Ultracold Fermi Gas with Repulsive Interactions
by

Ye-Ryoung Lee
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2012
c Massachusetts Institute of Technology 2012. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 10, 2012

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

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John Belcher
Professor of Physics, Associate Department Head for Education

2

Ultracold Fermi Gas with Repulsive Interactions
by
Ye-Ryoung Lee
Submitted to the Department of Physics
on August 10, 2012, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
This thesis presents results from experiments of ultracold atomic Fermi gases withrepulsive interaction.
Itinerant ferromagnetism was studied by simulating the Stoner model with astrongly interacting Fermi gas of ultracold atoms. We observed nonmonotonic behavior of lifetime, kinetic energy, and size for increasing repulsive interactions, whichis in good agreement with a mean-field model for the ferromagnetic phase transition.
However, later research showed the absence of enhanced spin fluctuation, which isdefinitive evidence against the ferromagnetic phase transition. Still, our work triggered a lot of research on repulsive interactions in ultracold Fermi gases.
A quantitative approach is taken to study ultracold Fermi gases with repulsive
interaction. This is done by careful measurements of density profiles in equilibrium.
First, Pauli paramagnetism is observed in trapped atomic samples which have an
inhomogeneous density due to the harmonic confinement potential. We experimentally measure the susceptibility of ideal Fermi gas. This research shows that ultracold
atoms can serve as model systems to demonstrate well-known textbook physics in amore ideal way than other systems. Then, Fermi gases with repulsive interactions arecharacterized by measuring their compressibility as a function of interaction strength.
The compressibility is obtained from in-trap density distributions monitored by phase
contrast imaging. For interaction parameters kF a > 0.25 fast decay of the gas prevents the observation of equilibrium profiles. For smaller interaction parameters, theresults are adequately described by first-order perturbation theory. A novel phase
contrast imaging method compensates for dispersive distortions of the images.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacAurthur Professor of Physics

3

4

To my Lord, Jesus
my love, Kyeong-Jae
and my dearest family

5

Acknowledgments“For I know the plans I have for you” declares the Lord,“plans to prosper you and not to harm you, plans to give you hope and a future.
Then you will call on me and come and pray to me, and I will listen to you.
You will seek me and find me when you seek me with all your heart.
I will be found by you” declares the Lord...
Jeremiah 29:11-14

Thank you Jesus, my joy and my strength.
First, I sincerely thank my advisors, Wolfgang and Dave. Your advice and support
has made this work possible. Thank you for giving me this research opportunity. I
thank Wolfgang for his dedication, passion, patience, and continuous encouragement
and support. You are truly the best at providing direction and motivation. I thankDave for his enthusiasm and passion for physics. This has always been a source ofinspiration to me. I have learned a great deal from both of you.
I am thankful that I shared the lab with great people.
It was such a blessing to work with Myoung-Sun during the last phase of my study.
His understanding of physics and calm and diligent personality encouraged me a lot
and helped me finally exit the revolving door.
Tout joined the group 3 years later than I joined. It was great to have an extremely
energetic and enthusiastic lab mate. I thank him for bringing energy to the lab andalso for his kindness, warm heart, patience, endless curiosity and optimism.
I have started grad school at the same time as Caleb. I am thankful that I had
such a nice friend as a lab mate. It was very comforting to have shared most of myjourney with Caleb, who is always calm and nice.
I thank Jae for helping me get through some hard times. His encouragement andadvice was a source of support. I shared great time in the lab at nights enjoying
discussion with him about physics and life.
My first physics project in the lab was the ferromagnetism project with Gyuboong. His enthusiasm for physics and his diligence inspired me. I thank him for6

supporting me joining this amazing group and guiding me through.
I thank Timur for bringing new energy into our team with his relaxed attitudeand charming personality. I was always amazed by Tonys ability to accomplish any
tasks perfectly. I am confident that he will go on to do amazing research. I also thank
Gregory for experimental assistance. I would also like to thank all past members of
BEC3 including Tom, Sebastian, and Michele.
Yong has been a mentor for me throughout my time in MIT. He not only answered
all my physics questions, but also provided guidance regarding all my concerns in andoutside the lab. I truly admire his intelligence, enthusiasm for physics, wisdom,
generosity, and diligence.
Andre, or in short An, was my office mate when I first joined Ketterle group.
We became close friends since. He was always there to give comforting conversation
whenever I had hard time at MIT. My life in the lab was enriched and brightened by
our friendship.
Whenever I needed help desperately at late nights in the lab, Christian was always
there to willingly share his wisdom. I really appreciated your help.
I thank Jongchul for his constant encouragement and support. He was kind enoughto lend his time and help out with any questions that I had. Thank you for all the
fun times we had outside the lab.
I thank all the other people in the hallway for always being kind and helpfulfriends. I thank BEC1 including Ariel, Mark, and Christian for sharing nights in thelab over the curtain with loud music and Feshbach field interference. I thank BEC2,
including Aviv, Ed, Jon, Jit kee, Yingmei, and Dan for providing all the equipments
and technical advices. I thank BEC4, Fermi1, and Vladans group including Hiro,David, Patrick, Gretchen, Chen, Jeewoo, Peyman, Vladan, Monika, Haruka, Ian, andMarko for their support. I also thank Joanna for her assistance and smiles.
Outside the lab, I express my gratitude to Beacon City Church and all my friends
at MIT and Harvard. I also thank my friends back in Korea.
I would like to thank my family, who has always loved, supported, and believedin me. Without them, I would not be where I am today. I love you all very much.
7

I am deeply blessed to be a child of my parents, Seunghee and Insun, and to be asister of Ye-Eun. I am secure because I know that they are always there to pour out
endless love, comfort and encouragement. Thank you for being the best parents inthe world who I respect the most, and for being the best sister whom I am so proudof. I would also like to thank my in laws, Jong-Kun, Soyeon, and Seong-Jae for theirlove, support, and prayer. Your encouragement was truly meaningful to me.
Finally, I give my special thanks to my husband, my guardian angel and my bestfriend, Kyeong-Jae. Without his prayer, support, caring, understanding, sacrifice,patience, comforts, and love, I would never have completed this thesis. I am sothankful to have shared 6 years at MIT with you. I must be the luckiest person
because you are my husband! I love you with all my heart.

8Contents
1 Introduction
1.1

17

Ultracold Fermi Gases . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.1.1

Ultracold Atomic Gases . . . . . . . . . . . . . . . . . . . . .

17

1.1.2

Quantum Simulator . . . . . . . . . . . . . . . . . . . . . . . .

18

1.2

Repulsive Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.3

Equation of States . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

1.4

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2 Quantum Statistical Mechanics for Ultracold Fermi Gases

25

2.1

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

25

2.2

Quantum Statistical Mechanics for Boson and Fermion . . . . . . . .

26

2.3

Fermi gases in a harmonic trap . . . . . . . . . . . . . . . . . . . . .

28

2.4

Interaction Effect : Perturbation limit . . . . . . . . . . . . . . . . . .

28

3 Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms

31

3.1

Itinerant Ferromagnetism and Stoner Model . . . . . . . . . . . . . .

31

3.2

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.3

Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms . . . . .

36

3.4

Further Development . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4 Pauli Paramagnetism in an Ideal Fermi Gas

41

4.1

Pauli Paramagnetism: Ultracold Atomic Gas vs Solid State System .

42

4.2

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

47

9

4.3

Susceptibility Measurement . . . . . . . . . . . . . . . . . . . . . . .

49

4.4

Susceptibility Measurement in Systems with Interaction . . . . . . . .

53

5 Compressibility of an Ultracold Fermi Gas with Repulsive Interactions

55

5.1

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.2

Loss Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.2.1

Loss Rate Estimation . . . . . . . . . . . . . . . . . . . . . . .

57

5.2.2

Loss Rate Measurement . . . . . . . . . . . . . . . . . . . . .

58

Compressibility Measurement . . . . . . . . . . . . . . . . . . . . . .

58

5.3.1

Measurement Method. . . . . . . . . . . . . . . . . . . . . .

60

5.3.2

Temperature Measurement . . . . . . . . . . . . . . . . . . . .

62

5.3.3

Interaction Effect . . . . . . . . . . . . . . . . . . . . . . . . .

63

Dispersive Effect in Phase-Contrast Imaging . . . . . . . . . . . . . .

65

5.4.1

Phase-Contrast Imaging . . . . . . . . . . . . . . . . . . . . .

66

5.4.2

Compensation for Dispersive Effect . . . . . . . . . . . . . . .

66

5.4.3

Imaging Focus Fine Tuning . . . . . . . . . . . . . . . . . . .

69

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.5.1

Molecular Fraction in the Density Profile . . . . . . . . . . . .

69

5.5.2

Interaction Strength Limitation . . . . . . . . . . . . . . . . .

71

5.5.3

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

72

5.3

5.4

5.5

6 Conclusion

73

A Tips for Precise Imaging

75

A.1 Imaging Beam Intensity . . . . . . . . . . . . . . . . . . . . . . . . .

75

B Exact Breit-Rabi Formula for the Ground States of 6 Li

77

C RF Antenna in BEC3’s Main Chamber

79

D Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms

81

10

E Compressibility of an Ultracold Fermi Gas with Repulsive Interactions

87

11

12

List of Figures
1-1 Feshbach resonance in 6 Li between the two lowest hyperfine states|F = 1/2; mF = 1/2i and |F = 1/2; mF = −1/2i . . . . . . . . . . . .

20

6

Li hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3-1 Simple mean-field model of a two-component Fermi gas . . . . . . . .

33

3-2 Mean-field prediction of ferromagnetism . . . . . . . . . . . . . . . .

34

3-3 The experimental procedure . . . . . . . . . . . . . . . . . . . . . . .

35

3-4 Atom loss rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3-5 Kinetic energy measurement . . . . . . . . . . . . . . . . . . . . . . .

38

3-6 Cloud size and molecular fraction . . . . . . . . . . . . . . . . . . . .

39

1-2

4-1 Comparison between a free electron gas and two-component Fermi
gases in the study of magnetism . . . . . . . . . . . . . . . . . . . . .

43

4-2 Comparison between a free electron gas and two-component Fermi
gases trapped in the spherical harmonic trapping potential . . . . . .

46

4-3 Simulated density profiles of two-component Fermi gases in a sphericalharmonic trap at zero temperature . . . . . . . . . . . . . . . . . . .

48

4-4 Double-shutter phase contrast imaging with two different imaging frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4-5 Experimentally obtained density profiles . . . . . . . . . . . . . . . .

51

4-6 Susceptibility at high magnetic field . . . . . . . . . . . . . . . . . . .

52

5-1 Characterizing atomic loss for increasing repulsive interactions . . . .

59

13

5-2 Determination of the compressibility of repulsively interacting Fermi
gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5-3 Measured temperature, normalized compressibility, and interaction effect on compressibility at various interaction strengths . . . . . . . .

64

5-4 Phase-contrast imaging of a balanced spin mixture in states |1i and |2i 67
5-5 Dispersion effect in phase-contrast imaging . . . . . . . . . . . . . . .

68

5-6 Imaging Focus Fine Tuning. . . . . . . . . . . . . . . . . . . . . . . .

70

C-1 Dimensions of RF antenna in BEC3’s main chamber . . . . . . . . . .

79

14

List of Tables
4.1

Comparison between a free electron gas and two-component Fermi
gases in the study of magnetism . . . . . . . . . . . . . . . . . . . . .

42

B.1 Constants for 6 Li Breit-Rabi formula . . . . . . . . . . . . . . . . . .

77

15

16

Chapter 1
Introduction
1.1

Ultracold Fermi Gases

1.1.1

Ultracold Atomic Gases

At very low temperature where the quantum coherence length is comparable to theinter-particle distance, particles obey quantum statistics. In quantum statistics, there
are two kinds of particle: bosons and fermions. Bosons follow the Bose-Einstein
statistics, and fermions follow the Fermi statistics.
According to Bose-Einstein statistics, a large fraction of the bosons occupy thelowest energy level at very low temperature. This state of matter is called BoseEinstein condensate (BEC), and it was first predicted by Satyendra N. Bose andAlbert Einstein in 1924 [9, 21]. The first experimental realization of Bose-Einsteincondensate was achieved in 1995 [2, 16]. This was done by cooling ultracold diluteatomic gases of 87 Rb [2] and 23 Na [16] to less than 1 µK above the absolute zerotemperature. The cooling was done by using laser cooling and magnetic trapping
techniques [20, 33].
The achievement of Bose-Einstein condensate in ultracold atoms not only enabled
the exciting researches on Bose-Einstein itself [1, 6, 7, 41, 42, 56], but also enabled
achieving the quantum degenerate Fermi gases [17].
Experiments with ultracold Fermi gases have explored many-body physics with17

strong interactions using Feshbach resonance or optical lattices. They have demonstrated long-predicted phenomena like the BEC-BCS crossover [34] and Lee-HuangYang corrections to the energy of degenerate gases [46, 47, 62]. Experiments havealso explored novel quantum phases like fermions with unitarity limited interactions [34, 36], population imbalanced Fermi gases [48, 75] and Hubbard models in
optical lattices [8, 23].

1.1.2

Quantum Simulator

Novel quantum materials such as high temperature superconductors, graphenes, andcolossal magneto-resistance materials are considered as potential driving force behindfurther development of technology. Even though understanding the physics behindthese materials is crucial for improving their properties and developing applications, it
is very difficult because of complexity in strongly interacting many-electron systems.
Theoretical studies suggests simple model systems in order to explain the physics,
but these systems cannot be verified directly by experiment on existing materials. In
addition, synthesizing new materials mimicking the simple models is based on trial
and error approaches.
Experiments using ultracold atomic gases are a new approaches to solve this problem. Using the tools and precision of atomic physics, ultracold atomic samples can beprepared to realize simple Hamiltonians. This concept of quantum simulator was first
proposed by Richard Feynman [22]. The main benefit of experiments with ultracold
atoms is the controllability. The density of atoms can be varied over three orders ofmagnitude. The atoms can be prepared in different hyperfine states, which in turncan represent different spin states of electrons. The trapping potential to hold theatomic gases can be manipulated and controlled easily, so that various lattice structures can be produced using optical lattices. The atomic sample is extremely pureunlike the materials containing impurities or defects used in condensed matter experiments. The interactions can be controlled using well understood Feshbach resonances.
Usually, interactions in ultracold gases are fully described by the scattering length,
which is a zero-range approximation greatly simplifying the theoretical description.
18

This approximation is valid since the diluteness of the atomic gases implies a particle
separation much larger than the short range of the van-der-Waals interactions. This
almost exact characterization of the interactions by a single parameter, the tunability of interaction strength, and precise experimental control over cold atoms systems
have made them an ideal testbed for many-body quantum calculations.
In this thesis, this controllability of ultracold atoms is used to simulate the simple
model system of ferromagnetism, the textbook model system of free electron gas withand without interaction.

1.2

Repulsive Interaction

The important recent developments in cold atom science reviewed in section 1.1.1
were realized in Fermi gases with strong attractive interactions. Fermi gases withrepulsive interactions didn’t capture much attention mainly because they are unstableagainst three-body recombination into weakly-bound molecules [52]. Nevertheless,fermions with repulsive interactions have been the focus of much recent work due tothe prediction of a phase transition to a ferromagnetic state for sufficiently strong
interactions [5,19,37,66,71]. Motivated by these predictions, experimental researches
on fermions with repulsive interactions were done, and they are presented in thisthesis.
As in Fig. 1-1, a Feshbach resonance couples a molecular bound state with anunbound state of two free atoms. A magnetic field changes the energies of two free
atoms relative to the molecular state and thereby controls the interatomic interaction
strength. The experiment with attractive interaction has been done on the groundstate branch (or so-called lower branch) in Fig. 1-1(b). The fermi gases with repulsive
interaction are realized on the the first excited branch (or so-called upper branch) ofa Feshbach resonance where decay is always possible into the lower branch which
consists of weakly bound molecular states with binding energy h̄2 /ma2 with a being
the scattering length. The molecules are formed through three-body collisions, andthe collision rate is proportional to n2 a6 max(T, TF ) = (kF a)6 n2/3 [18, 52].
19

(a)

(b)

Figure 1-1: Feshbach resonance in 6 Li between the two lowest hyperfine states|F = 1/2; mF = 1/2i and |F = 1/2; mF = −1/2i.(a) The red line shows the molecular binding energy, and the blue dash lines show the scattering length. (b) The blueline below the Feshbach resonance represents the excited state where the interaction
is repulsive.

20

1.3

Equation of States

A system is characterized by its equation of state. The equation of state can be expressed in different forms involving various thermodynamic variables including density, energy, pressure, temperature and entropy. For cold atom experiments, density,
chemical potential (through the trapping potential) and temperature are directly accessible to measurement.
In the first thermodynamic studies of ultracold Fermi gases, the thermodynamic
quantities of the whole trapped gas was measured. The trapping potential induces
inhomogeneity in ultracold experiments while most theoretical calculations assumes
homogeneous systems. Thus, the comparison between experiments and theoretical
calculations require integral over the whole trap. Integration smoothes sharp features,
so that it could conceal evidences of phase transitions and make different theoreticalpredictions indistinguishable.
However, using local density approximation, the ultracold gas can be locally described as a homogeneous gas. That is, each local point in the gas can be consideredas homogeneous systems with different conditions. Therefore, the inhomogeneity
problem in ultracold experiments can be resolved by accurate measurement of thethermodynamic variables at each point in the trap.
This new level of quantitative comparison between theory and experiment wasrecently reached by careful measurements of density profiles from which the equationof state could be determined. These techniques were first proposed by Chevy [12] andBulgac [10] and implemented by Shin [61]. Further improvements [27, 28, 36, 44–46]
resulted in impressive accuracy without adjustable parameters. These results hingeon accurate measurements of the equilibrium atomic density distribution. In thisthesis, we extend these work to ultracold Fermi gases with repulsive interaction. In
addition, we use ultracold Fermi gases as model systems to demonstrate well-known
textbook physics of free electron gas in a more ideal way than other systems.
21

1.4

Experiment

The details on the experimental setup and production of a strongly interacting degenerate gas of 6 Li can be found in Ref. [25, 34]. The design and construction of ourapparatus is thoroughly described in Ref. [13]. The most recent upgrade to from BEC
apparatus to Bose-Fermi mixture apparatus has been reviewed in Ref. [14, 30]. Here,we will only briefly review the procedure.

(a)

(b)

Figure 1-2: 6 Li hyperfine structure. (a) Energy level diagram for laser cooling. (b)A repulsive two-component Fermi gas is prepared in the two lowest hyperfine states, |1i and |2i states (corresponding to the |F = 1/2; mF = 1/2i and|F = 1/2; mF = −1/2i).
An atomic vapor is created in an heated oven containing solid 23 Na, a boson,and 6 Li, a fermion. This two-species atomic vapor is collimated by a nozzle anda collimation aperture. A Zeeman slower is used to slow down the thermal atomic
beam, and the slowed atomic beam is loaded into a dual species Magneto-Optical Trap
(MOT). In order to avoid spin-exchange collisions causing unwanted heating, 23 Na and22

6

Li atoms are optically pumped into their respective stretched states, |F = 2; mF = 2i

and |F = 3/2; mF = 3/2i, and they are microwave swept in the presence of bias field
for purification. The atoms are loaded into a magnetic trap of the Ioffe-Pritchard type.
Evaporative cooling of 23 Na using microwave sweep leads to sympathetic cooling of
6

Li. At this point, we routinely produce a spin-polarized Fermi gas of ∼ 5×106 atoms

in the stretched state of |F = 3/2; mF = 3/2i, and 23 Na atoms are fully evaporated.
The typical temperature of the sympathetic cooled 6 Li is T ∼ 0.5TF where TF is the
Fermi temperature.
The 6 Li atoms are then loaded into a single-beam optical dipole trap and transferred into the lowest hyperfine state |F = 1/2; mF = 1/2i by a Landau-Zener radiofrequency (RF) sweep. Additional axial confinement is provided by magnetic fieldcurvature. A desired population mixture of |1i and |2i spin states (corresponding tothe |F = 1/2; mF = 1/2i and |F = 1/2; mF = −1/2i states at low magnetic field) is
prepared by a Landau-Zener RF sweep at 300 G, followed by 500 ms wait time fordecoherence and evaporative cooling in the optical trap. The Feshbach resonance at
834 G [34] is used to tune the repulsive interactions between |1i and |2i. We increase
the magnetic field in 200 ms to 528 G, where the scattering length is zero and our
Fermi gas is non-interacting. The final trap has a depth of 4.1 µK and frequencies ofωx = ωy = 390 Hz and ωz = 34.7 Hz. The number of atoms per spin state is 8 × 105 ,which corresponds to a Fermi temperature TF of 1.4 µK. Note that the experiment
presented in chapter 3 was done in slightly different configuration. The mixture
preparation of |1i and |2i spin states was done at 590 G instead of 300G, and themagnetic field was lowered to 300 G for decoherence. The wait time for decoherence
was 1 s, and the field was ramped up back to 590 G. The final trap has a depth of
7.1 µK and frequencies of ωx = ωy ' 300 Hz and ωz ' 70 Hz.

23

24

Chapter 2
Quantum Statistical Mechanics for
Ultracold Fermi Gases
2.1

Bosons and Fermions

In classical mechanics, we can always distinguish particles under physical observationand measurements. In quantum mechanics, particles can be indistinguishable because
a detective’s observations inevitably and unpredictably alter the state. This featureof quantum mechanics determines two kinds of fundamental particles: bosons andfermions.
Let us consider a collection of N indistinguishable particles. The wave function
ψ(r1 , r2 , .., rN ) represents the probability amplitude for finding N particles at theposition of r1 , r2 , .., rN . Since particles are indistinguishable, exchange of any pair ofparticles keeps the probability density invariant:| ψ(r1 , .., ri , .., rj , .., rN ) |2 = | ψ(r1 , .., rj , .., ri , .., rN ) |2 .

(2.1)

This results in two degenerate wave functions under particle exchange.
ψ(r1 , .., ri , .., rj , .., rN ) = ±ψ(r1 , .., rj , .., ri , .., rN )
25

(2.2)

For simplicity, let us consider two identical particles. Suppose ψa (r1 )ψb (r2 ) andψb (r1 )ψa (r2 ) are two eigenstates of the two-particle system we consider. Then, thetwo wave functions satisfying Eq. 2.1 are
ψ± (r1 , r2 ) = A[ψa (r1 )ψb (r2 ) ± ψb (r1 )ψa (r2 )].

(2.3)

These two wave functions represent bosons (symmetric wave function, ψ+ ) andfermions (antisymmetric wave function, ψ− ). All particles with integer spin arebosons, and these particles obey Bose-Einstein statistics. All particles with halfinteger spin are fermions, and these particles obey Fermi-Dirac Statistics. One characteristic of fermions is that they cannot occupy the same state, the famous ”Pauli’s
exclusion principle.” If ψa = ψb in Eq. 2.3, ψ− = 0. That is, probability of two
identical fermions occupying the same state is zero.

2.2

Quantum Statistical Mechanics for Boson andFermionIn grand canonical ensemble, the probability that the system is in any state withparticle number N and energy Etot is given by the Boltzmann factor e−(Etot −µN )/kB T .
Let us consider a non-interacting many-particle system with single-particle energy
Ei and ni particles in energy state Ei . The grand canonical partition function Z for
this system is then
Z=

XY
ni

e−ni (Ei −µ)/kB T .

(2.4)

i

The average occupation number of state i is thenhni i = kB T

∂ ln Z1= (Ei −µ)/k T
B
∂µ
e
∓1

(2.5)

with the upper sign corresponding to bosons, the lower sign to fermions. Eq. 2.5
26

represents the Bose-Einstein and Fermi-Dirac distributions introduced earlier.

From the Fermi-Dirac distribution, we can obtain the average particle numberPN =i hni i. If we replace summations over i by corresponding integrations, thedensity becomes1n = − 3 Li3/2 (−eµ/kB T ),
λ dB

(2.6)

q
2πh̄2
where λdB = mk
is the thermal de Broglie wavelength and Lin (λ) is the nth -orderBT
Polylogarithm, defined as
1
Lin (λ) ≡ n
π

Z

1
.
d2n r r2e /λ − 1(2.7)

A useful formula for integrals over PolyLogarithms isZ ∞2

dxLin (λe−x ) =√πLin+ 1 (λ).
2

−∞

(2.8)

This formula is particularly useful when we obtain column or doubly integrated density profiles introduced later.

At zero temperature, the fermi occupation number (Eq. 2.5) is one for Ei < µ,and zero otherwise. The limiting value of µ is called the fermi energy, EF . In terms
2h̄
of density, EF = 2m
(6π 2 n)2/3 . From the definition, the density at zero temperaturein terms of µ becomes
1
n= 2
6π



2mµ
h̄2

3/2
.

(2.9)

At low temperature limit T  µ, one can use the Sommerfeld expansion to obtainthe temperature corrections to the density equation in terms of kEBFT [68]. At hightemperature limit T  µ, one can use the virial expansion to express the density
equation in terms of fugacity eβµ , where β = 1/kB T .
27

2.3

Fermi gases in a harmonic trap

The density obtained is for a homogeneous system. In the experiments with ultracold
atomic gases, atoms are held in a trapping potential and are thus inhomogeneous.
The trapping potentials can usually be approximated described by a harmonic trap.
Let us now consider particles confined in a harmonic trap, with trapping potential
1
V (r) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ),2

(2.10)

where m is atomic mass, and ωi is trapping frequency. In the local density approximation (LDA) the system can be described as locally homogeneous with a spatiallydependent chemical potential µloc = µ − V (r). This provides the density distribution
in a harmonic trap, given temperature T and the chemical potential µ:1n(r) = − 3 Li3/2 −e(µ−V (r))/kB T .
λ dB(2.11)In experiments with ultracold atomic gases, in-trap density profiles are obtained in
the form of column density. Column density ñ(y, z) can be obtained by integrating
this equation along one axis using Eq.2.8,
Z ∞n(x, y, z)dx = −ñ(y, z) =
−∞

2.4


m(kB T )2(µ−V (y,z)/kB T )
Li
−e
.
2
2πh̄3 ωx(2.12)

Interaction Effect : Perturbation limit

So far we considered non-interacting systems. Now let us consider a system with
spin-half fermions with balanced population in each spin state interacting througha short-range s-wave interactions (contact interactions) with scattering length a. If
the interaction is weakly repulsive, the ground-state energy can be exactly calculated
within standard perturbation theory. At zero temperature, the total energy is given28

to second order in a by the following [38]:10
4(11 − 2 ln 2)
3
2
kF a +
E = N EF 1 +(kF a) ,
5
9π
21π 2

(2.13)

where N is total number of particles, and kF = (3π 2 N)1/3 . Then, the chemicalVpotential is∂E44(11 − 2 ln 2)2µ== EF 1 +kF a +(kF a) .
∂N
3π
15π 2

29

(2.14)

30

Chapter 3
Itinerant Ferromagnetism in a
Fermi Gas of Ultracold Atoms
This chapter presents our research on itinerant ferromagnetism in a strongly interacting Fermi gas of ultracold atoms. This experiment was the first experimental
attempt at the repulsive side of Feshbach resonance. The research simulated a simple
Hamiltonian which can not be simulated by solid experiments. The experiment wasreported in the publication:• Gyu-Boong Jo, Ye-Ryuoung Lee, Jae-Hoon Choi, Caleb A. Christensen, Tony
H. Kim, Joseph H. Thywissen,David E. Pritchard and Wolfgang Ketterle
Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms
Science 325 , 1521-1524 (2009). Included in appendix D.
More details on this research can be found in Gyu-Boong Jo’s doctoral thesis [30].

3.1Itinerant Ferromagnetism and Stoner Model

Iron, nickel, and cobalt can form permanent magnets, or are attracted to magnets.
This kind of magnetism is caused when electron spins line up parallel with each otherand is called ferromagnetism. Here, the electrons whose spins aligned to create themagnetic state are the conduction electrons which are not localized to any particular
31

atoms in a metal. Thus, the magnetism of iron, nickel, and cobalt is called itinerant
ferromagnetism.
The Stoner model and its extensions [70] have been used to describe itinerant
ferromagnetism. Stoner assumed that the interaction between electrons can be seen
as screened Coulomb interaction, which in turn can be estimated as a short-range
repulsive interaction. This simplified model is not enough to explain real ferromagnets
in a quantitative way [72]. However, this model can be realized and tested using Fermi
gas of ultracold atoms instead of electrons [19].
Here we simulate the Stoner model using an ultracold Fermi gas of 6 Li atoms.
The 6 Li atom has the total nuclear spin I=1 and the total electron spin S=1/2.
Its ground state where angular momentum L=0 splits into six states under the external magnetic field. We use two lowest energy states, |F = 1/2; mF = 1/2i and|F = 1/2; mF = −1/2i, to simulate two spin-states of an electron. The screened
Coulomb interaction of electrons can be realized by the s-wave scattering caused
by Feshbach resonance of the two pseudo-spin states of 6 Li atoms. Due to the angular momentum conservation one state cannot be converted to the other state unlike
electrons. That is, the number of atoms in each state is fixed. Therefore, the net
spin polarization is zero in case of balanced spin mixture, which corresponds to zeroexternal magnetic field in solid state systems. More detailed comparison can be foundin chapter 4.1.
Intuitively speaking, the Stoner model predicts ferromagnetism based on competition between kinetic energy and interaction energy. In order to understand thismodel, let us consider a simple mean-field model of a two-component Fermi gas.
The total energy of a two-component Fermi gas of density nσ (for each spin state)in a volume V is
Etotal =

X 3
σ=↑,↓

where EF,σ nσ =5

V EF,σ nσ + gV n↑ n↓ ,

(3.1)

2
2
h̄2 kF,σaand g = 4πh̄
. The first part of the equation comes from kinetic
2mm

energy, and the second part comes from the interaction energy. Let us consider a32n + dnn - dnn - dnn + dnFigure 3-1: A simple mean-field model of a two-component Fermi gas. n is the densityof the gas. We introduce local density imbalance dn in order to check the instability
toward the ferromagnetic state.

locally polarized case while the net polarization is kept zero (see Fig. 3-1). We define, where dn = n↑ −n↓ and n = n↑ +n↓ .
the local magnetization of the Fermi gas, η = dnn
The total energy can be expressed using the local magnetization η as




3
2
5/3
5/3
Etotal = EF 2V n
(1 + η) + (1 − η)+kF a(1 + η)(1 − η) .
10
3π

(3.2)

The kinetic energy increases when the local magnetization increases. This increase
max out when the cloud is fully polarized. In contrast, the interaction energy ofthe balanced cloud continuously increases when the interaction parameter kF a increase, and the interaction energy of the fully polarized cloud becomes completely
zero. Therefore, the fully polarized case becomes energetically favorable when theincrease in interaction energy becomes greater than the increase in the kinetic energy.
This is the simple explanation behind Stoner model. The exact phase transition occurs when the curvature of the energy curve becomes zero at η = 0 (see Fig. 3-2 (a)),
which gives the critical interaction strength kF a = π/2.
We consider the thermodynamic quantities at constant volume for simplicity.
However, the atoms trapped in a harmonic potential is more relevant at constantpressure. At constat pressure, the system reaches equilibrium when minimizing theenthalpy H = Etotal − P V , where P = −dEtotal /dV . Fig. 3-2 (b) shows magnetiza33

tion, kinetic energy, volume, and enthalpy as a function of interaction parameter kF a.
At the phase transition, we can observe the onset of magnetization, the minimum inthe kinetic energy, and the maximum in the size of the cloud. Thus these features
can be interpreted as evidences for ferromagnetic phase transition.

Magnetization

1.0

1.3

Total Enthalpy

Increasing kfa

Energy at Constant Volume

Kinetic Energy(b)Volume

(a)

-1

-0.5

0

+0.5

0.6
0.2

1.1
0.9

1.45
1.35
1.25
1.35
1.25
1.15

+1

0.5

Magnetization

1.0

1.5

2.0

2.5

Interaction Parameter kFa

Figure 3-2: Mean-field prediction of ferromagnetism. (a) Total energy at different local magnetization η and interaction parameter kF a. (b)Magnetization, kinetic energy,
volume, and enthalpy as a function of interaction parameter kF a. The ferromagnetictransition occurs at the dotted line.

3.2

Experimental Procedure

An evaporative cooled equal mixture of |1i and |2i spin states is prepared at 590
G as described in chapter 1.4. We increase the magnetic field in order to increaseinteraction strength. Due to eddy current, the fasted ramp to the target field waslimited to 4.5 ms.
34

Magnetic Field [G]

(a)

Evap.
at 300G
834
820

Field
Ramp

Ballistic
Expansion

Hold

p Off
Trap

Imaging
at High Field

720
590
300
Imaging
at Zero Field
0

(b)

Lower
Trap
Depth

(i)

(ii)

(iii)

(iv)

(v)

1.0 s

600 ms

4.5 ms

0 - 14 ms

4.6 msIn-trap image at 812G

Time

4.6ms ToF at 0G

X
Z

Figure 3-3: The experimental procedure. (a) The sequence of the magnetic field
ramp. (b) Image samples.

In order to measure the loss rate, we recorded the atom number for 2 ms rightafter the ramp. Then, we obtained the loss rate by fitting linearly over time because
the initial loss rate can be approximated to be linear (see Fig. 3-4 (b),(c)). For
kinetic energy and cloud size measurement, the optical trap and the magnetic fields
were suddenly turned off. After 4.6 ms of ballistic expansion, we imaged the |1i state23mσ x, where σx
atoms at zero field. The kinetic energy was determined by Ekin = 24
2
tofis the Gaussian radial width and 4tof is the ballistic expansion time. The cloud size
was determined by the Gaussian radial width σz . We measured the temperature byfitting finite temperature in-trap density profile right after the ramp.
35

3.3

Itinerant Ferromagnetism in a Fermi Gas ofUltracold Atoms

As we reviewed in the section 3.1, ferromagnetic phase transition shows the onset ofmagnetization, the minimum in the kinetic energy, and the maximum in the size ofthe cloud. Even though these features are not sufficient conditions for ferromagnetic
phase transition, they are necessary conditions.
The onset of magnetization can be observed by measuring the loss rate because theloss rate is proportional to n↑ n↓ a6 max(T, TF )(∝ 1 − η 2 ) [52]. In Fig. 3-4, we observe
decreasing loss rate for k 0 F a > 2.2 at T = 0.12TF . If there is no magnetization,
the loss rate should increase as the interaction strength increases. Therefore, thismaximum in the loss rate indicates the onset of magnetization.
We also investigate thermodynamic quantities. In Fig. 3-5, we observe the minimum in the measured kinetic energy as in Fig. 3-2. In Fig. 3-6 (a), we observe themaximum in the measured size of the cloud as well.
Lastly the observed features depends on the temperature. As temperature of thesystem increase, the critical interaction strength k 0 F a increases. In addition, thecritical k 0 F a is consistent for all three evidences at the same temperature.
The critical value of k 0 F a we found is different from the value π/2 from the meanfield prediction in the section 3.1. The mean-field model assumed zero temperatureand homogeneous density. In the experiment, the temperature is finite and the density is inhomogeneous in the harmonic trap. In addition, we determined the k 0 F a
from total atom number assuming zero temperature without interaction, where thereal kF a depends on the local density at finite temperature with interaction. These
condition count for the discrepancy in the critical kF a. Theoretical works support
the discrepancy in the critical kF0 a by introducing the trap [37], Monte-Carlo simulation [11, 53], and the second-order correction to the mean-field interaction [15].
The most powerful evidence of the ferromagnetism would be the observation of
the spin domains. However, we were unable to observe the spin domains using in-situ
phase-contrast imaging. Based on the signal to noise ratio of ∼ 10 and the ∼ 3 µm
36Magnetic Field [G](a)Atom Loss Rate [1/s]

600

750

780

800

810

815

818

200
150
100
50
0
0

2

4

6

8

Interaction Parameter k˚F a

(c)
1.2

Atom Number

Atom Number

(b)
1.0

0.8

0.6

0.4

1.2
1.0
0.8
0.6
0.4

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

Hold [ms]Hold [ms]Figure 3-4: Atom loss rate. (a) Atom loss rate at different interaction strength k 0 F a
and at different temperatures of 0.55TF (dashed curve), 0.22TF (dotted curve), and
0.12TF (solid curve) Note that the curves are guides to the eye assuming that the loss
rate saturates. (b),(c) Loss rate determination.

37

Magnetic Field [G]
600

750

780

800

810

815

818

0.60

T/TF = 0.55

0.55
0.50
0.45

T/TF = 0.39

0.50

0.40

0.45

T/TF = 0.22

Kinetic Energy [ E˚F ]

0.45

0.40
0.35

Increasing Temperature

0.55

0.30
0.50

T/TF = 0.12

0.45
0.40
0.35

0

2

4

6

8

Interaction Parameter k˚Fa
Figure 3-5: Kinetic energy measurement. The kinetic energy at different interaction
strength k 0 F a and at different temperatures.

38

(a)

Magnetic Field [G]
600

750

780

800

810

815

818

1000

120
110

800

100
600

90
80

T/TF=0.55
105

750
700

100

650
95
600
90

550
500

T/TF=0.22

85

Axial Cloud Size (µm)

Chemical Potential [nK]

400

105

750
700

100

650
95
600

T/TF=0.12
0(b)Particle Number

600

2
750

780

4
800

6
810

815

90

8
818

1.0
0.8
0.6
0.4
0.2

Atoms+Molecules
Atoms

0.0
0

2

4

6

8

Interaction Parameter k˚Fa

Figure 3-6: Cloud size and molecular fraction. (a) Cloud size at different interaction
strength k 0 F a and at different temperatures. The chemical potential (b) Measured
molecular fraction in the sample we used.

39

resolution of our imaging system, the upper limit of the domain volume is ∼ 5 µm3
with ∼ 50 atoms.
The main difference between this experiment and the Stoner model is the existenceof the molecules in the sample. As in Fig. 3-6 (b), there is about 25 % of molecular
fraction in the sample. Even though the molecules exit, the molecular fraction of
the sample is constant from k 0 F a > 1.8 where we observed the evidences of thetransition. Thus, the molecular fraction is not the reason behind the sudden changes.
We confirmed this by conducting the kinetic energy measurement with about 60 %molecular fraction in the sample. We observed the minimum in the measured kinetic
energy at the same k 0 F a.
We measured the temperature by fitting finite temperature in-trap density profile
right after the ramp. Considering the trap frequency, 4.5 ms ramp is about a quarterof trap period, and it is only marginally adiabatic. Since the temperature fitting
should be done to a sufficiently equilibrated profile, this can be done more accurately.
In addition, at high field imaging, the atoms and molecules are indistinguishable [34].
Therefore, the density profile contains molecular density profiles resulting in errors.
Actually, the lowest temperature accurately measured at repulsive side was about 0.2
TF both in chapter 4 and in Ref. [57].

3.4

Further Development

A recent experimental study using 350 µs ramp to the target field observed that pairformation occurs on a very short time scale [57]. This work also showed the absence ofa dramatic increase in spin fluctuations, which is evidence against the ferromagnetictransition. A theoretical study supports this finding by showing that the pairing
growth rate is larger than that of ferromagnetic instabilities [51,65]. In addition, Ref.
[51] has shown that the pairing instability can lead to similar experimental signatures
as we have observed. In conclusion, it turned out that no ferromagnetic phase has
been formed in our experiment. Still, our work stimulated a lot of interest for ultracold
Fermi gases with repulsive interaction.
40

Chapter 4
Pauli Paramagnetism in an Ideal
Fermi Gas
In chapter 3, we focused on the qualitative features to observe the evidences of thephase transition. We measured thermodynamic quantities of the whole trapped gas
while the real atoms were held in a trapping potential and inhomogeneous. In thischapter, we focus on the more quantitative study of ultracold gases.
Using local density approximation, the ultracold gas can be locally described as ahomogeneous gas. That is, each local points in the gas can be considered as homogeneous systems with different conditions. Therefore, the trapped gas can provide lotsof useful information on the equation of state of the system.
A new level of quantitative comparison between theory and experiment was recently reached by careful measurements of density profiles from which the equation ofstate could be determined. These techniques were first proposed by Chevy [12] andBulgac [10] and implemented by Shin [61]. Further improvements [27, 28, 36, 44–46]
resulted in impressive accuracy without adjustable parameters. These results hingeon accurate measurements of the equilibrium atomic density distribution.
In these quantitative researches, ultracold atoms served as model systems for
exploring novel and unknown many-body physics. On the other hand, we can also
use the ultracold atoms as model systems to demonstrate well-known textbook physics
in a more ideal way than other systems.
41Here, we use the tunability of atomic interactions near Feshbach resonances tocreate a non-interacting Fermi gas with two components. This is an ideal realizationof a free electron gas (FEG) with spin up and spin down components, as assumed
in the simple theory of metals. We demonstrate how such an ideal Fermi gas will
respond to external magnetic fields, which is described by Pauli paramagnetism.
In all previous studies of paramagnetism, the magnetization was weak since theapplied magnetic field times the magnetic moment was much smaller than the Fermi
energy [32, 58–60]. In addition, the measurements even in simple metals revealed
major discrepancies to the predicted value due to interaction effects [32,58–60]. Withultracold atoms, we can easily realize the strong field case where the chemical potential
difference is larger than the Fermi energy and therefore fully polarizes the gas. In
addition, we can experimentally demonstrate Pauli paramagnetism in a truly noninteracting system which is exactly described by basic theory.

4.1

Pauli Paramagnetism: Ultracold Atomic Gas
vs Solid State System

Table 4.1: Comparison between a free electron gas and two-component Fermi gasesin the study of magnetism
Properties
Free electron gas
Two-component Fermi gases
MagnetizationµB (N↑ − N↓ )/V∆n = (N↑ − N↓ )/V

Magnetic field

External B field∆µ = µ↑ − µ↓Normalized SusceptibilityµB ∂(∆n/n)/∂(B/EF )∂(∆n/n)/∂(∆µ/EF )Pauli paramagnetism explains the magnetization of a free electron gas with two
spin states in the presence of an external magnetic field B neglecting the contribution
from the orbital motions. In the presence of an external magnetic field B, the energyof a particle is given byε=p2± µB B,
2m
42

(4.1)

(a)

(b)

Figure 4-1: Comparison between a free electron gas(a) and two-component Fermi
gases(b) in the study of magnetism. Curves show density of states for each spinstates g(E). (a) shows the response of free electron gases to the external magnetic
field B. (b) shows the imbalanced two-component Fermi gases.

43

with upper sign corresponding to the antiparallel spin state, the lower sign to theparallel spin state the magnetic field B. The total number N↑(↓) of electrons in eachspin state changes satisfying at zero temperature (T =0)Z µgg(E ± µB B)dE,

N↑(↓) =

(4.2)

∓µB B

where g(E) is the density of states and µg is the global chemical potential (see Fig.
4-1(a)). In an zero temperature ideal gas, the global chemical potential is the same2

2

as Fermi energy EF = h̄ (6π2mn)

2/3, where n = (n↑ + n↓ )/2. The magnetization M isthen given by
M = µB (N↑ − N↓ )/V = µB ∆n.

(4.3)

The susceptibility of the electron gas is defined as
χ=

∂∆n
∂M
= µB
.
∂B
∂B

(4.4)

In the limit of B → 0, the slice of energy thickness transferred ∆ is µB B. Thus, themagnetization isM = 2µB g(EF )∆ = 2µB g(EF )µB B.

(4.5)

Therefore, we obtain the low-field susceptibility
χ0 = 2µB 2 g(EF ) = 2µB 2

3 EF
.
2 n

(4.6)

In condensed matter physics, the normalized susceptibility χ is often used as a dimensionless quantity. The susceptibility is normalized using χ0 :1 EF ∂∆n1
χ = χ/χ0 == ∂3 n ∂µB B
3



∆n
n




/∂

µB B
EF


.

(4.7)

The last equation is valid because the average density n is constant.
Here we simulate this magnetism using an ultracold Fermi gas of 6 Li atoms. We
use two lowest energy states, |F = 1/2; mF = 1/2i and |F = 1/2; mF = −1/2i, to44

simulate two spin-states of an electron. As mentioned in chapter 3.1, one state cannot
be converted to the other state unlike electrons. That is, the number of atoms ineach state is fixed. Note that the real external magnetic field is not the source of the
population imbalance (magnetization). The spins here can be regarded nuclear spins,
hence the magnetic field here is only used to split the hyperfine state. Then, what is
the relevant variable in the ultracold atoms that corresponds to the external magnetic
field in the electron gas? In the two-component Fermi gases, the magnetization ∆n
is introduced by preparing different number of atoms in each state. The two different
pseudo spin states have different chemical potentials, µ↑ and µ↓ . In Fig. 4-1(b),we can observe the chemical potential difference between two states, ∆µ = µ↑ − µ↓ ,corresponds to the energy shift due to the external magnetic field in the electron gas(Fig. 4-1(a)).
If we compare the corresponding equations, the number in each spin state at T = 0
is given by

Z µg

Z µ̃g

↑(↓)

g(E ± ∆µ/2)dE,

g(E)dE =N↑(↓) =

(4.8)

∓∆µ/2

0

where µg↑(↓) is the global chemical potential of each component of gases. We defineµ̃g = (µ↑ + µ↓ )/2 and ∆µ = µ↑ − µ↓ . By comparing this with Eq.(4.2), we again findthat the chemical potential difference ∆µ corresponds to the energy shift due to themagnetic field in the electron gas.
Let us now consider the trapped two-component Fermi gases. As we discussed in
the previous section, we can treat each local position (~r) in the trap as homogeneous
systems with the local chemical potential µ↑(↓) (~r) = µg↑(↓) − V (~r) and the local densityµ↑(↓) (~r) using the local density approximation. Here, the chemical potential difference∆µ(~r) ≡ µ↑ (~r) − µ↓ (~r) is the same throughout the cloud as in Fig.4-3(b). However,
the meaningful measure of the chemical potential difference is its relative value tothe Fermi energy, or the global chemical potential. Therefore, we can still define andmeasure the normalized susceptibility of the ultracold atoms corresponding to theEq. 4.7 as

χ=∂

∆n
n




/∂

∆µ
EF



45



= ∂ ∆n /∂ ∆µ ,(4.9)

'HQVLW\

E

5DGLDOSRVLWLRQ
Figure 4-2: Comparison between a free electron gas and two-component Fermi gases
trapped in the spherical harmonic trapping potential. Figures at the top shows the
response of free electron gases to the external magnetic field B. g(E) is the densityof states. The figure at the bottom shows density distributions of two-component
ultracold Fermi gases at zero temperature, n↑ and n↓ . Each position in the trap
corresponds to the different magnetization under different strength of the field, Thisis illustrated in diagrams inside the plot, where the gaps between two states represent
non-interchangeability of spin state.

46

where we define ∆n = (n↑ − n↓ )/n and ∆µ = (µ↑ − µ↓ )/EF . For fully polarized gas,∆n = 2 and ∆µ = 22/3 . In the trapped gases, ∆n and ∆µ changes throughout thecloud as in Fig. 4-3. Therefore, we can determine the normalized susceptibility χ
by measuring the slope from ∆n versus ∆µ as in Fig. 4-3(d). The zero temperaturenormalized susceptibility in the limit of ∆µ → 0 (which is equivalent to ∆n → 0) isEFχ0 =∂(∆n)/∂
n

4.2

2/3

h̄2 (6π 2 )
2m

!

3
(n + ∆n)2/3 − (n − ∆n)2/3= .
2(4.10)

Experimental Procedure

An evaporative cooled imbalanced mixture of |1i and |2i spin states is prepared at
528 G where the interaction between two states is zero, as described in chapter 1.4.
In order to obtain the density profiles of each component of atoms, we use thedouble-shutter phase contrast imaging technique with two different frequencies. One
is tuned to the middle of the two optical transition frequencies for |1i and |2i and theother is tuned closer to the state |1i by 12 MHz.
The normalized phase-contrast signal (for negligible absorption) is√I= 3 − 2 2 cos(φ ± π/4),
Io(4.11)δω−ωo3 2where the phase shift φ = − ñσ2 o 1+δ
, and ñ is the column
2 , σo = 2π λ , δ =Γ/2density [33]. The sign depends on laser detuning and the sign of the phase shiftimparted by the phase plate. Using Eq. 4.11, we can determine the phase shift fromthe normalized phase-contrast signal. The phase shifts from two different frequencies
are
φ1 =
φ2 =

σo δo
(ñ1 − ñ2 )2 1 + δo 2


σo
δ1
δ2
ñ1 −
ñ2 .
2 1 + δ1 2
1 + δ2 247(4.12)(4.13)

Density (1012cm-3)

1.0

(a)

0.8
0.6
0.4
0.2
0.0
0.3

2

(10-6 K)

(10-6 K)

(b)
0.2

0.1

0

(1012 cm-3)

(c)

2

0.3
0.2

1

(1012 cm-3)

0.0

0.1
0.0
-20

0
20

0

(1012 cm-3)

Radial position (µm)
2.0

(d)

1.5
1.0
0.5
0.0
0.0

0.5

1.0

1.5

2.0

(10-6 K)

Figure 4-3: (a) Simulated density profiles of two-component Fermi gases in a sphericalharmonic trap at zero temperature. The total population imbalance ∆ ≡ (N↑ −N↓ )/(N↑ + N↓ ) = 21%. The shaded and white regions correspond to the partiallyand fully polarized regions respectively. (b) The local chemical potential differencewith and without the normalization. (c) The density difference with and withoutnormalization. (d) ∆n versus ∆µ from the density profile (a). The linear fit near theorigin (the dashed line) gives the low field normalized susceptibility.
48

e
Probe
beam

2
12 MHz1
Figure 4-4: The double-shutter phase contrast imaging with two different imaging
frequencies.
From this equation, we can find find the column density of each spin state.
δo
If we normalized the phase shift by σ2o 1+δ
2 , we can simplify the equation aso

φ1 =ñ1 − ñ2

(4.14)

φ2 =αñ1 − β ñ2 .

(4.15)

We know the value of α and β from calculation, but we can also determine themexperimentally by taking images of the pure |1i and |2i states.

4.3

Susceptibility Measurement

We measured line-of-sight integrated profiles of column density using in situ phasecontrast imaging [64]. The signal-to-noise was improved by averaging the four quadrants of the images around the center and by averaging the column density alongequipotential lines (which are ellipses for the anisotropic harmonic oscillator potential).
Fig.4-5(a) is the column density distributions in two states, where the ellipticaldensity profile is rescaled to be circular. These column density profiles are fitted with49

finite-temperature Fermi gas 2D distribution from Eq. 2.12,m(kB T )2µ − V (r).
ñ(r) = −Li2 − expkB T
2πh̄3 ωz

(4.16)First, we fit the column density profile of the majority component, | ↑i, and determine
the temperature T and the global chemical potential µg↑ . Since the temperature Tof the two components is the same, we fit the minority component to determine µg↓with the temperature constrained to be the temperature determined from the majority component fitting. Three-dimensional density profiles n(r) in Fig.4-5(b) werereconstructed by applying the inverse Abel transformation to the column densitiesñ(r) [64].
The temperature was about 0.2 TF . For finite but low temperatures, the low fieldsusceptibility isχ = χ0

π2
1−
12



T
TF


(4.17)

from the Sommerfeld expansion [49].
As we can see in Fig. 4-5(c), we cannot use the derivative of ∆n with respectto ∆µ at each position because of the noise. Thus, we use the linear fit of ∆n as afunction of ∆µ in order to determine the susceptibility. We accumulated ∆n vs ∆µplots from 20 images for averaging (see Fig. 4-5(c)). The linear fitting to the origin
near the center of the profile gives χ = 1.44, which is in a good agreement with thetheoretical value of 1.45 from Eq. 4.17 at T /TF = 0.2.
Note that the local T /TF and the local susceptibility changes within the fitting
region as in Fig. 4-3(d) because the density n and the normalized chemical potential
difference ∆µ changes within the fitting region. Thus, we limited the fitting region towhere the deviation of the susceptibility is small within our experimental resolution.
From simulation, the deviation in measured susceptibility from that of the center ofthe profile is about 4 %.
Even small difference in T /TF at the center of the profile gets amplified near thewings of the profile as TF decreases. In addition, the temperature dependence of thesusceptibility is beyond the second order correction from the Sommerfeld expansion.
50

Figure 4-5: (a) Column density ñ. The temperature determined from the fit (solid
curves) is about 0.2 TF . (b) Red line shows difference of three dimensional densities∆n. Blue line shows average three dimensional density n0 of the two states. Greenlines are ∆n and n0 from theoretical calculation using the chemical potentials andthe temperature obtained from (a). (c) The normalized three dimensional density
difference is plotted as a function of the normalized chemical potential difference.
The graph shows averaged plots from 20 profiles. We determine the susceptibility
from the slope of the linear fitting to the origin. The measured χ is 1.44.

51

Figure 4-6: Susceptibility at high magnetic field. Points are data from a single image.
Dashed line shows the plot from a simulation using the same parameters as theexperiment. Solid line is the linear fitting of the data for susceptibility measurement
around ∆µ = 1.

Therefore, we cannot use accumulated plots for susceptibility measurement near thewings of the profile, which corresponds to the susceptibility at strong magnetic field.
In order to measure the strong field susceptibility, we use a ∆n vs ∆µ plot from asingle profile (see Fig. 4-6). The linear fit around ∆µ/EF = 1 gives χ = 0.39. Within
the fitting range, T /TF changes from 0.45 to 0.7. The susceptibility obtained from asimulation with the same parameters is 0.48. About 6 % error comes from the T /TF
and ∆µ changing fitting region. We observe that the susceptibility approaches zeroas the chemical potential difference (corresponding the applied magnetic field times
the magnetic moment) becomes comparable to the Fermi energy of the system.
We measured the ideal Pauli susceptibility in ultracold Fermi gases. Since we
can control the interaction strength between two pseudo-spin states to be zero, it
is possible to measure the Pauli susceptibility accurately while the experiments incondensed matter physics observed major discrepancies due to interaction effects. In
addition, the strong field Pauli susceptibility can be measured where the chemical
potential difference is comparable to the Fermi energy. This shows that the ultracold
atoms can be a good tool to study basic physics of many-body systems.
52

4.4

Susceptibility Measurement in Systems with
Interaction

In the previous section, we measured the susceptibility of the ideal gas. Let us now
consider the interacting case. Recently, the susceptibility was measured for fermions
with attractive interactions [45]. The susceptibility measurements on the repulsive
side of interaction could address the possible existence of a ferromagnetic transition
in a repulsive Fermi gas (in chapter 3 for which the spin susceptibility would diverge
at the phase transition [54].
However, there are some complications with the repulsive side of interaction.
First, the two-component Fermi gases with repulsive interaction are unstable because they are realized on the so-called upper branch of a Feshbach resonance where
decay is always possible into the so-called lower branch which consists of weakly boundmolecular states. In order to observe the equilibrated density distribution, a favorable
ratio of lifetime to equilibration time is required. Since the lifetime depends on theinteraction strength, we have the upper limit for the interaction strength where weare able observe the equilibrated density profile.
Second, the achievable lowest temperature is much higher than that of attractive
interaction case. The interaction strength between atoms at the preparation stage
is much smaller in repulsive case, so that thermalization during evaporation is lesseffective. Then, why is this a problem? In Ref. [45], they used Thomas-Fermi radiusin order to determine the global chemical potentials for each state. It was possible
because the temperature is an order of magnitude lower than our case. Determining
Thomas-Fermi radius only requires finding the radius where the density becomes zero.
In finite temperature case, on the other hand, we need to fit at least some range ofthe profile in order to determine the global chemical potential. For non-interacting
case, it is easy because we know exact function of the density profile. However, forstrongly interacting case, the exact equation of state is unknown, so that we needto fit the wing of the cloud where the interaction is small enough for the known
perturbation correction. In addition, the temperature can not be measured at the53

fully polarized wings as in Ref. [45] because the finite temperature profiles don’t show
the fully polarized features clearly, due to the extended wings.
In the next chapter, we check the upper limit for the interaction strength where
we are able observe the equilibrated density profile. Then, we measure the compressibility of the Fermi gas with repulsive interaction. The compressibility measurement
does not require the population imbalance nor the absolute value of the global chemical potentials. Thus, the compressibility can be measured more simply than thesusceptibility when we introduce the repulsive interaction.

54

Chapter 5
Compressibility of an Ultracold
Fermi Gas with Repulsive
Interactions
As we reviewed in chapter 1.2, the two-component Fermi gases with repulsive interaction are unstable because they are realized on the so-called upper branch of a
Feshbach resonance where decay is always possible into the so-called lower branchwhich consists of weakly bound molecular states. In order to observe the equilibrated
density distribution, a favorable ratio of lifetime to equilibration time is required.
Since the lifetime depends on the interaction strength, we have the upper limit for
the interaction strength where we are able observe the equilibrated density profile.
Here we check the upper limit for the interaction strength where we are able
observe the equilibrated density profile. Through this we test the feasibility of experiments requiring equilibrated density profiles of ultracold Fermi gases with repulsive
interactions.
For weakly interacting gases, interaction effects can be explained by perturbative
corrections (see chapter 2.4). For bosons, the first beyond mean-field correction, theso-called Lee-Huang-Yang term, could be observed, but corrections were necessary to
account for the non-equilibrium profile since the time to sweep to strong interactions
was not long compared to equilibration times and inverse loss rates [47].
55Here we study fermions with repulsive interactions. Within a limited window
for metastability, we can observe equilibrated clouds and characterize the repulsive
interactions by obtaining the compressibility from observed profiles. We observe thelinear term in kF a corresponding to mean-field energy for the first time in density
profiles. In contrast to a Bose-Einstein condensate, the mean-field energy is smaller
than the kinetic energy and also competes with thermal energy, and is therefore much
more difficult to observe. The signal-to-noise ratio (and some heating) prevented
us from discerning the second-order interaction term which is the Lee-Huang-Yang
correction for fermions.
Our work features one technical novelty, a novel implementation of phase-contrast
imaging to address dispersive distortions of the cloud. All studies mentioned above,with one exception [61], were conducted using resonant absorption where dispersion(an index of refraction different from one) is absent. However, this severely limits
the cloud size and number of atoms to optical densities of a few. Phase-contrast
imaging has many advantages. It can be applied to clouds with much larger opticaldensities by adjusting the detuning. Due to coherent forward scattering, the heating
effect per detected signal photon is reduced by potentially a large number (which isequal to the resonant optical density divided by four [33]). This can e.g. be used for
repeated nondestructive imaging. However, for precision studies of density profiles,
small dispersive distortions of the density profile cannot be neglected. Previous work
including Ref. [61] was not sensitive to this effect. We have developed an experimental
technique to correct for dispersion.
The experiment was reported in the publication:• Ye-Ryoung Lee, Myoung-Sun Heo, Jae-Hoon Choi, Tout T. Wang, Caleb A.
Christensen, Timur M. Rvachov, and Wolfgang Ketterle
Compressibility of an ultracold Fermi gas with repulsive interactions
Phys. Rev. A 85, 063615 (2012). Included in appendix E.
56

5.1

Experimental Procedure

An evaporative cooled balanced mixture of |1i and |2i spin states is prepared at 528
G. In order to introduce repulsive interaction, we increase the magnetic field.
For loss rate measurements, the magnetic field is quickly ramped to the target
field. For compressibility measurements, the field is ramped up over 50 ms and heldfor 30 ms to ensure thermal equilibrium before imaging. The molecular fractionin the density profile is determined by dissociating molecules with a magnetic fieldjump to 20 G above the Feshbach resonance, and comparing with the atom number
after jumping to 528 G where the cross section for imaging molecules vanishes (seeRef. [74]).

5.2

Loss Rate

In this section, the feasibility of the experiment is tested through loss rate measurement. As expected we find only a limited window for metastability where we canobserve equilibrated clouds and characterize the repulsive interactions by obtaining
the compressibility from observed profiles. For interaction parameters kF a > 0.25
fast decay of the gas prevents the observation of equilibrium profiles.

5.2.1

Loss Rate Estimation

To obtain thermodynamic parameters from atomic density profiles requires equilibration. One time scale for equilibration is set by the longest trap period, which is 30 ms
for the axial direction. Ref. [47] studied the distortions of profiles of bosonic 7 Li due
to non-adiabatic ramping of the scattering length. The authors found that ramping
times longer than ≈ 6π/ω (which is 90 ms for our parameters) led to only small deviations from equilibrium profiles. Here, ω is the smallest of trapping frequencies, whichsets the timescale for equilibration. Assuming that losses sharply increase during thelast 5 ms of the ramp towards stronger interactions and limiting tolerable losses to
10 % leads to an estimate of a maximum tolerable loss rate of about 0.02/ms. The
57

fastest relaxation time for excitations created during a non-adiabatic ramp is 1/ω
(the damping time for a harmonic oscillator at critical damping). Allowing ∼ 10 %loss during this time, leads to an identical estimate for the tolerable loss rate of ∼0.02/ms.

5.2.2

Loss Rate Measurement

We measured loss rate as a function of interaction strength kF 0 a. Note that kF 0 is the
Fermi wave vector of the zero temperature noninteracting gas calculated at the trap
center using total atom number. The real kF is somewhat smaller because repulsive
interactions and non-zero temperature lower the density.
First, in Fig. 5-1(a) we measured the number of atoms right before and after the
fastest possible ramp (limited to 7 ms by eddy currents) to the target magnetic field.
During the ramp to the target magnetic field of kF 0 a ≈ 0.8, ∼ 35 % of the sample islost. Measuring the loss rate at higher kF 0 a is difficult because most of the sample islost even before reaching the target fields. The loss rate was determined by monitoring
the drop in the number of atoms immediately after the field ramp. The results in
Fig. 5-1(b) show that the measured loss rate reaches the maximum tolerable valueof 0.02/ms at kF 0 a ≈ 0.35, limiting our measurements of equilibrium density profiles
to smaller values of kF 0 a. Furthermore, at the same values of kF 0 a, the molecular
fraction when we measure equilibrium density profiles exceeded 10 % (see Fig. 5-1(c)).
As we shall discuss in section 5.5 the presence of a molecular component affects thecompressibility measurement.

5.3

Compressibility Measurement

A system is characterized by its equation of state. The equation of state can be expressed in different forms involving various thermodynamic variables including density, energy, pressure, temperature and entropy. For cold atom experiments, density,
chemical potential (through the trapping potential) and temperature are directly accessible to measurement. In the weakly interacting regime, the interaction manifests
58

Figure 5-1: Characterizing atomic loss for increasing repulsive interactions. (a) Remaining fraction of atom number immediately after the fastest possible ramp to thetarget field. (b) Atom loss rates at the target fields. Dashed line shows the estimatedmaximum tolerable loss rate of 0.02 /ms. (c) Molecular fraction after the 50 ms ramp
and 30 ms wait time, corresponding to when we measure equilibrium atomic density
profiles.
59itself as a perturbative term in the equation of state. Here we measure this perturbative interaction effect by measuring the derivative of density with chemical potential,
the isothermal compressibility of the gas. We prepare the system at the lowest temperature, but due to heating of the cloud by molecule formation, we have to apply acorrection for the measured finite temperature.

5.3.1

Measurement Method

Using the experimental procedure discussed above we prepared equilibrated clouds
at various magnetic fields and measured line-of-sight integrated profiles of column
density using in situ phase-contrast imaging [64]. The signal-to-noise was improvedby averaging the column density along equipotential lines (which are ellipses for the
anisotropic harmonic oscillator potential). The averaging region was restricted toan axial sector of ±60◦ to avoid corrections due to transverse anharmonicities [63].
Three-dimensional density profiles n(r) were reconstructed by applying the inverse
Abel transformation to the column densities ñ(r) [64].
The isothermal compressibility is obtained from a spatial derivative of n(r), sincein the local density approximation, the local chemical potential is µ = µo − mω 2 x2 /2,
where µo is the global chemical potential and ω the trap frequency. The compressibility is defined as
κ=

1 ∂n
.
n2 ∂µ

(5.1)

We normalize the compressibility κ by the compressibility of an ideal gas at the
1/3

3m
given density and zero temperature κo = nn2 h̄2 (6π, and obtain the normalized
2 )2/3compressibility,
κ̃ =

κ
=
κo

h̄2 (6π 2 )2/3 ∂n2/3
.
2m
∂µ

(5.2)

Here n is density, and m is the atomic mass. The normalized compressibility is
obtained as the slope in a plot of n2/3 vs. µ (Fig. 2). This plot is in essence the60observed density profile plotted with the central region to the left and the spatial
wings to the right. Experimentally, we find the slope to be constant over an extended
range of the density profile. Compressibilities were determined from fits to the slope
in the region of 90 to 50 % of the peak density. The region near the center of the cloud
was excluded since the center is singular for the inverse Abel transformation leading
to excessive noise. These compressibilities should be regarded as average values overthe density range used in the fit. The uncertainty of fitting the slope to a single profile
was 4.5 %. By averaging the profiles obtained from 20 images, the uncertainty was
improved to 1.3 %.

Figure 5-2: Determination of the compressibility of repulsively interacting Fermi
gases. The compressibility is the slope of a graph showing the density to the power
2/3 versus the square of the distance from the center. (a) Single shot density profile.
(b) Close-up of the region used for determining the compressibility for a plot showing
the average of 20 density profiles.

61

5.3.2Temperature Measurement

The normalized compressibility is a function of T /TF and kF a. At finite temperatureand scattering length a, T /TF and kF a change across a single density profile because
TF and kF depend on the local density. However, this density dependence is small
near the center of the cloud. Simulated density profiles showed that the averagecompressibility determined in the way described above agrees to within 0.6 % withthe compressibility at T /TF and kF a at the density in the center of the selected range.

Compressibility decreases for stronger repulsion, but also for higher temperature.
To identify the effect of repulsive interaction requires a careful consideration of finite temperature effects. First, the temperature of the cloud had to be accurately
determined. This can be done without any special assumptions by fitting the wingsof the cloud using a virial expansion [67], by thermometry with another co-trapped
atom [44], or for population imbalanced clouds by fitting the wings of the majority
component which is an ideal Fermi gas [73]. Here we chose to determine temperatureusing a virial expansion,p

λ3= eβµ + b2 e2βµ + O(e3βµ ),
kB T

(5.3)

q
2πh̄2
where λ= mk
is the thermal de Broglie wavelength, b2 is the virial coefficient, andBTeβµ is the fugacity. The virial coefficient for the Fermi gas with repulsive interaction
is b2 = −2−5/2 − a/λ [39]. Pressure p was obtained from the doubly-integrated
density profiles [27]. Temperature was determined in the wings of the profile where
βµ < −0.5. Here, the temperature measured with and without the interaction term−a/λ in b2 differ by about 3%. This suggests that higher-order corrections frominteraction term will be negligible. Note that the virial expansion up to second orderis valid to within 1% for the ideal gas at the density of the fitted wings.
62

5.3.3

Interaction Effect

The low temperature normalized isothermal compressibility of a non-interacting Fermi
gas is given by the Sommerfeld expansion [68],
π2
1−
12

κ̃o,T =



T
TF

"

2
+O

T
TF

4 #
,

(5.4)

where T is temperature and TF is Fermi temperature. To add the effects of interactions, it is useful to work with the inverse normalized compressibility,
32m∂µ(n, T, a)
.
n1/3
2
2
2/3
2 h̄ (6π )
∂n

1
=
κ̃

(5.5)

This is a derivative of the chemical potential, which has the following expansion intemperature and scattering length,"µ(n, T, a) = EF

π2
1−
12



T
TF

2

#
4
4(11 − 2 ln 2)+
kF a +(kF a)2 + CT 2 a2 ,
3π
15π 2

where C is constant, independent of density [50]. Therefore the inverse normalized
compressibility has additive correction terms for temperature and interactions up to
the second order of the interaction effect,
1
=
κ̃

1
κ̃o,T

+ Y (kF a).

(5.6)

This equation defines Y (kF a), the interaction correction to the inverse compressibility.
This term is the derivative of the interaction term of the chemical potential. In second(kF a)2 .
order perturbation theory, one obtains Y (kF a) = π2 kF a + 8(11−2ln2)
15π 2Fig. 5-3 shows the normalized compressibility, the temperature T /TF , and theinteraction correction to the inverse compressibility. The temperature increases withkF a due to the increase in three-body recombination where the binding energy of themolecules (h̄2 /ma2 ) is transferred to the remaining atoms. The measured temperature
is higher than that in previous experiments on the repulsive side [31, 57]. This differ63

Figure 5-3: Measured temperature, normalized compressibility, and interaction effecton compressibility at various interaction strengths. (a) The measured temperature
as a function of interaction strength. (b) Solid circles show the measured normalized
compressibility (κ̃) and open squares show the calculated normalized compressibility
at the measured temperature without interaction (κ̃o,T ). The difference between thetwo indicates the effect of interaction. Open triangles show the calculated normalized
compressibility using the second order perturbation theory, which is consistent with
our measured κ̃. (c) The measured interaction correction to the inverse compressibility(solid circle) is compared to a linear fit (solid line), and the first (dashed line) andthe second (dotted line) order perturbative result.

64

ence can be explained by a smaller kF a since the increase in T /TF is approximately
proportional to 1/kF a [57].
We perform a linear fit of the interaction effect Y (kF a) versus kF a (constrained
to pass through the origin) and obtain 0.680±0.147 for the slope, in agreement withthe perturbative prediction of π2 = 0.637. This is the first observation of the meanfield term for repulsively interacting fermions in a thermodynamic quantity. The
repulsive interaction has been seen as line shifts in RF spectroscopy experiments(which, in contrast to many thermodynamic quantities, are measured independentlyof the kinetic energy of the atoms) [24, 55]. In principle, it is possible to obtainthe mean-field term directly by fitting the density profiles with an extra mean fieldterm. In such fits, we obtained clear evidence for such a term, but with low accuracy.
It appears that the averaging over profiles for determining the compressibility(as inFig. 5-2) is superior. Figure 5-3 (c) shows the predicted effect of the second-orderterm on Y (kF a). With some improvements in signal-to-noise ratio, one should be
able to observe this term which is analogous to the Lee-Huang-Yang correction forbosons.

5.4

Dispersive Effect in Phase-Contrast Imaging

Phase-contrast imaging has several advantages over resonant absorption imaging.
However, for precision studies of density profiles, small dispersive distortions of thedensity profile cannot be neglected. Previous work including Ref. [61] was not sensitive to this effect. We have developed an experimental technique to correct for
dispersion.
Our work features one technical novelty, a novel implementation of phase-contrast
imaging to address dispersive distortions of the cloud. All equation of state studies,with one exception [61], were conducted using resonant absorption where dispersion(an index of refraction different from one) is absent. However, this severely limits
the cloud size and number of atoms to optical densities of a few. Phase-contrast
imaging has many advantages. It can be applied to clouds with much larger optical65

densities by adjusting the detuning. Due to coherent forward scattering, the heating
effect per detected signal photon is reduced by potentially a large number (which isequal to the resonant optical density divided by four [33]). This can e.g. be used for
repeated nondestructive imaging. Thus, the phase-contrast imaging has been applied
to many studies of cold Bose and Fermi gases [3,64,69]. The standard assumption has
been that dispersive imaging is quantitative when the phase shift φ across the cloudis less than π/4. The normalized phase-contrast signal (for negligible absorption) is√
3 − 2 2 cos(φ ± π/4), which is equal to 1 ± 2φ for small phase shifts [33]. The signdepends on laser detuning and the sign of the phase shift imparted by the phase plate.

5.4.1

Phase-Contrast Imaging

All equation of state studies, with one exception [61], were conducted using resonant
absorption where dispersion (an index of refraction different from one) is absent.
However, this severely limits the cloud size and number of atoms to optical densities
of a few. Phase-contrast imaging has many advantages. It can be applied to clouds
with much larger optical densities by adjusting the detuning. Due to coherent forward
scattering, the heating effect per detected signal photon is reduced by potentially alarge number (which is equal to the resonant optical density divided by four [33]).
This can e.g. be used for repeated nondestructive imaging. Thus, the phase-contrast
imaging has been applied to many studies of cold Bose and Fermi gases [3, 64, 69].
The standard assumption has been that dispersive imaging is quantitative when thephase shift φ across the cloud is less than π/4. The normalized phase-contrast signal√(for negligible absorption) is 3 − 2 2 cos(φ ± π/4), which is equal to 1 ± 2φ for small
phase shifts [33]. The sign depends on laser detuning and the sign of the phase shiftimparted by the phase plate.

5.4.2

Compensation for Dispersive EffectHere, we apply phase-contrast imaging for rather precise quantitative studies of ultracold Fermi gases and found that even for small phase shifts systematic dispersive66

distortions of the image cannot be neglected. Phase-contrast imaging relies on column density dependent phase shifts. However, if the object is not thin, but extended
along the line of sight, some lensing will affect the images. These distortions should
vary inversely proportional to the probe light detuning and become negligible for far
detuning.

e
Probe
beam
Ո<0

Ո>0
2

76 MHzՈ>01

Ո<0Figure 5-4: Phase-contrast imaging of a balanced spin mixture in states |1i and |2i.
White images (phase shift φ >0) were obtained for a probe beam red-detuned fromthe |2i → |ei transition, corresponding to δ < 0. Black images (φ <0) were obtained
for a probe beam blue-detuned from the |1i → |ei transition, corresponding to δ < 0.

We investigated positive and negative detuning of 40, 80, and 120 MHz (seeFig. 5-4). The normalized phase contrast image had a maximum signal of 0.35/1.85,
0.55/1.6, and 0.7/1.4 for the three positive/negative detuning. The lensing effect is
opposite (focusing vs. defocusing) for positive and negative detuning, and can therefore be identified by comparing profiles obtained with positive and negative detuning.
Fig. 5-5 shows that at 40 MHz, the two profiles show a visible difference, but for
profiles at 80 MHz and 120 MHz, the differences are smaller than the noise level.
However, the compressibility is determined by the slope of the profiles and very
sensitive to small distortions even if they are not perceptible in the profiles. Fig. 5-5(c)shows that even at 120 MHz detuning, the compressibilities obtained from profiles
with the two signs of the detuning differ by about 10 %. Since further detuning would
have resulted in a smaller signal we evaluated the average value of the compressibility
for positive and negative. When the dispersive distortions are small, the effect on
the compressibility should be a first-order effect in the phase shift and cancel for67

Figure 5-5: Comparison of column density profiles obtained from positive and negative
detuning of (a) 40 and (b) 80 MHz, respectively. (c) The measured compressibility (atkF a = 0) from positive (open triangle) and negative (open square) detuned profiles
and their averages (solid circle) are shown. The average value stays constant above
80 MHz detuning.

68the average. Indeed, the average value stays constant above 80MHz detuning. Our
conclusion is that for reasonable signal levels (i.e. 50 % of the baseline set by the probe
light) dispersive effects are relevant for quantitative studies, but can be eliminated by
performing averages over positive and negative detunings. The best solution would
be using double shutter imaging with the positive and negative detunings. Then, we
can find both positive and negative detuned profiles from a single atomic sample, sothat we can correct the dispersive effect in each profile. However, this is technicallycomplicated because it requires about 240 MHz imaging frequency change in about
10 µs. We use tandem AOM setup for double shutter imaging, but the frequency
range is about 80 MHz with 80 MHz AOMs.

5.4.3

Imaging Focus Fine Tuning

We could take advantage of the lensing effect for imaging focus fine tuning. Since
positive and negative detuning result in opposite lensing effect, we apply symmetrically positive and negative detuning from the resonance frequency to a single state
sample using double shutter technique. We used two steps to tune the focus. For
coarse tuning, we used the aspect ratio as a parameter. For our elongated cloud,negative and positive detuning gives different aspect ratio when the image is out offocus. The aspect ratio is the same when the image is in focus. For fine tuning, we
divided profiles obtained with negative and positive detuning. The divided profile isflat in focus and has slope or curvature out of focus. Note that the divided profile isnot perfectly flat when the detuning is large because the dispersive effect exists evenat the perfect focus position.

5.5

Discussion

5.5.1

Molecular Fraction in the Density Profile

We address now to what extent a small molecular fraction contributes to the observed
density profiles. The presence of molecules is unavoidable since they form during the69

Figure 5-6: Imaging Focus Fine Tuning. We divided the black and white image of asingle state sample. (a) shows the flat profile at the focus, and (b) shows slope outof focus.

ramping and equilibration time. At the highest magnetic field used in the experiment,
679 G, the molecular fraction was determined to be ∼ 10 % (Fig. 1). Ref. [4] reported
that molecules at 650G showed an absorption cross section of about half the value ofthe atoms for probe light at the atomic resonance. For phase-contrast imaging withlarge detuning molecules and atoms should contribute equally to the signal.

We performed simulations to address how the presence of molecules would affectthe compressibility measurements. We considered as possible scenarios (i) that themolecular fraction is constant throughout the cloud, (ii) that the molecular fraction is
proportional to the loss rate (n8/3 ), and (iii) that the molecular fraction is well equilibrated at the same temperature as atoms. The atomic profile is then the differenceof the measured density profile minus the simulated molecular density distribution.
Scenario (iii) is ruled out since it would result in a rather sharp peak in the density
profile which was not observed. The first two scenarios with a 10 % molecular fraction
resulted in a value for the normalized compressibility which was increased by 3.3 %and 4.4 % respectively. This shows that for our largest value of kF a the presenceof molecules starts to become a systematic effect. In addition to the contribution
to the density profiles, molecules can affect the atomic density distribution throughatom-molecule interactions.
70

5.5.2

Interaction Strength Limitation

Lower density
Our work shows that the interaction effect on the compressibility at the maximum
possible values of kF a is about 15 %. We could identify this effect only by carefulthermometry (to distinguish it from thermal effects) and by correcting small dispersive distortions of the cloud. It is desirable to study fermions for stronger repulsive
interactions where stronger and non-perturbative effects are predicted. The maximum possible kF a value for obtaining equilibrium density profiles is determined by
the loss rate which is proportional to n2 a6 max(T, TF ) = (kF a)6 n2/3 [18, 52]. Therefore, the maximum possible kF a for a given loss rate is proportional to n−1/9 andstronger interaction effects can be seen at lower density. This should be accomplishedby reducing the radial confinement and not the axial confinement which determines
the equilibration time. However, the weak density dependence will allow only modest
increases in kF a. A recent experiment used density ten times smaller than ours [57]and reported ramping from kF 0 a = 0 to kF 0 a = 0.35 in 500 ms losing only 5 % atoms.
Assuming the loss happened during the last 50 ms, we can roughly estimate a loss
rate of ∼ 0.001 ms−1 at kF 0 a = 0.3 which is lower than our measurement, consistent
with the lower density.

Narrow Feshbach ResonanceLonger lifetimes for a given kF a should be realized using narrow instead of broad
Feshbach resonances. For narrow Feshbach resonances the low-lying molecular state
has a dominant closed-channel character. Therefore, three body recombination of
atoms (which are in the open channel) has a smaller overlap to the molecular state
and therefore a reduced loss rate. Recent experiments using RF spectroscopy [26, 35]
confirm this. However, for such narrow resonances the zero-range approximation is
no longer valid, the interaction is no longer described by the scattering length alone
and becomes (through an effective range parameter) momentum dependent. As aresult, the narrow Feshbach resonances realize a different Hamiltonian.
71

5.5.3

Outlook

We have addressed the question to what extent Fermi gases with strong interactions
can be studied by observing equilibrium density profiles. The range of sufficiently longmetastability to reach equilibrium is limited to values of kF a < 0.25. In this range,interaction effects are comparable to thermal effects, but we were able to observe howinteractions reduce the compressibility and obtained quantitative agreement with thefirst-order mean field term. An observation of the second order Lee-Huang-Yang
correction is in experimental reach.
If experiments can be performed at stronger interactions, a natural extension of
our work would be a measurement of the spin susceptibility using population imbalanced Fermi systems. This was performed recently for fermions with attractive
interactions [45]. Such measurements could address the possible existence of a ferromagnetic transition in a repulsive Fermi gas [31] for which the spin susceptibility
would diverge at the phase transition [54].

72Chapter 6
Conclusion
Ultracold Fermi gases with repulsive interaction live on an excited branch of thestates coupled by a Feshbach resonance. This branch is unstable against molecularformation. Nevertheless, the theoretical predictions of itinerant ferromagnetism in a
two-component Fermi gas with repulsive interaction drew attention to this excited
branch. By simulating the Stoner model, we observed the nonmonotonic features in a
two-component Fermi gas near a Feshbach resonance, which agree with signatures offerromagnetic transition. These features were not definitive evidences, and we were
not able to observe ferromagnetic domains, the direct evidence of ferromagnetism.
Even though it turned out later that the features were not caused by ferromagnetictransition, this pioneering experiment attracted attention to the repulsive side of the
Feshbach resonance.
Since the spin susceptibility would diverge at the ferromagnetic phase transition,
quantitative measurement of spin susceptibility could provide a convincing evidence
for the ferromagnetic phase. This lead us to quantitative study of ultracold Fermi
gases with repulsive interaction. First, we started with a non-interacting case. We
could simulate the free electron gas system with ultracold atoms and measured susceptibility of ideal gas. This measured susceptibility showed excellent agreement withthe calculation. This couldn’t be done in condensed matter experiments because eventhe simplest materials still include defects or complicated interaction effect.
Then, we introduced weak repulsive interaction to the system. We checked possible73window for metastability, which turned out to be kF a < 0.25. Within this window,
we could observe equilibrated clouds and characterize the repulsive interactions by
obtaining the compressibility from observed profiles. We observe the linear term in
kF a corresponding to mean-field energy for the first time in density profiles. The
maximum possible kF a for a given loss rate is proportional to n−1/9 and stronger
interaction effects can be seen at lower density. The density dependence may be
too small to reach sufficiently strong interaction, but at least an observation of thesecond order Lee-Huang-Yang correction is in experimental reach. If it is possible toreach sufficiently strong interaction, measuring the spin susceptibility could provide
possible evidence of a ferromagnetic phase transition.
One of the important findings of this thesis work is the dispersive effect in thephase-contrast imaging. We found that the dispersive effect was inevitable even atvery large detunings for quantitative studies. The averaging method correcting forthe dispersion could be improved by double shutter imaging. This would require fastswitching technique between two probe beam frequencies which are far detuned toeach other.

74

Appendix A
Tips for Precise Imaging
I list a few tips for precise imaging which can be easily overlooked.

• Fine tweak the polarization of the imaging beam.
• Block all stray light along the beam/image path.
• Carefully find the center of the phase spot, so that the phase plate does notcreate any fringes.
• Tilt all the optics slightly, so that no back reflection creates any fringes.
• Be careful with the imaging beam intensity (see details in the following section).

A.1

Imaging Beam Intensity

For careful imaging of atom clouds, we need to be careful with the intensity of theprobe beam and the total number of photons absorbed by each atom during probing.
The intensity of the light has to be much smaller than the saturation intensity,
ISAT . For resonant probe beam, the saturation intensity is
ISAT =
75

h̄ωΓ
,
2σo

(A.1)

3 2
where σo = 2π
λ . Since Γ is 2π× 6 MHz for lithium, ISAT is about 2.6 mW/cm2 . Forprobe beam with detuning δ (as in phase-contrast imaging), the saturation intensity
is
ISAT =h̄ωΓ2δ
(1 + ( )2 ).
2σo
Γ

(A.2)

This assumes that the polarization of the light matches the transition polarization.
If not, only the useful portion of the polarization should be considered. In addition,
this is valid only for the cycling transition.
We need to consider the total number of photons absorbed by each atom because
the absorption of photons results in atoms’ recoil. The recoil velocity can be calculated
frommvrecoil =

h̄ω
.
c

(A.3)

For lithium, vrecoil is about 10 cm/s. The mean velocity in one direction is√v recoil =Nvrecoil ,3(A.4)

where N is the number of recoils. If we limit the recoil displacement of atoms to 1
µm during 10 µs of probing, 10 photon absorption per atom is allowed. Assuming
random walk, the number of recoil can be determined by
Nrecoil = I

σotprobe ,1 + ( 2δ)2
Γwhere I is intensity of the probe beam and tprobe is probing time.

76

(A.5)

Appendix B
Exact Breit-Rabi Formula for the
Ground States of 6Li
In order to know the exact rf frequencies for single-photon transitions between groundstates of 6 Li, we use the Breit-Rabi formula,
1 ∆E∆E
E(mF ) = −− gI µB BmF ±2 2I + 1
2

r
1+

4mF x
+ x2 ,
2I + 1(B.1)

where the + sign is from F = I + 1/2, and the − sign is for F = I − 1/2, ∆E is theI )µB Bzero field energy separation, ah 2I+1
, and x = (ge +g∆E.
2
For the ground states of 6 Li, useful numbers for the calculation are summarized inTable B.1. Physical constants from Ref. [43] and 6 Li magnetic moment from Ref. [29]are used.
Table B.1: Constants for 6 Li Breit-Rabi formula
Quantity Numerical valuegI

4.47701 × 10−4ge

2.0023193043622µB

1.399624604 MHz/G∆E

228.20527 MHz

77

78

Appendix C
RF Antenna in BEC3’s Main
Chamber
The main chamber of BEC3 was originally designed for RF evaporation. Thus, theantenna inside the chamber was designed with the resonant frequency being around
30 MHz. In order to optimize the antenna for lithium hyperfine state transitions, we
tried to match the resonant frequency of the antenna to around 80 MHz by impedance
matching. We could shift the resonant frequency using capacitors and inductors witha toy antenna which has the same dimension as the one in the chamber. However, thiswas not successful for the antenna in the vacuum chamber probably due to various
structures and connections in the chamber.

2.5 cm14 cm

5.1 cmFigure C-1: Dimensions of RF antenna in BEC3’s main chamber.
79

80

Appendix D
Itinerant Ferromagnetism in a
Fermi Gas of Ultracold Atoms
This appendix contains a reprint of Ref. [31]: Gyu-Boong Jo, Ye-Ryuoung Lee, JaeHoon Choi, Caleb A. Christensen, Tony H. Kim, Joseph H. Thywissen, David E.
Pritchard and Wolfgang Ketterle, Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms, Science 325 , 1521-1524 (2009).

81

REPORTS

References and Notes
1. D. Schurig, J. B. Pendry, D. R. Smith, Opt. Express 15,
14772 (2007).
2. A. V. Kildishev, W. Cai, U. K. Chettiar, V. M. Shalaev,
N. J. Phys. 10, 115029 (2008).
3. J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780 (2006).
4. Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, Science 315,
1686 (2007).
5. H.-T. Chen et al., Appl. Phys. Lett. 93, 091117 (2008).
6. J. N. Gollub, J. Y. Chin, T. J. Cui, D. R. Smith, Opt. Express
17, 2122 (2009).
7. H.-T. Chen et al., Nature Photon. 2, 295 (2008).
8. I. Gil et al., Electron. Lett. 40, 1347 (2004).
9. M. M. Qazilbash et al., Appl. Phys. Lett. 92, 241906 (2008).
10. W. J. Padilla, A. J. Taylor, C. Highstrete, L. Mark,R. D. Averitt, Phys. Rev. Lett. 96, 107401 (2006).
11. A. Degiron, J. J. Mock, D. R. Smith, Opt. Express 15, 1115
(2007).
12. T. Driscoll et al., Appl. Phys. Lett. 93, 024101 (2008).

13. I. V. Shadrivov, N. A. Zharova, A. A. Zharov, Y. S. Kivshar,Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70, 046615(2004).
14. T. Driscoll et al., Appl. Phys. Lett. 91, 062511 (2007).
15. M. Di Ventra, Y. V. Pershin, L. O. Chua, Proc. IEEE 97,
1371 (2009).
16. Materials and methods are available as supporting
material on Science Online.
17. A. Zylbersztejn, N. F. Mott, Phys. Rev. B Solid State 11,4383 (1975).
18. B.-G. Chae, H.-T. Kim, D.-H. Youn, K.-Y. Kang, Physica B
369, 76 (2005).
19. S. Lysenko et al., Appl. Surf. Sci. 252, 5512 (2006).
20. M. M. Qazilbash et al., Science 318, 1750 (2007).
21. T. Driscoll, H. T. Kim, B. G. Chae, M. Di Ventra,
D. N. Basov, Appl. Phys. Lett. 95, 043503 (2009).
22. R. Lopez, L. A. Boatner, T. E. Haynes, R. F. Haglund Jr.,
L. C. Feldman, Appl. Phys. Lett. 85, 1410 (2004).
23. G. V. Eleftheriades, O. Siddiqui, A. K. Iyer, Microwave
Wireless Compon. Lett. IEEE 13, 51 (2003).
24. J. D. Baena et al., Microwave Theory Tech. IEEE Trans. 53,
1451 (2005).
25. S. Tretyakov, http://arxiv.org/abs/cond-mat/0612247 (2006).
26. J. J. Yang et al., Nat. Nanotechnol. 3, 429 (2008).

Itinerant Ferromagnetism in a Fermi
Gas of Ultracold Atoms
Gyu-Boong Jo,1* Ye-Ryoung Lee,1 Jae-Hoon Choi,1 Caleb A. Christensen,1 Tony H. Kim,1
Joseph H. Thywissen,2 David E. Pritchard,1 Wolfgang Ketterle1
Can a gas of spin-up and spin-down fermions become ferromagnetic because of repulsive
interactions? We addressed this question, for which there is not yet a definitive theoretical answer,
in an experiment with an ultracold two-component Fermi gas. The observation of nonmonotonic
behavior of lifetime, kinetic energy, and size for increasing repulsive interactions provides strong
evidence for a phase transition to a ferromagnetic state. Our observations imply that itinerant
ferromagnetism of delocalized fermions is possible without lattice and band structure, and our data
validate the most basic model for ferromagnetism introduced by Stoner.
agnetism is a macroscopic phenomenon with its origin deeply rooted inquantum mechanics. In condensedmatter physics, there are two paradigms formagnetism: localized spins interacting via tunneling and delocalized spins interacting via an
exchange energy. The latter gives rise to itinerant ferromagnetism, which is responsible forthe properties of transition metals such as cobalt,
iron, and nickel. Both kinds of magnetism involve strong correlations and/or strong interactions and are not yet completely understood. For
localized spins, the interplay of magnetism withd-wave superfluidity and the properties of frustrated spin materials are topics of current research.
For itinerant ferromagnetism (1–7), phase transition theories are still qualitative.
We implemented the Stoner model, a textbook Hamiltonian for itinerant ferromagnetism(8), by using a two-component gas of free fer-

M

1

Massachusetts Institute of Technology–Harvard Centerfor Ultracold Atoms, Research Laboratory of Electronics,
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 2Department of Physics,
University of Toronto, Toronto, Ontario M5S1A7, Canada.
*To whom correspondence should be addressed. E-mail:gyuboong@mit.edu

mions with short-range repulsive interactions,which can capture the essence of the screened
Coulomb interaction in electron gases (8). However, there is no proof so far that this simple
model for ferromagnetism is consistent when thestrong interactions are treated beyond mean-fieldapproaches. It is known that this model fails in
one dimension, where the ground state is singlet
for arbitrary interactions, or for two particles inany dimension (3). In our work, cold atoms wereused to perform a quantum simulation of thismodel Hamiltonian in three dimensions, and weshowed experimentally that this Hamiltonian
leads to a ferromagnetic phase transition (2).
This model was also realized in helium-3 (9), but
the liquid turn into a solid phase and not into aferromagnetic phase at high pressure. It has also
been applied to neutrons in neutron stars (10).
To date, magnetism in ultracold gases has
been studied only for spinor (11, 12) and dipolar(13) Bose-Einstein condensates (BECs). In thesecases, magnetism is driven by weak spindependent interactions, which nevertheless determine the structure of the condensate becauseof a bosonic enhancement factor. In contrast,
here we describe the simulation of quantum magnetism in a strongly interacting Fermi gas.

www.sciencemag.org

SCIENCEVOL 325

27. Y. Muraoka, Z. Hiroi, Appl. Phys. Lett. 80, 583 (2002).
28. H.-T. Kim et al., N. J. Phys. 6, 52 (2004).
29. I. Shadrivov, SPIE Newsroom doi:10.1117/
2.1200811.1390 (2008).
30. V. M. Shalaev, Nature Photon. 1, 41 (2007).
31. This work is supported by the U.S. Department of Energy(DOE), the Air Force Office of Scientific Research(AFOSR), and ETRI. Work at UCSD on VO2 was supported
by DOE–Basic Energy Sciences and the metamaterials
work was supported by AFOSR and ETRI. M.D.
acknowledges partial support from NSF. H.K.
acknowledges research support from a project of the
Ministry of Knowledge Economy in Korea.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/1176580/DC1
Materials and Methods
Fig. S1
Table S1
References
20 May 2008; accepted 4 August 2009
Published online 20 August 2009;10.1126/science.1176580
Include this information when citing this paper.

An important recent development in cold
atom science has been the realization of superfluidity and the BEC–Bardeen-Cooper-Schrieffer(BCS) crossover in strongly interacting, twocomponent Fermi gases near a Feshbach resonance (14). These phenomena occur for attractive
interactions for negative scattering length and forbound molecules (corresponding to a positive
scattering length for two unpaired atoms). Verylittle attention has been given to the region of
atoms with strongly repulsive interactions. One
reason is that this region is an excited branch,which is unstable against near-resonant threebody recombination into weakly bound molecules. Nevertheless, many theoretical papers
have proposed a two-component Fermi gas neara Feshbach resonance as a model system for itinerant ferromagnetism (15–22), assuming that the
decay into molecules can be sufficiently suppressed. Another open question is the possibilityof a fundamental limit for repulsive interactions.
Such a limit due to unitarity or many-body physics may be lower than the value required for thetransition to a ferromagnetic state. We show that
this is not the case and that there is a window ofmetastability where the onset of ferromagnetism
can be observed.
A simple mean-field model captures many
qualitative features of the expected phase transition but is not adequate for a quantitative description of the strongly interacting regime. The
total energy of a two-component Fermi gas of
average density n (per spin component)
in an3
volume V is given by EF 2Vn 10½ð1 þ hÞ5=3 þ2
ð1 − hÞ5=3  þ 3p
kF að1 þ hÞð1 − hÞg, where EF
is the Fermi energy of a gas, kF is the Fermi wave
vector of a gas, a is the scattering length characterizing short-range interactions between the twocomponents, and h = Dn/n = (n1 − n2)/(n1 + n2)
is the magnetization of the Fermi gas. The local
magnetization of the Fermi gas is nonzero when
the gas separates into two volumes, where thedensities n1 and n2 of the two spin states differ

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ble spectra could easily push this effect to higher
frequencies that are beneficial for a variety ofpractical applications (30).

1521

by 2Dn. We studied an ensemble in which thenumber of atoms in each spin state is conserved.
This is equivalent to a free electron gas at zeroexternal magnetic field where the total magnetization is zero. The interaction term represents
any short-range spin-independent potential. Whenthe gas is fully polarized, it avoids the repulsive
interaction but increases its kinetic energy by afactor of 22/3. The phase transition occurs whenthe minimum in energy is at nonzero magnetization (Fig. 1A) at kFa = p/2. This onset waspreviously discussed in the context of phase separation in a two-component Fermi gas (15–18).
Figure 1B shows several consequences of thephase transition for a system at constant pressure. First, for increasing repulsive interactions,the gas expands, lowering its density and Fermi
energy; kinetic energy is therefore reduced.
When the gas enters the ferromagnetic phase,kinetic energy increases rapidly because of thelarger local density per spin state. Furthermore,
the volume has a maximum value at the phasetransition. This can be understood by noting thatpressure in our model is (2/3)Ekin/V + Eint/V,
where Ekin is kinetic energy and Eint is interactionenergy. At the phase transition, the system increases its kinetic energy and reduces its interaction energy, thus reducing the pressure. Thismaximum in pressure at constant volume turns
into a maximum in volume for a system held atconstant pressure or in a trapping potential. We
have observed three predictions of this model: (i)the onset of local magnetization through the
suppression of inelastic collisions, (ii) the minimum in kinetic energy, and (iii) the maximum inthe size of the cloud. These qualitative features
are generic for the ferromagnetic phase transitionand should also be present in more-advanced
models (19).
We start with an atom cloud consisting of an
equal mixture of 6Li atoms in the lowest two
hyperfine states, held at 590 G in an optical
dipole trap with additional magnetic confinement (23). The number of atoms per spin state isapproximately 6.5 × 105, which corresponds toa Fermi temperature TF of ~1.4 mK. The effective temperature T could be varied between
T/TF = 0.1 and T/TF = 0.6 and was determined
immediately after the field ramp by fitting the
spatial distribution of the cloud with a finite
temperature Thomas-Fermi profile. We definek∘F as the Fermi wave vector of the noninteracting gas calculated at the trap center. Applying
the procedure discussed in (24) to repulsive interactions, we estimate that the real temperature
is approximately 20% larger than the effective
one. The effective temperature did not dependon k∘F a for kFo a < 6. At higher temperatures,
additional shot-to-shot noise was caused bylarge fluctuations in the atom number. Fromthe starting point at 590 G, the magnetic field
was increased toward the Feshbach resonance at
834 G, thus providing adjustable repulsive interactions. Because of the limited lifetime of thestrongly interacting gas, it was necessary to ap-

1522

ply the fastest possible field ramp, limited to
4.5 ms by eddy currents. The ramp time is approximately equal to the inverse of the axial trap
frequency (23) and therefore only marginallyadiabatic. Depending on the magnetic field during observation, either atoms or atoms andmolecules were detected by absorption imaging
as described in fig. S1 (25).
The emergence of local spin polarization canbe observed by the suppression of (either elastic
or inelastic) collisions, because the Pauli exclusion principle forbids collisions in a fully polarized cloud. We monitored inelastic three-bodycollisions, which convert atoms into molecules.
The rate (per atom) is proportional to f(a,T)n1n2
or f(a,T) n2(1 − h2) and is therefore a measureof the magnetization h. For kFa << 1, the rate
coefficient f(a,T) is proportional to a6 max(T,TF)(26). This rate can be observed by monitoring
the initial drop in the number of atoms during
the first 2 ms after the field ramp. We avoided
longer observation times, because the increasing
molecule fraction could modify the properties ofthe sample.
A sharp peak appears in the atom loss ratearound kFo a ≅ 2.5 at T/TF = 0.12 (Fig. 2), indicating a transition in the sample to a state withlocal magnetization. The gradual decrease is consistent with the inhomogeneous density of thecloud, where the transition occurs first in thecenter. The large suppression of the loss rate
indicates a large local magnetization of the cloud.
The kinetic energy of the cloud was determined by suddenly switching off the optical trap
and the Feshbach fields immediately after thefield ramp and then imaging state |1〉 atoms atzero field using the cycling transition after aballistic expansion time of Dt = 4.6 ms. The kinetic energy was obtained from the Gaussianradial width sx as Ekin = [(3msx2)/(2Dt 2)] wherem is the mass of the 6Li atom. A minimum of
the kinetic energy at kFo a ≅ 2.2 for the coldest
temperature T/TF = 0.12 nearly coincided with

18 SEPTEMBER 2009

the onset of local polarization (Fig. 3). The peak in
the atom loss rate occurs slightly later than theminimum of kinetic energy, probably because
f(a,T) increases with a (22). Because the temperature did not change around kFo a ≅ 2.2, the increase in kinetic energy is not caused by heating
but by a sudden change in the properties of thegas, which is consistent with the onset of ferromagnetism. The observed increase in kinetic energy is approximately 20% at T/TF = 0.12, smaller
than the value (22/3 − 1) = 0.59 predicted for a
fully polarized gas. This discrepancy could be
due to the absence of polarization or partial polarization in the wings of the cloud. Also, it ispossible that the measured kinetic energy of thestrongly interacting gas before the phase transition
includes some interaction energy if the Feshbach
fields are not suddenly switched off. For the current switch-off time of ~100 ms, this should be
only a 5% effect, but the magnetic field decay
may be slower because of eddy currents.
Finally, Fig. 4 shows our observation of a
maximum cloud size at the phase transition, inagreement with the prediction of the model. The
cloud size may not have fully equilibrated, because
our ramp time was only marginally adiabatic, but
this alone cannot explain the observed maximum.
The suppression of the atom loss rate, theminimum in kinetic energy, and the maximum
in cloud size show a strong temperature dependence between T/TF = 0.12 and 0.22. The properties of a normal Fermi gas approaching theunitarity limit with k∘F a >> 1 should be insensitive
to temperature variations in this range; therefore,
the observed temperature dependence provides
further evidence for a transition to a new phase.
At higher temperature (e.g., T/TF = 0.39 asshown in Fig. 3), the observed nonmonotonic
behavior becomes less pronounced and shifts tolarger values of kFo a for 3 ≤ koF a ≤ 6. For all threeobserved properties (Figs. 2 to 4), a nonmonotonic
behavior is no longer observed at T/TF = 0.55 (27).
One interpretation is that at this temperature andFig. 1. Ferromagnetic phase transition at T = 0, according to themean-field model described in the
text. The onset of itinerant ferromagnetism occurs when the energy
as a function of magnetization flips
from a U shape to a W shape (A).
(B) Enthalpy, volume, and kinetic
energy, normalized to their values
for the ideal Fermi gas, and magnetization as a function of the interaction parameter kFa. kF is definedby the density of the gas. The dottedline marks the phase transition.

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REPORTSabove, there is no longer a phase transition. In amean-field approximation, a ferromagnetic phasewould appear at all temperatures but for increasing
values of kFo a. Our observations may imply thatthe interaction energy saturates around kFo a ≈ 5.

The spin-polarized ferromagnetic state should
not suffer from inelastic collisions. However,typical lifetimes were 10 to 20 ms, which were
probably related to a small domain size and threebody recombination at domain walls.

Fig. 2. Atom loss rate as a probe
for local spin polarization, fordifferent temperatures. T/TF = 0.55
(triangles, dashed curve), T/TF =0.22 (open circles, dotted curve),
and T/TF = 0.12 (solid circles, solid
black curve). The curves are guides
to the eye, based on the assumption of a loss rate that saturates forincreasing a in the normal state. The
shaded area around the phasetransition at T/TF = 0.12 highlights
the same region as in Figs. 3 and 4.

Fig. 3. Kinetic energy ofa repulsively interacting
Fermi gas determined for
different interaction parameters kFo a and temperatures. The measured
kinetic energy is normalized by the Fermi energy
EFo of the noninteracting
Fermi gas at T = 0, calculated at the trap centerwith the same number ofatoms per spin state. Eachdata point represents theaverage of three or fourmeasurements.

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SCIENCEVOL 325

We were unsuccessful in imaging ferromagnetic domains using differential in situ phasecontrast imaging (28). A signal-to-noise level of~ 10 suggests that there were at least 100 domains in a volume given by our spatial resolutionof ~3 mm and by the radial size of the cloud. Thisimplies that the maximum volume of the spin domains is ~5 mm3, containing ~50 spin-polarizedatoms. We suspect that the short lifetime prevented
the domains from growing to a larger size andeventually adopting the equilibrium texture of theground state, which has been predicted to havethe spins pointing radially outward, like a hedgehog(20, 22). All our measurements are sensitive only
to local spin polarization and are independent of
domain structure and texture.
The only difference between our experimentand the ideal Stoner model is a molecular admixture of 25% (Fig. 4). The molecular fraction
was constant for kFo a > 1.8 for all temperatures andtherefore cannot be responsible for the suddenchange of behavior of the gas at kFo a ≅ 2.2 at thecoldest temperature T/TF = 0.12. This prediction
was confirmed by repeating the kinetic energy
measurements with a molecular admixture of 60%.
The minimum in the kinetic energy occurred at thesame value of kFo a within experimental accuracy.
For a comparison of the observed phase transition at kFo a ≅ 2.2 to the theoretical predictions, theideal gas kFo has to be replaced by the value for theinteracting gas, which is smaller by ~15% becauseof the expansion of the cloud (Fig. 4), resulting in a
critical value for kFa ≅ 1.9 T 0.2. At T/TF = 0.12, thefinite temperature correction in the critical value for
kFa is predicted to be less than 5% (19). The
observed value for kFa is larger than both the meanfield prediction of p/2 and the second-order prediction of 1.054 at zero temperature (19). Depending on the theoretical approach, the phase transition
has been predicted to be first or second order. This
could not been discerned in our experiment becauseof the inhomogeneous density of the cloud.
It has been speculated (19) that earlier experiments on the measurement of the interaction energy (29) and radio frequency spectroscopy of Fermi
gases (30) showed evidence for the transition to aferromagnetic state at or below kFa = 1. This interpretation seems to be ruled out by our experiment.
Our work demonstrates a remarkable asymmetry between positive and negative scattering
length. Early work (15) predicted that for kF|a| =p/2, both an attractive and a repulsive Fermi gas
become mechanically unstable (against collapse
and phase separation, respectively). In an attractive Fermi gas, however, the mechanical instability does not occur [due to pairing (31)], in
contrast to our observations in a repulsive Fermi
gas. This suggests that the maximum total repulsive energy [in units of 3/5(2Vn)EF] is larger
than the maximum attractive energy |b| of 0.59
(32) that is realized for infinite a (23).
The interpretation of our results in terms ofa phase transition to itinerant ferromagnetismis based on the agreement with the predictionof simplified models [Fig. 1, (15–22)]. Future

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REPORTS

1523

Fig. 4. Maximum in volume at the phase transition. (A) Axial sizeand chemical potential of the cloud for various temperatures. The
chemical potential m is determined from the measured cloud size, sz,
as m = [1/2]mwz2sz2. (B) Number of particles including both atoms
and molecules right after the field ramp. This result shows that 25%of the atoms were converted into molecules during the field ramp,
and this fraction stayed constant for kFo a > 1.8, where the phasetransition was reached. This molecule fraction was independent of
temperature.

work should address how the observed signatures are modified by strong interactions andcorrelations. Additional insight can be obtainedby varying the magnetic field ramp time over awide range and studying the relaxation toward
an equilibrium state (33).
Heisenberg and Bloch’s explanation for ferromagnetism was based on exchange energy;
that is, the Pauli principle and spin-independent
repulsive interactions between the electrons.
However, it was unknown what other “ingredients” were needed for itinerant ferromagnetism.
It was not until 1995 (6, 7) that a rigorous proof
was given that, in certain lattices, spin-independent
Coulomb interactions can give rise to ferromagnetism in itinerant electron systems. Our finding
suggests that Heisenberg’s idea does not require alattice and band structure but already applies to afree gas with short-range interactions. Our experiment can be regarded as quantum simulation of a
Hamiltonian for which even the existence of aphase transition was unproven. This underlines thepotential of cold atom experiments as quantum
simulators for many-body physics.
References and Notes
1. F. Bloch, Z. Phys. 57, 545 (1929).
2. E. Stoner, Philos. Mag. 15, 1018 (1933).
3. E. Lieb, D. Mattis, Phys. Rev. 125, 164 (1962).
4. P. W. Brouwer, Y. Oreg, B. I. Halperin, Phys. Rev. B 60,
R13977 (1999).

1524

5. D. Vollhardt, N. Blumer, K. Held, M. Kollar, Metallic
Ferromagnetism–An Electronic Correlation Phenomenon.
vol. 580 of Lecture Notes in Physics (Springer,
Heidelberg, Germany, 2001).
6. H. Tasaki, Phys. Rev. Lett. 75, 4678 (1995).
7. A. Tanaka, H. Tasaki, Phys. Rev. Lett. 98, 116402 (2007).
8. D. W. Snoke, Solid State Physics: Essential Concepts(Addison-Wesley, San Francisco, CA, 2008).
9. D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).
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18 SEPTEMBER 2009

VOL 325SCIENCE

27. The interpretation of the loss rate is complicated because
f(a,T ) is unkown for kFa ≥ 1. The three-body rate f(a,T ) is
expected to be unitarity saturated for kFa >> 1 (34). Thelines in Fig. 2 indicate that the observed loss rate isconsistent with unitarity saturation and a sudden drop at
the phase transition, which occurs at large values of kFa
at higher temperature.
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W. Ketterle, Phys. Rev. Lett. 97, 030401 (2006).
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Phys. Rev. Lett. 91, 123201 (2003).
35. Supported by NSF and the Office of Naval Research, throughthe Multidisciplinary University Research Initiative program,and under Army Research Office grant no. W911NF-07-10493 with funds from the Defense Advanced Research
Projects Agency Optical Lattice Emulator program.
G.-B.J and Y.-R.L acknowledge additional support from theSamsung Foundation. We thank E. Demler, W. Hofstetter,
A. Paramekanti, L. J. LeBlanc, and G. J. Conduit for useful
discussions; T. Wang for experimental assistance; andA. Keshet for development of the computer control system.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/325/5947/1521/DC1
Materials and Methods
SOM Text
Fig. S1
References
1 June 2009; accepted 21 July 200910.1126/science.1177112

www.sciencemag.org

Downloaded from www.sciencemag.org on May 31, 2011

REPORTS

86

Appendix E
Compressibility of an Ultracold
Fermi Gas with Repulsive
Interactions
This appendix contains a reprint of Ref. [40]: Ye-Ryoung Lee, Myoung-Sun Heo, JaeHoon Choi, Tout T. Wang, Caleb A. Christensen, Timur M. Rvachov, and Wolfgang
Ketterle, Compressibility of an ultracold Fermi gas with repulsive interactions, Phys.
Rev. A 85, 063615 (2012).

87

PHYSICAL REVIEW A 85, 063615 (2012)

Compressibility of an ultracold Fermi gas with repulsive interactions
Ye-Ryoung Lee,1 Myoung-Sun Heo,1 Jae-Hoon Choi,1 Tout T. Wang,1,2 Caleb A. Christensen,1 Timur M. Rvachov,1 andWolfgang Ketterle1
1

MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
MIT-Harvard Center for Ultracold Atoms, Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Received 18 April 2012; published 21 June 2012)
Fermi gases with repulsive interactions are characterized by measuring their compressibility as a function of interaction strength. The compressibility is obtained from in-trap density distributions monitored by phase-contrast
imaging. For interaction parameters kF a > 0.25, fast decay of the gas prevents the observation of equilibrium
proﬁles. For smaller interaction parameters, the results are adequately described by ﬁrst-order perturbation theory.
We have developed a phase-contrast imaging method that compensates for dispersive distortions of the images.
DOI: 10.1103/PhysRevA.85.063615

PACS number(s): 03.75.Ss, 67.85.Lm, 05.30.Fk

I. INTRODUCTION

Experiments with ultracold atoms explore many-body
physics with strong interactions. They have demonstrated
long-predicted phenomena like the BEC-BCS crossover [1]and Lee-Huang-Yang corrections to the energy of degenerategases [2–4]. Experiments have also explored novel quantum
phases like fermions with unitarity-limited interactions [1,5],
population-imbalanced Fermi gases [6,7], and Hubbard models in optical lattices [8,9]. More recently, they have been used
to provide precision tests of many-body theories [5]. Usually,
interactions in ultracold gases are fully described by thescattering length, which is a zero-range approximation greatly
simplifying the theoretical description. This approximation isvalid since the diluteness of the atomic gases implies a particle
separation much larger than the short range of the van der
Waals interactions. This almost exact characterization of theinteractions by a single parameter, the tunability of interaction
strength, and precise experimental control over cold-atomsystems have made them an ideal test bed for many-bodyquantum calculations.
A new level of quantitative comparison between theory andexperiment was recently reached by careful measurements ofdensity proﬁles from which the equation of state could bedetermined. These techniques were ﬁrst proposed by Chevy[10] and Bulgac and Forbes [11] and implemented by Shin[12]. Further improvements [4,5,13–16] resulted in impressive
accuracy without adjustable parameters. These results hingeon accurate measurements of the equilibrium atomic density
distribution. Since all cold-atom systems are in a metastablephase, this requires a favorable ratio of lifetime to equilibration
time.
Long lifetimes and strong interactions were realized in
Fermi gases with strong attractive interactions since thedecay to lower-lying molecular states is suppressed by thePauli exclusion principle [17]. This is different for repulsive
interactions which are realized on the so-called upper branchof a Feshbach resonance, where decay is always possible into
the so-called lower branch, which consists of weakly boundmolecular states with binding energy h̄2 /ma 2 where a is thescattering length.
For bosons, the ﬁrst beyond-mean-ﬁeld correction, theso-called Lee-Huang-Yang term, could be observed, butcorrections were necessary to account for the nonequilibrium
1050-2947/2012/85(6)/063615(7)proﬁle, since the time to sweep to strong interactions was notlong compared to equilibration times and inverse loss rates [3].
Here we study fermions with repulsive interactions. They havebeen the focus of much recent work due to the predictionof a phase transition to a ferromagnetic state for sufﬁciently
strong interactions [18,19]. Recent experimental [20,21] andtheoretical studies [22] addressed the competition with strong
decay to the lower molecular branch.
As expected we ﬁnd only a limited window for metastability
where we can observe equilibrated clouds and characterize
the repulsive interactions by obtaining the compressibility
from observed proﬁles. We observe the linear term in kF a
corresponding to mean-ﬁeld energy in density proﬁles. kF
is the Fermi wave vector. In contrast to a Bose-Einsteincondensate, here the mean-ﬁeld energy is smaller than the
kinetic energy and also competes with the thermal energy, andis therefore much more difﬁcult to observe. The signal-to-noise
ratio (and some heating) prevented us from discerning thesecond-order interaction term which is the Lee-Huang-Yang
correction for fermions.
Our work employs phase-contrast imaging using two
different laser detunings to identify and correct for dispersive
distortions of the cloud. All studies mentioned above, with
one exception [12], were conducted using resonant absorption
where dispersion (an index of refraction different from 1)
is absent. However, this severely limits the cloud size andnumber of atoms to small optical densities. Phase-contrast
imaging has many advantages. It can be applied to clouds
with much larger optical densities by adjusting the detuning.
Due to coherent forward scattering, the heating effect per
detected signal photon is reduced by potentially a large number(which is equal to the resonant optical density divided by
4 [23]). This can, e.g., be used for repeated nondestructive
imaging. However, for precision studies of density proﬁles,small dispersive distortions of the density proﬁle cannot be
neglected. Previous work including Ref. [12] was not sensitive
to this effect. We have developed an experimental techniqueto correct for dispersion.

II. EXPERIMENTAL SETUP

A spin-polarized Fermi gas of 6 Li in the|F = 3/2; mF = 3/2 state is produced by sympathetic

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PHYSICAL REVIEW A 85, 063615 (2012)

cooling with bosonic 23 Na atoms in a magnetic trap as
described in [24]. The 6 Li atoms are then loaded into asingle-beam optical dipole trap and transferred into the lowest
hyperﬁne state |F = 1/2; mF = 1/2 by a Landau-Zenerradio-frequency (rf) sweep. Additional axial conﬁnement isprovided by magnetic ﬁeld curvature. An equal mixture of |1
and |2 spin states (corresponding to the |F = 1/2; mF = 1/2and |F = 1/2; mF = −1/2 states at low magnetic ﬁeld) is
prepared by a rf sweep at 300 G, followed by 500 ms wait
time for decoherence and evaporative cooling in the optical
trap. The Feshbach resonance at 834 G [1] is used to tune therepulsive interactions between |1 and |2. We increase themagnetic ﬁeld in 200 ms to 528 G, where the scattering lengthis zero and our Fermi gas is noninteracting. The ﬁnal trap has
a depth of 4.1 μK and frequencies of ωx = ωy = 390 Hz andωz = 34.7 Hz. The number of atoms per spin state is 8 × 105 ,which corresponds to a Fermi temperature TF of 1.4 μK. The
temperature of the atoms is 0.3TF at this point. For loss ratemeasurements, the magnetic ﬁeld is quickly ramped to thetarget ﬁeld. For compressibility measurements, the ﬁeld isramped up over 50 ms and held for 30 ms to ensure thermal
equilibrium before imaging. The molecular fraction in thedensity proﬁle is determined by dissociating molecules with amagnetic ﬁeld jump to 20 G above the Feshbach resonance,and comparing with the atom number after jumping to 528 G
where the cross section for imaging molecules vanishes (seeRef. [25]).
III. LOSS RATE

To obtain thermodynamic parameters from atomic density
proﬁles requires equilibration. One time scale for equilibrationis set by the longest trap period, which is 30 ms for theaxial direction. The authors of Ref. [3] studied the distortionsof proﬁles of bosonic 7 Li due to nonadiabatic ramping of
the scattering length. The authors found that ramping times
longer than ≈6π/ω (which is 90 ms for our parameters) led
to only small deviations from equilibrium proﬁles. Here, ω isthe smallest of the trapping frequencies, which sets the timescale for equilibration. Assuming that losses sharply increaseduring the last 5 ms of the ramp towards stronger interactions
and limiting tolerable losses to 10% leads to an estimate ofa maximum tolerable loss rate of about 0.02/ms. The fastest
relaxation time for excitations created during a nonadiabatic
ramp is 1/ω (the damping time for a harmonic oscillator at
critical damping). Allowing ∼10% loss during this time leads
to an identical estimate for the tolerable loss rate of ∼0.02/ms.
We measured loss rate as a function of interaction strength
kF0 a. Note that kF0 is the Fermi wave vector of the zerotemperature noninteracting gas calculated at the trap centerusing total atom number. The real kF is somewhat smaller
because repulsive interactions and nonzero temperature lower
the density.
First, in Fig. 1(a) we measured the number of atoms rightbefore and after the fastest possible ramp (limited to 7 ms byeddy currents) to the target magnetic ﬁeld. During the ramp to
the target magnetic ﬁeld of kF0 a ≈ 0.8, ∼35% of the sample islost. Measuring the loss rate at higher kF0 a is difﬁcult because
most of the sample is lost even before reaching the target
ﬁelds. The loss rate was determined by monitoring the drop

FIG. 1. Characterizing atomic loss for increasing repulsive interactions. (a) Remaining fraction of atom number immediately after
the fastest possible ramp to the target ﬁeld. (b) Atom loss rates at
the target ﬁelds. Dashed line shows the estimated maximum tolerable
loss rate of 0.02/ms. (c) Molecular fraction after the 50 ms ramp
and 30 ms wait time, corresponding to when we measure equilibrium
atomic density proﬁles.

in the number of atoms immediately after the ﬁeld ramp. The
results in Fig. 1(b) show that the measured loss rate reaches
the maximum tolerable value of 0.02/ms at kF0 a ≈ 0.35,
limiting our measurements of equilibrium density proﬁles tosmaller values of kF0 a. Furthermore, at the same values of
kF0 a, the molecular fraction when we measure equilibrium
density proﬁles exceeded 10% [see Fig. 1(c)]. As we discussin Sec. IV the presence of a molecular component affects thecompressibility measurement.
IV. COMPRESSIBILITY MEASUREMENT

A system is characterized by its equation of state. The
equation of state can be expressed in different forms involving
various thermodynamic variables including density, energy,

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PHYSICAL REVIEW A 85, 063615 (2012)

pressure, temperature, and entropy. For cold-atom experiments, density, chemical potential (through the trapping potential), and temperature are directly accessible to measurement.
In the weakly interacting regime, the interaction manifests
itself as a perturbative term in the equation of state. Here wemeasure this perturbative interaction effect by measuring thederivative of density with chemical potential, the isothermal
compressibility of the gas. We prepare the system at the lowest
temperature, but due to heating of the cloud by molecule
formation, we have to apply a correction for the measured
ﬁnite temperature.
Using the experimental procedure discussed above, we
prepared equilibrated clouds at various magnetic ﬁelds andmeasured line-of-sight integrated proﬁles of column density
using in situ phase-contrast imaging [26]. The signal-to-noise
ratio was improved by averaging the column density alongequipotential lines (which are ellipses for the anisotropic
harmonic oscillator potential). The averaging region wasrestricted to an axial sector of ±60◦ to avoid corrections due
to transverse anharmonicities [27]. Three-dimensional density
proﬁles n(r) were reconstructed by applying the inverse Abeltransformation to the column densities ñ(r) [26].
The isothermal compressibility is obtained from a spatial
derivative of n(r), since in the local density approximation,
the local chemical potential is μ = μ0 − mω2 x 2 /2, where μ0
is the global chemical potential and ω the trap frequency. The
compressibility is deﬁned as
κ=

1 ∂n
.
n2 ∂μ

(1)

We normalize the compressibility κ by the compressibility ofan ideal gas at the given density and zero temperature κ0 =n1/3
3m
, and obtain the normalized compressibility
2 2/3
n2 2
h̄ (6π )κ̃ =κh̄2 (6π 2 )2/3 ∂n2/3
.
=
κ0
2m
∂μ

FIG. 2. Determination of the compressibility of repulsively interacting Fermi gases. The compressibility is the slope of a graphshowing the density to the power 2/3 versus the square of the distancefrom the center. (a) Single-shot density proﬁle. (b) Closeup of theregion used for determining the compressibility for a plot showing
the average of 20 density proﬁles.

(2)

Here n is density, and m is the atomic mass. The normalized
compressibility is obtained as the slope in a plot of n2/3 vs μ
(Fig. 2). This plot is in essence the observed density proﬁleplotted with the central region to the left and the spatial wings
to the right. Experimentally, we ﬁnd the slope to be constantover an extended range of the density proﬁle. Compressibilities
were determined from ﬁts to the slope in the region of 90%
to 50% of the peak density. The region near the center ofthe cloud was excluded since the center is singular for the
inverse Abel transformation, leading to excessive noise. Thesecompressibilities should be regarded as average values overthe density range used in the ﬁt. The uncertainty of ﬁtting theslope to a single proﬁle was 4.5%. By averaging the proﬁles
obtained from 20 images, the uncertainty was improved to
1.3%.
The normalized compressibility is a function of T /TF andkF a. At ﬁnite temperature and scattering length a, T /TF andkF a change across a single density proﬁle because TF and kF
depend on the local density. However, this density dependenceis small near the center of the cloud. Simulated density proﬁlesshowed that the average compressibility determined in the way
described above agrees to within 0.6% with the compressibility

at T /TF and kF a at the density in the center of the selected
range.
Compressibility decreases for stronger repulsion, but also
for higher temperature. To identify the effect of repulsive interaction requires a careful consideration of ﬁnite temperatureeffects. First, the temperature of the cloud had to be accurately
determined. This can be done without any special assumptions
by ﬁtting the wings of the cloud using a virial expansion [28],by thermometry with another cotrapped atom [14], or for
population imbalanced clouds by ﬁtting the wings of the
majority component which is an ideal Fermi gas [29]. Herewe chose to determine temperature using a virial expansion,p

λ3= eβμ + b2 e2βμ + O(e3βμ ),
kB T

(3)


2πh̄2
is the thermal de Broglie wavelength, b2
where λ = mkBTis the virial coefﬁcient, and eβμ is the fugacity. The virial
coefﬁcient for the Fermi gas with repulsive interaction is b2 =−2−5/2 − a/λ [30]. The pressure p was obtained from the
doubly integrated density proﬁles [13]. The temperature wasdetermined in the wings of the proﬁle where βμ < −0.5. Here,the temperature measured with and without the interaction
term −a/λ in b2 differ by about 3%. This suggests that higherorder corrections from the interaction term will be negligible.
Note that the virial expansion up to second order is valid towithin 1% for the ideal gas at the density of the ﬁtted wings.
The low-temperature normalized isothermal compressibility of a noninteracting Fermi gas is given by the Sommerfeld

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PHYSICAL REVIEW A 85, 063615 (2012)

expansion [31]  4 π2 T 2T,κ̃0,T = 1 −
+O
12 TF
TF

(4)

where T is the temperature and TF is the Fermi temperature.
To add the effects of interactions, it is useful to work with theinverse normalized compressibility
32m∂μ(n,T ,a)
1
=
.
n1/3
κ̃
2 h̄2 (6π 2 )2/3
∂n(5)This is a derivative of the chemical potential, which has thefollowing expansion in temperature and scattering length: π2 T 24μ(n,T ,a) = EF 1 −
kF a
+
12 TF
3π

4(11 − 2 ln 2)(kF a)2 + CT 2 a 2 ,
+
15π 2

(6)

where C is a constant, independent of density [32]. Therefore
the inverse normalized compressibility has additive correction
terms for temperature and interactions up to the second orderof the interaction effect,1
1=+ Y (kF a).
κ̃
κ̃0,T

(7)

This equation deﬁnes Y (kF a), the interaction correction to
the inverse compressibility. This term is the derivative of
the interaction term of the chemical potential. In secondorder perturbation theory, one obtains Y (kF a) = π2 kF a +8(11−2ln2)(kF a)2 .
15π 2
Figure 3 shows the normalized compressibility, the temperature T /TF , and the interaction correction to the inverse
compressibility. The temperature increases with kF a due tothe increase in three-body recombination where the binding
energy of the molecules (h̄2 /ma 2 ) is transferred to theremaining atoms. The measured temperature is higher thanthat in previous experiments on the repulsive side [20,21].
This difference can be explained by a smaller kF a since theincrease in T /TF is approximately proportional to 1/kF a [21].
We perform a linear ﬁt of the interaction effect Y (kF a)
versus kF a (constrained to pass through the origin) andobtain 0.680 ± 0.147 for the slope, in agreement with theperturbative prediction of π2 = 0.637. Thus we have observed
the mean-ﬁeld term for repulsively interacting fermions in athermodynamic quantity. The repulsive interaction has been
seen as line shifts in rf spectroscopy experiments (which,in contrast to many thermodynamic quantities, are measured
independently of the kinetic energy of the atoms) [33,34]. In
principle, it is possible to obtain the mean-ﬁeld term directly
by ﬁtting the density proﬁles with an extra mean-ﬁeld term.
In such ﬁts, we obtained clear evidence for such a term, butwith low accuracy. It appears that the averaging over proﬁlesfor determining the compressibility (as in Fig. 2) is superior.
Figure 3(c) shows the predicted effect of the second-order termon Y (kF a). With some improvements in signal-to-noise ratio,
one should be able to observe this term, which is analogous to
the Lee-Huang-Yang correction for bosons.

FIG. 3. Measured temperature, normalized compressibility, andinteraction effect on compressibility at various interaction strengths.
(a) The measured temperature as a function of interaction strength.
(b) Solid circles show the measured normalized compressibility (κ̃)
and open squares show the calculated normalized compressibility at
the measured temperature without interaction (κ̃0,T ). The difference
between the two indicates the effect of interaction. Open triangles
show the calculated normalized compressibility using the secondorder perturbation theory, which is consistent with our measured κ̃.
(c) The measured interaction correction to the inverse compressibility(solid circle) is compared to a linear ﬁt (solid line), and the ﬁrst(dashed line) and the second- (dotted line) order perturbative results.
V. DISPERSIVE EFFECT IN PHASE-CONTRAST IMAGING

As mentioned in the Introduction, phase-contrast imaging
has several advantages over resonant absorption imaging,and it has been applied to many studies of cold Bose andFermi gases [26,35,36]. Absorption imaging is usually donewith absorptively dilute clouds, typically with 10% to 70%
absorption (or optical densitites below 1). The standard
assumption has been that dispersive imaging is quantitative
when the phase shift φ across the cloud is less than π/4. The
normalized√ phase-contrast signal (for negligible absorption)is 3 − 2 2 cos(φ ± π/4), which is equal to 1 ± 2φ for small
phase shifts [23]. The sign depends on the laser detuning andthe sign of the phase shift imparted by the phase plate.
Here, we applied phase-contrast imaging for rather precise
quantitative studies of ultracold Fermi gases and found that063615-4

COMPRESSIBILITY OF AN ULTRACOLD FERMI GAS . . .

PHYSICAL REVIEW A 85, 063615 (2012)

e
Probe
beam

>0

<0
76 MHz>02
1<0FIG. 4. Phase-contrast imaging of a balanced spin mixture instates |1 and |2. White images (phase shift φ >0) were obtained for aprobe beam red detuned from the |2 → |e transition, corresponding
to δ < 0. Black images (φ <0) were obtained for a probe beam blue
detuned from the |1 → |e transition, corresponding to δ < 0.

even for small phase shifts systematic dispersive distortions of
the image cannot be neglected. Phase-contrast imaging relies
on column-density-dependent phase shifts. However, if theobject is not thin, but extended along the line of sight, somelensing will affect the images. These distortions should varyin inverse proportion to the probe light detuning and become
negligible for far detuning.
We investigated positive and negative detunings (δ) of 40,
80, and 120 MHz (see Fig. 4). The normalized phase-contrast
image had a maximum signal of 0.35 (1.85), 0.55 (1.6),
and 0.7 (1.4) for the three positive (negative) detunings.
The lensing effect is opposite (focusing vs defocusing) for
positive and negative detuning, and can therefore be identiﬁed
by comparing proﬁles obtained with positive and negative
detunings. Figure 5 shows that at 40 MHz, the two proﬁleshave a visible difference, but for proﬁles at 80 and 120 MHz,the differences are smaller than the noise level.
However, the compressibility is determined by the slopeof the proﬁles and very sensitive to small distortions even if
they are not perceptible in the proﬁles. Figure 5(c) shows that,even at 120 MHz detuning, the compressibilities obtained from
proﬁles with the two signs of the detuning differ by about 10%.
Since further detuning would have resulted in a smaller signal,
we evaluated the average value of the compressibility for positive and negative detuning. When the dispersive distortions are
small, the effect on the compressibility should be a ﬁrst-ordereffect in the phase shift and cancel for the average. Indeed,
the average value stays constant above 80 MHz detuning. Our
conclusion is that for reasonable signal levels (i.e., 50% of thebaseline set by the probe light) dispersive effects are relevant
for quantitative studies, but can be eliminated by performing
averages over positive and negative detunings.
VI. DISCUSSION

We address now to what extent a small molecular fraction
contributes to the observed density proﬁles. The presence ofmolecules is unavoidable since they form during the ramping
and equilibration time. At the highest magnetic ﬁeld used in
the experiment, 679 G, the molecular fraction was determined
to be ∼10% (Fig. 1). The authors of Ref. [37] reported thatmolecules at 650 G showed an absorption cross section of
about half the value of that of the atoms for probe light at
the atomic resonance. For phase-contrast imaging with large
detuning, molecules and atoms should contribute equally tothe signal.

FIG. 5. Comparison of column density proﬁles obtained frompositive and negative detunings of (a) 40 and (b) 80 MHz, respectively.
(c) The measured compressibility (at kF a = 0) from positive- (opentriangle) and negative- (open square) detuned proﬁles and theiraverages (solid circle) are shown. The average value stays constant
above 80 MHz detuning.

We performed simulations to address how the presenceof molecules would affect the compressibility measurements.
We considered as possible scenarios (i) that the molecular
fraction is constant throughout the cloud, (ii) that the molecular
fraction is proportional to the loss rate (n8/3 ), and (iii) that themolecular fraction is well equilibrated at the same temperature
as atoms. The atomic proﬁle is then the difference of themeasured density proﬁle and the simulated molecular density
distribution. Scenario (iii) is ruled out since it would resultin a rather sharp peak in the density proﬁle, which was
not observed. The ﬁrst two scenarios with a 10% molecular
fraction resulted in a value for the normalized compressibility
which was increased by 3.3% and 4.4%, respectively. Thisshows that for our largest value of kF a the presence ofmolecules starts to become a systematic effect. In addition
to the contribution to the density proﬁles, molecules canaffect the atomic density distribution through atom-moleculeinteractions.

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Our work shows that the interaction effect on the compressibility at the maximum possible values of kF a is about 15%.
We could identify this effect only by careful thermometry (todistinguish it from thermal effects) and by correcting small
dispersive distortions of the cloud. It is desirable to study
fermions for stronger repulsive interactions where stronger
and nonperturbative effects are predicted. The maximum
possible kF a value for obtaining equilibrium density proﬁlesis determined by the loss rate, which is proportional ton2 a 6 max(T ,TF ) = (kF a)6 n2/3 [38,39]. Therefore, the maximum possible kF a for a given loss rate is proportional ton−1/9 and stronger interaction effects can be seen at lower
density. This should be accomplished by reducing the radialconﬁnement and not the axial conﬁnement which determines
the equilibration time. However, the weak density dependence
will allow only modest increases in kF a. A recent experimentused a density ten times smaller than ours [21] and reported
ramping from kF0 a = 0 to kF0 a = 0.35 in 500 ms, losing only
5% of the atoms. Assuming the loss happened during the last
50 ms, we can roughly estimate a loss rate of ∼0.001 ms−1 at
kF0 a = 0.3, which is lower than our measurement, consistent
with the lower density.
Longer lifetimes for a given kF a should be realized using
narrow instead of broad Feshbach resonances. For narrow
Feshbach resonances the low-lying molecular state has adominant closed-channel character. Therefore, three-bodyrecombination of atoms (which are in the open channel)
has a smaller overlap to the molecular state and therefore areduced loss rate. Recent experiments using rf spectroscopy[40,41] conﬁrm this. However, for such narrow resonances thezero-range approximation is no longer valid; the interaction is
no longer described by the scattering length alone and becomes(through an effective range parameter) momentum dependent.
As a result, the narrow Feshbach resonances realize a different
Hamiltonian.
In conclusion, in this paper we have addressed the question
to what extent Fermi gases with strong interactions can bestudied by observing equilibrium density proﬁles. The rangeof sufﬁciently long metastability to reach equilibrium is limited
to values of kF a < 0.25. In this range, interaction effects arecomparable to thermal effects, but we were able to observe
how interactions reduce the compressibility and obtained
quantitative agreement with the ﬁrst-order mean-ﬁeld term. An
observation of the second-order Lee-Huang-Yang correction
is within experimental reach.
If experiments can be performed at stronger interactions,a natural extension of our work would be a measurementof the spin susceptibility using population-imbalanced Fermi
systems. This was performed recently for fermions withattractive interactions [15]. Such measurements could address
the possible existence of a ferromagnetic transition in a
repulsive Fermi gas [20] for which the spin susceptibility
would diverge at the phase transition [42].

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ACKNOWLEDGMENTS

This work was supported by the NSF and ONR, an AFOSR
MURI grant, and by ARO Grant No. W911NF-07-1-0493 withfunds from the DARPA Optical Lattice Emulator program.
Y.-R.L. acknowledges support from the Samsung Scholarship.
T.T.W. acknowledges support from NSERC. We are thankful
to David E. Pritchard, Christophe Salomon, Christian Sanner,
Ariel Sommer, Mark Ku, and Martin Zwierlein for valuable
discussions, and Gregory Lau for experimental assistance.

063615-6

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