Studies of a Quantum Degenerate Fermionic Lithium Gas
byZoran Hadzibabic
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 2003
c Massachusetts Institute of Technology 2003. All rights reserved.
°

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 31, 2003

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomas J. Greytak
Professor of Physics, Associate Department Head for Education

Studies of a Quantum Degenerate Fermionic Lithium Gas
byZoran Hadzibabic
Submitted to the Department of Physics
on August 31, 2003, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
The work described in this thesis can be divided into two parts. The first part is concernedwith the production of degenerate Fermi gases and Bose-Fermi mixtures. The second partof the thesis describes studies of strongly interacting spin-mixtures of ultracold fermions.
We have cooled a magnetically trapped spin-polarized 6 Li Fermi gas into quantum degeneracy by sympathetic cooling with bosonic 23 Na. In order to do this, we have converted ourpreviously developed apparatus for Bose-Einstein condensation of 23 Na into a two-species
cooling facility. We have produced 6 Li samples with up to 7 × 107 atoms at temperatures
below 0.5 TF , and up to 3 × 107 atoms at temperatures as low as 0.05 TF . Further, we havealso produced stable quantum degenerate mixtures of 23 Na and 6 Li, with several million
atoms of each species.
In a second set of experiments, we have studied strongly interacting spin-mixtures of
6 Li. The strength of interactions was resonantly controlled by applying a magnetic field
and tuning to “Feshbach resonances”, which occur when the total energy of the colliding
particles coincides with the energy of a bound molecular state. These studies are relevant
for the continuing quest for a superfluid transition in a Fermi gas.
By studying the inelastic decay of the cloud, we have experimentally identified two
Feshbach resonances for the inter-state interactions of the two lowest ground states of 6 Li.
Most importantly, we found that the lifetime of the strongly interacting system is longenough to observe a superfluid transition in the future.
The effects of the coherent collisions in 6 Li spin-mixtures were studied using radiofrequency (RF) spectroscopy. We demonstrated the absence of mean field “clock shifts” ina two-state mixture of fermions, and have introduced a three-level method to observe theinteraction energy shifts. In the strongly interacting regime, near a Feshbach resonance, wehave observed universal behavior of the interaction energy, independent of the value of thescattering length.
Finally, we have theoretically studied the effects of Pauli blocking during a free expansionof a degenerate Fermi gas. We have concluded that a strongly interacting normal gas will
undergo a hydrodynamic expansion even at zero temperature, due to inefficiency of thePauli suppression of collisions. Therefore, the already observed hydrodynamic expansion
cannot be conclusively interpreted as a signature of superfluidity.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

2

To my parents, Borislav and Ljiljana

and my sister Selena

3

Acknowledgments
we took me and D. Boon’s and George’s stench
and put them up on stage
watt, Tour Spiel

During my years at MIT, I was indebted both professionally and personally to more
people than I could possibly list here. While some have directly contributed to my work,
others have made life outside lab worth living. I was particularly fortunate to have as myimmediate collaborators the sort of people who fall into both categories. Faced with theknowledge that I cannot do justice to everyone, I merely mention some of those whose
impact on my life was the greatest.
First of all, I would like to thank Wolfgang Ketterle for having faith in me and giving
me a chance to work in his lab at a time when it was not clear that I had any experimental
talents. Secondly, I would like to thank Wolfgang Ketterle for having faith in me in myearly days and months in the group, when it was still not clear that I had any experimental
talents. Over the years, Wolfgang has provided not only brilliant scientific advice, but also
moral guidance in times of hardship. Dave Pritchard has acted as an unofficial second
advisor, partially because he cannot really help it, and partially because I would havebullied him into it if he could. Dave has offered council beyond the call of duty, and hisshoot-from-the-hip approach to discussing anything, sometimes physics, has been greatly
appreciated.
Next I would like to express my eternal gratitude to the other graduate students on thelithium crew, with whom I have shared many a day and night - (Subha)Deep Gupta (whojams econo), Claudiu Stan, Martin Zwierlein and Christian Schunck. None of the work
described in this thesis would have been possible without them. Nor would all the hard
work be half as enjoyable without the sense of camaraderie that surpassed anything I could
have hoped for.
Deep and I arrived at MIT at the same time, taken all our exams at the same time, anddefended our theses within a day of each other. Over the last few years, we have shared
the office and worked together, or at least discussed both physics and life, almost everyday. This thesis is his as much as it is mine. I could not have had a better partner on
this journey. Deep is a constant source of ideas, both serious and bizarre and stupendously
funny. He also has an impeccable taste in food, a profound understanding of the genius
of Audrey Hepburn, and an agreeable taste in music. I should also admit that he is muchbetter than me at oil wrestling.
With Claudiu, I have started off the lithium project as a two-man show, and we sharea unique bond of entering a completely unknown territory together, uncertain whether our
4efforts would ever bear fruition. Those early days were frustrating more than not, andClaudiu’s unshakeable calmness has given me invaluable emotional support, probably moreso than he might be aware. Not to mention that he has built all the hardest parts of theexperiment, which would have taken me infinitely longer time.
Both Martin and Christian have initially joined us as “apprentices”, to do their undergraduate theses under our guidance. Within months, this apprenticeship had turned into an
equal partnership and a close friendship. We were fortunate to convince them to continue
working in our group as graduate students. It gives me a great peace of mind to know thatthe future of this experiment lies in the hands of two such enthusiastic and brilliant young
scientists.
Due thanks should also be given to the two experienced postdocs with whom we haveworked at different times - Kai Dieckmann and Axel Görlitz. They have not only offeredinvaluable advice, but have also taken equal part in the sometimes miserable life of graduate
students, doing the most mundane jobs and staying up with us all night, struggling withthe experiment.
Outside the lithium group, there is a large AMO community at MIT, and I am indebted
in various small and big ways to all of its members. Some of them simply cannot go
unmentioned.
Before starting the lithium project, I was guided in my first steps into experimental
atomic physics by the original BEC II group - Chandra Raman, Roberto Onofrio, MichaelKöhl, Dallin Durfee and Chris Kuklewicz. I have learned a great deal from each andeveryone of them, as well as from Dan Stamper-Kurn, Shin Inouye and Ananth Chikkatur,
who had been running the original BEC machine, now converted into our sodium-lithium
experiment.
Jamil Abo-Shaeer joined the group at about the same time as I did, and within 24
hours we had formed a friendship that has lasted and deepened over the years, and which
has enriched me for the rest of my life. His ability to cheer me up at work, when cheering
up is needed, and his selfless dedication to all of his friends, are second to none. Anotherveteran student whose stay at MIT has greatly overlapped with mine, and for whom I havedeveloped a fondness reserved only for the most wonderful and genuine people, is Aaron
Leanhardt. Aaron is also the single person, outside the lithium lab, whose physics help andadvice I have sought most often.
Among the many younger students who have joined the group in the recent years, I
would particularly like to mention Jit Kee Chin, Gretchen Campbell and Micah Boyd. I willcherish forever the times we have spent together, both in Cambridge and at an unforgettableconference in Orlando.
Last but not least, I would like to thank Carol Costa and everyone at RLE headquarters for keeping the show running. Carol in particular has helped me with all the little
organizational things I would have forgotten or failed to do in time, or at all.

5

Outside MIT, I have had a home away from home at 63 Henry Street in Cambridgeport.
There is no way for me to express how fortunate I was to live with Walter Rantner andMirko & Natasa Ristivojevic, whom I have known and loved for more than a decade, andwho have been my second family.
The part-time staff of the 63 is numbered in dozens. Everyone who has ever lived withus, visited us for a few weeks, or just came over for dinner and sat on our porch until thewee hours of the morning, has made me feel more at home. Bosiljka Tasic and Gojko Lalichave been over so many times, they could have just as well moved in. Particularly cherishedwere the visits by my sister Selena, who I have lately finally had a chance to spend moretime with, after seeing her way too infrequently for seven years.
While I am in many (predictable) ways happy that I am now done with my PhD, it
saddens me that the 63 crowd is being disbanded.
Finally, I would like to thank this lot for keeping it lit: Bluefin at Porter Sq., Will
Oldham, Bill Calahan, Chan Marshall, David Gedge, Howe Gelb, Joey Burns & John Convertino, Miracle of Science on Mass. Ave., Ethan & Joel Coen, Jeff Bridges, John Goodman, Larry Sellers, Philip Seymour Hoffman, Major Tom, the couple that used to run Mars
Records on Mass Ave., Guided By Voices, Yo La Tengo, Darkwood Dub, Jarboli, Edward
Norton, David Fincher, Paul Auster, Mira Sorvino, anyone who ever reads these pages,
John Lurie, Morrissey & Marr, Forster & Mclennan, Michael Rapaport, Gwyneth Paltrow,
Eliza Dushku, Nick Hornby, Jay McInerney, Dimitrios Cuisine & Pizza and Stefani’s (bothon Brookline St. in Cambridge), The Minutemen, Sacramento Kings, Dejan Bodiroga,
Tom Stoppard, D. C. Berman, Drag City, Frank Miller, Alan Moore, Garth Ennis, Adam
Green & Kimya Dawson, Other Music, Jack Rabid & the Big T, ventolin, Presing, Grant
Hart, Steve Wynn, Neil Gaiman, Destiny, Death, Dream, Destruction, Desire, Despair &Delirium.

This work was supported by the National Science Foundation (NSF), the Office ofNaval Research (ONR), the Army Research Office and the Joint Services Electronics Program (JSEP) of the Army Research Office (ARO), the National Aeronautics and Space
Administration (NASA), and the David and Lucile Packard Foundation.

6

Contents
1 Introduction
1.1

1.2

1.3

12

Of Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.1.1

Bose-Einstein condensates - a success story . . . . . . . . . . . . . .

13

1.1.2

Quantum degenerate fermions . . . . . . . . . . . . . . . . . . . . . .

14

1.1.3

Bose-Fermi mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Trapped ultracold gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.2.1

How cold is cold enough? . . . . . . . . . . . . . . . . . . . . . . . .

15

1.2.2

The µK containers for atoms . . . . . . . . . . . . . . . . . . . . . .

15

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2 Two-species apparatus for cooling of 6 Li and 23 Na
2.1

17

Sympathetic cooling of fermions . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.1.1

Why two species? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.1.2

Why a boson and a fermion? . . . . . . . . . . . . . . . . . . . . . .

18

2.1.3

Why 23 Na and 6 Li?

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

19

2.2

Outline of our cooling strategy . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3

Two-species oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4

Laser cooling of 6 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.4.1

Some laser cooling basics . . . . . . . . . . . . . . . . . . . . . . . .

23

2.4.2

Simultaneous Zeeman slowing of 6 Li and 23 Na. . . . . . . . . . . .

25

2.4.3

Differences between 6 Li and 23 Na MOTs . . . . . . . . . . . . . . . .

26

2.4.4

6 Li laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Two-species MOT for 6 Li and 23 Na . . . . . . . . . . . . . . . . . . . . . . .

32

2.5

2.6

2.5.1

Single-species 6 Li MOT. . . . . . . . . . . . . . . . . . . . . . . . .

32

2.5.2

Coexistence of the two MOTs . . . . . . . . . . . . . . . . . . . . . .

34

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Production of coexisting 6 Li Fermi seas and 23 Na BECs

36
37

3.1

Choice of 23 Na and 6 Li hyperfine states for sympathetic cooling experiments

37

3.2

Sympathetic cooling with sodium in the F = 1 state . . . . . . . . . . . . .

41

3.2.1

45

Some remaining confusions... . . . . . . . . . . . . . . . . . . . . . .
7

3.3

3.4

3.5

Condensation of 23 Na in the F = 2 hyperfine state . . . . . . . . . . . . . .

46

3.3.1

Magnetic trap loading . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.3.2

Avoiding three-body losses . . . . . . . . . . . . . . . . . . . . . . .

49

Sympathetic cooling with sodium in the F = 2 state . . . . . . . . . . . . .

50

3.4.1

Boost in the 6 Li atom number

3.4.2

Trading-in the 6 Li atom number for deeper quantum degeneracy

. .

51

3.4.3

Two-species degenerate mixture . . . . . . . . . . . . . . . . . . . . .

52

Summary of sympathetic cooling experiments . . . . . . . . . . . . . . . . .

52. . . . . . . . . . . . . . . . . . . . .

4 Interactions in 6 Li spin mixtures
4.1

4.2

4.3

50

54

Language of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.1.1

Inelastic, elastic, and coherent collisions . . . . . . . . . . . . . . . .

54

4.1.2

Elastic scattering amplitude . . . . . . . . . . . . . . . . . . . . . . .

55

4.1.3

Weakly interacting limit . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.1.4

Strongly interacting limit . . . . . . . . . . . . . . . . . . . . . . . .

56

Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.2.1

Scattering resonances . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.2.2

Tuning the scattering length with a magnetic field . . . . . . . . . .

59

4.2.3

Strongly interacting limit revisited . . . . . . . . . . . . . . . . . . .

61

4.2.4

Adiabatic creation of diatomic molecules . . . . . . . . . . . . . . . .

61

High temperature superfluidity of 6 Li?

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

5 Experiments with 6 Li in the vicinity of Feshbach resonances

62
64

5.1

Prediction of Feshbach resonance(s) in 6 Li . . . . . . . . . . . . . . . . . . .

64

5.2

Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.2.1

Generation of uniform high magnetic fields . . . . . . . . . . . . . .

65

5.2.2

Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5.2.3

Preparation of interacting spin mixtures . . . . . . . . . . . . . . . .

66

5.2.4

Spin selective imaging at high magnetic fields . . . . . . . . . . . . .

67

What to look for? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.3

6 Inelastic losses near a Feshbach resonance

70

6.1

Note on the experimental procedure . . . . . . . . . . . . . . . . . . . . . .

71

6.2

Two decay resonances of the |1i − |2i spin mixture . . . . . . . . . . . . . .

71

6.3

Relative positions of the decay and the Feshbach resonances . . . . . . . . .

72

6.4

The main good news . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

7 Elastic collisions and the expansion measurements

75

7.1

Hydrodynamic expansion of a classical gas . . . . . . . . . . . . . . . . . . .

76

7.2

Superfluid vs. classical hydrodynamics in a degenerate Fermi gas . . . . . .

79

7.3

Pauli blocking of collisions during the expansion of a degenerate Fermi gas .

81

8

7.4

7.3.1

Effects of the Fermi surface deformation on Pauli blocking . . . . . .

81

7.3.2

Deformation of the Fermi surface during the expansion . . . . . . . .

83

Conclusions and one final thought . . . . . . . . . . . . . . . . . . . . . . .

86

8 RF Spectroscopy of coherent collision energy shifts
8.1

8.2

87

Spectroscopic insensitivity to coherent collisions in a two-level system . . .

88

8.1.1

Bloch sphere representation of the RF transitions . . . . . . . . . . .

90

8.1.2

Comparison with bosons . . . . . . . . . . . . . . . . . . . . . . . . .

92

Spectroscopy of cold collisions in a three-level system . . . . . . . . . . . . .

92

8.2.1

95

Strongly interacting and high density limit . . . . . . . . . . . . . .

9 Conclusions and outlook

98

A 6 Li laser system baedeker

101

A.1 Diode Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

A.1.1 Free Running Diode Lasers . . . . . . . . . . . . . . . . . . . . . . .

101

A.1.2 Grating Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

A.1.3 Injection Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

Locking of the master laser to the 6 Li atomic resonance. . . . . . . . . . .

108

A.3 Imaging frequencies at high magnetic fields . . . . . . . . . . . . . . . . . .

111

Layout of the 6 Li laser system . . . . . . . . . . . . . . . . . . . . . . . . . .

113

A.2
A.4

B Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases

115

C Fifty-fold improvement in the number of quantum degenerate fermionic
atoms

120

D Radio-Frequency Spectroscopy of Ultracold Fermions

125

E Spectroscopic insensitivity to cold collisions in a two-state mixture offermions

130

9

List of Figures
1-1 Classical, Bose and Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . .

13

2-1 Outline of the 6 Li cooling strategy . . . . . . . . . . . . . . . . . . . . . . .

20

2-2

Two-species oven for 23 Na and 6 Li

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

22

2-3

Energy level diagram for laser cooling of 6 Li . . . . . . . . . . . . . . . . . .

23

2-4

Outline of the 6 Li laser system

29. . . . . . . . . . . . . . . . . . . . . . . . .

2-5 Frequencies of seeded slave lasers recorded by a Fabry-Perot spectrum analyzer 30
2-6 Tandem AOM for the 6 Li imaging light . . . . . . . . . . . . . . . . . . . .
2-7

Absorption monitor of a 6 Li MOT

2-8

Measurement of the 6 Li MOT temperature

2-9

Dependence of the 6 Li MOT temperature on the detuning of the cooling light. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .

31
32
33
34

2-10 Inter-species light assisted collisions in a two-species MOT . . . . . . . . . .

35

3-1 Ioffe-Pritchard magnetic trap coils . . . . . . . . . . . . . . . . . . . . . . .

38

3-2

Hyperfine structures of 23 Na and 6 Li

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

39

3-3

6 Li cooling path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3-4 Onset of Fermi degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3-5 Loading of |F = 2, mF = 2i sodium atoms into the magnetic trap . . . . . .

47

3-6 Frequencies for sodium optical pumping . . . . . . . . . . . . . . . . . . . .

48

3-7 Large and ultra-degenerate Fermi sea . . . . . . . . . . . . . . . . . . . . . .

52

3-8 Two-species mixture of degenerate Bose and Fermi gases . . . . . . . . . . .

53

4-1 Cartoon of scattering resonances. . . . . . . . . . . . . . . . . . . . . . . . .

58

4-2 Shape scattering resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4-3 Feshbach resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

5-1 Predicted Feshbach resonance in 6 Li . . . . . . . . . . . . . . . . . . . . . .

65

5-2

High field imaging and mutual cooling of an optically trapped 6 Li spin mixture

6-1 Magnetic field dependence of inelastic losses in a 6 Li |1i − |2i spin mixture

68
72

6-2 Updated prediction of the 6 Li a12 Feshbach resonances . . . . . . . . . . . .

73

7-1 Ballistic and hydrodynamic expansion of a classical gas . . . . . . . . . . .

77

10

7-2 Anisotropic expansion of a strongly interacting 6 Li spin mixture
7-3. . . . . .

Superfluid hydrodynamic expansion of a weakly interacting 23 Na BEC

78

. . .

80

7-4 Opening up of final states for elastic scattering at zero temperature. . . . .

82

7-5 Local momentum-space Fermi surface of an expanding gas . . . . . . . . . .

84

7-6 Allowed and forbidden elastic collisions during the expansion. . . . . . . . .

84

7-7 Fraction of Pauli allowed elastic collisions during the expansion. . . . . . . .

85

8-1 Equilibrium shifts of the energy levels in an interacting two-state mixture offermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

8-2 Absence of the clock shift in a pure fermionic state . . . . . . . . . . . . . .

89

8-3 Absence of the clock shift in a decohered two-state mixture of fermions . . .

90

8-4 Bloch sphere representation of the RF transitions . . . . . . . . . . . . . . .

91

8-5 RF spectroscopy in a three-level system . . . . . . . . . . . . . . . . . . . .

93

8-6 Spectroscopic measurement of the interaction energies . . . . . . . . . . . .

94

8-7 Spectroscopic frequency shifts as a function of the magnetic field . . . . . .

95

8-8 Expansion measurements of the interaction energy by Bourdel et al. . . . .

96

A-1 Grating stabilized laser system in a Littrow configuration . . . . . . . . . .

105

A-2 Summary of frequency scales in a grating stabilized laser . . . . . . . . . . .

106

A-3 Seeding of a diode laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

A-4 Saturated absorption spectroscopy setup . . . . . . . . . . . . . . . . . . . .

108

A-5 Optical pumping Doppler free spectroscopy . . . . . . . . . . . . . . . . . .

109

A-6 Absorptive and dispersive Doppler free spectrum . . . . . . . . . . . . . . .

111

A-7

Optical transitions in 6 Li at high magnetic fields. . . . . . . . . . . . . . .

112

A-8

Layout of the 6 Li laser system . . . . . . . . . . . . . . . . . . . . . . . . . .

114

11

Chapter 1

Introduction
I was one of seventeen children. We were all named Wilkinson...
I would’ve liked to’ve bin an only child.
That way when someone shouts Wilkinson, you know if it’s you or not.
Neil Gaiman, A Game of you

1.1

Of Bosons and Fermions

All particles in nature can be divided into two categories according to their “social behavior”.
While identical bosons like to stick together, identical fermions avoid each other at all costs.
Atoms fall into one of the two categories depending on the total number of protons, neutronsand electrons that they are made of. If this number is even, the atom is a boson; if it is odd,
the atom is a fermion. In this thesis we will be mostly concerned with two atomic species,
one fermionic - 6 Li, and one bosonic - 23 Na.
At a quantum mechanical level, the world is discrete, partitioned into unit cells of finite
size. Alternatively, we say that a particle can occupy only a discrete set of “quantum
mechanical states”, which correspond to different energies. While identical bosons likebeing in the same state, no more than one identical fermion can occupy the same unit cell.
Naturally, the question whether two particles like each other or not, arises only if they
actually meet. The probability that this happens distinguishes “classical” from “quantum
degenerate” gases. In a classical gas, such as the air around us, the meeting probability
is low, and hence it is not important whether the particles are fermions or bosons. In
this case, only a small fraction of energetically available quantum states in the system is
occupied by atoms. The issue whether two atoms would want to (bosons), or could not(fermions) occupy the same state then rarely arises.
However, if we cool the gas to very low temperatures, this greatly reduces the energy
available to the atoms, and hence the choice of states that they can occupy. The differences
12between Bose and Fermi atomic gases become pronounced once the number of energeticallyavailable states in the system is reduced approximately to the number of particles.
In the quantum degenerate regime, bosons pile up in the lowest energy state and forma “Bose-Einstein condensate” (BEC). Fermions on the other hand are forced to occupy
different states, and form a “Fermi sea”. This is illustrated in Fig. 1-1 for the case of atoms
trapped in a parabolic potential well.

energyBEC

Bosonsclassical gas

cold
Fermi Sea

Fermions

degeneracy
temperature

decreasing temperature

Figure 1-1: Classical, Bose and Fermi gases. The atoms occupy a discrete set of quantum
states corresponding to different energies. At high temperatures, there is no distinction
between Bose and Fermi gases. Here, the occupied states are few and far between. As welower the temperature of the gas, the atoms fall into lower and lower energy states. A fork
in the road occurs once neighboring states start being occupied. At this point, the atoms
have to decide whether they want to move in together or not. Bosons decide to all pile up
in the same state. Fermions on the other hand each keep a state to themselves, even though
this is much more energetically expensive.

1.1.1

Bose-Einstein condensates - a success story

In 1995, atomic physicists have successfully cooled bosonic gases into quantum degeneracy [10–12]. The story of BEC is one of great success, and has by now been told manytimes. What started as the quest for the demonstration of the BEC, has lent itself to muchbroader research, both in conventional atomic physics and in what has traditionally been
the area of condensed matter physics. A good summary of the BEC history is provided
by the 2001 Nobel Prize lectures of Eric Cornell and Carl Wieman [13], and Wolfgang
Ketterle [14].
Broadly speaking, atomic BECs are fascinating for two reasons. First, they give us
an almost textbook example of a weakly interacting superfluid, much more amenable totheoretical calculations than the strongly interacting 4 He liquid. Further, in atomic systems
13

we can relatively easily manipulate the strength of the particle interactions, as well as change
their nature from repulsive to attractive and vice versa. This makes BECs an ideal test bed
for fundamental many-body theories.
Second, atomic BECs are truly macroscopic coherent matter waves. This makes themextremely bright sources for a range of atom optics applications1 , from precision measurements of fundamental constants, to development of novel navigation instruments.

1.1.2

Quantum degenerate fermions

A natural complement to BEC research is to produce and study a quantum degenerate
gas of Fermi atoms. After the pioneering work by DeMarco and Jin in 1999 [15], thissubfield has gained its own momentum, and it is now becoming increasingly obvious that it
will provide (at least) as much excitement. So far, six experimental groups have achieved
quantum degeneracy in either 40 K [15, 16] or 6 Li [1, 17–19].
Like their Bose counterparts, these systems offer a great flexibility in tuning the experimental parameters such as particle number, density, strength of interactions, spin compositions or the temperature.
One particularly interesting prospect of this research is the attainment of fermionicsuperfluidity in a gaseous system. In analogy with superconductors or superfluid 3 He, at
low enough temperatures and for attractive effective interactions, fermionic atoms can pair
up into bosonic Cooper pairs [20], and undergo a phase transition into a superfluid state.
Due to the widely tunable experimental parameters, studies of superfluidity in an atomic
Fermi gas could provide a more general understanding of the pairing mechanisms in a range
of very different systems, from solids to neutron stars and the quark matter [21].
Experimental studies of ultracold Fermi systems in the regimes of strong and controllable interactions are already well underway [3, 5, 22–30]. According to some theories ofsuperfluidity in a Fermi gas [31–42], the combination of the temperature and the attractive
interactions required for the phase transition is already within experimental reach. Oneof the main current experimental challenges is to identify an effective way to observe thetransition.

1.1.3

Bose-Fermi mixtures

Finally, given that all particles obey one of the only two types of quantum statistics, thescientific motivation for producing and studying quantum degenerate mixtures of Bose andFermi atoms is obvious. A classic example of such a degenerate Bose-Fermi system is thestrongly interacting 4 He−3 He mixture.
A single experimental apparatus in which degenerate gases of either kind could be produced, as well as their mixtures, then seems like an omnipotent solution.
1

In the field of “atom optics”, we manipulate matter waves in the ways that we are used to manipulating
light - reflect them off mirrors, focus them with lenses, interfere them, or diffract them off gratings.

14

This thesis describes the production of a degenerate Bose-Fermi mixture of 23 Na and6 Li [1, 2]. In other current experiments, degenerate mixtures of 7 Li and 6 Li [17, 18], as well

as 87 Rb and 40 K [16], are also being produced.

1.2

Trapped ultracold gases

To get a sense of scale, let us introduce some basic experimental parameters of ultracold
atomic clouds.

1.2.1

How cold is cold enough?

We can quantify the level of quantum degeneracy in a gas in terms of the phase spacedensity:ρ = nλ3dB(1.1)√
where n is the gas density, and λdB = h/ 2πmkB T is the thermal deBroglie wavelength,
which measures the size of the quantum wave packets describing individual atoms. Forρ ¿ 1, there is no distinction between fermions and bosons. The cross-over to quantum

degeneracy occurs at a temperature for which ρ ∼ 1. At this point, the “size” of the atoms

becomes comparable to the typical distance between two particles in the gas, and the atoms
start “feeling each other” in real space. For an atomic gas with the same density as air, thedegeneracy temperature would be about 10 mK.
However, an additional problem in producing an atomic quantum gas is that at such
low temperatures, most substances will become solid, or at least liquid, before quantum
degeneracy is reached. We cannot really solve this problem, but we can postpone it. In
order to turn into a solid, an atomic gas starts off by creating molecules, which then combine
into larger and larger clusters. For this process to start, several atoms must be found in
the same place at the same time. If the gas density is very low, the probability of thishappening is greatly reduced and the process is slowed down.
In order to perform our experiments, we must make sure that the gaseous state has alifetime of seconds to minutes. This requires that the gas density is about a million times
lower than the density of air. The typical degeneracy temperature is then reduced to about
1 µK, i.e. one millionth of a degree above absolute zero.

1.2.2

The µK containers for atoms

Ultracold atomic clouds are too fragile to be put in contact with any material object.
The only place where they live long enough is in ultra high vacuum (UHV), levitating infree space electromagnetic traps. Even if it is not in contact with any material object, atrapped ultracold gas will eventually be destroyed by collisions with the background gas
15

in the chamber, which is at room temperature. In order to give our samples a respectable
lifetime, of about one minute, the pressure inside the vacuum chamber must be kept below
10−11 Torr, about 1014 times lower than the atmospheric pressure.
Usual traps are parabolic, or harmonic, like those illustrated in Fig. 1-1. The characteristic trapping frequency, with which the atoms oscillate around the bottom of thetrap, is ω ∼ 2π × 100 Hz. The typical size of an ultracold, µK atomic cloud is thenp
R = kB T /(mω 2 ) ∼ 50 µm.

In a cloud containing N atoms, the density in the center of the trap is approximatelyn ∼ N/R3 , and we can rewrite Eq. 1.1 as ρ ∼ N (~ω/kB T )3 . We thus most commonly

express the degeneracy criterion ρ ∼ 1, as:kB TDEG ∼ ~ωN 1/3(1.2)

where the number of atoms in a degenerate gas is typically N ∼ 106 .

To summarize, a typical quantum degenerate atomic cloud is a million times thinnerthan air, and contains about a million atoms, at a temperature of one millionth of a degree
above absolute zero.

1.3

Outline of this thesis

The first part of this thesis describes the production of degenerate Fermi gases and BoseFermi mixtures [1, 2]. In Chapter 2, I introduce our cooling strategy and all the technicaldevelopments which were needed in order to convert our previously built sodium BEC
apparatus into a joint 23 Na−6 Li experimental facility. Our results on simultaneous coolingof lithium and sodium into quantum degeneracy are summarized and discussed in Chapter 3.
The second part of the thesis, consisting of Chapters 4 - 8, is dedicated to interacting Fermi gases. Chapter 4 provides some theoretical background - I discuss the types ofcollisions which occur in spin-mixtures of fermions, how the interactions can be externallycontrolled near “Feshbach resonances”, and the possibility of achieving fermionic superfluidity. In Chapter 5, I briefly introduce the additional experimental techniques needed forstudies of 6 Li in the strongly interacting regime. Chapters 6 - 8 describe our experimental
and theoretical studies of different types of interactions in ultracold Fermi clouds [3–6].
Each of these three chapters is dedicated to one type of collisions - inelastic [3], incoherent
elastic [4], and coherent [5, 6].
Finally, some more technical aspects of the 6 Li laser system, as well as the reprints or
preprints of four publications [1, 2, 5, 6], are included in Appendices at the end of the thesis.

16Chapter 2

Two-species apparatus for coolingof 6Li and 23Na
This chapter describes the “development” phase of our two-species experiment, namely our
cooling strategy, the technical additions to the previously existing BEC apparatus, and thesimultaneous production of laser cooled 6 Li and 23 Na samples. This is the starting point forthe production of quantum degenerate Fermi and Bose-Fermi gases, which will be describedin Ch. 3.

2.1

Sympathetic cooling of fermions

2.1.1

Why two species?

All experiments with quantum degenerate alkali gases rely on two classes of cooling methods - laser cooling, and evaporative cooling [43, 44]. Evaporative cooling of Bose gases
is most commonly performed with magnetically trapped spin-polarized samples. As willsoon become apparent, this simplistic approach cannot be taken in cooling of fermions intoquantum degeneracy.
All laser cooling methods, such as Zeeman slowing, magneto-optical trapping, or polarization gradient cooling, are independent of the quantum statistics of the cooled atoms, andwork equally well for bosons and fermions. The reason for this is that all these methods
rely on fundamentally single-particle physics concepts - each atom interacts with the laser
fields independently. Further, currently produced laser cooled atomic samples are almost
always far away from quantum degeneracy, and the atoms are classical with respect to theirexternal degrees of freedom.
Evaporative cooling on the other hand is fundamentally a two-particle physics concept.
Most energetic atoms are selectively removed from the (magnetic or optical) trap, taking
away more than their share of energy, and leaving the remaining atoms to rethermalize
at a lower temperature. Both the population of the wings of the energy distribution,
17

and the subsequent thermalization of the sample, rely on binary collisions between atoms.
At ultralow temperatures, this process does depend on quantum statistics. Specifically,
evaporative cooling would not work in a spin polarized Fermi gas. The reason is obvious since all particles are in the same spin state, the two-particle wave function is necessarily
symmetrized in spin space, and hence must be anti-symmetric in real space. This preventss−wave collisions, while higher order partial waves are suppressed at low temperaturesby the centrifugal barrier. We can estimate the energy (or temperature) threshold forp−wave collisions to be kB Tp−wave ∼ ~2 /(2mR2 ). Here, R is the range of the inter-particle
interaction, and hence the maximum allowed impact parameter. Without knowing thedetails of the molecular potential curves, we can estimate R to be on the scale of a “garden
variety” s−wave scattering length, say a few nm1 . This implies a p−wave threshold on a mK
scale, while a typical temperature of quantum degeneracy is on the µK scale. Suppressionof higher order partial waves is even more drastic. The collisions thus freeze out long before
degeneracy can be reached. Recently, Regal and Jin [24] have demonstrated that p−waveinteraction could be enhanced near a Feshbach resonance (see Chapter 4) sufficiently toenable evaporative cooling. However, this enhancement is accompanied by strong inelastic
losses, and does not offer much promise for a practical cooling strategy.
The way to overcome these problems is to employ some form of “sympathetic” coolingbetween two distinguishable types of particles. In this scenario, thermalization of fermions ismediated by inter-species collisions. The two distinguishable species can be two spin statesof the same atom [15, 19], two different isotopes of the same chemical element [17, 18], or
two different chemical elements [1, 16]. Taking this approach, six experimental groups havethus far achieved quantum degeneracy in Fermi gases of either 40 K[15, 16] or 6 Li[1, 17–19].

2.1.2

Why a boson and a fermion?

In addition to the obvious scientific appeal of allowing for studies of degenerate Bose-Fermi(BF) mixtures, BF sympathetic cooling has several advantages over its Fermi-Fermi (FF)counterpart. In general, cooling of fermions becomes less time-efficient as the temperature
gets lower. The reason for this is that the scattering rate into low energy states is proportional to 1 − f , f being the Fermi-Dirac occupation number. As the occupation of the low

lying states becomes appreciable at T ¿ TF , the cooling rate is reduced. In a two fermion

experiment, both species rely on each other for cooling, and neither can easily find its way
down in temperature. On the other hand, if fermions are cooled in a bosonic thermal bath,which can rapidly achieve thermal equilibrium on its own, the problem should be reduced.
Note that time-efficiency of the cooling process is not just a matter of convenience. Coolingis in competition with the timescale for heating and decay of the system, set for example
by the collisions with the residual background gas in the vacuum chamber. Time efficiency
therefore has fundamental implications on the final temperatures and size of the system [45].
1

See Ch. 4 for a reasonable justification of this estimate.

18

Further, FF cooling is most efficient if the heat capacities of the two species are comparable at all times. This means that both species must be actively evaporated, and thedegenerate Fermi samples are much smaller than the initial samples prepared by laser cooling. In BF experiments, it is sufficient to evaporate the bosonic species. The sympathetic
cooling can thus, at least in principle, be conservative in the fermion atom number. Therefore, it is possible to produce large degenerate Fermi samples without huge demands on thelaser cooling of the fermionic species.

2.1.3

Why 23 Na and 6 Li?

The criteria for designing BF sympathetic cooling experiments include the heat capacityof the “refrigerator”, and the inter-species collisional properties, both elastic and inelastic.
23 Na offers the largest, stable Bose-Einstein condensates [46], which makes it a very appeal-ing choice for sympathetic cooling of fermions. Of course, in our case, the choice of 23 Na
was further motivated by the fact that we already had a functioning BEC apparatus.
Experiments with degenerate alkali fermions have thus far used either 40 K or 6 Li. Bothspecies are promising candidates for the observation of a BCS-like superfluid transition (seeChapter 4). However, while 6 Li is abundant in nature, 40 K is scarce. This makes only 6 Li
suitable for our particular cooling strategy. In our 23 Na experiments, the atomic source isbased on an oven and a Zeeman slower (see Fig. 2-1 below). This approach yields large
ultracold samples, but also uses the atomic resources inefficiently. Since we wanted to use
the same atomic source for both species, in order to minimize the modifications to our 23 Na
BEC apparatus, 40 K was not a viable choice for practical reasons.
Further, it is favorable that the fermion mass is lower than the boson mass. When the
two are held in the same magnetic trap, lower mass enhances the fermion trapping frequency,√∝ 1/ m, and hence the ratio of Fermi temperature to the BEC transition temperature.

Therefore, given the same characteristic temperature of the bosonic refrigerator, a lighter
fermion can be cooled deeper into quantum degeneracy.

Finally, the inter-species collisional properties are generally not predictable, and haveto be tested experimentally. In this regard, one often has to take a chance in designing a
sympathetic cooling experiment, but as we will see in Chapter 3, 23 Na-6 Li combination has
proven to be rather fortuitous.

2.2

Outline of our cooling strategy

Our 6 Li cooling strategy is designed around the previously established protocol for production of 23 Na BECs [47], outlined in Fig. 2-1. An effusive 23 Na atomic beam is produced byevaporating sodium metal in the oven. Zeeman slowed atoms are loaded from the atomic
beam into a dark-SPOT magneto-optical trap (MOT) [48]. The atoms are then loaded
into a Ioffe-Pritchard magnetic trap and evaporatively cooled to BEC. A detailed descrip19

tion of our 23 Na BEC apparatus, including the laser system, vacuum chamber, magnetic
trap, standard detection methods and computer control of the experiment, is given in theprevious PhD theses from this group [7].

:PO.Q=SRTFUBV=XW1IKQADFUFXH1I)JLNM
0/ #"! 1)2,  + 3:<;>=@?ACBEDFGFGH1IKJLNM
"!#$ !%! & #(')* !+, 
 
9Y'<K/Z

obp9qsrutXv0wVxKv0r
qCy"v%zx)q|Gv~}Kkr(v0{ rruv0.-[P\]^`_ba@ced@f0c)geh
h%iikj)gelNmn4

&57698  

Figure 2-1: Outline of the 6 Li cooling strategy.
The extension to a two-species cooling experiment is then conceptually straightforward(Fig. 2-1). A two-species atomic beam is produced in a common oven, and slowed in acommon Zeeman slower. The slowed atoms are then collected in a common two-species
MOT, and transferred into a common magnetic trap. At this point, only sodium atoms areactively evaporated and cooled to BEC in the same way as in our traditional single-species
experiments. At the end of the evaporative cooling cycle, in addition to a sodium BEC,
we get a lithium Fermi sea “for free”. During the evaporative cooling, the lithium number
ideally stays constant, which means that we need to load the magnetic trap with only as
many atoms as we want to, or can, cool into quantum degeneracy.
As indicated in Fig. 2-1, only two major hardware additions were needed in order toconvert our sodium BEC apparatus into a two-species experiment - a two-species oven which
produces a combined 23 Na-6 Li atomic beam, and a laser system for laser cooling of lithium.
Let us estimate the potential of this strategy, and the requirements for its implementation. In our 23 Na BEC experiments, the flux of hot atoms from the oven into the solid
angle subtended by the MOT is about 2 × 1012 s−1 . About 1010 Zeeman slowed atoms arecollected in the MOT, 30% of which are transferred into the magnetic trap. After evapo-

rative cooling, pure BECs of ∼ 107 atoms are produced. The number of atoms at the BEC
transition is higher, typically 0.5 − 1 × 108 .

Now, sympathetic cooling is efficient only as long as the heat capacity of the coolingbody is larger or equal to the heat capacity of the body to be cooled. Below the condensation temperature, heat capacity of a Bose gas starts dropping rapidly (∝ T 3 ) . In fact,

20

thermodynamic calculation with the constraint of constant particle number would tell usthat the BEC itself has a vanishing heat capacity, which would make it completely uselessas a refrigerant [17]. I believe this conclusion is a bit oversimplified, and is not triviallyapplicable to the case of an evaporatively cooled cloud, since in the latter case, heating
can always be converted into particle loss. For example, suppose we immerse a small andhot lithium cloud into a really large (zero temperature) sodium BEC. Each inter-species
collision will take energy out of the lithium cloud and create a high energy sodium particle. If such a particle stays in the trap, this would indeed lead to drastic heating of thesodium reservoir. However, if every thermal sodium atom is immediately removed fromthe trap, cooling of lithium can continue with the sodium thermal bath still being at zerotemperature. Nonetheless, it is reasonable to assume that cooling is efficient only as long as
the numbers of sodium and lithium atoms are at least comparable. We conclude that the
maximum number of 6 Li atoms we could possibly cool deep into quantum degeneracy is in
the range 107 − 108 . This is of course still very good news, as these numbers are comparable
to the largest alkali BECs ever produced [46].

From a practical point of view, this also means that our two-species apparatus does notneed to provide us with more than 107 − 108 laser cooled 6 Li atoms. In other words, on theway from the MOT to a BEC we lose about a factor of 200 in the sodium atom number. For

lithium, we can afford to do that much worse in the laser cooling stage 2 . Further, the large
sodium MOT is density limited [48], so we can do even worse in terms of the loading rate of
the lithium MOT. This moderate goal allowed for two major simplifications. Firstly, all thelithium laser cooling light could be generated by cheap, low power diode lasers. Secondly,
we could load the 6 Li MOT using a Zeeman slower which is actually optimized for 23 Na.

2.3

Two-species oven

The main difficulty in having a joint 6 Li and 23 Na atomic source is that the two metals haverather different vapor pressures. The conventional sodium oven is operated at ∼ 260 ◦ C.
At the same temperature, lithium has a vapor pressure lower by about three orders ofmagnitude, and needs to be heated to ∼ 400 ◦ C to achieve a decent atomic flux. (As a roughguide for both metals at these temperatures, a 20 ◦ C temperature increase corresponds toan increase in vapor pressure by about a factor of two.) Our solution to this problem issketched in Fig. 2-2. More technical details of the oven design will be given in the PhD
thesis of Claudiu Stan.
The oven has two chambers connected by a transfer nozzle of low conductance σ 1 =0.06
l/s (for 23 Na 3 ). In the first (left) chamber, sodium is heated to T1 =380 ◦ C. This produces
2
The 30% transfer efficiency of sodium atoms from the MOT into the magnetic trap is limited by the factthat optical pumping into a single Zeeman state is difficult in dense laser cooled samples. For small lithium
samples this factor of 3 loss is also absent.
3
Conductance expressed in l/s is a function of atomic velocity, and hence it depends on both the atomic21

an unusually high vapor pressure, but the small nozzle conductance limits the rate at which
sodium is evaporated. In the second chamber, sodium is mixed with lithium heated up
to T2 = 400 ◦ C, and they both leave through the main nozzle, which has a much higher
conductance σ2 =2.4 l/s for 23 Na, 4.5 l/s for 6 Li. Both nozzles are heated to 450 ◦ C. The
nozzles are kept hot even when the experiment is not running, in order to insure that theyremain the hottest spots in the oven at all times, and no clogging can occur.
At these settings, the atom fluxes into the solid angle subtended by the MOT region
are 3 × 1011 s−1 for 6 Li, and 2 × 1012 s−1 for 23 Na. Since σ2 /σ1 ≈ 40, practically all thelithium goes into the experiment, and the amount distilled into the sodium chamber can beneglected.

  !#"%$&(' )*+ (!,"-$


 . 0/' *
1 !!,"-$32(!!,"-$



4 '5 0/' 
16!#"%$32(!!,"-$

Figure 2-2: Two-species oven. 23 Na and 6 Li vapors are produced in separate chambers toallow for independent control of the atom fluxes. The two species are mixed in the lithium
chamber. The transfer nozzle has a conductance 40 times lower than the main nozzle, andlimits the undesirable diffusion of lithium into the sodium chamber.
However, it is important that the oven can also be operated in a single-species mode.
Lithium-only operation is trivial and consists of heating only the second chamber. The
small fraction of atoms that takes the wrong turn and ends up in the first chamber willcondense there, but that is a small loss. Sodium-only operation requires a bit more care. If
the fluxes of sodium through the two nozzles are to be equal, its equilibrium vapor pressurein the second chamber should be
P2 = P 1

1
σ1
≈ P1
σ2
40

(2.1)

where P1 is the vapor pressure in the first chamber.
The real vapor pressure is given by either this value, or the vapor pressure corresponding
to the second chamber temperature, whichever is lower. The factor of 40 in the vapor
mass, and the temperature. All conductances are calculated for the nozzle temperature of 450 ◦ C.

22

pressure corresponds to a temperature difference of about 100 ◦ C. Therefore, as long as
T2 ≥ 280 ◦ C, there will be no build up of sodium in the second chamber. For T2 < 280 ◦ C,

sodium will start condensing in the second chamber, rather than going into the slower andthe experiment. At this point, the flux of atoms is completely determined by the vapor
pressure given by T2 , and heating the first chamber even further would not increase it.
Therefore, when running in sodium-only mode we heat up the lithium reservoir to 280 ◦ C
as well. This means that some lithium atoms are wasted, but these numbers are negligible.
This oven has now replaced the sodium oven on our BEC apparatus, and in sodium-only
experiments it gives the same results as the old one.

2.4

Laser cooling of 6 Li

2.4.1

Some laser cooling basics

In our experiments we use standard techniques of atom beam slowing and magneto-opticaltrapping [43]. Here I will briefly mention the relevant experimental methods and parameters
only in as much detail as is needed to compare and contrast the laser cooling of 23 Na and
6 Li.

The ground state of alkali atoms has one unpaired electron in the S shell, 2 2 S1/2 inlithium, 32 S1/2 in sodium. The first excited state is of the (2, 3)2 PJ form, and is split by~ ·S~ coupling into J = 1/2 and J = 3/2 states. Laser cooling is done on the D 2 line,Lconnecting the ground state and the 2 P3/2 excited state. The wavelength of this transitionis 671 nm in 6 Li, and 589 nm in 23 Na. The diagram of relevant energy levels is shown for
6 Li in Figure 2-3.

Figure 2-3: Energy level diagram for laser cooling of 6 Li
Both the ground and the excited state are further split into different hyperfine states by
the I~ · J~ interaction between the electron and the nuclear spin I. At low magnetic fields,23

~ and its projection along the quantization
good quantum numbers are therefore F~ = J~ + I,
axis, mF . The value of the nuclear spin is I = 1 in 6 Li, and I = 3/2 in 23 Na. The hyperfine
splitting of the two ground states is 228 MHz in 6 Li, and 1.77 GHz in 23 Na.
The stretched ground and excited states, with mF = −F = −(I + J) = −(I + L + S),

connected by σ − polarized light, form a cycling transition which is used for laser slowing
and cooling in our experiments. In these states all the angular momenta are fully alignedwith the quantization axis, so at all values of the magnetic field the magnetic moment in

the ground state is simply one Bohr magneton, µB , and in the excited state it is 2µB . The
Zeeman shift of the cooling transition is thus −µB /h = −1.4 MHz/G, for both 6 Li and 23 Na.
The spontaneous light force on an atom absorbing photons with wavevector ~k at a rate
γs is simply:
F~ = ~~kγs(2.2)

For an atom moving with velocity ~v , in presence of a spatially varying magnetic fieldB(~r), the scattering rate is:
γs =

Γ
s
2 1 + s + (2δ/Γ)2(2.3)

where the effective detuning:δ(~v , ~r) = ω − ω0 − ~k · ~v + B(~r)µB /~(2.4)

includes the Doppler and Zeeman shifts. Here, Γ is the natural linewidth of the excited state,
and the saturation parameter s ≡ 2Ω2R /Γ2 is the dimensionless measure of the intensity of

the laser light, ΩR being the corresponding Rabi frequency.

The dependance of scattering rate γs on the velocity and the position of the atom isthe source of coupling between internal and external degrees of freedom, and the basis oflaser cooling and trapping. For example, a moving atom interacting with isotropic light
red detuned from the resonance will preferentially be kicked in the direction opposite to its
motion and hence slowed down.
Natural linewidths in lithium and sodium are 2π ×6 MHz and 2π ×10 MHz, respectively.

The recoil velocity, vrec = ~k/m, is about 10 cm/s in 6 Li, and about 3 cm/s in 23 Na. The
saturation intensity (corresponding to s = 1) is approximately 2.5 mW/cm 2 for lithium, and6.4 mW/cm2 for sodium. Saturation intensity sets the natural scale for all laser intensities
used for cooling. Therefore, we already see that, other things being equal, about 2.5 lesslaser power should be needed for lithium laser cooling.

24

2.4.2

Simultaneous Zeeman slowing of 6 Li and 23 Na

An important question in the design of our experiment was whether our existing sodium
Zeeman slower could be used for simultaneous slowing of lithium. Changing the slower itself
would have involved breaking the vacuum in the system, something that has not been done
since 1996. Keeping the same slower, but changing the currents running through it would
not have been as drastic, but could have compromised efficient loading of sodium into thedark-SPOT. Fortunately, neither was needed.
In our “increasing field” Zeeman slower (see Eq. 2.7), sodium atoms are slowed by acounter-propagating σ − polarized beam detuned ≈ 1 GHz from the cycling transition. The

capture velocity of our slower (for 23 Na atoms) is v0 ≈ 589 m/s, and the deceleration rate

is a ≈ 4 × 105 m/s2 . Longitudinal velocity distribution in an effusive beam has the form:2

ρ(v) ∝ v 3 e

mv
− 2k
T
B

(2.5)

The average velocity of atoms coming out of the oven at a temperature of 450 ◦ C is
3
vth =
4

r

2πkB T,m(2.6)

equal to ≈ 940 m/s for 23 Na, and 1840 m/s for 6 Li.

The spatial dependence of the DC magnetic field used to compensate for the changing

Doppler shift of slowing atoms, derived from the condition that δ = const in Eq. 2.4, has
the form:r¶µx
B = B0 1 − 1 −
L

(2.7)

where L = v02 /a ≈ 40 cm is the length of the slower, and B0 = (~/µB )kv0 ≈ 700 G corresponds to the Zeeman shift equal to the Doppler shift at v0 .

The atoms coming out of the oven are distributed over the two hyperfine ground stateswith probabilities proportional to their (zero magnetic field) degeneracies, 2(I ± 1/2) + 1.

In order to prepare the atoms for slowing, a σ − “repumping” laser beam, resonant withthe lower hyperfine state, is added to the slowing beam. At zero magnetic field, before theatoms enter the slower, the combination of the slowing and the repumping beam opticallypumps the atoms into the stretched ground state4 .
Using a slowing laser beam of 50 − 100 mW power, we typically slow a flux of few

times 1010 23 Na atoms/s, to a velocity of few tens of m/s. The repumping frequency,
detuned 1.77 GHz to the blue of the slowing frequency, is provided by a sideband generated
by passing the slowing beam through an electro-optical modulator. The repumping laser
power is about 1/3 of slowing power.
4

Repumping of the atoms which fall into the lower hyperfine state during the slowing cannot be very
efficient, as different transitions have different Zeeman shifts, and keeping the cycling transition on resonancewill in general not keep the repumper on resonance as well.

25

If two different alkali species are slowed using the same magnetic field profile (Eq. 2.7),
then:
v0 =

µ B B0
∝ 1/k
~k

and

For our case this gives:
aLi
=
aN a

µ

671
589

a=

¶2

v02
∝ 1/k 2
2L

= 1.3

(2.8)

(2.9)

On the other hand, the maximum deceleration rate, at infinite light intensity, is:amax = vrec Γ/2(2.10)

Na
equal to 18 × 105 m/s2 in 6 Li, and 9 × 105 m/s2 in 23 Na, so aLi
max /amax = 2.

This means that in a slower (magnetic field profile) optimized for 23 Na, 6 Li atoms will

be slowed at a smaller fraction of the maximum deceleration. A slower optimized for 23 Na
can thus be used for 6 Li, and it will only be more conservative. This also means that alower power of the slowing laser beam is needed to keep it working5 .
One drawback is that a smaller fraction of the thermal velocity distribution is efficiently
slowed down for 6 Li than for 23 Na. The capture velocities of the slower for the two species
are comparable (Eq. 2.8), while 6 Li thermal velocity at the slower entrance is almost twice
higher (Eq 2.6). Since the capture velocity of our slower is smaller than the most probablepvelocity, 3kB T /m, we can conservatively estimate that the successfully slowed fraction of

the hot atomic flux will be proportional to
µ

v0
vth

¶4

∝

m2
k4

(2.11)

This comes out to ∼ 9 times lower fraction for lithium than for sodium. This may seemrather inefficient, but even combined with the ∼ 7 smaller total flux of hot atoms, it still
keeps us on the right track for collecting ∼ 108 lithium atoms in a MOT.

2.4.3

Differences between 6 Li and 23 Na MOTs

We collect sodium atoms into a standard dark-SPOT MOT [48], which uses three pairsof counter propagating laser beams, and a three-dimensional quadrupole magnetic field,
created by two magnetic coils in anti-Helmholtz configuration.
Operation of the MOT fundamentally depends on constant optical repumping of atoms
out of the lower hyperfine state. Off-resonant excitations of the atoms into the F 0 < I + 3/2
excited hyperfine states allows the atoms to fall into the lower hyperfine ground state andbecome invisible to the cooling light. In a dark-SPOT, the repumping light is (mostly)5

A more practical upper bound on the deceleration rate is asmax = samax /(1 + s), the maximum deceleration for the available light intensity.

26

confined to the edges of the cloud, and the atoms in the center of the trap sit in the darkof the lower hyperfine state6 . The probability that the atoms are pumped into the upper
hyperfine is therefore high only if they attempt to escape from the trap, and need to bepushed back by the trapping light. This approach reduces the adverse effects of light-assisted
collisions between ground and excited state atoms, and the radiation pressure arising fromthe fact that a photon spontaneously radiated by one atom can be reabsorbed by another,leading to mutual repulsion between the two. This allows for storing of cold atoms in thedark at much higher densities, in our sodium case giving an enhancement by a factor oforder 10.
In our setup, the 1/e2 diameter of the 23 Na MOT beams is about 1 inch, and thelaser power is about 30 mW per beam, giving a peak intensity comparable to the saturation
intensity. The cooling laser frequency is tuned ∼ 20 MHz to the red of the cycling transition

F = 2 → F 0 = 3. Typical magnetic field gradients in our MOT are 20 G/cm in the axial,

and half that in the radial direction. The repumping is done with a single 30 mW beam
which is spatially tailored by a dark spot imaged onto the center of the trap. The diameterof the dark spot image, and hence the size of the high density cloud, is 10 mm. We typicallycollect ∼ 1010 23 Na atoms in the dark-SPOT, in about 4 s, at peak density of > 1011 cm−3 .
Typical temperature in our 23 Na MOT is ∼ 1 mK. Two-level theory of Doppler coolingpredicts a cooling limit of kB TD = (1/2)~Γ = 240 µK for 23 Na, and 140 µK for 6 Li. However,this temperature is (theoretically) achieved for a light detuning of δ = −Γ/2. Such a small

detuning leads to a small effective trap volume and is not compatible with maximizing theatom number. Historically, the dark-SPOT stage was followed by about 5 ms of polarization
gradient cooling, which resulted in sub-Doppler temperatures of ∼ 100 µK. We have since

found this cooling stage unnecessary, and have omitted it from the experimental procedure.
One peculiarity of 6 Li is that the excited state hyperfine splitting is smaller than theΓ = (2π)× 6 MHz natural linewidth (see Fig. 2-3). Different lines are thus unresolved, andwe can treat 6 Li as essentially a three level system, with both ground states being coupled
to the single excited state by light of any polarization.
One adverse consequence of the unresolved excited electronic state is that polarization
gradient cooling methods are inefficient in 6 Li [49]. Another important consequence is thatthe repumping of atoms out of the lower hyperfine state is more demanding. Off-resonant
excitations into the F 0 < 5/2 hyperfine state are quite likely, and the atoms “fall off”
the cycling transition more easily. In order to keep rescuing them, the best results are
obtained with about equal amounts of laser power in the cooling and repumping MOT
beams. Further, Schreck et al.[18] have found that the MOT functions properly only if therepumping light is arranged in a six beam MOT configuration. This also means that setting
up a dark-SPOT configuration for 6 Li would be technically more demanding than for 23 Na.
6
Some repumping light might be scattered into the center of the trap by the atoms on the edge of the
cloud

27

Since for the targeted lithium atom number in the MOT, ∼ 108 , density limitations do notplay a role anyway [49], we have not attempted to do this.

2.4.4

6

Li laser system

Outline of the System
The whole laser system for cooling and trapping of 6 Li is based on inexpensive low power
diode lasers in a Master-Slave configuration. To avoid obscuring the main storyline, wesummarize only the main points here. Appendix A is written up as a more comprehensive
companion to this section.
There are four principal frequencies needed for the experiment:
1. MOT trapping light, detuned ∼ 25 MHz to the red of the F = 3/2 → F 0 = 5/2

cycling transition.

2. MOT repumping light, detuned ∼ 25 MHz to the red of the F = 1/2 → F 0 = 3/2

transition.

3. Slowing light, detuned ∼ 1 GHz to the red of the F = 3/2 → F 0 = 5/2 cyclingtransition.

4. Slower repumping light, detuned ∼ 1 GHz to the red of the F = 1/2 → F 0 = 3/2

transition.

Figure 2-4 summarizes the “topology” of our laser system. One laser is used as themaster oscillator and is referenced to a 6 Li atomic line. Acousto-optic modulators (AOMs)
are used to generate four beams with the frequencies needed for the experiment. Each ofthese beams is then amplified in a dedicated slave laser by injection locking. Outputs ofthe slave lasers have the spectral characteristics, tunability, and long term stability of theinjection light, and the power of a free running laser diode, 30 − 50 mW. The total laser

power in the system is thus 120 − 200 mW 7 . To give a sense of scale, the dye laser systemused for our 23 Na experiment puts out about 1.6 W of power. Initial inspiration for our
diode laser system came from the work done in the group of Rudi Grimm [49].
Frequency stabilized Master Laser

The starting point of our laser system is a TuiOptics DL100 grating stabilized diode laser.
The grating setup is a Littrow configuration, based on a laser developed in the group of
Ted Hänsch [50]. A 30 mW diode placed in this setup gives us an output of 12 − 15 mW

with a laser linewidth of about 1 MHz (see Appendix A.1 for details).

Once the grating feedback, temperature, and the current of the laser are optimized forthe desired wavelength, frequency can be electronically scanned over up to 10 GHz withouta mode hop. This is achieved by applying voltage to the piezoelectric transducer mounted
7

Typical lifetime of a diode is on the order of few months, and the total power output of our laser system
fluctuates with the fluctuating market for 671 nm diodes.

28

Figure 2-4: Topology of the 6 Li laser system. (1) The master laser is locked to the 6 Li D2
resonance. (2) Three AOMs generate the four frequencies needed for slowing and magnetooptical trapping. Vertical ordering indicates the relative ordering of frequencies. (3) This
light is then amplified by injection locking of four slave lasers.
at the back of the grating. We found the slow drift of the laser frequency to be about 500
kHz/s. This is in fact enough to obtain a MOT, stable over several seconds. For long term
stability, the master laser is locked to the atomic resonance of a 6 Li vapor. The details of thelocking method, setup, and results are explained in Appendix A.2. Briefly, we use Doppler
free optical pumping (saturated absorption) spectroscopy, and lock to the crossover feature,
114 MHz to the blue of the cycling transition. We achieve long term frequency stability of
about 1 MHz.
Generation of frequencies
Master laser light is split into two beams using a polarizing beam splitter cube. The splitting
ratio is about 80/20%. The stronger beam is shifted to the red by about 770 MHz in a
double pass AOM with total efficiency of 25%. The (first pass) zeroth order of the AOM isused for locking the master, and for monitoring its spectral characteristics with a wavemeterand a Fabry-Perot spectrum analyzer.
The two beams, now carrying about equal laser powers, define the MOT and the slower
manifolds. Each of them is further split into two, using 228 MHz AOMs working at 50%
efficiency. Both orders of the AOMs are used, and the four beams then define our four
principal frequencies, each carrying about 1 mW of power.

29

Injection locked Slave Lasers
Once light at the four primary frequencies is generated, each of them is amplified in a
dedicated slave laser diode by injection locking, or seeding (see Appendix A.1). With 1 mWof seeding power, a 30 − 50 mW slave laser can be injection locked up to 3 nm away fromits free running wavelength.

For our slave lasers, we use commercial ThorLabs diode mounts and temperature andcurrent controllers. Single stage optical isolators used for protecting the diodes from unwanted feedback are purchased from Optics for Research, and provide about 40 dB isolation.
Small fractions of all four slave outputs are multiplexed into a single beam using beam
splitter cubes, so that the seeding can be constantly monitored with a single Fabry-Perot
spectrum analyzer. A typical spectrum is shown in Fig. 2-5.

Figure 2-5: Frequencies of the four seeded slave lasers recorded by a Fabry-Perot spectrum
analyzer. The analyzer has a free spectral range of 1 GHz, and thus the MOT and theslower manifolds are aliased almost on top of each other. The apparent laser linewidth islimited to about 10 MHz by the resolution of the spectrum analyzer.

Imaging and optical pumping light
In addition to the four slowing and trapping frequencies, two independently tunable frequencies are needed for imaging the F = 1/2 and F = 3/2 ground state lithium atoms. The
same light is used for optical pumping of atoms between the two hyperfine ground states.
Since only sub mW powers are needed, the light is split off the MOT and MOT repumper
slaves’ outputs. The two imaging light beams are combined on a polarizing beam splittercube with opposite linear polarizations and sent through a common “tandem AOM”, consisting of two AOMs shifting the light up by 200 MHz +δ, and then down by 200 MHz30

−δ. The frequencies can thus be shifted by an arbitrary amount 2δ. The two AOMs are

set up in such a way that changing the detuning δ has a minimal effect on the directionof the output beam (Fig. 2-6). One of the AOMs in the pair is also used for fast (< µs)externally controlled switching of the light. Polarization encoding of the frequencies allows
us to separate the two imaging light beams after the tandem AOM, again using a polarizing
beam splitter cube.

Figure 2-6: Tandem AOM for the 6 Li imaging light.
The complete setup of the lithium laser system is given in Appendix A.4.
Merging of the two laser systems
The laser systems for 6 Li and 23 Na are built on separate optical tables, and in separate
rooms. All the different laser beams are separately brought to the experiment via opticalfibers. The efficiency of light coupling into the fibers is 60 − 70%.

The two colors (for lithium and sodium) are merged on the experimental table using

dichroic beam splitters, which are transparent for 589 nm and reflective for 671 nm light,
or vice versa. However, due to relatively small difference in the two wavelengths, available
dichroics are not perfect and a power loss of about 10% is unavoidable. Therefore, in places
where the two overlapping beams (initially) have opposite polarizations, we combine themon polarizing beam splitter cubes.
After the two colors are overlapped, polarizations need to be manipulated simultaneously
and identically for both wavelengths. To this end, we have replaced all half and quarter
waveplates on the experiment with zero order waveplates designed for 633 nm. 8 To illustrate
the validity of this approach, we calculate the effect of such a quarter waveplate on linear
polarization, aligned at 45◦ to the plate’s axis. Projecting the result onto the “wanted”
and “unwanted” circular polarizations, we find that the fraction of light power in the wrong
polarization is (π/4)2 (1- 633 nm/λ)2 . Both for sodium and for lithium this is less than 1%.
Results for the half waveplate depend on the desired angle of rotation, but are generally
8

In general, a half (and likewise for quarter) waveplate has a retardation of n + 1/2 wavelengths. The
order of the plate, n, is an arbitrary integer, usually around 10. Except for specific, commensurate values ofn, such a plate can not be used for two different wavelengths since the error in retardation scales with theorder of the plate.

31

comparable or better. Operation of the MOT is too robust for this to present a serious
problem. Still, in order to have more tuning freedom, we also installed “full” waveplates
for one of the wavelengths. These basically consist of two consecutive half waveplates, with
their primary axes parallel. They are used to affect one wavelength without affecting theother one.

2.5

Two-species MOT for 6 Li and 23 Na

2.5.1

Single-species 6 Li MOT

After the optical fibers, the laser power available for each of the six lithium MOT beams
is about 5 mW, both for cooling and for repumping. The beams are collimated to an 1/e 2
diameter of about 1 cm. The powers in the slowing and slowing repumping laser beams are∼ 20 mW and ∼ 5 mW respectively.

The loading of the MOT is monitored either by an absorption monitor (Figure 2-7), or

simply by collecting the fluorescence of the MOT on a photodiode. Only in the development
stages of the experiment, we performed some more quantitative analysis of the MOT byabsorption imaging.
We can collect ∼ 2 × 108 6 Li atoms in a single species MOT in about 10 s. Our ovenwill allow an increase of the atom flux by at least another factor of 4. However, at present

we do not seem to be limited by the atom number. Our estimate of the peak MOT density
is ≤ 1010 cm−3 . At this density, we do not expect that implementing a dark-SPOT would

yield any benefits.

Figure 2-7: Absorption monitor of a 6 Li MOT. A weak probe beam is sent through the
MOT, and its frequency is continuously scanned over the resonance for one of the two
ground states. The transmission of the probe is recorded on a photo diode.

32

µ

 

µ



 

µ



µ











µ



µ







µ





µ








µ



µ





µ



 

µ    !#"$&% ' (*) %.*9/87
,

-

,

)

 +
*)

%- (/.1032465&) - 87,!:0;.9<)

,

N#O P


N#O N
~

}~
s

J

N#O Q
u

|w
z o{
xy

@? L

B
F

A

=

=
J D B
> A
E M F

C I

+ G

H
B
K

F

=

−F > A

RO S
RO T

vw
st u
qr
nop
m

h
]_^ i [
l k
j

RO P
[

RO N
Y

RO Q

N6Q6Q

P6QQ

TQQ


S6QQ
)2
,

.9

RUQQQ+ )  7=−j=

RVNWQQ

acbd

g

f

=
bd

bd

±
`

± µ
RVPWQQ

±

X

e b

e `

µ

µ

\

Z

RUTQ6Q

µ "Figure 2-8: Measurement of the 6 Li MOT temperature. Absorption images were taken aftera variable time of free expansion of the cloud, after the MOT light and magnetic field havebeen turned off. We also allow for an offset in the release time of the cloud.

33

Analysis of the MOT temperature from the free expansion of the cloud is shown in Figure
2-8. We have also studied the dependence of the MOT temperature with the detuning of thecooling light (Figure 2-9). The lowest temperature was achieved at δ = −2.5Γ. However,

the highest number of atoms was achieved at about δ = −4Γ.

radialaxial

900

T (mK)

800

700

600

500

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-d/G
Figure 2-9: Dependence of the 6 Li MOT temperature on the detuning of the cooling light.
Note that there is no good reason for the MOT temperature to be perfectly isotropic. Dueto the separation of time scales for atomic collisions and laser cooling, atoms are primarily
in thermal equilibrium with the laser beams, and not with each other.

2.5.2

Coexistence of the two MOTs

In loading 6 Li and 23 Na simultaneously into a two-species MOT, we encountered the firsttrue unknown in our experiment - do they like each other? It turned out that they do notmind each other too much.
The main additional source of trap loss in a two-species MOT are light assisted collisions
between ground and excited state atoms [51] (Figure 2-10). The small number of lithium
atoms does not affect the sodium MOT, while our largest sodium MOT clouds reducelithium atom number by not more than a factor of 4. The effect can be reduced if the
two MOTs are deliberately displaced with respect to each other. This is easily achieved
using the “full” waveplates for one of the species. Along one direction (horizontal in our
experiments), the power in the two counter-propagating MOT laser beams is deliberately
imbalanced in order to push the cloud away from the center of the quadrupole magnetic
field. In our experiments, we displace the 6 Li MOT. The reason is that atoms from such an“off-balanced” MOT are not optimally transferred into the magnetic trap, and the cloudis likely to be heated during the transfer. Therefore, we prioritize the 23 Na temperature,34

Atom Number (×107)

2.5

Lithium MOT
without sodium

2.0
1.5
1.0

Lithium MOTwith sodium

0.5
0.0
0

5

10

15

20

time (s)Figure 2-10: Inter-species light assisted collisions in a two-species MOT. Loading is monitored by collecting the fluorescence from the MOT on a photodiode. The absolute atom
number was calibrated by comparison with absorption images. This particular data wastaken using relatively small atom numbers of either species.
while the small 6 Li cloud thermalizes with sodium in the magnetic trap in any case. Notethat these results hold for a “DC” two-species MOT. In the complete cooling cycle, further
tricks can be played by varying the loading times of the two MOTs before transfer into themagnetic trap. We find that the characteristic timescale for loading of a well aligned sodium
MOT can be shorter than the timescale for light-assisted losses in a displaced lithium MOT.
By turning on the sodium light only during the last 2 s of lithium loading, we can have ∼ 10 8lithium atoms at the end of the MOT stage. This is already within a factor of ∼ 2 of the

maximum atom number in a DC single-species lithium MOT. Finally, note that the effect oflight -assisted collisions could be minimized further if lithium was collected in a dark-SPOT.
However, at this point, the potential benefits do not seem worth the effort.
Studying trap losses in two species MOTs is interesting in its own right (see [52] andreferences therein). However, this was not our primary area of interest, and for the mostpart we just concluded that the losses are perfectly acceptable, and moved on. Below I
present a crude semi-quantitative study of the inter-species inelastic collision rate. In thelimit of a small lithium cloud being immersed in a large and unaffected sodium cloud of
approximately uniform density, we can write down a simple heuristic model for loading rate
and final lithium atom number in the MOT:N˙Li = L − αNLi − βnN a NLi(2.12)

where L is the loading rate, α summarizes all the losses in a single species 6 Li MOT, andβ characterizes the additional losses in the presence of a large 23 Na MOT. The loading of
the MOT is then given by:35

NLi =³´L
1 − e−(α+βnN a )t
α + βnN a(2.13)

If the loading is interrupted, the atom number decays with the same time constant. An
example of such loading and decay curve is given in Figure 2-10. We estimate α ∼ 0.3 s −1 ,and assume that in a temperature limited MOT α is independent of the loading rate. (Inthis experiment L ∼ 107 s−1 .) Since our largest sodium MOTs have a peak density of≥ 1011 cm−3 and reduce lithium atom number by not more than a factor of 4, we canestimate an upper bound on β of about 10−11 cm3 /s. This crude estimate agrees quite wellwith the more detailed study reported in [51].

2.6

Conclusion

Modifying our 23 Na BEC apparatus to the point were we could simultaneously collect∼ 1010 sodium and ∼ 108 lithium atoms in a two-species MOT completed the development
phase of our experiment. These numbers were more than sufficient for the experiments on

sympathetic cooling of 6 Li into quantum degeneracy, which are described in the following
chapter. This is were real science began...

36

Chapter 3

Production of coexisting 6Li Fermi
seas and 23Na BECs
This chapter describes the sympathetic cooling experiments in which we produce quantum
degenerate 6 Li Fermi gases, and quantum degenerate Bose-Fermi mixtures of 23 Na and 6 Li
gases. The experiments discussed here were reported in the following publications: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) [1]. Included in Appendix B.
Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann,and W. Ketterle, “Fifty-fold improvement in the number of quantum degenerate fermionic
atoms,” cond-mat/0306050 (2003) [2]. Included in Appendix C.

3.1

Choice of 23 Na and 6 Li hyperfine states for sympathetic
cooling experiments

We perform evaporative (sympathetic) cooling of 23 Na (6 Li) atoms in a cylindrically symmetric Ioffe-Pritchard (IP) DC magnetic trap [47, 53] The coils of our trap are sketched inFig. 3-1.
The first issue that arises in starting a magnetic trapping experiment is the choice of(ground) hyperfine states to be trapped. Magnetic trapping is achieved by creating a freespace magnetic field configuration which has a minimum of the field strength. Choice of
atomic states is thus restricted to “low field seeking” states, in which the magnetic momentof the atom is anti-aligned with the magnetic field, and the ground state energy increases
with increasing field strength1 .
1High field seeking states cannot be trapped, because we cannot create a magnetic field maximum in37

Gradient coils

Anti-bias coil

Curvature coilxzyTrappedatomic cloud

Figure 3-1: Ioffe-Pritchard magnetic trap coils. “Gradient coils” produce a two-dimensional
quadrupole field in the radial plane. “Curvature” and “anti-bias” coils together produce alarge curvature and a small offset “bias” field along the axial direction. Anti-bias coils aremounted in a Helmholtz configuration and produce a uniform field which opposes the field
produced by the smaller curvature coils. This cancels most of the axial bias field, whileleaving the axial curvature intact. In this way, high trapping frequencies can be achieved
along both the axial and the radial direction. Technical details of our trap are described in
the Diploma thesis of Christian Schunck [8].
Hyperfine ground states of 23 Na and 6 Li, as a function of the magnetic field, are shownin Fig. 3-2. If we ignore the small nuclear contribution, all of the atom’s magnetic moment
comes from the one unpaired electron. At low magnetic field, the energy of an |F, m F i stateis:E = g F m F µB B

(3.1)

In the ground state of all alkali atoms, the g factor universally takes a simple form:
gF =I±1/2 = ±1(I + 1/2)(3.2)

and the states which are magnetically trappable are those in which gF mF > 0.
Since all of our work relies on being able to evaporatively cool sodium to BEC, let us
start from there. We have three low field seeking states to choose from, |F, m F i = |2, 2i,|2, 1i, and |1, −1i. State |2, 0i is only weakly magnetically trappable, due to the quadratic

Zeeman shift. Mixtures of states are unstable against a variety of spin exchange collisional

free space. Using index notation (i, j = 1, 2, 3), a stationary point of the magnetic field strength is givenby ∂i B 2 = 2Bj ∂i Bj = 0. At a local maximum, second derivatives in all directions are negative. However,the sum of second derivatives is ∂i2 B 2 = 2(∂i Bj )2 + 2Bj ∂i2 Bj = 2(∂i Bj )2 ≥ 0. The second term in the lastequation is zero because in free space we can describe the field by a scalar potential, B j = ∂j Ω, and hence~ ·B~ = 0. Since the sum of the second derivatives is positive, clearly∂i2 Bj = ∂i2 ∂j Ω = ∂j ∂i2 Ω = ∂j ∂i Bi = ∂j ∇they cannot be all negative.

38

(b)

(a)
23

Na

6

|2, 2
|2, 1

Li

| 32 , 32|2, 0| 32 , 12|2,-1| 32 ,- 12

228 MHz1772 MHz|2,-2| 32 ,- 32|1,-1| 12 ,- 12

|1, 0
|1, 1

0

| 12 , 12100
200Magnetic Field (G)

0

100
200
Magnetic Field (G)Figure 3-2: Hyperfine structures of 23 Na and 6 Li. The states are labelled in the low field,|F, mF i basis. (a) Sodium is magnetically trappable in the |1, −1i, and in the |F = 2, mF ≥0i states. Previously, sodium was evaporatively cooled to BEC only in the |1, −1i, lower
hyperfine state. (b) Due to finite trap depth of ∼ kB × 330 µK in the |1/2, −1/2i state,lithium can be efficiently loaded into the magnetic trap only in the upper, F = 3/2 hyperfine
state.
processes, so we need to consider only single states. Pure |2, 1i is also unstable against spin

exchange collisions2 , and hence not an option for evaporative cooling. This leaves us with
only two options. Traditionally, we produced sodium BECs in the |1, −1i state. The only
other state in which BECs could in principle be produced is |2, 2i. However, condensationof sodium in the upper hyperfine state is considerably more difficult than in the lower one,and had not been previously achieved. As we will see in Section 3.3, this route was actually
not completely blocked. However, we initially tried to avoid it.
Now let us look at 6 Li. The trappable states are |3/2, 3/2i, |3/2, 1/2i and |1/2, −1/2i.

At temperatures below the p−wave threshold, any single state is stable, since the particles

do not interact at all. However, any combination of these states will engage in drastic
inelastic collisions3 .
Taking into account the possible spin exchange collisions between sodium and lithium
atoms, i.e. ensuring that the total projection of the spin is conserved in the collision, andthat collisions which would require energies equal to the hyperfine splitting are energetically2e.g. |2, 1i + |2, 1i → |2, 1i + |1, 1i + h×1.77 GHz of released energy.
The spin exchange rate coefficients in 6 Li are anomalously high, in the 10−9 cm3 /s range [54]. The rateof spin flip collisions is given by the product of the rate coefficient and the gas density, and is independentof particle velocities.
3

39

forbidden, we arrive at a simple (and predictable) conclusion - the only two truly safe
combinations of states are:|1, −1i + |1/2, −1/2iand|2, 2i + |3/2, 3/2i
We initially favored the former.
However, for its many nice properties, 6 Li has one relatively serious problem, whichis indicated in Fig. 3-2. So far, I did not give a quantitative meaning to the term “lowmagnetic field”. The characteristic magnetic field at which the I~ · J~ hyperfine interaction isdecoupled, and Eq. 3.1 stops being valid, is set by the species dependent hyperfine constant.

Due to a rather low hyperfine constant, in 6 Li this field is only a few tens of Gauss, andthe quadratic Zeeman shift becomes important sooner than we would have liked. Most
importantly, we see that the |1/2, −1/2i state is low field seeking only at very moderate

field values. Since the external magnetic field couples |1/2, −1/2i only with |3/2, −1/2i,

the exact solution of this (Breit-Rabi [55]) problem involves simply diagonalizing a 2 × 2

matrix. This gives:1
E|1/2,−1/2i = const. − hνhf2

r2
1 − x + x2
3

(3.3)

where νhf = 228 MHz is the hyperfine splitting, and x = 2µB B/(hνhf ). We see that themagnetic moment, −∂E/∂B, vanishes in this state at x = 1/3, or B ≈ 27 G. The maximum

magnetic trap depth for this state is then given by E(x = 1/3) − E(x = 0) ≈ k B × 330 µK.
This presents a serious problem for magnetic trapping of 6 Li in this state. Both in ourown, and in experience of others [18, 49], it is hard to cool lithium optically below about
300 µK, even if the atom number is compromised. This means that most of the |1/2, −1/2i
atoms “loaded” into the magnetic trap actually immediately spill out. Further, atoms areusually transferred from the MOT into an IP trap with a finite offset magnetic field, making
the trap for the |1/2, −1/2i state shallower still. In practice, things can be even worse. For

example, the magnetic trap and the MOT are not perfectly mode-matched, and the transfer
leads to heating of the cloud. These concerns were supported by our experimental findings,
and after some initial failed attempts, we gave up on trying to magnetically trap a significant

number of laser-cooled 6 Li atoms in this state.
Now, allowing for slight oversimplification, the conclusions of our state analysis so far
could be summarized as follows:(1) 23 Na can be evaporated to BEC only in the F = 1, lower hyperfine state.
(2) 6 Li can be efficiently magnetically trapped only in the F = 3/2, upper hyperfine state.
(3) A mixture of F = 1 sodium and F = 3/2 lithium atoms is not stable.

40

On the face of it, this does not look good. In order to cool 6 Li into quantum degeneracy, we had to alter at least one of our three conclusions. To this day, we still believethat Conclusion 2 stands firm. Section 3.2 describes how we produced our first quantum
degenerate 23 Na-6 Li mixtures, by at least quantitatively working around Conclusion 3. In
Section 3.3, I will describe how we made Conclusion 1 completely obsolete. This allowed us
to produce far superior degenerate Fermi samples, and Bose-Fermi mixtures, as describedin Section 3.4.

3.2

Sympathetic cooling with sodium in the F = 1 state

Here I describe our first successful sympathetic cooling experiments, with sodium evaporated
in the F = 1, lower hyperfine state. In this way, we produced a respectable degenerate Fermi
gas, with > 105 atoms at a temperature of 0.5 TF [1]. The degenerate 6 Li samples were
produced in peaceful coexistence with 23 Na BECs of ∼ 2 × 106 atoms, with the lifetime ofthe degenerate mixture as long as 10 s. This was the first time that anyone had cooled amixture of two different chemical elements into simultaneous quantum degeneracy.

However, these achievements have been completely overshadowed by our later coolingexperiments (see Section 3.4 and publication [2]), and this cooling strategy is unlikely toever be used again. Here I describe these infant steps for two reasons. The first one ishistoric (or nostalgic). The second reason is the lingering sense of wonder - I am still notconvinced that these experiments were supposed to work. Some remaining confusions are
discussed at the end of this section.
In the early days of our sympathetic cooling experiments, just about the only thing
we were sure of was that we knew how to make large sodium condensates in the F = 1
state. Since the high heat capacity refrigerator for cooling of fermions was supposed tobe the main strength of our experiment, we were reluctant to compromise it by moving
sodium to the upper hyperfine state. Evaporation in the F = 2 state has been attemptedin the past, both in our group and at NIST, and was found to be much more difficult thanF = 1 evaporation. These efforts were abandoned without F = 2 condensates ever being
produced.
Therefore, we decided to bet on the 6 Li sympathetic cooling strategy depicted in Fig. 33. While we did not expect the combination of F = 1 sodium and F = 3/2 lithium atoms
to be stable against inelastic spin-exchange collisions at high densities and over long times,
we hoped that this mixture might allow us enough time for at least some initial cooling. If
we could cool lithium atoms in the upper hyperfine state down to at least ∼ 50 µK, then we

could transfer them into the |1/2, −1/2i state, thereby producing a stable 23 Na-6 Li mixture.

Cooling could then be safely continued.

At this point we knew absolutely nothing about the inter-species collisional propertiesof our mixture. However, while the rates for elastic and spin-exchange collisions are both41

Hyperfine transfer

Loading|3/2, 3/2>
|3/2, 1/2>
|3/2, -1/2>

End of cooling 228 MHz|3/2,-3/2>

Trapping limit, 27 G
|1/2, -1/2>

Trap bottom, 1 G
0|1/2, 1/2>
40

80

B field [G]Figure 3-3: 6 Li cooling path. The atoms are loaded into the magnetic trap in the upper
hyperfine state, at a temperature of ∼ 700 µK. After the initial cooling stage to 50 µK, theatoms are transferred into the |1/2, −1/2i state, and further cooled to a final temperatureof ∼ 330 nK.
proportional to the gas density, the former is proportional to particle velocities, and the latter is velocity independent. Therefore, it seemed at least plausible that the elastic collisions
dominate at high temperatures. It turned out that we were lucky enough with 23 Na-6 Licollisional properties for this approach to work.
In these experiments, we collected about 2 × 107 lithium atoms or 6 × 109 sodium atoms

in single-species MOTs. In a two-species MOT, lithium atom number was reduced by afactor of about 44 . The temperature of the lithium cloud was ∼ 700 µK.

After the MOT phase, we optically pumped lithium into the upper hyperfine state. At

the time, we did only hyperfine (F ) optical pumping, with a single laser beam resonantwith the F = 1/2, lower hyperfine ground state. We did not attempt any Zeeman (m F )pumping, for reasons I will touch upon below in Section 3.2.1. The transfer efficiency oflithium into the magnetic trap was (only) about 12 %. Part of the reason for such inefficient transfer was that at the time, the MOT stage was still followed by 4 ms of polarization
gradient “molasses” cooling of sodium. Molasses require a much more precise alignmentand balancing of light beams than the MOT. We could never simultaneously achieve perfect enough alignment for both species, and the lithium light was always damaging to thetransfer. To cut our losses, we simply turned off the lithium light, and let the lithium cloud
4

These numbers are clearly at odds with our current two-species MOT described in Ch. 2. Everything
was still sub-optimal in these early days. I hope that the system parameters I report today will appear as
bafflingly sub-par a year or two from now.

42

into free expansion for 4 ms. This reduced the transfer efficiency by about a factor of 2.
The transfer efficiency of sodium atoms into the magnetic trap was about 33%, without
any optical pumping, since in the dark-SPOT most of the atoms are collected in the (three)
F = 1 states. We have thus magnetically trapped ∼ 6 × 105 6 Li and ∼ 2 × 109 23 Na atoms.
At this point, we started the first cooling stage, trying to reach a temperature of ∼50 µK as quickly as possible. Being too conservative with the evaporation rate would cause
unnecessary additional spin-exchange losses - once the system has thermalized, waiting
around for the inelastic losses to kill it did not seem like a very good idea.
We evaporated sodium on the |F, mF i = |1, −1i → |2, −2i hyperfine transition near

1.77 GHz. In standard BEC experiments, sodium is evaporated on the |1, −1i → |1, 0i

RF transition. However, using this transition for evaporation would also cause unwanted

direct evaporation of lithium atoms. The 1.77 GHz microwave field is on the other handcompletely invisible to 6 Li.
After 5 s of evaporation, the mixture was cooled to ∼ 50 µK5 , with only about half thelithium atoms lost. At this point, we found that a substantial fraction of lithium atoms wasin the |3/2, 1/2i state, which was yet another stroke of luck. This allowed us to transfer
lithium atoms to the |1/2, −1/2i state on a single-photon RF transition at 228 MHz, whichis to first order independent of the magnetic field6 .

However, not all F = 3/2 atoms were in the |3/2, 1/2i state, or at least not all weresuccessfully transferred into the |1/2, −1/2i state. Consequently, spin-exchange collisions

between lithium atoms in different hyperfine states led to a rapid loss of atoms from thetrap. To avoid this, we followed the RF pulse with a light pulse resonant with the F = 3/2

state, which optically pumped the atoms into either |1/2, −1/2i, or the untrapped |1/2, 1/2i

state. In this way, the overall efficiency of our hyperfine transfer was ∼ 50 %.

The evaporation was then resumed for another 10 s. We observed efficient sympathetic

cooling of the |1/2, −1/2i atoms, and cooled both gases into quantum degeneracy without

further observable losses in the lithium atom number.

Fig. 3-4 displays the effect of sympathetic cooling on the 6 Li cloud. The three absorption images of the trapped 6 Li gas were taken after the 23 Na evaporation was terminatedat different trap depths. Cooling (from top to bottom) is seen in the shrinking of thedensity distribution, and an increase in the peak optical density. At this point, cooling isconservative in the lithium atom number.
In order to determine the temperature of the lithium clouds, we performed two-dimensional
5This temperature was determined from the sodium cloud. The density of the lithium cloud at this point
was too low to provide enough signal-to-noise for direct temperature measurement. We took it on faith thatlithium was also cooled. Our hopes were confirmed by efficient trapping upon transfer of lithium into thelower hyperfine state.
6
A similar two-stage cooling and hyperfine transfer of 6 Li was previously used by the group of Christophe
Salomon [18]. However, in their case 6 Li was sympathetically cooled with 7 Li, and in the first cooling stage
both atoms were trapped in the upper hyperfine states (|2, 2i for 7 Li). In this case, all 6 Li atoms were in
the |3/2, 3/2i state, and a two-photon transition had to be employed to transfer them into |1/2, −1/2i.

43

a)

b)

      
 

2 TF

1 TF

0.5 T F
-500

DF
0

500

Axial distance [µm]Figure 3-4: Onset of Fermi degeneracy. Three pairs of images (top to bottom) correspondto T /TF = 2, 1, and 0.5. (a) Column densities of the 6 Li cloud recorded by absorption
imaging. (b) Axial line density profiles and the Fermi-Dirac fits to the data are plotted.
The arrow indicates the size of the Fermi diameter, DF .
fits to the cloud column densities using a semi-classical (Thomas-Fermi) distribution fortrapped non-interacting fermions [56]. In this approach, the probability that an atom has
position ~r and momentum p~ is obtained from the Fermi-Dirac distribution for the total (kinetic and potential) energy of the particle. The fitting function for the spatial distributionof atoms in the trap is then obtained by integrating out the momentum degrees of freedom.
This approach is valid at all temperatures, as long as the number of particles in the systemis large [56], and we do not need to take into account the exact form of the eigenstates of
the harmonic trapping potential.
Fig. 3-4(b) shows projected line densities along the axial direction of the 6 Li cloud, andthe Thomas-Fermi fits to the data. The arrow indicates the size of the axial real-space
Fermi diameter
DF = 2s

2kB TFmωz2(3.4)

where ωz is the axial trapping frequency and TF is determined from the fitted atom number, NLi = 1.4 × 105 . DF represents the minimum size the cloud would assume at zerotemperature.

Encouraged by the seemingly low rate for spin-exchange collisions between F = 3/2
lithium and F = 1 sodium atoms, we also tried to cool this mixture all the way to degeneracy.
To our surprise even this worked, again leaving us with > 105 lithium atoms at T ∼ 0.5TF .
44

In this case all the surviving atoms were in the |3/2, 1/2i state.

3.2.1

Some remaining confusions...

We were very pleased with these results, but we never quite understood the population
dynamics of different lithium mF states in the magnetic trap. Before returning to the main
story line, I briefly discuss some remaining puzzles.
During the loading phase we made no deliberate bias towards |3/2, 3/2i or |3/2, 1/2i

state. Even the polarization of the optical pumping beam was not carefully controlled 7 .
Initially, part of the reason for this was that we did not know which state we preferred.
Later, the main reason for not studying this in detail was that the experiment seemed
to work, and we had more important things to do, such as to rush and start studying
interacting Fermi systems [3].
In the experience of Christophe Salomon’s group [57], and according to theoretical calculations [54], only one of the two states should have survived a fast (< 1 s) inelastic“clean-up”. In case of bosons, one can easily analyze possible spin-exchange collisions, andconclude that it is always the stretched state that survives. In the case of 6 Li, if we ignore
for now the presence of sodium in the trap, the winner of the clean-up should simply be thestate which is more populated to start off. So it could be either of the two, and the result
would depend on the uncontrolled outcome of optical pumping. The high shot-to-shot andday-to-day reproducibility of our experiments does not support complete randomness.
Now, at the 50 µK point, we discovered that the sample is sensitive to the RF frequency
corresponding to the zero-field hyperfine splitting of ∼ 228 MHz. Given the finite bias

magnetic field of B0 ∼ 1 G in our trap, this conclusively proved the presence of |3/2, 1/2i
atoms in the sample. The fact that in this way we could not transfer all the atoms into
the |1/2, −1/2i state indicates, but does not really prove that |3/2, 3/2i atoms were alsopresent in the sample. If theorists tell us that presence of one state excludes the presenceof the other one, we would have no hard proof to refute this.
On the other hand, if sympathetic cooling in the F = 3/2 state was continued forfurther 10 s, we were left with pure |3/2, 1/2i samples. At this point we could transfer all

the atoms into the |1/2, −1/2i state. The |3/2, 1/2i → |1/2, −1/2i transition is (at least to

first order) field independent and hence temperature insensitive. It is then hard to believethat the inefficiency of the same transfer at 50 µK is purely technical.
Now let us add sodium into the picture. Given the much higher density of the sodium
cloud, it is plausible that lithium spin-state dynamics are dominated by 6 Li-23 Na and not
6 Li-6 Li collisions.

It then seems quite reasonable that the large sodium cloud in |1, −1ipumped the lithium atoms towards lower mF values. However, if the sodium cloud actually
drove the |3/2, 3/2i → |3/2, 1/2i transition, then it is not clear why it did not further
7

By this I mean that the direction of the ambient magnetic field was not controlled.

45

convert |3/2, 1/2i → |3/2, −1/2i with similar efficiency, and expelled all lithium atoms

from the trap.

While these puzzles intrigue me personally, it would be dishonest to say that they “willbe the topic of further work”. We have since designed a far superior (and better understood)
cooling strategy, and I doubt that anyone will ever go back to study these issues.

3.3

Condensation of 23 Na in the F = 2 hyperfine state

Immediately after producing our first degenerate Fermi clouds [1], we started our first studiesof 6 Li in the strongly interacting regime [3]. However, it became increasingly obvious that
our modestly sized samples would not offer a sufficient signal-to-noise ratio for a variety
of standard techniques of BEC research, such as velocimetry using long expansion times
or Bragg spectroscopy. We therefore felt that all future experiments would benefit in aqualitative way if we stepped back to improve our cooling methods and produce larger, andif possible colder, Fermi samples.
To this end, we decided to reexamine the possibility of the evaporative cooling of sodiumin the F = 2, upper hyperfine state. Potential benefits for sympathetic cooling of 6 Li were
obvious - if we magnetically trapped both species in their stretched states, we would createa mixture which is stable against spin-exchange collisions at all densities. This would bring
us closer to the sympathetic cooling strategy promised in Section 2.2.
Condensation in the upper hyperfine state of sodium is more difficult than in the lower
state for the following two reasons:(1) Sodium atoms collected in the dark-SPOT are (essentially) all in the F = 1 state,
distributed approximately evenly among the three mF states. Efficient loading of the magnetic trap in the upper hyperfine state thus requires efficient optical pumping in a cloudwith a density ≥ 1011 cm−3 . This is notoriously hard. In fact, the dark-SPOT was invented
in order to create laser-cooled samples with optical density much larger than 1 [48] 8 .

(2) The experimentally measured three-body loss rate coefficient in the upper hyperfine
state is ∼ 1.6 × 10−29 cm3 /s [58], about an order of magnitude higher than in the lower

hyperfine state. This result was obtained by transferring optically trapped F = 1 BECs
into the upper hyperfine state, and then studying them.
Neither of these reasons was however really prohibitive, as we will see below.

3.3.1

Magnetic trap loading

Our method for loading sodium into the magnetic trap in the |2, 2i state is sketched inFig. 3-5.
8

In F = 1 BEC experiments, ∼ 33% of the atoms are transferred into the magnetic trap in the |1, −1istate, and we have traditionally been unsuccessful at increasing this transfer efficiency by optical Zeeman
pumping.

46

After the dark-SPOT laser light and magnetic fields were turned off, a guide magnetic
field of 3 G was applied, and the atoms were optically pumped for 2 ms, using σ + polarized
light. The polarization gradient cooling stage was eliminated from the experimental procedure, in order to enhance the efficiency of lithium loading into the magnetic trap (See
Section 3.2). We found that this did not have any adverse effects for sodium cooling.




σ






 










Figure 3-5: Loading of |F = 2, mF = 2i sodium atoms into the magnetic trap from a darkSPOT. Full lines: atoms are optically pumped from the F = 1 ground state using two σ +polarized light beams. Dashed lines: in the magnetic trap, the unwanted |F = 2, m F = 0, 1i
atoms are transferred into the untrapped |F = 1, mF = 0, 1i states in order to prevent
inelastic spin-exchange collisions with the |2, 2i atoms.
To achieve both F (hyperfine) and mF (Zeeman) pumping, two light beams were used,
resonant with the F = 1 and F = 2 ground states. Frequencies of the two pumping beams
were optimized empirically, and the highest fraction of atoms we could pump into the |2, +2i
state was about 30 %.

In optimizing the optical pumping frequencies we found a little surprise. The magnetic
trap loading was most efficient if both beams were resonant with F 0 = F transitions (Fig. 36), where F 0 denotes the total spin of the excited state, as usual.
It makes perfect sense that the mF -pumping “F = 2” beam is kept resonant with the
F = 2 → F 0 = 2 transition. In this way, |2, 2i is a dark state. If the cycling F = 2 → F 0 = 3
transition was used, this would ensure that F = 2 atoms do not fall back into the F = 1

ground state, but it would also make |2, 2i atoms visible to the light, and would lead to

unnecessary heating of the cloud.

47

 σ+ Figure 3-6: Frequencies for sodium optical pumping. Magnetic trapping was surprisingly
most efficient if the F = 1 → F 0 = 1 transition was used for hyperfine pumping. Zeeman
pumping was (sensibly) done on the F = 2 → F 0 = 2 transition.
However, keeping the hyperfine-pumping F = 1 beam resonant with the F 0 = 1 makes
very limited sense, since this also makes |1, 1i a dark state. Of course, light polarizations arenever perfect, and this state is not really dark. Still, for all F = 1 states, Clebsch-Gordan

coefficients tell us that optical pumping into the F = 2 ground state would be more efficient
if F = 1 → F 0 = 2 transition was used, in disagreement with our observations.

After the optical pumping stage, the atoms were loaded into the IP magnetic trap witha radial gradient of 164 G/cm, and axial curvature of 185 G/cm 2 . As I discussed earlier,sodium atoms in all three |F = 2, mF ≥ 0i states are (at least weakly) magnetically trappable (Fig. 3-2(b)), but only pure |2, 2i samples are stable against inelastic spin-exchange

collisions. A crucial step in preparing the samples for efficient evaporation was to actively re-move |F = 2, mF = 0, 1i atoms from the trap, before they engaged in inelastic collisions withthe |2, 2i atoms. The atoms were loaded into a weak magnetic trap, with a high bias field of

B0 = 80 G. This field splits the F = 2 Zeeman sub-levels by (1/2)µB B0 ∼ kB ×2.8 mK. Since

this splitting was larger than the temperature of the cloud, the different states could be

resolved in microwave spectroscopy, and the |F = 2, mF = 0, 1i atoms could be selectivelytransferred to the untrapped |F = 1, mF = 0, 1i lower hyperfine states (Fig. 3-5). Thistransfer was done with a microwave sweep near the 23 Na hyperfine splitting of 1.77 GHz.
In this way, all the |2, +2i atoms initially loaded into trap could be preserved.

Further, we were able to “recycle” some of the untrapped atoms by optically pumping

them out of the F = 1 ground states, and thus giving them a “second chance” to fall
into the |2, +2i state. The final setup consisted of two microwave sweeps, the first of 0.8 s

duration with the optical pumping light on, and the second of 2.4 s duration without thelight. In this way, the overall transfer efficiency from the dark-SPOT to the magnetic trap48

was improved to about 35 %, comparable to our standard F = 1 BEC experiments [47].
Starting with ∼ 1010 23 Na atoms collected in the dark-SPOT, we loaded > 3 × 109 |2, 2i

atoms into the magnetic trap.

At this point, the magnetic trap was tightened by reducing the bias field to 3.8 G in
2.4 s. Resulting trapping frequencies were 204 Hz radially, and 34 Hz axially. This provided
good conditions for forced runaway evaporation of sodium. Evaporation was done on the|2, +2i → |1, +1i microwave transition near 1.77 GHz. As in Section 3.2, the motivation

for using microwave evaporation was to ensure that 6 Li is not evaporated in sympathetic
cooling experiments.

3.3.2

Avoiding three-body losses

After 15 s of evaporation, the sodium atoms reached a temperature of T ∼ 10 µK. Up to thispoint, we did not observe any significant three-body losses. This was verified by interrupting
the evaporation, and observing that the lifetime of the cloud was still longer than 10 s.
However, if the evaporation was pushed further, three-body losses became significant.

We could still condense sodium after further 15 s of evaporation, but the BEC was limited
to ≤ 106 atoms, and had a lifetime of only ∼ 200 ms.

In order to eliminate the problem of three-body losses, at this point we reduced thecloud density by adiabatically reducing the trapping frequencies. The three-body loss rate
is proportional to the square of the gas density, n2 . Therefore, to increase the characteristic
decay time of the cloud from a few hundred milliseconds to a few seconds, it is sufficient todecrease the density by a factor of 3 − 5. In a classical gas, during adiabatic decompression,

density scales simply as the product of the inverse oscillator lengths in three dimensions,and so:ncl ∝ ω 3/2(3.5)

In a weakly interacting BEC, balancing the mean-field energy (∝ n), and the potential
energy in the trapping potential (∝ ω 2 n−2/3 ) gives:
nBEC ∝ ω 6/5(3.6)

In either case, reducing ω by a factor of about 3 suffices. We reduced the trapping
frequencies to 49 Hz radially, and 18 Hz axially. After this adiabatic decompression of thecloud and 15 s of further evaporation, we could produce almost pure |2, 2i BECs containing

up to 10 million atoms. The lifetime of the BEC in the weak trap was longer than 3 s.

49

3.4

Sympathetic cooling with sodium in the F = 2 state

Once we established evaporation of sodium in the F = 2 state, the benefits for sympathetic
cooling of lithium were immediate and dramatic.

3.4.1

Boost in the 6 Li atom number

As described in Ch. 2, we collected ∼ 1010 23 Na and ∼ 108 6 Li atoms in a two-species MOT.

After the MOT stage, during the 2 ms of optical pumping for sodium, we also optically

pumped 6 Li into the stretched |3/2, 3/2i state. Again, two σ + polarized laser beams wereused, resonant with the F = 1/2 and 3/2 ground states. In case of 6 Li, the excited state
is not resolved, so the discussion of dark states during the optical pumping does not arise.
Due to the small optical density of the 6 Li cloud coming out of a bright MOT, most of theatoms could be pumped into the stretched state.
This was pretty much all we had to do. At this point we simply evaporated sodium thesame way as in Section 3.3, and observed essentially ideal sympathetic cooling of lithium.
During the typical 30 s of evaporative/sympathetic cooling, we observed no significant inelastic loss of lithium atoms. Here, the residual losses could arise from three-body collisions
or dipolar relaxation, since spin-exchange collisions have been completely eliminated. However, in our case, the final number of degenerate 6 Li atoms was at least half of the number
initially loaded into the trap. We did not study this in detail, but such a small loss rate is notinconsistent with the only loss process being the unavoidable collisions with the backgroundgas in the vacuum chamber.
At the same time, we (again) observed a favorable rate of elastic collisions between thetwo species. The inter-species thermalization time was “shorter than 1 s” [2]. This loose
upper bound really means that, as long the number of sodium atoms was larger than thenumber of lithium atoms, at no time did we observe any lag in lithium temperature behindthe sodium one.
At the end of evaporation, we still had up to ∼ 7 × 107 6 Li atoms in the trap. Since ourprimary interest was cooling of fermions, we evaporated all sodium atoms in order to getlithium to the lowest possible temperatures. In this way, even in our largest 6 Li samples,
we achieved temperatures below 0.5 TF .
This meant that a year after our first sympathetic cooling experiments, we produced
6 Li samples with 500 times more atoms, at about the same temperature (measured relative

to TF ). The increase in atom number came from several sources, which I recapitulate
here. The balance roughly works out, even though some of these estimates are somewhat
provisional:(1) By getting better at simultaneously aligning and balancing the MOT laser beams
for the two species, we improved the number of atoms in the single-species 6 Li MOT by afactor of ∼ 5 − 10.
50(2) By learning how to minimize the light-assisted losses, we gained another factor of∼ 2 in the lithium atom number in the two-species MOT.

(3) Eliminating polarization gradient cooling of sodium from the experimental procedure

increased the transfer efficiency of lithium atoms into the magnetic trap by a factor of ∼ 2.

(4) We further increased the transfer efficiency of lithium into the magnetic trap by afactor of ∼ 3 − 5 by better optical pumping. In F = 1 sympathetic cooling strategy, opticalpumping probably could not have been improved, since (it seems that) the non-stretched|3/2, 1/2i state was preferred for magnetic trapping.

(5) By eliminating spin-exchange losses and the imperfect hyperfine transfer of lithium

atoms during the evaporative/sympathetic cooling, we gained another factor of ∼ 3 − 4.

3.4.2

Trading-in the 6 Li atom number for deeper quantum degeneracy

At this point, we had achieved more than we hoped for in terms of increasing the size ofour degenerate Fermi samples. However, we also wanted to reduce their temperature. With
our new sympathetic cooling strategy, we had 6 Li atoms to spare, and suddenly became
limited by the finite size of our 23 Na refrigerator.
In order to go deeper into quantum degeneracy, we deliberately reduced the number of
6 Li atoms, in order to increase the ratio of the heat capacity of our sodium refrigerator tothe heat capacity of the lithium cloud. This tradeoff turned out to be very favorable. By
reducing the lithium number only slightly, to ∼ 3 × 107 , we could achieve temperatures in

the range 0.05 − 0.2 TF . An example of our coldest Fermi samples is shown in Fig. 3-7.

As before, temperatures were extracted from absorption images of the clouds, using asemiclassical (Thomas-Fermi) fit to the Fermi-Dirac distribution [1, 56]. One difference isthat in contrast to Section 3.2, the clouds were now imaged after ∼ 10 ms of free expansion
from the trap. The quoted temperature range reflects both the shot-to-shot and day-to-day

reproducibility of our experiments, and the fact that the Fermi distribution is only weakly
sensitive to the temperature in the ultra-degenerate limit.
Comparing Fig. 3-7 with Fig. 3-4 should make it rather obvious how much we improved
our Fermi system since our early experiments, both in terms of the offered signal-to-noisefor absorption imaging, and in terms of the level of quantum degeneracy. Our new 6 Li
clouds had a high enough optical density for crisp absorption imaging even after ballistic
expansion to a size larger than one millimeter (Fig. 3-7(a)). While in Fig. 3-4(b), the sizeof the coldest cloud is “comparable” to the axial Fermi diameter DF [1], in Fig. 3-7(b) the
two are virtually indistinguishable.
Note that after time t of free expansion from the trap, the (axial) Fermi diameter of thecloud is given by:
DF (t) = DF (0)p

1 + ωz2 t2 = 2s

2kB TF(1 + ωz2 t2 ) = 2vF t
mωz2

51

s

1 + ωz2 t2
ωz2 t2

(3.7)

(a)

(b)

Fermi Diameter

3 mm

Figure 3-7: Large and ultra-degenerate Fermi sea. (a) Absorption image of 3 × 10 7 6 Li
atoms released from the trap and imaged after 12 ms of free expansion. (b) Axial (vertical)
line density profile of the cloud in (a). A semiclassical fit (thin line) yields a temperature
T = 93 nK= 0.05 TF . At this temperature, the high energy wings of the cloud do notextend visibly beyond the Fermi energy, indicated in the figure by the momentum-space
Fermi diameter.
where vF =p

2kB TF /m is the Fermi velocity. In absorption imaging, we always record

the spatial extent of the cloud. However, for long expansion times, this becomes the direct
measure of the momentum distribution.

3.4.3

Two-species degenerate mixture

In optimizing the sympathetic cooling of 6 Li we always evaporated sodium completely. Since
large lithium clouds present a severe heat capacity “payload”, we could not simultaneously
condense sodium and produce our best degenerate Fermi samples. However, we could
make a reasonable compromise and produce a two-species quantum degenerate Bose-Fermi
mixture with several million atoms of each species (Fig. 3-8). The mixture was stable, witha lifetime of several seconds.
In order to optimize the cooling strategy for either single- or two-species experiments,
we implemented a dual-imaging technique, in which both 6 Li and 23 Na clouds were imaged
onto the same CCD camera using separate light pulses. The times of free expansion of thetwo gases after the release from the magnetic trap could be varied independently.

3.5

Summary of sympathetic cooling experiments

There are two 23 Na and three 6 Li hyperfine ground states which are magnetically trappable
and, at least on their own, stable against spin-exchange collisions. During our search for theideal sympathetic cooling experiment, we have cooled all five of these states into quantum
52

23

Li

Na

1.7 mm6

Figure 3-8: Two-species mixture of degenerate Bose and Fermi gases. For the displayed
image, the expansion times were tLi = 8 ms and tNa = 25 ms and the atom numbers wereNLi ∼ 107 and NNa ∼ 6 × 106 . Sodium was cooled below the condensation temperature,corresponding to ∼ 0.2 TF for the lithium cloud. Condensation of sodium is seen in thecharacteristic anisotropic expansion of the BEC [10].
degeneracy.
While we have produced respectable degenerate Fermi samples in all three magneticallytrappable lithium states, the |3/2, 3/2i state, cooled with sodium in the |2, 2i state, gave

us by far the best results. In this state, we have now produced degenerate Fermi samples
comparable in size with the largest alkali BECs [46].
In our coldest lithium samples we reached a temperature of 0.05 TF , currently the lowest
ever achieved. At present, we would not know how to conclusively extract even lower
temperatures. Hence, while I believe that lower temperatures will soon be reached by all
experimental groups, we will also need to develop better fitting (or detection) methods inorder to extract them.
In achieving all this, at no point did we have to perform any detailed studies of thecollisions between 23 Na and 6 Li, either elastic or inelastic. To this day, the most accurate
statement we can make about the inter-species collisions in our experiments is that they
are favorable. Essentially everything we tried worked at least as well as we hoped. I cannot
stress enough how fortunate this position is for a sympathetic cooler.

53

Chapter 4Interactions in 6Li spin mixtures
The true long term interest in degenerate Fermi gases lies in the studies of the stronglyinteracting spin mixtures. For most part, we will be interested in the simplest interacting
Fermi system, involving two different spin states. The unique appeal of these systems
stems from the fact that the strength of the interactions is widely tunable, as we will seein this chapter. A particular “Holy Grail” of this field is the attainment of novel regimesof fermionic superfluidity [31–42]. However, with the number of new theoretical proposals
constantly growing, other interesting many-body states may be within reach.
As we will see in Ch. 5, production and studies of strongly interacting fermionic spin
mixtures involves a whole new set of experimental tools. Despite these difficulties, we haverecently witnessed a rapid progress in achieving and characterizing the strongly interacting
regime in both 40 K [22, 24, 27, 29] and 6 Li [3, 5, 23, 25, 26, 28, 59]. Our own contributions tothis research [3–5] will be described in Chs. 6 through 8.
In this chapter I briefly introduce the relevant theoretical background - the types ofcollisions occurring in the system, and the resonant control of interactions in the vicinityof Feshbach resonances. I conclude by briefly indicating why in these systems a superfluid
transition may occur at surprisingly modest levels of degeneracy.

4.1

Language of collisions

Collisions in ultracold gases are discussed in a host of review papers [60, 61] and books [62–65]. Here I discuss some main concepts without mathematically rederiving any well known
results.

4.1.1

Inelastic, elastic, and coherent collisions

We can distinguish three types of collisions in an atomic gas - inelastic, elastic, and coherent:(1) Inelastic collisions involve a transfer of energy between internal and external degrees
of freedom. Some examples of such collisions were mentioned in Ch. 3. In three-body54

recombination, two atoms form a molecule, and share the released binding energy with athird one. In spin-exchange collisions, the change in Zeeman and/or the hyperfine energyof one or both particles is converted into kinetic energy. As a rule of thumb, the internalenergy scale is much larger than the external one in ultracold gases. Products of inelastic
collisions are therefore most commonly lost from the trap, even when they are in principle
trappable. In rare cases when the released energy is still smaller than the trap depth,
inelastic processes lead to the heating of the sample.
(2) Elastic collisions conserve the total kinetic energy and momentum of the two colliding
particles, but can change the momenta of each particle. These collisions are responsible forestablishing thermal equilibrium in the sample. In a two-state mixture of fermions below
the p−wave threshold, the only interactions will be s−wave collisions between two atoms
in different spin states.
(3) Coherent collisions are a subset of elastic collisions in which the incoming and theoutgoing (two-particle) state are identical. These collisions do not change the momentum
distribution in the sample, but they do contribute to the (many-body) ground state energy.
In our case, the only allowed coherent collision process is forward scattering, with bothparticles preserving their momenta. The exchange interaction, in which the particles wouldinterchange their momenta, is forbidden because the particles are distinguishable and theoutgoing state would be different from the incoming one.

4.1.2

Elastic scattering amplitude

In the s−wave limit, two-body collisions at a finite energy are described by the scattering
amplitude:
f =−

a
a
=−
(1 − ika)
1 + ika
1 + k 2 a2

(4.1)

where a is the scattering length and ~k is the relative momentum of the colliding particles.
All the details of the scattering potential are encoded in the value of a. Postive/negativevalues of a correspond to repulsive/attractive interactions.
The total cross-section for elastic collisions is then given by:
σ = 4π|f |2 =

4.1.3

4πa2
1 + k 2 a2

(4.2)

Weakly interacting limit

In the weakly interacting limit, ka ¿ 1, Eqs. 4.1 and 4.2 reduce to the more familiar

expressions f = −a and σ = 4πa2 . In this limit, the deBroglie wavelength of the particles islonger than a, so it seems reasonable that the detailed structure of the scattering potential

is not important. Further, for fermions, the typical distance between particles in a gas
must be even larger than the deBroglie wavelength. We can then safely, though with some55

effort [60], model the interaction with an effective short range potential of the form:
V (~r) =4π~2 aδ(~r) = U0 δ(~r)
m

(4.3)

This result also agrees with the Born approximation. Intuitively, if the scattering amplitude is (proportional to) the Fourier transform of the scattering potential, and it has no k
dependence, then clearly the potential must be a δ function.
Now we can trivially see that the mean-field energy felt by a test particle moving througha sea of scatterers with density n is:
4π~2 anm(4.4)

In analogy with the index of refraction, or the interaction self-energy of a particle, we
intuitively expect that Re(f ) determines the change in the particle’s energy, while Im(f )determines its lifetime, or the total rate of scattering out of a state. The latter is formally
expressed through the optical theorem:
σ = 4π|f |2 =4πIm(f )k(4.5)

Therefore, even in this limit, we should formally preserve the small imaginary part off = −a(1 − ika).

In our language, Re(f ) describes the coherent collisions and the mean-field, while Im(f )describes all the other, velocity changing, elastic collisions. Therefore, Re(f ) controls theground state properties of a many-body system, while Im(f ) controls the dynamic properties
such as the thermalization rate.

4.1.4

Strongly interacting limit

As we will see in Section 4.2, a sensitively depends on the details of the interatomic potential
and can be orders of magnitude larger than the range of the potential R. Therefore, even in
the “low energy” s−wave limit, given by kR ¿ 1, it is still possible that k|a| À 1. In thislimit, scattering is still isotropic, but it does depend on the energy of colliding particles.
For k|a| → ∞, the elastic cross-section approaches a “unitarity limited” value of:
σ=

4π
k2

(4.6)

More intriguingly, we see that for k|a| À 1:
1
Re(f ) → − 2k aactually decreases with increasing |a|.
56

(4.7)

In a degenerate Fermi gas, atoms will have a spread of relative momenta set by the Fermi√
momentum ~kF = 2mkB TF . Therefore, obtaining both the total rate of elastic collisions
and the total interaction energy will involve averaging over the momentum distribution [9,26]. In this case, even for values of |a| up to ∼ 10/kF , the average value of Re(f ) is
comparable to −sgn(a)1/kF . Similar considerations apply to a classical gas, with the relative

momentum spread given by the temperature.

4.2

Feshbach resonances

One of the many flexibilities offered by dilute atomic gases is that the strength of interatomic interactions can be externally controlled. In the language of conventional condensed
matter systems, this is somewhat analogous to having an experimental knob which controls
the electron charge. In fact, the freedom offered to an atomic experimentalist is evengreater, as we can change not only the strength of interactions, but also their nature from repulsive to attractive. This tunability of interactions relies on the existence of socalled Feshbach resonances [66]. These scattering resonances occur if the total energy of
two colliding atoms is shifted into degeneracy with a quasi-bound molecular state. Thismechanism has been demonstrated in atomic Bose gases [67–70], and has been used todramatically alter their properties [71–75]. In experiments with degenerate Fermi gases,
the ability to control the inter-species interactions is a crucial component in the search fornovel regimes of superfluidity [37, 40, 42].
In this section I outline the main features of resonant scattering in general , and Feshbachresonances in particular.

4.2.1

Scattering resonances

In order to see how the scattering length can be strongly modified by small changes in the
scattering potential, consider the exactly solvable toy model shown in Fig. 4-1. Here, twoparticles of equal mass m interact via a spherical box potential of depth V0 and range R(Fig. 4-1(a)), and we assume that their kinetic energy in the center of mass frame is (closeto) zero. Discrete bound (molecular) states of the potential are indicated by dashed lines.
Fig. 4-1(b) shows the corresponding scattering length as a function of the potential depth
V0 [60]:a=R−
where k0 =tan(k0 R)
k0

(4.8)

√
mV0 /~. We see that for most values of k0 R (larger than ∼ 21 ), the scattering

length is roughly constant and has a typical “background” value of a ∼ R. In alkali atoms,
1

Real interatomic (molecular) potentials have many bound states, so we need not be concerned with the
low k0 R values.

57this is typically a few nanometers.
(a)
E

(b)
4

a/R

2

0

r

0
-2
-4

-V0

0

2

4

6

8

k0 R

R

Figure 4-1: Cartoon of scattering resonances. (a) A model scattering box potential of depth
V0 = ~k02 /m and range R. (b) The corresponding scattering length as a function of k 0 R.
Away from the discrete resonances, the dashed line reasonably approximates the plateaus
at a ≈ R.
However, we also see a number of discrete resonances at k0 R = (2n + 1)π/2, where n is
an integer. Each of these resonances exactly corresponds to an appearance of a new boundstate in the scattering potential. We can also see the general “topology” of a resonance a weakly bound state corresponds to a diverging positive a, while a virtual “about to bebound” state gives a diverging negative a.
To bring our discussion closer to reality, now consider the more realistic interatomic vander Waals potential shown in Fig. 4-2. Here I include the centrifugal barrier for partial waves
with angular momentum l > 0. We will eventually be interested in s−wave scattering, but
this slight dishonesty will ease the intuitive introduction of some resonance concepts. The
same qualitative analysis as for Fig. 4-1 applies here, and we can also make a few further
remarks:(1) Now we also have “bound” states of the potential with positive energies, giving riseto scattering resonances at finite kinetic energy. These states however are not truly bound,but have a finite lifetime given by the tunnelling rate through the potential barrier.
(2) Neglecting the interaction with the further detuned bound states, we know fromstandard second order perturbation theory that the virtual scattering leads to “repulsion”
of the incoming state and the closest bound state. This immediately reproduces our mathematical observation that a weakly bound state leads to repulsive interaction (a > 0), andthe reverse holds for a bound state slightly above the incoming energy.
(3) Exactly on resonance, non-degenerate perturbation theory breaks down. The proper
eigenstates of the system are then the symmetric and the anti-symmetric superposition of
the incoming and the bound state. The interaction splits these two states symmetrically.
Waving our hands, we can then argue that the incoming/outgoing state contains equal partsof these two states, and hence does not experience any energy shift. This can intuitively
58

explain why very close to resonance, while a diverges2 , Re(f ) → 0 (Eq. 4.7). We also
recover our analogy with the optical index of refraction, whose real part vanishes exactly

on resonance.

Er

Figure 4-2: Shape scattering resonance. Scattering is resonantly enhanced whenever theincoming energy coincides with the energy of a (quasi-)bound state. This mechanism occurs
for partial waves with l > 0.
This type of scattering resonance is know as the “shape” resonance, since it depends on
the shape of the scattering potential and (crudely) occurs when the deBroglie wavelengthof the incoming particle is commensurate with the range of the potential. Such a resonancefor p−wave scattering in 40 K was observed by DeMarco et al. [76].

4.2.2

Tuning the scattering length with a magnetic field

Figure 4-3 indicates how scattering resonances can also occur for s−wave scattering, andhow they can be reached by applying an external magnetic field. This effect is known as
the Feshbach resonance [65, 66].
So far, I have discussed collisions between two particles ignoring their internal degrees of
freedom. The molecular potential curves, such as shown in Fig. 4-2, actually depend on thespin configurations of the colliding particles. S−wave Feshbach resonances are mediatedby coupling between two different spin configurations, or collision “channels”. In Fig. 4-3,
molecular potential curves are sketched for two different (two-particle) spin configurations.
The atomic sample is prepared in the lower energy, “open” channel. The energy difference
between the two curves is set by some internal energy scale, and as such it is usually muchlarger then the typical external energies in an ultracold gas. This is why the higher energy
spin channel is “closed”.
A Feshbach resonance occurs if the energy in the open channel coincides with one of
the bound states of the closed channel. The coupling between the two channels takes placeat intermediate distances of order R. Generally, the molecular states (r ¿ R) and the free2

This is also not strictly true, if we introduce the finite width of the bound state.

59

Energyclosed channel

Eincident

resonant bound state

open channel

Atomic Separation

Figure 4-3: Feshbach resonance. Tunability of the scattering length by magnetic field.
particle states (r À R), have different good (magnetic) quantum numbers. In the crossover region, r ∼ R, the decoupled channels drawn in Fig. 4-3 do not represent an exact

diagonalization of the total hamiltonian. This channel-coupling interaction here takes therole which the tunnelling had for a shape resonance.

The beauty of the Feshbach resonances is that, since the two channels correspond todifferent internal atomic (or molecular) states, they respond differently to externally applied electro-magnetic fields. Specifically, if the two channels have different total magnetic
moments, we can tune their relative energies simply by applying an external magnetic field.
The energy distance to the closest bound state is then given by:
E − E0 = ∆µ(B − B0 )(4.9)

where ∆µ is the difference between the magnetic moments for the closed and the open
channel, and E0 and B0 are the energy, and the magnetic field position of the resonance.
In the vicinity of the resonance, we can assume that only the distance to the closest
bound state varies appreciably, while the contributions to a from the distant states areenergy independent. We then traditionally write the dependence of a on the magnetic field
B as [65, 77]:µa = anr 1 +∆B
B − B0¶(4.10)

where anr is the non-resonant, background contribution from the distant states. The mag-60

netic field width of the resonance:∆B ∝1
anr ∆µ(4.11)

is defined as the range of fields around the resonance for which the resonant contribution
to a dominates the background contribution.
Again, Eq. 4.10 can intuitively be understood as the result of non-degenerate second
order perturbation theory.

4.2.3

Strongly interacting limit revisited

Near the resonance, we can ignore the background contribution to a in Eq. 4.10, and consider
only the resonant form of the scattering length a = anr ∆B/(B − B0 ). Substituting thisform into Eq. 4.1, at finite k we obtain:

Re(f ) = −σ=

4πIm(f )k=

1 B − B0anr k 2 ∆B

4π
k2

1
1+

1
1+

³

B−B0
anr k∆B³

B−B0
anr k∆B´2´2(4.12)

These equations now indeed pleasingly resemble the dispersive (real) and absorptive(imaginary) parts of the optical index of refraction.
From an experimental point of view, we may also choose to redefine ∆B as the magnetic
field range in which the gas characterized by a typical k (or kF ) is strongly interacting,ka ≥ 1. This inequality holds for |B − B0 | < ∆B 0 , with ∆B 0 = kanr ∆B. Equation 4.12

than simplifies to:Re(f ) = −4πIm(f )
σ=
k

4.2.4

=

1 B − B01
¡ B−B0 ¢2
0k ∆B 1 +0∆B

4π
1¡ B−B0 ¢22
k 1+
0(4.13)∆B

Adiabatic creation of diatomic molecules

Finally, note that an adiabatic (Landau-Zener) sweep of the magnetic field through the
Feshbach resonance can be used to permanently transform an ultracold atomic gas into a
gas of diatomic molecules. The sweep should be done from the “atomic” to the “molecular”
side of the resonance, i.e. from the attractive (a < 0) to the repulsive (a > 0) side. Thisis a two-body process and does not involve any inelastic release of energy. The process isalso reversible - an adiabatic sweep in the opposite direction will convert the molecules backinto atoms.
61

Using this method, Regal and Jin [29] have recently created molecules from a 40 K Fermi
gas. Diatomic 6 Li molecules have subsequently been created in the group of Christophe
Salomon [30], and in our group (unpublished). In Bose atomic gases, the same method wasapplied to 87 Rb [78], 133 Cs (group of Rudi Grimm, unpublished) and 23 Na (our next door
neighbors, unpublished).

4.3

High temperature superfluidity of 6 Li?

As most papers concerning degenerate Fermi gases, and this thesis, keep reminding us, theHoly Grail of this field is the attainment of novel regimes of fermionic superfluidity. HereI very briefly outline why we, as the field, are hopeful that this transition may occur attemperatures which are already experimentally reachable.
In a conventional superconductor, electrons pair into spin singlets due to an effective
attractive s−wave interaction, which is mediated by the exchange of phonons. The maximum phonon energy is given by the Debye temperature TD , and hence the electron-electroninteraction can be attractive only within kB TD of the Fermi surface. The characteristicpvibrational energies in a crystal, or a molecule, are ∼ me /m times smaller than the electronic energies. Here, me and m are the electron and the atom (ion) mass, respectively. In a

metal, the Fermi energy of the electrons is of the same order as a typical electronic energy inp
an atom or a ground-state molecule, a few eV 3 Therefore, we can estimate TD ∼ me /mTF .
In the standard (Cooper) derivation of the superconducting gap parameter or pairing tem-

perature, we assume that at energies up to kB TD , the effective electron-electron attraction
is of constant strength V [20, 79]. The result for the superconducting critical temperature
can then be written as:
Tc ∼

r

me
− 1
TF e N 0 Vm(4.14)

where N0 = (me /2π 2 ~2 )kF is the density of states at the Fermi energy.
In ultracold Fermi gases, the most promising pairing mechanism is a direct, attractives−wave interaction between atoms in two different spin states. In this case, the interaction
is attractive at all energies up to EF . In the weak-coupling regime, the strength of theattractive interaction is given by (4π~2 /m)a, with a < 0. If we then blindly translate theCooper result (Eq. 4.14) into atomic physics, we obtain:
Tc ∼ T F e− 2k π |a|F(4.15)We see two reasons for the potential enhancement of Tc /TF . First, the prefactor inp
Eq. 4.15 is larger than in Eq. 4.14 by a factor m/me ∼ 102 . This will be true even for3

All of these energies correspond to the localization to approximately one Bohr radius. This is the typicalsize of the lattice spacing in a crystal, the size of an atom, or the size of a molecule.

62

more complicated pairing mechanisms, as atomic systems have only one characteristic massscale.
Second, the exponent in Eq. 4.15 is “widely tunable”. This statement should however
be taken with a grain of salt, since it not clear how the theory can be extended into thestrong-coupling regime kF a À 1. As we have already seen in this chapter, while a is widely

tunable, we should certainly not hope for the properly calculated exponent of Eq. 4.15 tobe smaller than unity, or even π/2. Nonetheless, we can see that it might be reasonable tohope for a transition on the 10−1 TF scale.

Several authors have theoretically studied the superfluid transition in the strongly interacting regime, near a Feshbach resonance [37, 40, 42], and have concluded that the transition
temperature may indeed be as high as 0.5 TF .
In this regime, a cross-over between a BCS state and a BEC of tightly bound diatomic
molecules could also be studied [42].

63Chapter 5

Experiments with 6Li in thevicinity of Feshbach resonances
In this chapter I briefly introduce the experimental methods and the types of measurements
we use for studying strongly interacting 6 Li spin mixtures. For a complementary discussion
and further technical details, I also refer the reader to the (concurrently written) PhD thesisof Subhadeep Gupta [9].

5.1Prediction of Feshbach resonance(s) in 6 Li

In order to design a Feshbach resonance experiment and access its feasibility, we need torely on theoretical predictions for magnetic field positions and widths of the resonances.
Examples of experimentally relevant questions are:(1) Do resonances occur at magnetic fields which can be relatively easily generated in
the lab?
(2) What internal atomic (ground) states are involved and how can they be trapped at
the relevant magnetic fields?
(3) What are the widths of the resonances, and hence what level of magnetic fieldstability and spatial uniformity is required in order to resolve them?
Theoretical predictions of Feshbach resonances crucially depend on the details of theinteratomic potentials, which are generally not well known. Such predictions are thus
constantly evolving, as they get further constrained by new experimental results. However,even preliminary calculations are usually accurate enough to design an experiment.
At the time we started our experiments, we were guided by the calculations of Houbiers
et al. [54] (Fig. 5-1). For the two lowest ground states of 6 Li, a broad inter-state s−waveresonance was expected to occur in the vicinity of 800 G. At high magnetic fields, these twostates correspond to |mJ , mI i = | − 1/2, 1i (lowest state) and | − 1/2, 0i (second to lowest).

For simplicity, we shall label them |1i and |2i at all magnetic fields (Fig. 5-1(b)).
64

(b)

Scattering Length a/a0(a)

Energy [MHz]

6
1000

5
4

0

-1000
0

1
500

3
2
1000

Magnetic Field [G]Magnetic Field [T]Figure 5-1: Predicted Feshbach resonance in 6 Li. (a) For the two lowest hyperfine states of
6 Li, |1i and |2i, the inter-state s−wave scattering length a
12 is predicted to have a broadresonance at ∼ 800 G (taken from [54]). a0 is the Bohr radius. (b) Electronic groundstates of 6 Li in the relevant range of magnetic fields. Since the I~ · J~ hyperfine interaction
is essentially completely decoupled by the external magnetic field, we see two mI triples
corresponding to mJ = ±1/2. The six states are labelled according to increasing energy.
Magnetic fields of ∼ 800 G are relatively moderate, and the (unusually) large width of

the resonance, of the order of 100 G, does not pose any stringent requirements for magnetic
field stability.

Similar Feshbach resonances should occur for mixtures of any two mJ = −1/2 states.

However, we intuitively expect that the combination of two lowest states is most stableagainst any inelastic processes [39]. This makes it particularly appealing for studies ofstrongly interacting Fermi systems, and possible observation of superfluidity.

5.2

Experimental methods

5.2.1

Generation of uniform high magnetic fields

Uniform magnetic fields up to ∼ 1000 G can be generated by the anti-bias coils of our IP
magnetic trap (Fig. 3-1). These coils generate a field of 2.06 G/A, and are powered by acommercial 500 A DC power supply.
We have recently built a new magnetic trap, taking extra care that the anti-bias coils
are mounted in an almost perfect Helmholtz configuration [8]. This ensures that the appliedmagnetic fields are essentially uniform across the few millimeter long atomic clouds. Ourestimate is that for the highest applied fields, the residual field curvature results in < 1 G
field variation across the atomic sample.

65

5.2.2

Optical trapping

States |1i and |2i are both high field seeking at high magnetic fields (Fig. 5-1). A stronglyinteracting mixture of these two states can thus be studied only in an optical dipole trap
(ODT) [80].
An optical dipole trapping potential is created by the negative AC Stark shift of theground state energy in the presence of a red detuned laser beam [43]. Our ODT is formed
by a focus of a single laser beam at 1064 nm, which is aligned with the axis of the magnetic

trap. Since the detuning of the trapping laser is much larger then the energy difference
between different ground states, all the states experience the same trapping potential.
The ODT light is produced by a commercial fiber-amplified Nd:YAG laser. The maximum laser power available for trapping is ∼ 4 W, and the beam is focused to 1/e 2 radiusof 16 µm. For 6 Li ground states, this yields a maximum trap depth of kB × ∼ 600 µK, andmaximum trapping frequencies of ∼ 18 kHz radially, and ∼ 280 Hz axially. This gives a
Fermi temperature as function of lithium atom number equal to:
TF ∼ 400 nK × N 1/3(5.1)

We usually operate our ODT at about 1/4 of the maximum power, which corresponds toa trap depth lower by a factor of 4, and trapping frequencies and Fermi temperature lower
by a factor of 2.

5.2.3

Preparation of interacting spin mixtures

Our degenerate Fermi samples are prepared in the magnetic trap in the state |6i, equal to|F = 3/2, mF = 3/2i at low magnetic field (see Ch. 3). At the end of the cooling cycle, all

the 23 Na atoms are evaporated away, and the 6 Li atoms are transferred into the optical trapby adiabatically ramping up the ODT laser power, and then turning off the magnetic trap
coils. We have not yet perfected this process, and the transfer is not perfectly adiabatic.
After the transfer, our single-state Fermi samples are usually only moderately degenerate,with a typical temperature of 0.5 − 1 TF .

After the transfer into the ODT, the cloud is transferred at (close to) zero magnetic field

into the |1i state, equal to |F = 1/2, mF = 1/2i at low fields. |6i → |1i is a single photontransition at ∼ 228 MHz. States |6i and |2i = |F = 1/2, mF = −1/2i are not coupled by asingle photon, since this transition requires ∆mF = −2. It is therefore easy to be sure thatan adiabatic RF sweep around 228 MHz produces a pure |1i sample.

At high magnetic fields, different mI states within the same mJ triplet (Fig. 5-1) aresplit approximately by |ahf mJ mI | ∼ 76 MHz. Here, ahf = νhf /(I + 1/2) is the hyperfine
constant, equal to 152 MHz in 6 Li. Starting with a pure |1i sample, we can create different
spin mixtures of states |1i and |2i (and |3i) at high magnetic fields using single photon RF

transitions near ∼ 76 MHz.

66

Once an interacting spin mixture is formed in a finite depth ODT, plain evaporative(re)cooling of the sample sets in [9] (see Fig. 5-2 below). Forced evaporative cooling could
also be implemented by gradually lowering the ODT power [19].

5.2.4

Spin selective imaging at high magnetic fields

In order to image lithium clouds at high magnetic fields, we have set up an additionalpart of the laser system which generates appropriately detuned imaging laser beams (seeAppendix A.3). Alternatively, magnetic fields could be switched off right before imaging [3].
However, the ability to image the clouds at high fields has several advantages.
Switching off our magnetic fields takes ∼ 50 − 100 µs, due to the relatively large induc-tance of our magnetic coils1 . This cannot be considered instantaneous, and the ability toprobe the dynamics of the system at shorter timescales could be important. For example,
this time is comparable to the inverse radial trapping frequency in the ODT, and hence
to the timescale on which the field-dependent interaction energy is converted into kinetic
energy if the gas is released from the trap [28]. In a strongly interacting system, otherrelevant timescales, such as the inverse mean field energy or the coherence time of a (hypothetical) superfluid state, could be even shorter. Hence, a superfluid state formed at somehigh magnetic field could decay after the field is turned off without allowing enough time
to image the system at low magnetic fields.
Further, at high magnetic fields, states |1i, |2i, and |3i are separated in energy bymore than the optical linewidth of 6 MHz. This allows for state selective imaging of spin

mixtures - spatially overlapping clouds in different spin states can be imaged simultaneouslywith separate light beams onto different parts of the same CCD camera. Examples of such
images are shown in Fig. 5-2.
Finally, high field imaging also provides a good calibration of our magnetic fields.

5.3

What to look for?

According to our classification of collisions in a gas (Ch. 4), there are fundamentally three
types of experiments one can perform in the strongly interacting regime - those that study(1) inelastic, (2) elastic, and (3) coherent collisions. Ultimately, from a condensed matterphysics point of view, the last type of experiments is the most interesting. However, studies of inelastic and (incoherent) elastic collisions are also essential for characterizing theinteraction properties of the system, and designing the optimal experimental conditions for
future studies of novel many-body states.
1
This is a consequence of the fact that the geometry of our steel vacuum chamber forces us to haverelatively large magnetic coils [8]. In contrast, in the group of Christophe Salomon, a small vacuum glass
cell allows for small magnetic coils and switch-off times shorter than 20 µs [28, 57].

67

900 G

3mm

530 G

10 ms waitz

100 ms waityState |1>

State |1>

State |2>

State |2>

Figure 5-2: Spin selective imaging of the optically trapped |1i − |2i mixture. Spatiallyoverlapping clouds in states |1i and |2i are simultaneously imaged onto different parts ofthe same CCD camera. Here we also see the effects of mutual cooling as a function of
the magnetic field and hold time in the ODT. At 530 G, a12 ∼ 0 [23, 59], and cooling isinefficient. At 900 G, cooling is seen in the shrinking size and increasing density of theclouds.
Each of the following three chapters is dedicated to one of the three types of collisions:(1) Inelastic collisions can be probed by studying the decay of a trapped cloud, as we
will see in Ch. 6.
(2) Rates of elastic collisions can be studied by perturbing the system in some way, andthen observing how the collisions drive it back towards the equilibrium. As we will see,this can be simply done by observing the expansion of the gas released from the trap. It
has recently been proposed that expansion measurements can also be used to observe thesuperfluid transition in a degenerate Fermi gas [81]. We have however argued that thiswould be very hard under current experimental conditions [4]. These two topics are coveredin Ch. 7.
(3) Interaction energy of the system caused by the coherent collisions can also be extracted from the expansion measurements [25, 26, 28], since it is converted into kinetic energy
during the expansion. This method was particularly nicely used by the group of Christophe
Salomon [28]. They have prepared |1i − |2i 6 Li spin mixtures at high magnetic fields, andthen let the gas expand, either in the presence of the magnetic field, or with the magnetic

field turned off at the beginning of the expansion. In the former case, the total kinetic
energy of the expanding gas is equal to the sum of in-trap kinetic and interaction energy. In

68the latter case, the interaction energy is taken out of the system, and only the in-trap kinetic
energy is measured. The interaction energy can then be extracted from the comparison of
the two measurements. Alternatively, coherent collision shifts of the atomic energy levels
can be measured spectroscopically. We have taken the latter approach, as I will describe in
Ch. 8.

69

Chapter 6

Inelastic losses near a Feshbachresonance
Our first experiment with the interacting |1i − |2i 6 Li spin mixtures was a study of inelastic
losses at magnetic fields up to 900 G [3]. This field range encompasses the predicted Fes-hbach resonance for the inter-species scattering length a12 (Fig. 5-1) [54]. Details of thisexperiment were reported in the following publication:
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) [3].
Enhancement of the scattering length near a Feshbach resonance generally also leads
to an increased rate of inelastic losses [82–84]. One loss mechanism which we expect tobe enhanced is the three-body recombination [85]. Near the resonance, atoms spend alarge fraction of their time in the off-resonant molecular state. This increases the chancesof a collision with a third atom, and subsequent quenching into a permanently boundmolecule. Further, the rate of collisions with a third atom is proportional to the elastic
cross-section, and hence also enhanced. In a two-state Fermi system, this process should
be at least somewhat suppressed, as it necessarily involves two identical fermions being inclose proximity [86, 87]. For the two lowest states of 6 Li, this is expected to be the only
resonantly enhanced loss process [39].
There are two reasons for starting off the studies of the strongly interacting regime bylooking at inelastic losses:(1) This is a very economic way to explore a wide range of magnetic fields and experimentally locate the resonances. All we have to do is to expose the spin mixture to different
magnetic fields, and observe the number of atoms that remains in the trap after some fixed
time. Historically, this method has proved very useful for the first observation of a Feshbach70resonance in a BEC [67].
(2) Based on our experience with BECs near Feshbach resonances, we know that strong
inelastic losses could preclude the studies of the coherent properties of an interacting systemat high densities [82]. It is therefore important to verify/ensure that a strongly interacting
Fermi system has a long enough lifetime compared to for example the expected time needed
for formation of Cooper pairs [34, 45].

6.1

Note on the experimental procedure

At the time when we performed these experiments, the experimental procedure was different
from the standard/current one described in Ch. 5. Sympathetic cooling of lithium was still
done with sodium in the F = 1 state (Section 3.2), and we still had not implemented spin
selective absorption imaging at arbitrary magnetic fields.
The 6 Li clouds were therefore sympathetically cooled and transferred into the ODT in
the |2i state, at a temperature T ∼ TF , and a typical peak density of ∼ 3 × 1013 cm−3 .

|1i − |2i mixtures were then created by non-adiabatic RF sweeps at low magnetic fields.

At low magnetic fields, the two states are nearly degenerate (on the scale of the opticallinewidth), but have different magnetic moments (Fig. 3-2). The spin composition of thesamples could thus be calibrated by spatially separating the two states with a Stern-Gerlachmagnetic field gradient [9]. In this way, the rate of the non-adiabatic RF sweep could be
adjusted so that an equal mixture was produced.
Either spin mixtures, or pure states, were then exposed to magnetic field pulses of duration 50 ms−2 s. After switching off the high magnetic field, the total number of remaining
atoms was recorded by absorption imaging.
Attempts to perform Stern-Gerlach analysis of the samples after the inelastic decay hadtaken place were flawed by poor signal to noise ratio. We therefore could not monitor the
spin compositions during the decay.
Absorption imaging of either state on the cycling mJ = −1/2 → −3/2 transition (Ap-pendix A.3) could be done only at select fields near 700 G, using a primitive setup which
redirected the red detuned light used for Zeeman slowing (Section 2.4.4) into the imaging
system. This method was used for calibrating the magnetic fields created by our coils.

6.2

Two decay resonances of the |1i − |2i spin mixture

The results of our measurements for an equal mixture of states |1i and |2i are summarized
in Fig. 6-1 [3]. In contrast to our expectations, we observed two decay resonances, one
narrow at ∼ 550 G, and one broad at ∼ 680 G.

Soon after we completed these measurements, an improved theoretical calculation showedthat a second narrow resonance is indeed expected to occur close to 550 G [23] (Fig. 6-2).

71

0.0

a)

500 ms hold time

1.0

Surviving Fraction

0.5

0.0

mixture |1> & |2>

200

400

600

b)

800

2 s hold time

1.0

0.5

0.0
500

600

700

800

Magnetic Field [G]Figure 6-1: Magnetic field dependence of inelastic losses in a 6 Li |1i − |2i spin mixture.
(a) The fraction of atoms remaining in the system after the magnetic field was applied for
500 ms. Two decay resonances are observed, at 550 G and 680 G. (b) The two resonances
are shown with higher density of data points, and for 2 s magnetic field pulses. Each data
point is an average of three measurements.

6.3

Relative positions of the decay and the Feshbach resonances

The updated calculations however still predicted the broad Feshbach resonance at ∼ 860 G,

relatively far from our inelastic loss peak. The new calculations included the measurementof the “zero crossing” of a12 at ∼ 530 G [23, 59] (Fig. 6-2). Note that the same experiment inwhich the zero crossing was measured [23], also observed a loss peak on the low field side of
the predicted resonance, at ∼ 650 G [23]. Later measurements by the group of Christophe

Salomon [28] also showed a similar discrepancy. In that experiment, both inelastic andelastic interactions were studied in more detail (See Ch. 7). While the elastic cross section
appeared to peak at a magnetic field in agreement with the predicted Feshbach resonance,
the losses peaked at a lower field in agreement with our measurements. The shift of the
decay resonance with respect to the elastic Feshbach resonance thus appears to be real.
72

Scattering Length (units of a_0)

Magnetic Field [G]Figure 6-2: Updated prediction of the 6 Li a12 Feshbach resonances, taken from [23]. The
calculations include the measured zero crossing of a12 at 530 G [23, 59]. The two resonances
correspond to two different molecular hyperfine sub-levels of the same vibrational state.
Recently, several authors have theoretically studied this somewhat surprising observation [86, 88–91]. However, we maybe should not be too worried as long as the shift does
not exceed the width of the resonance. This is particularly true if only the recombination
into the closest molecular level, i.e. the one responsible for the resonance, is resonantly
enhanced. For example, in that case the decay should occur only on the positive side of
the resonance (a > 0), since only then the molecular state lies below the incident collisionenergy, and the process is exothermic. The exact amount of released energy, comparedto the Fermi energy, or the trap depth, can also be relevant. For example, the productsof recombination may stay trapped or not, the other atoms may have enough energy to
break apart a weakly bound molecule, and so forth. The released energy is zero exactly on
resonance, and grows as the magnetic field is tuned to lower values. Therefore, as we moveaway from the resonance, the probability of the inelastic process decreases, but the amountof released energy increases. Without understanding the details, it then seems plausiblethat different tuning of these two relevant parameters may conspire to produce a shift ofthe measured loss peak of the order of the resonance width.
Finally, note that the ∼ 800 G 6 Li resonance is by far the broadest one ever studied. It

is therefore quite plausible that the observed effects are not particular to this resonance,
but its large width allowed us to resolve them for the first time.

6.4

The main good news

Remaining mysteries aside, the most important conclusion of these measurements was thatthe observed losses did not seem prohibitive for further studies of the strongly interacting73

6 Li spin mixtures.

Even at the resonance peak, losses took place on a timescale larger

than 100 ms. This time is longer than any other expected relevant timescale in the system,
such as the inverse trapping frequency, inverse Fermi energy, elastic collision rate, expected
inverse superfluid gap energy [37, 40], or the expected time needed for the formation of the
Cooper pairs [34, 45].

74

Chapter 7

Elastic collisions and the expansion
measurements
Enhancement of the elastic collisions near a Feshbach resonance can conveniently be characterized by releasing the gas from an anisotropic trapping potential, and monitoring the
expansion of the cloud [92, 93]. In the collisionally dense, classical hydrodynamic regime,
the spatial anisotropy of the trapped gas is reversed during the expansion [25, 27, 28, 94].
Such expansion measurements will be discussed in Section 7.1.
The first observation of an anisotropic expansion of a degenerate Fermi gas near a Feshbach resonance [25], has attracted a lot of attention because of a recent proposal that thismay also be a signature of the superfluid transition [81] (Section 7.2). The ability to experimentally differentiate the two physical effects strongly depends on the possibility that, at
low temperatures, the classical hydrodynamic behavior is suppressed by Pauli blocking ofelastic collisions. In Section 7.3, I discuss the role of Pauli blocking in a degenerate Fermi
gas in general, and during the expansion in particular.
Parts of this chapter are based on our theoretical paper:
S. Gupta, Z. Hadzibabic, J. R. Anglin, and W. Ketterle, “Collisions in zero temperature
Fermi gases,” cond-mat/0307088 (2003). [4]in which we have argued that making the distinction between the two types of hydrodynamic behavior may be very difficult under current experimental conditions. Mathematicaldetails of the calculations reported in [4] will be given in the PhD thesis of Subhadeep
Gupta [9].

75

7.1

Hydrodynamic expansion of a classical gas

Trapped atomic gases are usually “collisionless”, in the sense that the mean free path forclassical elastic collisions is much larger than the size of the cloud. For a classical gas, or amoderately degenerate Fermi gas, with peak density n0 , we can write this condition as:1
l0 = √
À Rr,z =
2n0 σ

s

kB T
2
mωr,z

(7.1)

where l0 is the mean free path, and Rr,z is the cloud size in the radial/axial direction.
Alternatively, condition 7.1 is equivalent to the rate of collisions being much lower than theptrapping frequencies, n0 σ kB T /m ¿ ωr,z .
The hydrodynamic parameter, which I define as the ratio:
Rr,zl0

ϕr,z =(7.2)

roughly corresponds to the number of collisions per particle during one trapping period. In
the collisionless regime, ϕ ¿ 1.

If the cloud density or the elastic cross section are increased, the gas may cross over

into the hydrodynamic regime, ϕ À 1. This regime has only recently been experimentallyrealized, both for Bose [94–96], and for Fermi gases [24, 25, 28, 97, 98].

For a gas trapped in an anisotropic potential, cross-over into the hydrodynamic regime,
and hence the rate of elastic collisions, can be studied by simply releasing the gas from thetrap, and observing the spatial distribution during the expansion [92, 93]. For the usualcase of cylindrically symmetric traps, the shape of an expanding cloud is characterized bythe (radial to axial) aspect ratio:
λ(t) =Rr (t)Rz (t)

(7.3)

ϕr
ωz
=
ωr
ϕz

(7.4)

where for a trapped gas, we have:
λ0 ≡ λ(0) =and for our cigar shaped optical trap (Section 5.2.2), λ0 ≈ 0.015.

In the collisionless regime, the gas will expand ballistically. For long expansion times,

ωr,z t À 1, we have:Rr,z (t) = Rr,z (0)q

2 t2 =
1 + ωr,z

r

kB T
t
m

s

1+

1
2 t2
ωr,z

→

r

kB T
t
m

(7.5)

The spatial distribution will thus become isotropic, λ∞ ≡ λ(t → ∞) = 1. This simply
76reflects the isotropic momentum distribution in the trap.
In the hydrodynamic regime, elastic scattering during the expansion will redistributethe particle momenta from the direction of greater collisional density (in our case axial) to
the directions of smaller collisional density (in our case radial). In this case, the spatial
distribution after long expansion times will be anisotropic, with the cloud aspect ratio
inverted [92, 93]. In the fully hydrodynamic regime (ϕ → ∞), and for λ0 → 0, we expect

λ∞ ∼ 0.38/λ0 [81]. For our optical trap, this corresponds to λ∞ ∼ 25. In principle,

we expect λ∞ > 1 even for infinitesimal values of ϕ. However, the effect is likely to beexperimentally observable only for ϕz ≥ 1.

(a)

(b)

(c)

Figure 7-1: Ballistic and hydrodynamic expansion of a classical gas. The ellipse indicatesthe cigar shape of the trapped cloud with λ0 < 1. We concentrate on 16 atoms whichstart the expansion from the center of the cloud, with velocities indicated by the arrows.
(a) Isotropic momentum distribution of a trapped gas. (b) Momentum distribution ofa ballistically expanding cloud. We always have 4 atoms moving in each direction. (c)Momentum distribution of an expanding cloud which is collisionless in the radial (horizontal)
direction, but hydrodynamic in the axial (vertical) direction. After one collision time, wehave 12 atoms travelling radially, and only 4 travelling axially.
The two types of expansion are simplistically illustrated in Fig. 7-1 for our trap geometry. In Fig. 7-1(a), the gas is still trapped, and the momentum distribution is isotropic,
locally and globally. In Fig. 7-1(b), the gas expands ballistically. Globally, the momentum
distribution remains isotropic. However, note that locally, the momentum distribution becomes anisotropic - on the top of the image, we have atoms moving upwards, and so forth.
In Fig. 7-1(c), the gas expands (partially) hydrodynamically. For simplicity, suppose thatϕz À 1 À ϕr . In this case, radially moving atoms leave the cloud without undergoing any

collisions. On the other hand, the atoms initially moving in the axial direction do undergo
collisions. Consequently, their momentum distribution becomes locally isotropic. This leads
to an anisotropic global momentum distribution, with more atoms moving radially thanaxially. Qualitatively, the same arguments will hold as long as ϕz /ϕr = 1/λ0 > 1.
77

The anisotropic expansion of our strongly interacting 6 Li spin mixture is shown in
Fig. 7-2 (unpublished). Our data lie in the intermediate, “cross-over” regime, between afully hydrodynamic (ϕ → ∞) and a collisionless (ϕ → 0) expansion. In the limits of ϕ → 0
or ∞, the expansion of the gas (obviously) does not explicitly depend on the elastic crosssection σ. However, in the intermediate regime, the aspect ratio of the cloud after a fixed

expansion time could be used to extract ϕ, and hence σ [93]. This method is unlikely to yielda “precision” measurement of σ, due to the systematic uncertainties in determining the gas
density. However, variation of ϕ with the applied magnetic field can be used to characterizea Feshbach resonance. This method was used for 6 Li in [28], and for 40 K in [27].

(a)

(c)

(b)

(d)

5

Aspect Ratio

4
3
2
1
0
0.0

1.0

2.0

3.0

Expansion Time (ms)Figure 7-2: Anisotropic expansion of a strongly interacting 6 Li spin mixture. An equal|1i − |2i mixture, with ∼ 1.5 × 106 atoms in each state, was released from the optical trap at
a temperature T ∼ 0.5 TF , and in the presence of an external magnetic field of 900 G. (a-c)
The two states are simultaneously imaged for expansion times t = 0, 900 µs, and 1500 µs.
(d) Evolution of the cloud aspect ratio λ(t), with λ0 = 0.015, and 1 ms corresponding to 13
radial trapping periods. The upper curve corresponds to a fully hydrodynamic expansion,
with λ∞ ≈ 25 [81]. The lower curve corresponds to a ballistic expansion, with λ∞ = 1. Ourdata lie in between these two limits.

78

7.2

Superfluid vs. classical hydrodynamics in a degenerate
Fermi gas

The hydrodynamic evolution of a classical gas plotted in Fig 7-2 is predicted from the scaling
solutions to the hydrodynamic equations of motion [81, 92, 93]:∂n = −∇ · (n~v )
∂t
µ
¶
1
∂
2
m ~v = −∇ µ + V + mv
∂t
2

(7.6)

where µ and V are the chemical and the external potential. The crucial assumption inwriting down these equations is that the gas is always in local equilibrium, and can be
described in terms of the local macroscopic variables - density n, and the velocity field ~v .
Let me make two comments about the applicability of the hydrodynamic equations of
motion:(1) We generally expect this approach to provide a good description of the long-wavelengthand low-frequency behavior of the system. At short distances and high frequencies, we needto describe the system in terms of the distribution functions for the (6N ) microscopic degrees of freedom. In a classical gas, the separation between “long” and “short” distances
is set by the mean free path. Likewise, “high” and “low” frequencies are separated by
the collision rate. In our discussion of the expansion measurements, we have qualified thegas as hydrodynamic if the lengthscale (cloud size) and the timescale (trapping frequency)
relevant for our specific problem could be considered long1 .
(2) The applicability of the hydrodynamic equations of motion hinges only on our ability
to describe the system in terms of the local macroscopic variables, and not on the exact
mechanism which ensures the local equilibrium. In a classical gas, the establishment of
the local thermodynamic equilibrium depends on the collisions between atoms. On theother hand, an atomic BEC is classically collisionless. Yet, in this case, the presence of themacroscopic (superfluid) order validates a hydrodynamic description [65, 99]. The system isfully described by the macroscopic wave function ψ = |ψ|eiϕ , which has only two degrees of

freedom, the amplitude and the phase. These two variables define the local density n = |ψ| 2 ,and the velocity field ~v = (~/m)∇ϕ. In this system, the role of the mean free path l 0 is
taken over by the healing length
ξ=

s

~2
2mµ

(7.7)

where µ = (4π~2 /m)na is the mean field chemical potential. In both systems, these length1

Of course, in our system, these are the largest length and timescale available. Therefore, our definitionof the hydrodynamic condition is the least stringent possible.

79scales (l0 , ξ) determine the distance over which the gas can respond to an external perturbation, and restore the equilibrium. Likewise, the role of the collision rate is taken over by
the frequency associated with the mean field, µ/~. Again, these are the timescales on which
the two systems can respond to an external perturbation, such as a change in boundary
conditions. Therefore, a BEC will expand hydrodynamically as long as the healing lengthis smaller than the size of the cloud. As in the classical case (Eq. 7.1), it is trivial to verify
that this criterion is identical to µ/~ being larger than the trapping frequencies.
If the requirements for the local equilibrium of the system are (somehow) fulfilled, theresulting hydrodynamic equations take a rather universal form. The only differences between different systems arise from different equations of state, which determine the local
pressure. For example, in a degenerate Fermi gas released from the trap, the chemical potential giving the local pressure is given by the (kinetic) Fermi energy, and scales as n 2/3 .
On the other hand, in an expanding weakly interacting BEC, the local chemical potential
is given by the mean field energy, and scales as n. The exact solutions thus might dependon the system, but some general features are ubiquitous. For example, as long as we canidentify a force field which is proportional to the density gradient, this will inevitably leadto the reversal of spatial anisotropy of the gas during the expansion.

Figure 7-3: Superfluid hydrodynamic expansion of a weakly interacting 23 Na BEC. The
cigar shaped (magnetic) trap is oriented as in Figs. 7-1 and 7-2. As in the case of classicalhydrodynamics, the aspect ratio of the cloud is inverted during the expansion.
The familiar anisotropic expansion of a weakly interacting BEC (Fig. 7-3) is used asa “smoking gun” for condensation. Since a fermionic superfluid is also predicted to obeyhydrodynamic equations [35, 81, 100, 101], it has recently been proposed [81] that the same
effect can be used as a qualitative and unambiguous signature of the superfluid transition ina degenerate Fermi gas. The potential simplicity of this detection method is very exciting.
One could for example observe a sudden onset of the anisotropic expansion as a function of
the temperature or the spin composition of the cloud.
However, for such a signature to be unambiguous, we would have to be able to distinguishit from classical hydrodynamics. This is difficult for two reasons:80(1) In this case, the two types of the hydrodynamic expansion are predicted to be
identical [81].
(2) All current efforts to achieve superfluidity are concentrating on the strongly interacting regime in the vicinity of a Feshbach resonance.
This issue has become very important now that hydrodynamic expansion of Fermi gases
has been observed. Further, hydrodynamic expansion of 6 Li was observed in an equal|1i − |2i mixture at T ∼ 0.1 TF , and a magnetic field of 910 G [25]. At least some theories
would suggest that these conditions satisfy all the criteria for a superfluid transition (seeSection 4.3). However, in the same experiment, hydrodynamic behavior was observed overa wide range of temperatures, up to ∼ 3.5 TF , without any observable changes. We, andother groups [27, 28], have since observed the same behavior at various temperatures andfor different spin compositions.

Classical and superfluid hydrodynamics could be distinguished at very low temperatures,
when the Pauli blocking suppresses the elastic collisions [25, 102]. However, this argument
does not apply to expansion measurements, as we will see in the next section.

7.3

Pauli blocking of collisions during the expansion of a degenerate Fermi gas

Pauli blocking prevents scattering of atoms into already occupied states, and limits thetotal rates of (incoherent) elastic scattering at low temperatures. Coherent collisions on
the other hand are not affected by Pauli blocking, since they do not involve any change inoccupation numbers of different momentum states.
In a trapped gas, Pauli blocking may have significant consequences for the rates of evaporative cooling [103–105], thermalization of an impurity in the gas 2 [106, 107], or dampingof collective excitations [98, 108].
In equilibrium at zero temperature, no elastic collisions occur even in an interacting(a12 6= 0) spin mixture. At a small but finite T , the total rate of collisions is determined by
the small number of energetically accessible unoccupied states.

However, if the Fermi surface is deformed from its equilibrium shape, collisions will occur
even at T = 0. This will have important implications on our interpretation of expansion
measurements in a degenerate Fermi gas. Before addressing that specific example, I discuss
this issue in a more general context.

7.3.1

Effects of the Fermi surface deformation on Pauli blocking

Two examples of the deformation (or excitation) of a T = 0 Fermi surface are illustrated inFig. 7-4. Such excitations necessarily lead to opening up of energetically accessible states,2

In a two-state mixture, an “impurity” can simply be an atom in the other spin state.

81

and to the elastic collisions which aim to restore the equilibrium. As in the case of a finite
temperature, the total rate of collisions will depend on the number of the available final
states, and hence on the extent of the deformation. For example, in the small amplitude
limit, we may find that the collective excitations of the system are not damped at all, because
the gas remains essentially collisionless at all times [108]. However, for larger deformations,the gas will behave increasingly more classically [4].

(a) Available(b)

unoccupied
stateskF

kF

E
Figure 7-4: Opening up of final states for elastic scattering at zero temperature. (a) The
momentum space Fermi sphere (shaded area) is displaced from its equilibrium position(dashed circle). All the states within the large thin-line circle are energetically accessible.
The unshaded area within this circle represents states which are also unoccupied and hence
available as final states for elastic collisions. This excitation is analogous to applying an
external electric field E on an electron gas in a metal. (b) The same analysis applies for an
arbitrary deformation of the Fermi surface.
As a simple example, let us consider the case of a linear displacement of the Fermi
surface (Fig. 7-4(a)). This situation may remind us of applying an external electric field
E on a (normal) metal at zero temperature. If the relaxation time due to elastic collisionsis τ , the drift velocity of the electrons will be limited to Eeτ /m, despite the constantly
applied acceleration. If the applied electric field changes sign on a timescale 1/ω ¿ τ ,collisions will not play any role. The electrons will then exhibit undamped oscillations, but
only because they never attempt to exceed the maximum drift velocity. In this case thetypical drift velocity is Ee/(mω) ¿ Eeτ /m. Now, if the collision rate is actually Paulilimited, τ will decrease with increasing E. Therefore, the same frequency oscillation may

be damped or not, depending on its amplitude. In this simple example, we considered adriven off-resonant oscillation. However, the same thinking will apply to damping of thenormal modes of the system.
Atomic systems are quite different from the electron gas in a metal, and most otherconventional condensed matter systems, in that drastic deformations of the Fermi surface82are easily accessible and quite common. For example, it is easy to displace a trapped gas
from its equilibrium position to a distance comparable to the size of the cloud, i.e. its real
space Fermi diameter. If the gas is then released to exhibit a dipole oscillation in the trap,
the common drift velocity of the atoms will be comparable to the Fermi velocity.
Some of these thoughts have been experimentally confirmed by DeMarco and Jin [98],in an experiment quite analogous to our electron gas discussion. There, two trapped clouds
in different spin states of 40 K were set into relative oscillation inside the magnetic trap. In
the degenerate regime (T ∼ 0.4 TF ), Pauli suppression of collisions was observed. However,if the amplitude of the oscillation was increased to ∼ 10 % of the cloud size, the collisional

behavior became essentially classical.

7.3.2

Deformation of the Fermi surface during the expansion

Free expansion of the gas released from an anisotropic trap is an extreme example of thedeformation of the Fermi surface. First, let us consider a collisionless expansion of a T = 0
gas. For example, let us assume that we have an equal mixture of two spin states, but for
now we set a12 = 0.
At t = 0, the system is in equilibrium, and the local momentum space Fermi surface isisotropic at all spatial positions:
X p2
i

i

2m

= EF −

X1
i

2

mωi2 x2i(7.8)

To find the local momentum distributions during the ballistic expansion, we replace x i →xi − (pi /m)t, and obtain:¶2
X 1 + ω 2 t2 µ
X 1 mω 2 x2
mxi ωi2 t2
i i
i
pi −
=
E
−
F
2
2m
t 1 + ω i t2
2 1 + ωi2 t2
i

(7.9)

i

At all positions, the momentum space Fermi surface evolves into ellipsoids, as illustrated
in Fig. 7-5. At long times, the center of the momentum ellipsoid at a particular position is
given by p~ = m~x/t. This is just the momentum which a particle starting from the center of
the trap would need in order to arrive at ~x in time t. The spread of the momenta at (~x, t)then reflects the spread of the initial positions of the atoms in the trapped cloud. Therefore,
for ωr,z t À 1, the local momentum space Fermi surface at all positions assumes the shapeof the original trapped cloud. The volume of the momentum space ellipsoid gives the local

real space density, and the overall shrinking of the momentum distributions describes the
decay of the gas density during the expansion.
Deformations of the Fermi surface shown in Fig. 7-5 clearly open up final states forelastic scattering, as shown in Fig. 7-6.
In [4], we have proposed a way to quantify the effectiveness of Pauli blocking during

83

(b)

(a)

10 3

1

2

0

2mEF

10 3

1

0

Figure 7-5: Local momentum-space Fermi surface of a ballistically expanding gas. As usual,
we consider a cigar shaped trap, here with λ0 = 0.2. Fermi surfaces are plotted for expansion
times ωr t = 0, 1, 3, and 10. Results of Eq. 7.9 are plotted for twop
different spatial locations- (a) the trap origin, and (b) a position radially displaced by EF /(mωr2 ). In (b), thechanging origin of the momentum space ellipsoid is subtracted.

p1 p4

p2

q3

p3

q2
q4

q1

Figure 7-6: Allowed and forbidden elastic collisions during the expansion. The ellipse
indicates the identical Fermi surfaces for the two spin states, labelled by momenta p~ and~q. Process (~p1 , q~1 ) → (p~2 , q~2 ) is allowed due to the Fermi surface deformation. (p~1 , q~1 ) →(p~3 , q~3 ) is classically allowed, but suppressed by Pauli blocking. (p~1 , q~1 ) → (p~4 , q~4 ) does notchange the momentum distribution, because the reverse process has the same rate.

84

(a)

Allowed Fraction ofCollisions

Collision Rate (Arb.)

(b)

1.0
0.8
0.6
0.4
0.2
0.0

0

5

10

15

20

0.0

0.2

0.4

0.6

0.8

1.0

Initial Real Space
Aspect Ratio

Expansion Time (1/ ω r )Figure 7-7: Fraction of Pauli allowed elastic collisions during the expansion. (a) Rates ofclassically (thin line) and Pauli (thick line) allowed elastic collisions. Only the collisions
which change the momentum distribution are included. The plotted rates are calculated for~x = p~1 = 0, and provide an upper bound for the effectiveness of Pauli blocking [4]. For thisparticular plot, the trap aspect ratio is λ0 = 0.03. (b) The two collision rates are integrated
over time, and the ratio is plotted as a function of λ0 .
the expansion. We allowed the Fermi surface to propagate ballistically, as in Fig. 7-5, andat each time t, we compared the number of allowed collisions in a T = 0 Fermi gas, andthe total number of energetically allowed collisions. In the notation of Fig. 7-6, the former
includes only the processes similar to (~p1 , q~1 ) → (p~2 , q~2 ), while the latter also includes theprocesses of the (p~1 , q~1 ) → (p~3 , q~3 ) kind, which are allowed only if we completely ignore thePauli blocking. Our results are plotted in Fig. 7-7. We found that for small values of λ 0 , thefraction of energetically allowed collisions which are not suppressed by the Pauli blocking
exceeds 1/2.
Our results show that, in the perturbative limit of small σ, only a fraction of collisions
is suppressed by Pauli blocking. This implies that for a larger σ, collisions will occur at ahigh rate, and transform the ballistic expansion into a more hydrodynamic expansion.
In the strongly interacting limit, the shape of the Fermi surface will not evolve ballistically, because collisions will occur and make it locally more isotropic. It may seem that
this would increase the level of Pauli suppression, and invalidate our results at least quantitatively. This is true, but in this case, “the deed is already done”, and the gas expands
anisotropically.
The point of our calculations is therefore that the idea that the gas expands ballisticallybecause of the Pauli blocking is not consistent, and can be reduced to a contradiction.
Therefore, if the gas is strongly interacting, it is hard to rule out classical hydrodynamics
as an interpretation of anisotropic expansion, even at T = 0. Certainly we cannot hope for85this approach to give us a simple qualitative tool for observing a superfluid transition.

7.4

Conclusions and one final thought

Expansion measurements provide a simple tool to observe the enhanced rate of elastic
collisions near a Feshbach resonance, and the resulting classical hydrodynamic behavior.
However, the exciting proposal that the hydrodynamic expansion can be used as a“smoking gun” signature of fermionic superfluidity does not seem easily applicable to current
experimental efforts.
On the other hand, if the hydrodynamic expansion was observed in a weakly interacting
gas, this would probably be accepted as an unambiguous signature of superfluidity. In
the limit of weak enough interactions, expansion will be classically collisionless even if wecompletely ignore Pauli blocking. However, this would pose unrealistic requirements on thecooling of the system. Using Eqs. 7.1 and 7.2, and applying them to a degenerate Fermi
gas, we find that ϕz ¿ 1 requires:kF a ¿

s

1/3

1/3

λ
3π
λ0
√
≈ 01/6
1/6
1/3
N
N
4 26

(7.10)

where N is the number of atoms in each state. For a pancake shaped trap (λ 0 > 1), ϕr ¿ 1

is the more stringent requirement, and similarly reads:
kF a ¿

1
1/6
λ0 N 1/6(7.11)

In our experiments, N −1/6 ∼ 0.07 − 0.1. Even if the number of atoms was drasticallyreduced, for all current and reasonable future experiments, N −1/6 < 0.2. The trap aspect
1/3

ratio constrains kF a further, in our case λ0≈ 0.25. This factor could be brought somewhat

closer to unity, however relatively significant trap anisotropy is needed in order for λ ∞ to
be observably different from 1.
Therefore, even most (unreasonably) optimistically, we would need kF a ¿ 0.2. In

this limit, we can estimate the resulting superfluid critical temperature from the standard
Cooper result:Tc ¿ 4 × 10−4 TF
This is very far from our current experimental ability.

86(7.12)

Chapter 8

RF Spectroscopy of coherent
collision energy shifts
This chapter focuses on our spectroscopic studies of the interaction energy arising fromcoherent s−wave collisions in spin mixtures of 6 Li. At the end of the thesis, I attach two
manuscripts which provide further details of our work:
S. Gupta, Z. Hadzibabic, M. W. Zwierlein, C. A. Stan, K. Dieckmann, C. H. Schunck,
E. G. M. van Kempen, B. J. Verhaar, and W. Ketterle, “Radio-frequency Spectroscopy of
Ultracold Fermions,” Science 300, 1723 (2003) [5]. Included in Appendix D.
M. W. Zwierlein, Z. Hadzibabic, S. Gupta, and W. Ketterle, “Spectroscopic insensitivity to cold collisions in a two-state mixture of fermions,” cond-mat/0306627 (2003) [6].
Included in Appendix E.
The interaction energy in a two-state mixture of fermions can shift the relative energy of
the two internal atomic levels away from its single-particle, vacuum value. This is illustrated
in Fig. 8-1.
The resulting shifts of the spectroscopic lines, suggested by Fig. 8-1, are known as “clock
shifts” [109], because they limit the accuracy of the current atomic clocks based on bosonicatoms1 [110, 111]. We would like to use the spectroscopic measurements to characterize theinteraction energy in the strongly interacting 6 Li spin mixtures. In this case, spectroscopy
is performed in the radio-frequency (RF) domain, near the |1i − |2i energy splitting of∼ 76 MHz.

However, as we will see in Section 8.1, in a two-state mixture of fermions, the clock

shift is absent [5, 112], even if the equilibrium energy levels of the two states are shiftedby different amounts [6]. While this is actually a problem for our (main) purposes, this1

In the case of bosons, both intra- and inter-state collisions occur. However, the basic idea illustrated inFig. 8-1 is still valid.

87

(a)

(b)

(c)

|2>

|1>
Figure 8-1: Equilibrium shifts of the energy levels in an interacting two-state mixture offermions. Full lines and arrow indicate the vacuum energy levels and their separation.
Dashed lines and arrows indicate the shifted energy levels and their separations. The mean
field energy of atoms in one state is proportional to the density of atoms in the other one.
Different populations in the two states are represented by differently sized circles. The
figures are drawn for an attractive interaction, a12 < 0. In (a), n2 > n1 , and hence the levelseparation is increased. In (b), the situation is reversed. In (c), n1 = n2 , and the difference
between the energy levels is equal to the vacuum value.
robust elimination of density dependent frequency shifts is potentially relevant for future
improvements of atomic clocks.
In Section 8.2, we will see how the interaction energy shifts can be observed if the spectroscopic methods are extended to three atomic levels. Using this approach, we have studied
6 Li spin mixtures in both the weakly and the strongly interacting regime. In the weakly in-teracting regime, our method reproduces the expected interaction energies calculated within
the mean field approximation. Our observations in the strongly interacting regime are atpresent not completely understood, and will be briefly discussed in Section 8.2.1.

8.1

Spectroscopic insensitivity to coherent collisions in a twolevel system

In the limit of low density and weak interactions, the energy shifts of different atomic states
are well described in the mean field approximation. In a two-state (|1i − |2i) statisticalmixture of fermions, the total interaction energy density is given by:
Eint = V12 n1 n2 ,

V12 =

4π~2
a12
m

(8.1)

The two equilibrium energy levels are then shifted by the interactions according to δµ 1 (2) =V12 n2 (1) , and the frequency (energy) separation of the two levels differs from the vacuum
value by:h∆ν = ∆Eint = δµ2 − δµ1 = V12 (n1 − n2 ).
88

(8.2)

Our measurements of the |1i − |2i transition frequency in the limits n1 = 0 and n2 = 0

are shown in Fig. 8-2 [5]. The experiments were performed at a gas density of ∼ 3 ×

1013 cm−3 , and a magnetic field of 570 G, where (theoretical) a12 ∼ 150 a0 . Equation 8.2

then predicts ∆ν = ±5 kHz for the two spectra, but the observed lines are separated only

by 0.04 ± 0.35 kHz. Therefore, within our experimental precision, no clock shift is observed.

As I outline below, this effect can be understood relatively easily, but this was the first time

Number of Atoms in Final State (arb)

it was experimentally confirmed.

-6

-4

-2
0
2
4
Frequency Detuning [kHz]6

Figure 8-2: Absence of the clock shift in a pure fermionic state. RF transitions were driven
between states |1i and |2i on a system prepared purely in state |1i (filled circles), andpurely in state |2i (open circles). Mean-field interactions would result in 5 kHz shifts forthe two curves in opposite directions. Gaussian fits to the data (solid lines) are separatedby (0.04 ± 0.35) Hz. This gives a (conservative) clock shift suppression factor of 30. The
widths of the spectra are dominated by the frequency broadening of the 140 µs RF pulses.
In these experiments, RF pulses were applied to pure single-state samples. Since theRF field causes a coherent evolution of the system, it simply acts as a rotation in the twodimensional Hilbert space, and does not change the purity of the state. In other words,
during the RF transfer, the superposition of the two spin states has the same relative phase
for all the atoms. The atoms thus stay indistinguishable, and cannot interact in the s−wave
regime. We then conclude that in the experiments shown in Fig. 8-2, there was no mean
field energy in the system, so it is not surprising that no clock shift was observed.
The interaction energy in the system is established only when the coherence of the twostate superposition is lost, and a purely statistical mixture of the two states is formed. Thisis an implicit assumption of Eqs. 8.1 and 8.22 .
2
We believe that the dominant source of decoherence in our system is the residual spatial inhomogeneityof the applied magnetic field. The (vacuum) energy difference between the two states depends on themagnetic field, albeit weakly (see Fig. 5-1). Therefore, the relative phase of the superposition state evolves
differently in different parts of the cloud, and atoms with different “trap oscillations histories” accumulate a
different phase. Once the uncertainty in the accumulated phase becomes of order π, the system acquires theproperties of a statistical mixture. Typical decoherence times in our experiments are on the ∼ 10 − 100 ms
scale. See [9] for a further discussion of this topic.

89However, in Fig. 8-3, we see that the resonance shifts are absent in a two-state systemof fermions even after the system has fully decohered, and the mean field energy has fully
developed [6]. As we have explained in [6], this is again a direct consequence of the coherent
nature of the RF excitation, and is indeed not dependent on the coherence of the sample

Population of State |2 After RF-Pulse [%]on which the spectroscopy is performed. Our argument is outlined in the following section.
80

60

40

20
-5
0
5
Frequency - 74,159.672 kHz

Figure 8-3: Absence of the resonance shifts in a decohered two-state mixture of fermions.
The resonance curves were measured for fully decohered 80%/20% mixtures of states |1i
and |2i. The experimental parameters were n1 + n2 ∼ 5 × 1013 cm−3 and B = 320 G. At thisfield, a ∼ −300 a0 , and Eq. 8.2 predicts the splitting of the two lines of ∼ 20 kHz. However,
the measured splitting is only (34 ± 146) Hz.

8.1.1

Bloch sphere representation of the RF transitions

In Fig. 8-4, the two experiments of Figs. 8-2 and 8-3 are (qualitatively) represented on aspin-1/2 Bloch sphere. The average properties of the many-body state are described by
the three coordinates of the Bloch vector m~ = mz êz + m~ ⊥ . The population difference inthe two states, or the “magnetization” of the sample, is given by mz = (n2 − n1 )/2, while

the coherence in the system is measured by m
~ ⊥ . Fully decohered statistical mixtures are
represented by vectors with polar angle θ = 0, or equivalently m~ ⊥ = 03 .
The length of the Bloch vector measures the purity of the state, and hence the entropyof the system. The key point is that the RF field just rotates the Bloch vector, and preserves
its length |m|.
~ In other words, a unitary transformation cannot change the entropy.

In Fig. 8-4(a), state B is created by applying an RF pulse on a pure sample A. This

corresponds to the experiment of Fig. 8-2. As I already explained, in this case there is nointeraction energy in the system during the RF pulse, and no frequency shift is expected [5].
3State A in Fig. 8-4(a) is a trivial special case of a pure state.

90





2



(a






2

θ





θ









Figure 8-4: Bloch sphere representation of the RF transitions. (a) corresponds to theexperiment in Fig. 8-2. An RF pulse rotates a pure state A into another pure state B.
Subsequently, the superposition state decoheres into a “ring” distribution, represented by
its average, C. (b) corresponds to the experiment in Fig. 8-3. Here, a second RF pulse
transforms the fully decohered state C into a partially coherent state D. The final state E
is reached only after further decoherence. (c) Transfers A→B and C→D are coherent andreversible. B→C and D→E are irreversible.
State C is formed through subsequent decoherence of state B. In B and C, the magnetization
mz is the same, but in C the mean-field has fully developed.
The experiment of Fig. 8-3 is performed on a C-like state. In this case, the equilibrium
energy levels are shifted according to Eq. 8.2. However, we now understand that the RF
field cannot connect two fully decohered states with different magnetizations, such as C and
E. Instead, in order to change mz , while preserving |m|,~ we must tilt the Bloch vector awayfrom the vertical axis, and reintroduce some level of coherence into the system. Therefore,

in Fig. 8-4(b), it is the energy of the transformation C→D, that needs to be calculated inorder to find the correct resonant RF frequency. So let us do that.
The inter-state s-wave interaction is described by the second-quantized hamiltonian
density:
Hint = V12 ψ1† (r)ψ2† (r)ψ2 (r)ψ1 (r).

(8.3)

The mean field expectation value of this hamiltonian is:
Eint = hHint i = V12 (n1 n2 − n12 n21 )(8.4)

where n1 = hψ1† ψ1 i, n2 = hψ2† ψ2 i, and we have introduced “coherences” n12 = hψ1† ψ2 i andn21 = hψ2† ψ1 i. In a fully coherent sample n12 n21 = n1 n2 , while in a fully decohered sample

n12 = n21 = 0.

91

In terms of the coordinates of the Bloch vector, we can rewrite:n1 n2 =µ

n1 + n 22

n12 n21 = m2⊥

¶2

− m2z
(8.5)

The interaction energy is then given by:
Eint = V12

µ

n2
− |m|
~ 2
4

¶

(8.6)

where n = n1 + n2 is the constant total particle density. We see that the interaction energy
depends only on |m|.
~ In a partially or fully decohered sample, |m|~ < n/2, and the system isinteracting. However, even then, the RF pulse does not change the total interaction energy.

Hence, the RF photons do not need to supply any extra energy, and the resonance frequency
is universally equal to the vacuum value.

8.1.2

Comparison with bosons

In [6], we have extended the above analysis to the RF spectroscopy of thermal bosons. This
calculation is included in Appendix E, and (even) further details will be given in the PhD
thesis of Martin Zwierlein.
In the case of thermal bosons, RF spectroscopy also measures frequency shifts different
from the equilibrium shifts of the energy levels. By now, this should not surprise us.
In a coherent superposition of two states, both the forward and the exchange interstate scattering is allowed. The contribution to the mean field energy is then twice larger
than in a decohered sample, where the particles are distinguishable, and the only coherent(inter-state) collision process is the forward scattering.
However, analogous to the Fermi case, RF methods always measure the coherent result,independent of the level of coherence in the sample. In this case, this corresponds toa “coherent factor of 2”. This effect has recently been observed by the group of Eric
Cornell [113], and was initially qualified as a “mystery” [114].
Using the Bloch sphere representation, we have provided an intuitive resolution of thispuzzle. One should however note that the mystery was formally resolved by more sophisticated calculations which have preceded our work [115–119].

8.2

Spectroscopy of cold collisions in a three-level system

In order to spectroscopically observe the effects of coherent collisions in 6 Li spin mixtures,
we have implemented a three-level method illustrated in Fig. 8-5. Here, a superpositionof states |1i and |2i is created by an RF drive, and allowed to decohere into a statisticalmixture [5, 9]. We then perform RF spectroscopy of the |2i → |3i transition. This technique92was independently introduced by the group of Debbie Jin for studies of interactions in
40 K [27].

a)
1000

b)

c)

Energy / h [MHz]

|6>
|5>
|4>

3
2
F1
2

|3>
|2>
|1>

75.185 MHz 84.754 MHz

|3>

νRF

-1000
0

|3>

|2>

|2>

|1>

|1>

500 G

500
Magnetic Field [G]desired field

Figure 8-5: RF spectroscopy in a three-level system. (a) Hyperfine structure of the groundstate of 6 Li. (b and c) Experimental scheme: (b) A statistical mixture of atoms in states |1i
and |2i is prepared at 500 G. (c) RF spectroscopy of the |2i → |3i transition is performed
at various magnetic fields.
The discussion of Section 8.1 is still completely applicable to the |2i − |3i two-level sys-tem, and the interactions between atoms in these two states do not affect the spectroscopic
measurements.

However, any superposition, or a statistical mixture, of states |2i and |3i is (equally andcompletely) orthogonal to state |1i. Hence, during the RF transfer, the interaction energyof the |2i−|3i atoms, due to forward scattering with |1i atoms, depends only on the relative

populations of the two states. In the mean field approximation, this contribution to theinteraction energy density is given by:
4π~2
n1 (a12 n2 + a13 n3 )
m

(8.7)

In contrast to Eq. 8.1, here we do not have to assume anything about the coherence of the|2i − |3i system.

In the language of the |2i − |3i Bloch sphere, this interaction energy now depends on m z

rather than |m|.
~ Since mz does change during the RF transfer, the effects of the interactions
are observable spectroscopically. We expect the |2i → |3i resonance to be shifted according

to:

∆ν =

2~
n1 (a13 − a12 )
m

93

(8.8)

Figure 8-6: Spectroscopic measurement of the interaction energy at 480 G. (a) Fraction of
atoms transferred from |2i to |3i in the center of the cloud, with |1i atoms absent (filled
circles), and present (open circles). Solid lines are gaussian fits to the data. (b) Absorption
images of state |3i for the interaction perturbed resonance. Images of states |2i and |1i areshown only for zero RF detuning. States |3i and |2i were imaged simultaneously to observe
their complementary spatial structure. State |1i was imaged after 760 µs expansion time torecord its population.
An example of a spectroscopic measurement using the three-level method is shown in
Fig. 8-6. Two spectra were taken at B = 480 G, one with n1 = 0, and one with n1 ∼ 2.4 ×
1013 cm−3 . The n1 = 0 measurement was used to calibrate the vacuum resonance frequency

ν0 . In the interacting system (n1 6= 0), the spectrum is both shifted and broadened. To

reduce the effect of the spatial inhomogeneity of n1 , only the central part of the cloud wasused to compute the fraction of atoms transferred from |2i to |3i [5].

Figure 8-6(b) displays the absorption images of the |3i cloud for n1 6= 0, and different

applied RF frequencies. While the radial density distribution is averaged out during the

140 µs RF pulse and the 120 µs expansion time, the axial distribution is clearly visible. Forzero detuning from ν0 , only the atoms in the low density wings of the cloud are transferred.
As we tune the RF frequency away from ν0 , the transfer efficiency in the wings decreases,
while the transfer efficiency in the center of the cloud initially grows and peaks for a detuningof ∼ −15 kHz.
94

Frequency Shift [kHz]

20

10

0

-10

-20

400

500
600
Magnetic Field [G]700

Figure 8-7: Spectroscopic frequency shifts as a function of the magnetic field. (a) Frequency
shifts of the |2i → |3i resonance due to the presence of atoms in state |1i. The fit at low
fields (solid line) uses Eq. 8.8 with n1 = 2.2 × 1013 cm−3 and the theoretical calculations of
the scattering lengths. The dashed line indicates the position of the predicted a 13 resonance.
(b) The latest calculations of the a12 and a13 s−wave scattering lengths as a function of
the magnetic field [120]. a12 has a narrow Feshbach resonance at 550 G and a wide one at
810 G. a13 has a wide Feshbach resonance at 680 G.
The summary of our measurements at different magnetic fields is shown in Fig. 87(a). At low fields, the data are fitted using the mean field prediction of Eq. 8.8, withthe latest calculations of a12 and a13 [5, 120], shown in Fig. 8-7(b)4 . However, in the mostinteresting region, above ∼ 630 G, our results are clearly strongly deviating from the mean

field description of a weakly interacting system.

8.2.1

Strongly interacting and high density limit

The mean field result of Eq. 8.8 relies on two assumptions:(1) That the two-body interactions are weak, k|a| ¿ 1. In this limit, the interactions

are energy independent, and Re(f ) ≈ −a.

(2) That the effective gas density is low, |na3 | ¿ 1.

Near the Feshbach resonance, both of these assumptions can be invalidated. Further, ina degenerate Fermi gas, the two conditions are qualitatively the same, since n ∼ k F3 .

In Ch. 4, we have seen how the strong two-body interactions can be properly describedfor all values of ka. However, we do not know how to properly account for the simultaneous
breakdown of the low-density approximation. This is a job for the many-body theorists [21].
4Note that this is the third plot of the |1i − |2i Feshbach resonance in this thesis (see Figs. 5-1 and 6-2).
As I mentioned at the beginning of Ch. 5, the theoretical predictions are constantly updated by the newexperimental inputs. The calculations displayed here make use of all the presently available 6 Li experimental
data.

95

Figure 8-8: Expansion measurements of the |1i − |2i interaction energy by Bourdel et al.,taken from [28]. The ratio of interaction to kinetic energy is plotted as a function of themagnetic field. Crosses and squares correspond to measurements in differently preparedsamples, while the solid curve shows a mean field calculation. See [28] for details.
It seems natural that in the limit |a| → ∞, this scale does not enter the problem explic-

itly, and is replaced by some real length scale, such as 1/k or the inter-particle separation.

In a degenerate Fermi gas, these two scales are the same, 1/kF . The question is whether inthis limit, the physics is still sensitive to the sign of a. If we just treat the two-body interaction properly in the unitarity limit, and average Re(f ) over the momentum distribution, we
will conclude that it is. As I mentioned in Ch. 4, we find < Re(f ) >∼ −sgn(a)(1/k F ) [9, 26].

Using this result, we can at least qualitatively explain our measurements in the 630 − 680 G

region. For these magnetic fields, both a12 and a13 are large and positive. Therefore, thetwo contributions to the frequency shift of the |2i → |3i transition saturate at a common

large value, and exactly cancel.

However, for fields above 680 G, a12 and a13 are large, and have opposite signs. Therefore, the two frequency shifts should add up to a large negative value. However, we stillobserve essentially no shifts. Our observations are at least qualitatively consistent withthe expansion measurements of the interaction energy in a |1i − |2i mixture by Bourdel
et al. [28], shown in Fig. 8-8. In that experiment, the interaction energy was found to beconstant and negative for magnetic fields up to 80 G below the a12 resonance. Also, thechange of the sign of the interaction energy was abrupt, similar to the sudden disappearanceof the frequency shift at ∼ 630 G in our measurements (Fig. 8-7(a)). In order to explain
our data, we would have to assume that the two contributions to the |2i → |3i frequency
shift have the same sign and magnitude at all fields above 630 G.

Some many-body theorists have suggested that in the high density regime, |na3 | À 1,

the interaction energy should indeed be universally negative and constant in the vicinityof a Feshbach resonance, on both of its sides [121, 122]. Other authors have proposedthat this effect could be related to the presence of molecules on the positive side of the96

resonance [28, 30]. However, my general feeling is that the experimental observations arestill not well understood.

97

Chapter 9

Conclusions and outlook
The bartender says we don’t serve robots
And the robot says, oh but someday you will
D.C. Berman, Frontier Index

The ongoing quest for fermionic superfluidity in a gaseous system is currently one of
the most exciting research topics in all of atomic physics. The eventual attainment of thisgoal will involve many demanding experimental steps. Ultimately, we need to create astable, deeply degenerate spin mixture of fermions, in which the interactions are strong andattractive. In this thesis, I have described our progress in achieving these tasks.
The first part of the thesis described our advances in the cooling of 6 Li fermions deep
into quantum degeneracy. The success of our lithium cooling strategy relied on the use ofa large 23 Na bosonic cloud as a refrigerant for sympathetic cooling. In order to improve
our refrigerator, we have also achieved first condensation of sodium in the upper, F = 2
hyperfine state by evaporative cooling. This allowed us to magnetically trap and coola mixture of sodium and lithium atoms which is stable against inelastic spin-exchange
collisions at all densities. At present, our system produces the largest and the coldest Fermi
samples, as well as the largest Bose-Fermi mixtures [2].
In the second part of the thesis I have discussed our studies of strongly interacting
fermions [3–6]. We have produced and studied optically trapped spin-mixtures of 6 Li in thevicinity of the magnetic field Feshbach resonances, where the phase transition to a superfluid
state is predicted to occur.
In our first study of this regime, we have investigated the stability of the system against
inelastic losses. We have shown that, in contrast to bosonic gases, a mixture of the twolowest ground states of 6 Li exhibits sufficient stability in the strongly interacting regime toallow further studies and the possible observation of superfluidity [3]. In these experiments,
we have also identified a new Feshbach resonance, which was not theoretically predictedprior to our work [23].
98In the second set of experiments, we have established RF spectroscopy as a method
to measure the interaction energy in 6 Li spin-mixtures arising from coherent collisions between atoms. As part of this work, we have also demonstrated the absence of mean field“clock shifts” in a two-state mixture of fermions [5, 6], which may be important for future
improvements of atomic clocks. Interaction energy shifts could be observed by extending
our scheme to a three-level system [5].
Our spectroscopic measurements of the interaction energy shifts in the strongly interacting regime have yielded some surprises, with the interaction energy seemingly being
universal to a surprisingly high degree, independently of both the strength and the repulsive or attractive nature of the two-body interactions. While our findings are consistent
with other experiments [28], they still present a challenge to the theorists.
Experimentally, the next challenge is to combine the deep degeneracy of our magnetically trapped 6 Li samples, with the tunability of interactions in the optically trappedspin-mixtures.
However, we will also need further theoretical insight in order to progress. Most importantly, we need to identify an effective and unambiguous way to detect the superfluid phasetransition. With several groups, including ours, already reaching the conditions at which
the transition may occur, this issue is becoming increasingly urgent. We are currently considering whether our RF spectroscopic methods can be used to observe the pair breaking
energy in the superfluid state [123, 124].
Finally, we should expect several related research avenues to open in parallel with theongoing efforts to achieve superfluidity in a (quasi-)uniform gas. Here I briefly mention
three topics which will almost certainly be important in the near future:
Bose-Fermi mixtures
So far, most Bose-Fermi experiments have concentrated on studies of the degenerate Fermi
gas, and bosons have mostly been used only as a refrigerant for fermions. Studies of nontrivial physics in Bose-Fermi mixtures have been limited to theory [125] and a single experimental paper, which explored collapse of a harmonically trapped 40 K degenerate Fermi gas
due to attractive interactions with a 87 Rb Bose cloud [126]. Future studies of the degenerate
mixtures may open a whole new parameter space for finding exotic many-body states.
Optical lattices
Ultimately, the most exciting studies of strongly correlated systems will almost certainly
involve optical lattice trapping potentials. In the case of bosons, this technique has already yielded fascinating results in the observation of the superfluid to Mott insulatortransition [127]. Optical lattices may soon become important both for the research with
Bose-Fermi mixtures, and for alternative ways of achieving fermionic superfluidity [41].
99Molecules
We have recently learned that diatomic 6 Li molecules created by sweeping the magnetic
field through a Feshbach resonance have a lifetime of almost a second [30]. This is a very
pleasant surprise, given that they are in the last bound, highly excited vibrational state, andwe would expect them to undergo a rapid inelastic relaxation. This surprising metastability
may offer a window of opportunity for evaporative cooling and Bose-Einstein condensationof molecules. Conversion of molecules back into strongly correlated atomic pairs may be
another way of achieving the superfluid state. Finally, it would be as exciting to condense
the molecules in their ground state. This is a serious experimental challenge for the future.

100Appendix A
6Li laser system baedeker
This appendix was written as a more technical companion to Section 2.4.4. As such, it
is rather a collection of loosely connected topics, than a coherent story. The first section
introduces the basics of diode lasers. The second section describes our scheme for referencing
the master laser to a 6 Li atomic resonance. In the third section, I calculate the frequencies
needed for imaging of 6 Li at high magnetic fields. Finally, I provide a technical drawing of
the actual setup of the laser system.

A.1

Diode Lasers

A.1.1

Free Running Diode Lasers

Laser diodes are the central ingredient of our laser system. Here we introduce some of theirbasic characteristics and share some practical experiences. Our experience is limited to670 nm laser diodes, and may not be more widely applicable. For a more general introduction to diode lasers and their uses in atomic physics, one can refer to the review article byCarl Wieman and Leo Hollberg [128]. A comprehensive source on lasers in general is thebook by Anthony Siegman [129].
Main Properties
When current is injected into the depletion region of a p−n semiconductor junction, electronhole pairs are recombined and photons corresponding to the band gap of the semiconductormaterial are emitted. The gain profile of the material can be up to 20 nm broad, but mode
competition will select one or a few modes near its peak. This is the basic principle behindthe operation of a diode laser, although more complicated layered semiconductor structures
are used in commercial products.
A diode laser basically consists of a semiconductor diode mounted on a temperaturestabilizing block. Mounts based on a Peltier element are commercially available and containa temperature monitor and contacts for current injection.
101

At this point only the cleaved surfaces of the semiconductor serve as the laser mirrors.
The size of this laser cavity is typically d ∼ 1 mm, corresponding to mode spacing of∆ν = c/2d ∼ 100 GHz. We measure the linewidth of the laser to be in the 100 MHz range,much larger than the 6 MHz natural linewidth of 6 Li. To make things worse, the diode may
not be lasing in a single mode, and will certainly not stay on a single mode over long times.
The maximum power that this type of laser can output in a continuous wave mode is about
50 mW.
The laser linewidth should be distinguished from the “cold” cavity mode width. The
latter measures the range of frequencies that the cavity will support and is given simply
by the single photon lifetime inside the cavity. A cavity that is as leaky as reflective has afinesse no better than 2 or 3, and the corresponding mode width should be several tens ofGHz.
Temperature Control
The band gap of the diode semiconductor material, and hence the wavelength of the emitted
light can be somewhat tuned by changing the temperature. In our experience, typicaltemperature dependence of 670 nm diodes is about 0.15 nm/K, with higher temperatures
corresponding to longer wavelengths. Without much complication, diodes can be operated
at least 30 K away from room temperature, giving a temperature tuning range of about±5 nm.

Commercially available controllers can stabilize the temperature stable to better than

0.01 K.
Current Control
A typical diode requires a threshold injection current of about 50 mA to start lasing.
This threshold marks the separation between the stimulated and the spontaneous emission regimes, and between a laser and a simple LED. The laser’s maximum output power
is reached at about 100 mA.
The ∼ 1 mW/mA scaling can be directly linked to the fact that a red photon has about

3 × 10−19 J of energy, while the charge of the electron is 1.6 × 10−19 C. The fact that thisrule of thumb does hold reasonably well also tells us that conversion of electron-hole pairs

into photons is a pretty efficient process, certainly in the several tens of percent ballpark.
This high efficiency can be characteristic only of direct gap semiconductors. In indirect gap
materials, photon emission relies on phonon assistance, and they should make rather poor
lasers.
The threshold current increases with increasing temperature. Therefore, in order toachieve a satisfactory power output, we generally operate heated diodes at a higher current
than is advised by the manufacturer. This is generally OK, if we keep the following warningin mind. The slope of power vs. current beyond the threshold point, which measures the102

conversion efficiency, generally does not change with temperature. We can keep increasing
the current as long as this slope remains constant. Eventual decrease in the slope withincreasing current can be interpreted as a “runaway threshold” behavior - higher current
increases the local junction temperature through Joule heating and hence increases thethreshold point even though the temperature is nominally set to the same value. We
interpret this as a warning to stop pushing the current further.
Since the injection current affects the local heating of the junction, varying its value
will tune not only the intensity but also the frequency of the light. This dependence is on
a GHz/mA scale.
Like their temperature counterpart, current controllers are also commercially available,
and will keep the chosen injection current stable to better than 0.1 mA.
Beam ShapeThe cross-section of a diode’s active region, perpendicular to the direction of propagation,is of rectangular shape, with an aspect ratio of about 3. In this way, the gain is bigger along
one axis and the emitted light has a well defined linear polarization along that direction.
Dimensions of the active region are rather small, of the order of 1 µm. Correspondingly,
the output beam has rather large diffraction limited divergence angles, of the order of 10 ◦ .
Thus the beam needs to be immediately collimated with a short focal length lens.
Our diodes have divergence angles along the directions parallel and orthogonal to thepolarization axis of about 8◦ and 20◦ respectively. Using a 4.5 mm lens, we collimate thelaser beams to an elliptical, 5 × 2 mm shape. By adjusting the position of the lens carefully

the beam can usually be collimated over a distance of several meters. Since usually acircular beam profile is preferred, we then use a combination of two prisms, known as ananamorphic prism pair, to expand the short axis of the beam by a factor of 2.5, without
affecting the long one.
Reliability and Lifetime
People much more experienced than us know that there are 1001 ways to kill a diode [128].

So far we have been fortunate not to discover too many of them.
In our experience, the fate of a laser diode has been determined in the first few hours ofoperation. Or attempted operation. Occasionally a diode would appear to work fine, butwould be impossible to collimate. This might be a sign that it has been electrocuted before
it was mounted. Sometimes a nicely collimated diode will turn out to have a bad mode
hop that makes it impossible to use at the desired wavelength. Further, sometimes the
manufacturer will specify only the nominal wavelength of the diode, and not the measured
value for the particular sample. The two can be different by several nanometers, and theexperimentalist can be in for a bad surprise. As a general rule, one should try to purchase
many replacement diodes and try them out. Given their low cost (usually below $100), this103

is not a big problem. Once a good diode is found, and then carefully operated, it can give
thousands of hours of reliable operation.
One safe way to kill a diode is to run too much current through it or to allow hightransients during switching. Fortunately, commercial current controllers have the option
to limit the maximum current, and have built in “soft” switches. Short of power failure
during operation, this should provide enough safety. If one is forced to heat the diode inorder to tune its wavelength, the threshold current will increase, and the output power at
the manufacturer recommended operating current will drop. It is then naturally tempting
to increase the current. We have certainly done this for diodes heated up to 50 ◦ C, andhave always managed to recover the nominal output power. We would however suggest thatthe current is increased carefully, and the power is constantly monitored, watching out for
the runaway threshold behavior we described earlier. Similarly, if cooling of the diode isneeded, it should probably be checked whether the maximum current should be reduced.
Another common wisdom is that heating a diode will shorten its lifetime “drastically”.
However the issue of what exactly are the time scales involved does not seem to be an exact
science.
Optical Isolators
Finally, it is important to note that laser diodes are very sensitive to optical feedback. Since
the internal feedback is weak due to a poor cavity, the laser will be affected by any external
light that falls within the very broad gain profile of the diode material. Controlled feedback
can be used to improve the laser characteristics, as we will discuss in the following sections.
However, it is crucial to protect the diode from any accidental uncontrolled feedback. Partof the laser’s own light reflected back into the diode from somewhere in the optics setup,
can disrupt and even kill the diode. Protection of the diode is conventionally done byplacing a “Faraday isolator” immediately in front of the laser. In brief, the central partof an isolator is a crystal, surrounded by a permanent magnet, which rotates the laser
polarization through 45◦ . This part is sandwiched between two polarizers positioned at
45◦ with respect to each other. The key point is that polarization is rotated in the same
sense independent of the direction of light propagation [130]. Therefore the isolator acts
as an optical diode. In the forward direction, all the light (with polarization aligned with
the input polarizer) is transmitted, whereas the backwards travelling light is blocked withisolation up to 40 dB.
Summary
To summarize, the basic (670 nm) diode laser setup consists of a diode mounted on acooling/heating block, temperature and current controllers, collimating and beam shaping
optics, and an optical isolator. Its output is a circular 5 mm beam, with linear polarization,104

power up to 30 mW distributed over several modes, linewidth of about 100 MHz, and nolong term stability.

A.1.2

Grating Stabilization

One way to improve the spectral characteristics of a free running laser diode is to put
it into an external cavity. A Littrow configuration illustrated in Fig. A-1 is an example
of a solution in this spirit. A collimated laser beam is incident on a holographic gratingaligned so that the first diffraction order is reflected back into the diode. The zeroth order,
containing about 50% of the diode power, constitutes the output of the laser. Therefore thecavity is now essentially formed by the back facet of the diode and the grating, and has atypical size of L ∼ 1 cm. Cavity modes are now separated by about 10 GHz, and are a few
GHz wide. The linewidth of our TuiOptics DL100 grating stabilized laser is about 1 MHz.

Figure A-1: Grating stabilized laser system in a Littrow configuration.
One new scale now appearing is the gain profile of the grating, which is broadened by
the fact that only a finite number of grating periods is illuminated by the laser beam. For a
5 mm beam, about 104 grating periods are used, and thus the grating resolution (essentiallygiven by the ratio of optical wavelength and the number of grating periods) is around
50 GHz.
Coarse wavelength selection of this type of a laser is done by rotating the grating, which
is mounted in such a way that the changes in the length of the cavity match the changes
in the wavelength selected by the grating angle. As long as the laser’s behavior is governedby the external cavity, its wavelength can be pulled up to 7 nm away from the free runningvalue. The quantitative measure of the grating feedback is the lowering of the thresholdcurrent from the free running value by up to 10 %. At the next level of precision, different
modes can be selected by varying the laser temperature and current. The summary ofdifferent frequency scales is given in Fig. A-2. Good lasing at the desired frequency isachieved when the different gain and mode profiles overlap as shown.
105

Medium Gainwidth ~ 10000 GHz

Grating Profile
width ~ 40 GHz

Internal Mode Profile
width ~ 25 GHz
External Mode~100 GHz~ 10 GHzemitted frequency

Frequency

Figure A-2: Summary of frequency scales in a grating stabilized laser.
Once the desired wavelength has been selected to within several GHz and the feedback
has been optimized, the final tuning can be done by applying voltage to the piezoelectrictransducer at the back of the grating mount. This method can also be used to scan thelaser frequency over the same range, at rates up to 1 kHz.
Generally speaking, grating (re)alignment needs to be done maybe twice a year. The
fact that it needs to be done at all is a sign of either (generally unpredictable) diode aging
effects, or the fact that the laser has been accidentally bumped into. Temperature of thediode is changed somewhat more often, probably in order to compensate for smaller aging
effects. Playing with the current and the grating voltage is a daily procedure, taking notmore than a few minutes.

A.1.3

Injection Locking

Another way to improve the spectral characteristics of a free running laser diode is toinject a small amount of high spectral quality light into it. As long as the injection lightis compatible with the (rather loose) constraints of the diode gain profile and internal
resonator, it will easily win the mode competition.
Optical isolators usually use polarizing beam splitter cubes as their input and exit polarizers. This provides an excellent entrance port for the seeding light. If light of appropriate
polarization is sent into the exit polarizer from the side, it will be reflected into the isolator.

106

Further, since it has the opposite polarization from the light that would have entered iten face, it will be rotated in such a way that it will exit the polarizer at the entrance end
without being attenuated. In addition, due to imperfect polarization, a small amount ofslave light is rejected by the exit polarizer, and can be used as a perfect counter propagating
tracer beam for alignment. We have replaced the exit polarizers of our optical isolators withfree standing polarizing beam splitter cubes to provide easier optical access, and overlap of
the seeding and the slave light. This arrangement is sketched in Fig. A-3.

Figure A-3: Seeding of a diode laser.
Effects of seeding are best monitored with a Fabry-Perot spectrum analyzer. The firstsign that the seeding beam is entering the slave diode is visible heating, pushing the slavefrequency to the red. If this effect is maximized, we can be pretty sure that the spatial
overlap of the two beams is optimized. At that point, a narrow spectral line will alreadyappear among the broad features of a free running slave. Varying the slave current slightly
will then transfer all the laser power into the seeded mode. As with the grating stabilization,
good injection locking will also manifest itself in a reduction of the threshold current, andhence an increase in the power output.
For long term seeding stability, it is favorable that the distance between the master laser
and the slave is minimal. At the same time, for good alignment of the seeding beam, it
is best that it is overlapped with the tracer beam over the longest possible distance. This
conflict can be resolved in the following manner. We align the slave laser output with somedistant object. An obvious choice is the tracer beam of the wavemeter, with which thelaser needs to be aligned anyway. Then we crudely align the seeding beam, so that it isgoes nicely through the optical isolator, and into the slave’s collimating package. At thatpoint, if we turn down the slave current, the small reflection of the seeding beam off thecollimating lens is visible and almost aligned with the wavemeter tracer beam. By then107

carefully aligning the two, we are certain that the seeding beam is also perfectly overlappedwith the slave beam. At this point simply turning up the slave current will almost invariably
reveal good seeding.

A.2

Locking of the master laser to the 6 Li atomic resonanceFor long term stability we lock our Master laser to the 6 Li atomic vapor resonance. At room
temperature, the Doppler width of the atomic vapor is on a GHz scale, and some form of
Doppler free spectroscopy needs to be employed. We use optical pumping spectroscopy,
which is a slight variation on the textbook saturated absorption method. The setup issummarized in Fig. A-4. The pump and the probe beam are counter propagated through a
6 Li vapor cell and the transmission of the probe beam as a function of the laser frequency

is recorded by a photodiode.

Figure A-4: Saturated absorption spectroscopy setup. Three Doppler free features are seen
on top of the broad Doppler profile.

Optical Pumping Spectroscopy in a Three Level System
The origins of the Doppler free features seen in Fig. A-4 are visually summarized in Fig. A-5.
At a general frequency, pump and probe beams are resonant with different velocity classes
and there is no effect on the probe absorption. This gives the ∼ GHz broad Doppler profile.

If the laser is resonant with one of the two atomic lines, both beams are competing for thezero velocity atoms and the absorption of the probe is reduced. These are the standard

Lamb dips, except that the absorption reduction comes not from the fact that atoms spend
a large fraction of the time in the excited state, but simply from the fact that they are
pumped over into the other ground state. When the laser is tuned half way between the two
resonances, the pump and the probe talk to the same velocity class, in two different ground108

states. This way, the pump beam will actually increase the population of the ground statethat the probe is resonant with, and the probe absorption will be enhanced. This feature
is known as the crossover peak. Since atoms in both ground states contribute to it, thecrossover is actually the most pronounced feature and is the one we use for locking. Atoms
contributing to this feature have a Doppler shift of 114 MHz, corresponding to a velocity
of about 70 m/s. Since this is much less than the width of the thermal distribution, their
population is essentially the same as that of the zero velocity atoms, and this does not resultin any relative reduction of the signal.

Figure A-5: Principle of saturated absorption (or optical pumping) Doppler free spectroscopy. (A) Broad Doppler background: The velocity class that is resonant with theprobe beam is not resonant with the pump. (B) Lamb dip: The laser is on one of the two
resonances - the probe and the pump both compete for the zero velocity class atoms. (C)Crossover peak: The laser is half way between the two resonances - the pump puts more
atoms in resonance with the probe.

6 Li Vapor Cell

Lithium vapor cells require a bit more effort than those of most other alkali gases. Firstly,in order to get a sufficient vapor pressure for a good locking signal, lithium must be heated
to about 350 ◦ C. Secondly, lithium is on rather bad terms with glass, and it is important tokeep the vapor away from the windows that are used for optical access. Our first cell, an
all-glass design based on work by Libbrecht et al. [131], had a lot of problems with coatingof the windows with lithium, and was eventually shelved. Our final design is a simple 40 cm
long metal cell with plain vacuum windows. A few grams of lithium metal is placed in the
middle part of the cell, which is wrapped in tape heaters. In order to keep lithium awayfrom the windows we load the cell with about 100 mTorr of argon buffer gas. The optimalargon pressure was empirically determined so that lithium fluorescence is well localized tothe central part of the cell, while the pressure broadening of the Doppler free features is not
too drastic, roughly within 20 MHz.

109

The Lock Point
The offset between the desired laser frequency and the crossover line is chosen by AOMs
inserted into both the pump and the probe beam. For tuning up to tens of MHz, it issufficient to change the frequency of one of the beams. The pump and the probe willthen have different frequencies, but all our previous arguments still hold, just with the“zero velocity” being redefined to mean the “velocity corresponding to half the difference
between the pump and the probe frequencies”. As long as this is much less then the widthof the thermal distribution, everything will be fine. To this end, the pump AOM is set up in
a double pass configuration, so that the lock point can easily be changed without changing
the alignment of the setup (see Fig. A-8).
As the lock point, we choose the frequency of the MOT cooling transition, about 25 MHz
below the F = 3/2 → F 0 = 5/2 cycling transition.
The error signal
In order to lock the master frequency, we need an ’error signal’, proportional to the difference
between the current frequency and the lock point. The absorptive signal shown in Fig. A-4
is not suitable for this. While one could choose to lock to the slope of the crossover peak,
the lock point would then depend on laser power and the vapor pressure in the gas cell.
Instead, we lock to the position of the crossover peak, by turning the absorptive signal intoa dispersive one. To achieve this, the pump frequency is modulated at a rate of 50 kHz, withan adjustable amplitude ∆ω ∼ 5 MHz. A (Stanford Research Systems) lock-in amplifier isused to provide the oscillator signal to the pump AOM. The probe absorption signal is thenfed into the lock-in amplifier which integrates out its 50 kHz Fourier component in phasewith the oscillator signal. The value of this Fourier component is the measure of the local
slope of the absorptive signal, and in the limit of small ∆ω, this essentially corresponds
to differentiation. Measured absorptive and dispersive signals are shown in Fig. A-6. Less
than 1 mW of master laser power is more than sufficient to obtain the displayed signal tonoise ratio. The lock-in integration time is 1 ms, and thus the error signal contains only
components up to 1 kHz. This is somewhat limiting, but we have found in practice that it
is sufficient.
Feedback to the Laser
Error signal from the lock-in amplifier is fed back to the laser, via a home built electronic
PI (Proportional - Integral) gain box with a bandwidth of 1 kHz. The “restoring force” onthe laser is mostly provided by the piezoelectric transducer attached to the grating. The
response time of the transducer is about 1 ms, in agreement with the bandwidth of our error
signal. We have however found that some feedback to the current controller increases thelaser stability, and tightens the lock from about 4 MHz to about 1 MHz.

110

Figure A-6: Measured absorptive and dispersive Doppler free spectral features. The absorptive signal from a photodiode, and the dispersive signal from the lock-in amplifier arerecorded on a digital oscilloscope.

A.3

Imaging frequencies at high magnetic fields

Ground, and 2P3/2 excited states of 6 Li at high magnetic fields are shown in Fig. A-7. In
the following calculations, all the frequencies are referenced to the frequency of the zero
field cooling transition F = 3/2 → F 0 = 5/2, hν0 = Ee − Eg . Further, all the calculations
are done with the precision of the natural linewidth, Γ = 6 MHz.

Let us start with the ground states. We are interested in imaging states |1i, |2i, and|3i of the ground state mJ = −1/2 manifold. I will label their energies E1,2,3 and reference

them to the zero field energy of the F = 3/2 state, Eg .

State |3i is stretched, so we simply have E3 = Eg − h × 1.4 MHz/G B. For the states|1i and |2i, we solve the standard Breit-Rabi problem, and get:r

2
1 + x + x2
3r2
E2 = Eg − h × 114 MHz − h × 114 MHz 1 − x + x2
3

E1 = Eg − h × 114 MHz − h × 114 MHz

where x = B×(1.4 MHz/G)/114 MHz. For best results, one should plot this one a computer.
However, for fields above ∼ 350 G, 114 MHz/x2 < Γ, so we can expand to order 1/x:r

1
4
2
1 ± x + x2 ≈ x ± +3
3 9xFurther, for fields above ∼ 350 G, we can, with Γ precision, replace 114 MHz × 4/(9x) with
6 MHz. So finally, for imaging purposes, we have:

E1 = Eg − h × 1.4 MHz/G B − h × 158 MHz
E2 = Eg − h × 1.4 MHz/G B − h × 82 MHz
E3 = Eg − h × 1.4 MHz/G B
111

(A.1)

υ [MHz]2µ B/h0υ 0+ 2000

mj = 3/2
υ 0+ 1000
2

2 P3/2υ0

mj = -1/2

υ 0− 2000-2µ0B/h(671nm)

υ1
1000

2

2 S1/2-2/3µ0B/h

mj = -3/2

υ 0− 1000

υ0

2/3µ0B/h

mj = 1/2

υ2µ0B/h

mj = 1/2

0

mj = -1/2-1000−µ0B/h

0

200

400

600

800

B [G]

Figure A-7: Optical transitions in 6 Li at high magnetic fields. Atoms in the three mJ =−1/2 ground states can be imaged on the transitions to the excited states m J = −3/2 (ν1 )or mJ = −1/2 (ν2 ). Transitions to the mJ = 1/2 states are too far detuned from all thefrequencies already available in the laser system.
Now let us look at the excited states. The hyperfine constant in the excited state oflithium is only ∼ 1 MHz, and the resulting hyperfine structure is not resolved at low fields(see Section 2.4). Likewise, at high fields, different mI states within the same mJ manifold
are not resolved. The two transitions which are easiest to reach with AOMs are those tomJ = −1/2 and mJ = −3/2. We label the energies of these states E−1/2 and E−3/2 , andreference them the the zero field energy of all F 0 states, Ee .

In 6 Li, the fine splitting of the 2P state is also rather small, about 10 GHz. Therefore, at~ S~ interaction already starts being decoupled. Hence, the mJ = −1/2
fields of interest, the L·

state has a small upwards quadratic shift because it is repelled from the mJ = −1/2 stateof the J = 1/2 manifold. So, again we solve a simple Breit-Rabi problem, and then expandin the small parameter, except now the small parameter is B, rather than 1/B. The
mJ = −3/2 state is stretched, so no effort is required there. This calculation gives:2
× 1.4 MHz/G B + h × 0.44 MHz/G2 B 2
3= Ee − h × 2 × 1.4 MHz/G B

E−1/2 = Ee − h ×
E−3/2

(A.2)

Subtracting Eq. A.1 from Eq. A.2, and remembering that (Ee − Eg )/h = ν0 is the MOT

cooling frequency (tuned to resonance), we obtain all the needed frequency with sufficient112

precision to be able to image the atoms. A further tuning of the imaging frequency, by afew MHz, can be done empirically.
Let us summarize all the six used frequencies:

−1/2

ν1

−1/2

ν2

−1/2

ν3

−3/2

ν1

−3/2

ν2

−3/2

ν3

1
× 1.4 MHz/G B + 0.44 MHz/G2 B 2 + 158 MHz
31= ν0 + × 1.4 MHz/G B + 0.44 MHz/G2 B 2 + 82 MHz
31= ν0 + × 1.4 MHz/G B + 0.44 MHz/G2 B 23= ν0 − 1.4 MHz/G B + 158 MHz= ν0 += ν0 − 1.4 MHz/G B + 82 MHz= ν0 − 1.4 MHz/G B(A.3)

where the subscript labels the ground state, and the superscript labels the excited state.

A.4

Layout of the 6 Li laser system

Figure A-8 shows the real space arrangement of the 6 Li laser system, drawn roughly to scale.
The dimensions of the optical table on which the system is assembled are 90 × 210 cm.

The part of the setup needed for imaging at high magnetic fields is not included, because

it is currently being changed.

113

l/4
0
-1

0
+1

Beam
Shape

l/2
l/4
-1

Beam
Shape
l/2

100mm

0
200MHz
-d
AOM

l/2

150mm

Imaging
Light

200mm

0

125mm
0

0

+1

+1
175mm

Beam
Split

0
+1

175mm

l/4

80 MHzAOM

75mm

Mechanical
Shutter

Mechanical
Shutter

Mechanical
Shutter

Beam
Shape
l/2
50mm

ISOLATORISOLATOR
l/2

LEGEND:l/2

SlowerLight
Amplification- Mirror

Mechanical
Shutter

50mm

SLOWERSLAVE- Lens

+1

400mm

l/2

SLOWERREPUMP
SLAVE

0

75mm

175mm

Beam
Shape

94 MHz
+/- d
AOM

500mm

150mm

Generationof Frequencies-1

Frequency Monitoring

200MHz
+d
AOM

50mm

228MHz
AOM
(50%)

200mm

WAVEMETER
Beam
Split
Fabry-Perot

MOT Light
Amplification

200mm
ISOLATOR

l/2

ISOLATOR

l/2

100mm

l/2

Flipper
MirrorSlave Monitoring Beam(s)228MHz
AOM(50%)

MASTER
LASER

MOT
SLAVE

ISOLATOR

Figure A-8: Layout of the 6 Li laser system.

200mm

MOT
REPUMPSLAVE

0375MHzAOMSLOWERFIBER
COUPLER

75mm

l/2

40mm
Pump

Probe

75mm

125mm

Photo
Diode
Li VAPOR CELL
40mm

MOT
FIBER
COUPLER

F=1/2 PROBEFIBER
COUPLER

F=3/2 PROBEFIBER
COUPLER

Locking to the Atomic ResonanceTransfer to the Experiment- Polarizing Beam Splitter Cube

+1

- Used AOM Diffraction Order0- Unused AOM Diffraction OrderAppendix B

Two-Species Mixture of Quantum
Degenerate Bose and Fermi Gases
This appendix includes the following paper [1]: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).

115

VOLUME 88, NUMBER 16

PHYSICAL REVIEW LETTERS

22 APRIL 2002

Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases
Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Görlitz,* and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
MIT, Cambridge, Massachusetts 02139(Received 26 December 2001; published 4 April 2002)
We have produced a macroscopic quantum system in which a 6 Li Fermi sea coexists with a large andstable 23 Na Bose-Einstein condensate. This was accomplished using interspecies sympathetic coolingof fermionic 6 Li in a thermal bath of bosonic 23 Na. The system features rapid thermalization and long
lifetimes.
DOI: 10.1103/PhysRevLett.88.160401

PACS numbers: 05.30.Fk, 32.80.Pj, 39.10. +j, 67.60. –gExperimental methods of laser and evaporative cooling,used in the production of atomic Bose-Einstein condensates (BEC) [1], have recently been extended to realize
quantum degeneracy in trapped Fermi gases [2–5]. Whatmakes gaseous fermionic systems particularly appealing toinvestigate is the relative ease with which their properties
can be varied. This allows the exploration of a vast rangeof experimental regimes, from noninteracting to strongly
correlated. In the first case, purely quantum statistical effects can be studied, such as the implications of Pauli exclusion on scattering properties of the system. In the otherextreme, exciting new regimes of BCS-like fermionic superfluidity may be within reach [6 –9]. An additional areaof interest is the production of a dilute quantum degenerate mixture of Bose and Fermi gases, akin to the stronglyinteracting 4 He-3 He liquid. This would extend the list of
possible experimental studies even further, to include effects such as interaction-driven phase separation [10] or
BEC-induced interactions between fermions.
In this Letter, we report the production of a new
macroscopic quantum system, in which a degenerate 6 Li
Fermi gas coexists with a large and stable 23 Na BEC. We
have achieved high numbers of both fermions .105  andbosons .106 , and 6 Li quantum degeneracy characterized
by a temperature of one-half the Fermi temperature TF .
This provides us with the starting point for studies ofthe degenerate 6 Li Fermi gas, which is regarded as aparticularly promising candidate for the BCS transition[6,7]. Further, favorable collisional properties makethe 6 Li-23 Na system very promising for studies of thedegenerate Bose-Fermi mixtures, including the limitations
to the cooling process [11,12]. It is also worth noting
that, in our experiment, a mixture of two different atomic
species has been simultaneously brought into quantum
degeneracy for the first time.
Our experimental approach is based on sympathetic
cooling of fermions in a large bosonic “refrigerator.” In
contrast to the bosonic case, two-body elastic collisions
are absent in a single-component Fermi gas at ultralow
temperatures due to the Pauli exclusion principle. This
lack of thermalization precludes direct implementationof forced evaporative cooling. Therefore, cooling offermions into the quantum degenerate regime must rely
on collisions between distinguishable atoms. In two
experiments which produced degenerate Fermi gases,
mixtures of two fermionic spin states were simultaneously
evaporated and mutually cooled [2,5]. Two groups havealso demonstrated sympathetic cooling of 6 Li by the 7 Libosonic isotope, thus also producing the first quantum
degenerate Bose-Fermi mixtures. However, this system
has a limitation that, in the upper hyperfine state, the 7 Li
condensate is unstable [3], while scattering properties
in the lower hyperfine state make sympathetic coolinginefficient, and limit the size of both 6 Li and 7 Li samples[4]. We have overcome both of these limitations by usinga large 23 Na cloud, instead of 7 Li, for sympathetic coolingof 6 Li. Our work provides the natural progression in the
search for an ideal Bose-Fermi system, where a “good”
Bose-Einstein condensate is chosen, and then combinedwith a favorable fermionic species. Similar two-species
experiments are currently being pursued by three othergroups [13 –15]. Given the vast variety of collisional
properties among alkali gases, and a limited choice offavorable Bose-Fermi combinations, the properties of the6
Li-23 Na mixture are truly fortuitous. In our experiment,
we have observed rapid interspecies thermalization,
while low rates for both intra- and interspecies inelastic
collisions result in a lifetime longer than 10 s. Both thepresence of sufficient “good” (elastic) collisions needed
for interspecies thermalization and the slow rate of “bad”(inelastic) collisions could not be taken for granted before
our studies.
For this experiment, we have upgraded our 23 Na BEC
apparatus [16] to allow for both lithium and sodium operation, while making minimal modifications to the original
setup. The additional laser light needed for optical coolingof 6 Li was generated by a low power, diode laser system[17]. The lithium (671 nm) and sodium (589 nm) laser
beams were overlapped using dichroic beam-splitters, andthe spatial arrangement of the laser and atomic beams used
to trap, cool, and detect lithium was identical to the original
sodium setup. Specifically, a two-species magneto-opticaltrap (MOT) was loaded from a single two-species atomic
beam, slowed in the same Zeeman slower previously used

160401-1© 2002 The American Physical Society

0031-900702 88(16)160401(4)$20.00

160401-1

VOLUME 88, NUMBER 16

PHYSICAL REVIEW LETTERS

in sodium-only experiments. The fact that the maximum
spontaneous light force deceleration is twice as large for6
Li as for 23 Na allowed us to slow lithium atoms withoutcompromising the slowing efficiency for sodium.
To implement our experimental strategy, we have developed a two-species oven in which the vapors of 6 Li and23
Na were mixed, and a single atomic beam containing
both species was produced (Fig. 1). The main difficulty indesigning such an oven is that, at the same temperature, thevapor pressure of lithium is 3 orders of magnitude lower
than that of sodium. To achieve comparable atomic fluxesof both species, the alkali vapors must be produced in separate chambers, and then delivered to a mixing chamber,at controllable rates. In our design, the lithium chamber
was also used for mixing. To operate the oven in either
single- or two-species mode, we tuned the atomic fluxes
independently by changing the temperatures of the alkali
reservoirs. The maximum atom fluxes into the solid angle
subtended by the MOT region were 3 3 1011 s21 for 6 Li
and 2 3 1012 s21 for 23 Na.
Under typical operating conditions, 5 s of loading resulted in single-species MOTs with 2 3 107 lithium atoms
or 6 3 109 sodium atoms. When both MOTs were operated simultaneously, interspecies light-assisted collisions reduced the number of lithium atoms by a factor of about
4, while the sodium atom number was not noticeably
affected. The number of 6 Li atoms in the MOT was
maximized when the trapping and the repumping lightfrequencies were tuned 25 MHz below the corresponding
resonances. The resulting temperature of the lithium atoms
was 700 mK.
Since the 23 Na BEC is produced in the jF, mF  
j1, 21 lower hyperfine ground state, to avoid inelastic
spin-exchange collisions, it is preferred to magneticallytrap 6 Li in the corresponding j12, 212 state. (Here, F isthe total angular momentum, and mF is its projection along
the quantization axis.) However, the maximum magnetic
trap depth in the j12, 212 state is only 330 mK (see22 APRIL 2002

Fig. 2), considerably lower than our MOT temperature.
Further, due to the inefficiency of sub-Doppler coolingmechanisms, it is not possible to optically cool lithium
to temperatures which would make magnetic trapping inthis state efficient [17]. Therefore, to avoid drastic losses
due to the limited trap depth, lithium atoms were optically
pumped and then magnetically trapped in the F  32
manifold. Before loading the magnetic trap, 4 ms wereallowed for the sub-Doppler polarization gradient coolingof sodium, during which the lithium cloud was in free
expansion. This reduced the transfer efficiency of lithium
atoms into the trap by a factor of 2, limiting it to about
12%. We have thus magnetically trapped 6 3 105 6 Li
atoms in the upper hyperfine state and 2 3 109 23 Na
atoms in the lower one. At low energies, our cloverleaf
magnetic trap is harmonic and axially symmetric. In
the lower hyperfine states, the trapping frequencies forlithium (sodium) are vz  2p 3 26 16 Hz axially andvr  2p 3 354 221 Hz radially.
Once the atoms were loaded into the magnetic trap,
we started the forced evaporative cooling of sodium [1].
A varying microwave field near 1.77 GHz was used togradually lower the trap depth for 23 Na, selectivelytransferring the most energetic atoms into the untrapped
j2, 22 state. This microwave field does not affect the 6 Li
atoms, which were therefore not evaporated. Cooling of
the lithium sample was instead achieved through thermal
contact with sodium. We observed efficient sympathetic
cooling of 6 Li in the upper hyperfine state by 23 Na in thelower one, and have successfully cooled this Bose-Fermi
mixture into simultaneous quantum degeneracy. This observation indicates a surprisingly favorable ratio between

Hyperfine transfer|3/2, 3/2>

Loading|3/2, 1/2>
|3/2, -1/2>

Transfer Nozzle
450 ºCEnd of cooling 228 MHz

Main Nozzle
450 ºC|3/2,-3/2>

Trapping limit, 27 G
Zeeman slower|1/2, -1/2>

Trap bottom, 1 G
0|1/2, 1/2>
40

80

B field [G]6

Na reservoir
200 ºC - 400 ºC

Li reservoir
280 ºC - 400 ºC

FIG. 1. Two-species oven. 6 Li and 23 Na vapors were produced
in separate chambers to allow for independent control of theatom fluxes. The two species were mixed in the lithium chamber.
The transfer nozzle has a conductance 40 times lower than themain nozzle, and limits the undesirable diffusion of lithium into
the sodium chamber.

160401-2

FIG. 2. Li in the magnetic trap. Ground state energy levels:
The six hyperfine states are labeled in the low magnetic field,
jF, mF  basis. The j12, 212 state becomes strong field seeking for fields above 27 G, limiting the trap depth to 330 mK.
Cooling path: Atoms in the j32, 12 upper hyperfine state
were loaded into the magnetic trap at a temperature of 700 mK.
After the initial cooling stage to 50 mK, the atoms were transferred to the j12, 212 state, and further cooled to a final
temperature of 330 nK.

160401-2

VOLUME 88, NUMBER 16

PHYSICAL REVIEW LETTERS

the good and bad interspecies collisions in this mixture.
The losses due to inelastic spin-exchange collisions took
place only on a time scale of several seconds, comparable
to the total evaporation time of 15 s.
In order to produce a collisionally stable Bose-Fermi
mixture, it is necessary to transfer the lithium atoms to thelower hyperfine state. To minimize the initial losses due
to spin-exchange collisions, this transfer should take place
as early in the cooling process as possible. On the other
hand, before lithium atoms can be efficiently trapped in
the lower hyperfine state, they must be cooled significantly
below 330 mK. Therefore, we implemented sympathetic
cooling in two stages (see Fig. 2). We optimized the initial evaporation stage to reach a temperature of 50 mK
in 5 s, while losing less than half of the lithium atoms, andmaintaining the conditions for efficient sodium evaporation. At this point, we found that a substantial fraction oflithium atoms was in the j32, 12 state. They could thus
be transferred to the j12, 212 state on a single-photon
rf transition at 228 MHz, which is, to first order, independent of the magnetic field. This simplification over asimilar hyperfine transfer previously employed in [4] was
not expected. After the rf pulse was applied, the remainingF  32 atoms were optically pumped into untrapped hyperfine states, and expelled from the trap. If this “cleanup”
light pulse was omitted, spin-exchange collisions between
lithium atoms in different hyperfine states led to a rapid
loss of atoms from the trap [18]. The overall efficiencyof our hyperfine transfer was 50%. The evaporation wasthen resumed for another 10 s. We observed efficient sympathetic cooling of the j12, 212 atoms, and cooled bothgases into quantum degeneracy without observable losses
in the lithium atom number.
Figure 3(a) displays the effect of sympathetic coolingon the 6 Li cloud. Absorption images of the trapped 6 Li
gas were taken after the 23 Na evaporation was terminatedat different trap depths, and the sample was allowed toequilibrate for 1 s. Cooling (from top to bottom) is seenin the shrinking of the density distribution and an increasein the peak optical density. In contrast to standard evaporative cooling, and the mutual cooling between two Fermi
species, the total number of atoms remains constant.
Quantitative analysis of the 6 Li clouds is depicted inFig. 3(b). We performed two-dimensional fits to therecorded column densities using both a simple Gaussianmodel and a semiclassical (Thomas-Fermi) distribution fortrapped noninteracting fermions. In the latter approach,the probability that an atom has position r and momentump is obtained from the Fermi-Dirac distribution for the
total (kinetic and potential) energy of the particle. The
fitting function for the spatial distribution of atoms in thetrap is then obtained by integrating over the momentum
degrees of freedom. While the Gaussian model gives a
valid description of the gas only in the classical, hightemperature limit, the Thomas-Fermi approach is valid
at all temperatures, as long as the number of particles in
160401-3

22 APRIL 2002a)b)

Optical Density
0

1.5

2 TF

1 TF

0.5 T F
-500

DF
0

500

Axial distance [µm]FIG. 3. Onset of Fermi degeneracy. Three pairs of images (topto bottom) correspond to TTF  2, 1, and 0.5. (a) Column
densities of the 6 Li cloud were recorded by absorption imaging.
(b) Axial line density profiles and the Thomas-Fermi fits to thedata are plotted. The arrow indicates the size of the Fermi
diameter, DF .

the system is large [19]. Indeed, at higher temperatures,the two fits performed equally well, and yielded the same
temperature. However, at a temperature of about 400 nK,
the classical fits started to fail. This was indicated by the
relative growth of the reduced x 2 values, by up to 20%
above the corresponding values for the fermionic fits.
For the coldest samples, Gaussian fits also overestimatedthe temperature by 15%. From the fitted number ofatoms in the system (NLi  1.4 3 105 , TF  670 nK),
we found that the noticeable inadequacy of the classicalfits occurred at 0.6 TF , which is a clear signature of
the Fermi degeneracy. Figure 3(b) shows projected line
densities along the axial direction of the 6 Li cloud and theThomas-Fermi fits to the data. Theparrow indicates thesize of the Fermi diameter, DF  2 2kB TF mvz2 , forthe fitted atom number. The spatial extent of the coldest
cloud T  330 nK  0.5 TF  is already comparable
to the minimum size the system would assume at zerotemperature. Typical densities of the coldest lithium
samples were 1012 cm23 .
In Fig. 4, the temperature of the 6 Li cloud is plotted asa function of the final 23 Na trap depth. The coldest lithium
samples were produced in coexistence with almost puresodium condensates with 2 3 106 atoms. The lifetimeof this degenerate mixture was limited to about 10 s by thethree-body decay of the BEC, while the lithium cloud had
a lifetime longer than 100 s. The lifetime of either species
was not detectably affected by the presence of the otherone. Even after sodium was lost from the trap, the energyof the lithium cloud did not increase noticeably ,5 nKsduring its slow decay. We also compared the 6 Li temperatures with the temperatures of the 23 Na cloud, extracted
from the thermal wings of the bosonic density distribution.
In hotter samples, the two agreed to within 10%. However,for the coldest samples we observed a discrepancy between
the two temperatures. The lowest measurable temperature
160401-3

VOLUME 88, NUMBER 16

PHYSICAL REVIEW LETTERS

tion to a fermionic superfluid state, for which 6 Li is a very
promising candidate.
We thank Florian Schreck for useful discussions andChristian Schunck for experimental assistance. This research was supported by NSF, ONR, ARO, NASA, and theDavid and Lucile Packard Foundation. M. W. Z. acknowledges the support of the Studienstiftung des deutschen
Volkes and the Ecole Normale Supérieure, Paris.

1.5

Lithium Temperature [ µK]

22 APRIL 2002

1.0

2

0.5

1

T/TF

0
0

5

10

15

0.0
0

5

10

15

Sodium Trap Depth [ µK]

FIG. 4. Temperatures of the 6 Li cloud as a function of the 23 Natrap depth. Each data point is an average of three measurements.
The error bars indicate the shot-to-shot fluctuations, while theuncertainties of the fits are comparable or smaller. In the casewhen sodium was completely evaporated from the trap (c.f. zero
trap depth data point), the very last stage of sympathetic coolingbecame inefficient due to the vanishing heat capacity of thebosonic reservoir. The inset shows the same temperature datascaled to the Fermi temperature.

of sodium was T  170 nK, about half the corresponding
lithium value. We verified that, in these samples, extending
the thermalization time at the end of evaporation up to 10 s
did not lower the lithium temperature any farther. The reasons for the observed temperature discrepancy are worthyof further investigation. Simple spatial separation of thetwo clouds due to different gravitational sags can be readily ruled out. However, some form of phase separation of
the two species [10], or constant heating of the fermioniccloud [12], could play a role.
In conclusion, we have produced a system in whicha 6 Li Fermi sea coexists with a 23 Na BEC. This provides us with a starting point for studies of the degenerate Bose-Fermi mixtures. In particular, the observed
temperature difference between the two spatially overlapped species might provide further insight into the limitsof sympathetic cooling. Further, by loading the degenerate Fermi cloud into an optical trap, effects of magnetically tunable interactions between lithium atoms in
different spin states can be studied [18]. A particularly
appealing prospect is the observation of the BCS transi-

160401-4*Current address: Universität Stuttgart, 5. Phys. Inst.,
Stuttgart, Germany.
[1] Bose-Einstein Condensation in Atomic Gases, Proceedingsof the International School of Physics “Enrico Fermi,”
Course CXL, edited by M. Inguscio, S. Stringari, and
C. Wieman (IOS Press, Amsterdam, 1999).
[2] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).
[3] A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B.
Partridge, and R. G. Hulet, Science 291, 2570 (2001).
[4] F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari,
T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev.
Lett. 87, 080403 (2001).
[5] S. R. Granade, M. Gehm, K. M. O’Hara, and J. E. Thomas,
Phys. Rev. Lett. 88, 120405 (2002).
[6] H. T. C. Stoof, M. Houbiers, C. A. Sackett, and R. G. Hulet,
Phys. Rev. Lett. 76, 10 (1996).
[7] M. Houbiers, R. Ferwerda, H. T. C. Stoof, W. I. McAlexander, C. A. Sackett, and R. G. Hulet, Phys. Rev. A 56, 4864
(1997).
[8] M. A. Baranov and D. S. Petrov, Phys. Rev. A 58, R801
(1998).
[9] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, andR. Walser, Phys. Rev. Lett. 87, 120406 (2001).
[10] K. Mølmer, Phys. Rev. Lett. 80, 1804 (1998).
[11] E. Timmermans and R. Côté, Phys. Rev. Lett. 80, 3419
(1998).
[12] E. Timmermans, Phys. Rev. Lett. 87, 240403 (2001).
[13] J. Goldwin, S. B. Papp, B. DeMarco, and D. S. Jin, Phys.
Rev. A 65, 021402(R) (2002).
[14] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni,and M. Inguscio, Science 294, 1320 (2001).
[15] M. Mudrich, S. Kraft, K. Singer, R. Grimm, A. Mosk, and
M. Weidemüller, physics/0111213.
[16] M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M.
Kurn, D. S. Durfee, and W. Ketterle, Phys. Rev. Lett. 77,
416 (1996).
[17] U. Schünemann, H. Engler, M. Zielonkowski, M. Weidemüller, and R. Grimm, Opt. Commun. 158, 263 (1998).
[18] M. Houbiers, H. T. C. Stoof, W. I. McAlexander, and R. G.
Hulet, Phys. Rev. A 57, R1497 (1998).
[19] D. A. Butts and D. S. Rokhsar, Phys. Rev. A 55, 4346
(1997).

160401-4

Appendix C

Fifty-fold improvement in thenumber of quantum degeneratefermionic atoms
This appendix includes the following manuscript [2]:Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann,and W. Ketterle, “Fifty-fold improvement in the number of quantum degenerate fermionic
atoms,” cond-mat/0306050 (2003).

120

Fifty-fold improvement in the number of quantum degenerate fermionic atoms
Z. Hadzibabic, S. Gupta, C.A. Stan, C.H. Schunck, M.W. Zwierlein, K. Dieckmann, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
MIT, Cambridge, MA 02139(June 3, 2003)
We have produced a quantum degenerate 6 Li Fermi gas with up to 7 × 107 atoms, an improvement
by a factor of fifty over all previous experiments with degenerate Fermi gases. This was achieved bysympathetic cooling with bosonic 23 Na in the F = 2, upper hyperfine ground state. We have alsoachieved Bose-Einstein condensation of F = 2 sodium atoms by direct evaporation.

cally trapped 23 Na-6 Li, Bose-Fermi mixture which is stable against spin-exchange collisions at all densities, anddramatically boosted our fermion atom number.

Over the last few years, there has been significant
progress in the production of quantum degenerate atomic
Fermi gases (40 K [1,2] and 6 Li [3–6]) and degenerate
Bose-Fermi mixtures (7 Li-6 Li [3,4], 23 Na-6 Li [6], and87
Rb-40 K [2]). These systems offer great promise forstudies of new, interaction-driven quantum phenomena.
The ultimate goal is the attainment of novel regimes of
BCS-like superfluidity in a gaseous system [7–10]. The
current efforts to induce and study strong interactions ina Fermi gas [11–20] are complemented with the ongoing
efforts to improve fermion cooling methods, which would
lead to lower temperatures and larger samples.
The main reason why studies of degenerate Fermi
gases are still lagging behind the studies of atomic BoseEinstein condensates (BECs), is the complexity of coolingmethods. The Pauli exclusion principle prohibits elastic
collisions between identical fermions at ultra-low temperatures, and makes evaporative cooling of spin-polarized
fermionic samples impossible. For this reason, coolingof fermions must rely on some form of mutual or sympathetic cooling between two types of distinguishable particles, either two spin states of the same atom [1,5], or two
different atoms [2–4,6]. A key element in fermion cooling is the design of better “refrigerators” for sympathetic
cooling.
In this Letter, we report the first production of degenerate Fermi samples comparable in size with the largest
alkali BECs [21]. We successfully cooled up to 7 × 107
magnetically trapped 6 Li atoms to below half the Fermi
temperature (TF ). This is an improvement in atom number by a factor of 50 over the largest previously reported Fermi sea [20]. Further, in samples containing
up to 3 × 107 atoms, we observed temperatures as low
as 0.05 TF , the lowest ever achieved. At these temperatures, the fractional occupation of the lowest energy state
differs from unity by less than 10−8 .
As in our previous work [6], 6 Li atoms were magnetically trapped in the F = 3/2, upper hyperfine groundstate, and sympathetically cooled by bosonic 23 Na. The
crucial improvement was our achievement of forced evaporation of sodium in the |F, mF i = |2, +2i, upper hyperfine ground state, producing large and stable BECs withup to 107 atoms. This allowed us to create a magneti-

(b)

(a)
6

Li

23

| 32 , 32

Na

|2, 2
|2, 1

| 32 , 12|2, 0| 32 ,- 12|2,-1

1772 MHz|2,-2

228 MHzarXiv:cond-mat/0306050 v1 3 Jun 2003

PACS numbers: 05.30.Fk, 32.80.Pj, 39.25.+k, 67.60.-g| 32 ,- 32|1,-1| 12 ,- 12

|1, 0
|1, 1

| 12 , 12

0

100
200
Magnetic Field (G)

0

100
200
Magnetic Field (G)FIG. 1. Hyperfine structures of 6 Li and 23 Na. The states
are labelled in the low field, |F, mF i basis. (a) Due to finite trap depth of ∼ kB × 300 µK in the |1/2, −1/2i state,lithium can be efficiently loaded into the magnetic trap onlyin the upper, F = 3/2 hyperfine state. (b) Sodium is magnetically trappable in the |1, −1i, and in the |F = 2, mF ≥ 0i
states. Previously, sodium has been evaporatively cooled to
BEC only in the |1, −1i, lower hyperfine state.

The criteria for designing sympathetic cooling experiments include the heat capacity of the refrigerator, andthe inter-species collisional properties, both elastic andinelastic. Large and stable 23 Na condensates are an appealing choice for sympathetic cooling of fermions. Further, a favorable mass ratio allows for simultaneous Zeeman slowing of 23 Na and 6 Li [6], and for simultaneous
magnetic trapping without large differences in the gravitational sag. The inter-species collisional properties are
generally not predictable, and have to be tested experimentally. In order to minimize all possible inelastic processes, the natural choice is to magnetically trap both
species in their lower hyperfine ground states. However,at temperatures reachable by laser cooling (≥ 300 µK),
6
Li can be efficiently magnetically trapped only in the1ber dropped to the point where the heat capacities of thetwo species become comparable. However, at this point it
appears unlikely that temperatures below 0.05 TF could
be conclusively extracted in order to differentiate the two
strategies.

upper hyperfine state, F = 3/2 [4,6] (Fig. 1(a)). On theother hand, until now sodium has been successfully evaporated only in the lower, F = 1 hyperfine state. This wasa limiting factor for sympathetic cooling of 6 Li, since themixture of sodium in the lower, and lithium in the upper
hyperfine state is not stable against spin-exchange collisions. The inelastic loss rate increases as the temperature
is lowered and the density grows. In our previous work[6], we partially overcame this problem by transferring
lithium atoms into the lower hyperfine state after an initial sympathetic cooling stage to ∼ 50 µK. By achieving
forced evaporative cooling and Bose-Einstein condensation of sodium in the F = 2 state, we have now realized amore robust sympathetic cooling strategy, and dramatically improved the size and temperature of a degenerate
Fermi system.
We loaded ∼ 3 × 109 sodium and up to 108 lithium
atoms in their upper hyperfine states from a two-species
magneto-optical trap (MOT) into the magnetic trap.
The adverse effect of light assisted collisions in a twospecies MOT [6,22] was minimized by slightly displacing the two MOTs with respect to each other. During
the typical 30 s of evaporative/sympathetic cooling, we
observed no significant inelastic loss of lithium atoms(by three-body collisions or dipolar relaxation), the final number of degenerate atoms being at least half ofthe number initially loaded into the trap. On the other
hand, we observed a favorable rate of elastic collisions
between the two species, with the inter-species thermalization time being shorter than 1 s. Therefore, sodium
atoms in the upper hyperfine state have ideal properties
as a refrigerant for 6 Li.
Since our primary interest was cooling of fermions, we
evaporated all sodium atoms in order to get lithium tothe lowest possible temperatures. Even in our largest 6 Li
samples, of ∼ 7 × 107 atoms, we achieved temperatures
below 0.5 TF . Temperatures in the range 0.05 − 0.2 TF
could be achieved by reducing the 6 Li atom numbers only
slightly, to ∼ 3 × 107 . Such big clouds had a high enough
optical density for crisp absorption imaging even after
ballistic expansion to a size larger than one millimeter(Fig. 2(a)).
Temperatures were extracted from absorption imagesof expanding clouds released from the trap, using a semiclassical (Thomas-Fermi) fit to the Fermi-Dirac momentum distribution [6,23] (Fig. 2(b)). The quoted temperature range reflects both the shot-to-shot and day-today reproducibility, and the fact that the Fermi distribution is very insensitive to the temperature in this ultradegenerate limit.
In these experiments, the 6 Li atom number was adjusted during the loading phase. Somewhat lower temperatures could possibly be achieved if the maximum
lithium atom number was loaded into the magnetic trap,
and then the hottest part of the cloud was selectively removed by direct evaporation once the sodium atom num-

(a)

(b)

Fermi Diameter

3 mm

FIG. 2. Large and ultra-degenerate Fermi sea. (a) Absorption image of 3 × 107 6 Li atoms released from the trap andimaged after 12 ms of free expansion. (b) Axial (vertical) line
density profile of the cloud in (a). A semiclassical fit (thinline) yields a temperature T = 93 nK= 0.05 TF . At this temperature, the high energy wings of the cloud do not extend
visibly beyond the Fermi energy, indicated in the figure by
the momentum-space Fermi diameter.

We also produced two-species degenerate Bose-Fermi
mixtures with several million atoms in each species(Fig. 3). The mixture was stable, with a lifetime of several seconds, limited only by the three-body decay of thesodium cloud.
Li

23

Na

1.7 mm

6

FIG. 3. Two-species mixture of degenerate Bose and Fermi
gases. After release from the magnetic trap, both 6 Li and23
Na clouds were imaged onto the same CCD camera using
separate light pulses. The times of free expansion of the twogases could be varied independently. This dual-imaging technique allowed for optimizing the cooling strategy for either
single- or two-species experiments. For the displayed image,the expansion times were τLi = 8 ms and τNa = 25 ms, andthe atom numbers were NLi ∼ 107 and NNa ∼ 6 × 106 .
Sodium was cooled below the condensation temperature, corresponding to ∼ 0.2 TF for the lithium cloud.

In the rest of the paper, we summarize the numerous
steps which were introduced to prepare sodium in the
F = 2 state as a refrigerant.

2

In contrast to 87 Rb, condensation of sodium by evaporative cooling was previously achieved only in the lower,|1, −1i hyperfine state. F = 2 sodium condensates could
thus be studied only by transferring optically trapped
F = 1 BECs into this state [24,25]. Condensation in theupper hyperfine state of sodium is more difficult than in
the lower state for two reasons: (1) The requirement for
efficient optical pumping in dense laser-cooled samples,and (2) an order of magnitude higher three-body loss rate
coefficient [24].
The basic setup of our experiment is described in [6].
In 10 s, we collected typically ∼ 1010 23 Na atoms, and ∼108 6 Li atoms in a magneto-optical trap (MOT). Typical
MOT temperatures were 0.7-1 mK. Sodium was collected
in a dark-SPOT variant of the MOT [26], and therefore
most of the atoms were in the F = 1 hyperfine state.
Lithium was collected in a “bright” MOT, with about
2/3 of the atoms in the F = 3/2 state.
Before the transfer into the magnetic trap, the atoms
were optically pumped into the stretched hyperfine
ground states, |2, +2i for 23 Na, and |3/2, +3/2i for 6 Li.
A magnetic guide field of 3 G was applied, and the atoms
were optically pumped for 2 ms, using σ + polarized light.
To achieve both F (hyperfine) and mF (Zeeman) pumping, two light beams where used for each species, resonant with the |F = I ± 1/2i → |F 0 = I ± 1/2i transitions. Here, I is the nuclear spin (I = 3/2 for 23 Na,
and I = 1 for 6 Li), and F 0 is the total spin in the excited electronic state. In this way, almost all the lithium
atoms could be pumped into the |3/2, +3/2i state. On
the other hand, the density of sodium atoms in the darkSPOT is ≥ 1011 cm−3 , and Zeeman pumping is notoriously difficult at such high densities. In our experiments,
the fraction of atoms pumped into the |2, +2i state waslimited to about 30 %, with most of the remaining atoms
distributed among the other mF sub-levels of the F = 2
manifold.
After the optical pumping stage, the atoms were loaded
into a Ioffe-Pritchard magnetic trap with a radial gradient of 164 G/cm, and axial curvature of 185 G/cm2.
Sodium atoms in all three |F = 2, mF ≥ 0i states are (atleast weakly) magnetically trappable (Fig. 1(b)). However, only pure |2, +2i samples are stable against inelastic spin-exchange collisions. A crucial step in preparing the samples for efficient forced evaporation was toactively remove |F = 2, mF = 0, +1i atoms from thetrap, before they engaged in inelastic collisions with the|2, +2i atoms. The atoms were loaded into a weak magnetic trap, with a high bias field of 80 G. This field
splits the F = 2 Zeeman sub-levels by ∼ kB × 2.8 mK.
Since this splitting was larger than the temperature of
the cloud, the different states could be resolved in microwave or rf spectroscopy, and the |F = 2, mF = 0, +1i
atoms could be selectively transferred to the untrapped|F = 1, mF = 0, +1i lower hyperfine states. This transfer
was done with a microwave sweep near the 23 Na hyperfine

splitting of 1.77 GHz. In this way, all the |2, +2i atoms
initially loaded into trap could be preserved. We wereeven able to “recycle” some of the untrapped atoms by
optically pumping them out of the F = 1 ground states,
thus giving them a “second chance” to fall into the |2, +2i
state. The final setup consisted of two microwave sweeps,
the first of 0.8 s duration with the optical pumping lighton, and the second of 2.4 s duration without the light. In
this way, the overall transfer efficiency from the MOT to
the magnetic trap was improved to about 35 %, comparable to our standard F = 1 BEC experiments [27].
After this purification of the |2, +2i sample, the magnetic trap was tightened by reducing the bias field
to 3.8 G in 2.4 s. Resulting trapping frequencies were
204 Hz (400 Hz) radially, and 34 Hz (67 Hz) axially for thesodium (lithium) stretched state. This provided goodconditions for forced runaway evaporation of sodium.
Evaporation was done on the |2, +2i → |1, +1i microwave transition near 1.77 GHz. In contrast to radiofrequency evaporation, this insured that 6 Li was far off
resonance. Further, microwave evaporation avoided any
undesirable aspects of “incomplete evaporation” into the|F = 2, mF = 0, +1i states, which could lead to inelastic
losses [28].
After 15 s of evaporation, the sodium atoms reached atemperature of T ∼ 10 µK. At this point, to avoid threebody losses in the |2, +2i state [24], the trap was weakened to frequencies of 49 Hz (96 Hz) radially, and 18 Hz(35 Hz) axially for sodium (lithium). The final evaporation to BEC took another 15 s. In this way, in the absenceof lithium atoms, we could produce almost pure |2, +2i
BECs containing up to 10 million atoms. The lifetime of
the BEC in the weak trap was longer than 3 s. In contrast
to our previous work [24,25], studies of F = 2 condensates are now possible without the added complexity ofan optical trap.
In conclusion, by creating a superior refrigerant for
sympathetic cooling of 6 Li, we have produced the coldest and by far the largest quantum degenerate Fermi gas
so far. With the number of atoms comparable with thelargest alkali BECs, and the temperatures reaching thepractical detection limit, we have fully exploited the potential of laser and evaporative cooling to engineer samples of ultracold fermions. In analogy with Bose-Einsteincondensates, we expect these large samples to insuresufficient signal-to-noise ratio for all the standard techniques of BEC research, such as velocimetry using longexpansion times, rf spectroscopy with Stern-Gerlach separation during ballistic expansion, direct non-destructive
imaging of the trapped clouds, and Bragg spectroscopy.
The next challenge is to maintain a similar combination of number and temperature for an interacting twocomponent Fermi gas [19].
We thank A. E. Leanhardt for critical reading of themanuscript. This work was supported by the NSF, ONR,
ARO, and NASA.
3

ski, Y. Shin, D. E. Pritchard, and W. Ketterle, Phys.
Rev. Lett. 89, 190403 (2002).
[26] W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, andD. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993).
[27] M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M.
Kurn, D. S. Durfee, and W. Ketterle, Phys. Rev. Lett.
77, 416 (1996).
[28] B. Desruelle, V. Boyer, S. G. Murdoch, G. Delannoy, P.
Bouyer, and A. Aspect and M. Lécrivain, Phys. Rev. A
60, R1759 (1999).

[1] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).
[2] G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys.
Rev. Lett. 89, 150403 (2002).
[3] A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B.
Partridge, and R. G. Hulet, Science 291, 2570 (2001).
[4] F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T.
Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett.
87, 080403 (2001).
[5] S. R. Granade, M. E. Gehm, K. M. O’Hara, and J. E.
Thomas, Phys. Rev. Lett. 88, 120405 (2002).
[6] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta,
M. W. Zwierlein, A. Görlitz, and W. Ketterle, Phys. Rev.
Lett. 88, 160401 (2002).
[7] M. Houbiers and H. T. C. Stoof, Phys. Rev. A. 59, 1556
(1999).
[8] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo,and R. Walser, Phys. Rev. Lett. 87, 120406 (2001).
[9] Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402(2002).
[10] W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D.
Lukin, Phys. Rev. Lett. 89, 220407 (2002).
[11] T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S.
Jin, Phys. Rev. Lett. 88, 173201 (2002).
[12] K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic,
C. H. Schunck, and W. Ketterle, Phys. Rev. Lett. 89,
203201 (2002).
[13] K. M. O’Hara, S. L. Hemmer, S. R. Granade, M. E.
Gehm, J. E. Thomas, V. Venturi, E. Tiesinga, and C. J.
Williams, Phys. Rev. A 66, 041401 (2002).
[14] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys.
Rev. Lett. 90, 053201 (2003).
[15] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R.
Granade, and J. E. Thomas, Science 298, 2179 (2002).
[16] M. E. Gehm, S. L. Hemmer, S. R. Granade, K. M.
O’Hara, and J. E. Thomas, arXiv:cond-mat/0212499(2002).
[17] C. A. Regal and D. S. Jin, arXiv:cond-mat/0302461(2003).
[18] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalhães, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov,
and C. Salomon, arXiv:cond-mat/0303079 (2003).
[19] S. Gupta, Z. Hadzibabic, M. W. Zwierlein, C. A. Stan,
K. Dieckmann, C. H. Schunck, E. G. M. van Kempen,
B. J. Verhaar, and W. Ketterle, Science (2003).
[20] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin,
arXiv:cond-mat/0305028 (2003).
[21] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Science 292, 476 (2001).
[22] V. Wippel, C. Binder, and L. Windholz, Eur. Phys. J. D
21, 101 (2002).
[23] D. A. Butts and D. S. Rokhsar, Phys. Rev. A 55, 4346
(1997).
[24] A. Görlitz, T. L. Gustavson, A. E. Leanhardt, R. Löw,
A. P. Chikkatur, S. Gupta, S. Inouye, D. E. Pritchard,
and W. Ketterle, Phys. Rev. Lett. 90, 090401 (2003).
[25] A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpin-

4Appendix DRadio-Frequency Spectroscopy of
Ultracold FermionsThis appendix includes the following paper [5]:S. Gupta, Z. Hadzibabic, M. W. Zwierlein, C. A. Stan, K. Dieckmann, C. H. Schunck,
E. G. M. van Kempen, B. J. Verhaar, and W. Ketterle, “Radio-Frequency Spectroscopy of
Ultracold Fermions,” Science 300, 1723 (2003).

125

REPORTS
Radio-Frequency Spectroscopy of
Ultracold Fermions
S. Gupta,1* Z. Hadzibabic,1 M. W. Zwierlein,1 C. A. Stan,1
K. Dieckmann,1 C. H. Schunck,1 E. G. M. van Kempen,2B. J. Verhaar,2 W. Ketterle1
Radio-frequency techniques were used to study ultracold fermions. We observed the absence of mean-field “clock” shifts, the dominant source of systematic error in current atomic clocks based on bosonic atoms. This absence
is a direct consequence of fermionic antisymmetry. Resonance shifts proportional to interaction strengths were observed in a three-level system. However,in the strongly interacting regime, these shifts became very small, reflecting thequantum unitarity limit and many-body effects. This insight into an interacting
Fermi gas is relevant for the quest to observe superfluidity in this system.
Radio-frequency (RF) spectroscopy of ultracold atoms provides the standard of time.
However, the resonance frequencies of ultracold atoms are sensitive to interactions between atoms, leading to the so-called clock
shifts of the unperturbed resonances (1).
These shifts limit the accuracy of current
atomic clocks (2, 3), but can also be used tocharacterize atomic interactions.
RF spectroscopy has previously been applied to cold atoms to determine the size andtemperature of atom clouds (4, 5). RF methods have also been used for evaporative cooling, for preparing spinor Bose-Einstein condensates (BECs) (6, 7), and as an output
coupler for atom lasers (5, 8). In all theseexperiments, shifts and broadenings due toatomic interactions were negligible. Recently, density-dependent frequency shifts of RF
transitions were observed in rubidium (9) andsodium (10) BECs. These frequency shifts
are proportional to the difference in meanfield energies of two internal atomic states
and allow scattering lengths to be extracted.
Mean-field shifts in BECs have also been
observed by optical spectroscopy (11, 12).
Here, we apply RF spectroscopy to ultracold clouds of fermions and demonstrate several phenomena: (i) the absence of a clock
shift in a two-level system because of fermionic antisymmetry, (ii) the emergence ofmean-field shifts in a three-level system after
the relaxation of pair correlations, (iii) the
limitation of mean-field shifts because of theunitarity limit, and (iv) the universality of the1

Department of Physics, Massachusetts Institute ofTechnology (MIT )–Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT,
Cambridge, MA 02139, USA. 2Eindhoven University ofTechnology, Post Office Box 513, 5600 MB Eindhoven,
Netherlands.
*To whom correspondence should be addressed. Email: deep@mit.edu

interaction energy in a dense cloud, independent of the attractive or repulsive nature of
the two-particle interactions.
Research in ultracold fermions has advanced rapidly, with six groups now having
cooled fermions into quantum degeneracy (13–18). A major goal of this research is to induce
strong interactions by tuning magnetic fields toscattering resonances (called Feshbach resonances). Under these conditions, Cooper pairsof fermions may form, leading to superfluidity.
This would establish a model system for studying Bardeen-Cooper-Schrieffer (BCS) pairing
at densities nine orders of magnitude lower thanin previous realizations in 3He and superconductors. We show that RF spectroscopy can beused to characterize interactions between fermions in the regime where superfluidity has been
predicted (19, 20).
Our experimental technique for preparing
ultracold fermions has been considerably
improved since our earlier work (17, 21).
Because the Pauli exclusion principle suppresses elastic collisions between identicalfermions at low temperatures and prevents
evaporative cooling, we cooled fermionic 6Li
sympathetically with bosonic 23Na loaded
into the same magnetic trap. In contrast to
previous work, we cooled both species intheir upper hyperfine states (23Na: F,mF 2,2, 6Li: F,mF  3⁄2,3⁄2, where F andmF are the quantum numbers for the total spin
and its z component, respectively. This led toa reduction of inelastic loss processes andboosted our final fermion atom numbers by
two orders of magnitude. We could produce
BECs that contained up to 10 million sodium
atoms in the 2,2 state by evaporatively
cooling pure bosonic samples in the magnetic
trap. For a Bose-Fermi mixture, the finite
heat capacity of the bosons limited the final
lithium temperature after the 30-s evaporation cycle to 0.3 TF for 10 million fermi-

ons and TF for 50 million fermions (22),where TF is the Fermi temperature.
The spin states of 6Li of most interest forsuperfluid pairing are the two lowest states
1 and 2 (1⁄2,1⁄2 and 1⁄2,1⁄2 at low
field), which are predicted to have an interstate s-wave (23) Feshbach resonance at
800 G (24, 25). However, both states are
high-field–seeking at these fields, whichmakes them unsuitable for magnetic trapping.
We therefore transferred the atoms into an
optical trap. For these experiments, 6 to 8
million 3⁄2,3⁄2 lithium atoms were loaded
into the optical trap at temperature T  TF 
35 K (26). The atoms were then transferredto the lowest energy state 1, with an adiabatic frequency sweep around the lithium
hyperfine splitting of 228 MHz. Magnetic
fields of up to 900 G were applied, a rangethat encompasses the 1  2 Feshbachresonance. Using RF-induced transitions near
80 MHz, we could create mixtures of states
1, 2, and 3 (3⁄2, 3⁄2 at low field) and
explore interactions between fermions inthese states.
Collisions between atoms cause a shift oftheir energy, which is usually described by themean-field effect of all the other atoms on theatom of interest. For example, atoms in state 2experience an energy shift, [(4 2 )/(m)]n1a12,that is due to the presence of atoms in state 1.
Here is Planck’s constant h divided by 2, mis the mass of the atom, n1 is the density of 1
atoms, and a12 is the interstate scattering length
between states 1 and 2. We use the convention that positive scattering length corresponds
to a repulsive interaction. Density-dependent
shifts of the resonance frequency for the transition that connects two states have been observed in laser-cooled (1) and Bose-condensed(9, 10) clouds.
In the case of ultracold fermions, onlyinteractions between different internal states
are allowed. For a system of density n, let uscompare the energy of a gas prepared purely
in state 1 to a gas in which one atom istransferred into state 2. The energy difference is h 12  [(4 2 )/(m)]na12, where 12
is the resonance frequency of the noninteracting system. Similarly, the energy difference
between a gas prepared purely in state 2 anda gas in which one atom is transferred into
state 1 is h 12  [(4 2 )/(m)]na12.
However, these energy shifts should notaffect the resonance for a coherent transfer
out of a pure state. For fermions in the initial
pure state, the pair-correlation function vanishes at zero distance because of the antisymmetry of the wave function. During any coherent transfer process, the state vectors of all
the atoms rotate “in parallel” in Hilbert space;i.e., the superposition of the two spin states

www.sciencemag.org SCIENCE VOL 300 13 JUNE 2003

1723

REPORTS
has the same relative phase for all atoms.
Thus, the atoms remain identical and cannot
interact in the s-wave regime. The mean-fieldenergy is thus established only after the coherence of the superposition state is lost andthe pair correlations have relaxed, forming apurely statistical mixture of the two states.
It is a consequence of Fermi statistics
that spectroscopic methods do not measure
the equilibrium energy difference between
the initial and final state of the system, butrather measure the unperturbed resonancefrequency. The expected absence of theclock shift has led to suggestions for theuse of fermions in future atomic clocks(27 ). Our work presents an experimental
demonstration of this phenomenon.
We determined the transition frequency between states 1 and 2, first starting with a purestate 1 and then with a pure state 2 sample.
The absence of a splitting between these twolines proves the suppression of the clock shift.
Fig. 1 shows an example of such measurements. The magnetic field was ramped up to
570 G with the cloud in state 1. At this field,
a12  150a0. Therefore, the expected equilibrium mean-field shifts were 5 kHz forour mean density of 3 1013 cm3 (28). Theinteraction between states 1 and 2 at thismagnetic field was also observed in the mutual
evaporative cooling of the two states in the
optical trap. RF pulses 140 s in duration wereapplied at frequencies near the unperturbed resonance 12  76 MHz. Atoms in states 1 and2 could be monitored separately by absorption
imaging, because they are optically resolved at
this field. We observed a suppression of theclock shift by a factor of 30 (Fig. 1). Using thesame method, we observed the absence of aclock shift at several other magnetic fields. In
particular, we observed a suppression of morethan three orders of magnitude at 860 G (29).

P-wave interactions (23) could lead to anonvanishing clock shift. However, at these
low temperatures, they are proportional to T
or TF, whichever is higher, and are therefore
strongly suppressed.
We can observe mean-field shifts andscattering lengths spectroscopically by driving transitions from a statistical mixture oftwo states to a third energy level. [While thiswork was in progress, use of a similar method
to measure scattering lengths in fermionic40
K was reported (30).] Specifically, we recorded the difference between the RF spectra
for the 2 3 3 transition in the presence
and in the absence of state 1 atoms. The
presence of atoms in state 1 is then expected
to shift the resonance by (31).
2n (a  a 12)
m 1 13(1)

In our experimental scheme to determine theinteraction energy at different magnetic fields(Fig. 2), the system was prepared by ramping
up the magnetic field to 500 G with the atoms

in state 1. Either partial or complete RFtransfer to state 2 was then performed. The
number of atoms in state 1 was controlled
by adjusting the speed of a frequency sweeparound the 1 3 2 resonance. A fast, nonadiabatic sweep created a superposition of thetwo states, whereas a slow, adiabatic sweep
prepared the sample purely in state 2. A
wait time of 200 ms was allowed for thecoherence between states 1 and 2 to decay
and the system to equilibrate.
Typical parameters for the decohered 1– 2 mixture were mean-density n1 
2.41013 cm3 and T  0.7 TF. The
magnetic field was then changed to thedesired value, and the transition from state
2 to state 3 was driven with 140-s RF
pulses (Fig. 2C). We monitored the appearance of atoms in state 3 and the disappearance of atoms from state 2, using
simultaneous absorption imaging. Fig. 2Dshows the unperturbed and perturbed resonances at the magnetic field B  480 G.
The position of the unperturbed resonanceFig. 1. Absence of the clock shift. RF transitions
were driven between states 1 and 2 on asystem prepared purely in state 1 (solid circles) and purely in state 2 (open circles).
Mean-field interactions would have resulted in
5-kHz shifts for the two curves in opposite
directions. Gaussian fits (solid lines) to the data
are separated by 0.04 0.35 kHz. This gives aclock-shift suppression factor of 30. Arb., arbitrary units.

1724

13 JUNE 2003 VOL 300 SCIENCE www.sciencemag.org

Fig. 2. Schematic of the meanfield measurement and representative spectra at 480 G. (A) Hyperfine structure of the groundstate of 6Li. (B and C) Experimental scheme: (B) Preparation of a
mixture of atoms in states 1and 2, and (C) RF spectroscopyof the 2 3 3 transition using
a variable radio frequency ( RF).
(D) The fraction of atoms transferred from 2 to 3, with 1
atoms absent (solid circles) andpresent (open circles). The
mean-field shift is computedfrom Gaussian fits to the data(solid lines). (E) Spatial images ofstate 3 for the perturbed resonance. The optical trap was
turned off immediately after the
RF pulse and absorption imagesof the atoms were taken after
120 s of expansion time. The
central section of 150-m vertical extent was used to extract
the transferred fractions in (D).
(E) also shows images of states
2 and 1 for zero RF detuning.
States 3 and 2 were imaged
simultaneously to observe theircomplementary spatial structure. State 1 was imaged after
760 s of expansion time torecord its density for normalization purposes.

REPORTS
23 also determines the magnetic field to an
accuracy of 0.1 G. Fig. 2E shows absorption images of atoms in state 3, obtainedfor different values of the applied radio
frequency. One can clearly see the spatial
dependence and thus the density dependence of the mean-field shift: Close to theunperturbed resonance, the low-density
wings of the cloud are predominantly transferred, whereas the high-density central
part of the cloud is transferred only at
sufficient detuning. To suppress spurious
effects from this spatial dependence, only asmall central part of the images was used to
extract the transferred atomic fraction.
To ensure that our mean-field measurements were performed on a statistical mixture, we measured the time scale for decoherence in our system. The decay of the 1  2
coherence at 500 G was observed by monitoring the 2 3 3 transfer at the measured
unperturbed resonance 23, as a function of
wait time (Fig. 3). For wait times that aresmall compared to the decoherence time ofthe 1  2 superposition, the 2 3 3 RF
drive places each atom in an identical threestate superposition. All mean-field shifts arethen absent and the resulting transfer is unchanged from the unperturbed case. For longer wait times, the 1  2 superposition
decoheres and mean-field interactions set in.
This shifts the resonance frequency of the 2
3 3 transition, reducing the transferredfraction at 23. The measured decoherencetime of 12 ms was attributed mainly to thesensitivity of 12 to magnetic field variations
across the cloud. These inhomogeneities
cause the relative phase of the 1  2
superposition in different parts of the trap to
evolve at different rates, given by the local
12. Atoms that travel along different paths
within the trap therefore acquire different
phases between their 1 and 2 components.
Being no longer in identical states, s-wave
interactions between them are allowed. The
inhomogeneities scale with B, whereas thesensitivity of the transition scales with  12/

Fig. 3. Emergence of mean-field shifts due todecoherence at 500 G. Decoherence leads to a
reduction of the 2 3 3 transfer at theunperturbed resonance 23. An exponential fit
to the data (solid line) gives a time constant of
12 ms.

B. We would thus expect the decoherencetime to vary inversely with the product ofthese two quantities. Our hypothesis is supported by our observation of longer decoherence times at higher fields, where B  12/
B is lower.
Fig. 4A summarizes the results of ourmean-field measurements for a wide range ofmagnetic fields up to 750 G. For magnetic
fields up to 630 G, our data can be explained
fairly well by using Eq. 1 with the theoretical
calculations of the scattering lengths shownin Fig. 4B and an effective density of n1 
2.21013 cm3, which is consistent with
the initial preparation of the system at 500 G.
A narrow resonance of a12 at 550 G (21,25, 32) is indicated by the data but was not
fully resolved. We also see additional structure near 470 G, which is not predicted bytheory and deserves further study.
For fields above 630 G, the measured shifts
strongly deviated from the predictions of Eq. 1,
indicating a different regime of interactions. In
the region between 630 G and 680 G, the two
scattering lengths are expected to be large andpositive, with a13  a12 (Fig. 4B). Eq. 1 would
thus predict large positive mean-field shifts. In
contrast, we observe very small shifts, indicating almost perfect cancellation of the two contributions. We also observe essentially no meanfield shifts between 680 G and 750 G, where thetwo scattering lengths are predicted to be verylarge in magnitude and of opposite signs, and ina simple picture should add up to a huge negative shift. These results are evidence for phenomena in a strongly interacting system, where

the scattering length becomes comparable toeither the inverse wave vector of interacting
particles or the interatomic separation.
Eq. 1 is valid only for low energies andweak interactions, where the relative wave
vector of the two particles, k, satisfies k 
1/a. For arbitrary values of ka, the s-wave
interaction between two atoms is describedby replacing the scattering length a with thecomplex scattering amplitude f.
fa(1ika)1k 2a 2(2)

The real part of f, Re( f ) determines energy
shifts, and hence the ground state properties ofan interacting many-body system. The imaginary part, Im( f ) determines the (inverse) lifetime for elastic scattering out of a momentum
state, and hence the dynamic properties of thesystem such as thermalization rates. For ka 3
, the elastic cross-section   4Im( f )/kmonotonically approaches the well known“unitarity-limited” value of 4/k2. On the other
hand, the two-particle contribution to the meanfield energy, proportional to Re( f )  a/(1 k 2a2 ) peaks at a  1/k and then, counterintuitively, decreases as 1/a for increasing a.
Averaging Re( f ) over a zero-temperature
Fermi distribution with Fermi momentum kF
limits its absolute value to 1.05/kF and markedly weakens its dependence on the exact value of
a in the kFa  1 regime (33). This results in a
prediction for the mean-field energy that issensitive to the sign of the scattering length,
remains finite for kFa  1, and never exceeds
0.45 EF, where EF is the Fermi energy. Hence,

Fig. 4. Spectroscopic measurement of interaction energy. (A) Frequency shift versus magnetic field
for the 2 3 3 resonance due to atoms in state 1. The shifts are computed by monitoring thearrival fraction in state 3 for 140-s RF pulses, except at 750 G. At 750 G, because of strong
inelastic losses between 3 and 1 atoms, we monitored the loss of atoms in state 2 after
applying RF sweeps 3 ms in duration and 2 kHz in width. All the data points are normalized to thesame atom number in state 1. The fit at low fields (solid line) uses Eq. 1 with n1  2.2 1013
cm3 and the theoretical calculations of the scattering lengths. The error bars reflect uncertainty
in the state 1 atom number and the uncertainty in the Gaussian fits to the spectra. The dashedline indicates the position of the predicted a13 resonance. (B) S-wave scattering lengths a12 and a13
as a function of magnetic field, obtained from a highly model-independent quantum-scattering
calculation. The calculation makes use of the presently available 6Li experimental data (40) in acoupled channel approach to deduce accumulated phases that characterize the less well-known,
short-range parts of the 6Li  6Li scattering potential (32). a12 has a narrow Feshbach resonanceat 550 G and a wide one at 810 G. a13 has a wide Feshbach resonance at 680 G.

www.sciencemag.org SCIENCE VOL 300 13 JUNE 2003

1725

REPORTS
this approach could qualitatively explain our
results in the 630 to 680 G region, but it is inclear contradiction with negligible resonanceshifts in the 680 to 750 G region (34).
We suggest that these discrepancies might
be due to the fact that we are in the high-density
regime, where na3 approaches unity. In a degenerate Fermi gas, the interparticle spacing iscomparable to the inverse Fermi wave vector,
kF3  62n. Hence, the unitarity limit coincides
with the breakdown of the low-density approximation (na3  1) and higher order manybody effects can become important. Some recent many-body calculations (35–37) suggestthat in the regime kFa  1 (or na3  1), theinteraction energy is always negative and independent of both sign and magnitude of a. Thissuggests that whenever the scattering length islarge, either positive or negative, the interactionenergy is a universal fraction of the Fermi energy (33). This is a possible explanation for thesmall line shifts that we observed for fields
higher than 630 G, where the interactions arestrong in both states.
This picture is consistent with other recent
experimental observations (30, 33, 38, 39).
Expansion energy measurements in a mixtureof states 1 and 2 of 6Li (39) showed anegative interaction energy at 720 G, which ison the repulsive side of the predicted Feshbach resonance. RF spectroscopy in 40K (30)
has also shown some saturation of the mean
field in the vicinity of a Feshbach resonance,
which may reflect the unitarity limit.
In characterizing an interacting Fermi gas by
RF spectroscopy, we have demonstrated absence of clock shifts in a two-level system andintroduced a three-level method for measuringmean-field shifts. For strong interactions, wehave found only small line shifts that may reflect both the unitarity limit of binary collisions
and many-body effects. It would be very important to distinguish between two-body and manybody effects by studying the gas over a broad
range of temperatures and densities. In a very
dilute and very cold gas, the weakly interacting
limit could be extended to very large values ofa, thus allowing for direct verification of molecular calculations. This presents experimental
challenges, because cooling changes the density
and the temperature together. It would also beinteresting to study similar phenomena inbosonic gases, in order to distinguish to what
extent the high density many-body effects depend on quantum statistics. This insight into thephysics of strongly interacting Fermi gases must
be taken into account in the search for superfluidity in these systems.

References and Notes
1. K. Gibble, S. Chu, Phys. Rev. Lett. 70, 1771 (1993).
2. C. Fertig, K. Gibble, Phys. Rev. Lett. 85, 1622 (2000).
3. Y. Sortais et al., Phys. Scr. T95, 50 (2001).
4. A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Lafyatis,
D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988).

1726

5. I. Bloch, T. W. Hänsch, T. Esslinger, Phys. Rev. Lett. 82,
3008 (1999).
6. M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998).
7. J. Stenger et al., Nature 396, 345 (1998).
8. M.-O. Mewes et al., Phys. Rev. Lett. 78, 582 (1997).
9. D. M. Harber, H. J. Lewandowski, J. M. McGuirk, E. A.
Cornell, Phys. Rev. A. 66, 053616 (2002).
10. A. Görlitz et al., Phys. Rev. Lett. 90, 090401 (2003).
11. T. C. Killian et al., Phys. Rev. Lett. 81, 3807 (1998).
12. J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999).
13. B. DeMarco, D. S. Jin, Science 285, 1703 (1999).
14. A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B.
Partridge, R. G. Hulet, Science 291, 2570 (2001).
15. F. Schreck et al., Phys. Rev. Lett. 87, 080403 (2001).
16. S. R. Granade, M. E. Gehm, K. M. O’Hara, J. E. Thomas,
Phys. Rev. Lett. 88, 120405 (2002).
17. Z. Hadzibabic et al., Phys. Rev. Lett. 88, 160401 (2002).
18. G. Roati, F. Riboli, G. Modungo, M. Inguscio, Phys. Rev.
Lett. 89, 150403 (2002).
19. M. Houbiers, H. T. C. Stoof, Phys. Rev. A. 59, 1556
(1999).
20. M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, R.
Walser, Phys. Rev. Lett. 87, 120406 (2001).
21. K. Dieckmann et al., Phys. Rev. Lett. 89, 203201(2002).
22. Z. Hadzibabic et al., preprint available at http://
arXiv.org/abs/cond-mat/0306050 (2003).
23. S-wave and p-wave interactions refer to two-bodyinteractions with relative angular momentum 0 and, respectively.
24. M. Houbiers, H. T. C. Stoof, W. I. McAlexander, R. G.
Hulet, Phys. Rev. A. 57, R1497 (1998).
25. K. M. O’Hara et al., Phys. Rev. A. 66, 041401 (2002).
26. The transfer efficiency is limited by finite optical-trap
depth and residual heating during the transfer.
27. K. Gibble, B. J. Verhaar, Phys. Rev. A. 52, 3370 (1995).
28. For our modest degeneracy, we can use a Gaussian
approximation for the density distribution. The meandensity is then lower than the peak density by afactor of 22.

29. Our estimate uses the direct mean field measurement of Bourdel et al. (39), scaled to our parameters.
30. C. Regal, D. S. Jin, preprint available at http://arXiv.
org/abs/cond-mat/0302461 (2003).
31. Because of fermionic antisymmetry, there is no clock
shift from interactions between atoms in states 2
and 3. Mean-field shifts arise only during a decoherence time (see later in the text) but are negligible
if the population of atoms in state 3 is small.
32. E. G. M. van Kempen et al., in preparation.
33. M. E. Gehm, S. L. Hemmer, S. R. Granade, K. M.
O’Hara, J. E. Thomas, preprint available at http://
arXiv.org/abs/cond-mat/0212499 (2002).
34. For fields higher than 630 G, number losses generally
reduced n1, and all the shifts in this region werescaled up by a factor of 1.1 to 2.7 but still remained
negligible.
35. J. V. Steele, preprint available at http://arXiv.org/abs/
nucl-th/0010066 (2000).
36. H. Heiselberg, Phys. Rev. A. 63, 043606 (2001).
37. R. Combescot, preprint available at http://arXiv.org/
abs/cond-mat/0302209 (2003).
38. K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R.
Granade, J. E. Thomas, Science 298, 2179 (2002).
39. T. Bourdel et al., preprint available at http://arXiv.
org/abs/cond-mat/0303079 (2003).
40. This includes the recent measurement of 800 40 G
for the resonance in a12 (39).
41. The work at MIT was supported by NSF, the Office ofNaval Research, the Army Research Office, and NASA.
The work at Eindhoven is part of the research programof the Stichting voor Fundamenteel Onderzoek derMaterie, which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek.
4 April 2003; accepted 25 April 2003
Published online 8 May 2003;
10.1126/science.1085335
Include this information when citing this paper.

The Interface Phase and the
Schottky Barrier for a Crystalline
Dielectric on Silicon
R. A. McKee,1* F. J. Walker,1,2 M. Buongiorno Nardelli,1,3
W. A. Shelton,1 G. M. Stocks1
The barrier height for electron exchange at a dielectric-semiconductor interface
has long been interpreted in terms of Schottky’s theory with modifications fromgap states induced in the semiconductor by the bulk termination. Rather, weshow with the structure specifics of heteroepitaxy that the electrostatic boundary conditions can be set in a distinct interface phase that acts as a “Coulomb
buffer.” This Coulomb buffer is tunable and will functionalize the barrier-height
concept itself.
When Schottky (1) and Mott (2) formulated thebarrier-height theory for a metal-semiconductor
junction and later when Anderson (3) formulated
the band-edge offset problem for semiconductorsemiconductor junctions, there was no consideration given to interface states as contributions
1
Oak Ridge National Laboratory, Oak Ridge, TN37831, USA. 2University of Tennessee (UT), Knoxville,
TN 37996, USA. 3North Carolina State University,
Raleigh, NC 27695, USA.

*To whom correspondence should be addressed. Email: mckeera@ornl.gov

to the electrostatic boundary conditions. The
charge distribution at the interface was treatedsimply as a superposition of the bulk-terminatedjunction. Certainly these theories have been insightful, but they consistently misrepresent thebarrier height or band-edge offsets because
real interfaces, apparently from interfacial
structure variations, modify the intrinsic bandalignment (4 – 6).
Although the bulk-termination view of the
problem has been enhanced over the years withan ever-increasing formalization of theoreticaltechniques (7–15), recent bond polarization ar-

13 JUNE 2003 VOL 300 SCIENCE www.sciencemag.org

Appendix E

Spectroscopic insensitivity to cold
collisions in a two-state mixture offermions
This appendix includes the following manuscript [6]:M. W. Zwierlein, Z. Hadzibabic, S. Gupta, and W. Ketterle, “Spectroscopic insensitivity
to cold collisions in a two-state mixture of fermions,” cond-mat/0306627 (2003).

130

Spectroscopic insensitivity to cold collisions in a two-state mixture of fermionsMartin W. Zwierlein, Zoran Hadzibabic, Subhadeep Gupta, and Wolfgang Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
MIT, Cambridge, MA 02139(Dated: June 24, 2003)

arXiv:cond-mat/0306627 v1 25 Jun 2003

We have experimentally demonstrated the absence of spectroscopic resonance shifts in a mixtureof two interacting Fermi gases. This result is linked to observations in an ultracold gas of thermalbosons. There, the measured resonance shift due to interstate collisions is independent of thecoherence in the system, and twice that expected from the equilibrium energy splitting between thetwo states in a fully decohered cloud. We give a simple theoretical explanation of these observations,which elucidates the effect of coherent radiation on an incoherent mixture of atoms.
PACS numbers: 03.75.Ss,05.30.Jp,32.30.Bv,34.20.Cf

The coherence properties of light and matter are intimately connected with the quantum statistics of theconstituent particles. One quantitative measure of thecoherence in a system is the two-particle correlation function at zero distance, g (2) , which measures the probability
that two particles are simultaneously detected. Intensity
fluctuations in the incoherent light emitted by a lightbulb lead to photon “bunching”, making this probability
twice higher than in the coherent light of a laser. Identical fermions on the other hand exhibit “anti-bunching”,
making such a probability zero.
Interactions in ultracold atomic gases crucially dependon the value of g (2) [1]. The reason is that s-wave scattering relies on particles overlapping in real space. Theinteraction energy in a many-body system is determinedby coherent, or “cold” collisions, for which the outgoing
and the incoming two-particle states are identical. Under
this constraint, the two colliding particles can at most do
two things - either preserve their momenta, or exchange
them. We can thus distinguish four cases: (1) Two identical bosons in a thermal gas can collide in both ways,
corresponding to g (2) = 2, in analogy with a light bulb.
(2) Two atoms in a Bose-Einstein condensate (BEC) havethe same momenta and cannot undergo the exchange interaction. Here, g (2) = 1, in analogy with laser light.
(3) Two distinguishable particles, fermions or bosons,also cannot exchange their momenta because that would
make the outgoing state different from the incoming one.
Again, g (2) = 1. (4) Two identical fermions cannot collide at all, so g (2) = 0. In all cases, the mean-field energyof a particle with mass m is given by g (2) (4π~2 /m)an,where a is the s-wave scattering length, and n is the density of atoms it interacts with.
In experiments on ultracold hydrogen, mean-field shiftsof the 1S-2S two-photon transition were used to prove theexistence of a BEC [2]. However, quantitative interpretation of the shifts led to a vivid theoretical discussion
about the coherence related “factors of 2” [3, 4, 5, 6].
More recently, Eric Cornell’s group at JILA performedRamsey spectroscopy in a two-component, thermal gas of87
Rb bosons to measure g (2) in the interstate collisional

shift [7]. Their measurements yielded g (2) = 2, independent of the degree of coherence between the two states.
The spectroscopic results thus seemed to correspond tothe case of all particles being in an identical coherent superposition of the two internal states, even though thebinary mixture was partially decohered and should have
had a mean-field energy corresponding to 1 < g (2) < 2.
The authors commented on this mystery [8]: “it is a pleasure to note that a two-level system can still yield surprises, 75 years after the advent of quantum mechanics.”
The mystery can be formally resolved using a quantum
Boltzmann equation [9, 10, 11, 12, 13].
Here, we experimentally address the relation between
coherence and spectroscopic measurements in a binary
mixture of ultracold fermions. We demonstrate thatshifts of spectroscopic lines are absent even in a fully
decohered binary mixture, in which the particles are distinguishable, and the many-body mean-field energy inthe system has developed. We theoretically show that
this is a direct consequence of the coherent nature of the
RF excitation, and is not dependent on the coherence ofthe sample on which spectroscopy is performed. Our calculation intuitively explains both our results for fermions,and the JILA results for bosons.
In a recent paper [14], we demonstrated the absence ofmean-field “clock-shifts” in a coherent two-state superposition of 6 Li fermions. In this case, RF spectroscopy
was performed on a gas prepared purely in one internal
state. Since an RF pulse acts as a rotation in the twostate Hilbert space, all the atoms stayed in an identical(superposition) state and could not interact. As longas the fermionic atoms were indistinguishable, g (2) = 0,
and the resonance was thus found to be unperturbed at
ν0 = Eh12 , where E12 is the energy difference between theinternal states |1i and |2i.
However, once decoherence sets in, for example due
to inhomogeneous magnetic fields across the cloud, thespatial overlap between atoms in different states grows
and mean-field energy density builds up:
Eint = g (2) V12 n1 n2 ,

V12 =

4π~2
a12 ,
m

(1)

2

∆Eint = δµ2 − δµ1 = V12 (n1 − n2 ).

(2)This suggests [7, 8, 14] that in a decohering sample,
the resonant frequency for population transfer between
the two states gradually changes from ν12 = ν0 to
ν12 = ν0 + h1 ∆Eint . Here, we show both experimentallyand theoretically that this conclusion is wrong, and thatthe spectroscopic resonance frequency ν12 is always theunperturbed frequency ν0 .
Our experimental setup was described in [14, 15].
Magnetically trapped 6 Li fermions were sympatheticallycooled with 23 Na bosons, and then transferred into an
optical dipole trap. About 107 atoms were trapped at atemperature of 35 µK. The two-level system under consideration is formed by the two lowest ground state hyperfine levels, |1i and |2i, corresponding to |F, mF i =|1/2, 1/2i and |1/2, −1/2i in the low field basis, respectively. A DC magnetic field of B = 320G was applied
to the sample in order to tune the interstate scattering
length a12 to a large value of ∼ −300a0, where a0 is theBohr radius [14]. At this field, electronic and nuclear
spins are mostly decoupled, and the two states experience approximately the same Zeeman shift. This greatly
reduces the effect of magnetic field fluctuations, although
the differential shift is still linear in B, unlike in [7].
We created a superposition of atoms in states |1i and|2i using a non-adiabatic RF sweep around the energy
splitting of 74MHz. As the sample decohered, efficientevaporative cooling set in, confirming a large elastic scattering length. After 1 second, we were left with a fully
decohered mixture at a mean density n = 5 × 1013 cm−3 .
The rate of the RF sweep was adjusted so that after decoherence and cooling, 80% of the atoms were in state |2i.
The mean-field interaction should thus have increased theenergy splitting of the two levels by hδν = δµ2 − δµ1 =V12 (n1 − n2 ) ≈ h × 10 kHz. Our experiments involving
a third state [14] have confirmed the presence of suchenergy shifts.
Rabi spectroscopy in the interacting binary mixture
was performed by applying 200µs RF pulses of different
frequencies, and recording the final populations in thetwo states by simultaneous absorption imaging (Fig. 1).
In order to eliminate the systematic uncertainty in thevalue of ν0 , we performed a second experiment with theroles of states |1i and |2i reversed. According to Eq. 2,
one would expect an opposite shift of the resonance.
Within our precision, no interaction shift of the resonance frequency was observed. Comparing the ex-pected difference in mean-field shifts for the two experiments, 2δν = 20 kHz, with the measured line separation of (34 ± 146) Hz, we arrive at an apparent value forg (2) = 0.002(7). This demonstrates the universal absenceof a resonance shift in a two-level Fermi gas, independentof the coherence in the system.
Population of State |2 After RF-Pulse [%]where n1 and n2 are the densities of particles in states|1i and |2i, and a12 is the interstate s-wave scattering
length. In a fully decohered cloud, we have a binary
mixture of two distinct species of atoms, with a meanfield energy density Eint = V12 n1 n2 . This interactionchanges the equilibrium energy level of atoms in state |1i(|2i) according to δµ1 (2) = V12 n2 (1) . The difference inequilibrium mean-field energy of the two states is then

80

60

40

20
-5
0
5
Frequency - 74,159.672 kHz

FIG. 1: Absence of mean-field shift of an RF transition in abinary Fermi system. The resonance curves were measured
for fully decohered 80%/20% two-state mixtures of fermions.
The measured frequency difference between the two lines is(34 ± 146) Hz, even though Eq. 2 would predict a splitting of
20 kHz.

Evidently, RF spectroscopy does not measure the expected difference in thermodynamic chemical potentials
for the two states. Experiments with thermal bosons
have posed a similar puzzle [7]. Here we explain that
this is a direct consequence of the coherent nature of the
RF excitation.
In Fig. 2, the average properties of the many-bodystate are described by the three coordinates of the spin1
repre1/2 Bloch vector m = mz êz + m⊥ : mz = n2 −n2sents the population difference in the two states, whereas
the transverse component m⊥ is a measure of the coherence in the system. In polar coordinates, the length of the
Bloch vector measures the purity of the state and hence
the entropy of the system. The polar angle θ encodes
the coherence of the sample. Fully decohered statisticalmixtures are represented by vectors with θ = 0, or equivalently m⊥ = 0, with state A being the special case ofa pure state. In Fig. 2a, state B is created by applying
an RF pulse on a pure sample A. In this case, there isno interaction energy in the system during the RF pulse,
and no frequency shift is expected [14]. State C is formed
through subsequent decoherence of state B. States B andC have the same number of particles in |1i and |2i, but
in C the mean-field has fully developed.
Our experiment is performed on a C-like state(Fig. 2b). Here we explain why Eq. 2 still does not give3
the correct resonance frequency for an infinitesimal transfer of atoms between |1i and |2i. The key point is thateven though the sample is fully decohered, the appliedRF pulse re-introduces coherence into the system. Let
us consider two fully decohered states, C and E. Eq. 2
correctly gives the energy of the transformation C→E.
However, these two states have different entropies, as indicated by Bloch vectors of different lengths. An RF
pulse is a unitary transformation of the system, and must
preserve entropy. The true effect of the RF pulse is thus
to change the relative populations of |1i and |2i by tilting the Bloch vector away from the z axis, into state D.
It is the energy of this transformation, C→D, that needs
to be calculated in order to find the correct resonant RF
frequency.
a)

b)

2
C

2

c)

B

C
E

B

C

dq D

D

q

E

A

1

1

A

FIG. 2: Bloch sphere representation of RF transitions. a) An
RF pulse rotates a pure state A into B. The superposition
state decoheres into a “ring” distribution, represented by its
average, C. b) A second RF pulse transforms the fully decohered state C into a partially coherent state D. The final
state E is reached only after further decoherence. c) Transfers
A→B and C→D are coherent and reversible. B→C and D→E
are irreversible.

In the case of fermions, we can prove very generallythat the resonance frequency will always be ν0 , by showing that the interaction hamiltonian is invariant underrotations of the Bloch vector. The interstate s-wave interaction is described by the second-quantized hamiltonian density
Hint = V12 ψ1† (r)ψ2† (r)ψ2 (r)ψ1 (r).

(3)

Under a general rotation, described by polar angles θ, φ,†transform according to:
the field operators ψ1,2
†
ψ1θ,φ

=

cos θ2 e−iφ/2 ψ1† + sin 2θ eiφ/2 ψ2††= − sin 2θ e−iφ/2 ψ1† + cos θ2 eiφ/2 ψ2†
ψ2θ,φ

(4)

Using the standard fermionic anticommutation relations(ψ1 ψ2 = −ψ2 ψ1 , ψ1 ψ1 = 0 etc. ), it is easy to show that:
θ,φ
Hint
=

=

†
†
(r)ψ2θ,φ (r)ψ1θ,φ (r)(r)ψ2θ,φV12 ψ1θ,φV12 ψ1† (r)ψ2† (r)ψ2 (r)ψ1 (r) = Hint

(5)

We therefore see that an RF-induced rotation on the
Bloch sphere commutes with the interaction hamiltonian,and hence does not change the energy of the many-bodystate. It is then obvious that the resonant frequency will
always be ν0 , independent of the coherence of the system.
We now present a more general calculation of themean-field frequency shifts, which holds for both
fermions and bosons. To reduce complexity and concentrate on the only controversial case of interstate interactions, we consider a fictitious boson with no intrastateinteractions (a11 = a22 = 0). The mean-field expectation
value of the hamiltonian in Eq. 3 is [16]Eint = hHint i = V12 (n1 n2 +  n12 n21 ),n12 n21g (2) = 1 + 
,
n1 n2

(6)

where n1 = hψ1† ψ1 i and n2 = hψ2† ψ2 i are the densities in the two states, we have introduced “coherences”
n12 = hψ1† ψ2 i and n21 = hψ2† ψ1 i, and  = ±1 forbosons/fermions. In a fully coherent sample n12 n21 =n1 n2 and g (2) = 1 + . As decoherence sets in, g (2) increases (decreases) from 0 (2) to 1 for fermions (bosons).
For the most general case of a partially decohered sample,
we can rewrite Eq. 6 in terms of the Bloch vector:
n1 = n2 − mz ,n12 = mx + i my ,n2 = n2 + mz
n21 = mx − i myn12 n21 = m2x + m2y = m2⊥ ,(7)

where n is the total particle density. This givesn2− V12 m2z + V12 m2⊥
4
n2
2
= V12
+ V12 |m| − (1 + ) V12 m2z .
4

Eint = V12(8)

Two states with same numbers of atoms in |1i and |2i,
but different levels of coherence, have the same mz , butdifferent |m⊥ | (e.g. states D and E in Fig. 2b). Again we
see that two such states indeed have different interactionenergies.
Now, let us evaluate the effect of coherence on the resonant RF frequency. A coherent RF excitation preserves
entropy (|m| = const.), and the total density n. In an
infinitesimal tilt of the Bloch vector, the density of atoms
transferred from |1i to |2i is dn2 = −dn1 = dmz . Therefore, the change of interaction energy per transferred particle, and thus the shift in the resonant frequency ∆ν,comes out to be∆ν =1 ∂Eint
1= (1 + )V12 (n1 − n2 ).
h ∂mz n, |m| h(9)In analogy with a spinning top which precesses in thegravitational field, the resonant frequency for an infinitesimal tilt of the Bloch vector is also equal to the frequency
of its free precession. In the traditional language ofatomic physics, this analogy just reiterates that Rabi [14]

4and Ramsey [7] spectroscopy measure the same characteristic frequency of the system. The striking result isthat in contrast to the interaction energy (Eqs. 6, 8), theprecession of the Bloch vector, or equivalently the RF
frequency shift (Eq. 9), does not dependent on the levelof coherence in the sample. Equation 9 explains both ourmeasurements with fermions, and the JILA experimentwith thermal bosons [7].

 2   
 

! 

2

θ
θ    "$#&% ')1 ( *&+-,

θ   θ     

./*$,-1*&+-,

FIG. 3: Mean-field represented as effective magnetic field.
a) Fermions: The exchange and direct interaction add up toform a magnetic field aligned with the average spin (V12 < 0in the drawing). The net torque vanishes and the Bloch vector m precesses at the unperturbed frequency ν0 . b) Bosons:
The exchange interaction has opposite sign than for fermions.
It exerts a torque on the average spin equal to the torqueinduced by the direct interaction, as can be seen by comparing the two cross products with m. The Bloch vector thus
precesses at ν0 plus twice the frequency shift due to directinteraction.

In order to further elucidate the role of coherences in
the precession of the Bloch vector, we employ the interpretation of the mean-field energy as the interaction of
the average spin with an effective magnetic field [12, 13].
Using Eq. 8, we obtain [17]1
Eint = const. − Beff · m
2
Beff = 2V12 (mz êz −  m⊥ ).

(10)

In this picture, the precession of the spin due to interactions is driven by the torque Beff × m. The magnetic
field along the z axis is induced by the direct interaction, and has the same sign for fermions and bosons(Fig. 3). The transverse magnetic field comes from theexchange interaction, and has opposite signs for fermionsand bosons. For fermions, Beff is parallel to m (Eq. 10)
and hence does not cause any precession. Equivalently,
the direct and exchange interaction exert torques equaland opposite to each other. For bosons, the two contributions add up to yield exactly twice the precession
frequency given by the direct interaction alone. During
decoherence, the exerted torque shrinks in proportion to
the decaying transverse spin. Therefore, the precession
frequency remains constant, no matter how small the coherences are.

In addition to the effects discussed here, interactionsin inhomogeneous samples can lead to spin waves [9, 10,
12, 18, 19] both in bosonic and fermionic gases.
In conclusion, we have demonstrated the absence of themean-field shift of RF transitions in a fully decohered, interacting binary mixture of fermions. This was explainedby proving the invariance of the interaction energy undercoherent Hilbert space rotations. This result is relevant
for the potential use of a fermionic atom supplying thefrequency standard in an atomic or optical clock, since
it implies a robust elimination of the systematic errors
due to density dependent frequency shifts. Previously,
the absence of such clock shifts was explained by theabsence of mean-field energy in a purely coherent superposition state [14]. Now we have shown that there is nospectroscopic shift even after decoherence has led to measurable mean-field energies. Further, we have presented
a simple theoretical framework for calculating the precession frequency of the Bloch vector which describes anarbitrary spin state of either fermions or bosons. This
resolves “The Mystery of the Ramsey Fringe that Didn’t
Chirp” [8] with a simple and intuitive picture.
We thank Claudiu Stan and Christian Schunck for experimental assistance, and Michele Saba for critical reading of the manuscript. This work was supported by the
NSF, ONR, ARO, and NASA.

[1] W. Ketterle and H.-J. Miesner, Phys. Rev. A 56, 3291
(1997).
[2] D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis,
S. C. Moss, D. Kleppner, and T. J. Greytak, Phys. Rev.
Lett. 81, 3811 (1998).
[3] R. Côté and V. Kharchenko, Phys. Rev. Lett. 83, 2100
(1999).
[4] M. Ö. Oktel and L. S. Levitov, Phys. Rev. Lett. 83, 6(1999).
[5] M. Ö. Oktel, T. C. Killian, D. Kleppner, and L. S. Levitov, Phys. Rev. A 65, 033617 (2002).
[6] C. J. Pethick and H. T. C. Stoof, Phys. Rev. A 64, 013618(2001).
[7] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, andE. A. Cornell, Phys. Rev. A 66, 053616 (2002).
[8] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, andE. A. Cornell, in Proceedings of the XVIII International
Conference on Atomic Physics, edited by H. R. Sadeghpour, E. J. Heller, and D. E. Pritchard (World Scientific,
Cambridge, Massachusetts, USA, 2003), pp. 3–10.
[9] M. Ö. Oktel and L. S. Levitov, Phys. Rev. Lett. 88,
230403 (2002).
[10] J. E. Williams, T. Nikuni, and C. W. Clark, Phys. Rev.
Lett. 88, 230405 (2002).
[11] A. S. Bradley and C. W. Gardiner, J. Phys. B: At. Mol.
Opt. Phys. 35, 4299 (2002).
[12] J. N. Fuchs, D. M. Gangardt, and F. Laloë, Phys. Rev.
Lett. 88, 230404 (2002).
[13] J. N. Fuchs, D. M. Gangardt, and F. Laloë, cond-5
mat/0304335 (2003).
[14] S. Gupta, Z. Hadzibabic, M. W. Zwierlein, C. A. Stan,
K. Dieckmann, C. H. Schunck, E. G. M. van Kempen,
B. J. Verhaar, and W. Ketterle, Science 300, 1723 (2003).
[15] Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck,
M. W. Zwierlein, K. Dieckmann, and W. Ketterle, condmat/0306050 (2003).
[16] This result can be derived using Wick’s Theorem [20].
Abnormal averages like hψ1 ψ1 i do not contribute in thecase of fermions or thermal bosons.

[17] The factor of 1/2 reflects the self-interaction.
[18] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, andE. A. Cornell, Phys. Rev. Lett. 88, 070403 (2002).
[19] J. M. McGuirk, H. J. Lewandowski, D. M. Harber,
T. Nikuni, J. E. Williams, and E. A. Cornell, Phys. Rev.
Lett. 89, 090402 (2002).
[20] A. L. Fetter and J. D. Walecka, Quantum Theory of
Many-Particle Systems (McGraw-Hill, New York, 1971).

Bibliography[1] 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).
[2] Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, “Fifty-fold improvement in the number of quantum degenerate fermionic atoms,” cond-mat/0306050 (2003).
[3] 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).
[4] S. Gupta, Z. Hadzibabic, J. R. Anglin, and W. Ketterle, “Collisions in zero temperature Fermi gases,” cond-mat/0307088 (2003).
[5] S. Gupta, Z. Hadzibabic, M. W. Zwierlein, C. A. Stan, K. Dieckmann, C. H. Schunck,
E. G. M. van Kempen, B. J. Verhaar, and W. Ketterle, “Radio-Frequency Spectroscopy of Ultracold Fermions,” Science 300, 1723 (2003).
[6] M. W. Zwierlein, Z. Hadzibabic, S. Gupta, and W. Ketterle, “Spectroscopic insensitivity to cold collisions in a two-state mixture of fermions,” cond-mat/0306627 (2003).
[7] D. M. Stamper-Kurn, Ph.D. thesis, Massachusetts Institute of Technology, 1999.
[8] C. H. Schunck, Diploma Thesis, University of Heidelberg 2002 .
[9] S. Gupta, Ph.D. thesis, Massachusetts Institute of Technology, 2003.
[10] 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).
[11] C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, “Evidence of Bose-Einstein
Condensation in an Atomic Gas with Attractive Interactions,” Phys. Rev. Lett. 75,
1687 (1995).
[12] 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).
[13] E. A. Cornell and C. E. Wieman, “Nobel Lecture: Bose-Einstein condensation in adilute gas, the first 70 years and some recent experiments,” Rev. Mod. Phys. 74, 875(2002).
136

[14] W. Ketterle, “Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser,” Rev. Mod. Phys. 74, 1131 (2002).
[15] B. DeMarco and D. S. Jin, “Onset of Fermi Degeneracy in a Trapped Atomic Gas,”
Science 285, 1703 (1999).
[16] 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).
[17] 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).
[18] 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).
[19] S. R. Granade, M. E. Gehm, K. M. O’Hara, and J. E. Thomas, “Preparation of adegenerate, two-component fermi gas by evaporation in a single beam optical trap,”
Phys. Rev. Lett. 88, 120405 (2002).
[20] L. N. Cooper, “Bound Electron Pairs in a Degenerate Fermi Gas,” Phys. Rev. 104,1189 (1956).
[21] H. Heiselberg, “Bosons and Fermions near Feshbach resonances,” cond-mat/0307726(2003).
[22] T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, “Resonant control ofelastic collisions in an optically trapped Fermi gas of atoms,” Phys. Rev. Lett. 88,
173201 (2002).
[23] K. M. O’Hara, S. L. Hemmer, S. R. Granade, M. E. Gehm, J. E. Thomas, V. Venturi,
E. Tiesinga, and C. J. Williams, “Measurement of the zero crossing in a Feshbachresonance of fermionic 6 Li,” Phys. Rev. A 66, 041401 (2002).
[24] 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).
[25] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas,“Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms,” Science
298, 2179 (2002).
[26] M. E. Gehm, S. L. Hemmer, S. R. Granade, K. M. O’Hara, and J. E. Thomas,“Mechanical Stability of a Strongly-Interacting Fermi Gas of Atoms,” Phys. Rev. A
68, 011401(R) (2003).
[27] C. A. Regal and D. S. Jin, “Measurement of Positive and Negative Scattering Lengths
in a Fermi Gas of Atoms,” Phys. Rev. Lett. 90, 230404 (2003).
[28] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalhães, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, “Measurement of the Interaction Energy
near a Feshbach Resonance in a 6 Li Fermi Gas,” Phys. Rev. Lett. 91, 020402 (2003).

137[29] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, “Creation of ultracold moleculesfrom a Fermi Gas of Atoms,” Nature 424, 47 (2003).
[30] 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,”
cond-mat/0308018 (2003).
[31] A. J. Leggett, “Cooper Pairing in Spin-Polarized Fermi Systems,” Journal de Physique41, C7 (1980).
[32] H. T. C. Stoof, M. Houbiers, C. A. Sackett, and R. G. Hulet, “Superfluidity of spinpolarized 6 Li,” Phys. Rev. Lett. 76, 10 (1996).
[33] M. Houbiers, R. Ferwenda, H. T. C. Stoof, W. McAlexander, C. A. Sackett, and R. G.
Hulet, “Superfluid state of atomic 6 Li in a magnetic trap,” Phys. Rev. A 56, 4864
(1997).
[34] M. Houbiers and H. T. C. Stoof, “Cooper-pair formation in trapped atomic Fermi
gases,” Phys. Rev. A. 59, 1556 (1999).
[35] M. A. Baranov and D. S. Petrov, “Critical temperature and Ginzburg-Landau equation for a trapped Fermi gas,” Phys. Rev. A 58, R801 (1998).
[36] R. Combescot, “Trapped 6 Li: A High Tc Superfluid?,” Phys. Rev. Lett. 83, 3766(1999).
[37] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, “ResonanceSuperfluidity in a Quantum Degenarate Fermi Gas,” Phys. Rev. Lett. 87, 120406(2001).
[38] J. N. Milstein, S. J. J. M. F. Kokkelmans, and M. J. Holland, “Resonance theory of thecrossover from Bardee-Cooper-Schrieffer superfluidity to Bose-Einstein condensation
in a dilute fermi gas,” Phys. Rev. A 66, 043604 (2002).
[39] S. J. J. M. F. Kokkelmans, J. N. Milstein, M. L. Chiofalo, R. Walser, and M. J.
Holland, “Resonance superfluidity: Renormalization of resonance scattering theory,”
Phys. Rev. A 65, 053617 (2002).
[40] E. Timmermans, K. Furuya, P. W. Milonni, and A. K. Kerman, “Prospects of creating
a composite Fermi-Bose superfluid,” Phys. Lett. A 285, 228 (2001).
[41] W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin, “High-temperature
superfluidity of fermionic atoms in optical lattices,” Phys. Rev. Lett. 89, 220407(2002).
[42] Y. Ohashi and A. Griffin, “BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance,” Phys. Rev. Lett. 89, 130402 (2002).
[43] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag,
New York, 1999).

138

[44] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, in Bose-Einstein condensation in
atomic gases, Proceedings of the International School of Physics Enrico Fermi, Course
CXL, edited by M. Inguscio, S. Stringari, and C. Wieman (IOS Press, Amsterdam,
1999), pp. 67–176.
[45] E. Timmermans, “Degenerate Fermion Gas Heating by Hole Creation,” Phys. Rev.
Lett. 87, 240403 (2001).
[46] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of Vortex
Lattices in Bose-Einstein Condensates,” Science 292, 476 (2001).
[47] M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and
W. Ketterle, “Bose-Einstein Condensaion in a Tighly Confining dc Magnetic Trap,”
Phys. Rev. Lett. 77, 416 (1996).
[48] W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High Densities
of Cold Atoms in a Dark Spontaneous Force Optical Trap,” Phys. Rev. Lett. 70, 2253
(1993).
[49] U. Schünemann, H. Engler, M. Zielonowski, M. Weidemüller, and R. Grimm,“Magneto-optic trapping of lithium using semiconductor lasers,” Optics Comm. 158,
263 (1998).
[50] L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic,
W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system foratomic physics,” Opt. Commun. 117, 541 (1995).
[51] V. Wippel, C. Binder, and L. Windholz, “Cross-section for collisions of ultracold Li-7
with Na,” Eur. Phys. J. D 21, 101 (2002).
[52] G. D. Telles, W. Garcia, L. G. Marcassa, V. S. Bagnato, D. Ciampini, M. Fazzi,
J. H. Müller, D. Wilkowski, and E. Arimondo, “Trap loss in a two-species Rb-Cs
magneto-optical trap,” Phys. Rev. A 63, 033406 (2001).
[53] D. E. Pritchard, “Cooling Neutral Atoms in a Magnetic Trap for Precision Spectroscopy,” Phys. Rev. Lett. 51, 1336 (1983).
[54] M. Houbiers, H. T. C. Stoof, W. I. McAlexander, and R. G. Hulet, “Elastic andinelastic collisions of 6 Li atoms in magnetic and optical traps,” Phys. Rev. A 57,
R1497 (1998).
[55] G. Breit and I. I. Rabi, “Measurement of the nuclear spin,” Phys. Rev. 38, 2082
(1931).
[56] D. A. Butts and D. S. Rokhsar, “Trapped Fermi gases,” Phys. Rev. A 55, 4346 (1997).
[57] F. Schreck, Ph.D. thesis, Université Paris VI, 2002.
[58] A. Görlitz, T. L. Gustavson, A. E. Leanhardt, R. Löw, A. P. Chikkatur, S. Gupta, S.
Inouye, D. E. Pritchard, and W. Ketterle, “Sodium Bose-Einstein Condensates in the
F = 2 State in a Large-Volume Optical Trap,” Phys. Rev. Lett. 90, 090401 (2003).

139

[59] M. B. S. Jochim, G. Hendl, J. H. Denschlag, R. Grimm, A. Mosk, and M. Weidemüller,“Magnetic Field Control of Elastic Scattering in a Cold Gas of Fermionic Lithium
Atoms,” Phys. Rev. Lett. 89, 273202 (2002).
[60] J. Dalibard, in Bose-Einstein condensation in atomic gases, Proceedings of the International School of Physics Enrico Fermi, Course CXL, edited by M. Inguscio, S.
Stringari, and C. Wieman (IOS Press, Amsterdam, 1999), pp. 321–350.
[61] D. J. Heinzen, in Bose-Einstein condensation in atomic gases, Proceedings of theInternational School of Physics Enrico Fermi, Course CXL, edited by M. Inguscio, S.
Stringari, and C. Wieman (IOS Press, Amsterdam, 1999), pp. 351–390.
[62] C. J. Joachin, Quantum Collision Theory (North Holland, Amsterdam, 1983).
[63] K. Huang, Statistical Mechanics (Wiley, New York, 1987).
[64] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Massachusetts,
1994).
[65] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge
University Press, Cambridge, 2002).
[66] H. A. Feshbach, “A unified theory of nuclear reactions,” Annals of Physics 19, 287(1962).
[67] 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).
[68] 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).
[69] J. L. Roberts, N. R. Claussen, J. B. Jr., C. H. Greene, E. A. Cornell, and C. E.
Wieman, “Resonant magnetic field control of elastic scattering in cold 87 Rb,” Phys.
Rev. Lett. 81, 5109 (1998).
[70] A. J. Kerman, V. Vuletić, C. Chin, and S. Chu, “Observation of Low-Field Feshbach
Resonances in Collisions of Cesium Atoms,” Phys. Rev. Lett. 82, 1406 (1999).
[71] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman,“Stable 85 Rb Bose-Einstein Condensates with Widely Tunable Interactions,” Phys.
Rev. Lett. 85, 1795 (2000).
[72] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E.
Wieman, “Dynamics of collapsing and exploding Bose-Einstein condensates,” Nature
89, 203201 (2002).
[73] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation andpropagation of matter-wave soliton trains,” Nature 417, 150 (2002).
[74] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y.
Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296,
1290 (2002).
140

[75] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, “Bose-Einstein condensation of cesium,” Science 299, 232 (2003).
[76] B. DeMarco, J. L. Bohn, J. J. P. Burke, M. Holland, and D. S. Jin, “Measurementof p-Wave Threshold Law Using Evaporatively Cooled Fermionic Atoms,” Phys. Rev.
Lett. 82, 4208 (1999).
[77] A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, “Resonances in ultra-cold collisionsof 6 Li, 7 Li and 23 Na,” Phys. Rev. A 51, 4852 (1995).
[78] S. Dürr, T. Volz, A. Marte, and G. Rempe, “Observation of molecules produced froma Bose-Einstein condensate,” cond-mat/0307440 (2003).
[79] J. Bardeen, L. N. Cooper, and J. R. Schreiffer, “Theory of Superconductivity,” Phys.
Rev. 108, 1175 (1957).
[80] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner,
J. Stenger, and W. Ketterle, “Optical confinement of a Bose-Einstein condensate,”
Phys. Rev. Lett. 80, 2027 (1998).
[81] C. Menotti, P. Pedri, and S. Stringari, “Expansion of an Interacting Fermi Gas,”
Phys. Rev. Lett. 89, 250402 (2002).
[82] 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).
[83] J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E. Wieman, “Magnetic Field
Dependence of Ultracold Inelastic Collisions near a Feshbach Resonance,” Phys. Rev.
Lett. 85, 728 (2000).
[84] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, “Three-body recombination at large scattering lengths in an ultracold atomic gas,” physics/0304052 (2003).
[85] P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, “Three-body recombinationof ultracold atoms to a weakly bound s level,” Phys. Rev. Lett. 77, 2921 (1996).
[86] D. S. Petrov, “Three-body problem in Fermi gases with short-range interparticle interaction,” Phys. Rev. A 67, 010703(R) (2003).
[87] B. D. Esry, C. H. Greene, and H. Suno, “Threshold laws for three-body recombination,” Phys. Rev. A. 65, 010705(R) (2002).
[88] R. Combescot, “Feshbach resonance in dense ultracold Fermi gases,” condmat/0302209 (2003).
[89] R. Combescot, “Shift of the molecular bound state threshold in dense ultracold Fermi
gases with Feshbach resonance,” New J. Phys. 5, 86 (2003).
[90] G. M. Bruun and C. J. Pethick, “An effective theory of Feshbach resonances andmany-body properties of Fermi gases,” cond-mat/0304535 (2003).
[91] T.-L. Ho and E. J. Mueller, “The Unitary Quantum Gas,” cond-mat/0306187 (2003).
141[92] Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, “Evolution of a Bose gas in anisotropic
time-dependent traps,” Phys. Rev. A 55, R18 (1997).
[93] P. Pedri, D. Guéry-Odelin, and S. Stringari, “Dynamics of a classical gas including
dissipative and mean field effects,” arXiv:cond-mat/0305624 (2003).
[94] I. Shvarchuck, C. Buggle, D. S. Petrov, K. Dieckmann, M. Zielonkowski, M. Kemmann, T. Tiecke, W. V. Klitzing, G. V. Shlyapnikov, and J. T. M. Walraven, “BoseEinstein Condensation into Nonequilibrium States Studied by Condensate Focusing,”
Phys. Rev. Lett. 89, 270404 (2002).
[95] D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews, and W. Ketterle,“Collisionless and hydrodynamic excitations of a Bose-Einstein condensate,” Phys.
Rev. Lett. 81, 500 (1998).
[96] M. Leduc, J. Leonard, F. P. dos Santos, E. Jahier, S. Schwartz, and C. CohenTannoudji, “Hydrodynamic modes in a trapped gas of metastable helium above theBose-Einstein transition,” Acta Phys. Pol. B 33, 2213 (2002).
[97] S. D. Gensemer and D. S. Jin, “Transition from collisionless to hydrodynamic behavior
in an ultracold Fermi gas,” Phys. Rev. Lett. 87, 173201 (2001).
[98] B. DeMarco and D. S. Jin, “Spin Excitations in a Fermi Gas of Atoms,” Phys. Rev.
Lett. 88, 040405 (2002).
[99] S. Stringari, “Collective excitations of a trapped Bose-condensed gas,” Phys. Rev.
Lett. 77, 2360 (1996).
[100] A. Minguzzi and M. P. Tosi, “Scissors mode in a superfluid Fermi gas,” Phys. Rev. A
63, 023609 (2001).
[101] F. Zambelli and S. Stringari, “Moment of inertia and quadrupole response functionof a trapped superfluid,” Phys. Rev. A 63, 033602 (2001).
[102] M. E. Gehm, S. L. Hemmer, K. M. O’Hara, and J. E. Thomas, “Unitarity-limited
Elastic Collision Rate in a Harmonically-Trapped Fermi Gas,” Phys. Rev. A 68,
011603(R) (2003).
[103] W. Geist, A. Idrizbegovic, M. Marinescu, T. A. B. Kennedy, and L. You, “Evaporative
cooling of trapped fermionic atoms,” Phys. Rev. A. 61, 013406 (1999).
[104] M. Holland, B. DeMarco, and D. S. Jin, “Evaporative cooling of a two-component
degenerate Fermi gas,” Phys. Rev. A. 61, 053610 (2000).
[105] W. Geist and T. A. B. Kennedy, “Evaporative cooling of mixed atomic fermions,”
Phys. Rev. A. 65, 063617 (2002).
[106] G. Ferrari, “Collisional relaxation in a fermionic gas,” Phys. Rev. A. 59, R4125 (1999).
[107] B. DeMarco, S. B. Papp, and D. S. Jin, “Pauli Blocking of Collisions in a Quantum
Degenerate Atomic Fermi Gas,” Phys. Rev. Lett. 86, 5409 (2001).
[108] L. Vichi, “Collisional Damping of the Collective Oscillations of a Trapped Fermi Gas,”
J. Low Temp. Phys. 121, 177 (2000).
142

[109] K. Gibble and S. Chu, “Laser-Cooled Cs Frequency Standard and a Measurement of
the Frequency Shift due to Ultracold Collisions,” Phys. Rev. Lett. 70, 1771 (1993).
[110] C. Fertig and K. Gibble, “Measurement and Cancellation of the Cold Collision Frequency Shift in an 87 Rb Fountain Clock,” Phys. Rev. Lett. 85, 1622 (2000).
[111] Y. Sortais, S. Bize, M. Abgrall, S. Zhang, C. Nicolas, C. Mandache, P. Lemonde,
P. Laurent, G. Santarelli, N. Dimarcq, P. Petit, A. Clairon, A. Mann, A. Luiten, S.
Chang, and C. Salomon, “Cold atom clocks,” Physica Scripta T95, 50 (2001).
[112] K. Gibble and B. J. Verhaar, “Eliminating cold-collision frequency shifts,” Phys. Rev.
A. 52, 3370 (1995).
[113] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, “Effect of cold
collisions on spin coherence and resonance shifts in a magnetically trapped ultracoldgas,” Phys. Rev. A 66, 053616 (2002).
[114] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, in Proceedings ofthe XVIII International Conference on Atomic Physics, edited by H. R. Sadeghpour,
E. J. Heller, and D. E. Pritchard (World Scientific, Cambridge, Massachusetts, USA,
2003), pp. 3–10.
[115] M. O. Oktel and L. S. Levitov, “Internal Waves and Synchronized Precession in a
Cold Vapor,” Phys. Rev. Lett. 88, 230403 (2002).
[116] J. E. Williams, T. Nikuni, and C. W. Clark, “Collisional frequency shifts of absorption
lines in an atomic hydrogen gas,” Phys. Rev. Lett. 88, 230405 (2002).
[117] A. S. Bradley and C. W. Gardiner, “Theory of Ramsey spectroscopy and anomalous
segregation in ultracold rubidium,” J. Phys. B: At. Mol. Opt. Phys. 35, 4299 (2002).
[118] J. N. Fuchs, D. M. Gangardt, and F. Lalo, “Internal State Conversion in Ultracold
Gases,” Phys. Rev. Lett. 88, 230404 (2002).
[119] J. N. Fuchs, D. M. Gangardt, and F. Lalo, “Large amplitude spin waves in ultra-coldgases,” cond-mat/0304335 (2003).
[120] E. G. M. van Kempen and et. al., “‘Scattering Calculations for Lithium’,” in preparation (2003).
[121] J. V. Steele, “Effective Field Theory Power Counting at Finite Density,” arXiv:nuclth/0010066 (2000).
[122] H. Heiselberg, “Fermi Systems with long scattering lengths,” Phys. Rev. A. 63, 043606(2001).
[123] P. Torma and P. Zoller, “Laser Probing of Atomic Cooper Pairs,” Phys. Rev. Lett.
85, 487 (2000).
[124] G. Bruun, P. Torma, M. Rodriguez, and P. Zoller, “Laser probing of Cooper-paired
trapped atoms,” Phys. Rev. A. 64, 033609 (2001).
[125] K. Mølmer, “Bose Condensates and Fermi Gases at Zero Temperature,” Phys. Rev.
Lett. 80, 1804 (1998).
143

[126] G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, “Collapseof a degenerate Fermi gas,” Science 68, 011601 (2002).
[127] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, “Quantum phasetransition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature
415, 39 (2002).
[128] C. E. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci.
Instrum. 62, 1 (1991).
[129] A. E. Siegman, Lasers (University Scienece Books, Sausalito, 1986).
[130] E. Hecht, Optics (Addison-Wesley, Reading, 1989).
[131] K. G. Libbrecht, R. A. Boyd, P. A. Willems, T. L. Gustavson, and D. K. Kim,
“Teaching Physics with 670nm Diode-lasers - Construction of Stabilized Lasers andLithium Cells,” Am. J. Phys. 63, 729 (1995).

144