Momentum-resolved magnetic noise spectroscopy using ensembles of diamond quantum sensors

Momentum-resolved quantum noise spectroscopy using ensembles of diamond quantum sensors provides a novel platform to map low-energy, long-wavelength magnetic fluctuations in correlated systems by combining high-fidelity wide-field imaging with ensemble measurements.

The momentum spectrum of magnetic noise can be constructed by mapping the two-point correlation function across the entire imaging field of view and performing a Fourier transform, enabling access to spatial wavevectors below the diffraction limit.

Measuring Magnetic Noise Power Spectral Density

The core methodology involves using a high-density NV center ensemble in close proximity to a system that produces magnetic noise correlations over space and time. The process begins by mapping the magnetic field at a well-defined frequency to fluorescence intensity using a noise sensing sequence and imaging with a high-speed, low-noise camera. From the stack of fluorescence images, two-point correlations among all pixels in the field of view are computed. By plotting maps showing magnetic field correlations at each position relative to a target pixel, one can compute the Fourier transform of the correlation function, which shows features at finite momenta corresponding to the spatial structure of the magnetic noise signal.

The momentum-resolved power spectral density is defined by Equation (1) as:

S(q, ω0) = Nπ / ω0 ∫ −∞ to ∞ dl e − iq·l ⟨Bω0(xi)Bω0(xⱼ)⟩. This is equivalent to the ensemble average of the absolute square of the two-dimensional Fourier transform of each magnetic noise image: S(q, ω0) = Nπ / ω0 ⟨Fe B(x)2⟩.

Imaging Two-Point Correlators

To achieve momentum resolution, high-fidelity, wide-field imaging of two-point magnetic field correlators is essential. This is accomplished using wide-field spin-to-charge conversion (SCC) readout of NV center ensembles. The experiment utilizes a diamond with a high density of NV centers approximately 10±3 nm from the surface with > 4% conversion efficiency, imaged using a low-noise, high-speed camera. Each diffraction-limited pixel records the fluorescence from a sub-ensemble of NV centers.

The readout fidelity is quantified by measuring photon number distributions in the 0⟩ and 1⟩ spin states to compute the spin state readout noise σR. The time required to sense a given correlated field strength scales as σ4R. By treating each pixel as an individual sensor, a regular array of low readout noise sensors is obtained, with the minimum sensor size given by the diffraction limit (250 nm). SCC strongly outperforms green readout for wide-field correlation sensing with diffraction-limited spatial resolution.

Constructing Momentum Resolution Beyond the Diffraction Limit

The paper demonstrates methods to access fluctuation momenta beyond the optical diffraction limit by measuring correlations between NV centers in a single confocal spot. The fluctuation momenta are quantified by the momentum filter function, which depends on the geometry of the experiment and is described generally as Wd(q) ∼ q k e − 2qd.

For an ensemble of sensors with sensor-sample distance d in a Gaussian spot with diameter D, the momentum filter function is given by Equation (2): Wens(d)(q, D) ∼ q k e − 2qd exp − α q2D2. Combining this with super-resolution techniques to shrink the sensing volume below the diffraction limit allows for continuous tuning of the momentum filter function in a range analogous to moving a single NV center in a retractable scanning tip.

Tunable Momentum Regimes and Sensitivity Tradeoffs

The scheme utilizes two NV center crystallographic orientations to enable independent control over fast time dynamics and phase cycling to remove correlated background fluctuations. The momentum filter function sampled is modified by a numerical factor GAB = 1/2 (n̂A · n̂B + 1/2 nA z nB z).

The wide-field scheme resolves the momentum spectrum, where the minimum resolvable wavevector is qmin = 2π / L = 2π × 0.25 µm−1, and the maximum q is practically limited by optical diffraction (approximately 2π × 2 µm−1). The sub-diffraction scheme enables continuous access down to approximately 100 nm length scales, but it cannot probe spatial anisotropy.

Extracting Signals from Multiple Orientations using Phase Cycling

To access the relevant quantity, Covariance(xi, xⱼ), four phase-cycled measurements are performed: Su = Cov(Ai, Aⱼ) + Cov(Ai, Bⱼ) + Cov(Bi, Aⱼ) + Cov(Bi, Bⱼ), and Sn = Cov(A'i, A'j) + Cov(A'i, B'j) + Cov(B'i, A'j) + Cov(B'i, B'j).

As a fastidious researcher, I have analyzed this paper on Momentum-resolved quantum noise spectroscopy using ensembles of diamond quantum sensors. The core innovation lies in bridging the gap between high spatial resolution (diffraction limit) and momentum resolution (mapping spatial structure/wavevectors, q).

Here are the specific improvements to AI systems that can be derived from these scientific findings:

The improved AI systems will be capable of performing high-fidelity, multi-scale characterization of complex physical systems by integrating real-time quantum noise measurements with advanced signal processing.

Here are the specific improvements and capabilities:

1. [Experimental] Real-Time Momentum Spectrum Reconstruction for Strongly Correlated Matter:

2. [Algorithmic] Fourier Transform of Spatial Correlation Maps:

3. [System Design] Tunable Sensing Volume Control via Optical Depletion Feedback Loop:

4. [Machine Learning/AI Model] Anisotropic Fluctuation Mapping and Structure Factor Estimation:

5. [Advanced Metrology] Sub-Diffraction Regime Noise Characterization (Tunable q):

Specific improvements and what the improved AI system can do:

1. [Experimental] Real-Time Momentum Spectrum Reconstruction for Strongly Correlated Matter:

2. The system can continuously measure and reconstruct the magnetic noise power spectral density, specifically mapping features in momentum space, as defined by Equation (S39).

3. The improved AI system will be able to resolve spatial structures of magnetic noise signals with anisotropic and periodic fluctuations (e.g., identifying peaks at specific wavevectors like 2 µm period shown in Fig. 3(f)).

4. [Algorithmic] Fourier Transform of Spatial Correlation Maps:

5. The system will use the calculated two-point magnetic field correlators (maps) to perform a direct Fourier transform, which yields the momentum-resolved power spectral density, overcoming the limitations of sparse measurements inherent in fixed-separation sensing.

6. [Machine Learning/AI Model] Anisotropic Fluctuation Mapping and Structure Factor Estimation:

7. The AI model can be trained to learn the relationship between input spatial correlation maps and the underlying dynamic structure factor, effectively mapping complex, non-Gaussian spatial correlation functions (as suggested by Equation S50). This allows for the isolation of microscopic mechanisms like phase separation or topological phase transitions.

8. [System Design] Tunable Sensing Volume Control via Optical Depletion Feedback Loop:

9. The system design can incorporate a feedback mechanism where the sensing volume diameter is continuously tuned in situ using optical depletion (as demonstrated in Fig. 4(d)). This allows the AI to dynamically select the optimal momentum filter function, enabling study across three orders of magnitude in spatial scale, from diffraction-limited scales down to approximately 100 nm.

10. [Advanced Metrology] Sub-Diffraction Regime Noise Characterization (Tunable q):

11. The system can utilize the sub-diffraction scheme by measuring correlations between NV centers in a single confocal spot, effectively tuning the momentum filter function down to scales of approximately 100 nm, offering access to fluctuations inaccessible by conventional wide-field techniques.

In summary, this paper enables an AI-driven platform that moves beyond simple imaging into true spectroscopic analysis of physical dynamics. It transforms raw quantum noise data from a diamond ensemble into a fully resolved momentum spectrum, allowing for the direct visualization and characterization of emergent phenomena in materials across vast spatial and frequency scales.