Precision measurements of electric-field-induced frequency displacements of an ultranarrow optical transition in ions in a solid
S. Zhang, N. Lu?i?, N. Galland, R. Le Targat, P. Goldner, B. Fang, S. Seidelin, Y. Le Coq
PPrecision measurements of electric-field-induced frequency displacements of anultranarrow optical transition in ions in a solid
S. Zhang, N. Luˇci´c, N. Galland,
1, 2
R. Le Targat, P. Goldner, B. Fang, S. Seidelin, ∗ and Y. Le Coq LNE-SYRTE, Observatoire de Paris, Universit´e PSL,CNRS, Sorbonne Universit´e, 75014 Paris, France Univ. Grenoble Alpes, CNRS, Grenoble INP and Institut N´eel, 38000 Grenoble, France Chimie ParisTech, Universit´e PSL, CNRS, Institut de Recherche de Chimie Paris, 75005 Paris, France (Dated: January 27, 2021)We report a series of measurements of the effect of an electric field on the frequency of theultranarrow linewidth F → D optical transition of Eu ions in an Y SiO matrix at cryogenictemperatures. We provide linear Stark coefficients along two dielectric axes and for the two differentsubstitution sites of the Eu ions, with an unprecedented accuracy, and an upper limit for thequadratic Stark shift. The measurements, which indicate that the electric field sensitivity is a factorof seven larger for site 1 relative to site 2 for a particular direction of the electric field are of directinterest both in the context of quantum information processing and laser frequency stabilization withrare-earth doped crystals, in which electric fields can be used to engineer experimental protocols bytuning transition frequencies. PACS numbers: 42.50.Ct.,76.30.Kg
Optically addressable ions as dopants in solids providean interesting framework for a wide range of contem-porary applications in physics. In particular, rare-earthions in a crystalline matrix are popular candidates for asteadily growing number of applications due to their out-standing coherence properties [1–5]. Applications includeboth classical [6, 7] and quantum [8–12] information pro-cessing schemes, quantum optical memories [13–15], andlaser stabilization techniques [16–18].In this work we use the 580 nm optical F → D tran-sition in Eu ions in an Y SiO host matrix (Eu:YSO).This transition has an ultranarrow linewidth, potentiallydown to 122 Hz at 1.4 K with 100 G magnetic field [2].The Eu ions can substitute for Y ions in two dif-ferent crystallographic sites, referred to as site 1 and 2,with vacuum wavelengths of 580.04 nm and 580.21 nm,respectively, and we here investigate both sites. Wehave recently demonstrated that the optical transitionfrequency of the ions can be modulated by applying me-chanical stress on the crystalline matrix [19, 20] whichmakes the system promising for optomechanics experi-ments in which a mechanical motion couples to the ionfrequency [21, 22]. An alternative way of controlling thefrequency is based on the Stark effect, in which an ex-ternal electric field interacts with the electric dipole mo-ment involved in the transition [23, 24]. In the Eu:YSOsystem, the Stark coefficients for the optical transitionhave so far been measured with a limited precision, andonly in one of the two different substitution sites [25, 26].Here, the technique of spectral hole burning (see later),combined with the use of an ultranarrow linewidth laser,allows us to obtain not only more accurate values for thealready measured site, but also provide measurementsfor the other much less sensitive site, which thus requiresa higher precision in order to assess the corresponding Stark coefficient.To study the influence of the electrical field (E-field),we use the technique of spectral hole burning, whichallows us to work with a large ensemble of ions with-out being limited by the inhomogeneous broadening ofthe collective absorption line, which is of the order of2 GHz in our case for 0.1-at.% europium doping. Moreprecisely, the F ground state consists of 3 hyperfinestates (figure 1a), with mutual separations in the 30-100 MHz range [1]. Spectral holes are formed by pump-ing the ions resonantly from the F to the D state,from where they decay radiatively, mainly to the F state [27], before decaying non-radiatively to the threehyperfine states in the F manifold. Optical pumpingprevents population build-up in the hyperfine level withwhich the pump beam is resonant, creating a transpar-ent window in the inhomogeneous profile at this exactfrequency. By subsequently scanning the probe laser inthe neighborhood of the pump-laser frequency, the shapeand frequency of the spectral hole can be recorded. Aspectral hole represents approximately 10 Eu ions.Our experimental setup has recently been described indetail elsewhere [18, 19, 28] and will be only briefly sum-marized in the following. We use two diode lasers (re-ferred to as master and slave) at 1160 nm. The masterlaser is frequency locked to an ultra-stable Fabry-Perotreference optical cavity by the Pound-Drever-Hall tech-nique, allowing for a frequency instability below 10 − for time scales of 1-100 s and a few Hz laser linewidth.The slave laser is phase locked to the master laser witha tunable frequency offset. The slave laser possesses thesame stability as the master, but its frequency is contin-uously tunable in a range of several GHz. They are bothsubsequently frequency doubled to reach 580 nm, with anoutput intensity of approximately 5 mW. Moreover, an a r X i v : . [ phy s i c s . a t o m - ph ] J a n + E ±5/2 F D F ±1/2±3/2±5/2 ~580 nm ±1/2±3/2 (a) (b) GNDV mm b D D D a c mm mm Figure 1: Ion level structure and crystal axes. In (a) theenergy diagram of the Eu ions. After resonantly pumpingat 580 nm, the ions decay radiatively primarily to F andthen non-radiatively to one of the hyperfine states which arenot resonant with the pump beam, leaving behind a sharphole in the absorption profile. In (b) we show the Y SiO crystal, which is cut parallel to the 3 dielectric axes, D , D and D (which is parallel to the crystalline b -axis, the a and c axes lying in the D , D plane), as well as the electrodesallowing the application of an electric field E (here, shownparallel to the D ) axis due to an electric potential V . acousto-optic modulator allows us to scan the slave-laserfrequency across the spectral structures with a range ofapproximately 1 MHz. A high stability (in fact, abso-lute) optical frequency measurement is provided whennecessary by comparing the slave laser against a pri-mary frequency reference via an optical frequency comb.Avalanche photodiodes are used to measure the inten-sity of the slave laser before and after passing throughthe crystal, thereby deducing the shape and position ofthe spectral hole. For maximum absorption [27] (bothfor burning and probing) we propagate the laser beamparallel to the crystallographic b -axis, and maintain thelaser polarization parallel to the dielectric D axis (fig-ure 1b). In this work, the pump and the probe beams arecollimated and diaphragmed by an iris of 2 mm diameter(a fourth of the crystal width and height), and kept inthe center of the crystal in order to minimize surface ef-fects. The crystal is maintained at 3.2 K in a closed-loopcryostat resting on an active vibration isolation platform.Under these conditions, we obtain spectral holes with afull width at half maximum (FWHM) of about 3 to 5kHz.To apply a homogeneous E-field, we use two circularelectrodes (31 mm in diameter) above and below the crys-tal as shown in figure 1b). As the electric permittivityof the crystal is a factor of ten larger than that of vac-uum, it is crucial to avoid gaps between electrodes andthe crystal. We use a thin ( < C (||b-axis)C C (||b-axis)C E (a) (b) Figure 2: The symmetry operations of the YSO crystal. In(a) we show the general case, where each δ µ i is related bysymmetry operations to 4 non-equivalent dipoles. In part (b)we show that when considering the projection of the dipolemoment on an axis perpendicular to the C rotation axis (thuscontained within the horizontal symmetry plane), this numberis reduced to 2. Note that the crystal has been rotated by 90 ◦ relative to figure 1 in order to follow the usual convention forwhich the C rotation axis vertical. of an E-field, then apply a voltage ranging from 0 to 28V, giving rise to an electric field in the range of 0 to 3500V m − , and observe the influence on the spectral hole.The YSO crystal is monoclinic and exhibits a C h sym-metry. The two nonequivalent crystallographic sites ofthe Eu ions are both of C symmetry. In the gen-eral case of an application of an E-field with an arbi-trary direction, the crystal symmetry gives rise to fourpossible non-equivalent orientations of the electric dipolemoment difference between the ground and excited state( δ µ i , i=1,2,3,4) for each site, arising from the order ofthe crystal symmetry group (4) divided by the order ofthe symmetry group of the crystal site (1), see figure2a. Thus, in the general case, the presence of an arbi-trary electric field will split a spectral hole into 4 compo-nents [30]. Because of the centrosymmetric crystal spacegroup, the splitting of the spectral hole will to first orderremain symmetric in frequency around zero.We will determine the Stark effect for an E-field par-allel to either the D or D axis, and thus in both cases,perpendicular to the crystal b -axis which coincides withthe C rotational symmetry axis. As the mirror symme-try plane of the crystal is perpendicular to the C rota-tion axis, the E-field is thus contained within this planein our case, as illustrated in figure 2b. This means thatthe component of a given electric dipole moment perpen-dicular to mirror plane is not influenced by our E-field,and thus the splitting arising from the mirror symmetryvanishes. Therefore, in all our measurements, the spec-tral hole only splits into two components, arising fromthe C rotation symmetry alone.This two-fold splitting into a positive and a negativefrequency branch is illustrated for an electric field appliedalong the D direction in figure 3. Here, the transmit-tance T of the crystal is plotted as a function of frequencydifference f with respect to the hole burning frequency,i.e. f ≡
0, where the initial spectral hole burning oc-curred, for 6 different values of the applied E-field forthe two different crystal sites. The center frequency of E l ec t r i c f i e l d [ V / m ] F r e q u e n c y [ k H z ] -100 -50 0 50 100 T r a n s m itt a n ce (b) S E l ec t r i c f i e l d [ V / m ] F r e q u e n c y [ k H z ] -200 -100 0 100 200 T r a n s m itt a n ce (a) S Figure 3: Transmittance of an Eu -doped crystal containing a spectral hole as a function of the electric field applied alongthe D -axis for site 1 (a) and site 2 (b). At zero field, the spectral hole exhibits a linewidth below 5 kHz. When an E-fieldis applied, the spectral structure splits into two components, of which the difference in frequency increases as the E-field isgradually incremented. each peak is then obtained by a double Lorentzian fit ofthe transmittance, T ( f ) = A + B/ (cid:0) f − f + ) /γ (cid:1) + C/ (cid:0) f − f − ) /γ (cid:1) , where f + and f − denote thecenter frequency of the two branches, γ is the (common)FWHM of the two Lorentzian, and A , B , and C take intoaccount the offset and normalization of T ( f ).These frequency shifts are then plotted as a functionof the E-field in figure 4, for an E-field aligned along ei-ther the D or D direction, and in each case for bothcrystal sites. Each point in this figure is the average oftwo separate measurement sequences or more (a data ac-quisition sequence lasts approximately 30 minutes). Foreach sequence, the shift for each data point is half thefrequency difference between the positive and negativebranch peaks centers, i.e. δf = ( f + − f − ) /
2, (therebyensuring negligible effect of slow frequency drifts of thereference Fabry-Perot cavity, as such drift would shiftboth peaks by the same amount in the same direction).The errorbars which are shown on the data and are usedin the fit algorithm are estimated individually for eachcurve, as the noise floor (data baseline) varies from scanto scan and adds to the uncertainty on the determinationof the central frequency. They correspond to approxi-mately one tenths of the FWHM of the spectral struc-tures in figure 3. In order to obtain the linear Starkcoefficients, we initially assume that the effect is purelylinear, and fit the data accordingly with a purely lin-ear model δf = A E using a least squares method. Thelinear Stark coefficients A are given in table I. Our mea-surement of the coefficient for site 1 corresponding to anE-field parallel to the D -axis is consistent with the ear-lier, less accurate measurement reported in ref. [26] (0.27kHz m V − with no explicit errorbar).The uncertainty on the electric field is composed ofthree elements: applied voltage, crystal dimensions and Site Axis Value Stat. Syst.1 D ± ± ± ∓ ± ± ± ∓ or D , and for crystal site 1 or 2. Thesecoefficients are given in Hz per (V/m). The ± and ∓ in thelast column indicate that the D and D systematic errors goin opposite directions. inhomogeneity of the electric field arising from an elec-trode geometrical misalignment and difference in permit-tivity between vacuum and YSO. The voltage and thecrystal dimensions were verified with SI-traceable cali-brated instruments, resulting in an error induced by thesequantities significantly below the stated uncertainty. Weperformed numerical simulations with a finite elementmethod software, utilizing the permittivity coefficient ofYSO expressed in ref. [29] and allowing for up to 1 mmmisalignment of the electrodes. These simulations con-firm that, averaged over the part of the crystal probed bythe laser beam, the relative error on the estimation of theelectric field is below 3 × − , implying negligible effecton our stated uncertainty. The statistical uncertaintyon the coefficients is obtained from the linear fit whichtakes into account the error-bars on each individual fre-quency measurement, arising from the lorentzian fits ofeach spectral pattern. The last column in table I showsthe systematic errors which arise from the uncertaintyon the orientation of the dielectric axis relative to thecrystal-cut ( ± ◦ , as set by X-ray orientation and preci-sion machining). Note that during all measurements, wesystematically probe the spectral structure without theE-field before and after the measurement sequence, toverify that no drift has occurred due to other independentphysical variables during the measurement sequence.For applications in which the narrow spectral feature isused for high precision laser frequency stabilization, thelinear shifts due to small fluctuations of an electric fieldonly broaden the peak symmetrically (an initial symmet-ric splitting) and thus does not change the frequency of alaser locked to this peak (only the slope of the frequencydiscriminator is changed, not its central position). Whatcan potentially be more detrimental in such applicationsis a higher order Stark effects which would introduce anon-symmetric behavior between the positive and nega-tive branches, and thus would result, for small electricfield fluctuations, in laser frequency fluctuations. Dueto the high precision in our frequency measurements, wecan search for a quadratic component at relatively weakfields, despite the non zero linear sensitivity, and set anupper limit of this type of non-linear Stark shift. Wenote that this kind of quadratic behavior physically cor-responds to a small electric polarizability effect, whereE-field-induced additional dipole moments in the groundand excited states, proportional to and along the E-field,produce a correction to the permanent δ µ i , in a direction(set by the E-field) which is identical for the classes ofions having opposite permanent dipole moments.Contrary to the linear (symmetric) case, it is de-sirable here to explicitly correct for possible slow fre-quency drift of the reference cavity, typically of the or-der of 0 . − , resulting in an apparent drift of fre-quency of the initial spectral hole, i.e. f = 0 no longerholds throughout the experimental sequence (which lastslonger than stability specification timescale of the cav-ity). This correction is done by using an optical frequency Applied Electric Field [V/m] F r e qu e n c y s h i f t [ k H z ] Center frequency shift for each spectrum D S D S D S D S Figure 4: The displacements of the center frequency of thespectral hole as a function of the amplitude of the E-fieldapplied parallel to D and D , for the two different crystalsites (S and S ), as well as linear fits to the data. Site Axis Value Stat. Abs. val. upper limit1 D − . × − ± . × − < × − . × − ± . × − < × − − . × − ± . × − < × − . × − ± . × − < × − Table II: Quadratic Stark coefficients for residues after sub-tracting the linear component. These coefficients are givenin Hz per (V/m) . The systematic error is, here, negligiblecompared to the statistic error. comb, referenced to primary frequency standard, which isconstantly monitoring the probe laser optical frequency.The results are shown, for both directions and sites, infigure 5 in which we have plotted δf = ( f + + f − ) / − f asa function of the applied E-field. We notice the absenceof any clear trend corresponding to an asymmetric shiftof the peaks in the data. Fitting (least squares method)with a purely quadratic function, δf = A E , we ob-tain the coefficients A for the quadratic Stark effect ex-pressed in table II. These quadratic coefficient can beconsidered compatible with zero given the measurementuncertainties, and, as conservative estimates, we set theupper limits stated in the last column of the table. Forsite 1, this is coherent with the significantly less stringentlimit of ± V − previously obtained [25, 31].When the peaks from the positive and negativebranches separate under the application of an E-field, afinal possibility of quadratic effect corresponds to a devi-ation from the linear behavior, that would be symmetricbetween the positive and negative branch. This wouldphysically require the field-induced part of the δ µ i to bealigned with the δ µ i rather than with the external E-field, but still be proportional to the amplitude of thefield. Although this situation is physically more difficultto interpret than the asymmetric quadratic Stark effectdiscussed above, we still investigate the data for the pres-ence of such behavior. For simplicity, we only considerthe positive branch, and examine the residue after sub-tracting the linear Stark shift determined previously, i.e. δf , + = f + − f − A E . These residues are then fittedwith a purely quadratic function, δf , + = A , + E . Fromthe fitting procedure, we obtain quadratic coefficients forthe D -direction of ( − . ± . × − Hz m V − forsite 1 and ( − . ± . × − Hz m V − for site 2, andfor the D -direction ( − . ± . × − Hz m V − forsite 1 and ( − . ± . × − Hz m V − for site 2 (withnegligible systematic uncertainties). Again, these coeffi-cients are compatible with zero within the measurementuncertainty and they are all below ± V − .The spectral holes prior to applying an E-field typi-cally exhibits FWHM of approximately 3 to 5 kHz. Whenthe E-field is increased, the spectral structures graduallybroaden, as observed in fig. 3. A minor part (approxi-mately 10 % by the end of 5 consecutive measurements) E-field [V/m] -3-2-10123 F r e q . s h i f t [ k H z ] E-field [V/m] -3-2-10123 F r e q . s h i f t [ k H z ] E-field [V/m] -3-2-10123 F r e q . s h i f t [ k H z ] E-field [V/m] -3-2-10123 F r e q . s h i f t [ k H z ] D S D S D S D S Figure 5: Mean central frequency of the positive and neg-ative branch peaks (with reference to the zero E-field casewhere the initial hole burning occurred) as a function of theapplied E-field parallel to D or D , and for site 1 (S ) orsite 2 (S ), as indicated by the labels in the insets. The datahas been corrected from the drift of the probe laser, which iscontinuously measured with a primary frequency referencedoptical frequency comb. The dashed lines are a quadratic fitto this data, showing an effect compatible with zero to withinthe measurement uncertainty. of this broadening is due to the probing beam, whichslowly pumps ions from the edges of the spectral holeinto dark states. In principle, the remaining part couldbe due to a spatial inhomogeneity in the applied field.Finite-element simulations show, however, that such ef-fect is very small ( <
1% of the linewidth), an assertioncorroborated by experimental findings that 1) using a6 mm diameter probe beam instead of 2 mm does notchange the broadening, and 2) highly E-field sensitivecases (eg. D S ) exhibit similar broadening as less sen-sitive ones (eg. D S ). However, a slight misalignemntof the field (within the stated precision of crystal cut-ting) producing a component along the b -axis may ex-plain most of the broadening (the positive and negativebranch broaden symmetrically under such effect). Goingto higher E-fields, we have indeed observed a secondarysplitting which, extrapolated, accounts for at least 50% of the residual broadening. However, no matter thereason behind the broadening, the corresponding uncer-tainty is largely included in our error-bars on the data,and thus in the uncertainty stated for the Stark coeffi-cients.In this letter, by means of direct probing of ultranar-row spectral structures, we have provided measurementsof the linear Stark effect for an optical transition inEu ions in an Y SiO matrix by applying a fieldparallel to the D or D axis, with an accuracy below the3 Hz m V − level. Surprisingly, for an E-field appliedparallel to the D -direction, the crystal site 1 is a factorof seven more sensitive to the field than site 2. This is useful information for applications using E-fields toelectrically tune the transition frequency. Moreover,our measurements have allowed us to provide an upperlimit for both an asymmetric and symmetric quadraticStark coefficient of the order of or below the 1 mHz m V − level, depending on the site, direction and effectconsidered, an important information for applicationsin which the electric field is a source of noise, such asmetrology applications, in which the transition is used asa frequency reference. For example, in order to reach arelative frequency stability of 10 − using a spectral holeas a frequency reference, an asymmetric quadratic Starkeffect ¡1 mHz m V − translates into a requirement ofthe E-field instability below 2 . − .The project has been supported by the EuropeanUnion’s Horizon 2020 research and innovation programunder grant agreement No 712721 (NanOQTech). Ithas also received support from the Ville de Paris Emer-gence Program, the R´egion Ile de France DIM SIRTEQ,the LABEX Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01) within the Program “Investissementsd’Avenir” operated by the French National ResearchAgency (ANR), and the EMPIR 15SIB03 OC18 programco-financed by the Participating States and the EuropeanUnion’s Horizon 2020 research and innovation program.The data that support the findings of this study are avail-able from the corresponding author upon reasonable re-quest. ∗ Electronic address: [email protected][1] Ryuzi Yano, Masaharu Mitsunaga and Naoshi Uesugi,
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