Tunable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond
S. Johnson, P. R. Dolan, T. Grange, A. A. P. Trichet, G. Hornecker, Y. C. Chen, L. Weng, G. M. Hughes, A. Auffèves, J. M. Smith
TTunable cavity coupling of the zero phonon line of anitrogen-vacancy defect in diamond.
S. Johnson , P. R. Dolan , T. Grange , A. A. P. Trichet , G.Hornecker , Y. C. Chen , L. Weng , G. M. Hughes , A.Auff`eves and J. M. Smith Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Universit´e Grenoble-Alpes & CNRS, Institut N´eel, 38000 Grenoble, FranceE-mail: [email protected]
Abstract.
We demonstrate the tunable enhancement of the zero phonon line of asingle nitrogen-vacancy color center in diamond at cryogenic temperature. An opencavity fabricated using focused ion beam milling provides mode volumes as small as 1.24 µ m . In-situ tuning of the cavity resonance is achieved with piezoelectric actuators.At optimal coupling to a TEM cavity mode the signal from individual zero phononline transitions is enhanced by about a factor of 10 and the overall emission rate ofthe NV − center is increased by 40% compared with that measured from the samecenter in the absence of cavity field confinement. This result is important for therealization of efficient spin-photon interfaces and scalable quantum computing usingoptically addressable solid state spin qubits.
1. Introduction
Coupling of fluorescence from nanoscale quantum systems to optical microcavitiesprovides a means to control the emission process and can be an essential element ofnanophotonic device applications. The negatively charged nitrogen-vacancy (NV − )defect in diamond is an example of a solid state system that has gained significantattention in recent years as a quantum spin register [1, 2] and nanoscale sensor [3, 4, 5]due to its long spin coherence times and capacity for optical manipulation and readout.The recent demonstration of quantum error correction in an NV − defect [6] providesa sound basis for using these systems in practical quantum information technologies.However a coherent spin-photon interface, necessary for many quantum technologies,can not be achieved if information is leaked via interaction with phonons, and so inthe NV − center only the ‘zero phonon line’ (ZPL) transition at 1.945 eV (637 nm)may be used. The low Debye Waller factor (DW (cid:39) .
04) of the center imposesa significant limitation, since most spontaneously emitted photons are useless. Theenhancement of this transition and its efficient coupling to external optics are thereforeimportant challenges to which microcavities are well suited. In particular is an essentialrequirement for generating large scale entangled states between spatially separated a r X i v : . [ qu a n t - ph ] J un unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. − ZPL to micro-ring resonators [11] and photoniccrystal cavities [12, 13] has been demonstrated to provide effective enhancements, butthe monolithic structure of these cavities prevents positioning of the emitter at theheart of the cavity mode in-situ. Tuning of the cavity mode to the ZPL resonancecan be achieved in these systems by progressive condensation of gas molecules ontothe structure, thus increasing the mode volume and red-shifting the resonance, but it isdifficult to optimise tuning using this procedure and it has limited scope for use in deviceapplications. Open cavities [14, 15, 16, 17] provide a flexible approach to cavity-coupleddevices that permit full in-situ alignment and tuning, combined with efficient couplingto external optics. Here we demonstrate control over the emission from a single NV − center by coupling its ZPL to the resonant mode of an open microcavity. The opencavity geometry allows direct comparison between the emission properties of the samedefect in and out of the cavity, thus providing unambiguous evidence of the effect ofthe cavity mode. We demonstrate clear tunable enhancement of the ZPL emission andreduction of the fluorescence lifetime of the defect in a controlled manner, and analyzeour results in terms of the Purcell effect acting on the ZPL and the phonon sideband(PSB). Our work builds on recent demonstrations of room temperature coupling of NV − m m xyz(a) (b) (c) m m (d) Figure 1.
Experimental configurations for characterisation of the same NV centerin (a) the ‘out-of-cavity’ and (b) the ‘in-cavity’ measurements. (c) Scanning electronmicrograph of a cross section through the Bragg coating of the concave mirror. (d)False color-scale plot of the electric field intensity distribution through a cross-sectionof the q = 4 TEM cavity mode calculated using FDTD modelling. The refractiveindex profile is included for clarity, with grey (purple) regions corresponding to thelow (high) index layers of the Bragg mirrors. unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond.
2. Experimental Method
The open microcavities are of a plano-concave design, the concave features producedby focused ion beam milling of a fused silica substrate [16]. These cavities support aHermite-Gauss mode structure with TEM modes that can be effectively mode-matchedto an external Gaussian beam. The radius of curvature of the concave mirror used hereis 7.6 µ m. The concave and planar dielectric mirrors have reflectivities of > µ m providing an additional 3 field intensity maxima of the TEM mode between the mirror surfaces with λ = 637 nm, as shown in figure 1(d). We labelthis mode with the longitudinal index q = 4, or the set of indices ( q, m, n ) = (4 , , − defects are located in nanodiamonds of average diameter 100 nm producedusing high pressure high temperature methods, which are spin cast onto the planarmirror. Registration marks on the planar mirror created using a focused ion beam allowindividual defects to be identified and characterised both in and out of the cavity. Thetwo experimental geometries used in this report are displayed in figure 1. The ‘out-of-cavity’ fluorescence is measured in the absence of a concave mirror and with the planarmirror as a substrate behind the nanodiamonds (fig 1a). For the ‘in-cavity’ experimentsthe planar mirror is inverted and both excitation of the NV − centers (at λ = 532 nm)and collection of fluorescence is carried out through the planar mirror. The microcavityassembly comprises a set of piezoelectric actuators that provide full control of the cavitylength and relative position of the planar and concave mirrors at cryogenic temperature(fig 1b). All measurements are carried out at 77K in a dry He exchange gas environmentwith the container supported in a liquid nitrogen bath cryostat [21]. Further details ofthe experimental apparatus can be found in section A of the supplementary information.
3. Results and Discussion
A low temperature fluorescence image recorded in the out-of-cavity configuration revealswell-dispersed nanodiamonds (fig 2a) with some instances of single centers recordedby the Hanbury Brown and Twiss method. The nanodiamond used in this study iscircled, and its uncorrected photon intensity correlation data shown in fig 2b, revealing g (2) (0) = 0 .
38. Subtraction of background due to autofluorescence from the mirrorsreduces this value to g (2) (0) = 0 .
05 implying that ≈
97% of the fluorescence is from asingle emitter. Fluorescence spectroscopy of this NV − center reveals a ZPL spectrumthat is dominated by a linearly polarised doublet (fitted peaks 2 and 3 in figure 2d)indicating a high level of strain that lifts the degeneracy of the orthogonal E x and E y excited state dipoles of the defect (fig 2c). The line widths of the individual ZPL unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. − defect axis is oriented at 49 ◦ relative to the optical axis of the cavity,and the two orthogonal transition dipoles for lines 2 and 3 are at angles of 39 ◦ and 24 ◦ to the plane of the mirror respectively (see supplementary information section B). TheZPL measured has DW = 0.044, as is typical for NV − centers. (b)(c) (d) (e)(a) Figure 2.
Characterisation of the NV − center on the planar mirror. (a) Fluorescenceimage showing point-like emission from single NV − centers, and registration linesgenerated by focused ion beam Ga + implantation that allow re-location of the same NVdefect in the two experimental configurations. (b) Photon correlation data from theNV − center measured, (c & d) Fluorescence spectra from the NV − defect at T = 77K,and (e) polarisation plots of peak 2 (blue) and peak 3 (purple) of the zero phonon linedoublet. unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. Wavelength (nm) T un i ng S t e p
625 630 635 640 645 65024681012141618 I n t e n s it y ( a . u )
634 635 636 637 638 639 640Cavity mode
Wavelength (nm)
NV ZPLs I n t e n s it y ( a . u ) Data Simulation
634 635 636 637 638 639 64001600320048000160032004800016003200480001600320048000160032004800 Cavity Mode
Wavelength (nm)
NV ZPLs I n t e n s it y ( a . u ) Data Simulation
Figure 3.
In-situ tuning of a cavity mode through the ZPL resonance. (a) Selectedspectra (blue) recorded at five different cavity positions, with a simulation of eachspectrum calculated using equation 5 (red). (b) 2D false color-scale plot of themeasured spectrum including all of the tuning steps recorded.
Figure 3 shows the measured fluorescence spectrum for the in-cavity configurationas a TEM cavity mode is tuned through resonance with the ZPL. Clear enhancementof the ZPL emission is observed, providing a fully saturated photon count rate of 15kc/s. This represents an experimental enhancement of the total ZPL signal by a factorof 2.5 compared with that recorded from the defect with the planar mirror alone (seesupplementary information section C).Figure 4 shows a side-by-side comparison of the optical properties of the NV − center out of the cavity, with that observed in the cavity at optimal tuning to theZPL. Figure 4a shows the spectra over the full range of NV- emission, illustrating theextent to which the coupled ZPL dominates the measured fluorescence, a result of thecavity having no other modes in this range that couple efficiently into the objectivelens. Figure 4b shows a comparison between the fluorescence decay characteristics.The lifetime of the out-of-cavity defect is measured as (30 . ± .
6) ns while that in thecavity is (22 . ± .
4) ns, corresponding to a (39 . ± .
7) % increase in the emission rate.The slight deviation from a single exponential decay in the in-cavity data at longerdelay time is suspected to be due to spectral instability of the cavity mode leading toinhomogeneous broadening. Figure 4c shows the photon autocorrelation data, whichreveal a reduced g (2) (0) correlation value of 0.28 suggesting a slight improvement in theisolation of the single emitter when only the cavity-coupled ZPL is measured.We separate the analysis of our results into two parts - a semi-analytic treatment ofthe coupling of the ZPL to the TEM mode and a numerical treatment of the couplingof the PSB to other cavity modes present. We begin with the ZPL coupling, for whichwe express the wavelength-dependent enhancement of the emission of a given dipole by unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond.
600 650 700 75002000400060008000 I n t en s i t y ( c oun t s / s ) Wavelength (nm) Out of Cavity In Cavity -150 -100 -50 0 50 100 1500.00.51.01.5
Time delay (ns) g ( ) ( t ) g ( ) ( t ) N o r m a li z ed I n t en s i t y Time (ns)
Figure 4.
Comparison of properties of the same NV − center in the in-cavity and out-of-cavity geometries. (a) fluorescence spectrum (b) fluorescence decay, and (c) photoncorrelation histogram. In (a) and (c) the excitation power is set to P sat as measuredin intensity saturation measurements. a single cavity mode as: F µ ( λ ) = ξ µ F max
11 + 4 Q ( λ/λ cav − (1)where F max = π (cid:16) λ cav n (cid:17) QV mode is the maximum rate enhancement assuming perfectspatial alignment and orientation, ξ µ = (cid:16) | µ. E || µ || E max | (cid:17) is the spatial overlap and orientationfactor between the emitting dipole µ and the cavity electric field E , λ cav is the cavitywavelength and Q is the cavity quality factor. As we are able to position the emitter atthe electric field maximum we assume that E = E max so that ξ µ = cos ( θ ) where θ isthe angle between the dipole and the plane of the mirror.The fractional increase in the total emission rate arising from the coupling with theZPL is given by F ZP L = (cid:88) µ n µ (cid:90) dλS µ ( λ ) F µ ( λ ) (2)where n µ are the branching factors of the two dipoles µ in the excited state and S µ aretheir normalized free emission spectra collected over all directions. For peaks 2 and 3 ofthe ZPL doublet n µ = 0.44 and 0.56 respectively (see supplementary information sectionB). F ZP L as defined includes the Debye Waller factor through the relative weight of S µ corresponding to the ZPL. We rewrite equation 2 as: F ZP L = F max (cid:34)(cid:88) µ n µ ξ µ (cid:35) (cid:90) dλ
11 + 4 Q ( λ/λ cav − S axial ( λ ) (3)where S axial ( λ ) = (cid:80) µ n µ ξ µ S µ ( λ ) (cid:80) µ n µ ξ µ (4)is the normalized spectrum emitted along the cavity axis, to which the spectrum shownin figure 2(c) is a good approximation when appropriately scaled. The spectrum emittedinto the cavity mode now reads S cav ( λ ) ∝ S axial ( λ )1 + 4 Q ( λ/λ cav − (5) unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. ( a )( c ) ( 4 , 0 , 0 )( 4 , 0 , 2 )( 4 , 2 , 0 )( 4 , 0 , 4 )( 4 , 2 , 2 )( 4 , 4 , 0 )( 5 , 0 , 0 ) ( b ) Intensity
W a v e l e n g t h ( n m )
Figure 5.
Comparison between (a) the measured in-cavity spectrum with the(5,0,0) mode optimally coupled to the ZPL, (b) the FDTD simulation of the Purcellenhancement, and (c) the semi-empirical calculation of the emitted power densityspectrum. Modes are identified by their ( q, m, n ) for longitudinal index q and transverseindices m, n . which, taking δλ cav = 0 . δλ cav used in equation 3 gives F ZP L = 0 .
25. This quantity isequal to the ratio of the overall ZPL emission rate into the cavity mode with the totalemission rate in free space.Based on the measured Debye Waller factor of 0.044 and the branching factor forpeak 3 of 0.56, the cavity-induced enhancement of emission from peak 3 is found to beabout a factor of 11. This compares with a value of 9.2 obtained using equation 1 withthe effective Q factor Q eff = λ/ ( δλ cavity + δλ emitter ), in which δλ emitter = 0.4 nm, thefitted line width for peak 3. unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. − center (see supplementary informationsection D for details). These calculations allow confirmation of the cavity parametersby matching the simulated mode spectrum to that measured, and a direct predictionof the change in emission rate that will occur between the in-cavity and out-of-cavityexperimental configurations. Figure 5 shows semi-logarithmic plots of three cavity-coupled spectra with the (5,0,0) mode tuned to the ZPL. This measured spectrum (fig.5a) is chosen as it reveals the positions of other modes coupling to the PSB and thereforeprovides a good reference by which to verify the geometrical parameters of the cavity.The relative Purcell factor (fig. 5b) and the resultant semi-empirical prediction for theemission power density spectrum of the NV − center in the cavity (fig 5c) are also shown.The PSB emission couples to cavity modes with longitudinal index q = 4 and transverseindices 0, 2, and 4. The absence of observed coupling to modes with odd transverseindices suggests that the NV − center is well positioned on the cavity axis of symmetrywhere the electric field intensities of these modes drops to zero.Integrating the emitted power density spectrum between 640 nm and 740 nm revealsthe Purcell enhancement of the NV − center PSB emission to be F P SBP = 0 .
93 (ie, a7% suppression of emission relative to the out-of-cavity geometry). The total change inemission rate for the NV − center is thus predicted to be F ZP L + F P SB = 1 .
14 significantlylower than the measured value of 1.395.Figure 6a shows the calculated Purcell enhancement factors of the ZPL and PSBfor different TEM modes tuned into optimal resonance with the ZPL. The ZPL datapoints correspond to the enhancement of the emission rate corresponding to the entireZPL: an estimate of the enhancement of peak 3 alone can be obtained by multiplyingthese values by 1 /n = 1 .
8. Figure 6b reveals that the theory above underestimatesthe experimentally observed lifetime changes for each of these cavity lengths. Thetotal emission rate is predicted to remain approximately the same for q = 4, 5 and 6because the increase in ZPL emission as the cavity length is reduced is compensated bya reduction in PSB emission.We attribute the difference between the predicted and measured behavior to thepresence of inhomogeneous broadening, both in the NV − center ZPL due to spectraldrift and in the cavity linewidth due to mechanical instability in our apparatus. Undersuch conditions the fastest component of the measured emission decay curve correspondsto near-resonant alignment and will primarily reflect the homogeneous line widths. Weintroduce inhomogeneous broadening of the cavity mode into our analytic calculationsabove by convoluting the mode line width with a Gaussian function g ( λ cav ). The changein the NV − emission rate due to the ZPL coupling is then given by (supplementaryinformation section E): F ZP L,inhom = (cid:82) dλ cav g ( λ cav )(1 + F ZP L ( λ cav )) F ZP L ( λ cav ) (cid:82) dλ cav g ( λ cav ) F ZP L ( λ cav ) (6)A cavity line width of δλ = 0.2 nm, corresponding to the value measured unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. q F t o t a l F Z P LP F P S BP
Figure 6.
Calculated and measured parameters as functions of the longitudinal modeindex q. (a) Calculated Purcell enhancement factors for the zero phonon line (openstars) and the phonon sideband (closed triangles). (b) Comparison of the measuredemission rate enhancement (solid squares) with the calculated values assuming a cavityline width of 0.7 nm and no inhomogeneous broadening (open circles) and with a cavityline width of 0.2 nm and inhomogeneous broadening of 0.5 nm (open triangles). by transmission spectroscopy in a nominally identical cavity, combined with aninhomogeneous broadening of 0.5 nm, gives F ZP L,inhom = 0.364, whilst leaving thedetuning spectra in figure 3 relatively unchanged. The overall changes in the NVemission rates are plotted in figure 6b, and are seen to agree more closely with, althoughstill consistently underestimate, the measured values. We attribute the remainingdiscrepancy between the measured and modelled lifetimes to inhomogeneous broadeningof the ZPL, which is more difficult to quantify and which the semi-empirical calculationmethod described above can not easily accommodate. A simple indication of thepotential effect to the lifetimes in this experiment can be obtained from equation 1however, in which a ZPL homogeneous line width of order 0.1 nm, consistent withvalues measured at this temperature in bulk materials [24], and a cavity line widthof 0.2 nm, give F µ = 33.6 when resonantly tuned to peak 3. The resultant emissionrate increase is F tot = 1 .
71, suggesting that the measured rate increase can indeed be unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. − center is its quantum efficiency (QE),which the above treatment assumes to be unity. Recent reports have demonstratedthat nanodiamonds of the size used here can have QE as small as 0.3 [25]. The effectof reduced QE would be to require higher still Purcell factors to achieve the measuredlifetime change, since the enhancement acts only on the radiative term.In conclusion we have shown the controlled coupling of the ZPL of a single NV − center in nanodiamond to an open cavity at cryogenic temperatures. This cavity systemshows significant potential for interfacing the NV − center with photonic networks andperforming quantum operations and measurements. The degree of enhancement iscurrently limited by the line width of the NV − centers in the nanodiamond, and bythe cavity line width as determined by scattering and instability of the low temperaturecavity assembly. Improvements in these areas are readily achievable and are expectedto produce much larger enhancements than those reported here. Open cavity qualityfactors exceeding 10 have been demonstrated [17], and modest further reductions incavity mode volume are clearly also possible. Single NV − defects implanted into bulkdiamond at depths of 100 nm can offer ZPL line widths as narrow as 27 MHz, [26]and electron spin coherence times > µ s. Resonant ZPL coupling of NV − centerssituated in diamond membranes to open cavities can thus result in enhancement ofthe emission rate into the ZPL by a factor of > , leading to an effective DebyeWaller factors approaching unity and indistinguishable photons with lifetimes of a fewhundred picoseconds. Such projections suggest that the experimental configurationdemonstrated here is an attractive route towards an efficient spin/photon interface andto the construction of scalable quantum processors.This work was funded by the European Union Seventh Framework Programme(FP7/2007-2013) under grant agreement no 61807. SJ acknowledges support from theUnited Kingdom Engineering and Physical Sciences Research Council. [1] M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R.Hemmer and M. D. Lukin, Science, , 316. 1312.[2] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R.Hemmer and M. D. Lukin, Science , 314, 281.[3] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A.Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko and J.Wrachtrup, Nature , 455, 648.[4] L. Rondin, J-P Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky and V. Jacques, Rep. Prog. Phys. , 77, 056503.[5] D. M. Toyli, C. F. de las Casas, D. J. Christle, V. V. Dobrovitski and D. D. Awschalom, Proc. Nat.Assoc. Sci. , 110, 8419.[6] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbr¨uggen, H. Abe, T. Ohshima, J. Isoya,J. F. Du, P. Neumann and J. Wrachtrup, Nature , 506, 204.[7] S. Barrett and P. Kok, Phys. Rev. A. , 71, 060310(R).[8] 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, 7447, 86.[9] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. , 86, 5188. unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. [10] S. C. Benjamin, B. W. Lovett and J. M. Smith, Laser Photon. Rev. , 3, 556-574.[11] A. Faraon, P. E. Barclay, C. Santori, K.-M. C. Fu and R. G. Beausoleil, Nature Photon. , 5,301.[12] A. Faraon, C. Santori, Z. Huang, V. M. Acosta and R. G. Beausoleil, Phys. Rev. Lett. , 109,033604.[13] L. Li, T. Schr¨oder,, E. H. Chen, M. Walsh, I. Bayn, J. Goldstein, O. Gaathon, M. E. Trusheim,M. Lu, J. Mower, M. Cotlet, M. L. Markham, D. J. Twitchen and D. Englund, Nature Commun. , 6, 6173.[14] M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka and M. Kraft,Appl. Phys. Lett. , 87, 211106.[15] T. Steinmetz, Y. Colombe, D. Hunger, T. W. H¨ansch, A. Balocchi, R. J. Warburton and J. Reichel,Appl. Phys. Lett. , 89, 111110.[16] P. R. Dolan, G. M. Hughes, F. Grazioso, B. R. Patton, and J. M. Smith, ‘Femtoliter tunableoptical cavity arrays’, Optics Lett. , 35, 3556.[17] D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. H¨ansch, and J. Reichel, New J. Phys , 12, 065038.[18] R. Albrecht, A. Bommer, C. Deutsch, J. Reichel and C. Becher, Phys. Rev. Lett. , 110,243602.[19] R. Albrecht, A. Bommer, C. Pauly, F. M¨ucklich, A. W. Schell, P. Engel, T. Schr¨oder, O. Benson,J. Reichel and C. Becher, Appl. Phys. Lett , 105, 073113.[20] H. Kaupp, C. Deutsch, H. C. Chang, J. 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Supplementary Information
A) Experimental Methods
Cavities are milled into fused silica substrates (UQG Optics) using a focused ion beam(FEI FIB200). High-reflectivity mirror stacks are then deposited on the substrates. Theplanar mirror is coated in-house using alternating λ/ n layers of SiO /TiO to achieve99.7% reflectivity at λ = 637nm. The planar mirror is terminated with the low indexmaterials such that there is an anti-node of the electric field at the surface for emitter-cavity coupling. The concave mirror was coated at LaserOptik GmbH, with 20 pairs ofSiO /Ta O to provide R¿99.99%, such that optical coupling is preferentially throughthe planar mirror. Nanodiamond solutions (100nm - 0.1mg/ml) are spin coated onto theplanar mirror substrates prior to cavity coupling. Optical characterisation is carried outwith a home-built beam-scanning confocal microscope employing a fast-steering mirror unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. Newport FSM300
Scanning Mirror Dichroic Mirror λ /2 λ /4 λ /2 !! Single Mode Fiber Filters Clean-Up Filter 532 CW 532 Pulsed 20kHz SPAD A SPAD B Spectrograph
Acton 2500i
Princeton Spec10 LN CCD Fiber Spliter 50:50
Scanning confocal microscope Excitation Detection & Spectroscopy
Flip Mirror Polarization Control
Optical Set-Up x y z z
77K Bath Cryostat
Wessington Cryogenics
Single Mode Fiber
Cavity Assembly
Figure 7.
Optical set-up for low temperature cavity experiments unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond.
B) Calculation of the ZPL dipole orientations
The orientations of the dipoles for the two strain-split ZPL transitions are calculatedby determining the polar angle of the NV axis relative to the optical axis ( θ ) and therotation of the dipoles about the NV axis ( β ) which will lead to the projections on theobservation plane.A thermal distribution between the populations of the two excited states comprisingthe ZPL doublet is also assumed, consistent with the findings of Fu et al [24]. Theenergy splitting of peaks 2 and 3 is 1.5 meV, so that the population ratio for the levelsresponsible for peaks 2 and 3 at 77K is 0.8:1. The measured intensity ratio betweenpeaks 2 and 3 due to the projection of this thermally distributed dipole pair onto themeasurement plane is 0.58:1, so the equivalent projection of a circle formed by twoperpendicular dipoles of equal strength would result in a ratio of R = 0 . θ is obtained from the sum of the polar intensity distributions ofpeaks 2 and 3, whereby the extrema have the following dependence on θ . I min = I max cos ( θ ) (7)The combined rotation matrix for the axial, then polar, rotation is A = cos ( β ) − sin ( β ) 0 cos ( θ ) sin ( β ) cos ( θ ) cos ( β ) − sin ( θ ) sin ( θ ) sin ( β ) sin ( θ ) cos ( β ) cos ( θ ) (8)Applying this rotation to unit vectors X & Y, and projecting the resultant vectorsonto the measurement plane, gives X (cid:48) = 1 − sin ( θ ) sin ( β ) Y (cid:48) = 1 − sin ( θ ) cos ( β )Where the square of the dipole vector have been taken to obtain the intensity.Knowing θ and the ratio R between these intensities, allows β to be determined. Finally,the angles φ X (cid:48) & φ Y (cid:48) between the rotated dipole vectors and the observation plane arefound by taking the dot product of the rotated dipole vectors and their projections,leading to φ X (cid:48) = arccos (cid:18)(cid:113) − sin ( θ ) sin ( β ) (cid:19) φ Y (cid:48) = arccos (cid:18)(cid:113) − sin ( θ ) cos ( β ) (cid:19) C) Comparison of excitation conditions for the two experimental geometries
For comparison of the emission intensities in the out-of-cavity and in-cavity geometriesit is necessary to establish equivalent excitation conditions in the two cases. To do thiswe measured the dependence of the emission intensity I on excitation power P to record unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. I = I sat PP sat + P The fitting parameters I sat and P sat are the fully saturated photon count rateand the characteristic saturation power for excitation respectively. In the out-of-cavitygeometry we find that I sat = 154 kc/s for the whole NV centre (6.78 kc/s for the ZPLonly) and P sat = 1 .
02 mW, whilst that in the q = 4 cavity (data in figures 3 and 4)yields I sat = 15.1 kc/s for the ZPL only and P sat = 1 .
89 mW. The spectra shown infigure 4 use an excitation power of P sat in each case. I n t en s i t y ( c oun t s / s ) Laser Power (mW)
Signal + Background Background Signal Fit I n t en s i t y ( c oun t s / s ) Laser Power (mW) a) Out of Cavity b) In Cavity P sat = 1.02mW I sat = 154kc/s P sat = 1.89mW I sat = 15100c/s
Figure 8.
Power Saturation measurement into the cavity coupled ZPL with modenumber q=4.
D) Finite Difference Time Domain simulations
We used Lumerical Solutions FDTD software to perform numerical simulations of adipole emitting into the cavity. Firstly, by reproducing the measured mode structureusing these simulations we confirm the cavity geometry. The concave mirror ismodelled as vertically stacked films of λ/ andTa O layers are 1.52 and 2.10 respectively. The concave mirror is situated as having 15Bragg pairs rather than the 20 in the real device, as this allows faster calculations withnegligible affect on the results. The dipolar source is positioned on the axis of symmetry,20 nm from the planar mirror surface, and within a dielectric sphere of diameter 100 nmand refractive index 2.4 in contact with the planar mirror to simulate the nanodiamond.Separate calculations are performed with dipoles at angles of 18 ◦ and 39 ◦ to the planarmirror representing peaks 3 and 2 respectively. We confirmed that the results of thecalculation are insensitive to the position of the source within the nanodiamond and onthe exact size of the nanodiamond. The finite element Yee cells have a minimum size unable cavity coupling of the zero phonon line of a nitrogen-vacancy defect in diamond. V mode = λz R L where z R = L (cid:114)(cid:16) βL − (cid:17) is the Rayleigh range, L is theoptical length of the cavity and β is the radius of curvature of the concave mirror. Theintegration used is V mode = (cid:82) Re [ (cid:15) ] | E | d V (cid:16) Re [ (cid:15) ] | E | (cid:17) dipole E) Calculation of inhomogeneous broadening
For an inhomogeneously broadened cavity mode, the total intensity is given by anaverage over the different cavity positions I ( t ) = (cid:90) dλ cav g ( λ cav ) A ( λ cav ) e − γ ( λ cav ) t (9)where g ( λ cav ) is the inhomogeneous distribution of the cavity position, taken as anormalized Gaussian. For a given cavity position, the fraction of light emitted in thecavity mode is proportional to the measured intensity integrated over a full decay event β ( λ cav ) = F ZP L ( λ cav )1 + F ZP L ( λ cav ) ∝ A ( λ cav ) /γ ( λ cav ) (10)The decay rate is then found by differentiating equation 9: γ inhom = − dIIdt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = (cid:82) dλ cav g ( λ cav ) γ ( λ cav ) β ( λ cav ) (cid:82) dλ cav g ( λ cav ) γ ( λ cav ) β ( λ cav ) (11)so that F inhom = γ inhom γ = (cid:82) dλ cav g ( λ cav )(1 + F ZP L ( λ cav )) β ( λ cav ) (cid:82) dλ cav g ( λ cav )(1 + F ZP L ( λ cav )) β ( λ cav ) (12)whereby substituting back in for β ( λ cav ) from equation 10 gives F inhom = (cid:82) dλ cav g ( λ cav )(1 + F ZP L ( λ cav )) F ZP L ( λ cav ) (cid:82) dλ cav g ( λ cav ) F ZP L ( λ cavcav