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Nonlinear spectroscopy is widely used for studying physical systems.Conventional nonlinear optical spectroscopy and magnetic resonancespectroscopy, which use classical probes such as electromagnetic waves, canonly access certain types of correlations in a quantum system. The idea ofquantum nonlinear spectroscopy was recently proposed to use quantum probes suchas entangled photons to achieve sensitivities and resolutions beyond theclassical limits. It is shown that quantum sensing can extract arbitrary typesand orders of correlations in a quantum system by first quantum-entangling asensor and the object and then measuring the sensor. Quantum sensing has beenapplied to achieve nuclear magnetic resonance (NMR) of single atoms and thesecond-order correlation spectroscopy has been adopted to enhance the spectralresolution. However, quantum nonlinear spectroscopy (i.e., the measurement ofhigher-order correlations) of single nuclear spins is still elusive. Here wedemonstrate the extraction of fourth-order correlations of single nuclear spinsthat cannot be measured in conventional nonlinear spectroscopy, usingsequential weak measurement via an atomic quantum sensor, namely, anitrogen-vacancy center in diamond. We show that the quantum nonlinearspectroscopy provides fingerprint features to identify different types ofobjects, such as Gaussian noises, random-phased AC fields, and quantum spins,which would be indistinguishable in second-order correlations. The measuredfourth-order correlation unambiguously differentiates a single nuclear spin anda random-phased AC field. This work constitutes an initial step toward theapplication of higher-order correlations to quantum sensing, to examining thequantum foundation (by, e.g., higher-order Leggett-Garg inequality), and tostudying quantum many-body physics.

Quantum nonlinear spectroscopy of single nuclear spins