This thesis analyses a lumped element circuit proposed for an analogue quantum simulation of opto-mechanics. The circuit consists of two resonators, a simple LC-resonator, and a similar resonator in which the inductor is replaced by a SQUID as a flux tuneable inductance. The interaction between the two resonators is established by mutual inductance between the inductor in the LC-resonator and the SQUID loop part of the other resonator. This makes the resonance frequency of the SQUID-resonator a function of the flux in the LC-resonator. It is shown that mutual and self inductances in the SQUID loop give rise to two transcendental constraints, for the magnetic flux in the SQUID loop, and the generalised flux across the SQUID. By an approximate solution for the constraints and under the assumption that the loop inductance is small compared to the SQUID inductance, we derive an approximate description for the circuit dynamics. We demonstrate that the circuit Hamiltonian contains the asymmetric opto-mechanical interaction, but in addition also a self-Kerr non-linearity in the analogue optical cavity, as well as a weak cross-Kerr interaction between the two resonators.

In this thesis, a we have designed and fabricated a Josephson Parametric Amplifier (JPA) using a new double-angle evaporation method without a Dolan bridge. We have found and resolved several issues in the fabrication procedure, but it requires further tuning before being fully functional. We have also simulated the behaviour of a general parametric amplifier with an additional Duffing non-linear term, and found that this term appears to limit the oscillation amplitude. We have attempted to characterize Josephson junctions fabricated with the new double-angle evaporation procedure, but without much success. Using a different fabrication method, a JPA was made and successfully characterized. The maximum measured gain is 16 dB, with a bandwidth of 1 MHz. The noise temperature is comparable to the cryostat temperature of 250 mK, but it was not characterized accurately.