Lissajous coherent states via projection

TL;DR

This paper constructs stationary coherent states on Lissajous figures for harmonic oscillators with arbitrary frequencies, clarifying their phase singularities, quantum interference, and vortex states, using projections of product coherent states.

Contribution

It introduces a method to generate Lissajous coherent states via projection, connecting classical trajectories with quantum states and analyzing their phase and interference properties.

Findings

States follow classical Lissajous trajectories

Established a link between probability flow and quantum interference

Provided a rigorous definition of vortex states in 2D harmonic oscillators

Abstract

We construct stationary coherent states concentrated on Lissajous figures of the isotropic and anisotropic harmonic oscillators, the latter having coprime frequencies, by projecting products of ordinary coherent states (one coherent state for each degree of freedom) onto sets of degenerate states. By performing these projections, we are deriving our states from sets of coherent states that are known to follow the classical motion of a two-dimensional harmonic oscillator for arbitrary frequencies. We clarify the nature of any singularities present in the phase of the wavefunction for each of the states we derive, and we establish a rigorous connection between the laminar flow of probability current and the emergence of quantum interference. Through this analysis, we are able to provide a clear and quantifiable definition for a vortex state of the two-dimensional harmonic oscillator (2DHO). In an appendix, we show that our stationary states are true coherent states as they can be used to resolve the relevant identity operators (the above mentioned projection operators) on their respective degenerate subspaces. In the special case of the isotropic oscillator, the states obtained are the SU(2) coherent states, and we derive from our formalism the familiar resolution of unity for these states.