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arXiv:1008.1548 · Physical Review A

Emergence of coherence in the Mott-insulator–superfluid quench of the Bose-Hubbard model

Patrick Navez, Ralf Schützhold

Published Dec 3, 2010• 65 citations
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Abstract

We study the quench from the Mott-insulator to the superfluid phase in the Bose-Hubbard model and investigate the spatial-temporal growth of phase coherence, that is, phase locking between initially uncorrelated sites. To this end, we establish a hierarchy of correlations via a controlled expansion into inverse powers of the coordination number $1/Z$. It turns out that the off-diagonal long-range order spreads with a constant propagation speed, forming local condensate patches, whereas the phase correlator follows a diffusionlike growth rate.

DOI
10.1103/physreva.82.063603

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We study the quench from the Mott-insulator to the superfluid phase in the Bose-Hubbard model and investigate the spatial-temporal growth of phase coherence, that is, phase locking between initially uncorrelated sites. To this end, we establish a hierarchy of correlations via a controlled expansion into inverse powers of the coordination number 1/Z1/Z1/Z. It turns out that the off-diagonal long-range order spreads with a constant propagation speed, forming local condensate patches, whereas the phase correlator follows a diffusionlike growth rate.
We study the Bose-Hubbard and Fermi-Hubbard model in the limit of large coordination numbers Z (i.e., many tunnelling partners). Via a controlled expansion into powers of 1/Z, we establish a hierarchy of correlations, which facilitates an approximate analytic solution of the quantum evolution. For the Bose-Hubbard model, we derive the growth of phase coherence after a quench from the Mott to the superfluid phase. For a quench within the Mott phase, we find that various local observables approach a quasi-equilibrium state after a finite period of time. However, this state is not thermal, i.e., real thermalisation -- if it occurs -- requires much longer time scales. For a tilted lattice in the Mott state, we calculate the tunnelling probability and find a remarkable analogy to the Sauter-Schwinger effect (i.e., electron-positron pair creation out of the vacuum due to a strong electric field). These analytical results are compared to numerical simulations for finite lattices in one and two dimensions and we find qualitative agreement. Finally, we generalize these studies to the more involved case of the Fermi-Hubbard model.
By means of an adapted mean-field expansion for large fillings n⪢1n⪢1n⪢1, we study the evolution of quantum fluctuations in the time-dependent Bose-Hubbard model, starting in the superfluid state and approaching the Mott phase by decreasing the tunneling rate or increasing the interaction strength in time. For experimentally relevant cases, we derive analytical results for the temporal behavior of the number and phase fluctuations, respectively. This allows us to calculate the growth of the quantum depletion and the decay of off-diagonal long-range order. We estimate the conditions for the observability of the time dependence in the correlation functions in the experimental setups with external trapping present. Finally, we discuss the analogy to quantum effects in the early universe during the inflationary epoch.
We study correlations in space and time after a quantum quench in the Bose Hubbard model. We obtain a two particle irreducible effective action within the Schwinger-Keldysh formalism that allows us to study dynamics in both the superfluid and Mott insulating phases. We derive equations of motion for both the superfluid order parameter and two-point correlation functions. We discuss the numerical solutions of these equations and calculate the quasimomentum distribution for quenches in the Mott phase. We demonstrate light-cone like spreading of correlations in the Mott phase in both one and two dimensions and calculate the propagation velocity in each dimension.
We study the sweep through the quantum phase transition from the superfluid to the Mott state for the Bose-Hubbard model with a time-dependent tunneling rate J(t). In the experimentally relevant case of exponential decay J(t) proportional variant e -gamma t, an adapted mean-field expansion for large fillings n yields a scaling solution for the fluctuations. This enables us to analytically calculate the evolution of the number and phase variations (on-site) and correlations (off-site) for slow (gamma<<mu), intermediate, and fast (nonadiabatic gamma>>mu) sweeps, where mu is the chemical potential. Finally, we derive the dynamical decay of the off-diagonal long-range order as well as the temporal shrinkage of the superfluid fraction in a persistent ring-current setup.
We investigate the time evolution of correlations in the Bose-Hubbard model following a quench from the superfluid to the Mott insulator. For large values of the final interaction strength the system approaches a distinctly nonequilibrium steady state that bears strong memory of the initial conditions. In contrast, when the final interaction strength is comparable to the hopping, the correlations are rather well approximated by those at thermal equilibrium. The existence of two distinct nonequilibrium regimes is surprising given the nonintegrability of the Bose-Hubbard model. We relate this phenomenon to the role of quasiparticle interactions in the Mott insulator.
Mott-insulator-to-superfluid quenches have been used by recent experiments to generate exotic superfluid phases. While the final Hamiltonian following the sudden quench is that of a superfluid, it is not a priori clear whether the final state of the system actually enters the superfluid phase. To understand the nature of the final state the temporal evolution of the momentum distribution is calculated following a Mott-insulator-to-superfluid quench. Using the numerical infinite time-evolving block decimation approach and the analytical rotor model approximation it is established that the one- and two-dimensional Mott insulators following the quench equilibrate to thermal states with spatially short-ranged coherence peaks in the final momentum distribution and therefore are not superfluids. However, in three dimensions a divergence in the momentum distribution indicating the emergence of true superfluid order is found.