Investigation of the Stark Effect on a Centrosymmetric Quantum Emitter in Diamond
Lorenzo De Santis, Matthew Trusheim, Kevin Chen, Dirk Englund
aa r X i v : . [ qu a n t - ph ] F e b Investigation of the Stark Effect on a Centrosymmetric Quantum Emitter in Diamond
Lorenzo De Santis, Matthew E. Trusheim, Kevin C. Chen, and Dirk R. Englund Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Quantum emitters in diamond are leading optically-accessible solid-state qubits. Among these,Group IV-vacancy defect centers have attracted great interest as coherent and stable optical inter-faces to long-lived spin states. Theory indicates that their inversion symmetry provides first-orderinsensitivity to stray electric fields, a common limitation for optical coherence in any host material.Here we experimentally quantify this electric field dependence via an external electric field appliedto individual tin-vacancy (SnV) centers in diamond. These measurements reveal that the permanentelectric dipole moment and polarizability are at least four orders of magnitude smaller than for thediamond nitrogen vacancy (NV) centers, representing the first direct measurement of the inversionsymmetry protection of a Group IV defect in diamond. Moreover, we show that by modulating theelectric-field-induced dipole we can use the SnV as a nanoscale probe of local electric field noise,and we employ this technique to highlight the effect of spectral diffusion on the SnV.
Quantum emitters in diamond have emerged as lead-ing solid-state quantum memories. The nitrogen vacancy(NV) center, in particular, has been used for basic quan-tum network demonstrations [1, 2] including on-demandentanglement [3]. Despite the NV’s excellent spin prop-erties, its optical interface is inefficient: due to strongelectron-phonon interactions, only a small fraction ofemission occurs into the spin-correlated zero-phonon line(low Debye-Waller factor) [4]. Integration into photonicnanostructures can improve this branching ratio, but atthe cost of linewidth broadening and spectral diffusionbecause of the NV’s sensitivity to electric field fluctua-tions, which are particularly strong near material inter-faces [5, 6]. These challenges have sparked strong interesttowards inversion-symmetric Group IV-Vacancy quan-tum emitters (SiV, GeV, SnV and PbV), whose Debye-Waller factor can be an order of magnitude greater thanin the NV. Additionally, these emitters have been shownto have low spectral diffusion and lifetime-limited emis-sion even in nanostructures [7]. Such robustness is at-tributed to the split vacancy configuration of the emit-ters with D d symmetry, which has been predicted toproduce a first order insensitivity to electric fields [8].This property enabled the demonstration of high QEDcooperativities for cavity-coupled emitters [9], wide spec-tral tuning of their emission energies [10], and large-scaleintegration of defects in waveguide arrays [11]. Despitebeing one of its key advantages, no measurement of thefirst-order electric field insensitivity of a Group IV emit-ter has been reported to date. In this work, we directlytest the expected insensitivity to electric fields by inves-tigating the Stark effect on a single negatively chargedSnV emitter. We confirm the absence of a significantpermanent electric dipole moment, resulting from the in-version symmetry of the defect, as well as an extremelylow polarizability — orders of magnitude lower than forthe NV center. Finally, we use the electric-field-induceddipole of the SnV defect as a nanoscale probe to estimatethe electric field noise in its vicinity.We consider the negatively-charged SnV in diamondas a prototypical Group-IV emitter. They share quali- tatively identical electronic structure consisting of fourelectronic orbitals connected by four optical transitions[8]. The D d symmetry that gives rise to the predicted FIG. 1. (a) Side view of the sample layout, showing twogold electrodes and the expected external electric field dis-tribution as simulated with COMSOL for an external biasof 200 V. The emitter location (expected 76 nm below thesuface) is indicated by a red circle. (b) Confocal photolu-minescence scan of the sample surface, showing an ensembleof defect centers in the exposed diamond, enclosed betweentwo electrodes. (c) Typical emission spectrum obtained undernon-resonant excitation. (d) PLE spectrum of the transitionto the lower ground state of an SnV, showing a linewidth of50MHz (e) second order correlation measurement on a singledefect, performed at high power (black dots) and low power(blue diamonds). first-order insensitivity is preserved [12–14]. Moreover,the large ground-state orbital splitting of the SnV (850GHz for SnV compared to 50 GHz for SiV) gives it the po-tential for long spin coherence above dilution refrigeratortemperatures, a key consideration for use as a quantummemory [15, 16].The SnV emitters analyzed here are created in a CVD-grown type IIa diamond through ion implantation andsubsequent vacuum annealing. To probe the electric fieldresponse, we deposit on the surface an interdigitated elec-trode structure, with an inter-electrode spacing rangingfrom 2 µm to 10 µm (see supplementary material for de-tails of the sample fabrication). This allows us to applyan electric field aligned with the cryostalline [110] axis asdepicted in Figure 1(a). We characterized the SnV emit-ters using a custom-built confocal setup, where the sam-ple is kept at a temperature of 4 K inside a closed-cycleliquid helium cryostat (Montana Instruments), while theelectric field applied to the emitters is controlled by anexternal DC voltage source, supplying up to ±
200 V tothe electrodes with a negligible leakage current (detailsin Supplementary). A spatial photoluminescence (PL)emission map obtained with a 515 nm pump laser isshown in Figure 1(b). The emitted PL spectrum (Fig-ure 1(c)) consists of many spectral lines, indicating anensemble of emitters within the optical diffraction limit.To investigate single SnV defects, we use photolumines-cence excitation (PLE), by scanning the frequency of anarrow-band laser across the SnV transition while de-tecting the emitted photons in the red-detuned phononsideband. Figure 1(d) plots the result of a PLE mea-surement, from which we can extract a single-transitionlinewidth of 47 ± g (2) (0) = 0 .
03, con-firming the single-photon nature of the collected PL, aswell as an optical T lifetime of 6 . ± . ± g (2) measurement reaveals Rabi oscil-lations with a phase lifetime T of 4 . ± . ± µ ind between the excited and groundstate dipole moments, and can be expanded as a powerseries in the applied electric field F [17]. Conventionally,Stark shift measurements focus on linear and quadraticshifts, and have been extensively used to investigate thepolarizabilities of atoms and molecules [18, 19]. Here weconsider shifts up to the fourth order in F , giving anexpected SnV transition energy shift ∆ E of∆ E = − ∆ µ ind ( F ) F == − ∆ µF −
12 ∆ αF −
13! ∆ βF −
14! ∆ γF (1)The first two coefficients of this equation allow us to ex- F r equen cy ( T H z ) -6-5-4-3-2-10 P ea k s h i ft ( G H z ) -50 0 50-0.2-0.10-250 -200 -150 -100 -50 0 50 100 150 200 250 Local field (MV/m) L i ne w i d t h ( M H z ) (a)(b)(c) FIG. 2. Stark shift of a single SnV center. (a) PLE spectraas a function of the local electric field at the defect. (b) Mea-sured shift of the center position of the transition (black dots).The dashed and solid red lines show a fit to a second and afourth order polynomial respectively. The inset is a close upof the shift for small applied fields, where the behaviour isquadratic. (c) Linewidth broadening caused by the inducedelectric dipole on the defect center. tract the difference between the ground and excited or-bital states in permanent dipole moment ∆ µ and in po-larizability ∆ α with the applied field, while the terms ∆ β and ∆ γ are related to the differences in the second andthird order hyperpolarizabilities of the electronic states.Using standard perturbation theory, the first-order Starkshift on the i th orbital is given by the matrix element h ψ i | µ | ψ i i where µ is the electric dipole operator, an oddfunction of position. Since the two ground (excited) stateorbitals maintain even (odd) symmetry, these matrix el-ements should vanish. The second-order correction isgiven by P j = i |h ψ i | µ | ψ j i| E i − E j . The only contributions to thissum are due to the matrix elements connecting groundand excited states, which should contribute little due tothe their large energy separation. We thus expect boththe first- and second-order Stark shifts to be vanishinglysmall in an ideal crystal.We approximate the local electric field F acting onthe defect from the Lorentz local field approximation F = F ext ( ǫ + 2) /
3, where F ext is the externally appliedfield, extracted from the COMSOL simulation of Figure1(a), and ǫ the dielectric constant of diamond. Owing tothe charge stability of the defect and the high dielectric (Å ) -1-0.500.51 ( D ) -3 FIG. 3. Experimental values of the change in dipole momentversus the change in polarizability for 11 different SnV cen-ters, describing respectively the first and second order Starkshift. The average polarizability is 0.23 ˚A , while the averagedipole moment is zero. strength of diamond, we can apply high electric fields ex-ceeding 200 MV/m and thereby detect even weak Starkeffects. Figure 2(a) presents the results, revealing thedependence of the SnV absorption spectrum on the elec-tric field F . Such dependence clearly shows a non-linearStark shift, a direct consequence of the absence of a per-manent electric dipole in a centrosymmetric defect suchas the SnV. The black dots in Figure 2(b) indicates thetransition energies of the SnV emitter, determined by fit-ting to the PLE spectrum. For this emitter, we extracta vanishing linear Stark shift with a slope of 6 . × − GHz/(MV/m), corresponding to a difference in dipolemoment of ∆ µ = 1 . ± . × − D. From the samefit, we can also extract a quadratic shift coefficient of − . × − GHz/(MV/m) , corresponding to a polariz-ability difference of ∆ α = 0 . ± . .By studying several individual SnV centers, as sum-marized in Figure 3, we confirm the absence of a signif-icant linear Stark shift in any of the observed defects,showing a distribution of ∆ µ between − × − D and1 × − D. The observed polarizability difference is in-stead always positive with a mean of 0.23 ˚A . The mea-sured values of ∆ µ and ∆ α are more than 3 orders ofmagnitude smaller than reported for the Stark shift ofNV centers [20, 21]. This result is expected from the in-version symmetry of Group IV-vacancy emitters, whichleads to the absence of a permanent electric dipole intheir electronic states. Additionally, the states withinthe ground and excited orbital manifolds share the samesymmetry and are strongly split by spin-orbit coupling,which suppresses the possible electric-field induced mix-ing of the orbitals. The exceptional electric field insen-sitivity thus arises both from the absence of a perma-nent electric dipole, as well as from the emitters’s lowpolarizability. At the same time, each emitter exhibitsa Stark shift of at least 1 GHz, which corresponds to atuning range hundreds of times larger than their naturallinewidth, without any noticeable quenching of emissionintensity. Electric fields can thus be used for the spectraltuning of the transition frequency of Group IV color cen- ters without noticeable degradation of optical properties.The lack of a permanent dipole and low polarizability,however, is not adequate to describe the Stark shifts athigh fields ( >
50 MV/m) where higher-order corrections(hyperpolarization) become significant. The observed de-pendence including these contributions is well reproducedby a fourth-order description (solid red line), showing afit to Equation 1. The third and fourth order coeffi-cients are respectively − . ± . × − GHz/(MV/m) and − . ± . × − GHz/(MV/m) , whose totalcontribution to the observed spectral trajectory is upto 25%. These higher-order effects, linked to higher-order moments of the optical transition [17], are typicallynegligible, but have been observed in systems such asmolecules[22] and hydrogen-like atoms, where only even-order terms are present due to their symmetry [23]. Thisstudy shows that centrosymmetric defects in diamondsuch as the SnV enable the observation and control ofhyperpolarization effects solid state systems.To extend the analysis above, we also investigate theeffect of the applied electric field on the emitter linewidth.We fit the PLE measurements of Figure 2(a) to a pseudo-Voigt profile, and extract the SnV linewidth as shown bythe black dots in Figure 2(c). In these measurementsthe laser scan across the transition in 2.5 s, thus theywill include all dephasing and spectral diffusion effectshappening up to that timescale. Without an externalfield, the emitter shows a narrow linewidth of 49 ± F DC will induce a higher dipole on the defect,thus a same field fluctuation will shift the optical transi-tion further. We assume here a root mean square electricfield noise at the defect location having a fixed magnitude F r.m.s ≪ F DC . By replacing F = F DC + F r.m.s in Equa-tion 1, we can see that F r.m.s will produce a root meansquare line shift of σ G = ∆ µ ind ( F DC ) F r.m.s . This resultsin a Voigt absorption lineshape on the emitter, whichcan be related to the constituent Lorentzian and Gaus-sian width as Γ V = Γ L + q ( Γ L ) + Γ G [24]. We identifyhere Γ L as the homogeneous linewidth of the SnV emit-ter, while Γ G = 2 √ σ G describes the stochastic Starkshift due to F r.m.s . The expected SnV linewidth can thenbe modeled as:Γ = Γ L s(cid:18) Γ L (cid:19) + 8 ln 2 ( F r.m.s ∆ µ ind ( F DC )) (2)where ∆ µ ind ( F DC ) can be deduced from the Stark shiftanalysis of the previous section. We fit this equation tothe measured linewidth values as shown in the red lineof Figure 2(c), from which we extract an average fieldfluctuation of F r.m.s = 2 . ± . L = 60 ± Frequency (THz) S c an Frequency (THz) S c an -200 -150 -100 -50 0 50 100 150 200 Local field (MV/m) L i ne w i d t h ( M H z ) -4 -3 -2 -1 Scan time (s) L i ne w i d t h ( M H z ) (a) (b)(c)(d) FIG. 4. Panel (a) and (b) show the SnV transition repeat-edly probed over 1.3 ms, for an electric field of 0 MV/m and250 MV/m respectively. (c) Black dots: Lorentzian linewidthobserved in an individual scan, averaged over 200 repetitions.Blue diamonds: expected Voigt linewidth by combining theLorentzian linewidth from a single scan and the Gaussianbroadening from the standard deviation of the peak positionsover the 200 repetitions. (d) Single-scan linewidth measuredat 0 MV/m as a function of the laser scan time across thetransition (black dots). The red dashed line shows the life-time limited value as extracted from the g (2) measurement ofFigure 1(e). sured F r.m.s is over four orders of magnitude greater thanreports for InGaAs QD devices [25], the SnV emittersstill show linewidths close to the lifetime limit, highlight-ing again their insensitivity to the charge environment.We note moreover that for F DC = 0 the extracted fieldfluctuation F r.m.s produces an average Stark shift muchsmaller than the natural linewidth of the SnV, thus theresidual broadening above the lifetime limit should notbe associated with electric field noise.We also investigated short-time SnV optical dynam-ics by fast sweeps of the PLE excitation laser at a rateof 20 GHz/s, scanning across the SnV transition in 1.3ms. At each bias value, the PLE scans are repeatedmultiple times to produce temporal series as in Figure 4(a) and (b). These measurements reveal a narrow andstable transition at zero bias field (Figure 4(a)), whilean increasingly strong spectral diffusion appears withlarger electric fields (Figure 4(b)). The mean of the PLElinewidths from individual high-speed scans as a functionof the electric field is shown by the black dots in Figure4(c) . In contrast to the observations of Figure 2(c), thelinewidth measured at this timescale is consistent witha zero-bias value of 45 MHz: it does not significantlydepend on the applied field. The broadening effect caninstead be reproduced from the standard deviation of thepeak positions in a temporal series, as seen in the bluediamonds in Figure 4(c). This data confirms our attribu-tion of the SnV linewidth broadening to the electric-fielddependent SnV dipole moment ∆ µ ind ( F ), which is re-sponsible for the stochastic Stark shift observed in Figure4(b). Moreover, we can confirm that this spectral diffu-sion effect happens at a timescale slower than 1.3 ms.To further investigate the residual broadening above thelifetime limit, we also measure the linewidth shown bythe emitter at smaller timescales. The results, obtainedunder zero bias, are shown in Figure 4(d). Here we seethat there is no clear dependence on the laser scan timeover the tested range, which span from 140 µ s to 2.3 s.Overall, the observed linewidth is consistently a factorof 1.7 above the lifetime limit of 27 MHz. The residualline broadening appears to originate from processes oc-curring above 7 kHz. Unfortunately, this timescale is notaccessible here through PLE measurements due to thelimited detector count rate. However, having confirmedthat the diffusion due to the stochastic Stark shifts hap-pens at a slower rate, we can exclude charge noise as asignificant contributor to broadening. The power broad-ening induced by the probe laser is also negligible, sothe deviation from the lifetime limited linewidth is likelydue to residual interaction with the acoustic phonon bath[13, 26].The techniques used here proved very useful to under-stand the effect of charge noise on SnVs. As an additionalexample, we observed a linewidth narrowing effect onsome SnV emitters at intermediate field values, detailedin Section 4 of the Supplementary material. We attributethis to the suppression of spectral diffusion from thesweep-out of charge traps, which is commonly observedin other quantum emitter systems such as self-assembledquantum dots [27] and silicon carbide defects [28]. Sec-ondly, we performed electric field-resolved spectroscopyon the 645 nm emission line that is commonly observedin Sn-implanted diamond. As detailed in section 5 ofthe Supplementary material, these measurements reveala linear Stark shift, indicating a permanent dipole mo-ment that is not consistent with the SnV’s inversion sym-metry.In conclusion, we have investigated for the first timethe Stark effect on a Group IV-vacancy diamond defect.Our measurements confirm the expected first-order in-sensitivity to electric fields and reveal suppressed secondorder effects, which we quantified by the ∆ µ and ∆ α parameters reported in Figure 3. Additionally, by mod-ulating the SnV electric dipole induced by the externalfield, we use its linewidth to probe the local electric fieldnoise. Using this ‘modulated atomic dipole’ technique,we show that spectral diffusion does not significantly con-tribute to the SnV linewidth. We expect this techniqueto be broadly useful for characterizing local electric fieldnoise, from studies on quantum emitters [29] to cold atomsystems [30] and superconducting qubits [31]. Our ex-periments also demonstrated the first Stark-shift controlof a Group IV-vacancy color center emission. Despitenot offering the same tuning range as strain fields [10],this Stark shifting enables precise and easy-to-implementspectral tuning of Group-IV centers. It is moreover com-patible with high-speed modulation, as demanded in pro-posals for photon-photon logic gates [32, 33]. Combinedwith large-scale integration of color centers on photoniccircuits [11], it opens the path to scalable precision con- trol of quantum memories in spin-photon quantum infor-mation processing systems.Note: during the writing of this manuscript we becomeaware of similar work investigating charge fluctuations onsingle molecules [34].L.D. acknowledge funding from the European Union’sHorizon 2020 research and innovation program underthe Marie Sklodowska-Curie grant agreement No 840393.M.T. acknowledges support through the Army ResearchLaboratory ENIAC Distinguished Postdoctoral Fellow-ship. 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