Superfluid insulator transition in a moving condensate

1. Superfluid insulator transition in a moving condensate Anatoli Polkovnikov(BU and Harvard) Ehud Altman, (Weizmann and Harvard) Eugene Demler, Bertrand Halperin, Misha Lukin (Harvard)

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

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

4. Equilibrium system. Interaction energy (two-body collisions): Eint is minimized when Nj=N=const: Interaction suppresses number fluctuations and leads to localization of atoms.

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

6. Classically the ground state has a 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.

7. Superfluid-insulator quantum phase transition. (M.P.A. Fisher, P. Weichman, G. Grinstein, D. Fisher, 1989) Strong tunneling Superfluid regime: Weak tunneling Insulating regime:

8. Superfluid Mott insulator M. Greiner et. al., Nature (02) Adiabatic increase of lattice potential

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

10. Explanation Revival of the initial state at

11. Fast sweep of the lattice potential wait for time t • Tuchman et. al., 2001, • cond-mat/0504762 Theory: A.P., S. Sachdev and S.M. Girvin, PRA 66, 053607 (2002), E. Altman and A. Auerbach, PRL 89, 250404 (2002)

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

13. Damping of a superfluid current in 1D C.D. Fertig et. al. cond-mat/0410491 Current damping below classical instability. No sharp transition. See also : AP and D.-W. Wang, PRL 93, 070401 (2004).

14. ~lattice potential possible experimental sequence: p ??? U/J SF MI p Unstable p/2 Stable MI SF U/J What happens if we there are both quantum fluctuations and superfluid flow? ???

15. Simple intuitive explanation Two-fluid model for Helium II Landau (1941) Viscosity of Helium II, Andronikashvili (1946) Cold atoms: quantum depletion at zero temperature. Friction between superfluid and normal components?

16. Physical Argument SF current in free space SF current on a lattice s–superfluid density, p – condensate momentum. 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. Not a coincidence!!! Wu and Niu PRA (01); Smerzi et. al. PRL (02).

17. Current state Fluctuation If I decreases with p, there is a continuum of resonant states smoothly connected with the uniform one. Current cannot be stable.

18. Include quantum depletion. With quantum depletion the current state is unstable at Equilibrium:  Current state:  p

19. 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. Need to coarse grain the system. After coarse graining we get both amplitude and phase fluctuations.

20. ( diverges at the transition) Stability analysis around a current carrying solution: p p/2 MI Superfluid U/J Time dependent Ginzburg-Landau: S. Sachdev, Quantum phase transitions; Altman and Auerbach (2002) Use time-dependent Gutzwiller approximation to interpolate between these limits.

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

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

23. 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)

24. Current decay far from the insulating transition

25. 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

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

27. semiclassical parameter (plays the role of 1/ ) Many body system, 1D – variational result Large N~102-103 Small N~1

28. Higher dimensions. Longitudinal stiffness is much smaller than the transverse. r Need to excite many chains in order to create a phase slip.

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

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

31. 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.

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

33. Dynamics of the current decay. Underdamped regime Overdamped 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).

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

35. 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)

36. p U/J SF MI Exact simulations: 8 sites, 16 bosons

37. Semiclassical (Truncated Wigner) simulations of damping of dipolar motion in a harmonic trap Quantum fluctuations: Smaller critical current Broad transition AP and D.-W. Wang, PRL 93, 070401 (2004).

38. p/2 U/J Superfluid MI Extrapolate Detecting equilibrium SF-IN transition boundary in 3D. p Easy to detect nonequilibrium irreversible transition!! At nonzero current the SF-IN transition is irreversible: no restoration of current and partial restoration of phase coherence in a cyclic ramp.

39. Quantum fluctuations mean field beyond mean field Depletion of the condensate. Reduction of the critical current. All spatial dimensions. Broadening of the mean field transition. Low dimensions asymptotical behavior of the decay rate near the mean-field transition p p/2 U/J MI Superfluid Summary Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition. Qualitative agreement with experiments and numerical simulations.

40. Quantum rotor model OK if N1: Deep in the superfluid regime (JNU) use GP equations of motion: Unstable motion for p>/2

41. p p/2 U/J MI Superfluid Time-dependent Gutzwiller approximation

42. Many body system At p/2 we get

43. Current decay in the vicinity of the Mott transition. 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: