  Ramsey interferometry is a widely used tool for precisely measuring
transition frequencies between two energy levels of a quantum system, with
applications in time-keeping, precision spectroscopy, quantum optics, and
quantum information. Often, the coherence time of the quantum system surpasses
the one of the oscillator probing the system, thereby limiting the
interrogation time and associated spectral resolution. Correlation spectroscopy
overcomes this limitation by probing two quantum systems with the same noisy
oscillator for a measurement of their transition frequency difference; this
technique has enabled very precise comparisons of atomic clocks. Here, we
extend correlation spectroscopy to the case of multiple quantum systems
undergoing strong correlated dephasing. We model Ramsey correlation
spectroscopy with $N$ particles as a multi-parameter phase estimation problem
and demonstrate that multiparticle quantum correlations can assist in reducing
the measurement uncertainties even in the absence of entanglement. We derive
precision limits and optimal sensing techniques for this problem and compare
the performance of probe states and measurement with and without entanglement.
Using one- and two-dimensional ion Coulomb crystals with up to 91 qubits, we
experimentally demonstrate the advantage of measuring multi-particle quantum
correlations for reducing phase uncertainties, and apply correlation
spectroscopy to measure ion-ion distances, transition frequency shifts,
laser-ion detunings, and path-length fluctuations. Our method can be
straightforwardly implemented in experimental setups with globally-coherent
qubit control and qubit-resolved single-shot read-out and is thus applicable to
other physical systems such as neutral atoms in tweezer arrays.
