  We explore the role of quantum fluctuations in the strong-coupling limit of
the dissipative Jaynes-Cummings oscillator driven on resonance. For weak
excitation, we derive analytical expressions for the spectrum and the intensity
correlation function for the photons scattered by the two-state atom coupled to
the coherently driven cavity mode. We do so by writing down a birth-death
process adding the higher orders in the excitation strength needed to go beyond
the pure-state factorization, following the method introduced in [H. J.
Carmichael, {\it Statistical Methods in Quantum Optics 2}, Springer, 2008, Sec.
16.3.4]. Our results for the first and second-order correlation functions are
complemented by the numerical investigation of the waiting-time distribution
for the photon emissions directed sideways, and the comparison with ordinary
resonance fluorescence. To close out our discussion, we increase the driving
field amplitude and approach the critical point organizing a second-order
dissipative quantum phase transition by depicting the excitation pathways in
the intracavity field distribution for a finite system size.
