# Time-independent approximations for time-dependent optical potentials

## TL;DR

This paper proposes an approximate method for solving time-dependent quantum systems with optical potentials by combining invariant-based approaches with perturbation or WKB approximations, demonstrated on solvable models.

## Contribution

It introduces a hybrid approach that applies approximate solutions to the invariant eigenvalue problem within the Lewis-Riesenfeld framework for optical potentials.

## Key findings

- The method yields accurate results compared to exact solutions.
- Perturbative and WKB approximations are effective in this context.
- The approach broadens the applicability of invariant-based techniques.

## Abstract

We explore the possibility of modifying the Lewis-Riesenfeld method of invariants developed originally to find exact solutions for time-dependent quantum mechanical systems for the situation in which an exact invariant can be constructed, but the subsequently resulting time-independent eigenvalue system is not solvable exactly. We propose to carry out this step in an approximate fashion, such as employing standard time-independent perturbation theory or the WKB approximation, and subsequently feeding the resulting approximated expressions back into the time-dependent scheme. We illustrate the quality of this approach by contrasting an exactly solvable solution to one obtained with a perturbatively carried out second step for two types of explicitly time-dependent optical potentials.