Spin-controlled generation of indistinguishable and distinguishable photons from silicon vacancy centres in silicon carbide

Quantum systems combining indistinguishable photon generation and spin-based quantum information processing are essential for remote quantum applications and networking. However, identification of suitable systems in scalable platforms remains a challenge. Here, we investigate the silicon vacancy centre in silicon carbide and demonstrate controlled emission of indistinguishable and distinguishable photons via coherent spin manipulation. Using strong off-resonant excitation and collecting zero-phonon line photons, we show a two-photon interference contrast close to 90% in Hong-Ou-Mandel type experiments. Further, we exploit the system’s intimate spin-photon relation to spin-control the colour and indistinguishability of consecutively emitted photons. Our results provide a deep insight into the system’s spin-phonon-photon physics and underline the potential of the industrially compatible silicon carbide platform for measurement-based entanglement distribution and photonic cluster state generation. Additional coupling to quantum registers based on individual nuclear spins would further allow for high-level network-relevant quantum information processing, such as error correction and entanglement purification.


Supplementary Note 1: Saturation behaviour of single h-VSi centre under pulsed laser excitation
To evaluate the optical excitation efficiency in the pulsed regime, we use a 780 nm picosecond laser diode (PicoQuant LDH-P-C-780). The laser operation regime was kept constant to maintain the same pulse shape and duration throughout all measurements. The pulse energy was subsequently varied and we observed the resulting photon count rate pulsed in the zero-phonon line, as shown in Supplementary Fig. 1.
In this experiment, we used a slow repetition rate of 2 MHz to assure that the electron comes back to the ground state before laser excitation (and is not trapped in the metastable state with ≈ 100 ns lifetime). Under this condition, we can interpret the background subtracted relative intensity as excitation probability. Since the laser pulse length (56 ps FWHM) is much shorter than the excited state lifetime of h-V Si centre (6 ns), the intensity saturation can be modelled by an exponential equation where 0,pulsed is the saturation intensity and 0 is the pulse energy at which the excitation probability is 1 −

transmissivity/reflectivity ratio of two beam splitters and fringe contrast
In this study, an unbalanced Mach-Zehnder type fibre-based interferometer ( Supplementary Fig. 2) is used for HOM type two-photon interference. To characterise the quality of the interferometer, the intensity transmissivity/reflectivity ratio ( / ) of the two beam splitters (BS1 and BS2) has to be measured. To this end, we excite a single h-V Si centre by the pulsed laser at 4.1 MHz repetition rate. We detect the rate of ZPL photons through the interferometer by two SNSPDs (D1 and D2) in early (0 ≤ ≤ 48 ns) and late (δ ≤ ≤ δ + 48 ns) time bins. Here, δ = 48.7 ns is the path travel time difference of two arms. As is much longer than the photon coherence time, there is no single-photon interference at the output. The photons in the early time bin take the shorter arm and the photons in the late time bin take the longer arm. The integrated photon counts of two detectors in early and late time bins are (D 1 , early) ≡ 11 = 1 1 2 0 , (D 1 , late) ≡ 12 = 1 1 2 0 , (D 2 , early) ≡ 21 = 2 1 2 0 , and (D 2 , late) ≡ 22 = 2 1 2 0 , where is the detection efficiency of D and 0 is the total number of input photons. From these relationships, the ⁄ ratios of two beam splitters are calculated to be 1 1 ⁄ = √ 11 21 12 22 ⁄ = 1.129 ± 0.006 and 2 2 ⁄ = √ 12 21 11 22 ⁄ = 1.046 ± 0.005.
The fringe contrast of the interferometer is measured with a highly coherent monochromatic laser (Toptica DLC DL pro 850) at ZPL wavelength of h-V Si centre. By optimising the polarisation rotation in the long interferometer arm with a fibre polarisation controller, the maximum interference fringe contrast obtained in this interferometer is (1 − ) = 0.995 . Since the theoretical limit of the fringe contrast with unbalanced / ratios is 2(√ 1 2 / 1 2 + √ 1 2 / 1 2 ) −1 = 0.9965 ± 0.0006, we consider the interferometer to be well aligned.

Ou-Mandel interference visibility
The HOM visibility gives the overlap integral of the wave packet of two photons in ideal conditions, but the experiment is affected by timing jitter of photon arrival time, background noise photons, and interferometer imperfections such as non-unity fringe contrast and unbalanced transmissivity and reflectivity. The timing jitter decreases the two-photon overlap integral. The existence of background noise photon decreases the probability of events to have two indistinguishable photons from the h-V Si centre at the beam splitter, resulting in the decrease of a raw HOM visibility. In this note, we extend the discussion on the HOM visibility by Santori et al. 1 by considering the effect of noise photons to estimate the correct photon overlap integral.
The signal photons are from 1 and 2 transitions of the h-V Si at the focus and the noise photons come from the ensemble of silicon vacancies on surface, bulk fluorescence, Raman scattering, laser breakthrough, etc.. We denote the probability to have a photon from the signal and noise sources per one laser pulse to be and , respectively. Since the pulse length of the excitation laser (56 ps) is much shorter than the excited state lifetime of h-V Si (6 ns), the probability to have two ZPL photons from the same h-V Si centre by one excitation laser is negligibly small. The noise photons can be modelled as a Poissonian photon source, however we can safely neglect the probability to have two noise photons per laser pulse since the average number of noise photon is much smaller than 1 per laser pulse. Under these assumptions, we write the probability to have photons per laser pulse, ( = 0, 1, 2), and the signal to noise ratio SN as Using these parameters and the other mentioned non-ideal parameters, the coincidence counts of five peaks in two-pulse HOM excitation scheme are calculated to be { +2⋅Δ = ( 1 + 2 2 ) 2 1 2 0 ⋅ 1 1 2 2 , −2⋅Δ = ( 1 + 2 2 ) 2 1 2 0 ⋅ 1 1 2 2 , where 0 is the number of repetitions of the experiment, (1 − ) is the interferometer's fringe contrast, and is the overlap integral of two photons from the h-V Si centre. The parameter = 2 2 ( 1 + 2 2 ) 2 ⁄ comes from the events in which two photons enter in the interferometer during one laser excitation pulse. Thus, it is equal to (2) ( = 0) when the autocorrelation measurement is performed under the same condition as the HOM experiment. When (2) ( = 0) degrades solely due to reduced signal-to-noise (SN), this parameter can be written as This equation gives the lower limit of the parameter and (2) ( = 0). Supplementary Fig. 3 shows the comparison of experimentally measured (2) ( = 0) (the same data as Fig. 1(d) in main text), and the SN limited value calculated from Supplementary Eq. (7). At pulse energies below 6 pJ, (2) ( = 0) is close to the SN limit.
The degradation of (2) ( = 0) at higher laser pulse energies is probably due to double excitation within one laser pulse. Note that in the HOM experiment, the effect of double excitation is greatly minimised by time-gating in the first several ns. the lower bound of (2) (0) (green triangles, left axis) calculated using Supplementary Eq. (7) from separately measured signal-to-noise ratio (red inverted triangle, right axis). Lines are guides to the eye. The difference between the experimentally measured (2) (0) and SN limited one corresponds to (2) (0) with background correction, i.e., the nonideality of the emitter.
Error bars represent one standard error.
Here, we assume an ideal interferometer ( = 0, = 1) and perfect two-photon overlap integral ( = 1). When SN is 28, which is a typical value for this study at the laser pulse energy of 5.5 pJ, the maximum achievable HOM visibility is upper bound at 81%. Note that for HOM experiments the repetition rate is lower compared to We estimate the timing jitter of the excitation laser to be about 55 ps (one standard deviation), most of which comes from the trigger pulse generation electronics. By adding the jitter coming from the finite pulse length (56 ps in FWHM), the laser related timing jitter is estimated to be 60 ps (one standard deviation). We assume that the timing jitter caused through the ultra-fast relaxation process in the excited state vibronic levels is negligibly small, thus we take jitter = 60 ps. As a consequence, the photon overlap integral decreases by 0.8 %. As a result, the HOM visibility after correction (including imperfection of the interferometer, SN ratio, and timing jitter) is Comparing our theoretical model with the time gated raw data shown in Fig. 4(b) in the main text, we find that the maximum achievable HOM visibility is upper bound at (80 ± 1)% by SN, interferometer imperfections and timing jitter. The experimentally extracted visibility parameter = 0.85 ± 0.04 underlines that essentially ideal contrast could be reached by further improving the setup and noise filtering strategy. Supplementary

Supplementary Note 4: Resonant Rabi oscillation experiment with short radiofrequency pulse for spin-controlled indistinguishable photon generation
To control the colour of photons via coherent manipulation of the ground state spin, a radiofrequency (RF) pulse is applied to the centre between two laser excitation pulses. The first optical transition initialises the spin state into one of the Kramers doublet subspaces s = ± 1 2 ⁄ or ± 3 2 ⁄ depending on the observed colour of the emitted zero phonon line photon ( 1 or 2 , respectively). The RF pulse coherently manipulates the spin state, and the resulting spin population in each subspace directly translated to the probability to observe the second photon in 1 or 2 . Considering the time difference of interferometer arms δ = 48.7 ns and the system's excited state lifetime ES = 6 ns, the allowed maximum pulse length is about 30 ns. Due to the short pulse length, we expect a frequency broadening exceeding the ground state zero-field splitting (ZFS) of 4.5 MHz. Thus, the RF field will drive all spin transitions simultaneously, leading to spin manipulation with non-unity fidelity. To determine optimal RF pulse length and the associated spin populations, we measured Rabi oscillation with resonant laser excitation 3 .
The h-V Si centre is irradiated by a laser resonant to the 2 ( 1 ) transition for 9 μs, which initialises the spin state applied. Subsequently, the population of the spin sublevels s = ± 3 2 ⁄ (± 1 2 ⁄ ) is read out by the same laser ns to induce a π/2-pulse, and 29 ns to induce a 3π/4-pulse.
Due to the high-power RF condition the sample heats up significantly, which causes optical line broadening. For proper interpretation of the visibility of the spin-controlled HOM interference experiments, we quantify the maximally achievable HOM visibility at those different RF pulse and temperature conditions. To this end, we always perform an additional HOM interference experiment in which the identical RF pulse is applied right before an experimental sequence, instead of applying it during the sequence. A typical measurement with uncorrected data is shown in Supplementary Fig. 6. Then we obtain the normalised HOM visibility as norm = (RF during sequence) (RF before squence) .
Here, (RF during sequence) and (RF before squence) are corrected for experimental imperfections by

Time evolution of ground state spin populations under pulsed strong RF drive
To evaluate the time-dependent spin populations in the ground states under strong RF drive, we start with the static Hamiltonian, describing the system in an external magnetic field aligned along the z-axis: Here, = 2 ⋅ 2.25 MHz is the ground state zero-field splitting, = 2π ⋅ 28 GHz ⋅ T −1 the electron gyromagnetic ratio, and ≈ 0.9 mT the externally applied field. Our RF drive is modelled by the interaction Hamiltonian: Here, Ω ≈ 2π ⋅ 14 MHz is the strength of the RF driving field, = 30.26911MHz is the RF frequency, the time, and is a (random) phase term, which accounts for the fact that the RF driving is here faster than the ground state level separation (2 ⋅ ) and not phase-synchronised, thus the rotating wave approximation might not be valid. With these Hamiltonian operators, the time evolution operator is given by The time evolution of the four spin states in the ground state is then Here, ( ) is the density matrix describing the system at time and 0 is the state at = 0. We obtain ( ) by solving Supplementary Eq. (15) numerically, averaging over the random phase term , and using small time steps ≈ 0.5 ns, which is significantly smaller than the typical time scale of spin state development (≈ 10 ns).

Supplementary Note 6: Vibronic interaction theory
The origin of dephasing in the optical signal for V1 center is the coupling to the V1' polaronic excited state mediated by acoustic phonons. As outlined in more details in Udverhelyi et al. 6 , at very low temperatures, only the acoustic phonons have significant occupation number. However, compared to the temperatures in the experiment, the polaronic gap between V1 and V1' is relatively large (4.4 meV), which excludes the consideration of two-phonon (Raman scattering) processes to be competitive with the single phonon absorption process. Thus we describe the dephasing with a resonant phonon coupling 7 . This can be formulated using timedependent perturbation theory with first order contribution in the linear vibronic interaction leading to Fermi's Golden Rule formula for the transition rate where is the index of phonon mode, is the acoustic phonon occupation number, is the linear vibronic interaction strength, ℏ is the acoustic phonon energy, and Δ is the energy gap between V1 and V1' levels.
For the density of acoustic phonon states ( ) we use the Debye-model as ( ) = 2 , where is a constant.
We can approximate | | 2 ̅̅̅̅̅̅̅ ≈ phonon mode average for the acoustic phonons, where is a constant. After this, the summation results in where we use the thermal occupation function of phonons (Δ , ). Since Δ is relatively large we find the low temperature limit of this function with exponential temperature dependence, as described in Eq. (2) in the main text.

Supplementary Note 7: Analysis of quantum beating with spin control via RF pulse
This note explains the analysis of quantum beating obtained with the HOM interference experiment with spinflip RF pulses. The time delay of the RF pulse from the first laser pulse is 18 ns. Therefore, the coincidence data is taken within the detection time window [ Start , Stop ] = [1.5 ns, 18 ns] with the time-gating technique described in Fig. 4(a) in the main text. This strategy rejects the laser related noise and ensures that the system is in the ground state while RF pulse is applied for the collected data. Due to non-unity spin flip fidelity in our conditions and the existence of noise photons, we consider three components in the coincidence data. photons is randomly distributed and the optical coherence time is expected to be much shorter than the timing resolution of our detection system and electronics (0.4 ns). Therefore, the corresponding coincidence counts ̅ 3 (2) is obtained from Supplementary Eq. (24) by averaging cosine term and taking a limit of → ∞ as The total coincidence counts ̅ tot (2) are obtained by summing up these three components ̅ (2) with associated coefficients . By considering < 0 region, a small time difference between two detectors 0 , and the finite detection timing resolution of the detectors and electronics (approximated by an Gaussian broadening with a standard deviation of det ), the total coincidence is calculated to be  In the first step, we characterise the main cavity parameters, i.e. its free spectral range (FSR) and transmission linewidth Δ cav . To this end, we scan a narrowband tunable diode laser (Toptica DLpro) over several cavity transmission peaks. The frequency of the laser is constantly monitored using a wavemeter (HighFinesse WS7-30). The result is shown in Supplementary Fig. 7. We find that the average separation between two consecutive peaks is FSR = 5.145 GHz, and that the Lorentzian transmission width of each individual peak is Δ = 29 ± 1 MHz.
Having characterised the main cavity parameters, we now use it as a spectrometer for inferring the linewidth of the ZPL emission in the 1 and 2 transitions under off-resonant excitation. We use continuous-wave offresonant excitation at 730 nm, and at a power of 150 µW, which corresponds to approximately three times the saturation power (half of maximum count rate is reached at sat = 57 ± 5 µW). We split the h-V Si centre emission into PSB and ZPL using a dichroic mirror. The ZPL emission is then fibre-coupled and sent to the cavity. We tune the cavity length (and thus the transmission frequency) via a piezoelectric actuator attached to one of the cavity mirrors. To monitor slow cavity drifts (e.g. due to temperature), we use the narrowband resonant excitation laser as a reference. The laser is frequency locked using the wavemeter at a frequency of 1-2 GHz below the h-V Si transition lines. A small fraction of the laser is sent through the setup, and through the cavity filter. A typical measurement result is shown in Supplementary Fig. 8.
Supplementary Figure 8. Typical cavity-recorded emission spectrum of the single h-V Si centre used for the HOM experiments.
Four peaks are observed, and associated with the narrowband frequency reference laser, and the 1 and 2 transitions, respectively.
The narrowband cavity is scanned in length for a frequency equivalent of about 7 GHz. The two high peaks correspond to the frequency-stabilised narrowband laser. From the previous cavity characterisation, the peak separation is known to be 5.145 GHz, which provides the necessary frequency calibration for the measurement.
The two smaller peaks are associated with the 1 and 2 transitions, respectively. Fitting the data with four Lorentzian peaks allows to extract the emission linewidths of the 1 and 2 transitions, respectively. Due to the finite resolution of the Fabry-Pérot cavity (Δ = 29 ± 2 MHz), the emission linewidths of the two transitions appear larger than they are. Thus, we deconvolute the measured linewidth with the cavity linewidth, using the fact that the convolution of two Lorentzian functions with widths 1 and 2 results in a Lorentzian function with width = 1 + 2 . For the defect that was used for the HOM experiments, we infer deconvoluted linewidths of Δ 1 = 57 ± 6 MHz and Δ 2 = 48 ± 6 MHz for the 1 and 2 transitions, respectively.
To show the repeatability of our results, we perform the same studies on six different h-V Si centres. The emission spectra are shown in Supplementary Fig. 9. The separation between two lines (mainly due to excited state zero field splitting) is also very consistent and in the range of 0.927 − 1.015 GHz. All solid lines are fits to the data using Lorentzian functions.