Ultrastrong waveguide QED with giant atoms

Quantum optics with giant emitters has shown a new route for the observation and manipulation of non-Markovian properties in waveguide-QED. In this paper we extend the theory of giant atoms, hitherto restricted to the perturbative light-matter regime, to deal with the ultrastrong coupling regime. Using static and dynamical polaron methods we address the low energy subspace of a giant atom coupled to an Ohmic waveguide beyond the standard rotating wave approximation. We analyze the equilibrium properties of the system by computing the atomic frequency renormalization as a function of the coupling characterizing the localization-delocalization quantum phase transition for a giant atom. We show that virtual photons dressing the ground state are non-exponentially localized around the contact points but decay as a power-law. Dynamics of an initially excited giant atom are studied, pointing out the effects of ultrastrong coupling on the Lamb shift and the spontaneous emission decay rate. Finally we comment on the existence of the so-called oscillating bound states beyond the rotating wave approximation.