Accelerated 2D magnetic resonance spectroscopy of single spins using matrix completion

Abstract

Two dimensional nuclear magnetic resonance (NMR) spectroscopy is one of the major tools for analysing the chemical structure of organic molecules and proteins. Despite its power, this technique requires long measurement times, which, particularly in the recently emerging diamond based single molecule NMR, limits its application to stable samples. Here we demonstrate a method which allows to obtain the spectrum by collecting only a small fraction of the experimental data. Our method is based on matrix completion which can recover the full spectral information from randomly sampled data points. We confirm experimentally the applicability of this technique by performing two dimensional electron spin echo envelope modulation (ESEEM) experiments on a two spin system consisting of a single nitrogen vacancy (NV) centre in diamond coupled to a single (13)C nuclear spin. The signal to noise ratio of the recovered 2D spectrum is compared to the Fourier transform of randomly subsampled data, where we observe a strong suppression of the noise when the matrix completion algorithm is applied. We show that the peaks in the spectrum can be obtained with only 10% of the total number of the data points. We believe that our results reported here can find an application in all types of two dimensional spectroscopy, as long as the measured matrices have a low rank.

Figure 1. Pulse sequence for the two dimensional ESEEM measurement used in our experiments.

See text for detailed description.

Figure 2. 2D ESEEM simulation and experimental data with a single NV when static magnetic field  G is applied at an angle of 34.1°.

(a) Simulation using the Hamiltonian (6). (b) Fourier transform of the complete set of the experimental data points . Fourier transform of the experimental data after applying matrix completion and using 20% (c) and 10% (d) of the time domain data. The main peaks are still observed even when 90% of the data is removed.

Figure 3. 2D ESEEM simulation and experimental data with a single NV coupled to a single ¹³C nuclear spin.

(a) Simulation using the spin Hamiltonian (6) and (7). (b) Fourier transform of the complete set of the experimental data points. Fourier transform of 20% (c) and 10% (d) of the data in the time domain. Here we again can recover the spectral information by keeping small amount of the experimental data.

Figure 4

(a) SNR of the 2D spectrum when the data is recovered with the matrix completion algorithm (red markers, right axis) and randomly subsampled matrix (black markers, right axis) as a function of the fraction of sampled elements (top axis) taken from the experimental data shown in Fig. 3. The matrix completion algorithm and subsampling were performed at least 10 times, here the errorbars denote the standard deviation. The blue curve (left axis) represents the singular values of the measured data (Fig. 3b) in descending order.

Figure 5

(a) Mean of fidelities of the matrix completion algorithm (red markers, right axis with ) and randomly subsampled matrix (black markers, right axis) as a function of the fraction of sampled elements (top axis). The matrix completion algorithm was performed 128 times with each time different random sampling, here the errorbars denote the standard deviation. The blue curve (left axis) represents the singular values of the measured data in descending order. (b) The fidelity of the matrix completion algorithm as a function of the fraction of matrix elements at different thresholds . Inset: Number of iterations required to run the matrix completion algorithm as a function of the threshold and the fraction of the matrix elements.