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arXiv:0711.2356

On a Random Matrix Models of Quantum Relaxation

Joel L. Lebowitz, Anna Lytova, Leonid Andreevich Pastur

Published Nov 15, 2007• 0 citations
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Abstract

Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level system interacting with a $n\times n$ reservoir and assuming that the interaction is modelled by a random matrix. We presented there a formula for the reduced density matrix in the limit $n\to \infty $ as well as several its properties and asymptotic forms in various regimes. In this paper we give the proofs of the assertions, and present also a new fact about the model.

DOI
10.48550/arxiv.0711.2356

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Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level system interacting with a n×nn\times nn×n reservoir and assuming that the interaction is modelled by a random matrix. We presented there a formula for the reduced density matrix in the limit n→∞n\to \infty n→∞ as well as several its properties and asymptotic forms in various regimes. In this paper we give the proofs of the assertions, and present also a new fact about the model.
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We consider a random matrix model of interaction between a small nnn-level system, SSS, and its environment, a NNN-level heat reservoir, RRR. The interaction between SSS and RRR is modeled by a tensor product of a fixed % n\times n matrix and a N×NN\times NN×N hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on RRR, the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by ρS(c)∼exp⁡{−βHS}ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}ρS(c)​∼exp{−βHS​}, where HSH_{S}HS​ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using % ρ_{S}^{(c)} to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.
We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.
We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment.We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso-and even nanosize.We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition.We then workout an analog of Born-Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord.We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits.In particular, we find the vanishing of the entanglement at finite moments and its subsequent revivals (entanglement sudden death and entanglement sudden birth).Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.
Abstract The model of a multilevel system interacting with several reservoirs is considered. The exact reduced evolution of a system’s density matrix can be obtained for this model without using the Markov approximation. Namely, this evolution is completely defined by the finite set of linear differential equations. The results obtained earlier for one Lorentz peak in the spectral density are generalized to the case of an arbitrary number of such peaks. The contribution of Ohmic spectral density is also considered.