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Communications in Theoretical Physics

Steady-state relation of a two-level system strongly coupled to a many-body quantum chaotic environment

Hua Yan, Jiaozi Wang, Wen-ge Wang

Published Dec 15, 2022• 0 citations
Publisher

Abstract

Abstract We study the long-time average of the reduced density matrix (RDM) of a two-level system as the central system, which is locally coupled to a many-body quantum chaotic system as the environment, under an overall Schrödinger evolution. A phenomenological relation among elements of the RDM is proposed for a dissipative interaction in the strong coupling regime and is tested numerically with the environment as a defect Ising chain, as well as a mixed-field Ising chain.

DOI
10.1088/1572-9494/acabb9

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Abstract We study the long-time average of the reduced density matrix (RDM) of a two-level system as the central system, which is locally coupled to a many-body quantum chaotic system as the environment, under an overall Schrödinger evolution. A phenomenological relation among elements of the RDM is proposed for a dissipative interaction in the strong coupling regime and is tested numerically with the environment as a defect Ising chain, as well as a mixed-field Ising chain.
We study the long-time average of the reduced density matrix (RDM) of a central system that is locally coupled to a large environment, under a Schr\"odinger evolution of the total system. We consider a class of interaction Hamiltonian, whose environmental part satisfies the so-called eigenstate thermalization hypothesis ansatz with a constant diagonal part in the energy region concerned. Relations among elements of the averaged RDM are derived. When steady states of the central system exist, these relations imply the existence of a preferred basis, which is given by the eigenbasis of a renormalized self-Hamiltonian that includes certain averaged impact of the system-environment interaction. Numerical simulations performed for a qubit coupled to a defect Ising chain confirm the analytical predictions.
I study the statistical description of a small quantum system, which is coupled to a large quantum environment in a generic form and with a generic interaction strength, when the total system lies in an equilibrium state described by a microcanonical ensemble. The focus is on the difference between the reduced density matrix (RDM) of the central system in this interacting case and the RDM obtained in the uncoupled case. In the eigenbasis of the central system's Hamiltonian, it is shown that the difference between diagonal elements is mainly confined by the ratio of the maximum width of the eigenfunctions of the total system in the uncoupled basis to the width of the microcanonical energy shell; meanwhile, the difference between off-diagonal elements is given by the ratio of certain property of the interaction Hamiltonian to the related level spacing of the central system. As an application, a sufficient condition is given, under which the RDM may have a canonical Gibbs form under system-environment interactions that are not necessarily weak; this Gibbs state usually includes certain averaged effect of the interaction. For central systems that interact locally with many-body quantum chaotic systems, it is shown that the RDM usually has a Gibbs form. I also study the RDM which is computed from a typical state of the total system within an energy shell.
We investigate decoherence in quantum systems coupled via dephasing-type interactions to an arbitrary environment with chaotic underlying classical dynamics. The coherences of the reduced state of the central system written in the preferential energy eigenbasis are quantum Loschmidt echoes, which in the strong coupling regime are characterized at long time scales by fluctuations around a constant mean value. We show that due to the chaotic dynamics of the environment, the mean value and the width of the Loschmidt-echo fluctuations are inversely proportional to the quantity we define as the effective Hilbert-space dimension of the environment, which in general is smaller than the dimension of the entire available Hilbert space. Nevertheless, in the semiclassical regime this effective Hilbert-space dimension is in general large, in which case even a chaotic environment with few degrees of freedom produces decoherence without revivals. Moreover we show that in this regime the environment leads the central system to equilibrate to the time average of its reduced density matrix, which corresponds to a diagonal state in the preferential energy eigenbasis. For the case of two uncoupled, initially entangled central systems that interact with identical local quantum environments with chaotic underlying classical dynamics, we show that in the semiclassical limit the equilibration state is arbitrarily close to a separable state. We confirm our results with numerical simulations in which the environment is modeled by the quantum kicked rotor in the chaotic regime.
As a model of decohering environment, we show that a quantum chaotic system behaves equivalently as a many-body system. An approximate formula for the time evolution of the reduced density matrix of a system interacting with a quantum chaotic environment is derived. This theoretical formulation is substantiated by the numerical study of the decoherence of two qubits interacting with a quantum chaotic environment modeled by a chaotic kicked top. Like the many-body model of environment, the quantum chaotic system is an efficient decoherer, and it can generate entanglement between the two qubits which have no direct interaction.
In this paper, a method is developed for the study of a generic small central quantum system, which is locally coupled to an environment as a many-body quantum chaotic system that satisfies the eigenstate thermalization hypothesis (ETH) ansatz. The approach is based on properties of environmental branches of the total system's state, the overlaps of which give the reduced density matrix (RDM) of the central system. To study evolution of the RDM within a finite time period, the period is divided into a series of short intervals, within each of which the RDM is computed by making use of a formal solution to the time evolution of the environmental branches. The expressions thus obtained are simplified by the ETH ansatz and, further, by decay of phase correlations among the environmental branches, the latter of which also originates from chaotic dynamics of the environment. This gives a generic method of deriving master equation. And, as an application, a master equation is derived in a simplest nontrivial case, which predicts a decoherence rate in agreement with that predicted by the random-matrix theory. Furthermore, the Born approximation, which is employed in the ordinary approach to master equation, can be justified within the proposed framework; and a Markovian feature is shown for the RDM's evolution in an effective sense.
As a model of decohering environment, we show that a quantum chaotic system behaves equivalently as a many-body system. An approximate formula for the time evolution of the reduced density matrix of a system interacting with a quantum chaotic environment is derived. This theoretical formulation is substantiated by the numerical study of the decoherence of two qubits interacting with a quantum chaotic environment modeled by a chaotic kicked top. Like the many-body model of environment, the quantum chaotic system is an efficient decoherer, and it can generate entanglement between the two qubits which have no direct interaction.