Abstract
The negatively charged silicon vacancy center (\({{\rm{V}}}_{{\rm{Si}}}^{}\)) in silicon carbide (SiC) is an emerging color center for quantum technology covering quantum sensing, communication, and computing. Yet, limited information currently available on the internal spinoptical dynamics of these color centers prevents us from achieving the optimal operation conditions and reaching the maximum performance especially when integrated within quantum photonics. Here, we establish all the relevant intrinsic spin dynamics of the \({{\rm{V}}}_{{\rm{Si}}}^{}\) center at cubic lattice site (V2) in 4HSiC by an indepth electronic fine structure modeling including the intersystemcrossing and deshelving mechanisms. With carefully designed spindependent measurements, we obtain all the previously unknown spinselective radiative and nonradiative decay rates. To showcase the relevance of our work for integrated quantum photonics, we use the obtained rates to propose a realistic implementation of timebin entangled multiphoton GHZ and cluster state generation. We find that up to threephoton GHZ or cluster states are readily within reach using the existing nanophotonic cavity technology.
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Introduction
Solidstate spin qubits based on color centers in wide bandgap semiconductors are one of the leading platforms for quantum networks, information processing, and sensing^{1,2,3,4} owing to their robust spinoptical properties and long coherence times. Among many available host materials^{5,6,7}, silicon carbide (SiC) particularly stands out as a waferscalable material with wellestablished isotopic engineering and compatibility with today’s complementary metaloxidesemiconductor (CMOS) microfabrication technology providing a path towards scalable systems. Rapid progress with spin qubits in SiC has already been made including the milestone demonstrations of millisecond spin coherence times at room temperature^{8,9}, highfidelity spin and optical control^{10}, coherent spinphoton interfaces^{11}, entanglement with nuclear spin registers^{12}, and singleshot charge readout^{13}. Capitalizing on the SiC’s material advantages, steps towards scalability have also been implemented through successful integration of negatively charged silicon vacancy color centers (\({{\rm{V}}}_{{\rm{Si}}}^{}\)) into nanophotonic waveguides^{14} and resonators^{15,16,17}, the latter one being compatible with SiConinsulator processing^{18}. These proofofconcept demonstrations identify the cubic lattice site \({{\rm{V}}}_{{\rm{Si}}}^{}\), i.e. V2 in 4HSiC as a strong contender for quantum applications based on a dense integration of multiple color centers on one chip. Further progress towards fully scalable integrated solutions with V2 is necessitated by a complete understanding of its intrinsic spinoptical dynamics to guide the engineering efforts of cavityemitter coupling and optimization of spin and optical properties. It will also provide the critical insights and metrics necessary for developing realistic quantum network applications.
In this paper, we focus on the V2 center in 4HSiC occupying the cubic lattice site having a much larger zerofield splitting^{8} (ZFS) in the ground state compared to the hexagonalsite V1 center^{10}. This larger ZFS leads to a much faster groundstate spin manipulation and higher state fidelities^{19}. We reveal the comprehensive internal spin dynamics of the V2 center by theoretical characterization of its electronic structure. The theory is confirmed by our experimental investigations, which include the measurements of spinselective excited state (ES) lifetimes, ground state spin initialization via resonant and offresonant laser excitation, as well as probing the intricate dynamics within the metastable state (MS) manifolds via spin manipulation combined with a delayed pulse measurement. In this way, we determine all the spindependent radiative and nonradiative transition rates and identify the intersystem crossing (ISC) mechanism which all play a crucial role in defining realistic protocols for scalable quantum network applications. To showcase this, we develop a protocol for generating timebin entangled multiphoton GreenbergerHorneZeilinger^{20} (GHZ) and cluster states^{21}, which are particularly important for quantum network applications, oneway quantum computation, and quantum repeaters. Using the involved intrinsic transition rates and ISC mechanism, we provide estimates for quantum efficiencies, state fidelities, optimal pulse timings, and minimum requirements for a cavity enhancement of radiative lifetimes. The approaches and insights developed here are also directly applicable to other color centers and their benchmarking for specific applications.
Results
Electronic fine structure of \({{\rm{V}}}_{{\rm{Si}}}^{}\) in 4HSiC
The crystal structure of 4HSiC allows for two nonequivalent lattice sites, socalled hexagonal (h) and cubic (k) sites, to be occupied by the deep center \({{\rm{V}}}_{{\rm{Si}}}^{}\) defect. Defects belonging to the h and ksites, referred to as the V1 and V2 centers, have distinct optical resonant excitation signatures at zerophonon line (ZPL) wavelengths of 862 nm and 916 nm, respectively. Five active electrons present in V1 and V2, originating from the four sp^{3} dangling bonds surrounding the vacancy and an additional captured negative charge, result with optically active ground and excited states in a Kramer’s degenerate S = 3/2 spin quartet configuration.
Both V1 and V2 have a local symmetry that belongs to the C_{3v} double point group that is only slightly distorted from the cubic T_{d} symmetry. In the case of V2, this distortion is stronger because of the additional next nearest neighbor silicon atom present along the caxis for ksites. As we will show in this work, this leads to a completely different spinoptical dynamics of the V2 center compared to the formerly studied V1 center^{22}.
To investigate the dynamics of the V2 center (from now on dubbed as \({{\rm{V}}}_{{\rm{Si}}}^{}\) center), we use the group theoretical framework, developed by Soykal et al. ^{23}, based on multiparticle symmetry adapted total wavefunctions built from singleelectron molecular orbitals (MOs) and linear combinations of localized many body sp^{3} orbitals. Combining the theory and our experimental measurements, we reveal all the spindependent radiative and nonradiative transition rates, and as a result the intrinsic optical and spin dynamics of \({{\rm{V}}}_{{\rm{Si}}}^{}\). Our results corroborate a complete picture of all the metastablestate doublets involved in the ISC and spin polarization of this defect with their relevant coupling mechanisms (spinorbit, dynamic pseudoJahn Teller, and optical deshelving) as explained in the following subsections. Although the possible involvement of higherlying doublets within the ISC was previously pointed out by Dong et al.^{24}, some of the crucial ISC doublet states were either missing or obtained incorrectly there, e.g. missing an extra uve doublet or not satisfying the orthonormality condition for the presented one. Coupled with lack of prior experimental data, critical mechanisms such as pseudoJahn Teller effect and involvement of a deshelving state were also not identified leaving the ISC still an open question. The comprehensive \({{\rm{V}}}_{{\rm{Si}}}^{}\) electronic fine structure is shown in Fig. 1a, in terms of MOs obtained by group theoretical analysis and considerations of manybody spinorbit, spinspin, and exchange interactions (see Supplementary Note 2).
The ground state (GS) manifold contains a spinquartet state in an orbitalsinglet (^{4}A_{2} : ve^{2}) configuration and five metastable spindoublet states. These doublets consist of two orbitalsinglet (^{2}A_{1}, ^{2}A_{2} : ve^{2}) and three orbitaldoublet (^{2}E : ve^{2}, e^{3}, v^{2}e) configurations (see ref. ^{23} and Supplementary Note 2). The first excited state manifold differs from the ground state solely in the u orbital, replacing v (ve^{2}→ ue^{2}). This leads to a spinquartet orbitalsinglet state (^{4}A_{2} : ue^{2}) and three metastable spindoublet states. These doublets consist of two orbitalsinglets (^{2}A_{1} : ue^{2}, ^{2}A_{2} : ue^{2}) and an orbitaldoublet (^{2}E : ue^{2}) configurations. It has been shown that the charge distribution localized on the nearest neighbor basal plane carbon atoms has an opposite parity for the u and v MOs, making the transition dipole moment (\({\mu}_{{ve}^{2} \rightarrow {ue}^{2}}\) less sensitive to fluctuations in nonaxial local electric field and strain, even though \({{\rm{V}}}_{{\rm{Si}}}^{}\) defect lacks an inversion symmetry^{25}. This explains the experimentally observed high spectral stability of \({{\rm{V}}}_{{\rm{Si}}}^{}\) center during a continuous resonant optical excitation^{14,16}. The second excited state manifold includes another optically active spinquartet orbitaldoublet (^{4}E : uve), which is split by the spinorbit coupling, as well as two metastable spindoublet orbitaldoublet states (^{2}E : uve). The symmetryallowed spinorbit coupling channels are indicated in Fig. 1a (dashed lines and arrows).
Based on the described fine structure and the allowed coupling channels, we develop a simplified and a fully equivalent energy level model of the \({{\rm{V}}}_{{\rm{Si}}}^{}\) center as shown in Fig. 1b. It consists of Kramer’s degenerate ground and excited states for the spin subspaces m_{s} = ±1/2 and m_{s} = ±3/2, and three effective metastable states, which can be further reduced to two as we show in the analysis below. The spinconserving optical transitions between ground and excited states are denoted by O_{1} and O_{2} for each spin subspace whereas the additional radiative and intersystemcrossing (ISC) rates are denoted by γ_{i}. In this work, we use resonant and offresonant optical excitation methods to extract all the transition rates shown in Fig. 1b, and we will show that the results are in excellent agreement with our theoretical calculations. The intrinsic spin dynamics of the \({{\rm{V}}}_{{\rm{Si}}}^{}\) defect under optical illumination is governed by the radiative transitions between states of the same spin multiplicity as well as nonradiative ISCs between states of different spin multiplicity. In the case of optical excitation of the first excited state, the ISC is established by two processes involving:

i.
Transitions from the optical (^{4}A_{2} : ue^{2}) first excited spinquartet state to energetically higher metastable spindoublet states (^{2}E : v^{2}e; ^{2}A_{1} : ve^{2}).

ii.
Transitions from the metastable spindoublet states (^{2}E : e^{3},v^{2}e) to the ground spinquartet (^{4}A_{2} : ve^{2}) state.
Both processes are mediated by a combination of spinorbit (spinlowering/raising) and electronphonon (spinconserving) interactions. The ISC mechanisms involved in these transitions can be explained intuitively using the simpler MO picture as shown in Fig. 1c.
ISC mechanism
The upper ISC mechanism is governed by the γ_{1,2} and \({\gamma }_{\mathrm{1,2}}^{{\prime} }\) rates (ES to MSs) in Fig. 1b. It consists of transitions from the ^{4}A_{2} : ue^{2} spin quartet state to the ^{2}E : v^{2}e and ^{2}A_{1} : ve^{2} spindoublets, being assisted by both the spinorbit coupling and the electronphonon interaction. Consider the system to be initially in the excited state ue^{2} spinquartet configuration, as shown in the inset (i) of Fig. 1c. From here, and as shown in the inset (ii) of Fig. 1c, the spinorbit coupling (α_{y}l_{y}s_{y}) can promote an electron from the v orbital to the \({\bar{e}}_{x}\) orbital. This is followed by a fast spinconserving decay of a second electron from the \(\bar{v}\) orbital to the \(\bar{u}\) orbital via either a photon emission or a phonon relaxation process. This then leads to the v^{2}e spindoublet configuration. The inset (iii) of Fig. 1c shows an alternative pathway from the configuration in (i). Here, the spinorbit coupling (α_{z}l_{z}s_{z}) can promote an electron from the e_{y} to the \({\bar{e}}_{x}\) orbital. Again, this is followed by the \(\bar{v}\to \bar{u}\) decay, resulting in the ve^{2} spindoublet. In fact, due to the stronger spinorbit coupling along the caxis (α_{z} > α_{x,y}), the latter process (iii) is expected to be faster than (ii), which will be confirmed by our experimental investigations. As shown in the inset (iv) in Fig. 1c, we must also consider that the ve^{2} and the e^{3} spindoublet states can be vibronically coupled together by the esymmetry acoustic phonons (^{2}A_{1} × e × ^{2}E) via the pseudoJahn Teller (PJT) effect. This allows the ve^{2} spindoublet of (iii) to transition into the e^{3} spindoublet state at a relatively fast dynamic relaxation rate (\(\gamma_{n1} \, \gg \, \gamma_{1,2}\)) which redistributes most of the population into the e^{3} spindoublet during an optical excitation cycle.
Deshelving mechanism
Subsequently, the nonradiative transition of the e^{3} spindoublet state to the ground ve^{2} spinquartet state forms the basis of the lower ISC mechanism represented by the effective γ_{3,4} rates (MS to GS). In the MS manifold, the ve^{2} doublets experience ultrafast relaxation to the e^{3} doublets (γ_{n1}), which is why we can combine them into a single effective spindoublet state. On the other hand, the v^{2}e state lacks a similar fast relaxation path (\({\gamma }_{{\rm{n}}2}\sim 0\)), thus resulting in low nonradiative decay rates \({\gamma }_{\mathrm{3,4}}^{{\prime} }\) to the ground state. Interestingly, the rather long lifetime of the v^{2}e state can result in a sizeable optical excitation rate into the higherlying uve spindoublet state (Fig. 1b). In this previously unknown deshelving process, an electron is promoted from the \(\bar{u}\) to the \(\bar{v}\) orbital (solid gray line in Fig. 1b). Here, the optical excitation is required to have the same polarization (π) and similar energy as the O_{1} and O_{2} transitions (only differing by the electron exchange correlations). The cycle is then completed by a spinorbit mediated nonradiative transition from the uve spindoublet to the ue^{2} first excited state (purple dashed arrows in Fig. 1b). This mechanism is expected to manifest itself as a laser powerdependent \({\gamma }_{\mathrm{3,4}}^{{\prime} }\) rates during a continuous resonant or offresonant optical excitation of the \({{\rm{V}}}_{{\rm{Si}}}^{}\) defect further evidenced by our experimental observations.
The effective rate model for V2 is summarized into a sixlevel rate model (see Supplementary Note 1) as: 1) the metastablestate doublets ve^{2} and lowest e^{3} are combined into one level due to ultrafast relaxation (\({\gamma }_{{\rm{n}}1}\)), and 2) the powerdependent deshelving mechanism is represented by the powerdependent decay rates \({\gamma }_{\mathrm{3,4}}^{{\prime} }\). Our theoretical calculations are based on this effective six levels to model the experimental data for inferring all the relevant rates.
Experimental determination of the spindependent excitedstate lifetimes
Firstly, we measure the spindependent excitedstate lifetimes using the experimental sequence depicted in Fig. 2a (see Methods and ref. ^{22}). A “repump” laser pulse is used to ensure that the \({{\rm{V}}}_{{\rm{Si}}}^{}\) center is in the desired negative charge state. This is followed by a sublifetime short (1.5 ns) laser pulse at 916.5 nm, resonant with either the O_{1} or O_{2} transition. The fluorescence decay signal is recorded and subsequently fitted using a single exponential function as shown in Fig. 2b. We determine the excitedstate lifetimes of bulk V2 centers as 6.1 ns and 11.3 ns for the O_{1} and O_{2} transitions, respectively.
Here, we note that the O_{2} lifetime is almost twice as long as O_{1}, which is in stark contrast with the two nearly identical spindependent excited state lifetimes of V1 center in 4HSiC^{22}. This indicates the V2 center has a much slower intersystem crossing for the ±3/2 spin sublevels associated with the O_{2} transition (compared to the spin ±1/2 sublevels belonging to the O_{1} optical transition). This also implies a significantly higher quantum efficiency for the O_{2} transition which has been experimentally observed in a recent work with the V2 centers integrated into nanophotonic resonators^{16}.
Probing the metastablestate dynamics of the V2 center
We now investigate the predominant decay processes out of metastable states. This requires pumping of the system into the MSs followed by the determination of subsequent spin populations \({p}_{1/2}\) and \({p}_{3/2}\) within the spin subspaces of \({m}_{{\rm{s}}}=\pm 1/2\) and \({m}_{{\rm{s}}}=\pm 3/2\), respectively. For these studies, we take advantage of the following three key features:
First, it is known that a prolonged offresonant excitation of \({{\rm{V}}}_{{\rm{Si}}}^{}\) center pumps the system into the MS which is followed by a partial, noncomplete spin polarization. It is assumed that the \({m}_{{\rm{s}}}=\pm 1/2\) subspace shows a slightly higher population compared to the \({m}_{{\rm{s}}}=\pm 3/2\) subspace^{26}.
Second, a prolonged resonant excitation along the O_{2} or O_{1} transitions can be used to achieve a (nearly) complete polarization into the \({m}_{{\rm{s}}}=\pm 1/2\) or \({m}_{{\rm{s}}}=\pm 3/2\) subspaces, respectively^{10,22}.
Third, the resonant excitation permits us to selectively read out the spin populations \({p}_{1/2}\) and \({p}_{3/2}\) in the subspaces \({m}_{{\rm{s}}}=\pm 1/2\) and \({m}_{{\rm{s}}}=\pm 3/2.\)
It is not straightforward to infer the relative population or population contrast of the ground states via the fluorescence intensities under the O_{1} and O_{2} excitation as they depend on the ISC rates and quantum efficiency of each spindependent transition. For this reason, we then measure spin Rabi oscillations and determine the groundstate population contrast for probing the metastablestate dynamics. The related experimental sequence is shown in Fig. 3a.
The sequence starts by predominantly initializing the system into the \({m}_{{\rm{s}}}=\pm 1/2\) (\({m}_{{\rm{s}}}=\pm 3/2)\) spin subspace using ~0.5 µs long resonant excitation along the O_{2} (O_{1}) transition. Subsequently, an offresonant laser pulse, with varying duration of T^{repump} and fixed power of 30 µW, is applied to increase the population in the MS, after which we allow the system to relax back to the ground states. To determine the absolute spin populations \({p}_{1/2}\) and \({p}_{3/2}\), we then perform spinRabi oscillations using a microwave drive at a frequency of 70 MHz, corresponding to the ground state splitting between the \({m}_{{\rm{s}}}=\pm 1/2\) and \({m}_{{\rm{s}}}=\pm 3/2\) subspaces. Finally, the spin population is read out by integrating the fluorescence intensity during a 0.5 µs short resonant laser pulse along the O_{2} (O_{1}) transition. We note that the short readout pulse duration ensures that we exclude any complex dynamics stemming from ISCs, so that the fluorescence intensity signal is proportional to the ground state population. Overall, this sequence allows us to determine the ground state population contrast \(\Delta p=({p}_{1/2}{p}_{3/2})/({p}_{1/2}+{p}_{3/2})\) as a function of the offresonant laser pulse duration T^{repump} (for more details, see Supplementary Note 3). The bottom inset of Fig. 3a shows the obtained experimental data. For short times of T^{repump}, the system has not yet reached an equilibrium (partial incomplete spin polarization into the \({m}_{{\rm{s}}}=\pm 1/2\) subspace^{26}). This is witnessed by a strong population contrast in both cases after initialization into the subspaces of either \({m}_{{\rm{s}}}=\pm 1/2\) (blue) or \({m}_{{\rm{s}}}=\pm 3/2\) (red). For T^{repump} >40 µs, the population contrast reaches a steady state value of \(\Delta p \sim 0.14\). Crucially, the behavior of the population contrast for both initializations is different. After initialization into \({m}_{{\rm{s}}}=\pm 1/2\), we find a monotonic decay of Δp. In contrast for initialization into \({m}_{{\rm{s}}}=\pm 3/2\), we observe a decay to \(\Delta p \sim 0\) at T^{repump} ~10 µs followed by an increase to the steady state value \(\Delta p \sim 0.14.\) This is explained by the initial population \({p}_{3/2} \sim 1\) dropping to \({p}_{3/2} \sim 0.5\) after \({T}^{\text{repump}} \sim 10\) µs, and further decreasing to \({p}_{3/2} \sim 0.43\) for \({T}^{\text{repump}} > 40\) µs. In other words, our experimental results unambiguously determine that the \({m}_{{\rm{s}}}=\pm 1/2\) subspace is preferably populated from the MSs with a ratio of 0.57/0.43 at an offresonant laser power of 30 µW. It is important to mention that optical reexcitation i.e. deshelving processes within the MSs are allowed by the selection rules (see Fig. 1b) and have been experimentally observed^{27,28}. Reexcitation can drastically alter the spin population dynamics and as we show in the Supplementary Note 3 we can achieve a ground state population ratio of 0.65/0.35 for an offresonant laser power of 4 mW.
Having already established that the system reaches a steady state at 30 µW offresonant pump power after \({T}^{\text{repump}} \sim 40\) µs, we now proceed to the highfidelity spin initialization using a resonant excitation along the O_{1} (O_{2}) transition. The related experimental sequence is shown in Fig. 3b (upper panel). An offresonant laser pulse of 40 µs duration and 30 µW power establishes a steady state (0.57/0.43 ground state populations). Thereafter, a resonant laser pulse of duration T and power 6 nW initiates the highfidelity spin pumping. The absolute ground state spin populations are read out as before (spin Rabi oscillations, followed by a 0.5 µs spinselective resonant laser excitation). The experimental results are shown in Fig. 3b (lower inset). For T > 20 µs, we reach high initialization fidelities of 95(1)% and 93(1)% into the spin subspaces \({m}_{{\rm{s}}}=\pm 1/2\) (blue) or \({m}_{{\rm{s}}}=\pm 3/2\) (red) under O_{2} and O_{1} optical pumping, respectively.
To further investigate the decay rates from the metastable (MS) states to the ground states (see Fig. 1b), we develop a delayed measurement scheme, as depicted in Fig. 3c. We use a 4 µs long offresonant laser excitation pulse at 4 mW power, to trap most of the spin population in the MSs at the end of the pulse. This is due to the considerably longer lifetimes of the MSs compared to the excitedstate lifetimes and the optical excitation rates. We then capture the MS depopulation dynamics by measuring the timedependent increase of the ground state populations. To this end, we permit the system to relax to the ground state for a duration of T_{Delay} before starting to integrate the fluorescence emission for 0.1 µs during a resonant excitation along the O_{1} and O_{2} transitions. We note that these measurements capture all the rates in and out of the MS states^{29} which are populated by the incoming ISC rates (\({\gamma }_{\mathrm{1,2}},{\gamma }_{\mathrm{1,2}}^{{\prime} }\)) and depopulated by the outgoing ones (\({\gamma }_{\mathrm{3,4}},{\gamma }_{\mathrm{3,4}}^{{\prime} }\)). The related experimental data is shown in Fig. 3c (bottom inset). The graph also includes the simulated curves using the ISC model and rates given in Table 1 showing an excellent agreement between the experiment and theory. We note that these rates have been inferred through the MS dynamics probed until here and subsequent powerdependent resonantexcitation measurements which are discussed in the next section.
Powerdependent resonantexcitation spin initialization measurements
To obtain a comprehensive understanding of the spin initialization process through resonant excitation, we use the resonant laser power as an additional probing parameter ranging from 6 nW to 20 nW. We limit the maximum power to 20 nW to avoid excessive photoionization^{30}, as well as power broadening, which would result with the loss of spin selectivity^{31}. As depicted in Fig. 4a, the measurement sequence consists of an offresonant excitation pulse (32 µW, 40 µs) to initialize the \({{\rm{V}}}_{{\rm{Si}}}^{}\) center into the negatively charged state and to initialize the ground state into slightly unbalanced spin populations (see Fig. 3a). Then, we selectively depopulate the spin subspaces \({m}_{{\rm{s}}}=\pm 1/2\) or \({m}_{{\rm{s}}}=\pm 3/2\) through continued resonant excitation along the O_{1} (or O_{2}) optical transition while recording the fluorescence intensity for 40 µs. At all power levels, 40 µs resonant excitation is sufficiently long to completely depopulate the respective spin sublevels, as witnessed by the fluorescence signals reaching the noise level of the singlephoton detectors. As shown in Fig. 4b, the timedependent fluorescence intensity shows a tail that extends over to several microseconds. This indicates the involvement of a longlived metastable state in the ISC, which is later corroborated by our rate results (see Table 1). The state initialization fidelity extracted from the resonant optical pumping measurement reaches \(\sim 99.5\left(1\right) \%\) which is higher than the value inferred from the spin Rabi oscillation which provides a lower bound of the state initialization fidelity. The lower fidelity in the latter case is caused by a small residual magnetic field originating from the magnetization of the instruments mixing the ground state spin sublevels and it can be further improved by applying a sufficiently large external magnetic field^{10,11}.
We now build a parameter optimization and finetuning algorithm that can be carried out over the density matrix master equation of the spin selective ISC model shown in Fig. 1b, constrained only by the measured excited state lifetimes (see Methods). The time dependent fluorescence decay data at all four powers and the delayed pulse measurement data are all simultaneously fitted using this algorithm. The resulting fit curves for the fluorescence decays are presented in Fig. 4b, showing excellent agreement with the experimental data. Individual analysis of the resonant initialization and delayed pulse measurement leads to a differing number of metastable states. Our ISC model presented in Fig. 1b leads to an excellent agreement with both measurements in Figs. 3 and 4, and involves a minimum number of metastable states for accurately describing the experimental data. The resulting transition rates are summarized in Table 1. In accordance with the model in Fig. 1b, we find two effective metastable states playing a significant role in the ISC. The first metastable state (MS_{1}) consists of the ve^{2} and e^{3} spindoublet states in which the fast dynamic relaxation (\({\gamma }_{{\rm{n}}1}\,\gg\, {\gamma }_{i},{\gamma }_{i}^{{\prime} }\)) of ve^{2} onto e^{3} is captured within. The second effective metastable state (MS_{2}) is formed from the v^{2}e spindoublet state. Due to the deshelving of v^{2}e spin doublet to uve spin doublet under optical excitation (see Fig. 1b), the MS_{2} lifetime shows a significant power dependence during resonant excitation measurements. Using the deshelving model and taking 20 nW as a reference power, from each of the remaining fit curves we theoretically infer the rest of the resonant excitation powers. The calculated powers are well within the experimental error tolerances showing excellent agreement (see Table 1) across all powers and providing further evidence of the deshelving mechanism.
At this point we highlight key differences between the V1 and V2 centers in 4HSiC. In our previous work, we showed that V1 center has a metastable state with ∼200 ns lifetime^{22}. This is very similar to the MS_{1} lifetime of the V2 center, as we have shown in this work. However, the V2 center shows an additional longlived MS_{2} metastable state (up to ∼3 µs at low excitation powers), for which no such evidence was observed for V1. We explain this difference by the PJT effect which results in a strong vibronic mixing between the MS_{1} (ve^{2}) and MS_{2} (v^{2}e) states. For the V1 center with nearT_{d} symmetry, this effectively results in a single metastable state due to the increased degeneracy between the v and e MOs. On the other hand, for V2 centers, such degeneracy is removed by the much more distorted local symmetry (C_{3v}) along the caxis^{32}, suppressing any mixing. The weaker PJT effect for V2 centers is additionally confirmed by our recent work, which showed that V2 centers maintain narrow optical linewidths at significantly higher temperatures compared to V1 centers^{32}. The very long lifetime of MS_{2} also affects the behavior of V2 color centers under offresonant excitation. Especially at high laser powers, Stokes excitation can lead to another depletion channel for MS_{2} which eventually reduces the effective lifetime of the entire metastable state manifold to the lifetime of the MS_{1}. This behavior is experimentally corroborated by our highrepumppower results in Fig. 3c, as well as previous roomtemperature investigations^{27,28}.
Discussion
From our measured rates for V2 centers in 4HSiC, the higher quantum efficiency of the O_{2} transition with spin \({m}_{{\rm{s}}}=1/2\) leads to a higher cooperativity (see Supplementary Note 4) when integrated in nanophotonic resonators, which has been recently observed^{16}. Here, we develop protocols for the generation of timebin entangled multiphoton states from a single V2 center that can take advantage of this high quantum efficiency of the O_{2} transition. We perform a detailed analysis of our protocols and consider multiple sources of imperfections including both spin conserving (i.e., excitedstate phonon scattering, imperfect excitation) and spinflip (due to ISC mechanism) errors.
Two groundstate spins \({{\rm{g}}}_{1}\rangle =1/2\rangle\) and \({{\rm{g}}}_{2}\rangle =3/2\rangle\) are chosen as entanglers for the GreenbergerHorneZeilinger (GHZ) and onedimensional cluster states. Our protocol is adapted from a similar concept initially developed for quantum dots^{33} and consists of periodic optical drive of O_{2} and coherent microwave control of groundstate spins. The Kramer’s degenerate groundstate spins are further split into four sublevels (see Fig. 5a) by applying a sufficiently large magnetic field (e.g., B = 5 mT) for longer spin coherence time^{34}. The entire protocol shown in Fig. 5b involves the following steps:

i.
The ground state spin is initialized to \({{\rm{g}}}_{1},0\rangle +{{\rm{g}}}_{2},0\rangle\) (normalization omitted w.l.o.g.) by resonant optical pumping of O_{1} transition during a continuous microwave driving (I_{MW}) of the spin \(+3/2\rangle \leftrightarrow +1/2\rangle\) transition followed by a microwave π/2 pulse (\({\text{S}}_{\text{MW}}^{{\rm{\pi }}/2}\)) resonant with \(3/2\rangle \leftrightarrow 1/2\rangle\).

ii.
An optical π pulse resonantly excites the O_{2} transition resulting with the spontaneous emission of the first ZPL photon into an early time bin: \({{\rm{g}}}_{1},0\rangle +{{\rm{g}}}_{2},{1}_{{\rm{E}}}\rangle\).

iii.
The spin states are swapped via \({\text{S}}_{\text{MW}}^{{\rm{\pi }}}\) pulse: \({{\rm{g}}}_{2},0\rangle +{{\rm{g}}}_{1},{1}_{{\rm{E}}}\rangle .\)

iv.
Upon resonant excitation of O_{2} with a second π pulse, another photon is emitted into a late time bin: \({{\rm{g}}}_{2},{1}_{{\rm{L}}}\rangle +{{\rm{g}}}_{1},{1}_{{\rm{E}}}\rangle\).

v.
For a GHZ state generation, a final \({\rm{R}}={\text{S}}_{\text{MW}}^{{\rm{\pi }}/2}\) pulse (i.e., Xgate) is applied resulting with \({{\rm{g}}}_{1},{1}_{{\rm{L}}}\rangle +{{\rm{g}}}_{2},{1}_{{\rm{E}}}\rangle\). Similarly, for a cluster state generation, the PauliX gate can be replaced by a Hadamard gate (\(\text{R}=\text{H}=\text{X}{\text{Y}}^{1/2}\)) resulting with a generator \({C}^{\dagger }={{\rm{g}}}_{+}\rangle \langle {{\rm{g}}}_{2}{a}_{\text{E}}^{\dagger }+{{\rm{g}}}_{}\rangle \langle {{\rm{g}}}_{1}{a}_{\text{L}}^{\dagger }\) for each period with \({{\rm{g}}}_{\pm }\rangle =({{\rm{g}}}_{1}\rangle \pm {{\rm{g}}}_{2}\rangle )/\sqrt{2}\). By repeating one period (from ii to v) of this protocol N times we obtain an Nphoton GHZ state or a cluster state depending on the final gate operation.
The dephasing of the excited states for V2 centers induced by acoustic phonons was shown to be negligible by the preservation of narrow PLE linewidths up to 20 K^{32}. Therefore, the fidelity related to phononinduced pure dephasing, defined^{33} as \({F}_{{\rm{p}}}^{{\rm{GHZ}},{\rm{C}}}=1N{\gamma }_{{\rm{d}}}/({\gamma }_{{\rm{r}}}+2{\gamma }_{{\rm{d}}})\), is close to 1 with \({\gamma}_{{\rm{d}}} \, \ll \, {\gamma}_{{\rm{r}}}\). We also consider excitation errors in the resonant driving pulses that induce undesired stimulated photon emissions (weak, long pulse) and detuned O_{1} transition (strong, short pulse) given^{33} as \({F}_{{\rm{ex}}}=1N\left(\frac{\sqrt{3}\pi }{8}\right){\gamma }_{{\rm{r}}}/\Delta\). With our measured radiative decay rate and ∼1 GHz separation between O_{1} and O_{2} transitions, the pulse timing for a square πpulse optimized upon 2πrotation of the detuned transition is found to be 0.9 ns for a threephoton GHZ/cluster state resulting in an excitation fidelity of 74.3%. The fidelity of the GHZ and cluster states with branching errors (emission into phonon sideband and ISC) are the same for the protocol shown in Fig. 5 and given as \({{F}_{\text{br}}}^{{\rm{GHZ}},{\rm{C}}}={\left({P}_{{{\rm{O}}}_{2}}\right)}^{N}\) conditioned upon a successful detection without any postprocessing. \({P}_{{{\rm{O}}}_{2}}\) is the ZPL emission probability for the O_{2} radiative transition and defined as \((P\alpha {\gamma }_{{\rm{r}}})/(\left(1\alpha \right){\gamma }_{{\rm{r}}}+P\alpha {\gamma }_{{\rm{r}}}+{\gamma }_{2}+{\gamma }_{2}^{{\prime} })\) in terms of DebyeWaller factor \(\alpha \sim 9 \%\)^{32}, Purcell factor P, and the rates inferred in Table 1. We find \({{F}_{\text{br}}}^{{\rm{GHZ}},{\rm{C}}}=0.06\) without any cavity Purcell enhancement with the above optimized pulse timing. The cavity enhancement of radiative lifetime improves \({F}_{{\rm{p}}}^{{\rm{GHZ}},{\rm{C}}}\) and \({F}_{{\rm{br}}}^{{\rm{GHZ}},{\rm{C}}}\), but it degrades F_{ex} slightly as the excited state radiative lifetime is modified as \({\gamma }_{{\rm{r}}}\to \left(1\alpha \right){\gamma }_{{\rm{r}}}+P\alpha {\gamma }_{{\rm{r}}}\). To maximize the overall fidelity defined as \({F}_{{\rm{t}}}={F}_{{\rm{p}}}{F}_{{\rm{ex}}}{F}_{{\rm{br}}}\), we calculate the optimized Purcell factors for each state size as shown in Fig. 5c. The primary limitation to the total fidelities shown here comes from the competition between the excitation and branching ratio errors as F_{br} increases while F_{ex} decreases with larger Purcell factors. To reach a total fidelity of 50%, GHZ or cluster states of maximum three photons are feasible. We calculate the minimum required Purcell factors to reach F_{t} = 50% (Fig. 5d) for various state sizes compatible with existing nanophotonic resonators^{15,16,35}. Photon states of larger size (>three photons) can be achieved by increasing the excitedstate ZFS (~∆) via applied strain allowing for much larger Purcell enhancements to be applied^{36}. In comparison to the quantum dot platform where the linear cluster state has been realized^{21}, the V2 centers additionally provide excellent quantum memories based on electron and nuclear spins^{8,10,14,37}. Thus, the system is in principle capable of connecting quantum memories with multiphoton states and promising a losstolerant demonstrator which requires one emitter, two memories and six photons^{38}.
In summary, we have established the complete electronic fine structure and intrinsic spin dynamics of the V2 silicon vacancy centers in 4HSiC unraveling all the previously unknown spindependent radiative and nonradiative decay rates, ISC, and deshelving mechanisms. The mechanisms identified here successfully explain several previous measurements done with V2 centers including antiStokes excitation^{39}, ODMR with offresonant readout^{28}, as well as autocorrelation of resonatorintegrated defects^{16}. Our work also explains the main differences between V1 and V2 centers in 4HSiC. The unraveled understanding of the complete spinoptical dynamics of the V2 center provides the critical engineering guidelines towards its integration into nanophotonic enhancement structures, such as waveguides and resonators. To showcase this, we additionally propose realistic protocols for generating timebin entangled multiphoton GHZ and cluster states, taking advantage of the high quantum efficiency of the O_{2} transition. We provide indepth analysis of state fidelities, optimization of pulse timings, and minimum Purcell enhancement requirements for generating GHZ or cluster states of various sizes. We show that twophoton GHZ and cluster states can be readily realized with existing SiC nanophotonic resonators, whereas higher photonnumber states would require further improvements. In this sense, we believe that phonon or strain engineering of V2 centers will become necessary to suppress excitation errors by increasing the excited state zerofield splitting, and to improve the overall branching ratio.
Overall, our studies provide a holistic summary on the intrinsic spinoptical dynamics of the V2 center in 4HSiC. This now permits defining ideal experimental protocols and routines for maximizing the performance in various quantum technology applications, as well as optimizing the optical performance of V2 centers via integration into nanophotonic resonators. Additionally, our methods can be straightforwardly adapted to improve the understanding of internal spin dynamics of other color centers.
Methods
Experimental setup
All the experiments are performed with a homebuilt scanning confocal microscope at 5.5 K in a closedcycle cryostat (Montana Instruments). The resonant excitation of the single V2 color center uses a tunable singlefrequency diode laser (Toptica DLC DL pro) at 916.5 nm. The acoustooptic modulator (Gooch&Housego) and electrooptic amplitude modulator (Jenoptik) enable continuouswave and pulsed excitation resonantly. A custommade diode laser is employed for offresonant excitation at 730 nm. A polarizationmaintaining fiber combines all excitation lasers which are focused onto the sample by a microscope objective (Zeiss EC EpiplanNeofluar ×100, NA = 0.9). The scanning of the sample is enabled by a faststeering mirror (Mad City Labs). The fluorescence emission is collected at phononside bands (940–1033 nm) by a superconducting nanowire single photon detector (Photon Spot). The fabrication of the 4HSiC sample, defect generation, and development of a solid immersion lens have been described in our previous work^{10}.
Density matrix master equation parameter optimization
The resonant PL decay of the V2 defect can be accurately modeled by using the fine structure and ISC model in Fig. 1b. The spinselective fluorescence signal corresponds to the timedependent excited state populations, that are calculated using the following master equation, \(\rho /{dt}=\frac{i}{\hslash }[{H}_{0},\rho ]+{\gamma }_{{\rm{r}}}{\sum }_{{\rm{i}}=1}^{2}L({A}_{{\rm{r}}}^{i})+{\sum }_{{\rm{i}}=1}^{4}{\gamma }_{i}L({A}_{{\rm{m}}{{\rm{s}}}_{1}}^{i})+{\sum }_{{\rm{i}}=1}^{4}{\gamma }_{i}^{{\prime} }L({A}_{{\rm{m}}{{\rm{s}}}_{2}}^{i})\). The radiative and nonradiative decay processes are represented by the Lindblad superoperators, L(O). The H_{0} is the Hamiltonian constructed from optically driven spin ±1⁄2 and ±3⁄2 ground and excited states, \({H}_{0}=({D}_{{\rm{g}}}{D}_{{\rm{e}}}{\delta }_{{\rm{L}}})({{\rm{gs}}}_{1/2}\rangle \langle {{\rm{gs}}}_{1/2}{{\rm{es}}}_{1/2}\rangle \langle {{\rm{es}}}_{1/2})({D}_{{\rm{g}}}{D}_{{\rm{e}}}{\delta }_{{\rm{L}}})({{\rm{gs}}}_{3/2}\rangle \langle {{\rm{gs}}}_{3/2}{{\rm{es}}}_{3/2}\rangle \langle {{\rm{es}}}_{3/2})+[{\Omega }_{{\rm{L}}}({{\rm{gs}}}_{1/2}\rangle \langle {{\rm{es}}}_{1/2}+{{\rm{gs}}}_{3/2}\rangle \langle {{\rm{es}}}_{3/2})+h.c.]\) in the rotating frame of the laser with power dependent Rabi frequency Ω_{L} and detuning \({\delta }_{{\rm{L}}}={{\omega }_{{\rm{L}}}\omega }_{{\rm{ZPL}}}\). The radiative decays, \({A}_{{\rm{r}}}^{i}={{\rm{gs}}}_{i}\rangle \langle {{\rm{es}}}_{i}\), are governed by the same radiative decay rates γ_{r} for both O_{1} and O_{2} transitions. The nonradiative ISC decays in and out of each metastable state (\({{\rm{ms}}}_{1}=v{e}^{2}\) and \({{\rm{ms}}}_{2}={v}^{2}e\)) are given by \({A}_{{{\rm{ms}}}_{1}}^{i}={{\rm{ms}}}_{1}\rangle \langle {{\varphi }}_{i}\) and \({A}_{{{\rm{ms}}}_{2}}^{i}=\left{{\rm{ms}}}_{2}\right\rangle \langle {{\varphi }}_{i}\) in which \({{\varphi }}_{\{i=\mathrm{1,2,3,4}\}}=\{{{\rm{es}}}_{1/2},{{\rm{es}}}_{3/2},{{\rm{gs}}}_{1/2},{{\rm{gs}}}_{3/2}\}\) after the effective rate simplification with \({\gamma }_{{\rm{n}}1}\gg {\gamma }_{\mathrm{3,4}}\) and \({\gamma }_{{\rm{n}}2}\ll {\gamma }_{\mathrm{3,4}}^{{\prime} }\). We use a custombuilt parameter optimization algorithm based on both Nelder–Mead and differential evolution numerical nonlinear optimization methods for simultaneously fitting the excited state population solutions of separate master equations at four different laser powers with the time dependent photoluminescence decay measurement data. At each trial, a secondary simplified master equation reflecting the 100 ns integration window for pulse sequence in Fig. 1c is used to also evaluate the fit quality of each rate solution with the delayed pulse measurement data.
Data availability
The data presented in this manuscript are available at the data repository of the University of Stuttgart under https://doi.org/10.18419/darus4226.
Code availability
The codes used in this manuscript are available from the corresponding authors on reasonable request.
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Acknowledgements
We acknowledge fruitful discussions with Petr Siyushev, Jianpei Geng, Naoya Morioka, Daniil Lukin and Melissa Guidry. D.L., F.K., V.B., E.H., T.S. and J.W. acknowledge support from the European Commission for the Quantum Technology Flagship project QIA (Grant agreements 101080128 and 101102140), the German ministry of education and research for the projects InQuRe, QR.X, Spinning (BMBF, Grants No. 16KIS1639K, No. 16KISQ013, and No. 13N16219) and INST 41/11091 FUGG, as well as the Ministerium für Wirtschaft, Arbeit und Tourismus BadenWürttemberg for the project SPOC (Grant Agreement No. QT6). F.K. acknowledges funding by the Luxembourg National Research Fund (FNR) (project: 17792569). F.K., J.U.H, and J.W. acknowledge support from the European Commission through the QuantERA project InQuRe (Grant Agreements No. 731473 and No. 101017733). J.U.H. acknowledges support from the Swedish Research Council under VR Grant No. 202005444. N.T.S. and J.U.H. acknowledge support from EU H2020 project QuanTELCO (Grant No. 862721), and the Knut and Alice Wallenberg Foundation (Grant No. KAW 2018.0071). Ö.O.S. acknowledges support from Booz Allen Hamilton Inc. T.O. acknowledges funding from the Japan Society for the Promotion of Science via the grant JSPS KAKENHI 21H04553, as well as the Japan Science and Technology Agency for funding within the MEXT QLEAP program via the grant JPMXS0118067395.
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D.L., F.K., Ö.O.S and J.W. conceived and designed the experiments. D.L. and F.K. performed the experiments. D.L. and Ö.O.S. analyzed the data. Ö.O.S. developed the theoretical modeling and simulations. D.L. and Ö.O.S. carried out the numerical simulations. J.U.H. provided the SiC sample. T.O. and N.T.S. irradiated the SiC with electrons for defect generation. F.K., V.B., E.H., T.S. and J.W. assisted the data analysis. D.L., F.K. and Ö.O.S. wrote the manuscript with helpful inputs from all the authors.
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Liu, D., Kaiser, F., Bushmakin, V. et al. The silicon vacancy centers in SiC: determination of intrinsic spin dynamics for integrated quantum photonics. npj Quantum Inf 10, 72 (2024). https://doi.org/10.1038/s41534024008616
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DOI: https://doi.org/10.1038/s41534024008616