  We derive effective Hamiltonians for a single dipolar emitter coupled to a
metal nanoparticle (MNP) with particular attention devoted to the role of
losses. For small particles sizes, absorption dominates and a non hermitian
effective Hamiltonian describes the dynamics of the hybrid emitter-MNP
nanosource. We discuss the coupled system dynamics in the weak and strong
coupling regimes offering a simple understanding of the energy exchange,
including radiative and non radiative processes. We define the plasmon Purcell
factors for each mode. For large particle sizes, radiative leakages can
significantly perturbate the coupling process. We propose an effective Fano
Hamiltonian including plasmon leakages and discuss the link with the
quasi-normal mode description. We also propose Lindblad equations for each
situation and introduce a collective dissipator for describing the Fano
behaviour.
