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1 Light Propagation in Nonlinear Waveguide and Classical Two-Dimensional Oscillator

A. Angelow

Published Jan 1, 1997• 9 citations

Abstract

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

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This paper has been withdrawn, as the explicit form of the propagator can be found in: V.V.Dodonov, I.A.Malkin and V.I.Man'ko, J.Phys.A 8 (1975) L19 V.V.Dodonov, I.A.Malkin and V.I.Man'ko, Int.J.Theor.Phys. 14 (1975) 37, Sec.3 V.V.Dodonov, I.A.Malkin and V.I.Man'ko, Theor.Math.Phys. 24 (1975) 746, Sec.2 I am grateful to Professor Victor Dodonov for bringing this to my attention.

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European Journal of Physics•Apr 1, 1989•10 citations

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

Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.

In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.

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.