Precise Determination of Excited State Rotational Constants and Black-Body Thermometry in Coulomb Crystals of Ca${}^{+}$ and CaH${}^{+}$

We present high-resolution rovibronic spectroscopy of calcium monohydride molecular ions (CaH${}^{+}$) co-trapped in a Coulomb crystal with calcium ions (${}^{40}$Ca${}^{+}$), focusing on rotational transitions in the $\mathrm{\mid }{X}^1{\mathrm{\Sigma }}^{+},\nu \mathrm{"}=0>\to \mathrm{\mid }{A}^1{\mathrm{\Sigma }}^{+},{\nu }^{\mathrm{\prime }}=2>$ manifold. By resolving individual P and R branch transitions with record precision and using Fortrat analysis, we extract key spectroscopic constants for the excited state $\mathrm{\mid }{A}^1{\mathrm{\Sigma }}^{+},{\nu }^{\mathrm{\prime }}=2>$, specifically, the band origin, the rotational constant, and the centrifugal correction. Additionally, we demonstrate the application of high-resolution rotational spectroscopy of CaH${}^{+}$ presented here as an in-situ probe of local environmental temperature. We correlate the relative amplitudes of the observed transitions to the underlying thermalized ground-state rotational population distribution and extract the black-body radiation (BBR) temperature.