Wigner Analysis of Particle Dynamics and Decoherence in Wide Nonharmonic Potentials

Abstract

We derive an analytical expression of a Wigner function that approximately describes the time evolution of the one-dimensional motion of a particle in a nonharmonic potential. Our method involves two exact frame transformations, accounting for both the classical dynamics of the centroid of the initial state and the rotation and squeezing about that trajectory. Subsequently, we employ two crucial approximations, namely the constant-angle and linearized-decoherence approximations, upon which our results rely. These approximations are effective in the regime of wide potentials and small fluctuations, namely potentials that enable spatial expansions orders of magnitude larger than the one of the initial state but that remain smaller compared to the relevant dynamical length scale (e.g., the distance between turning points). Our analytical result elucidates the interplay between classical and quantum physics and the impact of decoherence during nonlinear dynamics. This analytical result is instrumental to designing, optimizing, and understanding proposals using nonlinear dynamics to generate macroscopic quantum states of massive particles.