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arXiv:2003.09258 · Physical review. E

Closeness of the reduced density matrix of an interacting small system to the Gibbs state

Wen-ge Wang

Published Jul 13, 2020• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.102.012127

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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 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.
In this paper, we employ a semiperturbative theory to study the statistical structural properties of energy eigenfunctions (EFs) in many-body quantum chaotic systems consisting of a central system coupled to an environment. Under certain assumptions, we derive both the average shape and the statistical fluctuations of EFs on the basis formed by the direct product of the energy eigenbases of the system and the environment. Furthermore, we apply our results to two fundamental questions: (i) the properties of the reduced density matrix of the central system in an eigenstate, and (ii) the structure of the off-diagonal smooth function within the framework of the eigenstate thermalization hypothesis. Numerical results are also presented in support of our main findings.
Abstract In this paper, we employ a semiperturbative theory to study the statistical structural properties of energy eigenfunctions (EFs) in many-body quantum chaotic systems consisting of a central system coupled to an environment. Under certain assumptions, we derive both the average shape and the statistical fluctuations of EFs on the basis formed by the direct product of the energy eigenbases of the system and the environment. Furthermore, we apply our results to two fundamental questions: (i) the properties of the reduced density matrix of the central system in an eigenstate, and (ii) the structure of the off-diagonal smooth function within the framework of the eigenstate thermalization hypothesis. Numerical results are also presented in support of our main findings.
We study the problem of the basis of an open quantum system, under a quantum chaotic environment, which is preferred in view of its stationary reduced density matrix (RDM), that is, the basis in which the stationary RDM is diagonal. It is shown that, under an initial condition composed of sufficiently many energy eigenstates of the total system, such a basis is given by the eigenbasis of a renormalized self-Hamiltonian of the system, in the limit of large Hilbert space of the environment. Here, the renormalized self-Hamiltonian is given by the unperturbed self-Hamiltonian plus a certain average of the interaction Hamiltonian over the environmental degrees of freedom. Numerical simulations performed in two models, both with the kicked rotor as the environment, give results consistent with the above analytical predictions.
An explicit expression is derived for the statistical description of small quantum systems, which are relatively weakly and directly coupled to only small parts of their environments. The derived expression has a canonical form, but is given by a renormalized self-Hamiltonian of the studied system, which appropriately takes into account the influence of the system-environment interaction. In the case that the system has a narrow spectrum and the environment is sufficiently large, the modification to the self-Hamiltonian usually has a mean-field feature, given by an environmental average of the interaction Hamiltonian. In other cases, the modification may be beyond the mean-field approximation.