Abstract
Quantum devices for sensing and computing applications require coherent quantum systems, which can be manipulated in fast and robust ways^{1}. Such quantum control is typically achieved using external electromagnetic fields, which drive the system’s orbital^{2}, charge^{3} or spin^{4,5} degrees of freedom. However, most existing approaches require complex and unwieldy gate structures, and with few exceptions^{6,7} are limited to the regime of weak coherent driving. Here, we present a novel approach to coherently drive a single electronic spin using internal strain fields^{8,9,10} in an integrated quantum device. Specifically, we employ timevarying strain in a diamond cantilever to induce longlasting, coherent oscillations of an embedded nitrogen–vacancy (NV) centre spin. We perform direct spectroscopy of the phonondressed states emerging from this drive and observe hallmarks of the soughtafter strongdriving regime^{6,11}, where the spin rotation frequency exceeds the spin splitting. Furthermore, we employ our continuous strain driving to significantly enhance the NV’s spin coherence time^{12}. Our roomtemperature experiments thereby constitute an important step towards straindriven, integrated quantum devices and open new perspectives to investigate unexplored regimes of strongly driven multilevel systems^{13} and exotic spin dynamics in hybrid spinoscillator devices^{14}.
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Main
The use of crystal strain for the manipulation of single quantum systems (‘spins’) in the solid state brings vital advantages compared to established methods relying on electromagnetic fields. Strain fields can be straightforwardly engineered in the solid state and can offer a direct coupling mechanism to embedded spins^{8,15}. As they are intrinsic to these systems, strain fields are immune to drifts in the coupling strength. Also, strain does not generate spurious stray fields, which are unavoidable with electric or magnetic driving and can cause unwanted dephasing or heating of the environment. Furthermore, coupling spins to strain offers attractive features of fundamental interest. For instance, strain can be used to shuttle information between distant quantum systems^{15}, and has been proposed to generate squeezed spin ensembles^{14} or to cool mechanical oscillators to their quantum ground state^{16}. These attractive perspectives for straincoupled hybrid quantum systems motivated recent studies of the influence of strain on nitrogen–vacancy (NV) centre electronic spins^{9,10,17} and experiments on straininduced, coherent driving of large NV spin ensembles^{8}. Promoting such experiments to the singlespin regime, however, remains an outstanding challenge, and would constitute a major step towards the implementation of integrated, straindriven quantum systems.
Here, we demonstrate the coherent manipulation of a single electronic spin using timeperiodic, intrinsic strain fields generated in a singlecrystalline diamond mechanical oscillator. We show that such strain fields allow us to manipulate the spin in the strongdriving regime, where the spin manipulation frequency significantly exceeds the energy splitting between the two involved spin states, and to protect the spin from environmental decoherence. Our experiments were performed on electronic spins in individual NV lattice point defect centres, embedded in singlecrystalline diamond cantilevers. The negatively charged NV centres we studied have a spin S = 1 ground state with basis states {0〉, −1〉, +1〉}, where m_{s}〉 is an eigenstate of the spin operator along the NV’s symmetry axis, z (Fig. 1a). The energy difference between ±1〉 and 0〉 is given by the zerofield splitting D_{0} = 2.87 GHz. The levels ±1〉 are split by 2γ_{NV}B_{NV} (with γ_{NV} = 2.8 MHz G^{−1}) in a magnetic field B_{NV} applied along z. Hyperfine interactions between the NV’s electron and ^{14}N nuclear spin (I = 1 and quantum number m_{I}) further split the NV spin levels by an energy A_{HF} = 2.18 MHz into states m_{s}, m_{I}〉 (Fig. 1a)^{5}. In our experiments, we use optical excitation and fluorescence detection to initialize and read out the NV spin^{18} with a homebuilt confocal optical microscope^{9} (Methods). Furthermore, we use microwave magnetic fields to perform optically detected electron spin resonance (ESR; Fig. 1b) and manipulate the NV’s electronic spin states.
Coherent strain driving of NV spins is based on the sensitive response of the NV spin states to strain in the diamond host lattice. For uniaxial strain applied transverse to the NV axis, the corresponding straincoupling Hamiltonian takes the form^{14}
Here, ℏ is the reduced Planck constant, γ_{0}^{⊥} the transverse singlephonon straincoupling strength and and are the raising (lowering) operators for spin and phonons, respectively. Transverse strain thus leads to a direct coupling of the two electronic spin states −1〉 and +1〉 (ref. 9) and, in the case of nearresonant, timevarying (a.c.) strain, can coherently drive the transitions −1, m_{I}〉 ↔ +1, m_{I}〉 (ref. 8). For a classical (coherent) phonon field at frequency ω_{m}/2π, equation (1) can be written as , where Ω_{m} = γ_{0}^{⊥}x_{c}/x_{ZPF} describes the amplitude of the strain drive, with x_{ZPF} and x_{c} the cantilever’s zeropoint fluctuation and peak amplitude, respectively (here, with x_{ZPF} ∼7.7 × 10^{−15} m and γ_{0}^{⊥}/2π ∼ 0.08 Hz, Supplementary Information). Interestingly, strain drives a dipoleforbidden transition (Δm_{s} = 2), which would be difficult to access, for example, using microwave fields.
To generate and control a sizeable a.c. strain field for efficient coherent spin driving, we employed a mechanical resonator in the form of a singly clamped, singlecrystalline diamond cantilever^{9}, in which the NV centre is directly embedded (Fig. 1c and Methods). The cantilever was actuated at its mechanical resonance frequency ω_{m}/2π = 6.83 ± 0.02 MHz using a piezoelement placed nearby the sample. We controlled the detuning between the mechanical oscillator and the −1〉 ↔ +1〉 spin transition by applying an adjustable external magnetic field B_{NV} along the NV axis (Fig. 1a and Methods).
To demonstrate coherent NV spin manipulation using resonant a.c. strain fields, we first performed straindriven Rabi oscillations between −1〉 and +1〉 for a given hyperfine manifold (here, m_{I} = 1). To that end, we initialized the NV in −1,1〉 by applying an appropriate sequence of laser and microwave pulses (Fig. 1d). We then let the NV spin evolve for a variable time τ, under the influence of the coherent a.c. strain field generated by constantly exciting the cantilever at a fixed peak amplitude x_{c} ∼ 100 nm (Supplementary Information). After this evolution, we measured the resulting population in −1,1〉 with a pulse sequence analogous to our initialization protocol. As expected, we observe straininduced Rabi oscillations (Fig. 1e) for which we find a Rabi frequency Ω_{m}/2π = 1.14 ± 0.01 MHz and hardly any damping over the 30 μs observation time. Importantly, and in contrast to a recent study on NV ensembles^{8}, this damping timescale is not limited by ensemble averaging, because our experiment was performed on a single NV spin.
We obtain further insight into the strength and dynamics of our coherent straindriving mechanism from ESR spectroscopy of the straincoupled NV spin states, +1〉 and −1〉. For this, we employed a weak microwave tone at frequency ω_{MW}/2π to probe the 0〉 ↔ ± 1〉 transitions as a function of B_{NV} in the presence of the coherent strain field (Fig. 2b). This field has a striking effect on the NV’s ESR spectrum in that it induces excitation gaps at ω_{MW} − 2πD_{0} = ±ω_{m}/2, that is, for B_{NV} ∼ 0.9,1.6 and 2.3 G. At these values of B_{NV} the a.c. strain field in the cantilever is resonant with a given hyperfine transition, that is, the energy splitting \hslash {\omega}_{1,\text{}1}^{{m}_{I}} between −1, m_{I}〉 and +1, m_{I}〉 equals ℏω_{m}. The energy gaps which we observe in the ESR spectra under resonant strain driving are evidence of the Autler–Townes (AT) effect—a prominent phenomenon in quantum electrodynamics^{19,20}, which has previously been observed in atoms and molecules^{21}, quantum dots^{22} and superconducting qubits^{23}. Our observation of the AT effect was performed on a single electronic spin in the microwave domain, and to the best of our knowledge constitutes the first observation of the AT effect under ambient conditions.
The observed AT splitting can be understood by considering the joint energetics of the NV spin states and the quantized strain field used to drive the spin^{20} (Fig. 2a). The joint basis states i; N〉 consist of NV spin states i〉 dressed by N phonons in the cantilever. Strain couples +1; N〉 to −1; N + 1〉 and leads to new eigenstates ±(N)〉, which anticross on resonance, where are split by an energy ℏΩ_{m}. As expected, this splitting increases linearly with the driving field amplitude (Fig. 2c), which we control through the strength of piezo excitation.
To investigate the limits of our coherent, straininduced spin driving and study the resulting, strongly driven spin dynamics, we performed detailed dressedstate spectroscopy as a function of drive strength (Fig. 3a). To that end, we first set B_{NV} such that {\omega}_{\text{}1,1}^{{m}_{I=1}}={\omega}_{m} and then performed microwave ESR spectroscopy for different values of Ω_{m}. For weak driving, Ω_{m} ≪ ω_{m}, the dressed states emerging from the resonantly coupled states −1,1〉 and +1,1〉 split linearly with Ω_{m}. The linear relationship breaks down for {\omega}_{\text{}1,1}^{{m}_{I=1}} owing to multiphonon couplings involving states which belong to different subspaces spanned by ±(N)〉 and ±(M)〉, with N ≠ M (ref. 20). This observation is closely linked to the breakdown of the rotatingwave approximation^{24} and indicates the onset of the strongdriving regime we achieve in our experiment.
For even larger Rabi frequencies Ω_{m}, the dressed states evolve into a characteristic sequence of crossings and anticrossings. The (anti)crossings occur in the vicinity of Ω_{m} = qω_{m}, with q an odd (even) integer, and are related to symmetries of the Hamiltonian (1) (see ref. 20; Supplementary Information). Our experiment allows us to clearly identify the q = 1 and q = 2 (anti)crossings (circles and crosses in Fig. 3a) and thereby demonstrates that we reside well within the strongdriving regime (Ω_{m} > {\omega}_{\text{}1,1}^{{m}_{I}}) of a harmonically driven twolevel system. We have carried out an extensive numerical analysis (Fig. 3b and Methods), which shows quantitative agreement with our experimental findings. For the largest values of Ω_{m}, some discrepancies of the transition strengths between data and model remain; we tentatively assign these to uncertainties in microwave polarization, to possible variations of linewidths with Ω_{m} and to our particular ESR detection scheme^{25}. Our calculation further shows that, over our range of experimental parameters, Ω_{m} is linear in x_{c} and reaches a maximum of Ω_{m}^{max}/2π ∼ 10.75 MHz (at present limited by the maximally achievable piezo driving strength).
Continuous coherent driving can be employed to protect a quantum system from its noisy environment and thereby increase its coherence times^{12,26}. For NV centre spins, decoherence is predominantly caused by environmental magnetic field noise^{5}, which normally couples linearly to the NV spin through the Zeeman Hamiltonian H_{Z} = γ_{NV}S_{z}B_{NV} (Fig. 1a). Conversely, for the dressed states ±(N)〉 we create by coherent strain driving, 〈 ±(N) H_{Z} ±(N)〉 = 0 and the lowestorder coupling to magnetic fields is only quadratic (Fig. 2a). These states are thus less sensitive to magnetic field fluctuations and should exhibit increased coherence times relative to the undriven NV.
To demonstrate such coherence enhancement by continuous driving^{12}, we performed Ramsey spectroscopy on our straindriven NV spin and compared the resulting dephasing times T_{2}^{∗} against the undriven case (Fig. 4). For this, we adjusted B_{NV} such that {\omega}_{\text{}1,1}^{{m}_{I}=1} and mechanically drove the NV with Ω_{m}/2π = 1.68 MHz to induce phonondressing of the NV. We then used pulsed microwaves^{25} to perform Ramsey spectroscopy on the two Autler–Townes split dressed states emerging from m_{s} = +1, m_{I} = +1〉. The resulting coherence signal (Fig. 4a) decays on a timescale of T_{2}^{∗} = 16.4 ± 0.6 μs and shows beating of two longlived oscillations at 0.63 MHz and 1.05 MHz stemming from the two dressed states we address (Supplementary Information). Compared to the bare NV dephasing time of T_{2}^{∗} = 3.6 ± 0.1 μs (Fig. 4b), this demonstrates a significant enhancement of T_{2}^{∗} caused by our continuous, mechanical drive.
Although our protocol decouples NVs from magnetic field noise, it renders them vulnerable to fluctuations in electric field and strain. Most importantly, shallow NVs experience excess dephasing from fluctuating surface electric fields^{27}, which are likely to dominate the residual dephasing we observe. Additional dephasing mechanisms of lesser relevance to our experiment include cantilever thermal noise (Supplementary Information) or secondorder couplings^{28} of magnetic fields to the NV. Our decoupling protocol is readily tunable: for increasing Ω_{m}, we have observed an initially monotonic, approximately linear increase of T_{2}^{∗} (Supplementary Information), which saturates for Ω_{m}/2π ≳ 1 MHz. We assign this current limitation to the onset of technical noise^{12}, whose mitigation might lead to further improvements of T_{2}^{∗} in the future. Furthermore, tunability offers interesting perspectives to systematically study the still largely unexplored, electricfieldinduced dephasing processes for shallow NVs.
Our approach to strong coherent strain driving of a single electronic spin will have implications far beyond the coherence protection and dressedstate spectroscopy that we have demonstrated in this work. By combining our strain drive with coherent microwave spin manipulation, our NV spin forms an inverted threelevel ‘Δ’system, on which all three possible spin transitions can be coherently addressed. This setting is known to lead to unconventional spin dynamics^{29}, which here could be observed on a single, highly coherent spin and exploited for sensing and quantum manipulation of our hybrid device. Straininduced a.c. Stark shifts can furthermore be employed to dynamically tune^{30} the energies of the NV hyperfine states—an attractive perspective for the use of ^{14}N nuclear spins as quantum memories^{5}. The decoherence protection by continuous strain driving that we have demonstrated will have an impact for any quantum technology where pulsed decoupling protocols cannot be employed (such as d.c. electric field sensing). Further studies of the remaining decoherence processes under mechanical driving, which remain largely unexplored until now, offer another exciting avenue to be pursued in the future. On a more farreaching perspective, our experiments lay the foundation for exploiting diamondbased hybrid spinoscillator systems for quantum information processing and sensing, where our system forms an ideal platform for implementing proposed schemes for spininduced phonon cooling and lasing^{31} or oscillatorinduced spin squeezing^{14}.
Methods
Sample fabrication.
Our cantilevers consist of singlecrystalline, ultrapure [001]oriented diamond (Element Six, ‘electronic grade’), are aligned with the [110] crystal direction and have dimensions in the range of (0.2–1) × 3.5 × (15–25) μm^{3} for thickness, width and length, respectively. The fabrication process is based on recently established topdown diamond nanofabrication techniques^{32}. In particular, we use electron beam lithography at 30 keV to pattern etch masks for our cantilevers into a negative tone electron beam resist (FOX16 from Dow Corning, spun to a thickness of ∼500 nm onto the sample). The developed pattern directly acts as an etch mask and is transferred into the diamond surface using an inductively coupled plasma reactive ion etcher (ICPRIE, Sentech SI 500). To create cantilevers with vertical sidewalls, we use a plasma containing 50% argon and 50% oxygen (gas flux 50 sccm each). The plasma is run at 1.3 Pa pressure, 500 W ICP source power and 200 W bias power. NV centres in our cantilevers were created before nanofabrication by ^{14}N ion implantation with dose, energy and sample tilt of 10^{10} cm^{−2}, 12 keV and 0°, respectively. Based on numerical simulations (using the ‘SRIM’ software package), this yields an estimated implantation depth of ∼17 nm. To create NV centres, we annealed our samples at high vacuum (≲10^{−6} mbar) in a sequence of temperature steps at 400 °C (4 h), 800 °C (2 h) and 1,200 °C (2 h).
Experimental setup.
Experiments are performed in a homebuilt confocal microscope setup at room temperature and at atmospheric pressure. A 532 nm laser (NovaPro 532300) is coupled into the confocal system through a dichroic mirror (Semrock LM0155225). A microscope objective (Olympus XLMFLN40x) is used to focus the laser light onto the sample, which is placed on a micropositioner (Attocube ANSxyz100). Red fluorescence photons are collected by the same microscope objective, transmitted through the dichroic mirror and coupled into a singlemode optical fibre (Thorlabs SM600), which acts as a pinhole for confocal detection. Photons are detected using an avalanche photodiode (Laser Components Count250C) in Geiger mode. Scan control and data acquisition (photon counting) are achieved using a digital acquisition card (NI6733). The microwave signal for spin manipulation is generated by a SRS SG384 signal generator, amplified by a Minicircuit ZHL42W+ amplifier and delivered to the sample using a homebuilt nearfield microwave antenna. Laser, microwave and detection signals were gated using microwave switches (Minicircuit Switch ZASWA250DR+), which were controlled through digital pulses generated by a fast pulse generator (SpinCore PulseBlasterESRPRO). Gating of the laser is achieved using a doublepass acoustic optical modulator (Crystal Technologies 3200146). Mechanical excitation of the cantilevers was performed with a piezoelectric element placed directly below the sample. The excitation signal for the piezo was generated with a signal generator (Agilent 3320A). A threeaxis magnetic field was generated by three homebuilt coil pairs driven by constantcurrent sources (Agilent E3644A).
Measurement procedure and error bars.
ESR measurements were performed using a pulsed ESR scheme^{25}, where the NV spin is first initialized in m_{s} = 0〉 using green laser excitation, then driven by a short microwave ‘πpulse’ of length τ (that is, a pulse such that 2πΩ_{MW}τ = π) and finally read out using a second green laser pulse. Compared to conventional, continuouswave ESR, this scheme has the advantage of avoiding power broadening of the ESR lines by green laser light and was therefore employed throughout this work.
Our experiments were performed on three different NV centres in three different cantilevers: data in Figs 1 and 4 were obtained on NV #1, whereas Figs 2 and 3 were recorded on NVs #2 and #3, respectively. NVs #1–3 all showed slightly different values of D_{0} owing to variations in static local strain and transverse magnetic fields. The zerofield splittings for these three NVs were D_{0} = 2.870 GHz, 2.871 GHz and 2.8725 GHz, respectively. The values of ω_{m} for the cantilevers of NVs #1–3 were ω_{m}/2π = 6.83,9.18 and 5.95 MHz, respectively.
Throughout this paper, errors represent 95% confidence intervals for the nonlinear leastsquares parameter estimates to our experimental data. The only exception is the mechanical resonance frequency ω_{m}, where error bars represent the linewidth of the cantilever resonance curves, which we measured optically in separate experiments. The actual error bars in determining ω_{m} are significantly smaller than the linewidth and do not influence the findings presented in this paper.
Simulations.
Following ref. 11 we employ Floquet theory to treat the time dependence of the straininduced spin driving, , beyond the rotatingwave approximation (RWA), as it is expected to break down in the strongdriving limit Ω_{m} > {\omega}_{\text{}1,1}^{{m}_{I}=1}. The key idea here is to map the Hamiltonian with periodic time dependence on an infinitedimensional, but timeindependent Floquet Hamiltonian . We can then solve the eigenvalue problem with standard methods to obtain quasienergies ℏω_{j} and corresponding eigenvectors u_{j}〉.
Treating the weak microwave drive up to second order in drive strength we find the rate for the system to leave the initial state with Fermi’s golden rule as^{11}
where the sum over i and f runs over all the eigenstates of the Floquet Hamltonian and the microwave driving Hamiltonian is with drive frequency Ω_{MW}, assuming a linearly polarized microwave field. For the simulations shown in Fig. 3b we assumed an initial state u_{i}〉 = m_{s} = 0, m_{I}〉 and linewidths γ_{fi} = γ = 1 MHz, and summed the result incoherently over all nuclear spin quantum numbers m_{I} ∈ {−1,0,1}.
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Acknowledgements
We thank V. Jacques for fruitful discussions, P. Appel for initial assistance with nanofabrication and L. Thiel for support with the experiment control software. We gratefully acknowledge financial support from SNI; NCCR QSIT; SNF grants 200021_143697; and EU FP7 grant 611143 (DIADEMS). A.N. holds a University Research Fellowship from the Royal Society and acknowledges support from the Winton Programme for the Physics of Sustainability.
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A.B. and J.T. carried out the experiment and analysed the data. E.N. provided essential support in sample fabrication. A.N. provided theoretical support and modelled the data. All authors commented on the manuscript. P.M. wrote the manuscript, conceived the experiment and supervised the project.
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Barfuss, A., Teissier, J., Neu, E. et al. Strong mechanical driving of a single electron spin. Nature Phys 11, 820–824 (2015). https://doi.org/10.1038/nphys3411
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DOI: https://doi.org/10.1038/nphys3411
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