  Many-body correlations can yield key insights into the nature of interacting
systems; however, detecting them is often very challenging in many-particle
physics, especially in nanoscale systems. Here, taking a phosphorus donor
electron spin in a natural-abundance 29Si nuclear spin bath as our model
system, we discover both theoretically and experimentally that many-body
correlations in nanoscale nuclear spin baths produce identifiable signatures in
the decoherence of the central spin under multiple-pulse dynamical decoupling
control. We find that when the number of decoupling -pulses is odd, central
spin decoherence is primarily driven by second-order nuclear spin correlations
(pairwise flip-flop processes). In contrast, when the number of -pulses is
even, fourth-order nuclear spin correlations (diagonal interaction renormalized
pairwise flip-flop processes) are principally responsible for the central spin
decoherence. Many-body correlations of different orders can thus be selectively
detected by central spin decoherence under different dynamical decoupling
controls, providing a useful approach to probing many-body processes in
nanoscale nuclear spin baths.
