Nonperturbative Analysis of the Resonant Interaction between a Linear and a Nonlinear Oscillator

Abstract

The consideration of the resonant interaction between a linear harmonic oscillator (LO) and a (nonlinear) angular momentum oscillator (NO) is motivated by its relation to the interaction between a field mode and a number of atomic systems, its relation to the interaction between three field modes, and the appearance of qualitatively different results quantum mechanically and classically even in cases that can be regarded as macroscopic. The analysis is carried out both classically and quantum mechanically in the rotating-wave approximation, and the time dependence of the energies of the two oscillators is examined, mainly by means of a nonlinear differential equation for the NO energy. In the classical analysis, the problem turns out to be identical to that of a spherical pendulum, where the NO energy $L_3$ corresponds to the pendulum potential energy, the LO energy $n$ corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\gamma K}{\omega }$ is proportional to the angular momentum $K$ of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for $L_3\mathrm{}\left(t\right)\mathrm{}$ is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\equiv L_3+n)$ and $K$. $L_3$ oscillates periodically, except for the "unstable-equilibrium" solution and the "conical-pendulum" solution. The quantum-mechanical problem is solved exactly for the cases in which the NO is a two-level system and a three-level system, and certain aspects of the exact solution are derived in the case of a four-level system. The constants $E$ and $K$ become a complete set of quantum numbers. It is found that unstable equilibrium, in the sense that $〈L_3〉$ is constant, does not exist, because of spontaneous emission; it is also found that $〈L_3〉$ oscillates sinusoidally for the two-level system, oscillates periodically but not sinusoidally for the three-level system, and oscillates aperiodically for a four- or higher-level system, conclusions about the higher-level systems coming from general considerations and numerical solutions found in the literature. Approximate solutions for NO' s with arbitrarily large numbers of levels are obtained. One method of approximation displays spontaneous emission but not aperiodicity. Another approximate method, which contains the statistical spread of the initial phase difference between the oscillations of the two oscillators inherent in the uncertainty principle, displays both spontaneous emission and aperiodicity. It is shown that the aperiodicity - a property which does not exist classically for precise initial conditions - does not disappear even in the macroscopic limit and is due to the averaging performed, in taking the quantum-mechanical expectation value, over an ensemble of periodic oscillations with a spread in frequencies.