Dynamic nuclear polarization in diamond

We study the dynamic nuclear polarization of nitrogen-vacancy (NV) centers in diamond through optical pumping. The polarization is enhanced due to the hyperfine interaction of nuclear spins as applied magnetic fields vary. This is a result of the averaging of excited states due to fast-phonon transitions in the excited states. The effect of dephasing, in the presence of a vibronic band, is shown to have little effect during the dynamic polarization.


Introduction
Negatively charged nitrogen-vacancy (NV) centers in diamond are a promising candidate for solid-state quantum computation. Due to the long-life coherence of nuclear spin, optical coherent mapping of nuclear spin onto the electronic spin states of NV centers at room temperature, has attracted intensive research aimed at developing solid state systems for the study of spintronics and quantum computation. NV centers can be spinpolarized by optical pumping via intersystem states which transfers nonzero electronic spin of the excited states S 1 z =  to S 0 z = of the ground states. This is used to initialize the NV center in experiments. It is known that the nuclear spin of the NV centers can be polarized via hyperfine coupling of the electron/nuclear spin through Level Anti-Crossing (LAC) of sub-levels in the excited states at the magnetic field B=512 gauss, see equation (10) below [7][8][9]. This provides a method for controling the nuclear spin of the NV centers by varying magnetic fields during optical cycling between the ground and the excited states. LAC is the result of averaged states between two orbital branches E X and E Y in the excited states E 3 due to fast-phonon mediated transitions R 1 THz, see figure 1.
Although it is estimated that LAC is a result of the phonon-mediated transitions inducing the population transfer between the two branches, it has yet to be treated analytically [1]. In this paper, we analytically investigate the averaging in the excited states via the fast phonon-mediated transitions between the two branches and dynamic nuclear polarization of the NV centers in diamond, which is enhanced through LAC. This paper is organized as follows. In section 2, we describe the excited states model of the NV centers and show the averaging in the excited states due to the fast phonon-mediated transitions. In section 3, we derive the master equations for the whole systems during optical cycling, and in section 4 we discuss the results and the dynamic nuclear polarization of the NV centers.  [9]. The interaction in equation (1) represents a weak coupling between the phonon-bath and the NV centers. Assuming weak interaction and a Markovian approximation in the bath, we project out the bath in equation (1) to derive the master equations of the system of the NV centers [17,18].

Model
The master equations of the system density matrix can be written in terms of two variables (fast and slow variables, see below) due to the block diagonalized structure of the unperturbed basis S I , z z | ñ of electronic/ nuclear spin projection onto the symmetric axis of the NV centers in diamond in the order of m m , 1, , 1, , 1, , 1, , 0, , 0, 15 by assuming perfect alignment of the applied magnetic field B along the symmetric axis of the NV centers. This is well-defined at high strain in diamond, [3] and as a consequence of the negligible coherence between the two branches according to the Markovian approximation, the master equations of the two branches are ( where R is the rate of phonon transitions between the two branches, see figure 1. Separation of slow/fast variables in equation (5) as r r r = + + and  r r r = -+ to isolate the phonon related terms reads accordingly. It should be noted that the fast transition rate R 1 THz appears only in the second equation while the resonant frequency of the first equation is of the order of a few GHz.
The fast variables approach quasi-equilibrium in a short time due to large R, resulting in hich is the source of the dephasing in the excited states due to the fast-phonon mediated transitions between the two branches [1]. This can be shown by constructing an augmented matrix of d 2 -dimension from the d-dimensional matrix, the second term of equation (6b) on the right hand side, then diagonalizing the augmented matrix and showing the eigenvalues of the matrix are negligible in comparison with R. The augmented matrix has a form of Figure 1. Electronic excited states of the NV centers: two spin-triplet branches E X and E Y , and the phonon-mediated transition R between the two branches, which is to be averaged in this paper.
For N d , 6 15 = , and hence dim( 12 )  = . This structure remains intact when optical cycling between the ground and the excited states is included in section 3, since the optical transition is spin-conserving. Each submatrix in the block-diagonalized matrix, equation (8), can be arranged to have a structure of, for instance, i a a a a a a a a where a and ω are of the order of or less than GHz at given strain in diamond, corresponding to hyperfine coupling and resonant frequency, respectively. For the sub-matrix, there exists a similarity transformation T i , i.e. such that, for example, for detail). Thus, the equations of the fast variables equation (6b) reach quasi-equilibrium equation (7), accounting for leading order in R.
Substitution of the relation equation (7) into equation (6a) reads or slow variables r, corresponding to the averaged level states in the excited states with the introduction of dephasing due to phonons, the second term on the right hand side above has not received any attention in NV centers until recently [1].
which is explained empirically in several experimental papers [5,6,7,9] with the averaged energy splitting D 1.42GHz es = between m 1 S =  and m 0 S = in the excited states. This is the origin of LAC at 512 gauss ( 1.42 w » GHz).

Master equations: whole system
The Hamiltonian of the whole system, as a result of averaging the excited state branches (see equation (10) [8,9]. It is known that the electron g factor of the ground states is close to that of the excited states, g 2 » [4,6]. The master equations of the whole system are given by tracing out the bath, as done in section 2 [14,[17][18][19] ( where the second term on the right hand side represents the relaxation processes of the optical cycling through the phonon sideband, H int . We adopted a Markovian approximation in the bath of photon and phonon sideband. Here, we briefly discuss the adiabatic elimination of the phonon-sideband states mn ij s (superscripts related to the NV center states while subscripts to the phonon-states) [17,19]. For instance, where mn w is the resonant frequency between two phonon-sideband states, R s is the decay rate of the phononsideband state onto the NV center states, and 0 a is the absorption coefficients, see figure 2. This results in decreasing the magnitude of the coherence transfer between the ground states A

Excited state
The master equations for the excited states population E 3 after the averaging are (here, we labels diagonal terms represents the dephasing due to the fast phonon transition R between the two branches, see equation (7). The processes via the phonon-side band, see equation (15), are reflected on k for the absorption coefficients 0 a of spin-conserving transitions during optical pumping. where r  and r  are the populations corresponding to the nuclear spin up and down, and Γ is the decay rate of the intersystem states. As known, the selective transition is used to initiate the NV centers through optical pumping [3,5]. At this point, it should be noted that 0 * g g g g = + ¢ + + and 2 0 * g g g = + -for the decay rates in the excited states, where g¢ is a spin-selective transition rate from the electronic spin S 1 z =  in excited states to the intersystem states E 1 and * g is a small electronic spin mixing (see, equation (20)) [3].

Ground state
The master equations of the ground states population A mixing to address the same effects, most of all, analyzing the results of the phonon-mediated transitions in view of the exact basis. In addition, the dephasing originating via the phonon-sideband casts important questions relevant to the decoherence which attract further study by way of a polaron transformation [20] and/or the master equation approach [21] adopted in the study of resonant energy transfer (RET).
To conclude, we showed the averaging due to the fast phonon in the excited states and the dynamic nuclear polarization of the NV centers as a function of the applied magnetic field during optical cycling.