Bose polarons and rotating gases in an ultracold
Bose-Fermi gas mixture of 40 K and 87 Rb atoms
byMing-Guang Hu
B.S. & M.S. Theoretical Physics, Nankai University (China), 2009

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2015This thesis entitled:
Bose polarons and rotating gases in an ultracold Bose-Fermi gas mixture of 40 K and 87 Rb atoms
written by Ming-Guang Hu
has been approved for the Department of Physics

Deborah Jin

Eric Cornell

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iiiHu, Ming-Guang (Ph.D., Physics)
Bose polarons and rotating gases in an ultracold Bose-Fermi gas mixture of 40 K and 87 Rb atoms
Thesis directed by Dr. Deborah Jin

An ultracold Bose-Fermi gas mixture of 40 K and 87 Rb atoms has tunable interspecies interactions and therefore provides a fantastic platform for exploring not only few-body physics suchas Feshbach molecule formation and Efimov trimers, but also many-body physics including Bose
polarons, quantum Hall physics and so on. In this thesis, I present experimental evidence of Bose
polarons in cold atoms obtained using radio-frequency spectroscopy to measure the excitation spectrum of fermionic 40 K impurities resonantly interacting with a BEC of 87 Rb atoms. These Bose
polarons originate from the dressing of an impurity coupled to its environment, which is an important paradigm in quantum many-body physics. I also present initial work that launches an
exciting new direction for our experiment, which is exploring rotating quantum gases. Goals forthis work include studying both rotating Bose and Fermi superfluids with tunable interactions aswell as working toward rapidly rotating quantum gases in the quantum Hall regime. For thesegoals, a new all-optical trap for rotating gases was designed, implemented, and tested using a 87 Rb
Bose-Einstein condensate.

Dedication

I lovely dedicate this thesis to my wife Jinhua Li, who supported me each step of the way.

v

Acknowledgements

Before I came to CU, I was trained to be a theoretical physicist and had rare experience on
experimental physics. I want to thank my advisor, Debbie, for offering me such a great opportunity
to work in her group on cold atoms experiment. I also want to thank the former graduate students,
Ty and Ruth, who were patient to show me how to run our cold atoms machine. It’s their efforts thatmake my life easier. Started by Ty, two interesting science projects, Feshbach molecule associationand Efimov three-body physics, were successfully finished by Ruth and me. In the fourth yearof my Ph.D study (2013), Debbie and Eric planned to start a new joint project, rapidly rotating
ultracold gases. At the same time, I needed to figure out what my Comps III was about. With thehelp of Debbie, I proposed a new apparatus for rotating gases and simulated its feasibility, whichconvinced Eric to explore this novel direction in my lab. For me, I am excited that Eric joined ourlab and I can have more chances to talk with one of my heros.
Later, Vandy and Dhruv joined our lab. They are talented and easy to get along with. I
am impressed by how good Vandy is at finding right places to buy some unusual instruments. For
example, he found a good company to buy a broadband microwave amplifier with a reasonable
price and nice service. Dhruv gives me an impression that he has many science ideas, some ofwhich may be suitable to be tested by our machine in the future.
Finally, I want to thank the JILA electronics shop and instrument shop which help me alot. In particular, I want to thank Terry Brown from the electronics shop, who taught me a lot ofelectronics for stabilizing lasers.

Contents

Chapter
1 Introduction1

1.1

Many-body physics in ultracold atomic gases . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Bose gas and Fermi gas at ultralow temperature . . . . . . . . . . . . . . . . . . . .

4

1.3

Feshbach resonance for tuning interactions . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 Improved laser systems for cooling and trapping Rb and K atoms

11

2.1

Rubidium laser stabilization (FM spectroscopy) . . . . . . . . . . . . . . . . . . . . . 11

2.2

Potassium laser stabilization (modulation transfer spectroscopy) . . . . . . . . . . . 14

3 Bose polarons in the strongly interacting regime

18

3.1

Bose polaron theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2

RF spectroscopy of the Bose polaron . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3

Energy spectrum of the Bose polaron . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4

Coherent Rabi oscillation and damping of the Bose polaron . . . . . . . . . . . . . . 36

3.5

Data analysis in detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.1

A mean-field model of density inhomogeneity . . . . . . . . . . . . . . . . . . 40

3.5.2

Justification for inverse Abel transform . . . . . . . . . . . . . . . . . . . . . 43

3.5.3

Coherent and incoherent excitation spectrum . . . . . . . . . . . . . . . . . . 46

3.5.4

RF association and dissociation of K-Rb Feshbach molecules . . . . . . . . . 48

vii
3.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Quantum Hall physics: bizarre and interesting

50

4.1

Classical Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2

Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3

Integer and fractional quantum Hall effects . . . . . . . . . . . . . . . . . . . . . . . 56

4.4

Rapidly rotating atomic gases to quantum Hall regime: ideal cases . . . . . . . . . . 60

4.5

4.6

4.4.1

Fermion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.2

Boson case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Technical challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.1

An extremely smooth and harmonic optical trap . . . . . . . . . . . . . . . . 64

4.5.2

Rapid rotation close to 0.99 of trapping frequency . . . . . . . . . . . . . . . 64

4.5.3

Detecting local density by a quantum gas microscope

Other proposals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 A novel all-optical trap for rapidly rotating gases
5.1

5.2

. . . . . . . . . . . . . 65

66

A new apparatus for rotating gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.1

A painted trapping potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.2

A good imaging setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Optimizing the trap roundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1

Phase contrast imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.2

Inducing slosh of a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.3

Rounding out the trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3

Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4

Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5

Rotating a Rb BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6

Spin-up evaporation technique
5.6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

viii
5.6.2

Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.7

Rotation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.8

Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusions and outlook

97

Bibliography

99

Appendix
A RF spectroscopy of non-interacting atoms

105

B Time-averaged trapping potentials

107

ix

TablesTable
1.1

Useful formula for a weakly interacting BEC in 3D . . . . . . . . . . . . . . . . . . .

4

1.2

Useful formula for a weakly interacting BEC in quasi 2D . . . . . . . . . . . . . . . .

5

1.3

Useful formula for a degenerate Fermi gas in 3D

6. . . . . . . . . . . . . . . . . . . .

Figures

Figure1.1

A polaron formed by an electron moving in a crystal lattice and its counterpart, aBose polaron formed by an impurity atom immersed in a BEC . . . . . . . . . . . .

2

1.2

Similarity between a rotating BEC and a quantum Hall system . . . . . . . . . . . .

3

1.3

Few-body physics explored by tuning a Feshbach resonance . . . . . . . . . . . . . .

7

1.4

Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1

An improved rubidium laser system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

An improved potassium laser system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

Comparison between the modulation transfer spectroscopy and FM spectroscopy forthe potassium laser system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1

The regime where the Bose polaron behaves as a Fröhlich polaron. . . . . . . . . . . 21

3.2

Predicted energy spectrum of an impurity immersed in a Bose-Einstein condensate
with tunable impurity-boson interactions. . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3

Residue and effective mass calculated from the T-matrix method for 87 Rb and 40 K. . 26

3.4

Three-body recombination loss rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5

RF spectroscopy of the Bose polaron.

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

3.6

RF spectroscopy of the Bose polaron.

. . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.7

RF spectra of Bose polarons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.8

Summary of the RF spectra.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xi
3.9

Measured energy shift of the Bose polaron and weakly bound molecules. . . . . . . . 35

3.10 Coherent Rabi oscillation for the Bose polaron. . . . . . . . . . . . . . . . . . . . . . 38
3.11 Density profile of the initial impurity cloud and the BEC. . . . . . . . . . . . . . . . 41
3.12 Calculated RF lineshapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.13 Simulated data for testing the inverse Abel transform for extracting the Bose polaronsignal at the BEC core. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.14 Tan’s contact shown as a ω −3/2 tail in our RF spectroscopy data. . . . . . . . . . . . 47
3.15 RF association and dissociation of K-Rb Feshbach molecules. . . . . . . . . . . . . . 48
4.1

Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2

Lowest Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3

Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4

Understanding IQH effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5

Understanding FQH effect by the Laughlin’s wavefunction . . . . . . . . . . . . . . . 59

4.6

Realizing the IQH states by rapidly rotating a Fermi gas . . . . . . . . . . . . . . . . 61

4.7

Density distribution of a rotating Fermi gas in the quantum Hall regime . . . . . . . 62

5.1

A new apparatus for rotating ultracold gases . . . . . . . . . . . . . . . . . . . . . . 67

5.2

Control of laser beam pointing using an AOM . . . . . . . . . . . . . . . . . . . . . . 68

5.3

Time-averaged trapping potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4

Electronics of driving AOMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5

A good imaging setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.6

The lensing effect used when focusing our imaging system . . . . . . . . . . . . . . . 74

5.7

Using the lensing effect to focus our imaging system . . . . . . . . . . . . . . . . . . 75

5.8

Phase contrast imaging setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.9

Coarse alignment of the tiny phase dot . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.10 Stabilizing the time-averaged trapping potential . . . . . . . . . . . . . . . . . . . . . 79
5.11 Lab frame and trap frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xii
5.12 BEC slosh measurement for rounding out the trap . . . . . . . . . . . . . . . . . . . 81
5.13 Ellipticity of the trapping potential versus the painting trajectory . . . . . . . . . . . 83
5.14 Effect of the anharmonic trapping potential on slosh . . . . . . . . . . . . . . . . . . 85
5.15 Sheet beam information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.16 Detecting the roughness of the sheet beam using the density distribution of a BEC. . 88
5.17 Rotating a Rb BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.18 The idea of spin-up evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.19 Procedures of the Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . 91
5.20 Monte-Carlo simulation of the spin-up evaporation . . . . . . . . . . . . . . . . . . . 93
5.21 Experimental sequence of the spin-up evaporation. . . . . . . . . . . . . . . . . . . 94

5.22 Vortex lattice of a rotating BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Chapter 1

Introduction

With the development of laser trapping and cooling technologies, neutral atoms can be manipulated in an unprecedent way. Many applications have been achieved so far. One of the mostsuccessful directions is to study new phases of matter, which may only exist at ultralow temperatures near absolute zero (T ∼ 10−6 K). Here, the thermal de Broglie wavelength of atoms,√
λT = h/ 2πmkB T , becomes comparable to the interparticle spacing and quantum mechanicaleffects start to dominate. In this regime, quantum many-body physics can be engineered by tuning
the interactions between atoms. This thesis presents experiments that engineer and measure someinteresting quantum many-body physics by using an ultracold Bose-Fermi gas mixture of 40 K and
87 Rb.

1.1

Many-body physics in ultracold atomic gases
Condensed matter systems consist of many particles and many of these cannot be well de-scribed by single-body physics without interactions. Beyond few-body models, theoretical calculations and discussions of a many-body system can become extremely difficult. On the other
hand, some collective phenomena demonstrated by condensed matter samples, like superconductivity, ferromagnetism, and integer/fractional conductance, are intriguing for a physicist and crucial
for advance of technology. Understanding these interacting quantum many-body systems requiresconsiderations of hundreds and thousands of individual particles and particle-particle interactions,which cannot be treated exactly. As a result, new theoretical models are proposed. These theories

2

Figure 1.1: A polaron formed by an electron moving in a crystal lattice and its counterpart, a Bose
polaron formed by an impurity atom immersed in a BEC.

can not only explain observed collective phenomena but also can predict novel local properties, suchas a quasiparticle excitation with fractional statistics, known as an anyon. Furthermore, based on
the current understanding of many-body systems, people are proposing novel materials, of which
only a few have been realized in a condensed matter lab, while more of them remain unexplored.
Ultracold atomic gases with tunable interactions provide a promising platform to engineer these
exotic states.
The first many-body physics topic related to my research in this thesis is the Bose polaron.
An electron moving in an ionic crystal lattice is dressed by coupling to lattice vibration modes(phonons), which as a result forms a quasiparticle, known as a polaron (see Fig. 1.1a). Such amechanism is highly nontrivial and has become an important paradigm in quantum many-body
physics [1]. Recently, intense theoretical discussions focused on impurities immersed in a BoseEinstein condensate (BEC) with couplings between impurities and Bogoliubov phonon excitationsof BEC [2, 3, 4, 5, 6, 7, 8]. It is suggested that impurities dressed by phonons would form novel
quasiparticles, known as Bose polarons (see Fig. 1.1b), even in the strongly interacting regime[9, 10, 11, 12]. However, measurement of Bose polarons in the strongly interacting regime ischallenging due to the loss of atoms through three-body recombination and effects of density inhomogeneity in trapped atomic gases. In this thesis I report experimental evidence of Bose polarons
by using radio-frequency spectroscopy to measure the excitation spectrum of fermionic 40 K impurities resonantly interacting with a BEC of 87 Rb atoms. We find that the measured energy of the

3

Figure 1.2: Similarity between a rotating BEC and a quantum Hall system. Left: measured density
distribution of the gaseous Bose-Einstein condensate under rotation; an exotic vortex lattice forms
as the rotation speed increases. (credit: JILA TOP trap.) Right: illustration of moving electrons
when they are in quantum Hall states under strong magnetic field. (credit:UTokyo.)impurites immersed in the BEC matches well with that calculated for Bose polarons for a range ofinteraction strengths. In particular, we confirm that the polaron state exists even across unitarity
in the strongly interacting regime and approaches the weakly bound molecular state. Furthermore,
for attractive interactions, we show that Rabi oscillations can be observed when exciting Bose polarons. This work is the first measurement of the Bose polaron in a three-dimensional trapped atomgas, which helps better understanding of the quantum many-body physics of Bose-Fermi mixtures.
The second many-body physics topic related to the research in this thesis is the quantum Hall
effect. Quantum Hall states are exotic “topological” states of matter that were originally used todescribe states of electrons moving in a two-dimensional plane in a strong magnetic field (see rightpanel of Fig. 1.2). Understanding these states is important and has led to the proposal of novel
materials. A well-known example is the topological insulator, which is a popular research topic incondensed matter field at present [13]. Although the quantum Hall effect has been observed using
specialized semiconductors [14], there are very few experimental examples of quantum Hall states.
Recently it was proposed to realize quantum Hall states by rapidly rotating an ultracold atomicgas [15]. In this thesis, I will discuss the connection between quantum Hall states and rapidly
rotating ultracold gases. Then I will demonstrate a new apparatus that we designed with the goalof producing rapidly rotating ultracold gases into the quantum Hall regime. Technical challenges
as well as experimental progress will be shown.

4
In the past decade, many other examples of using ultracold atomic gases to study manybody physics were demonstrated. Topics include quantum magnetism, Harper Hamiltonian, Fermi
polarons, artificial gauge fields, and so on. More interesting experiments are on the way. Ultracold
atomic gases have become an indispensable test bed for exploring many-body physics.

1.2

Bose gas and Fermi gas at ultralow temperature
Many review papers and theses have comprehensive introductions on the ultracold Bose gas,

the Bose-Einstein condensate and the degenerate Fermi gas. Here I won’t repeat these and instead
I will just introduce the necessary knowledge for understanding the relevant many-body physics
that I will present in this thesis.
Table 1.1: Useful formula for a weakly interacting BEC in 3D at the Thomas-Fermi limit [15]. Here
M is the boson mass, n is the atom number density, g = 4πa~2 /M is the coupling constant with abeing the scattering length and M being the atomic mass of an atom.
Quantities
Oscillator length (non-interacting BEC size)
Healing lengthThomas-Fermi distribution
Chemical potential
Thomas-Fermi radii
Peak BEC density& Number†
Column density

Formulaqd0 = M~ω01
ξ = √8πan222n(r) = n0 1 − Rx 2 − Ry 2 − Rz 2z
 x 2/5 y
2/5N0 aµ = 52~ω0d0q
2µRx,y,z = M ω2x,y,zn0 = µ/g, N0 = 8πn0 R03 /15

3/2
2
2
n̄(r) = n̄0 1 − Rx 2 − Ry 2x

Peak BEC column density & Number
Averaged BEC density
RMS width & TF radiusyn̄0 = (4/3)Rz n0 , N0 = (2π/5)n̄0 Rx Ry
hni = (4/7)n0
σRMS = 0.437RTF† averaged Thomas-Fermi radius R = (R R R )1/3 .
0x y zWhen a thermal 87 Rb atomic gas in a harmonic trap is cooled down below a critical temperature, Tc = ~ω0 [N/ζ(3)]1/3 /kB , a phase transition occurs and a Bose-Einstein condensate is formed
with a BEC fraction given by, N0 /N = 1−(T /Tc )3 . Here the averaged trapping frequency is definedby ω0 = (ωx ωy ωz )1/3 , the Riemann zeta function ζ(3) = 1.2, kB is the Boltzmann constant, N0

5
is the number of atoms in the BEC, and N is the total atom number. When the BEC fraction is
big enough, the interaction between BEC atoms, characterized by the scattering length a, needs
to be considered. For our case of a weakly interacting BEC (ξ  1/n1/3 ) , the static BEC can be
described by the time-independent Gross-Pitaevskii (GP) equation [15]
 ~2 ∇22
−+g|Ψ(r)|
Ψ(r) = µΨ(r),
+
V
(r)
tr
 2M
{z } | {z } | {z } | trap
interaction(1.1)

kinetic

where M is the mass of the atom, µ is the chemical potential, Ψ(r) is the macroscopic condensate
wave function (or “order parameter”), and g is the coupling constant g = 4πa~2 /M . Here a = 100a0
is the scattering length between two Rb atoms with a0 being the Bohr radius [16]. The BEC
pdensity is given by n(r) = |Ψ(r)|2 , and sometimes it is useful to write Ψ(r) = n(r)eiφ(r) witha phase term φ(r). For this thesis, we are working in the Thomas-Fermi regime (N0 a/d0  1 orµ ≈ n0 g  ~ωx,y,z ) and thus the kinetic energy term in Eq. (1.1) can be ignored. As a result,the BEC density simply follows an inverse parabola distribution (see Table 1.1). A more detailed
introduction can be found in Ref. [15].
Table 1.2: Useful formula for a weakly interacting BEC in quasi 2D. Here g2 is the reduced coupling
constant in 2D.
Quantities
Thomas-Fermi distribution
Thomas-Fermi radii
Chemical potential
Column density&Number

Formula



2
2
2 2
n(r) = n0 1 − Rx 2 − Ry 2 e−z /lz
x
y
r
2µ0
Rx,y = M ω2D2x,y

2N0 g2µ2D = ~ωz /2 + µ02D , µ02D = πR
x Ry


y2
x2
π
n̄(r) = n̄0 1 − R2 − R2 , N0 = 2 Rx Ry n̄0 ,xyFor a rapidly rotating gas, a quasi-2D BEC is relevant in the regime of small chemical potential, where µ < ~ωz . Here, the BEC stays at the lowest energy level along z while it still
occupies many energy levels in the x and y directions. The wavefunction can be written asp22
Ψ(r) = π1/41√l e−z /2lz Ψ2 (x, y) with the oscillator length lz = ~/M ωz and Ψ2 (x, y) being thez

6√
2D condensate wavefunction. The critical temperature is Tc2D = ~ ωx ωy [N/ζ(2)]1/2 /kB with the
Riemann zeta function ζ(2) = 1.64 [17]. The GP equation is modified as [Rychtarik’s thesis [18]]
 2 2

~ ∇⊥
2
−
+ Vtr (r⊥ ) + g2 |Ψ2 (r⊥ )| Ψ2 (r⊥ ) = µ02D Ψ2 (r⊥ ),2M
where the 2D spatial coordinate r⊥ = {x, y}, the reduced coupling constant g2 =

(1.2)
√

8π~2 a/M lz andthe chemical potential µ2D = µ02D + ~ωz /2. Useful formula are listed in Table 1.2.
Table 1.3: Useful formula for a degenerate Fermi gas in 3D.
Quantities

FormulaFermi energy & Fermi temperature
Fermi-Dirac distribution∗

EF = kB TF = (6N )1/3 ~ω0
f () = e/kB 1T /ζ+1 , fugacity ζ = eµ/kB T

N = −(kB T /~ω0 )3 Li3 (−ζ)
1
Li3 (−ζ) = − 6(T /T3F)µ = kB T ln ζ
P∞ k n∗ Li (x) is Poly-Logarithmic function of order n defined as Li (x) =nnk=1 x /k .
NumberRelationship of fugacity and T
Chemical potental

For the Fermi gas of 40 K atoms in my system, most of the experiments in this thesis use a
relatively “hot” gas at T /TF ∼ 0.5. However, for completeness, I will list some frequently used
quantities that are important for discussing a Fermi gas. In general, the density of states in ad-dimensional harmonic potential isρ() =

d−1,
Q(d − 1)! di=1 ~ωi(1.3)

where ωi is the trapping frequency along i-th direction. The density distribution of a Fermi gas
trapped in a 3D harmonic trapping potential follows
n(r) = −(kB M T )3/2
− 2kM T (ωx2 x2 +ωy2 y 2 +ωz2 z 2 )
B
Li
−ζe
,
3/2
(2π)3/2 ~3(1.4)

which is reduced to the classic Maxwell-Boltzmann distribution at T /TF  0.1. Some basic quantities have been listed in Table 1.3 and more detailed discussions can be found in DeMarco’s thesis(P.233) [19].

7Figure 1.3: Few-body physics explored by tuning a Feshbach resonance. (a) Association of KRb
Feshbach molecules by ramping the magnetic field across a Feshbach resonance [20]. (b) Tests ofuniversal three-body physics in an ultracold Bose–Fermi mixture [21]. I was involved in previous
work on these two topics done in my lab; detailed discussions are included in Cumby’s thesis [22]and Bloom’s thesis [23].

1.3

Feshbach resonance for tuning interactions
One of our powerful tools in ultracold gases is a Feshbach resonance that can be used to tune

the atomic interactions [24]. The capability of controlling interactions is very important to study not
only few-body physics but also many-body physics [25]. Examples of few-body physics include twobody Feshbach molecule association [Cumby’s thesis [22]], three-body Efimov resonances [Bloom’s
thesis [23]], and so on (see Fig. 1.3). Examples of many-body physics include BCS-BEC crossoverof a degenerate Fermi gas [Regal’s thesis [26]], impurity-bath coupling [27, 28, 29, 30, 31], andso on. Simply speaking, when tuning an applied magnetic field B close to a specific value B0 ,the scattering rate between particles is enhanced. For the simplest situation of isotropic s-wave
scattering, interactions can be well described by a parameter known as the scattering length a. The
dependence of the scattering length a on B is given by [24]a(B) = abg 1 −∆B − B0


,

(1.5)

8

Figure 1.4: Scattering length aKRb versus magnetic field B around the interspecies Feshbach resonance between 40 K and 87 Rb atoms. The interaction is short-range. It is attractive for aKRb < 0
and repulsive for aKRb > 0.

with abg being the background scattering length and ∆ the resonance width. The coupling constantg is proportional to a.
For 87 Rb, there are no broad Feshbach resonances [16] and therefore we can simply treat it
to be a gas with constant background scattering length aBB = 100a0 . Here a0 is the Bohr radius.
For describing a 87 Rb BEC, a weak background interaction is included in the theories leading to
the Bogoliubov phonon excitations, which is relevant to the Bose polarons discussed in Ch. 3.
For 40 K, we work in the polarized limit in which all atoms are in a single hyperfine state|F, mF i = |9/2, −9/2i, where F corresponds to the total atomic angular momentum and mF is its
projection. Due to the quantum statistics of fermions, fermion-fermion interactions are strongly
suppressed in the ultracold gas. In my following discussions, fermion-fermion interactions are
ignored.
For a mixture of 40 K atoms and 87 Rb atoms, an s-wave Feshbach resonance is used to control
the interactions between bosonic 87 Rb atoms in the |1, 1i state and fermionic 40 K atoms in the|9/2, −9/2i state. The boson-fermion scattering length aKRb as a function of magnetic field B(Fig. 1.4) is given by aKRb =abg [1 − ∆/(B − B0 )], abg =−187 a0 , B0 =546.62 G, ∆=−3.04 G [32].
We use this Feshbach resonance to tune interactions between 40 K atoms and 87 Rb atoms to study9interesting many-body physics.

1.4

Overview
This thesis covers two main research topics that I was leading in the past two years. The

first one is a study of Bose polarons as described in Sec. 1.1. The main tools we used are theinterspecies Feshbach resonance and RF spectroscopy. This work helps elucidate the behavior of
impurities in a BEC, which is a paradigmatic problem in many-body physics. The second topic
is to design and construct a new apparatus for creating rotating ultracold atomic gases in thequantum Hall regime. Technical challenges are explored and experimental progress is discussed.
Besides studying quantum Hall physics, such a versatile apparatus is also promising to explore
other interesting physics such as the effect of dimensionality on the BEC/BKT transition, vortices
in a strongly interacting BEC, etc..
The remainder of the thesis is organized as follows.
Chapter 2 presents improvements I made to the frequency stabilization of the Rb and K laser
systems. A stable laser system is crucial for laser cooling and trapping atoms. It is also importantto have the ability to jump the laser frequency for atom detection. By using new electronics
technology, I set up a flexible beatnote locking that allows the frequency of a laser beam to be
shifted easily over a GHz range.
Chapter 3 is dedicated to the Bose polaron study. I include a general introduction to the Bose
polaron problem. Then I show our experimental results for the polaron energy spectrum, which iscompared to theoretical models. To probe the particle nature of polarons, we also investigate Rabi
oscillations when driving these states.
Chapter 4 discusses quantum Hall physics and its connection with rapidly rotating ultracoldgases. I introduce quantum Hall states and why they are bizarre. Their connection with rotatinggases as well as experimental challenges are discussed.
Chapter 5 introduces a new apparatus for rapidly rotating ultracold gases. A “painted” (ortime-averaged) optical trapping potential is a critical part of our approach. The time-averaging or

10“painting” can give us a near-perfect round trap geometry and can also reduce roughness of thetrapping potential, which is critical for rotating gases very rapidly.
Chapter 6 summarize the major results of this thesis and concludes with a discussion of future
experiments.

Chapter 2

Improved laser systems for cooling and trapping Rb and K atoms

A stable laser system is crucial for laser cooling and trapping atoms. In this chapter, I present
the improvement I made to stabilize our rubidium and potassium laser systems. In particular, I set
up a modulation transfer spectroscopy, an excellent way of stabilizing lasers developed recently, forthe potassium laser system [33]. This spectroscopy is compared against the frequency-modulation(FM) spectroscopy [34, 35]. In addition, with new electronics technology, I set up a flexible phaselock of a slave laser (trap laser) to a master laser (repump laser), which allows us to shift thefrequency of the slave laser over a GHz range. The spectroscopic properties of rubidium andpotassium can be found in [36] and [37], respectively.

2.1

Rubidium laser stabilization (FM spectroscopy)
To get a 87 Rb magneto-optical trap (MOT), two lasers, the repump laser and the trap laser,are needed. The repump laser is for hyperfine pumping using the (F = 1 → F 0 = 2) transition.
Here F (F 0 ) denotes the total atomic angular momentum of the ground (excited) states. In our
experiment, the repump laser is locked to the 87 Rb F = 1 → F 0 = 1, 2 crossover using FM
spectroscopy (see the next paragraph). Then, the frequency of the repump laser is shifted up by
78 MHz using a single-passed acouto-optical modulator (AO) (+1 order). The trap laser is forcooling and trapping using the cycling transition (F = 2 → F 0 = 3). Since the frequency of the
87 Rb (F = 2 → F 0 = 3) transition is less than that of the 87 Rb (F = 1 → F 0 = 1, 2 crossover1 )1

The crossover frequency is equal to the average of two involved atomic transition frequencies. For example,
fF =1→F 0 =1,2 crossover = (fF =1→F 0 =1 + fF =1→F 0 =2 )/2.

12Figure 2.1: An improved rubidium laser system. (a) The FM spectroscopy setup for the repump
laser. (EOM denotes an electro-optical modulator; PD is a photodetector; LF is a loop filter; AO
is an acousto-optical modulator; DBR is a distributed Bragg reflector laser.) A single-passed AO
is used to shift the laser frequency up by 78 MHz. (b) The error signal for locking the repump
laser. The upper panel shows the absorption spectroscopy (adapted from [38]) and the lower panel
is the error signal obtained using FM spectroscopy [34, 35]. The red dot indicates the reference
point, where we lock our repump laser’s frequency. (c) The phase-lock setup for the trap laser.
DDS denotes a direct digital synthesizer (Analog devices: AD9959-eval). “f /128” is a frequency
divider (Hittite microwave: HMC363). The phase frequency detector has the model number (Hittite
microwave: HMC440). (d) The RF spectrum of the beat between the trap laser and the repump
laser when the phase lock is on. The beat signal is measured by a RF spectrum analyzer. The
x-axis has the center at 6.618 GHz and the span of 20 MHz. The unit of the y axis is dBm.

transition by 6.4896 GHz [36], we lock our trap laser to the repump laser using a phase-lock loopwith a beat frequency of fb = 6.622 GHz. Then, the frequency of the trap laser is shifted up by
133.5 MHz using a double-passed AO (+1 order).
Fig. 2.1(a) shows the schematic of FM spectroscopy setup for the repump laser. There are
two overlapped counter-propagating beams passing through a Rb vapor cell. The beam probed
by a photodetector (PD) is called the probe beam and the other one is called the pump beam.

13
When we scan the laser’s frequency by ramping the laser current, the measured power of the probe
beam gives a Doppler-free absorption spectroscopy as shown in the upper panel of Fig. 2.1(b).
The red dot indicates the lock point, where we lock our repump laser. The lower panel of Fig.
2.1(b) shows the error signal that is input into a loop filter (LF) for stabilizing the repump laser
frequency. The error signal is the derivative of the absorption signal and is obtained using FM
spectroscopy [34, 35], which requires the laser beam’s phase to be modulated. An electro-opticalmodulator (EOM) shown in Fig. 2.1(a) is used to modulate the phase of the repump laser beamwith the modulation frequency, fm = 11.8 MHz. Although we modulate both the probe and pump
beams, only the probe beam needs to be modulated.
The error signal is crucial for stabilizing the laser’s frequency. We care about its signal-tonoise ratio (S/N) that is defined here as the ratio of the peak-to-peak amplitude over the backgroundnoise amplitude. The error signal of (F = 1 → F 0 ) is about ten times weaker than that of(F = 2 → F 0 ). To increase the S/N of the error signal at the F = 1 → F 0 = 1, 2 crossover,the vapor cell is heated to above the melting point of rubidium (38.9 ◦ C) to increase the vapor
pressure. The optimal temperature is determined by looking at the S/N and is set to be 58 ◦ Cin our experiment. Additionally, it is important to keep the light intensity of the probe beam lessthan the saturation intensity (Isat = 2.5 mW/cm2 [36]). For our case, the 1/e2 width of the probe
beam is about w = 1 mm and the power is P = 20 µW, which gives the peak intensity of the probe
2P
2beam, I = πw
2 ≈ 1.3 mW/cm .

Fig. 2.1(c) shows a diagram of the phase-lock setup for the trap laser. A 6.622 GHz beatsignal is detected by a fast PD (EOT Inc.: ET-4000) with a bandwidth of 12.5 GHz. Because
the phase frequency detector (PFD) (Hittite microwave: HMC440) only has a bandwidth of 10
to 1300 MHz, we divide the beat frequency by a factor of 128 (a factor of 8 using a frequency
divider (Hittite microwave: HMC363) and another factor of 16 by the PFD). By comparing against
a reference RF signal of 51.7335 MHz, the PFD converts the divided beat signal into an error
signal for the laser stabilization. The reference RF signal is provided by a direct digital synthesizer(DDS), which is controlled by a computer. We tune the trap laser’s frequency by changing the

14reference RF frequency. In this way, we can shift the trap laser’s frequency over a GHz range,
which is important for the phase contrast imaging discussed in Sec. 5.2.1. Fig. 2.1(d) shows the
RF spectrum of the beat between the trap laser and the repump laser when the phase-lock loop is
on. The delta function of the spectrum indicates a phase lock of the trap laser to the repump laser.
In the above discussion, I have focused on the spectroscopy of the Rb laser system that Ichanged for improving laser stabilization. The rest of the Rb laser system can be found in Chap.
4 of the previous thesis [23].

2.2

Potassium laser stabilization (modulation transfer spectroscopy)
For a 40 K MOT, both the repump and trap lasers are important for cooling and trapping.

The repump laser is red-detuned from the F = 7/2 → F 0 = 9/2 transition and the trap laser is
red-detuned from the cycling transition, F = 9/2 → F 0 = 11/2. Because of the small naturalabundance of 40 K (0.0117%), the spectroscopy signal of 40 K from a vapor cell is too weak to use
for laser stabilization. Instead we lock our repump laser to the 39 K F = 2 → F 0 = 3 transition
using modulation transfer spectroscopy [33] (see the next paragraph). The isotope 39 K has a large
natural abundance of 93.3%. The frequency of the 40 K F = 7/2 → F 0 = 9/2 transition is less thanthat of the 39 K F = 2 → F 0 = 3 transition by only 431.9 MHz [37], which can be easily reachedusing a double-passed AO. In our experiment (see Fig. 2.2(a)), a double-passed AO (+1 order)is used to shift the frequency of the repump laser up by 424 MHz before doing the modulationtransfer spectroscopy. For the trap laser, the frequency of the 40 K F = 9/2 → F 0 = 11/2 transitionis larger than that of the 39 K F = 2 → F 0 = 3 transition by 809.8 MHz [37] and we lock our trap
laser to our repump laser through a phase lock with the beat frequency 784.8 MHz.
Fig. 2.2(a) shows the schematic of the modulation transfer spectroscopy setup for the repump
laser. Different from FM spectroscopy, the probe beam in the modulation transfer spectroscopy is
not modulated, while the pump beam is frequency modulated by an EOM. However, to produce
the error signal, the probe beam needs containing modulated sidebands [39, 33]. It turns outthat the probe beam picks up the modulated sidebands from the collinear pump beam through a

15

Figure 2.2: An improved potassium laser system. (a) The modulation transfer spectroscopy setup
for the repump laser. A double-passed AO is used to shift the laser frequency up by 424 MHz. (b)The error signal for locking the repump laser. The yellow line shows the absorption spectroscopy
and the cyan line is the error signal obtained using the modulation transfer spectroscopy [33]. The
red dot indicates the reference point, where we lock our repump laser. (c) The phase-lock setup forthe trap laser. The beat frequency is 784.8 MHz.

nonlinear four-wave mixing mechanism [40, 41], which is a modulation transfer process. Due tothe nonlinear nature of the modulation transfer process, there are three main advantages using themodulation transfer spectroscopy [39, 33]. First, the error signal has a flat zero baseline, since themodulation transfer process only takes place at sub-Doppler resonances. Second, the zero-crossingsof the error signals correspond accurately to the atomic transitions [39]. Third, the error signal
for a cycling transition is largest, since the modulation transfer is strongest for cycling transitions,which allows the four-wave mixing to occur many times. In particular, as shown in [33], the error
signal corresponding to a cycling transition can be further enhanced by using σ + -σ + polarizations

16(i.e. right-hand circular probe and left-hand circular pump). This third feature is especially usefulwhen the spectrum contains several closely spaced transitions such as the spectrum of 39 K.
Fig. 2.2(b) shows the error signal (the cyan line) obtained using the modulation transfer
spectroscopy as well as the Doppler-free absorption spectroscopy (the yellow line). From the absorption spectroscopy, we see three dominant features. From the left to right, the first one is afeature containing three closely spaced 39 K (F = 2 → F 0 = 1, 2, 3) transitions, the second one
is a feature containing two 39 K ground-state crossovers, and the third one is a feature containing
three 39 K (F = 1 → F 0 = 0, 1, 2) transitions and two 41 K crossovers [33, 42]. The error signal iscrucial for stabilizing the laser’s frequency. To enhance the error signal at the cycling transition(F = 2 → F 0 = 3), I put two quarter-wave plates at two sides of the vapor cell to achieve σ + -σ +polarizations for the probe and the pump beams shown in Fig. 2.2(a). As shown in Fig. 2.2(b),the resulting error signal at the cycling transition (F = 2 → F 0 = 3) is particularly large, while theother non-cycling features are completely suppressed.
Additionally, to increase the S/N of the error signal, the K vapor cell is heated to above
the melting point of potassium (63 ◦ C) to increase the vapor pressure. The best value of thetemperature is determined by looking at the S/N and is set to be 70 ◦ C. The light intensity of theprobe beam is kept less than the saturation intensity (Isat = 1.75 mW/cm2 ). For our case, the1/e2 width of the probe beam is about 1.5 mm and the power is 60 µW, which gives the intensity
2P
2of the probe beam I = πw
2 ≈ 1.7 mW/cm .

Fig. 2.2(c) shows the diagram of the phase-lock setup for the trap laser. A 784.8 MHz
beat signal is detected by a fast PD (EOT Inc.: ET-3000A) with a bandwidth of 30 kHz to 1.5
GHz. The rest of the electronics are similar to the Rb phase-lock setup and can be understoodstraightforwardly.
For comparison, I put the FM spectroscopy signal and the modulation transfer spectroscopy
signal (without the two quarter-wave plates) together in Fig. 2.3. The FM spectroscopy is obtainedby moving the EOM so that it modulates both the probe and the pump beams (similar to the setup
shown in Fig. 2.1(a)). Generally, we set the frequency corresponding to the absorption peak as

17Figure 2.3: Comparison between the modulation transfer spectroscopy and FM spectroscopy forthe potassium laser system. The blue dashed line corresponds the absorption peak, where we used
to lock our K repump laser. The green dashed line marks the zero crossing of the error signal ofthe modulation transfer spectroscopy, where we lock our repump laser now. The green dashed line
also indicates the location of the (F = 2 → F 0 = 3) transition. The frequency difference between
the blue line and the green line is 11 MHz.

a lock point for FM spectroscopy (blue dashed line) and set the frequency at the zero crossingof the error signal as a lock point for the modulation transfer spectroscopy (green dashed line).
These two lock points are different by 11 MHz. From Fig. 2.3, we can see the three advantagesof the modulation transfer spectroscopy I mentioned above: (1) flat zero baseline; (2) accurate
correspondence between zero crossing and the atomic transition; (3) dominant error signal at cyclingtransitions. In addition, the error signal of the modulation transfer spectroscopy shows an antisymmetric shape around the zero crossing, which is good for laser stabilization. It also shows anarrower capture range (frequency range covered by the linear slope of the error signal) than FM
spectroscopy has.
In the above discussion, I have focused on the spectroscopy of the K laser system that Ichanged for improving laser stabilization. The rest of the K laser system can be found in Chap. 4of the previous thesis [23].

Chapter 3

Bose polarons in the strongly interacting regime

Ultracold atomic gases provide a fantastic platform for exploring many-body physics withtunable particle-particle interactions. In this chapter, I present an experimental study of the Bose
polaron, which is a quasiparticle created by a quantum many-body effect. We used RF spectroscopy
to measure the energy shift of the polaron state at different interaction strengths. We find that
our results match recent theoretical calculations very well. This work is the first measurementof the Bose polaron in a three-dimensional trapped atom gas and provides strong evidence of thepolaronic dressing due to quantum many-body physics.

3.1

Bose polaron theory
An electron moving in an ionic crystal lattice is dressed by coupling to lattice vibration

modes (phonons) and, as a result, forms a quasiparticle known as a polaron (see Fig. 1.1(a)). Sucha mechanism is highly nontrivial and has become an important paradigm in quantum many-body
physics [1]. Historically, such a mechanism was originally pointed out by Landau and Pekar [43, 44].
Then Fröhlich proposed an effective Hamiltonian to describe the polaron problem [45, 1]:
HFröhlich =

X
p,σ

†
εp Cp,σCp,σ + ω0Xqa†q aq +X†
V(q)Cp+q,σCp,σ aq + a†-q .

(3.1)

q,p,σ

†
In the first term, Cp,σ (or Cp,σ) is the annihilation (or creation) operator of an electron witha momentum p, spin σ, and the energy dispersion εp = p2 /2me . In the second term, aq (ora†q ) is the annihilation (or creation) operator of a phonon with a momentum q and the energy

19ω0 . Here phonons are the vibration modes of the crystal lattice. The third term describes thecoupling between the electrons and the phonons with the coupling strength V(q). The above
Fröhlich Hamiltonian has become a standard model for describing the original polaron problem
and a polaron that can be described by Eq. (3.1) is called a Fröhlich polaron. A large amount oftheoretical work has been done to solve the Fröhlich Hamiltonian and references can be found in[1]. The most accurate solution came from Feynman through a variational method based on path
integrals [46], although his theory is very lengthy [1].
Recently, intense theoretical discussions focused on impurities immersed in a Bose-Einsteincondensate (BEC) with couplings between the impurities and Bogoliubov phonon excitations of
BEC [2, 3, 4, 5, 6, 7, 8]. It was suggested that impurities dressed by phonons would form novel
quasiparticles, known as Bose polarons, even in the strongly interacting regime [9, 10, 11, 12]. To
see this, let’s consider a system of fermionic impurities immersed in a Bose-Einstein condensate.
The impurity-boson coupling can be tuned through a magnetic-field Feshbach resonance, which
was introduced in Sec. 1.3. The low-energy effective Hamiltonian [9, 10] is
H =

X
p

†
f1,p +
f (p)f1,pXqb (q)b†q bq +gbf X † †bk0 f1,q−k0 f1,q−k bk ,
V
0

(3.2)

k,k ,qwhere b,f (k) = k 2 /2mb,f are the kinetic energy terms for the majority bosons and the impurity
fermions respectively. Here mb,f denote the atomic mass of a boson and fermion, respectively. In†
the first term, fσ,p (or fσ,p) is the annihilation (or creation) operator of an impurity fermion in thehyperfine state σ with a momentum p. In the second term, bq (or b†q ) is the annihilation (or creation)
operator of a majority boson with a momentum q. The coupling constant gbf is determined by thescattering length abf between bosons and fermions. For the zero-range model, the coupling termµ1 P 2µbf
has an ultraviolet divergence, which is regularized by g1bf = 2πabf
2 − V
k k2 . Here the reduced
bf ~mass is µbf = mb mf /(mb + mf ) and V is the volume. The mean-field approximation consists of
M F = 2πa ~2 /µ , which can be either attractive (negative) or
taking the coupling constant as gbf
bf
bf

repulsive (positive) depending on the sign of the scattering length.
For a weak repulsion (n0 a3bb  1/6π 2 ) between bosons characterized by a boson-boson scat-

20
tering length abb , the ground state of the BEC can be well described by Bogoliubov theory. Heren0 is the BEC density. The elementary excitations in the condensate are the collective Bogoliubovpmodes, which have a dispersion of γb (k) = b (k)(b (k) + 2gbb n0 ). Here the mean-field coupling
constant given by gbb = 4πabb ~2 /mb and n0 is the BEC density. The Bogoliubov transforma†tion is bk = uk αk − vk α−kwith real and positive factors u2k = [1 + (b (k) + gbb n0 )/γb (k)]/2 andvk2 = [−1 + (b (k) + gbb n0 )/γb (k)]/2 [10]. Using the Bogoliubov transformation, the Hamiltonian
given in Eq. (3.2) becomes [10]:H = H0 + HI ,H0 = Eg +

X

(3.3)

†
f1,p +
f (p)f1,pX

γb (q)αq† αq ,

q

p


HI

=

p X
gbf  X †††f1,k (αp + α−p)+R(p)f1,k+pN0
f1,k f1,k + N0
V
k,p
k

X
†
†
,
D(k, k0 )f1,q+k
0 −k f1,q αk0 αkk,k0 ,q

where αq (or αq† ) is the Bogoliubov phonon annihilation (or creation) operator, Rp = up − vp ,D(k, k0 ) = uk uk0 + vk vk0 and uk , vk are the coefficients of the Bogoliubov transformation. An
PPapproximation, N0 = k hb†k bk i ≈ k b†k bk , has been used during the above derivation and N0 is
the BEC number. Eg is a constant ground-state energy. The first term in HI gives the conventional
mean-field interaction with the mean-field energy shift ∆M F = gbf n0 and the BEC density n0 =N0 /V . The second term in HI is identical to the coupling term in the Fröhlich Hamiltonian Eq.
(3.1). The third term is a scattering term.
To be in the Fröhlich polaron regime, it was pointed out that the condensate depletion in thevicinity of an impurity needs to be small, which gives rise to a constraint [47, 48]
2|gbf |  4~cs ξ =⇒

mb + mfmf

2

4π 3 a2bf abb n0  1,

(3.4)

p
where the speed of sound in the BEC is cs = gbb n0 /mb and the healing length of the BEC is√ξ = 1/ 8πabb n0 . Under this constraint, the third term in HI can be ignored and Eq. (3.3) is21
identical to Eq. (3.1). Further, Eq. (3.4) can be reduced to [48]


√
mb
3/2
1+
 ≡ 2π|abf | abb n0  1.
mfAs discussed in Ref. [48]1 , when choosing  ≤ 0.3 as the boundary for being in the Fröhlich regime,
one obtains the maximum BEC density

max
n0 =

2

1
,
3/2
2π (1 + mb /mf ) abb a2bf

below which the Bose polaron can be treated as the Fröhlich polaron. For our 40 K and 87 Rb system,
Fig. 3.1(a) shows the regime where it is valid to treat the Bose polaron as the Fröhlich polaron.
Another important quantity for discussing the Fröhlich polaron is the polaronic coupling constantFigure 3.1: The regime where the Bose polaron behaves as a Fröhlich polaron. For our system,
the boson is a 87 Rb atom and the fermion is a 40 K atom. (a)A plot that shows the valid regime
for treating the Bose polaron as a Fröhlich polaron. The red dashed line indicates the typical peak
BEC density in our experiment. (b)The maximum polaronic coupling constant achievable for theFröhlich polaron.

√

α ≡ 2 2π

r

n0 2
a .
abb bfAccording to Ref. [48], when the value of α is larger than one, the Fröhlich polaron is in aninteresting intermediate coupling regime. By substituting the maximum BEC density nmax
into
0
1

max
The numerical table given in this paper about nmax
and αRb-K
may be wrong. I suspect that they misused the0
Rb mass for the impurity mass, M , and the K mass for the majority mass, m.

22
the above equation, we get the maximum α allowed for the Fröhlich polaron as shown in Fig.
3.1(b). At a BEC density of n0 = 1.8 × 1014 cm−3 , the corresponding critical scattering length is
abf = 170 a0 and the maximal polaronic coupling constant is αmax = 0.12. Therefore, from Fig.
3.1(a), we can see that for our experimental conditions, the Bose polaron is outside the regime where
the Fröhlich approximation is valid and all interaction terms in Eq. (3.3) need to be considered.
From the phase boundary shown in Fig. 3.1(b), we note that reaching α > 1 in the Fröhlich regime
would require |abf | > 2, 500 a0 , which, as seen from the phase boundary shown in Fig. 3.1(a),corresponds to a BEC density more than two orders of magnitude smaller than in our experiment.
Bose polaron theory in the strongly interacting regime

In the regime of strong

impurity-boson coupling, beyond the Fröhlich approximation, one must consider Eq. (3.3). There
are two published analytical methods that were used to solve for the Bose polaron in the stronglyinteracting regime: the T-matrix method [9] and the variational method [10]:• The T-matrix method can be used to calculate the self-energy Σf (Ω, p) of the impurities [9].
The retarded Green’s function and the spectral function of the polaron can be calculatedusingGR
f (Ω, p) =1
p2~Ω − 2m− Σf (Ω, p) + i0+f

AI (Ω, p) = −2 ImGR
f (Ω, p),,(3.5)

where Ω is the energy and p is the momentum of the polaron. The quasiparticle dispersion
relation is defined as the solution ofE(p) −

p2− ReΣf [E(p), p] = 0,
2mf

where quantities depend only on the magnitude of momentum p = |p|. The spectral weightor quasi-particle residue, which describes how much the bare atom is involved in the polaronstate, is given byZ(p) =

1.
1 − ∂Ω ReΣf [Ω, p] Ω=E(p)/~23
The decay width is obtained from
γ(p) = −Z(p) ImΣf [E(p), p].
At the small momentum limit of p ∼ 0, the effective mass of the Bose polaron satisfies,2p
E(p) = E0 + 2m
+ O(p4 ), and can be obtained usingeff

meff =

mb /Z(p)p=∂p E(p) p=0
mb /mf + p1 ∂p ReΣf [Ω, p]|E(p),p=0with a simplification [9]meff = mf

1 + mf /mb
Z + mf /mb


,

(3.6)

for a noninteracting BEC.
• The variational method gives a trial wave function of polarons as [10]


X
†
†
|Φ(q)i ∼ φ0 (q)f1,q+
φk (q)f1,q−kαk†  |BECi,

(3.7)

k6=0

which has a total momentum q. The first term corresponds to bare impurity atom involvedin the polaron state and its coefficient has a connection with the quasi-particle residue
through |φ0 (q)|2 = Z(q). The second term describes a phonon excitation of the condensate
with wavevector k .
In our experiment, we use 40 K atoms as impurities in a 87 Rb BEC. To approach the impurity/polaron limit, we tune the populations of the two species so that the number ratio NK /NBEC ≈0.1 and estimated ratio of the peak densities is nK /nBEC <0.04. The fermionic impurity atoms have
T /TF >0.6 and therefore we expect that their quantum statistics is not important [49]. For ourweakly interacting BEC, the boson-boson scattering length is aBB = 100a0 [50]. Following otherpapers [9, 10], we define energy and momentum scales respectively by
En = ~2 (6π 2 nBEC )2/3 /2mRb ,kn = (6π 2 nBEC )1/3 .

(3.8)

Here a0 is the Bohr radius, ~=h/2π with h being the Planck constant and mRb is the mass of a Rb
atom. An s-wave Feshbach resonance is used to control the interactions through the impurity-boson

24
scattering length aIB for 87 Rb atoms in the |f, mf i=|1, 1i state and fermionic 40 K atoms in the|9/2, −9/2i state (see Sec. 1.3), where f corresponds to the atomic angular momentum and mf is
its projection.

Figure 3.2: Predicted energy spectrum of an impurity immersed in a Bose-Einstein condensate withtunable impurity-boson interactions. The red and blue lines represent two polaronic branches thatare called attractive polarons and repulsive polarons, respectively. For comparison, the green line
shows the two-body molecular bound state energy, −Eb /En = −mRb /µKRb /(kn aIB )2 , with µKRb
being the reduced mass of a Rb atom and a K atom. For comparison, we also plot energy shifts
based on the mean-field calculation (dashed lines) with ∆MF /En = (2mRb /3πµKRb )(kn aIB ).

In Fig. 3.2, we show the calculated energies of Bose polaron states as function of the interaction strength in the zero-temperature and zero-momentum limit. The theoretical lines are
based on a T-matrix approach presented in Ref. [9] and similar results have been obtained witha variational approach in Ref. [10]. The spectrum has two quasiparticle branches: the attractive
branch (red line) and the repulsive branch (blue line). The energies in the two branches shift in
opposite directions from the non-interacting case (∆ = 0 axis). In the weakly interacting limit, theenergy shift approaches the mean-field shift (dashed lines), ∆MF /En = (2mRb /3πµKRb )(kn aIB ).

25
The two branches have different properties. For the repulsive branch, the polaron is a welldefined particle state only in the weakly interacting regime or the large 1/(kn aIB ) regime, where
the lifetime is long and therefore the spectroscopic width of the polaron is narrow. In the stronglyinteracting regime, or the small 1/(kn aIB ) regime, the polaron lifetime becomes short and the energy
level is broadened as indicated by the shaded regime. For the attractive branch, the polaron stateis the ground state of the Hamiltonian and is well-defined in both the weakly interacting regime
and the strongly interacting regime. In particular, the attractive polaron state exists at unitarity(aIB → ∞) and is predicted to approch a weakly bound molecular state (green line) on the aIB > 0side of the Feshbach resonance.
Fig. 3.3 shows the corresponding residue Z and effective mass meff at different interaction
strengths. When there is no impurity-boson coupling, Z = 1 and meff = mK . For the attractive
branch, as the interaction strength 1/kn aIB increases towards the positive aIB side of the Feshbachresonance, Z → 0 and meff → (mK + mRb ), as the polaron approaches a two-body bound state ormolecule [9].
Three-body recombination

In the Bose polaron theory presented thus far, three-bodyrecombination has been ignored. However, this process, in which three atoms collide inelasticallyand form a diatomic molecule, and typically release enough energy to eject all three atoms from thetrap, exists in all Bose gases and Bose-Fermi gas mixtures. Moreover, with tunable interactions,the three-body recombination rate has an a4 dependence that results in a dramatic increase of
atom loss rates near a Feshbach resonance. Three-body recombination will limit the lifetime ofBose polarons. For our system, the three-body recombination happens between two Rb atoms andone K atom and can be described by [51]d1nK = − αnK n2BEC ,dt
2!
where nK (nBEC ) is the K density (BEC density) and the factor of 1/2! is included for a BEC
compared to a noncondensed ultracold gas of bosons. Here α is the three-body recombination
rate coefficient for uncondensed bosons, which scales as a4IB (ignore Efimov structure). For a large

26

Figure 3.3: Residue and effective mass calculated from the T-matrix method. (a) Residue Z at
different interaction strengths. (b) Effective mass meff /mK from Eq. (3.6) at different interaction
strengths. The red line shows the attractive polaron branch, while the blue line shows the repulsive
polaron branch.

population imbalance, the above equation can be simplified tod1
NK = − αn2BEC NK ,dt
2!
which has the solution NK (t) = NK (0) exp(−Γ3 t) with the loss rate Γ3 = 12 αn2BEC . To continue
our discussion, let me rewrite α as α = A3 a4IB [52] with A3 being a constant. The values of α havebeen measured at different values of aIB in our previous work [21]. We find A3 = 1.1 × 10−8 m2 /s
for aIB < 0 and A3 = 3.3 × 10−8 m2 /s for aIB > 0 by fitting our data to α = A3 a4IB . To compare

27

Figure 3.4: Three-body recombination loss rate. It is obtained using Eq. (3.9), which breaks down
at small |1/kn aIB |. As a consequence, Γ3 will saturate at small |1/kn aIB |. The dashed line is used
to illustrate this saturation behavior, although the saturation value is uncertain yet.

with the energy shifts of Bose polarons, we can express the loss rate Γ3 in terms of En and 1/kn aIB :~Γ3 /En = κ3 (kn aIB )4 ,(3.9)

A3 mRbwhere κ3 = (6π2 )2 ~ is a dimensionless quantity. We calculate κ3 = 0.004 for aIB < 0 and κ3 = 0.015

for aIB > 0.
Fig. 3.4 shows this three-body recombination loss rate of the impurity atoms based on Eq.
(3.9). However, a limitation of Eq. (3.9) is the assumption that α scales as a4IB , which breaks downin the regime where |1/kn aIB | is small. In this regime, α will saturate and a dashed line in Fig. 3.4
is used to illustrate this saturation behavior, although the saturation value is uncertain yet. From
Fig. 3.2 and 3.4, we can see that the three-body recombination rate becomes comparable with thepredicted energy shifts around |1/kn aIB | ∼ 0.5. Thus, a challenge for detecting Bose polarons inthe strongly interacting regime will be the finite lifetime due to three-body collisional loss.
Efimov effect

Ref. [12] theoretically considers the effect of Efimov trimers on the Bose

28
polaron. The authors consider an Efimov resonance at a− = −50 abb and predict that close to
1/kn a− , both the energy and residue of the Bose polaron would deviate from that calculated without
considering the Efimov effect. However, for our system, the location of the Efimov resonance ispredicted to be a− < −30, 000a0 [53], which is consistent with a measurement by our group [21].
The Efimov effect on the Bose polaron at such large scattering length would only affect an extremely
narrow regime in 1/kn aIB and we do not expect to resolve this.

3.2

RF spectroscopy of the Bose polaron
As discussed in the previous section, the Bose polaron is a rich quantum many-body effect thatis an important paradigm in condensed matter physics. Ultracold atoms, with their interactions
tunable with a Feshbach resonance, provide a unique opportunity to experimentally explore theBose polaron. In particular, we can go to regimes where the interparticle interactions are verystrong and where the physics goes beyond the often considered Fröhlich polaron. The regime ofstrong impurity-boson interactions has recently been treated theoretically; however, there are thus
far no experimental results probing the Bose polaron in a strongly interacting regime.
In contrast, impurities in a Fermi gas of atoms have been studied experimentally using RF
spectroscopy [27, 29, 30, 31]. Our experiment will employ similar techniques as these Fermi polaronmeasurements. However, there are some important differences between the Bose polaron and the
Fermi polaron. First, the Bose polaron problem is arguably richer because there are two relevantinteraction parameters: aBB and aIB , whereas there is no equivalent of aBB in a spin-polarizedFermi gas. Second, on the experimental side, we will see that three-body inelastic loss and therelatively small spatial extent of a BEC (compared to an impurity gas) both create challenges topolaron measurements.
To measure the energy spectrum of the impurities immersed in a BEC, we perform radiofrequency (RF) spectroscopy on 40 K impurty atoms. The impurity atoms are initially preparedin a weakly interacting state, |0i = |9/2, −7/2i and then, are driven into the strongly interacting|1i = |9/2, −9/2i state by a RF pulse at frequency ν (see Fig. 3.5(a)). The RF power is typically29

Figure 3.5: RF spectroscopy of the Bose polaron. (a) Energy diagram for the RF spectroscopy. (b)The spatial distribution of the trapped gases along with the imaging axis.

set to give a π-pulse for the bare atom transition. Impurity atoms in the |1i state are imaged
immediately after the RF pulse. This spectroscopy of RF injection into the strongly interacting
state minimizes the three-body losses. We denote the transition frequency of bare 40 K atoms
by ν0 . The RF pulse has a gaussian envelop with the Fourier transformed RF lineshape beingR(ν) ∝ exp(−2ν 2 /δν 2 ) (see appendix A). The Fourier limited linewidth δν is inversely proportional
to the pulse duration, δν = 1/(π∆t), with ∆t being the 1/e2 width of the gaussian pulse envelop
for the RF power. The RF pulse is applied from t = −2∆t to t = 2∆t.
An experimental challenge arises from the very different cloud sizes for the impurity gas
and the BEC as shown in Fig. 3.5(b). In our system, the measured trapping frequencies for K
atoms are ωρK = 2π × 50 Hz and ωzK = 2π × 281 Hz and the trapping frequencies for Rb atoms
are ωρRb = 2π × 39 Hz and ωzRb = 2π × 183 Hz. The two clouds also separate along the verticaldirection due to differential gravitational sag which is given by sag = g[1/(ωzRb )2 − 1/(ωzK )2 ] = 4µm. The cloud size of the impurity gas is about twice that of the BEC. As a result, only about
15% of the 40 K atoms overlap the BEC. By imaging the spin-flipped 40 K atoms in situ with aprobe beam propagating along z and with a resolution well below the BEC size, we can selectout signal that comes from 40 K atoms near the center of the BEC in the transverse direction.

30
However, the imaging integrates through the cloud in z and therefore the issue of signal from 40 K
atoms outside the BEC, as well as from regions of lower BEC density, remains. We deal withthis remaining density inhomogeneity issue by (1) using the transverse size of the imaged atoms toknow when the signal is dominated by 40 K atoms that overlap with the BEC, and (2) performing
an inverse Abel transform of this data to extract the central density. The inverse Abel transformallows one to find the central density where one assumes a symmetry between an imaged direction(transverse) and the integration direction (z). Differential gravitational sag breaks this symmetry,however, because of attractive mean-field interactions in the initial gas, this density profile of the
40 K atoms has approximate symmetry about the BEC center. We have modeled the effect of density

inhomogeneity and sag on RF spectroscopy in the mean-field regime and found that the inverse
Abel transform returns the correct central density of the cloud of spin-flipped 40 K atoms (see Sec.
3.5.2).
In Fig. 3.6(a-c), I show typical RF spectroscopy data at a weak interaction strength with
1/kn aIB = −2.3. The center optical depth (OD) and root-mean-square (RMS) width of the impurity
atom cloud are obtained here by fitting the image to a 2D gaussian distribution. To distinguishsignal that is dominated by 40 K atoms overlapping with the BEC, we use the cloud size. Opencircles are used to denote the bare atom signal, while the Bose polaron signal is represented by
red dots. Red dots are selected based on when the imaged cloud size is smaller than or equal tothe measured BEC size (see Fig. 3.6(c)). Fig. 3.6(d) shows the number of spin-flipped atoms
averaged over a circular annulus with the radius ρ = 28 µm. At this radius, the 40 K atom do not
overlap the Rb BEC and we obtain the RF spectra for the bare atom transition. The frequency
ν0 is obtained by fitting this data to a gaussian lineshape R(ν − ν0 ) (see Eq. (A.6) in appendix
A). The fit result is indicated by the black line. Fig. 3.6(e) shows the peak density (normalized)
obtained using an inverse Abel transform. We only show the data where the signal is dominatedby 40 K atoms overlapping with the BEC (red dots). We normalize the measured peak density ofthe spin-flipped atom cloud to the calculated peak density of the impurity atoms. This calculation
uses the measured 40 K atom number & BEC number, as well as the gas temperature measured

31

Figure 3.6: RF spectroscopy of the Bose polaron. (a-c) Typical RF spectroscopy data at a weakinteraction strength of 1/kn aIB = −2.3. Red dots mark data where the imaged cloud size is lessthan or equal to the BEC size. (b) Here the center OD data are normalized to its highest value. The
blue line is a fit to a mean-field model described by Eq. (3.20). (d) RF spectroscopy on the bareatom transition. Here the bare atom signal is obtained by averaging over a circular annulus withthe radius ρ = 28 µm (inset) and is normalized to its highest value. The line is a fit to the spectral
function, R(ν − ν0 ), from which we get the bare atom transition, ν0 . (e) RF spectroscopy showing
the signal from atoms located at the center of the BEC. We only use data where the imaged cloud
has a width smaller than or equal to the BEC size. A gaussian function, Eq. (3.10), is used to fit
our data (red line) with the fitting result of x0 = 0.139 and w = 0.045 (the Fourier width hδν/Enshould be compared to 2w).

from the tail of the initial 40 K cloud.
Comparing Fig. 3.6(a) and 3.6(d), one can see that when just looking at the number of spinflipped atoms, most of the signal comes from the large fraction ( 85%) of 40 K atoms that do not
overlap the BEC. The width of the main feature here, which corresponds the bare atom transition,
comes from the Fourier width of the RF pulse. The duration of the gaussian RF pulse for this datais 4∆t = 0.52 ms, which gives a Fourier width of δν = 1/(π∆t) = 2.4 kHz and hδν/En = 0.09.
Here, En (and kn ) are calculated from the measured peak BEC density of 87 Rb atoms in trap using

32
Eq. (3.8).
Looking at the central optical depth of the imaged (spin-flipped) atoms (Fig. 3.6(b)), we
see roughly equal contributions from atoms that overlap with the BEC (red dots) and atoms thatoutside the BEC in the vertical imaging direction. Finding the central 3D density of the imaged
atoms, by applying an inverse Abel transform to the images, allows us to extract an RF spectrum
from only impurity atoms that are located within the BEC (Fig. 3.6(e)). Here, we can fit to agaussian lineshapey(x) = A exp[−(x − x0 )2 /(2w2 )],(3.10)

to extract the center x0 = 0.139 and the RMS width w = 0.045 (red line). Note that here ourFourier limited width hδν/En should be compared to 2w from the fit.

3.3

Energy spectrum of the Bose polaronFig. 3.7 shows selected RF spectra at different interaction strengths obtained using an inverse

Abel transform to extract the signal from the center of the BEC. For cyan dots, we set the pulse
duration to be 4∆t = 0.52 ms except for 1/kn aIB = −2.9 and 4.4, where we set the pulse duration
to be 4∆t = 0.9 ms so that the Fourier width is δν = 1 kHz (or hδν/En = 0.035). The RF power is
set to give a π-pulse for the bare atom transition. The yellow dot corresponds to data taken with a4∆t = 0.52 ms RF 5π-pulse. Black dots show data taken with a 4∆t = 0.1 ms RF π-pulse (Fourier
width δν = 13 kHz and hδν/En = 0.5). For each RF spectrum shown in Fig. 3.7, we observe a peakstructure with a spectral width that broadens for larger interaction strengths. Using a gaussian
fit described by Eq. (3.10), we extract the energy shift, the peak height, and the spectral width,
which are shown in Fig. 3.8.
From Fig. 3.8, we see that the spectral width broadens for larger interaction strengths andthe peak height becomes correspondingly smaller. One thing that can cause a broadened spectral
width is the lifetime or the width of the energy level. The three-body recombination can cause
inelastic collision and therefore affect the lifetime of the energy level (see Fig. 3.4). In addition, the33

Figure 3.7: RF spectra of Bose polarons. Use data where the imaged cloud has a width smaller
than or equal to the BEC size. We use an inverse Abel transform to obtain the signal from impurity
atom within the BEC. For cyan dots, we set the pulse duration to be 4∆t = 0.52 ms except for
1/kn aIB = −2.9 and 4.4, where we set the pulse duration to be 4∆t = 0.9 ms. The RF power is
set to give a π-pulse for the bare atom transition. The yellow dot corresponds to data taken witha 4∆t = 0.52 ms RF 5π-pulse. Black dots show data taken with a 4∆t = 0.1 ms RF π-pulse. The
green circles represent data that may come from incoherent excitations (see Sec. 3.5.3), which do
not contribute to polaron signals.

repulsive polaron state can decay to the attractive polaron state, which broadens the energy widthof the repulsive polaron state. From the cyan dots as shown in Fig. 3.8, we see that the spectral
widths on the a > 0 side are broadened more than these on the a < 0 side. Although the density
inhomogeneity can cause a broadened spectral width, its effect is relatively small.
To extract the polaron energy shift from the RF spectra, we need to account for the mean-

34

Figure 3.8: Summary of the RF spectra. The bottom panel shows the energy shifts. The middle
panel shows the corresponding peak height. The top panel shows the spectral width.

field shift in the initial gas. The initial |0i state of the impurity atoms has a weak backgroundinteraction with the BEC atoms described by the background scattering length abg = −187 a0 [32].
This gives a mean-field energy shift of Ebg = gbg nBEC , where gbg = 2πabg ~2 /µKRb . The energy shiftof the Bose polaron is thus
∆ = h(ν0 − νrf ) + Ebg .

(3.11)For our case of averaged peak BEC density, nBEC = 1.8 × 1014 cm−3 , En /h = 28.4 kHz and Ebg =35

Figure 3.9: Measured energy shift of the Bose polaron (circles) and weakly bound molecules (triangles). The error bar of the energy shift corresponds to the RMS width of a gaussian fit, Eq. (3.10),
to the RF spectra (see Fig. 3.7). Yellow triangles show data for K-Rb Feshbach molecules. The
binding energy of K-Rb Feshbach molecules is extracted by fitting our K-Rb RF association data
to a well-established model [32].

−0.14En . Fig. 3.9 shows the measured energy shift, ∆, as a function of the interaction strength.
Cyan dots denote the data shown in Fig. 3.7. We also include measurements of the binding energy
of weakly bound molecules using RF association as shown by the yellow triangles. These molecule
data come from K atoms that do not overlap with the BEC, and spectra corresponding to the data
are presented in Sec. 3.5.4.
From Fig. 3.9, we can see that our measured energy shift of the impurities immersed in the
BEC matches well with that predicted recently for the Bose polaron (red and blue lines) in theregime of strong impurity-boson interactions. In particular, our data are consistent with a polaronstate (red line) that exists across unitarity (1/kn aIB → 0) with an energy that approaches that ofa weakly bound molecular state (green line) on the aIB > 0 side of the Feshbach resonance. Notethat the theory line for the molecular state comes from a two-body theory.
Ideally, RF spectroscopy of polarons can be used to extract the residue, Z, of the polaron

36
by integrating the spectrum over the whole frequency range. This has been used experimentally
to extract the residue of the Fermi polaron [27], which is a quasiparticle formed by an impurity
interacting with a Fermi gas. However, extracting Z in this way requires that the data is taken in thelinear regime, where the signal is proportional to the product of RF power and pulse duration. As
discussed in the next section, our data is not always taken in this linear regime. In addition, losses
could affect our data. While atom loss in a Fermi gas is suppressed due to the Pauli exclusion
principle, for our Bose-Fermi gas mixture, three-body recombination can cause significant atom
loss in the strongly interacting regime. This could result in a reduced signal strength in the RF
spectroscopy.

3.4

Coherent Rabi oscillation and damping of the Bose polaron
Another important property of Bose polarons is the quasiparticle’s residue Z, which describes

how much the excitation or quasiparticle looks like a non-interacting particle. This can be seen in
the trail variational wavefunction given in Eq. (3.7), where Z(q) = |φ0 (q)|2 . For our experiment,
the temperature T ≈ 150 nK, which corresponds to a momentum of the impurity atom pT =√mK kB T = 0.16 ~kn . The kinetic energy of kB T ≈ 0.12En is small compared to the measured
energy shift (see Fig. 3.8) so that the momentum of the impurity atom approximates to zero.
When performing a Rabi oscillation between the polaron state, |Φ(p)i, and the non-interacting
bare atom state, |0i, with zero momentum, the residue Z is related to the Rabi frequency Ω√through Z = Ω/Ω0 [29]. Here Ω0 is the Rabi frequency for non-interacting atoms. To understandthis relation and further extend it to a useful form for discussing our measurement, let me startwith the RF Hamiltonian in the Shrödinger picture described by (see Eq. (A.3))
Ĥrf (Ω0 ) = ~Ω0 X  †f1,k f0,k + H.c. ,2kwith the basis {|0i|BECi, |Φ(0)i} for the Rabi oscillation between the polaron state and the noninteracting bare atom state. Here we only consider the zero-momentum polaron state, |Φ(0)i, due
to the momentum conservation of the RF process. In the following discussion, I denote |0i|BECi

37
simply by |0i. The time-dependent wavefunction, |ψ(t)i = C0 (t)|0i + C1 (t)|Φ(0)i, can be expressed
as a vector, |ψ(t)i = (C0 (t), C1 (t))T (T denotes the transpose operation). The Schrödinger equationis thus written as





h0|Ĥrf (Ω0 )|Φ(0)i   C0 (t)  C0 (t)  h0|Ĥrf (Ω0 )|0ii~∂t  = C1 (t)
hΦ(0)|Ĥrf (Ω0 )|0i hΦ(0)|Ĥrf (Ω0 )|Φ(0)i
C1 (t)Ω00~ 2 φ0   C0 (t) 

= 

,
~ Ω20 φ∗0
0
C1 (t)(3.12)

with an initial condition of C0 (0) = 1 and C1 (0) = 0. To obtain Eq. (3.12), Eq. (3.7) has beenused and φ0 = φ0 (q = 0). Solving the above equation, we get the population of impurity atoms in
the polaron state as2

2|C1 (t)| = sin




dE0
|φ0 |t ,2(3.13)

where d is the induced electrical dipole moment of atoms and E0 is the electric field amplitude of
the RF pulse.
If we fix E0 and vary t, Eq. (3.13) describes a Rabi oscillation in time with
 
2
2 Ω
|C1 (t)| = sint ,2
where Ω = dE0 |φ0 | is the Rabi frequency of polarons. The Rabi frequency of non-interacting
impurty atoms is Ω0 = dE0 . The residue Z can be obtained using the relation of (Ω/Ω0 )2 =|φ0 |2 = Z. This relation has been used in Ref. [29] to measure the residue of the Fermi polaron.
Equivalently, if we fix t and vary E0 , Eq. (3.13) describes a Rabi oscillation in E0 with!
Ω̃
2
2
|C1 (E0 )| = sin
E0 ,(3.14)
2
where Ω̃ = d|φ0 |t is an equivalent Rabi frequency. The equivalent Rabi frequency for non-interacting
impurity atoms is Ω̃0 = dt. The quasiparticle’s residue Z can be obtained using
Z = |φ0 |2 = (Ω̃/Ω̃0 )2 .

(3.15)

This relation is particularly useful for measuring the Bose polaron’s residue using the Rabi oscillation described by Eq. (3.14), because the three-body recombination loss of impurity atoms

38

Figure 3.10: Coherent Rabi oscillation for the Bose polaron. (a) Rabi oscillation of Bose polarons
at different interaction strengths. The signal is from the center OD and normalized to the highest
value in each data set. Red lines are fits to Eq. (3.16). (b) The damping rate Γ and residue
Z = (Ω̃/Ω˜0 )2 from the Rabi oscillation. The inset shows the effective mass of the Bose polaronobtained by using Eq. (3.6). Lines are the calculated values based on the T-matrix method.

exponentially depends on t and is independent of E0 . By fixing t and varying E0 , the effect of thethree-body loss on the Rabi oscillation frequency is minimized, which helps accurately measuringZ.
In our experiment, we perform the Rabi oscillation between the polaron state and the weaklyinteracting initial state |0i. Two period of oscillation is achieved by varying the RF power, while
fixing the pulse duration to 4∆t = 0.4 ms and RF frequency at νp . The Rabi oscillation in afunction of the electric field amplitude, E0 , of the RF pulse has a form as shown in Eq. (3.14) and
we plot our Rabi signal versus E0 in Fig. 3.10(a).
The Rabi oscillation gets slower as expected at higher interaction strength. In addition, it39also shows a damping on the oscillation amplitude as well as a growing incoherent signal added tothe oscillation offset. Such decoherence behavior was also observed in Fermi polaron experiments[29, 30], in which it was explained due to two-body collisions. As indicated by the red lines in Fig.
3.10(a), we fit our data to an empirical model described by
"
!
#

Ω̃
y(x) = A e−2Γx sin2
x + 1 − e−2Γx ,2(3.16)

where x is E0 and Ω̃ is the equivalent Rabi frequency. The first term is the coherent part withthe damping rate Γ and the second term is the incoherent part. The Rabi frequency, Ω̃0 , of bare
K atoms is obtained by performing the same RF pulse on K atoms in the absence of Rb atoms.
When the interaction strength gets larger, the damping rate becomes higher and eventually it
washes out the oscillation pattern as 1/kn aIB > −0.7 in our experiment. For the repulsive polaron,
the damping is so fast that we cannot see any oscillation pattern until the interaction strength islowered to 1/kn aIB ≥ 3.57. This difference from the attractive case may be a comprehensive resultof the finite lifetime and the incoherent RF excitation (see Sec. 3.5.3) of the repulsive polaron.
In Fig. 3.10(b), the damping rate as well as the fractional Rabi frequency at different interaction strengths are shown. From the upper panel of Fig. 3.10(b), we see that the damping
rate Γ increases as the interaction strength 1/kn aIB increases and saturates at 1/kn aIB ≥ −1,
where the measured values of (Ω̃/Ω˜0 )2 deviate from the theoretical prediction (black line). When
1/kn aIB < −1, the measured values of (Ω̃/Ω˜0 )2 agree with the theoretical prediction pretty well
as shown in the lower panel of Fig. 3.10(b). The inset shows the effective mass of the polaronobtained using meff /mK = (1 + mK /mRb )/(Z + mK /mRb ) (see Eq. (3.6)). We can see that theeffective mass of the impurity atom becomes heavier at larger 1/kn aIB , which is due to the stronger
polaronic coupling between the impurity and the BEC.

40

3.5

Data analysis in detail

3.5.1

A mean-field model of density inhomogeneity

An experimental challenge arises from the very different cloud sizes for the impurity gas andthe BEC. This can be seen from the initial density distributions of the Rb BEC and K atoms in|0i state before a RF pulse


ρ2
z2
nBEC (ρ, z) = n0 1 − 2 − 2 ,
Rρ Rz


ρ2
(z − sag)2
nK,0 (ρ, z) = npk exp −−− gbg nBEC (ρ, z)/kB T ,
2Wρ2
2Wz2

(3.17)

where the size of the BEC is charaterized by the Thomas-Fermi radius {Rρ , Rz } and that of
the K cloud by the RMS width {Wρ , Wz }. Here we assume thermal equilibrium and ignore thequantum statistics of the fermionic 40 K atom. The RMS width is related to the temperature Tp
K . The gravitational sag between Rb and K cloud is given bythrough Wρ,z =kB T /mK /ωρ,zK are the trappingsag = g[1/(ωzRb )2 − 1/(ωzK )2 ] with g being the gravitational constant. Here, ωρ,zRb the trapping frequencies of Rb atoms. The last term in n comes
frequencies of K atoms and ωρ,zKfrom the mean-field potential due to the Rb BEC with the coupling constant gbg = 2πabg ~2 /µKRb
and abg = −187a0 [32].
Fig. 3.11 shows the calculated density distributions of the initial impurity cloud and the
BEC along z. The gravitational sag shifts the centers of the K and Rb clouds apart, which breaks
the ellipsoidal symmetry of the system along z. On the other hand, the background impurityboson interaction leads to an attractive mean-field potential for the impurity and thus results in an
enhanced impurity density near the center of BEC, which tends to restore the ellipsoidal symmetry
near the center of the BEC. In addition, to look at the symmetry of the impurity-boson system, it
is instructive to write down the energy density functional [54]:E[nBEC , nK,0 ] = gbb nBEC (ρ, z)2 +3(6π 2 )2/3 ~2
nK,0 (ρ, z)5/3 + gbf nBEC (ρ, z)nK,0 (ρ, z),
10mK+UBEC (ρ, z)nBEC (ρ, z) + UK (ρ, z)nK,0 (ρ, z),41

Figure 3.11: Density profile of the initial impurity cloud and the BEC based on Eq. (3.17). The
BEC density n0 = 1.8 × 1014 cm−3 , Rz = 3.4 µm, Wz = 3.2 µm, sag = 4.3 µm and T = 153µK. The gray region represents the BEC density. The red regime denotes the weighted impurity
density, nK,0 (0, z)nBEC (0, z)/n0 (see Eq (3.18)).

where UBEC (ρ, z) and UK (ρ, z) are the external trapping potentials of the BEC and the K atoms
respectively. In the limit of nBEC  nK,0 near the BEC center, the energy density functional issimplified to
E[nBEC , nK,0 ] = gbb nBEC (ρ, z)2 + gbf nBEC (ρ, z)nK,0 (ρ, z) + UBEC (ρ, z)nBEC (ρ, z), (3.18)
where the first and third terms have ellipsoidal symmetry, and the second term has approximate
ellipsoidal symmetry as shown by the red regime in Fig. 3.11. Thus the region near the center ofthe BEC has approximate ellipsoidal symmetry, which is important for us to use the inverse Abeltransform discussed in Sec. 3.5.2.
After the RF pulse, the number of spin-flipped K atoms in |1i state near the cloud center(ρ < ρ0 ) is modeled, based on Eq. (A.6), as
Z ∞

Z ρ0
NK,1 (ν, ρ ≤ ρ0 ) = A

dz nK,0 (ρ, z)R(ν − νp nBEC (ρ, z)/n0 ),ρdρ
0(3.19)−∞

in which A is a constant and νp is the mean-field energy shift at the center of the BEC. The effectof a non-uniform BEC density is modeled using mean-field energy shift in R(ν − νp nBEC (ρ, z)/n0 )42[55]. The effect of finite imaging resolution can be modeled by convolving the above equation with
2a point spread function, for which we choose a gaussian function, exp − 2σρ2 , with σres beingres

the resolution. Thus Eq. (3.19) is modified as
Z ∞

Z ρ0
NK,1 (ν, ρ ≤ ρ0 ) = A
Z ∞0

ρdρ
0



(ρ̃ − ρ)2×ρ̃dρ̃ exp −22σres

dz nK,0 (ρ̃, z)R(ν − νp nBEC (ρ̃, z)/n0 ) ,(3.20)−∞

which is a model that we use to understand the measured RF lineshape (see Fig. 3.6(b)) in the
weak interaction or mean-field regime.
Fig. 3.12 shows different calculated RF lineshapes that illustrate the effects of the initialbackground mean-field potential, the size of the integrated area, the temperature, and the imaging
resolution. There are two main features for the RF lineshape. The first one is the height of thepolaron signal relative to the bare atom signal. From Fig. 3.12, we see that this height is sensitive
to the initial background mean-field potential and the temperature. A deeper mean-field potential
or lower temperature generates a higher height of the polaron signal. The second feature is the
double-peak structure consisting of a bare atom peak and a polaron peak. The polaron peakstructure turns out to be sensitive to the initial background mean-field potential as well as theresolution as shown in Fig. 3.12. A shallower mean-field potential or a bad imaging resolution(large σres ) tends to flatten the polaron peak.
We fit our data shown in Fig. 3.6(b) to Eq. (3.20) with floating parameters {A, νp } and allother relevant quantities are obtained from the experimental measurements. The imaging resolution
is measured by imaging a tiny BEC as point source and we get σres = 3.4 µm. The fitting resultis shown in Fig. 3.6(b) as a blue line, which overlaps with our data very well at the negative aIBside. The fit also gives the polaron energy shift νp at the BEC center.
Eq. (3.20) is the model that we used for understanding our RF lineshapes at the relatively
weak interaction or mean-field regime with 1/kn |aIB | > 1.5. Once the interaction is strong enough,the above model is inadequate because the mean-field model breaks down. One needs to consider
the quasiparticle residue Z as well as the three-body recombination collision into Eq. (3.20), which

43

Figure 3.12: Calculated RF lineshapes based on Eq. (3.20). (a) RF lineshapes at different strengthsof the initial background mean-field potential, gbg n0 . Here T = 190 nK and σres = 0 µm. The actual
gbg n0 /h = −4 kHz. (b) RF lineshapes for different sizes of the integrated area. The radial size of
the BEC is chosen to be Rρ = 15.2 µm. Here T = 190 nK, σres = 0 µm and gbg n0 /h = −2 kHz.
The actual radius of the integrated area is 2.76 µm. (c) RF linesahpes at different temperatures.
T = 150 nK in our experiment. (d) RF lineshapes for different imaging resolutions. Here T = 190
nK and gbg n0 /h = −2 kHz. δν is set to be 1.4 kHz. The measured resolution is σres = 3.4 µm.
The dashed lines indicate the locations of νp .

is difficult.

3.5.2

Justification for inverse Abel transform

The inverse Abel transform can be used to calculate the three-dimensional density distribution
from an two-dimensional absorption image. When the system has the spherical symmetry, cylindrical symmetry, or ellipsoidal symmetry. In the following, I focus on a system with the ellipsoidal

44psymmetry. In this case, we can define a “radius”, r = x2 + b2 y 2 + c2 z 2 , for the three-dimensionalpspace and ρ = x2 + b2 y 2 , for the two-dimensional space. The two-dimensional absorption image
is denoted by F (ρ) and the three-dimensional density distribution f (r) is obtained using
1f (r) = −πZ ∞rdF (ρ)dρp.
2dρρ − r2(3.21)

With the inverse Abel transform, we can potentially solve the non-uniform density issue in ourBose polaron study. However, the gravitational sag between the K cloud and Rb BEC breaks theellipsoidal symmetry of our mixture system. In the following, we use a numerical simulation toshow that the inverse Abel transform nevertheless works well for us to extract the RF spectrumof Bose polarons at the BEC core. This is because the region near the center of the BEC, whichcontains the signal we wish to extract, has approximate ellipsoidal symmetry. We calculate theinverse Abel transform using a matrix approach that was adapted from Eric Mueller and that is
appropriate for an image with discrete points.
In our system, the measured trapping frequencies for K atoms are ωρK = 2π × 50 Hz andωzK = 2π × 281 Hz and the measured trapping frequencies for Rb atoms are ωρRb = 2π × 39 Hz
and ωzRb = 2π × 183 Hz. We estimate the gravitational sag using sag = g[1/(ωzRb )2 − 1/(ωzK )2 ] = 4µm. In our simulation, we start from the density distributions given by Eq. (3.17). The radiusppr = x2 + y 2 + (ωzK /ωrK )2 z 2 and ρ = x2 + y 2 . The RF spectroscopy is modeled based on themean-field theory and similar to Eq. (3.19), we use
nK,1 (r, ν) = nK,0 (ρ, z)R(ν − νp nBEC (ρ, z)/n0 ),(3.22)

to test how well the inverse Abel transform works for the purpose of extracting the RF spectroscopy
signal of Bose polarons at the BEC center. From Eq. (3.22), we can also obtain a calculated
absorption signal (column density)Z ∞dz nK,0 (ρ, z)R(ν − νp nBEC (ρ, z)/n0 ),n̄K,1 (ρ, ν) =−∞for a RF frequency ν. By performing the inverse Abel transform on the 2D column density n̄K,0 ,45

Figure 3.13: Simulated data for testing the inverse Abel transform for extracting the Bose polaronsignal at the BEC center. The blue stars represent the expected RF spectrum, nK,1 (r = 0, ν),based on Eq. (3.22). The red squares are the RF spectrum, nInv.Abel(r = 0, ν), extracted from theK,1
inverse Abel transform based on Eq. (3.23). (a) RF spectrum with sag. (b) RF spectrum withnoise added to the fake column density n̄K,1 (ρ, ν) (10% multiplicative noise and 0.05 additive noise;both numbers are Gaussian standard deviations). (c) RF spectrum at a larger interaction strength.

we getnInv.Abel(r, ν) = Inverse Abel transformation{n̄K,1 (ρ, ν)},K,1

(3.23)

which is the three-dimensional density distribution of the spin-flipped K atoms by the RF pulse.
Now we can compare nInv.Abel(r = 0, ν) against nK,1 (r = 0, ν) (Eq. (3.22)) at the BEC center.
K,1
Fig. 3.13 shows the simulation results. Around νp , the RF spectrum of Bose polarons overlaps
with the expected very well, although it overestimates the signal around the bare atom transition
ν0 . In Fig. 3.13(b), we add some noise to the fake column density n̄K,1 (ρ, ν) to test the robustnessof the inverse Abel method, and we find that noise does not change the results. In Fig. 3.13(c), wetest the RF spectrum at a larger interaction strength and it works likewise.

46
3.5.3

Coherent and incoherent excitation spectrum

As observed in Fermi polaron experiments [27, 29], a spectrally broad incoherent signal alongwith the narrow coherent signal are excited when performing a RF spectroscopy on the Fermi
polaron (a quasiparticle formed by an impurity interacting with a Fermi gas). This broad incoherent
signal would affect the extracted value of the polaron energy if it wasn’t treated correctly. The main
contribution to this incoherent signal stems from the molecule-hole continuum (MHC) existing in
the Fermi gas [29]. In contrast, for a Bose-Fermi gas mixture with the majority of bosons, thisMHC is not important. Instead, other decoherence mechanism may exist leading to an incoherent
signal [9, 56].
Based on the time evolution of impurities under the RF pulse, a generic feature of the RF
response is given in Sec. IV.A of Ref. [56] and is adapted for our system as
I(ω) = Icoh (ω) + Iincoh (ω),
Icoh (ω) = Zδ(ω − 2πνp ),Iincoh (ω  2πν0 ) ∝ C(2πν0 − ω)−3/2 ,(3.24)(3.25)(3.26)

where I is the RF response, Z is the quasiparticle residue, and C is the Tan’s contact [57, 58]. The
RF spectrum I(ω) consists of two parts, the coherent part Icoh and the incoherent part Iincoh . The
incoherent part is due to dephasing and exists only when ω < 2πν0 . Here a uniform BEC density is
assumed and the RF fourier limited lineshape R(ν) is not included. For the non-interacting case,
Icoh (ω) = δ(ω − 2πν0 ) and Iincoh = 0. By considering the RF fourier limited lineshape R(ν), Eq.
(3.24) becomes
˜
I(ω)
=

Z ∞

dω̃R[(ω̃ − ω)/2π]I(ω̃)
Z ∞= ZR(νp − ω/2π) +dω̃R[(ω̃ − ω)/2π]Iincoh (ω̃)−∞

(3.27)−∞

where I set R(ν) =

q

2 1
2
2
−3/2 .
˜π δν exp(−2ν /δν ). In particular, I(ω  2πν0 ) ∝ C(2πν0 − ω)Fig. 3.14 shows the observation of the Tan’s contact at both positive and negative interactions. As discussed above, the Tan’s contact as a ω −3/2 tail shows up only at the ν < ν0 side (green

47

Figure 3.14: Tan’s contact shown as a ω −3/2 tail in our RF spectroscopy data. Black lines indicate
the bare atom transition. Red dots denote the data with the cloud size smaller than or equal to the
BEC size. (a) RF spectroscopy at 1/kn aIB = 0.7. The RF pulse duration is fixed at 4∆t = 0.52 ms.
From the top to bottom, the RF power is increased to give a π pulse, 3π pulse, and 5π pulse. Greenlines indicate the Tan’s contact and come from fits of the RF tail to C(2πν0 − ω)−3/2 . Additionally,
K-Rb Feshbach molecules are formed from the Rb thermal cloud as shown on the right corner andthe pink line is a fit to a RF association model [32]. (b) RF spectroscopy at 1/kn aIB = −1.9. The
RF pulse duration is fixed at 4∆t = 0.52 ms. From the top to bottom, the RF power is increased
to give a π pulse and 5π pulse. The red dots on the left indicate the Tan’s contact tail, while thered dots on the right represent the attractive polaron signal.

lines). The fact that the cloud size of the contact signal is smaller than the BEC size (as indicatedby the red color) means that the contact signal is from the 40 K atoms overlapping with the BEC.
At relatively small RF power (e.g., π pulse), the contact signal is too small to be observable, which

48
is good for measuring the energy shifts of Bose polarons.
In conclusion, for the attractive polaron spectroscopy, the incoherent signal and the polaronsignal are well separated in frequency and are on the opposite sides of the bare atom transition.
For the repulsive polaron spectroscopy, although the incoherent signal and the polaron signal may
overlap with each other, the polaron signal can still be selected out by choosing a RF pulse withrelatively small RF power. Unfortunately, when performing a Rabi oscillation in terms of RF power,
this incoherent signal will wash out the oscillation pattern immediately for the repulsive polaron.

3.5.4

RF association and dissociation of K-Rb Feshbach molecules

Figure 3.15: RF association and dissociation of K-Rb Feshbach molecules. (a) RF associationof K-Rb Feshbach molecules at 1/kn aIB = 0.96, 0.72, 0.51, 0.37. The magenta line is obtained byfitting the molecule data to a well-established model [32]. The black line indicate the bare atomtransition. The green line is the contact tail as discussed in the previous section. (b) RF dissociationof K-Rb Feshbach molecules at 1/kn aIB = 0.72 (bottom panel). The magenta line is a fit to a wellestablished model [59]. As a reference, the association data is shown in the upper panel. The
dashed line indicates the location of the binding energy of molecules.

RF association of K-Rb Feshbach molecules has been well studied in [51] and [32]. Here49
we measure the binding energy of K-Rb Feshbach molecules in the strongly interacting regime,
where the binding energy follows a universal relation, Eb /En = mRb /µKRb /(kn aIB )2 . Fig. 3.15(a)shows the data of RF associating K-Rb molecules. The binding energy is extracted by fitting
the molecule data to a well-established model [32], which is indicated by the magenta line. The
binding energy of K-Rb molecules is shown in Fig. 3.9 as yellow triangles, which match theuniversal theory line (green) very well. The RF pulse we used for associating K-Rb Feshbach
molecules has a pulse duration 4∆t = 0.52 ms and the RF power is set to give a 7π pulse. In
addition, we also measured the binding energy by RF dissociating K-Rb molecules as shown in Fig.
3.15(b), which is in agreement with that measured by RF association. Here the K-Rb molecules
were created by magneto-association described in our previous paper [20]. The dissociated K
atoms are in |9/2, −7/2i state, which is directly imaged by using the non-cycling transition of|9/2, −7/2i → |11/2, −9/2i with the measured imaging efficiency of 80%.

3.6

Summary
I report experimental data probing Bose polarons using radio-frequency spectroscopy. We

measure the excitation spectrum of fermionic 40 K impurities strongly interacting with a BEC of
87 Rb atoms.

We find that the measured energy of the impurites immersed in the BEC matches

well with that recently predicted for Bose polarons [9, 10] in the regime of strong impurity-boson
interactions. In particular, our data are consistent with a polaron state that exists across unitarity(aIB → ∞) with an energy that approaches that of a weakly bound molecular state on the aIB > 0side of the Feshbach resonance. Furthermore, for attractive interactions, we observe Rabi oscillations when exciting Bose polarons. This work is the first measurement of the Bose polaron in athree-dimensional trapped atom gas and provides a starting point for understanding the quantum
many-body physics of Bose-Fermi mixtures.

Chapter 4

Quantum Hall physics: bizarre and interestingEver since Edwin Hall discovered the Hall effect in 1879, measuring electrical conduction in amagnetic field has become one of the standard ways to characterize conductive materials. In 1980,
Klaus von Klitzing discovered the integer quantum Hall (IQH) effect when he was measuring theHall conductivity of a 2D semiconductor sample in a strong magnetic field (B ∼Tesla) (the 1985
Nobel prize in physics). What he found is that the Hall conductivity was exactly quantized at
integer values of e2 /h. Here e is the charge of an electron and h is the Planck constant. Later thefractional quantum Hall (FQH) effect was discovered by Horst Störmer and Daniel Tsui and waspartially explained by Robert Laughlin (the 1998 Nobel prize in physics). What they found is thatthe Hall conductance of a 2D electron gas shows precisely quantized plateaus at fractional valuesof e2 /h. FQH states represent new states of matter that contain a completely new kind of order—
topological order, which cannot be described by the conventional theories. Thus they greatly enrich
our understanding of quantum many-body physics and quantum phase transitions. However, there
have been few experimental ways to locally detect these exotic states so far. Recently, ultracold
atomic gases have been proposed as systems where experiment would be able to simulate thequantum Hall states and furthermore be able to detect the local properties of these states [15].

4.1Classical Hall effect
The classical Hall effect is introduced and some important quantities are described here.

Considering a 2D conductor exposed to an external magnetic field B along z, one can measure the51

Figure 4.1: Classical Hall effect. (a) Measuring Hall conductance of a sample. An external electricalfield Ey is along y and an external magnetic field B is along z. Hall voltage VH as well as the normal
current I along y are measured. (b) The helical motion of an electron moving on the x-y plane dueto the Lorentz force based on Eq. (4.1). (c) The dependence of Hall resistance, RH , on B basedon Eq. (4.2).

conductance by applying an electric field Ey (or a voltage potential V ) across the sample along y(see Fig. 4.1 (a)). Due to the Lorenz force, electrons are doing helical motion inside the sample
and eventually reach the transverse edge (see Fig. 4.1 (b)). Thus, the transverse Hall voltage is
produced by the accumulated electrons on the edge. In experiment, the electric current I along yand Hall voltage VH along x are recorded. The normal resistance can be obtained using R = V /I,
while the Hall resistance is RH = VH /I.
In the classical case, the Lorentz force felt by electrons is F = −e(E + v × B). Here theelectric field E = Ex x̂ + Ey ŷ with Ey = V /Ly and Ex = VH /Lx . Lx and Ly are the lengths ofthe sample along x and y, respectively. The constant magnetic field B = Bẑ. Combined with the
Newton’s equation, F = me dv
dt , we can get the velocity of an electron asv(t) = v0 [cos(ωc t)x̂ + sin(ωc t)ŷ] +


eEy
eEx
x̂ −
ŷ ,
m e ωc
me ωc

(4.1)

where cyclotron frequency ωc = eB/me and v0 is the initial velocity. The first term in the above
equation describes the cyclotron orbit and the second term represents a drift motion with the drifteExvelocity vd = me ωyc x̂ − meEŷ. The electric current density is defined by j = −nevd with n beinge ωcthe electron density. Thus we have j = −(ne/B)Ey x̂ + (ne/B)Ex ŷ. According to the Ohm’s law,

52
j = σ · E 1 , we have the conductivity, σxx = σyy = 0 and σxy = −σyx = −ne/B. The resistivity,
as the inverse of the conductivity2 , is ρxx = ρyy = 0 and ρxy = −ρyx = 1/σxy = −B/ne. Herethe off-diagonal conductivity elements {σxy , σyx } are called Hall conductivities. The Hall resistance(Hall conductance) turns out to be the same as the Hall resistivity (Hall conductivity). This canbe seen from RH = VH /I = Ex Lx /jy Lx = Ex /jy = 1/σyx = ρyx . The value of the Hall resistancefollows
RH = ρyx =

B
.
ne

(4.2)

which depends linearly on B (see Fig. 4.1 (c)). Based on Eq. (4.2), the Hall effect has become an
important way to measure the charge carrier density n.
All above discussions made use of classical mechanics, which should be valid when the magnetic field is small enough that no energy gap is opened up. Next, I will discuss the case of strong
magnetic field, where the quantum mechanics is used.

4.2

Landau levels
Landau levels are a quantum mechanical effect of a magnetic field on the motion of charged

particles moving in a 2D plane. In the presence of a uniform magnetic field perpendicular to
the 2D plane, charged particles move along quantized orbits according to quantum mechanics andtherefore the energy of these particles are quantized into a series of equally spacing energy levels,
called Landau levels. For the case of a many-particle system, each of these Landau levels can beoccupied by multiple particles due to the degeneracy, which depends on both the strength of themagnetic field and the size of the system. What surprising is that L. D. Landau discussed thisphysical system in 1930 [60], 50 years before people discovered the integer quantum Hall effect.
Consider electrons moving in the 2D x-y plane, which has an area A = Lx Ly (see Fig. 4.1(a)).
 


jx
σxx σxyEx=.
jyσyx σyyEy
2−1
The resistivity ρ = σ , which means ρxx = σyy /(σxx σyy − σxy σyx ), ρyy = σxx /(σxx σyy − σxy σyx ), ρxy =−σyx /(σxx σyy − σxy σyx ), ρyx = −σxy /(σxx σyy − σxy σyx ).
1Here the Ohm’s law is a two-dimensional formula:

53
Applying an uniform magnetic field along z, it is well described by a single-body Hamiltonian [60]i21 hep̂⊥ + A⊥ (x, y)
2me
c


2
~ky 2
1
p̂x
+ me ωc2 x −
2me 2
me ωc
2
p̂⊥
1
+ me (ωc /2)2 ρ2 + (ωc /2)L̂z
2me 2

H =
=
=

(4.3)
(4.4)

where ωc is the cyclotron frequency with ωc = eB/me c (c is the speed of light). Here A(x, y) is thevector potential of the magnetic field with A⊥ = {Ax , Ay }, the momentum operator p̂⊥ = {p̂x , p̂y },and the radius ρ2 = x2 + y 2 . L̂z = xp̂y − y p̂x is the orbital angular momentum operator along z.
Eq. (4.3) is obtained by using the Landau gauge, A = {0, Bx, 0} and Eq. (4.4) is gotten by using
the symmetric gauge, A = 21 {−By, Bx, 0}, both of which are identical with each other and satisfy∇ × A = Bẑ. Their eigenenergy is
E = (n + 1/2)~ωc ,(4.5)

where n is an integer taking values of 0, 1, 2 · · · . Different values of n here represent different Landaulevels. The wavefunctions for different gauges are different from each other. Both situations areuseful for our discussions. The former is useful for explaining transport measurements and edge
states, while the latter is useful for discussing the connection with a rotating gas.
For Eq. (4.3) associated with the Landau gauge, the eigenfunction is [60]ψn,ky (x, y) = eiky y φn (x − x0 ),(4.6)

where n and ky are quantum numbers, φn is the well-known wavefunction of an harmonic oscillator.
The location of the harmonic oscillator is
x0 =

~ky
,
mωc

(4.7)

where ky is quantized due to the boundary condition and takes values ky = 2πny /Ly (ny =0, 1, 2, · · · , Ny ). Due to the constraint of 0 ≤ x0 ≤ Lx , we get Ny = Int(Φ/Φ0 ) (Int means integerpart) with Φ = BLx Ly being the magnetic flux and Φ0 = hc/e being the fundamental quantum of

54
flux. The degeneracy or the maximum number of particles per Landau levels is
D = (2s + 1)

Φ
,
Φ0

(4.8)

where s is the spin of an electron with s = 1/2. One important feature for Landau levels is thequantization of the cyclotron orbits. This can be seen by using a semi-classical picture: E =(n + 1/2)~ωc = 21 mωc2 Rc2 , which gives the quantized cyclotron orbits
Rc2 = (2n + 1)

~c
.
eB

(4.9)

For Eq. (4.4) associated with the symmetric gauge, the eigenfunction is [61, 62]22m ∂ n e−|w|
e|w| /2 ∂+−q,ψn,m (x, y) =2
πa⊥ n!m!

(4.10)

where n is the index of Landau levels, m is an “angular momentum” index with L̂z |ψnm i =p~(m − n)|ψnm i, w = (x + iy)/a⊥ , a⊥ =2~/me ωc , and ∂± = (a⊥ /2)(∂x ± i∂y ). Note thatp√a⊥ = 2lB with the magnetic length lB = ~/me ωc . The degeneracy of each Landau level also
follows Eq. (4.8).
√
2
2
For the lowest Landau level n = 0, the wavefunction is, ψ0,m (ρ, φ) = eimφ e−ρ /2a⊥ (ρ/a⊥ )m / m!π,which carries angular momentum m~. These states have a ring shape and the position of the max√
imum of |ψ0,m (ρ, φ)| is at ρpk = ma⊥ (Fig. 4.2). The ring of the mth state has an area of mπa2⊥ ,with an enclosed magnetic flux mπa2⊥ B = mΦ0 . Hence there is one flux quanta per state andthe number of states in the lowest Landau level corresponds to the number of flux quanta. The
many-body wavefunction for the lowest Landau level (n = 0) state has the form [63]
Ψ1 (z1 , · · · , zN ) = A[(z1 )0 (z2 )1 (z3 )2 · · · ]e−

= det

1

1

···

z1

z2

···

PN
k

e−

2
zk2 /4lB

PNk2
zk2 /4lB(z1 )2 (z2 )2 · · ·
..
..
..
.
.
.
=

N
PN 2 2Y(zi − zj ) e− k zk /4lB ,i<j(4.11)

55

Figure 4.2: Degenerate states of the lowest Landau level. (a) Circular and quantized orbits of thestates in the lowest Landau level. The mth state in the lowest Landau level has a ring shape with√the peak position at ρpk = ma⊥ . Solid lines represent the occupied states while the dashed lines
represent the empty states. (b) Amplitude squared of the wavefunctions of the states in the lowest
Landau levels. Each state has a single peak structure that spatially overlaps with neighboringstates. The 1/e2 width ofPeach peak is about lB . (c) Density profile of the partially filled lowest
Landau level with n2D = m |ψ0,m |2 . It is uniform near the center.

where A is the anti-symmetrization operator, N is the total number of electrons, and zj = xj +iyj =ρj eiφj is the complex coordinate in 2D for the jth electron.
The Landau levels are the foundation of the quantum Hall states. They have been usedsuccessfully for explaining the integer quantum Hall effect. However, to understand the fractional
quantum Hall effect, the picture of Landau levels is inadequate and one needs to include theparticle-particle interactions.

56

Figure 4.3: Integer and fractional quantum Hall effects (reproduced from [14]). The hallmark ofthese effects is the quantized Hall resistance, RH = νeh2 . The index ν = 1, 2, 3, · · · (IQH effect) andν = 13 , 25 , 73 , · · · (FQH effect).

4.3

Integer and fractional quantum Hall effects
Quantum Hall effects are the manifestation of quantum Hall states through electron transport.

To observe these effects, one needs a nearly ideal 2D electron system, which can be realized byconfining charge carriers in a narrow potential well. Such a system was originally realized by using
GaAs/AlGaAs heterostructures [14]. The carriers in such structures are free to move in a 2D
plane, but their motion perpendicular to the plane is frozen by cooling the sample (T ∼ 1mK) tothe lowest energy level along this direction. As a consequence, a quasi-2D metal emerges with thecarrier density (n ∼ 0.2 − 4 × 1011 cm−2 ) [14]. Quantum Hall effects have also been observed in57
other materials, for example, graphene [64].
Figure 4.3 shows the results of the measured normal resistance R and Hall resistance RH for
both the Integer quantum Hall (IQH) effect and the fractional quantum Hall (FQH) effect. The
hallmark of these effects is the quantized Hall resistance,
RH =

h
,
νe2

(4.12)

where the index ν = 1, 2, 3, · · · (IQH effect) and ν = 31 , 25 , 37 , · · · (FQH effect). At each plateau of
RH , the normal resistance R vanishes. According to
R=

Ey Ly
Ly
V
σxx
=
= ρyy (Ly /Lx ) =
,
I
jy Lx
σxx σyy − σxy σyx Lx

the conductivity σxx = 0, which means an insulator. On the other hand, the transverse conductivity
σxy 6= 0, which means a conductor. It turns out that the insulating behavior comes from the bulk
property of the material, while the conduction is due to edge states. Such kind of bizarre materials
are called the topological insulators [13].
The IQH effect can be understood by considering the filling of Landau levels (see Fig. 4.4).
For an electron gas trapped in a 2D uniform potential well, the density of states is homogenous
in energy (see Fig. 4.4(b)). At zero temperature, the energy states below the Fermi energy EF
are occupied by electrons. When B = 0, there are no Landau levels. Thus the system is a normal
conductor. As the magnetic field B is increased from zero, the system first shows the Hall effect.
As the value of B increases, the energy spacing ~ωc = ~eB/me c between Landau levels increases.
As a result, energy gaps are opened at large values of B and Hall plateaus start to show up. At thesame time, the density of states or the degeneracy of Landau levels, given by Eq. (4.8) increases
linearly with respect to B. Therefore, fewer and fewer Landau levels are filled. The number offilled Landau levels corresponds to the index ν of the IQH effect and ν is called the filling factor.
At each plateau, the Fermi energy lays in the energy gap, which means the system is an insulator
based on the traditional classification. What exotic is that this traditional insulator can conduct
electric current by edge states (Fig. 4.4(c)).

58

Figure 4.4: Understanding the IQH effect by filling Landau levels. (a) The integer quantum Hall
effect at different values of B. (b) The correspondent picture of filling Landau levels. (a) & (b)are adapted from (http://en.wikipedia.org/wiki/Quantum Hall effect). Red shaded regime means
occupied states below the Fermi energy (dashed line). The energy band is gapless at small values ofB, where the classical Hall effect dominates. Once B is large enough, energy gaps are opened andHall plateaus start to show up. Since the density of states or the degeneracy of Landau levels scales
linearly with B, fewer and fewer Landau levels are occupied at higher B. (c) Ideal Landau levels
for a device with boundaries (adapted from [65]). From Eq. (4.7), the location x0 is quantized as
represented by dots. The bulk states are insulating states, while the edge states contribute to the
conduction.

The FQH effect has much richer physics than the IQH effect. The picture of filling Landaulevels is not sufficient for describing the FQH effect and particle-particle interactions need to beconsidered. At fractional values of ν, the system has a partially filled Landau level. Furthermore,
the plateau structure of RH at the fractional ν means that an extra energy gap has opened in a
Landau level, which is not predicted by the single-body physics and is due to the particle-particle
interactions. The most exotic implication of the FQH effect is the existence of quasiparticles withfractional statistics and fractional charges. These composite quasiparticles are formed by attaching

59

Figure 4.5: Understanding the FQH effect by the Laughlin’s wavefunction. (a) Densities for 15,
20 and 25 particles in the ν = 1/3 quantum Hall regime (adapted from [66]). The densities are
obtained by classic Monte-Carlo calculations. Here ρ is the radius and lB is the magnetic length.
(b) A cartoon picture of the composite quasiparticles (reproduced from [67]). The electrons (reddots) are moving along cyclotron orbits around the magnetic flux Φ = 3Φ0 (lines) at the orbital
center.

magnetic fluxes to electrons and have become the underlying principle to describe the many-bodystates of the FQH effect.
In 1983, Laughlin proposed a many-body wave function for ν = 1/3 that is known as Laughlin’s wavefunction [61]
N
N
Y
X
2
3
Ψ1/3 =|zk |2 /4lB
(zi − zj ) exp −
i<j

!
.

(4.13)kThis wavefunction is similar to Eq. (4.11) except that it has a third-order zero (zi − zj )3 for any
pair of electrons. In addition, Ψ1/3 describes a circular droplet of uniform density (see Fig. 4.5(a)).
The exponent 3 in each factor expresses the attachment of three flux quanta to the position of eachelectron. It describes a composite quasiparticle that includes three flux quanta and one electron (seeFig. 4.5(b)). More generally, the exponent can be changed from 3 to q for the ν = 1/q (q = odd)
state and it then characterizes an electron dressed by q flux quanta. Only odd q are allowed,
since only they guarantee antisymmetry of this electron wave function. Based on the Laughlin’s
wavefuntion [61], it shows that an energy gap exists for the fractional quantum Hall state, which
explains the plateau structure of RH at the fractional values of ν.

60

4.4

Rapidly rotating atomic gases to quantum Hall regime: ideal cases~ do not feel any Lorentz force (for an electron,
Electricallly neutral atoms in a magnetic field B~ But if we put atoms in a rotating frame, they feel the Coriolis force, F = 2M~v × Ω,~
F = −e~v × B).
~ The identical structure between the Lorenz force and Coriolis force
due to a rotating field Ω.
indicates that we may be able to realize quantum Hall states by rotating ultracold gases of neutral
atoms. In this section, I will discuss the connection between a rotating gas and quantum Hall
states. Some experimental challenges will be reviewed in the next section.

4.4.1

Fermion caseLet me first discuss the case of rotating an ultracold Fermi gas. For a Fermi gas, the atoms
follow Fermi statistics and thus they naturally form an incompressible fluid due to Pauli blocking. It
requires that each quantum state can only be occupied by a single particle. This incompressibility
is important for observing the plateaus of quantum Hall effects, because it allows electrons tobe able to fill many Landau levels. To get the incompressibility for a bosonic gas, the repulsive
particle-particle interactions are in need.
Considering a 2D rotating Fermi gas in a harmonic potential, the single-body Hamiltonian
without interactions in the rotating frame is [62]H + ΩL̂z =

p̂2⊥
1
2 2
ρ + ΩL̂z ,+ M ω⊥
2M
2(4.14)

where M is the atomic mass of an atom and ω⊥ is the trapping frequency of the harmonic potential.
Assuming Ω = ω⊥ , Eq. (4.14) would have exactly the same form as Eq. (4.4). The eigenfunctions
and eigenvalues of Eq. (4.14) are [62]2un,m (x, y) =2m ∂ n e−|w|
e|w| /2 ∂+−q,
πa2⊥ n!m!

n,m = ~(ω⊥ + Ω)n + ~(ω⊥ − Ω)m + ~ω⊥ ,(4.15)(4.16)

where the quantum number n, m = 0, 1, 2 · · · with L̂z |un,m i = ~(m − n)|un,m i, w = (x + iy)/a⊥ ,pa⊥ = ~/M ω⊥ , and ∂± = (a⊥ /2)(∂x ± i∂y ). un,m is basically the same as ψn,m in Eq. (4.10) but

61

Figure 4.6: Realizing the IQH states by rapidly rotating a Fermi gas. (a) Shift of energy spectrum
due to the rotation based on Eq. (4.16) and (4.17). When Ω approaches ω⊥ , the Landau levels
are formed and the size of the un,m state is enlarged. (b)The gaps of energy bands are opened by
2 (see Eq. (1.3)). Red shaded
rotating a gas. The density of states for a static gas is ρ() = /~2 ω⊥
regime means occupied states below the Fermi energy (dashed line). The energy band is gapless at
small values of Ω. Once Ω is large enough, energy gaps are opened. Different from the 2D electron
system, the energy gap for a rotating gas is basically constant, independent of Ω. The number ofoccupied Landau levels can be tuned by varying the total number N of atoms (or εF ).

with different a⊥ . The quantized orbit for the un,m state can be seen byhρ2 in,m = hun,m |ρ2 |un,m i = (n + m + 1)a2⊥ ,(4.17)

which gives a good estimate of the area covered by the un,m state. Here ρ is the radius. At thelimit of Ω → ω⊥ , Eq. (4.16) reduces to the Landau levels
En = (n + 1/2)~(2ω⊥ ),(4.18)

with the energy spacing 2~ω⊥ . Note that the rotating rate Ω cannot be bigger than the radial

62

Figure 4.7: Density distribution of a rotating Fermi gas in the quantum Hall regime based on
Eq. (4.19).
Lower εF and lower T give more clear wedding cake structure. The oscillator lengthpa⊥ = ~/M ω⊥ = 2.5 µm for a 2π × 40 Hz trap (M is the atomic mass of a 40 K atom) andΩ = 0.99ω⊥ .

trapping frequency ω⊥ , because the centrifugal force gives effectively an anti-trapping potential.
P
For Ω < ω⊥ , the density of atoms at zero temperature is n2D (ρ) = n,m |un,m (ρ)|2 Θ(εF −Pn,m ) with Θ(x) being the step function and εF the Fermi energy. The total number N = n,m (εF −n,m ). At finite temperature T , the density of atoms becomesn2D (ρ) =X|un,m (ρ)|2 f (n,m ),(4.19)

n,mwith the Fermi-Dirac distribution function f () = 1/(e(−εF )/kB T + 1). Correspondingly, the totalP
number is N = n,m f (n,m ).
From Eq. (4.18), we can see that the integer quantum Hall states can be realized by rapidly
rotating an ultracold Fermi gas (Fig. 4.6). The degeneracy of the Landau levels at the limit ofΩ → ω⊥ isD=2ω⊥.
ω⊥ − Ω(4.20)

63
For the special case of Ω = ω⊥ , the atoms move in free space and there is no quantized energy
levels. To detect the Landau levels or the integer quantum Hall states, one way is to look at thedensity distribution [62]. In the quantum Hall regime, the density distribution will show a “weddingcake” structure (see Fig. 4.7), which looks similar to the plateau structure of the IQH effect. For√
the given εF , the size of the nth Landau level can be estimated by ρn = a⊥ n + m∗n + 1 with⊥ +Ω)nthe highest angular momentum state in the nth Landau level, m∗n = Int[ εF /~−ωω⊥⊥−(ω]. When−ΩΩ → ω⊥ , we have the size of the nth Landau level assεF /~ − (2n + 1)Ω
ρn = a⊥,ω⊥ − Ω
with the oscillator length a⊥ =(4.21)p~/M ω⊥ = 2.5um for a 2π × 40 Hz trap (M is the atomic mass of

a 40 K atom).

4.4.2

Boson case

The Eqs. (4.15) and (4.16) also work for a bosonic gas. The only difference from the fermioniccase is the compressibility of a bosonic gas. Due to the Bose-Einstein statistics, a single state canbe occupied by infinite number of bosonic atoms, which makes it hard to get a small filling factor
ν. In the past, the TOP trap experiment at JILA rotated a 87 Rb BEC into the lowest Landaulevel, but the filling factor was about 100. That is, there were 100 atoms occupying each u0,mstate. Assuming we have a Feshbach resonance (see Sec. 1.3), we can tune the particle-particle
interaction to be repulsive. In this case, the atoms would repel each other and eventually form
an incompressible fluid. By tuning the interaction strength, one can tune the number of occupied
Landau levels. Interactions of bosons potentially make them to be a fermion-like liquid [68, 69, 70].

4.5

Technical challenges
In the last decade, the technique of rotating ultracold gases has been developed for the original

goal of studying the superfluid property of a BEC [15]. However, due to technical challenges, noneof the existing schemes have been sufficient for observing quantum Hall states. In the following,

64
I discuss these challenges and in the next chapter I present an new apparatus for potentiallyovercoming these challenges.

4.5.1

An extremely smooth and harmonic optical trap

An ultracold atom experiment generally uses a harmonic trapping potential to confine thegas. To get fast rotation, the trap geometry needs to be as round as possible in the rotating plane,
which is relatively easy to achieve with careful design. Secondly, higher order anharmonic terms ofthe trapping potential could limit the rotation rate. This can be seen from the following potential(see Sec. 5.3)
U (x, y) = U0 + A2 (x2 + y 2 ) + A3 y 3 .
The y 3 terms can break the circular symmetry and slow down rotation. Thirdly, the smoothness ofthe potential is very important for a long-lived rotating gas. In particular, once the rotation rate isclose to the trapping frequency ω⊥ , any small roughness will have a big effect in damping rotation.
In order to get fast rotation, e.g. Ω ∼ 0.99ω⊥ , the roughness need to be reduced below 1%.
The harmonic trapping potential could be achieved with a magnetic trap or with an optical
trap. With a magnetic trap, it is easier to satisfy the conditions of smoothness and harmonicity.
However, a magnetic trap makes it very difficult to use a magnetic Feshbach resonance to tune theparticle-particle interactions, which is very important for reaching the quantum Hall regime. In
comparison, an optical trap is an ideal option for accessing a magnetic Feshbach resonance, but isdifficult to get as smooth and harmonic as a magnetic trap.

4.5.2

Rapid rotation close to 0.99 of trapping frequency

There are two main ways of rapidly rotating ultracold gases that have been demonstrated.
The first way is to use a blue-detuned laser beam to stir the trapped gas. This way has beenused by the MIT group for rotating a 23 Na BEC [71] and a 6 Li degenerate Fermi gas [72], and by
the Paris group for rotating a 87 Rb BEC [73]. However, because the center-of-mass motion of theharmonically trapped atom cloud is dynamically unstable as Ω → ω⊥ [74], the stirring method is

65
not fully appropriate for approaching the rapidly-rotating regime. The instability of center-of-mass
motion limits how close to ω⊥ the rotating rate Ω can be. The second way of rapidly rotating
ultracold gases is to use a deformable trap. The gas is stirred by deforming the trap to an ellipticalshape, then jumping the major axis by 45 degree to impart angular momentum to the atoms, andthen restoring the trap geometry to be round. After this stirring process, the rotation rate ofgas is generally up to about 0.4ω⊥ . To further speed up rotation, a special evaporation process,called spin-up evaporation, is used. The basic idea is to evaporate away atoms with lower angularmomentum so that the averaged angular momentum per atom is increased. As a result, the rotation
rate of the atom cloud is increased according to Lz = mΩ2 ρ. This way works great as demonstrated
by the JILA TOP trap experiment [75]. They also demonstrated a 87 Rb BEC that was rotated so
fast that all Rb atoms were in the lowest Landau level [76]. In the next chapter, we present ourscheme, which uses a similar spin-up evaporation technique to speed up the rotation of a gas.

4.5.3

Detecting local density by a quantum gas microscope

The wedding cake structure is an important feature of quantum Hall states. The plateau size
is about a⊥ = 2.5 µm. The atom number per unit area of πa2⊥ is only a few atoms. To detect sucha small feature, one needs a quantum gas microscope [77, 78, 79, 80, 81, 82, 83].

4.6

Other proposals
Beyond rotating gases, there are other proposals for simulating quantum Hall states using

ultracold atoms. These proposals include (i) rotating gases trapped in an optical lattice [84]; (ii)synthetic fields created by Raman beam coupling [85, 86, 87]; and (iii) laser assisted tunneling ina tilted optical lattice [88, 89].

Chapter 5

A novel all-optical trap for rapidly rotating gases

In this chapter, a new apparatus for rotating ultracold gases is introduced, and its optimization procedures are described. Some important features of this new apparatus include an all-opticaltrapping potential, a controllable potential geometry, and a good imaging system. To characterize
the apparatus, we used a 87 Rb BEC as a sensor to round out the trapping potential and further
to measure the roughness of potential. Ultimately, the apparatus was tested by rotating the 87 Rb
BEC and quantized vortices were detected. By counting vortices, we can estimate how fast theatomic gas is rotated. In particular, we used the rotating Rb BEC to test a spin-up evaporation
technique, which is a crucial part of our strategy for achieving rapid rotation.

5.1

A new apparatus for rotating gases
Fig. 5.1 shows the schematics of the new apparatus. It consists of two independent laser

beams that form an optical trapping potential. The first beam is a horizontally propagating sheet
beam that is designed to provide only vertical confinement for the atoms. The second beam is avertically propagating round beam, which confines the atoms horizontally. Both beams have thesame wavelength (1090nm) and generate a far red-detuned optical dipole trap. The linewidth ofthe lasers is 1 nm, which reduces the interference of the laser beams. An important feature of theapparatus is that the pointing direction of the vertical beam is controlled by a dual-axis acoustooptic modulator (AOM). By using the AOM, we can change the position of the vertical beam,
which makes it possible to create a time-averaged or painted trapping potential.

67

Figure 5.1: A new apparatus for rotating ultracold gases. It consists of two main laser beams. The
first beam is a horizontally propagating sheet beam with 1/e2 beam waist of 1 mm×30 µm, which
is designed to provide only vertical confinement for the atoms. The second beam is a vertically
propagating round beam, which confines the atoms horizontally. The B coils are used for accessing
a Feshbach resonance. An imaging setup is indicated by the blue line.

An AOM device can change the steering of a laser beam by imparting a phonon momentum~k to a photon that passes through the AOM [90]. Here phonons refer to the acoustic vibration
mode of the AOM crystal driven by an external RF signal with frequency ν. Photons come froma laser beam that passes through the AOM. Denoting the initial momentum of the photon by p~,
we get the final momentum of the photon as p~ + ~k. Since the initial momentum p~ is generally
perpendicular to the phonon momentum ~k, the laser beam deviates from its initial direction after
passing through the AOM. The deviation amplitude depends on the angle between p~ and p~ + ~k andthe magnitude of ~k is related to ν as, |~k| = hν/cs (h is the Planck’s constant and cs is the speed ofsound). Therefore we can change the propagating direction of a laser beam by changing the phonon
frequency ν. In the experiment, we use AOMs that use a longitudinal acoustic mode, which givesa rise time of 150 ns/mm(beam diameter). The AOMs we used are from Gooch & Housego with

68

Figure 5.2: Control of laser beam pointing using an AOM. (a) Deflection angle versus RF frequncy.
The central frequency of the AOM is 80MHz. The deflection angle is linearly proportional to the
RF frequency. The line shows a linear fit with the slope of 0.213 mRad/MHz. (b) The deflection
efficiency of the AOM vs. the RF frequency, using 33 dBm of RF power into the AOM.

the model number (46080-2-1.06-LTD). The AOMs have an aperture diameter of 2 mm and an
operating RF frequency ν tunable from 65 to 95 MHz. The dependence of the deflection angle on
the driving RF frequency is shown in Fig. 5.2(a). Fig. 5.2(b) shows the corresponding variationsof the deflection efficiency on the RF frequency. The deflection angle is determined by measuring
the transverse separation of the deflected and undeflected beams at a distance of 30 cm from theAOM. The deflection efficiency is the measured power in the deflected beam divided by the input
beam power.

5.1.1

A painted trapping potential

To make a painted trapping potential in the x-y plane, we use two AOMs oriented to give
perpendicular deflections to change the trap position in the x-y plane. The driving electronics
for the AOMs are controlled by a computer, which allows us to program an arbitrary trajectoryof the beam position within a period T . Because the period T is very short compared to thetime-averaged trapping period, the atoms feel a time-averaged trapping potential rather than the
instantaneous potential. For a static optical dipole trap, the trapping potential is described by
U (x, y) = U0 exp{−2[(x − x0 )2 + (y − y0 )2 ]/w02 } with the center position at (x0 , y0 ). After turning

69
on the painting, we can express the time-averaged trapping potential generally as


Z
U0 T
(x − xc (t))2 + (y − yc (t))2,
Upaint (x, y) =dt exp −2
T 0
w02

(5.1)

with a time-dependent center position (xc (t), yc (t)).
We want our time-averaged trapping potential to be axially symmetric around the z axis.
In the following discussions, it is convenient to work with the polar coordinates {x, y} → {ρ, φ}.
Thus we denote the center position of the instantaneous trapping potential by {ρc (t), φc (t)}. To
simplify the mathematical derivation, we assign two different time scales, T and T1 , for ρc (t) andφc (t) respectively with the condition of T  T1 . We then treat ρc as fixed when we discuss φc (t) at
the fast time scale. This is a reminiscent of a similar treatment that is often used in the descriptionof the motion of atomic nulei and electrons, known as Born-Oppenheimer approximation. To
maintain the axial symmetry, the trajectory of the polar angle φc (t) needs to follow a circle, i.e.,φc (t) = 2πt/T1 . We then have xc (t) = ρc cos(2πt/T1 ) and yc (t) = ρc sin(2πt/T1 ). By integrating
the fast time scale over the period of T1 , we get the time-averaged trapping potential based on Eq.
(5.1) asU0 −2ρ2 /w02
Upaint (ρ) =
e
T

Z T
dts e

−2ρ2c (ts )/w0204ρρc (ts )
J0 i
,
w02

(5.2)

where the only time-dependent quantity left is ρc (ts ) and J0 is the zeroth order of Bessel’s funcRπ
P∞
k 2
k
2
tion J0 (x) = J0 (−x) = π1 0 eix cos θ dθ =k=0 (−1) (x /4) /(k!) . The above equation is φindependent, as expected for axial symmetry. The geometry of the time-averaged trapping potential is then determined by the specific function form of ρc (ts ). In the following, I give functional
forms of ρc (ts ) for a ring trap, a homogeneous trap, and a harmonic trap.
A ring trap

The simplest functional form of ρc (ts ) = C, where C is a constant, gives aring trap with the time-averaged trapping potential given by
4ρ · C−2(ρ2 +C 2 )/w02
Upaint (ρ) = U0 e
J0 i
.
w02
The resultant trapping potential is shown in Fig. 5.3(a).
A uniform trap

It is impossible to get a uniform trapping potential over all space by

painting. Instead I focus on getting a uniform trapping potential around the center regime as

70

Figure 5.3: Time-averaged trapping potentials. The left is the trajectory of the trap center. The
middle is the calculated potential profile. The right is the measured time-averaged laser intensity.
(a) A ring trap. The functional form for the painted beam is ρc (t) = 2w0 . (b) A uniform trap.
Here, ρmax
is set to be 3w0 . At the center regime with ρ < ρmax
, there is a uniform trappingccmax
potential. (c) A harmonic trap. b is set to be 1 and ρc is set to be 3w0 . The center regime with
ρ < ρmax
shows a harmonic trapping potential.
cshown in Fig. 5.3(b). For this goal, ρc (ts ) = ρmaxcpts /T , which gives

1
Upaint (ρ) = U0
2



w0
ρmax
c

2
,

for ρ < ρmaxand ρmax≥ 3w0 . The analytical derivation of the above equation can be found inccAppend. B. From the above equation, we see that the bigger the painted size ρmaxis, the smallerc
the trap depth is.

71
A harmonic trap

For a harmonic trap, I found thatspb + 1 − (b + 1)2 − 4bts /Tmaxρc (ts ) = ρc ·
,
2b

(5.3)

with the dimensionless parameter 0 ≤ b ≤ 1. This function satisfies ρc (ts = 0) = 0 and ρc (ts =T ) = ρmax
. Eq. (5.3) is equivalent toc"ts = T · (b + 1) ·



ρc
ρmax
c

2


−b·

ρc

4 #

ρmax
c

.

Taking them into Eq. (5.2), we have
Upaint (ρ) =" #U0 w02U0 w02 b 2
w0 2
b
+
1
−
b
−
ρ ,
2(ρmax
)2
ρmax
(ρmax
)4
c
c
c

=

 max 2
 1
1
U0 w02
2
2
−
mω
(ρ
)
−
ω
+ mωρ2 ρ2 ,ρc
ρ
22(ρmax)42c≥ 3w0 . Here the trapping frequency ωρ depends on the b with a relationand ρmaxfor ρ < ρmaxccof 12 mωρ2 = −U0 w02 b/(ρmax)4 . The analytical derivation of the above equation can be found incAppend. B. The time-averaged trap depth is given by" #
U0 w02
w0 2
Udepth =b+1−b.
2(ρmax)2
ρmaxcc
Both the time-averaged trapping frequency and trap depth depend on not only ρmax
but also b.
cFig. 5.3(c) shows a time-averaged harmonic trapping potential.
To perform the painting trajectories discussed above in experiment, we use the electronics
shown in Fig. 5.4 to drive two AOMs. Two AOMs are oriented 90 degree from each other so thatthey can independently control the x and y positions of a laser beam. Each AOM is driven directly
by a voltage-controlled oscillator (VCO), whose output frequency is determined by an externallycontrolled voltage. We can change this external control voltage to change the position of the laser
beam. For getting a ring trap, I used a sine wave signal and a cosine wave signal for the external
control voltages to drive the two VCOs. The two control voltages are obtained from a direct
digital synthesizer (DDS) and share the same time-independent amplitude, which is controlled by
two variable gain amplifiers (VGAs). Based on the ring trap, a uniform trapping potential or a

72

Figure 5.4: Electronics for driving AOMs. A dual-axis AOM consists of two AOMs, whose orientations are perpendicular to each other. The driving frequency of the AOMs is centered at 80
MHz and is provided by two voltage-controlled oscillators (VCOs). A two-channel direct digitalsynthesizer (DDS, AD9959-eval) combined with two variable gain amplifiers (VGAs, AD8330-eval)
are used to control VCOs. In addition, the VGAs are controlled by an arbitrary-wave function
generator for painting ρc (ts ) from Eq. (5.3).

harmonic trapping potential can be obtained by setting a time-dependent amplitude for the sine andcosine wave signals. This time-dependent amplitude needs to follow the corresponding functional
form of ρc (ts ) discussed above.

5.1.2

A good imaging setup

To optimize our apparatus, we need an imaging system to be capable of resolving our BEC
in trap and resolving vortices after expansion. As shown in Fig. 5.5, I used a 1/2 inch diameter
objective lens with a numerical aperature (N.A.) of 0.12. Due to our limited optical access, it ischallenging to increase the N.A. of this lens. Our objective lens has a diffraction-limited resolutionof D = 0.61λ/N.A. = 3.9 µm (Rayleigh criterion). The depth of field is DOF = λ/N.A.2 = 54 µm.
To focus the imaging system, we fixed the position of camera about 400 mm away from a second,“eyepiece” lens with a focal length of f = 400 mm and moved the objective lens closer to or farther
from the atoms using a translation stage. The optimal position of the objective lens was found byusing the lensing effect.
A good introduction to the lensing effect can be found in Matthews’ thesis [91]. A trapped

73

Figure 5.5: A good imaging setup. Due to our limited optical access, I chose an 1/2” diameter
objective lens with N.A.=0.12. The objective lens has a diffraction-limited resolution of D =0.61λ/N.A. = 3.9 µm (Rayleigh criterion). The depth of field is DOF = λ/N.A.2 = 54 µm.

atomic gas is basically an optical medium and behaves like a lens when a laser beam passes throughit. The index of refraction of such an optical material depends on the frequency detuning of thelaser beam from the atomic resonance. For blue-detuned light, the index is smaller than 1 and thus
the trapped atomic gas with higher optical thickness in the center behaves as a concave lens. For
red-detuned light, the index is bigger than 1 and thus the trapped atomic gas with higher optical
thickness in the center behaves as a convex lens. For resonant light, the index equals to 1 and nolensing effect occurs. When we image an atomic gas, an out-of-focus imaging system distorts theimages differently for different detunings. As shown in Fig. 5.6(a), for the blue-detuned case, if the
focal plane of the imaging system is before (location A) the object plane, there is excess light in thecenter captured by the camera, which results in a negative dip in the measured optical depth; if the
focal plane of the imaging system is past (location B) the object plane, there are more light on theedge of the cloud captured by the camera, which results in negative edge in the measured opticaldepth. Similar but reverse lensing effects occur for the red-detuned case (see effects at locations C
and D). The lensing effect provides an accurate way for us to find the position of the object plane.

74

Figure 5.6: The lensing effect when focusing our imaging system. (a) Illustration of the lensing
effect when a laser beam passes through a trapped atomic gas (reproduced from Matthews’ thesis[91]). The optical index of the trapped atomic gas is less than 1 for the blue-detuned laser light andtherefore the trapped atomic gas behaves as a concave lens. On the other hand, the optical index
is bigger than 1 for the red-detuned light and the trapped atomic gas behaves as a convex lens.
(b) Measured lensing effect in our system. The position of the objective lens is measured from anarbitrary fixed point: lower values are farther from the atoms. The cloud size is from a gaussian
fit and increases on both sides because of the bad fittings caused by the saturated OD for B,C andthe dip structure for A,D.

Fig. 5.6(b) shows the lensing effect observed when we were focused our imaging system.
We imaged a dense Rb thermal gas (with the peak optical depth pkOD = 130) in trap whilemoving the objective lens to different locations. The cloud size in the images was obtained by

75

Figure 5.7: Using the lensing effect to focus our imaging system. The crossing point of blue-detuned
and red-detuned lines gives the optimal position.

fitting to a gaussian distribution. From the optical depth images, the lensing effect appears exactly
as we expect. The imaging system was focused by comparing the pkOD for blue-detuning andred-detuning, with the optimal position given by the crossing point (see Fig. 5.7).

5.2

Optimizing the trap roundness
To get a rapidly rotating gas, the trapping potential needs to be axially symmetric. Thatmeans the trapping frequencies along x and y need to be the same. An effective way to measure
the trapping frequencies is to look at the atoms sloshing in the trap. We use a non-destructive
imaging technique, called phase contrast imaging to monitor the slosh of a BEC to measure thetrapping frequencies.

5.2.1

Phase contrast imaging

Phase contrast imaging is a non-destructive imaging technique. It is generally used in theultracold atoms to image a BEC many times before destructing the BEC. We use it to monitor
the slosh of a BEC to measure the trapping frequencies. Right after passing through a BEC, the76

Figure 5.8: Phase contrast imaging setup. The dashed lines represent the scattered light by a BEC.
A phase plate with a 510 nm thick and 45 µm diameter phase dot (black mark) is placed at theback focus of the objective lens. The phase dot causes a π/2 phase shift in probe light relative tothe scattered light.

transverse electric field of the probe beam can be written as E(x, t) = E0 cos(ωt + φ(x)) with x
being the spatial coordinate along the transverse direction and φ(x) being the x dependent phaseshift caused by the BEC. For a small φ(x), the electric field can be written as [Matthews’ thesis[91]]
E(x, t) = E0 cos(ωt) + E0 sin(ωt)φ(x),in which the first term comes from the unscattered light and the second term comes from thelight scattered by the BEC. The probe light intensity, I(x) ∝ |E(x, t)|2 (the bar means the time
average), won’t generate any phase signal. But if we shift the second term by a phase of π/2, we
get E(x, t) = E0 cos(ωt) + E0 cos(ωt)φ(x) and the probe light intensity becomesI(x) = I0 (1 + 2φ(x)),(5.4)

where the higher order term of φ(x)2 is omitted. (The technique for applying this π/2 phase shift is
discussed in the next paragraph.) When imaging atoms in the experiment, we have three frames: ashadow frame that shows the probe beam after passing through the atoms (denoted by IS ), a light

77
frame that shows the probe beam (no atoms) (IL ), and a dark frame that shows only backgroundlight (no probe beam) (ID ). The final signal for phase contrast imaging is obtained using
S(x, y) =IS − ID− 1 = 2 |φ(x, y)| ,IL − ID

which is different from the usual absorption imaging where the optical depth is obtained using,D
OD(x, y) = ln IILS −I−ID . The signal φ is negative for a blue-detuned probe beam and positive for a
red-detuned probe beam [Haljan’s [92]]. The nondestructive property of the phase contrast imaging
can be seen from the relationship, φ ∝ 1/δ and the photon scattering rate Γ ∝ 1/δ 2 with δ being
the detuning. Therefore, at large detuning (δ ∼ 1GHz) with a BEC, the phase shift φ can bedetectable when Γ is negligible, which results in little atom loss in trap due to the scattering ofprobe photons. Additionally, for a large detuning δ  γ, absorption is related to the phase shiftby (see Eq. (4.11) of Matthews’ thesis [91])
OD ∝

γ
φ ,
δ

where γ is the natural line width. There is no detectable absorption at a large detuning. An
example of our phase contrast imaging can be found in Fig. 5.12.
To get a π/2 phase shift only on the scattered light, a piece of glass with a 510 nm thick and45 µm diameter phase dot is put at the back focus of the objective lens as illustrated in Fig. 5.8.
(The phase dot was made by depositing MgF2 on the glass by the JILA machining shop.) This
phase plate gives a π/2 phase difference between the probe light, which is focused on the phase dot
and then the glass, and the scattered light, most of which passes through the glass only. Thus thephase difference is
kn d − k0 d = k0 (n − 1)d = π/2,from which we get the thickness d of the phase dot. Here kn is the wave vector of the light in themedium (kn = 2πn/λ) and k0 is the wave vector of the light in the vacuum (k0 = 2π/λ). The indexof MgF2 is n = 1.38 and λ = 780 nm. At the position of the phase plate, the size of the phase dot
need to be larger than the size of the focused probe beam and much smaller than the size of thelight scattered by the BEC.

78

Figure 5.9: Coarse alignment of the tiny phase dot. Pictures show the diffraction pattern of thephase dot on the probe light without a BEC. At the optimal location (red regime), there should
be no diffraction, which means the whole probe beam is hitting the phase dot. A larger value of z
means a position closer to the objective lens.

For our case, the Thomas-Fermi radius of our BEC is about 16 µm (for 3 × 105 87 Rb atoms
in a harmonic trap with trapping frequencies 40 × 40 × 183 Hz) and the (1/e2 ) diameter of theprobe beam is 2 mm before passing the objective lens. Using the gaussian beam formulas, w(z) =p
w0 1 + (z/zR )2 and zR = πw02 /λ, we estimate the (1/e2 ) diameter of the scattered beam is 1.4 mm
and the (1/e2 ) diameter of the probe beam is 20 µm at the position of the phase plate. Therefore,a phase dot with diameter of 45µm is chosen to satisfy the condition, 20µm < 45µm  1.4mm. Toalign such a tiny spot, we do a coarse alignment without a BEC. We used a three-axis translation
stage to control all {x, y, z} positions of the phase plate. A beam profile camera (S-BC-XHR) was
put ∼ 10 cm after the focus point and before the “eyepiece” lens to monitor the diffraction patternof the phase dot on the probe beam as shown in Fig. 5.9. First, we put the phase plate very close
to the objective lens to align {x, y} positions by looking at the diffraction. Here the probe beam
has a large diameter, which makes the alignment easier. Next, we moved the phase plate along z
towards the focus point step by step and the {x, y} positions were corrected at every step. Finally,

79

Figure 5.10: Stabilizing the time-averaged trapping potential. (a) Setup of the feedback system.
The x and y positions of the trapping beam are monitored by a quadrant photodetector (SPOT9DMI) and are stabilized by driving the AOMs using a feedback loop. A 100Hz low-pass filter isused in the feedback loop to avoid affecting the 1kHz painting trajectory. (b) Position stability
with the servo system on and off measured by a beam profile camera. The center position of thetime-averaged beam is recorded as a dot after every time step. (c) The error spectrum with theservo system on and off measured by a SRS FFT spectrum analyzer.

we found the best location without any diffraction, which means the whole probe beam is hitting
the phase dot. After this, a fine alignment can be done by maximizing the phase signal of a BEC,
which is straightforward.

5.2.2

Inducing slosh of a BEC

To measure the roundness of the trap, we measure the trapping frequencies by exciting acenter-of-mass oscillation, or slosh, of a BEC in the x-y plane. One way of inducing slosh is to
apply a magnetic-field gradient to kick the atom cloud. However, we tend to work with a relatively
large bias magnetic field in the optical trap, which makes it difficult to induce significant slosh
amplitude along both x and y using additional small magnetic-field coils. Instead, we can induce
slosh by jumping the center position of the optical trapping potential. In fact, in our setup we
have a feedback loop to reduce the low-frequency drift (< 100Hz) of the center position of thetime-averaged trapping potential (see Fig. 5.10 (a)). Fig. 5.10 (b) and (c) show that the positionof the trapping potential can be well stabilized by the feedback loop. This also give us the freedomof jumping the positions of the optical trapping potential, simply by changing the control voltages

80

Figure 5.11: Lab frame and trap frame. x0 is defined as the minor axis of the trapping potential
and it corresponds to a higher trapping frequency f+ . y 0 is defined as the major axis of the trapping
potential and it corresponds to a lower trapping frequency f− .

that go into the servo electronics, to induce slosh of a BEC.

5.2.3

Rounding out the trap

To get a rotating gas, the trapping potential needs to be axially symmetric. That means thetrapping frequencies along x and y need to be the same. In this section, I discuss how to adjust
the painting parameters to make our trap as round as possible. Let’s consider with an elliptical 2D
trapping potential with principal axes denoted by {x0 , y 0 } that have an angle of θ from the imaging
frame {x, y} (see Fig. 5.11). The center-of-mass motion of a BEC in this 2D trap can be treated as
two independent 1D oscillations along the principal axes of the trapping potential. We can write
the center trajectory as x00 (t) = Amin e−t/τ cos(2πf+ t) and y00 (t) = Amaj e−t/τ sin(2πf− t + φ0 ) withf+ and f− being the trapping frequencies along x0 and y 0 respectively. Here the damping termis due to roughness of the time-averaged trapping potential. Usinng phase contrast imaging, wemeasure the BEC trajectory in the lab frame, or imaging frame, where the motion is connectedwith {x00 (t), y00 (t)} through the frame transformation

 


0
 x0 (t)   cos θ − sin θ   x0 (t) 

=

.
0
y0 (t)sin θ cos θy0 (t)(5.5)

To measure the trapping frequencies along both x0 and y 0 , a circular slosh trajectory is

81

Figure 5.12: BEC slosh measurement for rounding out the trap. The probe beam is 600MHz
red-detuned for phase contrast imaging. IS , IL and ID are the shadow frame, light frame andDdark frame, respectively. The final signal is obtained using S = IILS −I−ID − 1 . {x0 , y0 } is the center
position of the BEC cloud obtained by fitting the image(s) to a 2D gaussian. The lines throughthe data (points) show a fit to Eq. (5.5).

preferred. In the experiment, we first kicked the x position of the time-averaged trapping potential
and a quarter trapping period later, the y position was kicked. (To kick the x or y position, we
jump the center position of the trap and then immediately jump it back.) Limited by the camera
CCD size (1024×1024 pixels), we can take 18 pictures for each readout of the CCD camera usinga kinetics mode to shift the exposed pixels between pictures. As shown in Fig. 5.12, the trapping
frequencies {f+ , f− } are obtained by fitting the slosh data to Eq. (5.5). To describe the roundnessof the trapping potential, we define the ellipticity as εtrap = (f+2 − f−2 )/(f+2 + f−2 ) and the averaged
radial trapping frequency as fρ = (f+ + f− )/2 [Coddington’s thesis [93]]. A difference between f+
and f− causes a beat in the slosh with a beat frequency fb = f+ − f− .
As discussed in Sec. 5.1.1, to get a time-averaged harmonic trapping potential, we use apainting trajectory described byxc (t) = Aρc (t) cos(2πt/T1 + ϕ),yc (t) = Bρc (t) sin(2πt/T1 ),

(5.6)

82
where ρc (t) follows Eq. (5.3) with the period T . Note that xc is controlled by the vertical AOM andyc by the horizontal AOM (see Fig. 5.1). In our experiment, we chose 1/T1 = 100kHz, 1/T = 1kHz,
and b = 0.9. Parameters {A, B, ϕ} can be controlled independently in the experiment. Ideally,
they should satisfy A = B and ϕ = 0. In practice, due to the optical aberrations and imperfectalignment, the final painted trajectory can be different from the programmed function. This is why
we need to measure the slosh of a BEC to optimize the experiment input parameters {A, B, ϕ}.
In the experiment, ϕ is controlled by the DDS and {A, B} could be changed by either the VGAs
or the DDS (see Fig. 5.4). We chose to use the VGAs to round out the trap and the DDS wasreserved for the stirring process discussed in Sec. 5.5.
From the slosh data, we can measure {f+ , f− , θ} that describe the harmonic trapping potential. To connect such information to the painting trajectory in the lab frame, we can write anequipotential contour as
f+2 x02 + f−2 y 02





2
0
f
0
x
 +




x0 y 0 
0 f−2
y0



 


2
cos
θ
−
sin
θ
f
0
cos
θ
sin
θ

 +

 x 


 
x y 
sin θ cos θ
0 f−2− sin θ cos θ
y

 


2 cos2 θ + f 2 sin2 θ (f 2 − f 2 ) sin θ cos θ
f
+
−
−
 +
 x  
x y (f+2 − f−2 ) sin θ cos θ f+2 sin2 θ + f−2 cos2 θy
f+2 cos2 θ + f−2 sin2 θ x2 + f+2 sin2 θ + f−2 cos2 θ y 2 + [(f+2 − f−2 ) sin 2θ]xy= const= const= const= const= const. (5.7)

On the other hand, due to the optical aberrations and imperfect alignment, the final painted
trajectory can be different from the programmed function (see Eq. (5.6)). An effective painted
trajectory in the lab frame can be generally expressed as x̃c (t) = Ãρc (t) cos(2πt/T1 + ϕ̃) andỹc (t) = B̃ρc (t) sin(2πt/T1 ). To obtain values of {Ã, B̃, ϕ̃} from the measured {f+ , f− , θ}, we write
the effective painted trajectory into the following formB̃ 2 x̃2c + Ã2 ỹc2 +2ÃB̃x̃c · ỹc = const.
sin ϕ̃

(5.8)

83

Figure 5.13: Ellipticity of the trapping potential versus the painting trajectory. Curves are guides
to the eye. The x axes show the voltage going to the VGAs that we used for controlling A and B.
At ϕ = 5 deg, we get the minimum ellipticity.

84
By comparing Eqs. (5.7) and (5.8), we get the following relationships
f+2 sin2 θ + f−2 cos2 θÃ/B̃ =,
f+2 cos2 θ + f−2 sin2 θq2(f+2 −f−2 ) sin2 θ cos2 θ+f+2 f−2
if θ 6= 0, ±π/2, ±π, · · ·(f+2 −f−2 ) sin θ cos θϕ̃ ≈ sin ϕ̃ =
,



0
if θ = 0, ±π/2, ±π, · · ·(5.9)

which is valid when f+ 6= f− . Eq. (5.9) is useful in the sense that it tells us how we need to change
A, B and ϕ to reduce the ellipticity, namely, {A/B → A/B · (B̃/Ã), ϕ → ϕ − ϕ̃}.
Figure 5.13 shows some typical data for rounding out the trap. We need to optimize theellipticity in a 2D parameter space {A/B, ϕ}. Only when both A/B and ϕ are optimized, we findthat we can get the ellipticity below 1%. This measured minimal ellipticity may be limited by themeasurement time. When close to the optimal settings, the value of θ becomes very sensitive tothe value of A/B and θ jumps by 90 deg as A/B crosses the optimal value.

5.3

Anharmonicity
In addition to imperfect roundness of the trapping potential, anharmonicity and roughness

can also limit the rotating rate of our ultracold gases. In this section, I discuss the anharmonicity
for our time-averaged trapping potential. To discuss an anharmonic oscillation, it is good to startwith the Lagrangian [94]11
1
1
L = mẋ2 − mω02 x2 − mαx3 − mβx4 ,2234that includes x3 and x4 anharmonic terms with correspondent weights α and β. The trapping
potential is then111
U (x) = mω02 x2 + mαx3 + mβx4 .
234The corresponding equation of motion isẍ + ω02 x = −αx2 − βx3 .

85The solution of the above equation has been given in Ref. [94] and is given as follows
x(t) = x(1) + x(2) + x(3) ,ω = ω0 + ω (1) + ω (2) ,(5.10)

withx(1) = A cos ωt,αA2 αA2+
cos 2ωt,
2ω02
6ω02
 2

A3
α
1
(3)
x =
− β cos 3ωt.
16ω02 3ω02 2
x(2) = −ω (1) = 0


3β
5α2
(2)
A2
ω =
−
8ω0 12ω03

There are two features about this solution. The first is that the slosh frequency ω depends on theslosh amplitude A. The second is that the odd anharmonic term, x3 , can cause an offset shift ofthe slosh and this shift depends on the slosh amplitude.

Figure 5.14: Effect of the anharmonic trapping potential on slosh. Red and blue points aremeasured separately using 1D slosh. Along y, we see a clear amplitude dependence of ω(A) =ω0 − 0.031(Hz/px2 )A2 (blue line), while the slosh offset is basically unchanged. This indicatesa dominant x4 term along y. The offset along x is not shown because of the relatively small
anharmonicity. (1 px=1.38 µm).

Fig. 5.14 shows our measurement of anharmonicity of our time-averaged trapping potential.
In order to tell which of the anharmonic terms, x3 or x4 , is dominant, I consider the case where
the dominant anharmonicity is from the x3 term. In this case, by fitting our trapping frequency

86
2

5α
2
−1 s−2 , and therefore
data to the equation, ω(A) = ω0 − 12ω
3 A , we get the weight, α = ±973 px
0

the displacement, |∆x(A)| =

αA2
2ω02

= 0.009A2 , which does not match our slosh offset data in Fig.

5.14. This suggests that the x4 term is the dominant anharmonic term. Fitting our trapping
3β 2
A , we get the weight, β = −19.44 px−2 s−2 , andfrequency data to the equation, ω(A) = ω0 + 8ω0

the trapping frequency, ω0 = 2π × 37Hz. It further gives x(3) = 11.2 × 10−6 A3 cos 3ωt, which is sosmall compared to x(1) = A cos ωt that it won’t have a significant effect on our slosh data. The
fact that there is a bigger anharmonic effect on slosh along y than that along x suggests that theanharmonic source comes from the asymmetric part of the trapping potential, namely, the sheet
beam (see next section).

5.4

Roughness
In this section, I discuss the roughness of our optical trapping potential. Roughness refers

to high spatial frequency (short wavelength) defects in the trapping potential. We suspect thatthe sheet beam might have the dominant roughness. The setup of the sheet beam is shown in
Fig. 5.15(a). We want the sheet beam to provide the vertical confinement. Hence, its beamprofile was designed to have a small beam waist (30µm) along the vertical direction z and a large
beam waist along y as shown in Fig.5.15(b). To reduce the roughness and to further flatten thehorizontal confinement, we use an AOM to paint the sheet beam along y at 300 kHz; the resulting
time-averaged laser intensity is shown in Fig. 5.15(c).
One way to probe the roughness of the sheet beam is to expand the Rb BEC in the sheet beamby turning off the vertical painted trapping beam while keeping the sheet beam on. After sometime-of-flight (TOF), the density distribution of the expanded BEC is imaged using absorption
imaging. Fig. 5.16(a) shows the BEC distribution after expanding in the sheet beam for different
TOFs. The roughness of the trap results in parallel stripes seen at large expansion times (> 15 ms).
The orientation of these stripes is coincident with the propagating direction of the sheet beam.
To emphasize the effect of the trap roughness, we lowered the chemical potential of the BEC
to be closer to the quasi-2D regime, where the BEC density is more strongly affected by roughness

87

Figure 5.15: Sheet beam information. (a)Sheet beam configuration. The sheet beam enters thescience cell (cyan regime) with an incident angle of 10.6 deg, which was chosen to avoid overlappingwith the reflected beams. Roughness could be caused by imperfection in the glass wall of the cell.
(b) Static sheet beam. The vertical (1/e2 ) beam waist is 30µm and the horizontal waist is 700µm.
(c) “Painted” sheet beam. The central regime is flat as we expect.

in the trap potential. In the experiment, this was done by reducing the BEC number (from ∼ 105
to ∼ 104 ) as well as decreasing the radial trapping frequency (from 2π × 40 Hz to ∼ 2π × 2 Hz)
by lowering the vertical beam power. Fig. 5.16(b-d) shows the detected BEC density distributions
that reveal the trap roughness. In Fig. 5.16(b) and (c), the location of the stripes was monitored
when moving the vertical trap beam along y and x respectively. It turns out that the stripes stayat the same location independent of the vertical beam position. In addition, the stripes exist evenwhen the vertical beam power is weak. All these evidences suggest that the stripes are comingfrom the sheet beam. In Fig. 5.16(d), we see that the location of the stripes was unchanged whenmoving the sheet beam along y. This is consistent with the fact that our painting in y does notremove the roughness. A likely situation is that the cell glass has some fine structure that results

88

Figure 5.16: Detecting the roughness of the sheet beam using the density distribution of a BEC.
(a) BEC density distribution after expanding in the sheet beam for different TOFs. (b) Monitoring
the locations of stripes when moving the vertical beam along y. The green lines provide a reference
for ease of comparing the location of structure in the density profile. (c) Monitoring the locationsof stripes when moving the vertical beam along x. (d) Monitoring the locations of stripes whenmoving the sheet beam along y. The size of stripes was measured to be 16µm.

in shadows in the sheet beam. Because the cell wall is only ∼ 5 mm from the atoms, motion of
the sheet beam in y (with painting or by moving the center of the beam) is primarily translation(instead of a change in angle), which does not change the location of the shadow. We estimated

89
the roughness in Fig. 5.16(d) to be 15 nK by measuring the chemical potential of the BEC andestimated the trap depth to be 3 µK, which give a fractional roughness of the trapping beam of
0.5%. This roughness is suspected to limit the rotation rate (Ω/ω⊥ = 0.7) that we achieved so far(see Sec. 5.7), which is not enough for reaching the quantum Hall regime (Ω/ω⊥ > 0.9).
Based on the above analysis,we have designed a new sheet beam. The new beam has a similar
beam profile but with a much larger range of painting, which gives a significant angle change of thebeam at the cell that should wash out the roughness caused by the cell glass. Meanwhile, in thefollowing sections, I present rotation results using the existing setup.

5.5

Rotating a Rb BEC

Figure 5.17: Rotating a Rb BEC. The sheet beam power was lowered to evaporate Rb atoms to
get a BEC of about 105 atom; To stir the BEC, we changed the vertical time-averaged trapping
potential to be elliptical and then rotated the elliptical trap for 1 s. After stirring, the trap geometry
was restored to a round shape and the cloud was imaged after a 500ms thermalization time. The
picture on the left comes from the top imaging system introduced in Sec. 5.1.2, while the pictureon the right is from a side imaging system that has a magnification of 3 (not shown in Fig. 5.1).

90After the above optimizations, we tried rotating a Rb BEC and looked for BEC vortices forthe first time in my lab. An experimental timing diagram of the procedure is illustrated in Fig.
5.17. We started with a thermal Rb cloud around 2 µK. To get a BEC, the sheet beam power
was lowered to evaporate Rb atoms and achieve a BEC of about 105 atoms. To rotate the BEC,
we programmed the DDS to change the vertical time-averaged trapping potential to be ellipticaland rotated the elliptical trap for 1 s. After this stirring, the trap geometry was restored back toa round shape and we waited 500 ms for thermalization before taking an absorption imaging. To
be able to resolve the vortex structure, we expanded the BEC for 15ms before imaging. As shownin Fig. 5.17, vortices were detected, which is proof that we created a rotating BEC. The observed
small structure is consistent with the diffraction limited resolution of 3.9 µm. This experimentdemonstrates that our imaging system is capable of resolving vortex structure in a rotating BEC,
which is crucial for a rotating BEC experiment. However, the rotation rate achieved by this stirring
process is far from rapid rotation as indicated by the low vortex number. Next, we wanted to testout a spin-up evaporation technique to speed up the rotation.

5.6

Spin-up evaporation technique
A spin-up evaporation technique has been successfully demonstrated in a magnetic trap[75]. It consists of an evaporation process that can increase the average rotation rate per atom
by evaporating away atoms not just with high energy but also with low angular momentum. We
wanted to extend this technique to an optical trapping potential. The basic idea is based on
the angular-momentum or rotation-rate distribution of atoms in a harmonic trap. For an atomicgas rotating around the z axis, assuming rigid-body rotation, the angular momentum, Lz = mρ2 Ω,
indicates that the atoms with larger ρ have higher angular momentum as shown in Fig. 5.18. In theexperiment, when we evaporate the gas by lowering down the sheet beam power, the atoms aroundthe −z edge (indicated by the dashed line) are removed due to gravity. Most of these removed
atoms carry relatively low angular momentum. As a result, the average angular momentum per
atom increases.

91

Figure 5.18: The idea of spin-up evaporation. When we perform evaporation by lowering the sheet
beam power, atoms near the −z edge (indicated by the dashed line) are removed due to gravity.

5.6.1

Monte-Carlo simulation

To further explore this strategy, we did a numerical simulation of spin-up evaporation in the
optical trap. The simulation procedure is described in Fig. 5.19. At t = 0, we want to generate

Figure 5.19: Procedures of the Monte-Carlo simulation.

N = 9 × 103 atoms that were randomly distributed following a Maxwell-Boltzmann distribution in

92
a trapping potential U (~r) with an initial rotation rate Ω = 0.4ωρ and temperature T = 200 nK. (ωρis the radial trapping frequency.) To achieve this, we first generate N atoms without rotation in atrapping potential U (~r) − 21 mΩ2 ρ2 with the temperature T (see details of the random distribution
algorithm in Append. A of Cumby’s thesis [22]). The position and velocity, {~ri , ~vi }, of each particle
is recorded. Based on the fact that atoms moving in a potential U (~r) − 12 mΩ2 ρ2 in a rotating frame(with the rotation rate Ω) are identical to atoms that rotate at the rate Ω and feel a potential U (~r)
in the lab frame, the initial rotation rate of Ω was imparted to the atoms by a transformation of
coordinates xi = xi yi = yi, v x = v x − Ωyiii,(5.11)

 v y = v y + Ωxi
iiwhere ~ri = {xi , yi , zi } and ~vi = {vix , viy , viz } are used. Thus, we prepared N atoms that wererandomly distributed following a Maxwell-Boltzmann distribution in a trapping potential U (~r)
with an initial rotation rate Ω and temperature T . In our simulation, the trapping potential U (~r)
was described by


2P
2y 2
2z 2
1
2 2
2
exp −
−
+ mgz,
U (~r) = mωρ (x + y ) + α
2πwy (x)wz (x)wy (x)2 wz (x)2
where the first term is a round harmonic trap along the radial direction, the second term describes
the sheet beam with the beam power P , and the third term is due to gravity. The polarizabilitypα = −1.9 × 10−36 m2 s. wy (x) = wy0 1 + (x/xRy )2 with wy0 being the 1/e2 beam width along ypand the Rayleigh range xRy = π(wy0 )2 /λt . wz (x) = wz0 1 + (x/xRz )2 with wz0 being the 1/e2 beamwidth along z and the Rayleigh range xRz = π(wz0 )2 /λt . (λt = 1090 nm). For this simulation, I set
wz0 = 7 µm and wy0 = 300 µm.
The time evolution of the atoms during an evaporation process was simulated numericallyfor our optical trapping potential using the Monte-Carlo method described in [95, 96]. The boundary condition is determined by the border between the trapping and anti-trapping regimes (it isapproximately a plane at the −z edge). The collision includes only the elastic collision based ona hard core model [60]. Fig. 5.20(a) shows the numerically generated atoms (dots) in our optical

93

Figure 5.20: Monte-Carlo simulation of the spin-up evaporation. (a) Atom distribution in our
optical trapping potential. (b) The sheet beam power is exponentially lowered to evaporate theatoms. (c) & (d) The simulation results. The rotation rate increases significantly through thespin-up evaporation. The kinetic energy is lowered as expected.

trapping potential U (~r) at t = 0. The evaporation was done by lowering the sheet beam power,
which lowered the potential barrier on the −z edge due to gravity (see the lower panel of 5.20(a)).
The trajectory, {~ri , ~vi } of each atom was recorded after every time step. From the trajectory, we
can calculate the kinetic energy (K.E.), angular momentum Lz , inertia Iz , and rotation rate by
N=

X

1,

K.E. =i

Lz = mX1
i

X(xi viy − yi vix ),iIz = m2

mri2 /N,X(x2i + yi2 ),iΩ = Lz /Iz .
Fig. 5.20(c) & (d) show our simulation results. The evaporation trajectory is illustrated in94

Figure 5.21: Experimental sequence of the spin-up evaporation. The stirring process was performed
after we started evaporating but before a BEC formed. After that, we continued evaporating. The
voltages shown above are proportional to the sheet beam power. When evaporating to 0.85V, theimaged atom density shows that all atoms were in the thermal or uncondensed state. Once weevaporated to 0.8V, a BEC formed with vortices due to rotation. As we evaporated further, theBEC got larger. Meantime, more vortices showed up, which indicated a faster rotation rate. The
images were taken after a TOF=15ms expansion.

Fig. 5.20(b) and it can also be seen in the atom number decrease as a function of time and thecorresponding decrease in kinetic energy. At the same time, the rotation rate increases significantly.
Note that the kinetic energy along x is higher than that along z due to the rotation.

5.6.2

Experimental verification

In the experiment, spin-up evaporation can be realized simply by following the timing sequence shown in Fig. 5.17, but performing the stirring process earlier. As shown in Fig. 5.21,
we performed the stirring process after we started evaporating but before a BEC formed. Then
we continued evaporating until a BEC formed. As you can see from Fig. 5.21, vortices appeared
as soon as a BEC formed. As we evaporated further, the BEC number grew and more vortices
appeared, which indicates a faster rotation rate. This behavior demonstrates that the spin-up

95
evaporation works for our situation.

5.7

Rotation rate
Vortex lattice.

In a harmonic trapping potential, vortices in a low-temperature BEC forma triangular lattice (or “Abrikosov” lattice). However, imperfection of the trapping potential cancause partial melting of the vortex lattice, dislocations, and grain boundaries [71]. An example ofnear-perfect vortex lattice can be found in [97]. In our situation, shown in Fig. 5.22(a-b), we seesome dislocations in the vortex lattice, which we attribute to roughness of our trapping potential.
If we evaporate further, the density of vortices does not increase. Instead the shape of the whole
BEC deforms to be somewhat triangular, which indicates the excitation of surface modes; this canbe seen in Fig. 5.22 (c-d).

Figure 5.22: Vortex lattice of a rotating BEC. (a-b) Example images of the fastest rotating BEC
we have achieved. The vortex lattice is not perfect due to the roughness of our trapping potential.
(c-d) BEC vortex lattice for deeper evaporation. The vortex density didn’t increase. Instead theshape of the whole BEC deformed to be triangular, which indicates the excitation of surface modes.
The BEC was expanded in the sheet beam for 3 ms and then expanded in the free space for 15 ms
before the absorption imaging.

Rotation rate.

An effective way to measure the rotation rate is to use the aspect ratio of

the atom cloud (the axial size over the radial size) imaged in-situ [75]Ω/ωρ =p
1 − (λ/λ0 )2 ,(5.12)

where λ0 and λ are the initial and final aspect ratios of the cloud respectively. Here, the centrifugal
force increases the transverse cloud size compared to the axial size. However, because the BEC size

96
along z (Thomas-Fermi radius RTz F ∼ 3 µm) is below our imaging resolution, it is difficult for us tomeasure this aspect ratio accurately. Instead, we can estimate the rotation rate from the measured
aspect ratio of an expanded cloud. For this we need a model of BEC expansion [98, 99]. From the
expanded BEC cloud, we estimate the rotation rate of the BEC in Fig. 5.22(a-b) to be Ω/ωρ ≈ 0.7.
An alternative way to extract the rotation rate is to use the vortex density in-situ [15]nv = Nv /A = M Ω/π~
where Nv is the vortex number in the area A and M is the atomic mass of an atom. Since we countvortices on an expanded rotating BEC, to get the vortex density in-situ, we also need the model of
BEC expansion [98, 99].

5.8

Future directions
In this thesis, I presented our work developing and optimizing our apparatus by rotating a

87 Rb BEC. In future experiments, we would like to rotate a different atomic species such as bosonic
39 K, which has a convenient Feshbach resonance for tuning the interaction strength, or fermionic
40 K. Since our trap is all-optical, the techniques we have demonstrated should be readily applicableto these other species.

Chapter 6

Conclusions and outlook

In this Thesis, I explored some interesting many-body physics in an ultracold gas mixture of
87 Rb and 40 K atoms.

The first one is a study of Bose polarons as discussed in Chap. 3. I report experimental
evidence of Bose polarons by using radio-frequency spectroscopy to measure the excitation spectrumof fermionic 40 K impurities resonantly interacting with a BEC of 87 Rb atoms. We find that themeasured energy of the impurites immersed in the BEC matches well with that calculated for Bose
polarons for a range of interaction strengths. In particular, we confirm that the polaron state
exists even across unitarity in the strongly interacting regime and approaches the weakly boundmolecular state. Furthermore, for attractive interactions, we show that Rabi oscillations can beobserved when exciting Bose polarons. This work is the first measurement of the Bose polaron in athree-dimensional trapped atom gas, which helps better understanding of the quantum many-body
physics of Bose-Fermi mixtures.
The second topic is related to the quantum Hall effects as discussed in Chap. 4 and 5.
The interesting and bizarre quantum Hall effects are reviewed and their connection with a rapidly
rotating ultracold gas is discussed. To realize these quantum Hall states by using ultracold atoms,
we design and construct a new apparatus for creating rotating ultracold atomic gases. Technicalchallenges are explored and experimental progress is discussed.
Besides studying quantum Hall physics, such a versatile apparatus is also promising to explore
other interesting physics such as rotating a strongly interacting BEC and the effect of dimensionality

98
on the BEC/BKT transition.
Although rotating a BEC has been studied for more than a decade, as far as I know no one
has studied a rotating BEC with tunable interactions. With tunable interactions via the Feshbachresonance, the 39 K is the best candidate for us to study BEC vortices with tunable interactions.
Another interesting project that we are probably able to study by our unique apparatus is theeffect of dimensionality on the BEC/BKT transition. As you know, the BKT is a phase transition
from a well-ordered BEC superfluid to a partially-ordered BKT superfluid when changing dimensionality from 3D to 2D. As demonstrated in Sec. 5.4, we are able to control the dimensionality
from 3D to 2D through our unique apparatus. Thus it is possible for us to map out a phase diagramof the BEC/BKT transition.

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56:1424, 1997.

Appendix A

RF spectroscopy of non-interacting atoms

A RF photon has frequency ν0 typically around 80 MHz and has a negligible momentum. The
RF spectroscopy therefore induces momentum-conserving transitions. Let H0 be the noninteracting
Hamiltonian of K atoms and Hrf (t) be the time-dependent interaction between RF radiation andK atoms. we have
H = H0 + Hrf ,
i
Xh
†
†
H0 =
f (p)f1,p
f1,p + (f (p) + hν0 )f0,p
f0,p ,(A.1)

p

Hrf (t) = ~E(t)

X

†
(f1,k
f0,k + H.c.),k†where fσ,p (or fσ,p) is the annihilation (or creation) operator of an impurity fermion in the hyperfine

state σ with a momentum p. For a gaussian RF pulse during t ∈ [0, 4∆t], it is described by
E(t) = dE2 0 exp(−(t−2∆t)2 /∆t2 )(e−i2πν0 t +ei2πν0 t ), which corresponds to an electric field E(t) =E0 exp(−(t − 2∆t)2 /∆t2 ) cos(2πν0 t). Here d is the induced electrical dipole moment of atoms. The
fast-time-averaged RF power is P̄ (t) = 02c E02 exp(−2(t − 2∆t)2 /∆t2 ).
Let the initial state |Ii = |0, k = 0i and final state |F i = |1, ki. The time-dependent
Schrödinger equation for the RF spectroscopy isi~∂t |ψk=0 (t)i = [H0 + Hrf (t)]|ψk=0 (t)i.
To solve the above equation, I write a single-body wavefunction with zero momentum as|ψk=0 (t)i = C0 (t)e−i(f (0)+hν0 )t/~ |0, k = 0i + C1 (t)e−if (0)t/~ |1, k = 0i

106with the initial condition of C0 (0) = 1 and C1 (0) = 0. Solving the above differential equation,
we have C1 (t) = −i sin [Ω0 ∆t(1 + Erf ((t − 2∆t)/∆t))]. During the above derivation, the rotating
wave approximation has been used. Here the Rabi frequency Ω0 = dE0 . For our RF spectroscopy,
the gaussian pulse duration 4∆t is the same as t, which means C1 (t) = −i sin(Ω0 t/2). The fractional
population of the spin-flipped atoms in the final state |F (k)i is thusΩ0PF = |hF (k)|ψk=0 (t)i|2 = δ(k) sin2
t ,
2

(A.2)

which gives the Rabi oscillation with the oscillation frequency Ω0 . Here δ(k) is the Dirac delta
function.
In the Schrödinger picture, Eq. (A.1) can be written into a time-independent formĤrf (Ω0 ) = ~Ω0 X †(f1,k f0,k + H.c.),
2

(A.3)

k

with the corresponding time-dependent wavefunction at the momentum k being|ψk (t)i = C0 (t)|0, ki + C1 (t)|1, ki.
The Schrödinger equation is thus, i~∂t |ψk (t)i = Ĥrf (Ω0 )ψk (t)i, which gives the solution of C1 (t) =−i sin(Ω0 t/2). The Radi frequency has a relation
Ω20 =

2
4
h1, k|Ĥrf (Ω0 )|0, ki ,
2
~

(A.4)

which is useful when discussing the residue of a quasiparticle (see Sec. 3.4).
To get a lineshape for the gaussian pulse in the frequency domain, we need to do the Fouriertransformation of E(t), which has
Z ∞22Ẽ(2πν) =dt ei2πνt E(t) = 2~∆tΩ0 ei4ν/δν e−(ν−ν0 ) /δν ,

(A.5)

−∞

with 1/e width δν = 1/(π∆t). Therefore, the lineshape is R(ν − ν0 ) ∝ |Ẽ(2πν)|2 ∝ exp[−2(ν −ν0 )2 /δν 2 ]. After the RF pulse, the number of spin flipped K atoms in |1i state at the central regime(ρ < ρ0 ) can be modeled byZ ρ0
Z ∞NK,1 (ν, ρ ≤ ρ0 ) = Aρdρdz nK,0 (ρ, z)R(ν − ν0 ) = NK,0 (ν, ρ ≤ ρ0 )R(ν − ν0 ),0−∞where A is a constant and nK,0 (ρ, z) is the density distribution of the K atoms in |0i state.

(A.6)

Appendix B

Time-averaged trapping potentialspmod(ts , T )/T withp
T being the period. Let’s consider the range of ts ∈ [0, T ]. we have ρc (ts ) = ρmax
ts /T andc
An uniform trap

I choose the functional form of ρc (ts ) = ρmaxc

ts = T · [ρc (ts )/ρmax]2 . The Eq. (5.2) is thuscUpaint (ρ) =
=
=
=
=

=
=

Z T



4ρ · ρc (ts )
J0 i
dts e
w02
0


Z ρmaxc
U0 −2ρ2 /w02 2T
4ρ · ρc−2ρ2c /w02
J0 i
ρc dρc e
e
T
w02
(ρmax
)2 0
c
 2k
∞ Z max
U0 −2ρ2 /w02 2T X ρc2ρ−2ρ2c /w02 2k+1
e
dρc e
ρc
/(k!)222
maxTw(ρc ) k=0 0
0


Z
∞
2k
∞
U0 −2ρ2 /w02 2T X 2ρ22dρc e−2ρc /w0 ρ2k+1e/(k!)2
c
2
2
max
T
w
(ρc ) k=000∞ 
U0 −2ρ2 /w02 2T X 2ρ 2k 2 k+1e(w0 /2) /2/k!
T(ρmax)2 k=0 w02ck∞ 
U0 −2ρ2 /w02 2T X 2ρ2e(w02 /4)/k!
2
2
T
w
(ρmax
)
0
c
k=0

2
1
w0
U0
2
ρmax
c
U0 −2ρ2 /w02eT−2ρ2c (ts )/w02

R ∞ 2k+1 −x2 /a2
P
k 2
k
2
during which I have used the formula J0 (x) = ∞edx =k=0 (−1) (x /4) /(k!) and 0 x




2k
R ρmaxR∞2
22ρ
k! 2k+22−2ρ2c /w02 {· · · }. The approximation of 0 c dρc e−2ρc /w0 ρ2k+1
/(k!)
→
2
c
2a
0 dρc ew
0holds to be true when ρmax
≥ 3w0 (verified by a numerical calculation).
c
A harmonic trap

I choose a more complex functional form ofspb + 1 − (b + 1)2 − 4b · mod(ts , T )/Tmaxρc (ts ) = ρc ·
,
2b

108
with the unitless parameter 0 ≤ b ≤ 1. This function satisfies ρc (ts = 0) = 0 and ρc (ts = T ) = ρmax.
c
The above equation is equivalent to"
ts = T · (b + 1) ·



ρc
ρmax
c

2


−b·

ρc

4 #

ρmax
c

.

For the convenience of discussions, let me rewrite the above equation as ts = Aρ2c − Bρ4c with
A = T · (b + 1)/(ρmax)2 and B = T · b/(ρmax)4 . The Eq. (5.2) is thusccUpaint (ρ)
=
=
=
=

=

Z T



4ρ · ρc (ts )
dts e
J0 i
w02
0


 Z ρmaxZ ρmaxcc
4ρ · ρc4ρ · ρc
U0 −2ρ2 /w02−2ρ2c /w023−2ρ2c /w02
J0 i
J0 i
e
2A
ρc dρc e− 4Bρc dρc e
T
w02
w02
0
0




Z
Z
max
max
∞
ρc
ρc
2ρ 2k
U0 −2ρ2 /w02 X22−2ρ2c /w02 2k+3e
2Adρc e−2ρc /w0 ρ2k+1
−
4B
dρ
e
ρ
/(k!)2
c
c
c
2
T
w
0
0
0
k=0




Z
Z
∞
2k
∞
∞
2ρ
U0 −2ρ2 /w02 X
2
2
2
2
− 4B
dρc e−2ρc /w0 ρ2k+3
2Adρc e−2ρc /w0 ρ2k+1e/(k!)2
c
c
2
T
w
0
0
0
k=0
"  k+1
# 


∞
k+2
2k
w02
k!
w02
(k + 1)!
U0 −2ρ2 /w02 X
2ρ
2A
e
− 4B
/(k!)2
2
T
2
2
2
2
w0
U0 −2ρ2 /w02eT−2ρ2c (ts )/w02

k=0=

U0 w02(A − Bw02 − 2Bρ2 )T 2

R ∞ 2k+1 −x2 /a2
P
k 2
k
2
e
dx =
during which I have used the formula J0 (x) = ∞
k=0 (−1) (x /4) /(k!) and 0 x




2k
R∞
R ρmax2
2
2
22ρ
k! 2k+2/(k!)2 → 0 dρc e−2ρc /w0 {· · · }. The approximation of 0 c dρc e−2ρc /w0 ρ2k+1
c
2a
w2
0

holds to be true when ρmax
≥ 3w0 (verified by a numerical calculation).
c