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arXiv:nucl-th/9612011 · Chaos Solitons & Fractals

Quantum diffusion and tunneling with parametric banded random matrix hamiltonians

Aurel Bulgac, G. Do Dang, Dimitri Kusnezov

Published Jul 1, 1997• 7 citations
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Abstract

The microscopic origin of dissipation of a driven quantum many-body system is addressed in the framework of a parametric banded random matrix approach. We find noticeable violations of the fluctuation-dissipation theorem and we observe also that the energy diffusion has a markedly non-Gaussian character. Within the Feynman-Vernon path integral formalism and in the Markovian limit, we further consider the time evolution of a slow subsystem coupled to such a ‘bath’ of intrinsic degrees of freedom. We show that dissipation leads to qualitative modifications of the time evolution of the density matrix of the slow subsystem. In either the spatial, momentum or energy representation the density distribution acquires very long tails and tunneling is greatly enhanced.

DOI
10.1016/s0960-0779(97)00013-1

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The microscopic origin of dissipation of a driven quantum many-body system is addressed in the framework of a parametric banded random matrix approach. We find noticeable violations of the fluctuation-dissipation theorem and we observe also that the energy diffusion has a markedly non-Gaussian character. Within the Feynman-Vernon path integral formalism and in the Markovian limit, we further consider the time evolution of a slow subsystem coupled to such a ‘bath’ of intrinsic degrees of freedom. We show that dissipation leads to qualitative modifications of the time evolution of the density matrix of the slow subsystem. In either the spatial, momentum or energy representation the density distribution acquires very long tails and tunneling is greatly enhanced.
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 present a theory for the dynamical evolution of a quantum system coupled to a complex many-body intrinsic system (environment). By modeling the intrinsic many-body system with parametric random matrices, we study the types of effective stochastic models that emerge from random matrix theory. Using the Feynman-Vernon path integral formalism, we derive the influence functional and obtain either analytical or numerical solutions for the time evolution of the entire quantum system. We discuss thoroughly the structure of the solutions for some representative cases and make connections to well known limiting results, particularly to Brownian motion, Kramers classical limit, and the Caldeira-Leggett approach.
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 consider a quantum system coupled to a dissipative background with many degrees of freedom using the Monte Carlo wave function method. Instead of dealing with a density matrix which can be very highly dimensional, the method consists of integrating a stochastic Schrödinger equation with a non-Hermitian damping term in the evolution operator, and with random quantum jumps. The method is applied to the diffusion of hydrogen on the Ni(111) surface below 100 K. We show that the recent experimental diffusion data for this system can be understood through an interband activation process, followed by quantum tunneling.
Both in atomic and in mesoscopic physics it is interesting to consider the energy time dependence of a parametrically driven chaotic system. We assume an Hamiltonian H(Q,P;x(t))H(Q,P;x(t))H(Q,P;x(t)) where x(t)=Vtx(t)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}Vtx(t)=Vt. The velocity VVV is slow in the classical sense but not necessarily in the quantum-mechanical sense. The crossover (in time) from ballistic to diffusive energy spreading is studied. Dissipation is the associated irreversible growth of the average energy. It is found that a dimensionless velocity vPR{v}_{\mathrm{PR}}vPR​ determines the nature of the dynamics, and controls the route towards quantal-classical correspondence. A perturbative regime and a nonperturbative semiclassical regime are distinguished.