Geometric spin echo under zero field

Spin echo is a fundamental tool for quantum registers and biomedical imaging. It is believed that a strong magnetic field is needed for the spin echo to provide long memory and high resolution, since a degenerate spin cannot be controlled or addressed under a zero magnetic field. While a degenerate spin is never subject to dynamic control, it is still subject to geometric control. Here we show the spin echo of a degenerate spin subsystem, which is geometrically controlled via a mediating state split by the crystal field, in a nitrogen vacancy centre in diamond. The demonstration reveals that the degenerate spin is protected by inherent symmetry breaking called zero-field splitting. The geometric spin echo under zero field provides an ideal way to maintain the coherence without any dynamics, thus opening the way to pseudo-static quantum random access memory and non-invasive biosensors.

N itrogen 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 temperature [1][2][3][4] . Arbitrary state preparation, single-/two-qubit control, entanglement generation [5][6][7] and quantum teleportation 8 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 symmetrybreaking 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 m S ¼ 0 state far below the degenerate m S ¼ ± 1 states 9 . The zero-field split state provides a dynamic path to manipulate the geometric phase of the logical qubit based on the m S ¼ ± 1 states 10,11 . The same situation is seen in photon polarization 12,13 , which is also spanned by degenerate m S ¼ ± 1 states representing circular polarizations with spin-1 angular momentum. This correspondence allows spontaneous entanglement generation 14 and entangled absorption 15 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 polarizations 16 . Even when the subsystem is degenerate enough to avoid a field to drive the state, it is possible to geometrically control 17,18 the geometric phase or the Berry phase 19 . Geometric gate operation has been proposed for performing holonomic quantum computation with built-in noise resilience [20][21][22][23] . The geometric phase has been experimentally observed in molecular ensembles 24,25 , in a single superconducting qubit 26 , and in a single NV centre in diamond 10,11 . 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 sequences 17 , and even offers a universal set of quantum logic gates 10 .
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 ¼ (S x, S y, S z ) is the spin-1 operator of the vacancy electron spin, I z is the z component of the spin-1/2 operator of the 14 N nuclear spin, D ¼ (2p Â )2.87 GHz is the axial zero-field splitting, E x (E y ) is the x (y) component of the transverse zerofield splitting caused by a crystal strain, g e is the gyromagnetic ratio of the electron spin, B ext is the external magnetic field and A z ¼ À (2p Â )2.175 MHz is the z component of the hyperfine interaction between the electron spin and the 14 N 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 : for simplicity. If we take only the first term of equation (1) as a dominant term, the ground state forms triplet states consisting degenerate m S ¼ AE 1 j istates, which serve as logical qubit basis states, and a zero-field split m S ¼0 j i 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 AE 1 j i and 0 j i states is described as H MW ðtÞ¼ OðtÞ 2 S x , where O(t) denotes the Rabi frequency. We define the polarization of the microwave as a linear polarization oriented towards þ x. The spin 1 operator S  (Fig. 1b). After a round trip time T, defined as , the bright state B j i evolves as h j denotes the Pauli operator in the B j i À D j i subspace. Note that the prefactor À 1 is nothing but a global phase in the B j i À 0 j i subspace, whereas in the AE 1 j isubspace the global phase serves as a relative phase called the geometric phase for a geometric spin qubit. This geometric operation is represented as a p rotation around the B j i À D j i axis or the x axis in the AE 1 j i subspace as expð À i p 2 s f AE 1g x Þ¼ À is f AE 1g x as shown in Fig. 1c. The pulse sequences used in the demonstrations are summarized in Fig. 1d (Methods).
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 p pulse width required to flip the spin states between the B j i state and the 0 j i 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 ( 14 N) that comprises the NV. The Ramsey interference experiment or the free-induction decay indicates that the hyperfine coupling induces electron spin precession to alter B j i and D j i states at a frequency corresponding to twice the hyperfine coupling (Fig. 2b). The Gaussian decay of the envelope indicates the geometric spin coherence time T 2 * to be 0.61 ms. The origin of the decoherence would be the coupling of the electron spin to a spin bath consisting the proximate nuclear spins of the 13 C isotopes.
Geometric spin echo. The disappeared Ramsey interference signal seen in Fig. 2b recovers as a geometric spin echo by the insertion of 2p pulse after 35 ms 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 T 2 ¼ 83 ms under a zero magnetic field, which is about 140 times longer than the T 2 * of 0.61 ms. 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 T 2 in Fig. 3b is determined only by fitting with exp À t=T 2 ð Þ 3 Â Ã under the assumption that the population decay behaves as exp À t=T 1 ð Þ , where T 1 ¼ 700 ms is dominated by green laser leakage ( Supplementary Fig. 1).
The echo coherence time T 2 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 method 27 , which neglects electron spin flip owing to the large axial zero-field splitting 28 , for a spin bath consisting 13 C isotopes with a natural abundance of 1.1% (Methods). Single and dimer 13 C nuclear spins (nearest-neighbour nuclear spin pair) were taken into account as bath spins but the interaction between bath spins were neglected 29 . 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).

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 m S ¼ ± 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 B0.02 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 T 2 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 s AE 1 f g x to the second-order perturbation, the transverse zero-field splitting suppresses the axial hyperfine interaction, which causes a phase-flip error s   Fig. 1). The discrepancy between T 1 and T 2 can be compensated by excluding the effect of dimers not only by decreasing the abundances of 13 C isotopes but also by suppressing dimers into their singlet state to be spin transparent. Since the common dynamical qubit defined in the 0/ À 1 subspace uses only one component of the þ 1/ À 1 subspaces, which couples with the 13 C spin bath, the Ramsey coherence time T 2 * should be double of the geometric qubit. However, the geometric qubit should surpasses the dynamical qubit with the multi-pulse echo 17,31 even in the absence of magnetic field, as previously demonstrated in the presence of magnetic field 32 . Since the dynamical Hahn echo cannot inverse the þ 1/ À 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 þ 1/ À 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 freeinduction 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 13 C 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 h001i crystal orientation (from Sumitomo Electric) without any irradiation or annealing. This diamond has NV centres (B10 12 cm À 3 ) and nitrogen impurities called P1 centres (o1 p.p.m.), leading to relatively higher density than a chemical vapour deposition grown diamond. A negatively charged NV centre located at B30 mm below the surface was found using a confocal laser microscope. A 25 mm 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 B0.045 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 13 C 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 ms initializes the electron system into the ancillary state 0 j i, 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 B j i and the ancillary state 0 j i was first observed to determine the p pulse width. The Ramsey interference was then observed to confirm the coherence between logical qubit basis states þ 1 j i and À 1 j i by letting those superposition states revolve between B j i and D j i during the time between the two p pulses, instead of two p/2 pulses, as is used for the conventional Ramsey interference. Finally, we demonstrated the geometric spin echo by applying a 2p pulse in the middle of the free precession, instead of a p 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 ms of the 3-ms-green laser irradiation for the next initialization were used to measure the population in the |0i 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 T 2 echo time under a zero magnetic field, we neglected the T 1 relaxation time since it is sufficiently longer than T 2 . The decoherence is therefore dominated by the effects of single 13 C 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 spins 27,29 , where U (±) is the time evolution operator depending the electron spin state AE j i and r N is the spin bath state operator; at the high-temperature limit, r N ¼ 1 N 2 N (N: the number of bath spins). We considered only single and dimer 13 C nuclear spins as bath spins and neglected the interaction between bath spins 28 . The electron spin echo signal could thus be factorized into individual bath spins as follows: where S single,j (2t) is the jth single 13 C nuclear spin contribution and S dimer,k (2t) is the kth dimer 13 C nuclear spin contribution.
Electron spin decoherence. The dipole-dipole interaction between i spin and j spin is defined as where m 0 is the vacuum permeability, g i (g j ) is the gyromagnetic ratio of the i (j) spin (electron spin: g e ¼ À 1.76 Â 10 11 rad s À 1 T À 1 , 13 C nuclear spin: g c ¼ 6.73 Â 10 7 rad s À 1 T À 1 ), S i (S j ) is the spin operator of the i (j) spin and r i j is the displacement of the i spin from the j spin. In the following, we describe S (I j ) as a spin-1 operator of the electron spin (spin-1/2 operator of jth 13 C nuclear spin). Hyperfine interaction conditioned by the electron spin eigenstates n¼0; AE j iof the Hamiltonian in equation (1) can be represented as where A n ð Þ e j n h jS n j i Á A e j is the electron spin hyperfine field depending on the electron spin state n j i. Then, we can define the effective magnetic field, Hamiltonian and time evolution operator depending on the electron spin state n j i as H n ð Þ The echo signal given by the single jth 13 C spin is 33 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 I k0 , I k1 are 0th, 1st 13 C 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 13 C isotopes with natural abundance of 1.1% at a distance within 4 nm of the vacancy electron ( Supplementary Fig. 2). T 2 is determined by fitting with exp À t=T 2 ð Þ 3 Â Ã .
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 m I ¼ 0, ± 1 where y mI , f mI are the polar and azimuth angles of the Bloch sphere spanned by m S ¼ AE 1 j i , y mI ¼ arccos g e B ext;z þ A z m I h jI z m I j i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 x þ E 2 y þ g e B ext;z þ A z m I h jI z m I j i À Á 2 q 0 @ 1 A ; ð17Þ As the z component of the energy splitting g e B ext;z þ A z m I h jI z m I j i decreases compared with the transverse zero-field splitting ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi component of the energy splitting, the m S ¼ ± 1 states couple strongly and finally become completely coupled states AE j i¼ þ 1 j iAEe if À 1 j i À Á = ffiffi ffi 2 p . 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 A AE ð Þ e j ¼ AE h jS AE j iÁA e j . This protection effect explains the enhancement seen in Fig. 3b around 0 and 0.075  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 m I ¼ 0 state under a zero magnetic field, at which point the electron spin echo coherence time T 2 drastically increase.
Data availability. The data that support the findings of this study are available from the corresponding author upon request.