Strong mechanical driving of a single electron spin

Abstract

Quantum devices for sensing and computing applications require coherent quantum systems, which can be manipulated in fast and robust ways¹. Such quantum control is typically achieved using external electromagnetic fields, which drive the system’s orbital², charge³ 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 time-varying strain in a diamond cantilever to induce long-lasting, coherent oscillations of an embedded nitrogen–vacancy (NV) centre spin. We perform direct spectroscopy of the phonon-dressed states emerging from this drive and observe hallmarks of the sought-after strong-driving 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 time¹². Our room-temperature experiments thereby constitute an important step towards strain-driven, integrated quantum devices and open new perspectives to investigate unexplored regimes of strongly driven multilevel systems¹³ and exotic spin dynamics in hybrid spin-oscillator devices¹⁴.

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 systems¹⁵, and has been proposed to generate squeezed spin ensembles¹⁴ or to cool mechanical oscillators to their quantum ground state¹⁶. These attractive perspectives for strain-coupled hybrid quantum systems motivated recent studies of the influence of strain on nitrogen–vacancy (NV) centre electronic spins^9,10,17 and experiments on strain-induced, coherent driving of large NV spin ensembles⁸. Promoting such experiments to the single-spin regime, however, remains an outstanding challenge, and would constitute a major step towards the implementation of integrated, strain-driven quantum systems.

Here, we demonstrate the coherent manipulation of a single electronic spin using time-periodic, intrinsic strain fields generated in a single-crystalline diamond mechanical oscillator. We show that such strain fields allow us to manipulate the spin in the strong-driving 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 single-crystalline 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 zero-field splitting D₀ = 2.87 GHz. The levels |±1〉 are split by 2γ_NVB_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 ¹⁴N 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 spin¹⁸ with a homebuilt confocal optical microscope⁹ (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.

Figure 1: Experimental set-up and strain-induced coherent spin drive.

a, Energy levels of the NV spin as a function of magnetic field applied along the NV axis. Electronic spin states |m_s = ±1〉 each split into three levels owing to hyperfine interactions with the NV’s ¹⁴N nuclear spin (I = 1, A_HF = 2.18 MHz). Wavy lines indicate strain (red) and microwave (violet) fields of frequency ω and strength Ω. b, Optically detected electron spin resonance of a single NV centre. c, Experimental set-up. NV spins (red) are coupled to a cantilever, which is resonantly driven at frequency ω_m by a piezo-element. The NV spin is read out and initialized by green laser light and manipulated by microwave magnetic fields generated by a nearby antenna. d, Pulse sequence employed to observe strain-induced Rabi oscillations. e, Strain-driven Rabi oscillations. Data (blue) and a fit (black) to damped Rabi oscillations.

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 strain-coupling Hamiltonian takes the form¹⁴

Here, ℏ is the reduced Planck constant, γ₀^⊥ the transverse single-phonon strain-coupling 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 near-resonant, time-varying (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 = γ₀^⊥x_c/x_ZPF describes the amplitude of the strain drive, with x_ZPF and x_c the cantilever’s zero-point fluctuation and peak amplitude, respectively (here, with x_ZPF ∼7.7 × 10⁻¹⁵ m and γ₀^⊥/2π ∼ 0.08 Hz, Supplementary Information). Interestingly, strain drives a dipole-forbidden 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, single-crystalline diamond cantilever⁹, 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 piezo-element 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 strain-driven 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 strain-induced 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 ensembles⁸, 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 strain-driving mechanism from ESR spectroscopy of the strain-coupled 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 − 2πD₀ = ±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 molecules²¹, quantum dots²² and superconducting qubits²³. 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.

Figure 2: Mechanically induced Autler–Townes effect probed by microwave spectroscopy.

a, Eigenenergies of the joint spin–phonon system with basis states |m_s; N〉 as a function of the spin splitting ${\omega }_{1,\text{-}1}^{m_I}$. Strain couples |1; N〉 and |−1; N + 1〉 and, whenever the resonance condition ${\omega }_{1,\text{-}1}^{m_I}$ = ω_m is fulfilled, leads to new eigenstates |+(N)〉 and |−(N)〉 with energy splitting ℏΩ_m (see text). b, Microwave spectroscopy of phonon-dressed states using a weak microwave probe at ω_MW. For m_I = −1, 0 and 1, resonance is separately established at B_NV ∼0.9, 1.6 and 2.3 G. c, Dependence of the energy gap between |+(N)〉 and |−(N)〉 on mechanical driving strength. As expected, the gap scales linearly with Ω_m for each hyperfine state. Data was recorded over the parameter range indicated by white dashed lines in b.

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 spin²⁰ (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 anti-cross 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, strain-induced spin driving and study the resulting, strongly driven spin dynamics, we performed detailed dressed-state 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 multi-phonon couplings involving states which belong to different sub-spaces spanned by |±(N)〉 and |±(M)〉, with N ≠ M (ref. 20). This observation is closely linked to the breakdown of the rotating-wave approximation²⁴ and indicates the onset of the strong-driving regime we achieve in our experiment.

Figure 3: Dressed-state spectroscopy of the strongly driven NV spin.

a, Microwave spectroscopy of the mechanically driven NV spin at B_NV = 1.8 G (where ω_m = ${\omega }_{1,\text{-}1}^{m_I=1+}$) as a function of drive strength Ω_m. The resonantly coupled states (|±1, +1〉) at ω_MW/2π −D₀ = ±2.98 MHz first split linearly with B_NV and then evolve into a sequence of crossings and anti-crossings (green circles and crosses, respectively) with higher-order dressed states, indicative of the strong-driving regime. b, Calculated transition rates from |m_s = 0〉 to the dressed states obtained by Fermi’s golden rule (Methods). Blue, green and orange shaded transitions correspond to the hyperfine manifolds m_I = +1, 0 and −1, respectively. In both panels, black dots indicate the calculated dressed-state energies for m_I = +1. Grey lines in b show the same under the rotating-wave approximation. Deviations between black dots and the grey lines therefore indicate the onset of the strong-driving regime.

For even larger Rabi frequencies Ω_m, the dressed states evolve into a characteristic sequence of crossings and anti-crossings. 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 strong-driving regime (Ω_m > ${\omega }_{\text{-}1,1}^{m_I}$) of a harmonically driven two-level 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 scheme²⁵. 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 noise⁵, which normally couples linearly to the NV spin through the Zeeman Hamiltonian H_Z = γ_NVS_zB_NV (Fig. 1a). Conversely, for the dressed states |±(N)〉 we create by coherent strain driving, 〈 ±(N) |H_Z |±(N)〉 = 0 and the lowest-order 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 driving¹², we performed Ramsey spectroscopy on our strain-driven NV spin and compared the resulting dephasing times T₂^∗ 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 phonon-dressing of the NV. We then used pulsed microwaves²⁵ 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 T₂^∗ = 16.4 ± 0.6 μs and shows beating of two long-lived 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 T₂^∗ = 3.6 ± 0.1 μs (Fig. 4b), this demonstrates a significant enhancement of T₂^∗ caused by our continuous, mechanical drive.

Figure 4: Protecting NV spin coherence by coherent strain driving.

a, Spin coherence decay of |±(N)〉 (form_I = +1) as measured by Ramsey interferometry between the state |m_s = 0〉 and the |m_s = +1〉 manifolds using the pulse sequence depicted on the top. The probability for the NV to occupy the |m_s = 0〉 manifold after the sequence is denoted as P(|m_s = 0〉). The inset illustrates the NV spin’s eigenenergies as a function of B_NV and indicates the magnetic field and microwave frequencies employed (purple dot) with respect to the dressed-state spectrum shown in Fig. 2b. b, Measurement of NV spin coherence time in the undriven case, as determined by Ramsey spectroscopy between |m_s = 0〉 and |m_s = +1〉, in the absence of mechanical driving. Inset as in a. In both panels, the orange envelope indicates the coherence decay extracted from our fit.

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 fields²⁷, 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 second-order couplings²⁸ 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 T₂^∗ (Supplementary Information), which saturates for Ω_m/2π ≳ 1 MHz. We assign this current limitation to the onset of technical noise¹², whose mitigation might lead to further improvements of T₂^∗ in the future. Furthermore, tunability offers interesting perspectives to systematically study the still largely unexplored, electric-field-induced 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 dressed-state spectroscopy that we have demonstrated in this work. By combining our strain drive with coherent microwave spin manipulation, our NV spin forms an inverted three-level ‘Δ’-system, on which all three possible spin transitions can be coherently addressed. This setting is known to lead to unconventional spin dynamics²⁹, which here could be observed on a single, highly coherent spin and exploited for sensing and quantum manipulation of our hybrid device. Strain-induced a.c. Stark shifts can furthermore be employed to dynamically tune³⁰ the energies of the NV hyperfine states—an attractive perspective for the use of ¹⁴N nuclear spins as quantum memories⁵. 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 far-reaching perspective, our experiments lay the foundation for exploiting diamond-based hybrid spin-oscillator systems for quantum information processing and sensing, where our system forms an ideal platform for implementing proposed schemes for spin-induced phonon cooling and lasing³¹ or oscillator-induced spin squeezing¹⁴.

Methods

Sample fabrication.

Our cantilevers consist of single-crystalline, ultra-pure [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) μm³ for thickness, width and length, respectively. The fabrication process is based on recently established top-down diamond nanofabrication techniques³². In particular, we use electron beam lithography at 30 keV to pattern etch masks for our cantilevers into a negative tone electron beam resist (FOX-16 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 (ICP-RIE, 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 ¹⁴N ion implantation with dose, energy and sample tilt of 10¹⁰ 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 set-up.

Experiments are performed in a homebuilt confocal microscope set-up at room temperature and at atmospheric pressure. A 532 nm laser (NovaPro 532-300) is coupled into the confocal system through a dichroic mirror (Semrock LM01-552-25). 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 single-mode optical fibre (Thorlabs SM600), which acts as a pinhole for confocal detection. Photons are detected using an avalanche photodiode (Laser Components Count-250C) in Geiger mode. Scan control and data acquisition (photon counting) are achieved using a digital acquisition card (NI-6733). The microwave signal for spin manipulation is generated by a SRS SG384 signal generator, amplified by a Minicircuit ZHL-42W+ amplifier and delivered to the sample using a homebuilt near-field microwave antenna. Laser, microwave and detection signals were gated using microwave switches (Minicircuit Switch ZASWA-2-50DR+), which were controlled through digital pulses generated by a fast pulse generator (SpinCore PulseBlasterESR-PRO). Gating of the laser is achieved using a double-pass acoustic optical modulator (Crystal Technologies 3200-146). 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 three-axis magnetic field was generated by three homebuilt coil pairs driven by constant-current sources (Agilent E3644A).

Measurement procedure and error bars.

ESR measurements were performed using a pulsed ESR scheme²⁵, where the NV spin is first initialized in |m_s = 0〉 using green laser excitation, then driven by a short microwave ‘π-pulse’ of length τ (that is, a pulse such that 2πΩ_MWτ = π) and finally read out using a second green laser pulse. Compared to conventional, continuous-wave 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 D₀ owing to variations in static local strain and transverse magnetic fields. The zero-field splittings for these three NVs were D₀ = 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 least-squares 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 strain-induced spin driving, , beyond the rotating-wave approximation (RWA), as it is expected to break down in the strong-driving 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 infinite-dimensional, but time-independent Floquet Hamiltonian . We can then solve the eigenvalue problem with standard methods to obtain quasi-energies 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 as¹¹

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}.