Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
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
L3 corresponds to the pendulum potential energy, the LO energy
n corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\ensuremath{\gamma}K}{\ensuremath{\omega}}$ is proportional to the angular momentum
K of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for
L3(t) is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\ensuremath{\equiv}{L}_{3}+n)$ and
K.
L3 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
〈L3〉 is constant, does not exist, because of spontaneous emission; it is also found that
〈L3〉 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.
The choice of an axis of quantization automatically selects those species from a bulk which could become stationary in a field-free environment with related spatial orientations of their coupling angular momenta. But if this axis fails to coincide with the direction of a uniform external magnetic field, a field-imposed transverse torque forces these atoms' total angular momenta into steady nutation that cannot be abolished by coupling-related internal torques. Making any stationary state impossible for these atoms, it also causes the contributions to the energy expectation value to oscillate. This is the result of a rigorous solution to the time-dependent Schrödinger equation describing the coupling dynamics of two spins 1/2 in an external field. Demanding to solve a four-dimensional system of coupled linear differential equations of first order, the bulkiness of even this simplest case already brings the calculations to the verge of becoming infeasible analytically.