Introduction

Nitrogen vacancy (NV) centres in diamond provide promising platforms for quantum-related technologies. The coherence time of an electron spin in an NV centre has been shown to reach millisecond order even at room temperature1,2,3,4. Arbitrary state preparation, single-/two-qubit control, entanglement generation5,6,7 and quantum teleportation8 have also been achieved using the NV spin together with proximate nuclear spins. Quantum memory, however, must possess two contradictory qualities: noise resilience and controllability. Symmetric states are known to be fragile by themselves but become noise resilient with specific symmetry-breaking operations. Such symmetry breaking inherently occurs in an NV centre, where the ground-state electrons form triplet states with spin-1 angular momentum. The zero-field splitting due to the spin-spin interaction breaks the symmetry to push only the magnetic quantum number mS=0 state far below the degenerate mS=±1 states9. The zero-field split state provides a dynamic path to manipulate the geometric phase of the logical qubit based on the mS=±1 states10,11. The same situation is seen in photon polarization12,13, which is also spanned by degenerate mS=±1 states representing circular polarizations with spin-1 angular momentum. This correspondence allows spontaneous entanglement generation14 and entangled absorption15 between single photon and electrons in an NV centre in diamond. The correspondence also allows a microwave to be applied to manipulate an electron spin ±1 subsystem with arbitrary polarizations16. Even when the subsystem is degenerate enough to avoid a field to drive the state, it is possible to geometrically control17,18 the geometric phase or the Berry phase19. Geometric gate operation has been proposed for performing holonomic quantum computation with built-in noise resilience20,21,22,23. The geometric phase has been experimentally observed in molecular ensembles24,25, in a single superconducting qubit26, and in a single NV centre in diamond10,11. It has been shown that the geometric phase gate offers fast and precise control over the geometric phase, disrupts environmental interaction with multiple pulse echo sequences17, and even offers a universal set of quantum logic gates10.

Although those demonstrations introduced an energy gap to the qubit for controllability, here we show that it is possible to control a degenerate logical qubit, which we call a geometric spin qubit, by a purely geometric gate operation and that it can be protected by zero-field splitting with the help of a time-reversal operation, which we call geometric spin echo.

Results

System and scheme

Our experimental demonstration of geometric spin control is based on the application of resonant microwave to electron spin in a diamond NV centre under a zero magnetic field. The electron spin system in an NV centre is described by the following Hamiltonian.

where S=(Sx,Sy,Sz) is the spin-1 operator of the vacancy electron spin, Iz is the z component of the spin-1/2 operator of the 14N nuclear spin, D=(2π × )2.87 GHz is the axial zero-field splitting, Ex (Ey) is the x (y) component of the transverse zero-field splitting caused by a crystal strain, γe is the gyromagnetic ratio of the electron spin, Bext is the external magnetic field and Az=−(2π × )2.175 MHz is the z component of the hyperfine interaction between the electron spin and the 14N nuclear spin. The x, y components of the hyperfine interaction and magnetic field, which contribute to the second-order perturbation, are negligible owing to the large zero-field splitting. Note that in this paper we omit the Planck constant ħ for simplicity.

If we take only the first term of equation (1) as a dominant term, the ground state forms triplet states consisting degenerate states, which serve as logical qubit basis states, and a zero-field split state, which serves as an ancillary state for the geometric operation as shown in Fig. 1a. On the basis of the Jaynes–Cummings model, qthe interaction Hamiltonian with a microwave resonant to the energy gap between and states is described as , where Ω(t) denotes the Rabi frequency. We define the polarization of the microwave as a linear polarization oriented towards +x. The spin 1 operator indicates the state exchange between the bright state and , while the dark state remains unchanged (Fig. 1b). After a round trip time T, defined as , the bright state evolves as , where denotes the Pauli operator in the subspace. Note that the prefactor −1 is nothing but a global phase in the subspace, whereas in the subspace the global phase serves as a relative phase called the geometric phase for a geometric spin qubit. This geometric operation is represented as a π rotation around the axis or the x axis in the subspace as as shown in Fig. 1c. The pulse sequences used in the demonstrations are summarized in Fig. 1d (Methods).

Figure 1: Geometric operation and experimental sequence.
figure 1

(a) Energy level diagram of an electron spin in an NV centre under a zero magnetic field on the computational bases. (b) The logical bases are transformed into bright and dark states, as defined by the microwave polarization. (c) Bloch sphere representation of a 2π rotation starting from the bright state through the ancillary state returning to with an additional geometric phase factor −1. The geometric phase contributes to a π rotation around the axis in the logical qubit space. (d) Pulse sequence used for the Rabi oscillation, Ramsey interference, and echo recovery. Insets illustrate the logical qubit dephasing and refocusing after the 2π pulse.

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Rabi oscillation and Ramsey interference

A series of experiments for calibrating the condition to achieve the geometric spin echo are performed under a zero magnetic field. The Rabi oscillation experiment determines the π pulse width required to flip the spin states between the state and the state (Fig. 2a). The oscillation conforms to the theory considering hyperfine coupling between the electron spin at the vacancy and the nuclear spin at the nitrogen (14N) that comprises the NV. The Ramsey interference experiment or the free-induction decay indicates that the hyperfine coupling induces electron spin precession to alter and states at a frequency corresponding to twice the hyperfine coupling (Fig. 2b). The Gaussian decay of the envelope indicates the geometric spin coherence time T2* to be 0.61 μs. The origin of the decoherence would be the coupling of the electron spin to a spin bath consisting the proximate nuclear spins of the 13C isotopes.

Figure 2: Rabi oscillation and Ramsey interference.
figure 2

(a) The Rabi oscillation between the bright and ancilla states. The solid line shows the best theoretical fit to the data considering the hyperfine interaction with the nuclear spin of the nitrogen (14N) comprising the NV. (b) The Ramsey interference between and states. The solid line shows the best theoretical fit with a Gaussian envelope. The precession frequency of 4.35 MHz corresponds well to hyperfine splitting. Error bars are defined as the s.d. of the photon shot noise.

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Geometric spin echo

The disappeared Ramsey interference signal seen in Fig. 2b recovers as a geometric spin echo by the insertion of 2π pulse after 35 μs of time evolution, and the signal reaches a maximum when the second evolution time equals the first (Fig. 3a). The result indicates that the geometric spin echo rephases the geometric spin as the conventional Hahn echo does the dynamic spin, even under complete degeneracy of the qubit space. Figure 3b shows the signal decay of the geometric spin echo under external magnetic fields of 0 mT (red squares), 0.04 mT (green circles) and 0.12 mT (blue triangles) measured along the NV axis. The coherence time is extended by the echo process to T2=83 μs under a zero magnetic field, which is about 140 times longer than the T2* of 0.61 μs. The echo coherence time drastically increases as decreasing the magnetic field from 0.12 mT, within which both of the Zeeman-split electron spin states are equally driven by the microwave. The T2 in Fig. 3b is determined only by fitting with under the assumption that the population decay behaves as , where T1=700 μs is dominated by green laser leakage (Supplementary Fig. 1).

Figure 3: Geometric spin echo.
figure 3

(a) Refocusing of the geometric echo signal by fixing the first evolution time at 35 μs and sweeping the second evolution time ∼35 μs. The solid line shows the best theoretical fit with a Gaussian envelope. (b) Echo decays of the electron spin under magnetic fields of 0 mT (red squares), 0.04 mT (green circles) and 0.12 mT (blue triangles) measured along the NV axis. Solid lines show the best fitting curves integrated with T1 decay. (c) Echo coherence decay time T2 as a function of the magnetic field. The solid line shows the theoretical fits to the data (Methods). Error bars in a,b are defined as the s.d. of the photon shot noise. Error bars in c are given by least squares fitting.

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The echo coherence time T2 as a function of an external magnetic field measured along the NV axis (Fig. 3c) agrees relatively well with theory based on the disjoint cluster method27, which neglects electron spin flip owing to the large axial zero-field splitting28, for a spin bath consisting 13C isotopes with a natural abundance of 1.1% (Methods). Single and dimer 13C nuclear spins (nearest-neighbour nuclear spin pair) were taken into account as bath spins but the interaction between bath spins were neglected29.

Figure 4a decomposes contributions of single and dimer nuclear spins to the decoherence. Note that the single nuclear spins dominate decoherence under relatively high magnetic field, while the dimer nuclear spins dominate under a zero magnetic field. In the case of single nuclear spins, the quantization axis defined by the electron spin hyperfine field is deviated by the external magnetic field. In the other case of dimer nuclear spins, the quantization axis is deviate by the dipolar magnetic field within the dimer (Fig. 4b). In any case, the electron spin inversion in the logical qubit space cannot completely time reverse the nuclear spin dynamics during the geometric spin echo. In other words, the time evolution operators with orthogonal electron spin states become incompatible or irreversible by the additional field as in equation (9).

Figure 4: Single and dimer nuclear spins.
figure 4

(a) Calculated geometric spin echo decay induced by bath spins decomposed into single (green) and dimer (blue) 13C nuclear spin contributions. Solid (dashed) lines shows the decay under a magnetic field of 0 mT (0.4 mT) measured along the NV axis. (b) Schematic spin bath configuration around the vacancy electron spin under a zero magnetic field. Single 13C nuclear spins are quantized along the electron spin hyperfine field (grey arrows). Dimer 13C nuclear spins are quantized along the dipolar magnetic field within each of the dimers.

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Discussions

The deviation from the theory around 0 and 0.075 mT in Fig. 3c, on the other hand, is explained by the strain, or the transverse zero-field splitting, which couples the otherwise degenerate mS=±1 states to lift the degeneracy and further decouples the spin bath from the electron spin. The magnetic field to give maximum deviation 0.075 mT corresponds to the nitrogen hyperfine splitting 2.18 MHz. The range of the deviation ∼0.02 mT is explained by considering the strain splitting 0.23 MHz and inhomogeneous broadening 0.43 MHz of the optically detected magnetic resonance (ODMR) spectrum (Supplementary Fig. 3). The calculated magnetic field dependence of the echo T2 with the strain correction (Supplementary Fig. 3) agrees well with the experiments, indicating that the geometric spin qubit is protected against decoherence not only by the axial zero-field splitting but also by the transverse zero-field splitting. In contrast that the axial zero-field splitting suppresses the transverse hyperfine interaction, which causes a bit-flip error to the second-order perturbation, the transverse zero-field splitting suppresses the axial hyperfine interaction, which causes a phase-flip error , to the second-order perturbation30.

Although the achieved geometric echo coherence time T2 of 83 μs at the degeneracy is 140 times longer than the free-induction decay of 0.61 μs, it is several times shorter than the measured population decay time T1 of 700 μs for the geometric spin state relaxing from to (Supplementary Fig. 1). The discrepancy between T1 and T2 can be compensated by excluding the effect of dimers not only by decreasing the abundances of 13C isotopes but also by suppressing dimers into their singlet state to be spin transparent. Since the common dynamical qubit defined in the 0/−1 subspace uses only one component of the +1/−1 subspaces, which couples with the 13C spin bath, the Ramsey coherence time T2* should be double of the geometric qubit. However, the geometric qubit should surpasses the dynamical qubit with the multi-pulse echo17,31 even in the absence of magnetic field, as previously demonstrated in the presence of magnetic field32. Since the dynamical Hahn echo cannot inverse the +1/−1 subspace, it cannot reverse time to recover the original state even in the absence of magnetic field. In contrast, the developed geometrical qubit defined in the +1/−1 subspace is space-inversed by the geometrical echo and thus time-reversed to decouple the spin bath.

We demonstrated the geometric spin echo of a degenerate geometric spin qubit via the ancillary state in a diamond NV centre. The geometric spin echo recovered the coherence imprinted in the degenerate subspace after 140 times the free-induction decay time. The theoretical analysis indicates that the geometric spin qubit is three dimensionally protected against decoherence by the axial and transverse zero-field splittings with the help of time-reversal, leaving decoherence due to dimer 13C nuclear spins under a zero magnetic field. The purely geometric spin qubit is not only robust against noise caused by the spin bath but also robust against control error, and thus is suitable for a memory qubit used in quantum information and quantum sensing of magnetic, electric or strain fields for biomedical imaging.

Methods

Experimental setup

We used for the experiments a native NV centre in type-IIa high-pressure high-temperature grown bulk diamond with a 〈001〉 crystal orientation (from Sumitomo Electric) without any irradiation or annealing. This diamond has NV centres (∼1012 cm−3) and nitrogen impurities called P1 centres (<1 p.p.m.), leading to relatively higher density than a chemical vapour deposition grown diamond. A negatively charged NV centre located at ∼30 μm below the surface was found using a confocal laser microscope. A 25 μm copper wire mechanically attached to the surface of the diamond was used to apply a microwave for the ODMR measurement. An external magnetic field at an angle of 70° to the NV axis was applied to compensate the geomagnetic field of ∼0.045 mT using a permanent magnet. Careful orientation of the magnet was conducted with monitoring of the ODMR spectrum within 0.1 MHz. The Rabi oscillation and Ramsey interference were also used to fine-tune the field. The NV centre used in the experiment showed no hyperfine splitting caused by 13C nuclear spins exceeding 0.1 MHz. All experiments were performed at room temperature.

Pulse sequences

The pulse sequences used in the demonstrations are summarized in Fig. 1d. Green light irradiation for 3 μs initializes the electron system into the ancillary state , which is followed by microwave pulses resonant to the zero-field splitting D with pulse patterns depending on the experiment. The Rabi oscillation between the bright state and the ancillary state was first observed to determine the π pulse width. The Ramsey interference was then observed to confirm the coherence between logical qubit basis states and by letting those superposition states revolve between and during the time between the two π pulses, instead of two π/2 pulses, as is used for the conventional Ramsey interference. Finally, we demonstrated the geometric spin echo by applying a 2π pulse in the middle of the free precession, instead of a π pulse as is used for the conventional Hahn echo. Despite the differences in the scheme, the random phase shift caused by the spin bath rephased back into the initial state as is schematically shown in the inset of Fig. 1d. The photon counts during the first 300 ns normalized by those during the last 2 μs of the 3-μs-green laser irradiation for the next initialization were used to measure the population in the |0〉 state. All the pulse sequences and photon counts were managed by an FPGA-based control system developed by NEC communications.

Calculation model

For the calculation of the magnetic field dependence of T2 echo time under a zero magnetic field, we neglected the T1 relaxation time since it is sufficiently longer than T2. The decoherence is therefore dominated by the effects of single 13C nuclear spins and dimers (nearest-neighbour nuclear spin pair). Because of the large discrepancy in the Zeeman energy between the electron spin and the bath spins, we neglected the electron spin bit flip induced by the bath spins and attributed the coherence of the electron spin to that of the bath spins27,29,

where U() is the time evolution operator depending the electron spin state and ρN is the spin bath state operator; at the high-temperature limit, (N: the number of bath spins). We considered only single and dimer 13C nuclear spins as bath spins and neglected the interaction between bath spins28. The electron spin echo signal could thus be factorized into individual bath spins as follows:

where Ssingle,j(2τ) is the jth single 13C nuclear spin contribution and Sdimer,k(2τ) is the kth dimer 13C nuclear spin contribution.

Electron spin decoherence

The dipole–dipole interaction between i spin and j spin is defined as

where μ0 is the vacuum permeability, γi(γj) is the gyromagnetic ratio of the i (j) spin (electron spin: γe=−1.76 × 1011 rad s−1 T−1, 13C nuclear spin: γc=6.73 × 107 rad s−1 T−1), Si (Sj) is the spin operator of the i (j) spin and ri j is the displacement of the i spin from the j spin. In the following, we describe S (Ij) as a spin-1 operator of the electron spin (spin-1/2 operator of jth 13C nuclear spin). Hyperfine interaction conditioned by the electron spin eigenstates of the Hamiltonian in equation (1) can be represented as

where is the electron spin hyperfine field depending on the electron spin state . Then, we can define the effective magnetic field, Hamiltonian and time evolution operator depending on the electron spin state as

The echo signal given by the single jth 13C spin is33

These equations indicate that a single nuclear spin does not decohere the electron spin under a zero magnetic field after the spin echo, since the time evolution operators depending on the electron spin state are compatible, while it does decohere the electron spin under a non-zero magnetic field. On the other hand, the effective magnetic field, Hamiltonian and time evolution operator of the kth dimer are

where Ik0, Ik1 are 0th, 1st 13C spins consisting the dimer and we suppose that the effective magnetic field is applied equally to each nuclear spin. The echo signal given by the kth dimer is

The spin bath configuration is generated by randomly placing 13C isotopes with natural abundance of 1.1% at a distance within 4 nm of the vacancy electron (Supplementary Fig. 2). T2 is determined by fitting with .

Transverse zero-field splitting

If we neglect the transverse magnetic field for simplicity, the Hamiltonian in equation (1) can be rewritten as + thus generating the following eigenenergies depending on the nitrogen nuclear spin states mI=0, ±1

and eigenstates

where , are the polar and azimuth angles of the Bloch sphere spanned by ,

As the z component of the energy splitting decreases compared with the transverse zero-field splitting , where Ex (Ey) is x (y) component of the energy splitting, the mS=±1 states couple strongly and finally become completely coupled states . The transverse zero-field splitting under the condition thus decouples the interaction between the electron spin and bath spins to suppress the decoherence effect by decreasing the electron spin hyperfine field . This protection effect explains the enhancement seen in Fig. 3b around 0 and 0.075 mT, where the nitrogen hyperfine field cancels out the z component of the external magnetic field on the electron spin. The enhancement becomes prominent by initializing the nitrogen nuclear spin state to mI=0 state under a zero magnetic field, at which point the electron spin echo coherence time T2 drastically increase.

Data availability

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

Additional information

How to cite this article: Sekiguchi, Y. et al. Geometric spin echo under zero field. Nat. Commun. 7:11668 doi: 10.1038/ncomms11668 (2016).