Theory of the optical spinpolarization loop of the nitrogen-vacancy center in diamond
TTheory of the optical spinpolarization loop of the nitrogen-vacancy center in diamond
Gerg˝o Thiering
1, 2 and Adam Gali
1, 2, ∗ Wigner Research Centre for Physics, Hungarian Academy of Sciences, PO Box 49, H-1525, Budapest, Hungary Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki ´ut 8., H-1111 Budapest, Hungary
Nitrogen-vacancy (NV) center in diamond is of high importance in quantum information process-ing applications. The operation of NV center relies on the efficient optical polarization of its electronspin. However, the full optical spinpolarization process, that involves the intersystem crossing be-tween the shelving singlet state and the ground state triplet, is not understood. Here we develop adetailed theory on this process which involves strong electron-phonon couplings and correlation ofelectronic states that can be described as a combination of pseudo and dynamic Jahn-Teller inter-actions together with spin-orbit interaction. Our theory provides an explanation for the asymmetrybetween the observed emission and absorption spectra of the singlet states. We apply density func-tional theory to calculate the intersystem crossing rates and the optical spectra of the singlets andwe obtain good agreement with the experimental data. As NV center serves as a template for othersolid-state-defect quantum bit systems, our theory provides a toolkit to study them that might helpoptimize their quantum bit operation.
I. INTRODUCTION
The most known point defect in diamond is thenitrogen-vacancy (NV) center which acts as a quan-tum bit for solid state quantum information processingapplications . NV center is a negatively charged com-plex which consists of a substitutional nitrogen next toa vacancy in diamond [see Fig. 1(a)]. The defect hasan S = 1 ground state with milliseconds coherence timeat room temperature in C enriched diamond samples ,and can be optically excited in the visible . Under illumi-nation, the electron spin is preferentially populated at the m s = 0 spin state over the m s = ± . The ro-bust spin-selective fluorescence and photocurrent arethe most important features of this center that can beused for quantum bit initialization and readout schemes.Group theory considerations together with lumi-nescence and absorption measurements imply thattwo singlet levels, A and E , that are separated by1.19 eV, reside between the A ground state and E ex-cited state triplet levels. In the optical spinpolarizationcycle both singlet states play a role [see Fig. 1(c)]. Inthe upper branch, highly spin-selective intersystem cross-ing (ISC) occurs between the E triplet and A singletcaused by the phonon-mediated spin-orbit interaction.Combined photoluminescence excitation (PLE) measure-ments and perturbation theory on the ISC rates an-alyzed this process in detail. We note here that the ob-served multiple rates between the E triplet substatesand the singlet A goes against the selection rules that would allow only a single scattering channel [Γ A ,i.e., purple dotted arrow in Fig. 1(c)]. The multiplerates can be naturally explained by invoking the dynamicJahn-Teller effect on the E state which can account wellto the ratio of the observed ISC rates at cryogenic tem-peratures . The dynamic Jahn-Teller effect is a specialdescription of a strongly coupled electron-phonon sys-tem that mixes the pure electronic substates of E witheach other that results in vibronic states that are labeledby tilde in Fig. 1(c). The mixture of A into (cid:101) E , by phonons results in ISC from (cid:101) E , toward A [Γ E , , i.e.,green dotted arrow in Fig. 1(c)].However, the ISC process in the lower branch, i.e., be-tween E and A , is still not understood. By consideringsingle determinant E state built up from the e orbitals[see Fig. 1(b) and Refs. 15 and 16], group theory indi-cates that no ISC is allowed between this singlet and the A triplet. On the other hand, the measured lifetimeof the E state is T E = 371 ± and ISC toward the m s = 0 of A [Γ z , bluedotted arrow in Fig. 1(c)] should be effective to observespinpolarization in NV center. The measured T E is tem-perature dependent and decreases down to ∼
165 ns atroom temperature . The temperature dependence couldbe well understood by a stimulated phonon emission pro-cess with an energy of 16 . ± . . By means of spincontrol and pulsed optical excitation of NV center, thespin-dependent ISC rates of E were extracted at roomtemperature . Interestingly, it was deduced that theISC rates from E to m s = 0 and m s = ± ± and Γ ∓ ,red and orange dotted arrows in Fig. 1(c), respectively]are comparable. As the ISC is dominantly spin-selectivein the upper branch, this conclusion is not contradic-tory to the measured >
90% optical spinpolarization inthe triplet ground state. From these experimental datawe may conclude that E is linked to the m s = ± A state by spin-flipping transitions (rates of Γ ± and Γ ∓ ) and to the m s = 0 by the rate of Γ z , where T − E = Γ z + Γ ± + Γ ∓ =2.70 MHz at cryogenic temper-atures. In a previous measurement , a similar value, T − E =2.16 MHz was deduced. We emphasize that under-standing the mechanisms governing the ISC between E and A is very important as this ISC is responsible forclosing the optical spinpolarization loop of the NV cen-ter which is the base of quantum bit initialization andreadout.In this study we derive the electron-phonon assistedspin-orbit interaction between the E state and the A state. We show that the nature of | E (cid:105) is highly com-plex as it involves significant electron-phonon coupling to a r X i v : . [ qu a n t - ph ] J un F e r m i ene r g y ( e V ) spin down e ↑ a ↑ e ↓ a ↓ spin up [111] . Å (a) (b) VBCB (c) E n e r g y |0 〉 | A 〉 | A 〉 | + 〉 | − 〉 | E ~ + 〉 | E ~ − 〉 | E ′ + 〉 | E ′ − 〉 | E 〉 |± 〉 |0 〉 Γ A Γ E (ae)(ae)(ee) (ee)(ee) | E ~ 〉 ZPL:1.19eV Γ z Γ ± Γ ∓ ZPL:1.945eV | E ~ 〉 | E ~ x,y 〉 | A ~ 〉 | A ~ 〉 ~ FIG. 1. Properties of NV center in diamond. (a) Geometry structure showing the symmetry axis of C v symmetry. Thevacancy is the dotted circle whereas the solid circles depict carbon atoms in the diamond lattice. (b) Calculated HSE06 Kohn-Sham levels in the diamond band gap between the valence (VB) and conduction (CB) band. (c) Many-electron states areexpressed in two-particle Slater-determinants in the parentheses (see Eqs. (1)-(6)). The many-electron levels are also depictedwith the measured zero-phonon-lines (ZPL). The zero-field-splittings in the triplet manifolds are artificially scaled up by fiveorders of magnitude for the sake of clarity. We label the possible intersystem crossing rates (Γs) with colored dotted arrowsthat participate in the spin polarization cycle. We label the radiative transitions in the aforementioned cycle with solid blackarrows. We refer to the vibronic states (coupled electron-phonon wavefunctions) by tilde labels above the many-electron states.The nature of the vibronic (cid:101) E state is explained (blue and red solid curves and texts) where ee , PJT and DJT stands forelectron-electron, pseudo Jahn-Teller, and dynamic Jahn-Teller interactions, respectively. the | A (cid:105) and electron-electron interaction to the | E (cid:48) (cid:105) manifold. The former interaction can be described in theframe of pseudo Jahn-Teller (PJT) effect and is re-sponsible for closing the optical spinpolarization loop ofNV center. The latter one brings a dynamic Jahn-Teller(DJT) character to the | E (cid:105) , and explains the observedISC rate toward the m S = ± A ground state.We identify a novel interplay between PJT and DJT in-teractions that determine the phonon sideband of thephotoluminescence (PL) spectrum of the singlets . Ourresults also explain the appearance of a new feature in thePL phonon sideband upon the applied uniaxial stress and the asymmetry in the phonon sidebands of the PLand absorption spectra of the singlets. We use ab ini-tio wavefunctions and adiabatic potential energy surfaces(APES) to quantify the strength of interactions and thecorresponding temperature dependent ISC rates wherethe latter ones show good agreement with the experi-mental data.The paper is organized as follows. Sec. II describesthe electronic structure of NV center and establishes thenomenclature of the paper. Then we describe the abinitio methods in Sec. III. We present the theory ofpsedo Jahn-Teller effect and the dynamic Jahn-Teller ef-fect brought by electron-electron correlation on the shelv-ing singlet state in Sec. IV which contains the main ideaof the paper. Sec. V contains the main results of thepaper, where we apply ab initio calculations to calculatethe optical spectra and ISC rates based on the developed theories in Sec. IV. Finally, we summarize and concludeour results in Sec. VI. II. METHODOLOGY ON ATOMISTICSIMULATIONS
We apply ab initio wavefunctions and APES for de-termining the electron-phonon couplings, calculating theoptical spectra and ISC rates in the framework of spin-polarized density functional theory (DFT) as imple-mented in the vasp . We use the HSE06hybrid functional within DFT that technique repro-duces the experimental band gap and the charge tran-sition levels in Group-IV semiconductors within 0.1 eVaccuracy . We converged the electronic structure withself-consistent cycles on Kohn-Sham orbitals with a lowenergy cutoff (370 eV) within the applied projector-augmentation-wave-method (PAW) . The total en-ergies of the excited states were calculated within the∆SCF method that provides accurate zero-phonon-line(ZPL) energy and Stokes-shift for the optical excitationspectra of the triplets of NV center.The negatively charged NV defect is modeled in a512-atom supercell and Γ-point is applied to sample theBrillouin-zone. We determine the equilibrium position ofions by minimizing the quantum mechanical forces actingon them below the threshold of 10 − eV/˚A. In the APES,the C h distorted geometries exhibit the deepest energyconfigurations. The APES around high C v symmetryconfigurations toward the low C h symmetry configura-tions is calculated. The corresponding normal modes ofthe E -symmetry phonons participating in the distortionare calculated in the A ground state by using the quasi-harmonic approximation and finite difference method onthe quantum mechanical forces.In the calculation of ISC, the spin-orbit couplingshould be determined between the corresponding states.In our previous work, we determined the z -componentof the spin-orbit coupling ( λ z ) accurately by our DFTmethod that resulted in λ z =15.78 GHz. We also foundthat the calculation of the perpendicular component ofthe spin-orbit coupling ( λ ⊥ ) requires approximations (seeRef. 21 for discussion) that lead to a significant overesti-mation in λ ⊥ . Therefore, we use λ ⊥ here as a parameterwhich should be greater than the calculated λ z .The phonon sideband in the absorption spectrum isdescribed by Huang-Rhys (HR) theory that was pre-viously implemented for DFT supercell methodology .The contribution of the a and e phonons in the phononsideband of the PL spectrum for the triplets of NV centerwas obtained by this methodology where the latter isresponsible for the DJT effect in the E excited state.Finally, we note that our HSE06 DFT method can-not directly calculate the many-body A and E singletstates. Therefore, the energy gap between the E and A states (Σ) is a parameter. The full ab initio descrip-tion requires to go beyond Kohn-Sham DFT that can de-scribe strong correlation between localized electrons .On the other hand, we will show that HSE06 DFT singletstates within the ∆SCF framework can be employed toderive the parameters for the Jahn-Teller Hamiltoniansand estimate the strength of correlation between the E (cid:48) and E states. III. PRELIMINARIES
Here we define the basic nomenclature of the paper.We note that the orbitals and levels of NV center fromDFT calculations have been already published in severalpapers . Furthermore, the corresponding many-bodystates and the spin-orbit couplings between them werealso thoroughly analyzed . Instead of frequently re-ferring to these papers, we rather write explicitly the cor-responding wavefunctions and interactions that we use inthe entire paper.
A. Electronic structure
NV defect introduces an a and a double degenerate e level in the gap [ a and e orbitals in Fig. 1(b)] thatare occupied by four electrons in the relevant negativelycharged state. In the hole picture, two holes are left onthe e orbital in ground state electron occupation that wesimply label as ( ee ). The many-body ground state triplet state with labeling the m S = { + , , −} spin projectionscan be described as (cid:12)(cid:12) A +2 (cid:11)(cid:12)(cid:12) A (cid:11)(cid:12)(cid:12) A − (cid:11) = | e + e − (cid:105) − | e − e + (cid:105)√ ⊗ |↑↑(cid:105) √ ( |↑↓(cid:105) + |↓↑(cid:105) ) , |↓↓(cid:105) (1)where we introduced the | e ± (cid:105) = √ ( | e x (cid:105) ± i | e y (cid:105) ) com-plex combination of the e { x,y } real orbitals. In the ( ee )electronic configuration, a double degenerate E and anon-degenerate A state appear as (cid:12)(cid:12) E ∓ (cid:11) = | e + e + (cid:105)| e − e − (cid:105) (cid:27) ⊗ √ |↑↓(cid:105) − |↓↑(cid:105) ) (2)and (cid:12)(cid:12) A (cid:11) = 1 √ | e + e − (cid:105) + | e − e + (cid:105) ) ⊗ √ |↑↓(cid:105) − |↓↑(cid:105) ) . (3)The optically allowed triplet E excited state can bedescribed as an electron promoted from the a to the e or-bital in the spin minority channel [see the inclined arrowin Fig. 1(b)] which can be given an ( ae ) configuration inthe hole picture, (cid:12)(cid:12) E ± (cid:11) = √ ( | e + a (cid:105) − | ae + (cid:105) ) √ ( | e − a (cid:105) − | ae − (cid:105) ) (cid:41) ⊗ |↑↑(cid:105) √ ( |↑↓(cid:105) + |↓↑(cid:105) ) . |↓↓(cid:105) (4)Beside the triplet state, a double degenerate E (cid:48) statecan be constructed as (cid:12)(cid:12) E (cid:48)± (cid:11) = √ ( | e + a (cid:105) + | ae + (cid:105) ) √ ( | e − a (cid:105) + | ae − (cid:105) ) (cid:41) ⊗ √ |↑↓(cid:105) − |↓↑(cid:105) ) . (5)Both states are Jahn-Teller unstable because a single holeis left in the double degenerate e orbital. We note that E in the ( ee ) electronic configuration is not a Jahn-Tellersystem as closed-shell singlet states are formed in Eq. (2).We note that a high energy A (cid:48) may also exist as follows (cid:12)(cid:12) A (cid:48) (cid:11) = | aa (cid:105) ⊗ √ |↑↓(cid:105) − |↓↑(cid:105) ) . (6)We particularly focus on the interactions between thesinglet states in which an alternative description of thestates is useful. The 3 dimensional | E (cid:105) ⊕ | A (cid:105) can bealso expressed by these singlet wavefunctions, | xx (cid:105) = | e x e x (cid:105)| xy (cid:105) = √ [ | e x e y (cid:105) + | e y e x (cid:105) ] , | yy (cid:105) = | e y e y (cid:105) ⊗ √ |↑↓(cid:105) − |↓↑(cid:105) ) (7)where | xx (cid:105) is a single Slater-determinant and can be cal-culated by our HSE06 DFT method. Finally, the | E (cid:105) and | A (cid:105) in this basis are (cid:12)(cid:12) E x (cid:11) = √ ( | xx (cid:105) − | yy (cid:105) ) (cid:12)(cid:12) E y (cid:11) = | xy (cid:105) (cid:12)(cid:12) A (cid:11) = √ ( | xx (cid:105) + | yy (cid:105) ) (8)that are equivalent to Eqs. (2) and (3).The order of the corresponding levels can be cor-rectly computed by means of configurational interactionor Hubbard Hamiltonian numerical methods that re-sults in A , E , A , E , and E (cid:48) levels in ascendingorder. The A (cid:48) level resides far above that of E (cid:48) . Thisagrees well with the experimental data and previ-ous group theory considerations too. The ZPL en-ergies between the triplets and in-between singlets are at1.945 eV and 1.19 eV, respectively, are known from PLexperiments . The energy gap between the E and A levels [∆ in Fig. 1(c)] is estimated to be ∼ . That would result in Σ ≈ . E and A levels. Hubbard Hamiltoniancalculations within supercell method , that could nearlyreproduce the visible and near infared ZPL energies, in-deed yielded about 0.4 eV gap between the singlet-tripletlevels both in the upper and lower branches. B. Spin-orbit coupling between the states
We introduce the spin-orbit coupling (SOC) betweenthe electronic states that is responsible for ISC. The SOCmatrix elements between the possible two-particle many-body states can be derived by combining the group the-ory and the two-component spin-orbit operator on theSlater-determinants of orbitals where λ z and λ ⊥ compo-nents of SOC corresponds to the spin-projection conserv-ing and flipping mechanisms, respectively. By followingthe convention in Ref. 16, the A states are linked to A and E (cid:48) states as follows,ˆ W = 2 iλ z (cid:12)(cid:12) A (cid:11) (cid:10) A (cid:12)(cid:12) + iλ ⊥ (cid:2)(cid:12)(cid:12) E (cid:48) + (cid:11) (cid:10) A +2 (cid:12)(cid:12) + (cid:12)(cid:12) E (cid:48)− (cid:11) (cid:10) A − (cid:12)(cid:12)(cid:3) + c.c. , (9)where the triplet m s = { , + , −} manifolds are labeledas a subscript in A . The most important conclusion isthat E in the ( ee ) electronic configuration is not linkedto A . Thus, we seek a possible mechanism that mixes | A (cid:105) and | E (cid:48) (cid:105) characters into | E (cid:105) , otherwise therewould be no any allowed ISC from | E (cid:105) to | A (cid:105) . IV. THEORY ON THE NATURE OF THESHELVING SINGLET STATE
In the next sections we derive an approximate wave-function of the | E (cid:105) including the effects from electron-phonon coupling and many-body electron interaction.First, we derive the pseudo Jahn-Teller effect between the A and E states in Sec. IV A. Next, we determine thedynamic electron-electron correlation between | E (cid:105) and | E (cid:48) (cid:105) in Sec. IV B that induces a small but non-negligibledynamic Jahn-Teller effect in | E (cid:105) . We combine the twoeffects in Sec. IV C. Despite the small DJT effect, we willdemonstrate in Sec. V that only the combination of PJT and DJT accounts for the near infrared PL lineshape ofthe NV center. A. Pseudo Jahn-Teller effect between the lowestenergy singlet states
Since the lowest energy E and A states have dif-ferent irreducible representations only the symmetry dis-torting E vibration modes may couple the two states.This effect is known as pseudo Jahn-Teller (PJT) effectin the literature . We work out the PJT Hamiltonianin the basis of | xx (cid:105) , | xy (cid:105) , and | yy (cid:105) wavefunctions (seeEqs. (7) and (8)). By assuming an electronic gap of Λ e between the E and A before turning on the electron-phonon interaction and setting the energy of E to zerowe arrive atˆ H = Λ e (cid:124) (cid:123)(cid:122) (cid:125) = ˆ H e +¯ hω E (cid:88) α ∈{ x,y } a † α a α + 1 (cid:124) (cid:123)(cid:122) (cid:125) = ˆ H osc . + (cid:101) F (ˆ σ z ˆ x − ˆ σ x ˆ y ) , (cid:124) (cid:123)(cid:122) (cid:125) = ˆ H PJT (10)where ˆ H e , ˆ H osc . and ˆ H PJT are the electronic, harmonicoscillator and linear PJT Hamiltonian, respectively. Wenote that Λ e is not exactly the ZPL energy (Λ) betweenthe singlets because that will be corrected by the vibronicenergies coming from the electron-phonon interactions.The ˆ σ z and ˆ σ x operatorsˆ σ z = − ˆ σ x = 1 √ (11)are matrix representation of the L = 1 angular mo-mentum pointing to the electronic degree of freedomin the PJT interaction. The E vibration mode is de-scribed by a † x,y creation and a x,y annihilation operatorswith ω E frequency. For the sake of simplicity we usedimensionless coordinates where ˆ x = ( a † x + a x ) / √ y = ( a † y + a y ) / √
2. We show below that other statescontribute to the electron-phonon interaction in the E state. B. Dynamic electron-electron correlation betweenthe | E (cid:105) and | E (cid:48) (cid:105) states and the appearance ofdynamic Jahn-Teller effect The electron-electron correlation is possible among themany-body states with the same irreducible representa-tion as described in Eqs. (1)-(5).Thus it might be possi-ble that the | A (cid:105) and | A (cid:48) (cid:105) correlate at a certain degreeas well as the | E (cid:105) and | E (cid:48) (cid:105) do that similarly. We fo-cus on the mixture of | E (cid:105) and | E (cid:48) (cid:105) as this would allowΓ ⊥ = Γ ± + Γ ∓ ISC process between | E (cid:105) and | A ± (cid:105) .The mixing coefficient C describes a multi-determinantsinglet state ( (cid:12)(cid:12) ¯ E (cid:11) ) as (cid:12)(cid:12) ¯ E (cid:11) = C (cid:12)(cid:12) E (cid:11) + (cid:112) − C (cid:12)(cid:12) E (cid:48) (cid:11) , (12)where C can be chosen to be real without losing the gen-erality. Since (cid:12)(cid:12) E (cid:48) (cid:11) is an E ⊗ e DJT system (cid:12)(cid:12) ¯ E (cid:11) carriesa DJT character by the extent of (1 − C ). The DJTHamiltonian of (cid:12)(cid:12) E (cid:48) (cid:11) isˆ H JT = F (cid:0) ¯ σ z ˆ X − ¯ σ x ˆ Y (cid:1) , (13)where the electronic degree of freedom is expressed by¯ σ z and ¯ σ x Pauli matrices spanning the two dimensional | E (cid:48) (cid:105) space that can be written as ¯ σ z = | E (cid:48) x (cid:105)(cid:104) E (cid:48) x | −| E (cid:48) y (cid:105)(cid:104) E (cid:48) y | and ¯ σ x = | E (cid:48) x (cid:105)(cid:104) E (cid:48) y | + | E (cid:48) y (cid:105)(cid:104) E (cid:48) x | . The ef-fective DJT Hamiltonian in the basis of Eq. (7) used forthe PJT Hamiltonian isˆ H effJT = (1 − C ) F (cid:0) ¯ σ z ˆ X − ¯ σ x ˆ Y (cid:1) (14)with¯ σ z = 12 −
10 0 0 − ¯ σ x = 1 √ − − . (15) C. The effective electron-phonon Hamiltonian forthe shelving singlet state and the vibronicwavefunctions
The PJT effect was developed for | E (cid:105) , however, welearned in Sec. IV B that the lowest energy singlet is rather (cid:12)(cid:12) ¯ E (cid:11) that will reduce ˆ H PJT by C . Furthermore,one can estimate that the electron-phonon coupling inPJT ( ˜ F ) and in DJT ( F ) has a relation of ˜ F ≈ F .This relation can be envisioned from the two-particle | xx (cid:105) singlet wavefunction associated with its electron-phononcoupling ˜ F in which both orbitals are Jahn-Teller unsta-ble whereas only a single orbital is Jahn-Teller unstable in( ae ) electronic configuration associated with its electron-phonon coupling F . As a consequence, the Jahn-Tellereffect is twice stronger in | xx (cid:105) than that in ( ae ) elec-tronic configuration. The final effective electron-phononHamiltonian of the shelving singlet state isˆ H effel − ph = C F (cid:16) ˆ σ z ˆ X − ˆ σ x ˆ Y (cid:17) + (1 − C ) F (cid:0) ¯ σ z ˆ X − ¯ σ x ˆ Y (cid:1) .(16)The full Hamiltonian for the (cid:101) E ⊕ (cid:101) A system isˆ H = ˆ H e + ˆ H osc . + ˆ H effel − ph , (17)which results in the following (cid:101) Ψ vibronic wavefunctionsin the expansion of E phonon modes as follows, (cid:12)(cid:12) (cid:101) Ψ (cid:11) = ∞ (cid:88) n,m ( c xxnm | xx (cid:105) ⊗ | nm (cid:105) + c xynm | xy (cid:105) ⊗ | nm (cid:105) + c yynm | yy (cid:105) ⊗ | nm (cid:105) ) (18)where we limit the expansion in the Born-Oppenheimerbasis ( | nm (cid:105) = √ nm ( a † x ) n ( a † y ) m | (cid:105) ) up to 10 phononlimit n + m ≤
10 which is numerically convergent in ourparticular case. We span the electronic degrees of free-dom with | xx (cid:105) , | xy (cid:105) , | yy (cid:105) as we defined in Eq. (8). In thisform one can express the combined | (cid:101) A (cid:105) ⊕ | (cid:101) E ± (cid:105) stateswhich may transform as E , A and A . The (cid:101) A vibronicstates do not play a significant role, thus we only showthe expressions for the (cid:101) E ± and (cid:101) A vibronic states asfollows (cid:12)(cid:12)(cid:12) (cid:101) E ± (cid:69) = ∞ (cid:88) i =1 (cid:2) c i (cid:12)(cid:12) E ± (cid:11) ⊗ | χ i ( A ) (cid:105) + d i (cid:12)(cid:12) A (cid:11) ⊗ | χ i ( E ± ) (cid:105) + f i (cid:12)(cid:12) E ∓ (cid:11) ⊗ | χ i ( E ∓ ) (cid:105) + g i (cid:12)(cid:12) E ± (cid:11) ⊗ | χ i ( A ) (cid:105) (cid:3) (19a) (cid:12)(cid:12)(cid:12) (cid:101) A (cid:69) = ∞ (cid:88) i =1 (cid:20) c (cid:48) i (cid:12)(cid:12) A (cid:11) ⊗ | χ i ( A ) (cid:105) + d (cid:48) i √ (cid:0)(cid:12)(cid:12) E + (cid:11) ⊗ | χ i ( E − ) (cid:105) + (cid:12)(cid:12) E − (cid:11) ⊗ | χ i ( E + ) (cid:105) (cid:1)(cid:21) (19b)that govern the shape of the phonon sideband in the opti-cal spectra. We label the symmetry adapted vibrationalwavefunctions, e.g., | χ ( A ) (cid:105) = | (cid:105) , or in general, by | χ i ( . . . ) (cid:105) in the rest of the paper.We note that the g i coefficients are generally tiny andwill be ignored. On the other hand, the non-zero d i and c (cid:48) i ( f i and d (cid:48) i ) coefficients drive the Γ z (Γ ⊥ ) ISC process, andthey are also responsible for the shape of PL spectrumof the singlets. We further note that the analysis of strain depen-dence of the singlet states requires to extend the effec-tive electron-phonon Hamiltonian (Eq. (16)) with a strainHamiltonian that have similar matrix elements like theeffective electron-phonon Hamiltonian Hamiltonian thatmay explain the significant strain interaction of the (cid:101) E ± level . A detailed investigation on the interaction of the (cid:101) E ± ⊕ (cid:101) A vibronic levels with the strain is out of thescope of this study, and we rather concentrate on un-derstanding the role of the singlet vibronic states in theintersystem crossing processes of NV center. V. APPLICATION OF
AB INITIO
RESULTS ONTHE THEORETICAL MODELS
We first estimate the parameters in the developedelectron-phonon Hamiltonian from ab initio
DFT cal-culations for the singlet shelving state in Sec. V A andpresent vibronic electronic structure. We then applythe resulting vibronic wavefunctions to calculate the PLand absorption spectra for the singlets that will verifyour methodology in Sec V B. Finally, we determine theISC rates between | (cid:101) E (cid:105) into | A (cid:105) in Sec. V C that showgood agreement with the experimental data, includingthe temperature dependence. A. Derivation of the parameters of theelectron-phonon Hamiltionian of the singlet statesfrom DFT calculations and the resulting vibroniclevels
The true many-body singlet eigenstates of NV centercannot be exactly described by Kohn-Sham DFT meth-ods. Nevertheless, the closed-shell | xx (cid:105) state can be ex-pressed. Furthermore, the open-shell | a ↑ a ↓ e y ↑ e x ↓ (cid:105) can bealso calculated by ∆SCF Kohn-Sham DFT method. Wecalculated these singlet states by constraining the sym-metry into C v started from the optimized A groundstate geometry. We found that the geometry did notpractically change in the geometry optimization proce-dure of these singlet states which strongly hints that the A phonons do not contribute to the change of the ge-ometry of the true A and E states w.r.t. that of A ground state, thus the symmetry-breaking E phononsplay a role.In particular, the | xx (cid:105) is useful to derive the electron-phonon Hamiltonian of the singlet states as this stateappears in Eq. (16). By allowing free movement of atomsin the geometry optimization procedure, the | xx (cid:105) statespontaneously reconstructs to C h symmetry with a E JT Jahn-Teller energy of 316 meV and an effective phononmode ¯ hω E =66 meV where the latter is the solution of thequasi-Harmonic oscillator in the APES of | xx (cid:105) . This | xx (cid:105) state does not contain the strong electron correlation,e.g., the electronic gap Λ e ≈ Λ=1.19 eV between the E and A states. It can be easily shown by perturbationtheory (see Appendix A) that E JT will be damped by Λ e in E state [c.f., Figs. 2(a) and (b)]. The exact resultscome from solving the full electron-phonon Hamiltonianin Eq. (16). That requires to identify parameter C whichis associated with the contribution of E (cid:48) state in ¯ E state.We estimate parameter C from the character of theKohn-Sham wavefunctions of | xx (cid:105) in the C h APESglobal minimum. Although we calculated | xx (cid:105) by non- spinpolarized DFT, the symmetry distortion may resultin the contribution of the a Kohn-Sham orbital in thetwo-particle wavefunctions. By labeling the Kohn-Shamorbital in the distorted geometry by ξ , and the contri-bution of e x and a orbitals by p and s , respectively, onearrives at | ξξ (cid:105) = [ p | e x (cid:105) + s | a (cid:105) ] [ p | e x (cid:105) + s | a (cid:105) ] = p | e x e x (cid:105) + √ ps | ae x (cid:105) + | ea (cid:105)√ (cid:124) (cid:123)(cid:122) (cid:125)(cid:12)(cid:12) E (cid:48) x (cid:11) + s | aa (cid:105) (cid:124)(cid:123)(cid:122)(cid:125)(cid:12)(cid:12) A (cid:48) (cid:11) , (20)where (1 − C ) = 2 p s can be read out. By using pro-jector operators such as s = (cid:104) ξ | a (cid:105) we find that the con-tribution of A (cid:48) is minor and can be neglected, however,(1 − C )=0.1 is significant, and explains the Γ ⊥ ISC pro-cesses.This correlation effect also brings a DJT effect tothe electron-phonon Hamiltonian. Thus, the calculatedAPES of | xx (cid:105) contains both PJT and DJT effects thatshould be separated. This is established in Eq. (16) wherethe corresponding Jahn-Teller energy is E JT = (cid:0) C F + (1 − C ) F (cid:1) hω E . (21)By using the previously determined E JT , C and¯ hω E from DFT APES calculations of | xx (cid:105) , we obtain F =102.47 meV. Thus, we have all the parameters butΛ e to build up the full electron-phonon Hamiltonian ofthe singlet states.The measured ZPL energy between the singlet statesis 1.19 eV which is the energy difference between the vi-bronic ground states of the singlets. We fit the valueof Λ e to obtain the experimental ZPL energy after di-agonalizing the full electron-phonon Hamiltonian whichresulted in Λ e =1129.4 meV. The vibronic levels are de-picted in Fig. 2(c) where the corresponding coefficientsof the vibronic states are listed in Table I in Appendix B.One can see interesting features in the calculated vibronicspectra: (i) the vibronic levels of | (cid:101) E (cid:105) is very far fromthe solution of a quasi-harmonic oscillator and shows upa very complex feature; (ii) the vibronic levels of | (cid:101) A (cid:105) are equidistant, however, the PJT effect will increase theeffective phonon mode of 66 meV to 91.8 meV.It is important to highlight the complex interplay be-tween the PJT and DJT effects in the final vibronicspectrum of | (cid:101) E (cid:105) . Although, DJT effect is dampedby (1 − C ) factor, still it changes the spectrum at ≈
45 meV, and results in two split E levels that wouldnot be otherwise there. In addition, it changes the char-acter of these vibronic wavefunctions, so that it increasesthe optical transition dipole moments with the groundstate vibronic state of | (cid:101) A (cid:105) . Our results demonstratethat electron-electron correlation effect combined withelectron-phonon couplings of different nature and involv-ing three electronic states can only fully describe the A ~ | 〉 E ~ | 〉 xx | 〉 yy | 〉 xy | 〉 xy | 〉 xx | 〉 yy | 〉 E n e r g y (b) E n e r g y (a) (c) ZPL (d) E ~ | 〉 A EEEEE A +E+A +EA A +2E+A +2EA +A +2E2A A ~ | 〉 E PJT ≈ Exp. at T=300 KSim. (ab-initio results)Exp. at T=4.2 K +A A +A A A EA E n e r g y ( m e V ) E n e r g y ( m e V ) optically active transitionsactivated by unaxial stress ZPL E m i ss i o n i n t e n s i t y ( a . u . ) Shift from ZPL (meV) E JT =316meV tot . e V =0 ≈ FIG. 2. (a) Jahn-Teller nature of the | A (cid:105) ⊕ | E (cid:105) states as can be calculated by means of Kohn-Sham DFT which correspondsto Λ e =0 eV. The resulted Jahn-Teller energy E JT =316 meV. (b) After Λ e ≈ E PJT ≈
30 meV according to the solution of Eq. (17) that results in Λ=1.19 eV ZPL energy. (c) The vibronic levels of | (cid:101) E (cid:105) and | (cid:101) A (cid:105) . The selection rules for the photoluminescence spectrum is indicated. Here the ZPL energy of 1.19 eV between | (cid:101) E (cid:105) and | (cid:101) A (cid:105) is not scaled for the sake of clarity. (d) Experimental photoluminescence spectrum of the singlets at low(black solid line) and room (dotted black line) temperatures compared to the simulated spectrum from ab initio solution (redcurve). We note that the experimental spectra show a substantial and minor background at room and cryogenic temperature,respectively. The simulation curve does not include background signal. The ZPL energy is now set to zero in order to easilyread out the position of vibration features in the spectrum. We used 2 meV, 5 meV, and 10 meV gaussian smearing for thelinewidth of the ZPL, first and second vibronic emissions, respectively, where the width of the ZPL and vibration bands areread out from the experimental spectrum recorded at cryogenic temperature. electron-phonon system of the singlet states in NV center.We will show that this complex nature can only accountfor the measured optical spectra and ISC rates. B. Vibronic sideband of the 1.19-eVphotoluminescence and absorption spectrum
We discuss now the PL and absorption spectrum ofthe singlets. Several features of these spectrum can beunderstood by our vibronic wavefunctions that verify ourmethod in the calculation of the ISC rates. The relativeoptical transition dipole strength between the vibronic (cid:101) E and (cid:101) A ground states is derived in Appendix C.We first discuss the luminescence spectrum at low tem-peratures which is a radiative decay between the (cid:101) A ground state and the vibronic ground and n th excitedstates of (cid:101) E x [see Eq. (C6)], i.e., I (cid:16) (cid:101) A → (cid:101) E ( n ) (cid:17) = (cid:12)(cid:12)(cid:12)(cid:68) (cid:101) A (cid:12)(cid:12)(cid:12) ˆ d x (cid:12)(cid:12)(cid:12) (cid:101) E ( n ) (cid:69)(cid:12)(cid:12)(cid:12) . (22)We found from direct calculation of the intensities inEq. (22) that the optical transition to the first vibronic A state of (cid:101) E state is not allowed. However, there isa significant optical transition dipole toward the split E vibronic states around 45 meV. After switching offthe small DJT effect in the electron-phonon Hamilto-nian only a single E mode appears with a smaller opti-cal transition dipole moment. This clearly demonstratesthat the small DJT effect does play an important role inunderstanding the optical features of the singlet states.The simulated PL spectrum from ab initio wavefunc-tions is shown in Fig. 2(d) that can be directly comparedto the low temperature experimental PL spectrum . Clearly, the broad feature with the maximum intensityat ≈
43 meV can be reproduced (red curve). We findthat the broad feature consists of two close-level vibronicexcited states [see red text in Fig. 2(c)]. The experimen-tal intensity and the shape of this broad feature can bewell reproduced by invoking our electron-phonon Hamil-tonian (red curve). Our theory does not account for thefeatures at 133 meV and 221 meV. These features seemto disappear at room temperature PL spectrum, thus weconclude that they may not belong to NV center. Ourtheory is further supported by an uniaxial stress experi-ment on the PL spectrum which showed up the existenceof a forbidden state at ≈
14 meV . This can be naturallyexplained by our calculated A vibronic excited state [seegreen text in Fig. 2(c)]. This A state will play an im-portant role in the temperature dependence of the ISCrate where ≈
16 meV phonon mode was deduced from thetemperature dependent ISC rate measurements in non-stressed diamond samples that should be identical withthe optically forbidden vibronic mode.Now we turn to the absorption spectrum which isvery different from the PL spectrum [c.f. Fig. 2(d) andFig. 3(b)]. We can explain this feature by the presenceof simultaneous PJT and DJT effects. The PJT andthe DJT effects are separately create an axial symmet-ric APES about the symmetry axis of the defect, how-ever, the DJT will create a barrier energy for the freerotation about the symmetry axis in the PJT APES.This can be readily observed by comparing Eqs. (11)and (15) corresponding to PJT and DJT effects, respec-tively, which differently combine the wavefunctions uponthe same distortion. In the absorption process we as-sume that the photon absorption is a faster process thanthe quantum mechanical tunneling between the globalminima of APES, i.e., the Y -axis is frozen in Fig. 2(b)which leads to the APES in Fig. 3(a). The ground statevibronic wavefunction becomes localized in one of theAPES valleys at a distance of about 1.3 from the C v symmetry position. For the (cid:101) A excited state the DFTAPES predicted ¯ hω E =66.1 meV, however, the solutionof the full electron-phonon Hamiltonian revealed us thatPJT effect increases this energy to 91.8 meV. Therefore,we also created the corresponding harmonic APES [seeFig. 3(a)] and employed the Huang-Rhys theory to pro-duce, where R =1.3 results in an S ≈ hω E =66.1 meV (blue curve) and alsoshifted one, ¯ hω E =91.8 meV (red curve). We empha-size that these effective phonon frequencies create highpeaks at 55 meV and 81 meV in the absorption spec-trum with respect to the position of the ZPL. The latteris much closer to the experimental peak at 71 meV[see Fig. 3(b)]. It can be concluded that the electron-phonon Hamiltonian based effective phonon frequencyis slightly overestimated but the calculated broad fea-tures in the absorption phonon sideband are in agree-ment with the observed ones. The corresponding spec-tral functions [dotted curves in Fig. 3(b)] either deducedfrom experimental spectrum (black dotted line) or cal-culated (blue and red dotted lines) are also shown. Wenote that the very sharp feature at high energy in theexperimental spectrum is associated with the quasi-localnitrogen-carbon vibration modes . Our theory does notaccount to that feature that would require the exact cal-culation of APES of the singlet states. On the otherhand, the PJT-DJT theory (red curve) brings the resultsclose to the experimental values and explains the upwardshift in the first characteristic phonon peak w.r.t. that ofthe triplets ( ≈
64 meV).We conclude from these results that the combination ofPJT and DJT effects accounts for the observed asymme-try in the lineshape of the emission and absorption spec-trum. In the PL process, the selection rules are dictatedby the dynamic motion of ions combined with the elec-tron wavefunction which results in an optically forbiddentransition that becomes visible under uniaxial stress. Onthe other hand, this dark vibronic state of the shelvingsinglet can play an important role in the ISC process.In the absorption process, the dynamics of ions is frozenthus it shows up in the optical spectrum, and opticaltransition to all of the vibronic states of the upper sin-glet state is allowed. The dynamics of ions can be slowerthan the absorption of the photons because of the smallbut non-negligible DJT effect which produces an energybarrier for the motion of ions. This leads to the largeasymmetry in the corresponding phonon sidebands. Fur-thermore, PJT explains the enhanced effective phononenergy in the absorption spectrum of the singlets withrespect to that in the optical spectrum of the triplets.These results verify our theory on the singlets and serveas a good base to study the ISC process between theshelving singlet state and the triplet ground state. E n e r g y ( e V ) (b) Sim. (no shift) Sim. (shifted 26 meV)Exp. at T=10 K E m i ss i o n i n t e n s i t y ( a . u . ) Shift from ZPL (meV)
Conf. coord. X ħω e ff ==91.8 meV − (a)
26 meV
Spectral functions R JT ≈ Peak positions:
71 meV80 meV55 meV
FIG. 3. (a) Calculated APES of singlet states of NV cen-ter including the pseudo and dynamic Jahn-Teller effects si-multaneously but with Y = 0 constraint. Zero value at theconfiguration coordinates corresponds to C v symmetry. Thelocalization of the wavefunctions is depicted. We note thatthe potential energy is not axial symmetric as explained in thetext. (b) Low temperature experimental absorption spectrum(black solid curve) and the deduced spectral function (blackdotted curve) from Ref. 18 are compared with the calculatedabsorption spectra (solid blue and red curves) and spectralfunctions (dotted blue and red curves) with using the Huang-Rhys theory based on the APES of singlet states. We applied1.5 meV gaussian smearing on the theoretical lineshapes. C. Theory and ab initio results on the ISC processtoward the ground state
We determine the ISC rates from | (cid:101) E (cid:105) to | A (cid:105) basedon the vibronic states calculated from DFT wavefunc-tions and potentials. The ISC process is a spin-orbitdriven scattering of the electron that is mediated byphonons for energy conversation, as the expected energydifference between (cid:101) E to A levels is several orders ofmagnitude larger than the spin-orbit energy. In otherwords, the electron is scattered to the vibration levels( (cid:104) ... | ) of the A ground state. As we discussed previ-ously, the emission or absorption of A phonons is minorin the process, thus we rely on the contribution of the E phonons that are responsible for the PJT and DJTeffects. The ISC rate can be calculated using the Fermigolden-rule and assuming the strength of spin-orbit cou-pling does not change significantly upon the motion ofions in the process. This theory was developed for theISC process between E and A states in the upperbranch by Goldman and co-workers that we devel-oped further to take into account the vibronic nature of E state, i.e., (cid:101) E caused by DJT effect . By applyingthis theory to the Γ z ISC rate between | (cid:101) E (cid:105) and vibrationstate of A we arrive atΓ z = 2 πC ¯ h (cid:88) | ... (cid:105) (cid:12)(cid:12)(cid:12) (cid:104) ... | ⊗ (cid:10) A (cid:12)(cid:12) ˆ W (cid:12)(cid:12)(cid:12) (cid:101) E (cid:69)(cid:12)(cid:12)(cid:12) δ (Σ − E ( | ... (cid:105) ))= 2 πC ¯ h ∞ (cid:88) i λ z d i |(cid:104) ... | χ i ( E ± ) (cid:105)| δ (cid:0) Σ − n i ¯ hω E (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) ≈ S ( n i ) E (Σ) ≈ πλ z C ¯ h ∞ (cid:88) i d i S ( n i ) E (Σ) = 8 πλ z C ¯ h F E (Σ) (23)where the summation over all vibration wavefunctions of A collapses to the number of | χ i ( E ± ) (cid:105) vibration modesin the phonon overlap integral. Here, d i coefficient is re-sponsible to the contribution of electronic A state in | (cid:101) E (cid:105) that is connected to A by λ z . Now the energyconservation law is Σ = n i × ¯ hω E for some n i ( n i isthe phonon index of the i th | χ i ( E ± ) (cid:105) vibronic function).Here S E is the phonon overlap spectral function and F E is the modulated phonon overlap function caused by PJTeffect. Σ is the ZPL energy between | (cid:101) E (cid:105) and A [seeFig. 1(c)]. So far we used effective phonon energies withdiscrete quantum levels but this would lead to often azero overlap in F E . In reality, the diamond phonons in-teract with the quasi-local vibration modes found in PJTand DJT effects that can be described as a smearing ofthe energy spectrum of the quasi-local vibration modes.In order to incorporate this effect, we autoconvolute theelectron-phonon modes n i times by defining the followingrecursive formula, S ( n ) E ( x ) = (cid:16) S ( n − E ∗ S E (cid:17) ( x ) S (0) E ( x ) = δ ( x ) , (24)where ” ∗ ” labels the convolution, and δ ( x ) is the Diracdelta function. Similar considerations have been appliedrecently (see the Supplemental Material in Ref. 18).Beside Γ z ISC processes, the Γ ± and Γ ∓ ISC processescan take place governed by λ ⊥ because of the contribu-tion of E (cid:48) in | (cid:101) E (cid:105) . By applying the Fermi golden-ruleagain, we arrive atΓ ± = 2 π (1 − C )¯ h (cid:88) | ... (cid:105) (cid:12)(cid:12)(cid:12) (cid:104) ... | ⊗ (cid:10) A ± (cid:12)(cid:12) ˆ W (cid:12)(cid:12) (cid:101) E (cid:11)(cid:12)(cid:12)(cid:12) δ (Σ − E ( | ... (cid:105) ))= 2 π (1 − C )¯ h ∞ (cid:88) i λ ⊥ c i |(cid:104) ... | χ i ( A ) (cid:105)| δ (Σ − n i ¯ hω E ) ≈ π (1 − C ) λ ⊥ ¯ h ∞ (cid:88) i c i S ( n i ) E (Σ) = 2 π (1 − C ) λ ± ¯ h F (cid:48) E (Σ)(25) andΓ ∓ = 2 π (1 − C )¯ h (cid:88) | ... (cid:105) (cid:12)(cid:12)(cid:12) (cid:104) ... | ⊗ (cid:10) A ± (cid:12)(cid:12) ˆ W (cid:12)(cid:12) (cid:101) E (cid:11)(cid:12)(cid:12)(cid:12) δ (Σ − E ( | ... (cid:105) ))= 2 π (1 − C )¯ h ∞ (cid:88) i λ ⊥ f i |(cid:104) ... | χ i ( E ∓ ) (cid:105)| δ (Σ − n i ¯ hω E ) ≈ π (1 − C ) λ ⊥ ¯ h ∞ (cid:88) i f i S ( n i ) E (Σ) = 2 π (1 − C ) λ ⊥ ¯ h F (cid:48)(cid:48) E (Σ) ,(26)where F (cid:48) E and F (cid:48)(cid:48) E are the corresponding phonon overlapspectral functions caused by DJT effect.We previously calculated all the parameters from abinitio wavefunctions required to calculate the ISC ratesthat are plotted in Fig. 4 and compared to the observedinverse lifetime of the singlet . We find that the I S C r a t e ( M H z ) - triplet singlet splitting - (meV) = 386 meV E = =2.70 ± E =2.16 ± E = 402 meV FIG. 4. Calculated low-temperature ISC rates (Γ z , Γ ± , Γ ∓ from Eqs. (23)-(26) as a function of the energy gap (Σ) be-tween the shelving state singlet state and the triplet groundstate. Here we applied λ ⊥ = 1 . λ z by following Ref. 19. Twoexperimental data about the lifetime of the singlet state ( T E )is applied from Ref. 22 with T − E ≈ T − E ≈ energy gap between the shelving singlet state and thetriplet ground state is ≈ λ ⊥ = 1 . λ z , Γ z ISC rate (blue curve) is about 6 × largerthan the Γ ⊥ = Γ ± + Γ ∓ rate (green curve) at that en-ergy gap that would further strengthen the spinpolariza-tion process beside the strictly spin-selective process inthe upper branch. On the other hand, Robledo and co-workers deduced a smaller Γ z / Γ ⊥ ≈ . . . . where the common value was 1.20 by taking theuncertainty in the measurements into account. By us-ing the low-temperature simulation data, we varied the0 λ z /λ ⊥ and plotted Γ z / Γ ⊥ in Fig. 5 to analyze this issue,as there is uncertainty in the value of λ ⊥ . We conclude ( e V ) =1.2 =2.0 =1.0 =39.76 GHz =15.78 GHz=56.31 GHz=15.78 GHz FIG. 5. Γ z / Γ ⊥ is plotted as a function of λ ⊥ /λ z where λ z =15.78 GHz is our accurate DFT value. The λ ⊥ =56.31 GHz value approximated from DFT wavefunctionsis an overestimation. λ ⊥ =39.76 GHz yields Γ z / Γ ⊥ =1.2. that λ ⊥ /λ z ≈ λ ⊥ = 1 . λ z in the study of the temperaturedependence of the ISC rates that we plot in Fig. 6 (redcurve) and compare to previous experimental data takenon two single NV centers. Here we used the calculated vi- Temperature (K) L i f e t i m e ( n s ) FIG. 6. The calculated lifetime of the singlet shelving stateis plotted as a function of the temperature with the observedlifetimes for two single NV centers (dot and triangle datapoints with uncertainties) taken from Ref. 12. bronic states of the | (cid:101) E (cid:105) with the Boltzmann occupationof vibronic levels at the given temperature, in order tocompute ISC rates as defined in Eqs. (23)-(26). We founda very good agreement with the experimental data asthe calculated lifetime is reduced from 370 ns at cryo-genic temperatures down to 171 ns at room temperatureto be compared to 371 ± ±
10 ns, respectively.Our calculations reveal that the vibronic state associated with the optically forbidden phonon feature at ≈
14 meVin the PL spectrum plays a key role in the temperaturedependence of the ISC rates. The calculated Γ z / Γ ⊥ isonly reduced by ∼
5% going from cryogenic temperatureto room temperature which means that the spinpolariza-tion efficiency per single optical cycle does not degradesignificantly. These results demonstrate that our theorycan account for the intricate details of the ISC processesin NV center and reproduce the basic experimental data.
VI. SUMMARY AND CONCLUSIONS
In this work, we developed a theory on the nature ofthe singlet states including electron-electron correlationcoupled with phonons. We identified the strong electronphonon coupling between the singlet states that can bedescribed as a combination of pseudo Jahn-Teller effectand damped dynamic Jahn-Teller effect. We extendedthe theory of ISC rates of NV center to account for thiscomplex nature of the singlet shelving state that is re-sponsible for the ISC process toward the ground state.Our theory can explain several features in the opticalspectra of singlets. In particular, the presence of opti-cally forbidden state in unstressed diamond and the fea-tures in the phonon sideband of the PL spectrum waswell reproduced which are based on the vibronic natureof the singlet shelving state. The calculated ISC ratesand the deduced energy gap between the shelving sin-glet state’s level and the triplet ground state’s level areconsistent with the previous experimental data. The cal-culated temperature dependence of the lifetime of thesinglet shelving state is in very good agreement with theexperimental data. Our results complete the theoreticaldescription of the entire optical spinpolarization loop ofNV center.Our results may have an impact in the field, as NVcenter is a template for similar defects that act as solidstate qubits. The most obvious example is the neutraldivacancy in silicon carbide for which the first stepfor understanding the underlying mechanisms has beenrecently taken . Our ab initio toolkit can be extended toother defect systems including point defects in 2D mate-rials such as boron nitride, transition metal dichalgonidesand dioxides that have been attracted a great attention.Computing the ISC rates of these defects can contributeto understand their optical properties and optimize theirquantum bit operation. ACKNOWLEDGMENT
Support from the Hungarian Government and the Na-tional Research Development and Innovation Office (NK-FIH) in the frame of the ´UNKP-17-3-III New NationalExcellence Program of the Ministry of Human Capac-ities, NVKP Project Grant No. NVKP 16-1-2016-00431and the Quantum Technology National Excellence Pro-gram (Project No. 2017-1.2.1-NKP-2017-00001) are ac-knowledged.
Appendix A: Pseudo Jahn-Teller effect fromperturbation theory
We derive the PJT effect between E and A fromperturbation theory that provides an insight about thestrength of interaction where we concentrate on the vi-bronic ground state of the resulting (cid:12)(cid:12)(cid:12) (cid:101) E (cid:69) . The vibronicwavefunction from A coupled to the E should trans-form as E , thus the E phonon state should be occupiedin A state with an effective ¯ hω E phonon energy. Theenergy difference between the corresponding states willbe Λ e + ¯ hω E where Λ e is the energy gap between theelectronic levels of E and A . Now by choosing the x representation from the double degenerate states andlabeling the | E x E y (cid:105) vibration wavefunction by the occu-pation representation we arrive at (cid:12)(cid:12)(cid:12) (cid:101) E x (cid:69) = (cid:12)(cid:12) E x (cid:11) ⊗ | (cid:105) + χ PJT
Λ + ¯ hω E (cid:12)(cid:12) A (cid:11) ⊗ a † x | (cid:105) (A1)with the χ PJT is the coupling parameter and a † x is thecreation operator of the E x phonon. In the Kohn-ShamDFT calculations Λ e =0 for | xx (cid:105) singlet state resultingin relatively large Jahn-Teller energy, however, it can beseen in Eq. (A1) that the strength of the interaction issignificantly damped by Λ e ≈ χ PJT = (cid:10) E x (cid:12)(cid:12) ⊗ (cid:104) | ˆ H PJT (cid:12)(cid:12) A (cid:11) ⊗ a † x | (cid:105) , (A2)where ˆ H PJT is the PJT Hamiltonian from Eq. (10). Bysubstituting the ˆ H PJT into Eq. (A2), we arrive at χ PJT = ˜ F √ (cid:10) E x (cid:12)(cid:12) σ z (cid:12)(cid:12) A (cid:11) (A3)where we used (cid:104) | a y a † y | (cid:105) =0 and (cid:104) | a x a † x | (cid:105) =1 re-lations. As a next step we use the two-particle expressionof A in Eq.8 to arrive at χ PJT = ˜ F √ (cid:104) xx | − (cid:104) yy | ] σ z [ | xx (cid:105) + | yy (cid:105) ] (A4)and finally solve it in the matrix representation as χ PJT = ˜ F √ (cid:0) − (cid:1) − = ˜ F √ Appendix B: Vibronic wavefunctions
Here we show the calculated coefficients of the (cid:101) E ⊕ (cid:101) A vibronic wavefunctions in Table I. Appendix C: Transition dipole moment between thesinglet states
Here we determine the optical transition strengths be-tween (cid:12)(cid:12) A (cid:11) and (cid:12)(cid:12) E (cid:11) . Following the derivation of Hepp et al. in Eq. (13) in the Supplemental Material of Ref. 49,the transition dipole moments between the single particleorbitals in C v symmetry with polarization ” x ” are thefollowings” x ” (cid:12)(cid:12) e x (cid:11) (cid:12)(cid:12) e y (cid:11)(cid:10) e x (cid:12)(cid:12)(cid:10) e y (cid:12)(cid:12) (cid:34) d ⊥ − d ⊥ (cid:35) ” y ” (cid:12)(cid:12) e x (cid:11) (cid:12)(cid:12) e y (cid:11)(cid:10) e x (cid:12)(cid:12)(cid:10) e y (cid:12)(cid:12) (cid:34) − d ⊥ − d ⊥ (cid:35) . (C1)By using the relations in Eq. (C1), the dipole momentfor the two-particle wavefunctions can be expressed andapplied to get P (cid:16) A x ” ↔ E x (cid:17) = (cid:12)(cid:12)(cid:12)(cid:10) A (cid:12)(cid:12)(cid:124) (cid:123)(cid:122) (cid:125) A ˆ d (1) x + ˆ d (2) x (cid:124) (cid:123)(cid:122) (cid:125) E (cid:12)(cid:12) E x (cid:11)(cid:124) (cid:123)(cid:122) (cid:125) E (cid:12)(cid:12)(cid:12) = 4 d ⊥ (C2)where we introduced the transition optical dipole opera-tor ( ˆ d (1) x and ˆ d (2) x ) acting on particle 1 and 2, respectively.Our result is in agreement with a previous result (see Tab.A.4 in Ref. 16, where the 2 O b,x matrix element is thetransition dipole moment, thus the transition strength is4 O b,x that is corresponding to our 4 d ⊥ in Eq. (C2)). Thethree other possible transitions are P (cid:16) A y ” ↔ E y (cid:17) = 4 d ⊥ (C3)and P (cid:16) A x ” ↔ E y (cid:17) = P (cid:16) A y ” ↔ E x (cid:17) = 0. (C4)The dipole operator can be expressed in a similar formas the PJT Hamiltonian [see Eq. (10)],ˆ d x = 2 d ⊥ ˆ σ z ˆ d y = 2 d ⊥ ˆ σ x , (C5)where ˆ σ z,x matrices are defined in Eq. (11). As an exam-ple, we explicitly write the intensity of the ZPL transitionbetween the vibronically coupled singlet states as follows I ZPL = (cid:12)(cid:12)(cid:12)(cid:68) (cid:101) E x (cid:12)(cid:12)(cid:12) ˆ d x (cid:12)(cid:12)(cid:12) (cid:101) A (cid:69)(cid:12)(cid:12)(cid:12) = 4 d ⊥ (cid:32) ∞ (cid:88) i =1 c i c (cid:48) i + ∞ (cid:88) i =1 d i d (cid:48) i (cid:33) (C6)where the corresponding expansion coefficients( c i , c (cid:48) i , d i , d (cid:48) i ) are defined in Eqs. (19a) and (19b).2 TABLE I. Coefficients are defined in Eqs. (19a) and (19b) for the (cid:101) E and (cid:101) A vibronic states, respectively. The first columndefines the phonon index ( n ). The ”repr. of phonons” column shows the irreducible representation of states that can beconstructed from n phonons. The A modes are negligible and labeled in the parentheses. We determined the coefficients upto n = 10 phonon limit, but we present the rows only up to n = 6 since all of the n > n phonons by (cid:80) n i = n . The ground vibronic state of (cid:101) E that transformsas E is expressed by c , d , and f . The first excited vibronic wavefunction of (cid:101) E that transforms as A is expressed by c (cid:48) and d (cid:48) . (cid:101) E (cid:101) A n (cid:80) n i = n c i (cid:80) n i = n d i (cid:80) n i = n f i (cid:80) n i = n c (cid:48) i (cid:80) n i = n d (cid:48) i repr. of phonons0 c =0.645 - - c (cid:48) =0.017 - A d =0.029 f =0.063 - d (cid:48) =0.618 E c =0.090 d =0.004 f =0.089 c (cid:48) =0.045 d (cid:48) =0.042 A + E c =0.011 d =0.012 f =0.012 c (cid:48) =0.004 d (cid:48) =0.194 A + ( A ) + E c =0.015 d + d =0.002 f + f =0.016 c (cid:48) =0.016 d (cid:48) + d (cid:48) =0.018 A + 2 E c =0.002 d + d =0.003 f + f =0.002 c (cid:48) =0.002 d (cid:48) + d (cid:48) =0.032 A + ( A ) + 2 E c + c =0.002 d + d =0.000 f + f =0.002 c (cid:48) + c (cid:48) =0.003 d (cid:48) + d (cid:48) =0.003 2 A + ( A ) + 2 E ... ... ... ... ... ... ... ∗ [email protected] L. du Preez, Ph.D. thesis, University of Witwatersrand,Johannesburg (1965). A. Gruber, A. Drabenstedt, C. Tietz,L. Fleury, J. Wrachtrup, and C. v. Bor-czyskowski, Science F. Jelezko, T. Gaebel, I. Popa, A. Gruber, andJ. Wrachtrup, Phys. Rev. Lett. , 076401 (2004). T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura,C. Monroe, and J. L. O’Brien, Nature , 45 (2010). D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu,and J. R. Petta, Science , 1174 (2013). M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko,J. Wrachtrup, and L. C. Hollenberg, Physics Reports ,1 (2013). G. Balasubramanian, P. Neumann, D. Twitchen,M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard,J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko,and J. Wrachtrup, Nature Materials , 383 (2009). A. P. Nizovtsev, S. Y. Kilin, F. Jelezko, I. Popa, A. Gruber,C. Tietz, and J. Wrachtrup, Optics and Spectroscopy ,848 (2003). J. Harrison, M. Sellars, and N. Manson, Journal of Lumi-nescence , 245 (2004). N. B. Manson, J. P. Harrison, and M. J. Sellars, PhysicalReview B , 104303 (2006). L. Robledo, H. Bernien, I. van Weperen, and R. Hanson,Physical Review Letters , 177403 (2010). L. Robledo, H. Bernien, T. van der Sar, and R. Hanson,New Journal of Physics , 025013 (2011). E. Bourgeois, A. Jarmola, P. Siyushev, M. Gulka, J. Hruby,F. Jelezko, D. Budker, and M. Nesladek, Nature Commu-nications , 8577 (2015). A. Lenef and S. C. Rand, Phys. Rev. B , 13441 (1996). J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov,E. Kaxiras, and M. D. Lukin, New Journal of Physics , 025025 (2011). M. W. Doherty, N. B. Manson, P. Delaney, and L. C. L.Hollenberg, New Journal of Physics , 025019 (2011). L. J. Rogers, S. Armstrong, M. J. Sellars, and N. B. Man-son, New Journal of Physics , 103024 (2008). P. Kehayias, M. W. Doherty, D. English, R. Fischer,A. Jarmola, K. Jensen, N. Leefer, P. Hemmer, N. B. Man-son, and D. Budker, Physical Review B , 165202 (2013). M. L. Goldman, M. W. Doherty, A. Sipahigil, N. Y. Yao,S. D. Bennett, N. B. Manson, A. Kubanek, and M. D.Lukin, Physical Review B , 165201 (2015). M. Goldman, A. Sipahigil, M. Doherty, N. Yao, S. Bennett,M. Markham, D. Twitchen, N. Manson, A. Kubanek, andM. Lukin, Physical Review Letters , 145502 (2015). G. Thiering and A. Gali, Physical Review B , 081115(2017). V. M. Acosta, A. Jarmola, E. Bauch, and D. Budker,Physical Review B , 201202 (2010). I. Bersuker and V. Polinger,
Vibronic interactions inmolecules and crystals , Vol. 49 (Springer Science & Busi-ness Media, 2012). I. Bersuker,
The Jahn-Teller effect (Cambridge UniversityPress, 2006). N. Manson, L. Rogers, M. Doherty, and L. Hollenberg,e-print arXiv: quant-ph/1011.2840 (2010). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996). J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journalof Chemical Physics , 8207 (2003). A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E.Scuseria, The Journal of Chemical Physics , 224106(2006). P. De´ak, B. Aradi, T. Frauenheim, E. Janz´en, and A. Gali,Phys. Rev. B , 153203 (2010). P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994). O. Bengone, M. Alouani, P. Bl¨ochl, and J. Hugel, Phys.Rev. B , 16392 (2000). A. Gali, Phys. Rev. B , 241204 (2009). K. Huang and A. Rhys, Proceedings of the Royal SocietyA: Mathematical, Physical and Engineering Sciences ,406 (1950). A. Alkauskas, B. B. Buckley, D. D. Awschalom, andC. G. V. de Walle, New Journal of Physics , 073026(2014). P. Delaney, J. C. Greer, and J. A. Larsson, Nano Letters , 610 (2010). A. Ranjbar, M. Babamoradi, M. H. Saani, M. A. Vesaghi,K. Esfarjani, and Y. Kawazoe, Physical Review B ,165212 (2011). S. Choi, M. Jain, and S. G. Louie, Phys. Rev. B , 041202(2012). J. P. Goss, R. Jones, S. J. Breuer, P. R. Briddon, andS. ¨Oberg, Phys. Rev. Lett. , 3041 (1996). M. (cid:32)Luszczek, R. Laskowski, and P. Horodecki, Physica B:Condensed Matter , 292 (2004). J. A. Larsson and P. Delaney, Physical Review B ,165201 (2008). C.-K. Lin, Y.-H. Wang, H.-C. Chang, M. Hayashi, andS. H. Lin, The Journal of Chemical Physics , 124714 (2008). A. Gali, M. Fyta, and E. Kaxiras, Physical Review B ,155206 (2008). Y. Ma, M. Rohlfing, and A. Gali, Physical Review B ,041204 (2010). G. Davies and M. F. Hamer, Proceedings of the Royal So-ciety A: Mathematical, Physical and Engineering Sciences , 285 (1976). L. J. Rogers, M. W. Doherty, M. S. J. Barson, S. Onoda,T. Ohshima, and N. B. Manson, New Journal of Physics , 013048 (2015). A. Gali, Phys. Status Solidi B , 1337 (2011). W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine,and D. D. Awschalom, Nature , 84 (2011). D. J. Christle, P. V. Klimov, C. F. de las Casas, K. Sz´asz,V. Iv´ady, V. Jokubavicius, J. Ul Hassan, M. Syv¨aj¨arvi,W. F. Koehl, T. Ohshima, N. T. Son, E. Janz´en, A. Gali,and D. D. Awschalom, Phys. Rev. X , 021046 (2017). C. Hepp, T. M¨uller, V. Waselowski, J. N. Becker, B. Pin-gault, H. Sternschulte, D. Steinm¨uller-Nethl, A. Gali, J. R.Maze, M. Atat¨ure, and C. Becher, Physical Review Letters112