A new fast numerical method for the generalized Rosen-Zener model

TL;DR

This paper introduces a fast numerical method for solving the generalized Rosen-Zener model's linear ODE system, leveraging an infinite matrix equation and low-rank approximations to achieve linear scaling in computation time.

Contribution

It presents a novel numerical approach based on the $igstar$-product and matrix truncation techniques for efficiently solving large-scale generalized Rosen-Zener models.

Findings

Computing time scales linearly with model size.

Method efficiently approximates the solution operator.

Numerical experiments confirm the method's effectiveness.

Abstract

In quantum mechanics, the Rosen-Zener model represents a two-level quantum system. Its generalization to multiple degenerate sets of states leads to larger non-autonomous linear system of ordinary differential equations (ODEs). We propose a new method for computing the solution operator of this system of ODEs. This new method is based on a recently introduced expression of the solution in terms of an infinite matrix equation, which can be efficiently approximated by combining truncation, fixed point iterations, and low-rank approximation. This expression is possible thanks to the so-called $\star$-product approach for linear ODEs. In the numerical experiments, the new method's computing time scales linearly with the model's size. We provide a first partial explanation of this linear behavior.