High fidelity photon-photon gates by scattering off a two-level quantum emitter

We present a scheme for implementing a high-fidelity non-linear phase shift on a photonic state. The scheme is based on repeated scattering off a two-level quantum emitter embedded in a chiral or one-sided waveguide. The waveguide is equipped with elements inducing second-order dispersion and temporal phase shifts, which effectively form a harmonic trap and confine the photon pulses to a Gaussian shape. The same quantum emitter can be used for each scattering, and thus, only one quantum emitter is needed in this scheme. To illustrate the application of our scheme for photonic quantum computing and quantum communication, we analyze the implementation of a control-Z gate and a deterministic Bell-state analyzer for photonic qubits. Through numerical optimization, we show that we can reach a control-Z gate fidelity of $\mathcal{F} \sim 99.2\%$ ($\mathcal{F} \sim 96\%$) and a success probability of $P_s \sim 99.6 \%$ ($P_s\sim 98 \%$) for a Bell-state measurement with $N=17$ ($N=5$) scatterings.