Time-periodic Lindblad master equations for quantum systems with engineered interactions and dissipation

Floquet engineering describes the creation of exotic and correlated many-body quantum states by using time-periodic driving. However, driving usually generates heat in the quantum system which can eventually lead to thermalization and loss of coherence on long timescales. A pathway to circumvent these detrimental effects can be engineered dissipation that drains away part of the introduced energy and stabilizes the quantum system far away from equilibrium. One possibility to engineer dissipation and also interactions within the quantum system is by coupling it to bosonic modes. We will show how one can quite generally eliminate the bosonic modes in such a scenario and achieve a Lindblad master equation which includes the mediated interactions and dissipation. We apply this procedure to the time-periodic dissipative Dicke model, a workhorse for the recently observed dissipative time crystals, and confirm its validity. Our results pave the ways towards the theoretical description of many-body quantum systems with mediated interaction and dissipation in presence of periodic driving.