Slow dynamics in gapless low-dimensional Slow dynamics in gapless low-dimensional systemssystems

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

AFOSRAFOSR

Vladimir Gritsev – HarvardVladimir Gritsev – Harvard

Ehud Altman -Ehud Altman -WeizmannWeizmannEugene Demler – HarvardEugene Demler – HarvardBertrand Halperin - HarvardBertrand Halperin - HarvardMisha Lukin -Misha Lukin - HarvardHarvard

CMT Seminar, CMT Seminar, Yale, 11/08/2007 Yale, 11/08/2007

Cold atoms:Cold atoms:(controlled and tunable Hamiltonians, isolation from environment)(controlled and tunable Hamiltonians, isolation from environment)

1. Equilibrium thermodynamics:1. Equilibrium thermodynamics:Quantum simulations of equilibrium Quantum simulations of equilibrium condensed matter systemscondensed matter systems

2. Quantum dynamics:2. Quantum dynamics:

Coherent and incoherent dynamics, Coherent and incoherent dynamics, integrability, quantum chaos, …integrability, quantum chaos, …

2. Quantum dynamics:2. Quantum dynamics:

Coherent and incoherent dynamics, Coherent and incoherent dynamics, integrability, quantum chaos, …integrability, quantum chaos, …

Cold atoms:Cold atoms:(controlled and tunable Hamiltonians, isolation from environment)(controlled and tunable Hamiltonians, isolation from environment)

Qauntum Newton Craddle.(collisions in 1D interecating Bose gas – Lieb-Liniger model)

T. Kinoshita, T. R. Wenger and D. S. Weiss, Nature 440, 900 – 903 (2006)

No thermalization during collisions of two one-dimensional clouds of interacting bosons.

Fast thermalization if the clouds are three dimensional.

Quantum analogue of the Fermi-Pasta-Ulam problem.

3. = 1+2 Nonequilibrium thermodynamics?3. = 1+2 Nonequilibrium thermodynamics?

Cold atoms:Cold atoms:(controlled and tunable Hamiltonians, isolation from environment)(controlled and tunable Hamiltonians, isolation from environment)

1. Equilibrium thermodynamics:1. Equilibrium thermodynamics:Quantum simulations of equilibrium Quantum simulations of equilibrium condensed matter systemscondensed matter systems

2. Quantum dynamics:2. Quantum dynamics:

Coherent and incoherent dynamics, Coherent and incoherent dynamics, integrability, quantum chaos, …integrability, quantum chaos, …

Adiabatic process.Adiabatic process.

Assume no first order phase transitions.Assume no first order phase transitions.

Adiabatic theorem:Adiabatic theorem:

““Proof”:Proof”: thenthen

Adiabatic theorem for integrable systems.Adiabatic theorem for integrable systems.

Density of excitationsDensity of excitations

Energy density (good both for integrable and nonintegrable Energy density (good both for integrable and nonintegrable systems:systems:

EEBB(0) is the energy of the state adiabatically connected to (0) is the energy of the state adiabatically connected to

the state A. the state A. For the cyclic process in isolated system this statement For the cyclic process in isolated system this statement implies no work done at small implies no work done at small ..

Adiabatic theorem in quantum mechanicsAdiabatic theorem in quantum mechanics

Landau Zener process:Landau Zener process:

In the limit In the limit 0 transitions between 0 transitions between different energy levels are suppressed.different energy levels are suppressed.

This, for example, implies reversibility (no work done) in a This, for example, implies reversibility (no work done) in a cyclic process.cyclic process.

Adiabatic theorem in QM Adiabatic theorem in QM suggestssuggests adiabatic theorem adiabatic theorem in thermodynamics:in thermodynamics:

Is there anything wrong with this picture?Is there anything wrong with this picture?

HHint: low dimensions. Similar to Landau expansion in the int: low dimensions. Similar to Landau expansion in the order parameter.order parameter.

1.1. Transitions are unavoidable in large gapless systems.Transitions are unavoidable in large gapless systems.

2.2. Phase space available for these transitions decreases with Phase space available for these transitions decreases with Hence expectHence expect

More specific reason.More specific reason.

Equilibrium: high density of low-energy states Equilibrium: high density of low-energy states

•strong quantum or thermal fluctuations, strong quantum or thermal fluctuations, •destruction of the long-range order,destruction of the long-range order,•breakdown of mean-field descriptions, breakdown of mean-field descriptions,

Dynamics Dynamics population of the low-energy states due to finite rate population of the low-energy states due to finite rate breakdown of the adiabatic approximation.breakdown of the adiabatic approximation.

This talk: three regimes of response to the slow ramp:This talk: three regimes of response to the slow ramp:

A.A. Mean field (analytic) – high dimensions: Mean field (analytic) – high dimensions:

B.B. Non-analytic – low dimensionsNon-analytic – low dimensions

C.C. Non-adiabatic – lower dimensionsNon-adiabatic – lower dimensions

Some examples.Some examples.

1. Gapless critical phase (superfluid, magnet, crystal, …).1. Gapless critical phase (superfluid, magnet, crystal, …).

LZ condition:LZ condition:

Second example: crossing a QCP.Second example: crossing a QCP.

tuning parameter tuning parameter

gap

gap t, t, 0 0

Gap vanishes at the transition. Gap vanishes at the transition. No true adiabatic limit!No true adiabatic limit!

How does the number of excitations scale with How does the number of excitations scale with ? ?

A.P. 2003A.P. 2003

Perturbation theory (linear response). Perturbation theory (linear response). (A.P. 2003)(A.P. 2003)

Expand the wave-function in many-body basis.Expand the wave-function in many-body basis.

i Ht

Substitute into SchrSubstitute into Schrödinger equation.ödinger equation.

Uniform system: can characterize excitations by momentum:Uniform system: can characterize excitations by momentum:

Use scaling relations:Use scaling relations:

Find:Find:

Transverse field Ising model.Transverse field Ising model.

0 1z zi jg

1xig

There is a phase transition at There is a phase transition at g=g=11..

This problem can be exactly solved using Jordan-Wigner This problem can be exactly solved using Jordan-Wigner transformation:transformation:

† † †

1

2 1, ( 1) ( )x zi i i i j j j j

j i

c c c c c c

SpectrumSpectrum::

Critical exponents: Critical exponents: z=z===1 1 dd/(z/(z +1) +1)=1/2.=1/2.

Correct result (J. Dziarmaga 2005):Correct result (J. Dziarmaga 2005): 0.11exn

0.18exn

Linear response (Fermi Golden Rule):Linear response (Fermi Golden Rule):

A. P., 2003A. P., 2003

Interpretation as the Kibble-Zurek mechanism: Interpretation as the Kibble-Zurek mechanism: W. HW. H. . Zurek, U. Dorner, Peter Zoller, 2005Zurek, U. Dorner, Peter Zoller, 2005

Possible breakdown of the Fermi-Golden rule (linear Possible breakdown of the Fermi-Golden rule (linear response) scaling due to bunching of bosonic excitations.response) scaling due to bunching of bosonic excitations.

)( 2qq sq

Zero temperature.Zero temperature.

Most divergent regime: Most divergent regime:

Agrees with the linear response.Agrees with the linear response.

Assuming the system thermalizes Assuming the system thermalizes

Finite temperatures.Finite temperatures.

Instead of wave function use density matrix (Wigner form).Instead of wave function use density matrix (Wigner form).

ResultsResults

d=1,2d=1,2

d=1;d=1; d=2;d=2;

d=3d=3

Artifact of the quadratic approximation or the real result?Artifact of the quadratic approximation or the real result?

Non-adiabatic Non-adiabatic regime!regime!

Numerical verification (bosons on a lattice).Numerical verification (bosons on a lattice).

Use the fact that quantum fluctuations are weak and Use the fact that quantum fluctuations are weak and expand dynamics in the effective Planck’s constant expand dynamics in the effective Planck’s constant (saddle point parameter) (saddle point parameter) JnU 0/

Classical limit – use Gross-Pitaevskii equations with Classical limit – use Gross-Pitaevskii equations with initial conditions distributed according to the thermal initial conditions distributed according to the thermal density matrix.density matrix.

How do we add quantum corrections?How do we add quantum corrections?

We have two fields propagating in time forward and backward .We have two fields propagating in time forward and backward .

Idea: expand quantum evolution in powers of Idea: expand quantum evolution in powers of ..

Take an arbitrary observableTake an arbitrary observable

Treat Treat exactly, while expand in powers of exactly, while expand in powers of ..

Results:Results:

Leading order in Leading order in : start from random initial conditions distributed : start from random initial conditions distributed according to the Wigner transform of the density matrix and according to the Wigner transform of the density matrix and propagate them classically (propagate them classically (truncated Wigner approximationtruncated Wigner approximation):):

•Expectation value is substituted by the average Expectation value is substituted by the average over the initial conditions. over the initial conditions.

•Exact for harmonic theories! Exact for harmonic theories!

•Not limited by low temperatures!Not limited by low temperatures!

•Asymptotically exact at short times.Asymptotically exact at short times.

Subsequent orders: quantum scattering events (quantum jumps)Subsequent orders: quantum scattering events (quantum jumps)

Results (1d, L=128)Results (1d, L=128)

3/13/4 LTE PredictionsPredictions::

finite temperaturefinite temperature

2/1E

zero temperaturezero temperature

0.1 1

0.1

1

Ene

rgy

dens

ity

TWA Quantum correction

T=0.02T=0.02

3/13/4 LTE

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

t=0 t=3.2/ t=12.8/ t=28.8/ t=51.2/ t=80/ Thermall

a ja 0

L/ sin(j/L)

Correlation Functions

Thermalization at long times.Thermalization at long times.

2D, T=0.22D, T=0.2

3/13/1 LTE

Conclusions.Conclusions.

A.A. Mean field (analytic): Mean field (analytic):

B.B. Non-analyticNon-analytic

C.C. Non-adiabaticNon-adiabatic

Three generic regimes of a system response to a slow ramp:Three generic regimes of a system response to a slow ramp:

Open questions: general fate of linear response Open questions: general fate of linear response at low dimensions, non-uniform perturbations,…at low dimensions, non-uniform perturbations,…

Superfluid Mott insulator

Adiabatic increase of lattice potential

M. Greiner et. al., Nature (02)

What happens if there is a current in the superfluid?What happens if there is a current in the superfluid?

???

p

U/J

Stable

Unstable

SF MI

p

SF MI

U/J???

possible experimental sequence: ~lattice potential

Drive a slowly moving superfluid towards MI.Drive a slowly moving superfluid towards MI.

Include quantum depletion.Include quantum depletion.

Equilibrium:

Current state:coseffJ J J p

0.0 0.1 0.2 0.3 0.4 0.5

p*

I(p)

s(p)

sin(p)

Condensate momentum p/

( )sinsI p p

With quantum depletion the current state is unstable at

* / 2.p p

p

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=3

d=2

d=1

unstable

stable

U/Uc

p/

Meanfield (Gutzwiller ansatzt) phase diagram

Is there current decay below the instability?Is there current decay below the instability?

Role of fluctuations

Below the mean field transition superfluid current can decay via quantum tunneling or thermal decay .

E

p

Phase slip

1D System.

5/ 2

exp 7.12

JNp

U

7.1 – variational result

JNU

N~1

Large N~102-103

semiclassical parameter (plays the role of 1/ )

Fallani et. al., 2004C.D. Fertig et. al., 2004

Higher dimensions.

Longitudinal stiffness is much smaller than the transverse.

Need to excite many chains in order to create a phase slip.

12

r p

|| cos ,J J p

J J

r

6

2

2

d

d d

JNS C p

U

Phase slip tunneling is more expensive in higher dimensions:

expd dS

Stability phase diagram

3dS

Crossover1 3dS

Stable

1dS Unstable

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

unstable

stable

U/Uc

p/

Current decay in the vicinity of the superfluid-insulator transition

Use the same steps as before to obtain the asymptotics:

5

23

1 3 , expd

dd dd

CS p S

32

1 2

12

2

3

5.71 3

3.21 3

4.3

S p

S p

S

Discontinuous change of the decay rate across the meanfield transition. Phase diagram is well defined in 3D!

Large broadening in one and two dimensions.Large broadening in one and two dimensions.

Detecting equilibrium SF-IN transition boundary in 3D.

p

U/J

Superfluid MI

Extrapolate

At nonzero current the SF-IN transition is irreversible: no restoration of current and partial restoration of phase coherence in a cyclic ramp.

Easy to detect nonequilibrium Easy to detect nonequilibrium irreversible transition!!irreversible transition!!

J. Mun, P. Medley, G. K. Campbell, L. G. Marcassa, D. E. Pritchard, W. Ketterle, 2007

Conclusions.Conclusions.

A.A. Mean field (analytic): Mean field (analytic):

B.B. Non-analyticNon-analytic

C.C. Non-adiabaticNon-adiabatic

Three generic regimes of a system response to a slow ramp:Three generic regimes of a system response to a slow ramp:

Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition.

Quantum fluctuations

Depletion of the condensate. Reduction of the critical current. All spatial dimensions.

mean field beyond mean field

Broadening of the mean field transition. Low dimensions

Density of excitations:Density of excitations:

Energy density:Energy density:

Agrees with the linear Agrees with the linear response.response.