A quantum model for rf-SQUIDs based metamaterials enabling 3WM and 4WM Travelling Wave Parametric Amplification
A quantum model for Josephson-based metamaterials working in the Three-Wave Mixing (3WM) and Four-Wave Mixing (4WM) regimes at the single photon level is presented. The transmission line taken into account, namely Traveling Wave Josephson Parametric Amplifier (TWJPA), is a bipole composed by a chain of rf-SQUIDs which can be biased by a DC current or a magnetic field in order to activate the 3WM or 4WM nonlinearities. The model exploits a Hamiltonian approach to analytically work out the time evolution both in the Heisenberg and interaction pictures. The former returns the analytic form of the gain of the amplifier, while the latter allows to recover the probability distributions vs. time of the photonic populations, for multimodal Fock and coherent input states. The dependence of the metamaterial's nonlinearities is presented in terms of circuit parameters in a lumped model framework while evaluating the experimental conditions effects on the model validity.