A semiclassical analysis of dark state transient dynamics in waveguide circuit QED

TL;DR

This paper analyzes the transient dynamics of dark states in waveguide circuit QED systems with time delays, using semiclassical and quantum approaches, revealing conditions for dark state formation and differences based on impedance.

Contribution

It introduces a semiclassical analysis of dark state dynamics in waveguide circuit QED with time delays, connecting it to quantum optics methods and exploring impedance effects.

Findings

Dark states form when delay equals an integer multiple of oscillation periods.

Semiclassical and quantum methods are equivalent at low impedance.

Differences between approaches emerge at higher impedance.

Abstract

The interaction between superconducting qubits and one-dimensional microwave transmission lines has been studied experimentally and theoretically in the past two decades. In this work, we investigate the spontaneous emission of an initially excited artificial atom which is capacitively coupled to a semi-infinite transmission line, shorted at one end. This configuration can be viewed as an atom in front of a mirror. The distance between the atom and the mirror introduces a time-delay in the system, which we take into account fully. When the delay time equals an integer number of atom oscillation periods, the atom converges into a dark state after an initial decay period. The dark state is an effect of destructive interference between the reflected part of the field and the part directly emitted by the atom. Based on circuit quantization, we derive linearized equations of motion for the system and use these for a semiclassical analysis of the transient dynamics. We also make a rigorous connection to the quantum optics system-reservoir approach and compare these two methods to describe the dynamics. We find that both approaches are equivalent for transmission lines with a low characteristic impedance, while they differ when this impedance is higher than the typical impedance of the superconducting artificial atom.