Slide 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
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
Work at Penn State University with
Trevor Wenger
Prof. David S. Weiss
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
「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
量子力学特有な現象
多体効果
統計性の違い
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
位相空間密度
ldB n 3 ~ 1
ド･ブロイ波長 ~ 粒子間距離
ldB
(1/n)1/3
(n : density)
ldB n
3
~1
ド･ブロイ波長
∝ 1/√T
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 !
Laser Cooling & Trapping
Evaporative Cooling
Ref : 4He 2.17 K
・ドップラー冷却によるビームの減速
│e
>
│g
>
・磁気光学トラップ
（Magneto-Optical-Trap : MOT）
・偏向勾配冷却
到達温度：冷却能力＝加熱効果
・蒸発冷却 (from 低温物理学)
Density
１秒で数 mK ～数１０ mK
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
Energy
光双極子力
D
Induced Dipole :
Dipole in E :
光格子（Optical Lattice）
Udip ∝ −
I (r )
D
光双極子トラップ
D:
87
Rb 780 nm
1064 nm (YAG)
10.4 mm (CO2)
Uo
Uo : sub mK ~ mK
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
87
All-Optical BEC of Rb
-4
1s
Kinoshita, Wenger, Weiss,
Phys. Rev. A 71, 011602(R) (2005)
-2
-2
0
0
2
2
4
1.5 s
Evaporation times
2.0 s
3.5x105 BEC atoms every 3 s
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
全原子 in the Lowest Energy State
T=0K!
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
操作・制御
Achieved in 1995
2001 Nobel 賞
Time of Flight
吸収によるShadow Cast
原子集団の”運動量分布”
CCD
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.
破壊測定
In－situ Imaging
原子集団＝ Phase Object
“位相のシフト”
Phase Plate
(Phase Contrast)
‘Blocked’
(Dark Ground Imaging)
非破壊測定
Post BEC の1つの流れ
How to make strongly correlated system
with a dilute gas…..
Ekinetic
vs
≫
≪
Einteraction
Weakly
Interacting
Strongly
Interacting
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
平均場近似
強相関系
1D Bose gases with infinite
hard core interactions
Lewi Tonks, 1936: Eq. of state of a 1D
classical gas of hard spheres
Marvin Girardeau, 1960: 1D Bose gases
with infinite hard core repulsion
In 1D, if no two single particle wavefunctions overlap

ψbosons = ψfermions
“Fermionization”
1D Bose gases with variable pointlike interactions
Elliot Lieb and Werner Liniger, 1963: Exact
solutions for 1D Bose gases with arbitrary (z)
interactions
2
2
Solutions
m g1D
parameterized by  = 2
H1D = - 
all
atoms

+  g1Dδ  z 
2
2m z
all
n1D
>>1
Tonks-Girardeau
gas
<<1
mean field theory
(Thomas-Fermi gas)
kinetic energy
dominates
mean field energy
dominates
pairs
(g1D > 0)
large g1D
low density
small g1D
high density
1D Bose atomic gases
Maxim Olshanii, 1998: Adaptation to real atoms
a
2 a3D
 2
a n1D
a3D = 3D scattering length
a = transverse size
of wavefunction
 when a
3D,
n1D or a
1D waveguide
Optical Lattices
Calculable, versatile atom traps
Far from resonance,
no light scattering
UAC  Intensity
1D Bose gases
1D:
2D:
3D:
Adiabatic Loading from BEC into 2D 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
= 1D System
＋
チューブ内の軸方向に沿った緩やかな閉じ込め
2D Lattice Power
weakly
interacting
⇒ より強い閉じ込め
⇒
more strongly
interacting
In 1D
High density
Low density
Bundles of 1D Systems
detuning
3.2THz
w0~ 600 um
up to 85Erec
Blue-detuned Lattice
minimizes
spontaneous emission
For 1D: negligible tunneling;
all energies << ħω
Independently adjust
longitudinal and transverse
trapping
Recall:  when a or n1D
So  when the lattice power 
or the dipole trap power 
Expansion in the 1D tubes
0 ms
7 ms
17 ms
 up
50-300
atoms/tube
1000-8000
tubes
aspect ratio
150 ~ 700
Family of curves parameterized
by 
Science
Kinoshita, Wenger,
DSW,
305,
1125 (2004)
2
MF
1.5
1
TG
TonksGirardeau
gas
Exact 1D
0.5
1D
quasiBEC
0
0.4
weak
coupling
0.7
1
4

strong
coupling
Normalized Local Pair Correlations
By photo-association Theory: Gangardt & Shlyapnikov, PRL 90 010401 (2003)
g(2) of the
3D BEC is
1.
g
2
Expt: Kinoshita, Wenger, DSW, PRL 95 190406 (2005)
0.8
0.7
Strong coupling
regime
0.6
Pauli exclusion
for Bosons
0.5
0.4
0.3
0.2
0.1
0
Weak coupling .3
regime
1
 eff
3
10
Fermionized
Bosons !
g(3), higher
order
correlation
also
decreases
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
What happens when a 1D Bose gas is put
into a Non-Equilibrium state ?
Does it thermalize ?
Collisions in 1D
For identical particles, reflection looks just like transmission !

Two-body collisions between
distinct bosons cannot change
their momentum distribution.
Approach to a Thermal Equilibrium
It will ergodically sample the entire phase space (E = const.)
Integrable systems never reach
a thermal equilibrium
(too many constrains)
Does a Real 1D Gas Thermalize?
1D Bose gases with δ-fn interactions are
integrable systems  they do not:
ergodically sample phase space
≈ become chaotic
pa, pb, pc
≈ thermalize
pa, pb, pc
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.
1 standing
wave pulse
Optical thickness
Creating Non-Equlibrium
Distributions
2 standing
wave pulses
Wang, et al., PRL 94, 090405 (2005)
Optical thickness
Position (μm)
Position (μm)
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.
1D Evolution in a Harmonic Trap
ms
0
5
10
-500
Kinoshita, Wenger, Weiss
Nature 440, 900 (2006)
Position (μm)
0
500
40 μm
1st cycle average
15
30
195 ms
390 ms
Dephased Momentum
1 cycle average
Distributions
15 distribution
st
40 distribution
(30 in A)
Optical thickness
(normalized)
=18
=3.2
= 1.4
Position (μm)
Project the evolution
Negligible Thermalization
Optical thickness
(normalized)
Optical Thickness
Projected curves
and actual curves
at 30  or 40 
A
B
After dephasing,
=18
the 1D gases
th reach a steady
>390 state that is not
thermal
equilibrium
=3.2
>1910
C
= 1.4
>200
Spatial
Distribution
(mm)
Position
(μm)
Each atom
continues to
oscillate with
its original
amplitude
What happens in 3D?
Thermalization occurs in ~3 collisions.
0
2
4
9
Lack of Thermalization
A
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。
Optical Thickness
B
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。
C
This many-body 1D system
is nearly integrable.
Spatial Distribution (mm)
A New Type of Experiment : Direct Control of Non-Integrability
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 thermalize
Jx
Top view
JY
Allow tunneling among tubes  1D  2D and 3D behavior
Optical thickness
e. g. Ux = UY = 21 Erec
Ux=UY =60 Erec
=3.2
1st cycle average
15
40
1st
15
40
z (mm)
z (mm)
z (mm)
Thermalization in a 2D array of tubes
Lattice Depth (Erec)
20
30
40
50
0.02
1
0.8
0.015
0.6
0.01
0.4
0.005
0.2
0
equipartition
0
2
4
6
8
10
Lattice Depth (uK)
12
Fraction of energy in 1D
Thermalization Rate
(per collision)
10
no tails
2-body collisions are well
below threshold for
transverse excitation.
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?
Stories After our Experiments……
Do Integral Systems Relax ?
Approach to a Thermal Equilibrium
It will ergodically sample the entire phase space (E = const.)
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
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.
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)
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.
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)
初期宇宙
ブラックホール
Quantized Flux of Atoms
Quantum Gases Flowing in 2D Anti-Dot Lattices……
(some of them are ongoing projects)
Quantum Chaos
(Billiard of Quantum Gas)
Quantum Turbulence
Non-Equilibrium Phenomena
Creation of Macroscopic Coherence
Current Status of my Lab.,,,,,

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