Heralded entangling quantum gate via cavity-assisted photon scattering

We theoretically investigate the generation of heralded entanglement between two identical atoms via cavity-assisted photon scattering in two different configurations, namely, either both atoms confined in the same cavity or trapped into locally separated ones. Our protocols are given by a very simple and elegant single-step process, the key mechanism of which is a controlled-phase-flip gate implemented by impinging a single photon on single-sided cavities. In particular, when the atoms are localized in remote cavities, we introduce a single-step parallel quantum circuit instead of the serial process extensively adopted in the literature. We also show that such parallel circuit can be straightforwardly applied to entangle two macroscopic clouds of atoms. Both protocols proposed here predict a high entanglement degree with a success probability close to unity for state-of-the-art parameters. Among other applications, our proposal and its extension to multiple atom-cavity systems step toward a suitable route for quantum networking, in particular for quantum state transfer, quantum teleportation, and nonlocal quantum memory.

Figure 1

Pictorial representation of the quantum system for two noninteracting atoms confined into the same single-sided cavity, which has a decay rate of $2\kappa$. Each atom is described as a three-level system in a $\mathrm{\Lambda }$ configuration, the atomic transition $|1\rangle \leftrightarrow |3\rangle$ of which is resonantly coupled to the intracavity mode with coupling strength $g$. The rates ${\mathrm{\Gamma }}_{31}$ and ${\mathrm{\Gamma }}_{32}$ describe the spontaneous emission from the atomic excited state to the ground ones. Here, an input field (${\alpha }_{\text{in}}$) impinges on the partially transmitting mirror of the cavity, such that a detector can register a photon count in the output field (${\alpha }_{\text{out}}$) when there is no photon loss via atomic spontaneous emission.

Figure 2

(a) Concurrence ($\mathcal{E}$) of the atomic steady state (${\rho }_{\text{at}}^{\text{ss}}$) as a function of pulse duration (${\tau }_p$) considering different values of cooperativity ($C$) and using the initial state of Eq. (2). Also shown is the shape in the time domain of the input (${\alpha }_{\text{in}}$—dashed line) and output (${\alpha }_{\text{out}}$—solid line) fields for (b) short ($\kappa {\tau }_p=0.5$) and (c) long $(\kappa {\tau }_p=10$) input pulses, when both atoms are in the state $|2\rangle$ (empty-cavity-like case).

Figure 3

(a) Average number of photons outside the cavity ($P_s$) and (c) the atomic concurrence as a function of the cooperativity in the steady regime, considering the atom initially in the state given by Eq. (2) and $\kappa {\tau }_p=50$. In (b) we show the contribution of each one of the separable atomic states in $P_s$ when they are initially prepared.

Figure 4

Pictorial representation of the experimental setup when the atoms are trapped into locally separated cavities. A single photon crosses a 50:50 BS, virtually impinges on both cavities at the same time, and then is detected by one of the detectors ($D_1$ or $D_2$) after passing through the BS again.

Figure 5

(a) Concurrence associated to the atomic steady states ${\rho }_{D1}$ (solid black line) and $|{\mathrm{\Psi }}_{D_2}\rangle$ (blue dashed line), respectively, as a function of the cooperativity. (b) Detection probability of registering a photon count in $D_1$ (solid black line) and $D_2$ (blue dashed line), respectively, as a function of the cooperativity. He we consider a long pulse duration, $\kappa {\tau }_p=50$, and the initial state given by the left-hand side of Eq. (11) $[{\alpha }_{\text{in}}^k\left(t\right)={\alpha }_{\text{in}}\left(t\right)/2]$.