Cold Atoms Cold Atoms and and

Out of Equilibrium Quantum DynamicsOut of Equilibrium Quantum Dynamics

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

AFOSRAFOSR

Vladimir Gritsev – HarvardVladimir Gritsev – Harvard

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

UMASS, Boston, 11/21/2007 UMASS, Boston, 11/21/2007

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

Experimental examples:Experimental examples:

Optical Lattices:Optical Lattices:

OL are tunable (in OL are tunable (in real time): from real time): from weak modulations weak modulations to tight binding to tight binding regime.regime.

Can change Can change dimensionality and dimensionality and study 1D, 2D, and study 1D, 2D, and 3D physics.3D physics.

Both fermions and Both fermions and bosons.bosons.

I. Bloch, Nature Physics 1, 23 - 30 (2005)

Superfluid-insulator quantum phase transition in interacting bosons.Superfluid-insulator quantum phase transition in interacting bosons.

Repulsive Bose gas.

T. Kinoshita, T. Wenger, D. S. Weiss ., Science 305, 1125, 2004

Also, B. Paredes1, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T.W. Hänsch and I. Bloch, Nature 277 , 429 (2004)

M. Olshanii, 1998M. Olshanii, 1998

interaction parameterinteraction parameter

Lieb-Liniger, complete solution 1963.Lieb-Liniger, complete solution 1963.

Experiments:Experiments:

Energy vs. interaction strength: experiment and theory.

Kinoshita et. Al., Science 305, 1125, 2004

No adjustable parameters!

Local pair correlations.

Kinoshita et. Al., Science 305, 1125, 2004

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, …

In the continuum this system is equivalent to an integrable KdV In the continuum this system is equivalent to an integrable KdV equation. The solution splits into non-thermalizing solitons Kruskal equation. The solution splits into non-thermalizing solitons Kruskal and Zabusky (1965 ).and Zabusky (1965 ).

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 in1D.

Fast thermalization in 3D.

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

Example: crossing a QCP.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

Transverse field Ising model.Transverse field Ising model.

0 1z zi jg

1xig

Phase transition at Phase transition at g=g=11..

Critical exponents: Critical exponents: z=z===1 1 dd/(z/(z +1) +1)=1/2.=1/2. exnA.P. 2003, A.P. 2003, W. HW. H. . Zurek, U. Dorner, P. Zoller 2005, Zurek, U. Dorner, P. Zoller 2005, J. Dziarmaga 2005J. Dziarmaga 2005

Start at Start at ggand slowly ramp it to 0. Count the number of and slowly ramp it to 0. Count the number of domain wallsdomain walls

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

Hamiltonian of Goldstone modes: superfluids, phonons in solids, Hamiltonian of Goldstone modes: superfluids, phonons in solids, (anti)ferromagnets, …(anti)ferromagnets, …

In superfluids In superfluids is determined by the interactions. is determined by the interactions.

Imagine an experiment, where we start from a noninteracting Imagine an experiment, where we start from a noninteracting superfluid and slowly turn on interaction.superfluid and slowly turn on interaction.

Zero temperature regime:Zero temperature regime:

Assuming the system thermalizes at a fixed energyAssuming the system thermalizes at a fixed energy

EnergyEnergy

Finite TemperaturesFinite Temperatures

d=1,2d=1,2

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

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

Non-adiabatic regime!Non-adiabatic regime!dLTE 3/73/1

TSTE , d=3d=3

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

Use the fact that quantum Use the fact that quantum fluctuations are weak in the SF fluctuations are weak in the SF phase and expand dynamics in phase and expand dynamics in the effective Planck’s constant:the effective Planck’s constant:

JnU 0/

Nonintegrable model in all spatial dimensions, expect thermalization.Nonintegrable model in all spatial dimensions, expect thermalization.

)tanh()( 0 tUtU

Quantum expansion of dynamics:Quantum expansion of dynamics:

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)

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.

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