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Journal of the Physical Society of Japan

Entropy of the Generalized Coleman-Hepp Model

Sachiko Kitajima

Published Nov 15, 2000• 5 citations
Publisher

Abstract

Relaxation and decoherence properties of the exactly solvable Coleman-Hepp model are studied by determining the time evolution of the entropy. With the aid of our previous results, we can calculate the entropy by solving an eigenvalue problem for the reduced density matrices. Detailed numerical studies are performed to clarify the decoherence processes.

DOI
10.1143/jpsj.69.3477

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Relaxation and decoherence properties of the exactly solvable Coleman-Hepp model are studied by determining the time evolution of the entropy. With the aid of our previous results, we can calculate the entropy by solving an eigenvalue problem for the reduced density matrices. Detailed numerical studies are performed to clarify 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.
Time-evolutions of two-level and multi-level systems in phase relaxation processes are investigated in a systematic way. A general form of the decoherence function which determines phase relaxation of a quantum state is obtained by making use of a microscopic interaction Hamiltonian between the relevant system and reservoir. Furthermore the loss of the phase coherence is described in terms of the Pegg–Barnett phase operator. The decoherence function is explicitly calculated for the stochastic dephasing, the boson detector model and the Coleman–Hepp model.
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.