Quantum Lissajous Figures via Projection

TL;DR

This paper introduces quantum Lissajous states derived from projecting coherent states onto a degenerate subspace of a 2D harmonic oscillator, linking quantum probability densities to classical Lissajous figures and exploring quantum interference effects.

Contribution

It presents a novel class of quantum states associated with Lissajous figures, created through projection of coherent states, revealing new non-classical stationary states.

Findings

States have probability densities localized along classical Lissajous figures.

The work highlights the role of probability current density in quantum interference.

Discussion includes the emergence of vortex states in the quantum system.

Abstract

We present a new category of quantum Lissajous states for a 2DHO having commensurate angular frequencies. The states result from the projection of ordinary coherent states onto a degenerate subspace of the 2DHO. In this way, new, non-classical quantum mechanically stationary states arise from the classical but non-stationary coherent states. The connection to Lissajous figures is that our states all have probability densities that are localized along the corresponding classical Lissajous figures. We further emphasize the important interplay between the probability current density and the emergence of quantum interference in the states we examine. In doing so, we are able to present a consistent discussion of a class of states known as vortex states.