An ensemble of silicon vacancy ($SiV\u2212$) centers in diamond is probed using two-pulse correlation spectroscopy and multidimensional coherent spectroscopy. Two main distinct families of $SiV\u2212$ centers are identified, and these families are paired with two orientation groups by comparing spectra from different linear polarizations of the incident laser. By tracking the peak centers in the measured spectra, the full diamond strain tensor is calculated local to the laser spot. Measurements are made at multiple points on the sample surface, and variations in the strain tensor are observed.

## I. INTRODUCTION

Color centers in diamond and other crystalline host materials have shown potential in recent years as remote magnetic field sensors,^{1–4} room-temperature qubits,^{5} single photon sources,^{6–9} and even as candidates for use in novel particle detection schemes in dark matter direct detection experiments.^{10,11} One such color center is the negatively charged silicon-vacancy center ($SiV\u2212$) in diamond. Composed of an interstitial silicon atom located between two carbon vacancies in the diamond lattice, $SiV\u2212$ centers are, to a large extent, protected from first-order perturbations by the similarity of their ground and excited states.^{12} As a result, their electrostatic dipole moment vanishes, which reduces the $SiV\u2212$ sensitivity to first-order perturbations. In addition, $SiV\u2212$ centers exhibit weak electron–phonon coupling^{13–16} relative to other types of color centers in diamond. These properties combine to increase dephasing times of the $1.68eV$ zero-phonon line (ZPL) transition manifold to greater than $20ns$ in some cases.^{17} Due to the robustness and isolation that the diamond lattice provides, the $SiV\u2212$ is among the leading candidates for use in quantum information technology, remote sensing, and photonics applications.

In addition to these applications, it has recently been demonstrated that color centers in diamond can be used to sense strain in their environments^{18–21} by carefully tracking the frequencies of various optical or magnetic resonances as a function of sample position and then mapping that data to strain. This sensitivity is potentially useful in applications where locally measuring the deviation of structural components from expected stresses and strains is desired. For example, due to the ubiquity of diamond anvil cells as devices to apply strain to study a wide variety of material samples,^{22} having direct color-center strain imaging probes *in situ* would represent a tremendous advantage over traditional strain estimation techniques when studying small volume samples.

To this end, we demonstrate here a method employing silicon-vacancy centers in diamond to spectroscopically map all strain tensor elements. We employ two different coherent spectroscopic techniques to track the influence of strain on the $SiV\u2212$ zero-phonon line transitions in our sample. By combining linear spectra with nonlinear, multi-dimensional spectra^{23,24} as a function of polarization, we are able to provide an estimate of the ensemble averaged strain in the presence of inhomogeneity in our sample. Our methods are widely applicable and easily replicable with the recent advent of commercially available coherent spectrometers.

## II. EXPERIMENTAL METHODS

### A. Spectroscopic techniques

We study an ensemble of $SiV\u2212$ centers with two different coherent spectroscopy techniques. The first is a two-pulse correlation (TPC) measurement in which a titanium-sapphire (Ti:sapph) laser is fed through a Mach–Zehnder interferometer featuring tailored acousto-optic modulators to generate a pair of collinear, time-delayed, and frequency-tagged excitation pulses that are directed toward the sample as depicted in Fig. 1(a). Using a lock-in amplifier, we measure the modulated photoluminescence from the sample as a function of the temporal separation ($t$) between the pulses. Additionally, a continuous wave laser is co-propagated with the Ti:Sapph pulses to act as a phase reference for the lock-in amplifier and to remove the effects of mechanical fluctuations in the experiment. The photoluminescence intensity is Fourier transformed with respect to $t$ to yield a one-dimensional, coherently detected absorption spectrum. The phase coherent spectra reject signal contributions from long-timescale effects, as the spectral response is recorded as a function of time delays between pulses, which vary between $100fs$ and $1ns$.

For this sample, the linear TPC spectra can be difficult to interpret on their own, as there are many observed peaks to identify. Thus, we also employ a more powerful third-order nonlinear spectroscopic technique, multidimensional coherent spectroscopy (MDCS). Our collinear MDCS variant,^{25–27} depicted in Fig. 1(b), uses a nested Mach–Zehnder interferometer to separate four optical pulses from the Ti:Sapph laser. For the spectra used here, the first ($\tau $) and third ($t$) time delays are varied, and these data are Fourier transformed with respect to these time delays to obtain a two dimensional map of the third-order signal as a function of excitation ($\omega \tau $) and detection ($\omega t$) frequency axes.^{25–27} Note that because $\omega t$ has opposite phase evolution from $\omega \tau $, the vertical excitation axis is negative; thus, the diagonal at which the excitation and detection frequencies have equal magnitude is the downward sloping diagonal.

There are two main advantages to using multidimensional spectra. First, we can easily distinguish between homogeneous and inhomogeneous broadening, as these will broaden the MDCS peak along perpendicular diagonal directions. By measuring the widths of the peaks along and across the equal frequency diagonal, we can measure both the homogeneous and inhomogeneous linewidths, respectively. More important to this work, MDCS spectra can be used to identify if two spectral peaks are coupled. If coupling exists between two transitions, a coupling crosspeak will occur where the excitation and detection frequencies correspond to those of the two coupled transitions. Since a single $SiV\u2212$ center can emit at four different frequencies, a spectrum with several shifted families of $SiV\u2212$ centers can be difficult to interpret. Using a single multidimensional spectrum, we can determine which spectral peaks correspond to a single family.

While the information contained in a TPC measurement is not as rich as that in a full MDCS spectrum, the main advantage is that the acquisition time for a TPC spectrum is much shorter. This enables us to take a large number of linear spectra to observe position dependent trends, while only taking a few select MDCS spectra when needed to assign the peaks.

As mentioned previously, both the TPC and MDCS spectra used here utilize collinear geometries. This enables a smaller spot size than $k$-vector selection MDCS experiments^{26} and allows us to compare spectra taken at nearby locations on the sample. Studies of color centers often take measurements of a single center, with no guarantees that the center is representative of neighboring centers. In contrast, the high density of our sample and the ability of MDCS to untangle complicated spectra enable us to accurately measure the ensemble averaged optical response.

For both of these spectroscopic techniques, we linearly polarized the light incident on our sample. This was mainly so that the detected light could be cross-polarized to reduce laser scatter. Additionally, since the four electronic transitions of the $SiV\u2212$ center are polarization dependent,^{8} spectra taken at each polarization can be compared to understand the geometry of the sample.

### B. Sample information

Our sample is a chemical-vapor deposition grown, $(110)$-oriented mono crystalline diamond. An ensemble of $SiV\u2212$ centers was created by implanting silicon-29 ions with a focused ion beam at a depth of 0.5–2.4 $\mu $m and a number density of $7.5\xd71018cm\u22123$. The sample was then annealed at 1000–1050 $\xb0$C and tri-acid cleaned. A picture of the sample is included in Fig. 3(a). The sample has been cleaved, which explains its unusual shape. Additional information about the sample and its implantation parameters can be found in the supplementary material of Ref. 27.

$SiV\u2212$ centers can occur along four different orientations in diamond, corresponding to the four directions of carbon–carbon bonds in the diamond lattice. Figure 2(b) shows the four orientations of $SiV\u2212$ centers in our $(110)$-oriented sample, where the four vectors on the diamond correspond to the $SiV\u2212$ axis shown in Fig. 2(a). Note that we can group these peaks into in-plane (orange) and out-of-plane (purple) orientation families.

Figure 2(c) depicts the level structure of the $SiV\u2212$ zero-phonon line (ZPL). The ZPL has mean frequency $\Delta ZPL=407THz$, and the frequencies of the four optical transitions are also determined by the ground state and excited state splittings, $\Delta gs$ and $\Delta es$. It has been previously shown that two of the optical transitions are excited by light polarized perpendicular to the $SiV\u2212$ axis shown in Fig. 2(a), and the other two are excited by light polarized parallel to this axis.^{8}

The sample was placed in a closed-loop cryostat at a temperature of $11K$. Data were taken using $76$ MHz, $130$ fs pulses from a Ti:Sapph laser at a center wavelength of $736$ nm. The output of each interferometer branch had power $1.25$ mW, giving a total TPC power of $2.50$ mW and a total MDCS power of $5.00$ mW in a spot size of a few microns. At these powers, we do not expect laser heating to impact our results since diamond is transparent at this wavelength. This is corroborated by the relatively sharp peak widths observed in our measured spectra. Pulses were focused onto the sample face using a home-built scanning microscope with a $20\xd7/0.40NA$ objective of focal length $10mm$. The Rayleigh range within the sample is long enough that these measurements address the entire column of implanted $SiV\u2212$ centers. As shown in Fig. 1, the sample was tilted by $30\xb0$ from normal about the vertical axes in Figs. 2(b) and 3(a) to reject the reflected Ti:Sapph beam and any coherent scatter that could corrupt the PL measurements. Due to the high index of refraction of diamond, this $30\xb0$ tilt has a relatively small effect on the propagation direction of the laser within the sample. The two linear polarizations of light used here are depicted in Fig. 2(b).

## III. RESULTS AND ANALYSIS

### A. Peak identification

Previous experimental data taken of $SiV\u2212$ centers^{8,20,27,29} show the four spectral peaks corresponding to the four optical transitions in Fig. 2(c). Incoherent photoluminescence spectra taken from our sample have this structure as well.^{27} However, the TPC photoluminescence spectrum in Fig. 2(d) shows many more peaks, indicating the presence of multiple groups of color centers.

These results, while initially suggestive, are greatly clarified by comparison to the results of full-fledged MDCS. Figure 2(e) shows an example MDCS rephasing plot in which spectra taken with horizontally and vertically incident light have been summed together to better highlight all visible peaks. The insets of Figs. 2(f)–2(h) below this spectrum show more detail for some of the crosspeaks, both for the two linear spectra and the combined spectrum. Note that in the combined spectrum, the seven visible crosspeaks can be grouped into two squares, which are highlighted in Fig. 2(f). Each square is formed from two lower energy crosspeaks and two higher energy crosspeaks, and the peaks used in each square are distinct. This allows us to group the spectral peaks into two families, where peaks in a single family are coupled, and no coupling is observed between different families. Using this, we conclude that each family of peaks corresponds to a different type of $SiV\u2212$ center. Additionally, the MDCS spectrum shows some inhomogeneity, which could be due to microscopic strain fluctuations^{27} or interactions between nearby $SiV\u2212$ centers.^{30}

To identify the origin of the two families of $SiV\u2212$ centers, we can appeal to spectral peak polarization dependence. While the MDCS spectra can be used for this, it is easier to compare linear spectra, like the TPC spectra shown in Fig. 2(d). Note that the relative peak strengths of the two families of peaks under different polarizations of incident light are very different. This suggests that the two families of $SiV\u2212$ centers are oriented differently in the diamond lattice. By relating the possible orientations of $SiV\u2212$ centers in our sample to the polarization selection rules shown in Fig. 2(c), we find that the two sets of peak families correspond to the in-plane and out-of-plane orientations in Fig. 2(b). While it may be feasible to group the peaks into families based solely on polarization data, this would become increasingly difficult for higher amounts of strain, and the crosspeaks in the MDCS spectrum are easier to interpret and give a higher degree of certainty.

The peaks corresponding to these families are indicated in Figs. 2(d) and 2(e) using orange and purple vertical lines, corresponding to the in-plane and out-of-plane families. In the spectra seen here, we do see additional splitting of the in-plane peaks. The specific in-plane orientation can be determined with additional linearly polarized spectra.

### B. Strain calculation

While the analysis of Sec. III A justifies the necessity of having two different color-center families, it does not yet explain the origin of the different families’ peak shifts. We propose that the source of these shifts is due to strain intrinsic to our sample, mainly because the shifts are different for different $SiV\u2212$ orientations. By assuming that this hypothesis is true, we can solve for the full strain tensor in our sample.

Previous work completed by Meesala *et al.*^{20} derived equations relating four strain susceptibility parameters to the six strain tensor indices local to a given $SiV\u2212$ center. They combined these equations with experimental data and simulated strain tensor data to fit for the strain susceptibility parameters. In this work, we instead used experimental data and the strain susceptibility parameters to estimate the strain tensor.

The equations from Ref. 20 are

where the variables $\epsilon ij$ refer to the elements of the diamond strain tensor; $t\u2225$, $t\u22a5$, $d$, and $f$ refer to various strain susceptibility parameters whose values are given in Ref. 20; and as illustrated by Fig. 2(c), $\Delta ZPL$, $\Delta gs$, and $\Delta es$ represent the ZPL frequency and the ground and excited state splittings. Note that Eqs. (1b) and (1c) are linearly dependent, as the ground state and the excited state splitting respond similarly to strain. These equations are given in the $SiV\u2212$ reference frame, which is depicted in Fig. 3(b). Since there are four different orientations of $SiV\u2212$ centers in four distinct strain environments, we must first rotate these equations into a common basis. We choose the crystal basis depicted in Fig. 3(a). For instance, for one of the in-plane peaks, we have

This calculation is repeated for the three other orientations, and the results are substituted into Eq. (1), yielding 12 total equations, 8 of which are independent. Since (in the crystal basis) there are six strain tensor indices, we can use these equations to solve for the full strain tensor.

We took a series of TPC spectra at $23$ locations on our sample, as shown in Fig. 3(c). These locations lie along a single line, with a gap in the middle where the implantation density is lower and reliable peak locations and strain estimates could not be measured. Slight shifts in the frequencies of the spectral peaks can be seen.

To use these data to solve for the strain, we first used the techniques in Sec. III A to identify the peaks in our linear spectra (several MDCS spectra were collected at select locations to help with this). The peaks were fitted to find the locations of all $16$ spectral peaks for each scan, taking advantage of both vertical and horizontal polarization spectra, since some peaks are more visible on a given polarization. In cases where two peaks could not be resolved (which can occur for two in-plane or two out-of-plane peaks), a single frequency was reported.

Next, we used Eq. (1) to find analytical expressions for the frequencies of the $16$ spectral peaks as a function of the six strain tensor indices in the crystal basis, $picalc(\epsilon )$. We estimated the error (standard deviation) of our measured peaks $pimeas$ to be $\sigma =2GHz$. This gives us an expression for $\chi 2$,

Next, we found the strain tensor $\epsilon $, or equivalently the six strain tensor indices, such that $\chi 2$ is minimized. This tensor is our solved result.^{31}

To estimate the error in the calculation, we note that if the error in our peak locations is random, then the $\chi 2$ value from Eq. (3) should indeed follow a chi-square distribution with $16$ degrees of freedom for the true value of the strain tensor. Thus, there is a $90%$ chance that $\chi 2<23.5$. We explored the parameter space local to our solved strain tensor to find the range of strain tensors such that $\chi 2$ is less than the desired value.

The strain-solving algorithm was also tested on simulated data. Peaks were generated with Eq. (1) using random values for the strain tensor indices. Random errors were added to these peak values, and we attempted to recover the original strain tensor indices using the algorithm above. The algorithm behaved as expected and produced simulated results that were within the range of observed strain values. The algorithm begins to break down as the sheer strain tensor index $\epsilon xy$ becomes large (greater than $0.8\xd710\u22125$).

### C. Strain results

Extracted strain tensor results are shown in Fig. 4. Before drawing conclusions from these results, some limitations with the data should be noted. First, the values of the normal strain tensor indices are dependent on the mean ZPL frequency at zero strain or $\Delta ZPL,0$. We did not take a direct measurement of this, but we estimated it to be $406.795THz$. However, we find that changing this value results in a constant offset of the normal strain tensor indices $\epsilon xx$, $\epsilon yy$, and $\epsilon zz$ and that their relative values are preserved. Next, we encountered a sign ambiguity for small values of $\epsilon xy$; therefore, we have elected to plot the absolute value of this parameter in Fig. 4 instead of its sign-dependent form. Future work could attempt to resolve this sign ambiguity in a given sample by observing peak shifts due to $\epsilon zx$ strain values that have been intentionally applied to calibrate the measurement. For all indices, the error bars in Fig. 4 are only meant to represent errors due to random errors in the peak locations. They do not account for systematic errors, such as errors in the strain susceptibility parameters used in Eq. (1). We estimate that the error due to the strain susceptibility parameters is less than $0.1\xd710\u22125$ for the normal strain tensor indices and less than $0.03\xd710\u22125$ for the sheer strain tensor indices.

Despite these limitations, we are sensitive to strain differences of $2\xd710\u22125$–$3\xd710\u22125$, and we do see a varying strain in our sample. The normal strain indices are nonzero, and these values appear to vary across the sample. While we do not see statistically significant magnitudes for the sheer strain tensor indices $\epsilon xy$ and $\epsilon yz$, the sheer strain tensor index $\epsilon zx$ does show a statistically significant variation.

Previous strain measurements^{20} observe shifts in single $SiV\u2212$ centers. By contrast, these measurements are taken on an ensemble of $SiV\u2212$ centers, as mentioned in Sec. II A. This means that we measure the strain tensor averaged over the area and depth of the laser spot, rather than at specific centers. Note that this variation in macroscopic strain is also different from the microscopic strain fluctuations proposed in Ref. 27, as the latter concerns individual $SiV\u2212$ centers in highly strained environments.

The source of this strain is likely due to the implantation and annealing processes, although it has been established that diamond, natural or otherwise, has some degree of strain due to existing defects.^{32} As mentioned in Sec. II B, our sample has a rather high number density of implanted silicon of $7.5\xd71018cm\u22123$. This density corresponds to about one silicon atom for every $2.3\xd7104$ carbon atoms. If we assume, as an approximation, implanting the silicon does not increase the size of the sample, but merely pushes the atoms uniformly closer together, and then we can estimate the normal strain as $\u22121.4\xd710\u22125$. While this simple estimation does not take into account many complexities of the system, it does agree with our experimental values to within an order of magnitude.

## IV. CONCLUSION

By analyzing both MDCS and TPC measurements of $SiV\u2212$ centers, we were able to measure the full strain tensor of our sample. We measure a nonzero strain, which varies across the sample. This strain is most likely due to the large amount of implanted silicon in our sample. In future work, we hope to take measurements on a variable density sample to further measure the relationship between the implantation density and strain. In addition, the depth-dependent strain tensor could be studied using a sample containing thin layers of implanted $SiV\u2212$ centers or a more tightly focused beam spot. Modifications to the experiment could be implemented or other spectroscopic techniques could be used to improve sensitivity and accuracy. This work may also be useful in using $SiV\u2212$ centers in diamond, or other color centers with similar symmetries, as a strain gauge, potentially in a diamond anvil cell or an atomic force microscope tip.

## ACKNOWLEDGMENTS

C.L.S. acknowledges support from the National Science Foundation (NSF) under Grant No. 2003493. T.S. acknowledges support from the Federal Ministry of Education and Research of Germany (BMBF, project DiNOQuant, No. 13N14921). Ion implantation work to generate the $SiV\u2212$ centers was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the DOE’s National Nuclear Security Administration under Contract No. DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the DOE or the United States Government.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.