  We carry out a model study on the interaction of a $V$-type three-level
emitter with two quantized cavity modes which are weakly driven by two
classical fields. The emitter may be an atom or a molecule with nondegenerate
upper levels in general. The lifetimes of the two upper levels are assumed to
be much longer than the lifetime of the cavity photons. We calculate the
two-time second order coherence function, namely Hanbury Brown-Twiss function
$g^{(2)}(\tau)$ where $\tau$ is the time delay between the two modes. We
analyze the photon-photon correlations between the two cavity modes in terms of
$g^{(2)}(\tau)$. The variation of $g^{(2)}(\tau)$ as a function of $\tau$
exhibits collapse and revival type oscillations as well as quantum beats for
relatively short $\tau$ while in the limit $\tau \rightarrow \infty$,
$g^{(2)}(\tau) \simeq 1$. We further show that the two cavity field modes are
entangled when $g^{(2)}(0) > 2$. We develop a dressed state picture with single
photon in each mode to explain the results and use the negativity of the
partial transpose of the reduced field density matrix to show the entanglement
between field modes. The model presented in this paper may be useful for
generating entangled photon pairs, and also manipulating photon-photon
correlations with a cavity QED set up.
