Derivation of nonlinear single-particle equations via many-body Lindblad superoperators: A density-matrix approach

TL;DR

This paper derives a nonlinear, positive-definite single-particle density matrix equation from many-body Lindblad superoperators, ensuring physical consistency in quantum device modeling, and extends it to open boundary systems.

Contribution

It introduces a mean-field approximation leading to a nonlinear, non-Lindblad single-particle equation that preserves positivity, contrasting with traditional approaches.

Findings

The derived equation maintains positive-definiteness of the density matrix.

The approach extends to systems with open boundaries.

It provides a formal basis for density-matrix treatments in quantum devices.

Abstract

A recently proposed Markov approach provides Lindblad-type scattering superoperators, which ensure the physical (positive-definite) character of the many-body density matrix. We apply the mean-field approximation to such many-body equation, in the presence of one- and two-body scattering mechanisms, and we derive a closed equation of motion for the electronic single-particle density matrix, which turns out to be non-linear as well as non-Lindblad. We prove that, in spite of its nonlinear and non-Lindblad structure, the mean-field approximation does preserve the positive-definite character of the single-particle density matrix, an essential prerequisite of any reliable kinetic treatment of semiconductor quantum devices. This result is in striking contrast with conventional (non-Lindblad) Markov approaches, where the single-particle mean-field equations can lead to positivity violations and to unphysical results. Furthermore, the proposed single-particle formulation is extended to the case of quantum systems with spatial open boundaries, providing a formal derivation of a recently proposed density-matrix treatment based on a Lindblad-like system-reservoir scattering superoperator.