Cooling, Trapping, and Transport of Atom Clouds
in a New BEC Apparatus

A Thesis Presented
byStephan Gerhard Albert
to
The Graduate Schoolin Partial Fulfillment of theRequirements
for the Degree ofMaster of ArtsinPhysics
Stony Brook University
August 2007

Stony Brook University
The Graduate School

Stephan Gerhard AlbertWe, the thesis committee for the above candidate for the
Master of Arts degree, hereby recommend
acceptance of this thesis.

Dominik Schneble - Thesis Advisor
Assistant Professor, Department of Physics and Astronomy

Harold Metcalf
Distinguished Teaching Professor, Department of Physics andAstronomy

Thomas Bergeman
Adjunct Professor, Department of Physics and Astronomy

This thesis is accepted by the Graduate School.
Lawrence Martin
Dean of the Graduate Schoolii

Abstract of the Thesis
Cooling, Trapping, and Transport of Atom Clouds
in a New BEC Apparatus
byStephan Gerhard Albert
Master of ArtsinPhysics
Stony Brook University
2007

This thesis discusses laser cooling, magnetic trapping, and mechanical transport of ultracold atom clouds of rubidium in a new Bose-Einstein condensation(BEC) apparatus. After an introduction on different apparatus designs anda discussion of the techniques employed in our experiment, we describe thedevelopment of a preparation sequence for subsequent evaporative cooling andcharacterize the single stages.
We demonstrate a promising efficiency of the entire sequence: up to
1.2×1010 atoms can be caught from a rubidium vapor with a magneto-opticaliii

trap, then compressed, and further cooled by optical molasses. The cloud issubsequently transferred into a quadrupole magnetic field. The coils for thismagnetic trap are mounted on a translation stage, which moves the atoms to a10−12 torr vacuum glass cell where evaporative cooling will be performed. We
can transfer about 2.5×109 atoms to this glass cell. We measure parameters
important for successful evaporation and discuss the realizability of BEC in
our apparatus.

iv

parentibus carissimis

Contents

1

2

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

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

1

1.1

Bose-Einstein condensation - some quantum statistics . . . . .

2

1.2

Different designs . . . . . . . . . . . . . . . . . . . . . . . . .

6

Description of the apparatus and theory of the employed techniques 11
2.1

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

11

2.2

Rubidium and laser system . . . . . . . . . . . . . . . . . . . .

14

2.3

Atom-light interaction and laser cooling - a very short introduction 18
2.3.1

2.4

Time-dependent perturbation theory and two-level problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3.2

Spontaneous emission and optical Bloch equations . . .

21

2.3.3

Light shifts . . . . . . . . . . . . . . . . . . . . . . . .

24

The magneto-optical trap . . . . . . . . . . . . . . . . . . . .

26

vi

2.5

Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.6

Sub-Doppler cooling . . . . . . . . . . . . . . . . . . . . . . .

36

2.6.1

lin ⊥ lin polarization gradient cooling . . . . . . . . . .

37

2.6.2

σ + -σ − polarization gradient cooling . . . . . . . . . . .

39

Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . .

41

2.7
3

Methods, experiments, and results
3.1. . . . . . . . . . . . . . . . .

47

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.1.1

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

47

3.1.2

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

50

3.1.3

Choice of imaging method . . . . . . . . . . . . . . . .

53

3.2

Determination of temperatures. . . . . . . . . . . . . . . . .

54

3.3

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

56

3.3.1

Light induced atomic desorption . . . . . . . . . . . . .

59

3.4

Compressed magneto-optical trap . . . . . . . . . . . . . . . .

62

3.5

Molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.6

Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.6.1

Stern-Gerlach analysis of optical pumping . . . . . . .

70

Magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.7

3.7.1

Switching on the magnetic trap: the optimum catch point 74

3.7.2

Compression of the magnetically trapped cloud . . . .

80

3.8

Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.9

Feasibility assessment for BEC . . . . . . . . . . . . . . . . . .

91

3.9.1

Dependence of the transport efficiency on the backgroundpressure . . . . . . . . . . . . . . . . . . . . . . . . . .
vii

92

3.9.2Lifetime in the science cell . . . . . . . . . . . . . . . .

94

3.9.3

Overall efficiency of the entire sequence . . . . . . . . .

95

3.9.4

Rethermalization rate

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

98

3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

110

ADetailed explanation of lin ⊥ lin-polarization gradient cooling . .

111

BApparatus details . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

B.1 Vignetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

B.2 Laser frequency jitter . . . . . . . . . . . . . . . . . . . . . . .

116

B.3 Calibration of the analog channels . . . . . . . . . . . . . . . .

118

CSequence illustration for overall efficiency experiments . . . . . . .

120

DAbbreviations and rubidium properties . . . . . . . . . . . . . . .

121

viii

List of Figures

2.1

The apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Level structure of 87 Rb . . . . . . . . . . . . . . . . . . . . . .

17

2.3

Magneto-optical trapping . . . . . . . . . . . . . . . . . . . . .

28

2.4

MOT with quadrupole coils . . . . . . . . . . . . . . . . . . .

29

2.5

MOT loading curve and pressure dependence of atom number

32

2.6

Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.7

σ + -σ − -cooling . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.8

Field of a quadrupole trap . . . . . . . . . . . . . . . . . . . .

46

3.1

Absorption imaging principle . . . . . . . . . . . . . . . . . . .

48

3.2

Fluorescence imaging principle . . . . . . . . . . . . . . . . . .

51

3.3

Optimization of the MOT . . . . . . . . . . . . . . . . . . . .

58

3.4

Pressure drop after application of light induced atomic desorption 60

3.5

Effect of light induced atomic desorption on MOT atom number 62

3.6

Optimization of the CMOT . . . . . . . . . . . . . . . . . . .

64

3.7

Temperature in a MOT as a function of atom number . . . . .

66

3.8

Temperature after optical molasses . . . . . . . . . . . . . . .

67

3.9

Stern-Gerlach analysis sequence . . . . . . . . . . . . . . . . .

72

ix

3.10 Stern-Gerlach analyis example . . . . . . . . . . . . . . . . . .

73

3.11 Sequence for catch point optimization . . . . . . . . . . . . . .

75

3.12 Deformation of a symmetric linear potential by gravity . . . .

76

3.13 Effects of gravity on clouds in linear potentials . . . . . . . . .

77

3.14 Optimization of catch point for |1, −1i-atoms . . . . . . . . .

78

3.15 Optimization of catch point for |2, +2i-atoms . . . . . . . . .

80

3.16 Sequence for ramp speed optimization. . . . . . . . . . . . .

81

3.17 Effects of non-adiabatic compressing of the magnetic trap . . .

83

3.18 Adiabatic compression of a magnetic trap

. . . . . . . . . . .

86

3.19 Definition of lab coordinate system and path of the transport .

87

3.20 Transport optimization . . . . . . . . . . . . . . . . . . . . . .

89

3.21 Motion profile of the first path of the transport . . . . . . . .

90

3.22 Pressure dependence of transport efficiency . . . . . . . . . . .

93

3.23 Lifetime in the science cell . . . . . . . . . . . . . . . . . . . .

95

3.24 Overall efficiency of the preparation sequence. . . . . . . . .

96

3.25 Determination of the rethermalization rate . . . . . . . . . . .

99

3.26 Measurement of rethermalization time . . . . . . . . . . . . .

100

A.1 lin ⊥ lin-cooling . . . . . . . . . . . . . . . . . . . . . . . . . .

112

B.1 Vignetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

B.2 Calibration of the analog channels . . . . . . . . . . . . . . . .

119

C.1 Sequence for overall efficiency tests . . . . . . . . . . . . . . .

120x

List of Tables

3.1

Adiabatic compression . . . . . . . . . . . . . . . . . . . . . .

xi84

Acknowledgements

Two weeks before my return home, I am looking back on an interesting year
in my life, which will remain in my memory mostly for the work in Dominik
Schneble’s group. I want to express my gratitude to him first and foremost.
After three years of studies at my home school TUM that were characterized bytheoretical teaching, homework, and exam-preparation rather than research,he gave me the opportunity to conduct concrete experimental work for the firsttime in my life as a physicist. Not only to execute the routine work in the lab,
but also to develop and propose my own ideas was an especially motivating
experience. It was also Dominik who convinced me to write this thesis, which
had not been my initial intention, but yet definitely a good decision. He isa great teacher with an astonishing ability to give descriptive explanations ofcomplex phenomena and an almost inexhaustable patience. Knowing both theAmerican and the German academic system, he has also been an particularly
valuable advisor. Most important, however, his uncomplicated, pleasant, andhumorous personality made it fun to work in his lab, even in nights when thelasers were not turned off before dawn.
I also want to thank the other students in the group: David Sproles,
who started with Dominik two years ago and set up the laser and the vacuum

system, taught me the basic operation principles in my first weeks. Rebekah
Schiller proof-read large parts of the manuscript of this thesis. She has beena source of good mood for the entire lab, unflagging in mounting and aligning
optics, and explained to me what is so special about being a Southerner. She
has become a very good friend. Daniel Pertot designed the subtle electronics
for the TOP trap and is the expert for electronic issues. He tracked down andeliminated all kinds of frequency noise. Now finished, his TOP coils work perfectly. Daniel Greif, who started working in the group in January, reduced myconsiderable work load from the fall term and developed the radio-frequency
evaporation. He enriches the physical reflections of all group members withchallenging questions he encounters during his study of the literature, andgave me his notes from the class on General Relativity which I often missed
due to nights spent in the lab. Azure Hansen was always a helping hand forthe entire group and shares my passion for mountains. I was often amazed by
the wide knowledge of physics she has already acquired at her young age andwish her all the best for her graduate studies.
We did not achieve Bose-Einstein condensation during my stay, but I
am certain that the group will succeed very soon and thus open the way forthe actual experiments with optical lattices. During my time in the group, we
had to struggle with frequent recurrences of bad luck: as many as four laser
diodes had to be replaced within a period of eight months. The last one diedon the night when Rebekah, Daniel, and Dominik had very likely been justone or two steps away from condensation. I am convinced, though, that thegroup’s misfortune is now depleted and the progress will be much faster in thenext year.
I am also grateful for all teachers that I had at Stony Brook and wantto mention especially Hal Metcalf whose lecture on laser cooling helped myunderstanding of the lab work a lot.
The Fulbright program supported my exchange year at Stony Brook witha generous scholarship.
Vor allem möchte ich aber meiner Familie und besonders meinen Eltern danken. Ohne ihre Unterstützung, ihren Glauben an mich und ihren
Zuspruch in persönlich schwierigen Zeiten wäre die Anfertigung dieser Arbeit
nicht möglich gewesen. Ihnen ist sie deshalb gewidmet.

Chapter 1
Introduction

In 1924 the Indian physicist Satyendra Nath Bose derived the spectrum ofPlanck’s blackbody radiation by purely statistical arguments using the principle of indistinguishability of radiation quanta. He sent his article to Albert
Einstein, who translated it to German, published it [1], and extended the workwith considerations about the properties of the monatomic ideal quantum gas[2–4]. He predicted that below a certain (very low) critical temperature Tc ,a considerable (“macroscopic”) fraction of the gas atoms would gather in thecommon lowest energy level. The gas would undergo a phase-transition. Thisphenomenon became known as “Bose-Einstein condensation” (BEC).
Einstein’s derivation only relates to non-interacting particles. It had
been doubted that Bose-Einstein condensation could occur in reality becauseof the interaction between particles in any physical system. It was soon realized, however, that superfluidity in liquid helium, a strongly interacting system, discovered in 1937, could be understood with the theory of Bose-Einsteincondensation. The phenomenon of superconductivity can also be regarded as1

Bose-Einstein condensation of coupled electron pairs, so-called Cooper pairs.
In dilute gases, the first Bose-Einstein condensates were observed in 1995,
disproving the assumption that gases would solidify long before reaching theconditions for condensation. Three competing research teams were successful in that year: the group of Carl Wieman and Eric Cornell in Boulder [5]with the element rubidium, the group of Wolfgang Ketterle at MIT (sodium)[6], and the group of Randall Hulet at Rice University (lithium) [7–9]1 . The
years following the observation of Bose-Einstein condensate in gases in 1995
were marked by extensive research on a great variety of fascinating properties
Bose-Einstein condensates exhibit, such as observation of interference between
two condensates [10], the realization of the first “matter laser” [11–13], phasecoherent amplification [14, 15], vortices [16, 17], the observation of Feshbachresonances, BEC in optical lattices, and the study of strongly correlated systems [18–20].

1.1

Bose-Einstein condensation - some quantum statistics

In this section we will summarize briefly the results of the statistical theory
of Bose gases. For a more detailed treatment the reader is referred to thestandard literature, e. g. [21, 22].
In a system of non-interacting bosons the average occupation f (k ) of a state1

The analysis of 1995 was corrected in 1997: the originally reported number of 2 × 105
condensed atoms was corrected to 1000. The conclusion that BEC had been observed,
however, remained unchanged.

2

with single particle energy k is:f (k ) =1eβ(k −µ) − 1

,

(1.1)

where β = kB1T , with T the temperature and µ the chemical potential. The
total number of particles is then just given by summation of the above expression over all single particle states k, which can - for small energy spacings - beexpressed as the integral:Z ∞N (T, µ) =
0

g()
d ,
eβ(−µ) − 1(1.2)

where it was necessary to multiply by the density of states g() in order to gofrom summation to integration properly. If the particles of mass m are confinedin an anisotropic harmonic potential U (x, y, z) = 12 m ωx2 x2 + ωy2 y 2 + ωz2 z 2
as in most BEC experiments, the density of states, for example, is g() =√1 2, where ω := 3 ωx ωy ωz . For a gas confined in a box with volume V ,2 ~ω2m 3/2 √1g() is (2π)
. Bose-Einstein condensation occurs for a fixed atom
2V
~2number N at the critical temperature Tc , where the chemical potentialµ(T ), implicitly given by equation 1.2, becomes zero. The integral for thispoint µ(Tc ) = 0 can be expressed as:N(2πmkB Tc )3/2=
ζ(3/2) for the gas in the boxV(2π~)3

3

(1.3)

and

N=

kB Tc
~ω

where ζ(s) =P∞

3ζ(3) for the gas in the harmonic potential,

(1.4)

1
n=1 ns is the Riemann Zeta-function, ζ(3) ≈ 1.202, ζ(3/2) ≈ 2.612.

Defining the thermal de Broglie wavelength λT as

λT = √

2π~
2πmkB T,

(1.5)

we can give a criterion for BEC (for the gas in the box):
nλ3Tc ∼
= 2.612 ,(1.6)

with n = N
the (number) density. The product nλ3T is also called “phaseV
space density” D. For a harmonic potential2 , a straightforward calculation
gives the fraction of condensed particles from equation 1.23 :
N0
=1−
N



T
Tc

3
for T < Tc

,

(1.7)

where N is the total number of particles and N0 the number of condensed
particles. The results in equations 1.6 and 1.7 give criteria for condensationthat are very descriptive: BEC occurs at a critical temperature Tc when the2

For the gas in the box, the calculation is analog and the exponent in equation 1.7 is 3/2.
Because g( = 0) = 0, the integral does not count particles in the ground state, so for a
macroscopic occupation of the ground state N0 , it only gives the number of atoms in excitedstates Nex = N − N0 , the “uncondensed fraction”. By calculating, however, N with theparameters (T = Tc , µ = 0) and Nex with (T < Tc , µ = 0) separately, one retrieves thenumber of condensed atoms by subtraction: N0 = N − Nex .
34

thermal wavelength λT , i. e. the atoms’ de Broglie wavelength associated with
their thermal motion, becomes comparable to the typical distance between
particles (V /N )1/3 . The more the gas is cooled under the transition temperature Tc , the more the single particle wavefunctions will overlap with eachother. At T = 0, they form one big wavefunction which describes all of them.
The particles have lost their “individuality” and become a pure condensate.
The fact that BEC in gases is a metastable phenomenon (with respect
to the solid state) dictates the necessity of very low densities. In alkali gases,
they should not exceed ∼ 1014 cm−3 . This, in turn, results in extremely low
critical temperatures: the typical range is some hundred nanokelvins to a few
microkelvins. It was soon clear that this temperature regime could not bereached with traditional cooling methods. Instead, the realization of BEC inalkali gases is intimately linked with the emergence and development of laser
cooling in the 1980s [23], an entirely new field of atomic physics on its own.
But even the astonishing capabilities of laser cooling mechanisms alone are
not sufficient to produce the temperatures and densities needed. The final key
for the realization of BEC was surprisingly simple: evaporative cooling, first
proposed by Hess [24], could lower the temperature to and below the criticalpoint. This mechanism is just what everybody knows from daily-life experience about the cooling down of a hot cup of tea: the hottest molecules in theliquid have so much thermal energy that they are able to escape it, each ofthem carrying away higher-than-average thermal energy. Thus, the remaining
molecules rethermalize at a new, lower average energy. The necessary confinement of the atom cloud in a conservative potential during this final cooling5phase is provided by magnetic or optical trapping, which is the third major
tool for successful condensation.
In the following section we will give a quick overview on the different
experimental methods that successfully led to BEC. In the next chapter, we
will then explain our own apparatus, which employs many standard techniques,but also exhibits some novel technological aspects.

1.2

Different designs

Almost all BEC machines have the combination of laser cooling, magnetic
trapping, and evaporative cooling in common. Laser cooling (and trapping)
reduces the temperature to a regime where the atoms can then be magneticallytrapped. Once confined in a magnetic trap, the atoms are further cooled byevaporation. Different machine designs are different answers to some typicalchallenges that every new BEC experiment faces: one of them is the different
requirements of laser cooling and evaporative cooling on pressure: evaporative
cooling is a lengthy process and thus requires a very low gas pressure in order to
not lose the magnetically trapped cloud by collisions with the background gas
before condensation occurs. A common technique used for pre-cooling, the socalled vapor cell magneto-optical trap (MOT), needs, however, a significantly
higher background pressure in order to cool and trap a considerable amountof atoms. The operation principle of a MOT, first demonstrated by Raab etal. [25], will be discussed in more detail in section 2.4. For the purpose ofthis overview, the MOT may be described as a laser cooling mechanism in the6presence of a quadrupole magnetic field. The optical forces, in combination
with the field, not only slow down, i. e. cool the atoms, but also trap themat the minimum of the magnetic field. Alternatively to the method describedin this thesis, the aforementioned pressure issue, peculiar for the vapor cell
MOT, can be avoided by loading the MOT not from a vapor, but from a slow
atomic beam, produced by a Zeeman slower [26].
Another concern is the choice of the field geometry of the magnetic trap,
for it turns out that certain traps, namely those with a vanishing magnetic fieldat the “bottom of the trap” (the local minimum of the trapping potential),
have a “hole” at exactly that point: in a magnetic trap only atoms in certain
magnetic hyperfine states (“trapped spin state”) experience a confining potential while for atoms with an opposite spin quantum number the potential isrepelling. The degeneracy of those states at zero field is removed in a magnetic
field by the well-known Zeeman effect. Due to the possibility of spin flips tountrapped states (“Majorana losses”) in regions of negligible fields, magnetic
traps with a field zero at their bottom “leak out”. This is not dramatic for thetemperatures achieved by laser cooling, for which the relative size of that holewith respect to the total size of the cloud is small. However, the losses become
severe during evaporative cooling. In fact, it is not possible to get even close
to BEC with this obstacle. A typical example for traps with this problem isthe magnetic quadrupole field (section 2.7).
The atoms are usually evaporated by applying radio frequency (RF),which can flip the spin state of an atom from a trapped to an untrappedstate. Atoms in an untrapped state are repelled from the trap region and7thus removed from the rest of the trapped atomic cloud. In an inhomogeneous
magnetic field, as in a magnetic trap, the energy splitting of the spin states
is position-dependent. This allows to tune the RF so that only the outmost,i. e. hottest atoms are flipped. Evaporative cooling is then executed by slowly
ramping down the frequency of the “RF knife”.
The first BEC realized in Boulder [5] used a rubidium vapor cell magnetooptical trap (MOT). The atoms pre-cooled by the MOT were (after a compression of the MOT, see section 3.4) trapped with a quadrupole trap. Majorana
losses were eliminated by adding a rotating bias field to the quadrupole field.
This invention became known as TOP (“time-orbiting potential”) trap [27].
The setup characterized in this thesis uses this scheme as well4 . Evaporation was then performed by means of RF. Because the atoms were evaporated
in the vapor cell itself, the observed BEC was small compared to the later
experiments because of the pressure limitations described above.
The MIT experiment [6] took a different approach to BEC in some aspects. A Zeeman slower was used to load the atoms into a “dark SPOT”(“spontaneous-force optical trap”) [28], a modification of the MOT. In thisway, the problems of a vapor cell were avoided and the dark SPOT provided
sufficiently high densities as a starting point for evaporation. In this experiment, the hole in the magnetic trap was quite literally “plugged” by focusing alaser beam to the center of the trap. The resulting optical dipole force repelled
the atoms from the trap center.
A method used to solve the pressure dilemma in the vapor cell was the4

By the time this thesis was written the TOP trap had been designed, constructed, andextensively characterized, but not yet operated.

8

invention of the double MOT [29, 30]. This approach uses two MOTs: thefirst (“dirty” MOT) is placed in a chamber of the vacuum system where it
can quickly load from an atomic vapor. By multiple transfer of the trapped
atoms, the second (“clean”) MOT, installed in a region with a pressure morethan one order of magnitude lower than in the first chamber, is loaded. Also,
the quadrupole field was replaced in many experiments by more complicatedcoil geometries, described in reference [31], which have a non-vanishing magnetic field at the potential minimum. Among them, the Ioffe-Pritchard, first
implemented in reference [25], is the most prominent type. A variant of it wasused for BEC by the MIT group as early as 1996 [32]. The first condensate ina pure Ioffe-Pritchard trap was demonstrated in 1998 [33].
BEC has also been demonstrated without the use of magnetic trapping.
Instead, after conventional cooling with a MOT, the atoms were trapped by
two intersecting high power laser beams, employing the intensity dependent
optical dipole force [34]. The trapped cloud was then evaporatively cooledby lowering the intensity of the lasers, thereby decreasing the height of thepotential and not overly reducing the trap stiffness. The method was called“all-optical” BEC although this term is misleading, because the experiment
still employed a MOT, which does use a magnetic field. The final cooling was
also conducted by an evaporation procedure, and not by laser cooling, whichmight be suggested by the term “all-optical”.
Another innovation was the production of a BEC on a microelectric chip
[35, 36]. Here, the magnetic trap consists of fields generated by lithographic
conductors on the chip and external bias fields. The advantage of this ap-

9proach is the realizability of much higher magnetic field gradients than in theconventional scheme, leading to better confinement and faster rethermalization during evaporation. The entire evaporation stage on a microchip lasts
about 1 s, a factor of 20 less than in standard magnetic traps, and the requirements on background pressure are thus less stringent. The experiments also
demonstrated the transport of a condensate along an “atomic conveyor belt”
on the chip [35].
The idea of the double MOT design, namely spatial separation of thecapture and laser cooling part from the evaporation part can also be achievedby transport of magnetically trapped clouds. This was realized in two variants:
a chain of quadrupole coils was built to guide atom clouds from a MOT to
the magnetic trap in which evaporation was executed [37]. A second approachis the mechanical transportation of cooled and trapped atoms just with one
quadrupole trap that is mounted on a translation stage [38]. The trap movedby the stage can thus freely travel through the vacuum system. The design of
our experiment is inspired by this idea.

10Chapter 2
Description of the apparatus and theory of theemployed techniquesThis chapter will give an overview of our BEC apparatus. The apparatus isdesigned to produce condensates from the 87 Rb-isotope of the element rubidium. This overview will be followed by explanations of the theory and thebasic principles of the employed cooling and trapping techniques intended fora reader unfamiliar with this field.

2.1

Overview

Figure 2.1 shows a schematic view of the apparatus. The two core parts are thevapor cell (or “MOT cell”) and the science cell (“glass cell”). This realizes aspatial separation of the laser-cooling (vapor cell) and the evaporative cooling(science cell) stages of the experiment in regions with suitable pressures. The
vapor cell consists of Pyrex glass and the science cell of quartz glass. The
shape of the vapor cell is cylindrical with an outer diameter of 5.7 cm. Its
11

axis is oriented in the x-direction1 . The science cell is cuboidal. Its width(y-direction) is 1.0 cm and its height (z) 2.0 cm (inner dimensions). Both cells
are placed in different regions of a vacuum system, which is pumped by two
ion pumps, one for each region. The regions are connected by a differential
pumping tube (“thin tube”). The vapor cell is characterized by a non-rubidium
pressure of ∼ 2×10−10 torr and typical rubidium pressures of 109 -108 torr while
the pressure in the ultra-high vacuum science cell is less than 5×10−12 torr. We
will sometimes refer to the transfer chamber between the differential pumping
tube and the science cell as “square chamber”. The vacuum system is to betreated in more detail in reference [39]. Connected to the vapor cell is therubidium oven, which provides the rubidium gas in the vapor cell. The ovencan be manually opened and its temperature is controllable.
Up to 1.2 × 1010 Rb atoms are captured, laser-cooled, and trapped witha magneto-optical trap (MOT) directly from the vapor (section 2.4) in a timeof ∼ 10 s. At this point, the cloud has a temperature of a few hundred microkelvins. The magnetic field is generated by a fast-switching quadrupole
magnetic trap (section 2.7). Residual magnetic fields, such as Earth’s magnetic field, are eliminated by three bias field coils (not shown). Subsequent
to the MOT stage, a sudden compression of the MOT (section 3.4) within∼ 500 ms increases the density of the atomic cloud. This stage is immediately
followed by laser cooling via optical molasses (section 2.5) for ∼ 5 ms, which
lowers the temperature to ∼ 25 µK, a temperature that clearly indicates subDoppler cooling mechanisms (section 2.6). Optical pumping (section 3.6) for a
1

See the figure for definition of the lab coordinate system.

12

13
CCD

n

io

imaging optics

m
pu

p

quadrupole coils

lens (1’’ diameter)lens (2’’ diameter)
beam elevatorfiber
outcoupler
mirror (1’’)
mirror (2’’)

Legend:1

Rb oven

Rb

RepumpingLesker
GMPS075F1B

differential
pumping tube

“squarechamber”

Ti-subl.
pump

pump beamtranslational degrees of freedomzy lab coordinate system(not at origin)x

imaging beam

TOP trap

10mm (W) x 20mm (H)science cell (”glass cell”)

imaging/pump lightfrom cycling laser

ion pump

vapor cell(”MOT cell”) linear translation stages

Cycling

MOT/MT87

0.5m

λ/2-plate
λ/4-plate
shutterlaser beam (MOT height)
laser beam (table height)
polarizing beam splitter

3B

er 0F
sk 15
Le MPSG

Lesker
GMPS-SPL024

Figure 2.1: The apparatus: the figure shows a schematic drawing (to scale) of theparts of the apparatus that are relevant for the discussion of this thesis. The MOT
optics and laser system, however, are not shown in this figure. The apparatus also
includes: coils for bias fields, cooling hoses and holders for the quadrupole coils,
various stages for optical installations etc.

few milliseconds then increases the number of atoms in magnetically trappableatomic states. The cloud, having been prepared thusly, is then caught in amagnetic trap (section 2.7) produced by the same quadrupole coils that create
the magnetic field for the MOT. We can catch 1-3×109 atoms in the magnetic
trap. Immediately after the catch, the temperature of the cloud is ∼ 80 µK.
A quasi-adiabatic compression in the magnetic trap raises the temperature
to ∼ 350 µK. The quadrupole coils are mounted on a mechanical translation
stage that can freely travel in the x and y-directions, allowing movement of
the trapped atoms through the vacuum system. The transport of the trapped
cloud already starts during compression of the trap and guides the atoms ona z-shaped path to the UHV region. We can transfer the atoms with an efficiency of ∼ 70-80 % in ∼ 3.5 s from the vapor cell to the science cell (section
3.8), where the evaporation will then be performed. The TOP trap for theevaporation has already been designed and constructed, but not yet operated.
We do not use an additional magnetic trap in the science cell: the TOP trap
just consists of the same quadrupole coils used for the MOT and the transport
and two smaller coil pairs that create the rotating bias field. The TOP trap
will not be described in this thesis.

2.2

Rubidium and laser system

The set of elements which can be Bose-condensed is limited by some constraints: 1) Needless to say, they must be bosons. 2) Laser cooling must
(generally) be possible, which limits the choice to alkali, alkaline-earth atoms,
14and some metastable noble gases. 3) High-power laser light must be available.
4) The element must have a favorable ratio of so-called “good” collisions, which
support rethermalization, to “bad” collisions, which cause trap losses. Becauseof these limitations, BEC was originally pursued in alkalis and spin-polarizedhydrogen (which is an exceptional case, for it is not laser-cooled). After theobservation of BEC in 87 Rb [5], 23 Na [6], and 7 Li [7] in 1995, it was also reported in atomic hydrogen [40], 85 Rb [41], 41 K [42] 2 , metastable 4 He [43, 44],133

Cs [45], 174 Yb [46], and 52 Cr [47]. For this experiment, rubidium was chosen.

It can be cooled with low priced diode lasers (instead of expensive dye lasers),
and it also exhibits favorable properties for the optical lattice experiments thatwill be performed in this research group.
Laser cooling relies on the scattering of photons by the atoms that are
to be cooled. In the simplest picture, atoms moving in a certain direction
are slowed down, hence cooled, by a bombardment of photons traveling in the
opposite direction. The photons come from a laser that is tuned to a transition
frequency of the atomic level structure. They are absorbed by the atom, whichconsequently goes to an excited internal state. The momentum of a photon
and λ the laser light wavelength3 , is transferred to the atom~k, with k = 2π
λby a first kick slowing it down. The atom returns to its internal ground stateby spontaneous reemission of the photon in a random direction. To conserve
momentum, the atom experiences a second recoil kick in the direction opposite
2

BEC in 85 Rb and 41 K was indirectly obtained by evaporation of (87 Rb), which was inthermal contact with those gases during the process, a method called “sympathetic cooling”.
3
Definitions of abbreviations with a global meaning in the context of this thesis, such asλ, may not be repeated every time they appear. Instead, the reader can find a list of thoseabbreviations in appendix D.

15

to that reemission. Because of the random nature of this second recoil kick,its net effect after many scattering events is zero while every absorption event
slowed the atom down by ~k/m in its defined direction of travel. It can also
be easily shown that the energy of the emitted photons is higher than thatof the absorbed photons due to the Doppler effect. The energy difference is
supplied by the kinetic energy of the atoms. Thus, the atom’s velocity in its
direction of travel is reduced.
To slow down atoms from room temperature to close to 0 K, tens of
thousand scattering events are necessary. For this reason, one needs a closed
transition within the atomic level structure, i. e. the atom needs to return
after every excitation to the same ground state it came from. Atoms that havefallen into another ground state are lost because the laser light can only couple
one specific ground state to one specific excited state. Rubidium (as all alkalis) provides such a closed transition. Figure 2.2 shows the D2 -line of the levelstructure of 87 Rb. Important data for 87 Rb is found in appendix D. We use the
5 2 S1/2 , F = 2 → 5 2 P3/2 , F0 = 3-transition to cool the atoms. Because of theiterative nature of the process, it is usually called the “cycling transition”
and the laser used is referred to as cycling laser. This is a closed transition
because of the selection rules for angular momentum. For the hyperfine quantum number F only ∆F = 0, ±1 is allowed in a transition. Thus, atoms in
the F 0 = 3-state can only decay to the F = 2-state. Hence, the cycling laser
is tuned to that frequency. The cycling laser in our experiment is an external cavity diode laser built by Sacher Lasertechnik with an output power of∼ 1000 mW. It is frequency-stabilized with a polarization spectroscopy locking

16

F=3

gF = 2/3

F=2

gF = 2/3

F=1
F=0

gF = 2/3

266.650(9) MHz

5 2P3/2

72.9113(32) MHz
156.947(7) MHz
72.218(4) MHz

Cycling

Repumping

780.241 210 nm
384.230 484 GHz
1.589 049 eV

F=2

gF = 1/2

F=1

gF = – 1/2

2.563 006 GHz

5 2S1/2
4.271 677 GHz

Figure 2.2: Relevant part of the level structure of 87 Rb (D2 -line): the cycling F =2 → F 0 = 3 and the repump transition F = 1 → F 0 = 2 are marked. The drawing
is not to scale. The data is taken from reference [48].

technique described in reference [49]. Detuning from the locking frequency andpower can be controlled with acousto-optical modulators. A small amount of
the laser power is used for absorption imaging (chapter 3) and optical pumping (chapter 3). The cycling light is coupled into an optical fiber which guides
it to the (separate) table with the vacuum apparatus and the MOT optics,
for which ∼ 250 mW of laser power are available. The laser system and thelocking techniques will be described in more detail in reference [39].
17The cycling transition is not totally closed, however. Because of the small energy spacing in the 5 2 P3/2 between F 0 = 3 and F 0 = 2 of ∼ ~ × 2π × 267 MHz,
atoms can also be excited to the F 0 = 2 by the cycling laser with a small, but
considerable probability. From there, they can not only decay to F = 2, butalso to F = 1, where they are lost for the cooling process. For this reason one
needs a second laser which is commonly called the “repumper”. It is tuned
to the 5 2 S1/2 , F = 1 → 5 2 P3/2 , F 0 = 2-transition and excites the atoms to theF 0 = 2-state so that they can decay back to the F = 2-state, thus bringing
the “lost” atoms back into the cooling cycle. The repumper in this experimentis also an external cavity diode laser, made by Toptica Photonics. Its output
power is ∼ 120 mW. It is frequency-stabilized with the same method as thepumping laser. Its light runs directly to the vapor cell.

2.3

Atom-light interaction and laser cooling a very short introduction

The next paragraphs will be devoted to some basic aspects of the theory of
atom-light interaction which are necessary to understand the following sections
on magneto-optical trapping, optical molasses, and sub-Doppler cooling. Most
importantly, we will touch on the Rabi two-level problem, which serves as amodel in many descriptions of laser cooling mechanisms. Complete derivationsof the results are beyond the scope of this thesis and will not be given. Rather,
we will point out the important steps and the usual assumptions made in thederivations. We will also neglect all phenomena that do not directly relate to18

the techniques used in this experiment. The section follows largely the bookby Metcalf and van der Straaten [23]. An exhaustive theoretical treatment of
the subject can be found in reference [50].

2.3.1Time-dependent perturbation theory and two-levelproblem

In order to describe the interaction with the light field, we assume an atomic
Hamiltonian H that can be decomposed into a time-independent part H0 describing the internal atomic structure and a time-dependent perturbation H0 (t)that contains the interaction with the (classically described) radiation field.
The Schrödinger equation for the atom including the interaction with the lightfield is then:Hψ(~r, t) = (H0 + H0 (t)) ψ(~r, t) = i~∂ψ(~r, t)
∂t

(2.1)

Since the eigenfunctions φn (~r) of H0 form a complete set, we can make thefollowing ansatz for the solution of equation 2.1:

ψ(~r, t) =X

ck (t)φk (~r)e−iωk t

(2.2)

k

where Ek = ~ωk are the eigenvalues (so the unperturbed energy states) of H0 .
This is the standard approach in time-dependent perturbation theory foundin all quantum mechanics textbooks covering the subject. The ansatz leads to19

coupled differential equations of the form:i~dcj (t) X0=(t)eiωjk t
ck (t)Hjk
dt
k

(2.3)

0
(t) := hφj |H0 (t)| φk i and ωjk := ωj − ωk . Here, the first of threewhere Hjk

major approximations is made: the atom is treated as a two-level system
consisting only of a ground (g) and an excited (e) level, which is a reasonableassumption for an atom in a narrow-band laser field. The equations 2.3 thenlead to:dcg (t)0= ce (t)Hge(t)e−iωa tdtdce (t)0i~= cg (t)Heg(t)eiωa t
dt

i~

(2.4)

where ωa := ωe − ωg is the atomic resonance frequency. The time dependent
part of the Hamiltonian H0 (t) for a single electron in an electromagnetic fieldis~ r, t)~r ,H0 (t) = −eE(~(2.5)~ r, t) the electric field of the light at thewith e the elementary charge and E(~
position of the atom. At this point, two other assumptions are generally
made: the rotating wave approximation (RWA) neglects terms of the order
1/ωl compared to 1/δ where ωl is the laser frequency and δ is the detuningof the laser from resonance: δ := ωl − ωa . For a laser tuned close to theatomic resonance frequency ωa , this is again an adequate assumption. The
third important approximation is the electric dipole approximation. It

20states that the spatial variation of the light field over the the relevant regionis negligible. This approximation is also valid for the case discussed; thewavelengths in this context are usually in the range of ∼ 100 - 1000 nm whereas~ r, t) = E0 ˆ cos(kz −the typical size of an atoms is ∼ 1 nm. For a plane wave E(~0(t) of the time-dependent Hamiltonian can beωl t), the matrix element Heg0(t) = ~Ω cos(kz − ωl t), where the Rabi frequency is defined
expressed as Heg

by
Ω=

−eE0
−eE0
hφe |ˆ~r| φg i =hφe |r| φg i
~
~

4

.

(2.6)

With these approximations and some manipulation equations 2.4 become:d2 cg (t)dcg (t) Ω2−iδ+cg (t) = 0
dt2
dt
4dce (t) Ω2
d2 ce (t)+ iδ+cg (t) = 0 .
dt2
dt
4

(2.7)

These equations can be solved analytically. They yield oscillating probabilities
for the atom to be in the ground or in the excited state. The frequency of√these oscillations is Ω0 := Ω2 + δ 2 , the generalized Rabi frequency.

2.3.2

Spontaneous emission and optical Bloch equations

As pointed out in section 2.2, spontaneous emission plays a crucial role in(conventional) laser cooling schemes, for it provides the necessary dissipation.
To include spontaneous emission in the treatment of an ensemble of two-levelatoms, one has to make a transition to a statistical description because of the4

The last equality only holds for the electric dipole approximation, which assumes thatthe atom dipole moment e~r aligns with the polarization ˆ.

21

incoherence of the system. This is done by the introduction of the density
matrix ρ [21]ρ=ρee ρegρge ρgg

!
.

(2.8)

The equations 2.7 then lead to equations for the elements of the density
matrix, for example:
Ω∗
Ω
dρgg
= i ρ̃eg − i ρ̃ge
dt
2
2

(2.9)

where5 ρ̃ge := ρge e−iδt . Spontaneous emission is introduced by an exponential
decay of ρeg :dρeg
Γ= − ρeg + induced effects .
dt
2(2.10)

This leads to the so-called optical Bloch equations:
dρgg
dt
dρee
dt
dρ̃ge
dt
dρ̃eg
dt

i ∗
(Ω ρ̃eg − Ωρ̃ge )2i= −Γρee + (Ωρ̃ge − Ω∗ ρ̃eg )
2

Γ
i
= −
+ iδ ρ̃ge + Ω∗ (ρee − ρgg )
2
2


Γ
i
= −
− iδ ρ̃eg + Ω (ρgg − ρee ) .
22= +Γρee +(2.11)The optical Bloch equations can be easily solved for the steady-state where
the time derivatives on the left hand sides of equations 2.11 are zero. One

5

These equations are derived by assuming the relation of the density matrix of a purestate to the coefficients of the corresponding wavefunction ρeg = ce c∗g etc. In reference [51]however, it is shown that spontaneous emission and the effects of the light field on the atomfrom equations 2.7 can be treated separately, thus justifying the next step.

22

finds:
ρee =

s0 /2
1 + s0 +


2δ 2
Γ

.

(2.12)

This is the population ratio of the atoms in the excited state to the atoms in the
ground state in an ensemble of atoms. Here, we use the common saturation
parameter s0 which is defined ass0 :=2 |Ω|2I=
2
ΓIs(2.13)

where I is the intensity of the incident light beam and the saturation intensity Is is defined asIs =2π 2 ~c3λ3 τwith τ =1
Γ

.

(2.14)

This gives us finally the total scattering rate Γsc , which is simply the productof the population in the excited state multiplied by the decay rate of that state:Γsc = Γρee =s0
Γ
2 1 + s0 +


2δ 2
Γ

.

(2.15)

This result is important because of its direct applicability for the dissipative(i. e. cooling) effects of laser light: it gives the rate by which the atoms are
slowed down by portions of ~k/m. In the limit of weak intensities (s0  1)equation 2.15 becomesΓsc =Γ I Γ2 /4
2 Is Γ2 /4 + δ 2

23.

(2.16)

As a function of the detuning δ, this equation describes a Lorentzian with
FWHM = Γ, the natural linewidth of the transition. For high intensities(s0  1), Γsc saturates to Γ/2 on resonance. For intensities comparable toor greater than Is , s0 cannot be neglected with respect to 1, and Γsc (δ) is a√
Lorentzian with FWHM = Γ 1 + s0 :

Γsc =(1 + s0 )Γ2 /4
Γ s0
2 1 + s0 (1 + s0 )Γ2 /4 + δ 2.

(2.17)

This is the well-known effect of power-broadening: in a Doppler-free spectroscopy technique, spectral lines are broadened with respect to their naturallinewidth Γ if too much laser power is applied.
With these results, the scattering cross section of an atom σsc can beobtained with the relation Iσsc = ~ωΓsc . For low intensities, the dependence
on I drops out and one finds the important result that the scattering crosssection for atom light interaction is on the order of λ2 .

2.3.3

Light shifts

The interaction Hamiltonian H0 (t) also effects shifts of the energy eigenstates
Eg and Ee . After a suitable manipulation of the Hamiltonian that offsets
the energy of the excited state by ~δ (cg (t) → cg (t), ce (t) → ce (t)e+iδt ) andapplication of the rotating wave approximation, equations 2.4 become time24

independent and the new interaction matrix H̃0 can be written as6 :
~
H̃0 =
2

−2δ Ω
Ω∗ 0

!
.

(2.18)The eigenvalues λe,g of this matrix are:

λe,g = (−δ ∓p~δ 2 + |Ω|2 )2.

(2.19)

Expansion for low intensities (δ  |Ω| ∝ I) yields (after reversal of the manipulation of the Hamiltonian):∆Eg =|Ω|2 ~,
4δ

∆Ee = −|Ω|2 ~
4δ

.

(2.20)

These energy corrections are called light shifts. In a strong light field, theyδ ~|Ω|. For negative detunings (δ < 0), the energy levels areare ∆Eg,e = ± |δ|2

spread apart. An atom in the ground state is attracted to regions of highintensity. An important application of the light shift are optical lattices: in astanding wave the intensity varies spatially with a period of λ/2. Because of
the light shift, atoms in such a wave experience an effective potential and are
attracted to regions of higher or lower intensity depending on the detuning.
The resulting force is called the dipole force. If the scheme is extended
to three dimensions, a regular three-dimensional pattern of spots preferredby the atoms is created, just like in a crystal, and is, therefore, called an
optical lattice. The light shift also plays an important role in the theoretical
6

The notation is such that α |ei + β |gi →

 α 
β .

25

description of lin ⊥ lin-polarization gradient cooling which will be explained
below (section 2.6).

2.4

The magneto-optical trap

Almost all BEC experiments use a magneto-optical trap (MOT) for precooling. Since its first demonstration [25], the MOT has become the workhorseof ultracold atomic physics. It uses the combination of an inhomogeneous magnetic field and the dissipative radiative forces to cool and trap neutral atoms.
Thus, it confines the atoms in both momentum and position space. In thisrespect, it differs from e. g. optical molasses, which will be described in thenext section 2.5.
We will here discuss the simplest one-dimensional case of a hypotheticalatom with total angular momentum Jg = 0 in the ground state and Je = 1
in the excited state. In a linear magnetic field B(z) = bz z ≡ bz, the Zeeman
shift causes a position-dependent splitting of the three magnetic sublevels of
the excited state me = 0, ±1:∆Ee = ωZ (z)~ = gJ µB me B(z) = g1 µB me bz.

(2.21)Such a linear magnetic field can be generated by a quadrupole coil pair. Our
experiment uses for the magnetic trapping the same quadrupole coils as for
the MOT. The quadrupole field will therefore be discussed in section 2.7 aboutmagnetic trapping. Two circularly polarized counterpropagating laser beams
with opposite helicities are directed to the (spatial) region of the local min26

imum of the magnetic field strength. Figure 2.3 shows a schematic view ofthe situation discussed. We assume the laser light to be red-detuned, soδ = ωl − ωa < 0. As explained qualitatively in the figure caption, the setup
is such that the laser beams exert a repulsive force that drives atoms at restto the trap center, as well as a damping force that slows down atoms that are
moving away from the trap center. Quantitatively, we can calculate a force
F of a laser beam on an atom by use of equation 2.15 by multiplying thescattering rate with the momentum transfer per absorption:

F = ṗ = ~kΓsc =s0~kΓ
2 1 + s0 +


2δ 2
Γ

.

(2.22)

If F+ is the force from the σ + -beam and F− the force from the σ − -beam, then
the total force is FMOT = F+ − F− for low saturation. For F+ and F− , we justhave to replace the detuning δ in equation 2.22 by effective detunings δ± thattake the Zeeman effect ±g1 µB bz and the Doppler shift ∓kv into account:

δ± = δ ∓ kv ± g1 µB bz.

(2.23)The total force FMOT can be expanded if the Doppler shift and the Zeeman
effect are small compared to δ. One finds:

FMOT ≈ −βv − κz,

(2.24)

2|δ|s0where the damping constant is β = Γ(1+s8~k
2 2 and the spring constant is
0 +(2δ/Γ) )µB b
κ = g1~kβ. Those two contributions represent the dissipative, cooling and the27Position z

Energy [ħ]E(z)/ħ

ωamag. sublevel
me = +1δδ− + δ
ωlσ+me = 0

Energy [ħ]E(z)/ħ
effective red
detuning

v = 0 σ− Position zz´ > 0effective blue
detuning

+ kv2

− kv1

me = −1σ+mg = 0mag. sublevel
me = +1

a1 > 0v1 > 0a2 < 0v2 > 01

2

me = 0me = −1σ− Position z

mg = 0Figure 2.3: Magneto-optical trapping: the degeneracy of the magnetic sublevels isremoved by the Zeeman effect. A linear magnetic field with B(z) = bz is assumed.
The counterpropagating beams are circularly polarized with opposite helicities asshown. The laser light is red-detuned δ < 0.
The left figure shows an atom at rest at z 0 > 0. It is in the mg = 0-state. Because
both beams are circularly polarized, it can only be excited to the me = 1-state byabsorption of a photon from the σ + -beam or to the me = −1-state by absorption
from the σ − -beam. For z > 0 and δ < 0, the Zeeman effect, however, shifts theme = −1-state closer to resonance and reduces the detuning δ− for the transition
to this state while it increases the detuning δ+ for a transition to me = +1. Thus,
following equation 2.15, absorption from the σ − -beam coming from the right is more
likely, which pushes the atom back to the trap center. For z < 0 the situation is
analog with opposite signs.
The right figure shows an atom at z < 0 with positive velocity v (position 1). Its
finite velocity causes a Doppler shift: it “sees” the σ + -beam shifted to the red by
ωD = −kv1 . The atom is at a position where the Zeeman shift ωZ for me = +1
(δ+ > 0) is so large that the Doppler effect ωD = −kv shifts the atom right inresonance for the σ + -beam. This accelerates the atoms (a1 > 0) and drives it to thetrap center. At position 2 the situation is opposite: the atoms is traveling toward
the relevant beam (σ − ). The Doppler effect is therefore a blue shift ωD = kv2 .
Thus, absorption happens from the σ − -beam decelerating the atoms (a2 < 0). Bothsituations are, however, not totally symmetric: the acceleration for z < 0 causes anincrease of the Doppler shift, which does not correspond to the decrease of δ+ inthe direction of travel. The deceleration for z > 0 decreases the Doppler shift whichaccords with the decrease of (the absolute value of) δ− . Qualitatively spoken, thismeans that the effective range of deceleration is larger than the one of acceleration:
the atoms are decelerated by a larger velocity than they are accelerated. Becauseof that the MOT is actually able to effectively slow down and trap atoms.

28restoring, trapping force in a MOT. The trapping force is also called spontaneous forces because it originates from the spontaneous emission of the atoms.
This simple model can readily be extended to three dimensions so that one
needs a total number of six beams (figure 2.4). If one considers real atoms, themagnetic sublevel structure is more complicated, but it was shown in computer
simulations [25] and – more important, experimentally – that the technique
not only works in the simple example given above, but also in reality.

Figure 2.4: A real MOT consists of a total number of six beams in three pairs, onepair for every direction in space. The beams in a pair are counterpropagating with
opposite helicities (defined in an absolute frame). The schematic figure shows the
six beams and the two quadrupole coils with opposite currents, which generate themagnetic field.

The number of atoms N in a MOT after switch-on is described by a29

simple rate equation:
dN
=R−N
dt



1
1
+
τ1 τ2



Z
−χ

nMOT (~r)2 d3 r.

(2.25)

The first contribution R is the loading rate for an empty MOT. It depends
on the temperature of the vapor (which is usually a constant at room temperature), the mass of the species trapped, the capture volume (the volumeof space where atoms are caught), the capture velocity (the maximum speed
at which atoms can be trapped7 ), and the density nVC of the gas in the vapor
cell. Capture volume and capture velocity are dependent on the operating
parameters of the MOT: magnetic field gradient b, beam diameter, and detuning δ. The second term describes loss mechanisms: the time constant τ1gives the lifetime of an atom in the trap limited by collisions with atoms of
the same species as the trapped gas, so rubidium in our case. It is inversely
proportional to the density of that gas nVC . The time constant τ2 reflects thelosses due to collisions with atoms of other species that may exist in the vapor
τ2. If
cell. Both time constants can be summarized to a single lifetime τ = ττ11+τ
2

the density of the trapped gas nVC is much greater than the density of atomsof other species nal , which is equivalent to τ1  τ2 , then τ is approximately
equal to τ1 , and the influence of other species can be neglected. The third
term takes losses from inner-trap collisions into account. For this reason, it is7

Following an estimate expression from reference [52], the capture velocity for rubidium
is ∼ 20 ms , which means that a MOT collects the atoms from the low energy wing oftheq Boltzmann distribution. The most probable speed of an atom in a rubidium gas is
2kB Tm= 239 ms at T = 25 °C.

30

proportional to the integral over the square8 of the density inside the MOTnMOT . The constant χ is dependent on the nature of those collisions. If theχ-term is neglected, equation 2.25 has the simple solution9 :t

N (t) = Rτ (1 − e− τ ) .

(2.26)

So, the collision constant τ is also the characteristic time constant for theloading of the MOT. Figure 2.5(a) shows the fluorescence of a MOT during
the loading, exhibiting the exponential form of equation 2.26 (see the remark
in the footnote there). It can also be inferred from the solution that the
maximum atom number in a MOT, Nmax , is given by

Nmax = Rτ = R

τ1 τ2
τ1 + τ2.

(2.27)

Even though R can only be estimated in theoretical considerations, this result
has an important implication: for rubidium densities nVC much larger thenthe density of other gases, so nVC  nal or τ ≈ τ1 , the lifetime τ is inversely
proportional to nVC , while the initial loading rate R is (always) proportional
to nVC . This means that the maximum total atom number does not dependon the vapor cell density nVC or, using the ideal gas law p = nkB T , thebackground pressure pVC . In fact, Monroe et al. [54] observed a change ofNmax by only 30 % over a range of two orders of magnitude of pressure. For
8

Two collision partners, therefore the square.
For nMOT (~r) = const. = ns , the solution is the same, but the loss Rconstant τ has tobe replaced by a modified loss constant τs : τ → (1/τ + χns )−1 since χ nMOT (~r)2 d3 r =χn2s V = χns N [53]. This is an adequate approximation for MOTs far in the regime where
the density in a MOT is saturated because of photon reabsorption (see below).
9

31

2
increasing pressure, Nmax develops as a function of pressure p as τ2cτ
where
+c/p

c is some constant relating p and τ1 (see also figure 2.5(b)).

1 0
1 .2

n u m b e r

9n u m b e r N M O T [1 0 ]

9

1 .0
0 .8

0 .4

7
6
5
4
3

0 .2

2

a to mre la tiv e a to m

0 .6

8

0 .0

1
0

- 0 .2
0

2

4

6

8

0

1 0

5

1 0

1 5

2 0

2 5

3 0

3 5

tim e a fte r o p e n in g th e o v e n [m in ]tim e t [s ](a) MOT loading curve(b) Pressure dependence of the MOT

Figure 2.5: MOT properties: the two figures illustrate remarks made in the text
about the MOT. Figure (a) is a measurement of the fluorescence of a MOT duringloading. The fluorescence is proportional to the atom number in the MOT. A
function of the form in equation 2.26 is fitted to the data. This is an appropriate
procedure because we operate our MOT at parameters at which it enters the densitysaturated regime (see below) very soon after the start of the loading, and thus
the remarks in the footnote to equation 2.26 apply. The value for the modified
loss constant (defined in the aforesaid footnote) τs in figure (a) is 1.8 s. In theexperiments (chapter 3), the MOT is usually loaded for 10 s. Figure (b) shows thatfrom a certain background pressure on the maximal atom number in a MOT does
not depend on pressure any more as explained in the text. For the measurement,
the rubidium oven at a temperature of 55 °C was opened so that the Rb pressure
could rise, and the maximal atom number was determined over a time of thirty-fiveminutes. The upper limit for the atom number in our MOT is ∼ 1×1010 depending
on different other conditions. A typical size (4σ) of a MOT is 0.6-1.0 cm.

In the introduction we emphasized the decisive role of the density nfor the critical transition temperature Tc . Because of that dependence, wewant to give a short qualitative discussion on the density in a MOT. One
can principally distinguish three different regimes [55]. In the first regime,
the MOT contains only few atoms and the size of the cloud is limited by32

temperature. Since the repulsive force in equation 2.24 is linear, the MOT
has a harmonic potential. Therefore, we can use the equipartition theorem:
1/2 k

BT

= 1/2 κj hrj i where we want to indicate with the subscript j that the

spring constant κj is not necessarily the same in each direction. If more atoms
are added to the MOT, the size stays the same and the density increases.
This is, however, only true as long as laser beam photons scattered by atoms
inside the trap can leave the trap without reabsorption by other atoms inside
the trap. With more and more atoms in the trap, this reabsorption becomes
more likely, resulting in a repulsive force. The density in the MOT is thenlimited at a value where the repulsive force just balances the restoring force
in equation 2.24. The numerical value of this maximum density ns is on theorder of ∼ 1011 cm−3 for the parameters at which we operate our MOT (seechapter 3). The density has a linear dependence on the spring constant κwhich is exploited in the CMOT stage of our sequence (chapter 3). A nicediscussion of that effect is contained in reference [52]. The repulsive force
by rescattering causes a growth of the size of the MOT if more atoms areadded. Finally, the MOT extends even to regions where the absolute value ofthe magnetic field B is large10 . In those regions, sub-Doppler cooling, a term
which summarizes mechanisms that cool the atoms even under the Doppler
cooling limit (discussion in the next section, equation 2.29), does not work.
Those processes work only close to the magnetic field zero. The cloud thenenters the “two-component regime”, which means that two distributions arepresent in the MOT: one is characterized by a very low temperature, high10“Large” here means that the frequency associated with B, the Larmor frequency ωZ =gJ mJ µB B
, is larger than the scattering rate Γsc .
~33

density, and restricted to a small region around the trap center, the other ishotter, less dense, and extends much farther. The result of a measurementthat illustrates this behavior is found in chapter 3.

2.5

Optical molasses

The spontaneous force of laser light associated with the scattering rate Γsc
can be used in a simple scheme to cool atoms: as in a MOT, the atoms are
exposed to six red-detuned beams, two counterpropagating for each directionin space. Because of the negative detuning, an atom only experiences a large
spontaneous force from a beam if it is moving toward it, for the Doppler effectshifts the atom closer to resonance in that case. In one dimension, the total
force on a atom can be calculated, exactly as in the discussion for the MOT in
the previous section, by subtracting the forces F+ and F− from the two beams
with the only difference that the effective detunings δ± now solely consist of
the Doppler shift ωD,± = ∓kv. This leads to the same friction coefficient as in
the discussion for the MOT:
8~k 2 |δ| s0β=Γ(1 + s0 + (2δ/Γ)2 )2.

(2.28)

A figurative representation of the forces from the two beams and the resulting
force is shown in figure 2.6. It illustrates that the two forces combine to aforce FMol that is damping for a certain region around around v = 0 where it
can be approximated as F = −βv, thus acting like a viscous force. For thisreason this technique was given the name “optical molasses” and its cooling34effect was first observed in three dimensions in 1985 [56].

force [ħk/Γ]

0.4

0.2

0.0

-0.2

-0.4
-6

-4

-2

0

2

4

6

velocity [Γ/k]Figure 2.6: Doppler Optical molasses in one dimension: two counterpropagating reddetuned laser beams result in a damping force for velocities v . δ/k. The parameters
are chosen to be comparable with the situation in our experiment: δ = −3.5 Γ ands0 = 4.

Laser cooling cannot achieve temperatures arbitrarily close to 0 K. With
the example of the one-dimensional optical molasses, we want to discuss11
briefly the limits of laser cooling. With the friction coefficient β, the energycdissipated per unit time, the cooling rate, is given by dW= FMol v = βv 2 .
dt
2However, there is also heating by the recoil energy (~k)
acquired by the atom2mhduring absorption and during spontaneous emission. Its rate is dW
= 2×
dt
2

2 × (~k)Γsc . The two prefactors reflect that there are two beams involved and2mthat there are also two recoil kicks per scattering event. Cooling stops whenboth the heating and the cooling rate are equal: Ẇc = Ẇh . If one solves this11

The following derivation of the Doppler temperature was taken from [23].

35equation for v 2 and plugs the result into 1/2 kB Tfin = 1/2 mv 2 , one recovers an
expression for the final temperature Tfin as a function of δ. This expression
can be minimized, yielding the result for the minimum temperature:

TD =

~Γ
2kB

, where s0  1 was assumed.

(2.29)

This temperature limit is the so-called Doppler temperature limit TD . It
is the limit for a large group of cooling schemes and can be derived in amuch broader context than shown here. It is also the limit for the MOT
mechanism introduced in the previous section. For the D2 -line in 87 Rb, thecycling transition in our experiment shown in figure 2.2, it is 146 µK.
It had originally been assumed that the Doppler limit is the fundamental
limit of laser cooling. The measurement of temperatures significantly below
the Doppler limit [57] was, therefore, a big surprise. In the next section weare going to discuss two models of cooling below TD .

2.6

Sub-Doppler cooling

The observation of temperatures below the Doppler limit triggered the development of a model by Dalibard and Cohen-Tannoudji that could explain theeffect [58]. They identified polarization gradients in the light field as reason
for the observed cooling below the Doppler limit. Polarization gradients occur
when two counterpropagating beams of different polarization interfere. In a
one-dimensional or two-dimensional discussion with two, respectively four interfering beams, polarization gradients can be avoided because the orientations
36

of the electric fields of all four beams can be chosen to be the same. In thethree-dimensional case, this is no longer possible and polarization gradientsare always present. We want to restrict ourselves to the usual discussion of
two special cases in a one-dimensional scheme. The first case is the so-calledlin ⊥ lin-configuration where the two counterpropagating beams are both linearly polarized, but their polarization vectors are perpendicular with respect
to each other. The second case is the σ + -σ − -constellation. Both beams are
circularly polarized, but with different helicities.

2.6.1

lin ⊥ lin polarization gradient cooling

We consider two laser beams traveling along the z-axis in different directions.
√
For simplicity, we assume equal intensities, so equal amplitudes E0 / 2 of theelectric field. The beam traveling in positive direction is linearly polarizedalong the x-axis, the other beam along the y-axis. The electric fields of thetwo light fields can then be written as plane waves:E0~ + (z, t) = √E
ˆx cos(kz − ωt)2E0~ − (z, t) = √ˆy cos(kz + ωt) ,E
2(2.30)

where ˆx and ˆy are the unit vectors in x and y-direction respectively. Super~ t):position of the two beams gives the total electric field E(z,~ t) = E0
E(z,




1
1
√ (ˆx + ˆy ) cos(kz) cos(ωt) + √ (ˆx − ˆy ) sin(kz) sin(ωt).
22(2.31)
37

This result implies the existence of polarization gradients in the electric field:
for z = 0, the light is polarized along the unit vector √12 (ˆx + ˆy ), which weshall call the 45 °-direction. For z = λ/8, the light is right-circularly (σ − )polarized. For λ/4 the light is again linearly polarized, but in the −45 °direction. For 3λ/8, the light is left-circularly polarized, so σ + -polarized, forz = λ/2, the polarization is again linear in 45 °-direction. Between thesevalues it has elliptical polarization. The polarization thus changes within halfa wavelength, which is illustrated in figure A.1 (d) of appendix A.
The changing polarization is reflected in changing coupling strengths fordifferent magnetic substates in the ground level. Again for simplicity, we wantto assume that there are only two of them: mg = ±1/2. Following the discussion
in the section on the two-level atom, different coupling strengths are equivalent
to differing lightshifts. An atom that travels along the propagation direction of
the light beams is thus exposed to a potential rising and falling with a periodof λ/2. That means, however, that it permanently converts kinetic energy into
potential energy and vice-versa. The key mechanism for the cooling effect isnow optical pumping between the two substates, for potential hills of one
substate always coincide with potential valleys of the other: if the atom ispumped after it has reached the top of a potential hill, potential energy is notreconverted to kinetic energy. Instead, the atom finds itself in a valley andhas to climb a potential hill again. The difference in energy has been given
away to the light field. Hence, the atom always has to climb the hills, but
can never descend them and thus dissipates energy. Because of the analogyof that situation with the fate of a famous figure of Greek mythology, this38

cooling mechanism has been called Sisyphus cooling. The interested reader
finds some illustration along with more a detailed explanation in appendix A.
Similarly as in the remarks on optical molasses, one can derive a friction
coefficient for Sisyphus cooling. It is larger than the one for optical molasses,
but its validity is restricted to a smaller velocity range. Thus, the scheme cancool atoms pre-cooled by ordinary Doppler molasses to even lower temperatures than the Doppler temperature TD , which we identified as the cooling
limit in section 2.5.

2.6.2

σ + -σ − polarization gradient coolingIn the σ + -σ − -constellation the two beams can be written as:~ + (z, t) = E0 (ˆx cos(kz − ωt) + ˆy sin(kz − ωt))E
2E
0
~ − (z, t) =E
(ˆx cos(kz + ωt) + ˆy sin(kz + ωt))2.

(2.32)The total field is found to be:~ t) = E0 cos(ωt) (ˆx cos(kz) + ˆy sin(kz))
E(z,.

(2.33)

This field is always linearly polarized, but the direction of the polarization
performs a complete rotation within half a wavelength, which is shown infigure 2.7. This rotation also evokes a cooling effect which does, however, notallow a nice description as in the lin ⊥ lin-case. We want to give here only avery qualitative explanation. We assume an atom with ground state J = 1,
39

0


E
4λ

z

Figure 2.7: σ + -σ − -polarization gradient cooling: in the σ + -σ − -configuration thepolarization is always linear, but rotates with a period of λ/2. For an atom inmotion, this effects preferred scattering from one beam compared to the other,which - for negative detuning - slows the atom down.

so three magnetic sublevels mg = 0, ±1, and J = 2 in the excited state. For
atoms at rest in linearly polarized light, the population of the mg = −1 andthe mg = +1-state are the same. The motion of the atom in a field withchanging direction of polarization, however, gives rise to an additional term in
the Hamiltonian describing the response to the rapidly changing polarization.
This new Hamiltonian has different eigenstates than the Hamiltonian for theatom at rest. The relevant effect of the additional term in the Hamiltonian isthat it perturbs the populations of the old eigenstates: the populations of the
mg = −1 and the mg = +1-state are no longer balanced. Because transitions
strengths also differ (mg = −1 → me = −2 from σ − e. g. is stronger thanmg = −1 → me = 0 from σ + ), the imbalance in the population induced bymotion eventually causes a preferred scattering from one beam compared tothe other. For red detuning, the effect of this preferred scattering is such thatit slows the atoms down, regardless of their direction of motion. The complete
description is a nice exercise in perturbation theory [58].
40Although both mechanisms (the lin ⊥ lin and the σ + -σ − -scheme) described above work quite differently, both rely on polarization gradients, andtheir relevant effect is the same: temperatures well below the Doppler limit
are attainable even in ordinary optical molasses. The final temperature thatcan be reached by the processes described scales with |δ|−1 . For this reason,
one increases the red detuning during a molasses step; one starts with detunings suitable for Doppler molasses and then enforces the polarization gradientcooling by detuning the laser further from resonance. Sub-Doppler cooling exists both in optical molasses and in the magneto-optical trap. In the MOT it
is, however, restricted to a small region around the zero of the magnetic field.
For higher magnetic fields, other effects (e. g. velocity-selective resonance [59]),
which we will not discuss here, become important and the benefits from subDoppler cooling disappear. The three-dimensional case, so e. g. the opticalmolasses and the MOT in our experiment, can never be a pure case of thetwo schemes presented above and is little understood. The experimental results, though, confirm that polarization gradient cooling also works in three
dimensions: temperatures as low as 1.1 µK have been observed [54].

2.7Magnetic trapping

After pre-cooling with laser means, magnetic trapping provides the confinement necessary in a BEC experiment for two reasons: evaporative coolingneeds some form of containment for the atoms so that they do not drift apartand to generate the necessary density for a quick rethermalization after a cool41

ing step. As the atoms are cooled, the trap also compresses the atom cloudand increases the density that is critical for the BEC transition (equation 1.6).
Ions can be trapped only with a.c.-electric fields because trapping ofmonopoles with static fields is not possible as stated in Earnshaw’s theorem.
Neutral atoms, however, can be trapped by use of the interaction VZ of theirmagnetic dipole moment µ~ with a static magnetic field:~ ·µ
V Z = −B
~ .

(2.34)~ = Bˆz , andIf we assume that the magnetic field is oriented along the z-axis, B
that the magnetic moment is also quantized along ˆz , the quantum mechanicaloperator for this part of the Hamiltonian is:

HZ = gJµB ~ ~
B · J = gJ µB mJ B~,

(2.35)~ This result
where J~ is the total angular momentum and gJ is the g-factor for J.
~ is the magnetic field
can generally just be written as gJ µB mJ B where B := |B|
strength (regardless of its direction) if the magnetic moment µ~ has enoughtime to adjust to a (spatially or timely) changing field. The criterion for this“adiabatic following” is that the Larmor frequency ωZ = gJ µB mJ B/~ islarger than the rate at which the magnetic field changes:

ωZ 

~
1 dB
B dt

.

(2.36)

Except near field zeros, this condition is generally fulfilled in neutral atom42

traps. The right side of equation 2.35 represents energy shifts of the atomlevel structure by the magnetic field, the familiar Zeeman effect. But wecan also think about it as an effective conservative potential VZ (~r), whichis especially useful if the magnetic field spatially varies B = B(~r). From
this perspective, atoms can be trapped in two cases: a local minimum of themagnetic field means a potential minimum of VZ and thus confinement foratoms with gJ mJ > 0. Those are called low-field-seekers. A local maximumof the magnetic field would trap, in contrast, atoms with gJ mJ < 0 (which
are consequently called high-field-seekers) if it existed: it was shown by Wing[60] that in a region without currents or charges the magnitude of a magnetic(or electric) field cannot have a local maximum. Local minima, however, doexist12 . That means that only low-field seekers can be trapped.
The magnetic dipole interaction is much weaker than the interactionenergies of ions in electric fields. If the interaction energy of the order µB B
with realistic laboratory magnetic fields of B ∼ 0.1-1 T is compared with thethermal energy kB T , then atoms with T = 100-1000 mK can be trapped. Often,
this temperature is even much lower. It is common to specify this trap depth
in temperature units. This temperature range implies the necessity of some
sort of pre-cooling, e. g. laser cooling. It also explains the strict limitations on
background pressure that we have noted in the introduction: an atom fromthe background gas with T = 300 K can easily transfer enough momentum toa trapped atom in a collision to kick it (and many more) out of the trap. The
12Wing’s theorem is a nice curiosity from a historical perspective: although its proof justneeds Maxwell’s theory of electromagnetism which had been known since the 1880s, it took
more than a hundred years for someone to discover it – or to bother to write down [61].

43average lifetime of a trapped atom in a vapor cell with p ∼ 1 × 10−9 mbar ison the order of a few seconds.
The simplest magnetic field geometry with a local field minimum is thequadrupole field, which is also the configuration employed in this experiment
for magnetic trapping. This field can be generated by two identical circularcoils arranged coaxially (along the z-axis). The currents in the coils have
opposite directions. The configuration is, therefore, called anti-Helmholtz.
The field produced by the coils clearly has a vanishing magnitude on the axis
in the middle between the two coils. The magnetic field can be approximatedto first order just by two simple arguments: if we put the origin at the field zero,the fields in x and y-direction must be the same because of axial symmetry.
~
They are also radially orientated and have no z-components, so B(x,0, 0) =~ y, 0) = (0, by y,0) with bx = ∂B
= by = ∂B.
(bx x, 0, 0) and B(0,
∂x ~
∂y ~
r=(0,0,0)r=(0,0,0)~ r) = (bx x, bx y, bz z).
The magnetic field to lowest non-vanishing order is then B(~~ ·B~ = 0, it follows:
Because of the Maxwell equation ∇

bz :=

∂B
= −2bx = −2by∂z ~r=(0,0,0).

(2.37)In the notation of this thesis, the magnetic field gradient is the gradient inz-direction bz . It will sometimes just be abbreviated with b := bz . Its numericalvalue per current in the coils I for our setup was measured to be Ib = 0.88 cmGA
[39]. The magnetic field of the quadrupole configuration to lowest order is44thus:−x~ r) = b B(~ −y 
22z.

(2.38)

Its magnitude (strength) is~ r) =
B(~r) := B(~

bp 2
x + y 2 + 4z 22.

(2.39)

Respecting the presence of gravity in the experiments, the full potential is13 :VZ (~r) = mgz +gJ mJ µB b p 2
x + y 2 + 4z 22(2.40)The density n(x, y, z) of a thermal atom cloud in equilibrium as a function of
position in a quadrupole trap respecting the influence of gravity is given by abarometric height formula:g m µ b− mgz − J2kJ TBn(~r) = n0 e kB T e

B√

x2 +y 2 +4z 2.

(2.41)

Figure 2.7 shows a calculated contour plot of the magnetic field strength andthe plots of the calculated radial and axial fields in a quadrupole trap, which
is idealized as two current loops.
For later discussion, it is helpful to derive the scaling laws for temperature and volume in an adiabatic compression of the potential (neglecting the influence of gravity). We assume a compression by a factor ηof the potential, so b → ηb. The temperature may scale with a factor ξ:13

It is assumed that the strong axis z of the quadrupole field coincides with the directionof gravity as in our experiment.

45

magnitude of magnetic field a.u.

z-axisx-y-plane1

z-axis

0.8
0.6
0.4
0.2

(a) contour plot

trap depth

axis in x-y-plane2

1
01
distance from trap center R

2(b) field strength along axes

Figure 2.8: Field of a quadrupole trap, idealized as two current loops. The distance
between the coils is chosen to be equal to the diameter (2R) of the coils. Figure (a)shows a contour plot of the magnetic field strength. The coil axis is located at thecenter of the plane. The field zero in the center can be seen as well as the divergenceswhere the coils intersect the plane. Figure (b) shows a plot of the magnetic fieldalong the z-axis and along an arbitrary axis in the x-y-plane. The field gradientalong the z-axis close to the field zero is twice as large as in the x-y-plane.

T → ξT . The new density distribution with correct normalization is then:√ 3g m µ ηb
− J J Bx2 +y 2 +4z 2
n0 (~r) = ηξ n0 e 2kB ξT. The new average distance from the cenRR
ter in the cloud is given by hr0 i = rn0 (~r) d3 r = ηξ rn(~r) d3 r = ηξ hri. So, we 3have V 0 ∝ hr0 i3 ∝ ηξ . Using the adiabatic condition for a monatomic gas
T V 2/3 = const. ,(2.42)

we find:ξ = η 2/3⇒

T ∝ η 2/3 ,hri ∝ η −1/3 ,46

V ∝ η −1.

(2.43)

Chapter 3
Methods, experiments, and results

3.1

Imaging

Nearly all information about ultracold atomic clouds and their various properties can be obtained by analysis of optical images. For the results of thisthesis, two optical imaging methods were employed, which are presented in
the following paragraphs.

3.1.1

Absorption imaging

One technique to image an ultracold atom cloud is absorption imaging. Speaking in simple terms, one analyzes the shadow of the atoms which they cast in
an imaging beam. The darker the shadow is, the more atoms must have been
contained in the cloud to absorb and scatter the light of the imaging beam.
The geometry of the absorption beam in our setup is shown in figure 2.1: thebeam is delivered from the laser system [39] through an optical fiber, enters thevacuum system through a window, propagates through the differential pump47

ing tube, and hits the cloud in the vapor cell. The beam is turned on and off
using a combination of a mechanical shutter (slow, reaction time: 4 ms) andan acousto-optical modulator (AOM, fast, reaction time: 5 µs) inside the laserbox. The exposure timing is defined by the AOM. The shutter completely
extinguishes any leakage of light through the AOM from hitting the cloud
during the preparation sequence. The λ/4-waveplate in front of the entrance
window allows us to make the light circularly polarized if desired. The lightthen passes through a window at the front of the “square chamber” and runs
through a differential pumping tube to the vapor cell, where it hits the cloud.
It leaves the vacuum system again and is then scaled down with a telescopeand directed onto the chip of the camera. Figure 3.1 shows a schematic forthe beam path behind the vacuum system. The distances are chosen such thatboth absorption and fluorescence imaging (see section 3.1.2) can be done withthe same optical setup.

Shutter
Atom Cloud“Shadow” of cloud

CCD

Imaging light
f1 = 500 mm
f2 = 50/60 mm

Lens 2
Lens 1

f1

f1

f2

f2

Figure 3.1: Absorption imaging: the image shows the beam path in absorption
imaging. The second lens (initially f2 = 50 mm) was later replaced by a lens with
f2 = 60 mm. The magnification is thus 1:10 or 1:8.3 respectively. The shutter inthis picture is not necessary for absorption imaging, but needed for fluorescencemeasurements.

The CCD-chip is typically exposed for 100 µs to the imaging light by48

flashing on the AOM1 . In order to obtain quantitative information from absorption images and eliminate systematic errors, three pictures are acquired forevery measurement, each with the same exposure time: the first, Batoms , withthe atom cloud, the second one, Blight , as normalization reference without theatoms, but with the imaging beam turned on. The last picture, Bdark , is taken
without imaging light and serves to subtract the effects of stray light sources
and the dark current of the camera. The imaging software automatically creates a new image (Batoms − Bdark )/(Blight − Bdark ) from the three images.
With a short calculation, we now show how the three images can giveinformation about the atom number in a cloud. In an absorbing medium
the intensity of light decreases with Beer’s law as dI = −αIdx where theBeer absorption coefficient α is proportional to the imaginary part nI of theindex of refraction of the medium: α = 2knI . For a dilute atom cloud theenergy scattered per volume is ~ωn(x)Γsc , where n(x) is the density of atoms,
Γsc is the scattering rate (from equation 2.15), and ~ω is the energy of one
scattered photon. The scattering rate is related to the scattering cross section
σsc through ~ωΓsc = Iσsc . We get:dI = −αIdx = −~ωn(x)Γsc dx = −σsc n(x)I(x)dx.

1

(3.1)

The chip collects, however, light for a much longer duration because the internal shutterof the camera needs ∼ 60-100 ms to open and close.

49

With n(x) = dN/dV , taking dV = Adx, we can transpose and integrate:
A dI
A dI
1 dI
dV = −
dx = −
dN = −σsc I dxσsc I dx
σsc I

⇒

A
N =−
ln
σsc




Itrans.
I0(3.2)

The assumption dV = Adx is justified as long as A is small enough so thatthe density does not significantly change over the size of A. Equation 3.2 gives
us a formula with which the atom number in the cloud can be extracted fromthe absorption images:
Apixel X
Natom = −ln
σsc i,jBatoms− Bdark
i,j
i,j
dark
Blighti,j − Bi,j

!
,

(3.3)

where Apixel is the length calibration ([m2 /pixel]) and the sum runs over all
rows and columns of the images. It can be seen from the result that thismethod is independent of the intensity (as long as the intensity is well under
the saturation intensity Isat ), though care must be taken to control the polarization, which affects σsc . For this reason, the λ/4-plate is placed directly infront of the vacuum apparatus.

3.1.2

Fluorescence imaging

Observing the fluorescence of an atom cloud exposed to near-resonant light isanother way of imaging it. The cloud is exposed to the MOT beams (boththe cycling and repumping lasers) for a duration of 1 ms during which thescattered light is collected with the CCD camera. We typically use a detuningof δ = −3.5 Γ and the maximally available power in the MOT beams. Figure50

3.2 shows schematically the optics that are used to image the cloud on the
CCD-Chip. The objective lens (L1) is placed directly behind a window of

MOT Beams
repumping

relevant solid angle Ω
Shutter

cycling

Atom Cloud h

CCDdLens 2

f1 = 500 mm
f2 = 50/60 mm
d = 6 mmLens 1

f1

f1

f2

f2

Figure 3.2: Fluorescence imaging: the figure shows the beam path in fluorescenceimaging. A shutter controls the exposure time because the built-in shutter of thecamera does not work reliably.

the vacuum apparatus. It has an diameter of 200 and its focal length is f1 =500 mm. The ocular was exchanged during the measurements for this thesis.
Originally, it had a diameter of 100 and a focal length f2 = 50 mm, but it was
replaced by a 200 -lens with f2 = 60 mm. Since we use the same setup also forabsorption imaging, the optical pathlength between the two lenses is exactly
the sum of both focal lengths. The objective lens itself is f1 = 500 mm awayfrom the object, namely the cloud in the magneto-optical trap. In the actual
setup, a mirror is located between lens 1 and lens 2 for practical reasons (lackof space, see figure 2.1).
It was checked that vignetting [62], a common imaging problem for offaxis objects, is not a problem in our setup. For a detailed discussion see
appendix B.1.
The number of photons scattered per time t by a cloud with Natoms atoms
51

is Natoms Γsc t, where Γsc is the atomic scattering rate from equation 2.15. The
atom number Natoms can thus be extracted from fluorescence images accordingto
Natoms =4π Ncounts QEΩ ce− Γsc texp 0.96k

,

(3.4)

where Ω is the solid angle effectively contributing to the image, Ncounts is thenumber of counts from the CCD that constitute the image of the cloud, QE(0.75 for the camera used) is the quantum efficiency of the camera chip, ce−(216 /100000) is the number of counts that are registered per photo electron,
texp the exposure time of the chip to the fluorescent cloud (usually 1 ms), andthe factor 0.96k takes into account losses by reflection on glass surfaces withk being the number of surfaces without anti-reflection coating (k = 2). From
figure 3.2, it can be inferred that the relevant solid angle Ω is determined by2.
the diameter d = 6 mm of the shutter and the focal length f1 , so Ω = 4π π(d/2)
4πf 21Both the absorption and fluorescence images are processed with the computer algebra program Mathematica. The number of counts in a fluorescence
image is calculated after subtraction of a suitable background picture B0 fromthe image B to obtain a corrected image where potential counts from straylight have been eliminated. Two methods exist to extract Ncounts then fromthe corrected image: either all counts of the whole image array are simplyP
added up, so Ncounts = i,j Bi,j − B0i,j . Or alternatively, a Mathematica routine determines the 1-D column density n(z) and fits a model Gaussian dis2

2

tribution (n(z) = ae−z /(2σ ) ) to it, whose parameters give the count number:√
Ncounts = a 2πσ . Summation works better for large clouds where it is moreaccurate because in most cases a Gaussian function is only an approximation52

to the real shape of the cloud (e. g. an atom cloud in the linear potential ofa quadrupole magnetic trap is not a Gaussian). The strength of the second
method is that it is less sensitive to noise by stray light that would be counted
by the summation method. A particular source of stray light, which cannot
be removed by subtraction of a background image, are reflections from the differential pumping tube connecting the vapor cell and the “square chamber”.
This reflects so much of stray light from the fluorescing atom cloud that it
can be seen on all fluorescence measurements. Thus, the integration method
is usually applied only for small clouds.

3.1.3

Choice of imaging method

Principally, absorption imaging is superior to fluorescence imaging because
it provides a much better signal to noise ratio. Particularly for small atom
numbers, it is more accurate. It is also not dependent on intensity whilequantitative analysis of fluorescence images is. On the other hand, absorption
imaging needs a clear quantization axis as it depends on σsc , so all residual
magnetic fields must be eliminated and a bias field along the imaging axis must
be applied. Then, σ + light, e. g., can be chosen, for which σsc is well-defined.
For fluorescence imaging, in contrast, it is sufficient to assume the saturation
for the cycling transition in
intensity for unpolarized light, Isat,unpol = 4.1 mW
cm2
87

Rb. Fluorescence imaging can be only performed in the vapor cell, where

the MOT beams can be used as light source.
For all quantitative analysis in this thesis, fluorescence imaging was chosen for two reasons: 1) It allowed longer times of flight for temperature mea53

surements (see next section). For absorption imaging the cross sectional areaof the differential pumping tube limits the observable region for an atom cloud.
The maximum time of flight for a cloud with diameter 6 mm is then ∼ 20 ms.
In fluorescence imaging, the observable region is defined by the first imaging
lens, for which it was calculated that flight times up to 56 ms are possible
with our setup (see appendix B.1). 2) It was discovered that the frequencyof the cycling laser had a considerable jitter, even when locked. This jitter
influences the detected atom number in absorption images much more thanin fluorescence images due to the short exposure time and the low saturation(see appendix B.2). Because of these two reasons and knowing that the atom
numbers in MOT and MT are still large enough (in contrast to the shrinking clouds during an evaporation stage), fluorescence imaging was chosen as
primary imaging method. For analysis of clouds in the science cell, where nolight source for fluorescence imaging is available, we will change to absorption
imaging again.

3.2

Determination of temperatures

In order to determine the temperature of a cooled atom cloud, we observe
its thermal expansion after release from the trap as a function of time. For
this, we take a sequence of images of clouds that have been prepared underidentical conditions and have freely expanded for a varying time t. As allmeasurements described in this thesis have been taken for thermal clouds well
above Tc , we can treat the cloud classically. In this case, the expansion of a54

Gaussian thermal cloud with one-dimensional initial density profile for t = 0


x2
n0
exp − 2n(x, t = 0) = √2σx,02πσx,0,(3.5)√
where n0 is the peak density and σx,0 the 1/ e-width, can be expressed as:
ZZ

n0
1
√
n(x, t) = √2πσx,0 2πmkB Tx





x20
p2
exp − 2exp −2σx,02mkB Txpt
dp dx0(3.6)
δ x − x0 −mwhere δ is the Dirac δ-distribution, m the mass of the gas atoms, p the mo√mentum, and 1/ 2πmkB Tx exp [−p2 /(2mkB Tx )] the momentum distribution
at temperature Tx . The double integral represents a convolution of the initial
density profile with the momentum distribution. It has a descriptive meaning:
the two integrals express the expansion of every point x0 due to the thermal velocity distribution at that point, which yields after integration the newdensity distribution:"x2
n(x, t) = √ qexp −22 σx,0+ kBmTx t222π σx,0+ kBmTx t2
1

#
(3.7)

√
From this expression, the 1/ e-width after an expansion time t is clearly:r
σx (t) =2
σx,0+kB Tx 2
t
m

,

(3.8)

The temperature now is simply obtained by fitting the model function 3.8 tothe data (after generating a 1-D column density). This proves to be a robust55method, and we usually do not need more than four data points to obtain
trustable results. This discussion respected only one dimension, but can be
immediately expanded to three dimensions because of the separability of theinvolved integrals. It is important to note that the temperature Tx was introduced as a parameter describing the one-dimensional velocity distribution
in the x-direction. Under the condition of thermal equilibrium (which canbe reached in a magnetic trap), the value for Tj should be the same in eachdirection j ∈ {x, y, z}. However, for many experiments a full rethermalization
before detection is not possible. Therefore, we often determine two “temperatures” Ty and Tz (see section 3.7.2). Practically, we can calculate the column
densities only along the y or the z-axis. Expansion along the x-direction is
not observable as it is the optical axis of the imaging optics; we thus detect
only the projection of the cloud along this axis.

3.3

Magneto-optical trap

The collection of the atoms in the magneto-optical trap (MOT) is the firststage of any preparation sequence in our experiment. In various measurements
done prior to the time the author spent in the research group, the MOT was
optimized and characterized [39, 63]. It was found to work best at a magnetic
field gradient2 of ∂B/∂z = 8.8 G/cm, and a detuning of the pumping lasersof δMOT = −3.5 Γ. As discussed in [64], there is an optimal set of these2

At that time, the quadrupole coils were not the same as in the later experiments performed by the author: this field gradient now corresponds to a coil current of IMOT = 10 A
while at the time of the measurements a current of 12 A was needed.

56

parameters for every possible laser intensity and diameter of the trapping
beams. Increasing the field gradient means reducing the effective capture
range of a MOT. This effect is balanced by detuning the cycling frequency
further from resonance. Our values are roughly consistent with those reported
in literature ([38, 54, 64]).
Figure 3.3(c) shows the dependence of the maximum atom number collected in the MOT on detuning for a magnetic field gradient of ∂B/∂z =7.0 G/cm. As shown in figure 3.3(b), the shape of the cloud is a two-component
distribution, particularly for larger detunings: a central peak with a muchhigher atom density arises over a broader, less dense distribution (figure 3.3(b)).
This fact was predicted in [52] and first observed in [55, 65]. As explained inchapter 2, it is interpreted as manifestation of polarization gradient cooling in
the center of a MOT: for the small magnetic fields in the center of the trap,
sub-Doppler mechanisms are viable while they are impaired for the higher
field strengths in the outskirts of the cloud. We could also confirm the results in [55, 65] that for increasing field gradients the total number of atoms isapproximately constant while the number in the central distribution declines
reaching ∼ 50 % of the total atom number at a gradient of 20 G/cm (figure
3.3(d)). Since the atoms contained in the central distribution are much colder
and their density is higher, optimization includes maximization of the size of
this part of the MOT. This explains the choice of our MOT-parameters: fora detuning of −3.5 Γ and a field gradient of 8.8 G/cm, more than 90 % of theatoms were found to be in the center of the cloud, containing up to 1×1010
atoms (figure 3.3(d)).

57

400000
column density

1 cm

Data
Fit

30000020000010000000(a) Fluorescence image of a MOT
1 0

2 .0 x 1 0

1 0

9

1 .5 x 1 0

1 0

6 .0 x 1 0

94 .0 x 1 0

9

1 .0 x 1 0

1 0

2 .0 x 1 0

9

to ta l n u m b e r
c e n tra l fe a tu re

8 .0 x 1 0

200
Column300400(b) Two-component regime

to ta l n u m b e r
c e n tra l fe a tu re

N M O T (fo r b e s t d e tu n in g )N M O T

1 .0 x 1 0

100

0 .0

5 .0 x 1 0

9

0 .0
1 .5

2 .0

2 .5

3 .0

D e t u n i n g [ Γ]

3 .5

4 .0

4 .5

8

(c) Dependence on detuning

1 0

1 2

1 4

1 6

c o i l c u r r e n t [ Α]

1 8

2 0

(d) Dependence on field gradientFigure 3.3: Optimization of MOT loading parameters: figure (a) shows a fluorescence image of the magneto-optical trap. The atom number for this image is
1.2×1010 . The cut-outs that are visible are necessary because of reflections frompieces of the vacuum apparatus. The error caused by this correction is negligible.
Figure (b) is a graph of the one-dimensional column density in a MOT for a field
gradient of 7.0 G/cm. It shows the typical two-component distribution in a MOT.
Figure (c) shows the atom number in the MOT for IMOT = 8 A and various detunings. The filled dots in figure 3.3(c) give the total atom number in the MOT while
the circled ones represent the size of the central peak. Figure (d) shows the atom
number in a MOT for various field gradients where the result for the best detuning
was taken for each field gradient from measurements as in figure (c). The data infigures (b)-(d) was taken from references [39, 63]. At that time, the magnetic fieldcalibration was not the same as in the later measurements (different quadrupole
coils).

58

3.3.1

Light induced atomic desorption

Klempt et al. [66] showed a multiple increase in the atom number of a magnetooptical vapor cell trap by light-induced atom desorption (LIAD) of trappableatoms from the walls of the vacuum chamber. This effect is analog to the wellknown photoelectric effect: atoms that have deposited to the inside surfacesof the vacuum chamber can be dissolved by shining ultra-violet light fromoutside onto the chamber. This causes a sudden increase of the backgroundpressure in the vapor cell and consequently raises the maximum atom numberin the MOT Nmax following equation 2.27, unless the background pressure p
is in the regime where the MOT atom number does not depend on pressure
anymore and has saturated. When the light is turned off, the atoms stick to
the walls again and the pressure drops quickly. This phenomenon offers theopportunity to artificially raise the pressure during the loading phase of the
MOT and trap a high atom number by turning on a UV-source. The UV lightis then turned off while the MOT beams stay switched on. This allows the
background pressure to drop again. The pressure falls much faster than the
MOT atom number reacts to this decrease, and one can use the advantages ofa low pressure for the magnetic trap stage and the transport while having at
the same time the benefits from high pressure, i. e. high atom number.
The initial loading rate of the MOT R is directly proportional to therubidium background pressure in the vapor cell. We exploit this proportionality to measure the time constant with which the background pressure settles
back to its equilibrium value. For this purpose, the UV source is turned on
for 10 s, which is more than sufficient to solve the majority of absorbed atoms
59

from the wall, then turned off again. After a varied delay time t, the MOT
is loaded for 300 ms and the atom number in the MOT after this duration isdetermined. This number serves as a measure of the pressure. Note, however,that especially for short times t this is only an integrated (over 300 ms) value
because the pressure falls very quickly in the beginning. The actual pressuretime-dependence is probably even more pronounced than in our results in

MOT atom number after 300 ms

figure 3.4. This aspect may have been interpreted incorrectly in reference [66]3. ´ 108
2. ´ 108
1.5 ´ 108
1. ´ 108
7. ´ 107
5. ´ 107
3. ´ 107
0

2000

4000
6000
8000
delay time t @msD

10000

Figure 3.4: Atom number in a MOT after 300 ms loading with different delay times
t after turn-off of the LIAD UV-light. The atom number is proportional to thebackground pressure integrated over 300 ms at time t. The figure shows the dataon logarithmic scale: clearly two time constants, which suggest two different participating processes, are involved while the pressure is returning to its initial value.

as they employed the same measurement method (varying delay time with
300 ms loading time), but assumed the resulting curve to be proportional tothe pressure decay. We can, however, confirm their results that the pressure
drop involves two time constants: an initially very fast fall turns into a slow
convergence to the value received without the use of desorbing light. This canbe explained if adhesion of Rb atoms to a glass surface is more favorable than
to a rubidium layer. If this is the case, the first layer of atoms will build up
60

on the glass very quickly while the later layers will need more time to develop.
With a photodiode, we also measured the fluorescence of a MOT loaded with
LIAD after the UV-light had been turned off: it takes many seconds (∼ 30 s)
for the fluorescence to decay to a value equal to the fluorescence if loaded
without LIAD.
The enhancement factor of the atom number by use of LIAD is about
20-30 for very low background pressures. Similarly to the procedure in section
3.22, we also examined the enhancement as a function of background pressure:
the rubidium oven was opened and over a duration of two hours the atom
number in MOTs loaded without and then with LIAD was measured alongwith the lifetime of a magnetic trap, which is inversely proportional to thebackground pressure in the vapor cell. The data was taken in intervals of fiveminutes to allow enough time for readsorption of the dissolved atoms. It isshown in figure 3.5. It was found that for higher pressures the positive effect of
LIAD degrades dramatically and finally vanishes. This is not surprising since
the MOT atom number does not increase with rising background pressure
anymore in the regime where the rubidium density dominates the densityof other species in the MOT (see explanation in chapter 2). Consequently,
the additional increase of pressure by LIAD does not show any effects. The
enhancement for low background pressures is not large enough to operate
the apparatus in this regime: with the short transport times to the “squarechamber” realizable with the translation stage, it is still more efficient to loada MOT with a large atom number at high pressure than to load it with LIADat a low background pressure.

61

50

7 x 1 09n u m b e r N M O T

6 x 1 09

5 x 1 09

4 x 1 09

3 x 1 09

A to m2 x 1 09

1 x 1 09enhancement factor

8 x 1 0

9

20
10
5
2
1
0
20 40 60 80 100
time after opening the oven @minD(b) Enhancement by LIAD

0
0

2 0

4 0

6 0

8 0

1 0 0

tim e a fte r o p e n in g th e o v e n [m in ](a) MOT atom number NMOT with and without LIAD

lifetime Τ @sD

w ith L IA D
w ith o u t L IA D

7
6
5
4
3
2
1
0

0
20 40 60 80 100
time after opening the oven @minD(c) MT lifetime τVC

Figure 3.5: Light induced atomic desorption: figure (a) shows the atom numberin the MOT NMOT as a function of time after opening the oven, so as a functionof background pressure with and without the use of LIAD. Figure (b) shows theenhancement, i. e. the ratio of the two data sets in figure (a). Figure (c) shows
the decrease of the lifetime τVC in the vapor cell during the experiment. The runs
were taken in equal intervals over a time of two hours after the rubidium oven at atemperature of 22 °C had been opened.

3.4

Compressed magneto-optical trap

While the low field gradient of a MOT is suitable to capture a big number ofatoms and to reduce losses due to reabsorption of photons, it is not adequate
for the following sequence steps for two reasons. First, the density of the cloudis low, which is not desirable for evaporative cooling; second, the size of thecloud is large, which makes transfer into a magnetic trap difficult. Therefore, atypical element in any preparation sequence is the compression of the magnetooptical trap (CMOT), first demonstrated by Petrich et al. [65]. It exploits thedependence of the density n in a MOT on the spring constant κ in the regime
where the density is limited by reabsorption of scattered photons (see chapter622) [52]:n=3κc
Iσl2 (σr /σl − 1)

,

(3.9)

where σl is the cross section for absorption of photons from the laser fieldand σr the cross section for reabsorption of scattered photons. The spring
constant κ is proportional to the field gradient b. An increase of b, therefore,raises the density in the MOT. This is, however, only a transient effect as thecapture volume of the MOT, which influences the maximum atom number,is decreased by the compression. After a time comparable to the trap loss
constant τ (equation 2.26), the gain in density must be paid with a loss in atom
number. Therefore, the compression must be fast and immediately followedby the switch-off of the MOT and an efficient cooling stage by optical molasses
that reduces the high temperatures in the cloud (some hundred microkelvins)
after MOT and CMOT.
The CMOT was optimized with respect to the following conditions (inthis order): first, the total atom number should not decrease significantly during the compression3 . Second, the density is to be maximized. Similarly to[65], we found that a ramp of the magnetic field gradient is better than asudden change, during which atoms can be lost. The two parameters to bevaried in the CMOT step are essentially the ramp time and the maximum
field gradient after compression. In practice, we optimized the CMOT witha subsequent standard molasses stage (see figure 3.6(b) for illustration of thesequence). Because we observed that the cooling effect of any molasses is
3This follows the general agreement that the atom number is the most important single
parameter of a MOT/CMOT in predicting the chances for a BEC.

63

quite independent of the properties of the cloud that is loaded into it (such asatom number, density, and temperature), we think that this was an acceptableprocedure. Adding the molasses, however, allows us to determine the temperature more easily due to the low temperatures produced by it. Figure 3.6(a)shows the phase space density D (defined in chapter 1) for various final field
gradients after the compression and application of optical molasses. Because
there is no strong dependence visible, we decided on using the parameters forwhich we lose the least atoms. For a final field gradient of about 90 G/cm
(in lab units4 : A07 = 2.0 V and a compression time of 500 ms), we lost an
negligible amount of atoms. It was not necessary to adjust the detuning for
Phase

1 0

Duration [ms]

MOT
10000

-6

P h a s e s p a c e d e n s ity [1 0 ]

8

CMOT

Molassesvaried5

TOF
varied

Det.
1

10

10

10

10

Cycling
Power AO1

6

4

RepumpPower AO2

ra m p tim e : 5 0 0 m s
ra m p tim e : 2 0 0 m s

2Cycling
Detuning AO0

0.5
δ

9.74

7.76

7.76

0
4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

fin a l fie ld g ra d ie n t [G /c m ]Current
Coils AO7(a) Phase space density D

0.2

varied6.1(b) Sequence

Figure 3.6: Optimization of the compressed MOT: figure (a) shows the phase spacedensity D after MOT compression and molasses for different field gradients after thecompression. Figure (b) illustrates the employed sequence. The compression time
and the final field gradient are varied. The temperature is determined by time offlight measurements after a standard molasses. The atoms are detected (Det.) byfluorescence imaging. All values are in lab units.

the CMOT compared to the MOT stage. It must be mentioned, however,4

The calibrations defining the lab units are listed in appendix B.3.

64

that these measurements were taken at a time when the background pressurein the MOT cell as well as the absolute atom number in the MOT was very
low. In later measurements, we directly observed the losses in each single sequence step (see section 3.9.3). The CMOT still worked well for small clouds(negligible losses compared to MOT), but for high MOT-numbers of up to
10 × 109 atoms it had to be readjusted so that we could still hold about 85%of the initial atom number. Both the final field gradient and the detuning had
to be increased. For large clouds, the best parameters are a compression to∂B/∂z = 110 G/cm (A07 = 2.5 V) and a jump of the detuning after the MOT
stage from −3.5 Γ to −4.9 Γ. This confirms an assumption made in [65]: for
big MOTs the atom number is limited by reradiation of absorbed photons.
Because the dependence of the reradiation force on the detuning is stronger
than for the trapping forces, increasing the detuning helps hold the atoms.

3.5

Molasses

The CMOT is followed by an optical molasses stage for further cooling. Although in early BEC experiments [5] the clouds were directly loaded from the
CMOT into the magnetic trap, it was clear from the beginning that this could
not be realized with the high atom number in our experiment. We found temperatures after MOT and CMOT that were substantially higher (∼ 900 µK)
than the Doppler limit TD ≈ 146 µK. We tried to reduce this temperature
by varying different parameters with little success. However, a general dependence of the attainable temperature on the absolute number of atoms in65the cloud was observed (see figure 3.7). We first tried to lower the temper8 109

109
2 10 9

1000

4 10

8×10

400

6×10

9

4×109
2×109

200
0

Temperature T [μK]600

Atom Number NMOT

800

9Atom Number NMOT

Temperature T [μK]

1000

6

9

0

2

4

6

8

10

Repump power, AO2 [V](a) Temperaturedence

depen-

0

0

2

4

6

4 109

8

9

10

2 10 [V]Repump power, AO2(b) Atom number dempendence

800
600
400
200
0

8 109

0

2×109 4×109 6×109

8×109

Atom Number NMOT(c) Temperature vs. atom
numberFigure 3.7: Temperature in a MOT as a function of atom number: figure (a) shows
the dependence of the temperature in a MOT on the repump power. Figure (b)shows the atom number for the same experimental runs as in (a). The similar
trends suggests a direct relation of temperature and atom number, plotted in (c)ature in a MOT by changing the repump power, which could be a source of
unnecessary heating. We found that as soon as the temperature decreases,
the atom number also starts declining. Plotting the atom number against thetemperature though shows that there is a direct relation between these twoquantities for large MOTs, probably because of the increasing importance ofmultiple scattering with increasing atom number.
The temperatures attained in our MOT are a bad starting point formagnetic trapping. For this reason, one usually implements an optical molasses
stage, which further cools the atoms down. During the molasses, we sweep thedetuning of the cycling linearly from δ = −3.5 Γ to a final value of −6.0 Γ in
5 ms. During this interval, the cycling laser is kept at full power while therepump laser is switched to a value of ∼ 2% of its maximum power. This
did not affect the atom number compared to leaving the repumper at full66

power, but reduced the temperature by a factor of two. In combination with
the CMOT, the molasses reduces the temperature to 25 µK (figure 3.8). Thisvalue was very robust against fluctutations in atom number, changes in the
CMOT, etc. Detuning the laser during the molasses further from resonancewould produce even lower temperatures, but also increase the losses in atom
number. The PSD D after the molasses is on the order of 1×10−5 .
T = 24 ΜK
2.75

Data

Σ @mmD

2.5

Fit

2.25
2
1.75
1.5
1.25
0

10

20
30
40
Time of flight @msD

50(a) Analysis(b) Time-of-flight images (times in ms)

Figure 3.8: Determination of molasses temperature by time of flight measurements.
The phase space density D after optical molasses is ∼ 1×10−5 . The solid line infigure (a) is the model (equation 3.8).

3.6

Optical pumping

After cycling the rubidium atoms on the (5 2 S1/2 , F = 2 → 5 2 P3/2 , F0 = 3)transition during the laser cooling stages, we are principally able to magnetically trap three species, namely atoms in the mF = −1-substate5 of F = 1(|1, −1i) or in the mF = +1, +2-substates of F = 2 (|2, +1i, |2, +2i). A
5Note that for F = 1 the gF -factor is −1/2. Therefore, the state with negative mF is
low-field-seeking.

67

|1, −1i-cloud can be produced by turning off the repump laser for a few milliseconds after molasses while still cycling: all atoms will eventually gather in
the F = 1-state, which has only one magnetically trappable substate mF = −1.
We decided, however, on choosing the |2, +2i-substate since the trapping force
gF mF µB ∂B/∂z is twice as large, giving us a factor 23 = 8 in the initial density
before evaporation. The hyperfine level F = 2 has five substates. If the magnetic trap stage is placed directly after the molasses, only a certain fraction of
the atoms (less than 20 %), which is determined by the Clebsch-Gordon coefficients, would be in the |2, +2i-substate. In order to collect as many atoms
in the |2, +2i-state as possible, optical pumping of atoms in other states to|2, +2i is implemented following the molasses phase.
Because we observe that complete pumping to |2, +2i is not possible (seebelow), we choose the parameters for the magnetic trapping (see corresponding
section) such that the atoms in the other trappable substate of F = 2 (|2, +1i)
are also caught. They will function as a cooling agent during evaporation for
the |2, +2i-atoms because they are first evaporated in a magnetic trap due totheir weaker confinement. This “sympathetic cooling” was first demonstrated
by Myatt et al. [67]. In the pumping step, the cloud is exposed to on-resonanceσ + -polarized light from the pumping laser for several milliseconds. We couple
this light out of the linearly polarized, on-resonance absorption imaging beamwith a polarizing beam splitter (see figure 2.1). The power of this pumping
beam is on the order of 100-200 µW (on a area of 4 cm2 ). The efficiency ofthe pumping can be analyzed with the Stern-Gerlach method described below(section 3.6.1). Doing this, we found that complete pumping to the mF = 2-68state is not realizable. It was never possible to bring more than ∼ 65% in
the |2, +2i-state with ∼ 25% in the |2, +1i-state. Bad alignment, insufficientpumping power, mismatched polarization, and incomplete repumping wereruled out as reasons. Instead, it is very likely that reabsorption of photons
with random polarization scattered by the optically dense cloud limits thepumping efficiency. This is quickly shown by a rough estimate. The mean freepath l of a photon is given by l = nσ1sc , with n the atom density and σsc thescattering cross section. We use σsc I = ~ωΓsc :l=)
I 2 1 + s0 + ( 2δ
Is 2
1
Γ
=
≈
nσsc
n~ω Γ
s0
n~ω Γ

for I  Is and δ = 0.

(3.10)and typical values (Is = 1.67 mW/cm2 6 , N = 5 × 109 , V = 4/3πσ 3 , withσ = 3 mm ), this gives l = 0.1 mm  σ, so on average, a photon will experience
multiple rescattering. This is in agreement with the following observations: thepumping efficiency gets smaller for larger atom numbers in the MOT. We see
differences on the order of 10 % for clouds whose atom numbers differ by afactor of 5. Also, the pumping efficiency could be increased significantly by
first releasing the cloud and allowing the density to decrease before pumping.
The fraction of atoms in |2, +2i increased from ∼ 60 % to ∼ 80 % when thetime of flight before pumping was increased from 4 ms to 20 ms.
This is a basic limitation for our experiment, for we cannot afford a reduction of the density, which is decisive for the success of evaporation. Eventually,
we use pumping times of 4-8 ms. The relative gain in atom number (in the
mF = 2-state) depends strongly on the absolute atom number: for unpumped
6

This is the saturation intensity for σ + -light on the cycling transition. See appendix D.

69

clouds, we measured 41 % in the two trappable states (mF = 1: 24 %, mF = 2:
17 %, atoms in molasses ∼ 8 × 108 ), shown in figure 3.107 . With pumping,
we could bring as many as 90 % in trappable states for small clouds8 (24.5 %,65.5 %, 8 × 108 ). For the highest atom numbers possible with our apparatus,
this ratio goes down to9 ∼ 50 % (14.5 %, 35.5 %, 6.3 × 109 ).

3.6.1

Stern-Gerlach analysis of optical pumping

In order to determine the efficiency of the optical pumping phase described in
the previous section, we exploit the separation of the five magnetic substatesof 5 2 S1/2 , F = 2 in an inhomogeneous magnetic field by performing a SternGerlach measurment. Figure 3.9 shows schematically the sequence we employ.
After pumping, we let the cloud fall for 19 ms. After this time, the cloud is
at a position where the magnetic field gradient (and thus the force it exerts)
has a definite direction (whereas around the trap center the field lines goin all directions). Here the field of the quadrupole coils is suddenly turnedon at a value of 353 G/cm for 5 ms. This kicks the atoms of the different
substates in different directions, with different accelerations and with different
final speeds after this flash (or has no effect on them10 for mF = 0), thus
spatially separating them. To achieve full separation, we let the system evolvefor another 14.5 ms before detection with just a very weak bias field turned on7

run 37, 02/17/2007
run 40, 02/17/2007
9
run 38, 03/02/2007
10
After t = 19 ms time of flight, the atoms experience a magnetic field strenght of B =G~ J-coupling.
~
353 cm× 21 gt2 = 63 G. This magnitude is well in the low-field regime for the I2
The hyperfine Paschen-Back regime for the 5 S1/2 ground state starts around 1000 G (seereference [48]).
870

to avoid relaxation. Figure 3.10(a) shows a typical image for an unpumped
cloud consisting of five separated density peaks representing the five substates.
It can also be seen that for the “negative” substates (mF = −1, −2) the clouds
are expanded in the x-y-plane while the “positive” substates (mF = +1, +2)
are compressed. This is caused by the components of the field gradient in the
x-y-plane, which only vanish on the z-axis. The cloud, however, is not a pointon the axis, but an extended object, which explains the deformations. For the
analysis, we calculate the column density on the z-axis from the pictures anduse a fitting procedure to calculate the relative atom numbers in each of thestates. The model function is the sum of five Gaussians with equally spaced
centers and the same width. The relative atoms numbers are then given by
the ratios of the five amplitudes (compare figure 3.10(b)).

3.7Magnetic trap

Pumping to trappable states marks the last optical step on the road to BEC.
Following this, the atoms are transferred to a magnetic trap (MT), then transported to the science cell, where eventually the evaporative cooling will takeplace. The magnetic trap of our experiment is of the quadrupole type withthe strong axis in the direction of gravity (see chapter 2). The coils support
currents up to 450 A, which corresponds to a field gradient of 400 G/cm. The
field gradient per current is 0.88 G/(cm A). The coils can be switched on in
1.25 ms and completely switched off in 1.5 ms11 using an integrated gate bipo11

After these times, a steady state in current is reached again.

71

Figure 3.9: Stern-Gerlach analysis sequence: the schematic starts with optical molasses as the last step of the preparation sequence before optical pumping. The
MOT and CMOT stages are left out. Optical pumping is executed in the stage“pump 2” by flashing on the imaging AOM. The stage “pump 1” is necessary toallow the imaging shutter enough time to open and to bring the majority of atoms
into F = 2. Then the cloud is released (A) and falls for 19 ms to a position where thedirection of the field gradient is well defined for the entire cloud volume (B). During
this time of flight, a weak bias field prevents spin relaxation. Then the magnetic
field is suddenly turned on for 5 ms, thereby separating the five substates (C). The
system evolves for another time of flight of 14.5 ms to guarantee sufficient separationof the five substates at detection (D).

72


column density 1 mm

8  1076  1074  1072  1070

1 cm

column density [1/mm]

25


g

(a) Stern-Gerlach image

8 107data
fit
fit comp.

6 1074 1072 107
0
25

20

15

10
5
0
zposition [mm]510(b) Stern-Gerlach analysis

Figure 3.10: Figure (a) shows a typical result of a Stern-Gerlach measurment. Figure(b) shows the analysis: the resulting 1-D column density is fitted with a model
function consisting of the sum of five Gaussians (equally spaced, equal widths, butdifferent amplitudes).

lar transistors, which will be described along with the circuitry and the power
supply in reference [39].
Optimization of the magnetic trap stage involves several aspects: a reasonable value for the initial magnetic field (from now on referred to as “catch
field”) has to be found. Ramping up the field starting with ∂B/∂z= 0 or witha very low field gradient is unfavorable because the cloud could expand nonadiabatically during the ramp and gravity would cause a significant offset ofthe center of mass of the cloud with respect to the minimum of the potential,
which would lead to a sloshing motion, i. e. additional energy, heating thecloud during rethermalization. However, if the field is starting with a highvalue, large parts of the cloud will suddenly be placed in regions of a high potential energy, which will introduce additional kinetic energy, again resulting
in heating. There is consequently an optimum value for the catch field. Thisis sometimes referred to as mode matching [68]. The other concern in the MTphase is the compression of the trap. Here, a slow enough ramp speed needs tobe found that guarantees adiabaticity and thus conserves phase space density.
73The ramp also needs to be speedy since we want to move the MT as fast as
possible out of the vapor cell (where it is exposed to more trap loss collisions
than in the “good” vacuum in the square chamber and the science cell). We
also have to pass the narrow differential pumping tube (see the machine drawing 2.1) with the MT, imposing a size limitation. Therefore, the MT needs tobe (at least partially) compressed when it enters the tube in order to avoid
hitting its edges.

3.7.1

Switching on the magnetic trap: the optimum
catch point

The systematic measurements for the MT stage were done when it was still
intended to produce a |1, −1i-condensate. Later it was decided to seek a|2, +2i-condensate because the magnetic force is twice as strong for this state,yielding a density eight times larger in the MT compared to |1, −1i. The
optimization described in the following paragraphs was not redone in full scale,but the results can be immediately applied to the |2, +2i-state by rescaling
the magnetic field by a factor of 2. To optimize the catch point we used thesequence illustrated in figure 3.11. The cloud is captured with a constantmagnetic field, which stays turned on for 150 ms. This duration of the step
ensures that all atoms in the mF = 0-state have enough time to fall out of the
field of view. It is also needed to observe the effect of a field gradient thatcauses a force on the mF = −1-atoms just about as strong as gravity: if gravity
exceeds this force just a bit, the cloud should slowly fall during the holding
time. If, on the other hand, the magnetic force is slightly stronger than gravity,
74

PhaseDuration [ms]

3

1

2

MOT
10000

CMOT

4

Molasses Rep. off

500

4

5

5

6 7

Catch

TOF Det.

150

varied 1

10
Cycling
PowerAO1
10
RepumpPowerAO2
Cycling
Detuning AO00.5

δ9.74 ↔ δ = 0
7.76
6.1
0

Quadrupole
Coils AO7

2.0

0.2

variedFigure 3.11: Sequence used to optimize the catch point (not drawn to scale): the“Repump off”-step (no. 4) creates a F = 1-cloud. The magnetic field in the “catch”step (no. 5) is varied. For each value of this field, we also vary the time of flight inorder to determine the temperature. All values are in lab-units, i. e. the voltages
applied on the analog channels. (Decreasing the AO0-voltage means actually increasing the (absolute) value of the detuning; the detuning must be measured from
AO0 = 9.74 V as indicated in the schematic.)

the potential is deformed (as shown in figure 3.12), but its minimum stays at
the same position. In thermal equilibrium, the density of the atoms in thetrap is given by:√n(x, y, z) = n0 e−

V (x,y,z)
kB T

g

− F

= n0 emF µB bz /2

x2 +y 2 +4z 2 +gmz
kB T,

(3.11)

where V (x, y, z) is the full potential energy (consisting of the gravitationalenergy and the potential energy due to the magnetic field), bz = ∂B/∂z theaxial field gradient, g the earth acceleration, and m the mass of 87 Rb. It canbe easily seen that the position of the maximum density does not change due
to gravity. In our analysis, we do, however, not observe the actual density, but
75

w ith o u t g ra v .
m a x . fie ld g ra d .
ε=1 3 .3
ε=5ε=1 .3 3ε=0 .3 3
g ra v ity

p o te n tia l e n e r g y [ a .u .]

1 0

5

0
-4

-2

0

2

4

z - a x is [ a .u .]Figure 3.12: Gravity adds an additional term to the potential energy that breaks thepoint symmetry with respect to the field zero along the z-axis. This effect must be
taken into account when the catch field for the magnetic field is chosen. The figure
shows different situations: the pure magnetic potential (for maximal field gradient)and gravity is shown. The other curves represent the full potential for different field
F µB bz
gradients ∂B/∂z. The dimensionless quantity  is defined as  = mF gmg. The
ratios refer to mF = −1 (gF = −1/2).

we detect the projection of the cloud into the y-z-plane (which corresponds
to an integration along the imaging axis x) and we then sum up the columnsof the image with the analyzing routine again along the y-axis. Eventually,
only the column-density along the z-axis is discussed. After integration along
the x-axis one obtains for the two-dimensional density distribution (“column
density”) [69]:− mg zn(y, z) = 2n0 e kB T

p
1 gF mF µB ∂B p 2
y 2 + 4z 2 K1 (y + 4z 2 ) ,2 kB T ∂z(3.12)

where K1 is the modified Bessel function of the second kind and first order.
Figure 3.13(a) shows a column density plot for a magnetic force that is just76

16% stronger than gravity, both in experiment and theory. It can be seen

1 D column density [a.u.]that the cloud leaks out in the direction of gravity. Also, the highest column

1.0
0.8
0.6
0.4
0.2
0.0
-10

-5

0

5

10

15

20

25

5

0

5

z-position [mm]30

1 D column density a.u.

1.0
0.8
0.6
0.4
0.2
0.0
25

(a)

20

15

10

z position mm(b)

Figure 3.13: Effects of gravity on clouds in linear potentials: figure (a) shows adensity plot of an atom cloud in a linear potential that is just slightly stronger thanF µB bz= 1.16. The top picture is the experimental data (after a time
gravity,  = mF gmg
of flight of 5 ms) and the bottom one the theoretical calculation. The differencesin shape is due to incomplete rethermalization after 150 ms. Figure (b) shows the
1-D column density (numerical top, experimental bottom) after summation along
the y-axis. The maximum is shifted away to -2.16 mm from the minimum of thepotential at z = 0.

density is not at the minimum of the potential. This effect is enhanced further
by summation along the y-axis (which is done in the image analysis) as shownin figure 3.13(b). Consequently, when we extract the maximum of the column
density, there is no sharp transition for its position when the magnetic force77surpasses gravity. This is reflected in figure 3.14(a), where the vertical dashedline marks the critical field gradient of bcrit = 30.6 G/cm that is necessary tosupport the |1, −1i-atoms against gravity. Figure 3.14(b) shows the temper300

0250
Temperature @ΜKD

cloud position mm

2
4
6
8

mF g F μ B

10
25

50

∂B
= mg
∂z

75
100 125 150
Field gradient G cm

200
150
100

175

y-axis25200(a) Position of 1D peak column density after
150 ms of hold50

75
100 125 150
Field gradient @GcmD

175

200

(b) Temperature after catch
1.0
relative phase space density

1
0.8
NMTNMolz-axis

50

0.6
0.4
0.2

25

50

75
100
125
150
Field gradient @GcmD

175

0.8
0.6
0.4
0.2

25

200

50

75
100
125
Field gradient G cm

150

175

(d) Phase space density D(c) Transfer efficiency to MT

Figure 3.14: Optimization of catch point for |1, −1i-atoms. For discussion see text.

ature of the cloud after the hold in the trap (phase 5). The lowest values
are close to the temperature we measured for the pure molasses (section 3.5).
There is no significant heating by low catch fields. This changes for higher
catch fields. It can also be seen that the temperatures12 along the y- and thez-axis drift more and more apart for higher catch fields because the atoms ac12

Note that these are not actual temperatures, but rather parameters describing the widthof the velocity distribution along the axes. See the discussion in section 3.2.

78

200

quire more excess momentum along the z-axis where the field gradient is twice
as strong as on the y-axis. This also means that the rethermalization time forthe trapped clouds at those field gradients is longer than 150 ms, which is alsosuggested by the different shapes in figure 3.13(a). Figure 3.14(c) gives thefraction of atoms that are still present in the trap after the 150 ms hold duration. Above the critical field gradient this fraction first rises faster, but slows
down. This is because for low field gradients the potential is very shallow fornegative z (figure 3.12), allowing high-energetic atoms to escape as the trap
depth discussed in chapter 2 is lowered by gravity13 . Figure 3.14(d) shows
the deduced phase space density D in arbitrary units after the procedure. As
expected and just explained, the maximum of D is not found at the criticalfield gradient bcrit , but at a slightly higher value of ∼ 45 G/cm. For lower field
gradients (around bcrit ), non-adiabatic expansion (that reduces the density n),
and the thermal losses of fast atoms going over the hump of the potential(which diminishes the atom number N ) reduce the phase space density while
higher field gradients add excess potential energy that converts into kinetic
energy, thus raising T and again reducing D.
An attempt was made to redo this catch point analysis for a mixed|2, +1i, |2, +2i-cloud, but we had to deal with a quite unstable performance ofthe machine when the data was taken (appendix B.2). The results are shownin figure 3.15 and seem to confirm the earlier ones. As complete pumping tothe |2, +2i-state is not possible, it is not expected that the optimum catch
13

Compare figure 3.13 and figure 2.8(b). Gravity makes the potential shallow (figure3.13), but the linear potential, which is only an approximation around the field zero, does,of course, not extend to infinity. Rather, there is a hump (figure 2.8(b)), approximately at
a position comparable to the radius of the quadrupole coils.

79point would change significantly: if the catch point remains unaltered, thereis, of course, more heating due to the acquired potential energy for the |2, +2iatoms. The resulting reduction of D is, however, compensated above a certain
field gradient because the |2, +2i-atoms are caught as well as the |2, +1i-atoms.
The final catch point was chosen at ∂B/∂z= 48.5-53 G/cm (AO7 = 1.1-1.2 V),
1.0
relative phase space density

Temperature @ΜKD

400
300
200
100

z-axisy-axis

20

40

60
80
100
Field gradient @GcmD

120

140

0.8
0.6
0.4
0.2

20

40

60
80
100
Field gradient G cm

120

140

(b) Phase space density D(a) Temperature after catch

Figure 3.15: Optimization of catch point for pumped |2, +2i-atoms-clouds

slightly higher than the optimum determined above, in order to minimize theloss of atoms during the subsequent compression of the magnetic field (seenext section).

3.7.2

Compression of the magnetically trapped cloud

The next concern is to find a rate for compressing the magnetic field so thatadiabaticity is guaranteed and D stays unaltered. The effect of this ramp is
studied in the following way: the cloud is first caught with a catch field gradientof ∂B/∂z = 53 G/cm (which is the optimum catch point, see previous section),
which is immediately followed by a linear ramp of 125 ms to a varying peak field
gradient and then again ramped down to 53 G/cm (compare figure 3.16). The
80

PhaseDuration [ms]

MOT
10000

CMOT
500

Molasses Rep. off
4
5

Ramp
250

TOF Det.
varied 1

10
Cycling
PowerAO1
10
RepumpPowerAO2
Cycling
Detuning AO00.5

δ9.74 ↔ δ = 0
7.76
6.1
0

Quadrupole
Coils AO7

2.0

0.2

varied
1.2

Figure 3.16: Sequence for ramp speed optimization: the atoms are caught with acatch field of 53 G/cm (AO7 = 1.2 V). The field is immediately increased linearly toa maximum field gradient within 125 ms and ramped down again in the same time
to its initial value, followed by a temperature measurement.

“temperature” after this sequence is determined by the usual technique. With
this method, different ramp speeds can be realized. An alternative method
would be to ramp up to the same value and vary the ramp duration instead.
However, such a procedure would suffer from the effects of the relatively shortcollisional lifetime in the vapor cell. Different heating rates are found forthe z and the y-axis. This is expected because the field gradient along thez-axis is twice that on the y-axis and the procedure does not allow enoughtime for rethermalization. The heating rate due to ramping can, however, beestimated. The direction-dependent parameters Tx , Ty , and Tz are measuresof the average kinetic energy of an atom in the specific direction:

hEkin i ∝ vx2 + vy2 + vz2 ∝ Tx + Ty + Tz .
81

(3.13)

After rethermalization every degree of freedom carries the same energy according to the equipartition theorem, so:Tx + Ty + Tz = 3Treth⇒

Treth =

1
(Tx + Ty + Tz ) .
3(3.14)

From fluorescence measurements, Ty and Tz can be extracted while Tx may
be assumed to be the same as Ty because of the rotational symmetry of thetrap in the x-y-plane. We assume that the non-adiabatic heating in a first
approximation is not only dependent on the ramp speed ∆b/∆t, but also, fora fixed rate ∆b/∆t, on the ramp duration ∆t, which is then proportional to∆b = bfinal −bcatch . Therefore, figure 3.17 shows the dependence of temperatureon the product of ramp rate and height ∆b2 /∆t additionally to the measured
data in 3.17(a). The red line gives the rethermalization temperature calculated with equation 3.14. Its slope is κramp = (6 ± 2) 100 (G/cm)K(G/cm/ms) . This
calculation yields for a typical ramp of 350 G/cm in 500 ms a non-adiabatic
heating of ∼ 15 µK. It must be mentioned that the error for this value is large
as anticipated from the scatter of the data in figure 3.17(b). The error is
K ms±2 100 (G
2 /cm2 ) . However, even in an extreme case, a full ramp from the catch

point to the maximal field gradient with ∆b/∆t = 350 G/cm/300 ms and the
K msupper bound of κramp = 8 100 (G2 /cm2 ) , leads to a non-adiabatic temperature

increase of only14 ∼ 33 µK in contrast to ∼ 245 µK15 by adiabatic heating.
Perfect adiabaticity would require an infinitely slow ramp, which is, of course,

K ms
350 G
G
∆Tnon−adiab = κramp ∆b∆t ∆b = 8.0 100 (G2 /cm2 ) × 300 cm ms × 350 cm = 33 µK
15∆Tadiab = T0 (η 2/3 − 1) = 245 µK for η = 8.2 and T0 ≈ 80 µK. The value for T0 is takenfrom the data in figure 3.14(b).
14824 0 0

4 0 0

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

5 0

z -a x is
y -a x isre th e rm . te m p .

3 5 0

z -a x is
y -a x is

3 0 0

T e m p e r a t u r e T [ µK ]T e m p e r a t u r e T [ µK ]

3 5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0
2 5 0
2 0 0
1 5 0
1 0 0
5 0
0

3 0 0

0

R a m p s p e e d ∆b / ∆t [ G / c m / 1 0 0 m s ]242∆b / ∆t [ 1 0 0 G(a) Raw data

6
2

/c m

8
2

1 0

/m s ](b) Temperature as function of ramp speed
and height

Figure 3.17: Effects of non-adiabatic compressing of the magnetic trap: figure (a)shows the data taken. In figure (b) the final temperature is plotted in dependence
on the ramp speed ∆b/∆t and times the ramp height ∆b. The red line gives therethermalization temperature calculated with equation 3.14.

not viable due to the limited lifetime in the vapor cell. These measurements,
again, were taken with |1, −1i-clouds.
Other measurements also suggest that the typical ramp rates used byus come quite close to adiabaticity: instead of ramping the field gradientback down to its initial value (49 G/cm, AO7: 1.1 V) as in figure 3.16, theparameters Ty and Tz were measured immediately after reaching the peakfield bmax gradient. From these values the rethermalization temperature Trethcan be calculated with relation 3.14 and compared to the expected value in an
adiabatic compression with an initial temperature of T0 = 80 µK. The lattervalue is the rethermalization temperature in the trap for a field gradient of b0 =49 G/cm read off from figures 3.14(b) and 3.17(b). As derived in chapter 2, thetemperature in an adiabatically compressed linear trap scales as T = η 2/3 T0 ,where η is the compression factor: b = ηb0 . Table 3.1 gives the measured values
83as well as the derived quantitites and the theoretically expected temperature
Tth .
bmax [G/cm]
124
353
398

tramp [ms]
250
750
750

η
2.5
7.3
8.2

Tz [µK]
262
428
453

Ty [µK]
136
270
289

Treth [µK]
178
323
344

Tth [µK]
149
300
325

Table 3.1: Comparison of measured temperatures and expected temperatures in an
adiabatic process, assuming the initial temperature before compression was 80 µK.
The initial field gradient b0 was 49 G/cm.

The results listed in the table show that the measured temperatures are
in reasonably good agreement with theory and that for typical compression
rates in table 3.1 non-adiabatic heating is small compared to the adiabatic
temperature increase.
To minimize the time spent in the vapor cell, the compression of themagnetic field can be done simultaneously with the start of the mechanicaltransport (see next section)16 . In this case, one needs to make sure that thetrap is already compressed enough upon entering the differential pumping tube
that it fits through, with the inner diameter of this tube being d = 10.9 mm.
For the maximum acceleration of the translation stage, the trapped atoms
enter the tube at t = 225 ms after the start of the transport. To estimate
by what factor the magnetic trap must be compressed by this time so that16

By a simple argument it can be shown that atoms are not lost by the acceleration of
the translation stage: the field gradient bz = ∂B/∂z of the catch field is chosen so that it
can support the atoms against gravity. The field gradient along the x-axis, which is the
direction of the transport, is half of the one along the z-axis: bx = 21 bz . The maximum
acceleration with which we operate the translation stage is 450 cm/s2 ≈ g/2 (with g being
the earth acceleration). So, bx should be equivalent to a force that has the same magnitude
as the inertial force acting on the atoms because of accelerated motion and thus hold theatoms in the trap.

84

only a small ratio of the atoms is chopped off by the tube, we assume purely
adiabatic heating and calculate numerically with equation 3.12 the radius ρ ofa circle inside of which 95% of the trapped atoms are held. The center of thiscircle is chosen to be the center of mass in the two-dimensional distribution
3.12 (which does not coincide with either the peak 2-D-density or the peak
3-D-density because of the anisotropy due to gravity). Figure 3.18 shows theobtained values. The point at 155 G/cm is the field gradient reached at thefront edge of the tube with a ramp rate of 47 G/cm/100 ms (which is thesame as in the last row in table 3.1) and simultaneous motion with maximum
acceleration: 95% of the atoms are within a radius of 3 mm. This means thatthe cloud should be able to pass the tube without significant losses as longas the center of this circle is not significantly displaced from the center of the
tube and the path through the tube is parallel with respect to the axis of the
tube. From the results of other measurements (section 3.9.3), we can conclude
that the translation stages moves the trap apparently straight through thetube as we lose as little as 35 % on a two-way transport from the vapor cell tothe science cell and back.
We conclude, therefore, that the compression rate will be determinedby the speed of the first section of the transport since even for the maximum
acceleration the compression can still be slow enough to guarantee adiabaticity. If one chooses, however, to wait with the transport, leaving even moretime for the compression, one will apparently win only little in lowering thetemperature, but accept a loss of atoms due to background collisions which ismuch more severe than the loss of phase space density in a faster ramp.

85

4
2
0
2
4
6
6
= 1
F

m
F

[mm]

9 5 % - r a d iu s ρ[ m m ]

m

= 2

4

8
10

4

5

0

5

10

5

0 [mm]

5

10

2
2

0
2
0
0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4

4 5 0

F ie ld g ra d ie n t b z [G /c m ]10(a) Calculated cloud size ρ for different compression factors (adiabatic)(b) Cloud in MT for two compression factors

Figure 3.18: Figure (a) shows the dependence of the 95%-radius on the field gradient.
Figure (b) shows a visualization of the trapped cloud for the catch field gradient
49 G/cm (top) and 155 G/cm (bottom, the field gradient when the tube is reached).
The green circle gives the size of the tube that the cloud enters during the transport.
The red dashed circles are the 95%-radii ρ from (a) for mF = 1-atoms.

3.8

Transport

The magnetically trapped cloud is moved on a z-shaped path by the translation stage to the science cell with ultra-high vacuum suitable for evaporative
cooling. The path is shown in figure 3.19. The trap first passes the differential
pumping tube, enters the “square chamber”, makes a turn to the left, makes
another turn to the right, and comes to a stop in the science cell.
The optimization of the transport efficiency to the science cell comprises
two aspects: first, the effects of the acceleration on the temperature of thecloud have to be studied. Especially the first part of the motion, which is to86

bring the atoms to the square chamber, should be fast in order to leave thevapor cell as quickly as possible. Second, the correct path has to be found to
avoid accidental bumping of the cloud into pieces of the surrounding vacuum
apparatus. For the following discussion we use a coordinate system as definedin figure 3.19. Luckily, the very first path programmed did not lose its entire

y

0.5m

z

science cell

vapor cell(-2.600, 2.159)(-42.024, -6.351)

Path 2Path 3x(-2.159, 14.200)

Lesker
GMPS-SPL024

Lesker
GMPS075F1B

1

Le
GM sk
PS er
15
0F
3B

Path 1(-2.600, -6.351)

differential pumping tube(“thin tube”)

Figure 3.19: Defintion of the lab coordinate system and path of the transport. The
picture also contains the coordinates limiting the final transportation path.

load and it was possible to detect atoms after loading, compressing the trap,
and moving it all the way to the science cell and back. Only minor changes ofthis path were necessary. The different endpoints of the three (partial) paths
in figure 3.19 were determined by a simple procedure: the compressed trap
was moved to a specific point, which was varied in small intervals, moved backagain, and the atom number was determined. Figure 3.20 shows the results for87the adjustment of the endpoint of path 3 in the science cell. The large scatterin the results for the y-position are probably due to the problems with theabsorption imaging (see appendix B.2), which was used. The experiment for
the y-axis was performed for three different durations for which the trap washeld in the glass cell. From these results, it might be inferred that the curve
gets sharper for longer holding times. This can be explained by the assumptionthat even for a fully compressed trap (the field gradient was 353 G/cm, A07= 8.0 V) there are still some losses from collision with the walls, which willdiminish the number of atoms more for positions close to the walls than in
the middle of the glass cell, and thus a difference in the beginning will become
more pronounced. This implies that the lifetime depends on the position in
the glass cell. The figure also shows an analog measurement of the x-axis.
Obviously, the cloud reaches the end of the science cell at ∼ 18 cm. The final
endpoint of path 3 will be determined by the installation of the TOP coils andprobably be around 16 cm. Therefore, the dimensions of the glass cell in thisdirection will not be critical.
To measure the heating by the transport, we moved the trapped cloudwith different accelerations and velocity profiles to the square chamber andback and performed then temperature measurements. We did not find any
significant heating. This could, however, be due to the fact that for the shorttransfer times to the square chamber and back sufficient rethermalization is
not guaranteed. If one expects the heating by the transport to act mainly
along the direction of the transport (x), then this temperature increase could
not be detected as the x-direction is also the optical axis of our imaging system.

88

0 .8

0 .4 00 .7

0 .3 50 .6

fra c tio n

0 .2 50 .2 00 .1 5a to ma to mfra c tio n

0 .3 00 .1 0

0 .5
0 .4
0 .3
0 .2
0 .1

0 .0 50 .0 0

0 .0

1 .6

1 .8

2 .0

2 .2

2 .4

1 .6

2 .6

1 .8

2 .0

2 .2

2 .4

y -p o s itio n in s c ie n c e c e ll [c m ]y -p o s itio n in s c ie n c e c e ll [c m ](a) thold = 30 s(b) thold = 5 s

2 .6

0 .8

0 .8
0 .7

0 .6

fra c tio n

fra c tio n

0 .6
0 .5
0 .4

a to ma to m

0 .3
0 .2

0 .4

0 .2

0 .1
0 .0
1 .6

1 .8

2 .0

2 .2

2 .4

0 .0

2 .6

0

5

1 0

1 5

2 0

x -p o s itio n in s c ie n c e c e ll [c m ]y -p o s itio n in s c ie n c e c e ll [c m ](c) thold = 0 s(d) x-axis

Figure 3.20: Optimization of transport: figures (a), (b), and (c) show the atom
number after a full motion to the science cell and back, width the y-position in thescience cell varied. The cloud was held for different durations thold in the sciencecell. The inner width of the glass cell in y-direction is 1.0 cm. Figure (d) shows theresult of the same experiment with the x-position varied and the y-position held
constant at 2.16 cm. Obviously, the cloud hits the end of the cell for x-values of∼ 18 cm and larger.

89

Recent temperature measurements in the science cell, however, also suggestthat there is indeed no significant heating by the transport. In the science cell,the temperature is determined by the measurement of the cloud’s expansion
in the y- and z-direction, which allows to study the effects of the first very
fast transport motion to the square chamber. Given the racy motion of thetranslation stage in the experiments, it surprises that the transport does notcause additional heating. Though, if we apply the equivalence principle andcompare the motion with the compression of the magnetic field, we find thatthe acceleration and deceleration of the stage should be an adiabatic process forthe atom cloud. If the stage is moved with acceleration a, the atoms experiencea force Facc = ma. The change of this force in time can be compared with thechange of the trapping potential in a ramp of the magnetic field. Figure 3.21
shows the motion profile of the transfer with maximally realizable acceleration
to the square chamber. The change of the inertial force is Ḟacc = mȧ. In figurecms Velocity Profilecms2Acceleration Profile
400
200
0.2

0.4

-200
-400

0.6

0.8

s

80
60
40
20
0.2 0.4 0.6 0.8

(a)

(b)

s

cm Position Profile
35
30
25
20
15
10
5
s
0.2 0.4 0.6 0.8(c)Figure 3.21: Motion profile of the transport to the square chamber with maximum
acceleration. The transfer takes 0.84 s.

3.21(a), the acceleration is increased by ∆a = 450 cm/s2 within ∆t = 210 ms.
If we set the resulting change of the inertial force Ḟacc = m ∆aequal to the∆tchange of the magnetic force in a compression of the trap Ḟmag = gF mF µB ∆b
,
∆t
90

G
the motion is equivalent (for mF = 1) to a ramp rate of ∆b= 6.7 cm 100.
∆t
ms

The total acceleration increase of ∆a = 450 cm/s2 compares to a ramp heightof ∆b = 14 G/cm. The corresponding product of ramp rate ∆b/∆t and ramp
height ∆b as in figure 3.17(b) is 0.01 (100 G2 )/(cm2 ms). This is a very slow
rate, and inspection of figure 3.17 confirms that it should be in very good
approximation adiabatic. Thus, only extremely fast changes in acceleration,
which are mechanically impossible, could cause a heating of the cloud. We
also think that the transport is smooth enough to rule out the possibilityof bumpiness causing abrupt small changes of the motion that would heat
the cloud. We conclude that we can use motion profiles with the maximum
acceleration supported by the translation stage and its mechanical limitations,, in order to minimize the time spent in the vapor cell.
which is amax = 450 cm
s2
For the paths 2 and 3 in figure 3.19, speed is not critical and we chose slower
motions to avoid unnecessary stress of the translation stage.

3.9

Feasibility assessment for BEC

This section discusses a few general aspects of the efficiency of the entire preevaporation sequence as well as some figure of merits that will allow us toestimate the chances for successful evaporation.
913.9.1

Dependence of the transport efficiency on the background pressure

The background pressure in the vapor cell plays a critical role: it determines
the maximally possible atom number (see equation 2.27, as long as the partialpressure of non-rubidium atoms is not negligible), but also limits the lifetimeof a magnetically trapped cloud. Considering an entire sequence there are
two competing effects: increasing the background pressure raises the atom
number in the MOT, but increases the losses during the transport to thesquare chamber as well. Consequently, for a fixed transportation time (which
is 840 ms if the maximum acceleration is used) to the square chamber, there isan optimum pressure, respectively lifetime. An approximate orientation about
this parameter was obtained by opening the rubidium oven and thus increasing
the background pressure while measuring the development of the lifetime, thenumber of atoms trapped in the MT, and the number of atoms that could
be detected after the trap had been moved to the square chamber and back.
Our measurements suffered mainly from the time necessary to conduct them(∼ 3 hours for the full data set). On this time scale, laser drifts could not be
avoided, tampering the results. Figure 3.22 shows the data obtained. The
sudden jumps in the absolute atom numbers directly correlate with locking
problems of the lasers. The maximum number of atoms after transport seems
to be obtained around run 20. That means that one should aim to maintain alifetime of ∼ 2 s in the vapor cell. In a real sequence, one would only transport
one way, so the optimum would shift to an even lower lifetime.
92

4
lifetime Τ @sD

9

1 .6 x 1 0

w ith o u t tr a n s p o r t ( 1 0 0 m s h o ld , A O 7 = 3 .0 )1 .2 x 1 0

98 .0 x 1 0

8

4 .0 x 1 0

8

3
2
1
0
0

20

40 60 80 100
run number(b) Lifetime τVC

0 .0
0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

ru n n u m b e r(a) Absolute transport efficiency: atom number in reference run and after transport

ratio NmovNref

A to mn u m b e r

a fte r tra n s p o rt to s q u a re c h . a n d b a c k

0.6
0.5
0.4
0.3
0.2
0.1
0

0

20

40 60 80 100
run number(c) Relative transport efficiency

Figure 3.22: The absolute efficiency of the transport is pressure-dependent: increasing MOT atom number and rising transport losses are competing effects resulting
in an optimum background pressure (figure (a)). The lifetime (b) and the relativetransport efficiency (c) decrease with rising pressure. The runs were taken in equalintervals over a time of three hours after the rubidium oven had been opened. The
sudden jumps are caused by problems with holding the lock frequencies of the lasers.
The ratios of atoms before and after the transport cannot be compared to those presented in section 3.9.3 because the measurements were done for |1, −1i-atoms andthe vapor cell was not isolated from residual stray light coming from the laser boxes
at that time. The temperature of the rubidium oven was 30 °C.

933.9.2Lifetime in the science cell

One very crucial parameter for the lifetime of atoms held in the science cellis, of course, the vacuum, respectively the pressure. Because of the necessary
rethermalization processes, evaporation cooling is a (on the timescale of atomic
physics) lengthy procedure: one evaporation cycle in the first BEC produced
lasted 70 s [5]. A rule of thumb is that the lifetime of the trapped cloud in theenvironment where the evaporation is executed should be at least 150 times
the initial elastic collision time in the cloud [53].
We determine the lifetime τSC in the science cell by a similar method
as in the transport optimization: the trap is moved to the science cell, kept
there for a varying time, and moved back to the vapor cell where the atom
number is measured by absorption imaging. A simple exponential decay law
is fitted to the data obtained. The lifetime τSC was measured at two different
y-positions of the glass cell yielding two different values. This indicates againthat the compressed trap is still so big that the walls of the science cell dohave an influence on the lifetime, and that care must be taken to move theatoms exactly to the middle of the cell, found in section 3.8. The data again
refers to measurements with |1, −1i-atoms; clouds consisting of |2, +2i-atoms
are more compact, and thus the influence of the walls should be smaller. From
the results in figure 3.23(c), we deduce a lifetime in the glass cell τSC of ∼ 100 s.
A lifetime exceeding 100 s was reconfirmed for |1, −1i-atoms in more recent
measurements by means of (more reliable) fluorescence imaging. That value
for the lifetime should be long enough for evaporation until condensation underusual conditions.
94

0

20 40 60 80 100 120

Time thold [s](a) y = 2.36 cm,
177 G/cm

b =

1
0.8
0.6
0.4
0.2
0

τSC = 66 s

0 25 50 75 100 125 150 175

Time thold [s](b) y = 2.16 cm,
177 G/cmb =atom number [a.u.]τSC = 37 s

atom number [a.u.]

atom number [a.u.]

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0

τSC = 104 s

0 25 50 75 100 125 150 175

Time thold [s](c) y = 2.16 cm,
353 G/cmb =Figure 3.23: Lifetime in the science cell: the data in figure (a) was taken at y =2.36 cm. The trap was then moved to a better postion (y = 2.16 cm, figure (b)). In
figure (c) the steepness of the trap was twice as high as in (a) and (b). The scatterof the data is caused by the problems with absorption imaging, which was used,described in appendix B.2.

3.9.3

Overall efficiency of the entire sequence

As all the optimization steps described in the previous sections were done separately, an overall estimation of the entire preparation sequence is not possibleon the basis of the data from these sections. The fluctuations and changes ofparameters like vapor cell pressure, laser power, laser stability, beam alignment, etc. influence the results too much and do not allow comparison of thespecific data, especially regarding the values for the total atom numbers. For
this reason, an analysis of the overall sequence was performed by measuring
the total atom number after each single step, i. e. MOT, CMOT, molasses,
MT, and transport. This procedure allows also a “fine tuning” of, e. g., theCMOT or the catch point of the magnetic field, as it turns out that the optimaof these parameters depend on the total atom number.
Figure 3.24 shows the result of one of those measurements. The sequencethat was used for this measurement is illustrated in appendix C. In a MOT,
∼ 9×109 atoms can be captured if the rubidium pressure is sufficient. In
95

Atom number relative to MOT number

1.2
1.0

9.2×109

8.9×109
7.2×109

0.8
0.6

3.4×109

0.4

2.2×109

0.2
0.0

MOT

CMOT

Molasses

MT Two-way transport

Figure 3.24: Overall efficiency of the preparation sequence: the figure shows theatom number after each step. The last two columns are clouds consisting of a
mixture of |2, +1iand |2, +2i-atoms with a ratio of ∼ 70 % in mF = 2 to ∼ 30 %in mF = 1 (see section 3.6.1). The images of the clouds at each step are shown
above. It can be seen that they are often not regularly and nicely shaped. The oval
form of the cloud in the last image, however, reflects the shape of the quadrupole
potential as the time duration of the two-way transport provides enough time forrethermalization. The color scale for the five images is dynamic.

96

practice, this can be guaranteed if the oven is opened every day before theexperiments for a duration of ∼ 30 min so that the pressure can build up.
We have measured MOT atom numbers as high as 1.2×1010 . The next step,
the CMOT, is very sensitive to the atom number. While for small clouds
the value of the detuning δ during compression is not very critical, carefulattention must be given to it for large atom numbers. For the measurement
in figure 3.24, the detuning during the 500 ms CMOT had to be changed to alinear ramp from −4.9 Γ to −6.3 Γ while the magnetic field was ramped only
to a gradient of 66 G/cm (AO7 = 1.5 V). Large clouds extend to regions withlarge Zeeman shifts, which are scaled up during the compression. For thisreason, the detuning has to be readjusted during the compression in order to
not lose atoms in the outer regions of the cloud. After optical molasses andpumping, we are able to detect ∼ 50 % of the remaining atoms in the MT after
a catch with the magnetic trap and a compression of the trap to its maximum
value within 750 ms. This number is in good agreement with our analysis of
the optical pumping to trappable states in section 3.6, where we found an
efficiency of ∼ 50 % for large clouds. The last column in figure 3.24 represents
the number of atoms detected in the vapor cell after a round trip of the fullycompressed magnetic trap from the vapor cell to the science cell and back. It
shows that ∼ 65 % of the original load of the trap remained. This ratio wasincreased from ∼ 35 % to this value after we had eliminated every possibiltyof stray light by covering the laser boxes with black paperboard. We assume
the losses to be caused mainly by collision with background gas atoms. The√
efficiency of the one-way transport would then be 0.65 = 80 %. We can97thus expect to begin evaporation with an atom number of 2.5-3.0×109 atoms,
which is in comparison with other experiments a promising starting point forsuccessful Bose-Einstein condensation.

3.9.4

Rethermalization rate

The preparation in the vapor cell and the transport to the science cell will befollowed by evaporative cooling. The speed of an evaporation cycle is determined by the rethermalization rate in the magnetically trapped atom cloud
because after every evaporation “step”17 the atom cloud needs to be given
enough time to return to equilibrium. The lifetime of trapped atoms in thescience cell, thus the time in which the complete evaporation must be performed, is long, but not infinite. For this reason, the initial rethermalization
time is an important parameter to estimate the feasibility of Bose-Einsteincondensation. The initial rethermalization time18 is the duration which thethermal cloud (in the fully compressed magnetic trap) needs to return to equilibrium after a perturbation of its velocity distribution. It can be measured,e. g., by suddenly increasing the magnetic field gradient. This would cause ahigher average velocity increase along the strong magnetic axis (z) than along
a weak axis (e. g. y) and thus increase the width of the cloud along z, σz , inrelation to the width along the y-axis, σy . After a certain rethermalization
time τreth , the cloud will have re-equilibrated and the ratio of σz and σy will
have reached a constant value.
17
18

In reality evaporation is not executed in discrete steps, but continuously.
“Initial” because this time decreases during evaporation.

98

Practically, one can determine τreth by measuring the ratio of the two
widths σσyz (t)as a function of time t. The duration t is the delay time between(t)perturbation and detection. We employ the procedure illustrated in figure
3.25 to measure the rethermalization rate directly in the science cell. The
stage

MOT, CMOT,
catch
molasses, pumping & ramp
10.5 s
750 mstransport
4s

rep
ara
t

ion

duration

rethermalizationin science cell
10 s

9.0

adiabatic ramp
down
10 s

suddenturn-off & detection
swith-ont
2.5 ms turn-off(varied) 100 μs detection
9.0

clo
u

dp

4σy

Quadrupole
Coils AO7

0.8

0.8

4σz

Figure 3.25: Determination of the rethermalization rate: the thermal equilibriumin the cloud is perturbed by a sudden switch-on of the magnetic field, which causes
different heating along the strong and the weak axis of the trap. The time t thatis given the cloud to return to equilibrium in the fully compressed magnetic trap
is varied. By measuring the aspect ration σz (t)/σy (t) as a function of t, one candetermine the initial rethermalization time τreth .

cloud is prepared in the usual way and transferred to the science cell, where
it is held with the maximum field gradient for 10 s so that equilibrium isguaranteed. Then, the trap is slowly opened by ramping the field gradient
down to its catch value. This is immediately followed by a sudden switchon of the full magnetic field, which causes asymmetric heating of the atoms
along the strong and the weak axis of the field. We then vary the time t for
which the cloud is held at maximum field. Doing this, we can determine therethermalization time τreth by measuring the aspect ratio after a hold time t
in the compressed trap as explained above. The images for this procedure aretaken with absorption imaging in the science cell. The data obtained is shown99

in figure 3.26. We measure a rethermalization time of τreth ∼= 2.0 s. This timeA s p e c t r a t i o z ( t ) / y ( t )

0 .9

0 .8

r e t h = 1 . 9 8 s

0 .7

0 .6

0 .5
0

1 0

2 0

3 0

4 0

D e la y tim e t [s ]Figure 3.26: The figure shows how the aspect ratio of a thermal cloud of |1, −1iatoms returns to its equilibrium value. The initial rethermalization τreth time in thescience cell is 2.0 s.

constant is connected to the elastic collision rate τcoll by τreth = 2.7 × τcoll[70]. It has been shown with Monte Carlo simulations that for successful
evaporation the lifetime τSC of the magnetic trap should allow ∼ 150 or more
collision events [53]. With the lifetime τSC = 104 s (section 3.9.2), our values
yield ∼ 140 collision events.

3.10

Conclusion

We have implemented an efficient pre-evaporation sequence for our BEC apparatus. After collection of ∼1×1010 atoms in the MOT from a dilute rubidium
vapor, the atoms are compressed in the CMOT and cooled by optical molasses. Parameters were found for which the losses in the CMOT and molasses
100

stage could be minimized and efficient cooling well below the Doppler limit to∼ 25 µK was realized. For the molasses, we measure a phase space density D
on the order of 1×10−5 . Optical pumping brings about half of the atoms totrappable states. The subsequent catch with the quadrupole field was matchedto the cloud size after molasses and pumping. The temperature after the catchis ∼ 80 µK. The trap is quasi-adiabatically compressed to its maximum field
gradient while a simultaneous transport of the trapped cloud to the sciencecell is initiated. We measure a phase space density for the fully compressed
trap of ∼ 1.5-2×10−7 . More than 2×109 atoms could be detected in the vapor
cell after a round trip of the magnetic trap to the science cell and back. The
lifetime of the atoms in the compressed trap was measured to be ∼ 100 s in
the science cell, where they rethermalize within an (initial) time constant of
2 s. Thus, the lifetime multiplied by the elastic collision rate of the trapped
atoms is ∼ 140, which is a promising value for this parameter.
All these parameters indicate very good prospects for a close realizationof Bose-Einstein condensation. The phase space density in the trap as wellas the collision rate are comparable to the values in other experiments while
the atom number in the trap before evaporation is even slightly higher. Afterthe development of an adequate evaporation procedure for our experimental
situation, Bose-Einstein condensation should be achieved very soon. By thesubmission date of this thesis, a considerable increase in phase space density
has, in fact, already been observed after the first trials of evaporative cooling.

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110

Appendix A
Detailed explanation of lin ⊥ lin-polarization
gradient coolingThis appendix gives a more detailed treatment of the model for lin ⊥ lin
cooling proposed in [58] than the main chapter 2.
In chaper 2 it was discussed that the lin ⊥ lin-configuration effects aspatially varying polarization that changes from circularly to linearly polarized,
again back to circularly, but with opposite helicity and so on. The light shift0hφe |ˆ~r| φg i.
discussed in section 2.3.3 depends on the Rabi frequency Ω = −eE~

The matrix element in that expression describes the transition. For a realatom with magnetic substates, the dependence on the polarization yields thewell-known selection rules: circular σ + -light, e. g., couples transitions with
∆mJ = +11 , light with σ − -polarization transitions with ∆mJ = −1, andlinearly polarized light transitions with ∆mJ = 0. Spontaneous emission,
however, can happen with any polarization, thus the selection rule for it is∆m = 0, ±1. We want to consider atoms with angular momentum Jg = 1/2 in
1

The quantum number mJ here is a magnetic number of an arbitrary total angularmomentum, F for low fields. We do not consider atoms in the Paschen-Back regime.

111me = − ³⁄²(a)me = − ³⁄²

−½

+½

σ–

−½

(b)

+ ³⁄²

σ–

mg = − ½

σ–

+½

+½

+ ³⁄²

σ–

+½
mg = − ½
me = + ½(c)EnergyEg
mg = + ½v
mg = − ½

(d)

z
⁄8

0

λ

+45°

σ–

⁄4

⁄

λ

3λ8

σ+
–45°

⁄

⁄2

5λ8

+45°

σ–

λ

–45°

Figure A.1: lin ⊥ lin-polarization gradient cooling: figure (a) illustrates opticalpumping. The solid lines are the transitions possible with σ − -light, the dashed ones
are the possibilites for spontaneous emission. Figure (b) shows the light shifts for thesubstates in σ − -light. Figure (c) visualizes the Sisyphus effect in this configuration.
Figure (d) shows how the polarization changes within half a wavelength. The figurescontain much information. Full explanations are given in the text.

112

the ground state and Je = 3/2 in the excited state as an example. If such an
atom has mg = 1/2 in the ground state, then it cannot go to the mg = −1/2state: excitation can only drive the atom to me = +3/2, from where it canreturn only to me = +1/2 by spontaneous emission. In contrast, for atoms
with mg = −1/2 there is a chance to end up in the mg = +1/2-state: eventually,
all atoms will consequently gather in the mg = 1/2-state. This process iscalled optical pumping, illustrated in figure A.1 (a). The matrix element
also expresses the strength of a transition, which is directly proportional to
the light shift. In σ + -light, the matrix element and thus the light shift isthree times larger for the mg = +1/2 → me = +3/2-transition than for the
mg = −1/2 → me = 1/2-excitation (see figure A.1 (b)); the situation for σ − light is the same, but with opposite signs. For linearly polarized light, thetransitions are equally strong. In the light field of equation 2.30, this createasa spatially varying light shift shown in figure A.1. The cooling effect is now
based on the following model which is visualized in figure A.1 (c): suppose
an atom travels with velocity v in positive z-direction and is in the state
mg = −1/2 at z = λ/8 where the light is σ − -polarized. Because of the selection
rules, it cannot be pumped to mg = +1/2. We assume that it has traveled toz = 3λ/8 by the next absorption event. This means that it had to climb apotential hill and has converted kinetic energy to potential. At the top of thehill, the light is σ + -polarized, so the atom will be pumped to the mg = +1/2,
which has a lower energy at that position than the mg = −1/2-state. From
there, it can climb the next potential hill and again be pumped to a lower
state. To summarize in energy terms: by climbing the hill, the atom converts

113

kinetic energy to potential energy, which is then emitted to the light field.
The atom loses kinetic energy, thus it is cooled. The mechanism is often
called Sisyphus cooling because the repeated up-and-down in the potential
reminiscent of the famous Greek myth. One can easily identify two limitations
for this mechanism: if the atom is too fast, it will go over the potential hills
without being scattered. If the atom is too slow, one can no longer assume thatthe pumping is delayed until the atom has reached the top of the hill: pumping
then happens quasi-instantaneous as soon as the polarization contains a σ + part2 . A time lag of the optical pumping with respect to the motion of theatom is, therefore, important such that the population of the states does notinstantaneously follow the corresponding polarization at the atom’s position.
The optimum velocity vopt is given when one pumping event happens for every
climbing of a hill, so vopt ≈ λΓsc /4.

2Note that linear polarization can be thought of as superposition of circularly polarized
fields with opposite helicity.

114Appendix B
Apparatus details

B.1

Vignetting

Vignetting [62] is a common effect in optics. We will discuss its influence on
our measurements. In optical systems with more than one aperture, images ofobjects that are distant from the optical axis have a smaller luminosity thanthose of objects on axis. Consider the situation depicted in figure B.1 that
corresponds to the situation in our imaging setup. The gray shaded areas
symbolize the solid angle that can theoretically contribute to a point’s image
due to the finite size of the second aperture (a). If aperture (lens) 1 is notlarge enough, this angle will be diminished for off-axis objects resulting in adarker image than for a point on the axis of the same luminosity (b). We havecalculated that for our experiment vignetting becomes a problem for point-likeobjects that are 21.5 mm or more away from the optical axis. If one assumes
the diameter of a cloud to be 6 mm, this relates to a time of flight of 56 ms.
Since we have never used such long times, the effect does not influence our115measurements.
Another minor effect with similar consequence is the variation of theangle α(r) in figure B.1 (a). If l = f1 , α is constant. In our case l is ∼ 52 cm,
so almost equal to f1 = 50 cm: Even for objects at a distance where vignetting
becomes relevant, the variation of α(r) is less than 0.01% and can thus beneglected.
(a)second aperture (shutter)

α(r)

r

α

d = 6 mm
f1 = 500 mm
l = 520 mmdfirst aperture (lens 1)(b)

not contributing

r

d

f1

l

Figure B.1: Vignetting: in our setup, the variation of the angle α(r) in figure (a),respectively the solid angle Ω(r) that contributes to the image of a point is small
over the size of the cloud: α(r) varies by less than 0.01% (figure (a)). Figure (b)shows the effect of vignetting: the angle α(r) is cut down because the diameter ofthe first aperture (the lens) is not large enough. Four our values this becomes aproblem for r > 21.5 mm, which would only be relevant for very long times of flight.

B.2

Laser frequency jitterIn the course of the measurements for this thesis, a general inreproducibility for the atom numbers detected by absorption imaging was discovered. In
contrast, the values obtained by fluorescence measurements, were reproducible116within 5 %. This discrepancy could be modeled by a rms-frequency jitter of thecycling laser of ∼ ±0.5 Γ = ±3.0 MHz. The specifications of the manufacturer
for the linewidth of the laser is < 1.0 MHz. The frequency jitter influences theatom number obtained by absorption imaging more than the one obtained byfluorescence imaging for three reasons. The absorption imaging is on-resonancewhile the fluorescence imaging was performed with a detuning of −3.5 Γ. A
shift of the assumed frequency by a jitter weighs relatively more on-resonancethan off-resonance as can be seen from equation 2.15. Thus, the jitter has alarger influence on absorption imaging than on fluorescence imaging. It wasalso found that the jitter is periodical with a frequency of 2 kHz, which corresponds to a period length of 500 µs. The exposure time in absorption imaging
is 100 µs while we used 1 ms for fluorescence imaging. Consequently, the jitter effect averages out in fluorescence imaging, but not in absorption imaging.
This was confirmed by an increase of the exposure time for absorption imaging:
the fluctuations in atom number decreased for longer exposure times. Furthermore, fluorescence imaging is performed well above the saturation intensity,
but absorption imaging far below. The power broadening for the fluorescencemeasurments reduces the sensitivity to frequency fluctuations. It was also
possible to correct out the jitter of the frequency both for absorption andfluorescence imaging by monitoring and recording the frequency jitter during
detection from the spectroscopy signal and calculating correction factors fromthis data. The jitter originates from instabilities of the voltage applied to thepiezo-electric crystal of the laser. This was confirmed by measuring both thefrequency jitter from the spectroscopy signal and the piezo voltage separately.

117

The two curves show a clear correlation. The laser was sent back to its producer, but this insufficiency could not entirely be removed, which means thatthe light of the cycling laser is not usable for quantitative analysis. All resultsin this thesis with information about absolute atom numbers were, therefore,
obtained by fluorescence imaging.

B.3

Calibration of the analog channels

The calibration of the analog channels of the timing computer for cycling
detuning (AO0), cycling power (AO1), repump power (AO2), and quadrupole
coil current (AO7) define the “lab units” (in V applied to the channels) which
are sometimes used in this thesis. For reference, the calibration curves are
given in the following figure B.2.

118

1.0
Γ
= 1.62 × AO0 − 15.85 ΓV-2

relative intensity

detuning [Γ]

0

-4
-6
-8
-10
3

4

5

6

7

8

9

AO0-voltage [V]

0.8
0.6
0.4
0.2
0.0

10

0

relative intensity

1.0
0.8
0.6
0.4
0.2
0.0

0

2

4

6

4

6

AO1-voltage [V]810(b) cycling power

quadrupole coil current [A](a) detuning δ2

8

AO2-voltage [V]10(c) repump power

500
400
300

I = 50.1

A
× AO7
V

2

4

200
100
0

0

6

AO7-voltage [V]8(d) coil current

Figure B.2: Calibration curves of the analog channels that define the “lab units”.
The data was taken by H. Ruf. The field gradient b can be calculated from the
AO7-voltage by use of the regression in (d) and the field gradient per current b/I =0.88 G/(cm A).

119

10

Appendix C
Sequence illustration for overall efficiency
experiments}

optional

Phase
Duration [ms]

MOT

CMOT

10000

500

Molasses (Pump 1) (Pump 2) Magnetic Catch & Ramp
5

5

750

8

Transport (cont.)

TOF

Det.

6650 (two way)

4

1

250 mW
Cycling
PowerAO1
100 %

100 %

RepumpPowerAO2

2%0Γ

0Γ
Cycling
Detuning AO0

δ− 3.5 Γ− 3.5 Γ− 3.5 Γ− 4.9 Γ− 6.3 Γ− 6.0 Γ400 G/cm

Quadrupole
Coils AO7

8.8 G/cm

66 G/cm
49 G/cm

Pumping
Bias Field
Transport

Figure C.1: The figure illustrates the preparation sequence that was used for the last
data point (“two-way transport”) in the overall efficiency measurements described in
section 3.9.3. The atom numbers for the other stages (“Mot”, “CMOT”, “Molasses”,“MT”) were taken with the same parameters, but only the corresponding steps
activated.

120

Appendix D
Abbreviations and rubidium properties

The following symbols may be used with a global meaning in this thesis. If they
are not differently defined in the specific context of a section, they represent:cthe speed of light in vacuum: 299 792 458 ms~

reduced Planck constant: 1.054 571 628 × 10−34 kgsm

eelementary charge: 1.602 176 487 × 10−19 A sµB

Boor magneton: 9.274 009 15 × 10−24 A m2uatomic mass unit: 1.660 538 782 × 10−27 kg

kB

Boltzmann constant: 1.380 6504 × 10−23 kgK ms2

gstandard acceleration of gravity: 9.806 65 sm2

xvariable of a Cartesian coordinate system with variables x, y, zyvariable of a Cartesian coordinate system with variables x, y, zzvariable of a Cartesian coordinate system with variables x, y, zˆa unit vectorndensity of a gas, usually the density of a rubidium cloudN

the amount of something, usually the number of rubidium atoms

22

in a MOT, MT, etc.
Vσvolume√1/ e-width of a Gaussian distribution
121

D

phase space densitymmass of the isotope 87 Rb of rubidiumvvelocityωllaser light angular frequency [2π · Hz]ωaatomic transition angular frequency: ∆E/~ [2π · Hz]ωatomic transition or laser angular frequency when difference between ωl and ωa is not relevantδdetuning of the laser from resonance: ωl − ωa [2π · Hz]

λwavelength: 2πcωkwavenumber: 2πλΩ

Rabi frequency:Γnatural linewidth of transition [2π · Hz]Γsctotal scattering rate

σscscattering cross sectionI

intensity of laser beam(s)Issaturation intensity: 2π3λ~cΓ
3s0saturation parameter: IIs

σ+left-circular polarization, helicity: +1σ−~Eright-circular polarization, helicity: −1
an electric fieldBmagnitude of magnetic fieldbjmagnetic field gradient in ˆj -direction (j ∈ x, y, z) at field zero:√ω2 + δ22∂B∂j ~r=(0,0,0)bmagnetic field gradient in z-direction: b := bz

Jtotal angular momentum number in theoretical considerations

gJ

Landé g-factor of J

mJ

magnetic quantum number of J

F

~ +S
~ + I~
quantum number of total angular momentum F~ = L

gF

hyperfine Landé g-factor of F

122

mF

hyperfine magnetic quantum number of FT

temperatureTD~ΓDoppler temperature limit: 2k
B

BEC

Bose-Einstein condensate or Bose-Einstein condensationRbrubidium

MOT

magneto-optical trap

CMOT

compressed magneto-optical trapMT

magnetic trapTOP

time-orbiting potential

CCD

charge-coupled device

AOM

acousto-optical modulatora. u.

arbitrary units

FWHM

full width at half maximum

123Below some physical properties of 87 Rb are listed. The list contains only
data relevant in equations used in this thesis. All spectroscopic data relates tothe 5 2 S1/2 → 5 2 P3/2 -transition (D2 ). The values are taken from reference [48].

Z

Atomic number: 37

Z +N

total nucleons: 87matomic mass: 86.909 180 520 uTMmelting point: 39.31 °C

pVvapor pressure at 25 °C: 3.0×10−7 torrI

nuclear spin: 3/2

ωafrequency: 2π · 3.842 304 844 68 × 1014 Hz

λawavelength in vacuum 2πc= 780.241 209 686 nm
ωa~ωatransition energy: 1.589 049 439 eVkawavenumber 2π= 2π · 12 816.549 389 93 cm−1λaΓnatural line width (FWHM): 2π · 6.065 MHzτlifetime: 26.24 ns

TD

Doppler temperature limit: 146 µK

gF =2 (5 2 S1/2 )hyperfine Landé g-factor for 5 2 S1/2 , F = 2: 1/2

gF =1 (5 2 S1/2 )hyperfine Landé g-factor for 5 2 S1/2 , F = 1: −1/2

Isat (F = 2 → F 0 = 3, iso)effective saturation intensity for cycling transition
with isotropic polarization: 3.576 mW
cm2

Isat (F = 2 → F 0 = 3, σ + )saturation

intensityforcyclingtransition|F = 2, mF = +2i→|F 0 = 3, mF 0 = +3i: 1.669 mW
cm2124