1. Superfluid insulator transition in a moving condensate Anatoli Polkovnikov Harvard University Ehud Altman, Eugene Demler, Bertrand Halperin, Misha Lukin

2. Plan of the talk 1.General motivation and overview. 2.Bosons in optical lattices. Equilibrium phase diagram. Examples of quantum dynamics. 3.Superfluid-insulator transition in a moving condensate. Qualitative picture Non-equilibrium phase diagram. Role of quantum fluctuations 4.Conclusions and experimental implications.

3. Why is the physics of cold atoms interesting? It is possible to realize strongly interacting systems, both fermionic and bosonic. Parameters of the Hamiltonian are well known and well controlled. One can address not only conventional thermodynamic questions but also problems of quantum dynamics far from equilibrium. No coupling to the environment.

4. Interacting bosons in optical lattices. Highly tunable periodic potentials with no defects.

5. Equilibrium system. Interaction energy (two-body collisions): E int is minimized when N j =N=const: Interaction suppresses number fluctuations and leads to localization of atoms.

6. Equilibrium system. Kinetic (tunneling) energy: Kinetic energy is minimized when the phase is uniform throughout the system.

7. Classically the ground state will have uniform density and a uniform phase. However, number and phase are conjugate variables. They do not commute: There is a competition between the interaction leading to localization and tunneling leading to phase coherence.

8. Ground state is a superfluid: (M.P.A. Fisher, P. Weichman, G. Grinstein, D. Fisher, 1989) Superfluid-insulator quantum phase transition. Strong tunneling Weak tunneling Ground state is an insulator:

9. M. Greiner et. al., Nature (02) SuperfluidMott insulator Adiabatic increase of lattice potential Observe: Measurement: time of flight imaging

10. Nonequilibrium phase transitions wait for time t Fast sweep of the lattice potential M. Greiner et. al. Nature (2002)

11. Revival of the initial state at Explanation

12. wait for time t Fast sweep of the lattice potential A. Tuchman et. al., (2001) A.P., S. Sachdev and S.M. Girvin, PRA 66, 053607 (2002), E. Altman and A. Auerbach, PRL 89, 250404 (2002)

13. Two coupled sites. Semiclassical limit. The phase is not defined in the initial insulting phase. Start from the ensemble of trajectories. Interference of multiple classical trajectories results in oscillations and damping of the phase coherence. Numerical results: Semiclassical approximation to many-body dynamics: A.P., PRA 68, 033609 (2003), ibid. 68, 053604 (2003).

14. Classical non-equlibrium phase transitions Superfluids can support non-dissipative current. accelarate the lattice Exp: Fallani et. al., (Florence) cond- mat/0404045 Theory: Wu and Niu PRA (01); Smerzi et. al. PRL (02). Theory: superfluid flow becomes unstable. Based on the analysis of classical equations of motion (number and phase commute).

15. Damping of a superfluid current in 1D C.D. Fertig et. al. cond-mat/0410491 See: AP and D.-W. Wang, PRL 93, 070401 (2004).

16. What will happen if we have both quantum fluctuations and non-zero superfluid flow? p SFMI U/J ??? possible experimental sequence: ??? p U/J   Stable Unstable SF MI

17. Simple intuitive explanation Viscosity of Helium II, Andronikashvili (1946) Two-fluid model for Helium II Landau (1941) Cold atoms: quantum depletion at zero temperature. The normal current is easily damped by the lattice. Friction between superfluid and normal components would lead to strong current damping at large U/J.

18. Physical Argument SF current in free space SF current on a lattice Strong tunneling regime (weak quantum fluctuations):  s = const. Current has a maximum at p=  /2. This is precisely the momentum corresponding to the onset of the instability within the classical picture. Wu and Niu PRA (01); Smerzi et. al. PRL (02). Not a coincidence!!!  s – superfluid density, p – condensate momentum.

19. Consider a fluctuation If I decreases with p, there is a continuum of resonant states smoothly connected with the uniform one. Current cannot be stable. no lattice:

20. Include quantum depletion. In equilibrium In a current state: So we expect: With quantum depletion the current state is unstable at  p

21. Valid if N  1: Quantum rotor model Deep in the superfluid regime (JN  U) we can use classical equations of motion: Unstable motion for p>  /2

22. SF in the vicinity of the insulating transition: U  JN. Structure of the ground state: It is not possible to define a local phase and a local phase gradient. Classical picture and equations of motion are not valid. After coarse graining we get both amplitude and phase fluctuations. Need to coarse grain the system.

23. Time dependent Ginzburg-Landau: ( diverges at the transition) Stability analysis around a current carrying solution: p U/J   Superfluid MI S. Sachdev, Quantum phase transitions; Altman and Auerbach (2002) Use time-dependent Gutzwiller approximation to interpolate between these limits.

24. p U/J   Superfluid MI Time-dependent Gutzwiller approximation

25. Meanfield (Gutzwiller ansatzt) phase diagram Is there current decay below the instability?

26. Role of fluctuations Below the mean field transition superfluid current can decay via quantum tunneling or thermal decay. E p Phase slip

27. Related questions in superconductivity Reduction of T C 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)

28. Current decay far from the insulating transition

29. Decay due to quantum fluctuations The particle can escape via tunneling: S is the tunneling action, or the classical action of a particle moving in the inverted potential

30. Asymptotical decay rate near the instability Rescale the variables:

31. Many body system At p  /2 we get

32. Many body system, 1D – variational result – semiclassical parameter (plays. the role of 1/  ) Small N~1 Large N~10 2 -10 3

33. Higher dimensions. Stiffness along the current is much smaller than that in the transverse direction. We need to excite many chains in order to create a phase slip. The effective size of the phase slip in d-dimensional space time is

34. Phase slip tunneling is more expensive in higher dimensions: Stability phase diagram Crossover Stable Unstable

35. Current decay in the vicinity of the superfluid-insulator transition

36. In the limit of large  we can employ a different effective coarse-grained theory (Altman and Auerbach 2002): Metastable current state: This state becomes unstable at corresponding to the maximum of the current: Current decay in the vicinity of the Mott transition.

37. Use the same steps as before to obtain the asymptotics: Discontinuous change of the decay rate across the meanfield transition. Phase diagram is well defined in 3D! Large broadening in one and two dimensions.

38. See also AP and D.-W. Wang, PRL, 93, 070401 (2004) Damping of a superfluid current in one dimension C.D. Fertig et. al. cond-mat/0410491

39. Dynamics of the current decay. Underdamped regimeOverdamped regime Single phase slip triggers full current decay Single phase slip reduces a current by one step Which of the two regimes is realized is determined entirely by the dynamics of the system (no external bath).

40. Numerical simulation in the 1D case The underdamped regime is realized in uniform systems near the instability. This is also the case in higher dimensions. Simulate thermal decay by adding weak fluctuations to the initial conditions. Quantum decay should be similar near the instability.

41. Effect of the parabolic trap Expect that the motion becomes unstable first near the edges, where N=1 U=0.01 t J=1/4 Gutzwiller ansatz simulations (2D)

42. Exact simulations in small systems Eight sites, two particles per site SF MI p U/J

43. Semiclassical (Truncated Wigner) simulations of damping of dipolar motion in a harmonic trap AP and D.-W. Wang, PRL 93, 070401 (2004).

44. Detecting equilibrium superfluid- insulator transition boundary in 3D. p U/J   SuperfluidMI 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!

45. Summary New scaling approach to current decay rate: asymptotical behavior of the decay rate near the mean-field transition p U/J   Superfluid MI 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 Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition. Qualitative agreement with experiments and numerical simulations.