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1 岡山大学理学部附属量子宇宙研究センターセミナー 岡山大学 多成分ボースアインシュタイン凝縮体の混合ダイナミクス 学習院大学理学部物理学科 東條賢 Department of Physics, Gakushuin University Satoshi Tojo

2 学習院大学 増山雄太 (M1) 関根佐和子 (B4) 冨貴塚一規 (B4) 東條賢 ( 助教 ) Former members: 桑本剛 ( 日大 ) 田邉達良林太郎岩田正文濱崎康平 etc... 平野琢也 ( 教授 ) 田口義久 (M) 三輪いづ美 (B4)

3 多成分量子気体 量子縮退した原子気体の混合系における新しい物理 冷却原子系を用いれば 同じ空間に共存させた多成分超流動の研究が可能 (ex. 3 He- 4 He mixture) 成分間相互作用を外部から制御可能ダイナミクスを可視化して観測可能 密度が希薄なため 体問題へ帰着 理論と実験が高度に一致実験と理論が緊密に連携 1 成分では起こらない新現象を期待 相分離 準安定構造集団励起 不安定性 非線型励起 ( ベクトルソリトン ) スピン波 スピンテクスチャースピノル BEC( 磁性相 内部ジョセフソン振動など ) etc

4 多成分量子気体 量子縮退した原子気体の混合系における新しい物理 混ぜたら何が起こる?? 外部から操作 BEC1 BEC BEC1 BEC 光 磁場 水 & 酒 混合 or 相分離 水 & 油 表面波, 渦生成

5 Outline I. はじめに希薄原子気体の Bose-Einstein 凝縮 (BEC) II. III. IV. 成分 BEC 成分 BEC と相分離マルチドメイン形成 成分 BEC: hyperfine spin states 磁気トラップ中の 成分 BEC 光トラップ中の 成分 BEC 相互作用制御された 成分 BEC: Feshbach 共鳴 成分 BEC の対向超流動 スピン自由度を持つ BEC スピン交換衝突と内部ジョセフソン振動スピン スピノル BEC の磁性相 V. Summary

6 I. はじめに : 希薄原子気体の Bose-Einstein 凝縮

7 Energy scale superfluid 4 He (.17K) liquid Nitrogen (77K) room (300K) sun (core) evaporative cooling (<100nK) laser cooling (1µK ~1mK) liquid He (4K) ( K) [K] gaseous BEC!! (< 60 nk) magnetic refrigeration (1mK) dry ice (194K) sun (surface) (5780K) sun (corona) ( K) background radiation (3K) 60 nk 600 nk 1. µk and Wikipedia

8 ドップラー冷却 レーザー冷却 v 準位系 輻射圧弱 吸収係数 原子 輻射圧強 ω laser ω atom MOT ( 磁気光学トラップ ) x z ω laser ω atom J = 1 ( 励起準位 ) m -1 E 原子の遷移スペクトル周波数 ω 6 方向のレーザー光 y g N ~ 10 9 T < 1 mk +1 v σ + σ - J = 0 ( 基準準位 ) -1 z

9 磁気トラップ E m = +1 磁気トラップと蒸発冷却 原子のスピン状態を利用磁気コイルによって四重極磁場を作る x m = -1 m = 0 z y g 蒸発冷却 N E

10 Experimental setup 010 年 4 月 011 年 1 月

11 撮影方法 ~ 吸収イメージング法 ~ トラップ中の BEC 原子集団に共鳴光 ( プローブ光 ) を照射し 原子集団の影を撮影 数十 ms 程度落下させる 共鳴光を照射 CCD camera

12 Thermal atoms and BEC Thermal atoms (1µK) BEC 0ms ms µk 600 nk BEC 100 µm isotropic distribution anisotropic distribution

13 磁気トラップ 磁気トラップとスピン自由度 E m = +1 磁気トラップ : 原子のスピン状態を利用 スピン成分に依存 m = 0 スピン自由度がない m = -1 光双極子力トラップ Dipole force 1 ( ) = χe U x 0 感受率 χ 0 引力 共鳴 光双極子力トラップ : 光 - 原子間相互作用を利用スピン成分に依らない ν 斥力 U / U 0 Gaussian beam position x スピン自由度を持つ

14 Experimental procedure Energy level diagram of 87 Rb (ground hyperfine states) ~3x10 5 ~60nK rf magnetic trap Create BEC by evaporative cooling. Transfer to optical trap. m F = - BEC polarized. wait time 00ms Control spin-state by rf and MW transition. Zeeman splitting at B = 0 G It is possible to selectively prepare any spin-states. m F =+ F = +1 0 MW rf MHz 6.8 GHz rf -1 Δ=58 khz MHz Initial state - F = m F =+1

15 Experimental procedure 90 sec / shot TOF=15ms (free-fall) ~3x10 5 ~60nK Create BEC by evaporative cooling. Transfer to optical trap. m F = - BEC polarized. wait time 00ms g magnetic field gradient 30 G/cm 1~ ms Stern-Gerlah method m F = rf Control spin-state by rf and MW transition. Probe laser CCD atom cloud Time evolution Back to the first step! Transmission 0 1 absorption imaging Observe density distribution directly

16 一家に 1 枚周期表 011 年 1 月現在は第 5 版 レーザー冷却 BEC FD より

17 II. 成分 BEC

18 BEC の内部状態 BEC のスピン状態 good quantum number: hyperfine spin hyperfine spin F S I F = S + L + I S : electron spin L : electron orbital I : nuclear spin 87 Rb, 3 Na, 7 F =1, Li, 41 K 85 Rb F =, 3 unstable 133 Cs F =3, 4 5 Cr F =3 (S =3, I =0) 4 He*, 40 Ca, 84 Sr, 86 Sr, F =0 (S =0, I =0) 168 Yb, 170 Yb, 174 Yb, 176 Yb 光トラップを用いれば様々なスピン状態を用意できる 多成分 BEC 5 S 1/ m F F= F=1 =

19 多成分 BEC の実験 ( 相分離, 準安定構造 ) 多成分 BEC 多成分の凝縮体波動関数により記述 同じ原子の内部スピン状態を利用した多成分 BEC 3 Na F=1, m F = 0>, 1, 1> 87 Rb 1, -1>, 1, 0>, 1, 1> 87 Rb, ->,, -1>,, 0>,, 1>,, > H.-J Miesner et al., PRL 8, 8 (1999) M.D. Barrett et al., PRL 87, (001) L.E. Sadler et al., Nature 443, 31 (006) H. Schmaljohann et al., PRL 9, (004) M.S. Cheng et al., PRL 9, (004) T. Kuwamoto et al., PRA 69, (004) S. Tojo et al., PRA 80, (009) 87 Rb, +1>, 1, -1> 87 Rb, -1>, 1, +1> D.S. Hall et al., PRL 81, 1539 (1998) K.M. Mertes et al., PRL 99, (007) R.P. Anderson et al., PRA 80, 0363 (009) S. Tojo et al., PRA 8, (010) 異なる原子を利用した多成分 BEC 41 K- 87 Rb G. Modugno et al., PRL 89, (00) ( フェルミ縮退の混合実験は多数 : 40 K- 87 Rb, 6 Li- 40 K, 6 Li- 40 K- 87 Rb, etc ) 同位体を利用した多成分 BEC 85 Rb- 87 Rb S.B. Papp et al., PRL 101, (008) 174 Yb- 176 Yb 174 Yb- 173 Yb T. Fukuhara et al., PRA 79, 01601(R) (009)

20 Miscibility of BECs time-evolution BECs BECs or BECs Mixture of binary BECs miscible immiscible? 平均場によるエネルギー汎関数 1 E[ ψ, ψ ] = dr ψ + V ψ + g ψ + g ψ ψ ψ 1, ψ : 各成分の波動関数 g, g 4 1 i i,trap i i i 1 1 i= 1, mi Gross-Pitaevskii 方程式 1 g 1 : 同種成分間の相互作用パラメータ : 異種成分間の相互作用パラメータ ψ i 1 = + 1,trap t m V + g ψ + g ψ ψ 1 ψ = +,trap + ψ + 1 ψ1 ψ t m i V g g 参考 : 京大基研 008

21 Miscibility of BECs time-evolution BECs BECs or BECs Mixture of binary BECs miscible immiscible? 相分離の条件 (a) 体積 V のポテンシャル中に 種類の流体が一様に分布 E E 1 = + + V ( g N ) 1 gn g1 N N homo 1 1 (b) 種類の流体がそれぞれ V 1, V を占めて分布 inhomo 1 = + N1 N g1 g V 1 V これを N 1, N, V=V 1 +V が一定のもとで最小にすると E V 1 1 homo V 1, inhomo V 1 V 1 V 1 + V = V g N g1 N 1 = 1 + V, g1 N V1 = 1+ V 1 g N 1 ( = gn ) 1 + gn + gg 1 NN 1 参考 : 坪田先生のスライド V inhomo 1 京大基研 008

22 Miscibility of BECs time-evolution BECs BECs or BECs Mixture of binary BECs 相分離の条件 E homo > E inhomo のとき相分離する 1 1 V V miscible immiscible? ( gn ) ( ) gn + g1nn 1 > gn gn + g1gn1n 4π m これと相互作用パラメータ 同種粒子のとき m 1 = m = m であるので a g > i = a a 1 1 散乱長の大小関係がダイナミクスを決める i a i a i : 成分 i の散乱長 a a a 原子間距離

23 マルチドメイン形成のダイナミクス 成分 BEC 3 Na F =1, m F =1> & 1,0> H.-J. Miesner, et al., PRL 8, 8 (1999) 50%-50% mixture at 0ms a > aa 1 1 a 1 =.65 nm a = a 1 =.75 nm TOF 時間 ( トラップオフ後撮影 ): 15 ~ 30 ms ドメインサイズ : ~ 40±15 µm 相分離に関係する不安定性が生じた結果 このようなダイナミクスが起こる 成分 BEC の変調不安定性 K. Kasamatsu and M. Tsubota, Phys. Rev. Lett. 93, (004) ドメインサイズ : ~ 15 µm ( 計算はトラップ中 )

24 III. 成分 BEC: hyperfine spin states

25 成分 BEC in mixed hyperfine states 87 Rb high-field seeker m F low-field seeker F= F= 磁気トラップ可能 87 Rb 1,-1> &,+1> D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998) K. M. Mertes et al., Phys. Rev. Lett. 99, (007) 異なるhyperfine state R. P. Anderson et al., Phys. Rev. A 80, 0363 (009) > + 1>???

26 87 Rb 1, -1> &, +1> in 磁気トラップ 磁気トラップ中で 1>= 1,-1> と >=,+1> を用意 a 1 = a B, a = a B, a 1 = a B a > aa 1 1 > は非弾性衝突により減少 50%-50% mixture at 0 ms 30 ms 10 ms 5 ms D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998) 異なるhyperfine の相分離 80 ms K. M. Mertes et al., Phys. Rev. Lett. 99, (007) Ring 状に相分離

27 III. 成分 BEC: hyperfine spin states 光トラップ中の 成分 BEC

28 成分 BEC in mixed hyperfine states 87 Rb high-field seeker m F low-field seeker F= F= 磁気トラップ可能 87 Rb 1,-1> &,+1> D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998) K. M. Mertes et al., Phys. Rev. Lett. 99, (007) R. P. Anderson et al., Phys. Rev. A 80, 0363 (009) 磁場の影響が微小 ( 時計遷移 ) 87 Rb 1,0> &, 0> S. Tojo et al., LPHYS 09; T. Hayashi et al., JSPS 09 S. Tojo et al., (to be submitted) a 1 = a B, a = a B, (a 1 = 98.6 a B ) a a 1 1 > aa aa 1 1 > + 1>???

29 Spin manipulation between F = 1 & Energy level diagram of 87 Rb at 0 G Frequency dependence m F Microwave GHz clock transition (magnetic dipole transition) rf rf initial state F = F = 1 Relative population of 1,0> Rabi spectrum 50 mw (output) 800 mw (output) m F z Time dependence (Rabi oscillation) Rabi oscillation,0> 1,0> g 50 mw (output) 800 mw (output)

30 ドメインサイズは?? 87 Rb 1, 0> &, 0> in 光トラップ B = 0 G, 0> 100µm 0 ms 50 ms 100 ms 150 ms 00 ms K ( >, 1>) = 0 1, 0>,0> 原子は時間とともに減少, 0> 計算 1, 0> 0 ms 50 ms 100 ms 150 ms 00 ms GP 方程式で計算 ( 電通大斎藤さんのプログラム ) ψ i = +,trap 1 1 t m V + g ψ + g ψ ψ ψ i 1 = + 1,trap t m V + g ψ + g ψ ψ 1 相分離を再現

31 ボゴリューボフ励起の時間発展 ( ω) [ () t exp dt Im ω, 不安定波数 : 虚数のとき不安定性が生じる = ε ε + g n + g n 11 1 凝縮体の波数の評価 ( ) ± g11n1 gn + 4nn 1 g 1 Kasamatsu & Tsubota, PRL 93, (004) 00 ms 150 ms 100 ms 50 ms Bogoliubov GP 計算 ( 1,0> TOF) 実験 ( 1,0> のドメイン ) TOF 後ではトラップ中と比べてドメイン幅がやや大きくなる計算 ( 1,0> trap) (1.5~ 倍程度 ) 00ms 付近で理論と実験がよい一致

32 III. 成分 BEC: hyperfine spin states 異なる成分間の相互作用制御

33 Miscibility of BECs time-evolution BECs BECs or BECs Mixture of binary BECs 相分離の条件 E inhomo < E homo のとき相分離する 1 1 V V miscible immiscible? ( gn ) ( ) gn + g1nn 1 > gn gn + g1gn1n 4π m これと相互作用パラメータ 同種粒子のとき m 1 = m = m であるので a g > i = a a 1 1 散乱長の大小関係がダイナミクスを決める i a i a i : 成分 i の散乱長 直接変化させられないか? Feshbach 共鳴 を使えば可能!

34 成分 BEC in mixed hyperfine states 87 Rb high-field seeker m F low-field seeker F= F= 磁気トラップ可能 87 Rb 1,+1> &,-1> D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998) K. M. Mertes et al., Phys. Rev. Lett. 99, (007) R. P. Anderson et al., Phys. Rev. A 80, 0363 (009) a aa 1 1 磁場の影響が微小 ( 時計遷移 ) 87 Rb 1,0> &, 0> S. Tojo et al., LPHYS 09; T. Hayashi et al., JSPS 09 S. Tojo et al., (to be submitted) a > aa 1 1 Feshbach 共鳴が可能 87 Rb 1,-1> &,-> S. Tojo et al., Phys. Rev. A 8, (010) a aa 1 1 > + 1>??? 制御する similar experiment using 85 Rb- 87 Rb with controlled a 1, S.B. Papp et al., Phys. Rev. Lett. 101, (008)

35 Feshbach 共鳴 : 異なる成分間 Feshbach 共鳴 --- 原子と分子状態を共鳴 E 磁場強度を変化 resonance! 同じ成分 1, 1> & 1, 1007 G two-body loss rates scattering length r 原子間距離 異なる成分間の散乱長が変化 A. Merte et al., PRL 89, 830 (00). T. Voltz et al., PRA 68, 0170(R) (003). Prediction Experiment Magnetic field dependence of two-body loss rates 1.9 G 9.09 G 9.1 G E.G. van Kempen et al., PRL 88, (00). M. Erhard et al., PRA 69, (004). Widera et al., PRL 9, (004). Kaufman et al., PRA 80, (R) (009). 相分離は未報告

36 Spin manipulation between F = 1 & Energy level diagram of 87 Rb at 3.0 G m F Time evolution and imaging Microwave 6.8GHz + rf.0 MHz -photon transition (magnetic dipole transition) rf initial state F = F = 1 Stern-Gerlach method (SG) F = 1 and TOF 15ms for F = m F z Transmission ,-1>,-1> g 18ms for F =1 1,+1>

37 トラップ原子数の磁場依存性 Evolution time in a trap: 5, 50, 75 ms Extensional magnetic field time in TOF: 5ms Magnetic Feshbach resonance was observed at 9.10 G. B = 9.100±0.005G Theory 9.1 G Kempen et al., Phys. Rev. Lett. 88, (00) G Kaufman et al., Phys. Rev. A 80, (R) (009). Experiment (5) G Kaufman et al., Phys. Rev. A 80, (R) (009) 9.09 G 9.18(9) G Erhard et al., Phys. Rev. A 69, (004). Widera et al., Phys. Rev. Lett. 9, (004).

38 a a a eff Feshbach 75 ms bg = G aeff abg, -1> 1, +1>, -1> 1, +1> Number of atoms Phase-separation condition a a > a a 1 11 Miscible condition < a a 1 11 理論計算斎藤プログラム ( 約 1000 個のパラメータで計算, best fitを探す ) 相分離の様子が変化. 低磁場側 : 相分離がより顕著に enhanced phase separation 高磁場側 : 相分離しなくなる miscible Tojo et al., Phys. Rev. A 8, (010).

39 Tojo et al., Phys. Rev. A 8, (010) Estimation of scattering length from phase separation Minimum s indicates a. s =

40 Feshbach 共鳴近傍の相分離 : 散乱長と不安定波数の評価 相分離ダイナミクスより散乱長を評価 不安定波数の散乱長依存性,-1> Exp a 1 > a bg a 1 < a bg Bogoliubov 1,1> x 3,-1> Theory 1,1> 実験 Tojo et al., PRA 8, (010) TOF 後ではトラップ中と比べてドメイン幅がやや大きくなる. 測定解像度の下限 ( 回折限界 ) ( トラップ中の計算が ~3 倍程度大きい ) 理論と実験がよく一致

41 III. 成分 BEC: hyperfine spin states 対向超流動

42 成分 BEC in mixed hyperfine states 87 Rb high-field seeker m F low-field seeker F= F= 磁気トラップ可能 87 Rb 1,+1> &,-1> D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998) K. M. Mertes et al., Phys. Rev. Lett. 99, (007) R. P. Anderson et al., Phys. Rev. A 80, 0363 (009) a aa 1 1 磁場の影響が微小 ( 時計遷移 ) 87 Rb 1,0> &, 0> S. Tojo et al., LPHYS 09; T. Hayashi et al., JSPS 09 S. Tojo et al., (to be submitted) a > aa 1 1 Feshbach 共鳴が可能 87 Rb 1,-1> &,-> S. Tojo et al., Phys. Rev. A 8, (010) 磁場勾配の影響が正反対 87 Rb 1,-1> &,-> K. Hamasaki et al., JSPS 06; M. Iwata et al., JSPS 07 a a a < aa > aa 1 < aa 1 1

43 Spin manipulation between F = 1 & Energy level diagram of 87 Rb at 1.0 G m F F = マイクロ波を照射し磁場を挿引,-> と 1,-1> を 50:50 に用意 Microwave 6.8GHz B-field sweeping initial state F = 1 m F ,-> z 1,-1> g

44 磁場勾配による BEC の分離 磁場勾配を利用して 成分を分離 B 1,-1>,->, -> F =, m F = -> & F = 1, m F = -1> F= BEC の重心位置 (µm) 1, -1> z 反転させる! F= BEC の時間発展 B,-> 1,-1>, -> F=,m F =-> F=1,m F =-1> z 1, -1>

45 1, +1> & 磁場反転後の時間発展, ->, -> 0 ms Preliminary Relative 60 ~ 300 ms あたりまで非一様な密度分布が観測

46 ベクトルソリトン 成分 BEC におけるソリトン 複数の孤立波が結合した複合ソリトン i ψ,trap 1 1 V g g = + + ψ + ψ ψ t m i ψ 1 1,trap t m V g ψ g = ψ ψ 1 g 1 = g = g 1 のとき Manakov system ( 可積分 ) [S.V. Manakov Sov. Phys. JETP 38, 48 (1974)] ベクトルソリトンの分類 P.G. Kevrekidis, et al., Euro. Phys. J. D 8, 181 (004) Dark-antidark 一様系 Dark-gray a 1 = 0.5 a 1 = -0.5 a = 1 a = 1 a 11 (>0) で規格化 Bright-antidark a 1 = -0.5 a a = -1 1 = 0. トラップ系 Bright-gray a 1 = 0.5 a 1 = 0.86 a = -1

47 最近の混合 BEC の理論 成分 BEC の界面不安定性 BEC1 Generation from interference instability of two component BEC using counter flow Rayleigh-Taylor 不安定性 成分界面の微小な凹凸から擾乱が成長 流体運動が不安定化 Sasaki et al., Phys. Rev. A 80, (009) Kelvin-Helmholtz 不安定性 成分流体が異なる速度で運動 ( 剪断流 ) 接触界面が不安定に Takeuchi et al., Phys. Rev. B 81, (009) ソリトン誘起と量子乱流 成分界面からソリトンを誘起 (Dark soliton) ソリトンから量子渦を生成 Takeuchi et al., Phys. Rev. Lett. 101, (010) BEC 不安定性の誘起は散乱長に大きく依存

48 成分 87 Rb BEC の時間発展 Summary,1>& 1,-1> : 相分離の研究,0>& 1,0> : 異なる時間発展 散乱長の違い 不安定波数とドメイン構造の関係 変調不安定性の波数を用いて定量的に説明可 ( 過渡的な励起 ) 成分間の Feshbach 共鳴,-1>& 1,1> 間の Feshbach 共鳴を観測 : B = 9.100±0.005 G. BECs BECs Feshbach 共鳴付近で相分離が変化低磁場側 : 相分離が劇的に変わる高磁場側 : ほぼ混ざり合う Feshbach 共鳴による相分離の制御 BECs BECs a a > a a 1 11 < a a 1 11 成分間の対向流による混合,->& 1,-1> 間の混合ダイナミクスを観測通常とは異なる密度分布 ソリトンや界面不安定性の可能性,->& 1,0> 間の混合ダイナミクス w/ Feshbach 共鳴 (preliminary) 量子乱流へ?

49 IV. スピン自由度を持つ BEC

50 Spinor BEC Internal degrees of freedom Scalar BEC: spin state is fixed (magnetic trap) Spinor BEC: spin states are degenerate (optical trap) hyperfine spin 87 Rb, 3 Na, 7 Li, 41 K F =1, 85 Rb F =, 3 unstable 133 Cs F =3, 4 5 Cr F =3 (S =3, I =0) 4 He*, 40 Ca, 174 Yb, 176 Yb F =0 (S =0, I =0) Different spin states can be trapped in an optical trap at weak magnetic field spin exchange collision

51 IV. スピン自由度を持つ BEC 内部ジョセフソン振動

52 Optical trap (Far-Off Resonance Trap) Far-Off Resonant Trap (FORT) 5 deg. r (radial) g Energy level diagram of 87 Rb (ground hyperfine states) Zeeman splitting at B = 0 G B=0G FORT Beam (radial) Trap depth: ~ 1.0 µk z (axial) FORT Beam (axial) λ : 850 nm coil for magnetic trap beam waist radius radial : 90 µm axial : 3 µm It is possible to selectively prepare any spin-states. m F =+ +1 F = 0 rf MHz 6.8 GHz rf -1 Δ=58 khz MHz Initial state - F = m F =+1

53 Time-evolution of m F = 0 1.5G Time evolution m F = Trap time (ms) Total-spin-conserved spin-relaxation process Number of atoms ( 10 5 ) Total remained atoms total m F =0 m F =±1 m F =± quadratic Zeeman energy m F = F= weak magnetic field F= m F =±1 components appeared!! Trap time (ms) Relative population Elastic collision Decay to lower states of F= at finite magnetic field. Inelastic collision Decay to lower hyperfine states

54 Time-evolution of m F = , 0.3 G B = 0.75 G m F = 0 m F = ±1 m F = ± m F = 0 m = + 1 F B = 0.3 G m F = F= 0 m F =0 spin exchange m = 1 F m F =-1 +1 F=1 Kuwamoto, et al., Phys. Rev. A. 69, (004) Oscillation in spin B = 0.3, 0.75 G. cf. F = 1 Josephson Oscillation: Chang, Chapman et al., Nature Physics 1, 111 (005)

55 IV. スピン自由度を持つ BEC スピン スピノル BEC の磁性相 : 87 Rb F = BEC

56 Magnetic phases diagram of BEC Phase diagram of F = 1 and BEC at B = 0 Magnetic phases of F = 1 87 Rb F = 1 3 Na F = 1 0 c 1 Antiferromagnetic Ferromagnetic ferromagnetic antiferromagnetic ( ) c a a 1 0 /3 Magnetic phases of F = c antiferromagnetic Ferromagnetic Cyclic ferromagnetic cyclic c 1 ( ) c a a, 1 4 /7 ( ) c 7a 10a + 3 a /7 0 4 a 4, a, a 0 : s-wave scattering length Antiferromagnetic Ciobanu, Yip, & Ho, Phys. Rev. A 71, (000) Ueda & Koashi, Phys. Rev. A 65, (00)

57 Symmetry of the Order Parameters Shape of the order parameter in spin space Cyclic Phase headless triad 0 OP has T (tetrahedral) symmetry π rotation around (1,0,0), (0,1,0), (0,0,1) i /3 e π gauge transformation x π/3 rotation around (1,1,1) (-1,-1,1), (-1,1,-1), (1,-1,-1) 4 π /3 π /3

58 Symmetry of the Ground State Phases c Ferromagnetic: U(1) Symmetry Cyclic: Tetrahedral Symmetry c = 0c 1 Antiferromagnetic: Uniaxial Nematic: U(1) x Z Symmetry z c 1 Biaxial Nematic: D 4 Symmetry z y x

59 General order parameter Fractional vortex in cyclic phase Huhtamäki, Simula, Kobayashi, & Machida, Phys. Rev. A 80, (R) (009) Euler rotation in spin space Gauge transformation Huhtamäki, Simula, Kobayashi, & Machida, Phys. Rev. A 80, (R) (009) Kobayashi, Kawaguchi, & Ueda, arxiv: α, β, γ, φ can vary in space as long as the order parameter is single-valued Scalar BEC Cyclic BEC 1/3-1/3 vortex (fractional vortex) π/3 gauge transformation The operators in Tetrahedral group are non-commutable Vortices are also non-commutable Non-Abelian Vortex

60 Kobayashi, Kawaguchi, Nitta & Ueda, Phys. Rev. Lett. 103, (009) Non-Abelian vortices in cyclic phase Non-Abelian character becomes remarkable when two vortices collide with each other Numerical simulation of the Gross-Pitaevskii equation Initial state: two straight vortices in oblique angle vortex line When two vortices collide, they RECONNECT Abelian non-abelian rung

61 Magnetic phases in spin- system Phase diagram in spin- system c Ueda & Koashi, PRA 65, (00) Saito & Ueda, PRA 7, (005) Ferromagnetic Cyclic Measured coefficients of spin-dependent interaction 87 Rb c 1 c c ( π m) ( π m) ( + ± ) ( ± ) ab ab Widera et al., New J. Phys. 8, 15 (006) Magnetic phases of 87 Rb F = : New phase!! c 1 c 4π = m a a 7 4 π 7a0 10a + 3a4 4 = m 7 cf. 87 Rb F = 1 Antiferromagnetic ferromagnetic antiferromagnetic or ferro-magnetic cyclic

62 Time-evolution of m F = - & m F = + 45 mg magnetic field : 45mG initial spin-state: F =, m =+ + F =, m = Trap time (ms) 0 F m F =+ m F =- F Total remained atoms quadratic Zeeman energy m F = F= Stretched state Two-body inelastic loss rate cm -3 /s F=1 Relative population Strongly Evolve suggested to stable as anti-ferromagnetic... spin-states at almost zero magnetic field. Tojo et al., Appl. Phys. B 93, 403 (008). 300 No other spin states appeared However, Several problems should be considered!!

63 Determination of magnetic phases Phase diagram at B = 0 metastable 10 q c Cyclic Cyclic However, magnetic phase of spin- 87 Rb BEC has NOT been determined. Several problems should be considered. (A) High rate of spin-dependent inelastic collision m F F= = q : quadratic 4π azeeman 4 a energy c1 = c m 7 Ferromagnetic Antiferromagnetic π 7a0 10a + 3a4 4 = m q / 7 87 Rb Saito & Ueda, PRA 7, (005) c 1 F=1 (B) Displacement among other spin states B 50 0 ms B-field gradient (C) Effect of finite magnetic field B 0 ( B ~ 50 mg ) -> +> Tojo et al., Phys. Rev. A 80, (009) Tojo et al., Appl. Phys. B 93, 403 (008)

64 Future plan: vortex measurement (prediction) Kobayashi et al., arxiv: Quantum vortices can be generated by external rotation. Cyclic core will appear independently of whether the ground state is cyclic or antiferromagnetic. Madison et al., PRL84, 806 (000) cyclic

65 Future plan: vortex measurement (prediction ) Huhtamäki, Simula, Kobayashi, & Machida, Phys. Rev. A 80, (R) (009) Quantum vortices can be generated by external rotation. Fractional vortices are appeared in cyclic phase.,-1> and,+> states are initially populated. different angular momentum

66 Progress of experimental setup Magnetic shield room Installed on April 010 necessary condition : B ~ 30 mg Fluctuation of environmental magnetic field (probe: Flux gate sensor) B g 4[mG] inside of room 0 ~ 3 mg - 0:00 :00 4:00 6:00 8:00 10:00 time out of service in Yamanote line outside of room ~ 10 mg Reconstruction of BEC setup F m F F =, m F = 0 =, = 1 F =, m F = Optical spoon (prototype) AOM BEC August 010 rf AOM rf Total atoms: 3 x 10 5 laser AOM: acousto-optical modulator

67 Summary Spinor BEC Internal degrees of freedom: coherent spin exchange Magnetic phases: cyclic phase in spin- system ferromagnetic, anti-ferromagnetic, and cyclic phases 87 Rb F = 1 87 Rb F = Determination of magnetic phase of spin- BEC F = For m F = 0 BEC at 300 mg m F = ± via coherent spin dynamics For m F = ± BEC at 45 mg atoms remain in m F = ± Anti-ferromagnetic behavior F = However, NOT exclude a possibility of cyclic phase. high rates of inelastic collisions relative displacement or phase-separation?? insufficient magnetic field?? Future plan-1: controlled magnetic field gradient below 10 mg Future plan-: quantum vortices measurement

68 Thank you!!

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