  The biphotons, generated from a hot atomic vapor via the process of
spontaneous four-wave mixing (SFWM), have the following merits: stable and
tunable frequencies as well as linewidth. Such merits are very useful in the
applications of long-distance quantum communication. However, the hot-atom SFWM
biphoton sources previously had far lower values of generation rate per
linewidth, i.e., spectral brightness, as compared with the sources of biphotons
generated by the spontaneous parametric down conversion (SPDC) process. Here,
we report a hot-atom SFWM source of biphotons with a linewidth of 960 kHz and a
generation rate of 3.7$\times$ $10^5$ pairs/s. The high generation rate,
together with the narrow linewidth, results in a spectral brightness of
3.8$\times$ $10^5$ pairs/s/MHz, which is 17 times of the previous best result
with atomic vapors and also better than all known results with all kinds of
media. The all-copropagating scheme together with a large optical depth (OD) of
the atomic vapor is the key improvement, enabling the achieved spectral
brightness to be about one quarter of the ultimate limit. Furthermore, this
biphoton source had a signal-to-background ratio (SBR) of 2.7, which violated
the Cauchy-Schwartz inequality for classical light by about 3.6 folds. Although
an increasing spectral brightness usually leads to a decreasing SBR, our
systematic study indicates that both of the present spectral brightness and SBR
can be enhanced by further increasing the OD. This work demonstrates a
significant advancement and provides useful knowledge in the quantum technology
using photons.
