Continuous-variable entanglement of two bright coherent states that never interacted

We study continuous-variable entanglement of bright quantum states in a pair of evanescently coupled nonlinear ${\chi }^{\left(2\right)}$ waveguides operating in the regime of degenerate down conversion. We consider the case where only the energy of the nonlinearly generated fields is exchanged between the waveguides while the pump fields stay independently guided in each original waveguide. We show that this device, when operated in the depletion regime, entangles the two noninteracting bright pump modes due to a nonlinear cascade effect. It is also shown that two-color quadripartite entanglement can be produced when certain system parameters are appropriately set. This device works in the traveling-wave configuration, such that the generated quantum light shows a broad spectrum. The proposed device can be easily realized with current technology and therefore stands as a good candidate for a source of bipartite or multipartite entangled states for the emerging field of optical continuous-variable quantum information processing.

Figure 1

Sketch of proposed nonlinear directional coupler made of two identical waveguides a and b with second-order susceptibilities ${\chi }^{\left(2\right)}$. Phase matching is fulfilled only in the coupling region. In red are the signal waves, evanescently coupled. In blue are the noninteracting pump waves.

Figure 2

Negligible depleted-pump-field regime. From top to bottom: (a) dimensionless signal powers (solid), the elements of the correlation matrix $V(X^A,X^B)$ (dotted), $V(Y^A,Y^B)$ (dot-dashed) for the (b) signal and (c) pump fields, respectively, and (d) the logarithmic negativity $E_{\mathcal{N}}$ corresponding to the subsystem of signals (solid) and pumps (dashed), respectively. $\kappa =1.13$ and $P_s/P_p={10}^{-20}$. $\zeta$ is the normalized propagation coordinate. The vertical lines show the planes where the phase difference of the classical waves $\mathrm{\Delta }\theta$ ($\mathrm{\Delta }\varphi$) are multiples of $\pi$.

Figure 3

Highly-depleted-pump-field regime. From top to bottom: (a) dimensionless signal (solid) and pump (dash) powers, the elements of the correlation matrix $V(X^A,X^B)$ (dotted), $V(Y^A,Y^B)$ (dot-dashed) for the (b) signal and (c) pump fields, respectively, and (d) the logarithmic negativity $E_{\mathcal{N}}$ corresponding to the subsystem of signals (solid) and pumps (dash), respectively. $\kappa =1.13$ and $P_s/P_p=1$. $\zeta$ is the normalized propagation coordinate. The vertical lines show the planes where the phase difference of the classical waves $\mathrm{\Delta }\theta$ ($\mathrm{\Delta }\varphi$) are multiples of $\pi$.

Figure 4

Pump entanglement for different values of $\kappa$ and ratio $P_s/P_p$. (a) Logarithmic negativity $E_{\mathcal{N}}$ for a set value of $\kappa =1.13$ and different ratios $P_s/P_p$: $1/10$ (solid), $1/9$ (small-dashed), $1/4$ (large-dashed), $2/3$ (dotted), and 1 (dot-dashed). (b) Logarithmic negativity $E_{\mathcal{N}}$ for a set value of the ratio $P_s/P_p=1/4$ and different values of $\kappa$: 1.01 (solid), 1.13 (small-dashed), 1.6 (large-dash), 2.26 (dotted), and 3.2 (dot-dashed). $\zeta$ is the normalized propagation coordinate. Note that, in the panel (b), $\zeta$ is different for each plot.

Figure 5

Optimized van Loock–Furusawa inequalities. Simultaneous values under the threshold value imply CV quadripartite entanglement. Solid line shows the first and third inequalities. Dashed line shows the second inequality. Dotted line shows quadripartite entanglement threshold. In gray the area where the violation of the three inequalities is obtained. $\kappa =2.26$ and $P_s/P_p=1$. $\zeta$ is the normalized propagation coordinate.