Two-Photon Interference from Silicon-Vacancy Centers in Remote Nanodiamonds

In the emerging field of quantum technologies the distribution of entanglement is a key ingredient, for example to establish long-distance quantum state transfer and quantum networks [1]. One possible source of distributed entanglement generation is two-photon interference (TPI), commonly known as Hong-Ou-Mandel (HOM) interference [2]. A prerequisite are single photon sources that produce indistinguishable photons. Two-photon interference has been demonstrated with various sources of single photons, for example atomic vapors [3], quantum dots [4], molecules [5], coupled atom-cavity systems [6] and negatively-charged Nitrogen-Vacancy (NV^-) centers in bulk-diamond [7, 8]. Group IV color centers in diamond and, in particular, the negatively-charged Silicon-Vacancy center (SiV^-) are of great interest for the generation of indistinguishable photons, due to intrinsic spectral stability and narrow inhomogeneous line-distribution shown for bulk diamond [9, 10]. In recent years, there is an increasing effort to restrict the dimensions of the diamond host to very small size well below the optical wavelength. Such nanodiamonds (NDs) give rise to a large variety of new applications in the realm of hybrid quantum systems. Individually optimized [11, 12] and large-scale [13, 14] hybrid quantum photonic circuits [15, 16, 17] are constructed by means of nanomanipulation [18] and advanced fabrication techniques, respectively. Initially, the spectral properties of color centers in NDs were inferior compared to bulk diamond blocking further use in quantum optics applications. This deficiency was resolved over the past years by improved ND-production, sample preparation and control techniques [19].
In this letter, we demonstrate two-photon interference from SiV^- in remote NDs. Together with recently demonstrated access to electron spin [20] this work marks a further step towards applied hybrid quantum technology, such as the realization of scalable quantum networks, based on SiV^- in NDs.

The SiV^- is a point defect in the lattice of a diamond crystal where a silicon atom is located between two adjacent carbon vacancies as depicted in FIG. 1 a). At cryogenic temperatures four optically active transitions, resulting from spin-orbit coupling, can be observed. We refer to them as transitions A, B, C, D, as depicted in FIG. 1 b). To show two-photon interference, we excited two SiV^- in two remote NDs, separated by approximately 95 $\mu$m, off-resonantly and spectrally filtered the dominant transition C. The photons from both SiV^- interfered on a 50:50 beamsplitter, as schematically shown in FIG. 1 c). In the case of two identical photons entering the beamsplitter from two different input ports, the probability amplitudes for leaving at the same port will interfere constructively while the ones for leaving at different output ports interfere destructively. A second-order correlation measurement will therefore result in antibunching with vanishing correlations at zero time delay $g^{(2)}(\tau=0)=0$. In contrast, $g^{(2)}(\tau=0)=0.5$ is indicative of interference of two single but distinguishable photons.

The SiV^-s used for the experiment are located inside NDs with an avarage diameter of around 30 nm. They were coated onto a diamond substrate to ensure good thermal conductivity. The sample was investigated by off-resonant photo-luminescence (PL) and resonant photo-luminescence-excitation (PLE) measurements showing predominantly single SiV^- and a spectral distribution of $\approx$ 14 GHz for transition C (see Appendix B). To find a matching SiV^- pair we fixed the frequency of the scanning laser to the resonance of transition C of one SiV^- and scanned the sample laterally. This way only SiV^-s with spectral overlap were visible. The spectra of the two SiV^- chosen for the HOM measurement are shown in FIG. 2 a). Their ground-state splitting (GSS) differs by 12 GHz, but transition C shows a good overlap. The fact that transition C overlaps although the GSS of both emitters differs can be explained by different combinations of axial and transverse strain of the host crystal [21]. After filtering transition C (see Appendix A for details), only a single line was observed for each SiV^-, as exemplary shown for SiV 1 in FIG. 2 b). PLE measurements of the C transitions for both SiV^-s revealed linewidths of ($158\pm 5$) MHz and ($177\pm 4$) MHz with a detuning $\frac{\Delta}{2\pi}$ of ($83\pm 6$) MHz, as shown in FIG. 2 c). Single-photon emission from both SiV^-s was confirmed by off-resonant second-order correlation measurements which, after a normalization, resulted in $g^{(2)}_{1}(0)=0.33\pm 0.07$ and $g^{(2)}_{2}(0)=0.35\pm 0.08$ as depicted in FIG. 3 a) and b). We use the notation $g^{(2)}_{i}(\tau)$ for correlation functions including background noise, while the calligraphic $\mathfrak{g}^{(2)}(\tau)$ is used for the modeled correlation function without background. The measured data was fitted with

where $S_{i}$ is the signal from the emitter $i$, $I_{i}=S_{i}+B_{i}$ is the total signal including background counts $B_{i}$ and $\mathfrak{g}_{i}^{(2)}(t)=1-(1+a)\cdot\mathrm{exp}(-|t|/\tau_{1})+a\cdot\mathrm{exp}(-|t|/\tau_{2})$ is a three level model of the correlation function [22]. From the fit we determined the signal to noise ratio $S_{i}/B_{i}\approx 4$ for each individual SiV^- which sets a lower bound for the expected HOM dip, illustrated by the gray area in FIG. 3 c). To measure the two-photon interference we then off-resonantly excited both SiV^-s independently and let the emitted photons interfere on a 50:50 beamsplitter after the polarization of the photons was matched by half-wave plates. The resulting correlation function is shown in FIG. 3 c), where red dots show data for parallel polarization and blue triangles show data for perpendicular polarization. The data was fitted with [5]

where $c_{i}=I_{i}/(I_{1}+I_{2})$ and $g^{(1)}_{i}(\tau)=\exp(-\gamma_{i}|\tau|/2)$. The signal and noise for each emitter were fixed with the previously determined signal to noise ratio of the individual correlation measurements of both SiV^-. The variable $\eta$ in front of the interference term can be interpreted as an efficiency coefficient, where a value of 0 means no two-photon-interference and 1 means maximum interference. We determined a value of $\eta=0.61\pm 0.16$ for the case of parallel polarization.

As an additional figure-of-merit we calculated the coalescence time window (CTW) as described in [23]. It gives a time-window for which coalescence can occur on the beam-splitter. By integrating over the visibility function

we find

which is below the limit of 2$T_{1}$ expected for two ideal emitters with absolutely indistinguishable photons. Here $T_{1}\approx 1.9$ ns is the excited state lifetime. The result is limited by dephasing of the the emitters and background noise.
We demonstrated the generation of indistinguishable single photons from SiV^- in two remote NDs with an efficiency of $\eta=0.61$, the extracted CTW yielded 0.35 ns. The interference visibility is limited by technical imperfections such as polarization drifts or detector timing response, which can be improved in future experiments. Also minor spectral diffusion during the measurement can diminish the visibility. Our results establish SiV^- in NDs as a viable source for the generation of indistinguishable photons and open new possibilities for the integration into hybrid quantum photonics. The incorporation into photonic structures boosts the operation bandwidth and paves the way to establish remote entanglement of distant quantum nodes in an integrated fashion.
Furthermore, NDs much smaller than the wavelength of light, which contain individual quantum emitters with spectrally indistinguishable transitions offer new possibilities for the construction of cooperative quantum materials. Spatial indistinguishability can be achieved by positioning the NDs in a collective mode within a volume small compared to the third power of the radiation wavelength, $V\ll\lambda^{3}$. Thereby, collective states can be prepared in the Dicke regime [24] in a bottom-up approach by means of AFM-based nanomanipulation. Cooperative processes such as superradiance [25] and superabsorption [26] can be accessed with color centers in diamond. Pioneering work demonstrated superradiance effects with ensembles of NV^- center in the optical [27, 28] and microwave [29] domain and indicated a first onset of cavity-assisted superabsorption [30]. With the presented work, collective states could now be engineered atom-by-atom and with altered settings from spatially confined, interacting atoms to far distant non-interacting atoms [31].