Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime

We present the dipole induced transparency (DIT) of a diamond nitrogen-vacancy center embedded in a photonic crystal cavity coupled to two waveguides, and it is obvious with the robust and flexible reflectance and transmittance difference of circularly polarized lights between the uncoupled and the coupled cavities even in the bad cavity regime (the Purcell regime). With this DIT, we propose two universal hyperparallel hybrid photonic quantum logic gates, including a hybrid hyper-controlled-not gate and a hybrid hyper-Toffoli gate, on photon systems in both the polarization and the spatial-mode degrees of freedom (DOFs), which are equal to two identical quantum logic gates operating simultaneously on the systems in one DOF. They can be used to perform more quantum operations with less resources in the quantum information protocols with multiqubit systems in several DOFs, which may depress the resources consumed and the photonic dissipation. Moreover, they are more robust against asymmetric environment noise in the weak-coupling regime, compared with the integration of two cascaded quantum logic gates in one DOF.

Figure 1

The optical transitions of an NV center with circularly polarized lights. (a) A double-sided cavity-waveguide-NV-center system. (b) The optical transitions of an NV center. The photon in the state $|R^{\uparrow }\rangle$ or $|L^{\downarrow }\rangle$ corresponds to ${\sigma }^{+}$, and the photon in the state $|R^{\downarrow }\rangle$ or $|L^{\uparrow }\rangle$ corresponds to ${\sigma }^{-}$. $R^{\uparrow }$ ($R^{\downarrow }$) and $L^{\uparrow }$ ($L^{\downarrow }$) represent the right- and left-circularly polarized lights with their input (output) directions parallel (antiparallel) to the $z$ direction.

Figure 2

The reflection and transition coefficients of the double-sided cavity-NV-center system vs the normalized frequency detuning $(\omega -{\omega }_0)/\eta$ (${\omega }_c={\omega }_k={\omega }_0$). (a) $g=0$, $\gamma \sim 2\pi \times 80$ MHz [27], and $\eta =50\kappa \sim 2\pi \times 0.05$ THz ($Q\sim {10}^4$). (b) $g\sim 2\pi \times 0.06$ THz, $\gamma \sim 2\pi \times 80$ MHz, and $\eta =50\kappa \sim 2\pi \times 0.05$ THz. (c) $g\sim 2\pi \times 0.035$ THz, $\gamma \sim 2\pi \times 80$ MHz, and $\eta =50\kappa \sim 2\pi \times 0.5$ THz ($Q\sim {10}^3$).

Figure 3

Schematic diagram for a hybrid photonic hyper-cnot gate operating on a two-photon system in both the spatial-mode and polarization DOFs. X represents a half-wave plate which is used to perform a polarization bit-flip operation $X=|R\rangle \langle L|+|L\rangle \langle R|$. $Z_{\mathrm{n}}(n=1,2)$ represents a half-wave plate which is used to perform a polarization phase-flip operation $Z=|R\rangle \langle R|-|L\rangle \langle L|$. U represents a wave plate which is used to perform a polarization phase-flip operation $U=-|R\rangle \langle R|-|L\rangle \langle L|$. ${\mathrm{CPBS}}_m(m=1,2,3)$ represents a polarizing beam splitter in the circular basis, which transmits the photon in right-circular polarization $|R\rangle$ and reflects the photon in left-circular polarization $|L\rangle$, respectively. $i_{k1}$ and $i_{k2}$ represent the two spatial modes of photon $i$ ($i=a,b$), respectively. ${\mathrm{NV}}_k$ ($k=1,2$) represents a double-sided cavity-NV-center system. An optical switch is used in the merging point of $i_{kl}$ and $j_{kl}$.

Figure 4

Schematic diagram for the first step of our hybrid photonic hyper-Toffoli gate operating on both the spatial-mode and polarization DOFs of a three-photon system. $H_P$ represents a half-wave plate which is used to perform a polarization Hadamard operation on a photon. BS represents a 50:50 beam splitter which is used to perform a spatial-mode Hadamard operation on a photon $\left[\right|a_2\rangle \to \frac{1}{\sqrt{2}}\left(\right|a_1^{\prime }\rangle +|a_2^{\prime }\rangle \left)\right]$. DL represents a time-delay device.

Figure 5

Schematic diagram for the second step of our hybrid photonic hyper-Toffoli gate operating on both the spatial-mode and polarization DOFs of a three-photon system. S represents an optical switch.

Figure 6

Fidelity ($F$) of a hybrid spatial-polarization hyper-cnot gate vs the Purcell factor $F_P$ and the cavity decay rate $\lambda$.

Figure 7

Fidelity ($F$) of a hybrid spatial-polarization hyper-cnot gate on a two-photon system (red dashed line) and that of two identical polarization cnot gates on a four-photon system (blue dotted line) vs the Purcell factor $F_P$. Here the cavity decay rate is chosen as $\lambda =0.1$, and the construction of the polarization cnot gate is the same as that of the polarization part of the hyper-cnot gate in Ref. [13].

Figure 8

Fidelity ($F$) of a hybrid spatial-polarization hyper-Toffoli gate vs the cavity decay rate $\lambda$ in the case $\left|r\right|=|t_0|$ $[F_P=(1-\frac{{\lambda }^2}{4})/\lambda ]$, which is equal to the one for two polarization Toffoli gates on a six-photon system.