Lissajous coherent states via projection

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.

Figures (10)

• Figure 1: Probability density plots for the two independent standing wave Lissajous coherent states for the isotropic two-dimensional harmonic oscillator. Parts (a) and (b) show the surface and contour plots, respectively, for states satisfying the condition stated in Eq. (\ref{['realwfphasecond']}) with $n=0,\pm 2,\pm 4,...$, and Part (c) shows the contour plot for states having $n=\pm 1,\pm 3,\pm 5, ...$. For each of these cases the probability current density vanishes identically and, as discussed in the text, the interference fringes are maximally visible.
• Figure 2: Probability, (a) and (b), and probability current, (c), density plots for a Lissajous Coherent State at the so-called vortex limit for the isotropic two-dimensional harmonic oscillator.
• Figure 3: Surface (a) and contour (b) plots for the phase of the wavefunction $\chi\left(x,y\right)$ for $N=20$ of the vortex limit for Lissajous coherent states of the 2DHO along with the corresponding steady probability current density (c). The white 'dot' at the origin of (b) represents the non-physical, essential singularity in the phase at this point. The white rays extending radially from the essential singularity are jump discontinuities in the phase that are required to ensure the steady nature of the circulating probability current. The vertical axis of (a) gives the phase of the wavefunction $\mod2\pi$ between $-\pi$ and $\pi$.
• Figure 4: An intermediate vortex state for the isotropic 2DHO. Here the phase of the complex amplitude $\zeta$ is chosen to be midway between the condition for a standing wave state and that for the vortex limit as given by Eqs. (\ref{['realwfphasecond']}) and (\ref{['eq:vortexphase']}) (a) Surface plot of the probability density. (b) Contour plot of the probability density with the probability current density inlaid to emphasize the localization of each around the corresponding classical Lissajous figure (white ellipse). (c) Contour plot of the phase of the wavefunction $\chi\left(x,y\right)$ for $N=20$. The trivial essential singularity has stretched into a line segment along the major axis of the elliptical support for the probability and probability current densities. As in the case of the vortex limit, the rays of apparent jump discontinuities are simply artifacts of our restriction to the interval $[-\pi,\pi]\mod 2\pi$.
• Figure 5: Another intermediate vortex state for the isotropic 2DHO, this one close to the standing wave limit than the one in Fig.(\ref{['fig4:IsoVortexIntermediate']}). Specifically, here the phase of the complex amplitude $\zeta$ is chosen to be $\pi/8$. Parts (a), (b), and (c) are directly analogous to the same parts of Fig.(\ref{['fig4:IsoVortexIntermediate']}) for the new choice of complex amplitude.