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Chinese Journal of Physics

Generalized Master Equations with Nonhermitian Operators

C. F. Huang, Keh‐Ning Huang

Published Jun 1, 2004• 0 citations
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Abstract

By extending the mean-field Hamiltonian to include nonhermitian operators, the master equations for fermions and bosons can be derived. The derived equations reduce to the Markoff master equation in the low-density limit and to the quasiclassical master equation for homogeneuos systems.

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A quantum master equation is obtained for identical fermions by including a relaxation term in addition to the mean-field Hamiltonian. [Huang C F and Huang K N 2004 Chinese J. Phys. 42{\bf 42}42 221; Gebauer R and Car R 2004 Phys. Rev. B 70{\bf 70}70 125324] It is proven in this paper that both the positivity and Pauli's exclusion principle are preserved under this equation when there exists an upper bound for the transition rate. Such an equation can be generalized to model BCS-type quasiparticles, and is reduced to a Markoff master equation of Lindblad form in the low-density limit with respect to particles or holes.
For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced-density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non-Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time-evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.
In Chapter 9, we derived the master equation for a reduced density operator of a two-level atom coupled to a multi-mode vacuum field. We have seen that the master equation is an operator type equation that, in general, is not easy to solve. We have illustrated the solution of the master equation by representing the density operator in the basis of the two states of the two-level atom. The purpose of the representation of the master equation was to obtain c-number differential equations that are equivalent to the operator equations, but are more readily soluble.