Quantum Coherence in Electrical Circuits

Abstract

This thesis studies quantum coherence in macroscopic and mesoscopic dissipative electrical circuits, including LC circuits, microwave resonators, and Josephson junctions. For the LC resonator and the terminated transmission line microwave resonator, second quantization is carried out for the lossless system and dissipation in modeled as the coupling to a bath of harmonic oscillators. Stationary states of the linear and nonlinear resonator circuits as well as the associated energy levels are found, and the time evolution of uncertainty relations for the observables such as flux, charge, current, and voltage are obtained. Coherent states of both the lossless and weakly dissipative circuits are studied within a quantum optical approach based on a Fokker-Plank equation for the P-representation of the density matrix which has been utilized to obtain time-variations of the averages and uncertainties of circuit observables. Macroscopic quantum tunneling is addressed for a driven dissipative Josephson resonator from its metastable current state to the continuum of stable voltage states. The Caldeira-Leggett method and the instanton path integral technique have been used to find the tunneling rate of a driven Josephson junction from a zero-voltage state to the continuum of the voltage states in the presence of dissipation. Upper and lower bounds are obtained for the tunneling rate at the intermediate loss and approximate closed form expressions are derived for the overdamped and underdamped limits.