  Post-starburst or "E+A" galaxies are rapidly transitioning from star-forming
to quiescence. While the current star formation rate of post-starbursts is
already at the level of early type galaxies, we recently discovered that many
have large CO-traced molecular gas reservoirs consistent with normal star
forming galaxies. These observations raise the question of why these galaxies
have such low star formation rates. Here we present an ALMA search for the
denser gas traced by HCN (1--0) and HCO+ (1--0) in two CO-luminous, quiescent
post-starburst galaxies. Intriguingly, we fail to detect either molecule. The
upper limits are consistent with the low star formation rates and with
early-type galaxies. The HCN/CO luminosity ratio upper limits are low compared
to star-forming and even many early type galaxies. This implied low dense gas
mass fraction explains the low star formation rates relative to the CO-traced
molecular gas and suggests the state of the gas in post-starburst galaxies is
unusual, with some mechanism inhibiting its collapse to denser states. We
conclude that post-starbursts galaxies are now quiescent because little dense
gas is available, in contrast to the significant CO-traced lower density gas
reservoirs that still remain.
