Correspondence between open bosonic systems and stochastic differential equations

• URL: http://arxiv.org/abs/2302.01962v2
• Date: Fri, 30 Jun 2023 22:42:29 GMT
• Title: Correspondence between open bosonic systems and stochastic differential equations
• Authors: Alexander Engel and Scott E. Parker
• Abstract summary: 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. A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
• Score: 77.34726150561087
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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