Examination of classical simulations for Heisenberg-Langevin equations for spin-1/2

TL;DR

This paper explores classical simulations of Heisenberg-Langevin equations for spin-1/0 systems, benchmarking their accuracy against quantum dynamics in different temperature regimes.

Contribution

It introduces a classical ansatz for spin dynamics modeled by Heisenberg-Langevin equations and benchmarks it against quantum results using a modified Hamiltonian.

Findings

Classical simulations approximate quantum spin dynamics at T=0.

Benchmarking shows good agreement in high-temperature limit.

Modified Hamiltonian enables effective comparison with Weisskopf-Wigner theory.

Abstract

A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.