The relation and evolution of squeezing and instability for systems with quadratic hamiltonians

Authors propose a method that allows relating the quantum squeezing effect to the classical instability by establishing evolution equations for elements of the dispersion matrix directly in terms of elements of the stability matrix. The solution of these equations is written in terms of the evolution operator. Knowing this operator, authors can analyze the system instability at finite times. Based on the developed formalism, they investigate two physical systems: the degenerate and nondegenerate parametric amplifiers with external delta-shaped pulses. It is shown that one can either amplify or, on the contrary, weaken both the squeezing effect and the system instability using delta-pulses.