The ages, metallicities and element abundance ratios of massive quenched galaxies at z~1.6

We investigate the stellar population properties of a sample of 24 massive quenched galaxies at $1.25<z_\mathrm{spec}<2.09$ identified in the COSMOS field with our Subaru/MOIRCS near-IR spectroscopic observations. Tracing the stellar population properties as close to their major formation epoch as possible, we try to put constraints on the star formation history, post-quenching evolution, and possible progenitor star-forming populations for such massive quenched galaxies. By using a set of Lick absorption line indices on a rest-frame optical composite spectrum, the average age, metallicity [Z/H], and $\alpha$-to-iron element abundance ratio [$\alpha$/Fe] are derived as $\log(\mathrm{age}/\mathrm{Gyr})=0.04_{-0.08}^{+0.10}$, $\mathrm{[Z/H]}=0.24_{-0.14}^{+0.20}$, and $[\alpha/\mathrm{Fe}]=0.31_{-0.12}^{+0.12}$, respectively. If our sample of quenched galaxies at $\langle z \rangle = 1.6$ is evolved passively to $z=0$, their stellar population properties will align in excellent agreement with local counterparts at similar stellar velocity dispersions, which qualifies them as progenitors of local massive early-type galaxies. Redshift evolution of stellar population ages in quenched galaxies combined with low redshift measurements from the literature suggests a formation redshift of $z_\mathrm{f} \sim 2.3$ around which the bulk of stars in these galaxies have been formed. The measured [$\alpha$/Fe] value indicates a star formation timescale of $\lesssim 1$ Gyr, which can be translated into a specific star formation rate of $\simeq 1\,\mathrm{Gyr}^{-1}$ prior to quenching. Based on these findings, we discuss identifying possible progenitor star-forming galaxies at $z \simeq 2.3$. We identify normal star-forming galaxies, i.e, those on the star-forming main sequence, followed by a rapid quenching event, as likely precursors of the quenched galaxies at $\langle z \rangle = 1.6$ presented here.