  Single photons and single-photon pairs, inherently nonclassical in their
nature, are fundamental elements of quantum sciences and technologies. Here, we
propose to realize the nonclassical correlated deterministic photon pairs at
the single-photon level for a single atom trapped in bimodal cavities. It is
shown that the photon emissions for bimodal cavities exhibit a high single
photon purity and relatively large intracavity photon numbers, which are
ascribed to the combination of optical Stark shift enhanced energy-spectrum
anharmonicity and quantum interference suppressed two-photon excitation. Our
scheme generates photon-photon pairs with strong photon blockade for cavity
modes via constructing a highly tunable cavity-enhanced deterministic
parametric down-conversion process. Furthermore, we show that the
cross-correlation function between the two cavity modes vastly violates the
Cauchy-Schwarz inequality of the classical boundary, which unambiguously
demonstrates the nonclassicality of the single-photon pairs. Our result reveals
a prominent strategy to generate the high-quality two-mode single-photon
sources with strong nonclassical correlation, which could provide versatile
applications in distribution of quantum networks, long distance quantum
communication, and fundamental tests of quantum physics.
