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arXiv:1403.6578 · Journal of the Physical Society of Japan

General Relaxation Time of the Fidelity for Isolated Quantum Thermodynamic Systems

Takaaki Monnai

Published May 1, 2014• 16 citations
arXivHTMLPDF

Abstract

General evaluation of the relaxation time to equilibrium is usually considered as difficult, since it would strongly depend on the model of interest. In this paper, we provide a generic initial relaxation time of the fidelity for the isolated large systems. The decay of the fidelity is a combination of the Lorentzian and a sinusoidal oscillation. We calculate the relaxation time of the Lorentzian envelop, and the period of the oscillation. Remarkably, these two time scales are the same order when the energy range of the microcanonical state is larger than the thermal fluctuation. Also, the power law decay generally exists for long time regime.

DOI
10.7566/jpsj.83.064001

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General evaluation of the relaxation time to equilibrium is usually considered as difficult, since it would strongly depend on the model of interest. In this paper, we provide a generic initial relaxation time of the fidelity for the isolated large systems. The decay of the fidelity is a combination of the Lorentzian and a sinusoidal oscillation. We calculate the relaxation time of the Lorentzian envelop, and the period of the oscillation. Remarkably, these two time scales are the same order when the energy range of the microcanonical state is larger than the thermal fluctuation. Also, the power law decay generally exists for long time regime.
General evaluation of the relaxation time to equilibrium remains as a subtle unsolved problem, partially because of its possible dependence on the specific features of models. We evaluate the relaxation time of expectation values for generic isolated quantum many-body systems, and especially show that it at most polynomially depends on the system size. This provides a first model-independent quantitative analysis of the relaxation time. The estimation holds when the Hamiltonian is non-integrable, the initial deviation of the quantity of interest is order of its spectral norm defined on a finite space, and the relaxation process is essentially monotonic.
Despite being ubiquitous, out-of-equilibrium quantum systems are much less understood than systems at equilibrium. Progress in the field has benefited from a symbiotic relationship between theoretical studies and new experiments on coherent dynamics. The present work strengthens this connection by providing a general picture of the relaxation process of isolated lattice many-body quantum systems that are routinely studied in experiments with cold atoms, ions traps, and nuclear magnetic resonance. We show numerically and analytically that the long-time decay of the probability for finding the system in its initial state necessarily shows a power-law behavior ∝t−γ\propto t^{ - \gamma }∝t−γ. This happens independently of the details of the system, such as integrability, level repulsion, and the presence or absence of disorder. Information about the spectrum, the structure of the initial state, and the number of particles that interact simultaneously is contained in the value of γ\gammaγ. From it, we can anticipate whether the initial state will or will not thermalize.
Despite being ubiquitous, out-of-equilibrium quantum systems are much less understood than systems at equilibrium. Progress in the field has benefited from a symbiotic relationship between theoretical studies and new experiments on coherent dynamics. The present work strengthens this connection by providing a general picture of the relaxation process of isolated lattice many-body quantum systems that are routinely studied in experiments with cold atoms, ions traps, and nuclear magnetic resonance. We show numerically and analytically that the long-time decay of the probability for finding the system in its initial state necessarily shows a power-law behavior $\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\gamma}}$. This happens independently of the details of the system, such as integrability, level repulsion, and the presence or absence of disorder. Information about the spectrum, the structure of the initial state, and the number of particles that interact simultaneously is contained in the value of $\ensuremath{\gamma}$. From it, we can anticipate whether the initial state will or will not thermalize.
We initiate the study of relaxation to quantum equilibrium over long timescales in pilot-wave theory. We simulate the time evolution of the coarse-grained H-function Hbar(t) for a two-dimensional harmonic oscillator. For a (periodic) wave function that is a superposition of the first 25 energy states we confirm an approximately exponential decay of Hbar over five periods. For a superposition of only the first four energy states we are able to calculate Hbar(t) over 50 periods. We find that, depending on the set of phases in the initial wave function, Hbar can decay to a large nonequilibrium residue exceeding 10% of its initial value or it can become indistinguishable from zero (the equilibrium value). We show that a large residue in Hbar is caused by a tendency for the trajectories to be confined to sub-regions of configuration space for some wave functions, and that this is less likely to occur for larger numbers of energy states (if the initial phases are chosen randomly). Possible cosmological implications are briefly discussed.
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.