Evaporative cooling and Bose-Einstein Condensationof Rb-87 in a moving-coil TOP trap geometry

A Thesis Presented
byDaniel Guenther Greifto

The Graduate Schoolin Partial Fulfillment of theRequirements
for the Degree ofMaster of ArtsinPhysics
Stony Brook University
December 2007

Stony Brook University
The Graduate School
Daniel Guenther GreifWe, the thesis committee for the above candidate for the Master of Arts degree,
hereby recommend acceptance of the thesis.

Dominik Schneble, Thesis Advisor
Assistant Professor, Department of Physics and Astronomy
Stony Brook University

Harold Metcalf
Distinguished Teaching Professor, Department of Physics and Astronomy
Stony Brook University

Thomas Bergeman
Adjunct Professor, Department of Physics and Astronomy
Stony Brook University
This thesis is accepted by the Graduate School.

Lawrence Martin
Dean of the Graduate Schoolii

Abstract of the Thesis
Evaporative cooling and Bose-Einstein Condensationof Rb-87 in a moving-coil TOP trap geometrybyDaniel Guenther Greif
Master of ArtsinPhysics
Stony Brook University
2007

This thesis discusses evaporative cooling and Bose-Einstein Condensation of Rb-87 in the |1, −1i ground state. Quadrupole coils
mounted on a mechanical transporter are used to move the lasercooled atoms on a 2-dim. path into a glass cell to perform RF
evaporative cooling in a TOP trap. This trap is formed by themoving coil quadrupole field in conjunction with a rotating bias
field. Evaporative cooling simulations are carried out to provide a
guideline for the optimization in the experiment.
Typical condensate sizes are ∼ 1 × 106 atoms with a critical temperature of 150nK. Our data agrees well with the theoretical predictions of the phase transition and anisotropic expansion. The
condensate atom number is found to be very stable, while the position stability is very promising for condensation in a dipole trap
and future experiments in optical lattices.

iii

To my family and friends

Contents

List of Figuresvi

List of Tablesviii

Acknowledgementsx

1 Introduction

1

2 Experimental setup
2.1 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Rb-87 level scheme . . . . . . . . . . . . . . . . . . . .
2.1.2 Laser system . . . . . . . . . . . . . . . . . . . . . . .
2.2 Vacuum apparatus and quadrupole trap . . . . . . . . . . . .
2.3 TOP trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Trap desgin . . . . . . . . . . . . . . . . . . . . . . . .
2.4 RF evaporation coil . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Coil design . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 RF electronics . . . . . . . . . . . . . . . . . . . . . . .
2.5 Imaging of atom clouds . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Extraction of cloud parameters . . . . . . . . . . . . .
2.5.3 Imaging parameters . . . . . . . . . . . . . . . . . . . .

5
5
5
7
10
11
13
15
17
17
18
22
22
24
27

3 Achieving Bose-Einstein Condensation
3.1 Laser cooling and magnetic transport . . . . . . . . . . . . . .
3.2 Evaporative cooling simulations . . . . . . . . . . . . . . . . .
3.2.1 Stationary evaporation . . . . . . . . . . . . . . . . . .
3.2.2 Evaporation dynamics . . . . . . . . . . . . . . . . . .
3.2.3 Simulation results and experimental guidelines . . . . .
3.3 Optimization of the BEC machine . . . . . . . . . . . . . . . .

29
29
33
34
40
47
53

v

3.3.1
3.3.2

Optimization of the atom cloud parameters . . . . . .
Evaporation optimization . . . . . . . . . . . . . . . .

54
58

4 Bose-Einstein Condensate characterization
4.1 Condensate theory . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Anisotropic expansion . . . . . . . . . . . . . . . . . . . . . .
4.4 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Condensate atom number determination . . . . . . . . . . . .
4.6 Machine performance . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Position reproducibility . . . . . . . . . . . . . . . . . .
4.6.2 Atom number stability . . . . . . . . . . . . . . . . . .

69
71
75
77
77
80
82
83
84

5 Conclusion

86

A Evaporative cooling simulation details
A.1 Analytic solutions . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Additional formulas . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96
96
97
97

B Anisotropic expansion details
104
B.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

vi

List of Figures
1.1

Roadmap to BEC . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1 Rb-87 level scheme . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Scheme of optical setup . . . . . . . . . . . . . . . . . . . . . .
2.3 Optical system . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 BEC apparatus . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Average potential in TOP trap . . . . . . . . . . . . . . . . .
2.6 Design of TOP coils . . . . . . . . . . . . . . . . . . . . . . .
2.7 RF coil design . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Schematic design of RF electronics . . . . . . . . . . . . . . .
2.9 Magnetic RF field spectrum . . . . . . . . . . . . . . . . . . .
2.10 Imaging setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Length calibration . . . . . . . . . . . . . . . . . . . . . . . .

7
8
9
12
14
16
19
20
21
23
28

3.1 Old laser cooling sequence . . . . . . . . . . . . . . . . . . . .
3.2 Atom clouds in MOT cell . . . . . . . . . . . . . . . . . . . .
3.3 RF evaporation as Landau-Zener transition . . . . . . . . . . .
3.4 Constraints on RF power . . . . . . . . . . . . . . . . . . . . .
3.5 RF evaporation surfaces . . . . . . . . . . . . . . . . . . . . .
3.6 Dynamics of evaporation model . . . . . . . . . . . . . . . . .
3.7 Comparison of harmonic/linear trap . . . . . . . . . . . . . . .
3.8 Evaporation trajectory simulations . . . . . . . . . . . . . . .
3.9 Evaporation quality simulations . . . . . . . . . . . . . . . . .
3.10 Evaporation simulation without machine working region . . .
3.11 Example for (un)correlated parameter grouping . . . . . . . .
3.12 Evaporation simulation with machine working region . . . . .
3.13 New laser cooling sequence . . . . . . . . . . . . . . . . . . . .
3.14 Optimized evaporation sequence . . . . . . . . . . . . . . . . .
3.15 Evaporation in the experiment . . . . . . . . . . . . . . . . . .
3.16 Evaporation in the quadrupole trap . . . . . . . . . . . . . . .
3.17 Majorana losses in quadrupole trap . . . . . . . . . . . . . . .

30
31
36
38
39
45
48
50
51
53
55
56
57
58
59
60
61

vii

3.18 TOP loading optimization . . . . . . . . . . . . . . . . . . . .
3.19 Losses in TOP trap . . . . . . . . . . . . . . . . . . . . . . . .
3.20 Instantaneous mangetic TOP field . . . . . . . . . . . . . . . .
3.21 Evaporation in TOP trap . . . . . . . . . . . . . . . . . . . .

63
64
66
68

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

70
75
76
78
79
80
82
84
85

Images of first BEC . . . . . . . . . . . . . . . . . . . . . . . .
Theoretical anisotropic expansion . . . . . . . . . . . . . . . .
Trap frequencies measurement . . . . . . . . . . . . . . . . . .
Anisotropic expansion . . . . . . . . . . . . . . . . . . . . . .
Phase transition to a BEC . . . . . . . . . . . . . . . . . . . .
Condensate atom number . . . . . . . . . . . . . . . . . . . .
Atom number evaluation cross check . . . . . . . . . . . . . .
Condensate position reproducibility . . . . . . . . . . . . . . .
Condensate atom number stability . . . . . . . . . . . . . . .

viii

List of Tables
3.1
3.2

System parameter scaling during evaporation . . . . . . . . . .
Grouping of laser cooling knobs for optimization . . . . . . . .

ix

43
55

Acknowledgements
It is now only one week to my return back home to Germany. The last year
has been very interesting and exciting in many ways. I had the luck to join
the group at a time, when Bose-Einstein Condensation was close. Making acondensate for the first time is definitely one of the coolest experiences in mylife. Coming from a theoretical background, doing experimental physics was
a completely new experience for me. I would like to express my gratitude firstto Dominik Schneble, who gave me the fortunate opportunity to work in his
lab one year ago and made this experience unforgettable. He is a very patient
person, who spends as much time as possible in the lab to help the students.
Dominik is a relaxed person and manages to turn frustration in the lab into
curiosity and excitement, which makes him a great person to work with. I
would like to express my gratitude for proof-reading of my thesis and all theuseful corrections.
I would also like to thank the whole group: Rebekah Schiller was always asource of good mood and brightness in the often very dark lab room, showedme how to align optics and brought distraction by solving crosswords with me- although I could only rarely make a contribution. Daniel Pertot spent a lotof time helping me to set up the RF electronics. Without him I woulds still
have no clue what the RF box is doing. I had a great time spending numerous
nights in the lab with him, although we could never really agree on a goodradio station. His calm and gentle personality make Daniel a great person
to work with and his sodering skills are unchallenged. My thanks also go to
Stephan Albert, with whom I had a great road trip to Canada and who got
me interested in ultracold atomic physics. He showed and explained me thebasics of our apparatus. Azure Hansen helped to set up the RF shield andconstantly offered her helping hand to everbody in the lab. Yancey Sechrest
helped performing the TOP trap frequency measurement and is a very relaxedand easy-going person - and a very good soccer player. I wish them both goodluck for their graduate studies.
Dominik Schneble, Thomas Bergeman and Harold Metcalf, who were in mycommittee, gave me very constructive feedback on my thesis. My gratitudealso goes to the people in the administration, in particular Pat Peiliker, Sarah
Lutterbie and Maria Hofer, who are always very friendly and caring about
the many administrative problems graduate students have. I would also liketo thank Laszlo Mihaly and Peter Koch (Stony Brook) and Fakher Assaad(Würzburg) for giving me the opportunity to participate at the exchange program and to join the physics graduate school in Stony Brook.
My time in Stony Brook would not have been the same if I had not met somany woderful people, with whom I had a great time and will miss a lot.
Thank you for the many rides, invitation to parties and woderful discussions.
I am grateful to Claire, who gave me a place to stay in the last few months
and Jennifer, who I am very happy to have met and has become a special
person for me. I would also like to thank my friends in Germany, who kept
in touch with me over the last year and always showed me their support. At
this point I would like to mention Daniel, Florian, Jürgen, Barbara and AnneKatharina. I wish them the best for their future and hope to spend many
more years together with them. Cooking a turkey last Christmas together
with Thai, Volker, Alexander and Elena will always be a great memory. I also
wish Sebastian the very best with his family and Carsten good luck for his
graduation in Wuerzburg.
Special thanks go to my father Jakob, my mother Anna and my brotherThomas, who constantly support me and believe in me. They are a woderfulsource of encouragement and I am very grateful and happy to have such afamily. This thesis is dedicated to you.
This work was supported by the German government and the Research Foundation of the State University of New York.

1

Chapter 1
Introduction
The history of Bose-Einstein Condensation (BEC) reaches back to theyear 1924, when the Indian physicist S.N. Bose wrote an article [1] deriving Planck’s black body radiation spectrum from the indistinguishability ofparticles, a fundamental concept of quantum mechanics. A. Einstein, whotranslated the article and published it, applied this idea to an ideal gas of noninteracting bosonic particles [2–4] and predicted that the system undergoes aphase transition at temperatures below a critical temperature into a state, inwhich a macroscopic number of atoms is in the ground state. This macroscopic
occupation of the lowest energy state is referred to as Bose-Einstein Condensation.
It is remarkable that this phase-transition does not rely on interaction effectsof particles, as is the case for transitions into solid, liquid or gaseous forms for
instance. It is purely a result of quantum statistics and the fundamental difference between fermions (half-integer spin) and bosons (integer spin): while
it is forbidden for fermions to occupy the same ground state (Pauli-principle)
owing to the anti-symmetry of the wave function, bosons can occupy the sameground state and are characterized by symmetric wave functions. In an ensemble of many particles of the same species, this leads to two occupation
statistics - the Bose-Einstein distribution for bosons and the Fermi-Dirac distribution for fermions. Bose-Einstein condensation is a direct consequence ofthe bosonic distribution statistics.
Considering the wave nature of particles, a more intuitive interpretation ofcondensation is possible: a gas of atoms can be considered classical, if the average spacing between√the particles is much larger than the thermal de Broglie
wavelength λdB ∝ 1/ T . However, at very low temperatures (∼ 100nK for
dilute atomic gases at densities of ∼ 1014 cm−3 ), this wavelength exceeds theinter-particle spacing, resulting in an overlap of the wave functions, and the
many-particle state can be described by a single macroscopic wave function.

2This model gives a condition on the phase-space density nλ3dB (particles perunit volume of phase space), where n is the average particle density:
nλ3dB ∼ 1

(1.1)

Pushing a system to the quantum regime in the experiment is thus possible for high densities or for very low temperatures. This posed a big challenge
to the physics community for a long time. The main reason was that cryogenic
cooling limited the temperatures to ∼ 1mK, so that the densities had to be increased, which comes at the cost of increased interactions (e.g. van der Waals
interaction). After the discovery of the strongly interacting superfluid He-4 in
1937, F. London proposed that this could be interpreted as a Bose-Einstein
Condensate [5]. It was also found that the phenomenon of superconductivity,
which was discovered in 1957, was also a realization of Bose-Einstein Condensation - in this case of coupled electron pairs, called Cooper pairs (BCS
theory).
However, physicists still sought to achieve BEC in weakly interacting systems(in which the scattering length is much smaller than the inter-particle distances) to bring the ultimate proof that condensation is only a result of quantum statistics. Promising candidates were dilute atomic gases in a metastablestate: the idea is to bring a gas of weakly interacting bosonic atoms to densities where the gaseous state becomes metastable (three-body collisions) withlifetimes long enough for experiments (∼ 1s). This limits the densities to typically 1014 cm−3 , which makes ultra-low temperatures of 100 nK necessary more than 3 orders of magnitude colder than the temperatures achievablewith cryostats.
The situation changed suddenly with the invention of laser cooling in the 1980s[6], which earned S. Chu, C. Cohen-Tannoudji and W.D. Phillips the NobelPrize in 1997. This made efficient trapping and cooling of ∼ 1010 atoms at
10−100µK possible. In most cases the atoms have very simple level-structures,such as alkali atoms for example. One very prominent implementation of laser
cooling and trapping is the magneto-optical trap (MOT), which can capture
and cool a cloud of atoms at the same time using optical forces. The setup
basically consists of a magnetic field created by two anti-Helmholtz coils andsix counter-propagating laser beams resonant with a closed atomic transition.
In spite of this huge leap in phase-space density from 10−18 (thermal gas atroom temperature) to 10−6 (MOT), the phase space density was still not highenough for condensation due to photon reabsorption. The final breakthroughcame with the idea of evaporative cooling of the ”hot” atoms in a magnetic
trap, first proposed by Hess [7]: by removing the ”hottest” atoms from the

12

10

BEC
1

evaporative
cooling

OpticalMolasses

Fully compressedmagnetic trap

10-4Magnetic trap

1010

dB

1014

nλ 3

Peak Density [cm-3]

3

MOT

10-8
10-12

108
106

10-16

UHV background

10-20

10-7

10-5

10-3

10-1

101

103

Temperature [K]Figure 1.1: Roadmap to Bose-Einstein Condensation in our experiment. The graylines are region of particle density n and tempererature T with constant phase-spacedensity nλ3dB . Starting from a Rb background pressure in an UHV at room temperature, the atom cloud undergoes a tremendous increase of 18 orders of magnitude inphase-space density. The major job is done by laser cooling (MOT and Optical Molasses), while the magnetic trap and subsequent compression are necessary to form a
conservative trapping potential for evaporative cooling. In our setup, condensation
occurs at a critical temperature of 150nK leading to almost pure condensates oftypically 1.0 × 106 atoms at densities of 1...2 × 1014 cm−3

cloud, the average energy per atom is decreased and hence the ensemble temperature lowered. An overview of the different sequence steps and the scalingof the relevant parameters for our setup is given in Fig. 1.1.
The first Bose-Einstein Condensates were realized independently by threegroups in 1995 for the elements Rb-87, Li-7 and Na-23 [8–11], which earned E.
A. Cornell, W. Ketterle and C. E. Wieman the Nobel-Prize in 2001. Today,
BEC has been achieved with many more elements: H, Rb-85, K-41, metastable
He-4, Cs-113, Yb-174 and Cr-52 [12–19].
The successful demonstration of Bose-Einstein Condensation in diluteatomic gases in 1995 had a huge impact on the physics community.
Today, approximately 70 groups world-wide are currently doing research withultracold atomic gases. Among the many interesting experiments with degenerate bosonic gases are for example the study of excitations and propagation

4of sound [20, 21], observation of interference between two condensates [22], realization of an atom laser [23, 24], four-wave mixing and Bragg spectroscopy ofcondensates [25, 26], superfluidity and vortices [27, 28] or the quantum phasetransition from a superfluid to a Mott Insulator [29]. One of the many fascinating experiments with ultracold fermionic quantum gases are molecular condensation of atom pairs [30–32], the study of the BEC-BCS crossover [33, 34]
or Fermi gases with unequal spin populations [35, 36]. Thus, applications are
not only restricted to atomic physics - ultracold atomic physics has become
an interdisciplinary field ranging from many body theory (quantum simulators), precision measurements, ultracold collision physics, solid-state physics,
condensed matter model systems or even fundamental studies of quantum information/computation.
This thesis is divided into three main parts: the first chapter introduces
our Rb-87 BEC apparatus with a focus on the RF evaporation setup and the
TOP trap formed by the quadrupole coils of the mechanical transporter and arotating bias field. The second chapter explains RF evaporative cooling in the
dressed state picture and discusses a model for evaporative cooling simulations.
This model provided a guideline for the experimental optimization protocol forcondensation, which is also explained in this section. The last chapter shows
the data of our first BEC. In addition, the condensate is characterized bycomparing the data to the predictions of the phase transition and anisotropic
expansion. Finally, the reproducibility in atom number and position of thecondensate in the magnetic TOP trap is determined to estimate the prospects
for condensation in an optical dipole trap and optical lattice experiments.

5

Chapter 2
Experimental setup
General features of our BEC machine for dilute atomic gases are an
ultra-high vacuum with a bosonic atomic species, laser light for laser coolingand magnetic fields for trapping. Magnetic fields can also be used to transport
atoms, which is useful if the vacuum is divided into two pressure regions toprovide large MOT atom numbers and good lifetimes for evaporative cooling.
Since many system parameters, such as laser detuning and power, shutters,
magnetic fields or RF frequencies, have to be precisely controlled in time toachieve a BEC, a sequence generator on the computer is necessary, which controls several analog and digital output channels connected to the elements inthe experiment via an A/D card. Atom number and temperature of the atom
cloud are obtained by imaging the cloud onto a CCD chip. Fluorescence andabsorption imaging are destructive techniques, while phase-contrast imaging
is non-destructive.
In this chapter the experimental setup of our Rb-87 BEC apparatus is discussed. The first and second section explain the laser and vacuum system.
The magnetic TOP trap, which is formed by the quadrupole coils of the transporter in conjunction with a rotating bias field, is discussed in the third section.
Details of the RF coils for evaporative cooling in the TOP trap are explainedin the fourth section, whereas the last part deals with the imaging setup andextraction of cloud parameters.

2.1

Optical setup

2.1.1

Rb-87 level scheme

The basic idea of laser cooling and trapping is to use the atom-light interaction between a cloud of atoms and a laser to create a conservative and/or
cooling force on the atoms. The fundamental physical process is absorption

6and either spontaneous or stimulated emission of photons. Therefore, a laser
beam near the resonance of an optical transition of an atomic level structure
is required. A dissipative force can be created by many cycles of directional
absorption from one or more laser beams and subsequent spontaneous emission in random directions. The transition therefore has to be closed, i.e. loss
processes into other energy levels must not occur. All laser cooling schemes
have this very important requirement of a ”closed cycling transition” in common.
Since Rb-87 is an alkali atom, the level scheme is very simple due tothe single valence electron. The ground state 52 S1/2 splits into two hyperfine
states F=1 and F=2, because the nuclear spin of Rb-87 is 3/2. Fine splittingof the excited state leads to the two possible states 52 P1/2 (D1 -line) and 52 P3/2
(D2 -line). 52 P3/2 , which is relevant for our experiments, splits into the F=0,
F=1, F=2 and F=3 hyperfine states. An overview of the level scheme is given
in Fig. 2.1. Details of Rb-87 properties can be found in [37].
The transition 52 S1/2 , F = 2 → 52 P3/2 , F 0 = 3 is a good candidate forthe cycling transition, because F 0 = F + 1 as required by the MOT andspontaneous decay is only possible into the F=2 ground state. Unfortunately,
this transition is not entirely closed, as atoms can also be excited into the
52 P3/2 , F 0 = 2 state, which happens roughly every ten-thousand cycles. Under
this condition, the atom can decay into the F=1 ground state (∆F = 0, 1, −1).
A second laser is therefore necessary, which is resonant with the 52 S1/2 , F =1 → 52 P3/2 , F 0 = 2 transition to pump the atom back into the 52 P3/2 , F 0 = 2
state, where it can decay spontaneously into the F=2 ground state again. For
imaging an F=1 or F=2 ground state atom cloud, either the cycling or repump
beam can be used. Since closed transitions are also desirable for imaging, anadditional depump beam resonant to the 52 S1/2 , F = 2 → 52 P3/2 , F 0 = 2transition can be used for repump imaging to depump the F=2 ground state
atoms into F=1 ground state atoms.

7

F = 3 gF = 2/3
266.650(9) MHz

5 2P3/2

72.9113(32) MHz

F = 2 gF = 2/3

156.947(7) MHz
72.218(4) MHz

5 2P1/2

Cycling

794.978 850 9(8) nm
377.107 463 THz
1.559 590 99 (6) eV

780.241 210 nm
384.230 484 THz
1.589 049 eV

D1

D2

F = 1 gF = 2/3
F=0RepumpDepumpF=2

gF = 1/2

2.563 006 GHz

5 2S1/2
4.271 677 GHz

F = 1 gF = – 1/2

Figure 2.1: Fine- and hyperfine splitting of the ground and first excited state of
Rb-87. The marked transitions of the D2 -line are used for laser cooling and imagingof the atom cloud. The magnetic sub states are not shown here.

2.1.2

Laser system

For the cycling transition an external cavity diode laser is used with atapered amplifier chip (Sacher Tiger TEC 300-780, 1000mW). Since the hyperfine levels of the Rb-87 ground state are several GHz apart, an additional laser
system is necessary for the repump transition. A second external cavity diodelaser is used for this (Toptica DL100, 120mW). In both cases the external
cavity is formed by a grating mounted on a piezo in Littrow configuration andthe surface of the laser diode. The parameters temperature (slow timescales),diode current and piezo voltage (faster timescales) control the wavelength.
A small fraction of the laser output is used to perform spectroscopy on a Rb
cell. The method in our setup is Doppler-free polarization saturation spectroscopy [38], where the real part of the refractive index at the resonance ofthe transition is measured and converted into a voltage by photo diodes. The
voltage signal is then compared to a set voltage to extract an error signal,

8which is used to stabilize the laser frequency with a PID controller throughthe piezo voltage (active feedback loop). This way, the cycling laser is stablewithin ∼ 10MHz (limited by 60Hz noise from the lock box), while the frequency stability of the repump laser is approximately 1MHz. In both cases
the noise is in the 1-100kHz range (the line width of the Rb-87 D2 line is about
2π × 6MHz). A schematic view of the laser setup is given in Fig. 2.2.

Computer
Control

Cycling Laser

AnalogError
SignalError
SignalFrequency
Intensity

PID

Setpoint

Spectroscopy
SignalSpectroscopy
SignalSpectroscopy

DigitalSpectroscopy

Beam Preparation

AOM
Shutter

Cycling
MOT

Readout
Evaluation

PID

Repump Laser
Setpoint

Detection

Repump +Depump
MOT

Imaging
BEC cell

Pump
BEC cell

BEC Experiment

Figure 2.2: Scheme of optical setup. Both diode lasers are stabilized by an active
feedback loop controlling the grating mounted on the piezo. The beam preparation
section produces four main beams for the experiment, which are controllable infrequency and intensity from the computer via analog and digital channels.

The necessary laser wavelengths for the experiment are obtained bycontrolling frequency and intensity of the two laser systems with acoustooptical modulators (AOM). A double-pass AOM setup is used to avoid beam
shifts when changing the laser frequency. The cycling laser beam is split
into three parts: MOT, imaging and depump beam. The MOT light is detuned by typically ∼ 3.5Γ (controlled by an AOM), while both the imaging(F = 2 → F 0 = 3) and depump beam (F = 2 → F 0 = 2) are on resonance. The
repump laser system has one output beam, which is also frequency-controlled
by an AOM.
The entire laser setup is covered in boxes with shutters for the lasers to preventstray light from hitting the experiment. In essence, four main beam lines run
to the experiment:

9• MOT beam: detuned cycling laser beam (250mW behind fiber)• MOT cell pump: resonant repump (50mW) and depump (6mW) laser
beam• BEC cell imaging beam: resonant cycling or repump laser beam (250µW
behind fiber)• BEC cell pump: resonant repump (50mW with flip mirror) and depump(3mW) laser beam
To decouple the optics for the experiment from the laser system, single-mode
fibers are used (not the case for the pump beams). This way, beam drifts
are minimized and the beam profiles can be improved, which is important for
efficient laser cooling. The six MOT beams each carry about 40mW with beam
diameters of about 4-5cm, so that the total intensity of all beams is well above
the saturation intensity Is ∼ 4mW/cm2 of the cycling transition. A sketch of
the light production setup is shown in Fig. 2.3.
QWP/HWP

Shutter

PBS

Fiber

MirrorCycling

Flip mirror

Depump

Photodiode

RepumpRepump Imaging
250 µW

Cycling Imaging
250 µW
Depump MOT
6 mW
Repump MOT
50 mW
Depump BEC cell
3 mW
Repump BEC cell
50 mW

Beam Separation

diagnostic

diagnostic

Depump

AOM1

Toptica DL-100

AOM1

F=2 imaging

AOM

AOM

Rb

~120 mW

Repump Box

AOM

AOM

4

4

F=1 imaging

Rb

Cycling MOT
250 mW~1000 mW
Sacher TEC300

Cycling Box

Figure 2.3: Setup of optical system (not to scale). Both diode lasers are frequency
stabilized using polarization saturation spectroscopy with a feedback loop to theexternal grating. The light from both lasers is modified with AOMs. Four main
beam lines run to the experiment.

10

2.2

Vacuum apparatus and quadrupole trap

The vacuum system is split into two different parts: the MOT chamber, where the atom cloud is laser cooled, and the BEC cell, to which the
pre-cooled atom cloud is transported for evaporative cooling and condensation. This separation allows for long lifetimes in the BEC cell (∼ 130s) andhigh MOT loading rates R ∝ ρRb (vapor cell MOT) at the same time. Ultrahigh vacua are necessary to reduce the background pressure of other elements,
which knock out atoms from the MOT and reduce its steady-state atom number.
The MOT chamber is connected to a Rb oven, which can be opened manuallyand controlled in temperature. An ion pump provides the necessary UHV withpartial-pressures of 10−8 -10−9 torr for Rb and ∼ 10−10 torr of other elements.
The MOT chamber is made of Pyrex glass and has a cylindrical shape withan outer diameter of 5.7cm oriented in x-direction (lab coordinates). The six
MOT beams (cycling MOT laser) are along the vertical z-axis and in the x-yplane, but rotated by 45◦ towards the two axes. This allows optical accessfrom all directions, while the curved surface prevents etaloning of the MOT
beams. The helicity of all MOT beams is left-handed (in separate reference
frames), which is not significantly distorted by the curved surface of the MOT
cell, since the chamber diameter is much larger than the MOT size. The repump/depump beam enters along the y-axis.
The BEC cell is a cuboid made of quartz with a width of 1.0 cm and a height of
2.0 cm (excluding the wall thickness of 1 mm). It is connected via a round glass
tube to the ”square chamber”, which is part of the higher vacuum region. An
ion pump (75 l/s) and a titanium sublimation pump provides pressures in the
10−12 torr range for long trap lifetimes. The two vacuum parts are connectedby a thin tube of 1.0 cm diameter (differential pumping tube), which is large
enough that the atom cloud just fits through. A valve offers the possibility to
fully separate the two vacua from each other.
The magnetic trap is formed by a pair of coils in anti-Helmholtz configuration with the symmetry axis in the vertical z-direction. In the center ofthis geometry the magnetic field shows a linear behavior. Due to Maxwell’s~ ·B~ QP = 0, the axial field gradient b is twice the radial gradient.
equation ∇−x~ QP (x, y, z) = b  −y 
(2.1)
B
2
2z
The potential for the atoms is induced by the coupling between the atomic

11

magnetic moment and magnetic bias field, which results in a contribution toˆ·B~ QP . For low enough magnetic fields
the interaction Hamiltonian HI = ~µ(<< 1 T), the total atomic angular momentum F~ including hyperfine splitting
is still a good quantum number. Assuming the atomic spin can follow themagnetic axis adiabatically1 , the trap is linear in all directions (gF is the gfactor of F~ ):bp 2
V (x, y, z) = gF mF µB
x + y 2 + 4z 2
(2.2)
2
The coils are made of hollow copper tubes for water cooling and are connected
to a power supply, which can supply maximum coil currents of 450 A corresponding to an axial field gradient of ∼ 400 G/cm with a stability of < 10−3 .
The coils are mounted on a holder of aluminum on a translation stage, which
can move horizontally in the x-y plane. After the laser cooling sequences, theentire holder moves along a 2-dim. path to the BEC cell and carries the atom
cloud along. The stage provides acceleration of up to 0.5g, so that the magnetically trapped atom cloud only spends 250ms in the MOT chamber (τ ∼ 2s).
The 2-dim. path allows further optical access along the x-axis and prevents
hot atoms from the MOT chamber to spill into the BEC cell. The quadrupole
coils are used in three different ways: magnetic field for MOT, transport ofatoms and magnetic trap for evaporation. A picture of the real setup is shownin Fig. 2.4a) and an overview of the setup is given in Fig. 2.4b). Details of
the vacuum and laser system design are discussed in [39], and of the magnetictransport in [40].

2.3

TOP trap

The quadrupole coils mounted on the holder of the transporter allow
the atoms to be transferred into the BEC cell, so that the evaporative coolingcan be carried out in the quadrupole trap. A major problem in this case are
Majorana losses [41], which can occur near the trap center: if the absolute
magnetic field seen by an atom passing near the trap center is so low that theLarmor frequency is on the timescale of the relative change of the magnetic field
direction, the atomic spin can flip from a trapped to an untrapped state. As
discussed in Chapter 3.3.2, this Majorana loss rate increases with decreasing
temperature, so that for very low temperatures efficient evaporation is nolonger possible and Bose-Einstein condensation is prevented, cf. Chapter 3.2.2.
1 This condition is satisfied if the Larmor-frequency is much smaller than the rateof change of the magnetic field: ωL << B1 | d B|
dt

12

Figure 2.4: BEC apparatus. a) Picture of system. The quadrupole coils mounted
on the holder on the two perpendicular translation stages are visible. The laser
cooled atoms are transported along a 2-dim. path from the MOT cell through thedifferential pumping tube to the BEC cell for evaporative cooling in the TOP trap.
The MOT cell pressure is about 10−8 torr for high MOT loading rates, while the
BEC cell pressure is below 10−11 torr for lifetimes of ∼ 130 s. b) Schematic view
from above (along z axis). The MOT consists of six counter-propagating cycling
laser beams with an additional repump/depump beam and the quadrupole coils on
the holder. These coils are also used to transport the atoms to the BEC cell andare part of the TOP trap.

13

There are several ways to circumvent this problem. For example, the atom
cloud can be loaded into a different type of magnetic trap with a non-zerofield at the minimum so that the Larmor frequency does not become too small(Ioffe-Pritchard trap [42]). Alternatively, an optical trap can be used, which
also has the advantage that not only atoms of a certain magnetic sublevel aretrapped [43]. Instead of using a different trap, the ”Majorana hole” in thequadrupole trap can be ”plugged” with a blue detuned laser beam that repels
the atoms from the center [11]. The method chosen for our experiment is
the Time-Orbiting Potential (TOP) trap, first demonstrated by Cornell et al.,[41]. The basic idea is to rotate the Majorana hole around the trap center byapplying a rotating magnetic bias field BT OP . Hence, the troublesome part of
the quadrupole trap keeps hiding from the atoms, while they are still being
trapped.

2.3.1

TheoryIn a TOP trap the rotation frequency ωT OP needs to be much higher
than the trap frequencies ωtrap so that the atoms cannot follow the shiftedtrap minimum. In addition, ωT OP has to be much smaller than the Larmor
frequency ωlarmor , so that the spin can follow the rotating field and the atoms
see an averaged potential.
ωtrap << ωT OP << ωlarmor
The instantaneous potential is formed by the usual interaction energy of the~ QP (~r) + B~ T OP (t)|. Asatomic magnetic moment µ and total magnetic field |Bsuming that the stiff quadrupole axis with a field gradient b is in the z direction,
while the rotating bias-field is in the x-y plane, the potential is:
q
2
2
b
x + BT OP cos(ωT OP t) + 2b y + BT OP sin(ωT OP t) + (bz)2V (~r, t) = µ2cos(ωT OP t)~ T OP (t) = BT OP  −sin(ωT OP t) 
B
(2.3)
0
Time-averaging over a full period of a bias field rotation then gives the averagepotential V̄ (r, z) seen by the atoms. Near the trap center the potential can beapproximated by a harmonic potential, which can be seen by taking a Taylor
approximation for small distances. Neglecting the influence of gravity, thisleaves [44]:
s
√
1
1
µb2
2
2 2
V̄ (r, z) ≈ µBT OP + mωradr2 + mωax
z , ωax == 8ωrad (2.4)
22mBT OP

14

The aspect-ratio ωax /ωrad between the shallow and the stiff direction is thus√
8. The geometry is rotationally symmetric around the z-axis.

b)
4
2
0
-2
-4
-2

mF=-1
mF=0

TOP
mF=1
QP
-1
0
1
2
Horizontal x [rCoD]

Vaverage(x, y, z) [µBTOP]

Vaverage(x, y, z) [µBTOP]

a)

4
mF=-1

2

mF=0

0
-2
-4

-2

TOP
QP
-1
0
1
Vertical z [rCoD]

mF=1

2

Figure 2.5: Illustration of average potential Vaverage along different axes in TOPand quadrupole trap for Rb-87 atoms in the F=1 ground state. The natural unit
for the potential is µBT OP , while rCoD sets the length scale. This allows to omit
the dependencies on trap-specific parameters. The harmonic
approximation breaks√
down at rCoD . Neglecting gravity, the aspect ratio is 8.

The average trapping potential along the x- and z-axis is illustrated in Fig.
2.5 for Rb-87 atoms in the F=1 ground state, both with and without the
TOP trap. In the center the potential has an energy offset of Eof f = µBT OP .
The Majorana hole rotates in the x-y plane at a certain distance rCoD to thesymmetry axis (z-direction). Since atoms are lost if they cross this boundary,
this circle is referred to as ”circle of death” (CoD). This is also a typicaldistance where the harmonic approximation for small distances to the trap is
no longer valid and the potential approaches the quadrupole potential. If thebias field decreases, both the trap bottom and CoD radius decrease, while the
trap-frequencies increase (trap stiffening).
Eof f = µBT OP ,rCoD =

BT OP
b/2

(2.5)

The effect of gravity can be included by adding mgz to the instantaneous potential from Eq. 2.3. A Taylor expansion around the new equilibrium
position z0 , which is shifted downwards a little due to the ”gravitational sag”,
yields to lowest order a harmonic potential with slightly changed trap frequencies depending on a parameter ν = mg/µb. This also leads to a shift of the15trap aspect-ratio. The details of the calculation are carried out in [44].
2

2

√
z
−
z
1
r
0
2
2
3/2
1 − ν 2 (1+ν )+2(1−ν )]V̄ (r, z) ≈ µBT OP [ 1 − ν 2 +
4
rCoD
rCoD
r
√
1 − ν2rCoD ν
ωax /ωrad = 8(2.6),
z0 = −21+ν
2(1 − ν 2 )√In our case b∼ 350G/cm, so that ν∼ 0.1. The trap aspect-ratio changes by
only 1%, so that the effect of gravity is very small from that aspect. But theshift of the equilibrium position is approximately 5% of rCoD , which is about
10 condensate diameters for BT OP = 18 G.

2.3.2

Trap desgin

The TOP trap is formed by the quadrupole coils from the transporter anda rotating bias field. The rotating bias field is produced by two perpendicularcoil pairs each in Helmholtz configuration in the x-y plane (lab coordinates),through which an ac current is driven 90◦ out of phase. In addition to that,several specifications have to be fulfilled:• Large and tunable bias field, so that the initial cloud fits into CoD andtrap can be stiffened during evaporation• Very uniform field (∼ 10−4 ) on the length scale of the expected condensate and translation stage position reproducibility for stable trap bottom(RF-knife very close to trap bottom at onset of condensate)• Reasonably uniform field over initial cloud size (∼ 10−2 ) for clean trap
potential• Coil current stabilization (≤ 10−3 ) for stable trap bottom• Adiabaticity condition: ωtrap << ωT OP << ωlarmor• Optical access to BEC cell not blocked by coils
For axial field gradients of ∼ 100 G/cm typical TOP trap frequencies are(10 × 2π) Hz, while the Larmor frequencies are usually around (1 × 2π) MHz.
The bias field rotation frequency is chosen to be ωT OP = 10 × 2π kHz.
The geometry of each coil pair is such that they fit in between the quadrupole

16

coils of the translation stage, are in Helmholtz configuration and in the shadows of the edges of the BEC cell. The exact geometry is shown in Fig. 2.6a),while a picture of the real setup is given in Fig. 2.6b). Copper wires with
500 µm diameter (AWG 24) are used to form the rectangular coils - hollow
copper tubes would be very bulky and take away a lot of space around the
BEC cell. To ensure large enough bias fields at reasonable currents of a few
A, the coils have 20/25 turns. That way, a current of ∼ 8 A creates a 60 G
bias field, which corresponds to a CoD radius of 3 mm. The coils are glued
to coil holders made of glass fiber, which is very stiff and is not electricallyconducting (eddy currents).

Figure 2.6: Design of TOP coils. The rotating bias field is created by two pairs ofrectangular coils in Helmholtz configuration in the x-y plane of the lab coordinates.
The symmetry axis of the quadrupole trap is in z-direction. a) Schematic view withall measures. The geometry is chosen such that the magnetic field curvature of eachcoil pair vanishes in the center and the coils are in the shadow of the BEC celledges [45] b) Setup in the experiment. The TOP coils are mounted on a fiber glass
holder, which is connected to the vacuum chamber via 4 posts. The setup just fits
in between the quadrupole coils from the transporter.

Simulations of the magnetic field at the center of the BEC cell show that on
a length scale of 100 µm the field is uniform to ∼ 10−5 , while this reduces to∼ 10−4 for 1 mm [45]. This could be roughly confirmed experimentally. The
simulated bias field calibration is ∼ 7.5 G/A, which matches Hall probe measurements. Since only passive air cooling is available, serious heating occurs,
if too high currents flow through the coils for too long. For example, after
30 s of 8 A, the wire temperature increases from room temperature to 65 ◦ C.

17

Since the evaporation timescale is expected to be around 30-60 s, the current
should not be higher than 8 A - although the bias field is ramped down during
evaporation. The maximum bias field for our TOP trap is 60 G.
Matching capacitors in series with the coils minimize the impedance at 10 kHz
down to the ohmic resistance of the coils (impedance matching). A 800W
audio amplifier provides the necessary currents. In addition, active current
stabilization is necessary to compensate for the rise in resistance with increasing coil temperature. The current stabilization quality is found to be at least
10−3 . The timescale for switching the TOP on and off is ∼ 1...2ms. The bias
field is controlled by the computer (analog channel).

2.4

RF evaporation coil

2.4.1

Coil design

Our setup allows to perform evaporative cooling in both the harmonic
TOP and linear quadrupole trap. The process inducing the evaporation of thehottest atoms in the BEC cell is RF-forced evaporation. The basic idea is thata magnetic RF field induces spin-flips from trapped to untrapped magnetic
sub states on an evaporation shell around the trap center. This shell is givenby the resonance condition between the Zeeman splitting energy and the RF
photon energy hνRF . Evaporative cooling can then be performed by ramping
down the RF frequency. The higher the magnetic RF field amplitude, the moreprobable the spin-flip transition is and the cleaner the evaporation shell. But
for very cold clouds the RF power has to be reduced due to power broadening.
A detailed discussion of the involved mechanisms is carried out in Chapter
3.2.1. In this section only the design and characterization of the RF-coil are
explained. In principle, the experimental setup has to fulfil a few specifications:• Large oscillating magnetic field tunable in amplitude and frequency typically in the 1-100 MHz range• Amplitude only weakly dependent on frequency (no resonances)• Optical access to BEC cell not decreased• Magnetic field spatially uniform for well defined evaporation shell
As shown in Fig. 3.4, maximum RF amplitudes of 100 mG are definitely
sufficient. The requirement of a uniform amplitude-frequency spectrum is ofparticular importance, because resonances can cause the evaporation to be18very inefficient when ramping the RF-frequency over these resonances.
The RF field is created by a RF-coil, which is mounted directly in thesquare beneath the BEC cell formed by the coil holders of the TOP coils, sothat the optical access is not affected (3.0cm × 2.2cm). The orientation of
the magnetic RF field amplitude is in the z-direction of the lab coordinates,
so that the trap symmetry is conserved and the size of the evaporation holeis minimized, cf. Fig. 3.5. A Helmholtz-geometry is not chosen in order toreduce the amount of wires close to the BEC cell.
Fields in the range of 100 mG require coil currents on the order of ∼ 500 mA
for coils of this size. Under this condition, only ∼ 100 mW are directly dissipated by the coils. Thermal heating is thus negligible, so that active coolingis not necessary. Therefore, thin copper wires with 1.1 mm diameter (AWG
18) are chosen. Simulations show that for an ac voltage source of constant
amplitude the magnetic field at the center of a (round) coil is maximized, if
the coil turn number and the enclosed area are minimized. A picture of the
one-turn RF coil mounted in between the TOP coil holders directly beneaththe BEC cell is shown in Fig. 2.7a).
The initial cloud before evaporation in the BEC cell has a diameter ofabout 4 mm, which corresponds to about 400 µK in the quadrupole trap with
350 G/cm axial field gradient. Calculation results of the expected field profilein the near-field regime (distance to coil: ∼ cm, RF-wavelengths ∼ m) are
shown in Fig. 2.7b)+c): concerning the horizontal directions, the magnetic
field in the z-direction is uniform to within a few percent. In contrast the
profile in the axial direction shows a strong gradient resulting in a modulationof ±20%. This does not play a role in the beginning of the evaporation, because
the magnetic field amplitude is so high that the evaporation shell can still beconsidered ”sharp”. For very cold clouds, where the RF power needs to be
reduced (power broadening), the relative variation of the field amplitude in
the axial direction will have reduced to a few percent due to the small cloud
size.

2.4.2

RF electronics

The RF signal is created by a function generator (FG Agilent 33250A,
80MHz) connected to a class-A 40 dB 10 W RF amplifier (Delta RF, LA10-1-512-40) to compensate for the high RF coil impedance for frequencies in
the MHz range and to create large enough currents. This amplifier has an
input/output impedance of 50 Ω, so that at a maximum input voltage of 400

19

Figure 2.7: Design of RF coil for evaporative cooling (arrow). a) Picture of RF
coil mounted directly beneath the BEC cell in between the TOP coil holders. The
orientation of the magnetic field at the location of the atom cloud is in z-direction.
b)+c) Simulations of the spatial field amplitude profile in z-direction in units of theamplitude at the trap center. The calculation is based on the RF coil geometry used
for the experiment.

mV the average output power is still below 10 W for a 50 Ω resistor connectedto the output. For protection, attenuators convert the FG maximum output
voltage of 10 V to 400 mV. The amplifier itself is mounted on a heat sink,since class-A amplifiers dissipate a significant amount of the maximum output
power. An additional fan provides the necessary air circulation.
The function generator is connected via GPIB to the computer, from which
both frequency and amplitude can be controlled. Typical update times are∼ 100-200 ms. To have precise control over the RF, a digital switch is included with a reaction timescale of about 100 ns. This allows the RF-knife
to be precisely turned on/off at the critical point where the BEC sets in. To
prevent ground loops, an additional opto-coupler is installed. A schematic
overview of the setup is given in Fig. 2.8.
The amplifier output to the RF coil is governed by four main factors,
which cause the RF coil current amplitude not to be uniform over frequency:

20

Figure 2.8: Schematic design of RF electronics to create RF fields. The main partis in the RF box. A BNC cable (RG58) connects the RF amplifier output to the
RF coil and 50 Ω power terminator (75 W). Large distances d of around 50 cm
between connector and RF coil cause significant resonances in the B-field amplitude
spectrum of the RF coil. The inset shows a picture of the connector used in theexperiment, so that the distance d is kept as short as possible (∼ 5 cm).

• The RF amplifier has a variation of 1-2 dB amplification in the frequency
range of 1-80 MHz• The total output impedance is a function of frequency• Reflections in the cable from the RF Box to the RF coil cause frequency
dependent resonances of the coil voltage drop (RF wavelengths ∼ 1 m)
• Capacitive/inductive coupling to the environment (other coils, optical
table, cables) around the RF coil creates additional resonances
These effects typically cause B-field modulations of 1-2 orders of magnitude
for frequencies of 1-80 MHz.
To protect the amplifier a 50 Ω high power terminator (75 W) is connected inseries with the RF coil so that the total impedance is about 50 Ω and dominated by the ohmic resistance (in particular important for low frequencies,
where the reactance is much smaller than 50 Ω). A BNC cable (RG58) is used
to connect the amplifier to the RF coil. The coil ends are connected to thecenter conductor of the cable, creating a difference in path length between the
ground and center wire connected to the terminator, cf. Fig. 2.8, which is themain cause for resonances. Therefore, the terminator cannot cancel the cable21

resonances, if the connection between the RF coil and BNC-cable is too long.
A measurement of the magnetic field spectrum shows an order of magnitudeimprovement of the resonance behavior between a long (50 cm) and a short (5cm) connection, cf. Fig. 2.9a).
The remaining resonances due to capacitive/inductive coupling to the environment are reduced by ”shortening out” the contacts, as explained in [46]: allelements, to which the RF couples, are connected to each other and directly
grounded at only one point to avoid loops. The distances are kept as short as
possible to avoid resonances in the RF shield. The efficiency of the shields areobserved by measuring the induction voltage from a pickup coil at different
points in the lab. The major shielded elements are:
• BNC cables/connectors from RF box to RF coil
• Audio cable from TOP box to TOP coils• Quadrupole coil holder and power cables
In most cases aluminum foil was wrapped around the objects with duct tapefor isolation. This way, the RF resonances in the lab could be suppressedsignificantly.

a) 2.00b)
0.2 200

0.002
0.00175

1.50

0.15

0.00125

0.001

short
connector

1.00

0.000750.00050.50

0.00025

0.00
0

long
connector

10 20

30 40
νrf [MHz]

50 60

Brf [mG]

Brf [mG]

0.0015

150

0.1

100

0.05

50
0

0

current
measurement

inductionmeasurement

10 20

30 40 50
νrf [MHz]

60

Figure 2.9: Magnetic field amplitude Brf at BEC cell center as a function of the
RF frequency. a) Comparison between short (∼ 5 cm) and long connector (∼ 50cm) setup. The amplifier output was set to 1% of the maximum output. The longconnector data shows a resonance between 40-50 MHz, where the field drops to10%, while the field spectrum is very uniform for the short connector version. The
magnetic field was determined by measuring the current through the RF coil b)
Measurement of Brf (short connector) at maximum amplifier power. Two different
methods are chosen to cross-check - they agree at least within a factor of 2.

22

The magnetic field Brf at the center of the BEC cell as a function of RF
frequency νrf at full amplifier output power is determined by observing theinduced voltage of a pickup coil mounted above the RF coil (x-y plane). To
cross-check this measurement, the current through the terminator is measured,and the magnetic field created by this current through the RF coil is calculated.
The results for the short coil connector - as used in the BEC experiment - areshown in Fig. 2.9b).
The amplitude spectrum at full amplifier power is roughly uniform at around
100 mG with a systematic error of2 ∼ 50%. Possible reasons for deviations areresonances due to coupling to the environment, since the RF shields were notyet installed when the data was taken. In addition, the determination of thecoil current is not accurate due to remaining cable resonances in the RF coilcircuit. However, resonances causing field drops of 1-2 orders of magnitude
are clearly not present.

2.5

Imaging of atom clouds

2.5.1

Setup

Atom number and temperature of the atom cloud are obtained by imaging the cloud onto a CCD chip (PI PIXIS 1024B, pixel size (13µm)2 , 1024 ×
1024 pixels), which counts the number of arriving photons at every pixel witha quantum efficiency of ∼ 70%. One possibility for the cloud observation is
fluorescence imaging, where a certain solid angle of spontaneously emitted
light from the atoms is captured and guided into the camera. The emission
is induced by laser light resonant with a closed optical transition above thesaturation intensity Is (∼ 4mW/cm2 for the Rb-87 cycling transition), so thatthe decay rate of the excited atom is well defined. Due to the high recoils inthis regime, the atom cloud has to be excited from all six spatial directions.
This type of imaging is available in the MOT cell, where the six MOT beams
can be used to drive the cycling transition, while the repumper is still on to
make the transition closed. The details of the atom number and temperature
extraction with this method are explained in [40].
One of the disadvantages of this procedure is its sensitivity to reflections by
the glass of the MOT cell or the differential pumping tube. This reduces thesignal-to-noise ratio and hence limits the minimum atom number detectable
to typically 108 in our setup. An alternative method is absorption imaging,
2 The data shown here is for a different coil position and geometry than used in

the experiment - significant changes in amplitude are however not expected.

23

where ”darkness” instead of ”brightness” is measured. More precisely, a laser
beam resonant with an optical transition is shone along an axis on the atom
cloud and the relative intensity absorption is measured by imaging the shadowof the cloud onto the CCD chip. If the absorption cross section is known, theatom number can be calculated, while the temperature can be extracted fromeither the initial cloud profile or its expansion behavior. Quantitative details
can be found in the next section. One advantage is that absorption of theimaging beam due to other objects are normalized out, because the images
with and without the atom cloud are compared for the quantitative analysis.
Moreover, the evaluation is independent of the beam intensities in the regime
below saturation. This method is used to characterize the atom clouds in the
BEC cell.
Unzoomed100
CCD chip

750

ZoomedCCD chip
MirrorMirror
300

Shadow

100

imaging table
200
200

Fiber

z

150

750

HWP PBS

repump / cycling
imaging beam(~250µW)-75 200-25 7512550

MirrorMirror
BEC Cell
150

900

x

y

lattice table
depump (3mW) / repump (50mW)
pump beam

Figure 2.10: Schematic view of imaging setup (not to scale, some mirrors are notshown). Either the cycling or repump beam can be used as imaging beam. The firsttelescope behind the fiber increases the beam size for the zoomed imaging, while
the additional expander (dotted) is necessary for the unzoomed imaging, because alarger area in the x-z plane of the atom cloud is observed. The axial symmetry axis of
the atom cloud is in the vertical z-direction, so that the anisotropy is observable when
imaging along the y-axis. Switching between the zoomed and unzoomed imaging
requires the dotted optical elements to be inserted and the camera to be repositioned.
The repump/depump beam is necessary to close the optical transition.

A schematic view of the setup is shown in Fig. 2.10. The imaging
axis is along the y-axis (lab coordinates), so that the cloud anisotropy is observable. Since the cloud diameter decreases by about a factor of 100 during

24

evaporative cooling, two different optical pathways are possible: the unzoomed
imaging (dashed elements inserted) has a demagnification of ∼ 2 to observe
the initial cloud before evaporative cooling, while the zoomed imaging has amagnification of 5, so that the condensate extends over roughly 10 pixels in
one dimension on the CCD chip. The resolution in the zoomed case approximately corresponds to the diffraction limit (∼ 2 − 3 µm). Position holders
for the optical elements allow quick switching between the two possible beampaths (dashed elements in Fig. 2.10). Most optical imaging elements behind
the BEC cell are mounted on the imaging table above to increase free spacefor future lattice optics. The objective lens is mounted on two perpendiculartranslation stages to focus on the cloud and to correct for the effect of gravity
causing the cloud to fall out of the field of view of the CCD chip for ”timesof flight” (TOFs) larger than 20 ms. The atom clouds cannot be observed
while the magnetic trap is still on, because the Zeeman-effect shifts the atoms
out of resonance. After 0.5 ms the quadrupole coil current has decreased toalmost zero, but still exhibits minor oscillations for another 1.5 ms. Since the
TOP trap is already off after 1 ms, the imaging beam is turned on 2 ms afterswitching off the magnetic traps.
The imaging setup allows to observe both F=1 and F=2 Rb-87 atom clouds.
Either the F=2 (cycling transition) or F=1 (repump transition) imaging beam
can be used. For the imaging transitions to be effectively closed, cycling imaging requires an additional repump beam, as in the case of laser cooling in the
MOT cell, while a depump is necessary for F=1 repump imaging. This is accomplished by a second beam along the vertical z-axis providing either repump
or depump light at intensities around 1 mW/cm2 . All data for this thesis was
obtained by repump imaging of F=1 atom clouds without the depump light.

2.5.2

Extraction of cloud parameters

For every absorption image, the number of counts at every pixel of the
CCD chip is returned in form of a matrix I(x̃, z̃). The entries of this matrix
then correspond to the laser intensity at this point in relative units. If thelength calibration of the imaging setup is known, this matrix can be converted
into the real coordinates of the atom cloud, noted by I(x, z). To obtain thedensity distribution of the atom cloud, three pictures are taken: the first one
is with the atom cloud and imaging beam turned on (Iatoms (x, z)), while thesecond one is a reference picture without atoms (Iref (x, z)). The last picture isa background picture (Ibkg (x, z)) without the atom cloud and imaging beam,
which is used to subtract the background noise from the CCD chip as well
as residual stray light. The transmission T (x, z) of the imaging light is then25

given by:T (x, z) =Iatoms (x, z) − Ibkg (x, z)
Iref (x, z) − Ibkg (x, z)(2.7)

For imaging beam intensities I much smaller than the saturation intensity Is
and laser line widths much narrower than the optical transition line width,the on resonance absorption cross section σ becomes constant. Under thiscondition, the absorption is given by [6]:Z ∞−σn2D (x,z)
n(x, y, z)dy
(2.8)
I(x, z) = Ie,
n2D (x, z) =−∞Here n2D (x, z) is the 2-dim. column density and n(x, y, z) is the actual atom
cloud density. Introducing the optical density OD(x, z), the 2-dim. column
density can then be determined from the three images:

Iatoms (x, z) − Ibkg (x, z)
OD(x, z)(2.9)
, OD(x, z) = −Log
n2D (x, z) =σ
Iref (x, z) − Ibkg (x, z)
In essence, the total atom number N is proportional to the integrated
optical density on the CCD chip. The proportionality factor (atom number
calibration calatnum ) is then a function of the cross section σ and length calibration.
Z ∞Z ∞
N = calatnum
OD(x̃, z̃)dx̃dz̃
(2.10)−∞−∞

Since the imaging transition is not closed for F=1 repump imaging, the introduction of an effective cross section would be necessary in this case. As analternative, the atom number can be determined by fluorescence imaging in
the MOT cell before and after transport and compared with the integrated
optical density from absorption imaging of the same atom cloud in the BEC
cell. The resulting atom number calibration has a systematic error of ∼ 20%,
because it is not clear if the transport losses are one- or two-way.
To determine the temperature, the initial atom cloud profile right after
release from the magnetic trap can be extracted from the images. The 1-dim.
column densities can be obtained by integration of the 3-dim. profiles. This

26

yields for the quadrupole/TOP trap:√2
2
2− µb x2k+yT +4z

n3D,QP (x, y, z) ∝ e(2.11)B x

mω 2 x2− 2k xT
B x

mω 2 y 2− 2k yT
B ymω 2 z 2− 2k zT
B zen3D,T OP (x, y, z) ∝ ee
Z ∞ Z ∞ µb√x2 +y2 +4z2−
2kB Txen1D,QP (x) ∝dydz−∞(2.13)−∞mω 2 x2
B x− 2k xT

n1D,T OP (x) ∝ e(2.12)x2= e− 2σ2(2.14)In this case, µ is the projection of the magnetic moment along the magnetic
field axis, while b is the axial field gradient of the quadrupole trap. A fit
according to Eq. 2.14 or 2.13 to the integrated optical density OD(x, z) thengives the temperature. In most cases, the two temperatures Tx and Tz fromthe x- and z-axis fit are the same within 5%.
Alternatively, the expansion of the cloud can be observed directly. For the
harmonic trap, the cloud profile stays Gaussian for all times - with the only
difference that the width has increased [40]:
s
r
k
T
kB T
B
2
t2 ,
σx,0 =(2.15)σx (t) = σx,0+m
mωx2
The main idea is to observe the 1-dim. column density for different TOFs
and fit a Gaussian curve to each one. This gives the cloud widths at different
times, to which Eq. 2.15 can be fitted to obtain the temperature. Typically,
five different TOFs are sufficient to give robust results. Simulations show thatthe expansion behavior of an atom cloud initially in the quadrupole trap canalso be approximately described this way with an error of < 10%.
The temperatures determined from both methods agree well within 10%. Since
the first method requires fewer images and is therefore much faster, this approach is chosen when optimizing the BEC machine.
From the atom number N and temperature T, the maximum density nand phase-space density PSD in the center of the trap can be calculated. In
the case of the harmonic TOP trap, the cloud volume can be obtained fromthe Gaussian widths σx and σz of the 1-dim. column densities obtained by the
fits:
√
8N
h3 nT OP
nT OP = 3/2 2 ,P SDT OP =π σx σz(2πmkB T )3/2

27

In the case of the quadrupole trap, the temperature from the fit can be used:Z ∞ Z ∞ Z ∞
−1
√
2
2
2
h3 nQP− µb x2k+yT +4z
B xenQP = Ndxdydz, P SDQP =(2πmkB T )3/2−∞ −∞ −∞

2.5.3

Imaging parameters

Imaging laser intensity and exposure time are governed by several competing effects:• Intensities above saturation (I > Is ) cause intensity dependent crosssections and blow away the cloud• Signal-to-noise ratio improves with increasing intensity I and/or exposure time
• Exposure times greater than 150 µs cause the condensate to wash out
along the vertical axis at large TOFs due to gravity
Intensities of about 500µW/cm2 are used in the experiment, which is still
well below the saturation intensity, while the detection time is set to 150µs
3. The signal-to-noise ratio is optimum if the camera is almost saturated, i.e.
the counts/pixel are slightly below the maximum count value of 64,000 for
an absorption image. While this is the case for the unzoomed imaging, themaximum count for the zoomed imaging is reduced to only ∼ 3000, because
one pixel now corresponds to a much smaller area in the imaging plane of theatom cloud.
Apart from these imaging parameters, a few more points have to beconsidered:• Frequency: To increase the signal-to-noise ratio, zero detuning is chosen
to increase the absorption. Under this condition, the cross section - andhence the predicted atom number - can be frequency sensitive (transitionline width ∼ 6MHz). Since the jitter of the cycling laser is on the orderof 10-20MHz, many images of the same cloud are necessary for F=2
imaging to give a reliable atom number. In contrast, the line width of∼ 1MHz of the repump laser is sufficient to give reliable atom numbers
within 10%, as measured in Chapter 4.6.2.
3 For the unzoomed imaging this corresponds to an output power of ∼ 250µW

behind the imaging fiber, which reduces to about ∼ 25µW for the zoomed version(no beam expander before BEC cell, cf. Fig. 2.10).

28• Polarization: The cross section also depends on the imaging laser polarization and the population of the levels. The incident light is π polarizedin z-direction (labcoordinates) and the quantization axis is provided bya bias field of a few G in the same direction.
• Focusing: If the atom cloud is out of focus, blurring will wash out the
profile. If the laser frequency is not exactly on resonance, diffraction
patterns (e.g. Poisson spot) will occur due to the dispersive relation of
the real part of the refractive index. The optimum objective lens position
was found by minimizing the cloud size for a condensate without TOF.
However, for TOFs smaller than 10ms, interference fringes around thecondensate are still present.

The length calibration cal
Calibration 
2.61973 10 mm pix

6length of the imaging (conversion of pixel size tolength) can be obtained by observing the vertical condensate position z(t)on the CCD chip for different times of flight and fitting the function:z(t) = z0 −1
gt2T OF2callength(2.16)

z-position [pixel]

The center of the atom cloud for each image can be estimated by fitting a
Gaussian distribution to the 1-dim. column density n1D (z). Fig. 2.11 shows
the data for the zoomed imaging, which gives callength = (2.6 ± 0.01)µm/pixel.
The error stems from the fit.

1600

length calibration(2.61+/-0.01) µm/pix

1200
1000

800
800

400
0

5

10 15 20 25
TOF [ms]

Figure 2.11: Length calibration for the zoomed imaging obtained from free falling
BEC. By fitting a parabola according to Eq. 2.16, the length calibration can beextracted. The inset shows the very small size of the statistical error bars.

10

TOF


ms

15

20

25

29

Chapter 3
Achieving Bose-Einstein Condensation
Optimization of both the laser cooling and evaporative cooling sequences
are important for successful creation of a Bose-Einstein Condensate. Rapid
production of large condensates thus benefits from theoretical modeling of thevarious steps to guide the experimental sequence, as the overall optimizationof a BEC machine is a priori a high dimensional nonlinear problem.
In the first part of this chapter the sequence details of the laser cooling processes MOT, compressed MOT (CMOT), Optical Molasses and hyperfine state
preparation are briefly sketched as well as the magnetic transport of the atom
cloud into the BEC cell. The second part introduces the theory of RF-forced
evaporative cooling (static and dynamic case) and provides a guideline for the
optimization protocol by evaporation simulations. Experimental results of the
new cloud production sequence and optimized evaporation are presented in
the third and last part of this chapter.
All sequence values given are rough values and always require tuning on a
weekly-monthly basis. This is mostly due to drifts of the mirrors or changing background environment in the MOT cell. Reoptimization of the opticalsequences is thus required, while the dark part is very stable and requires notuning. Usually tweaking of the Molasses parameters is enough to achieve thedesired atom numbers and temperatures in the BEC cell.

3.1

Laser cooling and magnetic transport

The physics behind all the different laser cooling sequence steps are notdiscussed here and are assumed to be known. General explanations of theinvolved mechanisms can be found in [6, 47–49]. The details and influenceof system parameters in a MOT are discussed in [50–55] and for the Optical
Molasses and Sub-Doppler cooling mechanism in [56, 57]. Different magnetic

30

traps are explained in [58].
The laser cooling sequence illustrated here is a slightly modified versionof the original one developed by S. Albert [40]. The primary optimization
guideline was first atom number and then temperature. The sequence produces
atom clouds of 1.5 × 109 Rb-87 atoms in the F=1 hyperfine state in the BEC
cell at temperatures of ∼ 700µK in the fully compressed trap.

MOT

CMOT

Molasses

Preparation

Ramp

10s

500ms

5ms

3ms

750ms
374 G/cm

Axial field
gradient
Cycling
power
Cycling
detuning
Repumppower
Repumpdetuning

Digital

49 G/cm

62 G/cm
8 G/cm100%

100%

0 G/cm

0 G/cm

100%

100%

0%

3.5 Γ

3.5 Γ
6.5 Γ

100%

LIAD on

6.0 Γ

0%

0%

0%

>> Γ

>> Γ

Depump on(δ=0)

Translation
Stage Trigger

100%

5%
0Γ

>> Γ

6.0 Γ

0Γ

0Γ

Figure 3.1: Illustration of the sequence for an N optimized F=1 atom cloud production in the BEC cell. This sequence is a slightly modified version of the original
one developed in [40]. All given values are rough values and require tuning on amonthly basis. The repump beam is detuned far off resonance to prevent leaking
repump light in the preparation step. The absolute values for the beam intensities
are given in Chapter 2.1.2.

Concerning optimization, high atom numbers at very low temperaturesin the magnetic trap are desirable, as they enhance the evaporation efficiency
and thus the condensate size and production time. Consequently, the laser
cooling sequences create high atom number clouds while keeping it very dense
and small. The exact sequence settings are listed in Fig. 3.1, while the cloudparameters are illustrated in Fig. 3.2.

31

MOT

MolassesCMOT

N~1.5 x1010N~1.4 x 10

10

N~1.2 x 10

10

T~25µK

T~1mK
PSD~1x10-7-79Fully compressed
Magnetic Trap
N~2.3 x10

9

PSD~1x10

PSD<10-5

2-way Transport
N~1.0 x10

9

µ

~3cm
T~600-800 µK-8

PSD~5 x 10

T~700µK-8

PSD~2 x 10

Figure 3.2: Atom clouds after different stages of N-optimized F=1 atom cloud production in the MOT cell. All clouds have a TOF of 2ms and the atom number
was determined by fluorescence imaging. The temperature after the CMOT is too
hot to be determined. The shift of the molasses compared to the tube visible as aring suggests a slight misalignment of the magnetic trap center and cycling beams
center. The atom cloud is round before transport and thus not thermalized, while
showing the expected elliptical shape after transport (∼ 3s). The huge temperature
increase in the fully compressed trap is due to an already very high temperature
after magnetic catch of ∼ 150µK.

32• MOT: A very low field gradient with a moderate red detuning is chosen toincrease the capture range and loading rate from the background vapor.
Typically 1.5 × 1010 atoms are caught at ∼ 1 mK, so that the magnetooptical trap is in the density limited regime of photon-reabsorption. The
decreased lifetime due to the high Rb background density can be improved by using light induced atomic desorption (LIAD), which removes
adsorbed atoms from the glass of the MOT cell and causes a temporarypressure increase during the MOT loading phase [59].
• CMOT: For the compressed MOT stage the field gradient is ramped up
in 500 ms to reduce the cloud size by overcoming the repulsive photonreabsorption force. The time scale is short enough to use the transient
behavior of the steady state MOT atom number. The detuning is also
ramped so that the atoms do not spill out of the trap. Losses in atom
number are negligible, while serious heating occurs (several mK).
• Optical Molasses: The small but hot atom cloud is ”frozen out” byswitching off the magnetic trap and ramping the detuning in 5 ms. In
addition, the repump detuning is also ramped to increase the density.
Only very few atoms are lost during this procedure, so that typically 1010
atoms at around 25 µK are available in a mixture of magnetic substates.
The temperature is limited by the Sub-Doppler cooling mechanism 1 .
The efficiency depends critically on the alignment of the beams in respect
to the trap center while being very robust towards variations of the cloudparameters.
• Hyperfine preparation: Two ground states are available for Rb-87 (F=1,
F=2) with the trappable states are |1, −1i or |2, 1i and |2, 2i. For F=1
atoms, both the depump beam and the cycling beam (full power) areturned on while the repump beam is blocked away. The cycling detuning is the final ramp value of the previous sequence, so that a dark opticalmolasses is still present. Approximately 3 × 109 |1, −1i atoms are thentrappable. A preparation sequence is not necessary for F=2 atoms, as
the repump beam is still on during the Optical Molasses. Since a single clean magnetic substate is desirable for the evaporation to prevent
hyperfine spin changing collisions reducing the trap lifetime, F=1 atoms
are produced. Several groups report comparable successes in creating
F=1 and F=2 Rb-87 condensates [61–63].
1 For not too large detunings the Sub-Doppler temperature follows T[60].

SubD ∝

Icycl
δ

33

• Magnetic transport into BEC cell: The F=1 atoms are caught at a
sufficiently high field gradient to prevent spilling due to gravity so thatmost of the trappable atoms are caught - typically 2.3 × 109 . Due tothe trap asymmetry the cloud is initially not thermalized. This part isfollowed by an adiabatic ramp of the quadrupole trap up to 353G/cm
in 750ms, which causes adiabatic heating from 150µK to ∼ 700µK 2 .
To minimize the time spent in the bad background environment, thetranslation stage starts moving out of the MOT cell with a maximum
simultaneously with the ramp of the field gradient.
acceleration of 450 cm
s2
Heating due to the stage acceleration could not be observed. The twoway transport efficiency is ∼ 50%, leaving typical one way transportqualities of 75%.
The evaporation starting condition in the BEC cell is roughly 1 − 1.5 × 109
atoms at around 700µK with a phase space density of 2 × 10−8 , cf. Fig. 3.2.

3.2

Evaporative cooling simulations

After laser cooling and transport of the atom cloud into the BEC cell,
evaporative cooling in the compressed magnetic trap further increases thephase space density (PSD) to the phase transition. It is not a priori clear
what the best initial cloud parameters in the BEC cell are for large condensates, because the system is described by two parameters (atom number N andtemperature T). Ideally, N should be maximum and T minimum, so that theinitial phase-space density is large and the collision rate is high. In fact, there
is a trade off between these two values: for rather hot clouds the evaporation
is not very efficient and many orders of magnitude of phase space density haveto be overcome for condensation - but the final BEC atom number scales with
the initial atom number before evaporation, which can be large under thiscondition. On the other hand, the evaporation quality is better for rather cold
clouds and the initial phase-space density is closer to the phase transition.
However, since one starts out with fewer atoms, it is still possible that the
final condensate is smaller. The purpose of this chapter is to address questions regarding the evaporation optimization in the BEC cell by evaporation
simulations to extract a guideline for the experiment. In particular, the goalsare:• Compare the evaporation in the quadrupole and TOP trap
2 Since the Molasses temperature is much lower than 150µK, the final temperaturein the BEC cell is mainly determined by the size of the cloud after the CMOT stage.

34• Determine characteristic profiles of evaporation trajectories• Extract the typical scaling of N and PSD during evaporation• Compare the different BEC atom numbers for different initial (N,T)
values
The focus of the first part of this chapter is to examine the physical process
leading to the evaporation of the hot atoms and to compare the possible implementations. In the second part, the evaporation dynamics are discussed byapplying a simple single-step model. In the third part, this model is used toperform simulations for our trap.

3.2.1

Stationary evaporation

The main idea of evaporative cooling is very simple and can be foundin many examples in every-day life: if a liquid is very much hotter than its
environment, a physical process will take place to compensate for this imbalance. The hottest molecules in the Boltzmann distribution of the liquid will
escape, whereas the colder molecules cannot overcome the energy barrier ofleaving the surface. Therefore, the average energy per particle leaving the hot
liquid is higher than the actual average energy of the particles in the liquid.
Consequently, the average energy per particle is lowered and the temperaturedrops. The bottom line is that the system tries to adjust to the temperature
difference by evaporative cooling.
This idea equally applies to a cloud of trapped atoms: the hottest atoms are
selectively removed from the trap, while the cold atoms remain unchanged,
so that the temperature decreases. Most experimental implementations use aspatially-selective process for evaporation of the hottest atoms, [7, 64–69].
The idea to use evaporative cooling for further increase of the phase-spacedensity was suggested by Hess in 1986, [7]. The most common method is radiative evaporation, which was first proposed by Pritchard in 1989 [70] andlater shown experimentally [71, 72]. The basic idea is that atoms are resonantly
driven from a trapped to an untrapped state by an EM-field. If the conservative atom cloud trap is formed by the Zeeman shift (magnetic trapping), a
spatially selective removal of the atoms can be achieved using a RF-field resonant with transitions between different magnetic substates. This method isvery convenient: the trapping potential can remain unchanged and the evaporation surface is a clean shell around the trap center with easily adjustableparameters.

35(A) RF-forced evaporation in the dressed state picture
The physics behind the RF-induced transitions in a magnetic trap canbe understood in the dressed state picture of the atom and EM-field. Without
coupling, the atom and EM-fields decouple and the eigenenergies of the Hamiltonian are an infinite ladder of manifolds separated by hνrf , where νrf is the
RF frequency of the oscillating magnetic field. For a F=1 atom, each energy
manifold consists of three states: mF = 1, 0 (untrapped, |e >) and mF = −1
(trapped, |g >), where the gF -factor is -1/2. In this case, many crossings in
the energy spectrum versus magnetic field occur Fig. 3.3a).
The EM bias field oscillating at νrf creates coupling between the two systems
ˆatom · B
~ rf |g >. Coupling of the atomwith an interaction energy of ~ΩR =< e|~µto the electric field is negligible, as transitions are dipole forbidden between
magnetic sublevels. The magnetic interaction results in off-diagonal elementsin the coupled atom-light Hamiltonian, so that the eigenergies are shifted [47].
These shifts are large where the RF-photon energy hνrf is resonant with two
different Zeeman-levels in the magnetic trap, i.e. at the crossings in the energy
spectrum. A brief calculation shows that the shifts cause avoided-crossings between these levels [6], cf. Fig. 3.3b). Note that this situation is analogous to~ rf creates the offoptical transitions, for which the coupling ~ΩR = p~atom · E
diagonal terms stemming from the induced atomic polarization p~atom .
The atomic motion introduces non-adiabaticity into the problem, which canbe addressed in the Landau-Zener formalism [73, 74]. At a single avoided
crossing, the atom can be in three different states a,b and c. As the atom
moves, the Zeeman-splitting changes and thus the eigenergies of the uncoupled Hamiltonian, while the off-diagonal elements remain the same. Assuming
a linear atomic motion, the uncoupled eigenenergies also change linearly far
away from the resonance. This is exactly the case of the Landau-Zener formalism. With the atom velocity vatom and magnetic field gradient b, the 3x3
Hamiltonian then reads [75]:
 


√R
γt
− ~Ω
0
a
a
2 2
d  
 − ~Ω√R
~Ω
R 
√
0
−
b
b 
=  2 2
i~
2 2 
dt
~Ω
c
c
0
− 2√R2 −γtγ = µB mF gF bvatom

,

(3.1)

1
2
|~ΩR |2 = m2F gF2 µ2B Brf[F (F + 1) − mF (mF − 1)]
4

For t → −∞ the atom is in the trapped mF = −1 state. At t → ∞ the atom
is in general in a superposition of all three states. An analytical expression for
the transition probability Ptrans into the two untrapped states can be found

36

a)

hνrf

m F =-1hνrf

m F =0

Edr.st.

Edr.st.

b)m F =10hνrf

m F =-1hνrf

m F =0m F =10

BZeeman

BZeemand)c)m F=-1hΩRm F =0m F =1

BZeeman

Edr.st.

Edr.st.

m F =-1hΩR∆tm F =0m F =1

BZeemanFigure 3.3: Landau-Zener transitions in the RF domain: a) Landau Zener ladder
without coupling between rotating RF field and magnetic substates. The energy
manifolds are separated by hνrf . b) Effect of the coupling RF field. Light shifts
in the dressed state picture cause avoided crossings where angular momentum isconserved. c) zoom to relevant avoided crossings during RF evaporation. Slow atom
movement corresponds to adiabatic transition into the untrapped states, while fast
equals the non-adiabatic process and thus finite probability in the trapped state.
d) replacement of interaction region by black box using Heisenberg uncertainty
principle. If the energy uncertainty is larger than the energy spacing, tunneling into
the untrapped state can occur.

37

by a generalization of confluent hypergeometric functions [73–77]:2
π ~|ΩR |γ

Ptrans = 1 − e− 4

(3.2)

If the trapped atom is very slow, it follows the avoided-crossing adiabaticallyand therefore ends up in an untrapped (bare) state. The non-adiabaticity of afast atom results in a mixing, so that the transition probability is reduced, cf.
Fig. 3.3c). Additional insight is gained by using the Heisenberg uncertainty
principle: the energy separation at the avoided crossing is given by ~|ΩR |.
The physics at the avoided crossing can then be replaced by a black box
through which the atoms passes in the time ~|ΩR |/γ. The corresponding
energy uncertainty is γ/|ΩR |. If the atom is so fast that this energy uncertainty
is larger than the avoided crossing energy gap, the atom can tunnel between
the dressed states. The atom is then still in the trapped (bare) state andevaporation has not occurred. A significant change in the transition probability
is thus expected for γ∼ ~|ΩR |2 and the behavior should be an exponential2function of ~|ΩγR | .
(B) Boundaries on RF power
In the magnetic trap only those atoms with sufficiently high energy canreach the regions where the RF-field is resonant (evaporation shell). Hence,the cold atoms remain trapped, while the hottest atoms evaporate - if the
Landau-Zener transition probability is large enough so that a clean and sharp
evaporation shell is formed aroundpthe center of the trap. With Eq. 3.1 and
3.2 and the thermal velocity v = 8kB T /m, this basically sets a lower limit
to the RF magnetic field amplitude for the transition probability not to drop
considerably3 :
r
~b
8kB T
2
Brf
>
(3.3)
πgF µB [F (F + 1) − mF (mF − 1)]m
At a typical initial temperature of 400µK and a RF field of 100mG, the transition probability is very close to 1 (∼ 1 − 1 × 10−11 ), so that the evaporation
surface can be considered very clean4 . The surface radius itself is determinedby the RF frequency. However, the edge of the ”RF-knife” is not exactly”sharp”: as in the optical regime, the transition is power broadened, whichtranslates into a spatial width ∆x of the evaporation surface. This width can
3 For atomic trajectories perpendicular to the equipotential surfaces.
4 This is also true including realistic atomic trajectories.

38

be estimated by the spatial width of the avoided-crossing in the dressed-stateenergy ladder:
~ΩR
∆x∼
(3.4)
µB mF gF b
At the beginning of the evaporation, the width is negligible compared to theradius of the evaporation shell. But at the end of the evaporation the RFpower needs to be reduced, so that the linewidth remains much smaller thanthe evaporation surface radius and the edge of the RF-knife is still sharp. An
estimate of the upper and lower boundaries using Eqs. 3.3 and 3.4 is shownin Fig. 3.4.

Brf [mG]

100
50

B rf, max

20
10
5

B rf, min

TC

2
1
0.1

1

10

100

1000

T [µK]
Figure 3.4: Estimate for the constraints on the RF power during evaporation. The
upper limit stems from the power broadening of the transition, while the lower
boundary originates in the finite transition probability into an untrapped state.
The critical temperature from the experiment (cf. Chapter 4.4) is still above thepoint where the boundaries meet.

(C) Geometry of the evaporation shell
Size and shape of the evaporation shell are determined by the resonancecondition between Zeeman-splitting and the RF-field. For the quadrupole trap
this results in an ellipsoid around the center, cf. Fig. 3.5a)+b):r
p
x2 y 2~r
Bax mF µB gF 1 − (~er · ~ez )2++ z 2 = hνrf ,
~er =
2
2
|~r|
The dotproduct arises because only the RF amplitude perpendicular to thequantization axis of the magnetic trap field contributes. This leads to holes in39

Figure 3.5: RF evaporation surfaces for different traps and parameters. Only theperpendicular RF amplitude to the Zeeman-field can contribute to the transitions
leading to the formation of holes around the axial axis. a) quadrupole trap (Bax =350G/cm) with RF at 80MHz and 100mG. The hole is rather large as the atoms
are still hot. b) quadrupole trap with RF at 5MHz and 100mG, the situation at theend of the pre-evaporation, cf. Chapter 3.3.2. The hole has shrunk as the atoms
are much colder. c) 60G TOP trap with RF at 80MHz and 35mG. The shell isdeformed due to the bias field and does not correspond to the equipotential surface
any more. d) realistic situation in the experiment with 18G TOP trap and RF at
26MHz and 100mG, cf. Chapter 3.3.2. The red ring corresponds to the trajectoryof the rotating magnetic field zero (”circle of death” or CoD).

40

the evaporation surfaces, cf. Fig. 3.5.
The situation is a little more complicated for the TOP trap due to thepresence of the additional rotating bias field. However, the time scale of theatomic motion and thus the crossing of the Landau-Zener transition is muchslower than the rotation of the bias field, which is in turn much slower thanthe Larmor frequency. Therefore, non-adiabaticity due to changing magnetic
fields can be excluded. It is hence sufficient to estimate the size of the shellby taking the maximum magnetic field amplitude seen by the atom. Due tothe elevated trap bottom, cf. Eq. 2.5, the RF photon energy has to be atleast higher than µBT OP so that the evaporation occurs from the outside to
the inside when reducing the RF frequency. This leads to a flat football-likeevaporation surface, cf. Fig. 3.5c):q(3.5)|BQP (x, y, z) + BT OP (x, y, z)|max mF µgF 1 − (~ex,y,z · ~ez )2 = hνrf
The biggest advantage of RF evaporation is that the evaporation is easily
tunable and the trap geometry remains unchanged. An alternative scheme isthe ”circle of death” evaporation, where atoms are removed by the rotating
Majorana hole around the trap center [44]. This procedure is less efficient thanthe RF scheme, as the evaporation surface is a 1-dimensional ring, whereas theRF-evaporation is on a 2-dimensional shell, cf. Fig. 3.5d).

3.2.2

Evaporation dynamics

It is not sufficient to consider a stationary shell for evaporation, because
the RF frequency is ramped down in the experiment to evaporate the atoms
efficiently. A model for evaporation therefore needs to capture the dynamics
of the ramp-down. The basic assumptions and methods are discussed in thissection.
Considerable work on the simulation and optimization of evaporative
cooling has already been done by a large number of groups [78–83]. The
main problem is that the system is not strictly in thermodynamic equilibrium during the evaporation. The most accurate way would be to consider
the phase-space density function directly including the effects of many body
losses (background, spin-changing and inelastic three body collisions) and thechanging evaporation shell in three dimensions. This can be done by random
sampling of the phase space (Monte-Carlo method) [78, 79] or direct particle
trajectory simulations [83]. A simplification is to consider a truncated Bose-41

Einstein distribution at all times for the density function. This leads to set of
coupled differential equations which can be solved in a self-consistent way by astep-wise cutting treatment. The advantage of this method is that three-bodylosses at the onset of condensation can be taken into account [80–82].
It is in general very tedious to deal with the phase-space density function
directly, which can be circumvented by assuming ergodic mixing of the gas.
The distribution is then only a function of the total energy and not just thepotential energy. In the classical regime, this results again in a set of coupled
partial differential equations, which can either be integrated directly or solved
by a truncated Boltzmann distribution in a quasi-thermodynamic equilibrium[84, 85]. These models usually do not include three-body losses and energy
dependent scattering cross sections, nor the quantum mechanics at the phasetransition. But several simulation cross-checks of the classical density function evolution (Boltzmann equation) show that these simplifications are valid
at least in the classical regime and match very well with both the experimental
results and the Monte-Carlo simulations [84–86].
An even further simplification is the step-wise model, an iterative approachof evaporation and subsequent rethermalization [87, 88]. This model relies
purely on the scaling of characteristic parameters, but various loss processes,
collision rate enhancement by Feshbach resonances and crossovers to the hydrodynamical regime can still be included via rate constants [89].
(A) Model assumptions and derivation of differential equations
For the simulation of the evaporation dynamics in our setup, a modification of the simple step-wise model is used. The non-equilibrium dynamics
are replaced by a repetitive process of cutting the Boltzmann distribution at a
cutoff energy, waiting until the gas has rethermalized without forced evaporation and then cutting at a lower energy again. Repeating this procedure manytimes then yields an interpolation of the changing system parameters during
evaporative cooling. This is roughly the case for frequency controlled RF evaporation in both trap types (quadrupole and TOP) as used in our experiment,
because the evaporation shell approximately matches the equipotential surfaceof the trap, so that all atoms above a certain threshold energy are chopped off.
In addition, collisions with the background gas introduce a finite lifetime in
the trap. The details of the basic assumptions are comparable to those madeby Sackett et al. [84]:• Atom cloud in 3-dim. trapping potential (quadrupole or TOP trap)
parametrized by U (x, y, z) = A1 |x|s1 + A2 |y|s2 + A3 |z|s3

42• Energy-independent s-wave scattering for elastic collisions Γel = n̄σel v̄th ,where n̄ is the average cloud density, σel the elastic s-wave collision crosssection for Rb-87 and v̄th the average velocity• Characterization of system through atom number N and temperature T,
background collision rate Γbkg and ”collision ratio” Γel /Γbkg• Ergodic mixing and evaporation on equipotential surfaces of trap
• Piece-wise cutoff at Ecut = ηkB T followed by rethermalization in τretwith the truncation parameter η• Classical statistics (PSD<< 1)• Incorporation of one-body losses and Majorana losses in quadrupole trap
through a temperature dependent loss rate constant• Neglect two- and three body losses
After a single cut and rethermalization, the atom number and temperature
have decreased fromN to N’ and T to T’. The new atom number is defined byR ηkT
the cutoff integral 0 B ρ(E)dE of the density function, while the new total
R ηk T
energy can be calculated from 0 B Eρ(E)dE. The scaling of the temperature and atom number is then found to be only a function of the truncation
parameter η. The scaling of all other relevant system parameters after rethermalization can then also be calculated. The details of the calculation are
carried out in [6, 64] and only the main results are quoted.
To determine the rethermalization time after a single cut, it is necessary
to evaluate the propagation of a truncated Boltzmann distribution using theBoltzmann equations. This was done by Luiten et al. [85] for different systemparameters, and the authors found that after typically 2.7 collisions the distribution is almost in thermodynamic equilibrium. During this time the atom
number decreases exponentially due to collisions with the background at a rateΓbkg . For discrete steps of ∆t = Γ2.7el this leads to a set of coupled differential

43

System parameter
SymbolAtom number
N
TemperatureT
Total energy
E=( 32 + ξ)NkB T
VolumeV∝ Tξ
Densityn=N/V
Phase-space density PSD∝ nT −3/2
Collision rateΓel ∝ nT1/2

Scaling X 0 /X
ν
νγ
ν 1+γ
ν γξ
ν 1−γξ
ν 1−γ(ξ+3/2)
ν 1−γ(ξ−1/2)(ξ+3/2,η)ν = ΓincΓ(ξ+3/2)

inc (ξ+5/2,η)γ = Logν [ ΓincΓ(ξ+3/2,η)(ξ+3/2)]
ξ = s11 + s12 + s13
Rη
Γinc (a, η) = 0 xa−1 e−x dx
Γ(a) = Γinc (a, ∞)Table 3.1: Scaling of the thermodynamic quantities after a single cut and rethermalization as a function of truncation η and trap ξ.

equations under consideration of the scaling from Tab. 3.15 :Γel (t) Γinc (ξ + 3/2, η(t))Γinc (ξ + 3/2, η(t))
Ṅ (t)=[− 1] − Γbkg
N (t)
2.7
Γ(ξ + 3/2)
Γ(ξ + 3/2)
Ṫ (t)
Γel (t)Γinc (ξ + 5/2, η(t))
=
[
− 1]
(3.6)
T (t)
2.7 Γinc (ξ + 3/2, η)Γ(ξ + 3/2)N (t)Γel (t) ∝T (t)ξ−1/2(B) Discussion of the differential equations
The scaling of N and T after a cut and rethermalization is only determined by the trap, truncation and collision ratio Γel /Γbkg , cf. Eq. 3.6.
It is therefore sufficient to characterize the atoms in the trap at any given
time during evaporation only by the instantaneous collision ratio and truncation parameter. The relative scaling of all quantities after a single cut
and rethermalization can then be expressed as a function of these two values. In that sense, evaporative cooling with arbitrary initial atom numbers,
temperatures and evaporation trajectories is fully characterized by the curve(collision ratio(t), η(t)), where η(t) is controlled externally (corresponding to
the tunable RF frequency in the experiment). An example is shown in Fig.
5 For a Taylor approximation of small changes in N and T these exact equations

match with the approximate ones from [88].

443.6a), where the truncation is held constant/increases over time.
From an experimental point of view, it is relevant to determine regions ofcollision ratio and η, where either the PSD, density or elastic collision rate areconstant during evaporation (”break-even curves”). These implicit functions
can be found in the Appendix A.1. The corresponding curves for all three conditions are shown in Fig. 3.6a)+b) for a linear and harmonic trap, which differ
considerably from [88], as no Taylor approximation is made. An approximate
approach leads to wrong results at very low truncation parameters so that the”break-even” curves are shifted to the right.
If the evaporation starts above the collision rate curve (highest ”break-even”
curve in Fig. 3.6a)+b)), ”run-away evaporation” will occur, i.e. the elastic collision rate will increase making the evaporation very efficient. This is
only possible for sufficiently large trap lifetimes. On the other hand, below
this threshold the collision rate will decrease and the evaporation time will
increase. It is still possible to have increasing densities or PSD in this region,
but they will ultimately decrease after sufficiently long time, as indicated by
the example trajectory in Fig. 3.6a). The collision ratio break-even curveshows a minimum as a function of η because of two competing effects: fordeep truncations (small η) background losses do not play a role any more,since they decrease the total atom number by a very small amount. On theother hand, weak truncations (large η) increase the effect of the background.
For the extreme case of starting directly on the collision ratio ”break-even”
curve, all right hand sides of Eq. 3.6 become constant for all times and ananalytical solution is available showing an exponential behavior with respect
to time of all thermodynamic quantities as well as of the cutoff energy Ecut (t),Appendix A.1.
(C) Numerical integration and simulation
Obtaining an exact solution requires numerical integration of the differential equations with a step size of ∆t = 2.7τel while taking the background
losses into account. The time scale ∆t reflects the initial model assumption ofrepetitive cutting and rethermalization. The external control parameter of theevaporation is the truncation parameter η. The optimum truncation function
η(t) can be determined by testing all different possible curves and comparing
the expected condensate size. A simpler approach was suggested by Hess etal. [7, 64]: instead of global optimization of the evaporation trajectory, a local
method is sufficient: at any given time during evaporation, the system has acertain collision ratio with η as the only degree of freedom for the next cut.
The best truncation can then be found corresponding to a chosen ”strategy”.

100
80

n= const

Γel =const

Runaway
Evaporation

60
40

possible evaporation
trajectories

20

b)

500

collision ratio
Γel /Γbkg

a)collision ratio
Γel /Γbkg

45

400

Runaway
Evaporation300Γel =const

200

n=const100

PSD=const
0

c)

2

4

6

η

8

10

PSD=const

12

0

10

2

4

6

η

8

10

12

8

d)

9
evaporation
quality

7

7
η opt

η opt

8
6

evaporation
quality

6

5
4
0

100

200
300
Γel /Γbkg

collision ratio

5

collision ratio
400

500

200

400

600
Γel /Γbkg

800

1000

Figure 3.6: a)+b) show the break-even curves for constant Γel , n and PSD for a linear(a)) and harmonic (b)) trap. The scaling behavior of the thermodynamic parameters
is only a function of the trap, the truncation and the collision ratio. The lowest
possible collision ratio Γel /Γbkg for runaway evaporation is at η∼ 3.5 for the linear
trap and at η∼ 5 for the harmonic trap. The collision ratio threshold is much lower
for a linear trap because the cloud volume decreases much more than in a harmonic
trap during cooling. Example trajectories during evaporation are also indicatedin a). c)+d) show the optimum η for the two different strategies ”EvaporationQuality” and ”Collision-Quality” in the linear (c)) and harmonic (d)) trap. For low
collision ratios the best truncation parameter converges to the previously mentioned
minimum value for the two different strategies.

46• ”Evaporation quality” for large BEC: maximize

Log[P SD0 /P SD]−Log[N 0 /N ]

Γ0• ”Collision quality” for fast BEC: maximize Γelel˙• Alternative evaporation time: maximize PP SD
SD
The ”evaporation quality” measures how fast the phase-space density increases
as atoms are lost due to the evaporation. This strategy was used for the simulations.
In essence, every strategy yields an optimum η value for each collision ratio in
the runaway region, cf. Fig. 3.6c)+d). The collision quality strategy does notshow a variation in η for different collision ratios and stays at the minimumof the elastic collision rate break-even curve, while the evaporation quality
strategy prefers higher truncations with increasing collision ratio. The reason
is that for very high elastic collision rates, rethermalization occurs almost immediately and losses become negligible making it favorable to cut away only
the very hottest atoms. These curves are used to perform the integration of
Eq. 3.6. The code (Mathematica 6.0) is listed in Appendix A.3. The results
are discussed in the next section.
Some extensions of this basic simulation scheme are possible to describe
the experimental situation more accurately. For the evaporation in the linear
trap, the Majorana losses can be incorporated through a temperature dependence of the trap lifetime in each cutting step. This effect was measured in
Chapter 3.3.2 for the trap used in the experiment and compared with the theory. The loading of the atom cloud from the quadrupole to the TOP trap canbe predicted by assuming adiabaticity. The details can be found in Appendix
A.2.
To increase the evaporation quality, the TOP circle of death can be ramped
down simultaneously with the RF evaporation shell radius. Assuming rCoD =αrrf with rCoD = 2BT OP /b and using Eq. 3.5, this gives the condition:
rrf = (1hνcut
− BT OP )mF gF µB
Brad⇒

BT OP (νcut ) =

h
α
νcut
(3.7)
mF gF µB
α+1

The trap tightening can be included into the simulation by first performing
the cut and the scaling of the parameters and then adiabatically loading thecloud into the stiffer trap. Exact formulas can be found in the Appendix A.2.
One of the main flaws of this step-wise model is that the procedure does
not reflect the experimental conditions, since the atom cloud is actually never

47

in equilibrium. But comparisons with Monte-Carlo simulations by Hulet et al.
[84] show that both methods match very well. For very fast rethermalization
times compared to the total evaporation time scale, the time discretization
should be a good approximation. The simulations by Walraven et al. confirm
that the assumption of ergodic mixing is valid [85, 90]. In spite of this, deviations might still occur resulting from the assumption of a clean evaporation
shell on the equipotential surface of the trap. In addition, the absolute value
for the simulated condensate size will not be correct, because three body losses
set in at the critical density (n∼ 1014 cm−3 ).

3.2.3

Simulation results and experimental guidelines

In this section the evaporative cooling simulation is used to provide arough guideline for the procedure in the experiment. One of the main problems
is whether or not to perform evaporation in the linear trap and if yes, todetermine when to switch to the TOP trap so that the condensate is as large
as possible. This is a nontrivial problem, because the solution depends on
many parameters: initial atom number and temperature of the atoms, theevaporation trajectory of the evaporation in the linear trap or the switchon value for the TOP bias field. It is thus useful to only study the roughbehavior of the system through the simulations and extract the most promisingexperimental procedure.
(A) TOP loading
To have a guideline of what to expect in the experiment, the evaporation
Log[P SD0 /P SD]
quality (− −Log[N 0 /N ] ) during evaporation in the quadrupole and different
TOP traps (60G, 30G and 10G) is simulated, cf. Fig. 3.7. With decreasing
TOP bias field, the evaporation becomes more efficient. This is due to thestiffer harmonic trap increasing the elastic collision rate and thus the collision
ratio. Initially, evaporation in the linear trap is most efficient, but for lower
temperatures the Majorana losses decrease the lifetime of the cloud, which ismodeled through a change in Γbkg , so that the collision ratio decreases andultimately drops below the break-even threshold. This causes the crossing of
the evaporation quality in the linear and harmonic trap.
The reason why evaporation in the linear case is much better are the scaling
laws: for a certain decrease in temperature, the cloud volume shrinks down a
3/2
3
lot more than in the harmonic case (Vlin ∝ Tlin
while Vharm ∝ Tharm ). The
increase of the collision ratio is thus much higher in the linear trap.

48

PSD

1
0.01
10 -4
10 -6
0.1

B TOP =60G
B TOP =30G
B TOP =10G
Quadrupole trap
0.5 1.0

5.0 10.0
N [10 7]50.0 100.0

Figure 3.7: Comparison of evaporation quality in linear and harmonic trap. The
initial cloud before evaporation has 109 atoms at 200µK in the quadrupole trap.
Adiabatic loading into the TOP trap is assumed without cloud chopping from the
CoD. Majorana losses are taken into account for the quadrupole trap through themeasured dependence of the trap lifetime, cf. Chapter 3.3.2. The simulation in thelinear quadrupole trap stops at a cloud temperature of T= 30 µK (PSD ∼ 10−5 ),because the quadrupole trap lifetimes were only measured at higher temperatures.
The evaporation quality corresponds to the slope of the lines.

The loading of the TOP trap is determined by two major effects: evaporation in a very stiff harmonic trap can reach the quality of a linear trap.
But for too low bias fields the CoD is much smaller than the size of the cloud,causing considerable chopping. However, while evaporation in the linear trap
is superior to shallow TOP traps of large bias fields, this inverts for lower temperatures, because of Majorana losses. Therefore, the optimum strategy is to
do some evaporative cooling in the quadrupole trap beyond the point where
the cloud just fits into a 60G TOP trap and switch to the TOP trap at somelater time with a lower bias field. The procedure for finding this optimum
loading point will be discussed in Chapter 3.3.2.
(B) Evaporation trajectories
For typical evaporation trajectories in the experiment, several different
possibilities are considered, as illustrated in Fig. 3.8: evaporation in the linear
trap and harmonic trap at different bias fields, with and without a simultaneously following CoD. As the behavior of the evaporation trajectory is afunction of the initial cloud conditions, a set of typical experimental values ischosen to illustrate the influence.
All trajectories in the quadrupole trap look almost linear with slight curva-

49

ture downwards. In the break-even situation, cf. Fig. 3.6a), an exponential
dependence is expected - but since the initial collision ratio is far away fromthreshold, the rate constant of the exponential keeps increasing over time causing a straightening. In contrast, the trajectories for TOP evaporation at 60G
and 20G are almost pure exponential curves. This is due to the fact that thelowest collision ratio for run-away evaporation is much higher in the harmonic
than in the linear trap, cf. Fig. 3.6a)+b). Note that the evaporation timescale in the quadrupole trap is significantly faster (∼ 10s-20s) than in the TOP
trap (∼ 60s). Finally, the simulation with a following TOP bias field showsan almost linear behavior again. There are two reasons for this: as the bias
field decreases, the trap bottom shifts downwards, which the RF-knife has tofollow to maintain the same truncation parameter η. Also, the adiabatic trap
stiffening increases the collision ratio, thus making the evaporation more efficient and quicker over time. Therefore, the rate constant of the exponential
increases over time and the final curve looks approximately linear.
For evaporation in the linear trap or harmonic trap with a following CoD, asingle linear slope seems suitable for the experiment. In the case of the TOP
trap at constant bias field, a multi-linear interpolation of maybe two or threedifferent slopes should be appropriate for experimental optimization.
(C) Evaporation quality
The same major cases as before are used to determine typical evaporation
qualities in the different trap types, cf. Fig. 3.9. In the quadrupole trap,
the evaporation quality is ∼ 3 − 3.5 for not too long evaporations - a rather
high value stemming from the very low collision ratio threshold for runaway
evaporation, cf. Fig.3.6a). In the very shallow TOP trap of 60G, this thresholdis much higher and the reduced elastic collision rate causes the evaporation
quality to drop to ∼ 2, while in the much stiffer harmonic trap of 20G valuesof ∼ 3 − 3.5 are reached. These numbers reflect again the fact that it is best
to start with some evaporation in the quadrupole trap and then load into the
TOP trap. Finally, the evaporation in a TOP trap of an initial bias field of
60G and a following CoD shows an evaporation quality of ∼ 2. Note thatthe initial atom numbers are reduced by a factor of 10 compared to the casewithout the decreasing bias field so that the initial cloud is not even in the
runaway regime. Due to the adiabatic trap stiffening, the collision ratio still
increases over time and slowly drifts into the runaway regime. If the samestarting values are chosen, the evaporation quality in the case of a following
CoD would be ∼ 2.5 − 3. This makes sense, because the value lies in between
those of the stationary 60G and 20G case.

50

0.6
0.4
0.2
0.0

b)

Linear Trap

1.2
1.0
0.8

N=10 , T=700 K
N=109, T=500 K
N=109, T=300 K
9

0

5

10

15

20

Ecut[a.u.]

Ecut[a.u.]a)Btop=60G
1
0.5
0.1
0.05
0.01
0.005
0.001

25

N=109, T=180 K
N=109, T=150 K
N=109, T=110 K

020

Btop=20Gd)

N=10 , T=40 K
N=108, T=30 K
N=108, T=5 K
8

0 10 20 30 40 50 60 70t [s]Ecut[a.u.]

Ecut[a.u.]

2
1
0.5
0.2
0.1
0.05
0.02

60

80

t [s]

t [s]
c)

40

Btop=60G, CoD follow

1.00
0.70
0.50
0.40
0.30
0.20
0.15

N=108, T=50 K
N=108, T=30 K
N=108, T=20 K

0

10

20 30 40

50 60

t [s]

Figure 3.8: Simulation of evaporation trajectories for different traps. To observe
the dependence on the initial cloud, multiple temperatures are chosen for each trap
according to typical values from the experiment. Note that a) is on a linear scale,
while b), c) and d) are on a logarithmic scale. The simulations in all harmonic traps
were carried out until the phase transition, while the evaporation in the quadrupole
was only done to a temperature of ∼ 5µK, where the Majorana losses are definitely
too large to continue. The minimum bias field in d) was 10G. The general trajectory
shapes (linear or exponential) do not exhibit a significant dependence on initial cloudparameters.

51

Linear Trap

a)

b)

10

Btop=60G

-4

1

10-6

N=109, T=700 K
10-7
N=109, T=500 K-810 N=109, T=300 K
15 20 30

PSD

PSD

10-5

50

70 1007

N [10 ]

PSD

10

Btop=20G1
0.1
0.01 N=108, T=40 K
0.001 N=108, T=30 K
8
10-4 N=10 , T=5 K
1.0 1.5 2.0 3.0 5.0 7.0 10.0

N [107]

N=109, T=180 K
10 N=109, T=150 K
N=109, T=110 K
10-6
0.1
0.5 1.0 5.0 10.0 50.0 100.0-4

N [107]

d)

PSD

c)

0.01

Btop=60G, CoD follow
10
1
0.1
0.01
8
0.001 N=10 , T=50 K-4 N=108, T=30 K
10
10-5 N=108, T=20 K
0.05 0.10 0.50 1.00 5.00 10.00

N [107]

Figure 3.9: Simulation of evaporation quality for different traps. Different temperatures motivated from experimental data are chosen to determine the dependence
on the initial trap values (similar to Fig. 3.8). The slopes of the lines correspond to0
N0
the evaporation quality (−Log PPSD
SD /Log N ). They exhibit only a weak dependence
on initial cloud values, since they are all nearly parallel for each trap.

52(D) Optimum initial (N, T) in BEC cell
The last question to be addressed by simulations are the optimum initial(N, T) values in the BEC cell before evaporation to achieve the largest condensate. This is done by simulating the evaporation for various initial clouds,
cf. Fig. 3.10. For each (N, T) pair the evaporation in the quadrupole trap
is performed until the temperature is so low that the cloud just fits into the
CoD of a 60G bias field TOP trap. (A simulation for the optimum point toload the atom cloud into the TOP trap is omitted for simplicity.) Then theactual evaporation proceeds with a joint CoD/RF ramp until the condensate
sets in (PSD∼ 1). The TOP loading is assumed to be adiabatic as well as thestiffening of the harmonic trap.
In principle, the BEC machine will have a ”working region”, which meansthat atom number and temperature of the initial cloud in the BEC cell are
only in a certain area of this 2-dim. parameter space. An example of thisworking region is shown in Fig. 3.10, where the edge of this area is assumed
to be a straight line (N ∝ T ). Obviously, the optimum (N, T) value is thensomewhere on this edge. Under this condition, two cases can be identified:
if this line is very high in offset, N - rather than T - should be optimized(maximum N). In contrast, for a very low offset, this rule is inverted and it ismore promising to start in the lower temperature region with a smaller numberof atoms.
The physical explanation is that if the edge is in the upper left area of theparameter space (N, T), the collision ratio is very close to the threshold of
run-away evaporation. Hence, the collision ratio does not increase rapidly andthe evaporation quality does not increase as the evaporation proceeds. It istherefore better to start out with more atoms, because the evaporation is notmuch more efficient in the low (N, T) region of the edge of the working region.
On the other side, if the edge is in the lower right region of the parameter space(N, T), the collision ratio already starts out far off from threshold causing a
rapid increase of the collision ratio and thus an increase of the evaporation
quality. It is therefore more useful to start out in the low (N, T) region of theedge to increase this effect, although having lesser atoms initially.
To finally answer the question of what the best point to start out with is, theactual working region of the machine has to be determined experimentally, i.e.
what atom numbers and temperatures can be produced in the BEC cell. This
will be discussed in the next section.

53

100

50

50

20

20

10

10

5

5

2

2

1

1

800

Initial T

100

N BEC[a.u.]

600

Runaway
Possible
Runaway
ThRunawayreshold
Threshold machine region

Possible optimal
machine region

400
Edge of possible
working region

200
0.5

1

1.5
2
2.5
Initial N[109 ]3

Figure 3.10: Simulation of relative expected BEC atom number for different initialatom numbers and temperatures in the BEC cell before evaporation. A statement of
the absolute condensate size from the simulations is inaccurate, because three-bodylosses are neglected. The whiter the region, the more atoms in the condensate. The
marked area shows a possible ”working region” of the machine, while the edge isindicated by the dashed line. The dotted line indicates the runaway threshold. As
evaporation proceeds, the atom cloud parameters will move on a contour line.

3.3

Optimization of the BEC machine

The main purpose of the evaporation simulations is to provide a guideline for the optimum condensate production in terms of atom number. The
experimental optimization protocol is explained in this chapter. In the firstsection, the experimental determination of the machine working region is discussed to determine the best atom number and temperature to start out with,
before evaporative cooling in the BEC cell. In the second section the experimental optimization of the final evaporation sequence is illustrated using both
the guidelines from the simulation and experimental data.

54

3.3.1

Optimization of the atom cloud parameters

Determination of the machine’s working region in terms of possible initial(N,T) values in the BEC cell requires all different settings of all parameters
for the laser cooling part to be checked. Since there are at least a dozen
experimental parameters involved, this seems impossible to accomplish. In
addition, it would require the machine to run stable on a very long time scale(several days), which is difficult with our Rb oven.
A simplification is to subdivide the sequence into several temporal steps (MOT
loading, compression, cooling, loading into magnetic trap,...). Each block canthen be optimized for itself. The parameter space can be reduced this way
and the working region can be determined by varying the different blocks. To
clarify this idea, consider the following example: since the final temperaturein the BEC cell depends on the CMOT size before loading the atoms into themagnetic trap, the (N,T) region can be scanned by varying the cloud diameter.
This can be done by changing the atom number of the density-limited CMOT.
This general approach was actually first tried in the experiment and was also
the guideline for optimizing the sequence described in Chapter 3.1. However,this approach has some disadvantages:• The optimization time needed in the experiment is very long, because
the sequence settings of the different blocks have to be changed in acontrolled way and quantitative evaluation of the cloud in the BEC cellis necessary. Hence, the blocks were only optimized according to theirtemporal order - cross correlations between the blocks are therefore nottaken in to account.
• Long-term drifts of the oven on a time scale of several days prevents astraight-forward optimization procedure• Reoptimization of the laser cooling sequence is necessary after typicallyone month due to drifts of machine parameters (e.g. MOT mirrors)
The idea for an improved method is to group the various sequence knobs of thelaser cooling part not in a temporal order, but into functional groups, among
which the correlation is low. From experimental experience it was found thatsome knobs change the outcome a lot, whereas other knobs are irrelevant toatom number and temperature - they would rather influence other values suchas the production time. This results in three different knob categories: rough,intermediate and fine. For optimization, each category is then optimized for
itself. An overview of this classification can be found in Tab. 3.2.

55

Rough
Rb oven
MOT loading time

Intermediate
Molasses detuning ramp
Magnetic catch point
Molasses duration
CMOT field ramp+det.

Fine
Molasses repump ramp
Prep. cycling det.
CMOT duration

Table 3.2: Grouping of sequence parameters for atom cloud production in BEC cell.
The distinction stems from experimental experience and is guided by how much theinitial atom cloud parameters N and T are changed.

The advantage of this method is that cross correlations between different
sequence steps are taken into account, because the sequence is no longer optimized step by step, but treated as a single one. To increase the update timeof the iteration, the darkness and size of the atom cloud in the BEC cell are
estimated by visual inspection of the raw image, rather than by the evaluationof the data. The optimization time is therefore much lower and the influenceof long term drifts can be omitted. Also, if any drifts occur, a reoptimization
can be done very quickly.

a) Slow Methodb) Fast Method

group 11
2

3

4group 2 Optimumgroup 11

2group 2

Optimum

Figure 3.11: Comparison between a correlated grouping (a)) and an uncorrelatedgrouping (b)). The black lines indicate changes caused by optimized one groupwhile holding the other constant. The optimum point lies in the middle and only
two groups are considered here. Many iteration steps are necessary for a correlatedgrouping, while the optimization time is much faster in the uncorrelated case.

This idea can be illustrated in the case of a two-dimensional parameterspace. An example for a very correlated grouping is shown in Fig. 3.11a):

56

by varying the first group, the radius to the reference point in the bottom
left corner is varied, while changes in the second group correspond to varying
the angle. Therefore, when optimizing group by group, many iterations arenecessary, because the optimum point of one group depends on the positionof the other group (correlated grouping). In contrast, Fig. 3.11b) shows asituation, where the groups are uncorrelated, because the best position ofeach group is independent of the position of the other group. The amount of
iteration steps is therefore much lower.

100

50

50

20

20

10

10

5

5

2

2

1

1

800

Initial T

100

N BEC[a.u.]

600

400

200

Optimum
operating point
Maximum atom
number in BEC cell

0.5

1

1.5
2
2.5
Initial N[109 ]3

Figure 3.12: Determined edge of the machine working region with expected BEC
atom number in relative units for various cloud parameters before evaporation. The
dashed line serves as guide to the eye to indicate the working region. The maximum
amount of atoms transferable into the BEC cell is 1.6 × 109 .

The working region of the machine was determined by setting the tworough knobs (Oven open or closed and MOT loading time) to a certain value
and then optimizing first the intermediate and then the fine screws. This was
repeated for various positions of the rough knobs. Since the knobs within agroup are still correlated, the optimization for the intermediate/fine screws is
obtained by a manual random walk in the parameter space6 . Aside from that,
6 The most important knob in the intermediate group seems to be the molasses

detuning ramp.

57

there is no further guideline. It was found that the optimum settings of the intermediate/fine screws are the same for a broad range of rough screw positions.
The manual iteration time can therefore be even more decreased by shortening the MOT loading time to e.g. 1 s. With this method the temperature
could be reduced by 50% (compared to the a priori optimization of the laser
cooling sequence) in only ∼ 30 min while keeping the atom number at 1.6×109 .
The results for the edge of machine working region are shown in Fig. 3.12.
In our case, the machine does not make it into the area where the temperature
becomes more important (bottom right). The best operating point is thus
given by atom number optimization with N∼ 1.6 · 109 and T∼ 450µK. The
machine region shows a cutoff in atom number at this point, because this is
the maximum amount of atoms transferable into the BEC cell. The sequence
for the optimized atom cloud production in the BEC cell is illustrated in Fig.
3.13, which forms the starting point of evaporative cooling. Note that there is
no CMOT in this case, as the compression only caused significant heating.
MOT

Molasses

Preperation

Ramp

10s

5ms

3ms

750ms
353 G/cm

Axial field
gradient
Cycling
power
Cycling
detuning
Repumppower
Repumpdetuning

Digital

100 G/cm
8 G/cm100%

100%

0 G/cm

0 G/cm

100%

100%

0%
2.8 Γ

3.5 Γ

>> Γ
9.3 Γ

100%

100%

100%

0Γ

0Γ

0Γ

LIAD on

9.3 Γ

0%

0%

>> Γ

>> Γ

Depump on(δ=0)

Translation
Stage Trigger

Figure 3.13: Summary of the optimized F=1 atom cloud production sequence. The
given values are rough values and require tuning on a monthly basis due to mirror
drifts or changing Rb background environment in the MOT cell. The absolute values
for the beam intensities are given in Chapter 2.1.2.

58

3.3.2

Evaporation optimization

In this section the experimental optimization of the evaporation is discussed. The general guideline is to maximize the evaporation quality. It is
crucial that the initial atom number (1.6 × 109 ) and temperature (450 µK)
are reproducible during this procedure, since the best evaporation trajectory
depends critically on these values. A mismatch of initial (N,T) in the BEC
cell will cause falling off the narrow runaway path and condensation will notoccur. The full sequence consists of evaporation in the quadrupole trap, loading of the TOP trap and final evaporation in the TOP trap. The details of the
optimized evaporation sequence can be found in Fig. 3.14, while quantitative
results are summarized in Fig. 3.15.

Pre-Evap

5ms

14s

rfcut 1

rfcut 2Detection

10s

20s

150µs
80 MHz80 MHz80 MHzrf-knife

26 MHz10 MHz14.6 MHz

18 MHz100%

100%

rf power

40%
1%

0%

18 G

TOP

18 G
0G

0G

axial field
gradient

0%353 G/cm

0 G/cmFigure 3.14: Optimized evaporation sequence for initially ∼ 1.6 · 109 F=1 atoms
at ∼ 450 µK. The Pre-evaporation starts in the quadrupole trap, while rfcut1 andrfcut2 are in a 18 G TOP trap. The TOP switch-on time is 5 ms with a linear rampof the bias field. The trap bottom in this case is at 12.4 MHz. The critical point is
at around 15.0 MHz, while almost pure condensates appear at 14.6 MHz. The twolinear evaporation trajectories resemble the expected exponential from simulations,cf. Fig. 3.8.

59

Figure 3.15: Absorption images of atom cloud at different points of the evaporation
sequence. The first three pictures have a demagnification of ∼ 1:2, while the imaging
system has a ∼ 5:1 zoom for the last three ones. The aspect-ratio in the quadrupole
trap (images 1+2) is clearly lower than in the TOP trap (images 3-6). The cloud
expands a little upon loading into the harmonic trap (2+3), since the confining
potential is shallower. During this transfer, heating causes a loss of a factor of 2 inphase-space density, while almost no atoms are lost.

60(A) Slope of evaporation in quadrupole trap

Atom Number [a.u.]

As motivated by the simulations, after some evaporation in the quadrupole
trap, the TOP trap is switched on and the final evaporative cooling is performed there. The best trajectory for the linear trap is found by ramping
the RF-knife with different slopes to a common reference temperature while
noting the atom number. If the slope is too low, background losses become
significant. On the other hand, too steep slopes cause chopping of cold atoms
and the rethermalization is insufficient. The maximum occurs at a slope ofroughly 5MHz/s, cf. Fig. 3.16.

1.0
0.9
0.8
0.7
0.6
0.5
0.4

2

4

6

8

10 12 14

Pre-Evaporation Slope [MHz/s]
Figure 3.16: Pre-evaporation slope optimization in quadrupole trap. The RF-knife
was ramped down with different slopes to a common reference temperature of 200µK,
where Majorana losses are still rather low. The optimum slope of ∼ 5MHz/s is where
the atom number is maximum. The starting cloud parameters before evaporation
are N = 1.6 × 109 at T = 450µK. The solid line is a guide to the eye (polynomial)(B) Effect of Majorana losses
The incorporation of Majorana losses into the simulation requires themeasurement of the quadrupole trap lifetime τQP versus the atom cloud temperature T. This can be estimated in a very crude model developed by Petrich
et al. [41]: if an atom passes too close to the trap center, the Larmor precession time scale becomes comparable to the rate of the change of magnetic
field direction. The total loss rate is determined by the product √of the cloud−3
density n ∝ T , the flux proportional to the atom velocity v ∝ T and thesurface of the loss ellipsoid ∝ v. For large temperatures the lifetime is limited

61

1.6
1.4
1.2
1.0
0.8
0.6
0

b)
TOP

150
τ = 40+/-15s
QP-Trap

Lifetime [s]Atom number [10 8 ]

a)

100
50
0

10

20

30

40

QP

0

Hold time [s]50 100 150 200 250 300 350
Temperature [µK]

Figure 3.17: Atoms in the quadrupole trap. a) Typical data set for the determinationof the lifetime at a certain temperature in the quadrupole trap via an exponential
fit. b) Results of the quadrupole trap lifetime for multiple cloud temperatures. The
error bars stem from the exponential fit to the atom number decay for different holdtimes. The red solid curve is the theory fit to the data from Eq. 3.8, while the grayregion indicates the predicted range of the limiting background losses from the fit.

to τbkg by background collisions. The total loss rate is given by:
1
1
1
=
+
τQP (T )τbkg βT 2(3.8)

Results of the trap lifetime measurements are shown in Fig.3.17a)+b). A fitof the theory to the data then yields:
τbkg = (135 ± 35) s

β = (0.003 ± 0.001) s/(µK)2

For comparison, very cold clouds are loaded into the TOP trap and the lifetime is again determined, cf. Fig.3.17b). These lifetimes match the predicted
QP background lifetime from the Majorana losses fit within the error bars.

(C) Evaporation duration in the quadrupole trap and TOP loading
The optimum evaporation time in the quadrupole trap is the most fragile
parameter of the sequence, because it depends strongly on the cloud parameters. If the cloud is for instance too cold, Majorana losses can kill the cloudaway, while the CoD can chop off many atoms if the temperature is too high.
The longer the evaporation in the quadrupole trap is at a given slope, thelower the cloud temperature with which the TOP trap is loaded will be. The
bias field at which the TOP trap is turned on should be so low that the cloudjust barely fits in (mode matching). In essence, the minimum TOP bias field
is a function of the cloud temperature in the quadrupole trap. To measure62

this, the initial cloud in the BEC cell is evaporated to a certain temperature(5 MHz/s) and then the TOP trap is turned on at various bias fields. The
optimum bias field for a certain cloud temperature is then given by the point,where the relative number of atoms caught in the TOP trap saturates, cf. Fig.
3.18a)+b). A theoretical estimate of this dependence is possible by assumingthat the CoD radius (rCoD = 2BT OP /b) should be at least twice the 1/e radiusof the atom cloud after an adiabatic transfer into the TOP trap. Here b is theaxial field gradient of the quadrupole trap. With the trap parameters αlin (b)and αharm (b, BT OP ) defined in Appendix. A.2 one finds:
BT OP (TQP ) = b

4/3



3
2π

4/9

[αlin (b)αharm (b, 60 G)]2/9
TQP(60 G)1/3(3.9)

After loading the TOP trap at a certain temperature of the atom cloud,a combined RF/CoD evaporation (rrf = rCoD ) is performed. The optimum
TOP loading point is found by comparing evaporation sequences with different
evaporation times in the quadrupole trap to a common reference point chosen
to be at a later point in the overall evaporation sequence. A 15G TOP trap issuitable as a reference point7 .
In the experiment we chose different durations for the evaporation in thequadrupole trap at constant slope, switched on the TOP at the corresponding
optimum bias field and finally evaporated down linearly to the reference point.
Log[P SD0 /P SD]
The parameter to maximize is then the evaporation quality ( −Log[N 0 /N ] ) bycomparing the cloud at the reference point to the intial cloud before evaporation, cf. Fig. 3.18c). Over a long range of evaporation times in the quadrupole
trap, the evaporation quality does not vary much. But at 14.5 s, the quality
drops and kinks down at 15.5 s. Since the evaporation in the linear trap ismuch faster than in the harmonic trap, an evaporation time of 14 s in thequadrupole trap is used. After this time the temperature is ∼ 75 µK, whichcorresponds to an intitial TOP field of 18 G for mode matching, cf. Fig. 3.18b).
The optimum slope of 1 G/s for the employed evaporation of this measurement in the harmonic TOP trap is determined by maximizing the evaporation quality for different slopes in the TOP trap, cf. Fig.3.18d). The data does
not exhibit the expected maximum, because the bias field cannot be ramped
slower than 1 G/s (TOP coils get too hot).
To examine the time scale during which the offset field should be ramped up
to minimize losses, the atom number in the harmonic trap is measured for
7 As discussed later, lower values would lead to a decrease in the TOP lifetime.

63

a)

b)

Optimal TOP Switch-On: Sample

0.960
BTOP [G](turn on value)

0.8
NTOP /NQP

Optimal TOP Switch-On

0.7
0.6

T~280µK
BTOP~50G

0.5

50
40
30

0.4

20
0.3
30

35

40

45

50

55

100

60

150

BTOP [G]d)Pre-Evaporation - Slope
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0

250

300

(after pre-evap)

CoD+RF Evaporation -Slope
1.8

Evap-Quality

Evap-Qualityc)200
T [µK]

1.6
1.4
1.2
1

8

10
12
14
Pre-Evaporation Time [s]

16

2

4

6
8
10
TOP-slope [G/s]12

14

Figure 3.18: TOP loading optimization: a) Sample measurement of the relative
amount of atoms transferrable into the TOP trap from a 280 µK atom cloud after
evaporation in the quadrupole trap. The lowest bias field where the cloud barely fits
into the harmonic trap in this case is ∼ 50 G. b) Minimum offset field for different
cloud temperatures in the quadrupole trap. The blue solid curve is the theoretical prediction from Eq. 3.9. c) Measurement defining the evaporation time in thequadrupole trap: for different TOP loading points the cloud is always evaporated
further to a common reference point. The decrease for 14.5-16 s stems from Majorana losses, making 14 s the optimum value. d) Measurement to find the best slope
for CoD+RF evaporation in the TOP trap. The evaporation quality of 1.8 matchesroughly the prediction from the simulations in Fig. 3.9.

64

linear ramps from 1 ms to 20 ms. A dependence could not be observed. Since
the coil current overshoot is negligible for at least 5 ms, this value is used in
the experiment.
(D) RF/CoD evaporation in the TOP trap
With the atom cloud in the 18 G TOP trap, the further strategy is to simultaneously ramp the RF and CoD to a certain minimum offset field followed
by a final RF evaporation step at a constant offset field. The final ramp-downbias field is determined by the lifetime of the TOP trap as a function of BT OP .
This can be measured by evaporating the atom cloud down to different bias
fields and observing the lifetime after the ramp-down via different holding
times and an exponential fit, cf. Fig. 3.19.
For bias fields above 18 G, a dependence of the TOP lifetime on the bias fieldcannot be observed and the lifetime matches the expected background lifetime
from Fig. 3.17b). Below this value, the curve kinks down. Hence, it seems
to be best not to decrease the bias field below 18 G8 . Therefore, evaporative
cooling with a following CoD is not necessary, since the TOP trap is alreadyloaded at 18 G. The final evaporation is done at a constant TOP trap of 18 Gwith only the RF-knife.

Lifetime [s]

150
100
50
0
0

5

10

15

20

25

Btop [G]Figure 3.19: Losses in TOP trap. The data is obtained by cooling down to different
bias fields and then measuring the lifetime. The lower the offset field, the lower the
cloud temperature. The lifetime errors stem from the exponential fits. The red fit
function (solid line) is taken from Eq. 3.10. A background lifetime of 155s (dashedline) was assumed, which gives a fit constant of α = 0.45.
8 However, a lower bias field might still be better, since two effects contribute

when lowering the offset field: the lifetime decreases but the elastic collision rateincreases due to the stiffer trap, such that BEC can be reached faster.

65(E) Losses in the TOP trap
Losses at low bias-fields have been extensively investigated by Ensher et
al. [44]. There are several possible loss mechanisms, such as three-body losses,
CoD evaporation losses or even magnetic field noise in the kHz range inducing
spin flips into untrapped states for very low bias fields. These effects can all
be ruled out in our case, as discussed in [44].
Another possibility, which is also discussed in [44], are Landau-Zener transitions. If this is true, they might occur when the atom passes near the ”avoided
crossing” formed by the average potential of the trapped and untrapped stateat the center. Neglecting the temperature dependence, the transition probability should scale exponentially with the offset energy ∆Eof f = µBT OP of theharmonic potential in the center according to Eq. 3.2. With a proportionality
factor α, this leaves for the lifetime:−11−αBT OP(3.10)
τloss (BT OP ) = e+
τbkg
The fit in Fig. 3.19b) seems to confirm the presence of Landau-Zener losses.
However, it is still necessary to look at absolute values to see if the energy scales
match at all, because the fit only makes a statement about relative numbers:
the energy gap between the trapped and untrapped state in the center shouldroughly match the Heisenberg energy uncertainty associated with the atom
velocity v when passing near the center. This leaves the condition [44]:BT OP µ∼~Brad v
BT OP

At BT OP ∼ 10 G the data indicates a considerable reduction of the lifetime.
However, at this point the tunneling energy is about 6 orders of magnitude
lower than the energy gap 9 . So it seems that Landau-Zener losses are not thedominant mechanism, although the fit seems to support this idea.
An extension of this model is offered here: not only is the loss rate prediction 6 orders of magnitude too low, it also seems inadequate to consider
the average potential, because Landau-Zener transitions are a result of thepotential a particle sees over time. An atom off the axial symmetry axis of thetrap sees an oscillating instantaneous potential, Eq. 2.3. In that sense, theatom passes many times through an ”avoided crossing” formed by the trapped
and untrapped magnetic substates while the bias field rotates. Landau-Zener
9 The time spent in the avoided crossing is approximately 10ms

66transitions then might occur.
The energy separation of this ”avoided crossing” is given by ∆Emin ∼ µBmin ,where Bmin is the minimum magnetic field amplitude seen by an atom at a
certain position in the trap while the bias field is rotating. An illustration isshown in Fig. 3.20, which describes the situation for different atom positionsin the x-y plane of the rotating field.
a)

By

b)
BTOP
atom

c)

By

By

Bmin
Bx

atom

BQP

atom

Bx

Bx

Bmin
BQP

Figure 3.20: Magnetic field as seen by an atom at different positions in the x-yplane. a) Atom in the trap center. The bias field rotates around the atom in a circleof radius BT OP . b)+c) Average distance r of atom to symmetry axis in units of thequadrupole field (BQP ) for large (b)) and small (c)) bias fields. Because the RF-knife
is ramped simultaneously along the CoD in the experiment, BQP /BT OP ≈ const
during evaporation (i.e. r/rCoD ≈ const)

In general, the scaling of the losses with BT OP depends on the different positions of the atoms in the trap all seeing different minimum energy separations∆E. For simplification, the atoms are now assumed to be at a certain average relative distance r/rCoD from the axial symmetry axis (z-direction) in
the x-y plane. For the experimental data, this relative distance is approximately constant over the range of different bias fields, because the rf-knife wasramped simultaneously along the CoD radius and the truncation parameter ηis roughly constant (η relates the average potential energy of the atoms to thetrap depth) 10 . Under this condition, the average minimum magnetic field theatoms see is proportional to BT OP , i.e. to the trap offset energy Eof f in thecenter. The scaling of the trap lifetime is then given by Eq. 3.10.
The actual transition probability is higher than in the case of the average potential, because the time spent in the avoided crossing region is much shorter(0.01...0.1 ms depending on how large the amplitude of the oscillating potential seen by the atom is). This leaves for the transition probability roughly
10 The relative distance is only constant if the trap stiffening is neglected. However,

the modulation on the width of the cloud profile is negligible, cf. Appendix A.2.

67

10−4 ...10−3 per cycle (∼ 10−6 for the average potential). Since the atom passes
10000/s through the ”avoided crossing” (ωT OP = 10 kHz), the overall loss rate
is approximately 1...10 s−1 . This corresponds to lifetimes of 0.1-1 s, which ismuch lower than the measured lifetime of 30 s at a bias field of 10 G. However, the predicted lifetime in this model is definitely longer due to the lowamplitude of the oscillating potential seen by the atoms (compared to Eof f )and the 3-dim. trajectories of the atoms (the atoms are mostly in the trap
center, where the loss rate is lower).
Unlike the average potential, the instantaneous potential predicts a scale,
which is much closer to the lifetime measurement in the TOP trap and gives
an explanation for Eq. 3.10 as a good fit function. But further investigationis necessary to confirm this model. One example is to measure the lifetime in
the TOP trap at a constant bias field at different temperatures. If the model
is correct, the lifetime should increase for lower temperatures (limited by the
background lifetime), because the atoms are closer to the axial symmetry axis,
where the loss rate is smaller.
(F) RF evaporation in the TOP trap
The trajectory for the RF-evaporation in the 18 G TOP trap is determined by scanning different slopes, cf. Fig. 3.21. The initial RF evaporation
surface and the CoD match for ν = 26 MHz, Eq. 3.7. The steepest slopeof 0.8 MHz/s was chosen, where the evaporation quality saturates at around2-2.25. Since the evaporation trajectory simulations show a nearly exponential
behavior for a stationary offset field (cf. Fig. 3.8), two linear ramps are used.
After experimental determination of the RF value for the phase transition,
the first linear ramp (rfcut1) is extended to a few MHz above this point. For
the final evaporation step (rfcut2), both time and final RF value are varied
while maximizing the condensate size with the inverted aspect-ratio at typical
TOFs of 15 ms. Instead of evaluating the images, only the cloud darkness wasobserved. After rfcut1, the cloud temperature is ∼ 10 µK. As demonstrated
in Fig. 3.4, the RF-power needs to be decreased in the final evaporation step.
This was optimized by trying different power values for rfcut2 and observing the condensate size. An additional rapid ramp-down at the onset of thephase-transition increased the atom number even more. This finally gives the
optimized evaporation sequence from Fig. 3.14.

Evap-Quality

68

2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5

1

1.5

2

2.5

3

3.5

4

RF Evaporation Slope [MHz/s]

Figure 3.21: Optimization of RF-knife slope in rfcut1. The saturated evaporation
quality of 2-2.25 is a little lower than the simulations, cf. Fig. 3.9, since the starting
values are worse in the experiment. To minimize the production time, a slope of 0.8
MHz/s was chosen.

69

Chapter 4
Bose-Einstein Condensate characterization
A dilute gas Bose-Einstein Condensate displays remarkable experimental
signatures, such as the thermal phase transition: as the phase-space density
approaches one, the particle wave functions start to overlap and quantum mechanical effects have to be considered. The atom cloud then consists of two
components: a condensate parabola (cf. Chapter 4.1) and a thermal Gaussian.
As the temperature is decreased below Tc , this parabola appears and increases
in height, while the Gaussian background decreases (more atoms in condensate
fraction). For very low temperatures, the Gaussian background component is
no longer visible and only the parabola remains indicating the presence of analmost pure condensate.
A second signature is the anisotropic condensate expansion: instead of a thermal expansion to a round shape for large TOFs after release of the cloud fromthe trap, a rapid expansion along the stiff axis occurs. This causes the initialaspect-ratio in the anisotropic trap to invert after large TOFs. This is causedby the hydrodynamic behavior of the condensate and the ”quantum pressure”
stored in the interaction energy before the trap is switched off.
The first condensate was achieved at 5:55am on 14th July, 2007 after afull week of non-stop operation of the machine. The data obtained on this day
is shown in Fig. 4.1a) and Fig. 4.1b), clearly showing the phase transition
and anisotropic expansion.
In the first section of this chapter the theory for the BEC characterization is explained as well as the extraction of the condensate parameters fromthe absorption images, while the second section briefly discusses the measurement of the trap frequencies of the TOP trap. This is then used in the next
two parts to quantitatively compare the data from the experiment with thetheory predictions of the anisotropic expansion and phase transition of the

70

a)

500µm

g

T > TcT ~ TcT <<Tc (pure BEC)

Optical Density

b)

500µm

x
z
TOF 0ms

TOF 15ms

TOF 25ms

Figure 4.1: Images of first BEC. The imaging plane is in the x-z plane of thelab coordinates (gravity along z-axis). a) Observation of phase transition in thevelocity distribution with a TOF of 15 ms after release from the trap. As thetemperature is lowered (lower RF frequencies), the condensate emerges from thecenter of the thermal Gaussian distribution. Far below the critical temperature
Tc = 200 nK, the thermal part has disappeared and an almost pure condensate of
1.5 × 105 atoms remains (after sequence optimization the condensate size could be
increased to ∼ 1.0 × 106 ). b) Anisotropic expansion of almost pure condensate (withpixel-averaging of raw data). The inversion of the aspect-ratio is clearly visible. The
trap frequencies are ωax = 2π × (96.4 ± 0.1) Hz and ωrad = 2π × (34.2 ± 0.1) Hz inthis case.

71

condensate. In the fourth section the details for the evaluation of the condensate atom number are discussed. The machine performance is characterized
in the last section by measuring the reproducibility of the condensate atom
number and the position stability.

4.1

Condensate theory(A) Non-interacting model
For a non-interacting cloud of bosons in a harmonic trap, the Hamiltonian of the system can easily be diagonalized and yields for the eigenstates theusual energy ladder spectrum:1
1
1
U (x, y, z) = mωx2 x2 + mωy2 y 2 + mωz2 z 2 ,
2
2
2

~k =

X

~ωj (nj + 1/2)

j=x,y,zThe occupation is governed by the Bose-Einstein statistics, where n̄~k is theaverage probability of an atom to be in the energy state ~k and ~k is the quantum
number for the different trap states.
n̄~k =

1(~k −µ)/kB Te−1

In the grand-canonical ensemble of constant temperature T and chemical
potential µ, the other thermodynamic quantities, such as total energy E andatom number N, are given by a normalization condition. For non-interacting
systems, the chemical potential is never larger than the ground-state energy
0 . For decreasing temperature, the pole in the occupation statistics at  = µ
shifts towards the ground-state energy. At the critical point where µ = 0 , thenumber of atoms in the ground state N0 suddenly increases on a macroscopic
scale and a condensate forms, while the remaining atoms in the excited states
form a thermal background1 .
Under this condition, the condensed fraction exhibits the typical scaling of asecond order phase-transition, with NN0 as order parameter and a critical exponent of 3 (ω̄ 3 = ωx ωy ωz )[91]:
N0
=1−
N



T
Tc

3

~ω̄
, where Tc =kBN
ζ3 (1)

1/3
and ζn (z) =

∞
X
zj
j=1

jn

(4.1)

1 Instead of only the atoms in the ground state, the entire atom cloud is often

said to be the condensate.

72

The condition of a macroscopic population of the ground state can be rewritten
into the phase-space density criterion mentioned in the introduction:
nλ3dB = ζ3/2 (1)∼ 2.612(B) Interaction effects
The experimentally observed density profile of the thermal background inan ultracold atomic gas is estimated correctly in this simple ideal gas model.
But both size and shape of the condensate part are completely inadequate,
as discussed in [92]. The reason is the shift of the ground-state energy due
to atomic interactions, which become more significant the denser the cloud
gets. For typical critical temperatures of ∼ 100 nK, the atom-atom interactions are dominated by elastic s-wave scattering. Because of the diluteness of
the weakly interacting system2 , the complicated many-body correlations canbe replaced by scattering of two particles with a contact potential of gδ(x)2a. This leads to a ”mean-field potential”
with the coupling constant g = 4π~mseen by the atoms, so that the many particle wave function ψ(x1 , ..., xN ) canbe approximated by a direct product of single-particle functions as in the caseof a non-interacting Bose gas. Since a macroscopic phase is present in a condensate allowing for coherence to build up on the length scale of the cloud, amacroscopic wave function ψ(~r) can be introduced. This leaves the celebrated
time-independent Gross-Pitaevskii (GP) equation. The derivation details canbe found for example in [91]:~22∆ + V (~r) + gN |ψ(~r)| ψ(~r) = µψ(~r)
(4.2)
−
2m
(C) Thomas-Fermi approximation
The ground-state of this Hamiltonian for the condensate can be obtained analytically in the Thomas-Fermi (TF) approximation, [91]: the kinetic energy scale is given by the trap Ekin ∼ N0 ~ω. On the other hand,
the interaction energy scale is dominated by the local density and interac2
tion strength Eint ∼ gN0 n ≈ 4π~
N02 a3a , where the harmonic oscillator lengthmoscpaosc = ~/mω defines the length scale of the ground state in the harmonic
trap.
Ekin 4πaosc
∼
(4.3)
Eint
Na
2 Thismeans low quantum depletion into other states apart from the noninteracting ground state

73

Typical experimental values are a = 10 nm, aosc = 10 µm and N∼ 105 − 107 ,so that the scales differ by 2-4 orders of magnitude. This allows to neglect the
kinetic energy in the GP equation,. The chemical potential is then much larger
than the energy scale of the trap (µ >> ~ω), and quantum pressure builds
up in the trap. Under this condition, an analytical solution for the density
distribution ncond (~r) is available.
(1(µ − U (~r)) if U (~r) < µ,ncond (~r) = g
0else
In the harmonic trap the distribution is a 3-dim. inverted ”Thomas-Fermi
parabola”, which is chopped off on a discrete shell shaped as an ellipsoid characterized by the Thomas-Fermi radii Rx,y,z . For a given trap and condensate
size, the other thermodynamic quantities can be calculated, cf. [92]:1µ = mωj2 Rj2 ,2

ωj5 Rj5 m2
N=,
15~2 ω 2 ancond (0) =µm
4π~2 a(4.4)

To obtain the full cloud profile at finite temperatures, the non-condensate
fraction needs to be included, [93]. The GP equation cannot be used in thissituation, as it assumes no excitations. However, the density distribution canbe estimated by a sum of a thermal Gaussian and a 3-dim. TF parabola.
−n(~r) = agauss e

y2
z2
x2
2 + 2σ 2 + 2σ 2
2σx
y
z




+ aT Fy2z2x2
1 − ( 2 + 2 + 2)
Rx Ry Rz

(4.5)

This approach breaks down directly at the critical point, where large scale
phase fluctuations and quasi-condensates form [94, 95]. Since the interacting
system still has the same order parameter and critical exponent as in theideal gas case, the scaling at the critical point remains unchanged. But thecritical temperature is shifted to lower values by the interactions and many
body effects, which increases with condensate size. In addition, the finite
atom number can also decrease Tc , if the condensates are very small. These
two effects are given by [91, 92]:
δTc,int
a 1/6= −1.33
N ,
Tc
aosc

δTc,f in(ωx + ωy + ωz )/3 −1/3
= −0.73
N
(4.6)
Tc
ω

For typical trap frequencies of (10...100) × 2π Hz in the experiment and condensates of N0 ∼ 106 , the shift of the critical temperature due to interactions
is about 5%, while the finite size effect is < 1%.

74(D) Anisotropic expansion
The TF approximation cannot be used directly to describe the anisotropic
condensate expansion after release from the trap, because the ”quantum pressure” immediately translates into kinetic energy and Eq. 4.3 is no longer valid.
The solution to this problem is motivated by examining the expansion of a classical gas in the hydrodynamic regime for a time dependent trap, which can be
written as a time-dependent transformation of the atom cloud diameter. Thisidea can then be used for the time dependent GP equation by applying botha unitary and a gauge transformation on the many body wave-function. The
time dependence of the kinetic part of the Hamiltonian is then hidden in thepotential energy and interaction energy term. Under this condition, a TF approximation is valid. For a harmonic trap that is switched off at t=0, this leads
to an expansion of the TF parabola with RT F,x,y,z (t) = RT F,x,y,z (0)λx,y,z (t),where the scaling parameters λx,y,z evolve as [96–98]:
λ̈j (t) =ωj2,
λj (t)λx (t)λy (t)λz (t)λj (0) = 1 and λ̇j (0) = 0(4.7)

In the case of an anharmonic trap with two different frequencies ω1 and ω2 ,an expansion in ω1 /ω2 leads to an analytic solution. For a TOP trap with
ωx = ωy = ωrad and ωz = ωax , this leaves to lowest order [98]:p
1 + (ωax t)2(4.8)
λax (t) =pλrad (t) = 1 + 1/8 ωax tArctan(ωax t) − Log 1 − (ωax t)2
While the analytic solution accurately describes the situation in√ Ioffe-Pritchard
traps with ω1 << ω2 , the aspect-ratio in a TOP trap is only 2 2, which causes
the analytic solution to be inaccurate for large enough TOFs. This problem canbe overcome by transforming Eq. 4.7 into a first order system and performing
a numerical integration. The algorithm is shown in Appendix B.1. The results
are shown in Fig. 4.2a)+b)(E) Extraction of cloud parameters from velocity distributions
The bimodal clouds are observed with typical TOFs of 15ms after release of the trap to increase the resolution. The thermal Gaussian and TF
parabola from Eq. 4.5 are assumed to expand independently. Hence, thedensity distribution is still the same up to increased TF radii and Gaussian
widths (σx (t) ≈ kBmT t, cf. Chapter 2.5.2). Since the bimodal clouds are imaged
along the y-axis (lab coordinates), the optical density profile is obtained after

75

Ioffe-Pritchard Trap

Aspect-ratio
RTF, ax /RTF, rad

2.5

b)

ωx=ωy=(100 x 2π) Hzωz=(10 x 2π) Hz

2.0
1.5
1.0
0.5
0.0

0

5

10

15 20
TOF [ms]

25

1.5
1.0

ωx=ωy=(34 x 2π) Hzωz=(96 x 2π) Hz

0.5
0.0

30

TOP Trap

2.0

Aspect-ratio
RTF, ax /RTF, rad

a)

0

5

10 15 20
TOF [ms]

25

30

Figure 4.2: Comparison of exact numerical integration (solid line) and analyticalsolution (dashed line) of Eq. 4.7 for the anisotropic expansion of condensates indifferent traps. a) Ioffe-Pritchard trap with an initial aspect-ratio of 10 (cigarshaped). √b) TOP trap with a pancake-like condensate at the beginning. The aspectratio is 2 2 in this case. The low aspect-ratio in the TOP case causes significant
deviation of the analytic solution in the high TOF region, while both calculations
match well in the strongly anisotropic Ioffe-Pritchard trap.

a one-dimensional integration of Eq. 4.5. Including a possible background
offset, this leaves nine free parameters.
(x−x0 )2(y−y0 )2−−2σx (t)22σy (t)2n2d (~r) = a0 + aG es
+ aT F3(x − x0 )2 (y − y0 )2
1−
+
(4.9)
Rx (t)2
Ry (t)2

The cloud parameters are found by fitting this function to the optical density
distribution of the experimental data. The atom number of the thermal background is obtained from integration of the Gaussian fit and the atom number
calibration, cf. Chapter 2.5.2. In contrast, the condensate atom number andchemical potential are inferred from the expanded TF-radii: with the numerical integration of Eq. 4.7, the initial TF-radii can be calculated from the
expanded radii. These are then used to extract the condensate atom number
and chemical potential via Eq. 4.4. The temperature of the bimodal cloud is
obtained from the expansion behavior of the Gaussian widths.

4.2

Trap frequencies

Quantitative characterization of the condensate requires the measurement of the trap frequencies in the TOP trap. The basic idea is to give the
BEC a kick along one trap axis and to observe the motion. This can be realizedby suddenly adding a magnetic offset field in one direction, which shifts the76

trap minimum. As very strong fields would be necessary to resolve the spatial
oscillation directly (∼ 1 G), the trap is switched off after some oscillation time
and the cloud position is observed at a large TOF, which corresponds to ameasurement of the velocity of the cloud.

ωx = (25.0+/-0.1)x2π Hz

200 300 400 500 600 700
Time after kick [ms]Vertical Position [pixel]

160
140
120
100
80
60

b)

l

Horizontal Position [pixel]a)ωz = (70.4+/-0.1)x2π Hz

250
200
150
100
60
80
100 120
Time after kick [ms]Figure 4.3: Measurement of trap frequencies in the radial direction (horizontal x)
and the axial direction (vertical z). The trap bottom was shifted by an additional
magnetic offset field of ∼ 100 mG for 1 ms (switch-on time < 500 µs) causing thecondensate to oscillate along the direction of the magnetic shift field. After a variable
hold time, a TOF of 21 ms was used to observe the velocity oscillation after holding
the cloud for a variable time in the trap. The√ aspect-ratio inferred from the trap
frequencies (2.81 ± 0.01)is slightly lower than 8, because the rotating bias field has
an eccentricity of 3 − 5%.

A sinusoidal fit is applied to the data to extract the trap frequency. Twofull oscillation periods are measured at different times to establish the longterm behavior. Since the imaging beam is along the x-axis, only the verticaltrap frequency ωz and horizontal trap frequency ωx could be measured, cf.
Fig. 4.3a)+b).
ωx = (25.0 ± 0.1) × 2π Hz,

ωz = (70.4 ± 0.1) × 2π Hz

Simulations of the bias field profile (Chapter 2.3) predict trap frequenciesof ωx,th = 26.6×2π Hz and ωz,th = 75.3×2π Hz at a current of 2.4 A. A possible
reason for the deviation could be a possible eccentricity  of the rotating bias
field. This would lead to a shift of the trap aspect-ratio in the z-x direction[44]. A numerical calculation shows that our aspect-ratio corresponds to an
eccentricity of ∼ 1.03. This was roughly confirmed by directly measuring thecurrents flowing through the two pairs of Helmholtz coils, which differed by
∼ 5%.

77

4.3

Anisotropic expansion

To observe the anisotropic expansion, almost pure condensates are made,released from the trap and observed at different TOFs. A random order of
TOFs was picked to prevent systematic errors of the TF radii. Since the imaging is from the side (horizontal), the condensate falls out of the field of view for
TOFs larger than 20 ms, so that the objective lens height had to be adjusted.
During this procedure, the atom number drifted to lower values, so that thedata is divided into two groups (low TOFs: 1-20 ms and high TOFs: 20-27
ms). Every TOF is measured five times to estimate the statistical errors. The
results of the aspect-ratio and absolute TF-radii of both axes (axial and radial)
are shown in Fig. 4.4a)-c) along with the theoretical fits from Eq. 4.7.
The data for the aspect-ratio agrees very well with the prediction. The
free parameterof the√ fit function is the intial aspect-ratio, which gives a value√of 0.96 8, where 8 is the theoretical aspect-ratio of the TOP trap for a
perfectly round bias field rotation. There are two possible reasons for thisdeviation:• Due to the eccentricity of the trap, the real aspect-ratio is decreased byabout 0.5%, cf. Chapter 4.2• Since the detection time is 150 µs, the vertical cloud profile smears out.
This leads to an increased TF radius in the axial direction. Simulations
show that this effect is ∼ 1%
The data for the absolute TF-radii also agree well with the prediction on bothaxes. In both data sets for the low TOF section there seems to be a slightsystematic downwards trend with increasing TOF compared to the fit. This
is not the case for the high TOF region. It is possible that there is still someblurring or diffraction occuring.

4.4

Phase Transition

To measure the critical behavior of the order parameter N0 /N of thephase transition (ratio of condensate fraction to total atom number of cloud),the final rf-knife value in rfcut2, cf. Fig. 3.14, is varied between 15.3 MHz(thermal) and 14.5 MHz (almost pure condensate). After the evaporation
sequence the trap is immediatly switched off and the cloud is observed after 15
ms TOF. The theoretical prediction is given by Eq. 4.1. The atom number of
the thermal background is obtained by integration of the Gaussian fit function

TF, radial/R
TF, axial

Aspect-ratio R

200 µmh12 . spe , 1, 0

120

100

100

TOF=11ms

80

60

1
0.96 8

40

40

20

20

0

0
0

0

TOF=17ms

80

60

5

10

20

40

60

80

100

0

12020

40

60

25

15 20
TOF [ms]

80

100

120

30

c) 60

60
40

100
80
60
40
40

20

10

5

10

TOF

15
ms

20 25 30
TOF [ms]

15
TOF [ms]

20

50
40
30

R TF, rad [µm]

b) 80

0

analytic
solution

h11 . spe , 1, 0
1200.00005
RT F, r a d

0numerical
integration

TOF=1ms

R TF, rad [µm]

0

1.6
1.4
1.2
1.0
0.8
0.6
0.4

R TF, axial [µm]0a)R TF, axial[µm]0780.00004
0.000030.00002

5

0

50
40
30
15 20 25 30
TOF [ms]

10 5 10 TOF15ms
TOF [ms]

20

Figure 4.4: Anisotropic condensate expansion. a) shows the aspect-ratio, while b)
and c) illustrate the absolute TF-radii on the vertical and horizontal axes. The
error bars stem from multiple measurements in random order to prevent systematic
errors. The TF-radii cannot be measured accurately for TOFs smaller than 10 ms
because of diffraction around the condensate. This causes interference fringes inimages. The data for the absolute TF radii are divided into a low and high TOF
region, because the condensate atom number drifted to lower values while adjusting
the objective lens position (imaging from the side). This separation is not necessary
for the aspect-ratio, because they are atom number independent. The fit functions
to the data are obtained from integration of Eq. 4.7. For comparison, the prediction
obtained from the analytic functions of Eq. 4.8 are shown (dashed line). The√fit
parameter for the aspect-ratio data is the initial aspect-ratio, which gives 0.96 8.
For the absolute TF radii, the initial TF radius is the free fit parameter (atomnumber dependent).

15

20

79

to the bimodal distribution, while the condensate atom number number is
extracted from the TF radii. To omit the atom number dependence of thecritical temperature of each data point, the order parameter is not plottedversus T, but versus Trel = T /Tc (N ), where Tc (N ) includes the finite size andinteraction corrections to the critical temperature, Eq. 4.6. The fit function
rel)3 , where Tc,rel is 1 in theory, cf. Fig. 4.5b).
is then 1 − ( TTc,rel

1
0.8
0.6

b)
Tc~150nK

0.4
0.2
0

N BEC /NN BEC /N

a)

1
0.8
0.6

Tc, rel ~0.97

0.4
0.2
0

0

100 200 300
T [nK]

0

0.5 1 1.5 2
T/Tc

Figure 4.5: Phase transition from thermal cloud to a Bose-Einstein condensate. a)
Critical behavior of order parameter N0 /N . The data is plotted versus T /Tc (N ) torel 3
omit the atom number dependence of each data point. The fit function is 1−( TTc,rel) ,which gives Tc,rel = 0.97 in this case (solid line). The dashed curve corresponds tothe theoretical prediction of Tc,rel = 0.97. b) Raw data indicating Tc ∼ 150nK byfitting 1 − ( TTc )3 .

The data agrees very well with the theory and the fit gives Tc,rel = 0.97.
The statistical error is estimated to be ∼ 10% from the data scattering. The
temperature determination has an overall systematic error of ∼ 5 − 10% (evaluation of both axes). Temperatures lower than 0.5 Tc could not be determined,
because the thermal background is so much lower than the noise of the datain this case, that a Gaussian fit is not possible. The systematic uncertainty of
the atom number evaluation is discussed in the next chapter. Fig. 4.5a) shows
the condensate fraction as a function of the temperature with a fit function of
1 − ( TTc )3 . The critical temperature is roughly 150nK.

80

4.5

Condensate atom number determinationg

Optical Density

The condensate atom number is determined by extracting the TF radius
after TOF from the fit of Eq. 4.5 to the optical density distribution of theexperimental data. The theoretical anisotropic expansion of the condensate
obtained from the numerical integration of Eq. 4.7 is then used to calculate
the inital TF radius in the trap, which gives the atom number according to Eq.
4.4. Note that the atom number cannot be extracted directly from integrationof the parabola fit, because the optical density of the images is saturated in
the center (typically at OD of 2-3). This means that the absorption in thecenter is so high, that all incident photons are absorbed and the transmission
approaches 0. In this case, the height of the fit to the TF parabola is underestimated, while the TF radii are still accurate. Fig. 4.6 shows a typicalcondensate with ∼ 1.0 × 106 atoms in the F=1 Rb-87 ground state. The saturation of the condensate profile is clearly visible in this case.

VerticalHorizontal
500 µmFigure 4.6: Typical data obtained after optimization. The TOF is 13 ms with trap
frequencies ωax = 2π ×(70.4±0.1) Hz and ωrad = 2π ×(25.0±0.1) Hz. As the image
is saturated, the atom number can only be evaluated from the anisotropic expansion,
yielding ∼ 1.0×106 . The systematic error is estimated to be approximately +10%/−30%. Evaluation of the cloud at higher TOFs gives similar atom numbers.

Inspection of Eq. 4.4 reveals that N∝ RT5 F , which causes an overall errorof ∼ 50% for the condensate atom number determination (30% systematic
error, 20% statistical error). In principle, three error sources contribute:

81• In general the atom number from the expansion in axial direction is morereliable, because the relative change is much higher than on the radial
axis. But RT F,axial is overestimated due to the motion of the cloud during
the detection phase of 150 µs. This leaves a systematic error of ∼ 10%
after averaging the axial and radial atom numbers.
• The anisotropic expansion data from Fig. 4.4b)+c) suggests that blurring/diffraction causes systematic errors of < 5% for the TF radii. Thistranslates into a systematic error of ∼ 20% for the atom number.
• The TF radius determination from the image typically has an error of
about 1 pixel, which results in an additional statistical uncertainty of∼ 20% in the atom number
To cross check the atom number evaluation, the condensate can be evaluated at large TOFs. As long as the condensate size does not exceed 2 × 105
atoms, the images are not saturated in the TOF region of 25 ms or higher.
Under this condition, the height of the fit to the TF parabola can be trusted.
Since the atom number is already known from the anisotropic expansion, an
atom number calibration calae can be extracted in this case. On the other
hand, an atom number calibration calf l is already available, which was determined by fluorescence imaging in the MOT cell and estimating the amount of
atoms in the BEC cell, cf. Chapter 2.5.
Alternatively, the phase-transition diagram can be used as another cross-checkfor the atom number determination by anisotropic expansion. The basic idea
is to examine the influence of the atom number calibration on the fit to thetheoretical prediction: by changing the atom number calibration, the condensate fraction and absolute temperature are still the same each data point, but
Tc (N ) is shifted, as it depends on the absolute atom number. The quantity
rel)3 to theof interest is then how good Tc,rel from the fit function 1 − ( TTc,rel
phase diagram (Chapter 4.4) still matches the theoretical value of 1 for eachpossible atom number calibration, cf. Fig. 4.7. The lowest error is achieved
for a calibration of 325 /pix. For comparison, the atom number calibrations
from the anisotropic expansion and fluorescence imaging are also shown with
their error bars. Assuming a 10% confidence interval for the fit of the phasetransition curve to the data, a third calibration calpt can be extracted andcompared with the other two calibrations.
calf l = (310 ± 60) /pix
calae = (360 ± 140) /pix
calpt = (325 − 80 + 120) /pix

(4.10)(4.11)(4.12)

82

It is striking that the atom number calibrations match so well within theirerror bars, because three completely different types of physics were used todetermine them (spontaneous emission of photons, condensate expansion ofinteracting bosons and phase transition diagram). According to the threecalibrations, the systematic error of the atom number determination from theanisotropic expansion of the condensate seems to be about +10%/ − 30%,
which is in the range of the a priori estimate for the systematic error from the
beginning of this section.

|Tc, rel-1|

0.15
0.10
0.05
0.00

10% confidence
regioncalae
calfl
250 300 350 400 450 500
Calibration [1/pixel]

Figure 4.7: Dependence of the fit parameter Tc,rel as a function of atom number
rel 3
calibration using 1 − ( TTc,rel) as fit function for the phase transition data. Ideally,
Tc,rel is 1. Assuming a 10% confidence interval, an atom number calibration calpt
can be extracted (gray region).

4.6

Machine performance

For future experiments with condensates in optical lattices, it is very
important that the cloud properties are reproducibile to be able to investigate physical effects. For example, a clear signature of the quantum phasetransition from a Superfluid to a Mott-Insulator requires multiple images atdifferent lattice depths. As the dipole trap was not yet fully operable to characterize the stability directly in that sense, condensates in the TOP trap wereobserved. The atom number reproducibility was found to be as good as theactual imaging stability (σ/mean < 10%), while the position stability is better
than 5 µm.

83

4.6.1

Position reproducibility

Since a mechanical transporter is used to transfer the laser cooled atom
cloud from the MOT cell to the BEC cell, too large variations in the condensate
position can result in significant modulations of the TOP trap bottom, which
can cause the BEC to disappear. The stability is investigated by observing
the condensate position for multiple runs for TOFs < 1 ms. A Gaussian is
fit to every image to extract the cloud center, cf. Fig. 4.8. The runs weretaken without any major pauses to observe the short-term drifts. The axial
position shows a slow upwards drift with a σ = 1 µm and a range of 4 µm,
while the horizontal direction has a larger drift and a σ = 2 µm with a rangeof 8 µm. The second horizontal axis cannot be observed, as it coincides with
the imaging axis.
The slow systematic drift in the vertical data might be caused by the rise of
the power cable temperature for the quadrupole trap. This can release stresson the coil holder increasing over run time. But even with cooldown pauses,
the drift was still observable. An overall drift of the coil position due to thestress created by the scews holding them can also be excluded, as after very
long pauses the cloud resumes its old position. The most probable cause is anexternal field: the optical lattice table exhibits a small magnetization resulting
in a bias field of about ∼ 4 G at the condensate position in axial direction.
The reason is that the hole in the lattice table below the BEC cell breaks themagnetic shielding and allows magnetizations of 0.1 − 1% stemming from thequadrupole field. For multiple runs this magentic field slowly saturates to ∼ 4
G, while a remaining systematic drift and random variations of 1 − 10% might
still occur (∼ 100 mG). In that sense, this might also be one of the major
contributions to the scattering of the vertical data. Another reason for the
scattering of the data might be the camera fan, which creates oscillations bothof the CCD chip and on the imaging table. Since the evaporation takes 45 s,
oscillations of the coil holder are already damped out when the image is taken.
Finally, the spatial variations would correspond to drop time uncertainties of∼ 1 ms, so that unreproducible magnetic trap switch-offs cannot be the cause
either.
The manufacturer gives an accuracy of the translation stage of ±5 µm,
which is probably one of the limiting factors to the scattering of the horizontal data. Another possibility is again the fan of the camera. Kicks of thecondensate into the radial direction due to unreproducible switch-offs of themagnetic trap can be excluded as the cause for the scattering/drift. The kickvelocity would have to be around 10 µm/ms, in order to be observable after

100
95
90
85
80
75

b)

0

5

10

Run

15 20

Vertical position [µm]a)Horizontal position [µm]

84

90
85
80
75
70
65

0

5

10

Run

15 20

Figure 4.8: Condensate postition reproducibility in TOP trap. The position wasdetermined with 0ms TOF by gaussian fits to the cloud profiles. a) is the radial
variation, while b) is the axial direction. The third axis could not be observed forbeing parallel to the absorption imaging beam. For both data sets σ < 5µm, so thatloading of a 1µK thermal cloud into a crossed dipole trap causes negligible heating.

< 1 ms TOF. A cause for the drift in the horizontal data might be a small
error in the position stabilization of the stage building up over several runs.
Since the hole in the optical lattice table is not symmetric in x-direction (labcoordinates), the magnetic field from this table also has a component in thehorizontal x-direction. Similar to the case of the vertical axis, this could resultsin both a drift and scattering of the horizontal position. But as the creep of
the horizontal position disappears, if the stage settings are set to more stiffer
values, this contribution can probably be neglected in this case.

4.6.2

Atom number stability

To measure the atom number stability, many almost pure condensates
are made in the TOP trap and the atom number is observed, cf. Fig. 4.9.
On a timescale of more than 100 minutes, the data shows very good stability
with a scattering of σ/mean∼ 6% and a range of 34%. The drift of the datais caused by the repump laser slowly shifting in frequency. This causes (N, T)
not to be the ideal starting values in the BEC cell for the used evaporation
sequence, so that the final BEC is smaller. After relocking, the atom number
resumed the old value.
The scatter of the data is probably caused by the frequency jitter of the imaging laser (repumper), which is about 10 − 20% of the natural linewidth of
the F=1 imaging transition. To verify this, the integrated optical density ofunevaporated clouds is measured in the BEC cell, inset of Fig. 4.9. From fluo-

85

rescence measurements in the MOT cell before and after mechanical transport(which do not exhibit laser jitter problems due to the much longer exposuretime), the atom number is determined to be very stable within 1% or less,while the scattering in the absorbtion images is 8% with a range of 30%. Thismeans that the real condensate atom number is probably even more stable
than indicated by the data. This also means that the rotating bias field of
the TOP trap is so uniform over the length scale of the condensate position
variations that an influence of trap bottom shifts on the condensate size are
negligible.

0.8
Atom number [a.u.]

N BEC [a.u.]

1
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0 2 4 6 8 10 12 14
run

20

40
60
80
Run time [min]laserrelock100

Figure 4.9: Condensate atom number stability. Over more than 100 minutes runtime, the scattering is only σ/mean = 6%. This is roughly the quality of the imaging
due to frequency jitter of the imaging laser (σ/mean = 8%), which is shown in theinset by observing clouds of the same atom numbers. Therefore, the condensate
atom number is probably even more stable than 6%. The overall drift is caused by
the repump laser slowly falling out of lock.

86

Chapter 5Conclusion
We have created the first Rb-87 Bose-Einstein Condensate in a moving
coil TOP trap geometry. The condensates were made in the |1, −1i state. In
our setup, the laser cooled atoms are transported along a curved path into a
glass cell using a moving-coil quadrupole magnetic trap. RF-forced evaporative cooling is performed in a TOP trap, formed by the moving coil quadrupole
field in conjunction with a rotating bias field. Evaporative cooling simulations
were carried out to provide a guideline for the optimization in the experiment. This way, the initial condensate size of ∼ 1 × 105 could be increased to∼ 1 × 106 atoms with a critical temperature of 150nK. Our data agrees wellwith the theoretical predictions of the phase transition and anisotropic expansion. A consistency check was performed for the atom number determination,
which gives an estimated systematic uncertainty of +10%/ − 30%.
The condensate atom number is found to be stable within 6%, while the position stability is approximately 5 µm in the axial and 10 µm in the radial
direction of the magnetic TOP trap. These numbers are very promising forcondensation in a dipole trap and future experiments in optical lattices. The
position uncertainty corresponds to heating of ∼ 100 nK in an optical trapwith a depth of 5 µK, which is negligible if the trap is loaded at a cloud temperature of 1 µK. The final evaporation can then be performed by lowering
the laser power to reduce the trap depth.

87

Bibliography[1] S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für
Physik 26, 178 (1924).
[2] A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsbericht der Preußischen Akademie der Wissenschaften XXII, 261 (1924).
[3] A. Einstein, Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung, Sitzungsbericht der Preußischen Akademie der Wissenschaften
II, 3 (1925).
[4] A. Einstein, Zur Quantentheorie des idealen Gases, Sitzungsbericht derPreußischen Akademie der Wissenschaften III, 18 (1925).
[5] F. London, The λPhenomenon of Liquid Helium and the Bose-Einstein
Degeneracy, Nature 141, 643 (1938).
[6] H. J. Metcalf and P. van der Straten,(Springer, Berlin, 1999).

Laser Cooling and Trapping[7] H. F. Hess, Evaporative cooling of magnetically trapped and compressedspin-polarized hydrogen, Phys. Rev. B 34, 3476 (1986).
[8] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic
Vapor, Science 269, 198 (1995).
[9] C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, Evidence of
Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett. 75, 1688 (1995).
[10] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Bose-Einstein Condensation of Lithium: Observation of Limited Condensate Number, Phys. Rev.
Lett. 78, 985 (1997).

88[11] K. Davis, M. Mewes, M. Andrews, N. van Druten, D. Durfee, D. Kurn,and W. Ketterle, Bose-Einstein Condensation in a Gas of Sodium, Phys.
Rev. Lett. 75 (1995).
[12] D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss,
D. Kleppner, and T. J. Greytak, Bose-Einstein Condensation of Atomic
Hydrogen, Phys. Rev. Lett. 81, 3811 (1998).
[13] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E.
Wieman, Stable 85 Rb Bose-Einstein Condensates with Widely Tunable
Interactions, Phys. Rev. Lett. 85, 1795 (2000).
[14] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio, Bose-Einstein Condensation of Potassium Atoms by Sympathetic
Cooling, Science 294, 1320 (2001).
[15] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, and A. Aspect, A Bose-Einstein Condensate ofMetastable Atoms, Science 292, 461 (2001).
[16] F. P. D. Santos, J. Léonard, J. Wang, C. J. Barrelet, F. Perales, E. Rasel,
C. S. Unnikrishnan, M. Leduc, and C. Cohen-Tannoudji, Bose-Einstein
Condensation of Metastable Helium, Phys. Rev. Lett. 86, 3459 (2001).
[17] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, BoseEinstein Condensation of Cesium, Science 299, 232 (2003).
[18] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura,
T. Yabuzaki, and Y. Takahashi, Spin-Singlet Bose-Einstein Condensationof Two-Electron Atoms, Phys. Rev. Lett. 91, 0404041 (2003).
[19] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, BoseEinstein Condensation of Chromium, Phys. Rev. Lett. 94, 1604011 (2005).
[20] D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell, Collective Excitations of a Bose-Einstein Condensate in a DiluteGas, Phys. Rev. Lett. 77, 420 (1996).
[21] M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G. Townsend,
S. Inouye, and W. Ketterle, Propagation of Sound in a Bose-Einstein
Condensate, Phys. Rev. Lett. 79, 553 (1997).

89[22] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M.
Kurn, and W. Ketterle, Observation of Interference Between Two Bose
Condensates, Science 275, 637 (1997).
[23] M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend,
and W. Ketterle, Output Coupler for Bose-Einstein Condensed Atoms,
Phys. Rev. Lett. 78, 582 (1997).
[24] I. Bloch, T. W. Hänsch, and T. Esslinger, Atom Laser with a cw Output
Coupler, Phys. Rev. Lett. 82, 3008 (1999).
[25] L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. Luienne,
J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Fourwave mixing with, Nature 198, 218 (1999).
[26] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E.
Pritchard, and W. Ketterle, Bragg Spectroscopy of a Bose-Einstein Condensate, Phys. Rev. Lett. 82, 4569 (1999).
[27] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman,and E. A. Cornell, Vortices in a Bose-Einstein Condensate, Phys. Rev.
Lett. 83, 2498 (1999).
[28] C. Raman, M. Köhl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ketterle, Evidence for a Critical Velocity in a BoseEinstein Condensed Gas, Phys. Rev. Lett. 83, 2502 (1999).
[29] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas ofultracold atoms, Nature 415, 39 (2002).
[30] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta,
Z. Hadzibabic, and W. Ketterle, Observation of Bose-Einstein Condensation of Molecules, Phys. Rev. Lett. 91, 250401 (2003).
[31] M. Greiner, C. A. Regal, and D. S. Jin, Emergence of a molecular BoseEinstein condensate from a Fermi gas, Nature 426, 537 (2003).
[32] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin,
J. Hecker Denschlag, and R. Grimm, Bose-Einstein Condensation of
Molecules, Science 302, 2101 (2003).

90[33] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J.
Kerman, and W. Ketterle, Condensation of Pairs of Fermionic Atoms
near a Feshbach Resonance, Phys. Rev. Lett. 92, 120403 (2004).
[34] C. A. Regal, M. Greiner, and D. S. Jin, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett. 92, 040403 (2004).
[35] G. B. Partridge, W. Li, Y. A. Liao, R. G. Hulet, M. Haque, and H. T. C.
Stoof, Deformation of a Trapped Fermi Gas with Unequal Spin Populations, Physical Review Letters 97, 190407 (2006).
[36] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle,
Fermionic Superfluidity with Imbalanced Spin Populations, Science 311,
492 (2006).
[37] D. A. Steck, Rubidium 87 D Line Data, http://steck.us/alkalidata 2003.
[38] Y. Yoshikawa, T. Umeki, T. Mukae, Y. Torii, and T. Kuga, Frequency
stabilization of a laser diode with use of light-induced birefringence in an
atomic vapor, Appl. Opt. 42, 6645 (2003).
[39] D. E. Sproles, in preparation, Master’s thesis, Stony Brook University,
2007.
[40] S. G. Albert, Cooling, Trapping and Transport of Atom Clouds in a New
BEC Apparatus, Master’s thesis, Stony Brook University, 2007.
[41] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral
Atoms, Phys. Rev. Lett. 74, 3352 (1995).
[42] M. Mewes, M. Andrews, N. van Druten, D. Kurn, D. Durfee, and W. Ketterle, Bose-Einstein Condensation in a Tightly Confining dc Magnetic
Trap, Phys. Rev. Lett. 77 (1996).
[43] D. Stamper-Kurn, M. Andrews, A. Chikkatur, S. Inouye, H. Miesner,
J. Stenger, and W. Ketterle, Optical Confinement of a Bose-Einstein
Condensate, Phys. Rev. Lett. 80 (1998).
[44] J. R. Ensher, The First Experiments with Bose-Einstein Condensation of
Rb87, PhD thesis, University of Colorado at Boulder 1997.
[45] D. Pertot and D. Sproles, (private communication).

91[46] A. Rich, Shielding and Guarding, Analog Devices - Application Notes
AN-347.
[47] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon
Interactions (Wiley-VCH, Weinheim, 2004).
[48] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, Making, probing
and understanding Bose-Einstein condensates 1999.
[49] C. Wieman, G. Flowers, and S. Gilbert, Inexpensive laser cooling andtrapping experiment for undergraduate laboratories, American Journalof Physics 63, 317 (1995).
[50] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, Trappingof Neutral Sodium Atoms with Radiation Pressure, Phys. Rev. Lett. 59,
2631 (1987).
[51] C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, A. M. Steane,
P. Szriftgiser, H. Perrin, and J. Dalibard, Phase-space density in themagneto-optical trap, Phys. Rev. A 52, 1423 (1995).
[52] A. M. Steane, M. Chowdhury, and C. J. Foot, Radiation force in themagneto-optical trap, J. Opt. Soc. Am. B 9, 2142 (1992).
[53] H. Ruf, Progress report from the Schneble lab, unpublished, 2006.
[54] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Behaviorof atoms in a compressed magneto-optical trap, Journal of the OpticalSociety of America B Optical Physics 11, 1332 (1994).
[55] S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, ThreeDimensional Viscous Confinement and Cooling of Atoms by Resonance
Radiation Pressure, Phys. Rev. Lett. 55, 48 (1985).
[56] P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould,
and H. J. Metcalf, Observation of Atoms Laser Cooled below the Doppler
Limit, Phys. Rev. Lett. 61, 169 (1988).
[57] J. Dalibard and C. Cohen-Tannoudji, Laser cooling below the Doppler
limit by polarization gradients: simple theoretical models, J. Opt. Soc.
Am. B 6, 2023 (1989).
[58] T. Bergeman, G. Erez, and H. J. Metcalf, Magnetostatic trapping fields
for neutral atoms, Phys. Rev. A 35, 1535 (1987).

92[59] C. Klempt, T. van Zoest, T. Henninger, O. Topic, E. Rasel, W. Ertmer,
and J. Arlt, UV light-induced atom desorption for large rubidium andpotassium magneto-optical traps, Phys. Rev. A 73, 0134100 (2006).
[60] J. Dalibard and C. Cohen-Tannoudji, Dressed-atom approach to atomic
motion in laser light - The dipole force revisited, Journal of the OpticalSociety of America B Optical Physics 2, 1707 (1985).
[61] E. W. Streed, A. P. Chikkatur, T. L. Gustavson, M. Boyd, Y. Torii,
D. Schneble, G. K. Campbell, D. E. Pritchard, and W. Ketterle, Large
atom number Bose-Einstein condensate machines, Review of Scientific
Instruments 77, 023106 (2006).
[62] U. Ernst, A. Marte, F. Schreck, J. Schuster, and G. Rempe, Bose-Einsteincondensation in a pure Ioffe-Pritchard field configuration, EPL (Europhysics Letters) 41, 1 (1998).
[63] A. Marte, Feshbach-Resonanzen bei Stoessen ultrakalter Rubidiumatome,
PhD thesis, Max-Planck Institut für Quantenoptik 2003.
[64] W. Ketterle and N. van Druten, Evaporative cooling of trapped atoms,
Adv. At. Mol. Opt. Phys. 37, 181 (1996).
[65] H. F. Hess, G. P. Kochanski, J. M. Doyle, N. Masuhara, D. Kleppner,and T. J. Greytak, Magnetic trapping of spin-polarized atomic hydrogen,Phys. Rev. Lett. 59, 672 (1987).
[66] I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans,
and J. T. M. Walraven, Optical cooling of atomic hydrogen in a magnetic
trap, Phys. Rev. Lett. 70, 2257 (1993).
[67] C. L. Cesar, D. G. Fried, T. C. Killian, A. D. Polcyn, J. C. Sandberg,
I. A. Yu, T. J. Greytak, D. Kleppner, and J. M. Doyle, Two-Photon Spectroscopy of Trapped Atomic Hydrogen, Phys. Rev. Lett. 77, 255 (1996).
[68] M. D. Barrett, J. A. Sauer, and M. S. Chapman, All-Optical Formation of
an Atomic Bose-Einstein Condensate, Phys. Rev. Lett. 87, 010404 (2001).
[69] M. Hammes, D. Rychtarik, H.-C. Nägerl, and R. Grimm, Cold-atom gas
at very high densities in an optical surface microtrap, Phys. Rev. A 66,
051401 (2002).
[70] D. E. Pritchard, K. Helmerson, and G. A. Martin, Atom Traps, Atomic
Physics 11 11, 179 (1989).

93[71] A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and D. E.
Pritchard, rf Spectroscopy of Trapped Neutral Atoms, Phys. Rev. Lett.
61, 2431 (1988).
[72] O. J. Luiten, H. G. C. Werij, I. D. Setija, M. W. Reynolds, T. W. Hijmans, and J. T. M. Walraven, Lyman-alpha spectroscopy of magneticallytrapped atomic hydrogen, Phys. Rev. Lett. 70, 544 (1993).
[73] L. Landau, Phys. Z. Sowjetunion 2 (1932).
[74] C. Zener, Proc. R. Soc. London A 137, 696 (1932).
[75] C. E. Carroll and F. T. Hioe, J. Phys. A: Math. Gen. 19, 1151 (1986).
[76] E. T. Whittaker, Bull. Am. Math. Soc. 10 (1903).
[77] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1927).
[78] H. Wu, E. Arimondo, and C. J. Foot, Dynamics of evaporative coolingfor Bose-Einstein condensation, Phys. Rev. A 56, 560 (1997).
[79] J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, andT. J. Greytak, Evaporative cooling of atomic hydrogen: Theory of coolingand progress towards the Bose-Einstein transition, Physica B Condensed
Matter 194, 13 (1994).
[83] M. Holland, J. Williams, K. Coakley, and J. Cooper, Trajectory simulation of kinetic equations for classical systems , Quantum and Semiclassical
Optics 8, 571 (1996).
[80] M. Yamashita, M. Koashi, T. Mukai, M. Mitsunaga, and N. Imoto, Dynamics of evaporative cooling in magnetically trapped atomic hydrogen,Phys. Rev. A 62, 033602 (2000).
[81] M. Yamashita, M. Koashi, T. Mukai, M. Mitsunaga, N. Imoto, andT. Mukai, Optimization of evaporative cooling towards a large numberof Bose-Einstein-condensed atoms, Phys. Rev. A 67, 023601 (2003).
[82] T. Mukai and M. Yamashita, Efficient rapid production of a Bose-Einsteincondensate by overcoming serious three-body loss, Physical Review A(Atomic, Molecular, and Optical Physics) 70, 013615 (2004).

94[84] C. A. Sackett, C. C. Bradley, and R. G. Hulet, Optimization of evaporative cooling, Phys. Rev. A 55, 3797 (1997).
[85] O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Kinetic theory of
the evaporative cooling of a trapped gas, Phys. Rev. A 53, 381 (1996).
[86] D. W. Snoke and J. P. Wolfe, Population dynamics of a Bose gas nearsaturation, Phys. Rev. B 39, 4030 (1989).
[87] K. B. Davis, M. O. Mewes, and W. Ketterle, An analytical model forevaporative cooling of atoms, Applied Physics B 60, 155 (1995).
[88] C. J. Myatt, Bose-Einstein Condensation experiments in a dilute vapor of
Rubidium, PhD thesis, University of Colorado, Boulder, Colorado 1997.
[89] D. Comparat, A. Fioretti, G. Stern, E. Dimova, B. L. Tolra, and P. Pillet,
Optimized production of large Bose-Einstein condensates, Adv. At. Mol.
Opt. Phys. 73, 043410 (2006).
[90] E. L. Surkov, J. T. M. Walraven, and G. V. Shlyapnikov, Collisionlessmotion and evaporative cooling of atoms in magnetic traps, Phys. Rev. A
53, 3403 (1996).
[91] C. Pethick and H. Smith, Bose-Einstein condensates in dilute gases (Cambridge University Press, Cambridge, UK, 2002).
[92] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of
Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463(1999).
[93] L. Pitaevskii and S. Stringari, Bose-Einstein condensation (Oxford University Press, Oxford, UK, 2003).
[94] M. Köhl, M. J. Davis, C. W. Gardiner, T. W. Hänsch, and T. Esslinger,
Growth of Bose-Einstein Condensates from Thermal Vapor, Phys. Rev.
Lett. 88, 080402 (2002).
[95] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Scaling and Thermodynamics of a Trapped Bose-Condensed Gas, Phys. Rev. Lett. 78, 3987 (1997).
[96] J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A.
Cornell, Bose-Einstein Condensation in a Dilute Gas: Measurement ofEnergy and Ground-State Occupation, Phys. Rev. Lett. 77, 4984 (1996).

95[97] M. Holland and J. Cooper, Expansion of a Bose-Einstein condensate ina harmonic potential, Phys. Rev. A 53, R1954 (1996).
[98] Y. Castin and R. Dum, Bose-Einstein Condensates in Time Dependent
Traps, Phys. Rev. Lett. 77, 5315 (1996).

96

Appendix A
Evaporative cooling simulation details

A.1

Analytic solutions

The conditions for the three break-even curves of constant collision ratio,
density and phase-space density can be found by using the scaling behaviour
from Tab. 3.1 including the losses from the background:ξ+5/2 2.7Γbkg
ξ+3/2− ΓΓinc (ξ+5/2,η)Γinc (ξ+3/2,η)el=• Constant PSD:eΓ(ξ+3/2)
Γ(ξ+5/2)• Constant d:• Constant Γel :Γinc (ξ+3/2,η)Γ(ξ+3/2)

ξ+1

Γinc (ξ+3/2,η)Γ(ξ+3/2)−e

2.7ΓbkgΓelξ+1/2−e=

2.7Γbkg
ΓelΓinc (ξ+5/2,η)Γ(ξ+5/2)=ξΓinc (ξ+5/2,η)Γ(ξ+5/2)

ξ−1/2In the case of starting exactly on the break-even curve for the collision
rate in a harmonic trap, the differential equations for the evaporation simulation Eq. 3.6 become analytically solvable. In the Taylor approximation ofslowly changing parameters in a harmonic trap, the solutions read [? ]:  (1 + η + η 2 /2)e−η+ Γbkg t
N (t) ∝ Exp − Γel4η 3 e−ηt
T (t) ∝ Exp −Γel24
η 3 e−ηνrf (t) ∝ Exp −Γelt
24
The two relevant timescales τel and τevap are connected by τevap /τel = 24eη /η 3 ∼ 30.
Typical elastic collision rates are 1s, which sets the evaporation timescale to∼ 1min.

97

A.2

Additional formulas

In this section some additional relations for the evaporation simulation
are discussed to complete the discussion from Chapter 3.2.2. Assuming an adiabatic loading from the quadrupole to the TOP trap, the new temperature isgiven by the condition of equal phase space densities and the relation between
temperature T and volume V for both traps. This can be found by integrationof the density function in the ergodic regime using the classical Boltzmann
statistics. Even though the cloud distribution is not discrete, the density N/Vcorresponds to the maximal density in the center:3 Z ∞ √

2kB
2
2
2
3
e− x +y +4z dxdydz
Vlin (T ) = αlin T ,αlin =
gF mF µB Bax−∞!3/2 Z∞2k2
2
2
B
Vharm (T ) = αharm T 3/2 , αharm =e−x −y −8z dxdydz
2
mRb87 ωrad,T OP
−∞

1/3
1
αlin
1
3/2
=
⇒ Tharm =
Tlin
3/2
3/2
αharm
Vlin Tlin
Vharm Tharm
In the case of the circle of death following the rf-knife the trap stiffening
can be described again by the adiabatic assumption leading to a new cloud
temperature T 0 :
s

1/3

2/3
0
0
αharm (BT OP )Vharm
αharm (BT0 OP )
Γ0elαharm (BT OP )
Tharm
=
,
=
,
=
Tharm
αharm (BT0 OP )Vharm
αharm (BT OP )Γelαharm (BT0 OP )A.3

Algorithm

98

 Scaling Functions
In[1]:=<< C:\\BECIVTools\\PhysConst.m;ΗGammainc@a_, Η_D := à xa-1 Exp@-xD â x;
0

Nfactor@Η_, colratio_, Ξ_D :=Gammainc@Ξ + 3  2, ΗD

Efactor@Η_, colratio_, Ξ_D :=Gammainc@Ξ + 5  2, ΗD

Gamma@Ξ + 3  2D

* ExpB-

Gamma@Ξ + 5  2D

* ExpB-

2.7
colratio
2.7
colratioF;
F;

Tfactor@Η_, colratio_, Ξ_D := Efactor@Η, colratio, ΞD  Nfactor@Η, colratio, ΞD;
Nfactor@Η, colratio, ΞDΓnewtoΓoldfactor@Η_, colratio_, Ξ_D :=Tfactor@Η, colratio, ΞD ^ HΞ - 0.5L

nnewtonoldfactor@Η_, colratio_, Ξ_D :=Tfactor@Η, colratio, ΞD ^ HΞL

Nfactor@Η, colratio, ΞD

PSDnewtoPSDoldfactor@Η_, colratio_, Ξ_D :=;;Nfactor@Η, colratio, ΞDTfactor@Η, colratio, ΞD ^ HΞ + 3  2L

;

99

2

Evaporation Simulation.nb

 Truncation optimization
In[9]:=

evapqualityharmonic@Η_, colratio_D := -Log@PSDnewtoPSDoldfactor@Η, colratio, 3  2DD
Log@Nfactor@Η, colratio, 3  2DD;

evapqualityharmonicder@Η_, colratio_D := Derivative@1, 0D@ evapqualityharmonicD@Η, colratioD;
optimalΗharmoniclist =Table@8colratio,
If@colratio < 200, 4,
If@colratio < 5000,
FindRoot@evapqualityharmonicder@Η, colratioD  0, 8Η, 8<D@@1DD@@2DD,
If@colratio < 40 000,
FindRoot@evapqualityharmonicder@Η, colratioD  0, 8Η, 10<D@@1DD@@2DD,
FindRoot@evapqualityharmonicder@Η, colratioD  0, 8Η, 12<D@@1DD@@2DD
DDD<, 8colratio, 100, 10 000, 20<D;
optimalΗintharmonic = Interpolation@optimalΗharmoniclistD;
evapqualitylinearder@Η_, colratio_D := Derivative@1, 0D@ evapqualitylinearD@Η, colratioD;
optimalΗlinearlist =Table@8colratio,
If@colratio < 100,
FindRoot@evapqualitylinearder@Η, colratioD  0, 8Η, 5<D@@1DD@@2DD,
If@colratio < 500,
FindRoot@evapqualitylinearder@Η, colratioD  0, 8Η, 7<D@@1DD@@2DD,
If@colratio < 5000,
FindRoot@evapqualitylinearder@Η, colratioD  0, 8Η, 10<D@@1DD@@2DD,
If@colratio < 200 000,
FindRoot@evapqualitylinearder@Η, colratioD  0, 8Η, 14<D@@1DD@@2DD,
FindRoot@evapqualitylinearder@Η, colratioD  0, 8Η, 18<D@@1DD@@2DD
DDDD<, 8colratio, 30, 2000, 10<D;
optimalΗintlinear = Interpolation@optimalΗlinearlistD;

100

Evaporation Simulation.nb

 Trap parameters
In[16]:=

gF = 0.5;
mF = 1;b = 353.;H*Gcm*L
QP = b  2;H*Gcm*L
Α = 1; H*ration of CoD and rf-knife*L

CircleofDeath@top_D :=top
QPΩrad@top_D :=* 10^ H-2L;H*m*L

QP * 10^ H-4L * 100
top*ΜB * mF * gF;

87 * mu * 2Ωax@top_D := HSqrt@8D * Ωrad@topDL;
frad@top_D := Ωrad@topD  H2 * ΠL;
fax@top_D := HSqrt@8D * Ωrad@topDL  H2 * ΠL;
TOPOffset@top_D := ΜB * mF * gF * top;¥¥¥Αlin := NBà à à Exp@-Sqrt@x ^ 2 + y ^ 2 + 4 z ^ 2DD â x â y â zF *-¥ -¥ -¥¥¥¥

helper := NBà à à Exp@-x ^ 2 - y ^ 2 - 8 * z ^ 2D â x â y â zF;-¥ -¥ -¥

Αharm@top_D := helper *2 * kB^ 3;

87 * mu * Ωax@topD ^ 2

Vharm@T_, top_D := Αharm@topD * T ^ H3  2L;
Vlin@Tlin_D := Αlin * Tlin^ 3;
Τb@T_D := 1  J1  132 + 1  J0.00276277* I106 * TM NN;
22 * kBgF * mF * ΜB * Hb * 10^ H-2LL

^ 3;

3

101

4

Evaporation Simulation.nb

 Evaporation in linear Trap
PreEvaporation@8atomnumber_, temperature_, Tfinal_<D := ModuleB
8AtNum, T, Ξ, Τth, Τel, Τbkg, t, Η, Nthing, Tthing, databeginning, dataend, AtNuminitial,
Tinitial, Τthinitial, Τbinitial, Tscale, Nscale, PSDinitial, PSD, Ecut, colratio, data<,
AtNuminitial = 1. * 10^ 9; Tinitial = 500. * 10^ H-6L; Τthinitial = 2.; Τbinitial = Τb@TinitialD;
PSDinitial =Ξ = 3;

Ñ ^ 3 * H2 ΠL ^ 3J2 Π * 87 * mu * kB * Tinitial N ^ 3

AtNuminitial*;

Vlin@TinitialD

HT  TinitialL ^ HΞ - 0.5L

* Τthinitial;
AtNum  AtNuminitialΤel = Τth  2.7;
Τbkg = Τb@TD;Τbkg
colratio =;ΤelAtNum  AtNuminitialPSD =* PSDinitial;
HT  TinitialL ^ HΞ + 3  2L

Τth =t = 0;
Η = optimalΗintlinear@colratioD;
AtNum = atomnumber;
T = temperature;
Ecut = kB * T * Η;data = 88t, AtNum, T, Ecut, colratio, Τel, Τbkg, PSD<<;

IfBcolratio < 20, data = Join@data, 880, 0, 0, 0, 0, 0, 0, 0<<D,
WhileBT > Tfinal,Τbkg* 2.7;
colratio
If@colratio < 30, Η = 5, Η = optimalΗintlinear@colratioDD;
Nscale = Nfactor@Η, colratio, ΞD;
Tscale = Tfactor@Η, colratio, ΞD;
AtNum = Nscale * AtNum;
T = Tscale * T;
Tscale^ HΞ - 0.5L
* Τel;
Τel =NscaleΤbkg = Τb@TD;Τbkg
colratio =;ΤelNscale
PSD = PSD *;Tscale^ HΞ + 3  2L
t=t+

Ecut = kB * T * Η;data = Join@data, 88t, AtNum, T, Ecut, colratio, Τel, Τbkg, PSD<<DF;

F;

88atomnumber, temperature, Tfinal<, data<

F;

102

Evaporation Simulation.nb

Evaporation in TOP Trap
TOPTemp@TQP_, top_D :=^ H1  3L * HTQPL ^ H3  2L;

ΑlinΑharm@topD

Evaporationstiffening@8atomnumber_, temperature_, topbeginning_<D := ModuleB

8AtNumbeginning, Tbegninning, PSD, colratio, Η, dataend, AtNuminitial, Tinitial,
TOPfieldinitial, Τbinitial, Τthinitial, Τelinitial, colratioinitial, PSDinitial,
Ξ, T, AtNum, Τth, Τel, Τbkg, t, TOPfield, Ecut, data, Nscale, Tscale,TOPfieldold, TOPfieldnew, Tscale2, Vscale2, Τscale2, result , Τscale, Vscale<,

AtNuminitial = 1. * 10^ 9;
Tinitial = 100. * 10^ H-6L;
TOPfieldinitial = topbeginning;Τbinitial = 132;
Τthinitial = 2;Τelinitial = Τthinitial 2.7;
Τbinitial
colratioinitial =;ΤelinitialÑ ^ 3 * H2 ΠL ^ 3
AtNuminitial
PSDinitial =;*J 2 Π * 87 * mu * kB * Tinitial N ^ 3 Vharm@Tinitial, TOPfieldinitialD

Ξ = 3  2;
T = temperature;
AtNum = atomnumber;
AtNum  AtNuminitialPSD =* PSDinitial;
HT  TinitialL ^ HΞ + 3  2L
Tscale =ΑharmA60 * 10-4 E;

Αharm@topbeginningD
Vscale =Αharm@topbeginningDΑharmA60 * 10-4 E
VscaleΤscale =;;TscaleΤth =HT  TinitialL ^ HΞ - 0.5L

* Τthinitial* Τscale;
AtNum  AtNuminitialΤel = Τth  2.7;
Τbkg = Τbinitial;Τbkg
colratio =;Τel
t = 0;
Η = optimalΗintharmonic@colratioD;
TOPfield = topbeginning;

5

103

6

Evaporation Simulation.nb

TOPfield = topbeginning;
Ecut = kB * T * Η + TOPOffset@topbeginningD;
data = 88t, AtNum, T, Ecut, colratio, Τel, Τbkg, PSD, TOPfield<<;

IfBcolratio < 80, data = Join@data, 880, 0, 0, 0, 0, 0, 0, 0<<D,
WhileBPSD < 2.614,
Τbkg* 2.7;
colratioΗ = optimalΗintharmonic@colratioD;
Ecut = kB * T * Η + TOPOffset@TOPfieldD;
Nscale = Nfactor@Η, colratio, ΞD;
Tscale = Tfactor@Η, colratio, ΞD;
AtNum = Nscale * AtNum;
T = Tscale * T;
Nscale
PSD = PSD *;Tscale^ HΞ + 3  2L
t=t+

Τel =

Tscale^ HΞ - 0.5L

colratio =NscaleNscale* Τel;

Tscale^ HΞ - 0.5L

* colratio;

TOPfieldold = TOPfield;
Α
1
Ecut;
TOPfieldnew =1 + Α ΜB * mF * gF
IfATOPfieldnew > 10 * 10-4 , TOPfield = TOPfieldnew, TOPfield = 10. * 10-4 E;

Tscale2 =Vscale2 =TOPfieldold 32;

TOPfield
TOPfield

32;TOPfieldold
Vscale2Τscale2 =;Tscale2
T = T * Tscale2;
Τel = Τel * Τscale2;
Τbkg
colratio =;Τeldata = Join@data, 88t, AtNum, T, Ecut, colratio, Τel, Τbkg, PSD, TOPfield<<DF;F;

result = 88atomnumber, temperature<, data<

F;

104

Appendix B
Anisotropic expansion details

B.1

Algorithm

omRad 2
34.2;
omAx 2
96.4;
initialvector
1., 1., 0., 0. ;vector initialvector;
0;
t 0;
0.01;
result0, vector ;
While

t

50

10^

3 ,

;
t
omAx;
vector 1
senkrecht
vector 2 ;axvectorsenkrecht, dervectorax, der

4

;
3

;

;
omRad omAx ^ 2

senkrecht, dersenkrecht, der^3senkrecht
senkrecht

senkrecht;

senkrecht, der1ax, derax, der
senkrechtax, der;

vectorsenkrecht ,ax ,resultJoin result,

ax

ax

^2

ax ^ 2

senkrecht, der ,t, vector

ax

;

;

ax, der;;