Strong photon-photon interactions are a key component for realizing quantum gates. The fidelity of such gates in Rydberg atoms are known to be limited primarily by one parameter: the optical depth of the medium within the interaction range, denoted as OD_B. Despite the long-range interactions of Rydberg atoms, on the scale of 10 μm, the largest observed OD_B~10 is still very small compared to achievable optical depths of the full medium OD~1000, thus limiting the fidelity of Rydberg gates.

Here we show how this limit can be surpassed, by utilizing the OD of the whole medium. In our scheme, a uniform Bragg grating is induced by optically modulating the response of the medium. At the band-edge of the grating, where the transmission is high, the periodic modulation acts like a Fabry-Pérot cavity. A “gate” photon stored as a Rydberg excitation, can switch between transmission and reflection of a “target” photon. The effect of the Rydberg blockade is then enhanced by multiple passes in the grating.

Our scheme is most suitable for realizing a controlled-NOT gate, where the presence of the “gate” photon imprints a π-phase on the “target” photon. The switching between transmission and reflection of the target photon in the absence or presence of the control photon allows at most a π/2-phase. To facilitate a full π-phase, we position the grating inside a Sagnac interferometer. For specific parameters of the atomic scheme, the phase difference of the output photon between the two cases is π. The fidelity increases with OD, which effectively enhances OD_B.