Superfluid to insulator transition in a moving system of interacting bosons Ehud Altman Anatoli Polkovnikov Bertrand Halperin Mikhail Lukin Eugene Demler References: J. Superconductivity 17:577 (2004) Phys. Rev. Lett. 95:20402 (2005) Phys. Rev. A 71:63613 (2005) Physics Department, Harvard University

Outline Introduction. Cold atoms in optical lattices. Superfluid to Mott transition. Dynamical instability Mean-field analysis using Gutzwiller variational wavefunctions Current decay by quantum tunneling Current decay by thermal activation Conclusions

Atoms in optical lattices. Bose Hubbard model Theory: Jaksch et al. PRL 81:3108(1998) Experiment: Kasevich et al., Science (2001) Greiner et al., Nature (2001) Cataliotti et al., Science (2001) Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004), …

Equilibrium superfluid to insulator transition Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98) Experiment: Greiner et al. Nature (01) U m Superfluid Mott insulator What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition. Explain experiment. SF well described by a wavefunction in which the zero momentum state is macroscopically occupied. Mott state is rather well described by definite RS occupations. t/U

Moving condensate in an optical lattice. Dynamical instability Theory: Niu et al. PRA (01), Smerzi et al. PRL (02) Experiment: Fallani et al. PRL (04) v Related experiments by Eiermann et al, PRL (03)

??? This talk: How to connect the dynamical instability (irreversible, classical) to the superfluid to Mott transition (equilibrium, quantum) Unstable ??? p U/J p/2 Stable SF MI This talk U/t p SF MI ??? Possible experimental sequence:

Superconductor to Insulator transition in thin films Marcovic et al., PRL 81:5217 (1998) Bi films d Superconducting films of different thickness

Dynamical instability Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation Current carrying states Linear stability analysis: States with p>p/2 are unstable Amplification of density fluctuations unstable unstable r

Dynamical instability for integer filling Order parameter for a current carrying state Current GP regime . Maximum of the current for . When we include quantum fluctuations, the amplitude of the order parameter is suppressed decreases with increasing phase gradient

Dynamical instability for integer filling SF MI p U/J p/2 Vicinity of the SF-I quantum phase transition. Classical description applies for The amplitude and the phase of the order parameter can only be defined if we coarse grain the system on the scale ksi. Applying the same argument as before we find the condition that p ksi should be of the order of pi over two. Near the SF-I transition ksi diverges. So the critical phase gradient goes to zero. Dynamical instability occurs for

Dynamical instability. Gutzwiller approximation Wavefunction Time evolution We look for stability against small fluctuations Phase diagram. Integer filling

Order parameter suppression by the current. Number state (Fock) representation Integer filling N N+1 N-1 N+2 N-2 Distribution of the occupation numbers is determined by the competition of the kinetic and interaction energies. Kinetic energy favors wide distribution. Interaction energy favors distribution peaked at a single filling. The wider is the distribution, the larger is the order parameter. When there is a phase gradient, the effect of the kinetic energy is reduced and the distribution gets squeezed. This reduces the order parameter. N-2 N-1 N N+1 N+2

Order parameter suppression by the current. Number state (Fock) representation Integer filling Fractional filling N N+1 N-1 N+2 N-2 N-1/2 N+1/2 N-3/2 N+3/2 N-1/2 N+1/2 N-3/2 N+3/2 Distribution of the occupation numbers is determined by the competition of the kinetic and interaction energies. Kinetic energy favors wide distribution. Interaction energy favors distribution peaked at a single filling. The wider is the distribution, the larger is the order parameter. When there is a phase gradient, the effect of the kinetic energy is reduced and the distribution gets squeezed. This reduces the order parameter. N-2 N-1 N N+1 N+2

Dynamical instability Integer filling Fractional filling p p p/2 p/2 U/J U/J SF MI

Optical lattice and parabolic trap. Gutzwiller approximation The first instability develops near the edges, where N=1 U=0.01 t J=1/4 Gutzwiller ansatz simulations (2D)

Beyond semiclassical equations. Current decay by tunneling phase j phase j phase j Current carrying states are metastable. They can decay by thermal or quantum tunneling Thermal activation Quantum tunneling

Decay of current by quantum tunneling phase j phase Quantum phase slip j Escape from metastable state by quantum tunneling. WKB approximation S – classical action corresponding to the motion in an inverted potential.

Decay rate from a metastable state. Example

Weakly interacting systems. Quantum rotor model. Decay of current by quantum tunneling At p/2 we get For the link on which the QPS takes place d=1. Phase slip on one link + response of the chain. Phases on other links can be treated in a harmonic approximation

For d>1 we have to include transverse directions. Need to excite many chains to create a phase slip Longitudinal stiffness is much smaller than the transverse. The transverse size of the phase slip diverges near a phase slip. We can use continuum approximation to treat transverse directions

Weakly interacting systems. Gross-Pitaevskii regime. Decay of current by quantum tunneling SF MI p U/J p/2 Fallani et al., PRL (04) Quantum phase slips are strongly suppressed in the GP regime

Strongly interacting regime. Vicinity of the SF-Mott transition MI p U/J p/2 Close to a SF-Mott transition we can use an effective relativistivc GL theory (Altman, Auerbach, 2004) Metastable current carrying state: This state becomes unstable at corresponding to the maximum of the current:

Strongly interacting regime. Vicinity of the SF-Mott transition Decay of current by quantum tunneling SF MI p U/J p/2 Action of a quantum phase slip in d=1,2,3 - correlation length Strong broadening of the phase transition in d=1 and d=2 is discontinuous at the transition. Phase slips are not important. Sharp phase transition

Decay of current by quantum tunneling

Decay of current by thermal activation phase j phase Thermal phase slip j DE Escape from metastable state by thermal activation

Thermally activated current decay. Weakly interacting regime phase slip DE Activation energy in d=1,2,3 Thermal fluctuations lead to rapid decay of currents Crossover from thermal to quantum tunneling

Decay of current by thermal fluctuations Phys. Rev. Lett. (2004)

Conclusions Dynamic instability is continuously connected to the quantum SF-Mott transition Quantum fluctuations lead to strong decay of current in one and two dimensional systems Thermal fluctuations lead to strong decay of current in all dimensions