Exponential Integrators for Stochastic Schr\"odinger Equation

TL;DR

This paper introduces exponential integrators for solving the stochastic Schrödinger equation, enabling efficient and accurate simulations of open quantum systems through matrix exponentials and Krylov methods.

Contribution

The paper develops a new class of exponential integrators tailored for stochastic Schrödinger equations, incorporating Kunita's representation and third-order commutators for improved accuracy.

Findings

Methods achieve strong and weak convergence.

Local accuracy improved with third-order commutator.

Validated on quantum system example from literature.

Abstract

We present a class of exponential integrators to compute solutions of the stochastic Schr\"odinger equation arising from the modeling of open quantum systems. In order to be able to implement the methods within the same framework as the deterministic counterpart, we express the solution using the Kunita's representation. With appropriate truncations, the solution operator can be written as matrix exponentials, which can be efficiently implemented by the Krylov subspace projection. The accuracy is examined in terms of the strong convergence, by comparing trajectories, and the weak convergence, by comparing the density-matrix operator. We show that the local accuracy can be further improved by introducing a third-order commutator in the exponential. The effectiveness of the proposed methods is tested using the example from Di Ventra et al. [Journal of Physics: Condensed Matter, 2004].