On semi-classical spectral series for an atom in a periodic polarized electric field

TL;DR

This paper investigates the semi-classical spectral properties of an atom in a periodic polarized electric field, focusing on the tunneling effects and eigenvalue splitting of a magnetic Schr"odinger operator with a double well.

Contribution

It introduces a novel analysis of the spectral series for an atom under a periodic polarized electric field using semi-classical methods and magnetic Schr"odinger operators.

Findings

Construction of semi-classical ground states for the magnetic Schr"odinger operator.

Analysis of eigenvalue splitting in the double magnetic well scenario.

Insights into tunneling phenomena in quantum systems with periodic fields.

Abstract

In this report we present preliminary results about the tunneling problem for a magnetic Schr\"odinger operator. As a motivation we consider the 3-D time-dependent Schr\"odinger operator $H(t)=-h^2\Delta+V+E(t)\cdot x$ where $V$ is a radial potential and $E(t)$ a circularly polarized field with uniform frequency $\omega$. The quantum monodromy operator (QMO) that takes the system through a complete period $T=2\pi/\omega$, turns out to be unitarily equivalent to $e^{iTP_A(x,hD_x)/h}$, where $P_A(x,hD_x))$ identifies with a magnetic Schr\"odinger operator. When $V$ is sufficiently confining, $P_A(x,hD_x))$ presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.