Excitation of Wannier-Stark states in a chain of coupled optical resonators with linear gain and nonlinear losses

In this paper we theoretically study the nonlinear dynamics of Wannier-Stark states in the dissipative system consisting of interacting optical resonators, whose resonant frequencies depend linearly on their number. It is shown that the negative losses in some resonators can switch the system into a lasing regime with Wannier-Stark states serving as working modes. It is shown by extensive numerical simulations that there may be single-frequency stationary regimes as well as multi-frequency regimes. In the latter case Bloch oscillations can appear in the system. The possibility of selective excitation of Wannier-Stark states by the appropriate choice of the dissipation profile is investigated. A simple perturbation theory describing the quasi-linear regimes is developed and compared against the numerical results.