  Characterizing the physical conditions (density, temperature, ionization
state, metallicity, etc) of the interstellar medium is critical to our
understanding of the formation and evolution of galaxies. Here we present a
multi-line study of the interstellar medium in the host galaxy of a quasar at
z~6.4, i.e., when the universe was 840 Myr old. This galaxy is one of the most
active and massive objects emerging from the dark ages, and therefore
represents a benchmark for models of the early formation of massive galaxies.
We used the Atacama Large Millimeter Array to target an ensemble of tracers of
ionized, neutral, and molecular gas, namely the fine-structure lines: [OIII]
88$\mu$m, [NII] 122$\mu$m, [CII] 158$\mu$m, and [CI] 370$\mu$m and the
rotational transitions of CO(7-6), CO(15-14), CO(16-15), and CO(19-18); OH
163.1$\mu$m and 163.4$\mu$m; and H$_2$O 3(0,3)-2(1,2), 3(3,1)-4(0,4),
3(3,1)-3(2,2), 4(0,4)-3(1,3), 4(3,2)-4(2,3). All the targeted fine-structure
lines are detected, as are half of the targeted molecular transitions. By
combining the associated line luminosities, the constraints on the dust
temperature from the underlying continuum emission, and predictions from
photoionization models of the interstellar medium, we find that the ionized
phase accounts for about one third of the total gaseous mass budget, and is
responsible for half of the total [CII] emission. It is characterized by high
density (n~180 cm$^{-3}$), typical of HII regions. The spectral energy
distribution of the photoionizing radiation is comparable to that emitted by
B-type stars. Star formation also appears to drive the excitation of the
molecular medium. We find marginal evidence for outflow-related shocks in the
dense molecular phase, but not in other gas phases. This study showcases the
power of multi-line investigations in unveiling the properties of the
star-forming medium in galaxies at cosmic dawn.
