Quantum properties of high-$Q$ macroscopic resonators

An analysis is made of the quantum characteristics of the behavior of an electromagnetic (acoustic) resonator over intervals shorter than the relaxation time. As a result of the high $Q$ factor, weak coupling of various modes is assumed. By considering a single mode, it can be shown that evolution of all the moments (and not only of the first) is described by classical equations for any initial states. The probabilities of transitions between groups of energy levels in the quasiclassical case $(E_{cp}\gg\triangle E\gg\hbar\omega)$ can also be calculated using classical equations. The possibility of observing experimentally the quantum nature of Brownian fluctuations of a high-$Q$ resonator is discussed.