Lissajous dynamics of a quantum particle in a tilted two-dimensional discrete lattice

TL;DR

This paper explores how a quantum particle in a tilted 2D lattice exhibits Lissajous curve trajectories, linking quantum dynamics with classical oscillatory motion through parameter tuning.

Contribution

It demonstrates the classical-quantum correspondence by showing how lattice parameters influence quantum trajectories to follow classical Lissajous curves.

Findings

Probability distribution shape remains constant during evolution.

Particle's center follows classical Lissajous trajectories.

Parameters can be tuned to control quantum motion.

Abstract

The quantum dynamics of a single particle in a discrete two-dimensional tilted lattice is analyzed from the perspective of the classical-quantum correspondence. Utilizing the fact that tilting the lattice results in oscillatory dynamics, we show how the parameters of the lattice and the initial state of the particle can be tuned so that during evolution the probability distribution does not change its shape while its center follows the trajectory known in classical mechanics as Lissajous curves.