The Evolution of Environmental Quenching Timescales to $z\sim1.6$
  Using a sample of 4 galaxy clusters at $1.35 < z < 1.65$ and 10 galaxy
clusters at $0.85 < z < 1.35$, we measure the environmental quenching
timescale, $t_Q$, corresponding to the time required after a galaxy is accreted
by a cluster for it to fully cease star formation. Cluster members are selected
by a photometric-redshift criterion, and categorized as star-forming,
quiescent, or intermediate according to their dust-corrected rest-frame colors
and magnitudes. We employ a "delayed-then-rapid" quenching model that relates a
simulated cluster mass accretion rate to the observed numbers of each type of
galaxy in the cluster to constrain $t_Q$. For galaxies of mass $M_* \gtrsim
10^{10.5}~ \mathrm{M}_\odot$, we find a quenching timescale of $t_Q=$ 1.24 Gyr
in the $z\sim1.5$ cluster sample, and $t_Q=$ 1.50 Gyr at $z\sim1$. Using values
drawn from the literature, we compare the redshift evolution of $t_Q$ to
timescales predicted for different physical quenching mechanisms. We find $t_Q$
to depend on host halo mass such that quenching occurs over faster timescales
in clusters relative to groups, suggesting that properties of the host halo are
responsible for quenching high-mass galaxies. Between $z=0$ and $z=1.5$, we
find that $t_Q$ evolves faster than the molecular gas depletion timescale and
slower than an SFR-outflow timescale, but is consistent with the evolution of
the dynamical time. This suggests that environmental quenching in these
galaxies is driven by the motion of satellites relative to the cluster
environment, although due to uncertainties in the atomic gas budget at high
redshift, we cannot rule out quenching due to simple gas depletion.
