Superradiant and lasing states in driven-dissipative Dicke models

We present the non-equilibrium phase diagram of a model which can demonstrate both Dicke-Hepp-Lieb superradiance and regular lasing by varying the coherent and incoherent driving terms. We find that the regions in the phase diagram corresponding to superradiance and standard lasing are always separated by a normal region. We analyse the behaviour of the system using a combination of exact numerics based on permutation symmetry of the density matrix for small to intermediate numbers of molecules, and second order cumulant equations for large numbers of molecules. We find that the nature of the photon distribution in the superradiant and lasing states are very similar, but the emission spectrum is very different. We also show that in the presence of both coherent and incoherent driving, a period-doubling route to a chaotic state occurs.