Download Molecular Bose-Einstein Condensates and p

Molecular Bose-Einstein Condensates and
p-wave Feshbach Molecules of 6Li2
A thesis submitted for the degree of
Doctor of Philosophy
by
Jürgen Fuchs
Centre for Atom Optics and Ultrafast Spectroscopy and
ARC Centre of Excellence for Quantum-Atom Optics
Swinburne University of Technology, Melbourne
ii
Declaration
I, Jürgen Fuchs, declare that this thesis entitled:
“Molecular Bose-Einstein condensates and p-wave Feshbach molecules of 6 Li2 ”
is my own work and has not been submitted previously, in whole or in part, in respect of
any other academic award.
Jürgen Fuchs
Centre for Atom Optics and Ultrafast Spectroscopy
Faculty of Engineering and Industrial Sciences
Swinburne University of Technology
Melbourne, Australia
Dated this day, February 1, 2009
iii
Abstract
This thesis describes the production of molecular Bose-Einstein condensates (BEC) of 6 Li2
dimers and binding energy measurements of p-wave Feshbach molecules. A σ − Zeeman
slower is used to produce a continuous beam of isotopically enriched 6 Li atoms at speeds
low enough to load a magneto-optical trap (MOT). We currently have a flux of slowed
atoms of ∼ 5 · 106 atoms/s loading the MOT with more than 108 atoms. We then transfer
up to 106 atoms in an almost equal spin mixture of the two lowest hyperfine states into an
optical dipole trap.
We achieved condensates in three different crossed optical dipole trap geometries. The
initial low power optical dipole trap was formed using light from a 25 W VersaDisk Yb:YAG
laser at 1030 nm. It consisted of a 15 W beam crossed with a 13 W beam at about 80
degrees with a waist of approximately 30 µm in each beam. By translating the focus of
the second beam we could change the trap geometry from near symmetric to elongated.
Nowadays, we produce condensates in a crossed dipole trap formed by a 100 W fibre laser.
Both arms are focussed to a waist of 40 µm, cross each other at 14 degrees and have laser
powers of ∼80 W and ∼70 W, respectively.
Evaporative cooling is achieved by reducing the laser power near the broad s-wave
Feshbach resonance at 834 G. By tuning to the low magnetic field side (770 G) of the
Feshbach resonance molecules are formed through three-body recombination at sufficiently
low temperatures. Further evaporation leads to the creation of a molecular BEC. After
reducing the laser power by a factor of about 1000 in approximately 3 s we have observed
more than 30 000 condensed molecules. During the evaporation the temperature decreases
from about 100 µK to below 100 nK.
We present measurements of the binding energies of 6 Li p-wave Feshbach molecules
formed in combinations of the |F = 1/2, mF = +1/2i (|1i) and |F = 1/2, mF = −1/2i
(|2i) states. The binding energies scale linearly with magnetic field detuning for all three
resonances. The relative molecular magnetic moments are found to be 113 ± 7 µK/G,
111 ± 6 µK/G and 118 ± 8 µK/G for the |1i − |1i, |1i − |2i and |2i − |2i resonances,
respectively, in good agreement with theoretical predictions.
iv
Acknowledgements
The work presented in this PhD thesis would not have been possible without the help of
many others involved in the project.
First of all, I would like to thank my supervisors Wayne Rowlands and Peter Hannaford
for giving me the opportunity to work in this exciting research area of physics at Swinburne
University. Not only have they given me constant guidance and support throughout my
PhD they have also introduced me to the Australian culture which made my stay in
Australia a thoroughly enjoyable experience.
A huge thanks must be given to Grainne Duffy who worked with me for almost three
years. Her constant encouragement and working commitment has been invaluable for our
project from the very beginning. Working with us in the laboratory until late on her very
last day in Australia is a prime example showing her devotion.
Next, I would like to express my thanks to Chris Vale who has contributed so much to
the progress of our project since the moment he arrived at Swinburne. His experience in
the field has accelerated our progress tremendously.
I was very fortunate to work with my fellow graduate students Gopi Veeravalli, Paul
Dyke and Eva Kuhnle. It was a pleasure to work with all of them for which I am very
grateful. I thoroughly enjoyed the great atmosphere amongst us.
Our project has benefited enormously from the contributions of Chris Ticknor. On
many occasions he enlightened us with his wealth of knowledge in this field answering
patiently our most trivial and non-trivial type questions. For this I would like to say many
thanks.
Sharing the laboratory with Heath Kitson has been a great experience both scientifically
and personally. I have learned a great deal from him, for this I would like to say thanks.
Furthermore, I would like to thank Alexander Akulshin who introduced me to the field
of EIT and EIA. I enjoyed the countless discussions in the CAOUS tea room from which
I have learned so much. I am very grateful that he convinced us that it is worthwhile to
study sub-natural resonances in lithium.
Through the years there have been a number of people contributing to our project. I
would like to take the opportunity to say thanks to Markus Bartenstein, Michael Vanner,
Anthony Teal and Richard Moore.
v
I would like to acknowledge the big support from the Atom-Optics laboratory, in particular from Shannon Whitlock, Brenton Hall, Michael Volk, Mandip Singh, Holger Wolff,
Russell Anderson and Russell McLean. It was always fruitful to discuss both experimental
and theoretical issues. Without their help this project would not have moved as fast.
I would like to thank Mark Kivinen for the technical support he has given us. He
manufactured complex experimental components with an outstanding precision and his
working speed was just magnificent. A huge thanks also goes to Tatiana who helped us
with the administrative work load.
I would have suffered significant starvation without the great food from the restaurants
around the campus. I would like to thank Penang Coffee House, Nelayan, Red Bean,
Shanghai Tan, Curry Bazaar and HKSF for the delicious food and, more importantly, all
the people who joined me on those great occasions.
Over the last few years I have visited many laboratories worldwide. I wish to thank all
the people who welcomed me and showed me around their laboratories. In particular, I
would like to thank Andre Schirotzek and Silke and Christian Ospelkaus who even kindly
let me stay in their apartments.
Finally, I would like to express my thanks to my parents for their understanding and
their constant support.
vi
Contents
Declaration
ii
Abstract
iii
Acknowledgements
iv
Contents
vi
List of Figures
xi
1 Introduction
1
2 Interactions in an ultracold gas
5
2.1
Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Weakly bound Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
p-wave scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3 Quantum degenerate Fermi gases
21
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2
Ideal Fermi gas in a harmonic trap . . . . . . . . . . . . . . . . . . . . . .
23
3.3
Pairing and superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.3.1
Bose-Einstein condensation
. . . . . . . . . . . . . . . . . . . . . .
25
3.3.2
The BCS regime and the BEC-BCS crossover . . . . . . . . . . . .
28
3.3.2.1
The BCS regime . . . . . . . . . . . . . . . . . . . . . . .
28
3.3.2.2
BEC-BCS crossover and unitarity . . . . . . . . . . . . . .
29
3.3.3
p-wave superfluids
. . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
30
viii
CONTENTS
4 Experimental set-up
4.1
31
Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.1
Oven and oven pumping chamber . . . . . . . . . . . . . . . . . . .
31
4.1.2
Main vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . .
35
Laser system for 671 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.2.1
Saturation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . .
38
4.2.2
Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.3
Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.4
Feshbach coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.5
Experimental control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2
5 Sub-natural Resonances
5.1
49
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.1.1
Electromagnetically induced transparency . . . . . . . . . . . . . .
49
5.1.2
Zeeman coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.1.3
Coherent population oscillation . . . . . . . . . . . . . . . . . . . .
53
5.1.4
What is special about 6 Li? . . . . . . . . . . . . . . . . . . . . . . .
53
5.2
Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.3
Hyperfine EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.3.1
Vapour cell EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.3.2
Atomic beam EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Zeeman coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.4.1
EIT in the 6 Li D1 line . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.4.2
Sub-natural resonances for a pure four-level atomic system . . . . .
66
5.4
5.5
6
5.4.3
EIA in the Li D2 line . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.4.4
Modelling and discussion . . . . . . . . . . . . . . . . . . . . . . . .
70
Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6 Laser cooling of 6 Li
77
6.1
Spontaneous force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.2
Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.3
Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.3.1
Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.3.2
Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . .
83
ix
CONTENTS
6.3.3
Experimental realisation . . . . . . . . . . . . . . . . . . . . . . . .
7 Dipole traps
7.1
7.2
7.3
7.4
83
87
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
7.1.1
Single focussed dipole trap . . . . . . . . . . . . . . . . . . . . . . .
88
7.1.2
Crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Low power optical dipole trap set-up . . . . . . . . . . . . . . . . . . . . .
91
7.2.1
Intensity stabilisation of the dipole trap laser . . . . . . . . . . . . .
93
7.2.2
Trapping frequency measurement . . . . . . . . . . . . . . . . . . .
94
New dipole trap formed by a 100 W fibre laser . . . . . . . . . . . . . . . .
96
7.3.1
High power crossed dipole trap set-up . . . . . . . . . . . . . . . . .
97
7.3.2
Trapping frequency . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7.3.3
Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Loading the crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . . . 102
8 Bose-Einstein condensation of molecules
105
8.1
Theory of evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.2
MBEC in a Low Power Crossed Dipole Trap . . . . . . . . . . . . . . . . . 108
8.3
8.2.1
Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.2
Quantum degenerate Bose and Fermi gases . . . . . . . . . . . . . . 110
MBEC in a high power crossed dipole trap . . . . . . . . . . . . . . . . . . 114
8.3.1
Evaporation and realisation of a molecular BEC . . . . . . . . . . . 115
9 Binding Energies of 6 Li p-wave Feshbach Molecules
9.1
119
Inelastic losses at the |1i − |1i Feshbach resonance . . . . . . . . . . . . . . 119
9.2
Binding energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.3
Transition rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10 Conclusions
127
10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliography
131
Publications of the author
151
x
CONTENTS
List of Figures
2.1
Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
s-wave scattering length . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3
Binding energies of Feshbach molecules . . . . . . . . . . . . . . . . . . . .
12
2.4
Properties of p-wave Feshbach molecules . . . . . . . . . . . . . . . . . . .
18
3.1
Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.1
Photo of the vacuum set-up . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6
4.2
Schematic energy level diagram for Li . . . . . . . . . . . . . . . . . . . .
38
4.3
Saturation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.4
Doppler-free spectrum of 6 Li . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.5
Laser set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.6
Typical scan of the spectrum analyser . . . . . . . . . . . . . . . . . . . . .
43
4.7
Absorption imaging set-up . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.1
(a) Basic lambda scheme for electromagnetically induced transparency. The
bare atomic states |1i and |2i are coupled by laser light to state |3i. (b)
Atomic eigenstates in the presence of a weak probe and strong pump field,
5.2
resonant with the |1i → |3i and |2i → |3i transition, respectively. . . . . .
Simplified scheme of the experimental set-up.
51
. . . . . . . . . . . . . . . .
55
5.3
Vapour cell EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.4
Atomic beam EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.5
Comparison of the D1 and D2 lines . . . . . . . . . . . . . . . . . . . . . .
60
5.6
Fluorescence of the D1 line as a function of frequency difference . . . . . .
61
5.7
Zeeman coherence of the D1 line . . . . . . . . . . . . . . . . . . . . . . . .
62
5.8
Zeeman coherence in a magnetic field . . . . . . . . . . . . . . . . . . . . .
63
xi
xii
LIST OF FIGURES
5.9
Width of EIT fluorescence resonances
. . . . . . . . . . . . . . . . . . . .
64
5.10 Sub-natural resonances for the |2S1/2 , Fg = 1/2i → |2P1/2 , Fe = 1/2i transition 66
5.11 EIA in the 6 Li D2 line . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.12 Splitting of EIA resonances in an external magnetic field . . . . . . . . . .
69
5.13 Fluorescence for the |2S1/2 , Fg = 3/2i → |2P3/2 i transition . . . . . . . . . .
69
5.14 Numerical modelling of probe absorption spectra . . . . . . . . . . . . . . .
71
5.15 Set-up used to observe Ramsey fringes . . . . . . . . . . . . . . . . . . . .
73
5.16 Ramsey fringes in a medium of EIT and EIA . . . . . . . . . . . . . . . . .
74
6.1
Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6.2
Doppler temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
6.3
MOT temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
7.1
Crossed dipole trap gallery . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
7.2
Dipole trap potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
7.3
Set-up of the crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . . .
92
7.4
Elongated crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . . . .
93
7.5
Lifetime of the crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . .
94
7.6
Trapping frequency measurements . . . . . . . . . . . . . . . . . . . . . . .
95
7.7
High power crossed dipole trap set-up . . . . . . . . . . . . . . . . . . . . .
97
7.8
Radial trapping frequency measurements . . . . . . . . . . . . . . . . . . .
99
7.9
Axial trapping frequency measurements
. . . . . . . . . . . . . . . . . . . 100
7.10 Lifetime of the crossed dipole trap in different geometries . . . . . . . . . . 102
7.11 Absorption images of the crossed dipole trap . . . . . . . . . . . . . . . . . 103
7.12 Loading of the crossed dipole trap . . . . . . . . . . . . . . . . . . . . . . . 104
8.1
Evaporative cooling in the crossed dipole trap . . . . . . . . . . . . . . . . 108
8.2
Absorption images of the molecular gas durig evaporative cooling . . . . . 111
8.3
Integrated cross sections along the weakest trapping axis . . . . . . . . . . 112
8.4
In situ absorption images of a trapped molecular BEC and DFG . . . . . . 113
8.5
Observation of a degenerate Fermi gas . . . . . . . . . . . . . . . . . . . . 114
8.6
Evaporative cooling in the crossed dipole trap formed by the fibre laser . . 115
8.7
Absorption images during evaporative cooling in the high power dipole trap
and integrated cross sections along the weakest trapping direction . . . . . 117
LIST OF FIGURES
9.1
9.2
9.3
9.4
xiii
Atom loss at the |1i-|1i 6 Li p-wave Feshbach resonance . . . . . . . . . . . 121
Magneto-association spectrum for the |2i − |2i p-wave Feshbach resonance
122
Binding energies of p-wave Feshbach molecules . . . . . . . . . . . . . . . . 124
Magneto-association rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
xiv
LIST OF FIGURES
Chapter 1
Introduction
The discovery of superconducting mercury in 1911 by K. Onnes [Onn11] marked the starting point of the field of fermionic superfluidity and superconductivity. Subsequent experimental breakthroughs have been the realisation of liquid 3 He in 1972 [Osh72] and the
(at the time) surprising discovery of high-temperature superconductivity in cuprates in
1986 [Bed86]. However, not all aspects of these exciting phenomena are currently understood in depth.
Laser and evaporative cooling of neutral atoms has opened the way to studies of quantum degenerate bosonic and fermionic systems in table-top experiments. These dilute
gases have received enormous attention both theoretically and experimentally since the
first experimental realisation of Bose-Einstein condensates in 1995 [And95, Dav95, Bra95].
Experiments on ultracold Fermi gases lagged somewhat behind at that time. Certainly,
one reason was purely technical since evaporation of fermions requires more sophisticated
experimental set-ups. This is because collisions which are necessary for rethermalisation
during evaporative cooling are frozen out in an ultracold one-component fermionic gas due
to the Pauli exclusion principle. Thus, initially the interest in degenerate fermionic gases
was in the shadow of bosonic systems because “non-interacting” Fermi gases did not seem
attractive. It is an irony of life that experiments on ultracold fermionic gases nowadays
particularly focus on strongly interacting and strongly correlated systems.
Rapid progress has been made in fermionic systems through the use of magnetic field
Feshbach resonances which can dramatically alter the two-body interactions. These scattering resonances occur when the energy of two colliding atoms is Zeeman tuned to coincide with a bound molecular state. The stability of fermionic systems near such res1
2
CHAPTER 1. INTRODUCTION
onances came somewhat as a surprise and it is, in essence, due to the Pauli exclusion
principle [Pet04]. Bosonic systems, on the other hand, suffer from frequent inelastic collisions near Feshbach resonances and this ultimately limits the tunability of the interaction
strength. Feshbach resonances play a key role in studies of ultracold Fermi gases and have
led to the experimental realisation of long-lived bosonic molecules comprised of fermionic
atoms [Reg03a, Reg04a, Cub03, Str03, Joc03a] and the Bose-Einstein condensation of these
molecules [Joc03b, Gre03, Zwi03]. This thesis describes two different all-optical set-ups
employed to produce molecular condensates of 6 Li dimers in our laboratory.
The resonant superfluid phase which occurs in the crossover region between a molecular BEC and a Bardeen-Cooper-Schrieffer (BCS) state of Cooper pairs is of particular
interest [Hol01]. Since the first experimental investigations in this regime [Reg04b, Zwi04,
Chi04a, Bar04b, Bou04, Kin04a, Par05] progress to date has been extremely rapid with
studies of collective oscillations [Bar04a, Alt07, Kin04a, Kin04b], universal behaviour [Kin05,
Tho05a, Luo07, Ste06, Par06a], superfluidity [Chi06, Zwi05, Zwi06b, Sch07b, Sch08], polarised Fermi gases [Par06a, Zwi06a, Shi06, Par06b, Sch07a, Shi08] and the speed of
sound [Jos07].
An outstanding goal in cold atom physics is to extend the previous work on s-wave
paired condensates to superfluids of pairs with nonzero angular momentum. Recent experiments have shown the production of p-wave Feshbach molecules of
6
40
K2 [Gae07] and
Li2 [Ina08]. In this thesis we present our binding energy measurements of the three p-wave
Feshbach molecules formed by all three combinations of the two lowest hyperfine states.
The techniques and results presented in this thesis may provide a foundation towards the
production and observation of long-lived p-wave Feshbach molecules made from fermionic
atoms.
This thesis deals with the production of molecular Bose-Einstein condensates of 6 Li2
dimers in two different all-optical set-ups. Furthermore, we present the studies on the
binding energies of p-wave Feshbach molecules. The thesis is structured as follows: Firstly,
in chapters 2 and 3 we focus on the scattering and many-body properties of these ultracold
Fermi gases. A brief description of the vacuum apparatus and laser system is given in
chapter 4. Experiments on sub-natural resonances employing the atomic 6 Li beam is
the topic of chapter 5 which have been published in [Fuc06, Fuc07a]. Our laser cooling
and dipole trapping is presented in chapters 6 and 7. Then, the production of molecular
Bose-Einstein condensates in different geometries is described in chapter 8 which is partly
3
published in [Fuc07b]. Parts of the last experimental chapter 9 on p-wave molecules have
been published in [Fuc08]. The main results are summarised and a brief outlook is provided
in chapter 10.
4
CHAPTER 1. INTRODUCTION
Chapter 2
Interactions in an ultracold gas
In recent years there has been enormous progress in the study of ultracold Fermi gases.
Experimentally, the advances were mainly driven by the use of magnetic field Feshbach
resonances which can dramatically alter the two-body interactions. The breath-taking
experiments in the BEC-BCS crossover have utilised broad s-wave Feshbach resonances in
either 6 Li [Zwi04, Bar04b, Bou04, Kin04a, Par05] or
40
K [Reg04b]. Additionally, in our
experiments we are also interested in extending this work on s-wave pairing to pairs with
nonzero angular momentum.
Here, we briefly discuss basic scattering theory and aspects of Feshbach resonances
and Feshbach molecules which are relevant for our experiments. We only discuss twobody interactions because in typical experiments on ultracold gases the mean interparticle
separation (n−1/3 ∼ 1 µm) is much larger than the spatial range of the interatomic potential
(∼ 3 nm). This chapter is kept short as it is meant to be a reminder rather than an
introduction to the field. There have been a number of extensive reviews on this topic,
e.g. [Dal99].
2.1
Elastic scattering
Two-body scattering problems are commonly described in the centre of mass frame which
reduces the dimensionality from 6 to 3 as only the relative motion is relevant. Hence,
our scattering problem can be described by a particle with momentum ~k and reduced
mass mr (equal to half the atomic mass m) that is scattered by the interaction potential
V (r). The non-relativistic Schrödinger equation for a spherically symmetric potential, i.e.,
5
6
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
V (r) = V (r), is given by
~2
∇2 + V (r) Ψ(r) = EΨ(r).
2mr
(2.1)
(~k)2
E=
>0
2mr
(2.2)
In scattering processes V (r) is the interatomic potential and
is the kinetic energy of the incoming particle. Solutions with E = −EB < 0 give rise to
discrete bound states with a binding energy EB . Generally, eigenstates to this centrally
symmetric potential are of the form
Ylm (θ, φ)
uk,l,m(r)
,
r
(2.3)
where Ylm (θ, φ) are the spherical harmonic functions. The functions uk,l,m(r) obey the one
dimensional radial Schrödinger equation with the effective radial potential
Veff (r) =
~2 l(l + 1)
+ V (r).
2mr r 2
(2.4)
The centrifugal barrier Vl = ~2 l(l + 1)/(2mr r 2 ) suppresses collisions with l > 0 for low
energies. In lithium, for example, this centrifugal barrier is ∼ 7kB ×mK for l = 1 [Jul92],
where kB is Boltzmann’s constant. This barrier is significantly larger than the average
kinetic energy of laser cooled atoms. Hence, we typically only need to consider l = 0
collisions, also called s-wave collisions. One important exception to this will be discussed
in section 2.4.
In scattering processes it is usual to describe the wave function Ψk (r) as the sum of
the incoming plane wave Ψinc (r) and the scattered outgoing spherical wave for r → ∞
Ψsc (r). An arbitrary incoming wave packet can be described as a superposition of many
plane waves. This yields
Ψk (r) ∼ Ψinc (r) + Ψsc (r) = eik·r + f (k, θ)
r→∞
eikr
.
r
(2.5)
The function f (k, θ) is the scattering amplitude that for a given energy E only depends
on the scattering angle θ. Due to the azimuthal symmetry the scattering amplitude can
be expanded in spherical harmonic functions Ylm with m = 0 [Gur07]
f (k, θ) =
∞
X
p
l=0
4π(2l + 1)fl (k)Yl0 (θ).
(2.6)
7
2.1. ELASTIC SCATTERING
Here, fl (k) is the partial wave scattering amplitude which can be written in terms of the
partial wave phase shifts δl (k)
1
ei2δl − 1
=−
2ik
−k cot δl (k) + ik
k 2l
k 2l
∼
=
.
=
−k 2l+1 cot δl (k) + ik 2l+1
vl−1 − 21 k 2 kl + ik 2l+1
fl (k) =
(2.7)
(2.8)
The form of the partial wave scattering amplitude (equation 2.7) follows from comparing
asymptotic results to the Schrödinger equation for r → ∞ and equation 2.5. In the second
step of equation 2.8 we parametrise the scattering amplitude for low energies by Taylor
expanding k 2l+1 cot δl (k) in powers of k 2 (which is proportional to the energy). This is
meaningful due to physical restrictions on the scattering amplitude [Gur07]. Therefore, at
small k the scattering amplitude becomes
fl (k) ∝ k 2l ∝ E l .
(2.9)
In many experiments we are interested in the scattering cross section. The differential
cross section can be related to the scattering amplitude by
dσ
= |f (k, θ)|2 ,
dΩ
(2.10)
where Ω is the solid angle. From this, one can calculate the total cross section
σ(k) =
∞
X
σl (k)
with
σl (k) = 4π(2l + 1)|fl (k)|2 =
l=0
4π
(2l + 1) sin2 δl (k).
k2
(2.11)
From equations 2.9 and 2.11 it follows that the energy dependence of the cross section
σl (E) ∝ E 2l in the low energy limit. Hence, the s-wave scattering cross section becomes
constant at low energies whereas higher order cross sections vanish as E → 0. This
confirms that at low temperatures scattering processes with small l dominate as we have
already discussed previously. The different threshold behaviour for l = 0 and l = 1 at low
temperatures has been observed in reference [DeM99a].
If the two colliding particles are identical, quantum statistics has to be included in the
theory. At the low temperatures we are dealing with, this becomes particularly important
and changes the scattering properties significantly. Quantum mechanics requires the wave
function to be symmetric for bosons and anti-symmetric for fermions under interchanging
both particles. From this it follows that the cross section for a partial wave l of two identical
8
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
colliding fermions (bosons) vanishes for even (odd) l and doubles for odd (even) l. Hence,
for identical particles equation 2.11 becomes [Dal99]
Bosons :
Fermions :
8π X
(2l + 1) sin2 δl (k)
k 2 l even
8π X
σ(k) =
(2l + 1) sin2 δl (k).
k2
σ(k) =
(2.12)
(2.13)
l odd
The important consequence is that identical fermions do not scatter in the s-wave channel. Hence, due to the symmetry principle and the centrifugal barrier, ultracold identical
fermions are typically non-interacting.
s-wave scattering
The s-wave partial wave scattering amplitude f0 (k) can be parametrised by Taylor expanding k cot δ0 (k)
1
−k cot δ0 (k) ∼
(2.14)
= a−1 − k 2 r0 .
2
Here, we define the scattering length a and the effective range of interaction r0 . A negative
(positive) scattering length results in an overall attractive (repulsive) interaction. The
magnitude and sign of the scattering length for a single potential are mainly determined
by the highest bound molecular state. If it is just below the continuum the scattering
length is large and positive. The effective range of interactions is typically on the order
of the spatial range of the interatomic potential. In the s-wave limit for a point-like
interaction (r0 = 0) the scattering cross section for distinguishable particles is according
to equation 2.11
4πa2
.
(2.15)
1 + k 2 a2
In the weakly interacting limit (ka ≪ 1) this gives σ0 (k) = 4πa2 , whereas in the strongly
σ0 (k) =
interacting limit the scattering cross section σ0 (k) = 4π/k 2 becomes independent of the
scattering length. Instead, it is proportional to the spread of the wave packet of the atom
represented by the square of the de Broglie wave length
s
2π~2
,
λdB =
mkB T
where T is the temperature of the gas and m the mass of the particles.
(2.16)
9
2.2. FESHBACH RESONANCE
(a)
free atoms
(b)
∆E = ∆µ × B
energy
energy
closed channel
virtual bound state
Feshbach molecule
open channel
free atoms
magnetic field
interatomic distance
Figure 2.1: (a) Due to the different magnetic moments the closed channel can be shifted
relative to the open channel by applying a magnetic field. Tuning a bound state of the
closed channel to degeneracy with the continuum of the open channel leads to a Feshbach resonance. (b) Energy dependence of the molecular and atomic states with respect
to magnetic field. Due to the avoided crossing Feshbach molecules can be adiabatically
transferred into free atomic pairs and vice versa.
2.2
Feshbach resonance
It was an important breakthrough in ultracold atom experiments to discover the possibility to tune the scattering length, and hence the interactions, by applying magnetic
fields near Feshbach resonances. These scattering resonances [Fes62] have been predicted
by [Tie93] for ultracold atomic gases. Experimentally, they were first discovered by Inouye
et al. [Ino98] in
23
Na and Courteille et al. [Cou98] in
85
Rb.
To understand the underlying principle, we consider two molecular potentials for atoms
in different hyperfine states (see figure 2.1). The incoming atoms scatter in the lower
potential also named an open channel. The upper potential is typically referred to as the
closed channel because it is energetically not accessible at R = ∞. The highest lying
vibrational levels of the closed channel can lie above the continuum of the open channel.
The scattering properties are altered by the coupling of both channels which is typically
due to the Coulomb or hyperfine interaction [Bur02]. A bound state of the closed channel
just below the continuum of the open channel gives rise to a large positive scattering
10
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
length. Similarly, a virtual bound state just above the continuum yields a large and
negative scattering length. If the molecular state has a different magnetic moment to that
of the free atoms an external magnetic field shifts the potentials with respect to each other.
Hence, by applying a magnetic field it may be possible to bring a bound state of the closed
channel into degeneracy with the continuum giving rise to a Feshbach resonance. At the
position of the Feshbach resonance the scattering length diverges. Near the resonance the
scattering length behaves as
a(B) = abg 1 −
∆ ,
B − B0
(2.17)
where abg is the background scattering length, B0 the position of the resonance and ∆
the width of the resonance. It is usual to distinguish between broad and narrow Feshbach
resonances. Broad Feshbach resonances are characterised by kF |r0 | ≪ 1, where kF is
the Fermi number which will be properly introduced in equation 3.11. One finds for
broad resonances that the many-body properties of the ultracold gas are determined by
the dimensionless parameter kF a, whereas in narrow resonances (kF |r0 | & 1) the effective
range of interactions becomes crucial [Gio07].
Figure 2.2 shows the scattering length versus magnetic field of two lithium-6 atoms in
the two lowest hyperfine ground states [Bar05]. The unusual broad Feshbach resonance at
834 G has a width of 300 G and was predicted by Houbiers et al. [Hou98]. Furthermore, an
∼ 100 mG wide resonance at 543 G has been found and was first observed by Dieckmann
et al. [Die02]. The scattering length starts off near zero in zero magnetic field. With
increasing magnetic field the scattering length decreases to a local minimum of -300 a0
at 325 G before crossing zero at 530 G. At large magnetic fields the scattering length
approaches the very large triplet scattering length of 2200 a0 .
2.3
Weakly bound Feshbach molecules
Extending the work on ultracold atoms to ultracold molecules is of great interest. The
first production of cold dimers was demonstrated by photoassociating cold atoms [Fio98].
Although this is now a very established method the conversion efficiency was limited in
the experiments performed so far. In 1999, Abeelen et al. [Abe99] pointed out that it is
possible to create ultracold molecules near Feshbach resonances. The main idea is shown in
figure 2.1(b). The dependence of energy on the magnetic field is depicted for the free atoms
11
scattering length [1000 a0 ]
2.3. WEAKLY BOUND FESHBACH MOLECULES
4
2
0
−2
−4
0
500
1000
magnetic field [G]
1500
Figure 2.2: s-wave scattering length of two colliding 6 Li atoms in the two lowest hyperfine
states |F = 1/2, mF = +1/2i (|1i) and |F = 1/2, mF = −1/2i (|2i) as a function of
magnetic field. Of particular interest are the broad Feshbach resonance at 834 G and the
very narrow one at 543 G.
in the open channel and the bound state in the closed channel. Due to an avoided crossing
the molecular state below the resonance is connected to the free atom state above the
resonance. The Feshbach molecule above the resonance is unstable and is hence referred
to as a virtual bound state.
In 2002, Donley et al. were the first to observe a signature of molecules created near
a Feshbach resonance [Don02]. In a Ramsey type experiment they observed Ramsey
fringes between molecules and atoms. Regal et al. directly observed creation of molecules
by adiabatically sweeping across a Feshbach resonance in fermionic
40
K [Reg03a]. This
6
was followed by similar experiments on an ultracold gas of Li [Joc03a, Str03, Cub03].
Furthermore, creation of molecules has also been reported in bosonic systems, such as
Cs [Her03], Rb [Dür04] and Na [Xu03]. Detailed reviews on Feshbach molecules can be
found in [Koh06, Hut06].
It is characteristic that these Feshbach molecules in the s-wave channel are only very
weakly bound. The binding energy EB = −E of these molecules represents the en-
12
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
energy
molecules
(b)
atoms
|ν, ν′i
...
...
|0, 1i , |1, 0i
|νm i
|0, 0i
...
...
Eb
|1m i
|0m i
binding energy [kB µK]
(a)
1
10
0
10
−1
10
−2
10
650
700
750
800
magnetic field [G]
Figure 2.3: (a) Lowest vibrational levels of bound molecules and free atom pairs. The binding energy of Feshbach molecules corresponds to the energy difference between the lowest
molecular and atomic vibrating states. (b) Magnetic field dependence of the binding energy
of 6 Li2 Feshbach molecules below the broad s-wave resonance according to equation 2.18.
ergy difference between the lowest molecular and atomic vibrating states as depicted in
figure 2.3(a). It corresponds to the poles of the s-wave scattering amplitude f0−1 (k =
√
2mr EB ) = 0 which yields
r0 i
~2 h
1+
EB = −
2mr a2
a
for
r0 ≪ a.
(2.18)
As expected, the binding energy depends mainly on the scattering length a. Thus, by
varying the magnetic field near a Feshbach resonance it is possible to alter the binding
energy dramatically. In fact, the binding energy depends quadratically on the magnetic
field detuning (B0 − B). In figure 2.3(b) we show the binding energy of 6 Li2 Feshbach
molecules as a function of magnetic field below the broad resonance in the two lowest
hyperfine states. It was calculated using equation 2.18 with r0 ∼ 3.2 nm [Gio07].
Compared to tightly bound molecules the particle separation of weakly bound molecules
can be orders of magnitudes larger. This can be seen by considering the wave function.
Due to the large inter-particle separations compared to the classical turning point the wave
13
2.3. WEAKLY BOUND FESHBACH MOLECULES
function can be approximated by [Koh06]
Ψ(r) = √
1 exp(−r/a)
.
r
2πa
(2.19)
This gives a mean particle separation of hri = a/2.
To achieve a molecular Bose-Einstein condensate the scattering properties of the Fes-
hbach molecules are of importance. It has been shown that the scattering length for
dimer-dimer collisions add and dimer-atom collisions aad can be related to the atom-atom
scattering length a [Pet04, Pet05]. The theoretically calculated values
add = 0.6a
and
aad = 1.2a,
(2.20)
are in good agreement with experimental data [Joc03b, Cub03].
Collisional stability of Feshbach molecules
Weakly bound Feshbach molecules are in the highest rovibrational molecular state. Therefore, inelastic atom-molecule and molecule-molecule collisions may lead to relaxation into
deeply bound states releasing enough energy for the molecules to escape the trap. It has
been found that this relaxation process is quite rapid for Feshbach molecules comprised of
bosonic atoms. Typical lifetimes are on the order of 1 ms which is too short to achieve
rethermalisation of the ensemble. The high inelastic loss rate near Feshbach resonances also
limits the tunability of the atomic interactions in bosonic gases [Ste99a]. Inelastic collisions
can be avoided in a three dimensional lattice by loading exactly two atoms per lattice site.
In a subsequent magnetic sweep one molecule per lattice site can be formed [Vol06].
The situation is very different for weakly bound molecules made from fermionic atoms.
This has been theoretically investigated by Petrov et al. [Pet04] for systems where the scattering length is much larger than the range of the interatomic potential. They showed that
for molecules in long range states, e.g., obeying the wave function given by equation 2.19,
the atom-molecule and molecule-molecule inelastic collisions are suppressed due to Fermi
statistics. This is because the collisional relaxation is a three-body process where the three
atoms have to get as close to each other as the range of interactions. In a two species spin
mixture the third atom is necessarily identical to one of the atoms bound in the molecule.
Pauli blocking then prevents the third atom from getting as close to the molecule as required for inelastic collisions. In [Pet04] the atom-dimer αad and dimer-dimer αdd collision
14
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
rates that lead to transitions in tightly bound molecules have been calculated and have the
following dependence on the scattering length
αad ∝ a−3.33
and
αdd ∝ a−2.55 .
(2.21)
It is remarkable that the inelastic collision rates decrease as the scattering length a increases. This leads to very stable systems close to Feshbach resonances where the binding
of the molecule is particularly weak. The inelastic loss rates are in agreement with experimental observations [Reg04a, Bou04]. Reference [Pet04] also shows that the inelastic
collision rate is significantly smaller than the elastic scattering rate (see chapter 8.1 and
equation 2.20). This allows efficient evaporative cooling of weakly bound molecules near
Feshbach resonances which is the fundamental basis for nearly all our experiments.
Three-body recombination
Many groups produce weakly bound Feshbach molecules from a BEC or degenerate Fermi
gas by adiabatically sweeping over the Feshbach resonance making use of the avoided
crossing as depicted in figure 2.1(b). However, as was shown by Jochim et al. [Joc03a]
it is also possible to produce molecules by three-body recombination from a thermal gas
of atoms. When the scattering length is such that the binding energy EB ∼
= ~2 /ma2 is
higher than the atomic temperature (but lower than the depth of the potential confining
the cloud) a trapped mixture of atoms and molecules in thermodynamic equilibrium may
exist. As the temperature decreases below the binding energy of the molecules, the equilibrium favours molecules, which are formed by three-body recombination. This enables
the production of cold molecules by evaporative cooling (which will be explained in more
detail in section 8.1) of an incoherent mixture of atoms in states |F = 1/2, mF = +1/2i
(|1i) and |F = 1/2, mF = −1/2i (|2i), on the a > 0 side of a Feshbach resonance. This
molecule production process has been studied theoretically [Chi04b, Kok04, Wil04]. According to [Chi04b], the molecular phase space density Dmol and the atomic phase space
2 EB /kB T
density Dat (see section 3.1) are related by Dmol = Dat
e
. This suggests effective
molecule production at large binding energies. However, this is not an ideal situation in
cold atom experiments because the binding energy of the molecule is converted into kinetic
energy which heats up the cloud.
2.4. P -WAVE SCATTERING
2.4
15
p-wave scattering
For identical fermions s-wave scattering is prohibited due to quantum statistics. Hence,
p-wave collisions become the dominant scattering process. However, only close to p-wave
Feshbach resonances is p-wave scattering typically noticeable in cold atom experiments
due to the centrifugal barrier. These scattering resonances are very similar to s-wave
resonances; however, the closed channel involved in p-wave resonances is a molecular state
with finite angular momentum l = 1. It is unique to atom pairs with nonzero angular
momentum that a quasi-bound molecular state exists on the high magnetic field side of
the resonance. This state possesses positive energy and is only temporarily bound by the
centrifugal barrier [Kno08]. p-wave resonances are generally narrow as the colliding atom
has to tunnel through the centrifugal barrier to be affected by the bound state.
Initial experiments on p-wave Feshbach resonances in ultracold Fermi gases have focussed on 6 Li and
40
K. In 6 Li, three p-wave resonances have been identified in all com-
binations of atoms in the |1i and |2i states. Atom loss associated with these resonances
has been observed at fields of 159 G, 185 G and 215 G due to the |1i − |1i, |1i − |2i and
|2i − |2i resonances, respectively [Zha04, Sch05]. Inelastic and elastic collision rates for
the |1i − |2i and |2i − |2i resonances were calculated in [Che05c]. Evidence of molecule
formation via adiabatically sweeping the magnetic field across the |1i − |2i resonance was
seen in [Zha04], but no long lived trapped molecules were detected. Enhanced three-body
loss has also been reported for the 159 G p-wave resonance through interactions with a
second species (87 Rb) [Deh08]. Somewhat more progress has been made using
40
K includ-
ing measurements of the field dependent elastic scattering cross-section [Reg03b] and the
observation of the doublet corresponding to the different projections of the angular momentum [Tic04, Gün05], where the weak dipole-dipole interaction lifts the degeneracy of
the ml = ±1 and ml = 0 projections. In 2007, Gaebler et al. succeeded in creating p-wave
molecules from a gas of spin polarised
40
K [Gae07] using both magneto-association and
three-body recombination and measured the binding energies and lifetimes in the bound
and quasi-bound regimes. Unfortunately, these molecules experience rapid decay to lower
lying atomic spin states through dipolar relaxation and their lifetime was limited to less
than 10 ms. This presents a major impediment to creating a
40
K p-wave superfluid (see
6
section 3.3.3). In Li, however, the |1i − |1i p-wave resonance involves two atoms collid-
ing in their lowest spin state; hence molecules produced on this resonance would not be
16
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
susceptible to dipolar relaxation. These molecules therefore have the potential to be much
longer lived and may prove to be a viable avenue to studies of p-wave superfluidity with
ultracold atomic gases. Dipolar relaxation is also suppressed for the ml = 1 projections
of the |1i − |2i and |2i − |2i resonances due to angular momentum conservation. How-
ever, these states are degenerate with other unstable ml projections, unlike the |1i − |1i
resonance which is stable for all projections. Recently, Inada et al. reported the formation of p-wave molecules in all three combinations of the two lowest hyperfine states of
6
Li [Ina08]. Furthermore, elastic and inelastic collision rates were determined. However, it
is noteworthy that Inada et al. reported in recent experiments a 1/e lifetime of only up to
approximately 20 ms for a gas of |1i − |1i molecules.
The appropriate parametrisation of the p-wave scattering amplitude (see equation 2.8)
is found by Taylor expanding k 3 cot δ1 (k). This results in a p-wave scattering amplitude of
f1 (k) =
k2
−v −1 + 12 k0 k 2 − ik 3
for
1
− k 3 cot δ1 (k) ∼
= −v −1 + k 2 k0 .
2
(2.22)
Here, v is the scattering volume which is related to the p-wave scattering length v = a3p . The
characteristic wave vector k0 , which is negative everywhere, corresponds to the effective
range in s-wave scattering. Both v and k0 depend on the magnetic field but not on the
collision energy. For small k the scattering amplitude depends linearly on the scattering
energy, i.e.,
2mr E
.
(2.23)
~2
Near p-wave Feshbach resonances the scattering volume can be tuned by varying the
f1 (k) ∝ k 2 =
magnetic field and it diverges at the resonance and changes its sign. For low collision
energies the p-wave scattering amplitude just below the resonance (vk 2 ≫ 1 and k ≪ k0 )
becomes f1 (k) ≈
1
k0
which is typically on the same order of magnitude as non-resonant
2
s-wave scattering. For a collision energy E ≈ − 2m2~r k0 v the scattering amplitude becomes
unitary f1 (k) ≈ ki .
Properties of p-wave molecules
The binding energy of resonant states can be found by analysing the poles of the scattering
amplitude in equation 2.22. In the case of low energies one can neglect ik which results in
a binding energy of
EB ∼
=
~2
.
mr vk0
(2.24)
2.4. P -WAVE SCATTERING
17
For v > 0 this corresponds to a real bound state while for v < 0 we obtain a quasibound state. Even close to resonance one needs two parameters, in this case v and k0 , to
describe the binding energy accurately. This is in contrast to s-wave molecules, where the
scattering length a is sufficient to determine the binding energy near resonance, as seen
in equation 2.18. The reason for this is that the properties of s-wave bound states in the
vicinity of a Feshbach resonance are hardly affected by the short-range properties of the
potentials due to the large spatial extent of the molecule. Because of the centrifugal barrier
this is not the case for p-wave Feshbach molecules, as we will see shortly.
The binding energy of p-wave molecules scales linearly with respect to the magnetic
field detuning. This is also in contrast to weakly bound s-wave molecules which have a
squared dependence. The consequence of this is that the binding energy of p-wave molecules
increases much faster as we increase the magnetic field detuning. If the binding energy
of the p-wave molecule is greater than the trap depth the molecules will escape the trap
after being formed from free atoms. p-wave molecule production is therefore limited to the
vicinity of Feshbach resonances.
The properties of p-wave molecules are essentially constant across the resonance, in
strong contrast to s-wave molecules [Koh06]. To illustrate this we first consider the closed
channel amplitude, Z [Koh06, Gub07]. The wavefunction can then be written in terms of
the open channel component, ψo , and ψc which represents all closed channel components
yielding
|ψmol i =
√
Z|ψc i +
√
1 − Z|ψo i.
(2.25)
Physically, the closed channels correspond to channels with different (separate) atomic
hyperfine quantum numbers which are energetically forbidden at long range, but couple to
the open channel at short range via spin-exchange. In figure 2.4 we have plotted Z (solid
blue) for 6 Li p-wave molecules as a function of detuning (right vertical axis) on the bound
side of the resonance. Z was obtained directly from the full closed-coupled calculation
performed by C. Ticknor. It is roughly 0.82 for all detunings shown. This property is
essentially constant across the resonance until the detuning is extremely small (< 5 mG).
In figure 2.4 we have also plotted the size of the p-wave molecules (black) as a function
of detuning. In addition we have separately plotted the size of a dominant closed channel
(red dashed), the open channel (red), and an analytic open channel model (dashed brown,
see below). We can understand this behaviour by expressing the size of the molecule in
18
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
Figure 2.4: Properties of p-wave Feshbach molecules as a function of magnetic field detuning. The size of the molecule (black), open (red) and closed (dashed red) channel
components are shown. The closed channel amplitude, Z (blue, right vertical axis), is
shown as a function of detuning.
terms of open and closed channels:
rmol = Zrc + (1 − Z)ro ,
(2.26)
where ri = hψi |r|ψi i, and i = {mol, o, c}. This shows that the size of the molecule very
closely follows Z. In addition to this, both ro and rc are much smaller than those in the s-
wave molecules. This fact arises due to the centrifugal barrier confining the wavefunction to
short range. Near an s-wave resonance the molecules are open channel dominated because
Z → 0 as the magnetic field detuning, δB = B − B0 , goes to zero [Koh06]. This means
s
rmol
∼ ro (equation 2.26). From the asymptotic form of the s-wave radial wavefunctions
s
(equation 2.19) one finds ro is remarkably large for s-wave molecules: rmol
= a/2 where a
is the s-wave scattering length. In contrast, p-wave molecules carry much more amplitude
at short range due to the extra radial dependence in the asymptotic form of the radial
wavefunction
ψop (r)
e−kr
∝
r
1
1+
.
kr
(2.27)
One can use equation 2.27 to make an analytic approximation to the size of the open channel
p
√
using ~k = 2mr EB = mµm |δB|, where mr is the reduced mass and µm is the magnetic
2.4. P -WAVE SCATTERING
19
moment of the p-wave Feshbach molecule. The analytic model diverges as r → 0; so we
choose a cut off radius of 35a0 , such that the model and the full calculation have similar
ro values at large detuning. We show ro obtained from the analytic model as a dashed
brown line in figure 2.4. Note that even as the detuning becomes small (δB ≈ 5 mG) the
size of the open channels remains quite small, ro < 200a0 . Additionally, p-wave moleucules
are closed-channel dominated (Z ∼ 0.8), and rc ≈ 40a0 is constant for all δB. The small
size of the open channel, combined with the closed-channel character of p-wave molecules,
p
results in rmol
< 70a0 , which is much smaller than typical s-wave Feshbach molecules.
20
CHAPTER 2. INTERACTIONS IN AN ULTRACOLD GAS
Chapter 3
Quantum degenerate Fermi gases
At the densities and low temperatures we achieve in our experiments quantum statistics
becomes crucial. This chapter introduces quantum statistics and many-body properties of
ultracold Fermi gases. There are many excellent textbooks and review articles available
on this topic for further reading, e.g. [Pet01, Pit03, Gio07].
3.1
Introduction
At high temperatures a gas of indistinguishable particles can be described by classical
Maxwell-Boltzmann statistics. Here, the mean occupation number f (ǫν ) of the single
particle state ν is much smaller than one. As the gas is cooled down f (ǫν ) increases for
the low lying energy states. The indistinguishability of identical particles becomes crucial
when the phase space density
D = nλ3dB
(3.1)
approaches unity. Here, n is the density of the identical particles and
λdB =
s
2π~2
mkB T
(3.2)
is the thermal de Broglie wavelength which represents the spread of the atomic wave
function. Depending on the spin of the identical particles either Fermi-Dirac or BoseEinstein statistics describes the gas accurately.
21
22
CHAPTER 3. QUANTUM DEGENERATE FERMI GASES
nλ3dB ≪ 1
nλ3dB & 1
T =0
Bose-Einstein
Fermi-Dirac
Figure 3.1: We compare a gas of identical bosons and fermions, respectively, at different
temperatures. At sufficiently high temperatures (nλ3dB ≪ 1) quantum statistics is negligible
because the mean occupation number for all states is much less than one. However, as the
phase space density approaches unity the statistics becomes crucial. A gas of identical
bosons begins to occupy the ground state macroscopically for nλ3dB & 1 , whereas fermions
fill up the low lying energy states with at most one atom per state. At T = 0 all bosons are
condensed in the ground state and the fermions fill up all states up to the Fermi energy.
The mean occupation number for the three different cases are
1
exp[(ǫν − µ)/kB T ]
1
f BE (ǫν ) =
exp[(ǫν − µ)/kB T ] − 1
1
.
f FD (ǫν ) =
exp[(ǫν − µ)/kB T ] + 1
f MB (ǫν ) =
Maxwell-Boltzmann:
Bose-Einstein:
Fermi-Dirac:
(3.3)
(3.4)
(3.5)
The chemical potential µ is given by the normalisation condition
X
f (ǫν ) = Ntotal ,
(3.6)
ν
where Ntotal is the total number of particles and the sum is taken over all single particle
states. From equation 3.5 it follows that the mean occupation number for fermions f FD can
23
3.2. IDEAL FERMI GAS IN A HARMONIC TRAP
not exceed one. This agrees with Pauli’s exclusion principle which states that two identical
fermions can not occupy the same state. At T = 0 the fermions fill up all states up to the
so-called Fermi energy with exactly one atom per state. The situation is fundamentally
different for identical bosons which can multiply occupy the same state. This leads, at cold
enough temperatures, to a macroscopic occupation of the ground state. This phenomenon
is well known under the expression Bose-Einstein condensation and will be discussed in
section 3.3.1. Firstly, we will turn our attention towards degenerate fermions.
3.2
Ideal Fermi gas in a harmonic trap
In our experiments the gas is typically confined in an approximate harmonic trap with
potential
1
1
1
V (r) = mωx2 x2 + mωy2 y 2 + mωz2 z 2 .
(3.7)
2
2
2
A semi-classical approach, which is valid if kB T ≫ ~ωi , replaces summations - such as in
equation 3.6 - by integrals. The density of states in a harmonic trap is then given by
g(ǫ) =
ǫ2
,
2(~ω)3
(3.8)
where ω = (ωx ωy ωz )1/3 is the geometric mean of the three trapping frequencies ωx , ωy and
ωz . The normalisation condition that fixes the chemical potential is consequently
Z
N = g(ǫ) f FD (ǫ) dǫ.
(3.9)
For T = 0 the mean occupation number f (ǫ) is a step function. It is unity up to an energy
EF = µ(T = 0) which is called the Fermi energy and zero for energies above. In a harmonic
trap the Fermi energy EF is
EF = ~ω(6N)1/3 = kB TF = kB × 188
ω/2π N 1/3
nK.
100 Hz 104
(3.10)
In the second last step we converted the Fermi energy into the Fermi temperature. Commonly used is also the definition of the Fermi wave number
kF =
where m is the mass of the atom.
p
2mEF /~2 ,
(3.11)
24
CHAPTER 3. QUANTUM DEGENERATE FERMI GASES
Thomas-Fermi approximation
The density and momentum distribution of trapped gases is often calculated using the
semi-classical Thomas-Fermi approximation which assumes that locally the trapped gas
can be described by a uniform gas. In this approximation, the quantum states are labeled
by the continuous parameters position r and momentum p. The Fermi-Dirac distribution
is thus
1
.
+ Vext (r) − µ)/kB T ] + 1
Similarly to above, the chemical potential is obtained by normalising
Z
1
dr dp f FD (r, p),
N=
3
(2π~)
f FD (r, p) =
where
1
(2π~)3
exp[(p2 /2m
(3.12)
(3.13)
is the density of states. In the Thomas-Fermi approximation a local Fermi
wave vector kF (r) is defined for each point in space by
~2 kF2 (r)
+ V (r) = EF .
2m
(3.14)
This implies that the chemical potential is equal in space which is fulfilled in an equilibrium
as we would have a net flux of particles otherwise. For a fermionic gas at T =0 all momentum
states at position r are occupied up to ~kF (r). It follows that atoms further away from the
centre of the trap have smaller momenta. The number of atoms that fit into the momentum
sphere with radius ~kF (r) multiplied by the density of states yields the density at position
r
1
4
.
(3.15)
n(r) = π(~kF (r))3
3
(2π~)3
Combining equations 3.14 and 3.15 gives the density distribution for V (r) < EF yields
i3/2
1 h 2m
n(r, T = 0) = 2 2 EF − V (r)
.
(3.16)
6π ~
For V (r) > EF the density distribution vanishes. From this we also obtain the maximum
extent of the fermionic cloud RFi in direction i = x, y, z by setting V (RFi ) = EF
s
r
1/6 ω
2kB TF
~
i
i
RF =
or
RF =
48N
.
2
mωi
mω
ωi
The density profile (equation 3.16) then reads
h
x 2 y 2 z 2 i3/2
8N
n(r, T = 0) = 2 x y z 1 −
−
.
−
π RF RF RF
RFx
RFy
RFz
(3.17)
(3.18)
25
3.3. PAIRING AND SUPERFLUIDITY
The momentum distribution ñ(p) can be calculated similarly using the equivalent equations yielding
ñ(p) =
8N (p/~)2 1
−
.
π 2 kF3
kF2
(3.19)
The density and momentum distribution presented here are only exact for T=0. The
expressions for finite temperature can be found in [Geh03], for example.
3.3
Pairing and superfluidity
In 1995, the observation of Bose-Einstein condensation and hence superfluidity of bosonic
atoms was an amazing breakthrough in cold matter physics [And95, Dav95, Bra95]. Due
to the Fermi exclusion principle this phase transition is not possible for fermions. However,
pairing of fermionic atoms can lead to bosonic systems for which such a superfluid phase
transition is possible. Close to a Feshbach resonance, in particular, pairing may lead to
molecules for repulsive attraction and long range Cooper pairs for attractive interaction.
The region between both regimes is called the BEC-BCS crossover. In the following, we
shall discuss all three regions while special emphasis is laid on molecular Bose-Einstein
condensation as this is most relevant for our experiments.
3.3.1
Bose-Einstein condensation
As discussed in section 3.1 each quantum state can at most be occupied once in a gas of
indistinguishable fermions. In contrast, the bosonic mean particle distribution f BE (ǫν ) can
be greater than one. In particular, the population of the ground state ǫ = 0 can become
arbitrarily large for µ = 0 as T → 0. For a temperature below a critical temperature TC
the normalisation condition
N=
Z
g(ǫ) f BE(ǫ) dǫ
(3.20)
can only be fulfilled for an unphysical positive chemical potential. To fulfill the normalisation condition below the critical temperature a macroscopic fraction of bosons occupy the
ground state. The number of thermal particles NT is given by the normalisation condition
and the population of the ground state is thus N BEC = N − NT .
In a harmonic trap the critical temperature and condensate fraction for a non-interacting
26
CHAPTER 3. QUANTUM DEGENERATE FERMI GASES
Bose gas are given by
TC = 0.94~ωN 1/3 /kB
T 3
N BEC
= 1−
.
N
TC
(3.21)
(3.22)
All condensed particles occupy the ground state of the harmonic oscillator with wave
function φ0 (r). Thus, the density distribution of the condensate is given by nBEC (r) =
N BEC |φ0 (r)|2 . The geometric average of the width of the wave function φ0 (r)
aho =
~ 1/2
mω
(3.23)
is much narrower than the corresponding width of the thermal cloud. From this it follows
that the density distribution of the condensed part can easily be distinguished from the
thermal cloud by its narrow peak.
Interacting condensates
For non-interacting Bose-Einstein condensates the density distribution n(r) is proportional
to the single-particle density distribution. This is no longer true for interacting condensates. In principle, the many-body wave function of a Bose-Einstein condensate can be
found exactly by solving the many-body Schrödinger equation. However, this is rarely
practical as it requires enormous computer power even for fairly low particle numbers. Instead, interactions are often treated as a mean field potential g |n(r, t)|, where the coupling
constant g is given by g = 4π~2 a/m. This leads to a non-linear equation for the order
parameter (or condensate wave function) Ψ(r, t) [Pit61, Gro61]
i~
~2 2
∂
Ψ(r, t) = −
∇ + Vext (r) + g |Ψ(r, t)|2 Ψ(r, t)
∂t
2m
(3.24)
from which the density distribution can be calculated by n(r, t) = |Ψ(r, t)|2 .
This so-called Gross-Pitaevskii equation resembles a non-linear Schrödinger equation
and describes the macroscopic structure of the condensate for N ≫ 1 and T = 0. A
condensate in equilibrium is described by the time-independent Gross-Pitaevskii equation
~2 2
2
−
∇ + Vext (r) + g |Ψ(r)| Ψ(r) = µΨ(r),
2m
(3.25)
27
3.3. PAIRING AND SUPERFLUIDITY
where µ is the chemical potential. If the mean field energy dominates, which is often the
case, the kinetic energy term can be neglected . In this Thomas-Fermi approximation the
density distribution is given analytically by
1
n(r) = (µ − Vext (r)).
g
The chemical potential µ is determined by the normalising condition
yields
(3.26)
R
dr n(r) = N and
~ω 15Na 2/5
µ=
.
(3.27)
2
aho
The shape of the density distribution is an inverted parabola where the density goes to
zero at the Thomas Fermi radius
Ri = aho
15Na 1/5 ω
aho
ωi
.
(3.28)
We see that the shape of a Bose-Einstein condensate is very similar to the Fermi profile
(equation 3.16 and 3.17) with the main difference being the dependence of the radii on the
particle number N. In the case of identical fermions interactions can be neglected because
the quantum pressure term dominates.
The finite temperature generalisation is straightforward for non-interacting clouds as
the thermal and condensed parts can be treated separately. In interacting clouds the two
components are coupled by the interaction term. However, the coupling between both
components can often be ignored as the density of the thermal cloud is usually much lower
than the density of the condensed component. Hence, in this limit the condensed cloud
can be described by the T = 0 solution where the particle number N is replaced by the
number of condensed particles N0 .
In a cloud with repulsive interaction (a > 0) the peak density is lowered compared to a
non-interacting cloud (see equation 3.26). The reduced peak density leads to a reduction
of the critical temperature TC . The shift compared to the critical temperature of a noninteracting gas TC0 has been calculated to be [Gio96]
TC − TC0
a
.
= −1.32N 1/6
0
TC
aho
(3.29)
Below TC the number of condensed particles is also reduced due to repulsive interactions. Perturbatively, the condensate fraction has been found to be [Nar98]
T 3
T 3 2/5
N
ζ(2) T 2 =1−
1−
−η
N0
TC0
ζ(3) TC0
TC0
(3.30)
28
CHAPTER 3. QUANTUM DEGENERATE FERMI GASES
where ζ(2)/ζ(3) ≈ 1.37 and
η=
2/5
µ(T = 0)
1/6 a
≈
1.57
N
kB TC0
aho
(3.31)
is a scaling parameter.
3.3.2
The BCS regime and the BEC-BCS crossover
For completeness of this dissertation, we briefly introduce the BCS state and the BEC-BCS
crossover. Although our system allows to explore the crossover we have focussed on other
experiments during the time of my studies.
3.3.2.1
The BCS regime
A fermionic gas with weakly attractive interactions kF |a| ≪ 1 can undergo a phase transition into a superfluid phase at low enough temperatures. This is accompanied by the
formation of long range Cooper pairs [Coo56] which consist of two attractive fermions in
two different spin states which are correlated in momentum space rather than in real space.
In contrast to the formation of molecules, Cooper pairing is a many-body phenomenon relying on the presence of the “Fermi sea”. The size of the pairs is typically much larger
than the mean particle spacing n−1/3 , so that the pairs overlap [Tin66, Hou97].
In metals, the macroscopic formation of Cooper pairs of two electrons with opposite
spin gives rise to superconductivity. In 1957, this phenomenon led to the development of
a novel theory by Bardeen, Cooper and Schrieffer (BCS) [Bar57]. An introduction to the
theory of superconductivity is given in [Tin66].
Of particular interest is the critical temperature at which the phase transition occurs.
In terms of the Fermi temperature this is given by [Gor61]
kB TC ≈ 0.28EF exp −
π
2kF |a|
.
(3.32)
In the BCS limit the critical temperature is well below the Fermi temperature and already
for kF a = −0.2 we have TC = 10−4 TF which is experimentally unfeasible with current
set-ups using cold atoms.
The spatial density profile can be approximated in the Thomas-Fermi limit for kF |a| ≪ 1
(BCS limit). The energy of atoms in state |↑i shifts due to interactions by gn↓ where
29
3.3. PAIRING AND SUPERFLUIDITY
g = 4π~2 a/m is the coupling constant and n↓ is the density of atoms in state |↓i. The
density distribution (equation 3.16) then reads
i3/2
1 h 2m
.
n↑ (r) = 2 2 (EF↑ − V (r) − gn↓ )
6π ~
(3.33)
This can be greatly simplified assuming an equal spin mixture (n(↑) = n(↓) = n). The
attractive interaction in the BCS regime thus compresses the cloud. To first order perturbation theory, the Fermi radius reduces to [Gio07] (note a < 0)
RFint = RF0 1 +
256 0 k a
315π 2 F
(3.34)
where RF0 and kF0 are the Fermi radius and Fermi wave vector of the non-interacting gas,
respectively.
3.3.2.2
BEC-BCS crossover and unitarity
We have introduced molecular Bose-Einstein condensation and the Bardeen-CooperSchrieffer state of long-range Cooper pairs. These two form the two extremes of the BECBCS crossover which has recently been studied in several laboratories utilising Feshbach
resonances [Reg04b, Zwi04, Chi04a, Bar04b, Bou04, Kin04a, Par05]. The crossover is
characterised by a smooth transition from bosonic to fermionic behaviour of the gas. A
theoretical treatment of the underlying physics can be found, for example, in [Che05a].
Close to the Feshbach resonance the scattering length and hence the size of the weakly
bound Feshbach molecule can exceed the interparticle separation. In this regime the physics
becomes universal, depending only on the interparticle separation n1/3 and the Fermi energy
EF . In the unitarity limit (kF |a| → ∞) the scattering cross section σ0 (k) = 4π/k 2 (see
equation 2.15) is independent of the scattering length. Thus, the mean-field shift no longer
depends on the scattering length and is instead proportional to the Fermi energy
g = βEF .
(3.35)
The dimensionless proportionality factor β is universal to all Fermi gases at unitarity.
It has been measured in fermionic gases of both lithium [Tho05a, Par06a, Tar07] and
potassium [Ste06] yielding β ∼ −0.58. The shape of the trapped gas becomes equivalent
to the one of a non-interacting system. However, due to interactions the Fermi radius
reduces to [Gio07]
RFunitarity = (1 + β)0.25 RF0 .
(3.36)
30
CHAPTER 3. QUANTUM DEGENERATE FERMI GASES
3.3.3
p-wave superfluids
In section 2.4 we discussed the formation of p-wave molecules. Being composite bosons a
gas of p-wave molecules can, at least in principle, form a Bose-Einstein condensate. To
date such a superfluid has not been realised and remains one of the big challenges in
the field of cold atom physics. There are many properties that make p-wave condensates
an important object to study. Such gases display a BEC to BCS superfluid crossover
and a complex phase diagram with phase transitions between different projections of the
angular momentum [Bot05, Che05b, Gur05, Isk06]. These condensates may also give a
link with other pairing phenomena such as d-wave high-TC superconductors [Tsu00] and
liquid 3 He [Lee97]. Higher-order partial wave Feshbach resonances are intrinsically narrow,
because the interacting atoms have to tunnel through the centrifugal barrier before they can
interact with each other. This narrowness allows arbitrarily exact theoretical calculations
in the low collision energy limit [Gur07].
It is also interesting to consider what implications the molecular size (section 2.4) has for
the realisation of a BEC-BCS crossover regime for nonzero orbital angular momentum pairing. The crossover regime for s-wave pairs is comparatively smooth because the molecular
size grows appreciably as the detuning approaches zero from below. However, for p-wave
pairs the crossover will be much more abrupt as the molecular size barely changes at the
resonance. Additionally, on the BCS side of a p-wave resonance, the pair wavefunction may
have significant amplitude at short range because of the centrifugal barrier. We also expect
an increase in the rate of inelastic vibrational quenching collisions between molecules and
free atoms which release large amounts of energy and lead to rapid loss. Fermionic particle
statistics greatly suppresses this process for weakly bound s-wave molecules comprised of
two fermions [Pet04]. However, near a p-wave Feshbach resonance, fermions in the same
spin state interact resonantly so this suppression mechanism, and the considerations leading to it, will not apply. In fact, it has recently been shown that p-wave molecule-atom
collisions are expected to be largely insensitive to the two-body p-wave elastic cross-section
which can lead to short lifetimes of an atom-molecule mixture [D’I08].
Chapter 4
Experimental set-up
This chapter deals with the basic components of our experimental set-up. It describes the
vacuum apparatus in which the science takes place, the laser system used to laser cool and
trap 6 Li atoms, the absorption imaging set-up, the Feshbach coils which allow tuning the
atomic interactions and briefly how the experimental control is realised.
4.1
Vacuum
Nearly all our experiments are performed in an ultra-high vacuum environment which is
shown in figure 4.1. The vacuum apparatus can be divided into the oven, the oven pumping
chamber, the Zeeman slower region, the glass cell in which the science takes place and the
main pumping chamber. The total length of our vacuum system is ∼ 1.60 m and is mounted
at a centre height of 23.8 cm above the optical table. The vacuum chamber is supported by
1.5” outer diameter (OD) posts from Thorlabs. The posts are attached to vacuum flanges
via home made aluminium rings cut into two pieces.
4.1.1
Oven and oven pumping chamber
Since the vapour pressure of lithium at room temperature is negligible we heat up a chunk
of lithium metal in an oven from which an atomic beam originates. The oven consists of a
standard 2 34 ” steel tube. It is located at one end of the vacuum system and connected to
the oven pumping chamber via a 10 cm long nozzle with an inner diameter of 4.5 mm. The
nozzle provides the required pressure difference between the oven and the oven pumping
31
32
CHAPTER 4. EXPERIMENTAL SET-UP
Figure 4.1: This photo shows the vacuum apparatus explained in this chapter. The photo
was taken when we characterised the Zeeman slower in a test set-up (see section 6.2,
particularly figure 6.1). Before the titanium sublimation pump was attached to the oven
chamber a window was placed on top of the six-way cross allowing us to observe the atomic
beam.
chamber and collimates the atomic beam. Nickel gaskets are used to seal the oven as nickel
reacts less with lithium than copper.
The oven is filled with approximately 1 g of 95% enriched lithium-61 . Typically, the
oven is heated to a temperature of approximately 420◦ C, well above the lithium melting
temperature of 181◦ C. In order to recirculate some of the lithium a fine mesh2 is placed
inside the nozzle. A temperature gradient along the nozzle is designed to enforce condensed
1
2
Sigma Aldrich: #340421
Goodfellow: Stainless Steel 316, Wire diameter 0.066 mm, Nominal Aperture 0.103 mm
33
4.1. VACUUM
lithium on the mesh to flow back into the oven. The oven and nozzle are heated by a
thermocoax cable3 which is wound around both oven and nozzle. In order to insulate the
oven from the surrounding air it is embedded in two cement blocks. This reduces the
electric power consumption required for heating the oven to ∼130 W.
When running the oven at T = 420◦ C the vapour pressure in the oven is approxi-
mately [Lid05]
pLi = 108.061−8310/T [K] mbar = 1.2 · 10−4 mbar.
(4.1)
This corresponds to a number density n and a mean free path λ of [Vál77]
n=
p
atoms
= 1.2 · 1012
kB T
cm3
and
λ= √
1
= 7.6 m,
2nσ
(4.2)
where σ = 7.6 · 10−16 cm2 is the collisional cross section for lithium [Vál77]. The mean free
path is longer than the dimensions of the oven and collisions of atoms with each other are
thus irrelevant. The velocity distribution for atoms (or molecules) with mass m in thermal
equilibrium, e.g., in the oven, is given by the Maxwell-Boltzmann distribution
f (v) = 4π
−mv 2 m 3/2 2
.
v exp
2πkB T
2kB T
(4.3)
The average velocity of atoms in our oven at a temperature of 420◦C can thus be calculated
r
8kB T
m
v=
∼ 1560 .
(4.4)
πm
s
The number of atoms leaving the oven through the nozzle with area A per second is [Sco87]
1
Ṅ = Anv ∼ 5 · 1015 /s.
4
(4.5)
From this we can calculate an estimated lifetime of our oven giving approximately 250
days. However, due to the mesh we expect a somewhat longer lifetime of the oven.
The nozzle is attached to the six-way cross of the oven pumping chamber. The pumps in
this chamber consist of a 50 l/s Varian ion pump4 , a titanium sublimation pump (TSP)5 and
a turbomolecular pump which are all mounted to the six-way cross. The turbomolecular
pump, which is used for initial pumping when the two other pumps are inefficient, is
3
Thermocoax: SEI 20/200
Varian: RVA-VIP-50 l/s
5
Vacom: 360043
4
34
CHAPTER 4. EXPERIMENTAL SET-UP
connected to the system through an angle stainless steel valve. Furthermore, an edgewelded bellow is placed between the valve and the turbo pump to avoid any stress in the
glass cell when removing the turbomolecular pump. The titanium sublimation pump is
mounted on a 2 34 ” OD nozzle. To activate the titanium sublimation pump one heats a
filament of titanium by passing an electric current through it. This coats a thin layer of
titanium onto the surrounding walls which then acts as an efficient surface getter binding
most chemically active gases. The average pumping speed for walls coated with titanium
is approximately 5 ls−1 cm−2 for air. The surface coated with titanium in our experiment is
approximately 300 cm2 yielding a pumping speed of 1500 l/s which is, however, effectively
somewhat reduced by the conductance between the pump and the chamber. The pumping
speed for lithium vapour is unknown but it is supposedly on the same order of magnitude.
The use of the ion pump is necessary, particularly to pump chemically non-reactive gases,
such as noble gases, which are not absorbed by the titanium.
An in-house made mechanical shutter to block the atomic beam when needed is attached
to the six-way cross from underneath. It consists of a pivotable ∼10 cm long rod with a
diameter of 6 mm. The rod is attached to a blank flange which is connected to a bellow.
By closing a 6” long, 2 34 ” OD all-metal valve6 the oven pumping chamber can be isolated
from the main vacuum chamber.
Atomic beam
For atoms in an atomic beam the velocity distribution is somewhat modified from the
one in the oven which was discussed in equation 4.3. It takes the form [Ram89]
r −mv 2 v 3 m 2
.
exp
fab (v) =
2 kB T
2kB T
From this it follows that the lithium atomic beam has an average velocity of
r
9π kB T
m
∼ 1840
v ab =
8 m
s
◦
at the typical oven temperature of 420 C.
(4.6)
(4.7)
We are interested in the flux of atoms entering the Zeeman slower region per second Φ.
This can be calculated from [Hua87]
1
A′
v ab ,
Φ = nA ′
4
A + πd2
6
MDC: MIV-150-V
(4.8)
35
4.1. VACUUM
where n is the atom density in the oven, A and A′ are the areas of the oven aperture and
the Zeeman slower aperture, respectively and d is the distance between the two apertures.
The parameters of our system are T =420◦C, A = π · 1.752 mm2 , A′ = π · 22 mm2 and
d=45 cm. This yields a flux of
atoms
.
(4.9)
s
Here we used an inner diameter of the oven nozzle of only 3.5 mm because of the mesh
Φ ∼ 1 · 1011
which is placed inside the nozzle. Note that this is also the flux of atoms into the glass cell
(when the Zeeman light is turned off).
Without any Zeeman slowing present the atomic beam has an angular divergence of
approximately 8 mrad and a diameter of the atomic beam at the centre of the trapping
region of approximately 7 mm.
4.1.2
Main vacuum chamber
Experiments on ultracold atoms require an ultra-high vacuum (UHV) to minimise collisions
of cold atoms with the background gas. The pressure difference between the main chamber
and the oven pumping chamber is maintained in the Zeeman slower region. The Zeeman
slower coil is wound on a 30 cm long stainless steel tube and will be described in more
detail in section 6.2. The inner diameter of the tube increases in four sections from the
oven end to the main chamber end. The first two sections are each 10 cm long with an
inner diameter of 4 mm and 6 mm, respectively. In the next two 5 cm long sections the
inner diameter increases to 8 mm and 10 mm, respectively.
The Zeeman slower coil is directly connected to the glass cell which was custom-made
by Hellma. Both the glass cell and the Zeeman slower have 2 34 ” OD flanges on either
side. The centre part of the glass cell is 12 cm long with a square cross section and outer
dimensions of 3×3 cm2 . The glass cell is made from Vycor quartz7 . The glass cell is antireflection coated for both 1030 nm and 670 nm light on the outside surface. 10 cm long
glass-metal transitions on either side smoothly match the thermal expansion coefficients
from quartz to steel.
The glass cell is very sensitive to mechanical stress. Hence, at least one end of the
glass cell has to be mounted flexibly to allow some relaxation. In our set-up the glass
cell is therefore attached to the 4.5” six-way cross of the pumping chamber via an 2 34 ”
7
Base Vycor 7913
36
CHAPTER 4. EXPERIMENTAL SET-UP
OD edge-welded bellow. The flange between the glass cell and the bellow is supported by
an in-house made spring loaded mount that minimises the stress on the glass cell due to
gravity.
The UHV in the glass cell is provided by the main pumping chamber. A non-evaporative
getter (NEG) pump8 is placed inside a 4.5” OD steel tube which sits on top of the 4.5”
six-way cross of the pumping chamber. Its pumping speed is 240 l/s for CO. Noble gas
pumping is provided by a 50 l/s ion pump9 which is also directly attached to the 4.5”
six-way cross. A turbomolecular pump which was used to pump out the system initially is
also connected to this pumping chamber. It is connected to the six-way cross through an
all-metal angle valve and an edge-welded bellow. The pressure in the system is measured
by an UHV cold cathode gauge which is mounted at the bottom of the six-way cross.
The Zeeman slowing light enters the chamber through a viewport. Since lithium is very
corrosive to glass a sapphire window is used. Furthermore, the window is continuously
heated to 80◦ C to minimise lithium deposition on it. Because the sapphire viewport is
birefringent the polarisation of the Zeeman light had to be adjusted before assembling the
oven.
The ultimate vacuum pressure equilibrates when the pumping rate equals the flux of
atoms into the chamber. Leakage through the CF flanges used in our set-up is negligible
when assembled carefully. Contaminating gases are mainly due to outgasing from the
chamber walls. Outgasing can be greatly enforced initially by increasing the temperature
of the walls of the complete vacuum system. As a consequence, the vacuum improves
when the apparatus is cooled down again. Baking the sensitive glass cell, however, requires
extra care. It was baked in a cage made of 6 long bars along the main axis of the cell.
A thermocoax cable was wound around the bars providing homogeneous heating. We
increased or decreased the temperature in the system only by approximately 5 K/hour to
a maximum temperature of ∼150◦ C. By controlling the temperature along the glass cell
using 6 thermocouples we minimised the temperature gradient along the glass cell. The
system was baked at the maximum temperature for six days. Before cooling down the
non-evaporative getter and the titanium sublimation pump were degassed. Therefore, a
current of 50 A was applied for 30 s in each filament of the TSP and a current of 3.5 A for
45 min in the NEG. The titanium sublimation pump was activated again just before the
8
9
SAES getters: GP 100 MK5
Varian: RVA-VIP-50 l/s
37
4.1. VACUUM
system cooled down to room temperature. During the bake out the valve between the oven
pumping chamber and the main chamber was closed to reduce the risk of losing vacuum
in case of a turbomolecular pump failure.
Differential pumping
As discussed above, the lithium vapour pressure in the oven at 420◦ C is expected to be
∼ 1.2 · 10−4 mbar which is many orders of magnitude greater than the ultra-high vacuum
required in the science chamber. Differential pumping maintains the vacuum pressure difference between the different regions of our set-up. It is attained by a low conductance
between the chambers and high pumping speed. In our experiment, differential pumping
is provided by the oven nozzle and the Zeeman slower. In this section we estimate relevant
parameters for our vacuum system. The equations used here hold in the so-called molecular flow regime where interactions between particles in the gas are negligible due to the
long mean free path length.
First we consider a long round tube of length l and diameter d. The conductance Ctube
of this tube is given by [O’H89]
πvth d3
,
(4.10)
12l
is the average thermal velocity. The tube divides two vacuum chambers with
Ctube =
where vth
pressure p1 and p2 . If one vacuum chamber is pumped by a vacuum pump of speed Cpump
the resulting pressure in this chamber is
p2 =
Ctube
p1 .
Ctube + Cpump
(4.11)
Assuming an average thermal velocity of the lithium-6 atoms of 1560 m/s yields a conductance of the oven nozzle of ∼0.18 l/s. The vapour pressure in the oven of approximately
1.2 · 10−4 mbar and the pumping speed of the ion pump and TSP in the oven pumping
chamber result in a pressure of ∼ 10−8 mbar in the oven pumping chamber.
The differential pumping stage of the Zeeman slower consists of four sections with
different diameters and conductances Ci . This yields an overall conductance Ctotal of
1
Ctotal
=
1
1
1
1
+
+
+
C1 C2 C3 C4
=⇒
Ctotal = 0.12 l/s.
(4.12)
The 240 l/s non-evaporative getter pump and the 50 l/s ion pump in the main pumping
38
10.053 GHz
D2 670.977 nm
F′ =3/2
D1 670.979 nm
22 P1/2
F′ =5/2
22 S1/2
F′ =1/2
F=3/2
F=1/2
26.1 MHz
22 P3/2
228.2 MHz
F′ =1/2
F′ =3/2
4.6 MHz
CHAPTER 4. EXPERIMENTAL SET-UP
Figure 4.2: A schematic energy level diagram for 6 Li.
chamber reduce the pressure by a factor of ∼ 1000. This gives a vacuum pressure in the
glass cell end of ∼ 10−11 mbar which is consistent with measured values.
4.2
Laser system for 671 nm
In this section we describe the laser system used to laser cool 6 Li and for absorption
imaging. A schematic energy level diagram for 6 Li can be seen in figure 4.2. The laser
wavelengths required are approximately 671 nm.
4.2.1
Saturation spectroscopy
Saturation spectroscopy in a lithium vapour cell (figure 4.3) provides sub-Doppler reference
resonances for laser frequency stabilisation. An external cavity diode laser (ECDL) (Toptica DL 100) is used as the source of the resonant optical field which has an output power
of 15 mW and a linewidth of approximately 1 MHz. Using the so-called feed-forward technique, where both the diode current and the grating position are varied, a fine frequency
mode hop free tuning range of over 20 GHz can be obtained. The vapour cell consists of a
stainless steel crossed tube configuration with a flexible bellow construction and viewports
39
4.2. LASER SYSTEM FOR 671 NM
isolator
valve
PD
λ/4
ECDL
thermocoax
6 Li
vapour cell
Figure 4.3: Experimental set-up for saturation spectroscopy in a vapour cell.
on each end of the 30 cm horizontal arm. Using a thermocoax element the centre part of
the vapour cell is heated to 350 ◦ C yielding a maximum unsaturated absorption of about
20%. At this temperature the Doppler width for the lithium 671 nm line is 3.5 GHz.
Lithium vapour is limited by its mean free path to the centre part of the tube, protecting
the viewports from becoming coated with lithium. To further reduce this problem we heat
the viewports to 120◦ C.
In our saturation spectroscopy set-up (figure 4.3) the counter-propagating retroreflected
laser beam, the probe, is focussed onto a photodiode. Typically, the incident laser power
is 1.5 mW and has a beam diameter of 2 mm. Figure 4.4 shows typical Doppler-free
saturation spectra of 6 Li. A frequency scan over both the D1 and D2 lines is shown in
figure 4.4 (a) whereas (b) and (c) represent a close-up of the D1 and D2 lines, respec′
tively. The outer two peaks in (b) correspond to the D1 |F = 3/2i → |F = 1/2, 3/2i
′
and |F = 1/2i → |F = 1/2, 3/2i transitions whereas the outer peaks in (c) correspond
′
′
to the D2 |F = 3/2i → |F = 5/2, 3/2, 1/2i and |F = 1/2i → |F = 3/2, 1/2i transitions,
respectively. The reduction in absorption of the probe at these outer peaks is due to the
saturation of the transition by the pump beam. The enhancement in absorption of the
strong crossover peak is due to compensation of optical hyperfine pumping, which occurs
when the pump and probe laser are resonant with both ground state sublevels at the same
time. This crossover resonance is strong if the hyperfine splitting of the ground state is
smaller than the Doppler width [Vel80] and does not exist for other alkali atoms, such as
rubidium or cesium, where the Doppler width is typically much smaller than the ground
state hyperfine splitting. In figure 4.4 (b) the incident optical power was reduced to 150 µW
to increase the resolution. Although it was not possible to resolve the hyperfine splitting
of the |22 P1/2 i state, we could partly resolve the crossover between the transitions from
40
CHAPTER 4. EXPERIMENTAL SET-UP
Transmission (arbitrary units)
(a)
2 GHz
Frequency
(c)
Transmission (arbitrary units)
Transmission (arbitrary units)
(b)
100 MHz
100 MHz
Frequency
Frequency
Figure 4.4: Doppler-free spectrum of 6 Li. (a) Frequency scan over both the D1 and D2
lines. (b) and (c) A close-up of the scan across the D1 and D2 line respectively. The
′
outer two peaks in (b) correspond to the |F = 3/2i → |F = 1/2, 3/2i and |F = 1/2i →
′
|F = 1/2, 3/2i transitions, whereas the outer peaks in (c) correspond to the |F = 3/2i →
′
′
|F = 5/2, 3/2, 1/2i and |F = 1/2i → |F = 3/2, 1/2i transitions. The hyperfine structure
of the |22 P3/2 i state is unresolved.
4.2. LASER SYSTEM FOR 671 NM
41
the |F = 1/2i and |F = 3/2i ground states (figure 4.4 (b)).
Saturation spectroscopy of lithium vapour provides an excellent frequency reference
required for our experiments. To frequency stabilise the laser we modulate the laser frequency by dithering the frequency of an acousto-optical modulator (AOM) which is passed
in a double-pass configuration before the light enters the spectroscopy cell. Using a lock-in
amplifier the derivative of the absorption profile is obtained. In most experiments the
ECDL is locked to the crossover dip. To achieve a stable lock, we require a large crossover
amplitude. This is obtained by applying relatively high laser intensities in our saturation
spectroscopy set-up which ultimately broadens the width of the crossover dip. When applying these higher intensities the splitting of the crossover peak in figure 4.4 (b) is no
longer observed and resembles the spectroscopy of the D2 line (figure 4.4 (c)). The width
of the crossover peak in figure 4.4 (c) is ≈ 25 MHz. The ratio of the crossover peak to the
Doppler broadened resonance is ≈ 100%.
4.2.2
Laser system
This section describes the laser light generation required for the magneto-optical trap
(MOT) and absorption imaging. Other aspects of laser cooling and imaging will be explained in chapter 6 and section 4.3, respectively. A schematic diagram of the optical
set-up used for the MOT and absorption imaging is shown in figure 4.5. The laser set-up
was similar for the experiments on sub-natural resonances presented in chapter 5; however,
the frequencies and orders of some AOMs were changed at times to achieve the required
laser frequencies.
We frequency stabilise the ECDL approximately 200 MHz red-detuned from the D2
crossover dip. This is realised by sending the light first through an acousto-optical modulator in a double-pass configuration before it enters the spectroscopy cell. As the power
of the laser light from the ECDL is not sufficient for running a magneto-optical trap we
amplify the power in a master-slave configuration. Therefore, the light from the ECDL
injection locks another diode slave laser labelled S1 which in turn injection locks four more
slave lasers SMOT , SRP , SZS and SIm . The SMOT and SRP provide MOT laser light for trapping and repumping, the SZS is used for Zeeman slowing and SIm for absorption imaging.
To monitor the frequency of each slave laser we send a fraction of the laser light into a
Fabry-Perot spectrum analyser. Figure 4.6 shows a typical scan of the spectrum analyser
42
+110 MHz
CHAPTER 4. EXPERIMENTAL SET-UP
diode laser
DL 100
PD
vapour cell
SIm
S1
+110 MHz
low/high field imaging
−450 MHz
SZS
SMOT
+80 MHz
+80 MHz
SRP
Spectrum analyser
+80 MHz
to MOT
to Imaging
to ZeemanSlower
Figure 4.5: Schematic diagram showing the optics for the MOT. The set-up consists of a
frequency stabilised ECDL and five additional slave diode lasers used for amplification (S1 ),
repumping (SRP ) and trapping (SMOT ) the MOT, Zeeman slowing (SZS ) and absorption
imaging (SIm ). The frequency modes of the slave lasers are monitored on a spectrum
analyser. Depending on whether SIm is seeded by S1 or SZS the imaging light is resonant
with atoms at either zero magnetic field or near the 834 G Feshbach resonance. All beams
can be shut by commercial shutters.
43
4.2. LASER SYSTEM FOR 671 NM
228 MHz
Voltage [a.u.]
820 MHz
Frequency
Figure 4.6: In this figure we present a typical scan of the spectrum analyser showing the
Zeeman slower laser SZS , the main slave laser S1 , the trapping and repumping laser SMOT
and SRP , respectively (from left to right) . The sidebands of the Zeeman slower light are
due to the current modulation applied.
for the Zeeman slower laser, the main slave laser S1 , the trapping and repumping laser.
The set-up for the trapping and repumping light is very similar. We require a 228 MHz
frequency offset between both beams to match the ground state hyperfine splitting. We
achieve this by sending the injecting light for the repump slave laser through a double-pass
AOM which is operated at a frequency of half the hyperfine splitting. The amplified laser
light of both beams is subsequently blue-shifted by single-pass AOMs at 83 MHz. The
two AOMs also provide the possibility of controlling the intensities of both beams which
we make use of when compressing and cooling the atoms in the MOT. Both trapping
and repumping lasers are overlapped on a non-polarising beam splitter. The two outputs
are used for the vertical and the two horizontal MOT beams, respectively. Both arms
are expanded to a diameter of approximately 15 mm for which we use a 40/300 mm
and a 35/200 mm lens configuration, respectively. The laser intensities of the trapping
44
CHAPTER 4. EXPERIMENTAL SET-UP
and repump lasers are ∼5 mW/cm2 each which corresponds to approximately twice the
saturation intensity. The three counter-propagating beams required for the MOT are
simply obtained by retro-reflecting the incoming beam.
A detuning of about 1 GHz with respect to the MOT transition is required for Zeeman
slowing. In our experiment this is realised by double passing a part of the light from
S1 through an AOM at 410 MHz before injection-locking SZS . We have not seen any
improvement of the Zeeman slower characteristics by adding repump light detuned by
the hyperfine splitting of 228 MHz from the principal Zeeman slower light. However,
following the Innsbruck group [Joc04] we have found that modulating the current of the
Zeeman diode laser enhances the loading rate of the MOT significantly. The resulting
sidebands of our modulation at 120 MHz can be clearly seen in figure 4.6. The asymmetry
of the sidebands arise because the current modulation modulates both amplitude and
frequency of the Zeeman light simultaneously. The Zeeman light is then expanded and
slowly focussed using a 200 mm and 1000 mm lens. The diameter of the light at the
viewport is approximately 1” and it is focussed near the oven.
In the experiments presented in this thesis we have taken absorption images at different
values of the magnetic field such as 0 G, 159 G and in the range of 660-1100 G. In our setup we can quickly change between the required laser frequencies in the absorption imaging
beam. This is realised by having the possibility of injection locking the imaging slave laser
SIm by either S1 or SZS . A single pass AOM (and double pass AOMs, if necessary) shifts
the frequency to the required resonance. The light is then sent through a single mode,
polarisation maintaining fibre.
All slave laser diode mounts10 , temperature controllers11 and laser diode current drivers12
are commercial products purchased from Thorlabs. The laser diodes used for the Zeeman
slower laser and absorption imaging slave laser are 30 mW laser diodes with a centre wavelength of 675 nm available from Toptica. The three other diodes are manufactured by
Hitachi and have a specified maximum output power of 80 mW at 658 nm13 . To obtain
a stable injection lock at the required wavelength the temperature of these diode lasers is
increased to approximately 58 ◦ C. We typically run the diodes at the maximum specified
room temperature current of 130 mA yielding an output power of 45 mW. The laser current
10
Thorlabs: TCLDM9
Thorlabs: TED200
12
Thorlabs: LDC202 and LDC205
13
Hitachi: HL6526FM
11
45
4.3. ABSORPTION IMAGING
is chosen rather conservatively and no laser diode failed in three years.
4.3
Absorption imaging
Important experimental data can often be extracted from absorption images [Ket99]. In
this technique the atomic cloud partially absorbs a resonant laser beam. The shadow of
the cloud is subsequently imaged onto a CCD chip from which the column atomic density
R
n(x, y) = n(x, y, z) dz can be determined. To compensate for any inhomogeneities in the
laser profile we take a second image without any atoms approximately 400 ms after the
image of the atomic cloud. The intensities of both beams after passing the science cell
are I(x, y) and I0 (x, y), respectively. The ratio of the intensities is related to the atomic
density n(x, y, z) through
I(x, y)
= exp − σ
I0 (x, y)
Z
n(x, y, z) dz ,
(4.13)
where σ is the absorption cross section. For a two-level system the resonant cross section
is given by σ = 3λ2 /2π. In section 4.2.2 we discussed the laser light preparation for the
imaging beam including the possibility of imaging at various magnetic fields. The light
is then sent through an optical fibre to the experiment. Only one 60 mm lens is used to
collimate the laser beam to a diameter of approximately 20 mm. The cloud is imaged
onto the CCD chip using a pair of 2” achromatic lenses with a focal length of 75 mm and
150 mm, respectively, yielding a magnification of 2. The images are taken with a 12 bit
CCD camera from Q Imaging14 . The chip consists of 1392 × 1040 pixels, where each pixel
has a size of 6.45 µm both horizontally and vertically. Thus, one pixel corresponds to
3.225 µm. In this set-up we achieve a resolution of approximately 4 µm.
In the initial imaging set-up, the imaging laser light propagated horizontally and thus
perpendicular to the magnetic field provided by the Feshbach coils. In this case, the cross
section is reduced by a factor of two. Recently, we have changed the imaging optics allowing
us to image along the magnetic field. This has the advantage that the laser light can be
circularly polarised driving the closed |2S1/2 , m = −1/2i to |2P3/2 , m = −3/2i transition
(see figure 4.2). This improves the imaging contrast due to the larger absorption. More
important, though, is the gain in optical access because in the new set-up the imaging
14
Q Imaging: Retiga-Exi-F-M-12-IR
46
CHAPTER 4. EXPERIMENTAL SET-UP
imaging beam
CCD
camera
glass cell
λ/4
λ/4
MOT beam
f =75 mm
f =150 mm
Figure 4.7: Schematic diagram of the absorption imaging set-up.
lenses are mounted inside the Feshbach and MOT coils. The schematic diagram of the
absorption imaging set-up is shown in firgure 4.7. To be able to image along the magnetic
field it is necessary to overlap the beam paths of the imaging and vertical MOT beams.
This requires somewhat more optics in the imaging beam path which reduces the quality
of the images noticeably.
4.4
Feshbach coils
To tune the interaction of lithium-6 atoms we apply magnetic fields of up to 1.5 kG. Coils
with low inductance are desired as they allow fast switching times of the magnetic field.
Our coils thus consist of only a limited number of turns and a small diameter which in turn
requires high currents of up to 200 A. Such high currents need efficient cooling to avoid
any damage due to heating. Our coils are made of square hollow copper tubing which can
be wound more efficiently than round tubing. Pressurised water pumped through the hole
provides cooling. The copper tubing has outer dimensions of 1/8” with a 1/16” square
hole and is insulated with kapton tape15 .
Both coils have 51 turns each and are wound on a mount made from hard plastic. The
inner (outer) radius is 3.5 (5.5) cm with a thickness of 3 cm. The separation of both coils is
15
The wire was custom made from S & W Wire company.
47
4.4. FESHBACH COILS
3.5 cm. The coils are orientated horizontally producing a magnetic field along the direction
of gravity. The maximum current of 200 A yields a magnetic field of approximately 1500 G.
The inductance was measured to be 433(10) µH allowing relatively fast switching times.
The coils are slightly further apart than a perfect Helmholtz configuration. This has the
consequence that the second derivative of the magnetic field is finite. We measured and
calculated a curvature of 0.033(2)/cm2 multiplied by the magnetic field. Typically, we only
trap 6 Li atoms in the two lowest high-field seeking states |1i and |2i. For this state the
curvature provides a (anti-)trapping potential in the horizontal (vertical) plane.
The required flux of water (in ml/minute) which keeps the coils at constant temperature
when dissipating power P is given by
F = 14.3
P
,
∆TH2 O
(4.14)
where ∆TH2 O is the temperature change of the cooling water. Currently, the cooling water
is provided from a tap which provides a flux of 150 ml per minute through each coil when
the water flows in parallel through the coils. A temperature increase of 25 K limits the
average power dissipation to 250 W in each coil which is approximately one tenth of the
maximum power dissipation of P = 2 kW. Using an external water pump at a water
pressure of 12 bar we have obtained a flux of 650 ml per minute. However, we have not
found that this additional cooling is needed for the experiments performed so far since we
rarely apply high magnetic fields for longer than 5 s.
Magnetic field stabilisation
High magnetic field stability and low field noise are essential to probe narrow 6 Li p-wave
resonances. The current for the Feshbach coils is provided by a power supply with a
maximum output of 200 A at a voltage of 30 V16 . We control the current via a 0-5 V
analogue programmable input. We find that it is necessary to actively stabilise the current. We therefore measure the current in the coils by means of a current transducer17
which down-converts the current by a factor of 2000. To increase the sensitivity of the
current measurement the current carrying wire passes the transducer several times. We
thus measure an integer of the current flowing through the Feshbach coils. We convert the
16
17
Delta Elektronika BV: SM30-200
LEM IT 400 S, overall accuracy 0.0033%
48
CHAPTER 4. EXPERIMENTAL SET-UP
secondary current from the transducer into a voltage in the range from -10 V to +10 V.
The error signal is obtained by comparing this voltage to the set PC controlled voltage
output. A PI controller is used to adjust the current of the power supply accordingly. The
magnetic field switching time is on the order of 100 ms limited by the current fall time
of the power supply. We have connected a high power resistor with a resistance of 1/7 Ω
in series with the Feshbach coils18 . This significantly reduced the switching time of the
magnetic field and also improved the AC noise level.
For fine tuning and fast switching the magnetic field across the narrow Feshbach resonances we make use of a secondary pair of coils. The primary pair of Feshbach coils
produces a magnetic field of approximately 1 G below the corresponding Feshbach resonance and a secondary pair of coils produces magnetic fields of up to a few Gauss. The
current in the secondary coils is actively stabilised by means of a MOSFET-switch. Electronic feedback is provided by the voltage drop across a power wire-wound resistor which
is connected in series with the magnetic field coils. The electronic circuit consists of a
comparator and an integrator driving the MOSFET.
We achieve a magnetic field stability at the level of a few mG shot to shot and better
than 20 mG over the course of a few days.
4.5
Experimental control
Our experiments require precise control over many parameters such as laser powers and
frequencies, magnetic fields and camera trigger. We control all parameters which need to
be changed during the data run by means of control voltages. It is necessary to control
both output voltages and timing intervals precisely. For this task we use three National
Instruments PCI output boards housed in a standard personal computer. Each board has
eight analog outputs ranging from -10 V to +10 V. The high resolution 16 bit master
board19 triggers the two other 12 bit boards20 . An array of voltages is buffered for each
timing interval in the PC’s memory. Even though the boards allow a time resolution of up
to 1 µs we typically operate at a resolution of 10 µs due to limited memory. The boards
are controlled from a graphical user interface written in LabVIEW21 .
18
The resistor bank consists of seven 150 W power resistors in parallel.
National Instruments: PCI 6733
20
National Instruments: PCI 6713
21
National Instruments: LabVIEW 6
19
Chapter 5
Sub-natural Resonances
Along the way to producing ultracold atoms and molecules we performed spectroscopic
coherence experiments on the lithium atomic beam. We interpret sub-natural resonances
not only as electromagnetically induced transparency and electromagnetically induced absorption but also as coherent population oscillation. Furthermore, we studied Ramsey
spectroscopy in media of both electromagnetically induced transparency and absorption.
Most of the results presented in this chapter have been published in [Fuc06, Fuc07a].
5.1
5.1.1
Introduction
Electromagnetically induced transparency
When a resonant laser beam propagates through an atomic gas, the atoms will absorb
a substantial fraction of the photons. While this result is no surprise, the situation may
change dramatically when a second resonant laser beam is added to the system. Classically,
one would expect that at least the same number of photons are absorbed from the bichromatic laser beam. However, due to quantum interference the opposite can occur, namely
a decrease in absorption. When two Zeeman sublevels or two hyperfine levels in an atomic
ground state are coupled by light to a common excited state, the interference between amplitudes of alternative transition paths can substantially reduce the absorption. The atoms
are pumped by the laser light into a coherent superposition of the ground state sublevels.
The second light beam turns the opaque medium into a transparent one. This phenomenon
is commonly known as Electromagnetically Induced Transparency (EIT) [Har90, Har97].
49
50
CHAPTER 5. SUB-NATURAL RESONANCES
EIT is closely related to Coherent Population Trapping (CPT) which describes the phenomenon where a coherent superposition of atomic ground states exists that does not
couple to the laser radiation [Ari96].
By means of a multimode laser Alzetta et al. discovered in 1976 the effect of coherent
population trapping in sodium [Alz76]. This experiment showed a decrease of resonant
fluorescence when the multimode spacing equalled the hyperfine splitting. In 1989 Harris published a theoretical study on lasing without inversion [Har89]. This fundamental
work was followed by the first experimental observation of EIT in 1991 [Bol91]. This
breakthrough was realised in a strontium vapour by applying pulsed laser fields with a
wavelength of 570 nm and 337 nm. In subsequent experiments, EIT was realised employing CW lasers in a vapour of alkali atoms such as Rb [GB95] and Cs [Nag99]. These
alkali systems provide an ideal lambda level scheme by making use of the hyperfine ground
states. Although most EIT experiments are carried out in a gaseous medium EIT has
also been achieved in a solid medium. For example EIT was realised in ruby by means
of a microwave field [Zha97]. Reviews of experimental and theoretical work on EIT can
be found in [Mar98, Fle05]. EIT is of much interest due to its wide application in lasing
without inversion [Zib95], control of light propagation (slow light) [Mat01] and enhanced
Kerr nonlinearity [Har99, Aku03b]. Furthermore, the concept of non-absorbing dark states
is the key basis of light storage [Luk03].
To understand the quantum mechanical phenomenon of EIT in somewhat more detail
we consider a simple Λ scheme configurations as shown in figure 5.1 (a). The two bare
atomic states |1i and |2i are coupled by laser light to the common state |3i. It is of interest
to calculate the new atomic eigenstates in this bichromatic laser light field. The interaction
Hamiltonian for the atom-laser interaction in the dipole approximation takes the form
Hint = µ · E,
(5.1)
where µ is the transition electric dipole moment and E the electric field. It is common to
express the interaction in terms of the Rabi frequency Ω = µ · E0 /~ which is the frequency
at which stimulated absorption and stimulated emission cycles occur. Note that |E0 | is
the amplitude of the electric field. The atoms are placed in a bichromatic laser field with
Rabi frequencies Ω1 and Ω2 . In the following analysis it is assumed that the probe is on
resonance, e.g., ∆1 = ∆2 , where ∆i is the laser detuning to the relevant atomic transition
|ii → |3i. In this case the absolute frequency difference of the two lasers equals the
51
5.1. INTRODUCTION
(a)
|3i
∆1
(b)
∆2
|a+ i
|a− i
probe
|2i
|1i
|a0 i ∼
= |1i
Figure 5.1: (a) Basic lambda scheme for electromagnetically induced transparency. The
bare atomic states |1i and |2i are coupled by laser light to state |3i. (b) Atomic eigenstates
in the presence of a weak probe and strong pump field, resonant with the |1i → |3i and
|2i → |3i transition, respectively.
energy difference between states |1i and |2i divided by Planck’s constant h. Diagonalising
the Hamiltonian of the atom in the laser light field yields the new eigenstates which are
superpositions of the bare atomic states [Fle05]
1 Ω1
|1i + |3i +
|a+ i = √
2 Ωx
Ω2
Ω1
|a0 i =
|1i −
|2i
Ωx
Ωx
1 Ω1
|a− i = √
|1i − |3i +
2 Ωx
where Ωx =
Ω2
|2i
Ωx
Ω2
|2i ,
Ωx
(5.2)
(5.3)
(5.4)
p
Ω21 + Ω22 . It is remarkable that state |a0 i is independent of state |3i. This
has the consequence that there is no probability of finding this eigenstate in the upper
level |3i. Hence, the atoms are trapped in a non-absorbing state decoupled from the laser
field. From the analysis given here it is clear that |a0 i is the state involved in CPT. The
other two states |a+ i and |a− i, on the other hand, have an equal probability of being found
in the upper bare atomic state |3i or in one of the ground states. The physical picture
behind the dark state |a0 i is destructive interference which occurs between the two possible
ways that can result in an excitation of state |3i, e.g., |1i − |3i and |2i − |3i. Typically,
EIT experiments are carried out by using a weak probe beam and a stronger pump beam
(Ωprobe ≪ Ωpump ). The eigenstates for such a system are shown in figure 5.1 (b). In this
52
CHAPTER 5. SUB-NATURAL RESONANCES
case the dark state |a0 i essentially becomes the bare atomic state |1i, assuming that the
probe field is chosen to be tuned to the |1i − |3i transition.
5.1.2
Zeeman coherence
The coherence between magnetic sublevels belonging to the same ground hyperfine sublevel
may result not only in absorption reduction (Zeeman EIT), but also in absorption enhancement leading to Electromagnetically Induced Absorption (EIA) [Aku98, Lez99a]. According to [Aku98, Lez99a], a significant enhancement of absorption and, consequently, resonant
fluorescence due to EIA in an atomic medium driven by resonant laser radiation occurs at a
Raman resonance under the following conditions: (i) the optical transition is closed and (ii)
the degeneracy of the upper level is higher than that of the lower level, i.e., 0 < Fg < Fe .
Recently it has been shown that condition (i) is not very strict [Zib05]. EIA and Zeeman
EIT are complementary effects with many common features [Ren99, Val02, Aku03b, Val03].
Atomic coherence in both cases originates from two-photon Raman transitions between
ground-state magnetic sublevels. Consequently, the ultimate width of EIT and EIA absorption resonances is determined by the ground-state relaxation time, which is usually orders
of magnitude longer than the lifetime of the upper state. Thus the linewidth becomes subnatural which can be observed in the absorption signal of a probe laser when the resonance
condition is fullfilled, e.g., ∆ = 0. Coherent atomic media thus exhibit a giant nonlinearity
of the refractive index at very low light intensity [Har99, Bud02, Aku04]. Steep normal or
anomalous dispersion gives large variations in the group velocity of light, leading to slowlight or fast-light atomic media [Har92, Sch96, Aku03b, Boy02, Aku03a, Kim03, Gor03b].
Even the storage and retrieval of optical pulses in EIT and EIA media under the same
experimental conditions show remarkable similarities [Aku05].
A comprehensive numerical simulation of sub-natural EIT and EIA resonances obtained
in degenerate two-level atomic systems in the presence of a magnetic field was reported
in [Lez99a]. The transfer by spontaneous emission of the light-induced anisotropy, which
can be in the form of atomic coherence or population difference, is the physical origin of
EIA as suggested in [Tai99, Gor03a]. A detailed theoretical study of EIA magneto-optical
resonances is presented in [Bra05].
5.1. INTRODUCTION
5.1.3
53
Coherent population oscillation
Sub-natural absorption resonances in atomic media are normally associated with a ground
state coherence or, more precisely, with a long-lived light-induced anisotropy of the ground
state. However, the excitation of an open two-level atomic system by co-propagating drive
and probe waves with tunable frequency offset can result in a sub-natural resonance, as
shown by Baklanov and Chebotaev [Bak71, Let77]. The periodic modulation of the groundstate population at the beat frequency between the probe and pump fields and multi-photon
Raman scattering are responsible for this effect, known as Coherent Population Oscillation
(CPO). The effect was predicted for resonant interactions of an atomic vapour with laser
light and in the 1970s became the topic of a large body of research. It was studied in
connection with parameters of Doppler-free saturated absorption resonances, such as the
shape, contrast and width, for a probe wave in the presence of a strong counter-propagating
wave [Bak72, Har72]. Due to CPO an adequate description of coherent nonlinear processes
in strongly driven resonant systems and more precise evaluation of nonlinear susceptibility
became possible [Boy81, Boy84]. Recently CPO was employed to produce ‘slow light’ in
solid state media [Big03]. However, to the best of our knowledge, CPO is not conventionally
discussed in the context of sub-natural absorption resonances in atomic media. The reason
is that in a real atom with a complex structure of energy levels light-induced coherence
between ground-state sublevels can mask effects produced by a two-level CPO.
Here, we show that for a quasi-degenerate two-level system a variety of experimental
parameters allow simultaneous observation and separation of sub-natural resonances produced by ground-state Zeeman coherence and CPO. The most common way to produce
EIT for the D lines in a vapour of alkali atoms is to couple both ground-state hyperfine
sublevels to a common excited state by resonant laser light. However, we have found that
the separation of CPO and ground-state Zeeman coherence can be realised more clearly in
a degenerate two-level atomic system.
5.1.4
What is special about 6 Li?
Lithium has two stable isotopes, 6 Li and 7 Li (7.5 % and 92.5 % natural abundance, respectively). Magnus et al. [Mag05] reported the first study of EIT on the D1 and D2 lines of 7 Li
in a vapour. However, due to the unique level structure 6 Li offers new aspects for the study
of sub-natural resonances. Most importantly, EIT in alkali atoms with half-integer total
54
CHAPTER 5. SUB-NATURAL RESONANCES
angular momentum (F = J + I, J = L±1/2) have not, to our knowledge, been previously
investigated. A schematic diagram of the relevant atomic energy levels of 6 Li was shown in
figure 4.2. The level structure of 6 Li is somewhat different to other alkali atoms that have
been used to study EIT and other effects related to ground state coherence. There are
only three sublevels in the |22 P3/2 i state compared with four sublevels for all other alkali
|n2 P3/2 i states due to the 6 Li nuclear spin I = 1.
The strongest optical transitions of 6 Li are two electric dipole transitions |22 S1/2 i →
|22 P1/2 i (D1 line) and |22 S1/2 i → |22 P3/2 i (D2 line) which are separated by a fine structure
splitting of ∼ 10 GHz. Of all the alkali atoms 6 Li has the smallest hyperfine splitting
in both the ground and excited states. The hyperfine splitting of the |22S1/2 i ground
state, 228.2 MHz [Wal03], is much less than the Doppler width of the D lines near room
temperature. But, more interestingly, the hyperfine splitting of the |22 P3/2 i state is smaller
than the 5.9 MHz [McA96] natural width of the optical transition. Thus, every atom has
comparable probabilities, independent of the atomic velocity, of being excited via different
transitions on the D2 line. This unique situation for alkali atoms allows the analysis of
the influence of additional optical transitions, which do not contribute to the preparation
of a non-absorbing coherent state, responsible for EIT. This study could be useful for
optimisation of EIT resonances widely used for metrological applications [Kna05].
Furthermore, we show that both coherent effects, i.e., ground-state Zeeman coherence
and CPO, can produce EIT and EIA resonances in the fluorescence from a collimated
atomic beam of 6 Li. The hyperfine splitting of the excited state |2P3/2 i is smaller than the
natural width of the optical transition of 5.9 MHz. Thus, atoms can be simultaneously
excited by resonant light via transitions leading to ground-state coherences with opposite
absorbing properties at the same time, resulting in EIT or EIA.
To the best of our knowledge, EIA has been observed and studied only in Rb and
Cs atoms, which have very similar structure in their D lines [Ren99, Kim01]. Thus, an
extension of the range of optical transitions and conditions under which EIA might exist,
as well as the possibility to transform an EIT atomic medium into an EIA medium, is of
interest.
The ultimate width of sub-natural resonances also depends on the mutual coherence of
the probe and control laser fields [Ari96]. In the case of Cs and Rb, which have large ground
state splitting, several methods can be used to produce two phase-stable laser fields such
as high frequency acousto-optical modulators (AOM), two phase-locked lasers, applying
55
5.2. EXPERIMENTAL SET-UP
PMT
Laser
Frequency
offset
Polarisation
ν1 , ν2 optics
δ = ν1 − ν2
Frequency
stabilisation
B
magnetic
shield
6
Li
Figure 5.2: Simplified scheme of the experimental set-up.
current modulation to laser diodes and the use of electro-optic modulators [Wyn99]. High
frequency AOMs are expensive while phase locking at precise frequency offset and current
modulation at high frequencies are experimentally not simple. However, in the case of
lithium the ground state splitting is relatively small and mutually coherent probe and
control fields can be easily prepared from the same laser using a low-cost AOM in a double
pass configuration, thus eliminating laser linewidth contribution.
5.2
Experimental set-up
Figure 5.2 shows a simplified scheme of the experimental set-up which was used for the
experiments on sub-natural resonances. In initial experiments, EIT was obtained in a
vapour cell (see section 4.2.1) which was placed in the set-up instead of the atomic beam.
All subsequent experiments on hyperfine EIT, Zeeman EIT, EIA, CPO and Ramsey fringes
were performed in the atomic beam which was described in detail in section 4.1.1. In
brief, the atomic beam has a mean velocity of ∼ 1850 m/s, an angular divergence of
approximately 6 mrad and a beam diameter at the interaction region of approximately
7 mm1 . For the experiments in this chapter the glass cell was wrapped in a µ-metal foil
to reduce stray magnetic fields. A controllable magnetic field along the atomic beam is
produced by a solenoid.
In order to generate and analyse sub-natural coherent resonances we employed a spec1
Note that the vacuum apparatus employed for the EIT experiments was slightly different to the set-up
described in section 4.1 which resulted in a somewhat smaller beam divergence.
56
CHAPTER 5. SUB-NATURAL RESONANCES
tral method employing a bichromatic laser beam. In this approach a frequency offset
between the co-propagating probe and drive components is scanned to satisfy resonant
conditions for two-photon Raman transitions either between different hyperfine or different ground-state Zeeman sublevels [Aku03b]. We performed experiments on both D1 and
D2 lines. For the vapour cell experiments the master laser is not actively stabilised since
its long-term drift is small compared to the Doppler width. However, for the atomic beam
experiments the master laser is actively locked to the corresponding crossover of the saturated absorption lines. The probe laser frequency is scanned over ≈ 25 MHz which is
larger than the natural width of the optical transitions. The frequency of the pump laser is
kept fixed. Acousto-optical modulators were employed to produce mutually coherent drive
and probe components. The two beams are carefully combined into a bichromatic beam
on a non-polarising beam splitting cube. The beam diameter at the interaction region was
approximately 5 mm for the vapour cell and 18 mm for the atomic beam experiments.
The polarisations of the drive and probe components were independently adjusted with
polarisers, λ/4 and λ/2 waveplates. If not stated otherwise, the maximum intensity of
both components was approximately 1.5 mW/cm2 .
The collimated atomic beam is excited at right angles by resonant laser light in the
interaction region. The fluorescence of the excited atoms is detected using a photomultiplier
tube. The signal from the photomultiplier was averaged up to 128 times and stored on a
digital oscilloscope.
5.3
Hyperfine EIT
5.3.1
Vapour cell EIT
By tailoring the optical frequencies present we performed experiments on EIT in both
a vapour cell and an atomic beam. To obtain hyperfine ground state coherence we use
mutually coherent laser fields whose frequencies are separated by the 228 MHz hyperfine
splitting of the ground state. The pump and probe are sent through the vapour cell with
intensities of 8 mW/cm2 and 2 mW/cm2 respectively. To improve the signal to noise ratio
the pump is amplitude modulated by means of an mechanical chopper (1 kHz) and the
probe is detected using a lock-in amplifier. For the results shown here, both beams are
linearly polarised and orthogonal to each other.
57
5.3. HYPERFINE EIT
B=0
D1
(b)
(e)
(d)
Absorption (arbitrary units)
(c)
D2
~ ~k
B⊥
Absorption (arbitrary units)
(a)
215
220
225
230
235
Frequency difference [MHz]
240
215
220
225
230
235
Frequency difference [MHz]
240
Figure 5.3: Absorption of the D1 (a and b) and D2 (c and d) line as a function of frequency
difference between the pump and probe. (a) and (c): B = 0, (b) and (d) have a magnetic
field of ≈5 G perpendicular to the laser light. (e) shows the Raman transitions responsible
for coupling the ground state Zeeman sublevels for each of the resonances in (b and d). The
′
thicker lines represent the pump laser which is tuned to the |F = 1/2i → |F i transitions.
Results
Figure 5.3 shows absorption plots of the D1 and D2 lines as a function of frequency difference between the collinear pump and probe beams. With no externally applied magnetic
field a single absorption dip at a frequency difference of 228 MHz is observed (figure 5.3 (a)
and 5.3 (c)). The width of these resonances is approximately 4 MHz which is less than
the natural width of 5.9 MHz. The amplitudes for the D2 line are notably weaker than
for the D1 line. This is due to destructive excitations via cycling transitions which do not
contribute to the preparation of dark coherent states. This is in agreement with the results
of Stähler et al. [Stä02] and Magnus et al. [Mag05] which show greater contrast in the dark
resonance in the D1 line than in the D2 line of
85
Rb and 7 Li, respectively.
It is difficult to compare the width of the EIT resonances obtained for both lines be-
58
CHAPTER 5. SUB-NATURAL RESONANCES
cause any ambient magnetic field can introduce additional broadening. To investigate
the possible ambient field broadening an external homogeneous magnetic field (approx~ ~k. The magnetic field
imately 5 G) was applied perpendicular to the laser beam B⊥
removes the degeneracy of the Zeeman levels, splitting the sub-natural EIT resonance
(figure 5.3 (b and d)). The applied magnetic field also introduces a convenient quantisation axis.
The pump light with linear polarisation parallel to the magnetic field
can produce π transitions, while the probe with orthogonal linear polarisation excites
the σ ± transitions (figure 5.3 (e)). The m = 0 → m′ = 0 type Raman transition,
which is magnetic field insensitive in the linear Zeeman approximation, does not exist
′
′
for 6 Li; however, the Raman transitions |F = 1/2, mF = 1/2i → |F = 3/2, mF = −1/2i
′
′
and |F = 1/2, mF = −1/2i → |F = 1/2, mF = 1/2i are also magnetic field insensitive, be-
cause the Zeeman shift of the upper and lower magnetic sublevels are almost equal. Thus,
the EIT dip in absorption, which remains unshifted at 228 MHz frequency difference with
increasing magnetic field is due to the above mentioned Raman transitions. The width of
this EIT resonance for the D1 line is approximately 3 MHz. Both outer dips are symmetrically shifted by 2∆ from the unshifted centre dip, where ∆ = µB gF B/h, µB is the Bohr
magneton, h is Planck’s constant and gF is the Landé factor. The widths of the outer dips
are 25% larger than the width of the unshifted dip due to spatial inhomogeneities in the
magnetic field.
We reduced the intensities of the pump and probe lasers but observed no change in
the width of the EIT resonances which implies that the width is not limited by power
broadening. Although Li-Li collisions are negligible, collisions with residual background gas
atoms may cause spin depolarising collisions of the ground states which possibly contribute
to the EIT width. The resolution is further limited by the finite interaction (transit) time.
The spectroscopy of atoms in a metal vapour cell has the limitations of randomly directed
velocities (i.e. Doppler broadening), field inhomogeneities and high collision rates.
5.3.2
Atomic beam EIT
To improve the resolution of EIT resonances experiments were performed using the standard technique of a collimated atomic beam.
59
5.3. HYPERFINE EIT
220
Fluorescence (arbitrary units)
(b)
Fluorescence (arbitrary units)
(a)
230
240
250
Frequency difference [MHz]
260
200
210
220
230
Frequency difference [MHz]
Figure 5.4: D1 line fluorescence from 6 Li atoms. Probe laser is scanned over both the
′
′
|F = 3/2i → |F = 1/2i and |F = 3/2i → |F = 3/2i transitions. (a) Fixed pump laser
′
′
tuned to (a) |F = 1/2i → |F = 1/2i transition and (b) |F = 1/2i → |F = 3/2i transition.
Results
Resonant fluorescence in the collimated atomic beam showed a much narrower transverse
Doppler width of 20 MHz compared to ≈ 3.5 GHz obtained in the vapour cell. Therefore,
the hyperfine splitting of the 22 P1/2 state of 26.1 MHz can be resolved. Figure 5.4 shows a
plot of fluorescence of the D1 line as a function of frequency difference between the pump
and probe laser. In this figure the probe laser is scanned over both the |F = 3/2i →
′
′
|F = 1/2i and |F = 3/2i → |F = 3/2i transitions whereas the pump laser has a fixed
′
′
frequency tuned to the |F = 1/2i → |F = 1/2i (a) or |F = 1/2i → |F = 3/2i (b) tran′
sition. The EIT dip is more pronounced when the fixed laser is tuned to the |F = 3/2i
state which is due to the larger transition probability. For this reason we performed all
′
of the subsequent experiments with the pump laser tuned to the |F = 1/2i → |F = 3/2i
transition. It is remarkable that we observe EIT in spite of destructive excitations via a
cycling transition. The EIT resonances in the 6 Li beam shown in figure 5.4 have a width
of ≈ 300 kHz for the D1 line. This is a factor of 10 reduction compared to the vapour cell
experiments.
The EIT resonances obtained on the D1 and D2 lines are shown in figure 5.5. The
intensity of the resonant fluorescence on the D2 line is higher, but the contrast of the
60
CHAPTER 5. SUB-NATURAL RESONANCES
Fluorescence (arbitrary units)
D1
D2
′
F =3/2
(d)
F′ =1/2
(e)
F′ =3/2
(a)
F′ =5/2
F′ =1/2
(b)
(c)
F=3/2
227.5
228
228.5
Frequency difference [MHz]
229
F=1/2
F=3/2
F=1/2
Figure 5.5: Fluorescence of the D1 and D2 line as a function of frequency difference between
the pump and probe laser. (a) D1 resonance (width of 210 kHz) with applied magnetic
field, (b) D1 resonance (370 kHz) with no magnetic field, (c) D2 resonance (560 kHz) with
no magnetic field. The Raman transition responsible for the D1 and D2 resonance is shown
in (d) and (e) respectively.
sub-natural resonances with respect to the residual Doppler background is lower compared
to the D1 line. In this regard the observations in the atomic beam and the vapour cell
are very similar. However, better spectral resolution in the atomic beam allows us to
demonstrate that the width of EIT resonances in both cases is also essentially different.
The sub-natural resonances observed in the atomic beam on both lines under very similar
experimental conditions such as light intensity, polarisation and beam overlapping are
shown in figure 5.5. The curve in figure 5.5 (c) represents the EIT resonance observed on
the D2 line without applied magnetic field. The width of the resonance is approximately
560 kHz. The sub-natural resonance on the D1 line (370 kHz) (figure 5.5 (b)) reveals a
doublet structure due to residual magnetic field in the interaction region. This splitting
is hidden on the D2 line by the large width of the EIT resonance. The double structure
can be removed by applying a small magnetic field along the atomic beam resulting in the
210 kHz wide resonance (figure 5.5 (a)) on the D1 line. We believe that the EIT resonance
on the D2 line is wider because of a shorter lifetime of the ground state coherence destroyed
′
via the cycling transition |F = 3/2i → |F = 5/2i.
Figure 5.6 shows several EIT resonances obtained for different polarisations with and
without an applied magnetic field (≈2 G) along the atomic beam. In figure 5.6 (a) and
61
5.3. HYPERFINE EIT
⇑↑
(d)
215
Fluorescence (arbitrary units)
Fluorescence (arbitrary units)
⇑→
Fluorescence (arbitrary units)
B=0
(a)
(b)
B 6=0
(c)
(e)
220
225
230
235
Frequency difference [MHz]
240
215
(f)
220
225
230
235
Frequency difference [MHz]
240
Figure 5.6: Fluorescence of the D1 line as a function of frequency difference. Fixed pump
′
laser tuned to |F = 1/2i → |F = 3/2i. (a) and (d): B = 0, (b) and (e): B ≈ 2 G. (a)
and (b): linear perpendicular polarisation, (d) and (e): linear parallel polarisation. The
transitions shown in (c) and (f) are responsible for each of the corresponding resonances
to their left. The thicker lines represent the pump transitions.
figure 5.6 (b) the polarisation of both lasers is linear and perpendicular to each other. The
two outer peaks in figure 5.6 (b) are both shifted by 2∆ from the zero field resonance
which is consistent with the Raman transitions depicted in figure 5.6 (c). In figure 5.6 (d)
and figure 5.6 (e) the polarisation of both laser beams are aligned parallel to each other
and orthogonal to the magnetic field. Only two Λ-type Raman transitions are possible in
this configuration (figure 5.6 (f)). The two resulting EIT resonances are both shifted by
∆ to either side of the zero-magnetic field resonance. The amplitudes are much smaller
when parallel polarisation light was applied. This can be understood by considering that
the atoms will undergo transitions between all Zeeman sublevels, but for parallel polarised
pump and probe light, not all of these contribute to coherences (see figure 5.6 (f)). It is
notable that a magnetic field insensitive fluorescence resonance (in the linear approxima-
62
CHAPTER 5. SUB-NATURAL RESONANCES
Figure 5.7: Resonant fluorescence of the D1 line of 6 Li atoms as a function of the frequency
offset δ. The frequency of both the probe and drive optical components is tuned to the
|2S1/2 , Fg = 3/2i → |2P1/2 , Fe = 1/2i transition. The optical components have orthogonal
linear polarisations.
′
tion) is demonstrated despite the fact that the m = 0 → m = 0-type Raman transition
does not exist for 6 Li. Similar results were obtained for the D2 line.
5.4
Zeeman coherence
5.4.1
EIT in the 6Li D1 line
It has been shown both theoretically and experimentally that excitation of both hyperfine
ground-state sublevels for the Rb D1 line may result in a much higher contrast EIT resonance than for the D2 line [Tai06, Stä02]. If bichromatic light excites atoms from either of
the two hyperfine ground-state sublevels, the strongest EIT resonance on the 87 Rb D1 line
occurs for the |5S1/2 , Fg = 2i → |5P1/2 , Fe = 1i transition. In the case of the 6 Li D1 line we
have also found that the largest reduction of the resonant fluorescence occurs for the same
type of optical transition, i.e., |Fg i → |Fe = Fg − 1i. In this case atoms from all ground-
state magnetic sublevels can be connected by Λ-type links, which lead to non-absorbing
CPT states [Smi89].
Figure 5.7 shows the dependence of the resonant fluorescence on the frequency offset
between the drive and probe components. At Raman resonance, which takes place at zero
5.4. ZEEMAN COHERENCE
63
Figure 5.8: (a): Raman transitions responsible for the dark Zeeman coherence for different
polarisations of the drive and probe components. (b): Resonant fluorescence of the D1 line
of 6 Li atoms as a function of the frequency offset δ. The laser frequency is tuned to the
|2S1/2 , Fg = 3/2i → |2P1/2 , Fe = 1/2i transition. The probe and drive optical components
have orthogonal linear polarisations (i) or parallel polarisations (ii). A magnetic field of
approximately 2.5 G is applied in a direction orthogonal to the light beam and collinear
to the atomic beam.
offset in the absence of an external magnetic field, the intensity of fluorescence is reduced
by approximately 45% when both optical components are linearly polarised perpendicular
to each other. For parallel linear polarisations the EIT resonance is almost three times
weaker than in the case of orthogonal polarisations.
If the two-level atomic system is completely degenerate and the quantisation axis has
an arbitrary direction, all possible Raman transitions can contribute to the EIT resonance
at zero frequency offset. To separate the contributions caused by CPT and population
oscillation, we apply an external magnetic field. An external magnetic field B shifts the
ground state Zeeman sublevels by an amount ∆ = gF µB mF B/h that depends on the
magnetic quantum number mF and the strength of the magnetic field B.
The external magnetic field also defines a convenient basis. If the quantisation axis is
−
→
collinear with the wave vector of the linearly polarised light, k , the interaction with atoms
results in σ + and σ − transitions. This is the natural basis for zero magnetic field and for
→
−
→ −
a magnetic field which is collinear with the light wave vector ( B k k ). If the magnetic
→
−
→ −
field is orthogonal to the wave vector ( B ⊥ k ) and collinear with the light polarisation,
64
CHAPTER 5. SUB-NATURAL RESONANCES
Figure 5.9:
Width of EIT fluorescence resonances in the |2S1/2 , Fg = 3/2i →
|2P1/2 , Fe = 1/2i transition within the 6 Li D1 line as a function of the total light intensity.
A magnetic field (B ≈ 1 G) is applied along the atomic beam.
−
→
π transitions take place. When the magnetic field is orthogonal to both k and the light
polarisation, σ + and σ − excitations occur.
Thus, once a magnetic field is applied and the quantisation axis is defined, different
types of optical transitions between the ground and excited states produced by the probe
and drive components occur. For example, a drive optical component with linear polarisation orthogonal to the magnetic field can make σ ± transitions, while a probe component
with parallel polarisation produces π transitions, as shown in figure 5.8(a). In the case of
identical linear polarisations orthogonal to the magnetic field, both components produce
σ ± transitions.
When the degeneracy of the ground-state hyperfine levels is removed by an applied magnetic field, the resonant fluorescence splits into several resonances. In order to find resonant
conditions for a nearly degenerate two-level system all Raman transitions involving one photon from each optical component should be considered. The Raman transitions responsible
for the dark Zeeman coherence in the case of the |2S1/2 , Fg = 3/2i → |2P1/2 , Fe = 1/2i optical transition are shown in figure 5.8(a).
5.4. ZEEMAN COHERENCE
65
For orthogonal linear polarisations there are two high-contrast EIT resonances, curve
(i) in figure 5.8(b). As expected both fluorescence dips are shifted from the zero frequency
offset by ∆ = gF µB B/h.
The appearance of the unshifted or magnetically insensitive resonance for parallel polarisations, curve (ii) in figure 5.8(b), does not fit into a Λ-type coupling scheme. Different ground-state sublevels could not be coupled by any Raman transition and groundstate coherence can not be prepared at a zero frequency offset. The unshifted resonance
corresponds to Raman transitions from a given Zeeman sublevel to itself, as mentioned
in [Aku98, Lez99b]. Coherent oscillations between ground and excited states result in this
resonance, which is insensitive to the applied magnetic field.
The two magnetically sensitive fluorescence dips, curve (ii) in figure 5.8(b), are the
result of a light-generated ground-state Zeeman coherence. They are similar in shape to
the resonances observed when the light components have orthogonal polarisations, but are
shifted from the zero frequency offset by 2∆.
We have found that power broadening of the magnetically sensitive and magnetically
insensitive EIT resonances is very different. The power broadening of EIT resonances due
to hyperfine or Zeeman ground-state coherence has been studied in detail both theoretically and experimentally [Ari96, Aku98, Lez99b, Aku91, Jav02]. The width Γ of the EIT
resonances grows linearly with the drive intensity: Γ = Ω2D /γ, where γ is the homogeneous linewidth. However, our measurements reveal that the width of the unshifted EIT
resonance does not depend on light intensity within the uncertainty of experimental measurements, while the magnetically dependent resonance due to CPT of Zeeman sublevels
experiences expected power broadening. The experimentally measured widths of both
resonances at different light intensity are presented in figure 5.9.
Thus, we can conclude that the high-contrast EIT resonances are due to the combined effect of CPT amongst the magnetic sublevels from the ground-state hyperfine level
|Fg = 3/2i and CPO between the ground and excited states. For orthogonal polarisations
the contribution from Zeeman coherence is dominant. However, for optical components
with parallel polarisations both effects have comparable contributions to the EIT resonance.
66
CHAPTER 5. SUB-NATURAL RESONANCES
Figure 5.10: Sub-natural resonances observed with linear parallel (i) and orthogonal (ii)
polarisations for the |2S1/2 , Fg = 1/2i → |2P1/2 , Fe = 1/2i transition without (c) and with
(d) an externally applied magnetic field. The sensitivity for curves marked (ii) is four times
higher than for (i) curves.
5.4.2
Sub-natural resonances for a pure four-level atomic system
We have investigated sub-natural resonances for the 6 Li |2S1/2 , Fg = 1/2i → |2P1/2 , Fe = 1/2i
transition. According to the Clebsch-Gordan coefficients [Met99] this transition is eight
times weaker than the |2S1/2 , Fg = 3/2i → |2P1/2 , Fe = 1/2i transition and, consequently,
the resonant fluorescence must be much weaker, resulting in a lower signal to noise ratio.
However, a pure four-level scheme, which can be realised within this 6 Li transition, allows
better discrimination between the two above-mentioned mechanisms (CPT and CPO) responsible for sub-natural resonances.
For the drive and probe components with orthogonal linear polarisations, the groundstate Zeeman sublevels can be coupled by a combination of π and σ ± transitions, while for
parallel linear polarisations they are not coupled by any Raman transition, (figure 5.10(a,
b)). Sub-natural resonances, with reduced fluorescence, have been observed in both cases,
5.4. ZEEMAN COHERENCE
67
as shown on figure 5.10(c, d).
For the |2S1/2 , Fg = 3/2i → |2P1/2 , Fe = 1/2i transition, as discussed in section 3.1, the
EIT resonance is more pronounced in the case of linear orthogonal polarisations than for
linear parallel polarisations. However, for the |2S1/2 , Fg = 1/2i → |2P1/2 , Fe = 1/2i transition the sub-natural resonance for parallel linear polarisations of the drive and probe
components is much stronger.
In a collimated atomic beam of 6 Li a contribution to the resonant fluorescence from
the nearby |2S1/2 , Fg = 1/2i → |2P1/2 , Fe = 3/2i transition is not negligible because of the
small hyperfine splitting of the |2P1/2 i state. However CPT does not exist for this transition because Fg < Fe [Smi89]. Thus, the observed sub-natural resonance for parallel
polarisations is entirely due to CPO generated in an X-type atomic system, which consists
of two magnetic sublevels in the ground state (mg = −1/2; 1/2) and two in the excited
state (me = −1/2; 1/2), as shown in figure 5.10(b). As expected, the width and the am-
plitude of this resonance show no dependence on the external magnetic field, (curve (i) in
figure 6(c) and 6(d)). On the other hand the EIT resonance due to ground-state coherence
splits when a magnetic field is applied.
5.4.3
EIA in the 6 Li D2 line
In order to observe EIA the laser was tuned to the appropriate closed transition |Fg i →
|Fe = Fg + 1i on the D2 line, i.e., |2S1/2 , Fg = 3/2i → |2P3/2 , Fe = 5/2i. The fluorescence
spectrum of the lithium beam excited by the bichromatic laser light is shown in figure 5.11.
With no applied magnetic field, a sharp enhancement of fluorescence due to EIA occurs
at zero frequency offset (δ = 0). A maximum contrast of the sub-natural peak relative
to the total resonant fluorescence, of approximately 40%, is observed with orthogonal
linear polarisations. An approximately five times smaller EIA resonance was observed
with opposite circular polarisations. A 20 MHz wide broadening due to the divergence
of the atomic beam has been subtracted in the data presented later in this thesis (see
figures 5.12(a) and (b)). We have also found that the amplitude and the shape of the EIA
resonance do not noticeably depend on a red or blue optical frequency detuning from the
|2S1/2 , Fg = 3/2i → |2P3/2 , Fe = 5/2i transition.
An external magnetic field splits the EIA peak into two resonances for linear orthog-
onal polarisations, as shown in figure 5.12(a), in a similar way to EIT resonances for the
68
CHAPTER 5. SUB-NATURAL RESONANCES
Figure 5.11: Curve (i) represents the resonant fluorescence of 6 Li atoms excited on the
|2S1/2 , Fg = 3/2i → |2P3/2 i transition by a bichromatic field. The drive and probe optical
components have linear orthogonal polarisations. Curve (ii) shows the resonant fluorescence produced by the probe component only.
D1 line. The positions of the sub-natural peaks are consistent with Λ-type Raman transitions between the ground-state magnetic sublevels. In the case of orthogonal circular
polarisations, (figure 5.12(b)), the EIA peak splits into three resonances with sub-natural
width. The unshifted resonance does not correspond to any Raman transitions between
ground-state magnetic sublevels and, as in the case of the unshifted EIT peaks, this can be
attributed to CPO. Thus, for some transitions CPO can result in absorption enhancement
and, consequently, fluorescence enhancement if the ground state magnetic sublevels most
strongly coupled to the interaction with light are more populated.
For parallel linear polarisations an unexpected change of the sign of the EIA fluorescence
resonance occurs at zero frequency offset (figure 5.13). This sub-natural resonance is
approximately two times narrower than the EIA resonances observed with orthogonal linear
polarisations under the same experimental conditions including the same light intensity.
No splitting is observed with an external magnetic field.
Thus, we observe that CPO in 6 Li atoms excited on the |2S1/2 , Fg = 3/2i → |2P3/2 i
transition may lead to a fluorescence suppression or enhancement depending on the polarisations of the bichromatic light.
69
5.4. ZEEMAN COHERENCE
(a)
(b)
Figure 5.12: Splitting of EIA resonances in an external magnetic field for the drive
and probe optical components with orthogonal linear polarisations (a) and opposite
circular polarisations (b).
The frequency of the bichromatic beam is tuned on the
|2S1/2 , Fg = 3/2i → |2P3/2 i transition. The external magnetic field is (i) 0, (ii) 2.5 G,
(iii) 3.75 G and (iv) 5 G.
Figure
5.13:
Reduction
of
resonant
fluorescence
of
6
Li
atoms
for
the
|2S1/2 , Fg = 3/2i → |2P3/2 i transition without (i) and with (ii) a transverse magnetic field as a function of the frequency offset δ. The optical components have linear
parallel polarisations.
70
CHAPTER 5. SUB-NATURAL RESONANCES
5.4.4
Modelling and discussion
The model used in [Lez99a] which is based on a semiclassical treatment of the atom-light
interaction using the optical Bloch equations can not be directly applied for calculating
fluorescence spectra. Nevertheless, A. Lezama has undertaken a numerical simulation of
the probe absorption spectra of 6 Li atoms excited by a quasi-degenerate bichromatic light
in the presence of a magnetic field to compare with the experimental data. This model
employs the master equation to formulate an equation for the atomic density matrix ρ as
described in [CT77]. The free atomic evolution is then governed by a Hamiltonian that contains terms of the external magnetic and electromagnetic pump and probe fields [Lez99b].
Furthermore, the model incorporates atomic relaxation such as spontaneous decay of the
excited state, subsequent population increase of the ground states, and effects due to the
limited interaction time. The master equation is then solved to first order of the probe field
by employing a Liouville space formalism. The probe absorption is subsequently calculated
from the macroscopic atomic polarisation.
Probe absorption spectra were calculated under the assumption that the Rabi frequency
of the pump component is ΩD = 0.1Γ (Γ is the excited state width), while the probe
component is negligibly weak; the ground state relaxation is 0.001Γ and a Zeeman shift due
to an applied transverse magnetic field is 0.02Γ. The relative strengths of the transitions
and optical pumping to other hyperfine ground levels are included in the model. Some
results of the numerical modelling are presented in figure 5.14. The vertical axis represents
absorption normalised to maximum linear absorption for a given transition. This means
that the vertical scale of one plot relative to another does not represent directly the relative
strengths of the resonances.
Figure 5.14 (a) shows calculated EIT resonances for the |2S1/2 , Fg = 3/2i →
|2P1/2 , Fe = 3/2i transition for different polarisations of the bichromatic light. The cal-
culations reproduce the existence of the magnetically insensitive resonance for orthogonal
polarisations and the positions of the magnetically dependent resonances. However the
relative amplitudes of the calculated resonances differ substantially from the experimental
fluorescence observations.
The absorption spectra for the |2S1/2 , Fg = 1/2i → |2P1/2 , Fe = 1/2i transition, pre-
sented in figure 5.14(b), demonstrate an absence of magnetically dependent resonances for
parallel linear polarisations which are orthogonal to the direction of the magnetic field.
71
5.4. ZEEMAN COHERENCE
(a)
(b)
δ/Γ
(c)
δ/Γ
δ/Γ
Figure 5.14: Numerical modelling of probe absorption spectra for different transitions in
the 6 Li D1 and D2 lines. The optical components have linear orthogonal (i) or parallel (ii)
polarisations.
This result agrees with the experimental observations (figure 5.10).
EIT and EIA resonances calculated separately for the three |2S1/2 , Fg = 3/2i →
|2P3/2 , Fe = 5/2, 3/2, 1/2i transitions for the case of orthogonal linear polarisations are
shown in figure 5.14(c). Generally speaking, the experimental observation of EIA for
the 6 Li D2 line is somewhat surprising, due to competing processes taking place for
the |2S1/2 , Fg = 3/2i → |2P3/2 i transition. Firstly, there is incoherent hyperfine optical
pumping. This process can be considered as a leak to a non-resonant hyperfine sublevel,
which imposes an additional limit on the ground-state lifetime. Secondly, atoms may
be trapped in a non-absorbing dark coherent state prepared in the |2S1/2 , Fg = 3/2i →
|2P3/2 , Fe = 1/2, 3/2i transitions. Additionally, a more absorbing coherent superposition
can be generated for the cycling transitions |2S1/2 , Fg = 3/2i → |2P3/2 , Fe = 5/2i. The
bichromatic light can be almost equally resonant for all these transitions.
This is a unique situation in the case of alkali atoms. For Cs or Rb D2 lines, for
example, monochromatic resonant radiation tuned to transitions from the upper ground
state hyperfine sublevels interacts with atoms contained in a vapour cell in three different
velocity groups. Atoms from two velocity groups may be pumped into non-absorbing
dark states, while a more absorbing state may be prepared in the third group, as was
shown in [Aku03b]. The contributions from those groups to the total probe absorption
are opposite, but the absorbing properties of atoms from the same group are very similar.
This separation of EIT and EIA in velocity space is not possible in the case for the 6 Li
D2 line, because the hyperfine splitting of the |2P3/2 i state is smaller than the natural
linewidth. This means that laser light can be resonant with the three optical transitions
72
CHAPTER 5. SUB-NATURAL RESONANCES
|2S1/2 , Fg = 3/2i → |2P3/2 , Fe = 1/2, 3/2, 5/2i simultaneously and 6 Li atoms can be excited
by different transitions with comparable probabilities. This makes modelling of the atomic
response of 6 Li atoms more difficult.
It is worth noting that similar doublet and triplet structures of sub-natural resonances
in the probe absorption have been obtained for the
85
Rb D2 line both experimentally and
numerically [Lez99b]. However, the connection of the magnetically insensitive sub-natural
resonances with CPO was not mentioned.
There are strong grounds to believe that in atomic media excited by bichromatic resonant light a contribution of the optical coherence (population oscillations between the
ground and excited state) to the sub-natural resonance is essential in many experimental
realisations and its role has been underestimated. For example, experimentally observed
enhanced nonlinearity and efficient four wave mixing (FWM) in EIA and EIT media has
been attributed entirely to ground-state Zeeman coherence [Aku03b]. However, the spectrum of FWM obtained in a Cs vapour was previously found to be surprisingly insensitive
to an ambient magnetic field. Also the spectral region of the comb-like spectrum of FWM
was limited by the natural width of the optical transition. This suggests that CPO played
an important role in this nonlinear process. The effect of external magnetic fields on the
Kerr nonlinearity of an atomic medium enhanced at Raman transitions is currently under
detailed experimental study.
5.5
Ramsey spectroscopy
Further reduction of EIT and EIA resonance widths was achieved using Ramsey spectroscopy [Ram89]. In this technique atoms or molecules in a beam pass two spatially separated interaction regions. Prior to our work, optical Ramsey fringes had been observed on
beams of Na and Cs atoms pumped into coherent non-absorbing states [Tho82, Hem93].
Ramsey fringes in EIT
To observe Ramsey fringes in an EIT medium a coherent superposition of the |F = 1/2i
and |F = 3/2i ground states is prepared in a light field. The laser field consists of two
co-propagating laser beams which are resonant with the |F = 1/2i → |F ′ = 3/2i and
|F = 3/2i → |F ′ = 3/2i transitions. After some time the state is probed in a second in-
73
5.5. RAMSEY SPECTROSCOPY
PMT
6
Li
Figure 5.15: Simplified scheme of the experimental set-up used to observe Ramsey fringes
in EIT and EIA spectra. A bichromatic laser beam, in which the complete central vertical
portion was blocked, produces two interaction regions separated by L = 7.7 mm. Fluorescence of atoms in the second interaction region was measured on a photomultiplier tube
(PMT).
teraction region consisting of the same two frequencies. Similar to the EIT experiments
described previously, the laser resonant with the |F = 1/2i → |F ′ = 3/2i transition is kept
fixed while the other frequency is swept over the |F = 3/2i → |F ′ = 3/2i transition.
In our experiment the two interaction regions are separated by L = 7.7 mm. This is
achieved by using only one laser beam in which the complete central vertical portion was
blocked. A simplified scheme of the experimental set-up used for observing Ramsey fringes
in both EIT and EIA is shown in figure 5.15. Figure 5.16(a) shows a plot of fluorescence as a
function of frequency difference. The intensities for both pump and probe were 2 mW/cm2 .
The polarisation of both beams were linear and perpendicular to each other. To amplify
the Ramsey fringes the laser light in the first interaction region is chopped, while the
fluorescence of the probe region is detected by a photo multiplier tube (PMT) using a
lock-in amplifier. The obtained width (FWHM) of the resonance is narrower than 100 kHz
which is consistent with calculations based on a mean atomic velocity v ≈1850 ms−1 and
a laser field separation L: ∆ν = v/(3L) = 80kHz [Dem03].
Ramsey fringes in EIA
For the first time to our knowledge we have extended Ramsey spectroscopy to media
74
CHAPTER 5. SUB-NATURAL RESONANCES
100 kHz
Frequency difference
Fluorescence (arbitrary units)
(b)
Fluorescence (arbitrary units)
(a)
200 kHz
Frequency difference
Figure 5.16: Optical Ramsey fringes of two co-propagating laser beams in a Raman-type
configuration in a medium of (a) EIT and (b) EIA. The blue spectrum in (b) shows the
enhanced absorption when the first interaction region is blocked.
of electromagnetically induced absorption. Here, we prepared an EIA coherent state in
the first interaction region. The experimental set-up resembles the one used to observe
Ramsey fringes in EIT. However, the laser light in both interaction areas consisted of two
frequencies resonant with the |F = 3/2i → |F ′ = 5/2i transition and detuned from each
other by an adjustable frequency offset δ. The intensities of both beams were ∼ 1 mW/cm2
with orthogonal linear polarisation and the spacing between both interaction regions was
7.7 mm.
The blue line in figure 5.16(b) shows the enhanced absorption when the first interaction
region is blocked similarly to figure 5.11. The increased width in this case is due to transittime broadening [Fuc06]. After preparing the EIA coherence in the first interaction region
a significant narrowing of the width can be observed as shown by the red spectrum in
figure 5.16(b). It is noticeable that the contrast of the Ramsey fringes in EIA is reduced
compared to the fringes in EIT. We have considered three possible reasons for this. Firstly,
the relaxation time of the coherence could be shorter in the case of EIA. Secondly, the
unresolved hyperfine structure in the excited level |2P3/2 i may lead to a reduction of the
coherence. Lastly, in the EIA experiments presented here the master laser was free-running
due to technical reasons.
Despite the reduced fringe contrast the experiments presented here show that the co-
5.5. RAMSEY SPECTROSCOPY
75
herence can survive without interaction with light. Furthermore, the coherence time is
longer than the time of flight between both interaction regions. The experiments on Ramsey spectroscopy thus show striking similarities between EIA and EIT coherences similar
to references [Aku05, Lez06] which deal with the storage and retrieval of light pulses in
EIT and EIA media.
76
CHAPTER 5. SUB-NATURAL RESONANCES
Chapter 6
Laser cooling of 6Li
An important step towards the production of ultracold lithium atoms is a magneto-optical
trap (MOT). To obtain atoms slow enough to be captured in the MOT we slow down atoms
from the atomic beam (see section 4.1.1) by means of a Zeeman slower. Here, we discuss
briefly the theory of laser cooling and magneto-optical traps as well as our experimental
realisation. More details on theoretical aspects can be found in [Met99].
6.1
Spontaneous force
For simplicity, we consider a two-level atom with ground state |gi and excited state |ei
and a resonance frequency ω0 in a laser field with frequency ω ≈ ω0 . When the atom
absorbs a photon with wave vector ~k the momentum of the atom changes by p~ = ~~k due to
momentum conservation. If this is followed by a stimulated emission the momenta of both
photons are equal and hence the momentum of the atom after absorbing and emitting a
photon is unchanged. However, the isotropic distribution of spontaneously emitted photons
leads to an atom momentum change. The resulting force when averaged over many cycles
is
F~sp = ~~kΓS ,
(6.1)
where the scattering rate ΓS describes the rate at which photons are absorbed and spontaneously emitted. The scattering rate depends on the atom velocity ~v , the laser intensity
I and the velocity dependent detuning
δ ′ = δ − ~k · ~v,
77
(6.2)
CHAPTER 6. LASER COOLING OF 6 LI
78
where δ = ω − ω0 . It is given by [Met99]
I/Is
1
,
ΓS = Γ
2 1 + I/Is + (2δ ′ /Γ)2
(6.3)
where Γ is the natural linewidth and Is is the saturation intensity. The values for the closed
transition in 6 Li are Is = 2.5 mW/cm2 and Γ = 2π · 5.9 MHz [McA96, Geh03].
6.2
Zeeman slower
As described in section 4.1.1, the starting point of our experiments is an atomic oven at a
temperature of approximately 420◦ C. The atomic beam emitted from the oven has a mean
velocity of 1840 m/s. However, the magneto-optical trap (section 6.3) requires atoms with
velocities as low as tens of meters per second. By using a counter-propagating resonant
laser beam the thermal atomic beam can be decelerated. On average, each absorbed photon
reduces the momentum of the atom by ~k. The atomic lifetime τ = 1/Γ sets the limit for
the maximum deceleration to
~k
,
(6.4)
2mτ
where m is the mass of the atom. However, the atoms shift out of resonance when slowed
amax =
down due to the velocity dependence of the scattering rate in equation 6.3. This is generally
circumvented by either chirping the laser frequency (Chirp slower) [Ert85] or applying an
inhomogeneous external magnetic field (Zeeman slower) to keep the atoms in resonance.
The first Zeeman slower was realised in 1982 by Phillips and Metcalf [Phi82] and was used
to slow down sodium atoms.
Zeeman slowers rely on the fact that the resonance frequency depends on the magnetic
field. For circularly polarised light (σ ± transitions) equation 6.2 in a magnetic field becomes
δ± = δ − ~k · ~v ∓ (ge me − gg mg )µB B/~,
(6.5)
where ge and gg are the Landé g-factors of the excited and ground state, respectively.
Two types of Zeeman slowers can be realised depending on the direction of the circular
polarisation. In our experiment the Zeeman slower coil provides an increasing field for the
atoms which requires in turn σ − polarised laser light. However, many Zeeman slowers used
for cold atom-experiments have a decreasing magnetic field and σ + polarised light. The
advantage of such a slower is a very small magnetic field at the end of the slower, which
79
6.2. ZEEMAN SLOWER
makes it possible to place the end of the Zeeman slower very close to the MOT. The disadvantage though is that the laser light used to slow the atoms is not far detuned from the
zero-field resonance, which may limit the lifetime of the MOT. From equations 6.3 and 6.5
it follows that the magnetic field profile that keeps the atoms at a constant deceleration a
for an increasing field Zeeman slower is given by
B(z) = B0 1 −
The parameter
p
1 − z/z0 .
(6.6)
mvi2
η~kΓ
(6.7)
~kvi
(ge me − gg mg )µB
(6.8)
z0 =
is the length of the Zeeman slower,
B0 =
the maximum magnetic field, vi the maximum initial atomic velocity which is slowed down
and η = a/amax the parameter that characterises the deceleration efficiency.
Our Zeeman slower coil is wound on a 30 cm long tube and consists of a single tapered coil with approximately 2400 turns. Hollow copper tubing wrapped around the tube
provides water cooling. The flat surface of a very thin brass plate placed on top of the
copper tubing simplified winding the coil. It is necessary to water cool the outside of the
Zeeman coil which is also done by means of hollow copper tubing. A current of 2.94 A
through the 0.69 mm thick current carrying wire gives the required maximum magnetic
field of approximately 620 G. The magnetic field keeps the atoms at a deceleration close
to η = 0.5. Atoms with velocities of up to ∼650 m/s are slowed down to ∼50 m/s over the
30 cm length. The Zeeman slower was initially tested by observing fluorescence spectra
which is shown and described in figure 6.1.
The σ − Zeeman slowing light has a detuning of 920 MHz with respect to the |F = 3/2i →
|F ′ = 5/2i transition. This large detuning allows the Zeeman slowing light to pass through
the MOT with no visible effect on trapping. Sidebands are added to the Zeeman slower
light by modulating the laser diode current at 120 MHz which enhances the flux of atoms
trapped in the MOT by a factor of two.
When an atom is slowed down from vi to vf it absorbs and emits approximately
N=
vi − vf
∼ 6200 photons,
vrec
(6.9)
CHAPTER 6. LASER COOLING OF 6 LI
80
(b) 600
420 G
Fluorescence
378 G
336 G
294 G
252 G
velocity of slowed atoms [m/s]
(a)
500
400
300
200
100
210 G
0
0
100
Frequency
200
300
400
final B−field [G]
500
Figure 6.1: (a) Typical fluorescence spectra of atoms exiting the Zeeman slower for different
final magnetic fields. The laser light of a frequency scanning laser (∼ 1 GHz) is split into
two probe beams which intersect with the atomic beam at 45◦ and 90◦ , respectively. The
fluorescence signal was measured by a photo-multiplier tube. The large peaks at zero
frequency are due to the laser beam at 90◦ which is resonant to all atomic velocities at
zero detuning from the |F = 3/2i → |F ′ = 5/2i transition. The peaks due to the 45◦ beam
are velocity selective and shift with increasing final magnetic fields to smaller detunings
and hence lower velocities. (b) Plot of the velocities of the slowed atoms when exiting the
Zeeman slower versus the final magnetic field. Due to the larger radial spread of the atoms
at the interaction region the atomic flux could not be observed any more for velocities as
low as required for capture in the MOT.
where vrec is the recoil velocity. Due to spontaneous emission in random directions the
atomic velocity in the radial direction increases. Treating this process as a random walk
gives an average mean velocity at the end of the Zeeman slower in the radial direction
of [Jof93]
q
p
2 /3 = 4.5 m/s.
2
∼
vx,y
= Nvrec
(6.10)
It is straightforward to show that the transverse spreading is more severe for lighter atoms
such as lithium. The mean transverse velocity at the entrance of the Zeeman slower is
approximately 5 m/s. Including both sources of transverse heating yields a radial velocity
of approximately 7 m/s at the exit of the Zeeman slower. Furthermore, the radial spread
6.3. MAGNETO-OPTICAL TRAP
due to spontaneous emission can be calculated as
r
2 Nt
p
vrec
tr
hx2 i =
= 2 mm,
9
81
(6.11)
where ttr is the time an atom needs to fly through the Zeeman slower. The radial spread
of the atoms at the start of the Zeeman slower (∼1.5 mm) has to be taken into account as
well. This gives a total radial spread of ∼2.5 mm.
The magneto-optical trap is located approximately 18 cm from the end of the Zeeman
slower. Assuming a velocity of 50 m/s when leaving the Zeeman slower, which is approximately the capture velocity of our MOT, gives a radial spread of 3 cm at the MOT. Since
the diameter of our MOT beams is only about 15 mm, only 4% of all atoms reach the MOT
region and can thus be trapped. Implementing a two-dimensional laser cooling stage at
the glass metal transition facing the Zeeman slower would enhance the atom flux captured
in the MOT significantly.
6.3
6.3.1
Magneto-optical trap
Optical molasses
An optical molasses consists of three counter propagating laser beams at right angles with
each other [Chu85]. Such a set-up allows atoms to be slowed in all three dimensions. In
one dimension the resulting force is
F~total = F~+ + F~−
~~kΓ
I/Is
I/Is
~~kΓ
−
.
=
2
2 1 + I/Is + [2(δ+ /Γ]
2 1 + I/Is + [2(δ− /Γ]2
(6.12)
Disregarding terms of order (kv/Γ)4 this can be simplified to
F~total ≈
8~k 2 δ~vI/Is
≡ −β~v .
Γ(1 + I/Is + (2δ/Γ)2 )2
(6.13)
Choosing δ < 0 the constant β is positive and the force is opposing the motion of the
atoms. Hence the atoms are slowed down in all three dimensions.
CHAPTER 6. LASER COOLING OF 6 LI
82
Temperature [µK]
600
500
400
300
200
100
0
1
2
3
4
Detuning [δ/Γ]
Figure 6.2: Theoretical predictions for the temperature of a Doppler cooled 6 Li molasses
versus the detuning for different laser power intensities (equation 6.14). The solid line shows
I/Is = 0, dashed line I/Is = 1 or IT /Is = 6, dot-dashed line I/Is = 0.1 or IT /Is = 0.6.
According to equation (6.13) it is possible to cool down an atomic gas to T = 0 K.
However, heating processes due to the finite momentum transfer of ~k for each random
recoil kick lead to a minimum equilibrium temperature given by [Let88, Xu02]
TD =
~Γ2
[1 + IT /Is + (2δ/γ)2 ],
8kB δ
(6.14)
where IT is the total laser intensity, e.g., 6I in the case of three retro-reflected beams.
Theoretical predictions for the temperature of a Doppler cooled 6 Li molasses versus the
detuning for different laser power intensities based on equation 6.14 are shown in figure 6.2.
For very low laser intensities and a detuning of δ = − Γ2 this yields the Doppler temperature
TD =
~Γ
.
2kB
(6.15)
However, temperatures have been achieved for most alkali atoms well below the Doppler
limit due to a cooling mechanism called Sisyphus cooling. Here the atomic sublevels and
the electrical dipole shift (see chapter 7 on dipole traps) of the atoms in the laser light
have to be considered. However, due to the unresolved excited hyperfine states in 6 Li this
cooling mechanism is not applicable in our experiments. Thus, in our experiments the
Doppler temperature for 6 Li of 140 µK sets the lower limit that can be achieved in an
optical molasses.
83
6.3. MAGNETO-OPTICAL TRAP
6.3.2
Magneto-optical trap
Atoms can not be trapped in an optical molasses because the force has no spatial dependence. However, trapping can be achieved by applying a quadrupole magnetic field
and choosing appropriate laser polarisations and frequencies. Magneto-optical traps were
first realised in 1987 by the groups of S. Chu and D. Pritchard [Raa87]. The underlying principle is most simply explained for a one-dimensional two-level system where the
excited state and the ground state have a total angular momentum Fe = 1 and Fg = 0,
respectively. Two magnetic coils in an anti-Helmholtz configuration lead to a magnetic
field that vanishes in the centre and increases linearly in the vicinity of the centre of the
trap with B = Az. The external magnetic field lifts the degeneracy of the excited states
mF = 0, ±1. The resonance frequencies of the three transitions thus depend on the posi-
tion of the atom. The polarisation of the laser light is opposite circular giving rise to σ ±
transitions. Reformulating the laser light detuning in equation 6.5 gives
δ± = δ ∓ ~k · ~v ± (ge me − gg mg )µB B/~,
(6.16)
Expanding similarly to (6.12) we obtain
F~ = −β~v − κ~r
(6.17)
with the damping constant β and the spring constant κ given by
8~k 2 δI/Is
,
β =
Γ(1 + I/Is + (2δ/Γ)2 )2
(ge me − gg mg )µB A
κ =
β.
~k
(6.18)
(6.19)
The three dimensional MOT works in an analogous way. However, since the magnetic
gradient is twice as large in the axial direction as in the radial direction, a three-dimensional
MOT is anisotropy.
6.3.3
Experimental realisation
In our experiment, the MOT is realised in the glass vacuum cell by three retro-reflected laser
beams with intensities of approximately twice the saturation intensity Is for both trapping
and repumping. During the MOT loading the trapping and repumping lasers are both red
detuned by 4 natural linewidths Γ from the (unresolved) |F = 3/2i to |F ′ = 1/2, 3/2, 5/2i
CHAPTER 6. LASER COOLING OF 6 LI
84
(b)
(c)
(d)
(e)
(a)
(width of atomic cloud)2 [mm2]
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0
0.2
0.4
0.6
0.8
1
(time of flight)2 [ms2]
3.87 mm
Figure 6.3: (a) Temperature measurement of the MOT. The radial width squared is plotted
versus the time of flight squared. The linear fit gives a temperature of 280(15) µK. (b-e)
Absorption images of the MOT taken after (b) 100 µs, (c) 400 µs, (d) 700 µs and (e) 1 ms
time of flight.
and |F = 1/2i to |F ′ = 1/2, 3/2i transitions, respectively. The rather large detuning leads
to a large trapping volume which increases the minimum capture velocity of atoms from
the atomic beam. After 40 s loading from the Zeeman slower the MOT typically contains
more than 108 atoms at a temperature of ∼1 mK. The 1/e lifetime of the MOT is ∼ 60 s.
In order to maximise the number of atoms loaded into the crossed dipole trap the highest
possible MOT phase space density
2π~2 32
D=n
(6.20)
mkB T
is desired. For this reason the MOT is compressed and further cooled for 20 ms by increasing
the magnetic field gradient from 20 G/cm to 50 G/cm, decreasing the detunings of both
trap and repump laser to ∼ Γ/2 and reducing their intensities to well below Is . This reduces
the temperature of the cloud to approximately 280 µK which is twice the limiting Doppler
temperature of 140 µK. The rms radius of the cloud is ∼ 0.5 mm corresponding to a peak
number density on the order of 1011 . Under these conditions the MOT suffers from high
inelastic losses [Kaw93]. Optical pumping into the |F = 1/2i state is achieved by reducing
6.3. MAGNETO-OPTICAL TRAP
85
the repump laser intensity more rapidly than the trapping laser. After transferring the
atoms into the dipole trap all laser beams are switched off by commercial shutters avoiding
any resonant light on the trap.
In figure 6.3(b-e) we show absorption images taken of the MOT after varying time of
flights t between 100 µs and 1 ms. The width of the MOT σ(t) increases according to
r
kB T 2
t,
(6.21)
σ(t) = σ02 +
m
where σ0 is the initial MOT width. The images are taken perpendicular to the magnetic
field axis and it can be clearly seen that the cloud is asymmetric due to the different
magnetic field gradient along the two axes. In more recent experiments we image along
the magnetic field axis and thus obtain more symmetric images of the cloud. We have
plotted the square width of the MOT versus the time squared in figure 6.3 for this new
imaging configuration. A fit to equation 6.21 gives a temperature of 280(15) µK.
86
CHAPTER 6. LASER COOLING OF 6 LI
Chapter 7
Dipole traps
Dipole traps for cold atoms were first realised by S. Chu and co-workers in 1986 [Chu86].
Since then they have become an important tool for studies on ultracold atoms. In this
chapter we will first discuss theoretical aspects. This is followed by a description of the
different dipole traps which were set up during my studies. As an example we show in
figure 7.1 three different images of atoms trapped in distinct geometries which will be
explained later in this chapter. A detailed review on dipole traps can be found in [Gri00]
(a)
(b)
0.6 mm
(c)
0.6 mm
2.3 mm
Figure 7.1: Crossed dipole trap gallery. Absorption images taken in the (a) near symmetric,
(b) elongated and (c) IPG crossed dipole trap (after 300 µs time of flight). The different
geometries are described in this chapter.
87
88
CHAPTER 7. DIPOLE TRAPS
7.1
Theory
Dipole traps rely on the interaction of the electric field of the laser light E with the induced
dipole moment of the atom p. The strength of the induced dipole moment depends on the
polarisibility α of the atom (p = αE). This interaction gives rise to the interaction energy
Udip = − 12 hpEi. The atomic polarisibility α is calculated in many references, e.g., [Gri00].
The trapping potential is then given by
Udip (r) = −
3πc2 Γ
Γ I(r),
+
2ω03 ω0 − ω ω0 + ω
(7.1)
where c is the speed of light, ω0 the atomic transition frequency, ω the driving laser field
frequency, Γ the natural linewidth and I(r) the position dependent laser field intensity.
From the trapping potential it follows that for red-detuned laser fields the dipole force is
attractive. This means that atoms can be trapped in the maxima of red-detuned laser
fields or, equivalently, in the minima of blue detuned laser fields.
Dipole traps are usually far detuned from the atomic resonance. It is nevertheless
possible that atoms absorb and spontaneously emit photons from the dipole trap beam
which consequently leads to heating. The spontaneous photon scattering rate is given by
3πc2 ω 3 Γ
Γ 2
Γdip (r) = −
+
I(r).
2~ω03 ω0
ω0 − ω ω0 + ω
(7.2)
The scattering rate decreases with the square of the detuning while the potential depth
scales only linearly. Therefore, to minimise heating due to spontaneous scattering, it
is beneficial to trap atoms in far detuned dipole traps. Dipole trap potentials can be
considered as being conservative for far detuned laser light because spontaneous scattering
can be neglected. This is in contrast to laser trapping that relies on radiation pressure, e.g.,
magneto-optical traps, which are ultimately limited by the recoil energy. This property is
essential for evaporative cooling which will be described in section 8.1.
7.1.1
Single focussed dipole trap
The easiest realisation of a dipole trap is a single focussed red-detuned laser beam. In such
a trap atoms can be trapped in the vicinity of the focus of the beam. We will consider
an ideal Gaussian beam with power P , waist w0 and a Rayleigh length zR = πw02/λ. The
waist is defined as the 1/e2 minimum radius. The intensity distribution for a Gaussian
89
7.1. THEORY
beam is
r2 2P
exp
−
2
πw 2 (z)
w 2(z)
I(r, z) =
where
(7.3)
r
z 2
w(z) = w0 1 +
zR
(7.4)
is the beam waist along the beam axis. The potential formed by such a single focussed
laser thus takes the form
Udip (r) =
U0
1+
z 2
zR
exp
−2
x2 + y 2
w02 1 +
z 2
zR
!
.
(7.5)
The trap depth U0 can easily be calculated with equations 7.1 and 7.3 by setting r = z = 0.
Typically, the thermal energy of the cold atoms is much smaller than the trap depth.
Thus, atoms predominantly reside near the trap centre. This justifies the harmonic approximation of equation 7.5 for most cases which is simply obtained by the Taylor expansion
h
r 2 z 2 i
.
Udip (r) ≃ −U0 1 − 2
−
w0
zR
(7.6)
In this form the radial and axial trapping frequencies can easily be determined to be
s
s
4U0
2U0
ωr =
and
ωz =
.
(7.7)
2
mw0
mzR2
7.1.2
Crossed dipole trap
Although a single focussed laser beam confines atoms in all three dimensions the axial
confinement is typically rather weak. This may be a limiting factor when high atomic
densities are required. A nearly isotropic trap can be formed in a crossed dipole trap
which consists of two intersecting single focussed laser beams. In the standard crossed
dipole trap geometry the two beams have equal waist and intersect at their foci at right
angles. However, many variations have been realised leading to slightly different trapping
geometries. In our experiment we worked with three geometries which will be described in
sections 7.2 and 7.3. In figure 7.2 we compare the potentials and laser beams of a single
dipole trap with an ideal crossed dipole trap.
Equivalently to equation 7.6, we can formulate the harmonic approximation for an ideal
crossed dipole trap. Assuming that one laser beam propagates along the x-axis and the
90
CHAPTER 7. DIPOLE TRAPS
(a)
(b)
(c)
(d)
Figure 7.2: (a) and (b) show the potential the atoms experience due to the (a) single and
(b) symmetric crossed dipole trap. The beam profiles of the respective traps are depicted
in (c) and (d). Note, that the figures are not to scale.
other along the y-axis this yields
x2 + y 2 + 2z 2 UCDT (r) ≃ −2U0 1 −
.
w02
(7.8)
Obviously, the maximum trap depth is twice the depth of a single dipole trap laser. However, the effective depth is only U0 because atoms with a higher energy can leave the
region of the intersection along one of the arms of the trap. The trapping frequencies of
this crossed dipole trap are
ωr =
s
4U0
mw02
and
√
ωz = 2 ωr .
(7.9)
Similarly, the trapping potential for an arbitrary angle φ between both beams can be
91
7.2. LOW POWER OPTICAL DIPOLE TRAP SET-UP
computed [Gri00]
2x2 2y 2 2z 2 UCDT (r) ≃ −U0 1 − 2 − 2 − 2 ,
wx
wy
w0
where
wx2 =
cos2 (2φ)/w02
1
+ sin2 (2φ)/2zR2
and
wy2 =
sin
2
(2φ)/w02
(7.10)
1
. (7.11)
+ cos2 (2φ)/2zR2
The coordinate system is chosen so that both beams enclose an angle of φ/2 with the
x-axis.
Due to the phase coherence of laser beams, interference of the two beams may lead to
an intensity modulated crossed dipole trap with period D = λ/(2 sin(2φ)), where λ is the
wavelength of the dipole trap laser. This can be largely avoided if both beams have orthogonal linear polarisations. Nevertheless, even then the polarisation of the combined laser
field exhibits a modulation between circular and linear with period D. If both beams have
parallel linear polarisations U0 depends on y and takes the form U0 = Umax cos2 (πy/D).
The situation is somewhat different when both laser beams are frequency detuned from
each other, e.g., by means of an acousto-optical modulator. In this case the interferences
move much faster than the atoms which then experience a smooth time-averaged potential.
The multi-level structure of real atoms leads to a small intensity modulation even for
perfectly orthogonal polarised light beams [Gri00]. The amplitude of this modulation is
1
g m ∆ /∆,
3 F F HF
where ∆HF is the hyperfine splitting. Due to the far detuning of the lasers
in our experiments used and the small hyperfine splitting of 228 MHz this effect is negligible
for 6 Li atoms.
7.2
Low power optical dipole trap set-up
In this section we describe the set-up of our initial dipole trap formed by a 22 W Yb:YAG
thin disc laser (ELS VersaDisk 1030-20 SF). We have demonstrated the production of
molecular Bose-Einstein condensates of 6 Li dimers and degenerate Fermi gases in this
versatile low power crossed dipole trap which will be the topic of section 8.2.
Experimentally, the generation of such gases is highly complex and to date has generally
relied on very high power optical dipole traps (∼140 W) [Kin05] or resonant build-up cavities [Joc03b], evaporative cooling of two different hyperfine states [Gre03] or sympathetic
cooling with another isotope [Bou04, Par05] or another species [Zwi03, Roa02, Sil05]. Our
92
CHAPTER 7. DIPOLE TRAPS
f=250mm (L2)
NDF
λ/2
f=250mm
(L3)
Yb:YAG
beam
dump
0110 photodetectors
f=250mm (L1)
glass cell
λ/2
AOM
beam dump
Figure 7.3: Set-up of the crossed dipole trap. Before entering lens L1 the beam is near
collimated with a 1/e2 radius of 3.4 mm. By translating lens L3 we can tune the aspect
ratio of the trap over a wide range.
system includes a number of simplifications over previous experimental set-ups and allows
us to easily tune the trap geometry from near spherically symmetric to highly elongated
cigar shaped. Two beams of 16 W and 14.5 W are crossed at 80◦ at the centre of the MOT
(figure 7.3). To avoid interferences the polarisation of the second arm is rotated by 90
degrees with respect to the first. Both beams are focussed to a waist of approximately
30 µm in the glass cell. When the beams intersect at their foci, the crossed dipole trap is
nearly spherically symmetric, with an aspect ratio of 1.4. By translating the lens used to
focus the second arm along the beam we can tune the aspect ratio of the crossed dipole
trap. This reduces the trap depth of the crossed part inversely to the square of the beam
waist of the second arm in the overlap region. However, the total trap depth is reduced by
less than 50%. In figure 7.4 we show the laser beam profile and the dipole trap potential
of such an elongated crossed dipole trap.
7.2. LOW POWER OPTICAL DIPOLE TRAP SET-UP
(a)
93
(b)
Figure 7.4: (a) Laser beam profile and (b) dipole trap potential of an elongated crossed
dipole trap obtained by translating the focus as described in the text.
7.2.1
Intensity stabilisation of the dipole trap laser
To evaporatively cool these atoms (evaporation will be explained in detail in chapter 8)
it is necessary to precisely control the absolute intensity of the laser over approximately
three orders of magnitude. We achieve this by locking the laser intensity with a PID
controlled acousto-optical modulator. A small fraction of the transmitted dipole trap light
is measured on two photodetectors1 . The signals from the detectors are added together
and sent to one input of the PID controller2 . This measured voltage is compared to a
setpoint signal from our computer control system and used to derive an error signal which
determines the amount of radio-frequency power sent to the AOM in figure 7.3. Light
entering one photodetector is attenuated by a factor of ∼ 30 using a Neutral Density Filter
(NDF) so that with the same gain the signals from each detector will differ by this factor.
At high intensities, the output from the unattenuated detector is saturated and simply
provides a fixed offset on the combined signal. The signal from the other detector is below
saturation and detects changes in intensity at high powers. At lower intensities, the signal
from this photodetector decreases significantly and eventually becomes negligible compared
to the signal from the unattenuated detector which effectively takes over once the intensity
has been reduced by a factor ∼ 30. This allows us to precisely control the laser intensity
over the required range for evaporation using low noise linear detectors, without the need
1
2
Newfocus: Model 1623
SRS: SIM 960
94
CHAPTER 7. DIPOLE TRAPS
Number of atoms
10000
2 linear PDs
τ =6 s
5000
2500
1 linear PD
τ = 650 ms
1000
0
2000
4000
Hold time [ms]
6000
Figure 7.5: Lifetime of the crossed dipole trap with one and two photodetectors (PD) for
intensity stabilisation. Number of atoms in state |1i for different hold times in a very
shallow crossed dipole trap (∼ 1 µK). The lifetime increases from 650 ms to approximately
6 s due to the improved trap stability of the dual photodetector scheme.
for a logarithmic amplifier.
When only a single detector is used, the small (mV) level photodetector signals are
susceptible to electronic ripple at mains frequencies which can lead to significant intensity
noise on the trap laser [Geh98]. The dual photodetector scheme overcomes this problem
by always allowing larger control voltages (of order a few 100 mV) at the input to the PID.
To demonstrate this improvement, we compare the lifetime of cold atomic clouds near the
broad s-wave Feshbach resonance in very shallow traps using single and dual photodetector
configurations in figure 7.5. The lifetime is seen to increase from 650 ms to approximately
6 s due to the improved trap stability.
7.2.2
Trapping frequency measurement
For a systematic analysis of trapped atoms it is often necessary to know the trapping
frequencies of the dipole potential. We measured the trapping frequencies in the near
symmetric crossed dipole trap by parametric heating (figure 7.6 (a)) and by oscillations
of the atomic cloud (figure 7.6 (b)). Furthermore, we determined the weak axial trapping
frequency of the elongated trap by cloud oscillations (figure 7.6 (c)).
Sinusoidally modulating the dipole trap laser intensity can lead to parametric heating
95
7.2. LOW POWER OPTICAL DIPOLE TRAP SET-UP
(a)
(b)
12000
1.5
1
amplitude [pixel]
number of atoms
10000
8000
6000
0.5
0
−0.5
−1
4000
2
4
6
8
10
12
14
16
−1.5
0
0.5
modulation frequency [kHz]
(c)
1
1.5
2
2.5
3
3.5
time [ms]
165
160
Amplitude [µm]
155
150
145
140
135
130
125
0
100
200 300
1000 1100
Hold time [ms]
Figure 7.6: Trapping frequency measurements. (a) shows the data obtained by parametric
heating the cloud in the near symmetric crossed dipole trap. The double dip structure is
from exciting the radial and axial direction, respectively. The trapping frequencies have
also been determined via trap oscillations in the (b) axial direction of the near symmetric
CDT and (c) axial direction of the elongated CDT.
which eventually results in atom loss. In a harmonic trap the parametric resonance occurs
at multiples of twice the trap frequency. However, in a Gaussian potential the spacing
between energy levels becomes smaller at higher energies. This leads to an asymmetric
line shape of the loss resonance. On the high frequency side the resonance shows a sharper
drop than on the lower frequency side. Depending on the initial temperature of the atoms
the peak of the resonance is shifted to lower frequencies.
In our experiment (figure 7.6 (a)) we first evaporatively cool the atoms in the crossed
dipole trap to a temperature of approximately 10 µK. After evaporation, the majority of the
96
CHAPTER 7. DIPOLE TRAPS
atoms are trapped in the region of the crossed dipole trap. We then apply a modulation of
the laser intensity of approximately 3% for 1 s. Two atom loss resonances can be resolved
(figure 7.6 (a)) corresponding to parametric driving along the radial and axial direction,
respectively. The maximum loss occurs at 7.4 kHz and 9.8 kHz. This yields trapping
frequencies of 10.1 kHz radially and 13.4 kHz axially at full power. However, due to the
Gaussian trapping potential this is most likely an underestimation.
To measure cloud oscillations the atomic cloud is first displaced from the equilibrium
position by applying a magnetic field gradient. After switching off the magnetic field
gradient cloud oscillations can be observed by absorption imaging. In this way we have
measured the axial trapping frequencies in both the elongated and near symmetric trap.
We evaporatively cool the atoms to a temperature of 270 nK to measure axial trap oscillations in the near symmetric crossed dipole trap. After applying a magnetic field gradient
of 19 G/cm the cloud oscillates around the equilibrium position as shown in figure 7.6 (b).
The frequency of 736 Hz corresponds to a trapping frequency of 15.6 kHz axially at full
power. We have not observed radial cloud oscillation; however, from the axial measure√
ment a trapping frequency of ∼ 15.6/ 2 = 11.0 kHz can be inferred. The measurements
of the cloud oscillations are consistent with the results obtained by parametric driving.
From this, we can calculate the trap depths of both arms yielding 0.77 mK and 0.70 mK,
respectively.
The axial trapping frequency of the elongated crossed dipole trap were measured similarly. After reducing the laser power by a factor of 1120 we measured a trapping frequency
of 23.8 Hz. However, this trapping potential is due to a combination of a magnetic field
curvature and the crossed beam which will be discussed in more detail for the high power
dipole trap laser in section 7.3.2.
7.3
New dipole trap formed by a 100 W fibre laser
In the previous section we described the experimental set-up used to produce our first
molecular Bose-Einstein condensates. However, due to the limited optical power of the
ELS VersaDisk laser the number of atoms loaded into the dipole trap was rather small in
this configuration. We have therefore upgraded the dipole trap laser to a high power fibre
laser from IPG3 . This laser has an output power of over 100 W at a centre wavelength of
3
IPG: YLR-100, linear polarisation
97
7.3. NEW DIPOLE TRAP FORMED BY A 100 W FIBRE LASER
beam
dump
f=250mm
dipole trap
photodetector
f=250mm
λ/2
glass cell
f=250mm
λ/2
IPG fibre laser
AOM
beam dump
Figure 7.7: High power crossed dipole trap set-up. The dipole trap is formed by two laser
beams which intersect at an angle of ∼14 degrees. Both beams are focussed to a waist of
40 µm.
∼1075 nm. The laser light has linear polarisation and is characterised by an M 2 of less
than 1.05. Spectrally, the output consists of many longitudinal modes distributed over less
than 3 nm. In the new configuration we have increased the initial number of atoms loaded
into the dipole trap and, consequently, achieve larger molecular Bose-Einstein condensates.
In this chapter we present the set-up and characterisation of the new system.
7.3.1
High power crossed dipole trap set-up
Several modifications were made to the optical dipole trap set-up which can be seen in
figure 7.7. Similar to the previous dipole trap the new set-up consists of a crossed dipole
98
CHAPTER 7. DIPOLE TRAPS
trap. However, the two beams intersect at an angle of only approximately 14 degrees. We
have found that this provides a good compromise of high initial phase space densities and
a large trapping volume.
Both arms of the dipole trap have a beam waist of 40 µm. The beam waist was
somewhat limited by the high laser power used in the experiment. Counterintuitively,
a larger minimum beam waist at the trap centre would have resulted in a higher laser
intensity at the glass cell. In a test set-up we found that 50 W of the IPG dipole trap laser
focussed to a waist of 45 µm at the glass surface damages the coating of the cell. With
the beam waist of 40 µm at the centre of the glass cell, however, we are confident that no
damage of the coating on the cell can occur.
The available dipole trap power nearly doubles if both dipole trap arms are produced
from the same laser beam. Furthermore, we will discuss in section 7.3.3 that it is advantageous that the two laser beams are near co-propagating. To fulfill both requirements we
send the collimated beam after passing the glass cell back through the cell without passing
the atoms (see figure 7.7). This beam is subsequently focussed again to form the second
arm of the dipole trap. A beam path around the glass cell would have been considerably
longer which potentially would have limited the mechanical stability of the trap.
In section 7.2.1 we presented our technique to stabilise the laser light intensity of the
low power dipole trap. Two photodetectors in series enabled intensity stabilisation over
a few orders of magnitudes. In the new dipole trap set-up we make use of a similar
idea for which we only need to use one photodetector. When loading the dipole trap the
intensity is not yet actively stabilised. We subsequently reduce the laser power by a factor
of approximately 10 by simply decreasing the laser current in the fibre laser. We find that
the laser intensity of the IPG laser allows stable trapping down to an output power as low
as 12 W. By the end of the ramp a photo diode which detects a fraction of the dipole
trap laser light becomes unsaturated. From that moment on the optical power is actively
stabilised with a PID controlled acousto-optical modulator as discussed in 7.2.1. As shown
in figure 7.7 the laser light used for stabilisation is taken from a reflection from the glass
cell.
99
90
(a)
(b)
80
Number of atoms [103]
7.3. NEW DIPOLE TRAP FORMED BY A 100 W FIBRE LASER
70
Number of atoms [103]
80
70
60
50
40
30
60
50
40
20
10
0
1
2
3
Modulation Frequency [kHz]
30
1
4
600
4
1.5
4
700
(c)
(d)
650
Temperature [µK]
550
Temperature [µK]
1.5
2
2.5
3
3.5
Modulation Frequency [kHz]
500
450
400
350
600
550
500
450
300
0
1
2
3
Modulation Frequency [kHz]
4
400
1
2
2.5
3
3.5
Modulation Frequency [kHz]
Figure 7.8: Measurements of the radial trapping frequency in the (a) and (c) single dipole
trap and (b) and (d) crossed dipole trap at a final power of ∼1 W. (a) and (b) show the
trap loss after parametric heating as a function of the modulation frequency. In (c) and
(d) the temperature is plotted for both geometries. In the case of the single dipole trap we
can clearly observe two distinct extrema which, we believe, is due to astigmatism in the
dipole trap.
7.3.2
Trapping frequency
The radial trapping frequency was measured by trap loss and heating due to parametric
drivings which was described in section 7.2.2. Initially, the cloud was evaporatively cooled
by lowering the dipole trap to a final power of ∼0.5 W. After doubling the laser power
the cloud was well confined in the bottom of the dipole trap. At this power the dipole
trap intensity was modulated by 3% for 1 s. Figures 7.8 (a) and (b) show the trap loss in
100
Axial trapping frequency [Hz]
CHAPTER 7. DIPOLE TRAPS
120
100
80
60
40
20
0
0
200
400
600
800
Dipole trap laser power [mW]
1000
Figure 7.9: Measurement of the axial trapping frequency for four different laser intensities
by observing cloud oscillations. The trap frequency due to the curvature in the Feshbach
magnetic field was determined to be 20(2) Hz at a magnetic field of 694 G. The inset shows
the measurement for 230 mW yielding a trapping frequency of 59.7(6) Hz.
the single dipole trap and in the crossed dipole trap, respectively, versus the modulation
frequency. In 7.8 (c) and (d) the temperature is plotted for both geometries. In the case
of the single dipole trap we can clearly observe two distinct extrema which is, we believe,
due to astigmatism in the dipole trap. The two minima in the atom number are located
at 1.73 kHz and 2.27 kHz. The maximum heating when driving radial oscillations of the
crossed dipole trap has been determined to be at 2.35 kHz.
As the trapping frequencies obtained by the maximum loss is most likely an underestimation, our best guess value of the radial trapping frequency of the crossed dipole trap
is
p
ωrad = 2π · 1200 P/W Hz,
(7.12)
yielding ∼11 kHz at the full dipole trap laser power of 80 W. The total trap depth at full
power is then ∼ 3 mK.
We measured the axial trapping frequency for four different laser intensities by observ-
7.3. NEW DIPOLE TRAP FORMED BY A 100 W FIBRE LASER
101
ing cloud oscillations as shown in figure 7.9. This allowed us to determine the magnetic
potential due to the curvature in the Feshbach magnetic field. It results in a trapping
frequency of ωmag =20(2) Hz at a magnetic field of 694 G. The inset in figure 7.9 shows the
measurement for 230 mW yielding a trapping frequency of 59.7(6) Hz. The results can be
summarised by
ωaxial =
p
814(180) B/kG + 13.6(4) P/mW Hz.
(7.13)
From this it follows that the curvature of the magnetic field in the horizontal plane is
s
2 m
ωmag
d2 B
atom
=
= 0.024(3) · B cm−2 .
(7.14)
2
dz
µB
This is in reasonable agreement with the field curvature measured by the Gauss meter (see
section 4.4).
7.3.3
Lifetime
In our initial crossed dipole trap produced from the IPG fibre laser both beams intersected
at an angle of approximately 165 degrees. We found that the lifetime in this configuration
was considerably shorter than in the single dipole trap. In figure 7.10(a) we compare the
number of atoms in state |1i at full power (blue dots) and at 17% of the initial trap depth
(red dots). The lifetime was measured to be 1.7(1) s at full power and 1.2(1) s at 17% of the
initial trap depth. This lifetime is not sufficient for most of our experiments. Furthermore,
we found that the lifetime was very sensitive to the polarisation of both arms. The data in
figure 7.10(a) was taken for perpendicular linear polarisations for which we have achieved as expected - the longest lifetimes. Rotating the polarisation by only 5 degrees reduced the
number of atoms after 1 s hold time approximately by 50%. We believe that the relatively
short lifetime is due to interference of the two laser beams which arises because of the equal
spacing of the various frequency components.
Other groups with similar crossed dipole traps have made the observation that the trap
lifetime can be increased significantly when both beams intersect in a near co-propagating
geometry4 . We have successfully followed this idea and changed our geometry to be nearly
co-propagating as discussed in 7.3.1. In figure 7.10(b) we present our lifetime measurements
of a |1i − |2i spin mixture near the broad Feshbach resonance at 770 G and of a non-
interacting gas of spin-polarised atoms. The number of atoms in state |1i is plotted for
4
Private Communication with F. Schreck, University of Innsbruck.
102
CHAPTER 7. DIPOLE TRAPS
(a)
(b)
5
75000
Number of atoms
Number of atoms
10
50000
4
10
25000
0
1
2
3
4
0
5
10
Hold time [s]
15
20
25
30
35
Hold time [s]
Figure 7.10: Lifetime of the crossed dipole trap in a near counter-propagating and near
co-propagating geometry. We show the number of atoms in state |1i for different hold
times. (a) In the counter-propagating geometry the lifetime was measured to be 1.7(1) s
at full power (blue dots) and 1.2(1) s at 17% of the initial trap depth (red dots). (b) In
the co-propagating geometry we measure the lifetime of a |1i − |2i spin mixture near the
broad Feshbach resonance at 770 G and of a non-interacting gas of spin-polarised atoms.
At a laser power of ∼100 mW we obtain a lifetime of 7.6(3) s for the spin mixture (red
dots) and 37(2) s for the spin-polarised gas (blue dots).
both data runs. At a laser power of ∼100 mW we obtain a lifetime of 7.6(3) s for the spin
mixture (red dots) and 37(2) s for the spin-polarised gas (blue dots).
7.4
Loading the crossed dipole trap
In section 6.3.3 we described our experimental procedure to produce magneto-optical traps
with high phase-space densities. These conditions are ideal to load the optical dipole trap
efficiently. The loading procedure is equivalent for both optical dipole traps. The only
difference is that the ELS laser is switched on at full power during loading the MOT while
the IPG laser is switched on only 100 ms before compressing the MOT. Figure 7.11 shows
absorption images taken of the crossed dipole trap formed by the IPG fibre laser. The
alignment of the MOT with respect to the dipole trap can be controlled by taking the
image shortly after the MOT has been switched off (figure 7.11(a)). In this image we
103
7.4. LOADING THE CROSSED DIPOLE TRAP
(a)
(b)
3.225 mm
3.225 mm
Figure 7.11: Absorption images of the crossed dipole trap, taken (a) 1.3 ms after switching
off the MOT and (b) after 100 ms of plain evaporation. Both images are shown on the
same colour scale.
can see the expanding magneto-optical trap and the dipole trap simultaneously. After
plain evaporation with constant dipole trap power (figure 7.11(b)) the atom density in the
crossed region increases while the more energetic atoms leave the trap.
In figure 7.12 we compare the number of atoms loaded into the crossed dipole trap for
two different trapping geometries. We show the atom number versus the optical laser power
for the near symmetric low power crossed dipole trap in figure 7.12 (a) and for the high
power crossed dipole trap in figure 7.12 (b). We determined the atom number by means
of absorption imaging after sufficient plain evaporation to reach an approximate steady
state. In both plots we only count the atoms in the crossed part of the trap. We were
able to load &100,000 atoms into the low power trap and 106 atoms into the high power
dipole trap. The maximum number of atoms loaded into the elongated low power trap
(not shown here) was approximately 400,000 atoms. However, at the atomic temperature
in this trap the atoms do not accumulate in the shallow crossed part and hence the total
atom number in both beams was measured.
104
(a)
CHAPTER 7. DIPOLE TRAPS
(b)
70
600
Number of atoms [103]
Number of atoms [103]
60
50
40
30
20
400
300
200
100
10
0
500
6
0
8
10
12
14
Dipole trap laser power [W]
30
40
50
60
70
80
Dipole trap laser power [W]
90
Figure 7.12: Number of atoms in state |1i loaded into the crossed dipole trap after plain
evaporation. We compare atoms loaded into the (a) near symmetric low power crossed
dipole trap formed by the ELS laser and (b) high power crossed dipole trap. The total
number of atoms (state |1i and |2i) is approximately twice the numbers given here.
Atomic densities
Not only the number of atoms loaded into the crossed dipole trap is of interest, but also
the atom density. Making the typically valid assumption that the thermal atomic gas has
a Gaussian spatial distribution the peak density is given by
n=
N
,
σx σy σz π 3/2
(7.15)
where σi is the 1/e2 radius. In a harmonic trap the spread of a thermal cloud in direction
i is given by
σi =
s
2kB T
.
ωi2m
(7.16)
Therefore, the peak density can also be calculated from
n=N
mω 2 3/2
.
2πkB T
(7.17)
We achieve high initial atomic densities in the near symmetric low power and in the highpower crossed dipole traps due to the high trapping frequencies. In both cases we achieve
atomic peak densities as high as n ∼ 2 · 1013 atoms·cm−3 per state after plain evaporation.
Chapter 8
Bose-Einstein condensation of
molecules
We have achieved molecular Bose-Einstein condensates (MBEC) in all three different dipole
trap configurations. The dipole trap set-ups were described in detail in chapter 7. In this
chapter we first present theoretical aspects of evaporative cooling. Then, we focus on the
experimental realisation of the MBEC in the low power crossed dipole trap. The results
have been published in [Fuc07b]. In section 8.3 we show the more recent data obtained in
the high power crossed dipole trap.
8.1
Theory of evaporative cooling
As we will see shortly, we achieve phase space densities on the order of 10−3 after loading the
dipole trap from the magneto-optical trap. As discussed in chapter 3 quantum statistical
effects only become significant when the phase space density is approximately one. To
bridge this gap we use the well established technique of evaporative cooling.
The main idea of evaporative cooling is that the most energetic particles escape the
trap and the remaining fraction rethermalises at a lower temperature. Initial evaporation is
achieved in a plain evaporation stage at constant dipole trap power after loading the trap.
However, after a few 100 ms the evaporation process almost stagnates because fewer and
fewer high energetic atoms leave the trap. By actively removing the most energetic particles
continuously from the trap ongoing evaporation is possible. In optical dipole traps this
forced evaporation is achieved by lowering the potential depth in time, first demonstrated
105
106
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
in 1995 [Ada95]. For efficient evaporation a fast rethermalisation rate is required. This
depends on the number of elastic scattering processes per second per particle which is given
by the elastic collision rate γ.
As discussed in chapter 2, identical fermions do not elastically s-wave scatter. Thus,
elastic collisions are typically frozen out in an ultracold gas of identical fermions. To
evaporatively cool fermions two main approaches have been followed. Firstly, fermionic
atoms have been cooled sympathetically by evaporatively cooled bosons. The bosonic
coolant can either be a different isotope to the fermion [Sch01, Tru01] or another species
[Roa02, Had02, Sil05, Tag08]. The first quantum degenerate dilute Fermi gas, however,
was achieved by evaporatively cooling two different hyperfine states of
40
K [DeM99b].
Since then, this technique has also been applied to produce quantum degenerate 6 Li gases
which was pioneered by the Innsbruck group [Joc03a].
In our experiment we also evaporatively cool a near 50-50 spin mixture of the two lowest
hyperfine states of 6 Li. For this case, the elastic scattering rate is given by [O’H01]
γ = nσv =
4πNmσν 3
.
kB T
(8.1)
Here, n is the atomic density of each state, v is the mean velocity of the atoms, N is
the number of atoms in each state, ν = ω/2π the mean trapping frequency and σ the
elastic s-wave scattering cross section of colliding atoms in the two different hyperfine
states (equation 2.15). Equation 8.1 also holds for a single species bosonic gas with total
number N. However, due to quantum statistics the elastic scattering cross section then
doubles for the same scattering length a and relative scattering wave number k.
In the strongly interacting regime near the broad s-wave Feshbach resonance centred
at 834 G the scattering length becomes σ ≃ 4π/k 2 , where the wave number k is given by
the temperature dependent relative scattering velocity v by k = mv/2~. The temperature
after loading the dipole trap is typically on the order of 100 µK resulting in a scattering
cross section of ∼ 3 · 1011 cm−2 . In the near symmetric and high power crossed dipole traps
the initial elastic collision rate is consequently as high as ∼ 1.5 · 104 s−1 . Initial phase space
densities in both traps are
D=N
(~ω)3
∼ 0.003.
(kB T )3
(8.2)
The group led by J. E. Thomas developed scaling laws predicting the remaining atom
number N, the increase in phase space density D and the elastic collision rate γ as a
107
8.1. THEORY OF EVAPORATIVE COOLING
function of the dipole trap depth U [O’H01, Luo06] during evaporation. The scaling laws
assume that the cutoff parameter η = U/kB T is kept constant during the evaporation.
During evaporative cooling in dipole traps the cutoff parameter is typically η ∼ 10. The
scaling laws for this case are
N
=
Ninitial
D
=
Dinitial
γ
=
γinitial
U
0.19
Uinitial
U −1.3
Uinitial
U 0.69
.
Uinitial
(8.3)
(8.4)
(8.5)
Collisions with background atoms and other imperfections are not considered in these
equations.
Evaporation relies on atoms being scattered into low lying energy states. As the temperature decreases the lowest energy states become more densely populated with a probability
f FD . Thus, due to the Pauli principle, the scattering rate in fermionic gases decreases because of the reduced number of states that one of the atoms can be scattered into. This
effect becomes significant as the temperature approaches T = TF . It can be shown that
the evaporation rate scales as T /TF for T ≪ TF [Hol00, O’H01]. However, this does not
ultimately limit the lowest achievable temperature of fermionic gases. Particularly at low
temperatures evaporative cooling is limited by atom losses and heating. This is more severe
in fermionic gases because atom losses deep from the Fermi sea lead to significant heating
of the cloud [Car05].
As with other 6 Li experiments, we exploit the broad collisional Feshbach resonance at
834 G to tune the interaction between particles in states |1i and |2i. When the magnetic
field is tuned just below the resonance, where the atomic scattering length is large and
positive, a weakly bound molecular state exists, as discussed in section 2.3. As we evaporatively cool the cloud we form weakly bound Feshbach molecules by means of three-body
recombination (see section 2.3). In thermal equilibrium the temperature of the molecules
equals the temperature of the atoms. Furthermore, the molecules have the same trapping
frequencies as the atoms because doubling the mass is compensated by the doubled polar√
isibility. The spatial extent of the molecules is consequently reduced by a factor 2. Thus,
in thermal equilibrium predominantly free atoms evaporate from the dipole trap. It is remarkable that the statistics change from Fermi-Dirac to Bose-Einstein as we evaporatively
cool.
108
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
(a)
(b)
100
10
Eσ (µK)
Particle number 10
3
100
1
10
near symmetric
elongated
0.01
0.1
1
10
Final laser power (W)
0.1
near symmetric
elongated
0.1
1
10
Final laser power (W)
Figure 8.1: Evaporative cooling in the crossed dipole trap on the molecular side of the
Feshbach resonance at a magnetic field of 770 G (images taken at 694 G). (a) Number of
atoms in state |1i in the near symmetric (squares) and elongated (circles) crossed dipole
trap is plotted versus final dipole trap laser power. The solid line represents the scaling
law prediction for a cutoff parameter of η = U/kB T = 10 [O’H01]. (b) Release energy as a
function of final dipole trap laser power. The vertical dashed line indicates the temperature
below which molecule formation becomes significant. Below the vertical dashed line the
release energy no longer corresponds to the temperature of the mixture of atoms and
molecules present in the trap. The solid (dash-dotted) line corresponds to one-tenth of the
total trap depth in the near symmetric (elongated) trapping geometry.
8.2
8.2.1
MBEC in a Low Power Crossed Dipole Trap
Evaporative Cooling
In sections 7.2 and 7.4 we presented the set-up and loading of our low power crossed
dipole trap. With our high trapping frequencies, tunably large scattering lengths and
high initial phase space densities, the requirements for efficient evaporation, are readily
obtained. A pair of high field coils (described in section 4.4) allows us to subject the
optically trapped atom cloud to magnetic fields of up to 1500 G. This covers the entire
range of the broad Feshbach resonance between the lowest two hyperfine states of 6 Li at
834 G. Our evaporation consists of three stages: the first is a plain evaporation phase in
which the magnetic field is turned up to 770 G and the laser intensity is held fixed at the
8.2. MBEC IN A LOW POWER CROSSED DIPOLE TRAP
109
maximum trap depth for 1 s. During this stage atoms accumulate in the crossed part of the
trap via elastic collisions, locally increasing the phase space density. Next, the first ramp of
the laser intensity takes place in which the laser power is lowered linearly by approximately
a factor of 30 over 1.5 s to the point where the second photodetector is no longer saturated.
Then a second sweep of 1.5 s takes place, this time logarithmic in shape, in which the laser
intensity is reduced by another factor of approximately 30. We find that the evaporation
works well on either side of the Feshbach resonance when the magnitude of the scattering
length is & 2000a0 .
Figure 8.1(a) shows the number of atoms in state |1i as a function of the final laser
power after evaporation at a magnetic field of 770 G in two trapping geometries. Due to
the 50/50 spin mixture of states |1i and |2i this is approximately half the total number of
atoms. Absorption images are taken at 694 G after ramping the magnetic field in 100 ms.
The elongated trap (circles) was obtained by translating the lens L3 in figure 7.3 by 10 mm
resulting in an aspect ratio of 15 for atoms at the bottom of the crossed dipole trap. In this
trap, the cloud is initially much hotter than the trap depth of the crossed part of the trap
and a significant fraction of atoms reside in the arms, i.e., outside of the region where the
beams intersect. In the near symmetric trapping geometry (squares), however, the atoms
are well confined in the crossed part after 1 s of plain evaporation. The solid line represents
the scaling laws prediction (equation 8.3) for an ideal evaporation with a cutoff parameter
of η = U/kB T = 10. The atom numbers follow the scaling laws well for the first factor
of 30 in the evaporation for the elongated trap. After further evaporation, the number of
atoms drops below the ideal evaporation line. One reason for this is that the density of
atoms in the arms of the crossed trap becomes very low and it is possible that not all atoms
were included in the atom count. A second reason arises from the fact that below 1 µK a
significant number of molecules will be formed (when the molecular binding exceeds the
thermal energy of the cloud). Our efficiency for detecting molecules at the imaging field
is approximately 70% of that for detecting atoms, which would account for part, but not
all, of the reduction in signal [Joc04]. In the near symmetric trap (aspect ratio of 1.4), the
atom number decreases more rapidly than the scaling laws predict for η = 10. We believe
this is due to the lower initial number of atoms and a shorter trap lifetime at lower powers
as this data was obtained using only a single photodetector for trap intensity stabilisation.
Also shown in figure 8.1(b) is a plot of the measured release energy of the cloud as a
function of final trap depth when evaporating on the molecular BEC side of the Feshbach
110
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
resonance. The classical release energy is defined (in units of temperature) as
2
σ 2 ωrad
matom
,
(8.6)
2
t2TOF
kB 1 + ωrad
is the atomic mass, σ is the rms radius of the cloud in the radial direction,
Eσ =
where matom
ωrad /2π is the radial trapping frequency and tTOF is the time of flight before taking the
image. For final trap depths above 10 µK the release energy is equal to the temperature
of the atomic cloud. Below 10 µK a mixture of atoms and molecules will be present in
the cloud and the release energy measured no longer represents the atomic temperature as
the mass of the molecules is twice the mass of the atoms. Furthermore, as the molecules
become degenerate the cloud profile becomes bimodal and a Gaussian fit no longer provides
a representative width for a temperature measurement. The significant drop of the release
energy at 50 mW, particularly in the case of the elongated dipole trap, can be attributed
to the formation of a molecular BEC, as described in section 8.2.2. The solid line in
this figure represents one-tenth of the total trap depth in the near symmetric trapping
geometry, indicating that the evaporation is highly efficient with a cutoff parameter of
η = 10. The dash-dotted line shows one-tenth of the total trap depth in the elongated
trapping configuration. Here, the evaporation is less efficient as at higher temperatures a
significant fraction of the atoms reside in the arms of the crossed trap where the density is
lower and elastic collisions are less frequent. As the evaporation progresses, atoms are lost
from the arms and the coldest atoms collect in the dimple where the two beams intersect.
As the temperature is lowered molecules initially form in the region of the highest atomic
phase space density which is in the dimple.
8.2.2
Quantum degenerate Bose and Fermi gases
As seen in figure 8.1 temperatures well below 1 µK can be achieved for the lowest final
trap powers. When the magnetic field is tuned below the Feshbach resonance and the
temperature drops below the molecular binding energy, molecules are formed and remain
trapped. The molecules can elastically scatter from each other and due to the low inelastic
losses efficient evaporative cooling of molecules can take place [Pet04]. These molecules are
bosonic and can therefore undergo Bose-Einstein condensation at sufficiently high phase
space density as has previously been reported [Joc03b, Gre03, Zwi03, Bou04, Par05].
We have produced a molecular BEC of 6 Li2 in a near symmetric crossed dipole trap
(with an aspect ratio of 1.4) and in an elongated trap (aspect ratio of 15). In the elongated
111
8.2. MBEC IN A LOW POWER CROSSED DIPOLE TRAP
P =50 mW
Nmol =30 000
P =33 mW
Nmol =22 000
P =25 mW
Nmol =16 000
P =18 mW
Nmol =12 000
Figure 8.2: Three dimensional absorption images of the molecular gas during evaporative
cooling in the elongated trap taken after 1.5 ms time of flight. The images were taken at
B = 694 G after evaporation at B = 770 G.
trap configuration we can condense up to 9,000 molecules. This geometry has the advantage
of a smaller trapping frequency in one direction which means that the size of the BEC is
much larger than our imaging resolution of ∼4 µm, allowing us to discriminate the thermal
and condensed molecules more easily. In figure 8.2 we show three dimensional absorption
images of the molecular gas during evaporative cooling in the elongated trap. Following
evaporation at 770 G, the images are taken after 1.5 ms time of flight at B = 694 G.
Figure 8.3 shows integrated cross-sections along the weakest trapping direction through
expanded clouds for a near symmetric (left) and elongated crossed dipole trap (right) at
various final trap depths. Absorption images are taken 700 µs after release from the trap
and the emergence of a bimodal distribution becomes evident. As the temperature drops
below the critical temperature for condensation (∼250 nK) a high density peak appears
in the centre of the expanded clouds. When evaporating in the elongated trap we detect
12,000 molecules with a condensate fraction of 75%. However, due to the reduced detection
efficiency (70%) when imaging molecules the actual number of molecules will be somewhat
higher [Joc04]. For the weakest elongated dipole trap shown the trapping frequencies are
23 Hz axially and 340 Hz radially. The dashed, dash-dotted and solid lines are fits to
the Gaussian, Thomas-Fermi, and combined profiles showing the thermal and condensed
fractions, respectively. The images were taken at B = 694 G after evaporation at B = 770 G.
112
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
Near symmetric trap
Elongated trap
N=2700
P=15 mW
N=12000
P=14 mW
N=4000
P=26 mW
N=16000
P=25 mW
N=6200
P=39 mW
N=21000
P=39 mW
Figure 8.3: Integrated cross sections along the weakest trapping axis 700 µs after release
from the trap showing the transition to a molecular BEC in a near symmetric (left) and
an elongated (right) crossed dipole trap. The dashed, dash-dotted and solid lines are fits
to the Gaussian, Thomas-Fermi and combined profiles showing the thermal and condensed
molecules, respectively. When evaporating in the elongated trap we observe a condensate
fraction of 75% for a total of 12,000 molecules. All images were taken at B = 694 G after
evaporation at B = 770 G.
8.2. MBEC IN A LOW POWER CROSSED DIPOLE TRAP
(a)
113
(b)
150 µm
150 µm
Figure 8.4: In situ absorption images of a trapped molecular BEC and DFG, in identical
crossed dipole traps following (a) evaporation at 770 G (imaging at 694 G) and (b) evaporation and imaging at 1100 G. The difference between a MBEC and a DFG is clearly
evident in the density and size of the clouds.
Degenerate bosons and fermions behave very differently and with our system we can
readily compare the two cases. A striking example is shown in figure 8.4 where we compare
in situ absorption images of (a) a trapped molecular BEC on the a > 0 (694 G) side of the
Feshbach resonance and (b) a trapped degenerate Fermi gas on the a < 0 (1100 G) side of
the Feshbach resonance. These two cases represent two identical runs of the experiment
with identical evaporation ramps (final dipole trap power 9 mW), the only difference being
the magnetic field at which the evaporation and imaging take place. On the a < 0 side,
there is no bound molecular state and we simply have a DFG in an incoherent spin mixture.
Its distribution is wider and less dense due to the Fermi pressure.
The emergence of the Fermi pressure can be seen in figure 8.5 where we compare the
widths of trapped clouds on either side of the Feshbach resonance. In figure 8.5(a) we
2
have plotted the mean square radius σrad
versus the final dipole trap laser power. On
the BEC side of the resonance, the distribution of bosonic molecules becomes successively
narrower as the final trap depth is lowered while for fermionic atoms the width increases
(the trapping frequencies reduce with the square root of the laser intensity in both cases).
Figure 8.5(b) shows the same data relative to the Fermi radius (σrad /RF )2 where
RF =
s
2kB TF
2
matom ωrad
114
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
(a)
(b)
a<0
(c)
a<0
a>0
a>0
2
Figure 8.5: Observation of a degenerate Fermi gas. (a) Mean square width σrad
plotted
versus the final dipole trap laser power on both sides of the Feshbach resonance (b) Relative mean square width (σrad /RF )2 plotted versus the final dipole trap laser power (c)
(σrad /RF )2 plotted versus the degeneracy parameter T /TF . The solid line shows the theoretical prediction without taking into account atom-atom interactions. The dashed line
shows the prediction for a classical Boltzmann gas.
2
and TF = ~ ωax ωrad
6N
1/3
is the Fermi temperature. These are also plotted against the
degeneracy parameter, T /TF , in figure 8.5(c) at trap depths for which we have independent
time of flight temperature measurements. To determine the degeneracy parameter we
compared the width of Gaussian fits after 1 ms time of flight to Gaussian fits to Fermi
distributions. For a classical Boltzmann gas the width of the cloud decreases linearly
to zero as the temperature decreases (dashed line). This also holds for fermions at the
limit of high temperatures (T ≫ TF ). However, due to the Fermi pressure the theoretical
curve (solid line) vastly deviates from the classical law for low temperatures and σ 2 /RF2
approaches 0.177 as T goes to zero when integrated once over the imaging direction. The
theoretical curve was determined for a non-interacting gas. However, the data in figure 8.5
was obtained in a spin-mixture of the two lowest hyperfine states, and due to attractive
interactions on the BCS side of the Feshbach resonance the width of the cloud is expected
to be slightly lower than for a non-interacting gas [Gio07]. The highest degeneracy we have
observed in this system corresponds to T /TF = 0.25.
8.3
MBEC in a high power crossed dipole trap
In the previous section we presented the production of molecular Bose-Einstein condensates
in a crossed dipole trap formed by the 22 W ELS VersaDisk laser. We did not create large
115
8.3. MBEC IN A HIGH POWER CROSSED DIPOLE TRAP
(a)
(b)
100
Release energy [µK]
Number of particles [10 3]
500
100
50
10
1
0.1
20
0.1
1
10
Final dipole trap laser power [W]
100
0.1
1
10
Final dipole trap laser power [W]
100
Figure 8.6: Evaporative cooling in the high power crossed dipole trap formed by the fibre
laser. The magnetic field is kept on the molecular side of the Feshbach resonance at 770 G
while the evaporation takes place and is subsequently ramped to 694 G for imaging. (a)
Number of atoms in state |1i is plotted versus final dipole trap laser power. The solid line
represents the scaling law prediction for a cutoff parameter of η = U/kB T = 10 [O’H01].
(b) Release energy as a function of final dipole trap laser power. The vertical dashed line
indicates the temperature below which molecule formation becomes significant. Below the
vertical dashed line the release energy no longer corresponds to the temperature of the
mixture of atoms and molecules present in the trap. The solid (dashed) line corresponds
to one-tenth of the total (single) dipole trap depth.
condensates due to the small number of atoms initially loaded into this dipole trap. This
was the reason for investing in a more powerful laser. The set-up of our new dipole trap
formed by the 100 W fibre laser from IPG was described in section 7.3. We discussed
in section 7.4 that this increased the initial number of atoms loaded into the dipole trap
significantly. Furthermore, we achieved considerably greater initial phase space densities
than in the elongated crossed dipole trap.
8.3.1
Evaporation and realisation of a molecular BEC
The evaporation procedure is very similar to the one described previously (section 8.2) for
the low power crossed dipole trap. After loading the dipole trap we ramp up the magnetic
field to 770 G and keep the dipole trap power constant for 300 ms. We found that a longer
116
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
plain evaporation time, as in the low power dipole trap experiments, reduces the atom
number significantly. We believe that the high laser power increases the vacuum pressure
locally due to the ablation of lithium from the glass cell. Following the plain evaporation
we ramp down the laser power linearly from approximately ∼80 W to ∼11 W in 2 s. The
final evaporation stage is a 2 s logarithmic ramp to the final laser power.
In figure 8.6 (a) we show the number of atoms in state |1i during evaporative cooling
plotted versus the final dipole trap laser power imaged at 694 G. The solid line represents
the scaling law prediction for a cutoff parameter of η = 10. The experimental data fits
the scaling law predictions very well down to a laser power of approximately 100 mW.
The atom number, particularly at the lowest trap depths, is very sensitive to the precise
overlap of the crossed dipole trap and is in part responsible for the sharp drop in atom
number. The release energy as a function of final dipole trap laser power is depicted in
figure 8.6 (b). The vertical dashed line indicates the temperature below which molecule
formation becomes significant. Below the vertical dashed line the release energy no longer
corresponds to the temperature of the mixture of atoms and molecules present in the trap.
The solid line corresponds to one-tenth of the total dipole trap depth. However, as can
be seen the temperatures are significantly lower, indicating that this is not the relevant
trap depth. We therefore also plot the predictions based on the single dipole trap depth
(dashed line). This fits the data well showing that this is the effective trap depth.
We show typical absorption images during evaporative cooling in figure 8.7. All images
were taken at 694 G after evaporation at 770 G to a final trap depth of (a) 172 mW,
(b) 115 mW and (c) 57 mW. As the trap depth and hence the temperature decreases we
observe a shrinking of the cloud due to the drop in the release energy. In figure 8.7(d) we
show integrated cross sections along the weakest trapping direction for the same conditions
as above. The dashed, dash-dotted, solid lines are fits to the Gaussian, Thomas-Fermi and
combined profiles, respectively. The condensate fraction increases from ∼ 10% to ∼ 50%.
We typically cross the critical temperature for molecular Bose-Einstein condensation at a
final dipole trap power of approximately 250 mW which corresponds to a trap depth of
∼20 µK for the molecules. At this stage we trap approximately 150 000 molecules which
is a seven-fold enhancement over the experiments in the low power dipole trap. When we
evaporate to a final trap depth of 50 mW we nowadays observe ∼ 60 000 molecules and a
condensate fraction of over 60%.
117
8.3. MBEC IN A HIGH POWER CROSSED DIPOLE TRAP
(a)
(d)
1
0.5
Optical density
(b)
N=56 000
P=57 mW
0
0.4
0.2
(c)
N=77 000
P=115 mW
0
0.4
0.2
N=100 000
P=230 mW
0
326 µm
−200 −100
0
100
Position [µm]
200
Figure 8.7: (a)-(c) Typical absorption images during evaporative cooling in the high power
crossed dipole trap 2 ms after release from the trap. All images were taken at 694 G after
evaporation at 770 G to a final dipole trap power of (a) ∼172 mW, (b) ∼115 mW and
(c) ∼57 mW. (d) Integrated cross sections along the weakest trapping direction for the
same conditions as (a)-(c). The dashed, dash-dotted, solid lines are fits to the Gaussian,
Thomas-Fermi and combined profiles, respectively, showing the increase in condensate
fraction.
118
CHAPTER 8. BOSE-EINSTEIN CONDENSATION OF MOLECULES
Chapter 9
Binding Energies of 6Li p-wave
Feshbach Molecules
Superfluidity in dilute gases of atom pairs with non-zero angular momentum is still to be
achieved. There are many aspects that make such a condensate an interesting object to
study. Those systems provide the possibility of investigating the complex phase diagram
of superfluids of atom pairs with non-zero angular momentum [Gur05]. Furthermore, the
relation to other pairing phenomena such as d-wave high-TC superconductors [Tsu00] and
liquid 3 He [Lee97] is of great interest.
In this chapter, we present measurements of the binding energies of lithium p-wave
Feshbach molecules formed near the three Feshbach resonances of the two lowest hyperfine
states. A sinusoidally modulated magnetic field near a Feshbach resonance converts free
atoms into bound or quasi-bound molecules. The rate of conversion depends on the resonant properties of the scattering states which we compare with theoretical predictions. The
theory of p-wave scattering and p-wave Feshbach molecules was discussed in section 2.4.
The results shown in this chapter are published in [Fuc08].
9.1
Inelastic losses at the |1i − |1i Feshbach resonance
To perform measurements near the particularly narrow 6 Li p-wave resonances we require
high magnetic field stability and low field noise. Our experimental technique to achieve this
was described in section 4.4. Initially, we took an atom loss measurement at the |1i − |1i
p-wave Feshbach resonance located at approximately 159 G to characterise our system.
119
120 CHAPTER 9. BINDING ENERGIES OF 6 LI P -WAVE FESHBACH MOLECULES
For this experiment a mixture of atoms in states |1i and |2i was evaporatively cooled at a
magnetic field above the broad s-wave resonance to an energy of 150 nK (T ≈ 0.2TF ). Next
atoms in state |2i were blasted away with a 40 µs pulse of resonant laser light. To avoid
heating of the remaining state |1i atoms as much as possible we shift the magnetic field
well above resonance for this. The magnetic field was then switched to a value just above
the |1i − |1i p-wave resonance and the laser power lowered adiabatically in 20 ms to yield a
cloud energy of 100 nK. The magnetic field was subsequently ramped to a test value, Btest ,
over 40 ms and then held there for 60 ms to map out the atom loss across the resonance.
Figure 9.1 shows the atom number remaining after the hold time as a function of magnetic
field detuning, δB = Btest − B0 , where B0 is 159.14 G [Sch05]. Each data point is the
average of six measurements. The full width at half maximum of ∼ 25 mG, is primarily
limited by our field stability. The splitting of the ml = ±1 and ml = 0 projections is
predicted to be 10 mG for this resonance (compared to 500 mG for
40
K [Gae07]) which
we cannot resolve in our current set-up. The asymmetry of the loss feature may be due
to thermal or threshold effects and also the presence of the doublet. As the ml = ±1
projections are themselves degenerate, there are twice as many possible scattering states
for |ml | = 1 as for ml = 0, and the latter occurs 10 mG higher in field. This asymmetry
has also been observed by other groups [Sch05, Ina08].
9.2
Binding energies
To characterise p-wave Feshbach molecules a systematic study of the binding energies, EB ,
is of interest. We have measured the binding energies and obtained the magnetic moments,
µm , of p-wave Feshbach molecules formed by 6 Li atoms in the two lowest hyperfine states |1i
and |2i. We achieved this by means of magneto-association spectroscopy [Tho05b, Gae07].
By sinusoidally modulating the magnetic field close to a Feshbach resonance free atoms
can be associated to form bound or quasi-bound Feshbach molecules which are not seen
in our absorption images. The binding energies of p-wave molecules produced near the
|1i − |1i and |2i − |2i Feshbach resonances were measured using spin-polarised gases of
atoms at a temperature of ∼ 400 nK in the appropriate state. The binding energies of the
|1i − |2i molecules were measured at a temperature of ∼ 1 µK to avoid producing s-wave
molecules during the magnetic field ramp down to the p-wave field. Measurements for the
|1i − |1i and |2i − |2i molecules were performed in traps with final oscillation frequencies of
9.2. BINDING ENERGIES
121
Figure 9.1: Atom loss at the |1i-|1i 6 Li p-wave Feshbach resonance versus magnetic field
detuning δB = Btest − B0 (B0 = 159 G). The number of atoms remaining after 60 ms hold
time was measured via absorption imaging. Each point is the average of six images.
∼ 550 Hz radially and ∼ 60 Hz axially, while for |1i − |2i molecules a ∼ 870 Hz by ∼ 95 Hz
trap was used.
Atom-pair association occurs when the modulation frequency, νmod , corresponds to the
energy difference between the free atom and bound or quasi-bound molecular states. When
the resonance condition is fulfilled, significant atom loss can be observed. We apply the
modulation by means of a 2.5 cm diameter coil placed approximately 2 cm below the atomic
gas. It produces an oscillating magnetic field which is oriented along the same direction as
the primary magnetic field of the Feshbach coils. We vary the time, tmod , for which the field
modulation is held on from 200 ms to 2 s. At larger binding energies, longer modulation
times were required to compensate for the reduced association rates. The amplitude of the
magnetic field modulation was kept fixed at 180 mG for all experiments. A typical scan is
shown in figure 9.2 where a spin-polarised gas of atoms in state |2i is probed close to the
215 G Feshbach resonance. The magnetic field was varied while the modulation frequency
was fixed, in this instance at νmod = 650 kHz. The principal loss feature in the centre
of the scan is due to inelastic losses at the Feshbach resonance and its position coincides
122 CHAPTER 9. BINDING ENERGIES OF 6 LI P -WAVE FESHBACH MOLECULES
Figure 9.2: Magneto-association spectrum for the |2i − |2i p-wave Feshbach resonance
in 6 Li at νmod = 650 kHz. The central loss feature is due to three-body recombination
on resonance while the loss features to the left and right are due to resonant magnetoassociation of bound and quasi-bound Feshbach molecules, respectively.
with the magnetic field where the free colliding atoms are degenerate with the molecular
state. The two loss features on either side of this are due to magneto-association. Atom
loss on the low magnetic field side of the resonance is due to resonant conversion of atoms
into bound p-wave Feshbach molecules. The loss feature on the high magnetic field side
is unique to Feshbach resonances involving molecules with nonzero angular momentum
and is due to the production of quasi-bound molecules. Quasi-bound pairs have energies
above the free atom continuum and can tunnel through the centrifugal barrier, limiting
their lifetimes to a few ms. This leads to a broadening of the loss feature which was
observed in
40
K [Gae07] and from which the lifetime of the quasi-bound molecules could
be inferred. In our experiments, however, we could not detect any significant broadening
of the line shape. Furthermore, at the collision energies used in these experiments, we
did not observe any noticeable temperature dependence of the position of the loss feature.
The |ml | = 1 − ml = 0 doublet was again unresolved in these spectra. The solid line in
figure 9.2 is a fit of three Lorentzians from which we determine the central position of each
123
9.2. BINDING ENERGIES
loss feature.
By repeating these measurements using different νmod , it is possible to obtain the binding energy as a function of magnetic field (i.e., the magnetic moment of the molecules). We
have taken scans with up to eleven different modulation frequencies for each of the three
6
Li p-wave Feshbach resonances and the fields corresponding to the bound and quasi-bound
loss features are plotted in figure 9.3. The main panel, (a), shows the binding energies for
the |1i − |1i resonance and the left, (b), and right, (c), insets are for the |1i − |2i and
|2i − |2i resonances, respectively. The binding energies vary linearly with magnetic field
detuning as pointed out in section 2.4. Our measured gradients which are listed in table
9.1 are in good agreement with our theoretical predictions for this resonance and previous
work [Zha04].
[Zha04]
States
B (G)
∆B (mG)
µexp
m
µth
m
|1i − |1i
159
10
113
-
185
4
113 ± 7
116
117
|2i − |2i
215
12
118 ± 8
111
111
|1i − |2i
111 ± 6
µm
Table 9.1: Summary of the results of our binding energy measurements and calculations.
For each of the three p-wave Feshbach resonances we give the approximate absolute magnetic field, B, the calculated splitting of the |ml | = 1 - ml = 0 doublet, ∆B, our measured
and calculated relative magnetic moments, µexp,th
, and the magnetic moment calculated in
m
[Zha04]
reference [Zha04], µm
. All magnetic moments are expressed in µK/G.
Our theoretical binding energy slopes, µth
m , are found from a full closed-coupled calculation [Wei99] performed by C. Ticknor. The values for µth
m are the average of the slopes on
the bound and quasi-bound sides of the resonance. On the bound side, these are approximately 0.7 % steeper than on the quasi-bound side. We are unable to resolve this difference
within our experimental uncertainties.
The population of the closed channel can be estimated by means of the magnetic moment of the molecules obtained experimentally. This gives for the closed channel amplitude
(see section 2.4)
Z=
1 dEB
µm
=
,
∆µ dB
∆µ
(9.1)
124 CHAPTER 9. BINDING ENERGIES OF 6 LI P -WAVE FESHBACH MOLECULES
Figure 9.3: Binding energies of p-wave Feshbach molecules formed near the (a) |1i − |1i,
(b) |1i − |2i and (c) |2i − |2i resonances. All vertical (horizontal) axes are in units of µK
(G). Linear fits to these data yield gradients of (a) 113 ± 7 µK/G, (b) 111 ± 6 µK/G and
(c) 118 ± 8 µK/G.
where ∆µ is the relative magnetic moment of the open and closed channels [Koh06]. Using
only data from the molecular side of the resonance (δB < 0) yields µm = 115 ± 9 µK/G.
From our closed coupled calculations we find ∆µ = 2.12µB = 142 µK/G giving an experimental value of Z = 0.81 ± 0.07, in good agreement with the theory (0.82). Z is
important because it can be related to other molecular properties, such as the molecular
size as discussed in section 2.4.
As summarised in table 9.1 the magnetic moments of the three 6 Li p-wave Feshbach
molecules are approximately 115 µK/G. This value is close to twice the Bohr magneton
2µB = 134 µK/G. The interpretation of this is as follows: at these moderately high fields,
the open channels are mostly triplet in character and thus have a magnetic moment near
125
9.3. TRANSITION RATES
2µB . The closed channel, however, is predominantly spin singlet having a magnetic moment
of zero. As discussed above the closed channel amplitude Z is close to unity and, thus,
the molecular state is closed-channel dominated. From this, it follows that the relative
magnetic moment between the open channel and the molecular channel of the 6 Li p-wave
molecules is close to 2µB .
It is of interest to compare the findings for 6 Li p-wave Feshbach molecules to the case
of
40
K p-wave molecules. The closed channel amplitude of
40
K p-wave Feshbach molecules
6
is Z ≈ 0.7 [Gub07] which is similar to that of Li. However, the relative magnetic moment
between the open and closed channel is only 0.175µB [Gub07]. Consequently, the slopes
for the 6 Li p-wave resonances are approximately twelve times steeper than that measured
for 40 K p-wave molecules [Gae07]. This makes the requirements for magnetic field stability
even more stringent in the case of 6 Li.
9.3
Transition rates
The rate at which molecule formation from a gas of free atoms occurs using magnetoassociation depends on the magnetic field detuning. As the detuning increases the hold
time for the modulation, tmod , required to see significant atom loss due to resonant magnetoassociation increases. In the limit of low molecular conversion, the number of atoms remaining decays exponentially with time, Na (t) = Na0 e−Γt , where Na0 is the initial atom
number and Γ is the two-body association rate which varies with magnetic field. Fitting
this exponential decay, for known Na (t) and Na0 , to our magneto-association spectra yields
Γ at the different magnetic fields. We have done this for the data taken on the |1i − |1i
resonance and the results are plotted in figure 9.4. Note that we do not observe any Rabi
oscillations in the decay of the atom number.
Also shown in figure 9.4 is a scaled Fermi Golden Rule (FGR) calculation by C. Ticknor
of the transition rate, Γth ∝ |hψmol |µz |ψat i|2 where ψat is the multi-channel wavefunction
of the free atoms, ψmol is the multi-channel molecular wavefunction and µz is the magnetic dipole operator. The real time dependent conversion rate depends on the density
and temperature of the atomic cloud [Han07]. As these were the same for each run of our
experiments on a given resonance, these effects can be accounted for by a simple scaling
factor which is the same for all runs. The shape of the theoretical curve matches the experimental association rate extremely well over nearly two orders of magnitude, indicating
126 CHAPTER 9. BINDING ENERGIES OF 6 LI P -WAVE FESHBACH MOLECULES
Figure 9.4: Measured magneto-association rate, Γ, versus magnetic field detuning (points).
The solid line shows a scaled Fermi golden rule calculation for the transition rate from our
calculated atomic and bound molecular states. The scaling factor was chosen to fit the
data and accounts for the density and temperature dependence of the conversion rate.
that the states used in the FGR calculation are indeed a good representation of the true
states. The calculation was only performed on the bound side of the resonance as the
localised molecular wavefunction allows the FGR integral to converge.
The molecular properties are roughly constant across the resonance (see section 2.4)
and the FGR shows the change in the properties of the continuum scattering state near
resonance. We note that the experimental data is highly symmetric about δB = 0. We do
not observe any evidence of increased conversion on the quasi-bound side of the resonance,
even though the thermally averaged elastic scattering cross-section is known to be larger
for δB > 0 [Reg03b, Tic04].
Chapter 10
Conclusions
This thesis describes an experimental apparatus capable of producing molecular BoseEinstein condensates and quantum degenerate Fermi gases. It reports various experiments
performed, of which the main results have been published in references [Fuc06, Fuc07a,
Fuc07b, Fuc08]. In addition, the thesis describes the theory necessary for understanding
the experiments. This chapter summarises the most important results presented in the
thesis. Furthermore, an outlook of future experiments is provided.
10.1
Summary
Initial experiments were carried out in a collimated atomic beam of 6 Li. High contrast
EIT and EIA fluorescence resonances with sub-natural width have been observed for the
D1 and D2 line. The appearance of fluorescence peaks due to EIA is nontrivial taking
into account that the hyperfine splitting of the 2P3/2 state is masked by the natural line
broadening. We show that some EIT and EIA fluorescence resonances with sub-natural
width obtained for certain polarisations and magnetic field directions can not be explained
in terms of ground-state Zeeman coherence. In addition we show that coherent population
oscillations between ground and excited states also contribute to sub-natural resonances.
The resonances due to ground-state Zeeman coherence and population oscillations have a
very different power broadening. Additional experiments using the Ramsey separated field
technique showed further reduction in the width of both the EIT and EIA resonances.
The main result of this PhD project was the production of Bose-Einstein condensates of
6
Li2 molecules. We have achieved condensates in three dipole trap geometries. Evaporative
127
128
CHAPTER 10. CONCLUSIONS
cooling is achieved by reducing the dipole trap laser power near the broad Feshbach resonance at 834 G. By tuning to the low magnetic field side (770 G) of the Feshbach resonance,
molecules are formed through three-body recombination at sufficiently low temperatures.
Further evaporation leads to the creation of a BEC of 6 Li2 molecules. After reducing the
laser power by a factor of approximately 1000 we have observed condensates of up to 9,000
molecules in the low power dipole trap and 40,000 molecules in the high power dipole trap.
We have also produced a highly degenerate Fermi gas, as evidenced by the emergence of
the Fermi pressure.
Finally, we have presented measurements of the binding energies of the three p-wave
Feshbach molecules formed by 6 Li atoms in the two lowest hyperfine states. We find that
the binding energy scales approximately linearly with the magnetic field detuning. Our
data agrees well with theoretical calculations for the binding energy slopes, implying good
agreement for the closed channel amplitude and molecule size.
10.2
Outlook
BEC-BCS crossover
In recent years many fascinating experiments have extended our knowledge of superfluidity in the BEC-BCS crossover. Nevertheless, there are still many open questions which
motivate further studies. Our system is well suited for experiments of degenerate Fermi
gases in this exciting region.
Currently, our group is setting up a laser system to probe the gas by means of Bragg
spectroscopy [Ste99b]. This spectroscopic tool has been exploited to investigate the properties of a molecular Bose-Einstein condensate in the vicinity of the 834 G Feshbach resonance [Ina07]. Our plan is to extend the experiments across the Feshbach resonance. Bragg
spectroscopy of Cooper pairs may further our understanding of this pairing mechanism and
the transition from Cooper pairs to molecules [Cha07].
p-wave superfluidity
We also intend to employ Bragg spectroscopy for studies near p-wave Feshbach resonances.
Since the Bragg resonance frequency is sensitive to the mass of the deflected particle it can
10.2. OUTLOOK
129
be used to distinguish p-wave Feshbach molecules from free atoms. To date we have not
observed any evidence of 6 Li |1i − |1i p-wave molecules remaining trapped. Because of the
steepness of the binding energy slopes, molecules produced by three-body recombination
will easily be lost from the trap unless the field is very close to the resonance. Nevertheless,
in a recent experiment Inada et al. directly observed trapped p-wave Feshbach molecules
formed by atoms in the two lowest hyperfine states [Ina08]. They reported a 1/e lifetime
of only approximately 20 ms for the most stable gas of |1i − |1i molecules. This lifetime is
too short for rethermalisation and producing p-wave condensates. It is therefore crucial to
understand the inelastic collision processes in more detail. However, we still believe that
6
Li p-wave molecules produced on the |1i − |1i resonance are a promising candidate for
achieving superfluidity of atom pairs with nonzero angular momentum in dilute gases.
130
CHAPTER 10. CONCLUSIONS
Bibliography
[Abe99]
F. A. van Abeelen and B. J. Verhaar, Time-Dependent Feshbach Resonance
Scattering and Anomalous Decay of a Na Bose-Einstein Condensate, Phys. Rev.
Lett. 83, 1550 (1999).
[Ada95]
C. S. Adams, H. J. Lee, N. Davidson, M. Kasevich and S. Chu, Evaporative
Cooling in a Crossed Dipole Trap, Phys. Rev. Lett. 74, 3577 (1995).
[Aku91]
A. M. Akulshin, A. A. Celikov and V. L. Velichansky, Sub-natural absorption
resonances on the D1 line of Rb induced by coherent population trapping, Opt.
Commun. 84, 139 (1991).
[Aku98]
A. M. Akulshin, S. Barreiro and A. Lezama, Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate
levels in Rb vapor, Phys. Rev. A 57, 2996 (1998).
[Aku03a] A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford and G. I. Opat,
Light propagation in an atomic medium with steep and sign-reversible dispersion,
Phys. Rev. A 67, 011801 (2003).
[Aku03b] A. M. Akulshin, A. Cimmino, A. I. Sidorov, R. McLean and P. Hannaford,
Highly nonlinear atomic medium with steep and sign-reversible dispersion, Journal of Optics B: Quantum and Semiclassical Optics 5, 479 (2003).
[Aku04]
A. M. Akulshin, A. I. Sidorov, R. J. McLean and P. Hannaford, Enhanced atomic
Kerr nonlinearity in bright coherent states, Journal of Optics B: Quantum and
Semiclassical Optics 6, 491 (2004).
131
132
[Aku05]
BIBLIOGRAPHY
A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean and P. Hannaford,
Storage of light in an atomic medium using electromagnetically induced absorption, J. Phys. B. 38, L365 (2005).
[Alt07]
A. Altmeyer, S. Riedl, C. Kohstall, M. J. Wright, R. Geursen, M. Bartenstein,
C. Chin, J. H. Denschlag and R. Grimm, Precision Measurements of Collective
Oscillations in the BEC-BCS Crossover, Phys. Rev. Lett. 98, 040401 (2007).
[Alz76]
G. Alzetta, A. Gozzini, L. Moi and G. Orriols, An experimental method for
the observation of R.F. transitions and laser beam resonances in orientated Na
vapour, Nuovo Cimento B 36, 5 (1976).
[And95]
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).
[Ari96]
E. Arimondo, Coherent population trapping in laser spectroscopy, Progress in
Optics 35, 257 (1996).
[Bak71]
E. V. Baklanov and V. P. Chebotaev, Effects of fields in resonant interaction
of opposing waves in a gas, Zh. Eksp. i Theor. Fiz. (Sov. Phys. JETP) 60, 551
(1971).
[Bak72]
E. V. Baklanov and V. P. Chebotaev, Field effect in resonance interaction between counter-running waves in a gas I., Sov. Phys. JETP 34, 490 (1972).
[Bar57]
J. Bardeen, L. N. Cooper and J. R. Schriffer, Theory of Superconductivity, Phys.
Rev. 108, 1175 (1957).
[Bar04a]
M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag
and R. Grimm, Collective Excitations of a Degenerate Gas at the BEC-BCS
Crossover, Phys. Rev. Lett. 92, 203201 (2004).
[Bar04b]
M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag
and R. Grimm, Crossover from a Molecular Bose-Einstein Condensate to a
Degenerate Fermi Gas, Phys. Rev. Lett. 92, 120401 (2004).
BIBLIOGRAPHY
[Bar05]
133
M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin, J. H.
Denschlag, R. Grimm, A. Simoni, E. Tiesinga, C. J. Williams and P. S. Julienne,
Precise Determination of 6 Li Cold Collision Parameters by Radio-Frequency
Spectroscopy on Weakly Bound Molecules, Phys. Rev. Lett. 94, 103201 (2005).
[Bed86]
J. G. Bednorz and K. Mueller, Possible high-Tc superconductivity in the Ba-LaCu-O system, Z. Physik B 64, 189 (1986).
[Big03]
M. S. Bigelow, N. N. Lepeshkin and R. W. Boyd, Superluminal and Slow Light
Propagation in a Room-Temperature Solid, Science 301, 200 (2003).
[Bol91]
K. J. Boller, A. Imamoglu and S. E. Harris, Observation of electromagnetically
induced transparency, Phys. Rev. Lett. 66, 2593 (1991).
[Bot05]
S. S. Botelho and C. A. R. Sá de Melo, Quantum Phase Transition in the BCS
to BEC Evolution of p-wave Fermi gases, J. Low Temp. Phys. 140, 409 (2005).
[Bou04]
T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann,
L. Tarruell, S. J. J. M. F. Kokkelmans and C. Salomon, Experimental Study
of the BEC-BCS Crossover Region in Lithium 6, Phys. Rev. Lett. 93, 050401
(2004).
[Boy81]
R. W. Boyd, M. G. Raymer, P. Narum and D. J. Harter, Four-wave parametric
interactions in a strongly driven two-level system, Phys. Rev. A 24, 411 (1981).
[Boy84]
R. W. Boyd and S. Mukamel, Origin of spectral holes in pump-probe studies of
homogeneously broadened lines, Phys. Rev. A 29, 1973 (1984).
[Boy02]
R. Boyd and D. J. Gauthier, Slow and Fast Light, Progress in Optics 43, 497
(2002).
[Bra95]
C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Bose-Einstein
Condensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett.
75, 1687 (1995).
[Bra05]
D. V. Brazhnikov, A. M. Tumaikin, V. I. Yudin and A. V. Taichenachev, Electromagnetically induced absorption and transparency in magneto-optical resonances
in an elliptically polarized field, J. Opt. Soc. Am. B 22, 57 (2005).
134
[Bud02]
BIBLIOGRAPHY
D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk and
A. Weis, Resonant nonlinear magneto-optical effects in atoms, Rev. Mod. Phys.
74, 1153 (2002).
[Bur02]
K. Burnett, P. S. Julienne, P. D. Lett, E. Tiesinga and C. J. Williams, Quantum
encounters of the cold kind, Nature 416, 225 (2002).
[Car05]
B. T. Carr, L. D. and Y. Castin, Limits of sympathetic cooling of fermions by
zero-temperature bosons due to particle losses, Phys. Rev. A 69, 033603 (2005).
[Cha07]
K. J. Challis, R. J. Ballagh and C. W. Gardiner, Bragg Scattering of Cooper
Pairs in an Ultracold Fermi Gas, Phys. Rev. Lett. 98, 093002 (2007).
[Che05a] Q. Chen, J. Stajic, S. Tan and K. Levin, BCS-BEC Crossover: From High Temperature Superconductors to Ultracold Superfluids, Physics Reports 412 (2005).
[Che05b] C.-H. Cheng and S.-K. Yip, Anisotropic Fermi Superfluid via p-wave Feshbach
Resonance, Phys. Rev. Lett. 95, 070404 (2005).
[Che05c]
F. Chevy, E. G. M. van Kempen, T. Bourdel, J. Zhang, L. Khaykovich, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans and C. Salomon, Resonant
scattering properties close to a p-wave Feshbach resonance, Phys. Rev. A 71,
062710 (2005).
[Chi04a]
C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag and
R. Grimm, Observation of the Pairing Gap in a Strongly Interacting Fermi Gas,
Science 305, 1128 (2004).
[Chi04b]
C. Chin and R. Grimm, Thermal equilibrium and efficient evaporation of an
ultracold atom-molecule mixture, Phys. Rev. A 69, 033612 (2004).
[Chi06]
J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu
and W. Ketterle, Evidence for superfluidity of ultracold fermions in an optical
lattice, Nature 443, 961 (2006).
[Chu85]
S. Chu, L. Hollberg and J. E. Bjorkholm, Three-dimensional viscous confinement
and cooling of atoms by radiation pressure, Phys. Rev. Lett. 55, 48 (1985).
135
BIBLIOGRAPHY
[Chu86]
S. Chu, J. E. Bjorkholm, A. Ashkin and A. Cable, Experimental Observation of
Optically Trapped Atoms, Phys. Rev. Lett. 57, 314 (1986).
[Coo56]
L. N. Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Phys. Rev.
104, 1189 (1956).
[Cou98]
P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen and B. J. Verhaar,
Observation of a Feshbach Resonance in Cold Atom Scattering, Phys. Rev. Lett.
81, 69 (1998).
[CT77]
C. Cohen-Tannoudji, Atoms in strong resonant fields, in: R. Balian, S. Haroche,
and S. Liberman edts., Frontiers of Laser Spectroscopy, North-Holland, Amsterdam 1 (1977).
[Cub03]
J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov and
C. Salomon, Production of Long-Lived Ultracold Li2 Molecules from a Fermi
Gas, Phys. Rev. Lett. 91, 240401 (2003).
[Dal99]
J. Dalibard, Collisional dynamics of ultra-cold atomic gases, in: M. Inguscio, S.
Stringari and C. Wieman edts., Bose-Einstein Condensation in Atomic Gases:
Proceedings of the International School of Physics (Enrico Fermi), Varenna 1998
321 (1999).
[Dav95]
K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn and W. Ketterle, Bose-Einstein Condensation in a Gas of Sodium
Atoms, Phys. Rev. Lett. 75, 3969 (1995).
[Deh08]
B. Deh, C. Marzok, C. Zimmermann and P. W. Courteille, Feshbach resonances
in mixtures of ultracold 6 Li and
87
Rb gases, Phys. Rev. A 77, 010701 (2008).
[DeM99a] B. DeMarco, J. L. Bohn, J. P. Burke, M. Holland and D. S. Jin, Measurement
of p-Wave Threshold Law Using Evaporatively Cooled Fermionic Atoms, Phys.
Rev. Lett. 82, 4208 (1999).
[DeM99b] B. DeMarco and D. S. Jin, Onset of Fermi Degeneracy in a Trapped Atomic
Gas, Science 285, 1703 (1999).
136
[Dem03]
BIBLIOGRAPHY
W. Demtröder, Laser spectroscopy: Basic concepts and Instrumentation vol 3,
Springer-Verlag, New Delhi (2003).
[D’I08]
J. P. D’Incao, B. D. Esry and C. H. Greene, Ultracold atom-molecule collisions
with fermionic atoms, Preprint arXiv:0802:2553v1 (2008).
[Die02]
K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic, C. H. Schunck and W. Ketterle, Decay of an Ultracold Fermionic Lithium Gas near a Feshbach Resonance,
Phys. Rev. Lett. 89, 203201 (2002).
[Don02]
E. A. Donley, N. R. Claussen, S. T. Thompson and C. E. Wieman, Atommolecule coherence in a Bose-Einstein condensate, Nature 417, 529 (2002).
[Dür04]
S. Dürr, T. Volz, A. Marte and G. Rempe, Observation of Molecules Produced
from a Bose-Einstein Condensate, Phys. Rev. Lett. 92, 020406 (2004).
[Ert85]
W. Ertmer, R. Blatt, J. L. Hall and M. Zhu, Laser Manipulation of Atomic
Beam Velocities: Demonstration of Stopped Atoms and Velocity Reversal, Phys.
Rev. Lett. 54, 996 (1985).
[Fes62]
H. Feshbach, Unified theory of nuclear reactions. II, Annals of Physics 19, 287
(1962).
[Fio98]
A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws and
P. Pillet, Formation of Cold Cs2 Molecules through Photoassociation, Phys.
Rev. Lett. 80, 4402 (1998).
[Fle05]
M. Fleischhauer, A. Imamoglu and J. P. Marangos, Electromagnetically induced
transparency: Optics in coherent media, Rev. of Mod. Phys. 77, 633 (2005).
[Fuc06]
J. Fuchs, G. J. Duffy, W. J. Rowlands and A. M. Akulshin, Electromagnetically
induced transparency in 6 Li, J. Phys. B. 39, 3479 (2006).
[Fuc07a]
J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford and A. M.
Akulshin, Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6 Li, J. Phys. B. 40, 1117 (2007).
BIBLIOGRAPHY
[Fuc07b]
137
J. Fuchs, G. J. Duffy, G. Veeravalli, P. Dyke, M. Bartenstein, C. J. Vale, P. Hannaford and W. J. Rowlands, Molecular Bose-Einstein condensation in a versatile
low power crossed dipole trap, J. Phys. B. 40, 4109 (2007).
[Fuc08]
J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Rowlands, P. Hannaford and C. J. Vale, Binding Energies of 6 Li p-wave Feshbach Molecules, Phys.
Rev. A 77, 053616 (2008).
[Gae07]
J. P. Gaebler, J. T. Stewart, J. L. Bohn and D. S. Jin, p-Wave Feshbach
Molecules, Phys. Rev. Lett. 98, 200403 (2007).
[GB95]
J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin and M. Xiao, Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory
and experiment, Phys. Rev. A 51, 576 (1995).
[Geh98]
M. E. Gehm, K. M. O’Hara, T. A. Savard and J. E. Thomas, Dynamics of
noise-induced heating in atom traps, Phys. Rev. A 58, 3914 (1998).
[Geh03]
M. E. Gehm, Preparation of an optically-trapped degenerate Fermi gas of 6 Li:
Finding the route to degeneracy, PhD Dissertation, Duke University (2003).
[Gio96]
S. Giorgini, L. P. Pitaevskii and S. Stringari, Condensate fraction and critical
temperature of a trapped interacting Bose gas, Phys. Rev. A 54, R4633 (1996).
[Gio07]
S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of ultracold Fermi gases,
Preprint arXiv:0706.3360 (2007).
[Gor61]
L. P. Gorkov and T. K. Melik-Barkhudarov, Contribution to the theory of superuidity in an imperfect Fermi gas, Sov. Phys. JETP 13, 1018 (1961).
[Gor03a] C. Goren, A. D. Wilson-Gordon, M. Rosenbluh and H. Friedmann, Electromagnetically induced absorption due to transfer of coherence and to transfer of
population, Phys. Rev. A 67, 033807 (2003).
[Gor03b] C. Goren, A. D. Wilson-Gordon, M. Rosenbluh and H. Friedmann, Switching from positive to negative dispersion in transparent degenerate and neardegenerate systems, Phys. Rev. A 68, 043818 (2003).
138
[Gre03]
BIBLIOGRAPHY
M. Greiner, C. A. Regal and D. S. Jin, Emergence of a molecular Bose Einstein
condensate from a Fermi gas, Nature 426, 537 (2003).
[Gri00]
R. Grimm, M. Weidemüller and Y. Ovchinnikov, Optical dipole traps for neutral
atoms, Adv. At. Mol. Opt. Phys. 42, 95 (2000).
[Gro61]
E. P. Gross, Structure in Quantized Vortex in Boson Systems, Il Nuovo Cimento
20, 454 (1961).
[Gub07]
K. B. Gubbels and H. T. C. Stoof, Theory for p-wave feshbach molecules, Physical Review Letters 99(19), 190406 (2007).
[Gün05]
K. Günter, T. Stöferle, H. Moritz, M. Köhl and T. Esslinger, p-Wave Interactions in Low-Dimensional Fermionic Gases, Phys. Rev. Lett. 95, 230401 (2005).
[Gur05]
V. Gurarie, L. Radzihovsky and A. V. Andreev, Quantum Phase Transitions
across a p-Wave Feshbach Resonance, Phys. Rev. Lett. 94, 230403 (2005).
[Gur07]
V. Gurarie and L. Radzihovsky, Resonantly-paired fermionic superfluids, Ann.
Phys. (N.Y.) 322, 2 (2007).
[Had02]
Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Görlitz
and W. Ketterle, Two-Species Mixture of Quantum Degenerate Bose and Fermi
Gases, Phys. Rev. Lett. 88, 160401 (2002).
[Han07]
T. M. Hanna, T. Kohler and K. Burnett, Association of molecules using a
resonantly modulated magnetic field, Phys. Rev. A 75, 013606 (2007).
[Har72]
S. Haroche and F. Hartmann, Theory of Saturated-Absorption Line Shapes,
Phys. Rev. A 6, 1280 (1972).
[Har89]
S. E. Harris, Lasers without inversion - interference of lifetime-broadened resonances, Phys. Rev. Lett. 62, 1033 (1989).
[Har90]
S. E. Harris, J. E. Field and A. Imamoğlu, Nonlinear optical processes using
electromagnetically induced transparency, Phys. Rev. Lett. 64, 1107 (1990).
[Har92]
S. E. Harris, J. E. Field and A. Kasapi, Dispersive properties of electromagnetically induced transparency, Phys. Rev. A 46, R29 (1992).
BIBLIOGRAPHY
[Har97]
139
S. E. Harris, Electromagnetally induced transparency, Physics Today 50, 32
(1997).
[Har99]
S. E. Harris and L. V. Hau, Nonlinear Optics at Low Light Levels, Phys. Rev.
Lett. 82, 4611 (1999).
[Hem93]
P. R. Hemmer, M. S. Shahriar, H. Lamela-Rivera, S. P. Smith, B. E. Bernacki
and S. Ezekiel, Semiconductor laser excitation of Ramsey fringes by using a
Raman transition in a cesium atomic beam, J. Opt. Soc. Am. B 10, 1326 (1993).
[Her03]
J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nagerl and
R. Grimm, Preparation of a Pure Molecular Quantum Gas, Science 301, 1510
(2003).
[Hol00]
M. J. Holland, B. DeMarco and D. S. Jin, Evaporative cooling of a twocomponent degenerate Fermi gas, Phys. Rev. A 61, 053610 (2000).
[Hol01]
M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo and R. Walser, Resonance
Superfluidity in a Quantum Degenerate Fermi Gas, Phys. Rev. Lett. 87, 120406
(2001).
[Hou97]
M. Houbiers, R. Ferwerda, H. T. C. Stoof, W. I. McAlexander, C. A. Sackett
and R. G. Hulet, Superfluid state of atomic 6 Li in a magnetic trap, Phys. Rev.
A 56, 4864 (1997).
[Hou98]
M. Houbiers, H. T. C. Stoof, W. I. McAlexander and R. G. Hulet, Elastic and
inelastic collisions of 6 Li atoms in magnetic and optical traps, Phys. Rev. A 57,
R1497 (1998).
[Hua87]
K. Huang, Statistical Mechanics (2nd edition), Wiley (1987).
[Hut06]
J. M. Hutson and P. Soldán, Molecule formation in ultracold atomic gases,
International Reviews in Physical Chemistry 25, 497 (2006).
[Ina07]
Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata-Gonokami, M. Ueda and
T. Mukaiyama, Universal Thermodynamics of a Strongly Interacting Molecular
Condensate, Preprint arXiv:0712.1445v3 (2007).
140
[Ina08]
BIBLIOGRAPHY
Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata-Gonokami, M. Ueda and
T. Mukaiyama, Collisional Properties of p-Wave Feshbach Molecules, Preprint
arXiv:0803.1405v1 (2008).
[Ino98]
S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and
W. Ketterle, Observation of Feshbach resonances in a Bose-Einstein condensate,
Nature 392, 151 (1998).
[Isk06]
M. Iskin and C. A. R. Sá de Melo, Evolution from BCS to BEC Superfluidity
in p-wave Fermi Gases, Phys. Rev. Lett. 96, 040402 (2006).
[Jav02]
A. Javan, O. Kocharovskaya, H. Lee and M. O. Scully, Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium,
Phys. Rev. A 66, 013805 (2002).
[Joc03a]
S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. H. Denschlag
and R. Grimm, Pure Gas of Optically Trapped Molecules Created from Fermionic
Atoms, Phys. Rev. Lett. 91, 240402 (2003).
[Joc03b]
S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. H.
Denschlag and R. Grimm, Bose-Einstein Condensation of Molecules, Science
302, 2101 (2003).
[Joc04]
S. Jochim, Bose-Einstein Condensation of Molecules, PhD Dissertation, Innsbruck University (2004).
[Jof93]
M. A. Joffe, W. Ketterle, A. Martin and D. E. Pritchard, Transverse cooling
and deflection of an atomic beam inside a Zeeman slower, J. Opt. Soc. Am. B
10, 2257 (1993).
[Jos07]
J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov and J. E. Thomas, Measurement of sound velocity in a Fermi gas near a Feshbach resonance, Phys.
Rev. Lett. 98, 170401 (2007).
[Jul92]
P. Julienne, A. Smith and K. Burnett, Theory of collisions between laser-cooled
atoms, Adv. At. Mol. Opt. Phys 30, 141 (1992).
141
BIBLIOGRAPHY
[Kaw93]
J. Kawanaka, K. Shimizu, H. Takuma and F. Shimizu, Quadratic collisional loss
rate of a 7 Li trap, Phys. Rev. A 48, R883 (1993).
[Ket99]
W. Ketterle, D. Durfee and D. Stamper-Kurn, Making, probing and understanding Bose-Einstein condensates, in: M. Inguscio, S. Stringari and C. Wieman
edts., Bose-Einstein Condensation in Atomic Gases: Proceedings of the International School of Physics (Enrico Fermi), Varenna 1998 67 (1999).
[Kim01]
K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An and J. B.
Kim, Electromagnetically induced absorption spectra depending on intensities
and detunings of the coupling field in Cs vapour, J. Phys. B. 34, 4801 (2001).
[Kim03]
K. Kim, H. S. Moon, C. Lee, S. K. Kim and J. B. Kim, Observation of arbitrary
group velocities of light from superluminal to subluminal on a single atomic
transition line, Phys. Rev. A 68, 013810 (2003).
[Kin04a]
J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov and J. E. Thomas, Evidence
for Superfluidity in a Resonantly Interacting Fermi Gas, Phys. Rev. Lett. 92,
150402 (2004).
[Kin04b] J. Kinast, A. Turlapov and J. E. Thomas, Breakdown of hydrodynamics in the
radial breathing mode of a strongly interacting Fermi gas, Phys. Rev. A 70,
051401 (2004).
[Kin05]
J. Kinast, A. Turlapov, J. E. Thomas, Q. Chen, J. Stajic and K. Levin, Heat
Capacity of a Strongly Interacting Fermi Gas, Science 307, 1296 (2005).
[Kna05]
S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew and J. Moreland, A chip-scale atomic clock based on
87
Rb with improved frequency stability,
Optics Express 13, 1249 (2005).
[Kno08]
S. Knoop, M. Mark, F. Ferlaino, J. G. Danzl, T. Kraemer, H.-C. Nagerl and
R. Grimm, Metastable feshbach molecules in high rotational states, Phys. Rev.
Lett. 100, 083002 (2008).
[Koh06]
T. Kohler, K. Goral and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys. 78, 1311 (2006).
142
[Kok04]
BIBLIOGRAPHY
S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov and C. Salomon, Degenerate
atom-molecule mixture in a cold Fermi gas, Phys. Rev. A 69, 031602(R) (2004).
[Lee97]
D. M. Lee, The extraordinary phases of liquid 3 He, Rev. Mod. Phys. 69, 645
(1997).
[Let77]
V. S. Letokhov and V. P. Chebotaev, Nonlinear Laser Spectroscopy, Springer
Series in Optical Science, Vol. 4, Springer-Verlag, Berlin (1977).
[Let88]
P. D. Lett, R. N. Watts, C. I. Westbrook and W. D. Phillips, Observation of
Atoms Laser Cooled below the Doppler Limit, Phys. Rev. Lett. 61, 169 (1988).
[Lez99a]
A. Lezama, S. Barreiro and A. M. Akulshin, Electromagnetically induced absorption, Phys. Rev. A 59, 4732 (1999).
[Lez99b]
A. Lezama, S. Barreiro, A. Lipsich and A. M. Akulshin, Coherent two-field
spectroscopy of degenerate two-level systems, Phys. Rev. A 61, 013801 (1999).
[Lez06]
A. Lezama, A. M. Akulshin, A. I. Sidorov and P. Hannaford, Storage and retrieval of light pulses in atomic media with slow and fast light, Phys. Rev. A 73,
033806 (2006).
[Lid05]
D. R. Lide, Handbook of Chemistry and Physics (86th edition), CRC Press, Boca
Raton (2005).
[Luk03]
M. D. Lukin, Colloquium: Trapping and manipulating photon states in atomic
ensembles, Rev. Mod. Phys. 75, 457 (2003).
[Luo06]
L. Luo, B. Clancy, J. Joseph, J. Kinast, A. Turlapov and J. E. Thomas, Evaporative cooling of unitary Fermi gas mixtures in optical traps, New Journal of
Physics 8, 213 (2006).
[Luo07]
L. Luo, B. Clancy, J. Joseph, J. Kinast and J. E. Thomas, Measurement of the
Entropy and Critical Temperature of a Strongly Interacting Fermi Gas, Phys.
Rev. Lett. 98, 080402 (2007).
[Mag05]
F. Magnus, A. L. Boatwright, A. Flodin and R. C. Shiell, Optical pumping and
electromagnetically induced transparency in a lithium vapour, Journal of Optics
B: Quantum and Semiclassical Optics 7, 109 (2005).
BIBLIOGRAPHY
[Mar98]
143
J. P. Marangos, Topical review: Electromagnetically induced transparancy, J. of
Mod. Opt. 45, 471 (1998).
[Mat01]
A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov and
M. O. Scully, Slow, ultraslow, stored, and frozen light, Adv. At. Mol. Opt. Phys
46, 191 (2001).
[McA96]
W. I. McAlexander, E. R. I. Abraham and R. G. Hulet, Radiative lifetime of
the 2P state of lithium, Phys. Rev. A 54, R5 (1996).
[Met99]
H. J. Metcalf and P. v. d. Straten, Laser Cooling and Trapping, Springer-Verlag,
Heidelberg (1999).
[Nag99]
A. Nagel, C. Affolderbach, S. Knappe and R. Wynands, Influence of excitedstate hyperfine structure on ground-state coherence, Phys. Rev. A 61, 012504
(1999).
[Nar98]
M. Naraschewski and D. M. Stamper-Kurn, Analytical description of a trapped
semi-ideal Bose gas at finite temperature, Phys. Rev. A 58, 2423 (1998).
[O’H89]
J. F. O’Hanlon, A Users Guide to Vacuum Technology, Wiley, New York (1989).
[O’H01]
K. M. O’Hara, M. E. Gehm, S. R. Granade and J. E. Thomas, Scaling laws for
evaporative cooling in time-dependent optical traps, Phys. Rev. A 64, 051403
(2001).
[Onn11]
H. K. Onnes, On the sudden rate at which the resistance of mercury disappears,
Akad. van Wetenschappen 14(113), 818 (1911).
[Osh72]
D. D. Osheroff, R. C. Richardson and D. M. Lee, Evidence for a new phase of
solid 3 He, Phys. Rev. Lett. 28, 885 (1972).
[Par05]
G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack and R. G. Hulet,
Molecular Probe of Pairing in the BEC-BCS Crossover, Phys. Rev. Lett. 95,
020404 (2005).
[Par06a]
G. B. Partridge, W. Li, R. I. Kamar, Y. Liao and R. G. Hulet, Pairing and
Phase Separation in a Polarized Fermi Gas, Science 311, 503 (2006).
144
[Par06b]
BIBLIOGRAPHY
G. B. Partridge, W. Li, Y. Liao, R. G. Hulet, M. Haque and H. T. C. Stoof,
Deformation of a trapped Fermi gas with unequal spin populations, Phys. Rev.
Lett. 97, 190407 (2006).
[Pet01]
C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press (2001).
[Pet04]
D. S. Petrov, C. Salomon and G. V. Shlyapnikov, Weakly Bound Dimers of
Fermionic Atoms, Phys. Rev. Lett. 93, 090404 (2004).
[Pet05]
D. S. Petrov, C. Salomon and G. V. Shlyapnikov, Scattering properties of weakly
bound dimers of fermionic atoms, Phys. Rev. A 71, 012708 (2005).
[Phi82]
W. D. Phillips and H. Metcalf, Laser Deceleration of an Atomic Beam, Phys.
Rev. Lett. 48, 596 (1982).
[Pit61]
L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13,
451 (1961).
[Pit03]
L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University
Press (2003).
[Raa87]
E. Raab, M. Prentiss, A. Cable, S. Chu and D. Pritchard, Trapping of NeutralSodium Atoms with Radiation Pressure, Phys. Rev. Lett. 59, 2631 (1987).
[Ram89]
N. F. Ramsey, Molecular Beams, 2nd Edn, Clarendon, Oxford (1989).
[Reg03a] C. A. Regal, C. Ticknor, J. L. Bohn and D. S. Jin, Creation of ultracold molecules
from a Fermi gas of atoms, Nature 424, 47 (2003).
[Reg03b] C. A. Regal, C. Ticknor, J. L. Bohn and D. S. Jin, Tuning p-Wave Interactions
in an Ultracold Fermi Gas of Atoms, Phys. Rev. Lett. 90, 053201 (2003).
[Reg04a] C. A. Regal, M. Greiner and D. S. Jin, Lifetime of Molecule-Atom Mixtures
near a Feshbach Resonance in K, Phys. Rev. Lett. 92, 083201 (2004).
[Reg04b] C. A. Regal, M. Greiner and D. S. Jin, Observation of Resonance Condensation
of Fermionic Atom Pairs, Phys. Rev. Lett. 92, 040403 (2004).
145
BIBLIOGRAPHY
[Ren99]
F. Renzoni, A. Lindner and E. Arimondo, Coherent population trapping in open
systems: A coupled/noncoupled-state analysis, Phys. Rev. A 60, 450 (1999).
[Roa02]
G. Roati, F. Riboli, G. Modugno and M. Inguscio, Fermi-Bose Quantum Degenerate
40
K-87 Rb Mixture with Attractive Interaction, Phys. Rev. Lett. 89, 150403
(2002).
[Sch96]
O. Schmidt, R. Wynands, Z. Hussein and D. Meschede, Steep dispersion and
group velocity below
c
3000
in coherent population trapping, Phys. Rev. A 53, R27
(1996).
[Sch01]
F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles
and C. Salomon, Quasipure Bose-Einstein Condensate Immersed in a Fermi
Sea, Phys. Rev. Lett. 87, 080403 (2001).
[Sch05]
C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle,
A. Simoni, E. Tiesinga, C. J. Williams and P. S. Julienne, Feshbach resonances
in fermionic 6 Li, Phys. Rev. A 71, 045601 (2005).
[Sch07a]
C. H. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein and W. Ketterle, Pairing Without Superfluidity: The Ground State of an Imbalanced Fermi Mixture,
Science 316, 867 (2007).
[Sch07b]
C. H. Schunck, M. W. Zwierlein, A. Schirotzek and W. Ketterle, Superuid Expansion of a Rotating Fermi Gas, Phys. Rev. Lett. 98, 050404 (2007).
[Sch08]
C. H. Schunck, Y. Shin, A. Schirotzek and W. Ketterle, Determination
of the fermion pair size in a resonantly interacting superfluid, Preprint
arXiv:0802.0341v1 (2008).
[Sco87]
G. Scoles, Atomic and Molecular Beam Methods, Oxford University Press, New
York (1987).
[Shi06]
Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek and W. Ketterle, Observation of Phase Separation in a Strongly Interacting Imbalanced Fermi Gas,
Phys. Rev. Lett. 97, 030401 (2006).
146
[Shi08]
BIBLIOGRAPHY
Y. Shin, C. H. Schunck, A. Schirotzek and W. Ketterle, Phase diagram of a
two-component Fermi gas with resonant interactions, Nature 451, 689 (2008).
[Sil05]
C. Silber, S. Günther, C. Marzok, B. Deh, P. W. Courteille and C. Zimmermann, Quantum-Degenerate Mixture of Fermionic Lithium and Bosonic Rubidium Gases, Phys. Rev. Lett. 95, 170408 (2005).
[Smi89]
V. S. Smirnov, A. M. Tumaikin and V. I. Yudin, Coherent stationary states under the interaction of atoms with elliptically polarized light. Coherent population
trapping: (General theory), Sov. Phys. JETP 69, 913 (1989).
[Stä02]
M. Stähler, R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev
and V. Yudin, Coherent population trapping resonances in thermal
85
Rb vapor:
D1 versus D2 line excitation, Opt. Lett. 27, 1472 (2002).
[Ste99a]
J. Stenger, S. Inouye, M. R. Andrews, H.-J. Miesner, D. M. Stamper-Kurn and
W. Ketterle, Strongly Enhanced Inelastic Collisions in a Bose-Einstein Condensate near Feshbach Resonances, Phys. Rev. Lett. 82, 2422 (1999).
[Ste99b]
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).
[Ste06]
J. T. Stewart, J. P. Gaebler, C. A. Regal and D. S. Jin, Potential Energy of a
40
[Str03]
K Fermi Gas in the BCS-BEC Crossover, Phys. Rev. Lett. 97, 220406 (2006).
K. E. Strecker, G. B. Partridge and R. G. Hulet, Conversion of an Atomic Fermi
Gas to a Long-Lived Molecular Bose Gas, Phys. Rev. Lett. 91, 080406 (2003).
[Tag08]
M. Taglieber, A.-C. Voigt, T. Aoki, T. W. Hansch and K. Dieckmann, Quantum degenerate two-species Fermi-Fermi mixture coexisting with a Bose-Einstein
condensate, Phys. Rev. Lett. 100, 010401 (2008).
[Tai99]
A. V. Taichenachev, A. M. Tumaikin and V. I. Yudin, Electromagnetically induced absorption in a four-state system, Phys. Rev. A 61, 011802 (1999).
[Tai06]
A. V. Taichenachev, V. I. Yudin, V. L. Velichansky, A. S. Zibrov and S. A.
Zibrov, Pure superposition states of atoms generated by a bichromatic elliptically
polarized field, Phys. Rev. A 73, 013812 (2006).
BIBLIOGRAPHY
[Tar07]
147
L. Tarruell, M. Teichmann, J. McKeever, T. Bourdel, J. Cubizolles,
L. Khaykovich, J. Zhang, N. Navon, F. Chevy and C. Salomon, Expansion
of an ultra-cold lithium gas in the BEC-BCS crossover, Preprint arXiv:0701181
(2007).
[Tho82]
J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard and C. R.
Willis, Observation of Ramsey Fringes Using a Stimulated, Resonance Raman
Transition in a Sodium Atomic Beam, Phys. Rev. Lett. 48, 867 (1982).
[Tho05a] J. E. Thomas, J. Kinast and A. Turlapov, Virial Theorem and Universality in
a Unitary Fermi Gas, Phys. Rev. Lett. 95, 120402 (2005).
[Tho05b] S. T. Thompson, E. Hodby and C. E. Wieman, Ultracold Molecule Production
via a Resonant Oscillating Magnetic Field, Phys. Rev. Lett. 95, 190404 (2005).
[Tic04]
C. Ticknor, C. A. Regal, D. S. Jin and J. L. Bohn, Multiplet structure of Feshbach
resonances in nonzero partial waves, Phys. Rev. A 69, 042712 (2004).
[Tie93]
E. Tiesinga, B. J. Verhaar and H. T. C. Stoof, Threshold and resonance phenomena in ultracold ground-state collisions, Phys. Rev. A 47, 4114 (1993).
[Tin66]
M. Tinkham, Introduction to superconductivity, McGraw-Hill, New York (1966).
[Tru01]
A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge and R. G.
Hulet, Observation of Fermi Pressure in a Gas of Trapped Atoms, Science 291,
2570 (2001).
[Tsu00]
C. C. Tsuei and J. R. Kirtley, Phase-Sensitive Evidence for d-Wave Pairing
Symmetry in Electron-Doped Cuprate Superconductors, Phys. Rev. Lett. 85,
182 (2000).
[Vál77]
L. Vályi, Atom and Ion Sources, Wiley, New York (1977).
[Val02]
P. Valente, H. Failache and A. Lezama, Comparative study of the transient
evolution of Hanle electromagnetically induced transparency and absorption resonances, Phys. Rev. A 65, 023814 (2002).
148
[Val03]
BIBLIOGRAPHY
P. Valente, H. Failache and A. Lezama, Temporal buildup of electromagnetically
induced transparency and absorption resonances in degenerate two-level transitions, Phys. Rev. A 67, 013806 (2003).
[Vel80]
V. L. Velichansky, A. S. Zibrov, V. S. Kargopoltsev, O. R. Kachurin, V. V.
Nikitin, V. A. Sautenkov, G. G. Kharisov and D. A. Tyurikov, Spectroscopy
within the Doppler profile of potassium D lines using an external-resonator semiconductor injection laser, Sov. Journal of Quantum Electronics 10, 1244 (1980).
[Vol06]
T. Volz, N. Syassen, D. M. Bauer, E. Hansis, S. Dürr and G. Rempe, Preparation
of a quantum state with one molecule at each site of an optical lattice, Nature
Physics 2, 692 (2006).
[Wal03]
J. Walls, R. Ashby, J. J. Clarke, B. Lu and W. A. van Wijngaarden, Measurement of isotope shifts, fine and hyperfine structure splittings of the lithium D
lines, European Physics Journal D 22, 159 (2003).
[Wei99]
J. Weiner, V. S. Bagnato, S. Zilio and P. S. Julienne, Experiments and theory
in cold and ultracold collisions, Rev. Mod. Phys. 71, 1 (1999).
[Wil04]
J. E. Williams, N. Nygaard and C. W. Clark, Phase diagrams for an ideal gas
mixture of fermionic atoms and bosonic molecules, New Journal of Physics 6,
123 (2004).
[Wyn99]
R. Wynands and A. Nagel, Precision spectroscopy with coherent dark states,
Appl. Phys. B 68, 1 (1999).
[Xu02]
X. Xu, T. H. Loftus, M. J. Smith, J. L. Hall, A. Gallagher and J. Ye, Dynamics
in a two-level atom magneto-optical trap, Phys. Rev. A 66, 011401 (2002).
[Xu03]
K. Xu, T. Mukaiyama, J. R. Abo-Shaeer, J. K. Chin, D. E. Miller and W. Ketterle, Formation of Quantum-Degenerate Sodium Molecules, Phys. Rev. Lett.
91, 210402 (2003).
[Zha97]
Y. Zhao, C. Wu, B.-S. Ham, M. K. Kim and E. Awad, Microwave Induced
Transparency in Ruby, Phys. Rev. Lett. 79, 641 (1997).
149
BIBLIOGRAPHY
[Zha04]
J. Zhang, E. G. M. van Kempen, T. Bourdel, L. Khaykovich, J. Cubizolles,
F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans and C. Salomon,
P-wave Feshbach resonances of ultracold 6 Li, Phys. Rev. A 70, 030702 (2004).
[Zib95]
A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L.
Velichansky and H. G. Robinson, Experimental Demonstration of Laser Oscillation without Population Inversion via Quantum Interference in Rb, Phys. Rev.
Lett. 75, 1499 (1995).
[Zib05]
A. S. Zibrov and A. B. Matsko, Induced absorption resonance on the open Fg
= 1 → Fe = 2 transition of the D1 line of the
87
Rb, Zh. Eksp. Teor. Phys. 82,
529 (2005).
[Zwi03]
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).
[Zwi04]
M. 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).
[Zwi05]
M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck and W. Ketterle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature 435,
1047 (2005).
[Zwi06a]
M. W. Zwierlein, A. Schirotzek, C. H. Schunck and W. Ketterle, Fermionic
Superfluidity with Imbalanced Spin Populations, Science 311, 492 (2006).
[Zwi06b] M. W. Zwierlein, C. H. Schunck, A. Schirotzek and W. Ketterle, Direct observation of the superfluid phase transition in ultracold Fermi gases, Nature 442,
54 (2006).
150
PUBLICATIONS OF THE AUTHOR
151
Publications of the author
Contents of this thesis have been published in:
Chapter 5
J. Fuchs, G. J. Duffy, W. J. Rowlands and A. M. Akulshin, Electromagnetically induced
transparency in 6 Li, J. Phys. B. 39, 3479 (2006).
J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford and A. M. Akulshin,
Electromagnetically induced transparency and absorption due to optical and ground-state
coherences in 6 Li, J. Phys. B. 40, 1117 (2007).
Chapter 8
J. Fuchs, G. J. Duffy, G. Veeravalli, P. Dyke, M. Bartenstein, C. J. Vale, P. Hannaford
and W. J. Rowlands, Molecular Bose-Einstein condensation in a versatile low power
crossed dipole trap, J. Phys. B. 40, 4109 (2007).
Chapter 9
J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Rowlands, P. Hannaford and
C. J. Vale, Binding Energies of 6 Li p-wave Feshbach Molecules, Phys. Rev. A 77, 053616
(2008).
Other publications of the author:
S. Ospelkaus, C. Ospelkaus, R. Dinter, J. Fuchs, M. Nakat, K. Sengstock and K. Bongs,
Degenerate K-Rb Fermi-Bose gas mixtures with large particle numbers, Journal of
Modern Optics, advanced online publication (2006).