**2. Physical properties of NV center in diamond and their potential applications**

The NV center is a unique defect in diamond, made up of a substitutional nitrogen atom (N) and a vacancy (V) at one of the closest neighboring sites of the diamond crystal lattice (**Figure 1a**) [6]. The neutral state NV0 and the negatively charged state NV-, which have extremely different optical and spin characteristics, are the two distinct varieties of this defect that have been discovered so far [7, 8]. Only the negatively charged state of the defect (NV− ) is interesting for quantum computing and sensing, since it can provide a spin triplet ground level which can be initialized, coherently manipulated with long coherence time and readout by pure optical techniques. The followings will focus on the negative charge defect NV− (denoted as NV center defect).

This *C*3v symmetry defect behaves like an artificial atom nested in the diamond matrix and exhibits a broadband photoluminescence (PL) emission with a zerophonon line at 1.945 eV (λZPL = 637 nm), making it possible to detect individual NV center using optical confocal microscopy at room temperature [9, 10]. Furthermore, the NV centers exhibit perfect photostability without photobleaching or blinking, which can enable the construction of very reliable single-photon sources operating at room temperature and be applied in biology, where NV defects are used as fluorescent labels [11–13]. In addition, the ground level with a spin triplet state <sup>3</sup> *A*2, whose sublevels are divided in energy by spin–spin interaction into a singlet state of spin projection ms = 0 and a doublet ms = ±1, separated by *D* = 2.87 GHz in the absence of a magnetic field, is a crucial characteristic of the NV center (**Figure 1b**). Here, ms stands for the spin projection along the intrinsic quantization axis of the NV defect, which corresponds to the axis connecting the nitrogen and the vacancy ([111] crystal axis) [14]. The NV defect can be optically excited through spin-conserving transitions to a spin triplet <sup>3</sup> *E* excited state, which is an orbital doublet with a zero-field splitting *D*es = 1.42 GHz at room temperature, exhibiting the same quantization axis and gyromagnetic ratio as in the ground level [15]. After optically excited in the <sup>3</sup> *E* level, the NV center can relax to ground level through two various transition routes, including the same radiative transition path producing a broadband red PL and another transition path involving non-radiative intersystem crossing (ISC) to singlet states (**Figure 1b**). Since non-radiative ISCs to the <sup>1</sup> *E* singlet state are highly spin-selective, whereas optical transitions are mostly spin-conserving (ms = 0), the shelving rate from the ms = 0 sublevel is substantially less than those from ms = ±1. Furthermore, the NV defect prefers to decay from the lowest <sup>1</sup> *A*1 singlet state toward the ground state ms = 0 sublevel [16, 17]. Consequently, these spin-selective processes use optical pumping

#### **Figure 1.**

*(a) Atomic structure of the NV defect in diamond. (b) Schematic diagram of energy levels in NV defect. Spin conserving optical transitions from the 3 A2 spin triplet ground state to the 3 E excited state are shown with solid arrows. The dashed arrows indicate spin selective intersystem crossing (ISC) involving the singlet states 1 E and 1 A1. The infrared (IR) transition occurring at 1042 nm between the singlet states is also shown. Reproduced with permission from ref. [5]. Copyright 2014 IOP Publishing Ltd.*

to achieve substantial electron spin polarization into ms = 0. Moreover, due to the non-radiative ISCs, the PL intensity of the NV center will become higher when the state is ms = 0 populated. Indeed, by providing a resonant microwave (MW) field, a single NV defect that was initially prepared in the ms = 0 state by optical pumping can be driven to the ms = ±1 spin state, leading to a decrease in the PL signal. This property has been widely applied recently in the context of diamond-based quantum information processing, where the NV defect is investigated as a solidstate spin qubit [18].

Neglecting the hyperfine interaction with nearby nuclear spins in the diamond lattice, the spin Hamiltonian of the level can be written as:

$$H = \left(hD + + d^{\parallel}E\_x\right)\left[\mathbf{S}\_x^2 - \frac{\mathbf{S}\left(\mathbf{S} + \mathbf{1}\right)}{\mathbf{3}}\right] + \mathbf{g}\,\mu\_\mathbf{s}\mathbf{B}\cdot\mathbf{S} - d^{\perp}\left[E\_x\left(\mathbf{S}\_x\mathbf{S}\_y + \mathbf{S}\_y\mathbf{S}\_x\right) + E\_y\left(\mathbf{S}\_x^2 - \mathbf{S}\_y^2\right)\right] \tag{1}$$

where *z* is the NV defect quantization axis, *h* is the Planck constant, *d* is the ground triplet state permanent electric dipole moment, *E* is the electric field, *μ*B is the Bohr magneton, *g* ≈ 2.0 is the electron g-factor, *B* is the applied magnetic field, *S* is the electron spin operator, and *S*x, *S*y, and *S*z are the Pauli matrices [19]. The axial zero-field splitting parameter *D* is ~2.87 GHz resulting from spin–spin interaction between the two unpaired electrons of NV defect [5]. The measured axial ( *d h*/ ) and non-axial ( <sup>⊥</sup> *d h*/ ) components of the ground triplet-state permanent electric dipole moment are ~0.35 ± 0.02 and 17 ± 3 Hz cm V−1, respectively [19]. In addition, the axial splitting parameter D is related to the temperature due to the combination effects of thermal expansion and electron–phonon interactions [20]. Therefore, the NV center in diamond offers unique possibilities to be employed as a nanoscale sensor for detection and imaging of magnetic fields, electric fields, and temperatures (**Figure 2**). For instance, the measurement principle for magnetometry applications of NV defect is comparable to that of optical magnetometers based on the precession of spin-polarized atomic gases [22].

**Figure 2.**

*The possible applications of NV defect in the detection or imaging of temperature, electric fields, and magnetic fields. (a) Reproduced with permission from ref. [20]. Copyright 2014 American Physical Society. (b) Reproduce with permission from ref. [19]. Copyright 2011 Nature Publishing Group. (c) Reproduced with permission from ref. [21]. Copyright 2013 Nature Publishing Group.*

The applied magnetic field is evaluated through the observation of Zeeman shifts of the NV defect electron spin sublevels. In fact, the Zeeman effect occurs when a magnetic field is applied near the NV defect, lifting the degeneracy of ms = ±1 spin sublevels and causing the emergence of two resonance lines in the ESR spectrum. Therefore, a single NV defect can act as a magnetic field sensor with an atomicsized detecting volume [23, 24].

Additionally, the NV defect in diamond holds great promise for quantum communication applications, and its special structure enables spin states to be measured at room temperature [25]. Weakness of spin-orbit coupling and high Debye temperature of diamond ensure long spin–lattice relaxation time for NV-electron spins, [26, 27] making NV defect in diamond a promising candidate for quantum computing and quantum communication. After a single NV center in diamond absorbs a photon (532 nm), NV center will relax to emit a photon (637 nm). Subsequently, the emitted photon can be captured to optically readout the spin state. After a certain period of time, the NV center is reset, allowing the aforementioned processes to be repeated. Therefore, the controlled generation of single photons based on NV defect in diamond is significant for applications in quantum communication [28, 29]. Moreover, room-temperature quantum entanglement with the lifetime of milliseconds between two single NV defect spins in diamond can be realized for the future room-temperature quantum device [30].
