  We demonstrate a silicon-chip biphoton source with an unprecedented quantum
cross correlation up to ${\rm g_{si}^{(2)}(0) = (2.58 \pm 0.16) \times 10^4}$.
The emitted biphotons are intrinsically single-mode, with self correlations of
${\rm g_{ss}^{(2)}(0) = 1.90 \pm 0.05}$ and ${\rm g_{ii}^{(2)}(0) = 1.87 \pm
0.06}$ for signal and idler photons, respectively. We observe the waveform
asymmetry of cross correlation between signal and idler photons and reveal the
identical and non-exponential nature of self correlations of individual signal
and idler photon modes, which is a nature of cavity-enhanced nonlinear optical
processes. The high efficiency and high purity of the biphoton source allow us
to herald single photons with a conditional self correlation $\rm
g_{c}^{(2)}(0)$ as low as $\rm 0.0059 \pm 0.0014$ at a pair flux of $\rm 1.95
\times 10^5$ pairs/s, which remains below $\rm 0.026 \pm 0.001$ for a biphoton
flux up to $\rm 2.93 \times 10^6$ pairs/s, with a photon preparation efficiency
in the single-mode fiber up to 51%, among the best values that have ever been
reported. Our work unambiguously demonstrates that silicon photonic chips are
superior material and device platforms for integrated quantum photonics.
