Nonlinear Propagation of Coherent States through Avoided Energy Level Crossing

TL;DR

This paper investigates how wave packets in a nonlinear Schrödinger system behave near an avoided energy level crossing, showing that transitions can be predicted by linear formulas despite nonlinear effects.

Contribution

It demonstrates that nonlinear effects are negligible during the crossing, allowing transition probabilities to be computed using the linear Landau-Zener formula in a nonlinear setting.

Findings

Transitions occur at leading order during the crossing.

Nonlinear effects are negligible near the crossing point.

Transition probabilities match the linear Landau-Zener predictions.

Abstract

We study the propagation of wave packets for a one-dimensional system of two coupled Schr\"odinger equations with a cubic nonlinearity, in the semi-classical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an "avoided crossing": at one given point, the gap between them reduces as the semi-classical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point, there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point, the nonlinearity's role is negligible at leading order, and the transition probabilities can be computed with the linear Landau-Zener formula.