Spin-photon correlations from a Purcell-enhanced diamond nitrogen-vacancy center coupled to an open microcavity

Domain: physics
Published: 2025-11-27
Editorial outcome: minor_revision

Abstract
An efficient interface between a spin qubit and single photons is a key enabling system for quantum science and technology. We report on a coherently controlled diamond nitrogen-vacancy center electron spin qubit that is optically interfaced with an open microcavity. Through Purcell enhancement and an asymmetric cavity design, we achieve efficient collection of resonant photons, while on-chip microwave lines allow for spin qubit control at a 10 MHz Rabi frequency. With the microcavity tuned to resonance with the nitrogen-vacancy center’s optical transition, we use excited state lifetime measurements to determine a Purcell factor of 7.3 ± 1.6. Upon pulsed resonant excitation, we find a coherent photon detection probability of 0.5% per pulse. Although this result is limited by the finite excitation probability, it already presents an order of magnitude improvement over the solid immersion lens devices used in previous quantum network demonstrations. Furthermore, we use resonant optical pulses to initialize and read out the electron spin. By combining the efficient interface with spin qubit control, we generate two-qubit and three-qubit spin-photon states and measure heralded Z-basis correlations between the photonic time-bin qubits and the spin qubit.

Introduction
Optically active solid-state qubits are promising physical systems for quantum networking. Among several candidates, the nitrogen-vacancy (NV) center in diamond is one of the most-studied, with the realization of qubit teleportation across a multi-node quantum network and metropolitan-scale heralded entanglement. Furthermore, efficient quantum frequency conversion to telecom wavelength and hybrid entanglement of photons and nuclear spins have been demonstrated. The NV center offers excellent spin qubit properties and the nitrogen nuclear spin together with nearby carbon-13 spins can function as a multi-qubit register to store and process quantum states with high fidelities. The main challenges for the generation of spin-photon states with the NV center is the relatively low zero-phonon line (ZPL) emission of ~3% and the reduced optical coherence in the presence of charge noise hindering integration into nanophotonic devices. State-of-the-art quantum network nodes are thus realized with solid immersion lenses (SIL), in which the NV center retains lifetime-limited optical linewidth. In practice, such systems have shown ZPL detector click probabilities around 5×10−4 upon pulsed resonant excitation.

In this work, we realize the integration of a NV center into a fiber-based Fabry-Pérot microcavity in combination with coherent microwave spin control, enabling the generation of spin-photon states. This platform features bulk-like optical coherence of the NV center by incorporating a micrometer-thin diamond membrane into the cavity and a Purcell-enhanced emission into the ZPL combined with an efficient photon extraction. We quantify the coupling of the NV center’s readout transition to the cavity modes with the Purcell factor by measuring the reduction of the excited state lifetime. Further, we investigate the cavity-coupled readout transition with a photoluminescence excitation (PLE) measurement to determine the optical linewidth, a Hanbury Brown and Twiss (HBT) experiment to verify single-photon emission, and a saturation measurement, which reveals a ZPL detector click probability that is an order of magnitude higher compared to standard solid immersion lens systems. Moreover, we demonstrate coherent control of the NV center’s electron spin and characterize its coherence properties in a Ramsey and a Hahn-Echo measurement. Finally, we use our platform to generate spin-photon states of the electron spin qubit with one and two time-bin encoded ZPL photon qubits. We herald on the photon detection in their time-bin states and observe the correlation with the corresponding spin qubit readout.

Results
Interfacing diamond nitrogen-vacancy center spin qubits with an optical microcavity
In this section, we conceptually outline the optical interfacing of NV centers which are coupled to a fiber-based Fabry-Pérot microcavity as schematically depicted in Fig. 1a. At its heart, a sample mirror with a bonded diamond membrane faces a laser-ablated spherical mirror on the tip of an optical fiber forming the microcavity. Using the experimental setup presented in Fig. 1b, the cavity-coupled NV centers are optically addressed via the microcavity, which is mounted inside a closed-cycle optical cryostat. Details about the low-vibration cryogenic microcavity setup can be found in ref. 41, and the fabrication of the diamond device, which is also used in earlier work, is reported in ref. 39.

For the experiments in this study, the two orthogonal, linear polarization modes of the microcavity must be considered. Due to birefringence in the diamond membrane, the frequency degeneracy of these cavity modes is lifted, which is identified as a polarization mode splitting in the optical cavity response. In our lab frame, the low frequency (LF) and high frequency (HF) cavity mode is referred to as horizontal and vertical polarization cavity modes, respectively. The NV center’s optical dipole of the readout transition around 637 nm couples to both polarization cavity modes, enabling resonant excitation and detection in a cross-polarized fashion. Nanosecond-short optical excitation pulses are coupled into the vertical polarization cavity mode through the sample mirror via the reflection of a polarizing beamsplitter. Upon excitation, the NV center ZPL emission into the horizontal polarization cavity mode is coupled out through the sample mirror and detected by a fiber-coupled single-photon detector in transmission of the polarizing beamsplitter. This cross-polarization filtering reaches high suppression values of the excitation laser and is actively optimized during the experiments via piezo rotation mounted waveplates. In addition, a cavity lock laser at 637 nm is launched over the optical fiber mirror into the horizontal polarization cavity mode and is detected in cavity transmission by the same single-photon detector. This signal is used in the experiments to run a side-of-fringe lock that feeds back on the piezo that is controlling the fiber mirror position to maintain a constant cavity length. Moredetails about the experimental setup and the methods to maintain a high excitation laser suppression and to lock the cavity are described in the Methods section.

To prepare the NV center in its negatively charged state an off-resonant 515 nm charge repump laser is deployed over the fiber mirror next to a second, 637 nm laser to initialize the NV center in its ms = 0 spin ground state via the E transition (see Fig. 1c). A permanent magnet outside the cryostat creates a static magnetic field at the position of the NV center, which Zeeman-splits the NV center’s ms = ± 1 spin states, allowing us to define a qubit subspace consisting of the ms = 0 and ms = −1 states. Microwave pulses for qubit control are delivered via a gold stripline, which is embedded into the sample mirror.

All reported uncertainties in this work correspond to one standard deviation confidence intervals.

Coupling a single nitrogen-vacancy center to the microcavity
With the full spatial and spectral tunability of the microcavity, we select a position on the diamond membrane that is close to the mirror-embedded gold stripline, exhibits a high cavity quality factor, and a well-coupled NV center. At the selected position, a frequency splitting between the horizontal and vertical polarization cavity mode of (9.56 ± 0.02) GHz is observed, which we attribute to birefringence due to the presence of strain in the diamond membrane. The individual cavity polarization mode shows a linewidth of (1.69 ± 0.02) GHz, which corresponds to a quality factor of 280 × 10^3. With the determined cavity geometry, a cavity mode volume of 86 λ^3 is simulated. Accounting for our residual cavity vibration level of 22 pm, this results in a maximal vibration-averaged Purcell factor of 12 for a NV center that is perfectly coupled to a single cavity mode. In addition, a cavity mode outcoupling efficiency through the sample mirror of 39% is calculated by taking the mirror coating transmission losses into account (see Supplementary Note 1 for the cavity characterization details).

For the investigated cavity we find a NV center that is coupled with its Ey transition (readout transition) at a frequency of 470.377 THz (−23.34 GHz with respect to 470.4 THz). Figure 2a shows a PLE measurement of this transition in the LF cavity mode, resulting in a linewidth of (45 ± 3) MHz determined by a Gaussian fit. We attribute the additional broadening above the expected linewidth of 1/2πτP,LF ≈ 20 MHz for the Purcell-reduced lifetime τP,LF in the LF cavity mode as stated below to spectral diffusion. The determined transition linewidth is typical for NV centers with bulk-like optical properties, and the used experimental PLE sequence involving 515 nm repump laser pulses. Furthermore, we find the NV center’s Ex transition at a frequency of 10 GHz. This corresponds to a transverse strain with Ex-Ey excited state splitting of ~33 GHz, which is larger compared to typical bulk samples. We note that we also find less strained NV centers in the diamond membrane (see Supplementary Note 9 for measurements of a second cavity-coupled NV center). Next to the NV center transitions associated with the ms = 0 spin state, we find emission lines at −26.0 GHz as well as −24.5 GHz that are associated with the ms = ± 1 spin states. We attribute these to the NV center’s E1 and E2 transitions, where we use the former as our spin initialization transition. In all measurements, a static magnetic field of about 37.5 G is present along the NV center crystal axis. See Supplementary Note 2 for the PLE measurements of the other transitions and details about the used sequences.

In the sequences of all following measurements, we start by applying a 515 nm charge repump laser pulse to prepare the NV center with high probability in its negatively charged state, and a spin initialization laser pulse to initialize the NV center in its ms = 0 spin state (see Supplementary Note 6 for details). Figure 2c shows lifetime measurements of the NV center’s Ey excited state using 2 ns short resonant excitation pulses (see Supplementary Note 3 for details about the excitation pulse). The Purcell-reduced lifetime is measured in the LF and HF cavity modes with the cavity being on resonance with the readout transition. An off resonance lifetime is determined in the LF mode by detuning the cavity from the readout transition by −4 GHz. The lifetime measurement data is fitted with monoexponential curves. We observe that the data start to deviate from a monoexponential behavior for detection times ≳ 20 ns. We attribute this to signal of other (weaker coupled) NV centers or fluorescence of the cavity mirrors, and therefore restrict the fit window size. Details about the applied background subtraction and the influence of the fit window are further outlined in Supplementary Note 4. From the monoexponential fit, we determine an off resonance lifetime of τoff = (9.5 ± 0.4) ns and a Purcell-reduced lifetime of τP,LF = (7.8 ± 0.2) ns and τP,HF = (9.0 ± 0.2) ns for the LF and HF cavity mode, respectively. The off resonance lifetime differs from the expected excited state lifetime of 12.4 ns for low-strained NV centers. We attribute this to strain-induced mixing in the excited states. In Supplementary Note 4, we show a complementary investigation of this mechanism with time-resolved detection during continuous wave resonant excitation. The determined lifetimes allow to calculate the Purcell factor
FP = (τoff / β0 τP) − 1, (1)
with the NV center Debye-Waller factor β0 = 0.03. The Purcell factors for the LF and HF cavity mode are FP,LF = 7.3 ± 1.6 and FP,HF = 2.0 ± 1.4. In all the following measurements, we use the stronger coupling to the LF cavity mode to enhance the NV center emission and the HF cavity mode for excitation. Based on the determined Purcell factor the NV center coherently emits βLF = β0 FP,LF / (β0 FP,LF + 1) ≈ 18% into the LF cavity mode, which corresponds to a cavity outcoupled ZPL emission of about 7% and a maximal detector click probability of about 1.4% per excitation pulse for our setup efficiency and detection time window (see Supplementary Note 10 for details).

In Fig. 2b, the saturation behavior of the NV center readout transition is studied. For that, the peak power of the short excitation laser pulse is varied, and the ZPL detector click probability after the first excitation pulse is measured. A ZPL detector click probability of 0.5% per pulse is determined for the highest power applied, which outperforms the standard NV center quantum network node setups based on solid immersion lenses by an order of magnitude. In addition, the detector click probability still increases in a linear fashion for the investigated laser power regime showing that higher detector click probabilities can be reached if more laser power is applied. In the experiments, we are limited to the investigated laser power regime due to the efficiency of our short optical laser pulse generation setup.

Furthermore, we confirm single-photon emission by performing a pulsed resonant HBT experiment on the readout transition. In the experiment, we perform in total 55 × 10^6 measurement sequence repetitions in which we apply a train of short excitation pulses with a consecutive time separation of 126 ns. Figure 2d shows the second-order correlation function for 30 consecutive excitation pulses. The second-order correlation function exhibits clear antibunching at zero pulse number difference. Further, it shows two bunching-like features, which we attribute to the spin flipping probability into the ms = ± 1 states for larger pulse number differences and the probability to decay back into the ms = 0 state from the intersystem crossing singlet state for small pulse number differences. We fit the data with a triangular function as expected for the finite pulse train multiplied by a monoexponential function capturing the spin flipping process (see Supplementary Note 5 for details). The exponential decay for small pulse number differences is not included, since the amount of affected data points is insufficient for a reliable fit. We quantify an antibunching value of g(2)(0) = 0.034 ± 0.005 by the ratio of measured coincidences in the zero bin to the triangular fit amplitude.

Coherent microwave control of the nitrogen-vacancy center spin qubit
The NV center electron spin is coherently manipulated with microwave pulses, which are delivered via an about 10 μm distant gold stripline. Before microwave spin manipulation a 515 nm charge repump laser pulse prepares the NV center with high probability in its negatively charged state and the spin is optically initialized in its ms = 0 ground state with a spin initialization laser pulse and a fidelity of (93.5 ± 0.9)% (see Supplementary Note 6 for details about the initialization and qubit readout analysis). The applied magnetic field along the NV center crystal axis splits the ms = ± 1 states by about 210 MHz, enabling selective driving of the ms = 0 and ms = −1 qubit subsystem. After spin manipulation, the qubit is readout via the readout transition using the short excitation pulses and the same integration window as introduced in Fig. 2b. Figure 3a shows the statistics of detected photons for our spin qubit readout after spin initialization in the ms = ± 1 or ms = 0 state. This measurement is used as a readout fidelity calibration, and its dependency on the number of used short readout pulses is presented in the inset. The average fidelity Favg of the ms = ± 1 readout fidelity F±1 and the ms = 0 readout fidelity F0 plateaus for larger readout pulse numbers and is optimal around 200 pulses, which is used in the following experiments. These readout fidelities are used for qubit readout correction, where the finite ms = 0 spin state initialization fidelity is also taken into account. We note that the average readout fidelity is not limited by laser power, but rather the average number of emitted photons before the studied NV center spin flips. The outcoupled ZPL emission of about 7% for our system is comparable to the collected non-coherent phonon sideband (PSB) emission used for readout in a SIL setup, rendering high-fidelity qubit readout possible.

In Fig. 3b, coherent Rabi oscillations with a Rabi frequency of Ω = (10.73 ± 0.02) MHz are shown for our spin qubit at a microwave frequency of 2.773 GHz. Based on the fitted contrast of the Rabi oscillations and correcting for the finite ms = 0 spin state initialization fidelity, a microwave π pulse fidelity of (83 ± 9)% is extracted. For the next experiments, a π pulse and a π/2 pulse are calibrated by varying the amplitude of five consecutively applied π pulses that minimize the probability to readout ms = 0. The π pulse duration is fixed to 64 ns, and half of that duration is used for the π/2 pulse. As depicted in Fig. 3c, the qubit coherence is probed in a Ramsey experiment with an artificial detuning of 50 MHz by applying a delay time-dependent phase shift to the second π/2 pulse. The observed beating signature is attributed to the hyperfine coupling of the nitrogen nuclear spin, which is included in the fit model (see Supplementary Note 7 for details). By fitting the measurement data, a free-induction decay time of T2* = (170 ± 20) ns is determined, which is short compared to typical T2* times for these diamond devices (see Supplementary Note 9 for data of a second cavity-coupled NV center).

Using a Hahn-Echo experiment as presented in Fig. 3d, the qubit coherence can be largely recovered by choosing an appropriate total delay time τ, which mitigates quasi-static magnetic field fluctuations. In this measurement, three revivals are observed and a Gaussian decay curve with a time constant of 180 μs is displayed as a guide for the eye in Fig. 3d. The revivals occur around decoupling interpulse delay times τ/2 that correspond to integer multiples of the bare Larmor period of the carbon-13 nuclear spin bath of about 25 μs (γC13 = 1.0705 kHz/G and B = 37.5 G).

Generation of spin-photon states
In this section, we use our system to generate two-qubit and three-qubit spin-photon states and measure heralded correlations between the photonic and spin states. Figure 4a depicts the used pulse sequence, which combines spin qubit microwave control with the short resonant excitation pulses to generate spin-dependent photonic time-bin qubits. After optical initialization, the spin qubit is brought into equal superposition with a π/2 microwave pulse, followed by two resonant excitation pulses with an intermediate microwave π pulse. This generates the Bell state |NVspin, photon = (|1, E + |0, L) / √2 between the spin qubit states ms = 0 (|0) and ms = −1 (|1) and the photonic qubit in the time-bin basis of early (|E) and late (|L) as used for remote entanglement generation in quantum network demonstrations. The generated photon is measured by a single-photon detector, which heralds an early or late detection event. The spin qubit is optically read out in its Z-basis by 200 short readout pulses. The qubit readout probabilities conditioned on a photon heralding event in the early or late time-bin are shown in Fig. 4b (see Supplementary Note 6 for details about the applied qubit readout correction and Supplementary Note 8 for an X-basis spin qubit readout). A photon detection event in the late time-bin heralds the qubit in its expected |0 state. For an early time-bin photon detection event, the spin is readout with a probability of 82% in its expected |1 state. We attribute the lower readout probability of the |1 state to our limited microwave π pulse fidelity and an imperfect pulse calibration. In the Bell state correlation measurement, we record 27143 photon heralding events in 5 × 10^6 attempts, which correspond to a probability of 0.54% per attempt.

Furthermore, the Greenberger–Horne–Zeilinger (GHZ) state |NVspin, photon, photon = (|0, E, L + |1, L, E) / √2 can be generated by repeating the center sequence with the two short resonant excitation pulses before readout (see case N = 2 in Fig. 4a). The correlations after a successful double photon heralding event are displayed in Fig. 4c. After heralding a correct two photon state, the expected spin state is read out with similar probabilities as in the Bell state measurement. In total, we measure 5242 two photon heralding events in 220 × 10^6 attempts, which correspond to a probability of 2.4 × 10^−5 per attempt.

Methods
Cryogenic cavity setup and microwave pulse delivery
We use a floating stage closed-cycle optical cryostat (Montana Instruments HILA) with a base temperature of about 6 K and a sample mirror holder temperature of about 8 K. The spherical fiber mirror, which faces the sample mirror, is made out of a single-mode optical fiber with a pure silica core and a polyimide protection coating (Coherent FUD-4519, S630-P) and is mounted on a cryo-compatible positioning stage (JPE CPSHR1-a). A room temperature objective (Zeiss LD EC Epiplan-Neofluar), which is thermally shielded, is used to optically interface the sample mirror side of the cavity. The objective is positioned with three linear nanopositioning stages (Physik Instrumente Q-545) in a tripod configuration. A permanent neodymium disc magnet (Supermagnete S-70-35-N), which is mounted on the lid of the cryostat, is used to create a static magnetic field at the position of the sample mirror. Further details about the cryogenic setup can be found in ref. 41.

The setup is operated with a computer and the Python-based software QMI version 0.44. The data is stored and analyzed with the Python framework quantify-core. The actual measurement sequences are timed and executed by an interplay between a microcontroller (Jäger Computergesteuerte Messtechnik Adwin Pro II) and an arbitrary waveform generator (Tektronix AWG5014C). Microwave pulses are generated with a single sideband modulated vector signal generator (Rohde & Schwarz SMBV100A), amplified by a microwave amplifier (Mini-Circuits ZHL-50W-63+), and delivered to the cryostat via a home-built microwave switch. At the qubit frequency of 2.773 GHz we measure about 26 dB transmission loss through the cryostat. Further details about the microwave wiring and a picture of the sample mirror with mirror-embedded gold striplines can be found in ref. 41.

Laser pulse generation and delivery
We use one 637 nm continuous wave diode laser each for cavity locking (Newport Newfocus Velocity TLB-6300-LN) and spin initialization (Toptica DL Pro 637 nm). To modulate the laser intensity and generate microsecond-long optical pulses with high on/off ratios, two cascaded in-fiber acousto-optic modulators (AOM, Gooch and Housego Fiber-Q 633 nm) are used. Both lasers are combined free-space by a 50:50 non-polarizing beamsplitter (Thorlabs BS016). Then, a Glan-Taylor polarizer (Thorlabs GT10-A) and a half-wave and quarter-wave plate (Thorlabs WPH05M-633 and WPQ05M-633) are used to set and control the polarization. Finally, the 637 nm lasers are overlapped with the 515 nm repump laser (Hübner Photonics Cobolt 06-MLD) by a dichroic mirror (Semrock Di01-R532) and coupled into a single-mode fiber, which is connected to the cavity fiber.

A frequency-doubled, high-power tunable diode laser (Toptica TA-SHG Pro 637 nm) is used for readout as well as for cross-polarization control. For that, the laser is split by a 50:50 fiber beamsplitter (Thorlabs PN635R5A2) into two in-fiber paths. In the path for cross-polarization control, two in-fiber AOMs are used for modulation, whereas in the readout laser path, a temperature-stabilized electro-optical amplitude modulator (EOM, Jenoptik AM635) is deployed additionally. This allows for faster intensity modulation and the generation of nanosecond-short resonant excitation pulses. The EOM is DC biased by a programmable bench power supply (Tenma 72-13360) using a bias-tee (Mini-Circuits ZX85-12G-S+) and a DC block (Mini-Circuits BLK-89-S+) to control a constant transmission level. In addition, fast electrical pulses are generated with a home-built standalone pulse generator and applied to the EOM via the RF input of the bias-tee to generate nanosecond-short excitation pulses (see Supplementary Note 3 for their characterization). Finally, both in-fiber paths are combined with a 75:25 fiber beamsplitter (Thorlabs PN635R3A2) whose 75% output is connected to a polarization-maintaining fiber (Thorlabs P3-630PM-FC-2) before free-space launching. The 25% output is connected to a power meter (Thorlabs PM100USB with head S130VC) for laser power monitoring. To preserve a high polarization extinction ratio in both paths in-line fiber polarizers (Thorlabs ILP630PM-APC) are used.

All 637 nm lasers are stabilized on a wave meter (High Finesse WS-U).

Free-space cross-polarization optics
The cavity is interfaced from the sample mirror side with free-space cross-polarization optics as depicted in Fig. 1b. The collimated beam leaving the cryostat chamber is guided by broadband dielectric mirrors (Thorlabs BB03-E03) on the optical table and is reflected off a dichroic mirror (Semrock Di02-R635) before reaching the polarization optics. Next, the beam passes through half-wave and quarter-wave plates (Thorlabs WPH05M-633 and WPQ05M-633), which are mounted in piezo rotation stages (Newport AG-PR100 controlled by Newport AG-UC8). These waveplates map the excitation and detection polarization of the polarizing beamsplitter cube (PBS, Thorlabs PBS202) to the cavity polarization modes.

In the excitation path of the PBS the readout laser is launched from the polarization-maintaining fiber into free-space (Thorlabs KT120 with objective RMS10X) and passes a Glan-Thompson polarizer (Thorlabs GTH10M-A), followed by a quarter-wave and half-wave plate, which are mounted in piezo rotation stages as well.

In the detection path of the PBS, the light is filtered by a Glan-Thompson polarizer (Thorlabs GTH10M-A), an angle-tunable etalon (LightMachinery custom coating, ≈100 GHz full width at half maximum (FWHM) at 637 nm), and a bandpass filter (Thorlabs FBH640-10). Finally, the light is coupled (Thorlabs KT120 with objective RMS10X) into a single-mode fiber (Thorlabs P3-630A-FC-2) and detected by a single-photon detector (Picoquant Tau-SPAD-20). For the HBT experiment shown in Fig. 2d, a 50:50 fiber beamsplitter (Thorlabs TW630R5A2) is added, and a second single-photon detector (Laser Components Count-10C-FC) is used. The detectors are connected to a single-photon counting module (Picoquant Hydraharp 400) and the counter module of the microcontroller. The gating of the single-photon detectors is used to protect them against blinding during the application of the spin initialization laser.

Microcavity operation
For the resonant cross-polarization excitation and detection scheme used in this work, it is essential to keep the cavity on resonance with the NV center and to maintain a high excitation laser suppression during the measurements. To ensure these operational conditions, we probe the cavity resonance as well as the excitation laser suppression interleaved with the experimental sequences and stream the data to a computer in real-time to run an optimization routine next to the measurement. In both cases, the probe laser light is detected with the ZPL single-photon detector and recorded with the counter module of the microcontroller. This technique allows for live feedback on the order of Hertz, which is sufficient to compensate for drifts of the cavity resonance and the cross-polarization filtering. With the methods outlined in the following, it is possible to run measurements remotely for days.

To keep the cavity on resonance with the readout transition of the NV center, a side-of-fringe lock is deployed. For that, the cavity lock laser is frequency-tuned to the fringe of the cavity mode, and the single-photon detector counts are recorded during interleaved 100 μs long probe laser pulses. The recorded counts are streamed to a computer that uses a proportional control loop programmed in Python to optimize on a specified set value. The proportional control loop feeds back on the piezo that is controlling the fiber mirror position and, by that, locks the cavity on resonance. The cavity lock laser power is adjusted such that a count rate of about 700 kHz is measured when the cavity is on resonance with the cavity lock laser. For every specified set value, the cavity lock laser frequency can be adjusted such that the cavity is on resonance with the NV center.

For the cross-polarization filtering, interleaved 10 nW probe pulses of the cross-polarization control laser are applied for 400 μs while recording the single-photon detector counts. These counts represent the current excitation laser suppression and are streamed to a computer that runs a hill climbing type algorithm programmed in Python. The algorithm makes iterative changes to the four piezo rotation mounted waveplates as introduced in Fig. 1b to minimize the recorded counts. With this polarization suppression control loop, typical long-term count rates are <2 kHz for 10 nW probe pulses, which corresponds to excitation laser suppression values >60 dB.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript written by J. Fischer et al., the authors demonstrated an integration of a diamond membrane with an open\nFabry Perot cavity, showing an improvement of emitted photon detection and reduction of NV's optical lifetime. Moreover, the\nauthors showed that microwave manipulation can be incorporated into the device to achieve efficient spin manipulation and\nspin-photon interfaces. Although the idea of integrating diamond membranes with open cavities has been demonstrated by\nseveral groups in the past few years, most realizations are limited to cavity mode characterization, Q-factor improvements, or\nobservation of Purcell enhancements. By integrating MW control and creating spin-photon pairs, this work is one step\nforward towards the practical usage of such devices for quantum networking purposes. Therefore, I recommend the work for\npublication in Nature Communications, after some minor concerns are addressed.\n1. It seems that the birefringence of the diamond membrane is critical for the successful operation of the device, but it is not\nan intrinsic property of ideal bulk diamond. The authors mentioned that the strain in the diamond might be the reason. Is the\nstrain generation deterministic? Is it uniform across the membrane? What amount of strain is sufficient for the measurement\nprotocol?\n2. From the two NV centers measured in the manuscript, they seem to have varying strain profiles. I cannot tell if that is\nconsistent with the strain profile in diamond observed by the cavity mode, or if there are some local strains of NV itself in\naddition to the diamond membrane's \"global\" strain. It is also challenging to interpret the magnitude of the strains based on\nthe frequency shift and differences. Some further elaborations on the magnitude and orientation of the strain profile (such as\nwhich type of strain is more sensitive to what kind of measurements) are needed.\n3. The author aligned all optical frequencies to 470.4 THz. Does this frequency have a specific meaning?\n4. The measurement protocol involves numerous measurements using the Ey, Ex, E1, and E2 transitions of NV. Therefore, I\nrecommend adding Ex and E2 in Figure 1 (c), and some explanations for them for the readers' convenience.\n5. The Hahn echo in Figure 3 (d) is somewhat noisy. I wonder what may cause this to happen.\n6. In principle, when the cavity mode is detuned from the NV transition, the modified mode density will induce an extension\nof the optical lifetime. Maybe this effect is small, but it would be helpful to know how far the intrinsic lifetime is from \\tau_off in\nFigure 2 (c), based on the cavity parameters. This analysis may also explain the legitimacy of using 4 GHz detuning as the\nlifetime reference.\n7. What are the limiting factors of the current platform, and what optimizations can be made to improve the performance\nfurther? Some discussions on that would be helpful.", "reviewer_2": "This work builds on development of a fiber-based Fabry-Perot cavity and implements coherent microwave control of NV\ncenters embedded in a diamond thin film bonded to the Bragg mirror of the cavity. The small cavity mode volume\nconfinement of light enhances the emission of NV center relaxation into the ZPL which, without the cavity, is very low even\nat cryogenic temperature. The integration of coherent spin control of NV centers and the Purcell enhanced optical coupling\nis a key step towards scalable quantum networks and is of interest to researchers in quantum information sciences. The\nexperiments and system characterizations are well-known methods. An NV center is placed under a charge repump laser\nand a laser at ZPL wavelength for resonant excitation and spin initialization. Microwave is used for driving the group-state\nspin transitions. The group characterize linewidth of the readout transition (that is on resonant with the cavity) by PLE. More\nthan lifetime limited linewidth is seen which is attributed to spectral diffusion. The Purcell factor of the cavity is calculated\nfrom the lifetime reduction of the excited state, leading to an order of magnitude bigger click rate of the ZPL\ndetector. This is a significant improvement upon using standard solid immersion lenses milled out from a diamond thin film. The group\nreviews coherent control of the spin qubit by measuring Rabi oscillation, Ramsey, and Hahn-echo. The reported parameters\n(T_2* etc) are typical for NV centers. Finally, the system generates two-qubit and three-qubit spin-photon states. The system\nhas the capability of generating Bell state and GHZ state by using short optical pulses for spin-dependent photon-time-bin\nqubits.\nOverall, the incremental work reported in this paper is scientifically sound. While there is no fundamentally new component\npresented in this paper, the integration of two previously published systems requires significant engineering and fabrication\nefforts. The system has the potential of being a building block in quantum networking and gives inspiration to platforms\nbased on other solid-state spins. I tip slightly towards recommending the paper for publication in nature comm for its proof-of-\nprinciple work for fiber-based-cavity-enhanced NV center platform for quantum networking.\nThe method section (along with the supplementary notes) well documents the system parameters and provides sufficient\ninformation for other research groups that wish to develop upon such system.\nHowever, there is one thing I would like the authors to address before the editor publishes the paper:\nThe Hahn-echo experiment (Figure 3d) is weird to me:\n1. the horizontal axis looks non-linear. Is this on a log-scale? (Though it doesn’t look like one) Is there a reason to it or\nmerely a plotting mistake?\n2. Why at \\tau=0 we do not see probability 1 for m_s=0 state? Is the system not initialized to m_s=0 state?\nLastly, I wish the authors can discuss a few more things: what is the entanglement generation rate required for quantum\nnetworking? Is there possibility of improving the system (with sufficient rate budgets) to meet that level? The entangled state\ngeneration rate is 10x better than solid-immersion-lenses devices on NV centers, how does this compare to other platforms\nsuch as trapped ions, neutral atoms, quantum dots, or other solid-state spins? How would the authors address spectral\ndiffusion?", "reviewer_3": "This paper reports a coherently controlled diamond NV center optically coupled to an open microcavity. Purcell\nenhancement and an asymmetric cavity design enable efficient resonant photon collection, while on-chip microwave lines\nachieve 10 MHz Rabi spin control. Tuning the cavity to the NV’s optical transition yields a measured Purcell factor of 7.3 via\nexcited-state lifetime analysis. Pulsed resonant excitation gives a 0.5% photon detection probability per pulse—an order of\nmagnitude improvement over solid immersion lens systems. Combining efficient optical interfacing with spin control, two-\nand three-qubit spin-photon entangled states are generated, and heralded Z-basis correlations are measured\nbetween spin and photonic time-bin qubits.\nThe NV center is widely used in quantum information science demonstrations and is considered one of the most promising\nphysical systems. This study demonstrates remarkable progress by achieving a high Purcell factor and realizing a photon\ndetection probability of 0.5% per pulse under pulsed resonant excitation. I strongly believe the presented NV center, coupled\nto an open microcavity, is expected to become a key platform for future quantum network research.\nBefore the publication, I would like to ask questions as follows:\n(1) While the current results demonstrate a high Purcell factor and a photon detection probability of 0.5% per pulse under\npulsed resonant excitation, the main limiting factors at this stage remain unclear. Are these values approaching the\ntheoretical limits, or is there still room for technical improvement? If the latter, it would be important to identify what is\ncurrently limiting the performance in order to explore potential strategies for further enhancement. A more detailed\nexplanation on this point would be appreciated.\n(2) In my opinion, the explanation for the bunching behavior mentioned on page 3, line 136 is somewhat unclear. While\nfitting curves are presented in Figure 2(d), it would enhance the clarity of the discussion, for example, if the authors could\ndescribe the fitting function employed and elaborate on its physical interpretation.\n(3) What is the fidelity of the spin initialization? The reported fidelity of the MW π-pulse is 83%, which seems somewhat low.\nIt would be helpful if the authors could clarify the underlying reason for this limited fidelity and comment on possible\napproaches to improve it.\n(4) The reported values of T2*=170ns and Hahn echo T2=180 µs appear relatively short. For reference, significantly longer\nvalues—up to 1.5 ms and 2.4 ms at room temperature—have been reported (e.g., Nat. Commun. 10, 3766 (2019)). Could the\nauthors clarify the reasons for these shorter coherence times? Was an isotopically purified 12C-enriched diamond used in\nthis study? Is it possible that cavity fabrication introduced defects into the sample? Or are the limitations due to\nmeasurement-side issues such as optical noise, rather than the sample quality itself?"}, "author_response": "We thank the referee for their time reviewing our manuscript, and for their recommendation for\npublishing in nature communications.\n1. It seems that the birefringence of the diamond membrane is critical for the successful operation of the\ndevice, but it is not an intrinsic property of ideal bulk diamond. The authors mentioned that the strain in\nthe diamond might be the reason. Is the strain generation deterministic? Is it uniform across the\nmembrane? What amount of strain is sufficient for the measurement protocol?\nThe birefringence of the diamond membrane causes the cavity polarization modes to split in frequency.\nIn combination with the used cross-polarization resonant excitation and detection scheme, where the\ndetection mode is on resonance with the readout transition, the frequency splitting affects the excitation\nprobability. This is indeed relevant to consider for the operation, where saturating the NV center is\ndesired. However, in the end it is the interplay of frequency splitting of the cavity polarization modes,\ncavity linewidth, Purcell factor in the excitation mode, available laser power, among others, that\ndetermines the excitation probability. Hence, it is difficult to state a range of frequency splittings (strain),\nthat is sufficient for certain measurements.\nWe researched the diamond membrane on different cavity spots in a range of ~100μm along the gold\nstripline and always measured similar polarization modes splittings of ~10GHz. It would require the\nfabrication and characterization of multiple diamond membranes to evaluate deterministic strain\ngeneration. We added the information about the observed polarization mode splittings to\nSupplementary Note 9.\n“The frequency splitting of the polarization cavity modes ~10GHz is observed for all cavity spots in the\ninvestigated range of ~100 μm along the gold stripline.”\n2. From the two NV centers measured in the manuscript, they seem to have varying strain profiles. I\ncannot tell if that is consistent with the strain profile in diamond observed by the cavity mode, or if there\nare some local strains of NV itself in addition to the diamond membrane's \"global\" strain. It is also\nchallenging to interpret the magnitude of the strains based on the frequency shift and differences. Some\nfurther elaborations on the magnitude and orientation of the strain profile (such as which type of strain\nis more sensitive to what kind of measurements) are needed.\nThe second cavity-coupled NV center exhibits the E and E transition around 470.440THz with an E - E\nx y x y\nexcited state splitting of ~12GHz, which are typical values for low-strained NV centers in solid immersion\nlens devices (see for example NV B in reference 48; L. Robledo et al., High-fidelity projective read-out of\na solid-state spin quantum register, Nature 477, 574 (2011)). This indicates that the potential influence of\n“global” strain on the NV center transitions does not dominate the “local” strain NV centers experience\nin sold immersion lenses (bulk samples).\nFor our work the lateral strain level (E - E excited state splitting) of the NV center is the most relevant as\nx y\nit determines the mixing in the excited states and by that the spin flipping process, which in turn limits\nthe readout fidelity. In reference 48 (L. Robledo et al., High-fidelity projective read-out of a solid-state\nspin quantum register, Nature 477, 574 (2011)) high-fidelity readout has been demonstrated with a NV\ncenter that is similar to our second cavity-coupled NV center underlining the potential of our open\nmicrocavity system. We added this information to Supplementary Note 9.\n“The absolute transition frequencies as well as the lateral strain level are comparable to NV centers in\nsolid immersion lens devices, which also have been used to demonstrate high-fidelity readout [8]. This\nshows that low-strained NV centers similar to solid immersion lens devices are found in the diamond\nmembrane.”\n3. The author aligned all optical frequencies to 470.4 THz. Does this frequency have a specific meaning?\nThis frequency reference is commonly used in literature for NV centers, but besides being very close to\nthe zero-phonon line it has no special meaning.\n4. The measurement protocol involves numerous measurements using the Ey, Ex, E1, and E2 transitions\nof NV. Therefore, I recommend adding Ex and E2 in Figure 1 (c), and some explanations for them for the\nreaders' convenience.\nWe thank the referee for this suggestion. We see the benefit for the reader of adding the E and E\nx 2\ntransition in Figure 1(c) and changed the figure and caption. We note that in all measurements\npresented in the figures of the main text only the E (readout) and E (initialization) transition are used.\ny 1\n5. The Hahn echo in Figure 3 (d) is somewhat noisy. I wonder what may cause this to happen.\nIn dynamical decoupling measurements, the coherent interactions of the electron spin to individual\ncarbon-13 nuclear spins can cause sudden loss in coherence for specific decoupling times, which leads to\ndips in the observed coherence revivals. In this sense, the data is not noisy but rather shows a unique\nfingerprint of the carbon-13 nuclear spin environment of the investigated NV center (for measurements\nat a similar magnetic field see L. Childress et al., Coherent Dynamics of Coupled Electron and Nuclear\nSpin Qubits in Diamond, Science 314, 281 (2006)).\n6. In principle, when the cavity mode is detuned from the NV transition, the modified mode density will\ninduce an extension of the optical lifetime. Maybe this effect is small, but it would be helpful to know\nhow far the intrinsic lifetime is from \\tau_off in Figure 2 (c), based on the cavity parameters. This analysis\nmay also explain the legitimacy of using 4 GHz detuning as the lifetime reference.\nThe Purcell factor at a cavity detuning of -4GHz from the readout transition, taking our cavity linewidth\n(Voigt profile with Lorentzian cavity mode and Gaussian vibration contribution) into account, is about 4%\nfrom its value on resonance. Considering this contribution in the analysis would change the Purcell\nfactors determined by the excited state lifetime measurements from 2.0 ± 1.4 to 2.1 ± 1.5 and from 7.3 ±\n1.6 to 7.7 ± 1.7 for the HF mode and LF mode, respectively. The actual influence on the off resonance\nlifetime is by the Debye-Waller factor (β =0.03) smaller yielding a corrected off resonance lifetime of (9.6\n0\n± 0.4)ns compared to (9.5 ± 0.4)ns without correction. The residual contribution is small compared to the\nuncertainties and as done in our study (without correction) presents a conservative estimate on the\nPurcell factor.\n7. What are the limiting factors of the current platform, and what optimizations can be made to improve\nthe performance further? Some discussions on that would be helpful.\nThe main limiting factor of our microcavity are additional cavity losses (1260ppm) that contribute to the\n2260ppm total cavity losses. This reduces the Purcell factor and the cavity outcoupling in our\nexperiments. Improving the device quality to eliminate these additional losses suggests an almost 4\ntimes improvement in cavity outcoupled ZPL emission. We extended the discussion section of the main\ntext with this elaboration.\n“Furthermore, improving the device quality to eliminate additional cavity losses suggests a ZPL detector\nclick probability of about 5%, which corresponds to a cavity outcoupled ZPL emission of 26%.”\nWe thank the referee for their time and the positive feedback on our manuscript.\nHowever, there is one thing I would like the authors to address before the editor publishes the paper:\nThe Hahn-echo experiment (Figure 3d) is weird to me:\n1. the horizontal axis looks non-linear. Is this on a log-scale? (Though it doesn’t look like one) Is there a\nreason to it or merely a plotting mistake?\nThe x axis of Fig. 3(d) is indeed in log scale. In dynamical decoupling experiments with NV centers, where\ncoherence times can be improved by orders of magnitude, logarithmic tau scaling is often used and so\nalso in our study. We made the log scale more clear by adjusting the x axis ticks of Fig. 3(d).\n2. Why at \\tau=0 we do not see probability 1 for m_s=0 state? Is the system not initialized to m_s=0\nstate?\nThe x axis is in log scale, and the data starts at tau=30.176us. This is also made more clear by adjusting\nthe x axis ticks of Fig. 3(d).\nLastly, I wish the authors can discuss a few more things: what is the entanglement generation rate\nrequired for quantum networking? Is there possibility of improving the system (with sufficient rate\nbudgets) to meet that level? The entangled state generation rate is 10x better than solid-immersion-\nlenses devices on NV centers, how does this compare to other platforms such as trapped ions, neutral\natoms, quantum dots, or other solid-state spins? How would the authors address spectral diffusion?\nThe required entanglement generation rates in a quantum network depend on the specific network\napplication/experiment and the hardware platform. This makes it hard to give a general answer on the\nneeded entanglement rates and a comparison with other platforms. As outlined in reference 22 (C. E.\nBradley et al., Robust quantum-network memory based on spin qubits in isotopically engineered\ndiamond, npj Quantum Information volume 8, 122 (2022)), the active link efficiency can be used to\nquantify entanglement links under network operation with memory qubits. A ten times improvement on\nthe entanglement generation rate suggests an active link efficiency greater than one, which is a key\nenabling condition for scaling quantum networks. In terms of absolute rates, ~40Hz has been\ndemonstrated with NV centers in solid-immersion lenses (P. C. Humphreys et al., Deterministic delivery\nof remote entanglement on a quantum network, Nature volume 558, pages 268–273 (2018)), which\nwould translate to ~400Hz for the current system – this would surpass the current state of the art in all\nplatforms with demonstrated memory qubits under network operation (rates ~250Hz have been\nachieved with trapped ions; J. O’Reilly et al., Fast Photon-Mediated Entanglement of Continuously\nCooled Trapped Ions for Quantum Networking, PRL 133, 090802 (2024)).\nThe measured spectral diffusion is comparable to NV centers in solid-immersion lenses and typical for\nthe used experimental sequence, which involves off-resonant charge repump pulses. The effect of\nspectral diffusion is efficiently mitigated by using a charge-resonance check procedure – a standard tool\nin NV center network experiments. We adjusted the discussion section of the main text to add this\ninformation.\n“Moreover, the system can be further optimized by implementing a charge-resonant check procedure of\nthe optical transitions to mitigate spectral diffusion, improving charge state and qubit initialization\nfidelities [51], as well as microwave pulse shaping for high-fidelity qubit control [27].”\nWe thank the referee for their time reviewing our manuscript, and for their recommendation for\npublishing in nature communications.\nBefore the publication, I would like to ask questions as follows:\n(1) While the current results demonstrate a high Purcell factor and a photon detection probability of\n0.5% per pulse under pulsed resonant excitation, the main limiting factors at this stage remain unclear.\nAre these values approaching the theoretical limits, or is there still room for technical improvement? If\nthe latter, it would be important to identify what is currently limiting the performance in order to explore\npotential strategies for further enhancement. A more detailed explanation on this point would be\nappreciated.\nThe main limiting factor of our microcavity are additional cavity losses (1260ppm) that contribute to the\n2260ppm total cavity losses. This reduces the Purcell factor and the cavity outcoupling in our\nexperiments. Improving the device quality to eliminate these additional losses suggests an almost 4\ntimes improvement in cavity outcoupled ZPL emission. We extended the discussion section of the main\ntext with this elaboration.\n“Furthermore, improving the device quality to eliminate additional cavity losses suggests a ZPL detector\nclick probability of about 5%, which corresponds to a cavity outcoupled ZPL emission of 26%.”\n(2) In my opinion, the explanation for the bunching behavior mentioned on page 3, line 136 is somewhat\nunclear. While fitting curves are presented in Figure 2(d), it would enhance the clarity of the discussion,\nfor example, if the authors could describe the fitting function employed and elaborate on its physical\ninterpretation.\nWe thank the referee for this comment and added the explicit fit function with explanations to the\nmanuscript (Supplementary Note 5).\n(3) What is the fidelity of the spin initialization? The reported fidelity of the MW π-pulse is 83%,\nwhich seems somewhat low. It would be helpful if the authors could clarify the underlying reason for this\nlimited fidelity and comment on possible approaches to improve it.\nThe initialization fidelity into the m=0 state is (93.5 ± 0.9)% as stated in the microwave control section of\ns\nthe main text. We attribute the microwave π pulse infidelity deduced from the electron spin Rabi\noscillations to nonuniform driving of the three nitrogen nuclear spin transitions. This limitation can be\novercome by applying pulse shaped microwave pulses as suggested in the discussion section of the main\ntext.\n(4) The reported values of T2*=170ns and Hahn echo T2=180 µs appear relatively short. For reference,\nsignificantly longer values—up to 1.5 ms and 2.4 ms at room temperature—have been reported (e.g.,\nNat. Commun. 10, 3766 (2019)). Could the authors clarify the reasons for these shorter coherence\ntimes? Was an isotopically purified 12C-enriched diamond used in this study? Is it possible that cavity\nfabrication introduced defects into the sample? Or are the limitations due to measurement-side issues\nsuch as optical noise, rather than the sample quality itself?\nThe sample is fabricated from diamond with a natural abundance of carbon. The presented data in\nSupplementary Note 8 of a second cavity-coupled NV center in the same diamond membrane shows a\nfree-induction decay time of T2* = (3.7 ± 0.4)μs. This value is typical for diamond samples with a natural\nabundance of carbon and comparable to our solid-immersion lens devices (H. Bernien et al., Heralded\nentanglement between solid-state qubits separated by three metres, Nature 497, pages 86–90 (2013):\nT2* = (3.07 ± 0.06)μs for NV A and T2*= (0.96 ± 0.03)μs for NV B). Therefore, we attribute the shorter\ncoherence time of the NV center studied in the main text to an unfortunate environment in the diamond\nlattice."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I think the revised manuscript and the response are well written. Therefore, I recommend this manuscript for publication in\nNature Communications.", "reviewer_2": "The authors of this manuscript have addressed my concerns. I recommend publication of this work in Nature\nCommunications.", "reviewer_3": "I am satisfied that the authors have thoroughly addressed all of my comments and questions. The manuscript is clearly\nwritten and well structured. I therefore recommend it for publication in Nature Communications."}, "author_response": ""}]

Decision letter