  Discrete time-translation symmetry breaking can be observed in
periodically-driven systems oscillating at a fraction of the frequency of the
driving force. However, with the exception of the parametric instability in
period-doubling, multi-periodic driving does not lead to an instability
threshold. In this paper, we point out that quantum vacuum fluctuations can be
generically employed to induce period multiplication. In particular, we discuss
the period-tripled states in circuit QED and propose a microwave setup. We show
that for weak dissipation or strong driving, the system exhibits a
non-equilibrium phase transition in the sense that the time scale over which
the period-tripled state is generated can be arbitrarily separated from the
time-scale of the subsequent dephasing.
