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Journal of the Physical Society of Japan

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems. II

Sachiko Kitajima, 領子 山上, Fumiaki Shibata

Published Jul 15, 2000• 13 citations
Publisher

Abstract

Dynamics and decoherence are studied in a theoretical model in which an incident particle interacts with an array of harmonic oscillators. This system is analogous to the generalized Coleman-Hepp model treated in the previous paper and can also be formulated systematically and solved exactly. In spite of the similarities there exist certain differences between the two. We give detailed numerical calculations and discussions on the decoherence processes.

DOI
10.1143/jpsj.69.2004

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Dynamics and decoherence are studied in a theoretical model in which an incident particle interacts with an array of harmonic oscillators. This system is analogous to the generalized Coleman-Hepp model treated in the previous paper and can also be formulated systematically and solved exactly. In spite of the similarities there exist certain differences between the two. We give detailed numerical calculations and discussions on the decoherence processes.
Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).
Entropy dynamics of the Coleman–Hepp and the boson detector models are solved by calculating the von Neumann entropy exactly. This is done with the use of our previous results of the density matrices and newly obtained entropy formulae. Randomization and recoherence are found for spin entropy of an incident particle of the CH model when an interaction is strong whereas a staircase increase is observed for the weaker interaction. For the boson detector model, a monotonous time evolution of the entropy is found for arbitrary strength of interaction. Entropy dynamics for detector systems are also determined. Response of the detector depends on the interaction strength, magnitude of detector spins and number of constituent elements of the detectors. Reciprocity relation of entropy between an incident particle and a constituent detector element is found.
The modified Coleman-Hepp model proposed recently by Nakazato and Pascazio [Phys. Rev. Lett. 70, 1 (1993)] is further generalized so that an ultrarelativistic particle interacts with a linear array of N spins, each of generic magnitude s rather than 1/2. We study the evolution of an appropriately defined measure C(t) of quantum coherence, with t being time, and elucidate the roles played by N and s on the process of decoherence. It is shown that N and s appear in the form of the product Ns in C(\ensuremath{\infty}) at zero temperature, but not in general. It is also noted that the effect of temperature shows up only for s\ensuremath{\ge}1. We study the temperature dependence of C(\ensuremath{\infty}) in detail, and point out in particular that it is not necessarily a monotonic function of temperature. The limiting cases of s\ensuremath{\rightarrow}\ensuremath{\infty} and/or N\ensuremath{\rightarrow}\ensuremath{\infty} are also considered to investigate how the present model tends to a boson model, that is, a model in which the spins are replaced by harmonic oscillators.