Entanglement of electron spins and geometric phases in the diamond color center coupled to the P1 center

Impurity spins in semiconductors are potential quantum bits. Entanglement and topological phases are key resources in quantum computation. We prove that the coupled electron spins carried by a diamond nitrogen-vacancy color center (NV${}^{-}$) and a single substitutional nitrogen impurity (P1 center) are entangled in the immediate vicinity of the level anticrossing that appears in the Zeeman energy diagram at about 500 G. We also determine the Aharonov-Anandan, Berry, and marginal geometric phases that can be accumulated by the state vectors of this spin system when it is magnetically transported around a closed path. At the resonance where the gap between two energy levels is minimum, the geometric phases undergo discontinuities, and the entanglement of the two electron spins is maximal.

Figure 1

(top) Level anticrossings of the NV${}^{-}$-P1 coupled defect pair around 500 G.[5] The spin flips are shown. A single upwards (downwards) arrow stands for the spin state $\frac{1}{2}$ $(-\frac{1}{2}$), and a double upwards (downwards) arrow stands for state 1 ($-1$). (bottom) The von Neumann quantum entropy $S_{\frac{1}{2}}$ ($=S_1$) of the electron spins with respect to the strength of the magnetic field. The two electron spins are maximally entangled at the resonances.

Figure 2

Variations of the cyclic geometric phases (in units of $\pi$) with respect to the strength of the magnetic field. $\omega$ and $\theta$ are, respectively, the rotational frequency of the magnetic field about the $z$ axis and its inclination with respect to the same axis. ${\omega }_L$ is the Larmor frequency. (top) Nonadiabatic Aharonov-Anandan (AA) geometric phase. (bottom) The Berry phase.

Figure 3

Variations of the marginal geometric phases (in units of $\pi$) acquired by the electron spins with respect to the strength of the magnetic field. The sum of ${\gamma }_1$ and ${\gamma }_{\frac{1}{2}}$ gives the Berry phase shown in Fig. 2. $\omega$ and $\theta$ are defined in the text and in Fig. 2.