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Role of the Initial State in the Nonequilibrium Quantum Dynamics of Many-Body Systems

Eduardo Jonathan Torres-Herrera, Léa Ferreira dos Santos

Published Oct 18, 2013• 2 citations
OpenAlex

Abstract

We study isolated many-body quantum systems after a quench and show that their dynamics depends on the interplay between the initial state and the Hamiltonian dictating the evolution. The systems are in the nonperturbative regime. The relaxation process is controlled by the width of the energy distribution of the initial state and may be very similar for both chaotic and integrable Hamiltonians. We derive an analytical expression for the fidelity decay, which displays excellent agreement with our numerical results. This decay is Gaussian and may persist until saturation. Analytical expressions that describe very well the initial evolution of the Shannon entropy and of few-body observables are also provided. The analyses are developed for deterministic one-dimensional systems and initial states of interest to current experiments with cold atoms in optical lattices.

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Quantum Reports•Dec 15, 2019•16 citations
We study isolated many-body quantum systems after a quench and show that their dynamics depends on the interplay between the initial state and the Hamiltonian dictating the evolution. The systems are in the nonperturbative regime. The relaxation process is controlled by the width of the energy distribution of the initial state and may be very similar for both chaotic and integrable Hamiltonians. We derive an analytical expression for the fidelity decay, which displays excellent agreement with our numerical results. This decay is Gaussian and may persist until saturation. Analytical expressions that describe very well the initial evolution of the Shannon entropy and of few-body observables are also provided. The analyses are developed for deterministic one-dimensional systems and initial states of interest to current experiments with cold atoms in optical lattices.
We study numerically and analytically isolated interacting quantum systems that are taken out of equilibrium instantaneously (quenched). The probability of finding the initial state in time, the so-called fidelity, decays fastest for systems described by full random matrices, where simultaneous many-body interactions are implied. In the realm of realistic systems with two-body interactions, the dynamics is slower and depends on the interplay between the initial state and the Hamiltonian characterizing the system. The fastest fidelity decay in this case is Gaussian and can persist until saturation. A simple general picture, in which the fidelity plays a central role, is also achieved for the short-time dynamics of few-body observables. It holds for initial states that are eigenstates of the observables. We also discuss the need to reassess analytical expressions that were previously proposed to describe the evolution of the Shannon entropy. Our analyses are mainly developed for initial states that can be prepared in experiments with cold atoms in optical lattices.
We study isolated quantum systems with two-body interactions after a quench. In these systems, the energy shell is a Gaussian of width $\ensuremath{\sigma}$, and it gives the maximum possible spreading of the energy distribution of the initial states. When the distribution achieves this shape, the fidelity decay can be Gaussian until saturation. This establishes a lower bound for the fidelity decay in realistic systems. An ultimate bound for systems with many-body interactions is also derived based on the analysis of full random matrices. We find excellent agreement between numerical and analytical results. We also provide the conditions under which the short-time dynamics of few-body observables is controlled by $\ensuremath{\sigma}$. The analyses are developed for systems, initial states, and observables accessible to experiments.
We explore the role of the initial state on the onset of thermalization in isolated quantum many-body systems after a quench. The initial state is an eigenstate of an initial Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ and it evolves according to a different final Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$. If the initial state has a chaotic structure with respect to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens when ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ is a full random matrix, because its states projected onto ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$ are fully delocalized. The results for the observables then agree with those obtained with thermal states at infinite temperature. However, finite real systems with few-body interactions, as the ones considered here, are deprived of fully extended eigenstates, even when described by a nonintegrable Hamiltonian. We examine how the initial state delocalizes as it gets closer to the middle of the spectrum of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, causing the observables to approach thermal averages, be the models integrable or chaotic. Our numerical studies are based on initial states with energies that cover the entire lower half of the spectrum of one-dimensional Heisenberg spin-1/2 systems.
We provide an overview of our numerical and analytical studies of isolated interacting quantum systems that are abruptly quenched out of equilibrium instantaneously. We describe the relaxation process to a new equilibrium and obtain lower bounds for the relaxation time of full random matrices and realistic systems with two-body interactions. We show that the size of the time fluctuations after relaxation decays exponentially with system size for systems without too many degeneracies. We also discuss the conditions for thermalization and demonstrate that it can happen after local and global quenches in space. The analyses are developed for systems, initial states, and few-body observables accessible to experiments with optical lattices.
Numerically, we study the time fluctuations of few-body observables after relaxation in isolated dynamical quantum systems of interacting particles. Our results suggest that they decay exponentially with system size in both regimes, integrable and chaotic. The integrable systems considered are solvable with the Bethe ansatz and have a highly nondegenerate spectrum. This is in contrast with integrable Hamiltonians mappable to noninteracting ones. We show that the coefficient of the exponential decay depends on the level of delocalization of the initial state with respect to the energy shell.
We investigate the dynamics of a few bosons in an optical lattice induced by a quantum quench of a parameter of the many-body Hamiltonian. The evolution of the many-body wave function is obtained by solving the time-dependent many-body Schrödinger equation numerically, using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We report the time evolution of three key quantities, namely, the occupations of the natural orbitals, that is, the eigenvalues of the one-body reduced density matrix, the many-body Shannon information entropy, and the quantum fidelity for a wide range of interactions. Our key motivation is to characterize relaxation processes where various observables of an isolated and interacting quantum many-body system dynamically converge to equilibrium values via the quantum fidelity and via the production of many-body entropy. The interaction, as a parameter, can induce a phase transition in the ground state of the system from a superfluid (SF) state to a Mott-insulator (MI) state. We show that, for a quench to a weak interaction, the fidelity remains close to unity and the entropy exhibits oscillations. Whereas for a quench to strong interactions (SF to MI transition), the relaxation process is characterized by the first collapse of the quantum fidelity and entropy saturation to an equilibrium value. The dip and the non-analytic nature of quantum fidelity is a hallmark of dynamical quantum phase transitions. We quantify the characteristic time at which the quantum fidelity collapses and the entropy saturates.