Department of Physics
Cavendish Laboratory

Studies of a Homogeneous Bose Gas

Tobias Fabian Schmidutz
Trinity College
July 2014

This dissertation is submitted for the degree ofDoctor of Philosophy

Tobias Schmidutz, 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 other qualification
at any other university. No part of this thesis has alreadybeen 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.

9th of July 20143

Abstract
In this thesis we describe the first experimental realization of a homogeneous
atomic Bose-Einstein condensate in three dimensions. We also outline our first experimental studies of this new system, which exhibit textbook-like thermodynamicand quantum-mechanical behaviour.
First, we describe a simple and compact single-chamber apparatus for robust production of harmonically trapped 87 Rb Bose-Einstein condensates. The apparatus isbuilt from off-the-shelf components and allows the production of quasi-pure condensates of > 3 × 105 atoms in < 30 s. Our system provides an excellent compromise
between compactness, cost and performance.
We then detail the transfer of the atoms into our box potential, formed by a dark
optical trap. Next, we characterize the homogeneity of the trapped gas and deducethat its behaviour is very close to that of an ideal homogeneous gas. The trapping
potential is well approximated by a power-law of order 15, which constitutes only aminor deviation from a uniform box.
We conduct an in-depth thermodynamic study of the homogeneous gas. We characterize the critical point for condensation and observe saturation of the thermal
component in a partially condensed cloud, in agreement with Einstein’s textbook
picture of a purely statistical phase transition. Further, we observe the quantum
Joule-Thomson effect, manifesting itself as isoenthalpic cooling of an (essentially)
ideal gas. We extract a Joule-Thomson coefficient µJT > 109 K/bar, which is about
10 orders of magnitude larger than observed in classical gases.
Finally, we study the properties of the condensate ground state, using highresolution Bragg spectroscopy. We directly observe the Heisenberg-limited momentum uncertainty of the condensed atoms for a range of box sizes, up to 70 µm. We
measure the condensate interaction energy with a precision of kB × 100 pK andstudy the dynamics of its free expansion upon release from the box potential. All our
measurements are in good agreement with theoretical expectations for a perfectly
homogeneous condensate, establishing the uniformity of our optical-box system on
a sub-nK energy scale.
The main value of this work lies in adding the uniform trap to the toolkit of cold
atom experiments. One of the major goals in the field of cold atoms is to establish
a closer link to condensed matter problems. As many of these are inherently in a
homogeneous environment, this work should constitute a substantial step towards
this goal.

4

Acknowledgements
I would like to express my gratitude to everyone who helped and supported me throughthe years that lead to this thesis. Foremost, I would like to thank my supervisor Zoran
Hadzibabic, who took me on as a green summer intern and who has been a great mentor
throughout my PhD. His support, advice and enthusiasm have guided me through many
scientific crises during the last years.
The foundations of our lab were laid by its first students, Naaman Tammuz and Robbie
Campbell, whose friendliness and willingness to help was one of my reasons to join thisgroup. I would also like to thank Stuart Moulder and Scott Beattie, who were always in
the lab next doors, available for advice or a cup of coffee. Following in their footsteps as
the steward of the “old lab”, Richard Fletcher has been a formidable proof-reader in-chiefof this group and we have shared a quite a few cups of coffee discussing the lab exclusion
principle,1 Thanks also goes to Nir Navon and Martin Robert-de-Saint-Vincent, the mostrecent additions to our group, who brought a lot of knowledge and energy to the table.
My gratitude extends to everyone else who was part of the team along the way, especiallyPhil Richman, Alexandre Dareau and Aziza Suleymanzade.
Alex Gaunt’s computer genius and experimental skills contributed a huge amount tothis work. Beyond that, his service as the quiz master has done much to increase thehistory knowledge of the members of this group. I am hugely indebted to Robert Smith,whose knowledge, technical expertise and boundless helpfulness were always there to becalled upon when things went wrong.
My largest thanks goes to Igor Gotlibovych, who has been my lab buddy for the lastyears. Having done all of this work jointly with him, this thesis is as much his as it ismine. He was not only a brilliant colleague, but also a great friend. Our shared sense of
humour, the late-night discussions about the meaning of life and our common “affection”
for the small machine have kept me sane over the last years. Also, I have learned a thingor two about boats in the process.
Last but not least, I would like to thank our fantastic group secretary Pam Hadder,whose friendliness and competency is unsurpassed in the Cavendish.
1

It is impossible for both labs to be in the same (functional) state at the same time.

5Outside the lab, there are many people who made my time in Cambridge a great
experience. Unable to mention everyone by name, I would like to thank the “Team Youth
Hostel”, the Judo club and the Trinity crowd for many good times.
Special thanks goes to Janina for always being supportive and understanding during
the last years, even when I was stressed, struck by graduation panic or working late-night
shifts.
Most of all, I would like to thank my family, my parents Maria and Hubert, and mysiblings Sonja and Florian, for their unconditional support and endless belief.

6

Contents
1

2

Introduction

11

1.1

Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . .

11

1.2

The box and the harmonic oscillator . . . . . . . . . . . . . . . . . . . .

12

1.3

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Bose-Einstein Condensation

15

2.1

Bose-Einstein Condensation in a homogeneous potential . . . . . . . . .

15

2.2

Thermodynamics in Power-law potentials . . . . . . . . . . . . . . . . .

16

2.2.1

The local density approximation . . . . . . . . . . . . . . . . . .

17

2.2.2

Thermodynamics in power-law potentials . . . . . . . . . . . . .

18

2.2.3

Momentum and spatial distributions . . . . . . . . . . . . . . . .

19

2.2.4

The condensate wave function . . . . . . . . . . . . . . . . . . .

20

Expansion and Thermometry of cold gases . . . . . . . . . . . . . . . . .

21

2.3.1

Thermal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.3.2

Degenerate gases . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.3.3

Condensed gases . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3.4

Thermometry of partially condensed gases . . . . . . . . . . . .

24

2.3.5

Column densities . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.3

3

Atom Cooling and Trapping

27

3.1

Atomic properties of Rubidium . . . . . . . . . . . . . . . . . . . . . . .

27

3.2

Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3

Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.3.1

Evaporative cooling in Power-Law Traps . . . . . . . . . . . . .

31

3.3.2

Crossed-Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . .

34

3.3.3

Hybrid Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.3.4

Homogeneous Trap . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.3.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

7

CONTENTS

4

Experimental Setup and Methods

39

4.1

Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.2

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

41

4.2.1

Magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2.2

Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.3.1

Transfer to the magnetic trap . . . . . . . . . . . . . . . . . . . .

47

4.3.2

RF Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.3.3

Single-Beam Dipole Trap: “Hybrid Trap” . . . . . . . . . . . . .

50

Imaging and Image processing . . . . . . . . . . . . . . . . . . . . . . .

53

4.4.1

Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . .

53

4.4.2

Density calibration . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.4.3

High intensity imaging . . . . . . . . . . . . . . . . . . . . . . .

55

Experiment control and data processing . . . . . . . . . . . . . . . . . .

58

4.5.1

Sequence setup and control . . . . . . . . . . . . . . . . . . . . .

58

4.5.2

Imaging control . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.5.3

Analysis software . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.3

4.4

4.5

5

Creating a homogeneous Bose-Einstein Condensate

61

5.1

Dark Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.2

Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5.3

Loading of the Uniform trap . . . . . . . . . . . . . . . . . . . . . . . .

64

5.3.1

Transfer procedure . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.4

Evaporation and Condensation . . . . . . . . . . . . . . . . . . . . . . .

67

5.5

Volume and in situ shape . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.5.1

Harmonic density profile . . . . . . . . . . . . . . . . . . . . . .

68

5.5.2

Uniform density profile . . . . . . . . . . . . . . . . . . . . . . .

68

5.5.3

General power-law traps . . . . . . . . . . . . . . . . . . . . . .

69

Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.6.1

Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.6.2

Numerical implementation . . . . . . . . . . . . . . . . . . . . .

72

How uniform is the trap? . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.7.1

Conclusions from a single cloud profile . . . . . . . . . . . . . .

74

5.7.2

Conclusions from several profiles . . . . . . . . . . . . . . . . .

75

5.7.3

The critical point . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.7.4

Finding the critical point . . . . . . . . . . . . . . . . . . . . . .

80

5.7.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.6

5.7

8

CONTENTS

6

Thermodynamics in a uniform trap

83

6.1

Critical number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.1.1

Experimental procedure . . . . . . . . . . . . . . . . . . . . . .

84

6.1.2

Self-consistency . . . . . . . . . . . . . . . . . . . . . . . . . .

85

6.1.3

Dependence of volume on temperature . . . . . . . . . . . . . .

86

6.1.4

Critical density . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Saturation of the thermal component . . . . . . . . . . . . . . . . . . . .

87

6.2.1

Mean-Field theories . . . . . . . . . . . . . . . . . . . . . . . .

87

6.2.2

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . .

93

6.2.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Quantum Joule-Thomson effect . . . . . . . . . . . . . . . . . . . . . . .

94

6.3.1

Joule Thomson effect in a classical gas . . . . . . . . . . . . . .

96

6.3.2

Joule-Thomson effect in a condensed quantum gas . . . . . . . .

97

6.3.3

QJT in a degenerate thermal gas . . . . . . . . . . . . . . . . . .

99

6.3.4

Experimental observation of the QJT . . . . . . . . . . . . . . . 100

6.3.5

Experimental details . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2

6.3

7

Bragg spectroscopy
7.1

7.2

7.3
8

107

Theory of Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.1

Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1.2

Momentum distributions . . . . . . . . . . . . . . . . . . . . . . 112

7.1.3

Mean-Field corrections . . . . . . . . . . . . . . . . . . . . . . . 114

Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2.2

Heisenberg uncertainty relation . . . . . . . . . . . . . . . . . . 116

7.2.3

Mean-field shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2.4

Evolution in ToF . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Conclusions and Future Work125

A Polylogarithms and the Gamma function

127

B IGBT driver129

B.1 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.2 Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B.3 Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.4 Gate charging and protection . . . . . . . . . . . . . . . . . . . . . . . . 133
9

CONTENTS

C Design of a high-temperature superconducting magnet
137
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.1.1 Trapping fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.1.2 Water-cooled Magnets . . . . . . . . . . . . . . . . . . . . . . . 139
C.1.3 HTS Magnet requirements . . . . . . . . . . . . . . . . . . . . . 139
C.2 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
C.2.1 Current leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
C.2.2 Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.2.3 Filling system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C.3 Geometric design and fields . . . . . . . . . . . . . . . . . . . . . . . . . 150
C.4 Coil electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.4.1 Quench protection . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.4.2 Switching performance . . . . . . . . . . . . . . . . . . . . . . . 154
C.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.6 Technical drawings of the cryostat . . . . . . . . . . . . . . . . . . . . . 160

10Chapter 1
Introduction
1.1

Bose-Einstein Condensation

The advent of quantum mechanics in the early 20th century has revolutionized the way
we think about particles. Rather than considering them as rigid spheres bouncing around,they can be described as wave functions indicating the probability of finding them in agiven location. As the name indicates, the energies quantum-mechanical particles can
carry are not continuous, but discrete: they are quantized. A particle with a given energy and having the associated wave function is said to be in a quantum-mechanical state.
While those assumptions already lead to surprising behaviour, such as tunnelling throughseemingly impassable barriers, it gets even more confusing when considering groups ofparticles sharing a given space. Particles of the same type are fundamentally indistinguishable and exchanging them cannot alter the physical reality. This requirement leads
to the conclusion that the total wave function only can get multiplied by a factor of ±1
upon exchange of the particles. Particles can now be classified accordingly: Those witha symmetric wave function are called bosons, and those with an antisymmetric wave
function are called fermions. This difference in overall symmetry has far-reaching consequences. In simple terms, bosons like to be near each other, while fermions do not.
The Pauli exclusion principle states that two fermions cannot occupy the same state. If
we now consider the ground state of a system- the collective configuration with the least
total energy - with several particles of the same kind, we notice a fundamental difference
which is illustrated in Fig. 1.1. In the fermionic case, there is one particle in each energy
state, filling the states subsequently starting with the one with the least energy. Bosons,however, like to share and all of them jointly populate the lowest energy single-particle
state.
In many-particle physics, we are usually dealing not with a few particles as shown in
Fig. 1.1, but with thousands, millions or even more. At this point, it makes more sense to11

1.2. THE BOX AND THE HARMONIC OSCILLATOR

Figure 1.1: Bosons and fermions in their respective ground states.
use a statistical approach and not track each particle individually. From the temperatureof the ensemble, we can deduce a prediction of how many particles we expect in eachstate. For gases at high temperatures, the probability of finding an atom in a given stateis very small and the difference between fermions and bosons becomes negligible. At
low temperatures, we start to approach the zero temperature limit sketched in Fig. 1.1. In
the case of Bosons1 , once we cool them enough, they start to occupy the lowest energy
state in large numbers. Interestingly, we see significant occupation of the lowest state
before we reach zero temperature; it occurs already at the so-called critical temperature.
Under typical experimental conditions, this critical temperature can be hundred times
larger than the energy gap between the lowest and first energy level. This effect is called
Bose-Einstein condensation and was predicted as early as 1924 [1,2]. So how cold is coldenough to see this effect? We can estimate this by considering the spatial extend of the
wave function of a single atom. For a given temperature T this can be approximated by
the thermal wavelength Λ, which is defined as
Λ= √

h
,
2πmkT

(1.1)

where m is the mass of the atoms in question. Once Λ gets comparable to the inter-particle
spacing, the wave functions overlap. Several particles share the ground state and BoseEinstein condensation occurs. We can express this condition in terms of the phase-spacedensity D
D ≡ nΛ3 ≈ 1.
(1.2)
This crude condition is surprisingly accurate in predicting the onset of condensation andis correct up to a numeric factor.

1.2

The box and the harmonic oscillator

To produce atomic Bose-Einstein condensates (BECs), we usually need to cool atoms
into the nano-Kelvin regime. The technical challenges are enormous and it took until
1

While interesting things happen to Fermions as well, we are not concerned with them in this thesis.

121.2. THE BOX AND THE HARMONIC OSCILLATOR

1995 [3–5] to overcome them. The usefulness of atomic BECs stems from several factors:
Unlike in superfluid 4 He, the atomic interactions are weak and can be neglected in many
situations, or at least their effects can be described within simple mean-field frameworks.
Even better, Feshbach resonances allow tuning the interactions in many atomic species [6]to the experimenters liking. This makes them a great model system for understanding
many-body physics problems, such as superconductivity or two-dimensional systems.
We have not addressed one question yet: in which potentials are the atoms confined?
During an undergraduate course in quantum mechanics, the first and simplest potentialthat is inevitably solved is the particle in a uniform (or homogeneous) box. The next major
problem that pops up is the harmonic oscillator, which everybody will admit is muchharder to solve. The same is true for BECs: Einstein’s original prediction was made for ahomogeneous gas and theorists usually still prefer to solve problems in uniform problems.
Ironically, it is experimentally much easier to create a harmonic trap than to create a
uniform trap. Nature is generally very fond of smooth potentials. This is especially true
when working at extremely low temperatures, where the smallest deviations destroy thehomogeneity of the trap. For those reasons BECs have traditionally been produced inharmonic traps.
While the behaviour of an ideal quantum gas is well understood even in a harmonic
potential, it gets complicated once interactions are thrown into the mix. Also, most solidstate systems, which cold gas researchers are trying to emulate, are homogeneous in nature. Often, the effect of the harmonic trap can be addressed - sometimes even in a beneficial way - by using the local-density approximation (LDA) [7–15]. Within the LDA,
we treat the whole non-uniform trap as a collection of small homogeneous subsections ofdifferent depth. However, this is not always possible; integrating over the varying density
can smear out or even qualitatively change experimental signatures (see, e.g., [10,16–19]).
Moreover, the LDA breaks down close to phase transitions, where the correlation length
diverges [20]. In the drive towards a homogeneous trap, BECs have been loaded into
modified geometries [21, 22]. However, those traps are uniform along only one direction;
they are still harmonic along the other two directions.
In this thesis, we are going to describe the creation of the first cold atom experiment capable of producing three-dimensional homogeneous BECs. Apart from describing the technical challenges, we will describe the follow-up measurements we have madeto quantify the homogeneity of our trap. Further, we have already been able to prove theusefulness of the uniform trap by measuring effects that have been obscured in harmonic
traps.
13

1.3. THESIS OUTLINE

1.3

Thesis outline

The main results of this thesis have been published in several papers. Below, we give an
outline of the thesis and the respective publication for reference.
• Chapter 2: We summarize the essential theoretical foundations of the thermodynamics of quantum gases in power-law.
• Chapter 3: We introduce the theoretical foundation of cooling and trapping atomic
gases.
• Chapter 4: We give an overview of the experimental setup and procedures we use
to produce harmonic BECs. See also [23].
• Chapter 5: We describe the technical details of loading harmonic BECs in a uniform
trap. We also discuss the characterization of the trap, proving its uniformity. See
also [24].
• Chapter 6: We present our measurements of the thermodynamics in a box. We characterize the critical point for condensation and explore the influence of interactions
on the thermal fraction. We also observe the quantum Joule-Thomson effect. For
all of those effects, we recover ideal gas scalings despite weak interactions. See
also [25].
• Chapter 7: We use Bragg spectroscopy to study the ground state properties of a
uniform gas and measure the Heisenberg uncertainty relationship between spaceand momentum. Also, we measure the mean-field shift due to interactions. Finally,
we explore the evolution of the condensate in time-of flight. See also [26].
These results are also documented in Igor Gotlibovych’s thesis [27].

14Chapter 2
Bose-Einstein Condensation
This chapter will provide a brief overview of the thermodynamics of Bose gases, bothdegenerate and classical in power-law traps. We consider the static behaviour in trap aswell as the dynamics in time of flight (ToF) when released from the trap. Most of theseresults are well known and detailed derivations can be found in textbooks (e.g. [28]). We
will summarize the relevant results as they are instrumental for the thermometry of gases,
the analysis of the box trap, as well as for the following thermodynamic studies.

2.1

Bose-Einstein Condensation in a homogeneous potential

Here, we will briefly present the derivation of Bose-Einstein condensation in a box potential. This will mainly serve to introduce the basic concepts and variables, before moving
on to power-law potentials. The starting point for most calculations is the Bose distribution f () which predicts the occupation number of a state with energy  for non-interacting
Bosons in thermal equilibrium:f () =1eβ(−µ) − 1

,

(2.1)

with β = 1/kB T . µ denotes the chemical potential, which acts as a normalization constantfor the total atom number N . µ must always be equal or smaller than the ground stateenergy 0 . Unless stated otherwise, we usually set 0 = 0. The total atom number N isgiven by the sum over the occupation number of all states:N=X
f ().


15

(2.2)

2.2. THERMODYNAMICS IN POWER-LAW POTENTIALSThis can be approximated by converting the sum over the energy states to an integral overthe density of states
V m3/2
D() = 1/2 2 3 1/2 ,
(2.3)
2 π ~
where V is the volume of the box the particles are confined in. However, we need to treat
the ground state population N0 separately to avoid divergence for µ = 0:Z∞
N ≈ N0 +f ()D() d  ≡ N0 + N 0 .

(2.4)

0

Here, we introduced the thermal number N 0 , counting all atoms in excited states. Analysisof the integral shows that N 0 reaches its maximum value Nc (T ) for a given temperaturewhen µ = 0. So what happens when we increase the atom number further? The atoms
all have to accumulate in the ground state and N0 = N − Nc (T ). This sudden onset ofmacroscopic occupation of the ground state is called Bose-Einstein condensation. When
fixing the number instead of the temperature, we refer to the temperature at which thisphase-transition happens as the critical temperature Tc . Solving Eq. (2.4) for the criticalnumber yields
ζ(3/2),(2.5)
Nc (T ) = N 0 (T, µ = 0) = V
Λ3
where ζ(3/2) is the Rieman zeta function, which is defined as the polylogarithm function
g3/2 (1). Appendix A contains the definitions of those functions, as well as some usefulintegrals commonly used in the solution of equations such as Eq. (2.4). We remember theestimate Eq. (1.2) which predicts a critical density ofnc (T ) = Nc /V ≈ Λ−3 .

(2.6)

Comparison shows that the estimate is only off by a factor ζ(3/2) ≈ 2.61.

2.2

Thermodynamics in Power-law potentials

After having considered the onset of Bose-Einstein condensation in a uniform trap, we
will now generalize these methods to derive the thermodynamics in general power-law
potentials. Most importantly, we are going to introduce the semi-classical local-density
approximation, allowing us to relate the thermodynamics in arbitrary potential to the uniform case. Further, we are going to consider the equilibrium in-trap density and momentum distributions. More detailed information can be found in [28, 29].
16

2.2. THERMODYNAMICS IN POWER-LAW POTENTIALS

2.2.1

The local density approximation

We will discuss a degenerate Bose gas in an arbitrary power-law potential in three dimensions
3
s
X
xi i
U (r) =
,
(2.7)
Li
i=1
and derive thermodynamics scalings. For all calculations, we will neglect interactions
with the ground state. The occupation of levels with energy  is given by1f () =eβ(−µ) − 1

.

(2.8)

To calculate meaningful thermodynamic quantities from this, we need to extract the density of states by solving the HamiltonianH=

p2+ U (r)2m(2.9)

for the eigenstates. While this can be done easily for harmonic or uniform potentials - infact they are the first thing solved in most introductory textbooks - this is much harder for
arbitrary potentials. Even worse, if we want to evaluate the density distribution n(r) of
our atoms in trap, we need to add up all the wave functions:n(r) =Xf () |ψ |2 .

(2.10)This is rather cumbersome (see [27] for a quantum mechanical treatment of a box potential). The full quantum-mechanical treatment can be avoided by using the semi-classicalapproximation, which assumes the gas behaves locally as a bulk gas. Often it is also referred to as the local-density approximation. It holds if the thermal wavelength is small
compared to the length scale of the variation of the trap. Assuming the energy of classicalparticlesp2+ U (r),(2.11)
(p, r) =2m
we can now modify Eq. (2.8) tof (p, r) =1eβ((p,r)−µ) − 1= g0 (exp[−β((p, r) − µ)]) ,(2.12)

where we introduced the polylogarithm function of order zero g0 (). Its definition andproperties are detailed in Appendix A.
To obtain thermodynamic quantities, we have to integrate over phase-space, e.g. we
17

2.2. THERMODYNAMICS IN POWER-LAW POTENTIALS
can calculate the thermal number N 0 as
ZZ
dp
0
N =
dr 3 f (p, r)h(2.13)

Often, it is simpler to use the (semi-classical) density of states, which can be evaluatedas [29, 30]
ZZ1
δ( − U (r) − p2 /2m)drdp = Cα α−1 ,(2.14)
D() = 3hwhere Cα is a function of the trap dimensions. The scaling parameter α can be derived
from the trap-exponents si as
α = 3/2 + 1/s1 + 1/s2 + 1/s3 .

(2.15)For a uniform gas, we have si = ∞ and hence α = 3/2. The trap dimensions enter as
V m3/2
C3/2 = 1/2 2 3 ,2 π ~(2.16)

where V is the volume of the box. In a harmonic trap the potential is given byX
1
U (r) = m
ω 2 r2 ,2 i=x,y,z i i(2.17)

where ωi are the trapping frequencies and the scaling parameter α = 3. The prefactor isgiven by1
(2.18)
C3 = 3 3 ,2~ ω̄
where ω̄ is the geometric mean of the individual trapping frequencies. Note that we canget the same results by deriving the density of states quantum-mechanically.

2.2.2

Thermodynamics in power-law potentials

With the semi-classical density of states at hand, we can derive thermodynamic quantities
for arbitrary power-laws with the usual methods [28]. For now, we consider a partiallycondensed gas T < TC , i.e. we can set µ = 0. The critical number can then be calculated
as:Z
Nc = dD()f (, µ = 0)(2.19)
Z= dα−1 g0 e−β(2.20)= Cα ζ(α)Γ (α) (kT )α ,18

(2.21)

2.2. THERMODYNAMICS IN POWER-LAW POTENTIALS

where Γ (α) is the Gamma function (see Appendix A). Similarly we can obtain further
thermodynamic properties of the partially condensed gas:
Table 2.1: Thermodynamic properties of a (partially) condensed gas in a power-law
trap.
Property

Value

Critical number Nc
Condensed Fraction N0 /N

Cα ζ(α)Γ (α) (kTc )α α
1 − TTc

Energy E

Cα ζ(α + 1)Γ (α + 1) (kT )α+1

Entropy S

ζ(α + 1)Γ (α + 1) (kTT)
Cα (α+1)α

α+1We see that the parameter α = 3/2 + 1/s1 + 1/s2 + 1/s3 reflects the scaling of thedensity of states and will determine the relevant thermodynamic scalings of the gas. These
results can be easily extended to cover thermal gases with µ < 0 and T > Tc .

2.2.3

Momentum and spatial distributions

While the LDA is useful for determining thermodynamic properties of power-law potentials, this can be also achieved by solving the Hamiltonian to obtain the quantummechanical density of states. As mentioned before, its true strength lies in deriving local
properties, e.g. the density and momentum distribution of the atoms in trap. This is applicable for the thermal atoms N 0 only; the condensate has a macroscopic wave function
and behaves fully quantum-mechanically. Therefore, we will address the thermal density
n0 (r) and the condensed density n0 (r) separately.
The degenerate thermal gas
To calculate the in-trap density n0 (r), we integrate over the momentum in Eq. (2.12),
using the integrals from Appendix A. We obtainZ∞0n (r) =

dp  −(β/2m)p2
g0 e
A(r, T )h3(2.22)−∞=

g3/2 (A(r, T )),
Λ3

(2.23)

where we set A = exp(βµ−βU (r)). We notice that each component of the momentum p2i
increased the power of the polylogarithm by (1/2) and scales the equation with the thermal
192.2. THERMODYNAMICS IN POWER-LAW POTENTIALS

wavelength. This equation also underlines that condensation will occur locally as soon as
the local phase-space density reaches ≈ 2.61. Similarly, we can sequentially integrate thespatial components of Eq. (2.13) to obtain the in-trap momentum distribution:Z∞0n (p) =



−β

g0 e

x i sii=1 LiP3A(p, T ) dr

(2.24)−∞∝ (kT )δ gδ (A(p, T )) ,(2.25)

where A = exp(βµ−βp2 /2m) and δ = α−3/2 = 1/s1 +1/s2 +1/s3 . The prefactors were
dropped for readability. We see that trap -power anisotropy gets “averaged-out” when
considering the scaling of momentum distribution. This explains why a single parameter
α captures the interesting physics; it tells us all we need to know about the momentum
distribution.

2.2.4

The condensate wave function

The behaviour of the ground state of an interacting Bose gas can be described with theGross-Pitaevskii equation (GPE):∂Ψ(r, t)~2 2
∇ Ψ(r, t) + U (r)Ψ(r, t) + g |Ψ(r, t)|2 Ψ(r, t) = i~.
2m∂t(2.26)

This is a non-linear Schroedinger equation for the wave function Ψ(r, t) of the condensate.
The non-linearity is introduced by a mean-field term, which is proportional to the local
condensate densityn(r) = |Ψ(r, t)|2 ,(2.27)
where the interaction parameterg = 4π~2 a/m(2.28)

scales with the s-wave scattering length a. In the static case, this can be written similar to
the time-independent Schroedinger equation~2 2∇ Ψ(r) + U (r)Ψ(r) + g |Ψ(r)|2 Ψ(r) = µΨ(r),2m(2.29)

where the chemical potential µ = ∂E/∂N takes on the role of the energy eigenvalues.
Thomas-Fermi approximation
We usually operate in a regime where the mean-field energy dominates over the kinetic
energy. Neglecting the kinetic energy term is referred to as the Thomas-Fermi approxi20

2.3. EXPANSION AND THERMOMETRY OF COLD GASES

mation and allows us to determine the density asn0 (r) = maxµ − U (r),0 ,g(2.30)

which means that the condensate “fills up” the potential up to the chemical potential. Ina harmonic trap, the condensate has thus a parabolic profile, where the half-widths Ri are
related to the frequencies wi as
2µ
Ri2 =
,
(2.31)
mwi2
where µ can be found by equating the integral over the density to the total atom number.
In the uniform trap, the density is particularly simple to obtain:
n0 (r) = µ/g(2.32)and integration gives µ = gN0 /V . Hence, within the TF-approximation, the density of
an interacting condensate is homogeneous in a uniform trap at T = 0. This is in stark
contrast to a non-interacting gas, where the wave function of the condensate takes thesine-like solution of a single particle in a box.
At the edges of the box, the density of the condensates jumps from zero to the finite
value in Eq. (2.32). Therefore, the Thomas-Fermi approximation is clearly not applicableat the boundaries of the box as the kinetic energy ∝ ∇|Ψ(r)|2 is infinite there. We have totherefore introduce a region at the boundary where the density varies smoothly from zero
to its final value. This length scale is called the healing length ξ and can be estimated byequating the kinetic energy with the mean-field energy:
~2
= ng.
2mξ 2

(2.33)The healing length also allows us to estimate the validity of the Thomas-Fermi approximation: once the healing length is on the order of the length of the box, we need to solve
the full GPE (usually numerically) to obtain a density profile.

2.3

Expansion and Thermometry of cold gases

If we switch off our trap, the atomic gas expands. From the density distribution after
a given time of flight, we can extract the in situ momentum distribution. This not only
allows us to determine the temperature of the gas, but also gives us information about thetrap shape. Assuming a classical ballistical expansion of the atoms, we can determine
21

2.3. EXPANSION AND THERMOMETRY OF COLD GASES
from the in-trap distribution, f (r0 , p, t = 0), the density n(r, t) at any time t asZn(r, t) =δ(pt/m − (r − r0 ))f (r0 , p, t = 0)dr0 dp.

(2.34)In the following we will derive the appropriate expressions in various trap geometries inthe thermal, degenerate and condensed regimes.

2.3.1

Thermal gas

Generally, we are only interested in thermal clouds for the optimization of the experimental sequence on the road towards degeneracy. The distribution of a thermal gas obeys
Maxwell-Boltzmann statistics, i.e.
f (r, p) ∝ exp(−[p2 /2m + U (r)]β).

(2.35)

Using this distribution, it is easy to see that for long time of flight the density in Eq. (2.34)
is only determined by the momentum distribution, which is Gaussian. We see that for thermal gases, the cloud shape loses the initial trap-shaped aspect ratio and expands isotropically. If the trap can be approximated as an anisotropic harmonic trap, we can derive ananalytic form for any time of flight:
−mωi2 ri2
eµ/kB T Y
2 2 −0.5(1 + ωi t )exp.
n(r, t) =λ3
2kB T (1 + ωi2 t2 )
i

2.3.2

(2.36)

Degenerate gases

In degenerate gases, we have to use the Bose-Einstein distribution in Eq. (2.34) and obtainZ

0n (r, t) =g0m(r − r0 )2+ U (r0 ) − µ
−β
2
2t




dr0 .

(2.37)

It is obvious that the limiting case for t = 0 is the in situ distribution of Eq. (2.23), while
for large ToF the distribution is given by the momentum distribution in Eq. (2.25). Theintermediate regime, where the initial size and the distance travelled are comparable, is
not trivial though. While it is tempting to write the density distribution at any time as
the convolution of the in-trap density distribution with the momentum distribution, thefollowing equation is generally not true:0n (r, t) ∝Zn0 (p = (r − r0 )m/t) × n(r0 )dr0 .
22(2.38)

2.3. EXPANSION AND THERMOMETRY OF COLD GASES

The reason for this is that momentum and position of a single atom in trap are correlated
in a power-law potential, and we cannot treat the expansion of the cloud as the sum ofidentical point-sources. There are two notable special cases: we can find a closed form
for the harmonic trap which greatly simplifies fitting:
#!
"
3
2
X
1
x
i
,
n0 (r, t) ∝ 3 g3/2 exp µ −
2
Λ
R
i
i=1
where the thermal radii ares
Ri =1 + wi2 t2
βmwi2(2.39)(2.40)

determined by the trapping frequencies wi and the time of flight. It is no coincidence thatthe harmonic equation can be solved analytically; we exploit the fact that momentum andpotential enter the Hamiltonian with the same square power-law.
Second, in a uniform trap, the in-trap density is homogeneous. We use a potential0 r∈VU (r) = 

∞ r∈
/V

(2.41)

Therefore Eq. (2.38) does apply in this special case such that(r − r0 )2 mn (r, t) ∝ g0 −β
−µ
dr0
2t2
V



Z
(r − r0 )2 m
= g0 −β−µ× n0 (r, t = 0)dr0
2t2
0

Z

(2.42)(2.43)and so we can write the in-ToF distribution as the convolution of the momentum distribution and the in situ distribution.

2.3.3

Condensed gases

The situation is more complicated for BECs. Without interactions, their expansion can becalculated by propagating the ground state wave function with the free-space Schroedinger
equation. Once we include repulsive interactions (for 87 Rb), we need to use the GrossPitaevskii equation (Eq. (2.26)) in the same fashion. For most potentials, these solutions
have to be solved numerically.
Even though we never actually produce a non-interacting condensate, it is instructive
to consider the expansion from a harmonic potential without interactions. The ground23

2.3. EXPANSION AND THERMOMETRY OF COLD GASES

state is simply a Gaussian
Φ(r)20 ∝Y22e−xi /2∆xi(2.44)iwith ∆x2i = ~/(mwi ). The distribution in momentum space is given by the Fouriertransform, which is again Gaussian. The width in momentum space is hence given by∆pi ∆xi = ~/2. In an anisotropic potential, the tight directions will therefore have thegreater momentum width. Expansion will thus happen anisotropically, inverting the aspect ratio of the cloud after a long time of flight. The width σi of the Gaussian-shaped
cloud in ToF is [31]2~
(1 + wi2 t2 ).
(2.45)
σ(t)2i =mwi
This aspect ratio inversion is a common feature for non-interacting, anisotropic condensates irrespective of the trapping potential. In the ground state, the uncertainty principle
dictates ∆xi ∆pi & ~/2. Hence, if we compress one direction in situ, it will necessarily
expand more in ToF.
In our experiments, we cannot neglect interactions and we need to numerically invoke
the full GPE to calculate the expansion. An exception is the harmonic trap, for which thecloud retains its parabolic shape during expansion [32, 33]. Conveniently, the aspect ratioof the rather common cigar-shaped, harmonic traps (w1 = w2  w3 ) inverts as well.
However, this time the inversion is not driven by the ground-state momentum, but by theinteraction energy. In simple terms, the gradient in density and hence acceleration during
ToF will be strongest along the short directions cause the inversion after long enough ToF.
We conclude that the inversion of the aspect ratio is a strong indicator of Bose-Einsteincondensation; it can be driven by either the Heisenberg uncertainty of the ground state orby the mean-field acceleration originating from atomic interactions.

2.3.4

Thermometry of partially condensed gases

So far, we have only considered the thermal and condensed gases separately. When considering partially condensed gases, we usually operate under the assumption that the thermal and condensed components of the gas do not affect each other. We should note that
this assumption is not strictly true. Especially in harmonic traps, the expected atom numbers deviate from the ideal gas prediction due to the mean-field energy of the condensate.
However, we can still use the wings of the thermal component to deduce the temperatureof the sample. These effects of the condensate on the thermal atoms have been scrutinised
in a series of papers from our group [10, 16, 34].
24

2.3. EXPANSION AND THERMOMETRY OF COLD GASES

Harmonic Trap
In a harmonic trap, we obtain a bimodal distribution in ToF where the thermal polylogarithmic distribution (Eq. (2.39)) and the parabolic condensate profile (Eq. (2.30)) aresuperimposed on top of each other. Using a multi-pass fitting algorithm, we can use thethermal wings to deduce the temperature and the shape of the thermal and condensed fractions to determine the respective atom numbers. The procedure is well-established and is
described in depth in [27, 31, 35, 36].
General Power-Law Traps
For general power-law traps, the shape of the expanding condensate has no known analytical solution. While numerical solutions can be obtained using the GPE, using them for
fitting is highly impractical. We therefore resorted to fitting the thermal part of the wingsand extrapolating them into the region with a condensate. The condensate number canthen be obtained by counting the total number and subtracting the fitted thermal part. We
will discuss this procedure in detail in Chapter 5 for quasi-homogeneous traps.

2.3.5

Column densities

Experimentally, we cannot observe three-dimensional densities n(x, y, z); instead we observe the two-dimensional column density ñ(x, y), which is defined asZñ(x, y) =n(x, y, z) dz.

(2.46)

Fortunately, integration is pretty straightforward for most density profiles. Often they areseparable along the trap axis; in the case of polylogarithms they can be evaluated using
the integrals in appendix A. The experimental details of measuring the column densities
with absorptive imaging can be found in Section 4.4.

25

2.3. EXPANSION AND THERMOMETRY OF COLD GASES

26

Chapter 3
Atom Cooling and Trapping
Often, BECs are created using alkali atoms, due to their simple hydrogen-like atomic
structure. An atom source is placed in a vacuum system and the atoms are trapped with amagneto-optical trap (MOT). Laser cooling allows cooling down to the µK range. Further
cooling is achieved by forced evaporation down to the nK range until condensation occurs.
We will briefly review the theory for the cooling mechanisms with the focus on the relevant quantities for our experiments. In-depth explanations can be found in [28,31,37,38].

3.1

Atomic properties of Rubidium

Alkali energy levels As 87 Rb has only one electron in the outermost shell, it has ahydrogen-like level structure. Still, analytical calculation of its energy levels is not possible and exact results rely on numerical analysis and experimental measurements. Unlike
in hydrogen, the energy levels in alkalis are not degenerate in the angular momentum L,
even before taking fine structure corrections into account. Taking the interaction between
the spin S and angular momentum L into account, the energy levels are best describedusing
J = L + S,
(3.1)
pwhere |J | = J(J + 1)~ and similarly for L and S. Taking the atomic nuclear magnetic
momentum I into account, we obtain the hyperfine splitting which is parametrized by
F = J + I. The hyperfine levels are degenerate in the eigenvalues mF = −F, −F +1, ..., F − 1, F of the z-component of F , where we assume the z-axis is the quantisation
axis of the system. In the presence of small magnetic fields Bz along this quantisation
axis, the Zeeman effect breaks this degeneracy. The energy difference ∆E from theoriginal hyperfine level is given by∆E = µB gF mF Bz ,
27

(3.2)

3.2. LASER COOLING

Figure 3.1: The 87 Rb D2 energy level scheme. Data taken from [39], drawing takenfrom [40].
where µB = 9.27 × 10−24 J/T is the Bohr magneton gF are the numerical Lande factors.
87

Rb levels For experimental purposes, we are mainly interested in the D2 line (52 S1/2 →52 P3/2 ) at approximately 780 nm. The D1 line (52 S1/2 → 52 P1/2 ) is located at 795 nm
and is experimentally irrelevant. For most intents and purposes we can consider our atoms
as a two-level system, split only at the hyperfine structure level. For simplicity, we referto the levels of 52 S1/2 , F = 1, 2 simply as F = 1, 2 and add a dash F 0 = 0, 1, 2, 3 torefer to 52 P3/2 , F = 0, 1, 2, 3 . The D2 energy levels of 87 Rb are shown in figure 3.1.
An excellent compilation of data on 87 Rb can be found in [39].

3.2

Laser cooling

The first cooling step in BEC experiments is laser cooling. A photon with momentum ~ktransfers momentum to an atom when it is absorbed. If we consider a photon beam, we
obtain a net time-averaged force F SC = ~kΓSC , where we used the scattering rate ΓSC
and k is the wavevector of the light. The atom will decay back to its ground state by spon28

3.2. LASER COOLING

taneous emission at a rate Γ. Since the emitted photon will leave in a random direction,
this contribution will average out over time and only the absorption will contribute to theaverage force. By tweaking this scattering force so it opposes the direction of movement,
one can slow the atoms. This technique is called optical molasses. Adding a carefully
designed spatial dependency, the atom can be confined in a trap, as photons will pushthem back when they try to escape. This is done in a magneto-optical trap (MOT). For
simplicity, we will only consider an imaginary two-level F=0 to F’=1 energy scheme.

Optical molasses The absorption rate and thus the force of a low intensity laser beamon an atom can be derived by using the optical Bloch equations. We assume that thelaser light has a wavenumber k and the corresponding angular frequency ω = ck. If
the resonance frequency of the F=0 to F’=1 transition is ω0 , the detuning is given δ =ω − ω0 + kv, where v is the atom velocity and the term kv accounts for the Doppler effect.
The scattering force is given by [37]:
FSC = ~kI/IsatΓ,
2 1 + I/Isat + 4δ 2 /Γ2(3.3)

where the saturation intensity is given by Isat = 2π 2 ~Γc/3λ3 . In one dimension, overlapping two red-detuned (i.e. ω0 − ω > 0), counter-propagating beams ensures that the beam
opposing the velocity of the atom gets absorbed more due to the Doppler shift. An atom
moving through the beams will experience a velocity-dependent damping force, hence
the name optical molasses. The principle can be easily extended to three dimensions byusing six beams.

The Magneto-Optical Trap (MOT) In optical molasses, atoms are slowed down, butthey are not trapped and will eventually escape the area exposed to the laser beams. We
therefore introduce a restoring force, which pushes the atoms back towards the centre of
the trap. This is done by applying a magnetic quadrupole field, which has a magnetic zeroat the trap centre and increases linearly along the axes: Bx ∝ x, By ∝ y and Bz ∝ z (seeFig. 3.2a). Again, we will consider only the one-dimensional case for simplicity. The
excited level F’=1 splits due to the Zeeman effect (Eq. (3.2)) into non-degenerate mF =−1, 0, +1 states. The energy splitting ∆E between the states increases linearly with the
distance from the trap centre, ∆E ∝ z, this is illustrated in Fig. 3.2b. We use circularly
polarized light σ + and σ − , which addresses only transitions with ∆mF = ±1 respectively.
Away from the trap centre, predominantly the polarization corresponding to the mF state
which is closer to resonance will be absorbed. By choosing the polarization of the beams
correctly (see figure Fig. 3.2a for more details), we make sure that predominantly the29

3.2. LASER COOLING
(a)

(b)

RHCP
x

E

mF
-1

I

LHCP

+1

z
y

s+

mF
+1

F=1

0
B

LHCP

B

0
s-

LHCP

-1
LHCP

F=0

I
RHCP

z

Figure 3.2: Operation principle of a MOT. (a) A pair of coils (current direction ingreen) in anti-Helmholtz configuration creates a quadrupole magnetic field (blue) at
the atoms. We show the correct polarizations of the laser beams (red) needed to createa MOT. The fields and polarizations along the y-axis are identical to those shown
along the z-axis. Drawing taken from [40]. (b) The operation principle of a MOT.
The magnetic field splits the energy levels according to the Zeeman effect. Circularly
polarized light σ ± selectively drives mF = ±1 transitions. Combined with the spatial
dependence of the energy levels, this results in preferential absorption of the beam
which pushes the atoms back towards the trap centre. Drawing taken from [40].
Note that ’left/right-hand circularly polarized’ (LHCP/RHCP) refers to the directionof propagation of the light; σ ± is defined with respect to the quantization axis of theatoms (here z).
beams pushing the atoms back get absorbed. Thus, we have created a trap.
If the experimental parameters are chosen right, we can combine the cooling of themolasses with the magneto-optical trapping. Mathematically, we describe this by including the Zeeman shift in Eq. (3.2) in the detuning δ in Eq. (3.3). Near the trap centre, thisresults in a restoring force linear in the displacement z and opposed to the atom velocityv:
FM OT (v, z) ≈ −αv − βz,(3.4)
where α and β absorb experimental parameters [37]. Again, the principle is easily extendible to three dimensions.
In practice, the MOT beams and fields are switched on in a vacuum with a small,
residual 87 Rb background pressure. Slow enough atoms straying into the area where
Eq. (3.4) applies are cooled and trapped. Over time, the number of atoms in the MOT
grows until the capture rate is balanced by loss processes (e.g. collisions with hot nontrapped atoms).
Limits of cooling The theoretical limit of the laser cooling mechanism as described so
far is given by the Doppler temperature TD ,kB TD =
30

~Γ
2

(3.5)

3.3. EVAPORATIVE COOLING

and is caused by the statistic nature of the absorption and emission momentum kicks [28].
For Rubidium, this temperature is TD = 144µK. To the surprise of the first experimenters
and in violation of Murphy’s law, temperatures below this were achieved, thanks to aprocess called Sisyphus cooling. The process relies on having a Zeeman structure in the
ground state and cannot be explained in the simple F=0 and F=1 energy scheme employedso far. We are not going to go into detail here, see [28, 37, 41] for more information.
For us, it is important that this process allows us to beat the Doppler temperature and is(theoretically) limited by the recoil energy of a single photon. This corresponds to theso-called recoil temperature Tr , with
Tr =

(~k)2
.
2mkB

(3.6)

For 87 Rb, Tr ≈ 362 nK  TD . The Doppler limit gives a good estimate for experimentally viable MOT temperatures. Including a brief molasses stage after cooling the atoms
in a MOT, i.e. switching off the magnetic confinement, allows us to take advantage of theSisyphus cooling process. While experimentally we do not even get close to Tr , we docool significantly below TD .1

3.3

Evaporative cooling

3.3.1

Evaporative cooling in Power-Law Traps

After the MOT stage, the atoms are loaded into a new trapping potential for further cooling. A range of different trapping potentials exist, most of them based on optical ormagnetic confinement. As we are still far from degeneracy (where the phase-space density D = nΛ3 ≈ 1), we need to cool the atoms further. A trapping potential of finite
depth U0 allows the most energetic atoms with energy kB T > U0 to escape, truncating
the Boltzmann distribution. Elastic collisions then rethermalize the remaining atoms andthe average temperature drops. This process is called evaporative cooling.
However, once kB T  U0 , the cooling process will come to an asymptomatic standstill. Hence, to keep the cooling process going (and produce BECs in a finite time), welower the trapping potential continuously. The slower we reduce U0 , the fewer atoms weneed to expend to cool by a given amount. In a perfect system we could reduce the trap
height infinitely slowly, allowing us to cool the gas while losing only a negligible numberof atoms. However, unwanted atom loss and heating processes compete with the coolingand continuously reduce the phase-space density. Hence, to increase D, we are forced to1

Typically, we achieve T ≈ 15 µK post-molasses; see Chapter 4.

31

3.3. EVAPORATIVE COOLING

cool at finite rate. We will derive a simple model following [28, 42] to derive the principles which govern the process of evaporative cooling in a range of power-law trapping
potentials. Let us consider an ideal, non-degenerate gas in a power-law potential:
U (r) =

3
X

Ui |ri |si

(3.7)

i=1

We recall the semi-classical density of states for a power-law potential from Eq. (2.14)
D() = Cα α−1 ,(3.8)

as well as the power-law parameter α = 3/2 + 1/s1 + 1/s2 + 1/s3 . For a classical gas it
is convenient to consider the single-particle partition functionZZ

2e(−U (r)−p /2m)/kB T drdp

(3.9)

D()e−/kB T d

(3.10)= Cα Γ(α)(kB T )α .

(3.11)Z1 =Z=

Alternatively we can evaluate the spatial and momentum integrals separately and write
Z1 = V0 /Λ3 ,(3.12)

where we define the effective volume asZ
V0 (T ) =e−U (r)/kB T d3 r

(3.13)

and we find that the momentum integral can be expressed in terms of the thermal wavelength ΛZ
exp[(−p2 /2m)/kB T ]dp = Λ−3 .

(3.14)

We now consider the in situ density distribution for N particles
Rn(r) = N=

exp[(−U (r) − p2 /2m)/kB T ]dp
Z1

N −U (r)/kB Te.
V0(3.15)(3.16)The peak density in the centre of the trap is triviallyn(0) = N/V0 .
32(3.17)

3.3. EVAPORATIVE COOLING

The energy E of the gas can be calculated as
N
E=
Z1

Z

D()e−/kB T d

=αN kB T.

(3.18)(3.19)

We consider now a trap containing particles with an average energy  = E/N . The
trap is truncated in a way that particles with energies U0 = (β + 1) can escape, whereβ  1 2 . We assume that β is large enough to justify the assumption that all escaping
particles have energy U0 . We can write for the change in total energyĖ = Ṅ U0 = Ṅ (β + 1).

(3.20)

We can also write
Ė
Ṅ E
− 2.
N
N
Combing both equations and eliminating the time dependence yields
˙ =d ln 
= β.
d ln N(3.21)(3.22)

Eq. (3.19) states that the temperature is proportional to  and henced ln T= β.
d ln N

(3.23)To achieve degeneracy, we aim to optimize the peak phase-space density,
D0 ≡ n(0)Λ3 =N∝ N T −αZ1(3.24)

We can derive a scaling constant γ for the evolution of the phase-space densityγ=d ln D0= βα − 1.
−d ln N

(3.25)We see that a low power-law trap is advantageous to achieve large gains in phase spacedensity. Experimentally, γ is used as a figure of merit for the evaporation process, eventhough the derivation above uses very simplified assumptions. So far, we have not taken
losses into account and evaporating slowly maximizes efficiency.
To tackle this issue, we introduce the half-life time of the atoms due to evaporative
cooling τE , and a non-energy-selective loss-lifetime τL . Doing so modifies Eq. (3.23)
2

Note, that here β is a number and not 1/kB T .

33

3.3. EVAPORATIVE COOLING

to [28]d ln TτL=β= β 0.
d ln NτL + τE(3.26)

Similarly we have to replace β → β 0 in Eq. (3.25). We see that losses reduce our evaporation efficiency. A further requirement to keep in mind is that the evaporation speed islimited by the rate at which atoms are scattered into the untrapped states with  > U0 .
This rate is comparable to the rate of elastic collisions τC in the trap centre
1/τC = n(0)σvrel ∝ n(0)T 1/2 ,(3.27)√
where we used that the relative speed between atoms scales as vrel ∝ T . We also
assume a temperature independent s-wave scattering rate, which is applies in 87 Rb for
T < 25 µK [43].
A more demanding limitation on the evaporation efficiency is imposed by requiringthat the scattering rate increases throughout the evaporation process. In this regime, whichis called runaway evaporation, our efficiency γ increases throughout the evaporation process. The change in elastic scattering rate can be quantified byγcoll =ln d(1/τC )= β 0 (α − 2) − 1.
−d ln N

(3.28)

Runaway evaporation occurs for γcoll > 0. If we cannot achieve runaway evaporation, theevaporation efficiency γ is going to decrease over time and the process will become more
and more inefficient. From Eq. (3.28), we see that in three dimensions we cannot enter the
runaway regime for α < 2, i.e., for power-law potentials U (r) ∝ rs with s > 6. However,
we should note this is not prohibitive to achieving condensation for higher-order traps.
We simply have to make sure that we start with a sufficiently high collision rate and atom
number.
In the following sections, we consider a few traps relevant to our experiment. We
derive their respective coefficients β and γcoll , which allows us to predict the performanceof the different trap types for evaporative cooling.

3.3.2

Crossed-Dipole Trap

We will now consider the optical crossed-dipole trap. Commonly formed by two overlapping Gaussian beams, the AC stark shift introduces a potential to the atoms proportional
to the beam intensity [28]. This potential is approximately harmonic in the beam centre.
However, it has a finite depth and lowering it necessarily decompresses the trap. We willshow that runaway evaporation is not possible in a simple isotropic configuration. However, as mentioned above, this is not a deal-breaker; condensation has been achieved in34

3.3. EVAPORATIVE COOLING

all-optical trap configurations [44–46]. Often they are used as a final ’storage’ trap after
evaporation in a more efficient configuration. They offer many benefits, especially allowing us to use magnetic fields freely, e.g. for tuning interactions with Feshbach resonances.
The derivation below follows mainly [47], although with the notation in line with [28].
An optical crossed-dipole trap can be described as
U (r) = U0 exp[−2(x2 + y 2 + z 2 )/w02 ],(3.29)

where we neglected the weak longitudinal confinement along the beams and anisotropies.
The depth of this trap is given by U0 and the waist by w0 . For kB T  U0 the potential isapproximately harmonic1(3.30)
U (r) = mω 2 r2 ,2
q2
with ω = w0 Um0 . With Eq. (3.30), the effective Volume can be calculated asV0 =2πkB T
m

3/2

1
.
ω3

(3.31)In an infinitesimal evaporation step, two things happen at once. The total energy is reduced due to the evaporated atoms leaving the trap, and also decreased due to the adiabatic
decompression of the trap. The energy change due to evaporation is given by Eq. (3.20)
with α = 3
Ė1 = Ṅ (β + 1)3kB T.
(3.32)The decompression is adiabatic, and conserves entropy S and also peak phase space density D0 [48]


dD0
= 0.
(3.33)
dt
From the definition of D0 in Eq. (3.24) follows
3/2

D0 =

N 3  w 3 U0
∝
Λ ∝
.
V0
T
E

(3.34)

Using Eq. (3.33), we can derive the change in energy due to decompression
Ė2 =

E U̇0
.
2 U0

(3.35)

We assume that the temperature is proportional to the trap depth U0 ∝ T . The total energychange rate is given byĖ = Ė1 + Ė2 .
(3.36)

35

3.3. EVAPORATIVE COOLING

Solving this for N and T yields
d ln(T )= 2β.
d ln(N )(3.37)

We can now calculate the change in phase space density as in Eq. (3.25), but we need totake into account that the effective trap volume depends on ω. Lowering the trap depth
U0 throughout the evaporation process according to U0 ∝ kB T decompresses the trap as
1/2
ω ∝ U0 ∝ T 1/2 . The inherent temperature dependence of the effective volume now
cancels with the temperature dependence of the trapping frequency:
V0 ∝

T 3/2
∝ T 0.
ω(T )3(3.38)The gain in phase space density by lowering the number is thenγ=−d ln D0d ln Λ3 N V0−1=−
= 3β − 1,d ln Nd ln N(3.39)

which is the same result as in a harmonic trap without decompression. Given that adiabatic decompression conserves phase-space density, this is not surprising. However, thethe collision rate is reduced tod ln (T 1/2 N V0−1 )d ln (1/τC )=−= −β − 1.
γcoll = −d ln Nd ln N(3.40)We see that γcoll is negative and runaway evaporation is not possible.

3.3.3

Hybrid Trap

We consider now a harmonic trap, which decompresses only in two directions, but retains
its trapping frequency along the third direction. This is a good approximation to a hybrid optical-magnetic trap, consisting of a single beam optical-dipole trap with magnetic
confinement along the third direction. In our setup, we use this kind of trap (see Section 4.3.3) as the last evaporation stage to achieve a harmonically trapped BEC. As thedecompression only occurs along two axes, Eq. (3.31) is modified toV0 ∝ T 3/2 ω −2 ∝ T 3/2 U0−1 ,(3.41)

where U0 ∝ T is again the trap depth. Decompression is again adiabatic and γ does notchanged ln D0γ=−= 3β − 1.
(3.42)
d ln N
36

3.3. EVAPORATIVE COOLING

The coefficient γcoll is trivial to calculate
d ln Nd ln (T 1/2 N V0−1 )=−= −1γcoll = −d ln Nd ln N(3.43)and does not depend on the evaporation efficiency. Obviously run-away evaporation isstill not possible.

3.3.4

Homogeneous Trap

In a homogeneous trap, runaway evaporation is not possible. The trap volume of a box
does not depend on temperature and the atomic density cannot increase due to cooling.
The gain in phase-space density is given by Eq. (3.25)
γ = (3/2)β − 1(3.44)and the change in collision rate by Eq. (3.28) .
γcoll = −β/2 − 1(3.45)

While run-away evaporation is not possible, the change of collision rate is somewhere in
between the crossed-dipole trap and hybrid trap scenarios.

3.3.5

Conclusions

It is beneficial to evaporate in low power-law traps, such as linear magnetic traps, as muchas possible to maximize gains in phase-space density. When using optical traps, there areclear benefits to using the hybrid solution, which decompresses only along two directions
and maintains higher collision rates. While the homogeneous trap intuitively seems like a
bad choice, it is not much worse than optical traps, especially in terms of collision rates.
One should note that the considerations in this chapter are only valid for classical gases.

37

3.3. EVAPORATIVE COOLING

38

Chapter 4
Experimental Setup and Methods
This chapter will give an overview of the experimental setup and cooling methods to
produce 87 Rb BECs. The structure of the chapter mimics the order of the experimental
steps taken on the road to degeneracy. A more compact documentation of the setup has
been published previously [23]. Compared to most state of the art 87 Rb systems, our
vacuum system is exceptionally compact. It features a single vacuum chamber, where
both trapping of the atoms in a magneto-optical trap (MOT) and evaporative cooling takeplace. While often simplicity comes at the cost of reduced atom numbers, we manage to
produce quasi-pure BECs of over > 3 × 105 atoms in < 30 s. Here, we only discuss the
preparation of BECs in a harmonic trap. The transfer of the cold atoms into a uniform
trap is detailed separately in Chapter 5.
The prerequisite for trapping and cooling atoms is an ultra-high vacuum (UHV) environment. Section 4.1 describes the characteristics of the single chamber system used inour experiment. At the start of the sequence we create a magneto-optical trap (MOT) to
trap atoms from the background vapour and cool them to around 15 µK. We detail thenecessary magnetic coils and laser setup in section 4.2. The next stage is evaporative cooling and is described in Section 4.3. We transfer the atoms to a magnetic quadrupole trap,
where they are further evaporatively cooled using radio-frequency (RF) induced transitions. Before spin-flip (Majorana) losses become significant, we transfer the gas into a
hybrid trap [49], where we achieve condensation. At the end of our experimental sequence, the atoms are probed by absorptive imaging (Section 4.4). Finally, in Section 4.5,
we describe the experiment control and analysis soft- and hardware.

4.1

Vacuum System

We inherited the vacuum system from previous students of the group [35, 36, 50]. In its
previous incarnations, it was capable of producing BECs of around 105 atoms. However,394.1. VACUUM SYSTEM

Figure 4.1: Schematic view of the 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,11gradient compensation. All coils except 9, 10 and 11 are paired (only one coil perpair is shown).
a full reassembly and thorough scrubbing was needed to revive it. The single-chamber
vacuum system is rather unusual as many contemporary experiments are based on multichamber geometries [51–53], often in conjunction with Zeeman slowers [49,52]. Separating the MOT spatially from the trap in which evaporative cooling is done allows maximizing atom numbers and life times. Also, it increases optical access for more sophisticatedexperiments. While a single chamber system is more compact and also cheaper, it suffers from balancing two contradicting requirements: while a higher 87 Rb vapour pressure
facilitates loading of the MOT, a lower background gas pressure increases the lifetimeand evaporation efficiency during later cooling stages. While compact, simple systems
are nothing new [44, 54–59], we feel our system excels in the crossover region between
compactness, cost and performance.
Our vacuum system with its surrounding magnetic coils is shown in Fig. 4.1. A
quartz cell1 allows optical access via two 50 mm (top and bottom) and eight 25 mm antireflection coated windows. The vacuum is maintained by a single 45 l/s ion pump2 . In
1
2

Triad Technology
Oerlikon IZ50

40

4.2. MAGNETO-OPTICAL TRAP

−6
Pressure gauge reading
Extrapolationlog(p/mbar)

−7
−8
−9
p below sensitivity

−10

Room
temperature

−11
−12

2.2

2.4

2.6

2.8
1/T [K]

3

3.2

3.4
−3
x 10

Figure 4.2: Final pressure during bakeout. The red line is an extrapolation beyond therange of the pressure reading of our ion pump assuming 1/T ∝ log( p/mbar) [35].
order to remove residual outgassing of contaminants, the system was baked at 200 ◦ C fora week, following the procedures outlined in [35, 36]. We estimated that the pressure ofunwanted background gases is below 10−11 mbar (see Fig.4.2).
We use vapour sources3 to release pure 87 Rb into the system, rather than the natural≈ 3 : 1 85 Rb-87 Rb abundance. This increases lifetimes of trapped atoms, as it reduces
the rate of background-gas collision for a given 87 Rb pressure. We achieved magnetic
trap lifetimes of over 20 s for very low 87 Rb pressures. We found an optimum for BEC
production at a pressure corresponding to atomic lifetimes of ≈ 10 s. Comparing this tothe low-pressure lifetime shows that we are barely (if at all) limited by unwanted gas in
the system. Rubidium is released by heating the dispensers by running a current throughthem. Probing the resistance (and hence temperature) of the dispensers with a four point
measurement and varying times and currents showed that Rubidium is released above athreshold temperature in a very non-linear fashion. We optimized the 87 Rb pressure in
our system by trial and error, finding that activating the dispensers once every evening for
80 − 85 s with 6 A gives best results.

4.2

Magneto-optical trap

4.2.1

Magnetic coils

The magnetic quadrupole field for both the MOT and the magnetic trap is created by ananti-Helmholtz pair of 16-turn water-cooled coils wound from 4 mm copper tubing (“7”3Alvatec41

4.2. MAGNETO-OPTICAL TRAP
in Fig. 4.1). These create an axial gradient of B 0 = 400 G/cm when carrying 200 A. The
coil is switched using an IGBT, which is driven by a driver built by Igor Gotlibovych [27].
A more compact version of the driver is described in the Appendix B. The quantisation
axis for optical pumping and imaging is provided by a magnetic field along the imaging
axis. We use a low-inductance planar 10 turn coil wound from 5 mm copper wire (“9” inFig. 4.1), providing a field of 5 G at the atoms. A radio-frequency (RF) field for forced
evaporative cooling (see Section 4.3.2) is created by a three-turn 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.
A vertical and a horizontal compensation coil allow us to fine tune the magnetic field
gradient along the “orthogonal” and “axial” directions (as defined in Fig. 4.1).
Gravity compensation during time-of-flight measurements as well as during later trap
stages (i.e. hybrid and uniform) is achieved by combining a quadrupole field (createdby the MOT coils) with a homogeneous vertical bias field. The magnetic gradient field
Bg required to cancel the gravitational acceleration g is given by Bg = mg/(µB gF mF ).
Typically, our atoms are in the |F = 2, mF = 2i state, for which Bg ≈ 15.3 G/cm. We
apply a quadrupole field with our anti-Helmholtz pair near the atoms




 z 
B = Bg 
,
−r/2

(4.1)

where z is the distance from the magnetic zero along the “vertical” direction and r therespective distance along the radial (i.e “orthogonal” or “axial”) direction. We then add
a homogeneous field B0 = B0 êz along the vertical axis to suppress the radial gradient.
The absolute value of the field is given by




 
1 Bg2 2
 z 
1
r .
B = Bg  − B0   ≈ B0 − Bg z +
2 4B0
−r/2
0

(4.2)

We use two 50-turn coils in a Helmholtz configuration (“8” in Fig. 4.1) to provide thebias field B0 ; up to a maximum value of 70 G when run at 10 A. The residual curvaturepcorresponds to a trapping frequency of ω = 0.5Bg µB gF mF /B0 m. To minimize ω, it
is intuitively best to increase B0 as much as possible. We found, however, that this introduces an additional harmonic potential along x, which can be explained by not perfectly
centred bias coils. We therefore settled on running the bias coils at ≈ 5 A or 30 G. This
minimizes any curvature along z, but leaves ω = 2π × 1.6 Hz (measured with oscillating
clouds) trapping frequency along r.
42

4.2. MAGNETO-OPTICAL TRAP

4.2.2

Laser Cooling

Our MOT operates on the D2 F = 2 → F 0 = 3 transition of 87 Rb (see Fig. 4.3).
The spontaneous decay back to F = 2 is strongly favourable, which creates a nearlyclosed transition. Still, off-resonant transfer to F 0 = 2 occurs and can be followed byspontaneous decay into F = 1. To prevent atom accumulation in the F = 1 dark state, a
repump beam continuously drives the F = 1 → F 0 = 2 transition.
We generate D2 light with two grating-stabilized external-cavity diode lasers4 . Typically, these lasers can produce up to 80 mW of power. We generate cooling light close
to the |F = 2i → |F 0 = 3i cycling transition and repumping light resonant with the|F = 1i → |F 0 = 2i transition. They are referenced and frequency-locked to separate vapour cells, using Doppler-free spectroscopy. This stabilizes the laser frequencies to< 1 MHz, which is smaller than the natural line-width Γ ≈ 2π×6 MHz. These techniques
are well-established, for reference, see, e.g., [27, 35, 37].
The cooling light is amplified with a tapered amplifier5 , with a maximum output powerof 1.5 W. At various stages, light passes through acousto-optic modulators (AOMs),
which allow to dynamically shift frequencies and regulate intensities. An overview oflaser frequencies with respect to the atomic levels is given in Fig. 4.3. The corresponding
setup is shown in Fig. 4.4.
The cooling and repumping light are combined and split into six MOT beams using afree-space fibre cluster, built in-house from slot-in components.6
The six MOT beams are delivered to the glass cell via optical fibres (Fig. 4.5). A totalof 120 mW of cooling light and 10 mW of repumping light reaches the atoms. With thislarge amount of power, we can use large diameter (25 mm) beams while still being able
to operate near the saturation intensity Isat ≈ 1.67 mW/cm2 in Eq. (3.3). Using large
beams like this facilitates alignment of the beams; typically realignment is not needed formonths. The light is linearly polarized in the fibres and is converted to the appropriate
circular polarization by quarter-wave plates. Final condensate numbers are extremely
sensitive to the correct polarization. Occasional realignment (∼ months) of the wave
plate is necessary due to long-term drifts of the cooling light polarization in the fibres.
The light for optical pumping and absorption imaging of the atoms is derived from thecooling light. The pumping and imaging beams are controlled by additional AOMs andenter the chamber via the “back” viewport (“5” in Fig. 4.1).
We load the MOT with an axial field gradient of B 0 = 13 G/cm and a cooling-light4

Toptica Photonics, DL Pro
Toptica Photonics, BoosTA
6
Thorlabs, ‘Fiber Bench’ range. The fibre cluster is notoriously hard to adjust, as components creep fora long time after adjustment. In hindsight, it is probably not worth the savings compared to an integrated,
commercial solution.
543

4.2. MAGNETO-OPTICAL TRAP

variable detuningwith double-pass AOM

193.7408(46) MHz

double-pass AOM
2x -105 MHz

F=3cooling 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) GHz

52S1/24.27167663181519(6) GHz

F=1

Figure 4.3: The Rb87 level scheme, including the AOM transition frequencies. Data
from taken from [39], drawing taken from [40].

444.2. MAGNETO-OPTICAL TRAP

Figure 4.4: The optical setup. AOMs are used to shift the light frequencies if needed.
The transition frequencies are given in figure 4.3. Unfortunately, the splitting between
repump and cooling transition is 6.8 GHz. This is too large to be covered by AOMs,
which is why two separate lasers are used. The drawing is an update to [40].

454.3. EVAPORATIVE COOLING

Figure 4.5: The optical setup around the vacuum chamber. The top and bottom
MOT beams are not shown. The MOT fluorescence photodiode picks up a signal
proportional to the atom number in the MOT. Also shown is the optical dipole trap
(ODT). The ODT monitor photodiodes pick up a signal proportional to the opticalpower and allow us to stabilize the beam intensity with a feedback circuit.
detuning of −25 MHz. Typically, we load the MOT to ≈ 109 atoms in 15 s. Note, however, that 5 s of loading is sufficient to produce BECs with > 105 atoms. We use thefluorescence signal on a photodiode to monitor the loading of the MOT. Following theinitial loading stage, we compress the MOT by increasing B 0 to 64 G/cm for 20 ms, andthen detune the cooling light by −68 MHz for 1 ms [60]. We found that these steps reduce
the temperature and improve transfer into the magnetic trap. They are followed by 2 msof optical molasses during which the magnetic field is switched off and the cooling-lightdetuning of −68 MHz is maintained. This cools the cloud to ≈ 15 µK.

4.3

Evaporative cooling

The evaporative cooling to condensation in a harmonic trap can be divided into fourstages:• Transfer to the magnetic trap
• RF evaporation• Hybrid trap transfer
46

P HWLf HMHzL B' HG cm-1 L

4.3. EVAPORATIVE COOLING

I

II

III

200
0
15
0
5
0
0

1

2

Evaporation time HsL
3

4

5

6

Figure 4.6: 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.
• ODT evaporation
Fig. 4.6 summarizes the evolution of the relevant experimental parameters during theevaporation sequence, while Fig. 4.7 displays the evolution of the atom number N , thetemperature T , and the calculated phase-space density D. N and T are measured using time-of-flight (TOF) absorption imaging and D is calculated from a semi-classicalmodel [49], using an analytical 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 theoretical predictions. We omit the calculated values of D instage II, where they are unreliable because the trapping potential varies slowly over large
volumes and V0 diverges.

4.3.1

Transfer to the magnetic trap

After the molasses, we apply a σ + polarized, resonant |F = 2i → |F 0 = 2i light pulse.
This pulse drives only ∆mF = +1 transitions. Spontaneous decay transfers the atoms
back to |F = 2i, with ∆mF = ±1, 0. After several of those cycles, most atoms end up
in the |F, mF i = |2, 2i state, which is a dark state for σ + light. We also add repump
light to prevent accumulation in the |F = 1i dark state. We optimize the pump time tomaximize phase-space density, as heating due to photon absorption competes with transfer
efficiency. We pump > 80% of the atoms into the |2, 2i state in 40 µs, while heating thecloud to 40 µK.
47

4.3. 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 4.7: 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. 4.6 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) and3 (red)] and its integrated profile. Fits to the thermal and condensed components areindicated.

The atoms are then captured in a magnetic quadrupole field B = B 0 ×(z, −x/2, −y/2)
produced by the MOT coils. The Zeeman effect creates a potential U (r, mF ) in which theatoms are trapped
1(4.3)
U (r, mF ) = mF µ0 |B(r)|2where µ0 is the Bohr magneton and we used and gF =2 = 1/2. In practise, we suddenly
turn on the quadrupole field with B 0 = 64 G/cm (matching the gradient of the CMOT)
and then ramping B 0 to 80 G/cm over 200 ms. Finally, the trap is further compressed byraising B 0 to 200 G/cm over 500 ms.

4.3.2

RF Evaporation

We now evaporate by transferring the atoms from the trapped |F = 2, mF = 2i state
to the untrapped |F = 2, mF ≤ 0i states. We generate an RF-field with frequency ωRFwith the RF coil above the cell. The resonance condition is met on a ellipsoid around theorigin, where the Zeeman splitting between the mF -states ∆E = U (r, mF )−U (r, mF −1 )48

4.3. EVAPORATIVE COOLING

Figure 4.8: Radio-frequency driven evaporation. (a) The Zeeman effect causes a
linear splitting in the mF manifold. A resonant RF-field induces transition between
the states. (b) Sketch of potential energy in the trap. Where the splitting is resonantwith the applied RF-field, atoms are transferred to untrapped mF states and ejectedfrom the trap.
matches the transition frequency:
1~ωRF = ∆E = µ0 |B(r)| .
2

(4.4)

If initially in the |F = 2, mF = 2i state, only atoms with energy E > 2 × ∆E =2~ωRF can reach the transition area. The atoms are then shifted down in steps of ∆mF =−1 and are expelled. Fig. 4.8a illustrates this process. The resulting trap is approximately
a truncated linear power law (Fig. 4.8b. This method allows to control both collision rateby (de-)compressing the trap via tuning B 0 , as well as the trap depth by ramping ωRF .
During the initial evaporation stage in the compressed trap (I in Figs. 4.6 and 4.7),
we ramp the RF frequency linearly from 15 MHz to 5 MHz in 2 s. We achieve a 30-foldincrease in the peak phase space density, with evaporation efficiency
γ = −d[ln D]/d[ln N ] = 2.1.

(4.5)

At the end of this stage the cloud typically contains N = 50×106 atoms at T = 90 µK. At
this point further evaporation is rendered inefficient by spin-flips at the trap centre (Majorana losses), where the field is low. Here, the different Zeeman states become degenerate
and the quantization axis is not well-defined. Non-adiabatic transfer to other, possibly
untrapped, states occurs and atoms are lost. Even worse, this affects mostly low energy
atoms, as they spend more time near the trap centre.
To prevent these losses, we modify the trapping geometry so it does not include themagnetic zero. Various approaches exist, such as a TOP trap in which the first BEC was49

4.3. EVAPORATIVE COOLING

produced [4, 61], the Ioffe-Pritchard trap [62], or the use of optical trapping methods.
Another common method is shooting a blue detuned laser through the centre of the trap,
repelling the atoms. This technique is called an optical plug. Stuart Moulder achieved
successful condensation with this method with as little as 70 mW of plug-laser power[23, 50] in this setup.

4.3.3

Single-Beam Dipole Trap: “Hybrid Trap”

We use a hybrid approach, where we retain the linear magnetic trap, but also superimposed a red-detuned laser trapping beam below the magnetic zero. This approach is described in length in [49] and the derivations here closely follow it.
The total potential U (r) is given by the sum of a quadrupole trap, an optical dipole
trap (ODT) and gravity

p−2[x2 + (z − z0 )2 ]2
2
2
U (r) = µ0 B x /4 + y /4 + z − U0 exp+ mgz,
w02
0

(4.6)

where z0 is the distance of the beam centre from the magnetic zero. During the initial RF
evaporation (stage I) described above, the ODT is already on, but its effect on the potential is minor (see Fig. 4.9). We can neglect its contribution to Eq. (4.6) and the potential iswell approximated by a simple quadrupole trap, where gravity leads to minor corrections.
During stage (II), reducing the magnetic field gradient while cooling transfers the atoms
gradually into the centre of the ODT beam below the magnetic zero, preventing spin-flip
losses. For low temperatures and magnetic gradient forces similar to g (stage III), the trapping potential is dominated by the ODT along the x- and z-axis, with trapping frequenciespωx,z = 2 U0 /mw02 . Along the y-axis, the atoms are still trapped magnetically, however
the displacement from zero leads to a smooth potential. This can be approximated as apharmonic trapping potential with frequency ωy = 1/2 µB 0 /mz0 . Like in an all-opticalcrossed-dipole trap (CDT), the trap depth is given by U0 . Evaporation is now driven bylowering U0 , as atoms can escape downwards aided by gravity. As calculated in Section
3.3, a big advantage compared to a conventional CDT is that only two out of the three
trapping frequencies are lowered during evaporation, allowing for higher collision rates.7
The cross-over between the two regimes happens in the transition region (II). Here,neither the linear nor the harmonic approximations hold and the trap shape is best described numerically. Also, the phase space density will be affected due to the change in
effective volume V0 . At some stages, U (r) varies very slowly over large regions and thenumerical values for V0 and hence D are unreliable.
7
An extension of this is the tilted trap approach, where the magnetic field gradient is used to lower thetrap depth without changing the ODT trapping frequencies [58].

50

4.3. EVAPORATIVE COOLING

Figure 4.9: Trapping potential evolution throughout the evaporation process. The left
column shows the vertical direction which is affected by gravity. The right column
shows the horizontal direction along the beam, where the confinement is predominantly magnetic at all times. The red bars signify the resonance of the RF knife,that is where the potential is truncated. The “red liquid” level in the potential wellcorresponds to T . We see that during the inital stages of evaporation, the trap is predominantly linear and magnetic. Later, the confinement along z is optical and atoms
can evaporate by “falling out” thanks to gravity. The potential along y is still magnetic, however the confinement is harmonic, as the minimum of the trap is displaced
from zero. The potential along x (in the horizontal plane orthogonal to the ODT) isvery similar to the z-potential, except that at later stages gravity does not truncate the
trap.

51

4.3. EVAPORATIVE COOLING

Optical dipole trap laser
The optical-dipole-trap beam used for the hybrid trap is derived from a 20 W ytterbium
fibre laser8 . After passing through an AOM, the beam is focused by a concave-convex
pair of lenses, 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 lead toexcessive atom losses, which can be attributed to two-photon transitions between F = 1
and F = 2 [63]. These transitions are caused by the large linewidth of the trapping laser(1.8 nm in our case). The focal point is positioned approximately a beam waist below
the zero 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 an AOM. Throughout this range, we achieve an RMS
power noise below 1 mW over a bandwidth of 1 kHz.

Loading of the Hybrid Trap
In stage II, the magnetic field gradient is decompressed to just below the gravity-compensating
value, Bg0 = mg/µB = 15.3 G/cm. From this point on, the atoms are supported against
gravity by optical forces only, while the confinement along the ODT axis is dominatedby the magnetic forces [49]. Note that the ODT beam is on from the beginning of themagnetic trapping (see Fig. 4.6), but initially does 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 despite
the weak magnetic confinement. Finally, the magnetic field is decompressed from 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.
Sweeping B 0 in this manner changes the RF-limited trap depth approximately linearlywith time, with the approximation being exact in the case of no ODT and B20 = Bg0 . At
the end of this stage we typically have 20 × 106 atoms at 30 µK.

Optical Evaporation
For B 0 < Bg0 , evaporation is no longer driven by the RF field. Instead, the trap depth islimited by the potential at the saddle point vertically below the magnetic-field zero [58].
Evaporation can be forced by lowering the ODT power U0 , albeit at the cost of reducing
the trapping frequencies.
8

IPG Photonics, YLR-20-LP

52

4.4. IMAGING AND IMAGE PROCESSING

We ramp U0 exponentially to its final value with a time constant of 1.25 s and total
ramp duration of 7 s. Both parameters have been optimized empirically and the optimum
depends on the current vapour pressure. We achieve an evaporation efficiency of γ = 3.4,
leading to condensation at P = 0.16 W.
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-pure9 condensates of > 3 × 105
atoms. The lifetime of the quasi-pure BEC is ∼ 3 s, consistent with the expected 3-bodylosses [64, 65].

4.4

Imaging and Image processing

We image the atoms absorptively, using a circularly polarized beam resonant with the|F = 2, mF = 2i → |F 0 = 3, mF 0 = 3i cycling transition and measure the resulting absorption pattern on a CCD camera10 . We start by considering the basic principles
of absorptive imaging in the low intensity regime. After this we discuss the deviationof the measured absorption cross-section from its theoretical value, which are primarily
due to imperfections in the light polarisation. We calibrate this deviation by comparing
the measured atom number at the BEC critical point Ñc with the theoretically predicted
critical number Nc . Ñc was measured by extrapolating the non-saturation slope of thethermal component to the condensation point, as described in [16, 34]. Finally, we describe high-intensity imaging [66], which allows us to measure higher optical densities.
This technique also reduced the density correction factor.

4.4.1

Absorption Imaging

The absorption of a beam of intensity I propagating along the z-axis through a cloud of
atoms is given by [37]IdI
= −n(z)σ0,2
dz(2∆/Γ) + (1 + I/Isat ))(4.7)

where n is the density, σ0 = 3λ2 /2π is the resonant cross-section, Isat = (π/3)hcΓ/λ3
the saturation intensity and ∆ the angular detuning from resonance. We consider now the
simplest case with no detuning ∆ = 0 and low intensity I  Isat . After passing throughan atomic cloud with density n(x, y, z), we measure the following intensity pattern on the
910

We can only measure thermal fractions accurately down to about 50%.
PCO, Pixelfly USB

53

4.4. IMAGING AND IMAGE PROCESSING

CCD screen:I(x, y) = I0 (x, y) exp[−σ0 ñ],
(4.8)
R
where we introduced the the column density ñ(x, y) = n(x, y) dz. To analyse a cloud,
we first measure the intensity IA (x, y) in the presence of the atoms. Then we measure
the original light intensity IL (x, y) without absorption from atoms. Last we take one
background image IBG (x, y) without atoms or light to measure the dark count and otheroffset effects. We can extract the two-dimensional column density from the optical density
IA (x, y) − IBG (x, y)OD(x, y) = σ0 ñ(x, y) = − ln.
IL (x, y) − IBG (x, y)(4.9)

In practise, we apply a magnetic bias field for the shot with IA and tune the lightfrequency accordingly to compensate for the Zeeman shift. Before measuring IL (x, y),
the field is then quickly switched off, tuning the atoms out of resonance and thus making
them transparent to the imaging light. The background shot is less critical and fairly
reproducible. We also measure and correct for fluctuation in global light intensity between
IA and IL (corresponding to an offset in OD) by comparing an illuminated region far
from the atoms. We usually also remove a weak offset (a camera artefact) from the OD
by fitting a wide parabolic profile to the region of the image not containing atoms.

4.4.2

Density calibration

In most experiments it is necessary to introduce an empirical factor F (often referred toas the “fudge factor”) to take deviations from the theoretical cross section into account.
These deviations are usually caused by polarization imperfections, but also other effects
can reduce the imaging efficiency.11 We thus modify Eq. (4.9) toIA (x, y) − IBG (x, y)
× F.
OD(x, y) = σ0 ñ(x, y) = − ln
IL (x, y) − IBG (x, y)

(4.10)

F can be calibrated by measuring the apparent critical number according to Eq. (4.9)
in a harmonic trap and comparing it to the predicted critical number for the appropriate
temperature and trapping frequencies. To do so, we need to take into account that theideal gas model does not fully capture the behaviour of the condensed gas. We needto take the non-saturation of the thermal fraction into account, i.e., the thermal number
11
For instance, F increases if the frequency-lock of the imaging beam is not tight enough, i.e., if thelinewidth is too broad.

54

4.4. IMAGING AND IMAGE PROCESSING

behaves as [16, 34]:
2/5

4/5

N 0 = Nc + S0 N0 + S2 N0


2/5
ξ(2) kB T 2
15a
S0 =
2
~w̄
ah o


4/5
ξ(3) kB T
15a
S2 = (3 ± 0.7)
.
4
~w̄
ah o

(4.11)(4.12)(4.13)

Hence, the thermal number N 0 varies in a nonlinear fashion near the condensation point.
A more detailed treatment of this effect is given in Section 6.2.1. We measure a rangeof condensate sizes for a fixed temperature and use Eq. (4.11) to extrapolate to the condensation point. Multiplying the observed numbers N 0 and N0 with F allows us to relatethem to the theoretical critical number Nc
2/5

4/5

N 0 = F −1 (Nc + S0 N0 F 2/5 + S2 N0 F 4/5 ).

(4.14)

We now can fit Eq. (4.14) to experimental data, allowing us to extract F . We did thiscalibration whenever we had reason to believe that the factor might have changed, suchas after modifying imaging beam optics. We obtained values between 1 and 2, which isfairly typical [59, 67]. Figure Fig. 4.10 shows a calibration example. The main errors are
induced by the uncertainty of temperature and trapping frequencies. We measured thetrapping frequencies ωy and ωz with a precision of better than 1 Hz by exciting a small
amplitude oscillation. The main uncertainty in frequencies is due to ωx , as we cannot
image oscillations along the imaging axes. Assuming an isotropic ODT, it is similar toωy . We estimate the difference due to the magnetic field using Eq. (4.6). Depending on
the trapping potential and temperature, deviations from the harmonic trap shape need tobe taken into account when calculating the effective volume. However, these corrections
typically are only on the order of a few percent.

4.4.3

High intensity imaging

Using low imaging intensities comes with some drawbacks. Foremost, the dynamic rangeof counts per pixel is limited (12 bit in our case), leading to an upper cap in OD value.
The theoretical cap of the optical OD < ln(12) ≈ 9.7, is further reduced by backgroundcounts and other noise, rendering images with OD > 3 questionable. This limits us tomeasuring dilute clouds. While reducing the cross-section by increasing ∆ helps, this candistort images severely by lensing effects [66].
We now drop the requirement that I  Isat . We include a fudge factor F̃ in Eq. (4.7)
55

4.4. IMAGING AND IMAGE PROCESSING

Figure 4.10: Example of a fudge factor calibration. Using the theoretical variationof the thermal number with condensate number, as predicted by Eq. (4.13), we findthe best fit for the fudge factor F = 1.7 ± 0.2. The error is dominated by systematic
uncertainties in the trapping frequencies. Here, ωx,z,y = 2π × (106, 30, 102) Hz.
We also leave the prefactor of S2 as a free fit parameter and obtain 3.9 ± 0.6 inaccordance with the experimental value 3±0.7 given in [34]. We corrected for a linear
temperature drift between 245 and 255 nK using the correction suggested in [16]. The
critical number is corrected for finite-size effects [28] and the mean-field shift [10,59].

and integrate to obtain the corrected column density
σ0 ñ = F̃IL − IAIA (x, y) − IBG (x, y)+.
− ln
IL (x, y) − IBG (x, y)F̃ Isat(4.15)

We can identify the first term as Eq. (4.9), which holds for low intensities. The second
term dominates at high intensities when the transition is saturated. We distinguish thenew over-all fudge factor F̃ from the corresponding low-intensity fudge factor F . This isnecessary as F often includes a contribution from the high-intensity term, if I  Isat is
not fully met.
[66] uses Eq. (4.15) to find F̃ by varying I at constant atom number. For the correct
value of F̃ there is minimal variation in the column density. However, this requires anaccurate calibration of Isat via other means, which we found difficult to achieve. Instead
we use Eq. (4.15) to find effective saturation intensity F̃ Isat and then calibrate the fudge
factor using the non-saturation method.
We use the following relation between experimentally measured counts per pixel N
and intensity I
4qA,(4.16)
N = Ithf
where A = (2.51 µm)2 is the pixel area, q ≈ 10 − 20% the quantum efficiency of the56

4.4. IMAGING AND IMAGE PROCESSING

Figure 4.11: Calibration of the imaging intensity. (a) We vary the imaging intensity
I and exposure time t while keeping the total photon count constant. The images areef fevaluated with a different values for the saturation count Ṅsat. For the correct value,
there should be no variation in of the measured number N with I (or equivalently t).
(b) The average slope of the different curves in (a). We find minimum variation for
ef f
Ṅsat
≈ 280 µs−1 .
camera12 and t the exposure time. For reasons only known to the camera manufacturer,
the camera internally multiplies the counts by 4. We rephrase equation Eq. (4.15) byreplacing all the intensities with their respective countsNL − NA
NA (x, y) − NBG (x, y)+
σ0 ñ = F̃ − ln,ef f
NL (x, y) − NBG (x, y)tṄsat(4.17)

ef fdNsat , which is thewhere we introduced the effective saturation count rate Ṅsat= F̃ dt
counterpart of the effective saturation intensity F̃ Isat . We then varied the intensity over
two orders of magnitude while adjusting the exposure time to keep the average number
ef fof imaging photons constant. The pictures were post-processed using different Ṅsat(see
Fig. 4.11a). For the correct value, we expect a flat line, as the total atom count should notdepend on imaging intensity when processed correctly. In the example shown, we found
ef f
Ṅsat
≈ 280 µs−1 meets this criterion best (see Fig. 4.11b).

Repeating the density calibration with Eq. (4.17) with typical imaging intensities, wefound that F̃ ≈ 0.8F , i.e., 20% of our low-intensity imaging fudge factor could be attributed to operating to close to Isat .
ef fNote that this value for Ṅsatis roughly consistent with the estimated counts:
ef f
Ṅsat
≈ F̃ Isat

4qA≈ (240 ± 80)F̃ µs−1 .
hf(4.18)

12
Some measurements we conducted caused us to have some doubts concerning the efficiency quoted by
the manufacturer.

57

4.5. EXPERIMENT CONTROL AND DATA PROCESSING

It is important to consider the maximum amount of photons absorbed per atom during
imaging. The absorption of too many photons (followed by spontaneous reemission) canlead to acceleration of the atom towards the camera. If the resulting Doppler-shift is toolarge, this can tune the atom out of resonance. It can be shown that if we keep the total
photon count per pixel Nmax constant, the maximum number of photons is absorbed forI  Isat . In our case Nmax = 6000, and we obtain a maximum of 500 photons per atom.
This corresponds to a Doppler-shift of around 3 MHz  Γ/2 at the end of the imaging
pulse. Hence the scattering cross section at the end of the light pulse is halved for low
intensities; however the effect on the total counts is less than that. Still, in retrospect weshould have used fewer photons.
ef f
The calibration of the Ṅsat
allows us to increase I, which allows us to image dense
clouds reliably. This is essential in the medium-intensity range, which is useful for resolving thermal distributions originating from homogeneous traps, which are inherently“peaky” in the centre.

In the limit I  Isat , Eq. (4.15) is especially simple and we obtain a column density
profile, which is directly proportional to (IL − IA )/(Isat ). Even when Isat is not properly
calibrated, this allows us to extract the correct density profile, e.g., when considering insitu distributions.

4.5

Experiment control and data processing

4.5.1

Sequence setup and control

We control our experiment with a National Instruments system consisting of a chassis(PXIe-1082), two digital output cards (PXI-6536) and two analogue output cards (PXI6713, PXI-6733). This gives us a total of 64 digital TTL-compatible output channels,
which are isolated from each other with optocouplers to prevent ground loops. The analog
cards provide 16 output channels in total. These digital channels allow us to interface withthe various devices in our experiment in an on-off fashion. Devices which need gradualcontrol, such as the quadrupole power supply, are controlled by the analogue controls. Allof these channels are synchronized with a common clock.
The hardware is interfaced using Aviv Keshets “WordGenerator” software [68], whichallows us to design experimental sequences. Besides the digital and analogue channels, it
adds the possibility of outputting serial commands on the computers port via RS232 andGPIB. Besides communicating with instruments with the appropriate ports, we use thiscapability to interface with custom-made software, e.g. to implement functionalities notnative to “WordGenerator”. This is achieved by creating a pair of virtual serial commu58

4.5. EXPERIMENT CONTROL AND DATA PROCESSING

nication ports with the free “com0com” software. If WordGenerator sends a command to
one of those ports, it can be read out by any other program from the other port. We use thisfor two purposes. First, this allows us to save complementary files for each experimental
run with the most relevant parameters. This is immensely useful for evaluation and documentation. Second, by adding an additional software layer between WordGenerator andthe experiment, we can implement feedback. This means we can automatically modify
future measurements to compensate for drifts, without the need of manually adjusting theexperiment settings in WordGenerator.
To reduce the influence of noise from the mains power, we use WordGenerator’s builtin feature to synchronize the start of each experimental run with the phase of the mains
power. Using a simple zero-crossing circuit, we detect the phase of the 240 V AC to≈ 1 ms accuracy and use a digital trigger to synchronize the experiment with it.

4.5.2

Imaging control

The CCD camera is controlled by a custom software written in LabView. Images aretaken when the camera is triggered with a digital signal from the control hardware. The
imaging parameters (such as exposure time) are set within the imaging software. The
software has several operation modes:• Absorption imaging: Three subsequent images are taken and the OD is calculated
according to Eq. (4.9).
• Absorption imaging with double-shuttering: Double shuttering allows us to taketwo images within only 10µs. This allows us to minimize intensity fluctuations
between the images with and without atoms, reducing noise.
• Single shot: Takes a single image, e.g. to image the MOT for fluorescence imaging.
• “Live mode”: Continuously records images without waiting for a trigger and displays them as a live video. This is particularly useful for observing the MOT oraligning beams.
The camera has a resolution of 14-bit, but registers a single photon as 4 counts, resulting in a maximum count of 216 . We can thus save the unprocessed images as 16-bit
grey-scale “.tiff” files. When generating OD-images, we stretch their values according
˜ = 28 × (OD + 8), using the full bit depth to ensure efficient and lossless storto OD
age. When high-intensity imaging is used, we use additional programs to generate opticaldensity images according to Eq. (4.15) using the unprocessed images.
59

4.5. EXPERIMENT CONTROL AND DATA PROCESSING

While the exposure time of the camera is set by the software, we set it to be larger than
the actually desired exposure time. The effective exposure time is then determined by the
duration of the imaging pulse. This time is controlled to 0.5 µs precision by an Arduino
microcontroller, allowing us to improve beyond the time resolution of the WordGenerator
software.13 The required exposure time is sent to the Arduino via serial communication
at the beginning of the sequence; the pulse is digitally triggered when required.

4.5.3

Analysis software

Igor Gotlibovych has programmed an analysis software allowing us to view the images ina graphical user interface. Once loaded, various fits can be applied, extracting informationsuch as temperature. These fits can be saved to disk and contain a link to the original
images, fitting results and optional extra information from log files. More information on
the sofware architecture and its implementation can be found in [27].

13
Alternatively, this can be implemented with a variable time base as described in the WordGenerator
documentation.

60

Chapter 5
Creating a homogeneous Bose-Einstein
Condensate
The creation of homogeneous Bose-Einstein condensates has been a long-standing goalin the field. They offer a closer connection to conventional many-body systems, while
simplifying theoretical problems. In this chapter, we explain how we created the firstquasi-homogeneous BEC. We cover the technical challenges encountered and discuss thefirst thermodynamic studies we conducted to confirm the uniformity. The main results ofthis chapter have been published in [24].
We use blue-detuned light to confine our atoms in a steep-walled dark optical trap,
while cancelling gravity with a magnetic gradient (Sections 5.1 and 5.2). The trap isloaded with a gas precooled in a harmonic trap, followed by evaporation in the dark trap
(Sections 5.3 and 5.4). We characterize the trap by examining the in situ density distribution in Section 5.5. In Section 5.6, we approximate our trap by a high power-law potential
and derive the resulting momentum distribution. This model allows us to measure thermodynamic properties such as temperature and number. Comparing the predictions withthe measurements in Section 5.7 allows us to quantify the uniformity of the potential:
we find the trap is well approximated by a power-law of order 15, which is essentially
uniform. Finally, we confirm the theoretically expected critical point in a homogeneous
gas.

5.1

Dark Optical Trap

Like attractive optical traps, dark optical traps operate on the principle of the AC Stark
effect [69]. The light frequency is blue-detuned from resonance and the atoms are repelledby the light. We enclose our atoms with walls of light, while minimizing the amount oflight inside the enclosed trapping volume. This technique has several advantages over61

5.1. DARK OPTICAL TRAP

Figure 5.1: The beam configuration for the production of the green trap. Many of
the optics are dichroic and are shared with the MOT and ODT beams (compare with
Fig. 4.5).
attractive optical traps: First, there is only negligible heating due to light scattering, as the“contact area” between atoms and the walls is small. Second, it requires far less light, as
the wall thickness can be very thin (it is limited by the diffraction limit). Most importantly,
the trap is dark inside, so that intensity variations do not cause variations of the trapping
potential.
We use a hollow-tube beam (pseudo-Laguerre-Gaussian [70]) and intersect it with two
Gaussian sheet beams, creating a can-shaped dark region enclosed by repulsive walls. To
create those beams, we reflect a 532 nm Gaussian laser beam1 off a spatial-light modulator (SLM)2 . The spatial profile of the laser beam is very close to a true Gaussian anddoes not require cleaning up by passing it through a fibre. The SLM allows us to imprint
an arbitrary phase pattern onto the beam. In the far-field plane, which we emulate byfocussing the beam, this yields approximately the Fourier-transform of this phase pattern,allowing us to create arbitrary, two-dimensional patterns. In the focal planes, the beamprofile is approximately constant over a Rayleigh length (∼ 50 µm), allowing us to extrude these profiles into three-dimensional shapes. Details of the beam-shaping and thedigital holography in our experiment can be found in [70, 71].
1
2

Quantum Laser, Excel 2W
Hamamatsu, X10468-04

62

5.2. MAGNETIC FIELDS

Our setup to create the green beams is shown in Fig. 5.1. A Gaussian beam is incident on the SLM, where a phase pattern corresponding to the wall and tube beams are
imprinted. We also diffract the beams into different directions using phase “wedges”. We
focus the beams to create a first far-field image in the splitting focal plane, where thebeams are separated by 1.3 mm. This allows us to use a pick-off mirror to deflect only
the wall beams by 90 degrees. Further lenses recollimate the beams and focus them again
onto the atoms, creating the dark trap.
Alex Gaunt has assessed in [24] the steepness of the beam shaping apparatus, boththeoretically and on the bench, parametrizing the radial and axial directions as powerlaws. The tube beam can be parametrized by a radial power-law ∝ rs with an exponents & 7, while the sheet beams have a profile with s  10 . As the actual shape of thebeams are not power-laws, these numbers are only approximations to the actual profile ata given trap depth. Therefore these values should be taken with a pinch of salt and weshould not expect that these values give more than a ballpark estimate of how close ourtrap is to an ideal box. Instead, we will use thermodynamics later on to extract an overall
effective power-law that describes the box.

5.2

Magnetic fields

Even if the trap is perfectly dark inside, we still need to compensate the gravitational
potential, which introduces a linear gradient along z. We address this by adding a linear
magnetic field gradient Bg0 ≈ 15.3 G/cm, which induces a force compensating gravity as
described in Section 4.2.1. We assess the necessary flatness by considering the densityn(r) of fully condensed BEC in the Thomas-Fermi approximation (Eq. (2.30))
n(r) = [µ − U (r)]/g,(5.1)

where µ is the chemical potential, U (r) is the potential and g our usual interaction strengthparameter. We require that any potential variation across the trap satisfies ∆V  µ, tomake sure there are no significant variations in density.
Let us consider the restrictions this imposes on the accuracy of our magnetic field
gradients in a typical trap with length L = 63 µm and radius R = 15 µm, loaded with
50 000 atoms. We require that the variation of the magnetic energy according to Eq. (3.2)
across the trap is below µ . kB × 1 nK, which means we need to stabilize the gradient towithin 10−4 Bg0 . We accomplish this by releasing the cloud in ToF with the same gravity
compensating field. If gravity is perfectly compensated, the centre of mass of the cloud
does not move. We use up to 0.2 s ToF and can localise the centre of mass with a precision635.3. LOADING OF THE UNIFORM TRAP

Figure 5.2: Laser power levels and magnetic fields during the transfer of the atoms
from the harmonic to the uniform trap. The compensation fields are not included.
Figure taken from [27, 70].
of several pixels (2.5 µm) on the image. This allows us indeed to detect gradient variationsof < 10−4 Bg0 . Another correction to take into account is the curvature of the magnetic
field in the horizontal plane (see Section 4.2.1). The curvature of ω = 2π × 1.6 Hz gives
a potential variation of ≈ 0.5 nK across the box and is negligible for thermodynamics.
Note that we have been playing devil’s advocate in this calculation by choosing µ as
our energy benchmark. For the behaviour of thermal clouds (such as the critical point),
T ≈ 50 nK is the more appropriate and also more forgiving energy level. This changes in
Chapter 7, where we consider the ground state properties of our condensed gas.

5.3

Loading of the Uniform trap

After having considered the tools to create a uniform trap, we will now discuss the experimental procedure of loading atoms into the uniform trap. First, we produce degenerate
clouds in the harmonic trap as described in Chapter 4. We then transfer those precooled
atoms into the uniform trap.

5.3.1

Transfer procedure

Typically, we evaporatively cool the gas in the harmonic (hybrid) trap down to T ≈120 nK, when the cloud size is similar to the size of our optical box (see Fig. 5.3). We
then ramp up the green walls, followed by switching off the ODT beam. Typically, we
use a total laser power of P = 600 mW, corresponding to a trap depth U0 > 1 µK. The
power level is not critical; using excessive power, however, leads to heating. Too little
645.3. LOADING OF THE UNIFORM TRAP

Figure 5.3: Transfer of the could from the harmonic to the uniform trap. (a) Sketchof the harmonic trap surrounded by the uniform walls. (b) The cloud in the dark trap
after switching off of the harmonic confinement. (c) Experimental column density of
the harmonic trap before transfer. The line-densities (red) are shown next to the column density intensity image. The dashed blue lines are fits to the thermal component.
(d) Experimental column density of the cloud in the uniform trap. The line-densities(red) are shown below next to the column density intensity image. The dashed green
lines are fits to the line density assuming a uniform density distribution.(e) The coordinate system used for the rest of the chapter. Figure taken from [24].
power leads to incomplete transfer, as it is effectively truncating the thermal distribution.
Typically, we capture around 80% of the atoms. After the transfer into the dark trap
is complete, we ramp up the magnetic gradient to its gravity compensating value Bg0 .
Simultaneously, we ramp up the vertical bias field to cancel the magnetic gradient in thehorizontal plane. In addition to that, we use additional compensation coils (item “11” inFig. 4.1) to fine tune the magnetic fields. The magnitude and variation of the magnetic
fields produced by the compensation coils during transfer is optimized empirically. An
overview of the laser powers and magnetic fields during transfer is shown in Fig. 5.2.
Phase-space density during transfer
Before the transfer into the uniform trap, the gas is usually partially condensed, but the
BEC is lost during the transfer into the box trap. Even perfect adiabatic transfer does notretain phase space density, if the power-law is changed [48]. This effect has been exploitedto achieve condensation solely by deformation of the trap [72]. We will now derive thechange in phase-space density when transferring the gas between different power-law
65

5.3. LOADING OF THE UNIFORM TRAP
Uniform N0 •N
1.0
0.8
0.6
0.4
0.2

Harmonic N0 •N
0.0
0.2
0.4
0.6
0.8
1.0
Figure 5.4: Change in condensed fraction N0 /N when transferring a partially condensed gas from a harmonic to a uniform trap.

traps. We consider both a classical gas as well as a condensed gas, and apply our results
to the transfer from a harmonic to a uniform trap.
Classical gas Let us first consider a hot gas at T  Tc . The peak phase space density
D0 can be related to the entropy S as [48]S
D0 = exp α + 1 −
Nk


.

(5.2)

α is the power-law parameter introduced in Chapter 2. As a reminder, in a power-law
U (r) ∝ rs , we have α = 3/2 + 3/s. Transferring adiabatically (S = const.) from aharmonic trap with α1 = 3 to a uniform trap α2 = 3/2 leads to a significant decrease inpeak phase-space density:D2= eα2 −α1 ≈ 0.22.
(5.3)
D1
Since we also know that collision rates and evaporative cooling efficiency are worse in a
uniform trap than in a hybrid trap (Section 3.3), this suggests we should transfer as late as
possible.
Condensed gas For a partially condensed gas in a power-law potential, we can express
the entropy in terms of the condensed fraction (Table 2.1)
ζ(α + 1)S= (α + 1)
Nk
ζ(α)



N0
1−
N





N0
≡ f (α) 1 −
,
N

(5.4)

where we introduced the function f (α) for brevity. Changing the power-law from α1 to
α2 with constant entropy yields a new condensed fraction:


N0
N



f (α1 )=1−f (α2 )
2
66


 
N0
1−
.
N 1

(5.5)

5.4. EVAPORATION AND CONDENSATION

P/P0 = 1

P/P0 = 0.4

P/P0 = 0.2

50 µm

OD

3

300 µmOD0
60

Figure 5.5: Condensation by forced evaporation in a uniform trap. The power P of
the trapping potential is reduced to a fifth of its initial value P0 . Top panels: While thein situ shape (small insets) remains relatively unchanged, the momentum distribution
as observed in ToF shows a clear bimodality. Bottom panels: We show the cuts
through the respective column density distributions in ToF from the top panels. Figure
taken from [24].
This means that the condensate is only retained if the initial condensed fraction is large
enough, i.e. if f (α2 )
N0
≈ 64%.
(5.6)
>1−
N 1
f (α1 )We see in Fig. 5.4, that we need large condensed fractions in the harmonic trap to geta notable condensed fraction in the uniform trap. Due to residual heating and instability
at low powers, we found that large condensed fractions in the harmonic trap (> 50%) are
hard to achieve and even harder to measure. Hence, further evaporation after transfer inthe uniform trap is necessary to achieve condensation.
Also, early transfer into the uniform trap allows us to minimize three-body losses
in the harmonic trap. The three-body loss rate scales ∝ n3 , and is therefore strongly
enhanced in dense regions. This explains why these losses are prominent in the parabolic
condensate of a harmonic trap, but virtually non-existent in a uniform trap with similar
spatial extent and atom number.

5.4

Evaporation and Condensation

After the transfer into the box trap the cloud contains typically N ≈ 6 × 105 atoms at
T ≈ 130 nK. From this point, we cool the gas below Tc by forced evaporative coolingin the box trap, lowering the trapping power P in an exponential ramp with a 0.5 s time67

5.5. VOLUME AND IN SITU SHAPE

constant. Fig. 5.5 qualitatively illustrates the effects of evaporation and condensation in
the box trap. We show images of the cloud both in situ and after t = 50 ms of ToF
expansion from the trap.
While in a harmonic trap cooling results in a real-space bimodal distribution, as shownin Fig. 5.3, condensation in a uniform trap does not induce any dramatic change in real
space. Condensation occurs exclusively in momentum space (i.e., in ToF), and the emerging bimodality in the expanding density profile allows us to detect the formation of a BEC.
Besides the bimodality, we observe how the aspect ratio inverts, again a typical smoking
gun of condensation. Typically, we achieve condensates with up to 2 × 105 atoms.

5.5

Volume and in situ shape

The initially most striking signature of the transfer into the box is the change of the insitu density profile. Fig. 5.2 shows the density profiles before and after the transfer. The
clouds were imaged using high intensity imaging as described by Eq. (4.15).

5.5.1

Harmonic density profile

Integrating the density of the thermal component of a partially condensed gas in a harmonic trap Eq. (2.23), gives us the line density of the thermal component along any direction (here the x-axes)
n(x) ∝ g5/2 exp −(1/2)mw2 x2 .

(5.7)

Unsurprisingly, we find this describes our measured line density in Fig. 5.3c in the wings,
while a small condensate is visible in the centre.

5.5.2

Uniform density profile

For an ideal uniform box, we expect the same uniform profile for the thermal and condensed component (neglecting the healing length at the edges). We use the coordinate
system shown in Fig. 5.3, in which the x-axis is vertical (parallel to gravity) and the yaxis runs in the horizontal plane parallel to the rotation axis of the cylindrical box trap.
Integrating our can-shape trap of length L and radius R along z to obtain the column denp
sity, we expect a rectangle distribution along y. Along x we obtain 1 − (x/R)2 due tothe integration across the circular can-profile, and we can write the column density asn(x, y) ∝ Πyp
1 − (x/R)2 ,
L
68

(5.8)

5.5. VOLUME AND IN SITU SHAPE
where Π(y/L) is the rectangle function, which is one for |y| < L/2 and zero otherwise.
This is separable and the line density profiles are trivially given by the respective components of above equation. A measurement of our real box will deviate from this for two
reasons. First, the box potential is not perfect due to the diffraction-limited steepness of
the walls and other experimental imperfections. These imperfections are what we want toquantify. Second, the measurement of the density profile is not perfect. We need to take
into account the resolution of our imaging system which smears out the edges. Mathematically this can be modelled by convolving Eq. (5.8) with an assumed Gaussian point
spread function (PSF) 2
x + y2
1,
exp −P SF (x, y) =
2πσ 2
2σ 2

(5.9)

with a waist w0 = 2σ. The measured density ñ(x,y) is then given by the convolutionn(x, y) ∗ P SF (x, y)
 0




y p
(x − x0 )2(y − y 0 )202ñ(x, y) ∝Π
1 − (x /R) exp −exp −dx0 dy 0 .
L
2σ 2
2σ 2
(5.10)
Again, this expression is separable in x and y and can be easily integrated to obtain thecorresponding line density profiles. The uniform profile in Fig. 5.3d fits this prediction
quite well assuming a PSF with σ = 2.5 µm.
Z Z

5.5.3

General power-law traps

The description above is appropriate for an ideal box. Given that no box in nature is ideal,
we return to the power-law approximation in Eq. (2.23) to predict the in-trap thermal density for a degenerate gas at T = Tc . We will discuss to what extent measuring the in-trap
density profile allows us to extract a power-law estimate. In the spirit of the characterization of the trap on the bench mentioned earlier [70], we assume a radial power-law of
power n and infinitely steep walls for the sheets. We obtain a thermal density pnn0 (r) ∝ Π(y)g3/2 − (x2 + z 2 ) .

(5.11)

As we are only interested in the profile, we use dimensionless variables x, y, z. Calculating the line density profile along x is analytically not trivial, but can be done numerically.
In Fig. 5.6 we show the resulting profiles for various power-laws compared with the ideal
circular profiles. We see that for the range of power-laws in question, it becomes very hardto differentiate them for n > 6, especially when including the imaging resolution blur by
convolving with the PSF. Experimentally, shot noise and other imperfections make it very69

n(r) (a.u.)n(r) (a.u.)n(r) (a.u.)

5.6. THERMOMETRY(a) n=2(b) n=6(c) n=8n(r) (a.u.)x (a.u.)n(r) (a.u.)x (a.u.)n(r) (a.u.)x (a.u.)x (a.u.)x (a.u.)x (a.u.)(d) n=10(e) n=12(f) n=20

Figure 5.6: The theoretical column/line density profile (red) of various power-laws rn
at Tc along the radial direction of the box. They are compared to the ideal prediction
in an infinitely steep box (blue). Both are convolved with the same imaging resolution
PSF to simulate the observed profile. The densities are scaled to unity and the widthsof the power-laws and ideal box are matched for simplicity. We see that we can easily
distinguish power-laws . 6 from an ideal box profile, but it gets increasingly hardfor higher exponents. For very high power-laws, the profiles are identical to the ideal
box profile as expected.

hard to differentiate those profiles reliably. To add to the difficulties, we are never exactly
at Tc . This means, we have either a uniform density offset from the condensed density n0
for T > Tc , or we need to evaluate Eq. (2.23) for T < Tc with an unknown µ < 0. Given
these predictions, we see that obtaining a good estimate for the power-law exponent fromthe in situ density is not practical. For now, we approximate the real in situ density distribution as identical with the density in an ideal uniform trap. We will see from further
analysis that the power-law is steep enough to justify this assumption.

5.6

Thermometry

We now assess the flatness and uniformity of our potential by considering the thermodynamics of the trapped gas. By contrasting our findings with the prediction for an ideal
box and a harmonically trapped gas, we quantify the uniformity of our gas.
70

5.6. THERMOMETRY

5.6.1

Scaling laws

We recall from Section 2.2.2, that the thermodynamics of a power-law potential
U (r) =

3
X

si

i

xi
Li

X

1/si .

i=1(5.12)

scale with the parameter
α = 3/2 +(5.13)

i=1

For a uniform box, we have si = ∞ and thus α = 3/2; for a harmonic trap with si = 2
we have α = 3. From theoretical limitations and bench measurements of the trap, [70]predicts power-laws of s  10 axially and s & 7 radially. Hence, we expect α .
3/2 + 1/∞ + 2/7 ≈ 1.8. We have seen that the measurement of thermal in situ profiles
agrees with these predictions, but does not allow great accuracy. Instead, we consider thein-trap momentum distribution from Eq. (2.25)
nP L (p) ∝ (kT )δ gδ exp(−β(µ − p2 /2m)) ,(5.14)

which also depends directly on α = δ + 3/2.
The traditional way to extract the momentum distribution is to image the cloud in ToF.
Unfortunately, even with a long time of flight (t > 50 ms), we cannot use Eq. (5.14) to
describe the density distribution. At typical temperatures (T ∼ 100 nK), the distancetravelled and the initial size of the trapped cloud are comparable. While we can evaluate
Eq. (2.37) to describe the expansion at any ToF numerically, this is very unwieldy forfitting purposes.
We have established in Section 2.3.2 that we cannot generally write the distributionin ToF n(r, t) as the convolution of the in situ density with the momentum distribution unless the in situ distribution is uniform. We have established earlier that the in-trap density distribution is de-facto uniform. Hence, we approximate the ToF-distribution n(r, t)
as the convolution of the density of an ideal box nbox (r0 ) with the power-law momentum
distribution nP L (p) from Eq. (5.14)Zn(r, t) ∝nP L (p = (r − r0 )m/t) × nbox (r0 )dr’


 0


y
2 βm
×Π
Π
∝ gδ exp −βµ − (r − r’) 22t
L(5.15)√

x02 + z 02
R!
dr’. (5.16)

Integration along the line of sight z yields the column density ñ, which is observed directly715.6. THERMOMETRY

in our experimentsZñ(x, y, t) ∝



h
m
m i
gδ+1/2 exp −β µ − (x − x0 )2 2 + (y − y 0 )2 22t2t
 0  0pyx×Π
1 − (x0 /R)2 dx0 dy 0 .
Π
LR(5.17)(5.18)

Further integration along y yields the line density n̂Zn̂(x, t) ∝

h
i
0 2 m
gδ+1 exp −β µ − (x − x ) 2
2t
 0
p
x
1 − (x0 /R)2 dx0 .
×Π
R




(5.19)(5.20)

Both the column density and line density can be easily evaluated numerically. The convolution adds a small initial size correction to the ToF-expansion of the momentum distribution.

5.6.2

Numerical implementation

To implement this into a numerical fitting algorithm for the column density, we introduce
the following reduced
variables: the reduced chemical potential µ̃ = βµ, the thermalq2

kTx,y twidths Rx,y =, the centres of the cloud xc , yc and a scaling parameter A(T ).
mAdditionally, we use a normalized convolution matrix (kernel) K(x, y), which is proportional to the in situ column density of an ideal box given by Eq. (5.8). For simplicity
we assume that the measured optical densities have been converted to column densities
according to Eq. (4.9). Eq. (5.18) can now be written as the convolution




A
(x − xc )2 (y − yc )2ñf it (x, y) =gδ+1/2 exp µ̃ −−∗ K(x, y).
Rx Ry
2Rx2
2Ry2

(5.21)We have introduced different thermal widths for x and y to take a small harmonic magnetic trapping potential into account, which compresses the distribution along y during
ToF. We therefore extract two different temperatures Tx and Ty . As Tx is not compressed,
it is more reliable. Unless mentioned otherwise, we will set T = Tx .
Usually, we deal with a bimodal distribution including both a thermal component described by Eq. (5.21) and a condensate of unknown shape occupying the centre of theimage. To obtain a fit for the thermal component, we ignore the region in the centre
containing the condensate and consider the thermal wings only. Numerically, we implement this using a mask excluding an elliptical region around the centre of the condensate
72

5.6. THERMOMETRY(x0 , y0 )1 forM (x, y) = 
0 for



2
(x−x0 )20)
+ (y−y
> 1
r0 x
r0 y

.

(5.22)

2(x−x0 )20)
+ (y−y<1
r0x
r0y

We then use Matlab’s least-square fitting algorithm to find the best match of ñf it (x, y)
to the measured column density ñexp outside the masked area by minimizing the sum-ofsquares of the matrix|ñf it (x, y) − ñ(x, y)exp | × M (x, y).
(5.23)
Usually, we vary A, Rx , Ry , xc , yc , µ̃ as free parameters. The procedure can be easily
modified to work in one dimension on line densities.

Divergence issues
A fine point that can cause some numerical headaches is that gz (x) diverges for z ≤ 1 andx → 0. While this is not an issue for the line density gδ+1 (x) with finite power-laws with
δ > 0, the column density gδ+1/2 (x) diverges if δ < 1/2. It is important to keep in mind
that we do not sample a continuous distribution, but an averaged value for each pixel.
This is irrelevant for slowly varying distributions far away from the singularity. Near thedivergence, we need to integrate x and y over the pixel area. The integration along two
spatial directions increases the order of the polylogarithm by one to δ + 3/2 and ensuresa finite value.
Even when considering continuous functions on a theoretical level, the convolution
with a finite size kernel has the same effect. As we always deal with a finite-size, real
BEC and not a point source, this prevents an unphysical singularity in the density.

Extracting thermodynamic properties from the fit
After finding the best fit, we can easily extract thermodynamic properties from the fit
parameters. The total thermal number N 0 can be extracted by integrating Eq. (5.21)
N 0 = 2πAgδ+3/2 (µ̃) .

(5.24)

The condensate number N0 is calculated by summing over excess atoms within the condensate region:Z
N0 =ñexp (x, y) − ñf it (x, y) × (1 − M (x, y)) dx dy.
73(5.25)

5.7. HOW UNIFORM IS THE TRAP?

Figure 5.7: We search for the best fit to the line-density n̂ of a thermal, degenerate
cloud for various power-law parameters δ. In (a), we compare the experimental linedensity along x (red) to the uniform fit (green dashed) and the harmonic fit (blue). In(c), the fitting residues are shown. (b) shows the reduced fitting χ2 against δ. Figure
adapted from [24].
The temperature T = Tx is then

Rx2 m
,
kt2

(5.26)µ = µ̃kT.

(5.27)

T =while trivially

In practise, all these continuous functions and integrals need to be converted to discrete
matrices and sums, which is here omitted for readability.

5.7

How uniform is the trap?

5.7.1

Conclusions from a single cloud profile

As an initial test of our fitting procedure we scrutinize the line density of a non-condensed
degenerate cloud in ToF. We apply various fits (using the line density version of Eq. (5.21))
starting with a uniform profile δ = 0 to a harmonic profile δ = 3/2. We find that theuniform profile fits the experimental data substantially better than the harmonic profile.
Fig. 5.7 illustrates this in various ways. The difference ∆n̂ between the fitted profiles and74

5.7. HOW UNIFORM IS THE TRAP?

the experimental line densities shows a clear preference for the uniform profile. Whilethe harmonic profile has clear systematic deviations of ∆n̂ from zero, the uniform profile
fulfills ∆n̂ ≈ 0. To quantify this, we fitted the profile for various value of δ and calculated the reduced χ2 . We find that χ2 is approximately constant for values of 0 ≤ δ ≤ 0.7,
while values above that are strongly excluded.
This result is in line with our previous boundaries from theoretical predictions andin situ volumes. While it excludes a harmonic trap strongly, it does not place very strict
bounds on the effective power-law (s > 4).

5.7.2

Conclusions from several profiles

Looking at a single image is clearly not enough to determine the effective power-law
accurately. We therefore consider a series of images across a range of temperatures
90 nK < T < 135 nK, which is above or just below condensation. We exploit the factthat A as defined in the previous section is not an arbitrary scaling constant - even though
we treated it as a free fitting parameter. Evaluation of the column density with the relevant
prefactor yields
 αT.
(5.28)
A=θ
Here, we switched again to α = 3/2 + δ for compactness, as it is the natural variable formany thermodynamic relations in power-laws (see table 2.1). θ is a temperature independent constant that absorbs various numerical and physical prefactors. While we are notinterested in the absolute magnitude of θ, there should be no variation with temperature,i.e. dθ/dT = 0 for the correct power-law.
We fit the series of pictures with power-laws of 1.5 ≤ α ≤ 2 (corresponding to powerlaw exponents between s = ∞ and s = 6) and and extract θ(α) for all pictures. As we
can see in Fig. 5.8a, θα (T ) is quite flat for α ≤ 1.8, while for larger power-laws there
is a clear temperature dependence. We quantify the deviations in first order using linear
fits. The slope of the fits are shown in Fig. 5.8b, confirming that the effective power-law
is α ≤ 1.8, with the best candidate being α = 1.7 (s = 15) .
To underline that this analysis finds a good fit and not only selects the best out of abunch of bad options, Fig. 5.9 shows a two-dimensional cloud profile near condensation
fitted with δ = 0.2 and δ = 1.5. We can see by looking at the fitting residues that δ = 0.2
yields a good fit with no major deviations from the experimental data. δ = 1.5, however,shows significant deviations near the cloud centre, and does not represent the data well.
In the original publication [24], we used a slightly different approach. First of all, we
used the 1-D line-density for the analysis. The advantage of this is simplicity and fitting
speed. On the down side, a lot of information can be lost due to averaging prior to fitting.
75

5.7. HOW UNIFORM IS THE TRAP?
(b)dq•dT @a.u.D

q @a.u.D

(a)

2
1
90

100

110

120

130

0

1.5 1.6 1.7 1.8 1.9 2.0
aT•nK

Figure 5.8: Considering the momentum distribution over a range of temperatures
allows us to assess the uniformity of our trap. (a) The ideal gas scaling in Eq. (5.28)
predicts that the parameter θ(T ) is constant for a range of temperatures for the correct
value of α. We show θ(T ) for α = 1.5 (bottom red) up to α = 2 (top orange). The
data is approximated by linear fits. (b) The slopes of the fits in (a) are plotted against
the respective value of α.

Notably, to avoid the condensate region, we have to exclude a whole strip of the image,rather than just an elliptical region around the condensate. Also, we included temperatures
far below condensation, which has several drawbacks. First of all, the effective power-law
is an approximation and we cannot assume that one power-law value is appropriate overan arbitrary range of temperatures. In terms of Fig. 5.8a, this would imply that there is no“correct” α to ensure θ(T ) = const for all temperatures.
More importantly, we made the inherent assumption that the condensate does not
influence the thermal cloud and that we can use ideal gas theory to describe the gas below
the critical temperature Tc . While we are going to show in Chapter 6 that this is a good
approximation, we should not draw any premature conclusions at this stage. For instance,
it has been shown [16], that the presence of a condensate alters the thermal distribution inharmonic traps.

5.7.3

The critical point

Using α = 1.7 to fit the clouds, we will compare the observed critical temperature Tc withthe prediction for a uniform box. We will show that the quantum mechanical treatment
yields the same transition temperature as the semi-classical treatment in Section 2.2.2. In
addition to that, we consider corrections to the transition temperature due to the finite sizeof the trap as well as interactions.
76

5.7. HOW UNIFORM IS THE TRAP?

Figure 5.9: A two dimensional optical density image fitted with different values forδ. The cloud is near the critical point. The top row is fitted with the best valueδ = 0.2, while the bottom row shows the same cloud fitted with δ = 1.5, which
would be the right parameter for a harmonic trap. (a) An intensity plot of measured
cloud profile. The panels below and to the right show cross sections superimposedwith the respective fit (red). The colour map ranges from -0.1 to 1.29. (b) The fitting
residue. There are no significant deviations from zero, apart from the cloud centre
where the BEC starts forming in various locations. Note that the colour map ranges
now from -0.15 to 0.3. (c) Major deviations between the fitted and measured crosssections can be seen in the cloud centre which is most sensitive to the chosen valueof δ. (d) These deviations are clearly visible as coloured rings in the residue plot; thefit does not describe the cloud shape well.
Quantum mechanical density of states
Using the semi-classical description, we have already derived the critical number Nc fora given temperature as:ζ(3/2)= V ζ(3/2)
Nc = V
Λ3



mkT
2π~2

3/2
.

(5.29)

To do so, we approximated the density of states by the semi-classical distributionZN≈Zg()f ()d ≈77f (r, p)/h3 dr dp .

(5.30)

5.7. HOW UNIFORM IS THE TRAP?

We quickly recapitulate the derivation of the quantum-mechanical density of statesin the usual form, before considering additional corrections which give rise to finite size
corrections to Nc . The energy eigenvalues of a three-dimensional cube with length L aregiven byπ 2 ~2 2(n + n2y + n2z ) ≡ E1 (n2x + n2y + n2z ),(5.31)
(nx , ny , nz ) =2mL2 xwhere ni = (1, 2, 3, ...). We can consider those eigenvalues as points in a three-dimensional
space with coordinates (nx , ny , nz ). The total number of states G() with energy E <p(k, l, m) is approximated by the volume of a sphere with radius /E1
"   #
3/2
1 4π
G() =,
8 3 E1(5.32)

where the factor 1/8 makes sure that we only count the states in the positive octant of thecoordinate system. Differentiation yields the density of states g():g()d =π −3/2 1/2
E
 d.
4 1

(5.33)

If we use this in Eq. (5.30) to calculate Nc we obtain the same result as the in Eq. (5.29),
which has been derived semi-classically.

Finite size corrections
While the above derivation is correct in the thermodynamic limit, we need to take finite
size effects into account when evaluating the density of states of a finite sized box. So
far, we have ignored the states with ni = 0 in Eq. (5.32). Quantum-mechanically, these
are not accessible as all quantum numbers need to fullfill ni ≥ 1. We therefore need tosubtract the states on the coordinate planes to obtain a better estimate for the total numberof states
"  
#
3/2
1 4π


− 3π
,
(5.34)
G() =
8 3 E1
E1
yielding the corrected density of states"#
π 1/2
1
g()d =− 3πd.
4 E13/2
E1(5.35)The observant reader will have noticed that we are still neglecting over-counting on theaxes and the origin - these effects are of higher order and negligible [73].
Unfortunately, we cannot integrate this new density of states in a straightforward man78

5.7. HOW UNIFORM IS THE TRAP?
(a)

(b)

0.12

140

0.11
120

ideal
finite size

0.1

Tc [nK]

Tc/Tc0−1

100
0.09
0.08
0.07

80
60

0.06

40

0.05

20

0.04
0

2

4
N’/ 105

6

0
0

8

2

4
N’/ 105

6

8

Figure 5.10: Critical temperature in a box with L = 63 µm and R = 15 µm. (a)Relative finite size shift Tc /Tc0 − 1. (b) Critical temperature for a range of common
atom numbers.

ner, i.e. we cannot use

Z∞
Nc =

f (, µ = 0)g()(5.36)

0

to obtain a revised Nc , as the correction to the density of states causes the integral todiverge. This can be solved by introducing a lower cut-off energy on the order of theground state energy 0 = 3E1 . The details are rather intricate and are described in [73].
The final estimate for the critical temperature Tc is
Tc = Tc0




2
1 p
0
1− √
0 /kTc0 ln (0 /kTc ) ,3π ζ(3/2)(5.37)

where Tc0 is the critical number in the thermodynamic limit as obtain by inverting Eq. (5.29).
So far, we have assumed that our trap is an isotropic cube. The generalization of
Eq. (5.37) to an anisotropic box with the same aspect ratio as our trap has been done in[27]. This anisotropy scales the correction term in Eq. (5.37) with a shape factor of order
0.95, which is experimentally irrelevant. Also, we should note that exact summing overthe states (also done in [27]) introduces slightly larger deviations, which are still below ourexperimental accuracy. A more details analysis of the finite size corrections (especiallyin a cylindrical geometry) might become necessary once precision measurements of thephase transition are attempted. For our purposes, we estimate the leading order of thefinite size shift with Eq. (5.37), while using the ground state energy of a cylindrical box3
3
Its ground state energy is pretty much identical to the ground state energy of a cubic box with the same
aspect ratio and volume.

79

5.7. HOW UNIFORM IS THE TRAP?
(b)

(a)

1.0

N'•105N0 •105

1.5

0.5
0.0
-0.5

0

20

40

60

80

100 120 140

7
6
5
4
3
2
1
0
0

20

T•nK

40

60

80

100

120

140

T•nK

Figure 5.11: Determination of the critical point in a quasi-uniform trap. (a) Extrapolation of the condensed number N0 to zero to find the experimental critical temperature Tc . (b) The thermal number atom numbers N 0 (red dots) are plotted against
temperature T . Based on the critical temperature Tc = 113 ± 3 nK measured in (a),we determine the theoretical prediction for the critical number N (Tc ) (black). We
also indicate its errors due to experimental and theoretical uncertainties. The red line
is a T 1.7 fit to the N 0 below Tc , which is the expected scaling for an ideal gas. Above
Tc , µ < 0 and N 0 departs from this scaling.
as 0 . Fig. 5.10 illustrates the relative and absolute magnitude of the finite-size Tc shift.
Interaction shift The presence of small interactions in a homogeneous gas are assumed
to cause a beyond-mean-field shift of the critical temperature [10]:Tc− 1 = 1.3an1/3 .
Tc0(5.38)For typical atom numbers, N < 6 × 105 , this relative shift is smaller than 2.5% andtherefore small compared to the other theoretical and experimental uncertainties. For thetime being, we ignored the shift. In the future, a precision measurement of Tc using
tunable interactions might be used to verify the magnitude of this shift.

5.7.4

Finding the critical point

Now we can calculate Tc accurately, we compare the theoretical prediction with the experimental measurements. We use the same set of data that we have used to calibrate αearlier. In Fig. 5.11a, we extrapolate the condensed number N0 to zero and find the critical temperature Tcexp = 113 ± 3 nK. From this we deduce the expected critical number
Nc using Fig. 5.10b. In Fig. 5.11b, we compare this to the experimental findings. We
find that the expected critical number (black marker) matches the experimentally measured thermal number (red) at the transition (i.e., N0 = 0) within error. The error is80

5.7. HOW UNIFORM IS THE TRAP?
mainly introduce due to uncertainties in atom number calibration ≈ 10% and trap volume
measurement ≈ 14%.
Below Tc , we expect N 0 ∝ T α with α = 1.7 for an ideal gas. The fit to Fig. 5.11byields good agreement. This is a first indication that the gas behaves like an ideal gas
despite interactions, i.e. the thermal gas is not influenced by the condensate. Above Tc ,the thermal number N 0 drops below the N 0 ∝ T α line, as expected for a non-condensed
gas.4
A note on θ and N 0 Now it might be more obvious why we have considered θ only for
T & Tc . If we had applied this method below Tc , we would select the specific α forcing
the thermal number to follow the ideal gas scaling N 0 ∝ T α . More specifically, below
Tc , the chemical potential is zero, µ ≈ 0, and there are only two degrees of freedom in
the fits Eq. (5.21): T and α. If we were to select α such that θ(T < Tc ) is constant, we
automatically make sure that N 0 ∝ T α . This can be seen by inserting Eq. (5.28) into
Eq. (5.24).
In Chapter 6, we will properly scrutinize the saturation of the thermal gas. Hence,here we can only use T & Tc to analyse the trap or we risk falling for a self-fulfilling
prophecy.

5.7.5

Conclusions

We have created the first truly three-dimensional quasi-uniform trap. By approximating
the trap with an effective power-law potential, we have quantified the small deviation from
uniformity. All different forms of analysis put a lower bound on the power-law exponents ≥ 10, while s ≈ 15 describes the trap best. We found that the critical point can be wellestimated by approximating the trap with an ideal box of appropriate volume.

4
This illustrates the quantum-statistical character of Bose-Einstein condensation. For a given temperature, the total number of occupiable thermal states scales Nc ∝ T α . If the total number of atoms N exceeds
Nc , all thermal states are occupied, i.e., N 0 = Nc and the rest of the atoms occupies the ground state:
N0 = N − Nc . If we have fewer atoms than the critical number, N < Nc , we can accommodate all theatoms in the thermal states N 0 = N < Nc ∝ T α .

81

5.7. HOW UNIFORM IS THE TRAP?

82

Chapter 6
Thermodynamics in a uniform trap
In this chapter, put our uniform trap to use and conduct a range of thermodynamic measurements. We focus on measurements of a range of quantum-statistical effects, which
are obscured in harmonic traps, even when using only weakly interacting gases. First, inSection 6.1, we measure the critical number for a range of temperatures. In other terms,
we are mapping out the boundary in the T -N phase diagram, which separates the condensed from the thermal gas. Comparing the experimental data to the prediction allows
us to finalize the value of the effective trap power-law. In Section 6.2, we show that thethermal component in our trap is saturated, i.e. it follows ideal gas scalings even in thepresence of a condensed component. Finally, in Section 6.3, we observe the quantum version of the Joule-Thomson effect. Unlike the interaction-driven classical equivalent, thequantum-mechanical version is purely statistical. The main results of this chapter havebeen published in [25].

6.1

Critical number

In the previous chapter we analyzed a rather arbitrary trajectory (see Fig. 6.1) in the T -N
phase diagram to find α. This trajectory was determined by the evaporation process rather
than by theoretical considerations. While offering experimental ease, we have seen that
this causes two major theoretical headaches. Below Tc , we have to assume saturation, i.e.
N 0 = Nc , and above Tc we rely on correctly measuring the chemical potential µ.
Here, we will instead measure the critical number Nc at different temperatures, solving
both of these issues. In terms of the N -T phase diagram, we will trace the boundary
between the non-condensed and condensed phase (black in Fig. 6.1). We expect that thecritical number follows
Nc ∝ T α .
(6.1)
83

6.1. CRITICAL NUMBER

Figure 6.1: The evaporation trajectory from Chapter 5 is shown. The green points
were used to extract α, while we discarded the blue points due to their high condensed
fraction. The black line represents the phase boundary, which we trace and use foranalysis in this chapter.

Assuming that this ideal gas scaling holds, we will determine the power-law parameter α,which fits the experimental data best. We will find that this method offers better accuracy
than the one employed in the previous chapter.

6.1.1

Experimental procedure

To measure T and the number of thermal and condensed atoms, we image clouds after
50−70 ms of time-of-flight expansion from the trap. We vary T by changing the final trap
depth of the evaporation sequence, while we vary the total atom number N by adjusting
the MOT loading time. After forced evaporation we always raise the trap depth to U0 ∼kB × 0.4 µK, where evaporation is negligible. Also, ramping back up to a fixed trap
depth, i.e. the same trapping laser intensity, ensures the trap shape does not vary between
measurements. After changing the depth, we hold the gas for another second before ToF
to allow for thermalisation.
We determine the critical point by the appearance of small BECs. All images thathave small, but unmistakable, condensates in the range 1000 < N0 < 5000 are selected
and assumed to have a thermal number N 0 ≈ Nc .1 We then average the data points withinsmall temperature bins (3 nK), and calculate the errors of the mean. The resulting smoothseries of points can then be fitted with function Eq. (6.1) to obtain α as shown in Fig. 6.2.

1

For a saturated gas, this equation holds.

846.1. CRITICAL NUMBER
(a)

(b)

(c)

Figure 6.2: Critical point in a quasi-uniform box. (a) A fit to the critical number
Nc ∝ T α gives α = 1.65. (b) αf = αc (see text) is fulfilled within errors for1.6 . α . 1.7. (c) α = 1.5 is still a good fit, illustrating the smallness of thedeviation from perfect box.

6.1.2

Self-consistency

Unfortunately, this method has a slight complication: in order to extract N 0 , N0 and T
from an image, we need to fit the cloud column density with the appropriate fit function∝ gα−1 . Obviously, this requires prior knowledge of α, which we were hoping to extractby fitting Nc (T ). To resolve this issue, we fit all images with a sensible range of α values
1.5 ≤ αf ≤ 2.0. For each αf value, we therefore obtain a different Nc -T plot similar to
Fig. 6.2. To each of these plots, we fit Nc ∝ T αc , where αc is a free parameter. For each
assumed cloud profile αf , we extract a corresponding αc describing the phase boundary.
For an ideal gas, αc and αf are identical. Therefore self-consistency requires αc = αf .
For our trap, this requirement yields 1.6 . α . 1.7 (see Fig. 6.2b).
This value is consistent with the estimate in the Chapter 5, 1.5 . α . 1.8, but offers
smaller error bounds. It should be noted that the trap was not exactly identical and weoperated at lower temperatures, so we do not expect exact agreement. For the further data
analysis we fix αf = 1.65, which corresponds to an effective power-law r20 .
While in the above we were careful to characterize the small deviation of our trap froma perfect box, we note that simply assuming a perfectly homogeneous gas of constantvolume leads to very small errors. This is shown in Fig. 6.2c: we fix both αf and αc to85

6.1. CRITICAL NUMBER
(a)

(b)

−14

x 10

V [1000µm2]

5.5

5

4.5

4
20

30

40
T [nK]

50

60

Figure 6.3: (a) Dependence of measured volume on temperature (blue). A powerlaw fit gives V ∝ T 0.3±0.2 (red). (b) Comparison of the experimentally measured
critical density (blue band) with the theoretical prediction of the critical density ofan ideal box including the finite size effect (black dashed line). The blue band is
calculated from the power-law fit to the measured volumes in (a) and the criticalnumber measured in Fig. 6.2a. The width of the band represents the experimental
systematic errors.
the ideal-box value α = 3/2 and obtain a fit of Nc (T ), which is still quite good.
Method comparison It is interesting to note that this self-consistent method of finding
α is conceptually using the same procedure as the one based on θ(T ) = const, which weused in Section 5.7.2:• We measure the density distribution in ToF of a degenerate gas at different temperatures and numbers.
• We guess a range of values for the power-law parameter α and calculate the dependent thermodynamic quantities for each value of α.
• We then select the value for α which matches the ideal gas prediction N 0 (T, µ, α)(Eq. (2.13)) best in a self-consistent manner. We introduce a single-parameter toquantify the deviation. In Chapter 5.7.2 we required dθ/dT = 0. With the selfconsistent method above, the corresponding criterion is αf = αc .

6.1.3

Dependence of volume on temperature

In the spirit of the local-density approximation, we can also attribute the deviation fromthe perfect uniform scaling, Nc ∝ V T 3/2 , to an increase of the box volume V with temperature T . Small deviations of α from 3/2 correspond to a slight increase of the effective866.2. SATURATION OF THE THERMAL COMPONENT
trapping volume with temperature, V0 ∝ T δ ∝ T α−3/2 (see Eq. (2.25)). From in situ
images, we measure a (20 ± 10)% increase in V0 between T = 25 and 50 nK, (shownin Fig. 6.3a). A power-law fit V (T ) ∝ T α−3/2 gives α = 1.8 ± 0.2. While consistent
with the previous results, it should be clear from the scatter and systematic errors that thismeasurement is supplementary at best.

6.1.4

Critical density

Over the whole temperature range in Fig. 6.2a, we can use the effective volumes V0 (T )as interpolated in Fig. 6.3a to calculate the corresponding critical phase space densityρc (T ) = Nc (T )Λ3 /V0 (T ) . We obtain ρc (T ) ≈ 2.0 ± 0.2 (blue band in Fig. 6.3b), which
we compare to the theoretical prediction. In the thermodynamic-limit we expect ρc ≈2.612. The inclusion of the finite-size correction as in Fig. 5.10 reduces this value ρc ≈ 2.4(black dashed in Fig. 6.3b). The remaining discrepancy can be explained within oursystematic uncertainties of 10% in Nc and 20% in V0 . Also, possible disorder, i.e. noiseat the bottom of the potential could reduce the effective volume further. The influence ofdisorder remains to be studied.

6.2

Saturation of the thermal component

So far we have neglected the interactions in the gas. We have assumed that the thermal gas
acts like an ideal, non-interacting gas with the total number capped at the critical number:
N 0 = Ncid . No interaction occurs with the condensed component, which holds the surplus
atoms N0 = N − Ncid . In a harmonically trapped gas this picture is strongly violated evenfor weak interactions, such as in 87 Rb, and can be recovered only by extrapolating to thestrictly non-interacting limit [16], where direct measurements cannot be performed due
to absence of thermal equilibrium [74]. This strong deviation from ideal-gas behaviour
arises due to an interplay of interactions and the non-uniformity of the gas, and can be
explained using a mean-field theory that does not require violation of the saturation picture
at the level of the local thermal density (LDA) [16,34]. We will now examine the situation
experimentally and theoretically in a uniform gas.

6.2.1

Mean-Field theories

To first order, saturation effects are usually discussed in the framework of mean-field
theories, instead of considering every single atom-atom interaction. We have alreadyencountered the Gross-Pitaevskii equation, which addresses the influence of the meanfield gn on the ground state wavefunction at T = 0. Several frameworks exist to tackle
876.2. SATURATION OF THE THERMAL COMPONENT

the influence of the mean-field for T > 0 onto the thermal and condensed distributions.
Hartree-Fock approximation
In the harmonic trap, experimental studies revealed a strong non-saturation, N 0 (T .
Tc ) > Nc . For temperatures T . Tc , this can be explained within the Hartree-Fock approximation [33]. In this framework, we account for the extra interaction energy between
atoms in different energy levels, which arises from the exchange term of the bosonic
wavefunction. In a partially condensed gas, only the ground state is macroscopicallyoccupied. We can therefore approximate that adding a ground state atom increases themean-field energy by g(N0 + 2N 0 ), while adding it in any other state increases the energyby g(2N0 + 2N 0 ). Incorporating the LDA, we obtain the following set of equations forthe thermal distribution [28]:p (r) = p2 /2m + 2n0 (r)g + 2n0 (r)g + U (r)Z
dp
10
n (r) =.
exp[(p (r) − µ)/kB T ] − 1 h3

(6.2)
(6.3)

Within the Thomas-Fermi approximation, the local condensate density is related to thechemical potential as:µ = U (r) + [n0 (r) + 2n0 (r)]g.
(6.4)
Generally, these equations need to be solved self-consistently. Fortunately, in the caseof a harmonic trap, we can apply several approximations, which simplify the problem to apoint where we do not need numerical calculations. First of all, in harmonic traps, the condensate is concentrated in the centre of the trap and has typically densities n0 (r)  n0 (r).
We therefore neglect the mean-field of the thermal density, and only consider the repulsion originating from the condensed fraction. The condensed density is then described bya parabolic Thomas-Fermi profilen0 (r) = max([µ − U (r)] /g, 0).

(6.5)

We can thus approximate (p (r) − µ) in Eq. (6.3) as
p2+ Uef f (r),
2m

(6.6)

where the effective potential Uef f (r) = |µ − U (r)|. Hence, the thermal atoms behave as
if they were confined in a potential Uef f , which is shaped like a Mexican hat. Within the
LDA and at a fixed temperature, this effectively increases the number of thermal states
as the condensate grows. Therefore, when increasing N0 , N 0 grows as well and thus88

6.2. SATURATION OF THE THERMAL COMPONENT
N 0 > Nc .
In first order, this effect can be approximated as [34]:gn0 (0)N0≈ 1 + 1.37
.
Nc
kB T

(6.7)

This equation has been confirmed experimentally for small condensates [16]. For larger
condensates, including higher orders of gn0 (0)/kB T becomes necessary. Even when
doing so, the experimental measurements deviate from the MF-predictions and show thatthe full scope of the problem cannot be captured within MF-theory.
We now apply the Hartree-Fock approximation to a uniform gas. The local densities
are trivially related to the total number by:
n0 (r) = N0 /V(6.8)

n0 (r) = N 0 /V

(6.9)

Condensation occurs for µc = 2n0 g ≈ 0.05kB T (for typical numbers) ; the critical number
is not affected by this shift in µ. Below Tc , the chemical potential is given byµ = (n0 + 2n0 )g = µc + n0 g

(6.10)and the density can be evaluated as:
0

Z

n =

dp
1
exp [(p2 /2m + n0 g)/kB T ] − 1 h3

=g3/2 (exp[−n0 g/kB T ]) /Λ3(6.11)(6.12)

This expression is very similar to the thermal number of an ideal gas above Tc , whereµ < 0 in the polylogarithm is replaced by −n0 g < 0.
We see that within the HF-theory, we actually expect weak anti-saturation, i.e. a
decrease of the thermal number with increasing condensed number. Unlike in harmonic
trap, the deviation from the ideal gas is not driven by a change in effective volume, but
can be regarded as a result of the modification of the effective dispersion relation bymean-field energies.

Popov approximation
We now consider the situation in a general power-law potential with exponent s by solving
the local equations for the density numerically. Intuitively, we expect the saturation slope
to vary somewhere in between the limiting cases s = 2 and s = ∞. As we will solve the89

6.2. SATURATION OF THE THERMAL COMPONENT

1

0.8

N’/Nc

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

N0 /N

Figure 6.4: Hartree-Fock approximation for a uniform gas. The normalized thermal
number is plotted against the condensed fraction.

problem numerically anyway, we will use the slightly more complex Popov approximation [28, 33, 75]. The numeric solutions of these equations have been considered in detailin [76] for the uniform and harmonic trap. For completeness we will quickly reproduce
the methods and notation used for solving the uniform problem, before applying them toa range of power-laws. The equations for the uncondensed gas remain the same as in the
Hartree-Fock approximation Eq. (6.3). In the presence of a condensate we use [28]µ = U (r) + (n0 (r) + 2n0 (r))gZ
a (r, p) dp
10n (r) =
exp [(r, p)/kB T ] − 1 (r, p) h3
p2
a (r, p) =+ gn0 (r)2m
1/2
 2  2pp
(r, p) =+ gn0 (r).
2m 2m

(6.13a)(6.13b)(6.13c)(6.13d)Like in the Hartree-Fock approximation, we take the exchange energies into account. Additionally, the dispersion spectrum Eq. (6.13d) is modified to match the T = 0 Bogoliubov
dispersion spectrum [77]. This ensures we treat the low-energy phonon-like excitations
correctly.
Within the LDA, we can solve the equations locally. For each value of µ, U and T wecan calculate the densities n0 and n0 by solving Eqs. 6.13. We introduce the dimensionlessvariables
α=

µ−U
kB T

A=4

a
Λ

G0 =
90

2gn0
kB T

G0 =

2gn0
.
kB T

(6.14)

6.2. SATURATION OF THE THERMAL COMPONENT

1

0.8

N’/Nc

0.6

0.4

0.2

0

0

0.2

(a)

0.4
N0 /N

0.6

0.8

1

(b)

Figure 6.5: Numerical solution of the Popov equations in a uniform potential. (a)Graphical representation of Eq. (6.16) (figure from [76]). The value at the intersectionof F (x) and α − x/A gives the phase space density D0 of the thermal fraction. For
x > 0, condensation occurs and D0 > 0. (b) The normalized thermal number N 0 /Nc
is plotted against the condensed fraction for a uniform gas at T = 45 nK. The gap
at low condensed fractions is caused by the ambiguity between multiple solutions incase II.

The phase-space densities are given by
Di = Gi /A.

(6.15)

It can be shown that solving Eqs. 6.13 is equivalent to solving the following equation forx = α − AD0 :α−x= F (x),(6.16)
A
where




x
 g3/2 (e ):x≤0 
F (x) =.
(6.17)
1/2
2
2 1/2

 √2 R ∞ [(x +Q ) −x] dQ : x ≥ 0 

eQ −1
π 0
From the value x which solves Eq. (6.16), we can deduce
D0 = (α − x)/Aand


 D0 = 0:x≤0 

(6.18).

(6.19) D = 2Ax : x ≥ 0 
0Fig. 6.5a illustrates the problem graphically. We find solutions by searching for theintersection between the left- and right-hand sides of Eq. (6.16). There are three different916.2. SATURATION OF THE THERMAL COMPONENT

kind of cases:• Case I: One solution with x > 0, a BEC is present.
• Case II: Three solutions. We assume that the purely thermal solution at x < 0 iscorrect.
• Case III: One purely thermal solution with x < 0.
Case II is somewhat problematic. We are not aware of any justification for choosing
the purely thermal solution. This choice ensures that the transition in ni from Case III to
Case II is continuous. The transition from Case II to Case I when α/A > ζ(3/2) ≈ 2.612
introduces a discontinuity in the solution for x and thus ni .
Ignoring this issue, we see that the solutions for thermal and condensed density are
only a function of α and A. In terms of common parameters, it depends on chemical
potential, local potential and temperature. For a given temperature, we can now vary µ
to obtain the self-consistent solutions for a uniform potential with U = 0 as shown in
Fig. 6.5b.
With the uniform solutions ni (µ/kB T ) at hand, we can calculate the solutions for arbitrary potentials within the LDA. We sample the potential U (r) in small enough volume
elements dV to obtain the distribution f (U ). Each of these elements contributes a density
ni (µ − U (r)/kB T ). The total atom numbers N0 and N 0 for a given chemical potential arecalculated asX
Ni (µ) =f (U )ni (µ − U, T ) dV,(6.20)
where we can truncate the sum for marginally populated states U & 2kB T . The implementation of this is straightforward; convolution of the arrays containing the tabulated
values for ni and f gives Ni , while circular shifting of ni allows us to vary µ. In Fig. 6.6
the results for various power-laws are compared. We see that increasing the power beyonds = 2 quickly reduces the saturation slope. For s = 8, corresponding to α ≈ 1.9 , it isapproximately flat. Increasing the power-law further, causes the slope to slowly converge
towards the uniform anti-saturation. For our trap, at s = 20 or α = 1.65, we therefore do
not expect major deviations from ideal gas behaviour at low to intermediate condensed
fractions.
The influence of random variations of the potential in the trap centre can be treated ina similar manner. The saturation slope is affected in a similar way as when lowering thepower-law. Within the LDA, only the total number of volume elements f (U ) at a given
potential matters; it is not relevant how they are distributed spatially. We should keep inmind that mean-field theories do not always capture the full picture and beyond-meanfield results can be important. In the harmonic trap, deviations from the HF (and Popov)
926.2. SATURATION OF THE THERMAL COMPONENT

2

N’/Nc

1.5

1

0.5

0

0

0.2

0.4
N0 /N

0.6

0.8

Figure 6.6: Numerical calculations for isotropic three-dimensional power-law traps
U (r) = (kB T )rs . The condensed fraction is plotted versus thermal number
normalized to the critical number. The blue curves show, top to bottom, s =(2, 4, 8, 10, 15, 20), while the red curve shows the result for a uniform potential.
approximations at larger condensed fraction have been confirmed experimentally [16,34],
while beyond-mean-field shifts of the critical point have been observed [10].
Our theoretical considerations show that a uniform gas is not expected to be fully
saturated within mean-field theory. However, the deviations are expected to be minor
under typical experimental conditions.

6.2.2

Experimental Results

To study saturation experimentally, we prepared a condensed gas at T0 = 45 ± 1 nK witha varying total number. We found that the MOT loading time provides good control over
the total number at the end of the sequence, while the evaporation depth in the green trap
sets the final temperature. In Fig. 6.7a, we plot the condensed fraction N0 /N against thenormalized thermal number N 0 /Nc . We see that the gas is saturated within error.
The data shown is compensated for small temperature variations to first order by correcting the thermal number according to ideal gas scalings:0
Ncorr= N 0 (T0 /T )α .

(6.21)The data is then averaged within bins of size ∆N0 /N = 0.04. The x-y errorbars are thestatistical error of the mean. In Fig. 6.7b, we evaluate the same data at the edge of ourconfidence interval for α. We see that the variation in the thermal number is small forboth values.
For comparison, we also show analogous measurements for the harmonic trap fromwhich our box trap is loaded. Here the non-saturation of N 0 is prominent and agrees withthe prediction from [16, 34] (Eq. (4.13)).
93

6.3. QUANTUM JOULE-THOMSON EFFECT
(a)

(b)

h a r m o n ic
b o x

1 .6

N’/Nc

N ' / N c

1.1
1 .3

1 .0

1

0

0.9
1 0
2 0
C o n d e n s e d F r a c tio n ( % )

0

3 0

0.1
0.2
0.3
Condensed Fraction %0.4

Figure 6.7: (a) Saturation of the thermal component in a partially condensed gas.
In the box trap the gas follows the ideal- gas prediction N 0 = Nc , whereas in theharmonic trap the thermal component is strongly nonsaturated. The dashed red line
shows the theoretical prediction for the harmonic trap. (For the harmonic trap data,T ≈ 110 nK and Nc ≈ 65000). Figure taken from [25]. (b) Variation of the saturationin the uniform trap with α. The same data as in (a) evaluated with α = 1.6 (blue) and
α = 1.7 (red).

6.2.3

Discussion

We have found experimentally that the thermal component in our quasi-uniform trap issaturated. Within experimental uncertainties, the gas follows Einstein’s textbook picture
N 0 = Nc ∝ T α(6.22)

for N > Nc . This is consistent with theoretical calculations for high, but not infinite
power-laws. Further experimental and theoretical studies might be fruitful and could shedlight on the behaviour at higher condensed fractions and tunable interactions.
Eq. (6.22) also explains why the inclusion of condensed clouds in the power-law analysis in [24] did not distort the results for α.

6.3

Quantum Joule-Thomson effectDuring our saturation measurements, we observed unexpected cooling of condensed gases
while holding them in the uniform trap. Since atom numbers were dropping as well, we
first explained this by evaporation. However, this explanation did not hold, as we could
not stop the cooling by increasing the depth of the trap. Also, the effect was severely
reduced or even absent for thermal gases.
The cooling can instead be explained with the quantum Joule-Thomson (QJT) effect.
946.3. QUANTUM JOULE-THOMSON EFFECT

Figure 6.8: Isoenthalpic rarefaction. In a conventional Joule expansion process thevolume V is increased, whereas in our experiments the atom number N is reduced. In
an ideal gas both processes conserve the specific enthalpy and are thermodynamicallyequivalent.
Collisions with high-energy background gas atoms remove atoms slowly from the trap.
Crucially, collisions with the background gas are independent of the energy of the trapped
atoms, so the average energy per trapped particle, E/N , remains constant.
The pressure of an ideal gas is given - irrespective of statistics- by [28]:
p=

2E
3V

(6.23)The Enthalpy H is then:5
H = E + pV = E(6.24)
3
Keeping E/N constant throughout the process means that the specific enthalpy h = H/N
is conserved2 , which is the definition of a Joule-Thomson process.
The Joule-Thomson effect causes the cooling or heating of a gas when changing thepressure in an isoenthalipic manner. Its magnitude and sign are quantified by the JouleThomson coefficient [78] ∂T.
(6.25)µJT =∂p h
The JT process is exploited on an industrial scale for cooling and liquifaction of cryogenic
gases, by repeatedly throttling the gas through a valve. On an everyday level, it causes
compressed air to cool when leaving a pressured container.
Unlike in those cases, the effect we observe is not driven by a change in volume, but
a change in number. As all relevant thermodynamic properties in Eq. (6.25) are intensive(i.e. n, p, T and h), the processes are thermodynamically equivalent nevertheless. We
illustrate this equivalence in Fig. 6.8. In the left panel we illustrate free Joule expansion
into vacuum. In the right panel we show the loss-driven rarefaction happening in our
experiment.
Before describing our experimental measurements and analysis of the QJT, we aregoing to derive the theoretical framework to describe the Joule-Thomson effect:• In a classical (non-ideal) gas, T  Tc where the Joule-Thomson process is only2

Sometimes the specific Enthalpy is defined as H/m.

95

6.3. QUANTUM JOULE-THOMSON EFFECT

driven by interactions.
• In an ideal, condensed gas, T < Tc . Here, the Joule-Thomson process occurs due
to quantum-statistical effects.
• In an ideal, thermal but degenerate gas, T & Tc . Even without a BEC, quantum
statistics predicts a QJT effect.

6.3.1

Joule Thomson effect in a classical gas

Here, we will quickly recapitulate the classical Joule-Thomson effect. First, let us consider an ideal gas without interactions. The equation of state is given by
pV = N kB T.

(6.26)

T =

5 h
,
2 kB

(6.27)

It is easy to show thatand therefore
µJT =

∂T
∂p

= 0.

(6.28)h

There is no Joule-Thomson effect in a non-interacting (i.e. ideal) classical gas. Expansion
does not change the temperature, as each atom is not aware of the presence of the otheratoms.
The Joule-Thomson effect in real gases is driven by interactions between the atoms.
In the case of attractive interactions, expanding the gas requires energy to overcome theattractive forces - the gas cools.
The simplest model for a gas with interactions is the Van-der-Waals gas:


p+

a
(v − b) = RT.
v2(6.29)

Here, we introduced the specific Volume v = V Na /N and the constants a and b. a is ameasure of the attraction between particles, whereas b relates to the excluded volume perparticle. The gas constant R is related to the Avogadro constant Na by R = Na kB . The
Joule-Thomson coefficient can be expressed as [78]
"  
#
1
∂v
µJT =
T
−v ,
Cp,m
∂T p

(6.30)

where Cp,m ≈ (5/2)R is the molar heat capacity. Using Eq. (6.29), the Joule-Thomson
966.3. QUANTUM JOULE-THOMSON EFFECT

Figure 6.9: Microscopic origin of JT cooling in a saturated quantum gas. Removalof a thermal atom requires a zero-energy particle to come out of the BEC in orderto maintain N 0 = Nc , while energy conservation requires redistribution of atoms
between energy levels. Cooling is seen in the change of the relative populations of
the excited states. In this cartoon we neglect the fact that Nc also slowly decreases
with T .
coefficient can now be evaluated as
µJT =

1
Cp,m




(2a/RT )(1 − b/v)2 − b.
1 − (2a/vRT )(1 − b/v)2(6.31)

This expression is not very intuitive. For comparative purposes, we are mainly interested
in the magnitude of a positive µJT , which is appropriate for cooling gases. We approximate Cp,m ≈ (5/2)R and assume v  b, which allows us to estimate the value of theJoule-Thomson coefficient asa(6.32)µJT ≈ 2 .
R T
a is typically on the order of < 0.5 Pa m6 / mol2 . At room temperature, we expect a
Joule-Thomson coefficient on the order of < 3 K/bar. Typically, experimental values are
smaller than this, even at lower temperatures. One should also note that while the Vander-Waals model reproduces general trends and behaviour, it is of limited usefulness inpractice.

6.3.2

Joule-Thomson effect in a condensed quantum gas

We have seen that in a classical gas the Joule-Thomson effect is caused by interactions.
This is very different in a quantum gas, where the Joule-Thomson effect occurs without
interactions, purely due to quantum statistics. This effect has been predicted as early as
1937 [79].3
The cooling effect can be derived quickly without having to calculate the JouleThomson coefficient. Neglecting the interaction energy, the energy can be written as
E ∝ N 0 kB T.

(6.33)

3
The original paper considers the magnitude of the QJT in 4 He at 5 K. They estimate that the QJT
coefficient is on order 0.076 K/bar, while the expected Van-der-Waals JT coefficient is ≈ 0.7 K/bar. This
would make it difficult, but probably not impossible to measure the QJT coefficient in Helium.

976.3. QUANTUM JOULE-THOMSON EFFECT
We have shown earlier that the gas is saturated, i.e. N 0 = Nc . Also, we have shown thatthe critical number scales with temperature as Nc ∝ T α . We can therefore write
E ∝ Nc T ∝ T α+1 .

(6.34)As the loss process conserves E/N , the atom number N and temperature T can be relatedasT ∝ N 1/(α+1) .
(6.35)The microscopic origin of this cooling is qualitatively illustrated in Fig. 6.9. Removalof any thermal atom requires a zero-energy particle to come out of the BEC in orderto maintain the saturation of the thermal component. Energy conservation then requires
simultaneous redistribution of atoms between energy levels. The net cooling is seen in
the change of the relative populations of the excited states. Note that direct removal of
BEC atoms does not change the temperature, and also that in this simplified cartoon weneglect the fact that Nc (T ) gradually decreases as T drops.
In a uniform gas, it is simple to evaluate the coefficient µJT 4 . We rewrite Eq. (6.25)
as

µJT =

∂N
∂T

−1 
h

∂N
∂p


.

(6.36)hExpressing Eq. (6.23) as
5 pV
2 h

N=
allows us to calculate



∂N
∂p


=
h

3VN
.
2 E

(6.37)(6.38)

Now, we write Eq. (6.34) as E = B kB T Nc (T ), where B absorbs constants. Rearranging
yields5 BkB TN=
Nc (T ),
(6.39)
3 hand by differentiation we obtain


∂N
∂T




=

h

N
T



 
5
BkB
Nc (T ).
2

(6.40)

Combining the two derivatives in Eq. (6.36) allows us to calculate the Joule-Thomson

4

For a calculation ina power-law potential, the volume V is not defined. Therefore we need to redefine
the pressure p = T VS E,N by replacing V with the appropriate trap-shape parameter (e.g. the trap frequency in a harmonic trap). The resulting Joule-Thomson coefficient is not expressed in Kelvin/bar and is
not very intuitive.

98

6.3. QUANTUM JOULE-THOMSON EFFECT

60
T=50 nK

50
E/(α N k) [nK]

T=40 nK40
T=30 nK30
Isenthalp

20

BEC

10
0
3

3.5

4

4.5
log(N)

5

5.5

6

Figure 6.10: The quantum Joule-Thomson effect in a degenerate gas T > Tc in a
uniform trap (α = 1.5). As h ∝ E/N , isenthalps are horizontal lines. A range
of isotherms is depicted in blue. In this example, the isenthalp starts at the criticalpoint with 50 nK. Diluting the gas by two orders of magnitude, e.g. by loss-driven
Joule-Thomson rarefaction, cools it down to T ≈ 26 nK.
coefficient
µJT =

2
Λ3
.
5 ζ(5/2) kB(6.41)

It is interesting to note that in Fermi gases the loss-driven QJT is predicted to cause
heating rather than cooling [80].

6.3.3

QJT in a degenerate thermal gas

The quantum Joule-Thomson effect does not only occur in condensed gases at T < Tc . It
also affects degenerate, thermal gases with T & Tc . This is actually the effect predictedin [79].
The energy in a degenerate gas can be evaluated (using Eq. (A.11)) from the densityof states D() in Eq. (2.14) and the Bose-Einstein distribution with µ < 0 :ZE=

D()f (, µ)d(6.42)= Cα Γ(α + 1)gα+1 (exp[−µ/kB T ]) (kB T )α+1 .

(6.43)In the classical limit, this reduces to [28]E = αN kB T,
99(6.44)

6.3. QUANTUM JOULE-THOMSON EFFECT

while at the critical point with µ = 0 we have
E = αN kB T ζ(α + 1)/ζ(α).

(6.45)The maximum reduction of temperature under isenthalpic conditions, i.e. when conserving E/N , is given by the ratio of the classical and the critical energies, ζ(α + 1)/ζ(α) ≈51% in a uniform gas. Qualitatively, degeneracy enhances occupation of low energy
states, with  < |µ|, so E/N is lower than the classical value αkB T . In a harmonic gas
the ratio is ζ(4)/ζ(3) ≈ 0.9, which would make the QJT effect very hard to observe.
To illustrate the variation of energy per particle according to Eq. (6.43), we plot
E/αN kB against the atom number in Fig. 6.10. We display this variation for a rangeof temperatures. These isotherms curve smoothly from the classical value in Eq. (6.44)
to the critical value in Eq. (6.45). In this diagram, isenthalps with h = const are simply
horizontal lines. The QJT loss-cooling emerges now naturally by following an isenthalp
in the direction of lower atom numbers, which intersects the isotherms. The QJT effectpersists over a large range of numbers; in the example shown we would have to dilute thegas by an order of magnitude to reduce the cooling due to QJT notably.

6.3.4

Experimental observation of the QJT

To study the QJT, we prepared clouds at Ti ≈ 45 nK in a trap of depth U0 , where
evaporation is negligible. We constantly lose atoms due to background gas collisions
on an exponential timescale τ ≈ 10 s. Due to low atomic density in our uniform trap(< 5 × 1012 cm−3 ), the three-body recombination rate is negligible [65]. As N decays,
the elastic collision rate (among the trapped atoms) remains sufficiently high for the gas tocontinuously re-equilibrate on a timescale τeq . 2 s  τ [43, 81]. Using our previously
established thermometry methods, we observed the atom number and the temperatureof the trapped cloud and monitored their variation over time. In Fig. 6.11 we show theevolution of T with decaying N , with both quantities scaled to their initial values. We
show data for a partially condensed gas, with an inital atom number Ni ≈ 1.7 Nc (Ti ),and a thermal cloud, with Ni ≈ 0.6 Nc (Ti ). Note that in both cases we measure down
to approximately the same final N value, ≈ 0.3 Nc (Ti ). At this point N0 in the initiallycondensed sample vanishes, and our temperature fits become less reliable.
In the (partially) condensed sample we observe a drop in T by a factor of two, i.e.
by ≈ 22 nK. Meanwhile, the interaction energy per particle is always smaller than
(8π~2 a/m)Ni /V [28], corresponding to < 4 nK. The change in T therefore must predominantly be a quantum-statistical, rather than an interaction effect. We estimate thatthe total change in T due to the change in the interaction energy per particle could at most100

6.3. QUANTUM JOULE-THOMSON EFFECT

1 .0

T / T i

0 .8
0 .6
0 .4
0 .2

th e rm a l
c o n d e n s e d

0 .0
0 .0

0 .2

0 .4

0 .6
N

/ N

0 .8

1 .0

i

Figure 6.11: Measurements with a partially condensed and a thermal cloud. Solidand dashed blue line are predictions of Eq. (6.35)) with α = 1.65 and α = 3/2,
respectively. The dotted red line is a numerical calculation based on the equation ofstate for the degenerate, thermal atoms.
be ∼ 3 nK, which is comparable to our experimental data scatter. Moreover, note that for
a > 0 the drop in the interaction energy should lead to slight heating, rather than coolingof the gas.
Indeed, the scaling of the QJT in a condensed gas (Eq. (6.35)) fits the data very well.
We show predictions for α = 1.65 (solid blue line) and α = 3/2 (dashed blue line). The
two are almost indistinguishable, reaffirming that the behaviour of our clouds is very close
to that of a perfectly homogeneous gas. We also notice a slight cooling in the thermal gas(red points), which is consistent with a numerical calculation (dotted red line) based on
the QJT in a degenerate thermal gas (Eq. (6.43)). Unfortunately, we cannot explore athermal gas in the classical regime, where no QJT should be present. Diluting the gas
further reduces the signal level to a point where the determination of atom numbers andtemperatures is unreliable. Increasing the temperature is also not possible, as it is limitedby the maximum depth of our uniform trap.
We can now evaluate the JT coefficient µJT using Eq. (6.41). At our lowest temperature, we observe µJT ≈ 4 × 109 K/bar, which is about ten order of magnitude higher
than in classical gases.

6.3.5

Experimental details

For completeness, we describe some additional experimental details and analysis, whichwere not included in our original paper [25]. We focus on two topics: First, we detailthe preparation of the gas by quenching non-adiabatically through the transition. The
dynamics of condensate formation after quenching the atoms in a such a manner are very
101

100
50
0
0

5

10
time [s]

15

20

5

10
time [s]

15

20

5

10
time [s]

15

20

500
0
0

50

0

N /1000

N/1000T [nK]

6.3. QUANTUM JOULE-THOMSON EFFECT

00

Figure 6.12: Temporal evolution of a gas in a uniform trap after lowering the trap
depth abruptly. At first the gas rethermalizes in the condensed state; a condensate
forms. After around 2s the condensate has stopped growing. We identify the coolingafter 4 s (dashed line) as driven by the quantum Joule-Thomson effect.

interesting and are currently investigated in follow-up experiments. Second, we discuss
the scaling of the condensed atom number N0 throughout the QJT process, as opposed tothe total number N we have been considering so far.

Preparation of the cloud
We prepared the gas by lowering the depth of the uniform trap non-adiabatically to
U0 = kB × 0.4 µK, rather than by gentle lowering of the depth. This truncates thehigh-energy end of the thermal distribution. The following rethermalization cools thegas below its critical temperature and condensation occurs. We show the evolution of thegas in Fig. 6.12. Until 2 s after the quench, the condensate grows rapidly. Until then,the gas is not in equilibrium and evaporation-driven cooling is still ongoing. This is consistent with our previous estimate of the collision rate. At this point, the temperature isaround 45 nK ≈ U0 /(9kB ), and it is reasonable to assume that evaporation is negligible.
For good measure, we waited another 2s before considering data for Fig. 6.11.
102

6.3. QUANTUM JOULE-THOMSON EFFECT

Condensed atom number during the QJT
We have found in Fig. 6.11, that the total number and temperature follow the prediction in
Eq. (6.35). Combining Eq. (6.22) and Eq. (6.35) allows us also to predict the point where
the condensate vanishes. We define the initial total number Ni and initial thermal number
Ni0 . We define Nt as the number where the condensed fraction vanishes so that Nt = Nt0 .
We obtain
 0 α+1
Ni
Nt=,(6.46)Ni
Ni
which relates the initial thermal fraction Ni0 /Ni to the ratio Nt /Ni at which the condensate
disappears.
Our experimental data shows that the condensate persists down to Nt /Ni ≈ 0.25,
which is significantly longer than Nt /Ni ≈ 0.45, as suggested by the initial thermal
fraction Ni0 /Ni ≈ 0.74. This discrepancy can be explained because we underestimatedthe initial condensate number N0i . This undercounting is caused by “blacking out” in thecentre of the image for high optical densities, i.e. the cloud is so dense that barely any light
can reach the camera. This is illustrated in Fig. 6.13b. While this reduces the measured
condensate size, it does not affect the temperature reading, which is deduced from thelow-density wings of the thermal distribution. Also, the effect on the total number N 0 is
rather small.
In order to test this hypothesis, we rewrite Eq. (6.46), to predict the condensate size
N0theo for all atom numbers N , based on the atom number Nt where the condensate disappears :N0theo = Nt − (N Ntα )[1/(α+1)] .
(6.47)In Fig. 6.13a, we compare this prediction N0theo with the measured number N0exp . We findthat the prediction is correct for small numbers, but deviates from the measured number
N0exp for N > 130 000. This coincides for the point where the peak optical densities
exceed the trusted range, OD & 4, and the image blacks out. To check that this does notalter the results of Fig. 6.11, we correct the total number for the difference in measured
and predicted condensate size
Ncorr = N + (N0theo − N0exp )(6.48)

and replot Fig. 6.11. The corrected data points are shown in Fig. 6.13c. We see that thecorrected data fits the scaling Eq. (6.35) at least as good as original data. It even seems
to suggest it is possible to extend the range of data where QJT scaling applies, indicatingthat the gas is in equilibrium after 3 s rather than after 4 s (not shown in Fig. 6.13c).
There are also a couple of data points which do not have a condensate. Strictly speak103

6.3. QUANTUM JOULE-THOMSON EFFECT

ing, the condensed QJT scaling does not apply to them; however the effect of the thermal QJT near Nc has a very similar scaling, which allows us to plot them on the same
curve [27].

6.3.6Conclusions

In summary, we have experimentally confirmed the quantum Joule-Thomson effect in a
degenerate gas. Unlike in classical gases, interactions are not the driving force of thequantum manifestation of the Joule-Thomson effect. We are able to extract a coefficientµJT ≈ 4 × 109 K/bar, which is ten orders of magnitude larger than observed in classicalgases. It is also interesting to note that µJT in Eq. (6.41) explicitly vanishes in the classicallimit, ~ → 0, reiterating that this is a purely quantum effect.
Similar theoretical analysis applies to an ideal harmonically trapped gas, but in thatcase α = 3, implying weaker cooling in both condensed clouds and degenerate thermal
samples. Moreover, in practice harmonically trapped gases strongly deviate from thispicture, because N 0 is not saturated, much higher typical BEC densities enhance the roleof interactions, and three-body recombination continuously increases E/N .

104

6.3. QUANTUM JOULE-THOMSON EFFECT

(a)

(b)

4

12

x 10

10

N0

8
6
4
2
0
0

0.5

1

1.5
N

2

2.5

3
5
x 10

(c)
50

T [nK]

40

30

20

10

0
0

0.5

1

1.5
N

2

2.5

3
5

x 10

Figure 6.13: (a) Comparison of the expected and measured condensate numbers.
The dashed blue line shows the prediction N0theo given by Eq. (6.47). The blue circles
show the measured condensate number N0exp . (b) An image that is “blacked-out”
in the centre. For optical densities OD > 4, the condensate number N0 is underestimated by N0exp . This occurs at numbers N > 130 000, which is consistent with(a). (c) We correct the total number N for the difference N0theo − N0exp observed in(a). We compare the original data (red) with the corrected data (black). The difference is small for most data points. The continuous lines show the respective best fit
N ∝ T −(1.65+1) . The corrected data still agrees with this prediction. For clarity, error
bars were omitted.

105

6.3. QUANTUM JOULE-THOMSON EFFECT

106

Chapter 7
Bragg spectroscopy
Our studies of degenerate Bose gases in a box potential have so far focused on the criticalpoint for condensation and the thermodynamics of the gas close to the critical temperature. These previous experiments have established the uniformity of the thermal gas
from which the BEC forms. In this chapter, we probe a box-trapped Bose gas in the lowtemperature regime of quasi-pure condensates. These measurements require a precision
which can resolve a two orders of magnitude lower energy scale.
We study the coherence and energy of the gas, as well as its free expansion fromthe box trap, employing two-photon Bragg spectroscopy [82–89]. This method allows
us to obtain high resolution measurements of the momentum distribution and interactionenergy.
In Section 7.1, we establish the theoretical foundations of Bragg diffraction; see also
[83, 86, 90] for more complete treatments. In the experimental Section 7.2, we describe
the experimental setup, followed by a description of the measurements we conductedto characterize the condensate. First, for a wide range of box sizes (up to 70 µm), we
directly observe the Heisenberg-limited momentum uncertainty of the condensed atoms,
corresponding to a fully coherent macroscopic BEC wavefunction spanning the whole
box trap. From the interaction shift of the Bragg resonance we deduce the BEC groundstate energy (per atom) with a precision of kB × 100 pK, and find good agreement withthe mean-field theory for a perfectly uniform condensate.
Finally, we study the free time-of-flight (ToF) expansion of a BEC from the box trap.
We follow the evolution of the cloud shape and the gradual conversion of the interactionenergy into the width of the momentum distribution, and reproduce our observations innumerical simulations based on the Gross-Pitaevskii equation.
The results of this chapter have been published in [26].
1077.1. THEORY OF BRAGG SPECTROSCOPY

Figure 7.1: Bragg scattering in a two level system (not to scale). The energy of theground state |gi and the excited state |ei vary with momentum p according to thedispersion relation. The single photons are detuned from the transition by ∆ and only
two-photon transitions can occur via a virtual level (dashed line). The atom is still in|gi after the transfer, but has acquired momentum qr and kinetic energy hf .

7.1

Theory of Bragg spectroscopy

Bragg spectroscopy uses two off-resonant laser beams to transfer momentum and energy
to the atoms. The frequency dependence of this process allows us to probe the energy
levels of the atoms. Unlike in Raman spectroscopy, we are not transferring the atom
between different internal states (such as hyperfine levels), but we transfer the atom only
to a different momentum state. We illustrate the process in Fig. 7.1. Two beams withdifferent frequencies (f2 < f1 ) are overlapped at an angle 2θ on the atoms. The two
beams have wavevectors k1 and k2 . Absorbing one photon from one beam, and stimulated
emission into the other beam causes the atom to absorb energy and recoil momentum.
Energy conservation dictates that the atom acquires the energy
hf = h(f1 − f2 )(7.1)

qr = ~(k1 − k2 ).

(7.2)

and the recoil momentum

For now, we neglect mean-field effects and approximate the dispersion relation of theatoms by that of free, classical particles
Ekin =

p2
2m

(7.3)

with momentum p. We assume the momentum transfer qr is parallel to the z-componentof the momentum pz . It is now easy to derive the Doppler-shifted resonance frequency
108

7.1. THEORY OF BRAGG SPECTROSCOPY

Figure 7.2: Bragg spectroscopy of a uniform BEC. (a) Shape and orientation of our
box trap; in the lab frame the x direction is vertical. (b) A trapped atom with initialmomentum ~k absorbs recoil momentum qr aligned with z. The trap wall height(green) is lower than the energy of the recoiling atoms. The Bragg beams and lattice
are not drawn to scale; in reality the beams are much larger than the box and the latticespacing is much smaller. (c) Absorption image of the atoms, taken 170 ms after thestart of the Bragg pulse. After the end of a 35 ms light pulse, the non-diffracted atoms
were released from the trap. Figure adapted from [26].

fr (pz )
pz qr
qr2
+
.
(7.4)
2m
m
Scanning the resonance frequency allows us to measure the momentum distribution of thecloud. Usually, k1 ≈ k2 = k and we can approximate qr ≈ 2~k sin(θ). We will operate
in a regime where the recoil velocity vr = qr /m imparted on the atom is much larger than
the initial velocity of the atom pz /m. After a long enough ToF, the diffracted atoms willthus separate from the rest of the cloud as shown in Fig. 7.2. From the number of atoms
in the diffracted cloud we can deduce the fraction of atoms whose momentum matches
the applied frequency.
hfr (pz ) =7.1.1

Transfer function

So far, we have not discussed the time evolution of the Bragg diffraction process. It canbe addressed within the same mathematical framework as two-photon Raman transitions.
The two overlapping beams in the rotating-wave approximation produce a semi-classical109

7.1. THEORY OF BRAGG SPECTROSCOPY
(a)

(b)
20

0.14

18

0.12

16
∆ FWHM [Hz]0.16

Ndif /N

0.1
0.08
0.06

14
12
10

0.04

8

0.02

6

0
−500

0
∆f [Hz]40500

Total Broadening
1/t Broadening only

510Ω/2π [Hz]15

20

Figure 7.3: (a) Ideal box spectrum (blue) with a zero-to-zero width of 200 Hz compared with the broadened spectrum (dashed red) due to Ω = 2π ×8 Hz and t = 35 ms.
(b) Broadening of the FWHM (see text) due to Ω with fixed t = 35 ms (continuousblue). The dashed line indicates the baseline broadening due to the temporal 1/t
Fourier-broadening.

light field with amplitude|E| ∝ cos1[(q /~)r − 2πf t] .
2 r

(7.5)

This can be seen as an optical lattice of wavelength h/qr moving with velocity vr /2.
The name Bragg spectroscopy is derived from this picture. Unlike in the traditional Bragg
process, we are not scattering photons (e.g. x-rays) off a lattice made of matter (e.g. crystals), but we are diffracting a matter wave off a lattice made of photons. The perturbation
Eq. (7.5) causes a Rabi oscillation between the levels |g, pz i → |g, pz + qr i via a virtuallevel (see Fig. 7.1). The effective Rabi frequency Ω is given by [37]
Ω=

Γ2 I
,
4∆ Isat(7.6)

where Γ is the line width of the |ei level and ∆ is the detuning from the |gi → |eitransition. I ≈ I1 ≈ I2 are the beam intensities and Isat is the saturation frequency of thetransition. We define the detuning δ asq2
pz qr
δ = 2π(f − fr (pz )) = 2π f − r −
2mh
mh


110


.

(7.7)

7.1. THEORY OF BRAGG SPECTROSCOPY

The transfer probability P (pz ) for a given momentum pz is then given by [37]Ω2
P (δ, t) = 2
sin2Ω + δ2!
√Ω2 + δ 2 t
,
2

(7.8)

where t is the duration of the Bragg lattice light pulse.
It is convenient to use Bragg frequency units to describe the momentum pz
pz qr
f˜ =
.
hm

(7.9)

In these units, Eq. (7.8) can be written as

P (f˜) =



Ωt
2

2

q

2 + 4π 2 (f − f (0) − f˜)2Ωrt .
sinc2 2

(7.10)

We assume that all of our atoms N are in the ground state with momentum distribution
Ψ(pz ) along the z-axis, or equivalently Ψ(f˜) in frequency units. The diffracted atom
number Ndif for a given frequency can be calculated by summing the transferred fraction
over all momentum states
Z
Ndif (f ) = N0 |Ψ(pz )|2 P (pz )dpz
(7.11)Z= N0 |Ψ(f˜)|2 P (f˜)df˜.
(7.12)
This is simply a convolution of the transfer function Eq. (7.8) with the momentum distribution |Ψ(f˜)|2 centred around fr (0):Ndif (f − fr (0))
=
N



Ωt
2

2 Z


q
2 + 4π 2 (f − f (0) − f˜)2Ωrt df˜.
|Ψ(f˜)|2 × sinc2 2(7.13)In the limit of long Bragg times t and low Rabi frequencies Ω (i.e. low beam intensities), P (f˜) turns into a δ-function and we observe the momentum distribution directly.
However, finite times t and Rabi frequencies Ω result in broadening of the spectrum, as
illustrated in Fig. 7.3a. Considered separately, finite times t with negligible Ω cause atemporal sinc-squared Fourier-broadening with zero-to-zero width 2/t in frequency units.
Also, finite Rabi frequencies Ω cause additional power-broadening (see Fig. 7.3b). Experimentally, we observe the diffracted fraction, which scales ∝ (Ωτ )2 . Hence, to keept in a experimentally viable range (especially for measurements which are in ToF rather
than in trap), we have to accommodate a certain broadening. However, Eq. (7.13) gives1117.1. THEORY OF BRAGG SPECTROSCOPY
(a)

(b)
12

FWHM broadening [%]

10
8
6
4
2
0
0

0.02

0.04
ξ/L

0.06

0.08

Figure 7.4: (a) Wavefunction of an interacting BEC in a perfect box (blue). The
rounding-off effect of the healing length (here for a typical length ξ ≈ 0.03L) is
shown (green). (b) FWHM broadening in momentum space due to the healing length
for a range of typical experimental values of ξ/L.
us the analytic tools to predict this broadening.
To quantify the broadening in a simple manner, we use the full-width at half-maximum(FWHM)1 . Analytically, the implementation is trivial. To obtain the FWHM from experimental data, we use appropriate fitting functions to approximate the data; the functionused is not relevant as long as it reproduces the data well at the maximum and the halfmaxima. Typically, the sum of two Gaussians or a sinc-squared on top of a Gaussian work
well.

7.1.2

Momentum distributions

The momentum distribution of a non-interacting BEC in a box of length L is given by thequantum-mechanical ground state of a single particle. The spatial wave function is given
by a cosine in the box and is zero elsewhereΨ(z) ∝ Π(z/L) cos(πz/2L),(7.14)

which we expressed mathematically by truncating the cosine at the box edges using thetophat function Π(z/L), where Π(z/L) = 1 for |z| > L/2 and zero otherwise. We obtainthe corresponding momentum-space wave function by Fourier transformation
Ψ(kz ) ∝ cos(kz L/2)/(kz2 L2 − π 2 ).

(7.15)

1
We decided against using the variance as a measure of the broadening, as it diverges for a sinc-squared
distribution - which is the the momentum distribution in a perfect box.

112

7.1. THEORY OF BRAGG SPECTROSCOPY

The zero-to-zero width in k-space is 6π/L. According to Eq. (7.9), the corresponding
Bragg spectral line has zero-to-zero width 3vr /L (in frequency units).
On the other hand, the profile of an interacting BEC in the Thomas-Fermi regime is a
simple tophat function of width L (see Eq. (2.32))
Ψ(z) ∝ Π(z/L),(7.16)

where we are for now neglecting blurring at the edges of the box because of the healing
length (Eq. (2.33)). The corresponding wave function in momentum-space isΨ(kz ) ∝ sinc2 (kL/2),(7.17)

with a zero-to-zero of 4π/L in k-space and 2vr /L in Bragg frequency units.
We notice that these scalings of the widths in real and momentum space are manifestations of the Heisenberg uncertainty relation∆x∆k & 1.

(7.18)

Confining an atom in a box gives us information about its position, fixing the uncertainty2
to ∆x ≈ L. Hence, the momentum uncertainty has to scale as ∆k ≈ 1/L.
In practice, the edges of the box are going to be smeared out on the order of the healing
length ξ, which is defined as~2= ng,(7.19)
2mξ 2
where n is the atom density and we recall g = 4π~2 a/m is proportional to the s-wave
scattering length a. If ξ  L, we can approximate the blurred tophat wave function
with [28](L/2) − |z|√Ψ(z) = Ψ0 tanhΠ(z/L).
(7.20)
2ξ
A comparison of this shape and the original rectangle function is illustrated in Fig. 7.4a.
The broadening induced by the healing length can be determined by numerically calculating the Fourier transform. For typical values of ξ ≈ 0.05L ≈ µm, the FWHM broadening
is below 10%. The expression for the healing length in Eq. (7.20) is only valid for small
healing lengths; if ξ becomes comparable to L the underlying Thomas-Fermi approximation becomes invalid. The effects of the diffraction-limited steepness of the trap are hardto measure, as they cannot be resolved by our imaging resolution of ≈ 5 µm. Hence, we
can estimate the effects to be comparable to typical healing length broadening.
2
One should note that numerical factors enter the Heisenberg relation, depending on how the uncertainties ∆x and ∆k are defined.

113

7.2. EXPERIMENTAL REALIZATION

7.1.3

Mean-Field corrections

Until now, we have only discussed the effect of the interactions on the momentum distribution. However, they also affect the dispersion relation. In the Bogoliubov approximation, the dispersion relation is given by [28, 77]s
E = hf (q) =

q2
2m




2ng +

q2
2m

2
,

(7.21)

where q is the momentum. For large momentum transfers qr2 /2m > ng, we can approximate this asq2
hf (q) =+ ng.
(7.22)
2m

This introduces an additional mean-field shift ∆fM F = ng/h to the centre frequency
f0 = qr2 /2m in Eq. (7.4) [83, 87].
qr2
pz qr
+
+ ng
2m
m
≡ f0 + ∆fDoppler (pz ) + ∆fM F .

hfr (pz ) =(7.23)(7.24)

This shift can be explained by the additional mean-field energy gn an atom gains whenleaving the ground-state. This is caused by the “bosonic factor of 2”, which doubles theinteraction energy of an atom with atoms in different states. Thus, if all atoms are in
the ground state, the mean-field energy of an atom has to double when leaving it. In our
experiments, we are always deeply in the regime where Eq. (7.22) applies. The phonon
regime, where the dispersion relation Eq. (7.21) is linear, can be probed by reducing
qr [84, 91]. This is typically achieved by using small angles θ between the beams.

Besides the shift of the resonant frequency ∆fM F , any spatial variation of the density
n leads to inhomogeneous broadening of the Bragg spectrum. To extract the momentum
distribution from the measured spectra, we have to subtract the mean-field broadening. As
this can be easily comparable to the momentum width, it introduces quite large uncertainties [83]. In our case, n is essentially homogeneous, the broadening effect is absent andinteractions just shift the Bragg resonance. We can thus directly measure the momentum
distribution in the BEC.
114

7.2. EXPERIMENTAL REALIZATION
PD

PD"Lawrence"

-80+f

λ/2

-80

"Henry"Figure 7.5: The optical setup for Bragg spectroscopy. We split off light from therepump laser via an optical fibre. The beams are then sent through AOMs, shifting
the frequencies and stabilizing the intensity. Finally, they are delivered to the cloud
via further fibrers. Photodiodes (PD) are used for intensity feedback. Figure takenfrom [27].

7.2

Experimental realization

7.2.1

Setup

In Fig. 7.5, we outline our Bragg-spectroscopy setup. We intersect two collimated (3 mmwide) 780 nm laser beams on the atoms at an angle of 2θ = 30◦ . They are detuned fromthe atomic |F = 2i → |F 0 = 3i transition by 6.8 GHz. This is several orders of magnitude larger than typical Rabi frequencies, strongly suppressing single-photon transfers to|F 0 = 3i. The beam parameters result in a recoil velocity vr ≈ 3 mm/s. The diffracted
and non-diffracted atoms are thus well separated in velocity space, as qualitatively seenin Fig. 7.2c.
The beams are branched off from the repump light and split into two branches of
equal intensities. In honour of Bragg father and son [92] we named them “Lawrence”
and “Henry”. The beams are delivered via the top viewport of the MOT cell. To ruleout broadening of the spectra due to incoherences, we adjusted the optical path difference
between “Lawrence” and “Henry” to within a few cm.
Acousto-optical modulators (AOMs)3 are used to shift Henry’s and Lawrence’s frequencies by 80 Mhz and (−80 + f ) Mhz respectively. The AOMs are driven by twophase-locked function generators4 , which are amplified with broadband radio-frequency(RF) amplifiers. The phase-lock makes sure that the two beams are shifted coherently.
3
4

Crystal Tech
Agilent 33250A

115

7.2. EXPERIMENTAL REALIZATION
Two TTL-controlled RF switches5 allow us to switch the AOM with the required µs precision. The frequency of the “Lawrence” function generator is set via the GPIB protocol6 .
To ensure stable and reproducible beam intensities, we monitor weak reflection off
glass plates. As typical light pulses are very short, it is very difficult to stabilize themon-the-go. Instead, we pre-stabilize the beam intensities while the MOT is loading andthen switch them back on when needed to the same value. This allows us to counterat least long term drifts. To avoid having to use analogue sample-and-hold feedback
devices (such as special PIDs), we use digital feedback instead. The intensity signal fromthe photodiodes is read by an Arduino microcontroller. The microcontroller forwards
the intensity values to a custom-written MATLAB program on the experiment control
computer (via serial communication). This program then adjusts the output voltage ofthe function generators via GPIB until the intensities are stabilized. The target values arereceived at the beginning of the sequence from the WordGenerator software7 .

7.2.2

Heisenberg uncertainty relation

First, we measured the experimental Bragg spectrum and assessed how close it is to thesinc-squared momentum distribution S(f ) (Eq. (7.17)) of a uniform gas, as well as tothe corresponding time- and power broadened- distribution S̃(f ) given by Eq. (7.13).
We prepared clouds with condensed fractions & 80 % at a temperature T . 10 nK by
evaporative cooling in the box trap. We then applied a Bragg pulse of length t = 35 ms
and a Bragg Rabi frequency with Ω/(2π) ≈ 8 Hz, keeping the fraction of diffracted atoms
to . 15 %. We turned off the trap 25 ms after the end of the Bragg pulse and measured
the fraction of diffracted atoms after further 110 ms of ToF (see Fig. 7.2c).
The duration of the Bragg pulse was chosen to be t & L/vr . This ensures that the
Bragg spectrum is not significantly Fourier-broadened. Equivalently, a recoiling atom cantraverse the box during the pulse, and we thus probe phase coherence across the whole
BEC. To apply very long Bragg pulses to a trapped cloud, we set the trap depth below
the recoil energy hfr (0) ≈ 50 nK, so that the diffracted atoms can leave the box without
bouncing off the trap walls (see Fig. 7.2b).
In Fig. 7.6a, we show measurements for L = 30 ± 1 µm. As shown in the left 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. Within experimental errors, the measured L is consistent with the value we calculate from the settings of the5

Minicircuits ZYSWA-2-50DR
We use a USB to GPIB adapter (prologix, GPIB-USB Controller 6.0) allowing us to use the simpler
RS232 protocol.
7
Communication between software is implemented with a virtual serial port created by the freecom0com software.
6116

7.2. EXPERIMENTAL REALIZATION
(a )

(b )

0

W

(H z )

1 5 0

-2 0

0
z ( µm )

2 0

0
N

1 0 0
(1 0 3)

2 0 0

H

W
H

1 0 0

W

5

0

(H z )

L

H

W

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

f-f0 (H z )

0

2 0

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

Figure 7.6: Heisenberg-limited momentum spread in a uniform interacting BEC. (a)Main panel: Measured 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(f )and the power-time broadened S̃(f ), respectively. Left inset: L is determined byfitting the in-trap BEC density profile, accounting for the imaging resolution. Right
inset: FWHM of the spectrum, W , versus the atom number N , for the same L. The
filled square corresponds to the data 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(f ) and S̃(f ), 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 bluelines show WH , W̃H and W̃H0 , respectively. Figure taken from [26].

spatial light modulator used to create and align the trapping beams. In the main panel, weshow the measured Bragg spectrum, centred on fr (0) ≈ 1 kHz. The solid red and dashedblue lines show theoretical expectations for a Heisenberg-limited momentum spread. The
solid line is the sinc-squared function S(f ), while the dashed line represents the expectedbroadening S̃(f ) according to Eq. (7.13). The data is extremely close to the Heisenberg limit, corresponding to a fully coherent BEC wavefunction spanning the whole box.
(For corresponding measurements on a harmonically-trapped gas, in momentum and real
space, see [83] and [93], respectively.)
We quantitatively compare different (measured and calculated) Bragg spectra using
their full-width-at-half-maximum W . In the right inset of Fig. 7.6a, we plot the measured W versus the total atom number N , for L = 30 µm. The solid red and dashed bluelines show the two calculated Heisenberg-limited values, WH for S(f ) and W̃H for S̃(f ).
For comparison, we also calculate W̃H0 (dotted blue line) for the sine-like non-interacting
ground state of the box potential. We see that interactions reduce W below W̃H0 . Moreover, the measured W shows essentially no dependence on N , as expected for a BEC ofa spatially uniform density.
To confirm the Heisenberg-like scaling of the momentum-width with length as predicted in Eq. (7.18), we repeated the measurement for a range of box lengths, L =15 − 70 µm. While numbers and densities varied across those measurements, we have117

7.2. EXPERIMENTAL REALIZATION
(a)

(b)
60

Mean Field shift [Hz]

50
40
30
20
10
0
−10
0

0.2

0.4
0.6
Transferred fraction

0.8

Figure 7.7: (a) Interaction energy in a uniform BEC, for L = 30 µm. We plot fr (0)versus N for Bragg pulses applied in situ (solid squares) and after 50 ms of ToF (open
diamonds). Dashed line: fr (0) ≈ f0 ≈ 975 Hz, solid line: fr (0) = f0 + αN , withα ≈ 20 × 10−5 Hz/atom. Figure adapted from [26]. (b) Dependence of the mean-field
shift ∆fM F = fr (0) − f0 on the transferred fraction. To vary the transferred fraction,
we scanned Ω at constant t = 35 ms.
already established that the momentum-width is essentially number-independent as longas we are deeply in in the Thomas-Fermi regime. In Fig. 7.6b, we summarise our data.
Plotting W versus 1/L, we demonstrate the expected Heisenberg scaling of the momentum uncertainty. The deviations from the expected broadened width W̃H are below 10%
and can be explained by a combination of healing length (see Fig. 7.4b) and diffractionlimited steepness of the walls.

7.2.3

Mean-field shift

We now turn to the study of the ground-state energy of a uniform interacting BEC. From
Eq. (7.24), we see that the mean-field shift of the centre frequency of the Bragg spectrum,∆fM F , is directly proportional to the chemical potential µ = ng = ∂E/∂N . Thanks
to the unprecedented narrowness of our Bragg spectra, we measured such shifts with a
precision of 2 Hz, corresponding to an energy of kB × 100 pK.
In Fig. 7.7a we plot the central frequency fr (0) versus N , for a fixed L = 30 µm andfor two sets of Bragg spectra:• The in situ spectra (solid symbols) were taken as above, with the main cloud trapped
during the Bragg pulse. These spectra are obtained in the same way as the Heisenberg spectra above. Varying the atom number N changes the mean-field shift.
• The “in-ToF” spectra (open symbols) were taken by releasing the BEC from thebox and letting it expand for 50 ms before applying the pulse. After long ToF the118

7.2. EXPERIMENTAL REALIZATIONW(H z )

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

5 0

1 0 0

1 5 0

2 0 0

N (1 0 3)

Figure 7.8: Spectral width for Bragg pulses applied after 50 ms of ToF expansion,for L = 30 µm. The W values are extracted from the in-ToF spectra in Fig. 7.7a.
The solid line shows a numerical simulation based on the GP equation. Figure takenfrom [26].
atomic density is negligible, and ∆fM F ≈ 0. The N -independent central frequency
fr (0) provides a good measurement of the baseline resonance f0 ≈ 975 Hz. For
the measurement of f0 , it is quite important to measure both ±f0 diffraction orders.
Small potential gradients during long ToF accelerate the cloud to the left or right,
causing a global Doppler shift of the spectrum. Averaging the central frequencies
allows us to cancel this effect.
For the in situ spectra, within mean-field theory and for an infinitesimal Bragg transfer(Ωτ → 0), we expect∆fM F = fr (0) − f0 =2~aηN ≡ αN,Vm(7.25)

where η is the condensed fraction and V is the volume of the box. From in situ images wemeasured V = (25 ± 2) × 103 µm3 . We assess η = 0.8 ± 0.1 from the maximal fractionof Bragg-diffracted atoms (15% in Fig. 7.6a). This estimate is further supported by “BEC
filtering” introduced in [59].8 The uncertainty indicates variations between experimental
runs. We thus obtain a theoretical prediction αth = (24 ± 2) × 10−5 Hz/atom.
From a linear fit to the in situ data (solid line in Fig. 7.7a) we get αexp = (20 ±1) × 10−5 Hz/atom, slightly below αth . This small difference can be attributed to the
15% depletion of the ground-state population during the pulse; the effective mean-field
shift varies with time. To quantify this effect, we took additional measurements with arange of Ω for our largest N , with constant t = 35 ms. This corresponds to varying the8
We apply a short (4 ms) Bragg π-pulse to the cloud. This pulse has a Fourier-width of 500 Hz zeroto-zero, which transfers the BEC, but barely affects the much broader thermal distribution. In ToF, the two
clouds separate and the condensed fraction can be measured.

119

7.2. EXPERIMENTAL REALIZATION
transferred fraction ∝ (Ωt)2 . In Fig. 7.7b, we see a clear reduction of the mean-field shift∆fM F with increasing transferred fraction. By extrapolating the shift to zero transfer (reddashed line), we can estimate that ∆fM F (15%) ≈ ∆fM F (0%) × 0.87. We thus get aslightly revised α̃exp = (23 ± 1) × 10−5 Hz/atoms. The systematic uncertainty in α̃exp ,due to the systematic uncertainty in our absolute atom-number calibration, is 10%.
Complementary to Fig. 7.7a, we plot the measured width W for the in-ToF Bragg
spectra in Fig. 7.8. Qualitatively, W now grows with N because during ToF interactionenergy gets converted into kinetic energy. Quantitatively, the problem of the expansionof an interacting BEC from a box potential has not been solved analytically (see [32] forthe harmonic-trap case). However, we find good agreement between our data and numerical simulations (solid line) based on the GP equation (Eq. (2.26)) done by Alexander
Gaunt. Using the GPE, we find the momentum and density distribution in-trap and after expansion respectively; convolution with Eq. (7.24) then predicts the measured Bragg
spectrum including mean-field broadening as well as power- and time-broadening. In our
simulations, we neglect the small thermal component and use the measured in-trap BEC
energy (∝ α̃exp N ) to predict W in ToF. We also take into account the small curvature of
our gravity-compensating magnetic field. This has a noticeable effect only for long ToF(& 50 ms).

7.2.4

Evolution in ToF

Finally, we study the evolution of the BEC in ToF. We probe the cloud after 0 ms to 50 ms
free expansion with L = 30 µm and N ≈ 200 × 103 . In Fig. 7.9a, we show absorption
images of the expanding cloud (top), and the corresponding GP-based simulations (bottom). The simulations reproduce the characteristic diamond shape that emerges during
ToF. Qualitatively, this conversion to a diamond shape is the analogue of the inversion of
the aspect ratio of a BEC released from an anisotropic harmonic trap. In both cases theforce driving the initially accelerating expansion is given by the gradient of the atomic
density (i.e., the gradient of the interaction-energy density) and in both cases the fastest
moving wave fronts develop at the points where the curvature of the constant-density
surfaces is minimal during the early stages of the expansion.
We measured the Bragg spectrum at different times during the expansion, similarly tothe in-ToF spectra above. In Fig. 7.9b we show the gradual decay of ∆fM F and growthof W for in-ToF Bragg spectra, again finding good agreement with our simulations (solidlines). Note that in these experiments, and simulations, we reduced τ to 10 ms and increased Ω to 2π × 28 Hz. While this broadens our spectral resolution, we need to improve
the time resolution to be able to probe the distribution at a given ToF.
Alternatively, it is possible to use the position of the diffracted cloud to extract its
120

7.2. EXPERIMENTAL REALIZATION

Figure 7.9: Free expansion of a BEC released from the box trap with L = 30 µm
and N ≈ 200 × 103 . (a) Absorption images (top) and simulations (bottom) of theatomic distributions. (b) Evolution of ∆fM F and W during ToF. The solid lines show
numerical simulations based on the GP equation. The solid square corresponds to thein-trap data from Fig. 7.7a. The horizontal error bars indicate our temporal resolution,
limited by the Bragg-pulse duration. Figure taken from [26].

momentum pz at the time of the outcoupling. During the initial ToF, ToFA = 0 − 60 ms,
the cloud moves with a velocity of vz ≈ pz /m. This assumes a negligible (or fast enough)change of the initial velocity due to mean-field energy. The Bragg pulse then increases
the velocity from vz to vz +vr . The diffracted atoms are much faster than the rest of cloud,and separate quickly. In the process, the atoms “roll” down the mean-field potential of themain cloud and convert g × n(r, t = ToFA ) into additional kinetic energy. The velocity ṽ
is now1(pz + qr )2+ gn = mṽ 2 .
(7.26)
Ekin =
2m
2
If we now wait for another expansion time ToFB before imaging, the cloud will have
travelled a total distance zz(ToFA + ToFB ) ≈ vz × ToFA + ṽ × ToFB .
121(7.27)

7.2. EXPERIMENTAL REALIZATION
19:2?:379 on 2013−11−04

23:2?:383 on 2013−10−25

0.3

fit =

0.3

fit =

General model:22
fit(x) = ampG*exp(−(x−x0) /(2*SigG ))+ ampG2*exp(−(x−x0)
Coefficients (with 95% confidence bounds):
SigG =
48.37 (44.67, 52.06)
SigG2 =
232.5 (133.5, 331.5)
ampG =
0.232 (0.2154, 0.2487)
ampG2 = 0.02946 (0.01785, 0.04107)
off = 2.959e−08 (−0.004336, 0.004336)
x0 =−1005 (−1008, −1001)

0.25

General model:22
fit(x) = ampG*exp(−(x−x0) /(2*SigG ))+ ampG2*exp(−(x−x0)
Coefficients (with 95% confidence bounds):
SigG =
63.67 (55.58, 71.76)
SigG2 =
195.3 (157.9, 232.7)
ampG =
0.1343 (0.1166, 0.152)
ampG2 = 0.06501 (0.04716, 0.08285)
off = 0.001506 (−0.001446, 0.004457)
x0 =−1000 (−1003, −996.8)

0.25

0.2

0.2

N

0.15

N

0.15

SigG = 48.37 +/− 2
SigG2 = 232.5 +/− 5e+01
ampG = 0.232 +/− 0.008
ampG2 = 0.02946 +/− 0.006
off = 2.959e−08 +/− 0.002
x0 = −1005 +/− 2

0.1

SigG = 63.67 +/− 4
SigG2 = 195.3 +/− 2e+01
ampG = 0.1343 +/− 0.009
ampG2 = 0.06501 +/− 0.009
off = 0.001506 +/− 0.001
x0 = −1000 +/− 2

0.1

0.05

0.05

0

0

−0.05
−2000 −1500 −1000
f

−500

−0.05
−2000 −1500 −1000f0(a) ToFA = 0 ms

425:2?:761 on 2013−10−30
fit =

0.2

fit =

General model:
fit(x) = ampG*exp(−(x−x0)2/(2*SigG2))+ ampG2*exp(−(x−x0)
Coefficients (with 95% confidence bounds):
SigG =
51.26 (42.29, 60.23)
SigG2 =
162.8 (149.6, 176.1)
ampG =
0.06528 (0.05368, 0.07688)
ampG2 =
0.1048 (0.09331, 0.1163)
off = 0.004309 (0.002309, 0.00631)
x0 =−987 (−990, −984)

0.15

General model:
fit(x) = ampG*exp(−(x−x0)2/(2*SigG2))+ ampG2*exp(−(x−x0)
Coefficients (with 95% confidence bounds):
SigG =
60.81 (48.87, 72.75)
SigG2 =
154.1 (137.4, 170.8)
ampG =
0.06757 (0.04687, 0.08828)
ampG2 =
0.109 (0.08844, 0.1295)
off = 0.006169 (0.003715, 0.008624)
x0 =−984.9 (−987.9, −982)

0.15

N

0.1

N

0.1

SigG = 51.26 +/−5
SigG2 = 162.8 +/−7ampG = 0.06528 +/− 0.006
ampG2 = 0.1048 +/− 0.006
off = 0.004309 +/− 0.001
x0 =−987 +/−
2

0.05

SigG = 60.81 +/−6
SigG2 = 154.1 +/−8
ampG = 0.06757 +/− 0.01
ampG2 = 0.109 +/− 0.01
off = 0.006169 +/− 0.001
x0 = −984.9 +/−
1

0.05

0

−0.05

0

(b) ToFA = 30 ms

392:4?:732 on 2013−10−28
0.2

−500

0

−1500

−1000
f

−0.05

−500

(a) ToFA = 30 ms

−1500

−1000
f

−500

(b) ToFA = 50 ms

Figure 7.10: Comparison of the Bragg spectra (blue) and position spectra (red) for
various ToFs. The continuous red lines are bimodal-Gaussian fits to the Bragg spectra.
We see that the two methods produce similar results, although the Bragg offers abetter signal to noise ratio in the wings.
Clearly this is an approximation, neglecting changes in velocity vz during ToFA . The approximation holds in the limit of ToFA  ToFB . By fitting the position of the diffracted
cloud, we can extract z. As ToFA is small, we approximate vz ≈ ṽ − vr . Now, we
extract ṽ using Eq. (7.26), which allows us to reconstruct a velocity against diffracted
number spectrum that is equivalent to the mean-field broadened Bragg spectrum given by
Eq. (7.24).
In Fig. 7.10, we overlap the resulting position spectra (red) with their Bragg counterpart (blue). We see that we manage to obtain very similar spectra. However, the noiselevels are much higher for the position-based spectra. Especially in the wings, where
signal levels are low, one cannot obtain reliable position measurements. Further, theinfluence of the expansion during ToFA is unclear. Additionally, we need to calibrate
for potential variations along the flight path; in our case the weak harmonic curvatureinduced by the compensating magnetic field. There are several ways to improve this122

7.3. CONCLUSIONS

method; especially a differential measurement of z for various ToFB would allow us to
extract ṽ without having to worry about what happens during ToFA . We did not pursue
this idea, as the frequency method provided the precision we required.

7.3

Conclusions

In conclusion, we have characterized 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-result of our measurements is that they place the most stringent bound so far on the spatial uniformity of an
ultracold gas produced in our optical box. While earlier (thermodynamic) studies established uniformity on a 30 − 100 nK energy scale [24, 25], 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).

1237.3. CONCLUSIONS

124

Chapter 8
Conclusions and Future WorkIn this thesis we have demonstrated the first experimental realization of a homogeneous
Bose-Einstein condensate. We have discussed the experimental procedure and equipment
for trapping atoms in a uniform potential. Further, we have shown by verifying thermodynamic scalings that the trap is nearly uniform on an energy scale kB T ≈ kB × 100 nK.
In the process, we have observed the quantum Joule-Thomson effect for the first time. We
have also seen that the gas follows approximately textbook-like ideal gas scalings despite
atomic interactions; most notable, the thermal component is not altered by the presence ofthe condensate. This is in stark contrast to the situation in a harmonic trap, where interactions strongly modify the thermodynamic behaviour. After scrutinizing the thermal gas,
we have probed the ground state of the condensate. Using Bragg spectroscopy, we havemeasured the momentum and density distribution of the condensed fraction. We havemanaged to observe the Heisenberg uncertainty on a macroscopic length scale to within a
few percent.
Besides the creation of a homogeneous BEC, we have demonstrated the ability of
a comparatively simple and compact experimental system to produce BECs suitable forcutting-edge experiments. Still, the shortcomings of the system are not deniable: While
the size of the condensate is respectable, the accessible parameter range was often limited
by atom numbers. Further, the use of a single chamber for the MOT and evaporation process limits optical access to the cloud. At this stage, we have pretty much exhausted allfree space around the cell for additional optics. Also, the high background gas pressure
severely limits the life-time of BECs. Finally, the lack of a convenient Feshbach resonance in 87 Rb (and the physical space to accommodate coils) means we cannot tune theinteractions at will. Still, in the long run, it should be of interest especially for groups on
a tight budget or for teaching purposes.
For those reasons, a new experiment similar to the one described in [53] is currently
under construction. Together with the now established experimental toolkit for the pro125

duction of uniform gases, it should enable (or at least facilitate) a range of interesting
experiments:• Tunable interactions and a better imaging resolution will enable more accurate studies of the thermodynamics in the box and should allow us to observe beyond meanfield effects.
• Squeezing the trap along one dimension, will allow us the study of homogeneous
gases in two dimensional gases.
• Using two separate boxes, coupled with a thin channel should allow us the observation of transport phenomena similar to superfluid helium experiments.
• Currently our group is exploring the dynamics of condensate domain formation
depending on the quench rate through the Bose-Einstein phase transition. Thisshould allow us to establish a link to the Kibble-Zurek mechanism.
While we have measured some so-far elusive effects in our work with uniform traps,
none of the results presented in this thesis was unexpected from a theoretical point ofview. The main value lies in adding the uniform trap to the toolkit of cold atom experiments. One of the major goals of the field of cold atoms is to establish a closer link tocondensed matter problems. As those are inherently in a homogeneous environment, thiswork should be a substantial step towards this goal.

126

Appendix A
Polylogarithms and the Gamma
function
Function Definitions
The polylogarithm function is defined asgγ [A] =

∞
X
An

nγ
n=1

.

(A.1)

The Gamma function is defined asZ∞Γ(x) =

tx−1 e−t dt.

(A.2)

0

Relation to the Bose distribution
A very simple case is1= g0 (A) .
A−1 − 1
We can thus represent the Bose Distribution function as a polylogarithmf (r, p) =1
= g0 (A) ,A−1 − 1

(A.3)

(A.4)

where we set
A = exp[(µ − (r, p))β].

127

(A.5)

Power-law integrals
We often have to solve the following integral:
Z∞


s 
gγ Ae(−a|x| ) dx = 2

−∞

Z∞


s 
gγ Ae(−ax ) dx.

(A.6)

0

Now we use the definition of the polylogarithm to obtain:Z∞

(−a|x|s )

gγ Ae



dx = 2

−∞

Z∞ X
∞
0

An −anxs
e
dx.
nγ
n=1

(A.7)

The definition of the Gamma function allows us to evaluate the integral and collapse theresult into a higher-order polylogarithm:
Z∞
−∞

∞
X

An
s 
(an)−1/s Γ (1 + (1/s))
gγ Ae(−a|x| ) dx = 2
γ
n
n=1
∞
X

An

(A.8)

(a)−1/s(A.9)= 2a−1/s Γ (1 + (1/s)) gγ+(1/s) (A) .

(A.10)= 2Γ (1 + (1/s))nγ+(1/s)n=1This can easily be generalized to include a power-law prefactor
Z∞
gγ Ae

(−a|x|s )2
x dx = a[−(1+p)/s] Γ
s
p

−∞

128



1+p
s


g{γ+ 1+p } (A) .
s(A.11)

Appendix B
IGBT driver
We rely on an IGBT (insulated-gate bipolar transistors) to switch the high currents needed
to operate our quadrupole magnet. Crucially, it can also tolerate the inductive voltage
spikes created during switch-on and switch-off of the coils.
IGBTs require drivers to (de-)charge the gate of the IGBT to open or close the circuit.
Further, these drivers need to have the capability to interface with security interlocks(such as temperature sensors or flow meters) and shut down the IGBT under dangerous
conditions. The driver design we used in the experiments in this thesis are describedin [27].
Here, we are going to describe an improved design, which is functionally nearly identical, but uses significantly fewer components. This makes it possible to accommodate
the whole driver board on top of the IGBT. Also, assembly time and cost are decreased
significantly. The system consists of three major components (see Fig. B.1): The “IGBT
driver TS V4” is designed to slot onto the matching “contact board V4”, which is bolted
onto a Semikron Semix IGBT module (here SEMIX402GB066HDs). The design has
been tested and used to drive the superconducting coil in Appendix C.

B.1

Functionality

The driver state is set with a TTL-compatible 5 V input. Depending on the applied voltage,
the IGBT opens or closes. The driver also provides four multi-purpose error inputs thatcan be used e.g. for interfacing with temperature monitors or interlocks. One of those
inputs is directly connected to the integrated temperature sensor or the Semix module,
preventing overheating of the IGBT. The driver latches-up on error, shutting down thedriver until manual reset.
129B.2. INPUTS

Figure B.1: Assembly of an IGBT, contact board and driver.

B.2

InputsPower The driver requires 5V DC (I ≈ 500 mA) , which need to be applied to the “5Vin” and the “Gnd in” pins of the dual screw terminal (see Fig. B.6). A cheap plug-in-thewall supply is good enough, but the power supply should not be shared with other devices.
The power is internally converted and regulated to provide ±15 V to the driver stage.
TTL The ground and signal wires of the TTL are connected to the respective “GND”
and “TTL” screw terminals. The ground is shared with the shared with the 5 V input
power ground, so only floating ground TTL channels (e.g. isolated with an optocouplerbox) should be used. The accepted signal levels are approximately compatible with the
TTL standard.
Multi-purpose error inputs There are four multi-purpose error inputs, which can beused to detect a range of errors and shut down the IGBT. They are simple voltage bridges,
where the middle terminal voltage is compared against a reference voltage, which is set
by a simple adjustable resistor (trimpot). If the reference voltage is lower than the voltage
130B.3. OUTPUTS

Figure B.2: The input stage of an error input.
on terminal block (2), the IGBT will detect an error and shut down. If any of the error
inputs detects an error, a latch shuts down the IGBT and a manual reset is required toresume operation. Capacitors together with the (external) resistors act as a low-pass filter,rejecting noise on the input of the comparators. One of the inputs is already connect tothe internal PTC of the IGBT via the contact board. Fig. B.3 explains the use of the error
inputs.
Reset button Connect to the “GND” (shared with TTL) and the “Push Button” terminals. The error-latch resets, if the two terminals are short-circuited. It works best using a“push-to-make” button.
Trimpot adjustment Under non-error conditions, short-circuit the reset terminals (e.g.
by pressing the button continuously) and adjust the trimpot in question until the red LED
just goes out. Note that all the others error terminals need to be high (i.e. no other errors);
if in doubt short-circuit their terminals (2) and (3), or turn their trimpots to 5 V.

B.3

Outputs

LEDs The red LED illuminates if the IGBT is shut down due to an error and the latchis activated. A 500 Ω resistor in line is recommended, as the red LED pin is driven by 5 V.
Alternatively, the pin could be used provide a TTL error signal (5 V on error), e.g. to shut
down a power supply. The pin is not buffered, so not more than 10 mA should be drawn
and external buffering and/or isolation with a relay is recommended.
A green LED illuminates, if the IGBT is in the low-ohmic state. THE LED NEEDSTO BE CONNECTED for the driver to work. No resistor should be used in line.
The terminal names “LED A/K” refer to the anode and cathode of the LEDs, respectively.
131

B.3. OUTPUTS

(a)

(b)

(c)

(d)

(e)

Figure B.3: Suggested uses of the multi-purpose error inputs. (a) NTC thermistor.
Assume the negative temperature coefficient (NTC) thermistor has a resistance of
5k Ω at room temperature and 500 Ω when hot. Therefore terminal (2) will be at 2.5 V
at room temperature and at 4.5 V when hot. If the reference voltage is set to 3.5 V,
a hot NTC will produce an error and shut down the IGBT. The resistor R should
be chosen to have a similar value to the NTC under normal conditions so that thereference voltage is somewhere in the middle of the 5V range. Also, it should be notmuch smaller than a couple of hundred Ohms to prevent excessive current flowing
between terminals (1) and (3). (b) PTC thermistor. Use as in the NTC case, butswap the resistor and the thermistor. (c) Interlock. Set Vref to 2.5 V. If the interlock
is closed: Terminal (2) is low, no error. If the interlock is open: Terminal (2) ishigh, an error is detected. Swap interlock and R to invert behaviour. (d) (Floating)
TTL. Not tested or recommended. This might even work if swapping the TTL signal
and ground. Use a high Pull-up resistor (a floating TTL might already have a pullup/pull down resistor somewhere, which might limit their current sourcing abilities).
(e) Terminal not used. To prevent random charge oscillations, pull down terminal(2) to ground in all unused error input terminals. Set the corresponding trimpots toapproximately 2.5 V.

132

B.4. GATE CHARGING AND PROTECTION

B.4

Gate charging and protection

Gate power supply A ± 15 V DC/DC converter powers the gate driver power supply.
The gate driver IC is optically isolated from the inputs. There are power resistors in theline to the gate driver power supplies, limiting the DC current to the gate to a couple ofmA. This protects against overheating if the IGBT breaks so that uncontrollable current
can flow from the broken gate to the emitter. However, the driver power pins are bufferedwith 10 µF capacitors so that the driver can provide 1-2 A peak-current during typicalgate charging times. While it is perfectly capable of driving the gate at low switchingfrequencies, a high switching frequency (> 100 Hz) should not be attempted.
Gate charging The gate is charged to around +13 V (There is a small voltage drop in thedriver chip. The voltage can be increased by carefully increasing the “5 V” power supply
voltage). If the driver is switched off, the gate is charged to -15 V. This helps to discharge
the gate to below 0 V quickly, decreasing switching times and reducing heating of the
IGBT (negative gate drive). Peak currents are around 1 to 2 A, which helps fast switching
and reduces switching power losses. Typical gate charges are 1 µC. The maximum gate
current can be limited by the changing the gate resistor, which is per default 10 Ω (theinternal gate resistance of the IGBT is typically a couple of Ω). It can be omitted to speed
up switching, but that is usually not necessary. Do not increase the gate resistor beyond
50 Ω, as the IGBT might be damaged due to slow switching. If more information on
switching speed is needed, check the homepage of Semikron for the IGBT application
notes.
Gate protection The driver chip has (depending on model though) an undervoltage
lockout, that shuts down the driver if the supply voltages is too low to drive the gate
properly. Two back-to-back 18 V Zener diodes limit the maximum voltage on the gate
to ± 20V. A 50 kΩ pull-down resistor between gate and emitter prevents the gate to float
to random voltages if it is not powered. The resistors in the power line to the gate chip
prevent it from overheating incase the IGBT breaks and excessive gate-emitter current
flows.
Gate selection The semix module has two IGBT modules and therefore also two gates
controlling the modules. The driver can only drive one module at a time. To selectthe appropriate gate/emitter combination, move the jumpers on the board to connect the“driver out” pins (on the far left/ far right) to the IGBT gate, as shown in Fig. B.4. NEVER
short Gate and Emitter vertically.

133

B.4. GATE CHARGING AND PROTECTION

Figure B.4: Selection of IGBT module. The correct jumper setting to select module(1) is shown in red; for module (2) in blue.

Figure B.5: Schematic of the Analog/Digital circuit of the driver.

134B.4. GATE CHARGING AND PROTECTION

Figure B.6: Circuit board design of the Driver. Red is the top copper layer, blue
the bottom layer. Most components are mounted on the top (black), only a few are
mounted on the bottom (turquoise).

135

B.4. GATE CHARGING AND PROTECTION

136

Appendix C
Design of a high-temperature
superconducting magnet
C.1

Introduction

In this appendix, we describe a prototype of a high-temperature superconducting (HTS)
trapping magnet. While not relevant to the rest of this thesis, we included it here fordocumentation purposes.
We initiated this project hoping to achieve higher current densities than in conventional water-cooled copper coils. This would allow us to reduce the size of the magnets
or to increase the spacing between the individual coils. Both measures would increase
optical access to the science cell between the coils.
While we were able to construct a functional prototype, the increase in performance
was not large enough to justify the increased complexity and cost. We therefore did notuse it in our experiments.
The first prototype coil was built as my diploma thesis project [94]. The second prototype, which is described here, was then constructed during the first year of my PhD. The
following report is based on [95].

C.1.1

Trapping fields

High-strength quadrupole magnets are essential to achieve trapping of atoms during evaporative cooling. They are usually produced by a coil pair in Anti-Helmholtz configuration
as shown in figure C.1. We will evaluate how to calculate their fields and consider the
implications for coil design.
The field of a current loop (radius a) on the z-axis can be obtained by integrating the137C.1. INTRODUCTION

a

I

d

z

y
x

I

Figure C.1: Two current loops in Anti-Helmholtz configuration.
Biot-Savart law:
B=

µ0 I
a2
ez .
2 (a2 + z 2 ) 23

(C.1)

Differentiating and adding a second coil at distance d yields the following gradient at
B(z = 0) = 0 in the centre between the coils:
dBz
3µ0 I
a2 d
=−
.
dz
2 (a2 + d2 /4) 25(C.2)

Using ∇B = 0 and the rotational symmetry of the system around the z-axis, we obtain
dB
1 dB
dB
=
=−
.
dx
dy
2 dz

(C.3)

The z-axis gradient is usually referred to as the strong axis gradient (as opposed to theweak axis gradient in the x-y-plane). We will use this gradient as a benchmark for coilcomparison.
It can be seen that the field gets stronger for smaller d and a. Hence, the coils need to
be placed close to the atoms to obtain a sufficiently high gradient. For a given radius, themaximum gradient is given for d = a. In this case equation C.2 simplifies to:dB= −0.86µ0 Ia−2 .
dz

(C.4)

In this simple model, doubling the current allows us to extend the coil spacing by a factor
1.4 while keeping the gradient constant. In a real magnet, we have to take into account thewidth and height of the coils. This can be done numerically by adding up the contributionsof many thin loops. From Eq. (C.2), we can see now that stacking windings provides
only limited benefit. Each new winding is one wire cross section further away and withdiminishing contribution to the field. Building stronger magnets buy stacking up windings
leads to considerable bulk. Therefore, increasing the current density is the only viable way
to gain a significant amount of optical access.
138

C.1. INTRODUCTION

C.1.2

Water-cooled Magnets

It is impossible to build conventional high field magnets without proper cooling; without
it, the electric heating P = I 2 R quickly turns the magnet into a piece of modern art.
Generally, hollow copper tubes are used to form the windings of the coil and water ispumped through at high pressure, cooling the magnet.
The pressure drop ∆p in a pipe in the turbulent regime is according to the DarcyWeissbach equation∆p = 0.5Rf l Q2 ,(C.5)
where Rf l is the flow resistance of the pipe and Q is the volumetric flow rate. The exact
1/2
expression for Rf l is rather complicated, but it is approximately proportional to L/Af l ,where L is the length of the pipe and Af l its cross section. Assuming that the heat transferfrom the copper to the water is perfect, the cooling power is
Pcool = QCδ[Tcoil − TH2 O ],(C.6)

where C is the heat capacity of water, δ its density and Tcoil − TH2 O the temperature
difference between coil and water. Equating heating and cooling gives us:∆p ∝L3 I 4,
Af0.5l A2Cu

(C.7)

where ACu is the copper area of the tube.
We can see that increasing the pressure allows only moderate current gains. Runningthe cooling water through parallel layers helps quite a lot, however at the cost of increasing bulk due to the additional connections. Ultimately, current densities are capped bytechnically viable pressures. In our lab, the total current density I/(Af l + Acu ) is around
1.5 kA/cm2 at 2 bar. The extreme end of the scale is used in the labs in Innsbruck [96],with up to 12.8 kA/cm2 at 35 bar. One should note that a 35 bar cooling system requires
a lot of infrastructure and does not come cheap.

C.1.3

HTS Magnet requirements

The currently best commercially available HTS is YBCO (Yttrium barium copper oxide).
We bought the superconducting tape for the prototype coil from American Superconductor. We used a tape with a critical current of 90A and a cross section of 4 mm×0.2 mm.
This corresponds to a current density of 11.3 kA/cm2 at 77K (liquid nitrogen temperature).
This is comparable to the best copper cooled systems. However, this is not the full truth.
First of all, the self-field of a coil reduces the critical current of the superconductor. In ad139

C.2. CRYOGENICS

dition to that, the cooling structure and insulation around the coil are going to reduce theeffective current density. On the plus side, lowering the temperature below 77K increases
the critical current drastically. Taking all these factors into account, one can currently not
expect to surpass water-cooled designs in terms of current densities. However, expecting
future progress in the field of HTS made us explore their use in cold atom experiments.
A good design of a HTS coil system for magnetic trappings needs to take care of thefollowing things:• Cooling: Making sure that the superconductor stays cold, and that the outside of
the cryostat remains at room temperature to prevent condensation of air humidity.
• Optimizing the geometry: The gradients need to be sufficiently large, while reducing bulk to allow optical access to the cell with the atoms.
• Quench protection: Superconductors have to be protected against damage fromoverheating.
• Fast ramping: Ramping of the coil needs to be fast enough for initial trapping of
the atoms and for time-of-flight switch-off.
These requirements will be discussed in the following sections.

C.2

Cryogenics

The cryogenic requirements in high-temperature superconductors (HTS) are much lessstringent than for low-temperature superconductors. Pouring liquid nitrogen over a HTS
coil sends it quickly superconducting state. During my diploma thesis we built a cryostat
with conventional insulation instead of the customary vacuum insulation. Since nitrogenis cheap, we were not concerned about the extra consumption. However, we realizedthat the biggest challenge is not to get the liquid nitrogen into the cryostat, but to get thenitrogen gas out of it (expansion ratio ≈ 200). Evaporation rates were around 100 W
and bubbles formed in the cryostat. The created gas layer insulated the HTS against theliquid nitrogen severely encumbering cooling. This made the operation of this prototype
extremely unreliable and not suitable for experiments. Additionally, the outside of thecryostat got cold as well, leading to condensation air humidity.
For these reasons, we decided to use vacuum insulation for the prototype presented
in this report. Vacuum insulation renders direct heat conduction losses through the walls
negligible (≈ 1 W), solving the above problems. The other important heat leak (≈ 10W) is caused by the non-superconducting current leads connecting the coil to the outside
140C.2. CRYOGENICS

world. However, this only causes evaporation in the pipes leading to the cryostat, which
is non-critical to the coil stability.
As manual topping up the coil with liquid nitrogen is not practical in the long run, an
automated system has been designed.

C.2.1

Current leads

Metal current leads connect the HTS coil to the outside world. They need to carry thecurrent from the point where superconductivity ceases (around 77 K) to the outside world(300 K). The heat load causes by the leads is the sum of electrical heating and heat conduction. The Wiedemann-Franz Law (WFL) relates resistivity ρ(T ) and heat conductivity
λ(T ) for metals as:
ρ(T )λ(T ) = L0 T,(C.8)
where L0 = 2.44 × 10−8 W ΩK −2 is the Lorentz number. If we consider a current leadwith diameter A, length l carrying a current I, the heat flux Q̇ through an element dx
along the superconductor is given byQ̇ = λ(T )A

ρ(T )
dT
+ I2
dx = 0.
dx
A

(C.9)

Solving this equation with the boundary conditions Tl = 77 K and Th = 300 K yields
a minimum heat load Popt for a given current (see [97] for the calculation in Helium
cryostats, and [94] for HTS). Popt is approximately independent of the metal used :√Popt = I

L0 (Th2 − Tl2 )p.
Th2 − Tl2

(C.10)

This heat load is only achieved for an optimum length to area ratio f , which depends onthe metal as well as Tl and Th :Il
= f (λ, Th , Tl ).
A(C.11)The resulting values are tabulated in Tab. C.1.
For the critical current in the prototype coil of around 55 A, this yields a heat fluxof around 4.6 W for two leads. The heat load can be reduced by a factor of two by
vapour-cooling the leads with the evaporating nitrogen (see Tab. C.1). We paralleled
the leads with HTS tape until the filling pipe, so that the heat load is localized in thefilling pipe and does not cause bubbles in the cryostat. Therefore it is not critical forsuperconductivity stability. It might pay off to make the leads thinner than the optimum
steady-state ratio Imax l/A, as the coil current has typically a small duty cycle ratio in
141

C.2. CRYOGENICS

CopperBrass

Stainless Steel

3.5 × 106

6.5 × 106

1 × 105

42.7

45.5

45.5

3.5 × 106

5.7 × 105

1.1 × 105

Conduction cooled
Numerical(IL/A)opt[A/m](P/I)opt[10−3 W/A]

Semi-analytical(IL/A)opt[A/m](P/I)opt[10−3 W/A]

45.3

45.3

45.3

[A/m]

5 × 106

8 × 106

1.5 × 105

Vapour cooled
Numerical(IL/A)opt(P/I)opt[10−3 W/A]
23.2
25
25.5
Table C.1: Comparison of the optimal design parameters for different materials anddifferent sources. Semi-analytical results with thermal conductivities taken from [98]and [99] were done based on the calculations above. They are compared to theresults from [100] which take deviations from the WFL into account. The values
for vapor cooling are also taken from this reference. Both calculations agree very
well, except for brass which can be explained by the variety of different brass alloys(unfortunately the alloy type is not specified in [100]) and a lack of data points for its
thermal conductivity. The temperatures used were the boiling temperature of nitrogen
Tl = 77 K and room temperature Th = 300 K.

cold atoms experiments. However, the deviation should not be larger than a factor of two,otherwise the leads melt during continuous operation [100].

C.2.2

Insulation

After the poor results using polyurethane foam as well as with aerogel, we decided to use
vacuum insulation. Vacuum insulation can be made rather compact (e.g. in thermosflasks)
if designed carefully. We will show that the vacuum requirements to reduce insulation
losses below the losses in the current leads are not very demanding ( 1 × 10−3 mbar).
Our cryostat, shown in Fig. C.2), features a doubled wall vacuum space. Lids areneeded to insert the coils, while indium seals are used to seal the gaps. Multi-layer insulation (MLI) is used to shield to radiation losses.
142C.2. CRYOGENICS

(a)

(b)

Pump out port

Filling tube

Bottom lid

Indium seals

Top lid
VacuumNitrogen

HTS

SpacerFigure C.2: (a) A photo of the prototype. (b) Schematic cross section of the prototype.

143

C.2. CRYOGENICS

Theoretical predictions
Mean free path Heat conduction through a gas falls either into a low pressure or a highpressure regime, depending on whether the mean free path λ of the particles is smaller
or bigger than the distance between the walls [99]. The mean free path is the average
distance travelled between collisions of the particles. It can be estimated as
kT
,
λ= √
2σp

(C.12)

where T is temperature, p the pressure and σ the collision cross section of the gas molecules.
The cross section is related to the diameter as σ = πd2 . Air mainly consists of N2 andO2 , which both have two atomic shells. The Bohr model yields an estimated diameter of
2 × 10−10 m for each atom. Taking the second atom in the molecule into account, one
can estimate d = 3 × 10−10 m [101]. The resulting λ for room temperature is plotted infigure C.3. At pressures of around 1 × 10−3 mbar, we can be sure that λ is larger than thespacing between the walls.
2

10

0

Free molecular length [m]

10

−2

10

X: 0.1207
Y: 0.0008671

−4

10

−6

10

−8

10

−4

10

−2

10

0

10
Pressure [mbar]

2

10

Figure C.3: The mean free path of an average air molecule depending on pressure
at 300 K. The crossover between the high pressure regime C.13 and the low pressure
regime C.14 occurs when λ is comparable to the wall spacing, which is 5 mm for the
prototype coil.

High pressure At high pressures (such as atmospheric pressure), λ is shorter than the
distance between the walls. In this case the heat transfer between two plates (a good144

C.2. CRYOGENICS

approximation for a double walled cryostat) Q̇ can be calculated as
Q̇ = λ∆T A/d,(C.13)

where λ ≈ 0.025 W/m K for air [99]. A is the area between the plates and d theirseparation, ∆T is the temperature difference. The results for our cryostat are plotted inFig. C.4. It can be seen that the high pressure regime leads to unacceptably high heatlosses.
Low pressure We therefore need to operate in the low pressure regime (p < 1 × 10−3mbar) to achieve good enough insulation. There, the heat flux depends only on pressure:Q̇ = k∆T pA,(C.14)

where k is a gas and geometry dependent constant on the order of one [99]. Below 1×10−3
mbar, our heat load is clearly negligible compared to the current lead contribution (seeFig. C.5).

Figure C.4: Heat flux in the high pressure regime, calculated for ∆T = 230 K anda cyrostat area A = 0.07 m2 . Note that the flux is not pressure dependent. This is arather optimistic calculation, since convection currents are likely to increase the heattransfer. To reduce the heat load below 10 W, a huge air gap would be needed.

Radiative Losses In the prototype cryostat, vacuum pressures of 1 × 10−5 mbar were
easily obtained using a turbo and scroll pump combination. At these pressures, we areclearly in the free molecular regime and the heat load transmitted by particles is basicallyzero. However, radiative heat transfer becomes relevant. We estimate it with the Stefan145

C.2. CRYOGENICS
6

10

4

Heat Load [W]

10

2

10

0

10

−2

10

−4

10

−4

−2

10

10

0

10
Pressure [mbar]

2

10

Figure C.5: Heat flux in the low pressure regime, calculated for ∆T = 230 K and thearea of the cyrostat A = 0.07 m2 . At pressures below p < 1 × 10−3 mbar it becomes
negligible.

Boltzman equation:
qr = r σT 4 ,(C.15)

where qr is the heat flux from a surface at Temperature T , σ = 5.67 × 10−8 is the StefanBoltzman constant and r is the emissivity. For the flux between two walls at different
temperatures, an effective emissivity
ef f =

cold warm
cold + warm − cold ∗ warm(C.16)

can be derived. The flux between two walls is then [102]:44
qr = ef f σ(Twarm− Tcold).

(C.17)[102] gives tabulated values for ef f and qr . For polished stainless steel, ef f = 0.12 and
qr = 55 W/m2 for surfaces at room and liquid nitrogen temperatures. Using the approximate area of our cryostat, A = 0.07 m2 , this causes a radiative heat load of approximately
4 W. T This heat load can be reduced further by inserting multi-layered insulation (MLI)
between the walls. MLI is a very thin aluminium foil coated with plastic that acts as
floating radiation shield and is readily available from cryogenic suppliers. Using flux
conservation, we can show that N layers of MLI reduce the heat load to
qr =

al
4
4
σ(Twarm− Tcold
),
N +1
146

(C.18)

C.2. CRYOGENICS

where al is the emissivity of aluminium, which is similar to stainless steel. This effectcan be used to get a good emissivity even when the cryostat surface is not well polished.
More importantly, the heat load is reduced by more than a factor of N . We used 5 layers,
reducing the radiative heat load below 1W.

Spacer and connection pipe losses There are also conductive heat leaks from theplaces where the inner cryostat wall is connected to the outer wall. These heat leaks q̇
can be calculated as
ZT2(C.19)q̇ = A/L λ(T ) dT,
T1

where A is the cross-section, λ(T ) the thermal conductivity at temperature T and L thelength and Ti are 77 K and 300 K respectively. Values for the integral are tabulated in [99].
In the cryostat, we have several connections where losses occur. The stainless steel
pipe leading to the outside induces losses of only 0.5 W, as stainless steel is a bad conductor and the pipe is fairly thin. Three spacers between the walls prevent movement of
the inner cryostat. Using PTFE as a material, we expect another 0.5 W heat load.

Experimental results
To measure the total heat leak, we filled the cryostat with liquid nitrogen. We measure thetime until complete evaporation of the nitrogen, as well as monitoring the flow rate of theevaporated gas. Both of these measurements allow us to estimate the heat leak.

Vacuum The vacuum insulated cryostat (no MLI) filled with 0.3 L of liquid nitrogen
lasted 2h at an average gas flow rate of 1.5 l/min. [99] lists the liquid (2.25 × 10−2 L/h W)
and the gas volume boil-off rates (0.243 L/min W) for liquid nitrogen. Our measurements
correspond to heat leaks of 6.7 W and 6.2 W, respectively. Theoretically, we expect aheat leak of 4 W from radiative losses plus 1 W from spacers/pipe conduction, which iscompatible with our observations.

Air gap The measurement was repeated with an air-filled vacuum gap, measuring heatloads of around 40 W. Theoretical estimates using Eq. (C.13) range between 40W-80W.
The estimate depends strongly on which temperatures are assumed. This is difficult as theouter wall is cooling down over time. Also the heat conductivity of air varies considerablyfrom source to source. Given the uncertainties, this is a rather good agreement; it certainlyshows that air gap insulation is not practical.
147

C.2. CRYOGENICS

Conclusions It is certainly advisable to test the vacuum before inserting the coil witha helium leak tester. On the first test, this revealed a huge leak (actually too big forthe helium leak tester to resolve). Over-pressuring the cryostat and floating it in ethanol
revealed a bubbling gap where a weld connection was not done properly. After mending
the gap, the vacuum worked great and no measurable leaks were found with the leaktester.
We can conclude that the vacuum insulation performs as expected. The problems withthe previous non-vacuum insulated prototypes are resolved. The consumption of liquid
nitrogen is reduced by an order of magnitude (from 100W to below 10 W, corresponding
to less than 1 l/h ) and we found that the coils can be run continuously for hours at criticalcurrent.

C.2.3

Filling system

Manual valve
LiquidNitrogenDewar
1.5 bar

Solenoid valve

Flex. metal hoseTank
0.2 to 1 bar

ExhaustScrollpump

HTS coilPressure gauge:
0-4 barPressure relief
valveElectric Feedback

Figure C.6: Liquid nitrogen is pumped into the tank, from where gravity feeds it
directly into the coils. Electric feedback keeps the nitrogen level constant. Pumping
down the pressure of the gas phase above the liquid in the tank allows to reduce thetemperature of the bath.
We built a fully-automated liquid nitrogen filling system (see Fig. C.6). Liquid nitrogen is pumped from the dewar through transfer lines into the top filling tank. From there,
it is fed by gravity into the HTS cryostat. The evaporating gas is vented into an exhaustto prevent low air oxygen levels. For safety reasons, an oxygen detector is installed in
the proximity of the coil. Connections to the cryostat are made with vacuum CF-Flanges.
Standard BSP-threaded fittings also work sufficiently well when enough PTFE tape andexcessive amounts of torque are used.
Nitrogen level sensors Home-made nitrogen sensors in the tank sense the filling level.
They consist of PTC thermistors (B59721A0070A062, Epcos), which we found exhibit148

C.2. CRYOGENICS

a 1000-fold increase of resistance between room and liquid nitrogen temperature. They
are heated by 20 Ω resistors with around 0.05 W each. As the cooling power of liquid
nitrogen is much higher than of nitrogen gas, this leads to a steep change in resistanceof the PTC, depending if it is below or above the liquid level. A similar method withcarbon resistors is described in [103]. In [94], we introduced the dual Resistor-Thermistor
system to increase the resistance contrast ratio. Both PTC and resistors are surface mountcomponents, taking up minimal space as shown in Fig. C.7.

Figure C.7: Composite PTC-R nitrogen level sensor covered in epoxy.

Signal processing The sensor signals are processed by a Max32 Chipkit microcontroller (compatible to the Arduino), which is programmed in C++. It features digital I/O
channels as well as 10-bit analogue inputs. The analogue sensor readings are converted
into digital level signals. These level readings are processed and control the nitrogen flow
via an electric valve. The Max32 has pre-built libraries for LCD displays, allowing us tocreate a user-friendly visual interface. The control box is shown in Fig. C.8. The fillingsystem is fully automated. The filling behaviour can be reprogrammed in-place via a USB
port.

Vapor pressure pumping We found that inserting a scroll pump into the vent line
speeds up filling due to the increase in differential pressure between the tank and cryostat. More importantly, it lowers the vapour pressure of the liquid nitrogen. Lowering
the vapour pressure reduces the temperature of the nitrogen and thus increases the criticalcurrent of the coil. The pressure could be pumped down all the way to 0.2 bar, which
is close to the triple point of nitrogen, where it solidifies at a temperature of 67 K. This
yielded an increase of the critical current from 55A to 85 A.
149

C.3. GEOMETRIC DESIGN AND FIELDS

Figure C.8: The control interface of the filling system. It controls the filling valve
depending on the nitrogen level and shuts down the magnet if a fault is detected. As
the sensors are disconnected in the picture, several error messages are displayed. The
user can also manually control the filling process.

C.3

Geometric design and fields

Geometry of the coil The YCBCO tape is a flat 4 mm by 0.4 mm tape that is sold in
unspliced lengths of around 30 m. The main parameters for coils are their inner and outerradius and the number of layers. Since splicing different tapes together is difficult andcan damage the superconductor, the most common shape are coil stacks that are muchwider than high (’pancake coils’). The main considerations when choosing the design
parameters are:
• To create a large, uniform magnetic field gradient near the atoms.
• To reduce the size of the cryostat.
• To minimize the self-field, i.e. the magnetic field in the superconductor.

Figure C.9: A photo of YBCO tape next to a schematic cross section. The actual
superconductor constitutes only small part of the tape.
150C.3. GEOMETRIC DESIGN AND FIELDS

The first two aims are obvious; minimizing the self-field not so much. As any superconductor, YBCO has a critical Temperature-Magnetic Field-Current surface. On one
side of this surface, YBCO is superconducting; on the other side it is not. Crucially, ahigh self-field reduces the critical current and thus the field at the atoms. These requirements cannot be maximized at the same time, so a sensible compromise is needed. To
calculate the fields for a given configuration, we used calculated the fields for a range ofpancake coil configurations by decomposing them into current loops and adding up theircontributions using elliptical integrals [104]. By determining the maximum self-field in
the coil, we can estimate the critical current from Fig. C.11. An important result fromthese calculation is that more than two coil layers provide very limited benefits, as theincrease in field per current is eaten up by the self-field reduction of the critical current.
Taken these considerations into account, we designed a cryostat with an inner radiusof 26 mm, an outer radius of 67 mm and two layers of HTS rated at 90 A . A double wall
allows for vacuum insulation. The lids are sealed with 1mm indium wire (0.7 mm turned
out to be too thin). Indium seals seal well at cryogenic temperatures, do not take up muchspace and are not very difficult to use. U-formed PTFE spacers were used to clamp thecoils down. A pump out port (similar to the ones on nitrogen dewars) allows us to create
the vacuum between the walls. Detailed drawings can be found at the end of this report.

Critical Currents HTS are much more susceptible to parallel fields than to perpendicular ones [105]. Using our numerical simulation of the coils, we obtain a maximum parallel
self-field of around 46 G/A. Combining this with the manufacturer’s plot of the self-fielddependence of the critical current, we estimate a critical current of 45A (see Fig. C.11).
Experimentally, we observed 55-60A. Pumping the vapour pressure of the nitrogen close
to the triple point down to 0.2 bar (≈ 65 K) raised the current to around 85-90 A. Fig. C.11
predicts a critical current of around 80 A. We see that the predictions are generally around
10 A too low. This might be accounted for by the fact that our model is a worst-casescenario. Only 2% of the coil length exceeds 90% of the critical field. Current sharing as
shown in figure C.10 in these regions will suppress heat production and should increase
the viable current by a few Amperes.

Magnetic Field Measuring the magnetic field along the z-axis with a Gaussmeter allows
to check the numerical predictions. Fig. C.12 shows that they are in excellent agreement,
validating the coil model. This allows us to predict a gradient with a second coil of 505
G/cm as shown in figure C.13. Vapour pressure pumping increases this to 780 G/cm.
151

C.3. GEOMETRIC DESIGN AND FIELDS

I-Ic

I

CopperYBCO

Ic
Figure C.10: Once the critical current is exceeded, the copper cladding takes overpart of the current.
11

2x10

11

1010
8x10
10
6x10

µ0H||c

10

4x10

2

Jc (A/m )AMSC13 [T=85 K]
AMSC13 [T=77 K]
AMSC13 [T=64 K]
AMSC13 [T=50 K]
AMSC13 [T=40 K]
AMSC13 [T=20 K]
AMSC13 [T=10 K]
AMSC13 [T=4.2 K]

=90A

10

2x10

10

10 9
8x10
9
6x10
4x10

9

2x10

9

Prototype load curve]

9

108
8x10
8
6x10
4x10

8

2x10

8

0.01

0.1

1

B (T)

Figure C.11: Intersecting the manufacturer’s background field reduction data [105]and the numerical simulated maximum self-field per current (blue line) allows us toestimate the critical current for different temperatures. The maximum current density
at zero field and 77 K of the manufacturers curve has been normalized to the 90 A
capacity of our tape. The benefits of lowering the temperature are quite obvious.
1800
Experimental data
Numerical prediction

1600
1400

Bz [G]

1200
1000
800
600
400
200
0
−2

0

2

4

6
z [cm]

8

10

12

14

Figure C.12: A comparison of the measured and the numerically predicted fieldalong the z-axis. They agree very well, confirming the validity of model.

152C.4. COIL ELECTRONICS
300
200
100

grad Bz [G/cm]

0
−100
−200
−300
−400

X: 0
Y: −505.3

−500
−600
−5

−4

−3

−2

−1

0
z [cm]

1

2

3

4

5

Figure C.13: Predicted Gradient with a second coil. At 55 A, a gradient of 505 G/cm
at the trap centre can be expected.

C.4

Coil electronics

C.4.1

Quench protection

Superconductors are very prone to local overheating due to their high current densities
and the steep increase of resistance when transitioning to normal state. The process is
usually initiated by a local hot spot. Such a hot spot can be induced by a defect in the
superconductor or other sources of heating. If the temperature of the hot spot is above thecritical temperature of the superconductor, the section will go normal and ohmic heating
will start. This will raise the temperature even further and heat up neighboring sections
and eventually drive them normal as well. The normal zone will thus spread throughthe magnet. Such an event is called a quench. In big1 low-Tc magnets this is a rather
spectacular event, as quenches spreads with a velocity of several meters per second [106].
This event will convert the magnets energy into heat immediately and evaporate the liquid
helium.
In a small high-Tc magnet, such a quench is rather different. It spreads much slower(mm/s) and the developing heat will not evaporate the nitrogen instantly [97, 107]. Because of this slow propagation, the superconductor will melt locally in a thermal runaway. Therefore a quench needs to be detected and countermeasures need to be taken. In
our case, we decided to use the most common approach, a ’detect and dump’ protection
scheme. If a resistive voltage due to the quench is detected, the current is switched off
and the coil energy LI 2 is dumped into a varistor.
The same protection electronics as in [94] were used. Three voltage taps were attached to the coil, connected to an inductive Wheatstone bridge. This cancels inductive
1

The size of an NMR or a particle accelerator magnet.

153C.4. COIL ELECTRONICS

V
LC
Φ
LP
RP
Figure C.14: A simple circuit explaining eddy currents.
voltages as well as noise and detects quenches reliably. We established numerically andexperimentally [94] that the experimentally achieved detection voltage of 20 mV can beconsidered as absolutely safe. Having a good protection system is crucial to prevent damage when running the coil close to its critical current.

C.4.2

Switching performance

Theoretical predictions A HTS coil has typically many more windings n than a conventional coil. This means we get a huge inductance, as it scales as L ∝ n2 . The coil prototype has n ≈ 2∗100 and L ≈ 4 mH. The energy stored in the coil scales as E = 0.5LI 2 ,so energizing and deenergizing the coil quickly is difficult for large inductances. However, fast switching is crucial for trapping cold atoms. Fast switch-on is required to trap
atoms after the molasses stage before. Also, fast switch-off is required to produce accurate time of flight (TOF) measurements. Both situations can be represented within thesame mathematical framework, in which a voltage V0 is applied to an inductance. Duringswitching on, V0 is the driving voltage of the power supply (Delta SM15-200D) ranging
up to 15 V. During switching off, V0 is the (negative) clamping voltage of the varistor
2
and is limited by the maximum voltage rating 1200 V of the switching device (Semix
252GB126HDs IGBT). In both cases, the change in current is:˙ = V0 /L ⇒ Tswitch = Imax L
I(t)V0(C.20)An anti-Helmholtz coil pair composed of two HTS coils would have L ≈ 8mH andwould produce a maximum gradient of around 500 G/cm. For switch-on, we expect a2

A Varistor is a surge suppressing device, which nearly fulfils V (I) = const.

154C.4. COIL ELECTRONICS

switching time Tswitch = 30 ms corresponding to a magnetic gradient ramp of 17 G/(cmms). When switching off, we get (with V0 = 1000 V) a switching time Tswitch = 0.44 ms,
corresponding to a gradient ramp of 1100 G/(cm ms). These equations are only valid if noeddy currents are induced in the surrounding structures. Since the cryostat walls are madefrom stainless steel, eddy currents cannot be avoided. We will therefore derive a modelof the coil, which includes eddy currents. We assume that the cryostat walls contributing
to the eddy currents act as if they were a coil with a single winding nP = 1, inductance
LP , resistance RP , and current IP . Additionally, we assume that the coil and the wall
are close together and have a similar geometry. We represent our model in the circuit
diagram shown in Fig. C.14. Generally, the magnetic flux through a coil A is related to
its inductance by:
ΦAA = LA IA /nA .
(C.21)For two close and well coupled coils and a magnetic flux coupling coefficient 0 ≤ T ≤ 1,
we can write for the flux through coil A caused by coil B:
ΦAB = T LB IB /nB .

(C.22)The induction voltage in A is then:UAind = nAd(ΦAB + ΦAA ).
dt(C.23)

Using these equations for the HTS Coil C and the Plate ’Coil’ P, we obtain:−V−RIP= nT LP I˙P + LC I˙C= LP I˙P + T LC I˙C /n(C.24)(C.25)(C.26)

Solving these equation gives a change of coil current of:hitI˙C = − LVC 1 + T 2 e− ττ2)= LP (1−TRP(C.27)(C.28)

Looking at this, we see that in the limits of low coupling T → 0 or small eddy current
time constant τ → 0 , we recover Eq. (C.20).
We obtain an upper limit for τ by approximating the relevant cryostat parts with ad = 5 mm thick stainless steel plate (resistivity ρSS ) with the same inner and outerradii as the HTS coil. We approximate LP ≈ LC /n2 ≈ 1 × 10−7 mH and RP =155

C.4. COIL ELECTRONICS
ρSS d ln(rin /rout )/2π ≈ 6 × 10−4 Ω. With T = 0, this gives τ ≈ 0.17 ms. This timescaleis much shorter than the switch-on time; no eddy-current effects are to be expected.
However, the switch-off time is on the millisecond scale and eddy currents are expected to be significant. The decay of the field happens in two steps: First, the coilcurrent Ic decays, charging the plate current IP in the process. Integrating C.27 with thestarting condition IC (t) = Imax , we see that the coil current decays quicker than without
eddy currents. Once the coil current has dissipated, we can assume IC (t) = 0. After this,the current in the plate IP will decay exponentially with a time constant τ , causing themagnetic field to decay with a similar decay time.

Experimental results We checked the predictions above by measuring IC with a current clamp as well as using a pick-up coil to monitor the change in magnetic field. The
results are shown in Fig. C.16 for switch-on and Fig. C.15 for switch-off. Our measurements confirm the theoretically predicted behaviour and timescales. We achieve switch-on
ramps of 4.5 G/(cm ms) and switch-off times of below 1 ms.

Timescales in cold atom experiments Extrapolating the switching behaviour to a quadrupole
coil pair (with twice the inductance), we expect it to switch-on at a rate of 2×104 G/(cm s).
Switch-off has a nearly linear part which is negligible and a ringing decay time τ ≈ 0.15
ms. Switch-off should therefore be complete after ≈ 1 ms. Is this fast enough for trapping
and releasing atoms?
When releasing the atoms, this has to happen non-adiabatically, that means that thetrap has to be switched off faster than the atoms move in the potential. For a harmonic
trap with frequency ω this means:
tof f << ω −1 .

(C.29)

In typical experiments, the final trap frequency ω is on the order of 100 to 200 Hz [28],i.e.
we need to switch faster than 1 ms. This is the magnitude of our switch-off time, which
could cause problems. However, this is unlikely, as most of the field is already decayedat that time and thus distortions of the cloud shape should be small.
For trapping, Stuart Moulder has established optimized procedure for ramping themagnetic fields [50]. Immediate switch-on of the coils to 62 G/cm for mode-matching
is followed by an adiabatic ramp of 3.5 × 103 G/(cm m). The adiabatic ramp is easy
enough to match, but how fast is an ’immediate’ switch-on? Assuming that the atoms
with temperature T ≈ 0.2 mK expand freely for a time ton ≈ 3 ms, we can estimate the156

C.5. CONCLUSIONS
50

40

Coil current decay
Plate current decay
Plate current charges

Coil Current [A]

30

40A
30A
20A
10A

20

40A fit10

Pick−up Coil [mV]

0

−10

−20

−30
−1

0

1

2
Time [s]

3

4

5
−4

x 10

Figure C.15: Switch-off times at different starting currents. The varistor used has
VClamp = 550 ± 10V . The upper half shows the current in the HTS coil measured
with a current clamp. The shape of the curve is as expected the integral of equation
C.27. Fitting gives reasonable results. However, the relevant timescales: V /LC andτ vary considerably between curves and fitting methods. The most consistent resultsfor V /LC ≈ (1.75±0.5)×105 A/s can be extracted by fitting to the initial linear slopeof IC . The theoretical prediction is (1.4 ± 0.1) × 105 A/s. The discrepancy can be
attributed to the simple model or that the calculated inductance is a bit too high. The
lower half shows the induced voltage in the pick-up coil, caused by the magnetic fluxchange. This is nearly a pure exponential decay as the varistor is not in the saturation
region and IC (t) = 0. This allows easy fitting and extraction of τ ≈ (0.14 ± 0.01)
ms. This is in very good agreement with the estimate of 0.17 ms. One should note,that this excellent agreement is probably a lucky punch given the crude model.
gain in potential magnetic energy:∆E ≈= µB 0 v0 ton = B 0p
2kT /mRb ≈ 3 × 10−3 kT.

(C.30)We see that even in this rather pessimistic scenario, the additional energy is negligible.
The travelled distance d = v0 ton ≈ 40µm is much smaller than the diameter of a typical
MOT (≈ 1 mm). The initial ramp should be fast enough in order not to affect the phasespace density.

C.5

Conclusions

We built a functional HTS prototype coil. We achieve a magntic field gradient 25% above
the water-cooled coils we use in the experiment. When employing vapour-pressure pumping, the gain can reach up to 175%. However, the prototype is still bigger than the water157

C.5. CONCLUSIONS
60

50

Coil Current [A]4030

15.5 V12.5 V
10 V
7.5 V
5V
2.5 V
Fit to 15.5 V

20

10

0
0.38

0.4

0.42

0.44
Time [s]

0.46

0.48

0.5

(a) A plot of the coil current during switch-on with different driving voltage. The ramps are linear,as predicted by I˙C = V /LC . One exemplary fit is shown. The wiggles once the coils are ramped
up are substantial and need to be taken care of. Using a PID for feedback should allow to fix it.
5000
4500

Swtich−on current ramp [A/s]

4000
3500
Ramping speeds
Fit

3000
2500
2000
1500
1000
500

1.3 V voltage drops in leads+ IGBT
0

0

1

2

3

4

5

6

7
8
9
Driving voltage [V]

10

11

12

13

14

15

16

(b) A plot of the ramping speeds against the driving voltage. As expected, the relation is linear,with an 1.3 V offset that corresponds to the voltage drop in the leads and the IGBT. A fit gives thevalue of the ramp: 1/LC = 1/3.1 mH= 323 A/(V s). The inductance seems to be lower than the
numerical prediction of 4 mH.

Figure C.16: Switch-on behaviour of the coil.

cooled design used in the experiment. Even when taking into account that the prototype
has been over-dimensioned for ruggedness, one should not expect a substantial increaseof current density from nitrogen-cooled HTS coils compared to the water-cooled solution.
To increase performance further, liquid helium cooling or a close-circuit cryocooler arenecessary to lower the temperature. This is viable with the vacuum insulation and should
increase the current density by an order of magnitude.
However, it is questionable whether this increase in performance outweighs the increased cost and technological complexity imposed by the necessary cryogenic infrastructure of HTS coils. If better optical access is really needed, it can be achieved in manyother ways, e.g. transporting the atoms to another experimental chamber, removing the158

C.5. CONCLUSIONS

need for coils with all-optical-trapping methods or simply increasing the pressure of thecooling water through the coils.
While we originally intended to implement the coils in our current setup, we decided
against it due to the reasons listed above.

159

C.6. TECHNICAL DRAWINGS OF THE CRYOSTAT

C.6

Technical drawings of the cryostat

Figure C.17: The assembled cryostat.
160

C.6. TECHNICAL DRAWINGS OF THE CRYOSTAT

161

C.6. TECHNICAL DRAWINGS OF THE CRYOSTAT

162

C.6. TECHNICAL DRAWINGS OF THE CRYOSTAT

163

C.6. TECHNICAL DRAWINGS OF THE CRYOSTAT164

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