1. Non-equilibrium dynamics of cold atoms in optical lattices Vladimir Gritsev Harvard Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Bertrand Halperin Harvard Mikhail Lukin Harvard Eugene Demler Harvard Harvard-MIT CUA

2. Motivation: understanding transport phenomena in correlated electron systems e.g. transport near quantum phase transition

3. Superconductor to Insulator transition in thin films Marcovic et al., PRL 81:5217 (1998) Tuned by film thicknessTuned by magnetic field V.F. Gantmakher et al., Physica B 284-288, 649 (2000)

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

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

6. 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 quantum sine Gordon model v J

7. 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)

8. 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), …

9. Equilibrium superfluid to insulator transition t/U Superfluid Mott insulator Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98) Experiment: Greiner et al. Nature (01) U 

10. 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 by Eiermann et al, PRL (03)

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

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

13. 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

14. SFMI p U/J   Dynamical instability for integer filling Dynamical instability occurs for Vicinity of the SF-I quantum phase transition. Classical description applies for

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

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

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

18. SFMI p U/J   Dynamical instability Integer filling p   Fractional filling U/J

19. The first instability develops near the edges, where N=1 U=0.01 t J=1/4 Gutzwiller ansatz simulations (2D) Optical lattice and parabolic trap. Gutzwiller approximation

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

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

22. Decay rate from a metastable state. Example

23. At p  /2 we get 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

24. 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 Longitudinal stiffness is much smaller than the transverse.

25. SFMI 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)

26. This state becomes unstable at corresponding to the maximum of the current: 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 MIMI p U/J   Metastable current carrying state:

27. 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 SFMI p U/J   Strongly interacting regime. Vicinity of the SF-Mott transition Decay of current by quantum tunneling Action of a quantum phase slip in d=1,2,3

28. Decay of current by quantum tunneling

30. Decay of current by thermal activation phase j Escape from metastable state by thermal activation phase j Thermal phase slip EE

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

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

33. Dynamics of interacting bosonic systems probed in interference experiments

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

35. 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)

36. 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

37. Studying coherent dynamics of strongly interacting systems in interference experiments

38. 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

39. Coupled 1d systems J Relative phaseParticle number imbalance Conjugate variables Small K corresponds to strong quantum fluctuations

40. Quantum Sine-Gordon model Quantum Sine-Gordon model is exactly integrable Excitations of the quantum Sine-Gordon model Hamiltonian Imaginary time action soliton antisolitonmany types of breathers

41. 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

42. 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 Boundary Sine-Gordon Model space and time enter equivalently

43. 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 Boundary sine-Gordon model Coherence Matrix elements can be computed using form factor approach Smirnov (1992), Lukyanov (1997)

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

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

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

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

48. 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