  We investigate the stellar populations for a sample of 24 quiescent galaxies
at 1.5 < z < 2.5 using deep rest-frame optical spectra obtained with Keck
MOSFIRE. By fitting templates simultaneously to the spectroscopic and
photometric data, and exploring a variety of star formation histories, we
obtain robust measurements of median stellar ages and residual levels of star
formation. After subtracting the stellar templates, the stacked spectrum
reveals the Halpha and [NII] emission lines, providing an upper limit on the
ongoing star formation rate of 0.9 +/- 0.1 Msun/yr. By combining the MOSFIRE
data to our sample of Keck LRIS spectra at lower redshift, we analyze in a
consistent manner the quiescent population at 1 < z < 2.5. We find a tight
relation (with a scatter of 0.13 dex) between the stellar age and the
rest-frame U-V and V-J colors, which can be used to estimate the age of
quiescent galaxies given their colors. Applying this age--color relation to
large, photometric samples, we are able to model the number density evolution
for quiescent galaxies of various ages. We find evidence for two distinct
quenching paths: a fast quenching that produces compact post-starburst systems,
and a slow quenching of larger galaxies. Fast quenching accounts for about a
fifth of the growth of the red sequence at z~1.4, and half at z~2.2. We
conclude that fast quenching is triggered by dramatic events such as gas-rich
mergers, while slow quenching is likely caused by a different physical
mechanism.
