Characterization of the internal state of nitrogen-vacancy centerin diamond and second quantization formalism

Huerta, C. I.; Urrutia-Bañuelos, E.; Pedroza-Montero, M. R.; Riera, R.; Huerta, C. I.; Urrutia-Bañuelos, E.; Pedroza-Montero, M. R.; Riera, R.

doi:10.31349/revmexfis.66.814

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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.66 no.6 México nov./dic. 2020 Epub 31-Ene-2022

https://doi.org/10.31349/revmexfis.66.814

Research

Solid State Physics

Characterization of the internal state of nitrogen-vacancy centerin diamond and second quantization formalism

C. I. Huerta^a

E. Urrutia-Bañuelos^a

M. R. Pedroza-Montero^a

R. Riera^a^b

^{^a}Departamento de Investigación en Física, Universidad de Sonora, Apartado Postal 5-088, 83190 Hermosillo, Sonora, México.

^{^b}Centro de Investigación y Educación Superior CIES-UNEPROP. Blvd. José María Escribar 38 Col. Villa del Palmar, 83105 Hermosillo, Sonora, México. e-mail: rriera@cifus.uson.mx

Abstract

In this paper a general study of nitrogen-vacancy center internal state in diamond is presented. This study is based on experimental and theoretical findings found in literature. First, a Hamiltonian model for center nitrogen-vacancy internal state is proposed in terms of a complete set of commuting observables (CSCO), which consist of angular momentum L^, spin momentum S^, total angular momentum J^=L^+S^, and spin momentum on z-direction S^z. The second quantization formalism -in steady-state (conservative model)- is used. The creation and annihilation operators are used to described the steady spin-levels structure and in dynamic-state (non-conservative model) and can also describe the system dynamic between different spin-levels transitions. Finally, a discussion is presented about the application of this study in the photochromism phenomenon and solid-state quantum bit (qubit).

Keywords: NV Center; defect; photochromism; qubit

1.Introduction

The nitrogen-vacancy (NV) center is one of numerous point defects in diamond. It exhibits two electronic configurations: the neutral charged-state (NV^o) and the negatively charged-state (NV^-). The phenomenon called photochromism consists in the NV center interchange between two different charged-states by a possible mechanism involving nitrogen donors ¹. Electron spins at NV centers can be manipulated by applying a magnetic field, electric field, electromagnetic radiation (light), or a combination of them, resulting in a photoluminescent signal.

The photophysics of color NV centers in diamond have attracted interest during the last decade because of their possible application in quantum information processing, quantum optics, quantum metrology, nanotechnology, as biological and sub-diffraction limit imaging, to test of entanglement in quantum mechanics and nano-medicine (sensors) ²^-⁵. Currently, the technology tends to miniaturization, but it faces a problem: the integration techniques of circuits are not effective due to the quantum effects which become important in low dimensionality. Therefore, computational science has begun to research in quantum mechanics a new technological alternative with the aim to overcome the transistor technology.

The most promising systems are ions or atoms traps, which use their electronics states as representation of the quantum bit (qubit) states (e.g. ground and excited electronics states) ⁶; quantum dots in which its confined electron spins are controlled by tunneling effect via electric field and each different electron-spin configuration is used as qubit state ⁷^,⁸; setting diverse magnetics polarizations as qubit states by using NMR to manipulate nuclear spins of one or lots of molecules, generally in liquid state ⁹^,¹⁰; superconducting loops with persistent current form a qubit by superposition of two macroscopic quantum states ¹¹; and the defects present in diamond, more precisely NV centers of which their electronic spins system are taken advantage, allowing to be initialized, manipulated, and measured (readout) by optical means ¹².

At this moment, the available computing power is not enough. For this reason the Quantum Informatics become important. Apparently, the best candidate for solid-state qubit is the NV^- center in diamond since it works at room temperature, has long spin coherence time as well as the diamond lattice helps as an insulator for external magnetic effects because it is diamagnetic; in addition, thanks to its high Debye temperature, the vibrational modes of the lattice are not affecting the center because there is a low coupling between lattice phonons and center electrons ¹³^,¹⁴. In this paper a general study of internal state NV center in diamond is presented. From the second quantization formalism, -in steady-state (conservative model)- the creation and annihilation operators are used to describe the steady spin-levels structure and in dynamic-state (non-conservative model); these operators can also be used to describe different spin-level transitions. Finally, a discussion is presented about the applications of this study on different areas, especially in the photochromism phenomenon and solid-state qubit.

2.NV centers in diamond

The atomic configuration of NV center consists of two places that should be occupied with two carbon atoms but, instead, one carbon atom was replaced by a nitrogen atom and the other place is empty. In other words, there is a vacancy. The NV defect has point group C_3v, where the principal axis C₃ is along nitrogen-vacancy in crystallographic ¹¹ direction; and trigonal symmetry with point group ¹³.

For the creation of diamond NV centers, it is necessary to create vacancies inside diamond lattice by two methods: the first one is radiating with ions, electrons or neutrons to nitrogen-rich diamond (¹⁴N) type Ib; the second one is through the implantation of nitrogen (both ¹⁴N and ¹⁵N) inside a pure diamond type IIa which creates simultaneously vacancies in the crystal lattice. Upon creation of the vacancies, it is necessary to perform an annealing process of temperatures around 550^oC to 850^oC (oxygen atmosphere free) producing the vacancies migration. After this process the NV centers are created ¹⁵^-¹⁷.

3.Charged-states of NV center

Atomic structure of NV center in diamond consist of a carbon vacancy and an impurity of nitrogen which captures this vacancy in order to create the center, furthermore, exhibit two electronic configurations: the neutral charged-state (NV^o) and negatively charged-state (NV^-) ¹^,¹³^,¹⁸.

In the NV^o center, the nitrogen impurity has five valence electrons (being in group V of the periodic table). Three of them are shared with the three closer atoms of carbon and the other two are located in the direction of the vacancy. Thus, in the periphery of the vacancy, there exist three unsatisfied bonds, one for each of the surrounding carbons (see Fig. 1).

Figure 1 Schematic representation in two dimensions of the atomic configuration inside the NV0 center (where the marked circles with a C simulate the crystalline lattice), in which it can be observed that a carbon atom has been substituted for a nitrogen atom (in color blue), which captures a vacancy (marked as V) that remains with five electrons (black dots) on its direction.

The atomic configuration of NV^- center is the same as the NV^o center but with the only difference that it has six electrons (see Fig. 2).

Figure 2 Schematic representation in two dimensions of the atomic configuration inside the NV^- center (where the marked circles with a C simulate the crystalline lattice), in which it can be observed that a carbon atom has been substituted for a nitrogen atom (in color blue) which captures a vacancy (marked as V) that remains with five electrons (black dots) on its direction and a sixth electron positioned at the center of the vacancy.

4.Electronic configurations

4.1.NV^o center configuration

As shown in Fig. 1, the nitrogen directs its full orbital towards the vacancy as well as the three surrounding carbons their semi-full orbitals point towards it. Therefore, the total spin state (S) can be only 3/2 or 1/2 since the S = 5/2 state is not possible because it would be necessary for nitrogen electrons -which point towards the vacancy- to not be paired (see Fig. 3).

Figure 3 Schematic representation of the electron-spin configuration of NV⁰ center, in which it is observed the possible values for the total spin S (indicated with black arrows). The lines represent the atoms of nitrogen (N), surrounding carbons (C₁, C₂, and C₃) and the vacancy (V).

4.2.NV^- center configuration

As shown in Fig. 2, the nitrogen directs its full orbital toward the vacancy, the unsatisfied bonds of the surrounding carbons are oriented towards the vacancy, and probably the sixth electron is pushed towards some of the surrounding carbons by effect of the two electrons of the full orbital of the nitrogen; which would very likely result in the paring of the sixth electron with some of the carbons with unsatisfied bonds. Thus, the system could have S = 2, S = 1 or S = 0 states (see Fig. 4).

Figure 4 Schematic representation of the electron-spin configuration of NV^- center, in which it is observed the possible values for the total spin S (indicated with black arrows). The lines represent the nitrogen atoms (N), surrounding carbons (C₁, C₂, and C₃) and the vacancy (V).

5.Spin configurations

5.1.NV^o center

In the case of S = 1/2, there are two doublets: one of them being the groud-state with ms=±1/2 and the other one being the excited state as shown in Fig. 5.

Figure 5 Schematic representation of the spin-level structure of a NV⁰ center in which the possible S = 1/2 spin configuration is observed.

For the case of the S = 3/2 state, there are two multiplets: one excited and the other is the ground-state. These are made up of two doublets with ms=±1/2 and ms=±3/2, respectively, as shown in Fig. 6.

Figure 6 Schematic representation of the spin-level structure of a NV⁰ center in which the possible S = 3/2 spin configuration is observed.

5.2.NV^- center

In the case of S = 2, there are two multiplets: one excited and the other is the ground-state. These are made up of two doublets (ms=±2 and ms=±1) and one singlet (m_s = 0) as shown in Fig. 7.

Figure 7 Schematic representation of the spin-level structure of a NV^- center, in which the possible S = 2 spin configuration is observed.

Similarly, in the case of S = 1, there are two triplets: one excited and the other the ground-state. These are made up of a doublet with ms=±1 and one singlet with m_s = 0 as shown in Fig. 8. Finally, for the case S = 0, there are two singlets: one excited and the other the ground-state; both with m_s = 0 as shown in Fig. 9.

Figure 8 Schematic representation of the spin-level structure of a NV^- center in which the possible S = 1 spin configuration is observed.

Figure 9 Schematic representation of the spin-level structure of a NV^- center in which the possible S = 0 spin configuration is observed.

6.Experimental characterization

Experimentally, the NV centers in diamond are characterized by Electron Paramagnetic Resonance (EPR) or Optically Detected Magnetic Resonance (ODMR) techniques and the fluorescence signal emitted when an excited state is relaxed to a base state.

In the NV⁰ center, it has been detected a Zero Phonon Line (ZPL) at 2.156 eV (575 nm), which it is associated with a radiative transition (Δms=0, due to the selection rules present in the system) between the ground (2E) and excited (2A) spin doublets of the S =1/2 state. So far, in the spin-levels ²E and ²A a signal has not been detected using EPR; this is believed to be due to dynamic distortion caused by the Jahn-Teller effect, which causes the EPR lines to widen, preventing its detection ²⁰. In an experiment ²⁰, a paramagnetic resonance using continuous optical illumination in the S = 3/2 state was observed, which leads to conclude that it is an spin excited level (⁴A₂) of the NV⁰ center. The ⁴A₂ level consists of a multiplet of two doublets, one of them with ms=±3/2 spin-levels and the other with ms=±1/2 spin-levels (see Fig. 10); between the doublets of the multiplet, there exists a resonance at ∼1.685 GHz ²⁰. In conclusion, the ²E and ²A doublets are caused by a S = 1/2 state and the ⁴A₂ multiplet is caused by a S = 3/2 state. On the other hand, the change between the S = ½ and S = 3/2 states is a result of a non-radiative transition (in this case the mechanism is not spin-conserving, Δms≠0). This was demonstrated using the experiment described above and this stems from the fact that the system has to be polarized in the case of the ms=±1/2 spin-levels of the ²A excited levels making that the ms=±=1/2 and ms=±3/2 spin-levels of the ⁴A₂ state populate with greater and lesser probability, respectively (see Fig. 10) ²⁰^,²¹.

In the NV^- center, there exists a radiative transition with spin conservation due to the selection rules present in the system (Δms=0), from ground spin triplet (³A₂) to excited spin triplet (³E) of the S = 1 state. This transition presents a ZPL at 1.945 eV (637 nm) (see Fig. 11) and the spin triplet was demonstrated via ODMR study ²²^,²³ in which was observed a resonance signal at ∼2.88 GHz between the spin-levels m_s = 0 and ms=±1 of spin-level ³A₂¹³^,²⁴^,²⁵. Furthermore, a non-radiative transition (Δms≠0) from the ³E triplet to a spin excited singlet (¹A₁) of S=0 state occurs, and then a radiative transition between ¹A₁ singlet and the spin ground singlet (¹E) of S = 0 state is observed. That transition was confirmed by observation of an infrared ZPL at 1.185 eV (1046 nm) ²⁶^,²⁷. Finally, a non-radiative transition from the ¹E singlet to the ³A₂ triplet occurs (see Fig. 11) ²⁶^-²⁹. The transitions between S = 1 and S = 0 states are non-radiative because they are not spin-conserving. In addition, there is a competition between different non-radiative path of spin-levels of the S = 1 state triplets and the S = 0 state singlets, that is depicted in Fig. 11. The most likely path between the ³E to the ¹A₁ is from ms=±1 to m_s = 0, respectively and similarly, the transition between the ¹E to the ³A₂ is from m_s = 0 to m_s = 0, respectively (see Fig. 11 red wavy arrows). The latter allows the system to be polarized ²⁶^-²⁹.

The NV⁰ and NV^- centers are promising candidates for solid-state qubits because both centers have characteristics that make them utilizable. In the case of NV⁰, it is possible by using a ⁴A₂ excited level since it presents two doublets which can be used as qubit states. Other promising choice would be to use the transition between the ground ms=±1/2 of ²E level and excited ms=±1/2 of ²A level doublets via optical pumping at 2.156 eV following the remotion of the optical excitation, causing the system to decay from the ms=±1/2²A level to the ms=±1/2²E level. However, there exists a problem with this option because the state loses coherence because of the Jahn-Teller dynamic effect ¹⁹^,²⁰. On the other hand, in the case of a NV^- center the process can begin with the m_s = 0 ³A₂ ground level (S = 1) by using optical pumping cycles, manipulated via microwave fields and optical excitations. Measurements can be performed thanks to the fluorescence signal emitted by the decay between the excited and ground spin levels following the selection rules of the radiative transitions.

7.Theoretical characterization

The spin-orbital interaction, spin-spin interaction with electric and strain fields (see Eqs. (6), (8), (13), and (17)) are all Hamiltonians in terms of the Irreducible Representations (IRs) by, using Group Theory and are discussed in Ref. ³⁰.

The electronic Hamiltonian of the NV^- center (Eq. (1)) -corresponding to the ground electronic state- with an adiabatic approximation and considering the nuclei of the crystal to be fixed at their equilibrium coordinates is shown in ³¹. The interaction Hamiltonians of the center with electric (Stark shift), magnetic (Zeeman effect), and strain fields of Eq. (16) are shown in Ref. ³².

In Ref. ⁵ the ground state fine and hyperfine structures for both nitrogen isotopes can be described by the canonical spin-Hamiltonian of trigonal defects given by Eq. (1) and the potential of Eq. (2) describes the influence of static electric, magnetic, and strain fields on the NV^- ground state. Also, it is shown the effective Hamiltonian and potential that describes the influence of static electric, magnetic, and strain fields for excited state 3E to low-temperature and room-temperature (see Eqs. (3)-(6)).

7.1Steady-state (Conservative model)

In the present paper, the Hamiltonian of NV center, H^sys which only describes the internal state defect without external perturbation, is composed of the spin-spin interaction, H^S-S, the spin-orbital interaction (fine structure), H^L-S, and the electron-nuclear spin coupling (hyperfine structure) H^I-S. Notice that the nuclear-nuclear spin coupling has not been considered because, in the case of a nitrogen ¹⁵ isotope, the system does not have electric quadrupole moment. Consequently,

H^sys=H^S-S+H^L-S+H^I-S (1)

H^S-S=∑i∑jμ0γe24πr3S→i∙S→J-3S→i∙n→ijn→ij∙S→j, S→i=SixI^+SiyJ^+Sizk,^ (2)

H^L-S=-∑iγeB→∙S→i=-∑igsμBℏmeec2rL→i∙S→i=-∑γemeec2rL→i∙S→i, (3)

H^I-S=∑i∑jμ0γeγn4πr3×I→i∙S→j-3I→i∙n→ijn→ij∙S→j,S→j=SjxI^+Sjy+Sjzk^,I→i=IixI^+IiyJ^+Iizk^ (4)

where the index i, and j represent the NV center electrons, μ₀ represents the magnetic constant (vacuum permeability), y_e and γ_n are the electron and nuclear gyromagnetic constants, respectively, S indicates the magnitude of the total electron-spin of the system S⃗i and I⃗i represent the electron and nuclear spin vectors, respectively, m_e is the electron mass value, c is the speed light constant in vacuum, r symbolizes the distance between interacting electrons, and n⃗ij stands for a unit vector which shows the interacting direction between electron-spins.

On the other hand, the steady-state solution (conservative) will be,

H^sys|ΨNV⟩=E|ΨNV⟩, (5)

using the second quantization formalism, the form of a NV^o center wave can be written as follows

|ΨNV⟩=A∑iCi(r,σ,t)|ΨNV0(0)(r,σ,t)⟩, (6)

where the index i indicates the center electrons, (r, σ, t) represent the spatial coordinates, electron-spin and time, respectively. A symbolizes a normalization constant, C_i is a proportional constant, and |ΨNV0(0)(r,σ,t)⟩ denotes the non-interacting wave function, which is composed of products of single-electron functions, namely,

ΨNVO(0)r,σ,t=14!∑p-1NVΦC1(0)r1,σ1,t×ΦC2(0)r→3,σ3,t×|ΦV0(0)r→4,σ4,t (7)

Similarly, it is obtained the non-interacting wave function of the NV^- center,

where summation is done over all different possible p permutations and N_p represents the permutation index, C₁, C₂, C₃ represent the three unbonded carbons, and V represents an electron in the vacancy (see Fig. 2). Notice that the subindex on V in the function ΦVi(0), represent an empty vacancy (NV^o center) or an occupied vacancy for the values i = 0 or i = 1, respectively.

Now, if the wave functions of NV^o and NV^- centers are projected on the symmetrical occupation-number basis {|n⟩} in a Folk space, it leads to:

|Ψ´NV0=∑nn|ΨNV0r→,σ,t=∑nΨ´NVn0r→,σ,t|n, (9)

|Ψ´NV-=∑nn|ΨNV-r→,σ,t=∑nΨ´NV-nr→,σ,t|n (10)

and defining α^V†, α^V as, creation and annihilation operators, respectively, which acts only on the vacancy state, i.e.,

α^V†=∑nnV|nC1,nC2,nC3,nv-1×nC1,nC2,nC3,nV|, (11)

α^V=∑nnV+1|nC1,nC2,nC3,nV+1×nC1,nC2,nC3,nV (12)

When the operators given by Eqs. (11) and (12) are applied onto the wave functions of Eqs. (9) and (10), the result is

α^V†|Ψ'NV0⟩=|Ψ'NV-⟩; α^Vα^V†|Ψ'NV0⟩=|Ψ'NV0⟩, (13)

α^V|Ψ'NV-⟩=|Ψ'NV0⟩; α^V†α^V|Ψ'NV-⟩=|Ψ'NV-⟩. (14)

Thus, from (13) and (14), it can be concluded that [α^V†,α^V]=1 for both.

Since the previous wave functions, (7) and (8), depend on the spin σ, then they can be rewritten as the product of their orbital wave function |Φ(r⃗,t)⟩ and its spinorial function |X(σ,t)⟩, that is:

|ΨNV0(0)(r⃗,σ,t)⟩=|ΦNV0(r⃗,t)⟩|XNV0(σ,t)⟩, (15)

|ΨNV-(0)(r⃗,σ,t)⟩=|ΦNV-(r⃗,t)⟩|XNV-(σ,t)⟩. (16)

Therefore, the form of the spinorial functions of the NV^o and NV^- centers can be obtained from

|XNVoσ,t=13!∑p-1Np|χC10σ1,t×χC20σ2,tχC30σ3,t, (17)

XNV-σ,t=14!∑p-1NpχC10σ1,t|χC20σ2,t×|χC30σ3,t|χV(0)σ4,t (18)

where summation is performed over all different possible permutations p of the system, C₁, C₂, C₃ represent the three unbonded carbons, and V represents an electron to place in the vacancy (see Fig. 2). As the negatively charged-state center NV^- is the best candidate to implement as a solid-state qubit based on the DiVenzo Criteria ³³, it will be the focus of the present study. The electrons of NV^- center that are involved in the processes of the radiative and non-radiative transitions are the three of the carbons that have unsatisfied bonds and an extra that is transferred by some close donor impurity into the vacancy. Therefore, there are four electrons participating in the different S = 0 and S = 1spin states; S = 2 will be omitted because there is not experimental evidence for this spin configuration. Accordingly, the possible forms of the spinorial function of Eq. (18) for the two different spin states in ground state are

• S=0, ms=0

|χNV-σ,t=12αC1αC2βC3βV-αC1βC2βC3αV+βC1βC2αC3αV-βC1αC2αC3βV (19)

• S=1, ms=1

|XNV-(σ,t)⟩=αC1αC2αC3αV (20)

• S=1, ms=0

|χNV-σ,t=12αC1αC2βC3βV+αC1βC2βC3αV+βC1βC2αC3αV+βC1αC2αC3βV (21)

• S=1, ms=1

|XNV-(σ,t)⟩=γC1γC2γC3γV (22)

where C₁, C₂, C₃ represent the three unbonded carbons, and V represents an electron to place in the vacancy (see Fig. 2). Furthermore, the α_k, β_k, and γ_k states are characterized by the index k, which can take the values of C₁, C₂, C₃, or V and are defined as

α≡(100000000)β≡(000100000)γ≡(000000100). (23)

Well, if the |XNV-(σ,t)⟩ spinorial function is projected on the symmetrical occupation-number basis {|n⟩} in a Folk space, it leads to

|X´NV-σ,t=∑nn|χNV-σ,t=∑nχ´NVn-σ,t|n (24)

where X'NV-n(σ,t) represents a single-electron spinor on n state of the |X'NV-(σ,t)⟩ spinorial function. Now, if t = t₀ is fixed, it leads to steady-states as it follows

|X'NV-(σ,t0)⟩=∑n=18χ'NV-n(σ,t0)|n⟩ (25)

where the eight |n⟩ states (four ground and four excited) are represented by

• S=0

|1 is the ms=0 ground state and |2 is the ms=0 excited state (26)

• S=1

|3 is the ms=0 ground state and |4 is the ms=0 excited state

|5 is the ms=-1 ground state and |6 is the ms=-1 excited state

|7 is the ms=1 ground state and |28 is the ms=1 excited state (27)

The expressions of Eqs. (26) and (27) have been depicted in the Figs. 8 and 9 of Sec. 5.2 according to their spin-level structure.

7.2.Dynamic-state (Non-conservative model)

The dynamic Hamiltonian H^dyn, is composed of the NV center and electromagnetic field Hamiltonians denoted by H^sys and H^int, respectively (see Eqs. (1)-(4)); the resulting Hamiltonian has the following form:

H^dyn=H^sys+H^int, (28)

where

H^int=∑λℏωλa^λ†a^λ+12. (29)

Therefore, the dynamic-state solution (conservative) will be,

iℏ∂∂t|ΨNV⟩=H^dyn|ΨNV⟩. (30)

Well, substituting (10) into (30), it is possible to obtain an expression for the NV^- center, namely, center

iℏ∂∂t|Ψ'NV-⟩=H^dyn∑nΨ'NV-n(r⃗,σ,t)|n⟩. (31)

The creation and annihilation operators for the NV^- center between its electron-spin states, from (31), acquire the form

a^S,ms†=nS,ms|nS,ms+1⟩⟨nS,ms|, (32)

a^S,ms=nS,ms+1|nS,ms-1⟩⟨nS,ms|, (33)

where the |nS,ms⟩ are identified by its spin value S and spin momentum on z- direction m_s. Thus, the transitions between the ground-states and excited-states (see Fig. 11) can be obtained from

Tn⟶n*=2πℏn*H^intn2ρEn* (34)

Fermi´s Golden Rule

where |⟨n*|H^int|n⟩ is the matrix element of the interaction H^int between the final n* and initial n states and ρ(En*) is the density of states at the energy of the final states. The density of states for both centers NV is discrete and is defined as

ρ(E)=∑iδ(E-Ei), (35)

where i is the set of spin states for each different charged-state of NV centers. If the same energy interval is chosen, therefore the number of states of a NV^o center is greater than for NV^-, meaning that

NV^o center has two different total spin states, and these are S = 1/2 and s = 3/2.
NV^- center has two different total spin states, and these are S = 1 and S = 0.

Based on the above and reviewing the spin-level structures for the charged-state of each center (see Fig. 5-6 for NV^o and Fig. 8-9 for NV^- ), it is concluded that the density of states of NV^o p(E) is greater than the one for NV^-, p´(E). The density of states on the photochromism phenomenon changes from p(E) to p´(E) and viceversa. From this fact, a total density of states p_T(E) can be defined as

ρT(E)=aρ(E)+bρ'(E), (36)

which, when α = 0 and b = 1, represents the density of states of NV^-. In turn, if α = 1 and b = 0, it represents the density of states of NV^o. Accordingly,

α^S,msnS,ms=nS,ms+1nS,ms-1, nS,mS*,α^S,ms*†α^S,mS|nS,mS=nS,mSα^S,mS+1×|nS,mS-1,nS,mS*+1,n*H^intn∝nS,mS*|α^S,mS†a^S,mS|nS,mS (37)

Therefore, a^S,mS*† and a^S,mS work on different states, ground or excited, but always following the selection rule dictated by Δms=0. This is, if a^S,mS annihilates an electron on the S = 1, m_s = 0 ground state, a^S,mS*† creates an electron on the S = 1, ms*=0 excited state; in other words, it is a transition caused by the electromagnetic interaction.

8.Discussion of results

On the previous section, the different models, assumptions, and approximations reported in the literature for NV centers in diamond. By comparison with the present model, the main differences with respect of the models above are the use of the second quantization formalism and the photochromism phenomenon.

In this section, the most important results obtained for the characterization of the NV center internal state will be shown using the second quantization formalism, including the photochromism effect. First, the schematic representations of the two different NV center electronic configurations, neutral and negative charged-states, was designed in order to clarify how the different possible configurations of each electron-spin modify the system total spin S. The fact that diamond is the host lattice of the NV defect plays a very important role because if it was not for the diamond energy potential, the NV center would not exist. It must also be noted that the defect behaves as an internal system within the diamond matrix, since the spin-levels are in the middle, above the valence band which is populated and below the conduction band which is empty, both bands belong to the diamond; in other words, the electrons of the diamond lattice do not play a role within the dynamics of the NV center. Considering the previous assertions, the Hamiltonian modeling of the system internal state, H^sys (see Eqs. (1)-(4)) is justified. Therefore, it was applied the second quantization formalism on the H^sys model (see Eqs. (9) and (10)). From Eq. (10), the ground-levels for different spin-states in the steady state of the NV^- center were determined (see Eqs. (19)-(22)), which are also compatible with those shown in Figs. 8 and 9. Thus, it can be stated that this model reproduces the spin-levels structure of the system.

On the other hand, the NV center exhibits a switching charged-state process called photochromism ¹. There are two different ways to describe this phenomenon:

The one described in Ref. ¹ presents the hypothesis that the switch from NV^o to NV^- state is photoinduced. This process would be possible due to the ionization of nitrogen donors present in the surroundings of NV defect.
The other hypothesis suggested is by means of the existence of a weak interaction between NV^o and NV^- centers, which are created in almost the same proportion in diamond during the production process. This interaction is due to the interchange of vacancy-electron from NV^- to NV^o, resulting in a switching charged-state steady-process. Therefore, the creation and annihilation operators of Eqs. (11) and (12), respectively, are useful to describe the photochromism since they switch the defect from NV^o to NV^- center and vice versa.

Finally, from H^dyn (see Eq. (28)) it was possible to obtain the transitions between ground- and excited-states (see Fig. 11), which describe the system dynamics of the NV^- center. This description is really important because it is necessary for the design of solid-state qubits. A qubit is the quantum counterpart of the classical bit, which means it is the minimal unit of information in Quantum Informatics and essential difference of a qubit and a bit relies on the fact that the qubit can present a superposition between two states, whereas the classical counterpart can only represent one state at a time. Therefore, the spin-levels m_s = 0 and, m_s = -1 in ground-state S = 1 of the NV^- center (see Fig. 8) can represent such states of the qubit; that is,

|Ψ⟩=α|0⟩+β|-1⟩, (38)

where α and β are complex coefficients that satisfy |α|2+|β|2=1.

To make use of the NV^- center as a qubit, it will be necessary to use the spin-levels of its different states, ground and excited. For example, by taking advantage of the photoluminescent response that the system undergoes, when it passes from an excited level to a ground level, the system configuration can be found, meaning that the system is read-out. On the other hand, applying a magnetic field tuned to a suitable resonance frequency, the manipulation of the electron population of ground and excited states can be achieved. Finally, due to the existence of the two different spin configuration, S = 1 and S = 0, it is possible to initialize the system in a specific spin-level.

Now that the procedure has already been proposed, it is necessary to know what spin-levels are more convenient to represent the two qubit states. There are two possibilities to represent these qubit states. The first one would be to use the ground-state levels, m_s = 0 and m_s = -1 -the most advisable for energy efficiency- which would represent the qubit states. To avoid the mixing between the spin levels, a magnetic field would need to be applied to make use of the Zeeman effect, splitting the levels and, consequently, decreasing the mixing probability. The other possibility is to use the photochromism phenomenon by manipulating the configuration of two different NV centers, one with S = 1 and the other one with S = 0 as qubit states. In this case, the initialization can be represented by means of one of the two configurations of total spin S; the manipulation would be through electron population distribution in different configurations of spin using magnetic and electromagnetic fields. Finally, the measurement would be performed using the ZPL, which represents every spin configuration. This possibility could be deeply studied in another future paper.

9.Conclusions

Based on the above, the NV^- center is promising because of its characteristics. It is concluded that a possible candidate for solid-state qubit requires the following features:

The host lattice of the defect should show a low coupling between the system electrons and the lattice phonons (high Debye temperature) in order to work at room temperature. In addition, the lattice symmetry may be C_3v with the purpose to obtain a triplet-spin system.
To have radiative transitions between different spin levels and in the case that they are degenerate, it must be possible to split them by external means (i.e., a magnetic/electric field or stress application) in order to avoid mixed states. However, the radiative transitions are not necessarily the only way, since there might be alternatives on the condition that the read-out signal presents few difficulties during the measurement.
The system must allow to be initialized, manipulated, and measured via external means.

Acknowledgments

This work was supported by CONACyT.

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Received: January 21, 2020; Accepted: June 25, 2020

Conflict of Interest Statement The authors declare that the research was conducted in the absence of any commercial or financial relationship that could be construed as a potential conflict of interest.

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