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arXiv:1606.04117 · Physical Review A

Light-cone-like spreading of single-particle correlations in the Bose-Hubbard model after a quantum quench in the strong-coupling regime

Matthew Fitzpatrick, Malcolm P. Kennett

Published Nov 19, 2018• 20 citations
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Abstract

We study the spreading of correlations in space and time after a quantum quench in the Bose-Hubbard model. We derive equations of motion for the single-particle Green's function within the contour-time formalism, allowing us to study dynamics in the strong-coupling regime. We discuss the numerical solutions of these equations and calculate the single-particle density matrix for quenches in the Mott phase. We demonstrate light-cone-like spreading of correlations in the Mott phase in one, two, and three dimensions and calculate the propagation velocities in each dimension.

DOI
10.1103/physreva.98.053618

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We study the spreading of correlations in space and time after a quantum quench in the Bose-Hubbard model. We derive equations of motion for the single-particle Green's function within the contour-time formalism, allowing us to study dynamics in the strong-coupling regime. We discuss the numerical solutions of these equations and calculate the single-particle density matrix for quenches in the Mott phase. We demonstrate light-cone-like spreading of correlations in the Mott phase in one, two, and three dimensions and calculate the propagation velocities in each dimension.
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.
Lieb-Robinson and related bounds set an upper limit on the speed at which information propagates in nonrelativistic quantum systems. Experimentally, light-cone-like spreading has been observed for correlations in the Bose-Hubbard model (BHM) after a quantum quench. Using a two-particle irreducible (2PI) strong-coupling approach to out-of-equilibrium dynamics in the BHM we calculate both the group and phase velocities for the spreading of single-particle correlations in one, two, and three dimensions as a function of interaction strength. Our results are in quantitative agreement with measurements of the speed of spreading of single-particle correlations in both the one- and two-dimensional BHM realized with ultracold atoms. They also are consistent with the claim that the phase velocity rather than the group velocity was observed in recent experiments in two dimensions. We demonstrate that there can be large differences between the phase and group velocities for the spreading of correlations and explore how the anisotropy in the velocity varies across the phase diagram of the BHM. Our results establish the 2PI strong-coupling approach as a powerful tool to study out-of-equilibrium dynamics in the BHM in dimensions greater than one.
Experimental advances have made ultracold atoms in optical lattices a favourable setting to study out-of-equilibrium phenomena and attracted considerable attention in recent years.These systems are highly versatile in that experimental parameters can be tuned over a wide range of values in real time.This facilitates the study of quantum quenches, in which parameters in the corresponding Hamiltonian are varied in time faster than the system can respond adiabatically.Such protocols open the door to a rich range of many-body physics and have been studied intensely both theoretically and experimentally.The Bose-Hubbard model (BHM) has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model.The BHM is a particularly convenient context for studying quantum quenches as it displays a quantum phase transition between superfluid and Mott-insulator phases.In this thesis, we develop a strong-coupling approach that allows the study of correlations in space and time in both the superfluid and Mott-insulating phases of the BHM.Specifically, we obtain a two-particle irreducible (2PI) effective action within the contourtime formalism that allows for a description of both equilibrium and out-of-equilibrium phenomena.We derive equations of motion for both the superfluid order parameter and the single-particle many-body Green's functions.First, we assess the accuracy of our formalism by studying the equilibrium solution for the homogeneous BHM and comparing our results to existing strong-coupling methods as well as to exact methods where possible.We then consider homogeneous systems that are initially thermalized in the Mott phase, and which are then subjected to quenches.We solve numerically the equations of motion for this scenario and calculate the single-particle density matrix.We demonstrate a Lieb-Robinson-like maximal propagation velocity for the spreading of single-particle correlations in one, two, and three dimensions.We study the dependence of the maximal propagation velocity on the quench protocol, chemical potential, temperature, and dimensionality.We compare our results to exact methods, existing strong-coupling approaches, and experiments where possible.Lastly, we extend our strong-coupling approach to the disordered BHM and derive equations of motion for the disorder-averaged single-particle Green's function.We discuss how these equations of motion can be used to study the phase stability of many-body localization in the disordered BHM for dimensions higher than one.
We study the spreading of density-density correlations in Bose-Hubbard models after a quench of the interaction strength, using time-dependent variational Monte Carlo simulations. It gives access to unprecedented long propagation times and to dimensions higher than one. In both one and two dimensions, we find ballistic light-cone spreading of correlations and extract accurate values of the light-cone velocity in the superfluid regime. We show that the spreading of correlations is generally supersonic, with a light-cone propagating faster than sound modes but slower than the maximum group velocity of density excitations, except at the Mott transition, where all the characteristic velocities are equal. Further, we show that in two dimensions the correlation spreading is highly anisotropic and presents nontrivial interference effects.
We analyze the propagation of correlations after a sudden interaction change in a strongly interacting quantum system in contact with an environment. In particular, we consider an interaction quench in the Bose-Hubbard model, deep within the Mott-insulating phase, under the effect of dephasing. We observe that dissipation effectively speeds up the propagation of single-particle correlations while reducing their coherence. In contrast, for two-point density correlations, the initial ballistic propagation regime gives way to diffusion at intermediate times. Numerical simulations, based on a time-dependent matrix product state algorithm, are supplemented by a quantitatively accurate fermionic quasiparticle approach providing an intuitive description of the initial dynamics in terms of holon and doublon excitations.
Lieb-Robinson and related bounds set an upper limit on the rate of spreading of information in non-relativistic quantum systems. Experimentally, they have been observed in the spreading of correlations in the Bose-Hubbard model after a quantum quench. Using a recently developed two particle irreducible (2PI) strong coupling approach to out-of-equilibrium dynamics in the Bose-Hubbard model we calculate both the group and phase velocities for the spreading of single-particle correlations in one, two and three dimensions as a function of interaction strength. Our results are in quantitative agreement with measurements of the velocities for the spreading of single particle correlations in both the one and two dimensional Bose-Hubbard model realized with ultra-cold atoms. They also agree with the claim that the phase velocity rather than the group velocity was observed in recent experiments in two dimensions. We demonstrate that there can be large differences between the phase and group velocities for the spreading of correlations and also explore the variation of the anisotropy in the velocity at which correlations spread across the phase diagram of the Bose-Hubbard model. Our results establish the 2PI strong coupling approach as a powerful tool to study out-of-equilibrium dynamics in the Bose-Hubbard model in dimensions greater than one.