  The pairing ratio, a crucial metric assessing a biphoton source's ability to
generate correlated photon pairs, remains underexplored despite theoretical
predictions. This study presents experimental findings on the pairing ratio,
utilizing a double-$\Lambda$ spontaneous four-wave mixing biphoton source in
cold atoms. At an optical depth (OD) of 20, we achieved an ultrahigh biphoton
generation rate of up to $1.3\times10^7$ per second, with a successful pairing
ratio of $61\%$. Increasing the OD to 120 significantly improved the pairing
ratio to $89\%$, while maintaining a consistent biphoton generation rate. This
achievement, marked by high generation rates and robust biphoton pairing, holds
great promise for advancing efficiency in quantum communication and information
processing. Additionally, in a scenario with a lower biphoton generation rate
of $5.0 \times 10^4$ per second, we attained an impressive signal-to-background
ratio of 241 for the biphoton wavepacket, surpassing the Cauchy-Schwarz
criterion by approximately $1.5\times10^4$ times.
