Simulation of quantum spin dynamics by phase space sampling of Bogoliubov-Born-Green-Kirkwood-Yvon trajectories

Abstract

A numerical method, suitable for the simulation of the time evolution of quantum spin models of arbitrary lattice dimension, is presented. The method combines sampling of the Wigner function with evolution equations obtained from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Going to higher orders of the BBGKY hierarchy allows for a systematic refinement of the method. Quantum correlations are treated through both, the Wigner function sampling and the BBGKY evolution, bringing about highly accurate estimates of correlation functions. The method is particularly suitable for long-range interacting systems, and we demonstrate its power by comparing with exact results as well as other numerical methods. As an application we compute spin squeezing in a two-dimensional lattice with power-law interactions and a transverse field, which should be accessible in future ion trap experiments.

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

“…In this section, we show how to add corrections to the TWA by following the method proposed in [2], which consists of increasing the number of classical variables and evolving them with BBGKY-type equations. For this we go back to the Wigner-Weyl representation introduced in Section III and work in the Heisenberg picture, where the time dependence is in the Weyl symbols of operators instead of the Wigner function.…”

“…For this we study recently developed extensions of TWA [2,27], adapted to our spin-boson model, which allow one to add corrections order by order and hence estimate the error of the approximation. In this framework corrections are added by solving a truncated Bogoliubov-BornGreen-Kirkwood-Yvon (BBGKY) hierarchy of equations.…”

“…This makes the TWA prediction deviate significantly from the quantum solution already at early times. To cure the self-interaction and, more generically, to add corrections to the spin-boson TWA we follow the method proposed in [2], which is based on using an extended set of equations similar to a BBGKY hierarchy.…”

“…In this work we adapt the TWA to generic spin-boson models by making use of recently developed algorithms for discrete phase spaces [1]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations [2] to our coupled spin-boson model. This allows in principle to study how systematically adding higher order corrections improves the convergence of the method.…”

Spontaneous Beliaev-Landau scattering out of equilibrium

“…As an alternative approach we discuss how a truncated hierarchy of correlations can serve as a consistent expansion beyond the Bogoliubov approximation [37][38][39]. In particular, we will illustrate that the truncation of the hierarchy at the third-order correlation functions, i.e.…”