Experiments with Degenerate Bose and Fermi Gases
by

Subhadeep Gupta
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 29, 2003

Certiﬁed by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Pritchard
Cecil and Ida Green Professor of Physics
Thesis Supervisor
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

Experiments with Degenerate Bose and Fermi Gases
by
Subhadeep Gupta
Submitted to the Department of Physics
on August 29, 2003, in partial fulﬁllment of therequirements for the degree ofDoctor of Philosophy

Abstract
Two sets of studies are described in this thesis. In the ﬁrst set, an atom interferometry
technique was developed for the measurement of the ﬁne structure constant using a BoseEinstein Condensate. In the second set, degenerate Fermi gases were prepared and theirproperties explored in a regime of strong interactions.
We have developed an atom interferometer which is capable of measuring with highprecision the “photon recoil frequency” (ωrec ). ωrec corresponds to the kinetic energy of an
atom recoiling due to absorption of a photon. ωrec can be used to determine the quantity
h/matom and the ﬁne structure constant, α. A preliminary measurement using a 23 Na BoseEinstein Condensate yielded ωrec with a precision of 7 × 10−6 which deviated by 2 × 10−4
from the currently accepted value. Plausible upgrades to the apparatus should produce a
precision of 10−9 which would bring within reach a measurement of h/matom and α in the
10−9 range. Such accuracy would be of considerable scientiﬁc and metrological import.
A quantum degenerate gas of 6 Li fermions was produced by sympathetic cooling with
23 Na bosons in a two-species atom trapping apparatus. The cooling strategy was optimized
to enable production of fermions with atom numbers up to 7 × 107 at half the Fermi
temperature (TF ), or temperatures down to 0.05 TF ∼ 100 nK with 3 × 107 atoms. We canalso produce degenerate Bose-Fermi mixtures with several million atoms in each species.
We studied the behavior of mixtures of fermi gases in regimes of strong interactions near
“Feshbach” resonances. A study of system stability enabled the experimental observationof two such Feshbach resonances.
We carried out a theoretical study to interpret the observation (by many experimentalgroups) of hydrodynamic behavior of fermi gases during expansion out of an atom trap in astrongly-interacting regime. The study concluded that this behavior was not a qualitative
signature of fermionic superﬂuidity and could arise from classical collisions.
Finally, radio-frequency (RF) spectroscopy was used to probe interaction in 6 Li. We
demonstrated the absence of mean ﬁeld “clock” shifts of RF transitions in a two-(spin)state
fermion system. Using a three-state system, we measured the interaction strength between
diﬀerent spin states. The measurements near Feshbach resonances indicate a saturationof the interaction strength at a large negative value. This result is of relevance in thecontinuing quest for fermionic superﬂuidity in atomic gases.
Thesis Supervisor: David E. Pritchard
Title: Cecil and Ida Green Professor of Physics
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics
2

To Ma, Baba,

and my brother Rajdeep.

3

Acknowledgments
There are many individuals I wish to thank for making this work possible. This oﬃcial
forum for acknowledgements stands only as a small token of the gratitude I feel towards
them all.
I was attracted to atomic physics and MIT by Dave Pritchard. I owe him a huge thankyou for providing me with the opportunity to be a part of the atomic physics family on thesecond ﬂoor(s) of Building 26 (and 36). His enthusiasm and love for all aspects of physics
has always been inspiring to me. Dave’s hands-oﬀ attitude has allowed me to develop as aphysicist at my own pace. He has also provided a good ratio of encouragement and criticism
which kept me excited and challenged over the last several years.
I have been extremely fortunate to have had Wolfgang Ketterle as a co-advisor. Wolfgang’s skill at simultaneously grasping the nuts and bolts as well as the big picture of a
research endeavor has always been a key element for his group’s success. Even with theincreased burden of multiple research groups, he continues to perform this feat every day.
Just about every conversation I have had with him left me feeling excited and ﬁred up
about the task at hand as well as how it related to the overall goal we were aiming for. It
has been a privilege and a pleasure to work with him and learn from him.
I want to single out and thank the people I have worked most closely with over the last
few years. Their eﬀorts formed key elements in all the work presented in this thesis.
Zoran Hadzibabic has been much more than a colleague and a friend to me over thelast two-three years. Zoran and I were on the same time track at MIT and defended our
PhDs within a day of each other. Zoran’s infectious friendliness, his passion for science andhis obsession with understanding have been instrumental in making my life at MIT joyful
and productive. We spent many memorable nights wrestling the machine, fuelled by The
Minutemen and by the never-stale references to The Big Lebowski. In addition, Zoran’s
deep understanding of physics provided a fantastic resource to bounce ideas back and forth
several times a day.
Claudiu has set high “Stan”-dards of technical excellence in the lab from the very beginning. If someone wants to know the correct way to build some experimental component,
he can provide. His attention to details as well as aesthetics while building seminal piecesof the machine have made the lab more reliable as well as easier on the eye!
Martin Zwierlein’s amazing scientiﬁc capabilities are enhanced by his irrepressible enthusiasm towards everything in lab (and elsewhere). I have thoroughly enjoyed working
with him and learning from him during various projects. In him, I also found a partner for
the frequent good(?) pun.
Another amazing scientist and the newest addition to the Lithium team, Christian
Schunck, has become an eﬃcient and dependable lab companion in a very short time. I
have enjoyed good times with him from drilling holes in plexiglass to ﬁguring out the best
way to pack a lunch buﬀet take-out box. The Lithium lab now remains in the solid hands
4

of Claudiu, Martin and Christian. I have learned from all of you and will miss working withyou.
I thank post-doc Kai Dieckmann for all his eﬀorts towards ensuring the success of theatom interferometer project. His prior experience at building a BEC machine was alsobrought to bear on the building of the double-species sodium-lithium apparatus.
I now turn to thanking the people I learned from during my initial stages at MIT.
I ﬁrst joined Dave Pritchard’s beam atom interferometer group and learned about AMO
research from David Kokorowski, Tony Roberts, Rich Rubenstein, Al-Amin Dhirani and Ed
Smith. In particular, David introduced me to various facets of laboratory life including alove-hate relationship with dye lasers.
As Dave Pritchard’s interests changed towards ultracold-atom interferometry, I became a“cross-over” student in the Ketterle/Pritchard BEC collaboration. I learned about ultracold
atom research from Dan Stamper-Kurn, Shin Inouye, Ananth Chikkatur and Axel Görlitz.
Shin was the person I worked most closely with during my early period in the BEC I group.
Shin’s superb scientiﬁc skills and calm presence provided a terriﬁc example to learn
from. In addition, I have to thank him for providing copious quantities of cookies duringlate night runs as well as tasty Miso soup on more social occasions. I also had the good
fortune of collaborating closely with Tilman Pfau, Robert Löw and Till Rosenband. Eachan extraordinary individual, they provided physical insights, hard work and technical innovations which pushed the research forward. I also learned a great deal from interactions
with the BEC II group: Chandra Raman, Roberto Onofrio, Dallin Durfee, Chris Kuklewicz,
and Michael Köhl.
During the middle part of my graduate career, a major goal of the sodium BEC I
experiment was the building of BEC III - the “science chamber” machine. I have fondmemories of working with Todd Gustavson, Ananth Chikkatur, Aaron Leanhardt and Axel
Görlitz, during this phase. The expertise and work ethics which the post-docs Todd andAxel brought into the group sped everything up in the lab and also provided models toaspire towards. Ananth and I not only collaborated on building pieces of the new apparatus,
but also went through the ﬁrst-year problem sets together, and then shared an apartment
for three years. Our friendship has spanned DJ Shadow-inspired earth-compensation coilwinding to spiked-chai-enhanced drumming.
From the early days of winding a Zeeman slower (“it’ll only take a few hours”), Aaron
Leanhardt has remained a trusted collaborator and friend. His wide ranging physics skills
have been an invaluable asset to the whole group. Aaron has also been a staunch partnerin-crime to Jamil Abo-Shaeer - together, they have organized a steady stream of memorable
gatherings which tied the whole group together in a very festive manner. Jamil’s cheerfulspirit and resounding hallway high-5’s have become an integral part of the group. I will
also fondly remember a fun-ﬁlled conference trip to Kissimmee, Florida with Jamil, Aaron,
Zoran, Gretchen Campbell, Micah Boyd, and Jit Kee Chin. The high-spirits shared will5not easily slide from memory.
Huge thank you’s to various people in the AMO community for helpful advice, miscellaneous favors, loan of equipment, computer expertise and uplifting conversations at
various times: Mike Bradley, Alex Cronin, Joel deVries, Yong-Il Shin, Cort Johnson, David
Kielpinski, Jana Lehner, Lia Matos, Dan Miller, Takashi Mukaiyama, Tom Pasquini, Simon
Rainville, Michelle Saba, Andre Schirotzek, Dominik Schneble, Julia Steinberger, Jeﬀ Steinhauer, Erik Streed, James Thompson, Yoshio Torii, Kendra Vant, Mukund Vengalatorre,
Johnny Vogels, and Kaiwen Xu. It has been great to get to know all of you.
Thanks to Dr. James Anglin for explaining subtleties of hydrodynamics to me as wellas his continual interest in our work. I also thank Prof. Winn Smith with whom I had thepleasure to collaborate on several papers. Thanks also to Prof. Daniel Kleppner who was
my academic advisor during my time at MIT and also served on my thesis committee.
Carol Costa’s presence in the CUA hallway has been calming and her pleasant andeﬃcient dissemination of information has always served to quickly simplify bureaucratic
procedures at MIT. The entire RLE support staﬀ deserve thanks for providing a shadyadministrative umbrella - in particular I will point out Maxine Samuels, Gerry Power andAl McGurl who have helped a lot at critical times.
Life at MIT has to be balanced with life outside MIT. For this I have to thank several
friends around town. In particular, Zeenat Potia has been a frequent source of good cheer
and good food. Kinger Mallik’s chilling nature, skills at Scrabble, and (shared) love foranything Austin Powers, formed the basis of several entertaining evenings. Thanks to
Pradeep “The Champ” Sarin for his generosity with home built electronics as well as thefrequent use of his car for driving practice. Pradeep was also my apartment-mate forseveral years as was Augustine Urbas. I enjoyed many post 3 a.m. dinners with Pradeep
and Augustine at (the now sadly extinct) Deli-Haus. Several late-night experimental runs
found a tasteful conclusion in Virtua Fighter battles with Augustine. Thanks also to Sanjoy
and Doel Ghosh, Nick Gould and Adrian Quackenbush for the occasional food and wine.
I especially thank Munira Khalil for always being there for me and supporting me. Her
companionship is a constant source of joy and strength. I hope that her grad school life
was as much enriched by me as mine was by her.
Finally, I would like to thank my family. My parents Bhaskar and Ruby Gupta haveconsistently provided encouragement and support during my studies. Their love and blessings continue to energize me as I move on in the world. My brother Rajdeep has inspiredme from childhood and was a key factor in my decision to study abroad. Thanks to you all.
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

12

1.1

Degenerate Atomic Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.2

Diﬀerent Quantum Matters . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3

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

14

2 Atom Interferometry for h/m and α

16

2.1

Atom Interferometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2

Manipulation of Atoms with Standing Waves of Light . . . . . . . . . . . .

18

2.2.1

Bragg Scattering, Ref. 2 . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2.2

Kapitza-Dirac Scattering, Ref. 2 . . . . . . . . . . . . . . . . . . . .

25

2.3

A simple atom interferometer - the Mach-Zehnder conﬁguration . . . . . . .

26

2.4

Atom Optics with Bose-Einstein Condensates . . . . . . . . . . . . . . . . .

29

2.4.1

Standing Wave Diﬀraction of sodium Bose-Einstein Condensates . .

30

2.4.2

BEC Mach-Zehnder interferometer . . . . . . . . . . . . . . . . . . .

32

First Generation Recoil Measurement . . . . . . . . . . . . . . . . . . . . .

33

2.5.1

Motivation for measuring α . . . . . . . . . . . . . . . . . . . . . . .

33

2.5.2

Scheme of Contrast Interferometer . . . . . . . . . . . . . . . . . . .

34

2.5.3

Advantages of the contrast interferometer . . . . . . . . . . . . . . .

37

2.5.4

Contrast Interferometer Signal . . . . . . . . . . . . . . . . . . . . .

39

2.5.5

Measurement of the Recoil Frequency . . . . . . . . . . . . . . . . .

42

2.5.6

Insensitivity to Vibrations . . . . . . . . . . . . . . . . . . . . . . . .

43

2.5.7

N 2 scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.6.1

Scaling up the Contrast Interferometer - improving the precision . .

45

2.6.2

The Mean Field Contribution . . . . . . . . . . . . . . . . . . . . . .

46

2.5

2.6

3 Interacting Fermi Gases

48

3.1

Non-interacting Fermi Gases. . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.2

Interactions in Trapped Fermi Gases . . . . . . . . . . . . . . . . . . . . . .

50

3.2.1

Collision Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.2.2

Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

7

3.3

3.2.3

Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.2.4

Observations of Interactions . . . . . . . . . . . . . . . . . . . . . . .

55

Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.3.1

57

Strongly Interacting Fermi Gases . . . . . . . . . . . . . . . . . . . .

3.4

Ground State of 6 Li

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

59

3.5

Fermionic Superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.5.1

BCS Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.5.2

Resonance Superﬂuidity . . . . . . . . . . . . . . . . . . . . . . . . .

62

4 Experimental Techniques

63

4.1

Production of a Degenerate Fermi Gas . . . . . . . . . . . . . . . . . . . . .

64

4.2

Optical Trapping of Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.3

Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.4

Spin State Manipulation and Detection . . . . . . . . . . . . . . . . . . . .

71

4.5

Evaporative Cooling in the Optical Trap . . . . . . . . . . . . . . . . . . . .

74

5 Decay of an Ultracold Fermi Gas near a Feshbach Resonance
5.1

76

Resonant Decay of an Ultracold Fermi Gas . . . . . . . . . . . . . . . . . .

77

5.1.1

Observation of Field-Dependent Losses . . . . . . . . . . . . . . . . .

77

5.1.2

Decay Curves on resonance . . . . . . . . . . . . . . . . . . . . . . .

79

5.2

Loss Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.3

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

6 Collisions in zero temperature Fermi Gases
6.1

6.2
6.3

6.4

Free Expansion of Atomic Gases . . . . . . . . . . . . . . . . . . . . . . . .

83

6.1.1

Expansion of non-degenerate gases . . . . . . . . . . . . . . . . . . .

83

6.1.2

Expansion of Superﬂuid Gases . . . . . . . . . . . . . . . . . . . . .

84

Observation of hydrodynamic expansion in a strongly-interacting Fermi gas
mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Expansion of a zero temperature Fermi gas . . . . . . . . . . . . . . . . . .

86

6.3.1

Momentum Space Fermi Surface . . . . . . . . . . . . . . . . . . . .

86

6.3.2

Collisions in the Perturbative Limit . . . . . . . . . . . . . . . . . .

88

6.3.3

Perturbation from Hydrodynamic Behavior . . . . . . . . . . . . . .

90Concluding Thoughts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 RF Spectroscopy of Ultracold Fermions
7.1
7.2
7.3

82

91
92

Clock Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

7.1.1

Absence of the Clock Shift in Fermions . . . . . . . . . . . . . . . .

95

Emergence of Mean Field Shifts in a Fermi System . . . . . . . . . . . . . .

98

7.2.1

Decoherence in our system . . . . . . . . . . . . . . . . . . . . . . .

98

RF Spectroscopy of a 3-level system . . . . . . . . . . . . . . . . . . . . . .

100

8

7.4

Observations of Unitary and Universal Behavior. . . . . . . . . . . . . . .

102

7.5

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

8 Conclusions and Outlook

104

A Contrast Interferometry using Bose-Einstein Condensates to Measureh/m and α107

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

112

C Decay of an Ultracold Fermionic Lithium Gas near a Feshbach Resonance117
D Collisions in Zero Temperature Fermi Gases

122

E RF Spectroscopy of Ultracold Fermions

128

9

List of Figures
2-1 Bragg transition diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2-2 Mach-Zehnder interferometer schematic. . . . . . . . . . . . . . . . . . . . .

27

2-3 Schematic of the apparatus for the BEC contrast interferometer. . . . . . .

30

2-4 BEC diﬀracted by pulsed optical standing waves. . . . . . . . . . . . . . . .

31

2-5 Signal from a BEC Mach-Zehnder interferometer. . . . . . . . . . . . . . . .

33

2-6 Best measurements of α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2-7 Scheme of contrast interferometer to measure the photon recoil. . . . . . . .

35

2-8 Pulse 1 and pulse 2 of the interferometer sequence. . . . . . . . . . . . . . .

39

2-9 Typical single-shot signal from the contrast interferometer. . . . . . . . . .

40

2-10 Atom signal from the contrast interferometer. . . . . . . . . . . . . . . . . .

41

2-11 Measurement of ωrec in sodium. . . . . . . . . . . . . . . . . . . . . . . . . .

42

2-12 Vibration insensitivity of the contrast interferometer. . . . . . . . . . . . . .

43

2-13 Demonstration of the quadratic scaling of the recoil phase with additional
photon recoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2-14 Scaling up the contrast interferometer - vertical fountain conﬁguration. . . .

45

3-1 Scattering Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3-2 Schematic of Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . .

56

3-3 Real and Imaginary parts of (f0 ) averaged over a zero temperature Fermi
3-4

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Ground state hyperﬁne levels of 6 Li.

60. . . . . . . . . . . . . . . . . . . . . .

3-5 Prediction of interstate s-wave Feshbach resonances in the ground state of 6 Li. 61
4-1 Ground state hyperﬁne structures of 6 Li and 23 Na. . . . . . . . . . . . . . .

64

4-2 Magnetic coils for atom trapping and production of Feshbach ﬁelds. . . . .

65

4-3 Large and ultra-degenerate Fermi sea. . . . . . . . . . . . . . . . . . . . . .

66

4-4 Two-species mixture of degenerate Bose and Fermi gases. . . . . . . . . . .

67

4-5 Predictions of s-wave Feshbach resonances between in the 6 Li ground state.

68

4-6 Homogeneity of the Feshbach ﬁeld. . . . . . . . . . . . . . . . . . . . . . . .

70

4-7 Current turn on and shut oﬀ in Feshbach (antibias) coils. . . . . . . . . . .

71

4-8 |1 → 2 and |2 → 3 RF transition frequencies. . . . . . . . . . . . . . . . .

72

10

4-9 High ﬁeld imaging of 6 Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4-10 Simultaneous multiple state imaging. . . . . . . . . . . . . . . . . . . . . . .

74

4-11 Mutual cooling of states —1 and |2 in the optical trap. . . . . . . . . . . .

75

5-1 Prediction of s-wave Feshbach resonance between states |1 and |2. . . . .

77

5-2 Feshbach resonances identiﬁed by losses. . . . . . . . . . . . . . . . . . . . .

78

5-3 New prediction of s-wave Feshbach resonances between states |1 and |2. .

79

5-4 Decay curve at 680 G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

6-1 Hydrodynamic Expansion of 6 Li from the optical trap at 900 G. . . . . . . .

86

6-2 Evolution of aspect ratio during free expansion at 900 G - evidence for stronghydrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

6-3 Deformation of the momentum space Fermi surface . . . . . . . . . . . . . .

88

6-4 Geometrical representation of collisions in momentum space . . . . . . . . .

89

6-5 Perturbative calculation of collision rate during expansion of zero temperature Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

7-1 The diﬀerence in the clock shift for Bose and Fermi atoms. . . . . . . . . .

94

7-2 Shift in equilibrium energy in a 2-level Fermi system. . . . . . . . . . . . . .

95

7-3 Absence of clock shift in fermions. . . . . . . . . . . . . . . . . . . . . . . .

96

7-4 RF is sensitive to mean ﬁeld shifts in a 3-level system. . . . . . . . . . . . .

97

7-5 Expected variation of decoherence time with magnetic ﬁeld. . . . . . . . . .

98

7-6 Decoherence time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

7-7 Schematic of the 3-level experiment. . . . . . . . . . . . . . . . . . . . . . .

100

7-8 Representative 3 level RF spectra at 480 G. . . . . . . . . . . . . . . . . . .

101

7-9 a13 − a12 as a function of magnetic ﬁeld. . . . . . . . . . . . . . . . . . . . .

102

11

Chapter 1

Introduction
Over the last century, the scientist’s understanding of the world has undergone a profoundchange. The gradual realization of the underlying discreteness of nature has shaped a“quantum” picture of the world. In this picture, Nature is divided into very small but
discrete units called quantum states. The behavior of particles which cannot be describedwithout assuming this underlying disreteness is called quantum behavior. Typically, such
behavior is exhibited at very small microscopic scales of length, energy, etc. However,experimental advances can sometimes lead to spectacular macroscopic quantum phenomena.
One such incredible advance occurred in 1960 with the invention of the optical laser [13, 14].
A laser is created when particles of light, photons, occupy a single quantum state. Since
all the photons are in the same state, they share the same energy, polarization, direction of
propagation and so on. This is the reason for the spectacular diﬀerence between laser lightand ordinary light for example from a light bulb.
With the turn of the century, a new paradigm of macroscopic quantum phenomena is
rapidly establishing itself. Particles of matter, atoms, can also occupy a single quantum
state. This marks a phase transition into a new state of matter - the Bose-Einstein Condensate (BEC). This phase transition was observed in dilute alkali metal vapors in 1995
[15, 16, 17]. Since then, gaseous BECs have been in the limelight of science.
As with lasers, BECs can be applied towards spectroscopy and metrology[1]. The weakinteractions in dilute gas BECs make it an ideal testing ground for basic quantum theories.
This was not possible for the previous realization in 4 He[18]. In addition, gaseous atomic
systems are equipped with a whole range of experimental “knobs” which allow studies ina wide regime of temperatures, densities, interaction strengths, internal state composition,and so on.
The early achievements of BEC in 87 Rb[15], 7 Li[16], and 23 Na[17], have been followed by
the creation of BECs in other dilute atomic gases: 1 H[19], 85 Rb[20], metastable 4 He[21, 22],41 K[23], 133 Cs[24], and 174 Yb[25]. This list is constantly growing.

12

1.1

Degenerate Atomic Gases

There are only two types of particles in nature. Identical particles which can occupy thesame quantum state (such as photons and 23 Na atoms) are called bosons. The other typeof particles are called fermions. Two identical femions cannot occupy the same quantum
state.
Pronounced quantum behavior of atomic gases can occur when the number of available
quantum states approaches the number of identical particles in the system. Such conditions
induce behavior which begins to depart dramatically from the classical characteristics ofroom temperature gases. BEC is an example of such behavior. Let’s estimate the conditions
for this to happen.
The Heisenberg Uncertainty principle of quantum mechanics relates the size ∆x andmomentum ∆p of a quantum state:∆x∆p = .
Using the kinetic energy relation (for a particle of mass m at temperature T ):(∆p)2
= kB T2mthen gives the de-Broglie wavelength∆x ≈ λdB = √h.
2mkB TλdB describes the spatial extent of the quantum state at temperature T . For a gas ofparticles of mass m at temperature T , quantum behavior will ensue if the density n of thegas is large enough to cause overlap between the diﬀerent quantum states. This requirement
produces the condition:
nλ3dB  1⇒

3

nT − 2  (2mkB 3/2
) .
2

(1.1)

Under these conditions, quantum phenomena can be expected to occur. Very often, thephrase “quantum degenerate” is applied to a gas under these conditions.
At room temperature (∼ 300 K), Eqn. 1.1 gives a degeneracy density for 23 Na atomsof n ∼ 1027 cm−3 . At these densities, the system is no longer a gas at room temperature.
The phase transition to liquid and then solid will mar the approach to degeneracy. Several
technical advances have been required to keep the particles in a gaseous state while lowering
their temperature in order to reach degeneracy. In current experiments on alkali metal
vapors, degeneracy is reached at a temperature of  1 µK (a billion times colder than air).
Eqn. 1.1 then implies a density of ∼ 1014 cm−3 . This requirement makes the gas dilute andallows for simpliﬁed theoretical treatments.
13

1.2

Diﬀerent Quantum Matters

Bosons and fermions, behave diﬀerently near degeneracy. This is formally understood fromthe diﬀerence in their quantum statistics which relates equilibrium state occupation probabilities with thermodynamic quantities.
Bosonic degeneracy involves the formation of the Bose-Einstein Condensation - macroscopic occupation of a single quantum state. Fermionic degeneracy involves single occupation of “nearest neighbor” quantum states. This tight packing creates a Fermi Sea ofparticles. A minimum size is thus maintained giving rise to the so-called Fermi pressure.
Additional particles cannot penetrate into the Fermi Sea, giving rise to a “Pauli blocking”
behavior.
The ﬁrst observation of Fermi degeneracy in 1999 was in 40 K[26] by the group of D. Jinat the University of Colorado. Since then ﬁve other groups have made similar observationsin 6 Li[27, 28, 29, 3] as well as in 40 K[30].
Several groups are now actively trying to observe another kind of quantum behavior
which arises when degenerate fermions attract each other and form particle-pairs with longrange correlations. This state is known as the Bardeen-Copper-Schreiﬀer (BCS) superﬂuid[31]. For reasons similar to those mentioned in the case of BEC, in spite of previous
realizations in 3 He and superconductors, gaseous fermionic superﬂuids should provide awhole new range of tests and explorations.
This thesis reports on studies using both degenerate bosons and degenerate fermions
with particular emphasis on experiments towards creating the BCS superﬂuid.

1.3

Outline of this thesis

The work reported in this thesis can be divided into two parts. The ﬁrst part was performed
under the supervision of Dave Pritchard and involves an application of BECs to high precision measurements. We developed a BEC atom interferometer using optical standing waves
that is capable of precisely measuring ωrec , the recoil frequency (energy) of an atom due toabsorption of a photon. ωrec in turn can be used to determine h/matom and α, the ﬁne structure constant. This work is described in Chapter 2. Our publication of a “ﬁrst-generation”
measurement of ωrec in sodium is included as Appendix A.
The second part of the thesis was performed under the supervision of Wolfgang Ketterle
and involves the development of a new quantum system which holds promise for producing
a BCS superﬂuid of paired gaseous atoms. Chapter 3 describes some theoretical aspects
necessary for a motivation and understanding of the experiments while Chapter 4 describes
the apparatus used for the work. The remaining chapters discuss our experimental andtheoretical results.
Chapter 5 describes our studies of stability in strongly interacting fermion systems which
resulted in the ﬁrst observation of Feshbach resonances in 6 Li. Chapter 6 describes a theo14

retical study of the role of Pauli blocking of collisions during the expansion of an interacting
Fermi gas. Our results are relevant for understanding the observations (in many laboratories) of hydrodynamic behavior during expansion of a strongly-interacting Fermi gas.
Chapter 7 describes studies of fermionic interactions by observations of resonance shifts ofradio-frequency transitions in ultracold gases. The relevant publications are included as
Appendices B,C, D, and E.

15

Chapter 2

Atom Interferometry for h/m and α
We have demonstrated a new atom interferometer conﬁguration capable of measuring the“photon recoil frequency” to high accuracy. The photon recoil frequency is the frequency
corresponding to the kinetic energy of an atom recoiling due to absorption of a photon.
This frequency is an important ingredient for a high accuracy measurement of α, the ﬁne
structure constant. Our publication describing the ﬁrst generation measurement using
Bose-Einstein Condensates is included in Appendix A:• S. Gupta, K. Dieckmann, Z. Hadzibabic, and D.E. Pritchard, “Contrast Interferometry using Bose-Einstein Condensates to measure h/m and α” Phys. Rev. Lett, 89,
140401 (2002) [1]. Included in Appendix A.
A key aspect of our atom interferometer is the use of optical standing waves as diﬀraction
gratings. Methods of atom manipulation using optical standing waves are described in
Section 2.2. This section is taken from our review paper:• S. Gupta, A.E. Leanhardt, A.D. Cronin and D.E. Pritchard, “Coherent manipulationof atoms with standing light waves.” Special Issue of Compte rendus de l’académie
des sciences, Série IV - Physique Astrophysique 2, 479 (2001) [2].
Matter-wave duality is one of the fascinating features of quantum mechanics. This duality can also be disturbing since it departs so directly from everyday experience. Matter-wave
duality lies at the heart of the relatively new ﬁelds of atom optics and Bose-Einstein condensates (BECs). The combination of these two ﬁelds should then be even more fascinating.
In addition, the traditional experiments with atom interferometers: testing of fundamental
principles, studies of atomic properties, applications as inertial sensors, and measurementsof fundamental constants [32] can beneﬁt from the brightness (intensity and small momentum spread) of BEC sources. Finally, the coherence properties of condensates may allow
BEC based atom interferometers to approach the Heisenberg detection limit [33]. This cor√
responds to a measurement precision which scales like 1/N for N atoms and not like 1/ N
as for independent measurements on N atoms.
16

As techniques of atom manipulation and those for creating condensates have developed,
the last few years have seen the birth of BEC atom-optics. Free falling condensates havebeen coherently manipulated with standing waves of light creating “free-falling” atom interferometers [34, 35, 1]. “Trapped” BEC atom interferometers have also been created using
optical potentials [36]. This chapter deals with our eﬀorts towards developing a free-falling
atom interferometer for measuring the photon recoil frequency.
Section 2.1 contains a discussion of the requirements for making an atom interferometer. I then introduce the tools of standing wave diﬀraction gratings that can be used toconstruct a BEC atom interferometer in Section 2.2. Section 2.3 discusses a simple MachZehnder interferometer geometry and its realization with a BEC. In Section 2.5, I present
our scheme to measure the recoil frequency and the “ﬁrst-generation” results. Finally, in
Section 2.6.1 I discuss prospects for scaling up our interferometer for scientiﬁcally signiﬁcantmeasurements.

2.1

Atom Interferometers

Interference is an underlying principle of quantum mechanics embedded in the superposition
principle. An interferometer is a device designed to exploit this principle to measure some
useful physical quantity.
The working of an atom interferometer can be broken into 3 simple steps: (1) startingfrom a single atom source, coherently create multiple states1 (sometimes called “paths”), (2)let these multiple states evolve over some time during which the interaction to be measured
occurs, and (3) recombine the states and then detect the (interference pattern of) atoms.
Measurements are then sensitive to the diﬀerent evolution characteristics of the diﬀerent
states/paths. Various interferometer “architectures” or “geometries” based on atom beams
as well as laser-cooled atomic sources have been summarized elsewhere [32].
A basic requirement of atom interferometers is then state manipulation, in order toaccomplish steps (1) and (3). The usefulness of the interferometer is dependent on what
happens during step (2). For example, one can place diﬀraction gratings (∼ 200 nm) in thepath of a supersonic sodium atomic beam (∼ 1000 m/s). This results in a splitting into
diﬀerent transverse momentum states travelling at a small angle from each other. Thesefollow physically diﬀerent paths which can be bent with additional gratings and recombinedat an atom detector. An electric ﬁeld placed in a way that aﬀects the diﬀerent paths
diﬀerently will aﬀect the phase of the ﬁnal atom signal. This can then be used to measure
the electric polarizability of an atom [37].
Although diﬀerent atomic internal states can be used to make an interferometer[38],the use of diﬀerent external momentum states turns out to be advantageous[39]. Most of
the work towards making a meaningful measurement usually comes from trying to isolate
1Here “coherently” means that the diﬀerent states have a deﬁnite phase relationship with each other17the system into feeling only the interaction of interest (in the above example, the electricﬁeld) and nothing else. The use of a single internal state can reduce many perturbations
to “common mode” so that they do not shift the relative atom phases. Perturbations fromlight shifts and magnetic ﬁelds can be suppressed in this way[39, 1]. Optical standing waves
can be used to manipulate momentum states of atoms in a very controlled manner to create
arbitrary interferometer geometries.

2.2

Manipulation of Atoms with Standing Waves of Light

When an atom scatters a photon, it recoils due to the momentum exchanged with it.
Thus photons can change the momentum of atoms. The recoil from spontaneous scattering
processes is the key element of laser cooling[40].
The photon absorbed by an atom from a laser beam can be spontaneously emitted(scattered), making the atom recoil in a random direction. If the atom is exposed to two laser
beams, the photon absorbed from one beam can scatter in a coherent stimulated manner into
the other beam. If the stimulated process is faster than the spontaneous process, the atom
recoils in the direction given by the diﬀerence in wavevectors of the two laser beams. Thiscondition can be achieved with suﬃcient detuning of the laser beams from the electronicresonance. The two beams of course form a standing wave, and an equivalent description ofthis process can be given in terms of atom diﬀraction oﬀ the resulting standing wave. For
momentum state manipulation, counter-propagating laser beams are usually used, whichconstrains the atom to change its momentum in multiples of 2k, where k is the wavevector
associated with a single photon.
Two standard processes used in the manipulation of external states of neutral atoms
using optical standing waves are the Bragg and the Kapitza-Dirac processes. These areused in the construction of our atom interferometer. The rest of this section is taken from
[2] and describes these processes in some detail.

2.2.1

Bragg Scattering, Ref. 2

Although it can be diﬃcult to realize the physical conditions that assure its occurrence,
Bragg scattering is the simplest example of coherent momentum transfer to atoms by light.
Consider a standing wave light grating formed by two counter-propagating plane waves(travelling parallel to the z-axis) of equal amplitude, E0 , wavevector, k, frequency, ω, polarization vector, ê, and temporal envelope function, f (t): t) = E0 f (t) sin(kz − ωt)ê + E0 f (t) sin(kz + ωt)ê,E(z,= 2E0 f (t) sin(kz) cos(ωt)ê.

18

(2.1)
(2.2)

Figure 2-1: (a) First and (b) NBth order Bragg transition diagram. The atomic kinetic energy
lies on the parabola N 2 ωrec , N = 0, ±1, ±2, . . . where the associated atomic momentum
is given by N k.

19

We would like to work with momentum states as our basis, thus it is easiest to consider thedescription of the electric ﬁeld driving the transitions in terms of two counter-propagating
travelling waves of deﬁnite momentum (Equation 2.1), as opposed to the single standing
wave they jointly form (Equation 2.2).
Momentum is transferred by paired stimulated absorption and emission processes, resulting in a transfer of photons between the travelling waves. An NBth order diﬀraction
process transfers NB photons from one travelling wave to the counter-propagating travelling wave and changes the atomic momentum by 2NB k. Furthermore, atomic populationis transferred only between |g, −NB k and |g, +NB k, where |g(e), ±N k denotes a twolevel atom in its ground (excited) state with momentum ±N k parallel to the standing
wave axis. The excited state remains nominally unpopulated so long as the temporal envelope function, f (t), does not have strong frequency components near the laser detuning,δ = ω − ω0 , where ω0 is the unperturbed frequency of the atomic transition. Furthermore,
for a given initial state (|g, −NB k) the uniqueness of the ﬁnal state (|g, +NB k) comes
about because of the fundamental assumption that makes Bragg scattering so simple todescribe; the uncertainty in the photon energy driving the transitions is small compared tothe energy separation between neighboring momentum states. A quantitative discussion of
the validity of this assumption will be given later.
We now calculate the probability, P1B (τ ), of the ﬁrst order (NB = 1) Bragg process
taking an atom from |g, −k to |g, +k when the atoms interact with a constant lightintensity for a time τ (i.e. in Equations 2.1 and 2.2 f (t) is a square wave of unit amplitude
and duration τ ). The transition is depicted in Figure 2-1(a). In the electric dipole approx
µ · E(t).
By momentum conservation,
imation, the interaction Hamiltonian is Hint (t) = −only the plane wave travelling in the ±ẑ direction couples the |e, 0 ↔ |g, ∓1 transitions.
By using this argument, we are eﬀectively viewing the electric ﬁeld as a quantum mechanical operator. Expanding the sinusoidal variation of the electric ﬁeld in terms of complex
exponentials and treating the spatially dependent complex exponential terms as quantum
mechanical momentum translation operators (e±ıkẑ |g(e), nk = |g(e), (n ± 1)k) yields
the interaction Hamiltonian [41, 42, 43]:
Hint (t) = −ıe−ıωtωR(|e, 0g, −1| − |e, 0g, +1|) + h.c.,
2

(2.3)

where the depicted terms are responsible for absorption and the hermitian conjugate (h.c.)terms give rise to stimulated emission. The light-atom interaction is parameterized by thesingle-photon Rabi frequency,
ωR =

µE0
,


(2.4)

where µ = e|er|g · ê is the electric dipole matrix element connecting the ground (|g) andexcited (|e) states of the atom. Without loss of generality, we will take µ and hence ωR to
be positive, real-valued quantities. In formulating Hint , we have neglected any frequency
20

components associated with the sudden switch on of the ﬁelds and the ﬁnite duration of
the light-atom interaction. The total Hamiltonian, H(t) = H0 (t) + Hint (t), follows simply
by including the electronic and kinetic energy terms,
H0 (t) = ω0 |e, 0e, 0| + ωrec (|g, −1g, −1| + |g, +1g, +1|),(2.5)

where the single-photon recoil energy, Erec , of an atom of mass m is given by:
Erec = ωrec =2 k 2.
2m(2.6)

Making the ansatz for the solution wavefuntion as,|Ψ(t) = c−1 (t)e−ıωrec t |g, −1 + c0 (t)e−ıω0 t |e, 0 + c+1 (t)e−ıωrec t |g, +1,

(2.7)

and substituting into the Schrödinger equation yields the three coupled ﬁrst order diﬀerential
equations:ωRċ±1 (t) = ∓ eı∆t c0 (t),2ωR −ı∆te(c+1 (t) − c−1 (t)),
ċ0 (t) =
2

(2.8)
(2.9)

where ∆ = δ + ωrec . Diﬀerentiating Equation 2.8 and substituting Equation 2.9 into theresult yields the two coupled second order diﬀerential equations:c̈±1 (t) − ı∆ċ±1 (t) +2
ωR(c±1 (t) − c∓1 (t)) = 0.
4

(2.10)

With the initial conditions,c−1 (0) = 1,

(2.11)

c0 (0) = 0 =⇒ ċ±1 (0) = 0,(2.12)

c+1 (0) = 0,(2.13)

2 , the solutions to Equation 2.10 read:and the assumption ∆2  ωR
(2)
ω
R

(2)

ωR
t),
2
(2)
(2)
ω
ω
R
c+1 (t) = ıe−ı 2 t sin( R t),2
c−1 (t) = e−ı 2 t cos(

(2.14)(2.15)

where the two-photon Rabi frequency is:
(2)

ωR =

2
ω2
ωR
→ R , |δ|  ωrec ,
2∆
2δ

21

(2.16)

with both transitions driven at equal single-photon Rabi frequencies, ωR .
Substituting the solution of Equation 2.14 or 2.15 into Equation 2.8 yields an expression
for the excited state amplitude:c0 (t) = −ı

ωR −ı∆t −ıω(2) t
e
e R .
2∆

(2.17)

This will be important in calculating the rate of spontaneous emission events later.
The solutions for c−1 (t) and c+1 (t) oscillate with the interaction duration, τ , yielding
the result for the |g, −k → |g, +k transition probability:
P1B (τ ) = |c+1 (τ )|2 = sin2

(2)

ωR
τ
2


.

(2.18)

Thus, the system oscillates between the two momentum states |g, −k and |g, +k ina manner analogous to the Rabi oscillation of atomic population between two resonantly
coupled states. This solution with oscillatory probabilities for the two Bragg coupled states
is known as the Pendellösung and has been observed for atoms [44], neutrons [45], andx-rays [46].
Viewing Bragg scattering as a two-photon transition from the initial ground state to theﬁnal ground state with opposite momentum illuminates the close connection with a Raman
transition between two internal substates of the ground state manifold, each with its own
external momentum state. The formalism describing the Raman transition is basically thesame as that presented here, except the two transitions can be driven at diﬀerent singlephoton Rabi frequencies, ωR1 and ωR2 , so that the generic two-photon Rabi frequency is(2)given by ωR = ωR1 ωR2 /2∆, where ∆ is the detuning from the intermediate state.
An NBth order Bragg process (similar to a 2NB -photon Raman process) is a coherent
succession of NB two-photon transitions with 2NB − 1 intermediate states of the form|e, (−NB + 1)k, |g, (−NB + 2)k, . . . , |g, (NB − 2)k, |e, (NB − 1)k. Such a process ischaracterized by a 2NB -photon Rabi frequency given by [47]:(2NB )

ωR=[ωR ]2NB,
22NB −1 ∆1 ∆2 · · · ∆2NB −1(2.19)

where ∆n is the detuning from the nth intermediate state. Figure 2-1(b) shows what thisprocess would look like for an NBth order Bragg transition where the intermediate state
detunings are given by:∆n =δ + (2NB n − n2 )ωrec(2NBn − n2 )ω

rec

:

n odd

: n even.

(2.20)

Substituting these detunings into Equation 2.19 yields the NBth order Bragg transition 2NB -22

(2NB )

photon Rabi frequency, ωR[47]:(2NB )

ωR=[ωR ]2NB
NB −1
24NB −3 [(NB − 1)!]2 δ NB ωrec,(2.21)

where we have assumed |δ|  NB2 ωrec .
To ensure that the system truly undergoes Bragg scattering, and validate the assumptionthat only states of equal kinetic energy and opposite momentum are coupled, the overall
exposure time, τ , of the atoms to the ﬁelds must be limited both from below and above.
The lower bound on τ is necessary to resolve the ﬁnal momentum state (|g, NB k) fromneighboring momentum states (|g, (NB ± 2)k) two photon recoil momenta away. Thisbound also prohibits resonant transitions from the initial momentum state (|g, −NB k)
to its neighboring momentum states (|g, (−NB ± 2)k) two photon recoil momenta away.
For ﬁrst order Bragg scattering processes (NB = 1) the initial and ﬁnal state are only
separated by two photon recoil momenta, thus the nearest lying momentum states that
may be mistakenly populated are |g, ±3k. Avoiding population transfer into these statesrequires τ  π/4ωrec . For all higher order (NB > 1) Bragg scattering processes, the nearest
momentum states are |g, ±(NB − 2)k, which limits the interaction time to:τπ,2(NB − 1)ωrec(2.22)π.
2NB ωrec(2.23)

which for NB  1 reduces to:τ

The upper bound on the interaction duration is necessary to avoid spontaneous emission. The interaction duration must be short enough so that the expected number, Ns , ofspontaneous emission events per atom during the time τ is negligible. Ns is simply given by
the product of the excited state fraction (Equation 2.17) and the probability of spontaneous
decay given that the atom is in the excited state:
Ns = |c0 (t)|2 Γτ =

2
ωR
Γτ,
4∆2

(2.24)

where Γ is the natural decay rate of the excited state. Avoiding spontaneous emission(2NB )(Ns  1) while still having a signiﬁcant probability for transitions (ωRτ  π), is apractical requirement to maintain coherence. For a ﬁrst-order Bragg process, this requires∆  Γ.
Bragg scattering of atoms from a standing light wave was ﬁrst observed at MIT in
1988 [44]. A supersonic atomic beam was diﬀracted from a standing wave of near-resonant
laser light. The angle between the atomic beam (of thermal wavelength λdB ) and the light23

grating (of periodicity λL /2) was tuned to the appropriate Bragg angle, θB , where:
λdB = λL sin(θB ),(2.25)and population transfer corresponding to both ﬁrst and second order Bragg scattering wasobserved. The experiment required a sub-recoil transverse momentum spread of the atomic
beam in order to resolve the diﬀerent momentum states in the far ﬁeld and limit the ﬁnal
state to only one diﬀracted order. The Pendellösung was observed as an oscillation in(2)

population transfer as a function of standing wave intensity, I ∝ ωR , for a ﬁxed interaction
time, τ .
Atomic beam diﬀraction from an optical standing wave is a continuous-wave (CW) experiment in which the selectivity needed for the Bragg process is imposed by good angularresolution of the particle beam and a high degree of parallelism between the light crystal
planes. This ensures that of the various ﬁnal Bragg orders allowed by momentum conservation, only one conserves energy (energy conservation is exact in a CW experiment). For
atoms scattering from a light crystal, parallelism of the crystal planes requires highly parallel photon momentum that implies a minimum width of the standing wave (the diﬀraction
limit for the collimated photons). The transit time, τ , of the atoms across this width thenexceeds the lower bound given in Equation 2.22.
The excellent collimation required of the atomic beam to ensure resolution of the Bragg
scattered atoms reduces the intensity of the source by many orders of magnitude. A BoseEinstein Condensate (BEC) is an attractive alternative source of atoms because its momentum spread is typically an order of magnitude below a single-photon recoil momentum. To
Bragg diﬀract atoms initially in a stationary BEC, it is easier to move the light crystal thanto accelerate the condensate. This is done by simply frequency shifting one of the travellingwaves so that the resultant standing wave formed by its interference with the unshiftedtravelling wave moves with the proper velocity (NB k/m) relative to the stationary atoms
to impart the necessary momentum. The Bragg scattered atoms will then have momentum
2NB k in the laboratory frame. The resonance condition thus becomes a condition on relative detuning, δNB , between the two laser beams forming the diﬀraction grating. For NBthorder Bragg diﬀraction, the relative detuning is given by:
δ NB =

2NB k 2= 4NB ωrec .
m(2.26)

The ﬁrst demonstration of Bragg scattering in a BEC was at NIST in 1999 [48]. They
used Bragg scattering mainly as a tool to manipulate the momentum of the BEC, observing up to sixth order processes. At MIT the interaction time was lengthened (to ≈ 100
times the lower bound of Equation 2.22 for ﬁrst order Bragg scattering), creating a new
type of spectroscopy called Bragg Spectroscopy. This was used to observe the momentum
distribution of a BEC in a magnetic trap [49].
24

2.2.2

Kapitza-Dirac Scattering, Ref. 2

In 1933, Kapitza and Dirac predicted that an electron beam incident onto a properlyorientated standing light wave would undergo stimulated Compton scattering and be reﬂected [50]. Since then, scattering that can be properly described by neglecting particle
motion over the duration of the interaction (the Raman-Nath approximation) has become
known as Kapitza-Dirac scattering. Atoms may also undergo such scattering from a standing light wave.
To restrict the atomic motion during the interaction time to distances small comparedto the characteristic dimensions of the interaction potential requires short interaction times.
Mathematically, this regime can be treated by neglecting the atomic kinetic energy term in
the Hamiltonian (the Raman-Nath approximation). This is equivalent to the eikonal(thinlens) approximation for scattering(optics). For a standing wave interaction, this approximation is well satisﬁed if the atomic motion during the interaction time is small comparedto the wavelength of the illuminating radiation. As a result, Kapitza-Dirac scattering islimited (relative to Bragg scattering) to short interaction times, τ , generally much smaller
than the inverse recoil frequency (τ  1/ωrec ). To observe appreciable population transfer
at such short times, large intensities are needed. Since Kapitza-Dirac scattering is a coherent process, the interaction time must also be short enough to make spontaneous emission
negligible. Thus the constraint Ns  1 (Equation 2.24) holds in this regime as well, where
Ns is the expected number of spontaneous emission events per atom during the time τ .
The standing wave interaction may be treated by considering the standing wave (ACStark shift) potential resulting from the applied ﬁelds given in Equations 2.1 and 2.2:
U (z, t) =2
ωR
f 2 (t) sin2 (kz),δ(2.27)

where we have assumed δ 2  Γ2 /4.
The quickest route to the momentum distribution of the diﬀracted atoms in the KapitzaDirac regime is to use the eikonal approximation for treating the scattering of the incident
atomic waves after passing through the AC Stark shift potential of the standing wave.
Given the initial atomic wavefunction, |Ψ0 , the atomic wavefunction imediately after theinteraction is given by:ı|Ψ = |Ψ0 e− dt U (z,t )

2
ωR
−ı 2δ
τ

= |Ψ0 e


,

2
ωR
ı 2δ
τ cos(2kz)e(2.28),(2.29)

dt f 2 (t ) and the integral is over the interaction duration. With the use of thenınβ , the atomic
identity for Bessel functions of the ﬁrst kind, eıα cos(β) = ∞n=−∞ ı Jn (α)ewhere τ =25

wavefunction can be written as:2 ωRτ eı2nkz ,
2δ
n=−∞
 2 
∞
2

ωR
ωR
−ı 2δ τn
τ |g, 2nk,
ı Jn
= e
2δ
n=−∞
2
ωR

|Ψ = |Ψ0 e−ı 2δ τ

∞




ın Jn

(2.30)(2.31)

where the position space representation of the momentum states has been used (|g, p =p

N eı  z , with N an arbitrary normalization factor) and we have taken |Ψ0  = |g, 0.
It is now clear that states with 2N k of momentum are populated with probability [51,52]:2(θ), N = 0, ±1, ±2, . . . ,PN = JN

where
θ=

2
ωR
(2)
τ = ωR τ
2δ

(2.32)(2.33)

is the pulse area. This leads to a transverse rms momentum of the diﬀracted atoms that islinearly proportional to the pulse area [51]:prms =∞
(nk)2 Pn = 21/2 θk.

(2.34)

n=−∞

Kapitza-Dirac diﬀraction of atoms was ﬁrst observed at MIT in 1986 [51]. Diﬀraction ofa well-collimated (sub-recoil) supersonic atomic beam was observed after passage throughthe tightly focused waist of a near-resonant standing wave. Signiﬁcant diﬀraction into
momentum states |g, ±10k was observed [51]. Even higher diﬀracted orders should beobservable in the future using laser beams directed at small Bose-Einstein Condensates forsomewhat longer times.

2.3

A simple atom interferometer - the Mach-Zehnder conﬁguration

Steps (1) and (3) of creating an atom interferometer (Section 2.1) can be performed byeither Bragg or Kapitza-Dirac processes. If the momentum states used to construct theinterferometer physically separate between steps (1) and (3), an intermediate process toensure their spatial overlap during recombination is necessary. This is often the case withinterferometers based on optical gratings, since the photon recoil momentum ωrec can bequite large. In such situations, an intermediate “reversal” diﬀraction grating can be applied which reverses the relative momentum of the interferometer arms. The interferometer
would then consist of three diﬀraction grating. In this section, a simple 3-grating atominterferometer based on the Bragg process is discussed.

26

First consider the ﬁrst order Bragg process applied on a BEC (Eqn. 2.18, and Eqn. 2.26
(2)with NB = 1). A π/2−pulse (ωR τ = π2 ) splits the atom 50-50 into the two momentum(2)

states (a 50-50 beamsplitter). A π−pulse (ωR τ = π) puts the entire atom into the othermomentum-state (a mirror2 ). As we will see shortly, such beamsplitters and mirrors can beused to construct interferometers for atoms.
A simple interferometer geometry known as the Mach-Zehnder interferometer is shownin Fig.2-2. This was the conﬁguration of choice in MIT’s ﬁrst atom interferometer which
was constructed using material gratings to deﬂect thermal beams [53]. This conﬁguration
has also been used to demonstrate a BEC atom interferometer [34]. Understanding thissimple geometry elucidates several aspects of atom interferometry, so let’s go through it.
|2hk>

|0>

1

2

0

T

2T

1st order
Bragg
π/2-pulse

1st orderBragg
π-pulse1st order
Bragg
π/2-pulse

TOF

Figure 2-2: Schematic of a Mach-Zehnder interferometer. Paths 1 and 2 are in states |0 k
and |2 k for equal lengths of time T . This makes this interferometer insensitive to thephoton recoil.
Beginning with atoms at rest , the ﬁrst π/2−pulse creates an even superposition of
momentum states |0 k (at rest, path 2) and |2 k (in motion, path 1). The two paths
have a relative velocity of 2 photon recoils (6 cm/s for sodium). After time T , they are2
The analogy between a Bragg π−pulse and a mirror is not entirely correct because of their action
on a distribution of momenta. In the reference frame of a real mirror, all incoming atoms will reverse
their momentum katom → −katom . This means diﬀerent incoming momentum states receive diﬀerent
momentum kicks. For standing wave diﬀraction, all incoming atoms receive the same momentum kick =2kphoton .

27

“reﬂected” by the Bragg π−pulse and the momenta are swapped. The third pulse projects
the phase of the atomic interference pattern onto the fractional population of the two
momentum states.
I now derive the interferometer signal, trying to emphasize the relevant physics. I ﬁrst
present a derivation following the treatment of light pulse interferometers by Steve Chu
et. al. [54]. Gravity is neglected in this derivation, making this a 1-D problem, withthe only direction deﬁned by direction of the optical standing wave (vertical in Fig. 2-2).
Immediately before the ﬁrst pulse is applied, the atoms are at rest:|Ψ = |0
Let the 3 diﬀraction gratings have spatial phases φ1 , φ2 , φ3 relative to any ﬁxed position,with positive being upwards in this 1-D problem (Fig. 2-2). These phases can be manipulated by the RF synthesizers controlling the AOMs but may also vary randomly due tovibrations of the apparatus. The ﬁrst π/2-pulse rotates the statevector in Hilbert space
into an even superposition of the two states given by [54]:1|Ψ = √ |0 − ieiφ1 |2k2Path 2 acquires the energy phase −Et/ relative to path 1 during the time T . The wavefunction immediately before the second pulse:1|Ψ = √ |0 − iei(φ1 −4ωrec T ) |2k
2The action of the π-pulse is another coherent rotation:1|Ψ = √ −ei(φ1 −φ2 −4ωrec T ) |0 − ieiφ2 |2k
2Evolution for another time T produces the following wavefunction immediately before thethird pulse:1|Ψ = √ −ei(φ1 −φ2 −4ωrec T ) |0 − iei(φ2 −4ωrec T ) |2k
2The third pulse produces the ﬁnal wavefunction:|Ψ =1−ei(φ1 −φ2 −4ωrec T ) − ei(φ2 −φ3 −4ωrec T ) |021+ iei(φ1 −φ2 +φ3 −4ωrec T ) − iei(φ2 −4ωrec T ) |2k
2(2.35)The probabilities for being in the two states (which can be called the output ports of28

the atom interferometer) is just the amplitude squared of the respective components:
1 i(φ1 −φ2 −4ωrec T )1+ ei(φ2 −φ3 −4ωrec T ) |2 = (1 + cos(φ1 + φ3 − 2φ2 ))e2
2
1 i(φ1 −φ2 +φ3 −4ωrec T )1i(φ2 −4ωrec T ) 2= | e−e| = (1 − cos(φ1 + φ3 − 2φ2 ))(2.36)22

P0 = |P2k

The phase of the interferometer, φ1 + φ3 − 2φ2 , is a result of the interference of the twopaths with this phase diﬀerence at time 2T . Let’s represent the phase picked up by path i
as Φi . After pulse 1,
Φ1 = φ1 ,Φ2 = 0.

After time T free evolution,
Φ1 = φ1 − 4ωrec T,

Φ2 = 0.

After the second pulse,
Φ1 = φ1 − 4ωrec T − φ2 ,

Φ2 = φ2 .

After time 2T ,
Φ1 = φ1 − 4ωrec T − φ2 ,

Φ2 = φ2 − 4ωrec T.

After the third pulse, there are actually 4 paths. At the |0 k port:
Φ1 = φ1 − 4ωrec T − φ2 ,

Φ2 = φ2 − 4ωrec T − φ3 .

At the |2 k port:
Φ1 = φ1 − 4ωrec T − φ2 + φ3 ,

Φ2 = φ2 − 4ωrec T.

Thus, Φ1 − Φ2 = φ1 + φ3 − 2φ2 , as expected. Now, any interaction placed diﬀerentially on
one of the arms would be picked up by Φ1 − Φ2 and show up in the interferometer signal.
Thus a Mach-Zehnder interferometer can be used to for example measure rotational phases,
gravitational phases, electric polarizability, or index of refraction of gases [32].

2.4

Atom Optics with Bose-Einstein Condensates

In this section, we will see how the ideas of the previous sections can be experimentallyrealized with a BEC source.

29

2.4.1

Standing Wave Diﬀraction of sodium Bose-Einstein Condensates

To make and control the optical diﬀraction gratings, one needs control over the intensity,
timing and relative detuning of the two laser beams comprising the standing wave. All
these features are provided by acousto-optic modulators (AOMs)3 driven by standard RF
electronics. Phase control is another feature which can easily be incorporated and has
proven to be useful in the diagnosis of our atom interferometer. The switching time for
AOMs is ∼ 100 ns.
(a)from 899 dye laser

AOM #0(80 MHz)mechanicalshutter (MS1)

Bragg Fiber
input (60% eff.)(b)
PMTpinhole(at imageplane)

pick-offphotodiode 1

BS#2

z

AOM #1 (30 MHz)x
BS#1
AOM #2 (30 MHz)yvacuum windows(in recessed flanges)

BEC

Bragg Fiber
outputpick-offphotodiode 2

mechanicalshutter (MS2)stainless steel
vacuum parts

Figure 2-3: Top view of apparatus for the BEC contrast interferometer. (a) shows thebeam derived from the sodium dye laser and coupled into the Bragg ﬁber. (b) shows the
setup next to the vacuum chamber. The standing waves were horizontal and along thelong axis (z-axis) of the condensate in the trap. The reﬂected beam was separated fromthe ingoing beam by beamsplitter BS#2, sent through a 100 µm pinhole at an intermediate
image plane of the atoms and then imaged once more onto the photomultiplier tube (PMT).
The standing waves are applied along the z-axis while atoms are imaged along the x-axis.
The details of the apparatus to produce sodium BECs in the |1, −1 state can be found3
these are commercial devices in which an RF drive (∼ 100 MHz-∼ 1 GHz) applied to a crystal (usually
Tellurium Dioxide) sets up a phonon wave. A laser beam travelling through a well aligned crystal can be
Bragg diﬀracted oﬀ the phonon grating, resulting in a diﬀracted beam which is shifted both in frequency andin direction (usually only ﬁrst order diﬀraction is used). Note that AOMs also depend on Bragg diﬀraction!

30

in earlier theses [9, 10]. For the experiments described in this chapter, we used lighttuned close to the F = 2 → F  = 3 transition to create the diﬀraction gratings (ﬁg.23). This is ∼ 1.8 GHz (approximately the hyperﬁne splitting) red-detuned from the D2 line
(3 S1/2 →3 P3/2 ) for the F = 1 atoms in the BEC. We used an 80 MHz AOM (AOM#0)
as a switch for this light. The light was then sent though an optical ﬁber to clean up themode and transported to the experiment. The output of the ﬁber was split in two andsent through independent 30 MHz AOMs (AOM#1 and AOM#2). These were controlledwith Stanford Research Systems SRS DS-345 synthesizers. The two light beams were thenaligned counter-propagating and horizontal through the vacuum windows of the apparatus(z-axis in ﬁg.2-3).

zyFigure 2-4: Resonant absorption images of BECs diﬀracted by pulsed standing waves.
The images were taken with a resonant laser beam which propagated through the vacuum
chamber along the (vertical) x-axis (see Fig. 2-3). Pulses were applied a few ms after releaseof the BEC from the magnetic trap. The standing waves were aligned in the z-direction(see Fig. 2-3). Total time-of-ﬂight was 20 ms. (a) No pulse was applied. The position of the
BEC in the image corresponds to the state |0 k . (b) A 600 ns Kapitza-Dirac pulse wasapplied which split the BEC into 3 momentum states |0 k , |2 k , and | − 2 k . (c)A 1 µs Kapitza-Dirac pulse with similar intensity. The second orders |4 k and | − 4 k
can now be seen. (d) A 10 µs Bragg pulse with 200 kHz detuning was applied and caused anear-perfect transfer into the state | − 4 k .
Let’s estimate some numbers necessary for the implementation of Bragg and KapitzaDirac pulses on a stationary BEC. First we introduce the usual deﬁnition of the saturation
intensity:
Γ2 I
2
= ωR
,
2 Isat(2.37)For sodium, the saturation intensity Isat = 6 mW/cm2 , the natural linewidth Γ = 2π ×
10 MHz and the recoil frequency ωrec = 2π × 25 kHz. This gives, 1/ωrec ∼ 6µs, giving
us the timescale separating Bragg and Kapitza-Dirac processes. Assume 2 mm diameterbeams at 1.8 GHz red-detuning. Using Eqns.2.16, 2.37, and 2.18, the power needed for31

a 10 µs long, ﬁrst order Bragg π/2−pulse is ∼ 0.4 mW in each beam of a standing wave
with relative detuning 100 kHz. Using Eqn. 2.21, the power needed for a 10 µs second order
Bragg π−pulse is ∼ 1.7 mW, with 200 kHz relative detuning. A ∼ 3.3 mW 1 µs pulse with 0
relative detuning will operate in the Kapitza-Dirac regime and populate both states | ± 2k
with 25% of the original condensate (Eqn. 2.32).
The diagnosis of momentum states in atom trapping experiments is traditionally doneby turning of the atom trap suddenly and observing the ballistic expansion of atoms. Long
time-of-ﬂight (TOF) converts the momentum information into spatial information which canthen be directly detected using standard absorption imaging techniques. The sub-recoil momentum spread of BEC’s implies the complete separation of momentum states/diﬀraction
orders separated by 2k.
Fig.2-4 shows time-of-ﬂight images of BECs split into diﬀerent momentum states by
optical standing wave pulses. The standing wave pulses were applied after releasing the BEC
from the trap in order to reduce the density and associated mean-ﬁeld eﬀects. Combinationsof such gratings can be used to create atom interferometers.

2.4.2

BEC Mach-Zehnder interferometer

The extension of atom diﬀraction to an atom interferometer follows directly with the application of multiple diﬀraction pulses in succession.
A frequency diﬀerence of δ1 = 2π × 100 kHz (Eqn. 2.20) between AOMs 1 and 2 fulﬁlled the requirements for a ﬁrst order Bragg process. Using 2 phase locked SRS DS-345
synthesizers to control the 2 AOMs, we were able to control the relative phase of the RF
supplied to the AOMs from the front panel of the instruments. This in turn controlled thephase of the optical standing wave (φi ). Varying the relative phase of the second grating,
we obtained the interferometer signal shown in Fig. 2-5. The contrast or visibility is deﬁned
as (max-min)/(max+min) of an interference signal. The contrast in Fig. 2-5 is  90%, a
result of the extreme sub-recoil nature of the atom source. A large momentum distribution
would cause the Bragg resonance condition (Eqn. 2.26) to be diﬀerent for diﬀerent initialmomenta. This would mean that the π/2-pulse and π-pulse condition could not be fulﬁlled
simultaneously for all the atoms and lead to a reduced contrast. This makes the use of a
BEC with its narrow momentum spread particularly useful for atom interferometry withstanding light waves.
The Mach-Zehnder interferometer just described is an example of a “phase” interferometer, where the signal is sensitive to the phase of the atomic interference pattern. The phase
is of course sensitive to mirror vibrations (via the φi ’s) and interferometers constructedon this principle in general require considerable vibration isolation to achieve high accuracy measurements [55]. As I will show, a signal sensitive to the contrast of the atomic
interference pattern might not have this problem. The following section discusses a contrast interferometer scheme for measuring the photon recoil frequency. The scheme has
32

1.0

N2k fraction
N0 fraction

Atomic Fraction

0.8

0.6

0.4

0.2

0.0
0

20

40

60

80

100

120

140

160

180

dialed in phase, degrees

Figure 2-5: Signal from the BEC Mach-Zehnder Bragg interferometer of Fig. 2-2. The 1storder condition was met by applying a frequency detuning 100 kHz between AOM#1 andAOM#2. The time between pulses is 50 µs for this example. The pulse time τ was 10 µs
for each pulse. Pulse 2 was twice as intense as pulses 1 and 3.
additional advantages (beyond vibrational immunity) which make it a viable high accuracy
measurement scheme.

2.5

First Generation Recoil Measurement

The techniques for condensate manipulation with standing waves can be applied to measuring the ﬁne structure constant.

2.5.1

Motivation for measuring αα, the ﬁne structure constant, is a fundamental quantity of physics. It lies at the very
heart of quantum electrodynamics (QED), since it plays an important role whenever theelectromagnetic interaction is involved. This makes α essentially ubiquitous. Measurementsof α are therefore relevant for various subﬁelds of physics. In turn, a whole range ofindependent physical measurement methods have been developed with several competing
values below the ppm (part-per-million) level [56]. Fig. 2-6 shows some of these.
Our experiment is relevant to the atomic physics route to measure α, based on therelation [57]:α2 = 2 2
2R∞ h
2R∞ M h
e
.
=
=
c
c me
c Me m(2.38)

Here e is the electron charge, c is the speed of light, R∞ is the Rydberg constant, Me (me ) is
the electron mass in atomic (S.I.) units and M (m) is the mass of some test particle in atomic(S.I.) units. For a particular choice of test particle (atom), interferometry can provide ωrec ,33

which is related to h/m by the relation:
ωrec =

1 2
k2m(2.39)All the other quantities are either already known to sub-ppb (part-per-billion) accuracy orare accurately accessible for alkalis (see [1] for details). An atom interferometer developedin Steve Chu’s group at Stanford has measured ωrec in cesium to 15 ppb accuracy [58].
µhfs
ac Josephson
h/mn
He fs (prelim)
quantum Hall
h/mCs (prelim)
g-2
-150

-100

-50

0

{(α/α98) - 1} x 109

50

100

Figure 2-6: Best measurements of α shown as ppb deviation from the 1998 CODATA value
[59].

2.5.2

Scheme of Contrast Interferometer

First consider the phase interferometer of Fig. 2-7(a). It is a 2-path scheme, just like theMach-Zehnder, but has an important diﬀerence in how the 2 paths are recombined. Insteadof a ﬁrst order Bragg π−pulse, a second order one is applied. In the Mach-Zehnder, both
paths are in the |0 k and |2 k states for the same length of time T , though not simultaneously. This means that the phase diﬀerence between the 2 paths at 2T is not sensitive
to the photon recoil frequency ωrec (Eqn. 2.36). In contrast, path 1 in the interferometerof Fig. 2-7(a) is in the state |2 k for time T and state | − 2 k for time T , whereas
path 2 is always in the state |0 k . Around t = 2 T , a moving matter wave grating, with2π), is formed due to the overlap and interferspatial periodicity λ/2 (wavevector 2k = λ/2ence of the two paths. The phase of this grating at 2T is determined by the relative phaseΦ1 − Φ2 = 8 ωrec T , accumulated between paths 1 and 2 due to the diﬀerence in their kinetic
energies. A measurement of this phase for diﬀerent values of T will then determine ωrec .
Using the reasoning presented earlier, after the ﬁrst pulse,
Φ1 = φ1 ,Φ2 = 0.
34

After time T free evolution,
Φ1 = φ1 − 4ωrec T,

Φ2 = 0.

After the second pulse,
Φ1 = φ1 − 4ωrec T − 2φ2 ,Φ2 = 0(second order Bragg pulse gives −2φ2 ). After time 2T ,
Φ1 = φ1 − 8ωrec T − 2φ2 ,Φ2 = 0.

At time 2T , the phase diﬀerence between the two paths contains the recoil frequency. Thisis the element we want to exploit for our measurement. We note at this point that if N thorder and 2N th order Bragg processes had been used in the interferometer of Fig. 2-7(a),Φ1 − Φ2 = φ1 − 2φ2 − 8N 2 ωrec T,(2.40)

giving us a quadratic sensitivity to the number of photon recoils applied.
(b)

(a)
1

matter wave
grating

readoutpulse

1

contrast
signal223∆t=π/4ω rec z

z

0

T

1st orderBragg

2nd order
Bragg

2T

t

∆t=π/4ω rec

t

0

T

2T

KapitzaDirac

2nd order
Bragg

readout

Figure 2-7: Space-Time representation of the contrast interferometer. (a) shows a simple
2-path interferometer sensitive to the photon recoil phase. The 2k matter wave grating isshown at 2T and at 2T + π/4 ωrec . The extension to the 3-path geometry is shown in (b).
The overall 2k grating has large contrast at 2T and zero contrast at 2T + π/4 ωrec .
The extension of the phase interferometer into a contrast interferometer is shown in
Fig. 2-7(b). Three momentum states (paths 1, 2 and 3) are generated by replacing the ﬁrst
Bragg pulse with a short Kapitza-Dirac pulse. At t = 2T , there are now two matter wave
gratings with period λ/2, one from paths 1 and 2 and one from paths 2 and 3. These move35

in opposite directions at a relative speed 4k/m. If the maxima of the two gratings line
up to produce large contrast at time t, the maxima of one will line up with the minima of
the other at t + π/4 ωrec , to produce zero contrast. This results in an oscillatory growthand decay of the contrast of the overall pattern with time. The recoil induced phase can be
determined from this temporally oscillating contrast.
Extending the two path analysis to three paths, we ﬁnd at time 2T ,
Φ1 = φ1 − 2φ2 − 8ωrec T,

Φ2 = 0,andΦ3 = −φ1 + 2φ2 − 8ωrec T.

Let ai be the amplitudes of the three paths determined by the splitting in pulse 1. The
wavefunction at time 2T is then|Ψ = a1 eiΦ1 |2k + a2 eiΦ1 |0 + a3 eiΦ1 | − 2k.
This is a 1-D problem, so lets use |2 k = ei2kz , etc. The atomic interference due to thethree paths at time 2T .
Ψ |Ψ = a1 e−iΦ1 e−i2kz + a2 e−iΦ1 + a1 eiΦ1 ei2kz × a1 eiΦ1 ei2kz + a2 eiΦ1 + a1 eiΦ1 e−i2kz= 2a21 + a22 + 2a1 a2 (cos(2kz + Φ2 − Φ1 ) + cos(2kz + Φ3 − Φ2 )) + 2a21 cos(4kz + Φ3 − Φ1 )
Φ1 + Φ3) cos(2kz + Φ3 − Φ1 ) + 2a21 cos(4kz + Φ3 − Φ1 )= 2a21 + a22 + 2a1 a2 cos(Φ2 −23). At time
The contrast of the 2kz matter wave grating is proportional to cos(Φ2 − Φ1 +Φ
2

2T , we can probe this matter wave grating by reﬂecting a probe beam oﬀ it. The process isequivalent to Bragg scattering of an optical travelling wave oﬀ a matter wave grating. The
intensity of the reﬂected light is then proportional to the square of the density modulation(strength of the matter wave grating). Thus the reﬂected signal is proportional to:cos2 (Φ2 −Φ1 + Φ3) = cos2 (8ωrec T ).
2

(2.41)

Note that Eqn. 2.41 has a phase which evolves with ωrec , the recoil frequency and does
not contain any of the φi ’s. This makes it insensitive to vibrations (i.e, phase shifts) of
the pulses. Allowing for phase oﬀsets due to diﬀraction phase shifts, light shifts and othereﬀects, the reﬂected signal becomes cos2 (4ωrec T + φoﬀ ). At an arbitrary time t, the reﬂectedsignal can then be written as
S(T, t) = C(T, t) sin2 (8ωrec T + 4ωrec (t − 2T ) + φoﬀ ).

(2.42)

where C(T, t) is an envelope function which goes to zero for t much diﬀerent from 2T . The
extent of C in the variable t − 2T is the coherence time of the matter wave grating. It iscaused by the momentum spread of the atom source, and can be quite large for a BEC.

36

The diﬀraction phase shift contributing to φoﬀ has been modelled for this interferometer inthe group of J. Vigué in France [60]. They ﬁnd a value π/6 for our geometry. Note that
this phase oﬀset has no direct consequence since we only want to determine the slope of themeasured phase vs T , as I discuss in the next subsection.

2.5.3

Advantages of the contrast interferometer

The phase interferometer of Fig. 2-7(a) captures the essential spirit of our measurement.
Pulse 1 starts the clock and creates a component of the wavefunction on path 1 whichaccumulates phase at a multiple of the recoil frequency. Path 2 serves as the zero phasereference. Since optical wavelengths have considerable recoil momentum, pulse 2 is used toreverse the momentum of path 1, without disturbing its phase evolution. At time 2T , the2 paths are back together. The phase of their interference is then a direct measure of ωrecif the time T is known.
Lets see how an error in the phase measurement is related to the uncertainty in therecoil frequency determined. We expect a linear relationship between phase and time:Φ = 8ωrec N 2 T + φoﬀ
where φoﬀ is the sum of all possible oﬀset phases. The relative errors are then given by:
8N 2 (ωrec ∆T + T ∆ωrec ) + ∆φoﬀ∆Φ=.
Φ8ωrec N 2 T + φoﬀ
We will be measuring large phases, so 8ωrec N 2 T  φoﬀ :∆Φ(ωrec ∆T + T ∆ωrec )≈.
Φωrec TThe relative error in the phase will be much larger than that in T . Thus:
∆ωrec
∆Φ
≈
,
Φ
ωrec

(2.43)

the relative error in the recoil frequency is simply the relative error in the phase4 . To improve
the precision of the experiment, we should try to reduce the error in the measurement of
the recoil phase and try to increase the largest recoil phase we can measure.
This phase vs time measurement is very similar in concept to the way mass measurements
are performed in Dave Pritchard’s Penning trap experiments at MIT [62]. Extending thescheme into a contrast interferometer suppresses several “common mode” noise sources,
improving both the precision and accuracy of the measurement.
4

Even if ∆T
is comparable to ∆Φ, the linear relationship between phase and time allows Eqn. 2.43 to stillTΦhold [61].

37Using Eqn. 2.43, one can make an estimate of what to expect from the recoil measurement. For 105 atoms contributing to the reﬂected signal detected at the shot noise level, the√phase error ∆Φ is ∼ 2/ 105 ∼ 6.3 mrad (the factor of 2 comes from using the cos2 function).
To set the scale, take N = 1 and T = 1 ms. Then Φ is 8 × 12 × 2π × 25 kHz×1 ms∼ 1.3 × 103−6and ∆Φ
Φ ∼ 5 × 10 . Scaling this value with T = 50 ms and N = 10, one would estimate∆Φ−9 i.e, 1 ppb!
Φ ∼ 1 × 10

This is the result for a single shot! Averaging over many shots

would beat this value down even further. Allowing the very conservative 60 s per shot (dominated by the formation time of the BEC), one would still obtain < 10−10 with less than 2hours of data. This simple calculation demonstrates the inherent power of the method. I
will return to this point in the ﬁnal section of this chapter.
The precision is a measure of the sensitivity of the device to the eﬀect of interest.
The device can also be sensitive to other systematic perturbations which if not properly
taken into account may make the ﬁnal measurement inaccurate. Almost all high accuracy
measurements are limited not by precision but by accuracy. Even if a small accuracy isquoted, there could still be some perturbation that the experimenter has not thought ofyet! For a quantity as important as α, it is thus important to make diﬀerent types ofmeasurements to see how they compare against each other. This is well exempliﬁed inFig. 2-6.
I now summarize the properties of this scheme which make it a viable choice for thephoton recoil measurement:• The use of a single internal state reduces systematics from AC Stark shifts. This has
been a major source of systematic in the Stanford experiment, which relies on Raman
transitions between diﬀerent internal states [55].
• The contrast scheme is insensitive to mirror vibrations between pulses. This is immediately obvious from the form of Eqn. 2.41. The vibration phase picked up by path 1
is opposite in sign to that picked up by path 3. Path 2 is not diﬀracted at all and so
3= 0 for the vibrational contridoes not pick up vibrational phase. Thus Φ2 − Φ1 +Φ
2

bution. Vibrations during a diﬀraction pulse are equivalent to a phase modulation.
This may make the grating depart from the Bragg condition. However, the typicalpulse durations of 10 µs are much shorter than the typical timescales of vibrations in
the laboratory ( 1 kHz). T , the interferometer evolution time, should be scaleable tohundreds of ms. The Stanford scheme uses a fairly elaborate vibration isolation system to ﬁght this problem for a phase interferometer [55]. Our contrast interferometershould not require such an elaborate system.
• The recoil phase is proportional to kinetic energy. This makes the interferometer
quadratically sensitive to the number of recoils. This is useful for scaling up theinterferometer towards high precision. For comparison, the Stanford scheme is linearlysensitive to the number of recoils.
38• The symmetry of the 3-path geometry suppresses eﬀects from stray magnetic bias
ﬁelds and gradients. This scheme is sensitive only to ∂ 2 B/∂z 2 , the magnetic ﬁeld
curvature5 . The time dependence of ambient magnetic ﬁelds enters the phases Φithrough the time integrals over the energies. Thus time-dependent stray bias andgradients are also suppressed.
• Immunity to accelerations, including those from gravity. This follows from the same
reasoning as the vibrational immunity. Insensitivity to gravity forgives to a large
extent any accidental vertical component to the beam alignment.
• The use of Bragg processes allows for arbitrarily high diﬀraction orders to be generated(limited by heating due to spontaneous scattering). This issue will be dealt with inmore detail in the Outlook section.
• The contrast readout allows for a high S/N, since the entire photon signal comes out ofa small solid angle (diﬀraction-limited by the size of the BEC) and the corresponding
background is thus reduced by this small fraction ÷ 4π.

2.5.4

Contrast Interferometer Signal
(a)

-1.0

(b)

-0.5

0.0
t (µs)

0.5

1.0

-10

-5

0
5
time t - T (µ s)

10Figure 2-8: Pulse 1 and pulse 2 of the interferometer sequence. (a) Pulse 1 is ∼ 1 µs long,square. (b) Pulse 2 is near-Gaussian shaped. The best ﬁt Gaussian with 1/e full width of
7.6 µs is also shown. The vertical scale for (a) and (b) are diﬀerent and arbitrary.
The “ﬁrst-generation” experiment was performed on the “old” sodium BEC machine,with horizontal laser beams. We wanted to demonstrate that the scheme worked and obtain
the best measurement of ωrec without major changes to the BEC machine. Learning fromthis initial phase could then be incorporated into plans for a high-accuracy measurement
using a vertical fountain geometry (Section 2.6).
5

All odd spatial derivatives are suppressed by the symmetry. If someone up comes with a way to cancel
the sensitivity to curvature, then we would only be sensitive to ∂ 4 B/∂z 4 .

39The ﬁrst generation experiment demonstrated the concept of the scheme and produced
a value for ωrec in sodium to 7 ppm precision and 200 ppm accuracy. This is several ordersof magnitude away from a competitive value. However the demonstration of the usefulfeatures of the scheme bodes well for future high accuracy work.
Each interferometer shot measured Φ(T ), the phase of S(T, 2T ) in Eqn. 2.42 for a given
T . The shot consisted of making a BEC in the magnetic trap, releasing it for a variable
TOF, followed by the 3 pulse sequence forming the contrast interferometer and then readout
by Bragg back-scattering. We also monitored the atom number by imaging the atoms after
a long time-of-ﬂight (Fig. 2-10). The standard technique for |F = 1, mF = −1 BEC
production was followed[63, 9]. We loosened the trap at the end of evaporation to ﬁnal
trapping frequencies of about 50 Hz radially and 20 Hz axially. Almost pure BECs (negligible
thermal component) of a few million atoms were then released suddenly and allowed to
expand for ∼ 15 ms before the application of the interferometer pulses. Loosening the trap
eﬀectively “decompresses” the atoms thereby reducing the density. Both the decompression
and initial TOF stages were important to lower the atomic density (to ∼ 1013 cm−3 ), thereby
reducing superradiance [64] and mean-ﬁeld eﬀects. The initial decompression also lowered
the momentum spread and allowed for more eﬃcient Bragg diﬀraction (2nd pulse).

readout
pulse

contrast
signal
960

t=2T
980

1000

1020

1040

t [ µs]Figure 2-9: Typical single shot signal from the contrast interferometer. T = 502.5 µs, forthis example. Ten oscillations with ≈ 60 % contrast and ≈ 30 µs width are observed during
the 50 µs readout. A low-pass ﬁlter at 300 kHz (12dB per octave) was applied to the signal.
To form the interferometerof Fig. 2-7(b), we phase-locked two SRS DS-345 synthesizers
and applied the same frequency to both AOM#1 and AOM#2 for all pulses. Examplesof pulses 1 and 2 are shown in Fig. 2-8. We used laser beams with 1.8 mm waist. For
pulse 1, the Kapitza-Dirac symmetric splitting pulse, we obtained ∼ 25% of the condensate
40

z

y

pulse 1 and 2 only

pulse 1, 2 andstrong pulse 3

Figure 2-10: Atom signal from the contrast interferometer. When only pulses 1 and 2 wereapplied, the TOF atom signal was symmetric about the 0 momentum state. When a strong
readout pulse was applied, clear asymmetry is observed between the |2 k (upper) and|−2 k (lower) states, in keeping with the direction of reﬂection of the readout pulse (Fig. 23). These pictures are from the ﬁrst night that we observed the interferometer signal andprovide clear evidence that we had reﬂection from a matter wave grating. For actual datacollection we reduced the intensity of the readout pulse, and measured the backscattered
light.
in each of paths 1 and 3 (Eqn.2.32). For the second-order Bragg grating (pulse 2), we
used a near-Gaussian light pulse (to suppress other unwanted orders of diﬀraction) andobtained a diﬀraction eﬃciency greater than 90%. The timing control computer used in all
our BEC experiments has a resolution of 10 µs and is therefore inadequate to control thesepulses. We used Wavetek 50 MHz pulse generators, Model 801 to trigger the Bragg AOMs.
For the readout pulse, we simply switched oﬀ AOM#2 and back-reﬂected the light from
AOM#1 onto a photomultiplier tube (PMT, Fig. 2-3). A typical signal from the contrast
interferometer is shown in Fig. 2-9.
There was some eﬀort to obtain this “typical” signal. Since the probing beam andthe reﬂected beam counter-propagated and essentially overlapped each other, we used a
non-polarizing beam splitter cube (BS#2) to separate them (ﬁgure 2-3). However, this alsomeant that the probe beam reﬂected oﬀ BS#2 and other optics (including the vacuum
windows) could get into the PMT. Angling the beam relative to the optics by a small
amount (few degrees) was enough to get rid of most of the reﬂections by physical blocks.
The light due to the leakage RF driving AOM#2 was blocked with a mechanical shutter.
A pinhole (100 µm) was placed at an image plane of the atoms to allow only the light fromthe condensate to go through towards the PMT. This pinhole was in turn imaged onto the
PMT, to further improve the signal-to noise.
Fig. 2-10 shows the atom signal corresponding to the contrast interferometer. For astrong readout pulse, an obvious asymmetry was observed between |2 k and | − 2 k .
This was our ﬁrst sign (even before we obtained a PMT signal) that the contrast scheme

41was working. Such a strong readout quickly destroys the matter wave grating. We lowered
the readout pulse and measured perturbatively to obtain the recoil frequency.

2.5.5

Measurement of the Recoil Frequency

We ﬁt the reﬂected signal with a gaussian-weighted sinusoid and obtained the recoil phase
Φ(T ). The results of ﬁve iterations of the signal at diﬀerent time points T are shown in Fig. 211. We chose two ranges of times T for our measurements. The short time T ∼ 0.5 ms, wasrestricted by the closing time of a mechanical shutter (∼ 300 µs) which blocked the leakage
from AOM#2 (MS2 in Fig 2-3). The long time T ∼ 3 ms was limited by the atoms falling

3780
3770

Residuals[radians]

Recoil phase,φ (T)[radians]out of the (∼ 2 mm diameter) horizontal beam due to gravity.

(a)

650
640
630
0.50
0.5

(b)

0.51

3.00
0.52
Time T [ms]

3.01

0.0
-0.5

Figure 2-11: Measurement of ωrec in sodium. Two sets of recoil phase scans, around T =0.5 ms and T = 3 ms, are shown in (a). Each point is the average of ﬁve measurements. The
slope of the linear ﬁt gave ωrec to 7 ppm. The error bars (≈ 0.05 − 0.1 rads) are shown withthe ﬁt residuals in (b). The total time for this measurement was ∼ 2 hours.
Our measurement for sodium from these data is ωrec = 2π × 24.9973 kHz(1 ± 6.7 × 10−6 ).
The precision was limited by ∼ 200 mrad shot-to-shot variations of the ﬁtted value of Φ.
Our measured value is 2×10−4 lower than the sub-ppm value calculated using the publishedmeasurements of αg−2 [65, 66], R∞ [67, 68], MNa [62], Me [69], and λNa [70] in Eqs. 2.38 and2.39.
The shot-to-shot deviation of 200 mrad far exceeds the expectation from the shot noise
limit for ∼ 105 atoms (∼ 6.3 mrad, Section 2.5.3) which is approximately the atom number
contributing to our signal. It is possible that this is due to ﬂuctuating diﬀerential meanﬁeld shifts in the diﬀerent interferometer arms arising from ﬂuctuations in atom number.
This is discussed in Section 2.6.2. Another possibility is the diﬀerent diﬀraction phase shifts
from ﬂuctuations in light intensity as suggested by [60].
422.5.6

Insensitivity to Vibrations
0.8

(a)

(b)

2

Zero order fraction

Recoil phase φ [radians]

3

1
0
-1

0.6

θ
0.4

0.2

-2

T=502.5µs

T=50µs

-3

0.0
0.0

1.0

2.0

3.0

0.0

Applied phase, θ [radians]

1.0

2.0

3.0

Applied phase, θ [radians]

Figure 2-12: Vibration insensitivity of the contrast interferometer. (a) shows the measured
recoil phase at T = 502.5 µs from the contrast interferometer as a function of the applied
phase θ. The recoil phase is constant and demonstrates our insensitivity to phase noise fromthe gratings. (b) shows the fractional population of the |0 k state from the phase-sensitive
interferometer (inset) for a similar scan of θ at T = 50 µs. Also shown is the best-ﬁt sinusoidof the expected period.
To demonstrate the insensitivity of the measurement to phase noise of the light due
to mirror vibrations, we intentionally varied the phase θ of the second grating relative tothe ﬁrst one. The contrast signal was not visibly aﬀected by such phase variations (Fig. 212(a)). We compared this to a phase-sensitive readout method (Fig. 2-12(b), inset). Thiswas realized by replacing the readout pulse with a third pulsed 1 µs light grating in the
Kapitza-Dirac regime, phase-locked to the ﬁrst two pulses. This projected the phase of
the 2k pattern at t = 2T onto the fractional populations of the states |0 k, |2 k, and| − 2 k which leave this interferometer. The populations were measured by time-of-ﬂight
absorption imaging. The |0 k fraction is shown for the same variation of θ, in Fig. 2-12(b).
The oscillation demonstrates the phase sensitivity of any position-sensitive readout.
Both the Mach-Zehnder interferometer signal (Fig. 2-5) and the phase interferometersignal (Fig. 2-12(b)) became much more noisy and essentially random for T ≥ 3 ms. Thisis probably due to the mechanical vibrations of the apparatus. The contrast interferometeron the other hand retains a steady signal as can be judged from the small error bars of
Figure 2-11 at the large times.

43

2.5.7N 2 scaling
(b)

(a)

N=1

650
φ(T1) [radians]

N=2
N=1
T

N=2

640

x

0 T T
1
60µs 60µs

630
2T 2T+T1380

390
T1 [µs]400

Figure 2-13: Demonstration of the quadratic scaling of the recoil phase with additional
photon recoils. (a) shows the N = 1 (inner) and N = 2 (outer) interferometers used. (b)shows the recoil phase at the recombination time under variation of T1 .
The quadratic scaling with number of recoils N was demonstrated using the interferometer geometry shown in Fig. 2-13(a). For an intermediate time T1 , paths 1 and 3 weretransferred into states |4 k and | − 4 k respectively. The two paths were simultaneously
addressed by driving AOMs 1 and 2 with two frequencies (using an RF mixer), centeredat 29 MHz (near the AOM central frequency) but separated by 300 kHz. The pulses weresquare and 10 µs long and adjusted in intensity to match the 1st order Bragg π-condition.
Paths 1 and 3 were then resonant with only the frequencies (one from each beam) satisfying
the Bragg condition.
During the period T1 , paths 1 and 3 accumulate phase 22 = 4 times faster in the
N = 2 scheme than in the N = 1 scheme. Additional time T1 is required for the threepaths to overlap in the N = 2 scheme. For this geometry, the N = 2 recoil phase should
therefore evolve three times faster as a function of T1 than the N = 1 recoil phase. The
corresponding data sets are shown in Fig. 2-13(b). The linear ﬁts give a slope ratio of
3.06 ± 0.1, demonstrating the quadratic scaling.

2.6

OutlookThe ﬁrst-generation recoil measurement demonstrates the validity of the contrast interferometer scheme. Having accomplished this, we can move on to planning a future highaccuracy measurement. The purpose of this section is to outline our preliminary ideas inthis direction. Subsection 2.6.1 describes conceptually how a vertical geometry with an in44

creased number of diﬀraction pulses can scale up the current precision by a few orders ofmagnitude. Subsection 2.6.2 provides some estimates of the BEC mean ﬁeld contribution
to both precision and accuracy.

2.6.1

Scaling up the Contrast Interferometer - improving the precision

As argued earlier (Section 2.5.3), scaling up the precision of the contrast interferometer involves increasing the interferometer time T , increasing the number of recoils N and reducing
the shot-to-shot measurement error ∆Φ. The limitations on T are mainly from gravity andﬁnite beam size. This can be remedied by setting up a vertical geometry where the atoms
fall down the beams and not out of the beams. For a given vertical extent, even longer times
can be achieved by launching the atoms upwards before starting the interferometer pulse
sequence, as shown in Figure 2-14. The parameters of T = 50 ms and N = 10 chosen for theﬁgure should allow ppb precision for detection at the shot noise limit (Section 2.5.3). One
drawback of the vertical geometry is that Doppler corrections to the relative frequencies of
the Bragg gratings are needed to compensate for the changing velocities of the dropping
atoms. However, this is a small overhead if such an elaborate setup is indeed constructed.
40

gravity

Position (mm)

30

20

1

10

2

0

3

-10

0

20

40

60

80

100

time (ms)Figure 2-14: Scaling up the contrast interferometer - vertical fountain conﬁguration. The 3
path scheme in a vertical geometry allows long evolution times and multiple pulses without
the problem of the atoms “falling” out of the beams. The case of a fountain launch withvelocity 0.5 m/s, Bragg π-pulse acceleration up to N = 10 recoils and T = 50 ms, is shown.
This setup should approach the ppb-precision and is only 6 cm in vertical extent.
Fountain geometries can provide free fall times of a few hundred milliseconds[71]. Additional Bragg acceleration π-pulses to increase the interferometer order N can then be45

applied.

2.6.2

The Mean Field Contribution

The ﬁrst-generation experiment has yielded a result for ωrec,Na accurate to 2×10−4 . We havenot performed experimental studies of systematic eﬀects in the interferometer. However,
we believe the mean ﬁeld to be the main contribution to the deviation of our measurementfrom the accepted value. Here we provide some estimates of the contributions from themean ﬁeld due to atom-atom interactions in a BEC.
3
, whereAs shown in Eqn. 2.41, the phase of the reﬂected signal at time 2T is Φ2 − Φ1 +Φ2

Φi is the phase accumulated by path i. Any mechanism which can shift the relative phases
3
is aﬀected, is a possible source of a systematicof the three paths such that Φ2 − Φ1 +Φ
2

error. Atom-atom interactions (mean ﬁeld) can introduce errors into the measurement inthis manner.
Strong atom-atom interactions in dense sodium BECs can easily produce mean ﬁeld
energies as large as 2π × 5 kHz [72]. Since ωrec,Na ∼ 2π × 25 kHz, energy shifts due to meanﬁeld interactions can be an important systematic in our interferometer. A gas of ultracoldbosons at density n and interacting with s-wave scattering length a, has a mean ﬁeld energyof[73]6 :
Emf =

4π2
na
m

(2.44)For 23 Na in the |F = 1, mF = −1 state, a = 2.8 nm. This means that a density ofn = 3.4 × 1014 cm−3 produces Emf = 2π × 5 kHz. If diﬀerent arms of the interferometer3
may be aﬀected. If we have control overhave diﬀerent densities, then Φmeas = Φ2 − Φ1 +Φ
2

the interferometer pulses to allow a maximum fractional arm imbalance of x, the systematic
contribution will be:

Emf 2T x∆Φ=Φmeas
8ωrec N 2 T(2.45)

In our ﬁrst-generation measurement, we used a peak density of n ∼ 1013 cm−3 . For an
arm imbalance of ∼ 25%, this would contribute at a relative level of ∼ 4 × 10−4 , twice theobtained error. Further, if the mean ﬁeld is larger in path 2 than the sum of the mean ﬁeld
in paths 1 and 3, the interferometer will measure a lower value for ωrec , since the diﬀerence
in total energy between an extreme path (1 or 3) and path 2 is then smaller that 4ωrec .
This is consistent with our measurement being less than the currently accepted value.
We also note that atom number ﬂuctuations will change the density shot-to-shot andgive rise to variations of the measured recoil phase. Few percent variations in arm imbalance
could then explain our observed shot-to-shot variations of ∆Φ ∼ 200 mrad (Section 2.5.4).
The way to reduce the mean ﬁeld contribution is to control the interferometer imbalance
and reduce the atomic density. Further, even though this systematic scales with the inter6

a discussion of basic scattering processes can be found in Chapter 346

ferometer time T , it is insensitive to the number of recoils N . Using x = 0.05 and N = 20
in Eqn. 2.45, we get the requirement of n  1011 cm−3 in order to be at the ppb-level. Suchlow densities have recently been achieved in sodium BECs[74].
Another method to suppress the mean ﬁeld contribution would be to alter the scattering
length a to negligible values by means of a magnetic ﬁeld Feshbach resonance (Section 3.3).
This may be possible to do with the recently achieved cesium BEC because of the availabilityof such resonances at low magnetic ﬁeld [24].

47Chapter 3

Interacting Fermi Gases
From the late 70’s to the early 90’s, the prospect of creating a dilute gas superﬂuid fromatomic vapor was a holy grail in the ﬁeld of ultracold atoms. The realization of such asuperﬂuid - the Bose-Einstein Condensate in 1995 opened the doors for a rich variety of
exploration, both experimentally and theoretically. The previous chapter on atom interferometry with BEC’s represents one such avenue of research.
Fermionic superﬂuidity is the next holy grail which has emerged in the ﬁeld of ultracold
atoms. At very low temperatures, interacting fermions can undergo a pairing transition
resulting in a BCS type superﬂuid. The quest for this is now well underway with fermions
being routinely produced in the quantum degenerate state in 6 laboratories around theworld. This degeneracy is an essential ingredient towards superﬂuid pairing. Our groupat MIT is one of these six to produce Fermi degenerate gases. The next step is to induceinteractions between fermions. Several groups, including ours, have made steady progressin this endeavor over the last couple of years. The rest of this thesis deals with our eﬀorts
at inducing and studying strong interactions in ultracold fermions.
In this chapter, I discuss theoretical concepts which play a key role in understanding
the current state of the ﬁeld of interacting fermions. Although this chapter nominally deals
with theory, I try to make the connection with the relevant experimental aspects in various
parts of the presentation. The discussion is divided into ﬁve groups:(1) In Section 3.1, I introduce the non-interacting Fermi gas and its static properties ina harmonic trap. Strength of interactions can be gauged by the (observed) deviations of
the gas from this ideal behavior.
(2) In Section 3.2, diﬀerent types of collision processes are introduced with particularattention given to s-wave collisions, the dominant interaction mechanism in ultracold gases.
(3) In Section 3.3, I introduce the concept of a Feshbach resonance. This is our main
experimental tool to tune interaction strengths between fermions.
(4) In Section 3.4, I discuss the ground state hyperﬁne structure of 6 Li, the fermionic
atom that we use for experiments. Using the ideas from the previous sections, I outline thereasons why certain choices of internal states are more suited for studying interactions.
48

(5) The ﬁnal part of this chapter, Section 3.5, discusses the predicted transition of interacting fermions to a superﬂuid state.

3.1

Non-interacting Fermi Gases

An intermediate goal towards BCS is the production of degenerate Fermi gases. We canproduce a spin-polarized 6 Li Fermi gas in a harmonic trap at ultra high degeneracy. Sinces-wave interactions are forbidden by fermionic antisymmetry, this is essentially a noninteracting gas. A full description of such gases can be found elsewhere[75]. I present
here a few basic results. The Fermi-Dirac distribution function isf (3) =1eβ(−µ) + 1,

where β = kB1T . f (3) is the occupation probability of a state with energy 3. The chemical
potential µ is given by the normalization condition for the total number of atoms N in the
trap:
N=

d3f (3)g(3),(3.1)

where g(3) is the density of energy states. Only harmonic traps with cylindrical symmetry will be discussed (the usual experimental scenario). If the trapping frequencies are(ω1 , ω2 , ω3 ) = (ω⊥ , ω⊥ , ωz ), and deﬁning the aspect ratio λ = ωz /ω⊥ , the density of states2g(3) = 2λ(ω
3 has a quadratic dependence on energy. Integrating Eqn. 3.1 at T = 0 gives⊥)

the Fermi energy EF (N ) ≡ µ(T = 0, N ):EF = ω⊥ (6N λ)1/3(3.2)

Since the exact wavefunctions in a harmonic trap are well-known, the properties of a harmonically trapped gas can in principle be directly computed by summing over these states.
However it suﬃces (and is usually computationally necessary) to make the semi-classical“Thomas-Fermi” approximation which is valid in the large N limit [75] and consists of labelling the state of each atom by a position x and momentum p. This is equivalent to the
local density approximation where each position x is associated with local Fermi quantities- density, temperature, and the Fermi temperature. Essentially, the gas is considered locally homogeneous and all the formulas used to describe homogeneous Fermi gases can beapplied. The local Fermi energy is given by:
EF (x) =2 kF2 (x)= EF − Uh (x),2m49where kF (x) = pF (x)/m is the local Fermi wavevector and Uh (x) = 122 2i ωi xi .

This now

gives the local density as simply the volume of the momentum space Fermi sea:n(x, T ) = −where ΛT =

2π2
mkB T

1
g 3 (−e−(Uh (x)−µ)/(kB T ) ),
Λ3T 2

is the thermal de-Broglie wavelength and gα (x) =

(3.3)
∞

k
α
k=1 x /k is thepoly-logarithm function. The zero-temperature density reduces to:k 3 (x)4n(x; T = 0) = F 2 = π
6π
3



2mEF
h2

3/2

(1 −

m
Σi ωi2 x2i )3/2
2EF

(3.4)

We now have an expression for a direct experimental observable. Eqn. 3.3 forms the basis
of ﬁtting functions for characterizing our experimentally produced degenerate Fermi gas(Chapter 4). Our standard detection method of absorption imaging measures a 2-D “column” density distribution. The ﬁtting function is then a spatial integral of Eqn. 3.3 which
is an expression in the poly-logarithm function g2 . The 1-D projection involves another spatial integral and results in g5/2 . We use both 2-D and 1-D ﬁts to characterize our ultracoldgases.
Absorption imaging can also be applied after turning oﬀ the trap suddenly and allowing
the gas to expand ballistically for a variable time-of-ﬂight (TOF). The spatial distribution
in TOF from a harmonic trap is obtained by a substitution of coordinate xi by √ xi 2 2 ,accompanied by the necessary volume rescalingΠi√1

1+ωi2 t2

1+ωi t[76]. Thus, the density distri-bution for a non-interacting zero-temperature Fermi gas in time-of-ﬂight out of a harmonic
trap is (from Eqn. 3.4):4n(x, t; T = 0) = π
3

3.2



2mEF
h2

ωi2 x2i 3/2
m
Σ
)
i
2EF
1+ωi2 t22
Πi (1 + ωi t2 )1/2

3/2 (1 −(3.5)

Interactions in Trapped Fermi Gases

At the heart of every interaction is a collision process. In this section I introduce thediﬀerent collision processes that will be discussed in this thesis.
For the µK temperatures common in ultracold atom traps, interactions are dominatedby two-body s-wave scattering. This is because, for typical collisions, kR  1, wherek is the relative momentum (determined by the temperature) and R is the range of thepotential. This is equivalent to the de-Broglie wavelength being much larger than the rangeof the interatomic potential. The de-Broglie wavelength of 6 Li at 1 µK is ∼ 200 nm, while
the potential range R ∼ 50a0 ∼ 2.5 nm. Since a spin-polarized Fermi gas is non-interacting,all the s-wave interactions between 6 Li atoms will occur in mixtures of diﬀerent internal
states. Most of the activity in the ultracold atom ﬁeld takes place in the weakly interacting

50

(k|a|  1) and dilute (n|a|3  1) limit where a is the s-wave scattering length whichparametrizes the strength of the interaction and n is the gas density. This allows the useof an eﬀective contact interaction between particles [73, 77, 78]:
U (x − x ) =4π2 aδ(x − x ) = U0 δ(x − x ),
m

(3.6)

where m is the mass of an atom. In this description, particles only interact when they are“on top of” or “in contact with” each other. The determination of the scattering lengtha involves solving the Schrodinger equation for scattering from the molecular potentials.
Although straightforward in principle, this scattering problem is fairly diﬃcult to solve
accurately, principally because of insuﬃcient knowledge of the real molecular potentials.
The interaction potential of two ground state alkali atoms is relatively simple comparedto multi-electron atoms since alkalis possess only one valence electron. The interaction potential is composed of two parts - one corresponding to the electric dipole-dipole interaction,
often called the central part U c and the other corresponding to the magnetic dipole-dipole
interaction Umd [73]. The central part can be divided into two broad classes of interactions:
U c = Us P0 + Ut P1 ,(3.7)

the singlet Us and the triplet Ut interactions. P0 and P1 are the projection operators forthe two-electron singlet and triplet states respectively. The singlet interaction corresponds
to the valence electrons approaching in an antisymmetric spin combination, whereas thetriplet interaction involves them approaching in a symmetric spin combination. As may be
expected, the singlet and the triplet potentials are identical at large distances, decaying withthe usual attractive van der Waals 1/r6 behavior. The diﬀerence occurs only in the shortrange part where the spatial wavefunction is symmetric (singlet) or antisymmetric (triplet).
At very short internuclear distances, the potential is essentially hardcore repulsive.

3.2.1

Collision Channels

A set of internal quantum numbers (like the spin projection, total spin and so on) which
describes a two-particle state is referred to as a “channel”. In this language, collisions
which only couple the entrance channel to itself are single-channel processes whereas thosewhich can couple the entrance channel to a diﬀerent exit channel are called multi-channel
processes.
The kinetic energy of s-wave collisions is usually too small to allow endothermic processes
between diﬀerent channels which have energy separations on the order of hyperﬁne energies.
Thus, in ultracold atoms, the entrance channel is often called the “open” channel while
higher lying channels are called “closed” channels (Fig. 3-1). Atoms cannot leave each otherafter a collision in any closed channel.

51

Closed channel

Energy∆Ehf
Open channel

Internuclear Separation (r)Figure 3-1: Intermolecular potentials for 2 body scattering. The free atoms approach along
the open channel. The closed channel corresponds to a molecular state that can couple tothe open channel via U c or Umd . The diﬀerence in threshold (kinetic energy K.E. → 0)energies ∆Ehf between the two channels is on the order of hyperﬁne energies and typicallymuch larger than the kinetic energies available in ultracold atom clouds.

3.2.2

Inelastic Collisions

Inelastic collisions change the total kinetic energy of collision partners by changing concurrently some internal degree of freedom. Typically, these result in a decay of the system
under study. I review very brieﬂy the standard inelastic decay mechanisms that occur inatom trapping experiments.
Background gas
Collisions with the background gas in a vacuum chamber causes an exponential (singlebody) loss: Ṅ = −N/τ . The eﬀects of these collisions can be reduced by improving thepressure in the system. Pressures of ∼ 10−11 mbar, the ultra-high-vacuum (UHV) typical inultracold atom experiments, results in lifetimes of several tens of seconds (or even hundreds).
In our experiments, we take extra precautions to maintain these UHV conditions.
Spin Exchange Collisions
This process involves two-body collisions from the central potential U c which change theinternal states of the collision partners while preserving the total spin. The rate for thisprocess is proportional to atomic density (since it is a 2-body process) and independent52

of relative velocity. Therefore it becomes an increasing problem in an atom cooling experiment as the temperature is lowered and the density increases. The usual solution is tochoose atomic states in which either spin-exchange collision channels do not exist or are
endothermic (as shown in the example of Fig 3-1) and thus suppressed at low temperatures.
Dipolar Decay
This process involves two-body scattering from the interatomic magnetic dipole-dipole potential Umd . Total spin need not be preserved in such collisions. The potential can change
the relative angular momentum of the spatial wavefunction by 2[73]. The rate for thisprocess is also proportional to atomic density, however, as mentioned earlier, it becomessigniﬁcant only if spin-exchange collisions are suppressed. As usual, experimental statesshould be carefully chosen to eliminate or at least minimize this loss channel.
Three-Body Decay
Three-body recombination is a process in which two atoms form a bound state and a third
atom carries away the energy (including molecular binding energy) and momentum to satisfy the usual conservation requirements. Since three particles need to collide, the rate scaleswith the square of the density. For dense BECs, this is usually the loss mechanism that
limits the lifetime in an atom trap. Three-body decay can also be enhanced near scatteringresonances (Section 3.3) and form a strong decay channel in such regimes. However, thesymmetry/antisymmetry of the collision partners can suppress this rate since the process
does require multiple particles to be in close proximity. As one might expect, at low temperatures, the rate for fermions is indeed suppressed. Threshold (K.E. → 0) laws have been
derived for 3-body recombination in [79]. These calculations show that for a spin-polarizedFermi gas, the rate is suppressed as T 2 (or TF2 for T < TF ) and for a 2-spin gas, as T (orTF ). Bosons as well as distinguishable particles approach a rate independent of energy atthreshold.

3.2.3

Elastic Collisions

Elastic collisions refer to scattering from the central potential U c in which the incoming
and exiting channels are the same. They play an important role in the ﬁeld of ultracold
atoms because they preserve the internal state of the system and only change the external
state towards thermal equilibrium. Elastic collisions have to always compete against thedeleterious eﬀects of inelastic collisions, to preserve the stability of the system.
S-wave elastic collisions at low temperature in weakly interacting and dilute gases lead toenergy shifts in the form of mean ﬁeld interactions. The connection between thermalizing
collisions and interaction shifts will now be discussed. They are both important in thestrongly interacting regime that we want to explore.

53

The s-wave scattering amplitude
Eqn. 3.6 describes the eﬀective contact interaction that is used to describe elastic interactions
2

ain ultracold gases. It correctly predicts the mean ﬁeld shift as 4π
m n for interaction of aparticle with a gas of density n. However, it is only correct to lowest order in ka. This thesis
will be concerned with behavior of ultracold gases for ka  1. Therefore, I now introduce
the more correct form for the contact interaction where the scattering length a is replacedby −f0 (k), the s-wave scattering amplitude [77, 73]:
U (x − x ) = −4π2 f0
δ(x − x ).
m

(3.8)

The scattering amplitude can be derived from the usual partial wave analysis for a spherically symmetric potential and is related to the s-wave scattering length as [80]:f0 (k) = −aka2a(1 − ika)
a
+
i
.
=−
=−
1 + ika(1 + ika)(1 − ika)
1 + k 2 a2
1 + k 2 a2

(3.9)

The real part corresponds to the usual mean ﬁeld interaction, except now we havethe behavior appropriate for ka > 1. The imaginary part corresponds to the decay ofthe (relative) momentum state and is responsible for the thermalization of the system.
The optical theorem relates the imaginary part to the more common expression for thethermalization scattering cross-section:Im(f0 ) =

kσ
4π

=⇒

σ=

4πa2
1 + k 2 a2

(3.10)For small ka, this is recognizable as the s-wave cross-section for distinguishable particles, independent of a.
4πa2 . For large ka, this approaches the unitarity limit 4π
k2
The real part corresponds to collisions that do not change the relative momentum of thescatterers. These collisions are experimentally detectable as energy shifts. The imaginary
part contributes to what is usually meant by elastic collisions. Using Eqns. 3.8, 3.9, and 3.10,
the scattering cross-section can be used to rewrite the imaginary part of the interaction (fora gas density n) as:
ka2
4π2
n
= nσv = Γscat
m 1 + k 2 a2
where Γscat = nσv is the usual scattering rate for collisions at relative velocity v.
Eqn. 3.9 also suggests the analogy of an atom wave and a light wave propagating throughan atomic medium of density n. The real part corresponds to the refractive index while theimaginary part to the absorption in the medium. Thus, the real part gives rise to “coherent”
scattering while the imaginary part to “incoherent” scattering.
It is important to re-iterate that this form of the interaction is valid for any value of ka
provided kR  1. In particular, it may be used to understand (at least qualitatively) the54behavior near scattering resonances, where a diverges (Section 3.3).
Introducing the scattering amplitude thus draws the distinction between coherent andincoherent collisions and extends our interpretation of s-wave interactions into the ka  1
regime near Feshbach resonances.
Low temperature, Pauli blocking
Fermions at temperatures satisfying kR  1 are below the p-wave threshold and do notinteract in the same state1 . Another qualitative threshold occurs when the gas becomes
deeply degenerate and a large fraction of states below TF are occupied. As a rule of thumb,this is about 0.3−0.4 TF , since this is where observed collision rates start to deviate strongly
from the classical predictions [81]. The deviation arises due to the signiﬁcant populationof energetically accessible ﬁnal states for an elastic collision. This eﬀect, often called Pauliblocking, reduces the rate of incoherent collisions and decreases the eﬃciency of coolingfermions. The two energy scales corresponding to kR ∼ 1 (mK) and TF (µK) are separated
by about three orders of magnitude in typical atom trapping experiments. As the system
cools, the s-wave regime is reached ﬁrst. Substantial further cooling must take place before
the regime of Pauli blocking is reached.
Since coherent collisions do not involve an overall change in momentum of collisionpartners, they are not aﬀected by Pauli blocking of ﬁnal states. Thus, there are no incoherent
elastic collisions in a trapped zero-temperature two-spin Fermi gas with ﬁnite interstate
scattering length. However, mean ﬁeld energy exists in such a gas. As we will see in
Chapter 6, this zero-temperature Fermi gas may undergo elastic collisions when released
from the atom trap.

3.2.4

Observations of Interactions

Typically, inelastic collisions in a trapped atom cloud result in a release of kinetic energy
and/or the production of un-trapped states. Thus the experimentally observed gas (column)
density will correspond to an increased temperature as well as a reduced density. The system
will thus “decay” away. Generally, a stable system corresponds to a lifetime limited only
by background losses. A study of the stability of a strongly-interacting system 6 Li system
by monitoring inelastic losses is reported in Chapter 5 [4].
We have also experimentally observed eﬀects from both the real and the imaginary partsof the scattering amplitude. The eﬀect of incoherent elastic collisions was observed in themutual thermalization of a two-spin state mixture (Section4.5). Chapter 6 discusses eﬀects
that manifest due to a very large rate of incoherent collisions as well as the role of Pauliblocking.
1

p-wave interactions do contribute, but only with a cross-section which goes to zero according to the
Wigner threshold law[80]55The eﬀect of coherent elastic collisions can be observed in shifts of transition frequencies
between internal atomic states. Chapter 7 describes our use of radio-frequency techniques
to measure such shifts and thus scattering lengths.

3.3

Feshbach Resonance

Scattering resonances correspond to a divergence of the scattering length. One instance ofsuch a resonance occurs when a vibrational state in the open channel becomes just boundat threshold (K.E. → 0). This is a phenomenon familiar from the treatment of scatteringfrom a square well potential. The zero-energy scattering length diverges as a function ofwell depth, whenever the potential can just support a new bound state. Another example
is a “shape” resonance. A shape resonance occurs if the potential characterizing a collision
process can support a virtual bound state at the collision energy. This occurs for kR  1,
higher partial wave scattering.

Bound state

EnergyClosed channel

Open channel

Internuclear Separation (r)Figure 3-2: Schematic of a Feshbach resonance. The solid lines represent the potential
energy as a function of internuclear separation. The free atoms interact via the open channel.
The closed channel corresponds to the potential experienced by atoms in a diﬀerent twoparticle superposition of hyperﬁne states. The two channels are coupled by the hyperﬁneinteraction. A resonance occurs if a bound state in the closed channel crosses the open
channel asymptote. The eﬀective scattering length corresponding to the ﬁgure is large andnegative.
The above examples are single-channel eﬀects and can occur in atoms which have nointernal structure. However, the presence of internal structure allows multi-channel scatter-56

ing to occur via the electrostatic potential U c (Section 3.2). This allows the possibility ofFeshbach resonances which occur when a bound state in one of the closed channels comes
into resonance with the kinetic energy in the open channel (Figure 3-2). Far away fromscattering resonances, a will have a small non-resonant value. Typically, this is on the orderof the interatomic potential range (∼ 50a0 ). However, near a Feshbach resonance, one of
the molecular bound states becomes important and dominates the scattering behavior. If
a molecular bound state has a diﬀerent magnetic moment than the total magnetic momentof the free atomic collision partners, then the position of the bound state can be adjusted
relative to the open channel threshold by simply changing the external magnetic ﬁeld. The
scattering length undergoes a resonance when the bound state is tuned across this threshold. Such a resonance is known as a Feshbach resonance. Near a Feshbach resonance, thescattering length changes from a → ∞ for bound state below threshold (real molecule) toa → −∞ bound state above threshold (“virtual” molecule). Fig. 3-2 corresponds to a large
attractive interaction for atoms at zero kinetic energy.
The usual parametrization of a near a Feshbach resonance is[82]:a(B) = anr (1 +∆),B − B0(3.11)

where anr is the non-resonant scattering length far away from the resonance (of the orderof the range of the potential R). B0 is the position and ∆ is the width of the Feshbach resonance. The positions and widths of Feshbach resonances can be calculated by performing
coupled-channel calculations using potentials derived from up-to-date experimental data.

3.3.1

Strongly Interacting Fermi Gases

Resonant Enhancement of the Scattering Length, Unitarity Limit
Strongly interacting gaseous systems can be “engineered” by utilizing Feshbach resonances.
At a divergence of a, the behavior of the eﬀective scattering length aeﬀ = −f0 can beobtained by taking the a → ∞ limit of Eq.3.9:
a→∞⇒Re(aeﬀ ) →

1
k2 a

and

1
Im(aeﬀ ) → − .
k(3.12)The real part approaches zero near a scattering resonance, which may seem astonishingat ﬁrst. However, this is understandable on the basis of the analogy with the dispersive
light-atom interaction. The real part of the index of refraction goes to zero on resonance.
The imaginary part of aeﬀ reaches an a-independent plateau corresponding to the usual
unitarity limit of collisions. The behavior of the eﬀective scattering length aeﬀ is shown in
Fig 3-3 as a function of the bare scattering length a.
In a gas, the momentum-dependent aeﬀ should be averaged over the diﬀerent collision
momenta. For a two-spin gas, each with Fermi momentum kF , the momentum average at
57

(a)

(b)
Im(f0)-Re(f0)
1

2

0.8

1.5
0.6

1
0.4

0.5

0.2
5

10

15

5

20

Scattering Length a

10

15

20

Scattering Length a

Figure 3-3: The thin lines show (a) -Re(f0 ) and (b) Im(f0 ) as a function of a (Eq. 3.9 withrelative momentum k = 1). The thick lines show the averages over a zero temperature
Fermi distribution with Fermi momentum kF = 1 for each spin state. The thick line in (a)deviates from the thin line for a ≥ 1(1/kF ) and approaches zero much more gently. The
thick line in (b) has a similar rise as the thin line with a but saturates at the higher valueof 2.4.
T = 0 produces the thick lines shown in Fig. 3-3. Thus, the real part suppression at large
a becomes slower, since the average contains contributions from ka  1 as well. Theseka  1 contributions initially decrease (slightly) and eventually increase (factor > 2) the
imaginary part, proportional to σ, the scattering cross-section2 . However, the qualitative
behavior of both real and imaginary parts remain the same in spite of the averaging.
Universality of the Unitarity Limit in two-spin Fermi gases
It has been shown theoretically [83, 84, 85] that the unitarity-limited mean ﬁeld for interacting fermions is negative and proportional to the Fermi energy. Deﬁning β as the relevant
many-body parameter, we have:
Umf = −βEF ,kF a  1.

(3.13)

This is a surprising result because it implies that for very large and repulsive two-bodyinteractions, the eﬀective many-body interaction is attractive. In the context of achieving
fermionic superﬂuidity, this is good news because it increases the parameter range of strong
attractive interactions.
β has been calculated at zero temperature [83, 84] and measured at < 0.1 TF [86, 87]
yielding consistent numbers of 0.4 − 0.6. The fact that this attractive mean ﬁeld is neverstronger than the repulsive Fermi pressure, means that a two-spin interacting Fermi gas does
not undergo phase separation or implosion and is mechanically stable even at a Feshbach
2

Even though the scattering cross-section is ﬁnite, at T = 0, the scattering rate is zero because of Pauliblocking (see Section 3.2.3 and Chapter 6).

58

resonance. This is an important requirement towards producing a fermionic superﬂuid.
The experimental observation of this eﬀect[87] demonstrates another distinction from thebehavior of Bose gases where phase separation between two spin components has been
observed[88, 89]3 .
The density distribution of the interacting gas deviates from that of the non-interactinggas (Eqn. 3.3), due to the attractive (repulsive) mean ﬁeld shrinking (stretching) of thecloud. In the unitarity-limited regime, where Eqn. 3.13 is valid, the eﬀect of the mean ﬁeld
1
. Fitting the spatial distribution inis simply a rescaling of the trapping frequencies by √1+β

the trap by such a rescaling and using β as a ﬁt parameter can then be used to determine
β [86, 87].
Resonant Enhancement of three-body Decay
Near a Feshbach resonance, the three-body recombination rate is also enhanced. This was
the signal used for the ﬁrst observation of a Feshbach resonance in ultracold atoms [82]. As
shown by [90] and [91], the three-body decay rate for bosons for a  R (as near a Feshbachresonance) scales as a4 in the ultracold limit where the collision energy goes to zero. For an
ultracold two-spin fermion system, this rate gets an additional factor ∼ E/3bind [92] where
E is the kinetic energy of the collision and 3bind ∼ 2 /ma2 [93] is the binding energy of themolecular state. Thus the rate goes to zero at threshold (E → 0). This can be understoodas the requirement for at least two identical fermions to approach each other within the sizeof the bound molecular state. Thus, for an ultracold two-spin fermion system, the 3-bodydecay rate near a Feshbach resonance should scale as ∼ a6 EF . Our observations of inelastic
decay near a Feshbach resonance (Chapter 5) is probably due to this mechanism.

3.4

Ground State of 6 Li

In this section, I describe the hyperﬁne ground state of 6 Li, our fermion of choice. Fig. 3-4
shows the Breit-Rabi diagram for 6 Li (I = 1, J = 1/2). In Fig. 3-4(a), the six ground
hyperﬁne states are labelled with the low magnetic ﬁeld basis |F, mF . In Fig. 3-4(b), thenotation common to the ﬁeld is adopted, where the six states are labelled |1 through |6
in order of increasing energy.
As will be discussed in Chapter 4, we cool 6 Li to degeneracy in state |6 by sympathetic
collisions with 23 Na in a magnetic trap. We then transfer the 6 Li to an optical dipole trap
(ODT) which can trap any of the states |1 through |6. The most stable pair of states inwhich to try to induce interactions is the |1 − |2 combination. This has the least numberof inelastic decay channels. Most of the symmetry-allowed processes are endothermic at
ultralow temperatures and thus suppressed. The only possible two-body inelastic decay
3Note that a three state mixture of fermions may be mechanically unstable [84].

59

3
|3
2, 2

(a)

(b)

1
|3
2, 2

6

1
|3
2 ,- 2228 MHz

Energy (MHz)

1000

3
|3
2 ,- 2

0

-1000

1

1
|1
2 ,- 2

0

1
|1
2, 2

0

100

5
4

500

3
2
1000

200

Magnetic Field (G)Magnetic Field (G)Figure 3-4: Ground state hyperﬁne levels of 6 Li.
channel is the dipolar process:|1 + |2 −→ |1 + |1.

(3.14)

If a |1 and a |2 atom collide s-wave, the symmetry of the dipolar operator (Section 3.2.2)
can only allow the ﬁnal products to leave d-wave. Since, the ﬁnal states are identical, thisis not allowed. Thus dipolar decay corresponding to Eqn. 3.14 is forbidden for the s-wave
input channel. p-wave dipolar decay is allowed but highly suppressed at the typical ultralow
temperatures of our experiment.
In addition to the promise of stability, the |1−|2 mixture also features s-wave Feshbachresonances at accessible magnetic ﬁelds (Fig. 3-5). Thus, the |1 − |2 mixture is the ideal
choice of states to try to induce strong interactions. Feshbach resonances also exist in otherbinary combinations of |1, |2 and |3 (Fig. 3-5). This rich resonance structure opens up alarge parameter space to study interactions.

3.5

Fermionic Superﬂuidity

In this section, I brieﬂy outline the ideas of fermionic superﬂuidity. The realization of thesepredictions is a major goal for the ﬁeld.

60

Scattering Length (in units of a0)

Magnetic Field (Gauss)

Figure 3-5: Prediction of interstate s-wave Feshbach resonances in the ground state of 6 Li.
Broad Feshbach resonances exist in a12 at 812 G, in a13 at 680 G and in a23 at 843 G.
In addition, a narrow Feshbach resonance exists in a12 at 547 G. This calculations by thegroup of B.J. Verhaar at Eindhoven, the Netherlands and includes all currently available
experimental data to create the interaction potentials. Other (older) calculations will also
be presented in this thesis at some points to preserve the story-line.

3.5.1

BCS Pairing

BCS pairing of fermions occurs via attractive interactions between particles. In the standard
theory [31] which explains superconductivity in metals, electrons near the Fermi (momentum) surface of a metal can attract each other by phonon exchange. At a low enoughtemperature, this attraction can overcome Coulomb repulsion and electrons can undergo apairing transition into a superﬂuid state. For metals, BCS pairing occurs at temperaturesof ∼ 1 K, which is about 105 times lower than the Fermi energy. The gap energy ∆ characterizes the minimum excitation energy of the system. ∆ = kB Tc at zero temperatureand decreases smoothly to zero as T goes to Tc .
The basic idea of BCS pairing has been applied to the case of trapped weakly-interacting
atomic gases as well, with atomic mean-ﬁeld interactions taking over the role of the phononmediated interaction[94, 95, 96, 97]. In this theory, s-wave pairing of a spin-mixture offermions with inter-state scattering length a(< 0) has been predicted to occur at a temperature of [94]:Tc  TF exp(−π
2kF |a|),kF a  1(3.15)

Although other pairing mechanisms have been investigated[95, 96], this one has the highest
values of Tc /TF for a given a. However, the weakly-interacting criterion k|a|  1 immedi61

ately precludes an experimentally accessible Tc . Using some typical numbers of a ∼ 50a0
and TF ∼ 1 µK, we get Tc ∼ TF exp(−100), far below current experimental accessibility.
In addition to this very low Tc , the relevant timescale for formation of the superﬂuid
should be related to the gap as 1/∆ ∼ exp(+100) s. This is also far larger than typicalsystem lifetimes, or any other experimental timescale for that matter. A more sophisticatedcalculation performed by [98] also gives similar pair formation time in the weakly interacting
limit.
Achieving fermionic superﬂuidity will therefore require stronger interaction strengths.
With their wide tunability, Feshbach resonances seem the ideal choice of mechanisms toenhance interactions. However, when kF |a| becomes comparable to 1, Eqn 3.15 starts tobreak down. A modiﬁed BCS theory is therefore needed to determine whether fermionicsuperﬂuidity is possible for large kF |a|. A theory of “resonance superﬂuidity” has emerged
over the last few years which aims to answer exactly this question.

3.5.2

Resonance Superﬂuidity

Near a Feshbach resonance, the scattering length a becomes very large, invalidating thena3  1 assumption. Consequently, the appropriate interaction between atoms cannot
be the usual mean-ﬁeld, as assumed in Eqn. 3.15. However, an eﬀective mean-ﬁeld theory
can still be derived in the s-wave (kR  1) limit by explicitly including the bound state
responsible for the Feshbach resonance in the interaction. Such a theory named resonancesuperﬂuidity has been developed by Murray Holland and others [99, 100, 101]. Their predictions of Tc /TF ∼ 0.5 for both 40 K and 6 Li is very encouraging for the prospect of anear-future experimental realization of dilute gas fermionic superﬂuidity.
Although the BCS transition is yet to be achieved in dilute gases, several interaction
phenomena have been observed which are interesting in their own right. The results reported
in Chapters 5, 6, and 7 describe some of these.

62Chapter 4

Experimental TechniquesThis chapter describes several aspects of the apparatus on which the experiments describedin this thesis were performed. Although originally designed to Bose-condense 23 Na atoms[17], the apparatus has recently been upgraded into a two-species machine for 23 Na bosons
and 6 Li fermions. Using forced evaporative cooling of the bosons and sympathetic cooling of
the fermions by the bosons, we are now capable of producing degenerate bosons, fermions or
Bose-Fermi mixtures. Features have also been added to produce fermion spin-state mixturesof tunable interaction strengths.
Descriptions of all the elements of the machine that are required for Bose condensationof 23 Na can be found in previous theses [9, 10]. The basic BEC setup was used to construct
the atom interferometer described in Chapter 2. The major additions to allow two-species
operation are a combined 23 Na-6 Li atomic beam and laser light (at 671 nm) to cool andtrap 6 Li. Details of these additions can be found in the concurrently written thesis of Zoran
Hadzibabic [12].
In this chapter, I will ﬁrst brieﬂy discuss the sympathetic cooling part of the experiment
which produces a degenerate, spin-polarized Fermi gas that is essentially non-interacting.
An attached paper reports on the technique and results:• 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” [6]. e-print available at arXiv:cond-mat/0306050. Included in Appendix B.
I will follow the sympathetic cooling discussion with a description of the additional technical elements which have allowed us to produce and study a system of strongly interacting
fermions. These are: (1) an optical trap for 6 Li, (2) the creation of magnetic ﬁelds whichbring the system near Feshbach resonances and (3) the manipulation and detection of spincompositions.

63

4.1

Production of a Degenerate Fermi Gas

As with the production of degenerate alkali bosons, the initial cooling phase for alkali
fermions can also proceed via the well established techniques of laser cooling and magnetooptical trapping[40]. In the case of bosons, atoms from the magneto-optical trap (MOT)
are transferred into a magnetic trap1 , after which forced evaporative cooling is applied tocool the gas into degeneracy. The key ingredient for this last step to work is the existence ofa large collision rate which can thermalize the gas. The typical degeneracy temperature fordilute alkali gases of ∼ µK implies that the gas only interacts via s-wave collisions (kR 1). As mentioned earlier (Chapter 3), fermionic antisymmetry prohibits elastic collisions
between identical fermions at ultralow temperatures, and makes evaporative cooling of
spin-polarized fermionic samples impossible. The solution to this problem is to use someform of mutual or sympathetic cooling between two types of distinguishable particles, either
two spin states of the same atom[26, 29], or two diﬀerent atoms[27, 28, 3, 30]. Our solution
involves the use of bosonic 23 Na as a sympathetic cooling agent for fermionic 6 Li.
(b)

(a)
6

Li

23

| 32 , 32

Na

|2, 2
|2, 1

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

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

|1, 0
|1, 1

| 12 , 12

0

0

100
200
Magnetic Field (G)100
200Magnetic Field (G)Figure 4-1: Ground state hyperﬁne structures of (a) 6 Li and (b) 23 Na. The states arelabelled in the low ﬁeld, |F, mF  basis. Lithium (sodium) is loaded into the magnetic trapin the |F = 3/2, +3/2 (|F = 2, +2) state from the MOT. Sodium is evaporated on the|F = 2, +2 → |F = 1, +1 hyperﬁne transition near 1.7 GHz.
In our experiment, laser-cooled 23 Na and 6 Li atoms are transferred from a two-species
MOT [3, 12] into a common magnetic trap (ﬁg. 4-2). Forced evaporation cools 23 Na atoms,
1

Optical traps based on the AC Stark eﬀect are also used in a few laboratories such as [102].

64

which in turn thermalize and thus cool the 6 Li atoms “sympathetically”. Using this procedure, we can eﬃciently cool both species into simultaneous degeneracy. A critical requirement for our strategy to work is the dominance of cross-species elastic collisions overinelastic decay mechanisms. We are fortunate that this is indeed the case.

Gradient coils

Anti-bias coil

Curvature coilxzyTrappedatomic cloud
Figure 4-2: Schematic of current-carrying coils for magnetic trapping and production ofmagnetic ﬁelds near 6 Li Feshbach resonances. Running current through the gradient, curvature and antibias coils in the directions shown create a conservative trapping potential
about a ﬁeld minimum for atoms that are weak-ﬁeld seekers. The potential is harmonicnear the minimum and has symmetry about the z-axis. The optical trap is aligned along
the z-axis (Section 4.2). The Feshbach ﬁeld is applied by running current only through theantibias coils (Section 4.3).
Lithium is sympathetically cooled in the stretched |F = 3/2, +3/2 state2 by sodiumin the |F = 2, +2 state (Fig. 4-1). Evaporation from the |F = 2, +2 to the untrapped|F = 1, +1 state is carried out by applying a microwave ﬁeld at ∼ 1.7 GHz. This frequency
is far from any lithium resonance.
A schematic of the magnetic trap used for cooling is shown in Fig. 4-2. This design isbased on the clover-leaf conﬁguration ﬁrst demonstrated by our group in 1996[63, 9]. The
combination of gradient, curvature and antibias coils produce a magnetic ﬁeld minimum. At
the minimum, the ﬁeld points along +z. For a weak ﬁeld seeking state, there is conﬁnement
which is harmonic near the minimum with cylindrical symmetry about z. The conﬁnement
is stronger radially (x, y), making the atom cloud cigar-shaped. The real trap consistsof current-carrying wires wound in the shapes shown and packed together tightly using
epoxy. They are rigidly mounted outside the stainless steel vacuum chamber used forthe experiment. The shapes of the coils are chosen to simultaneously optimize magnetic
conﬁnement, vacuum requirements and optical access. A full description of our magnetic2

this was identiﬁed as state |6 in the previous chapter.

65

trap can be found in the diploma thesis of Christian Schunck [103]. Our typical trapping
currents give rise to a Fermi energy EF ∼ kB × N 1/3 600 nK where N is the atom number
expressed in millions (Eqn 3.2).
The mixture of stretched states has an extremely favorable ratio of elastic to inelastic
collisions. After about 30 s of cooling, we can produce up to 7 × 107 magnetically trapped
6 Li atoms at less than half the Fermi temperature (T ). Further, in samples containing up
F7
to 3 × 10 atoms, we observed temperatures as low as 0.05 TF . The ﬁnal temperature of

the fermions is ultimately limited by the ﬁnite heat capacity of the bosons. The numbers
quoted above represent the current upper limits of the system capability. On a day-to-day
basis, we can produce 30-50 million atoms at  0.5 TF or 10-30 million at  0.2 TF .

(a)

(b)

z

y

Fermi Diameter

3 mm

Figure 4-3: Large and ultra-degenerate Fermi sea. (a) Absorption image of 3 × 107 6 Li
atoms released from the magnetic trap and imaged after 12 ms of free expansion. (b) Axial(z-axis) line density proﬁle of the cloud in (a). A semiclassical “Thomas-Fermi” ﬁt (thinline) yields a temperature T = 93 nK= 0.05 TF . At this temperature, the high energy wingsof the cloud do not extend visibly beyond the Fermi energy, indicated in the ﬁgure by the
momentum-space Fermi diameter (1D projection of the momentum space Fermi surface).
The atoms are probed along the (usually) x-direction with resonant absorption imaging onto a CCD (charge-coupled device) camera. An example image is shown in Fig. 4-3.
The semi-classical ﬁt to the line density proﬁle (Section 3.1) yields a degeneracy parameter
T /TF = 0.05.
Although our (current) primary goal for sympathetic cooling is the production of large
numbers of degenerate fermions, the system can be adjusted to produce pure 23 Na BECs
with up to 10 million atoms3 , or simultaneously quantum degenerate Bose-Fermi mixtures,
each containing several million atoms (Fig. 4-4).
3

this also represents the ﬁrst production of evaporatively cooled sodium BECs in the upper hyperﬁnestate.

66

6

23

Li

Na

1.7 mmzyFigure 4-4: Two-species mixture of degenerate Bose and Fermi gases. Absorption images of
6 Li and 23 Na clouds on the same CCD camera using separate light pulses (see Section 4.4).
The times of free expansion of the two gases can be varied independently. This dualimaging technique allows 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
and the atom numbers were NLi ∼ 107 and NNa ∼ 6 × 106 . Sodium was cooled below thecondensation temperature, corresponding to ∼ 0.2 TF for the lithium cloud.
The sympathetic cooling discussed in this section has become standard in our laboratory
since the end of 2002. An older strategy, based on keeping 23 Na in the |F = 1, −1 state,
was used for about a year before that and is detailed in [3, 12]. The newer strategy increased
the atom number by two orders of magnitude and reduced T /TF by one order of magnitude.
The experiment described in Chapter 5 was performed using the older sympathetic cooling,
while the results of chapters 6,7 were obtained using the newer one.

4.2

Optical Trapping of Lithium

The most promising 6 Li spin mixture for inducing strong interactions is the |1 − |2 combination (Section 3.4). In particular, the interstate s-wave scattering length a12 exhibits avery broad Feshbach resonance at ∼ 810 G and a narrow one at 545 G (Fig. 4-5(b)). At
these high ﬁelds, both states are strong ﬁeld seekers (Fig. 4-5(a)). Therefore neither can bemagnetically trapped, necessitating the use of an optical dipole trap (ODT).
An atom exposed to a laser beam detuned to the red of an internal transition feels aforce which is directed towards higher optical intensity. This optical dipole force can beused to conﬁne atoms at a local intensity maximum such as the focus of a laser beam.
Atoms then feel a conservative potential which is generally known as the AC Stark shift.
This is the basis of the ODT.
67

(b)

5
4

6
1000
Energy (MHz)Scattering Length (in units of a0)

(a)

0

3
2

-1000

1
0

500

1000

Magnetic Field (Gauss)Magnetic Field (Gauss)

Figure 4-5: (a) Ground state hyperﬁne levels of 6 Li. (b) Prediction of interstate s-wave
Feshbach resonances in the ground state of 6 Li. Calculations by the group of B.J. Verhaar
at Eindhoven, the Netherlands. Broad Feshbach resonances exist in a12 at 810 G, in a13 at
680 G and in a23 at 845 G. In addition, a narrow Feshbach resonance exists in a12 at 545 G.
We use light at 1064 nm from a 20 W IPG Photonics ﬁber laser to trap lithium fermions(671 nm optical resonance). This large detuning from the optical resonance makes all 6
spin states of 6 Li (Fig. 4-5(a)) feel identical trapping potentials in our ODT. Although ourgroup has been trapping Bose-Einstein Condensates in ODT’s since 1998 [104], trapping of
ultracold fermionic clouds is more challenging simply because of its larger size/energy (seeFig. 4-4). To get a crude estimate of this diﬀerence, we can compare the chemical potentialµ of a BEC to the Fermi energy EF of a zero temperature Fermi gas (assuming the same
number of particles N and same mean trapping frequency ω̄ = (ω1 ω2 ω3 )1/34 ):(4 ω̄ 6 N 2 ma2 )1/5µ∼=EFω̄N 1/3

ω̄N 1/3(2 /ma2 )

1/5


∼

EF
(2 /mR2 )

1/5
(4.1)

for a ∼ R, the range of the interatomic potential, which is usually the case far away fromresonances. In the s-wave regime, EF  (2 /mR2 ). This simple argument shows thatthe optical potential will need to be deeper to accommodate degenerate Fermi clouds. By
assuming the same density for bosons and fermions, a relation similar to Eqn. 4.1 can be derived, except with a 1/3 power instead of 1/5 power on the right-hand side. Experimentally,
we have found that we need considerably more optical power to trap degenerate fermionsthan to trap BECs.
The 20 W output of the laser is sent through an acousto-optic modulator (AOM), whichacts as a switch and a variable attenuator for the 1064 nm radiation. In order to maximize
4
note that neither of these assumptions are quantitatively valid, but they are made only to emphasize
the role of quantum statistics in this technical problem.

68the available intensity, we do not send the beam through an optical ﬁber (typically 50%
eﬃciency). Instead, after passing through the AOM, the light goes directly through beam
shaping optics and enters the vacuum chamber along the z-direction (Fig. 4-2). The focusof the beam is aligned with the position of the atoms in the magnetic trap. The (< 100%)
eﬃciency of the AOM together with losses in the shaping optics restrict the total available
power at the atoms to about 5 W. The position and shape of the focus can be monitored
on a CCD camera placed at an imaging plane of the atoms. The size of the focus w (=1/e2
radius) varies from about 14 µm to 16 µm, depending on the exact alignment of all the2

optics. The Rayleigh range of the gaussian beam is then πw
λ ∼ 0.8 mm for w = 16 µm andλ = 1064 nm. Using the AC Stark shift potential for a gaussian laser beam [72]5 , we canderive simple relations for the trap depth Utrap , radial(axial) trapping frequencies ω⊥ (ωz ),Fermi temperature EF , and the zero-temperature peak density n0 in our system:
16 µm 2P
) ×w
1W
16 µm 2
P 1/2
) ×(
)
2π × 9.3 kHz × (w
1W
16 µm 3
P 1/2
) ×(
)
2π × 130 Hz × (w
1W
16 µm 7/3
P 1/2
N
) ×(
) ×(
)1/3
20.9 µK × (w
1W
1 million
16 µm 7/2
P 3/4
N
) ×(
) ×(
)1/2
1.9 × 1014 cm−3 × (w
1W
1 million

Utrap = 166 µK × (
ω⊥ =
ωz =
TF

=

n0 =

(4.2)

Ultracold atoms conﬁned in the magnetic trap are transferred into the optical trap byadiabatically (compared to trap frequencies) turning on the ODT. Typically, we remove all
the 23 Na atoms by evaporation and are left only with the non-interacting fermions before
turning on the ODT. After the optical power is ramped up to the desired value, the magnetic
trap is switched oﬀ. This procedure allows up to 20 million atoms to be loaded into the ODTat about the Fermi temperature. Atoms in state |3/2, +3/2 = |6 are then transferred to(now trappable) state |1 using a single photon transition at 228 MHz near zero ﬁeld. Thistransfer is close to 100% eﬃcient. Arbitrary spin states can then be prepared at arbitrary
magnetic ﬁelds (Section 4.4). Additional cooling can also be performed in the optical trap
(Section 4.5).

4.3

Magnetic Fields

A schematic of the magnet coils currently used in the experiment is shown in Fig. 4-2. An
optically trapped 6 Li spin mixture can be brought near a Feshbach resonance by sending
current only through the antibias coils (Fig. 4-2). To preserve the quantization axis of
6

5
it is important to include the counter-rotating term as it contributes about 18% of the trap depth for
Li atoms trapped in 1064 nm light.

69

the atoms in the z-direction, this current is run in the opposite direction than during themagnetic trapping phase (Fig. 4-2).

(a)

(b)
Axial Magnetic Field [G/A]

-4

-2

2.4

2.4

2

2

1.6

1.6

1.2

1.2

0

2

4

-4

-2

2

4

z [cm]

Figure 4-6: Homogeneity of the Feshbach ﬁeld. Comparison of the (calculated) antibias
ﬁelds from (a) old and (b) new magnetic traps. The new antibias coils are placed ina Helmholtz conﬁguration and produces larger bias and far less curvature than the oldantibias coils.
Full details of the design and construction of the magnetic trap can be found in [103].
This trap was built as a replacement to another one of similar design when the latter developed a short. The new magnetic trap improves on the old one in two important ways: (1)a high pressure water cooling system allows us to apply ﬁelds ≥ 1000 G continuously. Previously, we were restricted only to magnetic ﬁeld pulses (≤ 1 s at 1000 G). (2) an optimized“Helmholtz” placement of the antibias coils has dramatically reduced the ﬁeld curvatureaccompanying the application of large bias ﬁelds (Fig. 4-6). The curvature from the old
trap was anti-trapping at the Feshbach resonance ﬁelds (Fig. 4-5) and reduced the eﬀective
trap depth. This caused substantial spilling of atoms from the ODT. Very careful opticalalignment was required to match the centers of the ODT and the magnetic “anti”-trap, inorder to minimize this spilling. With the new trap, this issue is non-existent.
The antibias coils produce a magnetic bias ﬁeld of 2.06 G/A (Gauss/Amp) and a ﬁeld
curvature of ∼ 0.002 (G/cm2 )/A. The bias was calibrated by locating optical resonancesof 6 Li at large magnetic ﬁelds (500 − 1000 G) and comparing the resonance positions with
the Breit-Rabi predictions for the energy levels. The curvature was measured during the“bench”-test of the magnetic trap prior to installation, and is about 30 times lower thanthat of the older trap.
Magnetic ﬁelds of up to 1000 G can therefore be applied using a commercial 500 A power
supply. This range encompasses all the Feshbach resonances of Fig. 4-5. As shown in Fig. 47(a), we can switch on these ﬁelds in a few milliseconds (limited by the power supply). The
switch oﬀ time is a few hundred microseconds (Fig. 4-7(b)) and is fundamentally limited by
the inductance of the coils.
70

(b)

(a)
500

Current in Antibias coils (Amps)

Current in Antibias coils (Amps)

500

400

300

200

100

0
0

5

10
time (ms)

15

400

300

200

100

0
-200

0

200
400
time (µs)

600Figure 4-7: Current turn on and shut oﬀ in Feshbach (antibias) coils. Currents of 500 A(magnetic ﬁelds of 1000 G) can be (a) turned on in a few ms and (b) switched oﬀ in a few
hundred µs.

4.4

Spin State Manipulation and Detection

We have spent substantial eﬀort in developing ﬂexible methods of spin state manipulationand detection in the optical trap at arbitrary magnetic ﬁelds. This can be useful for several
reasons:(1) Studies of interactions by imaging at near zero-ﬁeld after turning oﬀ the Feshbachﬁelds may be inadequate because (a) the timescale for turning oﬀ the ﬁeld could competewith the timescales of dynamics that we wish to study or/and (b) the system could undergo
inelastic decay while going through a loss resonance at a lower ﬁeld during current turn-oﬀ.
These inadequacies are remedied by imaging the atoms at the magnetic ﬁeld of interest.
(2) As shown in Fig. 4-5, each binary mixture of states |1, |2 and |3 has at least one interstate Feshbach resonance. The ability to create and diagnose arbitrary spin compositions
near these resonances opens up a large parameter space for studying strongly interacting
fermions.
(3) Since fermionic superﬂuidity involves pairing of momentum states near the Fermi
surface, the particle numbers (determining the Fermi surfaces, Eqn. 3.2) in the spin components need to be controlled. The degree to which this “Fermi surface matching” will be
required for the superﬂuid transition is as yet unclear. However, tunability of the matchingprovides an experimental knob in the search for this transition.
(4) Mean ﬁeld interactions can be directly measured as resonance shifts in transitions
between spin states. These transitions can be monitored by detecting the spin state composition. Chapter 7 describes such experiments performed in our system.
71

Beginning with atoms in state |1 in the optical trap (Section 4.2), we can ramp up theﬁeld to any desired value (Section 4.3) and create an arbitrary composition of states |1, |2
and |3 using standard ∼ 80 MHz (Fig. 4-8) radio-frequency (RF) techniques. This involves
an RF synthesizer, an ampliﬁer and an RF coil - standard features of the experiment[105].
The new feature that was required was the implementation of simultaneous spin-selective
imaging at arbitrary magnetic ﬁelds.
(a)

(b)
240
2 --> 3 transition frequency [MHz]1 --> 2 transition frequency [MHz]

80

60

40

20

220
200
180
160
140
120
100
80

0
0

200

400
600
800
Magnetic field [G]

0

1000

200

400
600
800
Magnetic Field [G]1000

Figure 4-8: Energy diﬀerence between (a) states |2 and |1 and (b) states |3 and |2.
Imaging at arbitrary magnetic ﬁelds is accomplished by frequency shifting (with anAOM) the laser light used for cooling and trapping of 6 Li to the optical resonance at thedesired ﬁeld. The AOM has suﬃcient frequency tunability to encompass the Feshbachresonances shown in Fig. 4-5. More details can be found in [12].
Fig. 4-9 shows the ground 2 S1/2 state and excited 2 P3/2 state hyperﬁne structure of
6 Li.

In the excited state, the mI sub-levels are always unresolved within each mJ level(Fig. 4-9). Thus, optical transitions from states |1, |2 and |3 to any mJ level of the2P
3/2 state are separated only by the ground state hyperﬁne splitting.

At the Feshbachresonance ﬁelds (Fig. 4-5), these splittings are ∼ 80 MHz (Fig. 4-8), well above the naturallinewidth of 6 MHz. Thus, two beams separated in frequency by this splitting could eachbe selectively resonant with only one of the states. Both this frequency diﬀerence and thespatial separation required for independent imaging can be provided by an additional AOM.
An AOM which frequency downshifts light resonant with |2 at a particular magnetic
ﬁeld B, by the |2 − |3 splitting at B, simultaneously produces spatially separated laser
beams resonant with |2 (zeroeth order) and |3 (ﬁrst order). These beams can then bothbe directed onto a |2 − |3 binary mixture (either trapped or in free expansion). After
passing through the atoms the two beams can be sent to two diﬀerent parts of the CCD
chip on the imaging camera. In this way, overlapping atom clouds can be simultaneously
imaged onto diﬀerent parts of the camera. An absorption image using such a geometry is72υ [MHz]2µ B/h0υ 0+ 2000

mj = 3/2
υ 0+ 1000

2/3µ0B/h

mj = 1/2
2

2 P3/2υ0

mj = -1/2

υ0υ 0− 2000-2µ0B/h(671nm)

υ1
1000

2

2 S1/2-2/3µ0B/h

mj = -3/2

υ 0− 1000

υ2
µ0B/h

mj = 1/2

0

mj = -1/2-1000−µ0B/h

0

B [G]
200

400

600

800

Figure 4-9: High ﬁeld imaging of 6 Li. ν1 and ν2 are two optical transitions that we currently
use for high ﬁeld imaging. Each mJ level in the excited state has 3 unresolved mI states.
The optical frequencies from the ground mJ = −1/2 state are separated only by the groundstate Zeeman separations of ∼ 80 MHz at high ﬁelds. Within each of these optical lines, weselect the state to be imaged (|1, |2 or |3) with AOMs as described in the text.
shown in 4-10(a).
Using a charge shifting feature of a CCD chip, we can extend the above simultaneous
imaging capability to image all three states. It is possible to take multiple absorption images
in quick succession by shifting (fast: ∼ millisecond) the charge corresponding to an image
to a shielded part of the CCD chip before reading out the charge (slow: ∼ second). This isoften called the “kinetics” mode of operation of a CCD camera. Two images can then be
taken which are separated by a variable time. This time is only limited from below by the
charge shifting time of ∼ 600 µs for a typical 3 mm×3 mm exposure area. This separation
time can be used to shift the light to the frequency appropriate to image the third state(state |1 in Fig. 4-10(b)) as well as provide a variable expansion time for a released atom
cloud. Fig. 4-10(b) shows the image of a ternary mixture using this technique.
Using these imaging techniques, we have established a fairly ﬂexible and robust system
to prepare and study interacting fermions. Two consecutive images with an adjustable
separation/expansion time using the kinetics mode also allows us to take images of 23 Na6 Li mixtures as in Fig. 4-4.

In this case, the yellow (sodium) and red (lithium) imaging

73

(a)

(b)

3mm

z

y

State |2>

State |3>

State |1>

State |2>

State |3>

State |1>

Figure 4-10: Simultaneous multiple state imaging in kinetics mode. The dashed line separates two diﬀerent exposures in each image. Arbitrary spin compositions can be measured
independently at high magnetic ﬁeld. (a) shows a binary mixture of states |2 and |3. (b)shows a ternary mixture of states |1,|2 and |3.
beams are sent along identical imaging paths after being combined on a dichroic mirror.

4.5

Evaporative Cooling in the Optical Trap

A binary spin-mixture interacting with a ﬁnite interstate scattering cross-section will thermalize and evaporatively cool in a ﬁnite depth optical trap. Beginning with a 50-50 mixture,eﬃcient cooling can be performed in this way [29]. The thermalization rate or alternatively atom loss can be used to measure the scattering cross-sections in interacting fermions[106, 107].
We have observed cooling and atom loss for the |1 − |2 mixture in our ODT at various
magnetic ﬁelds. Fig. 4-11 shows the behavior of |1 − |2 mixtures at two diﬀerent magnetic
ﬁelds: 530 G and 900 G. At 530 G, the mixture shows no change on a 100 ms timescale whileat 900 G, there is a visible atom loss and cloud size shrinkage, reﬂecting the cooling. Thisis consistent with the magnetic ﬁeld dependence of a12 shown in Fig. 4-5.
A modestly degenerate (T  TF ) cloud loaded into the ODT has an evaporation parameter η given using Eqns. 4.2 by:
η=

Utrap
w 1/3
P 1/2
1 million 1/3
) ×(
) ×(
)
=8×(
TF
16 µm
1W
N

(4.3)

If we load 16 million atoms into a 4 W trap (near optimum conditions) and make a 5050 mixture, η = 8, small enough for atom loss and cooling to take place even without
lowering the trapping potential. This is consistent with our observations. Currently we canachieve temperatures of  0.5 TF in the ODT. Careful characterization and optimizationof this cooling process [29, 86] should produce degeneracies comparable to those observedin the magnetic trap. Such conditions would doubtless bring us even closer to fermionic74

900 G

3mm

530 G

10 ms waitz

100 ms waityState |1>

State |1>

State |2>

State |2>

Figure 4-11: Mutual cooling of states |1 and |2 in the optical trap. The images are takenusing the simultaneous imaging technique at an expansion time of 500 µs. The trap depth
is kept constant at ∼ 330 µK.
superﬂuidity.

75

Chapter 5

Decay of an Ultracold Fermi Gas
near a Feshbach ResonanceThis chapter focuses on the experiment reported in the following publication:• K. Dieckmann, C.A. Stan, S. Gupta, Z. Hadzibabic, C.H. Schunck, and W. Ketterle,“Decay of an Ultracold Fermi Gas near a Feshbach Resonance” Phys. Rev. Lett., 89,
203201 (2002) [4]. Included in Appendix C.
This chapter describes our ﬁrst experiments in an interacting regime with ultracold
lithium atoms. We successfully located two Feshbach resonances between the two lowest
Zeeman sub-levels of the ground state (states |1 and |2) by observing the magnetic ﬁelddependent decay of a mixture of these two spin components. We also determined the lifetimeof the |1 − |2 mixture in the Feshbach resonance regime where fermionic superﬂuidity is
predicted to occur[99, 108, 100]. The measured lifetimes are favorable compared to theestimated creation time of the predicted superﬂuid state.
As described in Section 3.4, for 6 Li, the natural choice of interacting states in an opticaltrap is the |1−|2 combination. This mixture has no energetically-accessible spin-exchange
channels. In addition, the only energetically-accessible dipolar decay channel is forbiddens-wave, while p-wave contributions are suppressed by our ultralow temperatures.
The question arises as to what would be the most direct signature of a Feshbach resonance. Prior to our experiment, the JILA group had measured such a resonance in fermionic
40 K using a thermalization time measurement [109]. However, the ﬁrst observation of Fesh-bach resonances in ultracold atoms was via the measurement of enhanced, ﬁeld-dependent
losses in a BEC in our group [82]. This is a straightforward measurement if the system does
indeed have a resonantly-enhanced decay mechanism.
Guided by this previous experience in our group, we simply followed the steps of opticaltrapping, mixture preparation and magnetic ﬁeld application and in true experimentalist
fashion tried to “see what happens”. We indeed saw atom loss as a function of magnetic ﬁeld.

76

The losses that we observed were the ﬁrst experimental signatures of scattering resonancesin the 6 Li |1 − |2 combination.
Since our publication [4], (Appendix C) details all the salient experimental aspects,
I will restrict myself to a short summary of the experimental method and results. Theinterpretation of the nature of the observed decay will also be discussed.

5.1

Resonant Decay of an Ultracold Fermi Gas

5.1.1

Observation of Field-Dependent Losses
(b)

(a)

Energy [MHz]

1000

5
4

a12 (in units of a0)

6

0

-1000

1
0

500

3
2

500 G

800 G

1000
Magnetic Field [Tesla]

Magnetic Field [Gauss]

Figure 5-1: (a) Ground state hyperﬁne levels of 6 Li. (b) Predicted s-wave scattering length
a12 between states |1 and |2 (thick lines in (a)). Feshbach resonances are predicted at∼ 800 G and ∼ 20, 000 G. a12 goes through zero near 500 G. At the experimentally accessible
Feshbach resonance at 800 G, states |1 and |2 are both high-ﬁeld seeking, making opticaltrapping a necessity. Figure (b) is from reference [110].
As shown in Fig. 5-1, states |1 − |2 of the 6 Li ground state were predicted to have aninterstate s-wave Feshbach resonance at ∼ 800 G. To achieve the corresponding experimental
situation, we ﬁrst created a degenerate gas in the |2 state in the magnetic trap using our ﬁrst
sympathetic cooling scheme (Chapter 4,[3]). After loading the gas from the magnetic trap
into the optical trap, we created a mixture of states |1 and |2 by an RF-induced spin ﬂip at
a ﬁeld of ∼ 1.5 G. The magnetic ﬁeld was then ramped up to a speciﬁc value where the atom
number was monitored for two diﬀerent wait times - 50 ms and 500 ms. We repeated thisprocedure at many diﬀerent magnetic ﬁeld values in order to map out the ﬁeld dependenceof loss processes. The number was measured using absorption imaging after turning oﬀ
the ﬁeld. Taking the ratio of the two obtained numbers served as a normalization process
guarding against systematics from loading losses. A second guard against systematics was
carried out by doing the same experiments on the pure states |1 and |2. As a function of77

magnetic ﬁeld, the pure states were mostly featureless (not shown) while the 50-50 mixture
showed pronounced loss features (Fig. 5-2).
0.0

(a)

500 ms hold time

1.0

Surviving Fraction

0.5

mixture |1> & |2>

0.0
200

400

600

(b)

800

2 s hold time

1.0

0.5

0.0
500

600

700

800

Magnetic Field [G]Figure 5-2: Feshbach resonances identiﬁed by losses. Beginning with a 50-50 mixture ofstates |1 and |2, the survival fraction is plotted as a function of the magnetic ﬁeld. (a) is
a scan over the range 200 G-850 G and shows the survival fraction calculated as the ratioof the atom number after wait times of 500 ms and 50 ms. The scan shows a pronounced
loss feature peaked at ∼ 680 G. In addition, a narrow loss feature at ∼ 550 G can also
be seen. The magnetic ﬁeld scan in (b) covers a narrower range, has a higher density ofmeasurements, and shows the survival fraction calculated from wait times of 2 s and 50 ms.
The 550 G resonance is now obvious.
We saw a broad decay resonance at 680 G, near the predicted Feshbach resonance (800 G)
and also discovered a narrow one at 550 G [4]. Although our decay peak for the broadfeature occurred more than 100 G away from the predicted value [110], it was still within
the predicted (unusually large) width (Fig. 5-1(b)). Our observation of a decay resonanceat 550 G was followed soon after by the prediction of a narrow Feshbach resonance at thisﬁeld value [106]. The older and the newer theoretical predictions are displayed in Fig.5-3.
Our measurements therefore experimentally located two s-wave Feshbach resonances in the|1 − |2 mixture.
The new theoretical prediction (Fig. 5-3(b)) beneﬁtted from a careful measurement of
the zero-crossing of a12 (at ∼ 530 G) by a group at Duke university[106]. The zero-crossing78was measured independently and with good agreement by a group at Innsbruck [107]. The
ﬁeld dependent losses for the wide resonance were also observed (almost simultaneously) in
the Duke group [106] as well as (later) by the Paris group [111].
(b)

Scattering Length a/a0

Scattering Length a/a0

(a)

500 G

800 G

Magnetic Field [Gauss]Magnetic Field [Tesla]

Figure 5-3: (a)Old prediction and (b)New prediction of s-wave Feshbach resonances between
states |1 and |2. Figure (a) from reference [110] and Figure (b) from reference [106]. Notethat (a) has a logarithmic x-scale, while (b) has a linear x-scale.

5.1.2

Decay Curves on resonance

The nature of the decay was studied by monitoring the system at various wait times forthe same magnetic ﬁeld. The decay curves obtained at 680 G is shown in Fig. 5-4. For a
two-body (three-body) process, the loss rate of atoms ṅ is proportional to n2 (n3 ), where
n is the atomic density. For our modest degeneracies (in this experiment T ∼ TF ), thedensity distribution is essentially classical and therefore should be proportional to atom
number N , provided the temperature does not change. Thus Ṅ should be proportional
to N 2 (N 3 ), for a two-body (three-body) process. This means that two-body processes
should show a linear dependence of 1/N with time, while three-body processes should show
a linear dependence of 1/N 2 with time. The measurements shown in Fig. 5-4 are consistent
with a two-body decay mechanism. Changes in temperature during decay which would
invalidate our analysis were hard to characterize because of the low signal-to-noise as thecloud decayed1 .

1

It can be shown that for a T = 0 Fermi gas, keeping track the changing volume due to atom loss actually
results in 1/N changing linearly in time for a 3-body process. This was pointed out to us by Jean Dalibard.

79

(a)

40

N

-1

-6

[10 ]

60

20

0

(b)

2

N

-2

-9

[10 ]

3

1

0
0

200

400

600

Time [ms]Figure 5-4: Decay of the atom number at 680 G. (a) The data plotted as 1/N show a linear
time dependency, consistent with two-body decay. (b) The same data plotted as 1/N 2
clearly show non-linear dependency, inconsistent with three-body decay. For the resonanceat 550 G, the comparison of least square ﬁts also revealed consistency with a two-bodydecay.

5.2

Loss Mechanism

Two questions naturally arise from our results:(1) what is the collision process responsible for the observed decay?
(2) why is the loss peak shifted such a large amount in magnetic ﬁeld (> 100 G) fromthe (predicted) divergence of the s-wave scattering length?
Let’s examine question (1). The |1 − |2 combination is immune to spin-exchange
losses as well as dipolar losses in an incoming s-wave channel (Section 3.4). The leading
inelastic 2-body collision which could contribute to the observed losses is then dipolar decay
in an incoming p-wave channel. Our ultracold temperatures would highly suppress such a
process. It is still possible that such a process contributes, for example in the presence of ap-wave Feshbach resonance [112]. However, theorists have concluded that there are no suchresonances in the relevant magnetic ﬁeld region[101]. The possibility of a non-resonant pwave process being enhanced by the s-wave resonance as a collisional cascade is also unlikely
because the shape of the loss curve deviates strongly from the predicted shape of the s-wave
scattering cross-section (Fig. 5-3 and Eqn. 3.10).
Resonant enhancement of 3-body decay near a 2-body Feshbach resonance is anotherpossibility. This mechanism may also answer question (2). As argued by Petrov [92], if the80binding energy released upon formation of the molecule does not exceed the trap depth, the2 collision products (molecule + 3rd atom) can remain trapped. For large and positive a12 ,the binding energy is positive and is given near resonance by the relation ([93]):3bind =

2
ma212

(5.1)

This means that the energy released is smaller closer to the divergence of a12 (as the boundstate tunes closer to threshold). The scattering length at which the loss should peak is thenrelated to the trap depth Utrap as:
aloss ∼ /

mUtrap .

(5.2)

For our experimental trap depth, this corresponds to a ﬁeld of ∼ 620 G, in the ballparkof the measured value. Currently, we believe that this 3-body loss mechanism is the mostprobable cause of the observed decay. The position of the same loss resonance observed
by a group in Paris shows agreement with Eqn. 5.2[111]. This group has also observed adependence of the position of the decay peak with trap depth [113].
The temporal behavior of the decay (Fig. 5-4) follows the characteristics of a two-bodyprocess rather than a three-body process. It is possible that near the resonance, long-lived|1−|2 quasi-bound states are formed (step 1) which can collide with either single spin state
in a three-body process (step 2). Since the quasi-bound state is long-lived, it would take along time to form. The timescale for this overall process would then be determined by step
(1) and the subsequent decay rate would have the characteristics of a two-body process.
Such a 2-step mechanism has been discussed in the context of recombination processes inspin-polarized atomic hydrogen [114].

5.3

Outlook

Our experiment proves that the |1 − |2 mixture is stable on a timescale of 100 ms for all
magnetic ﬁelds of interest and for typical densities of a few times 1013 cm−3 . If resonancehigh-Tc superﬂuidity (Section 3.5) were to occur, the gap energy ∆ would be ∼ (0.5kB TF ).
This should give the relevant timescale for the formation of the superﬂuid as 1/∆ < 1µ s,
for our typical parameters. The magnitude of this timescale relative to our loss timescale
bodes well for the realization of superﬂuidity in the 6 Li system.
The experiment reported in this chapter represents our ﬁrst observations of interacting
fermions. Since then, the apparatus has been upgraded in several ways. As of the end of
2002, we had implemented new magnetic coils, high-ﬁeld and multiple-state imaging, as
well as a new sympathetic cooling scheme which increased our atom number a hundred-fold(see Chapter 4). The experimental results reported in the rest of this thesis were performed
using the improved apparatus.
81

Chapter 6

Collisions in zero temperature
Fermi Gases
This chapter focuses on the theoretical results reported in the following paper:• S. Gupta, Z. Hadzibabic, J.R. Anglin and W. Ketterle, “Collisions in Zero Temperature Fermi Gases” [8]. Included in Appendix D. e-print available at arXiv:condmat/0307088.
A simple and useful tool to diagnose the behavior of trapped atomic gases is to release
them suddenly from their conﬁnes and observe the evolution of the spatial density. This
merely involves switching oﬀ the magnetic or optical trap and subsequently imaging theatom cloud with near-resonant light - features which usually come for free during setup of
an atom trapping experiment. This is a fundamentally new knob that atomic physicists
possess over their traditional condensed matter counterparts. Hydrodynamic expansion
from an anisotropic trap results in a reversal of anisotropy after a suﬃciently long time
(several radial trapping frequencies). This arises due to the diﬀerent pressure gradientsalong diﬀerent directions. The “smoking gun” demonstration of an interacting BEC is
the observation of anisotropic hydrodynamic free expansion. This chapter deals with the
behavior of interacting Fermi gases during free expansion.
Considerable excitement was generated in the ultracold atom ﬁeld with the observationof anisotropic hydrodynamic expansion of an interacting Fermi gas [86]. This had followed
rapidly in the wake of the theoretical prediction that a fully superﬂuid Fermi gas would
expand hydrodynamically (just like its bosonic counterpart) [115]. This idea necessarilyneeds to be modiﬁed with the idea of classical hydrodynamics in a strongly interacting
system [116]. Interacting Fermi gases realized near Feshbach resonances can be collisionallyhydrodynamic because of the large cross-section for incoherent collisions (Section 3.2.3).
This has been observed in various groups [86, 117, 111] as well as by us (unpublished,
Section 6.2). The distinction (if any) between classically collisional and superﬂuid behavior
hinges critically on the role of Pauli blocking for degenerate fermions.
82

A highly degenerate Fermi gas is of course almost collisionless in the trap, even in thevicinity of a Feshbach resonance. However this Pauli suppression of collisions breaks downwhen an interacting system is released from a spatially anisotropic atom trap. Even in azero temperature Fermi gas, the expansion deforms the Fermi surface and allows collisions
to take place. We have found theoretically that more than half the classically allowed
collisions can take place for highly anisotropic traps [8].
In this chapter, I ﬁrst introduce the standard theoretical treatment for analyzing the
expansion behavior of atomic gases. I then present our data exhibiting hydrodynamic
expansion of a strongly interacting Fermi gas. Finally, I discuss our theoretical results
regarding the collisional behavior of degenerate Fermi gases which are detailed in the paper
attached (Appendix D, [8]).

6.1Free Expansion of Atomic Gases

6.1.1

Expansion of non-degenerate gases

The expansion of classical Boltzmann gases, Bose gases above Tc , as well as Fermi gasesin the normal (non-superﬂuid) phase, can be described by means of a scaling formalism[116, 118]. This theory is valid in the semi-classical limit, where the thermal energy is large
compared to the separation between the energy eigenvalues of the potential.
The Boltzmann-Vlasov equation can be used to describe the dynamics of the phase-space
distribution f (x, p, t):p ∂f∂(Uh + Umf ) ∂f
∂f
+ .
−
.
= Γcoll [f ]
∂t
m ∂x
∂x
∂p

(6.1)

This is simply the standard Boltzmann equation [77] with the addition of the mean-ﬁeld
potential Umf . Uh is the harmonic trapping potential. The right hand side is the usual
Boltzmann collision integral. Free expansion distorts the local phase space distribution
while collisions try to preserve the local equilibrium. The local equilibrium corresponds toan isotropic thermal distribution where all the local quantities of Chapter 3 can be deﬁned.
In the limit of no collisions, Γcoll [f ] = 0, we obtain the equations of ballistic expansion. For
cylindrically symmetric traps (frequencies ω⊥ , ωz ), the aspect ratio (radial size/axial size)
is given by:ωzAballistic (t) =ω⊥

2 t2
1 + ω⊥
1 + ωz2 t2(6.2)

This will be derived in Section 6.3. An initially anisotropic cloud approaches spatial isotropy
as t → ∞. The ﬁnal spatial distribution mirrors the initial momentum space isotropy.
In the opposite limit, where the collision timescale is shorter than all the other timescales
in the system, the local phase space distribution f is always in equilibrium and isotropic.

83

One then obtains the classical hydrodynamic scaling solutions [116, 118]:
b¨i =

ωi2
(bx by bz )2/3 bi(6.3)

where bi is the scaling parameter in the i direction. The spatial distributions retain their
original form (Gaussian, Bose-Einstein, or Fermi-Dirac) but get scaled in size linearly by
the bi ’s. For cylindrically symmetric traps, the aspect ratio during expansion is:
Ahydro (t) =ωz b⊥ (t).
ω⊥ bz (t)(6.4)

The collision rate competes with the inverse trap frequency to determine whether thegas expands ballistically or hydrodynamically. Comparing the collision rate with the inverse
trap frequency is equivalent to comparing the collisional mean free path with the initial sizeof the cloud. Collisional hydrodynamics ensues when the mean free path is much smaller
than the cloud size.
The expansion behavior of gases in the (experimentally relevant) regime between ballistic and hydrodynamic has only recently been examined using an extension of the scaling
analysis [118]. However, the case of ultra-degenerate normal fermionic gases (where Pauliblocking becomes important) is not covered by this analysis. Our work (Section 6.3) provides some rudimentary results in this regime.

6.1.2

Expansion of Superﬂuid Gases

The hydrodynamic equations of superﬂuids are applicable if the healing length is muchsmaller than the size of the sample [115]. This implies that the whole system behaves similar
to a superﬂuid. The hydrodynamic description is based on the equation of continuity:
1 ∂∂n+(np) = 0,∂tm ∂x
and on the Euler equation:
∂
p2
∂p
+
[µ(n) + Uh +
]=0
∂t
∂x
2m
where µ(n) is the chemical potential of a uniform gas calculated at the density n. If theequation of state is a power law µ ∼ nγ , then the above equations have scaling solutions
during expansion. If the Thomas-Fermi radii evolve according to Ri (t) = Ri (0)bi (t), thenthe scaling parameters bi obey [115]:
b¨i =

ωi2
(bx by bz )γ bi

84

(6.5)

For a BEC,µ ∼ n1 ,γ = 1,[73]. For a Fermi superﬂuid, the authors [115] useµ ∼ n2/3 ,γ = 2/3,

the relation in a normal gas at zero temperature1 . The scaling prediction for a classical gas
and a superﬂuid Fermi gas in the hydrodynamic limit are then identical (Eqn. 6.3). Notethat the prediction for a BEC is diﬀerent. However, this diﬀerence only becomes signiﬁcantin pancake shaped traps and is negligible in cigar shaped traps [116].
For cylindrical geometries, deﬁning τ = ω⊥ t, λ = ωω⊥z (initial aspect ratio) and for
γ = 2/3, we get the non-linear diﬀerential equations:
d2 b⊥
dτ 2

=

d2 bz
dτ 2

=

1

(6.6)

7/3 2/3b⊥ bz

1
4/3 5/3
b⊥ bz

.

(6.7)

These equations can be solved numerically quite easily in Mathematica.

6.2

Observation of hydrodynamic expansion in a stronglyinteracting Fermi gas mixture

Anisotropic expansion of interacting Fermi gases out of a cylindrically symmetric trap havebeen observed recently by several groups [86, 117, 111]. We have also observed this phenomenon. Fig. 6-1 shows simultaneous absorption images of states |1 and |2 at diﬀerent
times-of-ﬂight after release from our cylindrically symmetric optical trap in a regime ofstrong interactions (900 G). The reversal of anisotropy during expansion is obvious.
The expansion data were ﬁtted with 2-D gaussians. The resulting aspect ratio is plotted in Fig. 6-2. Also shown are the expectations from ballistic and hydrodynamic scaling(Eqns. 6.2,6.4 and 6.6 solved for γ = 2/3 using Mathematica). Clearly, we are deep in thehydrodynamic regime.
The classical collision rate is given by ΓCl = nσv where n is the density, σ is the scatteringcross-section and v is the relative velocity. During the expansion, the density drops andthe expansion of strongly interacting systems falls out of the hydrodynamic regime very
quickly. However in the unitarity limit, the scattering cross-section also changes with thedensity. Eqn. 3.12 gives σ ∼ 1/kF2 ∼ 1/n2/3 , where kF is the local Fermi momentum.
1

The diﬀerence essentially stems from the fact that a BEC has essentially 0 kinetic energy and µ is
dominated by the mean ﬁeld interaction energy proportional to n and the fermion kinetic energy is set by
the Fermi energy proportional to n2/3 .

85

Figure 6-1: Anisotropic expansion of 6 Li from the optical trap at 900 G (about 1.5 million
atoms in each state). The atoms were cooled for 500 ms at 900 G. This lowered the temperature from ∼ TF to ∼ 0.5 TF . (a) in trap, (b) 900 µs TOF, (c) 1500 µs TOF. The reversalof the anisotropy is obvious. States |1 and |2 are imaged simultaneously as explained in
Section 4.4.
The relative velocity scales as v ∼ kF ∼ n1/3 . This gives ΓCl = nσv ∼ n2/3 . When thesystem is unitarity limited, the scattering rate actually drops slower during expansion andhydrodynamic conditions can be maintained for longer expansion times. This n2/3 drop is
maintained until the local Fermi momentum no longer satisﬁes akF  1. Thereafter, theusual scaling with n is restored.

6.3

Expansion of a zero temperature Fermi gas

I now discuss collisions in zero temperature Fermi gases. The manuscript attached as
Appendix D [8] presents our results in great detail. This section is meant to be a supplement
to that presentation. I only summarize the major aspects of the paper and detail some
subtleties.

6.3.1

Momentum Space Fermi Surface

The momentum space Fermi surface can be constructed by using the simpliﬁcation allowedby zero temperature Heaviside distribution functions and harmonic traps. The local Fermi
surface is then a sphere in the trap:
Σi p2i /2m = EF − Σi mωi2 x2i /2,(6.8)

which deforms into an ellipsoid during time-of-ﬂight:(1 + ωi2 t2 )
Σi
2m

2

mxi ωi2 t2mωi2 x2i
=
E
−
Σ
,
pi −
i
F
t 1 + ωi2 t22(1 + ωi2 t2 )

86

(6.9)

5

25

(a)

20

Aspect Ratio

Aspect Ratio

4

(b)

3

2

1

15

10

5

0

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

2000

4000

6000

8000

10000

Expansion Time (1/ ω )Expansion Time (ms)Figure 6-2: Aspect ratio vs time for the conditions of Fig.6-1. (a) shows the measured
aspect ratio as a function of time of ﬂight up to ∼ 3 ms for an initial aspect ratio λ = 0.015(our trap). For the longest expansion times, the aspect ratio climbs up to > 2.5. Thelines are theoretical predictions for 2W power, and 16 µm waist giving trapping frequencies of 13.2 kHz and 197.6 Hz. 1 ms then corresponds to ω⊥ t ∼ 83 inverse radial trapping
frequencies. (b) shows the asymptotic behavior expected from hydrodynamics. The calculated aspect ratio grows strongly even after several thousand inverse trapping times! The
asymptotic aspect ratio expected from full hydrodynamic scaling is ∼ 0.38/λ ∼ 25 [119].
via the ballistic property x(0) = x(t) − pt/m. For a cylindrically symmetric trap, themomentum space aspect ratio is:

1 + ωz2 t2
2 t2 .
1 + ω⊥(6.10)

Fig. 6-3(a,b) shows an example of this variable aspect ratio (or deformation).
This momentum space picture allows a direct calculation of the spatial density. Using

dxdp = h, the phase-space volume for a quantum state in 1-D, the volume for N fermionsis d3 xd3 pf (x, p, t) = h3 N . The spatial density is simply the integral of the phase space

distribution over momentum space with the appropriate 1/h3 factor. Thus:1d3 p f (x, p, t)
h3= volume of momentum ellipsoid4
π product of semi − axes=3
ωi2 x2i 3/2m2mEF 3/2 (1 − 2EF Σi 1+ωi2 t2 )
4
π
,
=
3
h2
Πi (1 + ωi2 t2 )1/2n(x, t) =87(6.11)

(b)

(a)

10

3

1

10

0

2

2mEF

3

1

0

Figure 6-3: (a) Deformation of the momentum space Fermi surface at x = 0, from asphere to an ellipsoid during expansion from an anisotropic harmonic trap. The case of
cylindrical symmetry is shown, where the three-dimensional distribution is symmetric about
the vertical axis. The parameters chosen are an aspect ratio λ = 0.2 and expansion times2.
ω⊥ t = 0,1,3 and 10. (b) The deformation at a position radially displaced by EF /mω⊥Note that the central momentum also changes with time for x = 0 (Eqn. 6.9), an eﬀect
which is not shown in the ﬁgure. However, this has no eﬀect on the collision rate.

as in Eqn 3.5. Notice that for long times, the spatial aspect ratio goes to 1 as expected forballistic expansion and with the scaling given by Eqn. 6.2.

6.3.2

Collisions in the Perturbative Limit

For a two-spin system interacting with a ﬁnite s-wave scattering cross-section σ, the collision
integral of Eqn. 6.1 can be written as:Γ(x, p1 , t) = −

σ
4πh3

(x,t)

d 3 q1 d 2 Ω|p1 − q1 |×m

(6.12)[f (p1 )f (q1 )(1 − f (p2 ))(1 − f (q2 )) − f (p2 )f (q2 )(1 − f (p1 ))(1 − f (q1 ))]Eq. 6.12 has a σ in front, integrates over the momentum ellipsoid and has the relative
velocity in the integrand. It thus has the general form nσv, a collision rate. The collision
integral takes into account the availability of ﬁnal states (the (1 − f ) terms), the basisof Pauli blocking. Fig. 6-4(a) shows the geometrical representation of elastic collisions in
the local phase space picture. Collisions have the tendency to restore the deformed Fermi
surface into a spherical shape corresponding to an equilibrium Fermi-Dirac distribution2 .
2

The concept of Pauli blocking removal by ballistic expansion can be applied also to other situations suchas p-wave collisions in a single-spin system.

88

(a)

(b)
p2

q1

p1
p2

p3

q3

q2

p1

q2
q1

Figure 6-4: Geometrical representation of collisions in momentum space. (a) The two spinstates have identical distributions and are labelled with p and q. The deformation of the
Fermi surface into an ellipsoid opens up ﬁnal states for scattering. The processes p1 + q1 →p2 + q2 and p1 + q1 → p3 + q3 are examples of allowed collisions in a zero-temperaturegas. (b) For collisions involving p1 at the√ center of the ellipsoid (calculation done in thepaper), a minimum deformation of 1 : 3 is needed before energetically accessible ﬁnal
states become available. The process p1 + q1 → p2 + q2 is among the ﬁrst contributing
collisions as a function of increasing deformation (all ﬁnal states on the horizontal ring
formed by rotating p2 − q2 about p1 − q1 as well as the symmetric ring in the lower
hemisphere start contributing simultaneously).
One can work out the collision integral of Eqn. 6.12 corresponding to ellipsoids for diﬀerent ballistic expansion times (Eqn. 6.9) using numerical integration in Mathematica. This
forms a perturbative result for the collision rate and the eﬀects of Pauli blocking. The resultfor Γ(0, 0) is shown in Fig. 6-5. This is compared to the result neglecting Pauli blocking(setting all the (1 − f ) terms to zero, ΓCl,p in the ﬁgure). Fig. 6-5(b) shows that as the initial aspect ratio λ approaches 0 (high anisotropy), the total number of collisions allowed byFermi statistics approaches ∼ 0.55 of the number neglecting Pauli blocking. This of course
shatters the assumption of high Fermi degeneracy suppressing collisions during expansion.
Zero temperature Fermi gases can collide strongly during expansion.
Two aspects of the calculation are now explained. The fact that the collision rate
abruptly rises from zero at a ﬁnite expansion time in Fig. 6-5(a) is because a minimum
deformation is needed before the center of the ellipsoid (0, 0) can collide with any otherpoint in the momentum ellipsoid (Fig. 6-4(b)). From the ﬁgure, one easily sees that this√point corresponds to an aspect ratio 3 (simple coordinate geometry). Using this relation

89

(a)

Allowed Fraction ofCollisions (F)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 )Figure 6-5: (a) Collision rate as a function of expansion time in the perturbative approximation for the initial aspect ratio λ = 0.03. Dashed line - total classical collision rate Γcl ,thin line - Γcl,p , thick line - Γ. Γ and Γcl,p are deﬁned in the text. Γcl is deﬁned as theﬁrst term on the right-hand-side of Eqn. 6.12 with the 1 − f factors artiﬁcially set to 1 andcorresponds to what is usually called the classical collision rate. The displayed rates were
evaluated at x = 0 and p = 0 and give an eﬀective upper bound on the Fermi suppression.
(See Appendix D for details). (b) Allowed fraction of collisions F (λ) for a zero-temperature
two-spin Fermi gas. For an initial aspect ratio λ = 0.05, F is 0.5. For large anisotropy(λ → 0), F approaches ∼ 0.55.
2in Eqn. 6.10, one can solve to get ω⊥ t = τ = 1−3λ2 . For λ = 0.03, this dimensionless time√is ∼ 2 ∼ 1.4, in agreement with the calculation (Fig. 6-5(a)). A similar reasoning explains

why the collision rate in Fig. 6-5(b) rises above zero only below a ﬁnite λ. The critical timeof ω⊥ t = τ =2is never reached when the denominator is imaginary i.e., for λ  0.58
1−3λ2

(in agreement with the ﬁgure). A calculation summing the contribution over all p1 wouldsmoothen the rise of both these curves from t = 0 (Fig. 6-5(a)) and λ = 1 (Fig. 6-5(b)).

6.3.3

Perturbation from Hydrodynamic Behavior

For a strongly-interacting system, one can look at the eﬀect of expansion deformation
as a perturbation on the isotropy required by local equilibrium. The expansion-driven
anisotropy is damped (strongly) by collisions. James Anglin, our local theorist, found tolowest order in the momentum space anisotropy, the zero temperature collisional responseof the Fermi-Dirac distribution to the deformations of time-of-ﬂight. His result is includedin the manuscript attached in Appendix D [8]. A more quantitative calculation with enough90

detail to diﬀerentiate between superﬂuid and collisional hydrodynamics in expansion may
warrant a re-examination of the current expansion data available in diﬀerent laboratories.
This remains a theoretical goal in the ﬁeld.

6.4

Concluding Thoughts

The easy to use diagnosis of time-of-ﬂight expansion and absorption imaging turns out
not to be a qualitative tool for distinction between superﬂuid and normal gases in thestrongly-interacting regime. However, if the collisional hydrodynamics contribution could bediscarded by working away from Feshbach resonances and thus lowering the collisional crosssection, then hydrodynamic behavior could be unambiguously interpreted as a signature ofsuperﬂuidity. However, as Section 3.5 discusses, the low temperatures that would then be
required are way beyond current experimental possibilities.
The last section alluded to the theoretical goal of obtaining a full description of thecollisional expansion of a normal gas in order to ﬁnd quantitative diﬀerences from thebehavior of a superﬂuid. This approach also requires a full understanding of the expansionof a superﬂuid. If for example, the pairing only takes place near the Fermi surface, then
the whole system may not behave hydrodynamically, thus negating the approach of [115]and modifying the hydrodynamic predictions for superﬂuids. These are hard questions
for the theoretical community to resolve. The experimentalists have moved on to explore
other possible signatures for fermionic superﬂuidity, such as those involving radio-frequency
techniques (next chapter).
Hydrodynamic behavior also manifests itself in the shift of collective oscillation frequencies [120, 116, 118]. This can also be used to measure the transition from collisionless tohydrodynamic behavior [121]. This is of course a more diﬃcult measurement than free
expansion. However, Pauli blocking is important in the trap. Evidence of in-trap hydrodynamic behavior at ultra-low degeneracy may therefore indeed be proof of superﬂuidity. At
least one experimental group is currently pursuing such experiments [122].

91

Chapter 7

RF Spectroscopy of Ultracold
Fermions
This chapter focuses on the experiments reported in the following publication:• 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, “RF Spectroscopy of Ultracold
Fermions” Science 300, 1723 (2003) [5]. Included in Appendix E.
Radio-frequency(RF) spectroscopy has been a valuable tool for atomic physicists forseveral decades since Fermi and Rasetti introduced it in 1925 [123]. The technology to
produce and manipulate these frequencies is now standard (Chapter 4).
The extension of RF spectroscopy to ultracold fermions (Boulder - 40 K [117], MIT 6 Li [5]) was motivated by the possibility of studying the strong interactions near Feshbachresonances with the sensitivity and ease of RF methods. In addition to being sensitive tointeractions in normal Fermi gases, RF probes may also be able to detect the superﬂuid
phase [124, 125, 126].
Using RF methods, we observed a number of eﬀects in ultracold Fermi gases which aredescribed in this chapter:(1) In the s-wave regime, an RF transition in spin-polarized fermions is unshifted infrequency from the single particle resonance. Our observation of this eﬀect veriﬁes thissimple yet startling hypothesis and also opens the door for future atomic clocks based on
fermions. The current standard based on bosons is limited by interaction “clock” shifts. We
have also demonstrated the absence of interaction shifts in a two-spin mixture of fermions[7, 12].
(2) Interactions between fermions can be measured by forming a statistical mixture oftwo spin states and then driving transitions to a third state. Using a mixture of the twolowest hyperﬁne states and driving transitions from the second to the third hyperﬁne state,
we were able to map out interactions in 6 Li as a function of magnetic ﬁeld.

92

(3) At ultracold temperatures (kR  1), these interactions are proportional to the swave scattering length a, for small a. At an s-wave Feshbach resonance, where a diverges,
we leave the dilute gas approximation. Using RF methods we observed saturating of theinteraction energy at a large negative value regardless of the sign of the two-particle scattering length a. Although this result is not yet fully understood, it is consistent with somemany-body theory predictions in the ka  1, kR  1 regime (section 3.3) [83, 84, 85]).

7.1

Clock Shifts

Atomic clocks are based on the notion that the energy diﬀerence between eigenstates ofisolated atoms are a constant of nature. The unit of time is then deﬁned from the energy(frequency) diﬀerence between any two atomic eigenstates. In practice, two states arechosen whose energy diﬀerence is amenable to easy measurement with the least amountof perturbation from external sources. Perturbations can change the atomic resonancefrequency resulting in a “clock shift”. Researchers have now studied the systematics of suchmeasurements to levels which allow probing of changes of fundamental constants, such as
the ﬁne structure constant α, which determines the separations between energy eigenstates[71].
Since 1967, the SI unit of time, the second, has been based on deﬁning the ground-state
hyperﬁne transition in 133 Cs to be 9 192 631 770 Hz. Currently, the best realization of thesecond is via Ramsey spectroscopy of this transition in laser cooled atomic fountains [71].
Laser cooled atoms are launched upwards and interact with a microwave ﬁeld twice - once
on its way up and once on its way down. This microwave ﬁeld (∼ 9 GHz) is generated
by an external oscillator, like a quartz crystal1 . This setup is analogous to the separated
oscillatory ﬁelds of the Ramsey method [127, 128, 129]. The atoms are detected after the twointeractions and the fractional populations in each state exhibit the usual Ramsey fringes
as a function of microwave detuning. An error signal generated from these fringes is thenfed back to the quartz crystal. The macroscopic oscillator (crystal) is thus locked to themicroscopic oscillators (atoms). The crystal can now provide a standard of time based on
an atomic transition.
Laser cooled clouds are of course composed of a large number of particles, rendering
the isolated atom assumption ﬂawed. Current experiments are sensitive to the mean ﬁeld
interaction arising from coherent collisions between atoms in the cloud. In fact, the meanﬁeld clock shift is the limiting systematic in current atomic clocks [130, 71]. Consider abosonic atom with the hyperﬁne ground states |1 and |2. At ultracold temperatures, onlys-wave interactions are allowed. Let the s-wave scattering lengths be a11 , a12 and a22 .
Adopting the usual convention that positive scattering length a corresponds to a repulsive
two-particle interaction, for a gas of |1 atoms at density n, the mean ﬁeld shift of the1

more recently sapphire cryogenic oscillators are the external oscillators of choice.

93

transition is given by
∆νB =

2
n(a12 − a11 ).
m

(7.1)

S-wave interactions between atoms are in eﬀect during the coherent drive to transfer state|1 to state |2. For a perturbative drive, we can neglect the density in state |2. Every
atom transferred therefore needs an additional energy given by the diﬀerence in mean ﬁeld
energy of a state |2 atom in a bath of state |1 and a state |1 atom in a bath of state |1,as described by Eqn.7.1.

(a)

(b)

2

2

νB = ν12 + ∆νBνF = ν12

1

1

BosonsFermions

Figure 7-1: The diﬀerence in the clock shift for Bose and Fermi atoms. (a) In bosons, thes-wave interaction between identical particles shifts the initial and ﬁnal states connected by
the RF unequally. (b) In fermions, s-wave interactions cannot take place between identicalparticles and an RF drive then connects two states which have the same energy diﬀerence
as in an isolated atom.
S-wave interactions are absent in spin-polarized fermions. A spin-polarized fermionicgas remains spin-polarized during a coherent drive and therefore no interactions take place
between atoms. This results in an absence of the clock shift for ultracold fermions, ∆νF = 0.
A schematic of this diﬀerence between the two types of fundamental particles is shown in
Fig.7-1.
The error from the clock shift for bosons is at the level of 10−16 in current cesium clocks[71, 130], constituting the dominant systematic. Experiments with rubidium have a much
lower contribution - ∼ 50 times less [130, 71]. Eﬀorts are underway to determine the overall
feasibility of rubidium atomic clocks.
Gibble and Verhaar have pointed out the use of fermion clocks to suppress the mean-ﬁeld
shift problem [131]. The experiment described in the following subsection constitutes theﬁrst observation of this feature of fermions - the absence of the clock shift in a spin-polarized
gas.

94

7.1.1

Absence of the Clock Shift in Fermions

For ease of applying RF methods, a transition must be chosen that is relatively insensitive toﬂuctuations of the magnetic ﬁeld. For alkali bosons, the popular choice is the (appropriately
named) “clock” transition |I −1/2, 0 → |I +1/2, 0, in the ground state hyperﬁne manifold,at near zero bias ﬁelds. The clock transition is ﬁrst order magnetic ﬁeld insensitive andtherefore less prone to systematics from magnetic ﬁeld ﬂuctuations both in space and intime2 .
For alkali fermions, since I is always an integer, the state |F, 0 does not exist. However,
one can ﬁnd ﬁrst order ﬁeld independent transitions. For example, in 6 Li, the transition|1/2, −1/2 → |3/2, +1/2 is ﬁrst order ﬁeld independent at low ﬁelds (Fig. 4-8). However,for this particular case, the two states are not stable against spin-exchange, and would
not be a practical system at high density. At large enough magnetic ﬁelds (above I.J
decoupling), the electron spin dominates the splitting and the sublevels due to the nuclear
spin are essentially parallel to each other within each electron spin level 3 (ﬁg.4-8). We usedthe |1 → |2 transition at high ﬁeld (see Chapter 4 for the magnetic ﬁeld sensitivity of thistransition). As argued in chapter 5, this is also a stable combination of states.

(a)

(b)

2

2

∆ν12 > 0∆ν12 < 01

1|1> --> |2>|1> --> |2>

Figure 7-2: Shift in equilibrium energy in a 2-level Fermi system for (a) all the atoms instate |1 (b) all the atoms in state |2.
For bosons, the clock shift is usually determined by extracting the microwave frequency
from the Ramsey fringes as a function of density [71]. Since this is linear in the density
2

other options include the |I −1/2, I −1/2 → |I +1/2, I −1/2 transition in ground state alkalis. Though
this is ﬁrst order ﬁeld insensitive, it is formally unstable against spin-exchange, the notable exception being87
Rb [132, 133, 134].
3
The Zeeman energy levels deviate from their simple linear form due to I.J decoupling which gives riseto the “Back-Goudsmit” linear levels at moderately high ﬁelds. These persist upto L.S decoupling, which
in turn gives rise to the linear “Paschen-Back” levels at even higher ﬁelds.

95

(Eq. 7.1), the intercept of a straight line ﬁt gives the unshifted value. The error bar on theslope then includes the error contribution from cold collisions.
Our approach to demonstrate the absence of the clock shift was based on measuring theresonance frequency of the |1 − |2 system using a single oscillatory ﬁeld (Rabi technique[135]). This technique is in general less sensitive than the two-pulse Ramsey method [127]but is quite adequate for our purpose. Varying the density of one of the states and measuring
the (lack of) change in the transition frequency to the other state is a rather cumbersome
task. We instead compared the transition frequency of the |1 → |2 transition and the |2 →|1 transition on identical pure state samples of |1 and |2 respectively. The expectationfrom assuming that the frequency is shifted by the equilibrium energies (for a perturbativetransfer) is shown in Fig.7-2. For any ﬁnite scattering length a12 , equilibrium energies of
the transferred atoms shift in the same direction (higher for positive scattering length).
The spectra thus might be expected to shift in opposite directions. Our measurement thendetermines a suppression of this eﬀect from the (small) diﬀerence in the two Rabi spectra.
Fig. 7-3(a) shows the Rabi spectra taken for the |1 → |2 and |2 → |1 transitions
at 570 G. The pulse length used was 140 µs and dominates the broadening of the spectra.
The scattering length a12 at this ﬁeld is ∼ 150 a0 . The measurements were performed ata density of 3 × 1013 cm−3 . The equilibrium energy shifts are ±5 kHz in this case. The
Gaussian ﬁts are separated by 0.04 ± 0.35 kHz. The deduced suppression is then ≈ 30.
0.30

(a)

(b)

1 -> 2 transfer
2 -> 1 transfer

transferred fraction

Number of Atoms in Final State (arb)

0.25

0.20

0.15

0.10

0.05

0.00

-6

-4

-2
0
2
4
Frequency Detuning [kHz]

-1.0

6

-0.5
0.0
0.5
Frequency Detuning [kHz]1.0

Figure 7-3: Absence of the clock shift in fermions. RF transitions were driven between
states |1 and |2 on a system prepared purely in state |1 (ﬁlled circles), and purely in state|2 (open circles). (a) Experiment at 570 G. Mean-ﬁeld interactions would result in 5 kHz
shifts for the two curves in opposite directions. Gaussian ﬁts (solid lines) to the data areseparated by 0.04 ± 0.35 kHz. This gives a clock shift suppression factor of 30. (b) Absenceof clock shift at 860 G with three orders of magnitude suppression. The pulse length used
is 1 msec which is responsible for the much narrower width than in (a). The magnetic ﬁeld
sensitivity ∂ν/∂B is 0.5 kHz/G about 10 times lower than at 500 G, allowing the applicationof longer pulses and the correspondingly smaller widths.
96

Fig. 7-3(b) shows the suppression of the clock shift in a regime of unitary-limited interactions - 860 G, near the divergence of the bare scattering length [106]. In this unitarity
limited interaction regime, one cannot use the bare scattering length a12 to estimate theinteraction energy, as will be shown later. However, we can estimate the equilibrium energy from expansion measurements. Using the measurements of [111], and scaling to ourparameters, we arrive at a suppression factor of more than three orders of magnitude.
For bosons, the intercept of the shift vs density line provides the clock transition frequency. Additionally, the slope of this line gives a relation for the scattering lengths in thesystem (Eq. 7.1). Such a measurement is of course impossible for fermions with coherent
drives on a pure state. Since, our initial reason to pursue RF spectroscopy of fermions was
the measurement of interactions (scattering lengths), we had to come up with a diﬀerent
system.
Our solution was to use a 3-level system (Fig. 7-4). After forming a superposition state
and waiting for it to decohere (next section), the equilibrium energy levels are as shown in
Fig. 7-4. Transitions to a third level are then independent of the |2 − |3 scattering lengtha23 (absence of clock shift out of a pure state) and are only by the interactions of state |1with states |2 and |3 (a12 and a13 ):∆ν =2
n1 (a13 − a12 )
m

(7.2)

3
∆ν = ∆ν13 - ∆ν12
2

1
Figure 7-4: The RF resonance is sensitive to mean ﬁeld shifts in a 3-level system. The
diagram shows the case for positive a12 and a13 with a13 > a12 .
97

Since the shift in this resonance takes place over a ﬁnite time, the decoherence time,
probing the |2 → |3 transition at varying times also provides a way to measure thedecoherence time of our system.

7.2

Emergence of Mean Field Shifts in a Fermi System

The evolution of a superposition state into a statistical mixture has been the subject ofnumerous discussions involving topics ranging from NMR to error correction in quantum
computation. The understanding of the mechanisms of decoherence presents another avenue of research with degenerate fermions. In this section, I will ﬁrst discuss the decohering
mechanisms in our system. Next, I will present an experiment which measured this deco-

B X dn/dB (arbitrary units)

herence time using RF methods.

0

200

400
600
B Field [G]

800

1000

Figure 7-5: Decoherence rate due to inhomogeneous magnetic ﬁelds. Plot of B × ∂ν12 /∂B.
The maximum of B × ∂ν12 /∂B is at ∼ 57.46 G.

7.2.1

Decoherence in our system

Since spin-polarized ultracold fermions are very near the epitome of a non-interacting system, all decohering mechanisms must be related to single atom eﬀects. These would theninclude any spatial variations of the energy diﬀerence of the two spin components across thephase space experienced by the particles. Other eﬀects might be collisions with backgroundgas particles or interactions with ambient photons. Without going into detailed study of allthese eﬀects, we can with some conﬁdence say that the most relevant mechanism is magnetic
ﬁeld inhomogeneity. The ﬁnite sensitivity of the transition to magnetic ﬁeld quantiﬁed by98∂ν∂B acts together with the imperfections of the magnetic ﬁeld (which scales with B, for stan∂ν× B is then thedard power supplies energizing the magnetic ﬁeld coils). The product ∂Bquantity which determines the decoherence time. A plot of this quantity for the |1 → |2
transition is shown in ﬁg.7-5.
This has to be contrasted with the case of ultracold bosons where s-wave interactions dotake place at all temperatures. This means that mean ﬁeld is an additional contribution todecoherence. A remarkable experiment in Eric Cornell’s group at JILA actually uses these
two eﬀects to cancel each other and increase the coherence time in a bosonic system [134].
There may be additional eﬀects coming in once partial decoherence has already occurred.
For example, elastic collisions do start to occur once any degree of non-orthogonality exists
in a system. The level of decoherence from these elastic collisions is still not fully understood.

Transferred Fraction

0.8

0.6

0.4

0.2

10

20

30
Wait Time [ms]40

50

Figure 7-6: Emergence of mean ﬁeld shifts due to decoherence at 500 G. Decoherence leads
to a reduction of the |2 → |3 transfer at the unperturbed resonance ν23 . An exponential
ﬁt to the data (solid line) gives the time constant of 12 ms.
We measured the decoherence time of our |1 − |2 system by driving the transition|2 → |3 (see ﬁg.7-4) at varying times after creating a superposition state of |1 and|2. For a pure superposition state, the resonance occurs at the unperturbed frequency
νcoh = ν23 . For a fully decohered state, the resonance is shifted by ∆ν (Eq. 7.2), i.e.,νdecoh = ν23 + ∆ν = ν23 + 2m n1 (a13 − a12 ). For a ﬁnite ∆ν and a broadening dominated bytrap inhomogeneity, the transfer at ν23 decays in time. We measured the transfer fraction
as a function of time at 500 G and ﬁt an exponential curve (Fig.7-6). We did not study theevolution dynamics from the pure superposition to the statistical mixture. 4 . Therefore ourchoice of the exponential curve may not be accurate.
We also measured longer decoherence times at higher ﬁelds (∼ 800 G), in agreementwith the expectation from ﬁg. 7-5. In principle, the decoherence times can be arbitrarily
extended by tuning the ﬁeld variation across the sample. A perfectly ﬂat bias ﬁeld would
4

this could form the subject of a future project!

99

give rise to extremely long decoherence times. Our ﬁeld curvature produces ∼ 2 Hz trapping
frequency at 500 G.

7.3

RF Spectroscopy of a 3-level system

Our experimental scheme to measure the interactions given by Eq.7.2 is shown in Fig.7-7.
The main idea is to measure ∆ν. This can be obtained from the diﬀerence in the |2 → |3
spectrum of the pure |2 state and a statistical mixture of |1 and |2 and the knowledge of
the |1 density n1 .
a)
1000

b)

c)

Energy / h [MHz]

|6>
|5>
|4>

3
2
F1
2

|3>
|2>
|1>

-1000
0

500
Magnetic Field [G]75.185 MHz 84.754 MHz

|3>

|3>

νRF

500 G

|2>

|2>

|1>

|1>

desired field

Figure 7-7: (a) Hyperﬁne structure of the ground state of 6 Li. (b and c) Experimental
scheme: (b) preparation of a mixture of atoms in states |1 and |2, and (c) RF spectroscopyof the |2 → |3 transition.
Starting from the pure state |1 at low ﬁeld in the ODT, we ﬁrst ramped up the ﬁeld
to 500 G in a few ms. Here, the choice of state was made. By applying a slow, adiabatic
Landau-Zener frequency sweep, a pure state |2 sample was created. By applying a fast, nonadiabatic sweep and letting the superposition decohere in 200 ms ( 12 ms), the statisticalmixture was prepared. The Rabi technique was used with pulses of 140 µs duration. Typicalparameters for the decohered |1 − |2 mixture were mean-density n1 ∼ 2.4 × 1013 cm−3 and√temperature T ∼ 0.7 TF . Here the mean density is lower than the peak density by 2 2 forour modest degeneracy, where the Gaussian approximation can be used.
Using the triple imaging technique described in Chapter 4, we monitored the appearanceof atoms in state |3 and the disappearance of atoms in state |2. Recording state |1
provided the required density n1 . Using the separation ∆ν of the two spectra for the3transferred fraction N2N+N
and n1 for normalization, (a13 − a12 ) can be determined from3

Eq. 7.2. A typical spectrum (at 480 G) is shown in Fig.,7-8. In addition to the easily resolved
separation of the two peaks (∼ 16 kHz), the spectra also show considerable diﬀerence in
100

(a)

Transferred Fraction

0.6

0.4

0.2

0.0
-50

(b)

-40

-30

-20

-10

0

10

Frequency Detuning [kHz]Figure 7-8: (a) Fraction of atoms transferred from |2 to |3, with |1 atoms absent (ﬁlled
circles), and present (open circles). The mean-ﬁeld shift is computed from gaussian ﬁts tothe data (solid lines). (b) Spatial images of state |3 for the perturbed resonance. The
optical trap was turned oﬀ immediately after the RF pulse and absorption images of theatoms were taken after 120 µs expansion time. The central section of ∼ 150 µm verticalextent was used to extract the transferred fractions in (a). (b) also shows images of states|2 and |1 for zero RF detuning. States |3 and |2 were imaged simultaneously to observe
their complementary spatial structure. State |1 was imaged after 760 µs expansion time torecord its density for normalization purposes.
their widths. For the pure state transfer (ﬁlled circles), the broadening is dominated by
the pulse width. For the incoherent mixture (open circles), the width is dominated by theinhomogeneous density distribution according to Eq. 7.2. The measured ∆ν for a wide rangeof magnetic ﬁelds 300 − 750 G, are shown in Fig. 7-9(a).
We compared our measurements with calculations from the Eindhoven group (Fig. 79)(b). The calculated scattering lengths and the measured density were used to computethe theoretical curve for ∆ν as a function of magnetic ﬁeld. We see that there is good
agreement at low ﬁelds. The RF method is sensitive to the narrow Feshbach resonance at
545 G and the sign change in Eq.7.2 at ∼ 600 G. Although the data tracks this theory quite
well for ﬁelds below 630 G, substantial deviations occur near the wide Feshbach resonances
at 680 G (a13 ) and 810 G (a12 ).

101

(a)

(b)

Frequency Shift [kHz]

20

10

0

-10

-20

400

500
600
Magnetic Field [G]700

Figure 7-9: Spectroscopic measurement of interaction energy. (a) Frequency shift vs. magnetic ﬁeld of the |2 → |3 resonance due to atoms in state |1. The shifts are computedby monitoring the arrival fraction in state |3 for 140 µs RF pulses, except at 750 G. Here,because of strong inelastic losses between |3 and |1 atoms, we monitored the loss of atoms
in state |2 after applying RF sweeps of 3 ms duration and 2 kHz width. All the data points
are normalized to the same atom number in state |1. The ﬁt at low ﬁelds (solid line) uses
Eq. 7.2 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 thegaussian ﬁts to the spectra. The dashed line indicates the position of the predicted a13
resonance. (b) s-wave Feshbach resonances in a12 and a13 . 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 6 Li experimental
data [136] in a coupled channel approach to deduce accumulated phases that characterize
the less well-known short range parts of the 6 Li + 6 Li scattering potential [137]. a12 hasa narrow Feshbach resonance at 550 G and a wide one at 810 G. a13 has a wide Feshbachresonance at 680 G.

7.4

Observations of Unitary and Universal Behavior

The Feshbach resonances in a12 and a13 are wide and overlap substantially. This distinguishes it from the 40 K system where the Feshbach resonances are narrow and well separated[109, 117]. Even though the width is in some sense forgiving - one does not need to havegreat magnetic ﬁeld stability to be in the vicinity of the resonance, the overlap of diﬀerent
resonances makes understanding the 3-level RF spectra more complicated.
As can be seen in Fig.7-9(a), the shifts for ﬁelds up to ∼ 630 G lie in agreement with the
expectation from Eq.7.2. Above this ﬁeld, there is a noticeable deviation from this simple
prediction. This is because the two assumptions of dilute gas mean ﬁeld theory: ka  1,
na3  1 start to break down. The broad structure of the Feshbach resonances allows us to
see the unitary behavior over large magnetic ﬁeld ranges.
102

The description of quantum mechanical unitarity in Chapter 6 helps us see some of thisdeviation. The mean ﬁeld, proportional to Re(f ) reaches a maximum near 1/kF and thenstarts to decrease gently. This means, that the shifts can never diverge like the scattering
length does. Once both interactions a12 and a13 are unitarity limited, they have the same
magnitude. This implies that they should cancel each other giving near zero spectroscopic
shifts when they are of the same sign. This explanation works below 680 G where a13
diverges. However between 680 G and 810 G, a13 and a12 are large and of opposite sign andshould then produce a large negative shift. However, we observe close to zero shifts in thisregion as well.
The explanation of this puzzle may lie in our having left the low density limit and entereda regime where higher order many-body eﬀects can become important. Some recent manybody calculations [83, 84, 85] do indeed suggest a universality of interaction energy in theunitarity limit, making the “mean ﬁeld” independent of the sign of the bare two-particle
interaction. This is a possible explanation of what we see and is consistent with othermeasurements in highly interacting Fermi systems - both by expansion energy measurements[138, 106, 111] and by RF spectroscopy methods [117]. The fact that these theories predicta universal negative (attractive) sign of the interaction is of course good news from the pointof view of achieving superﬂuidity in these strongly interacting systems.

7.5

OutlookThe use of RF methods to study ultracold fermions has generated considerable excitementin the ﬁeld. In addition to RF spectroscopy [117, 5], RF photodissociation has also beenused to detect molecules formed out of a mixture of ultracold fermions [139]. The RF tool
remains the best way to tailor the spin composition, an important component en route tosuperﬂuidity. Finally, the proposals of Zoller et. al. [125, 126] to use Raman methods
to couple the superﬂuid and normal states can be modiﬁed to RF methods and remains apossible method to detect the superﬂuid fraction.

103

Chapter 8

Conclusions and Outlook
Two sets of experimental studies were reported in this thesis. The ﬁrst involved the development of a new atom interferometer scheme which shows promise for high accuracy
metrology using Bose-Einstein condensed gases. The second involved the production andexploration of a strongly interacting Fermi gas mixture which is predicted to undergo asuperﬂuid phase transition under current experimental conditions.
We have demonstrated an atom interferometer using optical standing wave diﬀraction
gratings which has several desirable features for a high precision measurement of the photon recoil frequency ωrec [1]. ωrec can be used in conjunction with other measurements todetermine the ﬁne structure constant α, a fundamental constant of physics. Our preliminary result for ωrec,Na at 7 × 10−6 precision obtained using a 23 Na BEC and horizontal
standing waves deviates from the currently accepted value at the 2 × 10−4 level. We believe that this deviation arises from the mean-ﬁeld interaction between atoms. A future
measurement would involve converting to an atomic fountain setup with vertical standing
waves to increase the interaction times in the interferometer. The precision in such a geometry should approach ppb values. The mean-ﬁeld systematic should be suppressible tothis level using very low density BECs[74]. We hope to obtain a  1 ppb value for ωrec ina second generation experiment in which BECs are created elsewhere and transported into
the interferometer[140].
We have upgraded our 23 Na BEC machine with the capability to cool and trap 6 Li
fermions. By cooling 6 Li sympathetically with 23 Na in a magnetic trap, we can produce
quantum degenerate Bose-Fermi mixtures as well as very large and deeply degenerate Fermi
gases. Optimization of our sympathetic cooling strategy has established a (currently) unmatched system performance in terms of atom number (both in the mixture and purefermion operation) as well as degeneracy of pure fermions (0.05 TF ) [6]. Degenerate BoseFermi mixtures promise to be fertile research ground [141, 142]. We have however concentrated on studying the degenerate Fermi gas in a strongly interacting regime.
Using the variation of interaction strength oﬀered by magnetically-tunable Feshbach
scattering resonances, we have explored physical regimes in which a new type of BCS su104

perﬂuid is predicted to form [99, 100, 108] at modest degeneracies (high Tc ) of ∼ 0.5 TF . The
creation of this fermionic superﬂuid would establish a model system for studying superﬂuid
phenomena at densities which are a billion times lower than in previous realizations in 3 He
and superconductors.
Since suitable Feshbach resonances in 6 Li only exist in spin states that cannot be magnetically trapped, we transfer our degenerate gas into an optical trap. By monitoring themagnetic ﬁeld dependent stability of a two-spin mixture of 6 Li fermions, we have located
two s-wave Feshbach resonances [4]. The measured stability in the strongly interacting
regime is encouraging for the prospects of creating the superﬂuid state in 6 Li.
In the vicinity of Feshbach resonances, several groups [86, 117, 111] including us (unpublished) have observed strong hydrodynamic behavior in a fermion spin mixture during
expansion out of an asymmetric trap. The possible interpretation of this behavior as asignature of superﬂuidity (as in the case of BEC) has been considered [115, 86]. However,classical collision processes can also give rise to such behavior, mitigated only by the Pauli
suppression of collisions at low temperatures. By examining the deformation of the Fermi
surface during expansion out of anisotropic traps, we theoretically analyzed the collisional
behavior of an expanding, zero-temperature gas[8]. Our calculations indicate that unlikein the case of a BEC, hydrodynamic expansion of Fermi gases near Feshbach resonances
cannot be a dramatic, qualitative signal for the existence of the superﬂuid state, regardlessof the temperature of the gas.
Another proposed method to detect the BCS state is to monitor shifts of resonances
between atomic internal states due to the presence of the superﬂuid [125, 126] (corresponding
to a pair-breaking energy). Our eﬀorts in this direction (and those at JILA[117]) using radiofrequency (RF) transitions betweens 6 Li (40 K) spin states have not (yet) observed such ashift. However, we have measured shifts in RF transitions due to interactions between
diﬀerent spin states in a three-state system. Our measurements indicate a saturation of theinteraction energy near Feshbach resonances which has a negative sign regardless of the signof the bare (two-particle) interaction. Although this is consistent with other measurements[111], there is no clear agreement on the mechanism which would cause such behavior.
While some many-body theorists predict a universally negative interaction parameter inthe strongly interacting s-wave regime (n|a|3  1, nR3  1) of current experiments, others
propose an explanation based on molecule formation [111, 143]. Using RF techniques wealso observed the absence of interaction shifts in two-state fermion systems [5, 7]. This isencouraging for the prospects of using fermions in future atomic clocks. The use of the RF
tool to observe a fermionic superﬂuid remains on our experimental agenda.
The experimental observation of fermionic superﬂuidity is currently the foremost goal in
our research group. For a robust observation and characterization of the superﬂuid state, arelatively direct and easy-to-implement experimental diagnostic would be ideal. For BEC’stime-of-ﬂight expansion serves such a purpose. As indicated above, in the current regime105

of experiments, such a diagnostic cannot be a qualitative tool. On the other hand, RFdiagnostics still appears promising.
A new direction currently being followed by some groups involves the creation of moleculesfrom a two-spin Fermi gas by adiabatically varying the magnetic ﬁeld across a Feshbach resonance [139, 143, 144]. Trapped, long-lived (upto 10 s [144]) molecules have been created inthis way [143, 144]. These molecules could undergo Bose-Einstein condensation if they arecooled to below the critical temperature. The production of such a molecular BEC would
alone be an achievement of considerable scientiﬁc signiﬁcance. Additionally, the molecular
BEC could cross-over into a BCS state [145] by a further tuning of the interaction with themagnetic ﬁeld.
It seems quite safe to predict that the observation of fermionic superﬂuidity in dilutegases will cause an explosion of interest and activity in the ﬁeld as in the case of bosonicsuperﬂuidity (1995 onwards). The range of new and exciting physics that the creation ofsuch a system will produce is diﬃcult to overestimate. An important theme will be theunderstanding of the behavior of other Fermi systems such as high Tc superconductors.
Probably the most interesting studies will be the ones that we cannot predict at this point.

106Appendix A

Contrast Interferometry using
Bose-Einstein Condensates to
Measure h/m and αThis appendix includes the following paper [1]: S. Gupta, K. Dieckmann, Z. Hadzibabic,
and D.E. Pritchard, “Contrast Interferometry using Bose-Einstein Condensates to Measureh/m and α”, Phys. Rev. Lett. 89, 140401 (2002).

107

VOLUME 89, N UMBER 14

PHYSICA L R EVIEW LET T ERS

30 SEPTEMBER 2002

Contrast Interferometry using Bose-Einstein Condensates to Measure h=m and 
S. Gupta, K. Dieckmann, Z. Hadzibabic, and D. E. Pritchard
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge,
Massachusetts 02139(Received 19 February 2002; published 10 September 2002)
The kinetic energy of an atom recoiling due to absorption of a photon was measured as a frequency,
using an interferometric technique called ‘‘contrast interferometry.’’ Optical standing wave pulses wereused to create a symmetric three-path interferometer with a Bose-Einstein condensate. Its recoil phase,measurable with a single shot, varies quadratically with additional recoils and is insensitive to errors
from vibrations and ac Stark shifts. We have measured the photon recoil frequency of sodium to 7 ppm
precision, using a simple realization of this scheme. Plausible extensions should yield sufficientprecision to attain a ppb-level determination of h=m and the fine structure constant .
DOI: 10.1103/PhysRevLett.89.140401

PACS numbers: 03.75.Dg, 03.75.Fi, 06.20.Jr, 39.20.+q

where k is the wave vector of the photon absorbed by theatom. Recent experiments allow Eqs. (1) and (2) to be
applied to cesium. MCs is known to 0.17 ppb[9] and kCs to
0.12 ppb [10]. !rec;Cs has been measured at Stanford using
an atom interferometer based on laser-cooled atoms to
6 ppb [11–13]. Similar experiments are also possible withalkali atoms such as sodium and rubidium, where M has
been measured [9] and k is accurately accessible [10].

In this Letter, we demonstrate a new atom interferometer scheme which shows promise for a high precision
measurement of !rec . Our symmetric three-path configuration encodes the photon recoil phase in the contrast of
the interference fringes, rather than in their phase.
Because it is insensitive to the fringe phase, the method
is not sensitive to vibrations, accelerations, or rotations.
The symmetry also suppresses errors from magnetic field
gradients, and our use of only one internal state suppresses errors arising from differences in the ac Stark
shifts between different internal states. A crucial aspectof this new interferometer is the use of atomic samples
with subrecoil momentum distribution. We use a BoseEinstein condensate (BEC) as a bright subrecoil atom
source. This allows the contrast oscillations to persist
for many cycles, permitting precise determination of
the recoil phase in a single ‘‘shot.’’ This also allows for
extra photon recoils to be added within the interferometer, increasing the recoil phase shift and, hence, themeasurement precision quadratically.
The Stanford scheme to measure !rec uses different
internal states to separately address different interferometer paths, allowing a linear increase of measurement
precision by additional photon recoils. However, vibrations and ac Stark shifts have been of great concern in thisscheme [12]. An alternative interferometer to measure thephoton recoil using laser-cooled atoms in a single internal
state was demonstrated using rubidium atoms [14]. Like
ours, this scheme also incorporates a symmetric arrangement and operates by measuring contrast. This interferometer should also suppress vibration noise andsystematics arising from ac Stark shifts between different
internal states. However, different paths cannot be individually addressed in this scheme, making it difficult toextend to competitive precision. Our interferometer extends these previous schemes and combines their advantages. The precision of the Stanford scheme increases
linearly with additional recoils. Quadratic scaling
schemes have been proposed [15] and demonstrated in a

140401-1

 2002 The American Physical Society

Comparison of accurate measurements of the finestructure constant  in different subfields of physics
offers one of the few checks for global errors across thesedifferent subfields. The g2 measurement for the electron and positron, together with QED calculations, provides a 4 ppb measurement of  [1,2]. This has stood as
the best measurement of the fine structure constant since1987. The second most accurate published value of , at
24 ppb, comes from condensed matter experiments [3].
This is worse by a factor of 6, limiting the scientific valueof cross-field comparisons. A new and more robust routebased on atomic physics measurements has emerged in
the past decade [4]: 2 2e
2R h
2R M h
2 
:
(1)
 1
 1
h c
c me
c Me mThe Rydberg constant R1 is known to 0.008 ppb [5,6]
and the electron mass Me to 0.7 ppb [7]. M and m are themass of some test particle in atomic and SI units, respectively. Equation (1) offers the possibility of a ppb-levelmeasurement of  if M and h=m can be determinedaccurately.
h=m can be measured by comparing the de Broglie
wavelength and velocity of a particle, as demonstrated byKrüger, whose measurement using neutrons has yielded a
73 ppb value for h=mn [8]. For an atom, h=m can beextracted from a measurement of the photon recoil frequency,
1 h 2k;!rec 
(2)2m

0031-9007=02=89(14)=140401(4)$20.00

140401-1

VOLUME 89, N UMBER 14

PHYSICA L R EVIEW LET T ERS

multipath interferometer based on dark states [16].
However, the number of additional recoils in this scheme
is limited by the internal atomic structure.
Our scheme is based on the asymmetric interferometerof Fig. 1(a). At time t  0 a BEC is split coherently into
two momentum components, j2 hki
 and j0 hki, by a firstorder Bragg =2 pulse [17]. These are shown as paths 1
and 2 in the figure. At time t  T, a second order Bragg pulse reverses the direction of path 1, while leaving path 2
unaffected. Around t  2T, a moving matter wave gra2
ting, with spatial periodicity =2 (wave vector 2k  =2),
is formed due to the overlap and interference of the twopaths. The phase of this grating at 2T is determined by the
relative phase 1  2  8 !rec T, accumulated between
paths 1 and 2 due to the difference in their kinetic
energies. A measurement of this phase for different valuesof T will then determine !rec . If the momentum of path 1
is increased N times by additional photon recoils, thecorresponding grating phase will be1 2 
N 2 8 !rec T, leading to an N 2 -fold improvement in themeasurement precision. The fringes of all atoms will bein phase at 2T, forming a high-contrast matter wave
grating. This dephases in a time k 1 v , the coherencetime, where v is the atomic velocity spread.
Extension of this interferometer to a symmetric threepath arrangement is shown in Fig. 1(b). Three momentum
states (paths 1, 2, and 3) are generated by replacing thefirst Bragg pulse with a short Kapitza-Dirac pulse [17]. Att  2T, there are now two matter wave gratings withperiod =2, one from paths 1 and 2, and one from paths
2 and 3. These move in opposite directions at a relative
speed 4hk=m.

If the maxima of the two gratings line up to
produce large contrast at time t, the maxima of one willline up with the minima of the other at t  =4 !rec , toproduce zero contrast. This results in an oscillatory
growth and decay of the contrast of the overall pattern

FIG. 1. Space-time representation of the contrast interferometer. (a) A simple two-path interferometer sensitive to thephoton recoil phase. The 2k matter wave grating is shown at
2T and at 2T  =4 !rec . The extension to the three-path
geometry is shown in (b). The overall 2k grating has large
contrast at 2T and zero contrast at 2T  =4 !rec .

140401-2

30 SEPTEMBER 2002

with time. The recoil induced phase can be determinedfrom this temporally oscillating contrast.
The time evolution of this contrast can be monitored bycontinuously reflecting a weak probe beam from thegrating (the additional grating formed by paths 1 and 3
has period =4, which does not reflect the probe beam).
The reflected signal can be written ast  3 t
ST; t  CT; t sin2 1 2 t2
 CT; t sin2 8 !rec T  4 !rec t  2T ; (3)where CT; t is an envelope function whose width is thegrating coherence time, k 1 v . This motivated our use of a
BEC atom source. This allowed many contrast oscillations in a single shot. Using the phase of the reflection att  2T, T  8 !rec T, !rec can be determined by varying T. Vibrational phase shifts and the effect of magnetic
bias fields and gradients cancel in the evaluation of
1 t 3 t 2 t, due to the symmetry of our scheme.
2
In this experiment, we realized the scheme of Fig. 1(b)and measured !rec;Na to 7 ppm precision. We also demonstrated the insensitivity of the contrast signal to vibrations and the N 2 scaling of the recoil phase.
We used sodium BECs containing a few million atoms
in the jF  1; mF   1i state as our atom source. The
light pulses were applied 15 ms after releasing the
BEC from a weak magnetic trap. This lowered the peakdensity to about 1013 cm3 , thus preventing superradiance effects [18] and reducing frequency shifts frommean field interactions. Two horizontal counterpropagating (to 1 mrad) laser beams were used for the diffraction gratings. The light for the gratings was red-detuned
by 1.8 GHz from the sodium D2 line. Rapid switching ( <100 ns) and intensity control of the light pulses was doneby an acousto-optic modulator (AOM) common to thetwo beams. The phase and frequency of each beam were
controlled by two additional AOMs, driven by two phaselocked frequency synthesizers.
The interferometer pulse sequence was started with a
1 s, square Kapitza-Dirac pulse, centered at t  0. We
adjusted the beam intensity, to put 25% of the condensate in each of the j 2hki
 diffracted orders. This choiceyielded the best final contrast signal. The second order
Bragg pulse was centered at t  T and was close toGaussian shaped, with a width of 7:6 s. The intensity
was chosen to effect a  pulse between the j 2hkistates. The smooth pulse shape reduced the off-resonant
population of undesired momentum states, yielding atransfer efficiency of > 90%. The third pulse, used forreading out the contrast signal, was centered at t  2T and
was typically 50 s long. One of the Bragg beams wasused as the readout beam while the other was blocked.
The light reflected from the atoms was separated fromthe readout beam path using a beam splitter and directedby an imaging lens onto a photomultiplier. A typical
140401-2

VOLUME 89, N UMBER 14

PHYSICA L R EVIEW LET T ERS

interferometer signal is shown in Fig. 2. We observed the
expected contrast oscillations at frequency 8 !rec , corresponding to a 5 s period for sodium. We obtained therecoil phase T from the contrast signal by fitting to asinusoidal function as in Eq. (3).
The signal also contained a small pedestal of similar
width as the envelope. This consists of a constant offset
from residual background light and a smoothly varying
contribution from a small asymmetry between the j2hki amplitudes of <5%. This asymmetry creates a nonoscillating component of the 2k matter wave grating
which decays with the same coherence time. The uncertainty of the fitted phase is about 10 mrad, even if weneglect the envelope function, and assume a constant
amplitude extended over a few central fringes. Similar
uncertainty was obtained for large times T 3 ms. We
observe a shot-to-shot variation in the fitted value of thephase of about 200 mrad. We attribute this to pulse intensity fluctuations which randomly populated undesired
momentum states at the <10% level. This resulted inspurious matter wave gratings which shifted the observed
recoil phase.
The recoil frequency was determined by measuring therecoil phase around T  0:5 ms and around T  3 ms
(Fig. 3). An upper bound on T was set by the atoms fallingout of the 2 mm diameter beams. A straight line fit tothese data produced a value for the sodium photon recoil
frequency !rec;Na  2  24:9973 kHz 1 6:7  106 .
This is 2  104 lower than the sub-ppm value calculatedusing the published measurements of g2 , R1 , MNa [9],Me , and Na [19] in Eqs. (1) and (2). The systematic mean
field shift due to larger population in the middle path thanthe extreme paths probably explains this deviation.
Estimated errors from beam misalignment and wave
front curvature have the same sign as the observed deviation but several times lower magnitude.

30 SEPTEMBER 2002

To demonstrate the insensitivity of the measurement tophase noise of the light due to mirror vibrations, we
intentionally varied the phase  of the second grating
relative to the first one [20]. The contrast signal is not
visibly affected by such phase variations [Fig. 4(a)]. We
compared this to a phase-sensitive readout method[Fig. 4(b), inset]. This was realized by replacing thereadout pulse with a third pulsed 1 s light grating in
the Kapitza-Dirac regime, phase-locked to the first two
pulses. This projected the phase of the 2k pattern at t  2T
onto the fractional populations of the states j0 hki, j2 hki,and j2 hki which leave this interferometer. The populations were measured by time-of-flight absorption imaging. The j0 hki
 fraction is shown for the same variation of
, in Fig. 4(b). The oscillation [21] demonstrates the phasesensitivity of any position-sensitive readout.
These two interferometers respond differently to mirror vibrations. For large T, we have observed the effect ofthe mirror vibrations directly. At T 3 ms, the shot-toshot fluctuations of the phase-sensitive interferometer
was of the order of the expected fringe contrast. Thisagrees with observations with a standard Mach-Zehnderinterferometer constructed both by us and in [22]. In
comparison, the stability of the contrast interferometersignal is independent of T within our measurements. Thiscan be seen from the comparable statistical error bars for
short and long times in Fig. 3(b). In fact, the residuals andthe corresponding error bars are smaller at the longer
times. We attribute this to the decreased amplitude ofsome of the spurious gratings at longer times, due toreduced overlap of the contributing wave packets.
The quadratic scaling of the accumulated recoil phase,with the number of transferred recoils N, was demonstrated by comparing N  1 and N  2 interferometers.
An N  2 geometry, shown in Fig. 5(a), was realizedusing two additional first order Bragg  pulses spaced
T1 apart and affecting only the extreme paths. The acceleration pulse at t  60 s drove transfers from j 2hki
 toRecoil phase,φ (T)[radians]readout
pulse

980

1000

1020

1040

FIG. 2. Typical single-shot signal from the contrast interferometer. T  502:5 s, for this example. Ten oscillations with
60 % contrast and 30 s width are observed during the 50 sreadout. A low-pass filter at 300 kHz (12 dB per octave) wasapplied to the signal.

Residuals[radians]

t=2Tt [ µs]140401-3

(a)

650
640
630
0.50

contrast
signal
960

3780
3770

0.5

0.51

3.00
0.52
Time T [ms]

3.01

(b)

0.0
-0.5

FIG. 3. Measurement of !rec in sodium. Two sets of recoil
phase scans, around T  0:5 ms and T  3 ms, are shown in (a).
Each point is the average of five measurements. The slope of thelinear fit gave !rec to 7 ppm. The error bars ( 0:05  0:1 rad)are shown with the fit residuals in (b).

140401-3

VOLUME 89, N UMBER 14
0.8

(a)

30 SEPTEMBER 2002
(b)

(a)

(b)

0
-1

0.6

N=1

θ

T=50µs2.0

3.0

Applied phase, θ [radians]

0.0

1.0

2.0

2T 2T+T1380

390
T1 [µs]400

3.0Applied phase, θ [radians]

FIG. 4. Vibration insensitivity of the contrast interferometer.
(a) The measured recoil phase at T  502:5 s from the contrast interferometer as a function of the applied phase . The
recoil phase is constant and demonstrates our insensitivity tophase noise from the gratings. (b) The fractional population of
the j0 hki
 state from the phase-sensitive interferometer (inset)
for a similar scan of  at T  50 s. Also shown is the best-fit
sinusoid of the expected period.

j 4hki.
 The deceleration pulse at t  T1  60 s  T 
60 s drove transfers from j 4hki
 back to j 2hki.
During the period T1 , paths 1 and 3 accumulate phase
22  4 times faster in the N  2 scheme than in the N  1
scheme. Additional time T1 is required for the three paths
to overlap in the N  2 scheme. For this geometry, theN  2 recoil phase should therefore evolve 3 times fasteras a function of T1 than the N  1 recoil phase [Fig. 5(b)].
The corresponding linear fits give a slope ratio of 3:06
0:1. At present, we do not have sufficient control over thetiming and phase of the intermediate pulses to improve
our N  1 measurement precision by using N > 1.
In conclusion, we have demonstrated a contrast interferometer which has several desirable features for a highprecision measurement of the photon recoil frequency.
Such a measurement would involve converting to an
atomic fountain setup with vertical Bragg beams. In
this geometry, T can be extended by nearly 2 orders ofmagnitude. Our insensitivity to phase noise from mirror
vibrations should greatly alleviate vibration isolation requirements of the system for long T [12]. The order N of
the interferometer must also be increased, requiring improved timing and phase control of laser pulses. Direct
scaling of our current precision of 0:01 rad=shot results
in an estimated precision of <1 ppb=shot for T  100 ms
and N  20. A rigorous study of systematics will have to
be undertaken to increase the accuracy of our measurement. Estimates show that mean field effects can be suppressed to the ppb-level by reduction of atomic density to
1011 cm3 , together with pulse control for <5% imbalance between populations in the middle and extreme

140401-4

640

630
0 T T
1
60 µs 60 µs

0.0
1.0

N=2

x

0.2

T=502.5µs
0.0

T

0.4

-2
-3

φ(T1) [radians]

N=2

1

N=1

650

2

Zero order fraction

Recoil phase φ [radians]

3

PHYSICA L R EVIEW LET T ERS

FIG. 5. Demonstration of the quadratic scaling of the recoil
phase with additional photon recoils. (a) The N  1 (inner) andN  2 (outer) interferometers used. (b) The recoil phase at therecombination time under variation of T1 .

paths. In addition, our methods may provide a way to
study mean field effects with interferometric precision.
We hope to obtain a <1 ppb value for !rec in a second
generation experiment in which BECs are created elsewhere and transported into the interferometer [23].
We thank W. Ketterle for valuable discussions. This
work was supported by the NSF, ONR, ARO, NASA,and the David and Lucile Packard Foundation.

[1] R. S. Van Dyck, Jr. et al., Phys. Rev. Lett. 59, 26 (1987).
[2] T. Kinoshita, Phys. Rev. Lett. 75, 4728 (1995).
[3] M. E. Cage et al., IEEE Trans. Instrum. Meas. 38, 284(1989).
[4] B. Taylor, Metrologia 31, 181 (1994).
[5] C. Schwob et al., Phys. Rev. Lett. 82, 4960 (1999).
[6] Th. Udem et al., Phys. Rev. Lett. 79, 2646 (1997).
[7] T. Beier et al., Phys. Rev. Lett. 88, 011603 (2002).
[8] E. Krüger et al., Metrologia 35, 203 (1998).
[9] M. P. Bradley et al., Phys. Rev. Lett. 83, 4510 (1999).
[10] Th. Udem et al., Phys. Rev. Lett. 82, 3568 (1999).
[11] D. S. Weiss et al., Phys. Rev. Lett. 70, 2706 (1993).
[12] B. C. Young, Ph.D. thesis, Stanford, 1997.
[13] J. M. Hensley, Ph.D. thesis, Stanford, 2001.
[14] S. B. Cahn et al., Phys. Rev. Lett. 79, 784 (1997).
[15] Atom Interferomtery, edited by P. Berman (Academic,
New York, 1997), see pp. 281–282 and 379–381.
[16] M. Weitz et al., Appl. Phys. B 65, 713 (1997).
[17] S. Gupta et al., C.R. Acad. Sci. IV-Phys. 2, 479 (2001).
[18] S. Inouye et al., Science 285, 571 (1999).
[19] P. Juncar et al., Metrologia 17, 77 (1981).
[20] We scanned  by electronically shifting the phase of therf signal used to drive one of the two Bragg AOMs.
[21] The division of population into three output ports caused
74% ( <100%) contrast. We have seen 100% contrast ina standard Mach-Zehnder interferometer.
[22] Y. Torii et al., Phys. Rev. A 61, 041602 (2000).
[23] T. L. Gustavson et al., Phys. Rev. Lett. 88, 020401 (2002).

140401-4

Appendix B

Fifty-fold improvement in thenumber of quantum degeneratefermionic atoms
This appendix includes the following paper [6]: Z. Hadzibabic, S. Gupta, C.A. Stan, C.H.
Schunck, M.W. Zwierlein, K. Dieckmann and W. Keterle, “Fifty-fold improvement in thenumber of quantum degenerate fermionic atoms”, arXiV:cond-mat/ 0306050(2003), accepted in Phys. Rev. Lett.

112

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 signiﬁcantprogress 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 ongoingeﬀorts 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  = |2, +2, 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  basis. (a) Due to ﬁnite trap depth of ∼ kB × 300 µK in the |1/2, −1/2 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, −1, and in the |F = 2, mF ≥ 0
states. Previously, sodium has been evaporatively cooled to
BEC only in the |1, −1, 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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-

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.
(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, −1 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 foreﬃcient 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 hyperﬁneground states, |2, +2 for 23 Na, and |3/2, +3/2 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 ± 1/2 → |F  = I ± 1/2 transitions. Here, I is the nuclear spin (I = 3/2 for 23 Na,
and I = 1 for 6 Li), and F  is the total spin in the excited electronic state. In this way, almost all the lithium
atoms could be pumped into the |3/2, +3/2 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, +2 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 ≥ 0 states are (atleast weakly) magnetically trappable (Fig. 1(b)). However, only pure |2, +2 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, +1 atoms from thetrap, before they engaged in inelastic collisions with the|2, +2 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, +1
atoms could be selectively transferred to the untrapped|F = 1, mF = 0, +1 lower hyperﬁne states. This transfer
was done with a microwave sweep near the 23 Na hyperﬁnesplitting of 1.77 GHz. In this way, all the |2, +2 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, +2
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, +2 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, +2 → |1, +1 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, +1 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, +2 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, +2
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 insure
suﬃcient 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 C

Decay of an Ultracold Fermionic
Lithium Gas near a Feshbach
Resonance
This appendix includes the following paper [4]: 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).

117

VOLUME 89, N UMBER 20

PHYSICA L R EVIEW LET T ERS

11 NOVEMBER 2002

Decay of an Ultracold Fermionic Lithium Gas near a Feshbach Resonance
K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic, C. H. Schunck, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms,
and Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139(Received 1 July 2002; revised manuscript received 29 August 2002; published 24 October 2002)
We studied the magnetic field dependence of the inelastic decay of an ultracold, optically trapped
fermionic 6 Li gas of different spin compositions. The spin mixture of the two lowest hyperfine states
showed two decay resonances at 550 and 680 G, consistent with the predicted Feshbach resonances forelastic s-wave collisions. The observed lifetimes of several hundred ms are much longer than the
expected time for Cooper pair formation and the phase transition to superfluidity in the vicinity of the
Feshbach resonance.
DOI: 10.1103/PhysRevLett.89.203201

PACS numbers: 05.30.Fk, 32.80.Pj, 34.50.-s, 39.25.+kInteractions between atoms can be strongly modified
by tuning magnetic fields to Feshbach resonances where amolecular state has the same energy as the colliding
atoms. This mechanism has been used to dramaticallyalter the properties of ultracold bosonic gases [1– 4].
For degenerate Fermi gases, such control over the interaction strength is crucial in the search for a superfluid
phase transition. For dilute Fermi gases, the predicted
phase transition occurs at temperatures that are experimentally not accessible [5], unless the scattering length
is resonantly enhanced. In this case, as was pointed out
by [6 –10], the transition temperature can be comparable to the temperatures achieved in current experiments [11–16].
Promising candidates for an experimental observationof fermionic superfluidity are 6 Li and 40 K. For an optically trapped mixture of two spin states of 40 K, a
Feshbach resonance has been observed by measuring
the thermalization time of the gas [17]. For an opticallytrapped spin mixture of the two lowest Zeeman states of
6 Li, a wide s-wave Feshbach resonance has been predicted
first by [18]. Experiments with 6 Li have so far observed
only a magnetic field dependence of the elastic crosssection far away from the predicted resonance [14].
Near Feshbach resonances, the enhancement of the scattering length is usually accompanied by enhanced inelastic collisions which lead to rapid trap loss. This signature
was used to identify Feshbach resonances in bosonicgases [20 –22]. However, inelastic losses have also poseda severe limitation for experiments near Feshbach resonances, in particular, at high atomic densities. The superfluid phase transition for fermions will be observable only
if the time for the formation of Cooper pairs is shorter
than the decay time of the gas. For fermions, inelastic
decay in the s-wave channel can be suppressed due to thePauli exclusion principle. However, even in the zerotemperature limit the kinetic energy of the cloud is of
the order of the Fermi energy. Therefore, inelastic collisions for higher partial waves are expected to limit the
lifetime of the gas.

This Letter is the first report on the study of inelastic
collisions in a fermionic system near Feshbach resonances. We have observed resonant magnetic field dependent inelastic decay of an ultracold, optically trapped spin
mixture of 6 Li.
The ultracold lithium samples were prepared by sympathetic cooling of 6 Li by 23 Na as described previously[15]. The remaining 23 Na atoms were removed from themagnetic trap by rf induced spin flips to untrapped states.
This typically produced 3  105 lithium atoms in the
j1=2; 1=2i state at a temperature of 400 nK, equal tohalf the Fermi temperature. The atoms were transferred
into an optical trap formed by a single far detuned beamwith up to 1 W of power at 1064 nm. The beam had a
14 m waist and was aligned horizontally along thesymmetry axis of the magnetic trap. This generated a
175 K deep trapping potential, with 12 Hz radial and
200 Hz axial trapping frequencies. Prior to the transfer,the cloud was adiabatically decompressed in the radial
direction during 1 s to improve the spatial overlap withthe optical trap. After this stage, the trap frequencies in
the magnetic trap were 149 Hz radially and 26 Hz axially.
We then adiabatically ramped up the power of the opticaltrap over 500 ms. Subsequently, the magnetic trapping
fields were ramped down in 100 ms, leaving a 1.5 G
guiding field along the trap axis. After the transfer, thecloud contained 3  105 atoms at 3  1013 cm3 peakdensity and 22 K temperature, close to the 21 K Fermi
temperature. We attribute the rise in temperature relative
to the Fermi temperature to residual excitations during
the transfer into the optical trap. (We often observed axial
oscillations of the cloud after the transfer.)
We studied inelastic decay for three different spincompositions of the cloud. The lithium atoms were either
trapped purely in the lowest (j1i), or the second to lowest(j2i) energy state, or in a 50%-50% mixture of thesetwo Zeeman states. At low magnetic fields, the states j1i
and j2i correspond to the jF; mF i  j1=2; 1=2i
and j1=2; 1=2i states, respectively. A full transfer
j1=2; 1=2i ! j1=2; 1=2i was done at low magnetic

203201-1

 2002 The American Physical Society

0031-9007=02=89(20)=203201(4)$20.00

203201-1

PHYSICA L R EVIEW LET T ERS

203201-2

11 NOVEMBER 2002

At even higher magnetic fields, the decay persisted at aweaker but constant level. In a more detailed scan, shownin Fig. 1(d), a second, much weaker and narrower resonance was found at 550 G, with an approximate width of

a)
1.0

0.5
state |2>0.0

Surviving Fraction

field by applying a 1 s rf-driven adiabatic passage between
the two states, which was > 95% complete. The spin
mixture was produced by a faster, nonadiabatic rf sweepof 200 ms duration. The population of the states wasanalyzed by applying a 7 G=cm magnetic field gradientalong the trap axis with a 6.5 G offset field in the center,and reducing the strength of the optical confinement. This
resulted in full spatial separation of the two spin states in
the optical trap. Resonant absorption imaging was used todetermine the atom number in each of the spin states.
Using a full transfer we compared the absorption crosssections for circularly polarized light for the two spinstates and found a ratio of 1:1:2. Taking this into consideration, we were able to control the relative population of
the spin states by rf sweeps with an accuracy of 4 %.
In order to study the decay of the cloud near the
Feshbach resonance, predicted to occur at about 800 G[18], we applied magnetic fields up to 900 G using theantibias coils of the cloverleaf magnetic trap [23]. The
magnetic field strength was calibrated in two independent
ways to 2% accuracy. For calibration of magnetic fields up
to 100 G, we loaded 23 Na into the optical trap and drove rf
transitions to magnetically untrapped states. Resonanceswere observed by measuring the remaining atom number
after recapture into the magnetic trap. As a second
method at about 700 G, we used direct absorption imaging of 6 Li in the optical trap in the presence of higher
magnetic fields. The magnetic field values were thenderived from the frequency shifts of the observed resonances from the lithium D2 line. We also verified thatdrifts of the magnetic field during the pulses, occurringfrom thermal expansion of the coils due to the highcurrent load, were negligible.
We measured the magnetic field dependence of the
decay by measuring the atom number at two different
times, 50 and 500 ms, after switching on the magnetic
field within about 4 ms. For measuring the remaining
atom number, the magnetic field was rapidly switched off
within 100 s, and the cloud was probed by absorption
imaging at low magnetic field. Normalizing the number atlong time to the number at short time made the measurement less sensitive to atom number drifts and initiallosses from the optical trap. These losses can occur due
to the sloshing motion of the cloud and due to initial
evaporation.
For the cloud purely in state j2i, we observed nosignificant decay over the entire range of magnetic fields,
as can be seen in Fig. 1(a). This also confirmed that during
the measurement interval, one-body decay (e.g., due tocollisions with particles from the background gas) was
negligible.
The surviving fraction of the mixture is shown in
Fig. 1(c). No significant decay was observed at low magnetic fields. At higher magnetic field, we found two decay
resonances. A strong resonance occurred at 680 G withconsiderable losses over a range of approximately 100 G.

b)
1.0

0.5
state |1>

0.0

c)
1.0

0.5

mixture |1> & |2>

0.0
200

d)
Surviving Fraction

VOLUME 89, N UMBER 20

400

600

800

mixture |1> & |2>

1.0

0.5

0.0
500

600
700
Magnetic Field [G]800

FIG. 1. Magnetic field dependence of inelastic decay inclouds of fermionic 6 Li. The fraction of atoms remaining after
the 500 ms magnetic field pulse is shown for different spincompositions of the cloud. (a) For the state j2i, no significant
loss was observed. (b) The energetically lowest state j1i exhibits a weak decay resonance at 680 G. (c) The 50%-50%
mixture of two spin states shows two decay resonances, at 550
and 680 G. (d) The two resonances are shown with higher
density of data points and for 2 s magnetic field pulses. Eachdata point represents an average of three measurements.

203201-2

VOLUME 89, N UMBER 20

PHYSICA L R EVIEW LET T ERS

20 G. The weaker resonance became more pronounced
after 2 s of dwell time in the magnetic field, whereas thestronger resonance showed ‘‘saturation’’ broadening.
We also measured the time evolution of the atom number at the two resonances. For a two-body (three-body)process the loss rate of atoms N_ is proportional to N 2(N 3 ), where N is the number of trapped atoms. The decay
curves at 680 G are shown in Fig. 2. At both resonances
we found that the values for 1=N showed a linear dependence on time, characteristic for a two-body process. In
order to test for three-body decay we plotted the samedata as 1=N 2 . The nonlinear behavior is not compatible
with a simple three-body decay process.
Another experimental observation is the almost complete disappearance of the mixed cloud in Fig. 1(d). A
resonantly enhanced three-body process would involve
two atoms of opposite spin colliding, and a third in eitherof the spin states. Starting with a 50%-50% mixture, thedecay would stop when all atoms in state j1i (or in statej2i) are used up. Therefore, three-body decay can beconsistent with the observation of complete disappearance only if the decay rate does not depend on the spinstate of the third particle. In case of strongly different
rates for the two spin orientations, the surviving fraction
could not drop below 25%.
With the observation of two resonances and the position of the strongest decay of the main resonance deviat-

a)

40

N

-1

-6

[10 ]

60

20

0

b)

2

N

-2

-9

[10 ]

3

1

0
0

200

400

600

Time [ms]FIG. 2. Decay of the atom number at 680 G. (a) The dataplotted as 1=N show a linear time dependency, consistent with
two-body decay. (b) The same data plotted as 1=N 2 clearlyshow nonlinear dependency. For the resonance at 550 G, thecomparison of least square fits also revealed consistency with a
two-body decay.

203201-3

11 NOVEMBER 2002

ing from the theoretical prediction [18], the question
arises whether the observed decay of the spin mixture
can be interpreted as a signature of the Feshbach resonance for elastic s-wave collisions. After the submissionof this paper new improved theoretical calculations exhibited a second narrow Feshbach resonance for elastic
collisions in the s-wave channel at 550 G [19], in good
agreement with the position of the narrow decay resonance. The predicted magnetic field for the main resonance is 860 G. However, due to the huge width of theresonance it seems possible that the decay observed at
680 G is related to this s-wave resonance.
The measured decay curves suggest a two-body type of
decay. Because of the Pauli exclusion principle dipolar
relaxation is not possible in the s-wave channel [24].
Dipolar relaxation in the p-wave channel is possible, aseven in the zero-temperature limit the kinetic energy of
the cloud is of the order of the Fermi energy, and collisions in the p-wave channel do not completely freeze out.
However, no occurrences of resonances in the dipolar
decay are theoretically predicted [25].
Therefore, it is most likely that the observed decay is asignature of the Feshbach resonances for the elastic collisions, resulting in enhanced three-body decay. At
present no exact theoretical description for the threebody decay mechanism of fermions near a Feshbachresonance is available. Three-body decay is not supportedby the measured decay curves. However, one possibility isthat the decay curve is affected by a change in temperature. An accurate measurement of the temperature wasdifficult due to technical reasons and a low signal-tonoise ratio, as the absorption signal drops significantly
during the decay. If the sample had cooled down during
the decay (e.g., due to an energy dependence of the lossrate) it could speed up the decay in a way that three-bodyloss results in a decay curve similar to a curve for twobody losses at constant temperature. Another possibility
for the deviation from a three-body decay curve would beheating due to three-body recombination followed by trap
loss due to evaporation, or other processes involving
secondary collisions [26]. It should be noted that theobserved resonances do not resemble the predicted magnetic field dependence for elastic collisions [18]. Therefore, our decay data cannot be explained by elastic
collisions leading to evaporation.
We also observed resonant decay at 680 G of a cloud
purely in state j1i, as shown in Fig. 1(b). The fact that thisresonance is at the same magnetic field as for the mixture
suggests that the observed loss is due to a contaminationof the cloud with atoms in state j2i. For three-body of
decay, our measured > 95% purity of the preparation ofstate j1i allows for a maximum of 15% decay of thecloud, compared to the measured 21 %. Further measurements are needed to investigate whether there is an enhancement of losses by secondary collisions, or whether
there is a decay mechanism for atoms purely in state j1i.
203201-3

VOLUME 89, N UMBER 20

PHYSICA L R EVIEW LET T ERS

In conclusion, we observed two decay resonances forthe 6 Li spin mixture and one resonance in the lowest spinstate. Comparing our observations with recent theoretical
calculations which exhibit two s-wave Feshbach resonances suggests that the observed decay is a signatureof those resonances. Even on resonance, the observed
decay happened on a time scale longer than the trap
oscillation time, the time for elastic collisions, and the
expected submillisecond time needed for the formationof Cooper pairs [27,28]. Therefore, the 6 Li system is well
suited for the study of an interacting Fermi gas in thevicinity of an elastic Feshbach resonance, in particular,
for the search for the phase transition to a superfluid state.
This research was supported by NSF, ONR, ARO,
NASA, and the David and Lucile Packard Foundation.
C. H. S. acknowledges the support of the Studienstiftungdes deutschen Volkes.
Note added.—After the submission of this paper several groups reported related results. Measurements of theelastic cross section near the zero crossing associatedwith the Feshbach resonance have recently been performed by [19,29]. Inelastic decay of 6 Li fermionic clouds
near the Feshbach resonance was recently also observedin the groups of Thomas [19], and C. Salomon, and for40
K in the group of D. S. Jin.

[1] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner,
D. M. Stamper-Kurn, and W. Ketterle, Nature (London)
392, 151 (1998).
[2] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van
Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998).
[3] A. J. Kerman, V. Vuletić, C. Chin, and S. Chu, Phys. Rev.
Lett. 82, 1406 (1999).
[4] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell,and C. E. Wieman, Phys. Rev. Lett. 85, 1795 (2000).
[5] A. J. Leggett, J. Phys. (Paris) 41, C7 (1980).
[6] M. Houbiers, R. Ferwerda, H. T. C. Stoof, W. I.
McAlexander, C. A. Sackett, and R. G. Hulet, Phys. Rev.
A 56, 4864 (1997).
[7] E. Timmermans, K. Furuya, P.W. Milonni, and A. K.
Kerman, Phys. Lett. A 285, 228 (2001).
[8] M. J. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo,and R. Walser, Phys. Rev. Lett. 87, 120406 (2001).

203201-4

11 NOVEMBER 2002

[9] Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402(2002).
[10] J. N. Milstein, S. J. J. M. F. Kokkelmans, and M. J.
Holland, Phys. Rev. A 66, 043604 (2002).
[11] B. DeMarco, S. B. Papp, and D. S. Jin, Phys. Rev. Lett. 86,
5409 (2001).
[12] A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B.
Partridge, and R. G. Hulet, Science 291, 2570 (2001).
[13] F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari,
T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev.
Lett. 87, 080403 (2001).
[14] S. R. Granade, M. E. Gehm, K. M. O’Hara, and J. E.
Thomas, Phys. Rev. Lett. 88, 120405 (2002).
[15] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta,
M.W. Zwierlein, A. Görlitz, and W. Ketterle, Phys. Rev.
Lett. 88, 160401 (2002).
[16] G. Roati, F. Riboli, G. Modungo, and M. Inguscio, Phys.
Rev. Lett. 89, 150403 (2002).
[17] T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S.
Jin, Phys. Rev. Lett. 88, 173201 (2002).
[18] M. Houbiers, H. T. C. Stoof, W. I. McAlexander, andR. G. Hulet, Phys. Rev. A 57, R1497 (1998); see also
Ref. [19].
[19] K. M. O’Hara, S. L. Hemmer, S. R. Granade, M. E. Gehm,
J. E. Thomas, V. Venturi, E. Tiesinga, and C. J. Williams,
cond-mat/0207717.
[20] J. Stenger, S. Inouye, M. R. Andrews, H.-J. Miesner, D. M.
Stamper-Kurn, and W. Ketterle, Phys. Rev. Lett. 82, 2422
(1999).
[21] J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E.
Wieman, Phys. Rev. Lett. 85, 728 (2000).
[22] C. Chin, V. Vuletić, A. J. Kerman, and S. Chu, Phys. Rev.
Lett. 85, 2717 (2000).
[23] 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).
[24] H. T. C. Stoof, J. M.V. A. Koelman, and B. J. Verhaar,Phys. Rev. B 38, 4688 (1988).
[25] S. J. J. M. F. Kokkelmans and E. Tiesinga (private communication).
[26] D. Petrov (private communication).
[27] M. Houbiers and H. T. C. Stoof, Phys. Rev. A 59, 1556
(1999).
[28] E. Timmermans, Phys. Rev. Lett. 87, 240403 (2001).
[29] M. B. S. Jochim, G. Hendl, J. H. Denschlag, R. Grimm,
A. Mosk, and M. Weidemüller, cond-mat/0207098.

203201-4

Appendix D

Collisions in Zero Temperature
Fermi Gases
This appendix includes the following paper [8]: S. Gupta, Z. Hadzibabic, J.R. Anglin and W.
Keterle, “Collisions in Zero Temperature Fermi Gases”, arXiV:cond-mat/ 0307088(2003).

122Collisions in zero temperature Fermi gases
Subhadeep Gupta, Zoran Hadzibabic, James R. Anglin, and Wolfgang Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
MIT, Cambridge, MA 02139(July 3, 2003)
We examine the collisional behavior of two-component Fermi gases released at zero temperaturefrom a harmonic trap. Using a phase-space formalism to calculate the collision rate during expansion, we ﬁnd that Pauli blocking plays only a minor role for momentum changing collisions. As aresult, for a large scattering cross-section, Pauli blocking will not prevent the gas from entering thecollisionally hydrodynamic regime. In contrast to the bosonic case, hydrodynamic expansion at very
low temperatures is therefore not evidence for fermionic superﬂuidity.

arXiv:cond-mat/0307088 v1 3 Jul 2003

PACS numbers: 03.75.Ss, 03.75.Kk, 34.50.-s

The last few years have seen rapid progress in the ﬁeldof ultracold atomic Fermi gases [1–6]. Most recently,
regimes of strong interactions have been observed in thesegases near Feshbach resonances [7–10]. Studies of thesesystems are of particular importance because of the possibility of creating BCS-like superﬂuids [11,12]. Such a
realization would establish highly controllable model systems for studying novel regimes of fermionic superﬂuidity.
A unique feature of atomic systems is the ability toanalyze the gas by turning oﬀ the trapping potential
and observing the expansion. The expansion behavior
can reveal the momentum distribution and the eﬀects
of mean-ﬁeld interactions and collisions. Hydrodynamic
behavior can be easily detected when the gas is released
from an anisotropic atom trap. In that case, the spatial
anisotropy of the cloud reverses during free expansion.
This is caused by the larger pressure gradient along thetightly conﬁning direction, which leads to a faster expansion, and subsequent reversal of the spatial anisotropy.
This anisotropic expansion was used to identify the formation of the Bose-Einstein condensate [13,14].
A BEC obeys the hydrodynamic equations of a superﬂuid [15]. However collisional hydrodynamics arisingfrom a high elastic collision rate also results in anisotropic
expansion [16,17]. Thus, the normal component can also
expand anisotropically [18]. For the bosonic case, two key
points make the distinction between the two fractions obvious: (i) At the typical transition temperature, the BEC
has much less energy than the normal cloud, so the twocomponents are clearly separated in size. (ii) The scattering rate needed to achieve condensation is usually notlarge enough that the normal gas is in the hydrodynamic
regime. For these two reasons, the appearance of a dense
anisotropic cloud during expansion is considered to be
the “smoking-gun” for the formation of a Bose-Einsteincondensate.
A superﬂuid Fermi gas is predicted to obey the superﬂuid hydrodynamic equations of motion [19–22] andtherefore should show strong anisotropic expansion whenreleased from an anisotropic harmonic trap [22]. The recent observation of anisotropic expansion of an ultracold,

interacting, two-spin fermionic mixture [7–9] has createdconsiderable excitement and raised the question underwhat conditions is this expansion a signature of fermionicsuperﬂuidity and not of collisional hydrodynamics. There
are two major diﬀerences from the bosonic case: (i) Since
the energy of ultracold fermions always remains on theorder of the Fermi energy, the size in expansion for bothnormal and superﬂuid components will be similar. (ii)Current eﬀorts towards inducing BCS pairing all takeplace in strongly interacting systems. This results in alarge scattering rate modiﬁed only by the eﬀects of Pauliblocking at low temperatures.
The interpretation of the observed anisotropic expansion in strongly interacting Fermi gases is therefore critically dependent on the role of Pauli blocking of collisions during the expansion. The tentative interpretationof anisotropic expansion as superﬂuid hydrodynamics [7]was based on the assessment that collisions are strongly
suppressed at suﬃciently low temperatures [7,23–28].
Here we show generally that the collision rate becomes
independent of temperature and prevails even at zerotemperature, if the Fermi surface is strongly deformed.
This happens in an extreme way during ballistic expansion. In the small cross-section limit, we ﬁnd that lessthan half of the total number of momentum changing collisions is suppressed. For a large scattering cross-section,
the absence of suppression results in strong collisional
behavior of normal Fermi gases during expansion for all
initial temperatures. This result has the important consequence of rendering expansion measurements of Fermi
gases near Feshbach resonances ambiguous for diﬀerentiating between superﬂuid and normal components.
We ﬁrst consider the expansion of a single component
Fermi gas. At ultralow temperatures, fermionic antisymmetry prevents s-wave scattering in a single component and renders the gas completely collisionless. The
phase space occupation f (x1 , x2 , x3 , p1 , p2 , p3 ) = f (x, p)
at zero temperature in a harmonic trap with frequencies(ω1 , ω2 , ω3 ) can be written asf (x, p) = Θ(EF − Σi mωi2 x2i /2 − Σi p2i /2m)
where m is the particle mass, Θ is the Heaviside step1

function deﬁned as Θ(x) = 0(1) for x ≤ 0(x > 0) andEF = h̄(6N Πi ωi )1/3 is the Fermi energy for N particles.
At time t = 0, the trapping potential is turned oﬀ suddenly, allowing the gas to expand freely. At t = 0, themomentum space Fermi surface at x = (x1 , x2 , x3 ) is(1)
Σi p2i /2m = EF − Σi mωi2 x2i /2,a sphere of radius 2m(EF − Σi mωi2 x2i /2). In this noninteracting system, the evolution of the Fermi surface
can be derived from the simple evolution law for ballistic
expansion x(0) = x(t)−pt/m. Substituting this in Eq. 1,
we obtain:2mxi ωi2 t2mωi2 x2i(1 + ωi2 t2 )= EF − Σi
pi −
,
Σi
2
2
2m
t 1 + ωi t
2(1 + ωi2 t2 )(2)local-density approximation. These collisions have an appealing geometrical picture in the local phase-space description (Fig. 1(c)). Each elastic collision involves one
particle from each spin state. We label with p’s and q’s
the momenta of the two diﬀerent spin states. Consider
the collision p1 +q1 → p2 +q2 . Conservation of momentum and kinetic energy mandates p2 + q2 = p1 + q1 and|p2 − q2 | = |p1 − q1 |. These relations restrict p2 and q2
to lie on diametrically opposite ends of the sphere with
p1 −q1 as a diameter. The deformation of the Fermi surface during expansion opens up unoccupied ﬁnal states
p2 , q2 and therefore allows collisions to take place evenin a zero temperature Fermi gas (Fig. 1(c)).

which is an ellipsoid with generally unequal axesmω 2 x2
2m(EF − Σi 2(1+ωi 2 ti2 ) )( √ 1 2 2 , √ 1 2 2 , √ 1 2 2 ).
i

1+ω1 t1+ω2 t

2mEF
h2

3/2 (1 −10 3

1+ω3 t

ωi2 x2im
3/2
2EF Σi 1+ωi2 t2 ),Πi (1 + ωi2 t2 )1/2

p1 p4

p2

The anisotropy of the Fermi surface during expansion canbe understood generally by noting that for long times t,
at any position x, the spread in momentum ∆pi (t) canonly arise from the initial spread in position ∆xi (0). For
anisotropic traps this gives rise to an anisotropic momentum distribution during ballistic expansion. For a
mixture of two spin states, this deformation of the Fermi
surface from a sphere into an anisotropic ellipsoid removes Pauli blocking of ﬁnal states and allows collisions,as will be shown.
The momentum distribution at position x given by
Eq. 2, also allows us to simply calculate the spatial density distribution as the volume of the momentum-space
ellipsoid,4n(x, t) = π
3

(c)

(b)

(a)

1

10 3

0

1

q3

2

p3

0

q2
q4

q1

2mEF

FIG. 1. (a) Deformation of the momentum space Fermi surface at x = 0, from a sphere to an ellipsoid during expansion
from an anisotropic harmonic trap. The case of cylindricalsymmetry is shown, where the three-dimensional distribution
is symmetric about the vertical axis. The parameters chosen
are an aspect ratio λ = 0.2 and expansion times ω⊥ t = 0,1,3and
10. (b) The deformation at a position radially displaced2
EF /mω⊥. (c) Geometrical representation of collisions
byin momentum space. The two spin states have identical distributions. Three diﬀerent types of collisions are shown forparticles with initial momenta p1 and q1 - none, one or bothof the ﬁnal states are occupied.

(3)

The eﬀect of collisions can be formally calculated fromthe Boltzmann transport equation for the evolution of
the phase space distribution f (x, p, t). In the absenceof external potentials and neglecting mean ﬁeld we have[30]:

in agreement with other derivations [29]. For long expansion times t, the spatial distribution becomes isotropic,
mirroring the isotropic momentum distribution in thetrap.
Specializing to the experimentally relevant case ofa cylindrically symmetric trap, ballistic expansion deforms the local Fermi surface into a momentum
 ellip1+ω 2 t2
soid of cylindrical symmetry with aspect ratio 1+ω2z t2

∂f
∂f
+ v.
= Γcoll [f ]
∂t
∂x

(4)

where v = p/m and Γcoll [f ] describes the eﬀect of collisions. Collisions attempt to restore local equilibriumby countering the deformation of the momentum space
Fermi surface during free expansion (Eq. 2).
Γcoll [f ] can be written as the collision integral:⊥(Fig. 1(a,b)). Here ω⊥ (ωz ) is the radial (axial) trapping
frequency. For long times t, this deformation reaches theasymptotic aspect ratio ωz /ω⊥ = λ, the initial spatialaspect ratio in the trap.
Now consider an equal mixture of two spin states whichinteract via a ﬁnite s-wave scattering length. We assume
that the trapping frequencies are identical for the twostates (standard experimental conditions) and specialize to the usual case of two-body elastic collisions in theσΓ(x, p1 , t) = − 4πh3

3
2 |p1 −q1 |×m(x,t) d q1 d Ω[f (p1 )f (q1 )(1 − f (p2 ))(1 − f (q2 ))− f (p2 )f (q2 )(1 − f (p1 ))(1 − f (q1 ))]2(5)

where σ is the momentum-independent scattering crosssection, f (pi ) = f (x, pi , t), f (qi ) = f (x, qi , t) and Ω
points along p2 −q2 . The integral over q1 is over the momentum ellipsoid at position x and time t for one of thespin states. The ﬁrst term in the integrand is the collision
rate for the process p1 + q1 → p2 + q2 . The second term
corresponds to the reverse process p2 +q2 → p1 +q1 , andensures that only distribution changing collisions contribute.
Pauli blocking is expressed in the suppression factors
for the ﬁnal states (1 − f ) in Γ. The collision integral
neglecting Pauli blocking, ΓCl,p , is furnished by setting
the suppression factors all equal to 1 in Eq. 5. This is
the rate for classical collisions which change the momentum distribution. The total classical collision rate ΓCl
is the ﬁrst term on the right hand side of Eq. 5 without
any suppression factors. In addition to ΓCl,p , this also
contains the rate for collisions which do not change themomentum distribution: if both ﬁnal states are occupied, then the reverse process has the same rate. These
additional collisions do not aﬀect observables of the system. Fig. 1(c) shows examples of these diﬀerent types ofcollisions. p1 + q1 → p2 + q2 contributes to ΓCl , ΓCl,pand Γ. p1 + q1 → p3 + q3 contributes to ΓCl and ΓCl,p .
p1 + q1 → p4 + q4 contributes only to ΓCl . To determine the eﬀect of Pauli blocking, we compare Γ and ΓCl,pfor a small cross-section σ. The collision rate at a particular time t can then be calculated perturbatively, by
propagating the system ballistically for the time t andthen evaluating Eq. 5 with and without the suppression
factors.
Fig. 2(a) displays the numerically calculated collision
rates Γ, ΓCl,p and ΓCl , evaluated at x = p = 0, as arepresentative case, for an initial aspect ratio λ = 0.03.
Both Γ and ΓCl,p increase initially as the deformationof the Fermi surface becomes more pronounced. For
long times (ω⊥ t  1), they
are both suppressed be
cause both the density ( d3 q1 ) and the relative ve−q1 |locity ( |p1m
) drop. The two curves approach eachother with time since Pauli blocking becomes less effective with stronger deformation. The fraction of momentum changing collisions
 which are not aﬀected by
Pauli blocking, F (λ) = dtΓ(0, 0, t)/ dtΓCl,p (0, 0, t),
is shown in Fig. 2(b). The main result of our paper isthe ineﬃciency of Pauli blocking during expansion fromanisotropic traps. For λ < 0.05, F > 0.5, and approaches∼ 0.55 as λ approaches 0. Most experiments work in thisregime of trap aspect ratio.
The above results form an upper bound on the Fermi
suppression even if we consider all the possible collisions
occurring in the system, for arbitrary x and p. First, weobserve that for all x, at any time t, the Fermi surface
is identically deformed and diﬀerent only in size according to the local density (Eqs. 2,3, Fig. 1(a,b)). We havechecked numerically that to within 5%, the central mo-mentum provides a lower bound on Γ within a momentum ellipsoid, at all x and for all times t. Next, we notethat for x = 0, the density n(x, t) puts more weight atlonger times than n(0, t) (Eq. 3). Since Fermi suppression becomes less eﬀective with time, Pauli blocking ismost eﬀective at x = 0. The calculation for x = p = 0
thus provides an eﬀective upper bound for the overallcollisional suppression in the system. We conclude that
more than half of all the possible collisions are not Pauliblocked for typical experimental values of λ.
(a)Allowed Fraction ofCollisions (F)Collision Rate (Arb.)

(b)

1.0
0.8
0.6
0.4
0.2
0.0

0

5

10

15

Expansion Time (1/ ω )

20

0.0

0.2

0.4

0.6

0.8

1.0

Initial Real Space
Aspect Ratio

FIG. 2. (a) Collision rate as a function of expansion timein the perturbative approximation for the initial aspect ratio
λ = 0.03. Dashed line - total classical collision rate Γcl , thinline - classical rate for momentum changing collisions Γcl,p ,thick line - collision rate for fermions Γ. The displayed rates
were evaluated at x = 0 and p = 0 and give an eﬀective upperbound on the Fermi suppression. (b) Allowed fraction of collisions F (λ) for a zero-temperature two-spin Fermi gas. For
an initial aspect ratio λ = 0.05, F is 0.5. For large anisotropy(λ → 0), F approaches ∼ 0.55.

So far, we have not considered the eﬀect of the collisions themselves on the momentum distribution. Collisions drive the system towards equilibrium, which corresponds to an isotropic Fermi-Dirac distribution. If
this collisional relaxation (Eq. 5) is much faster than the
non-equilibrium perturbation due to ballistic expansion(Eq. 2), the momentum distribution maintains local equilibrium at all times. If local equilibrium is maintained,
the Boltzmann equation leads to the hydrodynamic equations [30]. For free expansion from anisotropic atom
traps, these equations lead to the reversal of anisotropy[16,17]. Even if equilibrium is not fully maintained, collisions always have the eﬀect of transferring momentum
from the weakly conﬁning axis to the strongly conﬁning
axis resulting in an eventual spatial aspect ratio > 1 [17].
We now want to reconcile our new result that Pauliblocking is ineﬃcient during free expansion, with previous results [26] which show that at low temperatures,
collisional damping of collective excitations is suppressed.
For this, we derive an equation of motion for the momentum space anisotropy α to leading order in α and T /TF
[31]:


nσpF
1
T
α̇ = (∂x vx + ∂y vy − 2∂z vz ) −
(6)
C α,
3
m
TF
3

where C describes the collisional relaxation and has theasymptotic forms:


 
 2
T
3π 2 T
T
=
C α,
α, α
TF
5
TF
TF
 
96 3
T
.
(7)
α ,
α
49
TF

and Claudiu Stan and Christian Schunck for critical reading of the manuscript. This work was supported by the
NSF, ONR, ARO, and NASA.

[1] B. DeMarco and D.S. Jin, Science 285, 1703 (1999).
[2] A.G. Truscott et. al., Science 291, 2570 (2001).
[3] F. Schreck et. al., Phys. Rev. Lett. 87, 080403 (2001).
[4] S.R. Granade, M.E. Gehm, K.M. O’Hara, and J.E.
Thomas, Phys. Rev. Lett. 88, 120405 (2002).
[5] Z. Hadzibabic et. al., Phys. Rev. Lett. 88, 160401 (2002).
[6] G. Roati, F. Riboli, G. Modungo, and M. Inguscio, Phys.
Rev. Lett. 89, 150403 (2002).
[7] K.M. O’Hara et. al., Science 298, 2179 (2002).
[8] C.A. Regal and D.S. Jin, Phys. Rev. Lett. 90, 230404(2003).
[9] T. Bourdel et. al., arXiv:cond-mat/0303079 (2003).
[10] S. Gupta et. al., Science 300, 1723 (2003).
[11] M. Houbiers and H.T.C. Stoof, Phys. Rev. A. 59, 1556
(1999).
[12] M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, andR. Walser, Phys. Rev. Lett. 87, 120406 (2001).
[13] M.H. Anderson et. al., Science 269, 198 (1995).
[14] K.B. Davis et. al., Phys. Rev. Lett. 75, 3969 (1995).
[15] S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).
[16] Y. Kagan, E.L. Surkov, and G.V. Shlyapnikov, Phys.
Rev. A 55, R18 (1997).
[17] P. Pedri, D. Guéry-Odelin, and S. Stringari, arXiv:condmat/0305624 (2003).
[18] I. Shvarchuck et. al., Phys. Rev. Lett. 89, 270404 (2002).
[19] M.A. Baranov and D.S. Petrov, Phys. Rev. A 58, R801
(1998).
[20] A. Minguzzi and M.P. Tosi, Phys. Rev. A 63, 023609(2001).
[21] F. Zambelli and S. Stringari, Phys. Rev. A 63, 033602(2001).
[22] C. Menotti, P. Pedri, and S. Stringari, Phys. Rev. Lett.
89, 250402 (2002).
[23] G. Ferrari, Phys. Rev. A. 59, R4125 (1999).
[24] W. Geist et. al., Phys. Rev. A. 61, 013406 (1999).
[25] M. Holland, B. DeMarco, and D.S. Jin, Phys. Rev. A.
61, 053610 (2000).
[26] L. Vichi, J. Low Temp. Phys. 121, 177 (2000).
[27] W. Geist and T.A.B. Kennedy, Phys. Rev. A 65, 063617(2002).
[28] M.E. Gehm, S.L. Hemmer, K.M. O’Hara, and J.E.
Thomas, arXiv:cond-mat/0304633(2003). The authors
point out that the suppression of in-trap collisions is not
suﬃcient to guarantee collisionless expansion.
[29] G.M. Bruun and C.W. Clark, Phys. Rev. A 61,061601(2000).
[30] K. Huang, Statistical Mechanics, (J. Wiley, New York
1987), 2nd. ed.
[31] J.R. Anglin et. al., in preparation (2003).
[32] The use of the Fermi-Dirac form neglects possible eﬀects
of strong interatomic correlations. Although such eﬀects
are not well understood, they are likely to shift a normal
degenerate Fermi gas even further away from the colli-

In terms ofα, the aspect ratio of the momentum space
1−αellipsoid is 1+2α
. pF , T, TF are the local Fermi momentum, temperature and Fermi temperature respectively.
This equation was derived from the second momentum
moment of the Boltzmann equation (Eqs. 4,5), using a
Fermi-Dirac distribution with an anisotropic Fermi surface as ansatz [32]. The numerical coeﬃcients in Eq. 7
were obtained by analytic integrations over momentum
space.
At zero temperature, there is no linear term in α inEq. 7. This shows that Pauli blocking is eﬃcient as longas the anisotropy is small. This is the case for small amplitude excitations in a trapped degenerate gas [26]. However, for the large anisotropies of ballistic expansion, theα3 term, which is independent of temperature and notaﬀected by Pauli blocking, is responsible for collisional
relaxation.
Eqs. 6,7 allow us to distinguish collisionless from hydrodynamic behavior in diﬀerent regimes. The driving
term involving v is on the order of the trap frequencyω⊥ and the damping term has a prefactor nσvF . Therefore, the dimensionless parameter characterizing the attainment of the hydrodynamic limit is Φ0 = nσvF /ω⊥ . If
1, then one can neglect collisions entirely, and theΦ0gas will expand ballistically. For small anisotropies, hydrodynamic behavior requires Φ0 (T /TF )2  1. For large
1/3
anisotropies, hydrodynamic behavior requires Φ0  1.
At ultralow temperatures, the expansion after releasefrom a highly anisotropic trap may be collisionless initially, but as α grows, the α3 term in Eq. 7 will become
important, and induce hydrodynamic behavior.
Our calculations clearly predict that for parametersof current experiments, Φ0 > 1, free expansion will not
be collisionless, but show behavior which is at least intermediate between collisionless and hydrodynamic [17].
Full hydrodynamic behavior may not be achieved, sincefor small values of α, Pauli suppression becomes eﬀective
again. More quantitative studies are necessary in order
to assess how much this behavior would diﬀer from superﬂuid expansion. This could be realized by extending
analytical studies [17] to high degeneracies or by MonteCarlo techniques [33]. Our main conclusion is clear, however, that the breakdown of Pauli blocking under free
expansion means that hydrodynamic expansion will not
be the dramatic, qualitative signal for superﬂuidity instrongly interacting fermions, the way it was for BEC.
We thank John Thomas, Sandro Stringari, Martin
Zwierlein and Aaron Leanhardt for valuable discussions,4sionless limit.
[33] H. Wu and E. Arimondo, Europhys. Lett. 43, 141 (1998).

5

Appendix E

RF Spectroscopy of Ultracold
Fermions
This 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).

128

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: ⱍF,mF典 ⫽ⱍ2,⫹2典, 6Li: ⱍF,mF典 ⫽ ⱍ3⁄2,⫹3⁄2典, 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 ⱍ2,⫹2典 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
ⱍ1典 and ⱍ2典 (ⱍ1⁄2,⫹1⁄2典 and ⱍ1⁄2,⫺1⁄2典 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 ⱍ3⁄2,⫹3⁄2典 lithium atoms were loaded
into the optical trap at temperature T ⬃ TF ⬃35 ␮K (26). The atoms were then transferredto the lowest energy state ⱍ1典, 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 ⱍ1典 ⫺ ⱍ2典 Feshbachresonance. Using RF-induced transitions near
80 MHz, we could create mixtures of statesⱍ1典, ⱍ2典, and ⱍ3典 (ⱍ3⁄2, ⫺3⁄2典 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 ⱍ2典
experience an energy shift, [(4␲ប2 )/(m)]n1a12,that is due to the presence of atoms in state ⱍ1典.
Here ប is Planck’s constant h divided by 2␲, mis the mass of the atom, n1 is the density of ⱍ1典
atoms, and a12 is the interstate scattering length
between states ⱍ1典 and ⱍ2典. 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 ⱍ1典 to a gas in which one atom istransferred into state ⱍ2典. The energy difference is h␯12 ⫹ [(4␲ប2 )/(m)]na12, where ␯12is the resonance frequency of the noninteracting system. Similarly, the energy difference
between a gas prepared purely in state ⱍ2典 anda gas in which one atom is transferred into
state ⱍ1典 is h␯12 ⫺ [(4␲ប2 )/(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 ⱍ1典 and ⱍ2典, first starting with a purestate ⱍ1典 and then with a pure state ⱍ2典 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 ⱍ1典. 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 ⱍ1典 and ⱍ2典 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 ⱍ1典 andⱍ2典 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 ⱍ2典 3 ⱍ3典 transition in the presence
and in the absence of state ⱍ1典 atoms. The
presence of atoms in state ⱍ1典 is then expected
to shift the resonance by (31).
⌬␯ ⫽2ប
n (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 ⱍ1典. Either partial or complete RFtransfer to state ⱍ2典 was then performed. The
number of atoms in state ⱍ1典 was controlled
by adjusting the speed of a frequency sweeparound the ⱍ1典 3 ⱍ2典 resonance. A fast, nonadiabatic sweep created a superposition of thetwo states, whereas a slow, adiabatic sweep
prepared the sample purely in state ⱍ2典. A
wait time of 200 ms was allowed for thecoherence between states ⱍ1典 and ⱍ2典 to decay
and the system to equilibrate.
Typical parameters for the decohered ⱍ1典– ⱍ2典 mixture were mean-density n1 ⬃2.4 ⫻ 1013 cm⫺3 and T ⬃ 0.7 TF. The
magnetic field was then changed to thedesired value, and the transition from state
ⱍ2典 to state ⱍ3典 was driven with 140-␮s RF
pulses (Fig. 2C). We monitored the appearance of atoms in state ⱍ3典 and the disappearance of atoms from state ⱍ2典, 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 ⱍ1典 and ⱍ2典 on asystem prepared purely in state ⱍ1典 (solid circles) and purely in state ⱍ2典 (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 ⱍ1典
and ⱍ2典, and (C) RF spectroscopyof the ⱍ2典 3 ⱍ3典 transition using
a variable radio frequency (␯RF).
(D) The fraction of atoms transferred from ⱍ2典 to ⱍ3典, with ⱍ1典
atoms absent (solid circles) andpresent (open circles). The
mean-ﬁeld shift is computedfrom Gaussian ﬁts to the data(solid lines). (E) Spatial images ofstate ⱍ3典 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
ⱍ2典 and ⱍ1典 for zero RF detuning.
States ⱍ3典 and ⱍ2典 were imaged
simultaneously to observe theircomplementary spatial structure. State ⱍ1典 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 ⱍ3典, 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 ⱍ1典 ⫺ ⱍ2典
coherence at 500 G was observed by monitoring the ⱍ2典 3 ⱍ3典 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 ⱍ1典 ⫺ ⱍ2典 superposition, the ⱍ2典 3 ⱍ3典 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 ⱍ1典 ⫺ ⱍ2典 superposition
decoheres and mean-field interactions set in.
This shifts the resonance frequency of the ⱍ2典
3 ⱍ3典 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 ⱍ1典 ⫺ ⱍ2典
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 ⱍ1典 and ⱍ2典 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 ⱍ2典 3 ⱍ3典 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 ⫽2.2 ⫻ 1013 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/ⱍaⱍ. For arbitrary values of ka, the s-wave
interaction between two atoms is describedby replacing the scattering length a with thecomplex scattering amplitude f.
f⫽⫺a(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 kⱍaⱍ 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 ⱍaⱍ ⫽ 1/k and then, counterintuitively, decreases as 1/ⱍaⱍ for increasing ⱍaⱍ.
Averaging Re( f ) over a zero-temperature
Fermi distribution with Fermi momentum បkF
limits its absolute value to 1.05/kF and markedly weakens its dependence on the exact value of
a in the kFⱍaⱍ ⬎ 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 kFⱍaⱍ ⬎⬎ 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 ⱍ2典 3 ⱍ3典 resonance due to atoms in state ⱍ1典. The shifts are computed by monitoring thearrival fraction in state ⱍ3典 for 140-␮s RF pulses, except at 750 G. At 750 G, because of strong
inelastic losses between ⱍ3典 and ⱍ1典 atoms, we monitored the loss of atoms in state ⱍ2典 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 ⱍ1典. 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 ⱍ1典 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 nⱍaⱍ3 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 (nⱍaⱍ3 ⬍⬍ 1) and higher order manybody effects can become important. Some recent many-body calculations (35–37) suggestthat in the regime kFⱍaⱍ ⬎⬎ 1 (or nⱍaⱍ3 ⬎⬎ 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 ⱍ1典 and ⱍ2典 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 ofⱍaⱍ, 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 andប, 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公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 ⱍ2典
and ⱍ3典. Mean-ﬁeld shifts arise only during a decoherence time (see later in the text) but are negligible
if the population of atoms in state ⱍ3典 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

Bibliography[1] S. Gupta, K. Dieckmann, Z. Hadzibabic, and D. E. Pritchard, “Contrast Interferometry using Bose-Einstein Condensates to Measure h/m and α,” Phys. Rev. Lett. 89,
140401 (2002).
[2] S. Gupta, A. E. Leanhardt, A. D. Cronin, and D. E. Pritchard, “Coherent Manipulation of Atoms with Standing Light Waves,” Cr. Acad. Sci. IV-Phys 2, 479 (2001).
[3] 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).
[4] 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).
[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, “RF Spectroscopy of Ultracold
Fermions,” Science 300, 1723 (2003).
[6] Z. Hadzibabic, S. Gupta, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, “Fifty-fold inprovement in the number of quantum degeneratefermionic atoms,” arXiv:cond-mat/0306050, Phys. Rev. Lett., in press (2003).
[7] M. W. Zwierlein, Z. Hadzibabic, S. Gupta, and W. Ketterle, “Spectroscopic insensitivity to cold collisions in a two-state mixture of fermions,” arXiv:cond-mat/0306627(2003).
[8] S. Gupta, Z. Hadzibabic, J. R. Anglin, and W. Ketterle, “Collisions in Zero Temperature Fermi Gases,” arXiv:cond-mat/0307088 (2003).
[9] D. M. Stamper-Kurn, “Peeking and Poking at a New Quantum Fluid: Studies of BoseEinstein Condensates in Magnetic and Optical Traps,” PhD Thesis (unpublished).
(2000).
[10] S. Inouye, “Manipulating Bose-Einstein Condensates with Laser Light,” PhD Thesis(unpublished). (2001).
[11] A. P. Chikkatur, “Colliding and Moving Bose-Einstein Condensates: Studies of superﬂuidity and optical tweezers for condensate transport,” PhD Thesis (unpublished).
(2002).

133

[12] Z. Hadzibabic, “Studies of a Quantum Degenerate Fermionic Lithium Gas,” PhD
Thesis (unpublished). (2003).
[13] T. H. Maiman, “Stimulated Optical Radiation in Ruby,” Nature 187, 493 (1960).
[14] A. Javan, W. R. Bennett, and D. Herriott, “Population Inversion and Continuous
Optical Maser Oscillation in a Gas Discharge Containing a He-Ne Mixture,” Phys.
Rev. Lett. 6, 106 (1961).
[15] 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).
[16] 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).
[17] 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).
[18] P. Nozires and D. Pines, The Theory of Quantum Liquids (Addison-Wesley, Redwood
City, CA, 1990).
[19] D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner, andT. J. Greytak, “Bose-Einstein Condensation of Atomic Hydrogen,” Phys. Rev. Lett.
81, 3811 (1998).
[20] 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).
[21] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, and A. Aspect, “A Bose-Einstein Condensate of Metastable Atoms,” Science
292, 461 (2001).
[22] F. P. D. Santos, J. Leonard, J. M. Wang, C. J. Barrelet, F. Perales, E. Rasel, C. S.
Unnikrishnan, M. Leduc, and C. Cohen-Tannoudji, “Bose-Einstein condensation ofmetastable helium,” Phys. Rev. Lett. 86, 3459 (2001).
[23] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio, “BoseEinstein Condensation of Potassium Atoms by Sympathetic Cooling,” Science 294,
1320 (2001).
[24] T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, “Bose-Einstein condensation of cesium,” Science 299, 232 (2003).
[25] Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T.
Yabuzaki, and Y. Takahashi, “Spin-Singlet Bose-Einstein Condensation of TwoElectron Atoms,” Phys. Rev. Lett. 90, 040404 (2003).
[26] B. DeMarco and D. S. Jin, “Onset of Fermi Degeneracy in a Trapped Atomic Gas,”
Science 285, 1703 (1999).
134

[27] 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).
[28] 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).
[29] 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).
[30] G. Roati, F. Riboli, G. Modungo, and M. Inguscio, “Fermi-Bose Quantum Degenerate
40 K - 87 Mixture with Attractive Interaction,” Phys. Rev. Lett. 88, 160401 (2002).
[31] J. Bardeen, L. N. Cooper, and J. R. Schreiﬀer, “Theory of Superconductivity,” Phys.
Rev. 108, 1175 (1957).
[32] Atom Interferometry, edited by P. Berman (Academic Press, New York, 1997).
[33] P. Bouyer and M. A. Kasevich, “Heisenberg-limited spectroscopy with degenerate
Bose-Einstein gases,” Phys. Rev. A 56, R1083 (1997).
[34] Y. Torii, Y. Suzuki, M. Kozuma, T. Sugiura, T. Kuga, L. Deng, and E. W. Hagley,“Mach-Zehnder Bragg interferometer for a Bose-Einstein condensate,” Phys. Rev. A.
61, 041602 (2000).
[35] J. E. Simsarian, J. Denschlag, M. Edwards, C. W. Clark, L. Deng, E. W. Hagley, K.
Helmerson, S. L. Rolston, and W. D. Phillips, “Imaging the Phase of an Evolving
Bose-Einstein Condensate Wave Function,” Phys. Rev. Lett. 85, 2040 (2000).
[36] Y. I. Shin, A. E. Leanhardt, M. Saba, T. Pasquini, W. Ketterle, and D. E. Pritchard,“Atom interferometry with Bose-Einstein condensates in a double-well potential,”
arXiv:cond-mat/0306305 (2003).
[37] C. R. Ekstrom, J. Schmiedmayer, M. S. Chapman, T. D. Hammond, and D. E.
Pritchard, “Measurement of the electric polarizability of sodium with an atom interferometer,” Phys. Rev. A 51, 3883 (1995).
[38] M. Kasevich and S. Chu, “Atomic Interferometry Using Stimulated Raman Transitions,” Phys. Rev. Lett. 67, 181 (1991).
[39] S. B. Cahn, A. Kumarkrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky,“Time-Domain de Broglie Wave Interferometry,” Phys. Rev. Lett. 79, 784 (1997).
[40] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer-Verlag
New York, Inc., 1999).
[41] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys.
Rev. Lett. 61, 826 (1988).
[42] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J.
Opt. Soc. Am. B 6, 2112 (1989).
135

[43] M. Marte and S. Stenholm, “Multiphoton resonances in atomic Bragg scattering,”
App. Phys. B 54, 443 (1992).
[44] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. Pritchard, “Bragg Scatteringof Atoms from a Standing Light-Wave,” Phys. Rev. Lett. 60, 515 (1988).
[45] C. G. Shull, “Obeservation of Pendellösung Fringe Structure in Neutron Diﬀraction,”
Phys. Rev. Lett. 21, 1585 (1968).
[46] P. Ewald, “Zur Begründung der Kristalloptik,” Ann. d. Phys. 54, 519 (1917).
[47] D. M. Giltner, R. W. McGowan, and S. A. Lee, “Theoretical and Experimental-Studyof the Bragg Scattering of Atoms from a Standing Light-Wave,” Phys. Rev. A 52,
3966 (1995).
[48] M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Coherent splitting of Bose-Einstein condensed atoms with
optically induced Bragg diﬀraction,” Phys. Rev. Lett. 82, 871 (1999).
[49] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, and
W. Ketterle, “Bragg spectroscopy of a Bose-Einstein condensate,” Phys. Rev. Lett.
82, 4569 (1999).
[50] P. L. Kapitza and P. A. M. Dirac, “The reﬂection of electrons from standing lightwaves,” Proc. of the Camb. Philos. Soc. 29, 297 (1933).
[51] P. L. Gould, G. A. Ruﬀ, and D. E. Pritchard, “Diﬀraction of Atoms by Light - thenear-Resonant Kapitza-Dirac Eﬀect,” Phys. Rev. Lett. 56, 827 (1986).
[52] Y. B. Ovchinnikov, J. H. Muller, M. R. Doery, E. J. D. Vredenbregt, K. Helmerson,
S. L. Rolston, and W. D. Phillips, “Diﬀraction of a released Bose-Einstein condensate
by a pulsed standing light wave,” Phys. Rev. Lett. 83, 284 (1999).
[53] D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, “An Interferometer
for atoms,” Phys. Rev. Lett. 66, 2693 (1991).
[54] B. C. Young, M. A. Kasevich, and S. Chu, “Precision Atom Interferometry with
Light Pulses,” Atom Interferometry, edited by P. Berman (Academic Press, New
York, 1997).
[55] J. M. Hensley, “A Precision Measurement of the Fine Structure Constant,” Ph.D.
thesis, Stanford (2001).
[56] T. Kinoshita, “The ﬁne structure constant,” Rep. Prog. Phys. 59, 1459 (1996).
[57] B. Taylor, “Determining the Avogadro Constant from Electrical Measurements,”
Metrologia 31, 181 (1994).
[58] A. Wicht, J. M. Hensley, E. Sarajlic, and S. Chu, “A Preliminary Measurement of
the Fine Structure Constant based on Atom Interferometry,” Physica Scripta T102,
82 (2002).
[59] P. J. Mohr and B. N. Taylor, “Codata recommended values of the fundamental constants: 1998,” Rev. Mod. Phys. 72, 351 (2000).
136

[60] M. Büchner, R. Delhuille, A. Mifre, C. Robilliard, J. Vigué, and C. Champenois,“Diﬀraction phases in atom interferometers,” Phys. Rev. A. 68, 013607 (2003).
[61] P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the
Physical Sciences (McGraw-Hill, 1992).
[62] M. P. Bradley, J. V. Porto, S. Rainville, J. K. Thompson, and D. E. Pritchard, “Penning Trap Measurements of the Masses of 133 Cs, 87,85 Rb, and 23 Na with Uncertainties≤ 0.2 ppb,” Phys. Rev. Lett. 83, 4510 (1999).
[63] 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 cd Magnetic Trap,”
Phys. Rev. Lett. 77, 416 (1996).
[64] S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and
W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,”
Science 285, 571 (1999).
[65] J. R. S. Van Dyck, P. B. Schwinberg, and H. G. Dehmelt, “New High-precision Comparison of Electron and Positron g factors,” Phys. Rev. Lett. 59, 26 (1987).
[66] T. Kinoshita, “New Value of the alpha 3 Electron Anomalous Magnetic Moment,”
Phys. Rev. Lett. 75, 4728 (1995).
[67] C. Schwob, L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O.
Acef, and A. Clairon, “Optical Frequency measurement of the 2S − 12D Transitionsin Hydrogen and Deuterium: Rydberg Constant and Lamb Shift Determinations,”
Phys. Rev. Lett. 82, 4960 (1999).
[68] T. Udem, A. Huber, B. Gross, J. Reichert, M. Prevedelli, M. Weitz, and T. W. Hnsch,“Phase-Coherent MEasurement of the Hydrogen 1S − 2S Transition Frequency withan Optical Frequency Interval Divider Chain,” Phys. Rev. Lett. 79, 2646 (1997).
[69] T. Beier, H. Hﬀner, N. Hermanspahn, S. G. Karshenboim, H.-J. Kluge, W. Quint, S.
Stahl, J. Verd, and G. Werth, “New Determination of the Electron’s Mass,” Phys.
Rev. Lett. 88, 011603 (2002).
[70] P. Juncar, J. Pinard, J. Hamond, and A. Chartier, “Absolute Determination of the
Wavelengths of the Sodium D1 and D2 Lines by Using a CW Tunable Dye Laser
Stabilized on Iodine,” Metrologia 17, 77 (1981).
[71] 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).
[72] 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.
[73] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge
University Press, Cambridge, U.K., 2002).
137

[74] A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski,
D. E. Pritchard, and W. Ketterle, “Adiabtic and Evaporative Cooling of Bose-Einstein
Condensates Below 500 Picokelvin,” Science, in press (2003).
[75] D. A. Butts and D. S. Rokhsar, “Trapped Fermi gases,” Phys. Rev. A 55, 4346 (1997).
[76] G. M. Bruun and C. W. Clark, “Ideal gases in time-dependent traps,” Phys. Rev. A
61, 061601 (2000).
[77] K. Huang, Statistical Mechanics (Wiley, New York, 1987).
[78] 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. 321–349.
[79] B. D. Esry, C. H. Greene, and H. Suno, “Threshold laws for three-body recombination,” Phys. Rev. A 65, 010705(R) (2001).
[80] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Massachusetts,
1994).
[81] 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).
[82] 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).
[83] J. V. Steele, “Eﬀective Field Theory Power Counting at Finite Density,” arXiv:nuclth/0010066 (2000).
[84] H. Heiselberg, “Fermi Systems with long scattering lengths,” Phys. Rev. A. 63, 043606(2001).
[85] R. Combescot, “Feshbach resonance in dense ultracold Fermi gases,” arXiv:condmat/0302209 (2003).
[86] 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).
[87] 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).
[88] H.-J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and
W. Ketterle, “Observation of metastable states in spinor Bose-Einstein condensates,”
Phys. Rev. Lett. 82, 2228 (1999).
[89] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, “The Dynamics of Component Separation in a Binary Mixture of Bose-Einstein Condensates,”
Phys. Rev. Lett. 81, 4531 (1998).

138

[90] P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, “Inﬂuence ofnearly resonant light on the scattering length in low-temperature atomic gases,” Phys.
Rev. Lett. 77, 2913 (1996).
[91] E. Nielsen and J. H. Macek, “Low-Energy Recombination of Identical Bosons by
Three-Body Collisions,” Phys. Rev. Lett. 83, 1566 (1999).
[92] D. S. Petrov, “Three-body problem in Fermi gases with short-range interparticle interaction,” Phys. Rev. A 67, 010703(R) (2003).
[93] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-relativistic Theory (Pergamon Press, Oxford, 1977).
[94] L. P. Gor’kov and T. K. Melik-Barkhudarov, JETP Lett. 13, 1018 (1961).
[95] A. J. Leggett, “Cooper Pairing in Spin-Polarized Fermi Systems,” Journal de Physique41, C7 (1980).
[96] M. A. Baranov, Y. Kagan, and M. Y. Kagan, “On the possibility of a superﬂuid
transition in a Fermi gas of neutral particles at ultralow temperatures,” JETP Lett.
64, 301 (1996).
[97] M. Houbiers, H. T. C. Stoof, and E. A. Cornell, “Critical Temperature of a Trapped
Bose Gas: Mean-Field Theory and Fluctuations,” Phys. Rev. A 56, 2041 (1997).
[98] M. Houbiers and H. T. C. Stoof, “Cooper-pair Formation in Trapped Atomic Fermi
Gases,” Phys. Rev. A 59, 1556 (1999).
[99] M. J. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, “Resonancesuperﬂuidity in a quantum degenerate fermi gas,” Phys. Rev. Lett. 87, 120406 (2001).
[100] M. L. Chiofalo, S. J. J. M. F. Kokkelmans, J. N. Milstein, and M. J. Holland, “Signatures of Resonance Superﬂuidity in a Quantum Fermi Gas,” Phys. Rev. Lett. 88,
090402 (2002).
[101] Private communication with S. J. J. M. F. Kokkelmans and E. Tiesinga.
[102] M. D. Barrett, J. A. Sauer, and M. S. Chapman, “All-Optical Formation of an atomic
Bose-Einstein Condensate,” Phys. Rev. Lett. 87, 010404 (2001).
[103] C. H. Schunck, “Study of an ultracold cloud of fermionic 6 Li atoms near a Feshbachresonance,” Diploma Thesis (unpublished). (2002).
[104] 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).
[105] M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W.
Ketterle, “Output coupler for Bose-Einstein condensed atoms,” Phys. Rev. Lett. 78,
582 (1997).
[106] 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).
139

[107] 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).
[108] 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).
[109] T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, “Resonant controlof elastic collisions in a optically trapped fermi gas of atoms,” Phys. Rev. Lett. 88,
173201 (2002).
[110] 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).
[111] 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).
[112] 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).
[113] Christophe Salomon, personal communication.
[114] See pp. 239 in Progress in Low Temperature Physics, Volume 10, edited by D. F.
Brewer (North-Holland Physics Publishing, a division of Elsevier Science Publishers
B.V., Amsterdam, The Netherlands, 1986).
[115] C. Menotti, P. Pedri, and S. Stringari, “Expansion of an Interacting Fermi Gas,”
Phys. Rev. Lett. 89, 250402 (2002).
[116] Y. Kagan and B. V. Svistunov, “Evolution of Correlation Properties and Appearanceof Broken Symmetry in the Process of Bose-Einstein Condensation,” Phys. Rev. Lett.
79, 3331 (1997).
[117] 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).
[118] P. Pedri, D. Guéry-Odelin, and S. Stringari, “Dynamics of a classical gas including
dissipative and mean ﬁeld eﬀects,” arXiv:cond-mat/0305624 (2003).
[119] For higly elongated traps (λ  1), the asymptotic value of the aspect ratio is given
by 0.38/λ [G.V. Shlyapnikov, private communication] - ref. 14 within [115].
[120] S. Stringari, “Collective Excitations of a Trapped Bose-Condensed Gas,” Phys. Rev.
Lett. 77, 2360 (1996).
[121] S. D. Gensemer and D. S. Jin, “Transition from collisionless to hydrodynamic behavior
in an ultracold Fermi gas,” Phys. Rev. Lett. 87, 173201 (2001).
[122] Duke group - John Thomas, talk at Gordon conference 2003.

140

[123] E. Fermi and S. Rasetti, “Eﬀect of an Alternating Magnetic Field on the Polarisationof the Resonance Radiation of Mercury Vapour,” Nature 115, 764 (1925).
[124] K. G. Petrosyan, “Fermionic atom laser,” JETP Lett. 70, 11 (1999).
[125] P. Torma and P. Zoller, “Laser Probing of Atomic Cooper Pairs,” Phys. Rev. Lett.
85, 487 (2000).
[126] G. Bruun, P. Torma, M. Rodriguez, and P. Zoller, “Laser probing of Cooper-paired
trapped atoms,” Phys. Rev. A. 64, 033609 (2001).
[127] N. Ramsey, “A Molecular Beam Resonance Method with Separated Oscillating
Fields,” Phys. Rev. 78, 695 (1950).
[128] M. Kasevich, E. Riis, S. Chu, and R. deVoe, “rf Spectroscopy in an Atomic Fountain,”Phys. Rev. Lett. 63, 612 (1989).
[129] A. Clairon, C. Salomon, S. Guellati, and W. D. Phillips, “Ramsey Resonance in a
Zacharias Fountain,” Europhys. Lett. 16, 165 (1991).
[130] 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).
[131] K. Gibble, S. Chang, and R. Legere, “Direct observation of s-wave atomic collisions,”
Phys. Rev. Lett. 75, 2666 (1995).
[132] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and E. A. Cornell, “Observation of
Anomalous Spin-State Segregation in a Trapped Ultracold Vapor,” Phys. Rev. Lett.
88, 070403 (2002).
[133] J. M. McGuirk, H. J. Lewandowski, D. M. Harber, T. Nikuni, J. E. Willeams, andE. A. Cornell, “Spatial Resolution of Spin Waves in an Ultracold Gas,” Phys. Rev.
Lett. 89, 090402 (2002).
[134] 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).
[135] I. Rabi, S. Millman, P. Kusch, and J. Zacharias, “The Molecular Beam Resonance
Method for Measuring Nuclear Magnetic Moments,” Phys. Rev. 55, 526 (1939).
[136] Our estimate uses the direct mean ﬁeld measurement of Bourdel et. al.[111] scaled toour parameters .
[137] E. G. M. van Kempen and et. al., in preparation (2003).
[138] 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).
[139] 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).

141[140] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt, A. Görlitz, S. Gupta, D. E.
Pritchard, and W. Ketterle, “Transport of Bose-Einstein Condensates with OpticalTweezers,” Phys. Rev. Lett. 88, 020401 (2002).
[141] K. Mlmer, “Phase collapse and excitations in Bose-Einstein condensates,” Phys. Rev.
A 58, 566 (1998).
[142] G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, “Collapseof a degenerate Fermi gas,” Science 68, 011601 (2002).
[143] 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).
[144] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. H. Denschlag, and R.
Grimm, “Pure gas of Optically Trapped Molecules Created from Fermionic Atoms,”
arXiv:cond-mat/0308095 (2003).
[145] Y. Ohashi and A. Griﬃn, “BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance,” Phys. Rev. Lett. 89, 130402 (2002).

142