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arXiv:quant-ph/0604054

On the quantum master equation for fermions

C. F. Huang, Keh‐Ning Huang

Published Apr 9, 2006• 0 citations
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Abstract

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. ${\bf 42}$ 221; Gebauer R and Car R 2004 Phys. Rev. B ${\bf 70}$ 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.

DOI
10.48550/arxiv.quant-ph/0604054

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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.
Microscopic master equations have gained traction for the dissipative treatment of molecular spin and solid-state systems for quantum technologies. Single particle approximations are often invoked to treat these systems, which can lead to unphysical evolution when combined with master equation approaches. We present a mathematical constraint on the system-environment parameters to ensure that microscopically-derived Markovian master equations preserve fermionic, NNN-representable statistics when applied to reduced systems. We demonstrate these constraints for the recently derived unified master equation and universal Lindblad equation, along with the Redfield master equation for cases when positivity issues are not present. For operators that break the constraint, we explore the addition of Pauli factors to recover NNN-representability. This work promotes feasible applications of novel microscopic master equations for realistic chemical systems.
An essential problem of the quantum information systems is the controllability and observability of the quantum states, generally described by Lindblad's master equation with phenomenological coefficients. However, in its general form, this equation describes a decay of the mean-values, but not necessarily the expected decaying transitions. We propose a quantum master equation with microscopic coefficients where these transitions are correctly described, and investigate the properties of the dissipative coefficients. Being of Lindblad's form, this equation preserves the quantum-mechanical properties of the density matrix during the whole evolution of the system and is in accordance with the entropy principle. On this basis, generalized Bloch-Feynman equations with microscopic coefficients for N-level systems are obtained.
The propagation of a fast particle in a low-density gas at thermal equilibrium is studied in the context of quantum mechanics. A quantum master equation in the Redfield form governing the reduced density matrix of the particle is derived explicitly from first principles. Under some approximations, this equation reduces to the linear Boltzmann equation. The issue of the positivity of the time evolution is also discussed by means of a Lindblad form. The Born and Markov assumptions underlying these equations, as well as other approximations regarding the bath correlation function, are discussed in detail. Furthermore, all these master equations are shown to be equivalent with each other if the density matrix of the particle is diagonal in the momentum basis, or if the collision rate is independent of the particle momentum.