Enhancement of the optomechanical coupling and Kerr nonlinearity using the Josephson capacitance of a Cooper-pair box

In this work, I explore several aspects of optomechanical systems coupled to their environments, nonclassical correlations created in such systems, and enhancing the very optomechanical coupling to the ultrastrong coupling limit. I also study micromechanical resonators revealing the de Haas—van Alphen effect in graphene. \n Two-level system defects are often found interacting with superconducting microwave setups warranting the study the of their effects on the dynamics of an optomechanical system. I derive a quantum Langevin equation formulation for a general system-environment interaction in terms of the system degrees of freedom, and apply this framework to a cavity surrounded by two-level system defects. Nonlinear dissipation and parametric effects, observed experimentally in other works, are found. \n I consider environmental noise effects, in a proposal for an optomechanical Bell inequality test, where nonclassical correlations between cavity fields are generated by a massive mechanical resonator. The violation of the CHSH Bell inequality is predicted with noise and system parameters compatible with modern technological capabilities. \nThe quantum nature of mechanical motion can further be explored in the ultrastrong optomechanical coupling regime, where the intrinsic nonlinearity of the coupling is observable. To this end, I present a scheme for a microwave circuit setup that is capable of enhancing the optomechanical radiation pressure and cross-Kerr single-photon couplings by several orders of magnitude. \n \nAdditionally, I study a graphene resonator setup operated as a magnetometer, where the quantum Hall effect and the de Haas—van Alphen effect are observed for the massless Dirac fermions of graphene via the mechanical motion. The general measurement scheme can also be generalized for other conducting two-dimensional materials.