Ionized and atomic interstellar medium in the $z=6.003$ quasar SDSS J2310+1855

Our ALMA detection of the [O I]_146μm and [N II]_122μm emission lines together with previous observations of the [O III]_88μm and [C II]_158μm lines in J2310+1855 allows us to constrain the atomic and ionized gas masses as well as the SFR. We follow the method in Ferkinhoff et al. (2011) to estimate the ionized gas mass from the [N II]_122μm and [O III]_88μm emission lines. Given the first and second nitrogen ionization energies of 14.5 and 29.6 eV, it is likely that in H II regions, nitrogen is mostly singly ionized and traces much of the fully ionized hydrogen, and its abundance relative to H ($\chi(\rm N^{+})$) can be estimated from the nitrogen relative abundance (N/H) through $\chi(\rm N^{+})$ $\lesssim$ N/H. For a local thermal equilibrium (LTE) level distribution, the line emission thus provides a lower limit on the ionized gas mass associated with the emission. The minimum ionized gas mass traced by singly ionized nitrogen is:

where $m_{\rm H}$ is the atomic hydrogen mass, $g_{2}=5$ and $A_{21}=7.5\times 10^{-6}\ s^{-1}$ are the statistical weight and Einstein coefficient of the upper energy level, $Q({T_{\rm ex}})=1+5e^{-188/T_{\rm ex}}+3e^{-70/T_{\rm ex}}=9$ (at the high-temperature limit) is the partition function, $h$ is the Plank constant, and $\nu_{21}$ is the line frequency. Adopting a solar nitrogen abundance $\chi(\rm N^{+})\lesssim N/H=6.76\times 10^{-5}$ from Asplund et al. (2009), we derive a minimum ionized gas mass of $M^{min}_{ionized}=6.2\times 10^{8}$ $\rm M_{\sun}$. The ionized gas mass can also be derived from the [O III]_88μm line by replacing the [N II]_122μm related terms with those of the [O III]_88μm line in Eq.1. The parameters for [O III]_88μm are $g_{2}=3$, $A_{21}=2.7\times 10^{-5}\ s^{-1}$, $Q({T_{\rm ex}})=1+5e^{-440/T_{\rm ex}}+3e^{-163/T_{\rm ex}}=9$ (at the high-temperature limit), and $\chi(\rm O^{++})\textless O/H=4.90\times 10^{-4}$ (solar abundance; Asplund et al. 2009). Adopting an [O III]_88μm luminosity of $2.44\times 10^{9}$ $\rm L_{\sun}$ (Hashimoto et al., 2019), we estimate a minimum ionized gas mass of $M^{min}_{ionized}=7.7\times 10^{7}$ $\rm M_{\sun}$.

In addition, we estimate the neutral gas mass from the [C II]_158μm and [O I]_146μm lines. In the optically thin regime, the singly ionized carbon mass (Weiß et al. 2005; Venemans et al. 2017b) is

where $Q(T_{\rm ex})=2+4e^{-91.2/T_{\rm ex}}$ is the C II partition function, and $T_{\rm ex}$ is the excitation temperature. Assuming LTE where $T_{\rm kin}\approx T_{\rm ex}$, and adopting a gas temperature of $T_{\rm kin}=228\ \rm K$ derived from the CO spectral line energy distribution (SLED) (Li et al., 2020), we estimate the singly ionized carbon mass of $M_{\rm C^{+}}=2.1\times 10^{7}\rm\ M_{\sun}$. With a $\chi(\rm C^{+})\lesssim C/H=2.69\times 10^{-4}$ (solar abundance; Asplund et al. 2009), we constrain the minimum atomic gas mass of $M^{min}_{neutral}=7.8\times 10^{10}\rm\ M_{\sun}$²²2If we adopted a temperature of $T_{\rm ex}=\rm 500,\ 100\ or\ 50\ K$, the estimated gas mass would be $\rm 0.9,\ 1.3\ or\ 2.3$ times of the value presented here.. Following a similar formula to [C II]_158μm, the atomic oxygen mass can be calculated by substituting the parameters with those of the [O I]_146μm line in Eq. 1 of Weiß et al. (2005):

where $Q(T_{\rm ex})=5+3e^{-228/T_{\rm ex}}+e^{-329/T_{\rm ex}}$ is the partition function of oxygen, $A_{01}$ is the Einstein coefficient, $T_{\rm 0}$ is the energy above the ground state, and $C$ is the conversion factor between $\rm pc^{2}$ and $\rm cm^{2}$. We derive an ionized oxygen mass of $M_{\rm O^{+}}=8.1\times 10^{6}\rm\ M_{\sun}$ (for $T_{\rm ex}=228\rm\ K$), which leads to $M^{min}_{neutral}=1.7\times 10^{10}\rm\ M_{\sun}$ adopting $\chi(\rm O^{+})\lesssim O/H=4.90\times 10^{-4}$ ³³3If we adopted a temperature of $T=\rm 500,\ 100\ or\ 50\ K$, the estimated gas mass would be $\rm 0.5,\ 5.3\ or\ 135.1$ times of the value presented here. (solar abundance; Asplund et al. 2009). The estimated neutral gas masses from the [C II]_158μm and [O I]_146μm lines are comparable to the molecular gas mass derived from the CO$(2-1)$ line of $4.3\times 10^{10}$ $\rm M_{\sun}$ (Shao et al., 2019), and about an order of magnitude larger than the ionized gas mass derived from the [N II]_122μm and [O III]_88μm lines.

The FIR FS lines are tracers of the SFR (e.g., Farrah et al. 2013; De Looze et al. 2014). We estimate the SFR based on the FS emission lines using Eq. 6 of Farrah et al. (2013):

The SFRs estimated from the [C II]_158μm, [O I]_146μm, [N II]_122μm, and [O III]_88μm lines are 1524, 11917, 2208, and 1058 $\rm M_{\sun}\ yr^{-1}$, respectively. We note that the [O I]_146μm line suggests an extremely high SFR. The error bars of the parameters in the above relations are large, which result in large uncertainties in the SFR estimation. ⁴⁴4The statistical uncertainties are 2.6, 2.0, 1.3, and 1.8 dex for the [C II]_158μm, [O I]_146μm, [N II]_122μm, and [O III]_88μm lines estimated from the relation (Farrah et al., 2013). Moreover, the [O I]_146μm line flux may not give a reliable estimate of the SFR, as (1) the high $L_{\rm[OI]_{146\mu m}/}L_{\rm FIR}$ ratio (see discussions in Section 4.2) suggests extra gas heating by the AGN in addition to the UV photons from young massive stars, and (2) the relation in Farrah et al. (2013) is based on samples with [O I]_146μm luminosity less than $10^{9}$ $\rm L_{\sun}$, which may not be valid when extrapolated to higher [O I]_146μm luminosity for SFR calculation. The SFR derived from the infrared dust ($10^{-11}\times L_{\rm IR}$) ranges from 1300 to 3600 $\rm M_{\sun}\ yr^{-1}$, depending on the different model parameters used in the dust continuum SED fitting (Shao et al., 2019). The SFRs traced by both the dust continuum and the [C II]_158μm, [N II]_122μm and [O III]_88μm FIR FS lines are consistent, while the [O I]_146μm line suggests a higher SFR, although all the derived values are consistent within statistical uncertainties. The derived gas masses and SFRs are listed in Table 1.

We compare the [N II]_122μm and [O I]_146μm emission in J2310+1855 with values found for local and high-$z$ systems by studying the line-to-FIR ratio as a function of FIR luminosity ($L_{\rm FIR}$), and as a function of FIR surface brightness ($\Sigma_{\rm FIR}$). The comparison samples include samples of local dwarf galaxies, (U)LIRGs, and AGNs (Farrah et al. 2013; Cormier et al. 2015; Spinoglio et al. 2015; Rosenberg et al. 2015; Díaz-Santos et al. 2017; Herrera-Camus et al. 2018a). We also include high-$z$ objects with available [O III]_88μm and [N II]_122μm detections for comparison: the $z=7.54$ quasar J1342+0928 (Novak et al., 2019), the quasar J0439+1634 at $z=6.52$ (Yang et al., 2019), the Cloverleaf, a lensed $z=2.56$ quasar (Ferkinhoff et al., 2015), the $z=4.22$ lensed dusty star-forming galaxy SPT 0418-47 (De Breuck et al., 2019), and the $z=4.69$ quasar BRI1202 QSO and its companion submillimeter galaxy (SMG) BRI1202 SMG (Lee et al., 2019). Detailed descriptions about the comparison samples are provided in the caption of Figure 3.

We show the results in Figure 3. For the [N II]_122μm line, all the local (U)LIRGs and AGNs show decreasing $L_{\rm[NII]_{122\mu m}/}L_{\rm FIR}$ ratios with increasing $L_{\rm FIR}$ for $L_{\rm FIR}\gtrsim 10^{11}$ $\rm L_{\sun}$. We adopt the J2310+1855 FIR luminosity of $L_{\rm FIR}=(1.03\pm 0.22)\times 10^{13}$ $\rm L_{\sun}$ from Shao et al. (2019). The quasar J2310+1855, together with other high-z objects (e.g., the Cloverleaf, BRI1202 SMG and BRI1202 QSO) show a slightly higher $L_{\rm[NII]_{122\mu m}/}L_{\rm FIR}$ ratio than the local (U)LIRG and AGN samples for $L_{\rm FIR}\sim 10^{13}L_{\sun}$, while the high-z objects with $L_{\rm FIR}\sim 10^{12}L_{\sun}$ (e.g., J1342+0928 and SPT 0418-47), follow the trend of the local (U)LIRG and AGN samples. Recent studies find that the “line deficit” is more closely correlated with $\Sigma_{\rm FIR}$ than $L_{\rm FIR}$ in samples of local (U)LIRGs and AGNs (e.g., Lutz et al. 2016; Díaz-Santos et al. 2017; Herrera-Camus et al. 2018a; Herrera-Camus et al. 2018b). For all the local comparison samples, FIR source sizes are only available in Díaz-Santos et al. (2017) and Herrera-Camus et al. (2018a). We compare the line-to-FIR ratio of the quasar with the above two samples of local (U)LIRGs and AGNs as a function of $\Sigma_{\rm FIR}$, and the results are shown in Figure 3. For the high-$z$ objects, we calculate $\Sigma_{\rm FIR}$ following the definition of Herrera-Camus et al. (2018a) and Díaz-Santos et al. (2017). All of the high-z objects including J2310+1855, lie within the range of (U)LIRGs and AGNs. This suggest that the $L_{\rm[NII]_{122\mu m}/}L_{\rm FIR}$ ratio measured from the nuclear region ($\lesssim 3$ kpc) of J2310+1855 is comparable to that of local (U)LIRGs, AGNs and other high-$z$ systems with a similar FIR surface brightness.

As for the [O I]_146μm line, all of the local (U)LIRGs and AGNs reveal decreasing $L_{\rm[OI]_{146\mu m}/}L_{\rm FIR}$ ratios with increasing $L_{\rm FIR}$ for $L_{\rm FIR}\gtrsim 10^{11}$ $\rm L_{\sun}$ (Figure 3). J2310+1855, together with J1342+0928 and SPT 0418-47, indicate higher $L_{\rm[OI]_{146\mu m}/}L_{\rm FIR}$ ratios compared to local samples at similar $L_{\rm FIR}$. As for J0439+1634, the uncertainty in the magnification factor makes it hard to conclude if values are consistent with the local relation (for $\mu_{\rm max}=6.6$) or if it has a comparable $L_{\rm[OI]_{146\mu m}/}L_{\rm FIR}$ ratio to J2310+1855 at similar $L_{\rm FIR}$ (for $\mu_{\rm min}=2.6$, Yang et al. 2019). As for the case of $\Sigma_{\rm FIR}$, similar to other high-$z$ quasars and galaxies, the [O I]_146μm line flux measured in the central $\approx 3$ kpc region of J2310+1855 exceeds the range of local systems and exhibits over two times the values in local (U)LIRGs and AGNs measured from the galaxy scale, suggesting no sign of [O I]_146μm deficit. The [O I]_146μm emission traces the warm and dense neutral gas (with temperature $\rm T\sim\Delta E/K=329\ K$ and critical density $\rm n_{cr}=9.4\times 10^{4}\ cm^{-3}$). The luminous [O I]_146μm detection of J2310+1855 suggests warm and dense gas in the nuclear region, which is possibly heated by the photons contributed by the AGN in addition to the UV photons from young massive stars. This is in agreement with spatially resolved observations of local AGNs, with higher [O I]_146μm fluxes detected in the central regions (which suggest warm and dense gas) compared to the mean values found in the outer star-forming disk (Herrera-Camus et al., 2018b). In addition, we compare the [C II]_158μm and [O III]_88μm lines of J2310+1855 with the $L_{\rm line}/L_{\rm FIR}$ versus $\Sigma_{\rm FIR}$ relation observed in local galaxies, (U)LIRGs and AGNs (Herrera-Camus et al., 2018a). The results suggest that the [C II]_158μm, [N II]_122μm, and [O III]_88μm lines exhibit a similar deficit as in local systems (Wang et al. (2013); Hashimoto et al. (2019)). The [O I]_146μm line is over two times brighter than in local (U)LIRGs and AGNs at its $\Sigma_{\rm FIR}$, suggesting that the absence of an [O I]_146μm deficit could be caused by a warm and dense neutral gas content, which is possibly heated by the photons from the AGN in addition to the UV photons from young massive stars.

The [C II]_158μm line traces both the neutral and the ionized ISM. The contribution to the [C II]_158μm emission from the HII regions can be constrained if tracers purely from ionized gas phase are observed. The [N II]_122μm and [N II]_205μm lines, as examples, are widely used to probe the [C II]_158μm emission from the ionized gas phase in the local universe to the highest redshift ($z\gtrsim 6$; e.g., Oberst et al. 2006, 2011; Herrera-Camus et al. 2016, 2018a; Díaz-Santos et al. 2017; Novak et al. 2019). The [N II]_122μm and [N II]_205μm lines are from the same ionization state of nitrogen, thus the ratio between these two lines has a negligible dependence on the radiation field and metallicity. The difference in the critical densities between the [N II]_122μm (310 $\rm cm^{-3}$) and [N II]_205μm (48 $\rm cm^{-3}$) lines enables the [N II]_122μm/[N II]_205μm ratio to trace ionized gas density. Once the gas density is determined, the [C II]_158μm/ [N II]_205μm ratio and the fraction of [C II]_158μm emission from the ionized gas phase will be constrained by comparing the observational data with radiative transfer models.

With only the [N II]_122μm line observed in J2310+1855, we estimate its [N II]_205μm luminosity from (1) the relation between the $L_{\rm[NII]_{205\mu m}}/L_{\rm FIR}$ ratio and the FIR color ($f_{70}/f_{160}$) in local (U)LIRGs (Eq. 1 from Zhao et al. 2016), and (2) the relation between the $L_{\rm[NII]_{205\mu m}}/L_{\rm[CII]_{158\mu m}}$ ratio and the FIR color ($f_{60}/f_{100}$) in local (U)LIRGs (Eq. 5 from Lu et al. 2015). Adopting a dust temperature of 40 K and an emissivity index $\beta=1.6$ from Shao et al. (2019), we estimate an $f_{70}/f_{160}$ ratio of 2.23 and an $f_{60}/f_{100}$ ratio of 0.92. The two methods give comparable estimated [N II]_205μm luminosities, with $L_{\rm[NII]_{205\mu m}}=4.2\times 10^{8}\ \rm L_{\sun}$ derived from the infrared luminosity (method 1), and $L_{\rm[NII]_{205\mu m}}=4.6\times 10^{8}\ \rm L_{\sun}$ based on the [C II]_158μm luminosity ⁵⁵5The estimated [N II]_122μm/[C II]_158μm ratio in J2310+1855 of 1/18 is consistent with values found for other high-z quasars, e.g., BRI 1335–0417 at z = 4.41 and BRI 1202–0725 at z = 4.70 (Lu et al., 2017, 2018), as well as other high-z galaxies, e.g., ID 141 at z=4.24 (Cheng et al., 2020). (method 2). Figure 4 (a) shows the [C II]_158μm/ [N II]_205μm and [N II]_122μm/ [N II]_205μm ratios in the H II region predicted by the radiative transfer code Cloudy (Ferland et al., 2017). Comparing the estimated [N II]_122μm/ [N II]_205μm ratio ⁶⁶6The [N II]_205μm luminosity is estimated from the relation of local (U)LIRGs (PDRs). In AGNs, the XDRs may contribute significantly to the [N II]_122μm emission, which should be subtracted when using the [N II]_122μm/ [N II]_205μm ratio as a density probe. In the case of J2310+1855, the observed [N II]_122μm/IR ratio is comparable to that found for local (U)LIRGs and AGNs, which disfavors a significant contribution of the [N II]_122μm emission from XDRs. Accordingly the estimated [N II]_122μm/[N II]_205μm ratio here traces the ionized gas density. with the model, we estimate that the ionized gas density is 45 $\rm cm^{-3}$ in J2310+1855. We emphasis that if we adopt a hotter dust temperature than those suggested by Wang et al. (2019b), the resulting estimated [N II]_205μm luminosity will be lower. Accordingly, we consider the derived ionized gas density as a lower limit. The ionized gas density $n_{e}\gtrsim$45 $\rm cm^{-3}$ of J2310+1855 is consistent with that found in local (U)LIRGs and AGNs (10 $\sim$ 200 $\rm cm^{-3}$ from Herrera-Camus et al. 2018a; a median of 46 $\rm cm^{-3}$ from Díaz-Santos et al. 2017), but lower than that in the $z=7.5$ quasar J1342+0928 of $n_{e}$ $\geq$ 180 $\rm cm^{-3}$. With the estimated ionized gas density of $n_{e}\gtrsim$45 $\rm cm^{-3}$, we derive an ionized gas contribution of $\lesssim$17$\%$ to the [C II]_158μm line in J2310+1855. This is in agreement with the values derived from local (U)LIRGs and AGNs (Díaz-Santos et al. 2017; Herrera-Camus et al. 2018a), and the z= 7.54 quasar J1342+0928.

The [O III]_88μm and [N II]_122μm lines have similar critical densities but different ionization potentials ($n_{cr}=510\ \rm cm^{-3}$, E = 35.1 eV for [O III]_88μm, and $n_{cr}=310\ \rm cm^{-3}$, E = 14.5 eV for [N II]_122μm). Thus, the flux ratio between these two lines could be used to trace the ionization parameters and metallicity. Utilizing the radiative transfer model Cloudy, Pereira-Santaella et al. (2017) and Rigopoulou et al. (2018) find that once the ionization parameter is determined, the ratio between the [O III]_88μm and [N II]_122μm line can be used as a gas-phase metallicity indicator in star-forming galaxies. The detection of both lines in J2310+1855 allows us therefore to constrain the ISM metallicity. However, several issues are noted before applying this method to quasars: (1) these models consider the galaxy stellar emission as the only radiation field. Pereira-Santaella et al. (2017) predict that in the presence of an AGN, the [O III]_88μm emission is barely affected but the [N II]_122μm line may have a significant contribution from XDRs (up to $\sim 90\%$) depending on the AGN power-law index; (2) the [O III]_88μm/[N II]_122μm ratio also has strong dependence on the ionization parameter ($U$) of the radiation field. Rigopoulou et al. (2018) proposed using the ratio of underlying continuum luminosity measured at the wavelengths of the [O III]_88μm and [N II]_122μm lines ($\rm 88\mu m/122\mu m$) for the ionization parameter estimation. However, it should be noted that the observed continuum $\rm 88\mu m/122\mu m$ ratio does not reliably probe the ionization parameter of the galaxy stellar radiation field for systems with powerful AGNs, and (3) the strengths of the [O III]_88μm and [N II]_122μm lines can be affected by the dust opacity effect. The local ULIRG Arp 220, as an example, was detected in absorption for the [O I]_63μm and [N II]_122μm FS lines (Rangwala et al. 2011; González-Alfonso et al. 2012). Optically thick dust for $\lambda_{rest}\lesssim 200\mu m$ has also been observed in high-z SMGs (Riechers et al., 2013). It is also possible that the dust in J2310+1855 is optically thick for $\lambda_{rest}\lesssim 200\mu m$. However, current dust continuum observations are insufficient to constrain the dust opacity, as well as to correct for the extinction effect for the FS lines in J2310+1855.

Following the method in Rigopoulou et al. (2018), we first estimate the ionization parameter of the galaxy stellar radiation field from the continuum $\rm 88\mu m/122\mu m$ ratio. To exclude continuum emission contributed by the quasar, we use the decomposed galaxy dust emission with a dust temperature $T_{\rm dust}=40\ \rm K$ and emissivity index $\beta=1.6$ (Shao et al., 2019) to estimate the continuum $\rm 88\mu m/122\mu m$ ratio. This leads to a continuum $\rm 88\mu m/122\mu m$ ratio of 1.4 and ionization parameter $\rm log(U)=-2.6\sim-2.1$ for typical H II region density of $n_{e}=\rm 10^{1-3}cm^{-3}$ (Figure 4, Rigopoulou et al. 2018). From the grid presented in Table A1 of Pereira-Santaella et al. (2017), the observed [O III]_88μm/[N II]_122μm ratio of 2.8 in J2310+1855 corresponds to a metallicity of $Z/Z_{\sun}$=1.5$-$2.1 ⁷⁷7We note that if the AGN contributes significantly to the observed [N II]_122μm flux, the estimated metallicity will be lower. For J2310+1855, the comparable $L_{\rm[NII]_{122\mu m}/}L_{\rm FIR}$ ratio with (U)LIRGs suggests no clear sign of significant XDR contribution to the [N II]_122μm line. relative to solar metallicity for the estimated $\rm log(U)=-2.6\sim-2.1$. The metallicity is consistent with that of other high-redshift quasars and galaxies ($Z/Z_{\sun}$=0.7$-$2.0 for J1342+0928, Novak et al. 2019; $Z/Z_{\sun}$=0.3$-$1.3 for SPT 0418-47, De Breuck et al. 2019).

Combinations of different FIR FS emission lines provide a wealth of information about the ISM physical and chemical conditions, e.g., temperature, density, ionization parameter, metallicity, and volume filling factor ratio between different gas phases. J2310+1855 has been detected in the [C II]_158μm, [O I]_146μm, [N II]_122μm, and [O III]_88μm FS lines, making it possible to compare its physical conditions of the neutral and ionized gas with that of local and high-redshift systems. The comparison samples are local dwarf galaxies, (U)LIRGs, and AGNs (Farrah et al. 2013; Cormier et al. 2015; Spinoglio et al. 2015; Fernández-Ontiveros et al. 2016; Herrera-Camus et al. 2018a), and the high-$z$ systems with available detections, including the quasars J1342+0928 and J0439+1634, as well as the lensed galaxy SPT 0418-47.

The majority of the [C II]_158μm emission is originating in the neutral gas phase. The [O I]_146μm line comes purely from the neutral gas phase and requires higher temperature and higher density to be excited compared to the [C II]_158μm line ($\rm T\sim\Delta E/k=329\ K$, $\rm n_{cr}=9.4\times 10^{4}\ cm^{-3}$ for [O I]_146μm and $\rm T\sim\Delta E/k=91\ K$, $\rm n_{cr}=2.8\times 10^{3}\ cm^{-3}$ for [C II]_158μm). Thus the [C II]_158μm/[O I]_146μm ratio depends on both temperature and atomic gas density. The [N II]_122μm and [O III]_88μm lines trace the ionized gas phase, and the [O III]_88μm/[N II]_122μm ratio is dependent on metallicity and ionization parameter (Section 4.4). Figure 4(b) shows the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ vs $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ diagnostic diagram. The $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio is found to be lower in the high-temperature and high-density regime (Díaz-Santos et al. 2017; Herrera-Camus et al. 2018a). The AGN samples display a wide spread in the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio, while the (U)LIRGs and dwarf galaxies show less scatter with a median value of $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}\sim 10$ for the (U)LIRGs and $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}\sim 20$ for the dwarf galaxies. The lowest $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ values are only found in AGNs, which indicate warm and dense gas components possibly heated by the AGNs (e.g., Herrera-Camus et al. 2018a). J2310+1855 exhibits a $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio of 6.5, which is among the lowest values found in local AGNs and smaller than local (U)LIRGs and dwarf galaxies. Similar $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratios are found in other high-$z$ systems (J1342+0928 and SPT 0418-47 on Figure 4(b), and J0439+1634 of 3.3, Yang et al. 2019) as well. The low $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio measured from the nuclear region (within $\approx 3$ kpc) in J2310+1855 suggests a warm and dense neutral gas component, which is likely to be heated by the luminous AGN (see also Section 4.2). The $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ ratio increases with increasing ionization parameter or decreasing metallicity (Nagao et al. 2011; Pereira-Santaella et al. 2017; Rigopoulou et al. 2018). The dust opacity could also influence the strengths of the $\rm[OIII]_{88\mu m}$ and $\rm[NII]_{122\mu m}$ lines (see Section 4.4). Local (U)LIRGs and AGNs reveal a wide range in the $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ ratio, which may result from different ionization parameters in different galaxies (considering negligible differences in metallicity). The dwarf galaxy sample, however, shows systematically higher $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ ratios compared to all local (U)LIRGs and AGNs, which can be attributed to the low metallicity in these dwarf galaxies. The $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ ratio of 2.8 in J2310+1855 is within the range of local (U)LIRGs and AGNs, consistent with the value found in J1342+0928, and smaller than the $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ ratio of 20.9 in SPT 0418-47. A considerable fraction of the [N II]_122μm emission can arise from the XDRs (see details in Sections 4.3 and 4.4). The comparable $L_{\rm[NII]_{122\mu m}/}L_{\rm FIR}$ ratio in J2310+1855 to the ratios found in local (U)LIRGs and AGNs, however disfavors a significant XDR contribution to the [N II]_122μm line. Accordingly, it is likely a result of (1) lower ionization parameter, (2) higher metallicity, or (3) higher dust opacity compared to SPT 0418-47 that causes the lower $L_{\rm[OIII]_{88\mu m}}/L_{\rm[NII]_{122\mu m}}$ ratio in J2310+1855. This is also in agreement with the derived smaller metallicity in SPT 0418-47 than J2310+1855 (see Section 4.4).

In Figure 4(c), we show the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ versus $L_{\rm[OIII]_{88\mu m}}/L_{\rm[OI]_{146\mu m}}$ diagnostic diagram. Because the [O III]_88μm/[O I]_146μm ratio has negligible dependence on metallicity, it is a direct probe of the volume filling factor ratio between the ionized and atomic gas. Local (U)LIRGs and AGNs have similar distributions of the $L_{\rm[OIII]_{88\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio, while the high $L_{\rm[OIII]_{88\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratios in dwarf galaxies are due to the low gas density that enables UV photons to reach the outer regions of galaxies. The high-redshift systems J2310+1855, J1342+0928 and SPT 0418-47 reside within the range of local (U)LIRGs and AGNs. J2310+1855 exhibits a $L_{\rm[OIII]_{88\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio of 1.9, which is over two times lower than J1342+0928 and SPT 0418-47, and approximately three times lower than the mean of the (U)LIRGs and AGNs, suggesting a slightly smaller ionized fraction in J2310+1855 compared to local (U)LIRGs, AGNs and other high-$z$ systems. This is also in agreement with the derived small contribution($\lesssim$17$\%$) of [C II]_158μm from ionized gas in Section 4.3. Figure 4(d) shows the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OI]_{146\mu m}}$ versus $L_{\rm[CII]_{158\mu m}}/L_{\rm[OIII]_{88\mu m}}$ diagnostic diagram. Assuming that the majority of [C II]_158μm emission arises from the neutral gas, the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OIII]_{88\mu m}}$ ratio could probe the atomic and ionized gas volume filling factor ratio as well. J2310+1855 reveals a $L_{\rm[CII]_{158\mu m}}/L_{\rm[OIII]_{88\mu m}}=3.4$ comparable with the mean of local (U)LIRGs and AGNs, and over five times of the values found in J1342+0928 and SPT 0418-47. That the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OIII]_{88\mu m}}$ and $L_{\rm[OIII]_{88\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratios deliver different results for J2310+1855 compared to other samples is likely due to the additional dependence of the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OIII]_{88\mu m}}$ ratio on metallicity and the variation of the [C II]_158μm emission from ionized gas in different galaxies. Therefore, the $L_{\rm[OIII]_{88\mu m}}/L_{\rm[OI]_{146\mu m}}$ ratio is a better tracer than the $L_{\rm[CII]_{158\mu m}}/L_{\rm[OIII]_{88\mu m}}$ ratio when probing the atomic and ionized gas volume filling factor ratio.