Nonequilibrium dynamics of cold atoms in optical lattices

Non-equilibrium dynamics of cold atoms in optical lattices Vladimir Gritsev Anatoli Polkovnikov Ehud Altman Bertrand Halperin Mikhail Lukin Eugene Demler Harvard/Boston University Harvard/Weizmann 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 Tuned by film thickness Tuned by magnetic field V. F. Gantmakher et al. , Physica B 284 -288, 649 (2000) Marcovic et al. , PRL 81: 5217 (1998)

Scaling near the superconductor to insulator transition Yazdani and Kapitulnik Phys. Rev. Lett. 74: 3037 (1995)

Breakdown of scaling near the superconductor to insulator transition Mason and Kapitulnik Phys. Rev. Lett. 82: 5341 (1999)

Outline v Current decay for interacting atoms in optical lattices. Connecting classical dynamical instability with quantum superfluid to Mott transition Phase dynamics of coupled 1 d condensates. Competition of quantum fluctuations and tunneling. Application of the exact solution of quantum sine Gordon model Conclusions 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 m Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98) Experiment: Greiner et al. Nature (01) U Superfluid Mott insulator t/U

Moving condensate in an optical lattice. Dynamical instability Theory: Niu et al. PRA (01), Smerzi et al. PRL (02) Experiment: Fallani et al. PRL (04) v Related experiments by Eiermann et al, PRL (03)

This talk: How to connect the dynamical instability (irreversible, classical) to the superfluid to Mott transition (equilibrium, quantum) p p/2 Unstable Stable ? ? ? SF This talk MI U/J p ? ? ? Possible experimental U/t sequence: SF MI

Dynamical instability Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation Current carrying states Linear stability analysis: States with p>p/2 are unstable Amplification of density fluctuations r

Dynamical instability for integer filling Order parameter for a current carrying state Current GP regime . Maximum of the current for When we include quantum fluctuations, the amplitude of the order parameter is suppressed decreases with increasing phase gradient .

Dynamical instability for integer filling p p/2 U/J SF Vicinity of the SF-I quantum phase transition. Classical description applies for Dynamical instability occurs for MI

Dynamical instability. Gutzwiller approximation Wavefunction Time evolution We look for stability against small fluctuations Phase diagram. Integer filling

Order parameter suppression by the current. Number state (Fock) representation Integer filling N-2 N-1 N N+1 N+2

Order parameter suppression by the current. Number state (Fock) representation Integer filling Fractional filling N-2 N-1 N N+1 N+2 N-3/2 N-1/2 N+1/2 N+3/2

Dynamical instability Integer filling Fractional filling p p p/2 U/J SF MI U/J

Optical lattice and parabolic trap. Gutzwiller approximation The first instability develops near the edges, where N=1 U=0. 01 t J=1/4 Gutzwiller ansatz simulations (2 D)

j phase Beyond semiclassical equations. Current decay by tunneling Current carrying states are metastable. They can decay by thermal or quantum tunneling Thermal activation Quantum tunneling j

phase Decay of current by quantum tunneling Quantum phase slip j j Escape from metastable state by quantum tunneling. WKB approximation S – classical action corresponding to the motion in an inverted potential.

Decay rate from a metastable state. Example

Weakly interacting systems. Quantum rotor model. Decay of current by quantum tunneling At p /2 we get 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 Longitudinal stiffness is much smaller than the transverse. The transverse size of the phase slip diverges near a phase slip. We can use continuum approximation to treat transverse directions

Weakly interacting systems. Gross-Pitaevskii regime. Decay of current by quantum tunneling p p/2 U/J SF MI Fallani et al. , PRL (04) Quantum phase slips are strongly suppressed in the GP regime

Strongly interacting regime. Vicinity of the SF-Mott transition p Close to a SF-Mott transition we can use an effective relativistivc GL theory (Altman, Auerbach, 2004) p/2 U/J SF M I Metastable current carrying state: This state becomes unstable at maximum of the current: corresponding to the

Strongly interacting regime. Vicinity of the SF-Mott transition Decay of current by quantum tunneling p p/2 U/J SF Action of a quantum phase slip in d=1, 2, 3 MI - correlation length 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

Decay of current by quantum tunneling

phase Decay of current by thermal activation Thermal phase slip j j DE Escape from metastable state by thermal activation

Thermally activated current decay. Weakly interacting regime DE Thermal phase slip Activation energy in d=1, 2, 3 Thermal fluctuations lead to rapid decay of currents Crossover from thermal to quantum tunneling

Decay of current by thermal fluctuations Phys. Rev. Lett. (2004)

Dynamics of interacting bosonic systems probed in interference experiments

Interference of two independent condensates Andrews et al. , Science 275: 637 (1997)

Interference experiments with low d condensates 1 D condensates: Schmiedmayer et al. , Nature Physics (2005, 2006) Transverse imaging Longitudial imaging trans. imaging long. imaging 2 D condensates: Hadzibabic et al. , Nature 441: 1118 (2006) z Time of flight x

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 finite or zero tunneling Measure time evolution of fringe amplitudes

Studying coherent dynamics of strongly interacting systems in interference experiments

Coupled 1 d 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 1 d systems J Conjugate variables Relative phase Particle number imbalance Small K corresponds to strong quantum fluctuations

Quantum Sine-Gordon model Hamiltonian Imaginary time action Quantum Sine-Gordon model is exactly integrable Excitations of the quantum Sine-Gordon model soliton antisoliton many types of breathers

Dynamics of quantum sine-Gordon model Hamiltonian formalism Initial state Quantum action in space-time Initial state provides a boundary condition at t=0 Solve as a boundary sine-Gordon model

Boundary sine-Gordon model Exact solution due to Ghoshal and Zamolodchikov (93) Applications to quantum impurity problem: Fendley, Saleur, Zamolodchikov, Lukyanov, … Limit enforces boundary condition Sine-Gordon + boundary condition in space Boundary Sine-Gordon Model Sine-Gordon + boundary condition in time two coupled 1 d BEC quantum impurity problem space and time enter equivalently

Boundary sine-Gordon model Initial state is a generalized squeezed state creates solitons, breathers with rapidity q creates even breathers only Matrix and are known from the exact solution of the boundary sine-Gordon model Time evolution Coherence Matrix elements can be computed using form factor approach Smirnov (1992), Lukyanov (1997)

Quantum Josephson Junction Limit of quantum sine-Gordon model when spatial gradients are forbidden Initial state Eigenstates of the quantum Jos. junction Hamiltonian are given by Mathieu’s functions Time evolution Coherence

Dynamics of quantum Josephson Junction Power spectrum power spectrum w E 2 -E 0 Main peak “Higher harmonics” Smaller peaks E 4 -E 0 E 6 -E 0

Dynamics of quantum sine-Gordon model Coherence Main peak “Higher harmonics” Smaller peaks Sharp peaks

Dynamics of quantum sine-Gordon model power spectrum w main peak “higher harmonics” smaller peaks sharp peaks (oscillations without decay)

Conclusions Dynamic instability is continuously connected to the quantum SF-Mott transition. Quantum and thermal fluctuations are important Interference experiments can be used to do spectroscopy of the quantum sine-Gordon model