  Chains of coupled two-level atoms behave as 1D quantum spin systems,
exhibiting free magnons and magnon bound states. While these excitations are
well studied for closed systems, little consideration has been given to how
they are altered by the presence of an environment. This will be especially
important in systems that exhibit nonlocal dissipation, e.g. systems in which
the magnons decay due to optical emission. In this work, we consider free
magnon excitations and two-magnon bound states in an XXZ chain with nonlocal
dissipation. We prove that whilst the energy of the bound state can lie outside
the two-magnon continuum of energies, the decay rate of the bound state has to
always lie within the two-magnon continuum of decay rates. We then derive
analytically the bound state solutions for a system with nearest-neighbour and
next-nearest-neighbour XY interaction and nonlocal dissipation, finding that
the inclusion of nonlocal dissipation allows more freedom in engineering the
energy and decay rate dispersions for the bound states. Finally, we numerically
study a model of an experimental set-up that should allow the realisation of
dissipative bound states by using Rydberg-dressed atoms coupled to a photonic
crystal waveguide (PCW). We demonstrate that this model can exhibit many key
features of our simpler models.
