Automatic Detection of Nuclear Spins at Arbitrary Magnetic Fields via Signal-to-Image AI Model

Quantum sensors leverage matter’s quantum properties to enable measurements with unprecedented spatial and spectral resolution. Among these sensors, those utilizing nitrogen-vacancy (NV) centers in diamond offer the distinct advantage of operating at room temperature. Nevertheless, signals received from NV centers are often complex, making interpretation challenging. This is especially relevant in low magnetic field scenarios, where standard approximations for modeling the system fail. Additionally, NV signals feature a prominent noise component. In this Letter, we present a signal-to-image deep learning model capable of automatically inferring the number of nuclear spins surrounding a NV sensor and the hyperfine couplings between the sensor and the nuclear spins. Our model is trained to operate effectively across various magnetic field scenarios, requires no prior knowledge of the involved nuclei, and is designed to handle noisy signals, leading to fast characterization of nuclear environments in real experimental conditions. With detailed numerical simulations, we test the performance of our model in scenarios involving varying numbers of nuclei, achieving an average error of less than 2 kHz in the estimated hyperfine constants.

Figure 1

Schematic representation of the SALI model. The neural network takes two input signals ($P_x$) obtained from CPMG sequences of different duration. In block (i) of the neural network, each input signal is independently processed through two one-dimensional convolutional blocks. Each block comprises two convolutional layers with $F$ filters of kernel size 3, followed by a batch normalization (BN) layer, a ReLU activation function, and a max-pooling layer with a window size of 2. The flattened outputs from these convolutional blocks are then concatenated and connected to block (ii), a fully connected block. The last dense layer in this block is reshaped into a rectangular image. Within block (iii), the second convolutional layer is a transposed convolutional layer with a stride of 2, effectively doubling the image’s width and height. The two-dimensional convolutional layers utilize $F$ filters with a kernel size of (3, 3). Block (iii) is subsequently connected to the output convolutional layer of the neural network, which is followed by an image postprocessing module to extract the number of nuclei ($n$) within the sample and the corresponding coupling constant pairs $(A_j^z,A_j^{\perp })$. See in-depth explanation in SM [23].

Figure 2

(a) True output image of the low-field neural network. Ten nuclei with random coupling constants in the ranges $A^z\in [-100,100]\mathrm{kHz}$ and $A^{\perp }\in [2,102]\mathrm{kHz}$ are represented. Each nucleus appears as a $5\times 5\mathrm{pixel}$ region in the image, as depicted in the highlighted circular area. The pixel values in this region are chosen to match a Gaussian distribution centered in $(A_j^z,A_j^{\perp })$. (b) Output image predicted by the neural network. (c) Output image after the image postprocessing. The white boxes represent predicted nuclei. (d) Evaluation of the model performance. The predicted nuclei (white boxes) are compared to the true nuclei (green boxes). In this example, we observe 9 TPs (true positives, i.e., correctly detected nuclei), 1 FP (false positive, i.e., predicted non-existent nucleus), and 1 FN (false negative, i.e., nonpredicted nucleus). In the highlighted circular area we examine a specific TP nucleus. The $5\times 5$ green box bounds the nucleus in the true output, while the white box bounds the nucleus in the predicted output.

Figure 3

(a) Precision ($P$) and recall ($R$) of the model with respect to the number of nuclei present in the true samples. (b) Mean absolute error (MAE) between the predicted and the true values of the coupling constants $A^z$ (green) and $A^{\perp }$ (red). This metric is calculated exclusively for TPs. (c) In yellow (blue), the MAE between the original and predicted input signals, corresponding to $\mathrm{N}=32$ ($\mathrm{N}=256$) pulses. The solid (dashed) lines in (a)–(c) correspond to the high (low) magnetic field scenario. (d)–(g) Signal comparisons from samples containing 16 nuclei. In black, the signals calculated with true nuclei. In yellow (blue), the $\mathrm{N}=32$ ($\mathrm{N}=256$) signals obtained with SALI’s predicted nuclei. (d) and (e) correspond to the low-field scenario (14 nuclei are detected), and (f) and (g) to high field (all 16 nuclei are detected). More details in [23].