  A two-level atom coupled to the quantized radiation field is studied. In the
physical relevant situation, the coupling function modeling the interaction
between the two component behaves like $|k|^{-1/2}$, as the photon momentum
tends to zero. This behavior is referred to as critical, as it constitutes a
borderline case. Previous results on non-existence state that, in the general
case, neither a ground state nor a resonance exists. Hasler and Herbst have
shown [10], however, that a ground state does exist if the absence of
self-interactions is assumed. Bach, Ballesteros, K\"onenberg, and Menrath have
then explicitly constructed the ground state this specific case [2] using the
multiscale analysis known as Pizzo's Method [13]. Building on this result, the
existence of resonances is considered. In the present paper, using multiscale
analysis, a resonance eigenvalue of the complex deformed Hamiltonian is
constructed. Neumann series expansions are used in the analysis and a suitable
Feshbach-Schur map controls the exponential decay of the terms in the series.
