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Most material properties of great physical interest are directly related tonuclear dynamics, e.g. the ionic thermal conductivity, Raman/IR vibrationalspectra, inelastic X-ray, and Neutron scattering. A theory able to compute fromfirst principles these properties, accounting for the anharmonicity and quantumfluctuations in the nuclear energy landscape that can be implemented in systemswith hundreds of atoms is missing. Here, we derive an approximate theory forthe quantum time evolution of lattice vibrations at finite temperature. Thistheory introduces the time dynamics in the Self-Consistent HarmonicApproximation (SCHA) and shares with the static case the same computationalcost. It is nonempirical, as pure states evolve according to the Dirac leastaction principle and the dynamics of the thermal ensemble conserves both energyand entropy. The static SCHA is recovered as a stationary solution of thedynamical equations. We apply perturbation theory around the static SCHAsolution and derive an algorithm to compute efficiently quantum dynamicalresponse functions. Thanks to this new algorithm, we have access to theresponse function of any general external time-dependent perturbation, enablingthe simulation of phonon spectra without following any perturbative expansionof the nuclear potential or empirical methods. We benchmark the algorithm onthe IR and Raman spectroscopy of high-pressure hydrogen phase III, with asimulation cell of 96 atoms. Our work also explores the nonlinear regime of thedynamical nuclear motion, providing a paradigm to simulate the interaction withintense or multiple probes, as in pump-probe spectroscopy, or chemicalreactions involving light atoms, as the proton transfer in biomolecules

Time-Dependent Self Consistent Harmonic Approximation: Anharmonic nuclear quantum dynamics and time correlation functions