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1 DOI:.38/NPHOTON.6.7 Multidimensionl Purcell effect in n ytterium-doped ring resontor Dpeng Ding, Lino M. C. Pereir, Jred F. Buters 3, Mrtijn J. R. Heck 3, Ges Welker, André Vntomme, John E. Bowers 3, Michiel J. A. de Dood nd Dirk Bouwmeester,4. Theory of multidimensionl Purcell effect A unified expression of the mximum Purcell fctor F D for two-level tom in n opticl structure with D- dimensionl mode confinement cn e written s F D = 3 4π D ( ) D λ Q D, () n V D where λ is the wvelength in vcuum, n is the refrctive index of the medium where the tom resides nd Q D nd V D re generlized qulity fctors nd volumes of the modes, respectively. For D = nd, Q D re given y Q = n p n g /n nd Q = n g /n, () where n p (n g ) is the effective phse (group) index of the guided mode. The D nd D Purcell effects re spectrlly rodnd nd therefore re not directly influenced y decoherence of the tom provided tht the resonnce frequency of the tom ω is well elow the cut-off frequency of the guided modes. The expression of Q 3 tht includes the influence of decoherence on the Purcell effect is Q 3 = ω cγ tot / γ tot +, (3) where ω c is the frequency of the cvity resonnce, = ω ω c is tom-cvity detuning nd γ tot is the totl decoherence rte of the tom-cvity system, given y γ tot = κ/+γ/+γ. (4) Here κ = ω/q c is the cvity field decy rte with Q c eing the cvity qulity fctor, γ is the tom relxtion rte nd γ is pure dephsing rte. In writing equtions (3) nd (4), we hve ssumed tht decoherence of the tom cn Huygens-Kmerlingh Onnes Lortory, Leiden University, Leiden, 333 CA, Netherlnds. KU Leuven, Instituut voor Kern- en Strlingsfysic, Leuven 3, Belgium. 3 Deprtment of Electricl nd Computer Engineering, University of Cliforni Snt Brr, Snt Brr, CA 936, USA. 4 Deprtment of Physics, University of Cliforni Snt Brr, Snt Brr, CA 936, USA. emil: ouwmeester@physics.ucs.edu e modelled s pure dephsing. This ssumption my not e vlid in some cses, e.g., for quntum dot coupled to photonic-crystl cvity s in ref. 3, 4. The generlized mode volumes V D re given y D V D = ɛ(r) E(r) d D r ɛ(r m ) E(r m ), (5) where ɛ(r) nd E(r) re the permittivity nd the electric field of the mode t position r, respectively nd r m is the position where the mode field energy density ɛ(r) E(r) is mximized. The ctul Purcell fctor F will e lower thn the mximum vlue s given y eqution () if the tom position r is different from r m or the tomic dipole moment µ is misligned with the locl electric field E(r ) such tht F = F D f(r ) cos θ, (6) where θ is the ngle etween µ nd E(r ) nd f(r ) is given y f(r )= ɛ(r ) E(r ) ɛ(r m ) E(r m ). (7) Decoherence processes depolrize the tomic dipole during the tom-mode coupling nd therefore indirectly influence the Purcell effect through the term cos θ in eqution (6). We introduce degree of polriztion Θ ( Θ ) to quntify the degree to which the tom mintins its initil dipole direction during its lifetime. Without loss of generlity, we ssume tht the dipole is initilly ligned with the y direction. For structure with D, D nd 3D Purcell effects nd with modes polrized long the x, y nd z directions, the totl Purcell fctor F cn e written s sum of contriutions over ll the dimensions nd modes : F = 3 +F D,y j 3 [ D= i,j,k f D,y j F D,x i f D,x i (r )( Θ) (r )( + Θ) + F D,z k f D,z k (r )( Θ) ], (8) where F D,x i is the mximum Purcell fctor for the i-thorder mode in D-dimension, polrized long the x direction. Eqution (8) is sed on the ssumption tht the D, D nd 3D Purcell effects re independent of ech NATURE PHOTONICS 6 Mcmilln Pulishers Limited. All rights reserved.

2 3. Opticl properties of the device other. This ssumption is vlid for our device ecuse the width of the Si3 N4 core nd the circumference of the ring resontor re significntly greter thn the wvelength. For our device, we neglect the Purcell effects of higherorder modes ecuse of their lrger VD nd smller QD s compred with the fundmentl modes nd pproximte the implnttion depth distriution u(y) s n verge position y for ech polriztion of modes. y cn e clculted s (y) E i (y) u(y)y dy (9) yi =, i = TM or TE, (y) E i (y) u(y) dy The ring resontor studied in the min rticle supports two trnsverse mgnetic (TM) modes nd three trnsverse electric (TE) modes t 976. nm s clculted y the finite element method (FEM) implemented in COMSOL. The clculted mode profiles re shown in Fig.. Becuse of the high-spect-rtio geometry of the Si3 N4 wveguide core, the ring resontor is single-mode long the core thickness direction (y direction) nd multimode long the core width direction (x direction). where (y) nd E i (y) re the vlues of (r) nd E i (r) t x = z =. Becuse of the high spect-rtio of the Si3 N4 core, E i (y) re pproximtely the sme for the modes with the sme polriztions in ll dimensions. With these pproximtions, eqution (8) cn e simplified s y (µm).. 3 Depth (nm) 4. e...4 c f.6.8 x (µm). Figure Cross-section nd mode profiles of the ring resontor., Schemtic cross-section of the ring resontor consisting of high-spect-rtio Si3 N4 core emedded in SiO cldding. The centre of the Y3+ depth distriution is out 7 nm ove the Si3 N4 top surfce nnotted y red line. f, Clculted mode profiles of TM (), TM (c), TE (d), TE (e) nd TE (f) y using the finite element method (FEM) implemented in COMSOL. The colour plots in nd c represent the y component of the electric field, E y, while in d f they represent the x component of the electric field, E x. Y density (tom %) Si nd O density (tom %) 4 Depth profiles of Y, Si nd O on reference smple s determined y Rutherford ckscttering spectrometry nd nlysed using NDF 5 re shown in Fig Ion implnttion O.4 Si Y.3.6 where p TM nd q TE. To model the dt, we set = for the 3D TM mode due to the resonnt excittion nd = 5. GHz for the 3D TE mode. Other prmeters in the model re explined in the min rticle. 8 SiO.5. d.8 Si3N4 () Y3+ Fm =(Fp + Fp + Fp3 )fp (yp )( + Θ)/3 + (Fq + Fq + Fq3 )fq (yq )( Θ)/3, DOI:.38/NPHOTON.6.7 The opticl properties of the ring resontor t 976. nm were first chrcterized t room temperture y using stndrd frequency scnning methods. A tunle nrrowlinewidth lser ws tuned to centrl wvelength of 976. nm nd its frequency ws scnned over 7 GHz round the centrl frequency. The output of the lser ws coupled to fire (the designtion of fires is shown in Fig. in the min rticle) through fire polriztion controller (FPC). This excited counter-clockwise trvelling modes in the ring resontor, which were coupled to nother stright wveguide nd susequently coupled to fire. The mesured trnsmission through fire s function of lser frequency detuning is shown in Fig. 3. By chnging the polriztion of the light coupled to the ring resontor using the FPC, the coupling to individul resonnces ws optimized nd their polriztions were identified. These polriztions re nnotted in Fig. 3. From the mesured dt we otined free spectrl rnges (FSRs) of 9.8, 9.3 nd 7.54 GHz corresponding to group in-. 5 Figure Depth profiles of toms. Depth profiles of Y (lue dots), Si (lck curve) nd O (green curve) s determined y Rutherford ckscttering spectrometry. Y ions with n energy of 36 kev were implnted into 3-nm-thick, thermlly-grown SiO lyer on Si sustrte. The depth profile of Y is fitted y Gussin function (red curve) resulting in FWHM of 86 nm nd centre 8 nm underneth the surfce. NATURE PHOTONICS 6 Mcmilln Pulishers Limited. All rights reserved.

3 DOI:.38/NPHOTON.6.7 Trnsmission (.u.) TE TM TM c TM Frequency detuning (GHz) d TM TE Figure 3 Trnsmission spectr of the ring resontor., Lser trnsmission round 976. nm s function of frequency detuning. Trnsmission ws mesured through fire while the frequency of the lser coupled to fire ws scnned over 7 GHz. The lser polriztion ws either set long the x direction (red curve) or the y direction (lue curve). The oserved resonnces re ssigned to the corresponding modes. d, Detils of the mesured resonnces (lue dots) of the TM (), TM (c) nd TE (d) modes, together with fits to Lorentzin functions (red curves). The qulity fctors derived from the fits re 4.8 6, 5. 5 nd 8.3 5, respectively. Intensity (.u.) F 5/.5..5 F 5/ F 7/ F 7/. 3 4 Wvelength (nm) Figure 4 Fluorescence spectr of Y 3+.,, Mesured fluorescence spectr of Y 3+ emitted from ring resontor t 95 K (lue curves) nd 5 K (red curves) nd their corresponding trnsition schemes. The spectrl fetures re nrrower t low temperture ecuse of the reduced homogeneous linewidth. In the rod pek centred t 976 nm is chrcteristic feture of Y 3+ corresponding to the trnsition from the lowest mnifold of the excited stte ( F 5/ ) to the lowest mnifold of the ground stte ( F 7/ ). In the fluorescence t longer wvelength corresponding to the trnsitions to the higher mnifolds of the ground stte is much weker thn tht for the trnsition s shown in. dices of.557,.556 nd.7 for the TM, TM nd TE modes, respectively. The corresponding qulity fctors Q c of these resonnces re 4.8 6,5. 5 nd tht were determined y Lorenztin fits s shown in Fig. 3 d, respectively. In the mesurement of Q c, the field strength in the ring resontor ws ove the sturtion level of Y 3+ nd therefore the mesured Q c were intrinsic vlues. We identified these resonnces with corresponding modes ccording to the mesured polriztions, FSRs nd qulity fctors. Resonnces of higher-order modes were not oserved in the frequency scnning mesurement ecuse of their reltively low qulity fctors. Figure 4 shows the mesured fluorescence spectr of Y 3+ emitted from the ring resontor t 95 K nd 5 K nd their corresponding trnsition schemes. The lser frequency ws tuned to 9. nm nd ws on resonnce with the TM mode, which excited Y 3+ to the higher mnifolds of the excited stte ( F 5/ ). The fluorescence ws collected through fire 4 nd nlysed with spectrometer through long-wvelength-pss filter with cutoff wvelength of 95 nm. The spectrl fetures re nrrower t low temperture ecuse of the reduced homogeneous linewidth. The rod pek centred t 976 nm in Fig. 4 is chrcteristic feture of Y 3+ corresponding to the trnsition from the lowest mnifold of the excited stte to the lowest mnifold of the ground stte ( F 7/ ). 4. Photon echo The homogeneous linewidth of Y 3+ ws mesured y using the two-pulse photon echo technique. A schemtic of the set-up is shown in Fig. 5. The output of n externl cvity diode lser t 976 nm wvelength ws modulted with two cousto-optic modultors in series nd then coupled to the ring resontor through fire polriztion controller nd fire. Frequency nd polriztion of the lser were djusted to mtch the 3D TM mode of the ring resontor y mximizing the trnsmission t fire. Photon echo signls were mesured t fire 3 using single photon counting module (SPCM). Figure 6 shows typicl mesured time trce t 4 mk with resolution of ns. The two excittion pulses were 6 ns nd ns wide, respectively. The SPCM ws highly sturted during the pulses such tht its ctive quenching mechnism cused low count rtes seen s verticl spikes nd the mesured pulses were significntly rodened s result of fterpulsing s shown in Fig. 6. The dely time etween the two pulses for this prticulr trce ws.79 µs. An idle time of 5 ms fter the pulses mde sure tht the ions completely relxed to the ground stte efore the next excittion cycle. The time trce ws otined y summing over 4 single excittion events with fixed dely time. An echo signl ppered fter 3 NATURE PHOTONICS Mcmilln Pulishers Limited. All rights reserved.

4 DOI:.38/NPHOTON.6.7 ECDL AOM FPC APD Fire Fire Cryostt Ring Fire 4 Fire 3 SPCM Figure 5 Set-up of photon echo experiments. ECDL, externl cvity diode lser; AOM, cousto-optic modultor; FPC, fire polriztion controller; Ring, Y 3+ -doped ring resontor; APD, vlnche photodetector; SPCM, single photon counting module. Blck curves denote opticl fires nd red lines denote light ems. the second pulse s shown in Fig. 6. The echo intensity ws mximized y djusting the lser power while keeping the pulse widths constnt. After this optimiztion, the two pulses idelly performed π/ nd π opertions for the ions. The first pulse excited the ions to superposition of the ground nd excited sttes. During the dely time, the ions dephsed due to their slightly different resonnce frequencies. The second pulse reversed the time revolution such tht fter the sme dely time the ions rephsed to form mcroscopic dipole nd emitted light in phse within short time. Counts PE intensity Time (µs) Dely time (µs) Figure 6 Photon echo signl nd its intensity decy., Mesured counts of the SPCM showing two lser pulses nd photon echo (PE) signl with temporl resolution of ns. The SPCM ws highly sturted during the lser pulses. The verticl spikes were due to the ctive quenching mechnism of the SPCM., Intensity of the PE signl s function of the dely time etween the two pulses (lue circles) nd fit with n exponentil function (red line). Rephsing of the ions is incomplete in the presence of homogeneous dephsing. Therefore the echo intensity decys with extended dely times nd the time constnt of the decy τ is mesure of the homogeneous linewidth of the ions Γ H (T ) t certin temperture. Figure 6 shows the echo intensity s function of dely time mesured t PE 4 mk. The dt re well fitted with single exponentil function resulting in τ =.5 ±.6 µs, where the uncertinty is two stndrd devitions s determined y the fit. Γ H (T ) t 4 mk is given y /4πτ = 53 ± khz. The vlues of Γ H (T ) in the temperture rnge of 5 mk re shown in Fig. 5 in the min rticle. Although Γ H (T ) strts to devite from T.3 elow 8 mk, our model is still vlid in this temperture rnge s the Purcell fctor is lmost independent of temperture with Γ H (T )(< 86 khz) much smller thn the linewidth of the 3D TM mode (64 MHz). We did not verify the power lw of T.8 t high temperture ecuse the 3D Purcell fctor is negligile nd the D nd D Purcell fctors re independent of the homogenous linewidth. 5. Dipole depolriztion Chopper ECDL SPCM Pol. Fire PBS APD BP+λ/+λ/4 BS FPC Fire Cryostt Ring Figure 7 Set-up of polriztion nlysis of resonnce fluorescence. ECDL, externl cvity diode lser; Pol., polrizer; BS, emsplitter; FPC, fire polriztion controller; Ring, Y 3+ - doped ring resontor; APD, vlnche photodetector; BP, ndpss filter; λ/, hlf-lmd wve plte; λ/4, qurter-lmd wve plte; PBS, polrizing emsplitter; SPCM, single photon counting module. Blck curves denote opticl fires nd red lines denote light ems. Dipole polriztion of Y 3+ ws mesured y nlysing the polriztion of the resonnce fluorescence emitted from the ring resontor. A schemtic of the set-up is shown in Fig. 7. The output of n externl cvity diode lser t 976 nm wvelength ws modulted with mechnicl chopper nd then coupled to the ring resontor through polrizer, emsplitter, fire polriztion controller nd fire. Frequency nd polriztion of the lser were djusted to mtch the 3D TM mode of the ring resontor y mximizing the trnsmission t fire. The fluorescence of Y 3+ ws collected through fire, then coupled to the reflective port of the emsplitter nd eventully nlysed using ndpss filter, wve pltes, polrizing emsplitter nd two single photon counting modules (SPCM). The wve pltes were djusted when the ring resontor ws cooled to mk with fully polrized dipoles of Y 3+ such tht the integrted signl of one SPCM denoted C ws mximized nd the integrted signl of the other SPCM denoted C ws simultneously minimized. Resonnce fluorescence with different polriztions my hve different collection efficiencies nd different detection efficiencies. These efficiencies were clirted t room temperture with fully depolrized dipoles. The efficiency 4 NATURE PHOTONICS 6 Mcmilln Pulishers Limited. All rights reserved.

5 DOI:.38/NPHOTON.6.7 fctor η is given y η = C /C, where C nd C re the vlues of C nd C t room temperture. The dipole polriztion Θ is given y Θ = (ηc C )/(ηc + C ). The vlue of Θ s function of temperture is shown in Fig. 5 in the min rticle. Becuse of the imperfection of the polriztion nlysis set-up, Θ is only given y reltive vlues (ritrry unit). References. Genov, D. A., Oulton, R. F., Brtl, G. & Zhng, X. Anomlous spectrl scling of light emission rtes in low-dimensionl metllic nnostructures. Phys. Rev. B 83, 453 ().. Auffèves, A. et l. Controlling the dynmics of coupled tom-cvity system y pure dephsing. Phys. Rev. B 8, 4549 (). 3. Ker, P., Nielsen, T. R., Lodhl, P., Juho, A.-P. & Mørk, J. Microscopic theory of phonon-induced effects on semiconductor quntum dot decy dynmics in cvity QED. Phys. Rev. B 86, 853 (). 4. Mdsen, K. H. et l. Mesuring the effective phonon density of sttes of quntum dot in cvity quntum electrodynmics. Phys. Rev. B 88, 4536 (3). 5. Brrds, N. P., Jeynes, C. & We, R. P. Simulted nneling nlysis of Rutherford ckscttering dt. Appl. Phys. Lett. 7, 9 93 (997). NATURE PHOTONICS Mcmilln Pulishers Limited. All rights reserved.