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Physical Review A

Generalized Coleman-Hepp model and quantum coherence

Kaoru Hiyama, Shin Takagi

Published Oct 1, 1993• 34 citations

Abstract

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.

DOI: 10.1103/physreva.48.2586

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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 →∞).

It is well known that environment may decohere a quantum bit (qubit) system immersed in it, making a quantum computation invalid. But the quantitative features of the decoherence seem to depend on both the constitution of the environment and the details of its coupling with the qubit system. In this paper, based on the dynamic approach for quantum measurement developed from the Hepp-Coleman model [K. Hepp, Helv. Phys. Acta 45, 237 (1972)], we generally model the environment as a collection of a large number of subsystems and then consider to what extent and in which way the environment and its coupling with the qubit system may affect a quantum computation process. In the weak-coupling limit, we find that as far as decoherence time is concerned, there is no essential difference between an environment of two-level subsystems and an environment of harmonic oscillators. This implies that there exists some universality independent of specific constitutions of environments. However, it is also shown that this is not true at finite temperature or in the case of strong coupling. So only if the coupling is weak and the temperature low does there exist the possibility of developing a universal scheme of controlling a qubit system such that the decoherence is avoided. The possible effect of environment on the efficiency of a quantum algorithm is also explicitly illustrated through the example of Shor's prime factorization algorithm.

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.

In the view of the quantum dynamic theory of measurement developed from the Hepp-Colemen (HC) model (K. Hepp, Hev.Phys.Acta, {\bf 45}, 237 (1972)), the quantum decoherence in neutrino flavor oscillation caused by an environment surrounding neutrinos is generally considered in this paper. The Ellis, Hagelin, Nanopoulos and Srednicki (EHNS) mechanism for solving the solar neutrino problem can be comprehended in a framework of the ordinary quantum mechanics. In the weak- coupling limit, a microscopic model is proposed to describe the transition of two neutrino system from a pure state to a mixed state. It gives the modified formula of survival probability of neutrino oscillation with two additional time-dependent parameters. For specified environments, this result shows that the oscillating phenomena of neutrino still exist even without a mass difference in free neutrino.