Strain engineering of the silicon-vacancy center in diamond

Srujan Meesala,1, ∗ Young-Ik Sohn,1, ∗ Benjamin Pingault,2 Linbo Shao,1 Haig A. Atikian,1 Jeffrey Holzgrafe,1

Mustafa Gundogan,2 Camille Stavrakas,2 Alp Sipahigil,3, 4 Cleaven Chia,1 Michael J. Burek,1 Mian Zhang,1 Lue

Wu,1 Jose L. Pacheco,5 John Abraham,5 Edward Bielejec,5 Mikhail D. Lukin,3 Mete Atature,2 and Marko Loncar1

1John A. Paulson School of Engineering and Applied Sciences,Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA

2Cavendish Laboratory, University of Cambridge,J. J. Thomson Avenue, Cambridge CB3 0HE, UK

3Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA4Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,

Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA5Sandia National Laboratories, Albuquerque, NM 87185, USA

We control the electronic structure of the silicon-vacancy (SiV) color-center in diamond by chang-ing its static strain environment with a nano-electro-mechanical system. This allows deterministicand local tuning of SiV optical and spin transition frequencies over a wide range, an essential steptowards multi-qubit networks. In the process, we infer the strain Hamiltonian of the SiV revealinglarge strain susceptibilities of order 1 PHz/strain for the electronic orbital states. We identify regimeswhere the spin-orbit interaction results in a large strain suseptibility of order 100 THz/strain for spintransitions, and propose an experiment where the SiV spin is strongly coupled to a nanomechanicalresonator.

I. INTRODUCTION

Solid state emitters such as color-centers and epitax-ially grown quantum dots provide both electronic spinqubits and coherent optical transitions, and are opticallyaccessible quantum memories. They can therefore serveas building blocks of a quantum network composed ofnodes in which information is stored in spin qubits andinteractions between nodes are mediated by photons1–4.However, due to the effects of their complex solid stateenvironment, most quantum emitters do not simultane-ously provide long coherence time for the memory, and fa-vorable optical properties such as bright, spectrally stableemission. The negatively charged silicon vacancy centerin diamond (SiV−, hereafter simply referred to as SiV)has been recently identified as a system that can over-come these limitations, since it provides excellent opticaland spin properties simultaneously. Its dominant zero-phonon-line (ZPL) emission and stable optical transitionfrequencies resulting from its inversion symmetry5–7 haverecently been used to realize single-photon switching8

and a fibre-coupled coherent single-photon source9 in ananophotonic platform. Further, recent demonstrationsof microwave10 and all-optical11 control of its electronicspin, as well as long (∼10 ms) spin coherence times atmK temperatures12, when electron-phonon processes inthe center are suppressed,10,13 make the SiV a good spinqubit.

Scaling up these demonstrations to multi-qubit net-works requires local tunability of individual emitters, aswell as the realization of strong interactions betweenthem. In this work, we control local strain in theSiV environment using a nano-electro-mechanical sys-

∗ These authors contributed equally

tem (NEMS), and show wide tunability for both opti-cal and spin transition frequencies. In particular, wedemonstrate hundreds of GHz of optical tuning, suffi-cient to achieve spectrally identical emitters for photon-mediated entanglement1,2. Further, we characterize thestrain Hamiltonian of the SiV and measure high strainsusceptibilities for both the electronic and spin levels.Building on this strain response, we discuss a schemeto realize strong coupling of the SiV spin to coherentphonons in GHz frequency nanomechanical resonators.While phonons have been proposed as quantum trans-ducers for qubits,14,15 experiments with solid-state spinshave been limited to the classical regime of large dis-placement amplitudes driving their internal levels16–24.The high strain susceptibility of the SiV ground statescan enable MHz spin-phonon coupling rates in existingnanomechanical resonators. Such a spin-phonon interfacecan enable quantum gates between spins akin to those inion traps25–27, and interfaces with disparate qubits28,29.

II. STRAIN TUNING OF OPTICALTRANSITIONS

The SiV center is an interstitial point defect in which asilicon atom is positioned midway between two adjacentmissing carbon atoms in the diamond lattice as depictedin the inset of Fig. 1(a). Its electronic level structureat zero strain is shown in Fig. 1(a). The optical groundstate (GS) and excited state (ES) each contain two dis-tinct electronic configurations shown by the bold hori-zontal lines. Physically, each of the two branches in theGS and ES corresponds to the occupation of a specificE-symmetry orbital by an unpaired hole.30 At zero mag-netic field, the degeneracy of these orbitals is broken byspin-orbit (SO) coupling leading to frequency splittings∆gs = 46 GHz, and ∆es = 255 GHz respectively. Due to

2

eg-↓〉eg+ ↑〉eg+ ↓〉eg-↑〉

eu-↓〉eu+ ↑〉

eu+ ↓〉eu-↑〉

B-eld

groundstates

excitedstates

406 THz(737 nm)

spin qubit

Δgs46 GHz

Δes255 GHz

ωs

AB

CD

y:[110]

(a)

(b)

Si

Si

xy

z

x:[112]

z:[111]

4 μm

FIG. 1. (a) Electronic level structure of the SiV center (molec-ular structure shown in inset) at zero strain showing groundand excited manifolds with spin-orbit eigenstates. The fouroptical transitions A, B, C, and D at zero magnetic field,and splittings between orbital branches in the ground state(GS) and excited state (ES), ∆gs and ∆es respectively areindicated. In the presence of a magnetic field, each orbitalbranch splits into two Zeeman sublevels. A spin-qubit canbe defined in the sublevels of the lower orbital branch in theGS. (b) Schematic of the diamond cantilever device and sur-rounding electrodes with a corresponding scanning electronmicroscope (SEM) image in the inset. Diamond crystal axesrelative to the cantilever orientation are shown. Four possi-ble orientations of the highest symmetry axis of an SiV areindicated by the four arrows above the cantilever. Underapplication of strain, these can be grouped into axial (red)and transverse (blue) orientations. Molecular structure of atransverse-orientation SiV as viewed in the plane normal tothe cantilever axis is shown below, and crystal axes that de-fine the internal co-ordinate frame of the color center are indi-cated. The z-axis is the highest symmetry axis, which definesthe orientation of the SiV.

inversion symmetry of the defect about the Si atom, thewavefunctions of these orbitals can be classified accordingto their parity with respect to this inversion center.5,30

Thus, the GS configurations correspond to the presenceof the unpaired hole in one of the even-parity orbitalseg+, eg−, while the ES configurations have this hole inone of the odd-parity orbitals eu+, eu−. Here the sub-

scripts g, u refer to even (gerade) and odd (ungerade)parity respectively, and +, − refer to the orbital angu-lar momentum projecton lZ . This specific level structuregives rise to four distinct optical transitions in the ZPLindicated by A, B, C, D in Fig. 1(a). Upon application ofa magnetic field, degeneracy between the SO eigenstatesis further broken to reveal two sub-levels within each or-bital branch corresponding to different spin states of theunpaired hole (S = 1/2). In this manner, a spin-qubitcan be defined on the two sublevels of the lowest orbitalbranch in the ground state.

To control local strain in the environment of the SiVcenter, we use a diamond cantilever, schematically shownin Fig. 1(b). Electrodes are fabricated, one on top ofthe cantilever, and another on the substrate below thecantilever to form a capacitive actuator. By applying aspecific DC voltage to these electrodes, we can deflect thecantilever to achieve a desired amount of static strain atthe SiV site. The fabrication procedure based on angledetching of diamond31,32 and device design are discussedin detail elsewhere33. The diamond sample with can-tilever NEMS is maintained at 4 K in a Janis, ST-500continuous-flow liquid helium cryostat. We perform op-tical spectroscopy on SiVs inside the cantilever via reso-nant laser excitation of the transitions shown in Fig. 1(a).Mapping the response of these transitions as a functionof voltage applied to the device allows us to study thestrain response of the SiV electronic structure.

The diamond samples used in our study have a [001]-oriented top surface, and the long axis of the cantileveris oriented along the [110] direction. There are four pos-sible equivalent orientations of SiVs - [111], [111], [111],[111] - in a diamond crystal, indicated by the four arrowsabove the cantilever in Fig. 1(b). Since the cantileverprimarily achieves uniaxial strain directed along [110],this breaks the equivalence of the four orientations, andleads to two classes indicated by the blue and red coloredarrows in Fig. 1(b). The blue SiVs, oriented perpendicu-lar to the cantilever long-axis, predominantly experienceuniaxial strain along their internal y-axis (see inset ofFig. 1(b)). On the other hand, the red SiVs are notorthogonal to the cantilever long-axis, and experience anon-trivial strain tensor, which includes significant strainalong their internal z-axis. For simplicity, we refer toblue SiVs as ‘transverse-orientation’ SiVs, and red SiVsas ‘axial-orientation’ SiVs. This nomenclature is usedwith the understanding that it is specific to the situationof predominantly [110] uniaxial strain applied with ourcantilevers.

Two distinct strain-tuning behaviors correlated withSiV orientation are observed as shown in Fig. 2. Orienta-tion of SiVs in the cantilever is inferred from polarization-dependence of their optical transitions at zero strain.30

With gradually increasing strain, transverse-orientationSiVs show an increasing separation between the A andD transitions with relatively small shifts in the B and Ctransitions as seen in Fig. 2(a). This behavior has beenobserved on a previous experiment with an ensemble ofSiVs.34 On the other hand, axial-orientation SiVs show

3

735.5 736.0 736.5 737.0 737.5 738 738.5

Nor

mal

ized

cou

nts

0V25V50V75V

100V125V150V175V200V220V

240V260V270V280V

Wavelength (nm)

A B C D

Si

736.2 736.6 737.0 737.4 737.8 738.2Wavelength (nm)

Nor

mal

ized

cou

nts

0V

50V

100V

150V

175V

200V

225V

250V

260V

270V

280VA B C D

z

Si

(a)

z

(b)

FIG. 2. Tuning of optical transitions of (a) transverse-orientation SiV (red in Fig. 1(b)), and (b) axial orientationSiV (blue in Fig. 1(b)). Voltage applied to the device isindicated next to each spectrum.

a more complex tuning behavior in which all transitionsshift as seen in Fig. 2(b).

In the context of photon-mediated entanglement ofemitters, typically, photons emitted in the C line,the brightest and narrowest linewidth transition are ofinterest8. Upon comparing Figs. 2(a) and (b), wenote that this transition is significantly more responsivefor axial-orientation SiVs. Particularly in Fig. 2(b),we achieve tuning of the C transition wavelength by0.3 nm (150 GHz), approximately 10 times the typ-ical inhomogeneity in optical transition frequencies ofSiV centers.6,35 Thus, NEMS-based strain control canbe used to deterministically tune multiple on-chip or dis-tant emitters to a set optical wavelength. In particular,integration of this NEMS-based strain-tuning with ex-isting diamond nanophotonic devices8,9,36–38 can enablescalable on-chip entanglement and widely tunable singlephoton sources. Besides static tuning of emitters, dy-namic control of the voltage applied to the NEMS can beused to counteract slow spectral diffusion, and stabilizeoptical transition frequencies39.

III. EFFECT OF STRAIN ON ELECTRONICSTRUCTURE

z Siz Si

0.0 0.5 1.0 1.5 2.0

50150250350450

GS

split

ting

(GH

z)250

350

450

550

650

ES s

plitt

ing

(GH

z)

Eg strain, ε (x 10-4)

ε

0.0 0.2 0.4 0.6 0.8 1.0

736.9

737.0

737.1

737.2

Mea

n ZP

L w

avel

engt

h (n

m)

A1g strain, ε|| (x 10-4)

ε||

SiSi

∆gs

∆es

ωZPL

(a)

Eg strain A1g strain

(b)

(c) (d)

(e) (f)

FIG. 3. (a) Dominant effect of Eg-strain on the electroniclevels of the SiV. (b) Dominant effect of A1g-strain on theelectronic levels of the SiV. (c) Normalized strain-tensor com-ponents experienced by transverse-orientation SiV (red in Fig.1(b)), and (d) axial orientation SiV (blue in Fig. 1(b)) in theSiV co-ordinate frame upon deflection of the cantilever. (e)Variation in orbital splittings within GS (solid green squares)and ES (open blue circles) upon application of Eg-strain.Data points are extracted from the optical spectra in Fig.2(a). Solid curves are fits to theory in text. (f) Tuning ofmean optical wavelength with A1g strain. Data points areextracted from the optical spectra in Fig. 2(b). Solid line isa linear fit as predicted by theory in text.

Following previous work on point defects,30,40,41 weemploy group theory to explain the effect of strain onthe SiV electronic levels, and extract the susceptibilitiesfor various strain components.

III.1. Strain Hamiltonian

In this section, we describe the strain Hamiltonian ofthe SiV center, and summarize the physical effects of var-ious modes of deformation on the orbital wavefunctions.A more detailed group-theoretic discussion of the resultsin this section is provided in Appendix A and in Ref.30.Based on the symmetries of the orbital wavefunctions, itcan be shown that the effects of strain on the GS (eg)and ES (eu) manifolds are independent and identical inform. For either manifold, the strain Hamiltonian in thebasis of |ex ↓〉, |ex ↑〉, |ey ↓〉, |ey ↑〉 states (pure orbitals

4

unmixed by SO coupling as defined in30) is given by

Hstrain =

[εA1g

− εEgxεEgy

εEgyεA1g

+ εEgx

]⊗ I2 (1)

The spin part of the wavefunction is associated withan identity matrix in Eq. (1) because lattice deforma-tion predominantly perturbs the Coulomb energy of theorbitals, which is independent of the spin character. Eachεr is a linear combination of strain components εij , andcorresponds to specific symmetries indicated by the sub-script r.

εA1g= t⊥(εxx + εyy) + t‖εzz

εEgx= d(εxx − εyy) + fεzx (2)

εEgy = −2dεxy + fεyz

Here t⊥, t‖, d, f are the four strain-susceptibility pa-rameters that completely describe the strain-response ofthe |ex〉, |ey〉 states. These parameters have differentnumerical values in the GS and ES manifolds. From theHamiltonian 1, we see that Egx and Egy strain causemixing and relative shifts between orbitals, and modifythe orbital splittings within the GS and ES manifolds asdepicted in Fig. 3(a). On the other hand, A1g strainleads to a uniform or common-mode shift of the GS andES manifolds, and only shifts the mean ZPL frequencyas depicted in Fig. 3(b).

By decomposing the strain applied in our experimentinto A1g and Eg components, we can confirm the ob-servations on tuning of transverse- and axial-orientationSiVs in Fig. 2. Strain tensors for transverse- andaxial-orientations of emitters obtained from finite ele-ment method (FEM) simulations are plotted in Figs.3(c), (d) respectively. As expected from the cantilevergeometry in Fig. 1(a), transverse-orientation SiVs pre-dominantly experience εyy and hence an Eg deformation.The Eg-strain response predicted in Fig. 3(a) leads tothe strain-tuning of mainly A and D transitions seen inFig. 2(a). On the other hand, axial-orientation SiVs ex-perience both εzz and εyz as shown in Fig. 3(d), whichleads to simultaneous Eg and A1g deformations. Indeed,a combination of the strain responses in Figs. 3(a), (b)qualitatively explains the strain-tuning behavior of thetransitions in Fig. 2(b).

III.2. Estimation of strain-susceptibilities

We now quantitatively fit the results in Fig. 2 withthe above strain response model. Adding SO cou-pling (HSO = −λSOLzSz) to the strain Hamiltonian inEq. 1, we get the following total Hamiltonian in the|ex〉, |ey〉 ⊗ | ↑〉, | ↓〉 basis.30

Htotal =

εA1g− εEgx

0 εEgy− iλSO/2 0

0 εA1g− εEgx

0 εEgy+ iλSO/2

εEgy+ iλSO/2 0 εA1g

+ εEgx0

0 εEgy − iλSO/2 0 εA1g + εEgx

(3)

Here, λSO is the SO coupling strength within eachmanifold - 46 GHz for the GS, and 255 GHz for the ES.Diagonalization of this Hamiltonian gives two distincteigenvalues

E1 = α− 1

2

√λ2SO + 4(ε2Egx

+ ε2Egy)

E2 = α+1

2

√λ2SO + 4(ε2Egx

+ ε2Egy) (4)

Each of these corresponds to doubly spin-degenerateeigenstates in the absence of an external magnetic field.Noting that Eqs. (4) are valid within both GS and ESmanifolds, but with different strain susceptibilities, weobtain the following quantities that can be directly ex-tracted from the optical spectra in Fig. 2.

∆ZPL = ∆ZPL,0 +(t‖,u − t‖,g

)εzz + (t⊥,u − t⊥,g) (εxx + εyy) (5)

∆gs =√λ2SO,g + 4 [dg(εxx − εyy) + fgεyz]

2+ 4 [−2dgεxy + fgεzx]

2(6)

∆es =√λ2SO,u + 4 [du(εxx − εyy) + fuεyz]

2+ 4 [−2duεxy + fuεzx]

2(7)

Here, the subscript g(u) refers to the GS (ES) mani- fold. ∆ZPL is the mean ZPL frequency, and ∆gs, ∆es are

5

the GS and ES orbital splittings respectively. ∆ZPL,0 isthe mean ZPL frequency at zero strain. Extracting allthree frequencies in Eqs. (5-7) as a function of strainfrom the optical spectra measured in Fig. 2, we fit themto the above model in Figs. 3(c), (d), and estimate thestrain-susceptibilities. The fitting procedure described indetail in Appendix B gives us

(t‖,u − t‖,g

)= −1.7 PHz/strain

(t⊥,u − t⊥,g) = 0.078 PHz/strain

dg = 1.3 PHz/strain

du = 1.8 PHz/strain

fg = −0.25 THz/strain

fu = −0.72 THz/strain (8)

We note that these values are subject to errors aris-ing from (i) imprecision in SiV depth from the diamondsurface (10% from SRIM calculations, and in practice,higher due to ion-channeling effects), and (ii) due to thefact that the device geometry cannot be replicated ex-actly in FEM simulations for strain estimation. In par-ticular, the values f and t⊥ are subject to higher error,since the Eg and A1g responses are mostly dominatedby the numerically larger susceptibilities d and t‖ respec-tively.

IV. CONTROLLING ELECTRON-PHONONPROCESSES

At 4 K, dephasing and population relaxation of the SiVspin qubit defined with the |eg+ ↓〉′, |eg− ↑〉′ states (′ de-noting modified SO eigenstates due to strain) is knownto be dominated by electron-phonon processes shown inFig. 4(a)10,13. In accordance with our observations onresponse to static Eg-strain in the previous section, weexpect that AC strain generated by thermal Eg-phononsat frequency ∆gs < kBT/h is capable of driving the GSorbital transitions. Since we can tune the splitting ∆gs byapplying static Eg-strain with our device, we have controlover these electron-phonon processes, and can engineerthe relaxation rates of spin qubit. In particular, by mak-ing ∆gs kBT/h, we have shown that spin coherencecan be improved significantly.33 Here, we elucidate thephysical mechanisms behind such improvement in spinproperties with strain control.

When a thermal phonon randomly excites the SiV cen-ter from the spin qubit manifold to the upper orbitalbranch, say from |eg+ ↓〉′ to |eg− ↓〉′ as shown by theblue upward arrow in Fig. 4(a), the energy of the ↓ pro-jection of the spin qubit suddenly changes by an amounth∆gs. After some time in the upper branch, the sys-tem randomly relaxes back to the lower manifold throughspontaneous emission of a phonon as shown by the bluedownward arrow in Fig. 4(a). In this process, the spinprojection is conserved, since phonons predominantly fliponly the orbital character. However, a random phaseis acquired between the ↓ and ↑ projections of the spin

qubit due to phonon absorption and emission, as well asfaster precession in the upper manifold. The dephasingrate is determined by the upward phonon transition rateγup(∆gs). Both this rate and the downward transitionrate γdown(∆gs) can be calculated from Fermi’s goldenrule and are given by -

γup(∆gs) = 2πχρ∆3gsnth(∆gs) (9)

γdown(∆gs) = 2πχρ∆3gs(nth(∆gs) + 1) (10)

where χ is a constant that encapsulates averaged interac-tion over all phonon modes and polarizations and nth(ν)is the Bose-Einstein distribution. It is instructive to viewthese rates as a product of the phonon density of states(DOS) and the occupation of phonon modes. In theabove expressions, the first part 2πχρ∆3

gs contains the

bulk DOS of phonons, which scales as ∼ ∆2gs. On the

other hand, nth(ν) is the number of thermal phonons ineach mode. Note that the +1 term in the downward ratein Eq. (10) corresponds to spontaneous emission of aphonon, a process that is independent of temperature.

Fig. 4(b) shows the theoretically predicted behaviorof upward and downward rates as a function of ∆gs attemperature T = 4 K. Here, we calculate both transitionrates with corrected exponent in Eqs. (9) and (10), ap-

-7.5 -2.5 2.5 7.5Freq. (MHz)

240260

280

Coun

t ra

te (H

z)

200 250 300 350 400 450

0.1

0.2

0.3

0.4

Spin

rela

xatio

n ra

te, 1

/T1 (M

Hz)

dephasing (T2*)population relaxation (T1)

Δgs

ωseg+↓ ‘〉eg-

‘〉

eg+‘〉

4

300 350 400 450 500

1

2

3

CPT

linew

idth

(MH

z)

Dip 1Dip 2

Phonon-emission

rate, γdow

n (Δgs )

Phon

on-a

bsor

ptio

n ra

te, γ

up(Δ

gs)

0

1

102

0 200 400 60010-1

100

10

10↑

eg-↓ ‘〉↑

GS splitting, Δgs (GHz)

(a) (b)

(c)(d)

GS splitting, Δgs (GHz)

GS splitting, Δgs (GHz)

FIG. 4. (a) Illustration of dephasing and population decayprocesses for spin qubit. Blue arrow shows a spin-conservingtransition responsible for dephasing. Red arrow shows a spin-flipping transition driving decay from |eg+ ↓〉′ to |eg− ↑〉′.Processes suppressed at high strain are crossed out. (b)Calculated rates for spin-conserving upward and downwardphonon processes. Both rates are normalized to their val-ues at zero strain. (c) Reduction in CPT linewidth with in-creasing GS splitting ∆gs. Inset shows an example of a CPTspectrum taken at ∆gs = 460 GHz. The two resonances inthe spectrum are due to the presence of a neighboring nu-clear spin33. Linewidths of both are plotted and indicatedas Dip 1 and Dip 2 in the main plot. (d) Reduction in spinrelaxation rate (1/T1) with increasing GS splitting ∆gs as ex-tracted from pump-probe measurements. Solid line is a fit totheory in Appendix C.

6

proximately 1.9 rather than 3, to take into account thegeometric factor associated with the cantilever33. Weobserve that the upward rate shows a non-monotonic be-havior, approaching its maximum value around h∆gs ∼kBT . In the h∆gs < kBT regime, the increasing DOSterm dominates, and causes γup to increase. However,when h∆gs kBT , thermal occupation of the modesis approximated by Boltzmann distribution nth(∆gs) =

exp(−h∆gs

kBT

), and this exponential roll-off dominates the

polynomially increasing DOS. Therefore, γup decreasesexponentially, when sufficiently high strain is applied.In contrast, the downward rate monotonically increaseswith the GS-splitting, because it is dominated by thespontaneous emission rate, which simply increases poly-nomially with the DOS. Fig. 4(c) shows experimentallymeasured improvement of spin coherence using coherentpopulation trapping (CPT) in this high strain regime33.Above ∆gs of 400 GHz, the dephasing rate saturates,indicating a secondary dephasing mechanism such as the13C nuclear spin bath in diamond. Our data is supportedby similar 1/T ∗2 measured at 100 mK where the thermaloccupation of relevant phonon modes is negligible12.

Population decay or longitudinal relaxation of the spinqubit shown by the red arrows in Fig. 4(a) is drivenby spin-flipping phonon transitions, which occur with asmall probability due to perturbative mixing of spin pro-jections. A detailed analysis of various decay channels ispresented in Appendix C. At high strain, it can be shownthat the decay rate is approximately 4 (dg,flip/dg)

2γup,

where dg,flip is the strain susceptibility for a spin-flippingtransition such as |eg+ ↓〉′ → |eg+ ↑〉′. Thus it is a frac-tion of the spin-conserving transition rate γup shown inEq. 9. The factor dg,flip/dg scales as ∼ 1/∆gs accord-ing to first order perturbation theory. As a result, weexpect exponential decrease in the population decay ratewith a different polynomial pre-factor compared to thespin decoherence rate. Fig. 4(d) shows this decreasingtrend with increasing ∆gs fit to this two-phonon relax-ation model.

V. STRAIN RESPONSE OF SPIN TRANSITION

So far, we have seen that static Eg-strain in the SiVenvironment can significantly impact spin coherence andrelaxation rates by modifying the orbital splitting in theGS. In this section, we discuss additional effects of thistype of strain on the SiV spin qubit that arise from SOcoupling. Particularly, we can tune the spin transitionfrequency, ωs by a large amount (a few GHz) at a fixedexternal magnetic field by simply controlling local strain.At the same time, we discuss how the magnitude of localstrain strongly determines the ability to couple or controlthe SiV spin qubit with external fields such as resonantstrain or microwaves at frequency ωs, and resonant laser-fields in a Λ-scheme.

The strain-response of the spin transition is measuredby monitoring the four Zeeman-split optical lines arisingfrom the C transition as shown schematically in Fig. 5(a).

eg- ↓〉eg+ ↑〉eg+ ↓〉eg-↑〉

eu- ↓〉eu+ ↑〉

eu+ ↓〉eu-↑〉

spin qubit

∆gs

∆es

ωs

C

(a)

C1C2

C3C4

45 55 65 75 85GS splitting, ∆gs (GHz)

-4

-2

0

2

4

Rela

tive

optic

al

tran

sitio

n fr

eque

ncy

(GH

z)

(b)

C1

C2

C3

C4

-4

-2

0

2

4

50 100 150 200 250 300 350 400 450 5002.5

3

3.5

4

4.5

5

Spin

tran

sitio

nfr

eque

ncy

(GH

z)

C1

C2C3

C4

GS spin transition, ωs

ES spin transition, ωs’

Rela

tive

optic

al

(c)

ωs’

B = 0.17 T along [001]

B = 0.17 T along [001]

spin-orbitregime

high strainregime

GS splitting, ∆gs (GHz)

tran

sitio

n fr

eque

ncy

(GH

z)

(d)

FIG. 5. (a) Splitting of the C transition into the four transi-tions C1, C2, C3, and C4 in the presence of a magnetic field.Spin transition frequencies on the lower orbital branches of theGS and ES are ωs, ω′s respectively. (b) Response of transitionsC1, C2, C3, and C4 upon tuning GS splitting ∆gs with Eg-strain. (c) Calculated response of optical transitions C1, C2,C3, and C4 to Eg-strain in presence of 0.17 T B-field alignedalong the [001] direction. Shaded regions on the left and rightends indicate the regimes in which the GS orbitals are deter-mined by SO coupling and strain respectively. (d) Strainresponse of spin transition frequencies upon tuning of groundstate orbital splitting ∆gs with Eg-strain. SO regime datapoints are extracted from the optical spectra in Fig. 5(b).High strain regime data points are obtained from CPT mea-surements on the SiV studied in Fig. 4. Solid (dashed) lineis calculated spin transition frequency on the lower orbitalbranch of GS (ES) from Fig. 5(c).

In Fig. 5(b), we apply a fixed magnetic field B =0.17T aligned along the vertical [001] axis with a perma-nent magnet placed underneath the sample, and gradu-ally increase the GS splitting of a transverse-orientationSiV by applying strain. With increasing strain, each ofthe four Zeeman-split optical transitions moves outwardsfrom the position of the unsplit C transition at zero mag-netic field. In particular, the spin-conserving inner tran-sitions C2 and C3 overlap at zero strain, but becomemore resolvable with increasing strain. Thus, all-optical

7

control of the spin11 relying on simultaneous excitationof a pair of transitions C1 and C3 (or C2 and C4) form-ing a Λ-scheme requires the presence of some local strain.The strain-tuning behavior of Zeeman split optical transi-tions can be theoretically calculated by diagonalizing the

GS and ES Hamiltonians in the presence of a magneticfield. Upon adding Zeeman terms to the Hamiltonian inequation 3, and switching to the basis of SO eigenstateseg− ↓, eg+ ↑, eg+ ↓, eg− ↑, we obtain

Htotal =

−λSO/2− γLBz − γsBz 0 εEgxγsBx

0 −λSO/2 + γLBz + γsBz γsBx εEgx

εEgxγsBx λSO/2 + γLBz − γsBz 0

γsBx εEgx0 λSO/2− γLBz + γsBz

(11)

Here we have discarded the A1g and Egy strain terms,since the transverse-orientation SiVs in our experimentsexperience predominantly Egx strain. We have also as-sumed that the transverse magnetic field is entirely alongthe x-axis of the SiV. The gyromagnetic ratios are γs= 14 GHz/T, γL = 0.1(14) GHz/T, where the pre-factor of 0.1 is a quenching factor for the orbital angularmomentum.30 The result of our calculation is shown inFig. 5(c). In the low strain regime indicated by the regionwith the shaded gradient, we reproduce the experimen-tal behavior in Fig. 5(b), and obtain good quantitativeagreement with the variation in the spin transition fre-quency ωs in Fig. 5(d).

Physically, this behavior of the spin transitions arisesas strain and SO coupling compete to determine the or-bital wavefunctions. From the Hamiltonian in equation11, we can see that the orbitals begin as SO eigenstateseg− ↓, eg+ ↑, eg+ ↓, eg− ↑ at zero strain, and end up asthe pure states egx ↓, egx ↑, egy ↓, egy ↑ at high strain(εEgx

λSO/2). At zero strain, the effective magneticfield from SO coupling quantizes the electron spin alongthe z−axis. In this condition, the off-axis B-field doesnot affect the spin transition frequency ωs to first order,so ωs ∼ 2(γs + γL)Bz = 3.1 GHz. As the strain εEgx

is increased far above the SO coupling λSO and the or-bitals are purified, the spin quantization axis approachesthe direction of the external magnetic field, and ωs ap-proaches 2γsB = 4.8 GHz. Since SO coupling in the ESis stronger, this limit is attained at higher values of strainthan in the GS as shown by the dashed line in Fig. 5(d).Once the orbitals in both the GS and ES are predomi-nantly dictated by local strain and SO coupling is merelyperturbative, the difference in GS and ES spin transitionfrequencies becomes vanishingly small, eventually lead-ing to converging C2 and C3 optical transitions as de-picted on the right hand side of Fig. 5(c). In the limit ofvery high strain, the transitions C2 and C3 also becomestrictly spin-conserving, and optical polarization10,42 andreadout of the spin qubit will be forbidden.

The rapid variation of the spin transition frequencyωs in the low-strain regime of Fig. 5(d) provides thefirst hint that the SiV spin-qubit can be very sensitiveto oscillating strain generated by coherent phonons. Theinteraction terms due to strain and the off-axis magneticfield predicted by the Hamiltonian in equation 11 are

depicted visually in Fig. 6(a). In particular, at zerostrain, the presence of the off-axis magnetic field perturbsthe eigenstates of the spin qubit to first order as

|eg− ↓〉′ ≈ |eg− ↓〉+γsBx

λSO|eg− ↑〉 (12)

|eg+ ↑〉′ ≈ |eg+ ↑〉+γsBx

λSO|eg+ ↓〉 (13)

This perturbative mixing with opposite spin-charactercan now allow resonant AC strain at frequency ωs todrive the spin qubit. For a small amplitude of such ACstrain εAC

Egx, we can calculate the strain susceptibility of

the spin transition dspin in terms of the GS orbital strainsusceptibility dg in Eq. 8.

dspin =〈eg− ↓′ |Hstrain|eg+ ↑′〉

εACEgx

dg =2γsBx

λSOdg (14)

Since dg is very large (∼1 PHz/strain), even with thepresence of the pre-factor γsBx/λSO, the spin qubit canhave a relatively large strain-response. For the presentcase of B=0.17 T along the [001] axis, we get dspin/d⊥ =0.085 yielding dspin ∼ 100 THz/strain. An exact calcu-lation of dspin for arbitrary local static strain using theHamiltonian in equation 11 is shown in Fig. 6(b). Asstatic strain in the SiV environment is increased far abovethe SO coupling, the AC strain susceptibility approacheszero. Thus we can conclude that coupling the SiV spinqubit to resonant AC strain requires (i) low static strainεEg λSO/2 and (ii) a non-zero off-axis magnetic fieldBx. The spin qubit can also parametrically couple to off-resonant AC strain with a different susceptibility tspin,and this is discussed in Appendix D. A similar analysispredicts the response of the spin qubit to resonant mi-crowave magnetic fields in Appendix E.

VI. PROSPECTS FOR A COHERENTSPIN-PHONON INTERFACE

Our results on the strain response of the electronic andspin levels of the SiV indicate the potential of this colorcenter as a spin-phonon interface. The diamond NV cen-ter spin, the most investigated candidate in this direc-tion has an intrinsically weak strain susceptibility (∼ 10

8

eg- ↓〉eg+ ↑〉

eg+↓〉eg-↑〉

∆gs

strain

eg- ↓〉eg+ ↑〉

eg+ ↓〉eg-↑〉

0 1 2 3 4Static Eg-strain (x10 )-5

0

0.02

0.04

0.06

0.08

0.1

Spin

str

ain

susc

eptib

ility

d sp

in(u

nits

of d

g)

50 60 70 80 90GS orbital splitting Δ gs (GHz) (a) (b)

B = 0.17 Talong [001]

5 um

(c)

730

nm1020 nm

strain

g-↓〉e

eg+‘〉‘

↑

ωs

ACεEgx ωs

FIG. 6. (a) Illustration of mixing terms introduced by Eg-strain and an off-axis magnetic field in the GS manifold. (b)Calculated susceptibility of the spin-qubit for interaction withAC Eg-strain resonant with the transition frequency ωs (in-teraction shown in inset). This AC strain susceptibility ismaximum at zero strain for the pure SO eigenstates. At highstrain, it falls off as 1/∆gs. Color variation along the curveshows the GS splitting ∆gs corresponding to the value of staticEg-strain at the SiV. Both the static and AC strain are as-sumed to be entirely in the β component. (c) SEM image ofan OMC nanobeam cavity43 along with an FEM simulationof its 5 GHz flapping resonance. Displacement profile anda cross-sectional strain profile of the mode are shown witharbitrary normalization.

GHz/strain) since the qubit levels are defined within thesame orbital in the GS configuration of the defect44.While using distinct orbitals in the ES can provide muchlarger strain susceptibility (∼ 1 PHz/strain)45,46, suchschemes will be limited by fast dephasing due to spon-taneous emission and spectral diffusion. In comparison,the SiV center provides distinct orbital branches withinthe GS itself. Further, the presence of SO coupling dic-tates that the spin qubit levels |eg− ↓〉, |eg+ ↑〉 correspondto different orbitals. As a result, one achieves the idealcombination of high strain susceptibility and low qubitdephasing rate.

The effects of various modes of strain and the richelectronic structure of the SiV allow a variety of spin-phonon coupling schemes. In this letter, we focus ondirect coupling of the spin transition to a mechanical res-onator at frequency ωs enabled by Eg-strain response ofthe spin discussed in the previous section. An alterna-tive approach utilizing propagating phonons of frequency∼ λSO coupled to the GS orbital transition is discussedelsewhere47. Our scheme would require diamond me-chanical resonators of frequency ωs ∼ few GHz, whichhave already been realized in both optomechanical43,48

and electromechanical platforms21–23,49. Fig. 6(c) shows

the strain profile resulting from GHz frequency mechan-ical modes in an optomechanical crystal cavity. Sincethis structure achieves three-dimensional confinement ofphonons on the scale of the acoustic wavelength, it pro-vides large per-phonon strain. For an SiV located ∼20nm below the top surface, when a magnetic field B = 0.3T is applied along the [001] direction, the spin qubit isresonant with the 5 GHz flapping mode, and has a single-phonon coupling rate g ∼0.8 MHz. At mK temperatures,given the low SiV spin dephasing rate γs ∼ 100 Hz12, evenmodest mechanical quality-factors Qm ∼ 103 measuredpreviously43 are sufficient to achieve strong spin-phononcoupling. At 4 K, despite the higher spin dephasing rateγs ∼ 4 MHz50,51 and thermal occupation of mechanicalmodes nth ∼ 20, high spin-phonon co-operativity canbe achieved if previously observed 4 K quality factorsfor silicon OMCs52, Qm ∼ 105 can be replicated in di-amond. This form of spin-phonon coupling can also beimplemented in other resonator designs such as surfaceacoustic wave cavities22,23,53, wherein piezoelectric ma-terials are used to transduce the mechanical motion withmicrowave electrical signals instead of optical fields.

VII. CONCLUSION

In conclusion, we characterize the strain responseof the SiV center in diamond with a NEMS device.The implications of our results are two-fold. First,the large tuning range of optical transitions we havedemonstrated establishes strain control as a technique toachieve spectrally identical emitters in a quantum net-work. Strain tuning is particularly relevant here sinceinversion-symmetric centers with superior optical prop-erties do not have a first order electric field response,thereby negating the feasibility of direct electrical tun-ing. Second, the intrinsic sensitivity of the SiV spin qubitto strain makes it a promising candidate for coherentspin-phonon coupling. This can enable phonon-mediatedquantum information processing with spins14,15. Thedevelopment of such a cavity QED platform with aphononic two-level system29,54 will also allow determin-istic quantum nonlinearities for phonons55, thereby over-coming inefficiencies in probabilistic schemes used to gen-erate single phonon states in cavity optomechanics56,57.Further, the use of optomechanical and electromechani-cal resonators towards this goal suggests the possibility ofcoherently interfacing diamond spin qubits with telecomand microwave photons respectively.

Appendix A: Group theoretical description of strainresponse

The response of the electronic levels of trigonal point-defects in cubic crystals to lattice deformations wastreated theoretically by Hughs and Runciman40. A solu-tion of this problem for the specific case of the SiV hasbeen previously carried out using group theory30 with

9

some errors. Here, we reconcile these two treatments,and present a model for the response of the SiV elec-tronic levels to strain (and stress). In what follows, weuse x, y, z to refer to the internal basis of the SiV (seeinset of Fig. 1(b). eg. for a [111] oriented SiV, we havex : [112], y : [110], z : [111]), and X,Y, Z to refer to theaxes of the diamond crystal, i.e. X : [100], Y : [010], Z :[001]. We use σ and ε for the stress and strain tensors inthe SiV basis, and σ and ε to refer to them in the crystalbasis. We also neglect the spin character of the states in-volved, since we are only concerned with changes to theCoulomb energy of the orbitals.

When the applied stress is small, in the Born-Oppenheimer approximation, the effect of lattice defor-mation is linear in the strain components and is capturedby a Hamiltonian of the form40 -

Hstrain =∑ij

Aijεij (A1)

Here i, j are indices for the co-ordinate axes. Vij areoperators corresponding to particular stress components,and act on the SiV electronic levels. Group theory canbe used to rewrite this Hamiltonian in terms of basis-independent linear combinations of strain componentsadapted to the symmetries of the SiV center. Each ofthese combinations can be viewed as a particular ‘mode’of deformation, and the effect of each mode on the orbitalwavefunctions, each with its own symmetries can be de-duced using group theory. More technically, such defor-mation modes are obtained by projecting the strain ten-sor onto the irreducible representations of D3d, the pointgroup of the SiV center.40 This transformation gives

Hstrain =∑r

Vrεr (A2)

where r runs over the irreducible representations. De-ducing the operators Vr simply requires computing thedirect products of irreducible representations.30 It can beshown that strain and stress tensors transform as the irre-ducible representation,A1g +Eg

30 which has even parityabout the inversion center of the SiV. Since the groundstates of the SiV transform as Eg (even), and the excitedstates transform as Eu (odd), lattice deformations do notcouple the ground and excited states with each other tofirst order. As a result, we can describe the response ofthe ground and excited state manifolds independently.

In particular, Hstrain is identical in form for both mani-folds, but will involve different numerical values of strain-response coefficients. Therefore, we drop the subscripts gand u used to refer to the ground and excited states, andsimply work in the doubly-degenerate basis |ex〉, |ey〉.The interaction Hamiltonian can be shown to comprisethree deformation modes -

Hstrain = α

[1 00 1

]+ β

[−1 00 1

]+ γ

[0 11 0

](A3)

The components α, β, γ corresponding to εr in Eq.A2 aregiven by the following linear combinations40

α = A1(εXX + εY Y + εZZ) + 2A2(εY Z + εZX + εXY )

β = B(2εZZ − εXX − εY Y ) + C (2εXY − εY Z − εZX)

γ =√

3B(εXX − εY Y ) +√

3C (εY Z − εZX)

The coefficients A1,A2,B,C completely determine thestrain-response of the |ex〉, |ey〉 manifold. It can beshown that α transforms as A1g, and β, γ transform asEgx, Egy.

To gain more physical intuition for these three defor-mation modes, we can write α, β, γ in the SiV basis us-ing the unitary transformation R = Rz(45)Ry(54.7),where Rz(θ), and Rx(φ) correspond to rotations by θand φ about the z- and x-axes respectively. Upon trans-formation, we get

α = t⊥(εxx + εyy) + t‖εzz ≡ εA1g

β = d(εxx − εyy) + fεzx ≡ εEgx(A4)

γ = −2dεxy + fεyz ≡ εEgy

Here t⊥, t‖, d, f are the four strain-susceptibility pa-rameters. They are related to the original stress-responsecoefficients of Hughs and Runciman40 according to theexpressions in Table I. Further, to explicitly indicatethe symmetries of these deformation modes, we hereafterswitch to the notation εA1g for α, εEgx for β, and εEgy

for γ in line with the description in Eq. (A2).At this juncture, we contrast Eqs. (A4) with the re-

cent results in Ref.30 (Eqs. 2.80-2.82). Our analysis pre-dicts a non-zero response to uniaxial strain along the highsymmetry axis εzz in A1g deformation, and to the shearstrains εzx and εyz in Eg deformations.

Appendix B: Extraction of strain susceptibilities

To extract all the values t⊥, t‖, d, f for both groundand excited state manifolds, in principle, strain needsto be applied at least in three different directions for agiven SiV. This procedure gives a set of overdetermined

equations in these parameters.40 However, the devicesin this study can only induce two types of strain profilesas shown in Fig. 3(c) and (d). In particular, for a givenSiV in either the ‘axial’ or the ‘transverse’ class, therelative ratio between strain-tensor components remainsconstant, when voltage applied to the cantilever is swept.This condition makes it difficult to estimate the relative

10

Strain term Susceptibility Relation to Hughes-Runciman coefficients

εxx + εyy t⊥ (c11 + 2c12)A1 − c44A2

εzz t‖ (c11 + 2c12)A1 + 2c44A2

εxx − εyy d (c11 − c12)B + c44C

εxy −2d

εzx f√

2 (c44C − 2(c11 − c12)B)

εyz f

TABLE I. Various strain-modes, and their susceptibilities in terms of the Hughs-Runciman stress-response coefficients40. Theconstants cij are the elastic modulus components of diamond.

contributions of t‖ and t⊥ to εA1g , and of d and f to εEg .

To get around this issue, we follow an approximate ap-proach. From Fig. 3(d), we observe that in the case ofan axial SiV, εzz (εxx + εyy) is always true. There-fore, we can use the response of the axial SiV in Fig.2(b) to approximately estimate

(t‖,u − t‖,g

)by neglect-

ing (εxx + εyy) in Eq. (5). Fig. 3(f) plots the mean ZPLfrequency of the axial SiV in Fig. 2(b) vs. εzz estimatedfrom FEM simulation. The slope of the linear fit yields(t‖,u − t‖,g

).

(t‖,u − t‖,g

)= −1.7 PHz/strain (B1)

Likewise, in the case of the transverse SiV in Fig. 3(c),we can conclude that (εxx − εyy) maxεzx, εyz. Withthis class of SiVs, we can approximately estimate dg, duby neglecting εzx, εyz in Eqs. (6,7). Fig. 3(e) plotsthe GS and ES splittings of the transverse SiV in Fig.

2(a) vs. ε⊥ =√

(εxx − εyy)2

+ 4ε2xy estimated from FEM

simulation. Fitting yields

dg = 1.3, du = 1.8 PHz/strain (B2)

Once we extract(t‖,u − t‖,g

)from an axial SiV, we

can use this value to further extract (t⊥,u − t⊥,g) by fit-ting Eq. (5) to the tuning behavior of the mean ZPLfrequency of the transverse SiV. This procedure yields

(t⊥,u − t⊥,g) = 78 THz/strain (B3)

We immediately note that(t‖,u − t‖,g

)is more than

an order of magnitude larger than (t⊥,u − t⊥,g). Thisimplies that εzz tunes the mean ZPL frequency muchmore effectively than (εxx + εyy). This can be intuitivelyexplained by examining the spatial profile of the GS andES orbitals (Table 2.7 of Ref.30). Since the GS and EScorrespond to even (g) and odd (u) eigenstates of SiV’sD3d point symmetry group respectively, the charge den-sity distributions of the orbitals egx, eux (and egy, euy)are similar in any transverse plane normal to the z-axis.As a result, we would expect that the common mode en-ergy shift resulting from the strain-mode εxx+εyy is verysimilar for the GS and ES manifolds, i.e. t⊥,u ≈ t⊥,g.

On the other hand, the energy shift from εzz is expectedto have opposite signs for the GS and ES manifoldsdue to the change in wavefunction parity along the z-axis.

As the last step, we extract the values fg, fu in Eqs.(6,7). We observe from table I that knowledge of d and Bcan allow us to determine f . The Hughs-Runciman co-efficients Bg=484 GHz/GPa and Bu=630 GHz/GPa canbe extracted based on uniaxial stress measurements car-ried out in Ref.30,34. Combining our estimates of dg anddu with this information, we predict

fg = −250, fu = −720 THz/strain (B4)

Appendix C: Spin relaxation (T1) model

∆gs

ν1 ν2

(a)

(b)

(c)

eg-↓ ‘〉eg+

‘〉

eg-‘〉

↑

eg+↓ ‘〉↑

∆gs ∆gs

∆gs

ν1 ν2

ν1 ν2

eg-↓ ‘〉eg+

‘〉

eg-‘〉

↑

eg+↓ ‘〉↑

eg-↓ ‘〉eg+

‘〉

eg-‘〉

↑

eg+↓↑

‘〉

eg-↓ ‘〉eg+

‘〉

eg-‘〉

↑

eg+↓↑

‘〉

FIG. 7. Various pathways for a phonon-mediated spin-flip(a) Direct relaxation via a single phonon resonant with the

|eg− ↓〉′ → |e↑g+〉′ spin-transition. (b) Two possible channelsfor a resonant two-phonon process involving the upper orbitalbranch. (c) Off-resonant two-phonon processes.

Eg-phonons predominantly drive spin-conserving tran-sitions between the GS orbitals of the SiV i.e. between|eg− ↓〉′, |eg+ ↓〉′, and |eg+ ↑〉′, |eg− ↑〉′ respectively.However, in the presence of an off-axis magnetic-field,

11

SpinRelaxation4K.pdf

FIG. 8. Rates of all three spin-relaxation mechanisms indi-cating their magnitudes and scaling with strain β.

and non-zero static strain, the eigenstates of the GS man-ifold are no longer pure SO or strain eigenstates, and alltransitions between the four states within the GS man-ifold become allowed for Eg-phonons. In this scenario,the various channels for spin-relaxation from |eg− ↓〉′ to|eg+ ↑〉′ are:

Direct single-phonon relaxation: Via a singlephonon of frequency ωs resonant with the spin-transition as shown in Fig. 7(a)

Resonant two-phonon relaxation: Via two phononsresonant with a level in the upper orbital branchas an intermediate state as shown in Fig. 7(b).The spin-flip can be caused by either the emittedphonon (left) or the absorbed phonon (right).

Off-resonant two-phonon relaxation: Via twophonons with a virtual level as an intermediatestate as shown in Fig. 7(c). The effective driv-ing strength will be reduced from its value in theresonant process by an amount corresponding tothe detuning from the upper orbital branch.

Using Fermi’s golden rule, the transition rates for theserelaxation channels can be calculated. The results aresummarized in Table II, and are plotted versus GS split-ting ∆gs in Fig. 8.

We see that spin relaxation at 4 K is dominated bya two-phonon process involving the upper ground stateorbital branches as intermediate states. In literature,this is frequently referred to as an Orbach process.58

The experimentally observed behavior of spin T1 in Fig.4(d) of the main text is well-explained by the scalingof such a process with the GS splitting ∆gs shown in

0 1 2 3 4-0.06

-0.04

-0.02

0

Static Eg strain (x 10-5)

Spin

str

ain

susc

eptib

ility

t sp

in (u

nits

of d

g)

50 60 70 80 90

GS orbital splitting ∆gs (GHz)

B = 0.17 Talong [001]

ωs

eg+↓〉

eg- ↑〉‘

‘

ACεEgx

FIG. 9. Calculated susceptibility of the spin-qubit for interac-tion with off-resonant AC Eg-strain that modulates the tran-sition frequency ωs (interaction shown in inset). Color varia-tion along the curve shows the GS splitting corresponding tothe value of static Eg-strain at the SiV. Both the DC and ACstrain are assumed to be entirely in the Egx-component.

Table II. Intuitively, we may understand the dominanceof the Orbach process in terms of the phonon DOS∝ ∆nexp (−h∆/kBT ) being maximized around the fre-quency ∆ ∼ kBT/h. We can similarly argue that thesingle and off-resonant two-phonon channels become rel-evant in other temperature regimes indicated in TableII, where the phonon DOS is maximized in a frequencyrange relevant for those processes.

Appendix D: Dispersive strain-coupling to spin qubit

From Eqs. (12, 13), we concluded that in the the lowstrain limit, the eigenstates of the SiV spin qubit |eg− ↓〉′,|eg+ ↑〉′ are linearly mixed by Eg-strain, and hence suit-able for resonant driving by AC strain at frequency ωs.This type of mixing also indicates that static Eg-strainwould cause a quadratic shift in the spin-transition fre-quency ωs. Such a quadratic response to an external fieldcan always generate a linear AC response in the presenceof a ‘bias’ field. Thus in the presence of non-zero staticEg-strain, ωs must also experience a linear modulationwith off-resonant AC strain. This is particularly usefulfor parametric coupling of the spin qubit to off-resonantmechanical resonators as demonstrated previously withNV centers18–20,24. A calculation of the magnitude ofmodulation in the spin transition frequency for a givenAC strain βAC yields the susceptibility tspin for disper-sive spin-phonon coupling, which can be of the same or-der of magnitude as dspin.

tspin =〈eg+ ↑′ |HAC

str |eg+ ↑′〉 − 〈eg− ↓′ |HACstr |eg− ↓′〉

βACdg

(D1)

12

Mechanism Rate Relevant regime Expected scaling of rate

Single-phonon 2π(

dspin

dg

)2

χρω3snth(ωs) kBT/h ωs B2

⊥∆−2gs ω

3sexp(−hωs/kBT )

Resonant two-phonon 4(

dg,flipdg

)2

γup kBT/h ∼ ∆gs B2⊥∆gs[exp(h∆gs/kBT )− 1]−1

Off-resonant two-phonon 8π3(

dg,flipdg

)2

χ2ρ2ω2s

(kBTh

)3

kBT/h ∆gs B2⊥∆−2

gs ω2sT

3

TABLE II. Summary of spin-relaxation mechanisms

0 1 2 3 40

0.5

1

1.5

2

2.5

g-fa

ctor

for

ωs

eg+↓

eg- ↑〉‘

‘Bx

AC

〉

B = 0.17 Talong [001]

50 60 70 80 90

GS orbital splitting ∆gs (GHz)

Static Eg strain (x 10-5)

FIG. 10. g-factor for a transverse microwave magnetic fieldresonantly driving the spin qubit (interaction shown in inset)calculated as a function of pre-existing static Eg-strain plot-ted along x− axis. Color variation along the curve shows theGS splitting corresponding to the value of static Eg-strainat the SiV. Both the DC and AC strain are assumed to beentirely in the Egx-component.

tspin is calculated as a function of pre-existing staticEg-strain, and plotted in Fig. 9. Its magnitude is maxi-mized at a moderately strained GS splitting of 50 GHz,and falls off as static strain is further increased. Thisnon-monotonic behavior arises from the fact that tspinis a result of linearizing the quadratic response due todspin, and therefore scales as the product of dspin andstatic strain in the environment. Thus there is an opti-mal static strain condition to maximize tspin.

Appendix E: Microwave magnetic response of spinqubit

At zero strain, the spin qubit cannot be driven by mi-crowave magnetic fields at frequency ωs. This is becausea magnetic field cannot flip the orbital character of the

pure SO eigenstates |eg− ↓〉, |eg+ ↑〉 that comprise thespin qubit as evinced by the Hamiltonian 11. However,just as a transverse B-field allows a strain susceptibilityfor the spin qubit as shown by Eqs. (12-14), we can arguethat the presence of strain induces a non-zero responseto transverse B-fields. Fig. 10 shows a calculation ofthe effective g-factor for transverse B-field BAC

x , whichdetermines the Rabi frequency ΩMW = gxµBB

ACx for

microwave control10 of the SiV spin. As expected, the g-factor approaches close to that of a free electron at highstrain, when the SO coupling can be neglected.

ACKNOWLEDGMENTS

This work was supported by STC Center for IntegratedQuantum Materials (NSF Grant No. DMR-1231319),ONR MURI on Quantum Optomechanics (Award No.N00014-15-1-2761), NSF EFRI ACQUIRE (Award No.5710004174), the University of Cambridge, the ERCConsolidator Grant PHOENICS, the EPSRC QuantumTechnology Hub NQIT (EP/M013243/1), and the MIT-Harvard CUA. B.P. thanks Wolfson College (Universityof Cambridge) for support through a research fellowship.Device fabrication was performed in part at the Cen-ter for Nanoscale Systems (CNS), a member of the Na-tional Nanotechnology Infrastructure Network (NNIN),which is supported by the National Science Foundationunder NSF award no. ECS-0335765. CNS is part ofHarvard University. Focused ion beam implantation wasperformed under the Laboratory Directed Research andDevelopment Program at the Center for Integrated Nan-otechnologies, an Office of Science User Facility operatedfor the U.S. Department of Energy (DOE) Office of Sci-ence. Sandia National Laboratories is a multi-missionlaboratory managed and operated by National Technol-ogy and Engineering Solutions of Sandia, LLC., a whollyowned subsidiary of Honeywell International, Inc., forthe U.S. Department of Energy’s National Nuclear Se-curity Administration under contract DE-NA-0003525.We thank D. Perry for performing the focused ion beamimplantation, and M. W. Doherty for helpful discussions.

1 H. J. Kimble, Nature 453, 1023 (2008).2 H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok,

L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen,L. Childress, and R. Hanson, Nature 497, 86 (2013).

3 A. Delteil, Z. Sun, W.-b. Gao, E. Togan, S. Faelt, andA. Imamoglu, Nature Physics 12, 218 (2016).

4 R. Stockill, M. J. Stanley, L. Huthmacher, E. Clarke,M. Hugues, A. J. Miller, C. Matthiesen, C. Le Gall, and

13

M. Atature, Phys. Rev. Lett. 119, 010503 (2017).5 A. Gali and J. R. Maze, Physical Review B 88, 235205

(2013).6 T. Muller, C. Hepp, B. Pingault, E. Neu, S. Gsell,

M. Schreck, H. Sternschulte, D. Steinmuller-Nethl,C. Becher, and M. Atature, Nature Communications 5(2014), 10.1038/ncomms4328.

7 A. Sipahigil, K. Jahnke, L. Rogers, T. Teraji, J. Isoya,A. Zibrov, F. Jelezko, and M. Lukin, Physical ReviewLetters 113 (2014), 10.1103/PhysRevLett.113.113602.

8 A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Bu-rek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L.Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho,F. Jelezko, E. Bielejec, H. Park, M. Loncar, and M. D.Lukin, Science 354, 847 (2016).

9 M. J. Burek, C. Meuwly, R. E. Evans, M. K. Bhaskar,A. Sipahigil, S. Meesala, B. Machielse, D. D. Sukachev,C. T. Nguyen, J. L. Pacheco, E. Bielejec, M. D. Lukin,and M. Loncar, Physical Review Applied 8 (2017),10.1103/PhysRevApplied.8.024026.

10 B. Pingault, D.-D. Jarausch, C. Hepp, L. Klintberg, J. N.Becker, M. Markham, C. Becher, and M. Atature, NatureCommunications 8, 15579 (2017).

11 J. N. Becker, B. Pingault, D. Groß, M. Gundogan,N. Kukharchyk, M. Markham, A. Edmonds,M. Atature, P. Bushev, and C. Becher, arXiv preprintarXiv:1708.08263 (2017).

12 D. D. Sukachev, A. Sipahigil, C. T. Nguyen, M. K.Bhaskar, R. E. Evans, F. Jelezko, and M. D. Lukin, Phys.Rev. Lett. 119, 223602 (2017).

13 K. D. Jahnke, A. Sipahigil, J. M. Binder, M. W. Doherty,M. Metsch, L. J. Rogers, N. B. Manson, M. D. Lukin, andF. Jelezko, New Journal of Physics 17, 043011 (2015).

14 M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, andP. Zoller, Physica Scripta T137, 014001 (2009).

15 P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris,P. Zoller, and M. D. Lukin, Nature Physics 6, 602 (2010).

16 O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent,and S. Seidelin, Nature Physics 7, 879 (2011).

17 S. Kolkowitz, A. C. Bleszynski Jayich, Q. P. Unterreith-meier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D.Lukin, Science 335, 1603 (2012).

18 P. Ovartchaiyapong, K. W. Lee, B. A. Myers, andA. C. B. Jayich, Nature Communications 5 (2014),10.1038/ncomms5429.

19 J. Teissier, A. Barfuss, P. Appel, E. Neu, andP. Maletinsky, Phys. Rev. Lett. 113, 020503 (2014).

20 A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, andP. Maletinsky, Nature Physics 11, 820 (2015).

21 E. R. MacQuarrie, T. A. Gosavi, A. M. Moehle, N. R.Jungwirth, S. A. Bhave, and G. D. Fuchs, Optica 2, 233(2015).

22 D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart,and H. Wang, Physical Review Letters 116 (2016),10.1103/PhysRevLett.116.143602.

23 D. A. Golter, T. Oo, M. Amezcua, I. Lekavicius, K. A.Stewart, and H. Wang, Phys. Rev. X 6, 041060 (2016).

24 S. Meesala, Y.-I. Sohn, H. A. Atikian, S. Kim, M. J. Burek,J. T. Choy, and M. Loncar, Physical Review Applied 5(2016), 10.1103/PhysRevApplied.5.034010.

25 J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).26 A. Sørensen and K. Mølmer, Phys. Rev. Lett. 82, 1971

(1999).27 D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett,

J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, and

T. Rosenband, Nature 422, 412 (2003).28 M. Schuetz, E. Kessler, G. Giedke, L. Vandersypen,

M. Lukin, and J. Cirac, Physical Review X 5 (2015),10.1103/PhysRevX.5.031031.

29 T. Aref, P. Delsing, M. K. Ekstrom, A. F. Kockum, M. V.Gustafsson, J. Goran, P. J. Leek, E. Magnusson, andR. Manenti, “Quantum acoustics with surface acousticwaves,” in Superconducting Devices in Quantum Optics,edited by H. R. and J. G. (Springer, Cham, 2016) pp. 217–244.

30 C. Hepp, Electronic structure of the silicon vacancy colorcenter in diamond, Ph.D. thesis, Universitat des Saarlan-des (2014).

31 H. A. Atikian, P. Latawiec, M. J. Burek, Y.-I. Sohn,S. Meesala, N. Gravel, A. B. Kouki, and M. Loncar, APLPhotonics 2, 051301 (2017).

32 M. J. Burek, N. P. de Leon, B. J. Shields, B. J. M. Haus-mann, Y. Chu, Q. Quan, A. S. Zibrov, H. Park, M. D.Lukin, and M. Loncar, Nano Letters 12, 6084 (2012).

33 Y.-I. Sohn, S. Meesala, B. Pingault, A. H. A., J. Holzgrafe,M. Gundogan, C. Stavrakas, M. J. Stanley, A. Sipahigil,J. Choi, M. Zhang, J. L. Pacheco, J. Abraham, E. Bielejec,M. D. Lukin, M. Atature, and M. Loncar, arXiv preprintarXiv:1706.03881 (2017).

34 H. Sternschulte, K. Thonke, R. Sauer, P. Munzinger, andP. Michler, Physical Review B 50, 14554 (1994).

35 R. E. Evans, A. Sipahigil, D. D. Sukachev, A. S. Zi-brov, and M. D. Lukin, Physical Review Applied 5 (2016),10.1103/PhysRevApplied.5.044010.

36 M. K. Bhaskar, D. D. Sukachev, A. Sipahigil, R. E. Evans,M. J. Burek, C. T. Nguyen, L. J. Rogers, P. Siyushev,M. H. Metsch, H. Park, F. Jelezko, M. Loncar, and M. D.Lukin, Phys. Rev. Lett. 118, 223603 (2017).

37 L. J. Zhang, S. Sun, M. J. Burek, C. Dory, Y.-K. Tzeng,K. A. Fischer, Y. Kelaita, K. G. Lagoudakis, M. Radu-laski, Z.-X. Shen, N. A. Melosh, S. Chu, M. Loncar, andJ. Vuckovic, arXiv preprint arXiv:1708.05771 (2017).

38 M. J. Burek, Y. Chu, M. S. Z. Liddy, P. Patel, J. Rochman,S. Meesala, W. Hong, Q. Quan, M. D. Lukin, and M. lon-car, Nature Communications 5, 5718 (2014).

39 V. M. Acosta, C. Santori, A. Faraon, Z. Huang, K.-M. C.Fu, A. Stacey, D. A. Simpson, K. Ganesan, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and R. G. Beau-soleil, Physical Review Letters 108 (2012), 10.1103/Phys-RevLett.108.206401.

40 A. E. Hughes and W. A. Runciman, Proceedings of thePhysical Society 90, 827 (1967).

41 J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov,E. Kaxiras, and M. D. Lukin, New Journal of Physics13, 025025 (2011).

42 L. J. Rogers, K. D. Jahnke, M. W. Doherty, A. Diet-rich, L. P. McGuinness, C. Muller, T. Teraji, H. Sumiya,J. Isoya, N. B. Manson, and F. Jelezko, Physical ReviewB 89 (2014), 10.1103/PhysRevB.89.235101.

43 M. J. Burek, J. D. Cohen, S. M. Meenehan, N. El-Sawah,C. Chia, T. Ruelle, S. Meesala, J. Rochman, H. A. Atikian,M. Markham, D. J. Twitchen, M. D. Lukin, O. Painter,and M. Loncar, Optica 3, 1404 (2016).

44 M. S. Barson, P. Peddibhotla, P. Ovartchaiyapong,K. Ganesan, R. L. Taylor, M. Gebert, Z. Mielens,B. Koslowski, D. A. Simpson, L. P. McGuinness, et al.,Nano Letters 17, 1496 (2017).

45 G. Davies and M. Hamer, in Proceedings of the Royal Soci-ety of London A: Mathematical, Physical and EngineeringSciences, Vol. 348 (The Royal Society, 1976) pp. 285–298.

14

46 K. W. Lee, D. Lee, P. Ovartchaiyapong, J. Minguzzi, J. R.Maze, and A. C. Bleszynski Jayich, Physical Review Ap-plied 6 (2016), 10.1103/PhysRevApplied.6.034005.

47 M.-A. Lemonde, S. Meesala, A. Sipahigil, M. J. A. Schuetz,M. D. Lukin, M. Loncar, and P. Rabl, arXiV preprintarXiV:1801.01904 [quant-ph] (2017).

48 D. P. Lake, M. Mitchell, Y. Kamaliddin, and P. Barclay,arXiv preprint arXiv:1712.01408 (2017).

49 E. R. MacQuarrie, T. A. Gosavi, S. A. Bhave, andG. D. Fuchs, Physical Review B 92 (2015), 10.1103/Phys-RevB.92.224419.

50 B. Pingault, J. N. Becker, C. H. Schulte, C. Arend,C. Hepp, T. Godde, A. I. Tartakovskii, M. Markham,C. Becher, and M. Atature, Physical Review Letters 113(2014), 10.1103/PhysRevLett.113.263601.

51 L. J. Rogers, K. D. Jahnke, M. H. Metsch, A. Sipahigil,J. M. Binder, T. Teraji, H. Sumiya, J. Isoya, M. D. Lukin,P. Hemmer, and F. Jelezko, Physical Review Letters 113

(2014), 10.1103/PhysRevLett.113.263602.52 J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and

O. Painter, Applied Physics Letters 101, 081115 (2012),https://doi.org/10.1063/1.4747726.

53 D. Lee, K. W. Lee, J. V. Cady, P. Ovartchaiyapong, andA. C. B. Jayich, Journal of Optics 19, 033001 (2017).

54 R. Ruskov and C. Tahan, Physical Review B 88, 064308(2013).

55 A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bial-czak, M. Lenander, E. Lucero, M. Neeley, D. Sank,H. Wang, M. Weides, J. Wenner, J. M. Martinis, andA. N. Cleland, Nature 464, 697 (2010).

56 R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, S. Juying,A. G. Krause, V. Anant, M. Aspelmeyer, and S. Grob-lacher, Nature 530, 313 (2016).

57 S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G.Hofer, R. A. Norte, M. Aspelmeyer, and S. Groblacher,Science (2017), 10.1126/science.aan7939.

58 R. Orbach, Proc. R. Soc. Lond. A 264, 458 (1961).