  We propose to use relative strengths of far-infrared fine structure lines
from galaxies to characterise early phases of the inside-out quenching by
massive black holes (BHs). The BH feedback is thought to quench star formation
by evacuating the ambient gas. In order to quantify the feedback effect on the
gas density in the galactic centres, we utilise the outputs of IllustrisTNG and
Illustris simulations, which implement different BH feedback models. We devise
a physical model of H$_{\rm ~II}$ regions and compute the intensities of
[O$_{\rm ~III}$] $52$ and $88~{\rm \mu m}$ lines. The line intensity ratio is
sensitive to the local electron density, and thus can be used to measure the
strength and physical extent of the BH quenching. If the BH feedback abruptly
operates and expel the gas when it grows to a certain mass, as modelled in
IllustrisTNG, the low-density gas yields relatively weak [O$_{\rm ~III}$] $52$
line with respect to $88~{\rm \mu m}$. In contrast, if the feedback strength
and hence the local gas density are not strongly correlated with the BH mass,
as in Illustris, the line ratio is not expected to vary significantly among
galaxies with different evolutionary stages. We find these features are
reproduced in the simulations. We also show that the line ratios are not
sensitive to the aperture size for measurement, and thus observations do not
need to resolve the galactic centres. We argue that the integrated line ratios
can be used to capture the onset of the inside-out quenching by BHs.
