  In this PhD thesis, we will present the theoretical and experimental work
that led to the realization of a radio-frequency circuit quantum
electrodynamics system (RFcQED). In chapter 2, we provide a detailed derivation
of the Hamiltonian of circuit QED formulated in the context of the Rabi model,
and extract expressions for the cross-Kerr interaction. The resulting
requirements for the coupling rate in RFcQED are discussed, one of them being
the need to dramatically increase the coupling rate compared to typical circuit
QED device. In chapter 3 we cover two experimental approaches to increasing the
coupling in a circuit QED system, one making use of a high impedance resonator,
the second utilizing a large coupling capacitor. In chapter 4, we combine these
two approaches to implement RFcQED. Through strong dispersive coupling, we
could measure individual photons in a megahertz resonator, demonstrate quantum
control by cooling the resonator to the ground state or preparing Fock states,
and finally observe with nanosecond resolution the re-thermalization of these
states. In chapter 5 we present QuCAT or Quantum Circuit Analyzer Tool in
Python, a software package that can be used for the design of circuit QED
systems such as the one presented in this thesis. In chapter 6 we discuss how
certain interplays between general relativity and quantum mechanics cannot be
described using our current laws of physics. In particular, we show how
radio-frequency mechanical oscillators are perfect candidates to perform
experiments in this regime. In chapter 7 we present the prospects for coupling
such mechanical oscillator to weakly anharmonic superconducting circuits such
as the transmon qubit or RFcQED systems.
