Non-Hermitian topological phases in traveling-wave parametric amplifiers

We study the phenomena of topological amplification in one-dimensional traveling-wave parametric amplifiers. We find two phases of topological amplification, both with directional transport and exponential gain with the number of sites, and one of them featuring squeezing. We also find a topologically trivial phase with zero-energy modes which produces amplification but lacks topological protection. We characterize the resilience to disorder of the different phases, their stability, gain and noise-to-signal ratio. Finally, we discuss their experimental implementation with state-of-the-art techniques.