An implicit split-operator algorithm for the nonlinear time-dependent Schr\"{o}dinger equation

TL;DR

This paper introduces high-order implicit split-operator algorithms for solving nonlinear time-dependent Schrödinger equations, overcoming limitations of explicit methods by ensuring norm conservation, time reversibility, and improved efficiency for separable Hamiltonians.

Contribution

It presents a novel family of implicit split-operator algorithms that are high-order, norm-conserving, and time-reversible, addressing inefficiencies of explicit methods in nonlinear Schrödinger equations.

Findings

Algorithms are proven to be norm-conserving and time-reversible.

Numerical demonstrations on a 2D retinal model show improved efficiency.

Applicable to separable Hamiltonians, outperforming some implicit midpoint methods.

Abstract

The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schr\"{o}dinger equations. However, when applied to certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithm loses time reversibility and second-order accuracy, which makes it very inefficient. Here, we propose to overcome the limitations of the explicit split-operator algorithm by abandoning its explicit nature. We describe a family of high-order implicit split-operator algorithms that are norm-conserving, time-reversible, and very efficient. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. Although they are only applicable to separable Hamiltonians, the implicit split-operator algorithms are, in this setting, more efficient than the recently proposed integrators based on the implicit midpoint method.