Bath-induced stabilization of classical non-linear response in two dimensional infrared spectroscopy

Classical response functions have shown considerable promise in computational 2D IR modeling; however, a simple diagrammatic description, analogous to that for open quantum systems, has been lacking. While a promising diagrammatic approach has recently been introduced for isolated systems, the resulting nonlinear response functions remain unstable at long times, a characteristic feature of integrable classical systems. Here, we extend this framework to incorporate system-bath interactions under the weak-anharmonicity approximation and explore the resulting conditions for bath-induced stabilization. The resulting expression for the weakly anharmonic response function is remarkably simple and exhibits a one-to-one correspondence with the quantum counterpart in the $\hbar\to 0$ limit, offering potential computational advantages in extending the approach to large, multi-oscillator systems. We find that (to lowest order in anharmonicity) the bath-induced stabilization of both linear and nonlinear classical response functions depends sensitively on the nature of spectral density, particularly on the balance between low-frequency and high-frequency components. Application of this classical diagrammatic approach to 2D IR spectroscopy of the amide I band captures the characteristic population-time-dependent dynamics associated with spectral diffusion, suggesting that the approach may prove useful in describing real experimental systems at ambient temperatures.