Method for stabilizing a non-linear parametric
amplifier array comprising N non-linear
resonator modes, at sites j = 1, ..., N, linear and non-linear couplings between sites, external
coherent control fields on each site and input / output ports to send and retrieve an amplified
signal; the method comprising: setting the phases of the external
coherent control fields to grow linearly with site index, dividing the N sites of the array in 3 regions, setting, in the
central region, an homogeneous steady-state solution, assuming
periodic boundary conditions, and setting the central quantities to produce directional topological amplification in an homogenous
system; in the left and right buffer regions, imposing to some quantities a smoothly increasing or decreasing profile function from the boundaries to the center; choosing external
coherent control frequencies or bare
resonator frequencies to locally compensate for the
photon number-dependent frequency shifts induced by non-linearities; and solving the exact steady-state mean-field displacements assuming open boundary conditions.