Controlled-phase Gate for Photons Based on Stationary Light

We propose a method to induce strong effective interactions between photons mediated by an atomic ensemble. To achieve this, we use the so-called stationary light effect to enhance the interaction. Regardless of the single-atom coupling to light, the interaction strength between the photons can be enhanced by increasing the total number of atoms. For sufficiently many atoms, the setup can be viable as a controlled-phase gate for photons. We derive analytical expressions for the fidelities for two modes of gate operation: deterministic and heralded conditioned on the presence of two photons at the output.

Figure 1

(a) Level diagram of $\mathrm{\Lambda }$-type atoms (levels $|a⟩$, $|b⟩$, and $|c⟩$) that can be switched to two-level atoms (levels $|d⟩$ and $|e⟩$) by the storage of a photon followed by a $\pi$ pulse. Green dots indicate the initial state of the atoms. (b) Level diagram of dual-$\mathsf{V}$ atoms that can be switched to $\mathsf{V}$-type atoms. (c) Dual-rail Bell-state measurement setup with the controlled-phase (cphase) gate as a part of it. An ensemble of atoms is placed inside a Sagnac interferometer, shown as a triangle in (c) and defined by (d). In the rail corresponding to state $|0{⟩}_B$, a beam splitter is added with transmission coefficient $t_b$. All the other beam splitters (BS) are $50:50$.

Figure 2

(a) Reflectances ($|r_0{|}^2$, $|r_1{|}^2$) and transmittances ($|t_0{|}^2$, $|t_1{|}^2$) of an ensemble of $\mathrm{\Lambda }$-type atoms without ($|r_0{|}^2$, $|t_0{|}^2$) and with ($|r_1{|}^2$, $|t_1{|}^2$) a stored photon for different frequencies (two-photon detunings) $\delta$. The vertical dotted line marks the operation point ${\delta }_{\mathrm{res}}$. The parameters are $N={10}^4$, ${\mathrm{\Gamma }}_{1\mathrm{D}}/\mathrm{\Gamma }=0.05$, ${\mathrm{\Delta }}_c/\mathrm{\Gamma }=-10$, and ${\mathrm{\Omega }}_0/\mathrm{\Gamma }=10$.

Figure 3

Numerically calculated $F_{\mathrm{CJ}}$ and $F_{\mathrm{CJ},\mathrm{cond}}$. For the $t_b<1$ curves, $t_b$ is optimized numerically such that $F_{\mathrm{CJ},\mathrm{cond}}$ is maximal. The dual-$\mathsf{V}$ scheme uses regular interatomic distance $d=0.266\pi /k_0$. The common parameters are ${\mathrm{\Gamma }}_{1\mathrm{D}}/\mathrm{\Gamma }=0.05$, and ${\mathrm{\Omega }}_0/\mathrm{\Gamma }=1$. Under EIT (storage and retrieval), $\mathrm{\Omega }(z)={\mathrm{\Omega }}_0$. Under stationary light (scattering), $\mathrm{\Omega }(z)={\mathrm{\Omega }}_0\cos (k_{0}z)$ and ${\mathrm{\Omega }}_{\pm }(z)={\mathrm{\Omega }}_0{e}^{\pm i{k}_{0}z}$ for $\mathrm{\Lambda }$-type and dual-$\mathsf{V}$, respectively.