  The discovery in deep near-infrared surveys of a population of massive
quiescent galaxies at $z>3$ has given rise to the question of how they came to
be quenched so early in the history of the Universe. Measuring their molecular
gas properties can distinguish between physical processes where they stop
forming stars due to a lack of fuel versus those where star-formation
efficiency is reduced and the gas is retained. We conducted Atacama Large
Millimeter/sub-millimeter Array (ALMA) observations of four quiescent galaxies
at $z=$ 3.5-4.0 found by the Fourstar Galaxy Evolution Survey (ZFOURGE) and a
serendipitous optically dark galaxy at $z=3.71$. We aim to investigate the
presence of dust-obscured star-formation and their gas content by observing the
dust continuum emission at Band-7 and the atomic carbon [C I]($^3P_1$-$^3P_0$)
line at 492.16 GHz. Among the four quiescent galaxies, only one source is
detected in the dust continuum at $\lambda_{\rm obs} = 870 {\rm \mu m}$. The
sub-mm observations confirm their passive nature, and all of them are located
more than four times below the main sequence of star-forming galaxies at
$z=3.7$. None of the targets are detected in [C I], constraining their gas mass
fractions to be $<$ 20%. These gas mass fractions are more than three times
lower than the scaling relation for star-forming galaxies at $z=3.7$. These
results support scenarios where massive galaxies at $z=$ 3.5-4.0 quench by
consuming/expelling all the gas rather than by reducing the efficiency of the
conversion of their gas into stars.
