Figure 5

Characteristics of nonreciprocal signal propagation along one edge. (a) Averaged steady-state amplitude of each oscillator in a lattice of 256 cavities. The coherent hopping strength is fixed, i.e., | J | = κ / 4 and | J | = 0.55 × κ / 4 for the amplification links if applicable, while the dissipative coupling strength is increased from the left to right column. Once the directionality conditions are matched, i.e., graphs (vi) and (ix), only the edge cavities have a finite occupation. However, to have a transmission close to unity, amplification stages have to be implemented. (b) Eigenvalues for an N = 8 oscillator lattice without and with amplification. The coherent hopping strength is set to | J | = κ / 4 and the dissipative rate Γ is varied. The dynamics at the point of directionality is described by purely real eigenvalues. Note, this fact is independent of the chosen propagation path and the lattice size; for details see Appendix pp3. (c) Transmission and added noise for propagation over one edge. By increasing Γ ̃ / κ the signal gets amplified, while the added noise is suppressed. For Γ ̃ / κ → 1 the transmission diverges and the noise reaches its minimal value of 3.5. In panel (c) (i) the suppression of the added noise is rather independent of the lattice size. The reason therefore is that a larger lattice size involves more amplification stages; thus, a larger coupling strength Γ ̃ results in a higher gain value. Comparing the added noise for various N and fixed transmission value we see that a larger lattice size requires a larger amount of gain to come close to the quantum limit; cf. graph (c) (ii).