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

MOSFIRE Spectroscopy of Quiescent Galaxies at 1.5 < z < 2.5. II - Star Formation Histories and Galaxy Quenching