  We investigate the two-photon transport through a waveguide side-coupling to
a whispering-gallery-atom system. Using the Lehmann-Symanzik-Zimmermann (LSZ)
reduction approach, we present the general formula for the two-photon processes
including the two-photon scattering matrices, the wavefunctions and the second
order correlation functions of the out-going photons. Based on the exact
results of the second order correlation functions, we analyze the quantum
statistics behaviors of the out-going photons for two different cases: (a) the
ideal case without the inter-modal coupling in the whispering gallery
resonator; (b) the case in the presence of the inter-modal coupling which leads
to more complex nonlinear behavior. In the ideal case, we show that the system
consists of two independent scattering pathways, a free pathway by a cavity
mode without atomic excitation, and a "Jaynes-Cummings" pathway described by
the Jaynes-Cummings Hamiltonian of a single-mode cavity coupling to an atom.
The free pathway does not contribution to correlated two-photon processes. In
the presence of intermodal mixing, the system no longer exhibit a free resonant
pathway. Instead, both the single-photon and the two photon transport
properties depend on the position of the atom. Thus, in the presence of
intermodal mixing one can in fact tune the photon correlation properties by
changing the position of the atom. Our formalism can be used to treat resonator
and cavity dissipation as well.
