  Negative nonlinear electron-phonon coupling involving an infrared-active
phonon mode can lead to an instability towards the formation of a polar lattice
distortion with ferrielectric (FE) moments accompanied by an electronic
charge-density wave (CDW). Analyzing a small model system in and out of thermal
equilibrium, we investigate the FE-CDW and its melting dynamics following an
ultrashort laser pulse that drives the ionic dipoles. We observe nonequilibrium
coherent phonon amplitude mode oscillations that soften towards the transition
to the normal phase. Our case study serves as a first step towards a
microscopic understanding of quantum nonlinear phononics as a basis for
nonequilibrium control in quantum materials.
