Statistically modeling optical linewidths of nitrogen vacancy centers in microstructures

We investigate the relationship between ion implantation and the optical linewidth of the nitrogen vacancy (NV) zero-phonon line (ZPL) in bulk and structured samples. We also propose a novel approach to ion implantation that we name postimplantation, in which nitrogen is implanted after all fabrication processes have been completed. We examine three post-implanted samples, one implanted with ${}^{14}\mathrm{N}$ and two with ${}^{15}\mathrm{N}$ isotopes. We perform photoluminescence excitation (PLE) spectroscopy to assess optical linewidths and optically detected magnetic resonance (ODMR) measurements to isotopically classify the NV centers. From this, we find that NV centers formed from nitrogen naturally occurring in the diamond lattice are characterized by a linewidth distribution peaked at an optical linewidth nearly two orders of magnitude smaller than the distribution characterizing most of the NV centers formed from implanted nitrogen. Surprisingly, we also observe a number of ${}^{15}\mathrm{NV}$ centers with narrow ($<500\mathrm{MHz}$) linewidths, implying that implanted nitrogen can yield NV centers with narrow optical linewidths. We further use a Bayesian approach to statistically model the linewidth distributions, to accurately quantify the uncertainty of fit parameters in our model, and to predict future linewidths within a particular sample. Our model is designed to aid comparisons between samples and research groups, in order to determine the best methods of achieving narrow NV linewidths in structured samples.

Figure 1

(a) Optical microscope image of sample B, showing the bulk area (bottom), membrane (center), and cantilevers (top). (b) Fluorescence map in the bulk area of sample B ($50\mu \mathrm{m}$ thickness). Each NV center we measured is indicated by a circle around it, with the color of the circle indicating the isotope. Some NV centers indicated did not exhibit measurable PLE. (c) Representative PLE measurement on an ${}^{14}\mathrm{NV}$, with the measurement pulse sequence shown above. The two lines have linewidths of $107\mathrm{MHz}$ and $186\mathrm{MHz}$ and are split by $4.4\mathrm{GHz}$. (d) Representative pulsed ODMR measurement on an ${}^{14}\mathrm{NV}$ center, with the pulse sequence shown above. Pulsed ODMR reveals the three peaks split by $2.2\mathrm{MHz}$ characteristic of ${}^{14}\mathrm{N}$. (e) Representative PLE measurement on an ${}^{15}\mathrm{NV}$ center. The two lines have linewidths of 3.4 and $5.1\mathrm{GHz}$ and are split by $6.8\mathrm{GHz}$. (f) Representative pulsed ODMR measurement on an ${}^{15}\mathrm{NV}$ center. Pulsed ODMR reveals the two peaks split by $3.1\mathrm{MHz}$ characteristic of ${}^{15}\mathrm{N}$. In (c)–(f), the gray lines indicate fits to the data.

Figure 2

The data for sample A are shown in the left column and for sample B in the right column. (a) A scatter plot showing the ZPL wavelength (in air) for each linewidth measured in sample A. The marker color indicates which area of the sample the datapoint was taken on. The ZPL wavelength was measured on a spectrometer with a resolution of $0.9\mathrm{nm}$ (corresponding to $70\mathrm{GHz}$ resolution at $637\mathrm{nm}$), whereas the linewidth was measured using PLE. (b) Stacked histograms of optical linewidths in sample A with the data labeled by sample location. There is evidence of two distinct populations of NV centers, and narrow NV centers can occur in the structured parts of the sample as well as the unstructured. (c) ECDFs of the linewidths in sample A. The plot shows that the median measured linewidth was $\approx 200\mathrm{MHz}$. (d) A similar scatter plot as in (a) for sample B. The marker color again indicates where the datapoint was taken, and the marker shape indicates which isotope the hyperfine structure indicated. The horizontal dashed lines demarcate the limits of the ZPL axis of (a), showing that sample B showed a much larger variation in ZPL wavelength, suggesting more variability in the local strain environment. (e) Stacked histogram of optical linewidths in sample B, with data labeled by sample location. Two populations are again evident, and they are not related to location in the sample. (f) ECDF for sample B. The median linewidth is $\approx 3.5\mathrm{GHz}$. The plateau indicates a clear separation between the two populations.

Figure 3

(a) Stacked histogram of optical linewidths for sample B, with data points labeled by color to indicate isotope. The histogram clearly shows that, with a few exceptions, the ${}^{14}\mathrm{NV}$ centers and the ${}^{15}\mathrm{NV}$ centers are completely separated into two distinct populations. The dashed lines indicate fits of log-normal sampling distributions $P\left(\left\{x_i\right\}\right|\mu ,{\sigma }^2)$ to the ${}^{14}\mathrm{NV}$ and (broad) ${}^{15}\mathrm{NV}$ datasets. (b) Isotopically classified ECDFs. The teal and purple dashed lines are the log-normal fits to the ${}^{14}\mathrm{NV}$ and ${}^{15}\mathrm{NV}$ datasets, respectively, showing excellent agreement between the ECDFs and the fits. The dashed orange line is obtained by fitting a sum of two log-normal distributions to the unclassified NV centers. The narrow ${}^{15}\mathrm{NV}$ data are included without a fit in this plot.

Figure 4

(a) The dashed lines reproduce the log-normal sampling distributions from Fig. 3. The solid lines indicate the posterior predictive distributions $P\left(\tilde{x}|\left\{x_i\right\}\right)$. They are slightly broader than the sampling distributions $P\left(x|\mu ,\sigma \right)$ because they account for the uncertainty in our estimates of $\mu$ and $\sigma$. The dotted lines are the posterior distributions for the median $P\left(\mu \right|\left\{x_i\right\})$, showing that the median ${}^{14}\mathrm{NV}$ and ${}^{15}\mathrm{NV}$ linewidths are well separated. (b) CDFs of the corresponding distributions in (a).

Figure 5

(a) Q-Q plot for the ${}^{14}\mathrm{NV}$ data, showing that a log-normal model fits the data well. (b) Q-Q plot for the ${}^{15}\mathrm{NV}$ data, again showing the model choice was appropriate. (c) The Q-Q plot for the narrow ${}^{15}\mathrm{NV}$ data shows that a log-normal model might be appropriate. (d) The Q-Q plot for the unclassified data shows that a single log-normal model is a poor fit to the data, as expected.

Figure 6

(a) Stacked histograms of data taken in sample C. The data are color coded to indicate which area of the sample they were taken on. (b) Fits (dashed lines) and posterior distributions for $\mu$ (dotted lines) for the two sample areas. The plots are color coded as in (a). Although the fits strongly overlap, the posterior distributions for $\mu$ do not, indicating that the thinner part of the sample is characterized by a higher median linewidth.