Time evolution of o-H$_2$D$^+$, N$_2$D$^+$, and N$_2$H$^+$ during the high-mass star formation process

Deuterium fractionation is a well-established evolutionary tracer in low-mass star formation, but its applicability to the high-mass regime remains an open question. The abundances and ratios of deuterated species have often been proposed as reliable evolutionary indicators for different stages of the high-mass star formation. We investigate the role of N$_2$H$^+$ and key deuterated molecules as tracers of the different stages of the high-mass star formation, and test whether their abundance ratios can serve as reliable evolutionary indicators. We conducted APEX observations of o-H$_2$D$^+$ (1$_{10}$-1$_{11}$), N$_2$H$^+$ (4-3), and N$_2$d$^+$ (3-2) in 40 high-mass clumps at different evolutionary stages, selected from the ATLASGAL survey. Molecular column densities ($N$) and abundances ($X$), were derived through spectral line modelling, both under local thermodynamic equilibrium (LTE) and non-LTE conditions. The $N$(o-H$_2$D$^+$) show the smallest deviation from LTE results when derived under non-LTE assumptions. In contrast, N$_2$D$^+$ shows the largest discrepancy between the $N$ derived from LTE and non-LTE. In all the cases discussed, we found that $X$(o-H$_2$D$^+$) decreases more significantly with time than in the case of $X$(N$_2$D$^+$); whereas $X$(N$_2$H$^+$) increases slightly. Therefore, the validity of the recently proposed $X$(o-H$_2$D$^+$)/$X$(N$_2$D$^+$) ratio as a reliable evolutionary indicator was not observed for this sample. While the deuteration fraction derived from N$_2$D$^+$ and N$_2$H$^+$ clearly decreases with clump evolution, the interpretation of this trend is complex, given the different distribution of the two tracers. Our results suggest that a careful consideration of the observational biases and beam-dilution effects are crucial for an accurate interpretation of the evolution of the deuteration process during the high-mass star formation process.