All-optical control of the spin state in the NV${}^{-}$ center in diamond

We describe an all-optical scheme for spin manipulation in the ground-state triplet of the negatively charged nitrogen-vacancy (NV) center in diamond. Virtual optical excitation from the ${}^3{A}_2$ ground state into the ${}^{3}E$ excited state allows for spin rotations by virtue of the spin-spin interaction in the two-fold orbitally degenerate excited state. We derive an effective Hamiltonian for optically induced spin-flip transitions within the ground state spin triplet due to off-resonant optical pumping. Furthermore, we investigate the spin qubit formed by the Zeeman sublevels with spin projection $m_S=0$ and $m_S=-1$ along the NV axis around the ground state level anticrossing with regard to full optical control of the electron spin.

Figure 1

Schematic depiction of the NV center in diamond with equivalent sp${}^3$-hybridized dangling bonds (blue) ${\sigma }_i$ ($i=1,2,3$) from the surrounding carbon atoms (black) and ${\sigma }_N$ from the substitutional nitrogen atom (orange) overlapping within the vacancy (center). The NV axis (long red arrow) defines the $z$ axis of the coordinate system and, due to the large ground state zero field splitting (compared to hyperfine flip-flop terms), the quantization axis of the electron spin. The magnetic field $B$ is applied along the NV axis.

Figure 2

Illustration of the optical pumping process with frequency $\omega$ from $m_s=0$ ground state spin level below the excited state. The degenerate $E_{x,y}$ levels are coupled to the $E_{1,2}$ levels by the transversal spin-spin interaction ${\Delta }^{\prime \prime }$.

Figure 3

The state $|s\rangle$ of the ground state spin qubit in the $m_S=0,-1$ subspace can be represented as a vector on the Bloch sphere where the two poles represent the two Zeeman-split eigenstates $\left|m_s=0\right.〉$ and $\left|m_s=-1\right.〉$. The vector $\mathbf{b}$ denotes the effective magnetic field acting on this pseudospin 1/2, with spherical angles $\theta$ and $\phi$.

Figure 4

Pairs of required magnetic field variation $\delta B$ and detuning $\delta \omega$ at fixed dipole coupling strength $\epsilon =500\mathrm{MHz}$ for different values of transversal strain ${\delta }_{\perp }=1,2,3,4$, and 5 GHz (from top to bottom) to fulfill the condition $\theta =\frac{\pi }{2}$ for precession around a rotation axis within the equatorial plane of the Bloch sphere for $\alpha =\beta =0$.

Figure 5

Precession frequency $b_{\perp }$ for different values of transversal strain ${\delta }_{\perp }=1,2,3,4$, and 5 GHz (from left to right) for polarization parallel ($\alpha =\beta$) and perpendicular ($\alpha -\beta =\frac{\pi }{2}$) to strain with dipole coupling $\epsilon =500$ MHz.

Figure 6

Pairs of detuning $\delta \omega$ and transversal strain ${\delta }_{\perp }$ that match the condition $\theta =\frac{\pi }{2}$ for precession around a rotation axis within the equatorial plane of the Bloch sphere for different dipole coupling $\epsilon =100,200,400,600,800$, and 1000 MHz (from bottom to top) for $\alpha =\beta =0$. The Zeeman splitting has been fixed at $g{\mu }_B\delta B=-100$ MHz.

Figure 7

Precession frequency $b_{\perp }$ around in-plane rotation axis at 10 GHz detuning for optical coupling $\epsilon =500$ MHz and polarization in $e_x$ ($\alpha =0$) and $e_y$ ($\alpha =\frac{\pi }{2}$) direction for different values of transversal strain ${\delta }_{\perp }=1,2,3,4,5$ GHz (increasing amplitude) over strain direction angle $\beta$.

Figure 8

Nonaxial orientation angle $\phi$ of precession axis for polarization parallel ($\alpha =0$) and perpendicular ($\alpha =\frac{\pi }{2}$) to $e_x$-orbitals symmetry axis in respect to strain orientation for different strengths of transversal strain.

Figure 9

Nonaxial orientation angle $\phi$ of precession axis for different polarization angles $\alpha$ in respect to strain orientation at transversal strain ${\delta }_{\perp }=5\mathrm{GHz}$.