Quantum Theory of Parametric Amplification. II

Abstract

The analysis of the quantum-mechanical parametric amplifier begun in the preceding paper is extended to describe the joint quantum state of the two interacting modes of oscillation of the system. Equations are derived for the time evolution of the Schr\"odinger density operator. Comparison with classical theory is made by describing the state of the quantum-mechanical system by means of the $P$ representation. The time evolution of the weight function $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ for this representation, and of the corresponding Wigner distribution function $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$, is found. A decoupled form for the equations of motion for the field variables is exhibited which simplifies the solutions for these functions. It is shown that the amplification process gives rise to a correlation between the two mode amplitudes which becomes exceedingly strong with increasing time. The relation of this correlation to the time development of the functions $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ and $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ is discussed.