Quantum nonlinear spectroscopy of single nuclear spins

The study investigates how to access higher-order correlations in quantum systems that are inaccessible to conventional nonlinear spectroscopy using classical probes. Correlations (moments) of measurement outcomes encode distinguishing features of different noise types and quantum objects, with higher-order correlations being especially informative because many different systems share similar first- and second-order statistics. Conventional nonlinear optical and magnetic resonance spectroscopies, driven by classical fields, measure response functions that involve only commutators and thus access a limited subset of correlations. Quantum nonlinear spectroscopy, by contrast, uses a quantum sensor entangled with the target object and read out in a chosen basis, enabling extraction of arbitrary types and orders of correlations, including those involving anti-commutators. The purpose of this work is to demonstrate quantum nonlinear spectroscopy of single nuclear spins via sequential weak measurements using a nitrogen-vacancy (NV) center in diamond, and to show that higher-order (fourth-order) correlations provide fingerprint features that distinguish quantum spins from classical fields (e.g., Gaussian noise or random-phased AC fields) that are otherwise indistinguishable by second-order correlations.

Prior works established that second-order correlation spectroscopy using NV centers enables high spectral resolution in nanoscale NMR and can probe quantum foundations (e.g., Bell and Leggett-Garg inequalities). Higher-order correlations have been used in cold atom systems to reveal non-Gaussian fluctuations and many-body physics. Conventional nonlinear spectroscopy (optical and magnetic resonance) with classical probes retrieves only specific commutator-based quantum correlations and classical products, missing anti-commutator-containing correlations. Quantum light spectroscopy with entangled photons demonstrated enhanced temporal and spectral resolution, motivating quantum-probe-based nonlinear spectroscopy. Theoretical proposals showed that sequential weak measurements by quantum sensors can characterize arbitrary-order correlations, separating quantum and classical contributions via measurement basis design. However, experimental quantum nonlinear spectroscopy of single nuclear spins and extraction of higher-order correlations had not been demonstrated before this work.

• Sensor-target system: An NV center electron spin (sensor) coupled to single 13C nuclear spins in diamond under a static magnetic field aligned with the NV axis (B0 ≈ 0.2502 T). Sensor states |+> = |0> and |-> = |1> of the NV spin triplet are used.
• Sequential weak measurement protocol: Each shot consists of (i) optical initialization of the NV to |0>, (ii) preparation of |x> via a microwave π/2 pulse, (iii) controlled weak interaction with the target via a dynamical decoupling sequence to induce weak entanglement, (iv) rotation to measure along a chosen axis at angle θ in the xy-plane (measuring σθ), and (v) storage of the sensor state in the 14N nuclear spin via a SWAP gate for repetitive readout. The sequence is repeated to build statistics for moments Sij, Sijk, and Sijkl reconstructed from photon counts.
• Dynamical decoupling and coupling parameters: A Knill KDD-XY5 sequence with N2 = 100 π pulses modulates the sensor-target interaction, yielding tunable weak coupling strength α ≈ 0.189π. Inter-pulse time ≈ 186.68 ns (π pulse length ≈ 68.67 ns). Under these controls, the effective nuclear field in the interaction picture is B(t) = A[cos(ν0 t) − i σx sin(ν0 t)], with nuclear Zeeman frequency ν0 = 2.6795 MHz.
• Readout enhancement: The NV electron spin state is transferred to the 14N nuclear spin via a SWAP composed of CNOTe – CNOTn – CNOTe gates. The 14N memory is repeatedly read out M times using spin-dependent fluorescence (limited to 40 repetitions to mitigate decoherence during laser illumination). Typical gate durations: CNOTe ≈ 4 µs, CNOTn ≈ 50 µs; each readout repetition includes one CNOTe and a 0.3 µs laser pulse.
• Photon-count-based reconstruction: Photon counts for spin states follow Poisson statistics with mean n+ and n−, yielding an observed count n = n̄ + δ σ d + w, where δ = (n+ − n−)/2, d is contrast, and w is intrinsic count noise. For different shots i ≠ j, intrinsic fluctuations wi are independent, giving ⟨δni δnj⟩ = d2 ⟨δσi δσj⟩ and ⟨δni δnj δnk⟩ = d3 ⟨δσi δσj δσk⟩, enabling reconstruction of moments of σ from photon count moments.
• Signals and correlation mapping: The first, second, and third statistical moments of measurement outputs are related to target field correlations. For classical fields B(t), S1 ≈ cosθ[1 − ⟨φ2⟩/2], S2 ∝ sin2θ ⟨φi φj⟩, and Sijk ∝ (1/2) sin2θ cosθ times combinations of phase correlations that map to fourth-order field correlations. For quantum targets (operator B), by choosing initialization and measurement bases, one can select contributions from commutators (quantum) or anti-commutators (classical-like). For high-temperature nuclear spins, second-order quantum correlations vanish, and the third moment contains both classical and quantum parts. For a spin-1/2 target, quantum correlations double the third moment contribution.
• Spectral analysis: Third-moment Sijk is acquired over varying time separations and 2D Fourier transformed in ti and tj to obtain characteristic peak patterns distinguishing quantum spins from classical fields (Gaussian noise or random-phased AC fields). The measurement angle was set to θ = 54.0037° to maximize sin2θ cosθ.
• Experimental setup and sample: Room-temperature confocal microscope in the bore of a 250 mT superconducting magnet; diamond sample is 12C-enriched (99.995%) (111)-oriented polished slice (2 mm × 2 mm × 80 µm) with single NV centers created by electron irradiation. Typical NV coherence: T1 ≈ 50 µs (Ramsey), T2 ≈ 300 µs (spin echo). Rabi frequency ≈ 7 MHz for electron spin.

• Demonstration of quantum nonlinear spectroscopy on a single nuclear spin: Using sequential weak measurement via an NV center, the experiment extracted fourth-order correlations of a single 13C nuclear spin—correlations inaccessible to conventional nonlinear spectroscopy with classical probes.
• Second-order equivalence between quantum spin and classical field: The second-order signal Sij ∝ cos(ν0 tij) e−tij/τc for the nuclear spin at ν0 = 2.6795 MHz (B0 = 0.2502 T) closely matches that from a random-phased AC field B(t) = B0 cos(ν0 t + φ), showing that second-order spectra cannot distinguish them.
• Distinct third-moment 2D spectra: The 2D Fourier spectrum of the third moment Sijk for a quantum spin shows four peaks at (±ν0, ±ν0) [diagonal and anti-diagonal], whereas a random-phased AC field exhibits six peaks at ±(2ν0, ν0), ±(ν0, −ν0), and ±(ν0, 2ν0). These are fingerprint features that distinguish a quantum spin from classical fields.
• Quantized peak-height ratio reveals spin number: For N uniformly coupled spins, the relative height n of eight specific peaks [(0, ±ν0), (±ν0, 0), ±(ν0, 2ν0), ±(2ν0, ν0)] to those at ±(ν0, −ν0) is n = 1 − 1/N. The measured average n ≈ 0.12 ± 0.10 indicates N ≈ 1, i.e., a single nuclear spin, analogous to photon g(2)-based emitter counting.
• Confirmation of quantumness via amplitude ratio: The third-moment amplitude relative to the square of the second moment yields factor r_c expected to be 1 for a quantum spin and 1/2 for a classical field. Fitting across datasets gives r_c = 1.13 with standard deviation 0.368, consistent with quantum behavior and inconsistent with purely classical noise.
• Methodological parameters: Measurement angle θ = 54.0037° maximized the third-order signal; KDD-XY5 with N2 = 100 pulses achieved effective coupling strength α ≈ 0.189π; inter-pulse time 186.68 ns.

The findings show that by entangling a quantum sensor with a target and choosing the measurement basis, one can extract higher-order correlations, including those containing commutators, that are not accessible with classical driving in conventional nonlinear spectroscopy. Although second-order signals cannot distinguish a single quantum spin from a random-phased AC field, the measured third-moment spectra reveal qualitatively different peak structures that unambiguously identify the quantum nature of the target. The quantized relation between peak heights and the number of coupled spins provides a discrete method to count spins, paralleling photon-count correlation techniques in quantum optics. This capability enables classical-noise-free detection of quantum objects and provides a tool to verify quantumness via higher-order correlations. The approach thus addresses the central question of whether higher-order, quantum-specific correlations can be measured for single nuclear spins and used to discriminate quantum targets from classical fluctuations.

This work experimentally demonstrates quantum nonlinear spectroscopy of a single nuclear spin using sequential weak measurements with an NV center in diamond, extracting fourth-order correlations and revealing quantum correlations absent in classical fields. The third-moment 2D spectra provide fingerprint features that differentiate quantum spins from classical noises and allow discrete counting of the number of spins. The method opens avenues for leveraging higher-order correlations in quantum sensing, noise characterization for quantum computing, studies of quantum many-body physics, and tests of quantum foundations (e.g., higher-order Bell or Leggett-Garg inequalities). Future directions include improving readout efficiency (e.g., resonant readout at low temperature) to reduce laser-induced decoherence, and implementing flexible initialization and measurement axes per shot to fully separate classical (anti-commutator) from quantum (commutator) contributions, enabling higher-dimensional spectroscopy and more precise quantumness verification.

• The measurement basis was fixed throughout the sequence (a single θ between x and y), which limited the spectroscopy to two dimensions because certain operators in consecutive measurements occur at the same time and prevented full separation of classical and quantum contributions; in the presented fourth-order signal, classical and quantum correlations contribute equally.
• Readout fidelity and system stability constrained the protocol; to mitigate nuclear spin decoherence from green laser excitation, readout repetitions were limited to 40, reducing overall signal-to-noise.
• The discrete spin-counting method assumes similar coupling strengths to multiple spins; additional work is needed to understand effects of dissimilar couplings, analogous to unequal emitter brightness in optical g(2)(0) measurements.
• Fast hopping between different sensor spin states can affect relative peak heights in the spectra.
• Room-temperature operation and optical pumping can induce unwanted nuclear spin decoherence; more efficient, lower-temperature readout could improve performance.