Abstract
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 nitrogenvacancy center in diamond based on photonelectron Bell state measurement by photon absorption. The carbon spin is first entangled with the electron spin, which is then permitted to absorb a photon into a spinorbit correlated eigenstate. Detection of the electron after relaxation into the spin ground state allows postselected 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 longhaul quantum communications, and distributed quantum computers for largescale quantum computation and metrology.
Introduction
Quantum teleportation^{1,2} is a widely used principle for quantum information technologies including quantum communication^{3,4,5} and quantum computing^{6}. Longhaul quantum communication requires quantum repeaters^{7,8,9,10,11,12,13} 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 nitrogenvacancy (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 nitrogenvacancy 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 spinhalf carbon nuclear spin under a zero magnetic field, in contrast to the spin1 nitrogen, which has a zerofield 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 thusprepared 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 postselective 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 Vtype threelevel system with two degenerate m_{s,I} = ±1 states (denoted \((\vert \pm 1\rangle_{\mathrm{e,N}}),\) 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 zerofield 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 twolevel system with two degenerate m_{I} = ±1/2 states (denoted \(\vert{\uparrow}\rangle_{\mathrm{C}},\vert {\downarrow} \rangle_{\mathrm{C}}\)) 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 \(\vert0\rangle _{\mathrm{e}}\vert {\uparrow} \rangle_{\mathrm{C}}\). 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 \({\vert} {+} 1\rangle_{\mathrm{{N}}}\), 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 spinorbit correlated eigenstate \( A_{2} \rangle = \frac{1}{{\sqrt 2 }}({ + 1,  1} \rangle _{l,{\mathrm{e}}} + {{  1, + 1}\rangle }_{l,{\mathrm{e}}})\)^{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 \({\vert} {+} 1\rangle_{\mathrm{{e}}}\)^{23}, which is then transferred into the nitrogen nuclear spin to polarize into \({\vert} {+} 1\rangle_{\mathrm{{N}}}\) (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 \({\vert} \!\downarrow \rangle_{\mathrm{{C}}}\) to polarize into \({\vert}\! \uparrow \rangle_{\mathrm{{C}}}\) (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 \( {{{\mathrm{\Phi }}^ + } \rangle }_{{\mathrm{e,C}}} = \frac{1}{{\sqrt 2 }}( { { { + 1, \uparrow }\rangle } _{{\mathrm{e,C}}} +  { {  1, \downarrow }\rangle }_{\mathrm{e,C}}} )\), 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 \({\vert} a \rangle\) and \({\vert} b \rangle\) while retaining \({\vert} d \rangle\) as \(X_{(\pi )}^{\left( {\left. {\left a \right.} \right\rangle ,\left {\left. b \right\rangle } \right.} \right)} = \left {\left. a \right\rangle } \right.\left\langle b \right. + \left {\left. b \right\rangle } \right.\left. {\left\langle a \right.} \right + \left {\left. d \right\rangle } \right.\left. {\left\langle d \right.} \right\). The Y gate applied to the spinhalf carbon nuclear spin stands for Pauli operator σ_{y}.
Next, we allow the electron to absorb an incoming photon with arbitrary polarization \(\vert \psi\rangle_{\mathrm{p}} = \alpha \vert +1\rangle_{\mathrm{p}} + \beta\vert1\rangle_{\mathrm{p}}\), which excites the electron into another spinorbit correlated eigenstate \( A_{1} \rangle = \frac{1}{{\sqrt 2 }}(  {{ + 1,  1}\rangle } _{l,{\mathrm{e}}}  { {  1, + 1}}\rangle _{l,{\mathrm{e}}})\)^{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 \({\vert} A_{1} \rangle\) state stops after nonradiative relaxation into \({\vert} 0 \rangle_{\mathrm{e}}\) of the orbital ground state. The projection of the prepared state composed of an arbitrary photon polarization \({\vert}\psi\rangle_{\mathrm{p}}= \alpha\vert+ 1\rangle_{\mathrm{p}}+ \beta\vert 1\rangle_{\mathrm{p}}\) and the electroncarbon entangled state \(\left {\left. {{\mathrm{\Phi }}^ + } \right\rangle } \right._{\mathrm{e,C}} = \frac{1}{{\sqrt 2 }}( {\left {\left. { + 1, \uparrow } \right\rangle } \right._{{\mathrm{e,C}}} + \left {\left. {  1, \downarrow } \right\rangle } \right._{{\mathrm{e,C}}}} )\) into the A_{1}〉 state is described as
where the photon polarization state \(\vert \psi\rangle_{\mathrm{p}}\) was substituted by the electron orbital state \(\vert \psi\rangle_{l}= \alpha \vert +1\rangle_{l} + \beta \vert 1\rangle_{l}\) 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 \(\frac{i}{2}\) implies that success probability of the partial Bell state measurement is \(\left {\frac{i}{2}} \right^2 = \frac{1}{4}\). However, the norm of the transferred nuclear spin becomes unity, which means that the nuclear spin state is in principle purified, after the postselection of the electron spin state. The resulting nuclear spin state corresponds to the photon polarization state with the additional unitary operation \(\sigma_{y}\).
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 \(\left {\left. \psi \right\rangle } \right._{\mathrm{p}} = \frac{1}{{\sqrt 2 }}\left( {\left {\left. { + 1} \right\rangle } \right._{\mathrm{p}} + e^{i\phi }\left {\left. {  1} \right\rangle } \right._{\mathrm{p}}} \right)\) should be transferred into the carbon nuclear spin polarization state \(\left {\left. \psi \right\rangle } \right._{\mathrm{C}} = \frac{1}{{\sqrt 2 }}\left( {\left {\left. \uparrow \right\rangle } \right._{\mathrm{C}} + e^{i\left( {\pi  \phi } \right)}\left {\left. \downarrow \right\rangle } \right._{\mathrm{C}}} \right)\) 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 \(\vert+\rangle_{\mathrm{C}} \vert \rangle_{\mathrm{C}}\) axis \(\left( {\left {\left. \pm \right\rangle } \right._{\mathrm{C}} = \frac{1}{{\sqrt 2 }}\left( {\left {\left. \uparrow \right\rangle _{\mathrm{C}}} \right. \pm \left {\left. \downarrow \right\rangle _{\mathrm{C}}} \right.} \right)} \right)\), 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 \(\vert E_{x}\rangle\) state (see Supplementary Fig. 5). Strong antiphase 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 mutuallyunbiased basis states of the photon polarization \(( { { { + 1}\rangle } _{\mathrm{p}}, { {  1} \rangle } _{\mathrm{p}}, { + \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( { { { + 1} \rangle } _{\mathrm{p}} +  { {  1} \rangle } _{\mathrm{p}}} ),  {  \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( { { { + 1} \rangle } _{\mathrm{p}}  \! { {  1} \rangle } _{\mathrm{p}}} ), { { + i} \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( { { { + 1} \rangle } _{\mathrm{p}} + i { {  1} \rangle } _{\mathrm{p}}} )} \\ {, { {  i} \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( { { { + 1} \rangle } _{\mathrm{p}}  i { {  1} \rangle } _{\mathrm{p}}} } ))\) (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 \({\mathrm{Tr}}[\chi _{{\mathrm{ideal}}}^\dagger \chi _{{\mathrm{exp}}}]\) 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), phase rotation during the Bell state 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 \(\vert 0\rangle_{\mathrm{N}}\) 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 decoherenceprotected 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 \(\vert 0\rangle_{\mathrm{e}}\) state and that of the \(\vert 0\rangle_{\mathrm{e}}\) 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 \(\vert 0\rangle_{\mathrm{e}}\) state, but the detection probability can be 100% with the singleshot 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}.
We demonstrated reliable quantum state transfer of photon polarization into a carbon nuclear spin coupled to an NV center in diamond based on photonelectron Bell state measurement by photon absorption^{23}. Detection of the electron relaxed in the spin ground state postselected the transfer of the photonic quantum state in the longlived 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 longhaul quantum communications and distributed quantum computers.
Methods
Sample and experimental conditions
We used a negatively charged NV center in a typeII 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 upconverting 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 (acoustooptic 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 (electrooptic 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.
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Acknowledgements
We thank Yuichiro Matsuzaki, Burkhard Scharfenberger, Kae Nemoto, William Munro, Norikazu Mizuochi, Nobuyuki Yokoshi, Fedor Jelezko, and Joerg Wrachtrup for their discussions and experimental help. This work was supported by Japan Society for the Promotion of Science (JSPS) GrantsinAid for Scientific Research (nos. 16H06326, 16H01052, 16K13818); by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) as an “Exploratory Challenge on PostK computer” (Frontiers of Basic Science: Challenging the Limits); by the Research Foundation for OptoScience and Technology; and by a Japan Science and Technology Agency (JST) CREST Grant (no. JPMJCR1773).
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The experiment was designed and analysed by K.T., R.K., H. Kano and H. Kosaka. Measurements were made by K.T., R.K. and H. Kano. Y. S. supported experiments in technical matters. H. Kosaka supervised experiments. K. T. and H. Kosaka wrote the paper.
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Tsurumoto, K., Kuroiwa, R., Kano, H. et al. Quantum teleportationbased state transfer of photon polarization into a carbon spin in diamond. Commun Phys 2, 74 (2019). https://doi.org/10.1038/s4200501901580
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