Non-Equilibrium1D Bose Gases Integrability and Thermaliz ation Work at Penn State University with Trevor Wenger Prof. David S. Weiss Graduate School of Human and Environmental ies (Course of Studies on Material Science) Kyoto University and JST PRESTO Toshiya Kinoshita アアアアアアアアアアアアアアアアアアアアア アアアアアアア 「」・ 2009 ア 10 ア 23 - 24 ア
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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. - PowerPoint PPT Presentation
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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
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)