Synchronization in coherently and dissipatively coupled spinor polariton time crystals

The spinor degree of freedom associated to exciton-polariton condensates can spontaneously self-oscillate breaking time translation symmetry, thus showing a continuous time-crystal (CTC) behavior. An open question in such driven-dissipative and non-linear quantum open systems is what happens when CTCs are brought together to interact. Here we experimentally study polariton condensates in coupled traps, evidencing mutual induction and synchronization of the pseudospin temporal GHz dynamics in the CTC phase. The individual and relative orientation of the (limit cycle) precessing pseudospins can be tuned by the optical excitation power, displaying both ferro and anti-ferro dynamical configurations. We theoretically show that the exciton reservoir, and both the coherent and long-range dissipative inter-trap coupling, play important roles in the CTC dynamics. The investigation of time-broken symmetry is thus extended here to more complex non-hermitian systems opening the path to study self-sustained collective dynamics in lattices of non-linear quantum condensates.