Emergent parametric resonances and time-crystal phases in driven BCS systems

We study the out-of-equilibrium dynamics of a Bardeen-Cooper-Schrieffer condensate subject to a periodic drive. We demonstrate that the combined effect of drive and interactions results in emerging parametric resonances, analogous to a vertically driving pendulum. In particular, Arnold tongues appear when the driving frequency matches $2{\mathrm{\Delta }}_0\mathrm{/}n$, with $n$ a natural number, and ${\mathrm{\Delta }}_0$ the equilibrium gap parameter. Inside the Arnold tongues we find a commensurate time-crystal condensate which retains the $U(1)$ symmetry breaking of the parent superfluid/superconducting phase and shows an additional time-translational symmetry breaking. Outside these tongues, the synchronized collective Higgs mode found in quench protocols is stabilized without the need of a strong perturbation. Our results are directly relevant to cold-atom and condensed-matter systems and do not require very long energy relaxation times to be observed.