Scattering of two distinguishable photons by a $\Xi$-type three-level atom in a one-dimensional waveguide

We investigate theoretically quantum scattering of two distinguishable photon wave packets from a $\Xi$-type three-level atom in a one-dimensional waveguide. The quantum state of scattered photons is solved analytically, and the solution indicates how the two photons become strongly correlated after the scattering. We determine the two-photon reflection and transmission properties, and analyze the quantum entanglement between the scattered photons. In particular, we show that the degree of entanglement can be enhanced by decreasing the spectral width of the incident photon wave packets.

Figure 1

Schematic diagram of the system. A $\Xi$-type three-level atom confined at the center of a 1D waveguide interacts with a photon A (red) and a photon B (blue).

Figure 2

Plots of dimensionless quantities: (a) $|\gamma C_{pq}^{RR}{|}^2$, (b) $|\gamma C_{pq}^{LL}{|}^2$, (c) $|\gamma C_{pq}^{LR}{|}^2$, and (d) $|\gamma C_{pq}^{RL}{|}^2$ as a function of normalized detunings of photons, ${\Delta }_p^{\left(A\right)}/\gamma$ and ${\Delta }_q^{\left(B\right)}/\gamma$, when ${\delta }_A={\delta }_B=0$. Other parameters are ${\gamma }_A={\gamma }_B\equiv \gamma$ and ${\epsilon }_A={\epsilon }_B=0.05\gamma$.

Figure 3

(a) Probability of two photons being reflected $\left(P_{LL}\right)$ as a function of ${\epsilon }_A$ and ${\epsilon }_B$ for ${\gamma }_A={\gamma }_B\equiv \gamma$. (b) Maximal probability $P_{LL}^{\max }$ as a function of coupling ratio ${\gamma }_A/{\gamma }_B$. ${\delta }_A={\delta }_B=0$ are used in both figures.

Figure 4

(a) Entanglement entropy of ${\Lambda }_{pq}^{\prime }$ as a function of $\gamma /\epsilon$ with ${\gamma }_A={\gamma }_B\equiv \gamma$ and ${\delta }_A={\delta }_B=0$. The inset shows the first 10 Schmidt eigenvalues for $\epsilon =0.1\gamma$. (b) Schmidt number $K$ of ${\Lambda }_{pq}^{\prime }$ as a function of $\gamma /\epsilon$; same parameters as in (a).

Figure 5

Photon loss probabilities (a) $P_{\mathrm{loss}}^{\left(3\right)}$ and (b) $P_{\mathrm{loss}}^{\left(2\right)}$ as functions of $\epsilon \equiv {\epsilon }_A={\epsilon }_B$ for the resonance case with ${\delta }_A={\delta }_B=0$ and ${\gamma }_A={\gamma }_B\equiv \gamma$. The decay rates are set to be ${\gamma }_3={\gamma }_2=0.1\gamma$ (solid line), ${\gamma }_3={\gamma }_2=0.05\gamma$ (dotted line), and ${\gamma }_3={\gamma }_2=0.01\gamma$ (dashed line).