Using dynamics for optical lattice simulations.Using dynamics for optical lattice simulations.

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

AFOSRAFOSR

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

gBECi and OLE/MURI MeetinggBECi and OLE/MURI Meeting

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

a)a) Simulation of phases and phase-transitions for complicated Simulation of phases and phase-transitions for complicated many-particle systems.many-particle systems.

b)b) Testing various analytical and numerical approaches to Testing various analytical and numerical approaches to many-body problems.many-body problems.

c)c) Better understanding and engineering strongly correlated Better understanding and engineering strongly correlated materials.materials.

2. Quantum dynamics:2. Quantum dynamics:

Quantum simulation of behavior of Quantum simulation of behavior of non-equilibrium many-body systems. non-equilibrium many-body systems.

Importance: current technology approaches quantum limits.Importance: current technology approaches quantum limits.

Challenges: Challenges: experimental:experimental: hard to realize solid state systems sufficiently hard to realize solid state systems sufficiently isolated from environment; isolated from environment; theoretical: theoretical: huge (exponentially large) Hilbert space, lack of huge (exponentially large) Hilbert space, lack of methods.methods.

Potential: Potential: • understanding fundamental problems related to integrability, understanding fundamental problems related to integrability, thermalization, quantum chaos, quantum measurement, …thermalization, quantum chaos, quantum measurement, …• using out of equilibrium effects as a tool to simulate using out of equilibrium effects as a tool to simulate equilibrium properties of interacting systems.equilibrium properties of interacting systems.

This talk.This talk.

1.1. Phase diagram of a moving condensate in an Phase diagram of a moving condensate in an optical lattice.optical lattice.

2.2. Response of generic gapless systems to slow ramp Response of generic gapless systems to slow ramp of external parameters.of external parameters.

3.3. Quench dynamics in coupled one dimensional Quench dynamics in coupled one dimensional condensates.condensates.

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

Related questions in superconductivity

Reduction of TC and the critical current in superconducting wires

Webb and Warburton, PRL (1968)

Theory (thermal phase slips) in 1D:

Langer and Ambegaokar, Phys. Rev. (1967)McCumber and Halperin, Phys Rev. B (1970)

Theory in 3D at small currents:

Langer and Fisher, Phys. Rev. Lett. (1967)

1D System.

5/ 2

exp 7.12

JNp

U

N~1

Large N~102-103

7.1 – variational result

JNU

semiclassical parameter (plays the role of 1/ )

Fallani et. al., 2004

Experiment: C.D. Fertig et. al., 2004

Numerical prediction: A.P. & D.-W. Numerical prediction: A.P. & D.-W. Wang, 2003Wang, 2003

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

5

23

1 3 , expd

dd dd

CS p S

3.43 S

Discontinuous change of the decay rate across the mean field 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

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.

Three regimes of response to the slow ramp: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. 2003)(A.P. 2003)

Transverse field Ising model (A.P. 2003, W. HTransverse field Ising model (A.P. 2003, W. H. . Zurek, U. Zurek, U. Dorner, P. Zoller 2005, J. Dziarmaga 2005).Dorner, P. Zoller 2005, J. Dziarmaga 2005).

exn

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

Example: harmonic system (e.g. superfluid)Example: harmonic system (e.g. superfluid)

Zero temperature. Start from noninteracting Bose gas) Zero temperature. Start from noninteracting Bose gas)

Finite temperatures.Finite temperatures.

d=1,2d=1,2 Non-adiabatic regime!Non-adiabatic regime!

Non-analytic regime!Non-analytic regime!

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

Use expansion in quantum fluctuations to do large scale Use expansion in quantum fluctuations to do large scale precise numerical simulations (TWA + corrections).precise numerical simulations (TWA + corrections).

3/13/4 LTE 2/1E

Results.Results.

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

Quench experiments in 1D and 2D systems:Quench experiments in 1D and 2D systems:

T. Schumm . et. al., Nature Physics 1, 57 - 62 (01 Oct 2005)

Study dephasing as a function of time. What sort of Study dephasing as a function of time. What sort of information can we get?information can we get?

Analyze dynamics of phase coherence:Analyze dynamics of phase coherence:

1 2( ( ) ( )) ( )( ) e ei t t i tf t

Idea: extract energies of excited states Idea: extract energies of excited states and thus go beyond static probes.and thus go beyond static probes.

Relevant Sine-Gordon model (many applications in CM):Relevant Sine-Gordon model (many applications in CM):

Analogy with a Josephson junction.Analogy with a Josephson junction.EE

nn

Higher breathersHigher breathersLowest breathers: Lowest breathers: massive quasiparticlesmassive quasiparticles

SimulationsSimulations

0 10 20 30 40 500.80

0.85

0.90

0.95

1.00N=1, U=0.5, J=1

Pha

se C

oher

ence

Time

Hubbard model, 2x6 sitesHubbard model, 2x6 sites

0.0 0.2 0.4 0.60.00

0.01

0.02

0.03

0.04

0.05

0.06N=1, U=0.5, J=1, J=0.1, M=6

Po

we

r S

pe

ctru

m

Frequency

bb0202

bb2424

bb4646 bb0404bb26262b2b0101 2b2b0202

Conclusions.Conclusions.

1.1. Phase diagram of a moving superfluid in optical lattices:Phase diagram of a moving superfluid in optical lattices:a)a) Quantum (and thermal) phase slips in low dimensions.Quantum (and thermal) phase slips in low dimensions.b)b) Accurate probe of the equilibrium phase diagram in Accurate probe of the equilibrium phase diagram in

high dimensions.high dimensions.

2.2. Slow dynamics in gapless systems:Slow dynamics in gapless systems:a)a) universal response near second-order phase transitions universal response near second-order phase transitions

(critical exponents, KZ mechanism, …).(critical exponents, KZ mechanism, …).b)b) possible breaking of the adiabatic limit in low possible breaking of the adiabatic limit in low

dimensions. It is crucial to have large scale isolated dimensions. It is crucial to have large scale isolated systems for experimental verification.systems for experimental verification.

3.3. Possibility of probing spectral properties of integrable and Possibility of probing spectral properties of integrable and weakly non-integrable models in quench experiments.weakly non-integrable models in quench experiments.