physical review applied 13, 054068 (2020) · 2020. 5. 28. · physical review applied 13, 054068...

PHYSICAL REVIEW APPLIED 13, 054068 (2020)

Acoustically Driving the Single-Quantum Spin Transition of DiamondNitrogen-Vacancy Centers

H. Y. Chen ,1 S. A. Bhave ,2 and G. D. Fuchs 1,*1Cornell University, Ithaca, New York 14853, USA

2Purdue University, West Lafayette, Indiana 47907, USA

(Received 6 March 2020; accepted 4 May 2020; published 27 May 2020)

Using a high quality factor 3 GHz bulk acoustic wave resonator device, we demonstrate the acousticallydriven single quantum spin transition (|ms = 0〉 ↔ |±1〉) for diamond nitrogen-vacancy (N-V) centersand characterize the corresponding stress susceptibility. A key challenge is to disentangle the uninten-tional magnetic driving field generated by the device current from the intentional stress driving within thedevice. We quantify these driving fields independently using Rabi spectroscopy before studying the morecomplicated case in which both are resonant with the single quantum spin transition. By building an equiv-alent circuit model to describe the device’s current and mechanical dynamics, we quantitatively model theexperiment to establish their relative contributions and compare with our results. We find that the stresssusceptibility of the N-V center spin single quantum transition is around

√2(0.5 ± 0.2) times that for dou-

ble quantum transition (|+1〉 ↔ |−1〉). Although acoustic driving in the double quantum basis is valuablefor quantum-enhanced sensing applications, double quantum driving lacks the ability to manipulate N-Vcenter spins out of the |ms = 0〉 initialization state. Our results demonstrate that efficient all-acoustic quan-tum control over N-V centers is possible, and is especially promising for sensing applications that benefitfrom the compact footprint and location selectivity of acoustic devices.

DOI: 10.1103/PhysRevApplied.13.054068

I. INTRODUCTION

Acoustic control of solid-state quantum defect centerspins [1–9], such as nitrogen-vacancy (N-V) centers indiamond [10], provides an additional resource for quan-tum control and coherence protection that is not availablewhen using an oscillating magnetic field alone [11,12]. Italso creates a path towards the construction of a hybridquantum mechanical system [13] in which a defect centerspin directly couples to a phonon mode of the host res-onator. For diamond N-V centers, there has been substan-tial effort to understand the phonon-driven double quantum(DQ) spin transition (|ms = +1〉 ↔ |−1〉) [1,4], includinga measurement of the corresponding stress susceptibility, b[3,12]. In contrast, the phonon-driven single quantum (SQ)spin transition, with its associated stress susceptibility, b′,remains a missing puzzle piece in the full characterizationof the N-V center ground-state spin-stress Hamiltonian. Arecent calculation [14] suggests that |b′/b| < 0.1, whichremains to be verified in experiments. Quantification ofthe phonon-driven SQ spin transition is important becauseSQ acoustic driving can enable all-acoustic quantum con-trol of the N-V center electron spin, which can have animpact in real-world sensing applications. Given that the

*[email protected]

solid-state phonon wavelength is 104 times shorter thanthat of a microwave photon at the same frequency, acous-tic wave devices also support higher resolution, selectivelocal spin control. Additionally, acoustic devices can beengineered with a smaller footprint and with potentiallylower power consumption.

Here we report our experimental study of acousticallydriven SQ spin transitions of N-V centers using a 3 GHzdiamond bulk acoustic resonator device [15]. The deviceconverts a microwave driving voltage into an acousticwave through a piezoelectric transducer, thus mechanicallyaddressing the N-V centers in the bulk diamond substrate.Because a microwave current flows through the devicetransducer, an oscillating magnetic field of the same fre-quency coexists with the stress field that couples to SQ spintransitions. To identify and quantify the mechanical contri-bution to the SQ spin transition, we use Rabi spectroscopyto separately quantify the magnetic and stress fields presentin our device as a function of the driving frequency. Basedon these results, we construct a theoretical model andsimulate the SQ spin transition Rabi spectroscopy to com-pare to the experimental results. From a systematicallyidentified closest match, we quantify the mechanical driv-ing field contribution and extract the effective spin-stresssusceptibility, b′. We perform measurements on both the|0〉 ↔ |+1〉 and |0〉 ↔ |−1〉 SQ spin transitions and obtain

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CHEN, BHAVE, and FUCHS PHYS. REV. APPLIED 13, 054068 (2020)

|b′/b| = √2(0.5 ± 0.2), around an order of magnitudelarger than predicted by theory [14].

II. EXPERIMENTAL SETUP AND THEORETICALMODEL

The ground-state electron spin of a N-V center can bedescribed by the Hamiltonian [14]

He/h = DS2z + γeB · S + Hσ /h, (1)where h is the Planck constant, D = 2.87 GHz is thezero field splitting, S = (Sx, Sy , Sz) is the vector of spin-1Pauli matrices. γe = 2.802 MHz/G is the spin gyromag-netic ratio in response to an external magnetic field, B =(Bx, By , Bz), and Hσ is a measure of the electron spin-stressinteraction.

As shown in Fig. 1(a), in the presence of an externalmagnetic field aligned along the N-V axis, B‖ = Bz, the|±1〉 levels split linearly with the magnetic field ampli-tude due to the Zeeman effect. This gives rise to threesets of qubits of two types: (1) {|0〉 , |+1〉} and {|0〉 , |−1〉}operated on a SQ transition; (2) {|+1〉 , |−1〉} operated onthe DQ transition. While the magnetic dipole transition isaccessible only for SQ transitions, phonons can drive bothSQ and DQ transitions [14].

In our experiments, uniaxial stress fields, σZZ , aregenerated along the [001] diamond crystal direction[see Fig. 1(b)]. This induces normal strain in the crys-tal, {�XX , �YY, �ZZ} = {−νσZZ , −νσZZ , σZZ}/E, where E isYoung’s modulus and ν is Poisson’s ratio. For simplic-ity, we work with the stress frame and the following stressHamiltonian applies:

Hσ = Hσ0 + Hσ1 + Hσ2, (2a)Hσ0/h = a1σZZS2z , (2b)Hσ1/h = 2b′σZZ{Sx, Sz}, (2c)Hσ2/h = 2bσZZ(S2y − S2x ). (2d)

Here a1, b′, and b are stress susceptibility coeffi-cients, where a1 = 4.4 ± 0.2 MHz/GPa and b = −2.3 ±0.3 MHz/GPa have been experimentally measured [16],while b′ has yet to be characterized. Theory predicts |b| �b′ = 0.12 ± 0.01 MHz/GPa [14]. After applying the rotat-ing wave approximation, the total Hamiltonian written inmatrix form on the basis {|+1〉 , |0〉 , |−1〉} is

He =

⎛⎜⎝

D + γeB‖ + a1σZZ 12�+1 e−iωt 12�2e−iωt12�

+1 e

iωt 0 12�−1 e

iωt

12�2e

iωt 12�

−1 e

−iωt D − γeB‖ + a1σZZ

⎞⎟⎠,

(3)

where ω = 2π f is the driving field frequency, �±1 =|�B ± �σ1| = 2|γeB⊥ · S ±

√2b′σZZ | is the SQ transition

(a) (b)

(c) (d)

(e) (f)

FIG. 1. (a) Ground-state spin levels of a N-V center as a func-tion of the axial magnetic field. The orange and green arrows rep-resent acoustically addressable qubit transitions and blue arrowsrepresent magnetically addressable qubit transitions. The greenarrows indicate the acoustically driven SQ spin transition understudy. (b) Schematics showing uniaxial stress along the diamond[001] crystal axis applied to a N-V center (orientation [11̄1̄]),generated from the longitudinal acoustic wave mode of the res-onator. (c) Photoluminescence scan of the device, consisting ofa semiconfocal diamond bulk acoustic resonator (center brightregion) and a 50-μm-radius microwave loop antenna. (d) Elec-tron spin resonance signal mapping of the device current fieldaround the resonator, showing current flow primarily along theelectrodes. The local contrast change in signal intensity is dueto fluorescence background variations. Crosses mark the loca-tions under investigation in the experiments. (e) Cross-sectionalview of the device. An NA = 0.8 microscope objective focusesthe 532 nm laser down into diamond at a depth close to thetransducer (around 3 μm in distance). The orange standing wavefield illustrates the acoustic wave mode. (f) Closeup of the areaunder study [orange cross in (d)]. The interrogated N-V ensembledistributes across the node and antinode of the acoustic wave.

Rabi field amplitude, �B and �σ are complex Rabi fieldsfrom the transverse magnetic field in microwave driving,B⊥, and the acoustic wave field, σZZ , respectively, and�2 = |�σ2| = 4bσZZ represents the acoustically drivenDQ transition Rabi field amplitude.

Experimentally, we fabricate a semiconfocal bulk acous-tic resonator device on a 20 μm thick optical grade

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ACOUSTICALLY DRIVING THE SINGLE-QUANTUM SPIN... PHYS. REV. APPLIED 13, 054068 (2020)

diamond substrate [(001) face orientation, N-V center den-sity approximately 2 × 1013 cm−3] to launch longitudinalacoustic waves along the diamond [001] crystal axis attwo mechanical resonance frequencies, fr = 3.132 GHzand 2.732 GHz. This allows phonon driving of both |0〉 ↔|+1〉 and |0〉 ↔ |−1〉 SQ transitions as well as the |+1〉 ↔|−1〉 DQ transition. At room temperature, we use a focused532 nm laser (0.5 mW) to excite N-V centers in the sub-strate and collect their fluorescence from phonon sidebands (greater than 675 nm) to detect their spin statesusing a home-built confocal microscope. A photolumines-cence scan of the device along the X -Y plane is shownin Fig. 1(c), featuring a micromechanical resonator (cen-ter bright circular area) encircled by a microwave loopantenna (radius approximately 50 μm). A detailed struc-tural description and fabrication process of the device canbe found in Ref. [15]. In Fig. 1(e) we show a schematicof the cross-sectional view of the device. The NA = 0.8objective in our confocal microscope has a depth resolu-tion around 2.8 μm inside diamond, which is comparableto half of the acoustic wavelength, λ/2, and much smallerthan that of the characteristic microwave magnetic fielddecay length.

Microwave (MW) driving of the piezoelectric trans-ducer creates a bulk acoustic wave confined in the res-onator and current-induced magnetic field throughout thedevice. On a spin resonance, fMW = f|0〉↔|−1〉 or fMW =f|0〉↔|+1〉, the mixing of |0〉 and |±1〉 spin states yieldsdecreased photoluminescence (PL) from the N-V centers.The PL signal from the electron spin resonance (ESR) canthus be used to spatially map the current-induced mag-netic field distribution in the device. In Fig. 1(d), withB‖ = 58 G, we drive the transducer off resonance usinga 2.715 GHz microwave source. The ESR measurementdelineates the current field flowing through the leads andthe resonator area. The acoustic wave field, however, ispresent only inside the resonator [see Fig. 1(e)] [15].

Given that the magnetic driving field, B⊥, and the acous-tic driving field, σ ZZ , coexist in the resonator and theyboth couple to the SQ spin transition, it is not possibleto acoustically drive the N-V center SQ spin transitionwithout magnetic driving field interference in our device.However, besides measuring the joint field effect on theSQ transition, �±1 , we can separately quantify the twofield contributions using independent measurements onthe two fields. More explicitly, the Rabi amplitude of thecurrent-induced magnetic field inside the resonator [orangecross in Fig. 1(d)], �B, is proportional to that outside theresonator due to current continuity, and can be quanti-fied by scaling the magnetic field �′B measured near theleads [blue cross in Fig. 1(d)] where the acoustic waveis absent, i.e., �B = β�′B. The scaling factor β can beexperimentally determined off a mechanical resonance fre-quency where the acoustic field is near zero. Because onlyphonon driving and not magnetic driving couples to the

DQ transition, the amplitude of the acoustic wave insidethe resonator can be inferred by measuring the DQ transi-tion Rabi field �σ2 [15], which is proportional to �σ1, i.e.,�σ1 = α�σ2 = |b′/(

√2b)|�σ2.

The phase properties of the two vector fields and theassociated complex Rabi fields are complicated and evolvewith frequency. The effect on �±1 depends on both res-onator electromechanical characteristics and field spatialdirections relative to the N-V atomic axis. We take intoaccount the first part by determining an equivalent cir-cuit of the device, and the second part by introducing anunknown constant parameter φ to describe the phase dif-ference of the two fields led by a spatial factor. The totalSQ Rabi field as a function of driving frequency f is then

�±1 (f ) = |β�′B(f )eiφ ± α�σ2(f )|, (4)

where �(f ) = �(f ) exp[iθ(f )], and θ(f ) depends onresonator electromechanical characteristics.

III. EXPERIMENTAL RESULTS AND DISCUSSION

A. SQ and DQ Rabi spectroscopy

We use SQ N-V center spin Rabi spectroscopy to mea-sure �′B(f ) at the marked blue cross location in Fig. 1(d).We first focus on the mechanical mode at fr = 3.132 GHz.In our experiment, we vary the N-V axial magnetic fieldB‖ around the point where the |0〉 ↔ |+1〉 transition fre-quency is close to fr [see Fig. 2(a1)]. At each valueof B‖, we correspondingly adjust our driving frequencyto the Zeeman splitting to maintain a condition of on-resonance driving. After spin state initialization into the|0〉 state using optical spin polarization, we drive Rabioscillations by applying varying length pulses to the piezo-electric transducer with a power at 10 dbm. We limit theprobed nuclear hyperfine levels to the mI = +1 subspace[see Fig. 2(b1)] by requiring �′B(f ) < 1.2 MHz, which isbelow the hyperfine coupling coefficient. We then measurethe spin population’s time evolution as a function of pulseduration τ through a photoluminescence measurement atthe end of the Rabi sequence [see the inset of Fig. 2(d1)].A series of measurements are done as we vary B‖ andthe corresponding resonant drive frequencies around fr. Toensure time stability, we keep track of the probed locationby active feedback to the objective position, and limit thedrift in the probed volume to less than (0.1 μm)3. The mea-surement results are shown in Fig. 2(c1). Rabi frequenciescan then be extracted from either fitting or by taking thepeak Fourier component of the Rabi spectroscopy results,as shown in Fig. 2(d1), and they are directly proportionalto the associated driving field amplitude, B⊥(f ). By com-paring the magnetically driven Rabi fields �′B and �B thatare measured off the mechanical resonance frequency, wecalculate β = 1.30 ± 0.01.

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(a1)

(b1)

(c1) (e1)

(f1)

(g1)

(h1)

(g2)

(h2)

(e2)

(f2)

(d1)

(c1)

(d1)(b2)

(a2)

FIG. 2. Spectroscopy study of N-V center spin under driving fields at frequencies near two acoustic modes at (1) 3.132 GHz and(2) 2.732 GHz. Ground-state spin level diagrams in (a1) and (a2) show the targeted SQ (blue arrow, |0〉 ↔ |±1〉) and DQ (orangearrow, |−1〉 ↔ |+1〉) transitions within a nuclear spin |mI = +1〉 hyperfine manifold. Transducer driving creates both magnetic, B⊥,and acoustic fields, σ ZZ , in the device, which can be measured independently by (c) SQ Rabi spectroscopy near the leads [blue cross inFig. 1(d)] and (e) DQ Rabi spectroscopy in the center of the resonator [orange cross in Fig. 1(d)]. In the resonator, the two vector fieldsadd coherently and drive the SQ transition together. Measurements of the resulting total field probed by SQ ESR and Rabi spectroscopyare shown in (b) and (g), respectively. (d),(f),(h) Fourier transforms of (c),(e),(g). The insets in (d1),(f1),(h1) show the correspondingmeasurement sequences. Norml. is short for normalized value. AP stands for adiabatic passage pulse.

We perform similar Rabi spectroscopy on the |−1〉 ↔|+1〉 DQ transition to measure �σ2(f ) at the markedorange cross location in Fig. 1(d), and target at a deptharound 3 μm from the transducer [see Fig. 1(e)]. From PLmeasurements of the N-V center ensemble in comparisonto the single N-V center PL rate in our setup, we estimatearound 150(20) N-V centers in the laser focal spot, amongwhich approximately 35% of N-V centers are aligned withthe external magnetic field B‖. As a result, there are 45–60N-V centers contributing to the signal, and they randomlyspan the node and antinode of the acoustic wave [seeFig. 1(f)]. The working axial magnetic field condition,B‖ ∼ 560 G, is close to the excited state level anticross-ing (ELAC) [17] and provides near-perfect nuclear spinpolarization in mI = +1. In each resonant Rabi drivingsequence [see the inset of Fig. 2(f1)], we use an adiabatic

passage pulse to prepare all spins into the |−1〉 state. Afterresonant acoustic field driving, another adiabatic passagepulse is used to shelve the residual spin |−1〉 populationinto |0〉 prior to PL measurement. The results are shown inFigs. 2(e1) and 2(f1).

For SQ Rabi spectroscopy of �′B(f ) measured at theleads, in Fig. 2(d1) we show two resonance features: themechanical resonance at fr = 3.132 GHz and an electri-cal antiresonance at fa = 3.134 GHz, which originate fromthe electromechanical coupling of the piezoelectric res-onator [18]. The result is consistent with the device admit-tance measurement using a vector network analyser (seeAppendix B), and is used later towards the constructionof a unique circuit model to describe the complex elec-tromechanical response of our device. For DQ Rabi spec-troscopy on �σ2(f ), Fig. 2(f1) reveals a single Lorentzian

054068-4


mechanical resonance at fr = 3.132 GHz, which corre-sponds to the longitudinal acoustic standing wave mode inthe resonator. The modal mechanical quality factor, Q, canbe directly calculated from the linewidth, which turns outto be around 1040(40). [Note that the measured �σ2(f )corresponds to the peak stress field amplitude, consider-ing that a N-V center located near the acoustic wave nodecontributes little to the Rabi oscillation signal.]

Having characterized the electromechanical response ofour device, we perform SQ Rabi spectroscopy inside theresonator at the same location as the DQ measurement[marked with an orange cross in Fig. 1(d)]. The total Rabifield �+1 (f ) is complicated because B⊥ and σ ZZ jointlyact on the transition, and the probed N-V center ensem-ble includes N-V centers coupled to both an antinode anda node of the acoustic wave [see Fig. 1(f)], introduc-ing spatial inhomogeneity in σ ZZ(f ). The result shown inFig. 2(g1) is distinct from the previous single field mea-surements, and its Fourier transform in Fig. 2(h1) dispersesinto multiple components due to the spatial inhomogene-ity of stress coupling to the ensemble. To understand thespectrum and characterize the mechanical contribution tothe SQ spin transition, we model our experiment in thefollowing sections to solve the problem. We also per-form similar measurements on a different acoustic modeat fr = 2.732 GHz to probe the |0〉 ↔ |−1〉 SQ transition.The results for �′B(f ), �σ2(f ), and �

−1 (f ) are shown in

Figs. 2(c2)–2(h2).

B. Modeling magnetic and stress fields

The electromechanical response of a piezoelectric res-onator like our device can be modeled using an electricalequivalent circuit known as the modified Butterworth–VanDyke (MBVD) model [18,19]. In the MBVD circuit model(Fig. 3 inset), the resonator acoustic mode is treated as adamped mechanical oscillator modeled using an electri-cal RmLmCm series circuit. The acoustic RmLmCm circuitis then coupled in parallel to an electrical R0C0 branch rep-resenting the electrical capacitance and dielectric loss inthe transducer. Lastly, a serial resistor, Rs, is introduced totake into account ohmic loss in the contact lines. Applyingan external voltage driving source near the resonance fre-quency, fr = 1/(2π

√LmCm), can excite both current and

acoustic fields in the circuit. More explicitly, the total cur-rent (or admittance, Y) in the circuit model is proportionalto the induced magnetic field, and the complex voltagevalue V across the capacitor Cm is proportional to the stressfield generated in the device.

With �′B(f ) and �σ2(f ) measured experimentally, weperform MBVD model fitting with all circuit parametersset free using the equations

�′B(f ) = A × |Y(Rm, Lm, Cm, R0, C0, Rs, f )|,�σ2(f ) = B × |V(Rm, Lm, Cm, R0, C0, Rs, f )|,

(5)

(a)

(b)

(c)

(d)

FIG. 3. Modeling device electrometrical characteristicsthrough the modified Butterworth–Van Dyke model (inset),using �′B(f ) and �σ2(f ) (blue and orange dots) as inputs. Afterfitting, the model yields both amplitude and phase informationof the (a),(b) magnetic field and (c),(d) acoustic field.

where A and B are free scaling parameters. To simplifythe modeling, we ignore other electrical components inthe device, such as wire bond electrostatics and waveguideresonance.

The model fitting results are shown in Fig. 3, with theoptimal fitting parameters as follows: {A, B, Rm, Lm, Cm, R0,C0, Rs} = {235 ± 155 MHz/S, 2.6 ± 0.1 kHz/V, 219 ±165 �, 13 ± 9 μH, 0.20 ± 0.14 fF, 46 ± 30 �, 0.20 ±0.13 pF, 98 ± 59 �} for the 3.132 GHz mode, and{624 ± 74 MHz/S, 4.9 ± 0.2 kHz/V, 69 ± 20 �, 10 ±4 μH, 0.33 ± 0.13 fF, 149 ± 11 �, 0.13 ± 0.05 pF, 732 ±86 �} for the 2.732 GHz mode. The large error bars area result of the small covariance between the large num-ber of fitting parameters; however, the model predictionsof amplitude and phase are insensitive to these uncertain-ties. Thus, our model accurately represents the physicalphenomena and is suitable for our purposes. The unifiedcircuit model not only reproduces the measured Rabi fieldamplitude, but also contains the requisite phase informa-tion. While the acoustic field goes through a 180◦ phasechange across a single mechanical resonance at fr, the cur-rent field undergoes two resonances, where the phase firstdrops around fr and then increases around fa. From thisanalysis, we have full information of the complex Rabifields, �′B(f ) and �σ2(f ).

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C. Simulation and extraction of SQ spin-stresssusceptibility

To interpret Fig. 2(g), evaluate α, and thus extract b′,we implement a quantum master equation simulation (seeAppendix A) as a function of the driving frequency f foran ensemble of six N-V centers. We treat the ensemble asevenly distributed from an antinode (z = 0) to a node (z =λ/4) of the acoustic wave, driven by the SQ Rabi field

�±1 (f , z) = |1.3�′B(f )eiφ ± α�σ2(f ) cos(kz)|, (6)

where k = 2π/λ, λ = 5.7 μm (6.7 μm) for the fr =3.132 GHz (2.732 GHz) mode. We sum the simulated timetraces of the spin population from each individual N-Vcenter and average to construct the final simulated Rabispectroscopy result.

We implement simulations for a range of {α, φ} val-ues and compare to the experimental data in Fig. 2(g)by calculating their structural similarity (SSIM) index(see Appendix A) [20]. The results are shown inFig. 4(a). A higher SSIM value indicates a bettersimulation-to-experiment match. We find the best match at{α, φ} = {0.5 ± 0.2, 10◦ ± 40◦} for the 2.732 GHz modeand {α, φ} = {0.5 ± 0.2, −60◦ ± 60◦} for the 3.132 GHzmode. The corresponding simulated Rabi spectroscopyresults are shown in Fig. 4(b), which agree well with exper-imental observation. Additionally, we obtain the same αresult from both modes, which is expected physically. Thedifference in φ implies a phase change of θ in electrome-chanical resonance or a direction change in the magneticvector field, B⊥, between the two resonance modes, whichcan be explained by the microwave resonance in the trans-mission line (see Appendix B). From our result that α =|b′/(√2b)| = 0.5 ± 0.2, we find that the spin-stress sus-ceptibility of the SQ transition compared to that of the DQtransition is |b′| = √2(0.5 ± 0.2)|b|. This finding furthercompletes our understanding of the N-V center ground-state spin-stress Hamiltonian. Note that b′ describes spincoupling to only a normal stress field. The spin coupling toshear stress wave coefficient for the SQ transition, c′, is notmeasured (see Appendix C).

The fact that b′ is comparable to b, about an orderof magnitude larger than expected, has important impli-cations for applications. It raises the possibility of all-acoustic spin control of N-V centers within their full spinmanifold without the need for a magnetic antenna. Forsensing applications, a diamond bulk acoustic device canbe practical and outperform a microwave antenna in sev-eral aspects: (1) Acoustic waves provide direct access to allthree qubit transitions, and the DQ qubit enables better sen-sitivity in magnetic metrology applications [21]. (2) Themicron-scale phonon wavelength is ideal for local selectivespin control of N-V centers. (3) Bulk acoustic waves con-tain a uniform stress mode profile and thus allow uniform

(a1) (a2)

(b1) (b2)

(c1) (c2)

FIG. 4. Quantum master equation simulation of single quan-tum spin transition under dual field driving using model inputsfrom Fig. 3. (a1),(a2) Heatmaps of structural similarity indexcalculated between experimental data and simulation in {φ, α}parameter space for fr = 2.732 GHz and 3.132 GHz mode,respectively. The dashed circles mark the locations of peak SSIMvalues. (b1),(b2) The corresponding simulated Rabi spectroscopyresults using the peak SSIM associated {φ, α} values match theexperimental data in Figs. 2(g1) and 2(g2). (c1),(c2) Fouriertransforms of (b1),(b2).

field control of a large planar spin ensemble, for example,from a delta-doped diamond growth process [22].

IV. CONCLUSION

In summary, we experimentally study the acousticallydriven single quantum spin transition of diamond N-V cen-ters using a 3 GHz piezoelectric bulk acoustic resonatordevice. We observe the phonon-driven SQ spin transi-tion, disentangled from the background driving microwavemagnetic field. Through device circuit modeling and quan-tum master equation simulation of the N-V spin ensemble,we successfully reproduce the experimental results andextract the SQ transition spin-stress susceptibility, whichis an order of magnitude higher than is theoretically pre-dicted and further completes the characterization of theN-V spin-stress Hamiltonian. This study, combined withpreviously demonstrated phonon-driven DQ quantum con-trol and improvements in diamond mechanical resonatorengineering, shows that diamond acoustic devices are apowerful tool for full quantum state control of N-V centerspins.

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ACKNOWLEDGMENTS

Research support was provided by the DARPADRINQS program (Cooperative Agreement #D18AC00024) and by the Office of Naval Research (Grant No.N000141712290). Any opinions, findings, and conclusionsor recommendations expressed in this publication are thoseof the author(s) and do not necessarily reflect the viewsof DARPA. Device fabrication was performed in part atthe Cornell NanoScale Science and Technology Facility,a member of the National Nanotechnology CoordinatedInfrastructure, which is supported by the National ScienceFoundation (Grant No. ECCS-15420819), and at the Cor-nell Center for Materials Research Shared Facilities thatare supported through the NSF MRSEC program (GrantNo. DMR-1719875).

APPENDIX A: QUANTUM MASTER EQUATIONSIMULATION

We perform numerical simulation of the experimentusing the Lindblad master equation [23],

ρ̇ = − i�

[He, ρ] +∑

i,j

γij

(LiρL

†j −

12{L†i Lj , ρ}

), (A1)

where ρ is the density matrix of the spin states, L isthe Lindblad operator, i, j = +1, 0, −1. For the short timescope studied here, τ ≤ 4 μs, we ignore the longitudi-nal spin relaxation process, and set phase coherence T2 =1/γii = 2 μs. We evolve the density matrix from a purestate of |0〉, i.e., ρ(τ = 0) = |0〉〈0|, and calculate the spin

population in ρ00(τ ) to simulate the PL signal collected inthe experiment.

Because the {α, φ} are unknown from direct experimen-tal measurement, we perform a series of simulations inthe range α = (0, 1.5) and φ = (−180◦, 180◦), as shownin Fig. 5(a), which can be directly compared to the exper-iment. The evaluation of an experiment-simulation matchis through the SSIM index, Si [20]:

Si(x, y) = (2μxμy + C1)(2σxy + C2)(μ2x + μ2y + C1)(σ 2x + σ 2y + C2)

. (A2)

Here μx, μy , σx, σy , and σxy are the local means, standarddeviations, and cross covariance for the two images, x andy, under comparison; C1 = C2/9 = (0.01L)2, where L =255 is the dynamic range value of the image; SSIM = 0indicates no structural similarity and SSIM = 1 indicatesidentical images. We choose to work with SSIM instead ofother error sensitive approaches, such as the mean squarederror, because SSIM is a perception-based model for imagestructural information, and the measurement is insensitiveto average luminance and contrast change in the images.

We are unaware of a formally defined measure of theuncertainty estimation using SSIM in image assessment.For quantitative evaluation, here we report the uncertaintyfor {α, φ} by the half width at half maximum in normalizedSSIM(α) and SSIM(φ). The results are shown in Figs. 5(b)and 5(c).

(a)

(b) (c)

FIG. 5. (a) Quantum masterequation simulation in {α, φ}space. (b),(c) Line cuts from thesimulation results in Fig. 4(a)for the two resonance modesat fr = 2.732 and 3.132 GHz,respectively. Half width at halfmaximum of the peaks in thenormalized curves are used toevaluate uncertainties in α and φ.

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(a)

(b)

(c)

FIG. 6. Electrical S-parameter measurements of the device(wire bonded to a printed circuit board) using a vector networkanalyser, showing the (a) voltage reflection S11, (b) amplitude,and (c) phase of admittance Y11 as a function of the driving fre-quency. The orange hue highlights the resonance modes understudy. The inset in (c) shows the measurement scheme.

APPENDIX B: ELECTRICALCHARACTERIZATION OF DEVICE

In our experiment, the device is wire bonded to a copla-nar waveguide (CPW) on a printed circuit board (PCB)for microwave driving. We electrically characterize thedevice-PCB complex using a vector network analyser(VNA). The results are shown in Fig. 6. The PCB waveg-uide quarter-wavelength resonance at around 2.66 GHz

distorts the device frequency response and introduces devi-ations from the standard MBVD model for the 2.732 GHzoperating mode. We attribute this as the main cause forthe small model fitting deviation from experimental dataforf < 2.72 GHz in Fig. 3(a).

We build our MBVD circuit model from Rabi fieldspectroscopy measurements instead of the electrical mea-surements because the electrical measurement results aremore sensitive to the microwave transmission line config-uration and calibrations. In contrast, N-V centers providedirect local measurement of both current and stress vec-tor fields, and thus serve as a more accurate report of thedevice behavior.

APPENDIX C: FULL SPIN-STRESSHAMILTONIAN AND STRAIN SUSCEPTIBILITY

MAPPING

The full spin-stress Hamiltonian for a [111] orientedN-V center is [14,16]:

Hσ0/h = MzS2z , (C1a)Hσ1/h = Nx{Sx, Sz} + Ny{Sy , Sz}, (C1b)Hσ2/h = Mx(S2y − S2x ) + My{Sx, Sy}, (C1c)

where

Mz = a1(σXX + σYY + σZZ)+ 2a2(σYZ + σZX + σXY), (C2a)

Mx = b(2σZZ − σXX − σYY)+ c(2σXY − σYZ − σZX ), (C2b)

My =√

3[b(σXX − σYY) + c(σYZ − σZX )], (C2c)

TABLE I. N-V center ground-state spin coupling to stress and strain field coefficients.

a1 a2 b c b′ c′(MHz/GPa) (MHz/GPa) (MHz/GPa) (MHz/GPa) (MHz/GPa) (MHz/GPa)

[16] 4.4 ± 0.2 −3.7 ± 0.2 −2.3 ± 0.3 3.5 ± 0.3 · · · · · ·[24] −11.7 ± 3.2 6.5 ± 3.2 7.1 ± 0.8 −5.4 ± 0.8 · · · · · ·[14] 2.66 ± 0.07 −2.51 ± 0.06 −1.94 ± 0.02 2.83 ± 0.03 0.12 ± 0.01 −0.66 ± 0.01

· · · · · · · · · · · · ±√2(0.5 ± 0.2)b · · ·λa1 λa2 λb λc λb′ λc′

(GHz/strain) (GHz/strain) (GHz/strain) (GHz/strain) (GHz/strain) (GHz/strain)

[24] −0.5 ± 8.6 −9.2 ± 5.7 −0.5 ± 1.2 14.0 ± 1.3 · · · · · ·[14] 2.3 ± 0.2 −6.42 ± 0.09 −1.425 ± 0.050 4.915 ± 0.022 0.65 ± 0.02 −0.707 ± 0.018

d‖ d⊥(GHz/strain) (GHz/strain)

[3] 5.46 ± 0.31 19.63 ± 0.40[12] 13.3 ± 1.1 21.5 ± 1.2

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Nx = b′(2σZZ − σXX − σYY)+ c′(2σXY − σYZ − σZX ), (C2d)

Ny =√

3[b′(σXX − σYY) + c′(σYZ − σZX )]. (C2e)

For the uniaxial stress field, σZZ , only a1, b, and b′ relatedcoupling are excited, and the above equation reduces toEq. (2). In order to probe c′, a shear stress field will berequired (note that b′ and c′ are respectively named d ande in Ref. [14]), which can be generated from, for example,a uniaxial stress applied along the [110] crystal direction.

To map spin-stress susceptibility to spin-strain suscep-tibility, we can apply the stress-to-strain tensor transfor-mation to Eq. (C2) using the elastic stiffness tensor C. Theresulting spin-strain coupling written in the N-V axis frame(x = [−1, −1, 2]/√6, y = [1, −1, 0]/√2, z = [1, 1, 1]/√3)is [24]

Mz = λa1�zz + λa2(�xx + �yy), (C3a)Mx = λb(�xx − �yy) + 2λc�xz, (C3b)My = −2λb�xy + 2λc�yz, (C3c)Nx = λb(�xx − �yy) + 2λc�xz, (C3d)Ny = −2λb′�xy + 2λc′�yz, (C3e)

where

λa1 = a1(C11 + 2C12) + 4a2C44, (C4a)λa2 = a1(C11 + 2C12) − 2a2C44, (C4b)λb = b(C11 − C12) + 2cC44, (C4c)λc =

√2[b(C11 − C12) − cC44], (C4d)

λb′ = b′(C11 − C12) + 2c′C44, (C4e)λc′ =

√2[b′(C11 − C12) − c′C44], (C4f)

C11 = 1079 ± 5 GPa, C12 = 124 ± 5 GPa, and C44 =578 ± 2 GPa [25].

Experimentally and theoretically characterized mechan-ical coupling coefficients are summarized in Table I. Notethat early experimental characterization [3,12] parameter-ized the coefficients as coupling to N-V axial strain d‖and transverse strain coupling d⊥, instead of the tensorcomponent representation as described above.

APPENDIX D: COMSOL SIMULATION OFMAGNETIC AND ELECTRIC FIELDS IN THE

DEVICE

Apart from magnetic and acoustic fields, an oscillat-ing electric field from the current discharge in our devicecan be a potential driving source for the spin transition.To quantify the electric field in the device, we performa COMSOL simulation where the transducer is modeled

(a) (b)

FIG. 7. COMSOL simulation of the (a) magnetic field and (b)electric field generated from the piezoelectric transducer mod-eled as a capacitor.

as a capacitor. We adjust the driving voltage source (f =3 GHz) such that the generated B field at the experi-mentally interrogated location is around 0.3 G and thecorresponding Rabi field is �B =

√2γeB ∼ 1.2 MHz, as

shown in Fig. 7(a).Under the same conditions, we plot the simulated elec-

tric field distribution in Fig. 7(b). The amplitude of theelectric field is around 2 × 103 V/cm. Using the known DQspin-electric field susceptibility, d⊥ = 17 Hz cm/V [26],the corresponding electric Rabi field amplitude is �E =2d⊥E < 0.1 MHz, which is negligible in our experiment.

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I. INTRODUCTIONII. EXPERIMENTAL SETUP AND THEORETICAL MODELIII. EXPERIMENTAL RESULTS AND DISCUSSIONA. SQ and DQ Rabi spectroscopyB. Modeling magnetic and stress fieldsC. Simulation and extraction of SQ spin-stress susceptibility

IV. CONCLUSIONACKNOWLEDGMENTSA. APPENDIX A: QUANTUM MASTER EQUATION SIMULATIONB. APPENDIX B: ELECTRICAL CHARACTERIZATION OF DEVICEC. APPENDIX C: FULL SPIN-STRESS HAMILTONIAN AND STRAIN SUSCEPTIBILITY MAPPINGD. APPENDIX D: COMSOL SIMULATION OF MAGNETIC AND ELECTRIC FIELDS IN THE DEVICE. References

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