Theory of terahertz pump optical probe spectroscopy of phonon polaritons in noncentrosymmetric systems

Hybrid lattice-light modes, known as phonon polaritons, represent the backbone of advanced protocols based on THz pumping of infrared modes. Here we provide a theoretical framework able to capture the different roles played by phonon polaritons in experimental protocols based either on Raman-like pump and probe schemes, typical of four-wave mixing processes, or on THz pump-visible probe three-wave mixing protocols. By using a many-body description of the nonlinear optical kernel, along with a perturbative solution of nonlinear Maxwell's equations, we discuss the limitations of all-optical four-wave mixing protocols and we highlight the advantages of exploiting broadband THz pumps to enlarge the phase space of the phonon polariton dispersion at low momenta accessible in a single experiment. Besides providing a quantitative description of existing and future experiments, our results offer a general framework for the theoretical modeling of the hybridization between light and lattice degrees of freedom in time-resolved experiments.