Quantum Geometric Injection and Shift Optical Forces Drive Coherent Phonons

We identify {\em injection} and {\em shift} rectified Raman forces, which are phononic counterparts of the photogalvanic effect, that drive lattice vibrations and trigger transient emergent properties. These forces are governed by the {\em quantum geometric tensor}, a {\em phononic shift vector}, and interband asymmetries in the electron-phonon coupling. The injection force acts displacively, while -- unlike conventional impulsive mechanisms -- the shift force emerges impulsively in the resonant interband absorbing regime when time-reversal symmetry is broken. Using the bilayer Haldane model, we quantify the injection and shift forces acting on interlayer shear phonons through both analytical and numerical methods. Strikingly, we reveal strong tunability, both in magnitude and direction, of the rectified forces by varying the driving frequency and magnetic flux, uncovering a distinct quantum geometric mechanism for ultrafast and coherent manipulation of quantum materials.