We study the out-of-equilibrium dynamics of a BCS system subject to a periodic drive in a wide range of frequencies and strengths, in the absence and presence of dissipation. The phase diagram is surprisingly rich, with at least four dynamical phases. For excitation frequency above the gap, we show the emergence of a new collective mode, dubbed Rabi-Higgs [1] which turns into a gapless regime following a second-order phase transition for large enough drive strengths. In the presence of dissipation, such Rabi-Higgs mode becomes a transient effect and interesting steady-states with strong nonlinearities appear [2]. On the other hand, for subgap excitation, we demonstrate that the combined effect of drive and interactions results in emerging parametric resonances, analogous to a vertically driving pendulum [3]. In particular, Arnold's tongues appear when the driving frequency matches 2Δ/n, with n being a natural number and Δ the equilibrium gap parameter. Inside Arnold's tongues, we find a time-crystal condensate that retains the U(1) symmetry breaking of the parent superconducting phase and shows an additional time-translational symmetry breaking. Outside the tongues, the synchronized collective Higgs mode found in quench protocols [4] is stabilized without the need for a strong perturbation. We show the parametric resonances are quite robust and do not need long coherence times to be observed. Our results are directly relevant to cold-atom and condensed-matter systems and may have applications in parametric amplification, frequency converters, and sensing.

References
[1]. H. P. Ojeda Collado, José Lorenzana, Gonzalo Usaj, and C. A. Balseiro, Phys. Rev. B 98, 214519 (2018).
[2]. H. P. Ojeda Collado, Gonzalo Usaj, José Lorenzana, and C. A. Balseiro, Phys. Rev. B 101, 054502 (2020).
[3]. H. P. Ojeda Collado, Gonzalo Usaj, C. A. Balseiro, Damián H. Zanette, and José Lorenzana, Phys. Rev. Research 3, L042023 (2021).
[4]. H. P. Ojeda Collado, Gonzalo Usaj, José Lorenzana, and C. A. Balseiro, Phys. Rev. B 99, 174509 (2019).