A Density Matrix Theory for Photothermal Two-Dimensional Infrared Spectroscopy

Two-dimensional infrared (2D-IR) spectroscopy can resolve vibrational anharmonicity, mode coupling, and energy transfer. The conventional 2D-IR detects nonlinear polarization that generates propagating photons after sample’s light matter interaction with an infrared pulse sequence. However, photon detection is typically limited by the diffraction limit. The newly developed photothermal AFM-based two-dimensional infrared (AFM-2DIR) spectroscopy changes this fundamental detection endpoint: a femtosecond infrared pulse sequence prepares vibrational excitations in the tip-enhanced nanoscale near field, after which nonradiative relaxation converts the deposited energy into local heat that is read out through AFM’s cantilever’s mechanical response, thereby bypassing the optical diffraction limit. Here, we establish a thermal action theory through propagation of density matrix that formulates photothermal 2D-IR as an action-detected nonlinear spectroscopy. The theory is anchored in a final-state population selection rule: only nonlinear optical pathways lead to excited state populations after the pulse sequence contribute to the detected photothermal 2D-IR signal. Based on this theory, we provide a unified description of the anharmonic ladder response of carbonyl, polariton-to-phonon relaxation in hexagonal boron nitride, and polariton-to-polariton conversion. This framework provides a testable theory for photothermal 2D-IR spectroscopy for nanoscale structural dynamics, interfaces, and energy transfer, as well as to serve as a basis for other photothermal 2D spectroscopy to be developed in the future.