Non-equilibrium dynamics of cold atoms in optical lattices

Vladimir Gritsev HarvardAnatoli Polkovnikov Harvard/Boston UniversityEhud Altman Harvard/WeizmannBertrand Halperin HarvardMikhail Lukin HarvardEugene Demler Harvard

Harvard-MIT CUA

Motivation: understanding transport phenomena in correlated electron systems

e.g. transport near quantum phase transition

Superconductor to Insulator transition in thin films

Marcovic et al., PRL 81:5217 (1998)

Tuned by film thickness Tuned by magnetic field

V.F. Gantmakher et al., Physica B 284-288, 649 (2000)

Yazdani and KapitulnikPhys.Rev.Lett. 74:3037 (1995)

Scaling near the superconductor to insulator transition

Mason and KapitulnikPhys. Rev. Lett. 82:5341 (1999)

Breakdown of scaling near the superconductor to insulator transition

Outline

Current decay for interacting atoms in optical lattices. Connecting classical dynamical instability with quantum superfluid to Mott transition

Conclusions

Phase dynamics of coupled 1d condensates.Competition of quantum fluctuations and tunneling.Application of the exact solution of quantumsine Gordon model

v

J

Current decay for interacting atoms in optical lattices

Connecting classical dynamical instability with quantum superfluid to Mott transition

References:

J. Superconductivity 17:577 (2004)Phys. Rev. Lett. 95:20402 (2005)Phys. Rev. A 71:63613 (2005)

Atoms in optical lattices. Bose Hubbard model

Theory: Jaksch et al. PRL 81:3108(1998)

Experiment: Kasevich et al., Science (2001) Greiner et al., Nature (2001) Cataliotti et al., Science (2001) Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004), …

Equilibrium superfluid to insulator transition

1n

t/U

SuperfluidMott insulator

Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)Experiment: Greiner et al. Nature (01)

U

Moving condensate in an optical lattice. Dynamical instability

v

Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)Experiment: Fallani et al. PRL (04)

Related experiments byEiermann et al, PRL (03)

This talk: This talk: How to connectHow to connect the the dynamical instabilitydynamical instability (irreversible, classical) (irreversible, classical)to the to the superfluid to Mott transitionsuperfluid to Mott transition (equilibrium, quantum) (equilibrium, quantum)

U/t

p

SF MI

???Possible experimental

sequence:

Unstable

???

p

U/J

Stable

SF MI

This talk

Linear stability analysis: States with p> are unstable

Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation

Current carrying states

r

Dynamical instability

Amplification ofdensity fluctuations

unstableunstable

GP regime . Maximum of the current for .

When we include quantum fluctuations, the amplitude of the order parameter is suppressed

Dynamical instability for integer filling

decreases with increasing phase gradient

Order parameter for a current carrying state

Current

SF MI

p

U/J

Dynamical instability for integer filling

0.0 0.1 0.2 0.3 0.4 0.5

p*

I(p)

s(p)

sin(p)

Condensate momentum p/

Dynamical instability occurs for

Vicinity of the SF-I quantum phase transition. Classical description applies for

Dynamical instability. Gutzwiller approximation

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/

Wavefunction

Time evolution

Phase diagram. Integer filling

We look for stability against small fluctuations

Order parameter suppression by the current. Number state (Fock) representation

Integer filling

N N+1N-1 N+2N-2

N N+1N-1 N+2N-2

Order parameter suppression by the current. Number state (Fock) representation

Integer filling

N-1/2 N+1/2N-3/2 N+3/2

N-1/2 N+1/2N-3/2 N+3/2

N N+1N-1 N+2N-2

N N+1N-1 N+2N-2

Fractional filling

SF MI

p

U/J

Dynamical instability

Integer filling

p

Fractional filling

U/J

The first instability develops near the edges, where N=1

0 100 200 300 400 500

-0.2

-0.1

0.0

0.1

0.2

0.00 0.17 0.34 0.52 0.69 0.86

Cen

ter

of M

ass

Mom

entu

m

Time

N=1.5 N=3

U=0.01 tJ=1/4

Gutzwiller ansatz simulations (2D)

Optical lattice and parabolic trap. Gutzwiller approximation

Beyond semiclassical equations. Current decay by tunneling

pha

se

jpha

se

jpha

se

j

Current carrying states are metastable. They can decay by thermal or quantum tunneling

Thermal activation Quantum tunneling

S – classical action corresponding to the motion in an inverted potential.

Decay of current by quantum tunnelingp

hase

j

Escape from metastable state by quantum tunneling.

WKB approximation

pha

se

j

Quantumphase slip

Decay rate from a metastable state. Example

0

22 3

0

1 ( ) 0

2 c

dxS d x bx p p

m d

2

1

1 2 cos

2j

j jj

dS d JN

U d

At p/2 we get

2

2 3

1 1

1 cos

2 3j

j j j jj

d JNS d JN p

U d

j jpj

Weakly interacting systems. Quantum rotor model.Decay of current by quantum tunneling

For the link on which the QPS takes place

d=1. Phase slip on one link + response of the chain.Phases on other links can be treated in a harmonic approximation

For d>1 we have to include transverse directions. Need to excite many chains to create a phase slip

The transverse size of the phase slip diverges near a phase slip. We can use continuum approximation to treat transverse directions

|| cos ,J J p

J J

Longitudinal stiffness is much smaller than the transverse.

SF MI

p

U/J

Weakly interacting systems. Gross-Pitaevskii regime.Decay of current by quantum tunneling

Quantum phase slips are strongly suppressed in the GP regime

Fallani et al., PRL (04)

This state becomes unstable at corresponding to the

maximum of the current:

13cp

2 2 21 .I p p p

Close to a SF-Mott transition we can use an effective relativistivc GL theory (Altman, Auerbach, 2004)

Strongly interacting regime. Vicinity of the SF-Mott transition

SF MI

p

U/J

Metastable current carrying state:2 21 ip xp e

Strong broadening of the phase transition in d=1 and d=2

is discontinuous at the transition. Phase slips are not important.Sharp phase transition

- correlation length

SF MI

p

U/J

Strongly interacting regime. Vicinity of the SF-Mott transitionDecay of current by quantum tunneling

Action of a quantum phase slip in d=1,2,3

Decay of current by quantum tunneling

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/

Decay of current by thermal activationp

hase

j

Escape from metastable state by thermal activation

pha

se

j

Thermalphase slip

E

Thermally activated current decay. Weakly interacting regime

E

Activation energy in d=1,2,3

Thermal fluctuations lead to rapid decay of currents

Crossover from thermal to quantum tunneling

Thermalphase slip

Phys. Rev. Lett. (2004)

Decay of current by thermal fluctuations

Dynamics of interacting bosonic systemsprobed in interference experiments

Interference of two independent condensates

Andrews et al., Science 275:637 (1997)

Interference experiments with low d condensates

2D condensates: Hadzibabic et al., Nature 441:1118 (2006)

x

Time of

flight

z

long. imaging

trans.imaging

Longitudial imaging Transverse imaging

1D condensates: Schmiedmayer et al., Nature Physics (2005,2006)

Studying dynamics using interference experiments Motivated by experiments and discussions with

Bloch, Schmiedmayer, Oberthaler, Ketterle, Porto, Thywissen

J

Prepare a system by splitting one condensate

Take to the regime of finiteor zero tunneling Measure time evolution

of fringe amplitudes

Studying coherent dynamics of strongly interacting systemsin interference experiments

Coupled 1d systems

J

Interactions lead to phase fluctuations within individual condensates

Tunneling favors aligning of the two phases

Interference experiments measure only the relative phase

Coupled 1d systems

J

Relative phase Particle number imbalance

Conjugate variables

Small K corresponds to strong quantum fluctuations

Quantum Sine-Gordon model

Quantum Sine-Gordon model is exactly integrable

Excitations of the quantum Sine-Gordon model

Hamiltonian

Imaginary time action

soliton antisoliton many types of breathers

Dynamics of quantum sine-Gordon model

Hamiltonian formalism

Quantum action in space-time

Initial state

Initial state provides a boundary condition at t=0

Solve as a boundary sine-Gordon model

Boundary sine-Gordon model

Limit enforces boundary condition

Exact solution due to Ghoshal and Zamolodchikov (93)Applications to quantum impurity problem: Fendley, Saleur, Zamolodchikov, Lukyanov,…

Sine-Gordon+ boundary condition in space

quantum impurity problem

Sine-Gordon+ boundary condition in time

two coupled 1d BEC

BoundarySine-GordonModel

space and timeenter equivalently

Initial state is a generalized squeezed state

creates solitons, breathers with rapidity

creates even breathers only

Matrix and are known from the exact solutionof the boundary sine-Gordon model

Time evolution

Boundary sine-Gordon model

Coherence

Matrix elements can be computed using form factor approachSmirnov (1992), Lukyanov (1997)

Quantum Josephson Junction

Initial state

Limit of quantum sine-Gordon model when spatial gradientsare forbidden

Time evolution

Eigenstates of the quantum Jos. junction Hamiltonian are given by Mathieu’s functions

Coherence

E2-E0 E4-E0

E6-E0

powerspectrum

Dynamics of quantum Josephson Junction

Main peak

Smaller peaks

“Higher harmonics”

Power spectrum

Dynamics of quantum sine-Gordon model

Coherence

Main peak

“Higher harmonics”

Smaller peaks

Sharp peaks

Dynamics of quantum sine-Gordon model

powerspectrum

main peak

“higher harmonics”

smaller peaks

sharp peaks (oscillations without decay)

Conclusions

Dynamic instability is continuously connected to thequantum SF-Mott transition. Quantum and thermalfluctuations are important

Interference experiments can be used to do spectroscopy of the quantum sine-Gordon model