Hyperfine-resolved optical spectroscopy of ultracold $^{87}$Rb$^{133}$Cs molecules: the $\mathrm{b}\,^3\Pi_0$ metastable state

Using an ultracold gas of $^{87}$Rb$^{133}$Cs molecules, we perform hyperfine-resolved spectroscopy of transitions from the vibronic ground state to the lowest rovibrational states of the electronic state $\mathrm{b}\,^3\Pi_0$, as a function of magnetic field. These transitions are spin forbidden, resulting in narrow linewidths, and feature near-diagonal Franck-Condon factors. We develop a model of the hyperfine and Zeeman structure that includes coupling between the $0^+$ and $0^-$ components of $\mathrm{b}\,^3\Pi_0$. We fit the spectra to obtain rotational and hyperfine coupling constants. We measure transition dipole moments associated with specific transitions by directly observing Rabi oscillations as a function of a resonant laser pulse duration. Using resonant $\pi$ pulses, we prepare molecules in the electronically excited state and directly measure the spontaneous emission rate.