  A Josephson junction laser is realised when a microwave cavity is driven by a
voltage-biased Josephson junction. Through the ac Josephson effect, a dc
voltage generates a periodic drive that acts on the cavity and generates
interactions between its modes. A sufficiently strong drive enables processes
that down-convert a drive resonant with a high harmonic into photons at the
cavity fundamental frequency, breaking the discrete time translation symmetry
set by the Josephson frequency. Using a classical model, we determine when and
how this transition occurs as a function of the bias voltage and the number of
cavity modes. We find that certain combinations of mode number and voltage tend
to facilitate the transition which emerges via an instability within a subset
of the modes. Despite the complexity of the system, there are cases in which
the critical drive strength can be obtained analytically.
