Phase-sensitive cascaded four-wave-mixing processes for generating three quantum correlated beams

Theoretical studies and experimental implementations of quantum correlation are the important contents of continuous variables quantum optics and quantum information science. There are various systems for the study of quantum correlation. Here, we study an experimental scheme for generating three quantum correlated beams based on phase-sensitive cascaded four-wave-mixing (FWM) processes in rubidium vapor. Quantum correlation including intensity difference or sum squeezing, two other combinatorial squeezing, and quantum entanglement among the three output light fields are theoretically analyzed in this paper. Also, the comparison of the quantum correlations have been made between the phase-sensitive cascaded FWM processes and the phase-insensitive cascaded FWM processes. By changing the phases and intensities of the input beams, it is interesting to find that the maximum degrees of various combinatorial squeezing are equal when the two FWM processes share a common intensity gain. When the common intensity gain of the two FWM processes changes, the maximum degrees of different combinatorial squeezing will be synchronously controlled. At last we discuss the genuine tripartite entanglement and steering in our phase-sensitive cascaded scheme, and compare them with the cases of the phase-insensitive cascaded scheme.

Figure 1

Phase-sensitive cascaded FWM processes in hot ${}^{85}\mathrm{Rb}$ vapor. (a) The experimental arrangement. (b) Double-$\lambda$ energy level of ${}^{85}\mathrm{Rb} D1$ line: $\mathrm{\Delta }$ and $\delta$ stand for the one-photon detuning and the two-photon detuning respectively. The interaction strength depends strongly on the one-photon detuning $\mathrm{\Delta }$ and the two-photon detuning $\delta$.

Figure 2

The relation among $G_{\mathrm{eff},\mathrm{probe}},{\phi }_1$ and ${\phi }_2$ under the ${\beta }_1={\beta }_2=1$ condition. (a) The entire picture for $G_{\mathrm{eff},\mathrm{probe}}$. (b) The local picture for $G_{\mathrm{eff},\mathrm{probe}}>1$. (c) The local picture for $G_{\mathrm{eff},\mathrm{probe}}<1$. Panel (b) shows the probe beam amplified. In addition, panel (c) shows the probe beam deamplified.

Figure 3

$\frac{1}{Z}$ plotted as a function of the input phase ${\phi }_1$ and ${\phi }_2$. (a) The entire picture for $\frac{1}{Z}$ varies with phase ${\phi }_1$ and ${\phi }_2$. (b) The local picture for $\frac{1}{Z}<0.1$.

Figure 4

The squeezing levels of intensity difference $D_{({\widehat{N}}_{a_2}-{\widehat{N}}_{b_2}-{\widehat{N}}_{b_1})}$ vary with ${\beta }_1$ and ${\beta }_2$ when ${\phi }_1={\phi }_2=0/2\pi$. (a) $G_1=G_2=2$: The squeezing level becomes maximal when ${\beta }_1=0.667$ and ${\beta }_2=0.333$. (b) $G_1=G_2=3$: The squeezing level becomes maximal when ${\beta }_1=0.75$ and ${\beta }_2=0.25$. (c) $G_1=G_2=4$: The squeezing level becomes maximal when ${\beta }_1=0.8$ and ${\beta }_2=0.2$. (d) $G_1=G_2=5$: The squeezing level becomes maximal when ${\beta }_1=0.833$ and ${\beta }_2=0.167$.

Figure 5

(a) $G_1=G_2=2$: The squeezing level of intensity difference $D_{({\widehat{N}}_{a_2}-{\widehat{N}}_{b_2}-{\widehat{N}}_{b_1})}$ varies with the phase ${\phi }_1$ and ${\phi }_2$ when ${\beta }_1=0.667$ and ${\beta }_2=0.333$. (b) $G_1=G_2=3$: The squeezing level varies with the phases when ${\beta }_1=0.75$ and ${\beta }_2=0.25$. (c) $G_1=G_2=4$: The squeezing level varies with the phases when ${\beta }_1=0.8$ and ${\beta }_2=0.2$. (d) $G_1=G_2=5$: The squeezing level varies with the phases when ${\beta }_1=0.833$ and ${\beta }_2=0.167$. All of the squeezing levels for different gains become maximal when ${\phi }_1={\phi }_2=0/2\pi$.

Figure 6

The squeezing levels of intensity sum $D_{({\widehat{N}}_{a_2}+{\widehat{N}}_{b_2}+{\widehat{N}}_{b_1})}$ in decibels as a function of ${\beta }_1$ and ${\beta }_2$ when ${\phi }_1={\phi }_2=\pi$. (a) $G_1=G_2=2$: The squeezing level becomes maximal when ${\beta }_1=0.667$ and ${\beta }_2=0.333$. (b) $G_1=G_2=3$: The squeezing level becomes maximal when ${\beta }_1=0.75$ and ${\beta }_2=0.25$. (c) $G_1=G_2=4$: The squeezing level becomes maximal when ${\beta }_1=0.8$ and ${\beta }_2=0.2$. (d) $G_1=G_2=5$: The squeezing level becomes maximal when ${\beta }_1=0.833$ and ${\beta }_2=0.167$.

Figure 7

(a) $G_1=G_2=2$: The squeezing level of intensity sum $D_{({\widehat{N}}_{a_2}+{\widehat{N}}_{b_2}+{\widehat{N}}_{b_1})}$ varies with the phase ${\phi }_1$ and ${\phi }_2$ when ${\beta }_1=0.667$ and ${\beta }_2=0.333$. (b) $G_1=G_2=3$: The squeezing level varies with the phases when ${\beta }_1=0.75$ and ${\beta }_2=0.25$. (c) $G_1=G_2=4$: The squeezing level varies with the phases when ${\beta }_1=0.8$ and ${\beta }_2=0.2$. (d) $G_1=G_2=5$: The squeezing level varies with the phases when ${\beta }_1=0.833$ and ${\beta }_2=0.167$. All of the squeezing levels for different gains become maximal when ${\phi }_1={\phi }_2=\pi$.

Figure 8

The squeezing levels of ${\widehat{N}}_{a_2}-{\widehat{N}}_{b_2}+{\widehat{N}}_{b_1}$ in decibels as a function of the intensity ratios when ${\phi }_1=\pi$ and ${\phi }_2=0/2\pi$. (a) $G_1=G_2=2$: The squeezing level becomes maximal when ${\beta }_1=0.988$ and ${\beta }_2=0.012$. (b) $G_1=G_2=3$: The squeezing level becomes maximal when ${\beta }_1=0.997$ and ${\beta }_2=0.003$. (c) $G_1=G_2=4$: The squeezing level becomes maximal when ${\beta }_1=0.998$ and ${\beta }_2=0.002$. (d) $G_1=G_2=5$: The squeezing level becomes maximal when ${\beta }_1=0.999$ and ${\beta }_2=0.001$.

Figure 9

(a) $G_1=G_2=2$: The squeezing level of ${\widehat{N}}_{a_2}-{\widehat{N}}_{b_2}+{\widehat{N}}_{b_1}$ varies with the phases when ${\beta }_1=0.988$ and ${\beta }_2=0.012$. (b) $G_1=G_2=3$: The squeezing level varies with the phases when ${\beta }_1=0.997$ and ${\beta }_2=0.003$. (c) $G_1=G_2=4$: The squeezing level varies with the phases when ${\beta }_1=0.998$ and ${\beta }_2=0.002$. (d) $G_1=G_2=5$: The squeezing level varies with the phases when ${\beta }_1=0.999$ and ${\beta }_2=0.001$. All of the squeezing levels for different gains become maximal when ${\phi }_1=\pi$ and ${\phi }_2=0/2\pi$.

Figure 10

The squeezing levels of ${\widehat{N}}_{a_2}+{\widehat{N}}_{b_2}-{\widehat{N}}_{b_1}$ in decibels as a function of the ratios when ${\phi }_1={\phi }_2=\pi$. (a) $G_1=G_2=2$: The squeezing level becomes maximal when ${\beta }_1=0.301$ and ${\beta }_2=0.699$. (b) $G_1=G_2=3$: The squeezing level becomes maximal when ${\beta }_1=0.564$ and ${\beta }_2=0.436$. (c) $G_1=G_2=4$: The squeezing level becomes maximal when ${\beta }_1=0.69$ and ${\beta }_2=0.31$. (d) $G_1=G_2=5$: The squeezing level becomes maximal when ${\beta }_1=0.761$ and ${\beta }_2=0.239$.

Figure 11

(a) $G_1=G_2=2$: The squeezing level of ${\widehat{N}}_{a_2}+{\widehat{N}}_{b_2}-{\widehat{N}}_{b_1}$ in decibels as a function of the phases when ${\beta }_1=0.301$ and ${\beta }_2=0.699$. (b) $G_1=G_2=3$: The squeezing level varies with the phases when ${\beta }_1=0.564$ and ${\beta }_2=0.436$. (c) $G_1=G_2=4$: The squeezing level of varies with the phases when ${\beta }_1=0.69$ and ${\beta }_2=0.31$. (d) $G_1=G_2=5$: The squeezing level varies with the phases when ${\beta }_1=0.761$ and ${\beta }_2=0.239$. All of the squeezing levels for different gains become maximal when ${\phi }_1={\phi }_2=\pi$.

Figure 12

Value of maximum squeezing degree as predicted by Eqs. (17), (19), and (22) plotted against the intensity gain $G$ when only $G\ge 1$ is considered.

Figure 13

Values of maximum squeezing degree followed by optical losses plotted against the intensity gain $G$.

Figure 14

Contour plot for $V_{123}$ when the intensity gains $G_1=G_2=2$ (a), $G_1=G_2=3$ (b), $G_1=G_2=4$ (c), and $G_1=G_2=5$ (d) respectively. The white regions ($V_{123}<1$) are the regions of the existence of tripartite entanglement.

Figure 15

Contour plot for $V_{123}$ as a function of $G_1$ and $G_2$ when ${\theta }_1^{\prime }={\theta }_2=0/2\pi$.

Figure 16

Contour plot for $V_{12}$ in Eq. (32) when the intensity gains $G_1=G_2=2$ (a), $G_1=G_2=3$ (b), $G_1=G_2=4$ (c), and $G_1=G_2=5$ (d) respectively. The white regions are the regions of $V_{12}<2$.

Figure 17

Contour plot for $V_{13}$ in Eq. (32) when the intensity gains $G_1=G_2=2$ (a), $G_1=G_2=3$ (b), $G_1=G_2=4$ (c), and $G_1=G_2=5$ (d) respectively.

Figure 18

The values of $V_{12}$ (a) and $V_{13}$ (b) in Eq. (32) vary with $G_1$ and $G_2$ when ${\theta }_1^{\prime }={\theta }_2=0/2\pi$. (c) The white region is the overlapped region of $V_{12}<2$ and $V_{13}<2$.

Figure 19

The values of $S{t}_{123}$ (a), $S{t}_{213}$ (b), $S{t}_{312}$ (c) in Eqs. (34, 35, 36), and $S{t}_{123}+S{t}_{213}+S{t}_{312}$ (d) vary with $G_1$ and $G_2$ when ${\theta }_1^{\prime }={\theta }_2=0/2\pi$.