Robust All-Optical Single-Shot Readout of NV Centers in Diamond
Dominik M. Irber, Francesco Poggiali, Fei Kong, Michael Kieschnick, Tobias Lühmann, Damian Kwiatkowski, Jan Meijer, Jiangfeng Du, Fazhan Shi, Friedemann Reinhard
11 Robust All-Optical Single-Shot Readout of NV Centers in Diamond
Dominik M. Irber , Francesco Poggiali , Fei Kong , Michael Kieschnick , Tobias Lühmann , Damian Kwiatkowski , Jan Meijer , Jiangfeng Du , Fazhan Shi , Friedemann Reinhard TU München, Walter Schottky Institut and Physik-Department, Am Coulombwall 4, 85748 München, Germany CAS Key Laboratory of Microscale Magnetic Resonance & Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China Applied Quantum Systems, Felix-Bloch Institute for Solid-State Physics, University Leipzig, Linnéstraße 5, 04103 Leipzig, Germany Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland
High-fidelity projective readout of a qubit’s state in a single experimental repetition is a prerequisite for various quantum protocols of sensing and computing . Achieving single-shot readout is challenging for solid-state qubits. For Nitrogen-Vacancy (NV) centers in diamond, it has been realized using nuclear memories or resonant excitation at cryogenic temperature . All of these existing approaches have stringent experimental demands. In particular, they require a high efficiency of photon collection, such as immersion optics or all-diamond micro-optics. For some of the most relevant applications, such as shallow implanted NV centers in a cryogenic environment, these tools are unavailable. Here we demonstrate an all-optical spin readout scheme that achieves single-shot fidelity even if photon collection is poor (delivering less than 10 clicks/second). The scheme is based on spin-dependent resonant excitation at cryogenic temperature combined with spin-to-charge conversion , mapping the fragile electron spin states to the stable charge states. We prove this technique to work on shallow implanted NV centers as they are required for sensing and scalable NV-based quantum registers . Much of the popularity of NV – centers in diamond is owing to the fact that the readout of its electron spin is straightforward, since fluorescence intensity correlates with the spin state . However, this simple approach is highly inefficient because the relative contrast between the spin states is short-lived (approx. 250 ns) and low (about 30 %) . This corresponds to a single-shot signal-to-noise ratio (SNR) of 0.05 (< 0.01) for a count-rate of 200 kcps (1 kcps). Thus, averaging over several hundred to several ten thousand of experimental repetitions is necessary to readout the spin state with an SNR of 1. One option to increase the single-shot SNR is spin-to-charge conversion (SCC) . This readout approach maps the fragile spin state to the more robust charge state of the NV center, which can be optically readout with close to 100 % fidelity even at room temperature . This mapping is typically achieved by first shelving the spin m s = |±1 ⟩ population to the meta-stable singlet state of the NV – center and subsequently ionizing the NV – center out of the triplet or the singlet state during the lifetime of the latter. So far, SCC has reached readout fidelities of up to 67%, limited by non-deterministic shelving to and storing in the singlet state. More sophisticated schemes for spin readout have achieved single-shot readout, i.e. a single-shot SNR > 1. A first method exploits repetitive readout from a nearby nuclear ancilla qubit . This method requires a strong and carefully aligned magnetic field, and efficient photon collection for readout to succeed within the lifetime of the nuclear qubit. A second scheme consists in tuning a narrow-linewidth laser in resonance to a cycling transition in the low-temperature excitation spectrum of the NV – center . In this configuration, the NV – is spin-selectively excited and thus producing fluorescence only if its spin state matches the used optical transition. This gives a high contrast signal for a finite time, limited by spin depolarization due to laser illumination. Therefore, the scheme requires efficient photon acquisition by means of all-diamond micro-optics, and is in particular not available for NV centers close to a planar sample surface. Fig. 1: Main idea of the readout scheme. (a) Energy levels (simplified) of an NV – center in diamond. Initially, the NV – center is in its (optical) ground state with spin state either |0 ⟩ or |±1 ⟩ . If the spin state is |0 ⟩ , a gated laser tuned to a |0 ⟩ transition (637 nm; light red) can excite the NV – center, while spin |1 ⟩ is protected against excitation. A second gated high-power laser (642 nm; dark red) can ionize from the excited state. (b) Schematic of the setup. Three individually gated lasers can illuminate the sample, which is mounted in a flow cryostat. In addition, the sample can be driven by two microwave (MW) frequencies. (c) Final pulse sequence. Red, dark red and green correspond to gated 637 nm resonant, 642 nm and 517 nm laser excitation. Blue refers to MW drive. During photon acquisition ( “APD”; rose shade) for postselection and final data acquisition, cw MW excitation at both ground state transitions is added to constantly mix the spin state during charge-state readout. (d) Lower panel: Photoluminescence excitation (PLE) spectrum of the deep NV – center that is also used for Fig. 2 and Fig. 3. Detuning is denoted from 637.20 nm. Off-axial strain is estimated to be 1.7 GHz The inset shows the used pulsed sequence. Upper panel: Simulated spectrum according to Doherty et al . Here, we present a single-shot readout scheme that eliminates the need for sophisticated optics. The key of our approach is spin-to-charge conversion at cryogenic temperature, where resonant excitation enables both high spin-selectivity of SCC and efficient readout of the charge state by poor collection optics. In detail, our protocol employs resonant excitation to only excite the NV – if it’s spin state matches the used optical transition, typically a spin |0 ⟩ transition. Simultaneous illumination with a high-power 642 nm laser ionizes the NV – from the excited state, while not causing internal excitation dynamics (Fig. 1a). Doing so, we lift the fidelity of SCC well above a single-shot SNR of 1 for a natural NV center microns deep in the diamond (‘deep NV’), and to the single-shot threshold for a shallow-implanted NV center closer than 100 nm to the diamond surface. The technique also promises to be robust against strong misaligned background magnetic fields. All measurements were performed in a homebuilt confocal microscope. The sample is in a Helium flow cryostat and can be illuminated through an air objective with numerical aperture of 0.95 simultaneously with three independently gateable lasers: a narrow-band red laser tuned to a strong cycling transition starting from spin state |0 ⟩ ( ‘resonant laser‘) ; a strong red diode laser for photoionization; and a green diode laser for initialization of the charge and (in some experiments) the spin (Fig. 1b). Besides, the NV – center can be excited by two gateable microwave (MW) sources, tuned to both directly allowed spin transitions within the ground state. A static magnetic field of ~1 mT is applied, which is not aligned along the NV axis (see supplementary information). These tools implement the final protocol (Fig 1c). Its most crucial components are spin readout by (1) a highly spin-selective photoionization step (‘spin-dep. ionization’) implemented jointly by the resonant and the ionization laser and (2) low-power detection of the charge state by the resonant laser (‘readout’), which is made agnostic to the spin state by a strong simultaneous microwave drive ( T Rabi = ~1 µs). Initialization of the charge state (‘charge init.’) is performed by the green laser and confirmed by a spin-agnostic probe for later postselection. The spin state is initialized in |+1 ⟩ by repeated resonant depletion of state |0 ⟩ followed by emptying of the |-1 ⟩ state by means of a microwave pulse. We first demonstrate the protocol on a deep natural NV center. At cryogenic temperatures line narrowing allows different spin states to be individually addressed and the NV – excited state reveals 6 sublevels. The spectrum is well described by the model of Doherty et al. (Fig. 1d upper panel) with a non-axial strain of 1.73GHz. See supplementary information for more details. Two of them have S z character , thus having allowed cycling transitions from ground state spin |0 ⟩ , one of which (-7GHz) serves as working transition for the red resonant laser. Driving the spin in state |0 ⟩ on this selected transition with 56 nW (0.08 P sat ; see supplementary information for saturation curves) induces fluorescence, which decays to almost zero within 20 μs (Fig. 2a, upper panel, dark blue curve), as the spin is pumped from spin |0 ⟩ to |±1 ⟩ due to spin mixing . During these 20 μs, we collect 0.17 photons on average. This low number compared to Robledo et al . is due to the fact that we do not use any photonic structures, and precludes direct single-shot readout of the spin by resonant fluorescence. This highly spin-selective fluorescence still enables benchmarking of the spin initialization. Using off-resonant excitation by a green laser for simultaneous charge and spin initialization, we obtain a mixture of |0 ⟩ :|+1 ⟩ :|-1 ⟩ = (70±1):(13±1):(16±1) percent, consistent with previous reports . The most effective way to improve spin initialization is to pump on an optical spin |±1 ⟩ transition. We decided for an experimentally simpler method; repeating optical depletion of the |0 ⟩ transition and a π-pulse on the |-1 ⟩ MW transition, which prepares the spin state |+1 ⟩ with improved purity ( |0 ⟩ :|+1 ⟩ :|-1 ⟩ = (0±1):(88±2):(12±2) percent; Fig. 2a lower panel). See supplementary information for more details on the spin initialization. We also use the resonant laser to read out the charge state, which is in contrast to the SCC publications so far, which used an orange laser for that purpose. We counteract spin depletion by simultaneously applying cw MW excitation at both ground state MW transitions to constantly mix spin population and in turn re-establish some population in |0 ⟩ . The charge state is stable under this combined excitation. Pumping on the transition with 13 nW (<0.02 P sat ) plus cw MW, the NV – gets ionized after a second timescale (Fig. 2b). This can be seen as a sudden decrease in count rate to almost zero, because NV has a higher-energy separation between ground and excited state, and is in turn protected against excitation by 637 nm. With the final readout power (56 nW, 0.08 P sat ), the charge state is stable for more than 10 ms (Fig. 2c). The photon statistics during a 1 ms readout pulse is presented in Fig. 2d. It displays clearly separated distributions for NV – and NV events. In the final readout, events with ≥3 clicks were assigned to be NV – . NV – events have been produced by initialization using the green laser at 1.4 mW for 2 μs, initializing into the negative NV – charge state in (46±1)% of repetitions. Post-selecting (≥6 clicks within 500 μs) on the charge state after the green illumination, as shown in Fig. 1c, improves initialization to NV – to (99.7±0.7)%. NV events have been produced by first initializing NV – as described, and appending a 20 µs long ionization pulse of 637 nm plus 642 nm, with cw MW added after 5 μs. Importantly, the high stability of the charge state under resonant excitation enables charge readout with near-perfect ((98.1±0.5)%) fidelity using inefficient collection optics. The heart of our readout protocol is the spin-dependent ionization, which is a two-photon process. The second photon is provided by the strong (17 mW) red laser, red-detuned (642 nm) against the NV – zero phonon line (ZPL). It ionizes from the E excited state on a fast (1 μs) time scale, but its energy is by itself insufficient to drive excitation into the E state. Besides, it causes negligible stimulated emission back into the A state . NIR lasers fulfill these criteria, too, however Fig. 2: Spin & charge stability of the deep NV center. (a) Average fluorescence under optical pumping with a 637 nm laser being resonant to the main |0 ⟩ transition. In the upper panel, the NV center was initialized to |0 ⟩ with (85.2±0.6)% fidelity by a simple green laser pulse, followed by no MW excitation (dark blue) or by a π -pulse on either the |0 ⟩ ↔ |+1 ⟩ or |0 ⟩ ↔ | -1 ⟩ transition (light blue and green curve). The lower panel displays the same measurement after initializing to |+1 ⟩ with (94.0±0.9)% fidelity using the explicit spin initialization protocol presented in Fig. 1c. The insets are sketches of the sequence used for the upper and lower panel. (b) Charge state stability under excitation with the 637 nm laser plus cw MW excitation at both ground state MW transitions. This is alternated with 1 s of green repumping. This data is not averaged, but just one repetition. (c) Average fluorescence of the NV being preferentially in the charge states NV – (upper curve) or NV (lower curve). This data (without using postselection) was taken simultaneously with the data presented in part d. (d) Distribution of the numbers of fluorescence photons that were detected during 1 ms of readout. The upper panel displays the distribution directly after the charge-state initialization to NV – , as shown in Fig. 1c, while the lower part includes a strong ionization pulse in between initialization and readout for conversion to NV . The insets are a zoom-in to the few-photon range. we observed much less efficient ionization at 980 nm (see supplementary information). Ionization is made spin-selective by simultaneously applying the resonant laser, which provides the first photon for excitation into the E state. As this laser only excites spin |0 ⟩ , the spin population in the excited state and, hence, ionization is highly deterministic for spectrally well-separated transitions. Fig. 3a shows the averaged fluorescence for charge state readouts after the NV – center was prepared in spin |0 ⟩ or spin |+1 ⟩ and spin-selectively ionized. Fluorescence is higher in the latter case, because spin |+1 ⟩ is protected against resonant excitation and in turn against ionization. We optimize the ionization time for highest fluorescence contrast between the two preparations (resulting in 2 μs and contrast 7.5 kcps vs 1.9 kcps). Fig. 3b is the statistics of photon counts for the whole initialization-ionization-readout protocol (Fig. 3d) for both spin preparations. The measured end-to-end fidelity of the scheme is (88.5±0.5)%. Correcting for imperfect spin initialization (fidelity of (94.0±0.9)%) and error of the MW π-pulse, the readout fidelity (comprising only the ionization and charge detection steps) is (96.4±2.2)%, which corresponds to a single-shot SNR of 3.5±1.2 (see supplementary information). Reducing the readout time for the final charge state from 1 ms to 100 μs, the end-to-end fidelity is only slightly degraded from (88.5±0.5)% to (82.3±0.5)%. Importantly, the same performance could be achieved under strongly reduced photon flux (0.5 kcps instead of 50 kcps), if a 10 ms readout window is used (see supplementary information). For a long (>10 ms) sensing sequence, this affords a speedup of 10 to 10 (for 50 kcps and 0.5 kcps) over standard readout. For a short sequence in typical (50 kcps) conditions, our method is still as fast as the standard technique (supplementary information). As an example, Fig. 3c shows Rabi nutations measured with 144 repetitions in 11 s, corresponding to a speed-up factor of one. Our method is applicable to “shallow” NV centers less than 100 nm close to the diamond surface. Fig. 4a-d present data recorded on a ~70 nm deep center (110 keV CN – implant; ). Spectral lines are inhomogeneously broadened to (0.43±0.02) GHz due to spectral diffusion (see supplementary information). We compensate for this challenge by increasing the resonant red laser power to 240 nW, which mainly compromises spin initialization fidelity (Fig. 4b). Without postselecting on the charge state, the useable contrast is best for 5 μs ionization time and yields 0.6 kcps and 1.5 kcps for spin |0 ⟩ and |+1 ⟩ , respectively (Fig. 4c). Including postselection on the charge state, the end-to-end single-shot fidelity as measured in Fig. 4d is (67.1±0.9)%. Correcting for the non-perfect spin initialization and π-pulse results in a fidelity of (78.6±2.5)%, corresponding to a single-shot SNR of 0.99±0.13. We note that sub-GHz optical linewidths have been reported for comparable implanted NV centers as shallow as 10 nm The protocol also promises to be resilient to strong misaligned magnetic fields, since it does not make use of spin-selective intersystem crossing into the A singlet state. It only requires a well-separated spin-selective transition, which can be found even in strong misaligned bias fields (Fig. 4e). Note that we expect the protocol to work with poorly cycling transitions by using a stronger ionization laser.
Fig. 3: Spin-dependent ionization. (a) Average fluorescence of the charge-state readout after charge initialization (without postselection), spin initialization according to Fig. 1c and spin dependent ionization with varying time. For parts b and c, 2 μs was used. (b) Distribution of the number of fluorescence photons that were detected during 1 ms of readout, after preparing the NV – in either spin |+1 ⟩ or |0 ⟩ and spin-dependently ionizing it for 2 μs. The as -measured fidelity is (88.5±0.5)%. The insets are a zoom-in to the few-photon range with full y scale, and sketches of the used sequences. (c) Rabi oscillation measured accordingly to part b. The y-axis is the fraction of experimental repetitions with detected photon number during readout above threshold. The data was measured within about 11 s. (d) Sequence as used for part b. The color-code is explained in Fig. 1. Fig. 4: Data for a representative shallow implanted NV center (110 keV CN – ). (a) PLE spectra as measured (lower panel) and simulated similarly to Fig. 1d (upper panel). Off-axial strain is estimated to be 12.6 GHz. The detuning is denoted from 637.20 nm. The 637 nm laser was tuned in resonance with the higher-energy transition at 19 GHz for all following measurements. (b) Average fluorescence under optical pumping on the selected transition after explicit spin initialization according to Fig. 1c, as presented in Fig. 2a in the lower panel for the deep NV center. The NV – center was initialized to |+1 ⟩ with (85.0±0.9)% fidelity. The background after 20 μs stems from finite excitation of the close-by |±1 ⟩ transition. (c) Similar to Fig. 3a. Average fluorescence of the charge-state readout after charge initialization (without postselection), spin initialization in |+1 ⟩ or |0 ⟩ according to Fig. 1c, and spin dependent ionization with varying ionization time. For part d, an ionization time of 5 μs was used. (d) Similar to Fig. 3b. Distribution of the number of fluorescence photons for 5 ms of readout. The as-measured fidelity is (67.1±0.9)%. (e) Simulation of the dipole transitions between optical ground and excited state for all three spin projection numbers for fixed strain-splitting of 3.5GHz and varying magnetic field 45° off-axis. The transitions highlighted in yellow are the most pronounced spin |0 ⟩ transitions. The coloring is a measure for the transition strength. There are several parameter ranges with spectrally well isolated transitions. Table 1: Overview of the initialization and readout metrics for both NV centers presented in this letter. deep natural NV center shallow implanted NV center NV – fraction (%) 99.7±0.7 92.9±1.3 spin init. fidelity (%) 94.0±0.9 85.0±0.9 MW error (%) 5.6±0.1 5.1±0.6 end-to-end fidelity (%) 88.5±0.5 67.1±0.9 readout fidelity (%) 96.4±2.2 78.6±2.5 We have pushed the fidelity of spin-to-charge conversion into the single-shot regime, by combining it with resonant excitation at cryogenic temperature. The resulting protocol can operate even on shallow implanted NV centers and eliminates the need for any optimized collection optics. We achieve a single-shot SNR of 3.5 and 0.99 on a deep and shallow center respectively, which provides a speedup in the range of 10 over standard readout. As its most important consequence, this technique will enable sensing experiments using long (ms) protocols. These are within the coherence time of shallow NV centers , but are currently precluded by acquisition speed. 1 ms of sensing time would enable coherent coupling to a single electron spin at 50 nm distance. In sensing, this would cover the entire thickness of a biological cryoslice , in computing it could enable coupling in scalable arrays of NV centers . The protocol is compatible with electric readout of the NV – spin state ; in combination with a single-electron transistor , single-shot electric readout might be possible. Our method could also enable single-shot readout of more challenging spin qubits, in particular in Silicon Carbide, where poor photon count rates currently hamper work for some centers with otherwise promising spin properties . Methods
The measurements were performed in a home-built confocal microscope, with the sample being in a CryoVac
KONTI flow cryostat using liquid Helium. Inside the vacuum chamber are a movable permanent magnet and a movable air objective (Nikon Plan Apo 40x NA0.95) to illuminate the sample and to collect fluorescence. The fluorescence was separated from the laser illumination with a 650 nm longpass dichroic mirror. Residual laser light was removed with a 650 nm longpass filter and a shortpass filters (800 nm; just relevant for the ionization with NIR, see SI). Photons were detected with an avalanche photo diode (APD). Three individually gated lasers are combined to a single excitation path, so that the sample can be illuminated simultaneously by all of them: A green 517 nm fiber-pigtailed laser diode (Thorlabs LP520-SF15) driven by a PicoLAS LDP-V 03-100 UF3, “cleaned-up” with a 540nm shortpass and combined to the common laser path with a 550 nm longpass dichroic mirror; a red 642 nm fiber-pigtailed laser diode (Thorlabs LP642-SF20) driven by an iC Haus iC-NZN and combined to the external cavity laser's path with a non-polarizing 90:10 beam splitter; a red external cavity diode laser (New Focus TLB-6704) stabilized with a HighFinesse WS6-600 wavemeter (absolute accuracy ±600 MHz) and gated by two AOMs in series. All laser beams were expanded to ~10 mm, which is approx. the back aperture of the used objective. For each beam, we can control the lateral alignment as well as the collimation. Two FPGAs were used to control all short-timescale pulses and to register the APD events. The deep NV center and the shallow implanted NV center are in two different pieces of diamond. Both diamonds are electronic grade from Element6 and have some spots where CN – molecules were implanted. Before implanting they were cleaned with a 1:1:1 mixture of sulfuric:nitric:perchloric acid. Afterwards, both diamonds were annealed at 900˚C for 3h and cleaned again with the 3-acid mixture. The diamond with the shallow implanted NV presented in the main text was additionally annealed for a second time at 1200˚C for 2h. Before measurements, both diamonds were treated in an Oxygen plasma. To remove high fluorescence in the surrounding of shallow NV centers, we illuminated the region with a high-power (~50 mW) green 532 nm laser after cooling down. Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
Acknowledgements
We thank Marcus Doherty for helpful discussion. This work has received support from the Deutsche Forschungsgemeinschaft (DFG) under grants RE3606/1-1, RE3606/2-1 and RE3606/3-1 and from the National Natural Science Foundation of China (NSFC) under grants 11761131011, 81788101, 91636217, 11722544.
Author contributions
D.M.I. and F.R. designed the experiment. D.M.I. built the setup. D.M.I., F.P. and F.K. conducted the experiment. D.M.I. and F.P. analyzed the data with support from F.R., F.P., F.K., F.S. and J.D. D.M.I. developed all numerical models with support from D.K. and F.P. M.K.. T.L. and J.M. prepared the sample. D.M.I., F.P. and F.R. wrote the manuscript. All authors commented on the manuscript.
Competing Interests
The authors declare no competing interests.
Additional Information
Supplementary information is available for this paper online. Correspondence and requests for materials should be addressed to F.R.
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Figure S.1: Hyperfine ODRM transitions measured at room temperature. Grey scatter and red linesare experimental data and Lorentzian fit, respectively.Standard pulsed Optically Detected Magnetic Resonance (ODMR) techniques were employed forthe evaluation of bias magnetic field strength and amplitude. By reducing the MW power used toexcite the NV − spin with a π -pulse, it is possible to selectively address the hyperfine energy levelsresulting from the interaction between the NV − electron spin and the N nuclear spin I = 1 thatforms the color center itself. The overall ground state Hamiltonian ˆ H g is:ˆ H g = ¯ h (cid:16) D g ˆ S z + γ e ˆ S · (cid:126)B + ˆ S ˆ A ˆ I + Q g ˆ I z + γ n ˆ I · (cid:126)B (cid:17) . The first term in the above Hamiltonian is dominant and can represent the system energy in theabsence of external fields, i.e., is the zero-field term, where ¯ h is the reduced Planck constant. ˆ S is theelectronic spin operator, and D g (cid:39) .
87 GHz the axial zero-field parameter. In the following terms, (cid:126)B is the external bias magnetic field, γ e (cid:39) π ×
28 MHz / mT and γ n (cid:39) π × − .
08 kHz / mT the electronicand nuclear gyromagnetic ratio, respectively, Q = − .
945 MHz the nuclear quadrupole interaction, and A the hyperfine spin tensor with axial A (cid:107) = − .
16 MHz and orthogonal A ⊥ = − .
62 MHz components.By solving the eigenvalue equation and considering the hyperfine transition energies as solution ofthe system, it is possible to determine the magnetic field amplitude B and orientation θ with respect tothe NV quantization axis, the only unknown parameters. In this way, we measured B = (0 . ± .
1) mTand θ = (39 ± (cid:176) for the deep NV center used in the main text. We want to stress that this was anon-optimized situation and a first indication that the field alignment requirements are not strict.For the shallow NV center also studied in the main text, pulsed ODMR measurements were not ableto solve a more complex hyperfine structure, which highlight interactions with multiple nuclear spinsrelated to other C or N impurities. Nevertheless, since both the rough magnet position and the | (cid:105)→ |± (cid:105) transitions frequency range correspond to the case related to the deep NV, we can assume B ≤ .2 Simulation of the Excited State Structure and OpticalTransitions d b , E | | E * + ] Figure S.2: Excited state structure of an NV − center at low temperature depending on the non-axialstrain. The magnetic flux density is 20 MHz and 45 (cid:176) misaligned. The coloring is just for the readersconvenience.We simulated the NV − excited state structure and the related optical transitions between groundand excited state according to Doherty et al. .We solve the Hamiltonian as follows for 14 energy levels: H = V opt + V ss + V so + ˆ O E,x (0 , ξ ⊥ ) + ˆ S z + ˆ S x + ˆ S y Here, V opt is a diagonal matrix with the rough energy differences without considering structure withinthe ground and excited state. V ss and V so are the spin-spin and spin-orbit interaction potential,according to Tab. 3 and Tab. 2 in Doherty et al. . The values for the entries are taken from Tab. 4 inDoherty et al. . ˆ O E,x is an orbital operator according to Tab. A.4 in Doherty et al. , where we setthe two different entries O a,x → √ (cid:104) a || V E || e (cid:105) to 0 and O b,x → √ (cid:104) e || V E || e (cid:105) to the non-axial strain ξ ⊥ . ˆ S x,y,z are the components of the total spin operator according to Tab. A.5 in Doherty et al. . Weset the entries S i to the component of the magnetic flux density (cid:126)B in the “respective” direction. B z is the component parallel to the NV center axis and determined as half the splitting from an ODMRspectrum.The transition matrix element is a measure for the transition strength between the inital state (cid:126)i and the final state (cid:126)f . M fi = (cid:12)(cid:12)(cid:12)(cid:68) (cid:126)f (cid:13)(cid:13)(cid:13) ˆ O E,x (1 ,
0) + ˆ O E,y (1 , (cid:13)(cid:13)(cid:13) (cid:126)i (cid:69)(cid:12)(cid:12)(cid:12) Similarly to ˆ O E,x , ˆ O E,y is an orbital operator according to Tab. A.4 in Doherty et al. , where we setthe two entries O a,y → √ (cid:104) a || V E || e (cid:105) to 1 and O a,y → √ (cid:104) e || V E || e (cid:105) to 0.For creating the simulated PLE spectra presented in Fig. 1d and 4a of the main text, we assumedLorentzian broadening, so that the expected fluorescence is f ( E ) = (cid:88) l A √ πγ · γ γ + ( E − E l ) We set the amplitude A to M fi and the FWHM γ to M fi / E i is the energy of the differenttransitions, i.e. ( E f − E i ). 2n literature, the lower-energy spin | (cid:105) transition ( E y branch) is reported to have more spin mixingwith the |± (cid:105) states and is accordingly less cycling. This statement is typically made with theexception of low strain, which is the case of the used deep NV center, where we measured the lower-energy transition to be more stable. This is in agreement with the simulation, where this transition iswell separated in energy as well as the related excited state level (Fig. S.2 green line).
S.3 Saturation Curves (a) (b) (c)(d) (e) (f) d ee p N V s h a ll o w N V green laser red laser (resonant) Figure S.3: Saturation curves for illuminating the deep (shallow implanted) NV center presented inthe main text with a green 517 nm laser (a,d) and a narrow-band laser tuned to an NV − spin | (cid:105) transition (b,e). Panels c and f are zoom-ins to the low-power regime of panels b and e, respectively.The red lines are fits to the data.Table S.1: Overview of the saturation powers and the related saturation fluorescence for the datapresented in Fig. S.3. f sat , g (kcps) I sat , g (mW) f sat , r (kcps) I sat , r ( (cid:181) W)deep NV 63 ± . ± .
04 58 ±
10 0 . ± . ± . ± .
07 20 ±
333 1 ± | (cid:105) transition (Fig. S.3b,c,e,f), we first initialized the NV centers charge and spin state (withoutpostselection). The actual illumination with the red laser is 100 ns short to acquire data without strongdepolarization effects.To determine the saturation power and fluorescence, we fitted the data with f = A I · I sat I + I sat . f sat = A · I sat .For the deep NV center, the saturation count rate under resonant illumination is similar to thesaturation value under green illumination. However, for the shallow implanted NV center, it differs a3ot, and we cannot observe a clear saturation behavior when illuminating resonantly. We attribute thesetwo effects to spectral diffusion of the shallow implanted NV − center’s optical transition; increasingthe laser power leads to power-broadening and in turn the laser line “hits” the optical transition moreoften. S.4 Spin Initialization
A major limitation concerning both quantum sensing and quantum computing with NV centers is theNV − spin state initialization. In this section, we present the spin dynamics under resonant opticalpumping on an optical transition. First, we exploit the depletion of the related spin state for improvingthe overall spin initialization. Second, based on these time average measurements, we modeled thespin dynamics with a rate equation that allows to extract the population of each spin state. S.4.1 Related Measurements d ee p N V c e n t e r s h a ll o w N V c e n t e r (a) (b) (c)(d) (e) (f) π π π [ ] π Figure S.4: Fluorescence during optical pumping on a spin | (cid:105) transition of the deep NV center (a-c)and the shallow NV center (d-f) used in the main text. Each panel shows four curves; either directlyafter optical initialization (dark blue), after a MW π -pulse on the | +1 (cid:105) and |− (cid:105) transition (light blueand green) as well as after two π -pulses on the | +1 (cid:105) transition (orange). In panels a and d, the spinwas initialized with a green laser pulse; in b and e, with a green laser pulse followed by a resonantpulse; and in c and f by the full spin initialization protocol. See the insets for the measurement pulsesequences. For better distinguishability, the measurement data are averaged over ten bins, with eachbin of 10 ns being the average of 500 000 (200 000) experimental repetitions for the deep (shallow) NVcenter. The red lines are fits to the non-averaged data, see Section S.4.2.Commonly in NV center research, the NV center’s charge and spin state is initialized into NV − with spin | (cid:105) by illuminating it off-resonantly with a green laser (typically 532 nm). According toDoherty et al. , “the degree of ground state optical spin- polarisation [into spin | (cid:105) ] is not consistentlyreported in the literature, with many different values ranging from 42%–96% reported”. Hopper et l. report a value of around 80 %.To determine the spin state distribution, we illuminate the NV center with the 637 nm laser at56 nW tuned into resonance with an optical transition with | (cid:105) character. Doing this directly afterinitialization, the fluorescence intensity correlates with the spin | (cid:105) population. To get the actual | (cid:105) fraction, the same measurement is necessary for the spin |± (cid:105) populations. To access them, we swapthe | (cid:105) population either with the | +1 (cid:105) or |− (cid:105) population by means of a MW π -pulse. As imperfect π -pulses can cause additional sources of errors, we also measure the fluoresce after two π -pulses on theMW | +1 (cid:105) transition. Fig. S.4a shows these four measurements for the deep NV center presented inthe main text. The red lines are fits to the curve to determine the actual values, as discussed below.Except for the π - π -pulse curve and the fits, these data are presented in Fig. 2a in the main text, too.Fig. S.4b displays the spin distribution after 20 (cid:181) s illumination with the resonant laser; that is thespin distribution after the first measurement without any MW excitation. Similarly to Fig. S.4a, thedifferent spin populations are measured by swapping the population to spin zero by MW π -pulses. Inaccordance with Robledo et al. , the first pulse mainly depleted the spin | (cid:105) transition and in turnspin |± (cid:105) is much higher populated.To exploit this for higher selectivity in spin initialization, after the first resonant pulse we swap theslightly lower-occupied |− (cid:105) population back to | (cid:105) by a MW π -pulse. Repeating this several times,more and more population accumulates in the other spin one state | +1 (cid:105) , as can be seen in Fig. S.4c.Fig. S.4d-f present the same measurements for the shallow NV center used in the main text. Thepronounced difference is the curves not decaying to zero but to a finite value. This stems fromsimultaneously also exciting an optical |± (cid:105) transition, which is spectrally close to the used optical | (cid:105) transition (compare Fig. 4a in the main text). This has implications on the used sequence for spinpolarization: As longer illumination will mix the spin populations more and more, we shortened eachoptical pumping pulse to 10 (cid:181) s. The resonant laser was operated at 170 nW, which is the power, wherejust a slight decrease in fluorescence happens during resonant illumination together with cw MW spinmixing, as used for the charge-state readout. S.4.2 Rate Equation Model
To determine the spin-state distribution and in turn the spin-initialization fidelity, we modeled the NVcenter dynamics as rate equations with five states. The states are 1) NV − | (cid:105) , 2) NV − | +1 (cid:105) , 3) NV − |− (cid:105) , 4) NV − singlet and 5) NV . n = n | (cid:105) n | +1 (cid:105) n |− (cid:105) n sing n NV n i is the time-average population of the NV center being in state i . At the beginning of each fit, thepopulations of the three different spin states add to one, while the probabilities to be in the singlet orin NV are set to 0.The dynamics while pumping on an optical transition with spin | (cid:105) character was modelled with5he transfer matrix as follows: T opt = p ts / p ts / p ts / p st , p st , p st , p ion p ion p ion To maintain probabilities, the empty diagonal entries are 1 minus the sum of the other elements in therespective column; e.g. the 5 th diagonal element is 1. p st , and p st , are the probabilities of having aninter-system crossing to the singlet in one time step, starting from spin | (cid:105) or |± (cid:105) , respectively. p ts is the probability of an inter-system crossing back to the triplet, where we assumed that of eventsend up in spin | (cid:105) and the other half is equally distributed to both |± (cid:105) states. Finally, p ion is theprobability of ionizing the NV − center to NV within one measurement time bin. We don’t includerecombination from NV to NV − because of the 637 nm laser’s energy per photon and its low intensityrender this process very unlikely.The excited state is excluded in the model, too, as it is implicitly described by the fluorescenceparameters f and f . These are used as link to fit the measured fluorescence decay curves, where thetotal fluorescence was modelled as follows for each time step: f ( t ) = n | (cid:105) ( t ) · f + n | +1 (cid:105) ( t ) · f + n |− (cid:105) ( t ) · f The MW π -pulses are modeled as follows (here for a π -pulse on the | +1 (cid:105) transition): T MW , +1 = E MW − E MW − E MW E MW As the overall fluorescence lowers after each optical pumping step (compare Fig. S.4a/d → b/e → c/f), we also include a “fluorLoss” parameter, which accounts for the reduced overall fluorescence inFig. S.4b,e (fluorLoss1) as well as d,f (fluorLoss6). We attribute this loss of fluorescence mainly toionization at the very first resonant illumination, when the spin | (cid:105) population is high. In turn, thereis a high population in the NV − excited state, which can get ionized comparably easy even by thelow-power 637 nm laser.Fig. S.5 shows all 12 measured fluorescence time traces under optical pumping. Tab. S.2 summarizesthe values from fitting all 12 curves simultaneously with the rate equation model described above. S.5 Charge Initialization
The NV charge state was initialized with a green 517 nm laser pulse of 2 (cid:181) s length at 1 . − spin state distribution according to fitting with the rate equation model described inSec. S.4.2 (upper two sub-tables) and global fitting parameters for all panels in Fig. S.5 together (thirdsub-table). In the upper two sub-tables, a-f refer to the panels in Fig. S.4 and S.5. The empty celldenote a lifetime that is several orders of magnitude larger, in the range of hours. Lifetimes (1/ e ) werecalculated as T i = − binwidth / ln (1 − p i ). deep NVa b c n | (cid:105) . ± . . ± . . ± . n | +1 (cid:105) . ± . . ± . . ± . n |− (cid:105) . ± . . ± . . ± . F spin . ± . . ± . . ± . n | (cid:105) . ± . . ± . . ± . n | +1 (cid:105) . ± . . ± . . ± . n |− (cid:105) . ± . . ± . . ± . F spin . ± . . ± . . ± . E MW (%) 5 . ± . . ± . f (kcps) 31 . ± . . ± . f (kcps) 0 . ± . . ± . T st , ( (cid:181) s) 4 . ± . . ± . T st , (ms) 0 . ± . . ± . − T ts ( (cid:181) s) 1 . ± .
09 11 . ± . T ion (ms) 0 . ± . . ± . . ± . . ± . . ± . R .
944 0 . >
17 mW for 1 (cid:181) s →
500 ns break → green 517 nm laser at 3 . (cid:181) s →
200 ns break → green517 nm laser at 3 . (cid:181) s (1 ms) directly after the initialization sequence for the deep (shallow implanted) NV centerand postselected on events with at least 6 (2) photons, which means a selection to about 37 % (22 %)of repetitions.To quantify the charge initialization and readout, we measured the photon count statistics for acharge readout both directly after the initialization and after a strong ionization pulse in between.7 a) (b) (c)(d) (e) (f) d ee p N V c e n t e r s h a ll o w N V c e n t e r Figure S.5: Same data than in Fig. S.4, including the same color code. Here each measurement bin isshown, as used for the fitting.This ionization pulse was 5 (cid:181) s of resonant laser together with the 642 nm laser, followed by 15 (cid:181) s withadditional cw MW to counteract depolarization. In this supplementary information, we present thedata taken for long photon acquisition time as this promises the least error due to charge readoutand in turn is the best estimate for the charge initialization to NV − . The resulting NV − and NV count statistics is presented in Fig. S.6, which includes fits with Poisson and Gauß distributions. Todistinguish between both charge states, we set a threshold that was determined by minimizing thesum of errors for both distributions, i.e. the percentage of events below (above) threshold for the NV − (NV ) distribution.For the deep NV center, the NV distribution for 5 ms readout duration can be fitted well ( R =0 .
999 96) with a Poison distribution centered around (0 . ± . − distributioncan be fitted ( R = 0 . − distribution has 0 .
25 % ofevents below threshold (5 photons), which is considered as NV fraction after the charge initialization.For the shallow NV center, the distributions were evaluated at the maximum readout time of 10 ms.Here, the fraction of the NV − distribution below threshold (4 photons) is 7 . distributioncan be fitted well with a Poissonian ( R = 0 . . ± . − distri-bution has to be fitted again with the sum of two Gaussians ( R = 0 . − ±
75) and (7 . ± .
6) photons with a standard deviation of (65 ±
19) and (5 . ± .
6) photons. Theamplitudes are (1 . ± .
6) and (1 . ± .
1) %.To determine the threshold for the spin-state distribution, we minimized the charge-state error asdescribed above. The only difference is that we used the same readout time than for the spin-statedistribution; compare Fig. 2d and 3b in the main text, which were both acquired with 1 ms of readouttime. 8 a) (b)(d)(c) d ee p N V s h a ll o w N V Figure S.6: Count statistics for the charge-state readout. For the deep NV center (a,b) the readouttime was 5 ms and for the shallow implanted NV center (c,d) it was 10 ms. The left column displaysthe distribution after the charge-state initialization, while the right column has a strong ionizationpulse included between initialization and readout.
S.6 Ionization with NIR (a) (b) (c)
Figure S.7: Fluorescence after ionization with different lasers and powers. The NV center was eithersubjected to no ionization (dark blue), to a combined pulse of the resonant + red + IR laser (lightblue), resonant + IR (green) and resonant + red (orange). The lasers were either (a) at their maximumpowers (red at 17 mW and IR at 33 mW), (b) at 17 mW and (c) at 10 mW.According to simple energy considerations, the second step for the NV − ionization might be possiblewith infrared (IR). The ionization energy for NV − , i.e. the energy difference between the NV − groundstate and the conduction band of diamond, was reported to be 2 .
60 eV . By just subtracting 1 .
95 eV(637 nm; zero-phonon line of NV − ) from the ionization energy, even a wavelength as long as 1900 nmmight serve as second photon for the ionization.To ionize a deep NV center, we compared using a red 642 nm and an IR 980 nm laser, which bothshould serve as source for the 2 nd photon. Both lasers are pigtailed laser diodes with single-mode fiber.Fig. S.7 presents measurements similar to Fig. 3a and 4c in the main text. We do see a reduction influorescence when illuminating the NV center simultaneously with the resonant laser (which provides9he first photon to excite the NV − ) and the IR laser (providing the second photon to ionize the NV − ).However, compared to the red 642 nm laser the ionization time needs to be more than one order ofmagnitude longer, indicating a much worse absorption cross section for 980 nm. We used the red642 nm laser as source for the second photon for all other data presented in this manuscript. S.7 Correcting the Readout Fidelity
The spin fidelity as measured in Fig. 3b and 4d of the main text is a measure of the end-to-endperformance of the used protocol. This includes the spin-dependent ionization and the readout stepitself, but also non-perfect charge and spin initialization as well as errors of the MW. These additionalerror sources due to initialization and MW are independent of the readout that is chosen and dependon the overall technical implementation. To correct for these effects and to get the actual fidelityrelated to just our readout scheme, we model the whole protocol as a multi-step process, see Fig. S.8. NV NV - NV P | - ⟩ P | + ⟩ chargeinit. spininit. MW spin-dep.ionization charge-statereadout E MW E MW E N V NV NV - NV NV highcountslowcounts E - E E E - E Figure S.8: Model of the whole protocol, consisting of charge and spin initialization, MW π pulse (orno MW), spin-dependent ionization and the final charge-state readout.Here, all variables are probabilities, which means that at any branching, the different leavingpaths add up to 1. E NV0 is the population in the neutral NV charge state after charge initialization(including postselecting on the charge state). P |− (cid:105) and P | +1 (cid:105) are the probabilities of having spin |− (cid:105) and | +1 (cid:105) , respectively. Accordingly, the probability of initializing into spin | (cid:105) is (1 − P |− (cid:105) − P | +1 (cid:105) ). E MW is the error of a MW π -pulse, meaning that with this probability, the π pulse does not flipthe spin. E is the error of not ionizing spin | (cid:105) despite having tuned the resonant laser to a | (cid:105) transition, and E the error of ionizing spin |± (cid:105) . The bold black arrows indicate the paths for perfectexperimental conditions. This includes the explicit spin initialization into | +1 (cid:105) , which is used for thespin-dependent count statistics presented in Fig. 3b (4d) for the deep (shallow) NV center.It is important to note, that the actual fidelity of our readout protocol just refers to the steps ‘spin-dependent ionization’ and ‘charge-state readout’, which are modeled together with the parameters E and E . On the other side, the errors that are used to determine the as-measured end-to-end fidelityare E , meas and E , meas . E , meas ( E , meas ) is the sum of all paths that end up in high (low) counts,while intending to prepare the spin in | (cid:105) ( | +1 (cid:105) ) by means of a MW π -pulse (no MW π -pulse) in step3 after the spin initialization. 10umming up all paths, we get a linear equation system with two unknown variables E and E . E , meas = (1 − E NV0 ) · P |− (cid:105) · (1 − E )+ (1 − E NV0 ) · P | +1 (cid:105) · E MW · (1 − E )+ (1 − E NV0 ) · P | +1 (cid:105) · (1 − E MW ) · E + (1 − E NV0 ) · (1 − P |− (cid:105) − P | +1 (cid:105) ) · (1 − E MW ) · (1 − E )+ (1 − E NV0 ) · (1 − P |− (cid:105) − P | +1 (cid:105) ) · E MW · E E , meas = (1 − E NV0 ) · P |− (cid:105) · E + (1 − E NV0 ) · P | +1 (cid:105) · E + (1 − E NV0 ) · (1 − P |− (cid:105) − P | +1 (cid:105) ) · (1 − E )+ E NV0
The variables and solutions for the deep and shallow NV center are summarized in Tab. S.3.Table S.3: Overview of the measured end-to-end spin fidelity F meas as well as the probabilities/errorsthat were estimated for the initialization with the explicit spin initialization and for the MW. Thesevalues were taken into account when solving the linear equation system with solutions E and E . Thecharge initialization was assumed to be perfect. Photons were collected for 1 ms (10 ms) for the deep(shallow) NV center. deep NV shallow NV E , meas (%) 17 . ± . . ± . E , meas (%) 5 . ± . . ± . F meas (%) 88 . ± . . ± . P |− (cid:105) (%) 12 . ± . . ± . P | +1 (cid:105) (%) 87 . ± . . ± . E MW (%) 5 . ± . . ± . E (%) 1 . ± . . ± . E (%) 5 . ± . . ± . F (%) 96 . ± . . ± . . ± . . ± . S.8 Photon Collection Statistics for Short Readout
The charge and spin count statistics (histograms) of the readout presented in Fig. 2d and 3b weremeasured with an actual readout duration of 10 ms. However, the software measured count statisticsnot only for the whole 10 ms, but also after the first 0 .
05, 0 .
1, 0 .
2, 0 .
5, 1, 2 and 5 ms. (In the maintext, they were evaluated for 1 ms.) In the first place, this allows to compare different readout timesfor otherwise exact same measurement parameters. In particular, it means that for the already quitelow saturation count rate of 50 kcps for this NV center in this setup we can speed up the measurementa lot by reducing the readout time to 100 (cid:181) s, while still maintaining an end-to-end fidelity of >
79 %,which corresponds to a single-shot SNR > . Tab. S.4 summarizes the end-to-end fidelities for readout11able S.4: End-to-end charge and spin fidelities of the deep NV center presented in the main text, aswell as fidelities corrected by spin-initialization and MW imperfections. The data for 1 ms refer to thedata presented in the main text. Note that the charge count statistics are not taken into account forcorrecting the spin fidelity—it is presented here to have a more clear estimate on the influence of thecharge readout fidelity. 10 ms 1 ms 100 (cid:181) s 50 (cid:181) s E NV0 w/o post-sel. (%) 46 . ± . . ± . . ± . . ± . E NV0 (%) 0 . ± . . ± . . ± . . ± . F charge , meas (%) 98 . ± . . ± . . ± . . ± . E , meas (%) 17 . ± . . ± . . ± . . ± . E , meas (%) 5 . ± . . ± . . ± . . ± . F meas (%) 88 . ± . . ± . . ± . . ± . E (%) 1 . ± . . ± . . ± . . ± . E (%) 5 . ± . . ± . . ± . . ± . F (%) 96 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . r ea dou t μ s r ea dou t μ s (a)(c) (d)(b) Figure S.9: Count statistics for 100 (cid:181) s and 50 (cid:181) s with the deep NV center presented in the main text.Parts a,c (b,d) were measured simultaneously with the data presented in Fig. 2d (3b) in the main text.12urations spanning two orders of magnitude and Fig. S.9 presents some related count statistics.In the second place, this simultaneous acquisition of count statistics for different readout timesallows to mimic the situation for poor collection optics: Evaluating just the first 100 (cid:181) s of readoutstill gives a good directly measured end-to-end fidelity of 82 . th the detectedfluorescence count rate and taking the count statistics for the whole 10 ms. With the NV center usedfor taking these data having a saturation count rate of 50 kcps, a single-shot readout would be possibleeven for a saturation count rate as low as 500 clicks per second. S.9 Speed-Up Factor V H Q V L Q J W L P H P V V S H H G X S Figure S.10: Estimated speed-up when using the readout protocol presented in this manuscript incomparison with conventional fluorescence-based readout.To determine the speed-up in comparison to the conventional off-resonant readout with a greenlaser, we estimated the time necessary to get an (average) SNR larger than one. For the single-shotprotocol it is always only one repetition with length of 1 . √ N · SNR, where N is the number of repetitions. Perrepetition, we estimated 1 . (cid:181) s plus the actual sensing sequence.Fig. S.10 shows the speed-up as function of the sensing sequence length. We assumed that theconventional readout can take advantage of the full saturation countrate of 50 kcps with a fluorescencecontrast of 30 % for 250 ns between the different spin states. Note that the speed-up factor is already2 for a zero-length sensing sequence. S.10 Spectral Diffusion
To determine to which extent the PLE linewidth of the shallow implanted NV used in the main text islimited by spectral diffusion, we fitted the used | (cid:105) transition both with a Gaussian and a Lorentziancurve (Fig. S.11). While the Lorentzian fit has an R of 0 .
69, the Gaussian fit performs better withan R of 0 .
87. This indicates that inhomogeneous broadening due to spectral diffusion is the mainreason for broadening, with a Gaussian FWHM of (0 . ± .
02) GHz. Note that the initializationprocedure for this measurement was different with green illumination for 500 ns at 1 . (cid:181) s) and13 O D V H U G H W X Q L Q J * + ] I O X R U N F S V G D W D * D X V V / R U H Q W ] Figure S.11: Section of the PLE spectrum presented in Fig. 4a in the main text. The used opticaltransition was fitted both with a Gaussian and Lorentzian curve in this section.stronger (3 . >
17 mW red pulse for 1 (cid:181) s. Thus, the inhomogeneouslybroadened linewidth in the relevant measurements is even broader.
S.11 SNR and Fidelity Calculation
The numbers for the single-shot SNR and the fidelity were calculated according to Hopper et al. . Forthe common off-resonant readout with a green laser, the single-shot SNR was estimated asSNR = ph − . · ph √ ph + 0 . · ph where ph = f sat ·
250 ns is the photon number per readout repetition and f sat is the saturation countrate when illuminating the NV center with a green laser. We assumed the contrast between spin statesto be 0 . − E − E (1 − E ) · E + (1 − E ) · E . References
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