  We discuss general properties of discrete time quantum symmetry breaking in
degenerate parametric oscillators. Recent experiments in superconducting
quantum circuit with Josephson junction nonlinearities give rise to new
properties of strong parametric coupling and nonlinearities. Exact analytic
solutions are obtained for the steady-state of this single-mode case of
subharmonic generation. We also obtain analytic solutions for the tunneling
time over which the time symmetry-breaking is lost above threshold. We find
that additional anharmonic terms found in the superconducting case increase the
tunneling rate, and can also lead to new regimes of tristability as well as
bistability. Our analytic results are confirmed by number state calculations.
