Modelling of flux-pumped Josephson Parametric Amplifiers with auxiliary circuits

Flux-pumped SQUID-based resonators can serve as parametric amplifiers. If the SQUID-based resonator is coupled to a frequency-independent environment such as a transmission line (TL), the bandwidth is limited. Inserting an auxiliary circuit into the transmission line can make the environment frequency-dependent and modify the susceptibility matrix by changing the self-energy. We show how the insertion of an auxiliary circuit affects the self-energy and thereby the susceptibility, thus facilitating the engineering of the amplifier bandwidth. We consider not only a classical model but also a quantum model, where we diagonalize the TL Hamiltonian by solving the TL normal modes subject to a boundary condition from the rest of the circuit. We also relate the self-energy to the environment impedance. We specifically discuss bandwidth engineering in the cases of in-series LC and parallel LC circuits as coupling circuits. In these cases the TL eigenmodes can be calculated, thus enabling us to find the exact Hamiltonian and coupling coefficients of the coupled system. Our results could also be useful for studying the influence of auxiliary circuits for other amplifiers.