1、Superfluid insulator transition in a moving condensate,Anatoli Polkovnikov,Harvard University,Ehud Altman, Eugene Demler, Bertrand Halperin, Misha Lukin,Plan of the talk,General motivation and overview. Bosons in optical lattices. Equilibrium phase diagram. Examples of quantum dynamics. Superfluid-i
2、nsulator transition in a moving condensate. Qualitative picture Non-equilibrium phase diagram. Role of quantum fluctuations Conclusions and experimental implications.,Why is the physics of cold atoms interesting?,It is possible to realize strongly interacting systems, both fermionic and bosonic.,Par
3、ameters 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.,Interacting bosons in optical lattices.,Highly tunable periodic potentials with no
4、defects.,Equilibrium system.,Interaction energy (two-body collisions):,Eint is minimized when Nj=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 th
5、e system.,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.,Ground state is a superfluid:
6、,Strong tunneling,M. Greiner et. al., Nature (02),Adiabatic increase of lattice potential,Nonequilibrium phase transitions,wait for time t,Fast sweep of the lattice potential,M. Greiner et. al. Nature (2002),Revival of the initial state at,Explanation,Fast sweep of the lattice potential,A. Tuchman e
7、t. al., (2001),A.P., S. Sachdev and S.M. Girvin, PRA 66, 053607 (2002), E. Altman and A. Auerbach, PRL 89, 250404 (2002),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 trajec
8、tories 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).,Classical non-equlibrium phase transitions,Superfluids can support non-dissipative current.,accelarate the la
9、ttice,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).,Damping of a superfluid current in 1D,C.D. Fertig et. al. cond-ma
10、t/0410491,See: AP and D.-W. Wang, PRL 93, 070401 (2004).,What will happen if we have both quantum fluctuations and non-zero superfluid flow?,?,Simple intuitive explanation,Viscosity of Helium II, Andronikashvili (1946),Two-fluid model for Helium II,Landau (1941),Cold atoms: quantum depletion at zero
11、 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.,Physical Argument,SF current in free space,SF current on a lattice,Strong tunneling regime (weak quantum fluctuations): s = const. Cu
12、rrent 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.,Consider a fluctuation,If I decreases with p, there is
13、 a continuum of resonant states smoothly connected with the uniform one. Current cannot be stable.,no lattice:,Include quantum depletion.,In equilibrium,In a current state:,So we expect:,With quantum depletion the current state is unstable at,p,Valid if N1:,Quantum rotor model,SF in the vicinity of
14、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 de
15、pendent Ginzburg-Landau:,( diverges at the transition),Stability analysis around a current carrying solution:,S. Sachdev, Quantum phase transitions; Altman and Auerbach (2002),Use time-dependent Gutzwiller approximation to interpolate between these limits.,Time-dependent Gutzwiller approximation,Mea
16、nfield (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 .,Phase slip,Related questions in superconductivity,Reduction of TC and the critical curren
17、t 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
18、,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,At p/2 we get,Many body system, 1D, variati
19、onal result, semiclassical parameter (plays . the role of 1/),Small N1,Large N102-103,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-dimensi
20、onal space time is,Phase slip tunneling is more expensive in higher dimensions:,Current decay in the vicinity of the superfluid-insulator transition,In the limit of large we can employ a different effective coarse-grained theory (Altman and Auerbach 2002):,Metastable current state:,This state become
21、s unstable at corresponding to the maximum of the current:,Current decay in the vicinity of the Mott 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 on
22、e and two dimensions.,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,Dynamics of the current decay.,Underdamped regime,Overdamped regime,Single phase slip triggers full current decay,Single phase slip reduces a
23、 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 ther
24、mal 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 in small sys
25、tems,Eight sites, two particles per site,Semiclassical (Truncated Wigner) simulations of damping of dipolar motion in a harmonic trap,AP and D.-W. Wang, PRL 93, 070401 (2004).,Detecting equilibrium superfluid-insulator transition boundary in 3D.,p,U/J,p/2,Superfluid,MI,Extrapolate,At nonzero current
26、 the SF-IN transition is irreversible: no restoration of current and partial restoration of phase coherence in a cyclic ramp.,Easy to detect!,Summary,Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition.,Qualitative agreement with experiments and numerical simulations.,
