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이학박사 학위논문Phase Transition Dynamics of Quenched
Spinor Bose-Einstein Condensates with
Antiferromagnetic Interactions퀜치된 반강자성 스피너 보즈-아인슈타인 응집체의 상 전이동역학2018년 8월서울대학교 대학원물리 · 천문 학부

강

세

지

Phase Transition Dynamics of Quenched Spinor
Bose-Einstein Condensates with Antiferromagnetic
Interactions퀜치된 반강자성 스피너 보즈-아인슈타인 응집체의 상 전이동역학지도교수 신 용 일이 논문을 이학박사 학위 논문으로 제출함
2018년 5 월서울대학교 대학원물리 · 천문 학부강 세 지강세지의 박사 학위 논문을 인준함
2018년 6 월

위 원 장
부위원장
위
원
위
원
위
원

제 원 호
(인)
신 용 일
(인)
민 홍 기
(인)
Uwe R. Fischer
(인)
최 재 윤
(인)

Phase Transition Dynamics of Quenched
Spinor Bose-Einstein Condensates with
Antiferromagnetic Interactions
by

Seji Kang, B.S.

Dissertation
Presented to the Faculty of the Graduate School ofSeoul National University
in Partial Fulfillmentof the Requirements
for the Degree ofDoctor of Philosophy

Seoul National University
August 2018

AbstractPhase Transition Dynamics of Quenched
Spinor Bose-Einstein Condensates with
Antiferromagnetic Interactions
Seji Kang
Department of Physics and Astronomy
The Graduate School
Seoul National University

Understanding non-equilibrium systems are challenging, because they areusually accompanied by either strong time-dependent perturbations or nonvanishing energy and mass flows. Many efforts have been made to study the farfrom-equilibrium dynamics by extending the statistical frame of an equilibrium
system. One promising progress has been made in the field of phase transition
dynamics near a critical point. The significant questions for phase transition
dynamics contain how the many-body system evolves into a newly ordered stateand how many excitations occurs. It is well known that the scaling behavior
in the vicinity of a critical point is determined by the correlation length, thedimension of the system, and the symmetry of an order parameter in the thermal
equilibrium. This scaling hypothesis can be extended to the dynamic scaling
hypothesis of an out-of-equilibrium system across a phase transition. At theend of the evolution, topological defects are formed as a result of excitations.
One typical protocol is a quantum quench that explores a non-equilibrium
system across a phase transition by suddenly changing the system’s Hamiltonian. However, for a many-body quantum system, the time-evolution of fari

from-equilibrium system is extremely complicated for study in either an analytical or numerical way, which requires an experimental approach. Ultracold
atomic gases provide a good platform to study these non-equilibrium dynamics
in a highly controllable and well-isolated manner with sufficiently long coherenttime.
In this thesis, we introduce some methods to induce non-equilibrium dynamics through quenched phase transition in a quasi-2D spinor Bose-Einsteincondensate (BEC) with antiferromagnetic interactions. We then discuss its evolution. There are two ground states for an antiferromagnetic spinor BEC depending on the quadratic Zeeman energy, q: the easy-axis polar (EAP) phase forq > 0 and the easy-plane polar (EPP) phase for q < 0. The microwave dressing
technique allow us to access both positive and negative values of q.
A BEC, initially prepared in the EAP state, becomes dynamically unstable when the q suddenly changes to negative. The EAP-to-EPP phase transition
occurs through the transverse magnon excitations according to the mean-field
theory. In addition, the dynamical instability is scaled with the quench depth of|q|. Then, the spin turbulence emerges, and the condensate enters a qualitatively
different stage of the quench evolution. Interestingly, the time-space scaling is
still preserved in the spin turbulence generation depending on q. As the spin
turbulence decays, half-quantum vortices (HQVs) can be identified as a resultof the EAP-to-EPP phase transition.
For the EPP state, the energy spectrum contains two magnon modes:
transverse and axial magnon modes. Without any gapless axial magnon excitations, the EPP-to-EAP phase transition occurs due to the gapped transverse
magnon excitations, similar to the EAP-to-EPP phase transition. Thus, the dynamical instability is also scaled with |q|. However, we can dynamically createiispecial solitons, named as magnetic composite solitons, which is terminated by
HQVs by using the EPP-to-EAP phase transition. We investigate the structure of a magnetic composite-soliton in terms of spin texture and supercurrent.
A magnetic composite-soliton splits into small pieces via the snake instability,
which is caused by the mass velocity. It is possible for a magnetic compositesoliton to shrink into a singly charged vortex with a ferromagnetic core. In
addition, we also observe the spin fluctuation effect on the dynamic instability,
where the spin fluctuations arise from the axial magnon excitations.
This thesis also includes a study of how the circulating condensate relaxes into a rotating condensate with a vortex lattice in an anharmonic trapping
potential. We drive the condensate to circulate around the trap center by circularly shaking the trapping potential. This method is one of the simplest ways toinject vortices in the condensate, which is analogous to what we typically do torotate wine in a glass. Furthermore, we investigate the chirality of geometricalHall effect by measuring the response of the condensate under various ellipticalpolarized driving. The geometrical Hall effect is caused by the 2D skyrmion spin
texture, which has a finite Berry curvature so that the neutral atoms feel theeffective Lorentz force.

Keywords : Antiferromagnetic Spinor Bose-Einstein condensate, Phasetransition, Scaling hypothesis, Spinor quantum turbulence, Topologicaldefects, Composite-soliton, Quantum fluctuations.
Student number :iii

2012-20353

Contents
Abstracti

List of Tables

viii

List of Figuresix

Chapter 1 Introduction

1

1.1

Phase transition dynamics . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Scaling hypothesis . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Creation of Topological defects . . . . . . . . . . . . . .

4Spinor Bose-Einstein condensates . . . . . . . . . . . . . . . . .

7

1.2.1

Scalar vs Spinor BECs . . . . . . . . . . . . . . . . . . .

8

1.2.2

Phase diagram of F = 1 spinor BEC . . . . . . . . . . .

11

1.2.3

The manifolds of order parameter of F = 1 spinor BEC .

14

1.3

Look over quantum phase transition studies in spinor BECs . .

16

1.4

The outline of the thesis . . . . . . . . . . . . . . . . . . . . . .

18

1.2

Chapter 2 Spin Manipulation & Imaging

20

2.1

Coil configuration . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2

Magnetic DC & AC fields control . . . . . . . . . . . . . . . . .

22iv

2.2.1

Magnetic field calibration . . . . . . . . . . . . . . . . .

23

2.2.2

Field inhomogeneity control . . . . . . . . . . . . . . . .

24

2.3

Microwave dressed fields . . . . . . . . . . . . . . . . . . . . . .

29

2.4

Imaging Spinor Bose-Einstein condensates . . . . . . . . . . . .

33

2.4.1

Absorption imaging & Stern-Gerlach imaging . . . . . .

33

2.4.2

Magnetization-sensitive phase contrast imaging . . . . .

36

Chapter 3 The EAP-to-EPP phase transition

38

3.1

Dynamic instability for the EAP phase . . . . . . . . . . . . . .

39

3.2

Experimental setup & Methods . . . . . . . . . . . . . . . . . .

42

3.3

Emergence of Spin turbulence . . . . . . . . . . . . . . . . . . .

45

3.3.1

Spin-wave excitation . . . . . . . . . . . . . . . . . . . .

48

Characterizing the quench dynamics & spin turbulence . . . . .

48

3.4.1

Time-Space scaling . . . . . . . . . . . . . . . . . . . . .

53

3.5

Spin population relaxation . . . . . . . . . . . . . . . . . . . . .

57

3.6

Topological Defects: Half-quantum vortex . . . . . . . . . . . . .

59

3.7

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

60

3.4

Chapter 4 The EPP-to-EAP phase transition

63

4.1

Dynamic instability for the EPP phase . . . . . . . . . . . . . .

65

4.2

Experimental setup & Methods . . . . . . . . . . . . . . . . . .

67

4.3

Characterizing the quench dynamics. . . . . . . . . . . . . . .

69

4.3.1

Measuring the transverse magnon excitation . . . . . . .

71

4.3.2

Time-Space scaling . . . . . . . . . . . . . . . . . . . . .

71

Dynamically formed topological objects . . . . . . . . . . . . . .

74

4.4

4.4.1

Magnetic composite-soliton: a domain wall terminated by
HQVs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v

78

4.5

Suppression of dynamical instability by the axial spin fluctuations 84

4.6

Creation of singly charged vortices . . . . . . . . . . . . . . . .

89

4.7

Sad history . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.8

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

94

Chapter 5 Rotating Bose-Einstein condensate

95

5.1

Anharmonic trapping potential . . . . . . . . . . . . . . . . . .

96

5.2

Generation of circular c.m. motion . . . . . . . . . . . . . . . .

97

5.3

Relaxation of the circulating Bose-Einstein condensate . . . . . 101

5.4

5.5

5.3.1

Surface mode excitation . . . . . . . . . . . . . . . . . . 102

5.3.2

Thermal distillation: re-condensation . . . . . . . . . . . 104

5.3.3

Rotating thermal clouds . . . . . . . . . . . . . . . . . . 109

Gauge-field effect in a magnetic quadrupole field . . . . . . . . . 112
5.4.1

Skyrmion Spin texture . . . . . . . . . . . . . . . . . . . 113

5.4.2

Chirality of Gauge potential . . . . . . . . . . . . . . . . 115

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 6 Conclusions

119

Bibliography

121

초

144

록

Publication List

147

Appendix A Scaling hypothesis

149

Appendix B Calculation of quantum depletion

151

vi

Appendix C Examples of SCPI commands

154

Appendix D Sad history

156

vii

List of Tables
1.1

Scaling of thermodynamic quantities in the thermal equilibrium.

3

1.2

Topological excitations described by the homotopy group. . . . .

7

1.3

The order parameter manifold of spinor condensates and theirhomotopy group. . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

15

List of Figures
1.1

Schematic phase diagram of the Ising model. . . . . . . . . . . .

4

1.2

Schematic classification of geometric structures. . . . . . . . . .

5

1.3

Schematic description of Homotopy classes. . . . . . . . . . . . .

6

1.4

Phase diagram of spin-1 Bose gas. . . . . . . . . . . . . . . . . .

12

1.5

The alignment of the nematic directors in an antiferromagnetic
spinor BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.1

Coil configuration around the main chamber. . . . . . . . . . . .

21

2.2

Rabi oscillation within the hyperfine manifold F = 1. . . . . . .

24

2.3

Background DC magnetic field calibration using Rabi oscillation.

25

2.4

Calibration of the DC magnetic field inhomogeneity. . . . . . . .

26

2.5

Spin dipole oscillation. . . . . . . . . . . . . . . . . . . . . . . .

28

2.6

Hyperfine structures of 32 S1/2 level of 23 Na atoms. . . . . . . . .

30

2.7

Calibration of the quadratic Zeeman shift with the microwave
dressed field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.8

Extract the numbers of BEC and thermal atoms from SG images. 35

2.9Magnetization-sensitive phase contrast imaging. . . . . . . . . .

37

3.1

Bogoliubov dispersion relations in the EAP phase. . . . . . . . .

41

ix

3.2

The EAP-to-EPP phase transition for q/h = −10 Hz. . . . . . .

3.3

The EAP-to-EPP phase transition for q/h = −5.4 Hz and q/h =46−20 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.4

Spin-wave excitations for q/h = −1.4 Hz and q/h = −2.7 Hz. . .

49

3.5

Temporal evolution of the EAP-to-EPP phase transition. . . . .

50

3.6

The effect of the field gradient on the dynamical instability. . . .

52

3.7

Spin turbulence generation and decay. . . . . . . . . . . . . . . .

54

3.8

Time-Space scaling in the EAP-to-EPP phase transition. . . . .

56

3.9

Decay of the fractional m = 0 population in the turbulence phase. 58

3.10 Creation of Half-quantum vortex as a result of the EAP-to-EPP
phase transition. . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.11 The EAP-to-EPP phase transition for |q|/h > 2c2 n. . . . . . . .

61

4.1

Bogoliubov dispersion relations in the EAP phase. . . . . . . . .

66

4.2

Power spectrum of a trapped thermal gas at high temperature. .

68

4.3

The EPP-to-EAP phase transition for q/h = 10.4 Hz. . . . . . .

70

4.4

Temporal evolution of the EPP-to-EAP phase transition. . . . .

72

4.5

Spatial scaling of the domain formation for the quenched EPP
condensate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6Spin texture of the topological defects with an antiferromagnetic
spin interactions. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

4.9

75

The formation of magnetic composite-solitons in a quenched EPP
condensate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

73

77

Dynamically formed topological defects in a quenched EPP condensate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Schematic of the Bragg spectroscopy and SG imaging sequence.

80x4.10 Bragg spectroscopy and spin separation imaging of a BEC with
MCSs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.11 Singly charged vortices having magnetized cores. . . . . . . . . .

83

4.12 Experimental sequence to enhance spin fluctuations in an EPP
condensate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.13 The EPP-to-EAP phase transition with the spin fluctuations for
q/h = 10.4 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.14 Dynamical instability of the quenched EPP condensate with spin
fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.15 Dynamically formed skyrmions and magnetic composite-solitons
in a quenched EPP sample with the spin fluctuations. . . . . . .

89

4.16 Creation of singly charged vortices depending on the number ofvacuum fluctuation seed. . . . . . . . . . . . . . . . . . . . . . .

91

4.17 The EPP-to-EAP phase transition for q/h = 2.0 Hz with slightspin fluctuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

4.18 The EPP-to-EAP phase transition for q/h = 10.6 Hz with slightspin fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.1

Circulating BEC in an anharmonic hybrid trapping potential. .

98

5.2

Generation of a circular c.m. motion of the condensate. . . . . . 100

5.3

Resonant generation of c.m. motion depending on the modulation
time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4

Surface mode excitations in the circulating condensate. . . . . . 103

5.5

Relaxation of the circulating condensate in the anharmonic trapping potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.6

Thermal distillation process. . . . . . . . . . . . . . . . . . . . . 107

xi

5.7

Spinning up the condensate by a rotating thermal cloud. . . . . 110

5.8

The evolution of vortex number, total atom number and the thermal fraction of the rotating BEC. . . . . . . . . . . . . . . . . . 111

5.9

Skyrmion spin texture configuration. . . . . . . . . . . . . . . . 113

5.10 Skyrmion spin texture configuration. . . . . . . . . . . . . . . . 114
5.11 Gauge-field effect. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.12 Generation of c.m. motion under differently polarized external
modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
D.1 Trying to figure out the magnetic field gradient of the grad.z andgrad.xy coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
D.2 Trying to observe the dynamics of Skyrmion decay. . . . . . . . 157

xii

Chapter 1
Introduction
Before we embark on our study-the phase transition dynamics in a spinor
Bose-Einstein condensate (spinor BEC) with an antiferromagnetic interaction,
we start to introduce why the phase transition dynamics has attracted attentions and what is the uniqueness of an antiferromagnetic spinor Bose-Einsteincondensate.

1.1

Phase transition dynamics
The field of the out-of-equilibrium dynamic is diverse in modern many-

body physics [1] from the cosmology [2] to condensed matter [3], but our understanding about the nonequilibrium system has been full of questions. Thisis because the free energy and the probability distribution are hard to definedin this system with, e.g., a time-dependent external driving or energy flows.
Many efforts to study of the far-from-equilibrium dynamics have been made by
the extension of statistical tools for the thermal equilibrium system. One of thepromising achievements is the scaling hypothesis near the critical point in the1system relaxing toward a newly ordered state along with symmetry breaking. A
quantum quench across the critical point is typical protocol to investigate thephase transition dynamics by suddenly changing the system’s Hamiltonian.

1.1.1

Scaling hypothesis

For a continuous phase transition in the thermal equilibrium, the thermodynamic quantities such as an correlation length, an order parameter, freeenergy, and specific heat of the system have singularities near the critical point
and these singularities are usually characterized by a certain power-law scaling
exponent, which is well known as the critical exponents [4]. These critical exponents are fascinating quantities to classify the systems into the universal classes,cexemplified by the Ising universality class. Let the relative distance as q = Q−Q
Qfrom the critical point, Qc , the free energy per particles as f , the magnitude ofa symmetry-breaking external field b, and the order parameter φ. Then, the fis generalized asf (q, b) = |b|2−α fˆ± (b/|q|∆ ),(1.1)

when q → 0 and b → 0, where distinct scaling functions fˆ+ for q > 0 and fˆ−for q < 0. The scaling behavior of various thermodynamic quantities can be
written as in Table 1.1.1. These critical exponents are related each other suchas α + β(1 + δ) = α + 2β + γ = 2 and γ = β(δ − 1)1 .
Astonishingly, the critical scaling hypothesis of a thermally static phasetransition can be extended to the dynamic scaling hypothesis with the fluctuationdissipation theorem and the concept of a relaxation time τ (q) ∼ ξ(q)z ∼ |q|−zν(reviewed on [5, 6]). The scaling hypothesis in a system far-from-equilibrium
1

Natural question arises: What is the origin of these scalings? The answer is the divergenceof the correlation length. Brief introduction is reviewed on Appendix. A.

2

Quantities
Specific heat

Singular behaviors
Cb=0 ∼ −(∂ 2 f /∂q 2 )b=0 = C± |q|−α(Order parameter

φ = −(∂f /∂qb ) = −|q|2−α−∆ fˆ−0 (b/|q|∆ )Susceptibilityχq ∼ (∂φ/∂b)q,b=0 = χ± |q|γ∼|q|β for b → 0
b1/δ for q → 0

Table 1.1: Scaling of thermodynamic quantities in the thermal equilibrium.
gives a powerful tool to investigate how a system evolves into a certain
state after the phase transition, even though it is not fully proved. Oneof the most well-known classical examples is a coarsening dynamics (or phaseordering kinetics) [7].
Consider the simple classical phase transition first. Figure 1.1 shows themagnetization as a function of temperature. Suddenly, the temperature change
to Tf below the critical temperature Tc and phase separation takes place, evolving into two equilibrium phases with magnetization ±M0 . The locally formed
domains subsequently grow and merge through a slow coarsening process as
the relaxation time τ → ∞ in the vicinity of critical point. The characteristic
domain size grows as L(t) ∼ t−z according to the scaling hypothesis. Near thecritical point, ξ replaces L and τ (q) ∼ ξ(q)z ∼ |q|−zν . For a quantum phase transition, the situation is similar but the dynamical scaling exponent z is associatedwith the energy scale ∆ ∼ |q|zν according to the uncertainty principle.
The scaling hypothesis provides a basis to investigate a phase transition
and manifest that the universality is found not only in the thermal equilibrium
but also in the far-from-equilibrium systems. In addition, it is supposed thatthe critical exponents depend only on the dimensionality of the system, d, andthe symmetry of the order parameter, and the range of interaction, ξ.
3Mphase unstable

M0

-M0

t=0
Tf

Tc

Ti

T

unifrom phase stable

Figure 1.1: Schematic phase diagram of the Ising model depending on temperature without an external field. Tc is the critical temperature. The arrow indicatesa temperature quench at time t = 0 from Ti to Tf .

1.1.2

Creation of Topological defects

What kind of excitations and how many excitations occurs forthe dynamics across the critical point? The Kibble-Zurek mechanism begins from these questions and whether the excitation energy depends on the rateof the driving in the quenched system similar to figure 1.1. When Q is close to aquantum critical point, Qc , the energy gap ∆ ∼ |q|zν in the spectrum vanishes(or the relaxation time τ diverges in classical manner), leading to the violation of adiabaticity. Kibble first reveal the topological domain formation withthe spontaneous symmetry breaking phase transition in the early universe [8,9],
and Zurek suggested the analogy between the long-lived cosmology strings in theearly universe and the vortex strings in the superfluid Helium [10]. The excitation density of the defects is described by a simple scaling with an assumption of
linear change rate. Let the external parameter, in this case temperature, changes
as T = Tc − vt with constant velocity v.
This KZ mechanism is generalized to the crossing of a continuous quantum phase transition (reviewed on [11]). The number of topological defects
4

(a)

A

(b)
B

A’
B’

x0

Figure 1.2: Geometric structure can be classified by a set of loop. (a) All loops, A,
winding the singular hole of the disk cannot be shrunk to a point continuously.
(b) Any kinds of loops, B, can be shrunk into a point, x0 without distortion.
reflects the length scale of the excitations and the number of the quantum noise
seeds.
Classification of topological excitations
The topological excitations have geometrically non-trivial structure, which
are invariant under the continuous deformation of the order parameter manifolds, and they are characterized by topological charges. One of the most wellknown example is a vortex having an integer phase winding in the U (1) manifold.
In other words, when we identify the symmetry of the order parameter manifolds, we can easily predict what kinds of topological excitations are allowed.
Firstly, we introduce homotopy group [12, 13].
A loop is a continuous path whose terminal point is identical to the initialpoint. Every loop can be shrunk to a point without any hole as Fig. 1.2(b). If
there exist a hole in a space, the loop of Fig. 1.2(a) cannot be shrunk to a point
through the continuous deformation. So, it is very natural that the two spaces
in Fig. 1.2 are characterized by a set of loops. In mathematical description, aloop is a continuous map defined as l : [0, 1] → Y such as l(0) = l(1) = x0 ,where Y is a topological space.

5

z

l1

R

l2
l1·l2
y
x0

x

Real space

M

Order parameter space

Figure 1.3: Loops l1 and l2 mapping from a real space R onto the order parameter manifold M . Only mapping of l1 is described. Green dashed loop also canbe the product of l1 and l2 .
Let’s consider two continuous maps f, g from X to Y , where X and Y
are topological spaces. f and g are homotopic equivalent, if the image f (X)is continuously deformed to g(X) in M 2 . The homotopy class of f is denoted
as [f ]. Generally, X is chosen to be well defined topological spaces such as then-sphere, denoted S n , where the n-sphere is the generalized n + 1-dimensionalsurface satisfying x21 + x22 · · · x2n = 1.
Then, we can consider the homotopy classes of loops, [l1 ] and [l2 ], mappingfrom a real space R to the order parameter manifold M . The product of two
homotopy classes is defined as [l1 ]·[l2 ] = [l1 ·l2 ], where [l1 ·l2 ] is a homotopy classin which l1 is traversed and then l2 is traversed [Fig. 1.3]. With the definition of
the product, a set of homotopy classes form a group, called as the first homotopy
group, is written as π1 (M, x0 ) or simply π1 (M ).
Extends the idea into the higher dimensions. A closed surface in real
space R is defined with the n-sphere and it is regarded as loops above3 . Then,
2

Mathematically, f and g are homotopic, if there exist a continuous mapping F (x, t) :X × [0, 1] → Y , satisfying F (x, 0) = f (x) and F (x, 1) = g(x). The homotopic equivalentrelation satisfies reflexive f ∼ f , symmetric (if f ∼ g, theng ∼ f ), and transitive(if f ∼ g andg ∼ h, then f ∼ h).
3
It can be regarded as a map from S n to a M that a map from an n-dimensional cube I n ={(t1 , t2 , · · · , tn )|0 ≤ ti ≤ 1, (1 ≤ i ≤ n)} to a M whose boundary ∂I n == {(t1 , t2 , · · · , tn )|ti =0 or 1} is shrunk to a base point.

6

we can consider a group of the homotopy equivalent classes, which consists ofthe maps from S n to the manifold, M .
πn
π0
π1
π2
π3
π4

Defects
Domain walls
Vortices
Monopoles

Solitons
Dark solitonsNonsingular domain walls
2D SkyrmionsSkyrmions, Knots
Instantons

Table 1.2: Topological excitations described by the homotopy group.
The possible topological defects are determined by the n-th homotopy
group of the order parameter manifolds. For example, a superfluid system isinvariant under the U (1) gauge transformation and π1 (U (1)) ∼= Z. Therefore,
a non-trivial geometrical structure is allowed with the integer winding number- the number of singularities in a loop. The non-trivial topological defect is aquantized vortex line. The homotopy classification of topological excitations issummarized as in Table. 1.2.

1.2

Spinor Bose-Einstein condensates
In case of a many-body quantum system, the generic time-evolution (|ψ(t)i =0

exp−iH t/~ |ψ(0)i) is extremely complicated under the quench through phasetransition (H → H 0 at t = 0) and the unitary time-evolution in quantum mechanics confuses us not allowing thermalization of a quantum system. Therefore,
many experimental attempts have been made. Over the past few years, ultracold atomic systems offered us good experimental platforms with sufficiently
long coherent time and the tunability of the system [17] since in 1995 [15, 16],
when a BEC was firstly realized. These properties of the ultracold atomic clouds
7

allow us to contact intriguing physics such as a history of early universe [18]. In
this section, we will briefly introduce our experimental platform, spinor BoseEinstein condensate, one of the ultracold atomic systems possessing the spin
degree of freedom.

1.2.1

Scalar vs Spinor BECs

In the limit of zero temperature, the ensemble of dilute and identicalbosonic particles forms a new phase, called a Bose-Einstein condensate (BEC),
accompanying U (1) symmetry breaking. A BEC can be simply understood by
the following description: A particle is usually described by a localized wave
hpacket with its size given by the thermal de Broglie wavelength, λdB = √2πmk,
BTwhere h is the Planck’s constant, kB is the Boltzmann constant, and T is thetemperature. As temperature lowers toward zero temperature, the spatial extent
of atoms increases and eventually λdB becomes comparable to the inter-particle
distance. The wave packets of indistinguishable bosonic particles overlap ending
up with a gigantic matter wave of Ψ(r1 , r2 , . . . , rN ) = ΠN
i=1 ψ(ri ), where ψ(ri ) isa single particle wave function [19]. The formation of macroscopic matter wave
begins when phase space density λ3dB ∼ 1. Penrose and Onsager suggested ageneralized criterion [20], which is given aslim
0|x−x |→∞hΨ̂† (x)Ψ̂(x0 )i = f ∗ (x)f (x0 ),(1.2)

where Ψ̂(x) = ψ(x)eiθ is the BEC order parameter, f (x) is a non-zero function.
The new thermodynamic properties of Bosons or Fermions can be explained by
the off-diagonal long range order of the density matrix [21].

8The Hamiltonian of a BEC is given as
H=

N 
X
i=1

 X
~2 2
∇ + Vtrap (ri ) +Vint (ri − rj ),
−
2m i
i<j

(1.3)

where Vtrap and Vint (ri − rj ) is the external potential and the interaction energy
between ith and j th atoms, respectively. Because the density of ultracold atoms
is extremely dilute, we consider the simplest contact interaction neglecting vander Waals interaction and three-body interaction [22]. In a low energy limit, thetwo-body interaction is written as
Vint (r1 − r2 ) = gδ(r1 − r2 ),(1.4)

where g = 4πas ~2 /m is the effective interaction strength and as is the s-wave√
scattering length. The condensate wave function, Ψ(r) = N ψ(r), satisfies
the Gross-Pitaevskii equation, which is derived from the non-linear schrödinger
equation with the particle number conservation:−~2 2∇ ψ(r) + Vex (r)ψ(r) + g|ψ(r)|2 ψ(r) = µψ(r),
2m

(1.5)

where µ = ∂E/∂N is the chemical potential of the system [23].
A single-component BEC is known as a scalar BEC. With the internal
hyperfine spin states for neutral atoms, a BEC have an extra spin degree of freedom. Thus, a special multi-component BEC, which consist of the spin components, is called as spinor BEC. In contrast to the conventional multi-component
BEC, a spinor BEC allows the spin-exchange collision among the spin states.
Spinor BEC shows rich physics with the coupling between the superfluidity andspin ordering. I shall restrict the case of F = 1 spinor BEC in my dissertation.
Similar to the dynamics in a scalar BEC, the low energy dynamics ina spinor BEC are also well described by the pairwise-interaction. This contact
9interaction preserves the individual atom hyperfine spin and it is invariant under
the hyperfine spin rotation [24, 25]. For two indistinguishable spin-F bosonicparticles, the total spin becomes Ftot = F1 + F2 with Ftot = 0, · · · , 2F −1, 2F . Only the even Ftot -channels are allowed, because the statistical symmetry
for two identical bosons does not change the spatial wave function 4 and weonly consider the s-wave scattering. Then, the interaction in the Hamiltonianis written as
Vint (r1 − r2 ) = δ(r1 − r2 )X

gF PF ,

(1.6)

F =0,22where gF = 4π~maF with aF is the scattering length for total spin-F channel,
P
and PF = mF |Ftot , mF ihFtot , mF | is the projection operator, satisfying theP
completeness relation of 1 = 2F
F =0, even PF . After simple algebra using F1 ·F2 =Ftot (Ftot +1)−2F (F +1), the interaction for spin-1 bosons is given as
2

Vint = δ(r1 − r2 )(c0 + c2 F1 · F2 ),(1.7)

where the c0 and c2 is the spin-independent and the spin-dependent interaction
coefficient, respectively, with
g0 + 2g2
4π~2 ā=3m
g2 − g04π~2 ∆a
=
,
=
3
m

c0 =
c2

(1.8)

F =0 5and ā = 2aF =23+aF =0 , ∆a = aF =2 −a. Therefore, we find that the total Hamil3

4(−1)2f = (−1)2f ×(−1)F +L should be satisfied for the many-body wave function, where F
is the total spin and L is the relative orbital angular momentum of the two identical particles.
For the s-wave scattering, L = 0.
5
The spinor system can described as a multi-spin component mixture of BECs. Then,the interaction between two BECs in hyperfine sublevels mF and m0F are characterized as
the s-wave scattering lengths of a1,1 = a−1,−1 = ā + ∆a = aF =2 , and a1,−1 = ā − ∆a =1
3 (aF =2 + 2aF =0 )10

tonian for a spin-1 spinor BEC is
H=

N 
X
i=1

 X
~2 2
−
∇ + Vex (ri ) +δ(ri − rj )(c0 + c2 Fi · Fj ).
2m i
i<j

(1.9)

The mean field order parameter of the spin-1 spinor BEC is represented byp
as a three-component order parameter, Ψ(x) = n(x)[ψ1 (x), ψ0 (x), ψ−1 (x)]T ,where the magnitude of the vector is normalized. It is not difficult that the orderparameter is invariant under the U (1) gauge transformation and the SO(3) spinrotation transformation. The mean field energy functional is given as
 2

Z
~
n2
2
2
E = dx|∇Ψ| + (Vex (x) − µ)n(x) + (c0 + c2 hF i) .
2m
2(1.10)

We can recognize that the spin dynamics in a spinor BEC is determined by thelast term of Eqn. 1.10. The ground state of a spin-1 condensate depends on thesign of c2 : for c2 < 0, ferromagnetic (|F| = 1), exemplified by Rb atoms [27] andantiferromagnetic (or polar, F = 0), exemplified by Na atoms [26] for c2 > 0.
The manifolds of order parameters are given by SO(3) for a ferromagnetic and[U (1) × S 2 ]/Z2 for an antiferromagnetic spinor Bose gases, respectively.

1.2.2

Phase diagram of F = 1 spinor BEC

With an external magnetic field, the rotational symmetry is broken and thelowest energy level is changed by the Zeeman shift. Simply, consider the systemwith spatially homogeneous magnetic field of B = B ẑ. Then, the Hamiltonian
for a spin-1 BEC is given as
H = HKE + Vtrap + HZ + Vint
(1.11)



Z
1
X
~2 ∇2
=
dr
Ψ̂†m (r) −+ Vtrap (r) + (pm + qm2 ) + Vint (r) Ψ̂m (r),
2M
m=−1
11

(b)
c2>0

(a)

p/c2n

p=q+c2n/2

Ferro.
1

EPP

q<0

Easy-Plane Polar(Antiferromagnetic)

Easy-Axis Polar(Polar)U(1)×S1_______
Ζ2

U(1)

0 0.5

EAP
q/c2n

-1

p2=2qc2nFerro.
q>0(c)

p=-q+c2n/2

p/c2n
p=q

Ferromagnetic ||Ferro. ⊥(Broken-Axis)

U(1)×Ζ2

U(1)×S1

p2=q2-2q|c2|nFerro.
0

Ferro.

c2<0

BA
2

Polar
q/|c2|np=-qFigure 1.4: (a) Phase diagram for F=1 spinor BEC with the external magnetic
field based on the mean-field theory from [28]. Ground states for spin-1 BEC
with (b) antiferromagentic (c2 > 0) and (c) ferromagnetic (c2 < 0) s-wave
scattering based on the energy per particle [12, 14]. The dotted lines indicate
the second-order phase boundaries.
where HKE is the kinetic energy, Vtrap is the trapping potential, HZ contains thelinear (p) and quadratic (q) Zeeman energies, and Vint is the interaction energyof Eqn. 1.6. The strength of the quadratic Zeeman energy is given by p = gµB B
and q = (gµB B)2 /Ehf , where gF is the Landé g-factor, and µB is the Bohrmagneton, and Ehf is the hyperfine energy splitting. For sodium, p = h × 700B0
kHz/G and q = (E+1 + E−1 − 2E0 )/2 ≈ h × 277B02 Hz/G2 , where EmF is
the energy shift of hyperfine sublevel mF . Then, the total energy functional isexpressed as
Z
ES =

dx n(

c2 n 2
hF i + phFz i + qhFz2 i).
2(1.12)The ground state obtained by the energy per particle is displayed in Fig. 1.4 (b)and (c) for ferromagnetic and antifferomagnetic spinor BECs, respectively.
R
Under the magnetization conservation of Mz = dx nhFz i, we can write
12δES − λδMz = 0 minimizing the quantity ES − λMz at fixed λ, where the λis the Lagrange multiplier, and λ = p. Thus, we can regard this spinor BEC
system in rotating frame with Larmor frequency, and the linear Zeeman energy
term is compensated by the magnetization conservation. This means the spin
dynamics is affected only by the competition of the spin-dependent interaction,
c2 n, and the quadratic Zeeman energy, q.
Figure 1.4 displays the phase diagram for a uniform F = 1 spinor BEC
in the presence of the external magnetic field based on the mean-field theory.
With antiferromagnetic interaction (c2 > 0) corresponding to Na atom case, theatoms are in either the mF = 0 state or the superposition of the mF = ±1 sates.
The former state is called the easy-axis polar (EAP) state and is preferred whenq > 0. The latter state is called the easy-plane polar (EPP) state, preferred whenq < 0. The EAP state is effectively same with the scalar BEC, having only U (1)gauge symmetry. On the other hand, for the EPP state, the spin rotation isbroken along the z-axis and the EAP has [U (1) × S 1 ]/Z2 . In simple description,
the ground states for an antiferromangtic spinor BEC is easily represented by
the nematic director, denoted as d: for the EAP state, d k ẑ, and for the EPP
state, d ⊥ ẑ, where ẑ is the direction of the background magnetic field. d
indicate the |mF = 0i states along the quantization axis. We will discuss thedetails of the order parameters and the nematic director in the next section.
In case of c2 < 0, a condensate prefers the magnetized states of mF = 1 or
mF = −1. Both aligned and anti-aligned states are immiscible. See Ref. [29–33]
and reviews [12, 14] if you want to know more about the ferromagnetic spinor
BEC system.

13

1.2.3The manifolds of order parameter of F = 1 spinor
BEC

A spin-1 BEC has both superfluidity and the spin degree of freedom. Thus,
the full symmetry of a spin-1 BEC is U (1) × S 2 . It means that the a generalorder parameters of a spin-1 BEC is given by


 
−iα
2 β
e cos 2
1


  √
√


 
ΨFerro =neiθ D(α, β, γ) 0 = neiθ−γ  √1 sin β 
 2

 
2 β
iα
e sin 2
0
 


−iα
0
− e√2 sin β
  √


√ iθ
 


ne D(α, β, γ) 1 = neiθ  cos β  ,
ΨAF =
 


e√i α
0
sin β
2

(1.13)(1.14)

where D(α, β, γ) is a rotation operator with Euler angles, α, β, and γ. The orderparameter of an antiferromagnetic (or polar) spin-1 BEC does not depend on γ,
yielding to the S 1 symmetry around the d = (sin β cos α, sin β sin α, cos β) [seeFig. 1.5]. This d is the nematic director aforementioned in previous section.
The nematic director seems unchanged under the rotation about the x or yaxis. This is because the single-valuedness of ΨAF is satisfied under the gaugetransformation by π (θ → θ + π) combined with a spin rotation about thex or y-axis by π (d = −d). Therefore, the order-parameter manifold of an
antiferromagnetic spinor condensate has discrete symmetry of Z2 and is given
by [U (1) × S 2 ]/Z2 [34].
Topological objects in F = 1 spinor BEC
In section 1.1.2, the symmetry of the order parameter determines the typesof possible topological excitations. One can easily predict the different topologi14

z

x
mF=0

z

x
mF=0

y

Easy Plane Polar(Antiferromagnetic)

0

y

q

Easy Axis Polar(Polar)Figure 1.5: The alignment of the nematic directors in an antiferromagnetic
spinor BEC based on the mean-field description. Green rods indicate the nematic directors and yellow pancake-shaped object indicate a BEC. For the EPP
phase, each nematic director can be randomly aligned in plane.
cal objects are allowed in antiferromagnetic and ferromagnetic spinor BECs. The
order parameter manifold of spinor condensates and their homotopy group isdisplayed in Table 1.2.3. We can predict that there exist singular line structures

Scalar BEC
Ferromagnetic spin-1 BEC
Antiferromagnetic spin-1 BEC

M
U(1)
SO(3)
S 2 ×U (1)
Z2

π1
Z
Z2
Z

π2
0
0
Z

π3
0
Z
Z

Table 1.3: The order parameter manifold of spinor condensates and their homotopy group.
from π1 (MAF ) ∼= Z, non-singular monopole-like structure from π3 (MAF ) ∼= Z,
and a Skyrmion-like structure from π3 (MAF ) ∼= Z in an antiferromagnetic spinor
BEC. For example, the singular line structure is a quantum vortex, having a halfquanta of h/2M 6 . When α = θ = nwind φ/2 and β = π/2, the order parameters
6

The unit quantization h/2M is half of a circulation for a singly-charged quantum vortex
in a scalar BEC.

15

are written as


−iφs
−e
r  −1  r


n
n




HQV
iφn
Ψ
=
e  0 , 0 =
2
2



iφs
inwind φee(1.15)

where nwind is an integer, φn and φs are the superfluid phase and spin orientationin the xy-plane, respectively. This is the order parameter of a half-quantum±φsvortex (HQV) confined in xy-plane and φn2π
should be integer because of thesingle-valuedness of order parameter 7 . HQVs are unique and energetically stablestructures for an antiferromagnetic spinor condensate, allowed by the discreteZ2 symmetry [35, 41]. Hedgehogs, Dirac monopoles, Shanker Skyrmions, andquantum knots can also be created in a spin-1 BEC with antiferromagnetic
interaction. A hedgehog is energetically unstable and deformed into an Alicering [42]. Dirac monopoles [36,37] and quantum knots [38] are created based on
the theoretical suggestions [39, 40].
In 2-dimensional antiferromagnetic spinor condensate, which we havetypically performed our experiments, we observe 2D Skyrmion [43], and halfquantum vortices [41, 44], experimentally.

1.3

Look over quantum phase transition studies in spinor BECs
Based on the theoretical suggestions [24], the ground state of a Na spinor

condensate and its metastability were first investigated in 1998 [26], and the7

In simple description, only one spin component either mF = 1 or mF = −1 has acirculation and other component is stationary.

16

ground state of a Rb BEC were observed in 2004 [27]. Since then, various spin
dynamics arising under rapid quantum phase transition has been investigated.
Quantum phase transition dynamics arises due to the amplification of
the vacuum fluctuations. The excitations of spin modes has been observed insmall condensates, where all spin states are regarded to have identical spatial
dependence [45, 46]. This assumption is known as the single-mode approximation (SMA) and one can ignore the spatial variance of wave function. The phase
diagram of spin-1 Na condensate with finite magnetization has been experimentally studied under SMA [47]. The amplification of the spin fluctuations in
one-direction reduces the uncertainty of the spin fluctuations in other-direction,
which is an example of spin-squeezing phenomena. This spin-squeezing is measured with the quenched spinor BEC [48].
For a condensate larger than the spin healing length, the spin dynamics accompanies spatial variance beyond the SMA. In 2006, the spin-domaincoarsening dynamics and polar-core vortices were observed in a quenched ferromagnetic spin-1 condensate through the polar-to-BA phase transition [49], andby extending this experiments, the vacuum fluctuations has been measured interms of the dependence of the quench depth [50]. After microwave dressing
technique has been developed [51], the deeper transition dynamics from the polar phase to the ferromagnetic phase [52] has been conducted by changing thequadratic Zeeman energy to negative. In case of an antiferromagnetic spinor
condensate, the instability rate and the spatial ordering has been measured inquenched 1D spinor condensates with F = 2 Rb [53] atoms and with F = 1
Na atoms [54–56]. Recent researches on the quenched spinor BECs have been
extended to a spinor BEC in an optical lattice [57, 58]. Using the methods of
optical lattice shaking or tuning the contact interaction by Feshbach resonance,17

the spin-domain coarsening dynamics has been measured with the pseudo-spinstates in terms of the universal scaling and its exponents [59–61].
In our experiments with a Na BECs, we focus on how the intrinsic timespace scaling of the initial dynamics near the critical point is inherited andtransformed in the subsequent evolutions. In addition, we explore what kindsof topological excitations are created during the phase transition dynamics inspin-wise 2D condensate.

1.4

The outline of the thesis
This thesis is organized as follows. In chapter 2, the methods of spin ordermanipulation are introduced, including the apparatus to control backgroundmagnetic fields.
Chapter 3 focuses on the spin turbulence generation and decay in aquenched spinor Bose-Einstein condensate through the easy-axis polar phaseto the easy-plane polar phase transition (the EAP-to-EPP phase transition) interms of the scaling on the quench depth, the quadratic Zeeman energy. After
reviewing the Bogoliubov dispersion relations in the EAP phase, the spin-wave
excitation and the emergence of spin turbulence are described. Interestingly,
the late-time dynamics beyond the mean-field description shows the same scaling behavior with the dynamics in the early stage. When the spin turbulence
decays, half-quantum vortices can be identified.
In case of the easy-plane polar phase to the easy-axis polar phase transition (the EPP-to-EAP phase transition), the initial dynamics also depends on
the quadratic Zeeman energy. Without any spin fluctuations, the quenched EPP
condensate coherently recovers its first state once. Then, the system evolves in18complex way. When the EPP condensate is prepared to have the spin fluctuations via the gapless magnon excitation, the coherent oscillation disappears. In
addition, the instability rate of the quench dynamics is affected by the amplitudeof the spin fluctuations. In the complex evolution stage, solitons terminated byhalf-quantum vortices (HQVs) are identified. We investigate the unique soliton
by measuring the mass flow around the endpoint and its spin textures. For a longhold time, singly charged vortices are created as a result of the EPP-to-EAP
phase transition. This results are introduced and discussed in chapter 4.
Chapter 5 describes the simple method to create vortex lattice in the rotating Bose-Einstein condensate (BEC) with the anharmonicity of the trapping
potential. The transferring procedure from the circular center-of-mass motionof BEC to the angular momentum of the BEC is studied. Then, the chirality of
the gauge potential, which is induced by the Skyrmion spin texture, is measured
with this rotating methods.
Finally, in chapter 6, the conclusions and the future outlooks are discussed.

19Chapter 2
Spin Manipulation & Imaging
Various coil pairs are needed for performing spinor condensate experiments, especially for controlling the magnitude, orientation, and inhomogeneity of background magnetic fields. Most of techniques are also discussed on
S.W. Seo’s thesis.

2.1

Coil configuration
Throughout the thesis, we define the x, y, and z-axes as described inFig. 2.1. Following this convention, there exist the 3 Helmholtz coil pairs constructed around the main chamber in x, y, and z-directions for compensationof the magnetic field of the Earth. We call these coil pairs as comp.x, comp.y,and comp.z coils each. The comp.z coils generate ∼ 0.7 G/V and ∼ 1.1 A/V.
On the top and bottom of the main chamber, we build 4 coil pairs of the
Feschbach coils for producing F = 2 Na and performing Li experiments, thequadrupole coil pairs for magneto-optical trap (MOT) and magnetic trap, thebias coil pairs for controlling z-directional bias magnetic field, and the grad.z
20

(a)

F

current direction

AB

60 mm

C
E
z

40.6 mmDyx

Side view

A: Feshbach coil pairs
B: Quadrupole coil pairs
C: Bucket window
Slower D: RF coil pairs
E: Bias field coil pairs
F: Comp. y coil pairs
G: Grad. z coil pairs

G

(b)

ODT

Comp. y coil pairsComp. x coil pairs
Bragg beam

RF coil pairs
y-direction
AtomicBeam

Ti-subpump

Slowery

RF coil pairs
x-directionxz
Top view

Bragg beamComp. z coil pairs
Grad. xy coil pairs

(c)

65 mm
15 turns
Grad. z coil
Quadrupole coilCompensate coilFigure 2.1: (a) Side and (b) top view of coils around the main chamber. Recessedbucket windows allow us to implant various coils close to the main chamber. A
blue colored pair of anti-Helmholtz coils is placed on the top and the bottom,
and another yellow colored pair of anti-Helmholtz coils is placed in the front andthe rear of the main chamber to reduce the DC magnetic field gradient. We namethese coils as the gradient z- and xy-coils, respectively. Two pairs of Helmholtz
coil pairs are placed inside the chamber, connected to the two electrical feedthrough the port, which are named as radio-frequency coil pairs. (c) A photo of
the 15-turn gradient z-coil on the top of the main chamber. The gradient z-coils
are fixed on the plastic frame with non-magnetic epoxy. The original figure ofcoil configuration is adopted from J. Choi’s publications [62, 63] and S.W. Seo’s
thesis [64]21

coil pairs for canceling out the residual magnetic field gradient. The Feschbachcoils generate 4.3 G/V and the quadrupole coils generates field gradient of 1.5
G/(cm A) along the z-axis. The grad.z coil pairs are anti-Helmholtz coil pairs
wound by 15 turns with circular nonmagnetic frame [Fig. 2.1 (c)] and generate
33 mG/(cm A) and 17 mG/(cm A) along the z- and residual directions. Anotheranti-Helmholtz coil pairs to cancel out the magnetic field gradient are woundalong the xy-direction (yellow coil pairs), which named as grad.xy coils.
Radio-frequency evaporation coils are placed inside the chamber, which
are called as x-directional RF coils (shortly RF coils). These RF coil pairs arealso used for spin manipulation within the hyperfine manifold F = 1. In the ydirection, there exist another Helmholtz coils are placed for AC field generation
and used for microwave dressing technique. These y-directional RF coils areusually called as microwave coils.

2.2

Magnetic DC & AC fields control
No matter how well we build the coil configurations, it is not possible tocontrol magnetic fields uniformly. Thus, the magnetic fields is expanded as

 
 
B
B
Bxy Bxzx 0,x   xx
 
~
∂B






~ = B  + B
B
,
where
B
=
x̂
·
.
(2.1)



B
B
y
ij
i
yy
yz
∂xj
 0,y   yx
 
B0,z
Bzx Bzy Bzzz~ = 0 and ∇ × B,
~ yielding
Maxwell’s equation requires ∇ · B

P

i=x,y,z Bii = 0 and

Bij = Bji . In our experiment, the spin dynamics is freezed in z-direction (Rz ξs ) and we ignore the z-directional field gradient. Without loss of generality, let
the magnetic field is oriented in plane such as B0,y = B0,z = 0 and B0,x = B0 .
22

For B0  Bij xj , Eqn. 2.1 is simplified as
q
~
|B| =
(B0 + Bxy y + Bxz z)2 + (Byx x + Byz z)2 + (Bzx x + Bzy y)2 (2.2)
2 
Bij xj≈ B0,x + Bxx x + Bxy y + O.
B0Firstly, we calibrate the DC and AC magnetic fields. Then, we discuss the methods to measure the field inhomogeneity and how we remove the field gradients.

2.2.1

Magnetic field calibration

The calibration of the magnetic fields is done by measuring the Rabi
oscillation. Typically, Na BEC is prepared in the |F = 1, mF = −1i state[Fig. 2.2.1(a)]. When we apply an oscillating electromagnetic field at a constant amplitude, atoms are transmuted into the other hyperfine sublevels intime. Consider a three level atomic system with the DC magnetic field, B ẑ, andAC field, B0 cos(ωt)x̂. In the rotating wave approximation, the dressed-state
Hamiltonian is expressed as


√
Ω/ 2



−δ
0
 √
√ 


~ Ω/ 2qΩ/ 2 ,


√
0
Ω/ 2
δ

(2.3)

where ~δ = ~(ω − ω0 ), ~ω0 = 12 µB |B|, q is the quadratic Zeeman energy, and~Ω = 21 µB |B0 | is the Rabi frequency.
Approximately, above the three-level system can be regarded as the two√level system with effective Rabi frequency, Ωeff = Ω2 + δ 2 . Ωeff is extracted
from the temporal evolution of populations. For example, the number of themF = −1 component oscillates as N−1 ∝ (1 − cos(Ωeff t)). To precisely calibrate
the background magnetic fields, one should find the minimum Ωeff [32]. However,23

Population(t)

(a) 1
0.8

(b)

mF=-1

mF=1

mF=-1

mF=1

0.6

initial preparation

0.4

(c)

mF=0

0.2
0

mF=0

0.2

0.4

0.6

0.8

1

t (2π/ω0)

on resonance

Figure 2.2: (a) Rabi oscillation within the hyperfine manifold F = 1 with (b)initial state, (ρ1 , ρ0 , ρ−1 ) = (0, 0, 1). Red, green , and blue lines represent the
populations of m√
F = 1, 0, and −1, respectively. We set the resonant Rabi frequency as Ω = 2ω0 . The solid and dashed lines display resonant and offresonant (δ = 0.3ω0 ) oscillations, respectively. (c) A BEC image with resonant
π−pulse.
atoms are perfectly transferred into the |F = 1, mF = 1i state from the |F =1, mF = −1i state for π−pulse only on resonance [Fig. 2.2.1(b,c)]1 .
We calibrated the background magnetic fields along the y, z−axes on
August 2017, displayed in Fig. 2.2.1 by applying the x-directional AC field(See Fig. 2.1): 0.72 G/V along the y-axis and 0.67 G/V along the z-axis2 . The
background magnetic fields become 0 V at Word Generator voltage of 0.15 V
and 5.01 V in y- and z-directions, respectively.

2.2.2

Field inhomogeneity control

Calibrating the magnetic field gradient
The Ramsey interferometry allow us to precisely measure relative phasefrom the magnetic resonance by interfering two-separately evolved interacting
1

When you prepare atoms in the |F = 1, mF = 0i state and apply π/2 rf pulse, 100% of
the population can be transferred into the superposition of the |F = 1, mF = ±1i with theoff-resonant frequency.
2~~
For F = 1 sodium, the Larmor frequency is ωL = gF µB |B|/~= 2π × 0.70 MHz/G × |B|.

24

(b)
250
-470 kHz/V
200
150

465 kHz/V

100
50

Bz

0
4.6

4.8

5.0

5.2

5.4

RF frequency, ω0/2π (kHz)RF frequency, ω0/2π (kHz)(a)

Word generator (V)

300
250
200
150

505 kHz/V

100
50

By

0
0.2

0.4

0.6

0.8

Word generator (V)Figure 2.3: Background DC magnetic field calibration using Rabi oscillation
along (a) the y-direction and (b) z-direction.
zones, and we use the method to calibrate the DC magnetic field inhomogeneity [65, 66]. Figure 2.2.2(a) displays the Ramsey interferometry sequence. The
states of atoms are illustrated with the nematic director, d.
Atoms are prepared in the |F = 1, mF = 0i state by using the LandauZener sweep sequence3 [29, 64] and the state is written as ζ~i = [ζ1 , ζ0 , ζ−1 ]T =[0, 1, 0]T . At this time, d aligns to z-axis (either ±ẑ direction). A π/2 pulse
rotates the nematic director around y-axis and the atoms are in the √12 [1, 0, 1]Tstates. The nematic director freely evolve for a time τ with an Larmor frequency
ωL = gF µB B/~. Because the magnetic field, B(r), varies in space, the atoms
have position-dependent phases of φ(r)s and the atomic states are given as√1 [1, 0, eiφ(r) ]T . Finally, we apply π/2 pulse and atoms in the|F = 1, mF = ±1i2states interfere [Fig. 2.2.2(b)].
yYou can easily estimate the number of fringe patterns∼ ω2πL τ dB
(2Ry )dyyfrom δφ ∝ ωL τ ∝ dByτ , when the patterns aligned in the y-direction, becausedy

the phase changes 2π across the nearest two density-holes (or density-peaks).
√
From Eqn. 2.3, the avoided crossings occur at δ ≈ ±q and δ = 0 with Ω−1,0 = ~Ω 2 andΩ−1,1 = ~Ω2 /q, respectively.
3

25

(a)

z

z

z

|0〉

y

|0〉

y

1
—
)
– ( |1〉+|-1〉
√2
x

y
x

Larmor precession (τ)
120
(c)

(b)

mF=1

π/2
pulse

1
φ —
– (|1〉+eiφ|-1〉)
√2

mF=0

Columndensity (arb. units)

x

π/2
pulse

mF=-1

80
40
0

-200 -100 0
100
Position (µm)

200

Figure 2.4: Calibration of the DC magnetic field inhomogeneity. (a) Schematic
diagram of a Ramsey interferometry. Spin evolution is displayed on the BlochSphere. The green arrows indicate the nematic directors. (b) Ramsey fringe
pattern with τ = 10 ms and B 0 = 6.2 mG/cm. (c) The red solid line is the 1D
column density of mF = 0 spin component in (b). The blue dashed line is thefitting curve by Eqn. 2.6.

26

Using the matrix algebra, consider a rotation operator D(α, β, γ), where α, β,and γ are the Euler angles [67]. Then, π/2 rotation is described as Dπ/2 ≡D(α = 0, β = π/2, γ = 0 such as
1(1 + cos β) − √12 sin β2Dπ/2 =  √1 sin β
cos β
2

1
(1 − cos β) √12 sin β21(1 − cos β)2− √12 sin β 

1
(1
+
cos
β)
2

,

(2.4)

β=π/2

and the Larmor precession during τ is given as D(α = φ(r), β = 0, γ = 0)
whose matrix elements are D(φ, 0, 0)mF ,m0F = exp(−imF φ(r))δmF ,m0F with φ =gF µB (B0 + B 0 y)τ /~. For the initial ζ~i = [0, 1, 0]T state, the atomic density as aresult of the Ramsey interferometry is1
sin2 (φ0 + gF µB B 0 yτ ~)
2




|ζ~f | = |Dπ/2 D(φ, 0, 0)Dπ/2 |2 =  cos2 (φ0 + gF µB B 0 yτ ~)  ,


2
1
0
sin (φ0 + gF µB B yτ ~)
2

(2.5)

where φ0 = (gF µB B0 )τ /~. The condensate is captured in the harmonic optical
dipole trapping potential. Thus, the density profile of the Ramsey fringe is givenas 2 2xñ1D (x, y) ∼ cos (φ0 + gF µB B yτ ~) × 1 −
and
Rx
x 2
y 2 3/2
ñ2D (x, y) ∼ cos2 φ(x, y) × 1 −
−
,
Rx
Ry
0

2

(2.6)

in case of 1D and 2D, respectively. An example of the gradient calibration isdisplayed in Fig. 2.2.2.
Cancelling out the magnetic field gradient
Using the grad.z and grad.xy coil pairs, we cancel out the field gradientaccording to Eqn. 2.2. Ironically or luckily, the quadrupole zero-field points of
27

0s

0.4 s

0.8 s

1.2 s

1.6 s

2.0 s

Figure 2.5: Dipole oscillation of an EPP condensate in the harmonic optical
dipole trapping potential induced by B 0 < 0.5 mG/cm.
the grad.z and grad.xy coil pairs are not placed the exact the center of the coils,
which allow us to remove the magnetic field gradient4 .
With the Ramsey method, it is not able to measure the extremely small
field inhomogeneity less than 0.3 mG/cm due to the limitation of τ ≤ 100 ms.
In our experiment, spin-dependent interaction is ∼ 10 Hz and unwanted spin
dynamics is possible to come in for long Larmor precessing time over 100 ms.
Therefore, we use spin-dipole oscillation [68, 69] to remove the magnetic field
gradient more completely. Figure 2.2.2 shows the spin-dipole oscillation withthe frequency of ωSD ≈ 2π × 0.8 ≈ 0.2ωx [69] under the field gradient B 0 <0.5 mG/cm. We can confirm that the field gradient is removed, when there isno spin-dipole oscillation and the mF = ±1 condensates completely overlap at
600 ms.
Another way to calibrate the field gradient is measuring the dynamical instability via the EAP-to-EPP phase transition. For low magnitude of |q|,
the influence of the field gradient is great [56]. We confirm this experimentally [Fig. 3.4].
4

The position quadrupole zero-field point can be adjust by the changing the bias fields.
We were trying to figure out the magnetic field gradient generated by the grad.z and grad.xy
coils. But, before that, we empirically cancel out the field inhomogeneity by tuning the bias
field and controlling the currents of both anti-Helmholtz coils (Appendix. D).

28

2.3

Microwave dressed fields
The hyperfine sublevels of a spin-F atom split due to the linear andquadratic Zeeman energies (Eqn. 1.11). The competition of the linear andquadratic Zeeman energies allow many researchers to investigate various spinor
dynamics such as spin-mixing dynamics-quantum pendulum [70]- and phasetransition dynamics [45–47, 49, 50]. In a DC magnetic field, q is typically positive. Thanks to the microwave dressing technique, one can contact negative q
region. According to the Breit-Rabi formula [71], the energy for each sublevels
is given as
Ehf
Ehf
EF,mF =− gI µB mF B + (−1)F2(2I + 1)
2

r
1+

4mF α
+ α2 ,
2I + 1(2.7)

where α = gJ µEBhf−gBI µI , Ehf is the hyperfine splitting, and gI , gJ are the Landég-factors for a nuclear and a valence electron, respectively. From Eqn. 2.7, we
can getE1,1 + E1,−1 − 2E1,0(gJ µB − gI µI )2
qDC =
≈
× B2,
2
16Ehf

(2.8)

ignoring higher orders. Under a DC magnetic field, q is typically positive. Thanks
to the microwave dressing technique, one can contact negative q region.
The microwave dressing technique use the AC stark effect. Consider thesimplest system consists of the ground state |gi and the excited state |ei. When
~ and frequency,
we apply an AC field with an magnetic field magnitude |B|
detuned by δ from the resonance frequency, we have the dressed-Hamiltonian,


δ
−iΩ
~
,
HD = 
(2.9)
2 iΩ −δ~where µ~ is a magnetic moment of the atom, and Ω = |~µ · B|/~is the Rabi
frequency. Then, according to the 2nd-order perturbation theory, the energy of29

mF=-2
mF=-1
mF=0

F=2

mF=1

σ−

mF=2
σ−
σ−

π

π
σ+

F=1

mF=1

π

σ+

σ+

q

mF=0

1.7 GHzmF=-1

Figure 2.6: Hyperfine structures of 32 S1/2 level of 23 Na atoms with an external
magnetic field. The magnetic sublevels of F = 1 and F = 2 states split by thelinear Zeeman shift. Allowed transitions depends on the polarization of an AC
fields. The dotted level is the energy of (E1 + E−1 )/2. Energy is not scaled.
|gi is shifted as
δE =

2
~
|he| − µ~ · B|gi|~Ω2=,Ee − Eg
4δ

(2.10)

where µ~ is a magnetic moment of the atom.
Expands above idea to Na atoms with F = 1 and F = 2 hyperfine states.
~ µ cos(ωµ t) = 1 gF µB F~ · B~ µ cos(ωµ t).
Let a microwave dressing field Vµ = −~µ · B~
Applying the 2nd-order perturbation, the energy shift is derived as
δEmF =

X |hF = 2, m0 |Vµ |F = 1, mF i|2
F

Em0F − EmF

m0F(2.11)

2~ X ΩmF ,m0F=,
4 0 δmF ,m0F
mF

where ΩmF ,m0F s are the Rabi frequencies between |F = 1, mF i and |F = 2, m0F iand δmF ,m0F is the detuning from the |F = 1, mF i ↔ |F = 2, m0F i transition. The
Rabi frequency ΩmF ,m0F is usually determined by the intensity of the microwave
30

|F=1, mF=1〉↔ |F=2, mF=2〉

atom number (x106)

(a)
2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5
0.0

0.0
0.1

atom number (x106)

(c)

0.2

0.3

0

0.4

0.2

0.4

0.6

0.8

(d) |F=1, mF=0〉↔ |F=2, mF=-1〉|F=1, mF=0〉↔ |F=2, mF=1〉

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5
0.0

0.0
0.1

atom number (x106)(b) |F=1, mF=1〉↔ |F=2, mF=0〉

0.2

0.3

0.4

0.5

0.1

0.6

0.2

0.3

0.4

0.5

(e) |F=1, mF=-1〉↔ |F=2, mF=0〉(f) |F=1, mF=-1〉↔ |F=2, mF=-2〉

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.6

0.0

0.0
0

0.2

0.4

0.6

0.1

0.8

time (ms)

0.3

0.4

time (ms)qnet/h (Hz)

(g)

0.2

(h)

qnet/h (Hz)

400

20
200
-800
-400

-200

200

400

∆ (kHz)-200∆ (kHz)

-400

10
0
-10
-20

-400

-30
-40

Figure 2.7: Rabi oscillations of |F = 1i ↔ |F = 2i transition. The measured
Rabi frequencies, ΩmF ,m0F are (a) 4.3 kHz, (b, e) 1.7 kHz, (c) 2.7 kHz, (d) 2.8 kHz,and (f) 4.2 kHz. (g) Calibrated quadratic Zeeman shift with above ΩmF ,m0F with
52 mG background DC field. (h) is the enlarged graph of the shaded area in (g).
31pulse and the transition strength, which is represented by Clebsch-Gordan coefficient. As illustrated in Fig. 2.3, the polarization of the microwave field allows
a certain transition of |F = 1, mF i ↔ |F = 2, mF + ki, where k = 0 and ±1 forthe π and σ ± -polarized microwave fields, respectively. The detuning δmF ,m0F is
expressed asδmF ,mF +k = ωµ − ωmF ,mF +k
gF µB B1
= ωµ − ω0,0 + [(mF + k) − (−mF )]2~
1
gF µB B= ∆ − [(mF + k) − (−mF )],2~(2.12)

where ωmF ,m0F is the resonance frequency between |F = 1, mF i and |F = 2, m0F istates, which can be estimated from Eqn. 2.7 as ωmF ,m0F = (E2,m0F − E1,mF )/~.
With simple calculation, the resultant quadratic Zeeman energy is given as
qnet = qDC + qµ(2.13)
δE1 + δE−1 − 2δE0
= qDC +,
2
Ω2mF ,mF +k~X
where δEmF =, (2.14)4∆ − [(mF + k)/2 − (−mF /2)]gF µB B/~
and qDC ≈ hB 2 × 277 GHz/G2 for Na atoms.
We measure the Rabi frequencies, ΩmF ,m0F , by using use circularly polarized microwave, which is equally mixed both σ ± , with frequency of 1.7 GHz.
Figure 2.3 displays the qnet as a function of ∆.
We tune qnet with 0.5 Hz accuracy by using this microwave dressing
technique. However, as ωµ close to the resonant frequency, the lifetime of thecondensate becomes shorter. This is because that the lifetime of F = 2 atoms
are much shorter than F = 1 atoms. In addition, the coupling strength between |F = 1, mF = −1(+1)i and |F = 2, mF i is larger than that between
32

|F = 1, mF = +1(−1)i and |F = 2, mF i for negative(positive) detuning fromthe on-resonant frequency, leading to the higher mF = −1(+1) atom loss (seeFig. 3.9(f)).

2.4

Imaging Spinor Bose-Einstein condensates
We generally measure the spin dynamics appearing in quenched spinor

condensate by the absorption imaging after a short spin separation and magnetizationsensitive phase contrast imaging (in short, magnetization imaging) in z-direction.
In this section, we shall briefly introduce these methods.

2.4.1

Absorption imaging & Stern-Gerlach imaging

Absorption imaging is one of the easiest imaging techniques to experimentally implement and interpret the signa]ls with high signal-to-ratio, because
it uses resonant light to the atomic transition line. When the probing beam
passes trough a BEC, the intensity of the beam is reduced by the absorptionof an atom. With atomic density of n and the resonant scattering cross section
σ = λ2 /2π, the imaging beam is attenuated asdI= −σnI,dx(2.15)

where λ is the wavelength of imaging beam and I is the intensity of the beam.
Therefore, we can obtain the density information of a BEC from n(r) = − σ1 ln(I/I0 ).
In reality, an absorption image is taken onto the camera, containing
Iimg (r) = I0 (r)e−σn(r) + S(r) + N (r),33

(2.16)

where S(r) is the photon-shot noise and N (r) represents other unwanted noises.
In order to remove the noises, we additionally take a bright probing beam image
without any atoms (Ibeam ) and a dark-field image (Idark ) without any lights:
Ibeam (r) = I0 (r) + S(r) + N (r),(2.17)

Idark (r) = N (r).

(2.18)

Thus, we can measure the column density of a BEC from a CCD image as
1
Iimg − Idarkn(r) = − ln.
(2.19)σ
Ibeam − Idark
For more information of our imaging system, see S.S. Kim’s publication [72] andS.W. Seo’s thesis [64].
In our experiments, atoms are possible to exist in all three spin states.
But, atoms in each spin state share the same space in the main chamber. Therefore, we use the Stern-Gerlach spin separation to spatially separate a condensate
by the spin components. When we apply an inhomogeneous magnetic fields in~ = gF µB mF ∇( µ~ · B),~ depending on theplane, atom feels a force of F = ∇(~µ · B)|~µ|mF . Then, we get rid of the magnetic field gradients and take absorption images
after a short time-of-flight time [Fig. 2.4.1(a)]. During the spin separation, theODT is turned off.
We crop an image as shown in Fig. 2.4.1(a) to extract the condensate
fraction by each spin. Then, the cropped images are fitted byY − x2i
X x2 3/2i,0+ne Wi ,
n2D (r) = nc max 1 −
th
2
R
i
i=1,2
i=1,2

(2.20)

where nc , nth are constants, and Ri , Wi are the widths of a condensate andthermal cloud, respectively. However, the quenched condensate has spatial structures due to the spin-wave excitations, disturbing the fitting. So, we obtain 1D34

(b) Column density (arb. units)
50

0

200

mz=-1

150

mz=0

100

(a) m =1
z

0

y

100

x

0

arb.
2 units

600

OD

200 300 400 500
x-directional pixel

200 µmtot
Σyn2D
(x, y)
c
n1D
(x)
tot
ΣOD<2, yn2D
fitting

Figure 2.8: (a) Example absorption images for each spin components. (b) displays the column optical densities (ODs) for mz = −1 component. Black solidline is 1D column density, n1D , obtained by integrating 2D column density, n2D
in y-direction. Red solid line is 1D column density, obtained by summing OD
value less than 2 from (a). Red dotted line is a fitting line for red solid line.
Green line is extracted by subtracting thermal parts from black solid line.
column density, ntot
1D , from Fig. 2.4.1(a) by integrating 2D column density iny-direction. Then, we cut a certain value off, which does not distort the bimodal
density profile and fit it with
2x2
x2
n1D (x) = nc max 1 − 2 , 0 + nth e− Wx .
Rx

(2.21)

We can get the information of thermal gas from the 2nd term and finally, theinformation of the condensate such as the number and size is extracted bysubtracting the thermal gas distribution from ntot
1D (x).
The Stern-Gerlach imaging is useful and simple methods to measure thespin dynamics. However, it distorts the condensate and the sizes of the condensate for each spin components varies, when the atoms are dragged. Thus, webuild magnetization-sensitive phase contrast imaging.

35

2.4.2Magnetization-sensitive phase contrast imaging

Magnetization-sensitive phase contrast imaging (PCI) enable us to detectspin turbulence and magnetized spin textures such as HQVs or spin turbulence.
We use σ − -polarized probe beam, detuned from the D2 atomic resonance. The
detuning is 20 MHz between |F = 1i and |F 0 = 2i5 .
The magnetization-sensitive PCI is homodyne detection by interfering
the scattered light from a sample with the unscattered light. Let an incident
electric field of the off-resonant laser beam, Ein = E0 . Then, the electric field
after passing a BEC is given as
Eout = eiφ E0 = Escatt + Eunscatt = ∆E + E0 ,(2.22)

where φ is a phase shift due to the sample6 . To develop the distinguishable phase
contrast signal for each spin component, we place a phase dot, which causes π/2phase shift of a light. The location of a phase dot is the lens focus prior to theimaging plane. Then, the electric field at image plane is rewritten asππ

Epc = Escatt + ei 2 Eunscatt = (eiφ − 1 + ei 2 )E0 .

(2.23)

Depending on the states of atoms, the interference between the Escatt andEunscatt is either constructive [Fig. 2.4.2(b)] or destructive [Fig. 2.4.2(a)]. For a
BEC in mz = −1(+1) state, we use constructive interference and get bright(dark)phase-contrast signal [Fig. 2.4.2(c)].
However, this phase-contrast imaging methods have limitations that itis difficult to distinguish the mz = 0 domains from the domains, where the5

Unfortunately, the detuning is too close to the resonance, and we can take only one image
for a single experimental sequence.
χ6
φ is well approximated as δij + 2ij , where χij is the dielectric susceptibility and it representχ
the state-dependent phase shift. Therefore, Eout,i = exp (δij + 2ij )Ein,j .

36

(a) (eiφ-1)E0=Escatt

(b)

E0

E0

iφ

e E0

ei∏/2E0eiφE0ei∏/2E0(eiφ-1)E0

Epc(eiφ-1)E0=Escatt(eiφ-1)E0

Epc

(c)

mz=1

mz=0

mz=-1

Figure 2.9: Electric field phasor diagram for (a) destructive and (b) constructive interference. (c) Magnetization-sensitive PCI signal. Raw images without
any analysis. The PCI signal of mz = +1(−1) spin component is dark(bright)
from the constructive(destructive) interference. The BEC in the mz = 0 spincomponent has almost identical signal with the background.
mz = ±1 atoms are equally populated7 . When we obtain the magnetization
distribution, the maximum magnetization is determined from the signal of HQV
cores. The magnetization profile of a HQV core is measured as a Gaussian22function of Mz (r) = ±Mz0 e−r /rcore and the peak magnetization |Mz0 | becomesmaximum magnetization of 1. Then, we transform three signals of ±1 and 0
to the magnetization by using the polynomial fit. For more information, see
S.W. Seo’s thesis [64].

7

In addition, this magnetization imaging depends on the density. Thus, it does not guarantee the linearity of the signals. It is slightly brighter in the denser region of mz = +1
image.

37

Chapter 3
The EAP-to-EPP phasetransition
One of the accessible far-from-equilibrium dynamics is phase transition
dynamics as aforementioned in chapter 1. A quench protocol is typically employed to explore nonequilibrium dynamics [73], where a system is prepared ina well-defined initial state and then its evolution is precisely examined after
the system’s Hamiltonian is rapidly changed. One of the key topics in current
research activities is quantum phase transition dynamics, which addresses theimportant question of how a many-body system evolves into a newly ordered
quantum state. Recently, it was theoretically proposed that scaling behavior
occurs in the phase-transition dynamics near quantum critical points [74–79],
and the existence of scaling and universality was indeed demonstrated in various
experiments of quantum phase transitions [60, 61, 80, 81].
For an antiferromagnetic spinor BEC, the spin state is parameterizedwith a spin nematic vector d~ = (dx , dy , dz ), such that the system is in the38

~ In an external magnetic|mF = 0i state for the quantization axis along d.
~ k ẑ, a uniaxial spin anisotropy is imposed on the system due to thefield, B
quadratic Zeeman energy Ez = qhFz2 i = q(1 − d2z ), and there exist two groundstates depending on the quadratic Zeeman shift, q: easy-axis polar (EAP) phasefor q > 0 [Fig. 3.2](a) and easy-plane polar (EPP) phase for q < 0. In thischapter, we introduce our experiments on the EAP-to-EPP phase transitionin terms of scaling behavior for spin turbulence generation and the creationof topological excitations. Interestingly, the phase transition between EAP and
EPP phases is 1st-order phase transition in the mean-field description, but thecontinuous spin-rotation symmetry in the xy−plane is spontaneously broken.
Finally, topological point defects, which are half-quantum vortices (HQVs) in
the EPP phase [41, 82–84], can be created in the spatially extended 2D system[Fig. 3.2(b)].
• Seji Kang, Sang Won Seo, Joon Hyun Kim, and Yong-il Shin, “Emergence and scaling of spin turbulence in quenched antiferromagnetic spinor
BECs”, Physical Review A 95, 053638 (2017).

3.1

Dynamic instability for the EAP phase
When the system in equilibrium changes across the critical point, thequantum fluctuations are amplified and leads to the quantum phase transition.
For the weak excitations, they are described by the Bogoliubov theory.
Let the field operator for each spin as ψ̂m = ψm + δ ψ̂m , where ψm is
the mean-field value of condensate with zero momentum, and the deviationof δ ψ̂m is the non-condensate field with non-trivial momentum. Therefore, the39

non-condensate field operator is expressed as δ ψ̂m = √1V

P

ik·r
, and
k6=0 âm,k e

the field operator is written as
1 X
ψ̂m = √
âm,k eik·r ,
V k

(3.1)

where V is the volume of the system and âm,k is the annihilation operator ofa boson with wave number k. Substitute Eqn. 3.1 to the total Hamiltonian(Eqn. 1.11), then we getH=X(k − pm + qm2 )â†m,k âm,k

(3.2)

m,k

+

1 X X
(c0 â†m,p â†m0 ,q âm,p+k âm0 ,q−k + c2 Fmm0 · Fll0 â†m,p â†l,q âm0 ,p+k âl0 ,q−k ),2V m,m0 ,l,l0 k,p,q

where k = ~2 k2 /2M and F is a set of the spin-1 matrices.
In case of the EAP phase, the field operator is given as

  
δ ψ̂
0
   +1   Ψ̂ = ψ0  +  δ ψ̂0  .
  δ ψ̂−10(3.3)

So, the non-condensed atoms in the |mF = 0i develop the density fluctuations
and the atoms in the |mF = ±1i develops the spin fluctuations. These density
and spin fluctuations are completely decoupled [85]. Using the number conserP
vation of N̂0 ≡ N̂ − m,k6=0 â†m,k âm,k , we can replace N̂ with a fixed scalar value.
Then, the Hamiltonian of equation 3.2 is given asH=X
c0 n2 Vc0 n † †+[(k + c0 n)â†0,k â0,k +(â â
+ â0,k â0,−k )(3.4)
2
2 0,k 0,−kk6=0X+{(k + mp + q + c2 n)(â†m,k âm,k )} + c2 n(â†1,k â†−1,−k + â1,k â−1,−k )],
m=±1

40

25
20

Phononυ=0Ek, υ/c2n15transverse
magnonυ=-110transverse
magnon
υ=1

5

0

0.5

1.0

1.5

2.0

2.5

3.0

kξs

Figure 3.1: Bogoliubov dispersion relations in the EAP phase. There exist threemodes of the excitation spectra-the phonon, and two transverse magnon modes.
Parameters: c2 n = 14.0h, c0 n = 882h and ξs = 2.2 µm corresponding to ourexperiment conditions, and p = 0.5c2 n, q = 0.8c2 n.
where V is the volume of the system and n = N/V . To diagonalize Eqn. 3.4,
apply Bogoliubov transformations:â0,k = hk b̂0,k − gk b̂†0,−k(3.5)

â±1,k = u±,k b̂±,k − v±,k b̂†∓,−k ,(3.6)

where hk , gk , u±,k and v±,k are real satisfying |hk |2 − |gk |2 = 1 and |u±,k |2 −|v±,k |2 = 1. After some calculation, we find that the Bogoliubov Hamiltonian isexpressed
H = Eg +X[E0,k b̂†0,k b̂0,k + E+,k b̂†+,k b̂+,k + E−,k b̂†−,k b̂−,k ],

(3.7)

k6=0

where Eg is the ground-state energy and the energy spectra are given bypk (k + 2c0 n) associated with the phonon excitationsp
E±,k = (k + q)(k + 2c2 n + q) ± p magnon excitations.
E0,k =

41

(3.8)
(3.9)

The two magnon excitations are degenerate when p = 0. These dispersion relations are described in Fig. 3.1.
When the quadratic Zeeman energy, q becomes negative, E±,k is imaginary and excites the two magnon modes, leading to the EAP-to-EPP phasetransition. The number of atoms in the |mF = ±1i grows exponentially [12]followinghn̂±1,k i = hâ†±1,k (t)â±1,k (t)i

(3.10)= f (E±,k , t)hn̂±1,k (0)i + c n 22|E±,k |

sinh(|E±,k |t/~)(hn̂∓,−k (0)i + 1)

∼ e2|E±,k |t/~ ,where the first term is related to the thermal fluctuations and the term ofhn̂∓,−k (0) + 1)i is associated with the quantum depletion and the quantum
vacuum fluctuations. For small |q|  c2 n, the magnon modes with k < kq ≡p2|q|m/~ are excited with an assumption that the thermal fluctuations are
insignificant. The instability rate, Γq , is given by the most unstable magnon
pmode and Γq = 2|E±,k=0 |/~ ≈ 8|q|c2 n/~. Thus, we can define the time and11length scales of tq = 2π/Γq ∝ |q|− 2 and lq = 1/kq ∝ |q|− 2 . We can expect that1

the scaling behavior depends on the quench depth, |q| 2 .

3.2

Experimental setup & Methods

Preparation of initial condensate in the EAP state
Our experiment begins by preparing a BEC of 23 Na atoms in the |F =1, mF = 0i state [41]. The condensate is confined in an optical potential withtrapping frequencies of (ωx , ωy , ωz ) = 2π × (3.8, 5.5, 400) Hz. The condensate
42contains N c ≈ 8.0 × 106 atoms and its Thomas-Fermi radii are (Rx , Ry , Rz ) ≈(232, 160, 2.2) µm. The sample preparation is carried out in a magnetic field of
Bz = 0.5 G along the z direction. Initially, the sample has only mz = 0 spincomponent and its thermal fraction is less than 10%. To initiate the transition
dynamics to the EPP phase, we first adiabatically ramp Bz to 33 mG for 0.2 s
and then, we suddenly turn on a microwave field to change the quadratic Zeeman
energy to a negative value q [51, 86]. The residual field gradient was measured
to be less than 0.1 mG/cm.
In our experiment, q/h ranges from −1.4 Hz to −20 Hz. The excitationenergy of the initial state after the quench is given by |q| with respect to the
EPP ground state, which is much smaller than the condensate chemical potentialµ = h × 880 Hz. Therefore, the subsequent evolution of the condensate can beapproximated as pure spin dynamics, not involving density excitations. For the
peak atomic density n of the sample, the spin interaction energy is c2 n/h =√14 Hz [87] and the spin healing length is ξs = ~/ 2mc2 n ≈ 4.0 µm. Because
the condensate thickness Rz is smaller than ξs , the spin dynamics in the oblate
condensate is effectively 2D.
SG imaging
The spin texture of the condensate is examined by taking an absorption
image after SG spin separation. After releasing the trapping potential, we apply
a magnetic field gradient pulse along the x direction and let the three mz =1, 0, −1 spin components are spatially separated for a 24-ms time of flight. The
image reveals the density distributions of the individual spin components. The
applied field gradient was slightly inhomogeneous to make the mz = ±1 spincomponents expand differently. We used F = 1 imaging and calibrated the43absorption coefficients for each spin component [Chap. 2].
Magnetization imaging
Spin-sensitive in-situ phase-contrast imaging is also employed to obtainfurther information on the spatial magnetization structure of the condensate [Chap. 2].
The probe beam frequency is tuned to give a signal proportional to the axial
magnetization Mz , i.e., the density difference of the mz = ±1 spin components.
The contribution of the mz = 0 spin component to the imaging signal is not
significant and we interpret the phase-contrast image as the axial magnetization
distribution Mz (x, y) of the condensate.
Measuring the spin turbulence: Power spectrum
We first obtained a reference image by averaging over ten images thatwere taken for the same experiment and subtracted the reference image fromindividual images to remove systematic fringes not related with the spin texR 2 i~k·~rM (~r) 2zture of the sample. Then, the power spectrum P (~k) =
was
d~r e
obtained by averaging the squared Fourier transforms of the subtracted images
and subtracting the photon shot noise level. We introduce a relative spectrumP̃ (~k) = P (~k)/Peq (~k) where Peq (~k) is the average spectrum of samples at thermal
equilibrium in the EPP phase. Because Peq is measured with the same imaging system, the relative spectrum P̃ provides spectral information free from thesystematic modifications of the imaging system. For the determination of theequilibrium spectrum Peq , we selected samples for q/h = −1.4 Hz at t = 5 s,in particular, without having HQVs because HQVs can affect the spectrum dueto their magnetized cores and magnon excitations generated by their collisional44

motions [84]. P̃ (~k) was isotropic and we obtained a 1D spectrum P̃ (k) by azimuthally averaging it over |~k| = k.

3.3

Emergence of Spin turbulence
Figure 3.2, 3.3 and 3.4 display image data sequences of the quenched

condensate for different q values from −1.4 Hz to −20 Hz. After a short holdtime, an irregular spin texture starts appearing in the condensate. It is clearlyshown that the mz = ±1 components are spatially separated from the mz = 0
component [Figs. 3.2(c, e) and 3.3(c, e)], which results from the immiscibilityof the mz = ±1 components from the mz = 0 component [26]. Spin domains
formed by an equal mixture of the mz = ±1 components have d~ ⊥ ẑ, where
the azimuthal direction of d~ is determined by the relative phase of the two
spin components. It is observed that the irregular spin texture emerges first in
the center region of the condensate and expands over the whole condensate. We
attribute it to the inhomogeneous density distribution of the trapped condensate√
because Γq ∝ n.
The appearance of the irregular spin texture is also observed in the magnetization image [Figs. 3.2(d, f) and 3.3(d, f)]. In the spin-exchanging process
where two mz = 0 atoms are scattered into a pair of mz = +1 and −1 atoms, thequadratic Zeeman energy is converted into the kinetic energy of the mz = ±1
atoms, imparting opposite momenta to the two atoms. Thus, spin currents are
generated in the mz = ±1 spin domains and axial magnetization develops at
the domain boundaries. The irregular structure in the Mz image constitutes anobservation of spin turbulence having a complex spin current pattern.
As the hold time t increases, the mz = 0 spin component keeps depleting45

(a) Easy-Axis Polarzyxq/h=-10.4 Hz(c) mz=1

mz=0

mz=-1

100 µm(d)t=20 ms

nematic director

Quenching

60 ms

100 ms

180 ms(b) Easy-Plane Polar

700 ms

HQVs

OD

2

3000 ms
0 arb.
units

Mz

1

-1 arb.
units

Figure 3.2: Phase transition from the EAP phase to the EPP phase of an antiferromagnetic spin-1 BEC. (a) The spin nematic director d~ is initially aligned withthe axial z direction, and (b) the phase transition to the EPP phase is triggeredby a sudden change in the quadratic Zeeman energy, q. (c) Absorption imagesof the mz = 1, 0, −1 spin components after 24 ms SG spin separation, and (d)in situ axial magnetization images Mz (x, y) for various hold times t after thequench to q/h = −10 Hz. HQVs are identified with density-depleted holes in
the SG images and their magnetized cores in the Mz images.

46

(c) mz=1

q/h=-5.4 Hz
mz=0

mz=-1

100 µmOD2(d)(e) mz=1q/h=-20 Hz
mz=0

mz=-1

t=40 ms

t=20 ms

100 ms40 ms

140 ms

100 ms

160 ms

160 ms

600 ms

600 ms3000 ms
0 arb.
units(f)

3000 msMz

1-1 arb.
units

Figure 3.3: Phase transition dynamics from the EAP phase to the EPP phaseof an antiferromagnetic spin-1 BEC after quench to (c) and (d) q/h = −5.4 Hz
and (e) and (f) −20 Hz. (a) and (c) Absorption images of the mz = 1, 0, −1
spin components after 24 ms SG spin separation, and (b) and (d) in situ axial
magnetization images Mz (x, y) for various hold times t.
and the spin texture becomes more complex. In particular, the length scaleof the spin texture becomes smaller, implying a direct energy cascade in thespin turbulence. The condensate eventually relaxes into the EPP phase withthe mz = 0 component vanishing. In the final state, HQVs are observed asmagnetized point defects in the Mz image [41] and also identified with densitydepleted holes in the SG image.

47

3.3.1

Spin-wave excitation

It is apparent in the comparison between the two image data sets in Fig. 3.2
and Fig. 3.3 for q/h = −5.4 Hz, −10 Hz and −20 Hz that when the system iscloser to the critical point, the time and length scales of the quench dynamics
become slower and larger, respectively, which is consistent with the theoretical
anticipation from the dynamic instability of the initial state. The length scaleof the spin texture was even larger with lower |q|. For |q/h| < 2 Hz, we observedthat the incipient spin texture shows a large ring-shaped pattern, which propagates toward the boundary and shortly becomes dismantled (Fig. 3.4(c,d)). The
ring-shaped pattern has the same ellipticity of the trapped condensate and webelieve that it corresponds to long-wavelength spin wave excitations induced by
the trapping geometry of the finite-size sample [45, 46].

3.4

Characterizing the quench dynamics & spin
turbulence
We first characterize the quench dynamics of the condensate by measur-ing the time evolutions of the fractional population, η, of the mz = 0 component and the magnetization variance, hδMz2 i, of the spin texture (Fig. 3.5).
P
Here η = N0c /N c and N c = i Nic , where Nic is the mz = i atom number ofthe condensate (i = 1, 0, −1) and determined from the SG absorption image[Fig. 2.4.1]. The thermal cloud contribution is subtracted using a Gaussian fit
to the outer thermal wing. For the measurement of hδMz2 i, we set the central
206 µm × 206 µm region of the condensate as the region of interest.
The quench evolution starts with a delayed, rapid decrease of η as ex48

(c) mz=1

q/h=-1.4 Hz
mz=0

mz=-1

100 µmOD2(d)(e) mz=1q/h=-2.7 Hz
mz=0

mz=-1

t=140 ms

t=120 ms

200 ms200 ms

300 ms

300 ms

300 ms

500 ms

350 ms2000 ms

600 ms0 arb.
units(f)

3000 msMz

1-1 arb.
units

Figure 3.4: (a) and (b) Spin-wave excitations for q/h = −1.4 Hz. A ring-shaped
oscillating spin texture appears and shortly becomes dismantled. (c) and (d)the EAP-to-EPP phase transition dynamics for q/h = −2.7 Hz. (a) and (c)Absorption images of the mz = 1, 0, −1 spin components after 24 ms SG spin
separation, and (b) and (d) in situ axial magnetization images Mz (x, y) for
various hold times t.

49

(a) 1.0
t1

0.6

q/h (Hz)
-1.4
-2.7
-5.4
-10
-20

0.4
0.2

t=3 s

|q|-0.53

0.1

0.8

η

(c)

2

6
4

1
10
|q|/h (Hz)XCamera image

0.0
6

0.01

2

4 6

0.1

2

4 6

1

2

4

(d)

(b)

30

OD (arb. units)<δMz2>

0.2

0.1

20
10
0

0.0
6

0.01

2

4 6

0.1

2

4 6

1

2

4

0

200

400

600

800

Pixel

Holdtime, t (s)Figure 3.5: Temporal evolutions of (a) the fractional population η of the mz = 0
spin component in the condensate and (b) the magnetization variance hδMz2 i forvarious q values. Each data point was obtained by averaging about five measurements, and its error bar denotes the standard deviation of the measurements.
The dashed lines are guide lines for eyes. The inset in (a) shows the time t1 forη(t1 ) = 0.8 as a function of |q|. The solid line denotes a power-law fit to thedata. (c) XCamera image taken for q/h = −5.4 Hz after hold time t = 3 s and(d) 1D optical density of (c).

50pected from the exponential growth of dynamically unstable magnon modes.
The early evolution of η is described as η(t) = 1 − b exp(Γq t), where b is a constant determined by the magnitude of magnetization fluctuations in the system.
In the inset of Fig. 3.5(a), we display the time t1 measured for η(t1 ) = 0.8 as afunction of |q|. A power-law fit to the experimental data gives an exponent of−0.53 ± 0.01, which is in quantitatively good agreement with the predicted scaling of Γq ∼ |q|1/2 . The measured t1 values give b ≈ 4.6 × 10−6 , which is slightly
higher than the value of b ≈ 3.3 × 10−6 estimated for quantum depletion [SeeAppendix B] at our peak atomic density [88], indicating thermal enhancementin the experiment.
The rapid decay of η is halted at a certain threshold value ηth and after
a short loitering period of a few tens of ms, η resumes its decay. Since hδMz2 i
rapidly increases when η undergoes this change, it is reasonable to infer that thecondensate enters a qualitatively different phase of the quench evolution, where
the role of the generated spin turbulence becomes significant. The thresholdvalue ηth monotonically increases from ≈ 0.4 for q/h = −1.4 Hz to ≈ 0.75 for
q/h = −20 Hz [Fig. 3.9(c)].
The spin turbulence develops further with maximizing its hδMz2 i andthen gradually relaxes with decreasing hδMz2 i. After a long hold time, t > 5 s,
the system is equilibrated with η ' 0 and stationary hδMz2 i. In our experiment,
the equilibrium value of hδMz2 i was insensitive to q because the final sample
temperature is mainly determined by the heating from microwave field dressing
and the evaporation cooling due to the finite trap depth. Note that for q/h =−1.4 Hz, hδMz2 i monotonically increases and saturates to the equilibrium value
over time. At the equilibrium, the thermal fraction was about 30% and thesample temperature was estimated to be T ≈ 100 nK. kB T  |q| and the
51

(b)

mF=0 fractional population

(a)

1.0

(c)

0.8

(d)

(a)
(b)

0.6

(a)

0.4

(b)

0.2
q/h=-2.1 Hz

q/h=-0.1 Hz

0.0
6

2

0.01

4 6

2

4 6

0.1

2

1

4

6

2

0.01

4 6

2

0.1

4 6

2

4

1

holdtime (s)Figure 3.6: The effect of the field gradient on the dynamical instability. (a)and (b) are taken during the spin-dipole oscillation at 500 ms. (c,d) Temporal
evolution of the phase transition dynamics. Darker and lighter data points aremeasured with the field gradients of (a) and (b), respectively. q/h =(c) -2.1 Hz
and (d) -0.1 Hz. The error bar is the standard error of the mean with at least
5 measurements.
thermal cloud was an equal mixture of the three spin components [89].
Interestingly, it seems that mz = 0 component condensate still exist forPc+long hold time t = 3 s [Fig. 3.5(c, d)]. N0tot /N tot = (N0c + N0th )/ mF (NmFth) ≈ 0.1. Because the condensate density of the mz = 0 component is too
NmF

dilute, it is not distinguishable from the thermal gas with the algorithm of extracting the fractional population (See section 2.4). In case of the EPP-to-EAP
phase transition (Chap. 4), we observe the uneven transverse magnetization distribution in the condensate for a long hold time. The observation manifests theexistence of the dilute EPP condensate. Thus, we can infer that the condensed
mz = 0 atoms still exist but the density is too dilute to spin-exchange collide.
Effect of the field gradient
The magnetic field gradient influences on the quench dynamics throughphase transition dynamics. The field gradients, illustrated in Fig. 3.4(a) and52

(b), are measured at 500 ms spin-dipole oscillating time. It seems that bothfield gradient is almost identical1 . But we can identify the magnitude of field
gradient (a) is larger than the magnitude (b) by comparing the dynamical instability [Fig. 3.4(c,d)].

3.4.1

Time-Space scaling

To investigate the spatial structure of the generated spin turbulence, weR~measure the power spectrum of the magnetization distribution, P (~k) = | dr2 eik·~r Mz (~r)|2 .
Figures 3.7(a,c) and 3.7(b,d) show the evolution of P̃ (k) for q/h = −10 Hz and−20 Hz, when hδMz2 i increases and decreases, respectively. As noticed in thevisual examination of the images in Figs. 3.2(d, f) and 3.3(d, f), the P̃ measurement results show that the spin turbulence develops from low to high wave
numbers k, i.e., from large to small length scales [Fig. 3.7(a)]. When hδMz2 idecreases, the power spectrum decays towards the equilibrium level, P̃eq = 1,
where the spectral strength subsides faster in the lower-k region [Fig. 3.7(b)].
Over the whole growth and decay evolution, the spectral center-of-mass of P̃ (k)keeps moving towards high k and this observation suggests a direct energy cascade of spin wave excitations in the spin turbulence.
In Fig. 3.7(a), it is noticeable that when the spin turbulence is generated,
the spectral slope is formed in P̃ (k) and propagates to high-k regions in aself-similar manner. In Fig. 3.7(c), we display P̃ (k) at the time when hδMz2 i
is maximum for various q values and we observe that the spectral slopes are
1

You might say that (b) seems to show stronger field gradient than (a). In my experience,
the field gradient of (a) is stronger. In (a), mz = +1(−1) condensate is dominant in theright(left) region, manifesting there exist subtle spin-dipole oscillation. But, there is no spindipole oscillation in (b). The black region in (b) appears due to the higher order of the field
gradient, O((Bij xj /B0 )2 ), from Eqn. 2.2.

53

(a) 6t (ms)
60-80
100
120-140
160-180

4
2

~
P(k)

10
6
4

(b) 6
2

10
6
4
2

2

1

1
6

6

8

1

2

t (ms)
4060
80-120140-180

4
2

10

~
P(k)

6

4

6

8

1

kξs

(c) 6

2

4

kξs

(d) 6

t (ms)
140-180
400-500
900-1000
2000
5000

4
2

10
6
4

6
4

2

2

1

1
6

6

8

1

2

6

4

8

1

2

q/h (Hz)
-1.4
-2.7
-5.4
-10
-20

4

2

2

10

10
6
4

6
4
2

2

1

1
6

8

1

2

6

4

kξs

4

kξs

(f) 6

4

6

6

kξs

(e) 6

~
P(k)

t (ms)
160-180
200-400
800-1000
2000
5000

4

6

8

1

2

4

kξs

Figure 3.7: Relative power spectra P̃ (k) of the axial magnetization Mz (x, y) of
the quenched BEC. Evolution of P̃ (k) for (a), (b) q/h = −10 Hz and (c), (d)−20 Hz when hδMz2 i (a), (c) increases and (b), (d) decreases [Fig. 3.5 (b)]. P̃ (k)(e) at the time when hδMz2 i is maximum and (f) at t = 5 s when the quenched
BEC is thermally equilibrated. The reference spectrum P̃ (k) = 1 was obtained
from the thermal equilibrium samples at T ≈ 100 nK, having no HQVs.

54

almost identical while the characteristic wave number increases with increasing|q|. Power-law scalings in the spin turbulence of antiferromagnetic BECs werepredicted by Fujimoto et al. [90, 91], but further analysis of our experimental
data is limited by a lack of quantitative understanding of Peq (k). In Ref. [92],Symes et al. calculated the static structure factor of the antiferromagnetic BEC
in the EPP phase at low temperatures. Our sample temperature of kB T /(c2 n) ≈150 is too high to be extrapolated from their results.
In Fig. 3.7(d), we display P̃ (k) at a long hold time t = 5 s, when thesample is thermally equilibrated. For high |q|, the spectral strength in k ≤ 1/ξs
is still noticeably higher than the equilibrium level P̃eq = 1. This is due to thepresence of HQVs and we confirmed it by correlating the deviation magnitudein the low-k region with the HQV number (Fig. 3.10).
Spin turbulence generation is seeded by the amplification of dynamicallyunstable magnon modes in the initial EAP state, in which the characteristic
time and length scales are given by tq ∼ |q|−1/2 and lq ∼ |q|−1/2 . To examine thescaling properties of the subsequent development of spin turbulence, in Fig. 5(a)we display the time evolution data of hδMz2 i as functions of the rescaled timet̃ ≡ t/tq . For all q, hδMz2 i starts its rapid increase at t̃1 ≈ 1.7 and becomesmaximized in a range of 5 < t̃ < 10. In Fig. 5(b), we display the relative power
spectra P̃ (k) for maximum hδMz2 i as functions of the rescaled wave numberk̃ ≡ k/kq . All the spectra collapse into a single line for 1.5 < k̃ < 4. Thisobservation demonstrates that the length scaling lq ∼ |q|1/2 of the initial stateis preserved in the subsequent turbulence generation dynamics.

55

(a)

<δMz2>

0.2

0.1

0.0
6

2

1

4

6

t/tq

6
4

(b)

2

4

6

q/h (Hz)
-1.4
-2.7
-5.4
-10
-20

2

10
6
4

~
P(k)

10

2

1
6
6

8

2

1

4

6

k/kq

Figure 3.8: Time-space scaling in the spin-turbulence generation. (a) p
Evolutionof hδMz2 i as a function of the rescaled time t̃ = t/tq , with tq = h 8|q|c2 n.
The vertical dotted line denotes the time, when η = 0.8, and the gray region
indicates the time window for hδMz2 i to be maximized. (c) Relative spectra P̃ (k)with maximum hδMz2 ip[Fig. 3.7 (d)] as functions of the rescaled wave numberk̃ = k/kq ˜, with kq = 2|q|m/h.

56

3.5

Spin population relaxation
As the spin turbulence develops, η exhibits different decay behavior forη < ηth [Figs. 3.9(a, b)]. In Fig. 3.9(d), we display the decay curves of η forvarious q values as functions of td = t − t2 , where t2 is the starting time of
the turbulence phase. η shows a nonexponential decay, where the relative decay
decreases as η decreases. In the turbulence phase, the mz = 0
rate γ = − η1 dηdt
spin component, being spatially separated from the mz = ±1 spin components,
forms axial polar domains in the condensate (Fig. 3.2-3.4). Since the density ofthe mz = 0 atoms in the spin domains is regulated by the condensate chemical
potential, a local two-body decay process would result in a constant decay of η,which cannot explain the observed nonexponential decay. The inset of Fig. 3.9(d)is a log-log plot of the same data with adjusted time offsets. The exponent
determination from a power-law fit to the data is not reliable due to its highsensitivity of the time offsets.
In our experiments, the decay rate γ is found to be insensitive to q, which
is different from a general expectation of higher γ for higher excitation energy,|q|. It might be speculated that small domain size for high |q| proportionallysuppresses γ, or it may be due to the high thermal energy of the system, kB T |q| [93, 94]. In Fig. 3.9(e), we characterize the relaxation of the thermal cloud
during the turbulence phase for q/h = −10 Hz. The mz = 0 spin fractionof the thermal cloud relaxes within 0.2 s to the equilibrium level of one third
and the thermal fraction of the sample increases from ≈ 20% to ≈ 30%. The
relaxation time is compatible to 1/γ = 0.2 s ∼ 0.6 s but the decrease of γ is
not in accord with the increase of the sample temperature. The spin turbulence
relaxation dynamics including the coupling to the thermal cloud [95] merits

57

0.5

(d) 1

0.8 (c)(b)q/h (Hz)
-1.4
-2.7
-5.4
-10
-20

6

0.6

4

0.0
0.01 0.1

1 0.01 0.1

0.4
1

1

10

η
(f) 1.0

0.1

td' (s)

0.1

1

4

η0
η−1

2

(e) 0.8

0.6

q/h=-5.4 Hz, -20 Hzηthermal

Fraction

Fractional population

0.1

6

η1

0.8

td' -1.4

2|q|/h (Hz)

t (s)

1

η

(a)

ηth

η

1.0

0.4
0.2
0.0

0.6

Thermal fraction

0.4
0.2
0.0

0

1

2

3

4

5

0.0

holdtime, t (s)

0.2

0.4

td (s)

0.6

0.8

1.0

Figure 3.9: Decay of η in the turbulence phase. η(t) for (a) q/h = −5.4 Hz and(b)−20 Hz; the threshold values ηth are indicated by the right arrows. (c) ηth
versus |q|. (d) Decay curves of η as functions of td = t−t2 , where t2 is the startingtime of the turbulence phase. The inset shows the same data with adjusted timeoffsets, and the solid line is a power-law fit to the data. (e) Relaxation of thethermal clouds. The solid symbols denote the thermal fraction of the sample,
and the open symbols denote the mz = 0 spin fraction of the thermal cloud.
Each data point is obtained by averaging about five measurements, and its error
bar denotes the standard deviation of the measurements. (f) The population
evolution of each spin components for q/h = −20 Hz. The inset images areabsorption images of mz = ±1 clouds for hold time 5 s.

58

2

10
|q|1

Nv

6
4
2

Hold time
3s
4s
5s

1
6
4
2

1

2

4

6

8

10

2

|q|/h (Hz)Figure 3.10: HQV number Nv versus |q|. Nv was measured by counting densitydepleted holes in the SG images of the mz = ±1 spin components. Each data
point is the average of 15 measurements for the same experiment, and its error
bars indicates the standard error of the mean. The blue dashed line denotes a
power-law fit to the data in the range of |q|/h ≤ 10 Hz for t = 3 s.
further investigation in future experiment.
You can recognize the different lifetime of atoms in |mF = ±1i for large
magnitude of |q| due to the different coupling strength. For t = 5 s, the fractional
population of mz = 1 and −1 components are 0.56 and 0.43, respectively. The
sizes of mz = ±1 clouds in Fig. 3.4(e) are also vividly distinguishable for t = 3 s.
The detuning ∆ = −150 kHz for q/h = −20 Hz (see section 2.3).

3.6

Topological Defects: Half-quantum vortex
At long hold times t ≥ 3 s, when the mz = 0 spin component almost

vanishes in the condensate, HQVs are unambiguously identified with densitydepleted holes in the SG images (Fig. 3.2). We measure the HQV number, Nv ,and find that it increases with increasing |q| and saturates for |q|/h > 10 Hz
59

(Fig. 3.10). When the spatial size of spin domains in the quenched condensate
scales with lq , the number of point defects created in the 2D system is expectedto scale as Nv ∝ lq−2 [7]. A power-law fit to the data for t = 3 s in the range
of |q|/h ≤ 10 Hz gives an exponent of 1.0 ± 0.2, which is consistent with thelength scaling lq ∼ |q|1/2 observed in the measurement results of P̃ (k). The
saturation of Nv for high |q| might indicate another length scale involving in
the defect creation, such as the spin healing length ξs [96], but its possible effectwas not observed in P̃ (k). Since HQVs can be pair-annihilated in the turbulence
relaxation process, leading to a non-exponential decrease of Nv [97], the HQVnumber measured at long hold times may not linearly reflect the initial Nv ofthe quenched BEC.

3.7

Outlook
We can extend this work to a deeper quench regime with |q| > 2c2 n. In

this case, the dynamic instability of the initial EAP state is driven by magnon
pmodes with finite wave numbers centered at kq = 2m(|q| − c2 n)/~ [98]. It
would be of high interest to examine how the energy injected at finite wave
numbers flows in the subsequent evolution of spin turbulence [99]. Since thelength scale ∼ 1/kq becomes comparable to and even smaller than the spin
healing length ξs , it might be speculated that qualitatively different turbulence
states would emerge in this high |q| regime. Figure 3.11 shows preliminary datafor q/h = −40(2) Hz and −80(10) Hz. The spin excitations occur in extremely
small length scales and the dynamical instability for |q|  c2 n is not measured
by the mean-field theory [Fig. 3.11(b)].
Another extension of this work is to study the effects of the quench rate60

(a) 1.0q/h=-80 Hz (b) 0.2

q/h=-40 Hz

0.8
0.6

η

t1

0.4

0.1
0.6

0.2

0.4

0.0
6

2 46

0.01

2 46

0.1

1

2 4 6

2 46

0.01

2 46

0.1

holdtime, t (s)

1

2 4

1

2

4 6

10

2

46

|q|/h (Hz)(c)

t=63 ms
(d)

mF=1

mF=0

mF=-1

t=57 ms

Figure 3.11: (a) Temporal evolution of η for q/h = −40(2) and -80(10) Hz. The
dotted lines are guide lines for eyes. (b) shows the time t1 , satisfying η(t1 ) = 0.8
as a function of |q|. Black circles are calculated t1 from the fitting function
1 − η(t) = exp(Γq t) and red asterisks are experimentally measured t1 from(a,b). The dashed line is the expected t1 from the mean-field theory. Absorption
images for each spin component at t = t1 with q/h = (c) 40 Hz and (d) 80 Hz,
respectively.

61

across the critical point [81, 96, 100]. In particular, the power-law scaling of the
HQV number after the quench is reminiscent of the Kibble-Zurek (KZ) mechanism [8–10]. We note that in the mean-field theory, the EAP-to-EPP phasetransition is described as first order; thus, the conventional KZ mechanism,
which involves a continuous phase transition, cannot be applied directly to oursystem. A generalization of the KZ mechanism could be studied with this system.
The turbulence of the spinor BECs represents a unique turbulence state
where the mass and spin superflows are entangled. Previous turbulence studies with spinor BECs were mostly focused on the case with ferromagnetic spin
interactions both experimentally [49, 52, 59, 101] and theoretically [102–104].
Our work demonstrates the peculiarities of the spin turbulence in an antiferromagnetic BEC, enabling a comparative study between the ferromagnetic andantiferromagnetic cases of the spin-1 BEC system [90, 91].

62

Chapter 4
The EPP-to-EAP phasetransition
In this chapter, we introduce the opposite directional phase transition to
the experiments in chapter 3, the EPP-to-EAP phase transition. Initially prepared multi-component BEC becomes a single component BEC through spin exchange collision losing the spin-degree of freedom, which is unique phenomenon
for spinor BEC. The symmetry breaking of the nematic director in the EPPto-EAP phase transition is similar to projected symmetry breaking in tachyon
condensation [111].
We observe a composite-topological defects, which consists of a domain
wall and HQVs. In this dissertation, the new type of a composite-soliton isnamed as a magnetic composite-soliton. Such a combined-topological defects
are usually predicted to be formed in the interface of two quantum phases.
During the inflation of early universe, complex interface physics can lead tocomplex walls bounded by strings and monopoles connected by strings [105].

63

It is extended to a He-3 superfluid system [106] and the experiments similar to the brane-antibrane annihilation have been performed [107]. Recently, acombined-topological defects in the interface between the ferromagnetic and theantiferromagnetic phases are suggested [109, 110].
Furthermore, according to Ref. [115], a magnetic composite-soliton mediates the KT transition from the normal gases to the exotic phase, called
the paired superfluid, in an antiferromagnetic spin-1 condensate. The paired
superfluid lies between the KT transition and the Ising transition. Thus, theobservation of a magnetic composite-soliton is significant itself.
The magnetic composite-solitons have analogy to strings connecting point
defects in the nematic liquid crystals [112] and θ-solitons combined with thespin-mass vortex in 3 He-B [113, 114]. The magnetic composite-solitons is allowed by the manifolds of order parameter, [U (1) × S 2 ]/Z2 , where the system isinvariant under the transformation of (θn , θs ) ↔ (θn +π, θs +π). We focus on the
structure of dynamically formed magnetic composite-soliton and its relaxation
process after quench.
• Seji Kang and Yong-il Shin, “Observation of magnetic composite-solitons
in an quasi-2D antiferromagnetic spinor Bose-Einstein condensate”, inpreparation.
• Seji Kang and Yong-il Shin, “Suppressing the amplification of the transverse spin fluctuations by the axial spin fluctuations”, in preparation.

64

4.1

Dynamic instability for the EPP phase
Basics of the Bogoliubov theory is identical to section 3.1 and the insta-

bility of the initial EPP state is expected to show power-law scaling behavior
1

of |q| 2 after the quench.
In case of the EPP phase, the field operator is given as

 

ψ
δ ψ̂
 +1   +1  Ψ̂ =  0  +  δ ψ̂0  .

 

ψ−1
δ ψ̂−1(4.1)

Due to the nature of antiferromagnetic interaction, the density fluctuation andspin fluctuations are coupled asâsx ,k = â0,k

(4.2)

1
âd,k = √ (â1,k + â−1,k )2
1âsz ,k = √ (â1,k − â−1,k ),
2

(4.3)
(4.4)

where âd,k describes the density fluctuation, and âsx ,k and âsz ,k describe thespin fluctuations around the x and z axes, respectively. In our experiments, weprepared the condensate with zero total magnetization (p = 0). So, the phonon
and magnon excitations are decoupled. The Hamiltonian is expressed asH=X
c0 n 2 V
c0 n † †+ qN +[(k + c0 n)â†d,k âd,k +(â â + âd,k âd,k )(4.5)
2
2 d,k d,k
k6=0
X
c2 n † †+{(k + c2 n)â†ν,k âν,k +(âν,k âν,k )} − qâ†sx ,k âsx ,k ],2
ν=s ,s
x

z

65

25
20
Phononυ=1

Ek, υ/c2n15transverse
magnon
υ=0

10

axial
magnon
υ=-1

5

0

0.5

1.0

1.5

2.0

2.5

3.0

kξs

Figure 4.1: Bogoliubov dispersion relations in the EAP phase. There exist threemodes of the excitation spectra-the phonon, the transverse magnon, and theaxial magnon modes. Parameters: c2 n = 14.7h, c0 n = 927h and ξs = 2.0 µm
corresponding to our experiment conditions, and p = 0.5c2 n, q = −0.8c2 n.
Apply a Bogoliubov transformations followingâsx ,k = u0,k b̂0,k − v0,k b̂†0,−k(4.6)

âd,k = u+,k b̂+,k − v+,k b̂†+,−k(4.7)

âsz ,k = u−,k b̂−,k − v−,k b̂†−,−k(4.8)

and we find that the energy spectra is given bypk (k + 2c0 n) associated with the phonon excitationspE0,k = (k − q)(k + 2c2 n − q) transverse magnon excitationspE−,k = k (k + 2c2 n) axial magnon excitations.

E+,k =(4.9)(4.10)(4.11)

These dispersion relation is described in figure 4.1. E−,k is gapless. Thus, thethermal population of the axial magnon modes generate the spin fluctuationsin the EPP condensate [41].

66When q > 0, E0,k becomes imaginary and leads to the transverse magnon
excitations. The amplification of the transverse magnetization fluctuation induces the EPP-to-EAP phase transition. Because the energy spectrum of thetransverse magnon mode is identical to that of the magnon modes in the EAP√
state (Eqn, 3.8), the instability rate Γq = 2|E0,k |/~ ∝ q. The characteristic√
time and length scales are also scaled as 1/ q for q  c2 n.

4.2

Experimental setup & Methods

Preparation of initial condensate in the EPP state
The typical number of total 23 Na atoms are ≈ 6.8 × 106 and the fractionof normal gases is around 15%. The condensate is confined in the pancakeshaped optical dipole trap with ωx,y,z = 2π × (4.5, 6.3, 460) Hz and its ThomasFermi radii are Rx,y,z = (201, 143, 2.0) µm. The condensate chemical potential µ = h × 927 Hz. Then, the DC magnetic field, Bz , adiabatically ramps
down from 0.5 G to 52 mG. Sodium atoms are prepared in the superposition of |F = 1, mF = ±1i states by π/2 spin rotation from the sample in the|F = 1, mF = 0i state [Chap. 2]. The nematic directors are spatially co-aligned
in x-direction [Fig. 4.3(a)]. We initiate the EPP-to-EAP phase transition bysuddenly increasing q. Initial phase transition dynamics in the EPP sample canbe explained by the mean-field theory in section 4.1. In order to keep the residual
field gradient less than 0.1 mG/cm, q is tuned by the microwave dressing technique. In our experiment, q/h ranges from 1.2 Hz to 17 Hz much smaller than thecondensate chemical potential and the spin healing length is ξs ≈ 4.5 µm< Rz .
Hence, the subsequent evolution after quench can be regarded as spin dynamics

67

Pth(k) (arb. units)

2
1
0.6
0.4
0.2
0.1
0.2

OD (arb. units)01

0.4 0.6

1

2

3 4 5

Spatial frequency, kξs

Figure 4.2: Right: Single in situ image of thermal gas. Left: Power spectrum ofa trapped thermal gas at high temperature. The solid line is a sigmoid function
fit with offset to the spectrum.
purely confined in 2 dimension.
Imaging methods
The spin texture of the condensate is examined by same methods in theformer experiments of 24 ms TOF-SG imaging and spin-sensitive in situ phasecontrast imaging. We use the spin-sensitive phase-contrast imaging to display
the axial magnetization distribution Mz (x, y). In addition, by applying π/2 rf
pulse, the transverse magnetization in one-direction, Mx (x, y), is also taken.
Measuring the length scales of the magnon modes

We measure the power spectrum of the mz = 0 density distribution,R
P (k) = | dreik·r δnc0 (r)|2 , where δnc0 (r) = nc0 (r) − n̄c0 (r) is the density fluctuation of the mz = 0 component. The power spectrum is obtained from at least 20
measurements. We subtract the average density distribution, n̄c0 (r), from the column density nc0 (r) to remove uncontrolled modulations in the trapping potential.
The power spectra for various q are extracted from the central 521 µm × 521 µm
68

regions of mz = 0 absorption images, when 0.2 < 1 − η < 0.4 [Fig. 4.5]. We
display the power spectra in the narrow range of k < 1/ξs , where P (k)s are notdistorted by the limitation of our system’s imaging sensitivity.
We measure the power spectrum of trapped thermal gas at high temperature [126, 127] and the noise power spectrum is displayed in Fig. 4.2. Whenthe thermal wavelength is shorter than the imaging resolution, the noise power
spectrum appear uncorrelated. Thus, the offset level in the high frequency region
is determined only by the photon shot noise. However, our imaging resolution
is at least 4.0 µm, less than 2 pixels(= 2 × 2.58 µm). Thus, the power spectrain the high frequency region contains not only the photon shot noise level, butalso the atomic density signals.

4.3

Characterizing the quench dynamics
Figure 4.3 display image data sequences of the quenched condensate for

q/h = 10.4 Hz. The EPP condensate becomes unstable for positive q, as thetransverse magnon excited. Thus, atoms are transmuted into mz = 0 spin state.
In the early stage, the transition occurs coherently and the finite-size trapping
geometry causes the ring-shaped transfer [45,46]. Then, the condensate recovers
close to its initial state in very short time about 60 ms. Then, the transition
becomes complex. For hold time around 100 ms, very sharp domain walls andskyrmions appear and split into small scales in the subsequent evolution. At
the end of the transition, singly-charged vortices are identified as the densitydepleted holes in mz = 0 absorption images.
The oscillatory behaviors in the early transition dynamics are not observed in the previous experiments on the EAP-to-EPP phase transition. We
69

(b) Easy-Axis Polar(a) Easy-Plane Polar
Quenchingzyx(c) 5 ms

nematic director

20 ms

Vortices

40 ms

60 ms

80 ms

100 ms200 ms

400 ms

800 ms

1s

mz=1

OD
2

mz= 0

mz=-1
100 µm

0
arb.
units

(d)
Mz, y
1

(e)

-1
arb.
units

Figure 4.3: Phase transition from the EPP phase to the EAP phase of an antiferromagnetic spin-1 BEC. (a) The initial spin nematic director d~ is coherentlyaligned in xy-plane and (b) the phase transition to the EAP phase is triggeredby a sudden change of the quadratic Zeeman energy, q. (c) Absorption imagesof the mz = 1, 0, −1 spin components after 24 ms Stern-Gerlach spin separationand (d) in situ transverse magnetization images in y-direction, My (x, y), and(e) in situ axial magnetization images,Mz (x, y), for various hold times t after
the quench to q/h = 10.4 Hz. Quantum vortices with single mass circulation
are identified with density-depleted holes in the mz = 0 optical density images.

70

conjecture that the spin turbulence suppress the coherent oscillation. Instead,
there exist a short loitering period at ηth (see section 3.4). Furthermore, when
the EPP sample initially has spin fluctuations, the transition dynamics show nocoherent oscillation. We will discuss this again in section 4.5.

4.3.1

Measuring the transverse magnon excitation

Small spin fluctuations in the transverse magnetization is amplified, inducing the EPP-to-EAP phase transition. These transverse magnon excitation
is identified by the transverse magnetization images in Fig. 4.3(d). It is notedthat the ferromagnetic spin components in the horizontal direction, Mx , aregenerated during the evolution. For a short time around 40 ms, the transition occurs coherently and the transverse magnetization is distributed in the
ring-shaped manner. Even for long hold time over t = 800 ms, the transverse
spin fluctuations still exist without vivid axial magnetization. This observation
demonstrates that there exist sprinkled BECs in the mz = ±1 states, where
their density is too dilute to occur spin-exchange collision.

4.3.2

Time-Space scaling

We characterize the evolution of the quenched condensate by measuring
the temporal evolution of the fractional population, 1 − η, of the mz = ±1
components. Here, η is the fraction of the mz = 0 population, η = N0c /N c ,P
and N c = i Nic , where Nic is the number of the condensate in the mz = istate (i = 0, ± 1). The number of the condensate is determined from the SG
images [Fig. 2.4.1].
We also measure the phase transition dynamics for extremely small q ≈
71

0.06

|q|-0.45

0.04

tq

0.8

1-ηosc(b) 1.0(a) 1.0

1

q/h (Hz)
2.6
5.1
10.4

0.4
0.2

0.01

2

0

10

q/h (Hz)

0.0
6

4 6

0.1

2

0.6
0.4

0.6

4 6

holdtime, t (s)

1

2 3

10

20

30

(c) 0.20
tosc (ms)

1-η

0.02

0.8

|q|-0.44

0.10
0.06
0.03
1

10

q/h (Hz)

Figure 4.4: (a) Temporal evolutions of the fractional population 1 − η of the
mz = ±1 spin component in the condensate. Each data point was obtained byaveraging at least five measurements, and its error bar denotes the standard
error of the mean. The dashed lines are guide lines for eyes. The inset shows thetime tq for 1 − η(tq ) = 0.7 as a function of q. The black line denotes a power-law
fit to the data in the range of h × 1 Hz < q < 2c2 n. (b) The peak value of 1 − ηoscillation and (c) its hold time are displayed as a function of q. The black solidline is a power-law fit to the data in the range of h × 1 Hz < q < 2c2 n.
h × 0.4 Hz and for q ≈ h × 30 Hz & 2c2 n, but we analyze the data only in therange of 1 Hz ≤ q/h ≤ 2c2 n. It is predicted that the excited magnon modes
for q > 2c2 n have finite wave numbers and the quench dynamics might become
different from that for other q values as we mentioned in previous section. For
extremely small q value, the energy barrier between the mz = 0 state and thesuperposition of the mz = ±1 states is too low and no one knows exactly what
happens near the critical point, q = 0. Hence, the spin dynamics near the criticalpoint should be carefully treated.
The 1 − η decreases rapidly to ≈ 0.3, then it increases to the peak valueof 1 − ηosc , keeping the coherence of the condensate. The early evolution of 1 − ηis described by the exponential growth of the mz = 0 population, represented
72

(a)

2

(b) 2

0.8

6

kcξs

10

4

0.6

|q|0.38

106

10

2

6
4

0.4

P(k)

2
1

5

10

q/h (Hz)

105

4
2

104
4

0.2

6
4

q/h (Hz)
1.2
10.4
2.6
17
5.1
30

2
4

0.4

0.6

1

10

0.3

0.6

1

k/kq

kξs

Figure 4.5: (a) Power spectra P (k) of the mz = 0 distribution of the quenched
BEC in the range of 150 ms ≤ t ≤ 250 ms are displayed for various q. The
inset shows the characteristic wave number, kc , satisfying P (kc ) = 5 × 104 , as afunction of q. The black line denotes a power-law fit. (b) Same√ power spectra to(a) are displayed as a function of rescaled k/kq , with kq = 2mq/~ for variousq.
as η(t) = b exp(Γq t), where b is a constant determined by the magnitude of themagnetization fluctuations.
From the excitation of the dynamically unstable magnon modes, the in√stability rate, Γq , is proportional to q. We fixed b ≈ 0.0053 from the leastsquares fitting. In the inset of Fig. 4.4(a), we display the time tq measured for1 − η(tq ) = 0.7 as a function of q. A power-law fit to the experimental data in
the range of 1 Hz ≤ q/h ≤ 20 Hz gives an exponent of −0.45 ± 0.04, which√is in quantitatively good agreement with the predicted scaling of 1/ q. The
oscillatory transfer is described by the amplitude of 1 − ηosc and the peak time,
tosc . The 1−ηosc increases from ≈ 0.6 for q/h = 1.2 Hz to ≈ 0.8 for q/h = 17 Hz.
The tosc shows identical power law of −0.44 ± 0.05 to t0.7 , manifesting that theearly spin dynamics is coherently sustained. Then, 1 − η decreases following asingle decay curve regardless of q.
73

To investigate the length scales of excited magnon modes, we measure
the power spectrum of the density-density correlations of the mz = 0 atom
2
R
dreik·r δnc0 (r) , when 1 − η ≈ 0.3. The spectral slopes areclouds, P (k) =almost identical and the characteristic wave number increases depending on thequench depth, q. The power spectra collapse into a single line with rescaled√
k/kq for q ≥ c2 n ≈ 14.7 Hz, where kq = 2mq/~. We characterize the characteristic wave number as the wave number, kc , satisfying P (kc ) = 5 × 104 . The
inset of Fig. 4.5(a) shows the scaling of kc with the exponent of 0.38 ± 0.05,
which is marginally smaller than the predicted power-law of 0.51 . This might
be the quenched condensate for long hold time over 100 ms contains various decay process such as the domain coarsening dynamics and the vortex-antivortex
annihilation. For q > c2 n, the power spectra overlaps. This is because that the
spin healing length limits the minimum size of mz = 0 domains [96].

4.4

Dynamically formed topological objects
Interestingly, we observe long-lived topological defects over 500 ms. Gener-

ally, the topological defects, created via the sudden phase transition, are understood by the quantum noise seeds [7,11,117,118]. This explains the formation of
singly charged vortices as a result of the quenched phase transition, which will be
discussed in section 4.6. However, our observation is beyond the understandingof either the phase-ordering kinetics or the symmetry of an antiferromagnetic
spinor BEC.
Thanks to the [U (1) × S 1 ]/Z2 , various topological defects can be formed
1

The exact exponent should be obtained without the technical issues such as the aliasing [116].

74

|0〉 β

→
d

eiβ
1
-1

eiπ|0〉

Figure 4.6: Schematic description of the topological defects such as skyrmions,line solitons and magnetic composite-solitons. The nematic directors are indicated as arrows. The tilted angle along z-axis, β, represents local spin phases.
⊗ ( ) indicates the clockwise (counterclockwise) superflow.
as in Fig. 4.4 such as skyrmions, HQVs, and domain walls. But, we also observe a
new kind of topological defects, named as magnetic composite-soliton. A magnetic composite-soliton consist of two different topological defects, a domain
wall and HQVs. The combined-topological defect is investigated in terms of its
formation process and structure between the spatially separated two quantum
phases [105, 107, 109, 110]. In our experiments, the manifolds of ferromagnetic
and antiferromagentic order parameters are mixed and a magnetic compositesoliton can be understood in the full symmetry of spin-1 BEC, which is given
by U (1) × SO(3).
We can imagine that the nematic director, initially aligned in x-direction,
stands vertically as d k ẑ, at the end of dynamics, choosing either negative or
positive rotational direction. When two domains in contact have the phase difference of π, the domain walls can be created in the interface. The domain walls
can be regarded as bright-dark-bright (BDB) solitons [119], where the dark75

soliton resides in the |mF = 0i state and the bright solitons in the |mF = ±1istates. The BDB solitons are formed in the 2D system, which are extended
from the realization of the three component solitons in 1D ferromagnetic spinor
BEC [120, 121]. The solitons are allowed by the nature of antiferromagnetic interactions. Thus, the alignment of the nematic directors across a soliton changes
continuously from d to −d. In principle, these solitons have no magnetization,
but they are also discernible in the axial magnetization-sensitive in situ images.
We attribute this slight phase-separation to the uncontrollable edge enhancement of phase-contrast imaging, the imperfect cancelation of magnetic field
gradient and the subtle discrepancy between the mz = ±1 atom numbers [122].
For smaller q/h, the length scale of magnon excitations, lq becomes large√
as lq = 2π~/ 2mq. Thus, we can selectively create a single soliton under thecondition of Rxy /lq ≈ 2.5 [Fig. 4.7]. The possible spatial spin distributions arealso displayed as cartoon images. The sharp magnetic composite-soilton andskyrmion images are taken by 1 ∼ 2 out of 10 runs. When the spin fluctuations
are enhanced by the excitation of the axial magnon modes in the EPP condensate (see section 4.5), magnetic composite-solitons are formed more frequently
with same q/h values, satisfying Rxy /lq ≈ 2.5 (Compare to Fig. 4.15). In the
future, we could investigate the stability of skyrmions.
A large BDB soliton across the whole condensate bends because of thesnake instability [123–125] and splits. At the end of the soliton, there should
exist HQVs, having both spin and mass currents. The existence of spin currents
is easily confirmed by the sharp domain walls in the spin-separated absorption
images and their spin texture in Fig. 4.4 and Fig. 4.7(g-l). To explore the masscurrent, we use Bragg spectroscopy techniques to the condensate, consist of the
mz = 0 component and discuss the result in the next section.
76

(a)

(b)

(c)

(d)

(e)

(f)

mz=+1

mz=0
OD
2.0

mz=-1

(g)

0.0
arbi.
100 µm units

100 µm

(h)

(i)

(j)

(k)

(l)

Figure 4.7: Dynamically formed (a-e) magnetic composite-solitons and (f) askyrmion with q/h = (a-c) 2.6 Hz and (d-f) 1.0 Hz. The evolution times are (a)300 ms, (b) 200 ms, (c, f) 600 ms, and (d) 400 ms, and (e) 700 ms, respectively.
Absorption images are taken after 24 ms spin separation. (g-l) Example of spin
textures. Experiments conditions: (a-c) initial µ ≈ h × 880 Hz, c2 n = 14h
and RT F = (232, 160, 2.2) µm. (d-f) initial µ ≈ h × 927 Hz, c2 n = 14.7h and
RTF = (201, 143, 2.0) µm.

77mz=+1

mz=0

mz=-1

(a)

(c)

100 µm

(b)

OD
2.0

(d)

0.0
arbi.
100 µm units

Figure 4.8: Dynamically formed hog-nosed spin structures with q/h = (a) 2.0 Hz
and (b) 5.1 Hz. The evolution times are 200 ms. Absorption images are taken
after 24 ms spin separation. (c, d) Example of spin textures. Experiments conditions: (a-c) initial µ ≈ h × 880 Hz, c2 n = 14h and RT F = (232, 160, 2.2) µm.
(d, e) initial µ ≈ h × 927 Hz, c2 n = 14.7h and RTF = (201, 143, 2.0) µm.
Before embark on the next section about the Bragg spectroscopy for amagnetic composite-soliton, I briefly mention about an intriguing spin structures in Fig. 4.8. Hog-nosed spin structures are formed for every single sequence
under the condition of Rxy /lq ≈ 5. It is alluring results that the formation of
topological objects is determined by the interwinding between the excitations of
unique magnon modes and the finite size of a condensate. I describe the nematic
director distribution, but it needs more theoretical works to understand thesestructures. For more images, see section 4.7.

4.4.1

Magnetic composite-soliton: a domain wall terminated by HQVs

Here, we concentrate on the evolution of a magnetic composite-soliton,
having HQVs at the endpoints. A HQV has the fractional supercurrent circulation of h/2m with the aid of the half spin-winding [Chap. 1]. In this section, we78

introduce Bragg spectroscopy techniques, referred to [128], and experimental sequence. Then, we discuss about the mass current encircling magnetic-composite
soliton.
Bragg spectroscopy with spin separation
When a BEC having mass flow, ~v , is irradiated by a pair of counterpropagating laser beams, a two-photon process imparting moment ~q and energy ε to2qan atom occurs under the resonant condition of ε = 2m
+ ~q · ~v . Here, ~q = 2~kL x̂0and ε = ~δ, where kL and δ are the wave number and frequency difference ofthe two beams, respectively. Because of the Doppler effect, the scattering response in a BEC having a circulation is asymmetric with respect to the Braggbeam, and the vortex sign can be determined from the position of the scattering region relative to the vortex core. For supercurrent with counterclockwise
half-circulation, the velocity is given by ~v = (~/2mr2 )ẑ × ~r, where ~r is the position relative to the vortex core as the origin, the resonant condition is given byδd = δ − δ0 = −(~kL y 0 /mr2 ), where δ0 = 2~kL /m. See Fig. 4.9(d) and Fig. 4.10.

We spatially overlap the two laser beams with frequencies ωL and ωL + δ.
The frequency difference δ is generated by the two independently controlled
acousto-optic modulator (AOM). Two pairs of Bragg beams are the red-detuned
laser beams from the |F = 1i to |F = 20 i transition. The 1/e2 width of eachbeam is 1.8 mm  Rxy and each beam has intensity of I = 2.9 mW/cm2 .
In order to identify the spin distribution and the mass current simultaneously, we inject Bragg beams with 15 ms spin separation. We first turn off the
optical dipole trapping potential and apply a residual field gradient for 200 µs.
Then, a pair of counterpropagating Bragg beams is applied to the sample for
79

Spin
dynamics(a)B’
200 µs

Bragg
beam
img.
beam

820 µsScattered fraction

ODT

x’ x

ω

y
y’

ω+δ

0.2
0.1

δd/2π (Hz)scattered
+x’
BEC
mz=1
BEC

BEC

(b)

0.3

0.0
-1.5 -1.0 -0.5 0.0 0.5 1.0

14 ms 30 µs

time

0.4

mz=0
BEC

mz=-1
BEC

ω+δ
(c)

B’

ω

(d)

-x’

scattered
BEC

Figure 4.9: (a) Schematic of the Bragg spectroscopy and SG imaging sequence.
Two atomic clouds are scattered outward from the mz = 0 BEC in the ±x0
directions. (b) Scattered-out atom number fractions measured for a stationary
mz = 0 BEC as a function of δd without SG imaging Sequence. δd = δ −δ0 , where δ0 = 99.0 kHz. The blue and red circles denote the atom number
fractions of the red and blue atom clouds in (d), respectively. (c) The directionsof two pairs of counterpropagating Bragg beams (blue and red arrows), and spin
separation (black arrow). (d) Schematic illustration of Bragg spectroscopy withspin separation imaging in CCD.

80

820 µs. After 14 ms time-of flight, an absorption image is taken. The experimental sequence is described in Fig. 4.9. The Bragg spectrum of a stationary
mz = 0 BEC is obtained by measuring the fractional population of the scatteredclouds as a function of δ [Fig. 4.9(b)]. The center frequency is experimentallymeasured as δ0 /2π = 99.0 kHz.
Measuring the mass current at the ends of MCSs
By using this methods, we can identify the position and the polarizationof HQVs at the endpoints of the MCS, simultaneously. We set the detuned
frequency at δd = −0.9 kHz from the resonance frequency, δ0 , for a stationary mz = 0 condensate. A Bragg signal is constructed with the SB (x0 , y 0 ) =n+ (x0 , y 0 ) − n− (x0 , y 0 ), where n± (x0 , y 0 ) are the ±x0 -scattered densities, respectively.
Figure 4.10 displays the Bragg signals corresponding to the respective
spin separated absorption images. Note that the clear counterclockwise supercurrent signal at the end of a MCS in Fig. 4.10(d). The density is dilute aroundthe endpoint of a MCS, because atoms are scattered by the Bragg beams. In
addition, there exist the opposite mass currents between the two MCSs in
Fig. 4.10(e,f). This is the evidence of soliton splitting process, inferring thatthese two MCSs originally make up of a single BDB soliton.
A sharp endpoint of a composite-soliton, transferring mass flows, is possible to lead the rotational motion and its shape becomes spiral-like. In recent
studies [129], it is suggested that a dark soliton carrying the effective fractional
supercurrent can be created in a scalar BEC. Due to the mass velocity, the darksoliton becomes spiral-shaped, leading to the snake instability. One can recognize the slight curvature at the end of MCSs in Fig. 4.7(d,f) and Fig. 4.10(a-c). It
81

mz=+1

mz=-1

mz=0

100 µm

(d)

(a)

ω

(e)

(b)

ω+δ
OD
1.2
0.0
arb.
units

(f)

(c)

SB
0.7
-0.7
arb.
units

Figure 4.10: Bragg spectroscopy and spin separation imaging of a BEC with
MCSs for q/h = 2.6 Hz for 600 ms hold time. (a-c) Absorption images for eachspin components and (d-f) the corresponding Bragg signals SB (x0 , y 0 ) = n+ −n− ,where n+ and n− are the density distributions of the +x0 - and −x0 -scatteredatom clouds, respectively. δd /2π = −0.9 kHz. The circles and crosses denote thesign of mass flow. Small arrows in (c) represent the direction of supercurrents.

82mz=+1

mz=0

mz=-1

100 µm

(b)

(a)

OD
1.2
0.0
arb.
units

(d)

(c)

Mz
1
-1
arb.
units

Figure 4.11: Singly charged vortices having magnetized cores. (a,c) Spin separated absorption images and (b,d) in situ are taken after (a,b) 800 ms and (c,d)
1 s hold times. The final q values are q/h = (a,b) 5.1 Hz and (c,d) 10.4 Hz,
respectively. Red boxes mark quantum vortices with magnetized cores. Dottedboxes mark a domain wall with unbalanced spin population.
is natural to derive the conclusion that the orientation of mass current is counterclockwise for Fig. 4.7(f). We also conjecture that the mass flows between
MCSs in Fig. 4.7(d) are identical to that in Fig. 4.10(e).
Singly charged quantized vortex with a magnetized coreAt the end of phase transition, singly charged quantum vortices are identified with density-depleted holes in the density of the mz = 0 atoms. Quantum
vortices are mainly formed by the phase differences in the interface of local
domains. The scaling of HQVs are mentioned in section 4.6.
Interestingly, some of quantum vortices have vivid ferromagnetic cores.
We can recognize these quantized vortices in both absorption images for eachspin components and axial magnetization images [Fig. 4.11]. The formationof a quantized vortex with a magnetic core which are not fully explained by
local quantum seeds. One possible scenario is that the shrinking of a soliton

83possessing imbalanced mz = ±1 spin populations in its domain wall part. When
the winding of the supercurrent encircling both edges of the soliton is 2π, the
MCS is able to become a singly charged vortex. The example is displayed inFig. 4.10(c,f).
Most of singly charged vortices have either empty cores or cores, which
is filled with the equal the mz = ±1 populations. This is because that we
prepared the sample with zero magnetization. We attribute the imbalanced
spin population in a vortex core to the discrepancy of the coupling strength ofthe microwave fields for each spin state. In our experiments, the lifetime of the
mz = +1(−1) atom is usually shorter than that of the mz = −1(+1) atoms forblue-detuned(red-detuned) frequency from the |F = 1i ↔ |F = 2i transition.
Spin phase separation, induced by a slight magnetic field gradient, might beanother reason.
In Fig. 4.11(a,c), a MCS has both spin components, but magnetic corevortices have either mz = 1 or −1 spin population in its cores. This might bebecause the system avoids the mz = ±1 states and the spin-exchange collisionoccurs lowering the system’s energy. As the same reason, a quantum vortex witha ferromagnetic core would be expelled in the boundary region of a condensate.

4.5

Suppression of dynamical instability by theaxial spin fluctuations

In contrast to the EAP condensate, the EPP condensate can be thermally
excited from the thermal population of the magnon modes. Right after the spin

84

(a)

(b) 0.07

spin rotation

0.06

EAP
A
A

qf>0

EPP

?

ts

th

B

<δMz2>

qi<0

B

0.05
0.04
0.03

qi/h (Hz)
-3.3
-5.6
-10.6

0.02
0.01

A

0
0.0

0.2

0.4

0.6

0.8

1.0

time, ts (s)Spin ﬂuctuations grow.

Figure 4.12: (a) Experimental sequence to enhance spin fluctuations in an EPP
condensate. A: An EPP condensate is prepared by the spin rotation. B: Spinfluctuations arise for ts from the thermal population of axial magnon modes withqi < 0. Then, we change the final quadratic Zeeman energy, qf , to the positive
value. (b) Variance of the axial magnetization, hδMz2 i, as a function of ts . hδMz2 i
is measured from the 157 µm × 157 µm center region of in situ magnetization
images. The error bars are the standard deviation of more than 5 measurements.
The growth of spin fluctuations are independent to the magnitude of qi .
rotation, the nematic directors are co-aligned to the x-direction as in Fig. 4.12 A
with zero temperature. We call the EPP condensate without any spin fluctuations as an either uniformly-aligned or coherent EPP condensate. The spin
fluctuations develop with the gapless magnon excitations. We characterize thespin fluctuations by the axial magnetization variance, hδMz2 i [Fig. 4.12(b)]. The
growth rates are irrelevant to the quadratic Zeeman energy, qi < 0, which is
obvious result. Then, the nematic directors tremble in xy-plane [Fig. 4.12 B].
We call the EPP condensate at B as a faltering EPP condensate. Because oursystem is in the spin-wise 2D, the spin fluctuations could be more enhanced.
It might be done by building a true 2D trapping potential and overcoming thelimitation of the condensate’s short lifetime due to the heating of the microwave
dressing.
We explore the properties of 2D spin dynamics by enhancing the spin85

t=20 ms

40 ms

40 ms

50 ms

50 ms

60 ms

60 ms

80 ms

100 ms

100 ms

400 ms

800 ms

1s

mz=1

mz= 0

mz=-1
80 ms

mz=1

OD
2

mz= 0

0
arb.
100 µm units

mz=-1

Figure 4.13: Temporal evolution of the quenched EPP condensate with the spin
fluctuations for q/h = 10.4 Hz. (a) The initial spin nematic director d~ is coherently aligned in xy-plane and (b) the phase transition to the EAP phase istriggered by a sudden change of the quadratic Zeeman energy, q. (c) Absorption
images of the mz = 1, 0, −1 spin components after 24 ms Stern-Gerlach spin separation and (d) in situ transverse magnetization images in y-direction, My (x, y),and (e) in situ axial magnetization images, Mz (x, y), for various hold times
t. Quantum vortices with single mass circulation are identified with densitydepleted holes in the mz = 0 optical density images.

86

fluctuations and measuring the dynamical instability of the EPP-to-EAP phasetransition. We generate the spin fluctuations in the EPP condensate for ts ,then suddenly change the quadratic Zeeman energy from qi < 0 to qf > 0.
The jump from qi to qf is implemented within 1 ms (for more information, see
Appendix C), which is much shorter than the spin-interaction time of 1/c2 n ≈68 ms.
In an intuitive thinking, one might assume that a faltering EPP condensate becomes more dynamically unstable at qf > 0, because its temperature
is higher than that of a coherent EPP condensate. In addition, the number ofquantum noise seeds might be larger for a faltering EPP condensate. Figure 4.13
shows the temporal evolution of a quenched EPP condensate with the spin fluctuations for q/h = 10.4 Hz. We recognize that the characteristic length of thetransverse magnon modes is shorter and the widths of solitons become sharper,
compared to Fig. 4.3. In addition, the coherent ring-shaped transfer is not clear
as in Fig. 4.3. This manifests that the number of local EAP domains increase incase of a fluctuating EPP condensate at positive qf . However, the delay time ofthe early dynamics increases as more spin fluctuations are generated in a EPP
condensate, where the first assumption appears to be wrong.
The suppression of the transverse magnon excitation is described by thechange of the fractional population in Fig. 4.14. The coherent oscillation isclearly suppressed. Instead, the rapid decay of 1 − η is halted at a certain
1 − ηth ≈ 0.5. Then, it follows a single decay line regardless of the condensate’s
initial condition. This behavior of 1 − η is much similar to the evolution of
the fractional population through the EAP-to-EPP transition in Fig. 3.5. In
addition, the dynamic instability is diminished for a faltering EPP condensate.
Recent study, referred to Ref. [130], suggests that the dynamical instability can87

(a) 1.0

no spin fluctuationt s=200 ms at qi/h=-5.6 Hz
t s=600 ms at qi/h=-5.6 Hz
t s=1 s at qi/h=-5.6 Hz

1-η

0.8
0.6
0.4
0.2
0.0
6

0.01

2

4

6

0.1

(b) 1.0

4

6

1

2

no spin fluctuationt s=600 ms at qi/h=-3.3 Hz
t s=600 ms at qi/h=-5.6 Hz
t s=600 ms at qi/h=-10.6 Hz

0.8

1-η

2

0.6
0.4
0.2
0.0
6

0.01

2

4

6

0.1

(c) 1.0

4

6

1

2

no spin fluctuationt s=200 ms at qi/h=-5.6 Hz
t s=200 ms at qi/h=-10.6 Hz

0.8

1-η

2

0.6
0.4
0.2
0.0
6

0.01

2

4

6

0.1

2

4

holdtime, t h (s)

6

1

2

Figure 4.14: Quantum vortex number, Nv , versus q. Nv was measured by counting density-depleted holes in the SG images of the mz = 0 spin components.
Each data point is the average of 13 measurements for the same experiment andits error bars indicates the standard error of the mean. The blue dashed line isa power-law fit to the data.
88mz=+1

mz=0

mz=-1

mz=+1

(a)

(c)

(b)

(d)

mz=0

mz=-1

OD
2.0
0.0
arbi.
100 µm units

Figure 4.15: Dynamically formed (a,b) skyrmions and (c,d) magnetic compositesolitons with q/h = 2.6 Hz. The spin fluctuation is enhanced for 600 ms andthe evolution times after quench are 150 ms. Absorption images are taken after
24 ms spin separation. Experiments conditions: initial µ ≈ h × 927 Hz, c2 n =14.7h and RTF = (201, 143, 2.0) µm.
be tuned by the spin-squeezing. However, in our experiments, the transverse
magnon modes are totally independent to the axial magnon modes. Instead, it
infers that the coherent EPP state is not a true ground state for many-bodyquantum system, even the coherent EPP condensate is well described by themean-field theory. Based on the inference, intriguing questions naturally arise:
what is the true ground state for the many-body antiferromagnetic spin-1 Bosesystem at negative q, and whether the dynamic instability lessens infinitely or
not near the critical point. We hope to find the theoretical and experimental
ways to answer the above questions in the future.

4.6

Creation of singly charged vortices
Because the coherence over the whole sample does not exist with thespin fluctuations, a magnetic composite-soliton is formed more frequently ina faltering EPP condensate across the EPP-to-EAP transition: skyrmions anddomain walls are formed for every experimental run. From the comparison of89

Fig. 4.15 to Fig. 4.7(a-d), we recognize that the domain walls are bendy andnarrow. The formation of MCS and skyrmions are influenced by the quantum
fluctuation seeds, in other words, the population of non-condensed mz = 0
component.
The topological defects, created as a result of a phase transition, reflect the number of the quantum noise seeds. At long hold times, t ≥ 1 s, asingly charged vortices are vividly identified with the density-depleted holes in
the mz = 0 absorption images. We measure the number of quantized vortices,
Nv [Fig. 4.16]. When we prepare the uniformly aligned EPP condensate, Nv increases with increasing q . c2 n with the power-law exponent of 1.3±0.22 , which√
is predicted from Nv ∝ lq−2 and the spatial size of the magnon modes, lq ∝ q.
For q > c2 n, Nv slowly increase with q [96]. When we prepare the faltering EPP
condensate, the number of quantum vortices is saturated for q ≤ 5.1 Hz, supporting that the temperature of a condensate increases the number of quantum
noise seeds.

4.7

Sad history
When I firstly conducted experiments about the EPP-to-EAP phase tran-

sition dynamics, the number of a condensate was about 8.0×106 and the thermal
fraction was less than 10 %. Thus, the sample size was large enough to contains
a lot of dynamically formed topological defects, including magnetic compositesolitons as displayed in Fig. 4.17 and Fig. 4.18. The life-time of a condensate
was long enough to measure the decay dynamics of quantized vortices for long2

It can be regarded as the quadratic Zeeman energy is converted into the kinetic energyof a condensate via the spin-change collision.

90

(a) 20(b)

Vortex number, Nv

10
1

|q|

6
4

|q|1.3

1

0.6
4

6

8 10

4

|q|1
2

10-4

2

2

no spin fluctuation
ts=600 at qi=-5.6h

6Vortex density, nv (/µm2)

no spin fluctuation
ts=600 ms at qi=-5.6h

2030

qf/h (Hz)

6
4
2

10-5

2

4

6

8 10

20

30

qf/h (Hz)Figure 4.16: Quantum vortex number, Nv , versus q. Nv was measured by counting density-depleted holes in the SG images of the mz = 0 spin components.
Each data point is the average of 13 measurements for the same experiment andits error bars indicates the standard error of the mean. The black dashed line
is a power-law guide line and the blue solid line is a power-law fit to the light
blue data for q/h ≤ 17 Hz.
hold times, th = 5 s.
However, the control switch, connecting between the microwave function
generator and RF coils, was malfunctioning and unwanted microwave dressed
fields disturbs the preparation of the uniformly aligned EPP condensate. So, I
did not know the amplitude of the spin fluctuations and these images show the
temporal evolutions of a quenched EPP condensate with the axial spin fluctuations. It was the reason why the sample contains a lot of quantized vortices.
Unfortunately, after I realized and fixed the switch problem, the numberof a BEC decreased and the lifetime became shortened for some reasons. I wish
I knew the reasons.

91t=50 ms

50 ms

100 ms

100 ms

150 ms

150 ms200 ms

250 ms

250 ms

300 ms

400 ms

mz=1

mz= 0

mz=-1
200 ms

mz=1

OD
2

mz= 0

0
arb.
100 µm units

mz=-1

Figure 4.17: Phase transition from the EPP phase for q/h = 2.0 Hz. Absorption
images of the mz = 1, 0, −1 spin components after 24 ms Stern-Gerlach spin
separation.

92t=30 ms

30 ms

60 ms

60 ms

100 ms200 ms

500 ms

700 ms

700 ms

1s

5s

mz=1

mz= 0

mz=-1
300 ms

mz=1

OD
2

mz= 0

0
arb.
100 µm units

mz=-1

Figure 4.18: Phase transition from the EPP phase for q/h = 10.6 Hz. Absorption
images of the mz = 1, 0, −1 spin components after 24 ms Stern-Gerlach spin
separation.

93

4.8

OutlookThe quenched EPP condensate across the EPP-to-EAP phase transition

provides us an awesome platform to access the fascinating, but still uncharted
physics area such as the formation and the evolution of a composite-topologicaldefects and the suppression of the dynamical instability. By using the dynamic
imaging such as absorptive spin-sensitive in situ imaging [33], one can capture
the subsequent evolution of a MCS. When the true 2D system is built, a paired
superfluid, having singlet pair superfluidity without spin ordering [115], can beobserved by amplifying the spin fluctuations.

94

Chapter 5
Rotating Bose-Einsteincondensate
In this chapter, we present the another type of non-equilibrium dynamics:
the relaxation dynamics for a circulating BEC. Being irrotational, a superfluid
shows a unique response to rotation, which involves quantized vortices. Since
its first realization, a BEC under rotation has been of great interest for the
study of superfluidity and various vortex dynamics have been investigated indifferent rotation regimes [131]. Many experimental methods were demonstrated
for rotating trapped BECs, which include phase imprinting techniques based on
optical transitions [132] or spin rotation [43,133], and applying rotating ellipticalpotentials [134–138]. Rotating BECs were also produced by evaporative coolingof rotating thermal gases [139]. It is anticipated that in a rapid rotation limit
a rotating BEC undergoes quantum phase transitions to strongly correlated
quantum Hall states [140], and there are ceaseless experimental efforts to reachthe fast rotation regime [141–143].

95

We present an experimental method for rotating a trapped BEC in ananharmonic trapping potential. A trapped BEC can have angular momentum
in two forms: the center-of-mass (c.m.) motion around the trap center and theinternal motion such as surface oscillations and quantized vortices. Contrary to acondensate in a harmonic potential, the anharmonicity of a trapping potential
provides the coupling between the c.m. motion and the internal motion of atrapped condensate [144–147].
• Jae-yoon Choi, Seji Kang, and Yong-il Shin, “Observation of a GeometricHall effect in a Spinor Bose-Einstein Condensate with a Skyrmion Spin
Texture””, Physical Review Letter 111, 245301 (2013).
• Seji Kang, Jae-yoon Choi, Sang Won Seo, Woo Jin Kwon, and Yong-il
Shin, “Rotating a BEC by shaking an anharmonic axisymmetric magnetic
potential ”, Physical Review A 91, 013603 (2015).

5.1

Anharmonic trapping potential
Our experiment uses Bose-Einstein condensates of sodium atom in the|F = 1, mF = −1i [148]. We prepare 2 × 106 sodium atoms in a pancake-shaped
optical dipole trap with trapping frequencies of ωx,y,z = 2π × (3.3, 4.1, 400) Hz
and apply to the condensate a three-dimensional quadrupole magnetic field,
B=

Bq(xx̂ + y ŷ − 2z ẑ) − Bx x̂ − By ŷ + Bz ẑ.
2

(5.1)

The position of the zero-field point is controlled by the bias field {Bx , By , Bz }as Q = {x0 , y0 , z0 } = {2Bx /Bq , 2By /Bq , Bz /Bq }. The overall external potential

96

can be described as
Vtrap (x, y, z) = VODT + VQuad + Vgravity
(5.2)
m 2 2
(ω x + ωy2 y 2 + ωz2 z 2 )=2 xµB Bq p+(x − x0 )2 + (y − y0 )2 + 4(z − z0 )2 − mgz,
4
where m is the atomic mass, µB is the Bohr magneton, and g is the gravitational
acceleration. The axial field gradient is set to be Bq = 7.6 T/m [63], almost
compensating the gravitational sag in the optical trap, and the position of thezero-field point is placed at Q = {0, 0, 32(2)} µm [Fig. 5.1 (a)]. The transverseq
B Bq
2 . Fortrapping frequency at the center of the hybrid trap is ω0 = µ8mz
+ ωx,y0dipole oscillations with small amplitude of 16 µm, the trapping frequency ismeasured to be ω0 = 2π × 42.5 Hz. The trap anisotropy due to the weak opticalpotential is estimated to be less than 0.2%. For the oblate condensate which is
confined to the z = 0 plane, we approximate the radial trapping potential as
Vrad (r) = 2mω02 z0
where r =q
r2 + z02 ,

(5.3)

p
x2 + y 2 . This potential behaves linear for large r ≤ z0 , representing

a highly anharmonic, axisymmetric trap for the condensate [Fig. 5.1 (b)]. The
Thomas-Fermi radius RTF ≈ 40 µm, which is comparable to z0 .

5.2

Generation of circular c.m. motion
In order to generate a circular c.m. motion of the condensate, we ap-

ply a rotating bias field as {Bx (t), By (t)} = δB(sin(ωm t), cos(ωm t). The trap
center correspondingly circulates around the origin as d(t) = {(x0 (t), y0 (t))} =

97

(b)

(a)

(d)

V(r)

yc.m.
Q(x0, y0, z0)-60 -300

30 60

r (μm)
y

(c)

z
y

yc.m.

x

xc.m.

x
xc.m.

Figure 5.1: (Color online) Circulating Bose-Einstein condensate in an anharmonic magnetic potential. (a) Schematic of the experimental setup. A condensate (yellow ellipsoid) is confined in an optical sheet (red plane) and aquadrupole magnetic field is generated by a pair of coils (blue rings). The redarrows indicate the current directions and Q is the zero-field point. (b) Anharmonic radial trapping potential from the quadrupole magnetic field. The dashedline shows a harmonic potential having the same curvature at the trap center.
(c) At an off-center position, the local curvatures of the anharmonic trapping
potential are different for the radial and azimuthal directions. When the condensate circulates around the trap center, it experiences a rotating anisotropic
potential. The center-of-mass motion (c.m.) of the condensate is coupled to its
internal motion. (d) The deformation of the condensate in the center-of-massframe.

98

d{sin(ωm t), cos(ωm t)} with d = 2δB/Bq . The turn-on sequence of the field modulations is displayed in terms of the trap position in Fig. 5.2(a). In the reference
frame of the circulating magnetic trap, the equation of motion for an atom inthe z = 0 plane is given asm

d2 r0dVr (r)dr0
2
× Bez ,
=
−
+
mω
d(t)
+
m
dt2
dr0
dt

(5.4)

where r0 = (x − x0 , y − y0 ), Bez is the effective magnetic field in z-direction
due to the Skrymion spin texture and the second term is the inertial force dueto the trap shaking. The last term is due to the Lorentz force by the Be (Seesec. 5.4). In the case of a harmonic trap with Vr (r) = 12 mω̄ 2 r2 , this rotating force
will drive the atom to have a circular motion, whose radial position oscillates as
2

ω̄−ωm
md
r0 = (ω̄2ω
t). The angular momentum can be resonantly injected into
2 −ω 2 ) sin(2m

the system. This is indeed analogous to what we typically do to rotate liquid ina glass without a spoon. This method is very simple, requiring no sophisticatedtechnique such as dynamically deforming the trapping potential.
Figure 5.2(b) shows the temporal evolution of the c.m. position of thecondensate for ωm = 2π × 39.5 Hz and d = 3 µm. An in-situ absorption image was taken every 4 ms of the driving and the c.m. position was determinedfrom a two-dimensional Gaussian fitting. The condensate develops a circular
c.m. motion and the radius of the circular motion oscillates with a period of
400 ms. For td = 200 ms driving, the radial position of the condensate increases
up to about 65 µm and the external angular momentum per particle, associatedwith the c.m. motion, increases as large as |Lz | ≈ 350~, where ~ is the Planck
constant divided by 2π. We see that the measured c.m. trajectory is reasonably accounted for by the single-particle simulation using Eqn. Eq:RotMotion[Fig. 5.2(c)]. The deviation for later time is understandable because the c.m.
99

(µm)
3

(a)
x0
y0

0
-3

C.M. postion (µm)
Lz / ĭ

ȟt(µm)
90 (b)
60
30
0
-30
-60
-90
-400

td

tr
x
y

(c)

-200
0
0

200

400

600

800

Driving time, td (ms)Figure 5.2: Generation of a circular c.m. motion of the condensate. (a) The
trapping potential is circularly shaken by applying a rotating bias field. The
driving frequency is ωm = 2π × 39.5 Hz close to the trapping frequency at thetrap center ω0 = 2π × 42.5 Hz. (b) The temporal evolution of the c.m. positionof the condensate (single measurements for every 4 ms) and (c) the externalangular momentum per particle, Lz , of the condensate. The c.m. velocity wasdetermined from the position data in (b) and Lz is displayed with 16 ms binning.
The solid line denotes the numerical result from the single-particle simulation
using Eqn. (4) and the dashed line shows the numerical result including theeffective Lorentz force (see Chap. 5.4).

100trel=0.00s
(a)

0.05s

0.1s

0.3s

0.5s

1s

1.5s

2s

2.5s

200 µm(b)(c)

Figure 5.3: (a-c) The whole development of vortex nucleation in a spinor condensate relaxed in a stationary anharmonic trap, after the condensate is modulated
circularly. Absorbtion images after 10 ms time-of-flight after various relaxation
times trel . The modulation times are td = (a) 200 ms, (b) 400 ms, and (c)1000 ms, respectively.
motion of the condensate having a finite spatial extent cannot be separablefrom its internal motion in the anharmonic potential [145–147]. In the following
experiment, we employ the same driving procedure.

5.3

Relaxation of the circulating Bose-Einsteincondensate
Figure 5.3 show the temporal evolution of the circulating condensate hav-

ing different angular momentum, Lz by varying td :(a, c) maximum |Lz | ≈ 350~,
and (b) minimum Lz ≈ 0~. The relaxation of the circulating condensate ischaracterized by the shape deformation, the fraction of BEC, and angular momentum or vortex numbers. We count the number of vortices by using computer
algorithm well described by Choi’s thesis [63].

1015.3.1

Surface mode excitation

At an off-center position in the anharmonic trapping potential, the local
curvature of the potential is not isotropic. For the external potential in Eqn. 5.3,
the local trapping frequencies along the radial direction and the azimuthal direction are given as
ωr =
ωθ =

q
q

2
1 ∂ 2 Vr
= ω0 1 + 4zr 2
m ∂r2
0





(5.5)

(ω02 −ωr2
ω02 +ωr2) ≥ 0. Therefore, the cir-

,


1 ∂Vr
r2
=
ω
1
+
,
2
0
mr ∂r
4z
0

respectively, and the local trap anisotropy  =culating condensate would experience a rotating anisotropic trapping potential
in its c.m. reference frame [Fig. 1(c)]. As the radial position of the condensate
increases under the driving, the local trap anisotropy gradually increases. The
rotating anisotropic potential would cause surface mode excitations [149, 150],
possibly leading to vortex nucleation [134–138, 151–158].
Figure 3(a) shows a series of in situ absorption images of the condensate
for td = 180 − 216 ms. In this period, the radial position of the condensate isr ≈ 65 µm, giving  = 0.34 with ωθ /ω0 = 0.84 and ωr /ω0 = 0.59. We see that thecirculating condensate is elliptically deformed and the axis of the deformation
rotates with the same frequency of the condensate circulation [Fig. 5.4(b)]. Thelong axis of the condensate is found to be along the azimuthal direction, which
is orthogonal to the weak confining direction of the local anisotropic trapping
potential. This is consistent with the theoretical prediction for the case where
the rotating frequency Ω of the elliptical potential is higher than the trapping
frequencies [152], which is true in our experiment for Ω ≈ ωm > ωθ,r . The steadystate solution for a condensate in a rotating elliptical harmonic potential predicts
the aspect ratio of the deformed condensate to be 1.16 for our experimental102

100 µm(a)180ms 184

(b)

188

192

196

200

204

208

212

216

π

Angle (rad)

θ
0

θ
θcm

θcm

-π

(x0,y0)-2π

Aspectratio θ-θcm

-3π
π

0
1.4
1.0
180

190

200

210

220

Driving time, t d (ms)

(c)

150µm

t d=200 ms

400 ms1000 ms

Figure 5.4: Surface mode excitations in the circulating condensate. (a) In situ
absorption images of the condensates for various driving times td . (b) Timeevolution of the shape deformation of the condensate. θ and θc.m. denote theorientation of the long axis of the condensate and the azimuthal angle of the c.m.
position, respectively. The aspect ratio shows small oscillations. The solid lines
are guide lines for the eye. (c) Images after time-of-flight expansion. Vortices
are nucleated in the boundary region of the condensate.

103condition with Ω/ωθ ≈ 1.11 and  = 0.3 [152]. We observe that the aspect ratioof the circulating condensate oscillates around 1.25 [Fig. 5.4(b)], which is inreasonable agreement with the prediction.
We identify vortex nucleation in the circulating condensate by taking anabsorption image after releasing the trapping potential and subsequent expansion for 16.25 ms [Fig. 5.4(c)]. We turn off the magnetic potential 6.25 ms earlier
than the optical potential, which is helpful to improve the visibility of the vortex core [97]. Vortices are generated in the condensate boundary and graduallydiffuse into the inner region. The vortex population increases for longer driving time. In previous experiments [134–138], which investigated condensatesin rotating elliptical potentials, the vortex nucleation mechanism was mainly
associated with the dynamic instability via the quadrupole surface mode resonance [152–158]. In our experiment, the rotating frequency Ω is almost two
times higher than the resonance frequency of the quadrupole surface mode,pΩ2 ≈ ωθ2 + ωr2 /2 = 2π × 21.7 Hz, and the dynamic instability is not likely
to be involved in the vortex nucleation. Furthermore, the observed oscillationsof the aspect ratio of the condensate indicate that the deformed condensate is
dynamically stable. This suggests that the vortex nucleation in the circulatingcondensate is driven by the energetic instability in the rotating potential.

5.3.2

Thermal distillation: re-condensation

After stopping the trap shaking, we investigate the relaxation of a freely
circulating condensate. Figure 5.5 and figure 5.3(a) show the time-of-flight images of the condensate for various relaxation times tr after 200 ms driving. In
the early phase of the evolution, the condensate keeps circulating, maintaining104

150 µm

tr=0.1 s

0.25 s0.5 s0.75 s

1.0 s

1.5 s

2.5 s

4.0 s

Figure 5.5: Relaxation of the circulating condensate in the anharmonic trapping potential. Time-of-flight images of the condensates for various relaxation
times tr after 200 ms driving. The circulating condensate evolves into a rotatingcondensate with a triangular lattice of vortices.
its elliptically deformed shape. It is known that a particle in a rotating ellipticalharmonic potential is dynamically unstable for ωr < Ω < ωθ [136, 152, 159]. Although Ω = ωθ in our case for the free circulation, close to the unstable regime,
the condensate remains stable without showing any turbulent behavior. Over500 ms, more vortices are nucleated up to Nv 40 and the radius of the c.m.
motion decreases gradually to about 45 µm.
As the relaxation proceeds, we observe that a faint cloud slowly forms
over a large central area. The cloud grows in density and reveals a large numberof vortices, which indicates that the newly formed cloud becomes a part of thecondensate. This process is remarkably gentle and the elliptical region of theoriginal circulating condensate is well preserved as a high density part. The
high-density part of the condensate gradually shrinks and becomes completely
dissolved at tr ≈ 1.5 s. At this moment, we have a large round condensate
containing many vortices whose spatial distribution is disordered. The vortex
number is counted up to Nv ≈ 200. This state represents a rotating condensate
105

with an irregular vortex lattice.
The development of the rotating condensate might be understood as athermal equilibrating process in the reference frame corotating with the circulating condensate (Fig. 5.6). In the rotating reference frame, the effective
trapping potential is given as V e (r) = V (r) − 12 mωθ2 r2 , including the centrifugal
potential. Because of the anharmonicity of V (r), V e (r) has a restricted trapping
region with a global potential minimum at the center. The circulating condensate is depicted to be located at the edge of the trapping region with the help of
the Coriolis force [Fig. 5.6(a)]. Since the circulating condensate has an effective
chemical potential higher than the potential minimum, it would thermodynamically drive thermal atoms to condense at the trap center, which would resultin a rotating condensate at the trap center.
This is a kind of a thermal distillation process where a condensate in ametastable state is transferred to the true ground state via thermal atoms [94].
Note that such a distillation process would not occur in a system using a harmonic potential, where the effective external potential for a circulating condensate is flat, having no potential minimum. When a quartic potential is employed,
which provides stronger confinement at a larger radius, the effective potential
has a ring geometry and a circulating condensate would relax into a giant vortex
state [160, 161].
In our experiment, the collision time of thermal atoms is estimated tobe τ ∼ 25 ms for the sample temperature T ∼ 300 nK 1 [162]. The growth of
the rotating condensate happens on a time scale of ∼ 20τ , which is comparable
to the results in previous studies on Bose condensation [163]. The formation of
The collision time τ = (nth σv̄)−1 , where nth = 2.612(mkB T /2π~2 )3/2 is the atom densityof a saturated thermal cloud, σ =p8πa2 denotes the elastic collision cross section (the scattering
length a = 2.75 nm), and v̄ = 4 kB T /πm is the thermal velocity.
1

106

(a)
Rc

(b)

(d)
Optical Density (arb. units)

Rc

Circulatingcondensate
Thermal distillation

(c)

1.2
1.0 s
1.5 s
2.5 s
4.0 s

0.8

0.4

0.0
0

100

200

300

400

Distance (µm)
Rotating condensate

Figure 5.6: Thermal distillation process. (a) In a reference frame corotating withthe circulating condensate, the condensate is located at the edge of the trapping
region of the effective trapping potential (solid line). The dashed line shows thetrapping potential in the laboratory frame. (b) The condensate thermodynamically drives thermal atoms to condense in the center region having the trapping
potential minimum. (c) A rotating condensate is formed at the trap center.
The right column displays the images corresponding to each step of the relaxation process. The dashed circles indicate the trap boundary determined by thecentrifugal effect. (d) Radial density distributions of the rotating condensates,
which are obtained by azimuthally averaging the images in Fig. 5.5. For tr = 1 s,
the high-density part, corresponding to the original circulating condensate, is
excluded in the averaging. The density modulations in the center region result
from density-depleted vortex cores.

107the rotating condensate over a large area implies that the atomic influx fromthe circulating condensate is higher than the internal equilibration rate of therotating condensate.
The round boundary of the whole condensate, which is determined by
the radial position of the circulating condensate, seems to reveal the trapping
radius of the effective potential in the rotating reference frame [Fig. 5.6(a)].
p
From dVre /dr = 0, the trapping radius is given as Rc = 2z0 (ω0 /ωθ )4 − 1.
When the spatial extent of a rotating condensate is limited by the centrifugal
effect, the vortex number of the condensate would bem 2 4mz02 ω04 − ω 4
ωRc =
,
(5.6)
~
~
ω3
H
which is estimated from the circulation relation r = Rc vs dr = (h/m)Nvc , whereNvc ≈vc is the superfluid velocity. For tr = 1 s, the in situ radius of the rotatingcondensate is measured to be about 50 µm, suggesting Nvc ≈ 200. This number
is comparable to the maximum vortex number observed in the experiment. The
rotating frequency of the newly formed rotating condensate is forced to be set
to the circulation frequency of the initial condensate.
After the high-density part in the condensate vanishes, the rotating condensate with an irregular vortex lattice undergoes an equilibrating process.
Crystallization into a triangular vortex lattice takes about 2 s, where the radialextent of the condensate decreases by about 25% and the atomic density at thecenter increases by a factor of 2 [Fig. 5.6(d)]. At tr = 4 s, the vortex number
is Nv ≈ 120. One may expect an increase of the rotating frequency due to the
reduction of the moment of inertia of the condensate, but the vortex areal density, i.e., the rotation frequency of the condensate, shows no significant change
during the evolution. The thermal fraction of the sample is as high as 50% at
108this phase and thus the interactions between the condensate and the thermal
cloud must be important. The rotating thermal cloud might lock the condensate
rotation in the equilibration process. With an initial state having a triangularlattice with about 100 vortices, the half lifetime of the vortices was measured
to be 9 s.

5.3.3

Rotating thermal clouds

We present the relaxation of the condensate after 400 ms driving. Thiscase is special in that the c.m. motion of the condensate almost stops after
one oscillation of the radial position [Fig. 5.2(c) and 5.3(b)]. Right after thedriving, the condensate shows a round shape and contains about 30 vortices
in its boundary region. As the hold time increases, we observe that more vortices come in and migrate to the inner region of the condensate, and, finally,a triangular lattice of about 60 vortices forms after 2 s [Fig. 5.7(a)]. In theThomas-Fermi approximation, we estimated the angular momentum per partiPcle of the condensate as Lz = ~ i [1 − (ri /R)2 ]5/2 , where ri /R is the ratio of thevortex radial position to the condensate radius, which was measured for eachvortex from the image [23]. Since the angular momentum of the condensate isincreased during the evolution [Fig. 5.7(b)], it is obvious that the thermal cloudsurrounding the condensate rotates faster than the condensate at the beginning.
During the driving period, the thermal cloud is not only subject to the appliedrotating force, but also dragged by the circulating condensate.We note that thetime scale of the vortex lattice crystallization is similar to that in the previous
case with td = 200 ms.
The equilibrium process of a condensate in a rotating thermal cloud was

109

(a)

150 µmtr=0.1 s

0.5 s

1.5 s

3.0 s

Angular momentum
per particle (ĭ)

(b)
25
20
15
10
5
0

0

1.0

2.0

3.0

Relaxation time, tr (s)Figure 5.7: Spinning up the condensate by a rotating thermal cloud. (a) Timeof-flight images of the condensate for various hold times tr after 400 ms driving.
There is no noticeable c.m. motion of the condensate [Fig. 5.2(b)]. The numberof vortices in the condensate gradually increases over the holding time and avortex lattice forms. (b) Evolution of the angular momentum per particle, which
is determined from the relative positions of vortices in the condensate.

110

(d)

80
40

circularlinear

0
2.4

(b)

2.0
1.6

Vortex number, Nv

Thermal Atom
Vortex number, Nv
fraction number

(a)

120

120
80
Relative phase
90º
60º
30º
linear

40
0

0

0.5

1.0

1.5

2.0

td (s)

0.5
0.3
0.1

(c)

0

0.5

1.0

1.5

Driving time, td (s)2.0

Figure 5.8: (a) Vortex number Nv vs driving time td . The vortex number wasmeasured after 3 s relaxation. The open circles show the result for linear driving, where the transverse bias fields, Bx and By , were modulated in phase witht = 0 [Fig. 5.2(a)]. Each data point consists of three measurements. The corresponding evolutions of (b) the total atom number and (c) the thermal fractionof the sample. (d) Temporal evolution of vortex number under various ellipticalpolarized driving. The error bars are not displayed. See also section 5.4.
theoretically investigated to study the role of a thermal cloud in vortex nucleation and lattice formation [164, 165]. Our rotating method provides an experimental way to prepare a nonequilibrium state where a thermal cloud rotates
faster than a condensate. We expect that the rotation frequency difference between the thermal cloud and the condensate may be controlled by the driving
parameters or the anharmonicity of the trapping potential. In future, we will
pursue a quantitative analysis of the equilibrium process, including measurements of the rotating frequency of the thermal cloud [137, 139].
Finally,we present the vortex number, Nv after 3 s relaxation as a function of the driving time, td in Fig. 5.8. As the driving time increases, the vortex
111number increases and becomes saturated to Nv = 100 with damped oscillations. The oscillations reflect the oscillating behavior of the external angularmomentum of the driven condensate [Fig. 5.2(c)]. The total atom number andthe thermal fraction of the sample also oscillate accordingly due to the atom
loss and heating associated with the relaxation process.
The angular momentum per particle of the stably rotating condensate
with 100 vortices is about 40~, which is only 10% of the maximum externalangular momentum of the circulating condensate. The poor efficiency of theangular momentum transfer to the internal rotation of the condensate is an
intrinsic aspect in the evolution of an isolated system. When the energy ofthe system is given as E = 12 L2z /I, where I is the moment of inertia of thesystem, the energy and angular momentum conservations require that I should
be preserved in the evolution. This means that when a rotating condensate forms
at the trap center, generation of thermal atoms having high angular momentum
is unavoidable. This argument suggests that in order to improve the transfer
efficiency of the angular momentum, i.e., to obtain a faster rotating condensate,
one needs to provide an additional work by dynamically deforming the trapping
potential during the relaxation.

5.4

Gauge-field effect in a magnetic quadrupole
field
When a particle with spin moves in a spatially varying magnetic field,

the particle acquires a quantum mechanical phase associated with the adiabatic
spin rotation, which is known as the Berry phase [166]. A gauge potential can

112

Q
β

β

0
π

Φ

Figure 5.9: Skyrmion spin texture configuration. When the zero-field point
moves by oscillating DC magnetic field, the skyrmion spin texture also moves.
be defined for the spatial distribution of the spin orientation and this resultsin generation of an effective magnetic field for the particle [167, 168]. When a
Bose-Einstein condensate is trapped in a magnetic potential, its spin texture is
imposed by the magnetic-field distribution. Because of the spin-gauge symmetry,the condensate would experience an effective magnetic field originated from thespin texture [169].

5.4.1

Skyrmion Spin texture

In the current setup, the condensate is radially confined by the quadrupole
magnetic field. Its spin texture exhibits a fountain-like configuration, displayedin Fig. 5.9 which is often referred to as a Skyrmion spin texture.
With non-uniform magnetic field, the Hamiltonian of scalar BEC is given
asp2
H=+ gF µB F · B,
2m

(5.7)

where B(r, t) is spatially varying magnetic field and its direction is definedas b̂ = B/|B|. Consider the local unitary transformation U = exp(−iβ n̂ · F),
satisfying U (b̂ · F)U † = Fz , where n̂ = b̂ × ẑ/|b̂ × ẑ| and β is tilting angle from
113

z
x

y

Local unitarytransformation

Magnetic ﬁeld,B

Gauge potential,A

Figure 5.10: By a local unitary transformation U (r), a spatially varying magnetic
field is mapped to an effective magnetic field, where the field direction is uniform
containing a gauge potential A = −iU ∇U † .
β to z-axis.
Using the unitary transformation, the Hamiltonian can be expressed in
the |ϕi = U |Ψi basis as∂|Ψi = H|Ψi∂t∂i~ |ϕi = (U HU † − i~U ∂t U † )|ϕi = H e |ϕi∂t1(p + ~A)2 − ~V + gF µB Fz |B|)|ϕi,
= (
2m

i~

(5.8)
(5.9)

where the effective vector potential A(rm, t) = −iU ∇U † and effective scalarpotential A0 (r, t) = iU ∂t U † . A comes from the non-uniform spin texture and A0
comes from the time-varying spin texture. Then, the effective electromagnetic
fields arises from Eqn. 5.4.1,
Eei = −~∂i A0 − ~∂t Ai = −mF ~b̂ · (∂t b̂ × ∂t b̂)mF ijkBei = −~ijk (∂j Ak − ∂k Aj ) = −~ b̂ · (∂j b̂ × ∂i b̂).
2

(5.10)(5.11)

Note that the effective Hamiltonian has same form to the Hamiltonian of acharged particle in the magnetic field. Therefore, the geometric force can exert
Lorentz force to a particle moving in the system with a non-uniform magnetic
field, whether the particle is charged or not.
114

In our setup, the resultant effective magnetic field is given as√
r02 + 4z 02 + 2z 0
0√A(r ) = −θ̂,r0 r02 + 4z 02
2~(−r0 + z0 ẑ).
Be (r0 ) = −~∇ × A = 02
(r + 4z02 )3/2

(5.12)

This gauge field is very weak and the corresponding cyclotron frequency, ωc =~/4mz02 , is three orders of magnitude smaller than the trapping frequency[Fig. 5.2(c)]. In our recent work [177], we succeeded in demonstrating the existence of the effective Lorentz force in the current setup by observing that acircular c.m. motion of the trapped condensate is generated with linear driving,leading to vortex nucleation [Fig. 5.8(a,d) and Fig. 5.12(a)].

5.4.2

Chirality of Gauge potential

We study the gauge-field effect by investigating the response of the condensate to the trap shaking for various elliptical polarizations of the driving. We characterize the gauge-field effect by the number of vortices nucleated through driving. In order to control the polarization of the driving in afine manner, we reduce By by a factor of 6 and apply a rotating bias field as[Bx (t), By (t)] = δB[sin(ωm t), 61 sin(ωm t + φ)]. The relative phase φ is controlled
by changing the turn-on time of By as ∆t = (3π − φ)/ωm [Fig. 5.2(a)].
Figure 5.11 displays the vortex number measured after 500 ms driving
and 3 s relaxation as a function of the relative phase. In the result, we see a
two-dome structure which is asymmetric with respect to the sign of the relative
phase φ. The right dome with clockwise driving is relatively more enhanced
than the left dome with counterclockwise driving. This is consistent with thesign of the effective magnetic field. The rotation direction of the condensate was
115

(a)

60
80

40

Nv

Vortex number

80

40

20
0
-0.15 -0.05 0.05

0
-π

0.15

Driving polarization
-π/2

0

π/2

Relative phase (φ)

π
Nv

6

100

By/Bx

(b) 1

0
-π

-π/2

0

π/2

π

Relative phase (φ)

Figure 5.11: Gauge-field effect. (a) The vortex number Nv was measured after
500 ms driving and 3 s relaxation for various elliptical polarizations of thedriving (see the top axis). The driving polarization is controlled by the relative
phase φ between the modulations of Bx and By (see text for details). Each data
point was obtained from five measurements. The inset displays the same result
as a function of the elliptical polarization of the driving. The blue (green) circles
denote the data with |φ| ≤ π/2 (|φ| > φ/2). (b) The vortex number is measured
with various ratio of By /Bx (The data are measured in different day from (a)).

116

x

y

-300

(d)

-200

Angular momentum, Lz/ћC.M. position (µm)

100
(a)
50
0
-50
-100
100
(b)
50
0
-50
-100
100
(c)
50
0
-50
-100
0

-100
0
-600

(e)

-400
-200
0
-600

(f)

-400
-200

200

400

600

800

0

0

200 400 600

800

td (ms)td (ms)Figure 5.12: (a-c) Temporal evolution of condensate position under differently
polarized external modulation of magnetic fields. The graphs are obtained data
by 2D Gaussian fitting from a single in-situ image. (d-f) display external angularmomentum per atom, Lz , and the data points are binned for 16 ms. Lz ismeasured from (a-c), each. (a,d) The condensate is modulated linearly in xdirection. The relative phases are (b,e) 60◦ and (c,f) 30◦ with By /Bx = 1,
respectively. Solid lines are lines of the single-particle simulation.
confirmed to be the same as the sign of the driving polarization at φ = ±π/2 bymeasuring the precession direction of the quadrupole excitations in the rotatingcondensate [170, 171].
The vortex generation is almost suppressed at nonzero negative valuesof φ. An interesting question would be whether we can derive a quantitative
relation between the magnitude of the gauge field and the critical driving polarization for nullifying the vortex generation. In the inset of Fig. 5.11, we replot them sin φ.
data as a function of the driving polarization, d = tan 21 sin−1 2XXm2Y+Y2mm

In our experiment, one driving polarization can be realized with two values of
φ and we see that the vortex numbers for two realizations are slightly different
117

from each other. This means that around the critical driving polarization, thecondensate dynamics is sensitive to the turn-on time of By , i.e., the initial condition of the condensate. A more elaborated setup is desirable for a quantitative
study of the gauge field with the critical polarization.

5.5

Outlook
One of the interesting extensions of this work would be introducing an

additional trapping potential during the relaxation to mitigate the centrifugal
effect and to increase the rotation speed of the condensate.We expect that our
trapping potential might be deformed into a ring potential after the driving
phase by applying an additional optical potential [172–174] or using RF-field
dressing techniques [175, 176], which would result in a giant vortex state [160,161]. Recently, Roncaglia et al. [143] showed that a giant vortex state having alarge angular momentum in a ring potential is adiabatically connected to thebosonic ν = 1/2 Laughlin state in a harmonic potential.

118

Chapter 6
Conclusions
Spinor Bose-Einstein condensates have several phases, depending on thenature of the interactions and the superfluidity, and offers scientists the opportunity to explore the unexplored worlds such as the spin turbulence and theearly dynamics of universe in the feasible ways. Me and my colleagues peek inat this uncharted worlds a little through the quenched phase transition in aspin-wise 2D spinor BEC with antiferromagentic interactions. In addition, we
also observe the chirality of the gauge potential by measuring the relaxation
dynamics from a circulating center-of-mass motion into the internal rotation of
a BEC in an anharmonic trapping potential.
For the EAP-to-EPP phase transition, spin turbulence arises while thespin domains are generated via the spin-exchange collision. The time-space scaling is preserved in subsequent dynamics, including complex spin turbulence
generation, and the spatial scaling is reflected by the number of HQVs.
Alluring phenomena happens with the EPP-to-EAP phase transition.
The evolution of a quenched EPP condensate is easily affected by the initial spin
fluctuations in contrast to that of a quenched EAP condensate. The thermally
119excited spin fluctuations lead to the number increase of quantum noise seeds,
but delay of the phase transition dynamics. Because of the sample lifetime,
we cannot measure whether the dynamical instability is saturated or not. In
addition, during the evolution, combined-topological defects such as magnetic
composite-solitons and skyrmions are formed. We investigate the structure of amagnetic composite-soliton by using both the Bragg spectroscopy and the spin
separated imaging, and observe its splitting process.
A magnetic composite-soliton might take the role of the topological order, mediating the KT-transition in 2D spin-1 system. Near the critical point,
the symmetry of the order space recovers the full symmetry of the polar spin1 order parameters, [U (1) × S 2 ]/Z2 , and thus it is quiet fascinating question
what quantum phenomena arise here. The paired-superfluid phase is one of the
expected exotic phases [115], which is analogous to the FFLO state [178]. In addition, the evolution of a composite-topological defects are usually consideredas the one of the candidates to reveal the scenario of the galaxy formation after
the inflation. In the near future, one could uncover the spinor dynamics near
the critical point and observe the singlet pair superfluids.

120

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초

록

비 평형 동역학은 시간에 따른 강한 외부 자극이나, 보존되지 않는 물리량,에너지의 흐름 등의 정의가 어려운 복잡한 요소로 인해 이론적인 접근이 쉽지
않다. 그렇기 때문에 비 평형 계에 대한 연구는 통계역학적 이해도가 높은 평형 계의 물리 법칙을 비 평형 계로 확장시키는 방향으로 연구되어 왔다. 이러한노력 중 진전이 있는 분야는 상 전이 동역학 분야이다. 상 전이 동역학 연구는어떤 과정을 통해 계의 새로운 안정적인 상태에 도달할 것인지, 어떤 들뜸 상태(excitations)가 얼마나 나타날지 등에 초점이 맞춰져 있다. 임 계점 근처의 평형계는 상관 길이(correlation length), 공간 차원(dimension), 질서 변수의 대칭성(the symmetry of order parameter)에 의해 모든 통계역학적 물리량의 특이점이스케일링 양상으로 나타난다는 스케일링 가정(scaling hypothesis)으로 기술된다.
이러한 스케일링 가정은 비 평형 상태의 상 전이 동역학에도 적용될 수 있음이 잘알려져 있고 더 나아가 계의 보편성(universality)를 찾으려는 노력이 계속되고있다. 또한, 상 전이를 일으키는 들뜸 상태는 그 계의 질서 변수 다양체의 대칭성에 의해 나타나는 위상적 결함 (topological defects)이 생성되는 것으로 확인 할수 있음이 잘 알려져 있다.
전형적인 상 전이 동 역학 연구 방법은 양자적 퀜치(quantum quench)로
이는 계의 해밀토니안을 갑자기 변화시켜 기저 상태였던 계를 들뜬 상태에서새로운 상태로 가는 것을 관찰하는 방법이다. 하지만, 다 쳬계 양자 시스템에서이러한 비 평형 상태의 이론적 연구와 계사는 너무 복접하기 때문에 실험적인방법이 필요하다. 이를 위해서는 주변 환경의 영향을 받지 않는 고립된 계가 필요하고, 외부 변수들을 통제할 수 있는 연구 환경을 갖춰야 한다. 극저온 양자기체계는 외부 자기장과 전자기파를 이용하여 변수들을 비교적 쉽게 조절할 수 있고충분히 오랜 시간동안 결 맞음이 유지될 정도로 외부 환경으로부터 영향을 받지144

않는 계로 비 평형 상 전이 동역학을 연구하는 최적의 연구 환경을 제공한다.
이러한 배경을 바탕으로 이 학위 논문에서는 반 강자성 상호작용(antiferromagnetic)을 갖는 준 2차원 스피너 보즈-아인슈타인 응집체(spinor BEC)에서
나타나는 두 방향의 상 전이 동역학에 대한 연구를 다룬다. 반 강자성 스피너 BEC
에서 나타나는 위상(phase)은 quadratic Zeeman energy에 의해 easy-axis polar
상(EAP phase)과 easy-plane polar 상(EPP phase)으로 결정된다. 우리는 microwave dressing 기술을 이용하여 이차 Zeeman energy를 조절하여 급작스러운상 전이 동역학 과정에 접근하였다.
EAP 상에서 EPP 상으로의 상 전이(The EAP-to-EPP phase transition)의
초반의 평균장 이론에 의해 수평방향의 마그논(transverse magnon)이 들뜨면서동적 불안정성에 의해 진행된다. 이러한 초반 동역학은 스핀 난류가 생성되면서
벗어나게 된다. 그러나 평균장 이론에 의한 초반의 공간적인 스케일링 양상은 스핀 난류의 생성 과정에서 계속 유지가 되며, 시간 상의 스케일링 또한 존재하였다.
계가 안정화됨에 따라 스핀 난류가 사라지고 상 전이의 결과로 위상적 결함인 반양자수 소용돌이 (half-quantum vortex, HQV)를 확인하였다.
EPP 상에서는 에너지 갭이 있는 수평방향의 마그논 이외에도 에너지 갭이 없는 자기장 축 방향의 마그논 들뜸(gapless axial magnon excitation)에 의해스핀 요동(spin fluctuations)이 생길 수 있다. 아무런 스핀 요동이 없는 경우,
EPP 상에서 EAP 상으로의 상 전이(The EPP-to-EAP phase transition)는 앞선 the EAP-to-EPP 상 전이와 마찬가지로 초반 동역학은 평균장 이론에 의해스케일링 양상이 나타나며, 상 전이 동역학의 결과로 순환이 1인 양자 소용돌이(singly charged quantum vortex)를 확인할 수 있다. 하지만, 특이하게도 2차원반 강자성 스피너 BEC에서만 나타날 수 있는 한 끝이 HQV로 끝나는 솔리톤(soliton terminated by HQVs)이 나타나는 것을 관측하였다. 이러한 위상 물체를 magnetic composite-soliton(MCS)이라 명명하고 이를 Bragg spectroscopy와
Stern-Gerlach imaging을 이용하여 구조를 분석하였다. MCS는 BEC 내부의 흐145

름에 의해 점점 작은 위상 물체로 쪼개지며 특정 조건에서 양자 소용돌이가 될 수
있다. 이 경우, 양자 소용돌이의 핵(core) 부분이 자성(magnetization)을 가질 수
있다. 또한, 마지막으로 gapless axial magnon excitation을 이용하여 스핀 요동이동 역학 불안정성에 어떠한 영향을 주는지 관측하였다.
2차원의 낮은 차원에서는 열적 요동이 먼 거리 질서도(long-range order)를 파괴하여 질서 변수가 정의되지 못하지만, 준 먼 머리 질서도(quasi long-rangeorder)에 의해 상 전이 현상이 존재할 수 있음이 알려져 있다. 이를 BerezinskiiKosterlitz Thouless(BKT) 상 전이라고 부른다. MCS는 spin-1 BEC에서 나타나는 BKT 상전이를 매개하는 매개체로서 우리의 관측결과는 paired superfluidity의 가능성을 보여주는 첫 증거라고 볼 수 있다.
이 논문은 스칼라 BEC에서 스커미온(Skyrmion) 스핀 구조를 이용한 QV
격자 구조 생성 과정에 관한 연구도 포함하고 있다. 스커미온 스핀 구조는 스핀들이 공간적으로 변화하고 있기 때문에 Berry 곡률이 유한한 값을 갖게 되고 중성원자는 기하학적인 전자기력을 받는다. 우리는 사중극 자기장을 이용해 스칼라
BEC 시료에 기하학적 자기력을 가하였고 비 조화 우물(anharmonic trapping
potential)에 갇힌 스칼라 BEC 전체가 회전하는 움직임이 초유체 내부의 회전으로 변화하는 과정과 주변의 열적 기체(thermal cloud)의 영향을 확인하였다.
또한, 기하학적 홀 효과의 방향 비대칭성 (chirality of geometrical Hall effect)을
관측하였다.

주요어 : 반강자성 스피너 보즈-아인슈타인 응집체, 상 전이, 스케일링 가설,스피 양자 난류, 위상적 결함, 양자 요동.
학번:

2012-20353

146

Publication ListSCI Journals[1] J. Choi, S. Kang, and Y. Shin, “Observation of a Geometric Hall Effect ina Spinor BEC with a Skyrmion Spin Texture”, Physical Review Letters 111,
245301 (2013).
[2] S. Kang, J. Choi, S.W. Seo, W.J. Kwon, and Y. Shin, “Rotating a BEC byshaking an anharmonic axisymmetric magnetic potential ”, Physical Review A
91, 013603 (2015).
[3] S.W. Seo, S. Kang, W. J. Kwon, and Y. Shin, “Half-Quantum Vortices in an
Antiferromagnetic Spinor Bose-Einstein Condensate”, Physical Review Letters
115, 015301 (2015).
[4] S. Kang, S.W. Seo, J.H. Kim, and Y. Shin, “Emergence and scaling of spin
turbulence in quenched antiferromagnetic spinor BECs”, Physical Review A 95,
053638 (2017).
[5] S. Kang, and Y. Shin, “Observation of magnetic composite-solitons in an
quasi-2D antiferromagnetic spinor Bose-Einstein condensate”, arxiv: (2018).
[6] S. Kang, and Y. Shin, “Suppressing the amplification ofthe transverse spin
fluctuations by the axial spin fluctuations”, in preparation (2018).

147

Appendix

148

Appendix A
Scaling hypothesis
Near the critical point, the order parameter susceptibility χ ∼ (hφ2 i−hφi2 )diverges and the spatial fluctuations become strong and long-range, assumingξ = ξ± |q|−ν as |q| → 0 with ν > 0. Then we may write f with the LandauGinzburg-Wilson Hamiltonian [180] as
Z
r

f
H[S]
1 ~
u
=
= dd x S(~x)2 + [∇S(~x)]2 + S(~x)4 − b(~xS(~x)]
kB T
kB T
2
2
4


li σ
∼ ξ −d A0 + A1
+ ... ,
ξ
where S(~x) is a local order parameter satisfying φ = hSi, r = a(Q − Qc ) witha > 0, σ is non-negative exponent, li s are certain local length scales and Ai s
are coefficients with weak temperature dependence in d dimensions. Therefore,
the two-point order parameter correlation function G(~x) = hS(~x)S(0)i − φ2 andthermodynamic susceptibilty becomes
Z
Z
dd x
dd x
−i~k·~x~
G(q, k) =G(q, ~x)e=|~x|(d−2−η) G˜± (~x/ξ)
(2π)d
(2π)d
= |~k|(−2+η) G˜± (~kξ),
χ(q, ~k) ∼ ξ 2−η ∼ |q|−γ as ~k → 0,149

yielding the hyperscaling relations of α = 2 − dν, β = ν(d − 2 + η)/2, and δ =(d + 2 − η)/(d − 2 + η). This manifests the salient point of the scaling hypothesis:
everything can be expressed in terms of ξ, invariant under scale transformationnear the critical point. The insensitivity to the length transformation identifies
the fixed point of the transformation and leads to the renormalization groupmethod [179].

150

Appendix B
Calculation of quantum
depletion
Quantum depletion represents the excitation number of quasi-particles,
containing even no real excitations, induced by the interaction. Consider thecase of an antiferromagnetic F = 1 neutral atoms (Eqn. 3.3). For the EAPphase, the quantum depletion is given as
N − N01 X=
hâ0,k a†0,k + â+1,k a†+1,k + â−1,k a†−1,k i(B.1)
N
N k6=0
√
q
p
m3
2
(
nc
+
2
n|c2 |3 ) × f ((q/c2 n − 1) + sgn(c2 )),=3π 2 ~Z3  0
3x2 + t + 1
where f (t) ≡ √dx p− 1 x2 .
(B.2)
2
2
2
(x + t)(x + t + 2)
For q < 0 the quantum depletion is given as√qp
N − N0m3= 2 3 ( nc20 + n|c2 |3 ) × (1 + f (|q|/c2 n)).
N
3π ~Above result is obtained from Ref. [85].
151

(B.3)

In our experiments, we are only interested in the part, which is related tothe transverse magnon modes. Therefore, the quantum depletion can be rewritten as
n±1

Z
1
=
dkhn̂±1 (k)i
(2π)3
Z
1
=
dkhδ Ψ̂±1 (k)† δ Ψ̂±1 (k)i.
(2π)3

(B.4)

According to Ref. [88], by taking account of the Bose statistics for normal gas,hn̂±1 (k, T )i =

εk
1
1
εk
+
− ,
βE
k
Ek e
− 1 2Ek 2(B.5)

where β = 1/kB T , kB is the Boltzmann constant, and εk = ~2 k 2 /2M . We only
consider the temperature-independent terms. Then, we getZ(x2 + 1)1 2
3n±1 = 4πk0 dx p−x,2 x2 (x2 + 2) 2
where k0 =

p

(B.6)

2|c2 |nm/~. In our experiments, we assume that the spin fluctua-tion is identical for each sequence. Mathematica code is written in below.

152

In[28]:=h = 6.626 * 10 -34;
hbar = h • 2 • p;

kb = 1.38 ´ 10 ^ -23;
m0 = 1.66 * 10 -27;
a0 = 5.292 * 10 -11;
u = 10 -6 ;mRb = 87 m0;
mNa = 23 m0;
a = 2.75 ´ 10 ^ -9;
c0 =
c2 =

4 p hbar 2 amNa
4 p hbar 2
3 mNa

;

2.47 a0;

TotalM = 8.8 ´ 10 6 ; H*total atom number*L
wx = 3.8 ´ 2 p; wy = 5.5 ´ 2 p; wz = 402 ´ 2 p;
M = TotalM 0.9; H*bec atom number*L

wbar = Hwx wy wzL1•3 ; H*average frequency of xyz axis*Lm=n=hbar wbar

hbar • mNa • wbar

2
m
c0

Rx =

2•5

15 M a;;2m
mNa wx 22m; Ry =

nNa@x_, y_, z_D := n 1 -x2

Tc = 0.94 hbar wbar HTotalM L

Rx

TNa = Tc H1 - 0.9L1•3 ;

2

-

1•3

2m

; Rz =

mNa wy 2

y2
2

-

‘ kb;
Ry

mNa wz 2

;

z2
Rz 2

nNa0 = nNa@0, 0, 0D;

b0 =

4p

H2 pL3K

2 mNa c2 nNa0

NIntegrate B

hbar 2O

3•2

Ix2 + 1M

x Ix + 2M

x2

2

2

1

ExpB

8x, 0, ¥<F “ nNa0;

4p

H2 pL

3

K

2

2 mNa c2 nNa022

Ir2 + 1M

constb0 = NIntegrate Br2b0 =x Ix + 2M
2

O

3•2

r Ir + 2M22

c2 nNa0
F-1
kb TNa-1
2

constb0 • nNa0

Ix2 + 1M+
2

x Ix + 2M
2

2

-

1
2

,

, 8r, 0, ¥<F;

H*quantum spin fluctuation *L
4p
2 mNa c2 nNa@x , y, zD 3•2
b@x_, y_, z_D :=K
O constb0 • nNa@x , y, zD
3
H2 pLhbar 24p
2 mNa c2 nNa@x , y, zD 3•2b2@x_, y_, z_D :=K
O constb0
3
H2 pL
hbar 2
Out[49]=3.29816 ´ 10 -6

hbar

153

Appendix C
Examples of SCPI commands
We use N5185A Function Generator from Agilent to generate microwave
pulses. There exist 5 ms delay to perform SPCI commands, after TTL signal.
It needs initializing commands before starting experimental sequence such as”TRIG:SOUR EXT ; :INIT:CONT ON ; :FREQ:MODE LIST ;LIST:RETR OFF ;”TRIGgger:SOURce EXTernal

Choose trigger source as external

TTL signal. When TTL signal is positive, a function generator transmits themodulating signal.
:INITiate:CONTiuous ON

This command selects restart sweep

automatically or wait until the trigger source is received, after the previous
sweep is finished. Only single sweep will be performed, if choose OFF.
:FREQuency:MODE LIST

Select frequency sweep mode. Whenappropriate trigger source is received, executes the commands to start the LIST
or STEP frequency sweep.
:LIST:RETRace OFF

The last sweep point is sustained, until thesweep is initiated and triggered again.
154

Because we generate rapid phase transition and study the relaxation dynamics by changing microwave frequency, the final microwave frequency should
be unchanged for long hold times. But, :LIST:RETR OFF command affects
the next sequence because of 5 ms signal delay by disturbing the initial preparation of the well-defined condensate before quench. Thus, at the end of sequences,
we commands the start point of the next sequence using GPIB control.
Example of sweep commands.
SWE:DWEL 0.002
SWE:POIN

4

FREQ:STAR

3

MHZ

FREQ:STOP

6

MHZ

Then, frequency changes as
Time after TTL signal Frequency Time Difference
5 ms

3 MHz

12 ms

4 MHz

7 ms

19 ms

5 MHz

7 ms

26 ms

6 MHz

7 ms

We confirm that these commands clearly works in the sweep range of a tens ofMHz. Minimum dwell time is 5 ms. Therefore, when we set the dwell time ofeach step as 0.0001 s(=100 µ s), the step dwell time becomes 5.1 ms. If the stepdwell time is set to 0.001 s(=1 ms), the actual step dwell time will be 6 ms.
:SWEep:DWELl

Set step dwell time.

:SWEep:POINts

Set the number of steps.

:FREQuency:STARt

Set the starting frequency of modulating sig-

:FREQuency:STOP

Set the final frequency of modulating signal.

nal.

155

Appendix D
Sad history
1.
We were trying to figure out the magnetic field gradient, generated by
the grad.z and grad.xy coils. But, since we empirically found the perfect conditions for removing the field inhomogeneity, we did not need the calibration.
We guess that the quadrupole zero fields, generated by each grad. coil pairs,
are placed as in fig. D(b). Empirically, we minimize the magnetic field gradient following the process: By using one gradient coils, make the Ramsey fringe
pattern aligns vertically. Then, remove the field inhomogeneity as possible, byusing other coils. Elaborately, tune the both gradient coil pairs to cancel out
the field inhomogeneity.
2. We were trying to measure skyrmion decay in a polar BEC. We expected that a skyrmion would be divided into two half-quantum vortices, where
the total mass current is zero. The orientation of two HQVs’ supercurrents
should be opposite. Figure D(d) shows an image of the desired result. However,
the experiments wouldn’t be completed, because of the technical issues.
A skyrmion spin texture has been imprinted by a quadrupole magnetic
156

(a) w/o grad. coils

y

(b)
x

Qxy
BEC
Qz

(c) grad.xy coils(d) grad.z coils

0.26 A

0.10 A

2.27 A

0.35 A

Figure D.1: Trying to figure out the magnetic field inhomogeneity of the (c)grad.xy and (d) grad.z coils using the Ramsey interferometry. The DC magnetic
field is 35 mG and the interrogation time is τ = 10 ms. The orientation of the
field gradient, generated by the grad.z coils are confirmed as red line in below
image. (b) illustrates predicted zero-field positions on top.

(a)

(c)

(b)

(d)

BEC

z

(e)
BEC

10 ms

40 ms

60 ms

100 msFigure D.2: Trying to observe the dynamics of Skyrmion decay. (a,b) Two
schematic experimental sequences. (c) SG image of created Skyrmion after 24 ms
spin separation. (d) Desired result. (e) Temporal evolution of (c), taken by Insitu magnetization image. The quadratic Zeeman energy, q/h = 0.3 Hz.

157

gradient. We used the quadrupole coils to generate quadrupole field gradient.
The z-directional bias coils and Feshbach coils are used to control ±z-directional
bias magnetic fields. The skyrmion was imprinted as in Fig. D(c). Then, thequadrupole fields was turned off and it went upward. Thus, for the sequence ofFig. D(a), the condensate felt the field inhomogeneity and skyrmion ring always
split in fixed direction as in Fig. D(e). However, the backflow current of the
Feschbach coils inhibited experiments with the sequence of Fig. D(b) at thattime. Hence, we put the skyrmion decay study in the future, and moved on the
study of the phase transition dynamics.

158