Manipulation of Semiclassical Photon States

TL;DR

This paper studies operators manipulating low-dimensional Hermite-Gaussian photon modes, analyzing their self-adjoint extensions and semiclassical propagator approximations, revealing detailed Hamiltonian flow and effective behavior on localized states.

Contribution

It extends the analysis of mode-manipulation operators by characterizing their self-adjoint realizations and developing semiclassical approximations of their propagators.

Findings

Operators have infinite deficiency indices and explicit self-adjoint extensions.

Semiclassical propagators are well-behaved on localized initial states.

Hamilton flow is described by elliptic functions, with trajectories on elliptic curves.

Abstract

Gabriel F. Calvo and Antonio Picon defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite-Gaussian modes {|0,0>,|1,0>,|0,1>}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And secondly we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite-Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.