Photophysics of single nitrogen-vacancy centers in diamond nanocrystals MartinBerthel1,2,OrianeMollet1,2,3,GéraldineDantelle1,2,4,ThierryGacoin4,SergeHuant1,2 andAurélienDrezet1,2 1 Université Grenoble Alpes, Inst. NEEL, F-38000 Grenoble, France 2 CNRS, Inst. NEEL, F-38042 Grenoble, France 3 LPN-CNRS, Route de Nosay 91460 Marcoussis, France 4 Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique, UMR CNRS 7643, 91128 Palaiseau, France 5 A study of the photophysical properties of nitrogen-vacancy (NV) color centers in diamond na- 1 nocrystals of size of 50 nm or below is carried out by means of second-order time-intensity photon 0 correlation and cross-correlation measurements as a function of the excitation power for both pure 2 chargestates, neutralandnegativelycharged,aswell asfor thephotochromicstate,wherethecen- n ter switches between both states at any power. A dedicated three-level model implying a shelving a level is developed to extract the relevant photophysical parameters coupling all three levels. Our J analysisconfirmstheveryexistenceof theshelvinglevelfortheneutralNVcenter.Itisfound that 5 itplaysanegligibleroleonthephotophysicsofthiscenter,whereasitisresponsibleforanincreasing 1 photonbunchingbehaviorofthenegativeNVcenterwithincreasingpower.Fromthephotophysical parameters, we infer a quantum efficiency for both centers, showing that it remains close to unity ] fortheneutralcenterovertheentirepowerrange,whereasitdropswithincreasingpowerfromnear l l unity to approximately 0.5 for the negative center. The photophysics of the photochromic center a reveals a rich phenomenology that is to a large extent dominated by that of the negative state, in h agreement with theexcesscharge release of thenegativecenterbeingmuchslower than thephoton - s emission process. e m PACSnumbers:42.50.Ar,42.50.Ct,81.05.ug,78.67.Bf . t a m I. INTRODUCTION proach, the photophysics of NV centers can be modeled by a two-level system with two photophysical parame- - d ters, the excitation rate, and the spontaneous emission n Withthedevelopmentofphotonicquantumcryptogra- rate. However,with increasing excitation power, the NV o phy and quantum information processes, there is need center, more particularly in the negative state NV-, can c for reliable and easy-to-use single photon sources. Such experience distinctive photon bunching at finite coinci- [ sources have been developed in recent years1, like single dence time, in addition to the expected antibunching at 1 molecules, colloidal or epitaxial semiconductor quantum zerodelay.Thiscanbeaccountedforwithinathree-level v dots, and color-centers in diamond.2–4 Here, we are in- systemwithadditionalphotophysicalparametersto des- 4 terested in the latter, namely the NV center, formed by cribe photon decays to, or from, the additional shelving 1 a substitutional nitrogen atom adjacent to a vacancy in level. 7 3 the diamond lattice. NV centers have found numerous The aim of this paper is to give a detailed description 0 applicationsrecentlythankstotheiruniquephysicalpro- oftheintrinsicphotophysicsofsingleNVcentersofboth . pertiessuchasexcellentphotostability2,5–7 andlongspin chargestatesinsurface-purified9nanodiamonds(NDs)of 1 coherencetimes8 aswellastoimprovedcontroloverboth sizearound50nm,orbelow,asafunctionoftheillumina- 0 5 their production9 and physical initialization protocols.10 tionpowerfromaCWlaser.Understandingthe intrinsic 1 NV centers can be made available both in ultra-pure photophysicsofNV centers is requiredbefore implemen- : bulk diamond8 and ultra-small crystals.11 Applications ting small fluorescent NDs in a complex electromagne- v range from high-sensitivity high-resolution magnetome- tic environment, such as practical single-photon devices, i X try12–18, to fluorescence probing of biological processes which will modify the photophysics.28 It is also useful to r 19,20, solid-state quantum information processing21,22, theapplicationsmentionedaboveasmostofthemexploit a spin optomechanics23,24, quantum optics25–27, nanopho- the single-photon emitter nature of isolated NVs. tonics28–31, and quantum plasmonics.32–34 The statistics ofbothNV chargestates hasbeen studied NV centers cantake two different chargestates with dif- previously in ND samples similar to those studied here ferentspectralproperties:theneutralcenterNV0,which and it was found to be size dependent, with a larger oc- has a zero-phonon line (ZPL) around 575 nm (2.16 eV), currence (> 80%) of the NV- center over the entire size andthe negativelychargedcenterNV-,whichhas a ZPL rangefrom20to80nm.9 Inthepresentstudy,wefurther around637nm(1.95eV).35 InadditiontoZPLs,thefluo- observedthatmostofthe single NVcentersbeing detec- rescence spectra of both centers exhibit a broad and in- ted as neutral from their fluorescence spectrum at low tense vibronic band at lower energy.A single NV center, illumination power (<0.5 mW) progressively gain with as a single-photon emitter, is characterized by a second- increasing power a photochromic character in that they order time-intensity correlation function that exhibits a also exhibit the NV- ZPL in addition to the NV0 ZPL. photon antibunching dip at zero delay. In a first ap- OnlyafewofNV0sremainpurelyneutraloverthe entire 2 power range, which we call pure NV0 behavior. On the otherhand,NVcentersdetectedasnegativelychargedat lowpowerremainsowithincreasingpower,whichwecall pure NV- behavior. We first focus our attention on such pure NV0 or NV- centers. In addition to such behaviors, we found that some rare NVs can see their charge swit- ching between neutral and negative36 already at the lo- westexcitationpower.Wealsodescribethephotophysics ofsuchaphotochromicNVoverthesamepowerrangeas the non-photochromic centers and show how both pho- tophysics can be linked. This allows us to find valuable information on the dynamics of photochromism. The paper is organizedas follows.The experimentalme- thods are described in Section II. The three-level model usedtointerpretthe experimentisdevelopedindetailin Section III. Section IV focuses on the experimental re- sults and on the extraction of the various photophysical parameters as a function of the excitation power for a NV0 and a NV- center. Section V describes the photo- physics of photochromism in a single NV. A summary is given in Section VI. Figure 1: (Color online) (a) Schematicof thetwo-levelsys- II. EXPERIMENTAL METHODS tem explaining the emission spectrum of the NV center. g : groundstate,e:excitedstate.Theseveralthinnerhorizontal linesrepresentthecouplingwiththephononbathinthedia- Preparation of the ND sample was achieved following mond matrix. The green arrow represents the excitation and a procedure reported previously.9,37 Commercial HPHT the red ones the optical decay at resonance (full arrow) and diamond nanocrystals are first irradiated using high- out of resonance (dotted arrow) (b) Typical emission spec- energy electrons, then annealed at 800˚C in vacuum to trum of an NV-center in a 25 nm ND. produce the fluorescentNV centers,andfinally annealed in air at 550˚C to remove surface graphitic compounds. Colloidal dispersion in water and further sonication al- whereP (t+τ|t)=P (τ|0)istheconditionalprobability 2 2 low us to obtain a uniform solution of NDs. We consider to detect a photon at time t+τ knowing that another hereNDswithatypicalsizeof25nmor50nmdeposited photon has been recorded at time t. This probability is on a fused silica substrate to minimize spurious fluores- normalized by the constant single photon detection rate cence.29SingleNVcenterswereopticallyaddressedusing P (t)=P (0). For a classical source of light g(2)(τ)≥1, 1 1 standard confocal microscopy at room temperature.29 A whereas the observation of an anti-bunching g(2)(τ) ≤ 1 CW laser light (wavelength λexc = 515 nm) falling wi- isaclearsignatureofthequantumnatureoflight.38–40 In thin the absorption band of the NVs is used to excite particular, at zero delay g(2)(0) = 0 for a single photon the NV fluorescence. Excitation light was focused onto emitter40, which means that the probability to detect the sample using an oil immersion microscope objective simultaneously two photons vanishes. with numerical aperture NA= 1.4. The NV fluorescence In practice, an HBT correlator (Fig. 2(a))5,6 is used to is collected through the same objective and is filtered measure g(2). Here, the fluorescence of the NV center is from the remaining excitation, i.e., with wavelength be- sentonabeamsplitter,whichseparatesthesignalintwo low 532 nm, by a dichroic mirror and a high-pass filter. equal parts sent to two avalanche photo-diodes (APDs) The collected fluorescence is subsequently sent either to named APD1 and APD2. The APDs are connected to a a Hanbury-Brown and Twiss (HBT) intensity correlator time-correlated single-photon counting module to build (see below) or to a spectrometer. An example of fluores- histograms of delays between photon events detected by cence spectrum is shown in Fig. 1(b) for the NV- case. theupper «Start»APD1 andthe lower«Stop»APD2. The ZPL corresponds to the resonant decay (Fig. 1(a)) Inordertoavoidunwantedopticalcrosstalkbetweenthe at λ = 637 nm while the wide fluorescence side-band is APDs,aglassfilteractingasashort-passfilterat750nm the phonon replica. and a diaphragm are added in both branches.41 In the The second-order time-intensity correlation function standardconfiguration,nobandpassfilterisaddedtothe containstheinformationontheclassicalversusquantum setup, in contrast with the configurations used to study nature of light. It reads in the stationary regime photochromism (Section V), so that it can be used for both charge states of the NV (provided that the NVs do g(2)(τ)=P (t+τ|t)/P (t) (1) not experience charge conversion, which is the case for 2 1 3 the selected NVs in the present work). their fluorescence spectra. A. Correlation function of a three-level system Inordertogiveaquantitativedescriptionofthethree- level model we first review briefly the interpretation of theg(2) functionusingEinstein’srateequations.Starting from Glauber quantum measurement theory50 we have h:I(t+τ)I(t):i g(2)(τ)= , (3) (hI(t)i)2 whereI(t)isthequantumoperatorequivalenttotheelec- tromagnetic energy flow absorbed by a detector at time t and :: represents normal ordering.38,39 Introducing the creationandannihilationphotonoperatorsa†(t)anda(t) Figure 2: (Color online) (a) Schematic of the HBT cor- in the Heisenberg representation, we obtain : relator (b) Time-intensity second-order correlation function g(2)(τ)ofaNV0centerexcitedat200µWandpresentingan ha†(t)a†(t+τ)a(t+τ)a(t)i antibunching at zero delay, (c) Time-intensity second-order g(2)(τ)= . (4) |ha†(t)a(t)i|2 correlation function g(2)(τ) of a NV- center excited at 5 mW and presenting a bunching at finite delay together with the For a two-level system with ground state g and excited antibunchingatzerodelay.SP750:short-passfilterat750nm state e,the creationoperatora†(t) attime t is to a good approximation proportional to the rising transition ope- rator |e,tihg,t|51, we deduce III. THREE-LEVEL SYSTEM : THEORETICAL p(e,t+τ|g,t) p(e,τ|g,0) g(2)(τ)= = (5) MODEL p(e,t) p(e,0) Fig. 2(b) shows a typical g(2) function measured for wherep(e,t+τ|g,t)istheconditionalprobabilityforthe a single NV at low excitation power. The experimental NV to be in the excited state e at time t+τ knowing data are compared with an equation stemming from a that it was in the groundstate g at time t. Like in Eq. 3 two-level model5,29 : this probability is normalized to a single event probabi- lity, i.e., the probability p(e,t) for the NV to be in the g(2)(τ)=1−e−(r+γ)τ, (2) excited state at the previous time t. The main interest of these equations is to link the probability of detection, where r and γ are the excitation and spontaneous emis- as given by Eqs. 1 and 3, to the emission probabilities sion rates, respectively. Within this model g(2)(τ) < 1, p(e,t+τ|g,t) and p(e,t) defined by the rate equations. which means that the emitted light is non-classical at Therefore,g(2) canbecompletelydeterminedifthetran- any delay τ. However, at higher excitation rate this sient dynamics of the emitter is known. simple model (Eq. 2) generally fails. This is particularly In the context of the NV center,we must use a two-level true for NV- centers. In this case, as shown in Fig. 2(c), systemwitha third metastablestate to explainthe bun- the g(2) function includes a bunching g(2)(τ)>1 feature ching observed in the correlation measurements.52 Since atfinitedelays,superimposedtotheantibunchingcurve. thepreviouscalculationsonlyconsideredtheexcitedand This kind of correlation profile, which contradicts Eq. 2, groundstatesinvolvedinthefluorescenceprocesswewill calls for a third level that traps the electron, preventing admit (see38,54 for a discussion and justification) that subsequent emission of a photon for a certain time.42–49 these results still hold with a three-level system if we re- Therefore, for these delays, the correlation function is placethe excitedandgroundstatesby the levels1 and2 higher than one. Though well known for the NV- center, in the Jablonski diagram, respectively (see Fig. 3). Here theexistenceofashelvinglevelislessdocumentedforits we neglect the channels 1 to 3 and 3 to 2, because the neutralcounterpartbut has been invokedtheoretically48 system is not supposed to be excited at these transition as well as in electron-paramagnetic resonance.49 From energies, in contrast with previous models.26,55 There, now on, we will describe phenomenologically the color channel 3 to 2 was taken into account and channel 3 to center dynamics by a three-level model and deduce the 1 was neglected, because the quantum yield associated intrinsic photo-physical parameters of pure NV- and withtheNVrelaxationwassupposedtobeclosetounity, NV0 centers (Section IV) and of a photochromic center while recent studies show that Q ≃ 0.6− 0.7.2,28,56–58 (section V). These NV centers have been selected from Therefore,we here take into account four channels. This 4 leads to four unknown parameters : the excitation rate, simple expression. More precisely, in addition to neglec- k , the spontaneous emission, k , andthe two parame- ting channels 1 to 3 and 3 to 2, we suppose that : 12 21 ters k and k of the additional decay paths involving 23 31 the shelving level. k21 being the only radiative channel, {k21,k12}≫{k23,k31}. (9) we can write the autocorrelationfunction as : Thisisjustifiedsince,evenifthethirdlevelisconsidered, g(2)(τ)= p2(τ) , (6) the associatedrates aresupposedly verysmallcompared p2(∞) to the singlet rates k12 and k21. We will see latter that this is not really true for the NV- center but that the re- where p2(t) is the population of state 2 at time t. p2(∞) sultsobtainedareactuallyrobustandkeeptheirmeaning represents the asymptotic limit of p2(t) when the tran- even outside of their a priori validity range. Within the sitory dynamics approaches the stationary regime. p2(t) abovementionedapproximationsthesecond-ordercorre- can be obtained by solving the system of rate equations lation function reads (see Appendix A for mathematical defining the three-level system presented in Fig. 3. With details) : g(2)(τ)=1−βe−γ1τ +(β−1)e−γ2τ, (10) where the parameters γ , γ and β are defined through 1 2 the relations : γ ≃k +k , (11) 1 12 21 k k 12 23 γ ≃k + , (12) 2 31 k +k 12 21 k k 12 23 β ≃1+ . (13) Figure 3: (Color online) Jablonski diagram of the three- k (k +k ) 31 12 21 levelsystem,includingthegroundstate(1),theexcitedstate (2) and a metastable state (3). Only the allowed transition channels taken intoaccount in themodel are shown. B. Determination of the k coefficients ij p , i = {1,2,3}, the population of state i we can write i The aim of this sub-section is to determine the k the following set of equations : ij coefficients for the specific measuredNV centers.The fit to the g(2)(τ) function allows us to determine γ , γ and p˙ =−k p +k p +k p , 1 2 1 12 1 21 2 31 3 β. From Eqs. 11-13 we deduce : p˙ =k p −(k +k )p , 2 12 1 21 23 2 p˙3 =k23p2−k31p3, k21 =γ1−k12, (14) γ 1=p1+p2+p3, (7) k31 = 2, (15) β where p˙i means time derivative of pi(t). The use of rate γ1γ2(β−1) k = . (16) equations instead of Bloch equations is here fully justi- 23 βk 12 fied since in ambient conditions, the coherence between levels is decaying very fast.55 Eqs. 7 show that the sys- tem is necessarily in one of the states at any time. The Wethenhavethreeequationsforfourunknownvariables. steady-state analysis of these equations permits to find Afourthequationisneededtosolvetheproblementirely. the explicitdefinitionofthefluorescencerateRatwhich Intheexperiment,wealsohaveaccesstotheradiationor the system emits photons.54 This definition allows us to fluorescencerateR,measuredins−1.This rateissimply explain the saturation behavior of the NV fluorescence, theaveragenumberofphotonsthattheAPDscollectper i.e., R tends towards a finite value for increasing excita- second. It represents, up to a multiplicative coefficient tion power. However, we are here looking for the time- associatedwiththe photonpropagationinthesetup,the dependent analysis to find p (τ) as a function of the k 2 ij probability for the system to be in level 2, multiplied coefficients. If we eliminate p from Eq. 7, we obtain : 3 by the transitionprobability k to relax (supposedly by 21 optical means) to the ground state. We have : p˙ =−(k +k )p +(k −k )p +k , 1 12 31 1 21 31 2 31 p˙ =k p −(k +k )p . (8) R=ξk p (+∞), (17) 2 12 1 21 23 2 21 2 The resolution of this pair of equations, with the initial where ξ is the collection efficiency of the system once conditions p (0) = 1 and p (0) = 0, leads to p (τ) = theNVcenterhasemitteditsfluorescence.Notethatthe 1 2 2 p(2,τ|1,0) and therefore to the expression of g(2)(τ). sameformulawasusedinrefs.26,55 withadifferentdefini- Important approximations can be made to obtain a tionofp (+∞)becauseofadifferentJablonskidiagram. 2 5 Figure 4: (Color online) Time-intensity second-order correlation functions g(2)(τ) for a given NV0 center (green lines) and NV-center(redlines).Theexcitationpowerisvariedfrom0.2mWto10mW.Thebluelinescorrespondtoathree-levelmodel fit. However,this has a physical meaning only if k is asso- 21 ciated with a pure radiative decay. Whereas the model of ref.26,55 implied a unity quantum yield, the quantum yield in our approach is defined by k 21 Q= . (18) k +k 21 23 In our phenomenologicalapproachthe third level is thus assumed to absorb all of the non-radiative transitions letting k be a pure radiative decay. Finally we remind 21 that the probability p (+∞) needed in Eqs. 17 and 6 2 shouldbecalculatedintheasymptoticstationaryregime, which can be obtained by canceling all p˙ in Eq.8 : i k 31 p (+∞)= . (19) 2 −k +k +(k +k )(1+k /k ) 21 31 21 23 31 12 Figure 5: (Color online) Evolution of the γ , γ and β pa- 1 2 Nowfourequationsareathandsothatthesystemcanbe rametersasafunctionoftheexcitationpower.Pointsarefits invertedtodetermineallcoefficients.Thiscalculationin- to the experimental g(2) functions using Eq. 10 and thelines volves the numerical resolution through Cardano’s algo- arelinearregressions.Thegreen(red)pointsandlinescorres- rithm59 ofathird-orderpolynomialasgiveninAppendix pond to the NV0 (NV-) center. Errors bars are shown where B. theyexceedthesymbolsize.Theyrepresentthestandardde- viationofeachvaluecalculatedfromthecovariancematrixas given by thefitting routine. IV. EXPERIMENTAL RESULTS AND EXTRACTION OF THE k PARAMETERS ij coefficients. We here consider two representativeexamples of NDs, Fig. 4 depicts the g(2) function for the two NV centers onehostingasinglepure NV- center,andthe secondone and for excitation powers P ranging from 200 µW to exc hosting a single pure NV0 center. The NV--center ND is 10 mW. The experimental curves are fitted with Eq. 10 about 25 nm in diameter, whereas the one hosting the taking into account the correction for the incoherent NV0 center is about50 nm in diameter.60 The g(2) func- background light collected by the APDs.5,6,29 The ex- tion was recordedfor both NDs with different excitation perimental antibunching dip does not drop to zero due powers P in order to study the evolution of the k to this incoherent background, which modifies Eq. 10 as exc ij 6 ge(2x)p.(τ) = g(2)(τ)ρ2 + 1 − ρ2 , where ρ = S/(S + B) contains the signal S and background B contributions from the NV fluorescence and the spurious incoherent light, respectively. By recording the average intensity fromtheAPD directlyonthe NVandatalocationclose to it, we experimentally determine ρ and the fit parame- ters of Fig. 4 as explained in refs.5,6,29. It is seen that the increase of P only induces a very small bunching exc for the NV0, contrary to what is observed for the NV-. However, the anti-bunching dip narrows with increasing power in both cases. Our observations are collected in Fig. 5, which shows the power evolution of the fit para- meters γ , γ and β. 1 2 The generaltrends seen in Fig. 4 are confirmed in Fig. 5 for both NVs since γ , which is associated with the an- 1 tibunching contribution, is clearly increasing with P , exc a fact which is reminiscent from Eq. 2. Furthermore, it is seenthatγ is alsoincreasingsignificantlyfor the NV- 2 which is clear signature of the third energy level. As far Figure 6: (Color online) Evolution of kij parameters with astheβ parameterisconcerned,itremainsataconstant powerexcitationwhenk isfixedfor(a)theNV0and(b)for 21 value β ≃ 1 for the NV0, while it increases up to β ≃ 7 theNV-.Errorsbarsareshownwheretheyexceedthesymbol fortheNV-.Thisbehavioragreeswithitsdefinitionfrom size. These error bars are estimated from those of the γ , γ 1 2 Eq. 10. Therefore, when β is very close to 1, there is no and β parameters, see Eqs. 14 to 16. significant bunching, and when β increases with power, the bunching turns on. Now that the three parameters have been found, the kij with the assumptions made in Eq. 9. However, for NV-, parameters can be traced back. This will be done in two the same parameters are no longer negligible compared steps. to the set of k , k values. They even overtake k for 21 12 21 P >6mW.However,weemphasizethatassumingk exc 21 constant, if natural, is actually not fully demonstrated. In order to check how robust this hypothesis is, we will A. k parameters with constant k ij 21 now try to approachthe values and evolutions of the k ij parameters that were obtained here by modulating the In Eq. 17 the collection efficiency ξ of the optical se- collection efficiency ξ using Eq. 17. tup must be known precisely to extract the various k . ij However, since this can only be estimated, we will first calculate the k parameters by assuming that k does ij 21 B. Modulation of the collection efficiency ξ not change with the excitation power. This hypothesis is intuitive because the parameter k , i.e., the sponta- 21 Now we use the fourth equation Eq. 17 to calculate neousemissionrate,issupposedtobesolelygovernedby the k parameters. As already stated, we do not know the Fermi’s golden rule, which in turn depends on the ij exactlythe ξ parameterbut,asitturns out,slightvaria- electromagneticenvironmentonly.In order to determine tions in ξ can produce significant changes in k . To find the value of k we observe that according to Eq. 14 we ij 21 the correctvalue of ξ, we adopt the following procedure. must have γ = k at zero excitation since in this case 1 21 We let its value vary continuously andcalculate the evo- k = 0. From the linear regression for γ (Fig. 5) we 12 1 lutions of the slope and Y-intercept of the linear regres- deduce γ(0)(P = 0) = k0 = 0.052 ns−1 for the NV0, 1 exc 21 sionsmadewiththeobtainedk (P )traces.Theslope and γ1(−)(Pexc = 0) = k2−1 = 0.046 ns−1 for the NV- should vanish because k21 is 2a1ssuemxced to be constant, (here the (0) and (-) exponents refer to NV0 and NV-, whereas the Y-intercept should reach the value obtai- respectively). These constants give radiative lifetimes of ned previously, i.e., the value of γ at zero excitation. 1 τ(0) = 19.2 ns and τ(−) = 21.7 ns, which are consistent Therefore, we calculate the evolution of the k parame- 21 21 ij with previous reports (see for example refs.6,55). ters as a function of the collection efficiency ξ. The re- Thanks to Eqs. 14 to 16, we deduce the three other sults are shown in Fig. 7, where the constant horizontal parameters as shown in Fig. 6 for both NVs. The first curves depict the values to be reached by the slope and point to notice is that for both NVs, the k parame- Y-intercept. For the NV0 (Fig. 7 (a)), there is indeed a 12 ter increases linearly with the pumping rate from zero value of ξ where the two parameters reach the assumed to a value exceeding k for P ≃ 2 mW. Moreover, values(blueverticalline).Thisgivesξ(0) =0.77×10−3,in 21 exc for NV0, we see that k increases, but keeps very small agreement with a rough estimate of our setup collection 31 values compared to the other parameters, in agreement efficiency taking into accountthe variousoptical compo- 7 nents. However, for the NV- (Fig. 7 (b)), it is seen that thesymbolsizeinFig.8(a),rightpanel),whichconfirms the very existence of the third metastable level for this charge state. Regarding the g(2) curves, it implies that the optical channel 2 to 1 is favored,which prevents any significant bunching. Furthermore, for the NV0 the nar- rowing of the antibunching dip is due to the increasing excitation rate as for a two-level system, i.e., Eq. 2 (see ref.29). The analysis of the k parameters for the NV- is more ij Figure 7: (Color online) Evolution of the slope and the Y- intersectoflinearfitofk21(Pexc)withthecollectionefficiency ξfortheNV0(a)andtheNV-(b).Straightlinesarethevalues obtained previously with k fixed. 21 the parametersdo notreachexactly the previous values. Yet, there is an optimum ξ for which the k approach Figure 8: (Color online) Evolution of the kij parameters ij with the excitation power taking into account the collection the previousvalues (blue verticalline). This corresponds efficiency for the NV0 (a) and the NV- (b) centers. Errors toξ(−) =1.6×10−3,whichdiffersonlybyafactor2from bars estimated as in Fig. 6 are shown where they exceed the the NV0 case. This appears reasonable since the measu- symbol size. rementswerenotcarriedoutthesameday(opticalalign- ments might be slightly different) and the collection effi- involved. Indeed, k increases very quickly to reach the 23 ciency ξ also depends on the unknown transition-dipole order of magnitude of k , while the latter is increasing 21 orientation in both NVs. as well (at zero excitation power we have 1/k ≃ 24 ns 21 and1/k ≃500ns).Theincreaseofk ,associatedwith 23 23 non-radiativetransitions,actuallyexplainsthegrowthof C. Comparaison of the photophysics of NV centers the bunching feature on the g(2) curves. Although the in both charge states physicaljustificationofthis findingisbeyondourpheno- menological treatment, it is likely that the variation of Fig. 8 depicts the k (P ) curves deduced from the k and k with excitation power is due to a change in ij exc 21 23 previous optimization. It is found that the order of ma- the local energy environment of the NV center at high gnitudeofthecoefficientsisthesameasforimposedk . power, in particular because the efficient coupling with 21 In particular, the relaxation rate k is unchanged be- the phonon bath in the diamond matrix is expected to 31 cause it only depends on the g(2) fit parameters, Eq. 15. betemperaturesensitive.Increasingtheexcitationpower For the two NV centers, the excitation rate vanishes in could thus correspond to an increasing effective tempe- theabsenceofanyexcitationpowerandlinearlyincreases rature, subsequently affecting the relaxationdynamics. withP asitshould.Themaindifferencebetweenboth It is worth pointing out that there is a limitation in the exc centers comes from the evolution of k , k and k . In- analysis done for the NV- center. Indeed, the very fact 23 31 21 deed, for NV0, k and k remain very small compared thatk and{k ,k }reachthesameorderofmagnitude 23 31 23 21 12 to k , which is almost constant (1/k ≈ 500 ns and contradictsthehypothesismadeinEqs.9-13.Actually,as 21 31 1/k ≈ 1000 ns if P tends to zero). Therefore, the already mentioned, the results obtained are much more 23 exc third level plays little role in the photodynamics of the robust that could be anticipated at first sight. This can NV0 center. However,it is worth stressing that although befiguredoutbyrelaxingtheconstraintofEq.9asdone verysmall,k isnotzeroforNV0(errorsbarsarewithin inthedetailedcalculationpresentedinAppendix C.The 23 8 at high power. This entails the fact that the NV- dyna- mics is strongly dependent on the excitation power as discussed before. V. PHOTOCHROMISM Photochromism of NV centers has been reported in ensembles of NV centers in CVD diamond films Figure 9: (Color online) Evolution of kij parameters with underadditionalselectiveillumination61,withsingle NV power excitation taking into account the collection efficiency centers in 90 nm NDs under femtosecond illumination, for the NV- in the approximation-free rate-equation model. which results in the photo-ionization of the negative ErrorsbarsestimatedasinFig.6areshownwheretheyexceed center to its neutral counterpart35, with ensembles thesymbol size. of NV centers in type Ib bulk diamond at cryogenic temperatures under intense CW excitation62, and with a single center in natural type-IIa bulk diamond under results obtained with the new rate-equation model are CW illumination.36 In this last report, a special scheme shown in Fig. 9 for the evolution of the kij coefficients of cross-correlation photon measurements was applied in the emission band of both charge states to show that the collected fluorescence in the NV0 and NV- states were correlated and originated form a single NV defect. Several studies have reported that charge conversion within the NVs critically depends on the illumination conditions.57,63,64 A complete understanding of NV photochromism is lacking but a widespread view is that the optical excita- tion, either CW or transient,tunes the quasiFermi level around a NV charge transition level, thereby inducing chargeconversion.61,65 This scenariohas been reinforced recently by electricalmanipulationof the charge state of NV ensembles and of single NVs by an electrolytic gate electrodeusedtotune the Fermienergy.65 Depending on the sort of diamond studied, photochromism is thought to be favored by the presence of electron donor or acceptor defects, such as nitrogen, in the neighborhood of the NV center.66 Recently, it was also found that re- Figure10: (Color online)Evolution ofthefluorescencerate sonantexcitationoftheNV0andNV-statesinultrapure (upper panel) and the quantum yield Q (lower panel) with synthetic IIa bulk diamond can induce reversible charge the excitation power for the NV- and NV0 center in the rate conversion in cryogenic conditions even at low power.10 equation model free of any approximation. Errors bars for This was taken as evidence that the charge conversion Q are shown where they exceed the symbol size. They are process is intrinsic in this sort of diamond, not assisted estimated from those of thekij parameters, see Eq.18. by an electron donor or acceptor state. The goal of this section is to give additional information on NV of the NV- center. The obtained values are very similar photochromism detected in surface-purified NDs, 25 nm tothoseoftheapproximatetreatment,therebyjustifying in size, subjected to a CW non-resonant excitation of thepreviousresults.FortheNV0thecouplingtothethird increasing power. By comparing the behavior of a single level is very weak and the modifications (not shown) are photochromiccentertothatofnon-photochromiccenters even smaller. in the same illumination conditions as described above, Tocomplete the analysiswe alsocomputedthe quantum we gainvaluable informationonthe richphotophysicsof yieldevolutionasgivenbyEq.18andcomparedwiththe photochromism. evolution of the fluorescence rate. Fig. 10 confirms that For the purpose of studying photochromism, we use within the Jablonskimodel sketchedin Fig. 3 and in the two additional configurations of the HBT correlator considered excitation regime, the quantum yield of the that differ only by the set of bandpass filters added NV0 center is approximately constant Q ≃ 1, in agree- in the interferometer branches. These configurations, ment with the intuitive factthat the third leveldoes not called NV-/0, and NV0/- respectively, are shown sche- play a significant role in the dynamics. In contrast, the matically in Fig. 11. In contrast to Fig. 2(a), these NV- quantum yield decreases dramatically with increa- two configurations add selective bandpass filters in the sing excitation power from a starting Q ≃ 1 to Q ≃ 0.5 interferometer branches. The NV-/0 configuration in 9 Pexc.(mW) 0.5 1 2 3 5 R(kHz) 11.8 17.1 23.0 27.7 32.0 g(2) (0) 0.25 0.32 0.48 0.45 0.6 exp. S(kHz) 10.2 14.1 16.6 20.5 20.2 B(kHz) 1.6 3.0 6.4 7.2 11.8 β 1.36 1.45 1.75 1.6 2.5 γ (ns−1) 0.03 0.036 0.044 0.052 0.054 1 γ (ns−1) 0.005 0.007 0.009 0.012 0.018 2 Figure11: (Coloronline)HBTconfigurationsusedtostudy Table I: Table summarizing the experimental parameters photochromism.(a)istheNV-/0configuration,whereaband- Pexc., R, ge(x2)p.(0), S, B measured on the photochromic NV pass filter (central wavelength 675 nm, bandwidth 67 nm) for the observation of the ge(x2)p. function. The fit parameters adapted to the negative NV- center is inserted in the upper γ1,γ2 and β using a 3 energy-level model are also given. start branch, whereas a bandpass filter adapted to the neu- tral NV0 (central wavelength 593 nm, bandwidth 46 nm) is inserted in thelower stop branch.(b)is theNV0/-configura- center. tion where both bandpass filters are interchanged compared TherelevantspectraareshowninFig.12(a).Itisfound to (a). thatboththe NV0 andNV- ZPLs areseenatanyexcita- tionpower.Thecorrespondingg(2) functionmeasuredin Fig. 11 (a) (respectively NV0/- configuration in Fig. 11 the standard configuration of the HBT (no bandpass fil- (b)) uses a filter selective to the NV- (respectively NV0) ter)isshowninFig.12(b).Itrevealsaclearantibunching fluorescence in the start (respectively stop) branch. dip at zero delay. A precise analysis taking into account Therefore, in the NV-/0 configuration, a single photon the background light (see Table 1) confirms this finding emitted by a NV- center and detected in APD1 gives since at high excitation power we observe that the back- the « Start » signal to the counting module, whereas ground B increases significantly with respect with the a single NV0 photon subsequently detected in APD2 NV fluorescence signal : ρ = S/(S +B) ≃ 0.6 while at produces the « Stop» signal. The NV0/- configuration low power ρ ≃ 0.9−1. With this value and using the works in just the complementary way. Note that our formula ge(2x)p.(τ)/ρ2 +1−1/ρ2 = g(2)(τ) we deduce the NV-/0 configuration is similar to the cross-correlation actual value of g(2)(0) ≃ 0.6,25,55,72 From these results, technique used in ref. 36. These schemes turn out to a natural interpretation for the observation of the NV0 be very powerful to study photochromism since cross and NV- ZPLs together with NV uniqueness is that this correlation can be expected only if NV- and NV0 particular ND is subjected to photochromism. In addi- photons originate from the same defect center. Note tion to the antibunching dip at zero delay, it is seen in that related techniques have also successfully been Fig. 12 that g(2) exceeds 1 at longer delays.55 In agree- applied to identify various excitonic species emitted by ment with the previous sections, this is evidence for the singlesemiconductorquantumdots,seeforinstance.67–71 presence of a trapping level and calls for a three-level description of the photochromic NV. The values of γ , 1 (2) γ2 and β parameters used for fitting the gexp. function agree qualitatively well with those obtained for the NV- consideredpreviouslyinagreementwiththefactthatthe system is acting more like a three-level system. AlthoughourstudyofNVphotochromismislimitedtoa particular example we suggest that the dynamics of the systeminvolvesprobablyallenergylevelsoftheNV-and NV0 with some possible hybridization. It could be inter- estingtoknow,whetherornot,thethirdlevelinvolvedin the photochromic case is identical in nature to the third level of the NV-. The role of the environment or of the Figure 12: (Color online) (a) Fluorescence spectra of the radiationpower73 onthe dynamics couldbe investigated photochromicNDnanoparticletakenat3differentexcitation in the future to clarify this point. powers of nominally 0.5 mW, 1 mW, 2 mW from bottom to Switchingnowtothecross-correlationregimeweshowin top,respectively.(b)Antibunchingcurveatshorttimedelays Fig.13theresultobtainedforP =3mW (similarfea- exc. and an excitation power of 5 mW. tures, not shown here, were observed at other excitation powers). It is worth emphasizing that cross correlations, With the abovesetup,wehavelocateda few rareNDs as described in e.g. refs.68,69,74, allow us to characterize hosting a single NV center that showed charge conver- the transitory dynamics between the two NV configura- sionatlowexcitationpoweralready.Inthe following,we tions. Using a formalism equivalent to the one leading consider such a ND hosting a single photochromic NV to Eqs. 3-5 we indeed obtain in the NV-/0 configuration 10 should present some symmetries. In the present case we used a three energy-levelfit, i.e. Eq. 10, for the theoreti- cal functions, Eqs. 20 and 21. The parameters obtained to reproduce the data are the same for Fig. 13 (a) and (b)uptoaninversionbetweenthe‘positive’and‘negati- ve’delayfor eachgraph.Therefore,we getfor Fig.13(a) γ =0.046 ns−1, γ =0.01 ns−1 and β =1.2 for τ <∆, 1 2 while we have γ = 0.052 ns−1, γ = 0.01 ns−1 and 1 2 β =1.5forτ >∆.Forthesecondcross-correlationcurve the parameters are identical but the roles of τ > ∆ and Figure 13: (Color online) Time-intensity second-order cor- τ < ∆ are inverted as it should be. We observe that relation functions measured for the photochromic NV center these values are very close to each other and also from in two different HBT configurations. (a) is the cross correla- theoneobtainedinTable1atthesameexcitationpower tion NV-/0 configuration; (b) is the cross correlation NV0/- P = 3 mW. This confirms that the system acts here configuration. The temporal window is [-100 ns; 500 ns] and exc. theexcitation power is 3 mW. mainly as a NV- center. Interestingly,the lastfinding impliesthatafter the emis- sion of a photon in the spectral fluorescence band of the sketched in Fig. 11 (a) NV-thedelayedemissionofasecondphotoninthe spec- tral band of the NV0, i.e., the conditional probability p(e,NV0,t+τ|g,NV-,t) given by Eq. 20, is also characterized by the dynamics g(2) (τ)= (20) −/0 p(e,NV-,t) of the NV- contrary to the intuition. Such behavior was reported in ref.36 for NV centers in bulk. In particular, where p(e,NV0,t+τ|g,NV-,t) is the conditional proba- inthis paper the time dependence ofthe conversionNV0 bility for the NV to be in the excited level e of the NV0 to NV- process (and its inverse) was studied using pulse state at time t +τ knowing that it was in the ground sequences. It was found that the relaxation from NV- to energy level g of the NV- charged state at the previous NV0 is a very slow process occurring with a decay time time t. This also means that a first photon emitted by ≃ 1 µs. This agrees with our finding in Fig. 13 since, the NV0 was detected at time t while a second photon even if the full dynamics of the photochromic NV cen- emitted by the NV- is detected at time t+τ . In a sym- ter is expected to depend on the energy levels of both metrical way using the NV0/- configuration sketched in the NV- and NV0 centers, it is clearly the NV- charac- Fig. 11 (b) we get teristicswhichdominateduringthetransitionassociated with Eq. 20. A similar qualitative analysis can be done p(e,NV-,t+τ|g,NV0,t) in the NV0 to NV- center conversion. The small dissym- g(2) (τ)= (21) 0/− p(e,NV0,t) metry between the two parts of the curves for positive and negative delays results from the presence of a small withsimilardefinitionsaspreviouslybutwiththe roleof NV0contributiontothedynamicsduringtheNV0toNV- NV0 and NV- inverted. transitionwhich is absentin the NV- to NV0 conversion. The experimental results corresponding to these two Clearly, this complex charged/uncharged transition dy- configurations are shown in Figs. 13(a) and (b), respec- namics would deserve systematic studies in the future. tively. Here, it is also important to have τ ≥ 0 in the calculation leading to Eqs. 20 and 21 in order to have a clear physical understanding. However, the electronic VI. SUMMARY delay ∆ = 100 ns included in the HBT correlator setup implies that sometimes even a photon emitted, say at To summarize, we have experimentally studied the timet ,byaNV-isrecordedafterasecondphotonemit- 0 fluorescence photodynamics of NV- and NV0 centers tedlaterbytheNV0state,i.e.,att +τ.Thiscorresponds 0 in diamond nanocrystals of 50 nm size or below using to the ’negative’ delay part of the graph,i.e., Fig. 13(a), HBT photon-correlation measurements as a function which is actually associated with the inverse dynamics of the excitation power. The dynamics was theoreti- NV0/-,i.e.,Eq.21.Hereforclaritywedidnotsubtractthe cally modeled using Einstein’s rate equations and the delayfromtheabscisesinFig.13.Thereforeforτ >∆in transition probability rates k entering the three-level Fig.13(a)wehaveg(2) (τ)=g(2) (τ−∆)ρ2+1−ρ2while ij exp. −/0 model developed to analyze the data were deduced and we haveg(2) (τ)=g(2) (∆−τ)ρ2+1−ρ2 for 0<τ <∆ used to infer a quantum efficiency to both charge states exp. 0/− with ρ = S/(S +B) as previously. In the same way for of the NV. It has been found that the shelving state, Fig. 13(b) we have g(2) (τ) = g(2) (τ −∆)ρ2 +1−ρ2 though present, plays a very small role on the neutral exp. 0/− center in those small diamond crystals. The narrowing for τ > ∆ and ge(2x)p.(τ) = g−(2/)0(∆−τ)ρ2 +1 −ρ2 for of the antibunching dip observed with increasing power 0<τ <∆. for this center is a simple power effect that does not As itisclearfromthe definitions these cross-correlations affect the near unity quantum efficiency. In contrast,