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