Higgs time crystal in a high-$T_c$ superconductor

We propose to induce a time crystalline state in a high-$T_c$ superconductor, by optically driving a sum resonance of the Higgs mode and a Josephson plasma mode. The generic cubic process that couples these fundamental excitations converts driving of the sum resonance into simultaneous resonant driving of both modes, resulting in an incommensurate subharmonic motion. We use a numerical implementation of a semi-classical driven-dissipative lattice gauge theory on a three-dimensional layered lattice, which models the geometry of cuprate superconductors, to demonstrate the robustness of this motion against thermal fluctuations. We demonstrate this light-induced time crystalline phase for mono- and bilayer systems and show that this order can be detected for pulsed driving under realistic technological conditions.