On the control by electromagnetic fields of quantum systems with infinite dimensional Hilbert space

TL;DR

This paper investigates how electromagnetic fields can control quantum systems with infinite-dimensional Hilbert spaces, providing conditions for finite-dimensional approximation and estimating the required dimension based on control field parameters.

Contribution

It offers a rigorous analysis of finite-dimensional approximation of infinite-dimensional quantum systems under electromagnetic control, with explicit error bounds and dimension estimates.

Findings

Finite-dimensional approximation is possible under certain conditions.

The dimension needed depends on the control field and desired accuracy.

Examples include rigid rotor and harmonic oscillator systems.

Abstract

We analyze the control by electromagnetic fields of quantum systems with infinite dimensional Hilbert space and a discrete spectrum. Based on recent mathematical results, we rigorously show under which conditions such a system can be approximated in a finite dimensional Hilbert space. For a given threshold error, we estimate this finite dimension in terms of the used control field. As illustrative examples, we consider the cases of a rigid rotor and of a harmonic oscillator.