Ultrafast Control of Excitonic Rashba Fine Structure by Phonon Coherences in a Metal Halide Perovskite CH$_3$NH$_3$PbI$_3$

We discover hidden Rashba fine structure in CH$_3$NH$_3$PbI$_3$ and demonstrate its quantum control by vibrational coherence through symmetry-selective vibronic (electron-phonon) coupling. Above a critical threshold of a single-cycle terahertz pump field, a Raman phonon mode distinctly modulates the middle excitonic states with {\em persistent} coherence for more than ten times longer than the ones on two sides that predominately couple to infrared phonons. These vibronic quantum beats, together with first-principles modeling of phonon periodically modulated Rashba parameters, identify a {\em three-fold} excitonic fine structure splitting, i.e., optically-forbidden, degenerate dark states in between two bright ones. Harnessing of vibronic quantum coherence and symmetry inspires light-perovskite quantum control and sub-THz-cycle "Rashba engineering" of spin-split bands for ultimate multi-function device.