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Journal of Modern Optics

Discrete time dynamical models and their quantum-like context-dependent properties

Andrei Yu. Khrennikov, Yaroslav Volovich

Published Apr 1, 2004• 13 citations
Publisher
DOI
10.1080/09500340408233646

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In this paper we demonstrate that a quantum-like interference pattern could appear as a statistical effect of interference of deterministic particles, i.e. particles that have trajectories and obey deterministic dynamical equations, if one introduces a discrete time. The nature of the resulting interference picture does not follow from the geometry of force field, but is strongly attached to the time discreetness parameter. As a demonstration of this concept we consider several numerical experiments of scattering of charged particles on the charged screen with a single slit. The resulting interference picture has a nontrivial minimum-maximum distribution which vanishes, as the time discreetness parameter goes to zero, that could be interpreted as an analog of quantum decoherence. This picture is qualitatively the same as the one one obtained in quantum experiments.
The feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spin-boson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration time-dependent Hartree context, is used to solve the time-dependent Schrödinger equations by explicitly including all the degrees of freedom (“system”+“bath”). Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of time-dependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.