Controllable effects of quantum fluctuations on spin free-induction decay at room temperature
Xin-Yu Pan, Gang-Qin Liu, Dong-Qi Liu, Zhan-Feng Jiang, Nan Zhao, Ren-Bao Liu
CControllable e ff ects of quantum fluctuations on spin free-induction decay at room temperature Xin-Yu Pan, ∗ Gang-Qin Liu, and Dong-Qi Liu
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Zhan-Feng Jiang, Nan Zhao, † and Ren-Bao Liu ‡ Department of Physics and Center for Quantum Coherence,The Chinese University of Hong Hong, Shatin, New Territories, Hong Kong, China
Fluctuations of local fields cause decoherence of quantum objects. It is generally believed that at high temper-atures, thermal noises are much stronger than quantum fluctuations unless the thermal e ff ects are suppressed bycertain techniques such as spin echo. Here we report the discovery of strong quantum-fluctuation e ff ects of nu-clear spin baths on free-induction decay of single electron spins in solids at room temperature. We find that thecompetition between the quantum and thermal fluctuations is controllable by an external magnetic field. Thesefindings are based on Ramsey interference measurement of single nitrogen-vacancy center spins in diamond andnumerical simulation of the decoherence, which are in excellent agreement. PACS numbers: 03.65.Yz, 76.70.Hb, 42.50.Lc, 76.30.-v
Quantum systems lose their coherence when subjected tofluctuations of the local fields ( b ). Such decoherence phenom-ena are a fundamental e ff ect in quantum physics [1–3] and acritical issue in quantum technologies [4–11]. The local fieldfluctuations can result from thermal distribution of the bathstates at finite temperature [12], formulated as a density ma-trix ρ E = (cid:80) J p J | b J (cid:105)(cid:104) b J | with probability p J for the local fieldof a certain eigenvalue b J . If the local field operator b does notcommute with the total Hamiltonian of the bath H E , a certaineigenstate | b J (cid:105) of b is not an eigenstate of the total Hamilto-nian and will evolve to a superposition of di ff erent eigenstatesof b , causing quantum fluctuations of the local field [13]. Itis generally believed that at high-temperatures (as comparedwith transition energies of the bath), the thermal fluctuationsare much stronger than the quantum fluctuations, though withcertain control over the quantum systems, such as spin-echoor dynamical decoupling control in magnetic resonance spec-troscopy [14–16], the decoherence e ff ects of thermal fluctua-tions can be largely suppressed.In this Letter, we show that in the case of strong system-bath coupling (as compared with the internal Hamiltonian ofthe bath), the quantum fluctuations can be comparable to thethermal fluctuations, and induce notable e ff ects even on free-induction decay of the central spin coherence. The competi-tion between the thermal and quantum fluctuations can be con-trolled by an external magnetic field, indicated by crossoverbetween Gaussian and non-Gaussian decoherence accompa-nied by decoherence time variation.The model system in this study is a nitrogen-vacancy center(NVC) electron spin coupled to a bath of C nuclear spins indiamond. This system has promising applications in quantumcomputing [6, 7] and nano-magnetometry [8–11]. The hy-perfine interaction between the NVC spin and the bath spinsis essentially dipolar and therefore anisotropic. Due to theanisotropy of the interaction, the hyperfine field on a nuclearspin is in general not parallel or anti-parallel to the externalmagnetic field and therefore the local Overhauser field b (as abath operator) does not commute with the Zeeman energy of the bath. This induces strong quantum fluctuations, when theexternal field is not too strong or too weak. The model systemis representative of a large class of central spin decoherenceproblems in which a central spin (such as associated with im-purities or defects in solids) has anisotropic dipolar interactionwith bath spins [17].The NVC has a spin-1, which has a zero-field splitting ∆ ≈ .
87 GHz between the states | (cid:105) and |± (cid:105) , quantized alongthe z -axis (the nitrogen-vacancy axis). Since the NVC spinsplitting is much greater than the hyperfine interaction withthe C spins, the center spin flip due to the Overhauser fieldcan be safely neglected [7]. We only need to consider the z -component of the local field fluctuation, b z = (cid:80) j A j · I j , where A j is the dipolar coupling coe ffi cients for the j th nuclear spin I j . The local field b z is a quantum operator of the bath. Withinthe timescales considered in this paper, the interaction be-tween the C nuclear spins, which has strength less than a fewkHz [18, 19], can be neglected. The only internal Hamiltonianof the bath is the Zeeman energy under an external magneticfield, H E = (cid:80) j γ C I j · B , where γ C = . × T − s − is thegyromagnetic ratio of C. To be specific, the magnetic field B is applied along the z axis in this paper, but the physics is es-sentially the same for field along other directions. The Hamil-tonian of the NVC spin and the bath can be written as [18, 19] H = ∆ S z + ( γ e B + b z ) S z + H E , (1)where γ e = . × T − s − is the electron gyromagneticratio, and S z is the NVC spin operator along the z -axis.At room temperature, the nuclear spins are totally unpo-larized. Thus the bath can be described by a density matrix ρ E = − N I , with N being the number of C included in thebath, and I is a unity matrix of dimension 2 N . When thebath contains a large number of nuclear spins (for example, N > σ = (cid:68) b z (cid:69) / = (cid:88) j A j / . (2) a r X i v : . [ qu a n t - ph ] O c t FIG. 1: (Color online) (a) A fluorescence image of single NVC’sin a type-IIa diamond. (b) Rabi oscillation of an NVC spin drivenby a microwave pulse with the same strength as used in the Ram-sey signal measurement. (c) Continuous-wave ODMR spectrum ofan NVC spin, measured with a relatively strong microwave field(such that di ff erent lines due to di ff erent N nuclear spin states arenot resolved). The two peaks (fitted with Lorentzian lineshapes indashed lines) correspond to the transitions | (cid:105) ↔ | ± (cid:105) . (d) PulseODMR spectrum near the | (cid:105) ↔ | − (cid:105) transition of an NVC spin,measured with a relatively weak microwave field (such that di ff er-ent lines due to di ff erent N nuclear spin states are resolved, fittedwith Lorentzian lineshapes in dashed lines). The magnetic field is10.3 Gauss in the measurement.
This so-called inhomogeneous broadening would cause aGaussian decay of the NVC spin coherence, e − ( t / T ∗ ) with thedephasing time T ∗ = √ /σ .The quantum fluctuation of the local field b z arises fromthe fact that in general (cid:2) b z , H E (cid:3) (cid:44)
0, especially when the nu-clear Zeeman energy is comparable to the hyperfine coupling γ C B ∼ A j [19]. In the weak field case γ C B << A j , the e ff ectof the quantum fluctuations would be negligible. In the strongfield limit, γ C B >> A j , the quantum fluctuation would also besuppressed, since the nuclear spin flips due to the o ff -diagonalhyperfine interaction (components of A j perpendicular to the z -axis) would be suppressed by the large Zeeman energy cost.In addition, the local field fluctuation under a strong exter-nal field should contain only the diagonal part, i.e., in Eq. (2)for the the inhomogeneous broadening, A j should be replacedwith the z -component A zj . Therefore, we expect the dephas-ing time in the strong field limit is longer than that in theweak field limit. In the transition regime, the quantum fluctu-ation e ff ect would be important, and the dephasing would bein general non-Gaussian. Such features of NVC center spindephasing have been noticed previously in numerical simula-tions [19].We use optically detected magnetic resonance(ODMR) [20] to measure the Ramsey interference ofsingle NVC spins in a high-purity type-IIa single-crystaldiamond (with nitrogen density (cid:28) FIG. 2: (Color online) (a), (b), and (c) in turn show three typical casesof experimentally measured Ramsey signals as functions of time forthree NVC’s A, B, and C under di ff erent magnetic fields. (d), (e),and (f) are numerical simulations corresponding to (a), (b), and (c)in turn. The red symbols are measured or calculated results, and theblack lines are fitting with Eq. (3). NVC’s in diamond are addressed by a home-built confocalmicroscope system [see Fig. 1(a) for a typical fluorescenceimage of the single NVC’s]. An external magnetic field isapplied along the z -axis. The field strength is tunable from0 to 305 Gauss. Under a weak field [10.3 Gauss as shownin Fig. 1(c)], the two NVC spin transitions | (cid:105) ↔ | ± (cid:105) arewell resolved in spectrum. Furthermore, due to the hyperfinecoupling to the N nuclear spin, each NVC spin transitionis split into three lines corresponding to the three states ofthe N nuclear spin-1 [7], which are resolved by weak-pulseODMR measurement [see Fig. 1 (d) for the | (cid:105) ↔ | − (cid:105) transition]. Fig. 1(b) shows the high-fidelity rotation of theNVC spin under a microwave pulse of di ff erent durations.The Ramsey interference measurement scheme is as follows:The single NVC spin is first initialized to the state | (cid:105) byoptical pumping with a 532 nm laser pulse of 3.5 µ s duration.Then a π/ | (cid:105) + | − (cid:105) ) / √
2. The pulse is tunedresonant with the central line (corresponding to the N spinstate | (cid:105) N ) of the | (cid:105) ↔ | − (cid:105) transition for each magneticfield. The pulse duration [34 ns, corresponding to π/ | (cid:105) ↔ | + (cid:105) transition and as short as to spectrally cover all the threehyperfine lines corresponding to di ff erent N nuclear spinstates. After the first microwave pulse, the spin is left tofreely precess about the magnetic field with dephasing. Aftera delay time t , a second π/ | − (cid:105) . FIG. 3: (Color online) Dependence on the magnetic field strength of(a) the dephasing time T ∗ and (b) the exponential decay index n forthree NVC’s (A, B, and C) measured in experiments (circle, square,and diamond symbols with error bars), compared with the numericalsimulations (solid, dashed, and dash-dotted lines). The fluorescence of the NVC, which is about 30% weakerwhen the spin is in |−(cid:105) than it is when the spin is in | (cid:105) , isdetected by photon counting under illumination of a 532 nmlaser of 0.35 µ s duration. Each measurement (for a certain B field and delay time t ) is typically repeated 0 . ∼ ffi cient signal-to-noise ratio.Typical Ramsey interference signals of single NVC spinsare shown in Fig. 2 (a-c). The oscillation is due to the beatingbetween di ff erent transition lines corresponding to the three N spin states [7]. As shown in Fig. 2 (a-c), the spin coher-ence represented by the fluorescence change as a function oftime, after subtraction of the background photon counting, iswell fitted with the formula S = Ce − ( t / T ∗ ) n (cid:34) +
23 cos ( A N t + φ ) (cid:35) , (3)in which A N is the hyperfine coupling constant to the Nnuclear spin, T ∗ gives the spin dephasing time, and the expo-nential index n characterizes the non-Gaussian nature of thedephasing ( n = T ∗ and the expo-nential decay index n as functions of the external magneticfield strength for three di ff erent NVC’s (labeled A, B and C).The increasing of the dephasing time with the magnetic fieldstrength and the non-Gaussian decay associated with the de-phasing time rising demonstrate the competition between thethermal fluctuations of the local fields and the quantum fluc-tuations. Since the C atoms (with abundance of 1.1%) arerandomly located around the NVC’s, the dephasing time T ∗ presents a random distribution depending on the C posi-tion configurations [19]. An NVC with longer dephasing timeshould have C atoms located farther away from the center en v e l ope (a) NVC AB=274G t ( μ s)(b) NVC BB=166G t ( μ s) en v e l ope (c) NVC CB=20G number of C FIG. 4: (Color online) Decay envelopes of the calculated Ramseysignals for various numbers of nearest C nuclear spins included inthe bath, shown as filled circles, stars, open circles, open squares,crosses, and solid lines for N =
1, 3, 5, 10, 30, and 100 in turn. (a),(b), and (c) are calculated under the same conditions as in Fig. 2 (d),(e), and (f) in turn. with weaker hyperfine interaction (as the hyperfine interac-tion is dipolar and decays rapidly with distance from the cen-ter). Therefore, we expect that the quantum fluctuations forNVC’s with longer dephasing times start to take e ff ect at lowermagnetic field. This is indeed confirmed by the three sets ofdata representing NVC’s with long, intermediate, and shortdephasing time (NVC A, B and C in turn).To further confirm the physical picture of the quantum-thermal fluctuation crossover, we carry out numerical simula-tions of the Ramsey signals with no fitting parameters. Sincethe positions of the C atoms are not determined and the de-phasing time depends on the positions of the nuclear spins, werandomly choose the spatial configurations such that the de-phasing times at zero field are close to the experimental valuesat the lowest field. The simulation is done with only single nu-clear spins dynamics taken into account (the interactions be-tween nuclear spins are neglected since they are not relevantin the timescales considered in this paper), which is an exactlysolvable problem. The Ramsey signal is given by [18, 19] S = (cid:88) m = , ± e imA t N (cid:89) n = Tr (cid:104) e i γ C BI zj t e i A j · I j t − i γ C BI zj t (cid:105) . (4)As shown in Fig. 2 (d-f), the calculated results are well fit-ted with Eq. (3). In the simulations, the nearest 500 nuclearspins are included ( N = ff er-ent distances to the NVC spin dephasing. The nearest few Cnuclear spins already contribute the major part of the localfield fluctuations. A close examination of the C positionsin di ff erent configurations reveals that the average hyperfinecoupling constants for the nearest 10, 5, and 3 nuclear spins(which contribute the major part of the dephasing) for NVCA, B, and C are ¯ A ≈ .
16, 0.51, and 1.7 µ s − in turn. Corre-spondingly, the quantum fluctuations should start to take e ff ectat magnetic field strength B ∼ ¯ A /γ C ≈
24, 76, and 260 Gaussfor NVC A, B, and C in turn. This is indeed consistent withthe experimental observation shown in Fig. 3.In conclusion, we demonstrate that even at room temper-ature (which can be regarded as infinite for the nuclear spinbaths considered here) and in free-induction decay of spin de-coherence, the quantum fluctuations of local fields can be asstrong as the thermal fluctuations in a mesoscopic spin bathwith anisotropic interaction with the central spin. The contri-bution of the quantum fluctuations can be tuned by an exter-nal magnetic field. In addition to revealing an aspect of thequantum nature of nuclear spin baths, the e ff ect can be usedto identify optimal physical systems and parameter ranges forquantum control over a few nuclear spins via a central elec-tron spin. Such control is relevant to quantum computing andnano-magnetometry [6–11].This work was supported by National Basic Research Pro-gram of China (973 Program project No. 2009CB929103),the NSFC Grants 10974251 and 11028510, Hong KongRGC / GRF CUHK402208 and CUHK402410, and CUHK Fo-cused Investments Scheme. ∗ Email: [email protected] † Present address: 3rd Physics Institute and Research CenterSCoPE, University of Stuttgart, 70569 Stuttgart, Germany ‡ Email: [email protected][1] W. H. Zurek, Phys. Today , 36 (1991). [2] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoherence and the appearance of a classicalworld in quantum theory (Springer, New York, 2003), 2nd ed.[3] M. Schlosshauer, Rev. Mod. Phys. , 1267 (2004).[4] J. Clarke and F. K. Wilhelm, Nature , 1031 (2008).[5] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe,and J. L. O’Brien, Nature , 45 (2010).[6] J. Wrachtrup and F. Jelezko, J. Phys. - Cond. Mat. , S807(2006).[7] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov,F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, Sci-ence , 281 (2006).[8] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Taylor,P. Cappellaro, L. Jiang, M. V. Gurudev Dutt, E. Togan, A. S.Zibrov, et al., Nature , 644 (2008).[9] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud,J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer,A. Krueger, et al., Nature , 648 (2008).[10] N. Zhao, J.-L. Hu, S.-W. Ho, T.-K. Wen, and R. B. Liu, NatureNanotech. , 242 (2011).[11] M. S. Grinolds, P. Maletinsky, S. Hong, M. D. Lukin, R. L.Walsworth, and A. Yacoby, Nature Phys. , 687 (2011).[12] I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. B ,205309 (2002).[13] N. Zhao, Z.-Y. Wang, and R.-B. Liu, Phys. Rev. Lett. ,217205 (2011).[14] E. L. Hahn, Phys. Rev. , 580 (1950).[15] J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R. B. Liu,Nature , 1265 (2009).[16] C. A. Ryan, J. S. Hodges, and D. G. Cory, Phys. Rev. Lett. ,200402 (2010).[17] V. V. Dobrovitski, A. E. Feiguin, R. Hanson, and D. D.Awschalom, Phys. Rev. Lett. , 237601 (2009).[18] J. R. Maze, J. M. Taylor, and M. D. Lukin, Phys. Rev. B ,094303 (2008).[19] N. Zhao, S.-W. Ho, and R. B. Liu (2011), arXiv:1108.2343.[20] A. Gruber, A. Dr¨oenstedt, C. Tietz, L. Fleury, J. Wrachtrup, andC. von Borczyskowski, Science276