Master thesis

Experimental Study on the
Dynamics of a Dual-Species
Bose-Einstein Condensate with
Tunable Interactions
Supervisor: Associate Prof. Shin Inouye

Department of Applied Physics
School of Engineering
The University of Tokyo
37-126537 Alto OSADA

Submitted in February, 2014

ABSTRACT
Since the first realization of a dilute gas multi-component Bose Einstein
Condensate (BEC), non-equilibrium properties of the gas have been investigated extensively, showing great success in unveiling the nature of solitons,
various spin textures, vortex rings, etc.. Until recently, however, important
ingredient in cold-atom experiments was not incorporated in these studies:
manipulation of interaction strength using Feshbach resonance.
Our experimental setup of 41 K and 87 Rb allows us to investigate thesuperfluid-superfluid interaction quantitatively with tunable interaction andto compare results to that of coupled Gross-Pitaevskii (GP) equations. As
an example, we examined the modulation instability of a strongly interacting dual BEC, by observing the time-evolution of a dual BEC using nondestructive imaging method. The spatial profile of the dual BEC in an elongated trap showed a characteristic pattern, and the change in the length scale
was studied as a function of the interspecies scattering length, and comparedwith the results from the coupled Gross-Pitaevskii equations. The observed
size of spatial structures (22(4) µm at aKRb = 80.8 a B ) agreed well with thetheoretical prediction of 23 µm.
.

iACKNOWLEDGMENTS
本論文は東京大学大学院工学系研究科物理工学専攻における 2013 年度の修士論文として執筆されたものです。本論文の執筆に際してお世話になった、
あるいは助力をいただいた方々へのお礼の言葉をここに述べさせていただきた
いと思います。
指導教官である井上慎准教授には、学部 4 年次より 3 年間お世話になりました。学部 4 年のころより私の素朴な質問にも親切に答えてくださり、ミー
ティングやその他の機会を通して沢山のことを学ばせていただきました。特に物理現象をなるべく平素な言葉で述べる姿勢、プレゼンテーションの際に聴衆の興味を引くためのテクニックなど非常に参考になるものが多く、今後の研究
生活の基盤をなすもっとも大きな構成要素といっても過言でない事柄を多く学ぶことができました。また、研究以外でも社会生活を営む上での作法や態度なども勉強をさせていただきました。たいへん世間知らずな私の性質でむしろご迷惑をお掛けすることも多々あり、私が問題を起こすたびにフォローをしていただき、私を諭してくださったことにはいくら感謝をしてもしきれません。まだまだ世間知らずが治ったとは言えない私ですが、今後も先生のお言葉を活かし、一角の人物となれるよう精進を怠らず邁進していく所存です。
助教である小林淳氏には学部 4 年次では ULE cavity を用いた光源の作成
で、修士課程進学後はミーティングやその他の機会でお世話になりました。とくに氏の豊富な経験からの助言は実験の方法や考察の随所随所で大変参考になり、修士２年次からの私単独での実験において、もっともたくさん助言をいた
だいた方の一人です。本論文 4 章で述べた TOF シークエンスの改善の折や 5 章
でのデータ解析において助言をいただいたことが特に印象に残っております。
また実験データや現象・物理の本質の理解に対して非常に積極的で、かつデー

iii

iv
タの妥当性に関して厳しくもあった氏の姿勢はぜひ見習いたく思います。また、実験中手が空くときにはよく他愛のない雑談等に付き合っていただきました。
ありがとうございました。
現在博士 3 年次の加藤宏平氏とは私が修士課程に進学してからかれこれ

2 年間、E1 グループの実験装置を共有してきました。初めの 1 年は氏とともに極低温極性分子の生成を目指した研究を行い、氏の指導の下実験装置に関する様々なことをはじめ電子デバイスやオプティクスの基礎的な扱い方から物品の発注の仕方まで一通りのことを学びました。修士 2 年に進級して実験テーマが別になったのちも私が投じるトンチンカンな質問にも真面目な議論にも根気強く答えてくださり、大変感謝しております。私の大雑把な性格とは対照的に念入りに実験を行う氏の姿勢は印象が深く、確認を怠ったことによるミスを犯す
たびに氏の慎重さを手本とし、改善の努力をしてきました。お手を煩わせることも多々あったとは思いますが、大変お世話になりました。
昨年井上研で修士論文を出された上原城児氏はアルカリ原子の光会合をテーマにされ、E1 グループの装置も用いて実験が行われていたのを覚えています。氏の実験テーマ、実験結果は当時 E1 グループに配属されて全容を把握しきれず、ようやく分子生成の話もつかみかけていた私には興味深くみえました。氏の夜遅くまで作業に取り組む姿に、私も頑張らねばと自分を奮い立たせていました。
昨年度卒業論文を井上研究室で執筆し、現在は情報理工学系研究科の数
理情報第 5 研究室に所属している鈴木皓博氏は、上原城児氏やその他の研究室メンバーと協力してずっとひたむきに作業をこなしておりました。試行錯誤をし、結果を残そうとしていた氏の姿が非常に心に残っています。これに並行してまったく分野の違う研究科を目指し、しっかり合格をもぎ取った氏の行動力にも驚きましたが、氏が修士課程に進学したのちに井上研に顔を出されたときに元気そうな顔を見せてくださり、私としても他愛もない話ができて大変楽し
く、非常に励みになりました。
現在修士 1 年次の早川悠介氏とは彼が修士課程で井上研に配属され、E1グループの実験に本格的に携わってから 1 年にも満たない期間の付き合いです
が、氏には実験上非常に多くの面で助けられました。とくに Fringe Cleaningの実行やカメラのデータ取得に関する問題点に関してはプログラミング言語にv
慣れ親しんだ彼の助力なしには成しえなかったと考えられます。また、Rb のZeeman シフトの計算も Phase Contrast Imaging には不可欠のものでした。現
在は加藤宏平氏とともに分子生成の実験及び Efimov 状態に関連した物理の研究を精力的に行っており、修士 1 年次ということで就職活動も始まっているようで大変でしょうが、ご健闘をお祈りしています。
同じく現在修士 1 年次の荻野敦氏は私が修士 1 年の時に学部 4 年生として井上研に配属され、以来現在までの 2 年弱を主に E2 グループにおいて過ごしてこられました。氏が学部 4 年次の時の研究テーマが私の学部 4 年次のそれと共通点が多くあったため当時はお手伝いをすることもありましたが、いまや小林淳助教の指導のもとで E2 グループの実験装置をコントロールしつつ、基礎定数の恒常性検証という問題に立ち向かう頼もしいメンバーとなっています。
夜遅くまで残って根気強く実験する氏の姿を見かけるたび自分の身が引き締まる思いでいました。氏も早川悠介氏とともに就職活動に身を投じることになります。なにかと大変なこともあるかとは思いますが、氏にとって良い結果となることを祈ります。
学部 4 年生として井上研究室に配属され、その後修士課程では物性研究所小林研に所属している現修士 1 年の大久保弘樹氏は氏の卒業研究において荻野敦氏とともに研究に励んでおられました。修士課程に進んでからも輪講でお会いした際に近況をお聞きすることができ、大変嬉しく思いました。氏の卒業研究は荻野敦氏と同じく冷却分子を用いた基礎定数の恒常性の検証でしたが、発表なども非常にわかりやすく、勉強をさせていただきました。氏のますますのご活躍を期待しております。
現在学部 4 年生として井上研に配属され、卒業研究をされている赤羽健二
氏、小野貴晃氏は小林淳氏の指導の下に着実に研究を進める姿が印象に残りました。特に両氏がお互いによく相談し、理解しあって分担して作業をこなす様
子や、作業中積極的に質問をしてくださる点にはある種の頼もしさを感じました。彼らの抱える技術的問題に対して私のアドバイスは的外れなことも多かったと思いますが、少しでも彼らにとって利する部分があれば幸いに思います。
両氏ともに修士課程進学後には別の研究室へと巣立っていくことになりますが、井上研で学んだことを糧に羽ばたいて行かれることを期待しております。
論文紹介セミナーでは実験、理論問わず冷却原子の研究を行っている研究vi
室の学生とスタッフが一堂に会して週ごとの論文を持ち回りで紹介しました。
論文紹介に参加された中でも理学系研究科上田研の遠藤晋平氏、理化学研究所
肥山研に所属されている作道直幸氏は理論家の観点から選ばれた論文の紹介と実験的な興味の両方をカバーした内容を常に提供してくださり、理解しきれないところはありつつも非常に興味深く新鮮に聞くことができました。とくにビリアル展開の物理的な意味と重要性や、フェルミ粒子のユニタリー極限に関する作道直幸氏の説明は明快で、いまだに記憶に残っております。また理学系研
究科五神研の冷却原子実験チームの堀越宗一助教は多様な実験を理解し、また知らないことに対する理解に貪欲でありました。同チームの学生である五神研所属の池町拓也氏、小芦研所属の伊藤亜紀氏は積極的に質問をして自らの理解を進めようとする姿に大変刺激を受けました。また小芦研の市田昌己氏はこのセミナーを通してご自身の知識を増やそうとされていて、見習いたい態度だと感じました。
私は 2013 年 1 月より統合物質科学リーダー養成プログラム（Materials

Education program for the future leaders in Research, Industry, and Technology、MERIT）に 2014 年 3 月まで所属させていただいております。MERIT
プログラムにおいては多彩なコースワーク、特別講義等を享受する機会をいた
だき非常に勉強させていただいたことに加え、奨励金の支給は私の生活の支えとなりました。MERIT プログラムにおいて副指導教員を担当していただいた中村泰信教授には四半期に一度の進捗報告の際には近しいとはいえ異分野の私の話を興味を持って聞いていただき、また中村研究室の学生、スタッフの方々との交流をさせていただいたりと貴重な体験をさせていただきました。とくに私の博士課程進学後に指導教員を担当していただく宇佐見康二准教授には初めて
中村研にお伺いした際に新しく赴任されるとご紹介いただき、それがきっかけ
で学会でお話を聞かせていただくなどして博士課程から異分野へ飛び込み、宇佐見准教授の下で新たな分野について学ぼうという決心をしました。現在の私の進路を決めるうえで中村教授、宇佐見准教授と出会うことができたことは非常に大きな影響を持っており、このような機会に恵まれたのはひとえに MERIT
プログラムの副指導教員制の恩恵であると強く感じています。博士課程進学後はほとんど一からのスタートにはなりますが、私の選択とご縁はきっと私の今後にとってかけがえのないものになると信じ、日々精進していく所存です。

vii
上記以外にも、私の家族をはじめとして学生およびスタッフの居室にいた様々な研究室の方々、物理工学専攻の教務関係の方々、6 号館工作室や 9 号館
事務の方々、教養学部時代からの友人、MERIT を通じて知り合った友人、数え切れないほど様々な方の助けを得て修士課程 2 年間を過ごすことができました。
この修士論文が完成をみることができたのは上記のすべての方々の助力あってこそのものです。最後に改めて心からの感謝の意を表し、この謝辞の結びとしたいと思います。

2014 年 2 月

長田有登Contents
1 INTRODUCTION
1.1

1.2

1.3
2

Historical Overviews . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Laser Cooling and Trapping of Neutral Atoms . . . . . . .

1

1.1.2

Realization of Bose-Einstein Condensation(1995) . . . . . .

4

1.1.3

Experiments Using BEC . . . . . . . . . . . . . . . . . . .

6

1.1.4

Experiments on Superfluid Helium . . . . . . . . . . . . .

7

Two-component Bose-Einstein Condensates . . . . . . . . . . .

8

1.2.1

Experiments on Two-component BECs . . . . . . . . . . .

8

1.2.2

Another Application: Ultracold Molecules . . . . . . . . .

10Contents of This Thesis . . . . . . . . . . . . . . . . . . . . . . .

12

THEORETICAL DESCRIPTION OF A DUAL-SPECIES BEC

13

2.1

Single-Component BEC . . . . . . . . . . . . . . . . . . . . . .

13

2.1.1

Gross-Pitaevskii Equation . . . . . . . . . . . . . . . . . .

13

2.1.2

The Transition temperature and The Condensate Fraction .

17

Two-Component BEC . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2.1

Coupled Gross-Pitaevskii Equations . . . . . . . . . . . . .

18

2.2.2

Tuning the Scattering Length by Feshbach Resonance . . .

24

2.2

3

1

EXPERIMENTAL SETUP
3.1

27

Toward In-situ, Nondestructive Observation of a Dual BEC . .

27

3.1.1

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

27

3.1.2

Elongated dual BEC in an Optical Dipole Trap . . . . . . .

28

3.1.3

Gravitational Sag and Magnetic Field Gradient . . . . . . .

29ix

CONTENTS

x
3.2

3.3

3.4

3.5

Interaction Between Electromagnetic Field and an Atom . . .

30

3.2.1

Density Matrix and Optical Bloch Equations . . . . . . . .

30

3.2.2

Refractive Index and Phase Shift . . . . . . . . . . . . . . .

32

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

34

3.3.1

Schematics of the Phase Contrast Imaging . . . . . . . . . .

34

3.3.2

Probe Lasers for Phase Contrast Imaging . . . . . . . . . .

37

3.3.3

Generating Pulse Trains for Probe Lights . . . . . . . . . .

43

Setup of Optical Dipole Traps . . . . . . . . . . . . . . . . . . .

45

3.4.1

Light Shift . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.4.2

Experimental Sequences . . . . . . . . . . . . . . . . . . .

47

3.4.3

Crossed ODT and Optical Lattice . . . . . . . . . . . . . .

48

3.4.4

Gravitational Sag and 809 nm ODT . . . . . . . . . . . . .

51

3.4.5

Setup for Generating Elongated BECs . . . . . . . . . . . .

52

Gravitational Sag and Magnetic Field Gradient . . . . . . . . .

56

3.5.1

Gravitational Sag in the Horizontal Direction . . . . . . . .

56

3.5.2

Effect of Magnetic Field Gradient on the Potential Minimum 57

4 IN-SITU OBSERVATION OF A DUAL BEC
4.1

4.2

4.3

Creation of an Elongated Dual BEC in an ODT . . . . . . . . .

62

4.1.1

Time-of-Flight Measurement . . . . . . . . . . . . . . . . .

62

4.1.2

Elongated Dual BEC in a New Crossed ODT . . . . . . . .

64

In-Situ Observation of a Dual BEC . . . . . . . . . . . . . . . .

65

4.2.1

Probe lasers for PCI and Phase Plate . . . . . . . . . . . . .

65

4.2.2

Phase Contrast Imaging of a Dual BEC . . . . . . . . . . .

69

Spatially Overlapped Dual BEC in 809 nm ODT . . . . . . . .

72

4.3.1

Cancellation of Relative Gravitational Sag in 809 nm ODT

72

4.3.2

Static Nature of Dual BEC . . . . . . . . . . . . . . . . . .

72

4.3.3

Tuning the Relative Position of Two BECs by Magnetic Field
Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Dynamical Motion of an Immiscible Dual BEC . . . . . . .

80

The Glitch in Time-of-Flight Imaging . . . . . . . . . . . . . . .

84

4.3.4
4.4

61

CONTENTS
5xi

OBSERVATION OF MODULATION INSTABILITY WITH A DUAL
BEC

89

5.1

Theoretical Description of the Modulation Instability . . . . .

91

5.2

Preparation of a New Optical Trap of an 809 nm ODT . . . . .

98

5.3

5.2.1

New Shape of an 809 nm Optical Dipole Trap

. . . . . . .

99

5.2.2

Radial Trapping Frequency of the New Optical Dipole Trap

100

5.2.3

Loading a dual BEC into the New Optical Dipole Trap . . . 104

Experimental Observation of the Modulation Instability . . . . 106
5.3.1

Experimental Conditions for Observing the Modulation Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6

5.3.2

Experimental Sequence for Observing Modulation Instability 107

5.3.3

Results at aKRb = 80.8 a B . . . . . . . . . . . . . . . . . . 108

5.3.4

Results at aKRb = 81.5 a B . . . . . . . . . . . . . . . . . . 114

5.3.5

Summary of the Results . . . . . . . . . . . . . . . . . . . 119

SUMMARY AND CONCLUSIONSA ALKALI D-LINE DATA

123
127

A.1 Properties of 87 Rb D2 Line . . . . . . . . . . . . . . . . . . . . . 127
A.2 Properties of 41 K D2 Line . . . . . . . . . . . . . . . . . . . . . . 128
B FRINGE CLEANING131

C DISCRETE FOURIER TRANSFORMATION

135

C.1 Discrete Fourier Transformation . . . . . . . . . . . . . . . . . . 136
C.2 Sampling Theorem and Aliasing . . . . . . . . . . . . . . . . . 137
C.3 Analysis of Experimentally Obtained Data . . . . . . . . . . . 138
D MEASUREMENT OF THREE-BODY LOSS COEFFICIENT141

E NUMERICAL SOLUTION OF COUPLED GP EQUATIONS147

F C++ PROGRAM FOR ANALYZING THE IMAGE

151

List of Figures
1.1

Schematic representation of a Magneto-optical trap (MOT). . .

1.2

False-color surface plots of the velocity distributions of atomic

3

clouds for various temperatures[11] . . . . . . . . . . . . . . .

5

2.1

Schematic plot of a Thomas-Fermi density profile . . . . . . .

16

2.2

Schematic plots of Thomas-Fermi density profiles of dual BECs
with variable interactions . . . . . . . . . . . . . . . . . . . . .

20

2.3

Schematic representation of scattering potentials . . . . . . . .

23

2.4

Schematic representation of an avoided crossing between atomic
and molecular states . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Theoretically calculated value of the interspecies scattering
length a RbK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

23
26

Theoretically calculated value of interspecies scattering lengtha RbK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.1

Schematic representation of the phase contrast imaging . . . .

34

3.2

Side imaging system . . . . . . . . . . . . . . . . . . . . . . . .

36

3.3

Top imaging system . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.4

Calculation of Zeeman shifts for K D2 line . . . . . . . . . . . .

40

3.5

Calculation of Zeeman shifts for Rb D2 line . . . . . . . . . . .

41

3.6

Preparation of probe lasers for PCI . . . . . . . . . . . . . . . .

42

3.7

Instruments for TTL pulse train generation . . . . . . . . . . .

44

3.8

Configuration of ODT (A) and (P). . . . . . . . . . . . . . . .

49

3.9

Configuration of optical lattice beams . . . . . . . . . . . . . .

50

xiii

LIST OF FIGURES

xiv

3.10 Modified evaporation sequence . . . . . . . . . . . . . . . . . .

53

3.11 Crossed dipole trap with ODT (P) and (B) . . . . . . . . . . . .

54

3.12 Calculation of potential surface by crossed dipole trap withODT (P) and (B) . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.13 Illustration of a tilted optical dipole trap and a plot of relative
position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.14 Illustration of the configuration of compensation coils and its
magnetic field/field gradient . . . . . . . . . . . . . . . . . . .

59

4.1

Time sequence of TOF and absorption imaging . . . . . . . . .

63

4.2

Absorption image of thermal clouds and dual BEC . . . . . . .

65

4.3

Optical path of probe beam before the glass cell . . . . . . . .

66

4.4

Optical path of probe beam after the glass cell . . . . . . . . .

66

4.5

Picture of optical path of probe beam after the glass cell . . . .

67

4.6

Schematic representation of focused Gaussian beam througha lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.7

Schematics of imaging sequence of PCI . . . . . . . . . . . . .

70

4.8

Observation of dual BEC generated in 1 µm ODT by PCI . . .

71

4.9

Time sequence of loading atoms into 809 nm ODT . . . . . . .

73

4.10 Two BECs in (a) an 1 µm ODT and (b) the 809 nm ODT . . . .

74

4.11 Cancellation of relative gravitational sag . . . . . . . . . . . . .

74

4.12 Observation of phase separation with dual BEC in 809 nm
ODT by absorption imaging. . . . . . . . . . . . . . . . . . . .

75

4.13 Miscible dual BEC observed by PCI . . . . . . . . . . . . . . .

76

4.14 Immiscible dual BEC observed by PCI . . . . . . . . . . . . . .

77

4.15 Tuning the relative position in horizontal direction . . . . . . .

79

4.16 Dynamical transition from a metastable configuration to thestable configuration . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.17 Time sequence of the loading into an 809 nm ODT . . . . . . .

82

4.18 Dynamical motion of a dual BEC undergoing the sudden changeof the interspecies scattering length. . . . . . . . . . . . . . . .

83

LIST OF FIGURES

xv

4.19 Typical images obtained with TOF measurement written in
Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.20 Modified TOF sequence . . . . . . . . . . . . . . . . . . . . . .

87

4.21 Comparison between dual BECs produced by the former andthe modified TOF sequence . . . . . . . . . . . . . . . . . . . .

88

5.1

Optical path of the tight 809 nm optical trap . . . . . . . . . . .

97

5.2

TOF measurement of rubidium BEC in (a) the former 809 nm
trap and (b) the new one . . . . . . . . . . . . . . . . . . . . . . 101

5.3

Measurement of the trapping frequency with a laser power of
84.6 mW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4

Measurement of the lifetime with a laser power of 84.6 mW . 102

5.5

Measurement of the trapping frequency with a laser power of
27.4 mW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.6

Measurement of the lifetime with a laser power of 27.4 mW . 103

5.7

TOF measurement of a dual BEC in the new 809 nm trap . . . 104

5.8

Calculated values of k max versus aKRb

5.9

Calculated values of Gmax versus aKRb . . . . . . . . . . . . . . 105. . . . . . . . . . . . . . 105

5.10 Hall probe measurement of magnetic field sweep . . . . . . . 107
5.11 Typical images of time evolution at aKRb = 80.8 a B . . . . . . . 109
5.12 Spectrum obtained at aKRb = 78.5a B . . . . . . . . . . . . . . . 110
5.13 Subtracted signals of two atomic species and Fourier transformation of them(aKRb = 80.8 a B ) . . . . . . . . . . . . . . . . . . 111
5.14 Spectrum obtained at aKRb = 80.8 a B .

. . . . . . . . . . . . . . 113

5.15 Typical images of time evolution at aKRb = 81.5 a B . . . . . . . 115
5.16 Subtracted signals of two atomic species and Fourier transformation of them(aKRb = 81.5 a B ) . . . . . . . . . . . . . . . . . . 116
5.17 Subtracted signals and fittings on them . . . . . . . . . . . . . 117
5.18 Spectrum obtained at aKRb = 81.5 a B .

. . . . . . . . . . . . . . 118

5.19 Spatial pattern imprinting on the BEC . . . . . . . . . . . . . . 119
A.1 Level structures of Rb and K D2 line . . . . . . . . . . . . . . . 129

xvi

LIST OF FIGURES
B.1 Processed images of a dual BEC with variable number of reference images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
C.1 Data points and sinusoidal wave fitting to them. . . . . . . . . 138
C.2 Data processing of images to obtain Fourier spectrum. . . . . . 139
D.1 Measurement of three-body loss coefficient in thermal mixture of K and Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
D.2 In-situ observation of dual BEC with time interval 200 ms. . . 146
E.1 Numerically calculated Thomas-Fermi profiles of dual BECs
with various interactions. . . . . . . . . . . . . . . . . . . . . . 150

Chapter 1
INTRODUCTION
1.1

Historical Overviews

Before proceeding to the introduction of a two-component BEC and studies on it, we will review the history of atomic physics and lasers. The inventions closely related to atomic physics are introduced step by step, fromlasers to a MOT. At the end of the section we mention the realization of asingle-species BEC and give a brief review on experiments of superfluid heliums, which are studies on the superfluid lasting from the beginning of the20th century. We will describe experiments and perspectives on a dual BEC
in the next Section, 1.2.

1.1.1

Laser Cooling and Trapping of Neutral Atoms

It was more than 50 years ago when the first amplification of electromagnetic wave was realized by A. L. Schawlow and C. H. Townes[2]. This
novel technique, the laser oscillation, is so ubiquitous that it is used very
commonly in various region of scientific research and in our daily life, e.g.,
optical writing and readout of the information to data disks, laser pointers
we use, and revolutionary fast optical communication.
By now the laser is a powerful tool to investigate the phenomena appearing in the nature. Since the laser light has spatial and temporal coher1

ence, light field with some specific frequency and wavevector can be applied
to a matter. This enables not only the detailed spectroscopy of the atomic,
molecular levels and the elementary excitations in solid-state materials, butalso the coherent control of them. In practice, for spectroscopic purpose weapply laser pulses to target matters and measure the frequency-dependenceof absorption or fluorescence with a high resolution. Simultaneously, thismeans that we can address elementary excitations in a selective manner,
or if the transition dipole moment and laser intensity are known, we canshine a π/2-pulse to prepare the superposed state of the ground and theexcited state. Besides, of course the light-matter interaction is not limited
only to the context of excitation, but can expose an influence on the groundstate without any excitations by making use of a off-resonant laser. Suchcontrol and spectroscopy are successfully realized in atomic gas systems because of an atom’s relatively simple electronic structure and its discrete energy levels. There have been experiments which investigate various physicalphenomena and quantities by utilizing these properties such as the Sagnac
effect[3], the measurement of the physical constants h̄/m[4] and the gravitational acceleration[5].
In order to provide an overview of how an atom interacts with light field,
we introduce two forces acting on an atom. The first one comes from theabsorption. When an atom absorbs a photon, momentum conservation saysthat it is kicked in the direction of photon. After a while the atom emits aphoton in a random direction (spontaneous emission) and it experiences akick in a random direction. For the absorbed photon has always the same
momentum (wavevector) and emitted one has its direction at random, thenet momentum that an atom acquires is (the number of absorbed photons)× (the momentum of a single photon) and thus an atom feels the net force
acting in a direction of the wavevector of photon. This net force is called theradiation pressure.
The second way of interaction is due to the refractive index of a single
atom. In this case an atom does not absorb photons, but it bends them to2

Figure 1.1: Schematics representation of a Magneto-optical trap (MOT),
which consists of six red-detuned lasers and a pair of coils in the antiHelmholtz configuration.

feel the net force, this time from the gradient of light intensity. As describedin detail in Chapter 3, this interaction is derived by regarding an atom as
an electric dipole whose orientation follows the direction of electromagnetic
wave (electric dipole approximation). This force is essentially different fromthe radiation pressure and called the dipole force.
Above, two types of forces acting on an atom by light have been introduced. The important fact is that the radiation pressure is dissipative while
the dipole force is conservative. Proper use of dissipative force can reduce
the total energy of, or cool, the ensemble. For the atomic gases, this scheme
3

works very well because they can be put in a highly isolated environment.
The most commonly used instrument is a vacuum chamber with gas pressure on the order of 10−9 to 10−11 Torr. Thus the cooling of matter by opticalforces has been developed successively for atomic gases and the techniqueof Zeeman slowing[6], optical molasses[7] and magneto-optical trap[8] wererealized. Optical molasses is based on the radiation pressure combined with
the Doppler effect. This is realized by counterpropagating two beams inthree directions, whose frequencies are slightly lower than the atomic resonance.
Magneto-optical trap (MOT) is constructed by just putting a pair of coilsin the anti-Helmholtz configuration in addition to optical molasses. Atoms
in MOT is dramatically cooled below 1 mK from room temperature. The
dipole force is also frequently used as an optical dipole trap (ODT). ODT isa very common technique comparable to the magnetic trap. The advantage
of using ODT is that this can trap any spin states and enables us to investigate properties of atoms in the magnetically-untrappable hyperfine states.
In addition, by counterpropagating two beams and letting them interfere toform optical lattice, we can study interaction of atoms and periodic potential
as a matter-wave gratings, or a lattice potential in which atoms mimics theparticles in condensed matter systems.

1.1.2

Realization of Bose-Einstein Condensation(1995)

Magneto-optical trap can cool an atomic ensemble below 1 mK. Even if
Sisyphus cooling works to lower the temperature down to µK regime, thephase space density ñ = nλ3dB is several orders of magnitude smaller than 1,
while the atomic gas undergoes the Bose-Einstein condensation (BEC) transition with phase space density ñ ∼ 1. In order to achieve lower temperature and higher spatial density, evaporative cooling was used. Roughly
speaking, this technique throw relatively hot atoms away and then remaining atoms are thermally redistributed to get cold. In 1995, researchers cooled4Figure 1.2: False-color surface plots of the velocity distributions of atomic
clouds for various temperatures, each corresponds to the one of 400 nK, 200
nK and 50 nK from left to right.[11].

atoms down to about 100 nK, especially the ones belong to Bose statistics,
and had found that they condensed into the quantum ground state of thetrapping potential: the BEC was first realized by E. Cornell, C. Wieman,and Wolfgang Ketterle’s group [9, 10]. In short, BEC phase is described as
the one that most atoms occupy the quantum ground state of the trapping
potential, sharing the same phase factor eiS on the wavefunctions. Therefore, there exists coherence among atoms and consequently BEC exhibitsrich properties which cannot be observed in ”hot” (or, thermal) atomic ensembles such as a superfluidity, an interference pattern formed by two separate BECs[16](Note that the interference pattern appears also in thermal(notBEC) but sufficiently cold atomic ensembles[17], though they observed aninterference of two thermal clouds separated from initially a single thermal
cloud), the formation of a vortex lattice[15] and so on.
51.1.3

Experiments Using BEC

Once a BEC is realized, it became not only an aim but also a tool to investigate many things that had never seen. For example, the matter-wave interferometry allowed us to examine the ”phase shift” experienced by atoms. If
the acquired phase shifts differs from one BEC to another, interference fringe
shifts by some amount of phase and we can measure the relative phase shift.
From this and due to its high fringe contrast, various phenomena can be observed, or some physical quantities have been precisely measured e.g., theStark shifts of the energy levels[13] and the constant h̄/m[12].
On the other hand, ultracold atomic collisions were also of interest, suchas the Feshbach resonance which was predicted in 1964 by the nuclear physicist Hermann Feshbach[69, 70]. For atomic physics experiment it is often referred to as the phenomenon that the coupling between free state and boundstate enhances the scattering amplitude and the scattering length varies dueto this. Due to different magnetic susceptibilities of these two states (channels), energy levels of two states degenerate at some magnetic field and thisresults in a divergence of s-wave scattering length at that magnetic field
and this resonant behavior is one of the consequences of Feshbach resonance. This had not been observed until it was observed in a BEC of 23 Na[14]. Around the resonance, the coupling can be tuned so that the scattering
length can also varied by changing a value of magnetic field.
This technique is now commonly used in experiments of ultracold atoms.
For example, by trapping a BEC in a two-dimensional optical lattice and tuning the scattering length at a very large positive value, Tonks-Girardeau gas
was studied[18]. By tuning scattering length from a large positive value to alarge negative value for the Fermi degenerate atomic cloud, a smooth transition from BEC regime to BCS regime was observed(BCS-BEC crossover)[19].
These experiments can be included in the attempts of so-called ”quantum
simulator”, originally dictated by R. P. Feynman. Note that we can tune thescattering length, namely the mean-field energy at will, and we can prepare
and tune the hight of lattice potential by optical lattice technique. These facts
6

supports the idea of quantum simulator. The first, impressive experimentappeared in Ref. [20]. In this experiment the superfluid-to-Mott-insulatortransition of a degenerate Bose gas in an optical lattice was observed andopened up a new way in atomic physics.

1.1.4

Experiments on Superfluid Helium

At this point, it has some importance to review experiments on the superfluidity using liquid helium and the relation to gaseous BEC experiments.
Helium consists of two isotopes, 4 He (99.999863%, boson) and 3 He (0.000137%,
fermion). It is very famous that 4 He was first liquefied at about 4.23 K in 1908
by Heike Kamerlingh Onnes, who discovered the superconductivity in mercury three years later. Later in 1930’s, researchers discovered and confirmedthat liquid helium-4 below 2.17 K does not behave as a classical fluid[34]. P.
Kapitza used a term ”superfluid” to refer to non-classical properties[37] andF. London suggested that superfluidity of helium originates in the behavior as a macroscopic liquid matter wave. Nowadays it has become almost
a common sense among researchers that superfluidity of helium-4 is related
to Bose-Einstein condensation. The hydrodynamic and excitation properties
of superfluid are well studied with superfluid helium-4.
Helium-4 is a bosonic species and can realize macroscopic occupation of
the ground state. For fermionic helium-3, this phenomenon does not occur
in principle. However, in 1970’s researchers found that helium-3 became
superfluid below a few mK[35, 36], which was three orders of magnitude
lower temperature than that of helium-4. This was thought to be simply
because fermionic helium-3 must form a pair to be a boson. Striking fact
was that this pair of atoms had nonzero angular spin ( S = 1 ), which could
not be explained by Bardeen-Cooper-Schrieffer(BCS) theory. A. J. Leggett
explained this mystery by considering an attractive interaction due to thespin fluctuation and won the Nobel prize in 2003.
In spite of these experimental successes, informations about BEC had not7been uncovered until the experimental realization of gaseous BEC of alkali
atoms. In a gaseous BEC, almost pure condensate with condensate fractionof almost 100 per cent( for liquid superfluid helium-4, condensate fractionis thought to be less than 10 per cent ) can be prepared. For its purity andoutstanding controllability, the properties of BEC and superfluidity from theentirely new approach has become possible.
Furthermore, using different spin states or atomic species we can easily investigate the superfluid-superfluid interaction with very pure condensates. This is difficult with helium-3, though it has spin-1, because of lowsolubility of at most ∼ 15 per cent in 4 He and the phase separation taking place between them when the concentration is higher than ∼ 15 percent[38, 39] at low temperature below 0.4 K. Therefore atomic physics experiment can offer the powerful tool to study the interaction and dynamical behavior of superfluid mixture. In the next section, we introduce twocomponent BEC experiments in detail.

1.2

Two-component Bose-Einstein Condensates

1.2.1

Experiments on Two-component BECs

As described above, investigations on a single-species atomic gas or a
BEC have been successful in many aspects and are now still the cutting-edgeof today’s physics, for example, a transport phenomena in 1D-disordered
lattice potentials across the fluid-insulator transition[22], an artificial gauge
field for neutral particles[23, 24, 25], and so on.
On the other hand, a few years later from the first realization of a gaseous
BEC, a two-component BEC(or a dual BEC) was generated for the first timeby mixing two hyperfine states of rubidium atoms[31] and subsequently of
sodium atoms[32]. This mixture of BECs is called a spinor BEC and it enabled us to succeed in measurement of the relative phase between a binary
component mixture of condensates by an interferometric method[33], and in8

observing a decay phenomenon of dark solitons into vortex rings[63]. Another impressive experiment with a two-component BEC is the one by C.
Hamner et al. [89] that observed the soliton train formation in the presenceof superfluid-superfluid counterflow. They also used a spin mixture of 87 Rb
which is only very weakly immiscible.
An experimental study on the mixture of BECs using different atomic
species was first done by S. Papp et al. [67]. In this experiment two different atomic species, 85 Rb and 87 Rb are sympathetically cooled to realize thesimultaneous condensation of both atomic species. The important point isthat since the intra-species scattering length of 85 Rb is a large negative value
with no magnetic field bias, they had to tune the scattering length of 85 Rb by
the intra-species Feshbach resonance. In other words, they can tune the ratio
between intra- and interspecies scattering lengths at will by making use of
the Feshbach resonance, so that their dual BEC is suitable for studies on spatial separation and all other phenomena where the ratio of scattering lengthis crucial. Actually, they investigated the droplet formation during the evaporative cooling with widely tunable interaction regime and observed an implication of the presence of the modulation instability in this process. This
result is supported by the theoretical analysis of this experiment by S. Ronenet al. [68].
There are also intensive theoretical studies with a dual BEC such as on themodulation instability[1], the Rayleigh-Taylor instability[55, 56], the generation of dark-bright solitons[57]. As can be seen with all these studies, and as
written above, a dual BEC is a system in which rich dynamical phenomena
are potentially available. This is why many researchers are paying attention
to the experiment using dual BEC, and the experimental investigation andunderstanding of quantum non-equilibrium nature of the superfluid mixture are expected.
In this thesis, we study on a dual BEC of 41 K and 87 Rb. By using dualspecies BEC, we can enjoy a few merits compared to a spinor BEC. First, we
use a dual-species BEC, both in the magnetic sublevel | F = 1, m F = 1 >9so that the atoms cannot exchange their spins with each other and thus thetwo-body inelastic collision is inhibited. For a spinor BEC, in spite of having an ability to study the spin-spin interaction, it suffers from the two-bodyinelastic loss due to the spin exchanging. This results into a relatively short
lifetime of about 100 ms[27]. On the other hand, As we observed in Chapter 4, our dual BEC of 41 K and 87 Rb survives for more than 800 ms, which
provides a great advantage for the investigation of various phenomena in adual BEC.
Second, a dual-species BEC can make use of an intra- or inter-species Feshbach resonance relatively easily. Although a Feshbach resonances is usedalso in a spinor BEC with two hyperfine states | F = 2, m F = −1 > and| F = 1, m F = 1 >[28], their use as an adjuster of the scattering length islimited, for its width is ∆B = 3 mG and requires the precise control of themagnetic field. For 41 K and 87 Rb, a Feshbach resonance with the width of 1.2
G is available at 78.67 G and this is suitable for tuning the scattering length
at an arbitrary value.
Third, different energy structures of the two atomic species are beneficial
in that if we apply a 780 nm-laser to excite or transfer Rb to another state,
the laser does nothing to K atoms and all the same in the opposite way.
Therefore we can access only one of the two species by an optical means andthis works well also for microwave transitions.

1.2.2

Another Application: Ultracold Molecules

By the way, we would like to mention to another application of a dual BEC.
There are growing interest in atomic gases as a quantum simulator, especially the one can not only mimic systems exhibiting the many-body effectin the ”beyond-mean-field” regime as in the condensed matter physics, butalso investigate a parameter region which is not available with bulk materials. However, there is a problem: electrons in solid state materials interact
with other electrons or ions mainly by Coulomb interaction proportional to10

1/r, where r is the distance between particles. But we have a BEC of neutral
atomic gas: they interact by short range Van der Waals interaction proportional to 1/r6 and this is negligible for the most dilute atomic gases and
BECs1 .
Therefore ultracold particles which have long range interaction are needed.
One of the candidate for this is the heteronuclear alkali-alkali dimers whichhave long range electric dipole-dipole interaction proportional to 1/r3 . For
instance the proposal on supersolid[43] is based on the long-range, dipoledipole interaction between particles in an optical lattice. Promising method
to create this ultracold polar molecules is to cool down two species of atoms
simultaneously and succeedingly convert them into heteronuclear molecules
adiabatically via the magneto-association using a Feshbach resonance. For
homonuclear molecules this method has been successfully done and themolecular BEC is realized[26], although this does not have a permanent
dipole.
For heteronuclear case the most successful experiment on the ultracold
molecule is the one demonstrated by JILA’s group, which uses fermionic
40 K and bosonic 87 Rb to generate fermionic molecules 40 K 87 Rb[29].

Re-

cently, this group observed the phase oscillation due to the dipole-dipole
interaction[30]. There have also been a few attempts aiming at ultracold heteronuclear bosonic molecules 85 Rb 133 Cs[45], 87 Rb 133 Cs[46], 23 Na 87 Rb[44],88 Sr 87 Rb[100] and so on.

For the sake of efficient generation of ultracoldbosonic molecules it is also important to understand the static nature of dual
BEC and let the spatial overlap as large as possible.

1 Researchers have developed method for producing BECs of atomic species with large

magnetic dipole moments such as 52 Cr[40]( 6 µ B ), 164 Dy[41]( 10 µ B ), and 168 Er[42]( 7 µ B ).
However, in lattice experiments their dipole-dipole interactions are said to be still small.

11

1.3

Contents of This Thesis

Currently there are two types of experiments running on our apparatus:
one is aiming at creating ultracold molecules and the other is the study on
non-equilibrium properties of a superfluid-superfluid mixture, which is themain contents of this thesis. Our purpose is an experimental study on the
dynamics of a dual BEC with tunable interaction strength using 41 K and
87 Rb, both bosonic species. Below we shall briefly summarize the contentsof this thesis.
To say our achievement first, we examined dynamical properties of adual-species BEC and especially we succeeded in the observation of spatial
patterns formed by the presence of the modulation instability, which occurs
when an immiscible dual BEC is initially overlapped. For this observation,
three things were needed to be realized: the generation of BECs spatiallyelongated in one direction, the complete overlap of two species of BECs
in an optical trap, and the in-situ, successive observation of a dual-species
atomic cloud. In order to implement these three, we prepared a pseudoone-dimensional optical trap for elongated BECs, made use of an 809 nm
optical trap and a pair of coils for overlapping two BECs, and prepared the
setup of the phase contrast imaging for each atomic species which enabled
the nondestructive observation of two atomic species in a single experimental run. Furthermore, we compared obtained results to the theoretical prediction by coupled Gross-Pitaevskii equations, so that it was confirmed thatthe observed spatial pattern originated in the modulation instability.
The quantitative analysis of the modulation instability using different
atomic species was done for the first time. The achievements in this thesis
imply the strong possibility of a 41 K-87 Rb dual BEC as a tool for investigating
non-equilibrium properties of a superfluid-superfluid mixture in a quantitative manner. This can tackle to other dynamical phenomena such as thevortices and dark/bright solitons, for example.

12

Chapter 2
THEORETICAL DESCRIPTION
OF A DUAL-SPECIES BEC
In this chapter we introduce coupled Gross-Pitaevskii equations which
explain the properties of a dual BEC very well and briefly discuss the properties derived from them. In particular, miscible / immiscible(phase-separate)transition, which is the most characteristic phenomenon in a dual BEC, isdiscussed in detail for the static case.

2.1

Single-Component BEC

Before we mention the coupled equations describing a spatially coexisting dual BEC, Gross-Pitaevskii equation for a single-component BEC isintroduced. Coupled ones are easily derived by analogy with the singlecomponent one.

2.1.1

Gross-Pitaevskii Equation

First we introduce a Gross-Pitaevskii equation for a single atomic species
in a single state, which has its mass m and the scattering length a. The
derivation is based on the variational method as in the Ref.[51]. In BEC
phase, all atoms in an ensemble share the same quantum ground state wave13

function ψ(r). For an ultracold atomic gas in an isotropic harmonic trap
V (r) = mω 2 r2 /2, the ground state is the lowest energy level among equidistantly aligned energy levels and the wavefunction is given as[ mω ]
1 ( mω ) 14
ψ0 (r) =exp −r2 ,4π πh̄
2h̄
where h̄ is the Planck constant divided by 2π. Note that this wavefunctionis the one for non-interacting particles and actually the ground state wavefunction is not like this due to the finite interaction between atoms. Let thisactual ground state be denoted by ψ(r). The energy E of a BEC can be written as the sum of the kinetic energy, the potential energy and the interactionenergy, which is
∫

(

E=

h̄2
g
|∇ψ|2 + V (r)|ψ|2 + |ψ|4
2m
2

)
dr,

where the coefficient g is an energy density of interaction between atoms.
√
Now the wavefunction squared have a unit of spatial density: ψ(r) = n(r)eiS ,where n is the spatial density and S the phase. Energy density g can be expressed by the s-wave scattering length of atoms, that is,g=

4πh̄2 a.
m

One would notice that we already assume a contact interaction for the twobody collision and applied the mean-field approximation. We can readilytake the functional derivative of the functional E[ψ(r)] to get the Euler equation
[

−

h̄ ∇2
2

2m

]

+ V (r) + g|ψ(r)|2 ψ(r) = µψ(r).

Here the chemical potential µ which has a unit of an energy is introduced.
This constant must be determined by the normalization condition∫N=|ψ(r)|2 dr.

14This condition says nothing but that the spatial density is integrated to givethe total number of atoms N. This differential equation is referred to as thetime-independent Gross-Pitaevskii equation, the most fundamental one fora single-component BEC.
One may notice that the Gross-Pitaevskii equation has a form very similar to the time-independent Schrödinger equation excepting the nonlinear,interparticle interaction term g|ψ(r)|2 . If the time evolution is under consideration, the Gross-Pitaevskii equation should take the time-dependent form
with the chemical potential µ being replaced by an operator ih̄∂/∂t (actually this procedure is a restoration) just as we can do in obtaining the timedependent Schrödinger equation:[]∂h̄2 ∇22ih̄ ψ(r, t). = −+ V (r) + g|ψ(r)| ψ(r, t).
∂t
2m
Third nonlinear term g|ψ(r)|2 on the left hand side makes it difficult tosolve this equation analytically, however, there exists a powerful approximation called the Thomas-Fermi approximation, which neglects the kinetic
energy term −h̄2 ∇2 /2m with respect to the interaction (mean-field) termg|ψ(r)|2 . This yields a quite simple form
µ = V (r) + gn(r)and together with the condition N =∫n(r)dr, the density and the chem-

ical potential can be determined self-consistently for the given number ofatoms. It is obvious that since µ is a constant, the spatial density of a BEC
obeys to the form of the trapping potential V (r), for example, when we apply an optical dipole trap, the central region of the trapping potential can be
regarded as a parabolic curve so that the density profile of a BEC is also aparabola. The boundary of a condensate with the Thomas-Fermi approachis estimated by setting n(r) = 0, namely,n (r) =

µ − V (r)
= 0.
g

15

Figure 2.1: Schematic plot of a Thomas-Fermi density profile. For comparison, a Gaussian profile f ( x ) = A exp (− x2 /2σ2 ) with σ = R/2 is shown inthe same region. Both of them are normalized by their peak densities.

For an isotropic harmonic trap V (r) = 12 mω 2 r2 we can easily calculate theboundary R as

√
R=

2µ
mω 2

which is often referred to as a Thomas-Fermi radius. Schematic representation of a density profile calculated by Thomas-Fermi approximation isshown by a red line in the Fig. 2.1. In this figure a Gaussian profile is also
shown for the sake of the comparison between interacting (Thomas-Fermi)
and non-interacting (Gaussian) case. As depicted in these two profiles, thedistribution in the vicinity of the central region becomes broader due to theinterparticle repulsion than that of the non-interacting one. From above formula the boundary size of a condensate depends on the interaction strength
through the chemical potential µ and the trapping frequency of the harmonic
trap we apply to. More concretely, the larger the trapping frequency, thesmaller the size of the condensate; the larger the interparticle interaction,
16

the larger the area that atoms spread over, in order to avoid excess increaseof the interaction energy. For the anisotropic potential, one would easily
derive

√
Ri =

2µ
mωi2

(2.1)

for three directions i = x, y, z.

2.1.2

The Transition temperature and The Condensate Fraction

Statistical physics gives a clear explanation of how a BEC emerges, andwe are not to explain the detail here, rather we only refer to some results.
Informations are found in Ref.[51]. According to this Ref.[51], the BEC transition temperature Tc and the condensate fraction N0 /N of trapped particles(bosons) in a harmonic trap are written as1

k B Tc =h̄ω̄N 31

= 0.94h̄ω̄N 3 ,(2.2)[ζ (3)]
( )3
N0
T
= 1−
,
N
Tc
here k B is the Boltzmann constant, ω̄ the geometric average of the trapping
1
3

frequencies, N0 the number of atoms in the ground state and ζ ( x ) the Riemann zeta function.
To have an intuitive understanding of the expression of critical temperature, it will be of great help to consider the density of states in a trap. Discreteenergy levels of a particle in a harmonic potential are written as h̄ω ( j + 1/2)
with integer j = 0, 1, 2, · · · . Therefore the interval of neighboring energy
levels is h̄ω and larger trapping frequency ω makes the energy interval getlarger. For equal number of atoms, condensation occurs when the thermalenergy k B T gets comparable to the energy interval, so that the large (small)
trapping frequency results in the high (low) critical temperature. Indeed thenumber of atoms also plays an important role in the respect of phase spacedensity and the correct formula is given as the equation (2.2).
17

2.2

Two-Component BEC

2.2.1

Coupled Gross-Pitaevskii Equations

Now we can deal with a dual BEC by directly extending the method
for a single-component BEC. The energy functional of two BECs is the direct
sum of the energy functional of each component and an additional, intercomponent interaction term, that is,
∫

E=

(

[

∑

i =1,2])gh̄2(∇ψi )2 + Vi (r)|ψi |2 + i |ψi |4 + g12 |ψ1 |2 |ψ2 |2 dr.
2mi2

The indices i = 1, 2 represent the components 1 and 2, and g12 is the intercomponent coupling coefficient, which is expressed using an intercomponent scattering length a12 and a reduced mass m12 = m1 m2 /(m1 + m2 )
as
g12 =

2πh̄2 a12.
m12

All we have to do is again taking functional derivatives of this energyfunctional with respect to ψ1 and ψ2 . Resulting Euler equations have theform similar to that of the single-component one, except for the term originates in the intercomponent coupling term g12 |ψ1 |2 |ψ2 |2 :]h̄2 ∇2+ V1 (r) + g1 |ψ1 (r)|2 + g12 |ψ2 (r)|2 ψ1 (r) = µ1 ψ1 (r),
−
2m1
[
]
h̄2 ∇2−+ V2 (r) + g2 |ψ2 (r)|2 + g12 |ψ1 (r)|2 ψ2 (r) = µ2 ψ2 (r).
2m2[

In this case the constraints are N1 =∫|ψ1 (r)|2 dr and N2 =

∫

(2.3)
(2.4)

|ψ2 (r)|2 dr. Im-

portant properties of the dual BEC are derived by this pair of equations. The
time-dependent form of this pair of equations can be written down again by18

restoring the chemical potentials by ih̄∂/∂t:[]h̄2 ∇2∂ih̄ ψ1 (r) = −+ V1 (r) + g1 |ψ1 (r)|2 + g12 |ψ2 (r)|2 ψ1 (r),
∂t
2m1
[
]
∂
h̄2 ∇2+ V2 (r) + g2 |ψ2 (r)|2 + g12 |ψ1 (r)|2 ψ2 (r).
ih̄ ψ2 (r) = −
∂t
2m2

(2.5)
(2.6)

Throughout this chapter we deal with time-independent Gross-Pitaevskii
equations and time-dependent ones are considered in Chapter 5 for analyzing the linear stability of a dual BEC.
First of all, Gross-Pitaevskii equations can be simplified by the ThomasFermi approximation, again neglecting kinetic energy terms −h̄2 ∇2 /2m1
and −h̄2 ∇2 /2m2 . This yields a pair of algebraic equations
g1 n1 (r) + g12 n2 (r) = µ1 − V1 (r),g2 n2 (r) + g12 n1 (r) = µ2 − V2 (r).
and their solutions are found to beg2 (µ1 − V1 ) − g12 (µ2 − V2 ),2
g1 g2 − g12g (µ2 − V2 ) − g12 (µ1 − V1 )n2 = 1.
2
g1 g2 − g12
n1 =

(2.7)
(2.8)

These expressions of density distributions imply that there are two phases
for the mixture of dual BEC depending on the values of intra-/inter-species
scattering lengths a1 , a2 and a12 . One is a miscible phase, which is character2 < g g . In this phase two BECs can coexist in theized by an inequality g12
1 2

same region in a trap and their density distributions satisfy formulae (2.7)
and (2.8). As it is obvious from these formulae, the existence of one component have an influence on the density distribution of the other component
and vice versa.
2 > g g . For thisAnother is an immiscible phase characterized by g12
1 2

parameter region, denominators of (2.7) and (2.8) are negative. Since thedensity distributions are always positive or zero, µ1 (µ2 ) gets large at some
19

Figure 2.2: Schematic plots of Thomas-Fermi density profiles of dual BECs
with variable interactions. For (a) aKRb = 50 a B , a dual BEC is miscible andwhen (b) aKRb = 230 a B > 74 a B it is in an immiscible regime.

20

position and µ2 (µ1 ) becomes small in order to keep n2 (n1 ) positive. However, in order to let n2 be positive, n1 becomes very small or zero due to largeµ1 and small µ2 . In other words, two components cannot coexist at the same
2 > g g .
position and phase separation occurs in the parameter region g12
1 2

In contrast to the miscible phase, the bulk regions of phase-separated BECs
does not have influence on each other.
Actual density profiles of a dual BEC can be calculated by a numericalmanner as it is summarized in Appendix. E. In short, numerical solutions
can be obtained by minimizing the energy functional without kinetic energy terms (Thomas-Fermi approximation) with respect to the density profiles. Since our system of K and Rb can be regarded to have constant intraspecies scattering lengths aK = 60 a B and a Rb = 100 a B , the free parameteris only the interspecies scattering length aKRb other than the trapping potential, which is fixed in the calculation. The immiscible phase emerges in
the region g2KRb > gK gRb , which is rewritten in terms of scattering length as
aKRb > 74 a B with parameters given above, and in the Fig. 2.2 (a) and (b) a
miscibility at aKRb = 50 a B and an immiscibility at aKRb = 200 a B are clearly
realized, respectively. More detailed informations are placed in Appendix.
E.
Moreover, an immiscible dual BEC system has interfaces between thecomponents, which are absent from the miscible one. BEC-BEC interface
physics has been intensively investigated theoretically[52, 53, 54]. According to these literature, properties of BECs at interfaces such as penetration
depths λi∗ and surface tensions σi change whether the intercomponent scat√deftering length is small (∆∗ = g12 / g1 g2 − 1 ≪ 1) or large (∆∗ ≫ 1). The
former case is referred to as the weak separation and the latter as the strong
separation. The difference of these two regimes is as follows[52]:

λi =



 ξ 1√+ξ 2

for ∆∗ ≪ 1ξ ifor ∆∗ ≫ 1

2 ∆∗

21


√

 P ∆∗ ξ 1 +ξ 2
4
σi =
√

 P 2 ξ 1 +ξ 2
3

Where ξ i = h̄/√for ∆∗ ≪ 1
for ∆∗ ≫ 1

2mi µi are healing lengths and P = µ21 /2g1 = µ22 /2g2 are

the pressures of trapped gases. A common feature is that both λi and σilose their dependence on the intercomponent interaction as ∆∗ gets larger.
This might be because very strong repulsive interaction let the BEC of one
component feel the other component like a very high potential wall, and thepenetration depth is determined by itself.
Theoretically, there are many references of studies on two-component
quantum fluids, especially the hydrodynamical instabilities such as the modulation instability[1], the Rayleigh-Taylor instability[58], the Kelvin-Helmholtz
instability [60], and so on.
Experimentally, the immiscible state of a two-component BEC which consists of two different hyperfine states of 87 Rb was observed by E. Cornell’sgroup in 1998[61]. They used magnetically trappable | F = 2, m F = 1 >state (state 1) and | F = 1, m F = −1 > state (state 2) in the ground state2S

1/2 of87 Rb. Scattering lengths are a

1 : a2 : a12 = 1.03 : 0.97 : 1 in propor-

tion, which indicate that the mixture is weakly separated. Some experiments
were done after this observation, for example, the observation of the decayof dark solitons into vortex rings[63], and the collapse and the recurrence ofRabi oscillations [64].
After the observation of a Feshbach resonance of 85 Rb [65] and the realization of an 85 Rb BEC[66], C. Wieman’s group realized a system of an 85 Rb87 Rb dual BEC with tunable proportion of intra- and inter-species interactions[67].

This experiment not only observed the miscible and the phase-separated
dual BECs with the same experimental apparatus, but also the formation ofdroplets when BECs nucleates during the evaporative cooling.
22

Figure 2.3: Schematic representation of scattering potentials. The triplet potential plays a role of a closed channel and the singlet potential does theopen channel.

Figure 2.4: Schematic representation of avoided crossing between atomic
and molecular states. Due to this, atomic and molecular states are connected
so that we can let atoms travel over atomic and molecular states back andforth by adiabatically ramping the magnetic field across the resonance.
232.2.2Tuning the Scattering Length by Feshbach Resonance

The term ”Feshbach resonance” was named after Herman Feshbach, whotheoretically predicted a resonant behavior of two-body scattering processes[69,70]. Feshbach originally predicted in the context of nuclear reaction, however, Feshbach resonance was observed for the first time by investigating thecollisions in an ultracold atomic gas[14, 72]. Observed Feshbach resonancesof various atomic species and the overview on the theory are summarized
in the review[71]. Since the magnetic susceptibility differs between atomic
level and molecular level, these two levels possibly degenerate at some finite
magnetic field B0 . If there is a finite coupling between atomic state and themolecular state, well-known avoided crossing occurs (see Fig.2.4) and theenergy splitting of two mixed states depends on the strength of coupling.
Thus, in short, the width of this resonance ∆B gets large when the atomic
state and the molecular state are similar. Resonant behavior of the scattering
length in the vicinity of a Feshbach resonance is expressed by a following
formula[51]:(a = abg∆B
1−
B − B0).

This means that the s-wave scattering length a of atoms can be tuned bymagnetic field to arbitrary values in the range −∞ < a < +∞. This has enabled the study of ultracold atomic gases in a very wide region of interaction
strengths, such as the observation of strongly correlated one-dimensional
quantum gas[73] (also referred to as the Tonks-Girardeau gas), the BCS-BEC
crossover[74], etc.
Feshbach resonance was first observed for the collision of the same atomic
species. In 2004, Feshbach resonance for the collision of different atomic
species was observed for the first time in Ref.[75], which was for 40 K(fermion)and 87 Rb(boson). LENS observed the heteronuclear Feshbach resonances[76] and heteronuclear Efimov resonances [77]1 for the system of 41 K and
87 Rb, both bosonic species.
1 Observation of an Efimov resonance for a homonuclear case is found in Ref. [59]24Our system can also generate a dual species BEC of 41 K and 87 Rb. Feshbach resonance we use is characterized by the parameters[79, 80, 81] B =78.67 G, ∆B = 1.2 G and abg = 284 a B . Actually there is another broad resonance with B = 38.4 G, ∆B = 37 G and abg = 284 a B , so that the scattering
length is not easily calculated. Accurate value of scattering length has to be
calculated by the more rigorous treatment such as the one considering scattering potentials. In Fig. 2.5 or Fig. 2.6, rigorously calculated interspecies
scattering length a RbK vs. the magnetic field for 41 K (| F = 1, m F = 1 >) and87 Rb (| F = 1, m

F = 1 >)[82] is shown.

25

Figure 2.5: Theoretically calculated value of the interspecies scattering
length a RbK is shown for the range of the magnetic field B < 100 G.

Figure 2.6: Theoretically calculated value of the interspecies scattering
length a RbK is shown for the range of the magnetic field 60 G < B < 100
G.

26

Chapter 3
EXPERIMENTAL SETUP
Experimental apparatus used in this thesis is described here, focusing
mainly on specific portions added or changed for generating elongated BECs,
overlapping them spatially and observing them nondestructively. Three basic ideas for these purposes are listed in the Section 3.1. More concrete discussions on these ideas are placed in the Sections 3.3, 3.4 and 3.5, which
utilize the knowledge of atom-light interaction reviewed in the Section 3.2.

3.1

Toward In-situ, Nondestructive Observation of
a Dual BEC

3.1.1

Phase Contrast Imaging

The purpose of this thesis is to observe a dual BEC in-situ. There is amethod to realize this: the phase contrast imaging (PCI). This method is
powerful because this probing method hardly make atoms absorb any photons, for the phase contrast imaging takes an advantage of the dispersive
property of an atom, that is, the refractive index. Namely, PCI enables us
to probe the time evolution of atomic cloud nondestructively, in contrast to
the absorption imaging which is a destructive one. Dispersive measurementof ultracold atomic cloud was first realized for a single-component BEC of27sodium atoms[83]. This method has been used for studies of the propagationof sound waves[84], the vortex precession[85] and other various aims.
For our case, it is desirable to observe 41 K and 87 Rb in the same experimental sequence. Of course this requires two different probe beams detuned
by few hundreds of MHz below or above each atomic resonance. In addition, we have to take the bias magnetic field for a Feshbach resonance (atB = 78.67 G) into account, because we should consider the Zeeman shiftsof both atomic species. Interaction between electromagnetic field and an
atom is reviewed in Section 3.2 and the preparation of probe lasers and theimaging system is described in Section. 3.3.

3.1.2

Elongated dual BEC in an Optical Dipole Trap

In atomic physics experiments people often use absorption imaging after
the time-of-flight(TOF) process, in which a trapping potential is turned off
suddenly and atoms freely expand for a few tens of ms. Contrary to this,we want to observe the time evolution of a dual BEC in-trap. With our typical trapping frequencies (ωx /2π, ωy /2π, ωz /2π ) = (40, 40, 100) Hz of thetrapping potential, an in-trap BEC is usually as small as 10 µm, which iscomparable to the resolution limit of our imaging system (∼ 4 µm). This
leads to a conclusion that the observation of the spatial pattern of the BEC isalmost impossible.
Our prescription for this problem is straightforward. We make the optical trap loose in one direction. By doing this, the size of a BEC gets larger
in the direction of loose potential as it can be seen from the formula (2.1).
There are two ways for loosening the trap: letting the power of the trapping
laser be smaller or the diameter of the one be larger. However, these two
finally result in totally different trapping potentials, especially the different
trap depths. The qualitative and quantitative discussions are placed in the
Section. 3.4.
28

3.1.3

Gravitational Sag and Magnetic Field Gradient

As mentioned later, if there is a potential gradient two different atomic
species possibly feel it in a different way. As a result potential minima of
the trap for the two atomic clouds can be different. In our experiment the
problematic ones are the gravity and an unwanted magnetic field gradientor curvature. The gravity is a special concern because the magnitude of thisfield gradient is proportional to the mass of an atom and the difference ofpotential minima due to the gravity is called the gravitational sag. Even
when we load a dual BEC in a 1080 nm optical trap with 100 Hz-trapping
frequency along the gravity, the gravitational sag is still comparable to thesizes of the BECs and the dual BEC is not overlapped. Our study on an overlapped dual BEC is done by transferring a dual BEC from 1080 nm opticaltrap to the one with different wavelength of light, the 809 nm optical trap.
We mention to this in the Section. 3.4.4.

The problem given just above is about the relative position of a dual BEC
along the gravity. Indeed, because of the loosened trap with trapping frequency of around 10 Hz in one of the horizontal direction, even a small
field gradient results in the difference of potential minima for each atomic
species. A small magnetic field gradient or curvature along the horizontal
direction and the tiny tilt of the optical trap with respect to the horizontal
plane might be origins of this problem. In Section. 3.5, we can rather takeadvantage of this effect to compensate this sag in the horizontal direction
using the magnetic field gradient by applying an additional pair of coils in
the anti-Helmholtz configuration.
29

3.2

Interaction Between Electromagnetic Field andan Atom

3.2.1

Density Matrix and Optical Bloch Equations

We shall now summarize a basic treatise of atom-light interaction. Formulation written below appears in some popular textbooks[48, 49]. For agiven Hamiltonian H = H0 + H I , where H0 gives atomic energy levels andwavefunctions without light field and H I is a perturbation term by the classical light field, we consider a time-dependent Schrödinger equationih̄∂Ψ = H Ψ.
∂t

(3.1)

Two-level treatment allows us to write wavefunction Ψ as a superpositionof unperturbed atomic states |1 > and |2 > with their eigenenergies E1 andE2 , respectively. Explicitly writing phase factors e−iEj t/h̄ = e−iω j t ( j = 1, 2),
Ψ = c1 (t)e−iω1 t |1 > +c2 (t)e−iω2 t |2 >and coefficients c1 and c2 satisfy the relation|c1 |2 + |c2 |2 = 1.
Substituting this expression of Ψ into eq. (3.1), we have a pair of equationsdc1= e−iω0 t H I c1
dt
dc2
ih̄
= e−iω0 t H I∗ c2dtih̄ω0 = ω2 − ω1 .
Here we let H0 include diagonal elements of matrix H I and denote offdiagonal elements of H I as H I . Classical electromagnetic wave as a perturbation can be written as H I = er · E0 cos (ωt) with the electron’s electriccharge e, the amplitude E0 and the angular frequency ω. Using this expression and rewriting cosine term by exponentials, we get the equations to30

solve:dc1Ω= eiδt c1 ,dt2dc2Ω∗ −iδt
i
=
e
c2 ,
dt
2
i

δ = ω − ω0 .
In these equations, two important parameters appear. One is the detuning δ which is related to the phase evolution of wavefunction. Another isthe Rabi frequency Ω which is nothing but the oscillation frequency of the
populations if the electromagnetic wave is applied to the two level system
continuously. This quantity is written asΩ=e < 2|r · E0 |1 >h̄which is proportional to the electric field amplitude of the incident light.
Here we introduce a density matrix 
|c1 |2 c1 c2∗
ρ11 ρ12=
ρ = |Ψ >< Ψ| = c1∗ c2 |c2 |2ρ21 ρ22on the pathway to reach at optical Bloch equations. Its diagonal and offdiagonal elements are called populations and coherences for their properties
respectively. Setting u = ρ12 e−iδt + ρ21 eiδt , v = −i (ρ12 e−iδt − ρ21 eiδt ) andw = ρ11 − ρ22 , and doing some maths we can get equationsu̇ = δv,v̇ = −δu + Ωw,ẇ = −Ωv.
The most frequently used form of optical Bloch equations contains decay
terms for the coherences u and v:
Γu̇ = δv − u,
2

(3.2)

Γ
v̇ = −δu + Ωw − v,
2

(3.3)

ẇ = −Ωv − Γ(w − 1),

(3.4)

31

where Γ denotes the full-width-half-maximum(FWHM) or natural linewidthcharacteristic of an excited state. These decay terms are introduced phenomenologically. Solution of these equations give the dynamics of a Blochvector (u, v, w) in the presence of the electromagnetic field and frequently
used in the analysis of optical transitions such as the Rabi oscillation and theadiabatic rapid passage. To investigate the steady state, we put the left handsides of equations (3.2), (3.3), (3.4) to zero, namely, u̇ = v̇ = ẇ = 0. Then optical Bloch equations become coupled algebraic equations that can be solved
easily. We focus on the population of the excited state ρ22 = (1 − w)/2 andwriting the solution for a stationary state, it reads
1Ω2 /2
2 δ2 + Ω2 /2 + Γ2 /4
1
I/Isat=.
2 1 + I/Isat + (2δ/Γ)2ρ22 =The quantity I/Isat = 2Ω2 /Γ2 is used here, where I is the intensity of incident light and Isat is the saturation intensity of the transition. From this we
can see that for fixed natural linewidth, the population of an excited stategets larger as the intensity of light gets larger or the detuning gets closer tozero. The scattering rate Rscatt is obtained by simply multiplying Γ:
Rscatt =I/IsatΓ.
2 1 + I/Isat + (2δ/Γ)2

In the frequency domain this has a Lorentzian shape and for large detuningδ ≫ Γ/2 it falls off as 1/δ2 .

3.2.2

Refractive Index and Phase ShiftWe also get the off-diagonal elements of the stationary solution for optical
Bloch equations given above. These are expressed as∗
ρ12 e−iωt = σ12 = σ21=−Ωδ + i (Γ/2)
22 δ + (Ω2 /2) + (Γ/2)2

32

(3.5)

µ ], where n is an atomic
The atomic polarization P is written as P = ntr[ρµ̂µ = µ (|1 >< 2| + |2 >< 1|) is the dipole moment of an atom.
density and µ̂
Together with the stationary solution (3.5), This reads−δ cos ωt + (Γ/2) sin (ωt)
δ2 + (Ω2 /2) + (Γ/2)2
On the other hand, atomic polarization can also be written by the complexµΩ
P = nµpolarizability α(ω ) as P = Re[α(ω )E0 e−iωt ]. Since the refractive index nr of√dilute atomic gas nα/ϵ0 ≪ 1 is given by nr = 1 + nα/ϵ0 ≃ 1 + nα/2ϵ0 , we
get the complex refractive index by comparison between two expressions of
the polarization, that is,
nr = 1 +−δ̃ + iσ0 nλ4π 1 + ( I/IS ) + δ̃2

where σ0 = 6πλ̄2 with λ̄ = λ/2π is the absorption cross section for the
resonant light and δ̃ = 2δ/Γ is a dimensionless form of the detuning. Notethat here we used the expression of natural linewidth
4 1 k30 µ2,Γ=
3 4πϵ0 h̄which can be derived as the Einstein’s A coefficient in a phenomenologicaltreatment. Thus, a dilute atomic gas is a matter with the complex refractive index nr . Generally the phase velocity of light running through a material becomes slow or fast with respect to that in a vacuum depending on
the refractive index of the material is higher or lower than 1. Consider wehave a coherent light wave and split it into the two light waves, one passes
through the atomic gas of thickness d and the other does not. Then one
passes through a gas travels effectively longer than the other by a length(Re[nr ] − 1)d and this generates a phase shift ϕ between two light waves.
This is given explicitly asω(Re[nr ] − 1)d
c
σ ñ
δ̃
= 0
.
2 1 + ( I/IS ) + δ̃2σ ñ 1
∼ 0
2 δ̃

ϕ=−

33

(3.6)
(3.7)

Figure 3.1: Schematic representation of the phase contrast imaging. Amplitude of the scattered light(red) does not change((a), left panel) but withthe rotation of the unscattered light(blue), the scattering vector(green) andthe unscattered light add up so that the phase shift is interpreted into theintensity of light after the interference of the scattered and the unscattered
light((b), right panel). The gray dotted circle shows the curve E = E0 .

Here the column density ñ =∫n(r)dz is introduced. As given above, therefractive index ( and phase shift ) scales as 1/δ̃ for the large detuning.

3.3

Phase Contrast Imaging

3.3.1

Schematics of the Phase Contrast Imaging

As derived above, an absorption rate ∝ 1/δ̃2 goes faster to zero than therefractive index ∝ 1/δ̃. Therefore for a moderate detuning, it is possible
to get a measurable phase shift and to keep the rate of absorption of light
negligible. If it is able to measure the phase shift between the unscattered
light and the refracted light, the column density of an atomic gas can be34

measured non-destructively. However, in general the amount of the refracted
light is very small.
To see what happens, it is convenient to consider the electric field E0 in acomplex plane. Given the transmission coefficient t and the phase shift ϕ, thescattered light field is tE0 eiϕ and the scattering vector is ∆E = E0 (teiϕ − 1) ≃iϕE0 . Here we assumed that the detuning is moderately off-resonant andthus t ≃ 1 and ϕ is small but finite. This means that if we measure theintensity of the scattered light, the information of phase shift is lost at least
for first order term1 ϕ1 . In order to extract the phase shift as the intensity oflight field, one prescription is to rotate E0 in the complex plane by an angle
π/2(Fig. 3.1). By doing this, the measured light field and intensity becomesfor small ϕ
E = E0 eiπ/2 + ∆E = iE0 + iϕE0 = iE0 (1 + ϕ),I=

ϵ0 cE02
ϵ0 c 2| E| =(1 + ϕ)2 ≃ I0 (1 + 2ϕ).
2
2

(3.8)

The phase shift ϕ can thus be detected by the change of the light intensity
which is proportional to ϕ.
What we need to realize this imaging technique, the phase contrast imaging(PCI), is the so-called phase plate which is a round glass plate with a dip
at the center, whose diameter is about 100 µm and the depth is about a few
hundreds of nm. By placing this phase plate at the focal point of probe lightin the imaging system, unscattered probe light goes through the dip whilemost of the scattered light passes outside the dip (but inside the glass plate).
A properly chosen depth of the dip shifts the phase of unscattered probe
light by π/2. In other words, optical path of the unscattered light wave isshorter than the scattered one by λ/4. Our phase plate has a dip with thedepth of 423 nm and the diameter of 140 µm[90], which causes π/2-phaseshift for 767 nm-wavelength light. This is designed to detect the potassium
1 Indeed there is an imaging method which is called the dark-field imaging where theunscattered light is blocked at its focus and only the scattered light is collected at the screen
to acquire signals proportional to the second order in ϕ.

35

Figure 3.2: Schematics of the side imaging system. The red area represents
the probe beam and the gray shaded area does the shadow of atoms. All
lenses are 2-inch, achromatic doublet lenses.

Figure 3.3: Schematics of the top imaging system. The red area represents
the probe beam and the gray shaded area does the shadow of atoms. All
lenses are 2-inch, achromatic doublet lenses, except for the lens with focal
length 500 mm, which is 1-inch, achromatic doublet lens.

36D2-line but also works almost the same for the rubidium D2 line whose resonant wavelength is 780 nm since this difference of wavelengths is sufficiently
small. In Figs. 3.2 and 3.3, experimentally used two imaging systems andthe place where the phase plate is introduced are indicated. One is called
the side imaging which probe light is incident in a horizontal direction andthe other, top imaging, is in a direction of the gravity.
For top imaging, the numerical aperture(NA) is designed[91] to be 0.13
and a measurement of the magnification revealed that the magnification is4.64. We use a charge coupled device(CCD) camera with a pixel of dimensions 16 µm × 16 µm, therefore 1 pixel in the top imaging corresponds to
3.451µm. The side imaging is somewhat complicated. Though the numerical aperture and the magnification are designed to be 0.11 and around 4,
measured magnification is 6.15 (1 pix. = 2.58 µm). This is probably because the side imaging path is partially the same as MOT beams and probe
beam passes through a 2-inch polarizing beam splitter(PBS). This large optics surely causes the non-negligible variation of rays and magnifications.
The resolution power dr of each imaging path is estimated with 780 nmprobe light to be 3.8 µm for the top imaging and 4.4 µm for the side imaging,respectively, by using the Rayleigh’s criterion dr = 0.61 × λ/NA. Phase
plate is placed at the focal point of probe beam in each imaging system,
however, being compatible with the absorption imaging, the phase plate ismounted on a flipper.

3.3.2

Probe Lasers for Phase Contrast Imaging

Remaining tools for the PCI are off-resonant probe lights for each atomic
species. We can calculate the phase shift by a formula (3.6). Since the trap
frequencies and the number of atoms are already known and cannot bechanged easily, we tune δ so that we can get detectable phase shifts andnegligible photon scattering events. As we describe later in the next section, typical trap frequencies are (40, 7, 100)Hz and combined with the typ37

ical number of atoms 3 × 104 , the measurable phase shift around 0.1 rad isavailable with the detuning of a few hundred MHz. For this range of detuning, the photon scattering probability by single probe pulse(300 µs duration,
which is determined by exposure time of a single strip of CCD camera) becomes at most a few per cent. Below we consider the frequency of atomic
transition in the presence of external field.
In order to determine the frequency of the probe light, we should note:• we use a magnetic Feshbach resonance between rubidium and potassium both in | F = 1, m F = 1 > state. The magnetic field bias is around78 G and the atomic ground state and the excited state experience thelinear and quadrature Zeeman shifts.
• because the probe beams for the side imaging passes through the optics
for MOT, their polarization become σ− . Thus the excited state we use
is limited to the one we can reach from the | F = 1, m F = 1 > state with
σ− light. We selected the | F = 0, m F = 0 > for an excited state, whichis the lowest-energy excited state among the 2 P3/2 states and we use
red-detuned lasers. The reason why we choose the lowest-energy stateis that we need to avoid other magnetic sublevels having influences on
the target transition.
Then in order to set the probe frequency, we should know Zeeman shiftsof the | F = 1, m F = 1 > in the ground state and the | F = 0, m F = 0 >in the excited state. Atomic properties needed to calculate the energy shifts
are listed in Appendix A. Ground state energy shift is easily calculated by
the Breit-Rabi formula and the problem is the excited state. The Breit-Rabi
formula is an analytical solution obtained by diagonalizing the interaction
Hamiltonian and in principle this procedure is also applicable to the excitedstate. However, in the case of excited state we have to diagonalize one 4 × 4
matrix, two 3 × 3 matrices and two 2 × 2 matrices, though in the case ofground state three 2 × 2 matrices appear.
So it is easy to let Wolfram Mathematica evaluate the matrices and get
38the numerical values of frequency shifts vs. the magnetic field. For example,
the evaluated values of frequency shifts of 41 K are plotted in Fig. 3.4. Thosefor 87 Rb can be calculated in the same way(Fig. 3.5). For potassium atom,
hyperfine constant of the excited state is so small that a diagonalization by
the |m I , m J > bases also works very well.
0 denote the master laser for potassium(locked to F=1 to
Let νK0 and νRb(F=1 and F=2) cross over of 39 K D2-line) and repump laser for rubidium(locked
to F=1 to (F=1 and F=2) crossover of 87 Rb D2-line). If we set the detuning from the transition | F = 1, m F = 1 >→ | F = 0, m F = 0 > as 200
MHz, we need the lasers with frequencies roughly νK = νK0 + 80 MHz and0 − 400 MHz, respectively. As we mentioned above, these frequenνRb = νRbcies are easily available by using acousto-optical modulators (AOMs).
Schematic of the PCI probe lasers is shown in Fig. 3.6. In short, two seedlasers are amplified by injection locking and detuned by AOMs. By insertingan RF switch (mini-circuits, ZYSWA-2-50DR) after the RF signal sent fromvoltage controlled oscillator (VCO) to RF amplifier, we can turn on probe
lasers by an external TTL signal from Labview program and suppress theleakage of RF signals to AOMs which causes the unwanted light scattering
during the imaging sequence. Two probe lasers are coupled to the opticalfiber after the non-polarizing beam splitter(NPBS). With this setup, we cantune the detuning from 200 MHz to 300 MHz.

39

Figure 3.4: Calculation of Zeeman shifts for K D2 line (2 S1/2 →2 P3/2 ). Redlines are the ground state | F = 1, m F = 1 > and the excited state | F =0, m F = 0 >. A red upright arrow represents the required probe light when
the detuning is 200 MHz.

40

Figure 3.5: Calculation of Zeeman shifts for Rb D2 line (2 S1/2 →2 P3/2 . Redlines are the ground state | F = 1, m F = 1 > and the excited state | F =0, m F = 0 >. A red upright arrow represents the required probe light when
the detuning is 200 MHz.
41

Figure 3.6: Schematic representation of the preparation of probe lasers for
the PCI. Seed lasers for K(master laser, locked to F=1 to (F=1 and F=2) crossover of 39 K D2-line) and Rb(repump laser, locked to F=1 to (F=1 and F=2)
crossover of 87 Rb D2-line) are amplified by injection locking and detuned
by AOMs. Two probe lasers are coupled to the optical fiber after the nonpolarizing beam splitter(NPBS). The frequencies shown next to AOMs are
the center frequencies of them.

42

3.3.3Generating Pulse Trains for Probe Lights

In this section, the detailed experimental setup which realizes the phase
contrast imaging of a dual species atomic cloud is introduced. Here we describe a method to generate TTL pulses from Labview, which controls theoverall experimental sequences, combined with function generators. Actual
observation of a dual BEC using instruments shown below is described in
the next chapter.
Let us consider the number of images we take in a single sequence. Of
course the larger the number of images, the better the imaging works as a”movie”. However, we want to take all images in one frame of CCD camera(row × column is 1048 pix. × 512 pix.) and the size of the BEC is at most
150 µm which corresponds to 60 pixels if seen by side imaging. Considering
these, we divide one frame into ten regions2 (row × column is 104 pix. ×
512 pix.) and assign five regions for the probe of each atomic species. Then
we need five pulses for each species.
For a single species, it is sufficient to send pulse train with desired intervals to take images one after another. However, we need to take a pairof images for Rb and K without delay while the separation between pairs
should be tunable (Fig. 3.7, gray shaded area).
In order to realize this, We constructed a system for the TTL pulse generation just summarized in Fig. 3.7. The reason why we generate TTL pulse
trains not by digital TTL signal using Labview(National Instruments), butby these instruments is that if we make pulse trains only with Labview, it
takes (4 time divisions) × (5 pairs of images) × (3 sets: shadow, light anddark ) = 60 time divisions. Actually it will overload the PC.
Using function generator 1 (Rigor, Function/Arbitrary Waveform Generator DG4062. FG1 in a shorthand), we can output 5 pulses with an arbitrary
interval τ. One may think that signals from FG1 can be used for the TTL of
K probe and three function generators are excess. However, it was observed
2 In both the top imaging and the side imaging, the elongated BEC appears with its longaxis in the column and the number of images taken in a single sequence is limited to 10.

43Figure 3.7: Instruments for TTL pulse train generation. Function generator 1 (Rigor, Function/Arbitrary Waveform Generator. FG1 in a shorthand)
can output 5 pulses with an arbitrary interval τ and pulse duration 300µmathrms. The pulse width 300 µs is determined by the exposure time ofa single strip of the CCD camera(Prinston Instruments, PhotonMAX 512),
and the Rb probe pulse should be send 300 µs from FG3 after the end of the
K probe pulse from FG2, which is determined by the frame transfer speedof the CCD camera. TTL pulses for CCD camera are generated by taking OR
signal of probe pulses for K and Rb.

44

that for a long interval τ > 100 ms, the leading shape of the pulse starts
to get distorted and it did not work correctly. So send the TTL signals to K
using FG2(Stanford Research Systems, DS345) triggered by FG1. The pulse
width 300 µs is determined by the typical acquisition time of a single strip of
the CCD camera(Prinston Instruments, PhotonMAX 512), and the Rb probe
pulse should be sent 300 µs after the end of the K probe pulse, which is determined by the frame transfer speed of the CCD camera. TTL pulses for the
CCD camera are generated by taking OR signal of probe pulses for K and
Rb.
Thus we can activate the phase contrast imaging sequence by only three
TTL pulses from Labview. Indeed we had to rewrite the Visual Basic 6.0
program code for acquiring the images but it is too messy to describe here.

3.4

Setup of Optical Dipole Traps

Optical dipole trap is very widely used in atomic physics experiments. It
makes use of the refractive index of atoms in the presence of a high-power,
far-off-resonant laser. As we described above, off-resonant light can be re′fracted by an atom by acquiring a momentum h̄k mostly orthogonal to thedirection of an original wavevector k. Then the action-reaction law(or, Newton’s third law of motion) suggests that the atom must be kicked in the oppo′

site direction to acquire the momentum −h̄k . In other words a photon feels
some force according to the spatial density of atoms and at the same time an
atom feels force by off-resonant light whose distribution directly reflects thespatial distribution of the laser beam. This force is called optical dipole force
and the trapping using this force be the optical dipole trap(ODT). Below we
describe the basics of ODT and the designs of ODT used in our experiment.
A review Ref. [94] is useful to calculate the trapping potential for a target
atom in practice.
45

3.4.1

Light Shift

Here the perturbation considered is also the electric field and its influence on the dipole moment of an atom. Energy shift ∆E is given as a scalar
product of these two quantities:∆E = −

1
1< P · E >= − Re[α]|E|2 .
22< · > represents a temporal average. Because an explicit form of Re[α] isavailable, we can write down the energy shift by light field in a detuningdependent form, that is, with the intensity of light I (r),
)
(
3πc2
Γ
Γ
∆E = −+I (r)2ω03 ω0 − ω ω0 + ω≃

3πc2 Γ
I (r).
2ω03 δ

Indeed, there are quantum states other than two states considered above,e.g., the fine and the hyperfine structures of atoms. The corrections comefrom this fact and results in the formula[94]()πc2 Γ 2 + Pg F m F 1 − Pg F m F∆E = −
+
I (r)
δD2
δD1
2ω03
especially in the case of considering D1 and D2 lines of alkali atom. HereP takes the values of 0 (linear pol.), +1 (σ+ ), and −1 (σ− ), g F is a g-factor,
m F is a magnetic quantum number. This formula gives the optical dipole
potential felt by atoms and as easily deduced from its dispersive origin, potential depth of ODT scales as ∝ 1/δ. Furthermore, if a red(blue)-detuned
laser is applied, an atom is attracted(pushed out) by the laser beam. Another important point is that the dipole potential is proportional to the lightintensity. For example, given the light intensity as
)
(
y2
z2
x2
I (r) = I0 exp − 2 − 2 − 2
2σx
2σy
2σz
(
)
y2
z2
x2
≃ I0 1 − 2 − 2 − 2 ,
2σx
2σy
2σz

46

(3.9)

trapping potential can be written in a quadratic form
V (r) =

1
1
1
mωx2 x2 + mωy2 y2 + mωz2 z22
22(3.10)

in the vicinity of the center of the laser beam. Here trapping frequencies ωi(i = x, y, z) contain almost all information of the trapping potential and are
related to the beam profile, the light intensity and the detuning.

3.4.2

Experimental Sequences

We use 1080 nm fiber laser(NP Photonics) amplified by fiber amplifiers(Nufern) for red-detuned optical trap. To explain the roles of 6 laser beams
used in our experiment, we will briefly itemize experimental sequences below:• First atomic gases of Rb and K are gathered and cooled by magnetooptical trap (MOT) and further cooled down to several tens of µK by
a compressed MOT (CMOT). The number of atoms are roughly from108 to 109 for Rb and 5 × 106 for K.
• Succeedingly both atomic gases are pumped to the | F = 2, m F = 2 >hyperfine state and cooled by forced evaporative cooling in the magnetic trap to reach about 1 µK with 106 rubidium atoms and 105 potassium atoms. In transferring atoms into the magnetic trap, we compress them first to get higher spatial density ∼ 1011 /cm3 for the sakeof an efficient evaporation. Hereafter rubidium atoms plays the role ofcoolant which sympathetically cool potassium atoms.
• Then these pre-cooled atomic clouds are cold enough to load into acrossed dipole trap made of (A) beam and (P) beam(described later),each has a power of 0.55 W and 2.0 W at most. Then rubidium andpotassium are transferred into the | F = 1, m F = 1 > hyperfine state
by adiabatic rapid passages(ARPs). By this moment the magnetic fieldbias for the Feshbach magnetic field should be ramped up to 72 G in
47

order to avoid unwanted lossy collisions due to the three body recombination and prepare for the evaporative cooling in an optical dipole
trap.
• By setting the bias field at 77.3 G which makes the interspecies scattering length aKRb to be 230 a B and starting ramping down the powerof ODT (P), atomic clouds are further evaporatively cooled below BEC
transition temperature. Finally we get dual BEC with about 3 × 104
atoms for both species and soon after this step various experiments
are to be done using 809 nm optical trap(described later), optical lattices((V), (B), (C) beams) and so on.
In this thesis the setup is not changed but the role of each beam is partly
changed. First we introduce the original roles of optical traps and later wewill mention the part changed for this thesis.

3.4.3

Crossed ODT and Optical Lattice

In Fig. 3.8, glass cell viewed from the top and two optical dipole traps(A) beam and (P) beam are illustrated. The (P) beam has an horizontally
flattened shape (465 µm and 164 µm in diameters), and this beam hold theatoms against the gravity. In the sequence of an evaporative cooling, thepower of the (P) beam is lowered. Another beam, (A) beam shaped roundwith diameter of 150 µm, is aligned orthogonally to (P) axis and confining
atoms along the y-direction. Final trap frequencies after the evaporation isroughly (ωx /2π, ωy /2π, ωz /2π ) = (40, 40, 100) Hz. Each axes (x, y and z)
are defined as in Fig. 3.8.
We have other 1080 nm laser beams for three dimensional optical lattice,(V), (B), and (C) beams(Fig. 3.9). Optical lattice is formed by retro-reflecting
three beams of 600 µm diameter and round shape. In this thesis opticallattice is not used but one of them, (B) beam, is used for another purpose
and the detail will be given later on.
48

Figure 3.8: Schematic representation of the configurations of ODT (A) and(P). Orange circle and ellipse in green shaded regions corresponds to theshape of trapping beams.

49

Figure 3.9: Configurations of optical lattice beams. All beams are retroreflected by dichroic mirrors. The shape of (V) beam is all the same as that of(B) and (C).

50

3.4.4

Gravitational Sag and 809 nm ODTThe last, 6th beam is most important in this thesis. Before introducing thistrapping beam, let us assign several lines for describing so-called the ”gravitational sag” in an optical trap. The idea is that if we make a trapping potential (1/2)mω 2 x2 along the direction of the gravity, total potential becomes1mω 2 z2 + mgz
2
1g= mω 2 (z + 2 )2 + const
2ω

V (z) =

due to the gravity. Here acceleration of gravity g = 9.80665m/s2 is a constant(this value is the definition of the standard gravity). We can easily see
the potential minimum is shifted downwards by g/ω 2 . This is the gravitational sag and obviously it depends on the trapping frequency. Actually weuse rubidium and potassium atoms and trap them in a 1080 nm-ODT simultaneously. Although this optical dipole trap offers the almost same trapping
2 z2 for two atomic species,
potentials U (z) = (1/2)m Rb ω 2Rb z2 = (1/2)mK ω Rb

the different masses of Rb and K result in the different trapping frequencies.
Now the problem with an 1080 nm dipole trap is clear: due to different
trapping frequencies, the potential minima for rubidium and potassium are
different from each other. For instance, final power of (P) beam after theevaporation is about 1.6 W and this gives trapping frequencies 100 Hz for
Rb and 135 Hz for K. This makes the potential minima different by roughly
10 µm and is larger than the Thomas-Fermi radius of BEC, which is 4 µm orless for each atomic species. Two BECs does not overlap at all in this trap.
In order to make them overlap, one way is to tighten the trap by increasing the power. But this method needs an order of magnitude larger laser
power to realize the relative gravitational sag to be less than 1 µm. Anotherway of tightening the trap is to make the radius of a trap beam smaller. These
are relevant, however, it can be said that even if we succeed in reducing therelative gravitational sag to be as small as 1 µm by doing so, the ThomasFermi radii of BECs also get smaller and it does not profit so much in therespect of the overlapping region of a dual BEC. Larger three-body loss due
51

to the higher spatial densities is also problematic. Our solution is to use
a laser beam of ”magic” wavelength for the relative gravitational sag[96].
What we want to realize is
ω Rb
=
ωK

√
VRb mK= 1.
·
VK m Rb

Then if we set the detuning of a laser so that VRb /VK = m Rb /mK is satisfied, trapping minima of each species are always the same, regardless of theincident power of a laser. Calculation in detail shows[97] that fortunately
our ”magic” wavelength exists and is 808.8 nm. We use this optical dipole
trap(this beam is also named (P) because 809 nm beam joins the same optical path as 1080 nm ODT via dichroic mirror) in order to do experimentswith spatially overlapped BECs. This 809 nm laser beam is also flattened
similar to the 1 µm (P) beam, whose diameters are horizontally 270 µm andvertically 150 µm.
In summary, for dual BEC experiments we first generate a dual BEC in
1 µm crossed ODT, where two BECs stay at different places. Then we load
the dual BEC into an 809 nm dipole trap to bring two species into the sameplace. After that we do experiments with that.

3.4.5

Setup for Generating Elongated BECs

As mentioned in 3.1.2, we want to generate elongated BECs in order to
get in-situ images of atomic clouds. However, Thomas-Fermi radii of a BEC
under trapping frequencies (ωx /2π, ωy /2π, ωz /2π ) = (40, 40, 100) Hz are
approximately ( R x , Ry , Rz ) = (10, 10, 4) µm with 3 × 104 atoms(there is no
significant difference between Rb and K). Therefore one solution is to makea cigar-shaped trap by loosening the trap in one direction. But if we simply
ramp down the (A) beam and lower the trapping frequency in y direction,
it will be problematic that trap depth gets low and atoms pour out of the
trap. This circumstance is qualitatively explained by considering the formulae (3.9) and (3.10). From these two expressions we can easily see that the52

Figure 3.10: Modified evaporation sequence. We ramp ODT (B) up in theearly stage of the evaporation and ramp ODT (A) down slowly in case thisramping causes the oscillation of a BEC. Total time of this sequence is 2000
ms.

53

Figure 3.11: Crossed dipole trap with ODT (P) and (B). (B) beam is originally used for optical lattice and here we use this as a dipole trap by simply
blocking the retro-reflected beam.

54

Figure 3.12: Calculation of potential surface by crossed dipole trap withODT (P) and (B). Power of (B) beam is 0.5 W.

trapping frequency ω is proportional to the quantity√

PODT /w2 and the trap

depth k B Ttrap is proportional to PODT /w2 , from which it follows that by lowering the power of ODT, the trap depth becomes too shallow before we get
the sufficiently small trapping frequency. Actually by lowering the powerof the (A) beam to 0.005 mW the trapping frequency is 11.2 Hz, whereas thetrap depth is k B × 74 nK. So we need a low trapping frequency with a larger
trap depth (higher laser intensity), namely the trapping beam with moderately large diameters. If we prepare a trap beam with its diameter 600 µm,
the power of 1.3 W gives the trapping frequency 11.4 Hz and the trap depthk B × 1.2 µK simultaneously.
According to above considerations, the solution to fulfill this requirementwith our setup is to use the (B) beam, which is originally designed for an optical lattice, without the retro-reflected beam. As illustrated in Fig. 3.11, (B)beam is aligned at angle 45 degree with (P) beam. We can think this as a√beam with diameter 2 × 600 µm is applied in y direction. Then the calcu55

Figure 3.13: Illustration of a tilted optical dipole trap and a plot of
relative position with respect to the angle θ.

Trapping frequencies(ωyRb /2π, ωyK /2π ) = (5.4, 7.2) Hz are used in the calculation.

lation indicates that 0.5 W of (B) beam forms a trapping potential with 5.4 Hz
trapping frequency (for Rb) and k B × 1.2 µK trap depth in y direction (see Fig.
3.12). Ramping up (B) in the early stage of evaporation and ramping down(A)(see Fig.3.10), an elongated dual BEC in crossed ODT (P) and (B) is realized. For this trap Thomas-Fermi radii of BEC are ( R x , Ry , Rz ) = (7.5, 33, 2.6)
µm.

3.5

Gravitational Sag and Magnetic Field Gradient

3.5.1

Gravitational Sag in the Horizontal Direction

Gravitational sag along the gravity is discussed above. However, thegravity also have an influence on the horizontal position of the potential
56

minimum when the trap beam is tilted with respect to the horizontal plane(xyplane in Fig.3.8) as illustrated in Fig.3.13. Actually, if the (P) beam is aligned
at angle θ with xy plane, the trapping potential becomes1mωy2 y2 + mgy sin θ2
1
g sin θ 2
= mωy2 (y +) + const.
2ωy2

V (z) =

(3.11)

Here, for simplicity we assumed that the beam is tilted only in the yz plane.
Though we use an 809 nm ODT for the (P) beam and the relative sag in zdirection is compensated, the (B) beam gives different trapping frequencies
in y-direction between Rb and K for its wavelength 1080 nm. Trapping frequencies (ωyRb /2π, ωyK /2π ) = (5.4, 7.2) Hz and an angle θ = 1 degree givethe difference of the potential minima in the y-direction of about 64 µm.

3.5.2

Effect of Magnetic Field Gradient on the Potential Minimum

Even if we have a perfectly aligned trap beam, namely θ = 0, anotherexternal field gradient for which Rb and K have different susceptibilities cancause all the same problem discussed in the Section 3.5.1. The most likely
candidate for this external field is the magnetic field.
Actually we use a pair of coils which generates the bias magnetic field
75 G or more, enabling us to make use of a Feshbach resonance. Of course
the magnetic field curvature is present and it is revealed experimentally thatthe position of atoms in an ODT is different from the center of this quadraticmagnetic field. Therefore an atom feels the magnetic field gradient and since
the magnetic susceptibilities of Rb and K differs from each other, the relative
position can be different for all the same reason with Section 3.5.1.
In order to discuss in an explicit form, we rewrite potential (3.11) in aform with the term containing mass replaced by a term containing magnetic
57

susceptibility:′1mωy2 y2 + µg J m J B y
2′µg J m J B 2
1
2= mω (y +) + const.
2mωy2

V (z) =

Coefficient µ is the magnetic susceptibility, g J the g-factor and m J the magnetic quantum number. Since the atomic spins are oriented to x direction
and the optical trap is loose only in the y direction, the problematic magnetic′

field gradient B = ∂Bx /∂y is considered here. Given the magnetic susceptibilities µ Rb = 0.7 MHz/G and µK = 1.1 MHz/G, the potential minima areseparated by several µm in the presence of 1 G/m field gradient.
Above we see that the magnetic field gradient makes the relative positionof potential minima different. It is true, but in other words we can compensate the horizontal sag due to any field gradient by applying magnetic field
gradient. However, usual setup of the anti-Helmholtz configuration of apair of coils makes ∂By /∂y and does not generate ∂Bx /∂y. We can obtain
non-zero ∂Bx /∂y by putting a pair of coils with their axes off just as illustrated in Fig.3.14(a). With these we can make more than 20 G/m magnetic
field gradient(see Fig.3.14(b)) by injecting several Amps of current and cancompensate the separation of atomic clouds up to a few tens of µm.

58

Figure 3.14: Illustration of the configuration of compensation coils and its
magnetic field/field gradient. (a): Configuration of coils. The red cloud
represents the atoms. (b): Magnetic field gradient ∂Bx /∂y by these coils(36turns) with current 4 A and is plotted. (c): Magnetic field distribution isshown in the vicinity of the position of atoms.

59Chapter 4

IN-SITU OBSERVATION OF A
DUAL BEC

Chapter 2 and 3 are devoted to the explanation of basic properties of adual BEC and major experimental challenges. Up to here three major solutions have been introduced: non-destructive imaging of a dual BEC, generation of an elongated dual BEC in ODT, and the compensation of the difference of potential minima due to field gradients. We show how to implement
these solutions in this chapter and we can see a fundamental nature of a dual
BEC at the end.

In this chapter we consider two topics: first, the in-situ observation of adual BEC in an 1µ m optical dipole trap is described. Second, a dual BEC
is loaded into an 809 nm dipole trap and we observe non-destructively the
famous miscible/immiscible phase. In addition to the observation of static
properties, that of a dynamical motion of a dual BEC in an immiscible phaseis given.
61

4.1

Creation of an Elongated Dual BEC in an ODT

4.1.1

Time-of-Flight Measurement

In this section we observe a dual BEC by time-of-flight (TOF) imaging,which is one of the most common methods. In this method, the atomic cloud
freely expands after the release from the trap and 30 ms later the resonant
probe light shines the cloud. Measuring the fractional decrease of the probe
light intensity at each position, we know the density of the cloud after the expansion. Since the atomic gas freely expands, observed density distribution
is a momentum distribution in the trap for a sufficient expansion time.
Observed shape of an atomic cloud after long TOF differs between a thermal cloud and a BEC. For the thermal cloud, kinetic energy is equally divided for three directions and the round cloud shape should be observed.
On the other hand, the cloud shape of a BEC after the TOF is not necessarily round, but rather elliptic depending on the anisotropy of the trapping potential. A condensate consists of macroscopic number of atoms in
the ground state of the trap and its momentum distribution depends on the
shape of the trap and directly reflects the anharmonicity of the trap. As you
can easily check, a tight (loose) confinement gives a sharp (blunt) spatial
distribution and its derivative, namely the momentum distribution is broad(narrow). If we neglect the interaction between atoms, this is the reason for
the anisotropic shape of a released BEC. Actually atoms interacts with eachother. When atoms interacts repulsively with each other, atoms expands dueto this repulsion and acquire about three times larger velocity than that ofa motion in a harmonic potential. Since the column density is largest in the
direction of the most tight potential, atomic cloud expands fastest in thisdirection. As a result we can observe a BEC wide in the direction of tightconfinement and narrow in that of loose one.
62

Figure 4.1: Time sequence of TOF and absorption imaging.
634.1.2Elongated Dual BEC in a New Crossed ODTFirst of all we have to create BECs in the optical trap introduced in Section 3.4 and to detect it by an absorption imaging after a TOF. In the newsetup of the ODT ((B)+(P), see Fig. 3.11), trapping frequencies are roughly(ωx /2π, ωy /2π, ωz /2π ) = (40, 7, 100) Hz. As mentioned earlier in the Section 3.4, we ramp (A) beam down to zero and ramp up (B) beam. At the endof the evaporative cooling we turn trap lasers off and release atoms. At thismoment atoms are in the | F = 1, m F = 1 > state and in order to make use′of the closed transition F = 2 → F = 3, weak repump beams resonant on′

F = 1 → F = 2 are shined for 100 µs to pump atoms in the F = 2 hyperfine
manifold. Schematics of the TOF and the absorption imaging process areillustrated in Fig. 4.1.

Thus we can observe a dual BEC generated in 1080 nm, new setup of
crossed optical dipole trap. Obtained images of BECs are shown in Fig. 4.6(b), where the number of atoms are 3.4 × 104 for K and 4.6 × 104 for Rb.
Nearly round shape of the cloud implies that 30 ms of free expansion is not
sufficient to see the anharmonicity of the trap. It is verified by changing the
expansion time of TOF that initially elongated cloud expands faster in x direction than in y. For comparison, thermal clouds of 6.4 × 104 K and 1.8 × 105
Rb with temperatures 87 nK and 114 nK is shown in Fig. 4.6 (a). It is easy todiscriminate BEC from thermal cloud by their optical densities(OD), where
OD ∼ 1 for BEC and OD ∼ 0.1 for thermal cloud for 4 × 104 atoms, while
the emergence of a BEC component is obvious from the bimodal distribution. The narrow width of the expanded cloud(less than a half of that of thethermal cloud) is also a comprehensive feature of BEC.
64

Figure 4.2: Absorption image of (a) the thermal clouds of 6.4 × 104 K and
1.8 × 105 Rb and (b) the dual BEC of 3.4 × 104 K and 4.6 × 104 Rb producedin 1 µm ODT.

4.2

In-Situ Observation of a Dual BEC

4.2.1

Probe lasers for PCI and Phase Plate

We succeeded in generating BECs of Rb and K simultaneously in a loose
1 µm ODT above. Next thing to do is the in-situ observation of a dual BEC
using instruments and sequences prepared in Chapter 3. Before describing
results of the observation of atoms, we mention actual parameters including
the intensity, the beam radius and the optical path, and the alignment of thephase plate.
First, two colors of 100 µW probe lights are coming out from the same
single-mode optical fiber in Fig. 4.3. The collimation lens with focal length
10 cm is placed 10 cm or less away from the edge of the fiber core. The
diameter of a collimated beam is measured to be 15.95 cm by the knifeedge method. This passes through a lot of optics such as the 2-inch PBS(for the compatibility with MOT), the dichroic mirror (for the compatibility
65

Figure 4.3: Optical path of probe beam before the glass cell. The probe beams
coming out of the single-mode optical fiber are collimated by the achromatic
doublet lens. Joining the same as a MOT beam and the ODT (A) beam, Probebeams finally reach the glass cell.

Figure 4.4: Optical path of probe beam after the glass cell. A MOT beam andthe ODT (A) beam are separated from probe beams by PBS and the dichroic
mirror respectively. Focused probe beams go through the phase plate at the
focal point.

66

Figure 4.5: Picture of optical path of probe beam after the glass cell.

67

Figure 4.6: Schematic representation of focused Gaussian beam through alens with focal length F.

with ODT(A) beam), and so on.
After the glass cell, the probe beam again pass through the 2-inch PBS for
MOT and then goes into the imaging optics (Fig. 4.4). Because the ODT (A)beam, which goes almost the same path with the probe light, is dangerous
for the CCD camera, three dichroic mirrors are inserted after the glass cell(one of them can be seen in Fig. 4.4). The phase plate is placed at the positionof the focal point of the probe beam after the dichroic mirror. It is ideal for
PCI that most of the focused probe beam goes through the hole at the centerof the phase plate. Gaussian beam optics says that when collimated beamwith radius w0 is focused by a lens of a focal length F, the resulting radius
w1 at the focus is calculated by the following formula[50]:w1 = √

F/z0
1 + ( F/z0 )2

w0 ,

(4.1)

where z0 = πw02 n/λ is the Rayleigh length. Here n is the refractive index
of the medium and λ is the wavelength of light. Given the beam diameter 15.95 mm and regarding two F = 250 mm lenses as one F = 125 mm
lens, the beam diameter at the beam waist is calculated to be 2 µm. Actually the beam diameter is not collimated to such a small diameter but only
about 100 µm (measured by beam profiler) and this can fall within the rangeof hole of diameter 140 µm. Remaining task for realizing PCI is the precise
68alignment of the phase plate. We mount the phase plate on the xyz-axes
stainless kinematic stage (SIGMA KOKI) which is equipped with micrometers in three directions. When a part of the probe beam hits the hole, an Airy
fringe shows up and if the beam is completely aligned inside the hole, thisfringe vanishes. Therefore the procedure is summarized as follows: at first
we put phase plate off the focus and observe the fringe. Second we adjust
the phase plate at the position of focus by moving it so that the fringe interval gets larger. When the fringe vanishes we move phase plate in transverse
directions to the probe beam back and forth and check whether we can see
fringes in both directions. If these are observed, the alignment is perfect.

4.2.2

Phase Contrast Imaging of a Dual BEC

Preparation for the phase contrast imaging is completed by now. We thenproceed to the description of an in-situ observation of a dual BEC. Because
we are now interested in taking a ”movie” of atomic clouds, the experimental sequence must be changed. In contrast to the imaging sequence shownin Fig. 4.1, in which the optical trap and the Feshbach magnetic field are
turnd off at the beginning of imaging sequence, for PCI we only have tohold them at constant values at the end of the sequence. Applying the probe
pulse trains just as in Section 3.3.3 during the hold time, we can realize PCI.
Schematic imaging sequence is depicted in Fig. 4.71 .
Observed dual BEC is shown in Fig. 4.8. From the top to the bottom
successive images of K and Rb are shown in chronological order. The images are constructed and displayed using the C++ program (Appendix. F)
and the plot region can be changed by tuning the parameters in the sourcecode. Total number of atoms calculated from the images are 7.3 × 104 for K
and 5.6 × 104 for Rb, which are consistent with the result obtained from theabsorption imaging. The more important thing is that we can image BECs
1 Tenth image is always shifted in the right in the image by 11 pixels and this is always

reproduced, so that we shift the tenth image to the left by 11 pixels. Consequently there are
not any signals in the small region nearby the right end.

69

Figure 4.7: Schematics of imaging sequence of PCI. Total time of this sequence is varied due to the interval τ.

over and over again without apparent heating or loss of atoms. As predictedby the formula for gravitational sag, Rb cloud sits just beneath the K cloud(Fig. 4.8 (a) ). Here we succeeded in the nondestructive observation of dual
BEC.
70

Figure 4.8: Observation of dual BEC generated in 1 µm ODT by PCI. Fieldof view of each image is 51 µm × 206 µm. The color bar indicates the phaseshift in radian.
71

4.3

Spatially Overlapped Dual BEC in 809 nm ODT

4.3.1

Cancellation of Relative Gravitational Sag in 809 nm
ODT

A dual BEC in an 1 µm ODT observed above has no spatial overlap as
can be seen directly from the images in 4.8. We transfer BECs from an 1 µm
ODT to the 809 nm ODT for the sake of the complete overlap. To realize this,the 809 nm ODT (P) is ramped up in the first half of the transfer sequence
while the 1 µm ODT (P) is kept constant. In the latter half, the 809 nm ODT(P) is kept constant and the 1 µm ODT (P) is ramped down(see the inset of
Fig. 4.11). Time sequence of this procedure is given in Fig. 4.9. We can also
take images of atoms during this transfer sequence and keep track of thepositions of BECs’ center. As a result we can directly see from Fig.4.11 that
relative gravitational sag between Rb and K gets small as we rump the 809
nm laser up and the 1 µm laser down, and two BECs get together at the endof the sequence.

4.3.2

Static Nature of Dual BEC

Section 2.2.1 says that a dual BEC put together in the same trap shows
miscible or immicible phase depending on the interspecies scattering length
aKRb . Revisiting in short, if aKRb < 75 a B , two BECs are mixed together andcalled miscible. If aKRb > 75 a B , it is not energetically favorable for both
species to be at the same place and two BECs get spatially separated. This iscalled immiscible phase. This miscible/immiscible nature can be observed
by TOF measurement. Acquired images are shown in Fig.4.12 and in theimmiscible regime we observed that K sits in the middle of trap and Rb ispushed away to sandwich K. It is known that various spatial distributionsof immiscible dual BEC are realized by the fine-tuning of the ratio of thenumber of atoms [98, 99].
As has been done in Section 4.2.2, we start PCI at the end of the loading72

Figure 4.9: Time sequence of loading atoms into 809 nm ODT. Total time ofthis sequence is 100 ms.

73

Figure 4.10: Dual BECs in (a) an 1 µm ODT and (b) the 809 nm ODT. Field ofview of each image is 51 µm × 206 µm.

Figure 4.11: Cancellation of relative gravitational sag during the transfer to809 nm ODT. Error bar indicates the statistical error of three data sets at each
time.

74

Figure 4.12: Observation of phase separation with dual BEC in 809 nm ODTby absorption imaging. (a) When aKRb = 39 a B (< 75 a B ), two BECs can bemixed together. (b) If we set interspecies scattering length to be aKRb = 230
a B (> 75 a B ), BECs avoid to be at the same position as a result of phaseseparation. In this case K sits in the middle of trap and Rb is pushed away
to sandwich K. It is known that various spatial distribution of immiscible
dual BEC are realized by fine-tuning of the ratio of number of atoms [98, 99].

75

Figure 4.13: Miscible dual BEC observed by PCI. Interspecies scattering
length is aKRb = 39 a B (< 75 a B ), in the miscible condition. In the rightcolumn the profiles along the horizontal axis at each time are shown. The
unit of horizontal axis is pixel, where 1 pixel corresponds to 2.58 µm.

76

Figure 4.14: Immiscible dual BEC observed by PCI. Interspecies scattering
length is aKRb = 230 a B (> 75 a B ), in the immiscible condition. In the rightcolumn the profiles along the horizontal axis at each time are shown. The
unit of horizontal axis is pixel, where 1 pixel corresponds to 2.58 µm.

77

of a dual BEC into an 809 nm ODT and continue acquiring images while
the dual BEC is held in the trap. Interspecies scattering length is changed in
the last 20 ms of the evaporative cooling by ramping up or down the bias
magnetic field and it is kept constant during the loading into an 809 nm
optical trap and after that.
Let us see the miscible phase first. We set aKRb = 39 a B by ramping
the magnetic field down to 74 G. In this interaction regime a dual BEC is
miscible and spatially overlapped BECs should be observed. Apparent from
Fig. 4.13, two BECs reside in the same place for most moments of images.
In the right column the profiles along the horizontal axis at each time areshown. The unit of horizontal axis is pixel, where 1 pixel corresponds to
2.58 µm. One might see the decrease of signal of K at the peak of Rb at t = 0
ms, however, this ”separation” does not last. It is not identified that where
this ”separation” comes from, and this observation does not oppose to themiscibility of dual BEC because from t = 20 ms to t = 80 ms the dip of K
cloud at the peak of Rb cloud vanishes.
Next, let us see Fig. 4.14. Now the magnetic field is set to 77.3 G and
the interspecies scattering length is aKRb = 230 a B , in the immiscible regime.
From the images one can easily see that in contrast to Fig.4.13, the spatial
separation of two BECs is clearly observed. This time the BEC of K sitting in
the right and that of Rb in the left are observed. In all shots with all the same
experimental parameter we observed this configuration of BECs, that is, K
in the right and Rb in the left. So we can conclude that this configuration isdue to a tiny difference of the potential minima along y direction discussed
in Section3.5. Here the compensation coils for dealing with this problem is
not used yet and used later in Section4.3.3. One might notice that both K
and Rb have ”tails” along the horizontal axis. This is considered to be thethermal components of each cloud and the bimodal feature is obviously seenfrom the plots in the right column of Fig. 4.14.
Thus we succeeded in observing miscible and immiscible phases of adual BEC nondestructively and obtaining the spatial density profiles of the78

Figure 4.15: Tuning the relative position in horizontal direction by compensation coils. By tuning the current at (a) 0 A, (b) 1.15 A and (c) 1.5 A various
initial and final states are observed. Here the interspecies scattering length
is aKRb = 230 a B , in the immiscible regime.

BECs. This is the first, but important step of this thesis and below we try tocontrol the overall experimental parameters in order to get the best experimental condition.

4.3.3

Tuning the Relative Position of Two BECs by Magnetic
Field Gradient

There is nothing to say about the importance of the spatial overlap between two BECs in the context of study on a dual BEC interacting with eachother. In our case the relative gravitational sag along the z axis is sufficiently
79small in 809 nm trap and problematic thing is the horizontal sag due to gravity and/or magnetic field gradient arising mainly from a part of wirings ofcoils without stranding, or possibly from the inhomogeneity of the bias field
and the magnetization of the vacuum chamber. We can cancel these influences by an additional magnetic field gradient by a pair of coils(see Section
3.5.2).
Actually, when we apply no current to the compensation coils K is on theright side of Rb at the early stage of the loading sequence into the 809 nm
trap and this configuration is conserved after the loading is finished(Fig. 4.15(a)). With current 1.5 A, we have K on the left and Rb on the right, conversely
to the case of (a) (Fig. 4.15 (b)). At the moderate value 1.15 A(Fig. 4.15 (c)),
the dual BEC shows a bit different behavior. For t = 0 ms and t = 40 ms
spatial separation is not apparent but at t = 80 ms we can see that two BECs
splits and settle in the seemly true ground-state configuration2 . From theseit can be said that when 1.15 A current is applied to the compensation coils
the relative sag in the horizontal direction is almost compensated.

4.3.4

Dynamical Motion of an Immiscible Dual BEC

By tuning the relative position of two BECs, we can prepare the initialconfiguration which does not equal to the true configuration with minimum
energy in the trap. Above we see that initially not completely separated
BECs gradually separates and turn out to be spatially separated with sometimescale (in our case this timescale is about several tens of milliseconds).
In addition to the observation described above, we observed clearly that the
2 Of course a dual BEC is always in its ground state when the transferring process isadiabatically implemented ( Fig. 4.15 (a) and (c) ). However the situation of Fig. 4.15 (b)does not seems that the dual BEC is in the ground state at t = 0 ms and t = 40 ms becausethroughout these images no conditions are observed to be changed. Therefore it can be said
that the dual BEC is in a metastable configuration at t = 0 ms and then subsequently turns
into the ground state. Moreover, it implies the possibility that the loading into 809 nm ODTbecomes non-adiabatic depending on the relative horizontal position where two immiscible

80Figure 4.16: Dynamical transition from a metastable configuration to thestable configuration. Field of view of each image is 21 µm × 206 µm. Herethe interspecies scattering length is aKRb = 230 a B , in the immiscible regime.

81

Figure 4.17: Time sequence of the loading into an 809 nm ODT. Total time ofthis sequence is 100 ms here and at the end of this sequence magnetic field
for Feshbach resonance is suddenly changed to the target value.

82

Figure 4.18: Dynamical motion of a dual BEC undergoing the sudden changeof the interspecies scattering length. Field of view of each image is 51.6
µm × 206 µm. Here the interspecies scattering length is suddenly changed
to aKRb = 230 a B at t = 0.

83

metastable configuration gradually turns into the stable one. Experimentallyobtained images and the plots of horizontal profiles are listed in Fig. 4.16.
At t = 0 ms, K stays at the center and Rb are outside to avoid K. As the time
passes, at t = 80 ms K settles in the left and Rb in the right. The images
from t = 20 ms to t = 60 ms shows the clear feature of the transition froma metastable configuration to the stable configuration. This is considered to
be one of a dynamical phenomena accompanied to a dual BEC, however, theanalysis of this phenomenon is not a scope of this thesis.
Another non-equilibrium phenomenon we observed is closely related tothe topic studied in the next chapter. We want to know how an initially
miscible dual BEC evolves to the most stable configuration of an immiscible
one when the interspecies scattering length is suddenly changed to the value
aKRb > 75 a B , which is an immiscible regime. There can exist an essentiallydifferent dynamics for this situation from the one described above where theinterspecies scattering length is kept constant for the whole sequence. The
experiment is done by the time sequence described in Fig. 4.17. We load adual BEC into the 809 nm ODT as indicated in Fig. 4.9 except for the bias
magnetic field for the Feshbach resonance being ramped up suddenly to 77.3
G ( at this value, aKRb = 230 a B ). Again as soon as the loading is completed
we start to observe atoms by the phase contrast imaging.
The result is shown in Fig. 4.18. In this case, at t = 0 ms the dual BEC
is completely overlapped and it can be seen from the images that two BECs
gradually moves to the opposite directions. Finally at t = 80 ms they settle
in the immiscible configuration.

4.4

The Glitch in Time-of-Flight Imaging

We now go back to the TOF measurement of a spatially overlapped dual
BEC. Indeed if we take images of a dual BEC in the way we described in
Section 4.1, we rarely get such nice images of a dual BEC as Fig. 4.12(these
BECs start to be in touch.

84

Figure 4.19: Typical images obtained with TOF measurement written in Section 4.1. Both atomic gases seem to be nearly thermal clouds from theseimages.

images are taken by modified method written hereafter). Typical images
obtained by this procedure is given in Fig. 4.19, where interspecies scattering
length is aKRb = 39 a B . These atomic clouds hardly seem to be BECs, rather
to be nearly thermal clouds.
We had seen the images like this almost always when we tried to generate a miscible dual BEC. This result made us believe that the three-bodyrecombination rate in our heteronuclear atomic gas mixture is very large sothat the dual BEC vanishes by the three-body loss as soon as two atomic
gases overlap. However, This contradicts with other two results we obtained so far: one is that we can observe spatially overlapped, miscible
dual BEC by means of the in-situ measurement. Another is the three-bodyloss rate measured by thermal clouds was an order of magnitude smaller(see AppendixD, in which we measured the three-body recombination rate
LKKRb = 7 × 10−27 cm6 /s and LKRbRb < 1 × 10−27 cm6 /s at aKRb = 250 a B .
This recombination rate gives a timescale of the three-body loss in a dual
85

BEC of roughly 100 ms for densities 3 × 1013 /cm3 for each species) .
Clues for solving this contradiction have been already mentioned up to
here. One is the TOF sequence (Fig. 4.1). When we released BECs fromthe optical trap, the Feshbach magnetic field was suddenly turned off. The
other is the magnetic field dependence of the interspecies scattering length(Fig. 2.5). Two things should be noted in this plot. First, there is a broad Feshbach resonance (center: 38.4 G, width: 37 G), and second, the interspecies
scattering length is more than 600 a B at B = 0 G. Broad resonance is crossedwith sweep time less than 1 ms, which is the switching-off timescale of thefeshbach magnetic field, and this is still comparable to the timescale of adiabatic conversion into Feshbach molecules. In addition, when the Feshbachresonance is crossed the three-body recombination rate is vastly enhanced
so that the loss can be considerable.
All these information implies that when the Feshbach magnetic field isturned off at the beginning of TOF, the interspecies scattering length suddenly becomes very large and the structure of BECs are not conserved, or destructed, over the 30 ms free expansion. Actually, as we will see in Chapter 5,
modulation instability, by which the structure with very large wavenumber
grows exponentially with timescale of a few hundreds of µs for aKRb ( B = 0
G ) > 600 a B , can be one of the reason for this destruction. Three-body loss
can also be one of the causes taking into account the fact that three-body loss
rate gets large as aKRb gets large.
In order to prevent BECs from experiencing such situation, we should
modify the sequence of the TOF measurement . The most easy prescription
is to keep Feshbach magnetic field until just before the imaging(Fig. 4.20).
Actually this modification enables us to obtain images of dual BEC with nosuch destruction as can be seen in Fig. 4.12.
Furthermore, we now apply strong magnetic field gradient in the horizontal direction for the first 300 µs of the TOF sequence to spatially separate
two expanding clouds, just in case a dual BEC evolves during the free fall.
Another solution is to take images after a TOF in the presence of magnetic86

Figure 4.20: Modified TOF sequence.
87Figure 4.21: Comparison between dual BECs produced by (a) the former and(b) the modified TOF sequence.

field. we did not adopt this because this needs another frequency-tunableprobe lasers.

88Chapter 5
OBSERVATION OF
MODULATION INSTABILITY
WITH A DUAL BEC
Phase contrast imaging is a powerful tool to acquire the information of
dynamical behaviors which is hardly available with the time-of-flight measurement. Up to here we mainly saw the static behavior of a dual BEC and in
this chapter we deal with the dynamical behavior, especially the modulation
instability.
Modulation instability is one of the universal phenomena accompanied
to the nonlinearity of materials. This has been studied for optical beams
inside crystals where the light is first slightly gathered by fluctuations of refractive index and consequently the increase of light intensity results in a
increase of the refractive index, then more light is gathered in this regionand so forth. This causes the interplay between diffractive and dispersive
effects, namely one that light expands as it propagates and one that light
is focused by a lensing effect. Then, triggered by fluctuations, initially uniform light intensity grows exponentially large (but saturates at some point)
to form solitons or some spatial patterns in the presence of a nonlinear optical effect. Indeed, the self-focusing effect[106], the filamentation or the89

breaking-up of continuous wave into short pulses [101, 102, 103, 104, 105]
are investigated in the context of nonlinear optical phenomena inside thebulk materials. All of these phenomena listed above are understood to be
sharing their origins in the modulation instability, which in general refers to
the spontaneous growth of fluctuations by the nonlinear effect. This anomalous effect is also applied to the routing of solitons in bulk matters[107], thegeneration of the ”space-time bullet train”[108] and so on.
This physics works almost the same for an atomic BEC in a sense that
Gross-Pitaevskii equation contains a nonlinear, or, a density-dependent termin its mean-field interaction part. Actually a dark-soliton was first generated
in a BEC by means of the phase imprinting [21], confirming the presence ofthe nonlinear phenomenon in a BEC. However, in a single component BEC
with repulsive interaction atoms do not tend to gather spontaneously even
if there is a density fluctuation because there is no attractive force among
them. So the study of nonlinear effects is done by externally manipulating
the phase of condensates as written above or tuning the scattering length toa negative value.
The situation is entirely different for a two-component BEC since there
takes place the phase separation. Phase separation can be interpreted asa circumstance that inter-component repulsion is so large compared to theintra-component ones that effectively atoms of the same component ”attract” each other in order to avoid mixing with the other component. In
other words, interplay between the tendency to spread over entire region
in the trap and tendency to be pushed away by the other component can
trigger nonlinear phenomena such as the modulation instability.
In the next section we explain how the modulation instability arises in adual BEC by applying linear stability analysis on coupled Gross-Pitaevskii
equations. This will replace above qualitative understanding by quantitative
one and gives the dependence on the inter-component scattering length forthe fastest-growing size of a spatial structure. Subsequent sections are devoted to the preparation of experiment and the analyses of experimentally90

obtained results.

5.1

Theoretical Description of the Modulation Instability

We first write down the theoretical analysis of the modulation instability
in detail. Outline of the analysis is as follows: we first integrate the coupled
Gross-Pitaevskii equations and obtain equations for elongated BECs. Then
we consider small periodic modulation for the linearized equation. Consideration on the solutions gives the wavenumber of the fastest-growing
modulation k max and this depends on the interspecies scattering length. The
analysis shown below follows the discussion in Ref.[1].
Situation we consider here is that we have two BECs in the same trapping potential and they are overlapping with each other. With this initialcondition and given the interspecies (or, intercomponent) scattering length,
we want to analyze what happens and what difference can be observed dueto the magnitude of the interspecies interaction strength. To understand this,first of all we start with coupled time-dependent Gross-Pitaevskii equations(2.5) and (2.6):[]h̄2 ∇2∂+ V1 (r) + g1 |ψ1 (r)|2 + g12 |ψ2 (r)|2 ψ1 (r),ih̄ ψ1 (r) = −
∂t
2m1
[
]
∂
h̄2 ∇2ih̄ ψ2 (r) = −+ V2 (r) + g2 |ψ2 (r)|2 + g12 |ψ1 (r)|2 ψ2 (r).
∂t
2m2
If we have a dual BEC in a cigar-shaped trap V (r, z) = (1/2)mi ωi2 (r2 + λ2 z2 )with sufficiently small λ, it is justified to regard the motion in radial direction
to be frozen and rewrite the wavefunction as
ψi (r) → ϕ(i) ( x, y)ψi (z, t)e−iωi t .
It is obvious from this substitution that ϕ(i) ( x, y) is the distribution of wavefunction in radial direction and does not contain time variable t. On the91other hand ψi (z, t) describes the axial wavefunction and has some information of the axial motion for its dependence on t. Substituting this representation in binary Gross-Pitaevskii equation and integrating with respect to x
and y, we get 1-dimensional coupled Gross-Pitaevskii equations[]
1∂h̄2 ∂2+ m1 λ2 ω12 z2 + u1 |ψ1 (z)|2 + u12 |ψ2 (z)|2 ψ1 (z),ih̄ ψ1 (z) = −∂t
2m1 ∂z2 2
[

ih̄

]

(5.1)

∂
h̄2 ∂21
ψ2 (z) = −+m2 λ2 ω22 z2 + u2 |ψ2 (z)|2 + u12 |ψ1 (z)|2 ψ2 (z).
∂t
2m2 ∂z2 2(5.2)

Here the interaction strengths u1 , u2 and u12 are defined as∫dxdy|ϕ(i) ( x, y)|4 =u i = gi∫

u12 = g12

where bi =

√

gi
,
2πbi2

dxdy|ϕ(1) ( x, y)|2 |ϕ(2) ( x, y)|2 =g12.
2
π (b1 + b22 )h̄/mi ωi are harmonic oscillator lengths of the trap in radial di-rection. Keep in mind that this is an approximation and the trapping potential is nearly 1-dimensional but not true 1-dimension, where Bose-Einsteincondensation does not occur.
In static Gross-Pitaevskii equations we can write wavefunctions as thesquare root of the spatial density of a gas. However, here we consider an
additional fluctuation of the wavefunction which is represented by the 1D
density of BEC ni0 as
ψi (z, t) =√ni0 + δψi (z, t).

Then we substitute these expressions into eqs. (5.1) and (5.2), and linearize
the equations with small parameters δψi (z, t). The results are calculated as92

follows:
ih̄

ih̄

√
∂
h̄2 d2δψ1 = −
δψ1 + u1 n10 (δψ1 + δψ1∗ ) + u12 n10 n20 (δψ2 + δψ2∗ ),2∂t
2m1 dz(5.3)√∂
h̄2 d2δψ2 = −
δψ2 + u2 n20 (δψ2 + δψ2∗ ) + u12 n10 n20 (δψ1 + δψ1∗ ).
2∂t
2m2 dz(5.4)

Eqs. (5.3) and (5.4) hold for small perturbations on the wavefunctions andwe further assume sinusoidal and co-sinusoidal functions as a form of this,namely,δψi (z, t) = ζ i cos (k i z − Ωt) + iηi sin (k i z − Ωt).

(5.5)

Substituting this into the linearized equations we obtain the dispersion relation between k i and Ω. This reads(Ω2 − Λ1 )(Ω2 − Λ2 ) = P2

(5.6)

where the parameters are defined as)(k2i
h̄2 k2i
Λi =+ 2ui ni0 ,2mi 2mi√u
P = √ 12n10 n20 k1 k2 .
m1 m2
We again define the parameters as Λ = Λ1 + Λ2 and ∆ = P2 − Λ1 Λ2 . With
these expressions we can write the solution of equation (5.6) as)√1(
Λ ± Λ2 + 4∆
Ω2± =
2

(5.7)

This is the final form of the dispersion relation we should discuss in detail.
Remind that Ω± is the solution which is to be substituted in the time dependence of perturbation (5.5). Since the amplitude of this perturbation canbe roughly written as δψi ∝ exp [i (k i z − Ωt)], whether Ω± has an imaginary
component or not is crucial in terms of the stability of the system. From theequation (5.7), Ω± takes only pure real or pure imaginary value dependingon whether the right hand side is positive or negative, respectively.
93

For the case of Λ > 0, Ω2+ > 0 always holds. However, we can easily
check that Ω2− takes negative value only if ∆ > 0. Then in this parameterregion there is a purely imaginary solution Ω− and the perturbation gets
exponentially large with time.
On the other hand, if Λ < 0, it can be said that Ω2− is always negativeand in this parameter region the fluctuation grows exponentially regardlessof the magnitude of the parameters. The table shown below summarizes thestable or unstable conditions against the spatial modulation for all combinations of the signs of parameters.

Λ

∆

Ω2+

Ω2−

+ +

+

− Unstable

+ −

+

+

− +

+

− Unstable− −−− Unstable

Stability

StableThis phenomena of spontaneously growing fluctuation is called modulation instability. This is one of the dynamical instabilities which appears
when a dual BEC undergoes the travel from an unstable configuration to thestable one. In our case of K and Rb mixture, intra-species scattering lengths
are aK = 60 a B and a Rb = 106 a B so that the parameter region is limited to
Λi > 0, that is, Λ > 0.
Let us now investigate further about this modulation√instability. For thesake of convenience, we introduce a lengthscale l =h̄2 /4m1 u1 n10 anddimensionless wavenumbers k̃ i = k i l. With these variables the parameters
read

(
)
2
m
h̄2
Λ=
k̃21 (k̃21 + 1) + 21 k̃22 (k̃22 + γ2 )
2
4
4m1 l
m2
(
)2
[
]
h̄2
2 2
2
2
2
∆=
k̃
k̃
γ
−
(
k̃
+
1
)(
k̃
+
γ
)
2
2
1 2
12
1
4m21 l 494where the γ2 and γ12 reflect the interaction strengths and their concrete
forms are given as
m2 u2 n20,
m1 u1 n10√
u12 m2 n20
γ12 =.
u1 m1 n10γ2 =As mentioned above we deal with the case Λ > 0 and the modulation instability appears for ∆ > 0. This condition can be expressed in terms of
k̃1 :

√
k̃1 <

2
γ12

k̃22 + γ2− 1.

There is necessary conditions derived by this inequality γ12 >√γ12 < − γ2 + k̃22 . The former reads
u12 >

√

u1 u2

√

γ2 + k̃22 or(5.8)

in the long wavelength limit k̃2 → 0. This condition is slightly different from√the condition of the occurrence of the phase separation g12 > g1 g2 because
u1 , u2 and u12 are renormalized by factors which include harmonic oscillator lengths b1 and b2 . Then this condition gives (b12 + b22 )/2b1 b2 times larger
value of aKRb for the modulation instability than that of the phase separation. The latter states that for sufficiently largely negative aKRb a dual BEC
collapses (this is not rigorous explanation because the condition of collapse√of dual BEC is usually expressed as g12 < − g1 g2 ).
One of the features which characterizes the modulation instability is thefact that the solution of Ω, namely Ω± , depends on the wavenumbers k̃1
and k̃2 . In our case modulation instability occurs when Λ > 0 and ∆ > 0,
since Ω2− is negative. The smaller the Ω2− (< 0) is, the faster the modulation
grows. So that the next thing we have to do is to find the wavenumbers
which minimize the value of Ω2− .
By the way, the modulation instability occurs when an immiscible dual
BEC totally overlaps with each other. Then it is natural to assume that when95the density of one BEC at some place increases, that of the other decreases.
From this assumption, we can observe the anti-correlation of spatial densities between two BECs. Moreover, we consider the situation that this occurs in most part of BEC and the modulation for each BEC has the same
wavenumber k̃1 = k̃2 = k̃. Then search for the k̃ with minimum Ω2− is reduced to find the solution of equation ∂Ω2− /∂(k̃2 ) = 0 and the solution k̃ max
is calculated to be1k̃ max =2[√]12 −γ −1(γ2 − 1)2 + 4γ12
2

2

.

(5.9)

By substituting back the definitions of parameters γ2 , γ12 and l, and doingdefsome maths we get an expression of λmax = 2π/k max as

2π
λmax = √
2η

[√

]1

(λ21 + λ22 )2 + 4ηλ21 λ22 + λ21 + λ22

2

.

√
def
def
Here we introduced the parameters λi = h̄/ 2mi ui ni0 (i=1, 2) and η =(u212 /u1 u2 ) − 1. Thus the behavior of λmax (and thus of k max ) becomes morecomprehensive: in the square brackets a length is given by the combinationof healing lengths of each components and this length is enhanced by a fac√
tor ∼ 1/ η to give the size of the fastest-growing modulation. If we assume
m1 = m2 = m ( of course, this is incorrect assumption for our system of K
and Rb), the growth rate is given as Gmax = h̄2 k̃2 /2ml 2 .
Note that the calculation done above is a linear analysis of the stabilityof the system. This is valid for sufficiently small perturbation and if themodulation gets deep this can break down so that there can be seen different behavior with dual BEC. But here we limit ourselves to investigate themodulation instability with linear regime.
96

Figure 5.1: Optical path of the tight 809 nm optical trap is indicated by the
red line. In (b), the area surrounded by the blue square line is expanded
and the 1080 nm optical trap is shown by the orange line. Two beams get
together at the long-wave-pass dichroic mirror.

97

5.2

Preparation of a New Optical Trap of an 809
nm ODT

In Section 4.3.4, we saw that a dual BEC undergoes the transition from ametastable configuration to the stable configuration ( see Fig. 4.16 ). The detail of this mechanism is not uncovered yet, however, one possibility is thatradial motions of BECs are not sufficiently suppressed so that condensates
can interchange their positions relatively easily. Then for what condition
we can neglect the motion in radial direction? For qualitative discussion,
one might compare the chemical potential of BEC with the first excitationenergy in the radial motion of a harmonic potential. This then reduces tothe comparison between trap frequencies in radial direction ω⊥ and axial
direction ω∥ . We now define the ratio ω⊥ /ω∥ as aspect ratio.
In experiments done in Chapter 4, the trapping frequencies are set to(ωx /2π, ωy /2π, ωz /2π ) = (40, 7, 100) Hz for Rb. The aspect ratio of thistrapping potential is less than 10 in x direction. For this aspect ratio, motion in radial direction in the trap is not suppressed sufficiently[99]. Thismay make it difficult to observe the modulation instability because the spatial structure has less difficulty in merging to form a larger structure, whichbasically lower the energy due to the spatial overlap.
Let us consider how large aspect ratio can suppress the radial motion in acigar-shaped trap. The necessary condition is expressed as a comparison between the chemical potential µ and the energy scale h̄ωr where (ωr , ωr , ωz )are the trapping frequencies with ωz ≪ ωr :µ < h̄ωr .

1

Aspect ratio is given here by A = ωr /ωz . Defining ω̄ = (ωr2 ωz ) 3 and inserting this into the formula for the chemical potential[51], it is written as98

follows:
(

)2
Na 5
aho
(
)2
1 ωr Na 5= h̄ 1.
2 A 3 aho1
µ = h̄ω̄
2Here the number of atoms N in BEC, the harmonic oscillator length aho =√h̄/mω̄ with mass of an atom m, scattering length a and the relation ω̄ =1

ωr /A 3 are used. Therefore the condition µ < h̄ωr reads
)2
Na 5
< 1.
aho
(
)6
1 Na 5⇐⇒ A >
.
8 aho

1
µ
=
1
h̄ωr
2A 3

(

(5.10)

Using the parameters N = 3 × 104 , a = 106 a B and ω̄ = 2π × 30 Hz for Rb,
the condition (5.10) becomes1 A > 25.
Therefore, in order to prevent BECs from the radial motion in a cigarshaped trap, it is sufficient to let the aspect ratio to be as large as 30. Thissection describes the new design of 809 nm optical dipole trap and observea dual BEC in it.

5.2.1

New Shape of an 809 nm Optical Dipole Trap

To obtain a higher aspect ratio, we should loosen the axial confinement
and tighten the radial one. However, we cannot lower the axial trapping
frequency because too low intensity of a trapping laser leads to a leakage ofatomic gases in the presence of small amount of the magnetic field gradient.
So we think of tightening the radial confinement.
The aspect ratio of more than 30 requires the radial trapping frequency
of about 300Hz. With our maximum intensity 84.6 mW we need to change
the beam radii of the trapping laser from 270 µm × 150 µm to 100 µm × 100µm. On the other hand, if we change beam radii of 1 µm ODT together
1 For K, N = 3 × 104 , a = 60 a

B and ω̄ = 2π × 40 Hz gives the condition A > 9.5

99whose shape is changed by the same optics as for 809 nm optical trap, we
have some difficulty in loading atoms from a magnetic trap to an 1 µm ODT.
Therefore we make an 809 nm beam of such shape by another optical pathand later put it together with a shaped 1 µm ODT by making use of thedichroic mirror.
The actual optical path is shown in Fig. 5.1. First, 809 nm linearly polarized, collimated beam with diameter 2.1 mm is coming out from the single
mode optical fiber. Then it is expanded by a telescope to ∼ 5 mm and focused by the lens of focal length of 500 mm. We confirmed that this optics
can make round-shaped, 100 µm-diameter beam at the expected position of
atomic clouds, consistent with the calculated value by formula (4.1). Finallya long-wave-pass dichroic mirror(coated by Lattice Electro Optics, Inc.) enables us to combine ODT beams with different colors. The optical path isindicated by the red lines.

5.2.2

Radial Trapping Frequency of the New Optical Dipole
Trap

We succeeded in preparing the optical beam and next we load the atoms
in this new optical trap. To show the difference clearly, we show two imagesof a rubidium BEC ( potassium is absent in these images ) in the former andthe newer 809 nm traps in Fig. 5.2. Since we tightened the trap in radial,that is, x and z directions, a BEC expands faster in these directions fasterthan before in the new trap. Resulting cloud shape after 30 ms from therelease from trap is an ellipsoid whose long-axis is in x direction when it isseen from the top.
We measured the radial trapping frequency of this trap. In order to do
this we first apply an short pulse (about a few hundreds of microseconds)of large magnetic field gradient to a BEC and let it oscillate for some time t
in the trap(Stern-Gerlach separation). After that we start TOF sequence andmeasure the position of the BEC, which is equal to the information of the mo100

Figure 5.2: TOF measurement of rubidium BEC in (a) the former 809 nm trapand (b) the new one. (a)In the former trap shape BEC is slightly elongatedin x direction and cloud after TOF has an elliptic shape whose long-axis is
in y direction. (b)In the new trap, BEC is largely elongated in y direction
and cloud after TOF has an elliptic shape whose long-axis is this time in x
direction.

mentum of the BEC during oscillation. By changing the holding time after
the magnetic field gradient pulse and observing the positions at each holding time, we can observe the position of the BEC oscillates with the trapping
frequency. Fig. 5.3 and Fig. 5.4 show the results of the measurements withfitting functions f (t) = y0 + A sin ( f t + phi) for determining the oscillation
period and f (t) = y0 + A exp (−t/(tau)) for the lifetime. The oscillation
frequency is measured to be 325(13) Hz, and at the same time, lifetime of399(9) ms due to one-body photon scattering is also measured. These agree
very well with the calculated ones with the power 84(1) mW and the diameter 100 µm.
We can also trap atoms with much lower trap laser power of 27(1) mW.
In this case the measured trapping frequency is 158(5) Hz ( result is shownin Fig. 5.5 ) and the lifetime is 1.1(1) s ( see Fig. 5.6 ) , which again show
agreement with the calculation. We succeeded in preparing not only thetight optical trap but also the trapping-frequency-tunable trap which ranges
from at least 158 Hz to at most 325 Hz.
101

Figure 5.3: Measurement of the trapping frequency with power of 84 mW.
Fitting function is f (t) = y0 + A sin ( f t + phi).

Figure 5.4: Measurement of the lifetime with power of 84 mW. Fitting function is f (t) = y0 + A exp (−t/(tau)), where we set y0 to zero.

102

Figure 5.5: Measurement of the trapping frequency with power of 27
mW.Fitting function is f (t) = y0 + A sin ( f t + phi).

Figure 5.6: Measurement of the lifetime with power of 27 mW. Fitting function is f (t) = y0 + A exp (−t/(tau)), where we set y0 to zero.

103

Figure 5.7: TOF measurement of a dual BEC in the new 809 nm trap. Theinterspecies scattering length is aKRb = 0.7 a B . The numbers of atoms are
typically 3 × 104 for each species.

5.2.3

Loading a dual BEC into the New Optical Dipole Trap

The aspect ratio of a BEC generated in 1 µm trap is less than 10, while
the one in an 809 nm trap is more than 30. This implies that we can nolonger transfer a BEC from the 1 µm trap to the 809 nm trap as fast as we did
before. It was observed that if we transfer BEC with 100 ms sequence, nonadiabaticity makes the BEC start to oscillate very hard and finally atoms get
rid of the trap. We solve this problem by two modifications on the loading
sequence: we assign 300 ms for the loading sequence (i.e. three times longer
than before ) and tighten the axial confinement by using twice the power
than before. Finally trapping frequencies were (ωx /2π, ωy /2π, ωz /2π ) =(285, 10, 285) Hz for Rb, where the radial trapping frequency was set tolower value than the maximum.
In Fig. 5.7, an image of a dual BEC with interspecies scattering length
aKRb = 0.7 a B in this trapping potential is shown. These images are taken by
30 ms-TOF measurement and indicate clearly the anisotropic expansion of
the atomic clouds. Here, we adopt the Stern-Gerlach separation method because even the modified TOF sequence can destroy a dual BEC for its higher
spatial density. As we have done for a single-species BEC, we succeeded in
104

Figure 5.8: Calculated values of k max versus aKRb . This plot is calculatedwith a set of parameters NK = NRb = 3 × 104 and trapping frequencies(ωx /2π, ωy /2π, ωz /2π ) = (285, 10, 285) Hz for Rb.

Figure 5.9: Calculated values of Gmax versus aKRb . We consider the reduced
mass of Rb and K here. This plot is calculated with a set of parameters
NK = NRb = 3 × 104 and trapping frequencies (ωx /2π, ωy /2π, ωz /2π ) =(285, 10, 285) Hz for Rb.

105

generating a dual BEC in the radially tightened 809 nm optical trap.

5.3

Experimental Observation of the Modulation
Instability

5.3.1

Experimental Conditions for Observing the Modulation
Instability

In the trap described above, we try to observe the modulation instability
with a dual BEC at variable scattering lengths. The aKRb -dependence of thefastest-growing size of the structure is derived in Section 5.1(see (5.9)). In
this formula we only have to consider γ12 which depends on aKRb , whileothers are constant. Since k max is monotonously increasing function of γ12and γ12 is the same as aKRb , k max is monotonously increasing function of
aKRb .
Given the trapping frequencies of the ODT, spatial densities calculated
from typical number of atoms, and intra-species scattering lengths, we canplot k max with respect to aKRb . The plots in Fig. 5.8 show the results of calculation with a set of parameters NK = NRb = 3 × 104 and trapping frequencies (ωx /2π, ωy /2π, ωz /2π ) = (285, 10, 285) Hz where the peak densities
are used. First, the modulation instability occurs in the parameter region√
aKRb > 80.5a B from the condition u12 > u1 u2 ( see ( 5.8) ), which canbe interpreted as the phase separation takes place for this condition in thequasi-1D configuration. Therefore k max is zero at aKRb = 80.5 a B and takes
finite positive value for aKRb > 80.5 a B . Second, the typical size of the structure which is observable with our experimental apparatus has both upperand lower limit. Upper limit comes from the uncertainty of the value of bias
magnetic field for Feshbach resonance, while lower limit takes into accountthe resolution and the magnification of the imaging system. As a result, we
expect to be able to observe the structure which has typical size from 8 µm
106Figure 5.10: Hall probe measurement of magnetic field sweep. Falling of
TTL (shown by green line, right axis) indicates the end of field ramping determined by Labview. Blue line (left axis) is the voltage signal obtained byHall probe sensor attached to the current path of the coils.

to 30 µm.

5.3.2

Experimental Sequence for Observing Modulation Instability

Experimentally, we first load a dual BEC into an 809 nm ODT to makespatial overlap between BECs. Interspecies scattering length is set to aKRb =0.7 a B expecting that three-body coefficient will be minimized and two BECs
overlap with each other. As soon as the loading sequence is finished, we
rapidly ramp the Feshbach magnetic field bias to the expected value whichcause the modulation instability. Just after the magnetic field ramp we nondestructively observe the time evolution of the dual BEC. Note that in Ref.
107

[1], Kasamatsu et. al. consider the situation that two BECs initially overlap and then evolve with interspecies scattering length which satisfies the√condition u12 > u1 u2 . Our setup resembles the theoretically proposed
situation[1] very well.
There is one problem with our instruments. The problem is that the magnetic field ramp cannot be sufficiently fast and if we change it very rapidly,
it overshoots. So we have to fine-tune the current going into the coils. By dividing the field ramp sequence into two and setting the final current of firsthalf of sweeping at a bit lower value, we realized this. Thus an optimized
sweep of magnetic field is realized and the result is shown in Fig. 5.10. Horizontal axis is time and the falling of TTL (shown by green line) indicates theend of field ramping determined by Labview. Blue line is the voltage signal
obtained by a Hall probe sensor attached to the current path of the coils.

5.3.3

Results at aKRb = 80.8 a B

We first describe results observed at aKRb = 80.8 a B . In this case k max ≃400 cm−1 corresponds to the typical size of the structure about 25 µm. Thiscan be observed in the image as the structure which has a size about 10 pixels. From Fig. 5.9, we set the total time of observation 80 ms accordingly tothe timescale 57 ms. Typical time evolution of a dual BEC in this experimentis shown in Fig. 5.11. As can be seen in these images, initially overlapped
clouds separates with some spatial pattern during the time evolution. Typical size of this spatial pattern seems to be about 10 pixels, which is consistent
with the prediction.
For the sake of not only qualitative but also quantitative discussion, we
need an analysis on these data. When we want to know the spatial size ofthe observed pattern, we have only to Fourier transform the observed horizontally summed signal. In addition to this procedure we use the differenceof the summed signals of K and Rb as a Fourier transformed signal. By using this signal we can extract the enhanced signal of spatial pattern if the108

Figure 5.11: Typical images of time evolution at aKRb = 80.8 a B .

109

Figure 5.12: Spectrum obtained at aKRb = 78.5 a B . Five experimental shots
out of 20 are selected and as a result 25 spectra are averaged. Error bars
indicate the standard error of the experimental shots. For this spectrum,
running average is taken over 3 points in order to smoothen the spectrum.

anti-correlation of the density distributions is present. We thus obtain fivespectra per single experimental sequence.
Obtained subtracted signals and spectra at each time are shown in Fig.
5.13 (a) and (b), respectively. Before proceeding into the understanding of
this spectrum, we first take a look at a spectrum taken in another experiment.
According to the condition (5.8), modulation instability does not occur if
aKRb is less than 80.5a B . Therefore if we bring aKRb from 0.7 a B to 78.5 a B ,density fluctuation does not grow and no peak is expected to be observed in
the spectrum.
The resultant spectrum at aKRb = 78.5 a B is shown in Fig. 5.12. Depending on the loading condition, the number of atoms in each species fluctuatesin each experimental run. Therefore this spectrum is obtained by selecting
110

Figure 5.13: (a)Subtracted signals of two atomic species( left, blue plots) and(b)Fourier transformation of them(right, purple plots) for aKRb = 80.8 a B . (c)Summation of 5 spectra in (b) is taken(yellow plot).

111

experimental sequences which satisfies that the ratio of peak number densities is from 0.9 to 1.1, and summing up all of these signals. Thus, 5 experimental shots out of 20 are used and by summing up 5 experimental shots,
we can get unwanted noises averaged. Error bars indicate the standard errorof the experimental shots. Below we always analyze data in this way when
we sum up the spectra. For this spectrum, the running average is taken over
3 points in order to smoothen the spectrum, which corresponds to the coarse
graining over 3 points in the real space..
There are two peaks in the spectrum in Fig. 5.12, one at 0 cm−1 and theother in the range from 250 cm−1 to 300 cm−1 . These two peaks have nothing
to do with modulation instability. First peak at 0 cm−1 means that these issome offset signal or very smoothly varying signal. This probably comesfrom the difference of the number of atoms. The other one at about 250
cm−1 corresponds to the typical fringe separation which is always present
even when atoms are absent. Although this fringe pattern is to some extent
suppressed by the fringe cleaning process (detail is given in Appendix B),
the remaining fringe is observed as this peak.

Similarly, we analyze the behavior at aKRb = 80.8 a B in this way. We got
11 experimental shots out of 60 trials and finally the spectrum is obtained as
the red line in Fig. 5.14. The red spectrum has an extra small peak at 450
cm−1 other than two peaks we mentioned above. This is less apparent in the
result at aKRb = 78.5 a B , indicated by the gray line. Thus it is justified that
this spectral feature originates in the modulation instability. Considering
the method of our analysis, it is natural to think that the position of the peakindicates the k max and we should take 1/e-width of the peak as a standard
deviation. However, it is difficult to extract this spectral feature as an apparent peak, simply due to the bad signal-to-noise ratio and this should only becompared with the theoretical prediction of 434 cm−1 with peak densities of
BECs (see Fig. 5.9).
112

Figure 5.14: Spectrum obtained at aKRb = 80.8 a B . Eleven experimental shots
out of 60 are selected and as a result 55 spectra are summed. Error bars indicate the standard error of the experimental shots. Again for this spectrum,
running average is taken over 3 points in order to smoothen the spectra. Redline indicates the result at aKRb = 80.8 a B and gray at aKRb = 78.5 a B .

113

5.3.4

Results at aKRb = 81.5 a B

We have succeeded in observing the modulation instability for interspecies scattering length aKRb = 80.8 a B and analyzed the size of the spatial
pattern which is consistent with the theory. Next we would like to investigate other interaction strength aKRb = 81.5 a B , which is slightly stronger thanabove experiment by only 0.7 a B . For this parameter, expected wavenumber of the fastest-growing spatial structure is 791 cm−1 . Surprisingly, this
only tiny variation of scattering length results in a large difference in thewavenumber k max .
Experimental sequence is totally the same as in Section 5.3.3. The only
differences are the Feshbach magnetic field, whose final value is changed
from 75.41 G to 75.43 G, and the time interval of probe pulse which is changed
to 8ms. Typical images of time evolution at aKRb = 81.5 a B is shown in
Fig. 5.15. The behavior of a dual BEC is now entirely different from thatof aKRb = 80.8 a B . Center of Rb BEC looks a bit dented in the presence of
K BEC and succeedingly, relatively small spatial structure seems to grow inthe region from 20 pixel to 40 pixel. At the same time, clear anti-correlationof spatial densities of K and Rb can be observed from these images.
This becomes clearer when we subtract two images (see Fig. 5.16, blue
plots). We can fit the subtracted signals by the sinusoidal function with thesecond-order polynomial as an envelope (Fig. 5.17). Envelope of the sinusoidal wave is taken to be the quadratic function of x just because the envelope can likely be given by the difference of two Thomas-Fermi profiles.
From these fitting results we can estimate that the periodic structure seen in
the images has a wavenumber of roughly (4 pixel)−1 = (10 µm)−1 = 1000
cm−1 .
Then let us analyze the results for aKRb = 81.5 a B in the same manneras in Section 5.3.3, that is, in terms of the Fourier spectrum. We have 56 experimental sequences and select 13 of them in accordance with the criterion
0.9 < NK /NRb < 1.1. Five Fourier spectra contained by each of these 13
shots are all summed to give 65-fold summed spectrum shown in Fig. 5.18.
114

Figure 5.15: Typical images of time evolution at aKRb = 81.5 a B . Field ofview of each image is 13 µm(vertical) × 227 µm(horizontal)

115

Figure 5.16: Subtracted signals of two atomic species and Fourier transformation of them(aKRb = 81.5 a B ).

116

Figure 5.17: Subtracted signals and fittings on them. Fit function is f ( x ) =s · sin (2πx/L + ϕ) + a + b( x − x0 )2 .

117

Figure 5.18: Spectrum obtained at aKRb = 81.5 a B . Thirteen experimental
shots out of 56 are selected and as a result 65 spectra are summed. Error
bars indicate the standard error of the experimental shots. Again for thisspectrum, running average is taken over 3 points in order to smoothen the
spectra. Red line indicates the result at aKRb = 81.5 a B and gray at aKRb =78.5 a B .

118

Figure 5.19: Spatial pattern imprinting on the BEC by two laser beams withan angle θ.

Again we can see the peaks at 0 cm−1 and 300 cm−1 which come from theoffset and the fringe on the whole image. One might well expect that there isa small peak at around 1000 cm−1 , which is consistent with the one observedin the real-space images above. However, this cannot be separated from theerror bars and it can be said that in this time no relevant peak which can beseparated from noises is observed.

5.3.5

Summary of the Results

In summary, we succeeded in an in-situ observation of the modulation
instability using a dual BEC by suddenly changing the interspecies scattering length from aKRb = 0.7 a B (in the miscible regime) to aKRb = 80.8 a B(in the immiscible regime). Not only the dynamical behavior of the spatial
structure of a dual BEC, but the qualitative analysis of over a dozen piecesof spectra was done. As a result for aKRb = 80.8 a B , observed feature of theFourier spectrum does not contradict to the theoretically calculated value
434 cm−1 .
119

We tried to observe the aKRb -dependence of the typical size of spatial
structure growing by the modulation instability by the same experimental
sequence, except for changing the interspecies scattering length from aKRb =0.7 a B to aKRb = 80.8 a B . Although the observed spatial structure implied
the spatial modulation with the wavenumber ∼ 1000 cm−1 , we could notobserve relevant peaks from the spectrum. It can be that the peak at 1000
cm−1 is so small that we cannot distinguish it from the noise floor, which isthe standard error of the experimental shots. In order to get this over, one
can prepare more and more experimental images and make the standard
error much smaller. If the signal-to-noise ratio is limited by the photon shot√noise, we can increase the ratio by a factor α with increased probe lightintensity by a factor α, in exchange for the absorption of photons by atoms.
In our experiment the photon absorption rate is thought to be negligible sothat this solution can be valid.
In addition, there is a fluctuation of magnetic fields for a Feshbach resonance. The 100 Hz, 17 mG (in a root mean square value) fluctuation of themagnetic field comes probably from the ripple noise of the current source.
This results in a fluctuation of the scattering length of 0.35 a B and thus of k max
ranging from 600 cm−1 to 900 cm−1 at the central value 81.5 a B . Therefore
the more reliable measurement requires the less fluctuating magnetic fieldrealized by, for example, the stabilization with the feedback to the current
injected into the coils. There is one more solution to this: the larger magnetic
field of more than 100 G gives the same scattering length with smaller sensitivity to the magnetic field which is an order of magnitude smaller. However, currently 100 G is not easily available with our setup of the coils andcurrent sources so that an additional coils for adding up the magnetic fieldbias is needed to implement this.
Another prescription might be the imprinting of spatial modulation by
two lasers with an angle θ just as in Fig. 5.19. In this situation, two light field′interfere and form aperiodic potential with its periodicity λ = λ/2 sin (θ/2),′′

where θ = 2 degrees give λ = 22 µm and θ = 4 degrees give λ = 11 µm.
120This can trigger the modulation instability if the fastest-growing size of the′structure and the wavelength λ coincide. Then with weakly applied continuous wave of two lasers or a weak pulse of two lasers, we can do a spectroscopy with respect to the interspecies scattering length by tuning it by a
Feshbach resonance, where there should appears the largest peak in spectra′

when λ = λmax . By doing so, it is expected that we can obtain more significant peaks in spectra and can observe the spatial pattern with a higher
wavenumber, which was difficult to observe in this thesis.

121Chapter 6
SUMMARY AND
CONCLUSIONS
Here we briefly summarize the achievements in this thesis below. First,
Chapters 1 and 2 are devoted to the introduction and the historical / theoretical overview on a dual BEC. Coupled Gross-Pitaevskii equations describedin Chapter 2 forms the basis of this thesis in a sense that we have dealt witha dual BEC regardless of the topics are static or not.
In Chapter 3, achievements can be divided into three major topics:• We prepared the probe lasers with moderate detunings and instrumental setup in order to realize the simultaneous observation of 41 K and
87 Rb by phase contrast imaging.

• Optical dipole traps for generating an elongated dual BEC was prepared.
• A pair of coils with anti-Helmholtz configuration was added in order
to compensate the horizontal position of two BECs.
These three were revealed to be equally important for the in-situ observationand study on the overlapped dual BEC. Actually, making use of these three
we successfully generate elongated BECs and observe both atomic species
non-destructively at a same experimental run. These results were achieved
123

in Chapter 4. In Chapter 4 we also observed the center-of-mass motion of adual BEC which implies the non-equilibrium nature.
Chapter 5 described the main issue of this thesis, the modulation instability, which is the dynamical phenomenon universal for the nonlinear system. We observed the spatial structure of the typical size ∼ 22(4) µm (or(447 ± 82) cm−1 in wavenumber) emerging for the condition aKRb = 80.8
a B . For aKRb = 81.5 a B we could not observe any significant peak in the
Fourier spectrum, however, in the spatial domain we observed the growthof spatial structure with typical wavelength ∼ 10 µm(or ∼ 1000 cm−1 inwavenumber). Observed two features at aKRb = 80.8 a B and aKRb = 81.5
a B do not contradict the theoretical values calculated from coupled GrossPitaevskii equations.
First of all, the biggest problem in the observation in Chapter 5 was thesignal-to-noise ratio of the obtained images. The larger S/N ratio can beobtained by shining stronger probe laser onto the atoms. This causes the absorption of the probe photons by atoms more, however, the photon scattering rate in the current setup is thought to be negligibly small and by increasing the light intensity we can get better signal-to-noise ratio proportional to√
I probe , where I probe is the probe intensity. Of course the photon scattering
rate scales as I probe , therefore we have to find the optimal strength of probe
light such that we can get sufficient S/N ratio and suppress the light absorption.
It is also important to suppress the fluctuation of the scattering length
by stabilizing the magnetic field or using larger values of magnetic field (∼100 G) which gives the same values of the scattering length but the smaller
sensitivity daKRb /dB ∼ 2.5 a B /G, the an order of magnitude smaller value
than the one considered in this thesis (∼ 3.7 a B /G). The latter prescription isrelatively easy but available if we utilize another pair of coils to add up thebias magnetic field.
There is a difficulty with observing modulation instability in the way we
did because the nucleation of density fluctuation is completely at random.
124

One can make it easy to study the domain formation by imprinting a spatial
modulation using two lasers with wavelengths λ incident to atoms with anangle θ as depicted in Fig. 5.19, which make a periodic potential with its
separation λ/2 sin (θ/2). By tuning the angle between two laser beams thewavenumber of imprinted spatial structure will change and this will enable
us to investigate the aKRb -dependence of the wavenumber of structures moreaccurately.
We have shown throughout in this thesis that the study on dynamical
behaviors in a dual BEC is available in our experimental system. Modulation instability is one of those non-equilibrium phenomena while there are a
lot of dynamical instabilities such as the Rayleigh-Taylor instability[55, 56],
Kelvin-Helmholtz instability[60] and others.
The study on solitons and vortices are also interesting issues. The generation of the dark-bright solitons[57] and the vortex-lattice dynamics[111] arealready observed experimentally, and theoretically the vortex-bright soliton
formation[109], the collective excitation in a binary BEC[110] and other phenomena are intensively studied. Our setup of a dual-species BEC combinedwith the nondestructive imaging will works as a powerful tool to investigate
these phenomena and we assign to uncover these non-equilibrium naturesof superfluid-superfluid interactions as a future works.

125

Appendix A
ALKALI D-LINE DATA
The properties of D-lines of alkali atoms listed below are quoted from D.
A. Steck’s document[92] and T. G. Tiecke’s thesis[93].

A.1

Properties of 87Rb D2 Line
2π · 384.230 484 468 5(62) THz

Frequency ω02π · 6.0666(18) MHz

Natural linewidth
Hyperfine constant A2 S1/2

3.417 341 305 452 145(45) GHz

Hyperfine constant A2 P3/2

84.7185(20) MHz

Electric quadrupole constant B2 P3/2

12.4965(37) MHz

Electronic g factor g J (2 P1/2 )2.002 331 13(20)Electronic g factor g J (2 P3/2 )1.336 2(13)In addition to these, the wavelength of the D1 transition of Rb, which is794.978 851 156(23) nm, is used for the calculation of trapping frequencies of
optical traps. The level structure is shown in Fig. A.1 (a).
127

A.2

Properties of 41K D2 Line
2π · 391.016 406 21(12) THz

Frequency ω02π · 6.035(11) MHz

Natural linewidth
Hyperfine constant A2 S1/2

127.0069352(6) MHz

Hyperfine constant A2 P3/2

3.363(25) MHz

Electric quadrupole constant B2 P3/2

3.351(71) MHz

Electronic g factor g J (2 P1/2 )2.002 294 21(24)Electronic g factor g J (2 P3/2 )4/3

In addition to these, the wavelength of the D1 transition of K, which is770.107 919 192(123) nm, is used for the calculation of trapping frequenciesof optical traps. The level structure is shown in Fig. A.1 (b).

128

Figure A.1: Level structures of Rb and K D2 line

129

Appendix B
FRINGE CLEANING
In experiment, images of atoms are generated by taking three frames ofpictures, namely the probe light with the atoms, the one without atoms andthe one without probe light nor atoms. These three are called ”shadow”,”light” and ”dark” images respectively. We take ten shadow images in the
shadow frame that consist of five images for each species. In the light anddark frames again ten images are taken in the same order with shadow
frame. The size of single frame is 1047 pixel × 512 pixel and let us denotesignals on shadow, light and dark frame S(i, j), L(i, j) and D (i, j), wherei = 0, 1, · · · , 1047 and j = 0, 1, · · · , 511. Signal F (i, j) is generated by following math:
F (i, j) =S(i, j) − D (i, j).
L(i, j) − D (i, j)

In terms of eq. (3.8), the fractional change of light intensity through the atom
I/I0 is directly related to this byS(i, j) − D (i, j)
I= F (i, j) =.
I0
L(i, j) − D (i, j)(B.1)

Actually, experimentally obtained images are polluted by some noises.
One is the fluctuation of light intensity or dark current at each position (i, j)which adds the white noise on the image. Another is the fringe, the spatiallyperiodic noise on the whole image. In this appendix we deal with the latter.
There might be some reasons for the occurrence of fringe, however, the main131reason can be the small shift of probe beam in the light image with respect tothe shadow image due to the mechanical fluctuation of the optics. One solution to this is the preservation of imaging optics from external fluctuations.
Even if we do this, often there remains the fringe and we have to do secondprescription: cleaning of fringe by image processing, which we call ”fringecleaning” here.
Fringe cleaning process consists of a few steps itemized below:• First we collect several tens of light images Lm (i, j) which has various
patterns of intensity fluctuations. These are usually taken the latest
several tens of light images before the on-going sequence.
• We find a region in S(i, j) where there is no atoms, and then fit thatarea with ∑m cm Lm (i, j) with fitting parameters {cm }. Detail of thisoptimization process is described later.
• Given the optimal parameters {c0m }, we replace L(i, j) in eq. (B.1) by′

L (i, j) = ∑m c0m Lm (i, j) to get clean images of atoms, that is,′S(i, j) − D (i, j)
I= F (i, j) = ′.
I0
L (i, j) − D (i, j)(B.2)

In short, we create the optimal light image by using the linear combinationof latest several tens of light images and replace the light image by an optimized light image.
Now let S0 (i, j) denotes a region in a shadow image where there is no
atoms. Similarly we take the same region for light images Lm (i, j) and letthese be L0m (i, j). Optimization of coefficients {cm } is done by the least square
method)2(
minimize∑i,j

S0 (i, j) − ∑ cm L0m (i, j)

def

= ∆.

m

∆ is quadratic in {cm } and this problem reduces to solve linear algebraic
equations as a consequence of conditions ∂∆/∂cn∑ Anm cm = Bn
m

132

(B.3)

where Anm and Bn are defined as
Anm = ∑ L0n (i, j) L0m (i, j),i,jBn = ∑ L0n (i, j)S0 (i, j).
i,jThe solutions {c0m } of eq. (B.3) is rigorously obtained by Gauss’s elimination
method and is numerically done very easily. Then we can calculate the righthand side of eq. (B.2) to get the clean image.
Processed images of dual BEC with variable number of reference images(equivalent to light images) are shown in Fig. B.1. Number of reference
images Nre f are 1, 10, 20 and 25 from top to bottom and it can be seen thatthe more the number of reference images, the cleaner the processed images.

133Figure B.1: Processed images of a dual BEC with variable number of reference images. Number of reference images Nre f are 1, 10, 20 and 25 from the
top to the bottom.

134

Appendix C
DISCRETE FOURIER
TRANSFORMATIONIn the context of Chapter 5, the spatial modulation on the density profile of BECs are to be observed. However, it was difficult to observe thespatial pattern directly because the actually emerging spatial pattern was
not purely a sinusoidal. This is possibly due to the growth of patterns withother wavenumbers caused by inhomogeneity of spatial densities and thefluctuation of the bias magnetic field. Even if we have a BECs with uniform
density and magnetic field with sufficiently small fluctuation, observed pattern might contain waves with wavenumbers around k max .

Under this situation, it is good to Fourier transform the signals and obtain a spectrum of it. In this case, we have 88 pixels in the horizontal direction and in order to find some spectral components of a series of discretely
sampled data, most commonly used method is discrete Fourier transformation(DFT). In this appendix the basics of DFT and some important featuresof DFT, namely sampling theorem and aliasing are introduced.
135

C.1

Discrete Fourier Transformation

Fourier transformation is one of the most commonly used mathematicaltechnique in science and technology. Given some function f ( x ) ∈ C suchthat square of this function is integrated over x ∈ (−∞, +∞) to give a finite
value:∫ ∞−∞| f ( x )|2 dx < ∞.

Then the Fourier transformation F (k) of f ( x ) exists. These are related toeach other by
F (k) =

∫ ∞−∞1
f (x) =
2πf ( x )e−ikx dx,∫ ∞−∞F (k )eikx dk.

This is what we usually refer to as the Fourier transformation(and its inverse) of f ( x ). However, this is not applied straightforwardly to the realistic problems because we can never have continuous data series f ( x ) but
have always discrete or sampled ones. Therefore the data is expressed as
f n = f (n∆X ), where n is an integer. ∆X ≡ X/N is the spatial or temporal
length of the data X divided by N, the number of data taken in the region
with length X. Given this expression, we can substitute this into the continuous form of Fourier transformation and its inverse, and discretize themas
N −1

Fm = ∆X ∑ f n exp [−i · 2πmn/N ],
n =0
N −1

fn =

1
Fm exp [i · 2πmn/N ].
N m∑=0Here n and m are integers and smallest step in the k-domain is ∆k ≡ 1/X =1/N∆X. With these formulae, we can deal with the realistic case which has
sampled data points. Note that the smaller the number of data points N is,
the larger the step of sample in k-domain ∆k is. This means that too roughsampling results in the loss of information of fine spectral structure.
136

C.2

Sampling Theorem and Aliasing

In the previous section we derived the formula for DFT. Of course, there
emerges some problems accompanied to the discretization of data points.
One of the biggest problem naturally appears when we think of how small
structure we can extract with sampling rate ∆X = X/N. Naively speaking, the maximum valid k can be detected when 1/k is equal to 2∆X, thatis, if there is sinusoidal wave with wavenumber k is present, one data point
hits the maximal and then nearest neighbor data point hits the minimal, succeedingly the next hits the maximal. This idea is supported by the famous”sampling theorem” which states that

”If the amplitude of Fourier component | F (k )| is always zerofor |k| < k c , signal f ( x ) is completely reproduced by sampling
rate x = n/2k c where n is an integer and k c is called Nyquist
frequency.”

In actual case we have a data set acquired with some sampling rate 1/k0 and
we can extract the Fourier components in the region |k| < k0 /2 as the valid
ones.
Then what will happen when we calculate the Fourier component in theregion |k | > k0 /2? The answer is in short expressed as the aliasing or folding
noise. Consider the case that we have five data points as in Fig. C.1(black
opened circle). For these points, sinusoidal wave with frequency 0.2 Hz(bluedash-dotted line) fits very well. However, one with 0.8 Hz(orange dashedline) also does quite nicely. Remind that in this case Nyquist frequency is
k c = 0.5 Hz, and frequencies 0.2 Hz and 0.8 Hz correspond to the region|k| < k c and |k| > k c , respectively. In terms of Fourier spectrum it looks
as if the spectrum in 0 < k < k c folds back at k = 0.5 to form spectrum ink c < k < 2k c . Therefore this phenomenon is called folding noise or aliasing.
137

Figure C.1: Data points(black opened circle) and sinusoidal wave fitting(blue and orange lines) to them.

C.3

Analysis of Experimentally Obtained Data

Actually we have a series of 88 signals in the horizontal direction within
a single image of atoms. As in Chapter 5, our analysis consists of three processes for obtaining Fourier spectrum we deal with. First we must integrate
the signals of 2D image in vertical direction to get horizontal profile. Second, horizontal profile of Rb is subtracted from that of K for enhancing theanti-correlation of horizontal profile. Finally we Fourier transform this subtracted signal to analyze the Fourier spectrum of anti-correlation. Outlineof this whole process is indicated in Fig. C.2, with actually obtained experimental images.
In Fig. C.2 (d) we plot the power spectral density as a Fourier spectrum.
When we Fourier transform some signal and obtain its Fourier component
F (k) ∈ C, we can always rewrite this as its modulus | F (k )| multiplied by
its argument ∠ F (k ): F (k ) = | F (k )|∠ F (k ). Squared modulus | F (k)|2 is called
energy spectral density. Power spectral density is defined as][
1
2| F (k)| ,P(k ) = limT →∞T
which we calculate this simply by P(k ) = F (k)/T for our analysis.

138

Figure C.2: Data processing of images to obtain Fourier spectrum.

139

Appendix D
MEASUREMENT OF
THREE-BODY LOSS
COEFFICIENTIn atomic physics experiment, even if we trap atoms they run away fromthe trap for some reasons. The reasons are classified by number of atoms
involved. One-body loss refers to the process involving one atom and its
interaction with surroundings. For example, in an optical trap the photon
scattering by laser for ODT can push atoms away from the trap. Or in themagnetic trap, spin flip results in the transfer of an atom to the high-fieldseeking state and thus atoms run away from the trap. Usually for experiments in an optical trap, the photon scattering lifetime is more than 1 second.
Two-body loss refers to inelastic collisions like spin exchange collisions
and the light-assisted collisions. Spin exchange collision takes place when,for instance, atoms in states | F = 1, m F = 1 > and | F = 1, m F = −1 >collides and then become two | F = 1, m F = 0 > atoms to get excess energy ∆E > 0. If this excess energy is negative, spin exchange collision rate
is forbidden at low temperature. Our atomic gas mixture, K and Rb, areboth prepared in | F = 1, m F = 1 > state so that this process is inhibited.
141

Another one, light-assisted collision is usually considered in the context of
MOT. When the near-resonant light is applied to atoms, collision between
ground state atom and excited state atom causes two types of inelastic collisions, namely fine-structure-changing collision and radiative escape. These
two are altogether called light-assisted collision. In an optical trap, nearresonant light is not applied in our experiment and we do not consider thisprocess here.
Then we arrive at the point to consider the three-body loss, what is also
called three-body recombination. Note that there can be no inelastic collision in two-body collision when the spins and quantum states are properly
prepared. However even in this situation, three-body collision is allowed to
be an inelastic collision by forming a diatomic molecule and obtaining thebinding energy as excess energy. This is always present in BEC experimentand the three-body recombination rate has been intensively studied as a candidate of tool to reveal few-body physics since the pioneering experiment ofRef.[86]. Recently three-body recombination rate of heteronuclear case is ofinterest and measured for 40 K and 87 Rb mixture[87]. This contributes to theunderstanding of universal three-body physics including Efimov effect[88],which is well investigated for homonuclear three-body collision[95].
Of course it is needed to let BECs of K and Rb overlap in our experiment and the three-body loss is always concerned. This is simply because
three-body loss rate is proportional to the spatial density cubed and spatial
density of BEC is typically an order of magnitude larger than that of thermalgas. Here in this Appendix we will take some pages for describing the measurements of three-body loss rate and briefly refer to on-going experimentin our lab.
Every time we measure the loss rate, whose origin depends on the situation, all we can do is to shed light on atomic cloud and count the number ofatoms at various time. As is usual we prepare thermal mixture of temperatures ∼ 200 nK and densities nK = 5 × 1012 /cm3 and n Rb = 6 × 1012 /cm3in 809 nm optical trap and hold it for a time t, then release them from trap142

and shine resonant light to measure the number of atoms. For extract the
information of three-body loss rate, we consider rate equations as follows:dnKn= −2LKKRb n2K n Rb − LKRbRb nK n2Rb − K
dt
τK
dn Rb
n
= − LKKRb n2K n Rb − 2LKRbRb nK n2Rb − Rb
dt
τRb

(D.1)
(D.2)

Here LKKRb and LKRbRb are three-body coefficients which have unit [cm6 /s]and τK and τRb are lifetime in optical dipole trap. As a whole, each of theserate equations contains two kinds of three-body loss channel and one-bodyloss due to the photon scattering by dipole trap laser. Factor 2 reflects thefact that for the loss channel which two K and one Rb are involved reduces
the number density of K two times faster than that which two Rb and one K
are involved.
The obtained decay curve of number densities calculated from measured
numbers of atoms after variable hold time is shown in Fig. D.1. Blue ( Red) circle indicates the measured density of K ( Rb ) and Blue ( Red ) solidline shows the simulated result of equations ( D.1 ) and ( D.2 ) with initial
densities nK = 5 × 1012 /cm3 and n Rb = 6 × 1012 /cm3 , and three-bodyloss coefficients LKKRb = 7 × 10−27 cm6 /s and LKKRb = 0 cm6 /s. Photon
scattering lifetimes are set to be τK = 24 s and τRb = 130 s. From the simulation with various parameters we concluded that LKKRb is less than 1 × 10−27
cm6 /s.
From this, our concern is now the LKKRb = 7 × 10−27 cm6 /s. Considering the density of BEC cloud n ≃ 3 × 1013 /cm3 , the loss rate is roughly
estimated by simply calculating LKKRb n2 ∼ 10 Hz, that gives the timescale
100 ms at this interspecies scattering length. This is not said to be sufficiently
long for various experiment. On the other hand we have a tool to observe
atomic clouds in-situ. By using this we observed that dual BEC survives
even if we hold it for 800 ms in 809 nm optical trap with interspecies scattering length aKRb = 0.7 a B (see Fig. D.2). This is very long, as is deducedby assuming the tendency of three-body recombination coefficient, that is,
LKKRb scales as a4KRb [77, 78].
143

Besides, in-situ observation of atoms by phase contrast imaging can extract the three-body loss coefficient with a single experimental shot. This is
strikingly efficient compared with the same experiment with TOF measurement. Furthermore, for destructive measurement we should deal with the
shot-to-shot fluctuation of number of atoms due to temperature excursion
of experimental apparatus. In-situ measurement can also neglect this fluctuation. Now some members of our group focuses on this and are measuring
the three-body coefficient of 41 K-87 Rb mixture. This is an on-going experiment.

144

Figure D.1: Measurement of three-body loss coefficient in thermal mixtureof K and Rb. Blue ( Red ) circle indicates the measured density of K ( Rb )and Blue ( Red ) solid line shows the simulated result of equations ( D.1 ) and( D.2 ) with initial densities nK = 5 × 1012 /cm3 and n Rb = 6 × 1012 /cm3 ,and three-body loss coefficients LKKRb = 7 × 10−27 cm6 /s and LKKRb = 0
cm6 /s.

145

Figure D.2: In-situ observation of dual BEC with time interval 200 ms. Dual
BEC appears to be survives even at 800 ms.

146Appendix E
NUMERICAL SOLUTION OF
COUPLED GP EQUATIONSIn the Fig. 2.2 in Chapter 2, we showed numerical solutions of coupled
Gross-Pitaevskii equations for interspecies scattering lengths aKRb = 50 a B
and aKRb = 200 a B . Actually what we do for obtaining these is to numerically find solutions which minimizes the energy functional in Thomas-Fermi
approximation
∫

(

E=
∫

=

(

[

]
gi
2
4
∑ Vi (x)|ψi | + 2 |ψi | + g12 |ψ1 |2 |ψ2 |2
i =1,2)dx)
gi 2 ]∑ Vi (x)ni (x) + 2 ni (x) + g12 n1 (x)n2 (x) dx,
i =1,2[where ni ( x ) (i = 1, 2) denote the spatial densities of two components andhere we limit the problem in 1-dimension. Let us further rewrite the integral
in the following way:)(∫[gi 2 ]∑ Vi (x)ni (x) + 2 ni (x) + g12 n1 (x)n2 (x) dx
i =1,2
[
(
]
)
∫
2
m
m
ω
1
a
a
/a
a
/a
p
i i 2
B 2
B
= 2πh̄2 B
x ni ( x ) + i
ni ( x ) + 12
n1 ( x )n2 ( x ) dx
2 a
m p i=∑
2
m
/m
m
/m
p
p
B
12
i
1,2 2πh̄
(
)
[
]
∫
∗
∗
a
a
a
= 2πh̄2 B × 10−40∑ ηi x2 ni (x) + mi∗ n2i (x) + m12∗ n1 (x)n2 (x) dx.
mp12i
i =1,2147

∗ = a /a are scattering lengths in the unit of a andHere ai∗ = ai /a B and a12
BB12∗ = m /m are scaled by the proton mass m .
the mass mi∗ = mi /m p and m12p
p
12

The factor ηi is written explicitly as follows:
ηi =m p mi ωi2
× 10−32 .
2πh̄2 a B 2
1

In this form a factor 10−32 is multiplied because in the third line in the above
formula x has a unit of µm and ni ( x ) have units of 1020 ×1/m3 . Assuming that two components 1 and 2 feels the same confining potential, that is,def

m1 ω12 /2 = m2 ω22 /2, we have η1 = η2 = η. Thus the integral which should
be minimized becomes, with parameters of Rb and K,∫ [ 2x n Rb ( x )10]2a∗KRb100 2x2 nK ( x ) 60 2+
n (x) ++ nK ( x ) +n ( x )nK ( x ) dx.
87 Rb
10
41
55.7 Rb

We used η ∼ 1/10 which can be obtained with a trapping frequency of ∼ 80
Hz. At last, we discretize this with respect to x and let Wolfram Mathematica
find the series of n Rb and nK . Here is the code:
Clear[f, c, d];aKRb = 50;
f = Sum[i*i*c[i]/10 + 100/87*(c[i])^2+ i*i*d[i]/10 + 60/41*(d[i])^2+ 2*aKRb/55.7*c[i]*d[i], {i, 0, 16}];cons = {Sum[c[i], {i, 0, 16}]/16 == 1,
Sum[d[i], {i, 0, 16}]/16 == 1,
Table[c[i] >= 0, {i, 0, 16}],
Table[d[i] >= 0, {i, 0, 16}]};vars = Join[Table[c[i], {i, 0, 16}], Table[d[i], {i, 0, 16}]];148

Sol = NMinimize[{f, cons}, vars,
Method -> {"SimulatedAnnealing", "PerturbationScale" -> 10 }]The obtained results for various interspecies scattering lengths are listed
in Fig. E.1 and the phase separation in the region aKRb = 75 a B can be clearlyobserved.

149

Figure E.1: Numerically calculated Thomas-Fermi profiles of dual BECs
with various interactions. The interspecies scattering lengths are set to be(a) aKRb = 0 a B , (b) 25 a B , (c) 50 a B , (d) 60 a B , (e) 65 a B , (f) 70 a B , (g) 80 a B and(h) 200 a B . A dual BEC becomes immiscible when aKRb > 75 a B .

150

Appendix F
C++ PROGRAM FOR
ANALYZING THE IMAGE
Here we put the whole C++ program for analyzing the image obtainedin a sequence, including the fringe cleaning, zooming up the atomic cloudand displaying the radially integrated, axial density profile of Rb and K in asame graph.

#include<iostream>#include<ctime>#include<time.h>#include<cstdlib>#include<string>#include<iomanip>#include<math.h>#include<typeinfo>#include<fstream>#include<stdio.h>#include<sstream>#include<cstdio>#define PGNUPLOT "C:\Program Files (x86)\gnuplot\bin\pgnuplot.exe"
using namespace std;151

string IntToString(int number){stringstream ss;
ss << number;
return ss.str();}

int main(){/*-------------------- Input file informations -------------------- */int i,j;int row = 1047;
int column = 511;
int n0 = 50; //NUMBER OF REFERENCES USED
int xc = 268; //X POSITION OF ATOMS
int r = 20; //MASKED REGION(HALF)
double e=2.71828182846;
double pi=3.1415926535898;/*--------------------*//* region of interest *//*--------------------*/
int x1 = 254;
int x2 = 279;
int y1 = 5;
int y2 = 97;
int d = 104;/*------------------- FILE NAME (SHADOW) ----------------------*/
string date="20131205";
string seqnum;
cout << "Input the sequence number(ex. 1, 17, 138, ...):";
int seqnumber;152

cin >> seqnumber;
if(seqnumber > 99)
seqnum = IntToString(seqnumber);else if( seqnumber <100 && seqnumber >9 )seqnum = "0" + IntToString(seqnumber);elseseqnum = "00" + IntToString(seqnumber);string sfile;
sfile = "C:\\Data\\aref\\shadow.txt";
cout << "Input file is ’" << sfile <<"’\n";clock_t start, end;
start = clock();/*---------------- LOAD SHADOW IMAGE -----------------------*/
int **shadow = new int*[row+1];
for(i = 1; i < row + 1; i++){shadow[i] = new int[column+1];}
for(i=1; i<row+1; i++){for(j=1; j<column+1; j++){shadow[i][j] = 0;}}

FILE* fp;
fp = fopen(sfile.c_str(), "r");if(fp==NULL) {printf("Cannot open the file. \n");return 1;}

153

for(j=1; j<column + 1; j++){for(i=1; i<row + 1; i++){fscanf(fp, "%d",&shadow[i][j] );}}/*---- LIMITING THE REGION OF IMAGE AND MASKING THE ATOMS -----*/
int xlim = 125;
int **s1 = new int*[row+1];
for(i = 1; i < row + 1; i++){s1[i] = new int[column-2*xlim-10+1];}//left halffor(i=1; i<row+1; i++){for(j=1; j<xc-r-xlim+1; j++){s1[i][j] = 0;s1[i][j] += shadow[i][j+xlim];}}//right halffor(i=1; i<row+1; i++){for(j=1; j<column-xlim-xc-(r-1)+1; j++){s1[i][xc-r-xlim+j] = 0;s1[i][xc-r-xlim+j] += shadow[i][xc+(r-1)+j];}}/*-------------------SPLIT INTO 10 IMAGES---------------------*/
int h;
int ***s2 = new int**[10+1];
for(h=1; h<10+1; h++){s2[h] =

new int*[d+1];154

for(i = 1; i < d + 1; i++){s2[h][i] = new int[column-2*xlim-10+1];}}
for(h=1; h<10+1; h++){for(i=1; i<y2-y1+1; i++){for(j=1; j<column-2*xlim-(2*r-1)+1; j++){s2[h][i][j] = s1[y1+i+(h-1)*d][j];}}}cout << "Shadow image for calculation is created. \n";/*------------- LOAD LIGHT IMAGES -----------------------*/
int n;
int **light = new int*[row+1];
for(i = 1; i < row + 1; i++){light[i] = new int[column+1];}
for(i=1; i<row+1; i++){for(j=1; j<column+1; j++){light[i][j] = 0;}}
int ***l1 = new int**[n0+1];
for(n=1; n<n0+1; n++){l1[n] =

new int*[row+1];

for(i = 1; i < row + 1; i++){l1[n][i] = new int[column-2*xlim-10+1];
}
}

155

int ****l2 = new int***[n0+1];
for(n=1; n<n0+1; n++){l2[n] =

new int**[10+1];

for(h = 1; h < 10 + 1; h++){l2[n][h] = new int*[d+1];
for(i = 1; i < d + 1; i++){l2[n][h][i] = new int[column-2*xlim-10+1];}}}
for(n=1; n<n0+1; n++){for(h=1; h<10+1; h++){for(i=1; i<y2-y1+1; i++){for(j=1; j<column-2*xlim-(2*r-1)+1; j++){l2[n][h][i][j] =0;}}}}

for(n=1; n<n0+1; n++){/*----------------- FILE NAME (LIGHT) ------------------------*/
string lfile;//cout << seqnumber <<"\n";int filenum = seqnumber-n+2;
string strn = IntToString(filenum);
lfile = "C:\\Data\\aref\\" + strn;
lfile += ".txt";//cout << " Input file is ’" << lfile <<"’\n";FILE* fp;
fp = fopen(lfile.c_str(), "r");156

if(fp==NULL) {printf("Cannot open the file. \n");return 1;}

for(j=1; j<column + 1; j++){for(i=1; i<row + 1; i++){fscanf(fp, "%d",&light[i][j] );}}/*--------- LIMITING THE REGION OF IMAGE(LIGHT) ------------*///left halffor(i=1; i<row+1; i++){for(j=1; j<xc-r-xlim+1; j++){l1[n][i][j] = 0;l1[n][i][j] = light[i][j+xlim];}}//right halffor(i=1; i<row+1; i++){for(j=1; j<column-xlim-xc-(r-1)+1; j++){l1[n][i][xc-r-xlim+j] = 0;l1[n][i][xc-r-xlim+j] = light[i][xc+(r-1)+j];}}for(h=1; h<10+1; h++){for(i=1; i<y2-y1+1; i++){for(j=1; j<column-2*xlim-(2*r-1)+1; j++){l2[n][h][i][j] = l1[n][y1+i+(h-1)*d][j];
}

157

}
}
} //n loop

cout << "Reference images for calculation are created.\n";//By now, elements for making coefficient matrix and vector are obtained.

/*---------- MAKING THE COEFFICIENT MATRIX -------------------*/
int m;
double ***A = new double**[10+1];
for(h=1; h<10+1; h++){A[h] =

new double*[n0+1];

for(n = 1; n < n0 + 1; n++){A[h][n] = new double[n0+1];}}

for(h=1; h<10+1; h++){for(n=1; n<n0+1; n++){for(m=1; m<n0+1; m++){A[h][n][m] = 0;
for(i=1; i<y2-y1+1; i++){for(j=1; j<column-2*xlim-(2*r-1)+1; j++){A[h][n][m] += l2[n][h][i][j]*l2[m][h][i][j];
}
}
}
}
}

double **B = new double*[10+1];
for(h = 1; h < 10 + 1; h++){158

B[h] = new double[n0+1];}

for(h=1; h<10+1; h++){for(n=1; n<n0+1; n++){B[h][n] = 0;
for(i=1; i<y2-y1+1; i++){for(j=1; j<column-2*xlim-(2*r-1)+1; j++){B[h][n] += l2[n][h][i][j]*s2[h][i][j];}}}}/*-----------------------GAUSS’ METHOD---------------------------*/
int k;
double c;for(h=1; h<10+1; h++){/*------ FORWARD ELIMINATION -------*/

for(i=1;i<n0+1;i++){for (j=i+1; j<n0+1; j++) {c = A[h][j][i] / A[h][i][i];for (k=i; k<n0+1; k++)A[h][j][k] -= c * A[h][i][k];B[h][j] -= c * B[h][i];}}/*---- BACKWARD SUBSTITUTION --------*/

B[h][n0] = B[h][n0]/A[h][n0][n0];159for (i=n0-1; i > 0; i--) {for (j=i+1; j<n0+1; j++)B[h][i] -= A[h][i][j] * B[h][j];B[h][i] /= A[h][i][i];}}for(h=1; h<10+1; h++){cout << "B[" << h << "]=";
for(n=1; n<n0+1; n++){cout << B[h][n] << " ";}cout << "\n";}/*------ MAKING AN OPTIMIZED REFERENCE --------*/

int **lighttemp = new int*[row+1];
for(i = 1; i < row + 1; i++){lighttemp[i] = new int[column+1];}
for(i=1; i<row+1; i++){for(j=1; j<column+1; j++){lighttemp[i][j] = 0;}}

double ***L = new double**[10+1];
for(h=1; h<10+1; h++){L[h] = new double*[d+1];
for(i = 1; i < d + 1; i++){L[h][i] = new double[column+1];
}

160

}
for(h=1; h<10+1; h++){for(i=1; i<d+1; i++){for(j=1; j<column+1; j++){
L[h][i][j] =0;}}}

for(n=1; n<n0+1; n++){/*----------- LOADING LIGHT IMAGE AGAIN ---------------*/
string Lfile;//cout << seqnumber <<"\n";int filenum = seqnumber - n+2;
string strn = IntToString(filenum);
Lfile = "C:\\Data\\aref\\" + strn;
Lfile += ".txt";

FILE* fp;
fp = fopen(Lfile.c_str(), "r");if(fp==NULL) {printf("Cannot open the file. \n");return 1;}

for(j=1; j<column + 1; j++){for(i=1; i<row + 1; i++){fscanf(fp, "%d",&lighttemp[i][j] );}}for(h=1; h<10+1; h++){161for(i=1; i<y2-y1+1; i++){for(j=1; j<column+1; j++){
L[h][i][j] += B[h][n]*lighttemp[y1+i+(h-1)*d][j];}}}} // n loop/*------------------------ FILE NAME (DARK) -------------------------*/
string dfile;
dfile = "C:\\Data\\aref\\dark.txt";//cout << "Input file is ’" << dfile <<"’\n";/*------------ LOAD DARK IMAGE ---------------------------*/
int **dark = new int*[row+1];
for(i = 1; i < row + 1; i++){dark[i] = new int[column+1];}
for(i=1; i<row+1; i++){for(j=1; j<column+1; j++){dark[i][j] = 0;}}

FILE* f2p;
f2p = fopen(dfile.c_str(), "r");if(f2p==NULL) {printf("Cannot open the file. \n");return 1;}162

for(j=1; j<column + 1; j++){for(i=1; i<row + 1; i++){fscanf(f2p, "%d",&dark[i][j] );}}/*----------------SPLIT DARK IMAGE INTO 10 IMAGES--------------------*/

double ***D = new double**[10+1];
for(h=1; h<10+1; h++){D[h] = new double*[d+1];
for(i = 1; i < d + 1; i++){D[h][i] = new double[column+1];}}
for(h=1; h<10+1; h++){for(i=1; i<d+1; i++){for(j=1; j<column+1; j++){D[h][i][j] =0;
D[h][i][j] = dark[y1+i+(h-1)*d][j];}}}double ***S = new double**[10+1];
for(h=1; h<10+1; h++){S[h] = new double*[d+1];
for(i = 1; i < d + 1; i++){S[h][i] = new double[column+1];}}
for(h=1; h<10+1; h++){for(i=1; i<d+1; i++){163

for(j=1; j<column+1; j++){S[h][i][j] =0;
S[h][i][j] = shadow[y1+i+(h-1)*d][j];}}}/*--------------- CALCULATION OF PHASE SHIFT ----------------*/
double ***phi = new double**[10+1];
for(h = 1; h < 10 + 1; h++){phi[h] = new double*[d+1];
for(i=1; i<d+1; i++){phi[h][i] = new double[column+1];}}

for(h = 1; h < 10 + 1; h++){for(i=1; i<d+1; i++){for(j=1; j<column+1; j++){phi[h][i][j] = 0;}}}
for(h = 1; h < 10 + 1; h++){for(i=1; i<d + 1; i++){for(j=1; j<column + 1; j++){phi[h][i][j] =acos((3-((S[h][i][j]-D[h][i][j])/(L[h][i][j]-D[h][i][j])))/(2*sqrt(2))) - pi/4;}}}

cout << "Calculation of phase shift is done.\n";164

for(h=1; h<10+1; h++){string image2;
string strn1 = IntToString(h);image2 = "C:\\C++\\data\\images\\" + date + "_"+ seqnum + "_" + "phaseshift_" + strn1 + ".txt";
ofstream image2out(image2.c_str());

if(!image2out){
cout << "cannot open the output file. \n";return 1;}

for (i =1; i < d-16+1; i++) {for( j = 1; j < column+1; j++){image2out << -1000*phi[h][i+4][j] << " " ;}image2out << "\n";}image2out.close();}/*---------------------SAVE --------------------------*/

double **a = new double*[row+1];
for(i = 1; i < row + 1; i++){a[i] = new double[column+1];}
for(i=1; i<row+1; i++){for(j=1; j<column+1; j++){a[i][j] = 0;
}

165

}

for(h=1; h<11; h++){if(h<10){for(i=1; i<y2-y1+1; i++){for(j=1; j<x2-x1+1; j++){a[i][j] = -1000*phi[h][i][j+x1];
}
}
}
else{
for(i=1; i<y2-y1+1; i++){for(j=1; j<x2-x1+1; j++){a[i][j] = -1000*phi[h][i+11][j+x1];}}}

string image1;
string strn1 = IntToString(h);image1 = "C:\\C++\\data\\temp\\image1_" + strn1 + ".dat";
ofstream image1out(image1.c_str());

if(!image1out){
cout << "cannot open the output file. \n";return 1;}

for (i =1; i < x2-x1+1; i++) {for( j = 1; j < y2-y1+1; j++){image1out << a[j][i] << " " ;}image1out << "\n";166}
image1out.close();string image;
image = "C:\\C++\\data\\images\\" + date+ "_" + seqnum + "_" + strn1 + ".dat";
ofstream imageout(image.c_str());

if(!imageout){
cout << "cannot open the output file. \n";return 1;}

for (i =1; i < x2-x1+1; i++) {for( j = 1; j < y2-y1+1; j++){imageout << a[j][i] << " " ;}imageout << "\n";}imageout.close();cout<< "’" << image << "’ is generated.\n";/*--------------- MAKING SUMMED SIGNALS ----------------*/

double **hor = new double*[10+1]; //HORIZONTAL PROFILEfor(n=1; n<10+1; n++){hor[n] = new double[y2-y1+1];}
double **ver = new double*[10+1]; //VERTICAL PROFILE

167

for(n=1; n<10+1; n++){ver[n] = new double[x2-x1+1];}for(i=1; i<y2-y1+1; i++){hor[h][i] = 0;}
for(i=1; i<y2-y1+1; i++){for(j=1; j<x2-x1+1; j++){hor[h][i] += a[i][j];}}for(i=1; i<x2-x1+1; i++){ver[h][i] = 0;
for(j=1; j<y2-y1+1; j++){ver[h][i] += a[j][i];}}string hplot;
hplot = "C:\\C++\\data\\temp\\hplot_" + strn1 + ".dat";
ofstream hout(hplot.c_str());

if(!hout){
cout << "cannot open the output file. \n";return 1;}

for (i =1; i < y2-y1+1; i++) {hout << i << " " << hor[h][i] << "\n";}hout.close();

168

/*
string vplot;
vplot = "C:\\C++\\data\\temp\\vplot_" + strn1 + ".dat";
ofstream vout(vplot.c_str());

if(!vout){
cout << "cannot open the output file. \n";return 1;}

for (i =1; i < x2-x1+1; i++) {vout << i << " " << ver[h][i] << "\n";}vout.close();for(n=1; n>10+1; n++){delete[] hor[n];}delete[] hor;for(n=1; n>10+1; n++){delete[] ver[n];}delete[] ver;*/} //n loop/*--------------------- IMAGE PLOTS -----------------------------*/FILE *Fp;
Fp = _popen("pgnuplot -persist", "w");fprintf(Fp, "set format y ’’\n");fprintf(Fp, "cd ’C:\\C++\\data\\temp’\n");fprintf(Fp, "set terminal png fontscale 3 size 1280,3000\n");169

fprintf(Fp, "set output ’C:\\C++\\data\\temp\\image.png’\n");fprintf(Fp, "unset contour\n");fprintf(Fp, "unset key\n");fprintf(Fp, "unset surface\n");fprintf(Fp, "set view map\n");fprintf(Fp, "set pm3d at b\n");fprintf(Fp, "set palette rgbformulae 22,13,-31\n");fprintf(Fp, "set multiplot layout 10,1\n");fprintf(Fp, "set tmargin 2\n");fprintf(Fp, "set bmargin 0\n");fprintf(Fp, "unset xlabel\n");fprintf(Fp, "set format x ’’\n");fprintf(Fp, "set ylabel ’’\n");

fprintf(Fp, "set xlabel ’’\n");

fprintf(Fp, "set cbr [0:200]\n");fprintf(Fp, "set cbr tics 0, 400\n");fprintf(Fp, "splot ’image1_1.dat’ matrix with pm3d\n");fprintf(Fp, "set tmargin 0\n");fprintf(Fp, "splot ’image1_2.dat’ matrix with pm3d\n");fprintf(Fp, "set tmargin 0\n");fprintf(Fp, "splot ’image1_3.dat’ matrix with pm3d\n");fprintf(Fp, "set tmargin 0\n");fprintf(Fp, "splot ’image1_4.dat’ matrix with pm3d\n");fprintf(Fp, "splot ’image1_5.dat’ matrix with pm3d\n");fprintf(Fp, "set tmargin 0\n");fprintf(Fp, "splot ’image1_6.dat’ matrix with pm3d\n");fprintf(Fp, "unset ylabel\n");fprintf(Fp, "set ylabel ’’\n");

fprintf(Fp, "splot ’image1_7.dat’ matrix with pm3d\n");fprintf(Fp, "set tmargin 0\n");fprintf(Fp, "splot ’image1_8.dat’ matrix with pm3d\n");fprintf(Fp, "splot ’image1_9.dat’ matrix with pm3d\n");fprintf(Fp, "set tmargin 0\n");170

fprintf(Fp, "splot ’image1_10.dat’ matrix with pm3d\n");fprintf(Fp, "unset multiplot\n");fprintf(Fp, "set out\n");fprintf(Fp, "set term pop\n");fprintf(Fp, "exit\n");fflush(Fp);_pclose(Fp);/*--------------- HORIZONTAL PLOTS ----------------*/

Fp = _popen("pgnuplot -persist", "w");fprintf(Fp, "set yrange [0:1500]\n");fprintf(Fp, "set ytics 500, 500, 1000\n");fprintf(Fp, "cd ’C:\\C++\\data\\temp’\n");fprintf(Fp, "set terminal png fontscale 3 size 1280,3000\n");fprintf(Fp, "set output ’C:\\C++\\data\\temp\\plot.png’\n");fprintf(Fp, "set multiplot layout 5,1\n");fprintf(Fp, "set tmargin 2\n");fprintf(Fp, "set bmargin 0\n");fprintf(Fp, "unset xlabel\n");fprintf(Fp, "set format x ’’\n");fprintf(Fp, "set ylabel ’’\n");

fprintf(Fp, "set xlabel ’’\n");

fprintf(Fp, "plot ’hplot_1.dat’ title ’K’

with lines linewidth 5,’hplot_2.dat’ title ’Rb’ with lines linewidth 5\n");fprintf(Fp, "set tmargin 0\n");fprintf(Fp, "plot ’hplot_3.dat’ title ’K’

with lines linewidth 5,’hplot_4.dat’ title ’Rb’ with lines linewidth 5\n");fprintf(Fp, "set ylabel ’Summed signal [arb. u.]’\n");fprintf(Fp, "plot ’hplot_5.dat’ title ’K’

with lines linewidth 5,’hplot_6.dat’ title ’Rb’ with lines linewidth 5\n");fprintf(Fp, "unset ylabel\n");fprintf(Fp, "set ylabel ’’\n");

171

fprintf(Fp, "plot ’hplot_7.dat’ title ’K’

with lines linewidth 5,’hplot_8.dat’ title ’Rb’ with lines linewidth 5\n");fprintf(Fp, "set xlabel ’Horizontal position [pix]’\n");fprintf(Fp, "set format x\n");fprintf(Fp, "set bmargin 3\n");fprintf(Fp, "plot ’hplot_9.dat’ title ’K’

with lines linewidth 5,’hplot_10.dat’ title ’Rb’ with lines linewidth 5\n");fprintf(Fp, "unset multiplot\n");fprintf(Fp, "set out\n");fprintf(Fp, "set term pop\n");fprintf(Fp, "exit\n");fflush(Fp);_pclose(Fp);/*--------------------------------------------------------------------------*//*-------------------------- SERIES OF DELETES -----------------------------*//*--------------------------------------------------------------------------*/
for(i = 1; i < row + 1; i++){delete[] a[i];}delete[] a;for(h=1; h<10+1; h++){for(i = 1; i < d + 1; i++){delete[] phi[h][i];}}delete[] phi;for(i = 1; i < row + 1; i++){delete[] dark[i];}delete[] dark;for(h=1; h<10+1; h++){for(i = 1; i < d + 1; i++){172

delete[] L[h][i];}}delete[] L;for(i = 1; i < row + 1; i++){delete[] lighttemp[i];}delete[] lighttemp;
for(n=1; n<10+1; n++){delete[] B[n];}delete[] B;for(h = 1; h < 10 + 1; h++){for(i=1; i<n0+1; i++){delete[] A[h][i];}delete[] A[h];}delete[] A;for ( n = 1; n < n0+1; n++) {for ( h = 1; h < 10+1; h++) {for(i=1; i<d+1; i++){delete[] l2[n][h][i];}delete[] l2[n][h];}delete[] l2[n];}delete[] l2;for ( i = 1; i < n0+1; i++) {for ( j = 1; j < row+1; j++) {delete[] l1[i][j];}173delete[] l1[i];}delete[] l1;for (int h = 1; h < 10+1; h++) {for (int i = 1; i < d+1; i++) {delete[] s2[h][i];}delete[] s2[h];}delete[] s2;for (int h = 1; h < 10+1; h++) {for (int i = 1; i < d+1; i++) {delete[] D[h][i];}delete[] D[h];}delete[] D;for (int h = 1; h < 10+1; h++) {for (int i = 1; i < d+1; i++) {delete[] S[h][i];}delete[] S[h];}delete[] S;for(i = 1; i < row + 1; i++){delete[] s1[i];}delete[] s1;
for(i = 1; i < row + 1; i++){delete[] light[i];}delete[] light;174

for(i = 1; i < row + 1; i++){delete[] shadow[i];}delete[] shadow;

end = clock();cout << "\n It took " << (end-start)/CLOCKS_PER_SEC << " seconds.\n";}175Bibliography[1] K. Kasamatsu and M. Tsubota.

”Modulation instability and solitary-wave formation in two-component Bose-Einstein condensates”

Phys.

Rev. A 74, 013617 (2006).
[2] A. L. Schawlow and C. H. Townes.

”Infrared and Optical Masers”

Phys. Rev. 112, 1940-1949 (1958).
[3] F. Riehle, Th. Kisters, A. Witte, Ch. J. Bord and J. Helmcke .

”OpticalRamsey spectroscopy in a rotating frame: Sagnac effect in a matterwave interferometer”

Phys. Rev. Lett. 67, 177-180 (1991).

[4] David S. Weiss, Brenton C. Young, and Steven Chu .

”Precision mea-

surement of the photon recoil of an atom using atomic interferometry”
Phys. Rev. Lett. 70, 2706-2709 (1993).
[5] Achim Peters , Keng Yeow Chung and Steven Chu .
of gravitational acceleration by dropping atoms.””Measurement

Nature. 400, 849-852(1999).
[6] John V.Prodan, William D.Phillips, and Harold Metcalf.

”Laser Pro-

duction of a Very Slow Monoenergetic Atomic Beam”

Phys. Rev. Lett.

49, 1149 (1982).
[7] Steven Chu, L.Jollberg, J.E.Bjorkholm, Alex Cable, and A.Ashkin.
”Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure”

Phys. Rev. Lett. 55, 48 (1985).
177

[8] E.L.Raab, M.Prentiss, Alex Cable, Steven Chu, and D.E.Pritchard.
”Trapping of Neutral Sodium Atoms with Radiation Pressure”

Phys.

Rev. Lett. 59, 2631 (1987).
[9] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E.
A. Cornell.

”Observation of Bose-Einstein Condensation in a Dilute

Atomic Vapor”

Science. 269, 198-201 (1995).

[10] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle.
Gas of Sodium Atoms””Bose-Einstein Condensation in a

Phys. Rev. Lett. 75, 3969-3973 (1995).

[11] Available online at http://www.colorado.edu/physics/2000/bec/what_it_loo
ks_like.html[12] S. Gupta, K. Dieckmann, Z. Hadzibabic, and D. E. Pritchard .

”Con-

trast Interferometry using Bose-Einstein Condensates to Measure h/m
and α.”

Phys. Rev. Lett. 89, 140401 (2002).

[13] Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E.
Leanhardt.

”Atom Interferometry with Bose-Einstein Condensates ina Double-Well Potential.”

Phys. Rev. Lett. 92, 050405 (2004).

[14] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. StamperKurn, and W. Ketterle .
Bose-Einstein condensate””Observation of Feshbach resonances in a
Nature. 392, 151-154 (1998).

[15] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle.

”Observa-

tion of Vortex Lattices in Bose-Einstein Condensates.”

Science. 292,

476-479 (2001).
[16] M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee, D.M. Kurn,and W. Ketterle.
densates.””Observation of interference between two Bose con-

Science. 275, 637-641 (1997).
178

[17] D.E. Miller, J.R. Anglin, J.R. Abo-Shaeer, K. Xu, J.K. Chin and W. Ketterle.

”High-Contrast Interference in a Thermal Cloud of Atoms.”

Phys. Rev. A 71, 043615 (2005).
[18] Belen Paredes , Artur Widera , Valentin Murg , Olaf Mandel , Simon
Fölling , Ignacio Cirac , Gora V. Shlyapnikov , Theodor W. Hänsch andImmanuel Bloch.
tical lattice.””Tonks-Girardeau gas of ultracold atoms in an op-

Nature. 429, 277-281 (2004).

[19] C.A.Regal, Markus Greiner, and D.S.Jin.

”Observation of resonancecondensation of fermionic atom pairs.”

Phys. Rev. Lett. 92, 040403(2004)[20] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hansch,
and Immanuel Bloch.

”Quantum phase transition from a superfluid

to a Mott insulator in a gas of ultracold atoms.”

Nature. 415, 39-44(2001)[21] J. Denschlag, J. E. Simsarian, D. L. Feder, Charles W. Clark, L. A. Collins,
J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S.
L. Rolston, B. I. Schneider, W. D. Phillips.

”Generating Solitons by

Phase Engineering of a Bose-Einstein Condensate.”

Science. 287, 97-

101 (2001)[22] Luca Tanzi, Eleonora Lucioni, Saptarishi Chaudhuri, Lorenzo Gori,
Avinash Kumar, Chiara D Errico, Massimo Inguscio, and Giovanni
Modugno.

”Transport of a Bose Gas in 1D Disordered Lattices at the

Fluid-Insulator Transition. ” Phys. Rev. Lett. 111, 115301 (2013)[23] Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William
Cody Burton, and Wolfgang Ketterle.

”Realizing the Harper Hamil-

tonian with Laser-Assisted Tunneling in Optical Lattices.”
Lett. 111, 185302 (2013)179

Phys. Rev.

[24] M. Aidelsburger, M. Atala, S. Nascimbe‘ne, S. Trotzky, Y.-A. Chen,
and I. Bloch.

”Experimental Realization of Strong Effective Magnetic

Fields in an Optical Lattice.”

Phys. Rev. Lett. 107, 255301 (2011)[25] Y.-J. Lin, R. L. Compton, K. Jimnez-Garca, J. V. Porto and I. B. Spielman.
”Synthetic magnetic fields for ultracold neutral atoms.”

Nature. 462,

628-632 (2009)[26] Markus Greiner, Cindy A. Regal, and Deborah S. Jin.

”Emergence ofa molecular Bose-Einstein condensate from a Fermi gas.”

Nature. 426,

537-540 (2003)[27] Satoshi Tojo, Taro Hayashi, Tatsuyoshi Tanabe, Takuya Hirano, Yuki
Kawaguchi, Hiroki Saito, and Masahito Ueda.

”Spin-dependent in-elastic collisions in spin-2 Bose-Einstein condensates.”

Phys. Rev. A.

80, 042704 (2009)[28] Satoshi Tojo, Yoshihisa Taguchi, Yuta Masuyama, Taro Hayashi, Hiroki Saito, and Takuya Hirano.

”Controlling phase separation of bi-nary Bose-Einstein condensates via mixed-spin-channel Feshbach resonance.”

Phys. Rev. A. 82, 033609 (2010)[29] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe er, B. Neyenhuis,
J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye.
Phase-Space-Density Gas of Polar Molecules.””A High

Science. 322, 231-235(2008)[30] Bo Yan, Steven A. Moses, Bryce Gadway, Jacob P. Covey, Kaden R.
A. Hazzard, Ana Maria Rey, Deborah S. Jin, Jun Ye.

”Observa-

tion of dipolar spin-exchange interactions with lattice-confined polar
molecules.”Nature. 501, 521-525 (2013)[31] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman.
”Production of Two Overlapping Bose-Einstein Condensates by Sympathetic Cooling.”

Nature. 396, 345-348 (1998)180

[32] J. Stenger, S. Inouye, D.M. Stamper-Kurn, H.-J. Miesner, A.P. Chikkatur,and W. Ketterle.
condensates.””Spin domains in ground state spinor Bose-Einstein
Phys. Rev. Lett. 78, 586-589 (1997)[33] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell.

”Mea-surements of Relative Phase in Binary Mixtures of Bose-Einstein Condensates.”

Phys. Rev. Lett. 81, 1543 (1998)[34] J. F. Allen and A. D. Misener.

Nature.

”Flow of Liquid Helium II.”

141, 75 (1938)[35] D. D. Osheroff, R. C. Richardson, and D. M. Lee.
Phase of Solid He3 .””Evidence for a New

Phys. Rev. Lett. 28, 885-888 (1972)[36] D. D. Osheroff*, W. J. Gully, R. C. Richardson, and D. M. Lee.
Magnetic Phenomena in Liquid He3 below 3 mK.”

”New

Phys. Rev. Lett. 29,

920-923 (1972)[37] P. Kapitza.

”Going with the flow.”

Nature. 141, 74 (1938)[38] D. O. Edwards, D. F. Brewer, P. Seligman, M. Skertic, and M. Yaqub.
”SOLUBILITY OF He3 IN LIQUID He AT 0K.”

Phys. Rev. Lett. 15,

773-775 (1965)[39] A. C. Anderson, D. O. Edwards, W. R. Roach, R. E. Sarwinski, and J. C.
Wheatley.

”THERMAL AND MAGNETIC PROPERTIES OF DILUTESOLUTIONS OF He3 IN He4 AT LOW TEMPERATURES” .

Phys. Rev.

Lett. 17, 367-372 (1966)[40] Axel Griesmaier, Jörg Werner, Sven Hensler, Jrgen Stuhler, and Tilman
Pfau.

”Bose-Einstein Condensation of Chromium.”

Phys. Rev. Lett.

94, 160401 (2005)[41] Mingwu Lu, Nathaniel Q. Burdick, Seo Ho Youn, and Benjamin L. Lev.
”Strongly Dipolar Bose-Einstein Condensate of Dysprosium.”
Rev. Lett. 107, 190401 (2011)
181

Phys.

[42] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F.
Ferlaino.

Phys. Rev. Lett.

”Bose-Einstein Condensation of Erbium.”

108, 210401 (2012)[43] B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, and G.
Pupillo.

”Quantum Phases of Cold Polar Molecules in 2D Optical

Lattices.”

Phys. Rev. Lett. 104, 125301 (2010)[44] Fudong Wang, Dezhi Xiong, Xiaoke Li, Dajun Wang and Eberhard Tiemann.

”Observation of Feshbach resonances between ultracold Na

and Rb atoms.”

Phys. Rev. A. 87, 050702 (2013)[45] Hung-Wen Cho, Daniel J. McCarron, Michael P. Kppinger, Daniel L.
Jenkin, Kirsteen L. Butler, Paul S. Julienne, Caroline L. Blackley, C. Ruth
Le Sueur, Jeremy M. Hutson, and Simon L. Cornish.

”Feshbach spec-

troscopy of an ultracold mixture of 85 Rb and 133 Cs.”

Phys. Rev. A. 87,

010703 (2013)[46] Tetsu Takekoshi, Markus Debatin, Raffael Rameshan, Francesca Ferlaino, Rudolf Grimm, Hanns-Christoph Ngerl, C. Ruth Le Sueur,
Jeremy M. Hutson, Paul S. Julienne, Svetlana Kotochigova, and Eberhard Tiemann.

”Towards the production of ultracold ground-state

RbCs molecules: Feshbach resonances, weakly bound states, and thecoupled-channel model.”[47] L. Pitaevskii, S. Stringari.

Phys. Rev. A. 85, 032506 (2012)”Bose-Einstein Condensation.”

Oxford Uni-

versity Press, 2003.
[48] Christopher J. Foot.

”Atomic Physics.”

Oxford University Press,

2005.
[49] H. J. Metcalf and P. van der Straten.

”Laser Cooling and Trapping.”

Springer; 1st ed. 1999.
[50] A. Yariv.

”Quantum Electronics.”
182

John Wiley & Sons, 1989.

”Bose-Einstein Condensation in Dilute Gases.”[51] C. J. Pethick, H. Smith.

Cambridge University Press, 2008.
[52] R. A. Barankov.

”Boundary of two mixed Bose-Einstein condensates”

Phys. Rev. A 66, 013612 (2002).
[53] B. V. Schaeybroeck.

”Interface tension of Bose-Einstein condensates”

Phys. Rev. A 78, 023624 (2008).
[54] I. E. Mazets.

”Waves on an interface between two phase-separated

Bose-Einstein condensates”[55] S. Gautam and D. Angom.
densates.”

Phys. Rev. A 65, 033618 (2002).
”Rayleigh-Taylor instability in binary con-

Phys. Rev. A 81, 053616 (2010).

[56] Kazuki Sasaki, Naoya Suzuki, Daisuke Akamatsu, and Hiroki Saito.
”Rayleigh-Taylor instability and mushroom-pattern formation in a twocomponent Bose-Einstein condensate.”

Phys. Rev. A 80, 063611 (2009).

[57] Middelkamp, S. Chang, J. J. Hamner, C. Carretero-Gonzlez, R.
Kevrekidis, P. G. Achilleos, V. Frantzeskakis, D. J. Schmelcher, P. andEngels, P..

”Dynamics of dark-bright solitons in cigar-shaped Bose-

Einstein condensates.”

Phys. Lett. A., Vol. 375, Issue 3, 642-646 (2011).

[58] K. Sasaki, N. Suzuki, D. Akamatsu and H. Saito.

”Rayleigh-Taylor

instability and mushroom pattern formation in a two-component BoseEinstein condensate”

Phys. Rev. A 80, 063611 (2009).

[59] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl1, C. Chin, B. Engeser,
A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl and R. Grimm.

”Evi-dence for Efimov quantum states in an ultracold gas of caesium atoms”
Nature. 440, 315-318 (2006).
[60] H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito and M. Tsubota.

”Quantum Kelvin-Helmholtz instability in phase-separated two-

component Bose-Einstein condensates”
183

Phys. Rev. B 81, 094517 (2010).

[61] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell.

”The Dynamics of Component Separation in a Binary Mixtureof Bose-Einstein Condensates”

Phys. Rev. Lett. 81, 1539 (1998).

[62] B. P. Anderson, P. C. Haljan, C. E. Wieman and E. A. Cornell.

”Vortexprecession in Bose-Einstein condensates: observations with filled andempty cores”

Phys. Rev. Lett. 85, 2857 (2000).

[63] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W.
Clark, and E. A. Cornell.

”Watching dark solitons decay into vortex

rings in a Bose-Einstein condensate”

Phys. Rev. Lett. 86, 2926 (2001).

[64] M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, M.J. Holland,
J.E Williams, C.E. Wieman, and E. A. Cornell.

”Watching a superuid

untwist itself: Recurrence of Rabi oscillations in a Bose-Einstein condensate”

Phys. Rev. Lett. 83, 3358 (1999).

[65] J. L. Roberts, N. R. Claussen, James P. Burke, Jr., Chris H. Greene, E.
A. Cornell, and C. E. Wieman.

”Resonant magnetic field control ofelastic scattering of cold 85 Rb”

Phys. Rev. Lett. 81, 5109 (1998).

[66] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell and C.E. Wieman.

”Stable 85 Rb Bose-Einstein Condensates with Widely Tunable

Interactions”

Phys. Rev. Lett. 85, 1795 (2000).

[67] S. B. Papp, J. M. Pino, and C. E. Wieman.

”Tunable Miscibility in a

Dual-Species Bose-Einstein Condensate”

Phys. Rev. Lett. 101, 040402(2008).
[68] S. Ronen, J. L. Bohn, L. E. Halmo and M. Edwards.

”Dynamical pat-tern formation during growth of a dual-species Bose-Einstein condensate”

Phys. Rev. A. 78, 053613 (2008).

[69] Herman Feshbach.

”Unified theory of nuclear reactions”

5, 357 (1958).
184

Ann. Phys.

[70] Herman Feshbach.

”A unified theory of nuclear reactions. II”

Ann.

Phys. 19, 287 (1962).
[71] C. Chin, R. Grimm, P. Julienne and E. Tiesinga.

”Feshbach resonances

Rev. Mod. Phys. 82, 1225-1286 (2010).

in ultracold gases”[72] J. L. Roberts, N. R. Claussen, James P. Burke, Jr., Chris H. Greene, E.
A. Cornell, and C. E. Wieman.

”Resonant magnetic field control ofelastic scattering of cold 85 Rb”

Phys. Rev. Lett. 81, 5109 (1998).

[73] Beln Paredes, Artur Widera, Valentin Murg, Olaf Mandel, Simon Flling,
Ignacio Cirac, Gora V. Shlyapnikov, Theodor W. Hnsch and Immanuel
Bloch.

”Tonks―Girardeau gas of ultracold atoms in an optical lattice”

Nature 429, 277 (2004).
[74] M. Greiner, C. A. Regal, and D. S. Jin.

”Emergence of a molecular

Bose-Einstein condensate from a Fermi gas”

Nature 426, 537 (2003).

[75] S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, and D. S.
Jin.

”Observation of heteronuclear Feshbach resonances in a mixtureof bosons and Fermions”

Phys. Rev. Lett. 93, 183201 (2004).

[76] G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani, F. Minardi, and M.
Inguscio.

”Double Species Bose-Einstein Condensate with Tunable

Interspecies Interactions”

Phys. Rev. Lett. 100, 210402 (2008).

[77] G. Barontini, C. Weber, F. Rabatti, J. Catani, G. Thalhammer, M. Inguscio, and F. Minardi.
Resonances””Observation of Heteronuclear Atomic Efimov

Phys. Rev. Lett. 103, 043201 (2009).

[78] J. P. D Incao and B. D. Esry.
body collision rates.””Mass dependence of ultracold three-

Phys. Rev. A. 73, 030702 (2006).

[79] C. Weber, G. Barontini, J. Catani, G. Thalhammer, M. Inguscio,and F. Minardi.
molecules””Association of ultracold double-species bosonicPhys. Rev. A 78, 061601 (2008).
185

[80] Andrea Simoni, Matteo Zaccanti, Chiara D Errico, Marco Fattori, Giacomo Roati, Massimo Inguscio, and Giovanni Modugno.

”Near-

threshold model for ultracold KRb dimers from interisotope Feshbach
spectroscopy”

Phys. Rev. A 77, 052705 (2008).

[81] G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani, F. Minardi, and M.
Inguscio.

”Double Species Bose-Einstein Condensate with Tunable

Interspecies Interactions”

Phys. Rev. Lett. 100, 210402 (2008).

[82] Private communication with Pascal Naidon.
[83] M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M.
Kurn and W. Ketterle.
Condensate””Direct, Nondestructive Observation of a Bose

Science 273, 84-87 (1996).

[84] M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G.
Townsend, S. Inouye, and W. Ketterle.
Bose-Einstein Condensate””Propagation of Sound in a

Phys. Rev. Lett. 79, 553-556 (1997).

[85] B. P. Anderson, P. C. Haljan, C. E. Wieman and E. A. Cornell.

”Vortexprecession in Bose-Einstein condensates: observations with filled andempty cores”

Phys. Rev. Lett. 85, 2857 (2000).

[86] E.A. Burt, R.W. Ghrist, C.J. Myatt, M.J. Holland, E.A. Cornell, and C.E.
Wieman.

”Coherence, Correlations, and Collisions: What One Learns

About Bose-Einstein Condensates from Their Decay”

Phys. Rev. Lett.

79, 337 (1997).
[87] Ruth S. Bloom, Ming-Guang Hu, Tyler D. Cumby, and Deborah S. Jin.
”Observation of heteronuclear Feshbach resonances in a mixture ofbosons and Fermions”[88] V. Efimov.

Phys. Rev. Lett. 111, 105301 (2013).

”Energy levels arising from resonant two-body forces in athree-body system”

Physics Letters B. 33, 563-564 (1970).
186

[89] C. Hamner, J.J. Chang, P. Engels and M. A. Hoefer.

”Generation ofdark-bright soliton trains in superfluid-superfluid counterflow”

Phys.

Rev. Lett. 106, 065302 (2011).
[90] 藤掛陽輔.

「ボース凝縮体の Phase Contrast Imaging」 東京大学 卒

業論文 (2008).

[91] 加藤宏平.
京大学「異核フェッシュバッハ分子の生成・制御に関する研究」 東修士論文 (2011).

[92] Daniel A. Steck,”Rubidium 87 D Line Data,”

available online at

http://steck.us/alkalidata (revision 2.0.1, 2 May 2008).
[93] Tobias Gerard Tiecke.

”Feshbach resonances in ultracold mixtures ofthe fermionic quantum gases 6 Li and 40 K,” PhD thesis, University of
Amsterdam, 2009[94] Rudolf Grimm, Matthias Weidemüller, Yurii B. Ovchinnikov.
tical dipole traps for neutral atoms””Op-

ADVANCES IN ATOMIC,

MOLECULAR, AND OPTICAL PHYSICS, VOL. 42, p. 92 , 1999
[95] T. Kraemer , M. Mark , P. Waldburger , J. G. Danzl , C. Chin , B. Engeser, A. D. Lange , K. Pilch , A. Jaakkola , H.-C. Nägerl and R. Grimm.
”Evidence for Efimov quantum states in an ultracold gas of caesium
atoms”Nature 440, 315-318 (2006).

[96] Silke Ospelkaus-Schwarzer.

”Quantum Degenerate Fermi-Bose Mix-tures of 40K and 87Rb in 3D Optical Lattices.” PhD thesis, UniversitätHamburg, 2006.
[97] 山陸芳之.
大学「３次元光格子に閉じ込めたフェッシュバッハ分子」

東京

修士論文 (2010).

[98] D. J. McCarron, H. W. Cho, D. L. Jenkin, M. P. Kppinger, and S. L.
Cornish.

”Dual-species Bose-Einstein condensate of Rb87 and Cs133”

Phys. Rev. A. 84, 011603 (2011).
187

[99] R. W. Pattinson, T. P. Billam, S. A. Gardiner, D. J. McCarron, H. W.
Cho, S. L. Cornish, N. G. Parker, and N. P. Proukakis.

”Equilibrium

solutions for immiscible two-species Bose-Einstein condensates in perturbed harmonic traps”

Phys. Rev. A. 87, 013625 (2013).

[100] Benjamin Pasquiou, Alex Bayerle, Slava M. Tzanova, Simon Stellmer,
Jacek Szczepkowski, Mark Parigger, Rudolf Grimm, and Florian
Schreck.

”Quantum degenerate mixtures of strontium and rubidium

atoms”

Phys. Rev. A. 88, 023601 (2013).

[101] V. I. Bespalov and V. I. Talanov.
Beams in Nonlinear Liquids.”[102] V. I. Karpman.
persive Media.””Filamentary Structure of Light

JETP Lett. 3, 307 (1966).

”Self-modulation of Nonlinear Plane Waves in DisJETP Lett. 6, 277 (1967).

[103] AKIRA HASEGAWA AND W. F. BRINKMAN.

”Tunable Coherent

IR and FIR Sources Utilizing Modulational Instability.”

J. Quant. Elec-tron. 16, 694 (1980).
[104] M. D. Iturbe-Castillo, M. Torres-Cisneros, J. J. Snchez-Mondragn, S.
Chvez-Cerda, S. I. Stepanov, V. A. Vysloukh, and G. E. Torres-Cisneros.
”Experimental evidence of modulation instability in a photorefractive
Bi12 TiO20 crystal.”

Opt. Lett. 20, 1853 (1995).

[105] M.I. Carvalho, S.R. Singh, D.N. Christodoulides.

”Modulational in-stability of quasi-plane-wave optical beams biased in photorefractive
crystals. ” Opt. Commun. 126, 167-174 (1996).
[106] P. Lallemand and N. Bloembergen.

”Self-Focusing of Laser Beams

and Stimulated Raman Gain in Liquids”

Phys. Rev. Lett. 15, 1010-1012(1965).
[107] Marco Peccianti, Claudio Conti, Gaetano Assanto, Antonio De Luca
and Cesare Umeton.

”Routing of anisotropic spatial solitons and188modulational instability in liquid crystals.”Nature. 432, 733-737(2004).
[108] Marco Peccianti, Ian B. Burgess, Gaetano Assanto and Roberto Morandotti.

”Space-time bullet trains via modulation instability and nonloOptics Express. 18, 5934 (2010).

cal solitons.”[109] K. J. H. Law, P. G. Kevrekidis, and Laurette S. Tuckerman.

”Sta-

ble Vortex―Bright-Soliton Structures in Two-Component Bose-Einstein
Condensates.”

Phys. Rev. Lett. 105, 160405 (2010).

[110] Th. Busch, J. I. Cirac, V. M. Prez-Garca, and P. Zoller.

”Stability andcollective excitations of a two-component Bose-Einstein condensed gas:
A moment approach.”Phys. Rev. A. 56, 2978-2983 (1997).

[111] V. Schweikhard, I. Coddington, P. Engels, S. Tung, and and E. A. Cornell.

”Vortex-Lattice Dynamics in Rotating Spinor Bose-Einstein Con-densates.”

Phys. Rev. Lett. 93, 210403 (2004).

189