Correspondence between open bosonic systems and stochastic differential equations

TL;DR

This paper establishes an exact correspondence between finite bosonic systems with environmental interactions and stochastic differential equations, providing insights into quantum-classical limits and potential quantum algorithms for stochastic nonlinear differential equations.

Contribution

It introduces a novel exact mapping between finite bosonic systems with environment interactions and stochastic differential equations, extending mean-field theory.

Findings

Exact correspondence at finite n between bosonic systems and stochastic equations

Stochastic terms vanish as n approaches infinity, recovering mean-field theory

Analysis of a stochastic discrete nonlinear Schrödinger equation

Abstract

Bosonic mean-field theories can approximate the dynamics of systems of $n$ bosons provided that $n \gg 1$. We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment and the mean-field theory is replaced by a stochastic differential equation. When the $n \to \infty$ limit is taken, the stochastic terms in this differential equation vanish, and a mean-field theory is recovered. Besides providing insight into the differences between the behavior of finite quantum systems and their classical limits given by $n \to \infty$, the developed mathematics can provide a basis for quantum algorithms that solve some stochastic nonlinear differential equations. We discuss conditions on the efficiency of these quantum algorithms, with a focus on the possibility for the complexity to be polynomial in the log of the stochastic system size. A particular system with the form of a stochastic discrete nonlinear Schr\"{o}dinger equation is analyzed in more detail.