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Journal of Physics A Mathematical and Theoretical

Lissajous coherent states via projection

Errico J. Russo, James Schneeloch, Edwin E. Hach, Richard J. Birrittella, Wanda Vargas, Christopher C. Gerry

Published Jul 9, 2026• 0 citations
Publisher

Abstract

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. 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.

DOI
10.1088/1751-8121/ae8895

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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. 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.
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.
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.
We demonstrate a formally exact quantum–classical correspondence between the stationary coherent states associated with the commensurate anisotropic two-dimensional harmonic oscillator and the classical Lissajous orbits. Our derivation draws upon the earlier work of Louck et al (1973 J. Math. Phys. 14 692) wherein they have provided a non-bijective canonical transformation that maps, within a degenerate eigenspace, the commensurate anisotropic oscillator on to the isotropic oscillator. This mapping leads, in a natural manner, to a Schwinger realization of SU (2) in terms of the canonically transformed creation and annihilation operators. Through the corresponding coherent states built over a degenerate eigenspace, we directly effect the classical limit via the expectation values of the underlying generators. Our work completely accounts for the fact that the SU (2) coherent state in general corresponds to an ensemble of Lissajous orbits.
The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU (2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.
The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)×O(N ) which implies a unique result for the energy spectrum of the system.Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter.This allows the construction of an infinite number of coherent states to be carried out.In the N = 1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case.Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.