Passive Quantum Phase Gate for Photons Based on Three Level Emitters

We present a fully passive method for implementing a quantum phase gate between two photons traveling in a one-dimensional waveguide. The gate is based on chirally coupled emitters in a three level $V$ configuration, which only interact through the photon field without any external control fields. We describe the (non)linear scattering of the emerging polariton states and show that for near resonant photons the scattering dynamics directly implements a perfect control phase gate between the incoming photons in the limit of many emitters. For a finite number of emitters we show that the dominant error mechanism can be suppressed by a simple frequency filter at the cost of a minor reduction in the success probability. We verify the results via comparison with exact scattering matrix theory and show that the fidelity can reach values $\mathcal{F}\sim 99\%$ with a gate success probability of $>99\%$ for as few as eight emitters.

Figure 1

Photonic phase gate for counterpropagating wave packets in a one dimensional waveguide, coupled to three level dipole emitters with distance $d$ to the next neighbors. The transition $|g⟩\to |e⟩$ ($|g⟩\to |f⟩$) in the three level systems couples chirally to the right (left) traveling waveguide modes with coupling rate ${\mathrm{\Gamma }}_R$ (${\mathrm{\Gamma }}_L$).

Figure 2

Real part of the two-dimensional output state of an initial two-dimensional Gaussian state with width $\sigma =0.05{\mathrm{\Gamma }}_0$ for both ${\omega }_1$ and ${\omega }_2$. For different emitter numbers (1,6,12) we compare our analytical approximation (left) with exact scattering matrix results (right).

Figure 3

(a) Infidelity $1-\mathcal{F}$ and (b) probability of (heralded) gate failure as a function of the width $\sigma$ of an incident Gaussian pulse for a perfectly chiral setup. The solid line is for in infinite chain while the dark symbols represent numerical results for 4 (diamonds), 8 (squares), and 12 (circles) emitters. The dashed line and lighter symbols in (a) result from not filtering out the inelastic scattering. (c) Imperfect coupling: results assuming that photons are emitted backward or to the side at identical rates rate ${\mathrm{\Gamma }}_B={\mathrm{\Gamma }}_S=0.01{\mathrm{\Gamma }}_0$. The success probability decays exponentially with the number of emitters while the fidelity still increases. We chose $k_{0}d=\pi$ to avoid Bragg scattering (see main text). (d) Gate operation for arbitrary phase $\phi$ by going off resonance [Eq. (12)]. The plot shows the lowest achievable infidelity for different numbers of 4, 8, and 12 emitters using the same symbols as in (a).