High-Fidelity Spin Measurement on the Nitrogen-Vacancy Center

Nitrogen-vacancy (NV) centers in diamond are versatile candidates for many quantum information processing tasks, ranging from quantum imaging and sensing through to quantum communication and fault-tolerant quantum computers. Critical to almost every potential application is an efficient mechanism for the high fidelity readout of the state of the electronic and nuclear spins. Typically such readout has been achieved through an optically resonant fluorescence measurement, but the presence of decay through a meta-stable state will limit its efficiency to the order of 99%. While this is good enough for many applications, it is insufficient for large scale quantum networks and fault-tolerant computational tasks. Here we explore an alternative approach based on dipole induced transparency (state-dependent reflection) in an NV center cavity QED system, using the most recent knowledge of the NV center's parameters to determine its feasibility, including the decay channels through the meta-stable subspace and photon ionization. We find that single-shot measurements above fault-tolerant thresholds should be available in the strong coupling regime for a wide range of cavity-center cooperativities, using a majority voting approach utilizing single photon detection. Furthermore, extremely high fidelity measurements are possible using weak optical pulses.

The negatively charged nitrogen-vacancy center (NV − ) in diamond is certainly an interesting candidate system for quantum technologies, with potential applications ranging from quantum metrology [88][89][90][91][92][93][94][95] to quantum communication [96][97][98], as well as simulation and computation [71,[98][99][100][101][102][103][104]. The NV − center contains electronic and nuclear spin components, with both optical and microwave transitions, and remains stable even at room temperature [105]. At low temperature (4 -8 K) the electronic spin coherence times are on the order of 1-100 milliseconds [106][107][108]. If the NV center is going to be used for tasks related to communication or computation, it needs to perform four core tasks: The CQED system is driven by a single-photon source while detection in the reflected mode is used to infer information about the state of the NV − center. The NV center is composed of an adjacent nitrogen atom (green) and lattice vacancy (red) in a tetrahedral carbon lattice (gray). (Right) The electronic energy level structure of the NV − center, at low temperature under a Bz = 20 mT external magnetic field set along the NV axis. The energy levels are represented by bold horizontal lines and grouped into three subspaces: ground state manifold (GSM), excited state manifold (ESM) and metastable state manifold (MSM). Allowed radiative transitions between the ground and optically excited states are horizontally segmented, with overlap indicating relative spin composition of the energy eigenstates. The optically excited states contain non-zero orbital angular momentum components +1 (red) and −1 (blue). Dashed arrows represent decay paths not resulting in reflection of a photon incident on the cavity (they represent transmission into the cavity mode, c, through the second cavity mirror (into b), spontaneous emission into the free field (d) and decay to a meta-stable state). The solid double-sided arrow represents the zero-phonon line transition in resonance with the optical cavity.
• Coherent manipulation of electronic and nuclear spins, • Entanglement generation between remote electronic spins, • Entanglement swapping between the electronic and nuclear spin states within an NV center, enabling the storage of remote entanglement in the nuclear spins, • Measurement of the electronic and nuclear spin states.
Though local operations (the coherent manipulation) of single centers has been performed with fidelities exceeding 0.999952(6) [82], sample fabrication and high fidelity measurements are technological challenges yet to be overcome [109,110]. The typical measurement scheme for an NV − center is based on a cycling transition between the m s = 0 levels in the ground and excited state manifolds. The presence of decay channels from the optically excited manifold through a meta-stable state is expected to limit the efficiency of this scheme to order 99%. The efficiency is therefore likely to be below the threshold required for larger-scale communication and computational tasks, though recent work utilizing repeated initialization of the electronic state between measurement trials [111] may prove to circumvent this limitation in cases not related to the generation of entanglement. Cavity-enhanced transitions have been exploited in Purcell-enhanced fluorescence measurement schemes [112][113][114] and in emissionbased entanglement schemes [115][116][117].
Recently an alternative approach based on dipoleinduced transparency has been proposed [118][119][120][121][122] wherein the state of the NV center changes the resonance properties of an optical cavity. More specifically, if the electronic spin in its |0 ms state, an incident photon is reflected from the cavity, while for the |+1 ms electronic spin state the photon enters the cavity where it is scattered but not absorbed by the NV center. Detection of the reflected photon is thus a definite signature that the electronic spin was in the |0 ms state, though the photon need not interact directly with the state of the NV center. In this work, we build upon the analysis of [121] and make use of recent improvements in the understanding of the NV's low-temperature optical characteristics [123] to refine the quantification of the scheme's potential. This paper is structured as follows: Section II begins with an overview of the main properties of the nitrogenvacancy center, followed in Section III by a description of the CQED system and the measurement scheme. Section IV analyzes the measurement process in detail and estimates its performance, while Section V considers the effect of replacing the single photon source with weak coherent laser pulses. Finally, Section VI summarizes the main conclusions of our analysis and briefly discusses some areas with the potential for improvement.

II. THE NITROGEN-VACANCY CENTER
The negatively charged nitrogen-vacancy center in diamond consists of the nearest-neighbor pair of a subsitutional nitrogen atom with a lattice vacancy (green and red respectively in Figure 1). Three dangling bonds from carbon atoms adjacent to the vacancy, two dangling bonds from the Nitrogen atom and an additional electron form the electronic structure of the center. Isotopes of both nitrogen (14,15) and carbon (12,13) allow us to tailor the number and properties of the nuclear spins. 14 N has a spin-1 nuclear spin while that of 15 N is spin 1/2. Similarly, 12 C has no nuclear spin while that of 13 C is spin 1/2. With isotopic engineering an NV center can be fabricated as one requires. For this article we consider an NV − center with the 15 N isotope and no nearby 13 C carbon atoms. This leads to the simplest NV center consisting of a spin-1/2 nucleus and an electronic structure (see Figure 1) is broadly classified by symmetry and/or multiplicity into three groups. These groups are the ground state manifold (GSM) 3 A 2 (three non-degenerate states with a 180 o -rotation symmetry about the principle axis), the optically excited state manifold (ESM) 3 E (three doubly-degenerate states), and the meta-stable manifold (MSM) containing the singlets 1 A 1 , one nondegenerate state with both 180 o -rotation and reflection symmetries, and 1 E, one doubly-degenerate state. The ground and optically excited states can be decomposed into two three-level subsystems. The ground states we refer to as the 'electron spin' S, with quantum number m s (and the nuclear spin I, with quantum number m n ). This will distinguish it from optical excitation, which we refer to as a change in the 'orbital angular momentum' L, with quantum number m l .
The structure of this center can be described, under minor approximations [127], by the Hamiltonian [105] where H GSM gives the structure of the ground state manifold and can be expressed as Here S and I representing the usual electronic spin-1 and nuclear spin 1 2 operators (with S z and I z being their respective z-components). The first term in H GSM represents an electronic spin zero field splitting of Here D  [125,126] and determine photon polarization selection rules on allowed optical transitions [128]. These are enumerated in Table I (along with free transition lifetimes [123,124]) for the zero-field and 20 mT external magnetic field cases [129]. The zero-phonon line between the |0 ms intrinsic spin ground state and the optically excited states at zero field is 637nm. As the nitrogen-vacancy center decays, emission into the phonon side-band collectively exceeds emission at the zero-phonon line [130][131][132]. Elastic scattering occurs approximately 3-5% of the time [133][134][135][136], linearly degrading the cooperativity of a coupled cavity.
Next (as seen in Figure 1), a decay channel couples the optically excited states to a meta-stable subspace. These decay rates (shown in Table I) can have a significant effect on the measurement fidelity as decay through a meta-stable state removes phase information and introduce a bit-flip error rate of 67-81% from the m s = +1 state, and 38-65% from the m s = 0 state [137,138].
Here we assume a polarizing sample with a bit-flip rate 81% from the m s = +1 state, and 38% from the m s = 0 state. The longer-lived 1 E meta-stable state has a lifetime of 462 ns at 4.4 K, while the shorter-lived 1 A 1 state (separated by a 1042 nm energy gap from its 1 E counterpart) has a lifetime of less than 1 ns [139]. Transitions to and from the meta-stable subspace do not conserve the intrinsic electron spin [123], they therefore degrade the measurement fidelity.
Having now outlined the properties of the NV center essential to the measurement process, let us move to a description of our photonic readout.

III. PHOTONIC READOUT OF THE NITROGEN-VACANCY CENTER STATE
The optically accessible transitions in an NV center provide a natural way to measure it, given the excellent frequency separation of allowed transitions from the m s = 0 state relative to those of the m s = ±1 states ( Table I). The absence of hyperfine interactions [140] and the smaller decay rates from the m s = 0 ESM levels to the meta-stable subspace leads to the selection of the |0 ms ↔ |M 3,4 ms transitions for the optical readout. The presence of decay through a meta-stable state, however, limits that measurement efficiency after a certain number of light pulses, as the excited states |M 3,4 ms will decay into the MSM (< 1% of the time). Dipole-induced transparency (giant Faraday rotation) [119][120][121][122]141] provides an elegant way to mitigate this effect, as we can use strong coupling between our NV center and an optical cavity to modify the resonance properties of that cavity. In this strong coupling regime, a σ + polarized photon near resonance with the empty cavity and with the |0 ms ↔ |M 4 ms transition would be reflected when the NV center is in the |0 ms state [142]. This means the photon is not absorbed by the NV center, mitigat- When the nitrogen-vacancy center is in the ms = 0 ground state, resonant coupling with the optical cavity causes the composite system to form dressed states detuned from the frequency of the incident photon -the photon does not enter the cavity, and is reflected. When the nitrogen-vacancy center is in the ms = +1 state, there is no optical transition close enough in energy to be able to couple with the cavity mode -the photon sees an empty cavity, enters, and is transmitted. In such ideal cases, no interaction occurs between the incident photon and the nitrogen-vacancy center, preserving the center's state.
ing the effect of decay through the meta-stable state and providing, in the ideal case, an interaction-free measurement. On the other hand, the photon will be transmitted through the cavity (and not absorbed or scattered) if the NV center is in its | ± 1 ms states as it is far off resonance with the | ± 1 ms ↔ |M 1,2,5,6 ms transitions. Measurement of the presence or absence of the reflected photon (or series of photons) thus allows us to infer the state of the NV's electron spin. How well this works required a detailed analysis of the entire measurement scheme.
Coupling between the cavity, external field modes and electron spin can be represented as shown in Figure 1 and by the Hamiltonian where a, b, c, d (a † , b † , c † , d † ) are the annihilation (creation) operators of the left-hand, right-hand, cavity and scattering (spontaneous emission) operators. Next, L x is the angular spin-1 X operator (see Appendix A) while g is the vacuum-Rabi coupling rate. Further, κ a and κ b are the left and right mirror cavity decay rates, which we assume are equal giving us a total decay rate κ = κ a +κ b . Finally, γ i is half the spontaneous decay rate of the ith optically excited state [143]. These three parameters allow us to define the cooperativity C i = g 2 /2κγ i , a use-ful measure for determining how strongly coupled one is [144]. On resonance, the coupling is related to the spontaneous decay rate by where f is the quantum efficiency of the transition (0.03 − 0.05 for the nitrogen-vacancy center), γ Rad is the radiative component of the spontaneous decay rate (2π × 13.2 MHz), ω is the angular frequency of the transition, c is the speed of light, and ρ ω denotes the energy density of the cavity-field per unit angular frequency [145].
With the NV and coupling Hamiltonians we can derive Langevin equations of motion for the various field and spin operators. These are nonlinear in nature due to coupling terms between the field and spin operators, which makes them difficult to solve in general. Two alternative approaches can be taken to simplify this situation: the first assumes that NV center is always in the GSM while the second is to work in a single excitation subspace [146] where the field probing the NV center contains no more than one photon (both approaches lead to the same answer). We take the second route and will use probe fields with at most one photon. This approximation results in a set of linear Langevin equations that are straightforward but tedious to solve. The solutions are those for the cavity mode and excited states and not what we measure. However by using input-output relations [147] a out (t) = a in (t) − i √ κc(t), we can express our output field a out (t) in terms of the cavity field c(t) and the cavity input field a in (t). This then enables us to calculate the mean photon number a † out (t)a out (t) over time and so determine whether an incident photon was reflected or not.
Source and detection efficiencies, as well as losses and imperfections, mean that one single-photon measurement will not be sufficient to determine the NV state with the accuracy we require. Instead our measurement here consists of a series of temporally spaced single-photon pulses which we individually attempt to detect. The time between photon pulses is chosen to be ∼ 165 ns, from the axial hyperfine Rabi period. By matching gate times to this period, the effect of electron dephasing on the nuclear state can be minimized. Though it is not possible to restrict pulse times exactly to a periodic point, it has been shown that error associated with decay channels following optical excitation can be reduced by centering the pulse on such a point [148]. This is much larger than the excited state lifetime of the |M 4 state (12.1 ns) [123], but significantly shorter than the GSM decoherence times.
The outcome at the detector of each measurement pulse is dependent on its initial state distribution and thus successive outcomes are dependent on the detection history preceding them. After n pulses therefore there are 2 n possible measurement paths. Not all branches in the outcome tree will be full-length, however, as outcome branches that are successful early in the procedure can be truncated. In particular, reflection from the |+1 ms state can be suppressed by polarization and detuning, so that a detection event very strongly indicates the |0 ms state and will terminate an outcome branch (especially as dark count probabilities can be exceptionally low [149][150][151]  Along with our description of the NV − center's photonic readout, let us now turn our attention to simulating it. Our model has a large number of parameters, but with the external magnetic (electric) fields set at 20 mT (0 V/m), our primary focus will be restricted to four experimentally relevant parameters: the cooperativity C, single photon source and detection efficiencies and the m s = 0 decay rates through the meta-stable subspace. Figure 4 shows example transmission and reflection spectra. It is useful to begin by considering the situation with perfect single photon sources and detectors but with finite cooperativity C and transitions through the metastable subspace (in the range 0-1% for the target transition, as indicated in Table I). When the electronic spin is in its |0 ms state, an incoming photon is reflected with a probability P R ≈ 4C 2 /(2C + 1) 2 , so that if we send n photons, the probability of detecting one or more photons is p s = 1 − (1 − P R ) n (equivalently we need n = log(1 − p s )/ log(1 − P R ) trials to have at least one photon detected with success probability p s ). There are two n's of interest here, n ave and n f t . n ave is associated with the average number of single photon trials to get a measurement 'click' while n f t is the number that guarantees at least one measurement click with probability above the fault tolerance threshold we set of 99.9%. For an initial state |+1 ms we have to perform n f t trials (we call this trial limit number the stop-limit) with no clicks to infer it was this state (with error probability 1 − p s ). Figure 3 shows the effect of the cooperativity on both n ave , n f t and the measurement fidelities F i∈{0,+1} = i|ρ|i ms (where ρ is the state of the system after the n trials) for varying decay rates to the metastable subspace. As the cooperativity, and therefore also the reflection probability, increases, the number of trials required to achieve p s decreases in integer steps. These steps manifest as negative discontinuous jumps in the measurement fidelity.
When we start with a general superposition state, each measurement pulse will introduce some error and so the reflection probability will not remain static as was assumed in Figure 3. Instead, the contrast between the initial |0 ms and |+1 ms states will degrade as the number of pulses increases, thereby increasing n f t . Rather than initially specifying a maximum number of trials, we can look at the performance over a range of such numbers to identify an optimum number of measurement trials and the sensitivity of the measurement performance to changes in this number. In Figure 5 we plot the measurement fidelity for the initial state |0 ms − 1 2 mn + |+1 ms + 1 2 mn / √ 2. For the ideal case we observe that the 99.9% threshold is met even at cooperativities as low as C = 2, and that the dependence of the measurement fidelity on the pulse number, while sensitive at low cooperativities, is as low as order 0.0001 for cooperativities greater than 5. Now including the effects of source and detection efficiency into our analysis, we show in Figure 6 the behavior of the measurement fidelity as a function of source and detection efficiencies. We observe that the performance depends slightly more sensitively on the detector than the source, but that the performance is robust for small (∼ 0.1) inefficiencies.

A. Realistic Performance Estimate
A natural question to ask is what our expected measurement fidelity would be with current technology. To this end let us specify a cooperativity of 0.2 [152], a single-photon source probability of 60% [153], a single- Loss of measurement fidelity, F , as function of the cooperativity, C, for an ideal single-photon source and detector, over a range of measurement pulses, n. The ms = 0 decay rate to the meta-stable subspace is 1%. While the optimal number of pulses decreases with increasing cooperativity, the error associated with moving away from this optimum is very small (∼ 0.0001) for cooperativities above 5.   photon detection efficiency of 92% [150,151,[154][155][156][157], and m s = 0 decay rates to the meta-stable subspace of 1% [123]. Our model then predicts a maximum measurement fidelity of F = 0.992 occurring (within rounding error) after 145 single photon pulses. The total time for the measurement is then ∼ 150 × 165 ns ≈ 25 µs. During this time we would expect an electron spin dephasing error between 1%-0.01% for coherence times in the region 1-100 ms. The upper range would have a consequential effect on remote entangling operations, though dephasing should not effect the nuclear spin measurements. Limiting the maximum number of pulses to 10 reduces the measurement fidelity to F = 0.686. In Table II for contrast we outline similar numbers for C ∈ {0.5, 1, 2, 5, 10} and η Source , η Detect ∈ {0.2, 0.6}. For a low source efficiency, it is apparent that high measurement fidelities can be maintained at the cost of greatly increasing the number of pulses.

B. Single Photon Pulses
In any realistic model we also need to consider the bandwidth of our atomic resonances compared to those of the photon. Here we are working in the regime where the photon bandwidth is much less than the bandwidth of the NV center's optical transitions. Our single photon pulses are 165 ns apart implying a bandwidth at least of order Γ P /2π ∼ 1/2 MHz (HWHM). The reflection TABLE II. Measurement fidelities, F , and pulse numbers, n, when the ms = 0 decay rate to the meta-stable subspace is 1%, varying the cooperativity and the single-photon source and detection efficiencies, ηSource and ηDetect. spectrum from |0 ms is not sensitive at those scales but reflection from |+1 ms sits in the trough of a narrow resonance (Γ R ∼ κ = 2π × 50 MHz), necessitating further consideration. We assume a Gaussian pulse shape, and set the standard deviation to σ t = 165/6 ns so that the width of the pulse is much less than the time between pulses (at three standard deviations the area truncated in the tails is 0.3%). By the time-bandwith product, σ 2 t σ 2 P = 1/4, the standard deviation in the frequency then becomes σ P /2π = 2.9 MHz. By integrating over the probability-weighted reflection spectrum we can estimate the realistic reflection probability to be The reflection here is a direct result of the relative magnitudes of the photon and cavity linewidths, and is not influenced by the nitrogen-vacancy center. Since the width of the reflection trough is set by κ, it also governs the transmission and scattering peaks, which are each reduced to 99.34% of their maximum values. A 0.66% reflection probability from the |+1 ms state, when we want to suppress our error to within 0.1%, means that detection is no longer a strongly-classifying event.
Let us assume our initial state is predetermined and restricted to the {|0 ms and |+1 ms } subspace, but that the fidelity decays exponentially in the number of photon pulses. The probability of each state after n pulses is then given by Bayes' theorem and a binomial distribution (similar to the treatment of a Poisson-distribution in [122]). The probability of a detection event from |0 ms and |+1 ms will be p 0 and p +1 respectively. With single-photon source efficiency, detection efficiency and cooperativity of 60%, 92% and 10 respectively, the photon bandwidth detection probabilities are p 0 = 0.50 and p +1 = 0.00364. We can estimate the error rates per pulse, as η 0 = 1.5 × 10 −4 (η 0 = 3.5 × 10 −4 ) for the m s = 0 decay to the meta-stable subspace of 0% (1%) respectively with η +1 = 1.2 × 10 −5 . This means we can achieve an expected measurement fidelity of F = 99.9% (F = 99.8%) using 13 (12) single photon pulses. Using more pulses than this decreases the resulting fidelity but at the levels indicated here we are right at the border of our 99.9% requirement.
At what pulse times does the error associated with false-positive detection events become significant? We can obtain a quick estimate by determining the error and the average number of additional trials that would need to be performed to distinguish between the two rates of reflection, assuming once more that the error follows an exponential decay. As above, the cooperativity is assumed to be 10, the source and detection probabilities are 60% and 92% respectively, and the m s = 0 metastable decay rate is 1%. In the point-frequency case this approximation predicts a fidelity of 0.99775 after 11 trials (this is a more conservative estimate than the numerical calculations represented in previous sections). Errors differing from this by 10 −5 , 10 −4 , and 10 −3 occur when the pulse times reach 455 ns (requiring 11 trials), 115 ns (requiring 12 trials), and 24 ns (requiring 19 trials) respectively.
Understanding that the above is a conservative approximation, we can also ask what values for κ or for σ t we would require to suppress reflection (rather than error) from the |+1 ms so that it is below our threshold of 0.1%. We find that either κ = 2π × 129 MHz for the current pulse-time or σ t = 71 ns for the current cavity decay rate (so that either is 2.58 times larger than its original value) satisfy this condition. Due to dephasing, for application to projective entanglement generation there is a fundamental tradeoff to be made between longer pulse times, related directly to σ t , and the number of such pulses required, as influenced by κ through the cooperativity.

V. WEAK COHERENT PULSES
Our previous considerations (and in [121]) assumed the use of single-photon sources that are technological challenging to realize and were found to be a key limiting factor in achieving higher measurement fidelities. Weak coherent laser pulses are a natural alternative [158] and offer a number of potential advantages including: • Ready availability at the appropriate wavelengths, • Easy tailoring of their mean photon number, • Ready pulse shaping to customize the state dependent reflection/absorption from the cavity.
Allowing more than one photon to be reflected from the cavity dramatically improves our detection efficiency, potentially turning this into a single-shot (pulse) measurement. There is however a potential issue here associated with ionization of the NV center, which can occur when the electronic state absorbs more than one 637 nm photon. Ionization converts the NV − center to the charge neutral NV 0 center (which is spin-0). Little is known about photo-induced ionization at this wavelength and temperature apart from the fact it must be a two-photon process [159][160][161]. Once more is known , it may become possible to manipulate the exciting pulse to suppress them. Our situation here is different as we are working with an NV in a CQED configuration at low temperatures, which will enhance the light-center interaction. We do however know that the probability of photons being reflected from the cavity mirror on resonance is P R ≈ 4C 2 /(2C + 1) 2 and for a weak coherent laser pulse |α the mean number of photons in that pulse entering the cavity is |α| 2 (1 − P R ). We can then estimate that over time of the entire pulse τ p , the probability P 2+ of having two or more photons within the cavity's line-width 1/κ should be for our typical parameters with |α| 2 ∼ 3. It is critical to remember that having two photons in the cavity does not that mean both are simultaneously absorbed and cause ionization. We can go a step further and calculate the probability that the NV center is in its ESM and that simultaneously at least one photon is in the cavity mode. The results of our master equation simulation are shown in Figure 7 and we immediately observe that this joint probability is nearly an order of magnitude lower than P 2+ . Unfortunately we do not know the coupling rate between the ESM and the conduction band. However, based on these preliminary estimates, for reasonable values of C and |α| 2 we expect that the ionization probability will be less than 10 −3 , and that therefore weak coherent laser pulses should be ideal for initial measurement experiments. Performance estimates for several cases are shown in Figure 8. In addition to increasing the ionization rate, a higher photon number also increases the coupling between the cavity and center, as well as the effective detection efficiency. In fact, with |α| 2 ∼ 3, cooperativities in the strong coupling regime and current detection efficiencies we should be able to achieve a single-shot measurement.

VI. DISCUSSION
The scheme investigated here uses a dipole-induced transparency to entangle the path of a photon with the spin state of a single nitrogen-vacancy center at cryogenic temperatures. This provides, through subsequent detection of such a photon, a projective measurement on the spin state of the center. The fidelity of this projective measurement forms the key figure of merit in our results. Typically, analyses of these approaches have used significant approximations (e.g. limiting the state space and assuming memoryless scattering distributions) to argue the initial case for their competitiveness, but it is important, before these devices are realized for scalable systems, to Loss of measurement fidelity, F , as function of the cooperativity, C, for a coherent light source and ideal detection, over a range of measurement pulses, n. The ms = 0 decay rate to the meta-stable subspace is taken to be 1%. Three cases are shown: C = 0.2, |α| 2 = 3 (light gray, dots), C = 0.2, |α| 2 = 10 (light gray, circles), C = 2.0, |α| 2 = 2 (dark gray, dots). determine just how scalable they are. This requires a more in-depth and complete model. In our work here: • The model of the energy level structure of the nitrogen-vacancy center incorporates all ground and optically excited states.
• Errors arising from evolution among optically excited states are considered, as well as the ability of the decay path through a meta-stable state to feed back into the correct subspace, when accounting for multiple single-photon pulses.
• We characterize the impact of external photon loss and variation in the decay rate to the meta-stable subspace.
• With the exception of Section IV B, scattering probabilities are dependent on the outcomes of preceding measurement pulses.
Incorporating these effects allows us to begin to make statements about the applicability of this approach to real, large-scale systems. Accounting for these additional factors, we expect that the two primary hurdles to the implementation of this scheme are the construction of high-fidelity, narrowbandwidth (σ t κ >> 1) single-photon sources and a cooperativity in the strong-coupling regime (recent work [152] reported a cooperativity C ≈ 0.2). The preliminary use of few-photon, weak coherent pulses may circumvent the issue of the single-photon sources, and obviously also increases the effective cooperativity. However, larger scale applications and projective entanglement generation between multiple color centres will require single-photon sources. For the larger applications, charge-state switching and its correction will cause temporal inhomogeneous broadening [162], while for entanglement generation, the possibility of stray photons, avoidable only with large cooperativities and high detection efficiencies, degrades the resultant entanglement fidelity. Contemporary difficulties in mind, however, with continuing development we do expect the parameters assumed here for cooperativity (10), and source (60%) and detection (92%) efficiencies to be experimentally achievable in the near future; we intend our characterization of the impact of individual error sources to assist experimental efforts to engineer high-fidelity projective operations with this system.
To summarize our view for the immediate future, setting aside the challenges of large-scale applications and projective entanglement generation between nitrogenvacancy center devices, we envisage that the preliminary use of weak coherent states should allow high-fidelity spin measurements in smaller-scale, contemporary settings. We have estimated an upper bound on the rate of charge-state switching for moderate photon-numbers on the order of 10 −3 . This bound, along with the fidelities depicted in Figure 5, suggests that the measurement scheme considered here sees an error rate improvement of, in principle, an order of magnitude over the traditional method of luminescence-detection. While our estimates could be improved with a finer characterization of the ionization rate and the m s = 0 decay rate through the meta-stable subspace, our results therefore suggest that dipole-induced transparency should provide high fidelity measurement of the spin state of the nitrogen-vacancy center.

ACKNOWLEDGMENTS
We are very grateful to Yuichiro Matsuzaki for helpful comments. This project/publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. KN and MH also acknowledge support from the MEXT KAK-ENHI Grant-in-Aid for Scientific Research on Innovative Areas Science of Hybrid Quantum Systems Grant No. 15H05870, while JS and MT additionally acknowledge the support of the WWTF project "PhoCluDi", the FWF project "SiC-EiC" (I 3167-N27) and TU Vienna research funds.