1 / 45

Non-Equilibrium1D Bose Gases Integrability and Thermalization

「アインシュタインの物理」でリンクする研究・教育拠点研究会　 2009 年 10 月 23 － 24 日. Non-Equilibrium1D Bose Gases Integrability and Thermalization. Toshiya Kinoshita. Graduate School of Human and Environmental Studies (Course of Studies on Material Science) Kyoto University and JST PRESTO.

lowell
Download Presentation

Non-Equilibrium1D Bose Gases Integrability and Thermalization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 「アインシュタインの物理」でリンクする研究・教育拠点研究会　2009年10月23－24日「アインシュタインの物理」でリンクする研究・教育拠点研究会　2009年10月23－24日 Non-Equilibrium1D Bose Gases Integrability and Thermalization Toshiya Kinoshita Graduate School of Human and Environmental Studies (Course of Studies on Material Science) Kyoto Universityand JST PRESTO Work at Penn State University with Trevor Wenger Prof. David S. Weiss

  2. Outline 1D Bose gas theory Equilibrium 1D Bose gas experiments - Total energy - 1D Cloud Size - Local Pair Correlations I will describe briefly in this talk. Non-Equilibrium 1D Bose gas experiments - the Quantum Newton’s cradle

  3. 「量子性・多体効果が顕著に現れる系を原子気体で創る」「量子性・多体効果が顕著に現れる系を原子気体で創る」 物性研究 ＝　量子多体系の研究 弱く相互作用するシステムを 創り、物事を単純化させ、 複雑な現象の背後に潜む 原理を抜き出す 量子力学特有な現象 多体効果 統計性の違い 純粋 ユニバーサル 位相空間密度 ldBn3~ 1 ド･ブロイ波長　~　粒子間距離 ldB (1/n)1/3 (n : density) 原子　→　クラスター、固体

  4. ド･ブロイ波長　　∝　1 / √ T ldBn3~ 1 原子　→　クラスター、固体 density 10 13 - 14 /cm3 真の熱平衡状態（固体・液体）への到達時間を長くする ～数分　≫　実験の時間スケール　10秒 気体状態が極めて長いLife Time をもつ準安定状態として存在 T < 1 mK ! Ref : 4He 2.17 K Laser Cooling & Trapping Evaporative Cooling

  5. Energy │e> ・　ドップラー冷却によるビームの減速 ・磁気光学トラップ 　　　　　　（Magneto-Optical-Trap : MOT） ・　偏向勾配冷却 │g> １秒で　数 mK ～数１０ mK 到達温度　：　冷却能力　＝　加熱効果 ・　蒸発冷却 (from 低温物理学) 高エネルギー原子の 選択的排除 Density 弾性衝突による 熱平衡化

  6. D 光格子（Optical Lattice） I (r ) Udip∝ D Uo 光双極子力 Induced Dipole : Dipole in E : − 光双極子トラップ • : 87Rb 780 nm 1064 nm (YAG) 10.4 mm (CO2) Uo : sub mK ~ mK

  7. 磁気トラップ 光双極子トラップ rf 磁場 U0 (final stage) < 20 mK ・　磁気サブレベルによらない ・　バイアス磁場の設定が自由 ・　トラップの変形・スイッチが容易

  8. -4 -2 -2 -2 0 0 0 2 2 2 4 All-Optical BEC of 87Rb 1 s 1.5 s 2.0 s Evaporation times Kinoshita, Wenger, Weiss, Phys. Rev. A71, 011602(R) (2005) 3.5x105 BEC atoms every 3 s

  9. 全原子 in the Lowest Energy State T = 0 K ! 量子縮退した気体＝“New Quantum State of Matter” ボーズ・アインシュタイン凝縮 位相のそろった 　　巨大な物質波集団！ マクロ（巨視的）な量子現象 「直接」観測 2001 Nobel 賞 操作・制御 Achieved in 1995

  10. Time of Flight 破壊測定 吸収によるShadow Cast CCD 原子集団の”運動量分布” Released Energy ＝　運動エネルギー +　 相互作用エネルギー 　　　　　　　　　　　　　　　　　　　　　 （Mean-field Energy） The mean-field energy is converted into the Kinetic energy immediately after the release.

  11. Phase Plate (Phase Contrast) ‘Blocked’ (Dark Ground Imaging) 非破壊測定 In－situ Imaging 原子集団＝Phase Object “位相のシフト”

  12. Post BEC の1つの流れ

  13. 重なりによる相互作用 エネルギーの上昇 局在による力学的 エネルギーの上昇 強相関系 How to make strongly correlated system with a dilute gas….. Ekinetic ≫ ≪ Einteraction vs Weakly Interacting Strongly Interacting 平均場近似

  14. Lewi Tonks, 1936: Eq. of state of a 1D classical gas of hard spheres 1D Bose gases with infinite hard core interactions Marvin Girardeau, 1960: 1D Bose gases with infinite hard core repulsion In 1D, if no two single particle wavefunctions overlap “Fermionization” 

  15. 1D Bose gases with variable point-like interactions Elliot Lieb and Werner Liniger, 1963: Exact solutions for 1D Bose gases with arbitrary (z) interactions Solutions parameterized by (g1D > 0) >>1 Tonks-Girardeau gas large g1D low density kinetic energy dominates <<1 mean field theory (Thomas-Fermi gas) mean field energy dominates small g1D high density

  16. 1D Bose atomic gases Maxim Olshanii, 1998: Adaptation to real atoms a 1D waveguide a3D = 3D scattering length a = transverse size of wavefunction  when a3D, n1Dor a

  17. Optical Lattices Calculable, versatile atom traps UAC Intensity Far from resonance, no light scattering 1D Bose gases 1D: 2D: 3D:

  18. Adiabatic Loading from BEC into2D Lattice a a = (ħ / mω) 1/2 ２次元光格子による動径方向の非常に強い閉じ込め 　　　　　　　　　　　　　　　＋ チューブ内の軸方向に沿った緩やかな閉じ込め = 1D System more strongly interacting ⇒より強い閉じ込め　　⇒ 2D Lattice Power In 1D High density Low density weakly interacting

  19. detuning 3.2THz w0~ 600 um up to 85Erec Blue-detuned Lattice minimizes spontaneous emission Bundles of 1D Systems For 1D: negligible tunneling; all energies << ħω Recall:  when a or n1D Independently adjust longitudinal and transverse trapping So  when the lattice power or the dipole trap power 

  20. 0 ms 7 ms 17 ms Expansion in the 1D tubes 50-300 atoms/tube 1000-8000 tubes aspect ratio 150 ~ 700  up

  21. Family of curves parameterized by  Kinoshita, Wenger, DSW, Science305, 1125 (2004) 2 MF 1.5 TG Tonks-Girardeau gas 1 0.5 Exact 1D 1D quasi-BEC 0 0.4 0.7 1 4 strong coupling weak coupling g

  22. Normalized Local Pair Correlations By photo-association Theory: Gangardt & Shlyapnikov, PRL 90 010401 (2003) Expt: Kinoshita, Wenger, DSW, PRL 95 190406 (2005) g(2) of the 3D BEC is 1. 0.8 0.7 Strong coupling regime 0.6 Pauli exclusion for Bosons 0.5 Fermionized Bosons ! 0.4 0.3 g(3), higher order correlation also decreases 0.2 0.1 0 Weak coupling regime .3 10 1 3

  23. What happens when a 1D Bose gas is put into a Non-Equilibrium state ? Does it thermalize ? Summary (1st Half) • Experiments with equilibrium 1D Bose gases across coupling regimes: total energy; cloud lengths, momentum distributions, local pair correlations Experiments agree with the exact 1D Bose gas theory, from Thomas-Fermi to Tonks-Girardeau. 1D systems are a test bed for modeling condensed matter using cold atoms. Other tests of 1D Bose gas theory : NIST(Gaithersburg), Zurich, Mainz

  24. Approach to a Thermal Equilibrium It will ergodically sample the entire phase space (E = const.) Collisions in 1D For identical particles, reflection looks just like transmission ! Two-body collisions between distinct bosons cannot change their momentum distribution.  Integrable systems never reach a thermal equilibrium (too many constrains)

  25. Does a Real 1D Gas Thermalize? pa, pb, pc pa, pb, pc 1D Bose gases with δ-fn interactions are integrable systems  they do not: ergodically sample phase space ≈ become chaotic ≈ thermalize Thermalization in a real 1D Bose gas has been a somewhat open question. Do imperfectly δ-fn interactions lift integrability enough to allow the atoms to thermalize? Do longitudinal potentials matter? Procedure: take the 1D gas out of equilibrium and see how it evolves.

  26. Creating Non-Equlibrium Distributions Optical thickness 1 standing wave pulse Position (μm) 2 standing wave pulses Optical thickness Wang, et al., PRL 94, 090405 (2005) Position (μm)

  27. Harmonic Trap Motion x A classical Newton’s cradle v We make thousands of parallel quantum Newton’s cradles, each with 50-300 oscillating atoms.

  28. ms 0 5 10 1D Evolution in a Harmonic Trap Kinoshita, Wenger, Weiss Nature 440, 900 (2006) Position (μm) -500 0 500 40 μm 1st cycle average 15 195 ms 30 390 ms

  29. Dephased Momentum Distributions 1st cycle average 15 distribution 40 distribution (30t in A) =18 =3.2 = 1.4 Optical thickness (normalized) Position (μm) Project the evolution

  30. th >390 >1910 >200 Negligible Thermalization Projected curves and actual curves at 30  or 40  After dephasing, the 1D gases reach a steady state that is not thermal equilibrium =18 =3.2 = 1.4 Optical thickness (normalized) Each atom continues to oscillate with its original amplitude Position (μm)

  31. What happens in 3D? Thermalization occurs in ~3 collisions. 0  2  4  9 

  32. Lack of Thermalization 初期に与えられた、平衡から大きく 離れた運動量分布を再分布させる 機構が存在しない。 軸方向の弱いトラップポテンシャルは可積分性を崩す ものの、熱平衡を引き起こすほどには十分でない。 This many-body 1D system is nearly integrable. A New Type of Experiment : Direct Control of Non-Integrability

  33. Is there a non-integrability threshold for thermalization? The classical KAM theorem shows that if a non-integrable system is sufficiently close to integrable, it will not ergodically sample phase space. Is there a quantum mechanical analog? Procedure: controllably lift integrability and measure thermalization. Ways to lift integrability Allow tunneling among tubes (1D  2D and 3D behavior); Finite range 1D interactions; Add axial potentials

  34. Ux=UY =60 Erec 40t 15t =3.2 1st cycle average 1st 15t 40t Optical thickness z (mm) z (mm) z (mm) Making 1D gases thermalize Top view Jx JY Allow tunneling among tubes 1D  2D and 3D behavior e. g. Ux = UY = 21 Erec

  35. 10 20 30 40 50 Thermalization in a 2D array of tubes Lattice Depth (Erec) Fraction of energy in 1D 0.02 Thermalization Rate (per collision) 1 0.8 0.015 0.6 no tails 0.01 0.4 0.005 0.2 2-body collisions are well below threshold for transverse excitation. 0 equipartition 0 2 4 6 8 10 12 Lattice Depth (uK)

  36. Summary • Experiments with equilibrium 1D Bose gases across coupling regimes: total energy, cloud lengths, momentum distributions, local pair correlations agree with the exact 1D Bose gas theory. • Non-equilibrium 1D Bose gases: quantum Newton’s cradle. Independent δ-int. 1D Bose gases do not thermalize! • Relaxed conditions allow 1D Bose gases do thermalize. We have a theory to test. • We can also lift integrability in other ways. Is there universal behavior?

  37. Stories After our Experiments…… Do Integral Systems Relax ? Approach to a Thermal Equilibrium It will ergodically sample the entire phase space (E = const.)

  38. Integrals of Motions (conserved quantities) other than the energy strongly restrict the sampling regions. Integrable systems never reach a thermal equilibrium (too many constrains) However, they may relax to a steady state (not a thermal equilibrium, but something else)

  39. Maximizing Entropy Rigol, Dunjko, Yurovsky and Olshanii, PRL, 98, 050405 (2007) Grand Canonical Distribution For Integrable system Maximize entropy S, subject to the constrains imposed by a full set of conserved quantities. Generalized Gibbs ensemble with many Lagrange multipliers.

  40. In 1D system, Discrete Momentum Sets are created by Periodic Potentials. Remove Potentials (Integrable system) Follow Time Evolution Relax to a steady state, but not a thermal equilibrium. “Memory” of initial states is left. Rigol, Dunjko, Yurovsky and Olshanii, PRL, 98, 050405 (2007)

  41. Control Non-Equilibrium process Understanding of Non-Equilibrium Dynamics is very important for Condensed Matter Physics and Statistical Physics Integrable System + Perturbation to control dynamics 1D Bosons (ongoing project) 1D Fermions Non-Integrable system, but some constrains what a kind of constrains, magnitude how to lift integrability quenched by suddenly changing parameters Cold Atom Experiments provide nice stages to study non-equilibrium dynamics.

  42. ブラックホール 初期宇宙 Quantized Flux of Atoms 1D System 1) 熱平衡に近づかない系 ＋ Ongoing project “擾乱” フェルミ＝パスタ＝ウーラムの実験 KAM理論 量子多体系で実験＆観測 2) Attractively Interacting 1D System 3) Atomic Flow in 1D Geometry (ongoing project)

  43. Quantum Gases Flowing in 2D Anti-Dot Lattices…… (some of them areongoing projects) Quantum Chaos (Billiard of Quantum Gas) Quantum Turbulence Non-Equilibrium Phenomena Creation of Macroscopic Coherence

  44. Current Status of my Lab.,,,,,

More Related