  Multiple emitters coherently interacting with an electromagnetic mode give
rise to collective effects such as correlated decay and coherent exchange
interaction, depending on the separation of the emitters. By diagonalizing the
effective non-Hermitian many-body Hamiltonian we reveal the complex-valued
eigenvalue spectrum encoding the decay and interaction characteristics. We show
that there are significant differences in the emerging effects for an array of
interacting anharmonic oscillators compared to those of two-level systems and
harmonic oscillators. The bosonic decay rate of the most superradiant state
increases linearly as a function of the filling factor and exceeds that of
two-level systems in magnitude. Furthermore, with bosonic systems, dark states
are formed at each filling factor. These are in strong contrast with two-level
systems, where the maximal superradiance is observed at half filling and with
larger filling factors superradiance diminishes and no dark states are formed.
As an experimentally relevant setup of bosonic waveguide QED, we focus on
arrays of transmon devices embedded inside a rectangular waveguide.
Specifically, we study the setup of two transmon pairs realized experimentally
in M. Zanner et al., arXiv.2106.05623 (2021), and show that it is necessary to
consider transmons as bosonic multilevel emitters to accurately recover correct
collective effects for the higher excitation manifolds.
