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arXiv:quant-ph/0008090 · Journal of Optics B Quantum and Semiclassical Optics

Effective Hamiltonian approach to the master equation

X X Yi, S X Yu

Published Nov 13, 2001• 24 citations
arXivHTMLPDF

Abstract

A method of exactly solving the master equation is presented in this paper. The explicit form of the solution is determined by the time evolution of a composite system including an auxiliary system and the open system in question. The effective Hamiltonian governing the time evolution of the composed system is derived from the master equation. Two examples, the dissipative two-level system and the damped harmonic oscillator, are presented to illustrate the solution procedure.

DOI
10.1088/1464-4266/3/6/304

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A method of exactly solving the master equation is presented in this paper. The explicit form of the solution is determined by the time evolution of a composite system including an auxiliary system and the open system in question. The effective Hamiltonian governing the time evolution of the composed system is derived from the master equation. Two examples, the dissipative two-level system and the damped harmonic oscillator, are presented to illustrate the solution procedure.
A method of exactly solving the master equation is presented in this letter. The explicit form of the solution is determined by the time evolution of a composite system including an auxiliary system and the open system in question. The effective Hamiltonian governing the time evolution of the composed system are derived from the master equation. Two examples, the dissipative two-level system and the damped harmonic oscillator, are presented to illustrate the solving procedure. PACS number(s): 05.30.-d, 05.40.+j, 42.50.Ct Typeset using REVTEX 1 The problem of open quantum systems has been around since the beginnings of quantum mechanics [1]. Important contributions to this general area have been made by researchers working in fields as diverse as cosmology [2], condensed matter [3], quantum optics [4,5], particle physics [6], quantum measurement [7], and quantum computation [8]. The problem can be described generally as interest in the effective dynamics of the open system surrounded
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the nonunitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped, driven two-level system.
By using the effective Hamiltonian approach, we present a self-consistent framework for the analysis of geometric phases and dynamically stable decoherence-free subspaces in open systems. Comparisons to the earlier works are made. This effective Hamiltonian approach is then extended to a non-Markovian case with the generalized Lindblad master equation. Based on this extended effective Hamiltonian approach, the non-Markovian master equation describing a dissipative two-level system is solved, an adiabatic evolution is defined, and the corresponding adiabatic condition is given.
It is known that a quantum master equation that describes dissipative relaxation (such as spontaneous decay) can be separated in a part that describes spontaneous jumps and a jump-free part. The jump-free part of the evolution is determined by a non-Hermitian Hamiltonian H, which acts on state vectors, rather than on density operators. We identify a class of systems for which the complete evolution is entirely determined by the jump-free evolution, which is governed by H. Although this class is restricted, it contains a sizable number of systems of physical significance. Among these are two-, three-, and four-state atoms. The method is used to find explicit expressions of time-dependent expectation values in terms of the complex eigenvalues of H. As a first application we evaluate exactly the photon statistics in the driven \ensuremath{\Lambda} system. As a second example of the method, we treat the configuration of a V system driven by two standing waves. This system constitutes an effective beam splitter.