Fig. 1. Chain of N photonic cavities with identical modes coupled to a nonreciprocal bus. The nonreciprocity of the bus can be enforced by inserting optical isolators between neighboring cavities. Γ denotes the coupling between each cavity and the bus, κ is the input/output coupling rate of each cavity, γ is the net pumping rate (including internal damping), and d is the spacing between neighboring cavities. Here, we take the natural unit of d=1.

Fig. 2. Scattering matrices for (a) γ=0.5 and (b) γ=0.2 with N=20. (c) Gain |SN1|2 for different system sizes as a function of the pumping rate γ. In all plots, Γ=1, κ=0.25, Δω=0, and ζ≫N.

Fig. 3. Gain |SN1|2 as a function of the pumping rate γ and the attenuation length ζ. The black dashed curve labeled by NAT corresponds to the nonreciprocal amplification transition, which agrees well with the topological phase transition (red dashed curve labeled by TPT) when ζ≪N. As ζ increases, the NAT starts to deviate from the TPT. The white dashed curve indicates the critical pumping beyond which the system becomes unstable. For the simulation, we take N=100, Γ=1, and κ=0.25.

Fig. 4. Gain |SN1|2 as a function of the pumping rate γ and the frequency detuning Δω for long-range coupling (ζ≫N). The black dashed curve corresponds to unidirectional amplification transition. The blue dash-dotted curve corresponds to half maximum amplification for different γ, which indicates the amplification bandwidth. For the simulation, we take N=100, Γ=1, and κ=0.25.