  We study the full time evolution of one- and two-mode bosonic quantum systems
that interact through single- and two-mode squeezing Hamiltonians. We establish
that the single- and two-mode cases are formally equivalent, leading to the
same differential equations encoding the full time evolution. These
differential equations can be easily employed in any application. We
analytically predict a dramatic transition in the population of the modes when
the coupling takes a specific critical value, leading to exponential growth of
the excitation population. We discuss the validity, scope and generality of our
results.
