# Nonequilibrium dynamics of spin-boson models from phase space methods

## TL;DR

This paper adapts phase space methods, specifically the Truncated Wigner Approximation and BBGKY hierarchy, to study the nonequilibrium dynamics of spin-boson models across various dimensions and system sizes.

## Contribution

It introduces a scalable phase space approach for spin-boson models and extends it with BBGKY hierarchy to systematically improve accuracy.

## Key findings

- Accurately reproduces time evolution of correlation functions in spin-boson models.
- Demonstrates applicability to large systems and higher dimensions.
- Shows systematic convergence with higher order BBGKY corrections.

## Abstract

An accurate description of the nonequilibrium dynamics of systems with coupled spin and bosonic degrees of freedom remains theoretically challenging, especially for large system sizes and in higher than one dimension. Phase space methods such as the Truncated Wigner Approximation (TWA) have the advantage of being easily scalable and applicable to arbitrary dimensions. In this work we adapt the TWA to generic spin-boson models by making use of recently developed algorithms for discrete phase spaces [Schachenmayer, PRX 5, 011022 (2015)]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations [Pucci, PRB 93, 174302 (2016)] to our coupled spin-boson model. This allows in principle to study how systematically adding higher order corrections improves the convergence of the method. To test various levels of approximation we study an exactly solvable spin-boson model which is particularly relevant for trapped-ion arrays. Using TWA and its BBGKY extension we accurately reproduce the time evolution of a number of one- and two-point correlation functions in several dimensions and for arbitrary number of bosonic modes.