Many-particle Sudarshan-Lindblad equation: mean-field approximation, nonlinearity and dissipation in a spin system

TL;DR

This paper explores the dynamics of a many-particle spin system interacting with a thermal reservoir, introducing mean-field approximations and analyzing nonlinearity and dissipation effects through analytical and numerical solutions of the master equation.

Contribution

It presents a pedagogical derivation of the nonlinear mean-field approximation for the Sudarshan-Lindblad equation in a spin system, including analytical solutions and discussion of higher-order correlations.

Findings

Derived the quantum BBGKY hierarchy for the system

Obtained analytical solutions for mean values and stationary states

Demonstrated numerical solutions of the nonlinear master equation

Abstract

A system of $N$ spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system $\mathcal{S}$ interacting with a thermal reservoir $\mathcal{R}$. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of $N-1$ equations ($N>1$). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of $\mathcal{S}$), although the superoperator responsible for the dissipation and decoherence of $\mathcal{S}$ remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected.