Continuous-Variable Triple-Photon States Quantum Entanglement

We investigate the quantum entanglement of the three modes associated with the three-photon states obtained by triple-photon generation in a phase-matched third-order nonlinear optical interaction. Although the second-order processes have been extensively dealt with, there is no direct analogy between the second and third-order mechanisms. We show, for example, the absence of quantum entanglement between the quadratures of the three modes in the case of spontaneous parametric triple-photon generation. However, we show robust, seeding-dependent, genuine triple-photon entanglement in the fully seeded case.

Figure 1

Evolution of the nonseparability criterion $S$ as a function of the seeding average photon number for $|\xi |=1.75\times {10}^{-6}$ and for different injection cases. The gray area highlights the quantum region of tripartite genuine entanglement. Full inseparability is fulfilled between 2 and 4. The triple-photon generation boxes show the different schemes with the seeded (full arrows) and nonseeded (dashed arrows) modes. The $S$ boxes indicate the modes concerned by the $S$ measurement. Magenta line with (plus) markers: single seeding case. Green line with (times) markers: double seeding case. Red line with (circle) markers for the fully seeded at ${\theta }_1=0$, ${\theta }_2={\theta }_3=\pi$, and for $\beta =\sqrt{2}$. The phase of the seeding coherent state is $\varphi =\pi /2$. Blue line with (square) markers: same as the red curve but for quadratures $\widehat{u}={\widehat{Q}}_1+{\widehat{Q}}_2$ and $\widehat{v}={\widehat{P}}_1-{\widehat{P}}_2$.

Figure 2

Gain (a) and ${\widehat{P}}_k$ (red) and ${\widehat{Q}}_k$ (blue) variances (b) as a function of the phase of the seeding fields, for $|\xi |=1.75\times {10}^{-6}$ and ${\overline{N}}_{\mathrm{in}}=4\times {10}^{10}$. The gray circles indicate the unity gain in (a) and the standard quantum limit in (b).