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Physical Review A

Diffusion and dissipation in complex quantum systems

Michael Wilkinson

Published May 1, 1990• 69 citations
Publisher

Abstract

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{} $^{2}$, 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{} $^{3/2}$.

DOI
10.1103/physreva.41.4645

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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.
Considers the effect of slowly varying the parameters X i of a finite-sized quantum mechanical system. The system is excited to higher energies by Landau-Zener transitions at avoided crossing; since this increases the energy of the system, it has the effect of dissipation of the driving motion. The rate of dissipation depends on the level spacing distribution of the system. When the spectral statistics are those of the Gaussian unitary ensemble, the rate of dissipation is proportional to X i 2 , i.e. there is viscous or ohmic damping. When the spectral statistics are of those of the Gaussian orthogonal ensemble, the rate of dissipation is proportional to X i 3 2/.
We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.
We study the perturbative response of a complex quantum system on time changes of an external parameter XXX. The driven dynamics is treated in adiabatic basis of the system's Hamiltonian H^[X]\hat{H}[X]H^[X]. Within a random matrix approach we obtained non--Markovian Fokker--Planck equation for the occupancy of given adiabatic state. We observed normal diffusion regime of the driven quantum dynamics at quite small values of the memory time defined by the time scales of the XXX--correlations and energy--distribution of the coupling matrix elements (∂H^/∂X)nm(\partial \hat{H}/\partial X)_{nm}(∂H^/∂X)nm​. Here the normal energy diffusion was found to drop out with the width of the matrix elements' energy--distribution and the diffusion may be significantly suppressed with the decrease of the correlations between the matrix elements. In the opposite limit of relatively large memory times we obtained ballistic regime of the dynamics.
We introduce a random-variable approach to investigate the dynamics of a dissipative two-state system. Based on an exact functional integral description, our method reformulates the problem as that of the time evolution of a quantum state vector subject to a Hamiltonian containing random noise fields. This numerically exact, nonperturbative formalism is particularly well suited in the context of time-dependent Hamiltonians, at both zero and finite temperature. As an important example, we consider the renowned Landau-Zener problem in the presence of an Ohmic environment with a large cutoff frequency at finite temperature. We investigate the ``scaling'' limit of the problem at intermediate times, where the decay of the upper-spin-state population is universal. Such a dissipative situation may be implemented using a cold-atom bosonic setup.
Within random matrix model, the response of a complex quantum system Ĥ(q[t]) on time variations in an external parameter q. In the limit of weak coupling of the external parameter to the quantum system, the different dynamical regimes of the diffusion in a space of the occupancies of eigenstates of Ĥ(q) are considered. The main focus is made on measuring the role of memory effects in the quantum diffusive dynamics. Possible macroscopic manifestations of the quantum mechanical diffusion are discussed in the context of the cranking approach, where the time variations in the classical parameter q provides the constancy of energy of the total system.
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.