Non-Equilibrium1D Bose Gases Integrability and Thermalization

Work at Penn State University with Trevor Wenger Prof. David S. Weiss

Graduate School of Human and Environmental Studies

(Course of Studies on Material Science) Kyoto University and JST PRESTO

Toshiya Kinoshita

「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

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

Outline

「量子性・多体効果が顕著に現れる系を原子気体で創る」

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

純粋ユニバーサル

物性研究 ＝　量子多体系の研究

量子力学特有な現象多体効果統計性の違い

ド･ブロイ波長　 ~ 　 粒子間距離

dB n 3 ~ 1

dB (1/n)1/3

(n : density)

位相空間密度

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 K ! Ref : 4He 2.17 K

Laser Cooling & Trapping

Evaporative Cooling

dB n 3 ~ 1ド･ブロイ波長　　∝　 1 / √T

原子　→　クラスター、固体

・　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

・　蒸発冷却 (from 低温物理学 )

│g >

│e >

Energy

高エネルギー原子の選択的排除

弾性衝突による熱平衡化

Den

sity

到達温度　：　冷却能力　＝　加熱効果１秒で　数 K ～数１０ K

Uo : sub K ~ mK

Rb 780 nm 1064 nm (YAG) 10.4 m (CO2)

Uo

光双極子力

I (r )Udip ∝

Induced Dipole :

Dipole in E :

−

光双極子トラップ

光格子（ Optical　Lattice ）

rf 磁場

磁気トラップ 光双極子トラップ

・　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

U0 (final stage) < 20 K

All-Optical BEC of 87Rb

-4 -2 0 2 4-2 0 2-2 0 2

1 s 1.5 s 2.0 sEvaporation times

3.5x105 BEC atoms every 3 sKinoshita, Wenger, Weiss, Phys. Rev. A 71, 011602(R) (2005)

量子縮退した気体＝“ New　Quantum　State　of　Matter”

全原子 in　the　Lowest　Energy　StateT = 0 K !

2001 Nobel 賞

位相のそろった　　巨大な物質波集団！

Achieved in 1995

ボーズ・アインシュタイン凝縮

マクロ（巨視的）な量子現象「直接」観測

操作・制御

Time　of　Flight

CCD原子集団の”運動量分布”

吸収による Shadow Cast

Released Energy 　

＝　運動エネルギー + 　 相互作用エネルギー　　　　　　　　　　　　　　　　　　　　　 （ Mean-field Energy ）

破壊測定

The mean-field energy is converted into the Kinetic energy immediately after the release.

In － situ Imaging

原子集団　＝　 Phase Object

Phase Plate(Phase Contrast)

‘Blocked’(Dark Ground Imaging)

非破壊測定

“ 位相のシフト”

Post BEC の 1 つの流れ

Weakly 　　 Interacting

Strongly 　　 Interacting

局在による力学的エネルギーの上昇

重なりによる相互作用エネルギーの上昇

≪≫

平均場近似 強相関系

EinteractionEkinetic vs

How to make strongly correlated system with a dilute gas…..

1D Bose gases with infinite hard core interactions

Marvin Girardeau, 1960: 1D Bose gases with infinite hard core repulsion

Lewi Tonks, 1936: Eq. of state of a 1D classical gas of hard spheres

“Fermionization”

In 1D, if no two single particle wavefunctions overlap

bosons f ermionsψ =ψ

1D Bose gases with variable point-like interactions

2 2

1D 1D2 all allatoms pairs

H =- + g δ z2m z

Elliot Lieb and Werner Liniger, 1963: Exact solutions for 1D Bose gases with arbitrary (z) interactions

Solutions parameterized by >>1Tonks-Girardeaugas

<<1mean field theory (Thomas-Fermi gas)

large g1D

low density

small g1D

high density

kinetic energy dominates

mean field energy dominates

1D

21D

m g=

n(g1D > 0)

1D Bose atomic gases

3D2

1D

2 aa n

Maxim Olshanii, 1998: Adaptation to real atoms

when a 3D, n1D or a

a3D = 3D scattering length

a = transverse size of wavefunction

a

1D waveguide

Optical LatticesCalculable, versatile atom traps

1D:2D:

UAC IntensityFar from resonance,no light scattering

3D:

1D Bose gases

a

Adiabatic　Loading　from　BEC　into 2D　Lattice

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

2D　Lattice　Power ⇒ 　より強い閉じ込め　　⇒　

more　strongly　interacting

　　　　　　　　　　　　　　　　　　　　　　In　1DHigh　density　　　　　　　Low　density

weakly　　　　　　　　　　interacting

=　1D　System

a = (ħ / mω) 1/2

Bundles of 1D Systems

Independently adjust longitudinal and transverse trapping

Recall: when a or n1D

So when the lattice power or the dipole trap power

For 1D: negligible tunneling; all energies << ħω

Blue-detuned Lattice minimizes spontaneous emission

detuning 3.2THzw0~ 600 um

up to 85Erec

Expansion in the 1D tubes

50-300atoms/tube

1000-8000 tubes

0 ms

7 ms

17 ms

up

aspect ratio 150 ~ 700

Family of curves parameterized by

Tonks-Girardeau gas

1D quasi-BEC

weak couplin

g

strong couplin

g

0

0.5

1

1.5

2

0.4 0.7 1 4

MF

TG

Exact 1D

Kinoshita, Wenger, DSW, Science 305, 1125 (2004)

Normalized Local Pair Correlations

2g

eff

g(2) of the 3D BEC is 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 10

Expt: Kinoshita, Wenger, DSW, PRL 95 190406 (2005)

By photo-association

.3 3

Theory: Gangardt & Shlyapnikov, PRL 90 010401 (2003)

Pauli exclusion for Bosons

g(3), higher order correlationalso decreases

Strong coupling regime

FermionizedBosons !

Weak couplingregime

Summary (1st Half) Experiments with equilibrium 1D Bose gases across coupling regimes: total energy; cloud lengths, momentum distributions, local pair correlations

What happens when a 1D Bose gas is put into a Non-Equilibrium state ?

Does it thermalize ?

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

Collisions in 1D

For identical particles, reflection looks just like transmission !

Two-body collisions between distinct bosons cannot change their momentum distribution.

It will ergodically sample the entire phase space (E = const.)

Approach to a Thermal Equilibrium

Integrable systems never reach a thermal equilibrium (too many constrains)

Does a Real 1D Gas 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.

1D Bose gases with δ-fn interactions are integrable systems they do not: ergodically sample phase space

≈ become chaotic≈ thermalize pa, pb, pc pa, pb, pc

Creating Non-Equlibrium Distributions

1 standing wave pulse

2 standing wave pulses

Wang, et al., PRL 94, 090405 (2005)

Op

tica

l th

ickness

Position (μm)

Position (μm)

Op

tica

l th

ickness

Harmonic Trap Motion

x

v

A classical Newton’s cradle

We make thousands of parallel quantum Newton’s cradles, each with 50-300 oscillating atoms.

1D Evolution in a Harmonic Trap

15

30

195 ms

390 ms

1st cycle average

-500 5000Position (μm)

ms

0

5

10

40 μm

Kinoshita, Wenger, WeissNature 440, 900 (2006)

Dephased Momentum DistributionsO

pti

cal th

ickn

ess

(n

orm

aliz

ed

)

Position (μm)

1st cycle average15 distribution40 distribution(30 in A)

=18

=3.2

= 1.4

Project the evolution

Negligible ThermalizationProjected curves and actual curves at 30 or 40

After dephasing, the 1D gases reach a steady state that is not thermal equilibrium

A

C

Op

tical

Th

ickn

ess

Spatial Distribution (m)

B

A

C

Op

tical

Th

ickn

ess

Spatial Distribution (m)

B

Op

tica

l th

ickn

ess

(n

orm

aliz

ed

)

　　　 Position (μm)

=18

=3.2

= 1.4

Each atom continues to oscillate with its original amplitude

th

>390

>1910

>200

What happens in 3D?Thermalization occurs in ~3 collisions.

0 2 4 9

A

C

Opt

ical

Thi

ckne

ss

Spatial Distribution (m)

B

A

C

Opt

ical

Thi

ckne

ss

Spatial Distribution (m)

B

This many-body 1D system is nearly integrable.

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。

A New Type of Experiment : Direct Control of Non-Integrability

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。

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

Making 1D gases thermalizeTop view

Allow tunneling among tubes 1D 2D and 3D behavior

e. g. Ux = UY = 21 Erec

Jx

JY

1st cycle average

z (m)

15

z (m)

40

z (m)

Ux=UY =60 Erec

1st

1540

=3.2

Opti

cal th

ickn

ess

10 20 30 40 50

0.005

0.01

0.015

0.02

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

Lattice Depth (uK)

Fractio

n o

f energ

y in

1D

Ther

mal

izat

ion

Rate

(per

col

lisio

n)

Lattice Depth (Erec)

Thermalization in a 2D array of tubes

2-body collisions are well below threshold for transverse excitation.

no tails

equipartition

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?

Summary

Do Integral Systems Relax ?　

It will ergodically sample the entire phase space (E = const.)

Approach to a Thermal Equilibrium

Stories After our Experiments……

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)

Maximizing Entropy

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.

Rigol, Dunjko, Yurovsky and Olshanii,PRL, 98, 050405 (2007)

Discrete Momentum Sets are created byPeriodic Potentials.

In 1D system,

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)

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.

Control Non-Equilibrium process

1D System

熱平衡に近づかない系＋

初期宇宙 ブラックホール

Quantized Flux of Atoms

“ 擾乱”

1)

3) Atomic Flow in 1D Geometry (ongoing project)

Ongoing project

2) Attractively Interacting 1D System

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

Quantum Chaos (Billiard of Quantum Gas)

Quantum Turbulence

Non-Equilibrium Phenomena

Creation of Macroscopic Coherence

Quantum Gases Flowing in 2D Anti-Dot Lattices……(some of them are ongoing projects)

Current Status of my Lab.,,,,,