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

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

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

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.

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

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.

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

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

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

Revival of the initial state at Explanation

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

Classical non-equlibrium phase transitions Superfluids can support non-dissipative current. Exp: Fallani et. al., (Florence) cond- mat/ 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).

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

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

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

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.

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

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

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.

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.

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

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

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)

Current decay far from the insulating transition

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

Asymptotical decay rate near the instability Rescale the variables:

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

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

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

Current decay in the vicinity of the superfluid-insulator transition

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.

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

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).

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.

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)

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

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

Detecting equilibrium SF-IN 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 nonequilibrium irreversible transition!!

Summary 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.

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

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

Many body system At p  /2 we get

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.