Nonreciprocal amplification transition in a topological photonic network

We studied the transport properties of a driven-dissipative photonic network, where multiple photonic cavities are coupled through a nonreciprocal bus with unidirectional transmission. For short-range coupling between the cavities, the occurrence of nonreciprocal amplification can be linked to a topological phase transition of the underlying dynamic Hamiltonian. However, for long-range coupling, we show that the correspondence between the nonreciprocal amplification transition and the topological phase transition breaks down as the transition conditions deviate significantly from each other. We found the exact transition condition for nonreciprocal amplification, supported by analytical calculation and numerical simulation. We also investigated the stability, the crossover from short-to long-range coupling, and the bandwidth of the nonreciprocal amplification. Our work has potential applications in signal transmission and amplification, and also paves the way to study other topological and non-Hermitian systems with long-range coupling and nontrivial boundary effects.