Precise assignment of excited-state bond length using branching ratio measurements: The B$^2\Sigma^+$ state of BaH molecules

Vibrational branching ratios in the B$^2\Sigma^+$ -- X$^2\Sigma^+$ and A$^2\Pi$ -- X$^2\Sigma^+$ optical-cycling transitions of BaH molecules are investigated using spectroscopic measurements and {\it ab initio} calculations. The experimental values are determined using fluorescence and absorption detection. The observed branching ratios have a very sensitive dependence on the difference in the equilibrium bond length between the excited and ground state, $\Delta r_e$: a 1 pm (.5\%) displacement can have a 25\% effect on the branching ratios but only a 1\% effect on the lifetime. The measurements are combined with theoretical calculations to reveal a clear preference for one particular set of published spectroscopic values for the B$^2\Sigma^+$ state ($\Delta r_e^{B-X}$ = 5.733 pm), while a larger bond length difference ($\Delta r_e^{B-X} = 6.3-6.7$ pm) would match the branching ratio data even better. By contrast, the observed branching ratio for the A$^2\Pi_{3/2}$ -- X$^2\Sigma^+$ transition is in excellent agreement with both the {\it ab initio} result and the spectroscopically measured bond lengths. This shows that care must be taken when estimating branching ratios for molecular laser cooling candidates, as small errors in bond length measurements can have outsize effects on the suitability for laser cooling. Additionally, our new calculations agree more closely with experimental values of the B$^2\Sigma^+$ state lifetime and spin-rotation constant, and revise the predicted lifetime of the H$^2\Delta$ state to 9.5 $\mu$s.