High-fidelity single-shot readout of single electron spin in diamond with spin-to-charge conversion

High fidelity single-shot readout of qubits is a crucial component for fault-tolerant quantum computing and scalable quantum networks. In recent years, the nitrogen-vacancy (NV) center in diamond has risen as a leading platform for the above applications. The current single-shot readout of the NV electron spin relies on resonance fluorescence method at cryogenic temperature. However, the spin-flip process interrupts the optical cycling transition, therefore, limits the readout fidelity. Here, we introduce a spin-to-charge conversion method assisted by near-infrared (NIR) light to suppress the spin-flip error. This method leverages high spin-selectivity of cryogenic resonance excitation and flexibility of photoionization. We achieve an overall fidelity > 95% for the single-shot readout of an NV center electron spin in the presence of high strain and fast spin-flip process. With further improvements, this technique has the potential to achieve spin readout fidelity exceeding the fault-tolerant threshold, and may also find applications on integrated optoelectronic devices.


EXPERIMENTAL APPARATUS
All the experiments in this work are performed on a home-built low-temperature ODMR setup. The sample is hosted at the temperature of 8 K in a closed-cycle optical cryostat (Montana Instruments Cryostation S200). The sample is positioned by a XYZ piezo stack (Attocube), at the focal point of a 0.9 NA objective (Olympus MPLFLN100x). The objective is mounted on the side wall of the vacuum chamber. A feedback loop is used to stabilize the temperature of the objective at room temperature. We use three lasers to excite the NV center: a traditional 532 nm laser (Changchun New Industries Optoelectronics Technology) to reset charge state and two tunable 637 nm lasers (New focus TLB-6704-OI, Toptica DLC DL PRO HP 637) to perform E y and E 1,2 resonant excitation respectively. The wavelengths of the 637 nm lasers are stabilized using a wavemeter (HighFiness WSU-10). We use a 1064 nm laser (Changchun New Industries Optoelectronics Technology MSL-III-1064) to ionize the NV center. A permanent magnet is positioned near the sample by another set of XYZ piezo stack (Attocube) to produce a static magnetic field. Microwave pulses are generated by an arbitrary waveform generator (Keysight M8190A). After amplification (minicircuit), the Microwave pulses are fed to a gold strip line fabricated on top of the sample.

SAMPLE INFORMATION
Here we use a chemical vapor deposition grown diamond with (111) surface. The NV center studied here is 9 µm below the diamond surface. The NV axis is perpendicular to the diamond surface. A solid immersion lens (SIL) is wrote with focused ion beam system to enhance the photon collection efficiency [1,2]. Gold microwave strip line and electrodes are deposited around the SIL (Fig. S1ab). Fluorescence saturation curves under resonance excitation are shown in Fig. S1cd. The fluorescence count rate (I) is the peak count rates of the photoluminescence (PL) decay curves (Fig. 1d of the main text) under different excitation powers (P ). The data are fit with I = I max /(1 + P/P 0 ), where P 0 is the saturation power and I max the maximum count rate.  SIL is known to enhance both the collection efficiency and laser pumping efficiency. In Fig. S2 we evaluate the pumping enhancement by comparing the saturation powers of NV centers with SIL and planar surface. The experiment is conducted with 532 nm excitation at room temperature. We measure the fluorescence saturation curves of NV centers on another diamond with (111) surface. The NV centers are all oriented along the [111] direction perpendicular to diamond surface. The NV depth is obtained by multiplying the diamond refractive index (2.4) by the moving distance of nanoscanner from the NV center to the diamond surface. The saturation power shows a negative correlation with the NV depth (Fig. S2b). For a 2.4µm-deep NV center the saturation power is 424 µW, 2.3 times of that with SIL. This enhancement should also hold for 1064 nm. According to the results in Fig. 2g of main text, the ionization rate is proportional to the NIR power density. This effect can be compensated by directly increasing the NIR power for applications on planar diamond. Nano-pillar is an alternative to maintain high pumping efficiency for shallow NV centers.

SPIN STATE INITIALIZATION FIDELITY
To estimate the spin initialization fidelity, we prepare |0 (| ± 1 ) state by a E 1,2 (E y ) pulse and record the resonance fluorescence time trace under E y (E 1,2 ) illumination. We fit the results with double exponential decay curves and extract the initial and final equilibrium count rate. The initial count rate for |0 (| ± 1 ) state is 166.7 kctps (39.4 kctps). Note that final count rate of the | ± 1 result is higher than the |0 result. This is due to a higher background count rate from the E 1,2 laser (2.8 kctps) than from the E y laser (0.5 kctps). After subtracting the background, we can estimate a remaining population of 0.17±0.06 % in |0 state after E y pumping, and 0.06±0.15 % in | ± 1 state after E 1,2 pumping, giving state initialization fidelity of 99.83±0.06 % for | ± 1 state and 99.94±0.15 % for |0 state. In the main text, to initialize to |0 state, we apply a 20 µs E 1,2 pulse, corresponding to a 99.82% initialization fidelity. To initialize to | + 1 state, along with the E y laser pumping |0 state, we apply an additional microwave pulse to flip the population from | − 1 back to |0 state. After 200 µs E y and microwave pulse, we can estimate the remaining population in both |0 and | − 1 as 0.17%, giving 99.66% initialization fidelity into | + 1 state. The charge readout fidelity mainly depends on the photon count rate and charge lifetime. Resonance fluorescence count of NV − is proportional to spontaneous emission rate and fluorescence collection efficiency, while NV 0 count is mainly affected by background fluorescence. Here we assume a background fluorescence of 2.5 kctps, and optimize the readout window by balancing the effects of photon shot noise and charge state lifetime, to obtain the best non-demolition charge readout fidelity. The black star marked our current level of charge readout (fidelity 99.96%). For single-shot charge readout, higher fidelity could be achieved by prolonging the reading time.

MODEL FOR SPIN AND CHARGE DYNAMICS
In order to understand the charge dynamics observed in experiment, we consider a 7 level energy diagram relevant to the SCC process, as depicted in Fig.S3. The dynamics among different levels can be described by the following rate equation d dt P 0 = −Γ ex,0 P 0 + ΓP Ey + Γ isc,0 P singlet + Γ f lip,E1,2 P E1,2 d dt P +1 = −Γ ex,±1 P +1 + 0.5ΓP E1,2 + αΓ f lip,Ey P Ey + Γ isc,+1 P singlet where the spontaneous emission rate Γ = 77 MHz [3], the excitation rate Γ ex,0 is estimated directly according to the fluorescence saturation curve, Γ ex,0 = PL PLsat−PL Γ = 480 1157−480 * 77 ∼ 54 MHz. Γ isc,+1 + Γ isc,−1 + Γ isc,0 = 1/τ singlet ∼ 0.3 MHz is the singlet decay rate at 8K [4]. The branching ratio from the singlet state to the ground state is Γ isc,+1 : Γ isc,−1 : Γ isc,0 = 1:1:8 according to the literature [5]. With these fixed parameters, we further determine the rates Γ ex,0 , Γ isc,Ey and Γ f lip,Ey by fitting the fluorescence decay under E y excitation. Noted that since | ± 1 is not excited under E y illumination, the rates  Fig. 2f. The derived ionization rates are given in Table I. The branching factor α of the spin-flip process from |E y to | ± 1 (Fig. S5) is determined by fitting the pulsed SCC curve in Fig. 3b. The origin of this spin-flip asymmetry is currently unclear and may be related to strain. All the parameters derived with our model are given in Table II.

FIDELITY OF THE RESONANCE FLUORESCENCE METHOD
The fidelity of the resonance fluorescence method can be modeled by considering the photon number statistics. Here we consider a readout threshold of one photon, which is adopted in the main text. The fidelity of the m s = 0 state readout is the probability of collecting at least one photon (n 1), conditional on the bright initial state, i.e. F 0 = P (n 1|m s = 0) = 1 − P (n = 0|m s = 0). Similarly, the fidelity of the m s = ±1 state readout is F 1 = P (n = 0|m s = ±1).
If the count rate is constant, i.e. spin flip is ignored, in a time duration t, the collected photon number for both the m s = 0 and m s = ±1 states yield a Poisson distribution. The probability of collecting zero photon during time t for m s = 0 and m s = ±1 state are given by Q 0,1 (t) = e −a0,1t , with a 0 and a 1 denoting the photon count rate of m s = 0 and m s = ±1 state respectively. Note that a 0 corresponds to the initial count rate of the PL decay curve (Fig.1d of the main text). The spin flip process is characterized by the PL decay rate γ, which shouldn't be confused with the spin-flip rate Γ f lip . The spin-flip rate Γ f lip characterizes the intrinsic spin-flip process, whereas the PL decay rate γ depends on both the spin-flip rate and the excitation strength. The dependence of decay rate on the spin-flip rate and the excitation strength can be inferred from the model given in the previous section. After an excitation of duration t, the probability of the spin remaining not flipped is given by P 0 (t) = e −γt . In the whole readout time window t, for the m s = 0 state, there is a finite probability of spin flipping, hence P (n = 0|m s = 0) consists of two parts: (i) the spin is not flipped during time t, and (ii) the spin is flipped at time τ ∈ [0, t]. This leads to For the dark state, the spin-flip can be ignored, giving FIG. S6. Optimal fidelity of the resonance fluorescence method. The orange line corresponds to the spin-flip rate observed in this work. The blue line corresponds to a previously reported NV center with a lower spin-flip rate [6]. The observed saturation counts and readout fidelity for both works are marked with asterisks.
The average fidelity is given by the average of bright and dark state readout fidelity, i.e. F avg = (F 0 + F 1 )/2. F avg is optimized at readout window The optimal fidelity is To compare optimal fidelity under different saturation counts and spin-flip rate, we take the NV centers in Fig. 4b  . We also assume that, for both NV centers, the counts in m s = ±1 state is a 1 = 1 Kcps. The optimal fidelity of the resonance fluorescence method is calculated using Eq. 1 and depicted in Fig. S6.
To have a better understanding of Eq. 1, we take the limit of F opt as a 1 → 0, since in our experiments, both a 0 and γ is in the order of MHz, while a 1 is in the order of kHz. This gives lim a1→0 F opt = 1 2 1 + a 0 a 0 + γ . (2) Note that the only independent parameter here is the ratio between a 0 and γ, hence we introduce n 0 = a 0 /γ, which gives the average total photon number of the m s = 0 state. Rewriting Eq. 2 gives This clearly shows that, for the resonance fluorescence method, the optimal fidelity is limited by n 0 , which is essentially determined by the saturation counts of the NV center and the intrinsic spin-flip rate Γ f lip .