Source: https://www.groundai.com/project/scaling-laws-of-the-cavity-enhancement-for-nitrogen-vacancy-centers-in-diamond/
Timestamp: 2019-04-21 22:51:21+00:00

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We employ a fiber-based optical microcavity with high finesse to study the enhancement of phonon sideband fluorescence of nitrogen-vacancy centers in nanodiamonds. Harnessing the full tunability and open access of the resonator, we explicitly demonstrate the scaling laws of the Purcell enhancement by varying both the mode volume and the quality factor over a large range. While changes in the emission lifetime remain small in the regime of a broadband emitter, we observe an increase of the emission spectral density by up to a factor of 300. This gives a direct measure of the Purcell factor that could be achieved with this resonator and an emitter whose linewidth is narrower than the cavity linewidth. Our results show a method for the realization of wavelength-tunable narrow-band single-photon sources and demonstrate a system that has the potential to reach the strong-coupling regime.
The negatively charged nitrogen-vacancy (NV) center in diamond stands out due to stable single photon emission Gruber et al. (1997); Kurtsiefer et al. (2000) and exceptional spin coherence of electronic ground-state levels Balasubramanian et al. (2009) . Together with spin-selective optical transitions, the NV center represents a particularly promising solid-state quantum bit with direct optical access Gruber et al. (1997); Balasubramanian et al. (2009); Maurer et al. (2012) . An important milestone yet to achieve is the realization of an efficient optical interface for this emitter. Ideally it would open the way for deterministic single photon sources Su et al. (2008) , non-destructive spin state detection Young et al. (2009); Robledo et al. (2011) , efficient spin-photon entanglement Togan et al. (2010) , and quantum coherent optical manipulation on the single photon level. A variety of optical microcavities has been studied in this respect, including photonic crystal cavities Wolters et al. (2010); Englund et al. (2010); van der Sar et al. (2011); Riedrich-Möller et al. (2012); Faraon et al. (2012) , microring resonators Faraon et al. (2011); Hausmann et al. (2012) , microdiscs Barclay et al. (2009a, b) , Fabry-Perot cavities Dumeige et al. (2011) , and plasmonic nanoresonators de Leon et al. (2012); Choy et al. (2011) . Considerable success has already been achieved, e.g. by demonstrating Purcell enhancement of the zero phonon line (ZPL) at cryogenic temperature, where 70% of the emission was channeled into a single cavity mode Faraon et al. (2011) . The broad emission spectrum and the small weight of the ZPL call for particularly large Purcell factors to achieve efficient cavity coupling. Furthermore, several issues remain problematic for the techniques used so far, which all rely on solid state cavities with limited flexibility to access the cavity mode and to tune the resonance frequency.
In this work we study the cavity enhancement of the phonon sideband emission of small ensembles of NV centers in individual nanodiamonds by means of a fiber based Fabry-Perot microcavity Hunger et al. (2010) . The cavity design relies on concave surface profiles machined on the end faces of optical fibers using CO2 laser processing Hunger et al. (2012) . The technique provides excellent surface quality to support ultra-low loss mirror coatings, and offers microscopic cavity mode volumes to achieve large Purcell factors. The open resonator geometry permits us to easily access the cavity field maximum, to study various emitters with one and the same cavity, and to fully tune the cavity resonance. These features have been beneficial for recent experiments with ultracold atoms Colombe et al. (2007) , trapped ions Steiner et al. (2013) , molecules Toninelli et al. (2010) , quantum dots Muller et al. (2009); Barbour et al. (2011); Di et al. (2012); Miguel-Sánchez et al. (2013) , and NV centers in diamond Albrecht et al. (2013) . Here we use these properties to explicitly demonstrate the scaling of the fluorescence enhancement for NV centers with the cavity parameters. Using direct comparison of free space and cavity enhanced emission, we determine the ideal and effective Purcell factors as a function of the mode volume and the quality factor. While narrow cavity resonances couple only to a small fraction of the broad emission spectrum, we observe an up to 300-fold enhancement of the emission spectral density, giving a direct measure of the ideal Purcell factor of the coupled system. Furthermore, we find that we can maintain a cavity finesse F=30000 in the presence of rather large nanocrystals.
Setup. In our experiments we use a cavity which consists of a mirror on a concave end face of a single mode fiber and a macroscopic plane mirror, schematically shown in Fig. 1(b). The effective radius of curvature in the center of the profile machined on the fiber is r=100 μm. Together, the two mirrors define a fully accessible microscopic cavity with a mode waist w0=2.2 μm and an effective cavity length as small as deff=4.3 μm, which includes the penetration of the cavity mode into the dielectric mirrors. The smallest mode volume we currently achieve in this way is V=16 μm3. The coatings are designed for highest reflectivity at 780 nm and are realized by multiple layer pairs of Ta2O5 and SiO2. The finesse reaches a value of F=30000 at 780 nm for small cavity length. It decreases continuously towards F=20000 at deff=39 μm and drops faster for larger distances, which we attribute to diffraction loss due to the finite mirror size. The largest quality factor we thus obtain is Q0=2×106 for deff=39 μm.
We introduce nanodiamonds containing NV centers into the cavity by directly spin coating an aqueous colloidal solution on the large mirror with low enough concentration (<0.01 μm−2) to isolate single nanocrystals with the cavity mode. For optimal coupling between the emitter and the cavity, the mirror is terminated with a λ/4 layer of SiO2 to shift the electric field maximum of the cavity mode slightly above the mirror surface. We employ two different samples of fluorescent nanodiamonds (FNDs) with size distributions peaking at 30 and 100 nm diameter. They were irradiated with He ions to increase the concentration of NV centers Mohan et al. (2010); Yu et al. (2005) . Surface treatment with acids and thermal oxygen etching removed graphitic surface shells and other contaminants. The resulting surfaces are predominantly oxygen terminated, which favors the negative charge state of the NV center Rondin et al. (2010); Hauf et al. (2011) . The emission spectrum of the NV center shows a zero phonon line at 637 nm with a room temperature linewidth of ∼2 THz Fu et al. (2009) , and a broad phonon sideband resulting from coupling to a phonon continuum with a distinct coupling strength maximum around a phonon frequency of 16 THz Davies (1974) . Transitions involving several phonons constitute the sideband fluorescence as schematically shown in Figure 1(a). The strength of the integrated k-th sideband can be estimated by the Frank-Condon factors for displaced harmonic oscillator states, ζk=e−DDk/k!, with a Huang-Rhys factor D=3.2 found for NVs in bulk. In our experiments, we observe spectra peaking at a central wavelength of 690 nm with a full width at half maximum of ∼70 nm [see Fig. 2(a)].
With our present mirror coatings, the NV emission spectrum is divided into two parts: For wavelengths shorter than 690 nm, the mirrors are transmissive and the color centers emit approximately under free space conditions. For longer wavelengths, the mirrors become increasingly reflective and cavity enhanced emission can be observed. This enables us to observe free space and cavity enhanced emission within one measurement, such that the cavity enhancement can be quantified from a direct comparison. Furthermore, it strongly facilitates the search for NV centers in the cavity, since a large fraction of their emission is transmitted. High transparency at the excitation wavelength provides nearly constant excitation conditions when varying the cavity length. The transmission of the plane mirror is chosen to be a factor 2 larger than that of the fiber mirror to optimize outcoupling efficiency into the detection channel used in the experiments. Finally, the strong change of mirror reflectivity across the fluorescence spectrum permits us to study the cavity enhancement as a function of the quality factor.
Figure 1: (Color online) (a) Level scheme of phonon assisted optical transitions in the NV optical cycle. (b) Schematic of the cavity. (c) Experimental setup combining a confocal microscope and a tunable microcavity (LP long pass, SP short pass, NF notch filter, λ/4 quarter wave plate).
The cavity setup is combined with a homebuilt confocal microscope as shown in Fig. 1(c). We use an excitation laser at 532 nm (Cobolt Samba) which is coupled to a singlemode fiber and short pass filtered (Thorlabs FES0550) to remove fiber fluorescence. The light is focused with a long working distance objective (Mitutoyo, NA=0.55) through the macroscopic mirror (diameter 0.5 in.) onto the surface supporting the FNDs. The mirror is mounted on a nanopositioning stage (PI 541.2 SL) to perform raster scans and to maximize the overlap of a nanodiamond with the cavity mode. The fluorescence is collected via the same objective, separated from the excitation light with a dichroic mirror (Thorlabs DMLP567), and spectrally filtered (Thorlabs FEL0600, FES0850, NF533-17). Thereafter, the fluorescence is either spatially filtered with a pinhole and led to a Hanbury Brown-Twiss setup with two avalanche photo diodes (Laser Components Count) behind a 50/50 beamsplitter, or coupled via a multimode fiber (Thorlabs HPSC25) to a grating spectrometer (Princeton Instruments Acton SP2500) with a CCD camera (Andor iKon-M). The cavity fiber is mounted on a three axis micropositioning mount. It can be withdrawn from the macroscopic mirror by some millimeters in order to use the setup as a conventional confocal microscope.
Figure 2: (Color online) (a) Free space spectrum of a 100 nm diamond on a suprasil substrate (in units kilocounts per second). (b) Calculated transmission spectrum of the macroscopic mirror. (c) Emission spectrum of a FND placed inside the cavity. The transmissive part can be explained by the free space spectrum and the mirror transmission (blue line). Within the mirror stop band, cavity enhanced emission into narrow resonances is observed. (d) Confocal scan with I=0.5 Isat showing a single nanodiamond, mirror background, and the effect of bleaching. (e) Saturation of free space NV emission (blue datapoints). For comparison we show the same measurement with linear background subtracted (red datapoints). Solid lines are fits to a saturation model.
Characterization. In a first step we characterize FNDs on Suprasil substrates replacing the macroscopic mirror with the fiber withdrawn. We confirm the presence of negatively charged NV centers by assessing fluorescence spectra, which show no significant contribution of NV0. We find that an appreciable fraction (>10%) of the fluorescent 30 nm diamonds contain single NVs by observing antibunching in the second order autocorrelation function of the photoluminescence. Comparing the average single emitter count rate with the fluorescence yield of 100-nm FNDs, we find that the latter host 30-300 NV centers per nanocrystal.
Next, we study FNDs prepared on a cavity mirror and with the fiber aligned to form a cavity. We perform confocal scans with the mirror surface facing the fiber and detect the partial transmission through the mirror as shown in Fig. 2(d). We observe strong background fluorescence within the detection spectral band, originating from fluorescence generated in the mirror and the fiber, as well as from Raman scattering in the fiber. To obtain signals much stronger than the mirror background, we use 100 nm nanodiamonds containing ensembles of NV centers. With a coating that prevents excitation light from entering the fiber, excessive background is avoidable and also permits single emitter experiments Albrecht et al. (2013) . The observed point spread function for individual FNDs is fit by a Gaussian with a 1/e2 radius of w=1.1 μm. This is larger than the value found when the sample faces the objective (w=0.75 μm) due to the reduced effective numerical aperture, NA′≈NA/nm when focusing through a dielectric medium with refractive index nm. For time resolved measurements with FNDs on the mirror we find fluorescence lifetimes ranging between 16 and 23 ns. This is on the short side of the typical range for nanodiamonds and can be attributed to a lifetime reduction due to the modified density of states on the mirror.
The observed fluorescence in the measurements shown below corresponds to roughly 50 NV centers, again estimated from the comparison with single emitter fluorescence yield. With the respective FND positioned in the focus of the microscope objective, we align the fiber to form a cavity such that the fundamental mode has optimal overlap with the FND. Figure 2(c) shows a typical spectrum obtained under such conditions. To understand the different parts of the spectrum, we show a typical free space spectrum taken on a Suprasil substrate with negligible background [Fig. 2(a)] and a calculation of the mirror transmission T2(λ) [Fig. 2(b)]. Multiplying the two contributions and scaling with a constant matches the modulated emission observed under cavity conditions for the short-wavelength part [Fig. 2(c)]. Within the stop band of the coating we observe enhanced emission into sharp cavity resonances. Subsequent fundamental modes at λ1=715 nm and λ2=755 nm and a series of weak higher-order transverse modes are visible. The separation of the fundamental modes gives direct access to the effective cavity length deff=λ1λ2/[2(λ2−λ1)]=6.75 μm. This length contains the part of the cavity mode penetrating into the mirror stack. From a simulation of the coating at the respective wavelengths we find that the penetration adds up to 1.5 μm. Evaluating the separation of the higher order transverse modes Δλnm=λ2arccos(√1−deff/r)/(2πdeff), we infer the effective radius of curvature of the fiber mirror, r=100 μm, in agreement with white light interferometric measurements of the fiber surface. This determines the mode waist w0 and volume V of the plane-concave cavity, given by w20=λ/π√rdeff−d2eff and V=πw20deff/4, respectively. Since deff≪r and w0≫λ/2, the paraxial approximation and thus the given formulas are valid to a high precision.
To characterize the background contribution, we retract the fiber and perform saturation measurements. We observe an intensity dependence that contains a saturating part and a linear contribution from background fluorescence, shown in Fig. 2(e). Fitting the integrated count rate of the transmissive part to the function P=P∞I/(Isat+I)+aI, with the intensity I and the fitting parameters P∞,Isat,a, we find a saturation intensity Isat=3.5 GW/m2 for the studied FND, which agrees with typical reported values Kurtsiefer et al. (2000) , and a background contribution of 20% in the linear regime of the NV fluorescence. To determine I, we measured the excitation mode waist to be 1.5 μm by scanning a FND through the focus and detecting the fluorescence without spatial filtering. Within the mirror stopband and for the cavity present we find a signal to background ratio of about 100 for strong cavity resonances. We correct for the background contributions in the subsequent evaluations. Taken together, the spectrum, saturation, and lifetime measurements confirm that we are indeed coupling NV emission to the cavity.
Figure 3: (Color online) (a) Comparison of the FND spectrum as inferred for free space conditions (Sfs, blue), emission through the mirror with the fiber retracted (Sm, gray), and a cavity configuration with deff=5.0 μm (Sc, red). (b) Normalized spectra for decreasing cavity length from deff=39 μm (top) to 5.4 μm (bottom). (c) Scaling behavior of the Purcell enhancement in the bad emitter regime: Effective Purcell factor evaluated for the strongest cavity resonance at λ0=710 nm as a function of mode volume (blue data points). The red line shows the prediction of Eq. (2), and the black line shows the ideal Purcell factor as given by Eq. (1) divided by a factor of 100 for easier comparison.
at the cavity resonance wavelength λ0, where ηλ=S(λ0)/max(S) accounts for a detuning from the maximum of the spectrum. In this regime, the effective Purcell factor no longer depends on the quality factor of the cavity but rather on the quality factor of the emitter, Qem=λ0/δλem.
For an experimental evaluation of the effective Purcell factor we compare background corrected spectra for three different configurations: a typical free space reference spectrum S0(λ) from an FND on a Suprasil substrate (sample facing the objective), and spectra Sc(λ) and Sm(λ) from the FND under study on the mirror (sample facing the fiber) with and without the fiber mirror being present to form a cavity, respectively. We calculate the expected free space emission spectrum Sfs(λ)=bS0(λ) of the FND under study from its emission spectrum obtained through the mirror by matching Sm(λ) to bT2(λ)S0(λ) via the scaling factor b determined by a least square fit. Here, T2(λ) is the transmission of the macroscopic mirror which we infer from a direct transmission measurement and simulate with a transfer matrix method Hood et al. (2001) .
With this at hand we can compare the integrated emission into the strongest cavity resonance Pc=∫λ0+Δλ0−ΔSc(λ)dλ with the integrated emission under free space conditions Pfs=∫Sfs(λ)dλ, where Δ denotes an integration range of 3 full linewidths (FWHM) around the resonance, and the free space spectrum is integrated from 590 to 770 nm to cover the entire emission. The Purcell factor compares the emission rate into a particular cavity mode with the free space emission into the entire solid angle, such that one has to account for the probability for emitted photons to reach the detector for both cases. For the free space situation in our case, the presence of the Bragg mirror and the substrate modifies the emission pattern. We perform analytical and finite difference time domain simulations to study this aspect. Since we collect without immersion liquid, increased emission into the dielectric beyond the critical angle remains trapped in the medium. Additionally, the angular distribution within the NA of our objective remains nearly unaffected by the refraction through the mirror layers and the substrate in the transmissive spectral range. Consequently, the collection efficiency can be determined from the free space dipole emission pattern, the effective solid angle of the objective as determined from the observed point spread function, and the angle dependent transmission coefficient. Weighting with a dipole orientation distribution (see below) yields a collection efficiency ηΩ=0.05. For a photon emitted into the cavity, we include the probability to leave the cavity through the respective mirror, ηc=T2/(T1+T2+L1+L2), where L1 and L2 are the loss of the fiber mirror and the macroscopic mirror respectively. At the wavelength of the cavity resonance, λ0=710 nm, we obtain ηc=0.68, where we use the measured values L1,2=(71,23) ppm, and T1,2(λ0)=(810,1900) ppm from the coating simulation. Losses on the way to the detector and detector efficiency are assumed to be equal for free space and cavity conditions and thus cancel out.
For the spectrum shown in Fig. 3(a) we obtain a value C=(6±2)×10−3. The error is estimated from systematic uncertainties of our evaluation method, mainly originating from differences between the spectra of different FNDs and from an alignment and wavelength dependent collection efficiency. Pulsed excitation measurements with and without cavity show no significant difference of the emission lifetime, as expected for the low value of C.
To obtain quantitative agreement between the prediction of Eq. (2) and our measurements evaluated according to Eq. (3), we estimate the influence of the term ηλ|→μ→E(λ0)/(μE0(λ0))|2. Over the studied spectral range, the position of the standing wave maximum shifts from the position of the emitters at the center wavelengh towards the coating for shorter wavelengths. This reduces the coupling in particular close to the edge of the stop band of the mirror, where a phase shift results from the dispersive character of the coating. For the resonance position (λ0=710 nm) and our coating parameters we calculate ηE=(E/E0)2=0.55. Next, the detuning from the maximum of the NV spectrum reduces the coupling by a factor ηλ=0.75, which we read off from the reference spectrum. Finally, we estimate the effect due to the fixed but random dipole orientations. The measurements were performed with circular polarization and excitation conditions below saturation, such that the angle dependence of the excitation probability (→μ→E)2∝cos2(θ) leads to a normalized distribution p(θ)=3/2cos2(θ) with θ being the angle between the mirror plane and the dipole axis. This reduces the observed average enhancement by a factor ηθ=∫p(θ)cos2(θ)cos(θ)dθ=0.8. Evaluating Eq. (2) for the parameters of the measurement (V=19 μm3,Qem≈10) we obtain C=3λ30/(4π2)Qem/V×ηθηληE=(5±2)×10−3, in good agreement with the experimental value.
To demonstrate the scaling of Eq. (2) explicitly, we perform measurements as shown in Fig. 3(a) and evaluate the effective Purcell factor for the emission into the strongest cavity resonance as a function of the cavity mode volume by changing the mirror separation. A few example spectra are shown in Figure 3(b), and the evaluation of the full data set is shown in (c). The observed behavior matches the prediction of Eq. (2) very well. Notably, C approaches unity only for very small mode volumes on the order of λ3. We expect that this regime can be approached with fiber-based Fabry-Perot cavities by further miniaturization.
Figure 4: (Color online) (a) Tuning of a single cavity resonance across the accessible NV spectrum starting at deff=4.3 μm. (b) Cavity spectrum with multiple resonances for deff=39 μm. (c) Ideal Purcell factor as a function of the quality factor for the data set partially shown in (a) (light blue) and (b) (dark blue) together with the predictions of Eq. (1) (solid lines). (d) Measurement (blue datapoints) and calculation (solid line) of the cavity quality factor as a function of the wavelength for deff=39 μm.
Ideal Purcell Factor. In contrast to the small fraction of the entire emission that is coupled to the resonator, we observe a strong enhancement of the emission spectral density on cavity resonance. In fact, comparing the spectral density for the cavity and free space case quantifies the ideal Purcell factor C0. This can be seen by considering a spectral element of the emission with Fourier limited width, which couples to the cavity resonantly and thereby experiences the full enhancement. With the strong variation of our coating properties across the emission spectrum, we can study the enhancement as a function of the quality factor of the cavity. By changing the mirror separation we can locate the cavity resonance at any wavelength and thereby sample quality factors spanning a range of almost three orders of magnitude. Figure 4(a) shows the stepwise tuning of a cavity resonance across one free spectral range for a short cavity corresponding to the mode with eight antinodes between the mirrors (smallest deff=4.3 μm). For comparison, we show a single spectrum for large cavity length deff=39 μm in Fig. 4(b), where the cavity geometry and the excitation conditions are fixed. There, several consecutive fundamental modes between 700 and 800 nm are observed within the mirror stop band, sampling different Q within a single measurement.
In Fig. 4(c) the values obtained from the data sets partially shown in Fig. 4(a),(b) are compared with theory, where we again include ηE and ηθ. Since Sc,max and Sfs are evaluated at the same wavelength, the same detuning factor applies for both and thus drops out. For deff=4.3 μm and Q=(3.5±0.3)×105, the measurements yield a maximum value of C0=300±100, one of the largest values reported to date. Notably, we observe only minor changes of the finesse (or the quality factor) measured at the largest value when introducing the nanocrystal (<20%).
For the ensemble, we replace cosθ with √ηθ to account for the orientation distribution. Finally, the energy decay rate of the cavity at the respective transition frequency is given by κk=c(T1+T2+L1+L2)/2deff√R1R2 and γ=1/τ=2π×8 MHz for typical nanodiamonds.
Here we account for all contributing transitions with respective frequencies ωk, coupling strengths g0k, total incoherent rates Γk=κk+γ0+γ∗k, and cavity detunings Δk=ωc−ωk. The effective Purcell factor is then given by C(ωc)=R(ωc)/γ0. One can see that C becomes independent of κk for γ∗k≫κk, supporting the previous definition in Eq. (2).
We note that the coupling of a number of N emitters to the cavity does not lead to a change of the coupling rate in the present situation. This is in contrast to the case of a coherent ensemble, where a collective dipole forms that leads to an increased coupling rate gN=√Ng0. Collective enhancement in the presence of strong dephasing can be quantified by the figure of merit α=NR/γ∗, which compares the enhanced emission rate under ideal conditions with the dephasing rate Temnov and Woggon (2005) . While modification of the dynamics becomes apparent for α>0.1, we find α∼10−3 for our parameters, such that no collective effects are expected.
To model the measurements of our experiment, we fit ζk and γk∗ with ωk=2π(470−k×16) THz to reproduce the free space spectrum S0(ω), and determine all other quantities from measurements. We find ζk=[0.02,0.25,0.44,0.24,0.06,0.01],γ∗k=2π[3,23,25,29,34,40] THz, contributions of transitions with larger k are negligible. The large difference in γ∗k for the ZPL and the sidebands reflects their slightly different origin. While for the ZPL, only thermally excited low frequency phonons contribute, each sideband is made up by the coupling to a structured phonon continuum extending up to the cutoff at ∼40THz Davies (1974) . Furthermore, the width of the bands increases with k due to the increasing number of possible phonon decay paths. We find good agreement between the predictions of the model and the data, e.g. for the parameters of the measurement shown in Fig. 3(a) we obtain C=(6±2)×10−3. The obtained transition strengths deviate from the expected Frank-Condon factors, which in part could be attributed to a wavelength dependent collection efficiency in our setup. This would imply an error in the evaluated Purcell factors of up to ∼20% in both models and the experimental data.
The generalized description of cavity enhancement confirms that the simple picture given previously is a sufficient approximation as long as the effective dephasing dominates all other rates and masks the structure of the phonon sideband. Beyond this, the model provides insight into the different roles of transition branching and dephasing.
Conclusion and Outlook. In our experiment we have studied the scaling laws for the cavity enhancement of the fluorescence of broadband emitters with a flexible architecture, offering high finesse and large ideal Purcell factors compatible with the presence of nanodiamonds. The results indicate a method for the realization of efficient single-photon sources with narrow bandwidth and wide tunability, and suggest that fiber-based microcavities could provide a route towards the strong coupling regime in a cryogenic environment.
While NV centers in nanocrystals show larger homogeneous linewidth and stronger spectral diffusion than in clean bulk samples, dephasing rates γ∗/2π<500 MHz for the ZPL appear feasible. Alternatively, diamond membranes or nanomachined high quality bulk diamond could be introduced to achieve better optical and spin properties. For an estimation of the achievable coupling strength we account for the ZPL branching ratio ζ0=0.04 and the splitting of the ZPL into six individual fine structure resonances. For the transitions connecting the mS=0 ground state with the Ex or Ey levels in the excited state, an overall branching ratio ζ0(Ex,Ey)≈0.02 remains for perfect state initialization Manson et al. (2006) . Optimization of the laser machining technique is expected to achieve mirror radii of curvature <10 μm, which would result in 2g00>2π×1 GHz for deff=2.0 μm. Applying a feasible coating with T+L=20 ppm at 637 nm yields F=150000 and κ/2π=500 MHz at this mirror separation. With a penetration depth ∼0.6 μm at the coating center, the mirror separation still amounts to ∼2λ, providing sufficient space for alignment and tunability. Together, the assumed parameters would reach the strong coupling regime, 2g00>(κ,γ,γ∗), and lead to an effective Purcell Factor C>130. This opens the perspective for a fully quantum-coherent spin-photon interface.
We acknowledge support from L. Costa, M. Mader, M. Weber, L. Liebermeister, and D. Burchardt, and helpful discussions with R. Albrecht, C. Becher, and A. Kubanek. This work was supported by the Nanosystems Initiative Munich. T.W.H. gratefully acknowledges support by the Max Planck Foundation.
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