  Molecular polaritons have gained considerable attention due to their
potential to control nanoscale molecular processes by harnessing
electromagnetic coherence. Although recent experiments with liquid-phase
vibrational polaritons have shown great promise for exploiting these effects,
significant challenges remain in interpreting their spectroscopic signatures.
In this letter, we develop a quantum-mechanical theory of pump-probe
spectroscopy for this class of polaritons based on the quantum Langevin
equations and the input-output theory. Comparison with recent experimental data
shows good agreement upon consideration of the various vibrational
anharmonicities that modulate the signals. Finally, a simple and intuitive
interpretation of the data based on an effective mode-coupling theory is
provided. Our work provides a solid theoretical framework to elucidate
nonlinear optical properties of molecular polaritons as well as to analyze
further multidimensional spectroscopy experiments on these systems.
