [64].
413.5. EVAPORATIVE COOLING

3.5.3

Optical dipole trap (ODT)

Optical dipole forces of a red-detuned laser beam can be used to confine the atoms
in the high-field region. Due to the AC Stark shift of the ground state energy, theatom experiences a potential of the form [41]
Uod (r) =~Γ Γ I(r),8 δ Isat(3.18)

where I(r) is the intensity of the confining light field. For negative (red) detuning,
atoms are thus attracted to regions of higher intensity. Multiple configurations withboth red and blue detuned beams have been studied [65].
A single Gaussian beam propagating along z with the waist at focus w0 createsa potential of the form
2


w0
−2r2
U (r, z) = U0
exp
,
w(z)
w2 (z)
pw(z) = w0 1 + (z/zR )2 ,zR = πw02 /λ.

(3.19)(3.20)(3.21)The beam waist w0 determines the radial confinement, while the the Rayleigh lengthzR determines the axial confinement. The potential near (r, z) = (0, 0) is approximately harmonic, with trapping frequencies given by ωr2 = 4U0 /w02 and ωz2 = 2U0 /zR2 .
Evaporation can be forced by reducing the trap depth U0 . However, in doing so we
also reduce the trapping frequencies and thus increase the effective volume - theresulting correction to the scalings presented in Table 3.2 is effectively descibed bysubstituting α → α − 3/2. Thus, our harmonic trap will only perform as well asa uniform one, with the effective volume remaining constant with temperature. As
discussed above, this will rule out efficient run-away evaporation. While crossing
two or more beams is possible to improve confinement, the unfavourable scaling
remains as long as the evaporation is forced by a simple scaling of the overall opticalpower.
Various techniques have been used to improve evaporation efficiency in opticaltraps [64, 66–70]. A simple method combining a single-beam ODT with a magnetic
quadrupole field to achieve efficient evaporations was proposed in [29]. It combines
the benefits of optical and magnetic traps, and allows efficient evaporation to degeneracy. This hybrid trapping technique is described next.
42

3.5. EVAPORATIVE COOLING

3.5.4

Magneto-optical hybrid trap

The hybrid trap uses a single-beam ODT to confine the cloud radially. The weakaxial confinement is enhanced by adding a magnetic quadrupole field, with themagnetic zero displaced vertically from the ODT axis. In our system the parameters
are chosen as follows: quadrupole field gradient B 0 = 14.8 ± 0.1 G/cm (just below the gravity-

compensating value of 15.3 G/cm)
 ODT beam waist w0 = 65 ± 5 µm
 vertical distance from the magnetic zero to the ODT axis z0 = −60 ± 5 µm

The resulting geometry improves the scaling of the average trapping frequency withtrap depth, as discussed in [29,70,71]. To emphasize this point, we plot the effective
trapping frequency f¯ = (fx fy fz )1/3 versus trap depth U0 in Fig. 3.7. Both are
obtained from a numerical model of our hybrid potential. The range shown in50
trap frequency @HzD

trap frequency @HzD

200
150
100
50
0

0

20

40
60
depth @mKD

80

40
30
20
10
0
0.0

100

0.4
0.6
depth @mKD

0.2

(a)

0.8

1.0

(b)

Figure 3.7: Effective trapping frequency versus trap depth in a hybrid trap. Solid
red line: our hybrid trap; dashed blue line: a purely optical trap with equivalent
1/2
1/3
starting conditions, f¯ ∝ U ; dashed red line: f¯ ∝ U(see text).
00Fig. 3.7 (a) corresponds to optical power of the ODT beam of 0 − 5 W, while panel(b) shows the detail for small powers of up to 60 mW (corresponding to trap depth
below 1 µK). We also show the evolution of the trapping frequency in a purely
optical trap with equivalent starting conditions. As explained above, each of the
1/2
three frequencies, and thus f¯, scale as U
in a pure ODT. In the hybrid trap, the0axial trapping frequency is determined by the quadrupole field, and thus remains
1/3
nearly constant. This results in a scaling of the form f¯ ∝ U (dashed red line). For
043

3.5. EVAPORATIVE COOLING

large depths, this agrees well with the calculated frequency. The evaporation willthus be described by α = 2, corresponding to s = 6 - the borderline case for runaway
evaporation. At low ODT powers the scaling improves further - because we choose amagnetic field gradient that does not fully compensate gravity, the trap depth goesto 0 while the trapping frequencies remain non-zero, as seen in Fig. 3.7 (b). The
behaviour thus approaches that of a (constant-f¯) harmonic trap, making runaway
evaporation possible. The performance of a hybrid trap is thus far superior to thatof a gravity-compensated single-beam optical trap4 .

3.5.5

Experimental sequence

Following molasses, the atoms are optically pumped into the |F, mF i = |2, 2i hyperfine ground state, using a combination of resonant |F = 2i → |F 0 = 2i σ + lightand repump light. We pump > 80% of the atoms into the |2, 2i state in 40 µs, whileheating the cloud slightly, to 40 µK.
The atoms are then captured in a magnetic potential by suddenly turning on thequadrupole field with B 0 = 64 G/cm5 and then ramping B 0 to 80 G/cm over 200 ms.
Finally, the trap is further compressed by raising B 0 to 200 G/cm over 500 ms.
From this point on, the evaporative cooling to condensation can be divided into
three stages: (I) RF evaporation, (II) transfer into the hybrid trap, and (III) ODTevaporation.
Fig. 3.8 summarizes the evolution of the relevant experimental parameters during
the evaporation sequence, while Fig. 3.9 displays the evolution of the atom number
N , the temperature T , and the calculated phase-space density ρ.
N and T are measured using time-of-flight (ToF) absorption imaging (see 3.6)
and ρ is calculated from a semi-classical model [29], using the analytic expression
for the hybrid trapping potential. We verified the parameters of the model (such as
beam waist) by measuring trapping frequencies and comparing them with theoreticalpredictions. We omit the calculated values of ρ in stage II, where they are unreliablebecause the trapping potential varies slowly over large volumes.
4

A single-beam ODT performs better without gravity compensation, but it still lacks the axial

confinement of our hybrid trap.
5
By matching the gradient of the CMOT, we ensure that the captured cloud is close to theequilibrium density distribution of the new trapping potential. Any “mode mismatch” leads toheating as the cloud relaxes to its new equilibrium shape.

44

P HWLf HMHzL B' HG cm-1 L

3.5. EVAPORATIVE COOLING

I

II

III

200
0
15
0
5
0
0

1

2

Evaporation time HsL
3

4

5

6

Figure 3.8: Evaporation sequence in the hybrid trap. We show the evolution of (topto bottom) the magnetic field gradient B 0 , RF frequency f , and ODT power P . The
sequence has three stages: (I) RF evaporation, (II) transfer into the hybrid trap, and(III) ODT evaporation.

RF Evaporation
During the initial evaporation stage in the compressed trap (I in Figs. 3.8 and
3.9), we ramp the RF frequency linearly from 15 MHz to 5 MHz in 2 s. We achievea 30-fold increase in the peak phase space density, with an evaporation efficiencyqρ = d[ln ρ]/d[ln N ] = −2.1. According to 3.5.1, this corresponds to η = 7.6.
At the end of this stage the cloud typically contains N = 50 × 106 atoms at
T = 90 µK.
Loading of the Hybrid Trap
In stage II the magnetic field gradient is decompressed to just below the gravitycompensating value, Bg0 = mg/µB = 15.3 G/cm, where m is the atom mass, g is thegravitational acceleration and µB is the Bohr magneton. From this point on, theatoms are supported against gravity by optical forces only, while the confinement
along the ODT axis is dominated by the magnetic forces [29]. Note that the ODTbeam is on from the beginning of the magnetic trapping (see Fig. 3.8), but initiallydoes not have a dominant role.
We decompress B 0 to 80 G/cm over 100 ms while keeping the RF frequency constant, then decompress further to B10 = 40 G/cm while sweeping the RF frequency
linearly to 2.5 MHz in 0.5 s; this provides a good gain in phase-space density de45

3.5. EVAPORATIVE COOLING
Evaporation time HsL
1
I

2

100

3
II

4 5 6 7 8 9III

Temperature HΜKL

Γ=3.4

10

1

Γ=2.1
100

10

Number of atoms H106 L

1
Phase Space Density

0

10-1
10-2
10-3
10-4
10-5
1

¬ 800 Μm ®

Figure 3.9: BEC production. We plot the temperature T (blue open circles, left axis)
and phase-space density D (red crosses, right axis) versus atom number N (bottom)
and time t (top) throughout the cooling sequence. See Fig. 3.8 and text for detailsof stages I - III. Also shown are an absorption image of a partially condensed cloudafter 50 ms of time-of-flight expansion [optical density varies between 0 (blue) and 3(red)] and its integrated profile. Fits to the thermal and condensed components areindicated.

spite the weak magnetic confinement. Finally, the magnetic field is decompressedfrom B10 to B20 = 14.8 G/cm. In this step we sweep the field gradient according to
B 0 (t) = B10 × (1 + t/τ )−1 , where τ = t2 B20 /(B10 − B20 ) and the sweep duration is
t2 = 800 ms. This changes the RF-limited trap depth approximately linearly withtime, with the approximation being exact in the case of no ODT and B20 = Bg0 .
As the magnetic trap is decompressed, the cloud shape is distorted, making
thermometry less accurate during this stage. Further, computing the peak phasespace density relies on integrating over all trapping volume, but convergence is very
slow for gently sloping potential in the downwards direction below the magnetic
zero. Thus, temperatures shown in Fig. 3.9 should be treated with caution, and ρ
is not shown for this stage.
At the end of the loading stage, we typically have 20 × 106 atoms at 30 µK.
46

3.6. IMAGING

Optical Evaporation
For B 0 < Bg0 , evaporation is no longer driven by the RF field. Instead, the trap depth
is limited by the potential at the saddle point vertically below the magnetic-fieldzero [70]. Evaporation can be forced by lowering the ODT power P , albeit at thecost of reducing the trapping frequencies. However, our hybrid trap has 3 times thetrapping frequency of an optical trap with the same starting depth and frequencies
when approaching condensation. Moreover, unlike in a purely optical trap, theeffective trapping frequency remains positive (in our case, near 30 Hz) as the trap
depth approaches zero, allowing efficient evaporation to very low temperatures.
We ramp P exponentially to its final value, Pend , with a time constant of 1.25 s
and total ramp duration of 7 s (both parameters have been optimized empirically).
We achieve an evaporation efficiency of qρ = −3.4, leading to condensation at Pend =0.16 W. Assuming the trap is harmonic with (nearly) constant frequencies in thelate evaporation stages, we find η = 7.4 (see 3.5.1).
At the critical point, N ≈ 106 and T ≈ 250 nK. At this point, the trapping
frequency is about 30 Hz along the ODT axis and about 90 Hz radially.
Lowering the ODT power further results in quasi-pure6 condensates of > 3 × 105
atoms. The lifetime of the quasi-pure BEC is ∼ 3 s, consistent with the expected3-body losses [72].

3.6

Imaging

All the thermodynamic properties of atomic clouds, i.e. the total atom number N ,the condensate number N0 , and the temperature T , were extracted by recording
and analyzing the density distribution of the atomic clouds, such as the one shownin Fig. 3.9. In studying BECs prepared in a harmonic trap, we typically record
the optical density (OD) of a cloud after long time of flight (ToF) using absorption
imaging [73]. More generally, we need a technique that would allow us to record
clouds in situ as well as after variable ToF, meaning a wide range of ODs (typically0-30). This can be achieved using high-intensity imaging [74], as explained below,
and is a necessary prerequisite for studying BECs in box-like potentials.
As shown below, optical density can be interpreted as the line density of theatomic cloud. While N can be obtained from these by simple integration, extracting
N0 and T relies on an analytic model of the cloud (either in situ or in ToF). Below,
6

we can only reliably detect thermal fractions down to about 30%

47

3.6. IMAGING

we review the well-known results for harmonically trapped clouds [73]. These will
form the basis for developing corresponding results for clouds in power-law anduniform potentials in subsequent chapters.
To conclude this section, we will briefly review the practical side of image analysis. Again, we will built on these techniques in subsequent chapters.

3.6.1

Absorption imaging

Absorption of light with intensity I by the atomic cloud of density n can be describedby the general Beer’s law
dI= −nσ(I)I,dz(3.22)

with absorption cross-section σ(I). For resonant light, we can write [74]
σ=

1
σ0
.
cσ 1 + I/Isat(3.23)

Here, σ0 is the resonant absorption cross-section, cσ accounts for any deviations
from the ideal resonant cross-section due to polarization imperfections, and Isat isthe (effective) saturation intensity. In our experiment, we use circularly polarized
light, and σ0 = 3λ2 /(2π).
In the low-intensity limit (I  Isat ), we find that the column density ñ can becalculated from the light intensity before and after the atoms (Ii and If , respectively)
via
Z
ñ =

n(z)dz = σ0−1 × OD = σ0−1


 
If
−cσ ln.
Ii(3.24)

Experimentally, the two intensities are obtained by recording the imaging beam
on a CCD or CMOS sensor with and without the atoms (in our setup, we use a
PCO Pixelfly CCD camera [75]). The maximum OD that can be recorded is limitedby the signal-to-noise ratio in the If . Increasing the intensity Ii to counter (anddecreasing the exposure time to keep the CCD from saturating) takes us into themid- or high-intensity regime.
The choice of imaging intensity is dictated by the trade-off between the signal-tonoise ratio and the available dynamic range (and level quantization) of the detector.
If we choose the intensity too high, the signal will be generally well above the noise
floor, but the relative variations across the image will be small and eventually theerror will be dominated by signal quantization. In practice, this means that for best
results the (raw) optical density must be on the order of 1-3.
48

3.6. IMAGING

3.6.2

High-intensity imaging

At high intensities, the absorption cross-section can no longer be treated as constant.
Using (3.23) to integrate (3.22), we find that (3.24) has to be modified to include a
linear term [74]:ñ = σ0−1



 

If
Ii − If−cσ ln+.
Ii
Isat(3.25)

While measuring ODs in the low-intensity limit only requires ratios of measured
CCD counts, high-intensity imaging requires absolute calibration of the recorded
intensities. We do this by recording thermal clouds produced at the end of thesame experimental sequence with different intensities Ii . All images are processedwith different assumed values of Isat , and we choose the one that yields the smallest
RR
variations in total atom number N =ñdxdy versus Ii . Finally, cσ is calibrated
by comparing N0 and Nth in partially condensed clouds with theoretical predictions,as described in [76, 77]. We find typically find cσ = (1.7 ± 0.2), and we repeat thecalibration after any major change to the experiment. This cσ is typical of othersimilar experiments [78, 79].

3.6.3

Condensate Density

Absorption imaging yields the column density ñ(x, y) =Rn(x, y, z)dz. Below, we

present a theoretical model for n and thus ñ. The parameters of the model are thethermodynamic variables such as T , N0 and Nth . Having an analytic model allows
us to determine these parameters by least-square fitting of the model function to
our measured ñ. Below, we review the theoretical description for clouds trapped inor released from a harmonic potential.
Trapped Gas
At temperatures 0 < T < Tc , the gas has two components: the condensate in
the ground state and the thermal cloud comprising atoms in the excited states.
Detecting the condensate experimentally requires a theoretical model for the spatial
density of each component.
The thermal component can be treated semi-classically [24] to obtain the density
profile1µ − V (rnth (r) = 3 g3/2 exp
,
λT
kB T
P
with the polylogarithm function gj (z) = i z i /ij
49(3.26)

3.6. IMAGING
The density of the condensate part is directly proportional to |Ψ|2 , where Ψ is the
many-body ground state wavefunction. For an ideal gas, this would be the Gaussian
ground state of the simple harmonic oscillator. However, even in dilute gases, thisis strongly affected by interactions. We only consider two-body scattering processes
described by the s-wave scattering length a. In the mean-field approximation, Ψobeys the Gross-Pitaevskii equation [23]
i~

∂Ψ
~2 2
=−
∇ Ψ + V (r)Ψ + U0 |Ψ|2 Ψ∂t2m(3.27)

with U0 = 4π~2 a/m As discussed in [80], the kinetic energy term can be neglectedunder conditions typical in atomic BEC experiments (Thomas-Fermi approximation), and (3.27) can be solved to yieldn0 (r) = maxµ − V (r),0 .
U0(3.28)In the harmonic potential V = 12 m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ), the density profile is thus
parabolic and can be clearly distinguished from the thermal profile (3.26). Notethat both components will have the same aspect ratio in an anisotropic trapping
potential, the extent in a given direction being inversely proportional to the thetrapping frequency. In general, imaging the trapped gas in-situ will yield a bimodal
profile, and fitting the density distribution with a weighted sum of nth (r) and nc (r)allows quantitative analysis of the condensate.
Free Expansion
While in-situ imaging provides a direct picture of the condensate in real space,
it is often convenient to study particle distributions in momentum space instead.
A distinguishing property of the condensate in an anisotropic trap is that, unlike
the thermal fraction, it is anisotropic in momentum space. Experimentally, this ispossible to observe in a time-of-flight (ToF) measurement, where the condensate isallowed to expand freely for a fixed time before imaging. For large expansion times
and negligible interactions, this technique provides a direct image of the momentum
distribution.
Considering the time-dependence of density distributions above, it can be shownthat both n0 and nth expand self-similarly [73]: the functional form of n0 remains
parabolic due to interactions, and nth follows the g3/2 distribution above (for a moredetailed review, see [27, 28]). Keeping the essential functional form only, we can50

3.6. IMAGING

write:"

Xµx2i
nth (x, y, z) ∝ g3/2 exp−
kB T i=x,y,z 2Ri2 (t)!
X x2in0 (x, y, z) ∝ max 1 −
,0 .
2
a
(t)
i
i=x,y,z!#,(3.29)(3.30)For a partially condensed sample, we can set µ = 0. Integrating (3.29) and (3.30)
along z, we find the column densities"ñth (x, y) ∝ g2 exp

x2i−
2Ri2 (t)
i=x,y
X

!#
,

(3.31)

!3/2
X x2
i
,0
.
ñ0 (x, y) ∝ max 1 −a2 (t)
i=x,y i(3.32)The thermal radii Ri , and thus T , as well as the condensed fraction, can be determined by least-square fitting of the measured ñ with a fitting function of the formf (x, y) = Ath ñth (x, y) + A0 ñ0 (x, y). The fitting algorithm will only be numericallystable if the measured distribution is bimodal, as shown in the inset of Fig. 3.9. Thisis generally the case for thermal fractions above 50% and long ToF. If the condensed
fraction is small or ill-defined, we can often extract Nth and T reliably by excluding
the central region of the image and fitting the thermal wings only. This technique
can be used to determine a good starting point for the bimodal fit [28]. Further, it
becomes essential when analyzing uniform BECs: while we shall derive a functional
form analogous to (3.31) for fitting the thermal wings, no analytic form is known
for ñ0 . Further, ñth peaks more steeply in the uniform trap case, meaning that thebimodality of the distribution is not clearly visible like it is in the harmonic case.

51

Chapter 4
Towards a Uniform BEC
Having created a Bose-Einstein condensate in a harmonic trap, our next goal wasto transfer it into a uniform potential. In this Chapter, we describe the necessary
steps. These include creating a suitable box potential, cancelling residual fields suchas gravity, and finally optimizing the experimental sequence to produce large BECs.
Further, we address the problem of detecting condensation in a uniform system,
as well as measuring thermodynamic properties of partially condensed clouds in a
uniform trap. We find that the experimental signature differs significantly fromthe well-known case of a harmonic trap, making an accurate theoretical description
essential for further studies.

4.1

Adding the box potential

Most potentials used in atom trapping experiments can be described by simple powerlaws. For example, we have described the linear quadrupole trap and the quadratic
optical dipole trap. Our ability to shape magnetic fields is somewhat limited: while
one can imagine a quadratic potential or, as in the case of a Helmholtz coil pair,
one with a leading term of order four, these generally offer very weak confinement,
limited by the size of current carrying components1 . Optical fields, on the other
hand, can be engineered much more freely, with the achievable spatial resolution
only limited by the diffraction limit of the optics used. The resulting light field withintensity I(r) will create a trapping potential U (r) ∝ I(r), as described in 3.5.3. Byreversing the sign of the detuning, δ, we can create attractive as well as repulsive
potentials. The main limitation of attractive (red-detuned) optical traps is their
1

This approach is used in “BEC on a chip” systems, aided by the small size of (often printed)

components. In our case, the vacuum system imposes a minimum size of ∼ 10 cm on all coils

534.1. ADDING THE BOX POTENTIAL

inherent heating [65]. Blue-detuned light, on the other hand, can be used to create
dark-field optical traps (see e.g. [65] and references therein), in which the atoms are
confined by repulsive walls in a region of low I, and thus experience little heating.
In this Section, we describe how these concepts were applied to create the firstever box-like trap for ultra-cold atoms. At the same time, we consider the effect ofother external fields on the trapped atoms. These include the gravitational force,
which creates a linear potential along the vertical direction. We thus address ways
to cancel the gravitational potential, to make full use of our dark-field optical trap.

4.1.1

Dark optical trap

The prospects of creating arbitrary potential for ultra-cold atom experiments havebeen studied extensively in [81, 82]. Using a spatial light modulator (SLM), it ispossible to imprint a phase pattern on a laser beam to create almost arbitrary
shapes in the desired plane perpendicular to the propagation direction. Shaping
light in three dimensions, on the other hand, is subject to constraints imposedby the Helmholtz equation (and ultimately, Maxwell’s equations). However, many
novel trapping geometries can be explored by intersecting two or more appropriately
shaped beams. Our implementation is shown in Fig. 4.1. A single SLM is used tocreate two spatially separate beams: one in the shape of a ring and another in the
shape of two sheets. The two beams are separated and guided to the vacuum cell,
where both are intersected at right angles. The details of the holographic methods
used to shape the beams can be found in [82]. The resulting dark-field region iscylindrical in shape and is aligned to overlap with the harmonic hybrid trap. We
use off-resonant blue-detuned light, with a wavelength of 532 nm, produced by an
Excel solid state laser from Laser Quantum [83].
The performance of the trap can be gauged in two ways: first, by the effective
trap parameter α which determines how closely the thermodynamics of the trappedgas resemble that of an ideal box system, and second, by the uniformity of thein-trap BEC density.
To estimate the effective parameter α, we recall that the effective volume scales
as V ∝ T 3/2−α , ord ln(V )= α − 3/2.
d ln(T )(4.1)

The temperature-dependent volume can be determined from the analytic model forthe light intensity I(r), as well as from I(r) measured in a bench environment.
Alexander Gaunt found that the largest contribution to α comes from the tube
54

200
200

200

45
200
AOM

200
Iris

60

Imaging beam

60
260

SLM PLANEFOCAL PLANE 1

200
200

𝜆𝜆
waveplate
2

3mm
waist

260

FOCAL PLANE 2

200

FOCAL PLANE 1

SLM

4.1. ADDING THE BOX POTENTIALFOCAL PLANE 2

tube beamtube beam

capping
beams

capping beams

Figure 4.1: Dark optical box trap (adapted from [82]). Top: experimental setup for
creating the walls of the box, showing distances and focal lengths in mm. Bottom
left: SLM phase pattern (phases from 0 to 2π are represented by colours from black
to white). Bottom center: tube and two sheet beams in the first focal plane, before
being separated by a half-mirror. Solid line shoes the mirror edge. Bottom right: tube
beam and sheet beams intersected at 90◦ in the second focal plane.

beam, with the effective power in the radial direction sr & 10 over a wide range ofT [82]. sr determines two of the three terms in (2), suggesting overall power s & 15
and thus 1.5 ≤ α ≤ 1.7.
To consider the uniformity of the condensed phase, we recall that the density
n is determined by the Thomas-Fermi approximation (under most experimental
conditions, see later),µ − U (r).
(4.2)
γ
We can easily calculate e.g. the standard deviation of the density from its mean hni
n(r) =

over the trapping volume (U (r) < µ). For the sake of argument, consider U (r) ∝ rs .
It is easy to show that the normalized root mean square deviation in density isr
1/2h(n − hni)2 i
3
=
(4.3)
hni
2s + 355

4.1. ADDING THE BOX POTENTIAL

where h· · · i denotes averaging over the trap volume. For s = 2, this results in a 43%
variation in density, while in an r15 potential it would be near 9%. On the other
hand, speckle noise dominates U (r) over large portions of our trap, with noise levels
consistently below 1% [82]. Further, the flatness of n is fundamentally limited by thehealing length ξ, typically of the same order as the extent of the sloping potential
walls (few µm).
While the predicted quality of the dark optical trap look promising, the above
considerations suggest that ultimately the flatness requirements will be determinedby the exact origin of the experimental signatures used to probe it. Examples presented in subsequent Chapters include thermodynamic studies, Bragg spectroscopyof the ground state as well as measurements of the mean field energy.

4.1.2

Gravity compensation

While our atoms may be confined in an excellent box-shaped trap, the potential
they experience is far from uniform until we take care of the gravitation potential.
A simple estimate shows that the gravitational potential, expressed in temperatureunits, equals to
1 µK
mg
=
kB
10 µm

(4.4)

for 87 Rb atoms. For a BEC of N0 = 105 atoms in a cylindrical trap with lengthl = 30 µm and radius r = 15 µm, the chemical potential is µ = γn0 = 1.7 nK × kB .
Keeping the linear contribution to the trapping potential below, say, 0.1µ requires
cancelling gravity to better than 0.6 × 10−4 g. A common approach is to combine agravity-cancelling quadrupole field with a strong homogeneous bias field along thevertical direction (x), resulting in a magnetic field of the form


 
x
1




 − B0 0 .
B = B0 
−y/2


 
−z/2
0

(4.5)

As described in 3.5.2, this creates a potential proportional to B = |B|. Expandingp
to second order in x and r = y 2 + z 2 , we find
B = B0 − B 0 x +1 B 02 r2
.
2 4B0

(4.6)

Thus, we can cancel the gravitational potential to 1st order by choosing B 0 =mg/µB = 15.3 G cm−1 and a large enough B0 . The radial term in (4.6) corresponds
56

4.2. LOADING THE BEC INTO THE BOX

to a harmonic potential with trapping frequency
ωr2 =µB B 02 r2
.
4B0

(4.7)

In practice, our coil arrangement is not perfectly aligned, which introduces a vertical
1/2

trapping frequency ωz . This frequency scales as B0- the inverse of (4.7). In our

experiments, we are therefore limited to B0 = 30 G, corresponding to a residual
potential of ωr = 2π × 1.6 Hz (as confirmed by measuring small center-of-mass
oscillations of the cloud). This corresponds to a residual potential of 0.1 nK near
the trap walls for a 30 µm trap - similar to that due to a relative change of 0.3 × 10−4
in B 0 . At the same time, we are able to stabilize B 0 to approximately 10−4 - we can
improve on that by post-selecting images based on the position of the cloud center
after long ToF in the more sensitive measurements.

4.2

Loading the BEC into the box

We have now described two major steps towards creating a uniform degenerate gas:
first, we have a source of ultra-cold atoms in a harmonic trap (Chapter 3), andsecond, we have created a uniform box-like trapping potential. Next, we describe
loading the atoms into the box. We describe our experimental procedure, and consider the evolution of the cloud as we transfer it between different trap types.

4.2.1

Experimental sequence

The typical transfer sequence is illustrated in Fig. 4.2. By ramping up the box trap
before ramping down the optical dipole trap, we can achieve efficient transfer. The
evolution of the trapping potential is kept slow to make the transfer as close toadiabatic as possible. The change in the in situ density profile is clearly seen in thehigh-intensity absorption images taken before and after the transfer. However, unlike
in a harmonically trapped cloud, in a uniform trap these images do not contain any
information on the temperature of degeneracy of the cloud - the density profile iswell fitted by a model assuming uniform density over the cylindric trapping region,
as shown in Fig. 4.2 (d). We first consider the theoretically expected evolution of
the cloud’s thermodynamic properties during the transfer, before discussing ways tomeasure these for a uniform gas using time-of-flight imaging.
57

4.2. LOADING THE BEC INTO THE BOX

B’

14.5 G/cm

B0

0G

15.3 G/cm
17.5 G

0.16 W

PODT

0W
2W

Pgreen0W
hybrid trap

100ms

100ms

450ms

ramp onbox trap

ramp offhybrid trap

ramp on gravity
compensation

(a)

(b)

uniform trap

70 µm
35 µm
6 µm

(c)

(d)

OD

150

50 µm

0

Figure 4.2: Loading sequence of the box trap(adapted from [82]). Top: evolutionof the quadrupole gradient B 0 , bias field B, ODT power PODT and green laser power
Pgreen . We also show a schematic cross-section of cloud relative to the green trapping
beams just before (a) and after (b) the transfer. Below, we show absorption images of
the cloud at the respective times (c,d), together with integrated density profile along
two directions. Dashed lines represent best fits described in the text.

4.2.2

Effect of transfer on phase space density and condensate fraction

Transferring either a thermal or a partially condensed cloud from the harmonic
hybrid trap into the uniform trap will affect the phase space density ρ in the former
case, and the condensate fraction N0 /N in the latter. To estimate these effects, weconsider an adiabatic transfer - that is, one at constant entropy S. We neglect atom
losses (due to both mismatched trap size and evaporation) during the transfer.
Thermal clouds
First, we consider the limit T  Tc . Using results and notation from Chapter 2, we
can write the cloud’s entropy in the formS
= (αi + 1) − ln ζ(αi ) − ln ρi + ln ζ(3/2),
kB N
58

(4.8)

4.2. LOADING THE BEC INTO THE BOX

where αi is the characteristic parameter of the respective trap type, i.e. α1 = 3 andα2 = 3/2 for a transfer from the harmonic to the uniform trap. From S = const we
can immediately deduceρ2ζ(α1 ) α2 −α1= 0.10.
=eρ1ζ(α2 )(4.9)

A 10-fold decrease in the phase space density (and an even bigger drop in the collision rate), combined with inefficient evaporation, suggests that the cloud should betransferred into the uniform box as late as possible in the cooling sequence. Further,limitations on the size and wall height of the uniform trap make it impossible toload large thermal clouds at high temperatures.
Partially condensed clouds
Instead, we consider transferring a degenerate cloud. The entropy in this case takes
the formSζ(αi + 1)= (αi + 1)
Nth,i .
kB
ζ(αi )(4.10)

Assuming the cloud remains condensed, the final thermal number can be determined
from
Nth,2
(α1 + 1) ζ(α1 + 1) ζ(α2 )== 2.81.
Nth,1(α2 + 1) ζ(α2 + 1) ζ(α1 )The condensed fraction after transfer can be written as


 
N0
N0
= max 2.81 ×− 0.644 , 0 .
N 2
N 1(4.11)(4.12)

This suggests that the condensate will disappear if the initial condensed fraction
is below 64%. In our experiments, we are unable to measure condensed fractions
above 70% reliably. We do, however, observe that samples with smaller condensates
are no longer condensed after loading into a sufficiently deep box trap. In shallower
traps, evaporation can drive further cooling and condensation.

4.2.3

Evaporative cooling in the uniform trap

We can reduce the temperature and increase degeneracy in the uniform trap further by lowering the optical power in the walls. As discussed in 3.5.1, evaporative
cooling in the uniform trap should be comparable to that in a conventional gravitycompensated optical dipole trap. We see the effect of cooling in the sharp increaseof the occupancy of the low-momentum states, as shown in Fig. 4.3. In contrast
594.3. THERMOMETRY ON A UNIFORM BEC

P/P0 = 1

P/P0 = 0.4

P/P0 = 0.2

50 µm

OD

3

300 µmOD0
60

Figure 4.3: (Reproduced from [20]) Evolution of the Bose gas in a uniform trap as
the optical power P is lowered from its initial value P0 . Absorption images taken after
50 ms ToF. Insets show corresponding in situ images. Bottom row shows cuts throughthe momentum distributions.

to the case of a harmonic trap, no dramatic effects of cooling are observed in situ.
However, the appearance of a BEC is clearly seen in the bimodality of the momentum distribution and the anisotropic expansion in time of flight. Extracting thecondensate fraction and temperature from these images relies on an analytic model,
which we develop below.

4.3

Thermometry on a uniform BEC

As explained in 3.6.3, determining T , Nth and N0 from an absorption image relies
on a theoretical model. Condensation in the harmonic trap can be observed insitu(i.e. in real space) as well as in ToF images (i.e. in momentum space). A
clear example of this was seen in Fig. 4.2 (c), where we show the fit to the thermal
wings of the condensate following 3.26. The excess atoms in the central region
can be identified with the condensate. In a box trap, both the thermal cloud andcondensate densities are uniform - Fig. 4.2 (d) shows no signature of condensationcomparable to the harmonic case. The variations in optical density are a result of
the box shape only. Consider a cylindrical box of length L and radius R, with its
axis at x = z = 0. Integrating along the imaging direction z, we expect x 2y  x rΠ1−,ñ(x, y, t = 0) ∝ Π
L
2R
R
60

(4.13)

4.3. THERMOMETRY ON A UNIFORM BEC

with the rectangle function Π(x) =(1,|x| ≤ 1/2

. In practice, we have to correct0, |x| > 1/2
for the finite imaging resolution, σ, to determine L and R. Assuming a Gaussian
point spread function, we write
x2 + y 2
exp −,K(x, y) = √
2
2σ 2
2πσ
ZZ
0
ñ (x, y, t = 0) =ñ(x0 , y 0 , t = 0)K(x − x0 , y − y 0 )dx0 dy 0 .
1

(4.14)(4.15)

Fig. 4.2 (d) shows a typical fit (dashed green lines) with σ = 2.5 µm, R = 15 ± 1 µm
and L = 63 ± 2 µm. It is clear that these results apply to both the thermal atoms
and the condensate. To distinguish the two, we must consider the behavior of thethermal cloud in time of flight.

4.3.1

ToF expansion from an ideal box(

First, consider an ideal box of volume V , such that U (r) =

0,

r∈V

. To
∞, r ∈/V
describe the thermal gas, we can consider the quantum-mechanical eigenstates jand their respective occupancies f [j ]. If we know the behaviour of the respective
wavefunctions Ψj (t), we can, in theory, deduce the condensate density (and anyother observable) at any time. While this approach is suitable for a simple potential
like a square box, finding the eigenstates and their time-dependence is a daunting
task for, say, a general power-law potential. It is thus helpful to consider the local
density approximation, which treats the gas as (locally) uniform at each point inspace [23].
Local density approximation
Within the LDA, we replace the sum over all macroscopic quantum states with asum over local plane-wave states of momentum p, as per
ZZX
d3 pd3 r0
f [j ] →f [(p, r0 )],(4.16)(2π~)3jZZ
X
p  d3 pd3 r022nth (r, t) =f [j ] |Ψj (r, t)| ≈f [(p, r0 )] δ 3 r − r0 + tm(2π~)3j(4.17)
Integrating over p, we getZnth (r, t) ∝

h 
i
m
f  p = (r − r0 ), r0 d3 r0t61(4.18)

4.3. THERMOMETRY ON A UNIFORM BEC

So far, we have not made any assumptions about the nature of the potential or
the statistical properties of our gas. We now recall that a non-relativistic Bose gas
follows Bose-Einstein statistics, defined by
p2
,
2m
1
f [] =
= g0 (Z exp(−β)).
Z exp(β) − 1

(p, r) = U (r) +Using U (r) = 0 (uniform potential), and defining the thermal radius rth = t
we findZ
nth (r, t) ∝

g0
V



(r − r0 )2
d3 r0 .
Z exp −
2
2rth

(4.19)(4.20)
p
kB T /m,(4.21)

Unsurprisingly, this can be written as a convolution of the g0 -like momentum distribution with the in situ density distribution nth (r, t = 0):Z(r − r0 )2nth (r, t) ∝g0 Z exp −× nth (r0 , t = 0)d3 r0
2
2r
3
R
th



Z
r0 2
=
g0 Z exp − 2
× nth (r − r0 , t = 0)d3 r0 .
2r
R3
th

(4.22)(4.23)

Integrating both sides over z is straightforward in this form, and allows us to evaluate
the z 0 integral (see Appendix E), yielding the column density

 02

Z
x + y 02
g1/2 Z exp −ñth (x, y, t) ∝ñth (x − x0 , y − y 0 , t = 0)dx0 dy 0 . (4.24)
2
2r
2
R
th
This is, again, a convolution of the (integrated) momentum distribution (mapped
to length units) with the in situ column density (4.13). Notice that the first term in
the integral above diverges at x = y = 0 as Z → 1 (µ → 0): we will return to thisproperty when we consider arbitrary power law traps below. The divergence is only
curbed by the initial size - the convolution is essential to obtaining a meaningfulmodel of the recorded optical densities. The striking difference between the uniform
and the harmonic case is illustrated in Fig. 4.4. In fact, the high occupancy of thelow-momentum states prompted us to scrutinize (4.17) by treating the problem fully
quantum-mechanically.
Quantum-mechanical description
The abundance of low-momentum states in our semi-classical distribution at µ = 0
raises the question whether these can be identified with the thermal cloud or mustbe attributed to the atoms in the ground state of the trap, i.e. the condensate.
624.3. THERMOMETRY ON A UNIFORM BEC
Ž
n

-2

-1

1

2

x•rth

Figure 4.4: Cross-section through the calculated optical density of a thermal cloudwith Z = 1 released from a harmonic (blue) and uniform (red) traps at long time offlight. Both profiles are normalized to the same total atom number.

This is essential for extracting N0 correctly from time-of-flight imaging. A related
problem arising from (4.17) is that of the effect of the LDA on the predicted criticalnumber. The effect on the critical number is discussed in 5.1.1.
Here, we study the expansion of a thermal cloud from our box trap using a
non-interacting quantum-mechanical model. First, we consider the momentum distribution of the cloud, corresponding to the long-ToF behaviour of the sample. For
an ideal rectangular box with dimensions lx × ly × lz , we can write the energy levels
2 2

2

n2

2

x

y

z

π nx
( l2 + l2y + nl2z ),Enx ,ny ,nz = ~2m

ni = 1, 2, . . ., with separable wavefunctions

Ψnx ,ny ,nz (r) = Ψnx (x)Ψny (y)Ψnz (z),
r


nπx
2
Ψnx (x) =sin, 0 ≤ x ≤ lx .
lx
lx

(4.25)(4.26)

The wavefunction is separable in both real and momentum space, and we can thus
consider one direction only. With l the relevant box size and p = ~k, the momentum
wavefunction takes the following form (up to a phase factor):
r 




1
l
lk − nπlk + nπnΨn (k) =sinc− (−1) sinc
.
2 π~
2
2

(4.27)We plot |Ψn (k)|2 for n = 1, 2, 3, 4 in Fig. 4.5. For n ≥ 2, the eigenfunctions contain a pair of peaks in momentum space, centered at k = ± nπ
. Importantly, thelmomentum distributions of different thermal states add incoherently. In the highT limit, i.e., assuming similar occupation of the relevant quantum states, we canapproximate the momentum distribution as
f (k) ≈

∞
X
n=1

2

|Ψn (k)| =

∞
X
n=−∞

63

2

sinc



lk − nπ2.

(4.28)

4.3. THERMOMETRY ON A UNIFORM BEC
ÈYÈ 2

ÈYÈ 2

→

-6 p

-4 p

-2 p

0

2p

4p

6p

kL

-6 p

(a)

-4 p

-2 p

2p

4p

6p

kL

(b)

Figure 4.5: (a) Lowest 4 eigenstates of the box potential in momentum space, and(b) an equivalent decomposition of the high-T momentum distribution.

The last sum is equivalent to a uniform distribution, as is best seen by applying
the convolution theorem to its Fourier transform (note this is not equivalent to thereal-space wavefunction).
While in the case considered above we assume a uniform occupation of all states(and, implicitly, a microscopic occupation of the ground state), the treatment canbe extended to an arbitrary (thermal Bose) envelope applied to the n-states. For all
k & 1/l, numerical calculations (by directly summing over n-states with a suitable
Bose occupation factor) show that the momentum distribution is indeed equivalent
to the LDA prediction.
Next, we turn our attention to the central region of the cloud and small time offlight, where both the LDA and the quantum-mechanical descriptions are affectedby initial-size effects, and we must solve the time-dependent Scrödinger equation toallow direct comparison. The evolution of the wavefunction (4.26) is easy to describe
in k-space, leading to⇒~2 k 2Ψn (k, t) = Ψn (k, 0) × exp −i
t
2m
 2 
1
xm
Ψn (x, t) ∝ Ψn (x, 0) ∗ √ exp i
t .
2~t
t

(4.29)(4.30)To quantify possible discrepancies between the LDA and the wavefunction approach,we consider a cloud released from a 30 × 30 × 30 µm3 volume trap, and calculate
the column density numerically using each approach for a wide range of T andµ. In Fig. 4.6 we illustrate the relative error in the column density at the box
center for different expansion times. Here, we show data for the case T = 100 nK,µ = −0.1 kT , and sum states with nx , ny ≤ 100 explicitly. Higher-lying states aretaken into account by including an analytic remainder term. The agreement between
644.3. THERMOMETRY ON A UNIFORM BEC
rel. error
0.01
10

20

30

40

t•ms

-0.01
-0.02

Figure 4.6: Relative error of the semi-classical approximation. lx = ly = 30 µm,
T = 100 nK, µ = −0.1 kT . States with nx , ny ≤ 100 are summed, and an analyticalestimate for the remainder term is used.

the LDA and the full quantum-mechanical sum is excellent, including at small timeof flight were we expect largest deviations. We thus use the LDA model for image
analysis throughout the experiments described in this Thesis.

4.3.2

ToF expansion from a quasi-uniform power-law trap

We now extend the LDA treatment of 4.3.1 to a general power-law potential U (r) ∝rs . The Bose factor becomesf (p, r) = g0



p2
s
Z exp −βas r − β.
2m(4.31)

Integrating f over p (see Appendix E), we find the in situ density distributionnth (r, t = 0) ∝ g 3 {Z exp [−βU (r)]} .

(4.32)

2

On the other hand, we can integrate f over all r to find the momentum distributionof the cloud (as measured in the very long ToF):



p2
nth (p) ∝ g 3 Z exp −β,s2m(4.33)

where 3s ≡ α − 32 , according to the notation introduced previously. Now, it is
tempting to write the ToF density as a convolution of the two. Formally, (4.18)
remains unchanged, andZ
nth (r, t) ∝g0

(r − r0 )2
Z exp −βU (r ) −
2
2rth


0



d3 r0 .

(4.34)

However, in the general case s < ∞, the integrand can not be factorized like we did
above. Thus,Z
nth (r, t) 6=m
nth p = (r − r0 ) × nth (r0 , t = 0)d3 r0 .
t
65

(4.35)

4.3. THERMOMETRY ON A UNIFORM BEC

The integrand does, however, factorize when the exponential factor is small andg0 (Ze(··· ) ) ≈ Ze(··· ) (i.e. the Bose factor takes the Boltzmann form). Moreover,RR
if the potential has sharp walls in one direction, the line density
n along thisdirection will be described precisely by the convolution with a box function, while
α will be determined by the “invisible” directions. Even in the most general case,numerical modeling shows that the difference between the exact distribution andR
one written as (4.35) is diminishing as one goes from n to column density ñ = n
RRR
to line density ñ =n.
In practice, the in situ density is indistinguishable from that of a uniform box due
to the limitations of our imaging system, and we can use the form (4.13). We always
analyze the region r & rth , and the time of flight is large to ensure that rth  R, L,
meaning that convolving the (integrated) momentum distribution (4.33) with (4.13)
introduces a very small correction outside the central region. In the central region,on the other hand, it is convenient because it ensures the correct normalization of
our distribution and allows us to count the thermal atom number without worrying
about the divergence of the fitting function. The details of our fitting routine aresummarized in the following section.

4.3.3

Image analysis

To analyze the column density extracted from the absorption images, we use twodimensional fitting functions of the form2



2πA
y2
x2
ñ(x, y) =gα−1 exp µ̃ − 2 − 2∗ K(x, y),rx ry
2rx 2ry(4.36)

RR
with the convolution kernel normalized toK(x, y)dxdy = 1. We perform leastPsquare fitting by minimizing i,j Mi,j |ñ(xi , yj ) − ñi,j |2 over the pixels (i, j) of thehigh-intensity absorption image, with the exclusion mask M defined by(
Mi,j =

1,2(xi −xc )2c)+ (xi −x
>1
r12
r22

0,

otherwise.

(4.37)

Exclusion radii r1 and r2 are chosen such that the quasi-pure condensates remain
within the excluded region throughout ToF. The fitting parameters (α, A, µ, rx , ry )2

We assume that ODs have been converted to atomic column densities.

664.4. HOW GOOD IS OUR TRAP?
relate to the thermodynamic properties of the condensate via3
Z = eµ̃ ,(4.38)

Nth = Agα (Z),(4.39)

2rx,ym
kB Tx,y =.
t2(4.40)We extract the condensate number N0 by counting the difference between the measured column densities and the thermal fit over the exclusion region:
N0 =X(1 − Mi,j )(ñi,j − ñ(xi , yj )).

(4.41)

i,j

On thermodynamic grounds (Chapter 2), the overall amplitude A is not an independent parameter. In fact, for an ideal Bose gas,α  α
kB T
T
Ath =
≡
Eα
θα
θα = const(4.42)(4.43)

We will use this equality in the analysis of our trap below, as well as in the context
of saturation in Chapter 5.

4.4

How good is our trap?

As discussed in [20], determining the value of α from the least-square fits is impractical. While it is easy to distinguish α = 3 (harmonic) from α = 3/2 (uniform), small
variations around the latter do not lead to a noticeable increase in the least-squares
residue. However, we can check the self-consistency of the fits according to (4.43).
Identifying the value of α to be used in fits is essential to determining N0 , Nth andT accurately. In Figure 4.7 (a), we plot θ versus T for 1.5 ≤ α ≤ 2.0. The data
is extracted by fitting our model to a range of clouds obtained by varying the final
evaporation depth in the box trap. The analysis presented here is similar to thatpublished in [20], with the following differences: We fit the 2D column density distribution ñ(x, y), while in [20] the data was

integrated and fitted in 1D.
3

We define two temperatures for fitting purposes, one of the two is typically affected by resid-

ual trapping during the expansion. We introduce the necessary corrections, as well as a modelindependent way to measure energy, in the next Chapter.

67

4.4. HOW GOOD IS OUR TRAP?
 We use an elliptic exclusion region, while [20] effectively excludes a strip region(since exclusion is done on the integrated 1D data).
 We restrict our analysis of θ to high-temperature clouds (70 nK ≤ T ≤110 nK).
Using a smaller temperature range solves several problems: first, as discussed in
section 4.1.1, the effective value of α could vary with T ; second, non-saturation
effects [76] can potentially make θ dependent on the condensate fraction.
We extract the slopes of the straight-line fits shown in Fig. 4.7 (a), and plot themagainst α in Fig. 4.7 (b). It is clear that the condition dθ/dT = 0 strongly excludes
the values α ≥ 1.7. Thus, we can say that the effective power of our potential iss ≥ 15. We use the value α = 1.6 (s = 30) to extract the atom numbers in thethermal cloud and in the condensate, and plot them versus T in Fig. 4.7 (c) and(d), respectively. The fit in (c) corresponds to Nth ∝ T α , and shows good agreementwith the data. We can use the N0 data to identify the transition point. From (d), we
observe Tc = 105 ± 5 nK. The predicted value for the corresponding ideal-gas criticalnumber is Nc = 6.5 ± 0.8 × 105 (using the measured box dimensions L and R). Bothvalues with their error bars are shown in Fig. 4.7 (c), and the thermal number at
the transition proves to be in excellent agreement with the ideal gas prediction.
In this Chapter, we have described the implementation of the first ever degenerate Bose gas in a uniform system. Understanding the inherent limitations of ourapproach is important for both developing the experimental setup and quantifying
deviations from the theoretical box-like system. Our estimates show that these are
extremely small, and first measurements suggest that the thermodynamics of thesystem closely follow the ideal-gas prediction made a century earlier. In subsequent
Chapters, we continue probing our system using a variety of techniques. Our focus
will be on testing the uniform-system behaviour while keeping in mind the influenceof imperfections in our trapping geometry.

684.4. HOW GOOD IS OUR TRAP?

(b)dq•dT @a.u.D

q @a.u.D

(a)
2

1
70

80

90

0

100 110

1.5 1.6 1.7 1.8 1.9 2.0
a

T•nK

(c)

(d)
1

6

N0 •105

Nth •105

8
4
2
0
0

0
0

20 40 60 80 100 120
T•nK

20 40 60 80 100 120
T•nK

Figure 4.7: Self-consistency of thermodynamic properties of a quasiuniform degenerate Bose gas. (a) Variations of θ with T for different assumed α (see text; data setscorrespond to the values of α shown in (b), i.e. between 1.5 (bottom, red) and 2.0(top, orange)). (b) Slopes of the linear fits to θ v.s T for different α, extracted from(a). (c) Thermal atom number versus T , extracted using α = 1.6. The solid red line
shows the expected T α scaling, and the black lines indicate the critical number andtemperature. (d) Condensed atom number versus T , extracted using α = 1.6. Blacklines show the same critical temperature as in (c).

69Chapter 5
Thermodynamics of a Uniform
BEC
5.1

Transition temperature of a uniform BEC

One of the most important characteristics of Bose-Einstein condensation is the critical atom number Nc (T ). Historically calculated for an ideal uniform system [3–5],
the treatment was later extended to consider the effects of interactions [84, 85] andfinite trap size [86]. However, since most experiments to date have been conductedin harmonic traps, the theoretical focus has shifted towards these. A large numberof theoretical [87–89] and experimental [16,90,91] studies have been conducted. Recently, the theory of condensation in more general power-law traps has also been
considered [92, 93]. Below, we review the key theoretical results. We then present
measurements of Nc in our box trap and compare these with theoretical predictions.

5.1.1

Theoretical description

Within the local density approximation (LDA), the thermal density, that is, thedensity of particles in the excited states, is limited to nc = ζ(3/2)λ−3
T , with the
1/2

thermal wavelength λT = (2π~2 /mkB T ), as defined in Chapter 2. This result

extends trivially to a uniform system by multiplying both sides by the volume V .
In a uniform system, condensation occurs simultaneously throughout the volume.
The LDA can be extended to treat interactions between the thermal and condensed
atoms, which we address in 5.2. Here, we only need to note that the interactions do
not change the transition temperature Tc in a uniform system [23].
In a harmonically trapped gas, on the other hand, the density at the centre of71

5.1. TRANSITION TEMPERATURE OF A UNIFORM BEC

the trap is higher and exceeds nc before other regions, and the interactions affectthe transition temperature (among other properties).
However, the LDA explicitly neglects the effects of the low-momentum cut-off
due to the finite size of the system: the lowest eigenstates of the trapping potential
are far from being plane-wave states of the LDA. Thus, to account for the finite trap
size, we must re-examine the density of states description.
Semi-classical description
As explained in 4.3.1, the main difference between the LDA and the full quantummechanical description lies in the way we count occupancies of all the energy eigenstates, as seen from the LDA
ZZ
X
d3 pd3 r0
f [j ] ≈f [(p, r0 )](2π~)3j(5.1)(with the Bose factor f ). Here, the left-hand side corresponds to the (exact) sum
over all eigenstates j, while the right-hand side illustrates the phase-space integral
used in the LDA . Evaluating the sum on the left is rarely possible directly, while
the LDA can not, by definition, describe any effects of the finite trap size. On theother hand, the semi-classical approximation is often employed to write the sum on
the left asXZ
f [j ] ≈f []g()d,

(5.2)

j

with the density of states g() defined byZ Xg(0 )d0 =
1.
0

(5.3)

j:j ≤To count the states with energies not exceeding , we start by considering a cubic
box with side length L. The energy levels are given by
Enx ,ny ,nz = E0 (n2x + n2y + n2z ),

(5.4)

2 2

π ~
with ground-state energy E0 = 2mL
2 and integer quantum numbers nx,y,z ≥ 1. E ≤ corresponds to a volume in (nx , ny , nz ) of approximately 3/21 4π
V0 () ≈ ×
.
8
3 E0

(5.5)

The corresponding density of states isπ
g0 ()d = dV0 ≈
4




E0

1/2

d
.
E0

This result is equivalent to the one quoted in Chapter 2 for s → ∞.
72

(5.6)

5.1. TRANSITION TEMPERATURE OF A UNIFORM BEC

Finite size effects
ny

ny

1
0

1

nx

nx

nz
(a)

(b)

Figure 5.1: Origin of finite size corrections in a uniform box. (a) Values of nx ,ny contributing to the sum. The states (small circles) are shaded according to their
thermal occupation number, from black (high) to white (low). Dashed circle :  =const. (b) Origin of the higher-order corrections to the density of states: three planes(nx = 0, ny = 0, nz = 0), three lines (nx,y = 0, ny,z = 0, nz,x = 0) and the origin(nx,y,z = 0) (see text).

We now consider the above standard derivation of g() in more detail. In counting 1/2in the (nx , ny , nz ) space, we
the states contained within a sphere of radius E0have implicitly included those points with one of the quantum numbers equal to 0.
This is illustrated in Figure 5.1 (a), where only the allowed states (nx , ny , nz ≥ 1)
are marked. To correct, we can subtract those “states” with nx = 0 etc., which
occupy the planes shown in Figure 5.1 (b). Now, those states with say nx = ny = 0
will have been subtracted twice, which can again be accounted for by introducing
corrections for the states along the three axis (red in Fig. 5.1 (b)). Formally, one
last term can be introduced for over-counting the state at (0, 0, 0). This yields thecorrected expression"  # 2/2 1/2
3/2
1
4π



V () = ×
−3×π
+3×2
−1 .
8
3 E0
E0
E0
The corresponding density of states is"   −1/2 #1/2π3π 3 
d
g()d = dV =
−
+
.
4 E0
8
2 E0
E0
73

(5.7)

(5.8)

5.1. TRANSITION TEMPERATURE OF A UNIFORM BEC

Now, evaluating the integral (5.2) is non-trivial. When using g0 (), we simply set
Z = 1 and integrate over 0 ≤  ≤ ∞. This does not work with the second termin g() - the integral diverges. The problem has been treated extensively in [86].
As can be expected on dimensional grounds, the lower bound must be replaced byapproximately1 E0 , leading to the following form for the finite-size correction to thewell-known result (2.6):
Tc
1
2− 1 ≈ −√
(β0 E0 ) ln(β0 E0 ),Tc,03π ζ(3/2)
E0≈ 2.1N −1/3 .
β0 E 0 =kB Tc,0(5.9)(5.10)We have extended this treatment to account for the anisotropy of the trap. For a
0.12

Tc•Tc,0-1

0.1
0.08
0.06
0.04
0

2

4

6

5

N•10

Figure 5.2: Finite size correction. Box size is a = 2.4b = 2.4c = 63 µm. We show theprediction of the equation (5.11) (solid line) and results obtained by direct summation
over states.

trap with sides a, b, c, we find that the finite-size correction takes the form
2
1Tc
− 1 ≈ −s(a, b, c) √(β0 E0 ) ln(β0 E0 ),(5.11)Tc,0
3π ζ(3/2)
with the shape factors(a, b, c) =(abc)1/3.
((a−1 + b−1 + c−1 )/3)−1(5.12)

The finite-size correction is positive for realistic N , and is typically of the order of
5%. s is typically very close to unity: for a = 2b = 2c, s − 1 ≈ 5%. In Fig. 5.2, wecompare the asymptotic expression (5.11) with the critical temperature obtained by
explicitly summing the left-hand side of (5.2) numerically, and find good agreement.
1This applies to the first term of the integral as well. The corresponding correction is small due

to fast convergence of the corresponding integral near 0.

74

5.1. TRANSITION TEMPERATURE OF A UNIFORM BEC

5.1.2

Measuring the critical point
2 0 0
1 0 0

N

c

(1 0 3)

2 0 0
1 5 0

0

3 0
T (n K )

6 0
2 .0

1 0 0

αc

N

c

(1 0 3)

0

5 0

1 .5
1 .5

0
0

2 0

4 0

αf

2 .0

6 0

T (n K )Figure 5.3: Critical point for condensation (reproduced from [21]). A power-law fit,Nc ∝ T αc , gives αc ≈ αf = 1.65 (see text). Bottom inset: Within errors, αc = αf(solid line) is satisfied for 1.6 . αf . 1.7. Top inset: Comparison with theory of aperfectly homogeneous gas, with fixed αc = αf = 3/2.

While detecting small corrections is beyond our measurement accuracy, showing
the Nc ∝ T α scaling of the critical number also requires extra care. Experimentally,
detecting the transition point relies on measuring N0 , which in turn depends onthe value of the trapping parameter α (define in Chapter 2) chosen for fitting themeasured momentum distribution. Thus, we perform the analysis for a range ofassumed values, αf . We image clouds close to the critical point after 50 − 70 ms
of time-of-flight expansion from the trap. We vary T by changing the evaporation
sequence, and the total atom number N by changing the initial loading of the
trap. After forced evaporation we always raise the trap depth to ∼ kB × 0.4 µK,
where evaporation is negligible, and hold the gas for another 1 s to ensure thermal
equilibration before ToF. We analyze the images by fitting the thermal wings using
the assumed value αf , and identify the critical point with the appearance of small
BECs on top of the thermal profile, with N0 ≤ 5000. Both N and T at the criticalpoint depend on the αf used. We plot Nc (T ) and fit the data with a function of
the form Nc ∝ T αc , deducing the best value for αc from least-squares fitting. The
bottom inset in Fig. 5.3 show αc vs αf . Self-consistency requires αc = αf , whichis the case for 1.6 . αf . 1.7. This suggests that our trap deviates slightly fromthe ideal box, with α = 1.65 corresponding to s = 20. At the same time, the dataagrees reasonably well with αf = αc = 3/2, as shown in the top inset of Fig. 5.3.
Deviations from the ideal box can be alternately attributed to a slight variation of75

5.2. SATURATION OF THE THERMAL GAS IN A BOX
the effective trapping volume V with temperature, V ∝ T α−3/2 . From in situ images,
we indeed measure a (20 ± 10)% increase in the cloud volume between T = 25 and50 nK, which is consistent with our α.
We now turn to the absolute value of Nc . In line with the ideal box assumption,
this is best described in terms of the critical phase space density ρc = Nc λ3T /V . The
data in Fig. 5.3 agrees with ρc = 2.0 ± 0.2, while the LDA value is ρc = 2.612. With
the finite size correction, we expect ρc ≈ 2.4. The small remaining discrepancy
is within our systematic uncertainties of 10% in Nc (resulting from the absolute
number calibration described in Chapter 3) and 20% in V (volume measurements
are described in Chapter 4).

5.2

Saturation of the thermal gas in a box

Next, we turn to studying partially condensed clouds. In Einsteins ideal-gas pictureof condensation, Nth saturates at the critical point and can never exceed Nc . In a
harmonically trapped gas, this picture is strongly violated even for weak interactions, such as in 87 Rb, and can be recovered only by extrapolating to the strictly
noninteracting limit [76], where direct measurements cannot be performed due tothe absence of thermal equilibrium [94]. This strong deviation from ideal-gas behavior arises due to an interplay of interactions and the nonuniformity of the gas,and can be explained by using a mean-field theory that does not require violationof the saturation picture at the level of the local thermal density [91].
While interactions have no effect in the critical point in a uniform system (within
LDA), they can change the number of available thermal states when a condensate
is present. We first consider the effects of mean-field interactions on the saturation
in a uniform system, before turning to experimental measurements of Nth below thecondensation point.

5.2.1

The Popov approximation

Within the LDA, we define the condensate and thermal densities, n0 and nth . A
2athermal atom experiences the mean-field energy 2γn, where γ = 4π~
and n =
mn0 +nth , while the condensate atoms are subject to the mean-field energy γn0 +2γnth[23]. Within the semi-classical Popov approximation, valid for all temperatures
below (but not too close to) Tc [23,95–99], the effect of interactions can be described76

5.2. SATURATION OF THE THERMAL GAS IN A BOX

by a set of equations for n0 and nth :p2+ γn0 ,2m
1/2
(p) = 2a (r, p) − (γn0 )2,(5.13)

a (p) =(5.14)µ = γn0 + 2γnth + U,Z a (p) d3 pnth = g0 e−β(p).
(p) (2π~)3(5.15)(5.16)(5.17)

These can be solved self-consistently for a range of µ using suitable numericals = 15

1

1

0.9

0.9

Nth •Nc

Nth •Nc

s=∞

0.8
0.7

0.8
0.7

0

0.2

0.4
N0•N

0.6

0.8

0

0.2

s=7

0.4
N0•N

0.6

0.8

0.6

0.8

s=2
1.6

1

Nth •Nc

Nth •Nc

1.8

1.4
1.2

0.9

1
0

0.2

0.4
N0•N

0.6

0.8

0

0.2

0.4
N0•N

Figure 5.4: Numerical solutions of the Popov equation in different power-law potentials. We mark thermal clouds with atom number up to Nc (red) and partiallycondensed clouds with increasing condensate fraction (blue). Nth decreases with N0
for large s and increases for small s. Only for s ≈ 7 is the cloud saturated (note thedifferent scale).

methods [100]. Once we have the µ-dependent LDA solution, we can easily determine
N0 and Nth for an arbitrary trapping potential V (r) by integration. In Fig. 5.4 weshow such solutions for a range of µ. An unfortunate feature of the equations
above is that their solutions are discontinuous near the transition point, as seen in77

5.2. SATURATION OF THE THERMAL GAS IN A BOX

the sudden increase in condensed fraction as µ exceeds 0. In any finite power-law
potential, the discontinuity disappears, because the onset of condensation is not
simultaneous throughout the trapping volume. In quasi-uniform traps (s  2),the theory predicts a decrease in Nth as N0 increases - in contrast to the s = 2(harmonic) trap [28].
The dominating effect in a uniform potential is due to the modification to thedispersion relation due to the finite condensate density, as described by (5.14). This
leads to a change in the density of states, and thus a decrease in the maximum
thermal density. In a harmonic trap, on the other hand, the effect is geometric:
the condensate, centered near the potential minimum, forces the thermal cloud intoa “mexican-hat” potential, thus modifying the density of states in the opposite
direction on a global rather than local level.
It has to be noted that the exact shape of the potential is irrelevant within the
LDA - the saturation behaviour is determined by the volume distribution of trappingR
potential values, i.e. PU (u)du = δ (U (r) − u) d3 rdu. Thus, the effect of finitely
steep walls is indistinguishable from floor noise of the same height in the central
region of the trap.
Further, it has been shown [28, 76] that beyond-mean-field effects contributesignificantly to the non-saturation in harmonic traps. Their influence in uniform
traps is unclear and remains to be studied in the future.

78

5.2. SATURATION OF THE THERMAL GAS IN A BOX

5.2.2

Experimental results

h a r m o n ic
b o x

2

1 .3

ac

N ' / N

c

1 .6

1.5
1 .0
0

1 0
2 0
C o n d e n s e d F r a c tio n ( % )

1.5

3 0

2

2.5

3

af

(a)

(b)

Figure 5.5: (a) Saturation in a uniform Bose gas (adapted from [21]). The quasiuniform clouds have Tc = 45 ± 1 nK. The red data points were taken in the harmonic
trap used to load our box, with T ≈ 110 nK and Nc ≈ 65 × 103 . The dashed redline is the theoretical prediction from the previous section (also described in [16, 76]).
(b) Self-consistency check for partially condensed clouds. For each αf used for image
analysis, we determine αc from the fit Nth ∝ T αc to the data (see text).

To study saturation, we use a variation of the experimental methods described in
5.1.2. First, we prepare degenerate samples with the condensate fraction of up to
40%. We then hold the cloud in the box trap at a constant temperature T . Thisis achieved by choosing a suitable box depth. Usually, our temperature is stable
to ±3%. We also post-select our data to exclude temperature outliers. As theatom number decays due to background collisions, we obtain clouds with a varying
condensed fraction N0 /N . Consistently with 5.1.2, we analyze the data assuming
α = 1.65. We plot the results in Figure 5.5(a) (blue points).
It can be seen that the gas is saturated to within our experimental errors. Detecting the mean-field effects described above would require a significantly higher
precision. For comparison, we show the strongly non-saturated thermal fraction in
the harmonic trap, which shows significantly non-saturated behaviour.
Using our results for the critical atom number, saturation is equivalent to Nth =Nc (T ) ∝ T α . Having checked this scaling at the transition, we now want to confirm
its validity for all partially condensed clouds. Again, we require self-consistency
between the value αf used for image analysis, and the value αc providing best leastsquares fit to the data. The result is shown in Figure 5.5 (b). We can see that thedata are consistent with ideal uniform gas theory, while power-law potentials with79

5.3. ENERGY AND HEAT CAPACITY

10

N0•N

c2

0.4

5
0
1.5

2

0.2

2.5

3

a

0
0.6

0.8

1

1.2

T•Tc

Figure 5.6: Condensed fraction of a partially condensed cloud in a quasi-uniform
trap. We show data analyzed with α = 1.5 (red), 1.73 (magenta) and 3 (blue). The
data points and error bars are deduced from a running average (raw data shown inthe background). Inset: χ2 values for a range of α. α = 1.73 produces excellent
agreement, while α = 1.5 is also very good.

s < 15 are strongly excluded.
A direct implication of saturation is the scaling of the condensed fraction withtemperature [23]:
Nth
Nc
N0
=1−
=1−
=1−
N
N
N



T
Tc

α
.

(5.18)

Keeping Tc constant while varying N0 is impractical, but we can still define Tc foreach N and plot the data accordingly. Again, we perform this analysis for different
α. Varying α changes both the data and the theoretical prediction, and we requireself-consistency between the two, as shown in Fig. 5.6.
Again, we see excellent agreement for α ≤ 1.75, seen both visually and by comparing χ2 . This is, of course, a direct consequence of the previously tested scalingof the critical number with temperature, and of saturation.

5.3

Energy and heat capacity

Next, we consider the energy E of the expanding condensate. The energy andits derivative, the heat capacity C = dE/dT |V , are of particular interest as their
behaviour near Tc changes qualitatively between uniform and harmonic systems:
while C is continuous in uniform traps (α = 3/2), it exhibits a discontinuity inharmonically trapped gases (α = 3, and, in fact, for all α > 2) [23]. ToF imaging
80

5.3. ENERGY AND HEAT CAPACITY

offers a way to extract the kinetic energy of the condensate in a model-free way. We
will consider how this relates to the total energy, and quantify the role of interactions.

5.3.1

TheoryWe start by considering an ideal gas. Using the results from Chapter 2, we can write
the energy of a non-interacting Bose gas as a function T /Tc , in the spirit of (5.18): αζ(α+1)Tα, TTc ≤ 1
E
ζ(α)
Tc
.
(5.19)=
α gα+1 (Z) ,N kB TT>1
gα (Z)

Tc

Note that this energy term corresponds to the kinetic energy of the thermal atoms
only (see below). In the case TTc > 1, Z has to be determined by numerically solving
−αTgα (Z) = ζ(α).
(5.20)Tc
At high temperatures, we recover the well-known classical result E = αN kB T . E
includes contributions from both kinetic and potential energy. In a typical experimental sequence, we switch off the external potential and let the cloud expand freely.
Thus, only the kinetic energy is accessible in our measurements. The ratios of the
kinetic and potential energies to the total energy E are given by the virial theorem:
3/2E,
αα − 3/2EU =E.
αEkin =(5.21)(5.22)In the limit α → 3/2, we can thus measure the total energy of the cloud as Ekin
(EU = 0).
Next, we consider the contribution due to mean-field interactions. Unlike thepotential energy, the interaction energy Eint is converted into kinetic energy as thecloud expands. By writing Eint /N kB = γn/kB , we estimate that the interactionenergy per atom corresponds to ∼ 1 nK for 105 atoms in a typical trap. This
contribution is negligible.
Finally, the kinetic energy of the condensate atoms is significantly smaller than
Eint - a key assumption within the Thomas-Fermi approximation.

5.3.2

Extracting energy from ToF images

In 4.3, we explained how the momentum distribution of the cloud maps to the atomic
density in ToF. We define the “point source” atomic density n∞ (r, t) ≡ n(p = mr/t),
81

5.3. ENERGY AND HEAT CAPACITY

which describes the expansion from a trap of negligible size (or at large t), and thenormalized in situ density K(r). The kinetic energy can be expressed as the integral
over the cloud’s momentum distribution n(p), or, equivalently, in terms of the second
moment of n∞ (r):Z
2m
Ekin = 2
n∞ (r, t)r2 d3 r.
(5.23)t
For sufficiently steep trapping potentials and finite times, the atomic density after
time t is given by the convolutionn(r, t) = n∞ (r, t) ∗ K(r).

(5.24)The second moment of the finite-t density distribution is not the sought energy.
However, we can make use of a general theorem about the 2nd moment of the
convolution 2 [101]: assuming all density distributions are centrally symmetric, wefind the exact result
Z
Z
Z
Z
Z
2 3
2 3
3
3
n(r, t)r d r = n∞ (r, t)r d r K(r)d r + n∞ (r, t)d r K(r)r2 d3 r (5.25)
Z
t2=
Ekin + n∞ (r, t = 0).
(5.26)
2m
Thus, correcting for the initial cloud size is possible by writing

Z
Z
2m
2 3
2 3
n(r, t)r d r − n(r, t = 0)r d r .
Ekin = 2t(5.27)

At fixed N , the initial size correction grows quadratically with the linear size of thetrap, and can be significant for larger boxes at typical ToFs.
Hidden direction
In practice, we can only measure the column density ñ(x, y). We can thus extract thesecond moment w.r.t. x and y, but not z. Luckily, our trapping geometry possessesRcylindrical symmetry around the x = z = 0 axis, and this gives n(r)z 2 d3 r =Rn(r)x2 d3 r. For each term in (5.3.2), we can thus writeZZ
2 3n(r)r d r = ñ(x, y)(2x2 + y 2 )dxdy.
(5.28)In this form, we can extract the kinetic energy by direct summation over the opticaldensity images.
2(h)Generally, for h = f ∗ g, the pth moment of h satisfies mp=

P (g) (f )p
0≤k≤p k mk mp−k . We use

the special case p = 2, and m1 = 0 for centrally symmetric distributions.

82

5.3. ENERGY AND HEAT CAPACITY

Correcting for residual potential curvature
Above, we have assumed that momentum p maps to position as p → r = (x, y, z) =tp/m. This is only true in absence of any external fields during ToF. As explainedin 4.1.2, our gravity compensating setup introduces a harmonic trapping frequency
ωr ≈ 1.6 Hz in the horizontal (y) direction3 . Thus, the mapping is modified top → r = (x0 , y 0 , z 0 ) with x0 = x, y 0 = sinc(ωr t)y and z 0 = sinc(ωr t)z. The corrected
expression for the kinetic energy thus becomes


Z
2m
y2
2
Ekin = 2dxdy−ñ(x, y, t) 2x +tsin(ωr t)
Z


2m
− 2
ñ(x, y, t = 0) 2x2 + y 2 dxdy.
t

5.3.3

(5.29)

Results

We now use the optical density images from 5.2.2 to extract energies using (5.29). In
practice, the noise far from the cloud center is amplified significantly due to the r2
scaling. We thus replace the pixel values in the thermal wings outside the exclusion
region (as defined in 4.3.3) by the best-fit model values to ensure convergence. At
t = 50 ms, the correction to energy due to the residual harmonic trapping is ≈ 10%
in the x-direction. We plot the kinetic energy per particle against T /Tc in Fig 5.7,
and compare it with the prediction (5.21). As in Fig. 5.6, the choice of α affects boththe theory and the data. The Ekin plot indicates that the analysis is particularly
sensitive to the value of α, and in fact excludes α = 3/2 for the data set shown,giving an estimate for α very close to that obtained in 5.1.2.
Our data is also consistent with the prediction that the heat capacity is continuous. At the same time, it has proven challenging to detect the discontinuity in thefirst derivative of energy in harmonic condensates [102]. Another way to approach
the problem is to note that dµ/dT = 0 when CV is continuous, and vice versa. However, determining µ from the shape of thermal clouds is extremely unreliable, andour data is not accurate enough to distinguish between µ approaching zero linearlyand quadratically.
3

Gravity compensation during long ToF is necessary due to the limited size of our CCD: thecloud falls outside the imaging area otherwise.

83

5.4. QUANTUM JOULE-THOMSON EFFECT IN A BOSE GAS
1.5

Ekin•HNk BTL

c2

20

1

10
0

1.5

2

2.5

3

a

0.5

0

0.5

1
T•Tc

Figure 5.7: Kinetic energy of a Bose gas in a quasi-uniform trap. We show data
analyzed with α = 1.5 (red), 1.73 (magenta) and 3(blue). The data points and error
bars are deduced from a running average (raw data shown in the background). Inset:
χ2 values for a range of α. α = 1.73 produces excellent agreement, while α = 1.5 isalso very good.

5.4

Quantum Joule-Thomson effect in a Bose gas

Finally, we consider the long-time evolution of a partially condensed Bose gas held inthe (uniform) box trap. We typically prepare clouds at Ti ≈ 45 nK in a trap of depth
U0 ∼ 10 − 20kB Ti , chosen such that evaporation is negligible. As the gas is held in
the trap, atoms are slowly removed from the cloud, with an exponential time scaleτ ≈ 10 s, by collisions with the background gas in the vacuum chamber. Becauseof low atomic density (n < 5 × 1012 cm−3 ), the three-body recombination rate is
negligible [30]. As N decays, the elastic collision rate (among the trapped atoms)
remains sufficiently high for the gas to continuously reequilibrate on a time scaleτeq . 2 s  τ [103]. Crucially, collisions with the background gas are independentof the energy of the trapped atoms, so the average energy per trapped particle,
E/N , remains constant. Within the ideal-gas approximation, pressure is simply
given by the energy density, so enthalpy H is simply proportional to E, and thespecific enthalpy h = H/N is a conserved quantity. The decay of the cloud is thus
equivalent to the Joule-Thomson isoenthalpic rarefaction. In a classical ideal gas, hdepends only on T , so temperature cannot change in an isoenthalpic process. Thisis, however, not true for a quantum degenerate ideal gas [104]. The resulting change
in temperature in a Joule-Thomson process is known as the Joule-Thomson effect.
We now show that the ideal Bose gas cools during isoenthalpic rarefaction, and84

5.4. QUANTUM JOULE-THOMSON EFFECT IN A BOSE GAS

compare the effect with that seen in classical gases.

5.4.1

Joule-Thomson effect in partially condensed clouds

To include both ideal box and power-law traps in our description, we first recall
the definition of enthalpy. In a uniform system, H = E + pV . While in a moregeneral trap we have to replace p, V by a suitable pair of conjugate variables (withV describing the trapping geometry), we can always write H = G + T S, where
G = N kB T ln(Z) is the Gibbs free energy. Using results from Table 2.1, we find
that
α+1
E.
(5.30)αIn a uniform ideal gas, this also follows directly from E = 23 pV . Thus, keepingH=

E/N constant does indeed correspond to an isoenthalpic process (h = const). For
partially condensed clouds, we can writeh=H
Nc kB T
T α+1∝∝
= const.
N
N
N(5.31)The last equation shows that, as N decreases, T follows a simple power law. Thisscaling is only valid while the condensed fraction is non-zero.
Consider a cloud with initial temperature Ti , total atom number Ni and thermal number Nth,i . While isoenthalpy implies N ∝ T α+1 , the thermal number issaturated, and therefore Nth ∝ T α . The thermal fraction thus follows
Nth∝ T ∝ N 1/(α+1) .
N(5.32)As the thermal fraction increases to 1, the condensate disappears. This happens
when
N
=
Ni

5.4.2



Nth,i
Ni

α+1
.

(5.33)

Experimental results

An experimental confirmation of the above scaling is shown in Fig. 5.8. We rescaletemperatures and numbers to their values at the beginning of hold time. We show theevolution of a partially condensed sample (blue points, Ni = 1.7 × 105 , Ti = 46 nK)
and a thermal sample (red points, Ni = 0.7 × 105 , Ti = 52 nK). Each point isobtained by averaging over five repeats of the experimental sequence. The datafor partially condensed clouds is indistinguishable from the uniform-trap scaling
T ∝ N 1/2.5 (dashed blue line in the figure). While our theory describes the data85

5.4. QUANTUM JOULE-THOMSON EFFECT IN A BOSE GAS

1.0

æ

æ

à

à

0.8

à
à
à

T•Ti

à

0.6

à

æ

æ

à

æ

æ
à

à

à
æ

à

à

à

à

à
à

0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

N•Ni

Figure 5.8: Quantum Joule-Thomson effect in a partially condensed (blue) andthermal (red) clouds. The solid and dashed blue lines are predictions of Eq. (5.31)
with α = 1.65 and α = 3/2, respectively.

well, we have to address several features for completeness. First, our partially condensed clouds start with a condensed fraction of 30 ± 5%. According to (5.33), thecloud should be purely thermal below N/Ni = 40 ± 6%, while we see continued
cooling. Second, the thermal sample prepared with N/Nc . 50% also exhibits somecooling, which could be mis-interpreted as a consequence of evaporation. Thus, wemust consider the isoenthalpic evolution of thermal samples, as well as the effect ofexternal cooling (or heating).

5.4.3

Joule-Thomson cooling in a thermal Bose gas

It is instructive to recall the energy plot from Fig. 5.7, where we plotted (up toa constant factor) E/(N T ) vs. T /Tc . With E/N = const and Tc ∝ N 1/α , thisbecomes a plot of 1/T vs. T /N 1/α . As N decreases, the cloud is thus constrainedby the energy curve in a way that requires an associated decrease in T , simply due
to the fact that E/(N kB T ) increases with T /Tc as the fraction of particles with zeroenergy decreases. At very high temperatures, the cloud behaves classically and (inabsence of interactions) the temperature remains constant along the isoenthalp.
Using the energy plot, we can predict the evolution of a thermal cloud by solving
E/N = Ei /Ni numerically. This allows us to extend our treatment to clouds bothabove and below condensation. We show the results of numerical calculations in
Fig. 5.9 for the starting conditions corresponding to our two samples. Interestingly,
while the condensate is depleted below N/Ni ≈ 0.45, the deviation from (5.31) is
minimal down to N/Ni ≈ 0.15. Further, our parameter-free model correctly predicts
86

5.4. QUANTUM JOULE-THOMSON EFFECT IN A BOSE GAS

the slight cooling of the thermal sample.
1.0

æ

æ

æ

æ
à

à

0.8

à
à
à

T•Ti

à

0.6

à

à

æ

æ
à

à

à
æ

à

à

à

à

à
à

0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

N•Ni

Figure 5.9: Quantum Joule-Thomson effect in a partially condensed (blue) andthermal (red) clouds. The solid blue line is as in Fig. 5.8, while the dashed blue andred lines are numerical calculations described in the text. The shaded areas show theeffect of heating or (evaporative) cooling between ±5 nK/lifetime (see text).

5.4.4

Effect of heating and evaporation

The starting point of our treatment has been the assumption that loss of atoms fromthe trap is independent of their energy. This can be written as a rate equation of
the form
EdN.
(5.34)
N
We now seek to estimate potential contributions due to external heating (e.g. due todE =glancing collisions [103] or scattered trapping light) or cooling (due to evaporation).
We add a heating term to the above equation and writedE =EdN − αN kB T̃ dN.
N(5.35)The last term corresponds to a constant heating rate per atom. Assuming that theatoms are lost on a time scale τ , we can write dN = −N dt/τ , and a sample far
from condensation (and thus with heat capacity αN kB ) would experience a change
in temperature of T̃ in time τ . Solving this equation, we find
 
E
Ei
Ni
.
=
− αkB T̃ ln
N
Ni
N(5.36)

This no longer describes isoenthalpic evolution, but allows us to extract T numerically, as described above. For partially condensed clouds, we find an analytic result875.4. QUANTUM JOULE-THOMSON EFFECT IN A BOSE GAS

modifying (5.31):


T
Ti

α+1

N
T̃ N
=
−
ln
Ni Ti Nth,i



N
Ni


.

(5.37)In Fig. 5.9, we show predictions derived from (5.36) for −5 nK ≤ T̃ ≤ 5 nK as shaded
areas. This corresponds to heating rates between ±0.5 nK/s. It can be seen thatour data strongly excludes higher heating (or cooling) rates. Especially for small
N/Ni , we expect our data to be extremely sensitive to heating. At the same time,
our temperature fits become less reliable due to decreasing optical densities of theclouds.

5.4.5

The Joule-Thomson coefficient

In classical gases, one commonly defines the Joule-Thomson coefficient


∂T
µJT =
.
∂P h
For an ideal partially condensed Bose gas in a uniform trap, we can write
3 N ∂TµJT = V.
2 E ∂N h(5.38)(5.39)

More conveniently expressed, the derivative of the scaling (5.31) (dashed blue line
in Fig. 5.8) corresponds to [23]
2
λ3T
µJT =
.
5ζ(5/2) kB(5.40)

At our lowest temperatures, µJT ≈ 4 × 109 K/bar, about 10 orders of magnitudelarger than observed in classical gases.
To understand the origins of this enhancement, on dimensional grounds we writeµJT = cJT T /P , so cJT is dimensionless and T /P ∼ 1/(nth kB ), where nth is thethermal particle density. The value of cJT depends on the equation of state. In
the ideal gas, cJT = 0 in the classical limit, while cJT = 1/(α + 1) for a saturated
partially condensed Bose gas. Interaction corrections to cJT are essentially given by
the ratio of interaction and thermal energy. In the partially condensed sample, we
observe a drop in T by ≈ 22 nK. Meanwhile, the interaction energy per particle isalways smaller than (8π~2 a/m)(Ni /V ), corresponding to under 4 nK. The change
in T therefore must predominantly be a purely quantum-statistical, rather thanan interaction, effect. Moreover, note that for a > 0 the drop in the interactionenergy should lead to slight heating rather than cooling of the gas. In our case88

5.4. QUANTUM JOULE-THOMSON EFFECT IN A BOSE GAS

the interaction corrections are unimportant and cJT ≈ 2/5, but for a classical gas
they are the only origin of the JT effect and typically cJT . 10−3 . Another large
3/2
enhancement of µJT comes from the fact that in a saturated gas nth ∼ λ−3
T ∼ Tdecreases with temperature, and our lowest nth is ∼ 107 times smaller than atambient temperature and pressure. Combining the differences in cJT and 1/nth ,we recover the ≈ 1010 -fold increase in µJT compared to classical gases. It is alsointeresting to note that µJT in Eq. (5.40) explicitly vanishes in the limit ~ → 0,
reiterating that this is a purely quantum effect.
On a final note, the sign of µJT in a bosonic gas is opposite to that in an ideal
Fermi gas [105].

89

Chapter 6
Spectroscopic Studies of a
Uniform BEC
Time-of-flight imaging allows measurements of the momentum distribution of weaklyinteracting thermal atoms in a partially condensed cloud, and is the standard technique for BEC thermometry. However, this method can not be applied to study
the BEC ground state in momentum space for two reasons: first, the achievable
resolution is limited by our imaging system (see 3.6), and second, the momentum
distribution evolves during the time of flight due to both interactions and the residual external fields (see 5.3.2). An alternative measurement technique described inthis Chapter is Bragg spectroscopy.
Bragg scattering of atoms by a standing light wave [106, 107] is a fundamental
prediction of quantum theory. Following the observations of light scattering byatomic lattices, i.e. crystals [108], as well as scattering of massive particles byatomic lattices [109], it strikingly complemented other evidence for wave-particle
duality.
Following the demonstration of Bose-Einstein condensation, experimental focus
shifted from the scattering of atomic beams by a standing wave towards the scattering of stationary ultra-cold atoms by a moving optical lattice [110–112].
Bragg spectroscopy has allowed measurements of the relative occupation of thedifferent momentum states in a degenerate gas [110] as well as the dynamic structure
factor [111]; it has also been used to directly measure the dispersion relation over awide range of momenta [113–115].
In a harmonically trapped condensate, the broadening of the Bragg spectrum due
to mean-field interactions exceeds by far the momentum width of the condensate
wave-function [110]. We shall see that this is not the case in our uniform BECs,
91

6.1. THEORY OF BRAGG SPECTROSCOPY

allowing a direct measurement of the momentum occupation function. We start bydeveloping a theoretical description of Bragg spectroscopy for large recoil momenta,
before presenting our measurements.

6.1

Theory of Bragg spectroscopy

In this chapter we present measurements of the momentum distribution of quasipure uniform BECs using Bragg scattering. Bragg-scattered atoms separate fromthe stationary BEC in time-of-flight if the recoil momentum acquired by scatteredatoms, qr , exceeds the width of the BEC in momentum space, ∆p ∼ ~/L. At thesame time, having a smaller recoil momentum improves the momentum resolution.
The Bogoliubov dispersion relation predicts essentially free-particle behaviour at
momenta & ~/ξ, where ξ is the healing length (∼ 1.5 µm for 105 atoms in a typicalbox trap). Thus, we choose qr > ~/ξ > ~/L to be able apply the free-particle
dispersion relation. On the other hand, measurements of the Bogoliubov spectrum
in future experiments would require ~/ξ > qr > ~/L. In the measurements describedin this Chapter, qr ≈ m × 3 mm/s ∼ ~/(0.2 µm) - about an order of magnitude
higher than ~/ξ. As we shall demonstrate below, the recoil momentum is mainly
determined by the beam geometry. In our system, this means that the range ofachievable qr is limited by the optical access available in our vacuum system.
We start by considering the interaction of atoms in the condensate with two
Bragg beams intersecting at an angle of 2θ (see Fig. 6.6 and Fig. 6.7 below). Assume
the two beams have wave vectors k1 and k2 = k1 − kr and frequencies f1 and
f2 = f1 − f . The fractional frequency difference f /fj = kr /kj is minimal (for 780 nm
light it is typically ∼ 1 kHz/384 THz = 2.6 × 10−12 ). We can write k = k1 ≈ k2 ,andkr is determined by the intersection angle 2θ:
kr = 2k sin(θ).

(6.1)

The corresponding recoil momentum acquired by an atom absorbing a photon frombeam one and re-emitting a photon into beam two is given by
qr = ~kr ∝ sin(θ).

(6.2)

Consider an atom with initial momentum pi and energy Ei . The scattered atom has
momentum pf = pi + qr , and thus energy
Ef =

p2f
pi · qr= Ei + Er +
,
2m
m
92

(6.3)

6.1. THEORY OF BRAGG SPECTROSCOPY
q1, hf1
-qr/2
qr
+qr/2

qr
q2, hf2

+qr/2

x
E

x
E

Δ

x
δ

E

δ

hf0
qr
(a)

qr

p
(b)

p

-qr/2

+qr/2

p

(c)

Figure 6.1: Left to right: three equivalent theoretical descriptions of Bragg scattering in real space (top) and in momentum space (bottom). (a) Single-atom picture.
An atom undergoes a Raman transition by scattering a photon from beam 1 into
beam 2 (top). Energy and momentum conservation must be satisfied in the transition (bottom). Both beams are far from resonance (∆  f0 , δ). Detuning δ isdue to the Doppler shift or frequency shift f1 − f2 − f0 (see text). (b) Schrödinger
picture in the lab frame. Two crossed beams create a moving lattice (top). Thiscan be seen as a time-dependent perturbation, causing scattering between momentum eigenstates, 0 → qr (bottom). (c) Schrödinger picture in the stationary latticeframe. The condensate is initially moving left, and is Bragg-reflected by the lattice(top). In the free-particle basis, momentum changes by qr (dashed lines, bottom).
In the basis of Bloch eigenstates, the momentum is restricted to the first Brillouinzone (shaded), and remains unchanged. We show the first two bands of the dispersion
relation. Bragg scattering is described by the free evolution of the Bloch vector in the
two-state basis, with the level splitting given by the band gap. Resonance condition(δ = 0) corresponds to the initial momentum being on the edge of the Brillouin zone(see text).

with recoil energy Er ≡ qr2 /(2m). The transition is resonant when the photon energy
difference equals

pi · qr(6.4)
m
The change in E and p due to the two-photon process is shown in Fig. 6.1 (a). We
hf = hf0 (pi ) ≡ Er +define the detuning δ ≡ f − f0 . As can be seen from the above resonance condition,
the detuning can be seen to arise due to either the Doppler shift [second term in93

6.1. THEORY OF BRAGG SPECTROSCOPY(6.4)] or the change in frequency f .

6.1.1

Raman picture

To calculate the transition rates for a particular δ, we must address the atomic
nature of the process. Bragg scattering is effectively a Raman transition - bothphotons are off-resonant, and the two-photon transition occurs via a virtual level,
as indicated by a dashed line in Fig. 6.1 (a). Thus, the transition can be describedby a single effective Rabi frequency [41]
Ω=

Ω1 Ω2
I Γ2
=
,
2∆
Isat 4∆(6.5)

where Ω1 = Ω2 are the Rabi frequencies of the single-photon transitions, ∆ is thedetuning of the individual beams from the atomic transition, I = I1 = I2 is theintensity in each beam, Isat is the saturation intensity and Γ the linewidth of theatomic transition. Thus, after subjecting the atoms to the Bragg beams for time τ ,we expect the transferred fraction to beΩ2
P (pi → pi + qr ) = 2
sin2Ω + δ2
with



1√ 2
Ω + δ2τ
2


,

(6.6)

δ = f − f0 (pi ).

(6.7)

Next, we extend our treatment to a cloud with the momentum distribution |Ψ(p)|2(Fig. 6.1 (b), bottom panel). We can predict the total number of transferred atoms
by integrating over all momenta; with the momentum wavefunction Ψ(p) normalized
to N , we expect the number of Bragg-diffracted atoms to beZ
NB = |Ψ(pi )|2 P (pi → pi + qr )d3 pi .

(6.8)

The integrand depends on the component of p parallel to qr only. The Doppler shiftterm in (6.4) suggests that we can write momentum in Bragg frequency units via
qr
p.
hm

p̃ =

(6.9)

Similarly, we introduce the recoil frequency fr = Er /h. Expression (6.8) becomes a
convolution:NB (f ) =

Ωτ
2

2 Z

2

2



|Ψ| (p̃) × sinc

1p 2
Ω + (f − fr − p̃)2 τ
2


dp̃.

(6.10)

So far, we have considered population transfer (wavefunction amplitudes) only, using
equations (6.5) and (6.6). These are standard results of time-dependent perturbation theory, and make use of the rotating wave approximation. Within the same94

6.1. THEORY OF BRAGG SPECTROSCOPY

framework, it is possible to find the phase evolution of each plane-wave eigenstate,
and thus the evolution of Ψ(p) during the Bragg pulse. We shall obtain these results
using a different approach, highlighting similarities with optical lattice experiments
and Bragg scattering in solid state physics.

6.1.2

Moving lattice picture

It is instructional to consider the (classical) light field created by the two beams.
Representing both beams by plane waves with equal amplitudes ε, and assuming
the polarization vectors of the two beams are parallel, we can write
Ej = εei(kj ·r−ωj t)with j = 1, 2,


k +k
ω +ω
k2 − k1
ω2 − ω1i( 1 2 2 ·r− 1 2 2 t)
E = E1 + E2 = 2εe
cos
·r−
t .
2
2

(6.11)(6.12)

Expression (6.12) can be seen to represent an optical field with the slowly moving
envelope|E| = 2ε cos1(kr · r − 2πf t) ,2

(6.13)

with kr = qr /~. The rotating wave term (the phase term in E) is off-resonant,
and we can describe the effect on the Hamiltonian by the AC Stark shift potential(3.5.3), which is proportional to the single-beam light intensity I ∝ ε2 . This is of
the form

I Γ2
U (r, t) = ~× 4 cos2
Isat 4∆
1(kr · r − 2πf t) .
2

(6.14)

Comparing with (6.5), we can write:
U (r, t) =~Ω
[1 + cos (kr · r − 2πf t)] .
2(6.15)We see that the potential is that of a moving 1D optical lattice with amplitude ~Ω,as shown in Fig. 6.1 (b) (top panel). The lattice period is Λ = 2π/kr , and the lattice
velocity is vl = 2πf /kr . By changing the frequency detuning f alone, we can vary
the lattice velocity, and even simulate BECs in accelerating frames [116, 117].

6.1.3

Bloch picture

Consider atoms initially at rest in the lab frame (pi = 0). The resonance condition(6.4) simplifies to f0 = Er , and the lattice velocity becomes vl = vr /2, where vr =qr /m is the recoil velocity. Upon scattering, the resonant atoms acquire twice thelattice velocity. This result is obvious in a frame of reference where the lattice is95

6.1. THEORY OF BRAGG SPECTROSCOPY

stationary: in this moving frame, the atoms, initially moving at −vr /2, are reflectedby the lattice and acquire an opposite velocity of vr /2, as shown in Fig. 6.1 (c).
In the lattice frame, we can treat the lattice as a time-independent perturbation
to the free-particle Hamiltonian. Moreover, it is periodic, and in our case contains
only few Fourier components:
X

U (x) =uG eiGx(6.16)~Ω.
4(6.17)

G∈Zkr

with u−kr = ukr = u =The eigenstates of the periodic Hamiltonian can be found explicitly. These are givenby the Bloch functions [118] of the formXΦ(x) =
ck+G ei(k+G)x(6.18)G∈Zkr

The lattice momentum k is restricted to the first Brillouin zone, −kr /2 ≤ k ≤ kr /2.
The coefficients c, as well as the energies , can be determined from the Schrödinger
equation, which takes the formX(k − )ck +uG ck+G = 0,

(6.19)

G∈Zkr

with k = ~2 k 2 /(2m). Because uG is only non-zero when G = ±kr , the Hamiltonianis tridiagonal in matrix notation, with coupling between k and k ± kr states only. In
practice, we only need to consider a small number of k. Initially, only k-states closeto −kr /2 are occupied. First-order transition can be describe by including stateswith momenta k and k + kr only. Including other k is equivalent to considering
four-photon (and higher-order) transitions.
Expressing momenta in units of kr /2 and energies in units of Er /4 = (kr /2)2 /(2m),the Hamiltonian (for k < 0) takes the form
H=

k2

u

u

(k + 2)2!
,(6.20)

with eigenenergiesr± = k 2 + 2 k + 1 ±u2+ (k + 1)24!
(6.21)and eigenvectors
c±
1

!

c±
2
96

.

(6.22)

6.1. THEORY OF BRAGG SPECTROSCOPY

These corresponds to the first two Bloch bands, with wavefunctionsikx
i(k+kr )x
Φ± (x, t = 0) = c±
+ c±
1e
2e

(6.23)

Now, we can describe the time evolution of the BEC wavefunction in the Blochbasis. The Bloch states evolve freely in time, according to1
Φ± (x, t) = Φ± (x, 0)e−i± t .

(6.24)The BEC wavefunction in the lab frame, Ψ(k), transforms to the stationary-latticeframe as Φ(k) = Ψ(k + k0 ), where ~k0 = −mvl is the momentum acquired by theatoms upon the Galilean transformation, which relates to the previously defined
detuning in the Raman picture via (k0 − kr )/kr = δ/f0 . After substituting vl insuitable units, the band gap near the Brillouin zone boundary takes the familiar√
form + − − = 21 Ω2 + δ 2 . We can invert (6.23) to write the initial state as
Z
Z


ikx
+
Φ(x, 0) = Φ(k)e dk = Φ(k) c−(6.25)1 Φ− (x, 0) + c1 Φ+ (x, 0) dk.
Subsequent evolution is described byZ+Φ(x, t) = Φ(k) c−
1 Φ− (x, t) + c1 Φ+ (x, t) dk
Z
 ikx

2 −i− t
2 −i+ t⇒ Φ(x, t) = Φ(k) (c−
+ (c+
e dk
1) e
1) e
Z
 − −i− t i(k+qr )x+ −i+ t+ Φ(k) c−+ c+edk
1 c2 e1 c2 e(6.26)(6.27)The second term in (6.27) describes the growth of the scattered cloud. Its norm
corresponds to the diffracted atom number,Z− −i(+ −− )t+ −i+ t 2
NB = |Φ(k)|2 c−+ c+dk.
1 c2 e1 c2 e(6.28)

This result turns out to be equivalent to (6.10). The key assumption in both cases
was that we consider first-order transitions between k-states separated by kr only.
In the Raman picture, we have neglected coupling via 4 and more photons, and in
the Bloch picture we have considered the two lowest bands only. The Bloch picture
has the advantage that it exposes a convenient basis that can be used as a starting
point for e.g. describing the effect of interactions. While we considered an infinite
lattice (neglecting the effect of box walls), the treatment can also be extended tofinite lattices [119].
1

We set ~ = 1 for ease of notation

976.1. THEORY OF BRAGG SPECTROSCOPY

6.1.4

Condensate wavefunction in momentum space

As shown above, Bragg spectroscopy naturally maps momentum p to frequency
qrp. The momentum distribution |Ψ(p)|2 maps to the scattered atomf = f0 + hm

number NB (f ), which is always broadened due to the finite effective linewidth of
the transition, as described by the convolution (6.10). The minimum achievable
linewidth is of the order of Ω (see below). For small enough Ω and long enough τ , we
expect this broadening to be minimal, and we recover the condensate wavefunction.
We start by considering how this wavefunction depends on interatomic interactions.
In a box of length L, several possible cases are conceivable: first, if the mean-fieldenergy is very low, the condensate will be described by the non-interacting “particle
in a box” groundstate. If, on the other hand, the condensate is in the Thomas-Fermi
regime, we expect the wavefunction to be approximately a top-hat function. The
corresponding momentum distribution is then simply a sinc2 function, S(L), of zeroto-zero width 2vr /L (in Bragg frequency units). In the intermediate regime, we must
consider the non-uniformity of the condensate within the healing length ξ from thewalls. In Figure 6.2 (a) we show the probability density |Ψ|2 (x) for the three cases(ξ = 0, 0.1, ∞). For ξ . 0.2, the deviations of the wavefunction from the uniform case are limited to the box edges, were the wavefunction follows Ψ(x) ∝ tanh√x2ξ[23]. We apply the Fourier transform to the analytic representation of Ψ(x) thus
obtained, and plot the momentum distribution |Ψ(k)|2 in Fig. 6.2 (b). In our studies,
ÈYÈ 2

ÈYÈ 2

-1•2

1•2

x•L

-3 p -2 p

(a)

-p

p

2p

3p

kL

(b)

Figure 6.2: Wavefunction in real space (a) and momentum space (b) for a noninteracting BEC (red), and for an interacting BEC with ξ = 0 (blue) and ξ = 0.1L
(purple, dashed).

we seek to quantify the width of the momentum distribution of our condensate. A
98

6.1. THEORY OF BRAGG SPECTROSCOPY

common way is to consider the root-mean-squared (RMS) width, given by
R 2
k |Ψ(k)|2 dk∝ Ekin .
RMS = R|Ψ(k)|2 dk
2(6.29)

Unfortunately, this is infinite for a perfectly uniform condensate with sharp walls.
We plot the change in the RMS width with ξ in Fig. 6.3 (a). It is easy to showthat the kinetic energy density term is proportional to 1/ξ 2 over a region of spaceof the width ∼ ξ, meaning that the kinetic energy scales as 1/ξ for ξ  L. We
demonstrate this asymptotic behaviour in the inset in Fig. 6.3 (a). The extreme
sensitivity to ξ, as well as the noise issues associated with extracting the second
moment from the distribution’s wings, make it unsuitably for quantitative studies.
While the distribution wings decay slowly, the width of the central peak in the

RMS•RMS0

4
3

1.3

5
0

2

0

FWHM•FWHM0

HRMS•RMS0L2

5

25
50
L•x

1
0

0

0.1
x•L

1.2
1.1
1

0.2

0

(a)

0.1
x•L

0.2

(b)

Figure 6.3: Measuring the width in momentum space: (a) root-mean-square width(rms), and (b) full width at half-maximum (fwhm) for different ξ/L. Red lines indicate
result for the non-interacting wavefunction. Inset in (a) shows that the energy of thecondensate (∝ rms width squared) increases as 1/ξ for small ξ (see text).

momentum distribution is well-behaved. We quantify this using the full width at
half-maximum (FWHM). Fig. 6.3 (b) shows that the variations in the FWHM is∼ 30% between ξ = 0 and ξ = ∞, with deviations from the ideal box on the orderof a few % in the experimentally relevant range.
The effect of finite wall steepness is similar to that of finite ξ. Generally, our
imaging resolution (∼ 5% in box length) is sufficient to distinguish ξ = 0 from
ξ = ∞, but it will largely mask the smaller effects of non-zero ξ and finite wall
width.
99

6.1. THEORY OF BRAGG SPECTROSCOPY

6.1.5

Effect of Rabi frequency and pulse length

Next, we consider the broadening of the measured momentum distribution during
Bragg spectroscopy, as described by the convolution kernel in (6.10). For small Rabi
frequencies (Ωτ  1), and thus small scattered fraction (NB /N  1), the convolution kernel is a sinc2 with zero-to-zero width of 2/τ in frequency units. However,even for large τ , we expect the measured momentum distribution to be broadenedby ∼ Ω. The measured Bragg spectrum for a given box length, S̃(L), will therefore
deviate from the Heisenberg-limited spectrum S(L). An illustration is shown in
Fig. 6.4.

NB•N

L = 30 µm, Ω = 8 Hz, τ = 35 ms

15%

FWHM/Hz
10%
5%

-400

-200

0

200

400

f - f0•Hz

realmeasured

ideal

118.9

124.5

uniform

88.6

94.0

Figure 6.4: Broadening of the measured momentum distribution in Bragg spectroscopy. We show the transfer fraction NB (f )/N vs f for ideal (non-interacting, red)and uniform (Thomas-Fermi, blue) BECs. Dashed lines are Heisenberg-limited momentum spectra S in frequency units, solid lines are predicted experimental spectra S̃
calculated using (6.10). The FWHM is tabulated for each of the four spectra shown.

To minimize the broadening, we should therefore use the smallest possible Ω. The
transferred fraction at resonance scales ∼ (Ωτ )2 , and we are thus forced to increaseτ to obtain a detectable signal. While this approach works well for stationary
clouds, the pulse length τ limits the time resolution when performing spectroscopy
on freely expanding clouds, or studying time dependence in general. It is intuitively
clear that short Bragg pulses will lead to excessive broadening for τ . L/vr , e.g.
10 ms for a 30 µm box. We demonstrate this dependence in Fig. 6.5, where we plot
the broadening of the Thomas-Fermi momentum distribution versus τ while keepingΩτ same as in Fig. 6.4.
The choice of suitable τ and Ω will ultimately depend on the required time resolution and the detectable scattered fraction, as limited by the initial atom number
and the signal-to-noise ratio in the imaging system. While the broadening can not
be fully eliminated in our system, the theoretical understanding developed in thepreceding sections provides us with a quantitative, no-free-parameters model of the100

6.1. THEORY OF BRAGG SPECTROSCOPY

FWHM•Hz

180
160
140
120
100
5

10

15

20
t•ms

25

30

35

Figure 6.5: FWHM after Bragg spectroscopy vs τ for a uniform (strongly interacting)cloud, computed using (6.10). L = 30 µm, Ωτ = 8 Hz × 35 ms. For large τ , thebroadening does not disappear, but approaches a constant ∼ Ω.

Bragg spectroscopy precess. Both τ and Ω are well known in our experiments, thelatter from direct measurements of Rabi oscillations on resonance.

6.1.6

The mean-field shift

So far we have considered the effect of interactions on the condensate wavefunction
only. However, interactions will also affect the Bragg spectroscopy process. As a
direct consequence of the bosonic exchange symmetry, an atom scattered out of
a BEC will experience twice the mean-field energy compared to an atom in thecondensate ground state - often described as the “bosonic factor of two” [23]. Thischanges the energy conservation condition (6.4) to
hf = hf0 (pi ) + γn0 ,(6.30)

with γ = 4π~2 a/m and n0 = N0 /V . The same result can be obtained by considering
the Taylor expansion of the Bogoliubov dispersion relation at large momenta.
For a typical box (L = 30 µm, R = 16 µm), this shift equals 32 Hz for each 105
atoms in the condensate. The Bragg spectroscopy peak will therefore be centeredon fc = fr + γn0 . Bragg spectroscopy can thus be used to measure the mean-fieldenergy of a uniform BEC directly. In a non-uniform system, on the other hand,
the mean-field shift varies spatially due to variations in density. This can lead tobroadening of the spectroscopic peak, which in harmonic BECs has been seen to
greatly exceed the Heisenberg width [110].
Note that n0 , and thus the shift (6.30), changes as n0 changes during the Bragg
pulse itself - another reason to keep Ωt small, and introduce suitable corrections.
1016.2. EXPERIMENTAL SETUP

6.1.7

Time evolution and the GPE

Finally, we briefly mention the time evolution of freely expanding clouds, and especially their width in momentum space. The Thomas-Fermi approximation statesthat the in-trap kinetic energy is negligible compared to the interaction energy.
However, the interaction energy is converted into kinetic energy upon expansion. It
is easy to see that the kinetic energy acquired per particle will be proportional tothe in-trap density. Unlike for BECs in harmonic traps [120], we are not aware ofan analytic form describing the expansion of a BEC from a cylindrical box trap.
The temporal evolution of the condensate wavefunction is described by the GrossPitaevskii (GP) equation [23]:
i~

dΨ
~2 2
=−
∇ Ψ + U Ψ + γ|Ψ|2 Ψ
dt
2m(6.31)As described above, the non-interacting part of the Hamiltonian is best solved in
momentum space, while the interacting part has to be solved in real space. We
thus employ a split-operator method, implemented by Alexander Gaunt. This provides numerical results for the real-space as well as momentum wavefunction of thecondensate after different expansion times. Combining the momentum wavefunction obtained by integrating the GP equation with our model of the Bragg transfer
process [equation (6.10], we can obtain a theoretical prediction for the width of the
Bragg spectrum, while considering the peak density of the cloud in real space allows
us to predict the mean-field shift of the Bragg peak. This proves extremely usefulin analyzing our experimental results, as shown below.

6.2

Experimental setup

To perform Bragg spectroscopy, we split a beam derived from the 780 nm repump
laser in two parts (affectionately referred to as “Henry” and “Lawrence”), as shownin Fig. 6.6. The beams are thus detuned by 6.8 GHz from the nearest resonanttransition. Each beam passes through an AOM, and the frequency of one of the
AOMs is adjustable. Although the coherence length of our laser source is typicallyon the order of meters, we took care to keep both optical paths (from the polarizing
beam splitter to the atoms) equal in length to within a centimeter. Finally, bothbeams are intersected at an angle of 2θ ≈ 30◦ , giving a recoil velocity of vr ≈ 3 mm/s.
Each beam is collimated to a width of approximately 3 mm, making the individual
beam size, as well as the intersection region of the two beams, much larger than102

6.2. EXPERIMENTAL SETUP
PD

PD"Lawrence"

-80+f

λ/2

-80

"Henry"Figure 6.6: Optical arrangement for Bragg spectroscopy.

our box trap, and thus providing an optical field of uniform intensity across thecondensate, both in situ and in time-of-flight.
Further, we stabilize the optical power in each of the beams using feedback
signals from independent photodiodes (see Fig. 6.6). This allows us to keep the Rabi
frequency Ω constant. We calibrate Ω by observing Rabi oscillations at resonance(f ≈ f0 ). Since Ω is directly proportional to the optical power in the beams [equation(6.5)], we can use the photodiode signal to measure Ω. For example, we adjust Ωwhen we change the duration of the Bragg pulse τ to keep the scattered fraction
approximately constant in different experiments, as explained above.
We generally distinguish two different measurement regimes of Bragg spectroscopy:
in situ and in time-of-flight.
In the former case, the momentum distribution does not change with time, allowing us to probe it using long pulses with τ & L/vr . This ensures that the
Bragg spectrum is not significantly Fourier-broadened. Equivalently, it ensures thata recoiling atom can traverse the box during the pulse, and we thus probe phasecoherence across the whole BEC. To apply very long Bragg pulses on a trapped
cloud, we set the trap depth below the recoil energy hfr [see Fig. 6.7(a)], so thediffracted atoms can continuously leave the box without bouncing off the trap walls.
In the case of ToF measurements, on the other hand, the wavefunction evolves intime, and τ places an upper limit on our temporal resolution. We thus use shorter
pulses and increase Ω accordingly. We address in situ measurements first, before
discussing measurements in ToF.
103

6.3. GROUND-STATE WAVEFUNCTION IN MOMENTUM SPACE

(b)

(a)

diffracted
atoms

pi+qr
pi

0.1 mmFigure 6.7: Bragg spectroscopy of a uniform BEC. (a) A trapped atom with initialmomentum pi undergoes Bragg scattering (see text). The trap wall height (green) is
lower than the energy of the recoiling atoms. (b) Absorption image of the main cloud
and the diffracted atoms, after ≈ 150 ms of separation. After the Bragg pulse, thenon-diffracted atoms are released from the trap.

6.3

Ground-state wavefunction in momentum space

6.3.1

Spectrum width

In Fig. 6.8 we present our measurements of the momentum uncertainty of the condensed atoms in a box potential. Here τ = 35 ms and the Bragg Rabi frequency wasΩ/(2π) ≈ 8 Hz, keeping the fraction of diffracted atoms to . 15 %. We turn off thetrap 25 ms after the end of the Bragg pulse, and measure the fraction of diffracted
atoms after a further 110 ms of ToF [see Fig. 6.7(b)].
In Fig. 6.8 (a), we show measurements for L = 30 ± 1 µm. As shown in theleft inset, L is determined by fitting the in-trap BEC density profile with a top-hat
function convolved with a Gaussian that accounts for our 5-µm imaging resolution(as described in 4.3). In the main panel we show the measured Bragg spectrum,
centered2 on fc ≈ 1 kHz. The solid red line shows the sinc2 function S(L). The
dashed blue line shows the predicted spectrum S̃(L), taking into account the small
corrections due to the non-infinite τ and non-zero Ω, as described above. The data
is clearly extremely close to the Heisenberg limit, corresponding to a fully coherent
BEC wavefunction spanning the whole box.
We quantitatively compare different (measured and calculated) Bragg spectrausing their full-width-at-half-maximum W . In the right inset of Fig. 6.8 (a), we plot
the measured W versus the total atom number N , for L = 30 µm. The solid red and2Here, we take fc to be the experimentally determined center of the distribution. It is close to

fr , and we address small deviations from fr below.

1046.3. GROUND-STATE WAVEFUNCTION IN MOMENTUM SPACE
(a )

(b )

0

W

(H z )

1 5 0

L
-2 0

0
z ( µm )

2 0

0
N

1 0 0
(1 0 3)

2 0 0

H

W
H

W
H

1 0 0

W

5

0

(H z )

W

1 0

2 0 0

1 0 0

lin e d e n s ity

d iffr a c te d fr a c tio n ( % )

1 5

5 0
0

2 v r /L
-3 0 0

0

0

3 0 0

0

f-fc (H z )

2 0

4 0
1 / L ( m m -1 )6 0

Figure 6.8: Heisenberg-limited momentum spread in a uniform interacting BEC.
(a) Main panel: Bragg spectrum of a trapped BEC of length L = 30 µm. The solid
red and dashed blue lines show the theoretical Heisenberg-limited spectra S(L) andS̃(L), respectively (see text). Left inset: L is determined by fitting the in-trap BEC
density profile, accounting for the imaging resolution. Right inset: Spectral width
W versus the atom number N , for the same L. The filled square corresponds to thedata in the main panel, with W = (100 ± 3) Hz. The solid red and dashed blue lines
show WH = 0.89 vr /L = 87 Hz and W̃H = 93 Hz, corresponding to S(L) and S̃(L),respectively. The dotted blue line shows W̃H0 = 123 Hz, expected for a non-interacting
BEC. (b) W versus inverse box length, 1/L, showing the expected Heisenberg scaling.
Solid red, dashed blue and dotted blue lines show WH , W̃H and W̃H0 , respectively.

dashed blue lines show the two calculated Heisenberg-limited values, WH for S(L)and W̃H for S̃(L). For comparison, we also calculate W̃H0 (dotted blue line) for thesine-like non-interacting ground state of the box potential. We see that interactions
reduce W below W̃H0 . Moreover, the measured W shows essentially no dependenceon N , as expected for a BEC of a spatially uniform density.
In Fig. 6.8 (b) we summarise data for a range of box lengths, L = 15 − 70 µm.
Plotting W versus 1/L, we observe the expected Heisenberg scaling of the momentum uncertainty.

6.3.2

Mean-field shifts

We now turn to the study of the ground-state energy of a uniform interacting BEC,
which is seen in the shift of the Bragg resonance, fc , from the recoil frequency fr .
Thanks to the extreme narrowness of our Bragg spectra, we can measure such shiftswith a precision of ≈ 2 Hz, corresponding to an energy of kB × 100 pK.
In Fig. 6.9 we plot fc versus N , for a fixed L ≈ 30 µm and for two sets of Bragg
spectra, taken “in-trap” (solid symbols) and “in-ToF” (open symbols). The in-trap
105

6.3. GROUND-STATE WAVEFUNCTION IN MOMENTUM SPACE

fc (H z )1 0 2 0in - s itu
in - T o F

1 0 0 0

9 8 0

0

5 0

1 0 0

1 5 0

2 0 0

3

N (1 0 )Figure 6.9: Interaction energy in a uniform BEC, for L = 30 µm. We plot f0
versus N for Bragg pulses applied in-situ (solid squares) and after 50 ms of ToF(open diamonds). Dashed line: f0 ≈ fr ≈ 975 Hz, solid line: f0 = fr + αN , withα ≈ 20 × 10−5 Hz/atom.

spectra are taken as above, with the main cloud trapped during the Bragg pulse.
For the in-ToF spectra we release the BEC from the box and let it expand for 50 ms
before applying the Bragg pulse. After long ToF the atomic density is negligible
and the N -independent fc provides a good measurement of fr ≈ 975 Hz. According
to 6.30, we can write∆fc ≡ fc − fr = αN,(6.32)

with α = 2η~a/(mV ), where η is the condensed fraction3 and V is the volume of thebox. From in-situ images we get V = (25 ± 2) × 103 µm3 . We assess η = 0.8 ± 0.1
by numerically simulating the fraction of Bragg-diffracted atoms for f = fc [seeFig. 6.8 (a)]; here the uncertainty indicates variations between experimental runs.
This estimate is further supported by “BEC filtering” introduced in Ref. [121], i.e.,
using a short (4 ms) Bragg π-pulse to separate the BEC from the residual thermal
component in ToF4 . We thus predict αth = (24 ± 2) × 10−5 Hz/atom.
From a linear fit to the in-trap data (solid line in Fig. 6.9) we get αexp = (20±1)×
10−5 Hz/atom, slightly below αth . This small difference is indeed expected for ourΩR τ , due to the ≈ 15% depletion of the ground-state population during the pulse.
For N ≈ 200 × 103 we took additional measurements with a reduced ΩR [122] and3

Both before and after undergoing Bragg scattering an atom initially in the BEC feels thesame “double” interaction with any thermal component of the cloud. Bragg spectroscopy is thus
sensitive only to the ground-state interactions.
4Note that the small thermal fraction is not directly measurable from either standard ToF
images or the Bragg spectra such as shown in Fig. 6.8(a).

1066.4. EXPANSION OF A QUASI-PURE BEC FROM A UNIFORM TRAP
extrapolating to ΩR = 0 we get a slightly revised5 α̃exp = (23 ± 1) × 10−5 Hz/atom.

F W H M(H z )

2 0 0
1 5 0
1 0 0
5 0
0
0

5 0

1 0 0
N

1 5 0

2 0 0

3

(1 0 )

Figure 6.10: Measured FWHM of the momentum distribution after 50 ms TOF fordifferent N (open diamonds). Simulated width shown (solid line).

Complementary to Fig. 6.9, in Fig. 6.10 we plot W values for the same in-ToF
Bragg spectra. Qualitatively, W now grows with N because during ToF interactionenergy gets converted into kinetic energy. Quantitatively, the problem of the expansion of an interacting BEC from a box potential has not been solved analytically(see [123] for the harmonic-trap case). However, we find good agreement between
our data and numerical simulations based on the GP equation (solid line). In our
simulations, we neglect the small thermal component and use the measured in-trap
BEC energy (∝ α̃exp N ) to predict W in ToF6 .

6.4

Expansion of a quasi-pure BEC from a uniform trap

Finally, we study the evolution of the BEC in ToF, for L ≈ 30 µm and N ≈ 200 ×
103 . In Fig. 6.11 (a) we show absorption images of the expanding cloud (top), andthe corresponding GP-based simulations (bottom). The simulations reproduce thecharacteristic diamond shape that emerges during ToF, also seen in Fig. 6.7 (b). In
Fig. 6.11 (b) we show the gradual decay of ∆f0 and growth of W for in-ToF Bragg
5

The systematic uncertainty in α̃exp , due to the systematic uncertainty in our absolute atom-

number calibration, is 10%.
6
We also take into account the small curvature of our gravity-compensating magnetic field (see5.3.2). This has a noticeable effect only for very long ToF (& 100 ms).

107

6.4. EXPANSION OF A QUASI-PURE BEC FROM A UNIFORM TRAP

spectra, again finding good agreement with our simulations (solid lines). Note thatin these experiments, and simulations, we reduced τ to 10 ms and increased ΩR /(2π)
to 28 Hz, reducing our spectral resolution in order to improve the time resolution.
(a)

(b)

(c)

0 ms

25 ms

50 ms

Figure 6.11: Evolution of the momentum distribution and mean-field energy of a
BEC released from a uniform trap (N = 125000). (a),(b) FWHM and f0 vs TOF:
measured (open diamonds) and simulated (solid line,see text). The solid square indicates the value obtained from data in Fig. 6.9 by interpolation. (c) Measured (toprow) and simulated (bottom row) optical density images of the expanding cloud at 0,
25 and 50ms.

108

6.4. EXPANSION OF A QUASI-PURE BEC FROM A UNIFORM TRAP

In conclusion, we have characterised the ground-state properties of an interacting
homogeneous Bose gas, including the Heisenberg-limited momentum distribution,
the interaction energy, and the free-expansion dynamics. An important by-resultof our measurements is that they place the most stringent bound so far on thespatial uniformity of an ultracold gas produced in our optical box. While earlier(thermodynamic) studies established uniformity on a 10 − 100 nK energy scale [20,21], all our present measurements indicate that our gas behaves as a homogeneous
system down to a sub-nK energy scale (corresponding to 20 Hz in frequency units).
Such a high degree of uniformity offers great promise for future studies of correlation
physics in a homogeneous gas in the T → 0 limit, for example for the preparationand detection of topologically protected states [124].

109

Chapter 7
Conclusions and Outlook
The system we chose to study in the experiments presented here - a uniform gas ofidentical bosons - is a textbook example of a quantum many-body system. Its appeal
lies first and foremost in its fundamental importance and its apparent simplicity.
Nonetheless, as this Thesis hopefully demonstrates, the system’s behaviour is richand complex. Both the theoretical description and the experimental studies present
unique challenges. Importantly, however, it is the experimental implementation thatwas lagging decades behind the theorists’ efforts. In the present work we tried toprovide the missing experimental tools, and thus went to some length in documenting
the details of technical nature.
The reader is probably aware that few of the experimental results presented here
were unexpected. In fact, we have stressed excellent agreement with theoreticalpredictions throughout. Yet, none of these results could have been directly confirmedprior to the implementation of the tools we describe. Thus, one important aspect of
this work is that it confirms existing mean-field theories in a textbook experimental
system. Yet another way to look at it is that all our experiments confirm thequality of our system, in other words, we have shown that our system is indeed
close to the textbook system we set out to emulate, beyond current experimental
accuracy. Specifically, our implementation of a box trap is suitable for a variety ofhigh-precision measurements.
Several applications can be envisioned. Some are in the planning or construction
stage, while others are already producing promising results at the time of writing. As
typical for evolutionary progress, many of these rely on combining several techniques
previously explored in independent settings. Examples include: Uniform BECs with tunable interactions: A system producing BECs of 39 K for111

loading into a uniform box trap is currently being built. Interaction strength
can be tuned over a wide range using a Feshbach resonance [28], thus allowing
further studies of interaction effects (including the potentially present beyondmean-field contributions).
 BECs in two dimensions: By reducing the length of our box trap, the gas canbe forced to effectively enter a 2D regime, where the reduced dimensionality
leads to qualitatively different behaviour.
 Domain dynamics near the phase transition: By quenching the ultra-cold gas,

we can observe the evolution of incoherent domains before coherence is established throughout the volume. Bragg spectroscopy allows us to measure
the spatial correlation function, either by Fourier-transforming the momentum distribution, or directly, by using interferometric methods. We envision
fundamental links to the Kibble-Zurek mechanism.
 Interferometric measurements using flexible geometries: Our light-shaping

methods allow the creation of multiple boxes simultaneously. Degenerate samples can be split and overlapped, and various implementations are imaginable.
 Transport phenomena: We can also create a pair of weakly coupled reservoirs,with a narrow channel (either 2D or 1D) linking the two. Thermal and superfluid flow across the channel can be studied, in analogy with superfluid helium
experiments.
Overall, the work presented here will hopefully prove to be a useful step towards
further developments in the field of ultra-cold atoms. The enthusiasm and dedication
exhibited by every member of our group as they commit to these new challenges
leaves me certain that new results will be plentiful in the very near future.

112

Appendices113

Appendix A
Performance of Grating-Stabilized
Diode Lasers
A.1

Grating-Stabilized Tunable Diode Laser Basics

One of the experimental appeals of 87 Rb is the availability of relatively cheap andmaintenance-free laser diodes at 780 nm. Laser diodes utilize various semiconductor
geometries to provide a gain medium and a lasing cavity. Pumping is achievedby passing a current through the semiconductor structure. For a review of basic
operating principles and available types of laser diodes we refer to [125]. The internal
cavity of a laser diode provides frequency selectivity due to its typical Fabry-Pérot
gain profile. However, the free spectral range of typically ∼ 50 GHz is much smaller
than the frequency range over which the lasing medium provides a significant gain.
The laser diode will thus typically emit a multi-mode signal at a range of frequencies.
To provide frequency selectivity, an external cavity with a frequency-selective output
coupler can be used. A diffractive grating adjusted such that it reflects a particular
non-zero order back into the laser diode allows us to select one of the internal cavity
modes, and the narrow width of the external cavity modes can further reduce theline width of the output. The mode selection process is illustrated in Fig. A.1.
Multiple variants of the grating feedback mechanism have been successfully implemented [125]. The two most widely used schemes are the Littman/Metclaf [42]and the Littrow [45]. Most atomic physics experiments use variations of the latter,
mainly due to its simplicity and higher output power. A wide-spread design proposed in [45] (Hänsch) is shown in Fig. A.2 (a). Several mechanisms allow tuning115

A.1. GRATING-STABILIZED TUNABLE DIODE LASER BASICS
medium gain
temperature
grating
feedback

50 Ghzgrating angle

internal
cavity5 Ghz
current
external
cavitygrating position
emitted frequency

Figure A.1: Mode selectivity in a grating stabilized diode laser: main contributions
and tuning mechanismsof the diode laser frequency. The internal cavity modes are tunable by adjusting
the temperature and/or the current. The external cavity is tunable by displacing
the grating, and the preferred grating feedback wavelength depends on the grating angle. For continuous frequency tuning, we must ensure that the same mode
receives maximum gain throughout, thus avoiding mode hops. It is important tocontrol grating angle, external cavity length and the diode current simultaneously.
The first two are usually controlled using a single piezoelectric actuator. Modulating
the diode current with a signal proportional to the piezo control voltage (“current
feed-forward”) improves the mode hop free scanning range and facilitates locking(see below).

laser diode

PZT stack1 cm

grating

micrometer
adjustment
screw

adjustment
screw

mirror

LM35 temp
sensorPZT element
mount forcollimation optics(a)verticaladjustment
screw

gratinglaser diodePZT disk

thermistor(b)verticaladjustment
screwsthermistor
mount for diodecollimation package

grating
mountmirrormount(c)Figure A.2: ECDLs in Littrow configuration: Hänsch design [45] (a), SussexMelbourne design [47] (b) and Alex Gaunt design (c)116A.2. LASER DESIGN AND PERFORMANCE

A.2

Laser Design and Performance

The Hänsch design has two drawbacks: firstly, the output beam direction changes
if tuning over a wide range, and secondly, the grating feedback and external cavity
modes do not move at the same rate in frequency space (Fig. A.1), causing mode
hops. The first problem can be eliminated by introducing a mirror moving with thegrating [47], as shown in Fig. A.2 (b). Same design addresses the second issue by
introducing an additional piezoelectric actuator (“disk”) for cavity length control.
Another way is to choose a pivot point which maximizes the mode hop free tuning
range [44]. Alex Gaunt built a diode laser combining the benefits of an optimized
pivot point and fixed beam output direction by mounting a mirror in the moving
cantilever arm, as seen in Fig. A.2 (c). However, since the mode hop free range
required to resolve the hyperfine structure of Rb is very small (∼ 7 GHz), these
design features turned out to be far less important than the overall mechanicalstability, power and locking bandwidth.
Diode lasers can typically provide ∼ 20 − 100 mW of power. Since this is clearly
not sufficient for cooling, it was decided to purchase a Toptica DLpro diode laser
with a BoosTA tapered amplifier, providing up to 100 mW before and 1.5 W after
the amplifier. The laser is supplied with an FPGA controlled digital locking system
and is highly stable and virtually maintenance free.
Following some previous in-house laser development by Phil Richman [126] andAlex Gaunt, it was decided to use one of their designs for the repump laser. We
first address laser operation issues common to both lasers, and then discuss the
drawbacks of each design.

A.2.1

Laser Diodes and Alignment

Crucial to laser performance is the choice of diodes. Most important in our experience was to choose a diode with a free-running wavelength closest to 780 nm. It
is advantageous to tune the free-running wavelength of the diode (without grating
feedback) by changing the temperature prior to further alignment. A tunability of+0.3 nm/K can be expected. Most laser diodes have a higher wavelength which
necessitates cooling, we are however restricted by the dew point, only allowing a 6 K
reduction in temperature. This corresponds to typically 2 nm wavelength change,
which is insufficient for some diodes. We made good progress with ADL-78901TL
diodes from Laser Components, achieving output powers of 60 mW at the desired
wavelength. Other high power diodes we tested included L785P090 and L785P100
117

A.2. LASER DESIGN AND PERFORMANCE

from Thorlabs. They lased at 785 ± 1 nm and where unsuitable form our applications. Both lasers utilize a diode collimation package from Thorlabs. The first stepis to collimate the output beam by adjusting the lens. Output of most commercial
laser diodes is elliptic and plane polarized, with the polarization axis parallel tothe semi-minor axis of the ellipse. For improved resolution, we want to illuminate
as many grating lines as possible. This requires the laser diode to be rotated suchthat the polarization plane is parallel to the grating lines. Both designs use the-1st order reflection, with both the incidence and reflection angle close to 45◦ . A
wavelength of ≈ 780 nm is achieved in this configuration if a 1800 mm−1 grating
is used. A small diffraction efficiency (< 0.3) is required to avoid diode damage
at high powers. The grating must be aligned such that it is perpendicular to thebeam plane. Both laser designs have a vertical adjustment screw. Coarse alignmentis achieved by observing the reflection off the diode facet and overlapping it withthe main beam. Further feedback optimization is best performed by measuring thethreshold current and minimizing it using both vertical and horizontal adjustment
screws. The grating position can then be scanned with an effective scanning rangeof ∼ 10 GHz while tweaking current and temperature to achieve best mode hop
free performance near the desired absorption spectroscopy peaks (spectroscopy andlocking are discussed in B). Multi-mode lasing is often an issue. It leads to dramatic
deterioration in the error signal and can be easily spotted by observing the FabryPérot spectrum using a wavemeter. It is best corrected by making small adjustments
to the current. The performance of both lasers improves dramatically when using
a mechanically shielded enclosure, as shown in Fig. A.3. Heavy aluminium blocks
aluminium enclosure
insulating foam

laser

aluminium block
Sorbothane padaluminium block
optical table

Figure A.3: Components for mechanical shielding and thermal stabilization ofgrating-stabilized diode lasers

118

A.2. LASER DESIGN AND PERFORMANCE

provide mechanical and thermal stability. A viscoelastic Sorbothane pad is used
for additional vibrational isolation. A layer of foam provides acoustic shielding and
impedes convection currents.

A.2.2

Alex Gaunt design

The first laser we used in the optical setup was built by Alex Gaunt with high modehop free scanning range in mind, necessitating a large distance between the pivot
point and the grating. The additional mirror also increased the oscillating mass
significantly. The long, thin cantilever deformed plastically on multiple occasions
and was not able to provide the restoring moment to follow the piezo stack at highfrequencies. In scanning mode, the effects can be reduced by avoiding discontinuous
sawtooth modulation and using a low-frequency triangle or sine wave instead. The
main drawback of the laser turned out to be its susceptibility to external noise. To
investigate the problem further, we performed eigenmode analysis of the cantilever
design using Autodesk Inventor Simulation. The two lowest eigenmodes are shownin Fig. A.4.

f1 = 34 Hz

f2 = 230 Hz

Figure A.4: Two lowest eigenmodes of the cantilever

The first eigenmode at f1 ≈ 34 Hz corresponds to the desired cantilever motion
during laser tuning. Its frequency however is very low due to the increased massand length compared to other common designs. For the grating to respond to thepiezo with a high enough bandwidth, the cantilever has to be pre-bent strongly,
approaching the limit of elastic deformation. The effect of reduced bandwidth isclearly visible in the spectroscopy signal as ringing when driving the piezo with a
discontinuous signal like a sawtooth (see below).
The second lowest eigenfrequency of the cantilever is f2 ≈ 230 Hz and corre119

A.2. LASER DESIGN AND PERFORMANCE

sponds to a torsional mode. It appears due to the large mass of the mirror positioned out of the cantilever plane. Like the first eigenfrequency, it lies in the acoustic
range and the cantilever will be susceptible to noise. Unlike the bending however,
the torsional mode will lead to loss of feedback altogether, and thus severe mode
hopping.
The eigenfrequencies of a standard Hänsch design are much higher, and especiallythe torsional modes only appear in the kHz region.
Using Alex Gaunt’s design in a saturated absorption spectroscopy setup, we wereable to resolve the hyperfine structure of the 87 Rb D2 line in a single scan, i.e. witha scanning range of at least 7 GHz. We where however unable to achieve stable
locking.

A.2.3

Phil Richman design

The laser built by Phil Richman is based on the Sussex-Melbourne design shown
above, however it lacks the second piezoelectric actuator (PZT disk). Its greatest
advantage is the ease of manufacture from standard parts. The heavy, spring-loaded
lever provided mechanical stability far superior to the laser by Alex Gaunt. Due tothe large lever length, the pizo stack only allowed a scanning range of ∼ 10 GHz,
which could easily be solved by replacing the stack. We only managed to achieve amode hop free scanning range of a few GHz, this was however sufficient to lock thelaser and use it in a MOT experiment. The laser often remained locked for hours,
was however subject to mechanical creep following alignment. Currently, we areusing this laser for MOT experiments with varying degrees of success. Due to thecompromised stability if the laser we decided to replace it with another DLPro.

120

Appendix B
Saturated Absorption
Spectroscopy and Locking
Atomic physics experiments require laser sources which are stable to within a fractionof the natural linewidth of the atomic transition. While the linewidth of an external
cavity diode laser is usually sufficiently small, drifts in the central frequency exceed
the 87 Rb linewidth of 6 MHz by far. The frequency of the diode laser must thus beactively stabilized using a feedback loop.

B.1

Principles of Absorption Spectroscopy

Absorption of a single probe beam passing through an atomic vapour will peak when
it is on resonance with an atomic transition. Absorption can be easily measured byscanning the frequency of a tunable laser over the range of interest and recording
the transmitted beam intensity with a photodiode. Due to the Doppler effect, a
simple two-level atom moving at velocity v along the beam will only be on resonanceif the laser frequency ω satisfies ω = ω0 + kv, where ω0 is the actual transition
frequency and k the wavenumber of the laser. Since the velocities in a gas follow athermal distribution, every absorption peak will be Doppler-broadened by as much as
1 GHz in 87 Rb at room temperature. While transitions in the D2 line with different
ground states can still be resolved, the hyperfine splitting of the excited state is notdetectable. A common method is to introduce a strong counterpropagating pumpbeam derived from the same laser source. A moving atom sees the pump and theprobe beam at frequencies ω ± kv respectively. If we make the pump beam strong
enough (such that it exceeds the saturation intensity), it will deplete the groundstate population of resonant atoms, i.e. those satisfying ω = ω0 − kv. This will not121B.1. PRINCIPLES OF ABSORPTION SPECTROSCOPY

affect the probe beam unless the latter is resonant with the same group of atoms,
which will only happen if v = 0 and ω = ω0 . Is this the case, the absorptionof the probe beam will be greatly reduced and we will see a “Lamb dip” in theabsorption signal. The position and width of the Lamb dips is no longer affected bythe (first-order) Doppler effect.
The observed saturated absorption spectrum is more complicated for a multilevel atom. If two atomic transitions at ω1 and ω2 share a common ground state,
we will see a drop in absorption for any of the four cases
ω − kv = ω1 ,

ω + kv = ω1

(B.1)

ω − kv = ω2 ,

ω + kv = ω2

(B.2)

ω − kv = ω1 ,

ω + kv = ω2

(B.3)

ω − kv = ω2 ,

ω + kv = ω1

(B.4)

Solving these, we see that a Lamb dip will appear at frequencies ω1 , ω2 as well as(ω1 + ω2 )/2. The last one leads to the appearance of what is known as a cross-overpeak.
Finally, if the two transitions share a common excited state, the pump beam
resonant with the transition at ω1 will deplete the corresponding ground state andincrease the occupancy of the second ground state through spontaneous decays.
If the pump beam is resonant with the second transition at ω2 at the same time,absorption will increase, showing as an inverted cross-over peak in the frequencyscan.
A typical spectroscopy setup is shown in Fig. B.1. The pump beam is retroreflected and used as the probe beam. A λ/4 wave plate before the mirror is used
to make the light circularly polarized. Its handedness changes upon reflection, andafter passing through the wave plate a second time the probe light’s polarization is
at 90◦ to that of the pump beam, enabling us to divert it to a photodiode.
A typical absorption spectroscopy signal obtained with this setup is shown in
Fig. B.2 (a). The F = 1 and F = 2 Doppler peaks in 87 Rb are clearly visible,
as well as the peaks originating from 85 Rb contained in the same vapour cell. The
hyperfine structure within each Doppler peak is visible through saturation peaks, as
shown for the repump transition in Fig. B.2 (b). All six peaks corresponding to the
F = 1 ground state can be clearly identified, including the three stronger crossover
peaks. The F = 1 → F 0 = 0 peak appears inverted, suggesting multiple closely
spaced ground states. This can be explained if one remembers that any external
122

λ/2

frequency
tuningλ/4B.2. FREQUENCY MODULATION SPECTROSCOPY AND LOCKING

ECDL

PD

phase
shift

Integrator

PID

Lock-in amplifier
scan

lock

oscilloscope

Figure B.1: A simple setup for saturated absorption spectroscopy and AC locking

magnetic field will split the F = 1 state into three Zeeman sub-states. There are
therefore three distinct transitions: σ + (|F = 1, mF = −1i → |F 0 = 0, mF 0 = 0i),π(|F = 1, mF = 0i → |F 0 = 0, mF 0 = 0i) and σ − (|F = 1, mF = +1i → |F 0 =0, mF 0 = 0i). Each pair of these can produce an inverted cross-over peak. The six
peaks are not resolved for the small fields in question - typical Zeeman splitting in
the Earth field is 0.5 MHz. Note that for most magnetic field directions both the σ +and the σ − transitions couple to either vertically or horizontally polarized light, inparticular for a B field along the laser beam. This is also the field direction for whichthe π transition is not permitted with either polarization. The strength of each of
the corresponding peaks is therefore strongly dependant on the magnetic field direction, and thus add up to a peak of variable strength. This is easily confirmed
experimentally by moving a bar magnet in proximity to the vapour cell. Experimentally, we always observed the peak to be inverted, meaning that the cross-overpeaks dominate for all field directions.

B.2

Frequency Modulation Spectroscopy and Locking

To lock the laser frequency to one of the 87 Rb D2 transitions we must derive an error
signal, that is a signal which is monotonous near the peak centre and zero when
the laser is on resonance. Using the spectroscopy signal s(ω), it is only possible
to lock to the side of one of the peaks (after introducing an offset to make it zerocrossing). What we really need is its derivative. This can be achieved using frequency
123

F’=(0,1)F’=0F’=(1,3)

F’=3

Rb87 F=2

F’=(1,2)

Rb87 F=1 Rb85 F=2

F’=1

B.2. FREQUENCY MODULATION SPECTROSCOPY AND LOCKING

Rb85 F=3(a)(b)

Rb87 F=1

Rb87
0.10

0.05

voltage @VD

voltage @VD

0.05

0.00

0.00

-0.05
-0.05

-0.10

-0.05

0.00

0.05

-0.10

0.10

-0.10

time @sD

-0.05

0.00

0.05

0.10

time @sD(c)(d)

Figure B.2: Saturated absorption signals (red) and deriver error signals (blue). (a)Scan over ∼ 10 GHz, showing the hyperfine structure of two Rb isotopes; (b) scan overthe 87 RbF = 1 feature, with theoretical peak positions fitted to the data; (c) sameas (b) with modulation and the corresponding error signal; (d) magnified view of theerror signal slope centred on the (0,1) cross-over peak. (Alex Gaunt’s laser was used
for spectroscopy)

modulation spectroscopy [49]. Consider sinusoidal modulation of the laser frequency
around the centre frequency ω0 : ω(t) = ω0 + A sin ωm t. For small modulation
amplitude A, we can write s(t) ≈ s(ω0 ) + As0 (ω0 ) sin ωm t. Extracting the spectral
component at ωm therefore yields s0 , which can be used as the error signal. Thisis achieved using a lock-in amplifier shown schematically in Fig. B.1. Fig. B.2 (c)shows the spectroscopy signal from Fig. B.2 (b) with added modulation in red, andthe derived error signal in blue. Optimizing the error signal quality is paramountfor stable locking. The error signal slope shown in Fig. B.2 (d) is indicative of
the maximum noise level for which stable locking is possible. The error signal isprocessed using a proportional-integral-derivative controller (PID) which provides
124

B.2. FREQUENCY MODULATION SPECTROSCOPY AND LOCKING

the feedback signal for adjusting the laser frequency.

125

Appendix C
IGBT driver electronics
Our experimental sequence requires switching quadrupole coil currents of up to
200 A on the ms timescale. Insulated-gate bipolar transistors (IGBTs) are among
the few solid state devices capable of operating at such high currents (in our setup,
we use IGBTs of the type SEMiX402GB066HDs). In the process of building our
experiment, we have developed custom driver electronics with the intention of providing a reliable standard solution for the current as well as future BEC machines
and incorporating various safety features.

Figure C.1: IGBT driver curcuit: IGBT interface.

The gate capacity of an IGBT is typically high, requiring high driving currents.
A typical driving circuit is shown in Fig. C.1. At the input, it takes a TTL signal
derived from the logics circuit shown in Fig. C.2. The latter implements safety
features by combining inputs from external temperature sensors, coolant flow sensors
and interlocks. The safety shut-off is latching, and a reset must be forced to resume
operation. Further, the circuit provides facilities to set a maximum “on” time.
All components are assembled on a two-sided printed circuit board, as shownin Fig. C.3. The interface to the IGBT is via a ribbon cable, and the system was127

designed with the ease of small-series assembly in mind. Figure C.4 shows two IGBT
drivers, together with the front panel, the IGBTs, water cooling, and high current
leads, assembled for use in a new BEC experiment currently under construction in
our group.

128

Figure C.2: IGBT driver curcuit: TTL layout and external inputs.

129

Figure C.3: IGBT driver implementation, showing the printed circuit board layout(top, two copper layers shown in red and blue) and the assembled circuit(bottom).

Figure C.4: Two IGBT drivers in use. Front panel, two water-cooled IGBTs, andribbon cable leads carrying driving signals and temperature sensor signals are clearlyvisible.

130

Appendix D
Analysis Software
Ultra-cold atom experiments rely on analyzing thousands of optical density images.
We have developed custom software (“Image Analysis GUI”) for this purpose. The
software is implemented in MATLAB, and a clear separation between the front-end(the graphical user interface, GUI) and the back-end (all processing and analysis
routines) is maintained. This allows us to use the back-end code in external batchprocessing routines to provide functionality not implemented in the GUI. Below, we
present the main components of the software.

D.1

Graphical user interface

The GUI is shown in Fig. D.1, and includes the following key components:
 Toolbar (topmost): provides import, export, auto-load and image sorting but-tons.
 Thumbnail panel (top): provides a quick view of the open images.
 Properties tab (left): displays file information, allows the input of physicalparameters and the region of interest (ROI) to be used for image analysis.
 Inspect tab (left): shows properties of the ROI of individual pixels.
 Image tab (center): displays the selected image, its cuts and the fitting func-tion.
 Summary tab (center): shows a table of all images with analysis results.
 Fitting tab (right): gives access to different analysis routines, allows fine-tuning of fitting parameters and batch fitting.
131D.2. SOFTWARE ARCHITECTURE

Figure D.1: Image analysis GUI window, showing the main components of the userinterface (see text).

 Advanced tab (right): controls down-sampling and filtering of images.

D.2

Software architecture

We use object-oriented implementation. The following files and classes are provide
the core functionality AnalysisGUI62.m (executable): defines the user interface; maintains an arrayof loaded images of the type ODImage.
 ThumnailViewer.m (class): defines the functionality of a scrollable panel ofthumbnails.
 ODImage.m (class): stores file path, name, image data, fitting data and allother image information. ODImage.FitPlugin is used to store a fitting plugin
object implementing the mathematical fitting routines.
132

D.2. SOFTWARE ARCHITECTURE Fitting plugins, e.g. Gauss2D.m, g alpha 2D ell mu0 excl count.m etc. (eacha handle class): implement the respective fitting routines. Their methods arecalled from the respective ODImage object.
An important consideration was efficient memory use. Thus, ODImage.Compress()and ODImage.Expand() routines are implemented to remove image data from RAM
when not needed for plotting or analysis and load it from the hard drive whennecessary.

133

Appendix E
The polylogarithm function
E.1

Definition

The polylog function is commonly defined as∞Xxkgα (x) =
kα
k=1

(E.1)

For α > 1 and x = 1, this becomes the Riemann zeta function:gα (0) = ζ(α)(E.2)

Trivially, gα (x) converges as x → 1 for α > 1 and diverges otherwise.

E.2

Integrals

Within the local density approximation, we often need to evaluate integrals of the
form
Z ∞

−a|x|s

gα ze
−∞



Z ∞X
∞

s

z k e−kaxdx = 2
kα
0
k=1
Z ∞
∞
X
zk
s
=
2
e−kax
α
k
0
k=1
∞
X
z k Γ(1 + 1s )2αk(ak)1/n
k=1


∞
X
1
zk
−1/n
= 2Γ 1 +ask α+1/nk=11= 2Γ 1 +a−1/n gα+1/n (z)
s

=

 √
In the commonly encountered case n = 2, the pre-factor is 2Γ 1 + 21 = π.
135

(E.3)
(E.4)
(E.5)
(E.6)
(E.7)

Appendix F
Efficient numerical evaluation of
the polylogarithm functions
Polylogarithm functions gα (x) are used extensively in our image analysis routines.
Least-square fitting of a parametrised model to an optical-density can involve ∼ 108
evaluations of gα per image. Processing thousands of experimental runs in realtime requires an efficient algorithm. Unfortunately, few standard libraries provide
the polylogarithm functions of arbitrary order. Wolfram Mathematica provides thearbitrary-precisions function PolyLog[α, x]. In our applications, precisions well shortof machine precision is acceptable, while evaluation time is often critical. As astarting point, we used the MATLAB implementation by M. Kuhnert [127], whichis in turn based on the numerical approximations derived in [128]. The relative
numerical error of this method is illustrated in Fig. F.1 for g0 .5(x) and g1 .5(x),0 ≤ x < 1. It can be seen that the relative error never exceeds 10−5 - this is always
significantly lower than experimental errors in the optical density measurements.
While this method provides a huge improvement in speed compared to the built-in
Mathematica function (see Table F.1), we found that it was not allowing real-time
image processing (i.e., fitting a single image took longer than one repetition of theexperimental sequence).
We achieved more than a 10-fold improvement in evaluation speed by optimizing the evaluation order, without changing the mathematical approximations used
(Table F.1). The general principle is best illustrated by a simple example: assume
we are trying to evaluate a polynomial approximation of the form
p(x) = a0 + a1 x + · · · + an xn
137

(F.1)

rel. error in gaHxL

10-6
10-8

a=0.5

10-10

a=1.5

10-12

10-14
0.0

0.2

0.4

0.6

0.8

1.0

x
Figure F.1: Relative error of our numerical approximation for gα (x), plotted for0 ≤ x < 1 for α = 0.5 (blue) and α = 1.5 (red)methodPolyLog [129]polylog [127]polylog [130]evaluation time

40 s

0.2 s

0.018 srelative speed

1/400

1

11

Table F.1: Benchamarks for different implementations of the polylogarithm function.
We evaluate gα (x) for n uniformly spaced points on [0, 1]. Data shown was obtained
for n = 105 , α = 0.5. Test rig: Sony Vaio laptop, Intel(R) Core(TM) i7-3632QM
2.20GHz CPU, 8GB RAM, 64-bit Windows 7, Mathematica 9.0 64-bit, MATLAB
R2012n 64-bit.

Evaluating terms of the form xn requires (in the worst case) n multiplications1 , fora total of ∼ n2 /2 multiplications to evaluate all terms in the above expression. On
the other hand, we can writep(x) = a0 + x(a1 + x(a2 + x(· · · )))(F.2)

for a total of only n multiplications. Further optimizations can be made, generally
by minimizing repeated calls to recursive sub-routines. A comparison between theoriginal and the improved code can be seen in the (shortened) listing below.

1

Optimized algorithms allow an evaluation in ∼ log2 (n) multiplications, by noticing that e.g.

8x = ((x2 )2 )2 . However, the difference is small for small n.

138

1

function [y errors] = polylog(n,z)

2%% polylog − Computes the n−based polylogarithm of z: Li n(z)3

% Usage:[y errors] = PolyLog(n,z)

4% Input:z < 1: real/complex number or array5%n > −4: base of polylogarithm

6% Output: y... value of polylogarithm

7%... number of errors

errors89...%error checking etc.

10 version = 2;
11 switch n
12 case 113

y = −log(1−z);

14 otherwise
15 switch version
16 case 1
17 % original, slower version
18 alpha = −log(z);19 y = zeros(size(z));
20 j = find(abs(z)>0.55); % if | z | > 0.55
21 preterm = gamma(1−n)./alpha(j).ˆ(1−n);
22 nominator = b(0) + ...
23− alpha(j).*( b(1) − 4*b(0)*b(4)/7/b(3) ) + ...

24+ alpha(j).ˆ2.*( b(2)/2 + b(0)*b(4)/7/b(2) + ...

25− 4*b(1)*b(4)/7/b(3) ) − alpha(j).ˆ3.*( b(3)/6 + ...

26− 2*b(0)*b(4)/105/b(1) + b(1)*b(4)/7/b(2) + ...

27− 2*b(2)*b(4)/7/b(3) );28 denominator = 1 + alpha(j).*4*b(4)/7/b(3) +...
29

+ alpha(j).ˆ2.*b(4)/7/b(2) +...

30

+ alpha(j).ˆ3.*2*b(4)/105/b(1) +...

31

+ alpha(j).ˆ4.*b(4)/840/b(0);32 y(j) = preterm + nominator ./ denominator;
33 j = find(abs(z)<=0.55); % if | z | <= 0.55
34 nominator = 6435*9ˆn.*S(n,z(j),8) − 27456*8ˆn*z(j).*S(n,z(j),7) + ...
35 + 48048*7ˆn*z(j).ˆ2.*S(n,z(j),6) − 44352*6ˆn*z(j).ˆ3.*S(n,z(j),5) + ...
36 + 23100*5ˆn*z(j).ˆ4.*S(n,z(j),4) − 6720*4ˆn.*z(j).ˆ5.*S(n,z(j),3) + ...
37 + 1008*3ˆn*z(j).ˆ6.*S(n,z(j),2) − 64*2ˆn*z(j).ˆ7.*S(n,z(j),1);
38 denominator = 6435*9ˆn − 27456*8ˆn*z(j) + ...
39 + 48048*7ˆn*z(j).ˆ2 − 44352*6ˆn*z(j).ˆ3 + ...
40 + 23100*5ˆn*z(j).ˆ4 − 6720*4ˆn*z(j).ˆ5 + ...
41 + 1008*3ˆn*z(j).ˆ6 − 64*2ˆn*z(j).ˆ7 + ...
42 + z(j).ˆ8;43 y(j) = nominator ./ denominator;

139

44 case 2
45 % new version optimized for array evaluation
46 y = zeros(size(z));
47 j = (z>0.55); % if | z | > 0.55
48 alpha = −log(z(j));
49 alphan = alpha;50
51 preterm1 = gamma(1−n)./alpha.ˆ(1−n);
52 den1 = ones(size(alpha));53 nom1 = b(0)* den1;
54 nom1 = nom1 − alpha.*( b(1) − 4*b(0)*b(4)/7/b(3) );55 den1 = den1 + alpha.*4*b(4)/7/b(3);56 alphan = alphan.*alpha; %alpha.ˆ2
57 nom1 = nom1 + alphan.* ...
58( b(2)/2 + b(0)*b(4)/7/b(2) − 4*b(1)*b(4)/7/b(3) );59 den1 = den1 + alphan.*b(4)/7/b(2);60 alphan = alphan.*alpha; %alpha.ˆ3
61 nom1 = nom1 − alphan.* ...
62( b(3)/6 − 2*b(0)*b(4)/105/b(1) + b(1)*b(4)/7/b(2) + ...

63− 2*b(2)*b(4)/7/b(3) );64 den1 = den1 + alphan.*2*b(4)/105/b(1);65 alphan = alphan.*alpha; %alpha.ˆ4
66 den1 = den1 + alphan.*b(4)/840/b(0);67 y(j) = (preterm1 + nom1./den1);
68 coeffs = [6435*9ˆn, −27456*8ˆn, 48048*7ˆn, −44352*6ˆn, ...
23100*5ˆn, −6720*4ˆn, 1008*3ˆn, −64*2ˆn, 1];
70 j = ˜j; % if | z | <= 0.55
69

71 z = z(j);72 den2 = zeros(size(z));
73 nom2 = den2; %zeros
74 s = den2; %zeros
75 % last term first76 den2 = den2 + coeffs(end);
77 zn = ones(size(z));
78 for i = 1:( length(coeffs)−1 )79

zn = zn.*z;

80s = s + zn./i.ˆn;81

nom2 = nom2 + coeffs(end−i)*s./zn;

82

den2 = den2 + coeffs(end−i)./zn;

83 end
84 % at the end, zn = z.ˆ8
85 nom2 = nom2.*zn;
86 den2 = den2.*zn;

140

87 y(j) = nom2./den2;
88 end89 end
90 % define b
91 function out = b(i)92out = zeta(n−i);

93 end
94 % define partial sums S
95 function out = S(n,z,j)
96

out =0;97

for i=1:j

98

out = out + z.ˆi./iˆn;

99

end

100

end

101

...%define zeta(x) etc.

102

end

141

Bibliography[1] E. Fermi, “Sulla quantizzazione del gas perfetto monoatomico,” Rendiconti
Lincei, vol. 3, pp. 145–149, 1926.
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A, vol. 112, no. 762, pp. 661–677, 1926.
[3] S. N. Bose, “Plancks Gesetz und Lichtquantenhypothese,” Zeitschrift f\”{u}Physik, vol. 26, p. 178, 1924.
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