Light propagation in nonlinear waveguide and classical two-dimensional oscillator

The quantum optical problem of the propagation of electromagnetic waves in a nonlinear waveguide is related to the solutions of the classical nonstationary harmonic oscillator using the method of linear integrals of motion [Malkin et al., Phys. Rev. D 2 (1970) 1371]. An explicit solution of the classical oscillator with a varying frequency, corresponding to the light propagation in an anisotropic waveguide is obtained using the expressions for the quantum field fluctuations. Substitutions have been found which allow to establish connections of the linear and quadratic invariants of Malkin et al. to several types of invariants of quadratic systems, considered in later papers. These substitutions give the opportunity to relate the corresponding quantum problem to that of the classical two-dimensional nonstationary oscillator, which is physically more informative.