Interaction between a Nonlinear Oscillator and a Radiation Field

Abstract

The behavior of a nonlinear oscillator (NO) coupled to a radiation field is investigated. The NO considered is an angular momentum oscillator, of energy $\hslash \omega L_3$, that describes the collective effect of a number of identical two-level (or spin-½) systems under given idealized conditions, a large total angular momentum quantum number $L_0$ corresponding to a large number of two-level systems. The field is described by a set of modes. Free decay—with the NO initially excited and the radiation field in the ground state—and forced oscillation—with the NO subject to a prescribed resonant driving field—are studied. The analysis is performed both classically and quantum mechanically, using the classical and Heisenberg equations of motion, respectively, so as to display explicitly the difference in the results. Interest in the comparison between the two formalisms is motivated by the expectation that for large $L_0$, the NO should behave essentially classically, except near its highest-energy state in the absence of a driving field. The general equations of motion are reduced to equations for the NO variables only. In the classical analysis of free decay, expressions for the energy and oscillating coordinates are derived. The decay time is shown to approach infinity as $L_3\mathrm{}\left(0\right)\mathrm{}$ approaches $L_0$ (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to $L_3\mathrm{}\left(t\right)\mathrm{}$. It is also shown that use of the rotating-wave approximation alters qualitatively the expression for the frequency shift. In the classical analysis of forced oscillation, approximate results are obtained for a weak driving field and a strong driving field, the NO being initially in the ground state. The weakfield results exhibit a monotonic approach of $L_3\mathrm{}\left(t\right)\mathrm{}$ to a constant (negative) value—or steady state—at which the power absorbed equals the power radiated; in the strong-field case, $L_3\mathrm{}\left(t\right)\mathrm{}$ oscillates periodically between the limits $\pm L_0$, the coupling to the radiation field having negligible effect on the frequency of this oscillation. In the quantum-mechanical analysis, the equations of motion become simplified for $L_0=\frac{1}{2}$, and this case is treated first. The free-decay results are essentially similar to those of the Weisskopf-Wigner theory, exhibiting an exponential decay of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ and a radiative frequency shift in the oscillating coordinates. Under forced oscillation, with $〈L_3\mathrm{}\left(0\right)\mathrm{}〉=-L_0, 〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ approaches a constant value either monotonically, if the driving field is sufficiently weak, or by means of a damped oscillation, if the driving field is strong. For $L_0>\frac{1}{2}$, the free decay is treated by a method that involves the derivation of a set of expressions for the $k\mathrm{th}$ derivative of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ as an expectation value of a polynomial in $L_3$ of order $k+1\mathrm{}$. This set, together with the eigenvalue equation for $L_3$, is shown to lead to a solution for all the moments of $L_3$. The method is used to obtain complete solutions for several low values of $L_0$, and also to obtain initial derivatives of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ as polynomials in $L_0$ for $〈L_3\mathrm{}\left(0\right)\mathrm{}〉=L_0$. For large $L_0$, a comparison of classical and quantum-mechanical equations shows that only the condition $〈L_3\mathrm{}\left(0\right)\mathrm{}〉\approx L_0$ requires quantum-mechanical treatment, and that for a short time only. The free-decay problem with initial condition $〈L_3\mathrm{}\left(0\right)\mathrm{}〉=L_0$ solved quantum mechanically up to such a time $t_1$ by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for $t\ge t_1$. The statistical aspects introduced by the quantum mechanics are preserved in the classical solution, and their significance is discussed, with several examples, comparison is also made with a nonstatistical approximation. In the case of forced oscillation, the behavior of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$—for arbitrary $L_0$—is examined in detail for a strong field, and turns out to be described by a damped oscillation, the existence of the damping being independent of $L_0$. The apparent inconsistency of this result with the expectation that a large system subject to strong forces should behave classically is discussed, and $〈L_2^3\mathrm{}\left(t\right)\mathrm{}〉$ is examined. It is concluded that in a single experiment, the NO energy oscillates approximately like the classical energy, without damping; quantum mechanics introduces, however, a slight randomness (or unpredictability) in the frequency of this oscillation, because of the coupling with the radiation field. Since $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ describes the average over an ensemble of which each member consists of a NO coupled to a radiation field, the random frequency variation among the members accounts for the damping of the average. The usefulness of the combination of classical and quantum-mechanical analyses in achieving and interpreting theoretical results for a class of phenomena involving the collective interaction between a number of atoms and a radiation field is pointed out.