In this thesis observations on the application of parametric drives to superconducting quantum circuits in disparate parameter regimes are presented. By the nonlinear inductance of the Josephson junction, a variety of interactions in circuit quantum electrodynamical systems comprised of strongly, moderately, and weakly nonlinear oscillators are realized.

Chapter 1 contains an introduction to classical and quantum information and introduces superconducting circuits as a platform for quantum information processing. An outline of the contents of the thesis is also provided.

In Chapter 2 a theoretical foundation for the later chapters is established, spanning from the classical harmonic oscillator to circuit quantum electrodynamical systems and parametric driving. The transmon qubit, junction-embedded coplanar waveguide, tunable coupler, and Josephson junction array resonator are introduced and some methods for realizing parametrically activated interactions in such systems are discussed.

Chapter 3 focuses on the steps necessary for constructing a superconducting quantum circuit. The design, simulation, and fabrication methods necessary for creating the experimental devices of later chapters are discussed.

In Chapter 4 results of the parametrically activated interactions between two tunably coupled transmon qubits by flux modulation of a SQUID are presented. When the coupling SQUID is modulated at the sum or difference frequencies of the transmons, level repulsion and attraction are observed spectroscopically. The viability of the platform for analog quantum simulations is discussed and the experimental results are compared to analytical models and numerical simulations of the quantum master equation.

In Chapter 5 spectroscopic signatures of a few-photon Kerr parametric oscillator are observed upon the application of an all-microwave bichromatic drive to a Josephson junction-embedded coplanar waveguide resonator. Semiclassical analytical, numerical, and quantum master equation simulations are performed and compared with the experimental results. An effective model based on semiclassical methods proves insufficient in modelling the behaviour of the system, indicating the presence of quantum effects.

In Chapter 6 a weakly nonlinear Josephson junction array resonator is bichromatically driven into a parametric phase state. Stochastic switching between the two non-equilibrium stationary states of the system is observed and the time between stochastic switching events is determined for a range of drive strengths. An additional microwave drive resonant with the frequency of parametric response is applied and the system is biased into one of the phase states. The biasing and change in switching time as a function of drive power and phase is shown. The contributions of classical and quantum effects to the occurrence of switching events is discussed.

In Chapter 7 measurements of a strongly parametrically driven Duffing oscillator are presented. As the system is strongly driven at a variety of large negative detunings, signatures of chaotic behaviour are observed in the output field spectrum and quadrature histograms. The observed features are discussed and compared to known markers of chaotic behaviour in classical parametrically driven Duffing oscillators.

Chapter 8 concludes the thesis, providing a review of the contents and findings of the previous chapters. The thesis ends with an outlook and suggestions for potential future topics of study.

Quantum information processing requires fast manipulations of quantum systems in order to overcome dissipative effects. We propose a method to accelerate quantum dynamics and obtain a target state in a shorter time relative to unmodified dynamics, and apply the theory to a system consisting of two linearly coupled qubits. We extend the technique to accelerate quantum adiabatic evolution in order to rapidly generate a desired target state, thereby realizing a shortcut to adiabaticity. Further, we address experimental limitations to the rate of change of control parameters for quantum devices which often limit one’s ability to generate a desired target state with high fidelity. We show that an initial state following decelerated dynamics can reach a target state while varying control parameters more slowly, enabling more experimentally feasible driving schemes.