Designing a cavity-mediated quantum cphase gate between NV spin qubits in diamond

While long spin coherence times and efficient single-qubit quantum control have been implemented successfully in nitrogen-vacancy (NV) centers in diamond, the controlled coupling of remote NV spin qubits remains challenging. Here, we propose and analyze a controlled-phase (cphase) gate for the spins of two NV centers embedded in a common optical cavity and driven by two off-resonant lasers. In combination with previously demonstrated single-qubit gates, cphase allows for arbitrary quantum computations. The coupling of the NV spin to the cavity mode is based upon Raman transitions via the NV excited states and can be controlled with the laser intensities and relative phase. We find characteristic laser frequencies at which the scattering amplitude of a laser photon into the cavity mode is strongly dependent on the NV center spin. A scattered photon can be reabsorbed by another selectively driven NV center and can generate a conditional phase (cphase) gate. Gate times around 200 ns are within reach, nearly two orders of magnitude shorter than typical NV spin coherence times of around 10 $\mu \mathrm{s}$. The separation between the two interacting NV centers is limited only by the extension of the cavity.

Figure 1

Two nitrogen-vacancy centers in diamond located in an optical cavity and coupled to a common cavity mode with frequency ${\omega }_C$ (shown schematically). The NV centers are excited by off-resonant laser fields (frequencies ${\omega }_{Li}$). Spin-dependent scattering of laser photons off the NV center into the cavity mode and back allows for a coupling of the two NV spins which produces the universal cphase quantum gate.

Figure 2

(a) GS and ES energy levels as a function of the magnetic field $B$ applied along the NV axis. In the ES, only one orbital triplet is shown. The effect of the spin-spin interactions ${\mathrm{\Delta }}_{1,2}$ is shown schematically by the dotted lines. (b) Simplified energy level scheme. Here, ${|0\rangle }_g$ and ${|-1\rangle }_g$ denote orbital ground-state levels with spin projections $m_s=0$ and $m_s=-1$. Similarly, ${|0\rangle }_e$ and ${|-1\rangle }_e$ stand for the corresponding excited-state levels. The scattering of a laser photon (blue) into a cavity photon (red) via the intermediate excitation of the NV center is suppressed in the $m_s=-1$ state by destructive quantum interference when ${\delta }_L=\mathrm{\Delta }+{\delta }_C/2$.

Figure 3

Time dependence of the Makhlin invariants $G_1$ and $G_2$ during the operation of a cavity-mediated two-qubit gate. Before the lasers are turned on, $G_1=1$ and $G_2=3$, which corresponds to the identity operation. When the lasers are turned on, the two NVs start to interact through the cavity, which leads to the appearance of entanglement and change in Makhlin invariants. After the lasers are turned off, the final state of the system is related to the initial one by a cnot operation, characterized by Makhlin invariants $G_1=0$ and $G_2=1$. The parameters chosen for this plot are $g_C=100\mathrm{MHz},{\delta }_C=400\mathrm{MHz},{\delta }_L=1640\mathrm{MHz}$. Inset: Laser pulse shape with maximum $g_{L,\max }=24\mathrm{MHz}$.