Multitime responses from a generalized cumulant algebra

The ability to predict the nonlinear response of matter across timescales is central to understanding transport in and the spectroscopy of molecular, condensed matter, and quantum information systems. Yet, the theories used to describe these processes remain limited: Kubo’s generalized cumulant expansion—a cornerstone of Gaussian field theories—provides exact quantum dynamical solutions for a system’s one-time response but its generalization becomes difficult when responses depend on expectation values of the products of operators evaluated over multiple timescales, limiting its ability to tackle a range of dynamical problems involving correlation across timed interactions. To address this problem, we introduce a generalized cumulant algebra (GCA) that provides a simple framework for constructing multi-time responses. Our GCA resolves the ambiguities in Kubo’s formulation by introducing a hierarchy of ordering prescriptions that enable consistent treatment of forward, backward, and asymmetric operator orderings. The resulting formalism yields closed-form generating functions for multi-time cumulants that yield exact results in the Gaussian limit and exactly reduces to Kubo’s celebrated result in the single-time limit. Applying our GCA to two-dimensional electronic spectroscopy, we demonstrate how to distinguish between vibronic and non-Condon fluctuations that produce nearly identical linear and nonlinear spectra. These results establish a unified and computationally efficient theory for predicting and interpreting a broad class of complex dynamical responses of open quantum systems.