Sequential Bayesian Experiment Design for Optically Detected Magnetic Resonance of Nitrogen-Vacancy Centers

In magnetometry using optically detected magnetic resonance of nitrogen vacancy (N-$V^{-}$) centers, we demonstrate speedup of more than 1 order of magnitude with a sequential Bayesian experiment design as compared with conventional frequency-swept measurements. The N-$V^{-}$ center is an excellent platform for magnetometry, with potential spatial resolution down to a few nanometers and demonstrated single-defect sensitivity down to nanoteslas per square root hertz. The N-$V^{-}$ center is a quantum defect with spin S = 1 and coherence time up to several milliseconds at room temperature. Zeeman splitting of the N-$V^{-}$ energy levels allows detection of the magnetic field via photoluminescence. We compare conventional N-$V^{-}$-center photoluminescence measurements that use predetermined sweeps of the microwave frequency with measurements using a Bayesian-inference method. In sequential Bayesian experiment design, the settings with maximum utility are chosen for each measurement in real time on the basis of the accumulated experimental data. Using this method, we observe an order of magnitude decrease in the N-$V^{-}$-center-magnetometry measurement time necessary to achieve a set precision.

Figure 1

(a) Crystal structure of the N-V⁻ center inside a diamond lattice. Green spheres denote carbon atoms, the yellow sphere is a nitrogen atom, and the purple sphere is a vacancy. Each white line corresponds to an sp³ bond created by a pair of electrons. (b) Schematic structure of the transitions between energy levels of the N-V⁻ center. The N-V⁻ center in the ground state can be excited by laser light (green arrows represent transitions due to the absorbed photons); the process preserves spin projection m_S. From the excited state, the N-V⁻ center can relax back to the ground state by emitting a red photon (m_S = ±1 or m_S = 0 excited states; red arrows represent transitions due to the emitted photons) or can relax nonradiatively through the dark state (only m_S = ±1 excited states; dashed gray arrow). Transition between the states with m_S = ±1 and m_S = 0 can be induced by microwaves (blue arrow). (c) Schematics of the photoluminescence spectrum of the N-V⁻ center under application of microwave irradiation and the external magnetic field B. The six dips are present due to the Zeeman splitting and hyperfine interaction. (d)–(g) Averaged data from (d) one scan, (e) five scans, (f) 30 scans, and (g) 140 scans (the inset shows an enlarged signal area) of conventional N-V⁻-center magnetometry using photoluminescence detection under sweeping of the microwave frequency. The magnetic field is calculated with use of the position of the signal (central dip) in the photoluminescence spectrum. (h) Dependence of the standard deviation ${\sigma }_f$ of the signal frequency f_B on the number of photoluminescence measurements n. Each solid purple circle corresponds to a unique number of averaged frequency-sweep scans; each scan consists of 8000 measured data points. Black symbols correspond to the data from (d)–(g). The solid black line shows inverse-square-root scaling. (i)–(l) The data from (i) 10, (j) 50, (k) 200, and (l) 1000 photoluminescence measurements by N-V⁻-center magnetometry using sequential Bayesian experiment design. (m) Dependence of the standard deviation ${\sigma }_f$ of the signal frequency f_B on the number of photoluminescence measurements n. Each solid orange circle corresponds to a unique number of photoluminescence measurements. Black symbols correspond to the data from (i)–(l). The solid black line shows inverse-square-root scaling.

Figure 2

Dependence of the probability distributions for signal-frequency and hyperfine-splitting parameters on the number of the measurements in N-V⁻-center magnetometry using sequential Bayesian experiment design. Probability distributions are shown after (a) 0, (b) 1, (c) 10, (d) 20, (e) 30, (f) 40, (g) 100, (h) 120, (i) 140, (j) 160, (k) 200, and (l) 1000 measurements. Each probability distribution consists of 10 000 points in parameter space with weights adding up to 1. Color represents the weight: less than 10⁻⁴, cyan; 10⁻⁴ or greater, red. Insets show enlarged areas of the probability distributions. All insets have the same size (1  × 1 MHz²), and span the same parameter space [(3102.5 MHz, 3103.5 MHz); (1.7 MHz, 2.7 MHz)].

Figure 3

(a) Dependence of the measurement frequency on the number of measurements for the conventional-N-V⁻-center-magnetometry microwave-frequency-sweep scan (solid purple circles) and N-V⁻-center magnetometry using sequential Bayesian experiment design (solid orange circles). The inset shows an enlarged view of the area enclosed by the dashed rectangle. Photoluminescence data for the first 1000 measurements from (b) the conventional N-V⁻-center-magnetometry microwave-frequency-sweep scan and (c) N-V⁻-center magnetometry using sequential Bayesian experiment design. (d) Distribution of the measurement frequency for the first 1000 measurements by N-V⁻-center magnetometry using sequential Bayesian experiment design. Dependence of (e),(h) the average normalized photon count $\overline{\mu }$, (f),(i) logarithm of the standard deviation of the normalized photon count ${\sigma }_{\mu }$, and (g),(j) number of measurements ν(f) on the measurement frequency for the first 24  000 measurements. (e)–(g) Data from the conventional-N-V⁻-center magnetometry scan (purple); (h)–(j) data from N-V⁻-center magnetometry using sequential Bayesian experiment design (orange). The solid black line in (e),(h) shows fitting using the function $\mu$ of all the measured data: 140 scans (1 120 000 measurements) of conventional N-V⁻-center magnetometry and 330 000 measurements by N-V⁻-center magnetometry using sequential Bayesian experiment design. The inset in (g) provides an enlarged view of the data.

Figure 4

Dependence of the standard deviation of the signal frequency ${\sigma }_f$ on the number of photoluminescence measurements. Each filled orange circle corresponds to a unique number of photoluminescence measurements using sequential Bayesian experiment design. Each filled purple circle corresponds to a unique number of averaged frequency-sweep scans; each scan consists of 8000 measured photoluminescence data points. Black symbols correspond to equal standard deviation of the signal frequency for sequential Bayesian experiment design (black circle) and conventional sweep measurement (black triangle). Solid black lines show inverse-square-root scaling.