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Tracking the precession of single nuclear spins by weak measurements

Abstract

Nuclear magnetic resonance (NMR) spectroscopy is a powerful technique for analysing the structure and function of molecules, and for performing three-dimensional imaging of their spin densities. At the heart of NMR spectrometers is the detection of electromagnetic radiation, in the form of a free induction decay signal1, generated by nuclei precessing around an applied magnetic field. Whereas conventional NMR requires signals from 1012 or more nuclei, recent advances in sensitive magnetometry2,3 have dramatically lowered the required number of nuclei to a level where a few or even individual nuclear spins can be detected4,5,6. It is unclear whether continuous detection of the free induction decay can still be applied at the single-spin level, or whether quantum back-action (the effect that a detector has on the measurement itself) modifies or suppresses the NMR response. Here we report the tracking of single nuclear spin precession using periodic weak measurements7,8,9. Our experimental system consists of nuclear spins in diamond that are weakly interacting with the electronic spin of a nearby nitrogen vacancy centre, acting as an optically readable meter qubit. We observe and minimize two important effects of quantum back-action: measurement-induced decoherence10 and frequency synchronization with the sampling clock11,12. We use periodic weak measurements to demonstrate sensitive, high-resolution NMR spectroscopy of multiple nuclear spins with a priori unknown frequencies. Our method may provide a useful route to single-molecule NMR13,14 at atomic resolution.

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Fig. 1: Scheme for tracking single nuclear spin precession.
Fig. 2: Experimental observation of single 13C precession.
Fig. 3: Frequency synchronization.
Fig. 4: High-resolution NMR spectroscopy of weakly coupled 13C nuclei.

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Data availability

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

References

  1. Hahn, E. L. Spin echoes. Phys. Rev. 80, 580–594 (1950).

    Article  ADS  Google Scholar 

  2. Poggio, M. & Degen, C. L. Force-detected nuclear magnetic resonance: recent advances and future challenges. Nanotechnology 21, 342001 (2010).

    Article  CAS  Google Scholar 

  3. Wrachtrup, J. & Finkler, A. Single spin magnetic resonance. J. Magn. Reson. 269, 225–236 (2016).

    Article  ADS  CAS  Google Scholar 

  4. Mamin, H. J. et al. Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor. Science 339, 557–560 (2013).

    Article  ADS  CAS  Google Scholar 

  5. Staudacher, T. et al. Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample. Science 339, 561–563 (2013).

    Article  ADS  CAS  Google Scholar 

  6. Müller, C. et al. Nuclear magnetic resonance spectroscopy with single spin sensitivity. Nat. Commun. 5, 4703 (2014).

    Article  ADS  Google Scholar 

  7. Korotkov, A. N. Output spectrum of a detector measuring quantum oscillations. Phys. Rev. B 63, 085312 (2001).

    Article  ADS  Google Scholar 

  8. Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys. 82, 1155–1208 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  9. Gefen, T., Khodas, M., McGuinness, L. P., Jelezko, F. & Retzker, A. Quantum spectroscopy of single spins assisted by a classical clock. Phys. Rev. A 98, 013844 (2018).

    Article  ADS  CAS  Google Scholar 

  10. Colangelo, G., Ciurana, F. M., Bianchet, L. C., Sewell, R. J. & Mitchell, M. W. Simultaneous tracking of spin angle and amplitude beyond classical limits. Nature 543, 525–528 (2017).

    Article  ADS  CAS  Google Scholar 

  11. Shiga, N. & Takeuchi, M. Locking the local oscillator phase to the atomic phase via weak measurement. New J. Phys. 14, 023034 (2012).

    Article  ADS  Google Scholar 

  12. Jordan, A. N. & Buttiker, M. Quantum nondemolition measurement of a kicked qubit. Phys. Rev. B 71, 125333 (2005).

    Article  ADS  Google Scholar 

  13. Ajoy, A., Bissbort, U., Lukin, M. D., Walsworth, R. L. & Cappellaro, P. Atomic-scale nuclear spin imaging using quantum-assisted sensors in diamond. Phys. Rev. X 5, 011001 (2015).

    Google Scholar 

  14. Perunicic, V. S., Hill, C. D., Hall, L. T. & Hollenberg, L. A quantum spin-probe molecular microscope. Nat. Commun. 7, 12667 (2016).

    Article  ADS  CAS  Google Scholar 

  15. Blok, M. S. et al. Manipulating a qubit through the backaction of sequential partial measurements and real-time feedback. Nat. Phys. 10, 189–193 (2014).

    Article  CAS  Google Scholar 

  16. Bloembergen, N. & Pound, R. V. Radiation damping in magnetic resonance experiments. Phys. Rev. 95, 8–12 (1954).

    Article  ADS  Google Scholar 

  17. Sidles, J. A. Folded Stern-Gerlach experiment as a means for detecting nuclear magnetic resonance in individual nuclei. Phys. Rev. Lett. 68, 1124–1127 (1992).

    Article  ADS  CAS  Google Scholar 

  18. Smith, G. A., Silberfarb, A., Deutsch, I. H. & Jessen, P. S. Efficient quantum-state estimation by continuous weak measurement and dynamical control. Phys. Rev. Lett. 97, 180403 (2006).

    Article  ADS  Google Scholar 

  19. Boss, J. M. et al. One- and two-dimensional nuclear magnetic resonance spectroscopy with a diamond quantum sensor. Phys. Rev. Lett. 116, 197601 (2016).

    Article  ADS  CAS  Google Scholar 

  20. Jordan, A. N. & Korotkov, A. N. Qubit feedback and control with kicked quantum nondemolition measurements: a quantum Bayesian analysis. Phys. Rev. B 74, 085307 (2006).

    Article  ADS  Google Scholar 

  21. Liu, G. Q. et al. Single-shot readout of a nuclear spin weakly coupled to a nitrogen-vacancy center at room temperature. Phys. Rev. Lett. 118, 150504 (2017).

    Article  ADS  Google Scholar 

  22. Kalb, N. et al. Experimental creation of quantum Zeno subspaces by repeated multi-spin projections in diamond. Nat. Commun. 7, 13111 (2016).

    Article  ADS  CAS  Google Scholar 

  23. Slichter, C. P. Principles of Magnetic Resonance 3rd edn (Springer, 1990).

  24. Taminiau, T. H. et al. Detection and control of individual nuclear spins using a weakly coupled electron spin. Phys. Rev. Lett. 109, 137602 (2012).

    Article  ADS  CAS  Google Scholar 

  25. Schmitt, S. et al. Submillihertz magnetic spectroscopy performed with a nanoscale quantum sensor. Science 356, 832 (2017).

    Article  ADS  CAS  Google Scholar 

  26. Boss, J. M., Cujia, K. S., Zopes, J. & Degen, C. L. Quantum sensing with arbitrary frequency resolution. Science 356, 837–840 (2017).

    Article  ADS  CAS  Google Scholar 

  27. Laraoui, A. et al. High-resolution correlation spectroscopy of C-13 spins near a nitrogen-vacancy centre in diamond. Nat. Commun. 4, 1651 (2013).

    Article  ADS  Google Scholar 

  28. Maurer, P. C. et al. Room-temperature quantum bit memory exceeding one second. Science 336, 1283–1286 (2012).

    Article  ADS  CAS  Google Scholar 

  29. Loquet, A., Lv, G., Giller, K., Becker, S. & Lange, A. 13C spin dilution for simplified and complete solid-state NMR resonance assignment of insoluble biological assemblies. J. Am. Chem. Soc. 133, 4722–4725 (2011).

    Article  CAS  Google Scholar 

  30. Zopes, J. et al. Three-dimensional localization spectroscopy of individual nuclear spins with sub-angstrom resolution. Nat. Commun. 9, 4678 (2018).

    Article  ADS  CAS  Google Scholar 

  31. Pfender, M. et al. High-resolution spectroscopy of single nuclear spins via sequential weak measurements. Nat. Commun. 10, 594 (2019).

    Article  ADS  Google Scholar 

  32. Babinec, T. M. et al. A diamond nanowire single-photon source. Nat. Nanotechnol. 5, 195–199 (2010).

    Article  ADS  CAS  Google Scholar 

  33. Momenzadeh, S. A. et al. Nanoengineered diamond waveguide as a robust bright platform for nanomagnetometry using shallow nitrogen vacancy centers. Nano Lett. 15, 165–169 (2015).

    Article  ADS  CAS  Google Scholar 

  34. Unden, T. et al. Coherent control of solid state nuclear spin nano-ensembles. npj Quant. Inform. 4, 39 (2018).

    Google Scholar 

  35. Zopes, J., Herb, K., Cujia, K. S. & Degen, C. L. Three-dimensional nuclear spin positioning using coherent radio-frequency control. Phys. Rev. Lett. 121, 170801 (2018).

    Article  ADS  CAS  Google Scholar 

  36. Ma, B. et al. Recent development in bonded ndfeb magnets. J. Magn. Magn. Mater. 239, 418–423 (2002).

    Article  ADS  CAS  Google Scholar 

  37. Rosskopf, T., Zopes, J., Boss, J. M. & Degen, C. L. A quantum spectrum analyzer enhanced by a nuclear spin memory. npj Quant. Inform. 3, 33 (2017).

    Article  ADS  Google Scholar 

  38. Taminiau, T. H., Cramer, J., van der Sar, T., Dobrovitski, V. V. & Hanson, R. Universal control and error correction in multi-qubit spin registers in diamond. Nat. Nanotechnol. 9, 171–176 (2014).

    Article  ADS  CAS  Google Scholar 

  39. London, P. et al. Detecting and polarizing nuclear spins with double resonance on a single electron spin. Phys. Rev. Lett. 111, 067601 (2013).

    Article  ADS  CAS  Google Scholar 

  40. Taylor, J. M. et al. High-sensitivity diamond magnetometer with nanoscale resolution. Nat. Phys. 4, 810 (2008).

    Article  CAS  Google Scholar 

  41. Glenn, D. R. et al. High-resolution magnetic resonance spectroscopy using a solid-state spin sensor. Nature 555, 351 (2018).

    Article  ADS  CAS  Google Scholar 

  42. Abobeih, M. H. et al. One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment. Nat. Commun. 9, 2552 (2018).

    Article  ADS  CAS  Google Scholar 

  43. Biercuk, M. J., Doherty, A. C. & Uys, H. Dynamical decoupling sequence construction as a filter-design problem. J. Phys. B 44, 154002 (2011).

    Article  ADS  Google Scholar 

  44. Degen, C., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work has been supported by the Swiss National Science Foundation through project grant numbers 200020_156100 and 200020_175600 and through the NCCR QSIT, and by the European Commission through DIADEMS grant number 611143 and ASTERIQS grant number 820394. We thank R. Liu, A. Retzker and T. Taminiau for discussions, K. Chang for experimental support and M. Palm for proofreading the manuscript.

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Nature thanks Dieter Suter and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations

Authors

Contributions

C.L.D. conceived the project. K.S.C., J.M.B. and K.H. carried out the experiments with the support of J.Z. and analysed the data. K.S.C., C.L.D. and J.M.B. performed the simulation and theoretical analysis of weak measurements. All authors discussed the results and participated in writing the manuscript.

Corresponding author

Correspondence to C. L. Degen.

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Extended data figures and tables

Extended Data Fig. 1 Extended pulse-timing diagram.

a, Protocol used for Figs. 2, 3. The sensor is initially polarized by means of an approximately 532-nm laser pulse. We subsequently apply a polarization transfer gate38 in a repetitive fashion. From the nuclear spin perspective, the polarization sequence consists of a π/2 conditional x rotation, followed by a π/2 Z-rotation and a subsequent π/2 conditional x rotation. We implement the conditional X-rotations as a resonant CPMG decoupling sequence applied on the electronic spin. The CPMG sequences consist of a periodic train of microwave π pulses with alternating phases and an interpulse delay of 2τ. The CPMG sequence is resonant when the interpulse delay matches the effective Larmor frequency of the nuclear spin, τ ≈ π/(2(γnB0 + 0.5a||)), where a|| is the parallel dipolar hyperfine coupling between the sensor and the nuclear spin. This condition leads to an effective interaction between the sensor and nuclear spin of the form \(g2{ {\hat{S}} }_{z}{\hat{I}}_{x}\) (ref. 19), that is, a simultaneous conditional rotation, where ga/π is a coupling rate determined by the transverse dipolar hyperfine coupling between the sensor and the nuclear spin. We implemented z rotations as a waiting time of duration τ0 = π/(2γnB0) if the sensor was polarized into mS = 0, or of duration τ if it was not. z-rotations can alternatively be implemented as non-resonant CPMG sequences, which effectively decouple the evolution of the nuclear spin from the sensor38. To initiate precession, we apply a π/2 y-rotation on the nuclear spin, implemented as another π/2 x rotation followed by a π/2 z rotation. We then probed the nuclear state x projection, \(\left\langle {\hat{I}}_{x}\right\rangle \), at intervals of a sampling time ts by means of weak measurements. Each weak measurement instance was implemented as a resonant CPMG decoupling sequence of duration tβ, sandwiched between two π/2 pulses whose axes were orthogonal, here x and y. We used a laser pulse to readout the sensor \({ {\hat{S}} }_{z}\) state upon each weak measurement instance. An additional delay time td was used to adjust the sampling time ts. b, Protocol used for Fig. 4 and Extended Data Figs. 4, 5. These experiments probed a bath of 13C spins whose hyperfine couplings were not known a priori nor directly measured, and the nuclear spins were directly manipulated by means of an external radio-frequency coil. We first polarized the host 15N spin using c-NOT gates on the electronic and 15N spin, implemented as a selective π pulse on the electron spin (conditional on the state of the 15N spin), followed by a selective π pulse on the 15N spin (conditional to the mS = 0 state of the electron spin), and a final laser pulse. Subsequently, we applied a NOVEL polarization transfer sequence consisting of a π/2 x rotation on the electron spin followed by a linear-ramp spin-lock pulse along its y axis. The relative amplitude increment of the spin-lock pulse was typically 10% around the resonant amplitude value and the duration 30 μs leading to a bandwidth of about 100 kHz. This procedure was repeated up to M = 1,200 times. To initiate precession, we applied a π/2 pulse on the bath of 13C nuclear spins using the radio-frequency coil. We additionally included a π pulse on the electron spin during the delay time td in order to recover information about the dipolar couplings of the bath spins. MW/RF, microwave, radio frequency.

Extended Data Fig. 2 Polarization of nuclear spin by repeated initialization.

The associated protocol is explained in Extended Data Fig. 1. The plot shows the degree of nuclear spin polarization \(\left\langle {\hat{I}}_{z}\right\rangle \) versus the number of repetitions of the initialization protocol. We measured \(\left\langle {\hat{I}}_{z}\right\rangle \) using spin tomography38. Different colours represent different angles for the conditional x rotations, varied from π/2 (green dots) to 0.125(π/2) (red dots). The plot demonstrates that even for incomplete x rotations, polarization transfer from the NV centre to the nuclear spin can still be achieved. This is relevant for a very weakly coupled nuclear spin, where the electron coherence time is too short to perform full π/2 x rotations.

Extended Data Fig. 3 Statistical analysis of Fig. 3a, demonstrating that the precession frequency is not modified by sufficiently weak measurements.

a, Signal frequency extracted from Lorentzian fits to each spectrum in Fig. 3a. We fit a linear function ax + b and find a = 0.999305 ± 0.000789 and b = 0.002171 ± 0.000916. A χ2 test yields χ2 = 0.000325 and a corresponding P value of 1.0 according to the χ2 distribution for k = 50 measurement points. b, Residuals for the linear fit in a. Frequency synchronization is absent in this plot because of the weak measurement strength (β ≈ 8°).

Extended Data Fig. 4 Weak measurement versus nuclear Ramsey spectroscopy.

a, Nuclear Ramsey spectroscopy of a 13C spin bath. The plot shows the normalized power spectrum of a 4-ms-long time trace after 22 h of signal integration. As described in Extended Data Fig. 1b, we first polarize the 13C spin bath, apply a π/2 radio-frequency pulse to initiate precession and finally perform a strong measurement at the end of a variable free evolution time, which we increase in steps of ts = 8 μs. We included a π pulse on the electron spin halfway through the free evolution time. We also tuned the strong measurement to maximize the signal dynamic range. The peaks in the power spectrum are associated with individual 13C spins that are relatively strongly coupled to the NV sensor, such that a strong measurement is possible. b, Weak measurement spectroscopy of the same 13C spin bath (blue). The plot again shows the normalized power spectrum of a 4-ms-long time trace, this time integrated over 2.5 h. The presence of many more peaks around the bare Larmor frequency (vertical grey dashed line) highlights a feature of continuous weak measurements: strongly coupled spins rapidly dephase, allowing for weaker signals (that would otherwise be submerged by a strong background) to be detected. The couplings of these nuclei can be estimated from the spectral shift with respect to the bare Larmor frequency. For fast optical readout and under our measurement sequence, the observed shifts correspond to a||/(4π). The y axis in both plots indicates the signal-to-noise ratio (SNR) calculated by normalizing the power spectrum amplitude to the standard deviation of the noise baseline (the portion of the power spectrum where no signals are present). Both measurements were performed under identical initialization, sampling time and readout parameters (see Supplementary Data 1).

Extended Data Fig. 5 Dependence of weak measurement spectra on measurement strength, polarization time and sampling rate.

a, Weak measurement power spectra for varying interaction time tβ, which defines the measurement strength β = gtβ. PSD, power spectral density. Increasing tβ (bottom to top) allows for more weakly coupled spins to be probed (peaks close to dashed line), while signals arising from nuclear spins with larger couplings g become increasingly dephased (peaks far from dashed line). The dashed line indicates the nuclear Zeeman frequency. b, Weak measurement power spectra for different durations of nuclear polarization, increasing from bottom to top. M indicates the number of iterations of the NOVEL polarization transfer sequence (see Extended Data Fig. 1b). The contact time for one iteration was 30 μs. c, Weak measurement power spectra as a function of sampling time ts. Peak locations do not shift when varying the sampling time, indicating that no signals are folded because of aliasing. Spectra are vertically offset for clarity.

Extended Data Fig. 6 Detection bandwidth of weak measurement spectroscopy.

a, Calculated filter function43 of a dynamical decoupling sequence with N = 8 pulses and 2τ = 232 ns interpulse delay; the dashed line is equation (7). The centre frequency is fc = 1/(4τ) = 2.154 MHz; equation (68) in ref. 44. The nominal bandwidth of the dynamical decoupling sequence is 1/(tβ) = 1/(4) ≈ 538 kHz; equations (68) and (69) in ref. 44. The Nyquist bandwidth is 1/(2ts) ≈ 254 kHz, where we chose tstβ for the simulation. (In a real experiment, ts > tβ because of readout overhead). b, Simulated weak measurement spectrum for three nuclear spins with parallel hyperfine parameters a||/(2π) = −20 kHz (spin 1), a||/(2π) = 200 kHz (spin 2) and a||/(2π) = −600 kHz (spin 3). The transverse hyperfine parameter was a/(2π) = 5 kHz for all spins. Note that the spectral shift for our scheme is 0.5a||/(2π) (not a||/(2π); for further details see Extended Data Fig. 1). For spin 3, aliasing leads to folding of the signal peak back into the Nyquist bandwidth. Simulations were implemented using density matrices. The spin system included the central NV meter spin and three nuclear spins. The NV centre spin was implemented by a quasi spin-1/2 system consisting of the mS = 0, −1 sublevels, and was simulated in the rotating frame of reference. c, Simulated weak measurement spectra for a single spin whose parallel hyperfine parameter was increased from a||/(2π) = 0 to 800 kHz in steps of 100 kHz. The transverse hyperfine parameter was a/(2π) = 5 kHz. The peak amplitudes clearly follow the profile of the filter function in a, demonstrating that the detection bandwidth of weak measurement spectroscopy is determined by the wide dynamical decoupling filter function. Arrows indicate back-folding of the peak due to aliasing. We note that for our specific experimental implementation, the detection of nuclear spins with strong a|| couplings becomes difficult, owing to inhomogeneous broadening caused by a residual hyperfine interaction during optical readout (see Methods).

Extended Data Fig. 7 Qualitative scaling of signal-to-noise ratio, spectral resolution, and receiver bandwidth with coupling g.

a, Log–log plot of the signal-to-noise ratio per unit time as a function of the coupling parameter g. Γe = (T2,DD)−1 is the decoherence rate of the electronic sensor spin and \({\Gamma }_{{\rm{n}}}={({T}_{2,{\rm{n}}}^{\ast })}^{-1}\) is the dephasing rate of the nuclear spin, and we assume Γe Γn. The plotted curve is based on Supplementary Note 2. b, Log–log plot of spectral linewidth (solid green curve) and receiver bandwidth (dashed red curve). For weak measurement spectroscopy, the linewidth is approximately Γn (Supplementary Note 2). The receiver bandwidth is \({t}_{\beta }^{-1}\) owing to the dynamical decoupling filter function. The frequency dynamic range of the measurement is given by the factor Γntβ and can be very large, which is important for NMR spectroscopy applications.

Supplementary information

Supplementary Information

Supplementary Notes 1 (Derivation of Measurement Back-action) and 2 (Signal-to-Noise Ratio), Supplementary Data 1 (Experimental Parameters), Supplementary Figures 1-19, and additional references.

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Cujia, K.S., Boss, J.M., Herb, K. et al. Tracking the precession of single nuclear spins by weak measurements. Nature 571, 230–233 (2019). https://doi.org/10.1038/s41586-019-1334-9

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