An understanding of microscopic interactions in liquids is of fundamental importance in chemistry. However, the structure and dynamics of complex systems in the condensed-phase, especially far from thermal equilibrium, is masked by broad, and often features, absorption and emission spectra. Nonlinear optical spectroscopy has proven to be a powerful and general approach to disentangling congested spectra by spreading information across multiple dimensions, revealing features oftentimes hidden in lower-order projections. As the dimensionality of the measurement increases, the better the microscopic interactions are revealed as spectral bands disperse in the large hyper-spectral volume. This, however, comes at a steep price as the signal decreases exponentially with increasing noise and experimental complexity. Here, we discuss a four-dimensional coherent spectroscopy that reveals coupling between electronic and vibrational transitions in complex, condensed-phase systems ranging from organic molecules to semiconductor nanocrystals. We reveal that high-resolution spectra may be extracted from these systems even in the presence of severe spectral broadening, both homogeneous and inhomogeneous in origin. The theoretical and experimental underpinnings of this method are discussed. Increasingly higher-order and higher-dimensionality spectroscopies are needed to understand the microscopic interactions that connect structure to dynamics to function.

Compressive sensing allows signals to be efficiently captured by exploiting their inherent sparsity. Here we implement sparse sampling to capture the electronic structure and ultrafast dynamics of molecular systems using phase-resolved 2D coherent spectroscopy. Until now, 2D spectroscopy has been hampered by its reliance on array detectors that operate in limited spectral regions. Combining spatial encoding of the nonlinear optical response and rapid signal modulation allows retrieval of state-resolved correlation maps in a photosynthetic protein and carbocyanine dye. We report complete Hadamard reconstruction of the signals and compression factors as high as 10, in good agreement with array-detected spectra. Single-point array reconstruction by spatial encoding (SPARSE) Spectroscopy reduces acquisition times by about an order of magnitude, with further speed improvements enabled by fast scanning of a digital micromirror device. We envision unprecedented applications for coherent spectroscopy using frequency combs and super-continua in diverse spectral regions.