Extracting Spectral Diffusion in Two-Dimensional Coherent Spectra via the Projection Slice Theorem

TL;DR

The paper tackles the challenge of quantifying spectral diffusion from two-dimensional coherent spectroscopy by embedding the Frequency-Frequency Correlation Function (FFCF) into the Projection Slice Theorem (PST). It derives time-domain projections that include $C(T)$ for arbitrary inhomogeneity, and converts them to 1D frequency-domain slices via numerical Fourier transforms, enabling simultaneous extraction of the homogeneous dephasing rate $\gamma$, the inhomogeneous width $\sigma$, and the FFCF value $C(T)$ from diagonal and cross-diagonal slices. The method is validated on GaAs quantum wells, showing that neglecting the FFCF leads to nonphysical results at longer waiting times, while including $C(T)$ provides physically meaningful parameters and good fits ($R^2 \approx 0.997$–$0.999$). The approach is computationally efficient, does not rely on full 2D fitting, and is broadly applicable to various systems undergoing spectral diffusion, with discussion of finite-pulse effects beyond the impulsive limit.

Abstract

A robust and streamlined method is presented for efficiently extracting spectral diffusion from two-dimensional coherent spectra by employing the projection-slice theorem. The method is based on the optical Bloch equations for a single resonance that include a Frequency-Frequency Correlation Function (FFCF) in the time domain. Through the projection slice theorem (PST), analytical formulation of the diagonal and cross-diagonal projections of time-domain two-dimensional spectra are calculated that include the FFCF for arbitrary inhomogeneity. The time-domain projections are Fourier transformed to provide frequency domain slices that can be fit to slices of experimental spectra. Experimental data is used to validate our lineshape analysis and confirm the need for the inclusion of the FFCF for quantum wells that experience spectral diffusion.

Figures (4)

• Figure 1: (a) Analyzing the cross-diagonal slice of both expressions with and without the Frequency-Frequency Correlation Function (FFCF) while setting C(T) = 1. (b) Comparison of diagonal slices.
• Figure 2: The sequence of 2D spectra (left column) and slices (right column) for a two-level system as $C(T)$ decreases from 1 to 0, arranged from top to bottom. The frequency detuning from the resonance frequency is normalized the the dephasing rate. The inhomogeneous distribution width is 5 times the homogeneous linewidth. For the two-dimensional spectra, the horizontal axis is the frequency corresponding to Fourier-transforming with respect to $t$ and the vertical axis with respect to $\tau$.
• Figure 3: Fit of the experimental data for a short population waiting time of T = 0.2 ps. (a) 2D rephasing spectrum of the heavy-hole exciton. (b) Cross-diagonal and (c) diagonal slices (dots) with the fit results (solid lines). At this short delay, where $C(T) = 1$, the fitting procedure accurately extracts the lineshape parameters, and the inclusion of the FFCF term does not change the result.
• Figure 4: Fit of the experimental data at a long population waiting time of $T = 60~\mathrm{ps}$. (a) The 2D rephasing spectrum, showing a more symmetric lineshape due to spectral diffusion. (b, c) Fits (solid lines) to the cross-diagonal and diagonal data (dots). At this longer delay, including the FFCF is essential for a physically meaningful result, yielding $C(T) \approx 0.44$. Ignoring the FFCF leads to an unphysical determination of the system parameters.