Master’s Thesis;

Bose-Einstein Condensates
in an
Optical Lattice（光格子中のボース・アインシュタイン凝縮体の量子的振舞い）

Supervisor: Associate Professor Shin Inouye
Kai Noda
76524
Department of Applied Physics,
School of Engineering,
the University of Tokyo.
February, 2009

Dedicated to the memories of87

Rb atoms

which were released from the dispenser,
captured in the MOT,
evaporatively cooled in the magnetic trap
until they Bose-Einstein condensed,
conﬁned in the optical lattice,shined by resonant lightto produce a shadow on the CCD,
anddissapeared into the walls of the ion pumps.

謝辞 (Acknowledgements)
本論文は東京大学 工学系研究科 総合研究機構 井上研究室において行った研究をまとめたものです．何よりまず同研究室の主宰者である井上慎准教授に感謝します．井上先生は私を冷却原子の分野に導いて下さいました．私は学部 4 年で行う卒業研究の時点で初めて井上研究室に参加しましたが，参加して間もなくに冷却原子について井上先生から初めて習った知識は，光子の運動量と原子の遷移の自然幅を基にした MOT 中の原子の冷却レートの見積もりでした．以来，冷却原子の研究を通じて様々な知識と経験を蓄えることができましたが，それらはいずれも井上先生の導きによるものです．
また東京大学 理学系研究科の上田正仁教授にも感謝します．上田教授は JST, ERATO の上田マクロ量子
制御プロジェクトの総括でおられ，井上研究室はそれを構成する一グループとして極低温極性分子生成を目指した実験を行っています．そのような時代の最先端を行く研究環境に身を置くことができたのは上田教授のご尽力のおかげです．またミーティング等を通じて上田教授から厳しくも示唆に富んだ指導をいただけたことは大きな財産となりました．
現在は電気通信大学 特任助教でおられる岸本哲夫博士には，岸本博士が井上研助手／助教でおられた時期
から現在に至るまで，実験のあらゆる側面について多くを学ばせていただきました．ボール盤，旋盤，フライス盤の使用方法を始めとする私の金属加工の技能はいずれも岸本博士から習い，また岸本博士を目標として修練したものです．他にも挙げればきりがありませんが，そもそも本研究の実験を行う真空槽自体や，本研究の主題である光格子の光学系の配置が岸本博士の設計によるのを始め，手間のかかる光路の調整を効率化するための示唆等，本研究の多くを岸本博士に負うています．日頃から岸本博士の豊富な経験に基づく豊かな発想に触れることで優れた実験家たるには何を目指すべきか学ばせていただきました．
本研究を含む，私が井上研究室で行った実験の大部分は，現在の井上研助教でおられる小林淳博士の指導の下，BEC を扱う井上研究室 “E1” チームにおいて行われました．卒業研究生の頃に行った 39 K の飽和吸収分光から始め，岸本博士と共に行った翌夏の 41 K BEC の生成および翌春にかけての |F = 1, mF = 1⟩ 状態の
41

K 間の Feshbach 共鳴の観察，そして本研究に至るまで，いずれも小林博士と共に追求してきた結果です．

当初はノートに実験系の設定パラメタを書き取る程度しかできませんでしたが，現在幾分なりとも系を把握して動かせるようになったのも，小林博士との研究活動を通じ揉まれたおかげです．実験上の困難を打開する
糸口が見えない中でも様々な方策を提案する鋭い頭脳と，確固たる結果を求めて粘り強く実験に取り組む集中力には，常に驚嘆させられてきました．また両面テープがいかに実験系構築を簡略化するかについても学ばせていただきました．
学術振興会研究員として井上研究室に参加しておられた赤松大輔博士は量子光学のエキスパートでおられ，その最先端についての知識に触れることは，今だ多くの謎を湛える量子力学の本質に関し考えを深めるよい契機となりました．一見初等的な疑問をないがしろにせず追求する姿勢からは物理学徒の後輩として大いに薫陶を受けました．さらに埼玉県と浦和レッズに関する私の貧弱な理解を幾分か正していただきました．
現在，博士課程に所属し井上研究室で研究をなさっている 1 年先輩の相川清隆氏は，私が卒業研究生として立ち上げ間もなくの井上研究室に参加した時点よりやや早くから井上研究室で研究を行っておられました．
当初から LD の準備や 39 K MOT 構築，また原子間ポテンシャルの計算においてたった 1 学年上とは思えぬ才覚を振るっていた様子を目にし，大学院生とはかくも水準が高い方々なのか，と衝撃を受けたのを記憶しています．相川氏の頭脳の回転の速さには圧倒されるばかりでしたが，そればかりでなく実験にも熱意と深い理解をもって取り組む姿を示していただくことで，気を引き締めて研究生活を送ることができました．
修士課程での同期生であった林正泰氏は井上研究室での所属チームこそ違うものの，物理を学ぶ学生としての苦楽を共にし，時おり根津の中華料理屋「楽翠荘」で夜食を取った同志でした．そして今も学生居室のi

謝辞 (ACKNOWLEDGEMENTS)

ii

向かい合った席で修士論文を共に執筆しています．“E2” チームにおける分光実験や，それに引き続く分子のSTIRAP において信号がノイズに埋もれ，そもそも広い範囲のどこに位置するのか分からないという逆境に置かれても，常に笑顔を絶やさない林氏の精神力は，共に過ごした 2 年間を通じ模範的なものとして私の目に映りました．それは今後も折に触れて思い出されることでしょう．
井上研究室での 1 年後輩である藤掛陽輔氏と山陸芳之氏は “E1” チームに対する熱意ある貢献振りで研究を加速させてくれました．藤掛氏は DDS を中核とする周波数生成・掃引システムの実装により，87 Rb BEC の実験をより完成度の高いものにしてくれました．第 3.5 節で触れたように，本研究でも 87 Rb 原子気体の蒸発冷却において当該システムを利用しています．山陸氏は勤勉な働きぶりで実験に不可欠な装置の構築に力を振るってくれました．87 Rb BEC 生成には各レーザーに対応して多数の AOM が必要とされますが，その制御回路の多くを山陸氏が製作してくれました．また本研究の次のステップに当たる， 87 Rb-41 K 二原子種 BEC の
効率的な光格子への閉じ込めに必要なパンケーキ形光双極子力トラップの実装に尽力してくれました．

2007 年度の卒業研究生として井上研に参加していた田野岡祐介氏，同じく 2008 年度の大島淳氏と森圭輔氏からは，溢れる若さのエネルギーを補充させていただきました．果たして私は，過去に私が知遇を得た方々のような良き先輩たり得たでしょうか？
上田マクロ量子制御プロジェクト 強相関量子制御グループのリーダーでおられる向山敬博士は，常に笑顔でハード・スケジュールの実験をこなす超人振りで，様子を横目で見ながら奮い立たせていただいたものです．また我々が現在光格子に使っているファイバ・レーザーの納品以前，向山グループ所有のファイバ・レー
ザーから数 W の出力を 40 m の光ファイバを通じて分けていただき，光学系のテストに使わせていただきました．他にも装置を融通していただくなど，多大な恩恵を受けました．
同グループの堀越宗一博士はアトムチップ上の BEC 干渉計に熟練した実験家でおられ，特に 1 次元光格子
中の BEC に関してご自身の経験に基づく貴重な助言をいただきました．
「s 波ハロー」についての議論は堀越博士からの助言が基になっています．また堀越博士にはしばしば 19 時前後に皆で夕食を食べに出かけるときのイニシアチブを取っていただきました．
同グループの Swarupananda Pradhan 博士からは 1 次元光格子中の原子の Kapitza-Dirac 散乱について，同グループにおける実験結果を基にアドバイスをいただきました．
同グループに参加なさっていた工学系研究科 五神研究室の稲田安寿氏は豊富な知識と豊かなユーモアを兼ね備えた尊敬すべき先輩でした．稲田氏は本研究と同時期に極低温 6 Li 原子気体に関する博士論文を執筆しておられ，その姿を目標とさせていただきました．また稲田氏はしばしば我々の実験装置の置かれた奥の部屋を訪れ食事の誘いを伝えて下さいました．
同グループに参加なさっている理学系研究科 上田研究室の中島秀太氏からは冷却原子や超流動・超伝導の理論的側面について教えをいただきました．中島氏の修士論文は本論文の執筆において大いに参考とさせていただきました．またコヒーレンスの消失と復活に関し，ヘアスタイルを通じて示唆をいただきました．
上田マクロ量子制御プロジェクト 不確定性制御グループのリーダーでおられる東京工業大学の上妻幹旺准教授には定例ミーティング後や学生居室で議論に加わらせていただき，量子力学の基礎原理に関し様々な教えをいただきました．同時にそのフランクなお人柄にも感銘を受けました．また我々が現在 87 Rb 原子の蒸発冷却に使っている 6.8 GHz 帯の rf 増幅器とスペクトラム・アナライザは上妻研究室からお借りしています．
同グループの竹内誠博士からは議論を通じ有益な情報をいただきました．本論文に関連する Mott 転移に関する私の理解の基礎となったのは竹内博士による講義でした．
現在は自然科学研究機構 分子科学研究所の助教でおられる武井宣幸博士からは，武井博士が同グループの研究員でいらした時期に量子光学の実験等に関し様々な教えをいただきました．
電気通信大学で冷却原子の実験をなさっている中川賢一准教授からも 6.8 GHz 帯の rf 増幅器をお借りしています．その rf 増幅器は我々の実験系における初の 87 Rb BEC 生成の鍵となりました．
上田マクロ量子制御プロジェクトの技術参事でおられる榎本陽一博士，また前任の技術参事でおられる金沢工業大学の山口敦史教授，同プロジェクト事務局の佐藤敏彦氏，塩田美奈子氏，国府田愛氏には研究の円滑な進行のためにご尽力いただきました．
最後に，両親に感謝します．

Contents
謝辞 (Acknowledgements)i

1 Introduction

1

1.1

Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Developments of quantum many-body theory . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Quests for BEC in a dilute atomic gas . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.3

Optical lattice: an ideal tool for simulating crystalline matters . . . . . . . . . . . .

5

1.2

Contribution of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2 Theoretical Backgrounds
2.1

8Interactions of Atoms and Electromagnetic Radiations . . . . . . . . . . . . . . . . . . . . .

8

2.1.1

Optical Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.2

Rabi frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Bose-Einstein condensation of ideal bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3

Optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.1

14

Periodic potential formed by a standing wave of two counter-propagating lasers . . .

3 Experimental Setup and Procedure
3.1

20

Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.1.1

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.1.2

Preparation of ultra-high vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.2

Frequency-locked lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.3

Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.3.1

1st MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.3.2

2nd MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.4

Magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.5

RF generator and controller for evaporative cooling . . . . . . . . . . . . . . . . . . . . . . .

29

3.6

Optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.6.1

1 µm, 5 W ﬁber laser and an auxiliary optical system . . . . . . . . . . . . . . . . .

31

3.6.2

Dimensions of the optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.6.3

Properties of the optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.7

4 Results and Discussions
41

39

4.1

BEC ofK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.2

Bragg diﬀraction of the condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.3

Interference patterns by condensates in 1D optical lattices . . . . . . . . . . . . . . . . . . .

42

4.4

Interference pattern of the condensate in 2D lattice . . . . . . . . . . . . . . . . . . . . . . .

42

iii

iv

CONTENTS

5 Conclusion and Outlook

46

A Theoretical Techniques47

A.1 Classical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 ABCD-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.2 Eﬀect of thin lens on Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
47
47

A.1.3 Numerical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Time-dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48
50

A.3 n-dimensional Laplacian and Laplace-Beltrami operator . . . . . . . . . . . . . . . . . . . .
A.3.1 Gradient operator on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3.2 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
51
53

A.4 Contour Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5 Mathieu equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54
55

B Converter program from SPE image for WinView to JPEG image

58

Bibliography

60

List of Figures
2.1

Udip (I; ω, Rb) and Γsc (I; ω, Rb) plotted for several laser frequencies. . . . . . . . . . . . . .

9

2.2Plot of λ(I) ≡ 2πc/ω(I, G; ωX ) and Udip (I; G) for G = 0.1 s−1 and X = Rb. . . . . . . . . .

10

2.3

Schematic view of spectroscopic line data on 87 Rb known so far. . . . . . . . . . . . . . . .

10

2.4

Geometric interpretation of the eﬀect of broadening of a retroreﬂected beam. . . . . . . . .

16

2

2.5

Plot of 1/e diameter of forward- and backward-propagating Gaussian beams. . . . . . . . .

17

2.6

Calculated visibilities at a few points on the standing wave corresponding to Figure 2.5. . .

17

2.7

The eﬀect of broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.8

Diﬀraction relation of atom and the principle of Bragg diﬀraction. . . . . . . . . . . . . . .

18

3.1

Design of the vacuum chamber by Dr. Kishimoto. . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2

Detailed view of Figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.3

A typical sequence of baking the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . .

23

3.4

Decrease of the pressure during gradual cooling of the chamber. . . . . . . . . . . . . . . . .

23

3.5

Schematics of the optical system to provide frequency-locked lasers.

. . . . . . . . . . . . .

24

3.6

The dependence of OD of an atomic cloud of 87 Rb on detuning of the probe beam. . . . . .

25

3.7

Schematics showing “MOT operation” of the coils of the Ioﬀe-Pritchard magnetic trap. . .

25

3.8

A typical signal from a etalon coupled with the injection laser of the TA for the 1st MOT. .

26

3.9

A typical loading of pushed

87

Rb atoms into the 2nd MOT. . . . . . . . . . . . . . . . . . .

27

3.10 Magnetic ﬁeld produced by a pair of coils sharing a common axis. . . . . . . . . . . . . . . .

28

3.11 The circuit for Cloverleaf coils of the Ioﬀe-Pritchard magnetic trap. . . . . . . . . . . . . . .

29

3.12 Circuit for Curvature and Anti-bias coils of the Ioﬀe-Pritchard magnetic trap. . . . . . . . .

29

3.13 A typical time sequence of a rf sweep for rf-forced evaporative cooling of

87

Rb. . . . . . . .

30

Rb. . . . . . . . . . . . . . . . . . . . . . .

30

3.15 Schematics of the optical system for providing laser beam with an optical lattice. . . . . . .

31

3.16 A typical conﬁguration of the telescope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.14 A trajectory of a typical evaporative cooling of

87

2

3.17 The measured 1/e width of collimated beams by three kinds of ﬁber couplers. . . . . . . .

33

3.18 Shematics of the optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.19 Detailed design of telescopes of the optical lattice. See Table 3.8 for actual lengths. . . . . .

35

3.20 An example of measured divergence of output beam from F230FC-C. . . . . . . . . . . . . .

35

3.21 Measurement of falling atomic cloud. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.1

Combination of CMOT and following Doppler cooling for 41 K. . . . . . . . . . . . . . . . .

40

4.2

Transitions used for rf-forced evaporative cooling of 41 K. . . . . . . . . . . . . . . . . . . . .

40

4.3
4.4

A typical time sequence of a rf sweep for rf-forced evaporative cooling of
Typical trajectory of evaporative cooling of
41

41

41

K. . . . . . . . .

41

K. . . . . . . . . . . . . . . . . . . . . . . . .

41

4.5

ToF images of

K BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.6

Interference pattern of a 1D array of 2D BEC. . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.7

Curve ﬁtting on interference pattern by Path (A). . . . . . . . . . . . . . . . . . . . . . . .

43

4.8

Interference pattern of a 2D arrays of 1D BEC. . . . . . . . . . . . . . . . . . . . . . . . . .

43

v

vi

LIST OF FIGURES
4.9 Example of curve ﬁtting on the absorption image obtained above. . . . . . . . . . . . . . . .
4.10 Typical time sequence of ramping up and down of the optical lattice. . . . . . . . . . . . . .

44
44

A.1 Coordinate systems to derive the lens formula for Gaussian beams. . . . . . . . . . . . . . .
A.2 Plot of Eqs. (A.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 A one-to-one correspondance between a parametrized surface and its tangent plane. . . . .

48
49
51

A.4 Calculation of a real deﬁnite integral through a contour integral. . . . . . . . . . . . . . . .
A.5 Plot of stable regions of the Mathieu equation by the C proglam written by Jones [1]. . . .

55
56

List of Tables
1.1

Historical developments of “old” quantum theory (“pre-Schrödinger” era.) . . . . . . . . . .

2

1.2
1.3
1.4

Historical developments towards modern theory of quantum mechanics. . . . . . . . . . . .
Achievements of quantum degeneracy in a dilute atomic gas (mixture,) sorted by Year. . . .
Observations and theoretical predictions of Feshbach resonance in a dilute atomic gas (mixture,) sorted by Year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2
5

2.1
3.1

Comparison between our and Michigan group’s experimental conditions. . . . . . . . . . . .
Components used in the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18
20

3.2
3.3

Properties of the 1st MOT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of the 2nd MOT. See also remarks on the Table 3.2. . . . . . . . . . . . . . . . .

26
26

3.4
3.5
3.6

Components used in the magnetic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of the magnetic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical time sequence of a rf sweep for rf-forced evaporative cooling of 87 Rb. . . . . . . . .

28
28
29

3.7
3.8

Fitted parameters for data shown in Figure 3.17. . . . . . . . . . . . . . . . . . . . . . . . .
Actual lengths (measured in milimeters) in Figure 3.19, for each path. . . . . . . . . . . . .

33
35

3.9 Components used in the optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Properties of the optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Components used in our imaging system. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36
37
38

3.12 Magniﬁcation of the imaging system for each operation mode. . . . . . . . . . . . . . . . . .

38

Comparison between MOT of 41 K and 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.1
4.2
4.3

41

A typical time sequence of a rf sweep for rf-forced evaporative cooling of K. . . . . . . . .
Obtained values for ﬁtting parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

5

40
45

Chapter 1

Introduction
1.1

Historical overview

1.1.1

Developments of quantum many-body theory

In 1995, the ﬁrst achievements of Bose-Einstein Condensation (BEC) in dilute atomic gases were reported
almost simultaneously by three American groups (Anderson et al. [2], Davis et al. [3], Bradley et al. [4].)Since then, more and more atomic species have been cooled to reach quantum degeneracy (BEC for bosons
and Fermi degeneracy for fermions.) Realization of BEC in a dilute atomic gas is often phrased as the “holygrail of physics,” since it oﬀers quite many opportunities in study of the quantum many-body systems. In
order to appreciate its signiﬁcance, we take a quick look at the history of the quantum theory of matters.
The eﬀorts to study physical properties of solids and liquids (collectively termed as condensed matters) from the microscopic view of points dates back to the 1920’s. At the turn of the 20th century, acompletely novel branch of physics describing energy levels of electromagnetic radiations and atoms wasborn; quantum mechanics. Before its birth, the majority of physicists and mathematicians thought thatphysics is a “mature,” or just “dead” branch of science. Ironically, among those scientists was Michelson,
whose experimental work would later become the basis for the another revolution in physics; that is, thetheory of relativity by Einstein (but we will not go further in this topic.) Feuer [5, Ch. 3] illustrates suchan atomosphere by quoting memoirs by Millikan [6], who experienced the revolution in physics as a young
physicist and himself made great contributions to it through measurement of the elementary charge:“My ﬁrst view of Michelson was at the convocation [of the Ryerson Laboratory in 1894]. Hegave the address on the place of very reﬁned measurement in the progress of physics—an addressin which he quoted someone else, I think it was Kelvin, as saying that it was probable that thegreat discoveries in physics had all been made, and that future progress was likely to be foundin the sixth place of decimals.”
However, the triumph of Bohr’s atomic theory showed that there was large amount of problems to beinvestigated in the microscopic world. As shown in Table 1.1 (adapted from Brown et al. [7],) the “old”
quantum theory, whose heart is Bohr’s atomic theory based on simple quantization rule of electron orbits,
successfully explained most of the phenomena found at that time with regard to atomic structure. Amongthem, the Stark shift, the (normal) Zeeman eﬀect and the formation rule of periodic table of the elements
are the most signiﬁcant (the last one lead to ﬁnding of Pauli exclusion principle.) Note also that the basis
for Bose-Einstein and Fermi-Dirac statistics was already found. However, as noticed by Pauli, the failureof Bohr theory in explanation of anomalous Zeeman eﬀect had so deep root that the whole theory needed
to be rebuild on a new solid foundation.
Schrödinger presented what is now called Schrödinger equation which describes time evolution of a
wavefunction of a quantum state. It had a structure of wave equation, so his theory was called “wavemechanics.” It correctly explained quantum phenomena discovered so far, including level structures ofcomplex atoms such as sodium and the anomalous Zeeman eﬀect. At the same time, Heisenberg formulated
his quantum theory of electron orbits in terms of non-commuting position and momentum variables (atﬁrst deﬁned in terms of Fourier coeﬃcients.) With the help by Jordan, Born reformulated Heisenberg’s
theory into inﬁnite-dimensional matrix algebra. Then their theory was called “matrix mechanics.” Finally,
Heisenberg realized that both of matrices and waves are representation of quantum states. That is, twoformulations were equavalent. The success of uniﬁcation of two diﬀerent-looking theories convinced us thecorrectness of both of two theories.
1

2

CHAPTER 1. INTRODUCTION
Researcher

Year

Achievement

M. Planck

1900

A. Einstein

1905

E. Rutherford

1910

N. Bohr1913

J. Stark
J. Franck and G. Hertz

1913
1914

P. Epstein /K. Schwarzschild
A. Einstein
W. Kossel

1916

Use of discrete energy levels in explaination of blackbody
radiation spectrum
Introduction of “energy quanta” in explaination of thephotoelectric eﬀect
Rutherford model of an atom consisting of a nucleus andorbiting electrons
Quantum theory of hydrogen energy spectrum; Bohr’s
quantization rule
Observation of Stark shift
Franck-Hertz experiment of electron-atom inelastic scattering
Explanation of Stark shift in terms of Bohr theory

A. Sommerfeld / P. Debye
L. de Broglie
S. BoseA. Einstein

1916
1923
1924
1925

W. Pauli
C. J. Davisson /G. P. Thomson

1925
1927

1916
1916

The concept of spontaneous emission
First trial to relate periodic table of the elements to Bohr
theoryExplanation of Zeeman∗1 eﬀect in terms of Bohr theory
The concept of matter waves
Bose-Einstein stastistics of photons
Bose-Einstein statistics of massive particles; prediction of
BEC for ideal bose gas
Pauli exclusion principle
Electron beam diﬀraction

Table 1.1: Historical developments of “old” quantum theory (“pre-Schrödinger” era.) Their subject waselectromagnetic radiation and energy structure of simple atoms, such as hydrogen. Note that this andsubsequent tables slightly overlap in time.
Researcher

Year

Achievement

O. Stern and W. Gerlach
G. Uhlenbeck and S. Goudsmit
W. Heisenberg
M. Born and P. Jordan

1922
1925
1925
1925

G. Uhlenbeck and S. Goudsmit

1925

Stern-Gerlach experiment of spin projection
The concept of electron spin
Quantum theory in terms of non-commuting variables
Reformulation of Heisenberg theory in terms of inﬁnitedimensional matrices
The concept of electron spin

Table 1.2: Historical developments towards modern theory of quantum mechanics. The main subjects wasenergy structure of complex atoms. Formulations developed through these works endured the test of timeand still plays the central role in modern physics.

As the correctness of quantum theory become more and more convincing, reseachers began to apply it
to condensed matter systems. Matters should be in principle described by quantum mechanics, just like
its ingredients, electrons and atoms. The application of quantum mechanics was ﬁrst limited to one-body
approximation. However, systematic approach to quantum many-body systems were gradualy developed.
Recent topics attracting interests from many researchers are systems such that quantum nature of particles
dominates the characteristics of the system.
In particular, a classical picture of matters breaks down if the phase space of the ensemble of particles
is densely ﬁlled (see textbook by Leggett [8];) in other words, quantum mechanics
becomes important in√describing a many-body system if the thermal de Broglie length λdB ≡ h/ 2πmkB T of particles is comparable or larger than the interparticle distance. In addition, we should take into account indistinguishabilityof particles and resultant bosonic and fermionic statistics if particles can exchange their positions and canoverlap. In fact, in metals, that is the principal object in study of condensed matter systems, both of theabove conditions are satisﬁed by electrons. Electrons in metals behave very much like an ideal degenerate
Fermi gas (see classical monographs by Pines and Nozières [9, 10, 11].) They can move almost freely in a
crystal, so statistical treatment is necessary. The density of electron gas is so high that they are almost
always in a Fermi degeneracy. In fact, some of the properties of metals which could not be explained
within the framework of classical physics were successfully understood by proper applications of quantum

1.1. HISTORICAL OVERVIEW

3mechanics. The ﬁrst successes of quantum mechanics in condensed matters include derivation of the linear temperature dependence of the electronic heat capacity, and ﬁndings of the electronic band structure
calculations of the kinetic coeﬃcients in the transport phenomena, just to name a few.
However, as the research progresses, the novel form of the matter which did not allow applicationof simple quantum theory was discovered: the superﬂuidity in liqueﬁed 4 He discovered by Kapitsa. For
normal liquids at suﬃciently low temperature T , the thermodynamic free energy F = E − T S is minimized
when kinetic energy E is minimized at the cost of low entropy S. However, 4 He atoms are so light thatquantum ﬂuctuations due to Heisenberg’s uncertainty principle does not allow to form a crystalline structure.
Although the ingenuity of Onsager captured its essence that the superﬂuidity is the result of BEC (1940), thehigh density of superﬂuid helium (or He-II) and resultant strong interactions between 4 He atoms prevented
further theoretical investigations. Theoretical methods which successfully described its properties such as
Ginzburg-Landau theory were to some extent phenomenological. There were Bogoliubov’s theory for weakexcitations of BEC, but it was not to meant to be a foundation of complete microscopic theory; it only
predicts resultant various phenomena provided that macroscopic occupation will occur. Various methods
based on perturbative techniques were developed, but they are hard to verify experimental agreements. In
order to develop further understandings of the quantum nature of matter, reseachers dreamed of an ideal
test bed for quantum many-body theories for long time.
The principal question is: how properties of a many-body system of interacting particles diﬀer from thoseof non-interacting ones? First, ground state of each of them are diﬀerent. Secondly, one may well lookat elementary excitations from the ground states. They can be classiﬁend into two; that is, quasiparticles
and collective modes. Moreover, one may hope to understand properties which are deﬁned in thermal
equilibrium states and are related to temperature of the system, or those whose manifestation are observedin irreversible phenomena to recover thermal equilibrium from inequilibrium states.
The historical remark was the BCS theory which describes a fermionic many-body system. Nextly,
atomic nuclear theories accelerated its progress. Only a few phenomenology were known at that time,such as the shell-model describing the nuclear structure and a few kinds of collective modes of nucleons.
However, applications of multiple scattering theory revealed the fact that strong short-range interactions
between nucleons do not alter excitation spectra very much, and thus nucleons are best described as
quasiparticles obeying the shell model. And ﬁndings in theories of superconductivity and electron gas were
imported to the nuclear theory. Then collective modes were proved to occur from weak interaction between
quasiparticles which originate from bare nucleons and interaction between them and are described by theshell model,
Lastly, theoretical models of many-body systems were developed to improve understandings of theessential characterics of many-body phenomena. Under appropriate assumptions, they give exact solutions
whose analytic properties can be thoroughly investigated and be compared with experiments. An example
is Tonks-Girardeau gas; a problem ﬁrst raised for 1D classical bose gas by Tonks [12], and for quantum
bose gas by Girardeau [13] more than 40 years ago. The physics of such a gas is dominated by the repulsive
interactions between bosonic particles. Due to their mutual repulsion, the bosons cannot occupy the same
position in space; thus the repulsive interaction between bosons is shown to have similar eﬀect on thephysics of the system as the Pauli exclusion principle for fermions (sometimes termed as “fermionization”
of bosons.) As another example, we refer to the review by Tasaki [14] on mathematical results obtainedfor original (or, Fermi-) Hubbard model. A modiﬁed version of Hubbard model for bosons (Bose-Hubbard
model) is also presented for superconducting currents in Josephson Junction Arrays (JJA) and it is also
used for description of behavior of atoms conﬁned in optical lattice. Lastly, we cite recent achievements in2002 by Lieb and his coworkers which proved the existence of BEC in a weakly interacting (i.e. non-ideal)
bose gas. For long time it has been believed that perturbation from ideal bosons works for interacting
bosons, but no proof has been done before. However, starting from the expression of ground state energy
for interacting bose gas, Note that, according to Baumgartner [15], there exists an interatomic potential
such that its scattering length is positive and it can hold no two-body bound state, but leads to a many-bodybound state. Such a complexity add a twist in the study of quantum many-body systems.

∗1 In fact, the phenomena observed by Zeeman was, in modern terminology, the anomalous Zeeman eﬀect in which D1 andD2 lines of sodium split into four and six, respectively (Brown et al. [7].) However, Zeeman thought that the number ofsplit lines was three. That erroneous observation accidentaly agreed well with the prediction based on the classical picture of
motion of electrons and polarization of light. Later, Cornu correctly observed that the number of split lines was four, and thecase in which splitting is more than three came to be termed as “anomalous” Zeeman eﬀect. Today the case of three split
lines is understood as a more special case than “anomalous” case, because such a splitting occurs only for atoms posessing
electronic ground state with zero nuclear spin.

4

CHAPTER 1. INTRODUCTION

1.1.2

Quests for BEC in a dilute atomic gas

Here we look back at the history of quests for BEC in a dilute atomic gas, following Kleppner et al. [16]. In
1976, based on a phenomenology, Stwalley and Nosanow [17] predicted that a gas of spin-polarized hydrogen(H↑) might have a remarkable property in views of low temperature physics. Because there is no boundstate in the triplet molecular potential between two H↑ atoms, a H↑ gas behaves like a simple monatomicgas. And because of the weak interaction between them and the atom’s low mass, they would not liquefy
even at absolute zero temperature. Consequently, it might be possible to cool H↑ to the quantum regime
and achieve BEC. That paper essentially launched the search for BEC in an dilute atomic gas. Moreover,H↑ has three advantages over other atoms:• BEC can be understood as overlapping√ of matter waves of atoms, whose scale of spacial length isthermal de Broglie length λdB ≡ h/ 2πmkB T . As it depends on the mass m of the atom as m−1/2 ,use of the lightest atom H↑ is favorable.
• H↑ has anomalously small s-wave scattering length and thus can be regarded as a quite nearly ideal
gas.
• The hydrogen plays a major role in fundamental physics as its energy structure can be calculated
from ﬁrst principle.
However, it was soon understood that the large dipolar loss in hydrogen gas was the major obstacle to coolhydrogen to reach BEC. Other reseachers began to seek other ways to cool atoms. In 1975, Hänsch andSchawlow [18] published the idea of laser cooling of neutral atoms, and in the same year, Wineland andDehmelt [19] also published the same idea for trapped ions. Then it evolved into Magneto-Optical Trap
(MOT) that not only cool atoms, but also trap them at the center of quadrupole magnetic ﬁeld formed infree space. The ﬁrst MOT was demonstrated by groups at Bell labs in 1987 (Raab et al. [20].) See Metcalf
and van der Straten [21, Sec. 6.2 and 11.4], Weiner [22, Sec. 10.2] and references therein. Alkali and alkali
earth atoms have only one electron in the outer shell, and thus have simple electronic energy structure.
Laser cooling depends on repeated absorptions and spontaneous emissions of laser photons forming a closed
cycle of electronic states, and thus the simple electronic energy structure is advantageous for other species.
In fact, uses of so-called “repump” laser which put atoms which got out of the closed cycle again into thecycle are doable for them. However, the highest phase space density reachable by MOT is not enough toreach BEC. This is because a random walk of an√atom caused by repeated scattering of photons balancesat the velocity termed as Doppler velocity vD ≡ ~γ/2m.
Thus one need another method for further cooling of atoms. The evaporative cooling, ﬁrst proposed by
Hess [23] in 1986, was soon demonstrated by the joint group of MIT and Bell labs in 1988 (Masuhara et al.
[24].) So far all achievements of producing BEC in dilute atomic gases use evaporative cooling as the ﬁnal
step of the procedure.
After the ﬁrst achievement of BEC, various kinds of its properties were investigated. Among them, thethermodynamic properties were the most basic. The agreement with theory was quite good for BEC of
weakly interacting atoms. Collective modes which comes from interaction were also studied.
See Table 1.3 for list of the ﬁrst achievements of quantum degeneracy for each atomic specie. Note that,most of them are alkalis and alkali earthes that have simple electronic energy structure and are suitable forlaser cooling, but quantum degeneracy in other species is realized as well. The ﬁrst success for a non-alkali
atom is hydrogen[25] which was pre-cooled by a criogenic liquid helium chamber and then evaporatively
cooled.
In 2003, reseachers at Innsbruck successfully produced BEC of 133 Cs (Weber et al. [26].) To circumventthe unwanted loss of atoms due to the atom’s peculiar resonant scattering properties, they utilized Feshbachresonance (Chin et al. [27]) along with the usual optical trapping methods. The signiﬁcance of cesium inmodern physics and metrology is best understood when we look at atomic clocks. Modern atomic clocks
that serve as frequency standard has cesium atoms in their heart to use its hyperﬁne transition as an
oscillator. Currently the deﬁnition of second, established at the 13th meeting of CGPM∗1 in 1967 [28], isbased on such cesium atomic clocks. Likewise, in 2007, Italian group successfully produced 39 K BEC (Roatiet al. [29].) They circumvent problems due to the negative and small scattering length of 39 K (−33 a0 )which leads to unstability of the condensate by utilizing Feshbach resonance (D’Errico et al. [30].) By
inducing Feshbach resonance during evaporative cooling, they changed the scattering length to the large
positive value (150 a0 .) In fact, this is larger than that of 87 Rb, leading to a good eﬃciency of evaporative∗1 CGPM (Conférence Générale des Poids et Mesures) is one of the three international organizations created by the Metre
Convention.

1.1. HISTORICAL OVERVIEW

5
BosonsIsotope
87

Rb
Li
23
Na
1
H
4
He∗
41
K
133
Cs
174
Yb
52
Cr
170
Yb
39
K
7

Group
JILA and Colorado [2]Rice [4, 31]
MIT [3]
MIT [25]
Orsay [32]LENS [33]Innsbruck [26]Kyoto [34]
Stuttgart [35]Kyoto [36]LENS [29]YearNn [cm−3 ]T [nK]

1995
1995
1995
1998
2001
2001
2003
2003
2005
2007
2007

3

2 × 101 × 105
5 × 105
1 × 109> 5 × 103
1 × 104
1.6 × 104
5 × 103
5 × 104
1 × 104
3 × 10412

2.5 × 10
3.7 × 1012
> 1014
4.8 × 1015
1013
6 × 1011
1.3 × 1013
4.7 × 1014

170
300
2000
5000
160
160
21
460
700
200
100

Year

N

n [cm−3 ]

T /TF

1995
1995

7 × 101 × 103

0.25

Fermions
Isotope
40
6

K
Li

Group
JILA and Colorado [37]
Rice [38]5

0.5

Molecular BEC
Isotope

Group

Year

23

MIT [39]
JILA, Colorado, and NIST [40]

2003
2003

6

Na2
Li2

Table 1.3: Achievements of quantum degeneracy in a dilute atomic gas (mixture,) sorted by Year.
Heteronuclear resonance

Homonuclear resonance
Isotope
Bosons

Group

Year

Cs
Li
23
Na
85
Rb
87
Rb
52
Cr
41
K

[27]
[41, 42]
[43]
[44]
[45]
[46]
[47]

2000
2002
1998
1998
2002
2005
2009

6

[48]
[49, 50]

2002
2002

133

7

Fermions

40

Li
K

Isotopes
BF

85
7

FF

6

Group

Year

Rb-87 Rb
Li- 87 Rb

[51]
[52]

2006
2009

Li- 40 K

Innsbruck,
Amsterdam,
Eindhoven,
NIST,
and
Maryland [53]2008

Table 1.4: Observations and theoretical predictions of Feshbach resonance in a dilute atomic gas (mixture,)sorted by Year.
cooling. Feshbach resonances observed for other atomic species such as also have similar possibilities to“tame” undesirable collisional properties for eﬃcient evaporative cooling. Moreover, heteronuclear Feshbachresonances found for Bose-Fermi mixtures such as

1.1.3

Optical lattice: an ideal tool for simulating crystalline matters

The research of optical lattice was developed along with that of BEC. Various methods to form various
kinds of periodic lattices are proposed using standing waves. See the reviews for “crystallography” of an
optical lattice[54, 55]. Some of them have various advantages, such as insensitivity to phase ﬂuctuationsof the laser and approximate uniformness of the potential[56]. Moreover, an optical lattice can be used forcooling atoms[55, 57, 58] when combinated with optical methods producing dissipative force, such as PGC(polarization gradient cooling.)
Then two ﬂows of researches combines as one; in 2002, the German groups announced their success inobservation of the phase transition of dilute gas of atoms conﬁned in an optical lattice. This is a mile-stone
in the ﬁeld of ultracold atoms, since this opens the way to simulate condensed matter systems in the real

6

CHAPTER 1. INTRODUCTION

world which consists of periodic lattice of ions and electrons spread inside it.
There are many possibilities for use of optical lattices exept for simulation of crystalline materials. For
example, the variation of eﬀective mass is observed in an optical lattice. Lastly, our group seeks the way
towards the realization of ultracold polar molecules in the absolute (both electronic and rovibrational)
ground state.
Our method goes as follows: ﬁrst, we simultaneously create pure samples of 87 Rb and 41 K BECs by
evaporative cooling in a magnetic trap. Then we adiabatically transfer them into a FORT (far-oﬀ-resonanceoptical dipole trap) to reduce their density to about 1013 cm−3 . Finally, we superimpose an optical lattice
onto them. Here, the reduced densities of BEC assure unit ﬁlling factors of the lattice for each specie. Then
we apply magnetic ﬁeld to induce the heteronuclear Feshbach resonance to bind them into a single molecule.
Such a kind of molecules, called Feshbach molecules, is not suitable for experiments in which interactions
between electric dipole moments play dominant roles, because they are so shallowly bound that their electric
dipole moments are small. Thus we use STIRAP (stimulated Raman adiabatic passage) to adiabaticallytransfer them into the absolute ground state. Note that, the fact mentioned above explains the need for an
optical lattice during combination process by Feshbach resonance. Collisions between atoms and Feshbach
molecules lead to dissociation of Feshbach molecules, since the kinetic energy of atoms in BEC is comparable(about tens of kilohertz times h, Planck constant) than the binding energy of Feshbach molecules (about
hundreds of kilohertz.) Thus we need to conﬁne BEC in the combined potential of a magnetic trap and an
optical lattice to Then we will obtain a pure sample of BEC of ultracold polar molecules.

1.2

Contribution of this work

We constructed a 3D optical lattice which can be superimposed onto a mixture of 87 Rb and 41 K BECs ina magnetic trap. We used each axis of the 3D optical lattice as a 1D optical lattice to obtain interference
patterns formed by a 87 Rb BEC released from the combined potential of the magnetic trap and the 1D
optical lattice.
Moreover, we deepened the 1D optical lattice so that the interference patterns could be diminishedand ﬁnally lost. This loss of the interference patterns may be attributed to suppression of tunneling ofatoms between neighboring sites of the 1D optical lattice. We conﬁrmed that the possibility of heating theatoms could be denied by shallowing the 1D optical lattice from the level at which there was no interference
pattern, and then observing the recovery of the interference patterns. This assures that the whole procedure
was done in an adiabatic manner.
Nextly, by using two of the three axes of the 3D optical lattice as a 2D optical lattice, we again obtainedinterference patterns that was 2D in this case. By adiabatic tuning of the depth of the 2D lattice, we
observed similar loss of the interference pattern and its recovery.
These result suggests that the 3D optical lattice can be used to observe the superﬂuid to Mott insulatortransition of 87 Rb BEC when we properly reduce the density of the BECs (an ongoing project in ourgroup) and activate all axes of the 3D optical lattice simultaneously. Moreover, it can be adiabaticallysuperimposed onto a mixture BEC of 87 Rb and 41 K that is already realized in our group so that for the
mixture BEC with unit ﬁlling factors for each specie. Then we will be able to combine them into ultracold
polar molecules in the absolute ground state using Feshbach resonance and STIRAP.

1.3

Structure of this thesis

First, the basic theoretical concepts are introduced (Chap. 2.) The main object of this ﬁeld of research is
an atom. Some formulas on interactions between an atom and electromagnetic ﬁelds used in subsequent
sections are summarized (Sec. 2.1.) By cooling atomic gas, and thus reducing its degrees of freedom, we
can observe a transition from the classical to the quantum degenerate state, namely, the Bose-Einsteincondensation. Its properties relevant to our experiment are summarized (Sec. 2.2.) The macroscopic
quantum nature of BEC is by itself so rich and attractive, but its signiﬁcance in modern condensed matterphysics can be much more enhanced by the combination with an optical lattice; that is why we choosen
it as the main topic of this master’s thesis. Physics governing a BEC in an optical lattice is explained,
including important consequenses used in analysis of the experimental results (Sec. 2.3.)
Nextly, our experimental system is described (Chap. 3.) We describe the structure of our vacuum
chamber, in which our experiments have been done (Sec. 3.1.) There we also describe a process to obtainultra high vacuum to make a lifetime of the condensate long enough. As another basis of our experimental
system, we describe optical system to provide frequency-locked lasers with various amounts of frequency

1.3. STRUCTURE OF THIS THESIS

7

shift and having enough powers (Sec. 3.2.) We also describe our setup of MOT and the procedure of CMOT(Compressed MOT) for eﬀective increase of phase space density (Sec. 3.3.) The procedure of optical pumping to prepare atoms in desirable spin states are also described there. After CMOT, further cooling ofatomic cloud is done by rf-forced evaporative cooling. We use a magnetic trap in Ioﬀe-Pritchard conﬁguration to conﬁne atomic clouds. Its parameters and subsystems attached for various purposes are described(Sec. 3.4.) As its name suggests, rf-forced evaporative cooling needs rf (radio-frequency electromagnetic
radiation) with a high power. The system to apply rf on atomic clouds are described (Sec. 3.5.) At theﬁnal stage of evaporative cooling, BEC is created and the optical lattice is superimposed on it. In order tosuppress hopping of atoms between neighboring lattice sites, we need a deep potential. Our experiments
heavily rely on visual image of the atomic cloud for measurement of its properties. The imaging system forabsorption imaging of the cloud is described (Sec. 3.7.)
Then we describe our experimental results (Chap. 4.) Our ﬁnal objective is creating ultracold polar
molecules of 41 K-87 Rb and we are each step for its realization. Implementation of a 3D optical lattice is
one step; and as one of other sub-goals, we created 41 K BEC.

Chapter 2Theoretical Backgrounds
2.1

Interactions of Atoms and Electromagnetic Radiations

Here we summarize the theory of interaction of atoms and electromagnetic radiations which are relevant
to use for the subsequent analysis of experiment results. Though one may work out in a fully quantummechanical treatment in which both atoms and electromagnetic waves are quantized and expressed as
so-called “dressed” states, the semiclassical theory is known to give correct results (see, e.g. Grimm et al.
[59].) So the following discussion is based on the semiclassical theory.

2.1.1

Optical Dipole Traps

To a ﬁrst approximation, the interaction between a neutral atom and electric ﬁeld E is the dipole interaction:
1∆E = − d · E,
(2.1)
2
where d denotes the induced dipole of the atom. This energy shift ∆E is called (ac) Stark shift. Let αdenote the polarizability of the atom: d = αE. Assuming the classical model of an electron bound to theatomic nucleus by a harmonic potential, α generally depends on the driving frequency ω, and has complex
value to describe absorption of photons by the atom. The energy shift ∆E, corresponding to the real partof α, give rise to the optical dipole potential Udip felt by an atom. And the imaginary part corresponds tothe photon scattering rate Γsc , which is the rate of spontaneous emission of a photon absorbed by the atom.
The expression for the depth of the dipole potential for the laser with intensity I is
(
)
3πc2
Γ
Γ
Udip (I; ω, ωX ) = − 3+I,2ωX ωX − ω ωX + ω(2.2)

where c is the speed of light, ω = 2πc/λ is the (angular) frequency∗1 of the laser of wavelength λ, ωX is theresonant frequency of atom X (X = K or Rb in our case,) and Γ is the natural linewidth of that transition,
which is the measure of expectation value of the dipole matrix element. The values of Γ are tabulated for
major species∗2 and for alkali atoms, but the approximated value Γ/2π ≈ 6 MHz usually suﬃces. Then we
can use the well-known rotating-wave approximation in which Γ/(ωX + ω) ∼ 10−8 is set to zero, obtaining
Udip (I; ω, ωX ) ≈3πc2 Γ
3 ∆ I ≡ Udip (ω, ωX ) · I.
2ωX

(2.3)

Here, ∆ ≡ ω −ωX is the detuning of the laser frequency from the resonant frequency of the atom. From thisdeﬁnition of the detuning, we see that the red-detuned laser, for which the detuning is negative, creates anattracting potential for atoms. And in the last formula, we deﬁned the propoptional coeﬃcient Udip (ω, ωX )to separate dependence of the potential depth on ω from linear dependence on I.
Similarly, the expression for the photon scattering rate is
(
)3 (
)2
(
)3 2
ω
Γ
Γ
3πc2
ω
Γ
3πc2
+
I
≈
I
Γsc (I; ω, ωX ) =
3
3
2~ωX
ωX
ωX − ω ωX + ω
2~ωX
ωX
∆2
≡ Γsc (ω, ωX ) · I,

(2.4)

∗1 We use s−1 as the unit of angular frequency and Hz as frequency. We also use s−1 as the unit of rate of events inprobabilistic phenomena; Hz is only used for periodic phenomena.
∗2 See, e.g. Steck [60] for 87 Rb, 85 Rb, 23 Na, and 133 Cs. And see Gehm [61] and cLeBlanc [62] for 6 Li and 40 K, respectively.

8

104

103

103

102

102

101

101

100

100

10-1

10-1

10-2

10-2

10-3
10-1

100
Laser intensity [kW/cm2]

Photon scattering rate [s-1]Trap depth [µK]2.1. INTERACTIONS OF ATOMS AND ELECTROMAGNETIC RADIATIONS

9

Depth(781 nm)
Depth(800 nm)
Depth(900 nm)
Depth(1064 nm)
Sc.rate(781 nm)
Sc.rate(800 nm)
Sc.rate(900 nm)
Sc.rate(1064 nm)

101

Figure 2.1: Udip (I; ω, Rb) and Γsc (I; ω, Rb) plotted for several laser frequencies.
where we again used the rotating-wave approximation and separated linear dependence on I. The spontaneous emission is an incoherent process leading to the unwanted heating of the atomic cloud. In order tosupress the photon scattering rate to the level at which it does no harm during experiments, we can use
the far-detuned (|∆| ≫ Γ) laser for the optical dipole traps. It is worthwhile to compare dependences ofUdip on ∆ and I, and that of Γsc :IIvs. Γsc ∝ 2 ,
(2.5)
∆
∆
where ω/ωX is approximated to unity. As |∆| increases, Γsc decreases faster than |Udip |. Thus we canﬁnd the optimal point in the ∆-I plane at which |Udip | is high enough (about 10 µK) and Γsc is low
enough (1/Γsc ≫ texp , where the length of experiment texp is about 0.1 s.) In Figure 2.1, we plotted thedependencies of Udip (I; ω, Rb) and Γsc (I; ω, Rb) on laser intensity I for several values of laser frequencies
ω = 2πc/λ corresponding to λ = 781, 800, 900, 1064 nm. Here we took into account of D1 line (explainedlater,) though it does not signiﬁcantly alter the shapes of the curves. We clearly see that the photon
scattering rates (dashed lines) decays faster than the potential depth (solid lines) as we increase detuningfrom 781 nm to 1064 nm. Moreover, under the constraint that the photon scattering rate is kept at aconstant value G, we can numerically solve Γsc (I; ω, ωX ) = G for ω to obtain laser frequency ω(I, G; ωX )which gives the necessary amount of detuning for that particular value of the photon scattering rate. The
resultant function Udip (I, G) ≡ Udip (I; ω(I, G; ωX ), ωX ) is plotted in Figure 2.1 for G = 0.1 s−1 and X =Rb. As we increase the intensity, we need larger (red) detuning, thus longer wavelength, to keep the photon
scattering rate constant. But resultant depth of the optical dipole potential increases almost linearly withI in the region shown in the Figure.
For application to the trapping of alkali atoms, we need to take into account both of D1 (2 S1/2 → 2 P1/2 )and D2 ( 2 S1/2 → 2 P3/2 ) transitions, because our optical lattice uses a far-detuned laser with wavelengthof 1064 nm. The detuning is larger than the diﬀerence between the frequencies of two lines, as can be seenin Figure 2.3. In the ﬁgure, the lower and upper horizontal axis shows wavenumbers and corresponding
wavelengths of transitions, respectively. The area of the bars indicates the relative intensity of observed
signal with wavenumbers ranging from 9000 cm−1 to 17000 cm−1 , adapted from NIST database[63]. The
largest and the second largest bars are D2 and D1 line, respectively. The vertical broken line shows
the wavenumber of the optical lattice. We clearly see that both of the D1 and D2 lines have dominant
contribution to optical dipole interaction potential of the optical lattice, and that contributions from othertransitions are negligible. Other alkali atoms have similar wavenumbers for D1 and D2 transitions. Thenthe depth of the optical dipole trap with linear polarization is calculated from the following formula:)(Γ
πc2 2Γ
+,(2.6)
Udip (ω, X) =
3
2ωX
∆2
∆1
Udip ∝

where ∆i denotes the detuning from the Di line of the atom X. Here, the hyperﬁne structures of both theground and the excited states are assumed to be unresolved because of the large detunings. For example,

CHAPTER 2. THEORETICAL BACKGROUNDS
20
10
5

980
U(I, G=0.1 s)
λ

930

2
1
0.5

880

0.2

780

830

0.01

0.03

0.1

0.3

1

3

Laser wavelength [nm]Trap depth [µK]1010Laser intensity [kW/cm2]

Figure 2.2: Plot of λ(I) ≡ 2πc/ω(I, G; ωX ) and Udip (I; G) for G = 0.1 s−1 and X = Rb.

11

12

13

600

900

10

700

1000

9

800

1100
1064

Wavelength [nm]

14
3

15

16

17

-1

Wavenumber [10 cm ]Figure 2.3: Schematic view of spectroscopic line data on 87 Rb known so far.

in the case of 87 Rb and the laser with wavelength of 1064 nm, we have
c
c
∆2
=
−
= (282 − 384) THz = −103 THz,
2π1064 nm 780 nm
∆1
c
c
=
−
= (282 − 377) THz = −95 THz,
2π1064 nm 795 nm

(2.7)
(2.8)

for respective lines (remember that negative values of ∆ means red-detuning.) The two terms in brackets
of eq. (2.6) represent the contributions of two lines to the total dipole potential. Likewise, the expression
for the photon scattering rate isΓsc (ω, X) =

πc2
3
2~ωX

(

2Γ2
Γ2
+
∆22
∆21

)
.

(2.9)

Assuming again the laser with wavelength λ = 1064 nm (corresponding to ω = 1.77 × 1015 s−1 ,) we set(Rb)

Udip ≡ Udip [(ω, ωRb,1 , ωRb,2 )/2π = (282, 377, 384) THz]
= kB · [4.45 µK/(W/cm2 )]= h · [3.68 MHz/(W/cm2 )]2(Rb)= [46.2/(W/cm )] · Erec,

(2.10a)(K)
Udip ≡ Udip [(ω, ωK,1 , ωK,2 )/2π = (282, 389, 391) THz]

= kB · (4.01 µK/W)= h · (3.31 MHz/W)

For the deﬁnition of the recoil energy Erec , see Sec. 2.3.

(K)
= 19.6 Erec

(2.10b)

2.1. INTERACTIONS OF ATOMS AND ELECTROMAGNETIC RADIATIONS

2.1.2

11

Rabi frequency

Basics: Rabi frequency of the two-level system (without decay)
Here, we consider the two-level system coupled to a coherent monochromatic eletromagnetic wave. The
wavefunction of the system |ψ(t)⟩ is expressed by two time-dependent coeﬃcients cg (t) and ce (t), whosesquare represent the probability of ﬁnding a system in the ground state |g⟩ or the excited state |e⟩, respectively; that is,
cg (t) ≡ ⟨g|ψ(t)⟩,ce (t) ≡ ⟨e|ψ(t)⟩.

(2.11)We are going go use the time-dependent perturbation theory explained in App. A.2. The unperturbed
Hamiltonian H0 is represented by the following matrix:
(
)
0
H0 =
(2.12)~ω0
The upper-left zero means that we measure energies from that of the ground state |g⟩. The perturbation
Hamiltonian V is represented by the following matrix:()~Ω cos(ωt)
V =(2.13)~Ω cos(ωt)
Here we used the so-called electric dipole approximation in which the spatial dependence is neglected in,e.g.,⟨e|V |g⟩ = ⟨e|[ϵE0 cos(kz − ωt) · (−er)]|g⟩(2.14)and thus−eE0 cos(ωt)ϵ · ⟨e|r|g⟩ = ~Ω cos(ωt)(2.15)

is used instead. Here, we deﬁned the (one-photon) Rabi frequency Ω such that~Ω ≡ −eE0 ϵ · ⟨e|r|g⟩.

(2.16)

Suppose that the two-level system under consideration is a hydgrogen atom. Then the numerical value for
the Rabi frequency can be computed from the ﬁrst principle. Denoting the wavefunctions of the 1 2 S1/2
state and the 2 2 P3/2 state by ϕg (r) and ϕe (r), respectively, we have∫⟨e|r|g⟩ = ϕ∗e (r)rψg (r)dr.
(2.17)
However, we rarely do such a calculation and alwasy use the experimental value. The state vector |ψ(t⟩can be represented by the following 2-dimensional column vector:() ()⟨g|ψ(t)⟩c (t)|ψ(t)⟩ == g.
(2.18)⟨e|ψ(t)⟩ce (t)
Then we go to the interaction picture. From Eq. (2.12), we have()
1
iH0 t/~e=.
eiω0 t(2.19)

Thus, in the interaction picture, the state vector |ψ I (t)⟩ and the perturbation Hamiltonian V I (t) is respectively represented as
(
)(
) (
) (
)
1
cg (t)
cg (t)
dg (t)|ψ I (t)⟩ ==≡,(2.20)
eiω0 tce (t)
eiω0 t ce (t)de (t)()()()
1~Ω cos(ωt)
1
V I (t) =eiω0 t~Ω cos(ωt)
e−iω0 t()~Ωe−iω0 t cos(ωt)=.
(2.21)iω0 t~Ωecos(ωt)Here, for each of i = g, e, we deﬁned di (t) ≡ eiϵi t/~ ci (t), where ϵi ≡ ⟨i|H0 |i⟩ is 0 for |g⟩ and ~ω0 for |e⟩.
Moreover, we use the so-called rotating-wave approximation (RWA); that is, we neglect oscillating term with

12

CHAPTER 2. THEORETICAL BACKGROUNDS

a high frequency such as ei (ω + ω0 ), as its frequency is so high that its eﬀects is averaged out and it never
have inﬂuence on physically observable quantities. We can do such an approximation for the matrix-elementof V I (t). For example, we have
] 1
1 [ i(ω−ω0 )t1e−iω0 t cos(ωt) =e+ e−i(ω+ω0 )t ≈ ei(ω−ω0 )t ≡ ei∆t .
(2.22)
222Here, we deﬁned the detuning ∆ as ∆ ≡ ω − ω0 . Then the perturbation Hamiltonian can be written as
)
(
~Ω
ei∆t
I
.
(2.23)
V (t) ≈
2 e−i∆t
Then the explicit solution of the coupled linear diﬀerential equations
(
)
(
)(
)
iΩ
d˙g (t)ei∆tdg (t)
i~∂t |ψ I (t)⟩ = V I (t)|ψ I (t)⟩ ⇐⇒=−.
de (t)d˙e (t)2 e−i∆t(2.24)

can be obtained by standard techniques. Here, we denote the time-derivative by the dot over a variable.
First, we apply simple substitutions:][][iΩ −i∆t ˙iΩ −i∆tiΩ i∆tiΩ −i∆t˙¨dg (t) −edg (t) = −i∆de (t) −e· − e de (t) .
de (t) = −i∆ − e
2
2
2
2
2
Ω
de (t) = 0.
(2.25)∴ d¨e (t) + i∆d˙e (t) +4
The uncoupled equation for dg (t) is a second-order linear diﬀerential equation with constant coeﬃcients.
Thus it can be solved by looking at the characteristic equation:
α2 + i∆α +Ω2i= 0 ⇐⇒ α = − (∆ ± Ω′ ) ,42(2.26)√
where we deﬁned the eﬀective Rabi frequency Ω′ under the detuning ∆ as Ω′ ≡ Ω2 + ∆2 . Using the above
α’s, we can write down the general solution using two coeﬃcients D+ and D− :′′de (t) = D+ e−i(∆+Ω )t/2 + D− e−i(∆−Ω )t/2 .

(2.27)The values of these two coeﬃcients are determined by the initial conditions, typically at t = 0. For example,
we specify that dg (0) = 1 and de (0) = 0; that is, all the population accumlates in the ground state. Then,
using the latter condition, we have( ′ )Ωtde (0) = D+ + D− = 0 ∴ de (t) = −2D+ e−i∆t/2 sin.
(2.28)
2
Then, using the former, we have
¯
¯
2i
2i
dg (0) = ei∆t d˙e (t)¯¯= − D+ Ω′ = 1ΩΩ
t=0∴ D+ =i Ω.
2 Ω′(2.29)

Finally, we have( ′ )
Ω −i(ω+ω0 )t/2
Ωt
ce (t) = ede (t) = −i ′ e
sin
,
Ω
2
(
)
Ω′
∆
Ω′ t
2i i∆t ˙i∆t/2
cos t − i ′ sin.
cg (t) = dg (t) = e de (t) = eΩ2Ω2−iω0 t(2.30a)(2.30b)Thus, the population of each state is as follows:Ω2
1 − cos Ω′ t,
Ω2 + ∆22Ω2
1 + cos Ω′ t∆2
|cg (t)|2 = 2+Ω + ∆22Ω2 + ∆2|ce (t)|2 =(2.31a)(2.31b)The dynamics of this system is illustrated in Metcalf and van der Straten [21]. Note that, though the
population for each state goes back to the the initial state after the time of 2π/Ω′ passed, the wavefunctionof the system acquires the extra factor of −1. This is the remarkable feature of the spin-1/2 system.
Note that, if we take into account of the decays from the excited states to lower states, then the processneeds to be described by the density matrix, since the decay is an incoherent process and the resultant
statistical mixture of the states cannot be written by a state vector in any Hilbert space. We do not discusssuch cases; refer to the standard textbooks on quantum mechanics.

2.1. INTERACTIONS OF ATOMS AND ELECTROMAGNETIC RADIATIONS

13

Advanced: Rabi frequency of the symmetric V-shaped system (without decay)
Now we calculate the three-level case. However, we restrict ourselves to the case of symmetric V-shaped
structure. There are a ground state |g⟩ and two excited states |e1⟩ and |e2⟩ whose energies are degenerate.
Let ~ω0 denote such energies. Then the unperturbed Hamiltonian H0 and perturbation Hamiltonian V are
respectively represented as
H0 = 



~ω0

,

0

(2.32a)~ω0
V = ~Ω cos(ωt)

~Ω cos(ωt)~Ω cos(ωt) ,~Ω cos(ωt)(2.32b)in the Schrödinger picture. Here, Ω again denotes the one-photon Rabi frequency. In the following, we see
how the shape of the level structrue makes the eﬀective Rabi frequency diﬀers from Ω. From Eq. (2.32a),
we have iω te 0.
1(2.33)
eiH0 t/~ = eiω0 tThus, in the interaction picture, we have
  iω t⟨e1|ψ(t)⟩e 0IiH0 t/~ ⟨g|ψ(t)⟩ =|ψ (t)⟩ = e⟨e2|ψ(t)⟩



 

ce1 (t)de1 (t)  cg (t)  ≡  dg (t)  .
1
iω0 t
e
ce2 (t)de2 (t)(2.34)

And the perturbation Hamiltonian is iω te 0I
V (t) =
 ~Ω cos(ωt)

1
iω0 t



e

iω0 t

= ~Ω e−iω0 t cos(ωt)

e

iω0 t

e

~Ω cos(ωt)~Ω cos(ωt)

cos(ωt)cos(ωt)  −iω te 0~Ω cos(ωt)
~Ω  i∆te−iω0 t cos(ωt) ≈
e
2


1


−iω0 t

e
−i∆t

e

−i∆t



ei∆t 

(2.35)e

Again, we used RWA in the last equation. Then we again try to solve the coupled linear diﬀerential
equations with respect to di (t)’s (i = g, e1, e2.) However, to simplify subsequent calculations, we assume
the special initial condition: dg (0) = 1 and de1 (0) = de2 (0) = 0. Then the time-evolution is symmetric withrespect to i = e1 and e2. Thus we rename those coeﬃcients as follows:d0 (t) ≡ dg (t),d1 (t) ≡ de1 (t) = de2 (t).

(2.36)

Then the coupled diﬀerential equations are rather simpliﬁed:()(
iΩd˙0 (t)=−d˙1 (t)2 e−i∆t

2ei∆t

)(

)
d0 (t).
d1 (t)(2.37)

It is worthwhile to compare the above quation with Eq. (2.24). The factor of two in the matrix element
appears because there are in fact two ﬁnal states for the transition from the ground state. Solving thecoupled equations with the above initial condition, we have( ′ )Ωt
Ω −i(ω+ω0 )t/2
e
sin
,
Ω′
2
(
)
2i
Ω′
∆
Ω′ t
cg (t) = d0 (t) = ei∆t d˙1 (t) = ei∆t/2 cos t − i ′ sin
.
Ω
2
Ω
2

ce1 (t) = ce2 (t) = e−iω0 t d1 (t) = −i(2.38a)(2.38b)√
These solutions are same with Eq. 2.30 except the deﬁnition of the eﬀective Rabi frequency, Ω′ ≡ 2Ω2 + ∆2 .
Note that, even under the resonance condition ∆ = 0, the maximal population in the excited states is 1/2.

14

CHAPTER 2. THEORETICAL BACKGROUNDS

2.2

Bose-Einstein condensation of ideal bosons

Here we present a few properties of BEC which are relevant to the subsequent analysis of their behavior ina optical lattice.
The condensate fraction N0 /N can be expressed as a function of temperature T :
( )3
N0
T
=1−
.
(2.39)N
Tc
Thus by the measurement of the condensate fraction, we know the temperature of the atomic cloud consisting of the condensate and thermal gas.
Quite many aspects of BEC can be explained in terms of Gross-Pitaevskii equation:
[
]
∇2
4πa~2
∂
2
+ U (r) +|Ψ| Ψ(r, t).
i~ Ψ(r, t) = −∂t
2mm(2.40)

This is a time-dependent nonlinear Schrödinger equation. By separation of variables Ψ(r, t) ≡ eµt/(i~) Φ(r),we have time-independent one:[]∇2
4πa~2 2µΦ(r) = −+ U (r) +|Φ| Φ(r).
(2.41)
2mm
Here µ have the meaning of the chemical potential of the condensate.
In Thomas-Fermi approximation, one ignores the kinetic term of the Gross-Pitaevskii equation, obtaining[]
4πa~2 2µΦ(r) = U (r) +|Φ| Φ(r)(2.42)
mfor the time-independent case. Then we can deﬁne the so-called “Thomas-Fermi radius” l of the condensate
for which trapping potential U (l) equals the chemical potential µ and thus density term (∝ |Φ|2 ) vanishes.
Due to the anisotropy in the trapping potential, Thomas-Fermi radius generally depends on the direction.
Thus, by li we denote Thomas-Fermi radius for the direction i ( = r or z.)
For each direction of the trap, Thomas-Fermi radius of the condensate can be calculated from thecorresponding trap frequency using the following formula:()1/5 {}[]1/5
1.72 N a~2 ω̄ 3
100 (ω̄/2π)3 N
5.62 (87 Rb)
li =
=
×
·
µm.
(2.43)
6.82 (41 K)
ωim2
ωi /2π 5 × 105 105
Here, ω̄ ≡ (ωx ωy ωz )1/3 is the geometric mean of trapping frequencies for three directions of the trap.
Coeﬃcients for respective atoms are calculated using these values:aRb = 100 a0

aK = 60 a0−25mRb = 86.9 u = 1.44 × 10

2.3mK = 41.0 u = 6.80 × 10kg

(2.44)−26kg(2.45)

Optical lattice

By utilizing a standing wave formed by two counter-propagating laser beams, one can form an opticalanalogue of the periodic lattice of ions in a crystal. Here we discuss the behavior of atoms conﬁned inside
it.

2.3.1

Periodic potential formed by a standing wave of two counter-propagating
lasersFor simplicity, our discussion begins with the case of plane waves. Suppose that two plane waves E andE ′ have same frequency ω and opposite direction of propagation k and −k, respectively. Then the electricﬁeld Etot formed by superposition of the two is
Etot (r, t) = E(r, t) + E ′ (r, t)
(
) (
)
∑
∑
′
=
ϵi Ei cos(ωt − k · r + δi ) +ϵi Ei cos(ωt + k · r + δi + ∆i ) ,ii(2.46)

2.3. OPTICAL LATTICE

15

where ϵi ’s are unit vectors pointing to the direction of polarization (hereafter i varies over two symbols “s”
and “p” representing s- and p-polarization, respectively,) δi ’s are initial phases for each linearly polarizedwave with wave vector k, and ∆i ’s are additional terms to represent initial phases for waves with wave
vector −k. Such treatment implies that the waves with −k originates from retroreﬂection of those with k.
In the following, we will discuss in a very general manner in that both amplitudes and initial phases aretreated as independent eath other, unless otherwise stated.
According to the classical theory of interaction between atoms and electric ﬁelds, the energy shift Udip
due to an electric dipole moment induced by the electric ﬁeld is proportional to the squared amplitude ofthe electric ﬁeld:∑[
Udip ∝ |Etot |2 =Ei2 cos2 (ωt − k · r + δi ) + Ei′2 cos2 (ωt + k · r + δi + ∆i )i

+ 2Ei Ei′ cos(ωt − k · r + δi ) cos(ωt + k · r + δi + ∆i )]∑ { E2
E ′2
E ′2
E ′2
i
+ i cos[2(ωt − k · r + δi )] + i + i cos[2(ωt + k · r + δi + ∆i )]
=
2
2
2
2
i
)
′
+ Ei Ei [cos(2ωt + 2δi + ∆i ) + cos(2k · r + ∆i )] .
(2.47)
Here, we used the identity
cos A cos B =1[cos(A + B) + cos(A − B)]2(2.48)and its corollary cos2 A = (1 + cos 2A)/2, obtained by setting A = B.
The frequency of light as an electromagnetic wave is so large as several hundreds of THz∗3 , so only theirtime-average can physicaly aﬀect atoms, or in particular, the electrons which are its ingredients. Denoting
the time-averaged value of a physical quantity A by ⟨A⟩, we have⟨|Etot |2 ⟩ =∑ [ E2
i

2

i

+

]
Ei′2
+ Ei Ei′ cos(2k · r + ∆i ) ,2(2.49)

since all terms containing cos 2ωt are averaged out to zero.
Idealized case
In an idealized case in which ∆i = 0, Es′ = Es and Ep′ = Ep , we have⟨|Etot | ⟩ =
2

(
∑

)
Ei2

[1 + cos(2k · r)] .

(2.50)

i

That is, we have a periodic (in particular, sinusoidal) potential for atoms with period of 2π/(2k) = λ/2.
Here, λ is the wavelength of the laser. The origin of the coordiate system is understood to lie on the planeof retroreﬂection, since the boundary condition∗4¯⟨|Etot (r, t)|2 ⟩¯r·k=0 = 0(2.51)
is satisﬁed for any t. Note that the (anti)nodes for the both directions of polarization agrees regardless of
the initial phases δi . Thus we set Ep = 0 and write Es as just E. Shifting the origin by multiples of λ/2,that is, to one of the antinodes at which the potential takes its minimum (for red-detuned case,) we expand
the above expression to obtain its harmonic approximation:[]
2E 2 cos(2k · r) = 2E 2 1 − (k · r)2 + O(k · r)4 .

(2.52)

This result will be used to calculate the frequency of a small oscillation of atoms conﬁned in an opticallattice.
∗3 For example, IR (infrared) light with wavelength of 1064 nm has frequency of 282 THz.

∗4 Here we assumed the use of metallic mirrors made of metallic material having a nearly inﬁnite conductivity (σ = 4.9 × 107(Ω · cm)−1 for gold, for example) for the boundary condition. The same also applies for dielectric mirrors that are used inpart of our experimental system.

16

CHAPTER 2. THEORETICAL BACKGROUNDSE ′⃗e2E⃗e1

θθ ≡ 2k · r mod 2π
d2 = ⟨|Etot |2 ⟩dLoss in visibility {Figure 2.4: Geometric interpretation of the eﬀect of broadening of a retroreﬂected beam.
Considerations on non-ideal case I: broadening of beams
For evaluation of the eﬀect of misalignment, consider the case in which Ep = Ep′ = 0 and ∆i = 0, but
Es′ ≡ E ′ < E ≡ Es . The physical meaning of this is as follows: we assume Gaussian beams which havenonzero curvature for their wavefront. Thus their intensity decays like 1/z 2 as it propagates through freespace (z → ∞.) In order to compensate this decay, we use a concave mirror for retroreﬂection. The radiusof curvature R is chosen so that foci of the two beams coincide at the center of a magnetic trap. That is,the distance d between the beam waist of the forward beam and the concave mirror should be set as d = R(or equivalently, d = 2f , if we use the focal length f .) Nevertheless, a real experiment cannot be free fromdeviations from ideal conditions. One need to evaluate eﬀects of mismatch between d and R on visibilityof an optical lattice.
First, we give an intuitive image of the problem under consideration. Under the above assumpiton, we
have⟨|Etot |2 ⟩ =1 2
1[E + E ′2 + 2EE ′ cos(2k · r)] = |E⃗e1 + E ′⃗e2 |2 .
2
2

(2.53)In the last equation, we used two abstract 2D geometric vectors ⃗e1,2 of unit length such that the mutual
angle between them is 2k · r modulo 2π (as depicted in Figure. 2.4.) Imagine that the root of E ′⃗e2 is set
at the tip of E⃗e1 , and E ′⃗e2 changes its orientation in a circular motion. Then the distance between theroot of E⃗e1 and the tips of E ′⃗e2 changes accordingly, and the square of the distance gives the depth of theperiodic potential of a non-ideal optical lattice.
For example, we analyze experimental data taken at our lab. An example plot is shown in Figure 2.5.
The cross-shaped markers are measured values of 1/e2 diameter of intensity of an output beam from a ﬁber
coupler F230FC-C used at Path (A) of the optical lattice. Least-square ﬁt of those by a phenomenologicalﬁtting function√()2z − z0f (z; w0 , z0 , zR ) = 2 × w0 1 +(2.54)zR
yielded parameters of w0 = 0.14(1) mm, z0 = 520(4) mm, and zR = 61(6) mm. The ﬁtting functionf (z; w0 , z0 , zR ) given above is “phenomenological” because it treats beam radius w0 at the beam waist andRayleigh length zR of the beam as two independent parameters, though the latter is given by πw02 /λ, where
λ is the wavelength. Such a treatment is often nesessary for better prediction of far-ﬁeld properties of thebeam because of existence of higher order TEM modes. In this case, however, we obtained zR = 61 mm.
This agrees well with zR = 58 mm predicted from w0 = 0.14 mm, and thus the inﬂuence of higher order
TEM modes can be ignored. The broken curve is the most important entity in the ﬁgure, the diameter ofthe retroreﬂected beam. The position of its beam waist and its radius at the waist is calculated using the“thin lens equation for Gaussian beams” (→ Appendix)ã(ã − 1) + z˜R 2,(ã − 1)2 + z˜R 2
w0
w0′ (ã, zR ; f ) = √(ã − 1)2 + z˜R 2b̃(ã, zR ; f ) =′⇐⇒ zR(ã, zR ; f ) =zR,(ã − 1)2 + z˜R 2

where ã ≡bzR
a, b̃ ≡ , and z˜R ≡,fff(2.55a)(2.55b)′where zR≡π(w0′ )2.
λ(2.55c)For concave mirrors, a relationship f = R/2 is used[64]. Then we can estimate visibility of the opticallattice for a given distance d between the beam waist and the concave mirror. Assuming that the waist

2.3. OPTICAL LATTICE

17

1/e2 diameter of intensity [µm]1600
forward(measured)forward(fit)retroreflected(calculated)
mirror position

1400
1200
1000
800
600
400
200
0
200

300

400

500

600

700

800

Distance from a fiber coupler [mm]Figure 2.5: Plot of 1/e2 diameter of forward- and backward-propagating Gaussian beams.
Figure 2.6: Calculated visibilities at a few points on the standing wave corresponding to Figure 2.5. (a)z = z0 , that is, at the beam waist. (b) z = z0 + 0.1zR . (c) z = z0 − 0.1zR .
Figure 2.7: The eﬀect of broadening.
of the forward beam is made coincide with the center of the magnetic trap within error whose order ofmagnitude is Rayleigh length, we substitute E with
[
]
[
(
)
]¯
¯
w0
r2
r2
E0
exp − 2exp −ik z ++ iζ(z) ¯¯ .
(2.56)
w(z)
w (z)
2R(z)
z=0
Note that the origin of coordiate system is shifted to make it coincide with the beam waist. This is justiﬁedby the fact that the intensity of Gaussian beams decays very slowly compared with the period of a standing
wave formed by superposition of them. Thus we may safely forget about the boundary condition (2.51).
And simultaneously we substitute E ′ with
[
]
[
(
)
]¯
¯
r2
r2
w0′
exp − ′2
exp −ik z − b ++ iζ(z − b) ¯¯ ,E0 ′′w (z − b)w (z − b)
2R (z − b)z=dwhere b ≡ f · b̃(d/f, zR , R/2),(2.57)and variables with primes are understood as transformed ones. By computing numelical values for realistic
experimental conditions, we can estimate the eﬀect of power imbalance between two beams. The resultant
decrease in visibility is about 10–20 %.
Considerations on non-ideal case II: extra reterdation
Dielectric mirrors are formed by many thin layers of dielectric materials with diﬀerent values of dieletricconstants. Their values are designed so that destructive interference causes transmitted light to vanish,leading to reﬂectivity as high as 99.9 %. Such an high reﬂectivity is not attainable with metal mirrors thatcannot be free from absorption, and leads to the great advantage of dieletric mirrors over metal mirrors.
However, this subtle mechanism often leads to unwanted eﬀect of retardation of phase∗5 . That is, dieletricmirrors works as reterdation plate with small but unspeciﬁeld retardation δ. Therefore when the incident
wave is linearly polarized but whose direction is neither parallel nor orthogonal to the plane of incidence,then the reﬂected wave is no longer linearly polarized. (As a corollary, the ellipticity of the ellipticallypolarized wave changes upon reﬂection by dielectric mirrors.)∗5 There exist phase retarders (such as half-wave plates or quarter-wave plates) which utilize this eﬀect positively. Suchproducts are called “geometric” or “reﬂection-mode” phase retarders and diﬀers from traditional phase retarders that are
made of birefrengent crystals having diﬀerent indeces of refraction for “ordinary” and “extraordinary” light in that even when
the direction of polarization of the incident wave is rotated along the axis of propagation, its functionality is not aﬀected.

18

CHAPTER 2. THEORETICAL BACKGROUNDS
Quantity
4

Total number of atoms [10 ]Radial and axial trap frequency (ωr , ωz )/2π [Hz]Density of atoms [cm−3 ]Wavelength of the optical lattice beam [nm]Maximal power of the 1D lattice beam [mW]
Diameter of the 1D lattice beam [µm]Ours

Michigan40(225,10)

5–8(45,15)1064
800
280 (e−2 )852
200
80 (FWHM)

Table 2.1: Comparison between our and Michigan group’s experimental conditions. Note that, our value
for the beam diameter in the last column refers to the e−2 diameter at which the intensity falls to e−2 ofthe peak value, while that of Michigan’s group refers to the FWHM (full width at the half maximum) at
which the intensity falls to a half of the peak value. For a gaussian beam, the former is about 1.700 times
wider than the latter.
Figure 2.8: Diﬀraction relation of atom and the principle of Bragg diﬀraction.
Similar eﬀect is observed for the AR- (anti-reﬂection) coatings on lenses and surfaces of the glass cell of
our vacuum chamber. The AR-coating has almost similar structure as dielectric mirrors except that constr
the transmitted wave is no longer linearly polarized in the situation stated above.
Therefore we need to treat ∆i ’s as nonzero, or speciﬁcally, ∆ ≡ ∆p − ∆s as nonzero. Assuming, in turn,that Ei′ = Ei for simplicity, we have∑⟨|Etot |2 ⟩ =Ei2 [1 + cos(2k · r + ∆i )]i= Es2 + Ep2 + Es2 cos(2k · r + ∆s ) + Ep2 cos(2k · r + ∆s + ∆),(2.58)and in the worst case such that Es = Ep ≡ E and ∆ = −π/4, we have⟨|Etot |2 ⟩ = E 2 + E 2 + E 2 cos(2k · r + ∆s ) + E 2 sin(2k · r + ∆s ) = 3E 2 .

(2.59)

That is, there is no standing wave at all! This circumstance rather resembels P olarization Gradient C ooling(PGC), or Sisyphus cooling, in which two counter-propagating, circularly polarized lasers form a “standingwave” of polarization.
Michigan group (Sapiro et al. [65]) exploited the Kapitza-Dirac KD diﬀraction to calibrate the depthof the lattice potential. They observed the population oscillation of the undiﬀracted (p = 0) state, andﬁtted the oscillation by the zeroth-order Bessel function J0 (x), thereby obtaining the Rabi frequency Ω0of the Bragg pulse from the ﬁrst zero z0 = π/∆ ≈ 2.405 of J0 (x). Alternatively, one might use theratios between populations of diﬀraction peaks of higher orders. However, Michigan group’s method isadvantageous in practice, because the populations of the peaks of higher orders often have large deviations
from the theoretical prediction. And they also observed the reversible loss of superﬂuidity of BEC. As they
deepen the 1D lattice, the sharp interference peaks which was originally present dissapeared around thedepth of 30 Erec . Their experimental conditions (adapted from Sapiro et al. [65] and Zhang et al. [66]) aresummarized and compared with ours in Table 2.1.
Moreover, was obtained experimentaly counterintuitive result on interference patterns from periodic
arrays of BECs (Hadzibabic et al. [67].)
Ideally, ToF imaging of trapped atoms corresponds to the Fraunhofer diﬀraction (FD) in classical optics.
However, Gerbier et al. [68] points out that the FD condition is often not met in current BEC experiments.
They propose the criteriontToF ≫ τFF ≡mlcoh R0,~

(2.60)

where tToF denotes the expansion time of ToF, m the mass of the atom, lcoh the characteristic coherence
length, and R0 the characteristic size of the cloud before expansion.
Last but not least, the dissertation by Greiner [69] contains much useful information.
One may consult Weiner [70, Sec. 5.7] for a table summarizing a number of scattering lengths, especiallythose between two identical alkali atoms. As of February, 2009, the newest reports for rubidium, potassium
and cesium are Falke et al. [71] and Lange et al. [72], respectively. For an accurate determination of
scattering lengths, PA (photoassociation) spectroscopy is widely used (see, for example, Jones et al. [73],2.3. OPTICAL LATTICE

19

Stwalley and Wang [74], and references therein.) Especially it can produce the so-called purely-long-range
molecules in which the Born-Oppenheimer (or, adiabatic) potential of the relative vibrational motion of two
nuclei has a minimum located an order of magnitude further than that of ordinary molecules (Stwalley et al.
[75].) Therefore the entire vibrational motions for low-lying vibrational levels take place in a long-range
manner; in particular, the inner classical turning point is located at a distance of more than 50 a0 (a0 is
the Bohr radius.) Then the energy levels for such molecules are solely determined by the van der Waals
interaction and one can neglect the interactions due to exchange and overlap of valence electrons that leads
to relatively lower accuracy of theoretical analysis.

Component

Company

Product Name

Ion pump (1st; 20 ℓ/s)
(2nd; 75 ℓ/s)
87
Rb dispenser
41
K dispenser

VarianVarianStarCell
StarCellHomemade

Table 3.1: Components used in the vacuum chamber.

Chapter 3

Experimental Setup and Procedure
3.1

Vacuum chamber

3.1.1

Structure

Our BEC machine employs a double-stage MOT design for eﬃcient loading of atoms into MOT. Thus our
vacuum chamber can be regarded as composed of two sections, connected by a thin pressure-diﬀerential
tube. A vapor-loaded MOT in the 1st section of the chamber (hereafter “1st MOT”) serves as a high-ﬂux
atom source for a MOT in the 2nd section (“2nd MOT.”) The 1st section has relatively higher vapor
pressure (low vacuum.) On the other hand, the 2nd section is kept at an UHV (ultra-high vacuum) Low
conductance of the pressure-diﬀerential tube, via which atoms are transferred, is responsible for keepingpressure diﬀerence between the two section.
Our vacuum chamber is illustrated in Figure 3.1 and 3.2. On each of the two opposite side of the 1st
section of the chamber, An AR- (anti-reﬂection) coated glass cell (10 cm × 2.5 cm × 2.5 cm) and a pressurediﬀerential tube are mounted. Moreover, a hot-ﬁlament ionization vacuum gauge and several dispensers of
atomic vapor are contained in the 1st section. Vacuum is drawn by a 20 ℓ/s ion getter pump dedicated forthe 1st section. Inside the ion pumps, high voltage (usually 2 kV; 7 kV at maximum) is applied so thatjunk molecules are ionized and absorbed into titanium walls of the pump.
The dispensers release atomic vapor by chemical reaction when heated by current of 4.2 A. To minimize
release of junk molecules when they are heated by 4.2 A, they are always moderately heated by 3 A;otherwise junk molecules might be absorbed into the dispensers when the dispensers are as cold as thechamber at the room temperature, rather than drawn by the ion pump. The vapor pressure of the 1st
section is kept relatively high (about 1 × 10−9 torr) by vapor from the dispenser so that it works as ahigh-ﬂux atomic source (→ Sec. 3.3.1.) On the other hand, that of the 2nd section is kept as low as possible(lower than the detection limit of a gauge, 2 × 10−11 torr,) to make lifetime of a condensate longer.
We can understand the eﬀect of the pressure-diﬀerential tube as follows: For dry air at the room
temperature in the molecular ﬂow regime, conductance C of a long tube is given byC=

12.1D3 /Lℓ/s,
1 + (4/3)D/L(3.1)

by taking account into ﬁnite sizes of openings[76]. Here, D and L are numerical values of the diameter andlength of a pipe, respectively, measured in cm. In our case, D = 0.8 [cm] and L = 14 [cm], yielding C = 0.36ℓ/s. When we make pressure diﬀerence between 1st and 2nd section, the ratio P1 /P2 between pressuresat those sections are related to the ratio between pumping speed P at the 2nd section and conductance C
20

3.1. VACUUM CHAMBER

21

A
g ra v ity

F la n g e s m o u n te d to o p tical ta b le

1 5 0 .5

2 4 2 .1

310
IP :7 5 l/s

g ra v ity
O D /ID 1 7 /1 5

150

B

270

In n e r d im . 6 0

C
133

224

121

20

8 /7
O D /ID 1 2 /1 0 1 0 /8 7 /5

In n e r d im . 6 0

60

IP :2 0 l/s

GVO D /ID = 5 8 /5 0 m m

C e ll O D /ID =3 0 /2 5

17

110

GV
125

240
140

40

140

125

51

40
218
1 4 5 = >1 5 0

C e n te r to to p :2 5 0 = >***

55

670

45

Figure 3.1: Design of the vacuum chamber by Dr. Kishimoto.

A

2 4 2 .1

1 5 0 .5

1 3 6 .5

IP :7 5 l/s

g ra v ity

B

IC F 1 52

135

220

150

g ra v ity

150

150

C

152

133

g ra v ity

IC F 3 4

IP :2 0 l/s

150
270
110

110

IC F 7 0table top

GV

180

65

75

125

125

IC F 7 065180table toptable top
145=>150

Figure 3.2: Detailed view of Figure 3.1. (A) 2nd section except a glass cell. (B) 1st section, six-way blockaround atomic vapor dispensers. (C) 1st section, connection gate to a TMP via a bellows.

22

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE

between the two as P1 /P2 = P/C. In our case, we haveP1
10−9 torr
= −11
= 102
P2
10
torr
P
75 ℓ/s== 210.
C
0.36 ℓ/s(3.2a)(3.2b)

Thus we can expect that the desired pressure diﬀerence is safely maintained.

3.1.2

Preparation of ultra-high vacuum

Here, we brieﬂy summarize procedures to draw ultra-high vacuum of the chamber.
First, the chamber was roughly drawn by rotary pumps until the pressure went down below 1 × 10−7
torr. Then TMPs (turbomolecular pumps) were operated and the whole chamber was baked for higher
vacuum.
The temperature was slowly raised to 200 ◦ C for a day to avoid leakage at ﬂanges or angle valves. Thosecomponents are made of stainless-steels that has relatively low thermal conductivity (about 16 W/m · K)
compared with other metals like Duralumin or steel (about 140 W/m · K and 50 W/m · K, respectively.)
When one side of such a component is heated, which is unavoidable because of limited number of heaters,
and the temperature raise is too fast, they may be strained due to acute temperature gradient and maylead to leakage. Therefore we need to wait long until temperature becomes almost uniform over the whole
component. In our case, the angle valve located between the two section determined the upper limit of rateof heating. We baked the chamber for a week, so that we surely removed most of the molecules absorbed
into inner walls of the chamber by enhanced detachments of them at the high temperature. Then thetemperature was lowerd, again slowly, to the room temperature.
The resultant pressure after baking (5 × 10−11 torr) was an order of magnitude higher than the ideal
value. So, as a ﬁnal stage, we ran a TSP (titanium sublimation pump) for several times. The 2nd section
has a large pipe dedicated to the TSP, in order to provide wide surface near ﬁlaments of the TSP. We
passed the current of more than 40 A to the ﬁlament for about 30 seconds. Then the ﬁlament sublimates
due to Joule heat, and the resultant titanium vapor coats nearby surfaces. The fresh surface coated withtitanium absorbs junk molecules. Thus it operates as a pump (getter pump.)
After the pressure at the 2nd section reached 3 × 10−11 torr, we closed the gate valve to isolate the 1st
section from the 2nd, and passed current of 5 A to the dispenser for degassing. Then the pressure of the
1st section increased to 8 × 10−7 torr at maximum. However, because of the operation of the ion pump,
the pressure decreased to 3 × 10−11 torr within an hour. The degassing was repeated for a few times foreach dispenser.
During the time, we sprinkled methanol on the chamber several times to locate the leakage, especiallyfrom ﬂanges. When we sprinkled methanol on leakage, it was absorbed into the chamber and raised thepressure inside the chamber. So we could know that the place where we sprinkled methanol had a leakage.
Then we tighten bolts of that ﬂange again with care that all bolts are tighten by equal torque (20 N · m.)
For this purpose, we used a digital torque wrench.
The chamber have been kept closed for one and half year. We have maintained stable ultra-high vacuum
though we had several planned power outage of the building. In such a case, we used portable gasoline
generator of AC 20 A instead of mains power outlet of the building.

3.2

Frequency-locked lasers

In Figure 3.5, we show a schematics of the optical system to provide lasers whose frequency is locked to87
Rb D2 line. We apply SA (saturated absorption) spectroscopy with FM sideband technique for frequency
locking. We separately prepare two subsystems (lower left of the ﬁgure) of SA spectroscopy for each oftrap and repump laser. For trap laser, the frequency of the master laser is locked to the center of thecrossover signal of |5 2 S1/2 , F = 2⟩ → |5 2 P3/2 , F ′ = 3⟩ and |5 2 S1/2 , F = 2⟩ → |5 2 P3/2 , F ′ = 2⟩ transitions.
The master laser is emitted from an ECLD (external cavity laser diode) whose frequency is tunable via
current and angle of a PZT (piezoelectric transducer) attached to a diﬀration grating which is a partof the ECLD (details are described elsewhere[77].) A PI- (proportional-integral) control circuit is used
to stabilize the frequency against its random ﬂuctuation (typical timescale being several kHz) and drift(typical timescale being an hour.) Likewise, the frequency of the repump laser is locked to the crossover
signal of |5 2 S1/2 , F = 1⟩ → |5 2 P3/2 , F ′ = 2⟩ and |5 2 S1/2 , F = 1⟩ → |5 2 P3/2 , F ′ = 1⟩ transitions.

3.2. FREQUENCY-LOCKED LASERS

7/10
12:00

7/11
12:00

7/12
12:00

23

Date and time
7/14
7/15
12:00
12:00

7/13
12:00

7/16
12:00

7/17
12:00

7/18
12:00

300

Temperature [C]

250

200

150

t21
t5
t9
t23
t17
t22
t27

100

50

t2
t6
t10
t14
t18
t24
t28

t3
t7
t11
t15
t19
t25
goal

t4
t8
t12
t16
t20
t26
mean

0

7/10
12:00

7/11
12:00

7/12
12:00

7/13
12:00

Date and time
7/14
7/15
12:00
12:00

7/16
12:00

7/17
12:00

7/18
12:00

Pressure [torr]

1.E-06
1.E-07
1.E-08

TMP(A)
TMP(A)-2

1.E-09

TMP(B)

1.E-10Figure 3.3: A typical sequence of baking the vacuum chamber.

050

Temperature [C]
100
150

200

Pressure [torr]

1.E-06
1.E-07

TMP(A)
TMP(B)
cell 1

TMP(A)-2
B 6crosscell 2

1.E-08
1.E-09
1.E-10Figure 3.4: Decrease of the pressure during gradual cooling of the chamber.

250

24

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE

EO

EOM

Fiber coupler

Mirror

PBS

HWP

QWP
Beam
damperIris

Faraday
isolator
Planoconvex
lens

1000

To (#)
2nd repump

Cyrindricalplano-convex
lens

100

100
AO

AO

Anamorphic
prism pair

100
AO

AO

F&mF-pumping

50
300

100

AO

100

Used
at E2

probe

Slave
(2)

1000

300

SA
EO

50

probe

Repumpmaster

Slave
(4)
100

AO

AO

100 100 50

2nd TA

probe

100

100
Trap
master

AO

Slave
(1)

50

1000
500

40

300

pump

blast

pump

100

SA

300

Slave
(3)

From (#)

1st TA

100

EO

100

300
50

50

Used
at E2

750 50750

Rb vapor
cell

push

1000

PhotodiodeSolenoid
shutter

probe

AOM

AO

Figure 3.5: Schematics of the optical system to provide frequency-locked lasers.

The master laser is ﬁrst split into two: one for probe beam and the other for further ampliﬁcation.
Considering the importance of the probe beam, we used a beam directly emitted from the master ECLD
and did not use injection locking method (described later) here. Possible imperfection of injection locking
may leads to self-sustained pulsation with broad spectrum from a slave ECLD. If such a pulsation light ismixed with an ampliﬁed laser whose spectral width is narrow enough (≈ 8 MHz,) OD (optical density∗1 ) ofimages taken by absorption imaging method is reduced by a few times smaller. The split beam for probe
passes a double-path AOM (acoustico-optical modulator) and its frequency is shifted by +67 × 2 = +134
MHz. The amount of shift is experimentally determined to maximize absorption by the atomic cloud in
MOT. As we tune the detuning, OD of the cloud varies as shown in Figure 3.6. The maximizing value∗1 OD is deﬁned to be logarithm of ratio of transmitted power to incident power; OD ≡ ln P /P .
ti

3.3. MAGNETO-OPTICAL TRAP

25

Figure 3.6: The dependence of OD of an atomic cloud of 87 Rb on detuning of the probe beam.
Radial gradient:~0.3 G/cm/A

Axial
gradient:~0.6 G/cm/A

Curvature A
Anti-bias B

Figure 3.7: Schematics showing “MOT operation” of the coils of the Ioﬀe-Pritchard magnetic trap.
agrees well with the theoretical value
133.3= 66.5 MHz.
2

(3.3)

The other beam is injected to a slave ECLD for ampliﬁcation by injection locking method. In theﬁgure, injection locking is denoted by incidenting of a beam on the side of the second PBS (polarizing
beamsplitter) of a Faraday isolator. Faraday rotation is a phenomenon in which plane of polarizing is
rotated because of circular birefrengence (diﬀerence in the index of refraction for two circular polarizedrays) of the crystal. Circular birefrengence is caused by ferromagnetic resonance and thus depends on thestrength of bias magnetic ﬁelds. Remembering that magnetic ﬁeld is antisymmetric under time-reversal, we
can use reciprocity theorem to conclude that, in order to have eﬀect on only one of the two rays of opposite
directions, we need a phenomenon which involves magnetic ﬁeld. Utilizing Faraday rotation, we can injecta beam from the master ECLD into the slave ECLD at normal incidence, without a beam emitted fromthe slave ECLD going back to the master ECLD. If it happens, lasing of the master ECLD may become
unstable, because the modes other than one selected by a diﬀraction grating may become dominant by theinﬂuence of the incoming beam in the competition to consume population inversion of the laser medium.
On the other hand, the repump
For each of trap (|5 2 S1/2 , F = 2⟩ → |5 2 P3/2 , F ′ = 3⟩) and repump (|5 2 S1/2 , F = 1⟩ → |5 2 P3/2 , F ′ = 2⟩)
transition,

3.3

Magneto-optical trap

As stated in the last section, our BEC machine employs a double-stage MOT design. Here we describe thedetailed properties of each MOT.

3.3.1

1st MOTThe 1st MOT is used as a high-ﬂux atom source for the 2nd MOT. First we made MOT with N = 7 × 109 .
However, it was tuned to optimize the loading rate of the 2nd MOT under the inﬂuence of push beam.
Thus, currently, the number of atoms, density, and temperature of the 1st MOT itself is not even measured
precisely, much less optimized.
Besides the 1st cell, we mount a pair of anti-Helmholtz coils for MOT with their symmetry axis oriented
parallel to gravity. They are rigidly ﬁxed to cope with Lorentz force to attract the coils.
The coils were capable of producing a quadrupole magnetic ﬁeld with linear gradients per 1 A of
(Br , Bz ) = (15, 7) G/cm in the radial and horizontal direction, respectively. We usually operate the MOT
coil at 1.5 A.
As a result, the time constant of loading into the 1st MOT is so fast as about 0.5 s.
Atoms were initially collected from the background vapor into the 1st MOT. Each of the MOT laser
beams has a diameter of about 3 cm, an peak intensity of 10Isat , and was tuned 9 MHz to the red of
the |5S1/2 , F = 2⟩ → |5P3/2 , F ′ = 3⟩ transition of 87 Rb. Here, Isat is the saturation intensity for the|F = 2, mF = 2⟩ → |F ′ = 3, m′F = 3⟩ transition. Another laser beam of intensity 0.2Isat , tuned 2.5 MHz
below the F51to F852 tran- sition, was used to repump atoms. A 4-mm-diam dark spot in the center ofthis beam, lled with a second beam tuned to the F52to F852 transition, was used to mitigate

26

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE
Loading time
Beam Diameter
Total power

0.4 s
5 cm
Trap
Repump

200 mW
5 mW

1.6

6

1.4

5

1.2

4

1

3

0.8

2

0.6

1

0.4

0

0.2

-1

0

-2
0

2

4
6
Time [ms]8

PZT Control Voltage [V]Fabry-Perot Signal [V]Table 3.2: Properties of the 1st MOT.

10

Figure 3.8: A typical signal from a etalon coupled with the injection laser of the TA for the 1st MOT.
Loading time
Beam Diameter
Total power

50 s
5 cm
Trap
Repump

250 mW
5 mW

Table 3.3: Properties of the 2nd MOT. See also remarks on the Table 3.2.

We summarize properties of the 1st MOT in Table 3.2. In the table, total power is measued just after the
optical ﬁber from which six beams for each direction are split. Isat is the saturation intencity 3.2 mW/cm2
for D2 line of 87 Rb. Note that, direct measurement of each of trap and repump beam is impossible forthe 1st MOT, because they are ampliﬁed by the TA as a single combined beam and cannot be split formeasurement after ampliﬁcation. Their powers shown in the table are obtained from scaling of total powerby relative magnitude of signals of a photodiode located after a Fabry-Perot etalon to which part of the
1st MOT beam is coupled. See Figure 3.8 for a typical signal of the etalon. In the ﬁgure, larger (smaller)
peaks correspond to the trap (repump) laser, respectively. The width of the etalon is scanned by varying
the voltage applied to a PZT on which one of the high-reﬂectivity mirrors of the etalon is attached. The
FSR (free spectral range) of the etalon is estimated to be about 1.24 GHz. Note that, the etalon and thescanning system are homemade and they are not intended to realize exactly linear scanning.

3.3.2

2nd MOTThe 2nd MOT is created inside the ultra-high vacuum chamber in order to prevent collisions from thebackground gas with the condensate. The velocity of atoms pushed from the 1st MOT is about 20 m/s.
In Figure 3.9, we show a typical loading into the 2nd MOT. The curve is shown with oﬀset (≈ 0.02 V)
subtracted. Curve ﬁtting with exponential yielded the time constant of 8.23 s. However, one can observea strong saturation of the curve after 12 s of loading. In fact, the curve becomes completely ﬂat after 12 swith height of 395 mV. This is far lower than the estimated ﬁnal value (522 mV) of the loading part of thecurve.

3.4. MAGNETIC TRAP27

PD Signal [mV]

400
300
200

y(t) = y0*(1-exp(-t/tau)),y0 = 522(1) mV,
tau = 8.23(3) s
395 mV

100
0
0

4

8
Time [s]12

16

Figure 3.9: A typical loading of pushed 87 Rb atoms into the 2nd MOT.

3.4

Magnetic trap

The magnetic trap used for evaporative cooling were constructed as a part of the author’s undergraduate
research project. For convenience of the readers, we describe its properties here.
Wing[78] showed that, in a free space, the strength of a static magnetic ﬁeld can have local minima but
not local maxima. Thus, by using a dc magnetic ﬁeld,
One of the most successful design for such traps is Ioﬀe-Pritchard type[79][80, Sec. 3.5], which we employ
for our magnetic trap. It is made of four pairs of cloverleaf coils, a pair of curvature coils, and a pair ofanti-bias coils. The roles of each set of coils are as follows:
Cloverleaf coils for radial conﬁnement. They are four pairs of coils. Each pair consists of two coils
sharing a common axis in which current ﬂows in the opposite direction. Axes of those pairs are all parallel.
Centers of those pairs forms a square around the center of the Ioﬀe-Pritchard trap in a plane parpendicular
to the center axis. Current ﬂows in the same direction for the two pairs in a diagonal line of the square, but
in the opposite direction for the other two pairs. Thus they produce a quadrupole magnetic ﬁeld in thatplane: Bclv ≈ B ′ (xx̂ − y ŷ) = B ′ r[cos(2ϕ)r̂ − sin(2ϕ)ϕ̂]∗2 . Here, B ′ is a proportional constant to express
linear dependence of magnitude on r: |Bclv | ≈ B ′ r. The dependence of B ′ on coil current is shown in
Table 3.5. Note that Bclv does not produce axial conﬁnement.
Curvature coils for axial conﬁnement. They are just a pair of coils which share a common axis
with the whole trap itself and in which current ﬂows in the same direction. However, they do not form
so-called “Helmholtz” pair. When we look at only one of these, the magnetic ﬁeld decays as z −2/3 as the
distance z from the center increases along the center (z-)axis. Let Bcurv denote the combined magnetic
ﬁeld by the pair. Since the center of the Ioﬀe-Pritchard trap is the middle
point of the centers of two∑
coils (each having radius R and located at ±a,) we see that |Bcurv | ∝ ± [(z ± a)2 + R2 ]−3/2 along thecenter axis, and thus axial conﬁnement is realized. In Figure 3.10, we plotted |Bcurv | for a unit current
versus z in the unit of R. Precisely [speaking, the combined magnetic ﬁeld ]of two coils is written as′′′′′′
Bcurv (r, z) ≈ −(Bcurv
z/2)(xx̂ + y ŷ) + Bcurv,0 + Bcurv
z 2 /2 − Bcurv(x2 + y 2 )/4 ẑ. Here the typical valueof B0,curv is so large as hundreds of Gauss.
Anti-bias coils to compensate for loosening eﬀect by curvature coils on radial conﬁnement.
The combination of the above two sets of coils do not suﬃce for strong conﬁnement of the atomic cloud.
By noting that the value of the combined magnetic ﬁeld Bclv+curv at the center is commont for both r andz directions, we see that the dependence of Bclv+curv on r is signiﬁcantly alterd√ from that of Bclv by thepresence of B0,curv . Actually it can be approximated as |Bclv+curv (r, z)| ≈ BB ′ r+0,curv . The magnetic
ﬁeld produced by curvature coils for axial conﬁnement have the side eﬀect to loosen radial conﬁnement.
The strength of magnetic ﬁeld is proportional to r From calculation, radial conﬁnement can be shown to
1/3
be proportional to B ′4/3 /B0 . Therefore it is better to lower B0 for strong conﬁnement. However, it mustbe lager than 1 or 2 gauss to avoid Majorana spin ﬂipping that leads to loss of the atoms.
The components used to build the trap is listed in Table 3.5 for convenience of the reader.
∗2 Here, for example, x̂ is the unit vector parallel to the x-axis.

28

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE

1d|B| / dz

0.75
0.5
0.25
0
-0.25
-0.5
-0.75

|B|

0.8
0.6
0.4
0.2
0
-1.5

-1

-0.5

0
z

0.5

1

z0=0.5z0=0.8z0=1.0z0=1.5z0=0.5z0=0.8z0=1.0z0=1.5

1.5

Figure 3.10: Magnetic ﬁeld produced by a pair of coils sharing a common axis.
Component

Company

Product ID

DC power supply
IGBT
IGBT driver
Diode
Heatsink for IGBTHeatsink for diode
Liquid cooling system
Copper tube

Agilent Technology
Mitsubishi ElectricIsahaya Electronics Industry
International Rectiﬁer
Takagi Manufacturing
Takagi Manufacturing
3R-Systems
Hitachi Cable

Epoxy adhesive
Coil holders
Water pressure booster pump

Konishi

6691A
CM600HA-24H
M57962CL-01
SD400N/R
M-400W
P-100S
Poseidon WCL-04
1/8” square tube with 1/16”
square hole
E250 for concrete
Glass epoxy

Table 3.4: Components used in the magnetic trap.
Magnetic ﬁeld at full operation

B′
B ′′
B0

270 G/cm
67 G/cm2≈2G

Voltage drop at full operation

Cloverleaf
Curvature
Anti-bias

27 V
12 V
12 V

Trapping frequencies for 87 Rb

radialaxialradialaxial

223 Hz
10 Hz
325 Hz
15 Hz

Trapping frequencies for 41 K

Table 3.5: Properties of the magnetic trap.
In Table 3.11, we summarize properties of the magnetic trap. All of them are based on numericalcalculation by Mathematica, exept the dependence of the bias ﬁeld at the center of the trap on Anti-bias
current. It is experimentally measured by varying the Anti-bias current and observing the minimal rf
frequency of evaporative cooling which does not completely remove atoms from the trap.
In Figure 3.11, we show the circuit for Cloverleaf coils of the Ioﬀe-Pritchard magnetic trap. In order toimprove the rising transients of the DC power supply, we attached a large capacitor (15 mF) resistant ofhigh voltage (250 V) as a fast DC power supply. As the capacitor emits its charge by passing current into
the coils, the voltage of the capacitor decays. However, it is slow enough that Agilent DC power supply
can response and manage to keep the constant voltage (therefore constant current.)

3.5. RF GENERATOR AND CONTROLLER FOR EVAPORATIVE COOLING

Agilent
30V,220A

29

Cloverleaf

damper
2Ω
B

+
15 mF

A

Kikusui
250V,0.25A

IGBT w/ocoolerfor rising

IGBT box (B)for Cloverleaf

Figure 3.11: The circuit for Cloverleaf coils of the Ioﬀe-Pritchard magnetic trap.

IGBT box (D)for Curvature & Anti-bias

Kikusui
16V,30A

damper0.72 Ω

Curvature

inner
B

IGBT box (C)for capture

Kikusui
16V,50A
for MOT

outer
inner

A

outer

+
15 mF

outer

B

Anti-bias

IGBT box (A)for Anti-bias

inner

A

Agilent
30V,220A

Air-cooled
IGBT
for MOT

inner
outer

Kikusui
250V,
0.25A

IGBT w/ocoolerfor rising

Figure 3.12: Circuit for Curvature and Anti-bias coils of the Ioﬀe-Pritchard magnetic trap.
Time [s]Frequency [MHz]

0
87

1
62

1.1
87

2
62

2.1
87

3
62

Time [s]Frequency [MHz]

5
147

13
75

21
38

29
15

35
4

37
1.2

39
0.6

40
0.38

41
0.36

Table 3.6: Typical time sequence of a rf sweep for rf-forced evaporative cooling of 87 Rb.
In Figure 3.12, we show the circuit for Curvature and Anti-bias coils of the Ioﬀe-Pritchard magnetic
trap. They are also used for producing a quadrupole magnetic ﬁeld for MOT (→ Fig. 3.7.) Moreover,Anti-bias coils can be used to produce uniform bias magnetic ﬁeld (≈ 60 G) to induce Feshbach resonanceof 41 K.

3.5

RF generator and controller for evaporative cooling

Here we brieﬂy describe the system for applying rf for evaporative cooling. Its details are described elsewhere. Typical sequence of rf sweep is shown in Figure 3.13 and corresponding numerical values are shownin Table 3.6. We use linear interpolation in the region between speciﬁed points. Values are separately
shown for the ﬁrst and latter half of the sweep. The ﬁrst half having zig-zag form is the “cleaning” rf for|2, 1⟩ atoms that cause unwanted inelastic loss during evaporative cooling. By three ramping down of therf, |2, 1⟩ → |1, 0⟩ transition is induced. The latter half is a ordinal rf sweep, as found in JILA’s guide[81].

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE

Frequency [MHz]

30

160
140
120
100
80
60
40
20
0
0

10
10

2020
Time [s]

30 30

40 40

Figure 3.13: A typical time sequence of a rf sweep for rf-forced evaporative cooling of 87 Rb. Actual
frequencies are the above values plus 6.842 GHz, which is the HFS (hyperfine splitting) of 5 2 S1/2 state of
87
Rb.

1

10

500 nK: log(PSD) = a + b*log(N),a = 21.4(5), b = -3.33(6): ζ(3) = 1.202

0Phase Space Density

10

-1

10

700 nK

-2

10

14 µK

2.4 µK

-3

10

-4

10

150 µK

-5

10

-6

10

730 µK

-7

10

-8

10

6

10

2

5

7

10

2

5

8

10

2

5

No. of atoms
Figure 3.14: A trajectory of a typical evaporative cooling of 87 Rb.

The resultant trajectory showing the dependence of PSDon decrease of the number of atoms are shownin Figure 3.14. In the ﬁgure, the red broken line indicates the theoretical threshold of BEC transition
PSD = ζ(3) ≈ 1.202 for ideal Bose gas conﬁned in a harmonic trap. As “hot” atoms, or atoms having thehighest kinetic energy among the cloud, are removed from the magnetic trap by spin-ﬂipping caused by rf
coupling the initial, weak-ﬁeld-seeking state that can be tightly trapped by the magnetic trap, to the ﬁnal,
strong-ﬁeld-seeking state that is repelled from the trap. The trajectory forms an almost straight line. The
incline of the line is a measure of eﬃciency of evaporative cooling[82]. When the points The ﬁnal frequency
at which BEC is produced sometimes fructuates by a few tens of kHz. Especially, the temperature of thelaboratory and the magnetic trap aﬀects the ﬁnal frequency.
For 87 Rb, we succeeded to produce BEC by both of the above two methods. In particular, as for thetransition between Zeeman sublevels, the loss of atoms due to the inelastic collision between 87 Rb atoms in
the |2, 1⟩ state is essential factor for this success. Burke et al. [83] ascribes the exceptionally small inelastic
loss rate to the coincidence between the singlet and triplet scattering length. Scattering lengths for otherrubidium isotopes are summarized in the dissertation by Burke [84].

3.6. OPTICAL LATTICE

31

AO AOM

Faraday isolator
(EOT)

PBS
HWP

MirrorBeam
Iris
damperPower meter for
retrorefrected beam
F230FC-C
fiber coupler

1div = 25.4 mm

Beamprofiler fortelescope

Planoconvex
lens

PAF-X-5-C
adjustable fiber coupler
26

-110
AO

+110
AO

21

-80
AO

16

AO
+80

11

(A)

400

(B)

6

750

(C)
(ODT)

500
26

21

16

11

1
6

1

Figure 3.15: Schematics of the optical system for providing laser beam with an optical lattice.

3.6

Optical lattice

3.6.1

1 µm, 5 W ﬁber laser and an auxiliary optical system

We use a DFB (distributed feedback) ﬁber laser with MOPA (master-oscillator power ampliﬁer) produced
by Furukawa Electric as a laser source. Its linewidth is less than 10 kHz and its output power is 5 W at
maximum. We split the emitted laser into four; three for each axis of a 3D optical lattice that is describedin later sections, and one for other purposes.
The optical system for splitting is shown in Figure 3.15. In order to show the way how lengths of thefour paths are matched, the ﬁgure is correctly drawn with respect to lengths between optical components.
The grid have distances of 25.4 mm, which is the distance between threaded holes on the optical table.
The output from ﬁber laser is emitted from a optical ﬁber attached to a adjustable ﬁber coupler. It
is located at the upper right corner in the ﬁgure, directing to the bottom. The adjustable ﬁber coupler
have multiple nobs. Two of them are used to tune transversal position of an aspheric collimation lens withrespect to an endpoint of a optical ﬁber for changing direction of the output beam. Three of them areused to tune longitudinal distance between the collimation lens and the endpoint for changing diameter ofthe output beam, and the angle of the collimation lens for changing divergence. They are optimized tominimize divergence of the output beam.
Faraday optical isolators
The beam is ﬁrst coupled to the Faraday optical isorators. When the depth of a standing wave formed by
the laser beam is maximized, a considerable portion of the retroreﬂected beam is coupled to the opticalﬁber and the AOM, eventually reaching the ﬁber laser itself. Therefore isolators are necessary to avoid

32

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE

Figure 3.16: A typical conﬁguration of the telescope.
instability. We use two isolators produced by Electro-Optics Technology with designed isolation of 30 dB
each, yielding the total isolation of 60 dB (= 10−6 .) Assuming that 1/3 of the emitted power is reﬂectedback to the isolator, we have
5 W×

1
× 10−6 = 2 µW
3

(3.4)

as a maximum power of the retroreﬂected beam reaching the ﬁber laser. Moreover, the ﬁber laser itself
has built-in optical isolators. So we may safely conclude that we are free from instabilities caused by theretroreﬂected beam.
“Telescope”
Nextly, we use lenses (a “telescope”) for ﬁne tuning of the diameter and the divergence of the beam, because
the eﬃciency of the AOMs located after it depends on them. The wide space reserved for the telescope isfor assurance of its ﬂexibility. A typical conﬁguration is shown in Figure 3.16.
Currently, only a single lens (f = 400 mm) is suﬃcient for maximizing the eﬃciencies of the AOMs.
That is, it is not a telescope in a strinct sence! The apertures of all the AOMs are about 1 mm square incommon, so we tune the telescope so that the beam waist agrees with the locations of the AOMs and thediameter at the waist is 0.8 mm. Then Rayleigh length πw02 /λ (where w0 is the beam radius at the waistand λ is the laser wavelength) becomes
3.14 · (0.4 mm)2
≈ 470 mm.
0.001064 mm(3.5)

The minimum distance between the AOMs and the telescope is about 500 mm and is comparable to Rayleighlength. So we cannot ignore broadening due to diﬀraction and need ﬁne tunings of the telescope to avoid
power loss at the aperature of the AOM. The radius at the waist is related to divergence θ as
θ=

λ
πw0

(3.6)

By the way, we have a choice in the future to replace the ﬁber laser with a new one with 1080 nm
wavelength and 10 W output power. In such a case, we may utilize the ﬂexibility of the telescope to adapt
to the new one that emits beam with possibly diﬀerent diameter and divergence.
When we tune the telescope, we can put a mirror before the iris to reﬂect to the left. Then we canobserve the far-ﬁeld cross section of the beam with a beam proﬁler. The space between AOMs and ﬁber
couplers is reserved for that purpose.
A pair of irises
In case of replacement of the ﬁber laser, we have a pair of irises which indicates the optimal optical path for
AOMs and ﬁber couplers. They can be useful when the ﬁber laser or mirrors before AOMs are accidentaly
moved and need to be ﬁxed to the original conﬁguration.
A series of PBSs and HWPs
We split the beam into four by these PBSs. Relative powers between the four pathes can be tuned byrotating these HWPs. Note that they are not intended for ﬁne tunings. PBSs are air-gapped and therefore
resintant to high input power up to about 10 W. They are designed to realize the maximum extinction ratioof 1:1000.
AOMs
The AOMs are used to avoid interference between diﬀerent axis of the optical lattice and an optical dipole
trap for reduction of the density of condensates. We used two kinds of AOMs which are designed for afrequency shift of 80 and 100 MHz, respectively. We use 1st diﬀracted beam whose frequency is shifted ±80

3.6. OPTICAL LATTICE

33

4500

F240FC-C h
F240FC-C v
F240APC-C h
F240APC-C v
F230FC-C h
F230FC-C v
F240FC-C h(fit)
F240FC-C v(fit)
F240APC-C h(fit)
F240APC-C v(fit)
F230FC-C h(fit)
F230FC-C v(fit)

4000

3000
2500
2000

2

1/e width [um]

3500

1500
1000
500
0
0

100

200 300 400 500 600 700
distance from fiber coupler [mm]800

Figure 3.17: The measured 1/e2 width of collimated beams by three kinds of ﬁber couplers. Here suﬃxes“h” and “v” mean horisontal and vertical width, respectively.
Coupler
F240FC-C (h)F240FC-C (v)
F240APC-C (h)F240APC-C (v)F230FC-C (h)F230FC-C (v)Fitting resultsa [µm/mm]
3.85(52)
3.83(46)
1.95(2)
1.73(10)0.885(49)0.942(25)b [mm]

1558(251)
1533(218)
1277(11)1338(48)763(22)697(11)

Table 3.7: Fitted parameters for data shown in Figure 3.17. Note that, a is so small that it can be used asdivergence θ of the ray measured in mili-radian, since tan θ ≈ θ when θ ≪ 1.

MHz or ±110 MHz for each of the four beams. Therefore the minimal frequency diﬀerece between any twoof the four beams is 110 − 80 = 30 MHz, which is enough.
The eﬃciency of those AOMs are maximized with collimated input beams rather than focused ones.
The maximum eﬃciency ranges from 90 % to 93 %. The dependency of the eﬃciency on the input power
is not observed.
We should note that the maximum eﬃciency stated above is measured for the stationary case. The
eﬃciency strongly depends on the temperature of the AOM and the temperature itself is aﬀected by the
input laser power and the input rf power. So when one changes the rf power to control the power of thediﬀracted beam, a strong hysteresis is observed with typical period of several seconds. Or we may say thatthe large-signal slew rate of the AOM is several orders of magnitude smaller than the one for small-signal.
Especially, we may need to take care so that the temperature of the AOM is kept high by maintaining
input rf power maximum for ﬁner control of the power of the diﬀracted beam. In such a case we may need
to attach mechanical shutters to cut the diﬀracted beams until they are used in the particular step of theexperiment.

Fiber couplers
We choose F230FC-C over F240FC-C and F230APC-C because of its smaller diverngence and easiness topurchace. Those ﬁber couplers are designed for wide range of wavelength (800–1100 nm) and not adjustable.
As a result, the divergence is not optimal for our 1064 nm laser. The measured width of collimated beams
by each of those ﬁber couplers are shown in Figure 3.17. The results of least-square ﬁt of them by linear
function f (x) = ax + b is summarized in Table 3.7.

34

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE

380
(B)
(C)

(A)
250
210

Fiber coupler F230FC-C

80
150
160
160

195
55

50
(C)

350
250

φ1” mirror
φ1” mirror with micrometerφ2” dichroic mirror;
reflects 1064 nm andtransmits 780 nm

50

(B)
(B)

(C)

Concave mirror, R=6”

280
(A)

Figure 3.18: Shematics of the optical lattice.

3.6.2

Dimensions of the optical lattice

Our design of the optical lattice is somewhat complicated because of the limited capacity for optical components. The laser beams of the optical lattice must pass through the spaces between optical componentsaligned around the glass cell and surrounding magnetic trap. Moreover, they need to cross each other atright angle. In order to satisfy these requirements, we used a dichroic mirror which transmits the MOT
lasers (780 nm for 87 Rb and 767 nm for 41 K) and reﬂects the lattice laser. We inserted them at 45 degrees
into each of the three optical path of MOT which are aligned orthogonal to each other. the lattice beams
are shined orthogonally to the MOT beams onto the dichroic mirrors and upon reﬂection their optical paths
overlap to those of the MOT beams. Fine tunings of the optical path are done by mirrors inserted before thedichroic mirrors. Their holders are attached with micrometers and their angles can be tuned with accuracyof 0.01 rad per one division of the micrometer. The longest distance between the mirror with micrometerand the center of the optical lattice is about 940 mm of Path (B). In this case, the beam is parallelly movedby
2 × 0.01 rad × 940 mm = 18 µm.
The beam diameter at the beam waist is tuned to 280 µm and Thomas-Fermi diameter of the BEC is 72 µm(→ Sec. 4.2.) Therefore we can safely rely on them for reproducibility of ﬁne tunings. We use retroreﬂection
by a concave mirror to form standing waves. Thus we ﬁrst split the lattice beam from the MOT beam byusing a dichroic mirror again after it passed the center of the optical lattice, and then place the mirror to
retroreﬂect it.
These three paths are named Path (A), (B), and (C). Schematics of them are shown in Figure 3.18. The
lengths in the ﬁgure are all measured in milimeters. Note that they have error of several milimeters due todiﬃculty in measurements because of complex structure of the system. The beam width are conﬁgured bytelescopic conﬁguration of lenses. Design of telescopes used in each optical path of the optical lattice areshown in Figure 3.19. Calculated lengths for each of three axis are shown in Table 3.8. They are calculatedby Maxima based on the formula of Gaussian beams. Here w0′ is the 1/e2 radius of intensity at the beam
waist inside the telescope. However, the thickness of the lenses are ignored in calculation. Their typical
thickness ranges from 5 to 10 mm. We needed ﬁne tunings of the conﬁguration of lenses based on location
and width of the beam waist measured by a digital laser beam proﬁler.

3.6. OPTICAL LATTICE

35

f1

2w0

f2

2w0'

2w0'' (= 0.28)

F230
FC-C

F

F1

F2

W

M

C

Figure 3.19: Detailed design of telescopes of the optical lattice. See Table 3.8 for actual lengths.

Path

F–C

f1

f2

F–L1

L1–W

W–L2

L1–L2

L2–C

w0′

(A)
(B)
(C)

530
940
730

50

75

51

49

99

148

300

0.042

50

100

52

49

124

173

505

0.030

Table 3.8: Actual lengths (measured in milimeters) in Figure 3.19, for each path.

1/e2 diameter of intensity [µm]

1600
1400
1200
1000
800
600
400
exp
fit

200
0
0

200

400

600

800

1000

1200

Distance from a fiber coupler [mm]Figure 3.20: An example of measured divergence of output beam from F230FC-C.

√
f (z) = 2 × w0(
1+z − z0zR)2

36

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE
Component

Company

Product ID

Fiber laser
Faraday isolator
MirrorLens
Half wave plate
PBS
AOM (80 MHz)(110 MHz)
RF Ampliﬁer
DBM
VCO
Optical ﬁber
Fiber coupler
Dichroic mirror
Concave mirror

Fitel (Furukawa Electric)
Electro-Optics Technology
Thorlabs
Thorlabs

ASF15403
HP-04-I-1064-000
BB1-E03
LA1***-B
ZOWP-1064-15.00mm-2mntd
STR12.7B-LT110
3080-197
3110-197
ZHL-1-2W
ZAD-1H+
ZOS-150
PMJ-3A3A-980-6/125-3-5-1
F230FC-C
SWP-45-Rs-1064-Tp-780-PW-2025-C
Techspec SPH 1.00IN F/6.0 Gold TS

WZW Optics
Crystal Technology
Mini Circuits
Mini Circuits
Mini Circuits
Oz Optics
Thorlabs
CVI Laser
Edmund Optics

Table 3.9: Components used in the optical lattice.

3.6.3

Properties of the optical latticeHere we consider a Gaussian beam. The peak intensity Iadv,0 at the center of its beam waist is related to
its total power Padv as
Iadv,0 =

2Padv
Padv
= 0.637 2 ,
πw02
w0

(3.7)

where w0 is the 1/e2 radius of intensity at the beam waist. When we make a standing wave, the squareof the amplitude of its electric ﬁeld, |E(r, t)|2 , reaches its maximum at antinodes around the beam waist.
It is four times larger than that of the original Gaussian beam because of constructive interference. We
express this fact by saying “a standing wave has a ﬁctious power P = 4Padv and a ﬁctious peak intensity
I0 = 4Iadv,0(3.8)

at its antinodes,” even though a standing wave carries no power (it is superposition of two waves advancing
in opposite directions.) Such a treatment can be justiﬁed by the fact that power and intensity (power per
unit area) are proportional to the square of the ﬁeld amplitude.
Now we consider experimental conditions to estimate the upper limit for I0 . The transmittance eﬃciencyof Faraday optical isolators used in our group was about 96 %. We used two isolators in serial, yielding
the total transmittance of (96 %)2 ≈ 92 %. The eﬃciency to produce 1st diﬀracted beam at the AOM(acousto-optic modulator) is a little more than 90 % at maximum. Fiber coupling eﬃciency ranges from 80% to 85 %. Therefore when we split the beam emitted by the ﬁber laser into four∗3 beams of equal powers
and couple them to optical ﬁbers, we have()
1
1
Padv = 5 W × 92 % × × 90 % × 80 % = 5 W ×× (92 % × 90 % × 80 %)4
4≈ 1.25 W × 66 % ≈ 0.82 W(3.9)

for each beam emitted from a ﬁber. Then the intensity at the center of the beam waist of the emitted beamis
Iadv,0 = 0.637 ×0.82 W= 2.66 × 107 W/m2 = 2.66 kW/cm2 ,(280 × 10−6 m)2(3.10)and the corresponding ﬁctious intensity at antinodes around the center of the standing wave is
I0 = 4Iadv = 10.7 kW/cm2 .

(3.11)

Here we used the measured value 2w0 = 280 µm for our optical lattice.
∗3 The fourth beam is saved for other use than the optical lattice. If more than (5 × 1/4) W is needed for the fourth beam,
the above calculation needs to be adapted accordingly.

3.7. IMAGING SYSTEM

37

Power emitted from each ﬁber
Beam diameter at the waist
Beam intensity at the waist
Equivalent peak intensity of the standing wave

Padv
2w0
I0,adv
I00.65 W
280 µm1.86 kW/cm2
7.44 kW/cm2

Potential depth, divided by kB

87

Rb
K
87
Rb
41
K

10.9 µK
9.85 µK(Rb)
10.9 Erec(K)
9.85 Erec

87

Rb
K

0.04 s−10.03 s−1

Rb
K
87
Rb
41
K

96.5 nK
205 nK
2.02 kHz4.28 kHz41

in the unit of Erec
Photon scattering rate

41

Recoil energy, divided by kB

87
41

divided by hTable 3.10: Properties of the optical lattice.
Using the above value for the maximum intensity of the lattice beam and formulas derived in Sec. 2.3,
we have
(Rb)

(Rb)

U0

(Rb)
,
≡ Udip · I = kB · (11.7 µK) = h · (220 kHz) = 123 Erec

(3.12a)

(K)
U0

(K)
≡ Udip · I(3.12b)(K)= kB · (10.5 µK) = h · (210 kHz) = 51 Erec,(X)

for the maximum depth of the optical lattice. Here, the recoil energies Erec for atom X and a laser withwavelength λ are deﬁned as
(X)
Erec
≡

(h/λ)2.
2mX(3.13)The interpretation of this quantity is as follows: Initially there was an atom X with mass mX which
was at rest, and then it is kicked by absorption of a photon with momentum ~k = h/λ (where k is thewavenumber.) The atoms begins to move with momentum h/λ and the corresponding kinetic energy is(h/λ)2 /2mX ; the above deﬁnition can be understood to refer to this quantity. The formulae below is usefulfor calculation of the numerical value for recoil energies:()[h/(λ̃ µm)]2
1.32 × 10−28
9.57 × 103
Erec (λ̃, A) ==J = kB ·nK(3.14a)
2Au
Aλ̃2
Aλ̃2
(
)
200
=h·
kHz .
(3.14b)Aλ̃2Here, λ̃ is the numerical value of the laser wavelength measured in µm and A = Z + N is the mass number
for the atom X. Then Aλ̃2 is usually between 101 and 102 . For example, for the lattice beam used in thisresearch, we have λ̃ = 1.064 (1064 nm,) thus
87
41

Rb:

λ̃2 ARb = 1.0642 · 87 = 98.5

(Rb)
∴ Erec= kB · (97.2 nK) = h · (2.03 kHz),(3.15a)K:

λ̃ AK = 1.064 · 41 = 46.4

(K)
∴ Erec= kB · (206 nK) = h · (4.31 kHz).

(3.15b)22Note that, the accurate mass of these atoms are not exact multiples of atomic mass unit u, or 1.660 538 73(13)×10−27 kg[85]. However, deviations from exact multiples∗4 can be safely neglected for our purpose, as donein the above calculations. For example, the accurate mass of 87 Rb atoms is 1.443 160 60(11) × 10−25 kg[86].
This amounts to 86.909 180 520(15) u and the fractional diﬀerence is only 86.909 180 520/87 = 0.1 %.

3.7

Imaging system

Our imaging system can be operated in three ways; magniﬁed vertical imaging, reducted vertical imaging,and magniﬁed horizontal imaging (Fig. 3.21.) We can switch between those operation modes by insertion∗4 These quantities corresponds to the binding energy of the nucleus of an atom, which is the basis for the nuclear ﬁssion(and possibly, in the future, nuclear fusion) reactors.

38

CHAPTER 3. EXPERIMENTAL SETUP AND PROCEDURE
Component

Company

Product Name

CCD camera
CCD camera driverCompiler for ﬁtting program∗1

Princeton Instruments
Princeton Instruments
Microsoft

PhotonMax
WinView
Visual Basic 6

Table 3.11: Components used in our imaging system.
Operation mode

Magniﬁcation

Magniﬁed vertical
Reducted vertical

4.2
0.3

Table 3.12: Magniﬁcation of the imaging system for each operation mode.

200

Yc [px]160
2y(t) = K2 t + K1 t + K0
K0 = 205.3(9)K1 = -0.41(29)K2 = -0.590(17)

120
80
40
0

2

4

6

8
10
ToF [ms]

12

14

16

Figure 3.21: Measurement of falling atomic cloud.
and removal of mirrors attached on a magnetic mount. The accuracy of position of the magnetic mounts
are well within tolerance, judging from the fact that images are quite reproducible over repeated insersion
and removal of those mirrors.
The magniﬁcation factors of two vertical imaging mode was measured by imaging thin scale made of
paper and comparing its periodicity. That of horisontal imaging was measured by multiple images of falling
atomic cloud and comparing its accelaration with gravitational accelaration.

∗1 The program for curve ﬁtting and overall control of the imaging system originates from JILA. It uses Microsoft Automation
technology to control WinView. The API (application programming interface) of WinView necessary for programming is
described in the manual[87].

Chapter 4

Results and Discussions
4.1

BEC of 41 K

After rf-forced evaporation, we obtained a pure sample of 41 K BEC[47]. The number of atoms in thecondensate NBEC was determined by ﬁtting Thomas-Fermi density proﬁle on the OD data integrated along
the radial (tight) axis of the magnetic trap. For a pure condensate, we obtained NBEC = 3 × 105 .
We started from a MOT with N = 1.6 × 109 , n = 1.6 × 1010 cm−3 and T = 5 mK, corresponding tophase space density of ρPS = 10−9 (Tab. 4.1). This is two orders of magnitude smaller than that of 87 Rb,
because of the ineﬃciency of MOT. The hyperﬁne splitting of 4 2 P3/2 of 41 K has an anomalously small valueof only 14 MHz. Therefore if we detune trap and repump lasers of MOT for eﬃcient cooling, the hyperﬁnestates are not resolved and closed transition is violated, leading to the ineﬃciency mentioned above. To
circumvent this undesirable feature, we employed a version of CMOT technique[89], in which the prodecudeis split into two stages for diﬀerent purposes[77].
Compression (usual CMOT) The duration of this stage is 40 ms. We ramp up the magnetic ﬁeld
gradient from about 3 G/cm to about 30 G/cm. Simultaneously, we lower intensity of the trap laserfrom 5.6Isat to 0.1Isat to avoid pressure from spontaneously emitted photons, and lower detuning from34 MHz to 31 MHz to enhance trapping eﬀect.
Doppler cooling The duration of this stage is 12 ms. At the beginning of this stage, we suddenly lower
the magnetic ﬁeld gradient to 2 G/cm and keep this value until the end of this stage.
After these stagse, phase space density of 41 K has a comparable value (1 × 10−6 ) to that of 87 Rb (6 × 10−6 ).
The data of 87 Rb is adapted from experimental report from MIT group who are regarded to have thecurrent state-of-the-art BEC machine. Thus our result is quite satisfactory.
Then we adiabatically transfere the 41 K cloud into the Ioﬀe-Pritchard magnetic trap (→ Sec. 3.4.) For
41
K, we need to scale trap frequencies described in Sec. 3.4 because of the diﬀerent masses of 41 K and 87 Rb.
Since the mass of 41 K is 40.961 825 4(12) u ≈ 41 u (0.09 % error,) the trap frequency diﬀers by a factor of√
41/87.
As we lower the rf frequency for evaporative cooling, bimodal structure of a ToF image was observed.
Figure 4.5 shows such images and vertical sum of OD. Plots (orange dots) below images are OD integrated
41

K in
stableMOT

87

Rb∗1 instableMOT

Splitting between F ′ = 2, 3 [MHz]Natural linewidth of D2 line [MHz]

14
6.2

267
6.1

Temperature [µK]
Density [1010 cm−3 ]Number of atoms [109 ]Phase space density [10−7 ]

5000
1.6
1.6
0.01

150
1
40
100

K after
two-stage
CMOT

Rb∗1
after
CMOT

100
5.1
1
10

10
10
≈ 10
60

41

Table 4.1: Comparison between MOT of 41 K and 87 Rb.
∗1 Adapted from report of MIT[88].

39

87

40

CHAPTER 4. RESULTS AND DISCUSSIONS

Figure 4.1: Combination of CMOT and following Doppler cooling for 41 K. (a) Loading into MOT (≈ 10s.) (b) CMOT for higher density. (c) Doppler cooling for lower temperature.

Figure 4.2: Transitions used for rf-forced evaporative cooling of 41 K for successful (blue) and unsucsessful(red) methods.
Time [s]Frequency [MHz]

0
300

16
300

24
269.2

28
264.3

32
260.3

34.9
259.4835
311

Table 4.2: A typical time sequence of a rf sweep for rf-forced evaporative cooling of 41 K.

along the vertical axis (the axis of tighter conﬁnement of magnetic trap) in arbitrary units. Results ofleast-square ﬁtting by the bimodal density proﬁle function are also shown in red curves.
As a conculusion of this section, we summarize keypoints of our realization of 41 K BEC.
• CMOT followed by Doppler cooling was eﬀective to cope with peculiar hyperﬁne level structure of41
K• A hyperﬁne transition |2, 2⟩ → |1, 1⟩ was eﬀective to avoid unwanted inelastic collision |2, 1⟩ + |2, 1⟩ →|2, 2⟩ → |2, 0⟩ which releases energy due to quadratic Zeeman eﬀect.

4.2. BRAGG DIFFRACTION OF THE CONDENSATE

41

Frequency [MHz]

450
400
350
300
250
0

10

20
Time [s]30

40

Figure 4.3: A typical time sequence of a rf sweep for rf-forced evaporative cooling of 41 K. Note that the
HFS of 4 2 S1/2 state of 87 Rb is 259.0 GHz.

Figure 4.4: Typical trajectory of evaporative cooling of 41 K, for successful (blue) and unsucsessful (red)methods.

4.2

Bragg diﬀraction of the condensate

After rf-forced evaporation, we obtained a pure sample of 87 Rb BEC. By the use of absortption imaging,
the number of atoms N in the condensate was determined to be N = 4 × 105 .
In our case, N = 4 × 105 and (ωr , ωz )/2π = (223, 10.3) Hz for 87 Rb atoms, so Thomas-Fermi radii are
{ }
{
}
{
}
lr
1/2.233.34= 5.62 ·· 1.005 · 1.32 =µm
lz
1/0.103
72.3

(4.1)

for the radial and axial direction of the Ioﬀe-Pritchard magnetic trap. Thus the condensate is split into
{
}
{
}
/
1.064 µm
13 (radial)
3.34
≈
(4.2)
2·
µm
270 (axial)
72.3
2
components for each direction. Then the density of the condensate is estimated to ben = 4 × 1014 cm−3 .

(4.3)

As a preliminary experiment, we performed diﬀraction experiments of the condensate using opticallattice. First, we worked out in the short-interaction (Kapitza-Dirac) regime, in which an opital pulse gives
only phase diﬀerence to the atoms, without signiﬁcant displacement of the atoms. Next, we worked out inthe long-interaction (Raman-Nath) regime, in which
By this experiment, we conﬁrmed that the Rabi frequency is about 4 kHz.

42

CHAPTER 4. RESULTS AND DISCUSSIONS

(a)

(b)

(c)

Figure 4.5: Images of 41 K BEC, taken after ToF of 40.3 ms. Below those images, vertically integrated OD
are plotted. (a) Thermal cloud. (b) Mixture of thermal cloud and BEC. (c) Pure condensate.

(A) 15Erec(B) 15Erec(C) 15Erec

Figure 4.6: Interference pattern of a 1D array of 2D BEC.

4.3

Interference patterns by condensates in 1D optical lattices

In Figure 4.6, we show interference pattern of a 1D array of 2D BEC formed by each of the 1D opticallattices (A), (B), and (C), respectively. These images are taken along the vertical imaging axis.

4.4Interference pattern of the condensate in 2D lattice

In Figure 4.8, we show interference pattern of a 2D arrays of 1D BEC formed by the 2D optical lattice.
Here axes (B) and (C) were used. We can observe 2D interference in ﬁgures (b)–(e).
By curve ﬁtting, we can extract the number of atoms in each of the center peak, four diﬀracted components, and background component. We used the following function:[]∑(x − Xbg )2 + (y − Ybg )2z(x, y) =Ai · TF2 (x − Xi , y − Yi ; li ) + Abg exp −
(4.4)
2
wbg
i=c,1,2,3,4
Here, TF2 (x, y; l) is a density proﬁle of BEC in a harmonic trap obtained using Thomas-Fermi approximation:
)]3/2
[
(
x2 + y 2,0(4.5)
TF2 (x, y; l) ≡ max 1 −l2Integrated OD [a.u.]

4.4. INTERFERENCE PATTERN OF THE CONDENSATE IN 2D LATTICE

43

Ar = 10840 ± 595
wr = 6.233 ± 0.408
Xr = 348.06 ± 0.275

Ac = 56042 ± 591
wc = 6.5971 ± 0.085
Xc = 259.82 ± 0.0548

Abg = 14316 ± 226
wbg = 75.934 ± 1.75
Xbg = 254.17 ± 0.805

Al = 13416 ± 619
wl = 5.8341 ± 0.324
Xl = 171.43 ± 0.215

y0 = -6764.8 ± 127

100

200

300

400

x [px]
Figure 4.7: Curve ﬁtting on interference pattern by Path (A). 1 px (pixel) corresponds to 4.0(2) µm.

(a) (0, 0)Erec(b) (6.5, 8.7)Erec(c) (11.6, 16.3)Erec(d) (18.2, 26.6)Erec(e) (22.8, 33.2)Erec(f) (34.4, 49.3)Erec

Figure 4.8: Interference pattern of a 2D arrays of 1D BEC.
where l is a parameter of the function TF2 denoting the circular radius of the region ﬁlled with a condensate.
Here, we assumed that the harmonic trap is isotropic in xy-plane. The derivation of this function is
as follows; we have a anisotropic parabolic density proﬁle n(x, y, z) = n0 · TF3 (x, y, z; lx , ly , lz ), wheren0 = µ/U0 is the peak density∗1 of the condensate and the function TF3 is deﬁned as(( )2 ( )2 ( )2 )xyz
TF3 (x, y, z; lx , ly , lz ) = max 1 −
−
−
,0
(4.6)
lx
ly
lz
where three parameters dx , dy and dz denotes the radius of the region in the parabola ﬁlled with a condensate. In Thomas-Fermi approximation, kinetic energies of atoms are approximated to zero because it is
far smaller than the mean-ﬁeld energy due to interatomic interactions. By absorption imaging, the density
proﬁle is integrated along the axis of propergation of a probe laser beam. Therefore we integrate TF3 along(for example) z-axis,( )2 ( )2 ( )2 ]∫ ∞∫ lz √1−(x/lx )2 −(y/ly )2 [
x
y
z
1−
−
−
dz
(4.7)
TF3 (x, y, z; lx , ly , lz )dz =
√
lx
ly
lz
−lz 1−(x/lx )2 −(y/ly )2−∞∗1 in fact, n can be probed to be equal to µ/U , where µ is chemical potential of the atomic cloud and U is the strength000of δ-like pseudopotential resulting from interatomic interaction.

44

CHAPTER 4. RESULTS AND DISCUSSIONS

400
350

400

300
250

300

200
150

200

100
50

100

0

0

200
180
160
140
120
100
80
60

280
240
200
160
120
80
40

160
120 140

180 200

220 240260

Potential Depth [Erec]

Figure 4.9: Example of curve ﬁtting on the absorption image obtained above.

40
30

(A)

(B)

20
10
0
0

100

200

300

400

500

Time [ms]Figure 4.10: Typical time sequence of ramping up and down of the optical lattice.
One can observe that the curve (surface) agrees well with the OD except in the middle-right region.
There S/N ratio is not good and one cannot expect good results of curve ﬁtting. The four diﬀracted
components have almost same sizes, with only a slight diﬀerence in the lower right peak.
In Figure 4.9, we show three examples of curve ﬁtting, corresponding to (b), (d), and (e) in Figure 4.8.
The middle, colored surface is obtained by averaging over nearest 5 × 5 pixels around a particular pixel inorder to reduce noise. As one can see in Figure 4.8, the typical length scale in the original images is morethan 10 pixels, so the averaging process does no harm. The lower plane shows a contour plot of the middle
surface. The seven countours correspond indicates levels with OD of {40, 80, . . . , 280}. The upper mesh isa plot of the result of curve ﬁtting. For visual clarity, the mesh is shifted upwards from the surface. The
top of the surface is cropped because the center peak is so high that other lower peaks become hard to seein the ﬁgure if it is entirely shown.
A typical sequence is shown in Figure 4.10. The images shown in Figure 4.8 are taken by ToF of 30.7
ms just after the sequence (A), with peak potential depth (30Erec in the above) varied. We also observed
a sharp peak like (a) in Figure 4.8 by ToF after the sequence (B). The peak potential depth was 100Erec
at that time.
The TOF image obtained after raising the barrier between sites in the optical lattice shows that there isno coherence between sites. Therefore we conclude that Mott insulator phase is realized. The characteristic
time for quantum tunneling τ is estimated to be order of 10 to 100 seconds.

4.4. INTERFERENCE PATTERN OF THE CONDENSATE IN 2D LATTICE
Parameter

Value
(d’)

Ac
lc
(Xc , Xc )A1
l1
(X1 , Y1 )A2
l2
(X2 , Y2 )A3l3
(X3 , Y2 )A4l4
(X4 , Y2 )A23
l23
(X23 , Y23 )Abg
wbg(Xbg , Xbg )y0692(5)
16.85(4)(182.2(2),134.3(2))
125(2)
13.4(1)(218.62(9),157.6(9))
141(2)
11.6(1)(162.42(8),159.30(8))
135(2)
11.4(1)(161.61(8),112.33(8))
115(2)
14.0(1)(216.7(1),110.6(1))64(2)
11.3(2)(137.1(2),137.4(2))
420(3)
21.66(9)(190.96(5),135.97(4))
15.7(2)

Table 4.3: Obtained values for ﬁtting parameters.
Here we compare these results with those observed by Bloch et al.[90].

45Chapter 5

Conclusion and Outlook
By the standard method, we obtained 41 K and 87 Rb BEC with the numbers of atoms of NK = 3 × 105
and NRb = 4 × 105 , respectively. Then we superimposed the one- and two-dimensional optical lattice on
the 87 Rb condensate in the magnetic trap. After release from the combined trap, the condensate formedinterferrence patterns. The shape of the patterns agreed well with the prediction from band theory. As thedepth of the optical lattice is increased, the visibility of the patterns decreased and ﬁnally vanished. Thisis explained from the suppression of tunneling of atoms between neighboring sites in the optical lattice.
Moreover, when we subsequently lowered the depth, the atomic cloud went back to the BEC. This is theproof that the dissapearance of the patterns is not caused by the heating up the atomic cloud. Therefore
we conclude that our optical lattice can be used for its ﬁnal objecitive, namely, conﬁning the mixture of
BECs of 87 Rb and 41 K to produce ultracold heteronuclear molecules by combination of Feshbach resonanceand STIRAP.

46

Appendix A

Theoretical Techniques
Here we describe some of the relevant points to analyze properties of BEC in a optical lattice.

A.1

Classical Optics

A.1.1

ABCD-matrix

Wide range of optical systems can be described in terms of the so-called ABCD-matrices. See standard
textbooks such as Yariv [91] for its details.
We deﬁne the co-called q-parameter q(z), also known as the complex curvature, of a Gaussian beam via
1
λ
1
1
≡
−i 2
=
.
q(z)
R(z)
πw (z)
z + izR(A.1)(The last equality follows from the deﬁnitions of R(z) and w(z).) Then the eﬀect of an optical apparatus
located at z0 can be written as()
A · q(z − z0 ) + B
1
D/q(z − z0 ) + C
q ′ (z − z0 ) =or, equivalently, ′
=
,
(A.2)
C · q(z − z0 ) + D
q (z − z0 )B/q(z − z0 ) + A
where q ′ (z) is the q-parameter of the transformed beam, and A, . . . , D are elements of the ABCD-matrix
for that apparatus. That is, by requiring continuity of the q-parameter q(z) for the original beam and q ′ (z)for the transformed beam at z0 , we obtain the complete speciﬁcation, e.g., both of the radius w0′ at the′beam waist and the Rayleigh length zR, of the transformed beam. Let ML denote an ABCD-matrix for alens with focal length f . Its elements are given as follows:
(
) (
)
AL BL
1
ML ≡
=
.
(A.3)
CL DL
−1/f 1Moreover, a concave mirror can be treated as a lens, when we deﬁne the coordinate system for the reﬂectedbeam so that z-axis points the opposite direction of that of the original system.

A.1.2

Eﬀect of thin lens on Gaussian beam

The elementary folmula of the eﬀect of lens is as follows: We are familiar with the
1
1 1
+ = .
a b
f

(A.4)

Here, a and b are respectively the distance between foci and the lens for the incident and transmitted beams.
This may be called the ray approximation, since this formula assumes that the wave propagates as ray andignores the ﬁnite diameter of the beam waist. By introducing “normalized” lengths
ã ≡

a
,
f

b̃ ≡
47

b
,
f

(A.5)

48

APPENDIX A. THEORETICAL TECHNIQUES

Figure A.1: Coordinate systems to derive the lens formula for Gaussian beams.
we have a more symmetric form of Eq. (A.4) as
b̃ =

ã
.
ã − 1(A.6)

Additionally, further transformation leads to b̃ − 1 = 1/(ã − 1), revealing that it is a hyperbola with its
origin at (ã, b̃) = (1, 1).
We want a similar formula for Gaussian beams which propagates through free space under the eﬀect ofdiﬀraction. This can be treated using ABCD-matrices as follows. First, we set up coordinate systems forthe original and transformed beams, as shown in Figure A.1. Then at the location of a lens, where bothz = a and z ′ = −b are satisﬁed, we haveq ′ (z ′ = −b) =A · q(z = a) + B
a + izR′⇐⇒ −b + izR=,
C · q(z = a) + D(−1/f )(a + izR ) + 1

(A.7)

′
where zR
represents the Rayleigh length of the transformed beam. Dividing both sides by f , we obtain−b̃ + iz˜R ′ =ã + iz˜R(ã + iz˜R )(1 − ã + iz˜R )ã(1 − ã) − z˜R 2 + iz˜R
=
=
,
2
2
1 − ã − iz˜R(ã − 1) + z˜R(ã − 1)2 + z˜R 2

(A.8)

′
where z˜R ≡ zR /f and z˜R ′ ≡ zR
/f . Comparing real and imaginary parts of both sides yields the following
two equations:ã(ã − 1) + z˜R 2,(ã − 1)2 + z˜R 2
zR
w0
′
′
√
=
zR
.
2 ⇐⇒ w0 =2(ã − 1) + z˜R(ã − 1)2 + z˜ 2b̃ =(A.9a)(A.9b)

R′= λ/(πw0′2 ).
Here, the beam waist size w0′ of the transformed beam satisﬁes zR
By comparing Eqs. (A.6) and (A.9a), we see the eﬀect of Gaussian beam, or eﬀect by Fraunhofer
diﬀraction. The diﬀerence becomes signiﬁcant when a → f ⇐⇒ ã → 1. In the ray approximation,
we have b̃ → ∞; i.e., the focal point can be as far as possible. However, in the Gaussian case, we haveb̃ → 1 ⇐⇒ b → f because of z˜R in both of the numerator and the denominator. In Figure A.2, Note that
this plot does not depend on wavelength of the Gaussian beam. Note also that in the limit of long focal
length (z˜R ≡ zR /f → 0,) the eﬀect of diﬀraction is negligible, thus Eq. (A.9a) reduces to Eq. (A.6).

A.1.3

Numerical evaluation

Since the equation we want to solve is a general cubic equations in z, we should use numerical rather thananalytic (symbolic) solutions. Though the latter may be computable by the use of CAS (compter algebra
systems,) it is slower and do not have advantage over the former, since we eventually need a numericalvalue for z. However, there is one thing we care about; among multiple solutions for z, we need to ﬁnd
a “physical” one which leads to positive values for distances between various lenses and foci. Numericalsolutions often ﬁnds only one of multiple solutions for z. Even if multiple solutions can be simultaneously
obtained, there is no diﬀerence in that we can often use only one of these solutions.
In order to do so, we begin with parameters for which an obviouslly correct solution can be obtained.
Then we slightly vary parameters towards the ﬁnal value and calculate the corresponding solution. The
solution can be used for the seed value for the next calculation with parameters which are more closer to theﬁnal values. By repeatedly varying parameters and the seed value for z, we may ﬁnally obtain a solution
for the condition under consideration.

A.1. CLASSICAL OPTICS

49

10

w0’ / w0

5
2
1
0.5
0.2
0.1
10
5

b/f

2
1
(ray approx.)zR/f = 13%
25%
50%
100%
200%

0.5
0.2
0.1
5

4.5

4

3.5

3

2.5
a/f

2

1.5

1

0.5

0

Figure A.2: Plot of Eqs. (A.9).
The following codes can be used to calculate the optimal value for z under ﬁxed parameters. We used
GNU Maxima which is a free software. Visit the website of GNU Maxima http://maxima.sf.net/ fordetails. However, there are less information on this CAS, so we present the code with comments.
← GNU Maxima denotes asignment by ‘:’ and end of a command by ‘$’.
( a/f * (a/f-1)← ‘:=’ is used to deﬁne a function.
+ (%pi * w0^2 /lambda /f)^2 )/ ( (a/f-1)^2← Spaces and line-breaks may be inserted anywhere you like.
+ (%pi * w0^2 /lambda /f)^2 ) $new_wid(a,f,w0):= w0 / sqrt( (a/f-1)^2+ (%pi * w0^2 /lambda /f)^2 ) $load(mnewton) $← Loading package for Newton’s iteration method with multiple variables
fpprintprec:5 $← ‘‘floating-point-number print precicision’’ is set to 5 digits.

lambda:1064e-3 $new_loc(a,f,w0):=

block(← The block clause introduces local variables∗1 .
[a:93, w0:0.36, L:730, f1:50, f2:100, w02:0.125], ← Asigning values to the local variables.
mnewton([ b+c+d+e - L,
← Execution of the mnewton function (See below.)
c/f1 - new_loc(b-a,f1,w0),
w01 - new_wid(b-a,f1,w0),
e/f2 - new_loc(d,f2,w01),
w02 - new_wid(d,f2,w01)],[b, c, d, e, w01],[0, 50, 130, 510, 0.05]));← To print a value, use ; at the end of a command instead of $.
The function mnewton([e1 , . . . , en ],[v1 , . . . , vn ],[i1 , . . . , in ]) executes Newton’s method to ﬁnd the values of variables v1 , . . . , vn for which all expressions e1 , . . . , en vanish, with the initial values of variables
set to i1 , . . . , in . To solve the coupled equations fk (v1 , . . . , vn ) = gk (v1 , . . . , vk ) (k = 1, . . . , n), choose∗1 The variables which are ‘‘local’’ to a block cannot be referred outside of that block.
They are advantageous in that they are less likely to be confused with variables which accidentally have the same name but are used with different meanings.

50

APPENDIX A. THEORETICAL TECHNIQUES

fk (v1 , . . . , vn ) − gk (v1 , . . . , vk ) as ek .

A.2

Time-dependent Perturbation Theory

Details are found in standard textbooks on quantum ﬁeld theory. We recommend those written for application for condensed matter systems Feynman [92].
We work out in the convenient interaction picture which eliminates obvious dependence on the unperturbed states from calculations. Remember the two diﬀerent pictures of quantum mechanics: theSchrödinger picture in which state vectors are responsible for all time-dependences, and the Heisenberg
picture in which states are time-independent and instead observables carries all time-dependences. Both of
the two pictures yield the same value for the expectation values ⟨A⟩ψ of an observable A for the state ψ:⟨A⟩ψ = ⟨ψ|A|ψ⟩.

(A.10)

If the introduction of the time-dependence is necessary, we can use both of the above two pictures:⟨A⟩ψ (t) = ⟨ψ(t)|A|ψ(t)⟩= ⟨ψ|A(t)|ψ⟩.

(Schrödinger)(A.11a)(Heisenberg)(A.11b)Here, |ψ(t)⟩ is deﬁned as the vector which experienced the time-evolution due to (total) Hamiltonian H
from the initial value |ψ0 ⟩; that is, |ψ(t)⟩ ≡ e−iHt/~ |ψ0 ⟩. However, in performing the time-dependent
perturbation for the Hamiltonian H = H0 + V , there is a third way to introduce the time-dependence.
Beginning with, for example, the Schrödinger picture,⟨A⟩ψ (t) = ⟨ψ(t)|e−iH0 t/~ eiH0 t/~ Ae−iH0 t/~ eiH0 t/~ |ψ(t)⟩ ≡ ⟨ψ I (t)|AI (t)|ψ I (t)⟩,(A.12)

where we deﬁned|ψ I (t)⟩ ≡ eiH0 t/~ |ψ(t)⟩,(A.13a)AI (t) ≡ eiH0 t/~ Ae−iH0 t/~ .

(A.13b)The deﬁnitions of |ψ I (0)⟩ can be understood as follows: In the simplest case of commuting H0 and V , wehave|ψ I (t)⟩ = eiH0 t/~ e−i(H0 +V )t/~ |ψ0 ⟩ = e−iV t/~ |ψ0 ⟩.

(A.14)

Thus this can be seen as a solution of the Schrödinger equation with the Hamiltonian V but without H0 .
Of cource, things are more complicated in the general case, because [a, b] ̸= 0 implies ea eb ̸= ea+b . However
it remains correct to understand its essence as removing the obvious dynamics due to the unperturbed
Hamiltonian.
We can translate the Schrödinger equation into the interaction picture by substitution of |ψ(t)⟩ with
e−iH0 t/~ |ψ I (t)⟩:i~∂t e−iH0 t/~ |ψ I (t)⟩ = (H0 + V )e−iH0 t/~ |ψ I (t)⟩ ⇐⇒ e−iH0 t/~ i~∂t |ψ I (t)⟩ = V e−iH0 t/~ |ψ I (t)⟩⇐⇒ i~∂t |ψ I (t)⟩ = V I (t)|ψ I (t)⟩.

(A.15)

Incorporating the boundary condition |ψ I (t = t0 )⟩ = |ψ0 ⟩, this diﬀerential equation can be rewritten to theintegral equation:∫ t∂t |ψ I (t)⟩dt
t0
∫
1 t
I
= |ψ (t)⟩ − |ψ0 ⟩ =dtV I (t)|ψ I (t)⟩.
(A.16)
i~ t0
In order to solve this equation, we introduce a unitary operator U (t, t0 ) called propagator such that |ψ I (t)⟩ =U (t, t0 )|ψ0 ⟩. By substitution, we have the following integral equation for the propagator:∫
1 t
dt V I (t)U (t, t0 )|ψ0 ⟩[U (t, t0) − 1]|ψ0 ⟩ =i~ t0⇐⇒ U (t, t0 ) − O[U (t, t0 )] = 1 ⇐⇒ U (t, t0 ) = (1 − O)−1
(A.17)

A.3. N -DIMENSIONAL LAPLACIAN AND LAPLACE-BELTRAMI OPERATOR

51

Figure A.3: A one-to-one correspondance between a parametrized surface and its tangent plane.
where we have deﬁned the integral operator O as∫
1 t
O[f (t)] ≡dt V I (t)f (t).
i~ t0(A.18)

Here we used t both as the upper limit and as the formal variable of the integration.
∑ Their meanings should
be understood from the context. Using the familiar expression (1 − a)−1 = n≥0 an for the geometricseries, we have the following formal expansion:
∫ tn
∫ t2
∑ 1 ∫ tU (t, t0 ) =dtn
dtn−1 · · ·dt1 V I (tn )V I (tn−1 ) · · · V I (t1 )(A.19)
n
(i~)
t0
t0
t0
n
Usually we truncate the inﬁnite sum at the ﬁrst few terms. Alternatively, we can obtain this formal
expansion by iterated substitution. Beginning with Eq. (A.17), we have
U (t, t0 ) = 1 + O[U (t, t0 )] = 1 + O{1 + O[U (t, t0 )]} = 1 + O(1 + O{1 + O[U (t, t0 )]}) = . . . ,(A.20)

and we can go ad inﬁnitum. Again, after a few iteration of substitution, we truncate the inﬁnite sum by
substitution of U (t, t0 ) with 1. These formal series expansion has the advantage of being able to control
how many orders of perturbative term we incorporate in the calculation, and t need not be small. For atheoretical interest, we explain how to use the superoperator of time-ordering product T .
∫ t
∫ tn
∫ t2
dtn
dtn−1 · · ·dt1 V I (tn )V I (tn−1 ) · · · V I (t1 )
t0
t0
t0
∫
∫ t
∫ t
1 t
=
dtn
dtn−1 · · ·dt1 T [V I (tn )V I (tn−1 ) · · · V I (t1 )](A.21)
n! t0
t0
t0(A.19)

A.3

n-dimensional Laplacian and Laplace-Beltrami operator

For the arbitrary n-dimensional space (n ≥ 2,) the Laplacian can always be split into the radial part1
rn−1

∂r rn−1 ∂r = ∂r2 +

n−1
∂r
r

(A.22)and the angular part (the Laplace-Beltrami operator) which is free of r and ∂r . Though this may be provenby a diﬀerential geometric calculation based on a metric tensor which gives a complete characterizationof an arbitrary surface embedded in the n-dimensional space, it is somewhat lengthy. Instead we give a
concise proof focusing on the above relation, following Matano and Jimbo [93].

A.3.1

Gradient operator on a surface

First, we deﬁne the gradient operator on a surface. Let S denote an arbitrary (n − 1)-dimensional smoothsurface embedded in the n-dimensional space. For each point a on S, there corresponds a pair (Ta , Pa )where Ta is a (n − 1)-dimensional tangent plane of S at a and Pa is a projection from S onto Ta . Because

52

APPENDIX A. THEORETICAL TECHNIQUESS is “smooth,” there is a ﬁnite region around a where Pa : S → Ta is not only a projection but also a
one-to-one correspondance, and we can go back from c ≡ Pa (b) ∈ Ta to the original b ∈ S. (Pa is not a
one-to-one correspondance only in a region where S and Ta are vertical or where a part of S away from ais folded to lie between a and Ta ; See Figure A.3.) Then for any function f : S → R and any point a on
S, one can associate a composite function f˜ : Ta → R deﬁned asf˜(c) ≡ f (b) ≡ f (Pa−1 (c)).

(A.23)

Since Ta is just a (n − 1)-dimensional Euclidean space, there is no diﬃculty in calculating the gradient off˜ at a ∈ Ta :)t(f˜(a + heaj ) − f˜(a)˜∇T a f (c = a) ≡ . . . , lim, ... .
(A.24)h→0h
Here, the right hand side is a (n − 1)-dimensional vector with its j-th element corresponding to directional
derivative along j-th basis vector eaj of Ta . Then this is used to deﬁne the gradient operator for u : S → R;
i.e., for each a in S (the domain of f ,) we construct¯ a tangent space Ta and deﬁne the gradient of f on S,
or ∇S f (a), as the gradient of f˜ on Ta , or ∇T a f˜(c)¯c=a .
Example: We choose the one-dimensional unit sphere S 1 as S. Then S as a subset of R2 can beparametrized by u ∈ (−1 . . . 1) as{√t(u, ± 1 − u2 ) (−1 < x < 1 and y ≷ 0),t
S ∋ a(u) = (x(u), y(u)) = t √(A.25)(± 1 − u2 , u) (−1 < y < 1 and x ≷ 0).
Each of these four ways of parametrization covering its own region of S is called a chart of S, and the whole
set of charts is collectively called the atlas of S. For the case of the ﬁrst chart, the tangent space Ta isparametrized by v ∈ R as
√
(
) (
)
c (v)
u − 1 − u2 v
Ta ∋ c(v) = 1= √.
(A.26)c2 (v)
1 − u2 + uv−→ −→This parametrization indeed satisﬁes Oa · ac = 0. Since Ta is a one-dimensional space, there is only one
vector ea1 in a basis of Ta . For ea1 , we choose)( √− 1 − u2.
(A.27)ea1 ≡uNote that c(v) can be expressed as a + vea1 . Then the gradient operator on Ta is expressed as√g(c + hea1 ) − g(c)∂g∂g
= ea1 · ∇g(c1 , c2 ) = − 1 − u2(c1 , c2 ) + u(c1 , c2 ). (A.28)
h→0
h
∂c1
∂c2

∇T a g(c) = lim

In order to express the gradient operator on S using the above result, we ﬁrst express the projection from
S onto Ta . That can be done by inner product with ea1 . For another point b in S whose parameter is w,
the projection is
[( √
)(
)]
t
w
−
u
− 1 − u2√√
Pa b ≡ a + [ea1 · (b − a)]ea1 = a +ea1u
1 − w 2 − 1 − u2√√
= a + (u 1 − w2 − 1 − u2 w)ea1 .
(A.29)
Its inverse Pa−1 is obtained by equating the above expression with c(v) and solving for w. That amounts
to equating coeﬃcients of ea1 :√√√√∴ w = u 1 − v 2 − 1 − u2 v.
(A.30)
v = u 1 − w 2 − 1 − u2 w
Then we have

√
) √
( √
√
√
2−
1 − u2 v
u 1 − v√
Pa−1 [c(v)] = b(w = u 1 − v 2 − 1 − u2 v) = √= 1 − v 2 a + vea1 .
1 − u2 1 − v 2 + uv

(A.31)

This is a function of the parameter v of c. Using c(v) = a + vea1 , we can express v as the function of c asin v = ea1 · c. Then we can express Pa−1 as the function of c:√(A.32)
Pa−1 c = 1 − (ea1 · c)2 a + (ea1 · c)ea1 .

A.3. N -DIMENSIONAL LAPLACIAN AND LAPLACE-BELTRAMI OPERATOR

53

Finally, using the results obtained so far, we have
−1

∂Pa
∇S f (a) = ∇T a f˜(c = a) = ea1 · ∇f˜(c = a) = ea1 ·(c = a) · ∇f (a)
∂c
)(
(
)
)
t( √
2
0
∂f (a1 , a2 )/∂a1
2 − 1−u√= ea1 · −aea1 + ea1 ea1 · ∇f (a) = |ea1 |∂f (a1 , a2 )/∂a2u
1 − 02∂f∂f= −a2(a1 , a2 ) + a1
(a1 , a2 ).
(A.33)∂a1∂a2
During calculation, we used the dyadic notation; e.g., aea1 is a second-rank tensor which maps a vector v
to another vector a(ea1 · v). They originates as follows:
]
∂ [√
∂Pa−1
ea1 · c(c) =1 − (ea1 · c)2 a + (ea1 · c)ea1 = − √aea1 + ea1 ea1 .
∂c∂c
1 − (ea1 · c)2(A.34)The gradient operator on the one-dimensional surface S 1 (A.33) can be seen as derivative with respect toθ when we parametrize S 1 as {(cos θ, sin θ) | θ ∈ [0 . . . 2π)}.

A.3.2

Proof of the theorem

The Laplace-Beltrami operator ∆S on a arbitrary surface S can be deﬁned as the operator such that theequation∫∫(∇S ψ) · (∇S u)dσ = − ψ∆S u dσ
(A.35)
S

S

holds for arbitrary functions u and ψ on S. Here, dσ denotes the surface element of S. Note that ∆S isthe derivative operator involving coordinates on the surface S only. This deﬁnition is delibarately made tolook like the following identity (Green’s theorem) involving the ordinary Laplacian:∫∫∫∫(∇ψ) · (∇u)dV =ψ(∇u · n)dS −ψ∇2 u dV = −ψ∇2 u dV.
(A.36)V∂V

V

V

Here, ∂V denotes the boundary of the region V , which is large enough, and n denotes the normal vector
pointing towards the outside of V . Moreover, ψ is assumed to have a ﬁnite support∗2 ; this leads to the lastequality, since V is so large that it contains supp f .
A point r in the n-dimensional space Rn can be written as r = (r, σ), where r is the distance fromthe origin and σ is the direction of the point from the origin; e.g., all the angular coordinates (coordinates
except for r) in the n-dimensional spherical coordinates are collectively denoted by σ. Thus any function
f (r) in Rn can be seen as a function of r and σ; i.e., f (r) = f (r, σ). Note that σ can be seen as a point on
the (n − 1)-dimensional sphere S n−1 ≡ S. Based the point of view stated above, we consider the following
integral involving two arbitrary functions ψ and u, where we assume that ψ has a ﬁnite support:
∫
∫
∫ ∞
∫
2
n−1
ψ∇ u dV = −(∇ψ) · (∇u)dV = −dr rdσ(∇ψ) · (∇u)
Rn
0
∫Rn ∫ ∞∫ ∞ S∫
1
= − dσdr rn−1 (∂r ψ)(∂r u) −dr rn−1 2dσ (∇S ψ) · (∇S u).
(A.37)r S
S
00
Here we used the fact that n-dimensional volume element dV is written as rn−1 drdσ, and the identity(∇ψ) · (∇u) = (∂r ψ)(∂r u) +1(∇S ψ) · (∇S u).
r2

Using integration by parts, we have∫ ∞∫ ∞dr rn−1 (∂r ψ)(∂r u) = [rn−1 ψ(∂r u)]∞−dr ψ∂r [rn−1 (∂r u)].
r=00

(A.38)(A.39)

0Here, the ﬁrst term vanishes because of the assumption that ψ has a ﬁnite support. Thus, for the ﬁrst termof Eq. (A.37), we have
∫
∫ ∞
∫
∫ ∞
∫
1
− dσ
dr rn−1 (∂r ψ)(∂r u) =dσ
dr ψ∂r [rn−1 (∂r u)] =ψ∂r [rn−1 (∂r u)]dV. (A.40)
n−1
S
0
S
0
Rn r

∗2 The support of a function f is the subset of the domain of f such that f is nonzero; supp f ≡ {x | f (x) ̸= 0}. “A function
f with a ﬁnite support” means that f vanishes outside of some ﬁnite region.

54

APPENDIX A. THEORETICAL TECHNIQUESFor the second term of Eq. (A.37), we use the deﬁnition of the Laplace-Beltrami operator (A.35) to obtain
∫
∫
∫ ∞
∫ ∞
∫
1
n−1 1
n−1 1
ψ∆S u dV.
(A.41)−dr r
dσ (∇S ψ) · (∇S u) =dr rdσ ψ∆S u =
2
2
2
r
r
r
0
S
0
S
Rn
Using the above results, we have
[
]
∫
∫
1
1
2
n−1
ψ∇ u dV =ψ n−1 ∂r (r∂r u) + 2 ∆S u dV.
r
r
Rn
Rn

(A.42)

Since ψ is an arbitrary function, the terms in [· · · ] must coincide with ∇2 u. Therefore we have proven themain theorem (A.22) of this appendix.

A.4

Contour Integral

Here we summarize how to calculate certain kinds of integrals using the technique of contour integration.
See standard
textbooks on complex analysis such as Ahlfors [94] for details. We racall that the contour
H
integral C f (z)dz of a function f deﬁned on a closed path C in the complex plane C is deﬁned throughparametrization of C; i.e., when we write each point z on C as z(t) in terms of t ∈ (a . . . b), where theupper and lower limits a and b may be inﬁnite, thenI∫ bdzf (z)dz ≡f (z(t)) (t)dt.
(A.43)dt
Ca
We often split C into multiple curves (lines) {Ik | k = 0, 1,⊔. . . , n} such that each curve Ik has its own
parametrization zk (t) : (ak . . . bk ) → Ik and the disjoint sum k Ik coincides with C. Then we haveI∑ ∫ bkdzk
f (z)dz =f (zk (t))(t)dt.
(A.44)
dt
Cakk

The main reason for physicists to∫be interested in contour integration is that, it is often possible to calculateua diﬃcult real deﬁnite integral l f (x)dx through a related contour integral which can be more easily
calculated. In order to do so, we ﬁnd an appropriate closed path C satisfying the following conditions:
1. C can be split into multiple curves {Ik } such that(a) One of Ik ’s (say Im ) coincides with the domain of integration of the original integral: Im =(l . . . u).
(b) Complex integrals over curves other than Im can be somehow calculated by standard techniques.
2. C contains only poles ∗3 of f , rather than its essential singularities.
H
Then, because the contour integral C f (z)dz can be calculated using the residue theorem explained below,∫u
we know the deﬁnite value of the integral l f (x)dx as
∫
I
∫ u
∑∫
f (x)dx =f (x)dx =f (z)dz −f (z)dz.
(A.45)
l

Im

C

k̸=m

Ik

The residue theorem is a strong theorem which states that a contour integral
closed path C) can be calculated asIf (z)dz = 2πi
C

N∑Res(f, pk ),HCf (z)dz (for a simple∗4(A.46)

k=1

where Res(f, pk ) is the residue of f at a pole pk of f enclosed by C, and N is the total number of poles off enclosed by C. Moreover, the residue Res(f, p) of f at its pole p with order n can be calculated asRes(f, p) =dn−1
1
lim n−1 [(z − p)n f (z)].
(n − 1)! z→p dz(A.47)∗3 A pole p of a function f with order n is a point in the domain of f such that f behaves as f (z) ∼ C/(z − p)n around p
for some C. However, at an essential singularity a of f , the behavior of f is something like “C/(z − a)∞ .” For example, thefunction e1/z has an essential singularity at z = 0.
∗4 A simple curve is a curve which does not intersect itself.

A.5. MATHIEU EQUATION

55
∫ u

I3
l

I2

C
l

I
f (z)dz −f (x)dx =I1 ≡ (l . . . u) uC∫ b2
f (z2 (t))
a2

−

x

∫ b3

dz2
(t)dt
dt

f (z3 (t))
a3

dz3
(t)dt
dt

Figure A.4: Calculation of a real deﬁnite integral through a contour integral.
∑
This can be understood if we assume that f is expandable into the Laurent series n∈Z an (x − p)n aroundp. Those theorems stated above can be proven by Cauchy’s integral theorem, but we omit the proofs.
Various examples of the uses of the residue theorem can be found as excersises of complex analysis; see,e.g., Spiegel [95]. Some of them have signiﬁcance also in physics. For example, the Airy’s integral used
for decription of the Fraunhofer diﬀraction can be calculated in this way. Moreover, using the technique ofcontour integration, Wittig [96] gave a proof for the Landau-Zener formulaP = exp (−2πΩτ )(A.48)

describing the probability of the diabatic transition in an time-dependent process. In the context of ultracold
atomic physics, the Landau-Zener formula is important in ARP (adiabatic rapid passage) used to ﬂip spinstates of an atomic ensemble.

A.5

Mathieu equation

Here we explain mathematical properties of the Mathieu equation
d2 f(z) + [a − 2q cos(2z)]f (z) = 0.
dz 2(A.49)

Its details are found in the standard references like Sin [97, Sec. 10], Barrett [98], and Meixner and Schäfke[99].
The Mathieu equation is a homogeneous linear second-order diﬀerential equation characterized by thefunctional coeﬃcient [a − 2q cos(2z)] of f (z) with two parameters a and q. This form∗5 is particularly
called the normal form of the equation. It was introduced by Mathieu [100] in 1868 in order to analyze thevibration of elliptical membranes. The equation is notorious for its diﬃcultness in the ﬁeld of diﬀerential
equations.
But by setting q = 0, it reduces to the familiar equation of a harmonic oscillator with frequency√
a. Therefore we see that the diﬃculties arise from the “modulation” 2q cos(2z).
In the context of ultracold atomic physics, its signiﬁcance is twofold: (a) the classical equation of motion
describing parametric oscillation of an atom conﬁned in a time-dependent harmonic potential∗6 , and (b) the
Schrödinger equation of an atom moving within an optical lattice (mimicking that of an electron moving
within a crystal lattice.) The case (a) is described by the time-domain Mathieu equationmd2 x(t) + [mω 2 + K1 cos(ω1 t)]x(t) = 0,
dt2

(A.50)

where m is an atom’s mass, x(t) the location of the atom at time t, and ω the trap frequency of the harmonic
trap U (x) = mω 2 x2 /2 felt by the atom in the unmodulated case. Moreover, K1 and ω1 are respectively the
amplitude and the frequency of amplitude modulation of the trap depth; i.e. the modulated trap potential
U1 (x) is written as
U1 (x) = U (x) +K1 2
1
x = [mω 2 + K1 cos(ω1 t)]x2 .
2
2(A.51)

Eq. (A.50) is obtained from the normal form (A.49) by the substitutions
f (z) = x(t),

z=

ω1
t,
2

(
a=

2ω
ω1

)2
,

q = −2K1.
mω12∗5 Unfortunately, diﬀerent authors use diﬀerent conventions for the Mathieu equation and related functions.
∗6 Jones [1] discusses its application in design of the Paul trap (a trap for ions.)(A.52)

56

APPENDIX A. THEORETICAL TECHNIQUES

10
8
6

0.75
0.5
0.25
1e-008

4
2
0

a

-2
-4
-6
-8
-10
0

2

4

6

8

10

q

Figure A.5: Plot of stable regions of the Mathieu equation by the C proglam written by Jones [1].
The case (b) is described by the space-domain Mathieu equation
[
]
[
]
~2 d2
~2 d2 ϕ
U
2
′
−
+
U
sin
(kx)
ϕ(x)
=
Eϕ(x)
⇐⇒
(x)
+
E
+
cos(2kx)
ϕ(x) = 0,2m dx22m dx2
2(A.53)

where m is an atom’s mass, ϕ(x) is Schrödinger’s (time-independent and, in this case, 1D) wavefunction,
E ≡ E ′ +U/2 is the energy eigenvalue, U and 2k = 2π/(λ/2) are respectively the depth and the wavenumberof the sinusoidal potential with period λ/2 felt by the atom. Eq. (A.53) is obtained from the normal form
(A.49) by the substitutions
f (z) = ϕ(x),z = kx,

a=

2mE ′
,
(~k)2

q=−

mU
.
2(~k)2(A.54)

Observe that a and q can be respectively seen as the ratio of the eigenenergy and the depth of the latticeto the recoil energy of the lattice laser (→ Sec. 2.3.)
In the following, we follow the notation of Meixner and Schäfke [99], except that we use a and q
instead of λ and h2 , respectively. Exploiting the periodicity of the modulation in the Mathieu equation, weapply Bloch’s theorem in solid state physics, or equivalently, Floquet’s theorem in the theory of diﬀerential
equations, which restricts forms of its solutions. Those theorem states that, for the ﬁxed values of a andq > 0∗7 , the equation has complex-valued solutions me±ν (z) called the Floquet solutions which have theform ofme±ν (z) ≡ me±ν (z; a, q) = e±iνz uν (z).

(A.55)

Here, ν is a function of a and q called the characteristic exponent (also known as the crystal momentum or
the quasimomentum in the ﬁeld of solid state physics; see Ashcroft and Mermin [101, Ch. 8],) and uν (z)is a periodic complex-valued function with period π which corresponds to the Bloch function, and can be
determined only up to a multiplicative factor. The periodicity of uν (z) can be understood to originate
from that of the modulation 2q cos(2z). The imaginary part of the characteristic exponent determines thestability of the Floquet solution, and thus has been subject to intense research by mathematicians. See, e.g.,Sträng [102] for algorithms to compute it. Its implementation by Jones [1] in C language yields Figure A.5
which shows regions in the a-q plane corresponding to the real ν and hence stability of the Floquet solution(Contours shows the iso-ν points.) The blank region corresponds to complex ν (instability of the Floquet
solution.) We can observe that, in the context of a classical particle moving in a modulated potential,
strong parametric heatings occur at a = ±n2 for an integer n; that is, the minimal value for the strength qof the potential leading to the stable orbit of the particle approaches zero under such a condition.
As with the case of linear second-order diﬀerential equations, one can choose two independent solutions(a basis of the solution space) such that general solutions of the Mathieu equation can be obtained by∗7 The case of q < 0 is easily handled by substitution of z with z + π/2; 2q cos[2(z + π/2)] = −2q cos(2z). The trivial case ofq = 0 is omitted.

A.5. MATHIEU EQUATION

57

linear combination of them. Except for the special case in which e±iπν is either 1 or −1 (which respectively
corresponds to ν = 2n or 2n + 1 for an integer n,) we can choose {meν (z), me−ν (z) = meν (−z)} for thebasis. Alternatively, two real-valued functions called the Mathieu (or elliptic) cosine yI (z) and sine yII (z)may be chosen∗8 . They are characterized by the normalization conditionsyI′ (z = 0) = 0,′yII(z = 0) = 1,

yI (z = 0) = 1,
yII (z = 0) = 0,

and in fact, reduced to the corresponding trigonometic functions for the special case of q = 0:√√sin( az)√yI (a, q = 0; z) = cos( az),yII (a, q = 0; z) =.
a

(A.56a)(A.56b)(A.57)

However, the Mathieu cosine and sine are generally not periodic in z. In fact, they can be related to the
Floquet solutions by formulae such asyI =1(meν + me−ν ).
2(A.58)

There are countably many values of a (that depend on q; thus we may denote the n-th of them byan (q) and bn (q),) leading to the special cases e±iπν = 0, 1. Then meν (z) and me−ν (z) coincides, have true
periodicity or antiperiodicity with period π, and become an even or odd function, corresponding to ν = 2n
or 2n + 1.
Mathematical analysis shows that am (q) ∼ bm+1 (q) when q → ∞, i.e., the m-th “allowed” energy
range in the band structure gets narrower and narrower in the limit of a deep periodic potential (it indeed
approaches a ﬂat “band,” which is obtained in the case of a simple harmonic oscillator.) However, there
always remains a ﬁnite width for the m-th band which can be proven to be√()
24m+5 2 m/2+3/4 −4√q
6m2 + 14m + 7
bm+1 (q) − am (q) ∼qe
1−(q → ∞).
(A.59)√m!
π
32 q(A standard reference by Abramowitz and Stegun [103] cites Meixner and Schäfke [99] for this expression,
though the latter only presents the result without a proof. The reader may found Dingle and Müller[104] and Liang and Müller-Kirsten [105] helpful.) Applying this expression to the spacial-domain Mathieu
equation Eq. (A.53) for an optical lattice yields an approximated formula for the hopping matrix element
J in the limit of deep lattice (See, e.g., Bloch et al. [106, Sec. II.B])√
4
J˜ ∼ √ (Ũ )3/4 exp(−2 Ũ )π(Ũ → ∞).

(A.60)

Here, we wrote the formula in terms of the “normalized” energies by the recoil energy Erec ; J˜ ≡ J/Erecand Ũ ≡ U/Erec . This formula is based on the fact that the hopping matrix element J for the atoms in thelowest band can be obtained from the width W ≡ b1 − a0 of the lowest (m = 0) band as J = W/4 (Jaksch[107, Sec. 7.2].) Then Eq. (A.60) immediately follows from substitution q = U/(4Erec )∗9 and m = 0.
In addition to the rich mathematical literature on analytic properties of solutions of the Mathieu equation, we may utilize CAS (computer algebra systems) such as Mathematica and Maple which have built-infunctions for the Mathieu functions, as well as the characteristic exponent.

∗8 Note that diﬀerent parameter regions require diﬀerent elements for a basis for the solution space. See Barrett [98] fordetails.
∗9 Forget about the sign of q! It is inessential, as explained in p. 56, ft.

Appendix B

Converter program from SPE image
for WinView to JPEG image
It is convenient when the SPE image ﬁles obtained are shown in thumbnail list during a series of runs inwhich we change one of the parameters to ingestigate the dependence on the parameter. If we convert
the SPE image to standard JPEG image, Windows Explorer or Microsoft Image Viewer can show them as
thumnail image list. So we made a converter from SPE to JPEG.
Here we show the source code in Java. The program is in the public domain.

58

59
package spe2jpg;
import java.io.*; import java.util.*; import java.util.regex.*; import java.awt.*; import java.awt.event.*;
import java.awt.dnd.*; import java.awt.datatransfer.*; import java.awt.image.*; import javax.swing.*;
import javax.imageio.*; import javax.imageio.stream.*;/*** @author NODA Kai*/
public class Main extends JFrame {DropTargetAdapter dta; JTextArea area;

static Pattern pat;/**
* @param args the command line arguments*/
public static void main(String[] args) {if (args.length == 0) { new Main(); }else { convertAux(args, System.out); }}public Main() {super("SPE to JPG converter");
setSize(new Dimension(600, 400)); setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
area = new JTextArea("Use Drag-and-Drop to convert SPE files. "+ "If you DnD a directory, its all entries are processed.\n"+ "(Altenatively, you can specify files/directries with command-line arguments.)\n"+ "Please note that, though the pseudo color used in this program looks almost same to that used in WinView, "+ "they are a bit different.");area.setEditable(false); area.setLineWrap(true);
JScrollPane sp = new JScrollPane(area); getContentPane().add(sp);dta = new DropTargetAdapterImpl();pat = Pattern.compile(".*\\.spe", Pattern.CASE_INSENSITIVE);
setVisible(true);}/*** A method that does the actual work.
* Refer to Appendix B of WinView 2.5 User Manual for the binary format of SPE file.
* @param f SPE file.
* @throws java.io.FileNotFoundException*/
public static void convert(File f) throws FileNotFoundException {try {BufferedInputStream is = new BufferedInputStream(new FileInputStream(f));
byte[] header = new byte[4100]; is.read(header);int xDim = ((0xff & header[43]) << 8) | (0xff & header[42]);
int yDim = ((0xff & header[657]) << 8) | (0xff & header[656]);
int frames = ((0xff & header[1449]) << 24) | ((0xff & header[1448]) << 16)| ((0xff & header[1447]) << 8) | (0xff & header[1446]);
int type = header[108];// System.out.printf("%d %d %d %d\n", xDim, yDim, frames, type);
if (frames == 1) {byte[] raw = new byte[xDim * yDim * 8]; is.read(raw);
BufferedImage img = new BufferedImage(xDim, yDim, BufferedImage.TYPE_INT_RGB);
DataBufferInt db = (DataBufferInt) img.getRaster().getDataBuffer();for (int i = 0; i < xDim * yDim; ++i) {long val = ((0xffL & raw[8 * i + 7]) << 56) | ((0xffL & raw[8 * i + 6]) << 48) |((0xffL & raw[8 * i + 5]) << 40) | ((0xffL & raw[8 * i + 4]) << 32) |((0xffL & raw[8 * i + 3]) << 24) | ((0xffL & raw[8 * i + 2]) << 16) |((0xffL & raw[8 * i + 1]) << 8) | (0xffL & raw[8 * i]);// NOTE: Put your own scaling factor here (default: 1e3 == 1000)
db.setElem(i, pseudoColor(Double.longBitsToDouble(val) / 1e3));}
Iterator writers = ImageIO.getImageWritersByFormatName("jpg");
if (writers.hasNext()) {ImageWriter iw = (ImageWriter) writers.next();
String origName = f.getAbsolutePath(), jpgName;int pos = origName.lastIndexOf(’.’);
jpgName = origName.substring(0, pos >= 0 ? pos : origName.length()) + ".jpg";
ImageOutputStream ios = ImageIO.createImageOutputStream(new File(jpgName));iw.setOutput(ios); iw.write(img); iw.dispose(); ios.close();}} else {for (int fi = 0; fi < frames; ++fi) { /* TODO: the case when there are multiple frames */ }}}catch (IOException ex) { ex.printStackTrace(); }}/*** This method is responsible for the pseudo-color.
* You might well to change the constants to modify the color curve.
* @param gray gray-scale value.
*/
static int pseudoColor(double gray) {gray = Math.min(Math.max(gray, 0), 1d);
float h = 2f / 3 - (float) (Math.min(5d / 6 * gray, 2d / 3));float s = (float) (Math.min(15 * (gray - 1) * (gray - 1.1), 1));
float b = (float) (0.4 + 3 * Math.min(gray, 0.2));
return Color.HSBtoRGB(h, s, b);

60

APPENDIX B. CONVERTER PROGRAM FROM SPE IMAGE FOR WinView TO JPEG IMAGE}
static void convertAux(String[] paths, PrintStream out) {File[] files = new File[paths.length];for (int i = 0; i < paths.length; ++i) { files[i] = new File(paths[i]); }PseudoPrintStream pps = new PseudoPrintStream() {PrintStream strm;
public void print(String s) { strm.print(s); }public void setDestination(Object o) { strm = (PrintStream) o; }};pps.setDestination(out); convertAux2(files, pps);}
static void convertAux(java.util.List<File> files, JTextArea area) {File[] arr = new File[files.size()]; int cnt = -1;
for (File f : files) { arr[++cnt] = f; }PseudoPrintStream pps = new PseudoPrintStream() {JTextArea ta;
public void print(String s) {ta.append(s);ta.setCaretPosition(ta.getText().length());}
public void setDestination(Object o) { ta = (JTextArea) o; }};pps.setDestination(area); SwingWorker worker = new SwingWorkerImpl(arr, pps);}

worker.execute();static void convertAux2(File[] files, PseudoPrintStream out) {try {for (File f : files) {if (f.isDirectory()) {out.print("\n");File[] entries = f.listFiles();for (File e : entries) {if (pat.matcher(e.getName()).matches()) {out.print("\nConverting " + f.getName() + "/" + e.getName() + "...");
convert(e); out.print(" done.");}}} else {if (pat.matcher(f.getName()).matches()) {out.print("\nConverting " + f.getName() + "...");
convert(f); out.print(" done.");}}}}catch (Exception ex) { ex.printStackTrace(); }}private class DropTargetAdapterImpl extends DropTargetAdapter {public DropTargetAdapterImpl() {DropTarget target = new DropTarget(area, DnDConstants.ACTION_COPY_OR_MOVE, this, true);}@Override
public void dragEnter(DropTargetDragEvent e) { e.acceptDrag(DnDConstants.ACTION_COPY_OR_MOVE); }public void drop(DropTargetDropEvent e) {e.acceptDrop(DnDConstants.ACTION_COPY_OR_MOVE);
try {if ((e.getDropAction() & DnDConstants.ACTION_COPY_OR_MOVE) != 0) {Transferable tr = e.getTransferable();java.util.List<File> lst =(java.util.List<File>) tr.getTransferData(DataFlavor.javaFileListFlavor);
convertAux(lst, area); e.dropComplete(true);} else { e.dropComplete(false); }}catch (Exception ex) { ex.printStackTrace(); e.dropComplete(false); }}}private interface PseudoPrintStream {public void print(String s);
public void setDestination(Object o);}private static class SwingWorkerImpl extends SwingWorker {private final File[] arr; private final PseudoPrintStream pps;
public SwingWorkerImpl(File[] arr, PseudoPrintStream pps) {this.arr = arr; this.pps = pps;}
@Override
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