  We conduct a quantitative analysis of the star formation history (SFH) of the
Milky Way's bulge by exploiting the constraining power of its stellar [Fe/H]
and [Mg/Fe] distribution functions. Using APOGEE data, we confirm the
previously-established bimodal [Mg/Fe]--[Fe/H] distribution within 3 kpc of the
inner Galaxy. Compared to that in the solar vicinity, the high-$\alpha$
population in the bulge peaks at a lower [Fe/H]. To fit these observations, we
use a simple but flexible star formation framework, which assumes two distinct
stages of gas accretion and star formation, and systematically evaluate a wide
multi-dimensional parameter space. We find that the data favor a three-phase
SFH that consists of an initial starburst, followed by a rapid star formation
quenching episode and a lengthy, quiescent secular evolution phase. The
metal-poor, high-$\alpha$ bulge stars ([Fe/H]<0.0 and [Mg/Fe]>0.15) are formed
rapidly (<2 Gyr) during the early starburst. The density gap between the high-
and low-$\alpha$ sequences is due to the quenching process. The metal-rich,
low-$\alpha$ population ([Fe/H]>0.0 and [Mg/Fe]<0.15) then accumulates
gradually through inefficient star formation during the secular phase. This is
qualitatively consistent with the early SFH of the inner disk. Given this
scenario, a notable fraction of young stars (age<5 Gyr) is expected to persist
in the bulge. Combined with extragalactic observations, these results suggest
that a rapid star formation quenching process is responsible for bimodal
distributions in both the Milky Way's stellar populations and in the general
galaxy population and thus plays a critical role in galaxy evolution.
