ボーズ凝縮原子を用いた量子原子光学の研究

課題番号：17340120

平成 17－19 年度科学研究費補助金
基盤研究 B
研究成果報告書

平成 20 年 4 月

研究代表者：中川賢一(電気通信大学レーザー新世代研究センター准教授)

平成 17－19 年度科学研究費補助金
基盤研究 B

ボーズ凝縮原子を用いた量子原子光学の研究
課題番号：17340120
研究組織
研究代表者

中川賢一(電気通信大学レーザー新世代研究センター准教授)

研究費
平成 17 年度

7,100 千円

平成 18 年度

4,900 千円

平成 19 年度

3,100 千円

計

15,100 千円

研究発表
論文
1. Munekazu Horikoshi and Ken'ichi Nakagawa, Suppression of dephasing due to a trapping
potential and atom-atom interactions in a trapped-condensate interferometer, Phys. Rev.
Lett. 99, 180401 (2007).
2. Mark Sadgrove, Munekazu Horikoshi, Tetsuo Sekimura1 and Ken'ichi Nakagawa,
Coherent control of ballistic energy growth for a kicked Bose-Einstein condensate, Eur.
Phys. J. D 45, 229-234 (2007).
3. Mark Sadgrove, Munekazu Horikoshi, Tetsuo Sekimura, and Ken'ichi Nakagawa ,Rectified momentum transport for a kicked Bose-Einstein condensate, Phys. Rev. Lett. 99,
043002 (2007).
4. Munekazu Horikoshi, Ken'ichi Nakagawa, Dephasing due to atom-atom interaction in a
waveguide interferometer using a Bose-Einstein condensate, Phys. Rev. A 74, 031602(R)(2006).
5. Munekazu Horikoshi, Ken'ichi Nakagawa, Atom Chip based fast production of
Bose-Einstein condensate, Appl. Phys. B 82, 363-366 (2006).
6. K. Nakagawa, Y. Suzuki, M. Horikoshi, and J. B. Kim, Simple and efficient magnetictransport of cold atoms using moving coils for the production of Bose-Einsteincondensation, Appl. Phys. B 81, 791-794 (2005).

国際会議発表1. Ken'ichi Nakagawa, Munekazu Horikoshi : Bose-Einstein-Condensate Interferometer on
an Atom Chip with a Long Coherence Time, 18th International Conference on Laser
Spectroscopy (ICOLS), Telluride, Colorado, June 25-29 (2007).
2. Ken'ichi Nakagawa and Munekazu Horikoshi : Atomic interferometer on an atom chip
(invited), International Conferernce on Lasers and Nanomaterials (ICLAN), Kolcata, Nov.
30 - Dec. 2 (2006).
3. M. Horikoshi, Y. Suzuki, K. Nakagawa : Fast and Efficient Production of Bose-Einstein
Condensate Atoms based on an Atom Chip, International Quantum Electronics
Conference 2005, Tokyo, July 11-15 (2005).
4. K. Nakagawa, M. Horikoshi : Towards coherent atom optics in an atom waveguide, 14th
International Laser Physics Workshop, Kyoto, July 4-8 (2005).
5. Ken'ichi Nakagawa, Munekazu Horikoshi, Yusuke Koda : Atomic interferometer withcold rubidium atoms for the precision measurements of gravirational acceleration, 17th
International Conference on Laser Spectroscopy, Scotland, June 19-24 (2005).
6. Munekazu Horikoshi, Ken'ichi Nakagawa : Atom-chip-based fast production of
Bose-Einstein condensate atoms, 17th International Conference on Laser Spectroscopy,
Scotland, June 19-24 (2005).

国内学会発表１. 福泉美穂、玉木嘉人、桂川眞幸、中川賢一：87Rb 原子の二光子遷移によるリュードベルグ状態の分光 II、日本物理学会第 63 回年次大会、23aQD-8, 2008 年 3 月．
２. Phase defects in an optical lattice: effect on coherent atomic dynamics, 日本物理学会第 63 回年次大会、25pQD-7, 2008 年 3 月．
３. 少数個レーザー冷却 Rb 原子における Rydberg 励起の双極子ブロッケード、日本物理学会第 63 回年次大会、26aQD-1, 2008 年 3 月．
４. Mark Sadgrove, Munekazu Horikoshi, Tetsuo Sekimura, Ken'ichi Nakagawa :Bose-Einstein Condensates in pulsed optical lattices, 日本物理学会第 62 回年次大会、22pRG-1, 2007 年 9 月．
５. 関邨哲雄、杉江宏太、中川賢一：Rb 原子の 2 波長光双極子トラップ、日本物理学会第 62 回年次大会、24pRH-8, 2007 年 9 月．
６. 福泉美穂、玉木嘉人、Mark Sadgrove、中川賢一：87Rb 原子の二光子遷移によるリュードベルグ状態の分光、日本物理学会第 62 回年次大会、 21pRH-13, 2007
年 9 月．
７. Mark Sadgrove、堀越宗一、関邨哲雄、中川賢一：Shift in fundamental quantum
resonance for atoms in a pulsed optical lattice、日本物理学会 2007 年春季大会、
20aXK11, 2007 年 3 月．
８. 堀越宗一、中川賢一：長いコヒーレンス時間を持った調和ポテンシャル中でのBEC 干渉計、日本物理学会 2007 年春季大会、20aXK10, 2007 年 3 月．
９. 中川賢一、江村真史、後藤亮彦：電場を用いたアトムチップによるボーズ凝縮体の操作 II、日本物理学会 2007 年春季大会、 20aXK9, 2007 年 3 月．
１０.

玉木嘉人、Mark Sadgrove、中川悠輔、中川賢一：単一原子トラップを用いた単一光子源、日本物理学会 2007 年春季大会、19aXK10, 2007 年 3 月．
１１.

堀越宗一、中川賢一：磁場ガイド BEC 干渉計における位相シフトとコントラスト低下、日本物理学会第 61 回年次大会、29pTA12, 2006 年 3 月．
１２.

江村真史、堀越宗一、中川賢一：電場を用いたアトムチップによるボーズ凝縮体の操作、日本物理学会第 61 回年次大会、29pTA11, 2006 年 3 月．
１３.

後藤亮彦、豊永匡利、中川賢一：フェルミ縮退に向けての 40K と 87Rb の同時トラップ、日本物理学会第 61 回年次大会、28aRE11, 2006 年 3 月．

１４.

小田悠介、後藤亮彦、中川賢一：原子干渉計を用いた重力加速度測定Ⅲ、日本物理学会 2005 年秋期大会、 20aWA14, 2005 年 9 月．
１５.

中川悠輔、中川賢一：磁気光学トラップによる 単一原子トラップⅡ、日本物理学会 2005 年秋期大会、19aWA2, 2005 年 9 月．
１６.

堀越宗一、江村真史、中川賢一：磁場ガイド中のボース凝縮体の操作、日本物理学会 2005 年秋期大会、19aWA3, 2005 年 9 月．

研究会発表等
１. 中川賢一：アトムチップによるボース凝縮原子の操作とその応用、原子・分子・光科学(AMO)第３回討論会、招待講演、東大駒場、2006 年 6 月 16 日．
２. 中川賢一：光でつくるコヒーレントな原子、2007 年度日本物理学会科学セミナー、
電気通信大学、2007 年 8 月 24 日．

目次
１. はじめに1

２. アトムチップを用いた高速 BEC 生成装置の開発

3

３. アトムチップ上の BEC 干渉計

7

４. BEC 原子を用いた量子ラチェット効果

13

５. 単一原子トラップの開発

16

６. 電場・磁場ハイブリッド型アトムチップの開発

19

７. まとめ

22

参考文献

23

A 付録

25

A.1 原著論文

26

A.2 国際会議プロシーディング等

52

A.3 研究会資料等

62

1. はじめにレーザー冷却によって得られる温度 1μK 程度の極低温原子においてはそのド・ブロイ波長は 100nm 以上にもなり粒子としてよりも波としての性質が顕著に現れる。このような極低温原子の温度をさらに下げ、また原子密度を上げて隣り合う原子同士の原子波が重なり合うようにすると多数の原子が全て同じ量子状態に落ち込むいわゆ
るボース・アインシュタイン凝縮(Bose-Einstein Condensation: BEC)が実現される。このボース凝縮状態の原子は光におけるレーザーと同様に多数の原子の波動関数の位相が揃ったコヒーレントな物質波とみなすことができ、このボース凝縮から取り出された原子はレーザー光と同様に高い干渉性を示すため原子レーザーと呼ばれる。このようなコヒーレントな物質波を光と同様に操作する研究はコヒーレント原子光学と呼ばれ、1995 年に Cornell 等によって最初の原子のボース・アインシュタイン凝縮が実現されて以来この分野の研究は劇的な発展を遂げてきた 1)。また最近では光における量子光学と同様に原子の波動および粒子の二重性をフルに用いて非古典的な量子状態を原子において実現する量子原子光学の研究が行われるようになり、これは光と同様に量子コンピューターなどの量子情報処理への応用が期待されている 2)。
本研究はこのようなボース凝縮原子によるコヒーレントな物質波を用いた原子光学の基礎技術および物理の確立およびこの様々な分野への応用をめざしたものである。本研究ではまず原子光学の研究において重要となるコヒーレントな物質波の発生源であるボーズ凝縮原子の生成装置の開発を行った。アトムチップと呼ばれる集積化
原子回路の手法を用いたボース凝縮生成装置を開発し、従来より一桁以上高速にボース凝縮原子を生成することが可能になり、様々な原子光学実験を効率良く行うことを可能にした。次にこのアトムチップを用いたボース凝縮生成装置を用いてチップ上でのボース凝縮原子を用いた原子干渉計の研究を行った。従来の自由空間中の原子に代わってトラップされた原子を用いた原子干渉計が実現されれば重力による自由落下に制限されずに長い相互作用時間が得られるため、高感度なジャイロスコープや加速度センサーなどへの応用が期待されている 3),4)。またボース凝縮原子を用いたコヒー
レント原子光学の研究の一つとしてパルス状の光定在波ポテンシャル中の原子の運動の研究を行った。この系は量子キック回転子と呼ばれ今までにレーザー冷却原子を用いて実験において量子カオスの研究が行われてきたが本研究では単一の量子状態にあるボース凝縮原子を用いることにより精密にこの系の性質を調べることが可能になる。本研究ではボース凝縮原子を用いた量子キック回転子系において量子ラチェット効果の実現を予想してこれを実験によって実証した。近年注目されている量子コンピューターなどの量子情報処理を中性原子で行うためにはボース凝縮原子のようなマクロな原子集団だけではなく、原子を 1 個レベルで操作する必要がある。このた1

め本研究ではレーザー冷却原子を光双極子トラップ中に 1 個レベルでトラップおよび検出するための基礎技術の開発を行った。この単原子操作および検出の技術を先のア
トムチップによる原子回路の技術と組み合わせることにより原子の粒子性および波動性を究極的に追求する量子原子光学の研究が可能になるとものと期待される。

2

2. アトムチップを用いた高速ボース凝縮原子生成装置の開発
アトムチップと呼ばれる基板上に集積化された原子回路を実現する方法が近年盛んに研究されている 5)。これはリソグラフィー技術を用いて基板上に微細な電線パターンを作成し、この電線に電流を流すことによって作られる微細な磁場ポテンシャルを用いて極低温原子を電線パターンに沿ってガイドして自由自在に操作しようというものである。このアトムチップを用いると原子を 1 個ずつトラップしたり操作することが可能になるため、中性原子を用いた量子コンピューターの実現という観点から非常に興味が持たれている。またアトムチップ上の原子導波路によって極低温原子やボース凝縮原子をガイドして光の干渉計と同様の原子干渉計が実現可能となり、光の干渉計を超える高い感度のジャイロスコープや加速度計が実現可能になるものと期待されている。
我々はこのアトムチップ上でボース凝縮原子を用いたコヒーレント原子光学の研究を行うため、先ずアトムチップを用いたボース凝縮原子生成装置を開発した 6)。ア
トムチップによって作られる微細な磁場ポテンシャルはボース凝縮の実現に必要な高い原子密度を比較的容易に実現できることから高速で高効率な蒸発冷却が可能となりボース凝縮原子の生成に好都合である。図 1(a)に本研究で作成したアトムチップの基板を示す。基板材料としては絶縁体で電線からの発熱を効率良く放熱するため熱伝導性が高いことが求められ、シリコン、サファイヤ、窒化アルミ(AlN)などが用いられる。電極材料としては電流による発熱を小さくするため電気伝導率が高い銅、金、アルミなどが用いられる。本研究ではシリコン基板上にリソグラフィー技術を用いて電線パターンのマスクを作成して金蒸着を行い、さらに金めっきを用いて電極の厚さを 10μｍまで増やした(図 1(b))。
(a)

(b)

(c)
原子導波路
原子
50~200μm

Si

原子~100μm
SiO2

金
銅

2 cm

Bx
電線
電流 I

Bϕ =

μ0 I
2πr

図 1 アトムチップ。(a) アトムチップの基板。(b) 電極の構造。(c) 電線を流れる電流が作る磁場による原子導波路。
3

このアトムチップ上の電線に電流を流すと電線の周りには円形の磁場 Bφ=μ0I/2πr が生じるが、これに電線に垂直方向に一様な磁場 Bx を加えると電線の中心から距離
μ0I/2πΒx の位置で磁場がゼロになり原子を電線に沿って閉じ込める磁場ポテンシャルが作られ、スピンを持った原子を電線に沿ってガイドする原子導波路が実現される(図 1(c))。実際にはさらに電線に沿った方向にも磁場を加えて磁場の最小値がゼロでないようにしてマヨナラ遷移による原子のスピンフリップを防ぐ。このように作られる磁場ポテンシャルによってスピンを持つ原子は電線に沿ってその近くに強く閉じ込められるため、電子回路と同様に原子を電線パターンに沿って操作して一種の原子回路が実現される。様々な電線パターンを用いることにより電線に沿って閉じ込める
導波路以外にも 3 次元的に原子を閉じ込める 3D トラップ、またこれを一列に並べたアレー型トラップなども実現することができる(図 2)。

(a) 導波路I(c) 1Dアレー型トラップ(b) 3Dトラップ

原子

電線

I

B

B

図 2 アトムチップ上の原子光学素子。(a) 導波路、(b) 3D トラップ、(c) 1D アレー型トラップ

このようなアトムチップをガラスセル内

アトムチップに配置して 87Rb 原子のボース凝縮原子を磁場トラップ用コイル

生成する装置を実現した(図 3)。ガラスセル
はイオンポンプおよびチタンサブリメンションポンプにより約 10-10 トル(10-9 Pa)以下の超高真空に排気されている。先ずセル中

レーザー
光

Rb原子
MOT
ガラス
セル

UV
LED

温度約 100μΚ程度に冷却して磁気光学トラップに 107 個以上の冷却 Rb 原子を集める。
次にこの冷却 Rb 原子は磁場トラップに移行され、複数の磁場トラップコイルを用い

4cm
図 3

の室温 Rb 原子をレーザー冷却法によってアトムチップを用いた小型・高速て 2cm 上方のアトムチップの基板近くまで輸送される。最後にアトムチップの電線にBEC 生成装置
4電流を流してチップ上のマイクロ磁場ポテンシャルに原子がローディングされる(図4)。チップにローディングされた冷却 Rb 原子は圧縮され、RF 磁場によるスピンフリップを用いた蒸発冷却を用いてさらに原子の温度を冷やす。温度が凝縮の相転移温度
約 2μK 程度まで下げると原子の速度分布に速度がほぼゼロの成分が現れてボース凝縮が始まり、さらに温度を下げるとほぼ全ての原子が凝縮する(図 5)。
このようなアトムチップを用いた方法によりチップ上に最大 15000 個のボース凝縮

Rb 原子を 10 秒以内に生成することができる。これは従来の大型の磁場コイルを用いた磁場トラップによるボース凝縮生成に比べて生成される凝縮原子の個数は一桁程
度少ないが生成時間は 1/3 以下に短縮される。これはアトムチップで作られる微小な磁場トラップにより高い原子密度が得られ、原子の弾性衝突確率を高くすることが可能となり、蒸発冷却が高効率かつ高速に行われるためである。

10mm

基板

図 4 レーザー冷却 Rb 原子のアトムチップへのローディングの様子

サーマル原子

サーマル原子+BEC

1.86MHz

1.83MHz

BEC
1.78MHz

Nc=15000

N=50000

光学密度

0.6

T=2μK > Tc

0.6

T =1.8μK ~ Tc

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0

0.0

−2.0 −1.0 0

1.0 2.0v [cm/s]−2.0 −1.00

1.0 2.0v [cm/s]

T < Tc−2.0 −1.0 0

1.0 2.0v [cm/s]

図 5 アトムチップで生成されたボース凝縮原子の TOF 吸収像(上段)および速度分布(下段)。

初期の原子密度および RF 周波数の掃引時間等を最適化することにより蒸発冷却時
5間を 0.7 秒まで短縮することができた(図 6)。レーザー冷却による予備冷却や磁場トラップへの移行に必要な時間を合わせても 3 秒以内にボース凝縮原子を生成することが可能になり、これは従来の大型の磁場コイルを用いた方法に比べて一桁以上短いものである。またこのアトムチップを用いた方法は従来の大型の磁場トラップを用いた方法に比べて磁場トラップに最初にローディングされる原子数に依存せずに安定に非常に再現性良くボース凝縮原子を生成することができるため、この凝縮原子を用いた様々な応用実験の非常に効率良く行うことができる。

35
Rf周波数[MHz]

30
25
20
15
10
5

従来型磁場トラップ（QUIC）

アトムチップ

10

20

30

40

蒸発冷却時間[秒]
図 6 蒸発冷却における RF 磁場の周波数掃引63. アトムチップ上のボース凝縮原子を用いた原子干渉計
原子干渉計とは光の代わりに原子を物質波として用いて干渉計を実現したものである。原子は光と異なり質量を持つことから原子自身が外力に対するセンサーとして働き、ジャイロスコープや加速度計が実現可能となる3),4)。アトムチップによって実現される原子導波路やマイクロ磁場トラップを用いることによりトラップされた極低温原子やBEC原子を用いた原子干渉計が実現可能となる。原子干渉計の検出感度は通常原子と外力との間の相互作用時間を長くするほど高くすることができるが、自由空間中では重力による自由落下によりこの相互作用時間は１秒程度が限界である。これに対してトラップされた原子を用いれば重力による制限なしに相互作用時間を1秒以上に延ばすことが可能となり高い検出感度が期待できる。またレーザー冷却によって得られる極低温原子の温度は通常マイクロケルビン(μK)程度であるが相互作用時間が長くなるにつれて高い干渉フリンジのコントラストを得るにはより低い温度が必要となる。これに対してボース凝縮原子の温度はナノケルビン(nK)台と低く、また空間的に位相が揃ったコヒーレントな物質波であるため高いコントラストの干渉信号が期待できる。このためアトムチップ上のボース凝縮原子を用いた原子干渉計によって長い相互作用時間を実現することに多くの関心が寄せられていた。
最初のアトムチップ上のボース凝縮原子を用いた干渉計はコロラド大学のJILA研究所のCornellおよびAndersonのグループによって実現された7)。彼らはチップ上のRb
原子のボース凝縮体に光定在波パルスを入射して原子のビームスプリッターを実現
して干渉計を構成した。この実験によって100nK以下のボース凝縮原子が室温のチップ基板から100μm程度の近傍にあるにも関わらず基板の影響をほとんど受けずに20％以上の高いコントラストの干渉信号が得られることを初めて示された。この実験
では相互作用時間が10ms以上になると干渉信号のコントラストが大きく低下することが確認されていたが、これは基板の影響ではなく磁場ポテンシャルの影響であることが示唆されていた。
我々はこのチップ上の干渉計の干渉信号のコントラスト低下の原因を探るため、JILAと同様のアトムチップ上のBEC干渉計を実現してこの原因を詳細に調べた8)。本実
験では周波数差Δωの対向する2つの光による原子のブラッグ回折を用いてπ/2-π-π/2の3つ光パルスによってマッハツェンダー型原子干渉計を構成している(図7(a))。2番目のπ/2光パルスの位相差φに応じて二つのパスを通った物質波が干渉して終状態の2つの運動量成分の分布数が変化する(図7(b))。この実験の結果、JILAの実験と同様に相互作用時間が短い場合には高いコントラストの干渉信号が観測されるが、相互作用時間を長くするにつれてコントラストが大きく低下するのが確認された(図8)。

7

(a)

(b)

アトムチップ

レーザー光

レーザー光

BEC

ω

ω+Δω

p=0

2hk

π/2
π

t

φ= 120 °φ= 240 °π/2
位相差φp = 0 p = 2hk
図 7 (a) アトムチップ上のマッハツェンダー型原子干渉計。(b) 各位相差φにおける干渉計の終状態の TOF 吸収像。
1.0
T = 1ms

T = 0.25ms

P2hk

0.8
0.6
0.4
0.2
0.0
-360

-180

0

180

360

位相[deg]
図 8 観測された干渉フリンジ信号。

この干渉信号のコントラスト低下はボース凝縮原子がチップ上の磁場ポテンシャ
このコントラスト低下は原子がポテンシャル内で運動することによる速度変化によるものであることが理論的な考察により明らかになり、これを実験で検証した。実験では軸方向のトラップ周波数(νz)を変えることにより磁場トラップ中の原子の速度変化の大きさを変化させてこれに伴う干渉信号の位相シフトおよびコントラストを測定したところ実験結果は理論的な解析結果とよく一致することが分かった(図9)。
このためこの原子の速度変化による空間的に不均一な位相分布(デフェージング)を何らかの方法で回避もしくは低減することができれば干渉計の相互作用時間を十分長くすることが可能であることが分かった。

8

(a)

(b)

P2 hk

(c)
位相シフト

干渉信号

コントラスト0

1.0

1.0
N=6,000

0.8
M

0.4

M

Δφ [rad]

−π/2

0.6

0.5

−π

N=8,000

0.2
0.0
Δφ

π
φL [rad]2π

N=10,000

0.0
0

10
20
νz [Hz]

30

0

10
20
νz [Hz]30

図 9 (a) 干渉信号の位相シフト(Δφ)とコントラスト(M)。(b) 位相シフトのトラップ周波数(νz)

そこでトラップ中の原子の速度変化の影響を回避する新奇な原子干渉計を考案した(図10)9)。これは調和振動ポテンシャル中の原子の速度は振動周期Tごとに再び同じ値に戻るということを利用しようしている。このためトラップの振動周期に合わせて原子のπ/2パルスを入射して原子を分割して再び結合すると先の速度変化の影響を受けずに高いコントラストで干渉信号が得られる可能性がある。実際にはボース凝縮原子を用いた場合、原子間の相互作用の影響は無視できず、もう少し状況は複雑になる。
調和振動ポテンシャル中のボース凝縮体の波動関数はポテンシャルの非調和項が無視できる場合、凝縮体の重心運動と内部運動(集団運動)に分離して考えることができる。重心運動は古典的な調和振動 Z (t ) = Z 0 sin(ω z t ) となり、凝縮体は振動周期T( =2π/ωz)で再び同じ位置および速度に戻る。このため相互作用があるボース凝縮体においても振動周期Tに合わせて2つのπ/2パルスを入射して凝縮体を分割および再結合することにより2つの凝縮体はほぼ完全に重なり合い、高いコントラストの干渉信号

図 10 調和振動ポテンシャル中の原子干渉計
9が得られることになる。厳密には原子間相互作用の影響は皆無ではなく、2 つのπ/2
パルスによって凝縮体が分割または再び重なり合う際に原子間相互作用による両凝縮体の斥力が働き、これが内部運動を励起する可能性がある。しかし二つの凝縮体が重なり合っている時間は数 ms と十分短いため、凝縮体の内部運動はほとんど励起されないものと予想される。
以上の理論的な予想のもと、実際に実験を行ってみたところ振動周期が約 59ms のとき、30％以上の高いコントラストの干渉信号が得られ、相互作用時間を大きく改善
することができた(図 11(a))。さらに振動周期を 97ms にまで長くすると、0.5 を中心にランダムに分布した干渉信号が得られた(図 11(b))。これが先の速度変化によるデフェージングによる影響だとすると信号は毎回 0.5 付近になるものと予想される。このため観測されたランダムな信号分布は外乱等によるランダムな位相揺らぎによって干渉信号が変調されていることが予想される。この予想を確かめるため光パルス間の間隔を振動周期からずらして同様の干渉信号の測定を行った(図 12)。

(a) T=58.66ms
1.0

(i)

(i)

0.8

2hk

0.6

P2 hk 0.4

0hk

(ii)

0.2

(ii)

0.0

π
φL [rad]02π(b) T=97.21ms
1.0
(i)
0.8

(i)
2hk

0.6

P2 hk 0.4

0hk

(ii)

0.2

(ii)

0.0
0

π
φL [rad]2π

図 11 各相互作用時間 T における干渉信号。右図は各干渉信号に対応する終状態の TOF 吸収像。

10

図 12 (a) 光パルスの間隔を振動周期からずらした場合のボース凝縮体の重なり合い。(b) パル
ス間隔をΔT だけ振動周期からずらした場合の干渉信号。

パルス間隔が振動周期に完全に一致しないと二つの凝縮体が完全に重なり合わな
いため可干渉性が低下して干渉信号は 0.5 近傍に収束することが予想されるが(図12(a))、実際の実験結果もこの予想通りとなった(図 12(b))。これより振動周期が 97
ｍs の場合の干渉信号のばらつきは外部からの摂動によるものである可能性が高くなったが、これは主に床からの装置全体の振動によるものであることが推測される。これはこの原子干渉計が共振型の加速度計(速度計)として働いていることからも分かる。
周波数ω/2πにおける加速度が a(ω)で与えられるとするとこの干渉信号には

Δφvib = 2ka (ω )T02

1
sin (δωT0 / 2)
2π (1 + ω / ω z ) δωT0 / 2で与えられる位相シフトが現れる。これより周波数ω/2πが振動周波数に一致するとき(δω = ω−ωz = 0)、外力に対する信号感度が最大となる。振動周期が 97ms の場合、振動周波数は約 10Hz となり、これは光学定盤の共振周波数約 2Hz に対して近く床からの振動を大きく受ける周波数である。このため先の干渉信号のランダムな変調は床からの振動が主な原因である容易に想像できる。
一方、ボース凝縮原子を用いた原子干渉計においては原子間相互作用により生じる
本質的な位相の揺らぎが存在することが知られている 10)。これは干渉計のビームスプ
リッターでボース凝縮原子をコヒーレントに 2 つに分割する場合、分割された各凝縮体の原子数に差が生じ、これが毎回ばらつくことによる。凝縮原子の最初の原子数をN 個としてこれをビームスプリッターで分割した後の両者の原子数差ΔN = N1−N2 の揺らぎの幅はδN ~ N1/2 で与えられる。一方、分割された 2 つの凝縮体の化学ポテンシ
11

ャルをμ1, μ2 とすると分割後時間 t における両者の位相差は

δφ = Δμ t h = ( μ 2 − μ 1 )t / h ~ (∂μ ( N ) ∂N )δ N t hと与えられ、位相差はδN および時間ｔに比例して大きくなる。一方、化学ポテンシ

⎛
mω ⎞
1
⎟
μ = hω ⎜⎜15 Na s
2
h ⎟⎠
⎝

2/5

ャルは
と与えられ、原子間相互作用の大きさを表す s 波散乱長 as およびトラップの幾何学的平均周波数 ω 、原子数 N によって決まる。本実験における実験条件よりこの原子間相互作用による位相揺らぎの大きさを求めてみると t = 100ms においてδφ ~ 1rad 程度となり、先のランダムな位相変調を与えるには若干小さい値である。相互作用時間を100ms よりさらに長くするとこの原子間相互作用による影響が無視できなくなるものと考えられる。この原子間相互作用の影響を低減する方法として最近原子数スクィーズド状態を用いる方法が提案および実演されており 11)、これを用いれば相互作用時間をさらに長くすることが可能になると考えられる。

12

4. BEC 原子を用いた量子ラチェット効果
古典力学的な運動がカオスとなる系においてこの運動を量子力学的に考えた場合、ある特有の量子力学的な性質が現れることが知られておりこのような性質を理論および実験でしらべる研究は量子カオスと呼ばれている。近年、このような量子カオスの研究を極低温原子を用いた系で実験的に研究することに注目が集まっており、その中でも原子にパルス状の光定在波ポテンシャルを一定周期で与えた系は量子キック
回転子と呼ばれ、今まで多くの研究が行われてきた 12)。レーザー冷却およびボース凝縮によって得られる極低温原子の物質波の波長は光の波長またはそれ以上に大きく、原子の運動は量子力学的な波束として扱うことができるため、この量子カオスの問題を扱うのに最適な系である。特にボース凝縮原子を用いた場合、初期状態として単一の量子状態または重ね合わせ状態が実現できるため量子力学的な性質がより顕著に現れ、この性質を詳細に調べることが可能になるものと期待される。この量子キック
回転子系においては周期的なキックによる運動エネルギーの増加が古典的な系とは異なりキックの回数を増やすと徐々に減少することが明らかになっており、これはダ
イナミカルローカリゼーションと呼ばれ、固体物理における電子のアンダーソン局在と同様の現象であることが指摘されている。
本研究では量子キック回転子の系において初期状態に量子力学的な重ね合わせ状
態を用いることにより量子ラチェット効果が現れることを予想し、これをボース凝縮
原子を用いた系で実験的に示した 13)。ラチェット(爪車)とは非対称の歯をもつ歯車とこの回転をとめる爪からなり、歯車は特定の一方向にのみ回転するような機構のことをいう。量子ラチェット効果は非対称な周期ポテンシャル中の粒子の運動において粒子に一方向の流れが生じるというものである。波長λの単色光の光定在波ポテンシャ
ルの場合、ポテンシャルは空間的に対称であるため原子がどちらか一方向に選択的に加速されることはない。しかしボース凝縮原子の様に原子の温度が 100nK 以下で光の反眺速度以下の速度を持つ原子においては原子の波動関数は光の波長以上に広がっているため、この波動関数と光定在波ポテンシャルとの間の相対的な位置関係によって一方向の加速が可能となる。
パルス状の 1 次元光定在ポテンシャル中の原子の運動は以下のようなハミルトニア
ンを用いて記述される。

p2
H =+ V 0 cos( 2 kx ) ∑ δ (t − nT )2mn
ここで k( = 2π/λ)はレーザー光の波数、T はパルス間の時間間隔を表す。原子は光定在
波ポテンシャルが与えられるごとに 2 光子の反眺運動量 2hk の整数倍の運動量を持つ

13

状態 2mhk (m = , , -2, -1, 0, 1, 2, , ,)に回折される。パルス間隔 T を Talbot 時間と呼ばれる TT = 1/νr = h/Er ( E r = h 2 k 2 2m :反眺エネルギー)に選ぶと、回折された各運動量状態
2mhk の位相は次のパルスのタイミング時に全て 2πの整数倍となって位相が再び揃う。ここで原子の初期状態として 2 つの運動量状態の重ね合わせ状態を考える。

ψi =

(

1
0 − ie iφ − 2hk2)

ここでφは両状態の間の相対的位相差を表す。N 回のパルス後の p = 2mhk の運動量状
態の振幅は

1
2

[

e − imπ / 2 J m ( KN ) − e iφ J m +1 ( KN )]ここで K = 2πV0 E r と与えられる。このとき位相差φの値によって N 回のパルス後の各運動量状態の分布は図のように与えられる。これより位相差φ = π/2 の時には運動量
分布は左右に対称に分布し、特定の一方向への加速は生じないが、φ = 0 またはπの時には運動量分布は左右非対称となり、特定の方向に加速されいわゆるラチェット効果が現れるの分かる。これは先の各運動量成分の振幅を与える式より、初期の 2 つの運動量状態の N 回の定在波パルスによって回折された成分が位相差φに応じて干渉して強めあう、または弱めあうことによりもので一種の原子干渉計を構成していることが分かる。このためこのような一方向の加速は純粋に量子力学的な干渉効果によって生じたもので量子ラチェットであるといえる。ここで加速の方向を決めているのは 2 つの運動量成分の相対位相差φで、この値によって原子の密度分布が定在波ポテンシャ
ルに対して非対称性を与え、一方向の加速を与えることになる。
このような量子キック回転子における量子ラチェット効果をアトムチップを用いた生成したボース凝縮原子による実験によって実証した(図 14)。

φ=0

-50

0

φ = π/2

50 -500

運動量(m)

図 13 量子キック回転子の N 回のパルス後の運動量分布

14

φ=π

50 -50050

アトムチップ上に生成されたボース凝縮原子は磁場トラップから解放されて自由落下する。初期状態の 2 つの運動量状態の重ね合わせ状態は対向するレーザー光の周
波数差をずらしてブラッグ回折の条件にして光パルスを入射して実現している。また

2 つの運動量状態の位相差φはその後の自由時間発展の時間を変えることによって与えられる。自由発展後に周波数差をゼロにした短い光定在波パルスを時間間隔 TT で N回入射した後の最終的な運動量分布を CCD カメラで測定したものを図 15 に示す。これより位相差φに応じた運動量分布の非対称性が現れることが確認でき、量子ラチェット効果が実証された。このようにラチェット効果が明確に確認できたのはボース凝
縮原子を用いることによりほぼ単一の量子状態を初期状態として用いたことによる。

図 14 量子ラチェット効果の検証実験

図 15 N 回キック後の原子の運動量分布。
15

５. 単一原子トラップの開発
近年、極低温原子を量子コンピューターなどの量子情報処理に応用することに多くの関心が集まっている 14)。これは中性原子の内部状態を量子ビットとして用いること
により周囲の環境の影響を受けずに長いコヒーレンス時間が得られるものと期待されることによる。また原子は電場、磁場および光を用いてその内部状態および外部運動状態を自在に操作することが可能であるため、多数の量子ビットを多彩に操作することができる。このような量子情報処理においては量子ビットを単位とした量子操作が基本となるため、原子を 1 個ずつ独立に操作および検出することが必要とされる。
また量子コンピューターにおいては複数の量子ビットの間に量子相関(エンタングル
メント)を実現する必要があり、これには原子の内部および外部自由度を完全に操作
して原子の粒子としての性質と波としての性質をフルに用いる必要がある。先のアト
ムチップの技術を用いてマイクロ磁場トラップを実現し、これに原子を 1 個ずつトラップして独立に操作することが期待されているがまだこの実現には技術的に解決すべき点が多くある。そこでアトムチップの磁場ポテンシャルに代わって光ポテンシャ
ルを用いて原子を 1 個ずつ操作および観測する技術を確立するため、光双極子トラッ
プを用いた単一原子トラップの開発を行った。
図 16 に開発した単一原子トラップを示す。ガラスセル中の室温 87Rb 原子はレーザ
ー冷却によって 100mk 程度まで冷やされ磁気光学トラップ(MOT)にトラップされる。
トラップされた 87Rb 原子からの蛍光はセル中の高 NA(0.5)の非球面レンズによって集光され、冷却 CCD および APD によって観測される(図 16)。

図 16

単一原子トラップの実験装置
16photon counts/100ms

原子数＝34atoms3atoms原子数＝22atoms

原子数＝1

1atom
原子数＝0

0

20

40

60
80
Time ［sec]

0atom
100

120

図 17 APD で測定した MOT 中の原子からの蛍光強度。

80μm

原子数＝0

原子数＝1

原子数＝2

図 18 冷却 CCD で観測されたトラップ原子の蛍光像。

高磁場勾配(350G/cm)の四重極磁場コイルおよび MOT 用のレーザー光のビーム径を1mm 程度に絞ることにより、MOT にトラップされる原子の数を非常に少なくすることができ、トラップされる原子数を 1~５程度の範囲で調整することができる 15)。ト
ラップされた原子の数は APD を用いて測定された離散的な蛍光強度より決定することができる(図 17)。また CCD カメラによって得られる原子からの蛍光像よりトラッ
プされた原子は約 20μm 程度の領域に閉じ込められていることが分かる(図 18)。
このようにして高磁場勾配 MOT を用いて単一 Rb 原子をトラップしたら次にこれを光双極子トラップに移行する。光双極子トラップには共鳴線より大きく負に離調した波長 1.064mm の Nd:YAG レーザーの光をレンズで 25μm 程度に集光したものを用いる。Yb 添加光ファイバー増幅器を用いてレーザー光のパワーを 2W 程度に増幅して用いることにより 1ｍK 以上のトラップの深さが得られるため、温度 100μK 程度のレーザー冷却原子を十分長い時間トラップすることが可能となる。MOT 中の冷却 Rb 原子を光双極子トラップに移行して移行後の存在確率を測定したところほぼ 100％の効率で両者の間で移行できることが分かった(図 19)。また光双極子トラップ中の原子の寿命を測定してみたところ 12～15s の寿命が確認され、これは主に残留ガスとの衝突によるトラップロスであると考えられ、これよりガラスセル内の真空度は 10-10 トル程度であると推定される。このようにして光双極子トラップ中に単一 Rb 原子を長時17間安定に閉じ込めることができたので、今後はこの内部状態および外部運動状態を操作して量子ゲート操作を実現することが可能になるものと考えられる 16)。またアトム
チップの技術と組み合わせてチップ上で単一原子のトラップおよびその観測に応用することも考えられる。

(a)

500ms
500ms
50ms

MOT

MOT

MOT
FORT

FORT
(b)
x 103
Natom = 3

photon counts/100ms

14
12

Natom = 2

108Natom=1

6
4
2
0

0

10

20

30

40

50

60

Time [sec]図 19 単一原子の光双極子トラップ。(a) タイミングチャート。MOT：磁気光学トラップ、
FORT：光双極子トラップ。(b) MOT－光双極子トラップ間の原子の移行

再トラップ確率 [%]10080
定在波 U=3.6mK60

τ=12.4±0.6 s
40
進行波 U=1mK
τ=15±4 s

20
0
0

10

20

トラップ時間 [sec]

図 20 光双極子トラップ中の原子の寿命。

1830

6. 電場・磁場ハイブリッドアトムチップの開発
本研究で用いた電流が作る磁場ポテンシャルを用いたアトムチップはボース凝縮
原子の生成に非常に有用であるが、チップ上のマイクロ磁場ポテンシャルで極低温原
子や凝縮原子を操作して様々な原子光学の実験を行うにはまだ多くの解決すべき問題がある。例えば光ファイバーのようにアトムチップ上の原子導波路を用いてボース
凝縮原子を伝播しようとすると磁場ポテンシャルの精度が問題になり、凝縮原子はフラグメンテーションを起こしてスムーズに伝播させることができない ( 図 21) 。幅200μm の電線に電流を流して原子導波路を作って極低温原子を伝播させると原子の分布は不均一にゆらいだものになり、これは電線パターンの幅の精度や電極表面の粗さが電流密度の揺らぎを与え、作られる磁場ポテンシャルがスムーズでないことが主な原因である。
(a)w = 200 μm

(b)

0.5mm
δU/kB = 1.4μK
δB = 20mG

図 21 原子導波路上の原子のフラグメンテーション。(a) アトムチップの電極。(b) 導波路
上の原子の空間分布と対応するポテンシャルの揺らぎ。

アトムチップの本来の特徴であるミクロン(μm)程度の任意のマイクロ磁場ポテン
シャルを実現してボース凝縮原子を自由自在に操作するにはまた別の問題がある。特にアトムチップを量子コンピューターに応用しようとすると原子を 1 個レベルでトラップおよび操作することが必要で、これにはμm 程度の微細な構造のトラップを実現する必要がある。原子をμm 程度の微小なポテンシャルで操作するには原子を電極からμm 程度まで近づける必要があり、これにはいわゆる表面問題を解決する必要がある 17)。これはスピンを持つ原子が金属電極の表面に近づくにつれて原子のトラップ寿
命が短くなったり原子の温度の上昇やデコヒーレンスを受けるもので、これは主に金属中の電子の熱雑音によるものである。この表面の影響を回避するため本研究では原子はアトムチップから 100μm 以上離れた位置にトラップして実験を行ってきたが、これではμm 程度の微細なポテンシャルによる操作は不可能である。この表面問題を19

解決するため、超伝導磁石を用いた方法や永久磁石を用いた方法が最近報告されている 18).19)。これに対して本研究では磁場ポテンシャルに電場ポテンシャルを併用したハ
イブリッドポテンシャルを用いる方法を考案し、これを実現するための二層構造の新しいアトムチップの試作を行った。
表面問題は金属電極中に原子のスピンの鏡像が生じ、この鏡像からの磁場が再び原子のスピンに影響を与えるもので、これは金属表面からそのスキンデップス程度の領域の金属中の電子がこれに寄与する。このため金属のスキンデップスの深さ数μm より薄い電極を用いることによりこの表面の影響をある程度低減することが可能である。このため本研究では 100nm 程度の薄い電極を用いて表面の影響を低減し、また電流による磁場の代わりに電場を用いて原子を操作する方法を用いることにした。電場の場合、電流を流す必要が無いため電極を十分薄くすることができる。また電場のみ
では基底状態の原子を 3 次元的に閉じ込めることができないため、磁場ポテンシャルを併用したハイブリッドポテンシャルを用いることとする。図 22 に本研究で試作したチップおよびその構造を示す。チップはシリコン基板の上に厚さ 50μm のガラス基
板を貼り付けた 2 層構造になっている。ガラス基板上には電場用の微細な電極が作られており、5μm 間隔の電極により 10μm 周期の周期的なポテンシャルを作ることができる。また周期構造の電極の中央の 5 つの電極は独立に電場を与えることができるため、二重井戸やベルトコンベア等が実現可能になっている。

(a)

25ｍｍ

(b)

5μm

ガラス基板
d=50μm

原子電場用金電極d=100nm
電流用金電極d = 10μm

シリコン基板

5μm
(c)

(d)

図 22 電場・磁場ハイブリッドアトムチップ。(a) 試作したチップ基板。(b) 2 層電極構造。
(c) 2 層部分の拡大図。(d) 電場電極部分の拡大図。

20

今回設計・製作したアトムチップによって得られる電場と磁場の合成ポテンシャルを計算してみたものを図 23 に示す。電流による磁場ポテンシャルにより電線から

10μm 程度の距離にポテンシャルの最低の場所が生じ、これに周期的な電場ポテンシ
ャルを加えると 10μm 周期の深さ 5μK 程度の周期的ポテンシャルが実現できることが分かる。このような周期ポテンシャルを用いれば原子数スクィーズド状態やジョゼフソン接合などを実現することが可能になると考えられる 11)。
本研究ではこの新しいアトムチップを用いたボース凝縮の生成および操作をするための真空装置を作ってその準備を進めてきたが実際にチップに原子をトラップするには至らなかった。今後は実際に開発したアトムチップを用いてハイブリッドポテ
ンシャルによる原子の操作を実証したいと考えている。

磁場のみ

10

10 μm

ΔU ~ 1.4 μK

ΔU=5μK
磁場+電場

9
8
7

基板から
の距離
(μm)
7

軸方向の位置(μm)

8

9

10

11

基板からの距離 (μm)

図 23 電場・磁場ハイブリッドポテンシャル(計算)21127. まとめ
本研究は極低温原子およびボース凝縮原子を用いた量子原子光学の基礎を確立する目的でアトムチップの技術を用いた原子導波路上でのコヒーレント物質波の生成およびそのコヒーレント操作を中心に研究を行ってきた。本研究で開発したアトムチ
ップによるボース凝縮の生成法は高速かつ非常に再現性が高いもので、様々な原子光学の実験を行うのに有用なものであることが分かり、今後さらに広く利用されることが期待される。またアトムチップの導波路中にトラップされたボース凝縮原子を用いた原子干渉計の研究においては 100ms 以上のコヒーレンス時間が得られる可能性を示し、今後この長いコヒーレンス時間を活かした高感度な物理計測への応用が期待される。ボース凝縮原子を用いた原子光学の応用として量子系と古典系の中間領域を扱う量子カオスの分野の研究を行い、量子ラチェット効果を実験で実演することができた。これは純粋に物質波の干渉効果によるもので一種の原子干渉計とみなすことができることが分かった。一方、ボース凝縮原子のようなマクロな数の原子による物質波だけでなく原子を 1 個単位で操作するため、光双極子トラップによる単一原子トラップの開発を行い、原子の波動性および粒子性を極限まで追求する量子原子光学の研究に必要な技術の一つを確立することができた。
本研究において取り組んだ主な研究対象は一粒子(原子)のコヒーレント操作および干渉効果である。これに対して現在多くの興味が持たれている量子コンピューターなどの量子情報処理においては多粒子(原子)の干渉効果、すなわち多粒子間の量子力学的な相関であるエンタングルメントの実現が重要となる。またボース凝縮原子だけでなくフェルミオン原子にも多くの関心が寄せられており、このフェルミオンの系においても原子間の相互作用によるクーパー対の実現など多粒子間の量子的な相関が重要となっている。このため今後はこのような多体系の物理を含む幅広い分野を扱う道具として原子光学の研究はますます重要なものとなるであろう。

謝辞
本研究に関わった多くの大学院学生および共同研究者に感謝する。特にアトムチップおよび BEC 干渉計の研究は堀越宗一氏(現 JST ERATO 研究員)の並々ならぬ努力とアイデアによるものでここに感謝と敬意の意を表する。また Mark Sadgrove 氏(JSTCREST 研究員)は原子光学の研究に量子カオスという新しい風を吹き込み、多くの新
しいアイデアと共に新しい展開をもたらしたことに感謝したい。

22

参考文献

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D. Phillips, Phys. Rev. Lett. 96, 160403 (2006).
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Rev. Lett. 99, 043002 (2007). 付録
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Meschede, Phys. Rev. A 67, 033403 (2003).
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24

Ａ

付録

A.1 原著論文
1. “Simple and efficient magnetic transport of cold atoms using moving coils for the
production of Bose-Einstein condensation”, Appl. Phys. B 81, 791-794 (2005).

p. 26

2. “Atom Chip based fast production of Bose-Einstein condensate”, Appl. Phys. B 82,
363-366 (2006).

p. 30

3. “Dephasing due to atom-atom interaction in a waveguide interferometer using a
Bose-Einstein condensate”, Phys. Rev. A 74, 031602(R) (2006).

p. 34

4. “Rectified momentum transport for a kicked Bose-Einstein condensate”, Phys. Rev. Lett.
99,043002 (2007).

p. 38

5. “Coherent control of ballistic energy growth for a kicked Bose-Einstein condensate”, Eur.
Phys. J. D 45, 229-234 (2007).

p. 42

6. “Suppression of dephasing due to a trapping potential and atom-atom interactions in atrapped-condensate interferometer”, Phys. Rev. Lett. 99, 180401 (2007).

p. 48

A.2 国際会議プロシーディング等
1. ICOLS05(2005 年 6 月、 Scotland ) “ATOMIC INTERFEROMETER WITH COLD
RUBIDIUM ATOMS … “、

p.52

2. ICOLS05(2005 年 6 月、Scotland )”ATOM-CHIP BASED FAST PRODUCTION OF
BOSE-EINSTEIN CONDENSATE ATOM”

p.54

3. IQEC05(2005 年 7 月、東京) ” Fast and efficient production of Bose-Einstein condensate
atoms based on an atom chip”

p.56

4. 14th International Laser Physics Workshop (2005 年 7 月、奈良) “Towards coherent atom
optics in an atom waveguide”

p. 58

5. International Conferernce on Lasers and Nanomaterials (2006 年 11 月、コルカタ) “Atomic
interferometer on an atom chip”

p.59

6. ICOLS07(2007 年 6 月、Denver) “BOSE-EINSTEIN-CONDENSATEINTERFEEOMETER ON AN ATOM CHIP WITH A LONG COHERENCE TIME” p.60
A.3 研究会資料等
1. 2007 年度日本物理学会科学セミナー(2007 年 8 月、電通大)

25

配布資料

p. 62

Appl. Phys. B (2005)

Applied Physics B

DOI: 10.1007/s00340-005-1953-8

Lasers and Optics

k. nakagawa1,uy. suzuki1
m. horikoshi1
j.b. kim2

Simple and efficient magnetic transportof cold atoms using moving coils
for the productionof Bose–Einstein condensation
1 Institute for Laser Science and CREST, University of Electro-Communications,

1-5-1 Chofugaoka, Chofu 182-8585, Japan
2 Korea National University of Education, Cheongwon-Gun, Chungbuk, 363-791 Korea

study on a Bose–Fermi mixture with 87 Rb and 40 K atoms [6].
But the performance of this method has not been fully reported so far.
In this paper, we describe a simple but efficient magnetic transport of cold 87 Rb atoms to load cold atoms into an
UHV chamber. We use a linear actuator for industrial use to
move magnetic trap coils for the magnetic atom transport [5].
We have investigated the optimum condition for reducing theheating of atoms and the atom loss during the magnetic transport. In order to reduce the collisional atom loss during thetransport, we employ a light induced atom desorption (LIAD)
technique for the MOT loading [7]. These two methods help
to realize an efficient loading of cold 87 Rb atoms and a Bose–
Einstein condensation in a QUIC type magnetic trap [8].

Received: 11 May 2005/Revised version: 26 July 2005
© Springer-Verlag 2005
ABSTRACT We have developed a simple magnetic transport
method for the efficient loading of cold atoms into a magnetic
trap. Laser-cooled 87 Rb atoms in a magneto-optical trap (MOT)
are transferred to a quadrupole magnetic trap and they are thentransported as far as 50 cm by moving magnetic trap coils witha low excess heating of atoms. A light induced atom desorption
technique helps to reduce the collision loss during the magnetic transport. Using this method, we can load cold 87 Rb atoms
into a magnetic trap in an ultra high vacuum chamber with highefficiency, and we can produce 87 Rb condensate atoms.
PACS 39.25.+k; 32.80.Pj; 03.75.Pp

1

Introduction

2

Recent developments in the field of atom optics
have opened up the possibilities of various applications ofultra cold atoms, such as precision measurements, and quantum information processing [1, 2]. For these applications ofultra cold atoms, it is important to realize a coherent atom
source based on the Bose–Einstein condensation of atoms [3].
A widely used method for the production of condensate atoms
is a combination of a magneto-optical trap (MOT) for the precooling of atoms, and a magnetic trap for further cooling of
the atoms using a RF-induced evaporation. In this method, theMOT and the magnetic trap are usually formed in the sameplace to realize an efficient loading of cold atoms into a magnetic trap. However, this configuration makes it rather difficult
to realize a small magnetic trap with a tight confinement, as
well as an optical trapping and manipulation of condensate
atoms using a three dimensional optical lattice [2]. Recently,
an alternative method, a magnetic transport of cold atoms
from the MOT to a spatially far separated ultra high vacuum(UHV) chamber, has been demonstrated using a row of multiple magnetic trapcoils [4]. But this requires sophisticatedswitching electronics for the many trap coils to avoid heating of atoms during the transport. A similar but rather simpler
method has also been demonstrated using moving coils instead of multiple coils [5], and it was applied to the recent

Experimental setup

A schematic diagram of the experimental setup is
shown in Fig. 1. The vacuum system was identical to the one
which was previously used for the double MOT system. Twoglass cells (32 mm × 32 mm × 100 mm) are separated byabout 50 cm and they are connected by a stainless-steel tube
with a inner diameter of about 1 cm. The first glass cell is connected to an ion pump and a Rb reservoir and it is used forthe cooling and trapping of 87 Rb atoms in the vapor cell MOT.
The Rb vapor pressure in the cell is controlled by slightly heating the Rb reservoir and a valve between the reservoir anda chamber. The second glass cell is evacuated to ultra highvacuum (UHV) at a pressure of 2 – 3 × 10−11 Torr by using an
ion pump and a Ti-sublimation pump and it is used for producing condensate atoms in the magnetic trap.

Schematic diagram of the experimental setup: MT’s, magnetic
traps; TSP, Ti sublimation pump

FIGURE 1

u Fax:+81-424-85-8960, E-mail: nakagawa@ils.uec.ac.jp
26Applied Physics B – Lasers and Optics

The laser system for the MOT consists of two extendedcavity laser diodes (EDLD) and a tapered amplifier diode.
Using a master-oscillator power amplifier (MOPA) configuration, we typically obtain 100 mW of laser power from a singlemode optical fiber, and this laser power is divided into six laser
beams with a diameter of 20 mm for the MOT. For the repumping, we use about 5 mW of the second ECLD power, andthis light is combined with the above MOT beams and they aresent into the first cell.
Two coils in an anti-Helmholtz configuration are used for
the MOT and a quadrupole magnetic trap at the first cell. Eachcoil has 64 turns with a diameter of 66 mm, and can produce
a magnetic field gradient of 100 G/cm in an axial direction
with a current of 20 A. These coils can be moved along a linear
track by a linear actuator (THK, VLA-ST-45-R-500). A maximum velocity and acceleration of this actuator is 0.68 m/s
and 2.9 m/s2 respectively. The position repeatability of thisactuator is within 20 µm. The motion of this linear actuatorcan be controlled by a host personal computer.
The magnetic trap at the UHV cell is of the Ioffe–Prichard
type using three coils in a quadrupole coil Ioffe configuration(QUIC) [8]. Two quadrupole coils and an Ioffe coil are madeby copper tubes with an outer diameter of 2.5 mm and 1.5 mm,
respectively, and are water cooled. Running with a current of
100 A in these coils, the radial and axial trap frequencies forthe 87 Rb F = 2, m F = 2 state are 314 Hz and 20 Hz, respectively. A bias field at the potential bottom is about 0.5 G.
3Experimental results

3.1

Light induced atom desorption

Time variation of the fluorescence from the Rb atoms in the
MOT. At first, the MOT magnetic field was switched on at t = 2 s, and next
the UV LED’s were switched on at t = 8 s. After 10 s, the UV LED’s were
switched off, and finally the MOT field was switched off at t = 20 s

FIGURE 2(Fig. 2). Under this low Rb vapor pressure, we typically collect 2 – 3 × 107 atoms in the MOT. Then we switch on the UV
LED light. The Rb atoms are desorbed, and the number oftrapped atoms increases and reaches about 3 × 108 after 5 s.
When we switch off the UV light, the number of atoms decreases but with a slower decay rate than a loading rate. In
order to examine the fast recovery of the Rb pressure afterswitching off the UV light, we transferred the atoms from the
MOT to the quadrupole magnetic trap after the LIAD MOT
loading. We measured the 1/e lifetime of the magneticallytrapped atoms to be about 2.8 s. We also measured the lifetime
for the case without using the UV light as a comparison andit was nearly the same 2.8 s for this case. This result implies
that the Rb pressure immediately returned to the initial value
after switching off the UV light. This lifetime is long enoughfor the atoms to escape from the high pressure glass cell in themagnetic transport.

In our present experimental setup, we need morethan 108 atoms in the magnetic trap at the UHV cell prior to
the evaporation cooling to produce Bose–Einstein condensate
atoms.We can typically collect 3 × 108 cold Rb atoms in a vapor cell MOT with a Rb pressure of about 10−9 Torr. When
we transport cold Rb atoms from a high pressure first cell to
the UHV second cell, it is important to reduce the atom loss
during the transport. In the case of magnetic transport of cold
atoms, the atom loss is mainly due to the collision to the Rb or
background gases in the first cell [4]. But if we increase the Rbpressure in the first cell to collect more atoms in the MOT, it
induces the increase of loss during the magnetic transport andthe following evaporation cooling. Thus we employ a light induced atom desorption (LIAD) method to temporally control
the Rb vapor pressure in the first cell [7]. We found that a lightfrom ultra violet (UV) light emitting diodes (LED) efficiently
desorbs Rb atoms on an inner surface of a pyrex glass cell [9].
We use a hundred UV LED’s (a peak emission wavelength ofabout 400 nm ) with a maximum total optical power of 1 W.
These UV LED’s shine directly onto the glass surface near the
MOT. The LIAD efficiency of these UV LED’s (1 W total) ismuch higher than that of the white light from a visible halogen
lamp (150 W,) and is comparable to that of the UV halogen
lamp (75 W) [9]. The advantage of the UV LED light is its
small heating effect on the glass cell and long lifetime.
We control the Rb vapor pressure in the first cell around
5 × 10−10 Torr by controlling the temperature of the Rb
reservoir. The number of atoms in the MOT is monitored
by detecting the fluorescent light from the trapped atoms

3.2

Magnetic transport of cold atoms using moving coils

Cold Rb atoms collected in the MOT are compressed and further cooled in optical molasses, and then optically pumped into the F = 2 and m F = 2 state before being trapped in a quadrupole magnetic trap. The magneticallytrapped Rb atoms are transported to the second UHV cell byusing a linear actuator to move the magnetic trap coils. A highinitial phase space density of atoms prior to the evaporation
cooling is crucial for the production of Bose–Einstein condensation. In order to avoid the excess heating of atoms during
the magnetic transport, the velocity of the actuator is smoothly
varied in a S-shaped curve and the actual velocity variation of
the actuator can be monitored by the rotation encoder (Fig. 3).
We evaluated the heating effect of atoms during the magnet
transport. The atoms were first moved by the moving coils in
the above scheme with various acceleration and velocity, andthen they were returned to the original position in the same
scheme. The final velocity distribution of atoms was measured
by the time of flight images. A typical velocity distribution of
atoms after the transport is shown in Fig. 4. In this case, the ac27

NAKAGAWA et al.

FIGURE 3

Simple and efficient magnetic transport of cold atoms

Measured time variation of the velocity of the linear actuator

Absorption images (a) and the horizontal (axial) density profiles (b) of the atomic cloud after 20 ms of expansion for the various final RF
frequencies

FIGURE 5

three magnetic trap coils are gradually changed to move theatoms smoothly from the initial quadrupole trap to the final
quadrupole trap [4]. The atoms are then adiabatically compressed by increasing the coil current of the quadrupole trapfrom 36 A to 100 A, and the magnetic potential is finally converted into an Ioffe–Prichard type potential in a quadrupuleand Ioffe configuration (QUIC) by gradually increasing thecurrent of the Ioffe coil [8]. As a result, we typically load
0.8 – 1.0 × 108 87 Rb atoms in the QUIC type magnetic trapwith a temperature of about 400 µK. The 1/e trap lifetime of
the atoms in the UHV cell is measured to be about 75 s. We apply an RF-induced evaporation cooling to the trapped atoms.
The RF frequency is swept from 30 MHz to about 650 kHz
within 50 s. Around a critical temperature of about 200 nK,
condensate atoms are formed and pure condensate atoms of
about 4 × 104 are produced below the critical temperature(Fig. 5).

Time of flight absorption images of the atomic cloud before themagnetic transport (a), and after the magnetic transport (b). Both images
were taken after 5 ms of expansion

FIGURE 4celeration and the maximum velocity is 2 m/s2 and 0.6 m/s,
respectively, and the one way transport time is about 1 s. The
initial temperature of atoms is about 100 µK (Fig. 4a). Afterthe transport, the velocity distribution along the moving direction increases, and it corresponds to the temperature of about
300 µK (Fig. 4b). We expect that the temperature increaseduring the one way transport is about 100 µK, which is halfof the measured temperature increase of 200 µK for the roundtrip transport. We found that the final temperature increased as
the maximum acceleration during the transport increased, but
the temperature increase was independent of the maximum
velocity. As the elastic collision rate in a magnetic trap is estimated to be 1 s−1 , the increased kinetic energy of atoms is
not redistributed into other directions within a transport timeof 1 s. Thus the loss in the phase space density of atoms after
the transport is estimated to be about 30 % for the temperature increase in one dimension from the initial 100 µK to thefinal 200 µK. As the loss in the number of atoms is within
20% using the LIAD method described above, the total lossof the phase space density during the magnetic transport iswithin 50%.
3.3

4Discussion

The present method described here is simple but
is useful for the production of BEC atoms. Compared witha conventional double MOT method [11], a careful alignmentof the laser beams for the pushing of atoms is not needed, andthe ultra high vacuum needed for the long evaporation time
is easily realized. It is also useful for the applications of produced condensate atoms for various experiments such as an
optical trap in a 3D optical lattice and a magnetic atom waveguide because the trapping laser beams for the MOT are notneeded to be introduced into a magnetic trap. Compared withthe similar method using multiple magnetic trap coils [4], thepresent method is much simpler and the atom loss during thetransport has been improved by using a LIAD technique.
Further improvements in the present method include theuse of a larger MOT beam size to increase the number ofatoms in the MOT, and the use of a smaller magnetic trap
to increase the initial atomic density at the evaporation cooling stage. These improvements will increase the number ofcondensate atoms and also the speed of evaporation cooling
due to the higher elastic collision rate. A micro magnetic trap
on a substrate, so called “atom chip”, is a good candidate to

Producing a Bose–Einstein condensate

The atoms transported by the moving magnetic
coils are transferred to the QUIC type magnetic trap in the
UHV glass cell. We additionally use transfer quadrupole
magnetic coils which partially overlaps two quadrupole magnetic coils as shown in Fig. 1 [4]. The coil currents of28

Applied Physics B – Lasers and Optics

realize fast production of condensate atoms due to its tightmagnetic confinement of atoms [12]. The present method canbe easily applied to the loading of cold atoms into this atom
chip.
Instead of the present LIAD method, a pulsed thermal Rb
atom source using an alkali dispenser may be also used forthe fast control of the Rb pressure [10], however, it simultaneously emits some amount of hydrogen gases which can not
be quickly pumped out by a small ion pump [9]. The hydrogengases can be effectively pumped out by the additional use ofa Ti sublimation pump or a non evaporating getter pump. We
found that LIAD using the UV LED works effectively for the
Rb atoms on a Pyrex glass surface [9]. A Pyrex glass contains
small amount of Na atoms, and these Na atoms may play animportant roll in increasing the efficiency of the LIAD of Rb
atoms on a Pyrex glass surface. But the detailed mechanism
has not been well understood and it is rather outside the scopeof this paper.
5

sation, and it will be also helpful for the various application ofcondensate atoms.
ACKNOWLEDGEMENTS This work was partly supported by
a Grant-in-Aid for Science Research (14047210) from the Ministry of Education, Science, Sports and Culture, and the 21st Century COE program on“Coherent Optical Science”.

REFERENCES
1 See, e.g., Atom Interferometry, P. Berman (ed.) (Academic, San Diego1997)
2 M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Nature 415,
39 (2002)
3 M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269, 1989 (1995)
4 M. Greiner, I. Bloch, T.W. Hänsch, T. Esslinger, Phys. Rev. A 63, 031401(2001)
5 H.J. Lewandowski, D.M. Harber, D.L. Whitaker, E.A. Cornell, J. Low
Temp. Phys. 132, 309 (2003)
6 J. Goldwin, S. Inouye, M.L. Olsen, B. Newman, B.D. DePaola, D.S. Jin,
Phys. Rev. A 70, 021 601(R) (2004)
7 B.P. Anderson, M.A. Kasevich, Phys. Rev. A 63, 023 404 (2001)
8 T. Esslinger, I. Bloch, T.W. Hänsch, Phys. Rev. A 58, R2664 (1998)
9 S. Du, M.B. Squires, Y. Imai, L. Czaia, R.A. Saravanan, V. Bright,
J. Reichel, T.W. Hänsch, D.Z. Anderson, Phys. Rev. A 70, 053 606(2004)
10 J. Fortagh, A. Grossmanna, T.W. Hänsch, C. Zimmermann, J. Appl.
Phys. 84, 6499 (1998)
11 C.J. Myatt, N.R. Newbury, R.W. Ghrist, S. Loutzenhiser, C.E. Wieman,
Opt. Lett. 21, 290 (1996)
12 W. Hänsel, P. Hommelhoff, T.W. Hänsch, J. Reichel, Nature 413, 498(2001)

Conclusions

In summary, we have demonstrated that an efficient loading of cold atoms into a magnetic trap is realized bya magnetic transport of cold 87 Rb atoms using moving magnetic coils. Using this method, we have successfully demonstrated a production of Bose–Einstein condensate atoms of87
Rb in a magnetic trap. The present method will allow realization a simple and fast production of Bose–Einstein conden-

29

Appl. Phys. B (2005)

Applied Physics B

DOI: 10.1007/s00340-005-2083-z

Lasers and Optics

m. horikoshiu
k. nakagawa

Atom chip based fast productionof Bose–Einstein condensate
Institute for Laser Science and CREST, University of Electro-Communications,
1-5-1 Chofugaoka, Chofu 182-8585, Japandensity into a small-volume chip potential. Up to now, twokinds of methods have been developed. The mirror MOT [6]is the most popular method to prepare cold atoms near a chipsurface. By using the chip surface as a mirror, we can prepare cold atoms just below the chip, and then capture themby the chip potential. This method is simple, but it needs
a flat metallic reflective surface, which may limit the designof an atom chip. Considering the MOT efficiency of a mirror
MOT, the capture volume of a MOT is smaller than a standard
six-beam MOT with the same trapping laser beam diameter,
and also the trapping beam size is limited by the size of thechip. On the other hand, the method of combining a standard six-beam MOT with an adiabatic magnetic transfer [2]has the advantage of a large MOT volume and a high loading
efficiency and is surface independent. However, this systemneeds a complex wire configuration in order to change themagnetic potential from quadrupole field to Ioffe–Pritchard(IP) field adiabatically. Therefore, it is important to develop
an alternative method of loading cold atoms into a chip potential to improve the total efficiency for fast BEC production.
In this experiment, we have improved the MOT efficiency andthe initial number of cold atoms in the chip potential by using
a combination of a standard six-beam MOT and a simple magnetic transfer with recatching, which enable us to realize fastand efficient production of a BEC with an atom chip in a small
glass cell.

Received: 4 August 2005/Revised version: 24 October 2005
© Springer-Verlag 2005
ABSTRACT We have developed a simple method for the fast andefficient production of a Bose–Einstein condensate (BEC) on an
atom chip. By using a standard six-beam magneto-optical trap
and light-induced atom desorption for loading, 3 × 107 87 Rb
atoms are collected within 1 s and loaded into a small-volume
magnetic potential of the chip with high efficiency. With thismethod, a BEC of 3 × 103 atoms is realized within a total timeof 3 s. We can realize a condensate of up to 2 × 104 atoms by
reducing three-body collisions. The present system can be used
as a fast and high-flux coherent matter-wave source for an atominterferometer.
PACS 03.75.Be; 32.80.Pj; 39.25.+k

1

Introduction

The production and manipulation of Bose–Einsteincondensates (BECs) on a substrate with integrated conductor patterns, a so-called ‘atom chip’, has raised considerable
interest recently [1–3]. The atom-chip technique can be employed for developing various quantum devices using coherent matter waves for precision measurements and quantum
information processing [3]. Another application, an atomic
interferometer on a chip, was demonstrated very recently [4].
Furthermore, atom chips allow us to realize a compact experimental apparatus for various practical applications of
BECs [1, 5].
A high repetition rate for the production of a condensate
and a large number of condensed atoms are preferable for precision measurements using an atomic interferometer in orderto improve the sensitivity. A total BEC production time fromthe magneto-optical trap (MOT) loading phase to evaporative
cooling mainly depends on the MOT loading time and theevaporation time. The MOT loading time is determined by theRb background pressure and the capture volume of the MOT,
and the evaporation time is determined by the elastic collisionrate, which is proportional to the atomic density and inversely
proportional to the trap volume.
In order to increase the efficiency of evaporative cooling,
it is necessary to load a large number of cold atoms with high2

Experimental setup

The experimental setup is shown in Fig. 1.
A Pyrex glass cell of inner size 30 mm × 30 mm × 90 mm
is connected with a 45-l/s ion pump, a titanium sublimation pump, and electrical feedthroughs. All processes fromcooling and trapping of a 87 Rb atom by the MOT to the realization of a BEC by rf evaporative cooling on the atom chipare operated in this glass cell. In order to realize efficientloading of many atoms into the MOT as well as ultra-highvacuum for a long lifetime in the magnetic trap, we use a lightinduced atom desorption (LIAD) technique that can increase
the pressure temporarily during the MOT loading phase, but
the ultra-high vacuum quickly recovers after loading [7]. We
use a UV halogen lamp (75 W) and UV LEDs (1 W) to desorb Rb atoms from the inner surfaces of the Pyrex glass cell.
Two pairs of anti-Helmholtz coils for MOT and transportationu Fax: +81-424-85-8960, E-mail: hori@ils.uec.ac.jp30Applied Physics B – Lasers and Optics

In the present experiment, we only use a Z-shaped wire (Zwire), which produces a Ioffe–Pritchard (IP)-type magnetic
trap with an external bias field Bx , and a crossed wire (Cwire), which produces a small dimple at the center of the IP
trap [10]. Other wire patterns have been prepared for the manipulation of condensate atoms, but they are not used in thepresent experiment. To minimize current noise, the currents
for these wires are provided by a regulated power supply withbatteries.
Experimental setup. The atom chip is inserted into a glass cellfrom upper electrical feedthroughs, and all coils are mounted outside the cell.
87 Rb atoms are cooled by a six-beam MOT, and magnetically transported
to the atom chip by the MOT coils and the T coils. The MOT coils and the
T coils overlap in order to transport the atoms adiabatically
FIGURE 1

3

Experimental results

The experimental procedure consists of threestages, a MOT loading stage, the magnetic transfer, and themagnetic trap on a chip. Initially, 87 Rb atoms are cooled
and trapped in a six-beam MOT. The laser power densitiesof the cooling and repumping beams are 4 mW/cm2 and1.6 mW/cm2 per beam, respectively, with a beam diameter of
15 mm. During MOT loading, the UV halogen lamp and UVLEDs are turned on. Although the efficiency of the LIAD issufficient to increase the pressure temporarily with only UVLEDs [9], we also use the UV halogen lamp to increase the
Rb pressure further for fast MOT loading. In this way, we cancollect up to 1 ×108 cold atoms within 5–10 s. However, we
typically collect 3 ×107 cold atoms within 1–5 s. After loading the MOT, the UV light is turned off and the atoms are heldin the MOT for 3 s to improve the pressure in the glass cell.
Then, we can realize a magnetic trap lifetime of about 15 s.
After compression MOT (CMOT) and polarization gradient cooling (PGC), the atoms are optically pumped into the|F = 2, m F = 2 hyperfine ground state and cold atoms arecaptured in a quadrupole (QP) magnetic field produced bya pair of MOT coils, with 73 G/cm in the axial direction. Magnetically trapped atoms are transferred close to the atom chipby using the transportation (T) coils. After a 100 ms magnetic
trap with the MOT coils, the T coils with 80 G/cm in the axial
direction are turned on within 250 ms, and then the current of
the MOT coils is decreased within 250 ms. Then, the atomic
cloud is transferred to 0.5 mm below the chip surface by adjusting the currents of the MOT and T coils. The position of
the atomic cloud in the x, y direction is adjusted by applying
homogeneous bias fields Bx and B y . The number, temperature, and phase-space density of the atoms are N ∼ 3 ×107 ,T = 110 µK, and ρps = 9 ×10−8 , respectively.
After the cold atoms are transferred close to the surface,
they are recaptured in the chip potential when the trapping potential is switched from QP potential to IP potential (Fig. 2c)within 1 ms. During the transfer, it is important to preserve
the phase-space density. The deviation between the positionof the transported atoms and the potential minimum of thechip trap causes serious heating and loss. Therefore, we adjust the position of the atomic cloud to the optimum position within 100 µm in all x, y, z directions. Also, we apply
weak homogeneous magnetic fields during the loading process in order to follow the atomic magnetic spin adiabatically.
Then, we can transfer 50% of the atoms from the QP potential to the chip potential. The loss is mainly caused by themode mismatch between the transported atomic cloud andthe relatively small atom-chip potential. The number, temperature, and phase-space density of the loaded atoms are(T coils), and two pairs of Helmholtz coils which generate homogeneous magnetic fields Bx and B y , are located outside theglass cell. The center of the MOT is located 2 cm below thechip surface in order not to block the cooling lasers. The MOT
and the T coils overlap so that trapped atoms are transferredadiabatically [8].
The atom chip used in this experiment is shown in Fig. 2a.
The atom chip is bonded to a copper block, which is mounted
downward in a glass cell. The gold pattern on a Si substrate was made by standard photolithographic and electroplating techniques [2]. The configuration of the wires used
to trap cold atoms and make the BEC is shown in Fig. 2b.

Atom chip and magnetic potential. (a) Atom chip used inthis experiment. A gold pattern is deposited on a Si substrate of 20 mm ×
20 mm × 0.5 mm. (b) Wire configuration used to trap and produce the BEC
at the center of the chip. The widths of the Z- and C-wires are 200 µm and100 µm, respectively, and the gold layer is 10-µm thick. The Z-wire and theC-wire support currents of 4 A and 1 A, respectively. Magnetic potential produced by an atom chip, (c) in a radial direction, (d) in the axial direction. The
dashed lines represent the magnetic potential for loading (Iz = 4 A, Ic = 0 A,
Bx = 0.5 G, B y = 15 G), and the solid lines that for compression (Iz = 4 A,
Ic = −0.6 A, Bx = 11 G, B y = 60 G)

FIGURE 2

31

HORIKOSHI et al.

Atom chip based fast production of Bose–Einstein condensate

N ∼ 1.5 ×107 , T = 100 µK, and ρps = 7 ×10−8 , respectively,
and the phase-space density is nearly preserved in this transfer
process.
A first set of experiments was optimized to maximize
the number of condensate atoms. Immediately after loading,the atomic cloud is radially compressed by increasing bias
fields within 200 ms, and then a small dip potential is realized at the center by introducing current in the crossed wire
within 300 ms (Fig. 2c). At this point, the potential minimum
is 130 µm from the chip surface, and the trapping frequencies are (νx , ν y , νz ) = (740, 4000, 4000) Hz. After this compression phase, 40% of cold atoms are tightly compressed ina dip shown in Fig. 2d, and the initial elastic collision rateis estimated to be 50 s−1 . Then, the rf field is applied to theatomic cloud for evaporative cooling. We use a DDS (direct
digital synthesizer) oscillator which realizes a rapid and arbitrary frequency sweep. The first rf linear sweep starts from37 MHz to 11 MHz for 3.2 s. Then, all of the atoms enter thedip of the cross-wire trap (Fig. 3a), and the estimated elastic collision rate reaches 6 × 103 s−1 . Then, we increase therf sweep rate to prevent three-body collision loss. We drivea second linear sweep from 11 MHz to 4 MHz linearly for
400 ms. At the end of the second sweep, the three-body recombination rate is estimated to be 5 s−1 [11]. This rate cannot
be neglected, and it causes serious loss and heating of thehigh-density atomic cloud, therefore limiting the number ofcondensed atoms [12, 13]. In order to reduce three-body collisions, we decrease the atomic density by decompressing thetrapping potential adiabatically. Then, the three-body recombination rate is reduced to 0.5 s−1 . After a final sweep from
4 MHz to around 1.8 MHz for 600 ms, the BEC phase transition occurs at T = 2 µK for N = 5 ×104 atoms. This is ingood agreement with the theoretically estimated critical temperature Tc = 1.8 µK. As a result of the optimization of theevaporative cooling to increase the number of the condensate
atoms, 2 ×104 condensate atoms are produced below the critical temperature with a total evaporative cooling time of 4.3 s(Fig. 3b). The BEC lifetime is about 30 ms, which is limited
due to three-body recombination.
A second set of experiments was optimized to minimize
the BEC production time. In order to increase the evaporative cooling speed, the initial elastic collision rate should be

increased. Thus, the atomic cloud is compressed further after
the loading into the chip. The potential depth of a dip is increased 1.5 times compared to the first set of experiments andabout 60% of the atoms are transferred into the dip. As a result, the initial atomic density increases twice and the initialelastic collision rate is increased to 100 s−1 . Since the evaporation cooling period is as short as 1 s, we simply sweep
the rf frequency linearly from the start frequency of about
35 MHz to the final frequency of around 2.8 MHz. After optimizing the start and final rf frequencies and the sweep time,
we could produce 3 × 103 condensate atoms with a rf evaporation period of 0.7 s. With such a short evaporative coolingtime the requirements for the ultra-high vacuum are not so
serious. After 1 s of MOT loading, the cold atoms are transferred into the magnetic trap without waiting for recoveryof the vacuum. The atomic cloud is transferred and loaded
into this small-size chip potential within 1 s. After this rapid
MOT loading and the transfer, the same amount of condensate
atoms are realized after this rapid evaporative cooling. A total
time from the MOT loading to the production of the BEC iswithin 3 s. Thus, we realize the shortest BEC production time
yet reported.
In this fast BEC production procedure, the maximum
number of condensate atoms is 8 × 103 where the period of thelinear rf sweep is around 1 s. On the other hand, a longer evaporation period resulted in the decrease of condensate atoms
mainly due to the three-body collision loss, which was on
the order of 5 s−1 at the final stage of the evaporative cooling. The lifetime of the condensate in this tight potential is
5 ms, whereas the estimated three-body recombination loss
rate of this condensate is about 100 s−1 . This result suggests
a non-negligible high three-body collision loss rate during theevaporative cooling.
4

Discussion

The increase of phase-space density with the number of atoms and evaporation time is shown in Fig. 4 in orderto evaluate the efficiency of evaporative cooling. We estimate
the efficiency of the evaporative cooling from the relative in-

The increase of phase-space density with the number of atoms
and evaporation time. ρps and N are the phase-space density and the number of atoms, respectively. (a) Phase-space density (circles) and peak density(triangles) vs. the number of atoms under the conditions of the maximum condensate (closed) and the fastest production (open). Each point
corresponds to MOT, chip trap, first evaporation, second evaporation anddecompression, and third evaporation from the right-hand side of the figure. (b) Phase-space density during evaporative cooling under the conditions of the maximum condensate (circles) and the fastest production(squares)
FIGURE 4

Evaporative cooling in a cross-wire trap and BEC phase transition. (a) Absorption images of atomic cloud during first rf sweep. Fromleft to right: νrf = 30 MHz, νrf = 20 MHz, νrf = 11 MHz. Cold atoms of
N = 5.5 ×105 , T = 90 µK, ρps = 1 ×10−3 are in the dip of the potential at
the end of this rf sweep. The TOF (time of flight) is 1.6 ms. (b) Final rf sweep.
From left to right: νrf = 1.86 MHz, νrf = 1.83 MHz, νrf = 1.78 MHz. We observe a sequence of thermal cloud, bimodal cloud, and pure condensate. The
TOF is 10 ms
FIGURE 332

Applied Physics B – Lasers and Optics

crease in the phase-space density ρps with the decrease of
atom number N as γ = −log ρps /log N in Fig. 4a. In the first
set of experiments, which realizes a large number of condensed atoms, the efficiency γ is 3.4 at the beginning of theevaporation and 2.4 at the end, and the average γ of the entire evaporative cooling is 3.0, which is almost close to thebest value that has been reported up to now [14]. Though γ
is slightly decreased by three-body loss during the final rf
sweep period, decreasing the atomic density helps to suppress
the influence of three-body collisions successfully. The initial increase of the phase-space density with time as shown in
Fig. 4b is limited by the initial atomic density. In the second
set of experiments, which realizes the fast BEC production,
the average γ is decreased to 2.2; however, the phase-spacedensity increases by a factor of eight within 1 s as shown in
Fig. 4b. This demonstrates the high efficiency of chip-based
evaporative cooling. This decrease of average efficiency iscaused by extra evaporation at low density and three-body loss
at high density, which is proven by the experimental result. We
find that it is essential for the fast production of a large condensate to increase an initial atomic density for quick rethermalization at the first stage of evaporative cooling, and to decrease
an atomic density for suppression of three-body loss at the
final stage.
Next, we discuss the optimization to increase the numberof atoms with a modest evaporative cooling period of about
3 s. As pointed out before, the efficiency of the evaporative
cooling is almost at maximum. Thus, the increase of the condensate atoms can only be realized by the increase of theinitial cold atoms. Under the same experimental conditions,
we could collect 108 atoms in the MOT with a rather longer
loading period of about 5 s, and we could load 5 × 107 cold
atoms into the chip trap. A higher density results in the increase of the three-body collision loss and a lower density
results in the decrease of the efficiency of the rf evaporation. Therefore, according to the increase of the initial numberof cold atoms, we should modify the size of the dip potential in order to keep the atomic density nearly the same as in
the dip potential during evaporative cooling. Thus, by optimizing the MOT loading and the dip potential size, we caneasily increase the number of condensate atoms to 6 × 104
with a total time within 10 s. Further increase of the condensate number is in principle possible, but it may require
the modification of the glass cell for accommodating larger
MOT trapping beams and also modified chip electrodes fora larger and deeper trap potential. We will be able to produce

more than 105 condensate atoms every 10 s using the same
method.
5

Conclusions

In summary, we realized a Bose–Einstein condensate of 3 × 103 atoms within a total time of 3 s by using
a combination of a standard six-beam MOT with LIAD, a simple magnetic transfer technique, and a tight confinement on
an atom chip. Efficient MOT loading increases the numberof cold atoms, which enhances the efficiency of evaporative
cooling, and also decreases the MOT loading time. The loading process using this method from a MOT into a chip trap is
simple and efficient. Since the efficiency of evaporative cooling in the tight confinement on an atom chip is very high,fast production of a BEC is realized. In addition, the present
method enables the use of various atom-chip configurations
toward atomic interferometers and other applications.
ACKNOWLEDGEMENTS We thank Taro Eichler for his criticalreading of our manuscript. This work was partly supported by a Grant-in-Aid
for Science Research (14047210) from the Ministry of Education, Science,Sports and Culture, and the 21st Century COE program on ‘Coherent Optical
Science’.

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Phys. Rev. Lett. 87, 230 401 (2001)
3 R. Folman, P. Krüger, J. Schmiedmayer, J. Denschlag, C. Henkel, Adv.
At. Mol. Opt. Phys. 48, 263 (2002)
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94, 090 405 (2005)
5 S. Du, M.B. Squires, Y. Imai, L. Czaia, R.A. Saravanan, V. Bright, J. Reichel, T.W. Hänsch, D.Z. Anderson, Phys. Rev. A 70, 053 606 (2004)
6 J. Reichel, W. Hänsel, T.W. Hänsch, Phys. Rev. Lett. 83, 3398 (1999)
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031 401(R) (2001)
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Phys. B 81, 791 (2005)
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14 B.P. Anderson, M.A. Kasevich, Phys. Rev. A 59, R938 (1999)

33

RAPID COMMUNICATIONSPHYSICAL REVIEW A 74, 031602共R兲 共2006兲

Dephasing due to atom-atom interaction in a waveguide interferometerusing a Bose-Einstein condensate
Munekazu Horikoshi and Ken’ichi Nakagawa
Institute for Laser Science and CREST, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu 182-8585, Japan
共Received 20 February 2006; published 13 September 2006兲
A Mach-Zehnder-type atom interferometer with a Bose-Einstein condensate has been investigated on an
atom chip by using optical Bragg diffraction. A phase shift and a contrast degradation, which depend on theatomic density and the trapping frequency of the magnetic-guide potential, have been observed. Also, theoutput wave packets were found to exhibit a spatial interference pattern. The atom-atom interaction and theguide potential induce the spatially inhomogeneous phase evolution of the wave packets in each arm of theinterferometer. The observed contrast degradation can be quantitatively explained as a dephasing due to thisinhomogeneous phase evolution.
DOI: 10.1103/PhysRevA.74.031602

PACS number共s兲: 03.75.Dg, 34.20.Cf, 32.80.Qk

Recently, a micron-sized atom waveguide has been developed by using microfabricated structures on a chip, known as
an “atom chip,” and many kinds of atom-wave manipulation
have been performed using this method 关1–4兴. An on-chip
atom interferometer presents a great possibility to realize asmall-scale and highly sensitive inertia sensor such as a gyroscope 关5兴, and progress has been made towards practicalapplications 关2–4兴. Although numerous difficulties exist regarding heating, decoherence, and fragmentation due to thechip surface and imperfect wire patterns 关1兴, these problems
can be solved by separating the atoms from the surface. A
high density of the condensate atoms has, however, beenargued to influence the interference signal due to atom-atominteractions. An atom interferometer with a Bose-Einsteincondensate 共BEC兲 in a magnetic-guide potential was demonstrated by using optical Bragg diffraction recently, and contrast degradation of the interference fringes was reported 关2兴.
Additionally, a theoretical calculation of the interference
fringe was suggested in Ref. 关6兴 and the influence of theatom-atom interaction was shown qualitatively.
In order to investigate how the atom-atom interaction influences the interference signal, we realized a BEC interferometer in a guide by using optical Bragg diffraction similar
to that reported in Ref. 关2兴. In this work, experimentally measured interference signals are given for various atomic densities and trapping frequencies and we report that the atomatom interaction plays an important role in inducing thedephasing of an interference fringe.
We performed the experiment with a condensate of about
8 ⫻ 103 87Rb atoms in the 5S1/2, F = 2, mF = 2 state. The condensate is prepared in a small dimple-shaped magnetic trap
produced by an atom chip following the method of Ref. 关7兴.
The magnetic potential is then changed into the guide potential, which has trapping frequencies of ␻r = 2␲ ⫻ 210 Hz and␻z = 2␲ ⫻ 10 Hz, where ␻r and ␻z are the radial and axialtrapping frequencies, respectively. The axial trapping frequency is adjusted by controlling the current in the crossed
wire on the atom chip 关7兴.
A Mach-Zehnder-type atom interferometer is realized bythree optically induced Bragg-diffraction pulses 关8–10兴.
Each pulse consists of two counterpropagating linearly polarized laser beams whose frequencies differ by the two1050-2947/2006/74共3兲/031602共4兲

photon recoil frequency of about 15 kHz in order to correlate
the two momentum states 兩0 ប k典 and 兩2 ប k典, where k is thewave vector and the beam waists are 200 ␮m. They are detuned by about −130 GHz from the atomic resonance共␭ = 2␲ / k = 780 nm兲 so that spontaneous emission is negligible. Two laser beams are incident on the condensate, which
is trapped at 300 ␮m below the chip surface, and they arealigned parallel to the axial direction of the guide potential
关Fig. 1共a兲兴. A ␲ / 2 pulse and a ␲ pulse are realized by asingle Bragg pulse of duration 50 and 100 ␮s, respectively.
The ␲ / 2-␲-␲ / 2 sequence of three Bragg pulses is applied tothe trapped BEC in the magnetic guide 关Fig. 1共b兲兴. In thisexperiment, the time interval T between pulses is fixed to
1 ms. After the second ␲ / 2 pulse, the magnetic-guide potential is turned off. After a 15 ms time of flight, the interference

FIG. 1. 共a兲 Schematic drawing for the atomic interferometer on
an atom chip. 共b兲 Space-time diagram for the interferometer. Three
optically induced Bragg-diffraction pulses form the Mach-Zehnder
atom interferometer as shown. The first ␲ / 2 pulse splits the atom
wave packet in two along the paths A and B, respectively. The next
␲ pulse reflects both wave packets and the second ␲ / 2 pulse acts asa second beam splitter.

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RAPID COMMUNICATIONSPHYSICAL REVIEW A 74, 031602共R兲 共2006兲

MUNEKAZU HORIKOSHI AND KEN’ICHI NAKAGAWA

FIG. 3. The phase shift 共a兲 and the fringe contrast 共b兲 as afunction of the trap frequency ␻z along the waveguide. 共a兲 The
observed and calculated phase shifts are plotted by a solid circle
with an error bar and a solid line, respectively. 共b兲 The observed
contrasts are plotted by a solid circle with an error bar. The calculated contrasts are plotted for the initial number of atoms N = 104
共long-dashed line兲, 8 ⫻ 103 共solid line兲, and 6 ⫻ 103 共short-dashedline兲, respectively. The number of atoms for the present experimentis N = 8 ⫻ 103.

FIG. 2. 共a兲 Typical time-of-flight absorption image of the condensate atoms after the second ␲ / 2 pulse for ␻z / 2␲ = 15 Hz and␾L = ␲ / 2. Two black areas correspond to the atoms in the momentum state 兩p = 0 ប k典 共right兲 and 兩p = 2 ប k典 共left兲, respectively. There
are also s-wave scattered atoms around two black peaks within theelliptical area. The density profile of the atoms along the axial direction is shown in 共b兲. This density profile is fitted with threeGaussians and the fitted profile is shown by a dashed line. The
population oscillation of the condensate in the state 兩p = 2 ប k典 as afunction of the phase ␾L of the second Bragg pulse is shown in 共c兲.

In order to explain the present experimental results, weevaluate the phase shift by a simple model 关6兴. The wave
function at t = 2T, but slightly before the final ␲ / 2 pulse, isgiven by a superposition of two wave functions, one representing path A and the other path B as in Fig. 1共b兲, namely,⌿ = 共␺Aei⌬␾ + ␺B兲 / 冑2. Here ⌬␾ means the relative phase shift
between path A and path B. The interfered output momentum
state after the final ␲ / 2 pulse, is determined by the relative
phase shift ⌬␾ as P2បk = 21 关1 − cos共⌬␾兲兴 at ␾L = 0. This relative phase shift is given by the classical path integration as⌬␾ = ប1 共兰 path ALdt − 兰 path BLdt兲, where L is the atomic
Lagrangian. When the experimental condition satisfies
␻zT ⬍ 1, the kinetic energy term hardly contributes to therelative phase shift. Then we may approximately evaluate therelative phase shift as

signal is observed using a standard absorption imaging technique.
A typical absorption image of the output wave packets isshown in Fig. 2共a兲. There are two peaks corresponding to the兩0 ប k典 共right兲 and 兩2 ប k典 共left兲 momentum components
共coherent components兲. There are s-wave scattered atoms
共decoherent components兲 around the two peaks 共inside thedotted elliptical area兲 in Fig. 2共a兲. In the present analysis, we
assume that the decoherent components do not influence
the coherent components 关11兴. In order to extract thecoherent components from the image, the density profile of
the atoms along the axial direction is fitted with three Gaussian functions 关Fig. 2共b兲兴, then the number of atoms in兩0 ប k典共N0បk兲 and 兩2 ប k典共N2បk兲 are evaluated 关11,12兴. Thus weevaluate the population of atoms in state 兩2 ប k典 as P2បk
= N2បk / 共N0បk + N2បk兲.
The relative phase of the second ␲ / 2 pulse, which is ␾L,is experimentally varied in the range 0 ⬉ ␾L ⬉ 2␲ at an incremental interval of ␲/6 to observe the interference fringe.
We varied the ␾L by changing the phase of the radio frequency of the acousto-optic modulator for one of the Braggdiffraction laser beam before the second ␲ / 2 pulse. At eachrelative phase ␾L, we have made five measurements to reduce statistical errors and we have calculated the mean andthe statistical error of the population P2បk 关Fig. 2共c兲兴.
We fitted the observed fringe to the expected form
1 M
P2បk = 2 − 2 cos共⌬␾ + ␾L兲 and we determined the fringe contrast M and the phase shift ⌬␾.
We investigated interference signals that depend on boththe atom-atom interaction and the magnetic-guide potential
simultaneously by changing the axial trapping frequencies,
since the strength of the atom-atom interaction depends onthe chemical potential, which itself depends on the axialtrapping frequency as described below. In this way, the phaseshift and the contrast, which depend on the trap frequency,
have been obtained as shown in Fig. 3.

⌬␾ ⯝ − ប1冉冕2T

UAdt −

0

冕

2T

0

冊

UBdt ,

共1兲

where UA,B are the effective potentials that the wave functions of ␺A and ␺B experience during propagation, respectively.
The effective potential is given by 关6兴UA = Ug„r,z + z̄A共t兲… + Us共r,z兲 + Um„r,z + ⌬z̄共t兲…,
UB = Ug„r,z + z̄B共t兲… + Us共r,z兲 + Um„r,z − ⌬z̄共t兲….

共2兲mHere Ug = 2 共␻2r r2 + ␻2z z2兲 is the magnetic-guide potential,
Ng
Us共r , z兲 = 2 n共r , z兲 is the self-mean-field potential, which iseffective within the wave packet of each condensate, andUm共r , z兲 = Ngn共r , z兲 is the mutual-mean-field potential thatacts between the two wave packets, respectively, and␮
n共r , z兲 = Ng max共1 − ␮ , 0兲, which is normalized by兰n共r , z兲dV = 1, represents the spatial atomic-density distribuUg共r,z兲tion of the condensate, g =031602-235

4 ␲ ប 2a s
mis the coupling constant,

RAPID COMMUNICATIONSPHYSICAL REVIEW A 74, 031602共R兲 共2006兲

DEPHASING DUE TO ATOM-ATOM INTERACTION IN A¼¯ 共15Nas冑 mប␻¯ 兲 is the chemical potential, where␮ = 21 ប ␻
2¯ = 共␻␳␻z兲1/3, and N is the number of the condensed atoms.
␻The peak strength of the mean and mutual mean-field poten␮
tials are given by Us共0 , 0兲 = 2 and Um共0 , 0兲 = ␮, respectively.
z̄A,B共t兲 are the center-of-mass 共c.m.兲 positions of the two
wave packets and ⌬z̄共t兲 is defined as ⌬z̄共t兲 = z̄A共t兲 − z̄B共t兲. The
c.m. motion is considered as
2/5z̄A共t兲 =

再

v rt
v rT

,

z̄B共t兲 =再0共0 ⬉ t ⬉ T兲

vr共t − T兲

共T ⬍ t ⬉ 2T兲,

共3兲where vr = 2mបk is the two-photon recoil velocity.
We evaluate the three contributions to the relative phaseshift by substituting Eqs. 共2兲 and 共3兲 for Eq. 共1兲, and weinvestigate each effect in terms of the atom-atom interaction
and the magnetic-guide potential. The first contribution to
examine is that due to the magnetic-guide potential, which is
given by ⌬␾g共z兲 = − ប1 兰 共UgA − UgB兲dt = −agz − bg. The second
one is that due to the self-mean-field potential, which is
given by ⌬␾s共z兲 = − ប1 兰 共UsA − UsB兲dt = 0. The third one is that
due to the mutual-mean-field potential, which is approximately given by ⌬␾m共z兲 = − ប1 兰 共UmA − UmB兲dt ⬃ 2agz whenvrT ⬍ Rz is satisfied, where Rz is the Thomas-Fermi radius of
the BEC in the axial direction. Therefore the total relative
phase shift is given by⌬␾共z兲 ⬃ ⌬␾g共z兲 + ⌬␾s共z兲 + ⌬␾m共z兲 = agz − bg ,FIG. 4. Absorption images of the condensate in the 兩p = 0典 momentum state for trapping frequencies of ␻ / 2␲ = 10 Hz 共a兲 and
30 Hz 共b兲. The calculated spatial-density distribution n0បk共r , z兲 ofthe atoms in the 兩p = 0 ប k典 momentum state for a trapping frequencyof ␻ / 2␲ = 10 Hz 共c兲 and 30 Hz 共d兲. The dashed line corresponds tothe size of the Thomas-Fermi profile of the condensate for 8 ⫻ 103
atoms.

gration convolves the spatial phase information into the following factor:

共4兲

where
P2បk =

ag = m␻2z T2vr/ ប ,bg = m␻2z T3v2r /2 ប .

n共r,z兲
兵1 − cos关⌬␾共z兲兴其.
2n2បk共r,z兲dV =1 M eff−
cos共⌬␾eff兲,
2
2

共7兲

where the effective phase shift ⌬␾eff and the contrast M eff are
Bgiven by ⌬␾eff = arctan共 A 兲 = −bg and M eff = 冑A2 + B2, andA = 兰n共r , z兲cos关⌬␾共z兲兴dV and B = 兰n共z兲sin关⌬␾共z兲兴dV. If⌬␾共z兲 is not uniform, this integration gives a dephasing andit results in the contrast degradation of the interference fringe关8兴. Due to the symmetric property of the spatial relative
phase, the effective phase shift does not depend on the volume of the condensate and is given only by −bg. However,
the effective contrast depends on the condensate volume,
which depends on the number of condensed atoms. These
parameters are related to the experimentally evaluated quantities.
The phase shift and the contrast calculated numericallyare shown in Fig. 3. The calculations were performed forvarious numbers of condensed atoms in order to investigate
the number dependence. The figures show that the effective
phase shift depends on the guide curvature and the effective
contrast decreases at a larger guide curvature.
Comparing the observed phase shift and the calculatedone 关Fig. 3共a兲兴, we see good agreement between the two inboth the qualitative potential curvature dependence and thequantitative values. The observed contrast degradation also
tends to follow the theoretical value. Furthermore, we haveobserved an absorption image with a unique spatial distribution, which is reflected in the spatial phase ⌬␾共z兲 关Fig. 4共b兲兴关10兴. This spatial profile is similar to the calculated n0បk共r , z兲关Fig. 4共d兲兴. Consequently, we conclude that the effective potential causes the relative phase and the density distribution

共5兲This estimate shows that the spatially inhomogeneous attractive magnetic-guide potential induces a positiondependent relative phase shift ⌬␾g共z兲 and that the spatiallyinhomogeneous repulsive mutual-mean-field potential due tothe inhomogeneous density distribution of the condensate
also induces a position-dependent relative phase shift⌬␾m共z兲. These two mechanisms work destructively and reduce the spatial relative phase, but they do not cancel eachother because the mutual-mean-field potential is twice aslarge as the magnetic-guide potential. Therefore, the total
relative phase shift has a spatial dependence given by ⌬␾共z兲
关8,10兴. The offset of the relative phase shift bg comes fromthe magnetic potential difference between two paths.
Since the relative phase shift has some spatial dependence
as discussed above, the output momentum states after thesecond ␲ / 2 pulse acquire this spatial dependence, which isn2បk共r,z兲 =冕共6兲Equations 共4兲 and 共6兲 show that the output momentum state
has a spatial interference pattern that has a spatial frequency
of ag and the phase of bg. In the present experiment and theexperiment reported in Ref. 关2兴, the number of atoms in thestate of 兩0 ប k典 and 兩2 ប k典 have been counted without spatial
information. For comparison, we integrate n0បk共r , z兲 andn2បk共r , z兲 over the whole volume numerically, and this inte-

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MUNEKAZU HORIKOSHI AND KEN’ICHI NAKAGAWA

of the output momentum state to depend on the position, andthat the contrast degradation may be explained as a dephasing effect.
The dephasing is suppressed at small axial-trap frequencies because it is the result of the position-dependent relative
phase characterized by ag, which is proportional to thesquared axial-trap frequency. Although the dephasing can besuppressed by preparing a BEC in a flat two-dimensionalguide potential, this arrangement has some drawbacks suchas the phase fluctuations observed in an elongated BEC
关13–15兴. Even if the potential is changed into a twodimensional guide potential from the three-dimensional one
nonadiabatically, the density distribution of the condensate
would lead to an inhomogeneous relative phase. As pointed
out in Ref. 关6兴, in the case of vrT ⬍ Rz, the spatial-phaseevolution can be suppressed by increasing the axial-trap frequency by a factor of 冑2 just before the first ␲ / 2 pulse. Then⌬␾共z兲 ⬃ 0 and the fringe contrast increases.
Here we also suggest that this contrast degradation due todephasing can be improved by the following method. We canevaluate the interference fringe from the observed spatial
pattern of the atomic density shown in Fig. 4共b兲. Although
the spatial resolution was not enough in the present experiment, if one observes the density distribution of the output
condensates with high spatial resolution, one can analyze thespatial interference pattern with Eq. 共6兲 and a high-contrast
interference fringe can be obtained 关10兴.
In the present analysis, we eliminated the influence of therepulsion effect 关10兴 and the s-wave scattering on the interferometer because the atomic density is low enough 共2⫻ 1013 cm−3兲. When we operated the interferometer with theatomic density up to 8 ⫻ 1013 cm−3, the fraction of scatteredatoms increased to 70%. In this case we observed a density
or scattering-dependent additional phase shift of 0.3 radcompared with the interference fringe at low density shownin Fig. 2共c兲. In the present analysis, the phase shift does notdepend on the density or the fraction of the scattered atoms.
However, when one operates an atomic interferometer with ahigh atomic density to realize a high signal-to-nose ratio,high atomic density is an inevitable condition. Therefore, thephase shift of the interferometer using high atomic density
should be investigated more carefully for precision measurements.
In conclusion, we have realized an atomic interferometerusing a Bose-Einstein condensate in a magnetic waveguideon an atom chip. We have observed the phase shift and contrast degradation of the interference fringe signal, which depend on the magnetic-guide potential and the atom-atom interaction. We have shown that the observed phase shift andthe contrast degradation are caused by dephasing, which is a
result of the spatial inhomogeneous phase evolution of the
wave packets. We also observed a fringe pattern in the density distribution of each wave packet due to the inhomogeneous phase evolution. A spatial analysis of the wave packet
will allow us to recover the fringe contrast from the degradation due to the dephasing.

关1兴 Ron Folman et al., Adv. At., Mol., Opt. Phys. 48, 263 共2002兲.
关2兴 Ying-Ju Wang et al., Phys. Rev. Lett. 94, 090405 共2005兲.
关3兴 Y. Shin et al., Phys. Rev. A 72, 021604共R兲 共2005兲.
关4兴 T. Schumm et al., Nat. Phys. 1, 57 共2005兲.
关5兴 T. L. Gustavson et al., Phys. Rev. Lett. 78, 2046 共1997兲.
关6兴 Maxim Olshanii and Vanja Dunjko, e-print cond-mat/0505358
共2005兲.
关7兴 M. Horikoshi and K. Nakagawa, Appl. Phys. B 82, 363
共2006兲.

关8兴 E. W. Hagley et al., Phys. Rev. Lett. 83, 3112 共1999兲.
关9兴 Yoshio Torii et al., Phys. Rev. A 61, 041602共R兲 共2000兲.
关10兴 J. E. Simsarian et al., Phys. Rev. Lett. 85, 2040 共2000兲.
关11兴 N. Katz et al., Phys. Rev. A 70, 033615 共2004兲.
关12兴 A. P. Chikkatur et al., Phys. Rev. Lett. 85, 483 共2000兲.
关13兴 S. Richard et al., Phys. Rev. Lett. 91, 010405 共2003兲.
关14兴 D. Hellweg et al., Phys. Rev. Lett. 91, 010406 共2003兲.
关15兴 S. Dettmer et al., Phys. Rev. Lett. 87, 160406 共2000兲.

We thank S. Watanabe and M. Sadgrove for their criticalreading of the manuscript. This work was partly supported
by Grants-in-Aid for Science Research 共Grants No.
14047210 and No. 17340120兲 from the Ministry of Education, Science, Sports, and Culture, and the 21st Century COE
program on “Coherent Optical Science.”

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PHYSICAL REVIEW LETTERS

week ending
27 JULY 2007

Rectified Momentum Transport for a Kicked Bose-Einstein Condensate
Mark Sadgrove,1,2,* Munekazu Horikoshi,3 Tetsuo Sekimura,1 and Ken’ichi Nakagawa1,21Institute for Laser Science, The University of Electro Communications, Chofushi, Chofugaoka 1-5-1, Japan
2
Japan Science and Technology Agency, Kawaguchi, Saitama, 332-0012, Japan
3
The University of Tokyo, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan.
(Received 16 March 2007; published 25 July 2007)
We report the experimental observation of rectified momentum transport for a Bose-Einstein condensate kicked at the Talbot time (quantum resonance) by an optical standing wave. Atoms are initiallyprepared in a superposition of the 0 and 2@kl momentum states using an optical =2 pulse. By changing
the relative phase of the superposed states, a momentum current in either direction along the standing
wave may be produced. We offer an interpretation based on matter-wave interference, showing that theobserved effect is uniquely quantum.
DOI: 10.1103/PhysRevLett.99.043002

PACS numbers: 32.80.Qk, 03.75.b, 05.45.Mt

The current interest in rectified atomic diffusion, or
atomic ratchets, may be traced back to fundamental thermodynamical concerns [1] and also the desire to understand the so-called ‘‘Brownian motors’’ linked to directeddiffusion on a molecular scale [2,3]. Abstractly, the ratchet
effect may be defined as the inducement of directed diffusion in a system subject to unbiased perturbations due to abroken spatiotemporal symmetry.
Given the scale on which such microscopic ratchets
must work, it is not surprising that the concept of quantum
ratchets has recently augmented this area of investigation.
The addition of quantum effects such as tunneling gives
rise to new ratchet phenomena such as current reversal [4].
Whilst early quantum ratchet investigations, both theoretical and experimental, have focused on the role of dissipative fluctuations in driving a ratchet current [5], recent
theory has considered the possibility of Hamiltonian ratchets, where the diffusion arises from Hamiltonian chaosrather than stochastic fluctuations [6]. This has led toproposals [7,8] and even an experimental realization [9]for ratchet systems realized using atom optics, in the context of the atom optics kicked rotor [10] where periodic
pulses from an optical standing wave kick atoms into
different momentum states.
It is generally accepted that a ratchet effect cannot be
produced without breaking the spatiotemporal symmetryof the kicked rotor system. In Ref. [9], a rocking sine wave
potential was combined with broken time symmetry of the
kicking pulses to effectively realize such a system in anexperiment. Other schemes involve the use of quantum
resonance (QR) to drive the ratchet effect. At QR, atoms
typically exhibit linear momentum growth symmetricalabout the initial mean momentum. However, it has beensuggested that merely breaking the spatial symmetry of the
kicked rotor at QR may be sufficient to produce a ratchetcurrent [11]. In this Letter we present the first experimental
evidence of such a resonant ratchet effect in which the0031-9007=07=99(4)=043002(4)underlying mechanism is purely quantum. Our system uses
a Bose-Einstein condensate (BEC) kicked by an opticalstanding wave [12], but there is no asymmetry in either the
kicking potential or the period of the kicks (which is set to
the Talbot time TT corresponding to quantum resonance[13] ). Rather, the observed directed diffusion is a propertyof the initial atomic wave function (which we prepare
before kicking) in the presence of a resonantly pulsed
optical lattice. The experiment cannot be performed withthermal atoms, as it requires an initial atomic momentum
spread much less than a photon recoil in order to avoid
dephasing effects. Our work presents analytical, simulation, and experimental results for a specific realization of aratchet at QR similar to that proposed in [11]. We also offera clear physical interpretation in terms of matter-wave
interference.
As shown in Fig. 1, our experiment is comprised of a
BEC which is subjected to pulses from an optical standing
wave. The experimental configuration has been explained
elsewhere [14,15] and thus we provide only a summary
here. A BEC of 3  103 87 Rb atoms is realized andloaded onto an atom chip [15]. The atoms are trapped in
the 5S1=2 , F  2, mf  2 state by the magnetic field generated by the chip and sit 700 m below the chip surface.
Typically, the axial trapping frequency for the BEC is!z  2  17 Hz and the axial and radial ThomasFermi radii are dz  17 m and d  3 m, respectively.
The BEC is prepared in an initial superposition state usinga Bragg =2 pulse and then kicked using light from a diodelaser. Figure 1(a) shows the configuration used to control
the intensity and frequency of the two beams used to create
the Bragg and kicking pulses. A free running 100 mW
diode laser, red detuned 4 GHz (i.e.,   780:233 nm)
from the 87 Rb 52 S1=2 ! 52 P3=2 transition, enters a 50=50
beam splitter and the output beams are passed throughseparate acousto-optic modulators (AOMs) to control theirfrequency and amplitude after which they intersect with

043002-1
38© 2007 The American Physical Society

PRL 99, 043002 (2007)

week ending
27 JULY 2007

PHYSICAL REVIEW LETTERS

Bragg =2 pulse. We will assume the BEC starts in an
initial 0 momentum eigenstate j0@kl i. This is not a bad
approximation, since the atoms in the BEC have a thermal
spread which is much less than 2@kl . The =2 pulse creates
an equally weighted superposition state j B i  p12 
j0@kl i  ij2@kl i. After the Bragg pulse has been applied,a period  of free evolution is allowed. During this time,
the j2@kl i state accumulates a phase   4!r  , where  2 corresponds to   TT . The initial state justbefore kicking starts is then1
j i i  p j0@kl i  iei j2@kl i:2(1)

The dynamics, due to sharp periodic momentum kicks
applied to this state, are governed by the Hamiltonian [16]Xp^ 2
^ t0  t;H^ 
 K cos2kl x
2t

FIG. 1 (color online). Diagrams of the experimental setup andsequence. In (a), the laser configuration used to control Bragg
diffraction and the kicking beam is shown. The beam is split by a
50=50 beam splitter (BS) and the output light passes throughseparate acousto-optic modulators (AOM) which control the
beam intensities I1;2 and frequencies !1;2 . The atom chip BEC
setup is shown schematically in (b) along with the optical latticecreated by the two intersecting beams. The three different phasesof Bragg diffraction, phase evolution, and kicking are shown in(c) and are explained further in the text.

(2)

where p^ and x^ are the atomic momentum and position
operators, respectively, K  @V0 Tp =@ is the kicking
strength for an optical potential of height V0 , t0 is time, t
is the kick counter, and   4T=TT is the scaled kicking
time. The associated Floquet operator for the case of QR^ . Applied(  4) is [17] U^ QR t  exp iKt cos2kl x
to j i i, the output wave function o and momentum distribution Pm are [18]the BEC [Fig. 1(b)]. The experimental sequence of laser
pulses is shown in the diagram in Fig. 1(c).
For the Bragg pulse, the intensity of one beam is
dropped to 3% of its maximum power using the amplitude
modulation (AM) mode of one function generator while
the frequency of the counterpropagating beam is increasedby 4!r  15 kHz (where !r  2:37  104 Hz is the recoil frequency of 87 Rb) relative to the other beam. After the
Bragg pulse, a period  of free evolution is used to adjust
the quantum phase of the j2@kl i state relative to j0@kl i,and the beam intensity and frequencies are made equal forkicking. The overall pulse envelope and timing were controlled by another pulse generator. The Bragg or kicking
beams have an optical power of about 5 mW. For a =2
pulse, a duration B of 60 s was used. For the kicking
pulses, a width of Tp  5 s was used with a pulse period
T equal to the Talbot time TT  =2!r  66:3 s for87
Rb. Like other groups performing kicked BEC experiments [12], we have found that neither the energy due toatom-atom interactions nor the harmonic potential affects
our results for the time scales used here, given the relatively much greater energy due to kicking of the atoms. We
simulate the system by calculating the evolution of theinitial wave function subject to the single atom
Hamiltonian 2 (i.e., simulation of the Gross-Pitaevskii
equation is not necessary).
We now provide a theoretical treatment of our system.
First we consider the preparation of the initial state by aei 2 mi Jo m  p Jm Kt  em1 Kt ;
2

(3)

2
2
Pm  12 Jm
Kt  Jm1
Kt  2 cosJm KtJm1 Kt :(4)
Equations (3) and (4) have a particularly interesting property: for general phase , the wave function and thus themomentum distribution grow asymmetrically with time.
This property is seen in Figs. 2(a) and 2(b) which show
the wave function after 5 kicks for   . The change innet momentum may be seen to be due to interference
between the diffraction orders of the two initial wave
functions which is mostly destructive below m  1 but
constructive above this initial mean momentum, leading toan asymmetric distribution of atoms [Fig. 2(c)]. The dramatic nature of this induced asymmetry is demonstrated
even more clearly in [Fig. 2(d)] which shows the theoretical probability distribution after 100 kicks. We note thatthe directed transport of atoms has been caused by the interference of diffracted matter waves, that is, the observed‘‘ratchet’’ effect is entirely quantum (indeed, our experiment may be viewed as a type of atom interferometer[19] ). Experimental confirmation is presented in Fig. 3
which shows absorption images of a kicked BEC after

043002-2
39

PRL 99, 043002 (2007)

week ending
27 JULY 2007

PHYSICAL REVIEW LETTERS(a)| -2h / λ〉φ = π/2(c)φ= π| 0h / λ〉

N

t=7

t=6

t=5

t=4

t=3

t=2

t=1

t=0

(b)

φ= 0

1
0.5
0
0

0

100
x (pixels)

200

0

100
x (pixels)

200

mean momentum calculated. To check repeatability wetook another set of data for the same parameters as thosein Fig. 3, for t > 2 (since very little diffusion occurs in thefirst two kicks). Average currents for the two data sets areshown in Fig. 4, along with error bars showing the difference between the measurements. The extraction of verysmall mean momenta (hpi  @kl ) from the distributions in
Fig. 3 is hampered by experimental imperfections such as
CCD noise and scattered light, and laser frequency drift,
which may lead to occasional changes in experimental
parameters. This is the most likely cause of the large error
bar seen in the case of    when t  6. Increasing theaccuracy of the measurements would require a larger atom
number and ideally a separate laser for Bragg diffraction
and kicking. Nonetheless, Fig. 4 clearly demonstrates themomentum current effect and a current reversal for   0compared with   . The data show a general linear
trend as predicted by Eq. (5), with fitted lines shown inboth cases. For   =2, although individual momentum
distributions are not perfectly symmetrical, the current isnear 0 on average. The control case for an initial j0@kl i
distribution is also shown and seen to exhibit near 0
average momentum current. Note that the dotted anddashed lines are not fits to the data, since there are nofree parameters in either of these cases. Theoretically, themomentum current should persist indefinitely. In an experimental setting, however, imperfections such as thefinite pulse width and any small difference between thepulse period and the Talbot time will reduce the ratchet

preparation into state i . The behavior seen matches thatpredicted by Eq. (4). In particular, for   0 the atomic
momentum distribution increases in asymmetry towards
negative momentum, whereas for   , the asymmetry is in the opposite direction. For   =2 the distribution is almost symmetrical (allowing for experimental
fluctuations).
We may also find the momentum current it  d=dt
hpti by calculating the P
first moment of
P the 2momentum distribution hpi  m mPm  12 m mJm
Kt 
2
Kt  2 cosmJm KtJm1 Kt . The first two
mJm1
terms give the momenta of the two superposed initialstates, e.g., 0 and 1
P (in 2@kl units), respectively. The
term of interest is m mJm KtJm1 Kt, which may besummed by applying the standard Bessel recursion formulaand the Neuman sum rule [20] to give Kt=2. ThusdK
hpti   cos :
dt
2

200

FIG. 3 (color online). Sequences of absorption images for theratchet BEC experiment from t  0 to t  7 kicks for (a)   0,(b)   =2, and (c)   . The t  0 case shows the initial
distribution after the Bragg =2 pulse. In the top panel of (a), the0 and 2@k momentum states are shown. At the bottom of eachcolumn, a sum over the rows of the image for t  7 is shown
giving the distribution of atom number N with position. In theseplots, the dashed line marks the position of the mean initialmomentum @k. The images clearly show the presence of aratchet current which reverses direction when the phase crosses   =2 (for which phase the current is seen to vanish).

FIG. 2 (color online). Wave functions and momentum probability distributions for kicked atoms with K  0:6. The momentum m is in units of 2@kl . In (a) and (b), respectively, the real andimaginary parts of atomic wave functions after 5 kicks for   are shown. The wave function evolving from an initial j0@kl istate is shown with x’s (simulations) and a dot-dashed line
[theory of Eq. (3)], while that which evolved from an initialj2@kl i state is shown with ’s (simulations) and a dotted line
(theory). The lines are merely to guide the eye, and the theoretical wave function is only nonzero at multiples of m  2@kl .
In (c) asymmetry is seen to arise in the final momentum
probability distribution corresponding to    (dashedline —theory, squares—simulations) while for   =2 (solidline —theory, triangles —simulations) there is symmetry aboutm  @kl . In (d) the same system is shown after 100 kicks
emphasizing the extreme asymmetry of the momentum distribution.

it 

100
x (pixels)(5)

Equation (5) offers a useful way to summarize the data.
The atomic momentum distribution was reconstructedfrom the absorption images shown in Fig. 3 and the043002-3
40

PHYSICAL REVIEW LETTERSPRL 99, 043002 (2007)

week ending
27 JULY 2007

M. S. would like to thank Scott Parkins and Andrew
Daley for discussions regarding this work.
Note added in proof.—After submission of this Letter,
similar experimental results were reported by Dana et al.
[21].

0.6
0.4
0.2
0

〈p〉

-0.2
-0.4

*mark@ils.uec.ac.jp[1] R. P. Feynman, The Feynman Lectures on Physics(Addison-Wesley, Reading, MA, 1963), Vol. 1, Chap. 46.
[2] R. D. Astumian and P. Hänggi, Phys. Today 55, No. 11, 33(2002).
[3] P. Riemann, Phys. Rep. 361, 57 (2002).
[4] P. Riemann, M. Grifoni, and P. Hänggi, Phys. Rev. Lett.
79, 10 (1997).
[5] M. Schiavoni et al., Phys. Rev. Lett. 90, 094101 (2003);
G. C. Carlo et al., ibid. 94, 164101 (2005); S. Flach,O. Yevtushenko, and Y. Zolotaryuk, ibid. 84, 2358 (2000).
[6] H. Schanz et al., Phys. Rev. Lett. 87, 070601 (2001);
J. Gong and P. Brumer, Phys. Rev. Lett. 97, 240602(2006).
[7] C. Mennerat-Robilliard et al., Phys. Rev. Lett. 82, 851(1999); T. S. Monteiro et al., Phys. Rev. Lett. 89, 194102(2002).
[8] G. C. Carlo et al., Phys. Rev. A 74, 033617 (2006);
D. Poletti et al., arXiv:cond-mat/0609535.
[9] P. H. Jones et al., Phys. Rev. Lett. 98, 073002 (2007).
[10] F. L. Moore et al., Phys. Rev. Lett. 75, 4598 (1995).
[11] E. Lundh and M. Wallin, Phys. Rev. Lett. 94, 110603(2005); D. Poletti, G. C. Carlo, and B. Li, Phys. Rev. E 75,
011102 (2007); S. Denisov, L. Morales-Molina, and
S. Flach, arXiv:cond-mat/0607558.
[12] G. J. Duffy et al., Phys. Rev. A 70, 041602(R) (2004);
S. Wimberger et al., Phys. Rev. Lett. 94, 130404 (2005);
G. Behinaein et al., Phys. Rev. Lett. 97, 244101 (2006).
[13] C. Ryu et al., Phys. Rev. Lett. 96, 160403 (2006).
[14] M. Horikoshi and K. Nakagawa, Phys. Rev. A 74,
031602(R) (2006).
[15] M. Horikoshi and K. Nakagawa, Appl. Phys. B 82, 363(2006).
[16] R. Graham, M. Schlautmann, and P. Zoller, Phys. Rev. A
45, R19 (1992).
[17] F. M. Izrailev and D. L. Shepelyansky, Sov. Phys. Dokl.
24, 996 (1979) [Theor. Math. Phys. (Engl. Transl.) 43, 553(1980)].
[18] D. Cohen, Phys. Rev. A 44, 2292 (1991).
[19] M. R. Andrews et al., Science 275, 637 (1997).
[20] G. N. Watson, A Treatise on the Theory of Bessel
Functions (Cambridge University Press, Cambridge,
England, 1996).
[21] I. Dana, V. Ramareddy, I. Talukdar, and G. S. Summy,
arXiv:0706.0871v1.

-0.6
-0.8
-1
-1.2
0

2

4

6

t (kicks)

FIG. 4 (color online). The experimentally measured hpi (inunits of 2@kl ) is shown along with theoretical curves for various
initial conditions. Experimental data are shown by x (  0),(  =2), 䊐 (  ), and + (no initial =2 pulse). The solidand dash-dotted lines are fits to the data for   0 and   ,
respectively. Dashed and dotted lines show hpi  0:5 andhpi  0, respectively (note that these lines are not fits to thedata).

current. Because of a low signal to noise ratio at higher
kick numbers in the current experiment, it was not possible
to probe these effects with our current setup. We note thatthe effects seen here require a well-defined quantum phasebetween the initial states in superposition. Therefore, theexperiment must be performed using a BEC as a thermal cloud typically has a large spread of initial momenta(and therefore quantum phase after free evolution), destroying the directed diffusion effect. It may be possible
to exploit any sensitivity of the ratchet current to pulse
timing and phase variations to make accurate interferometry measurements.
In summary, we have demonstrated a novel quantum
ratchet effect, in which directed momentum transport occurs in a system subject to a pulsed potential with no net
bias. The effect has no classical analogue, unlike previous
such systems studied experimentally. The direction of theratchet current varied with the initial quantum phase as
predicted, showing complete reversal for   0 comparedwith   . This realization of directed momentum transport suggests new possible mechanisms for directed motion on any scale where quantum interference effects are
non-negligible and resonant transport exists.

043002-4
41

Eur. Phys. J. D 45, 229–234 (2007)
DOI: 10.1140/epjd/e2007-00277-3

THE EUROPEAN
PHYSICAL JOURNAL D

Coherent control of ballistic energy growth for a kicked
Bose-Einstein condensate
M. Sadgrove1,a , M. Horikoshi2,b , T. Sekimura2 , and K. Nakagawa1,2
1
2

CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
Institue for Laser Science, The University of Electro Communications, Chofushi, Chofugaoka 1-5-1, Japan
Received 8 June 2007
c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2007
Published online 28 September 2007 – 
Abstract. We consider a Bose-Einstein condensate which is split into two momentum components and then“kicked” at the Talbot time by an optical standing wave. The mean energy growth is shown to be suppressedor enhanced depending on the quantum phase between the two momentum components. Experimental
veriﬁcation is provided and we discuss possible implications of our results for recently suggested applicationsof kicked atoms.
PACS. 05.60.Gg Quantum transport – 05.45.Mt Quantum chaos; semiclassical methods – 03.75.Nt Other
Bose-Einstein condensation phenomena

1 Introduction

rangement does not allow non-trivial control of the ballistic energy growth for the atoms.
Here we use a BEC in a diﬀerent initial conﬁguration to [8,9] in order to examine control of the ballisticenergy growth rate. Our system is comprised of a BEC
which is coherently split into two momentum components
separated by 4k. This initial state is then subjected to
sharp momentum “kicks” from a periodically pulsed optical standing wave. The two initial momentum components
in superposition evolve in principle as separate dynamicalsystems but because they have phase coherence, quantum
interference occurs leading to changes in the overall dynamics. In particular the rate at which the mean energyof the system increases may be suppressed or enhanced
compared with the standard case by adjusting the quantum phase between the two systems.

Interference is a fundamental eﬀect in quantum mechanics. In particular, the possibility of interference for objects which are classically point particles is one of the chiefwell-springs of so-called “quantum weirdness”, that is, thecounter-intuitive diﬀerences between classical and quantum mechanics. Moreover, quantum interference, alongwith entanglement, also oﬀers an extra resource absent inclassical physics which may be employed to improve measurement sensitivity [1] or to gain speed-ups over classicalinformation processing algorithms [2,3].
In this letter we consider the possibilities that suchinterference presents for the coherent control of energy
growth in the system known as the Atom Optics Kicked
Rotor [4]. This system consists of cold atoms which receive
sharp momentum kicks from a far-detuned optical standing wave with wave number k. The possibility of taking
advantage of quantum interference eﬀects in this setting
has arisen due to the introduction of Bose-Einstein condensates (BEC) in kicked atom experiments [5]. Gong andBrumer have predicted that preparing the rotor in an initial superposition of momentum eigenstates allows quantum coherent control of the ensemble mean energy in bothchaotic regimes [6] and at quantum resonance [7]. Indeed,
it has already been demonstrated experimentally, that an
initial superposition of 0 and 1 momentum eigenstates (inthe units of the optical standing wave momentum quanta2k) can lead to a ratchet eﬀect [8,9]. However, this ar-

2 Theory
The Hamiltonian for an atom which experiences δ-kicks
with period T , is given by [10]
Ĥ =


p̂2
+ K cos(2kx̂)δ(t − tτ ),
2
t

(1)

where p̂ and x̂ are the atomic momentum and position
operators respectively, k is the kicking strength parameter,t is time, t is the kick counter and τ = 4πT /66.3 × 10−6is the scaled kicking time.
The one kick evolution operator for atoms subject toa standing wave pulse followed by a period of free evolution is [11] Û = exp(−iK cos(2kx̂)) exp(−iτ p̂2 /2). In
this paper, we are interested in what happens when theae-mail: mark@ils.uec.ac.jp
Present address: Errato, c/o Institute of Engineering Innovation, The University of Tokyo, 2-11-16, Yayoi, Bunkyo-ku,
Tokyo 113-8656, Japan.
b

42

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The European Physical Journal D

Fig. 2. (Color online) The mean energy of a kicked superposition state after 7 kicks as a function of the quantum phase
φ for k = 0.14. The solid line shows the theory given in equation (5), and the squares show data from quantum simulations.
Crosses show experimental data for three values of φ and thecircle marks the value measured in the case of no initial Bragg
diﬀraction (which was used to determine k for these experiments). The inset shows simulation energies as a function of
the kicking period T demonstrating the suppressed height ofthe quantum resonance peak for the φ = π case (solid line)
compared with the φ = π/2 case (dashed line) and the enhancement which is possible in the φ = π case (dotted line).

Fig. 1. (Color online) Theoretical wavefunctions ((a) and (b))
and energy vs. time curves for interfering kicked rotor systems
with k = 0.14. In (a), the unﬁlled bars show the initial wavefunction for which the quantum phase is φ = π. The ﬁlled bars
show the absolute value of the wave function (i.e. the squareroot of the momentum distribution) after 7 kicks. In (b) thecase for φ = 0 is shown for the same parameters. In (c) themean energy of the atomic ensemble is shown (minus the initial energy) for φ = π (dashed line), φ = 0 (dotted line) andthe standard result (i.e. where there is only one kicked rotor
system and no quantum interference) (solid line).

quantum resonance condition is fulﬁlled, namely that theatoms start in an initial momentum eigenstate (or superposition thereof) and that the pulse period is equal to the
Talbot time (or τ = 4π). In this case, it may be shownthat the second exponential term reduces to unity andthe evolution operator for quantum resonance becomesÛq.r. (t) = exp(−iKt cos(2kx̂)).
We now extend the analytical results of reference [11]to the case where the initial state is of the form√√
|ψi  = 1/ 2(|0k − eiφ | − 4k) ≡ 1/ 2(|0 − eiφ | − 2),(2)where the superposed states are momentum eigenstatesdeﬁned by p̂|n = n|n. Cohen’s result [12] ψo (m) =m|Ûq.r. |n = (−i)m−n Jm−n (Kt) may be used to ﬁnd theoutput wavefunction ψo after kicking and thus the theoretical momentum distribution P (m) = |ψo |2 , giving

redistribution of atoms to higher momenta and a consequent increase in the mean energy. The mean energy may
be calculated by evaluating the second moment of the momentum distribution 2E = p2  = m m2 Pn (m). Substituting 3 gives
E(k, t) =
1 22
K t + cos φm2 Jm (Kt)Jm+2 (Kt)
4
m

1 22
K t + cos φC(Kt),(4)
4

where C(Kt) = m m2 Jm (Kt)Jm+2 (Kt). Cn (z) may beevaluated by successive applications of the Bessel recursion formula mJm (z) = z/2(Jm−1 (z) + Jm+1 (z)) alongwith Graf’s theorem [13] giving Cn (z) = z 2 /4. Thus
the mean energy of the interfering kicked rotor system
is given by
=

1 2
1 2
J (Kt)+ Jm+2(Kt)+cos φJm (Kt)Jm+2 (Kt).
2 m
2
(3)
The ﬁrst two terms in this equation are the usual momentum distributions for atoms starting with momenta 0 and−4k respectively and kicked at quantum resonance [11].
The ﬁnal term is an interference term which modiﬁes theoverall momentum distribution. Predicted wavefunctions
for the cases where φ = π and φ = 0 are shown in Figures 1a and 1b respectively. For the φ = π case, the momentum spread of the atoms is suppressed due to constructive interference of atoms in the −2k momentum
state and corresponding destructive interference at higher
momenta. Conversely when the phase is set to 0, destructive interference in the −2k momentum state leads toP (m) =

E(Kt) =

1 22
1
K t + cos(φ) K 2 t2 .
4
8

(5)

Equation (5) shows that in principle, the energy growthrate at quantum resonance can be suppressed or enhanced
by up to 50% by varying the relative phase between thetwo momentum states as shown in Figure 2. This coherent control of the dynamics is aﬀorded by quantum interference which redistributes atoms into higher or lower
momentum orders depending on the quantum phase. The
inset to Figure 2 shows simulated energies as the kicking
period T is swept over the quantum resonance at T = TT
for three diﬀerent values of the quantum phase φ. It may
be seen that the interference eﬀect gives rise to broadening and narrowing of the peak for φ = π and φ = 0

43

M. Sadgrove et al.: Coherent control of ballistic energy growth for a kicked Bose-Einstein condensate

respectively. Figure 2 also shows quantum simulation results for comparison with the theory. The simulations areof the single particle Hamiltonian (1), since on the timescales of the experiments performed here, atom-atom interactions in the BEC and also the eﬀect of the harmonic
trap potential may be ignored [5]. We note from the outset that the increase in the energy growth rate found herecannot be gained just by redeﬁning the time at which theﬁrst kick occurs in the normal kicked rotor. For example,
deﬁning the ﬁrst j kicks as a state preparation phase gives
E = (1/4)K 2 (t + j)2 , i.e. the energy increases faster thanthe usual case due to redeﬁnition of the time origin. However the coeﬃcient of ballistic growth cannot be changed
by a mere shifting of the time origin in the standard kicked
rotor experiment. It turns out that for the rectiﬁed diﬀusion experiment performed previously [8], any increase in
the energy growth rate due to the initial superposition ismerely of this trivial kind. The eﬀect seen in this paper is
new precisely because it leads to a non-trivial increase in
the ballistic growth coeﬃcient itself (conversely, no ratchet
eﬀect is observed for the conﬁguration considered here).
It is also worth noting that, until now, the maximum
energy growth rate for kicked atoms was k 2 t/2, which generally occurs for atoms in an initial momentum eigenstate
kicked at quantum resonance1. However, this growth rateis also found in the semiclassical limit of the kicked rotor,
as shown in reference [18]. Therefore, it is fair to say that
until now, the quantum kicked rotor has not been able toexceed the classical maximum momentum diﬀusion rate.
The eﬀect seen here, however, uses quantum coherence topush the diﬀusion rate above the classical maximum.

231

Experimentally, preparation of the initial state isachieved through the combination of a Bragg pulse
followed by a period ∆φ of free evolution to adjust
the quantum phase. Two counterpropagating beams arealigned along the z-axis of the condensate to provide theBragg/kicking beams for the experiment. The beams havea Gaussian proﬁle with a half width of ∼1.0 mm. The
optical power is 5 mW and the detuning from the 87 Rb
52 S1/2 → 52 P3/2 resonance is ∼4 GHz. We use the sametwo beams to provide both the Bragg pulse and the kicking
pulses. However the parameters required in each case arequite diﬀerent. To solve this problem we use two synchronised Agilent 33250A function generators to provide driving signals for the AOMs which control each Bragg beam.
We control the frequency of one generator using frequency
shift keyed (FSK) modulation to provide the required frequency diﬀerence ∆ω between the two Bragg beams, anduse externally gated amplitude modulation √of the othergenerator to ensure that the geometric mean I1 I2 of the
Bragg beams is suﬃciently low to give coherent Bragg
diﬀraction over a pulse interval of ∆B ∼ 100 µs (for a
π/2 pulse). In the experiments described here, the amplitude of beam 1 is reduced to IB = 0.12I0 for the Bragg
pulse. The envelope of the pulses, (including the quantum
phase accumulation time and kicking) ∆φ and the timingof the FSK and amplitude modulation is controlled by anadditional function generator (NF Wavefactory 1966).
Because the BEC is so cold (≈10 nK) it may be considered, to a good approximation, to be in a |0k momentum
eigenstate before the state engineering phase [14]. This assertion is supported by the recent measurements of highorder quantum resonances made using a BEC as reported
recently by Ryu et al. [16]. In that study, ballistic momentum growth was observed for up to 20 kicks — almost
three times as many as studied here — and no significant variation was observed due to the small non-zero
quasimomentum of the atoms2 . If we choose ∆ω = 8ωrwhere (where ωr ≈ 2.37 × 104 Hz is the recoil frequency of87Rb) the state after application of the Bragg π/2 pulse
is given by

3 Experimental results
We now consider the experimental veriﬁcation of equation (3). In order to realise the system experimentally, we
use a combination of Bragg diﬀraction of atoms to create
the initial superposition state required, followed by theusual kicking procedure used to realise the atom optics
kicked rotor (AOKR) with atoms in a pulsed standing
wave [4]. Our basic experimental setup and method has
been explained in detail elsewhere [14,15]. The main details are as follows: A BEC of ∼3 × 103 87 Rb atoms isrealised and loaded onto an atom chip [15]. The atoms aretrapped in the 5S1/2 , F = 2, mf = 2 state by the magnetic ﬁeld generated by the chip and sit 700 µm below
the chip surface. Typically, the axial trapping frequency
for the BEC is ωz ≈ 2π × 17 Hz and the axial and radial Thomas-Fermi radii are dz = 17 µm and dρ = 3 µm
respectively.
As discussed in reference [8] our experiments consistof an initial state engineering phase, to prepare the BEC
in a desired momentum superposition state, followed bya kicking phase to induce the resonant transport of theatoms. This is followed by standard absorption imagingof the BEC after a 25 ms ﬂight time [14].

11|ψi  = √ |0k − √ | − 4k.
2
2

(6)

The atoms are then allowed to evolve freely for a time ∆φgiving rise to a relative phase shift between the superposed
states of φ = (8kl2 )t/MRb ≈ 9.48 × 104 ∆φ , and the state
just before kicking occurs is then1
|ψi  = √ (|0k − eiφ | − 4k),2(7)

as assumed by the theory in equation (2). After the initial preparation of the atomic wavefunction, the frequency
and intensity of the beams is made equal to provide theoptical standing wave necessary for kicking. As in other2

In line with the predictions of reference [20], however, it
would be expected that after a large number of kicks, the nonzero momentum width of the condensate would result in subballistic energy growth.

1Away from quantum resonance, the growth rate oscillates
about the quasilinear value k2 /4 before dynamical localisation
sets in.

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Fig. 3. Images of initial state preparation and kicking of the
BEC with corresponding image cross sections showing the relative atomic population N . (a)√shows the BEC after preparationof an approximate |ψi  = (1/ 2)(|0k + | − 4k) initial stateusing a Bragg π/2 pulse. The |0k and | − 4k are indicated.
In (b) atoms were diﬀracted by 3 kicks starting from an initial|0k eigenstate, whereas in (c) the kicks were applied after the
preparation of the BEC in state ψi . The noise ﬂoor has been
suppressed in the images only, allowing a clearer picture of theatoms. The cross sections were created by summing over therows of the images.

Fig. 4. Experimental (left column) and theoretical (right column) atomic momentum distributions are shown for the caseof no initial superposition state. The vertical axis of the experimental data is in arbitrary units, whilst the horizontal axis
shows the momentum m in units of 2k. The simulations arefor k = 0.14. Atoms are coupled symmetrically into higher
momentum orders about the 0 momentum eigenstate.

note that as more kicks are delivered, the signal-to-noise
ratio of the data becomes noticeably worse as the condensate is diﬀracted into higher momentum orders, which
prevents us from probing high kick numbers or kicking
strengths for these experiments.
Figures 4–6 show experimentally obtained momentum
distributions after 7 kicks for the experimental realisationof the interfering kicked rotor system alongside theoretically predicted momentum distributions. Figure 4 showsthe “control” case in which the initial state is |0k. From
the observed diﬀraction, the kick strength parameter k
was estimated to be 0.14 and this value has been used
to create the theoretical distributions shown in the righthand column. In Figure 5 the initial state was preparedin the superposition state |ψi  with a relative quantum
phase φ = π between the states. As can be seen this leads
to a marked increase in the number of atoms in the central
momentum state (which is |−2k) leading to a lower momentum spread and hence mean energy than that foundin the control case.
For φ = 0, as seen in Figure 6, the momentum spread
about −2k is greatly enhanced compared with the φ = πcase. Most notably, the population of the −2k state remains relatively small compared with the other states forthe ﬁrst few kicks, unlike the case for φ = π. We notehowever, that although this larger momentum spread isin qualitative agreement with equation (3) the observed
momentum distributions do not agree so well quantitatively with the theoretical predictions. In fact, although
we were able to show suppression of energy growth consistently in our experiments, enhancement of the energy
growth coeﬃcient was not clearly observed in all cases.
The enhancement eﬀect relies on the precise cancellationof the atomic momentum-space wavefunction at −2k,which can easily be ruined by slight ﬂuctuations in the

recent reports, we have found that performing the kicking
experiments in trap is not problematic since the energy
gained by the atoms due to kicking far exceeds the trapping frequency and atom-atom interactions [5]. Therefore
a single particle analysis is suﬃcient to describe the dynamics as we now show. For the analytical solution of theatomic dynamics, we assumed that the kicking pulses wereideal δ functions. Although this is not well approximatedin most experiments, the analytical predictions turn out
to be remarkably robust against non-zero pulse widths
and non-rectangular pulse shapes so long as the numberof kicks and mean atomic energy remains low [17,18].
In our experiments, the atoms are imaged after kicking
using standard absorption imaging techniques [14]. The
resulting atomic distribution can then be analysed to calculate the atomic momentum distribution and also themean energy of the kicked atomic ensemble. Figure 3ashows an absorption image of the BEC after applicationof a Bragg π/2 pulse with ∆ω ≈ 30 kHz and with no
kicking applied. The atoms are clearly separated into two
distinct momentum classes of roughly equal amplitude.
The lower plot of Figure 3a shows a 1D atomic density
distribution calculated by summing over the rows of theabsorption image. Because the momentum separation ofeach group of diﬀracted atoms is known to be 2k, the ﬁnal momentum distribution of the atoms may be inferredfrom the normalised density distribution. In Figure 3b a
kicked condensate is shown (starting from an initial |0k
eigenstate) after 3 kicks have been delivered demonstrating the usual diﬀraction of atoms into higher momentum
orders. In Figure 3c the same thing is shown for a condensate which has been prepared in the initial state |ψi . We

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M. Sadgrove et al.: Coherent control of ballistic energy growth for a kicked Bose-Einstein condensate

233

momentum distribution mainly due to the low atom number and the resulting low signal to noise ratio. Future experiments with increased atom number should enable thereliable measurement of the theoretical curves seen in Figure 2.

4 Discussion
It seems pertinent to ask whether any other easily preparable √momentum states oﬀer the same beneﬁts as the |ψ =(1/ 2)(|0k − | − 4k) state. After all, in a standard
kicked rotor experiment, starting from a single momentum
eigenstate, the ﬁrst kick creates a momentum superposition state whose components then interfere during subsequent kicks. However, although the standard kicked rotor experiments essentially involve a kicked superposition
state after the ﬁrst kick, there is no way to control thequantum phase or amplitude of each momentum state in
the superposition. Engineering the initial state to have a
speciﬁc relative quantum phase and amplitude is the feature that allows the ballistic diﬀusion rate to be controlled
in this investigation. It has been shown that a superposition of the 0k and −2k eigenstates leads to directional
momentum transport [8,9], although, as noted earlier, nocontrol over the ballistic growth rate is available in thiscase. Furthermore, numerical simulations suggest that initial superpositions of two states separated by more than4k in momentum space do not exhibit quantum phasedependent changes in the diﬀusion rate.
We now speculate as to the possible advantages of thesystem studied here in applications compared with thenormal kicked atom system. In the ﬁrst place we consider the utility of the standard quantum resonance effect as augmented by our technique. As ﬁrst shown inreference [19] for the two frequency kicked rotor system,
the kicked rotor system contains resonances which exhibit
sub-Fourier narrowing with time, potentially allowing thediﬀerentiation of two frequencies in a time which beats theFourier limit. This feature is due to the nonlinear natureof the quantum kicked rotor. The quantum resonance inmean energy employed in this paper has also been shown
to exhibit such sub-Fourier scaling of the resonance widthwith time (a property which can be explained using asemi-classical description of the quantum resonance) [20].
Therefore, one could in principle distinguish the diﬀerence between a kicking signal’s frequency and exact quantum resonant frequency in sub-Fourier time by monitoring the width of the atomic momentum distribution of
the kicked atoms. Our method can increase the sharpnessof the quantum resonance curve (see the inset of Fig. 2)
thereby providing a purely quantum increase to the sensitivity of any sub-Fourier frequency measurement schemes(although it should be noted that the overall scaling of theresonance peaks is expected to remain the same).
More generally, the increase in energy diﬀusion overthe classical value may give rise to a quantum-overclassical speed-up in any process which relies on momentum diﬀusion. To use a contemporary example, recent

Fig. 5. Experimental (left column) and theoretical (right column) atom distributions are shown for the case an initial superposition state with φ = π. The simulations are for k = 0.14.
Suppression of momentum growth by constructive interferencein the −k momentum state is clearly seen.

Fig. 6. Same as Figure 5 but with φ = 0.

power and detuning of the free running Bragg laser whichlead to slight inequalities in the populations of the 0kand −4k states or even slight population of the −2kmomentum order in the initial state. Indeed, some asymmetry is apparent in the initial superposition state. Suchexperimental imperfections ruin the interference required
to reliably demonstrate the enhancement eﬀect in the lab.
This situation could be improved by introducing a separate laser for performing Bragg diﬀraction which could befurther detuned from atomic resonance, thus reducing theeﬀect of small ﬂuctuations in wavelength or power of thelaser. Nonetheless, the experimental results demonstratethat the interference eﬀect occurs and leads to the alteration of the energy growth rate compared with the standard result. In the current experiments, it proved diﬃcult
to accurately measure the mean energy from the atomic

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The European Physical Journal D

studies suggest links between quantum resonance and discrete time quantum random walks (QRW) [21]. QRW areof interest partially because they can be used to implement
quantum search algorithms [22]. Very recently, there has
been a proposal to implement QRW using a BEC in a momentum superposition state such as that used here [24].
Additionally, there is a separate proposal for implementing QRW in kicked cold atoms using a superposition ofinternal atomic electronic states to create the “quantumcoin” eﬀect needed for QRW [23]. Because the quantum
random walk takes place in momentum space, it may bepossible to use an additional superposition of momentum
states to enhance the momentum diﬀusion, hence speeding up the random walk through momentum space andplausibly any algorithm based on the random walk.
Lastly, we note the possibility of using the eﬀect seen
here is to distinguish the phase of momentum eigenstates in a superposition. In principle, for superpositionsof several diﬀerent eigenstates, the interference properties we have explored for a two state superposition should
uniquely identify a single phase ﬂipped state amongst a superposition otherwise consisting of states with 0 relative
phase, since the output momentum distribution depends
sensitively on interference between the states in superposition (a similar eﬀect has also been noted by Gong [7]).
Assuming that it is possible to set the phase of each eigenstate independently, this suggests the possibility of storage
and retrieval of information using the quantum phase of
the eigenstates in a similar way to that performed with
Rydberg atoms in reference [25] (which are also promising due to the existence of stable quantum wave packets
for Rydberg atoms [26]). That is, starting from an initialsuperposition state, the operations aﬀorded by applying
pulses from an optical lattice are conceivably suﬃcient toperform some quantum information processing algorithms
on the states in superposition (although such a scheme
would certainly not be a universal quantum computer).
This idea is the subject of ongoing research.
In conclusion, we have presented a method of coherently controlling the ballistic quantum transport of atoms
subject to Talbot pulses from an optical lattice. The
method uses quantum interference between two coherent
kicked atom systems to adjust the ballistic growth rateof the kicked atoms by up to 50%. The observed eﬀect
may be explained in terms of the interference between thematterwaves diﬀracted from the two initial sources. We expect that recent proposals for making use of the propertiesof quantum resonant dynamics will beneﬁt from similar
quantum interference methods.

This work was partly supported by a Grant-in-Aids for Science
Research (17340120) from the Ministry of Education, Science,Sports and Culture, and the 21st Century COE program on“Coherent Optical Science”.

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409 (2002)

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Suppression of Dephasing due to a Trapping Potential and Atom-Atom Interactions
in a Trapped-Condensate Interferometer
Munekazu Horikoshi1 and Ken’ichi Nakagawa1,21Institute for Laser Science, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu 182-8585, Japan
2
CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan(Received 10 October 2006; published 29 October 2007)
We propose and demonstrate a novel trapped-condensate interferometer using optical Bragg diffractions in a harmonic magnetic potential, which can realize a long coherence time with low dephasing.
Dephasing of wave packets due to the magnetic potential is canceled by setting the interrogation timeequal to the oscillation period of the harmonic potential. The harmonic potential also helps to suppressdephasing due to condensate atom-atom interactions. An interference signal with a fringe contrast of 30%
is observed at an interrogation time of 58 ms. For a longer interrogation time about 100 ms, the spatial
coherence of the condensate is still maintained with low dephasing, although the interference fringe iswashed out by external vibrational noise.
DOI: 10.1103/PhysRevLett.99.180401

PACS numbers: 03.75.Dg, 32.80.Qk, 34.20.Cf

In this Letter, we present a novel trapped-condensate
interferometer in a harmonic magnetic potential which can
suppress dephasing due to the trapping potential and atomatom interactions and can realize a long coherence timeand a large separation. We consider a system shown in
Fig. 1, where a BEC is trapped in a harmonic magnetic
potential given by Ur; z  m2 !2r r2  !2z z2  and optical
Bragg diffraction is used for a coherent atomic beam
splitter along the z direction. We present a principle of
suppression of dephasing using the harmonic potential andshow experimental results which confirm this principle.
Finally, we propose a trapped Sagnac atom interferometerbased on the principle.
In the case of trapped-atom interferometers, momentum
changes of the wave packets due to the trapping potential
cannot be avoided for a long interrogation time, and this isa major source of dephasing due to the trapping potential.
However, if the wave packets evolve in a closed system

Atom interferometry [1] has been successfully developed for precision measurements of acceleration [2 – 4],
rotation [5], fundamental constants [6,7], and other applications. The sensitivity of atomic interferometers increases
with longer interrogation time in general. In the case of afree space interferometer, the maximum interrogation time
is limited due to the free falling of atoms in the Earth’s
gravitational field. In order to improve the measurement
sensitivity, trapped-atom interferometers have been investigated with great interest recently, because the trapping
potential holds atoms in the Earth’s gravitational field andmakes it possible to realize a long interrogation time easily.
In particular, the coherent manipulation of Bose-Einsteincondensates (BECs) on a substrate, known as an atom chip[8], has provided the realization of a trapped-condensate
interferometer. Therefore, many experiments are currently
proceeding to investigate trapped-condensate interferometers on an atom chip [9–17].
Although a trapped condensate interferometer has anadvantage over the free space one in its long interrogation
time, it generally suffers from undesirable effects of thetrapping potential and the atom-atom interaction.
Decoherence due to the scattering or heating of atoms
can be reduced by lowering the atomic density in the
trap. However, dephasing which arises from the trapping
potential and atom-atom interactions [10,11,15], and also
phase diffusion due to atom-atom interactions [17,18],
limits the coherence time. Very recently, the coherencetime of up to 1 s was realized with trapped thermal atoms
by reducing the trap frequency [16]. Additionally, thecoherence time of trapped-condensate interferometers
was increased to up to 200 ms by using atom-number
squeezed states [17,18]. However, a trapped-condensate
interferometer compatible with all the requirements of along coherence time, large separation, and a large enclosed
area, has not been realized so far. Therefore, the development of a novel trapped-condensate interferometer is asubject of great interest for precision measurements.
0031-9007=07=99(18)=180401(4)FIG. 1. Schematic drawing for the dephasing-free atom interferometer. The interferometer is realized in a harmonic potential.
The interrogation time T is set equal to T0 .

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The experimental operation of the interferometer issimilar to our previous experiment [11]. A BEC of 87 Rb
in the 5S1=2 , F  2, mF  2 state is realized on an atom
chip following the method of Ref. [25] and loaded into amagnetic guide potential adiabatically. Each Bragg pulse
consists of two counterpropagating linearly polarized laser
beams with a frequency difference of about 15 kHz. The
Bragg laser beams are aligned along the axial direction
within 20 mrad so as not to excite any transverse motions
and are incident on the condensate which is trapped at
700 m below the chip surface in order to prevent thecondensate from experiencing surface effects [26,27]. The
relative phase of the second =2 pulse, which is L , isexperimentally varied in the range 0 % L < 2 to observe the interference fringe. Following the second =2
pulse, the magnetic guide potential is turned off. After a
25 ms time of flight, the interference signal is observed
using a standard absorption imaging technique. The population of the atoms in the j2@ki momentum state is evaluated from the absorption images as a function of L .
To realize the dephasing suppression condition, the oscillation period of the magnetic potential should be evaluated with high accuracy. In order to evaluate the oscillation
period, we applied a  Bragg pulse to the trapped condensate and observed the resulting oscillations. After an arbitrary delay time, the magnetic guide potential was turnedoff, and the c.m. position was observed after a 25 ms timeof flight. The observed oscillations for two trapping potentials are shown in Fig. 2. The oscillation frequencies
are evaluated by fitting a sinusoidal function to the data.
They have axial oscillation periods of 58.657(6) ms[97.21(1) ms], whose frequencies are !z  217:0481 Hz [10.287(1) Hz], respectively. Their trapping
frequencies and the recoil velocity result in the amplitudesof z0  110 m and z0  180 m. The radial confinement was fixed to !z  2 60 Hz in this experiment. In
these magnetic potentials, the BEC including 3
103 atoms has Thomas-Fermi radii of dz  17 m andd  3 m in the axial and radial directions, a tempera-

without energy dissipation, the initial state can be revived
periodically. In particular, any wave function of all of thenoninteracting atoms trapped in a harmonic potential returns to the initial state at t  mT0 , where T0  2=!z isthe oscillation period and m is an integer. This mechanism
is analogous to collapse and revival of the quantum field[19,20]. Therefore, if two =2 pulses are applied to theatoms with a time interval set equal to the revival time mT0 ,dephasing due to the potential can be canceled completely
for noninteracting atoms trapped in a harmonic potential.
Next, we consider weakly interacting condensate atoms
trapped in a harmonic potential, which is initially in p
nr; z  j0@ki. Here nr; z  max  Ur; z; 0=g is
an atomic density distribution of a BEC, where  is thechemical potential of a BEC and g  4@2 as =m is thecoupling constant characterized by the scattering length as .
The BEC is changed into a superposition of the two
momentum states by a =2 Bragg pulse along the z direcp
p
tion at t 
0, that is t  0   nr; z  i nr; zp
eimv2r z=@ = 2, where v2r  2@k=m is the two-photon recoil velocity, and then the excited BEC propagates along
the z direction. When the two BECs do not overlap spatially, the center-of-mass (c.m.) motion of the BECs isdecoupled with the internal dynamics, namely, collective
shape oscillations [21–24]. Consequently, the c.m. motions of the BECs can be considered as equivalent toclassical oscillation of an atom. Although their density
distribution should be slightly deformed by the influence of the condensate mean fields, time evolution of thep
BECs can be approximately given by t
f nr; z pp
i nr; z  zc:m: teimvc:m: tz=@ g= 2, where zc:m: t 
z0 sin!z t and vc:m: t  v2r cos!z t is the c.m. position
and the c.m. velocity. The oscillation amplitude is given byz0  v2r =!z . Since the wave function can almost return tothe initial wave function at t  mT0 , namely, 0
mT0 , the two BECs can be made to interfere with lowdephasing by applying the second =2 pulse at the revival time and can show the output state of outp
i nr; zeimv2r =@  j2@ki. Therefore, it is possible tosuppress dephasing for the trapped-condensate interferometer by using a harmonic potential and appropriate
timing of the Bragg pulses.
To this end, we propose a novel trapped-condensate
interferometer which can suppress dephasing, as depicted
in Fig. 1. The two optical Bragg pulses are applied in the z
direction with the interrogation time T set equal to T0 . The
first =2 pulse is applied to the trapped condensate at t 
0, and the excited BEC oscillates in the harmonic potential.
The two BECs are made to interfere by the second =2
pulse at the revival time of T0 . The resultant population of
atoms in the j2@ki momentum state is given by P2@k L, where P2@k is the population of atoms in thecos2 
2
j2@ki momentum state,  is the atomic relative phaseresulting from the interferometry, and L is the optical
relative phase of the second =2 pulse.

FIG. 2. Evaluation of the axial trapping frequencies for eachmagnetic guide potential. The solid circles represent a single
measurement and the solid lines are the fitted sinusoidal function. (a) T0  58:6576 ms, !z  2 17:0481 Hz, z0 
110 m, (b) T0  97:211 ms, !z  2 10:2871 Hz, z0 
180 m.

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PHYSICAL REVIEW LETTERS

ture of =kB  7 nK, and an energy of =h  150 Hz,
respectively.
Next we estimate the anharmonicity of the magnetic
potential. The potential is approximately given by Uz 
c0  12 c2 z2  4 13 2 c4 z4 due to its symmetry, and the fourthorder term is the main source of the anharmonicity. Since
the size of the condensate is determined by the potential
2 Uz
c4 2
curvature given by d dz
 c2 1  2cz , a small variation2
2of the potential curvature is required so as not to excite
internal modes. It leads to the requirement given by
c4 2
1 with jzj % z0 . Based on a numerical calculation
2c2 zof the magnetic potential, the change of curvature is estimated at less than 5 102 , and the requirement is satisfied. In fact, we did not observe any collective modes in
the oscillating condensate with our current imaging
resolution.
The observed interference signals are shown in Fig. 3.
The signals are fitted by a sinusoidal function. The interference fringe shows a contrast of 30% with the interrogation time of 58.66 ms [Fig. 3(a)]. This high contrast fringewith such a long interrogation time proves that the interferometer realizes the cancellation of dephasing due to thetrapping potential and atom-atom interactions. On theother hand, the data for an interrogation time of 97.21 ms
are distributed around 0.5 with peak-to-peak amplitudes ofabout 0.5 at each L , and the interference fringe is hardly
seen in Fig. 3(b). However, if the fringe contrast is limited
by dephasing, the signals cannot be largely distributed.
Therefore, the result shows that the dephasing was suppressed successfully for an interrogation time of nearly
100 ms.
Here we show that a spatial coherence of the condensate
is maintained by the dephasing suppression mechanism
even at T 100 ms. We performed an additional measurement with an interrogation time given by T  T0  T,
where T is a deviation time from the oscillation period.
Since a larger deviation time induces larger degree ofdephasing, the peak-to-peak amplitude of the signals
should become smaller than that for T  0. We set thetrapping frequency equal to 10.29 Hz (T0  97:21 ms) and
fixed the optical relative phase to L  0 and measured thesignals at each deviation time. The experimental results areshown in Fig. 4. We can see that the variance of the signals
is maximum at T  0; on the other hand, it decreases at
T  0. This experimental result clearly shows a highdegree of spatial coherence of the condensate withoutdephasing even for the long interrogation time of almost
100 ms. Therefore, we conclude that the contrast degradation of the interference fringe is not caused by dephasing
due to the potential and atom-atom interactions.
One possible source for the degradation of the fringe
contrast is the phase diffusion due to the atom-atom interaction in a condensate [17,18]. In our present experimental
condition, the maximum coherence time limited by thisphase diffusion is estimated to be about 400 ms, which isstill longer than T 100 ms. Then we next consider ex-

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FIG. 3. Interference signals from the atom interferometer at(a) T0  58:66 ms, !z  2 17:05 Hz and (b) T0 
97:21 ms, !z  2 10:29 Hz. The populations of the atoms
in the momentum state of j2@ki are plotted as a function of the
optical relative phase of L . The solid circles represent a single
measurement and the solid lines are the fitted line by a sinusoidal
function.

ternal vibrational noise as the other source for the contrast
degradation. We have evaluated the expected interference
phase change due to the external vibrational noise. If theposition of the magnetic potential and the laser field of the
Bragg laser are oscillating at a frequency ! with a same ac^celeration of a!simultaneously, the resultant phase shiftsin!T0 =2
1
2^
is given by   2ka!T
0 21!=!z  !T0 =2 , where!  !  !z . When !  !z , the phase change becomes^ 02 =2. In the case of
maximum and is given by   kaT
T  100 ms, therefore, the present interferometer is highly
sensitive to the vibration noise at the frequency around
10 Hz. In the present experiment, both of the vacuum
chamber and the mirror mounts for the Bragg laser beams
are mounted on an air damped optical table. The vibrational noise on the table, as measured using
an acceleromp
eter, was found to be a  2 105 ms2 = Hz level around
10 Hz. It increases rapidly for lower Fourier frequency andit has a peak around the natural frequency of about 2 Hz of
the optical table. The expected phase variation due to thisvibrational noise is at least 1 rad. Thus the vibrational noise
can be one of the main source of causing the random phaseshift and washing out the interference fringe at T 100 ms.

FIG. 4. The interference signals depend on the deviation timesof the second =2 pulse. The solid circles represent a single
measurement.

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In conclusion, we proposed and demonstrated a novel
trapped-condensate interferometer in a harmonic magnetic
potential by using a two =2 Bragg pulses with a timeinterval set equal to the oscillation period of the harmonic
potential, which can suppress dephasing caused by thetrapping potential and condensate atom-atom interactions.
An interference signal with a high fringe contrast of 30%
was observed at an interrogation time of 58 ms. With alonger interrogation time of nearly 100 ms, the separated
condensates still showed high spatial coherence with lowdephasing, but the interference fringe was washed out by
external vibrations.
We thank M. Sadgrove for his critical reading of themanuscript. This work was partly supported by a Grant in
Aid for Science Research (No. 17340120) from the Ministry of Education, Science, Sports, and Culture, ‘‘GroundBased Research Program for Space Utilization’’ promoted
by Japan Space Forum, and the 21st Century COE programon ‘‘Coherent Optical Science’’.

FIG. 5. Schematic drawing of the dephasing-free trapped-atom
Sagnac interferometry. A  pulse is applied to a BEC in the x
direction at t  0. A following =2 pulse in the y direction att  T=4 separates the BEC into two momentum states ofj2@kiy and j2@kiy [10,29], and the two condensates start torevolve with a radius of r0  vr =!r . A second =2 pulse isapplied at t  5T=4.

By further improving vibrational isolation of the apparatus,
the fringe contrast can be increased up to 50%, and furthermore it should be possible to increase the interrogation
time up to the maximum coherence time. Further extensionof the interrogation time can be realized by using atomnumber squeezed states [17,18] or ultracold fermions [28].
We now note a possible source of dephasing in thepresent interferometer. One of the most likely sources ofdephasing is the mutual mean-field between the two condensates. Even if the trapping potential is harmonic, theeffective trapping frequency which the condensates experience changes due to the mutual mean-field while the twocondensates separate and overlap [11,15]. Therefore, theinternal and the external degrees of freedom should couple
with the time dependent trapping frequency and then thecondensate wave packets will change during these processes [23]. Also, the size of the two condensates should
shrink by a factor of 0.87 after the first =2 pulse because
the number of atoms decreases to half that of the original
condensate. Therefore, some internal excitations, which
are collective shape oscillations, can be induced during
these processes. If collective shape oscillations are excited,
the wave function does not revive at a time T0 and dephasing is induced, because the shape oscillation frequencies
are not equal to an integral multiple of the axial trapping
frequency [21]. Further theoretical and experimental investigations are required to explain the real behavior of thecondensates during separating and overlapping and theresidual dephasing.
Finally, we can extend the present concept to a 2D
harmonic potential. If the trapping potential is given byUr  m2 !2r r2  !2z z2 , where r2  x2  y2 , the quantum field in x-y plane revives at t  T, where the interrogation time is redefined by T0  2=!r . By using such asystem, it is possible to realize a trapped-atom Sagnac
interferometer with a large enclosed area given by S 
v2r T02 =4, which is depicted in Fig. 5.

[1] Atom Interferometry, edited by P. R. Berman (Academic,
New York, 1997).
[2] A. Peters et al., Nature (London) 400, 849 (1999).
[3] A. Peters et al., Metrologia 38, 25 (2001).
[4] M. J. Snadden et al., Phys. Rev. Lett. 81, 971 (1998).
[5] T. L. Gustavson et al., Phys. Rev. Lett. 78, 2046 (1997).
[6] D. S. Weiss et al., Appl. Phys. B 59, 217 (1994).
[7] Holger Müller et al., Appl. Phys. B 84, 633 (2006).
[8] Ron Folman et al., Adv. At. Mol. Opt. Phys. 48, 263(2002).
[9] W. Hänsel et al., Nature (London) 413, 498 (2001).
[10] Ying-Ju Wang et al., Phys. Rev. Lett. 94, 090405 (2005).
[11] M. Horikoshi and K. Nakagawa, Phys. Rev. A 74,
031602(R) (2006).
[12] O. Garcia et al., Phys. Rev. A 74, 031601(R) (2006).
[13] Y. Shin et al., Phys. Rev. A 72, 021604(R) (2005).
[14] T. Schumm et al., Nature Phys. 1, 57 (2005).
[15] Maxim Olshanii and Vanja Dunjko, arXiv:cond-mat/
0505358.
[16] Edward Su, Saijun Wu, and Mara Prentiss, arXiv:physics/
0701018.
[17] G.-B. Jo et al., Phys. Rev. Lett. 98, 030407 (2007).
[18] Wei Li et al., Phys. Rev. Lett. 98, 040402 (2007).
[19] R. W. Robinett, Am. J. Phys. 68, 410 (2000).
[20] Markus Greiner et al., Nature (London) 419, 51 (2002).
[21] S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).
[22] M.-O. Mewes et al., Phys. Rev. Lett. 77, 988 (1996).
[23] Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996).
[24] H. Ott et al., Phys. Rev. Lett. 91, 040402 (2003).
[25] M. Horikoshi and K. Nakagawa, Appl. Phys. B 82, 363(2006).
[26] J. Fortaǵh et al., Phys. Rev. A 66, 041604(R) (2002).
[27] M. P. A. Jones et al., Phys. Rev. Lett. 91, 080401 (2003).
[28] G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004).
[29] Saijun Wu et al., Phys. Rev. A 71, 043602 (2005).

180401-4
51

ATOMIC INTERFEROMETER WITH COLD RUBIDIUM
ATOMS FOR THE PRECISION MEASUREMENTS OFGRAVITATIONAL ACCELERATIONKEN’ICHI NAKAGAWA, MUNEKAZU HORIKOSHI, AND YUSUKE KODA
Institute for Laser Science, University of Electro-Communications, 1-5-1 Chofugaoka,
Chofu 182-8585, Japan

We have developed an atomic interferometer using a stimulated Raman transition andlaser-cooled 87Rb atoms. Using this interferometer, we demonstrated a measurement of gwith a ∆g/g = 5 x 10-6. We discuss the future improvements toward the precision
measurements of g and also a Newtonian gravitational constant G using an atomic
interferometry.

Precision measurements using an atomic interferometry have become
important for the metrology and fundamental physics1. As the interactions
between an atom and external fields can be theoretically well characterized, one
can expect small systematic errors in the measurements using atomic
interferometers. We are interested in the developments of atomic interferometers
for the precision measurements of a gravity acceleration g and a Newtonian
gravitational constant G. The Newtonian gravitational constant G has been
determined with less precision than those of other fundamental physicalconstants until now2. Thus it is a challenge to determine this G with higher
precision using atomic interferometers. Until now, high precision measurementsof a gravity acceleration g and a gravity gradient have been demonstrated byusing atomic interferometers with laser cooled Cs atoms3-4. We have developed
a similar atomic interferometer with laser cooled Rb atoms. Compared to cesium,rubidium has an advantage of smaller collisional phase shift which willcontribute to the smaller systematic errors. And we can use Bose-Einsteincondensate atoms which will contribute to the higher sensitivity. Recently an
atomic interferometer with Rb atoms has been demonstrated by using a rather
different principle5.
Our atomic interferometer is basically same as the one reported by Peter et.
al using a stimulated Raman transition between hyperfine ground states toimplement beam splitters and mirrors for an atom3. We collect cold 87Rb atoms
1

52

2

from a vapor cell MOT and these atoms are further cooled down in a σ+-σoptical molasses. Using a velocity selective Raman pulse, we prepare about 106
atoms in the F=1, mF=0 state with a vertical velocity distribution of about 600
nK. Three π/2-π-π/2 stimulated Raman laser pulses are applied to cold atoms toform a Mach-Zehnder-type atom interferometer. Using this atomic
interferometer, we could measure the interference fringe signals (Fig. 1). From
this fringe signal, we could determine the gravity acceleration g with ∆g/g = 5 x
10-6, which was mainly limited by the available interrogation time between
Raman pulses of about 10 ms. We will use an atomic fountain technique toincrease this interrogation time up to 100 ms and this may give a 100-foldincrease of the sensitivity. We will discuss the future improvements towards theprecision measurement of the Newtonian gravitational constant G.
T=10.0ms

25.11

25.13
25.12
Sweep rate (MHz/s)

Figure 1. Typical interferometer fringe signal

References
1.
2.
3.
4.
5.

Atom Interferometry, edited by P. Berman (Academic, New York, 1997).
P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 72, 351 (2000).
Peters A., Chung K. Y., Chu S., Nature, 400, 849 (1999).
M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich,
Physical Review A, 65, 033608 (2002).
Sebastian Fray, Cristina Alvarez Diez, Theodor W. Hänsch, and Martin
Weitz, Phys. Rev. Lett. 93, 240404 (2004).

53

54

55

QFG1-1
Fast and efficient production of Bose-Einstein condensate atoms based on
an atom chip
Munekazu Horikoshi1, 2, Yusuke Suzuki1, 2, Ken’ich Nakagawa1, 2

1Institute for Laser Science, University of Electro-Communications, Japan;
2JST-CREST

hori@ils.uec.ac.jp
Abstract
We developed a fast and efficient method to
produce Bose-Einstein condensation on an atom
chip in a small glass cell. 87Rb condensate atoms
are realized within 3s and the maximum numberof condensation was 20,000.
Makingandmanipulating
Bose-Einsteincondensation (BEC) on the substrate withintegrated conductor patterns, so called ‘atom
chip’, is of considerable interest recently1. Using
the atom chip technique, it has been expected todevelop various quantum devices using coherent
matter wave for sensitive and precision
measurements and also quantum information
processing2. Now the compact experimental
apparatus based on this atom chip has beenrealized for these practical applications3.
For the various practical applications of coherent
atom optics, such as atomic interferometer andquantum information processing4-5, a simple
experimental apparatus for the production ofcondensate atoms has been developed using amirror MOT method6, in which the laser-cooled
atoms are collected near the chip surface.
However, this method requires the metal surface
on a chip substrate and it causes a trap loss anddecoherence for the atoms close to the surface
and it also limits the various design of atom chipusing integrated optical elements, opticaltransparent substrates, and superconducting wires
so on. Thus we have developed a reliable andwidely applicable method to produce condensate
atoms on an atom chip without using a mirror
MOT method.

Figure 1: Experimental setup
75W halogen lamp and 36 of 1mW UV-LEDs for
LIAD are located outside of the glass cell. The
atom chip is mounted in a glass cell downward.
Two pair of coils for MOT and transportation (Tcoil), and three pair of coils which generatehomogeneous magnetic field are located outsideof the glass cell. The distance from MOT to thechip is 2cm. The atom chip used in this researchis shown in figure 2. The gold pattern on Si
substrate was made by standard lithographic andelectroplating techniques. Magnetic potentials
produced by current carrying through these wires
manipulate cold atoms near the chip. The
configuration of the wires used to trap and make
BEC is shown in figure 2. This wire patternconsists of a Z-shaped wire (Z-wire) and acrossed wire (C-wire).

The experimental apparatus consists of threeparts, standard MOT, transportation, and theatom-chip (figure 1). A Pyrex glass cell of innersize 30×30×90mm is connected with a 25l/s ion
pump, a titanium sublimation pump, andelectrical feedthroughs. All processes fromcooling and trapping by MOT to the realizationof BEC are operated in only this glass cell. We
use light induced atom desorption (LIAD)
technique7 in order to modulate Rb pressure. A

Figure 2: Atom chip and wire configuration
At first, Rb87 atoms are cooled and trapped bystandard 6-beam MOT. During MOT loading,
LIAD is turned on for 1s, then Rb pressure
increases and the number of cold atoms increases.
Then LIAD is turned off and atoms are held in
the MOT for 1s to improve pressure in the glass

1464
56

cell. After compression MOT (CMOT) andpolarization gradient cooling (PGC), cold atoms
are optically pumped into the |F=2, mF=+2>
hyperfine ground state and captured inquadrupole magnetic trap (QP) with MOT coils.
3 x 107 atoms are captured in magnetic trap of
10s lifetime.
Magnetically trapped atoms are transported to the
optimum position near the atom chip within
600ms by the balance of the current of MOT and
T coils. Though our configuration of T coils andatom-chip is simple and versatile, it is impossible
to move magnetic minimum point adiabaticallyfrom QP field to Ioffe-Prichard (IP) field of
atom-chip in our configuration. Therefore twomagnetic potentials are switched within 1.1ms
nonadiabatically, that is, IP potential of atom-chiprecatches cold atoms released from QP trap of T
coils. At that time, two magnetic fields areslightly superimposed so that a magnetic spin canfollow adiabatically. The number of atoms,
temperature, phase space density of the atoms
captured by the potential generated from 4A of
Z-wire and 15Gauss of bias field are N = 1.5 x
107, T=100uK, phase space density = 7 x 10-8
respectively, and the loading total efficiency froma MOT to an atom-chip is 50%. The loading
efficiency is limited by mode mismatching.
Immediately after loading, the potential ischanged to cross wire trap of (Iz=4A, Ic=-0.6A)
and (Bx=11G, By=60G) in order to compress
atomic cloud enough. The trapping frequency is(740, 4000, 4000) Hz, and this tight confinement
help to make BEC efficiently. A purecondensation of 20,000 Rb87 atoms in|F=2,mF=+2> state is observed after 4.3s rf
induced evaporation cooling (figure 3).

Here we consider the utility of our system. Our
system doesn’t need mirror MOT. And our
method of transportation and loading is simple
and versatile. If only a chip has a wire to trap, it
is possible to produce BEC with almost any atom
chip after transportation cold atoms from normal
MOT. Though the maximum number of BEC is20,000 in this experiment, it is improved byincreasing the initial number of atoms andoptimizing evaporative cooling. The number of
MOT is limited by the size of cooling lasers andRb pressure. In this experiment laser diameter is
15mm, and we can expand up to size of glass cell.
Rb pressure during MOT loading is limited by
the efficiency of LIAD. And by inserting
something to increase area to adsorb inside theglass cell, and applying more strong light for
LIAD, Rb pressure can increase more. Finallyabout freedom of access to BEC optically, it issolved by transparent substrate. Considering
things described above, we can use chip based
BEC as well as coil based one.
In conclusion, we realized 8,000 of purecondensation within total time of 3s or 20,000 ofpure condensation within total time of 7s in a
glass cell. The number of BEC and total
production time can be improved more. Thissystem is useful for atom interferometer andquantum information processing due to fast cycle
and simplicity and also it gives more possibilityof atom chip research because the chip doesn’t
need high reflective surface for mirror MOT.
References
1) W. Hansel, P. Hommelhoff, T. W. Hansch & J.
Reichel : Nature 413, 498 (2001).
2) R. Folman, P. Kruger, J. Schmiedmayer, J.
Denschlag, C. Henkel: ADVANCES INATOMIC, MOLECULAR, AND OPTICALPHYSICS 48, (2002) 263.
3) S. Du, M. B. Squires, Y. Imai, L. Czaia, R. A.
Saravanan, V. Bright, J. Reichel, T. W. Hansch,and D. Z. Anderson : Phys. Rev. A 70, 053606(2004).
4) Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A.
Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R.
A. Saravanan, S. R. Segal, and S. Wu :cond-mat/0407689 (2004).
5) T. Calarco, E. A. Hinds, D. Jaksch, J.
Schmiedmayer, J. I. Cirac, and P. Zoller : Phys.
Rev. A 61, 022304 (2000).
6) J. Reichel : Appl. Phys. B 75, 469–487 (2002)
7) B. P. Anderson and M. A. Kasevich : Phys.
Rev. A 63, 023404 (2001).

0.6mm
1.86MHz

1.83MHz

1.78MHz
0.6mm

Figure 3: BEC phase transition
And also we realized BEC within total time of 3s.
After 1s of MOT loading, cold atoms aretransferred to magnetic trap without waiting torecover the vacuum. Then cold atoms aretransported and loaded into an atom chip andcompressed within 1s. Tight confined atomic
cloud in the potential of (Iz=4A, Ic=-0.8A) and(Bx=17G, By=70G) is applied single rf ramp
from 30MHz to 2.85MHz within 1s, and 8,000 of
BEC is realized.

1465
57

58

59

BOSE-EINSTEIN-CONDENSATE INTERFEEOMETER ON AN
ATOM CHIP WITH A LONG COHERENCE TIME*
KEN’ICHI NAKAGAWA, MUNEKAZU HORIKOSHI†
Institute for Laser Sciencet, University of Electro-Communications,1-5-1 Chofugaoka,
Chofu, 182-8585, JapanWe demonstrate an atom interferometry with a long coherence time using Bose-Einsteincondensates of 87Rb atoms on an atom chip. Dephasing of wave packets due to the trap
potential is reduced by applying two Bragg laser pulses with an time interval T equal tothe oscillation period of the magnetic trap. We observe a high contrast (~ 30 %)interference fringe signal for T = 58 ms.

Recently many attentions have been paid to atomic interferometry using
Bose-Einstein condensates in a trap or waveguide because of its inherent longinterrogation time [1-4]. However, the various dephasing effects due to theatom-atom interaction and/or the trap potential limit the maximum interrogation
time [1-4]. Although the atom-atom interaction can be reduced by lowering theatomic density or using a number squeezed state of atoms [2,3], the dephasing
due to the trap potential is still a problem for the realization of longinterrogation time [4]. In this paper, we present a novel interferometer using
Bose-Einstein condensates with a long interrogation time of more than 50 ms.
The experimental setup and procedure are similar to those of our previous
experiment [4][5]. Bose-Einstein condensates of about 3 x 103 87Rb atoms in the|F = 2, mF = 2> state is loaded into a micro magnetic trap on an atom chip [5].
The axial trap frequency (νaxial) is about 17 Hz (or 10 Hz) and the radial trap
frequency is 60 Hz. We use a Bragg diffraction of atoms using two offresonance counter-propagating laser beams as a beam splitter for the atom [4]. A
first π/2 Bragg laser pulse splits a condensate into two momentum components
0hk and 2hk (k=2π/λ, λ ~ 779 nm) (Fig. 1). The 2hk component starts tooscillate in a trap potential. If the anharmonicity of the trap potential is enoughsmall, the collective mode motion of the condensate is hardly excited during the*This work was partly supported by a Grant-in-Aids for Science Research (17340120) from the
Ministry of Education, Science, Sports and Culture, and a Ground-based Research Program forSpace Utilization promoted by Japan Space Forum.
†
Present address: The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan.
1

60

2

oscillation and the dephasing due to the atom-atom interaction can be
minimized. When the 2hk component returns to the initial position, a second
π/2 pulse is applied to recombine these two components. If the time interval T
between two pulses is equal to the trap oscillation period ( = 1/νaxial), thedephasing due to the trap potential is cancelled and a high contrast (~ 30 %)interference signal is observed for T = 58 ms (Fig. 2 (a)). When we increase thetime interval T to 97 ms (νaxial = 10 Hz), an interference fringe is washed out(Fig. 2 (b)). We attribute this contrast degradation to the external vibrational
noise which randomly modulates the optical phase of the Bragg pulses. Further
improvements in the vibrational isolation will allow to realize an interrogation

time of longer than 100 ms.
Figure 1. A schematic diagram of the interferometer.
1.0

P2 hk

(b)

(a)

0.8
0.6
0.4
0.2
0.0

0

π

φL [rad]

2π

0

π

2π

φL [rad]Figure 2. Interference fringe signals for T = 58 ms (a) and T = 97ms (b).

References
1.
2.
3.
4.
5.

O. Garcia et al., Phys. Rev. A 74, 031601(R) (2006).
G.-B. Jo et al., Phys. Rev. Lett. 98, 030407 (2007).
Wei Li et al., Phys. Rev. Lett. 98, 040402 (2007).
M. Horikoshi et al., Phys. Rev. A 74, 031602(R) (2006).
M. Horikoshi et al., Appl. Phys. B 82, 363 (2006).

61

第 1 章 光で作るコヒーレントな原子
電気通信大学レーザー新世代研究センター 中川 賢一

1.1

はじめに

レーザーは位相が揃ったコヒーレントな光を発生する光源ですが、これは光子が量子力
学的にはボース粒子 (ボソン) と呼ばれるボース・アインシュタイン統計に従う粒子であることからこのような性質が生まれます。ボース粒子は整数のスピンを持つ粒子で、これは同一の粒子の交換に対して対称な波動関数となるという性質を持っています。この性質から複数の同一粒子が同じ量子状態になることが許され、ある条件の下では多数の粒子が同一の状態を占有するいわゆるボース・アインシュタイン凝縮 (BEC: Bose-Einstein condensation)と呼ばれる現象が現れます。このためレーザーは単一のモード (状態) に多数の光子が存在する状態、すなわち光子がボース・アインシュタイン凝縮した状態とみることができます。
このボース・アインシュタイン凝縮は 1924 年に Bose と Einstein によって予言されましたがこれが原子を用いて実験的に確かめられたのは実に 70 年以上経った 1995 年で、今から 12年ほど前と比較的新しいことです。この原子のボース・アインシュタイン凝縮の実現によってレーザー光と同様に位相が揃ったコヒーレントな原子が得られ、これは光のレーザーの原子版ということから原子レーザー (atom laser) と呼ばれています。この原子レーザーはもはや粒子としてよりも物質波として波の性質が顕著に現れるため、光と同様に容易に反射、回折、干渉を行うことが可能になり、このような研究は原子光学と呼ばれています。この
原子光学の研究はボース凝縮の出現によって近年目覚しく発展した研究分野で、次々と興味ある新しい現象が発見され、また光のレーザーと同様に多くの分野への応用が期待されています。このような原子光学の研究においては光のレーザーは非常に重要な役割を果たしています。それは原子のボース凝縮を実現するには 1 マイクロケルビン (K) 以下の極低温に原子を冷却する必要がありますが、これにはレーザー光を用いて原子の運動を制御するレーザー冷却 (laser cooling) と呼ばれる方法が用いられています。また原子レーザーの操作にもレーザー光の利用が欠かせないものになっています。ここではこのような光のレー
ザーを用いてコヒーレントな原子を生成および操作する原子光学の最前線における研究の一端を紹介していきます。

1.2

ボース・アインシュタイン凝縮

ボース・アインシュタイン凝縮 (以下ボース凝縮または凝縮) は多数のボース粒子が同一の量子状態を占有するものですが、これを気体原子において実現するには原子の温度を通常 1K 以下の極低温まで下げる必要があります。これはボーズ凝縮が以下で示される条件

562下においてのみ実現されるからです。量子力学においては気体原子のような粒子は波束として捉えられ、この波束の広がりは熱的ド・ブロイ波長、

 



 



(1.2.1)

 

によって与えられます。ここで  は原子の温度、 は原子の質量です。室温 ( )

における原子の熱的ド・ブロイ波長の大きさは 0.1nm(10 m) 以下ですが、原子の温度を1K 程度まで下げるとこれは 100nm 以上に広がります。さらに温度を下げてこの熱的ド・ブロイ波長が原子間距離と同程度にまで大きくなると、隣り合う原子の波束が重なり始めます。このとき、区別がつかない同一粒子における量子力学的な性質によりボース粒子の場合には波束が重なり合った部分が強め合うことになり凝縮が始まります (図 1.1)。このときの原子の熱的ド・ブロイ波長   と原子の密度 との間には   


 



  

(1.2.2)という関係が成り立っています。この式 1.2.2 から気体原子のボース凝縮を実現するには低温かつ高密度の両方を満たす必要があることが分かります。

図 1.1: ボース・アインシュタイン凝縮。

気体原子の温度を 1K 程度の極低温にまで下げる方法にレーザー冷却と呼ばれる方法があります。これは光が運動量を持っていることを利用したもので、原子が光を吸収 (放出) する際に原子が光の運動量を得る (失う) ことを利用して原子の速度を減速して原子の温度をほぼゼロ近くまで下げる方法です。このような原子のレーザー冷却は 1980 年代に W. Phillips、
S. Chu 等によって最初に実演され、その後 C. Cohen-Tannoudji 等による冷却機構の解明やさらに低温に冷却可能な新しい方法の開発が行われ、原子の温度を 1K 以下にまで下げることが可能になりました。この 3 人には 1997 年にノーベル物理学賞が与えられています。
当初はこのレーザー冷却法を用いることによって先の原子のボース凝縮が実現されるもの

663

と思われていましたが、この方法のみでは原子間の衝突により原子密度が上がらないことが分かり、凝縮は実現されませんでした。
これに対してコロラド大 JILA 研究所の E. Cornel および C. Wieman 等はレーザー冷却によって予め 100K 程度に冷却された  Rb 原子を磁場ポテンシャルに移し、レーザー光を切ることによって原子衝突の影響を回避して原子密度を上げることを可能にしました。このためこの磁場ポテンシャル中で蒸発冷却と呼ばれる方法によって原子の温度を数 100nK程度にまで下げることにより 1995 年に遂に  Rb 原子のボース凝縮を実現しました。その直後に MIT の W. Ketterle 等が Na 原子においても同様の方法によってボース凝縮を実現し、先の 2 人と共に 2001 年のノーベル物理学賞を受賞しました。この最初の BEC が実現されてから、世界中でこのボース凝縮の研究が競って行われるようになり、現在では 100 近い
グループが実験を行っています。このボース凝縮が実現されている原子種も当初の Rb、Na
から、Li、H、K、Cs、He、Yb、Cr と多種に渡っています。また凝縮原子の生成方法も当
初は磁場ポテンシャル中の蒸発冷却のみでしたが、その後レーザー光によって作られる光
ポテンシャル中の蒸発冷却を用いた全光学的な方法によっても凝縮が実現されました。この全光学的な方法は凝縮原子を数秒以内に生成可能とするもので、これは従来の磁場ポテンシャルを用いる方法に比べて生成時間を 1/10 以下に短縮することが可能になりました。
このようにして生成された  Rb 原子のボース凝縮体の様子を図 1.2 に示します。磁場ポ
テンシャル中の凝縮原子を磁場を切って解放し、一定時間後の原子雲の広がりを CCD カメラで測定したもので、原子の速度分布が求められます。これより原子の温度が相転移温度
T 以上では等方的に広がっていた速度分布が、転移温度付近では中心に非等方的な狭い速度幅を持った凝縮成分が現れ、さらに温度を下げると完全に凝縮原子のみになっているのが分かります。

図 1.2: ボース凝縮原子の速度分布。

このボース凝縮原子は室温原子とはもちろん、レーザー冷却によって得られる極低温のサーマル原子とも異なる性質を持っています。それは凝縮体の原子は全て同じ量子状態に

764あり同じ波動関数で表されるということです。凝縮原子はレーザー光と同様に位相が揃ったコヒーレントな物質波とみなすことができ、これをポテンシャルの外に取り出すことにより光のレーザーと同様の原子レーザーが実現されます。このような原子の物質波が光と唯一異なるのは光子間には相互作用が無いのに対し、原子間には相互作用が存在することです。凝縮原子においては通常原子間に斥力が働いており、凝縮による圧縮する力と斥力により膨張する力がつりあって凝縮体は安定に一定の大きさを保っています。これに対して原子間に引力が働いていると、凝縮体は不安定になり一定の大きさ以上の凝縮体は収縮し始め、しまいには崩壊していまいます。この原子間の相互作用は後で詳しく述べますがコヒーレントな物質波の多彩な振る舞いを引き起こすことになります。

1.3

原子干渉計

ボース凝縮原子または原子レーザーが位相が揃ったコヒーレントな物質波であることを先に述べましたがこれを確認できる最も良い例が原子干渉計です。原子干渉計とは光の代わりに極低温原子を物質波として用いてその干渉効果を観測するもので、光の干渉計と同様に様々な物理量の高感度・高精度な測定が可能になります。このような原子干渉計は最初
はオーブンから出されるサーマル原子線を用いて実演され、その後レーザー冷却によって得られる極低温原子を用いて実現されました。しかしこのようなサーマル原子は光で言うと白熱電球の光のようなもので、物質波としてのコヒーレンスが低く、高いコントラストの干渉信号を得ることが難しいという問題がありました。これに対して MIT の Ketterle のグループは二つの凝縮体を重ね合わせることにより高いコントラストの原子分布の干渉フ
リンジパターンを 1996 年に初めて示し、凝縮体がレーザー光のように高い空間的なコヒーレンスを持っていてことを示しました。このため凝縮原子を原子干渉計に用いることにより従来のサーマル原子を用いた場合よりも高い検出感度が期待されます。
このようなボース凝縮原子を用いた原子干渉計の一例を図 1.3 に示します。この干渉計においては対向する二つのレーザー光が原子に対するビームスプリッター (半透鏡) および鏡になっており、原子は最初に二つに分割されて異なる経路を通って再び重ね合わすことによりマッハチェンダー (Mach-Zehnder) 干渉計を構成しています。原子の最終運動状態は二つの経路の位相差に応じて変化することが分かります (図 1.3 右)。
このような原子干渉計は高感度なジャイロスコープや重力加速度計の実現や、基礎物理定数の高精度な決定などに応用されています。このような原子干渉計を用いた計測においては一般的に原子と外場との相互作用時間を長くするほど検出感度が高くなりますが、地上においてはこの相互作用時間は重力による自由落下によって 1 秒以下に制限されます。このため飛行機による放物線飛行や地下の長いトンネル中の自由落下を用いて実現される微小重力環境下で原子干渉計の実験を行うことが検討されており、この場合、相互作用時間を数秒から 20 秒程度まで長くすることが可能となります。
一方、ボース凝縮原子は通常のサーマル原子に比べて原子密度が高いため、原子間相互作用が干渉信号に与える影響が無視できなくなります。このためボーズ凝縮原子を用いた
原子干渉計を高精度な物理計測に利用する場合、この原子間相互作用の影響をいかにして回避するかが課題となります。その一つの解決策としては後で紹介するボーズ凝縮原子の865

図 1.3: BEC 原子を用いた原子干渉計。

代わりにフェルミオン原子を用いることが挙げられます。

1.4

光格子

光格子 (optical lattice) とは 2 つの対向する光の定在波によって作られる周期的なポテン
シャル中に極低温原子を捕捉およびその運動を制御するものです。このような光定在波を 3
次元全ての方向に実現することにより結晶格子のように原子を一定間隔に並べて配置することが可能となります。ポテンシャルの周期は用いる光の波長の半分となるため、可視または近赤外光を用いれば 1m 以下の周期のポテンシャルが実現されます。
2002 年、Greiner 等はボース凝縮 Rb 原子をこの光格子に入れることによりに超流動状態と絶縁体状態の間の相転移現象を観測することに成功しました。これは固体物理における結晶格子中の電子の振る舞いとある意味で同じもので、電子の代わりにに極低温原子が、結晶格子の代わりに 3 次元の光の定在波ポテンシャルがそれぞれ置き換わったものと考えることができます。光で作った格子状のポテンシャルサイトの各サイト間の障壁が低い場合には原子は障壁を越えて自由に別のサイトに行き来することができるため超流動状態となります (図 1.4a)。これに対して光の強度を上げてポテンシャル障壁を高くしていくと原子は障壁を越えることができなくなり絶縁体状態になります (図 1.4b)。このとき 2 個以上の原子がポテンシャルサイトに入ると原子間には互い反発するような相互作用が存在するた
め 1 個の場合より高いエネルギーとなり、トンネル効果で他のサイトに抜けていく確率が高くなり、ある程度時間が経つと全ての原子が各サイトに 1 個ずつ入った状態になります。

966

図 1.4: 光格子中の極低温原子。

このような光格子に原子が 1 個ずつ並べられたものは量子コンピューターに応用可能だと期待されています。量子コンピューターとは量子力学の原理を基にした新しいコンピュー
ターで、量子力学の重ね合わせ状態を用いた並列計算によりある種の計算に対しては従来のコンピューターよりはるかに高速に計算が実行可能となると考えられています。この量子コンピューターでは量子ビットと呼ばれるものを情報の単位として用いて計算を行いますが、この量子ビットは原子の 2 つの内部状態を用いて実現することできます。光格子においては 1 万個以上の原子が光の波長の半分の間隔で並んでいるため、これらの原子全てが量子ビットとして利用可能となります。この光格子中の原子を用いた量子ビットの特徴
は周囲の環境の影響を受けずに長いコヒーレンス時間が得られることにあります。しかしこれらの量子ビットを用いて量子計算を行うには各原子ごとに独立にその量子状態を制御
し、またその状態を読み出す必要があり、これをどのようにして実現するかが課題となっています。
またこの光格子を用いることにより超高精度な原子時計が実現可能と期待されています。
現在、時間および周波数はセシウム (Cs) 原子の超微細準位間のマイクロ波遷移を用いたセシウム原子時計によって定義されていますが、これはレーザー冷却 Cs 原子を用いた原子泉
法によってその周波数精度 Æは 10  が得られています。一方、原子やイオンの可視や紫外域の吸収遷移を周波数基準として用いた光周波数標準はこのセシウム原子時計を越える高い周波数精度が得られるものと期待されており、これは可視光の周波数がマイクロ波に比べて 1 万倍以上高いことによるものです。しかし重力や原子間の相互作用の影響によりその潜在的に高い周波数精度を実現するのはそう容易ではありません。これに対して光格子ポテンシャル中にトラップされた極低温 Sr 原子を用いた光周波数標準が東大の香取によって実現され、セシウム原子時計をはるかに超える 10  程度の周波数精度が将来得られるものと期待されています。これは光格子ポテンシャルを作るレーザー光の波長をある
特定の波長に設定することによりこのレーザー光が原子時計の遷移周波数に影響を与えなくすることが可能になり、また光格子中に原子をトラップすることにより重力の影響を受けずに長い相互作用時間が得られることによるものです。また 3 次元の光格子を用いて原子を 1 個のみを各サイトにトラップする、もしくは後で詳しく述べるフェルミオン原子を1067用いることにより原子間の衝突を無くすことができ、原子間相互作用が周波数精度に与える影響を取り除くことができます。このような光格子を用いた原子時計は現在、日欧米で競って研究が進められ近い将来その高い周波数精度が確かめられるでしょう。

1.5

フェルミオン

今まで主にボース凝縮原子の話題を取り上げてきましたが最近ではフェルミ統計に従う
フェルミ粒子 (フェルミオン) の極低温原子に関しても多くの関心が集まっています。電子や陽子などのフェルミ粒子は半整数のスピンを持つ粒子で、これは同一の粒子の交換に対してその波動関数が反対称となる性質を持っています。このためフェルミ粒子の原子は極低温においてはボース粒子とは全く異なる振る舞いを示し、その一つにフェルミ粒子は複数の同一粒子が同じ量子状態になることが許されないというパウリの排他律があります。ボー
ス粒子では極低温になると基底状態に多数の原子が集まるボース凝縮が起きますが、フェルミ粒子ではエネルギーが一番低い基底状態から高いエネルギーに向かって一つずつ埋まっていき、このような状態をフェルミ縮退と呼びます (図 1.5)。

図 1.5: ボース粒子とフェルミ粒子。

このフェルミ縮退は原子の温度がフェルミ温度 T と呼ばれる温度以下になると現れ始め、さらに温度を下げるに従って徐々に縮退が進行します。このフェルミ温度 T は先のボース凝縮の相転移温度とほぼ同じ大きさで与えられ、典型的な実験条件下では数 100 nK 程度です。このフェルミ粒子においては先のパウリの排他律により極低温において同種の原子間の衝突が起きなくなるため、原子間の弾性衝突を用いた蒸発冷却法が働かなくなり、フェルミ縮退に近づくにつれて原子の温度がこれ以上下がらなくなるという問題があります。これに対してコロラド大 JILA 研究所の Jin 等は 2 つの異なるスピン状態の原子を混ぜてこの異状態間の衝突を利用して蒸発冷却を行うことにより 1999 年に初めて  K 原子においてこのフェルミ縮退の兆候を初めて観測し、その後  Li 原子や  Yb 原子においても同様のフェルミ縮退が観測されました。
フェルミ縮退原子においては先のボース凝縮原子と異なり、原子同士の衝突が無くなる
ため、原子間相互作用がその性質に与える影響はあまり大きくありません。しかし原子間に強い引力相互作用が働いているとその性質は一変し、超伝導体において電子がクーパー対を形成して超伝導状態になるのと同様に、二つのフェルミオン原子が対をなして超流動

1168状態となります。これは二つの異なる運動量を持つフェルミ粒子が空間的には離れているのにも関わらず相関を持って運動し、あたかも一つのボース粒子であるかのように振る舞
う現象です。このような超流動状態を実現するには原子の温度をフェルミ温度よりもさらにもっと低くする必要がありますが、2004 年に  K、 Li 原子においてこの超流動状態の兆候が確認されました。このような超流動状態は、フェッシュバッハ共鳴という方法を用いて原子間の相互作用の大きさや符合 (斥力または引力) を制御することによって実現されました。興味深いのはこのフェッシュバッハ共鳴を用いて相互作用の大きさを大きくすると、二つのフェルミオン原子が一緒になって分子が形成されます。このとき分子はボース粒子となるため、分子のボース凝縮が実現されることになります。このようにフェッシュバッハ共鳴を用いて原子間相互作用の大きさを制御することによりフェルミ縮退原子は BCS 理論で説明される超流動状態から分子の BEC 状態に連続的に移行され、これは BCS-BEC クロス
オーバーと呼ばれています。
このような超流動が極低温原子の系において実現されたことにより、これを高温超伝導などのメカニズムの解明に利用することが検討されています。この極低温原子の系の特徴
は相互作用の大きさなどこの系を特徴付ける各種のパラメータを磁場やレーザー光などによって自由に変えられることにあります。このため新たな材料を合成しなくてもパラメータを変えてその性質を調べることが可能になり、これは実際の物理系を利用したシミュレー
ションとみることができます。

1.6

アトムチップ

電波や光は同軸ケーブルや光ファイバーのような導波路を用いて伝送することができま
すが、極低温原子やボース凝縮原子も原子導波路を用いて伝送することが可能です。このような原子導波路は例えば図 1.6 a) に示すような電線を流れる電流が作る磁場によって実現することができます。電線に電流を流すとその周りに磁場が発生しますが、これにさらに外部から一様なバイアス磁場を加えると電線に沿ってその近傍に磁場がゼロとなります。
スピンを持った原子は磁場が最小の位置に集まろうとする力を受けるため、原子は電線に沿って磁場のゼロの付近にガイドされ原子導波路を作ることができます。シリコンなどの半導体の基板上にフォトリソグラフィー技術を用いて微細な金属電線パターンを作成する
ことにより原子を基板から 10 100m 離れた位置に電線パターンに沿ってガイドするマイ
クロ原子導波路が実現可能となります (図 1.6 b),c),d))。このような基板上のマイクロ原子導波路を用いて極低温原子やボース凝縮原子を操作する研究が近年盛んに行われており、これはアトムチップと呼ばれています。
このアトムチップ上に原子の導波路はもちろん、ベルトコンベヤー、ビームスプリッター、
干渉計、検出器などの多くの原子光学素子が今のところ実現されています。またアトムチッ
プを用いるとボース凝縮原子の生成も非常に高速かつ高効率で行うことが可能となり、ボース凝縮原子を用いた様々な応用においても非常に有用です。現在このようなアトムチップ
を用いて原子干渉計を用いた高感度なセンサーを実用しようとしたり、先の光格子と同様に原子を一個ずつマイクロトラップに捕捉して量子情報処理に応用することが検討されています。先の光格子においては基本的に周期的なポテンシャルしか作ることができません

12

69

が、このアトムチップを用いれば任意のパターンのマイクロトラップが実現可能で、さらに個々のマイクロトラップを独立に制御することも可能となるため、多数の原子を一個ずつその量子状態を制御する必要がある量子コンピューターの研究への応用が期待されています。

図 1.6: アトムチップ

1.7

おわりに

ここ 20 年ぐらいの間に私たちはレーザー光を用いて原子の運動をその量子力学的なレベ
ルでほぼ意のままに制御することができるようになりました。しかしまだ完全に原子の運動およびその量子状態を制御できているわけではなく、例えば多数の原子を原子一個レベルで制御したり、多数の原子の間にお互いに複雑な量子力学的な相関を持たせるといったことはまだ実現できていません。現在のこの分野の研究の劇的な進展を見ているとこのようなことが実現されるのもそう遠い将来のことではないかもしれません。
レーザーが多くの研究者のインスピレーションを刺激して多くの創造的な研究や発明を産み出してきましたが、コヒーレントな極低温原子もレーザーと同様に多くの幅広い分野の研究者を魅了し続けています。この分野の研究の劇的な進展を支えているのは、この極

1370低温原子に魅せられて日夜研究に取り組んでいる多くの若い学生や研究者達です。このセ
ミナーの受講者の中からもこの不思議な世界に興味を持ち、この研究に加わる人が出てくることを期待しています。
参考文献
1) Ketterle 氏のノーベル賞講演
Wolfgang Ketterle: WHEN ATOMS BEHAVE AS WAVES: BOSE-EINSTEIN CONDENSATION AND THE ATOM LASER, Nobel Lecture, 2001
http://nobelprize.org/nobel_prizes/physics/laureates/2001/index.html
2) ボース・アインシュタイン凝縮 (BEC) に関する解説論文
久我隆弘: 日本物理学会誌 第 55 巻 第 2 号 (2000) pp.90-98.
3) 極低温フェルミオン原子に関する解説論文
大橋洋士: 日本物理学会誌 第 59 巻 第 9 号 (2004) pp.591-600.
4) MIT の Ketterle のグループのホームページ。
http://cua.mit.edu/ketterle_group/5) コロラド大学の JILA 研究所の BEC ホームページ。
http://jilawww.colorado.edu/bec/

1471