Quantum-mechanical parametric amplification

Parametric amplification may be obtained in a harmonic oscillator if its spring constant is replaced by a time-dependent function. It is shown that the quantum transition probabilities may be expressed in a simple closed analytic form which involves only the classical amplification factor of the oscillator. The authors also compute the time evolution of the density matrix which initially describes a thermodynamic ensemble at temperature $T$ and show that this matrix can similarly be expressed in such a closed form.