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Exact time-dependent decoherence factor and its adiabatic classical limit

Shen, J Q, Chen, P, Hong Mao

Published Feb 15, 2003• 0 citations
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Abstract

The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what obtained by other authors, and futhermore we obtain the more general results concerning the time-dependent non-adiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor.

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The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what obtained by other authors, and futhermore we obtain the more general results concerning the time-dependent non-adiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor.
The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the Lewis–Riesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what is obtained by other authors, and furthermore we obtain more general results concerning time-dependent nonadiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor. PACS Nos.: 03.65.Ge, 03.65.Bz
There exist a number of typical and interesting systems or models which possess three-generator Lie-algebraic structure in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl(2,C)sl(2,C)sl(2,C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schrödinger equations governing various three-generator quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, we show that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors in time-dependent systems is briefly discussed.
An alternative concept of the basic invariants is introduced. The Lewis-Riesenfeld invariant theory is extended to obtain a generalized invariant formulation. The formulation is then used to establish four facts: (i) Any invariant for a quantum system can be constructed in terms of the basic invariants. (ii) It is possible to introduce a solution-generating technique by making use of the basic invariants. (iii) The path integral in the generalized invariant formulation reduces to an ordinary integral. (iv) The study of noncyclic evolution of a quantum system reduces explicitly to the study of the cyclic evolution. Finally, phase factors and general solution for the driven generalized time-dependent harmonic oscillator are studied as an illustrative example.
The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.