Fig. 1: Directional amplification in a driven-dissipative chain.

a Driven-dissipative chain of N bosonic cavity modes a_j evolving according to Eq. (2). Neighboring modes are coupled both coherently with strength J, see Eq. (1), and dissipatively at rate Γ through non-local dissipators \({\mathcal{D}}[{z}_{j}]\) with z_j ≡ a_j + e^−iθa_j+1. Each mode is coupled to a wave guide, which allows probing with a coherent input signal 〈\(a_{\ell, \, {\mathrm{in}}}\)〉 and introduces losses with rate γ. Incoherent pumping of photons at rate κ enables an amplifying regime for which local dissipation overcomes non-local dissipation, see Eq. (3). b Topologically non-trivial regimes of the chain correspond to non-reciprocal amplification of a coherent input signal. In the topologically non-trivial regime ν ≠ 0 (here ν = +1), according to Eq. (11), an input at one end (right) exits amplified at the opposite end (left). This is quantified by the scattering matrix S(ω) (center) given by Eq. (4). From the structure of S(ω), we see that amplification is directional, i.e., if input and output fields are exchanged, the transmission is strongly suppressed. We have chosen N = 10, \({\mathcal{C}}=2\Gamma /(\gamma +2\Gamma -\kappa )=2\), Λ = 4J/(γ +  2Γ − κ) = 2 and \(\theta =\frac{3\pi }{2}\).