Optically addressable universal holonomic quantum gates on diamond spins

The ability to individually control the numerous spins in a solid-state crystal is a promising technology for the development of large-scale quantum processors and memories. A localized laser field offers spatial selectivity for electron spin manipulation through spin–obit coupling, but it has been difficult to simultaneously achieve precise and universal manipulation. Here, we demonstrate microwave-driven holonomic quantum gates on an optically selected electron spin in a nitrogen-vacancy centre in diamond. The electron spin is precisely manipulated with global microwaves tuned to the frequency shift induced by the local optical Stark effect. We show the universality of the operations, including state initialization, preparation, readout and echo. We also generate optically addressable entanglement between the electron and adjacent nitrogen nuclear spin. High-fidelity operations are achieved by applying amplitude-alternating pulses, which are tolerant to fluctuations in microwave intensity and detuning. These techniques enable site-selective quantum teleportation transfer from a photon to a nuclear spin memory, paving the way for the realization of distributed quantum computers and the quantum Internet with large-scale quantum storage. Microwave-driven holonomic quantum gates on an optically selected electron spin in a nitrogen-vacancy centre in diamond are demonstrated. Optically addressable entanglement is generated between the electron and adjacent nitrogen nuclear spin.

I ndividual spins in solid-state materials are promising candidates for quantum memories in large-scale quantum storage as their weak interaction with the environment enables long coherence times [1][2][3][4][5][6] . However, it is difficult to apply microwaves, which are typically used for spin control, locally on submicrometre scales due to their long wavelength, making it challenging to selectively manipulate the individual spins. Several techniques have been developed to manipulate the individual spins resonant to a frequency that is spatially non-uniform owing to a gradient magnetic field [7][8][9] or their own random quantization axes determined by the crystal axes 10, 11 . All-optical techniques [12][13][14][15][16][17][18] for spin manipulation via spin-orbit coupling have also been developed to increase the spatial resolution into the diffraction limit. For example, an optical Stark effect 19,20 (or stimulated Raman 21 ) and geometric phase gate [22][23][24][25][26][27] on colour centres in diamond and quantum dots in semiconductors have been extensively examined. However, the fidelity of control over those spins is not high enough for practical use owing to their fast orbital relaxation.
In this Article, we demonstrate optically addressable high-fidelity manipulations of an electron spin by combining global microwave manipulation and local optical Stark shift using a nitrogen-vacancy (NV) centre in diamond. We also show that this optically addressable manipulation of the electron spin can be used to selectively manipulate the nuclear spin of the nitrogen atom constituting the NV centre, and even to control the interaction between the electron and nuclear spins. Finally, we demonstrate the generation of quantum entanglement between the electron and nuclear spins to prepare for the quantum state transfer 28,29 of a photon state into the nuclear spin state of the nitrogen quantum memory. In addition to offering fundamental quantum control within the NV centre, this scheme is compatible with interconnection with photons and could contribute to faster quantum networks by integrating independent NV centres 30 .

System and scheme
The negatively charged NV centre has the hybridized level structure of an electron orbital with a spin-1 electron spin. The orbital ground state has a degenerate V-shaped three-level spin structure. At zero magnetic field, |±1〉 S states with magnetic quantum numbers m S = ±1 are degenerate, and the |0〉 S state with m S = 0 energetically splits from |±1〉 S , known as zero-field splitting, of about 2.88 GHz. We here define |±1〉 S as the computational bases for a qubit. There is no direct magnetic transition between the two qubit bases. Instead, there is a transition between a bright state |B〉 S and an ancilla state |0〉 S depending on the polarization of the microwave field, while a dark state |D〉 S remains unchanged. A cyclic evolution in a bright space {|B〉 S , |0〉 S } induces a geometric phase shift γ in |B〉 S depending on the trajectory of the space. That allows the qubit to rotate by γ around the |B〉 S -|D〉 S axis to realize a universal single-qubit gate 31, 32 .
The key idea of this work is to induce an optical Stark shift in the ancilla state |0〉 S to correlate the microwave frequency resonant to the bright space with optical intensity (Fig. 1a). One of the orbital excited states |E y 〉 is allowed to make an optically coherent transition from |0〉 S at low temperature. Detuning the laser frequency from the energy gap between |E y 〉 and |0〉 S suppresses their optical transition to induce repulsive energy shifts owing to the optical Stark effect (Fig. 1b). The shift results in the modulation of the resonance frequency between |B〉 S and |0〉 S in the bright space, which is observed by the optically detected magnetic resonance spectrum (Fig. 1c) or Ramsey interference (Extended Data Fig. 1) (Methods) of the electron spin. The optical Stark shift Δ S depends on the laser detuning Δ L and the optical power or Rabi frequency Ω L , and is approximated by Δ S ≈ Ω L 2 /4Δ L , provided that Ω L ≪ Δ L (Fig. 1d). Note that the optical Stark shift does not change the degeneracy of the qubit space and thus we could not directly manipulate the qubit. However, a microwave tuned to the frequency shift in the bright space enabled selective manipulation of the qubit. Although a microwave Rabi frequency Ω MW ≪ Δ S is required for high frequency selectivity, in practice there is an upper limit for Δ S imposed by the condition Ω L ≪ Δ L to suppress the optical absorption. Moreover, Ω MW cannot be reduced because of the inhomogeneous broadening of |±1〉 S due to hyperfine interactions with environmental nuclear spins.

Optically addressable universal holonomic quantum gates on diamond spins
Yuhei Sekiguchi 1,2 , Kazuki Matsushita 3 , Yoshiki Kawasaki 3 and Hideo Kosaka 1,2,3 ✉ The ability to individually control the numerous spins in a solid-state crystal is a promising technology for the development of large-scale quantum processors and memories. A localized laser field offers spatial selectivity for electron spin manipulation through spin-obit coupling, but it has been difficult to simultaneously achieve precise and universal manipulation. Here, we demonstrate microwave-driven holonomic quantum gates on an optically selected electron spin in a nitrogen-vacancy centre in diamond. The electron spin is precisely manipulated with global microwaves tuned to the frequency shift induced by the local optical Stark effect. We show the universality of the operations, including state initialization, preparation, readout and echo. We also generate optically addressable entanglement between the electron and adjacent nitrogen nuclear spin. High-fidelity operations are achieved by applying amplitude-alternating pulses, which are tolerant to fluctuations in microwave intensity and detuning. These techniques enable site-selective quantum teleportation transfer from a photon to a nuclear spin memory, paving the way for the realization of distributed quantum computers and the quantum Internet with large-scale quantum storage.
We therefore developed an amplitude-alternating microwave pulse that works either as an identity operation or a rotation operation, depending on whether the microwave is on or off the resonance frequency, even by a small frequency difference (Extended Data Fig. 2,3) (Methods). The amplitude-alternating pulse reverses the amplitude (or shifts the relative phase by π) at the middle point of a square-envelope pulse. In the absence of the optical Stark shift without the addressing laser, the microwave is on resonance and thus |B〉 S rotates along a great circle in the Bloch sphere representing the bright space in the first half of the pulse, and then reverses the rotation to cancel the first rotation and returns to |B〉 S in the second half of the pulse (Fig. 1e). In the presence of the optical Stark shift with the addressing laser, the microwave is off resonance and thus |B〉 S rotates along two adjacent small circles in a figure of eight and returns to |B〉 S (Fig. 1f). As a result, |B〉 S obtains a geometric phase γ proportional to the solid angle enclosed by the trajectory of the Bloch vector as where the pulse length is

Optical addressability
We first demonstrated that the addressing laser switches between the two gate operations on the electron spin. The details of the experimental set-up are given in the Methods. The targeted optical Stark shift was Δ S = 4 MHz given by Ω L = 160 MHz and Δ L = 1.6 GHz, which were optimized so that Δ S T 2 Stark (which indicates the quality of the frequency shift, where T 2 Stark is the coherence time) became as large as possible (Methods). The amplitude-alternating microwave pulse was set to show Ω MW = 4√3 MHz with t MW = 250 ns so the qubit rotated by π in the qubit space. If the irradiated laser power is adequate, the initial state |+〉 S should rotate to |−〉 S . On the other hand, if the laser power is sufficiently small, the state should return to the original |+〉 S (Fig. 2a,b). Figure 2c,d shows the populations of the final state as functions of the laser and microwave powers. The measurements agree well with the simulations. Note that even when the microwave power was not necessarily optimal, the gate operation was inactive as long as the laser power was small enough. This scheme is notable in practical situations where microwave antennas cannot generate spatially uniform fields.
We next demonstrated gate contrast depending on the NV position. We measured the spatial distribution of the optical Stark shift by displacing the focal point of the addressing laser off the NV centre, resulting in a steep reduction of the shift (Fig. 2e). The distribution was well fitted by a Gaussian function with a standard deviation of 247 nm, which is consistent with the spatial distribution of the laser power. With this distribution, the spatial distributions of |±1〉 S probabilities after the gate operation ( Fig. 2f) were inferred from the dependence of the probability on the laser Rabi frequency shown in Fig. 2c,d.

Optically addressable universal single-qubit operation
We then demonstrated the universality of single quantum gates by showing the Pauli-X, Y and Z gate operations conditioned by  laser irradiation. The Pauli-X, Y and Z gates required γ = π, thereby requiring a laser power corresponding to Ω MW = √3Δ S , and pulse width of t MW = 1/Δ S . We set Δ S = 2 MHz, Ω MW = 2√3 MHz and t MW = 500 ns due to the upper limits of Ω MW in our current microwave set-up. The gate fidelities were estimated by quantum process tomography with (active) and without (inactive) the addressing laser. Figure 3a,b,c guarantees fidelities over 90% in any situations that show Pauli-X, Y and Z gate and identity operations. The spin coherence after the gate operation was well maintained even when the gate time was not sufficiently short compared with the spin coherence time T 2 * ≈ 3 µs, even in the inactive case. This is because the driving microwave suppresses the non-commutable phase relaxation induced by a 13 C nuclear spin bath, except for the Pauli-Z gate, where the drive Hamiltonian was commutable with the phase relaxation. We also show optically addressable preparation (Fig. 3d) and readout ( Fig. 3e) of the X, Y and Z basis states in the qubit space by the amplitude-alternating microwave pulse, as well as the gate operation. Although the success probabilities were not particularly high in the active case, the error probabilities were extremely low in the inactive case, which enables high-fidelity initialization conditioned by the addressing laser irradiation (Fig. 3f) (Extended Data Fig. 4) (Methods). We also show that the coherence can be maintained by a spin echo using an optically addressable gate (Fig. 3g).

Optically addressable quantum entanglement generation
We finally demonstrated optically addressable entanglement generation between the electron and nitrogen nuclear spins as a component of a quantum state transfer 28,29 from an entangled photon [33][34][35] to a nuclear spin quantum memory. We utilized optically addressed electron spin to activate or deactivate the nuclear spin gate operations. Conditioned by the electron spin to be in |0〉 S , the nuclear spin was universally manipulated with a polarized radiowave within the same three-level structure as that of the electron spin (Extended Data Fig. 5) (Methods). The pulse sequence is shown in Fig. 4a,d. The initial state prepared in |+1, 0〉 S,N , where subscript N denotes the nuclear spin state, was optically selectively transferred to |0, 0〉 S,N by the manipulation shown in Fig. 3e. After the nuclear spin rotation to |+〉 N conditioned by the |0〉 S state, the double controlled NOT (DCNOT) gate 28 generated an entangled state |ψ + 〉 = (|+1, −1〉 S,N + |−1, +1〉 S,N )/√2 (Fig. 4b). The DCNOT gate used an optimized waveform generated by the GRAPE algorithm to increase the gate fidelity 31 . If the addressing laser was not irradiated, the nuclear spin state remained unchanged as the electron spin was in the qubit space (Fig. 4e). We estimated the nuclear spin state using quantum state tomography with (active) and without (inactive) the addressing laser under the same microwave and radiowave conditions. Figure 4c,f shows that the fidelities of |+1, 0〉 S,N and |ψ + 〉 were respectively 83.4% and 91.1% in the inactive and active cases. The degradation of the fidelity was mainly due to unexpected dynamics of the electron spin even when the nuclear spin was in |0〉 N at the DCNOT gate. Although a specific |+1, 0〉 S,N state was prepared in the demonstration, we could apply the optically addressable entanglement generation scheme to any arbitrary states, including entangled states with other qubits.

Discussion
The amplitude-alternating microwave pulse developed in this study can be applied not only to the optical Stark shift, but also to the  gradient magnetic field scheme to individually manipulate qubits with different resonance frequencies. It also serves simply as a quantum gate that is extremely robust against Rabi frequency errors (Methods). The reason for the improved fidelity of our scheme over all-optical schemes is the use of an excess degree of freedom that is easier to control. Other techniques such as composite pulses 36   |-1, -1〉 |+1, +1〉  numerically optimized waveforms 37 could also further improve the fidelity of two-qubit gates, which is much harder to achieve than in single-qubit gates.
As the laser irradiation itself does not change the state of the qubit, unlike in the conventional all-optical approach, the spatial resolution may be further increased by using a strong gradient spot or by introducing additional detuned lasers to modify the Stark shifts. By increasing the resolution to several tens of nanometres, it would be possible to selectively control multiple NV centres that are coherently coupled by magnetic dipole-dipole interactions 38 or optical dipole-dipole interactions 39,40 . The demonstration of optically addressable manipulations in combination with optically coherent NV centre formation methods [41][42][43] and real-time optical deflection techniques [44][45][46] will enable selective access to more than 10,000 qubits in a 10 × 10 × 10 μm 3 image volume, paving the way to large-scale quantum storage.

Online content
Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/ s41566-022-01038-3.

Methods
Experimental set-up. The experiments in this work used a single naturally occurring NV centre in a high-purity type-IIa electric grade diamond with crystal orientation <100> grown by Element Six. The diamond was cooled to 5 K for coherent control of the electron orbitals. A purpose-built confocal microscope system was used to address the single NV centre in the following the method used by Sekiguchi et al. 35 . A green laser probabilistically reset the charge state. Spin pumping by the resonant excitation to |E 1,2 〉 initialized the electron spin state to |0〉 S . To read out the electron spin qubit state, the phonon sideband emissions induced by the resonant excitations to |E y 〉 were detected by an avalanche photodiode after a transformation between the readout basis and |0〉 S using a microwave π pulse. The power of the addressing laser for the optical Stark shift was monitored before the objective lens and stabilized to within 0.1% by proportional-integral-differential (PID) control. Two orthogonal wires were attached over the diamond surface to utilize the polarization degrees of freedom of the microwave and radiowave. The environmental magnetic field was nullified by a three-dimensional coil to achieve a zero magnetic field. To reduce the inhomogeneous broadening of the resonance frequency of the electron spin due to the hyperfine interaction with the nuclear spins, the nitrogen nuclear spin was initialized to |m I = 0〉 N , where m I is the magnetic quantum number of the nitrogen nuclear spin, in all experiments in this work.
Optimization of the optical Stark shift. As the optical Stark shift utilizes the orbital perturbation, the accompanying spin state is affected by the orbital relaxation to cause a phase relaxation. Hence, it is important to select the experimental parameters appropriately to achieve the best performance of the gates. To suppress the phase relaxation due to the orbital relaxation, the ratio of the detuning Δ L to the optical Rabi frequency Ω L must be sufficiently large. There are also other requirements: Δ L should be smaller than the energy gap of the orbital fine structure, and Ω L should be large enough to induce a larger frequency shift than the inverse of intrinsic spin relaxation time. However, because it is difficult to solve the optimal parameters analytically, we experimentally evaluated the quality of the optical Stark shift, which was defined as Δ S T 2 Stark . The optical Stark shift Δ S and coherence time T 2 Stark were obtained by fitting the Ramsey interference prepared in the initial state (|+1〉 S + |0〉 S )/√2 with an exponentially decaying sinusoidal wave (Extended Data Fig. 1a,b). Using the obtained two-dimensional colour map (Extended Data Fig. 1c-e), reasonable values of Δ L and Ω L were selected according to the required Δ S in the main text. In practice, to precisely adjust Δ S to the target value, the laser power or Ω L was swept while Δ L was fixed.
Amplitude-alternating microwave pulse. The rotating frame Hamiltonian of the V-shaped, three-level system under the microwave irradiation with detuning Δ and Rabi frequency Ω MW is given by where Ω eff = √ Ω 2 MW + Δ 2 is the effective Rabi frequency, is a unit vector indicating the rotation axis, , σ {0,B} 0,x,y,z are the Pauli operators and identity operator based on |0〉 S , |B〉 S , and |B⟩ S = cosθMW |+1⟩ S + e −iϕMW sinθMW |−1⟩ S is a bright state depending on the microwave polarization |ψ⟩ MW = cosθMW |+1⟩ MW + e −iϕMW sinθMW |−1⟩ MW , where θ MW and ϕ MW are a polar and an azimuth angles in a Poincaré sphere spanned by a right |+1〉 MW and a left |−1〉 MW circular polarization. The ± indicates a positive or negative microwave amplitude or a microwave phase of 0 or π. The time evolution operator as a function of time is The total time evolution of the amplitude-alternating pulse, which reverses the amplitude in the middle of the pulse with a pulse time width tMW = 4π Ωeff is This means that the qubit spanned by |±1〉 S was rotated by an angle 2iπΔS Ωeff around the |B〉 S -|D〉 S axis. Therefore, if Δ ≠ 0, arbitrary rotation was enabled by choosing the appropriate microwave polarization and pulse width. On the other hand, if Δ = 0, the identity operator was strictly implemented. Extended Data Fig. 2 shows the numerical calculations of the fidelity of frequency-dependent unitary evolution with the amplitude-alternating pulse and the simple square-envelope pulse. The calculations did not take into account the effects of phase relaxation induced by the 14 N and 13 C nuclear spin bath and the orbital relaxation. The amplitude-alternating pulse was extremely robust against Rabi frequency error, especially where the Rabi frequency was too large.
We simulated single-qubit gates with Ω L = 0.16 MHz, Δ L = 1.6 GHz and Δ S = 4 MHz), which yielded a relatively high-quality Stark shift. To calculate the fidelity, the dynamics of the quantum state were simulated by a master equation with the well-known Hamiltonian of the NV centre assuming the excited state relaxation time of 12 ns. The initial states in {|±1〉 S , |±〉 S = (|+1〉 S ± |−1〉 S )/√2, |±i〉 S = (|+1〉 S ± i|−1〉 S )/√2} were performed quantum state tomography after the time evolution with a 250 ns amplitude-alternating pulse. From these results, the χ matrix of the evolution was reconstructed by quantum process tomography. The simulations included an electron orbital, an electron spin and a carbon nuclear spin. Extended Data Fig. 3 shows the fidelities of Pauli-X, Y and Z gates as a function of the hyperfine splitting induced by the nuclear spin. In the weak hyperfine case, the fidelity was about 99% for each gate. However, in the inactive (laser-off) case of the Z gate, the fidelity was lower than that of the X and Y gates. This means that phase relaxation (entanglement with the nuclear spin) was suppressed by a non-commutative driving Hamiltonian. In the active case, the fidelity was limited by unintentional Stark shifts caused by non-target excited states excluding |E y 〉. This effect is reduced in other physical systems where the excited states are far from each other owing to a large spin-orbit interaction. It may also be reduced by introducing additional detuned lasers to compensate for the Stark shift caused by the non-target levels.
Optically addressable electron spin initialization. The electron spin of the NV centre at low temperature was typically initialized by spin pumping with spin-selective optical excitation. In this method, the resolution is limited to the spot size of the light, since the initialization is applied over the entire area irradiated by light. We here present an initialization method using a pulse sequence combining light and microwaves that, in principle, can achieve a resolution below the diffraction limit of light. Extended Data Fig. 4 shows the pulse sequence used in Fig. 3f. |+1〉 S was selectively transitioned to |0〉 S by optically addressable manipulation. Then, the resonant excitation to |E y 〉 relaxed the spin state |0〉 S to |±1〉 S in a half-and-half ratio. This process resulted in pumping only half of the population of |+1〉 S into |−1〉 S . In the experiment, this cycle was repeated ten times. For comparison, the initial state was prepared in a completely mixed state, and the spin state was measured after the initialization sequence was run. The bias of the spin state in the inactive case was kept small so that the error of the optically addressable transition in the inactive case was very small, as shown in Fig. 3e.
Optically addressable nuclear spin manipulations. We controlled whether or not the nitrogen ( 14 N) nuclear spin operation was activated via optically addressable electron spin because the light did not interact with the nuclear spin. When the electron spin was in the ancilla state |0〉 S , the nuclear spin constituted a degenerate Λ-shaped three-level structure induced by a nuclear quadrupole splitting of 4.95 MHz. In that subspace, an arbitrary single-qubit gate was feasible with the resonant radiowave in a similar way to the electron spin. When the electron spin was instead in the qubit space |±1〉 S , the nuclear spin was split by the hyperfine interaction and thus did not respond to a radiowave with the same frequency. Extended Data Fig. 5a shows the experimental sequence for demonstration of the optically addressable nuclear spin operations. We first initialized the nuclear spin in the ancilla state |m I = 0〉 N to show that the subsequent state preparation into qubit state |m I = ±1〉 N was activated by the light (Extended Data Fig. 5b). The fidelities of the prepared states were comparable to those in previous work 31 . Next, we applied holonomic quantum gates on the prepared states to perform quantum process tomography (Extended Data Fig. 5c). The averaged fidelity of the gates was 98%, which was also comparable to previous work. Only the activated states were taken into account in the quantum process tomography. In our demonstrations, although the inactive nuclear spin state was |0〉 N and the electron spin was conditioned to be in |+1〉 S , in principle, unconditional gate operations for the electron-nuclear spin qubit system were feasible. However, owing to the long manipulation time of the nuclear spin, it was necessary to combine the holonomic gate with a dynamical decoupling to preserve the coherence of the electron spin.

Data availability
The data that support the findings of this study are available from the corresponding author upon request.

code availability
All codes used to produce the findings of this study are available from the corresponding author upon request.  Fig. 4 | experimental sequence of an optically addressable initialization. First, the optically addressable manipulation transfers |+1〉 S into |0〉 S by the addressing laser for the optical stark shift of Δ S = 2 mHz (given by Ω L = 80√2 mHz and Δ L = 1.6 GHz) and the amplitude-alternating microwave pulse with Rabi frequency of Ω mW = 2 mHz. Next, a spin pumping by a resonant excitation to |E y 〉 equally distributes the |0〉 S population into |±1〉 S . this process is repeated 10 times.