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Physical Review B

Emergence of diffusion in finite quantum systems

Massimiliano Esposito, Pierre Gaspard

Published Jun 14, 2005• 54 citations
Publisher

Abstract

We study the emergence of diffusion for a quantum particle moving in a finite and translationally invariant one-dimensional subsystem described by a tight-binding Hamiltonian with a single energy band and interacting with its environment by an interaction energy proportional to some coupling parameter. We show that there exists a crossover between a nondiffusive relaxation regime for small sizes or low values of the coupling parameter and a diffusive regime above a critical size or for higher values of the coupling parameter. In the nondiffusive regime, the relaxation is characterized by oscillations decaying at rates independent of the size and proportional to the square of the coupling parameter and the temperature of the environment. In the diffusive regime, the damped oscillations have disappeared and the relaxation rate is inversely proportional to the square of the size. The diffusion coefficient is proportional to the square of the energy bandwidth of the subsystem and inversely proportional to the temperature of the environment and the square of the coupling parameter. The critical size where the crossover happens is obtained analytically.

DOI
10.1103/physrevb.71.214302

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We study the emergence of diffusion for a quantum particle moving in a finite and translationally invariant one-dimensional subsystem described by a tight-binding Hamiltonian with a single energy band and interacting with its environment by an interaction energy proportional to some coupling parameter. We show that there exists a crossover between a nondiffusive relaxation regime for small sizes or low values of the coupling parameter and a diffusive regime above a critical size or for higher values of the coupling parameter. In the nondiffusive regime, the relaxation is characterized by oscillations decaying at rates independent of the size and proportional to the square of the coupling parameter and the temperature of the environment. In the diffusive regime, the damped oscillations have disappeared and the relaxation rate is inversely proportional to the square of the size. The diffusion coefficient is proportional to the square of the energy bandwidth of the subsystem and inversely proportional to the temperature of the environment and the square of the coupling parameter. The critical size where the crossover happens is obtained analytically.
We study the transport property of diffusion in a finite translationally invari-ant quantum subsystem described by a tight-binding Hamiltonian with a sin-gle energy band. The subsystem interacts with its environment by a coupling expressed in terms of correlation functions which are delta-correlated in space and time. For weak coupling, the time evolution of the subsystem density matrix is ruled by a quantum master equation of Lindblad type. Thanks to the invariance under spatial translations, we can apply the Bloch theorem to the subsystem density matrix and exactly diagonalize the time evolution superop-erator to obtain the complete spectrum of its eigenvalues, which fully describe the relaxation to equilibrium. Above a critical coupling which is inversely pro-portional to the size of the subsystem, the spectrum at given wave number con-tains an isolated eigenvalue describing diffusion. The other eigenvalues rule the decay of the populations and quantum coherences with decay rates which are proportional to the intensity of the environmental noise. An analytical expres-sion is obtained for the dispersion relation of diffusion. The diffusion coef-ficient is proportional to the square of the width of the energy band and inversely proportional to the intensity of the environmental noise because diffu-sion results from the perturbation of quantum tunneling by the environmental fluctuations in this model. Diffusion disappears below the critical coupling. KEY WORDS: Quantum transport; environment-induced diffusion; quantum
We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is put forward as a quantitative measure of the degree of equilibration in the system, from the standpoint of the chosen basis. This approach is illustrated on the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.
Abstract We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is shown to measure the degree of equilibration of the system launched from a typical state, from the standpoint of the chosen basis. This approach is substantiated by an in-depth study of the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.
In this thesis, we investigate the emergence of kinetic processes in finite quantum systems. We first generalize the Redfield theory to describe the dynamics of a small quantum system weakly interacting with an environment of finite heat capacity. We then study in detail the spin-GORM model, a model made of a two-level system interacting with a random matrix environment. By doing this, we verify our new theory and find a critical size of the environment over which kinetic processes occur. We finally study the emergence of a diffusive transport process, on a finite tight-binding subsystem interacting with a fast environment, when the size of subsystem exceeds a critical value.
The Hamiltonian of a quantum-mechanical system contains a time-dependent parameter X. The system is prepared in a highly excited eigenstate of the instantaneous Hamiltonian at t=0, and we compute the amplitudes 〈n(t)\ensuremath{\Vert}\ensuremath{\psi}(t)〉 to be in the eigenstates \ensuremath{\Vert}n〉 of the instantaneous Hamiltonian H^(X(t)) at later times. It is found that for generic systems, with no symmetries yielding additional quantum numbers, the occupation probability spreads diffusively away from the initial state. This diffusion is an irreversible process, and can be related to a model for dissipation. The diffusion constant D is investigated numerically in a random matrix model as a function of X\ifmmode \dot{}\else \.{}\fi{}, the rate of change of the parameter. Good agreement with theoretical predictions is found in two limiting regimes. When X\ifmmode \dot{}\else \.{}\fi{} is large, D is proportional to X\ifmmode \dot{}\else \.{}\fi{} 2^{2}2, corresponding to the Ohmic dissipation predicted by the Kubo formula. When X\ifmmode \dot{}\else \.{}\fi{} is small, Landau-Zener transitions are the mechanism for diffusion, and D is proportional to X\ifmmode \dot{}\else \.{}\fi{} 3/2^{3/2}3/2.
The process of diffusion constitutes the dissipative motion of a classical particle or group of particles subjected to many-body interactions with their surroundings. When the surroundings consist of an infinitely large system that can rapidly recover from any deviation from thermal equilibrium such that its temperature remains roughly uniform, the system qualifies as a heat bath. Further, when the degrees of freedom of the heat bath are integrated out because of the implied timescale separation, the dynamics of the subsystem appear dissipative, characterized by parameters such as friction coefficient, diffusion constant, and others.