We present Atacama Large Millimeter/submillimeter Array (ALMA) Band 5 observations of a galaxy at $z=1.91$, GDS24569, in search of molecular gas in its vicinity via the [C I] $^3$P$_1$-$^3$P$_0$ line. GDS24569 is a massive ($log M_*/M_odot=11$) passively evolving galaxy, and characterized by compact morphology with an effective radius of $sim0.5$ kpc. We apply two blind detection algorithms to the spectral data cubes, and find no promising detection in or around GDS24569 out to projected distance of $sim320$ kpc, while a narrow tentative line ($4.1 sigma$) is identified at $+1200$ km/s by one of the algorithms. From the non-detection of [C I], we place a $3sigma$ upper limit on molecular hydrogen mass, $sim 7.1 times 10^9 M_odot$, which converts to an extremely low gas-to-stellar mass fraction, $< 5 %$. We conduct a spectral energy distribution modeling by including optical-to-far-infrared data, and find a considerably high ($sim0.1%$) dust-to-stellar mass ratio, $sim10$-$100times$ higher than those of local early-type galaxies. In combination with a previous result of an insufficient number of surrounding satellite galaxies, it is suggested that GDS24569 is unlikely to experience significant size evolution via satellite mergers. We discuss possible physical mechanisms that quenched GDS24569.
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