Geometric spin echo under zero field

Transcription

1 Geometric spin echo under zero field Yuhei Sekiguchi, Yusuke Komura, Shota Mishima, Touta Tanaka, Naeko Niikura and Hideo Kosaka * Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama , Japan *kosaka@yun.ac.jp 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 well 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. We here 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 center in diamond. The demonstration revealed that the degenerate spin is protected from both the amplitude and phase noises by inherent symmetry breaking called axial and transverse zero-field splittings. The geometric spin echo under zero field provides a way to ideally maintain the coherence without any dynamics, thus opening the way to pseudo-static quantum RAM and non-invasive biosensors. 1

2 A nitrogen-vacancy (NV) center in diamond provides a promising platform for quantum related technologies. The coherence time of an electron spin in an NV center has been shown to reach millisecond-order even at room temperature 1-3. Arbitrary state preparation, single-/two-qubit control, entanglement generation 4,5 and quantum teleportation 6,7 have also been achieved using the NV spin together with proximate nuclear spins. Quantum memory, however, must demonstrate two contradictory qualities: noise resilience and controllability. It is known that a symmetric state is known to be fragile by itself but becomes noise resilient with some specific symmetry breaking. Such symmetry breaking inherently occurs in an NV center, where the ground-state electrons form triplet states with spin-1 angular momentum. The axial zero-field splitting due to the spin-spin interaction breaks the symmetry to push only the ms=0 state far below the degenerate ms= 1 states 8. The zero-field split state provides a dynamic path to manipulate the geometric phase of the logical qubit based on the ms= 1 states 9. The same situation is seen in photon polarization 10,11, which is also spanned by degenerate ms= 1 states representing circular polarizations with spin-1 angular momentum. This correspondence allows spontaneous entanglement generation 12 and entangled absorption 13 between a photon and an electron in an NV center in diamond. The correspondence also applies to a microwave for manipulating an electron spin 1 subsystem with arbitrary polarizations 14. Even when the subsystem is degenerate enough to avoid a field to drive the state, it is possible to geometrically control 15,16 the geometric phase or the Berry phase 17. Geometric gate operation has been proposed for performing holonomic quantum computation with built-in noise resilience The geometric phase has been experimentally observed in molecular ensembles 22,23, in a single superconducting qubit 24, and in a single NV center in diamond 9,25. They have proved that the geometric phase gate offers fast and precise control over the geometric phase, disrupts environmental interaction with multiple pulse echo sequences 15, and even offers a universal set of quantum logic gates 9. Although those demonstrations introduced an energy gap to the qubit for controllability, we here show that it is possible to control a degenerate logical qubit, which we call a geometric spin qubit, by purely geometric gate operation and that it can be three-dimensionally protected by both the axial and transverse zero-field splittings with the help of time reversal. 2

3 Results System and scheme. Our experimental demonstration of geometric spin control is based on the resonant microwave application to an electron spin in a diamond NV center under a zero magnetic field. The electron spin system in an NV center is described by the following Hamiltonian. H = DS z 2 + E x (S x 2 S y 2 ) + E y (S x S y + S y S x ) + γb ex S + A z S z I z (1) where D = (2π ) GHz is the axial zero-field splitting, E x (E y ) is the x (y) component of the transverse zero-field splitting caused by a crystal strain, γ is the gyromagnetic ratio of the electron spin, B ex is an external magnetic field and A z = (2π ) 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 contributing 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 consider only the first term of equation (1), the ground state forms spin-triplet states consisting of degenerate ms = 1 states, which serves as logical qubit basis states, and a zero-field split ms = 0 state, which serves as an ancillary state for the geometric operation as shown in Fig. 1a. Based on the Jaynes-Cummings model, the interaction Hamiltonian with a microwave resonant to the energy gap between the 1 and states is described as H MW (t) = Ω(t) 2 S x, where Ω(t) denotes the Rabi frequency. We define the polarization of the microwave as a linear polarization oriented toward +x. The spin 1 operator S x = 0 B + B 0 indicates state exchange between the bright state B = 1 ( ) and, while the dark state D = 1 ( 1 1 ) remains 2 2 T unchanged (Fig. 1b). After a round trip time T, defined as Ω(t)dt = 2π, the bright state B T evolves as exp ( i H MW (t)dt) B 0 0 = exp( iπσ x (B,0) ) B = B, where σx (B,0) = 0 B + B 0 denotes the Pauli operator in the B subspace (Fig. 1c). Note that the prefactor 1 is nothing but a global phase in the B subspace, whereas in the 1 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 rotation around the B D axis or the x axis in the 1 subspace as 3

4 exp ( i π σ (±1) (±1) 2 x ) = iσx as shown in Fig. 1d. The pulse sequences used in the demonstrations are summarized in Fig. 1e (see methods). Figure 1 Geometric operation and experimental sequence. (a) Energy level diagram of an electron spin in an NV center under a zero magnetic field on the computational bases. (b) The logical bases ±1 are transformed into bright B and dark D states, as defined by the microwave polarization. (c) Bloch sphere representation of a 2 rotation starting from the bright state B through the ancillary state 0 returning to B with an additional geometric phase factor -1. (d) The geometric phase contributes to a rotation around B D axis in the logical qubit space. (e) Pulse sequence used for the geometric spin Rabi oscillation, Ramsey interference, and echo recovery. Insets illustrate the logical qubit dephasing (left) and refocusing (right) after the 2 pulse. Rabi oscillation and Ramsey interference. The series of experimental results measured under zero magnetic field at room temperature are shown in Fig. 2. The Rabi oscillation (Fig. 2a) determines the pulse width required to swap the spin states between the B state and the state. 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 geometric spin Ramsey interference (Fig. 2b) indicates that the hyperfine coupling induces electron spin precession to alter the B and D states at a frequency corresponding to twice the hyperfine 4

5 coupling. The Gaussian decay of the envelope indicates the geometric spin coherence time T2* to be 0.61 s. The origin of the decoherence would be the coupling to a spin bath consisting of the proximate nuclear spins of 13 C or electron spins of the P1 center as schematically shown in the inset of Fig. 2b. Figure 2 Rabi oscillation and Ramsey interference. (a) Geometric spin Rabi oscillation between the bright B and ancilla 0 states. The solid line shows the theoretical fit of the data considering the hyperfine interaction with the nuclear spin of nitrogen ( 14 N) comprising the NV. (b) Geometric spin Ramsey interference between B and 0 states. The solid line shows the theoretical fit with a Gaussian envelope. The precession frequency of 4.35MHz corresponds well to hyperfine splitting. The inset shows a decoherence model having an NV electron spin at the center coupled with a spin bath, which are correlated with each other. Geometric spin echo. The disappeared interference signal in Fig. 2b recovers as the geometric spin echo by inserting a 2 pulse after 35 s of free 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 axial magnetic fields of 0 mt (red squares), 0.04 mt (green circles), and 0.12 mt (blue triangles). The coherence time is extended by the echo process to T2 = 83 s under zero magnetic field, which is about 140 times longer than the T2 *. The echo coherence time drastically decreases as the magnetic field increases up to 0.12 mt, within which both the Zeeman split states are equally driven by the microwave. 5

6 The non-exponential decay in Fig. 3b agrees well with the following theoretical analysis (solid lines), not only for the direct coupling of the central electron spin with the spin bath (star-like coupling) but also for intra-bath correlation (inset of Fig. 2b). In contrast to the compensation of the star-like coupling by the geometric spin echo process, the intra-bath correlation cannot be compensated. When we assume Gaussian noise from the spin bath field, the intra-bath correlation function is characterized by so called Ornstein-Uhlenbeck process as B(0)B(t) = b 2 exp( R t ), where b describes the characteristic coupling strength of the central spin to the spin bath and R denotes the spin bath transition rate, which is much smaller than the coupling strength b. The intrabath correlation induces a noise to the central spin, even after the echo process, leading to the echo signal decay as exp( i B(s)ds)exp (i τ 0 2τ τ B(s)ds) = exp [ b 2 R(2τ) 3 12], where 2τ denotes the total evolution time 26. The time dependence of the echo signal is well reproduced by this function. We also considered population decay behaving as exp ( t/t 1 ), where T 1 is dominated by green laser leakage. Figure 3 Geometric spin echo. (a) Refocusing of the geometric echo signal by fixing the first evolution time at 35 s and sweeping the second evolution time around 35 s. The solid line shows the 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). Solid lines show fitting curves exp [ (t T 2 ) 3 ] integrated with T1 decay. (c) Echo coherence decay time T2 as a function of the external magnetic field. The solid line shows the theoretical fits of the data. 6

7 External magnetic-field dependence. Figure 3c shows the echo coherence time as a function of an external magnetic field measured at axial direction. The solid line shows the fitting curve using the following model. First, suppression of the star-like coupling by transverse zero-field splitting is represented as b = 1/T 2 (B) and T 2 (B) = κ(b)(t 2 (0) T 2 ( )) + T 2 ( ), where κ(b) is the mixing ratio of +1 and 1 on the Zeeman splitting of an electron spin of the NV T 2 (0) and T 2 ( ) are the coherence times for zero and the given maximum axial magnetic field 27. Second, we represent the promotion of the spin bath transition rate induced by an external magnetic field 28 as R = R 0 + αb 2 ex, where R 0 is the rate under the zero magnetic field and α is a fitting parameter. The quadratic term introduces the contribution of an external magnetic field to the spin bath transition. The agreement between the experiment and the theory indicates that the enhancement of the geometric spin echo T2 as the magnetic field decreases is due to the transverse strain and the suppression of spin bath dynamics. The T2 time critically degrades at a magnetic-field of 0.01 mt, which corresponds to the Zeeman splitting of 0.28 MHz. On the other hand, the transverse zerofield splitting caused by the transverse strain is estimated as half of the splitting in the ODMR resonance corresponding to a nitrogen nuclear spin state of mi = 0 (Fig. 4a) to be 0.23 MHz, which coincides reasonably with the critical magnetic field. The coincidence indicates 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. While the axial zero-field splitting protects the geometric spin qubit against a transverse field noise, which causes a bit-flip error σ x (±1), the transverse zero-field splitting protects it against axial field noise, which causes a phase-flip error σ z (±1). The overall fitting of the data in Fig. 3c, however, requires external magnetic-field dependence not only of the star-like coupling b but also of the intra-bath correlation R, which introduces an irreversible phaseflip error to the central spin. The origin of the external field contribution to R is considered to be a transverse Zeeman field to flip the spin bath due to a misalignment of the applied magnetic field or the transverse hyperfine coupling of 13 C. The small kink around mt is the effect of a nitrogen nuclear spin with mi = 1. 7

8 Discussion The achieved geometric echo coherence time T2 of 85 s at the complete degeneracy is several times shorter than the measured population decay time T1 of 700 s for the geometric spin state relaxing from B to 0 (Fig. 4b). Nevertheless, the geometric echo T2 is 17 times longer than the conventional Hahn echo T2 of 5 s for an HPHT diamond under magnetic field 29. The discrepancy between the T1 and T2 can be compensated by introducing a multiple pulse echo sequence 15, as in the conventional CPMG sequence, at least up to the limit of T2 0.5 T1, which is explained by spinphonon coupling at room temperature 30. Since the geometric spin qubit at complete degeneracy is not affected by the transverse hyperfine interaction or transverse field noise, which leads to a population change of the geometric spin qubit, application of twice the same geometric spin echo pulses will result in the recovery of an arbitrary geometric spin state to the original state. Furthermore, polarization of the microwave and modulation of phase would allow us universal gate control of a single geometric spin qubit and a coupled geometric spin qubits. We experimentally demonstrated the geometric spin echo of a completely degenerate geometric spin qubit via the ancillary state in a diamond nitrogen vacancy center. The geometric spin echo recovered the coherence imprinted in the degenerate subspace after 140 times the free induction decay. The longer coherence time is the consequence of the fact that the direct coupling to the noisy spin bath affecting the T2 * is canceled via the geometric spin echo, leaving only intra-bath correlation to affect the T2, and this correlation has a much smaller effect on T2 than on T2 *. The magnetic field dependence of the echo coherence time indicates that intra-bath correlation is the major decoherence source and is minimized under a zero magnetic field, thus revealing the importance of the degeneracy for long-lived memory. The theoretical analysis clarified 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. 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 field for bio-medical imaging. 8

9 Figure 4 Supporting experiments. (a) Frequency dependence of the pulsed ODMR signal showing the strain-originated split in the dip corresponding to the nuclear spin mi = 0 state. Solid lines are Lorenzian curve fittings. Inset shows full spectrum of the pulsed ODMR, where the center part (grey area) corresponds to the mi = 0 state. (b) Time dependence of the pulsed ODMR signal prepared in the bright state (red sqares) and the ancillary state (blue circles). Solid lines show exponential decay fittings. The decay rate corresponds to the population decay time T1 used in the fitting in Fig. 3c. 9

10 Methods Experimental setup. We use for the experiments a native NV center in type-iia high-pressurehigh-temperature (HPHT) grown bulk diamond with a 001 crystal orientation (from Sumitomo Electric) without any irradiation or annealing. This diamond has NV centers (~10 12 cm -3 ) and nitrogen impurities called P1 centers (less than 1ppm), leading to relatively higher density than a chemical-vapor-deposition (CVD) grown diamond. A negatively charged NV center located at about 30 m below the surface is found using a confocal laser microscope. A 25- m copper wire mechanically attached to the surface of the diamond is used to apply a microwave for the optically detected magnetic resonance (ODMR) measurement. The external magnetic field is applied to the sample using permanent magnet. Careful orientation of the magnet also allow to compensate earth magnetic field by monitoring the ODMR spectrum within 0.1 MHz. The Rabi oscillation and Ramsey interference are also used to fine-tune the field. When we need a completely zero magnetic field, the geomagnetic field of about 0.03 mt and the transverse zero-field splitting are compensated for by an external magnetic field. The NV center used in the experiment shows no hyperfine splitting caused by 13 C nuclear spins within 0.1 MHz. All experiments are performed at room temperature. Pulse sequences. The pulse sequences used in the demonstrations are summarized in Fig. 1e. Green light irradiation for 3 s 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 B and the ancillary state is first observed to determine the pulse width. The geometric spin Ramsey interference is then observed to confirm the coherence between logical qubit basis states +1 and 1 by letting those superposition state revolves between B and D during the time between the two pulses, instead of two pulses, as is used for the conventional Ramsey interference. Finally, we demonstrate the geometric spin echo by applying a 2 pulse in the middle of the free precession, instead of a pulse as is used for the conventional Hahn echo. Despite the differences in the scheme, the random phase shift caused by the spin bath rephases back into the initial state as is schematically shown in the inset of Fig. 1e. The photon counts during the first 300 ns normalized by those during the last 10

11 2 s of the 3-μs green laser irradiation for the next initialization is used to measure the population in the state. All the pulse sequences and photon counts are managed by an FPGA-based control system developed by NEC communications. 11

12 References 1. Balasubramanian, G et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mat. 8, (2009). 2. Ryan, C. A., Hodges, J.S. & Cory, D. G. Robust Decoupling Techniques to Extend Quantum Coherence in Diamond. Phys. Rev. Lett. 105, (2010). 3. Naydenov, B. et al. Dynamical decoupling of a single-electron spin at room temperature. Phys. Rev. B 83, (2011). 4. Neumann, P. et al. Multipartite entanglement among single spins in diamond. Science 320, (2008). 5. Pfaff, W. et al. Demonstration of entanglement-by-measurement of solid-state qubits. Nat. Phys. 9, (2013). 6. Bernien, H. et al. Heralded entanglement between solid-state qubits separated by three metres. Nature 497, (2013). 7. Pfaff, W. et al. Unconditional quantum teleportation between distant solid-state quantum bits. Science 345, (2014). 8. Maze, J. R. et al. Properties of nitrogen-vacancy centers in diamond: the group theoretic approach. New J. Phys. 13, (2011). 9. Zu, C. et al. Experimental realization of universal geometric quantum gates with solid-state spins. Nature 514, (2014). 10. Kosaka, H. et al. Coherent transfer of light polarization to electron spins in a semiconductor. Phys. Rev. Lett. 100, (2008). 11. Kosaka, H. et al. Spin state tomography of optically injected electrons in a semiconductor. Nature 457, (2009). 12. Togan, E. et al. Quantum entanglement between an optical photon and a solid-state spin qubit. Nature 446, (2010). 12

13 13. Kosaka, H. & Niikura, N. Entangled absorption of a single photon with a single spin in diamond. Phys. Rev. Lett. 114, (2015). 14. London, P., Balasubramanian, P., Naydenov, B., McGuinness, L. P. & Jelezko. F. Strong driving of a single spin using arbitrarily polarized fields. Phys. Rev. A 90, (2014). 15. Morton, J. J. L. et al. Bang bang control of fullerene qubits using ultrafast phase gates. Nat. Phys. 2, (2006). 16. Duan, L. M., Cirac, J. I. & Zoller, P. Geometric manipulation of trapped ions for quantum computation. Science 292, (2001). 17. Berry, M. V. Quantal phase-factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, (1984). 18. Zanardi, P. & Rasetti, M. Holonomic quantum computation. Phys. Lett. A 264, (1999). 19. Falci, G., Fazio, R., Palma G. M., Jens Siewert, J. & Vedral, V. Detection of geometric phases in superconducting nanocircuits. Nature 407, (2000). 20. Solinas, P., Zanardi, P., Zanghi, N. & Rossi, F. Holonomic quantum gates: a semiconductorbased implementation. Phys. Rev. A 67, (2003). 21. Johansson, M. et al. Robustness of non-adiabatic holonomic gates. Phys. Rev. A 86, (2012). 22. Jones, J. A., Vedral, V., Ekert, A. & Castagnoli, G. Geometric quantum computation using nuclear magnetic resonance. Nature 403, (1999). 23. Feng, G., Xu, G. & Long, G. Experimental Realization of Nonadiabatic Holonomic Quantum Computation. Phys. Rev. Lett. 110, (2013). 24. Abdumalikov, A. A. et al. Experimental realization of non-abelian non-adiabatic geometric gates. Nature 496, 482 (2013). 25. Arroyo-Camejo, S., Lazariev, A., Hell, S. W. & Balasubramanian, G. Room temperature highfidelity holonomic single-qubit gate on a solid-state spin, Nat. Comm. 5, 4870 (2014). 13

14 26. de Lange, G., Wang, Z. H., Ristè, D., Dobrovitski, V. V. & Hanson, R. Universal dynamical decoupling of a single solid-state spin from a spin bath. Science 330, (2010). 27. Dolde, F. et al. Electric-field sensing using single diamond spins. Nat. Phys. 7, 459 (2011). 28. Zhao, N., Ho, S. W. & Liu, R. B. Decoherence and dynamical decoupling control of nitrogen vacancy center electron spins in nuclear spin baths. Phys. Rev. B 85, (2012). 29. Bar-Gill, N. et al. Suppression of spin-bath dynamics for improved coherence of multi-spinqubit systems. Nat. Comm. 3, 858 (2012). 30. Bar-Gill, N., Pham, L. M., Jarmola, A., Budker, D. & Walsworth, R. L. Solid-state electronic spin coherence time approaching one second. Nat. Comm. 4, 1743 (2013). Acknowledgements We thank Yuichiro Matsuzaki, Fedor Jelezko, Burkhard Scharfenberger, Kae Nemoto, William Munro, Norikazu Mizuochi, and Joerg Wrachtrup for their discussions and experimental help. This work was supported by the NICT Quantum Repeater Project, by the FIRST Quantum Information Project, and by a Grant-in-Aid for Scientific Research (A)-JSPS (No ). Author contributions The experiment was designed and analysed by Y. S., Y. K. and H. K. Measurements were made by Y. S. and Y. K. S. M. and T. T. and N. N. supported experiments in technical matters. H. K. supervised experiments. Y. S., Y. K. and H. K. wrote the paper. Competing Financial Interests The authors declare no competing financial interests. 14