  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.
