Extremely Low Molecular Gas Content in the Vicinity of a Red Nugget Galaxy at $z=1.91$

We present Atacama Large Millimeter/submillimeter Array (ALMA) Band 5 observations of a compact 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 ($\log M_*/M_\odot=11$) and 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 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] in these deep data cubes, we place a $3\sigma$ upper limit on molecular hydrogen mass, $\sim 1.5 \times 10^9 M_\odot$, which converts to an extremely low gas-to-stellar mass fraction, $< 1 \%$. 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 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, $\sim10$-$100\times$ higher than those of local early-type galaxies. We discuss possible mechanisms that quenched GDS24569.