Dispersive Superconducting Circuit Quantum Electrodynamics, the Time-continuous Feynman’s Light Microscope and Non-Markovian dynamics in multi-mode Superconducting Circuits

Abstract

This thesis is divided into three parts, with each part concerned with one of the three pillars of Theoretical Quantum Optics: ‘Microscopic models of Light-Matter Interaction’, ‘Quantum Measurements’ and finally, ‘Driven Non-Markovian systems’. In the first part, I discuss in detail a widespread assumption, yet still unproven, in the description of Superconducting Circuit QED that: ‘electromagnetic modes above the superconducting gap have short lifetimes and contribute weakly to the qubit dynamics’. It is well known that superconducting waveguides strongly attenuate the propagation of electromagnetic waves with frequencies beyond the superconducting gap. The interaction between Transmon qubits and superconducting resonators invariably involves the Transmon coupling to a large set of resonator modes. So far, strong dispersion effects near and beyond the superconducting-gap have been ignored in quantization models. Rather, it is assumed that the superconducting resonator behaves ideally across the large frequency intervals. In this thesis I present a quantization approach which includes the superconducting frequency-dependent surface impedance and demonstrate that superconducting dispersion plays a role in determining the effective light-matter interaction cut-off. Going ahead, in the next part I present a new formulation for the emergence of classical dynamics in a quantum world by considering a path integral approach that also incorporates continuous measurements. This program is conceptually different from the decoherence program as well as the quantum-to-classical transition framework with coarse-grained measurements. The path-integral formulation provides the joint statistics of a sequence of measurements with each Feynman path picking up an additional random phase due to measurements. The magnitude of this phase is proportional to the measurement strength, and we give conditions under which the dominant contribution to the probability amplitude comes from the trajectories in the vicinity of the classical paths. The proliferation of this information across the environment, an essential feature of quantum Darwinism, takes place via scattering of plane-wave probes by the system. Extending to repeated measurements, I show that in the continuous limit, each system trajectory picks up an additional phase due to the momentum kicks from the probes - origin of the back-action force. We provide conditions for which the measurement provides sufficient “which-path” information and keeps the wave packet sufficiently localized. This allows for description of quantum-to-classical transition at the level of individual trajectories in contrast to the statistical ensemble interpretation provided by density matrices in the decoherence program. Finally, I show the connection of continuous quantum measurements to purely classical quantities, namely, the Hamilton-Jacobi equations and the Poisson Bracket. The final part concerns itself with a novel experimental regime that has been made possible due to the rapid progress in fabrication methods and microwave electronics, ushering in an entirely new regime of Quantum Optics. The multi-mode regime of Circuit Quantum Electro/Acoustodynamics involves the simultaneous coupling of a Qubit to a large set of discrete bosonic modes. The structured environment, as opposed to an infinite heat bath, in the multimode regime offers a versatile platform for investigating non-Markovian dynamics as well as the possibility of tailoring interactions between the modes for technological applications. In this work, I study a monochromatically driven Qubit interacting with a multi-mode, engineered cavity using coherent state path integrals. I calculate the contributions of the various multi-photon virtual processes to the self-energy corrections and demonstrate that resonance conditions between the Rabi frequency and the mode frequencies leads to strong qubit-mode and qubit mediated mode-mode interactions. Finally, this work proceeds towards explaining some of the key features observed in experiments (PRX 5, 021035 (2015)). Further, I also discuss the driven non-Markovian dynamics in such discrete environments by formulating an integro-differential equation for the Rabi-flopping amplitude and demonstrate that the dynamics are modified due to higher order environmental correlators, previously not considered in other works.