## Introduction

Quantum entanglement between remote qubits is an essential element for not only quantum networks1,2,3 but also quantum computers as well as quantum metrology. In these technologies, a quantum medium (degrees of freedom or material) that carries quantum information must be converted or interfaced between functional elements, such as registers, memories, and transmission qubits, just as in classical information technology. A hybrid quantum system is thus attractive as a quantum interface4. A candidate for such a hybrid quantum system is a color center in diamond5, such as nitrogen-vacancy (NV)6,7,8, silicon-vacancy (SiV)9,10, germanium-vacancy (GeV)11,12,13, and tin-vacancy (SnV)14,15,16 centers, which provides spin memories accessible by light17,18,19,20 and enables entanglement generation between a photon and a spin21,22,23,24,25. An electron spin in those color centers, surrounded by a number of carbon isotope spins, is commonly manipulated under a strong magnetic field26,27,28,29,30,31, which hinders the quantum medium conversion. For example, superconducting qubits, which are promising for quantum computing32, work under a weak magnetic field to avoid superconducting breakdown. A zero magnetic field is further desired to achieve high homogeneity in scaling up the system.

Geometric spin manipulation under a zero magnetic field33,34,35,36,37 has been demonstrated by utilizing the degenerate subspace of the spin-triplet state of a negatively charged NV center with a polarized microwave38,39,40 or with a polarized light41,42. Quantum state transfer from a photon polarization into a nuclear spin near an NV center43,44 has also been demonstrated based on entangled absorption45. In this work, we demonstrate entanglement generation between spin and photon polarizations with spontaneous emission under a zero magnetic field, which will enable the spontaneous generation of entanglement between remote spins when combined with the quantum state transfer (Fig. 1a).

## Results

### System and scheme

A negatively charged NV center has a spin-triplet electronic structure. In the orbital ground state, it forms a V-shaped spin-1 three-level structure with degenerate |mS = ±1〉S states and a zero-field split |mS = 0〉S state due to spin−spin interaction. These states are coupled with the degrees of freedom of the microwave polarization, resulting in transitions selective to the right- and left-circular microwave polarization. Similarly, the degenerate |±1〉S states and one of the excited states |A2〉, which is split from other excited states by the spin−orbit and spin−spin interactions, constitute a Λ-shaped three-level structure, resulting in transitions selective to left- and right-circular light polarization, respectively46 (Fig. 1b, c). Therefore, the degenerate spin subspace, called a geometric spin qubit, is accessible with an arbitrarily polarized light or microwave. Universal holonomic quantum gates47 are defined on the geometric spin qubit with a geometric phase given via a cyclic transition between the qubit space and ancillary state |0〉S or |A240,41. The correlation between the spin and photon polarizations also exists in the process of absorbing and emitting a single photon: when the state prepared at |A2〉 relaxes by emitting a photon with the polarization entangled with the spin polarization. In contrast, when a photon is absorbed to excite the electron orbital into the |A2〉 state, the photon polarization needs to be correlated with the spin polarization45, thus enabling the Bell state measurement conditioned on the absorption event.

### Geometric operation

A microwave π-pulse consisting of an arbitrary polarization state $$| \psi \rangle _{{{{{{{{\mathrm{MW}}}}}}}}} = \cos \theta _{{{{{{{{\mathrm{MW}}}}}}}}}| { + 1} \rangle _{{{{{{{{\mathrm{MW}}}}}}}}} + {{{{{{{\mathrm{e}}}}}}}}^{ - {{{{{{{\mathrm{i}}}}}}}}\phi _{{{{{{{{\mathrm{MW}}}}}}}}}}\sin \theta _{{{{{{{{\mathrm{MW}}}}}}}}}| { - 1} \rangle _{{{{{{{{\mathrm{MW}}}}}}}}}$$ excites a geometric spin qubit state $$\left| \psi \right\rangle _{{{{{{{\mathrm{S}}}}}}}} = \cos \theta _{{{{{{{{\mathrm{MW}}}}}}}}}\left| { + 1} \right\rangle _{{{{{{{\mathrm{S}}}}}}}} + {{{{{{{\mathrm{e}}}}}}}}^{ - {{{{{{{\mathrm{i}}}}}}}}\phi _{{{{{{{{\mathrm{MW}}}}}}}}}}\sin \theta _{{{{{{{{\mathrm{MW}}}}}}}}}\left| { - 1} \right\rangle _{{{{{{{\mathrm{S}}}}}}}}$$ from the ground ancilla state |0〉S. Here, θMW and ϕMW respectively denote the polar and azimuthal angles in the Poincaré sphere based on the right- and left-circular polarization states, |+1〉MW and |−1〉MW. In reverse, the microwave π pulse rotates the arbitrary qubit state to |0〉S; the state rotated to |0〉S can be read out by spin-dependent excitation to |Ey〉. We first evaluate the intensity and polarization of microwaves applied from two wires (Fig. 1d) to determine the parameters of the microwave voltage and phase that generate the arbitrary microwave polarization40. We then generate modulated waveforms with the GRAPE algorithm48 to increase the manipulation fidelity within the limited Rabi frequency, 2.5 MHz (see Supplementary Note 1). Any geometric spin qubit states are prepared and read out with the corresponding microwave polarization (Fig. 2a, b). Figure 2c, d show two-dimensional tomography on six prepared states $$\{ | + \rangle _{{{{{\mathrm{S}}}}}} = ( {| { + 1} \rangle _{{{{{\mathrm{S}}}}}} +| { - 1} \rangle _{{{{{\mathrm{S}}}}}}} )/\sqrt 2 ,| - \rangle _{{{{{\mathrm{S}}}}}} = ( {| {+ 1} \rangle _{{{{{\mathrm{S}}}}}} - | { - 1} \rangle _{{{{{\mathrm{S}}}}}}} )/\sqrt 2 ,| {{{{{\mathrm{i}}}}}} \rangle _{{{{{\mathrm{S}}}}}} = ( | \!+ 1 \rangle _{{{{{\mathrm{S}}}}}} +\, {{{{{\mathrm{i}}}}}}| { - 1} \rangle _{{{{{\mathrm{S}}}}}} )/\sqrt 2 ,\,| { - {{{{{\mathrm{i}}}}}}} \rangle _{{{{{\mathrm{S}}}}}} = ( {| { + 1} \rangle _{{{{{\mathrm{S}}}}}} - {{{{{\mathrm{i}}}}}}| { - 1} \rangle _{{{{{\mathrm{S}}}}}}} )/\sqrt 2 ,\,| { + 1} \rangle _{{{{{\mathrm{S}}}}}},\,| { - 1} \rangle _{{{{{\mathrm{S}}}}}} \}$$. The resulting average fidelity including the state preparation and measurement is 97% (Fig. 2) (see Supplementary Note 2).

### Spin-photon entanglement

To observe the photon emission process from |A2〉 to the ground state, we measure the zero-phonon line (ZPL) emission after resonant excitation. Since the wavelengths of the excitation light and the emitted photon are the same, it is necessary to separate the excitation light reflected at the optical element and the diamond surface from the emitted photon in space and time. In the present setup, although the reflected light is temporally eliminated by a two-stage electro-optic modulator (EOM), the overall extinction ratio is not sufficient. Therefore, we make the polarization of the excitation light orthogonal to the polarization of the photon measurement. Note that the polarization of the excitation light does not affect the measurement results because the excitation is only for preparing the quantum state to |A2〉. For practical use, since the polarizer cannot be used to eliminate the excitation light, either an AR coating of the diamond surface, a more elaborate optical design, or an improvement of the extinction ratio of the optical modulator is needed. Another possible way to eliminate the excitation light is nonresonant excitation with a wavelength filter to extract only the |A2〉 photon.

Figure 3a shows the ZPL emission intensity depending on the avalanche photodiode (APD) gate delay time, where the gate width is fixed at 10 ns. Since the probabilities of detecting a photon and a spin in the present setup are 3 × 10−6 and 0.15, respectively, simultaneous detection rarely occurs. We thus measure the spin state conditioned on the photon detection (Fig. 3b). Since the reflected excitation light is slightly included in the detected photons, as shown in Fig. 3a, the completely mixed spin state is probabilistically measured. In Fig. 3c, the estimated error detection count is subtracted to show only the intrinsic entangled state. The obtained results are in good agreement with the entangled state written as $$| {{{\Psi }}_ + } \rangle = \frac{{| { + 1} \rangle _{{{{{{{\mathrm{p}}}}}}}}| { - 1} \rangle _{{{{{{{\mathrm{S}}}}}}}} + | { - 1} \rangle _{{{{{{{\mathrm{p}}}}}}}}| { + 1} \rangle _{{{{{{{\mathrm{S}}}}}}}}}}{{\sqrt 2 }} = \frac{{| + \rangle _{{{{{{{\mathrm{p}}}}}}}}| + \rangle _{{{{{{{\mathrm{S}}}}}}}} - | - \rangle _{{{{{{{\mathrm{p}}}}}}}}| - \rangle _{{{{{{{\mathrm{S}}}}}}}}}}{{\sqrt 2 }} = \frac{{ - {{{{{{{\mathrm{i}}}}}}}}| { + i} \rangle _{{{{{{{\mathrm{p}}}}}}}}| { + i} \rangle _{{{{{{{\mathrm{S}}}}}}}} + {{{{{{{\mathrm{i}}}}}}}}| { - i} \rangle _{{{{{{{\mathrm{p}}}}}}}}| { - i} \rangle _{{{{{{{\mathrm{S}}}}}}}}}}{{\sqrt 2 }}$$, which indicates odd, even, and even parity along with the z-, x-, and y-axis measurements, where |+1〉p and |−1〉p denotes right- and left-circular polarization of the optical photon. The estimated state fidelity is 86.8%, which is limited mainly by NV-axis misalignment (6%), spin phase rotation (2%), spin measurement error (2%), and hybridization of excited states due to non-axial strain (1%) (Fig. 3d) (see Supplementary Note 3). The axis misalignment can be corrected by balancing the optical or spin systems since it is a coherent error (discussed later). The spin phase rotation is attributed mainly to the residual magnetic field and hyperfine interaction between electrons and the nitrogen nuclear spin during the indeterminate photon emission time (~12 ns), which can be suppressed by initializing the nitrogen nuclear spin. The spin measurement error can also be suppressed by high-fidelity manipulation with a stronger microwave.

### Polarization coincidence

To investigate the NV-axis misalignment in more detail, we perform microwave optical double resonance (Fig. 4a). We first excite a geometric spin qubit state with a polarized microwave, followed by the |A2〉 excitation with a polarized light (Fig. 4b). The PSB emission from the |A2〉 state is maximized when the bright state against the microwave coincides with that against the light (Fig. 4c). By fitting the emission curve with a sinusoidal function, the correspondence between the microwave and light polarizations is estimated (Fig. 4d). The nonlinearity is attributed to the tilting of the NV axis. The relationship between microwave and light polarization angles can be expressed as ϕMW = arctan{αtan(ϕoptϕNV)} + ϕNV, where α is the attenuation coefficient of the angle of the NV axis ϕNV depending on the angle of optical polarization ϕopt (see Supplementary Note 4). We estimate α = 0.60, ϕNV = 20.1° by fitting the experimental results shown in Fig. 4d. From the attenuation coefficient α, the tilt angle of the NV axis against normal to the surface is found to be 53.0°, which roughly corresponds to 54.7° as expected from a crystal orientation of 100. The attenuation can be eliminated by using a normally oriented NV center in <111>-oriented diamond. It can also be adapted by optical elements or spin manipulation to compensate for the amplitude ratio of the photon state biased by polarization distortion.

## Discussion

This demonstration differs from that in Togan et al.21, where a static magnetic field is applied to distinguish the spin up/down states. The magnetic field lifts the degeneracy to make the light polarization time-dependent although the quantum eraser technique erases the frequency information, while the degeneracy remains under a zero magnetic field in our case to keep the light polarization time-independent as well as the geometric spin qubit state. This is why the geometric spin qubit is well manipulated by a polarized microwave.

Although the present experiment did not use a single-photon microwave, it could in principle work with a one-photon process. The photon−spin entanglement measured with microwave polarization shown in Fig. 3 can be interpreted as a teleportation-based quantum state transfer of a polarized microwave photon into a polarized optical photon heralded by the detection of the ancilla state |0〉S occupation. On the other hand, the polarization coincidence of a photon and microwave shown in Fig. 4 serves as a phase transfer49 or conditional Bell measurement (see Supplementary Note 5). These schemes can be applied as a quantum transducer with quantum memories that interfaces superconducting or ion qubits with an optical photon via a microwave photon. Efficient transduction is expected, owing to the strong coupling of the orbital excited state with an electric field or mechanical vibration50,51 in an NV center. Other possible applications include multiphoton entangled resource for quantum networks and computation52,53,54,55.

In conclusion, we demonstrated geometric photon−spin entanglement via spontaneous emission with the help of the spin−orbit entanglement inherent in the molecular structure of the diamond NV center under a zero magnetic field. The combination of the demonstrated scheme with the heralded quantum state transfer of a photon polarization into a nuclear spin43,44 will establish a new scheme for generating heralded remote entanglement of quantum memories based on emission and absorption. The scheme is insensitive to the mode matching in time, frequency, and space, in contrast to the conventional scheme using Bell state measurement of two indistinguishable photons with a beam splitter in the middle56,57,58,59,60. These robustness are inherent in the physical system with complete special symmetry by excluding the unnecessary external fields. Our work paves the way for a mode-matching error-tolerant quantum network that interconnects heterogeneous quantum computers.

## Methods

We use a single naturally occurring NV center in a high-purity type IIa chemical-vapor deposition-grown diamond with a crystal orientation <100> produced by element six. The diamond is cooled to 5 K to control the electron orbitals coherently. In order to achieve a zero magnetic field, the residual magnetic field including the geomagnetic field is canceled out by a three-dimensional coil. The currents of the three coils are adjusted by monitoring spin-echo coherence time, which reaches its maximum at a zero magnetic field33. Optically detected magnetic resonance measurement confirms that there are no carbon nuclear spins with hyperfine interaction above 0.1 MHz. Two orthogonal copper wires are attached to the sample surface (Fig. 1d) to apply microwaves with arbitrary polarization, which is the same configuration as ref. 40. The optical system consists of a homemade confocal microscope system (Fig. 1d), which is similar to the system used in ref. 44. The green laser is used for charge and spin initialization by nonresonant excitation, and the two red lasers are used for preparation for the |A2〉 state and spin measurement using the |Ey〉 state by resonant excitation. The experimental sequences are controlled by a master homemade field-programmable gate array (FPGA) with a clock frequency of 100 MHz, which outputs digital signals to a directly modulated green laser, two acousto-optic modulators (AOM) for the red lasers, as well as the triggers for two slave arbitrary waveform generator (AWG) modulating two EOM for the red laser and two IQ mixers for the microwaves. In the spin-photon correlation experiment shown in Fig. 3, the signals from the two APD for PSB and ZPL photons are counted and processed by the FPGA in real time in order to make the spin measurements conditioned on the ZPL photon measurement.