  Many complex systems can spontaneously oscillate under non-periodic forcing.
Such self-oscillators are commonplace in biological and technological
assemblies where temporal periodicity is needed, such as the beating of a human
heart or the vibration of a cello string. While self-oscillation is well
understood in classical non-linear systems and their quantized counterparts,
the spontaneous emergence of periodicity in quantum systems without a
semi-classical limit is more elusive. Here, we show that this behavior can
emerge within the repeated-interaction description of open quantum systems.
Specifically, we consider a many-body quantum system that undergoes dissipation
due to sequential coupling with auxiliary systems at random times. We develop
dynamical symmetry conditions that guarantee an oscillatory long-time state in
this setting. Our rigorous results are illustrated with specific spin models,
which could be implemented in trapped-ion quantum simulators.
