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arXiv:quant-ph/0402100 · Progress in optics

Phase space correspondence between classical optics and quantum mechanics

Daniela Dragoman

Published Jan 1, 2002• 37 citations
arXivHTMLPDF
DOI
10.1016/s0079-6638(02)80029-4

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In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of the coherent state from | z 〉 to | sz − rz *〉 corresponds to the motion from a point z ( q , p ) to another point sz − rz * with | s | 2 − | r | 2 = 1. The evolution is governed by the so-called Fresnel operator U ( s , r ) that was recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation, and obeys group product rules. In other words, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.
We develop the mathematical properties of the ABCD -transformation from the classical and quantum mechanical points of view. First, we list the four types of generating function which generate the ABCD -transformation in classical mechanics. Second, we introduce the unitary operator of the ABCD -transformation. Next, we calculate the normal ordering of this unitary operator and derive the kernels in coordinate-momentum phase space. The kernels are comprised of the generating functions, which generate the ABCD -transformation in classical mechanics. This reveals a new correspondence between classical and quantum mechanics. As an application of these kernels, we show the propagation of coherent and Gaussian beams in the context of quantum optics which corresponds to the ABCD -transformation in matrix optics.
Abstract The two‐dimensional paraxial equation of optics and the two‐dimensional time‐dependent Schrödinger equation, derived as approximations of the three‐dimensional Helmholtz equation and the three‐dimensional time‐independent Schrödinger equation, respectively, are identical. Here the free propagation in space and time of Hermite–Gauss wave packets (optics) or harmonic oscillator eigenfunctions (quantum mechanics) is examined in detail. The Gouy phase is shown to be a dynamic phase, appearing as the integral of the adiabatic eigenfrequency or eigenenergy. The wave packets propagate adiabatically in that at each space or time point they are solutions of the instantaneous harmonic problem. In both cases, it is shown that the form of the wave function is unchanged along the loci of the normals to wave fronts. This invariance along such trajectories is connected to the propagation of the invariant amplitude of the corresponding free wave number (optics) or momentum (quantum mechanics) wave packets. It is shown that the van Vleck classical density of trajectories function appears in the wave function amplitude over the complete trajectory. A transformation to the co‐moving frame along a trajectory gives a constant wave function multiplied by a simple energy or frequency phase factor. The Gouy phase becomes the proper time in this frame. Key Points This paper builds a bridge between quantum mechanics (QM) and classical optics in that the identity of the paraxial equation of optics and the Schrödinger equation of QM is shown, the Bohmian trajectories of QM are defined in optics, the Gouy phase of optics is defined in QM and given a new interpretation and The space and momentum wave functions are equivalent along a trajectory.