Quantum Coherence and Magnetism in Bosonicand Fermionic Gases of Ultracold Atoms
by

Gyu-Boong Jo
B.S., Physics and Mathematics (2003)
Seoul National University
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
January 2010
c Massachusetts Institute of Technology 2010. All rights reserved.
°

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
January 29, 2010
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Pritchard
Cecil and Ida Green Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Krishna Rajagopal
Professor of Physics, Associate Department Head for Education

Quantum Coherence and Magnetism in Bosonic andFermionic Gases of Ultracold Atoms
by
Gyu-Boong Jo
Submitted to the Department of Physics
on January 29, 2010, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
In this thesis, two sets of experimental studies in bosonic and fermionic gases aredescribed. In the first part of the thesis, itinerant ferromagnetism was studied in astrongly interacting Fermi gas of ultracold atoms. 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 latticeand band structure, and our data validate the most basic model for ferromagnetism
introduced by Stoner.
In the second part of the thesis, the coherence properties of a Bose-Einstein condensate (BEC) was studied in a radio frequency induced double-well potential implemented on a microfabricated atom chip. We observed phase coherence between theseparated condensates for times up to 200 ms after splitting, a factor of 10 longer
than the phase diffusion time expected for a coherent state for our experimental conditions. The enhanced coherence time is attributed to number squeezing of the initialstate by a factor of 10.
Furthermore, the effect of phase fluctuations on an atom interferometer was studied in an elongated BEC. We demonstrated that the atom interferometer using thecondensates is robust against phase fluctuations; i.e., the relative phase of the split
condensates is reproducible despite axial phase fluctuations.
Finally, phase-sensitive recombination of two BECs was demonstrated on an atomchip. The recombination was shown to result in heating, caused by the dissipationof dark solitons, which depends on the relative phase of the two condensates. This
heating reduces the number of condensate atoms and provides a robust way to readout the phase.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics
Thesis Supervisor: David E. Pritchard
Title: Cecil and Ida Green Professor of Physics

To my parents,
Young Ki Jo and Ok Hwa Kim
and to Helen.

Acknowledgments
There are many people who I would like to thank for making this thesis possible.
This acknowledgement is a small token of my appreciation for their supports.
First of all, I owe my deepest gratitude to Wolfgang and Dave for giving me an
opportunity to work in their group. In past 5 years, I could learn how to find out aninteresting problem in physics, approach the problem in an efficient way, and extract
meaningful interpretation out of the observation. In almost every group meeting,
Wolfgang and Dave guided us to focus on the core of the problem we faced in the lab.
Their keen and deep insight into physics kept us seeing a big picture hidden behindthe problem we considered.
Also, I would like to thank Wolfgang and Dave for inspiring excitement about thephysics in past 5 years. When I met Wolfgang during the open house in 2004, he told
me as follows, “My students often spend whole night because they love to see a newphenomenon in the lab.”. During my Ph.D. course, I realized what Wolfgang meant.
The moment when I observed unexpected phenomena in the lab was unforgettable
experience.
When I arrived at MIT in 2004, I joined the third generation BEC machine (BEC
III), sometimes called “the science chamber ”. I had an extreme fortune because all
the people in the science chamber are very nice as well as smart. Without them, thisthesis would not have been possible. Here, I would like to thank the people I haveworked most closely with in the BEC III lab over the past few years.
I had spent about one year with Yong-il Shin before he graduated and started his
new experiments in BEC I. I could learn almost everything that I need to know inthe lab from Yong-il. To me, he has been a brother and a good friend as well as an
excellent mentor. His valuable advice has always helped me to be in the right trackin many respects. The atom chip experiments reported in this thesis would not havebeen possible without his contribution. I wish all the best for his research in Korea.
I also thank Michele for his endless helps and advices in the lab. I often foundtreasures in the discussion with him about physics and life. Whatever problem we
faced in the lab could be solved smoothly with Michele. I am grateful to Tom Pasquini
as well. He had made available his support in a number of ways. In my first andsecond year at MIT, I could manage my research life with the help of his invaluableadvices. I also benefited from an energetic diploma student, Sebatian Will. His carefulassessment of the problem we faced in the lab allowed me to solve them efficiently
and focus on the physics itself. Without his contribution, the “long phase coherencetime experiment” would not have been successful.
Also I had a great fortune to work with Jae. Jae was a great labmate who
could compensate for the absence of Yong-il and Michele. He brought his valuable
experiences into the science chamber team, and we could improve the apparatus
allowing us to run a successful experiment with sufficient stability in many respects.
In addition, Caleb and Ye-Ryoung have contributed to most of experiments describedin this thesis. Caleb has deep insight into electronic devices as well as physics andour lab still benefits from his presence. Ye-Ryoung brought a new energy into thescience chamber team. She has provided assistance in numerous ways. I wish themgood luck for their coming research. Finally, I also benefited from Tony Kim andTout Wang. Tony could make any electronic circuit with high standard. Tout has a
lot of curiosity about physics leading to broad knowledge. His enthusiasm in physics
would become an invaluable asset in the science chamber .
It is a pleasure to thank those who made this thesis possible outside the sciencechamber. I specially thank the members in BEC I for sharing the lab space and many
experimental tools with endless patience and kindness. I thank Martin Zwierleinfor sharing his deep understanding of the physics, Claudiu Stan for his super-stableelectronic boxes, Christian Schunck for playing the harp, and Andre Schirotzek for“nice” personality. Also, I would like to thank Kaiwen Xu, Daniel Miller, Jit Kee
Chin, Yingmei Liu , Christian Sanner, Aviv Keshet, Edward Su, Ralf Gommers,
Peter Zarth, Luis Marcassa, Micah Boyd, Gretchen Campbell, Jongchul Mun, Patrick
Medley, Hiro Miyake, David Hucul, and David Weld. Finally, I thank all membersof the Fermi I lab including Peyman Ahmadi, Ariel Sommer, Cheng-Hsun Wu, Ibon
Santiago, Jeewoo Park, and Mark Ku.
CUA (Center for Ultracold Atoms) is a wonderful place for me to have a frequentdiscussion with other people, interact with other physicists, and finally learn the
cutting-edge physics. I could benefit from many other labs in CUA in many respects;
I could get helpful advices from colleagues in CUA whenever I encountered a problem.
I would like to thank all CUA members for their supports. Especially, I must express
my gratitude to Joanna Keseberg for her support.
I am indebted to many of my colleagues to support me outside the lab. I owe mydeepest gratitude to Sang-il Lee, Seung-Hyuck Hong, Lin Han, Heechul Park, SungWhan Oh, Yeoreum Yoon, Wonyong Lee, Jaemyung Ahn, Seungbum Hong, Jungwoo
Joh, Joonsung Lee, Daniel S. Park, Eun Gook Moon, Gyuwon Hwang, Hyung-Gyu
Kwon, Jichul Woo, Sang-jin Kim, Hanlim Choi, and Jerin Gu. I also would like tothank Woojong Baek, Gangsang Lee, Gangwon Lee, Yongsuk Lee, Michael Min, ChrisBae, and Woojae Kim.
I am grateful to Prof. Wonho Jhe for giving me an opportunity to work in his
group before I started graduate school. His encouragement motivated me to explore
a new physics without being afraid of a failure.
Finally, and most importantly, I would like to thank my parents. Their support,
encouragement, patience and endless love have kept me feeling constantly comfortable in my Ph.d. course. Without their devotion and love, this thesis would not havebeen accomplished. Also, I owe my deepest gratitude to Helen. Her tolerance of myoccasional nervousness is a testament of her devotion and love. I really appreciate
her endless support again.
This work was supported by NSF and the Office of Naval Research, through theMultidisciplinary University Research Initiative program, and under Army Research
Office grant no. W911NF-07-1-0493 with funds from the Defense Advanced Research
Projects Agency Optical Lattice Emulator program. I also acknowledge additional
support from the Samsung Foundation.

Contents
1 General Introduction
1.1 The Quest for Low Temperatures . . . . . . . . . . . . . . . . . . . .
1.1.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . .
1.1.2 Ultracold Dilute Atomic Gas . . . . . . . . . . . . . . . . . . .
1.2
1.3
1.4

Next Challenges in the field of Ultracold Atomic Gases: Quantum
Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Identical Particles in Quantum Theory: Bosons and Fermions . . . .
Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
13
14
15
17
19

2 Apparatus and Basic Procedures
22
2.1 Versatile Apparatus: “Science Chamber” . . . . . . . . . . . . . . . . 22
2.2 F=1 23 Na Bose-Einstein Condensates in the Science Chamber . . . . 22
2.2.1 Magnetic Microtraps on an Atom Chip in the Science Chamber 24
Apparatus for Producing a Degenerate gas of 23 Na and 6 Li Atoms . .
2.3.1 Producing a Degenerate 6 Li Fermi Gas . . . . . . . . . . . . .
2.3.2 Li Laser System . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Tools for Experiments . . . . . . . . . . . . . . . . . . . . . .

26
26
29
31

3 Ferromagnetism in a Fermi Gas
3.1 Quantum Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 What is magnetism? . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Origin of Ferromagnetism: Exchange interactions . . . . . . .

35
35
35
36

2.3

3.2

3.1.3 Two Paradigms of Magnetism . . . . . . . . . . . . . . . . . . 38
Itinerant Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Long-standing Problem in Condensed-matter Physics . . . . . 39
3.2.2 Stoner Model in a Free Electron Gas with Short-range Interactions 41
3.2.3
3.2.4

Ferromagnetism in Hubbard Models . . . . . . . . . . . . . . .
Similar Studies in Normal Liquid 3 He . . . . . . . . . . . . . .
7

43
45

3.3

Itinerant Ferromagnetism in Ultracold Atomic Gases . . . . . . . . .
3.3.1 A Very Simple Mean-field Prediction for Stoner’s Instability in

47

an Ultracold Atomic Gas . . . . . . . . . . . . . . . . . . . . .
Domain Structure of the Ferromagnetic Ground State . . . . .

49
52

3.3.2

4 Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms
4.1
4.2
4.3
4.4

55

Two-component Fermi gas with strong repulsive interactions as a Model
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implementation of the Stoner model in an Ultracold Atomic System .
4.2.1 Experimental Technique . . . . . . . . . . . . . . . . . . . . .
Evidences for Itinerant Ferromagnetism in a Fermi Gas of Ultracold

55
60
61

Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 The Critical Value of kF a Observed in the Experiment . . . .
4.4.2 Estimation of the Maximum Total Repulsive Energy . . . . .

64
69
69
70

4.4.3

Comparison with Ferromagnetism in Spinor Bose-Einstein Condensates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrepancy of the Critical Value of kF a . . . . . . . . . . . .
Correlation Effects . . . . . . . . . . . . . . . . . . . . . . . .

71
71
73

Outlook: New Experiments . . . . . . . . . . . . . . . . . . . . . . .

74

5 Quantum Phase Coherence in Degenerate Bosonic Gases
5.1 Phase Coherence in a Bose-Einstein Condensate . . . . . . . . . . . .
5.2 Decoherence in Interacting Bose Gases . . . . . . . . . . . . . . . . .

77
77
81

4.4.4
4.4.5
4.5

5.3
5.4

Measurement of the Relative Phase between Two Condensates . . . .
Tools for the Study of Coherence Properties of a Bose gas . . . . . .

83
87

6 Long Phase Coherence Time via Number Squeezing and Phase Fluctuations of BECs in a Double-well Potential
6.1 Long Phase Coherence Time and Number Squeezing . . . . . . . . . .
6.1.1 Splitting Process in a Double-well Potential: a Bosonic Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

91
92
93

6.1.2 Preparation of a Number-squeezed State on an Atom Chip . . 98
6.1.3 Phase Diffusion and Number Squeezing . . . . . . . . . . . . . 103
6.1.4 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . . . 105
Matter-wave Interferometry with Phase Fluctuating Bose-Einstein Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8

6.2.1
6.2.2

Phase Fluctuations in an Elongated Bose-Einstein Condensate 107
Effect of Phase Fluctuations on Atom Interferometry . . . . . 109

7 Development of Atom Optics Elements
112
7.1 Phase Sensitive Recombination of Two Bose-Einstein Condensates . . 112
7.2 Trapping of Ultracold Atoms in a Hollow-core Photonic Crystal Fiber 117
8 Conclusion and Outlook
120
8.1 The future in BEC III . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A Properties of 23 Na and 6 Li.

122

B Designs for the Apparatus

125

C Itinerant ferromagnetism in a Fermi gas of ultracold atoms

131

D Long Phase Coherence Time and Number Squeezing of two BoseEinstein Condensates on an Atom Chip
143
E Matter-Wave Interferometry with Phase Fluctuating Bose-Einstein
Condensates
148
F Phase-Sensitive Recombination of Two Bose-Einstein Condensateson an Atom Chip
153
G Trapping of Ultracold Atoms in a Hollow-core Photonic Crystal
Fiber
158
H Interference of Bose-Einstein condensates split with an atom chip 163
Bibliography

168

9

List of Figures
1-1 Concept of the quantum simulator . . . . . . . . . . . . . . . . . . . .

16

2-1
2-2
2-3
2-4
2-5

Microfabricated atom chip mask . . . . . . . . . . . . . . . . . . . . .
A schematic of the dual-species oven . . . . . . . . . . . . . . . . . .
Dual maganetic optical trap . . . . . . . . . . . . . . . . . . . . . . .
A schematic of magnetic coils in the main chamber . . . . . . . . . .
Circuit diagram for producing a large bias magnetic field . . . . . . .

25
27
28
30
32

2-6 A schematic of a newly designed science chamber . . . . . . . . . . .

34

3-1 Classical electron gas in a box with external magnetic field . . . . . .
3-2 Density of states of a metal in the presence of a magnetic field . . . .
3-3 Hartree-Fock phase diagram for the two-dimensional Hubbbard model

36
42
44

3-4
3-5
3-6
3-7

Itinerant ferromagnetism in second-order perturbation calculations . . 48
Mean-field model at T = 0: Itinerant ferromagnetism . . . . . . . . . 50
Mean-field predictions for ferromagnetic phase transition at T=0 . . . 51
Schematic illustration of the ferromagnetic transition in the two-component

Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-8 Possible configurations of the ferromagnetic ground state in a twocomponent Fermi gas trapped in a spherical potential. . . . . . . . . .

53

4-1 The many-body branches near the Feshbach resonance. . . . . . . . .
4-2 The energy per particle and the thermodynamic pressure of the two-

57

component Fermi gas with repulsive interactions near Feshbach resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3 The sequence of the experiment to observe itinerant ferromagnetism.
4-4 The effect of eddy currents on the magnetic field ramp . . . . . . . .
4-5 Atom loss rate as a probe for local spin polarization . . . . . . . . . .
4-6 Kinetic energy of a repulsively interacting Fermi gas determined for
different interaction parameters kF◦ a and temperatures . . . . . . . .
10

54

60
62
63
65
67

4-7 Maximum in volume at the phase transition . . . . . . . . . . . . . . 68
4-8 The entropy difference between the normal and the ferromagnetic phase. 71
4-9 Schematic illustration of the RF quench experiment . . . . . . . . . .
4-10 Schematic illustration of the single domain experiment . . . . . . . .

75
76

5-1 Matter-wave interference and relative phase read-out . . . . . . . . .
5-2 Three different methods for the phase read-out of two Bose-Einstein85

condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3 Adiabatic RF-induced splitting . . . . . . . . . . . . . . . . . . . . .
5-4 Operation of a coherent beam splitter . . . . . . . . . . . . . . . . . .

86
88
89

6-1 Schematic illustration of the ground and the first excited state in a
double potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

6-2 Three different regimes in the description of the dynamic splitting ofcondensates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6-3 Schematic of the atom chip interferometer . . . . . . . . . . . . . . . 98
6-4 Phase evolution of the relative phase . . . . . . . . . . . . . . . . . . 100
6-5
6-6
6-7
6-8
6-9

Long phase coherence of two separated condensates . . . . . . . . . .
Quantitative study of phase coherence: Rayleigh test . . . . . . . . .
Phase diffusion and number squeezing . . . . . . . . . . . . . . . . . .
Confined atom interferometry with enclosed area . . . . . . . . . . . .
Phase fluctuations in an elongated condensate . . . . . . . . . . . . .

101
102
104
105
107

6-10 Measurement of phase fluctuations in a single condensate. . . . . . . 108
6-11 Spatial phase fluctuations and the reproducibility of of the relative phase.110
7-1 Phase-sensitive recombination . . . . . . . . . . . . . . . . . . . . . . 114
7-2 Oscillations of condensate atom loss after recombination reflecting thecoherent phase evolution. . . . . . . . . . . . . . . . . . . . . . . . . . 115
7-3 Schematic illustrations for trapping of atoms in a hollow-core fiber . . 118
7-4 Images of atoms in the optical dipole trap during the experiment . . 119
A-1 Sodium and Lithium energy levels and relevant atomic parameters. . 123
A-2 23 Na and 6 Li hyperfine structure in an external magnetic field. . . . . 124

11

List of Tables
2.1

Source of the magnetic bias field . . . . . . . . . . . . . . . . . . . . .

3.1

Density dependence in Long-range Coulomb interactions and shortrange contact interactions . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between cold atoms and solid state in the study of itinerant
ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

4.1

Prediction of critical interaction strength for itinerant ferromagnetism

12

30

40
49
73

Chapter 1
General Introduction
1.1

The Quest for Low Temperatures

The quest for low temperatures has a long history over 200 years. After Gaspard
Monge liquefied the first gas producing liquid sulfur dioxide in 1784, major efforts had
been directed toward liquefying oxygen (1877), hydrogen (1898), and helium (1908).
Helium-4, first liquified by Dutch physicist Heike K. Onnes in 1908 [140], was crucial
for the developments of two major concepts in modern physics, superconductivity andsuperfluidity. In 1911, Onnes used liquid helium-4 as a refrigerant, and discoveredthat the resistivity in solid mercury abruptly disappears at the temperature of 4.19
K [140]. He called this phenomenon as “superconductivity”. Later in 1937, Pyotr
L. Kapitza, John F. Allen, and Don Misener discovered that below the famous λtransition point at 2.17 K, liquid helium-4 becomes superfluid in which a liquid flows
through narrow capillaries or slits without exhibiting viscosity [3, 106]. Both concepts,
superconductivity and superfludity, have been actively researched in the 20th century
leading to the enhancement of our understanding of nature.

1.1.1

Bose-Einstein Condensation

The first idea of Bose-Einstein condensation (BEC) can be traced back to 1925 when
Satyendra N.Bose [28] and Albert Einstein [53, 54] predicted the occurrence of phasetransition in noninteracting bosonic particles such as photons. The phase transitionis accompanied by the condensation of atoms in the ground state. Unfortunately,
despite the novelty of this prediction, there had not been much attention paid toBose-Einstein condensation until the discovery of superfluidity in helium-4.
In 1938, Fritz London pointed out that Bose-Einstein condensation may be the13underlying phenomenon in superfluid helium-4 [130, 129]. Superfluid helium-4 was
the prototype Bose-Einstein condensate, and intense theoretical works had been developed in order to understand the relation between superfluidity and Bose-Einsteincondensation. Later, a microscopic theory for interacting Bose gases was developed
by Bogoliubov in 1947 [26]. In the 1950s, Penrose and Onsager introduced the concept
of off-diagonal long-range order and discussed the relation to a generalized criterionof Bose-Einstein condensation [149].
In parallel to theoretical works to understand the phenomenon of Bose-Einsteincondensation, the experimental efforts had been made to realize Bose-Einstein condensates in spin-polarized hydrogen without success until 1998 [61]. Recently, BoseEinstein condensation was also observed in metastable helium-4 in 2001 [165, 171].
The first experimental realization of Bose-Einstein condensate was achieved in 1995
in ultracold dilute atomic gases at JILA by Eric A.Cornell and Carl E.Wieman [6] andat MIT by Wolfgang Ketterle [47] independently. So far, BEC phase transitions havebeen achieved in dilute atomic gases of 23 Na, 7 Li, 87 Rb, 85 Rb, 39 K, 41 K, 52 Cr, 133 Cs,
170

Yb, 174 Yb, 1 H, and 4 He∗ (metastable helium) [152]. The success in dilute atomic
gases relied on the developments of laser cooling and magnetic trapping techniques in
the 1980s [12, 92]. Recently, atomic Fermi gases of 6 Li and 40 K atoms also could becooled to the degenerate temperature. This allowed for Bose-Einstein condensationof molecules consisting of a pair of fermionic atoms. The field of ultraclold diluteatomic gases keeps growing as we shall see in the following sections.

1.1.2

Ultracold Dilute Atomic Gas

Ultracold dilute atomic gases have several impressive features with respect to othersystems such as superfluid helium. Firstly, ultracold atomic gases have typically weakinteratomic interactions so that mean-field description of the system is well-suited [92].
This is one of the advantages for investigating quantum phenomena in atomic gases
which is not the case in superfluid helium.
Another feature is diluteness of the systems. The atomic density in condensatesof trapped atomic gases is typically ∼ 1013 cm3 which is much lower than the densityof atoms in liquids or solids, ∼ 1022 cm3 . The low-density of atomic gases requires
the low temperature of 10−5 K or less for observing quantum phenomenon. Liquid
helium, in contrast to atomic gases, shows quantum phenomenon such as superfluidity
near 1 K.
Despite diluteness of atomic gases, the strength of interatomic interactions can

14

be tunable either by exploiting the Feshbach resonance [108, 38] or by introducing
optical lattices [25]. Feshbach resonance occurs when a bound state is coupled toa free unbound state of colliding atoms allowing for tuning interaction strength directly in a broad range from negative to positive values. This technique opened a
new way to investigate Fermi atoms in the presence of weak attractive interactions
which is relevant to the famous theory of superconductivity developed by Bardeen,
Cooper, and Schrieffer (BCS theory) [17]. Further experiments realized the system
of molecular Bose-Einstein condensates consisting of two Fermi atoms, fermionic paircondensates in the “BCS” side, and finally an intermediate regime, so-called a BCSBEC crossover [212, 210], where the pair size of Fermi atoms is comparable to theinterparticle disctance. For comprehensive discussions, I refer to the Ref. [91]Alternatively, the effect of atom-atom interactions can be tuned in an opticallattice. An optical lattice is a periodic potential produced by a standing-wave laser
beam [25]. In this artificial crystal, ultracold atoms can mimic the strongly correlated
behavior of electrons in condensed matter. One of the most prominent experiments
in optical lattices is the observation of superfluid-Mott transition insulator transitionof bosonic gas [66]. The sufficiently strong repulsive interactions change the manybody ground state from the superfluid to the insulating phase in which the tunnelingof bosonic atoms between adjacent sites is suppressed and therefore each atom islocalized in one specific lattice site.

1.2

Next Challenges in the field of Ultracold Atomic
Gases: Quantum Simulator

In 1982, Richard Feynmann [57] proposed a quantum device, a so-called quantum
simulator, for efficient simulation of other quantum systems. The goal of quantum
simulator is to simulate a target quantum system with a controllable laboratory system governed by the same Hamiltonian. The quantum simulation is expected tosimulate quantum systems that cannot be efficiently simulated on a classical computer. There have been much attention paid to this idea because the understandingof many phenomena in various field including condensed-matter physics and highenergy physics would be enhanced through quantum simulations. To this end, major
efforts have been made to implement this idea with real quantum systems consistingof neutral atoms, ions, photons, and electrons.

15

Real physics
systmeCotrollable
knobs

Quantum
simulator

Model
Hamiltonian

Theoreticalunderstanding

Experimental
observations

Figure 1-1: Concept of the quantum simulator. The quantum simulators are controllable quantum systems in the laboratory simulating a theoretical model Hamiltonian
describing the real system.

The rapid developments in the field of dilute atomic gases over last 15 years after the first realization of Bose-Einstein condensates have featured the possibility of
studying strongly correlated phenomena of condensed-matter physics in the context
of quantum simulation. The seminal paper in 1998 by Jaksch [95] proposed the observation of the superfluid-Mott insulator transition in cold atoms in an optical lattice,which was experimentally observed in 2002 [66]. In the 2000s, the field of ultracold
atoms entered the area of strongly correlated systems. The great successes have beenachieved in the study of superfluid-Mott insulator transition of bosonic atoms in an
optical lattice [25], BCS-BEC crossover in a Fermi gas with tunable interactions [91],and Anderson localizations of bosonic atoms in a disordered potential [164, 22]. Remarkably, some of experimental investigations in those systems have been validated
through numerical studies.
Recently, the field of ultracold atomic gases keeps broadening the scope of research direction toward the realization of quantum magnetism in cold atomic gases
which can serve as a tunable model system for understanding the physics behindreal condensed matter systems. The presence or suppression of magnetic ordering isimportant not only for ferromagnetism and antiferromagnetism, but also for phenomena like high-temperature superconductivity, quantum magnets, and spin liquids instrongly correlated electron system [167]. In the system of ultracold atoms, the similar magnetic ordered states are expected to be realized when cold bosonic or fermionic
atoms are prepared in optical lattices or in a bulk system with magnetic interactions
such as exchange or superexchange interactions [123]. The magnetic interactions can
16be controlled by tuning the interaction strength via Feshbach resonance and/or by
quenching kinetic energy of the atom with an increase of the lattice depth. Of particular interest is the study of the repulsive Fermi-Hubbard model with two-component
Fermi gases trapped in optical lattices since the high-temperature superconductivity
usually occurs near the antiferromagnetic phase in complex cuprate materials. Some
promising works, demonstrating the metallic and insulating behaviors in a Fermi gas,
have been reported [173, 103], and major efforts are being directed toward the observation of the antiferromagnetic ordered state. In related works, the ferromagnetic
ordered state in a Fermi gas is also very important because it may compete withsuperconductivity near quantum phase transitions [128]. Recent experimental [102]and theoretical progresses [86, 170, 4, 182, 52, 19, 206, 114] in ferromagnetism in a
repulsive Fermi gas shall be discussed in the first part of this thesis.

1.3

Identical Particles in Quantum Theory: Bosons
and FermionsTwo particles are identical if they are non-distinguishable under physical observations
and measurements. The nondistinguishability of identical particles plays a secondaryrole in classical mechanics, but it becomes important in quantum mechanics characterizing the nature of the fundamental particle.
We now consider N identical particles governed by the Hamiltonian H. The
wavefunction ψ(r1 , r2 , ..., rN ) represents the probability amplitude for finding N particles at the position of r1 , r2 , ..., rN . Under the exchange of any pair of particles, theprobability density of finding N particles is invariant since all particles are identical:|ψ(r1 , .., ri , ..., rj , ..rN )|2 = |ψ(r1 , .., rj , ..., ri , ..rN )|2(1.1)

This derivation implies that the wavefunction ψ(r1 , r2 , ..., rN ) has an exchange degeneracy under particle exchange.
ψ(r1 , .., ri , ..., rj , ..rN ) = ±ψ(r1 , .., rj , ..., ri , ..rN )(1.2)

As a clear example illustrating the exchange degeneracy, let us consider two identical particles with their mass m governed by the Hamiltonian
H(1, 2) =P12P2+ 2 + V (|r1 − r2 |)
2m 2m
17

(1.3)

where the potential V depends only on the distance between two particles. Theenergy engenvalue equation now reads,
H(1, 2)ψ(r1 , r2 ) = Eψ(r1 , r2 )H(2, 1)ψ(r2 , r1 ) = Eψ(r2 , r1 )(1.4)

where ψ(r1 , r2 ) and ψ(r2 , r1 ) are two eigenstates of the two-particle system by recognizing the invariance of the Hamiltonian H(1, 2) = H(2, 1). Therefore, the eigenstateof the system is the superposition of ψ(r1 , r2 ) and ψ(r2 , r1 ). Without loss of generality,
the eigenstate of the Hamiltonian 1.3, ψ0 , is expressed byψ0 = α (|ψ(r1 , r2 ) + ψ(r2 , r1 )ψ(r1 , r2 ) − ψ(r2 , r1 )√√) +β (
)
2
2
|
{z
}
{z
}
ψ (S) ,symmetric(1.5)

ψ (A) ,antisymmetric(1.6)
where |α|2 + |β|2 = 1. Thus far, the eigenstate contains both symmetric (ψ (S) ) andantisymmetric (ψ (A) ) contribution. However, in contrast to above mathematical approach, only totally symmetric or antisymmetric states are observed in nature. Thisdifficulty is removed by the famous “symmetrization postulation” in quantum mechanics:“The states of a system with N identical particles are necessarily either all symmetrical (α = 1, β = 0 in the above example.) or all antisymmetrical (α = 0, β = 1) withrespect to particle exchanges”
Following this postulate, particles described by the symmetric state and the antisymmetric state are called Bosons and Fermions respectively.
Systems consisting of identical particles of integral spin (0,1,2, ...) 1 are describedby symmetric wavefunctions, and such particles obey Bose-Einstein statistics and arecalled bosons. In nature, photons, π-mesons, and K-mesons are examples of bosons.
On the other hand, identical particles of half-integral spin (1/2,3/2, ...) are called
fermions and obey Fermi-Dirac statistics. Since the system consisting of fermions is
described by the antisymmetric wavefunction, two identical fermions cannot occupy
the same quantum state leading to the “Pauli exclusion principle”.
1

The connection between spin and statistics is explained by the relativistic quantum mechanics.
Further discussions can be found in Ref [150].

18

1.4

Outline of Thesis

The experimental works in this thesis have been performed by my colleagues and me
in the BEC III lab at MIT. Our team has been supervised by Professor Wolfgang
Ketterle and Professor David.E. Pritchard.
In this thesis, I mainly discuss: (1) itinerant ferromagnetism in a Fermi gas ofultracold atoms [102], (2) apparatus upgrade for producing a Bose-Fermi mixture,(3) studies of quantum coherence in a Bose gas [101, 99], and (4) development of
atom optics elements [100, 39]. I was also involved in the other projects including“quantum reflection of BECs” [145], “optical weak link between two BECs” [176],
and “interference of BECs split on an atom chip” [179] which are described in the
Ph.D. theses of former members of BEC III [89, 147].
Outline
Chapter 2 describes the experimental procedures for producing a degenerate BoseFermi mixture. Preliminary efforts for this apparatus upgrade are discussed in the
Ph.d. thesis by Thomas A. Pasquini [147]. Here, I briefly summarize the basic
procedures and tools which are used in the current experiment.
Chapter 3 and chapter 4 discuss quantum magnetism in a two-component Fermi
gas of ultracold 23 Li atoms. In particular, the long-standing question of “itinerant
ferromagnetism” in condensed matter physics is addressed. The concept of quantum
simulation using cold atoms may apply to our work reported in this thesis. In chapter 3, I introduce some theoretical tools necessary to understand magnetic propertiesof a repulsive Fermi gas with short-range interactions. Chapter 4 mainly discusses
our experimental efforts and relevant theoretical understandings to observe itinerant
ferromagnetism in a Fermi gas. At the end of this chapter, I propose some future experiments which may overcome technical limitations and enhance our understandingof itinerant ferromagnetism in a Fermi gas.
In chapter 5, I deal with the basic concepts of Bose-Einstein condensates anddemonstrate experimental tools necessary to study the coherence properties of a Bosegas. A coherent splitter using adiabatic RF-induced potential and phase read-out
methods shall be described.
Chapter 6 discusses the quantum phase coherence of a Bose gas in a doublewell system implemented on an atom chip. The observation of number squeezing incondensates shall be described. Also, I provide a theoretical model describing the
dynamic splitting of the condensate. At the end of this chapter, I discuss the effect19

of phase fluctuations on the coherence properties.
Finally, chapter 7 deals with newly developed techniques in the field of atomoptics. First, I demonstrate the phase-sensitive recombination of two BECs which isa new method of measuring the relative phase. Secondly, I briefly describe a proof-ofprinciple experiment for trapping of ultracold atoms in a hollow-core photonic crystal
fiber.
• Publications of the Ph.D. work
• Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms
Gyu-Boong Jo, Ye-Ryuoung Lee, Jae-Hoon Choi, Caleb A. Christensen, Tony
H. Kim, Joseph H. Thywissen, David E. Pritchard and Wolfgang Ketterle
Science 325 , 1521-1524 (2009)• Trapping of Ultracold atoms in a hollow-core photonic crystal fiber
Caleb A. Christensen, Sebastian Will, Michele Saba, Gyu-Boong Jo, Yong-Il
Shin, Wolfgang Ketterle, and David Pritchard
Phys. Rev. A 78, 033429 (2008)• Matter-wave Interferometry with Phase Fluctuating Bose-Einstein Condensates
G.-B. Jo, J.-H. Choi, C.A. Christensen, Y.-R. Lee, T.A. Pasquini, W. Ketterle,
and D.E. Pritchard
Phys. Rev. Lett. 99, 240406 (2007)• Phase Sensitive Recombination of Two Bose-Einstein condensates on an atomchip
G.-B. Jo, J.-H. Choi, C.A. Christensen, T.A. Pasquini, Y.-R. Lee, W. Ketterle,
and D.E. Pritchard
Phys. Rev. Lett. 98, 180401 (2007)• Long phase coherence time and number squeezing of two Bose-Einstein condensates on an atom chip
G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, D. E.
Pritchard, M. Vengalattore, and M. Prentiss
Phys. Rev. Lett. 98, 030407 (2007)• Low Velocity Quantum Reflection of Bose-Einstein Condensates
T. A. Pasquini, M. Saba, G.-B. Jo, Y. Shin, W. Ketterle, D. E. Pritchard, T.

20

A. Savas, and N. Mulders
Phys. Rev. Lett. 97, 093201 (2006)• Optical Weak Link between Two Spatially Separated Bose-Einstein Condensates
Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard,
Phys. Rev. Lett. 95, 170402 (2005)• Interference of Bose-Einstein condensates split with an atom chip
Y. Shin, C. Sanner, G.-B. Jo, T. A. Pasquini, M. Saba, W. Ketterle, D. E.
Pritchard, M. Vengalattore, and M. Prentiss
Phys. Rev. A 72, 021604(R) (2005)

21

Chapter 2
Apparatus and Basic Procedures
2.1

Versatile Apparatus: “Science Chamber”

The third generation machine at MIT, which is called the science chamber (or BECIII), is very unique in that one produces a Bose-Einstein condensate (BEC) in themain chamber and then manipulates and diagnoses the BEC in the auxiliary chamber.
This is implemented by transporting a BEC trapped in an optical dipole trap [37, 74].
The two-step experiment - first producing the BEC and then transporting it into theauxiliary chamber - makes it possible to study the properties of the BEC in various
systems such as a double-well potential on an atom chip [112, 178, 101, 99], a doublewell optical potential [177, 166, 101, 99], or an atom-surface system [146, 145]. In
contrast to a conventional BEC machine in which optical and physical access tocold atoms are restricted due to the presence of magnetic coils and optics near thechamber, the science chamber machine has allowed us to change the experimental
system without perturbing the production of a BEC.

2.2

F=1 23Na Bose-Einstein Condensates in the
Science Chamber

In this section, we demonstrate the basic operation procedure for producing a bosonic
Na F=1 BEC. The condensate consists of the sodium atoms in the hyperfine stateof |F = 1, mF = −1i. Before the apparatus was upgraded to have a Bose-Fermi
23

mixture (bosonic 23 Na and fermionic 6 Li) in 2007, most of the experiments had been
performed with a F=1 condensate. Typical procedure for the production of the F=1
BEC includes:22• Pre-cooling atoms in magneto-optical trap (MOT) (∼3 s) : A thermal atomic
beam slowed by a Zeeman slower is continuously loaded into a dark-SPOT typemagnetic-optical trap (MOT). The atomic flux into the MOT is ∼ 1011 atoms/s
with a mean velocity of ∼ 30 m/s. In the dark-spot MOT, most of atoms lie in
the F=1 manifold.
• Forced rf-evaporation in the magnetic trap (∼30 s) : Atoms in the |F = 1, mF =−1i state are tranferred from the dark-spot MOT into a Ioffe-Pritchard magnetic trap. Subsequently, forced evaporative cooling is applied using a radio
frequency (rf) transition from the |F = 1, mF = −1i to the |F = 1, mF = 0, +1i
states. We routinely produce a F=1 condensate of 107 atoms after 30 s evaporation in the main chamber. The lifetime of the F=1 condensate in the magnetic
trap is longer than 60 s which is longer than the one for a F=2 condensate 1 .
• Trapping a BEC in an optical dipole trap (ODT): For the transport of the condensate into the science chamber, we load the condensate into an optical dipole
trap (ODT). The loading process starts with the decompression of the magnetic
trap containing the atoms for the mode-matching between the magnetic trap
and the ODT. Then, the atoms are transferred from the nearly spherical magnetic trap into the ODT adiabatically. The ODT typically contains 2 ∼ 4 × 106
atoms. The decompression and loading processes take 2 s and 0.5 s respectively.
• Transport into the science chamber (∼ 2 s): To transport the condensate
trapped in the optical dipole trap into the science chamber, the focal spotof the laser beam is spatially moved [74]. For more detailed information forthe design of the transportaion system, I refer to the Ph.D. thesis of the former
member of the Ketterle/Pritchard group [37].
• Bose-Einstein condensates in the science chamber : Once the condensate is
located in the science chamber, we initialized the condensate for a designed
experiment in the ODT. Additional cooling of the condensate in this stage
causes loss in the atom number. Finally, the atom number of the condensate is
typically 1 ∼ 2 × 106 in the scicence chamber.
For more detailed description of F=1 BEC production, I refer to previous theses [89, 147].
1

A F=2 condensate consists of atoms in the hyperfine state of |F = 2, mF = 2i.

232.2.1

Magnetic Microtraps on an Atom Chip in the Science
Chamber

Atom optics on an atom chip The development of atom chips [60, 45] has led tounprecedented control of a Bose-Einstein condensate with sub-micron precision. In
contrast to a conventional magnetic or optical trapping potential, the chip wires on an
atom chip are very close to the atoms (e.g. 10 ∼ 1000µm from the chip surface) providing much tighter and complex traps. Encouraged by this, an atom chip has evolved
into a platform integrated with atom-optics elements. Following the experimental
demonstration of magnetic microtraps and beam splitters with thermal atoms, atomoptics elements using Bose-Einstein condensates have been successfully integrated
with an atom chip. For example, an atom Michelson interferometer [196], coherent
dynamic splitting [174, 101], phase read-out through in-trap recombination [100], anddiffraction from magnetic lattices [71] were experimentally demonstrated.
In parallel to these efforts, an atom chip has been also used to study quantum
coherence in a double-well system [101], the non-equilibrium properties and phasefluctuations in one-dimensional Bose gases [83, 99, 85], and vortex dynamics in acondensate [112, 178]. More recently, atom chip technology has been combined withhigh finesse optical resonators [161].
Atom chip in BEC III Trapping neutral atoms in a magnetic potential is a wellknown technique, but it usually needs several coils generating magnetic fields. In
a conventional experiment, changing magnetic trap geometry requires to redesign
the coils , which sometimes makes it difficult to study various system with a Bose
condensate. Combining the idea of the science chamber with an atom chip, we haveroutinely redesigned the magnetic microtraps generated by wires on an atom chip 2 .
This has allowed for the studies of double-well atom interferometry [177, 166, 101],
the quantum reflection of the atoms from a solid surface [146, 145], vortex in a Bose
condensate [112, 178], integrated atom optics on an atom chip [179, 100, 101], andphase fluctuations in an elongated condensate [99]. The atom chip setup installed tothe science chamber in BEC III 3 is shown in figure 2-1.

2

Typically, a newly designed atom chip is installed in the place of the old one every 6 - 12 months.
This is the 5th generation atom chip in BEC III built in 2005. The previous chip designs can befound in Ref. [89].
3

24

(a)

Chip Mask Deisgn(b)Hollow-core Fiber

Figure 2-1: Microfabricated atom chip mask and the atom chip mounted on theholder. (a) The mask design for lithography technique is shown. The black lines
describe the pattern of current-carrying wires on the substrate. The width of the wireis 100 ∼ 200µm. The green line describes a hollow-core fiber mounted on separate
supporting material. The conducting wires are established on the substrate throughevaporation or electroplating deposition after lithography, subsequently followed by
final chemical process. More details can be found in Ref. [120, 31] (b) An atom chip
setup consisting of an atom chip, aluminium mounting structure (2” × 2” × 0.5”),and electric connections, is shown. In contrast to previous chip designs [89], electric
wires are soldered directly onto the conducting wires on an atom chip. In order toprevent solder from melting during the baking stage, silver solder with a high melting
point was used (purchased from “Amtech solder products, Inc.”). All chip-based
experiments in this thesis have been performed with the atom chip shown here.

25

2.3

Apparatus for Producing a Degenerate gas of23

Na and 6Li Atoms

In 2007, our appratus was upgraded to produce a degenerature Bose-Femri mixture.
Based on the successful experiences by other labs of the Ketterle group at MIT (BEC I
and II), we natuarally chose 6 Li atoms as fermions. In our lab, a degenerate Fermi gas
was firstly produced in October 2007. When I was preparing for this thesis, major
efforts in BEC III had been directed toward two projects including (1) the studyof itinerant ferromagnetism in a Fermi gas and (2) the production of heteronucler
ground-state molecules consisting of 23 Na and 6 Li atoms. Both experiments havebeen perforemd in the main chamber, but the idea of the science chamber (i.e. thetwo-chamber system) is still attractive and we are planning to trasport 23 Na and/or
6
Li atoms into the science chamber for the future experiments.

2.3.1

Producing a Degenerate 6 Li Fermi Gas

Cooling fermions appears to be standard now. Especially, three different machines(BEC I, II, and III) at MIT have a degenerate Fermi gas with a Bose-Einstein condensate as a coolant. In this section, a series of different techniques, implemented in
BEC III, for the reliable production of a degenerate Fermi gas will be summarized.
For more detiled descriptions of the procedure, I refer to the previous Ph.d. thesis offormer members of the Ketterle group [77, 183, 213].
New features of the dual-species apparatus in BEC III
Dual-species oven The dual-species oven in BEC III adopted the same oven design
implemented in BEC I and BEC II [183] as described in Figure 2-2. The only difference is an additional 45◦ elbow between the oven nozzle and the half-nipple lithium
cup to prevent lithium and sodium from spilling-over. The is necessary because thezeeman slower has an angle (about 45◦ ) relative to the plane perpendicular to gravity.
Except for the half-nipples for sodium and lithium, all parts including mixing/ovennozzle and mixing chamber is constantly kept at a temperature of 450◦ C. The typical operation temperature of the half-nipple cup is 370◦ C and 320◦ C for lithium andsodium respectively. Due to the small vapor pressure of lithium, the lithium source
lasts more than a few years with only 25 g. For sodium, however, the operation lifetime with 25 g sodium is rather short, less than 1000 hours. The single-species ovenwith sodium allowed for ∼2000 hours in BEC III.
26

450°C

450°C

3
4

5

6

450°C

2
7

330°C

Na

Li

1

1

320°C

370°C

380°C

8
To mainchamber

1. Half nipple
2. Sodium resovior (90 deg. elbow)
3. Mixing nozzle
4. Mixing chamber (Tee)
5. Intermediate
6. Intermediate elbow (45 deg.)

7. Oven nozzle
8. Reducer (4.5” to 2.75”): Heating band
: Flange (Ni gasket): Flange (Cu gasket)

Figure 2-2: A schematic of the dual-species oven. The dual-species oven is built from2.75 inch ConFlat fittings, made from 316SS. Customized nickel gaskets are used toseal the flanges because normal copper gaskets are likely to bond to the knife edge at
the operating temperature of 450◦ C. The nickel gasket allows us to disassemble theoven and reuse the flange. For more technical details, please see Ref. [183].

Dual-species magneto-optical trap (MOT) To implement a new Li MOT in
the main chamber, each sodium beam in the MOT optics was combined with a newlithium beam delivered through PM optical fibers4 . To this end, one mirror in eachpath of sodium MOT light was replaced by a dichroic beam splitter, which are transparent for the wavelength of 589 nm and reflective for 671 nm. After lithium andsodium lights are overlapped, all the half and quarter waveplates are replaced byzeroth-order waveplates designed for the wavelength of 633 nm. This makes it possible to manipulate the polarization for both wavelengths identically. The wrong polarization effect due to non-ideal wavelength is typically less than a few percent [77].
In contrast to the dark-spot MOT for sodium, the lithium MOT is bright; each MOT
beam for lithium contains both MOT and repumping lights. The typical dual-species
MOT is shown in figure 2-3.
New skimmer on the cold plate An old skimmer on the cold plate was built forthe single species oven (with sodium). After replacing the old single-species oven by
a new dual-species one, the unexpected deposition of sodium on the skimmer leaded
to the shorter lifetime (less than 500 hours) of the sodium in the oven. To avoid thedeposition, we decided to redesign a skimmer as described in appendix B. At the4

PM means “polarization-maintaining”

27

Na MOT

Li MOT

Na MOT

Li MOT

F=2 Na BEC

Li Cloud

Figure 2-3: Dual-species magneto-optical traps (MOT) are shown for 23 Na and 6 Li
respectively. A CCD camera with color fiters monitors the dual-species MOT fromthree different angles. Typical absorption images of the 23 Na and the 6 Li cloud
released from the magnetic trap are shown with the time-of-flight of 45 ms and 6 ms
respectively.

moment when I wrote this thesis, a similar deposition of sodium on the new skimmer
was observed after ∼800 hours running and the problem is still being assessed.
Sympathetic cooling with a F=2 sodium Bose-Einstein condensate The
rule of thumb in sympathetic cooling in a magnetic trap is that both 23 Na and 6 Li
must lie in the low-field seeking state. This allows us to simply add fermionic 6 Li
atoms in the process of evaporation cooling for 23 Na. To this end, two possiblecombination of hyperfine state are:(a) |F = 1, mF = −1iN a + |F = 1/2, mF = −1/2iLi(b)|F = 2, mF = 2iN a + |F = 3/2, mF = 3/2iLi
The other combinations suffer from spin-exchange collisions causing unwanted
heating. In BEC III, we decided to choose the combination of (b) based on theexperiences of BEC I and II. In this case (stretched state), 6 Li atoms in the state of|F = 3/2, mF = 3/2i can be stably trapped in a magnetic potential allowing effecient
sympathetic cooling of hot thermal 6 Li atoms 5 .
5Here, 6 Li atoms in the state of |F = 1/2, mF = −1/2i are low-field seeking only below 28 G.

28First, a spin-polarized sodium cloud in the state of |F = 2, mF = 2i (F=2 condensate hereafter) is achieved by optical pumping from the F = 1 manifold to theexcited manifod (F pumping). The hyperfine pumping light with σ + polarization illuminates tha atom cloud for 2 ms right after switching off the MOT light. For 6 Li, two
pumping beams resonant with the F=1/2 and F=3/2 states are used for 400 µs toprovide both F (hyperfine) and mF (Zeeman) pumping. During the pumping stage,
the polarization gradient cooling was omitted to improve the efficiency of lithium
loading into the magnetic trap.
Before the Ioffe-Pritchard trap is turned on, the 23 Na atoms in the other hyperfine
states are optically pumped into the stretched state. For the purification, the 23 Na
atoms are illuminated by F pumping light for 2.5 s in the presence of a 1.77 MHz
microwave sweep at the bias field of 80 G. Subsequently, the second purificaton isdone only with a microwave wave sweep for 0.5 s. Finally, ∼ 30% 23 Na atoms in the
MOT are transferred into the magnetic trap. For 6 Li atoms, without the purification
stage, the transfer efficiency fom the MOT into the |F = 3/2, mF = 3/2i is about∼ 40%.
Preparation of a repulsive two-component Fermi gas of 6 Li atoms After
sympathetic cooling, we routinely produce a spin-polarized Fermi gas of ∼5×106
atoms in the stretched state of |F = 3/2, mF = 3/2i (hereafter |6i state, also see
appendix A) in a magnetic trap. The typical temperature of the 6 Li cloud is T ∼0.5TF where TF is the Fermi temperature. Then 6 Li cloud is transferred into an
optical dipole trap, followed by an RF transfer into the lowest hyperfine state |F =1/2, mF = 1/2i. Finally, a Landau-Zener RF sweep is applied in order to prepare aequal mixture of |F = 1/2, mF = 1/2i (|1i state) and |F = −1/2, mF = 1/2i (|2istate) spin states. Additional evaporation cooling in the mixture of |1i and |2i atoms
has been implemented at 300 G after decoherence for 1 s.

2.3.2

Li Laser System

The new 6 Li laser system consists of a master laser and four slaver lasers. The
master laser is a commercial grating stabilized diode laser purchased from TOPTICA(model: TOPTICA DL100). The master laser gives us an output of ∼30 mW, andthe linewidth is reduced to a typical value of 1 MHz (measured in a millisecond) whichis reasonably smaller than the lithium natural linewidth of Γ = (2π) × 6 MHz. The
master laser locked to the 6 Li D2 -resonance is amplified by injection locking of fourslave lasers. The slave laser diode placed in the TEC cooled mount (Thorlabs: model
29

(a)

(b)

To Science
Chamber

14 cm diameter

C
C

M
Cuv

A

ODT
Atoms

Vacuum+Z (gravity)+y (BEC I
side)

Air

Cloverleaf (C)Anti-bias (A)MOT(M)
Curvature (Cuv)

Anti-Bias Cloverleaf Curvature
6 turns 4 turns
6 turns
3 layers 3 layers
8 layers

MOT
6 turns
7 layers

Figure 2-4: A schematic of magnetic coils in the main chamber is shown. (a) Geometryof the magnetic coils is described. The coils are implemented into the re-entrant
bucket port. A photo of one of coil setes is shown in (b). A large magnetic field for
the Feshbach resonance is produced by the current carried through the anti-bias coil.

TCLDM9) produces an output of ∼150 mW at desired wavelength of 670.977 nm
while retaining the narrow spectral characteristics of the master laser. The slave
diode6 operates at a temperature of ∼70◦ C and at a current of 370∼390 mA depending
on injection locking.
Power supply unit
curvaturebias
y-compy-trap bias
spin-flip
Feshbach

Connected coils
curvature and antibias
only curvaturecompensationcompensationcompensation
antibias

Magnetic field direction
(+)
(+)
(+)
(-)
(+)
(-)

Table 2.1: The direction of the magnetic bias field is described with the informationof the coil and the power supply generating the field. The (+)-direction points in thesame direction as +y in figure 2-430

2.3.3Tools for Experiments(1) Generating a large bias field for the Feshbach resonanceOne of the main tools in our dual-species apparatus is a Feshach resonance with which
one can tune the atom-atom interaction strength by changing a bias magnetic field. A
Feshbach resonance occurs when the scattering state in the open channel approaches
the bound molecular state in the closed channel energetically [108] leading to strong
mixing between two channels. By changing a magnetic field, the energy difference
associated with difference in magnetic moments can be controlled, which is called a“magnetically tuned Feshbach resonance”.
A simple form describing a magnetically tuned Feshbach resonance was introduced
in Ref. [132] as follows:∆a(B) = abg (1 −)(2.1)
B − B0where B is a magnetic field, a denotes the s-wave scattering length, abg is the background scattering length which is associated with the open channel potential, ∆ isthe resonant width, and B0 is the resonant magnetic field where a diverges. In BEC
III, we are interested in two homonuclear and hetroneclear s-wave resonaces: 834 G
for 6 Li atoms 7 and 795.6 G for 6 Li-23 Na atoms.
A large bias field for the Feshbach field is generated by the currents carried throughthe antibias coil (see figure 2-4 and table 2.1). Tha Feshbach field can be ramped
upto 1500 G with the maxium current of 500 A. The geometry of magnetic coils in
the main chamber is descirbed in figure 2-4, and the field direction generated fromthe coils are summarized in table 2.1. A stabilization scheme for Feshbach fields is
illustrated in figure 2-5.
(2) Optical dipole trap
In 6 Li experiments, Li atoms in states |1i and |2i are both high-field seeking athigh magnetic field where the two states are strongly interacting with each other.
Unfortunately, it was shown by Wing [200] that the magnitude of static electric or
magnetic field in free space cannot have local maxima; therefore, they cannot trap
strong-field seeking atoms. Thus, the strongly interacting mixture can be stored onlyin an optical dipole trap (ODT) as described in figure 2-4. The ODT is produced by
a commercial fiber-amplified Nd:YAG laser from IPG (model:YLR-30-1064-LP-SF).
6

The slave diode (model:ML101J27, Mitsubishi Laser Diodes) has a typical lifetime of a few
months,7This Feshbach resonance occurs between |1i and |2i state.

31

Curvature coils

Bias
Power
SupplyCurv.
Power
SupplyAnti-bias coils

IGBT
Feshbach
Power
SupplyFeedback
loop

PID

Hall Probe

Figure 2-5: Circuit diagram for producing a large bias magnetic field (called the
Feshbach field). A Feshbach power supply is connected to anti-bias coils to generatea large bias field. The Feshbach power supply run in voltage control mode providing
an arbitrary current depending on the resistance. The current is PID-controlled via
an IGBT operating in the linear resistance regime. The direction of the Feshbach
field is opposite to one of curvature/bias as described in table 2.1. In this diagram,the “de-bounce” circuit is omitted.

32

The laser light has single frequency at 1064nm and fixed polarization out of the laser.
The maximum output power of 30 W will be sufficient for future experiments in boththe main and science chamber.
(3) Science chamber designed for optical lattice experiments
Until summer 2007, the science chamber had been replaced several times when theexperiments required special features as described in other theses [89, 147]. For example, it was required to have a rather spacious science chamber with 6” OD windows
in order to facilitate various atom optics experiment including atom interferometry
and quantum reflection. The spacious science chamber allowed us to assemble twodifferent experiments at the same time without any conflict. Two experiments - coherent splitting of the condensate and trapping of cold atoms in a hollow-core fiber
covered in this thesis - were performed in the same setup built in the science chamberwith 6” OD windows.
Despite flexibility of the spacious science chamber, it was difficult to image atoms
with high resolution since the closest lens for imaging cannot be close enough to theatoms due to the 6” OD window of the chamber. This would be a potential problem
in future experiments which may require single-site imaging in the optical lattice.
Markus Greiner group at Harvard solved this problem by preparing an atomic cloudright below the window surface using a combination of evanescent wave, standing
wave, and magnetic potential [64, 16].
To solve this imaging problem in BEC III, we adopted an idea similar to the oneused in the main chamber as described in figure. 2-6 and appendix B. We installeda re-entrant “bucket” ports so that the closest lens of the imaging systems can beclose enough to the atoms. The distance between the first lens and the center ofthe chamber (or the position of atoms) is about 1.5“ which will allow us to improve
imaging resolution. The bucket ports were built by UKAEA who welded the 1.33” OD
viewport window onto the bucket ports supplied to them by Sharon vacuum. The
viewport window was polished to lambda/4. A detailed description of the sciencechabmer is available in appendix B.

33

(a)

(b)

Metal bellow

SciencechamberMainchamber

Figure 2-6: (a) The science chamber is newly designed solely for flexible optical access.
The chamber is connected to the main chamber through a metal bellow as shown in(b). The bellow mechanically isolates the main chamber from the science chamber.

34

Chapter 3
Ferromagnetism in a Fermi Gas
Can a gas of spin-up and spin-down fermions become ferromagnetic due to repulsive
interactions?
This question exhibits a long-standing problem which has not found a definitive theoretical answer. In the following two chapters, we shall discuss the theoretical andexperimental efforts to find a rigorous answer in the system of a two-component Fermi
gas. To begin with, we review some basic aspect of quantum magnetism in a Fermi
gas.

3.1

Quantum Magnetism

3.1.1

What is magnetism?

Magnetism is a macroscopic phenomenon with its origin deeply rooted in quantum
mechanics. In the early 1910’s [27], Bohr pointed out that a collection of classicalelectrons in a box can never be sensitive to the external magnetic field; therefore any
solid consisting of electrons should be non-magnetic in the classical sense. This isclearly illustrated by calculating the classical partition function Z of N electrons:
Z
Z∝

d3N xd3N p exp(−35~ 2β(~p − A)− βV (~x))
2m

(3.1)

Figure 3-1: Classical electron gas in a box with external magnetic field is shown. The
red dot describes the classical electron and the black circle denotes the electron’s orbit.
In the presence of magnetic field, the electrons move in a circle generating magnetic
moments. The total magnetic moment of the electron gas in the box, however, isalways zero.

~ = A(x)~As the vector potential A
is independent of p~ classically 1 , the partition
function splits into two parts:
Z
Zclassical ∝

Z
d

3N

x exp(−βV (~x))

d3N p exp(−
|
{z

~ 2
β(~p − A)
)
2m
}

(3.2)

constant~ is constant leading to a conclusion that the parThe second integral containing A~ ≡ Z(A~ = 0).
tition function Zclassical is insensitive to the vector potential A: Z(A)Therefore, classical description is insufficient to explain the magnetic properties in asolid. A collection of classical electrons in a box is a good example to emphasize thisas described in figure 3-1.

3.1.2

Origin of Ferromagnetism: Exchange interactions

The phenomenon of “spontaneous magnetization” or ferromagnetism, is the consequence of magnetic interactions between the discrete sources of magnetic moment.
Now, we have a question: “Where do the magnetic interactions come from?”. Our
first naive expectation is that the dominant magnetic interactions arise from dipole1

In quantum mechnics, the vector potential is not independent of p~ due to [x, p] 6= 0.

36

dipole interactions between discrete magnetic moments. These interactions between
dipole moments, however, are too weak to induce the magnetic structure. The magnitude of dipole-dipole interaction energy, Udipole can be estimated by considering an
atomic magnetic dipole moments m1 and m2 separated by r. For a typical solid,
U=

1(gµB )2
[m
·
m
−
3(m
·
r)(m
·
r)]
∼
∼ 10−4 eV
1
2
1
2
r3
r3

(3.3)

The value of 10−4 eV is equivalent to 1K in thermal energy. Therefore the dipolarinteraction is typically too weak to induce spontaneously magnetization or ferromagnetism in a solid whose ferromagnetic transition temperature is lager than 10K, even
1000K for iron.
What is the origin of spontaneous magnetization in a solid? Interestingly, magnetic fields are irrelevant to magnetic ordering of adjacent magnetic moments. Instead, Coulomb repulsion between electrons plays the important role. This seems
to be unreasonable because the Coulomb interactions are spin-independent (or nonmagnetic interactions). The problem is, however, reconciled by the Pauli exclusion
principle imposing a strict correlation between (spin-independent) Coulomb interactions and the total spin of the system. The emergence of magnetic interactions
between electrons from Coulomb repulsion becomes more clear in the calculationsof the energy of a two-electron system 2 . Since the total wavefuction of the twoelectron system must be antisymmetiric under particle exchanges, a spin singlet anda spin triplet states have different symmetries of the spatial wavefunction. For example, a spin triplet state, φT , has a antisymmetric spatial wavefunction and theenergy of the triplet state, εT , can be calculated by εT = hφT |Ĥ|φT i where Ĥ is the
spin-independent Hamiltonian of the system including Coulomb interactions. The
Hamiltonian acts only on spatial degree of freedom. Similarly, a spin singlet energy
becomes εS = hφS |Ĥ|φS i. Now, the difference between triplet and singlet energies
determines the tendency toward ferromagnetism.
εT − εS = −J

(3.4)

When J is positive, the groud state of the two-electron system becomes ferromagnetic.
Here, we call J the exchange energy leading to magnetism from electrostatic Coulomb
repulsion. The exchange interactions are nothing but the interplay of electrostatic
Coulomb interactions and the Pauli exclusion principle. The exchange energy denotes
2This calculation was done by Heitler and London in 1927. For more details, I refer to Ref. [49,134, 109, 198]37

the contribution to the Coulomb interaction energy of electrons associated with theuse of symmetric and antisymmetric wavefunctions.
Heisenberg model So far, we have seen that the Pauli exclusion principle correlates the symmetric and the antisymmetric spatial wavefuction with the spin wavefunctions leading to the difference of Coulomb repulsion energy for the spin singlet
and the triplet states. Now, the idea of two-electron system gives us construction
blocks for many-electron system. The exchange interaction can be expressed as aneffective spin Hamiltonian acting only on the spin degree of freedom. Since the spinspin interaction depends only the relative direction between two spins, the effective
spin Hamiltonian, Ĥef f , which is identical to the original Hamiltonian acting only on
the spatial degree of freedom becomesĤef f = −J Ŝ1 · Ŝ2(3.5)

where J is exchange energy. For a many-electron system in a lattice, the effective
Hamiltonian takes the formĤef f = −X

Jn,n0 Ŝn · Ŝn0

(3.6)

n,n0

where n denotes the nth lattice site. This model, so-called Heisenberg model, is thestarting point for further investigation of ferromagnetism and antiferromagnetism ininsulators.

3.1.3

Two Paradigms of Magnetism

In condensed matter physics, there are two paradigms for magnetism: localized spinsinteracting via tunnelling, and delocalized spins interacting via an exchange energy.
For localized spins, the magnetic phenomenon arises from the interplay of localized
electrons. In this case, the exchange interactions we have just seen above come fromeither the direct Coulomb interaction among electrons from different ions or indirect
coupling between magnetic ions mediated by nonmagnetic ions. The latter typeof magnetic interaction is called superexchange. Major open questions for localized
spins include the interplay of magnetism with d-wave superfluidity and frustrated
spin materials [115].
The delocalized spins gives rise to itinerant ferromagnetism which is responsible forthe properties of transition metals like cobalt, iron and nickel. Among the conduction38

electrons in metals, the electrons are no longer localized but rather itinerant. The
exchange interaction among the free electrons is called itinerant exchange, and this is
the main ingredient for itinerant ferromagnetism in a free electron gas as we will see
later.
Note that both kinds of magnetism involve strong correlations and/or strong interactions and are not yet completely understood. For example, phase transition theories
are still qualitative for itinerant ferromagnetism [24, 187, 124, 32, 194, 190, 188]. The
experimental and theoretical works for itinerant ferromagnetism shall be discussed inthis chapter and the next chapter 4.

3.2

Itinerant Ferromagnetism

3.2.1

Long-standing Problem in Condensed-matter Physics

Magnetism in a free electron gas
Itinerant ferromagnetism in a free electron gas has been a long-standing problem
in condensed-matter physics. Although the theory of magnetism in a free electron
gas is inadequate to describe magnetism in a real metal (due to a over-simplified
model), it is still of importance because of several reasons: (1) the model for itinerant
ferromagnetism does not require band structure and spin-dependent interactions, (2)it represents itinerant aspect of exchange interactions, and (3) it may demonstrate
the ferromagnetic behavior in transition metals 3 .
The origin of ferromagnetism in real material has been one of the mystery for a longtime [49]. Heisenberg first suggested that ferromagnetism is a quantum many-bodyeffect solely arising from the interplay between the spin-independent Coulomb repulsion and the Pauli exclusion principle, generating “exchange interaction” [201]. In theearly 1930’s, Stoner proposed that the magnetism may coexist with one-electron band
theory [187]. He pointed out that exchange interactions between overlapping electrons
favor ferromagnetism in an itinerant electron gas. A fer years earlier, Bloch [24] firstpointed out in the Hartree-Fock approximation that a free electron gas with long-range
Coulomb interactions may become ferromagnetic when rs > 4 where rs ∝ 1/n(1/3)
is the Wigner-Seitz radius , depending only on the density(n) of the electron gas.
However, it was pointed out that Bloch’s calculation may ignore the strong effect of
3

Note that it is widely believed that the ferromagnetism in some transition metals arises fromthe interplay between itinerant ferromagnetism, orbital degeneracy associated with band structure,and Hund’s rule [49, 134, 109, 198]

39

Kinetic energy (Ekin )Interaction energy(Eint )Long-range Coulomb interaction∝ n2/3∝ n1/3

Short-range contact interaction∝ n2/3∝nTable 3.1: Density dependence in Long-range Coulomb interactions and short-range
contact interactions is shown. For long-range Coulomb interactions, itinerant ferromagnetism is expected to occur at low density n as Eint /Ekin > 1 only when n ∼ 0.
The density dependence for short-range interactions is opposite; itinerant ferromagnetism is expected to occur at high density n À 1.
correlations which reduce the gain in interaction energy in a ferromagnetic state [199].
Recently, it was shown by quantum Monte Carlo (QMC) calculations [209, 141] thatthe ferromagnetic phase transition in a three-dimensional electron gas at zero temperature occurs only at the very low density of rs ∼ 50, suggesting that the effect ofcorrelations should be taken into account for calculating the critical density.
A free electron model has often been improved by including the screening effectbetween electrons [49, 134, 109, 198]. For more simplicity, the screened Coulomb interactions are sometimes approximated by short-range contact interactions (or hardsphere interactions). For such interactions, a rigorous calculation of the true groundstate does not exist yet. Variational approximations [203], however, support the
Hartree-Fock results. In contrast to bare Coulomb interactions, itinerant ferromagnetism may occur at high atomic density of the electron gas with short-range contact
interactions (see table 3.2.1).
In parallel to these efforts, ferromagnetism in Hubbard models [98] has been actively studied [127, 136, 125, 189]. The Hubbard model allows us to describe thequantum correlations in a proper way, leading to rigorous calculations of the true
ground state. For example, Elliott H. Lieb and F. Y. Wu showed that the true groundstate in the one-dimensional lattice system should have antiferromagnetic configurations [127]. Some other mathematical proofs have been achieved in the Hubbard-typemodel under certain conditions. For more discussions, please see section 3.2.3. But,
the question of “whether spin-independent Coulomb interaction can induce ferromagnetism in an itinerant electron system” has not yet been rigorously answered for thefree gas.

40

3.2.2

Stoner Model in a Free Electron Gas with Short-range
Interactions

Itinerant Ferromagnetism often occurs in a “metallic magnetic material”. Here, themetallic nature captures the itinerancy of electrons in a conduction band. In thissection, we will discuss the Stoner model explaining the ferromagnetic instability
in a free electron gas. The metallic state is characterized by the kinetic energy of
conduction electrons, imposed by the Pauli principle.
Short-range contact interations
The Stoner model can be derived by estimating total kinetic and interaction energy ofthe electron gas. Since electrons are itinerant, each single electron does not feel bare
Coulomb repulsion produced by other electron. Instead, the metallic state profoundly
modifies the Coulomb interactions between electrons, leading to a rapid decrease in
the Coulomb repulsion beyond some characteristic distance, the Thomas-Fermi length
λT F . The phenomenon, called screening, introduces a simple estimate of the Coulombe2
r between electrons is screened
interaction. The bare Coulomb potential U (r) = − 4πε0
by other electrons, and becomes exponentially weaker beyond λT F ,Uscr (r) = −e2 exp(−qT F r)
4πε0
r

(3.7)

where λT F = 2π/qT F . In typical transition metals such as iron, cobalt, and nickel,
Thomas-Fermi length λT F is less than the mean distance between electrons; therefore,qT F À q for short-range of the potential. In the form of the screened Coulomb
potential in momentum space
Uscr (q) =e2
12
ε0 q + qT2 F(3.8), one can simply approximate the screened Coulomb potential Uscr (q) to a constantneglecting the term q 2 . Then, the screened Coulomb potential reduces to a contact
interaction
U (r) = Iδ(r).
(3.9)
To evaluate the response of the electron system in the presence of a magnetic
field, we now calculate the magnetic susceptibility χ. The behavior of the magnetic
susceptibility clearly illustrates the role of short-range repulsive interactions in the
ferromagnetic phase transition.
41

(a)

(b)

N (E)

N (E)

δε

EF

EF

E

N (E)

E

N (E)

δε

Figure 3-2: Density of states of a metal. (a) In the presence of a magnetic field, thedensity of states for spin-up and spin-down states shift in the opposite directions. This
leads to a paramagnetic behavior. (b) With introduction of exchange interactions,the system of electrons are unstable toward a spontaneously magnetized state.

(1) Pauli paramagnetism (without Coulomb repulsion)
In the presence of the external magnetic field B, the electrons in the slice of energy
thickness δε = µB B are transferred from one spin-state to other [see figure 3-2 (a)].
The net magnetic moment in equilibrium becomesM = (N (εF )µB B × 2µB = 2(µB )2 N (εF )B(3.10)

where N (εF ) is a density of state (DOS) at the energy of εF Therefore, Pauli spin
susceptibility at T = 0 4 becomesχP auli = 2µ0 (µB )2 N (εF )(3.11)

clearly showing that there is no instability.
(2) Stoner model (with Coulomb repulsion)
Introducing Coulomb repulsion, we can examine the instability of the paramagnetic
state towards the ferromagnetic state. Here, we assume that Coulomb interactions
reduce to short-range contact interactions as describe in the relation 3.9. Thus theinteraction energy can be written as Eint = In↑ n↓ , where n↑ (n↓ ) is the density ofelectrons with the spin-up (spin-down) state and I denotes the reduction in interactionenergy caused by reversing a spin. When electrons in a slice of energy thickness δε4

The finite temperature effects change the density of state and displace the Fermi level. This
gives: χ(T ) = χ(T = 0)(1 + O(T 2 ))42

are transferred as in figure 3-2 (b), the kinetic energy of the electron gas increases:∆Ekin = N (εF )(δε)2(3.12)The interaction energy, however, decreases:∆Eint = I[

NNN
+ N (εF )δε][ − N (εF )δε] − I( )2 = −IN 2 (εF )(δε)22
2
2(3.13)

Therefore, total energy changes with introduction of the perturbation,∆Etotal = N (εF )(δε)2 (1 − IN (εF ))(3.14)

This implies that the paramagnetic state is energetically stable only when IN (εF ) <1. When interaction strength I is strong enough satisfying the criterion of IN (εF ) >1, the electron system undergoes a ferromagnetic phase transition. The spin susceptibility becomes:
χStoner =χP auli.
1 − IN (εF )(3.15)

This corresponds to the many-body correction to the Pauli susceptibility leading to
the Stoner’s criterion,
IN (εF ) > 1(3.16)

where itinerant ferromagnetism occurs in a free electron gas. This Stoner’s criterion involves microscopic aspects of itinerant exchange, through the term I, and thenumber of electrons, through the N (εF ). Stoner’s model is universal in that it depends only on the two parameters. Note that two major ingredients for itinerant
ferromagnetism are the Pauli principle and the Coulomb repulsion.

3.2.3

Ferromagnetism in Hubbard Models

The stoner model is generally expected to apply to systems with broad bands orpartially filled bands. As the bands become narrower, atom-atom correlation effects
become more important. In this case, it is much more convenient to describe the system in the Wannier basis. For this purpose, the Hubbard model [98] was proposed by
British physicist, John Hubbard, in the early 1960s to describe narrow band metals
in which the repulsive Coulomb interaction between two electrons at the same site
is likely to suppress the hopping between adjacent sites 5 . The model address both5

At about the same time, Martin Gutzwiller proposed a very similar model as well.

43

<n>
Figure 3-3: Hartree-Fock phase diagram for the two-dimensional Hubbbard model.
A, F, and P denote the antiferromagnetic, ferromagnetic, and paramagnetic groundstate respectively. From Ref. [82]the electronic quantum correlations and some degree of electron mobility leading tomagnetism which are not fully described in the tight binding model or classical Isingmodel [109]. The Hubbard model has been widely believed to be the basic ingredient in order to explain two major phenomena : quantum magnetism (for example,
ferromagnetism and antiferromagnetism ) and electrical conductivity (for example,
high-Tc superconductivity) in condensed matter.
The model has a simple Hamiltonian,H = −tXa†iσ ajσ + U

Xni,↑ nj,↓(3.17)

ihi,jiσwhere aiσ (a†iσ ) is an fermionic annihilation (creation) operator of spin σ on the sitei and n = a†iσ aiσ denotes the occupation number of the Wannier state φ(Ri , r). The
Hamiltonian has two competing terms which are described by the hopping amplitude
t between neighboring lattice sites and the repulsive on-site interaction U between
atoms with opposite spin. For low filling factor, the Hubbard model reduces to theStoner model when one uses the Bloch basis φ(k, r) and 2I in place of the Wannierbasis and U .
44

Can the Hubbard model favor particularly ferromagnetism or antiferromagnetism?
The answer is “YES in one-dimensional (1D) systems, but NOT always in othercases.”. In spite of the richness in quantum phases, an exact solution only exists
in one-dimensional system [127]. Lieb and Wu proved that the ground state in the
1D system must have antiferromagnetic configuration. In the 2D system, an antiferromagnetic ground state is expected for half-filling hni = 1 in the Hartree-Fockapproximation as described in figure 3-3.
On the other hand, the Hubbard model possesses a ferromagnetic ground state
under certain conditions. Nagaoka rigorously proved that the very strong repulsive
interactions induce ferromagnetism if the occupation number hni is slightly shortof half-filling [136]. Even with a single vacancy, the ferromagnetism occurs when
U 7→ ∞. Another rigorous model was discovered by Lieb [125] showing that a dispersionless flatband supports ferromagnetism. Recently, Tasaki has concluded that aferromagnetic ground state is favored in the Hubbard model when the electron can
hop farther than the nearest-neighbor sites [190].

3.2.4

Similar Studies in Normal Liquid 3 He

The seminal discovery of the exotic superfluid phases of 3 He in 1971 [143] stimulated
both experimental and theoretical researches on the normal phase of 3 He as well as
the superfluid phases [121, 193]. The normal liquid 3 He has served as a model systemof strongly interacting fermions. Due to the high density, the interatomic interactions
are approximated by short-range contact interactions neglecting an attractive Van der
Waals tail.
The properties of liquid 3 He can be described by Landau Fermi liquid theory [157]which deals with an interacting Fermi liquid phenomenologically. The theory of Landau assumes that the ground state of an interacting Fermi liquid consists of quasiparticles (or elementary excitations) characterized by an effective mass m∗ and an
effective interaction parameterized by Landau parameters. Here, quasiparticles obey
the same distribution function of a non-interacting Fermi liquid. In this frame work,
several physical quantities such as the specific heat, the spin susceptibility, and thecompressibility are given as a function of Landau parameters and m∗ . For example,
the spin susceptibility χs becomes,
χs =

m∗
m

1 + F0a45

χ0(3.18)

where χ0 denotes the spin susceptibility for a noninteracting system and F0a is one of
the Landau parameters.
Major efforts have been made toward an understanding of the properties of normal 3 He [121, 193]. In particular, the enhancement in the spin susceptibility, χs ,of normal 3 He has stimulated many theoretical and experimental discussions. In
the expression 3.18, F0a has a negative value indicating the enhancement of the spin
susceptibility due to the ferromagnetic contribution arising from the Pauli exclusion
principle. When F0a 7→ −1, the diverging spin susceptibility implies a phase transition
to a ferromagnetic state. The experimental measurement of χs [68], however, showsthat the spin susceptibility is enhanced only by F0a ' −0.75 for all pressures 6 .
The enhancement of the spin susceptibility can be described in the “paramagnon
theory” [111, 7]. In this theory, 3 He is considered as “almost ferromagnetic” and its
properties are associated with the incipient ferromagnetic phase caused by low-energy
spin fluctuations. On the other hand, the same enhancement of the spin susceptibility can be understood as a result of “localization due to strong correlations” [8].
In this picture, 3 He is considered as “almost localized” (and therefore stays in theparamagnetic phase) since the effective mass m∗ increases with pressure whereas theLandau parameter F0a is almost constant in the experiment.

6

At a pressure of ∼34 bars, liquid 3 He solidifies.

46

3.3

Itinerant Ferromagnetism in Ultracold Atomic
Gases

A Fermi gas of ultracold atoms as a model system
An important recent development in cold atom science has been the realization of superfluidity and the BCS-BEC crossover in strongly interacting two-component Fermi
gases near a Feshbach resonance [91]. These phenomena occur for attractive interactions for negative scattering length and for bound molecules (corresponding to a
positive scattering length for two unpaired atoms). Very little attention has been
given to the region for atoms with strongly repulsive interactions. One reason is that
this region is an excited branch, which is unstable against near-resonant three-bodyrecombination into weakly-bound molecules [153]. Nevertheless, many theoretical papers have proposed a two-component Fermi gas near a Feshbach resonance as a model
system for itinerant ferromagnetism [86, 170, 4, 182, 52, 19, 206, 114] assuming thatthe decay into molecules can be sufficiently suppressed.
In contrast to the solid state system, the study of itinerant ferromagnetism ina strongly interacting Fermi gas should address the effect of trap geometry and theimposed spin polarization defined by the population imbalance for each spin state assummarized in Table 3.2. For sufficiently large clouds (therefore neglecting surface
effects), trapped Fermi atoms can be simply treated as if each atoms are governed
by local Fermi energy and interactions imposed by other atoms. This approximation,so-called “local density approximation (LDA)”, allows us to use the result obtained
for the homogeneous Fermi gas.
Another feature of the atomic system is the pseudo-spin associated with two different hyperfine states. In an atomic Fermi gas, two lowest energy states are generally
selected as pseudo-spin states; therefore spin-spin relaxation collision is prohibited energetically. In this configuration, the two-component Fermi gas is also stable against
dipolar relaxation due to the Pauli exclusion principle [91].
The imposed population imbalance for each pseudo-spin state realizes a canonicalensemble with the fixed number of particles whereas solid state systems usually are
represented by a grand canonical ensemble with the chemical potential. In the case ofbalanced Fermi mixture, the net spin polarization is zero which corresponds to zeroexternal magnetic field in solid state systems (see Table 3.2).
One of the attractive feature in the cold atomic system is the unprecedented control of two-body interactions via the Feshbach resonance [38, 108]. There had been47

Figure 3-4: Itinerant ferromagnetism in second-order perturbation calculations. Magnetization is calculated as a function of kF a, for various temperatures. From left toright T /TF =0, 0.1, 0.15, 0.2, and 0.25. The magnetization undergoes the first-orderphase transition at low temperatures (the dashed line in the graph). The criticaltemperature for the ferromagnetic phase transition is shown as a function of kF a in
the inset. The solid line (the dotted line) denotes first-order transitions (second-ordertransitions ). From Ref. [52]many theoretical works studying a two-component Fermi gas with repulsive interactions as a model system of itinerant ferromagnetism. The phase separation in a
two-component Fermi gas has been studied by Salasnich et al. [170] and Sogo andYabu [182]; they treated harmonically trapped atoms in the Thomas-Fermi approximation. Duine and McDonald characterized the ferromagnetic phase transition in a
homogeneous Fermi gas, showing that the transition changes from the second orderto the discontinuous first order at the very low temperature [52] as summarized infigure 3-4. They indeed treated the effect of fluctuations in second-order perturbation
theory, and pointed out that Hartree-Fock theory may underestimate the tendency
towards ferromagnetism. Recently, the ferromagnetic phase transition and the spin
structure in the ferromagnetic state have been investigated in a Fermi gas in a harmonic trap [114, 19, 44]. In addition, some aspects of nonequilibrium dynamics ina Fermi gas have been studied when a non-interacting Fermi gas undergoes a rapid
quench to the repulsive side of a Feshbach resonace [14, 44]. Itinerant ferromagnetismin a Fermi gas has been also investigated in optical lattices. By engineering the band
structure to have a flat band in a two-dimensional honeycomb lattice, one can realize
a ferromagnetic phase in a two component Fermi gas [207, 195].

48

Properties
Thermodynamic ensemble
B Field

Cold atomic system
N1 ,N2 ,E fixed
Balanced |1 > −|2 > mixtureInter-particle
interactions
Tunable interactions
Charges-wave scattering(short-range interaction)
Feshbach resonance
Neutral system

Solid state systemµ,B,T fixedElectron gas withzero external magnetic field
Coulomb interactions(but screened)pressure
Charged system

Table 3.2: Comparison between cold atoms and solid state in the study of itinerant
ferromagnetism.

3.3.1

A Very Simple Mean-field Prediction for Stoner’s Instability in an Ultracold Atomic Gas

A simple mean-field model captures many qualitative features of the expected ferromagnetic phase in equilibrium. In the strongly interacting regime, however, thisHartree-Fock type approximation is not sufficient for a quantitative study since higher
order terms leading to quantum fluctuations play the important role. Nevertheless,mean-field model would be a starting point to understand the properties of a twocomponent Fermi gas.
We consider a uniform two-component Fermi gas interacting through a short-range
s-wave interactions (contact interactions). The main feature of the phase transition
in a homogeneous trap is expected to be maintained in a harmonic trap within thelocal density approximation (LDA).
The Hamiltonian describing the system is given by
H=

XZ
σ

d3 k
Ek c†kσ ckσ + g
3
(2π)

Z
d3 rc†r,↑ c†r,↓ cr,↓ cr,↑

(3.19)

2 2

where Ek = ~2mk is the kinetic energy of atoms with mass m and momentum ~k, and2ag = 4π~denotes the contact interactions. Here, the interaction term represents any
mshort-range spin-independent potential.
For a two-component Fermi gas trapped in a homogeneous trap with volume V
and average atomic density n (per spin component), the total energy is given by in amean-field model=
ETnormal
otal

X 3
5
σ=↑,↓

V (EF,σ nσ ) + gV n↑ n↓with density nσ = Nσ /V of the spin σ and Fermi energy EF,σ =49(3.20)
2~2 kF,σ.
2mMean-field model at T=0|1> n+dn|1> n-dn|2> n-dn|2> n+dn

Volume : V
Density : n
Local dendsity imbalance : dn

Figure 3-5: Mean-field model at T = 0 for itinerant ferromagnetism in a Fermi gas ofultracold atoms. With introduction of local density imbalance dn in a two-component
Fermi gas, the instability toward a ferromagnetic state can be described in a meanfield model.

Now, we introduce the local magnetization of the Fermi gas, η = dn/n, as described in figure 3-5. The local magnetization of the Fermi gas is non-zero when thegas separates into two volumes although the total number of atoms in each pseudospin
state is conserved. The total energy with local magnetization η becomesETη otal = EF 2V n[32{(1 + η)5/3 + (1 − η)5/3 } +kF a(1 + η)(1 − η)]
10
3π(3.21)For a system at constant volume, the shape of the total energy ETη otal as a functionof the magnetization η implies that the Landau-type second-order phase transition
occurs at critical value of kF a. Figure 3-6 (a) shows the total energy at different valueof kF a. It is clear that the system with minimum in energy spontaneously devolops
non-zero magnetization when kF a is larger than π/2. The phase transition occurs
when the curvature of the energy curve becomes zero at η = 0:
∂ 2 ETη otal
|η=0 = 0∂η 2(3.22)

giving the critical value of kF a = π/2.
To compare this model with a cold atomic system, it is more convenient to consider
the thermodynamic quantities at constant pressure because the atoms are trapped in

50

(a)

Magnetization

1.3

Volume1.0

Total Enthalpy

Increasing kfa

Energy at Constant Volume

Kinetic Energy

(b)

-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 3-6: Ferromagnetic phase transition at T=0, according to the mean-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). Figure (b)shows the enthalpy, volume and kinetic energy (normalized to their values for theideal Fermi gas), and magnetization as a function of the interaction parameter kF a.
Note that kF is defined by the density of the gas. The dotted line marks the phasetransition.

a harmonic potential 7 . For this, we consider the enthalpy, H, of the system:
H η = ETη otal − P V(3.23)

with the pressure P = −dETη otal /dV .
Similar to the constant volume case, one can calculate the magnetization η ofthe system at a given kF a by minimizing the enthalpy. Subsequently, kinetic energy,
volume, and total enthalpy are known as a function of kF a (see figure 3-6 (b)).
• Kinetic energy: For increasing repulsive interactions, the gas expands, lowering
its density and Fermi energy; kinetic energy is therefore reduced. When the
7

In a harmonic potential, the pressure is actually not constant but this simple approach is very
useful to capture qualitative features in a ferromagnetic phase transition.

51

gas enters the ferromagnetic phase, kinetic energy increases rapidly due to thelarger local density per spin state. When the gas is fully polarized, it avoids
the repulsive interaction, but increases its kinetic energy normalized by kF 2 bya factor of 22/3 .
• Volume: The volume has a maximum value at the phase transition. This canbe understood by noting that pressure in our model is given by(2/3)Ekin /V + Eint /V(3.24)

where Ekin is kinetic energy and Eint interaction energy. At the phase transition,
the system increases its kinetic energy and reduces its interaction energy, thus
reducing the pressure. This maximum in pressure at constant volume turns intoa maximum in volume for a system held at constant pressure, or in a trapping
potential.
• Magnetization: In the mean-field model, the phase transition is second-order;
the magnetization, therefore, changes continuously. The phase transition, however, may become first-order at the very low temperature due to quantum fluctuations [52].
In summary, we have obtained three predictions of this mean-field model:(1) the onset of local magnetization(2) the minimum in kinetic energy(3) the maximum in the size of the cloud
Note that the qualitative features are generic for ferromagnetic phase transition andshould be present in more advanced model.

3.3.2

Domain Structure of the Ferromagnetic Ground StateAs described in figure 3-7, there are several configurations for the ground state of
the ferromagnetic state in the uniform gas. When the number of atoms in each spinstate is not conserved (i.e. in the grand canonical ensemble), the true ground stateof the ferromagnetic phase is the homogeneous fully-polarized state with a constant~ 0 as described in figure 3-7 (b). In a trapped atomic gas, however,magnetization M
this is not valid since the total spin population in each state must be conserved. For
the gas of equal spin populations, the total magnetization of the system is alwaysR~ (r) = 0. Therefore, the configuration of the ferromagnetic ground state
zero: drM52

Increasing Interaction Strength (kFa)
Normal State

Ferromagnetic State

(a)

(b)

( Note: spin-up: spin-down )Figure 3-7: Schematic illustration of the ferromagnetic transition in the uniform twocomponent Fermi gas. As the repulsive interaction strength is increased, the gasof ultracold atoms becomes ferromagnetic at the critical value of kF a = π/2 in themean-field model. The kinetic energy of the gas increases by the factor of 22/3 when(a) the state is phase-separated or (b) the state becomes fully coherent.

53

(a)

(b)

Hedge Hog

Domain Wall

Figure 3-8: The possible configuration of the ferromagnetic ground state in a twocomponent Fermi gas trapped in a spherical potential. The hedhog state and thedomain wall state are shown in (a) and (b) respectively.

in the trapped gas must change to satisfy the above constraint. In a spherical threedimensional trap, two candidates, the hedge hog and the domain wall state, for the
ferromagnetic ground state are described in figure 3-8 (a) and (b) respectively. It has
been shown that the hedge hog state has lower energy than the domain wall state ina three-dimensional trap [19, 114]. However, the energy difference between the two
configurations turns out to be very small compared to the Fermi energy EF or thetemperature kB T of the system [114].

54

Chapter 4Itinerant Ferromagnetism in a
Fermi Gas of Ultracold Atoms
This chapter describes our research on itinerant ferromagnetism in a strongly interacting Fermi gas of ultracold atoms. Our experiment can be regarded as quantum
simulation of a Hamiltonian for which even the existence of a phase transition was
unproven. The experiment was reported 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 C

4.1

Two-component Fermi gas with strong repulsive interactions as a Model System

Overview
Recent developments in the field of ultracold Fermi gases rely on the ability of tuning the scattering length a from zero to the value larger than mean interatomic distance [65, 38, 108]. This is done by simply changing the magnetic field near a Feshbachresonance. In this section, we mainly discuss the properties of the system near the
Feshbach resonance. In particular, the repulsive gas is of interest since it would be amodel system for the study of itinerant ferromagnetism.
As described in figure 4-1, the Feshbach resonance couples a molecular boundstate (referred to as closed channel) with an unbound state describing a colliding pair55

of atoms (referred to as open channel). In a two-particle picture, the energy spectrum
in the system of Fermi gases near the Feshbach resonance shows that there are several
energy branches for the macroscopic state of the Fermi gas (see figure 4-1). Here, thetwo lowest-energy branches are of importance for our purpose.
1. The ground state branch : The ground state branch connects two different
regimes: a dilute gas of dimers (on the left-hand side of figure 4-1) and a weakly
attractive gas (on the right-hand side of figure 4-1). In the dilute Fermi gas withnegative scattering length (kF |a| ¿ 1), the many-body state can be describedin the BCS picture [17] both at T = 0 and at finite temperature. In thisregime, the Cooper paring instability toward the BCS state exists even in thepresence of extremely weak attraction. For the gas of dimers (kF a ¿ 1), thebehavior can be described by the BEC picture (hereafter called BEC limit).
Experimentally, the molecular Bose-Einstein condensation has been observedby several groups [211, 67, 30, 18, 144]. For both BCS and BEC limits, the gas
remains stable and shows superfluidity. The many-body physics becomes much
more challenging in the intermediate regime connecting the BCS and BEC limits
because the scattering length is larger than mean interatomic distance kF |a| À
1. For this regime, so-called BCS-BEC crossover, many relevant experiments
and theoretical calculations have indicated that the gas exhibits superfluidity
and remains stable [65, 91]. In addition, the universal behavior at the unitary
regime (kF a → ∞) has been studied experimentally in good agreement with
relevant numerical calculations. For more details, I refer to Ref. [65, 91].
2. The first excited branch : The repulsive gas of atoms is realized in thefirst excited branch. An attractive potential with a weakly bound state has
a positive scattering length describing repulsive interactions in the continuum.
In the dilute atomic gases, the true ground state in the presence of repulsive
interactions is the gas of the dimers since the long-range atom-atom interactions
are attractive. In the excited branch, therefore, the gas of atoms with positive
scattering length a is metastable since three-body collisions between atoms tends
to form a molecule [153] and populates ground state branch. The collision rate is
proportional to a6 for small scattering length, kF a < 1, and the binding energyof the produced dimer is carried away by the third atom. When kF a → 0, thegas corresponds to the weakly repulsive gas.
As we have seen above, two different branches near the Feshbach resonance allow us to prepare both an attractive and a repulsive gas of atoms. Experimentally,56

Two-particle energy

First excited
brancha>0a<0Ground state
branch

|m>
s
er
m
i
d

Feshbachresonance
Magnetic field

Figure 4-1: Schematic illustration of the energy spectrum of the two-particle system
in a spherical harmonic potential near a Feshbach resonance.

the atomic gas in the ground state branch has been achieved by (1) producing andsimultaneously cooling down a gas of dimers via three-body collisions with a fixed(positive) scattering length or (2) crossing the Feshbach resonance adiabatically starting from the negative scattering length. On the other hand, the repulsive gas canbe realized by adiabatically increasing the scattering length, starting from the smallvalue, a = 0+ [29]. For the repulsive atomic gas, the strongly interacting gas suffersfrom molecule formation leading to a short lifetime. To minimize deleterious molecule
formation in the atomic gas, one should increase the scattering length from zero to
the large value as quickly as possible. This quench could be done by ramping up themagnetic field [102] or by changing the hyperfine state of the atom with an RF pulse
as discussed in section 4.5.
Before we discuss the properties of the repulsive gas of atoms, it is useful to distinguish two different universal states in the unitary limit |a| → ∞. In the ground state
branch, the effective interaction in the unitary limit is attractive, whereas it is repulsive in the first excited branch. This different behavior of the gas is clearly identified
in the calculation of the thermodynamic pressure in a toy model (see figure 4-2).

57

Strongly interacting repulsive atomic gas
For an weakly repulsive Fermi gas, the ground-state energy be exactly calculated
within standard perturbation theory. The atom-atom interactions are approximatedby the psedopotential with a scattering length a. At nearly zero temperature, theenergy per particle is given by the following asymptotic expansion [88, 116, 126]:
E
3
2
4= EF ( +kF a +(11 − 2ln2)(kF a)2 + 0.23(kF a)3 + ...)
N
5 3π
35π

(4.1)

~2 k2

where EF = 2mF is Fermi energy and kF is the Fermi wavenumber, which should bemeaningful either for weakly interacting case (a → 0) or low density (kF → 0). Notethat the energy in the equation 4.1 denotes the energy of the true ground state forpurely repulsive potentials such as the hard-sphere model. In the dilute atomic gas,
this result describes the energy of metastable repulsive gas in the first excited branch.
Each term in the expression 4.1 has a physical physical interpretation:• The first term corresponds to the kinetic energy of the noninteracting Fermi
gas.
• The second term proportional to kF a implies both direct and exchange interaction energy associated with the two-particle forward scattering. Indeed, this isrelated to the definition of the s-wave scattering length a for low-energy collisions in a low-density gas. Neglecting the effect of the presence of other particles
in the Fermi sea, free-particle scattering amplitude f (k, k 0 ) has a constant value−a as k = k 0 → 0 (low-energy limit).
• The term proportional to (kF a)2 indicates the particle-hole excitation in thepresence of the Fermi sea. For example, this term includes the physical processthat an incident particle excites a particle in the Fermi sea, leaving a holebehind. Subsequently, the same two particles collide with each other again,
removing a hole in the Fermi sea and bringing the system back to the initialstate.
• A correction of order (kF a)3 represents the three distinct collisions associatedwith three-particle correlations [50].
The many-body problem for a large positive scattering length, kF a ≥ 1, is much
more challenging. To date, an exact solution of the many-body ground state in thisregime is not available and one has to rely on the numerical calculations or advanced
Hartree-Fock theories including the second-order fluctuations [133, 1, 52, 43].
58

We may now ask for the thermodynamic property of the Fermi gas when the
scattering length is larger than the mean distance between particles. In particular,
the ground-state energy, the compressibility, and the thermodynamic pressure of therepulsive gas near the Feshbach resonance are of interest.
For this purpose, we introduce a very simple model capturing relevant physics
in the strongly repulsive Fermi gas [160]. Although the many-body effects such ascorrelation effects are not well taken into account in this model, it describes someproperties of strongly interacting gases.
A simple model: A two-component Fermi gas with arbitrary values of the scattering length a can be simply modeled by considering a single fictitious particle in athree-dimensional spherical box interacting with a fixed scatterer at the center. Theinteractions between two atoms in different spin states are captured by the interactionof the fictitious particle with the scatterer. The boundary of the sphere box represents the Pauli blocking in the presence of other fermions and, indeed, the radius ofpthe sphere, R, is determined by the density of the gas: kF R = 5/3π. For N/2 spin
+1/2 and N/2 spin −1/2 fermions, the total energy of the system, E(a), at arbitrary
scattering length a is given by [160],
1
E(kF a) = N ε
2

(4.2)

2 2

where ε = ~ mk represents the energy of fictitious particle. The wavefunction ϕ(r) of
the fictitious particle satisfies−~2 2∇ ϕ = εϕ
m

(4.3)

(rϕ)
with the contact interaction condition limr→0 ∂rrϕ
= − a1 [139, 154]. For low-energy
limit, the only relevant partial wave is the s-wave leading to rotational symmetricsolution of equation 4.3. A simple calculation gives the energy of the fictitious particle
2 2
2 2
as follow; for positive (negative) energy ~ mk (− ~ mk ), k is determined by the solutionoftan kR = ka (tanh kR = ka).

(4.4)

In figure 4-2, the total energy of the gas and the thermodynamic pressure areshown as a function of kF a. Here, since we are interested in the two lowest branches,
only the two smallest eigenvalues in 4.4 are considered in figure 4-2. In figure 4-2(a), the upper branch corresponds to the first excited branch in figure 4-1 whereas

59

(b)

Total energy of the gas

Thermodynamic pressure

1.5

2.5

1.0

2.0

0.5

P (P0)

E/N (EF)

(a)

0
-0.5

1.5
1.0

-1.0

0.5

-1.5

0
0

2

4

6

8

0

10

2

4

K Fa

6

8

10

KFa

Figure 4-2: The energy per particle and the thermodynamic pressure of the repulsive
interacting two-component Fermi gas in the simple model. (a) The energy per particle(in (a)) and the thermodynamic pressure (in (b)) are shown as a function of kF a. They
are normalized by the Fermi energy, EF , and the pressure, P0 , respectively for the
non-interacting ideal Fermi gas.

the lower one to the ground state branch. The total energy of the repulsive gas (theupper branch) gradually increases with the interaction parameter kF a and eventuallysaturates near the Feshbach resonance. On the other hand, the lower branch shows
the feature of dimers; the binding energy of a dimer near the resonance is given by∼ ~2 /ma2 .
The difference between the two branches becomes more prominent in the thermoF a)
dynamic pressure P (a) = − ∂E(kwhere V is the volume of the system. After a∂Vsimple algebra, the pressure for the first excited branch, Pe (a), is given by
Pe (kF a) = P0 ×

10mε
1
(
)
2
3~2 kF 1 − √ 5 kF a2ma2 ε
3

π(1+

~2

(4.5)

)

~2 k 5where P0 = 15π2Fm is the Fermi pressure of the non-interacting gas. The result infigure 4-2 (b) shows that the thermodynamic pressure at large kF a À 1 implies the
behavior of the repulsive (Pe (kF a) > P0 ) and attractive (Pg (kF a) < P0 ) gas for the
first excited and the ground state branch respectively.

4.2

Implementation of the Stoner model in an Ultracold Atomic System

Motivation: So far, we have seen that the two-component Fermi gas near the Feshbach resonance may offer an opportunity to study the property of the repulsive
60Fermi gas. Encouraged by this, we now implement the Stoner model [181] using a
two-component gas of free fermions with short-range repulsive interactions which cancapture the essence of the screened Coulomb interaction in electron gases [181]. However, there is no proof so far that this simple model for ferromagnetism is consistent
when the strong interactions are treated beyond mean-field approaches. It is known
that this model fails in one-dimension where the ground state is singlet for arbitrary
interactions, or for two particles in any dimension [124]. Here, cold atoms are used
to perform a quantum simulation of this model Hamiltonian in 3D and provide experimental evidence that it leads to a ferromagnetic phase transition [187]. A similar
model is also realized in helium-3 [193], but it turns into solid and not into a ferromagnetic phase at high pressure. It has also been applied to neutrons in neutronstars [156].

4.2.1

Experimental TechniquePreparation of the ultracold 6 Li cloud
The first step is the production of a spin-polarized Fermi gas in the |F = 3/2, mF =3/2i state by sympathetic cooling with bosonic 23 Na atoms in a magnetic trap as
described in Ref. [78]. The 6 Li cloud was then loaded into a deep optical dipole trapwith a maxium power of 3W and radial trap frequency of ∼3.0 kHz, followed by an
RF transfer into the lowest hyperfine state |F = 1/2, mF = 1/2i. Additional axial
confinement was provided by magnetic fields. An equal mixture of |1i and |2i spinstates (corresponding to the |F = 1/2, mF = 1/2i and |F = 1/2, mF = −1/2i states
at low magnetic field, see also appendix A) was prepared by a Landau-Zener RF sweep
at a magnetic field of 590 G, followed by 1 s for decoherence and further evaporative
cooling at 300 G. Finally, the optical trapping potential was adiabatically reduced
over 600 ms, and the field increased back to 590 G. The trap had a depth of 7.1 µK
and was nearly cigar shaped with frequencies νx = νy ' 300 Hz and νz ' 70 Hz.
Magnetic field ramp
Our current coil geometry does not allow very fast magnetic field ramping. After
improving the ramp rate of our power supplies to better than 1 ms, we observed thatthe magnetic field was still lagging 4.5 ms behind, probably due to eddy currents as
described in figure 4-4. The actual magnetic field at the position of the atom cloud
was determined in a spectroscopic way. In the experiment, the magnetic field wasramped up from 590 G to the field of interest within 4.5 ms.
61

Magnetic Field [G]

(a)

Evap.
at 300G
834
820

Field
Ramp

Ballistic
Expansion

Hold

ap 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 4-3: (a) The schematic shows the time sequence of the experiment. The sample
was exposed to the magnetic field of interest for 0 - 14 ms and analyzed in-situ for loss
measurement or after 4.6 ms time-of-flight for the measurement of kinetic energy andthe axial size of the cloud. The Feshbach fields were suddenly switched off at a rate of
1G/µs, preventing the conversion of interaction energy into kinetic energy during the
expansion. (b) This absorption image shows the |1i component of the cloud trappedat 812 G (left), and after 4.6 ms ballistic expansion imaged at zero field (right). The
field of view is 840µm×550µm. The magnetic field ramp was limited by eddy currents
to 4.5 ms. Spectroscopic measurements of the magnetic field showed that the field
was trailing behind the current which was controlled with a time constant faster than
1 ms.

62

Magnetic Field [G]

800
775
750
725
Fast ramp-up
Normal ramp-up

700

0

1

2

3

4

5
6
Time [ms]

7

8

9

10

11

Figure 4-4: The effect of eddy currents on the magnetic field ramp was measured.
Though the current from the power supply was ramped up within 1 ms (red solid
curve), the magnetic field at the position of the atom cloud requires at least 4.5 ms(red solid circles). The magnetic field is also shown for the slower ramp rate (bluedotted curve and blue open circle) .

Molecule formation and the effective temperature
We start with an atom cloud consisting of an equal mixture of 6 Li atoms in the lowest two hyperfine states, |1i and |2i, held at 590 G in an optical dipole trap withadditional magnetic confinement. The number of atoms per spin state ∼ 6.5 × 105
corresponds to a Fermi temperature TF of ∼1.4 µK. The effective temperature T
could be varied between T /TF = 0.1 and T /TF = 0.6 and was determined right after
the field ramp by fitting the spatial distribution of the cloud with a finite temperature
Thomas-Fermi profile. Note that kF◦ is the Fermi wavevector of the non-interacting
gas calculated at the trap center. Applying the procedure discussed in Ref. [107] torepulsive interactions, we estimate that the real temperature is ∼20% larger thanthe effective one. The effective temperature did not depend on kF◦ a for kF◦ a < 6. At
higher temperatures, additional shot-to-shot noise was caused by large fluctuations in
the atom number. From the starting point at 590 G, the magnetic field was increased
towards the Feshbach resonance at 834 G, thus providing adjustable repulsive interactions. Due to the limited lifetime of the strongly interacting gas, it was necessary
to apply 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 and therefore
only marginally adiabatic. Depending on the magnetic field during observation, either atoms or atoms and molecules were detected by absorption imaging as described63

in figure 4-3 [211].

4.3

Evidences for Itinerant Ferromagnetism in a
Fermi Gas of Ultracold Atoms

In this section, we introduce the experimental characterization of the ferromagnetic
phase transition in a Fermi gas of ultracold aotms. We have observed three different
evidences for ferromagnetic phase transition. The observations are qualitatively inagreement with predictions in mean-field and more advanced models.
1. Local spin polariztion
The emergence of local spin polarization can be observed by the suppression of (either
elastic or inelastic) collisions, as the Pauli exclusion principle forbids collisions in a
fully polarized cloud. We monitor inelastic three-body collisions which convert atoms
into molecules. The rate (per atom) is proportional to f (a, T )n1 n2 or f (a, T )n2 (1−η 2 )and is therefore a measure of the magnetization η. For kF a ¿ 1, the rate coefficientf (a, T ) is proportional to a6 max(T, TF ) [48]. For kF a À 1, the f (a, T ) is unknown
and the rate coefficient may be suppressed at a very low temperature due to thePauli blocking. However, the effect of the Pauli blocking has not yet been studied
quantitatively. This rate can be observed by monitoring the initial drop in the numberof atoms during the first 2 ms after the field ramp. We avoided longer observation
times since the increasing molecule fraction could modify the properties of the sample.
Figure 4-5 shows a sharp peak in the atom loss rate around kF◦ a ' 2.5 at T /TF =0.12 indicating a transition in the sample to a state with local magnetization. The
gradual decrease is consistent with the inhomogeneous density of the cloud where thetransition occurs first in the center. The large suppression of the loss rate indicatesa large local magnetization of the cloud.
2. Kinetic energy of the cloud
The kinetic energy of the cloud was determined by suddenly switching off the opticaltrap and the Feshbach fields right after the field ramp and then imaging state |1i
atoms at zero field using the cycling transition after a ballistic expansion time of
4tof = 4.6 ms. The kinetic energy was obtained from the Gaussian radial width σx
3mσx 26
as Ekin = 24
Li atom. Figure 4-6 demonstrates a
2 where m is the mass of thetof◦minimum of the kinetic energy at kF a ' 2.2 for the coldest temperature, T /TF = 0.12,64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 4-5: (a) Atom loss rate as a probe for local spin polarization, for different
temperatures: T /TF = 0.55 (dashed curve), T /TF = 0.22 (dotted curve), and T /TF =0.12 (solid black curve). The curves are guides to the eye, based on the assumptionof a loss rate which saturates for increasing a in the normal state. The shaded areaaround the phase transition at T /TF = 0.12 highlights the same region as in figure 4-6
and 4-7. Atom loss rates at kF a = 2.5 and kF a = 4.9 are obtained by measuring theinitial drop in the atom number as shown in (b) and (c) respectively.

65

nearly coinciding with the onset of local polarization. The peak in the atom loss rateoccurs slightly later than the minimum of kinetic energy, probably because f (a, T )increases with a [114]. Since the temperature did not change around kF◦ a ' 2.2,
the increase in kinetic energy is not caused by heating, but by a sudden change inthe properties of the gas, consistent with the onset of ferromagnetism. The observed
increase in kinetic energy is ∼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 absenceof polarization or partial polarization in the wings of the cloud. Also, it is possiblethat the measured kinetic energy of the strongly interacting gas before the phasetransition includes some interaction energy if the Feshbach fields are not suddenly
switched off. For the current switch-off time of ∼100 µs, this should be only a 5%
effect, but the magnetic field decay may be slower due to eddy currents.
3. Cloud size
Finally figure 4-7 shows our observation of a maximum cloud size at the phase transition, in agreement with the prediction of the model. The cloud size may not havefully equilibrated since our ramp time was only marginally adiabatic, but this alone
cannot explain the observed maximum.
4. Temperature dependence
The suppression of the atom loss rate, the minimum in kinetic energy, and the maximum in cloud size show a strong temperature dependence between T /TF of 0.12 and0.22. As the properties of a normal Fermi gas approaching the unitarity limit withkF◦ a >> 1 should be insensitive to temperature variations in this range, this provides
further evidence for a transition to a new phase.
At higher temperature (e.g. T /TF = 0.39 in figure 4-6), the observed nonmonotonic behavior becomes less pronounced and shifts to larger values of kF◦ a for3 ≤ kF◦ a ≤ 6. For all three observed properties (figure 4-5∼ 4-7), a nonmonotonic
behavior is no longer observed at T /TF = 0.55. The interpretation of the loss rateis complicated since f (a, T ) is unkown for kF a ≥ 1. The three body rate f (a, T ) is
expected to be unitarity saturated for kF a À 1 [197]. The lines in figure 4-5 indicatethat the observed loss rate is consistent with unitarity saturation and a sudden dropat the phase transition, which occurs at large values of kF a at higher temperature.
One interpretation is that at this temperature and above, there is no phase transition any more. Note that in a mean-field approximation, a ferromagnetic phase66

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 4-6: Kinetic energy of a repulsively interacting Fermi gas determined for different interaction parameters kF◦ a and temperatures. The measured kinetic energy isnormalized by the Fermi energy EF◦ of the noninteracting Fermi gas at T =0, calculated at the trap center with the same number of atoms per spin state. Each data
point represents the average of three or four measurements.

67

(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˚F aFigure 4-7: Maximum in volume at the phase transition (a) Axial size and chemical
potential of the cloud for various temperatures. The chemical potential µ is determined from the measured cloud size, σz as µ = 21 mωz 2 σz 2 (b) Number of particles
including both atoms and molecules right after the field ramp. This result shows that
25 % of atoms were converted into molecules during the field ramp, and this fraction
stayed constant for kF◦ a > 1.8, where the phase transition was reached. This molecule
fraction was independent of temperature.

68

would appear at all temperatures, but for increasing values of kF◦ a. Our observations
may imply that the interaction energy saturates around kF◦ a ≈ 5.
5. Limitations in the experiment
The spin-polarized ferromagnetic state should not suffer from inelastic collisions.
However, typical lifetime were 10 - 20 ms, probably related to a small domain size(see below) and three-body recombination at domain walls.
We were unsuccessful in imaging ferromagnetic domains using differential in-situ
phase-contrast imaging [180]. A noise level of S/N ∼10 suggests that there wereat least 100 domains in a volume given by our spatial resolution of ∼ 3 µm andthe radial size of the cloud. This implies that the maximum volume of the spin
domains is ∼ 5 µm3 , containing ∼ 50 spin-polarized atoms. We suspect that the
short lifetime prevented the domains from growing to a larger size, and eventuallyadopting the equilibrium texture of the ground state, which has been predicted to havethe spins pointing radially outward, like a hedgehog [19, 114]. All our measurements
are sensitive only to local spin polarization, independent of domain structure andtexture.
The only difference between our experiment and the ideal Stoner model is a molecular admixture of 25 % (see figure 4-7). The molecular fraction was constant forkF◦ a > 1.8 for all temperatures and therefore cannot be responsible for the suddenchange of behavior of the gas at kF◦ a ' 2.2 for the coldest temperature T /TF = 0.12 .
This was confirmed by repeating the kinetic energy measurements with a molecular
admixture of 60 %. The minimum in the kinetic energy occurred at the same kF◦ a
within experimental accuracy.

4.4

Discussion

4.4.1

The Critical Value of kF a Observed in the Experiment

The observed phase transition at kF◦ a ' 2.2 must be rescaled in order to compare it to
the theoretical predictions. For this, we have to replace the ideal gas kF◦ by the value
for the interacting gas, which is smaller by ∼ 15% because of the expansion of thecloud (figure 4-7), and obtain a critical value for kF a ' 1.9 ± 0.2. At T /TF = 0.12,
the finite temperature correction in the critical value for kF a is predicted to be lessthan 5% [52]. The observed value for kF a is larger than the mean-field prediction of
π/2 and the second order prediction of 1.054 at zero temperature [52]. Depending on
69

the theoretical approach, the phase transition has been predicted to be first or second
order. This could not been discerned in our experiment due to the inhomogeneousdensity of the cloud.

4.4.2

Estimation of the Maximum Total Repulsive Energy

Our work demonstrates a remarkable asymmetry between positive and negative scattering length. Early work [86] predicted that for kF |a| = π/2, both an attractive
and a repulsive Fermi gas become mechanically unstable (against collapse, and phaseseparation, respectively). In an attractive Fermi gas, however, the mechanical instability does not occur (due to pairing [138]), in contrast to our observations in a
repulsive Fermi gas. This suggests that the maximum total repulsive energy (in unitsof 3/5(2V n)EF ) is larger than the maximum attractive energy |β| of 0.59 [34] realized
for infinite a.
Full phase separation at zero temperature requires a total repulsive energy of(22/3 − 1) = 0.59 in units of 3/5(2V n)EF . At finite temperature T , one has to add T S
where S = (2V n)kB ln2 is the entropy difference between the two phases as describedin figure 4-8. We can estimate the maximum repulsive energy by considering freeenergy F ,F = E − TS
(4.6)
where F is free energy of a two-component Fermi gas, T the temperature, and S theentropy. Across the ferromagnetic transition, the chagne in free energy satisfies
3∆F = FF M − FN ormal = (22/3 − 1) N EF + kb T N ln2 − ∆Erepulsive < 0
5

(4.7)

This gives us the maximum repulsive energy,
3T5∆Erepulsive > N EF ((22/3 − 1) +
ln2 )
5
TF
3

(4.8)

Our tentative observation of a ferromagnetic phase at T = 0.39TF implies a repulsive
energy of ∼ 1.04 assuming full phase separation, larger than the maximum attractiveenergy of 0.59.

70Normal state

Ferromagnetic state

demixing/ mixing

Entropy:SN

S N − k B N ln 2
( N : total number of particles)

Figure 4-8: The entropy difference between the normal and the ferromagnetic phase.

4.4.3

Comparison with Ferromagnetism in Spinor Bose-Einstein
Condensates.

So far, magnetism in ultracold gases has been studied only for spinor [184, 168]and dipolar [110] Bose-Einstein condensates. In these cases, magnetism is driven by
weak spin-dependent interactions which nevertheless determine the structure of thecondensate due to a bosonic enhancement factor. Especially in spinor condensates,
the long-range dipole-dipole interactions are not negligible in contrast to a stronglyinteracting Fermi gas in which the short-range exchange interactions are dominant.

4.4.4

Discrepancy of the Critical Value of kF a

The mean-field calculation predicts that itinerant ferromagnetism occurs at kF a =π/2 in a homogeneous Fermi gas. The experiment reported in this thesis, however,has observed that the critical value of kF a is rather larger. More advanced models
predicts even smaller critical interaction strength (see table 4.1). This discrepancy
raises a question if our theoretical predictions are not precise enough or other physicalmechanism plays a role. In any cases, it is very important to resolve the discrepancy
in the critical kF a since it would improve our theoretical understanding of a stronglyinteracting many-body system. Recently, two rather opposite physical mechanisms
affecting the critical kF a have been proposed: quantum fluctuations enhancing interacting energy and three-body losses damping out quantum fluctuations.

71

The role of quantum fluctuations
The energy density, ε, of the Stoner Hamiltonian 3.19 can be calculated up to secondorder in kF a by either perturbation theory [133, 1, 52] or field-theoretical approach [43]:‘ε=

2 2kF a 2 X n↑ (²k1 )n↓ (²k2 )[n↑ (²k3 ) + n↓ (²k1 )]
1 X
2kF a
²k nσ (²k ) +n↑ n↓ − 3 (
) ×
V k,σ
πν
V
πν
²k1 + ²k2 − ²k3 − ²k4|{z}mean−f ield

(4.9)
where nσ (²) denotes the Fermi distribution at the energy of ², ν is the density of stateat the Fermi surface, and a is the s-wave scattering length. The underbraced leading
terms simply recovers the mean-field Stoner model. The quantum fluctuations are
encoded into the term proportional to (kF a)2 leading to a first-order phase transition
at low temperature. Physically, the second-order term represent particle-hole excitations which are neglected in mean-field calculations. Also, the unitarity limit close to
the Feshbach resonance is taken into account by renormalizing (or regularizing) theinteraction strength g [148]:
‘

g −→

1
2 2kF a 2 X
2kF a
− 3(
)
πν
V
πν
²k1 + ²k2 − ²k3 − ²k4(4.10)

Note that this treatment regularize the unphysical divergence of the second-orderterms close to the Feshbach resonance.
The presence of the quantum fluctuations enhances the interaction energy so thatit decrease the critical interaction strength kF a∗ = 1.05. Here, the second-ordercontribution in 4.9 is actually positive definite 1 because of the renormalization ofg [148].
Recently, the effect of the atom loss via the three-body collisions has been studied
in Ref [42] in a Fermi gas of ultracold atoms. It was shown that the atom loss hinders
the ferromagnetic phase transition by damping out the quantum fluctuations.
In summary, quantum fluctuations have an important role in a ferromagnetictransition. Recently, however, it has been pointed out that the effect of quantum
fluctuations - first-order phase transition and lower critical interaction strength - may
be preempted by the formation of an inhomogeneous magnetic phase [43] and threebody loss respectively [42]. In particular, the three-body loss damps quantum fluctuations and thus increases the critical interaction strength needed for a ferromagnetic
1

Therefore, the second-order term increases effective interaction strength.

72

Predictions
MIT experiment [102]Analytic method [52]including quantum fluctuations
QMC [43]
Mean-field model [114]kF a
1.9 ± 0.3
1.05
0.85
1.84

Table 4.1: Prediction of critical interaction strength for itinerant ferromagnetism ina Fermi gas of ultracold atoms.

transition.

4.4.5

Correlation Effects

The interpretation of our results in terms of a phase transition to itinerant ferromagnetism is based on the agreement with the prediction of simplified models basedon the Hartree-Fock approximation. In a strongly interacting regime, however, theHartree-Fock approximation does not take into account correlation effects [56].
In the Hartree-Fock method, the parallel spins are kept apart by the Pauli exclusion principle whereas the antiparallel spins are spatially uncorrelated. Thus, theantiparallel spins possess a relatively large energy via Coulomb repulsion or othershort-range interactions. In an exact treatment without any approximation, however,
it is likely that the antiparallel spins are somehow correlated due to strong repulsive
interactions. The correlation eventually would reduce the interaction energy, Uint , inthe system. The difference between the exact ground state of an strongly interacting
system and the approximated ground state in the Hartree-Fock method is caused byso-called “correlation effects” [56].
The possible importance of correlations in the Hubbard model was pointed out
by Kanamori [105]. He argued that the short-range correlations reduce the effective
interaction energy, Uint ; therefore Uint must be finite even with infinitely large bareinteractions. Furthermore, Gutzwiller [75, 76] approximately calculated that the nonmagnetic state in the presence of strong correlations can have lower energy than aferromagnetic state at low density. For this, he constructed his famous projected
wavefunction describing the reduction of on-site interaction energy in a lattice. However, both arguments by Kanamori and Gutzwiller are based on the lattice-modelso that further theoretical developments are required to treat correlation effects in ainteracting free Fermi gas.
Recent work [205] has shown, within a phenomenological model, that correlations73

can lead to similar experimental signatures as we have observed [102]. However,
the model does not quantitatively agree with our data, and has some qualitative
discrepancies as well; for example the extrema of lifetime, kinetic energy and cloud size
do not occur at the same value of the parameter kF a in contrast to our observations.
It would be interesting to see if further development of the alternative theories could
lead to quantitative agreement with experiments.

4.5

Outlook: New Experiments

RF quench experiment with |1i and |3i atoms: studying of non-equilibrium
properties of the strongly interacting system.
As we have seen in the previous section, the MIT experiment shows some aspectsof both equilibrium and non-equilibrium dynamics. Our theoretical understandingof Stoner’s model basically relies on equilibrium properties whereas the finite experimental time scales (for example, field ramp time of 4.5 ms) suggests that thesystem is marginally adiabatic after rapid ramps of the magnetic field; therefore,
non-equilibrium.
In the previous experiment, we have prepared a strongly interacting mixture of|1i and |2i state atoms near the Feshbach resonance located at 834 G. For this, themagnetic field is ramped towards the resonance (834 G) within 4.5 ms. Unfortunately,
our current coil geometry did not allow sufficiently fast field ramp and the times scaleof the field ramp is not fast enough to study the quantum dynamics of systems in
non-equilibrium state, e.g. how magnetic domains grow, how itinerant ferromagnetic
phase transition occurs, and how clock shifts are suppressed.
Here, we propose a rapid switching of interaction strength using a quick RF
spinflip, which transforms a weakly interacting |1i-|2i mixture of fermions into astrongly interacting |1i-|3i one (see figure 4.5). Preparing the |1i-|2i mixture neara |1i-|3i Feschbach resonance for near 670 G, and applying RF photons to transfer|2i atoms into the |3i state. The RF spinflip may take a few hundred µs which
is marginally fast compared to the time scale of non-equilibrium dynamics (about∼ (EF /10)−1 ' 300µs) [14]. Note that a |1i-|3i mixture of fermions does not sufferfrom “final-state effect” in the RF spectroscopy [175]. Therefore, the measurementof clock shifts in the |1i-|3i mixture offer another tool to characterize the system.
Single-domain ferromagnetic state: Ramsey-type measurement of coherence The observation of ferromagnetic domains is considered as an experimental
74

|3>
|2>

3

20x10

Scattering Length [a0]

|1>

RF

RF

10

0

-10

|1>-|2> resonance|1>-|3> resonance

-20
500

600

700

800

900

Magnetic field [G]Figure 4-9: Schematic illustration of the RF quench experiment. A radio-frequency
pulse is applied in order to transform a weakly interacting |1i-|2i mixture of fermions
into a strongly interacting |1i-|3i one. The scattering length near the Feshbach resonance in the |1i-|2i (red solid curve) and |1i-|3i (blue dashed curve) mixture isshown.

smoking gun in itinerant ferromagnetism of cold Fermi gas since the ground state in
the stable ferromagnetic state consist of a single domain. The ferromagnetic domains,
however, were not observed in the previous MIT experiment [102] probably due tothe short lifetime of the systems which did not allow the domains to grow to larger
sizes [14]. Although non-equilibrium dynamics after quenching, such as the growthrate of the ferromagnetic domain, may offer an opportunity for experimental studies,
this limits us to characterize and test the ferromagnetic equilibrium state.
Here, we propose to use an alternative approach preparing the system directlyin the ground state. First, the absolute ground state can be created by preparing a√
coherence mixture of atoms in the (|1i+|2i)/ 2 state as pointed out in Ref [52]. In thepresence of magnetic field inhomogeneities 2 , the coherent mixture rapidly dephases
in about ∼ 10 ms for weak interacitons [72]. However, in the ferromagnetic region,dephasing is suppressed since it costs energy to break up one domain into many. The
suppression of dephasing in this regime may be characterized by (1) monitoring clock
shifts in the sample or (2) measuring atom-atom collisions (lifetime of the sample). In
the ferromagnetic ground state, we expect that clock shifts and atom-atom collisions
2

In our experimental setup, magnetic field inhomogeneities are inevitable as the axial confinementin the trapping potential mainly rely on the magnetic trap

75

π /2
Pulse

π /2
Pulse
Hold

Time

+z

+z

+z

+z

(a) Spin-polarizedgas(B) Single
domain(c) Domain
formation(d) Imaging

Figure 4-10: Schematics illustration of the single domain experimentare highly suppressed.
The interplay between magnetic inhomogeneities and the ferromagnetic stabilization energy would be more prominent in the Ramsey-type measurement of coherence(see figure 4.5). Preparing a single ferromagnetic domain (in x − y plane in the Bloch
sphere) via the first π/2 pulse, we can resolve spatial spin structures after some dephasing by selectively imaging the atomic cloud in two spin states (+z and -z in the
Bloch sphere) via the second π/2 pulse.

76Chapter 5
Quantum Phase Coherence inDegenerate Bosonic Gases
This chapter describes phase coherence of Bose-Einstein condensates (BECs) andthe decoherence dynamics. The experimental tools for a phase read-out and a phasecoherent splitting of Bose-Einstein condensates shall be discussed at the end of thechapter.

5.1

Phase Coherence in a Bose-Einstein Condensate

What is the phenomenon of Bose-Einstein condensation? To understand it, one needs
to formulate some of fundamental concepts such as long-range order, spontaneously
broken gauge symmetry , and order parameter [9]. In this section, I summarize
those concepts in the context of a weakly interacting Bose gas. From the following
discussions, we shall see that a many-body ground state of weakly interacting Bose
condensed atoms can be described by a condensate wavefunction (or a macroscopic
wavefunction) which possesses a well-defined phase.
Criterion for Bose-Einstein Condensation
In an ideal Bose gas in three dimensional potential, Bose-Einstein condensation occurs
when the particle density in the zero momentum state has a finite value. In this case,the condensation is a first-order phase transition, but this is not generally valid in a
weakly interacting Bose gas [87]. The weakly interacting Bose gas requires a moregeneral definition of Bose-Einstein condensation, and surprisingly it had taken long77

time about 40 years from the first theoretical work about BEC to the generalized
criterion for Bose-Einstein condensation [92]. The generalized criterion for BoseEinstein condensation was first proposed by Penrose and Onsager [149] considering
the one-particle density matrix ρ1 (x, y) = hψ † (x)ψ(y)i;hψ † (x)ψ(y)i −→ hψ(x)i∗ hψ(y)i when |x − y| −→ ∞(5.1)

where hi denotes ensemble average and ψ(x) is a quantized boson field operator
at the position of r. Here, the complex number hψ(x)i is called as “condensate
wavefunction”, and non-vanishing hψ(x)i implies the existence of a Bose-Einsteincondensate. In this case, the condensate wavefunction is characterized by a moduluspn0 (x) and a phase φ(x)hψ(x)i = Φ(x) =pn0 (x)eiφ(x)(5.2)

This criterion is often referred to as “off-diagonal long-range order ” [204] because it
involves the off-diagonal matrix element of the one-particle density matrix.
In the above criterion, however, one immediate question is raised up as follows.
Given a “global gauge invariance” of the Hamiltonian describing the system, hψ(x)i√must be always zero in the ensemble average because every value of ψ = n0 eiφ is√canceled by a value of ψ = n0 eiφ+iπ after a constant phase change of ψ. Therefore,
Bose-Einstein condensation can never occur in this mathematical argument.
The resolution of this conceptual problem lies in the recognition that the symmetryof the original Hamiltonian may be spontaneously broken in nature. In this picture,
the symmetry of the Hamiltonian is no longer applicable to the ground state. The
concept of spontaneously broken gauge symmetry (SBGS) is of importance nowadays
in condensed-matter physics, and generally believed to be the key to explore the exotic
phenomena such as superfluidity and superconductivity [9]. The idea of SBGS is often
adopted to account for a phase transition as it typically occurs between phases withdifferent symmetry. In most cases, the phase transition brakes the symmetry during
the process 1 . Following this concept of SBGS, the ensemble average of ψ, hψ(x)i ,has a non-vanishing value even in the presence of the global gauge invariance.
Finally, the criterion of 5.1 reveals another major phenomenon of Bose condensates, “phase coherence”. Phase coherence implies that the phase of the condensate
1This is not always true. For example, the infinite-order phase transitions are continuous
but break no symmetries. One of the famous example is Kosterlitz-Thouless transition in twodimensional XY model. In 2D, many phase transitions in electron gases belong to this class.

78

wavefunction, φ(x), must be spatially correlated over the length scale of the wholesystem. This enables us to treat Bose condensed atoms as being governed by thesame wavefunction ψ(x), which is actually the condensate wavefunction.
We are now ready to identify an order parameter describing the Bose-Einsteincondensed phase. The order parameter is proposed by Landau in his theory of thesecond-order phase transition [9]. He pointed out that one can define the order parameter for the phase transition accompanied by symmetry breaking. The order parameter is simply any parameter which is zero in the symmetric state and has a nonzero
value in the unsymmetrical state after the phase transition. In the case of Bose gases,a suitable choice of the order parameter is the condensate wavefunction 5.2. Its modulus determines the density of the condensate through n20 (x) = |ψ(x)|2 and the phaseof the condensate is well-defined through φ(x).
Mean-field description of a weakly interacting Bose gas.
So far, we have assumed that all particles stay in the zero momentum state when
Bose-Einstein condensation occurs. Starting from the generalized criterion 5.1, theorder parameter 5.2 characterizing the condensate wavefunction is identified for the
non-interacting Bose gas. This approach still retains validity when we introduce weakinteractions between atoms. To this end, the boson field operator is decomposed in
the form of
ψ(x) = Φ(x) + ψ 0 (x)(5.3)
pwhere Φ(x) =n0 (x)eiφ(x) denotes the order parameter and ψ 0 (x) describes the
non-condensed particles which occupy non-zero momentum states which is referred
to as the “quantum depletion” of the condensate. By treating ψ 0 (x) as a small perturbation, Bogoliubov formulated a mean-field description of a dilute Bose gas withweak interactions, and developed the first-order theory for the excitations of interacting Bose gases [26]. In his approach, the boson field operator is replaced by themean-field term Φ(x) with a small fluctuations characterized by ψ 0 (x). For weaklyinteracting alkali condensates, the quantum depletion is less than 1% whereas liquid
helium has large quantum depletion of 90%.
Bogoliubov’s idea separating out the condensate contribution from the boson field
operator (see equation 5.3) becomes particulary useful when the quantum depletionis negligible. In this case, we can derive the equation for the condensate wavefunction
Φ(x) as a result of the zeroth-order perturbation theory. To this end, we first introduce79

a many-body Hamiltonian describing N interacting bosons trapped by an externalpotential Vext in second quantization:
Z
Ĥ =

~2 2
1
drψ (r)[−∇ + Vext (r)]ψ(r) +
2m
2
†

Z
drdr0 ψ † (r)ψ † (r0 )V (r − r0 )ψ(r0 )ψ(r)(5.4)
where V (r − r ) describes an arbitrary two-body interaction potential and m denotes
the mass of particles.
For a weakly interacting dilute alkali condensate, the two-body interaction potential V (r − r0 ) reduces to the effective potential characterized by a single parameter,0

the s-wave scattering length a:V (r − r0 ) = gδ(r − r0 )

(5.5)

2

a
where g = 4π~.
mNow, nonlinear Schrodinger equation for the condensate wavefunction Φ(x) is
derived in the mean-field description:

(−

~2 2
∇ + Vext (r) + gΦ2 (x))Φ(x) = µΦ(x)
2m

(5.6)

where µ is the chemical potential. The equation 5.6 is called “Gross-Pitaevskii equation” [69, 70] which is one of the main tools for investigating ultracold Bose condensed
atoms trapped in a nonuniform potential. The generalization of Bogoliubov’s meanfield description to time-dependent configurations is given by the Heisenberg equationi~∂Φ(x, t) = [Φ, Ĥ]
∂t

(5.7)

giving the equation for the condensate wavefuction Ψ(x, t).
For more comprehensive discussions on the description of Bose condensed atoms,
I refer the reader to many excellent books, for example by Pethick and Smith [152]and Pitaevskii and Stringari [159].

80

5.2

Decoherence in Interacting Bose Gases

Decoherence can be defined as the loss of information from the system that we areconsidering into the environment. The major sources of decoherence are either theinteraction of the system with the environment or the intrinsic quantum and/or thermal fluctuations in the system. In the system of cold atomic gases, the information weare interested in is the phase of the condensate. By definition described in previous
section, the condensate has phase coherence over the whole system. However, thesystem of condensates inevitably undergoes decoherence process due to quantum andthermal fluctuations associated with atom-atom interactions. Of particular interestis the loss of phase coherence with time (so-called phase diffusion) and the loss of
spatial coherence along the condensate (phase fluctuation).
Phase diffusion
For two separated Bose-Einstein condensates, a state of well-defined relative phase(so called phase coherent state), |φi, is a superposition state of many relative number
states, |N1 , N2 i (or |k, N − ki), where N1 and N2 are the occupation of each well for
N = N1 + N2 atoms. Initially (t = 0), a phase coherent state takes the form of:|φ(t = 0)i =N
1 Xp

2N/2 k=0

N Ck eiφk|k, N − ki

(5.8)

Because of atom-atom interactions in the condensates, the energy of number
states, E(N1 , N2 ), have a quadratic dependence on the atom numbers N1 and N2 :dµ
N + (N1 − N2 )|k=N/2 (k − N/2)2 where k =
(5.9)
dk
2
where µ is the chemical potential of condensates. As a result, the different relative
number states have different phase evolution rates [35, 96] leading to a spread ofevolution rates, causing “phase diffusion” or “decoherence” of the relative phase withtime. A phase coherent state at time t is given by:E(N1 , N2 ) =|φ(t)i =

N
1 Xp

2N/2

N Ck eiφk −iR(k−N/2)2 t

e|k, N − ki

(5.10)

k=0

The phase diffusion rate, R, is proportional to the derivative of the chemical
potential of condensates, µ(Ni ) (i = 1, 2), with respect to the atom number and thestandard deviation of the relative atom number, ∆Nr [35, 122, 202, 96, 117, 97]:81

R = (2π/h)(dµ/dNi )Ni =N/2 ∆Nr(5.11)A typical decoherence time 1/R in a condensate is less than 50 ms. A number√
squeezed state with sub-Poissonian number fluctuations (∆Nr = N /s), where s > 1
is the squeezing factor, will exhibit a reduced phase diffusion rate relative to a phase√coherent state with ∆Nr = N . Now, the question is how to prepare a number
squeezed state in condensates. Indeed, it turned out that atom-atom interactions
tends to localize atoms during splitting process leading to number squeezing as weshall discuss in the next chapter 6.
In contrast to normal diffusion processes, the phase uncertainty, ∆φ, increases
here linearly. The phase uncertainty evolves with time as follows:∆φ(t)2 = ∆φ20 + (Rt)2(5.12)

where ∆φ0 ' (s/∆Nr ) is the initial phase uncertainty. Interestingly, the term “phasediffusion” is not really appropriate because the phase diffusion has a time dependence√of ∝ t, not ∝ t.
Phase fluctuation
So far, we have seen that a Bose-Einstein condensate should have a phase coherence
over a long-range length scale which is equal to the size of the condensate. This longrange coherence in a condensate is generally valid in three-dimensional systems, but
not in one- or two-dimensional systems where phase fluctuations play the importantrole.
Phase fluctuations cause the condensate to break up into several quasi-condensateswith random phase, i.e. long range coherence is lost. This usually happens in elongated geometries when the temperature is sufficiently high to excite such modes [155,51], or in interacting one-dimensional condensates even at zero temperature due toquantum fluctuations [23].
A typical elongated trap geometry, for example realized by an atom chip, has
an aspect ratio of ∼200 [179, 174, 101], sufficient to induce phase fluctuations in aquasi-condensate along the axial direction [155] already at very low temperatures (orin the 1D case, even at zero temperature). When the temperature of a condensate is82

above the characteristic temperature,T∗ =15N (~ωz )232µ(5.13)

where µ is the chemical potential, N total atom number, ωz axial trap frequency, and~ the Planck’s constant divided by 2π [155], then thermal excitations of low energy
axial modes lead to longitudinal phase fluctuations. For temperatures above T ∗ , thecoherence length L∗ of a phase-fluctuating condensate is shorter than the length L of
the condensate L∗ /L = T ∗ /T [155].
When Bose condensates are used for atom interferometry, spatial phase fluctuations have two major consequences. First, they speed up phase diffusion, since∆N/N refers now to the atom number in a single quasi-condensate. Second, they
make the atom interferometer much more sensitive to random relative displacementsof the split condensates, which have to be smaller than the coherence length, which,for condensates with phase fluctuations, can be much smaller than the size of thecondensate.

5.3

Measurement of the Relative Phase between
Two Condensates

Bose-Einstein condensates have phase coherence over the whole atomic cloud unlessphase diffusion and phase fluctuations emerge and destroy it. Of particular interest
has been the investigation of phase coherence properties in various systems such as
the Josephson junction in a double-well potential, optical lattices, and elongated onedimensional system.
In related efforts, access to a phase-coherent condensate has brought a new familyof interferometry technique in atom optics, so-called confined atom interferometry.
Until our work reported in chapter 7, the only method to read out the relative phasehas been the “ballistic expansion” method. In this scheme, two spatially separated
condensates with the well-defined relative phase are released after some hold timefrom their confining potential, and finally the phase shift is obtained by imaging andanalyzing the interference pattern formed in the overlapped atomic cloud. To date,
the relative phase has been reliably measured after ballistic expansion of two separated
condensates to decrease the atomic density [177]; therefore atom-atom interactions are
negligible during phase read-out. Nevertheless, the ballistic expansion method lacks

83

the simplicity and robustness of in-trap recombination. In addition, the condensates
cannot be recycled after phase read-out because of the destructive expansion of thecondensates.
In this section, we demonstrate three different phase read-out methods; (1) ballistic expansion method (2) phase sensitive in-trap recombination method and (3)optical read-out method (summarized in figure 5-2). Recent developments including
the method (2) and (3) suggest that the confined atom interferometer can circumventthe deleterious effects of atom-atom interactions during phase read-out.
(1) Ballistic expansion method
In this method, the relative phase between the two separated condensates is determined by the spatial phase of their matter wave interference pattern formed after
ballistic expansion as described in figure 5-1. The ballistic expansion reduces theatomic density so that the effects atom-atom interactions are neglected when two
condensate are overlapped. For a ballistic expansion time t À 1/ωr where ωr is thetrap frequency along the axis connecting two condensates, each condensate had aquadratic phase profile [46],→ψ± (−r , t) =

q

−
→
m −
→
n± (−
r , t) exp(i|→r ± d /2|2 + φ± )2~t(5.14)

where ± indicates the left or right condensate, n± the condensate density, m atomic−→
mass, d the vector connecting two condensates, and φ± the condensate phase. The
overlapped interference pattern now becomes,
md
√
→
n(−
r , t) = |ψ+ + ψ− |2 = [n+ + n− + 2 n+ n− cos(x + φr )]~t(5.15)

Here, the relative phase (φ+ − φ− ) can be extracted from the spatial phase shift
φr = φ+ − φ− in the modulated pattern of eq. 5.15. In the experiment, one can fit
an integrated cross section of the matter wave interference pattern with sinusoidallymodulated Gaussian curve (see figure 5-1),
I(x) = α exp(−(x − x0 )2 /σ 2 )[1 + β cos(2π(x − x0 ) + φr )].
λ(5.16)~tobtaining the relative phase, φr , of two condensates. Note that fringe spacing, λ = md,
is proportional to the expansion time and the inverse of the separation between two
condensates.

84

(b)
Optical density(a)Position

Figure 5-1: Matter-wave interference and relative phase read-out. (a) Absorption
image of an overlapped condensate released from double-well potential is taken after
10 ms time-of-flight. The optical density profile (open circle in (b)), obtained byintegrating the the optical signal between two dashed lines, is fitted to a sinusoidallymodulated Gaussian curve (blue curve in (b)) as showns in section 5.3 from which
the relative phase is measured.

(2) Phase sensitive in-trap recombination method
In contrast to the ballistic expansion of atom clouds, in-trap recombination of two
condensates is inherently simple and robust, and moreover it allows one to store thecondensates in a confined potential during the measurement process. This is one
definite advantage of a confined atom interferometer without precedent in optics.
The challenge, however, is how to extract the relative phase after connecting two
originally separated condensates. It was pointed out that the recombination process
may produce the excited mode with exponential growth rate, leading to an unstablemeasurement in phase [60]. Recently, however, we demonstrated that in-trap recombination leads to heating of the atomic cloud. The heating is phase-sensitive and canbe used as a robust phase read-out of the atom interferometer. This method potentially gives us an opportunity to recycle condensates after resetting the temperature
through evaporating cooling. Experimental details shall be discussed in chapter 7.1.
(3) Optical read-out method
Both phase read-out methods we have seen above rely on the merger of two separated condensates with or without ballistic expansion. This requires coherent control
of external (motional) state in order not to perturb the phase. Recently, a newphase-readout method based on stimulated light scattering was proposed [158], anddemonstrated experimentally [166]. Two counter-propagating laser beams outcouple
85

(a) Ballistic Expansion
BEC 1

Ballistic
Expansion (t)Fringes
SpacingdBEC 2(b) In-trap Recombination

Recombination

Heating(c) Non-destructive Optical Read-out
Atomic Flux

Figure 5-2: Schematics for measuring the relative phase of two separated condensates
are shown: (a) ballistic expansion method, (b) in-trap recombination method, and(c) non-destructive optical read-out method. Additional details are discussed in thetext.

a small fraction of atoms from the two spatially separated condensates leading to ainterference between outcoupled atomic clouds. By monitoring the rate of outcoupledatoms or the number of scattered photons, the relative phase between two original
condensates can be determined. In this process, the phase measurement establishes
the relative phase between two (originally) independent condensates. Note that thismethod is nondestructive similar to the in-trap recombination method.

86

5.4

Tools for the Study of Coherence Propertiesof a Bose gas

For the study of coherence properties in Bose gases including phase diffusion [177,174, 101], phase fluctuations [80, 163, 99], and non-equilibrium dynamics in onedimensional system [83, 23], interferometric measurements provide a very robust andprecise way to extract the physical quantities that we want to measure. For allthese purposes, it is imperative that the relative phase between two Bose condensates
is not perturbed by the splitting process itself. Recently, the coherent control ofexternal (or motional) [177, 174, 101] and internal [21] quantum states 2 in a Bose
condensate has been demonstrated. Here, we focus on the external dynamics assumingthat internal quantum states (usually hyperfine state in cold atom experiments) areconserved. By deforming a single trapping potential into double wells, the wavepacket
of a condensate is spatially delocalized into two wells adiabatically; we call this process“coherent splitting”. For the coherent splitting, there are two important time scales,
namely the oscillation period in a trap τtrap and the collapse time τcollapse . The
time scale τ at which the splitting ends up should be adiabatic with respect to theexternal dynamics (i.e. τ À τtrap ), but non-adiabatic the internal time scale related
to collapse-revival process (i.e. τ ¿ τcollapse ). In a typical BEC experiment, τcollapse
is on the order of 5∼50 ms.
The coherent splitting in a confined interferometer has been firstly demonstrated
by Shin et al. in an optical double-well potential [177], and then by Schumm et al.
in a magnetic trap combined with oscillating radio frequency field [174].
Coherent splitter using an adiabatic RF-induced potential
Adiabatic RF-induced potential was first proposed in order to trap neutral atoms in a
two-dimensional geometry [208]. Combining static magnetic trapping field with oscillating radio frequency (RF) near fields, the original magnetic potential adiabatically
evolves into the effective dressed potential as described in figure 5-3 (a)-(c). Here, anRF field couples internal atomic state with different magnetic moments characterized
by the quantum number mF 3 . In a dressed state basis, the effective potential Vef f (r)2Note that coherent control of internal states (in state-dependent fashion) is a crucial ingredient
to realize quantum entanglement in combination with controlled collisions.
3
Adiabatic RF-induced splitting sometimes can be considered as “anti-evaporation”. Whereas
the evaporation process removes atoms with higher kinetic energy in an untrapped internal state byadiabatically coupling two different internal state, RF-induced splitting adiabatically keeps atoms
in trapped internal state. In evaporation, the RF resonance is near the outer turning point; in87

Adiabatic rf-dressed Potential
(a)

(b)

m= -1

m= -1

(c)

|+>

RF
m= 0

m= 0

m= +1

m= +1

RF

|->

(d)

rf frequency [kHz]Figure 5-3: Schematic for an adiabatic RF-induced splitter used in MIT experiments.
Three different spin states are shown in (a). In the presence of oscillating radio
frequency fields (b), the effective potential becomes double wells in a dressed statebasis (c). (d) By adjusting the RF frequency, a single well is smoothly transformed
into double wells allowing for coherent splitting of a condensate. Note that coherent
splitting can also be possible by varying RF amplitude at a fixed RF frequency [174].

at position r is determined by
Vef f (r) = mFp[µB gF Bd.c (r) − ~ωRF ]2 + [µB gF BRF ⊥ (r)/2]2(5.17)

where gF is the Lande g-factor, µB the Bohr magneton, Bd.c the magnitude of thestatic magnetic field, BRF ⊥ the magnitude of the component of the RF field perpendicular to the direction of static field, and ωRF the RF frequency.
Recently, the adiabatic rf-induced potential has been implemented into an atom
chip realizing coherent splitting of condensates [174]. By turning on the RF field
adiabatically, the single trapping potential smoothly transforms into double wells.
Here, the conditions for adiabatic switch-on of the RF field are given by LandauZener theory.
The wires on the atom chip carry both d.c current and oscillating current. Depending on the relative geometry between the static trapping potential and the RF
field, the direction of splitting can be arbitrarily controlled as described in figure 5-4(a) [84].
RF-induced splitting, it is near the inner turning point.

88Horizontal splitting(b)

Atom Chip

Gravity
RF field

Atom Chipd.c.
trapping

RF wire +d.c.
Gravity

Bias field

RF wireVertical splitting

RF field

Trapping
field

Bias field(a)Trapping
field

Figure 5-4: The spatial orientation of the double-well potential is determined by the
relative orientation of the static magnetic field and the oscillating magnetic RF field.
Potential minima form where both the static and the oscillating fields are parallel as
described in (a) and (b). The red circle in (a) represents the position of the trapping
potential where a double-well potential is formed in the horizontal direction.

The adiabatic RF-induced splitting has several advantages [119] over the conventional two-wire scheme [179] which requires two independent wires generating
quadrupole trap. During the merging and splitting process in the two-wire scheme,
the trapping potential becomes very weak. Confinement is no longer parabolic, butcaused only by higher-order multipoles (usually hexapole) leading to a significant weak
confinement. This often induces unwanted atom loss during splitting, and therefore
makes the splitting region unstable. The adiabatic RF-induced splitting, however
keeps the confinement tight during splitting or merging and does not suffer from theatom loss. In addition, the RF-induced splitting is less sensitive to magnetic bias
fields and realizes high trap-frequencies far from a surface.
Figure 5-4 demonstrates the coherent beam splitter implemented at MIT (seechapter 6 for experimental details).
Towards precision measurement with Bose-Einstein condensates.
So far, we have discussed the phase coherence between two separated condensates.
This property allows us to build a new type of atom interferometer with BECs. In
atomic physics, precision measurements are usually performed at low atomic densities
because of collisional shifts and dephasing. Since the atom-atom interaction energy
results in so-called clock shifts [63] which are proportional to the high density, mostprecision measurements with neutral atoms are performed with free-falling atoms in89atomic beams [118, 73] or in fountain geometries [151]. Major efforts are currently directed towards atom interferometry using confined geometries, such as atom traps or
waveguides, often realized by using atom chips [59, 45]. These geometries are promising in terms of compactness and portability, and also offer the prospect of extending
interrogation times beyond the typical 0.5 s achievable in the atomic fountains [151].
Is spite of these advantages, confined atom interferometer has a few intrinsic limitations; e.g. phase diffusion [35, 122, 202, 96, 117, 97] and spatial phase fluctuations(see section 5.2). In the following chapter, we shall show that a chip-based atom interferometer can be operated even in the presence of both phase diffusion and phasefluctuations.

90

Chapter 6
Long Phase Coherence Time via
Number Squeezing and Phase
Fluctuations of BECs in a
Double-well Potential
This chapter describes the experimental studies of phase coherence and the decoherence
dynamics of Bose-Einstein codensates in a double-well potential. The experiments
were reported in two publications:• G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, D. E.
Pritchard, M. Vengalattore, and M. Prentiss
Long phase coherence time and number squeezing of two Bose-Einstein condensates on an atom chip
Physical Review Letters 98, 030407 (2007) Included in appendix H.

• G.-B. Jo, J.-H. Choi, C.A. Christensen, Y.-R. Lee, T.A. Pasquini, W. Ketterle,
and D.E. Pritchard
Matter-wave Interferometry with Phase Fluctuating Bose-Einstein Condensates
Physical Review Letters 99, 240406 (2007) Included in appendix E.
In this chapter, we discuss the phase coherence of a Bose-Einstein condensate
trapped in an elongated atom-chip trap. Two main questions are addressed in section
6.1 and 6.2 respectively:(1) How long can the phase coherence between separated condensates be maintained
91and how can one enhance the phase coherence time?
(2) What is the effect of spatial phase fluctuations on atom interferometry?

6.1

Long Phase Coherence Time and Number Squeezing

The distinguished property of Bose-Einstein condensates (BECs) is their phase coherence. High-contrast interference between condensates [11] clearly revealed thisproperty. Observations of many interesting phenomena, including Josephson oscillations [36, 5], number squeezing [142], and the transition from superfluid to Mott
insulator [66] rely on phase coherence between two spatially separated condensates.
As we have seen in the previous chapter 5, an RF beam splitter allows us to
study and/or exploit the phase coherence in condensates in many different ways.
Here, our system consists of a double-well potential on an atom chip with adjustablebarrier between two wells. This tunable barrier in a double-well system allows for the
preparation of coherent state of condensates, the realizatoin of Josephson junction, thestudy of decoherence dynamics, and the development of confined atom interferometry.
Furthermore, the geometry of the magnetic trap generated by an atom chip introduces
low-dimensional effects including phase fluctuations as follows.
• Highly nonequilibrium system :In contrast to equilibrium properties of a Bose condensate, the relaxation dynamics from nonequilibrium states is not well-understood [90]. One reason isthat it is relatively difficult to prepare such a nonequilibrium state in a coherent
manner. For the split-condensate system built in MIT experiments [99], a phasecoherent state is highly out of equilibrium, and the condensates inevitably suffer from both quantum and thermal decoherence over time. In particular, therelaxation process is of interest and importance in low-dimensional systems inwhich quantum decoherence effects are more prominent [99, 83, 85].
• Bosonic Josephson Junction (BJJ) :The Josephson effects, predicted by Brian D. Josephson in 1962 [104], occur
when two macroscopic quantum states are weakly connected. First experimental realization was made in the system of two superconductors coupled via an
insulating thin film and later the effects were observed in superfluid helium-3
and helium-4. Recently, the Josephson effects was also studied in Bose-Einstein92

condensates [36, 2]. Bose-Einstein condensates in a double-well potential withtunable barrier becomes available allowing for unprecedented quantitative investigation of the Josephson effects.
• Low-dimensional physics :As we have discussed in the section 5.2, the effect of phase fluctuations associated with quantum and thermal fluctuations becomes important in elongatedquasi-1D condensates realized on an atom chip. The coherent dynamics in degenerate quasi-1D Bose atoms can be investigated in a interferometric way using
coherent the RF splitter.
• Towards chip-based atom interferometry :A coherent beam splitter is one of the essential parts for confined atom interferometry. In general, a good confined atom interferometer requires the following
conditions: (1) high atomic density leading to large sinal-to-noise ratio, (2)long
phase coherence time for high sensitivity, (3) efficient and reliable phase readout, and (4) uniform phase along the condensate. Bose-Einstein condensates, avery bright source of atoms, have been considered to be used in a confined atominterferometer because of their phase coherence and high atomic density (condition (1)). In this chapter, we experimentally demonstrate that a chip-based
atom interferometer installed at MIT is promising to maintain phase coherence over a sufficiently long time (condition (2)). The robustness of the systemagainst phase fluctuations is also described in section 6.2 (condition (4)). For
phase read-out methods, conventional and recently developed methods are described in section 5.3 and chapter 7. All recent developments including MIT
experiments suggest that it is possible to operate confined atom interferometerwith Bose condensates at high atomic density.

6.1.1

Splitting Process in a Double-well Potential: a Bosonic
Josephson Junction

Splitting of the condensate results in two separated condensates which cannot besuitably described by the Gross-Pitaevskii equation (GPE) [69, 70] which describes
atoms only in a single-mode. This can be resolved by taking an approximate form of
the bosonic field operator in the two-mode basis which naturally consists of the ground[symmetric, figuure 6-1 (a)] and the first excited state [antisymmetric, figuure 6-1 (b)]in a double-well system. In this section, I will derive a relevant Josephson Hamiltonian
93

(a)

Symmetric state(b)Antisymmetric state

Figure 6-1: Schematic illustration of the ground and the first excited state in a doublewell potential. The symmetric ground state (a) and the antisymmetric excited state(b) are shown.

describing the splitting of the condensates in the two-mode approximation.
Two-mode approximation and a Josephson Hamiltonian
The relevant Hamiltonian describing weakly interacting bose condensed atoms in a
double-well potential is given byZb=H

d3 r[−~2 †g
Ψ (r)∇2 Ψ(r) + U (r)Ψ† (r)Ψ(r) + Ψ† (r)Ψ† (r)Ψ(r)Ψ(r)]
2m
2

(6.1)

where Ψ(r) is a bosonic field operator, U (r) the external potential describing a doublewell, and g = 4πa~2 /m the interaction strength with the s-wave scattering length a
and atomic mass m. In two-mode approximation, Ψ(r) becomes the superposition of
the ground state ψg and the excited state ψe with appropriate creation (or annihilation) operators:
Ψ(r) = âg ψg + âe ψe(6.2)
Here, â†g (â†e ) is the creation operator for a particle in the ground state (excited state).
Now, the observable quantities such as the atom number and the relative phase canbe conveniently described by changing the basis as follows:11âL = √ (âg + âe ) and âR = √ (âg − âe )
2
2

(6.3)

In this new basis, the relevant quantum state become ψR (ψL ) denoting the wavefunction of localized atoms in the right (left) well. More intuitive form can be obtainedby inserting equations 6.2 and 6.3 into equation 6.1. Neglecting two-particle tunneling94

process 1 , the two-mode Hamiltonian takes a form ofb = ec n̂2 − ej α̂
H
2

(6.4)

where

â†R âR − â†L âLâ† âR + â†R âL, α̂ = L(6.5)
2
2
Here, two conjugate operators n̂ and α̂ are the atom number difference (or relative
atom number) and the tunneling operator related to the relative phase respectively.
R
The coefficient ec = 4g dr|ψg |2 |ψe |2 describes the local interaction within each welland ej tunneling energy of particles. In the splitting process, ej is a function of time.
n̂ =The equation 6.4, sometimes known as Bose-Hubbard Hamiltonian, captures essential
physics of many phenomena including ac/dc-Josephson effect and self-trapping.
What is the ground-state wavefunction of the Hamiltonian 6.4? In the limitingcase of large tunneling energy ej À ec , the ground state wavefunction takes the form
Lof ψR√+ψ
containing the well-defined relative phase between two wells. On the other2
hand, the ground state beccoms a Fock state minimizing the atom number difference
when the local interaction ec is dominant.
The equation 6.4 derived from the two-mode approximation describes relevant dynamical properties of Bose Josephson junction as a function of two relevant wavefunctions ψg and ψe . Two conjugate observables, however, the atom number difference∆N and the relative phase φ, can be calculated only when ψg and ψe are exactly
known. For a sufficiently low temperature, the many-body Schrödinger equation reduces to Gross-Pitaevskii equation (GPE) and two wavefunctions ψg and ψe can bederived. Now, a Josephson Hamiltonian in terms of ∆N and φ is given byb Josephson = Ec Nr 2 − Ej cos(φ) ' Ec Nr 2 + Ej φ2
H
2
2

(6.6)

where Nr = NL − NR is the atom number difference of atoms in the two condensates 2 . Here, Nr and φ are conjugate variables satisfying [Nr , φ] = i. Indeed, thedµcharging energy Ec is given by Ec = dN
where µ is the chemical potential. Notethat the equation 6.6 is equivalent to the Hamiltonian of a harmonic oscillator withpa characteristic frequency of ω = Ec Ej when Nr and φ correspond to position andmomentum variables.
1
2

Two-particle tunneling process term is proportional to (â†L âR + â†R âL )2 .
Here, we assume that ~ = m = 1 for convenience.

95

Description of the splitting process: Rabi, Josephson, and Fock regime
To describe the splitting process of the condnesates, it is useful to study the propertyof the ground state of the Josephson Hamiltonian 6.6. First, comparing the equation 6.6 with a harmonic oscillator, the relative number fluctuation is simply givenEby ∆Nr = ( Ejc )1/4 strongly depending on the ratio of the charging and tunneling energies. Note that this expression is not suitable in Rabi regime. In Rabi regime, thetwo-particle tunneling process we have neglected in the expression 6.4 is not negligible. In this case, the tunneling energy Ej must be renormalized to obey Poissonian
distribution.
For our purpose, it is convenient to divide three different regimes as follows (seefigure 6-2):√• Rabi regime: Ec /Ej ¿ 1/N 2 and ∆Nr ≈ N√• Josephson regime: 1/N 2 ¿ Ec /Ej ¿ 1 and 1 ¿ ∆Nr ¿ N• Fock regime: 1 ¿ Ec /Ej and ∆Nr ¿ 1
The ground states in these regimes have different statistics in the atom number andthe relative phase. In Rabi regime, the relative atom number fluctuates with ∆Nr ≈√N , obeying Poissonian distribution. The system corresponds to the non-interacting
limit in this regime, and the relative phase is well-defined. In the Josephson regime,
the distribution of atom number is sub-Poissonian leading to a number-squeezed state√(1 ¿ ∆Nr ¿ N ). The relative phase can be well-defined in this regime although
there is a small amount of quantum noise in the phase. In a Bose condensate, theuncertainty in the phase is negligible as the atom number, N , is large enough. In
contrast to other regimes, the ground state in the Fock regime have only a well-definedatom number, not a relative phase. The uncertainty in the phase diverges infinitely.
The description of three different regimes in the Josephson junction may guide
us to obtain a simplified description of the splitting process. The tunneling energy
Ej (t) changes from infinity to zero within the time scale of τs when a condensatepsplits into two parts. By noting that two time scales ω(t) = Ec Ej and τs exist
in the system, we can distinguish two steps of the splitting process (see figure 6-2).
When ω(t) > 1/τs (step 1 in figure 6-2), the system of the double-well potential
evolves adiabatically so that the ground state of the equation 6.6 should describe
the actual state of the system in a good approximation. If adiabatic condition is nolonger valid at t∗ (therefore ω(t) < 1/τs ), the rest of the splitting is approximated to“instantaneous splitting” (step 2 in figure 6-2). In other word, one assumes that the96

Splitting

Step 1

Step 2t*Josephson

Rabi (SF)

Fock ( Mott )1

1/ N 2~ N∆N ~Ec / Ej

1 ≤ ∆N ≤∆N < 1N

Figure 6-2: Three different regimes in the description of the dynamic splitting ofcondensate: Rabi, Josephson, and Fock regimes.

tunneling energy Ej becomes zero at t∗ . Therefore, the final state when the splitting
ends is determined by the ground state of the system at the moment t∗ . Note thatslow (or adiabatic) splitting results in the Fock state in which ∆Nr = 0 whereas thefast splitting 3 the final state lies in the Rabi regime.
The condition ω(t∗ ) = 1/τs implies that the uncertainty in the relative atom
number after the splitting process is given by
r
∆Nr =

1
dµ
|t=t∗ where Ec =
Ec τs
dN

(6.7)

This result implies that one can prepare a number-squeezed state in a double-wellpotential via the dynamic splitting. The main knob is the speed of the splitting
process, τs , which will determine the critical time t∗ (see figure 6-2). In a harmonic
trap, a squeezing factor ξ becomes:
√

N
=
ξ=
∆Nr

r

2
µτs
5

(6.8)

As we have seen in the previous chapter 5, Bose condensates suffer from intrinsic
phase diffusion leading to the potential problem with atom interferometry. In the following section 6.1.2, we will discuss how we prepare a number-squeezed state throughthe dynamic splitting and the effect of number squeezing on the phase diffusion. In
the MIT experiment [101], this simple model gives a squeezing factor of ∼13 withτs =75 ms and µ = ~×6 kHz.
3

We shall see the condition for “fast splitting” in the next paragraph: τs ¿ 1/µ.

97

(a)

(c)

(b)
Atom chip
RF wire Trapping wire(~ 70 mA)(~ 1 A)0 ms

U
By~ 10.5 G

dYZ

194 msBz~ -2.2 G

Figure 6-3: (a) Schematic of the atom chip interferometer is shown. (b) Double-wellpotential. The separation d between the two wells and the barrier height U were
controlled by adjusting the frequency or amplitude of the RF field. (c) Matter wave
interference. For various hold times after splitting, absorption images of condensates
released from the double-well potential were taken after 10 ms time-of-flight. The
field of view is 260 × 200 µm.

6.1.2

Preparation of a Number-squeezed State on an Atom
Chip

So far, we have discussed the possibility of preparing a number-squeezed state throughsplitting precess. The result (see equation 6.8) may encourage us to operate an atominterferometer with high atomic density source even in the presence of deleterious
phase diffusion. Indeed the repulsive atom-atom interactions make it energetically favorable for the two condensates to split with equal numbers in a double-well potential,
whereas number fluctuations , such as in a coherent state, cost energy [131, 33]. The
relation 6.8 implies this process; the higer the atomic density (or chemical potential)
is, the less the relative atom number fluctuates. Interaction-induced squeezing reduces
the phase diffusion caused by the same interactions [35, 122, 202, 96, 117, 97, 131, 33]
Coherent splitting and matter-wave interference
Bose-Einstein condensates of ∼ 4 × 105 23 Na atoms in the |F = 1, mF = −1i state
were transported into the science chamber and subsequently transferred into a magnetic trap generated by atom chip wires and external bias field [179]. A double-wellpotential in the horizontal plane was formed using adiabatic RF-induced splitting as
described in figure. 6-3 (a) [208, 174] and in section 5.4.
Atoms were confined radially by the combined magnetic potential of a currentcarrying wire and external bias field as described in figure 6-3. Axial confinement in98

x direction was provided by a pair of endcap wires (not shown) [179]. By dressing theatoms with an oscillating radio frequency (RF) field from a second wire, the single
minimum in the magnetic trapping potential was deformed into a double-well [174].
If the trapping position lies on the circle containing the trapping wire and centeredon the RF wire, the splitting occurs in the horizontal plane. Condensates were placed
185 µm away from the chip surface. For the single well, the radial (axial) trap
frequency was fr = 2.1 kHz (fz = 9 Hz) and the Larmor frequency at the trap centerwas ≈ 190 kHz (Bx ≈ 0.27 G). Splitting was performed over τs =75 ms by linearlyramping the frequency of the RF field from 143 kHz to 225 kHz. Gravity points in the+z direction. Typical double-well potential is shown in figure. 6-3 (b). The separationd between the two wells and the barrier height U were controlled by adjusting thefrequency or amplitude of the RF field. Typically, the separation of the two wells wasd ∼ 8.7 µm, the height of the trap barrier was U ∼ h × 30 kHz, and the chemical
potential of the condensates, measured from the trap bottom, was µ ∼ h × 6 kHz (seefigure 6-3(b)).
The lifetime of the atoms at the splitting position was ∼1.8 s, significantly longer
than in our previously demonstrated two-wire splitting method [81, 41, 179].
Atoms were held in the double-well potential for varying hold times, released byturning off the trapping potential within 30 µs at a known phase of the RF field,
overlapped, and interfered in time-of-flight (see figure 6-3 (c)). The relative phase of
the two condensates was measured as the spatial phase of the interference pattern [177,179] as described in chapter 5.
When the trapping potential was switched off, we observed strong correlation
between the population distribution of the spin components and the phase of the RF
field at the moment of release, probably since the strength of the RF field (∼0.35G)
was comparable to the local static field (∼0.27G) [13]. The trapping potential was
switched off at a value of the RF phase chosen to release atoms predominantly in the|F = 1, mF = −1i state.
Atom interferometry in a double-well potential
Phase evolution : An atom interferometer requires two independent condensates
without any weak link which may lock the phase [177, 179]. To demonstrate this independence, we monitored the relative phase over short intervals right after and after
up to 190 ms delay time. During each of these intervals, the phase evolved linearlywith time at ∼ 2π × 200 Hz, the signature of independent condensates (figure 6-4).
The non-vanishing phase evolution rates are attributed to small asymmetries in the99Relative Phase(radian)

(b)

(a)

(c)

0

0

2

4

6

71 73 75 77 79

190 192 194 196

Hold Time (ms)Figure 6-4: Phase evolution of the relative phase during three different time intervals.
The evolution rate of the relative phase are determined from the linear fit to be (a)191 Hz, (b) 198 Hz, and (c) 255 Hz. The data points represent the average of ten
measurements for (a) and (b), and fifteen for (c).

two trapping potentials, which lead to slightly different chemical potential after the
splitting process. The time variation of this rate is attributed to axial motion of thetwo separated condensates. Note that the observed drift of the phase evolution rateof ∼ 60 Hz is only 1% of the condensates’ chemical potential. In principle, the phasedrift could be zeroed by a compensation field, but this has not been attempted.
Phase imprinting and read-out To rule out the possibility that any weak link
existed during the 200 ms time evolution and reset the relative phase, we demonstrated that an applied phase shift could be read out 200 ms after its appliance(figure 6-5). This proves that two independently evolving condensates have preserved
phase coherence up to 200 ms, a factor of 10 longer than the phase diffusion time,τc = 1/R '20 ms, for our parameters.

100Phase Shift at 191 ms (radian)

π

2π/3

π/3

0

π/3

0

2π/3

π

Phase Shift at 7 ms (radian)Figure 6-5: Applying various phase shifts to the condensates at 2 ms after splitting,the shifts of the relative phase were measured at 7 ms and 191 ms, showing strong
correlation. The dotted line denotes the ideal case of perfect phase coherence. Phase
shifts were applied by pulsing an additional magnetic field in the z direction for 50 µs
with variable amplitude.

Quantitative analysis of phase coherence: Rayleigh test
What is the experimental criterion for phase coherence between two condensates aftersome hold time? Most conveniently, one can measure the relative phases from the interference patterns formed after some ballistic expansion, and obtain the distributionof phases for different experimental runs [177, 174] which is typically characterized by
the standard deviation. For the quantitative study of phase coherence, however, thestandard deviation of the phase does not provide the best characterization because
the phase is measured modulo 2π (so-called circular data). In the limit of a large dataset, a completely random distribution has a phase variance of ∼ (3π/5)2 . Alreadyfor smaller variances, the overlap of the tails of the Gaussian distribution can cause
ambiguities.
To measure correlations in circular data appropriately, we perform a Rayleightest [58] on the measured relative phases as described in figure 6-6. Here, we represent
each measurement of the relative phase as a phasor with unit length and compare√
the length of the sum of N measured phasors with the expectation value of N for
N random phasors. The larger the difference, the smaller is the probability thatthe data set is compatible with a random phase distribution. This probability of101

Rayleigh Test
(1)

(2)

90
0.2
135

45
0.15
0.1

Phasor Sum= |

0.05
180

0
0

225

∑e |
iφ

0.05 0.1 0.15 0.2

315
270

: Phasor Vector(3)

Measured
Phasor Sum(4)

Randomness =Figure 6-6: Quantitative study of phase coherence. In a circular data (1), the Rayleightest compares the phasor sum of the data (2) with the distribution of phasor sums for
random data (3) and returns the randomness (4), the probability that uncorrelated
measurements would produce a phasor sum larger than the measurement.

102

uncorrelated phases is called randomness [58]. Uncorrelated data have an expectedvalue of randomness near 0.5, while strongly correlated data would have a smallvalue, e.g. ten data points drawn from a distribution with variance (π/5)2 have aprobability of only 10−4 to be compatible with uncorrelated phases. If a series ofphase measurements has a randomness value of 0.01 (0.1), the relative phase is nonrandom with a probability of 99% (90%).

6.1.3

Phase Diffusion and Number Squeezing

To study phase diffusion in our system, we analyze the distribution of ten measurements of the relative phase at various times after splitting, as shown in figure 6-7.
The data show a well-defined phase (with probability > 90%) for times shorter than∼ 200 ms. In contrast, the simulation for a coherent state in our experimental conditions, shown as a blue dotted line, predicts the same scatter of phase measurements
already after ∼20 ms. Fitting a phase diffusion model to the data points with randomness probability > 0.1, gives a phase diffusion time of 200 ms. The solid line is afit which includes the initial variance ∆φ20 :∆φ(t)2 = ∆φ20 + (Rt)2(6.9)

The variance of the initial state, ∆φ20 = (0.28π)2 is dominated by technical noise
including fitting errors and non-ideal trap switch off. The contribution due to initial√
number fluctuations, ∆φ20 ' (s/ N )2 , is ∼ (7.1 × 10−4 π)2 for a coherent state (s=1),
and remains small unless the squeezing leads to number fluctuations on the order√of a single atom, s ∼ N . The fitted value for the phase diffusion rate of R =5 s−1 includes technical shot-to-shot variations in the relative atom number of two
condensates after splitting and thermal fluctuation. Therefore, the inferred squeezing
factor s=10 represents a lower bound. It implies that our relative atom number
fluctuations were smaller than ±0.03% corresponding to ±50 atoms.
Locally the interference pattern of two pure condensates should always have 100%contrast, where contrast is defined as the density amplitude of the interference fringes
over the mean density. Since in our experiment the contrast is derived from interference pattern integrated along the line-of-sight, it decreased gradually with time andexhibited fluctuations most likely due to asymmetric axial motion (figure 6-7(b)).
Except for small regions near 110 and 300 ms hold time, the contrast was above 10%,
sufficient for accurate determination of the phase. The small windows with poor
contrast have a large probability for random phases.
103

Probability of Random Phases

1
8
6

2

0.1
8
6
4

2

0.01
8
6

25

Contrast (%)

(a)

4

(b)

20
15
10
5
0

0

50

100

150

200

250

300

350

400

Hold Time (ms)Figure 6-7: (a) The randomness probability of ten measurements of the relative phaseis displayed up to 400 ms after splitting. The blue dotted curve (red dashed curve)
shows a simulation for a phase-coherent state (number squeezed state with s = 10),which have negligible initial phase uncertainty. The solid line includes an initial phase
uncertainty of 0.28π (see text). The shaded region represents the window where ten
data points from the sample with the given phase uncertainty would fall with 50%
probability. (b) Contrast of the interference pattern. Since the endcap wires generate2za field gradient ∂B
as well as a field curvature ∂∂xB2x at the position of the condensates,∂x
the two wells are not parallel to the trapping wire and consequently have slightly
different axial trapping potential. This difference induces relative axial motion of thetwo condensates, which periodically reduces the contrast.

104

(b)

(a)
Shift

Z

X
0

80

430

0

16

26

22ms

Time (ms)204 kHz
154 kHzRF frequency

Figure 6-8: a) A single trapped condensate at rest was accelerated by shifting the trap
center by ∼ 430 µm in the axial direction (t = 0 ms). The frequency of the RF field
was linearly ramped from 154 kHz to 204 kHz during 16 ms after launch, splitting thecondensates and separating them by 5−6 µm. (b) Phase measurements were done forup to 26 ms after the launch. The probability for random phases was determined fordata sets of ten measurements. Until 16 ms, this probability was extremely low (lessthan 10−12 ) and the relative phase was constant, implying that the two condensates
were still connected. For t > 16 ms, the relative phase evolved (similar to figure 64), and the probability for a random phase distribution was smaller than 10%. This
demonstrates that phase coherence was preserved after full splitting. The figure shows
the interference pattern for t = 22 ms. The field of view is 260×200 µm.

6.1.4

Conclusion and Remarks

So far, we showed that we can operate a BEC interferometer at high atomic density,with mean-field energies exceeding h × 6 kHz. Using a radio frequency (RF) induced
beam splitter [208, 41, 174], we demonstrate that condensates can be split reproducibly, so that even after 200 ms, or more than one thousand cycles of the mean-fieldevolution, the two condensates still have a controlled phase. The observed coherencetime of 200 ms is ten times longer than the phase diffusion time for a coherent state.
This work is a major advance in the coherence time of confined atom interferometers, which have operated at interrogation times below ∼50 ms [177, 174, 62] due totechnical limitations. This work also advances the preparation of number squeezed
states to much higher atom numbers [101, 172]. Previous experiments in optical lattices [142, 66, 55] and in an optical trap [40] were limited to very small populations(∼ 1 − 1000 atoms). In addition, the fact that the clouds could be prepared on an
atom chip with dc and RF electric currents, but without any laser beams, is promisingfor future applications.

105

Rotational sensitive interferometry: “Sagnac effect”
The work presented so far, and also all previous work on interferometry of confinedor guided atoms, featured geometries without any enclosed area between the twopaths of the interferometer. An enclosed area is necessary for rotational sensitivity[169, 118, 73] and requires moving atoms. As described in figure 6-8, we were able
to extend the coherent beam splitter to condensates moving on an atom chip. The
observed coherence time (10 ms) and propagation distance after splitting (∼ 350 µm)
were only limited by the chip geometry. This corresponds to an enclosed area of∼1500 µm2 , and a response factor 4πmA
∼ 7.9 × 10−5 rad/Ωe for rotation sensing,hwhere m is the probe particle mass and Ωe the earth rotation rate [20].
Atom loss and Number squeezing
In the experiment, the condensates after coherent splitting exhibit losses with the
lifetime of 1.8 s. Thess stochastic atom losses introduce the relative atom number
uncertainty ∆Nloss during the hold :r∆Nloss ≈

N0

∆t
τloss

(6.10)

where N0 ∼ 4×105 is the total number of atoms right after splitting, ∆t ' 200 ms−1
denotes hold time, and τloss∼ (1.8 s)−1 is an initial atom loss rate. According to therelation 6.10, the atom number fluctuation due to atom losses becomes ∆Nloss ≈ 200
atoms. This estimation is inconsistent with our estimation of relative atom number
fluctuations of 50 atoms in the experiment.
The discrepancy in the atom number fluctuations may be resolved in two ways.
First, the stochastic nature of the atom losses spontaneously suppresses the relative
atom number fluctuations [93]. The dependance of the atom loss rate on the atomic
density tends to reduce the relative atom number; therefore suppresses the atom
number fluctuation. For the loesses via three-body collisions, it has been shown thatthe atom losses induce atom number squeezing by a factor of ∼ 2 [93]. Second, thelifetime of the condensates after splitting may be longer than ∼1.8 s since the atom
loss rate is proportional to n2 where n is the atomic density. In the experiment, thelifetime of ∼1.8 s was measured at the splitting position without RF fields.

106

(a)

Energy Scales in the System
100nk

500 pk

1 µk

60 nk 100 nk

Axial Freq.

300 nkRadial Freq.

T*(Phase Fluct. Temp)(b)

650 nkChem. Pot.

1 µk

Tc (BEC)
T(Temp.)

Phase Fluctuating
Condensate

Beam
SplittingInterference Fringe

Figure 6-9: Phase fluctuations in an elongated condensate. (a) Relevent energy scale
is described in an elongated condensate of the MIT experiment. The characteristic
temperature T ∗ for phase fluctuations and the typical temperature T of the condensate are shown. (b) In the splitting process, the effect of the phase flucuations arecommon-mode to both condensates. Therefore, the interference pattern formed in
the overlapped atomic cloud after ballistic expansion must be straight even in thepresence of the phase fluctuations. However, the relative motion of the condensates
associated with asymmetries in the double well potential may cause the degradation
the contrast in the interference fringe.

6.2

Matter-wave Interferometry with Phase Fluctuating Bose-Einstein Condensates

6.2.1

Phase Fluctuations in an Elongated Bose-Einstein Condensate

The phase coherence in elongated Bose-Einstein condensates (BECs) suffers fromspatial phase fluctuations even well below the BEC transition temperature. In thissection, I briefly demonstrate that atom interferometers using such condensates arerobust against phase fluctuations, i.e. the relative phase of the split condensate isreproducible despite axial phase fluctuations. For comprehensive discussion on theexperimental details, I refer to Ref. [99].
Previous experiments [179, 174, 101] including the one in the previous sectoin 6.1.2
on atom interferometry have operated in a regime, where phase fluctuations are predicted to be present. As described in figure 6-9 (a), the characteristic temperature107

(a) Absorption Image(b) Density Profile
Absorption
Signal [a.u.]

x

1.0
0.8
0.6
0.4
0.2
0.0
0

100

200

300

400

500

Spatial Postion(x) [pixel](d) Bandpass Filter
Magnitude of FFT [a.u.]

Magnitude of FFT [a.u.](c) FFT
2
1
0
-1
-2
0.0

0.1

0.2

0.3

0.4

0.5

2

Phonon energy~ 100 nk

1
0
-1
-2
0.0

Momentum [1/pixel]

0.1

0.2

0.3

0.4

0.5

Momentum [1/pixel]

Absorption
Signal [a.u.](e) Inverse FFT
40
20
0
-20
-40

(f ) Calculate the root-mean-square
average
0

100

200

300

400

Spatial Postion(x) [pixel]

Figure 6-10: Measurement of phase fluctuations in a single condensate. (a) Absorption image of the expanded atomic cloud after 7 ms time-of-flight was taken.
For quantitative analysis of the spatial phase fluctuations in a condensate, the density profile in (b) obtained from an absorption image is Fourier transformed as in(c). After filtering out high and low momentum components as in (d), the profile istransformed back (e) and its root-mean-square average is determined (f). The highmomentum limit of 0.2 pixel−1 in (d) corresponds to the phonon energy of 100 nK
which is eaual to the characteristic energy determined by the radial trap frequecy.
Note that the phonons with low energy less than 100 nK mainly contributes to thephase fluctuations in the elongated trap.

T ∗ for phase fluctuations (see equation 5.13) is much lower than the temperature, T ,of the system. However, their presence has not been observed because the interferometer was read out by integrating the interference fringes along the axial direction.
Other experiments characterized phase fluctuations by interferometric techniques [80]and Bragg spectroscopy [163], but didn’t study the effect of phase fluctuations on an
atom interferometer.
In this section, we demonstate the axial phase fluctuations spatially resolved andcharacterize their effect on the atom interferometer. We show explicitly, that atominterferometry can be performed in the presence of phase fluctuations. This has been
expected [23], since for sufficiently short times after splitting, those fluctuations are
identical for both condensates and therefore don’t affect the measurement of the relative phase (see figure 6-9 (b)). For the same reason, atom interferometry is possible
108with thermal clouds of atoms [73]. However, already at short times, phase fluctuations degrade the contrast and can limit the performance of the atom interferometer.
This degradation is not due to the quantum effect of the increased relative number
fluctuations in each quasi-condensate because of the high degree of number squeezing,but is rather caused by asymmetries in the double well potential leading to relative
motion of the condensates.

6.2.2

Effect of Phase Fluctuations on Atom Interferometry

Quantifying phase fluctuations
To study the effect of phase fluctuations, we first characterized the amount of phasefluctuations in the condensate before splitting by observing density modulations of
the expanded atomic cloud after 7 ms time-of-flight (figure 6-10 (a)). In trap, themean-field interaction energy suppresses density fluctuations, but ballistic expansion
converts phase fluctuations into density modulations [51] since the initial velocity
field is proportional to the gradient of the phase. The number of observed density
striations of around ten is consistent with the ratio of the measured temperature of∼ 650 ± 100 nK and the calculated value of T ∗ ' 60 nK. Since the barrier height iscomparable to the temperature, we assume that both condensates interact with thesame heat bath. However, we don’t expect any difference to the case of two separated
thermal clouds.The longitudinal phase fluctuations were quantified by measuring theroot-mean-square average of the density fluctuations as described in figure 6-10.
The amount of phase fluctuations was controlled by changing the atom number
and the temperature with rf-evaporation. The rf field generated by the rf wire [figure 5-4 (b)] was swept down from ∼ 20 kHz above the Larmor frequency at the trap
center to a variable final value, leading to a variable chemical potential and temperature of the condensate (figure 6-11 inset) 4 . The variation of the spatial phasefluctuations with chemical potential is shown in figure 6-11 (a).
Effect of phase fluctuation on the reproducibility of the relative phase
The effect of phase fluctuation on an atom interferometry was characterized by measuring the reproducibility of the relative phase of split condensates as described infigure 6-11 (b). To this end, we split the condensates coherently and observe thereproducibility of ten interference fringes (“randomness”) obtained by recombining
4

In the final stage of the evaporation, the ratio of trap depth to temperature was rather small(≈2), possiblely due to heating from the atom chip potential.

109

(b)

Probability of Random Phases

Temperature (nK)Spatial Phase Fluctuations (a.u.)

(a)
18
16
14

600
400
200
0

12

T*
2.0

3.0

4.0

5.0

10
8
6
4

0.25
0.20
0.15
0.10
0.05
0.00
2.5

3.0

3.5

4.0

4.5

Chemical Potential (kHz)

Figure 6-11: (a) Spatial phase fluctuations in a single condensate. The phase fluctuations were characterized by observing the density modulations in an absorption imageof the expanded cloud after 8 ms time-of-flight (see inset) and calculating the rms
fluctuations as described in reference [51]. The chemical potential (or atom number)
was controlled by additional rf-evaporative cooling. The temperature of the condensate is shown in the inset graph. For chemical potentials less than 3.5 kHz, we could
not measure the temperature of a condensate due to the lack of discernable thermal
atoms. The observed phase fluctuations do not decrease monotonically, but show aminimum at the chemical potential of ∼ 3.5kHz, probably because the effect of thelower temperature was more than offset by the loss in the atom number. In the inset
graph, T ∗ displays the characteristic temperature for the onset of the phase fluctuations. (b) Effect of spatial phase fluctuations on the reproducibility of the relative
phase right after splitting. The probability of random phases was measured withvariable longitudinal phase fluctuations immediately after splitting (0 ms hold time).
The phase was determined by analyzing the central ∼40% of the axial length of theinterference pattern.

110the condensates during ballistic expansion. For values of the chemical potential larger
than 3.0 kHz, the randomness is less than 0.1 which implies a reproducible phase with90% confinence.
Comparing (a) and (b) in figure 6-11, one clearly recognizes the degradation of
reproducibility of the relative phase with increasing spatial phase fluctuations. We
cannot rule out that the condensate had some weak collective excitations after preparation. However, the amount of the excitation should not depend on the final temperature. Therefore, we attribute the temperature dependence of the fringe contrast
to phase fluctuations.

111

Chapter 7
Development of Atom Optics
Elements
This chapter focuses on the experiments reported in two publications:• G.-B. Jo, J.-H. Choi, C.A. Christensen, T.A. Pasquini, Y.-R. Lee, W. Ketterle,
and D.E. Pritchard
Phase Sensitive Recombination of Two Bose-Einstein condensates on an atomchip
Physical Review Letters 98, 180401 (2007) Included in appendix F• Caleb A. Christensen, Sebastian Will, Michele Saba, Gyu-Boong Jo, Yong-Il
Shin, Wolfgang Ketterle, and David Pritchard
Trapping of Ultracold atoms in a hollow-core photonic crystal fiber
Phys. Rev. A 78, 033429 (2008) Included in appendix G

7.1

Phase Sensitive Recombination of Two BoseEinstein Condensates

Coherent splitting of a Bose-Einstein condensate and resulting long phase coherencetime have allowed us to study merger dynamics of two separated condensates. Encouraged by the long phase coherence time for the time up to 200 ms (see chapter 6), we
tried to coherently split a condnesate and reversely recombine two separated condensates in a trap. Many discussions of confined atom interferometers, inspired by opticalfiber interferometers, propose a read-out by merging the two separated clouds as we
tried [79, 185, 137]. These discussions usually assume non-interacting atoms [79, 10]112

and don’t address the deleterious effects of atomic interactions, including dephasing,collisional shifts, and phase diffusion [35, 122, 202, 96, 117, 97]. A recent study showedthat the recombination process is much more sensitive to atomic interactions than
the splitting process since merging clouds with the opposite phase involves excitedmodes of the recombined potential and can lead to exponential growth of unstablemodes [186]. To circumvent these problems, previous realizations of confined atominterferometry usually used ballistic expansion of the two spatially independent condensates, which decreases the atomic density before overlap [177, 179, 174, 101] orworked at very low atom densities and pushed the clouds into each other with photon recoil [62]. While this avoids the deleterious effects of atom-atom interactions
during the recombination, it lacks the inherent simplicity and robustness of in-trap
recombination.
In this section, I briefly introduce a new phase read-out method using in-trap
recombination of two separated condensates. It turns out that in-trap recombination
leads to heating of the atomic cloud which is phase-dependent. Our result demonstrated that it can be used as a robust and sensitive read-out of the atom interferometer. The resulting oscillations of the condensate atom number are dramatic (typically∼25% contrast), occur over a wide range of recombination rates, and permit highsignal to noise ratios since they simply require a measurement of the total number ofcondensate atoms in the trap.
Phase-sensitive recombination of two condensates
The recombination of two condensates was implemented using a radio frequency induced double well potential as described in figure 6-3. By changing the RF frequency,
two separated condensates with well-defined relative phase were prepared and thenthe double well potential was deformed into a single well as described in figure 7-1(a).
The phase-sensitive excitations after recombination lead to an increase in temperature based on our working assumption that the phase-sensitive excitation of thecloud decays quickly, on the order of ∼1 ms in our system, and leads to an increasein temperature on the order of ~ω/kB ' 100 nK for the case of 4φ = π, and less forother values of 4φ where ω is the radial trap frequency and 4φ denotes the relative
phase.
The increase in temperature of a merged cloud can be monitored by measuring thecentral atom density during ballistic expansion. Phase-sensitive collective excitations,in addition to mechanical excitations from the splitting and merging processes, heat113

RF frequency (kHz)

(a)

240
200
160

Split
0

Hold
75

Time (ms)(b)(c)
5π / 2

Loss of Condensate
Atoms (%)

Relative Phase (radian)

Merge

2π
3π / 2

π
π /2
0
0.0

0.5

1.0

1.5

2.0

2.5

Hold Time (ms)∆φ = π102030 ∆φ = 0
40
0

π /4

π /2

3π / 4

Relative Phase (radian)Figure 7-1: Phase-sensitive recombination of two separate condensates. (a) The
phase-coherent condensates were prepared using a radio frequency induced double
well potential on an atom chip [101]. The splitting was done within 75 ms by ramping up the rf frequency from 140 kHz to 225 kHz. During the hold time, the relative
phase of two independent condensates evolved with time at ∼500 Hz. After a variable
time, the double well potential was deformed into a single well and the two trapped
condensates were merged by decreasing the rf frequency by 33 kHz over a variable“recombination time”. The condensates started to spill over the barrier after ≤10%of the recombination time or ∼3 kHz decrease of the rf frequency. (b) The relative
phase of two split condensates was monitored for various hold time after splitting by
suddenly releasing the two condensates and observing interference fringes. For the
independent condensates (solid circle), the evolution rate of the relative phase were
determined from the linear fit to be ∼500 Hz. For the weakly coupled condensates(open square), the relative phase did not evolve. At 0 ms hold time, the relative
phase was set to zero for both cases. (c) For the same range of delay times as in(b), the condensate atom loss after in-trap recombination was determined. The relative phase (x-axis) was obtained from interference patterns as in (b). The merging
time was 5 ms. The matter-wave interference patterns (after 9 ms time-of-flight) andabsorption images of merged clouds (after 8 ms time-of-flight) show the correlation
between phase shift and absorption signal (see inset). The field of view is 260 × 200µm and 160 × 240 µm for matter-wave interferences and merged clouds respectively.

114

10

(a)

100 ms

(b)

10 ms

(c)

5 ms

(d)

1 ms

30

Loss of Condensate Atoms (%)

50
10
30
50
10
30
50
10
30
50
1

2

3

4

5

6

Hold Time (ms)Figure 7-2: The condensate atom loss was monitored during a variable hold time forthe two split condensates whose relative phase evolved at ∼500 Hz. The merging
was done for different values of the recombination time: 100 ms (a), 10 ms (b), 5 ms(c), and 1 ms (d). The dotted lines are sinusoidal curves fitted with fixed frequency∼500Hz. The reproducible phase shift for the 5 ms and 10 ms data occurred during
the recombination process. The data points represent the average of 6 measurements.

the cloud and lower the condensate fraction and, therefore, reduce the central density.
In the experiment, the split condensates were held in the double well potential forvarying hold times, merged into a single potential, and released by turning off thetrapping potential.
After 8 ms time-of-flight, we measured the number of atoms in a fixed area whichis comparable to the size of (expanded) Thomas-Fermi radius [dotted box in figure 71(c)]. While the total atom number was conserved, the number within the fixedarea decreased, indicating that the temperature had increased. The fractional loss ofcondensate atoms was obtained as the ratio of atom number after recombination to
the atom number before splitting.
To confirm that this oscillatory heating was associated with the relative phase115

of the split condensates, we measured the relative phase as the spatial phase of theinterference pattern when the split condensates were suddenly released and interferedduring ballistic expansion [figure 7-1 (b)] [177]. Figure 7-1 (b) and (c) show the strong
correlation between the relative phase and the heating of atom clouds. As the relative
phase increased from 0 to π, the atom loss after recombination increased [figure 71(c)]; π-relative phase (0-relative phase) difference leads to maximum (minimum) lossof condensate atoms.
The use of phase-sensitive recombination as a read-out for an atom interferometeris demonstrated in figure 7-2. The separated condensates accumulate relative phase
for an evolution time of up to 6 ms which is read out after in-trap recombination.
The phase-sensitive recombination signal showed high contrast over a wide range ofrecombination times (figure 7-2). The observed largest amplitudes of condensate atom
loss correspond to a change in temperature on the order of ∼100 nK, in agreement withthe estimate in the introduction. This is testimony to the insensitivity of the energy ofphase-dependent excitations against changes in the exact recombination parameters,
and is promising for further applications of chip-based atom interferometry.

116

7.2

Trapping of Ultracold Atoms in a Hollow-core
Photonic Crystal Fiber

In this section, I briefly describe a proof-of-principle experiment showing the possibility of trapping of ultracold atoms in a hollow-core photonic crystal fiber. This
project was originally initiated by Michele Saba in BEC III. The hollow-core fiber was
implemented into the atom chip setup described in figure 2-1. For more experimental
details, I refer to Ref. [39].
Ultracold atoms in waveguides are being used for studying quantum optics [191],
performing atom interferometry [196, 101], and implementing schemes for quantuminformation science [192]. Of particular interest is the ability of waveguides andmicrotraps to strongly confine atoms, providing high optical densities, strong interactions with light, and mechanisms for transporting atoms for further experiments.
Recently major efforts has been directed toward trapping of ultracold atoms in thehollow-core optical fibers because of its intrinsic feature: simultaneously confining
both a small number of ultracold atoms and photons inside the small volume of thehollow-core.
In particular, recently developed hollow-core photonic crystal fibers (model: Blaze
Photonics HC-1060-02) which propagate a single Gaussian mode light (1064 nm
wavelength) confined to a hollow core allow us to overcome the uncontrolled guiding of atoms in the conventional hollow core fibers used in the previous experiments [162, 94, 135]. Here, we have succeeded in trapping of ultracold atoms in
the hollow-core photonic crystal fiber. This preliminary experiment has shown thatthe ensemble of cold atoms can be loaded into the hollow core fiber with tight confinement.
In the experiment, the Bose-Einstein condensates trapped in the red-detuned optical dipole trap (ODT) is positioned 1 mm away from the end of a fiber (see figure 7-3).
The atoms trapped in the ODT is transferred into the core of the fiber by coupling
the red-detuned light into the fiber from the other side of the fiber [hereafter called
the hollow-core trap(HCT)]. By adjusting the intensities of two lights, the atom canbe loaded into the fiber or be retrieved controllably as described in figure 7-4. The
number of atoms after retrieval is about 5% of the 106 atoms initially prepared in the
ODT.
This system, consisting of tightly confined ultracold atoms in the core of the fiber,would open a new area of nonlinear quantum optics if a small number of photons are
introduced in the same region. To date, few-photon nonlinear optics has been only
117

(a)
Hollow-core photonic
crystal fiber

10 µm

Fiber
HCT

2cm

(b)

HCT

Figure 7-3: Schematic illustrations for trapping of atoms in a hollow-core fiber. (a)The atoms are transferred from a free-space ODT into a trap formed by a red-detuned
Gaussian light mode confined to the core of the fiber (HCT). The fiber used in theexperiment is 2 cm long. (b) Diagram of the optics setup. The light for the ODT and
the HCT is produced by a 1064 nm Spectra-Physics J201-BL-106C diode pumped
solid state multimode laser. Intensities are controlled by using acousto-optical modulators. The beams are frequency shifted such that they have 50 MHz relative detuning
in order to prevent static interference fringes in regions where the traps overlap.

118

(a)

(c)

(b)

(d)

Figure 7-4: Images of atoms in the ODT during the experiment. (a) Atoms are heldin the ODT near the fiber, with no light coupled into the fiber. The dashed line
indicates the position of the 100 µm thick fiber. (b) Light is coupled to the fiber, andas the ODT intensity is ramped down, atoms are depleted from the ODT until (c) no
atoms remain outside the fiber when the ODT power reaches zero. (d) After ramping
the ODT back up, atoms that were trapped in the HCT return to the ODT.

possible in the system of the cavity in which atoms strongly interact with photons.
This enhancement between atoms and photons would be also feasible in the hollowcore fiber system. For example, the recent seminal work has demonstrated the opticalswitching with cold atoms in the hollow core fiber [15].

119Chapter 8
Conclusion and Outlook
Transition to strongly interacting physics in BEC III
This thesis contains two different experimental studies of bosonic and fermionic gases.
Over 6 years after producing the first Bose-Einstein condensate in the BEC III lab(June 2001), major efforts had been directed toward understanding the propertiesof condensates and demonstrating atom interferometry with Bose condensed atoms.
To this end, a series of proof-of-principle experiments had been successfully demonstrated as summarized in the Ph.d. theses [113, 89, 147] of former members of Ketterle/Pritchard group. These highly productive results benefit from the concept of“Science Chamber” in our lab. In summer 2007, the apparatus was upgraded to generate a degenerate Bose-Fermi mixture of 23 Na and 6 Li. Adding fermions opened newopportunities to explore quantum magnetism in a repulsive Fermi gas and heteronuclear molecules consisting of bosonic 23 Na and fermionic 6 Li atoms. Also, from thispoint, we broadened the scope of research direction toward the understanding of thestrongly interacting regime in a gas of ultracold atoms. My Ph.D. research describedin this thesis is an example showing how the field of ultracold dilute gases evolves
into the area of strongly correlated physics.

8.1

The future in BEC III

At the moment when I was writing this thesis, the major efforts were directed toward
two projects simultaneously in the science chamber: (1) the study of itinerant ferromagnetism in a Fermi Gas of ultracold atoms and (2) the production of fermionic
heteronuclear molecules. For itinerant ferromagnetism, the study can be extended
into a low-dimensional system with or without optical lattices. In a lattice system,120

there are many fundamental questions relevent to magnetism, waiting to be answeredby experiments with cold atoms [123]. Also, the simulation of a model Hamiltonian ina repulsive Fermi gas such as the Stoner model would contribute to the realization of
the quantum simulator in atomic physics. The production of fermionic heteronuclear
molecules would also open a new era to study the properties of the dipolar atomicgas. In addition, the flexibility of the science chamber will allow us to explore systems
consisting of fermi atoms, fermionic heteronuclear molecules, or Bose-Fermi mixtures.
I am leaving the science chamber lab in capable hands and I hope that the sciencechamber will continue to contribute to the field of cold atom physics. Given theadvances and successes we have already accomplished in the science chamber, I believe
the future is bright.

121

Appendix A
Properties of 23Na and 6Li.
This appendix contains summarizes properties of 23 Na and 6 Li.

122

Sodium

Lithium

1.7 MHz

15.8

34.3

F‘=2

2P3/2

F‘=1
F‘=0

F=2 probe (on resonance) F=1 probe (on resonance)
Trapping light ( -13 MHz) Repumping light
Molasses light (-26 MHz)F‘=5/2

Trapping light(cycling transition)
F=3/2 probe (on resonance)

Repumping light
F=1/2 probe (on resonance)

Slower
(1 GHz detuned)

F=2
2S1/2
F=1

228.2 MHz

F=3/2
152.1 MHz

1771.626 MHz

664.36

Slowing light(1 GHz detuned)

3S1/2

2.8 MHzF‘=1/2F‘=3/22P1/2
D2 line(670.977 nm)

42.4

58.3 MHz

3P3/2

D2 line(589.158nm)

(b)

F‘=3

Linewidth = 6.0 MHz

(a)

F=1/2

Figure A-1: Sodium and lithium D2 transition hyperfine structure, with frequency
splittings between the hyperfine energy levels. The frequency of the light used fortrapping, repumping, slower, and imaging is shown.

123

Sodium

Lithium

1000

F=2

|4>

500

Energy [MHz]

|6>
|5>

ms=+1/2

F=3/2
0

-500

F=1/2
F=1

|3>

-1000

ms=-1/2

0

50

100
150
200
Magnetic field [G]

250

|2>
|1>

300

Figure A-2: 23 Na and 6 Li hyperfine structure in an external magnetic field. For
23
Na with a nuclear spin I = 3/2, there are (2S + 1)(2I + 1) = 8 hyperfine states.
For 6 Li with I = 1, six hyperfine states are available. A F=1 and F=2 sodium
condensate is produced in the hyperfine state |F = 1, mF = −1i (black solid line)and |F = 2, mF = 2i (red solid line) respectively. Sympathetic cooling with a F=2
condensate is performed in the stretched state |F = 3/2, mF = 3/2i (labeled as |6i,
black solid line) for lithium atoms. A repulsive two-component Fermi gas of lithium
atoms is prepared in the two lowest hyperfine states |F = 1/2, mF = −1/2i and|F = 1/2, mF = 1/2i (labeled as |1i and |2i repectively) near the Feshbach resonancelocated at 834 G.

124

Appendix B
Designs for the Apparatus
This appendix contains designs for the apparatus.

• Atomic beam skimmer• Oven nozzle design• Mount design for an atom chip experiment• Design for a new science chamber• Design for a re-entrant port

125

0.9”

a p pr o x 1 / 8 ”

0 .4 8” †

0
0.5
5” †

machine down to knife edge at tip(as thin as possible)

0.3”(to lera nc e 0 . 0 5 ” )1 0 o (tol er an c e 2 o )10-32 thru holes on
0 9” diameter circle
0.9

inside surface needs to be atconstant angle from tip to baseapprox 1/16” thick

1 0 o ( tol er an c e 2 o )(† tolerance set by angle)

SIDE VIEW

3D VIEW

1-1/4” OD stainlesssteel disk, 1/8” thickraised conical surface

0.3
centered 0
3” holeFRONT VIEW

10-32 thru holes on0.9” diameter circle(a) Atomic Beam Skimmer: material: non-magnetic 304 stainless steel(Image courtesy : Tout Wang)(b) Oven Nozzle Design* All CF flanges are for Ni gaskets (with less indentation),nominal diameter 2.75”* All the parts are made of 316SS** All the parts are made of 316SS.

0.021”Oven nozzle B

1.300

1.500

* All dimensions in inches* All dimensions in inches

0.200

0.200

0.500

45.000deg

8-32 tapped hole

0.100

0.425

0.150

2.750

0.300

0.400

0.350

Oven nozzle A

(c) Mount Design for an Atom Chip Experiment built in 2005
Material: 1/2" thick aluminum stock
All dimensions inches unless otherwise noted
2 pieces

0.050
0.250

0.250

0.500

0.650

0.500

0.900

1.000

(2) 6-32 tapped(2) 6-32 tapped

0.950
0.800
0.650
0.500

(4) 6-32 tapped(4) 9/32" clearance holes (thru)

0.150
0.150

grooves 0.035 wide X 0.050 deep

0.500

0.250

0.250

(d) Design for a New Science Chamber

(e) Design for a re-entrant Port

Appendix C
Itinerant ferromagnetism in a
Fermi gas of ultracold atoms
This appendix contains a reprint of Ref. [102]: Gyu-Boong Jo, Ye-Ryoung 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).

131Appendix DLong Phase Coherence Time andNumber Squeezing of two
Bose-Einstein Condensates on an
Atom Chip
This appendix contains a reprint of Ref. [101]: G.-B. Jo, Y. Shin, S. Will, T. A.
Pasquini, M. Saba, W. Ketterle , D. E. Pritchard, M. Vengalattore, and M. Prentiss,Long Phase Coherence Time and Number Squeezing of two Bose-Einstein Condensates on an Atom Chip, Phys. Rev. Lett. 98, 030407 (2007).

143Appendix E
Matter-Wave Interferometry with
Phase Fluctuating Bose-Einstein
Condensates
This appendix contains a reprint of Ref. [99]: G.-B. Jo, J.-H. Choi, C. A. Christensen,
Y.-R. Lee, T. A. Pasquini, W. Ketterle, and D. E. Pritchard, Itinerant Ferromagnetism, Phys. Rev. Lett. 99, 240406 (2007).

148Appendix F
Phase-Sensitive Recombination of
Two Bose-Einstein Condensates on
an Atom Chip
This appendix contains a reprint of Ref. [100]: G.-B. Jo, J.-H. Choi , C.A. Christensen, T.A. Pasquini, Y.-R. Lee, W. Ketterle, and D.E. Pritchard, Phase-Sensitive
Recombination of Two Bose-Einstein Condensates on an Atom Chip, Phys. Rev. Lett.
98, 180401 (2007).

153Appendix G
Trapping of Ultracold Atoms in a
Hollow-core Photonic Crystal
FiberThis appendix contains a reprint of Ref. [39]: C.A. Christensen, S. Will, M. Saba,
G.-B. Jo, Y.Shin, W. Ketterle, and D.E. Pritchard, Trapping of Ultracold Atoms in
a Hollow-core Photonic Crystal Fiber, Phys. Rev. A. 78, 033429 (2008).

158Appendix H
Interference of Bose-Einsteincondensates split with an atom
chipThis appendix contains a reprint of Ref. [179]: Y. Shin, C. Sanner, G.-B. Jo, T. A.
Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, Interference of Bose-Einsteincondensates split with an atom chip, Phys. Rev. A 72, 021604(R) (2005).

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