Abstract
Dynamical coupling with highquality factor resonators is essential in a wide variety of hybrid quantum systems such as circuit quantum electrodynamics and opto/electromechanical systems. Nuclear spins in solids have a long relaxation time and thus have the potential to be implemented into quantum memories and sensors. However, state manipulation of nuclear spins requires highmagnetic fields, which is incompatible with stateoftheart quantum hybrid systems based on superconducting microwave resonators. Here we investigate an electromechanical resonator whose electrically tunable phonon state imparts a dynamically oscillating strain field to the nuclear spin ensemble located within it. As a consequence of the dynamical strain, we observe both nuclear magnetic resonance (NMR) frequency shifts and NMR sidebands generated by the electromechanical phonons. This prototype system potentially opens up quantum state engineering for nuclear spins, such as coherent coupling between sound and nuclei, and mechanical cooling of solidstate nuclei.
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Introduction
Dynamical coupling to a highquality factor (highQ) resonator, i.e., an interaction with a rapidly oscillating degree of freedom in a resonator, is essential to implementing a wide variety of quantum hybrid devices. For instance, in circuit quantum electrodynamics (QED), which implements dynamical coupling between a superconducting microwave resonator and an artificial atom; quantum nondemolition measurements of the twolevel system via the resonator’s frequency shifts have been developed^{1,2}. On the other hand the parametric coupling between photons and phonons (phonons and phonons) in optomechanics^{3} (in coupled mechanical systems) have been used to measure the position of the mechanical resonator below the standard quantum limit^{4}. The resultant sidebands generated on the photonic (phononic) resonance have also been harnessed for sideband cooling the thermal phonons down to the quantum ground state when pumping the red sideband^{5,6,7}, and to generate an interresonator quantum entanglement^{8,9,10,11,12}.
Nuclear spins in solids have long relaxation times because of their weak interaction with the surrounding environment, and thus are expected to be utilized for various applications such as quantum information processing^{13,14,15} and quantum magnetic sensors^{16}. Consequently incorporating the abovementioned concept of coupling a resonator to nuclear spins has the potential to open another class of manipulation and detection methods for the nuclear spins. The concept of circuit QED has already been implemented with electron spins using a superconducting microwave resonator^{17,18,19,20}. In contrast, a similar approach based on superconducting resonator (necessary for their highQ) is difficult for nuclear spins, because the nuclear magnetic resonance (NMR) used to detect and manipulate their states requires highmagnetic fields of a few Tesla. To address this, a natural alternative would be to couple the nuclei to an electromechanical resonator that is compatible with highmagnetic fields, but despite this promise, an experimental demonstration of such a hybrid system remains unexplored.
In the first step towards such a resonatorbased hybrid system with a nuclear spin ensemble, we have developed an electromechanical system that imparts dynamical strain to nuclear spins naturally located within it. In this realization, the nuclear spins interact with the fundamental mode of the resonator at 1.7 MHz via mechanical strain in the quadrupole interaction^{21,22}. The NMR of the local nuclei in the resonator can be selectively detected by resistively detected NMR method using the integer quantum Hall effect through a nanofabricated transport channel at the clamping point of the resonator^{15}. The quadrupole interaction of nuclear spins in solids is important and has a long history in standard NMR and nuclear acoustic resonance experiments^{23}. We note however that most of the work reported so far has focused on static strain^{21,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37} or weak oscillating strain induced by ultrasound applied to bulk specimens^{38}. In contrast, here we achieve larger strain regime of 10^{−3} at 1.7 MHz with the aid of a highQ mechanical resonator that can enhance the mechanical displacement (which induces strain) by a factor of Q. As a consequence of this resonatorenhanced dynamical strain effect, we observe both red and blue sidebands on the NMR which correspond to nuclear spin transitions assisted by the absorption and emission of electromechanical phonons. Moreover, frequency shifts in the NMR are also observed, which scaled with the strain from the mechanical motion thus enabling the nuclear spin states to be mechanically manipulated.
Results
Electromechanical resonator and resistively detected NMR
A 50μmlong doubly clamped electromechanical resonator shown in Fig. 1a was fabricated from a GaAs/AlGaAs heterowafer using sacrificial layer etching (see Methods and Supplementary Fig. 1). Figure 1b details the motional piezovoltage V_{d} measured at different actuation voltages V_{a}, showing a resonance peak which corresponds to the fundamental flexural mode^{39,40} with frequency ω_{M}/2π = 1.716 MHz and quality factor Q = 5 × 10^{4}. With increasing V_{a}, the mechanical amplitude is deformed into a sawtoothshape reflecting the emergence of a nonlinearity in the underlying harmonic potential^{41}. This nonlinear feature is further enhanced in the strong driving regime, where the normal and shear strains can become as high as ε_{zz} ~ 1 × 10^{−3} and ε_{xy} ~ 2 × 10^{−3}, respectively (see below and Supplementary Note 1).
This GaAsbased resonator hosts three isotopes ^{75}As, ^{69}Ga, and ^{71}Ga, all of which are quadrupole nuclei having spin I = 3/2. Among them ^{75}As has the largest quadrupole moment and the strongest coupling to strain and therefore the effects generated from electromechanical strain are most favorable for this element. The fundamental flexural mode of this resonator creates strain that is magnified at the clamping points (see Supplementary Fig. 2) and thus detection of NMR for the strainmodified nuclei is made at this location. To this end, a resistively detected NMR method is adopted that uses breakdown of the integer quantum Hall effect (see Methods). Figure 1d shows the NMR measured from the ^{75}As nuclei at the clamping point, which is unstrained as the mechanical vibration is not activated and the peak frequency ω_{0} measured at different external field B_{0} shows a linear dependence (see Fig. 1e). The corresponding gyromagnetic ratio γ/2π = 7.3 MHz T^{−1} extracted from a linear fit confirms that the observed peak originates from the NMR of ^{75}As nuclei.
Sideband NMR assisted by electromechanical phonons
When the electromechanical resonator is driven, it can periodically modulate the quadrupole term associated with the nuclear spins via the motion induced strain. As a consequence of this offresonant parametric modulation, nuclear spins can form phonondressed states, which emerge as sideband NMR peaks as schematically depicted in Fig. 2a. A red (blue) transition can be induced with the absorption (emission) of phonons from (into) the mechanics via the quadrupole interaction and hence the additional NMR peak appears at ω_{0}−(+)ω_{M}. Figure 2b–d (2e) show the observation (simulation) of red and blue sideband NMR peaks for the nuclei under the driven electromechanical oscillation with V_{a} = 40 mV (point 5 in Fig. 3a). The strongest sideband NMR peaks are observed with ^{75}As whilst the weakest with ^{71}Ga, which reflects the magnitude of the quadrupole interaction between the nuclear spins and mechanical strain. We note that the apparently narrower NMR peak of ^{71}Ga is due to the limited signaltonoise ratio for this nuclear species, which is relatively scarce and has a weaker quadrupole interaction.
To better understand the observed sideband NMR peaks, the spectra were numerically simulated using a model Hamiltonian that included an oscillating quadrupole interaction associated with the driven mechanical motion:
Here the first term describes the Zeeman effect, \(\hat H_{\mathrm{Z}} =  \hbar \omega _0\hat I_z\), where ω_{0}/2π = γB_{0} with γ being the gyromagnetic ratio of the relevant nucleus, \(\hat I_z\) is the zcomponent of the Pauli spin operator \(\hat I\), and ħ the reduced Planck constant. For a nucleus with spin I = 3/2, four spin states \(\left m \right\rangle\) are Zeeman split to form equally spaced energy levels separated by ħω_{0}. The effect of strainmediated dynamical coupling from a driven mechanical resonator on nuclear spins is captured by the dynamic quadrupole interaction described by the second term \(\hat H_{\mathrm{Q}}(t) = x(t)\hat J_{\mathrm{Q}}\), which is proportional to the timedependent mechanical displacement x(t). Here the quadrupole operator \(\hat J_{\mathrm{Q}}\) is
where A_{1} and A_{2} are the quadrupole coupling constants at the clamping point and \(\hat I_ +\) and \(\hat I_ \) are the spin ladder operators (see Supplementary Note 1). The NMR transition triggered by absorbing a photon ω_{RF} irradiated by the antenna gate is described by the third term with \(\hat H_{{\mathrm{RF}}} =  \gamma \hbar B_x\hat I_x\), where \(\hat I_x\) is the xcomponent of the Pauli spin operator and B_{x} is the corresponding transverse magnetic field. Equation (1) can simulate the magnetization \(\left\langle {\hat I_z} \right\rangle\) driven by RF irradiation and its amplitude reflects the NMR response of the nuclei under the presence of an oscillating strain field.
Figure 2e shows the numerical simulation extracted from this framework within the rotatingwave approximation of both the main and sideband NMR spectra associated with ^{69}Ga nuclei (see Methods). The simulation not only reproduces the sideband NMR peaks but also the ratio of the signal amplitude between the main peak and the sidebands as observed in the experiments. However the simulated sidebands are narrower than the main NMR peak and this feature is not seen in the experimental spectrum as this is acquired from an ensemble average (in contrast to the simulation) that is inhomogeneously broadened by the spatial distribution of the conduction electrons and the strain. This agreement with the theory confirms that the observed sideband peaks in Fig. 2b–d are indeed associated with nuclear spin transitions aided by electromechanical phonons via the quadrupole coupling.
Dispersive shifts induced by a driven mechanical resonator
From the overall similarity between our system and a circuit QED device embedded with an electromechanical resonator^{42}, one can predict frequency shifts in the NMR caused by a driven mechanical resonator. This feature resembles the acStark shift in a circuit QED system^{2,43} but now the driven mechanical motion that is offresonant with the qubit transition frequency is treated as Floquet perturbation resulting in an energy shift, that was dubbed as the mechanical/phonon acStark shift^{42}.
In order to investigate this effect in the present system, the NMR spectra of ^{75}As nuclei were measured under the influence of intense AC electromechanical strain as shown in Fig. 3b. The bottom spectrum shows the NMR at zero mechanical displacement with a doublet peak structure as indicated by the solid circle (main peak) and open square (sub peak). However this figure also reveals that as the mechanical displacement is increased the main peak undergoes a blue shift whereas the sub peak remains fixed. To elucidate the underlaying physics of these observations, the NMR spectra are simulated using the same model Hamiltonian in Eq. (1) that describes nuclear spins under the influence of a rapidly oscillating quadrupole interaction arising from the driven electromechanical motion (see Methods) and are shown in Fig. 3c. The simulated NMR spectra (generated using the same parameters as in Fig. 2e but now with a variable displacement x(t)) reproduce the observed frequency shift, i.e., blue shift of the main peak (solid circle) and no shift of the sub peak (open square), again confirming that the experimental observations can be accommodated by the oscillating quadrupole Hamiltonian. To further quantify the simulated spectra, the experimental NMR frequencies of the main and subpeaks extracted from the measured NMR spectra are collated as a function of the physical displacement of the fundamental flexural mode x_{1}. The resultant plot in Fig. 3d shows that the experimental frequency shift agrees with the simulated frequency shift.
In order to decipher the mechanisms underlying the observed frequency shift, the model Hamiltonian is further analyzed. In the absence of a quadrupole interaction between nuclei and mechanics, the bare nuclei with I = 3/2 shows degenerate Zeemann split states that are equally spaced by ħω_{0} as shown in Fig. 4. Next the quadrupole interaction is treated using Floquet perturbation theory (see Methods), which notably indicates that the dynamic (oscillating) component of the perturbation term has no contribution at the first order while it has a nonzero value at higher order. Consequently the firstorder energy shift δ can simply be induced by static mechanical strain. The analytical expression of δ (see Methods for the derivation) indicates that its sign is positive (negative) for \(\left {m = \pm 1{\mathrm{/}}2} \right\rangle\) \(\left( {\left { \pm 3/2} \right\rangle } \right)\), which in turn lifts degeneracy as depicted in Fig. 4 and is experimentally observed as split NMR peaks even in the absence of ac strain in the lowest plot in Fig. 3b. Indeed, such NMR peak structure is commonly observed in GaAs heterostructures^{15} and is due to residual static strain that is incorporated during crystal growth and nano fabrication. Similarly, the sign of the secondorder energy shift χ is positive (negative) for \(\left {  1{\mathrm{/}}2} \right\rangle\) and \(\left {  3{\mathrm{/}}2} \right\rangle\) (\(\left {1{\mathrm{/}}2} \right\rangle\) and \(\left {3{\mathrm{/}}2} \right\rangle\)) and the dynamical electromechanical strain only blue shifts the resonant transition \(\left {1{\mathrm{/}}2} \right\rangle \leftrightarrow  \left {1{\mathrm{/}}2} \right\rangle\), whereas the other transitions remain unchanged as shown in Fig. 4. This is in contrast to the case with static strain where the transition \(\left {3{\mathrm{/}}2} \right\rangle \leftrightarrow \left {1{\mathrm{/}}2} \right\rangle\) \(\left( {\left {  3{\mathrm{/}}2} \right\rangle \leftrightarrow \left {  1{\mathrm{/}}2} \right\rangle } \right)\) is blue (red) shifted while the \(\left {1{\mathrm{/}}2} \right\rangle \leftrightarrow \left {  1{\mathrm{/}}2} \right\rangle\) transition remains unchanged. Consequently in the NMR spectra shown in Fig. 3b, d, the blue shifting main peak corresponds to the transition \(\left {1{\mathrm{/}}2} \right\rangle \leftrightarrow \left {  1{\mathrm{/}}2} \right\rangle\) stemming from nonadiabatic dynamical strain originating from electromechanical phonons, whilst the sub peak (open square) corresponds to the \(\left {  3{\mathrm{/}}2} \right\rangle \leftrightarrow \left {  1{\mathrm{/}}2} \right\rangle\) transition which arises from the residual static strain given by −2δ.
Discussion
The NMR spectra of ^{75}As, ^{69}Ga, and ^{71}Ga nuclei under the influence of intense periodic lattice strain induced by the large displacements of a stronglydriven doubly clamped electromechanical resonator were investigated. The resultant resistively detected NMR showed sideband peaks, which indicated the parametric coupling between nuclei and mechanics with the nuclear spin transitions assisted by electromechanical phonons. In addition, NMR frequency shifts were also observed that stemmed from nonadiabatic dynamics that scaled with the mechanical displacements. The agreement with theoretical modeling in both experiments confirms that our system faithfully implements dynamical coupling between a nuclear spin ensemble and an electromechanical resonator via mechanical strain.
In the present system, frequency shifts in the mechanical resonance from the NMR have yet to be observed, which indicates that the strainmediated coupling observed in this system corresponds to a weak coupling regime. For more advanced dynamic effects such as coherent state transfer between a nuclear spin ensemble and electromechanical phonons and electromechanical detection of the NMR requires stronger coupling between nuclei and mechanics. In case of an ensemble, the effective coupling can be scaled with \(\sqrt {N_{\mathrm{P}}N_{{\mathrm{NS}}}}\) due to collective effects^{44,45}, where N_{NS} is the number of nuclear spins equally coupled to the mechanical resonator which contains N_{P} phonons. Indeed such collective enhancement of coupling enables quantum entanglement to be generated between an ensemble and a resonator as demonstrated with an electron spin ensemble^{17,18,46}. To address the future possibility of accessing the strongcoupling regime in this platform, the coupling rate between a single ^{75}As nucleus and the zeropoint motion of the resonator is estimated to be g_{0} ~ A_{1}x_{ZP} = 0.15 mHz, where x_{zp} = 3 fm is the amplitude of the zeropoint fluctuations in the present mechanical resonator^{40}. The strongcoupling regime requires \(\sqrt {N_{\mathrm{P}}N_{{\mathrm{NS}}}} g_0\) > max(Δf_{M}, Δf_{NS}), where Δf_{M} (Δf_{NS}) is the frequency bandwidth of the mechanical resonance (NMR peak). In the present work, Δf_{M} = 25 Hz and Δf_{NS} = 5 kHz, where the NMR spectrum is inhomogeneously broadened. However N ~ 10^{10} nuclear spins can be obtained in a volume of ~(100 nm)^{3}, which is much smaller than the electromechanical resonator, in which case the strongcoupling regime is achievable with N_{P} ~ 10^{4} phonons, where these requirements would be accessible by simply optimizing the design of the resonator.
Further enhancement of this parametric coupling is necessary to achieving quantum coherent coupling, i.e., strongcoupling with N_{P} < 1. This objective could be attained by using strainengineered softclamped ultrahighQ mechanical resonators^{47,48}, which can have a narrow linewidth down to mHz, and by improving the crystal quality to suppress the inhomogeneous broadening of the NMR spectrum as well as using quantum dots for highly polarized nuclear spin states^{49}. On the other hand, a resonator with the frequency greater than the NMR frequency would be required for sideband cooling^{3,4,5,6,7,50} of the nuclear spins which would necessitate changing the material of the resonator and/or making it shorter and lighter where both could be achieved with semiconductor nanowires^{51,52}. For future application to a quantum coherent hybrid system, the decoherence induced by the unpolarized nuclear spins, which are coupled to mechanical motion at the clamping points (but unpolarized by the electron transport) needs to be suppressed. An optimized device structure where the mechanically strained clamping points completely overlap with the nuclei polarized from the electron transport can in principle achieve this objective. Ultimately optical access to the sample will enable the nuclear bath in the entire resonator to be polarized^{22,53}, thus enabling quantum coherent coupling between nuclei and phonons. Our observation of dynamical coupling between an nuclear spin ensemble and electromechanical phonons combined with these improvements is a prototype for electromechanical resonatornuclear spin hybrid systems.
Methods
Sample fabrication
A doubly clamped electromechanical resonator with a length of 50 μm, width of 6 μm, and a thickness of 1 μm was fabricated from highquality modulationdoped GaAs/Al_{0.3}Ga_{0.7}As heterowafer grown by molecular beam epitaxy on an undoped GaAs [100] substrate^{39,40}. The beam resonator was defined along the [011] crystal axis. A single heterojunction located 90 nm below the surface sustains a twodimensional electron gas (2DEG) with density n = 2.7 × 10^{11} cm^{−2}. The main heterolayer that constitutes the beam resonator was grown on an aluminumrich Al_{0.65}Ga_{0.35}As sacrificial layer to enable it to be suspended from the substrate. The fine mesa structure on the clamping points was defined by electronbeam lithography and wetetching. The antenna gate and the Schottky electrodes formed thereon were fabricated via electronbeam lithography and deposited with 1 nm/25 nm of Cr/Au. The final step to suspend the beam resonator was defined by photo lithography and wetetched to expose the sacrificial layer. The sacrificial layer was then selectively removed with dilute hydrogen fluoride (10 wt%).
Measurement setup
Supplementary Figure 1a shows the experimental setup. All measurements were performed in a highvacuum insert (<10^{−5} Pa) in a dilution refrigerator with a base temperature (<80 mK). Sourcedrain current I_{sd} is measured in a twoterminal configuration using a source measure unit with a bias voltage V_{sd}. In order to irradiate the RF transverse magnetic field in the NMR measurements, a signal generator is connected to one end of the antenna gate, while the other is grounded at low temperature so as to apply RF current I_{RF} with frequency ω_{RF}. The input RF power was fixed at −35 dBm throughout the measurements. The transverse rf field B_{x} is about 0.8 mT estimated from the input power and the resistance of the antenna gate. On the other hand, variable actuation voltage V_{a} was applied to the upper Schottky gate electrode fabricated on the left clamping point using a function generator while the 2DEG beneath the gate is grounded as shown in Supplementary Fig. 1a, b. The resulting piezovoltage arising from the motion of the beam at the lower gate was amplified by a homemade HEMTbased cryogenic amplifier with the input load resistor of R_{L} = 1 kΩ followed by a commercially available amplifier (NF SA220F5) with a total voltage gain of ~200 VV^{−1}. The amplified voltage signal was then measured with a lockin amplifier phaselocked to the function generator. The details for this amplification setup are also available in refs.^{39,40}.
Resistively detected NMR
The resistively detected NMR method relies on contact hyperfine interaction between conduction electrons and nearby nuclei, thereby enabling currentinduced dynamical nuclear polarization (DNP), as well as resistive detection^{15,21,54,55,56}. In our device, a shallow mesa sustaining a 2DEG was patterned onto the right clamping point. A static magnetic field B_{0} was then applied perpendicular to the 2DEG. Supplementary Figure 1c shows the twoterminal resistance measured as a function of B_{0}, showing the integer quantum Hall plateaus. In the quantum Hall regime, current is carried by conduction electrons flowing along two counterpropagating edge channels. In order to selectively polarize nuclear spins and detect their spin states solely at the clamping point, negative bias voltage V_{g} is applied to the Schottky electrode to form a narrow constriction at the right clamping point as shown in Supplementary Fig. 1a, b. When a large V_{sd} is applied, a large electric field owing to a high current density at the narrow constriction locally breaks down the quantum Hall state that entails spin flipflop events between hyperfinecoupled nuclei and electrons. As a consequence of these spin flipflop events, local nuclei at the clamping point can be spinpolarized, known as DNP^{56,57,58,59}. The number of nuclei in this interaction is estimated to be ~10^{9} or less from the effective volume of the constriction, which is roughly 1 μm × 1 μm × 20 nm.
The hyperfine field generated by the spinpolarized nuclei modifies the Zeeman energy of the electron and thereby alters the backscattering probability which is manifested as a change in I_{sd}. Supplementary Figure 1d shows the real time evolution of I_{sd}, after the abrupt application of V_{sd} = 13.2 mV at B_{0} = 3.5 T (ν ≈ 3.15) with V_{g} = −0.45 V to induce breakdown, which reveals a slow decrease for the first 30 s followed by a saturation. The observed decrease in I_{sd} can be attributed to currentinduced DNP resulting in a 20% spin polarization estimated from the numerical simulations of the NMR spectra. Since this DNP was observed only when V_{g} value is large enough to completely deplete the 2DEG underlying the electrode, it ensures that the DNP and its resistive detection are taking place solely in the vicinity of the narrow constriction, i.e., at the clamping point. After waiting for I_{sd} to saturate, ω_{RF} is slowly swept at a rate of 3 kHz min^{−1}. In order to resolve the fine peak structures in the NMR, (ΔI_{sd})^{2} is plotted as shown in Fig. 3b. When ω_{RF} matches the transition frequency of nuclear spin states, a coherent oscillation is driven that leads to the depolarization of the nuclei. The resultant change in the hyperfine field due to the depolarized nuclei modifies I_{sd} that manifests itself as an NMR peak as shown in Figs. 1d and 3b. The NMR detection via the electronic transport measurements has a fluctuation of about 0.3 nA that originates from charge fluctuation in the heterostructure. Because of this current fluctuation, the signaltonoise ratio in the NMR signal ranges from 2 to 10.
In this work, an NMR signal was observed in the magnetic field range of 3.1 < B_{0} < 3.55 T, which corresponds to a transition from filling factor ν = 4 to ν = 3 quantum Hall states as highlighted in Supplementary Fig. 1c.
Sideband NMR spectra
The NMR spectra on the red/blue sidebands shown in Fig. 2e can in principle be obtained by calculating the time evolution via the master equation for the Hamiltonian in Eq. (1). However, the full numerical time evolution without approximation was slow to execute due to the low transition rate of the sideband NMR. Instead a numerical simulation of the Hamiltonian, approximated using the rotatingwave approximation, is employed. Equation (1) is evaluated with x(t) = x_{1}cos(ω_{M}t) and by moving to a nonuniform rotating frame via a unitary transformation:
This yields a transformed Hamiltonian:
The second term in the right hand side describes the main NMR peak at ω_{RF} = ω_{0}, while the third (fourth) term corresponds to the blue (red) sideband. Here \(\hat I_{{\mathrm{side}}}\) describes the transition of the nuclear spin states at the sidebands and is defined as
where the first (last) two terms describe the transition between \(\left {1{\mathrm{/}}2} \right\rangle\) and \(\left {3{\mathrm{/}}2} \right\rangle\) (\(\left {  1{\mathrm{/}}2} \right\rangle\) and \(\left {  3{\mathrm{/}}2} \right\rangle\)) states. In order to calculate the time evolution from Eq. (4) at the red (blue) sideband, a transition to the frame rotating at frequency ω_{RF} ± ω_{M} along the zaxis is made using the transformation \(\hat U_2\) = \({\mathrm{exp}}\left\{ {i\left( {\omega _{{\mathrm{RF}}} \pm \omega _{\mathrm{M}}} \right)t\hat I_z} \right\}\) followed by the rotatingwave approximation to get
whose time evolution can then be easily calculated. It should be noted that, because of the rotatingwave approximation, this simulation does not reproduce the frequency shift induced by the ac strain, which is observed in Fig. 3b. Hence the main peak in Fig. 2e shows no frequency shift and this can only be reproduced in the full numerical simulation without the rotatingwave approximation as shown in Fig. 3c, d.
Numerical simulation of the NMR spectra
For the numerical simulations, the quadrupole coupling constants A_{1} and A_{2} in Eq. (2) are determined from a finite element method analysis of the strain associated with the fundamental mode (see Supplementary Note 1). We obtain A_{1} = 0.49 meV m^{−1} and A_{2} = −0.6 meV m^{−1} for ^{75}As, A_{1} = 0.16 meV m^{−1} and A_{2} = −0.14 meV m^{−1} for ^{69}Ga. The displacement is measured at the center of the beam (see point x in Supplementary Fig. 2a) and is decomposed into static and dynamic parts as x(t) = x_{0} + x_{1} cos(ω_{M}t). The time evolution for the density matrix ρ of the nuclear spin is then calculated using the von Neumann equation^{13,57}:
The temporal evolution of this firstorder ordinary differential equation is extracted using the fourthorder Runge–Kutta method with a time step of Δt = 10^{−9} s. The upper panels in Supplementary Fig. 3a, b show the simulated temporal evolution of \(\left\langle {\hat I_z} \right\rangle = {\mathrm{Tr}}\left\{ {\rho \hat I_z} \right\}\) with two different mechanical displacements (a) x_{1} = 0 and (b) x_{1} = 1.1 μm. Both plots show the first 1.5 periods of the coherent oscillation. From this time evolution, the normalized oscillation amplitude in \(\left\langle {\hat I_z} \right\rangle\) can be determined as shown in the lower panels in Supplementary Fig. 3a, b. In this simulation, the shape of the measured NMR spectra (Fig. 3b) is reproduced by adjusting x_{0} and ρ(0) values. The experimental spectrum exhibits two features: (i) the peak splitting in the absence of driven mechanical strain x_{1} = 0 is ~5 kHz, and (ii) the prominent peak’s blue shift (solid circle) whilst the sub peak (open square) is stationary. The former feature is reproduced with x_{0} = −22 nm, whereas the latter feature is reproduced by ρ(0) = {ρ_{3/2}, ρ_{1/2}, ρ_{−1/2}, ρ_{−3/2}} = {0.15, 0.2, 0.27, 0.36}, which is determined from a Boltzman distribution with a negative spin temperature of −4 mK. The negative temperature indicates that the nuclear spin state is population inverted due to the negative gfactor of the conduction electrons in GaAs^{57}. Figure 3c displays the resultant normalized oscillation amplitude extracted from the time evolution of \(\left\langle {\hat I_z} \right\rangle = {\mathrm{Tr}}\left\{ {\rho \hat I_z} \right\}\) at different mechanical displacements.
Floquet perturbation analysis
The quadrupole interaction experienced by the nuclei can be described as a timeperiodic perturbation that induces a level shift and thus lifts the degeneracy of the Zeeman split energy states. To quantify this level shift a Floquet perturbation analysis^{60,61} of the above Hamiltonian is employed to derive the energy diagram in Fig. 4 by decomposing it to: \(\hat H = \hat H_{\mathrm{Z}} + x(t)\hat J_{\mathrm{Q}}\) with x(t) = x_{0} + x_{1} cos(ω_{M}t). In this perturbative approach the first term is the bare Hamiltonian whose eigenenergy \(E_m^{(0)}\) is exactly solved as \(\hat H_{\mathrm{Z}}\left m \right\rangle = E_m^{(0)}\left m \right\rangle\), while the second term is treated as a periodic timedependent perturbation that causes an energy shift for \(\left m \right\rangle\). The eigenenergies E_{m} for the full Hamiltonian are then expanded as a perturbative series: E_{m} = \(E_m^{(0)} + E_m^{(1)} + E_m^{(2)} + \cdots\). The striking conclusion from the Floquet theory is that the oscillating components (∝x_{1}) have no contribution in the first order but a finite contribution in higher order. The firstorder energy shift is then calculated as
with a positive (negative) sign for \(\left { \pm 3{\mathrm{/}}2} \right\rangle\) \(\left( {\left { \pm 1{\mathrm{/}}2} \right\rangle } \right)\). Here only the diagonal part (real part) of \(\hat J_{\mathrm{Q}}\) contributes to the energy shift. Similarly, the secondorder energy shift is calculated as
The numerators in this equation can be evaluated as
and are independent of m. In the second order, the offdiagonal part (imaginary part) of \(\hat J_{\mathrm{Q}}\) has a nonzero contribution. This leads to
where the negative (positive) sign is for \(\left {3{\mathrm{/}}2} \right\rangle\), and \(\left {1{\mathrm{/}}2} \right\rangle\) (\(\left {  3{\mathrm{/}}2} \right\rangle\), and \(\left {  1{\mathrm{/}}2} \right\rangle\)) with approximately: x_{0}/x_{1} ≈ 0, and ω_{M}/ω_{0} ≈ 0.
Calibration of mechanical displacement and strain
In the experiments, the mechanical motion can be detected via the piezovoltage V_{d}, from which the corresponding displacement x_{1} at the center of the beam resonator (point x in Supplementary Fig. 2a) is determined via x_{1} = V_{d}/(5 × 10^{3} V m^{−1}). This calibration was derived by fitting the experimentally observed NMR frequency shift to the simulated frequency shift as shown in Fig. 3d. The electromechanical strain can be determined from this displacement calibration via a finite element method simulation (see Supplementary Note 1). To ensure validity of this calibration, the mechanical displacement is also quantified from the onset of nonlinearity. The onset of nonlinearity in a doubly clamped beam resonator can be estimated analytically from x_{c} ≈ t\(\sqrt {2{\mathrm{/}}0.528Q\left( {1  \nu ^2} \right)}\) ≈ 12 nm, where t = 1 μm is the thickness of the beam and ν = 0.31 is the Poisson ratio of GaAs^{41,52}. Experimentally, nonlinearity emerges at V_{d} ~ 0.2 mV as shown in Fig. 1b and in the strong driving regime at point 5 in Fig. 3a, the mechanical amplitude is roughly 30 times larger than this onset, and is estimated to be 30x_{c} ≈ 0.4 μm.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Change history
04 September 2018
This Article was originally published without the accompanying Peer Review File. This file is now available in the HTML version of the Article; the PDF was correct from the time of publication.
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Acknowledgements
The authors thank H. Chudo, K. Chida, K. Harii, Y. Hirayama, J. Ieda, K. Koshino, Y. Matsuzaki, W. J. Munro, and M. Ono for fruitful discussions. This work was partly supported by JSPS KAKENHI Grant No. 23241046, JP18H01156, and JP18H05258 and GrantinAid for Scientific Research on Innovative Areas No. JP15H05869.
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Y.O. fabricated the device, performed the measurements, and analyzed the data and theoretical model. K.O. grew the GaAs heterostructure. S.S. supported the device fabrication. S.N. and NH.K supported the numerical simulation. Y.O. I.M., and H.Y. wrote the paper.
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Okazaki, Y., Mahboob, I., Onomitsu, K. et al. Dynamical coupling between a nuclear spin ensemble and electromechanical phonons. Nat Commun 9, 2993 (2018). https://doi.org/10.1038/s41467018054633
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DOI: https://doi.org/10.1038/s41467018054633
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