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We study the breaking of the discrete time-translation symmetry in smallperiodically driven quantum systems. Such systems are intermediate betweenlarge closed systems and small dissipative systems, which both display thesymmetry breaking, but have qualitatively different dynamics. As a nontrivialexample we consider period tripling in a quantum nonlinear oscillator. We showthat, for moderately strong driving, the period tripling is robust on anexponentially long time scale, which is further extended by an even weakdecoherence.

Multiple-period Floquet states and time-translation symmetry breaking in quantum oscillators