  A quantum master equation model for the interaction between a two-level
system and whispering-gallery modes (WGMs) of a microdisk cavity is presented,
with specific attention paid to current experiments involving a semiconductor
quantum dot (QD) embedded in a fiber-coupled, AlGaAs microdisk cavity. In
standard single mode cavity QED, three important rates characterize the system:
the QD-cavity coupling rate g, the cavity decay rate kappa, and the QD
dephasing rate gamma_perpendicular. A more accurate model of the microdisk
cavity includes two additional features. The first is a second cavity mode that
can couple to the QD, which for an ideal microdisk corresponds to a traveling
wave WGM propagating counter to the first WGM. The second feature is a coupling
between these two traveling wave WGMs, at a rate beta, due to backscattering
caused by surface roughness that is present in fabricated devices. We consider
the transmitted and reflected signals from the cavity for different parameter
regimes of {g,beta,kappa,gamma_perpendicular}. A result of this analysis is
that even in the presence of negligible roughness induced backscattering, a
strongly coupled QD mediates coupling between the traveling wave WGMs,
resulting in an enhanced effective coherent coupling rate g = sqrt(2)*g0
corresponding to that of a standing wave WGM with an electric field maximum at
the position of the QD. In addition, analysis of the second-order correlation
function of the reflected signal from the cavity indicates that regions of
strong photon antibunching or bunching may be present depending upon the
strength of coupling of the QD to each of the cavity modes. Such intensity
correlation information will likely be valuable in interpreting experimental
measurements of a strongly-coupled QD to a bi-modal WGM cavity.
