  We utilized the all-copropagating scheme, which maintains the phase-match
condition, in the spontaneous four-wave mixing (SFWM) process to generate
biphotons from a hot atomic vapor. The scheme enables our biphotons not only to
surpass those in the previous works of hot-atom SFWM, but also to compete with
the biphotons that are generated by either the cold-atom SFWM or the
cavity-assisted spontaneous parametric down conversion. The biphoton linewidth
in this work is tunable for an order of magnitude. As we tuned the linewidth to
610 kHz, the maximum two-photon correlation function, $g_{s,as}^{(2)}$, of the
biphotons is 42. This $g_{s,as}^{(2)}$ violates the Cauchy-Schwartz inequality
for classical light by 440 folds, and demonstrates that the biphotons have a
high purity. The generation rate per linewidth of the 610-kHz biphoton source
is 1,500 pairs/(s$\cdot$MHz), which is the best result of all the sub-MHz
biphoton sources in the literature. By increasing the pump power by 16 folds,
we further enhanced the generation rate per linewidth to 2.3$\times$10$^4$
pairs/(s$\cdot$MHz), while the maximum $g_{s,as}^{(2)}$ became 6.7. In
addition, we are able to tune the linewidth down to 290$\pm$20 kHz. This is the
narrowest linewidth to date, among all the various kinds of single-mode
biphotons.
