Superfluid insulator transition in a moving condensate

1. Superfluid insulator transition in a moving condensate Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman - Weizmann Eugene Demler - Harvard Bertrand Halperin - Harvard Mikhail Lukin - Harvard

2. Plan of the talk • Bosons in optical lattices. Equilibrium phase diagram. • Superfluid-insulator transition in a moving condensate. • Mean field 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. 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

9. 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).

10. ~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? ???

11. 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).

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

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

14. 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.

15. ( 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.

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

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

18. 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)

19. Current decay far from the insulating transition

20. 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

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

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

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

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

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

26. 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.

27. 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)

28. 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)

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

30. 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).

31. 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.

32. 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.

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