Photonic controlled-phase gates through Rydberg blockade in optical cavities

We propose a scheme for high-fidelity photonic controlled-phase gates using a Rydberg blockade in an ensemble of atoms in an optical cavity. The gate operation is obtained by first storing a photonic pulse in the ensemble and then scattering a second pulse from the cavity, resulting in a phase change depending on whether the first pulse contained a single photon. We show that the combination of a Rydberg blockade and optical cavities effectively enhances the optical nonlinearity created by the strong Rydberg interaction and makes the gate operation more robust. The resulting gate can be implemented with cavities of moderate finesse, allowing for highly efficient processing of quantum information encoded in photons. As an illustration, we show how the gate can be employed to increase the communication rate of quantum repeaters based on atomic ensembles.

Figure 1

Schematic outline of the phase gate. (a) An input single-photon pulse along with a driving field induces a two-photon transition to the Rydberg state $|r\rangle$, which is subsequently transferred to another Rydberg state $|r^{\prime }\rangle$. Due to Rydberg interactions ${\mathcal{V}}_{kl}$ between the atoms, other Rydberg states $|r\rangle$ within the range of the interaction potential, given by the blockade radius of $r_b$, become off resonant, allowing no further excitation. (b) When an initial photon pulse is stored in the Rydberg ensemble, the second incoming photon cannot enter the cavity and is scattered off, which ideally induces a phase flip of $\pi$ on the scattered photons. (c) Dual-rail implementation of a cp gate (dotted box). A Bell state measurement can be implemented by combining the cp gate with Hadamard gates (HG).

Figure 2

Choi-Jamiolkowski fidelity (thin line) and postselected swap fidelity (thick line) as functions of the blockaded cooperativity ${\mathcal{C}}_b$ for a spectrally narrow pulse $\mathrm{\Delta }\omega \to 0$. We assume $|{\mathcal{C}}_b^{\prime }|={\mathcal{C}}_b$.

Figure 3

Secret key rate per repeater station $\left(r_{\text{secret}}\right)$ as a function of the blockaded cooperativity $\left({\mathcal{C}}_b\right)$ for a communication distance of 1000 km. We compare the protocol of Ref. [50] (original) with a modified protocol where the entanglement swapping is performed with the Rydberg cp gate (Rydberg). We consider an optimistic source repetition rate of 100 MHz and a more modest one of 1 MHz, as well as a perfect single excitation state created in the atomic ensembles, e.g., using a Rydberg blockade [29]. We assume an attenuation length of 22 km in the fibers and an optical signal speed of $2\times {10}^5$ km/s. The ensemble readout efficiency and photodetector efficiency are both assumed to be 90%. The steps in the curves reflect where the fidelity of the cp gate allows additional swap levels to be employed.