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Optics Letters

Optical realization of quantum-mechanical invariants

H. M. Moya-Cessa, M. Fernández‐Guasti, Victor M. Arrizon, S. Chávez-Cerda

Published Apr 28, 2009• 9 citations
Publisher

Abstract

We study the problem of paraxial propagation in two grade-index media by using invariant techniques that allow a continuous solution of the problem. By using the well-known fact that this problem is analogous to the time-dependent harmonic oscillator in quantum mechanics, known methods there may be imported producing on the one hand a solution to the propagation problem and on the other hand a realization of a quantum-mechanical invariant.

DOI
10.1364/ol.34.001459

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European Journal of Physics•Apr 1, 1989•10 citations
We study the problem of paraxial propagation in two grade-index media by using invariant techniques that allow a continuous solution of the problem. By using the well-known fact that this problem is analogous to the time-dependent harmonic oscillator in quantum mechanics, known methods there may be imported producing on the one hand a solution to the propagation problem and on the other hand a realization of a quantum-mechanical invariant.
This contribution has two main purposes. First, using classical optics we show how to model two coupled quantum harmonic oscillators and two interacting quantized fields. Second, we present classical analogs of coupled harmonic oscillators that correspond to anisotropic quadratic graded indexed media in a rotated reference frame, and we use operator techniques, common to quantum mechanics, to solve the propagation of light through a particular type of graded indexed medium. Additionally, we show that the system presents phase transitions.
It has been found a solution of the equation for classical nonstationary harmonic oscillator, with time dependent frequency in integral form. It has been shown, that the quantum optical problem for propagation of electromagnetic waves in nonlinear waveguide is related to the solutions of classical nonstationary harmonic oscillator. This method could be used for obtaining other solutions of the classical harmonic oscillator, with other time dependent frequency, if we have already determined the fluctuations σq, σp and covariance cqp by some different methods. There have been found the substitutions, which allow to establish a connection between the general method of linear invariants [Malkin et.al., Phys Rev.2D (1970) 1371] and some particular cases in later papers. This gives the opportunity to present the problem with equation of two- dimensional nonstationary harmonic oscillator, which is physically more informative. 1
Review of the paraxial approximation to the problem of light beams propagating in optical fibers is presented. The analogy of time-dependent integrals of motion for the quantum system and space-dependent invariants for the radiation in inhomogeneous optical waveguides is elucidated. A specific tomographic probability distribution associated with the light-pulse profile is considered analogously to tomographic probabilities in quantum mechanics. The new inequality is presented for the tomographic probability.
The authors discuss the harmonic oscillator problem in the domains of classical mechanics and the parabolic index optical waveguide problem in the domains of ray optics. While discussing the classical mechanics and the ray optics of the two problems the authors have used the Lagrangian formalism which allows them to see the close relationship between the two problems. In part II of the paper the authors study the corresponding quantum mechanical and wave optical problems. In particular whilst looking at the time evolution of a coherent state the authors demonstrate that it has a close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide.
Paraxial light and atom optics are compared. To lowest order the slowly varying amplitude of a light field in a dielectric medium with a spatially dependent refractive index satisfies an equation which has the form of a Schr\"odinger equation: the ``optical Schr\"odinger equation.'' The unsystematic procedure of neglecting certain second-order derivatives is replaced by a systematic expansion which allows the calculation of consecutive corrections. The general theory is applied to harmonic motion in a graded index fiber and to tunneling between coupled fibers. The physical relations between wave evolution of massive particles and paraxially propagating waves are elucidated.
For pt.I see ibid., vol.10, p.136-43 (1989). In the preceding paper the authors used the Lagrangian formalism to show the close relationship between the classical mechanics of the harmonic oscillator and the ray optics of the parabolic index optical waveguide. The authors have now studied the corresponding quantum mechanical and wave optical problems and, in particular, have studied the time evolution of a coherent state and have shown its close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide. The corresponding transitions to classical mechanics and ray optics have been shown. The authors have also presented an analysis of the recently developed squeezed states and have shown its correspondence to the waveguide problem. They have demonstrated that the relationship between classical and quantum mechanics is the same as between ray optics and wave optics. All such analyses should be of considerable pedagogical value.