Mapping Single Electron Spins with Magnetic Tomography

Mapping the positions of single electron spins is a highly desired capability for applications such as nanoscale magnetic resonance imaging and quantum network characterization. Here, we demonstrate a method based on rotating an external magnetic field to identify the precise location of single electron spins in the vicinity of a quantum spin sensor. We use a nitrogen-vacancy center in diamond as a quantum sensor and modulate the dipolar coupling to a proximate electron spin in the crystal by varying the magnetic field vector. The modulation of the dipolar coupling contains information on the coordinates of the spin, from which we extract its position with an uncertainty of 0.9 Å. We show that the method can be used to locate electron spins with nanometer precision up to 10 nm away from the sensor. We discuss the applicability of the method to mapping hyperfine coupled electron spins and show that it may be applied to locating nitroxide radicals. The magnetic tomography method can be utilized for distance measurements for studying the structure of individual molecules.

Figure 1

The simulation of the dipolar coupling frequency modulation by a tilted magnetic field. The curves show distinct dipolar coupling oscillations for two target spin positions. The inset depicts the positions relative to the sensor. The simulation is done for a magnetic field tilted at a polar angle ${\theta }_B={29}^{\circ }$ and a varying azimuth ${\phi }_B$. The amplitude, offset, and phase of the oscillation contain information on the target spin position.

Figure 2

Measurement of the dipolar coupling of proximate spins with DEER. (a) The top panel depicts the DEER pulse sequence to measure the electron spin resonance of spins proximate to the sensor. The blue dots in the plot are the sensor signal as a function of the DEER frequency ${\omega }_e$ and the orange curve is a fit to squared sinc function. Here, $B_0=35\mathrm{G}$ and is aligned with the axis of the sensor. (b) The top panel shows the DEER pulse sequence to measure the dipolar coupling strength. We set ${\omega }_e$ to the resonance frequency [obtained in (a)] and the duration of the spin echo $\tau$ is varied. The orange line is a fit to a decaying sine.

Figure 3

Locating a spin by magnetic tomography. (a) Examples of dipolar coupling measurements for two different magnetic field azimuths ${\phi }_B$, exhibiting two distinct dipolar coupling frequencies. The lines are fits to $f\left(\tau \right)=A\sin (\frac{1}{2}{\omega }_{dd}\tau +\varphi )e^{-\frac{\tau }{T_2}}$. (b) The dipolar coupling frequency as a function of the magnetic field azimuth ${\phi }_B$, oscillating over a single period. The black line is a fit to Eq. (3), incorporating variations in ${\theta }_B$. We extract the target spin coordinates $(r,\theta ,\phi )$ from the fit. (c),(d) Probability maps for the position of the target spin, as extracted from the data in (b), presented in a $\rho$-$z$ plot ($\rho \equiv \sqrt{x^2+y^2}$) and an $x$-$y$ plot.

Figure 4

The location precision of the electron spins. The location uncertainty $\mathrm{\Delta }R$ versus the target spin coordinates ($x=r\sin \theta ,z=r\cos \theta$), assuming $\mathrm{\Delta }{\omega }_{dd}=20\mathrm{kHz}$. The contours denote equal levels of uncertainty. The blue star marks the spin position of the experiment.

Figure 5

The magnetic azimuth sweep of nitroxide spin labels. A comparison of numerically calculated azimuth sweeps of a nitroxide radical (${}^{14}$$\mathrm{N}$) at different field magnitudes. The electron spin is arbitrarily positioned, with an arbitrary orientation of the hyperfine axes. The curves are compared to the theoretical model for ${\omega }_{dd}(\mathbf{r},\hat{\mathbf{b}})$. At $B_0\gtrsim 100\mathrm{G}$, the calculated curve is well approximated by the theoretical model.