<i>UltraStat</i>: Ultrafast Spectroscopy beyond the Fourier Limit Using Bayesian Inference

The discrete Fourier transform (dFT) plays a central role in many ultrafast experiments, allowing the recovery of spectroscopic observables from time-domain measurements. In resonant experiments when population relaxation and coherence components of the signal coexist, the dFT is usually preceded by multiexponential fitting to remove the large population term. However, this procedure results in errors in both the recovered decay rates and the line shapes of the coherence spectral components. While other methods such as linear prediction singular value decomposition fit both terms simultaneously, they are limited to specific models that may not represent the true signal. These methods do not allow for systematic noise analysis or error estimation and require <i>a priori</i> knowledge of the signal rank. Here, we describe a general approach to parameter estimation in ultrafast spectroscopy─<i>UltraStat</i>─grounded in Bayesian analysis without the limitations set by Fourier theory. Using simulated, but realistic data, we demonstrate in a statistical sense how <i>UltraStat</i> provides accurate parameter estimation in the presence of many experimental constraints: noise, signal truncation, limited photon budget, and nonuniform sampling. <i>UltraStat</i> provides superior resolution compared to the dFT, up to an order of magnitude in cases where the line shapes are well-approximated. In these cases, we establish that primarily noise, not sampling, limits spectral resolution. Moreover, we show that subsampling may reduce the number of acquired points by 90% compared to the Nyquist-Shannon criteria. <i>UltraStat</i> greatly improves parameter estimation by providing statistically bound spectral and dynamics analysis, pushing the limits of ultrafast science.