Analogous to the precession of a Foucault pendulum observed on the rotating Earth, a precessing spin observed in a rotating frame of reference appears frequency-shifted. This can be understood as arising from a magnetic pseudo-field1, 2 in the rotating frame that nevertheless has physically significant consequences, such as the Barnett effect3. To detect these pseudo-fields, a rotating-frame sensor is required4. Here we use quantum sensors, nitrogen–vacancy (NV) centres, in a rapidly rotating diamond to detect pseudo-fields in the rotating frame. Whereas conventional magnetic fields induce precession at a rate proportional to the gyromagnetic ratio, rotation shifts the precession of all spins equally, and thus primarily affect 13C nuclear spins in the sample. We are thus able to explore these effects via quantum sensing in a rapidly rotating frame, and define a new approach to quantum control using rotationally induced nuclear spin-selective magnetic fields. This work provides an integral step towards realizing precision rotation sensing and quantum spin gyroscopes.
At a glance
A spin measured by an observer in a rotating frame appears to precess faster or slower depending on Ω, the rotational angular frequency of the frame. This can be thought of as arising from an effective magnetic field1, 2 BΩ = Ω/γ in the rotating frame, with γ the spin gyromagnetic ratio. Despite being referred to as ‘fictitious’ fields, rotationally induced magnetic pseudo-fields have measurable effects, in the same way that spin-state-dependent vector light shifts5 and artificial gauge fields6 have real effects. In the Barnett effect3, for example, the effective magnetic field generated by physically rotating an initially unmagnetized rod of iron leads to polarization of the constituent electron spins along the rotation axis, and magnetization of the iron sample. In this work, we explore for the first time quantum sensing of pseudo-fields in the physically rotating frame, using solid-state qubits that detect rotational pseudo-fields and simultaneously are uniquely suited to exploring quantum control with rotation.
Exploring rotational pseudo-fields imposes considerable experimental challenges, as the sensor must be in the rotating frame4, 7. Nuclear spin gyroscopes operate on a similar principle, where it is the sensing apparatus that executes rotations about a gas of nuclear spins8. Magic-angle spinning9 nuclear magnetic resonance (NMR) experiments routinely study rapid rotations of more than 10 kHz in nuclear spin systems, and recent work has used a pickup coil rotating with the sample to measure the rotational pseudo-field4, 10, 11. However, NMR-based experiments require a strong polarizing magnetic field (much larger than the pseudo-fields) to obtain a signal, limiting these experiments to detection of small perturbations due to rotation. Both nuclear spin gyroscopes and pickup coils are operated essentially classically, with limited scope to fully study the effects of rotational pseudo-fields on quantum systems.
Solid-state spin systems, such as the nitrogen–vacancy (NV) centre in diamond12, 13, 14, have attracted considerable attention as robust quantum sensors, and have innate advantages for the study of rotational pseudo-fields. NV centres are intrinsic to the rotating sample, with an electron spin that is easily controlled and measured with microwave and optical fields, and are also amenable to quantum measurement and control protocols15. Nuclear spins (such as spin-1/2 13C) located within a few lattice sites of the NV spin precess at kilohertz rates in laboratory fields of several gauss, rotation rates achievable with modern electric motors and magic-angle spinners. The electron spin coherence time T2 ~ 0.1–1 ms of the NV centre16, 17 can be made comparable to the nuclear spin precession period at low fields, allowing a substantial time window for quantum sensing during rotation. The NV electron spin directly measures the nuclear spin magnetic dipole field, so we do not need strong conventional fields to polarize nuclear spins to attain a measurement signal.
A conventional magnetic field induces precession at a rate proportional to the spin gyromagnetic ratio, but rotational pseudo-fields shift the precession of all spins equally, the effective field being inversely proportional to the gyromagnetic ratio. In addition to quantum detection, a rotating NV–nuclear spin system allows our experiments to investigate a regime denied to previous studies, where rotational pseudo-fields are large enough to cancel a conventional magnetic field for the nuclear spins. The NV electron spins are essentially unaffected by rotation, and retain a significant Zeeman splitting from the conventional field. The NV remains an independent, controllable and incisive probe of the zero-field nuclear spin dynamics.
Although the NV has been used as a quantum sensor in a variety of noisy, real-world conditions, such as within biological cells18, 19, it is not immediately apparent that its abundant sensing advantages can be accessed when executing rapid rotation. A physically rotating NV centre is predicted to accumulate a geometric phase20, which forms the basis of a proposed nanoscale gyroscope21. Theoretical work has also proposed a nuclear-spin gyroscope using the intrinsic nitrogen nuclear spin of the NV centre to sense rotationally shifted precession21, 22. To date, experiments on NVs in moving diamonds have considered the quasi-static case, where standard experimental protocols can be applied18. In our experiments, we employ NV centres as quantum magnetometers in a frame rotating with a period comparable to the spin coherence time, and establish the means of extracting quantum information from rapidly rotating qubits.
An outline of our experiment is depicted in Fig. 1. For simplicity, we show the case for single NV sensors, but the physics is equally valid for the ensembles of NVs used in our experiments: multiple identical NV sensors greatly enhance the measurement signal relative to noise, and considerably simplify the experimental procedure. A synthetic diamond with an ensemble density of NVs and a 1.1% natural abundance of 13C is mounted to a high-speed electric motor, with one of the four orientation classes of NVs approximately parallel to the rotation axis (denoted as ). With a magnetic field B0 also parallel to , an effective two-level system is formed from the mS = 0 and mS = −1 states of the NV ground state. The experimental procedure, summarized in Fig. 1d, consists of standard optical preparation, microwave state manipulation and analysis of time-dependent photoluminescence emitted by the ensemble to determine the relative phase between the states of the NV two-level system accumulated in the spin-echo sequence. We employ several modifications to accommodate the rotation of the diamond (see Methods and Supplementary Methods).
The first part of this work concerns detection of rotational pseudo-fields. Spin-1/2 13C nuclei in the diamond lattice (precessing at ω13C = 2πf13C = γ13CB in a magnetic field of strength B = | B |, γ13C/2π = 1.0715 kHz G−1) generate a time-varying magnetic field at the NV that is detected in a spin-echo experiment23. With the NV in the mS = 0 state and the magnetic field parallel to the NV axis, the nuclear spins precess at f13C. When the NV is in the mS = −1 state, its dipole field interacts with the 13C spin, rapidly modulating the spin-echo signal. The many different configurations of NV–13C pairwise interactions (when averaged over the ensemble) result in a spread of oscillation frequencies that in turn beat against each other. The spin-echo signal collapses with a magnetic-field-strength-dependent characteristic time23τC(B), S(τ) ∝ exp(−(τ/τC)n), for typical lab fields of B > 1 G, n = 4. When the spin-echo measurement time τ equals 2/f13C, the total phase accumulated is zero, and the spin-echo signal revives. Any change in the nuclear spin precession frequency changes the time at which the echo signal revives. We measure this revival time to infer the overall 13C precession frequency f13C and then quantify the rotationally induced shift (Fig. 1d–f).
The gyromagnetic ratio of the 13C nucleus is positive: in the presence of a magnetic field oriented along , the 13C dipole moment precesses in a negative direction (clockwise, looking from above along , as shown in Fig. 1b, c). The 13C spin-state populations are thermally distributed in our experiments, but it is the relative direction of spin precession and physical rotation that is significant. The precession direction of the 13C dipole moment is the same, regardless of spin state. Inverting the magnetic field direction changes the precession direction of the 13C dipole relative to the imposed rotation Ω = 2πfrot. There are thus four possible configurations we investigate: and anticlockwise (frot > 0) or clockwise (frot < 0) rotations.
The total field B = B0 + BΩ experienced by the 13C spins is measured by determining the time when the spin-echo signal revives. The echo signal around the revival typically appears Gaussian in spin-echo time, and for each rotation speed and magnetic field configuration we fit a Gaussian to the echo signal and extract the revival time. Example revivals are shown in Fig. 2a. The extracted 13C precession frequency for an applied field of (B0 = 37 G, f0 = 40 kHz) is shown in Fig. 2b as a function of rotation speed for the four possible configurations of rotation and magnetic field. The results closely match the expected linear shift f13C = f0 ± frot characteristic of rotational pseudo-fields.
We then considered the case where the rotational pseudo-field is comparable to the external magnetic field, allowing us to cancel the conventional magnetic field for the nuclear spins in the rotating frame, and thus control the NV electron spin coherence. We studied the initial collapse of the spin-echo signal, where the field-dependent collapse time τC(B) is indicative of the dominant spin bath interactions24: for B > 1 G, the NV-13C hyperfine interaction dominates and the collapse time is expected to increase with lower magnetic field strengths. Below 1 G, nuclear spin flips mediated by the internuclear dipole–dipole interaction dominate, with τC saturating to some maximum value limited by the uncorrelated magnetic noise in the spin bath.
In Fig. 3a we show the initial collapse of the echo signal for an applied B0 = 4.8 G field along . With the diamond rotated at frot = 5.167 kHz, the induced BΩ is along and cancels the magnetic field (top). When rotated at frot = −5.167 kHz, BΩ is parallel to B0, resulting in B = 9.6 G, the measured spin-echo signal closely matches the echo signal observed for a stationary diamond and an applied 9.6 G field. We then observed how the echo signal changes for a range of total fields, from B0 − BΩ = −1.34(4) G to 4.84(1) G (Fig. 3b). We modelled the normalized spin-echo signal as16
with free collapse time τC and decay exponent n. The τC extracted from fits to the data is shown in Fig. 3c. The collapse time is well described by the power-law function τC ∝ B−k, with k = 0.42(2) for B < 1 G, before saturating at B = 0 to around 70 μs. The value of k in the power-law model is theoretically predicted24 to be 0.5 (for a single NV in a nuclear spin bath), but depends on the specifics of the bath environment. The collapse time is relatively independent of the decay exponent. The value of n is also expected to change with field strength, from n = 4 to n = 2 as the field is reduced from moderate strength to near zero25. Although we did observe this general trend, improved statistics would be needed for more detailed investigation of the behaviour of the decay exponent.
The NV electron spin coherence is intrinsically linked to the dynamics of the surrounding bath of nuclear spins. We have shown in Fig. 3 that pseudo-fields allow us to control the spin bath independently of the NV electron spin, which retains its sensing utility even when the total field experienced by the nuclear spins is zero. This nuclear-spin selectivity stems from a unique property of rotational pseudo-fields: BΩ, e = Ω/γe, experienced by the NV (with gyromagnetic factor γe/2π = 2.8 MHz G−1) is a factor of γ13C/γe less than BΩ, 13C, that experienced by the 13C spins. For the maximum rotation frequency used in this work, 5.5 kHz, BΩ, e = 2 mG compared to BΩ, 13C = 5.13 G for the 13C spins, and for all rotation speeds considered we did not observe any additional Zeeman splitting of the NV energy levels.
The nuclear-spin selectivity of pseudo-fields invites application in other schemes to selectively manipulate nuclear spins in diamond. For instance, much higher rotation speeds (111 kHz has been demonstrated in NMR magic-angle spinning experiments26) could create 13C-specific fields of more than 100 G while still minimally perturbing the NV electron spin. Large pseudo-fields potentially offer alternatives to existing decoupling schemes, controlling different nuclear spin bath elements independently of the NV electron spin and enriching zero-field nanoscale NMR experiments with NV centres27, 28, 29 and schemes of quantum information processing between coupled electron and nuclear spins30.
In addition to using quantum sensors to demonstrate the profound connection between magnetism and physical rotation, our findings advance quantum sensing and measurement into the physically rotating frame, and form a significant step towards NV-based rotation sensing and gyroscopic applications. Our results also establish a unique, highly selective method of controlling the nuclear spin bath surrounding an ensemble of electron spin qubits, and we anticipate interesting new directions for employing rotation as a quantum control.
State preparation and readout.
The diamond used in this work contains an ensemble density of NV centres (approximately 2 × 1015 cm−3) equally distributed between the four possible orientation classes, one of which is normal to the diamond surface. The -oriented magnetic bias field is aligned parallel to the rotation axis, so that only the NV centres normal to the surface of the diamond are resonantly addressed with microwaves. The preparation laser is aligned close to the centre of rotation to maximize optical pumping and readout efficiency. For rotational speeds above 1.667 kHz the NVs are illuminated with green light for one rotational period, optically preparing a ring of NV centres. At lower speeds the NVs are prepared using a 3 μs laser pulse; at these speeds drift of the rotational centre during experiments is less severe. We verified the efficacy of optical preparation for both pumping schemes. Since the NV axis is parallel to the rotation axis, each microwave pulse then addresses the entire ring of optically prepared NVs, and precise synchronization to the motor rotation is not necessary.
Magnetic field alignment.
The NV orientation class we probe is not precisely aligned with the rotation axis. When the magnetic field is also tilted from the rotation axis, the Zeeman shift of the mS = ±1 states changes during rotation. This leads to an effective AC field as experienced by the NVs of magnitude Beff = B0 sinθB sinθNV, with B0 the total field strength and θB and θNV the misalignment of the magnetic field and NV axis from the rotation axis, respectively. A spin-echo measurement is sensitive to these fields, which in the case of large misalignments suppress the 13C revival or result in faster initial collapse, since the experimental pulse sequence is not synchronous with the rotation of the diamond. We use the amplitude of the spin-echo signal to diagnose the elimination of magnetic field components transverse to the rotation axis and minimize θB, as detailed comprehensively in the Supplementary Methods.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
- Gyromagnetic and electron-inertia effects. Rev. Mod. Phys. 7, 129–166 (1935).
- Theory of gyromagnetic effects and some related magnetic phenomena. Rev. Mod. Phys. 34, 143–165 (1962). &
- Magnetization by rotation. Phys. Rev. 6, 239–270 (1915).
- Observation of Barnett fields in solids by nuclear magnetic resonance. Appl. Phys. Express 7, 063004 (2014). et al.
- Effective operator formalism in optical pumping. Phys. Rev. 163, 12–25 (1967). &
- Synthetic magnetic fields for ultracold neutral atoms. Nature 462, 628–632 (2009). , , , &
- Rotational Doppler effect in magnetic resonance. Phys. Rev. B 82, 174418 (2010). , &
- Nuclear magnetic resonance gyroscopes. IEEE Sensors 2010 Conf. 17–22 (IEEE, 2010).
- Magic angle spinning in solid state NMR spectroscopy. Phil. Trans. R. Soc. Lond. A 299, 505–520 (1981).
- Line splitting by mechanical rotation in nuclear magnetic resonance. Jpn. J. Appl. Phys. 54, 050302 (2015). et al.
- Rotational Doppler effect and Barnett field in spinning NMR. J. Phys. Soc. Jpn 84, 043601 (2015). et al.
- Single defect centres in diamond: a review. Phys. Status Solidi a 203, 3207–3225 (2006). &
- The nitrogen-vacancy colour centre in diamond. Phys. Rep. 528, 1–45 (2013). et al.
- Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65, 83–105 (2014). , , &
- Quantum Information Processing with Diamond: Principles and Applications (eds Prawer, S. & Aharonovich, I.) (Woodhead Publishing, 2014).
- Coherence of nitrogen-vacancy electronic spin ensembles in diamond. Phys. Rev. B 82, 201201 (2010). et al.
- Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 8, 383–387 (2009). et al.
- Quantum measurement and orientation tracking of fluorescent nanodiamonds inside living cells. Nat. Nanotech. 6, 358–363 (2011). et al.
- Nanometre-scale thermometry in a living cell. Nature 500, 54–58 (2013). et al.
- Measurable quantum geometric phase from a rotating single spin. Phys. Rev. Lett. 108, 240403 (2012). , , &
- Gyroscopes based on nitrogen-vacancy centers in diamond. Phys. Rev. A 86, 052116 (2012). , , , &
- Stable three-axis nuclear-spin gyroscope in diamond. Phys. Rev. A 86, 062104 (2012). &
- Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281–285 (2006). et al.
- Decoherence and dynamical decoupling control of nitrogen vacancy center electron spins in nuclear spin baths. Phys. Rev. B 85, 115303 (2012). , &
- Analytic solutions to the central-spin problem for nitrogen-vacancy centers in diamond. Phys. Rev. B 90, 075201 (2014). , &
- Structure of fully protonated proteins by proton-detected magic-angle spinning NMR. Proc. Natl Acad Sci. USA 113, 9187–9192 (2016). et al.
- Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume. Science 339, 561–563 (2013). et al.
- Magnetic resonance detection of individual proton spins using quantum reporters. Phys. Rev. Lett. 113, 197601 (2014). et al.
- Nuclear magnetic resonance detection and spectroscopy of single proteins using quantum logic. Science 351, 836–841 (2016). et al.
- Quantum register based on individual electronic and nuclear spin qubits in diamond. Science 316, 1312–1316 (2007). et al.
We acknowledge valuable discussions with L. P. McGuinness and J.-P. Tetienne, and thank D. A. Simpson and A. D. Stacey for discussions and providing the diamond sample. This work was supported by the Australian Research Council Discovery Scheme (DP150101704).
- Supplementary information (430 KB)