Quantum teleportation-based state transfer of photon polarization into a carbon spin in diamond

Quantum teleportation is a key principle for quantum information technology. It permits the transfer of quantum information into an otherwise inaccessible space, while also permitting the transfer of photon information into a quantum memory without revealing or destroying the stored quantum information. Here, we show reliable quantum state transfer of photon polarization into a carbon isotope nuclear spin coupled to a nitrogen-vacancy center in diamond based on photon-electron Bell state measurement by photon absorption. The carbon spin is ﬁ rst entangled with the electron spin, which is then permitted to absorb a photon into a spin-orbit correlated eigenstate. Detection of the electron after relaxation into the spin ground state allows post-selected transfer of arbitrary photon polarization into the carbon memory. The quantum state transfer scheme allows individual addressing of integrated quantum memories to realize scalable quantum repeaters for long-haul quantum communications, and distributed quantum computers for large-scale quantum computation and metrology.

Q uantum teleportation 1,2 is a widely used principle for quantum information technologies including quantum communication [3][4][5] and quantum computing 6 . Long-haul quantum communication requires quantum repeaters 7-13 based on quantum teleportation to transfer a quantum bit or qubit into a distant site without revealing the qubit state. Quantum computation 6 also requires quantum teleportation not only for oneway quantum computing but also for quantum blind computing 14 to securely transfer input and output data via quantum communication. Finally, quantum teleportation is also in demand for the quantum storage of highly confidential data, such as DNA data, that are securely transferred by photons into quantum memories.
We have recently demonstrated the transfer of photon polarization into a nitrogen nuclear spin in a single nitrogen-vacancy (NV) center in diamond 11 . Delteil et al. have also demonstrated heralded absorption of a single photon color qubit in a single quantum dot 12 . However, the stored information is limited to only one nuclear spin per one NV center or one electron spin per one quantum dot, which is a big obstacle for scaling up the memory size. On the other hand, there exist a large number of carbon isotopes ( 13 C), which also have nuclear spins, within the reach of a nitrogen-vacancy center via a hyperfine interaction even in diamond with a natural abundance of 13 C (1.1%), and the number can be increased by isotope enrichment technology 15 . The stored qubits in carbon nuclear spins must be independent of each other and individually addressable [16][17][18][19][20][21] . We have also recently demonstrated that the quantum state of isolated carbon nuclear spins weakly coupled to the NV center's electron can be ideally maintained by the geometric spin echo based on time reversal under a zero magnetic field 22 . However, it is hard to initialize and manipulate a spin-half carbon nuclear spin under a zero magnetic field, in contrast to the spin-1 nitrogen, which has a zero-field split state to allow manipulation by a resonant microwave.
In the present work, we successfully initialize and manipulate a carbon nuclear spin using nitrogen as a nanomagnet to lift the degeneracy of the electron while maintaining a magnetic field of zero at the carbon nuclear spin, then transfer the polarization state of photons into the spin quantum state of the thus-prepared carbon nuclear spin. We experimentally demonstrate that the state transfer process shows quantum nature and thus the transfer fidelity exceeds the classical limit, based on the quantum process tomography technique.

Results
Principle. Quantum teleportation consists in the preparation of an entanglement and the measurement in the Bell basis, resulting in post-selective transfer of the quantum state (Fig. 1a). In the current demonstration, we first prepare an entanglement between electron spin and carbon nuclear spin, and then measure photon polarization and electron spin in the Bell basis by photon absorption 23 to transfer the photon polarization state into the carbon spin state (Fig. 1b, c). In the practical protocol of the oneway quantum repeater system with an NV center at each node, the photon is emitted from one node, leaving an electron entangled with the emitted photon 24 (Fig. 1d). The success of the photon storage in the other node establishes the entanglement between two adjacent nodes.
A negatively charged NV center in diamond consists of a nitrogen impurity ( 14 N) and an adjacent vacancy (V), where the triplet state electron (e) is localized (Fig. 1b). Both the electron and the nitrogen nucleus show a spin 1 property constituting a V-type three-level system with two degenerate m s,I = ±1 states (denoted ðj ± 1i e;N Þ; which constitute a logical qubit called a geometric spin qubit 22,23,[25][26][27][28][29] , and an m s,I = 0 state (denoted |0〉 e,N ), which constitute an ancilla. These are split by a zero-field splitting of around 2.87 GHz for the electron and a nuclear quadrupole splitting of around 4.95 MHz for the nitrogen. On the other hand, a carbon nuclear spin ( 13 C), weakly coupled to the electron via a hyperfine interaction (0.9 MHz in this demonstration), shows a spin half property constituting a two-level system with two degenerate m I = ±1/2 states (denoted j"i C ; j#i C ) under a zero magnetic field (Fig. 1c). We utilize the nitrogen as a nanomagnet localized at the vacancy.
To prepare the spin entanglement between the electron and carbon nuclear spin, we first initialize them into j0i e j"i C . Although it is hard to initialize the carbon nuclear spin under a zero magnetic field, the nuclear quadrupole splitting of the nitrogen nuclear spin with the help of the polarized electron spin enables the polarization into jþ1i N , which is used as a nanomagnet to apply a local magnetic field on the electron to initialize the carbon nuclear spin (Fig. 1c) with the following sequence (see Supplementary Note 1). The coherent population trapping (CPT) using a red light resonantly exciting the electron into a spin-orbit correlated eigenstate jA 2 i ¼ 1 ffiffi 2 p ðjþ1; À1i l;e þ jÀ1; þ1i l;e Þ 23,24 (l and e, respectively denote orbital and spin angular momentums of an electron) with the right circular polarization |+1〉 p first polarizes the electron into jþ1i e 23 , which is then transferred into the nitrogen nuclear spin to polarize into jþ1i N (purple line in Fig. 2a). The degeneracy of the electron spin is therefore lifted via the hyperfine interaction with the nitrogen, allowing selective flip of the carbon nuclear spin j #i C to polarize into j "i C (green line in Fig. 2a). Initialization processes of the nitrogen and carbon nuclear spins are shown in Fig. 2b. The electron is again initialized into |0〉 e with the red light resonant to the |A 1 〉 state. The electron and carbon are then manipulated with a microwave optimized by GRAPE (GRadient Ascent Pulse Engineering) 30 and a radiowave to create entanglement between them into jΦ þ i e;C ¼ 1 ffiffi 2 p ðjþ1; "i e;C þ jÀ1; #i e;C Þ, which is one of four Bell states, along with the quantum circuit shown in Fig. 2c, where we define the SU(3) X gate that applies the X gate in the partial space spanned by jai and jbi while retaining jdi as X The Y gate applied to the spin-half carbon nuclear spin stands for Pauli operator σ y .
Next, we allow the electron to absorb an incoming photon with arbitrary polarization jψi p ¼ αj þ 1i p þ βj À 1i p , which excites the electron into another spin-orbit correlated eigenstate jA 1 i ¼ 1 ffiffi 2 p ðjþ1; À1i l;e À jÀ1; þ1i l;e Þ 23 . The photon absorption projects the polarization state of the photon and the spin state of the electron into one of the Bell states as demonstrated in Kosaka et al. 23 Although the projection is probabilistic and allows only partial Bell state measurement, we repeat the transfer process until absorption into the jA 1 i state stops after non-radiative relaxation into j0i e of the orbital ground state. The projection of the prepared state composed of an arbitrary photon polarization jψi p ¼ αj þ 1i p þ βj À 1i p and the electron-carbon entangled state Φ þ i j e;C ¼ 1 ffiffi 2 p ð þ1; "i j e;C þ À1; #i j e;C Þ into the |A 1 〉 state is described as where the photon polarization state jψi p was substituted by the ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-019-0158-0 electron orbital state jψi l ¼ αj þ 1i l þ βj À 1i l since the photon polarization state p corresponds to the electron orbital state l via the angular momentum conservation or the polarization selection rule. The coefficient i 2 implies that success probability of the partial Bell state measurement is i However, the norm of the transferred nuclear spin becomes unity, which means that the nuclear spin state is in principle purified, after the post-selection of the electron spin state. The resulting nuclear spin state corresponds to the photon polarization state with the additional unitary operation σ y .
Experiments. We first measure the phase correlation between the input photon and the transferred carbon to show that the transfer operation conserves the quantum coherence. In principle, the photon polarization state ψi should be transferred into the carbon nuclear spin polarization state ψi with the additional unitary transformation σ y determined by Eq. (1). Figure 3a shows the photon polarization dependence of the carbon nuclear spin population measured in jþi C À jÀi C axis ± i , which is obtained by measuring the photon count after the irradiation of a radiowave and a microwave followed by the red light resonant to the jE x i state (see Supplementary Fig. 5). Strong anti-phase correlation is observed as expected, indicating the quantum nature of the transfer.
To evaluate the fidelity of the quantum process during the state transfer, we prepare the six mutually-unbiased basis states of the photon polarization ðjþ1i p ; jÀ1i p ; (Fig. 3b) and estimate the carbon nuclear spin state after the transfer based on the quantum state tomography. The Bloch vectors for the carbon nuclear spin states transferred from the six photon polarizations are reconstructed as shown in Fig. 3c. The fidelities projected into the ideal state reach 78 ± 2% on average, which well exceeds the classical limit of 67% (Fig. 3d). With the reconstructed Bloch vectors, we evaluated the quantum channel of the transfer based on the quantum process tomography as shown in Fig. 3e. The fidelity of the transfer process is estimated from the χ matrix defined as Tr½χ y ideal χ exp to be 76%, indicating that the transfer channel maintains the quantum coherence.
The fidelity of the quantum state transfer degrades due to the imperfection of entanglement generation and Bell state measurement, which is caused by the incomplete initialization of spins (Fig. 3f), the mixing of orbital excited states due to the crystal strain (Fig. 3g)

Spin-orbit interaction
Hyperfine interaction e Fig. 1 Schematics of the quantum state transfer. a Protocol of the transfer scheme. We first prepare an entanglement between electron spin (e) and carbon nuclear spin ( 13 C) (bottom), and then measure photon polarization (p) and electron spin (e) in the Bell basis by photon absorption 23 (middle), which announces the success of the quantum state transfer from the photon into the carbon (top). b The lattice structure of diamond contains a nitrogen-vacancy (NV) center with surrounding carbon nuclear spins. An electron locates in the vacancy. c Energy level diagram of the photon based on the right/left ðjþ1i p =jÀ1i p Þ circular polarizations, the electron based on up/down ðj þ 1i e =j À 1i e Þ spin polarizations, and the carbon nuclear spin based on up/down ðj"i c =j#i c Þ nuclear spin polarizations. The nitrogen impurity ð 14 NÞ polarized in j þ 1i N nuclear spin polarization serves as a nanomagnet to apply a magnetic field only for the electron, allowing the preparation of entanglement between the hyperfine coupled electron and carbon spins under a zero magnetic field. The electron is then resonantly excited to a spin-orbit correlated eigenstate ðjA 1 iÞ by photon absorption, allowing the measurement of the photon polarization and electron spin in the Bell basis to result in quantum state transfer from the photon into the carbon. d The supposed protocol of the one-way quantum repeater system with an NV center at each node. A photon is emitted from one node (left), leaving an electron entangled with the emitted photon. The success of the photon storage in the other node (right) establishes the entanglement between two adjacent nodes measurement (see Supplementary Note 2), and gate error. The initialization fidelity can be improved by repeating the initialization sequence. The effect of the crystal strain can be calibrated by identifying e x and e y , which are the x and y components of the crystal strain 23,26 . Phase rotation can be avoided by initializing the nitrogen nuclear spin into j0i N before the transfer. Gate error can be improved by the development of a robust gate against inhomogenous broadening by the environmental nuclear spins.

Discussion
The hyperfine coupling of the nitrogen nuclear spin with the electron spin takes a constant value of around 2.2 MHz, while that of the carbon nuclear spin largely depends on its position, becoming smaller with increasing distance from the electron. By transferring the photon polarization state to the nuclear spin state of the carbon instead of the nitrogen, not only can the number of quantum memories which store the photon polarization state be increased but also the decoherence caused by the orbital excited state can be reduced by selecting a more distant carbon nuclear spin with smaller hyperfine coupling. Coupled carbon nuclear spins can also be used for a quantum memory that is robust against noise by utilizing a decoherence-protected subspace (DPS) 20 . The demonstration is independent of the photon number since we transferred the photon polarization state into the nuclear spin polarization state of a single carbon. However, the single photon transfer 12 is important for the application of quantum key distribution with a single photon. Although around 10 4 photons are contained in a single pulse (200 nW, 20 ns), the probability of absorbing more than two photons is about 2.5% (see Supplementary Note 3).
Quantum efficiency of the state transfer is decomposed into the probability of photon absorption followed by relaxation into the j0i e state and that of the j0i e state detection. The overall efficiency demonstrated here is estimated to be 3 × 10 −6 since the incoming light contains about 10 4 photons and the detected signal contains 3 × 10 −2 photons in average (PL intensity: 500 counts per second; gate width: 60 μs). Both the absorption and emission probabilities can be increased at least seven times by implementing a solid immersion lens and much more by implementing an optical cavity. In principle, the absorption probability is limited to 25% by the partial Bell state measurement in addition to 40% by the relaxation into the j0i e state, but the detection probability can be 100% with the single-shot projective measurement. Even though the efficiency is not perfect, the state transfer scheme allows postselection of the succeeded transfer, leading to high fidelity transfer independent of the absorption efficiency. The transfer rate can be drastically increased by introducing NV integrated memories controlled with optical holonomic quantum gates [26][27][28][29] . The transfer can be also deterministic in principle by introducing a feedback loop with a tunable waveplate 31 .   Fig. 2 Detailed scheme of the quantum state transfer. a Optically detected magnetic resonance (ODMR) spectra before and after the initialization process. The red, purple and green lines respectively show the spectrum before the initialization, after the initialization of only the nitrogen, and after initialization of both the nitrogen and carbon. The splittings of 2.2 MHz and 0.9 MHz are caused by the hyperfine interaction of the electron with the nitrogen and carbon, respectively. The achieved initialization fidelities are 94% for the nitrogen and 90% for the carbon. Error bars defined as the standard deviation of photon count are within point size. b Level diagrams showing populations for the nitrogen and the carbon nuclear spin corresponding to three steps in a. c Quantum circuit representing the transfer scheme. The controlled NOT(CNOT)-like symbols represent two or three qubit SU(3) gates, where the gates shown in the box are operated on the target qubit conditioned by the control qubit if the state is as given in the black circle or not as given in the white circle (double-sign corresponds). Below the circuit, energy level diagrams show the gate-induced transitions of the population in the ideally initialized case. Microwave (MW) 1 performs Toffoli-like π-gate to introduce the electron-carbon interaction, radio frequency (RF) performs controlled π/2 gate to generate a superposition of the carbon nuclear spin, MW2 performs controlled π-gate to eliminate the electron-carbon interaction, and MW3 again performs Toffoli-like π-gate to convert the superposition into an electron-carbon entanglement. Detection in the |0〉 e state relaxed after resonant excitation into |A 1 〉 post selects the photon polarization and electron spin states on a certain Bell state 23 We demonstrated reliable quantum state transfer of photon polarization into a carbon nuclear spin coupled to an NV center in diamond based on photon-electron Bell state measurement by photon absorption 23 . Detection of the electron relaxed in the spin ground state post-selected the transfer of the photonic quantum state in the long-lived carbon memory without revealing or destroying the stored quantum information. This demonstration shows that the state transfer scheme can be applied not only for a spin 1 system but also for a spin half system. The scheme allows integrated quantum memories to be individually addressed in order to realize scalable quantum repeaters for long-haul quantum communications and distributed quantum computers.

Methods
Sample and experimental conditions. We used a negatively charged NV center in a type-II a bulk diamond made by Element Six. The sample has a <001> crystal orientation. We adjusted the position of a permanent magnet in the direction along the geomagnetism to achieve a zero magnetic field by reducing the Zeeman splitting in the ODMR spectrum within 0.1 MHz. The NV center was located about 30 μm below the surface and has a crystal strain of about 1.8 GHz, which was estimated by the splitting between the |E x 〉 and |E y 〉 transitions in the photoluminescence spectrum. The polarization of the photon was calibrated for the NV center to be properly applied by considering the Hamiltonian including the offalignment between the optical axis and the crystal orientation 26 .
Microwave setup. A 25 μm copper wire mechanically attached to the surface of the diamond was used to apply a microwave and a radiowave. A shaped microwave around 2877 MHz optimized by GRAPE 30 for manipulating the electron spin was generated by up-converting a local oscillator (2776 MHz) with a radiowave around 100 MHz generated by an arbitrary wave generator (Keysight 33622A). A radiowave around 0.9-7 MHz for manipulating the nuclear spin was directly generated by an arbitrary wave generator (Keysight 33522A).
Laser setup. A green laser (wavelength: 532 nm; power: 150 μW) was pulsed by an AOM (acousto-optic modulator) with an extinction ratio of about 10 3 . A red tunable laser, whose wavelength (~637.2 nm) was set to be resonant to |A 1 〉 was branched into three beams. Two of them were pulsed by both an AOM and EOM (electro-optic modulator) with an extinction ratio of about 10 9 and were used for initialization of the electron into |0〉 e (power:~30 nW) and for the transfer of photon polarization into carbon nuclear spin (power:~200 nW), respectively. The other beam was pulsed only by an AOM with an extinction ratio of about 10 4 together with an EOM as a phase modulator to set its frequency to resonant to |E x 〉 and used for the readout of |0〉 e (power:~5 nW). Another red tunable laser (wavelength:~637.2 nm; power:~500 nW) was pulsed by both an AOM and EOM with an extinction ratio of about 10 9 and a frequency set to be resonant to |A 2 〉, and was used for the CPT of the electron into | + 1〉 e . Arbitrary polarization was set using a pair of variable wave plates placed at the end of each beam line.

Data availability
The data that support the findings of this study are available from the corresponding author upon request. a Correlation between the photon phase angle (twice the polarization angle) and the carbon nuclear spin population measured in jþi C À jÀi C axis, which is converted into the population of the electron spin ancilla state. b The Poincare sphere representation of the polarization state of the input light. c The Bloch sphere representation of the quantum state stored in the carbon nuclear spin reconstructed by the quantum state tomography. Six basis polarization states of the incoming photon are transferred into the memory carbon. The meshed frame shows the distorted Bloch sphere estimated by the process tomography measurements shown in e. d The state fidelities of the carbon for the six basis polarization states defined by Tr½ρ y ideal ρ exp . The averaged state fidelity is 78%, which well exceeds the classical limit of 67%. e Matrix representation of the quantum process tomography for the state transfer. The matrix elements χ exp are calculated based on the transferred states shown in c. The transfer process fidelity defined by Tr½χ y ideal χ exp is 76%. Error bars defined as the standard deviation of photon count in a, c-e are within point size (<2%). f Simulated dependence of the transfer fidelity on the initialization fidelities for the carbon and nitrogen nuclear spins for the photon polarizations (i), jþi p ; jÀi p ; j þ ii p ; j À ii p ; (ii), j þ 1 i p , and (iii), j À 1i p . Solid circles represent the ranges of the initialization errors for the demonstration. g Simulated dependence of the transfer fidelity on the two vector components of the crystal strain for the photon polarizations (i), | + 〉 p , (ii), | + i〉 p , and (iii), | + 1〉 p . Dashed circles represent the ranges of the crystal strain for the demonstration