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arXiv:1007.5238 · Physical Review Letters

Quantum Quenches and Off-Equilibrium Dynamical Transition in the Infinite-Dimensional Bose-Hubbard Model

Bruno Sciolla, Giulio Biroli

Published Nov 22, 2010• 152 citations
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Abstract

We study the off-equilibrium dynamics of the infinite-dimensional Bose-Hubbard model after a quantum quench. The dynamics can be analyzed exactly by mapping it to an effective Newtonian evolution. For integer filling, we find a dynamical transition separating regimes of small and large quantum quenches starting from the superfluid state. This transition is very similar to the one found for the fermionic Hubbard model by mean field approximations.

DOI
10.1103/physrevlett.105.220401

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•Jul 26, 2018•0 citations
We study the off-equilibrium dynamics of the infinite-dimensional Bose-Hubbard model after a quantum quench. The dynamics can be analyzed exactly by mapping it to an effective Newtonian evolution. For integer filling, we find a dynamical transition separating regimes of small and large quantum quenches starting from the superfluid state. This transition is very similar to the one found for the fermionic Hubbard model by mean field approximations.
We develop a generic method to compute the dynamics induced by quenches in completely connected quantum systems. These models are expected to provide a mean-field description at least of the short-time dynamics of finite-dimensional systems. We apply our method to the Bose–Hubbard model, to a generalized Jaynes–Cummings model, and to the Ising model in a transverse field. We find that the quantum evolution can be mapped onto a classical effective dynamics, which involves only a few intensive observables. For some special parameters of the quench, peculiar dynamical transitions occur. They result from singularities of the classical effective dynamics and are reminiscent of the transition recently found in the fermionic Hubbard model. Finally, we discuss the generality of our results and possible extensions.
We show that the nonequilibrium time evolution after interaction quenches in the one-dimensional, integrable Hubbard model exhibits a dynamical transition in the half-filled case. This transition ceases to exist upon doping. Our study is based on systematically extended equations of motion. Thus it is controlled for small and moderate times; no relaxation effects are neglected. Remarkable similarities to the quench dynamics in the infinite-dimensional Hubbard model are found, suggesting that dynamical transitions are a general feature of quenches in such models.
The exact description of the time evolution of open correlated quantum systems remains one of the major challenges of condensed matter theory, especially for asymptotic long times where most numerical methods fail. Here, the post-quench dynamics of the NNN-component Bose-Anderson impurity model is studied in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The equilibrium phase diagram is similar to that of the Bose-Hubbard model in that it contains local versions of Mott and Bose phases. Using a numerically exact procedure, we are able to study the real-time evolution including asymptotic long-time regimes. The formation of long-lived transient phases is observed for quench paths crossing foreign phases. For quenches inside the local Bose condensed phase, a dynamical phase transition is reported that separates the evolution towards a new equilibrium state and a regime characterized at large times by a persistent phase rotation of the order parameter. We explain how such nondecaying modes can exist in the presence of a dissipative bath. We discuss the extension of our results to the experimental relevant finite-NNN case and their implication for the existence of nondecaying modes in generic quantum systems in the presence of a bath.
We study the nonequilibrium dynamics in the fermionic Hubbard model after a sudden change of the interaction strength. To this scope, we introduce a time-dependent variational approach in the spirit of the Gutzwiller ansatz. At the saddle-point approximation, we find at half filling a sharp transition between two different regimes of small and large coherent oscillations, separated by a critical line of quenches where the system is found to relax. Any finite doping washes out the transition, leaving aside just a sharp crossover. In order to investigate the role of quantum fluctuations, we map the model onto an auxiliary quantum Ising model in a transverse field coupled to free fermionic quasiparticles. Remarkably, the Gutzwiller approximation turns out to correspond to the mean-field decoupling of this model in the limit of infinite coordination lattices. The advantage is that we can go beyond mean field and include Gaussian fluctuations around the non-equilibrium mean-field dynamics. Unlike at equilibrium, we find that quantum fluctuations become massless and eventually unstable before the mean-field dynamical critical line, which suggests they could even alter qualitatively the mean-field scenario.
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.
In the context of out of equilibrium dynamics of isolated interacting quantum many-body systems, we study the mean-field discretized and nonlinear Gross-Pitaevskii equation (GPE) of motion for a fully connected (or all-to-all coupled) Bose-Hubbard model describing a large population of bosons on a lattice of generic dimension V, with potential experimental applications ranging from ultra-cold atomic gases in optical traps to systems of Josephson junctions. Considering a quench on the system Hamiltonian, we describe the time evolution of a system initialized to the Mott insulating phase and taken in the strong superfluid regime, finding a sharp change in the dynamical behaviour when varying the quench intensity across a critical value, that is a dynamical phase transition. Our results generalize the study presented in some previous papers focusing on the particular cases of V = 2, V = 3 and 1D lattice, and place particular emphasis on the relaxation of some dynamical order parameters in the large V limit. On the side of the dynamical phase transition corresponding to the strong superfluid regime, we find such relaxation to consist of a π-synchronization of the mean field bosonic phases, that we find to compete against the eventual site-dependent disorder of the Hamiltonian in what, in analogy with the notorious Kuramoto model for nonlinear coupled oscillators, can be referred to as a synchronization transition. Additionally, in support of the validity of the mean field approximation far from the transition, we report a study beyond mean field based on the Bogolioubov-de Gennes method.