  The origin of charge density waves (CDW) in TiSe$_2$ has long been debated,
mainly due to the difficulties in identifying the timescales of how and when
the excitonic pairing and electron-phonon coupling (EPC) come into play.
Without a proper time resolution and microscopic mechanism, one has to assume
simultaneous appearance of CDW and periodic lattice distortions (PLD). Here, we
accomplish a complete separation of exciton and PLD dynamics and unravel their
interplay in the ultrafast time domain in our real-time time-dependent density
functional theory simulations. We find that laser pulses knock off the exciton
order and induce a homogeneous bonding-antibonding transition in the initial 20
fs, then the weakened electronic order triggers ionic movements antiparallel to
the original PLD. The EPC comes into play after the initial 20~fs, and the two
processes mutually amplify each other leading to a complete inversion of CDW
ordering. The self-amplified dynamics reproduces the evolution of band
structures in excellent agreement with ultrafast photoemission experiment.
Hence we resolve the key processes in the initial dynamics of CDW that help
elucidate the mechanism underlying the long debated problem.
