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Journal of the Physical Society of Japan

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems: III

Sachiko Kitajima, Fumiaki Shibata

Published Aug 15, 2001• 11 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.70.2273

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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.
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 →∞).
Quantum mechanical decoherence process is studied with the use of a boson detector model. Generalizing the previous method, thermal effect is taken into account: Coherence of an incident particle is destroyed in interaction with its surroundings at finite temperature. Vanishing of off-diagonal elements of a reduced density matrix is rigorously shown when the detector (surroundings) is composed of a large number of constituents and/or its temperature is very high. The temperature effect reveals itself through a renormalization of interaction strength between an incident particle and the detector. Time evolution of the von Neumann entropy is explicitly determined.
Dynamics of quantum information channel is studied with the use of a boson detector model. Explicit results are given for quantum dense coding process on the basis of an equivalence relation between a classical capacity and von Neumann mutual information. Reduced density matrices for an entangled spin degree of freedom of an incoming particle are exactly solved to give complete time evolution under influence of fluctuations due to a detector. Quantum mechanical channel matrix and the classical capacity are determined as functions of time and temperature. Decoherence properties of the entangled states under quantum and thermal noises are fully discussed.
The entanglement of the N-harmonium system (i.e., a completely-integrable model of N particles where both the confinement and the two-particle interaction are har-monic) is shown to be analytically determined in terms of N and the relative interaction strength. First, the entanglement of the N-boson system via the von Neumann entropy is computed by use of the corresponding natural occupation numbers. Below a critical value Nc of N it is found that, for positive values of the coupling constant the entan-glement grows when the number of particles is increasing; opposite situation occurs for N> Nc. Then the entanglement of the N-fermion system is investigated by means of the linear entropy, which is a linearized form of the von Neumann entropy of the one-body reduced density matrix. We have shown that the entanglement grows when the number of particles is increasing in the region of negative coupling. For positively increasing values of the coupling the entanglement as a function of N varies according to various regimes. Moreover, in the strong coupling regime the entropy of the system decreases when N is growing. For fixed N, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Finally, let us point out that the spatial and spin contributions to the total entanglement are of comparable size.