Enhancement of quadripartite quantum correlation via phase-sensitive cascaded four-wave mixing process

Multipartite quantum correlation is important for both fundamental science and quantum information processing. Its enhancement is critical to improve the performance of physical systems. Therefore, it is of great practical significance to study the enhancement of multipartite quantum correlation. Here, we theoretically investigate a scheme for enhancing quadripartite quantum correlation by utilizing a phase-sensitive cascaded four-wave mixing (CFWM) process. We find that the intensity-difference squeezing (IDS) among the four output beams generated by the phase-insensitive CFWM process can be largely enhanced by introducing the phase-sensitive CFWM process. We also find that our phase-sensitive CFWM process can be used to generate intensity-sum squeezing (ISS), which cannot be generated by the phase-insensitive CFWM process. Moreover, we find that the maximum squeezing value of ISS is equal to that of IDS when the intensity gains of the three four-wave mixing processes are equal in the phase-sensitive cascaded scheme. In the end, we discuss the effects of the losses and the phase fluctuations on the squeezing values of our phase-sensitive CFWM process and the corresponding phase-insensitive CFWM process with the same intensity gains. It can be seen that the squeezing enhancement of our phase-sensitive CFWM process holds also in presence of the losses and the phase fluctuations. Our results pave the way for experimental implementation and may find applications in quantum metrology and quantum communication.

Figure 1

The phase-sensitive CFWM process in a ${}^{85}\mathrm{Rb}$ vapor cell for enhancing quadripartite quantum correlation. (a) Schematic view of the phase-sensitive CFWM process. PBS, polarizing beam splitter; ${\mathrm{cell}}_1$, ${\mathrm{cell}}_2$, and ${\mathrm{cell}}_3$, ${}^{85}\mathrm{Rb}$ vapor cells; ${\widehat{a}}_0$ and ${\widehat{a}}_1^{\prime }$, coherent probe fields; ${\widehat{b}}_0$ and ${\widehat{b}}_1^{\prime }$, coherent conjugate fields; ${\widehat{a}}_1$ and ${\widehat{b}}_1$, output fields of ${\mathrm{cell}}_1$; ${\widehat{a}}_2$, ${\widehat{b}}_2^{\prime }$, ${\widehat{a}}_2^{\prime }$, and ${\widehat{b}}_2$, output fields of the phase-sensitive CFWM process; ${\widehat{c}}_1$, ${\widehat{c}}_2$, and ${\widehat{c}}_3$, pump fields. All these field operators are annihilation operators associated with the corresponding optical fields. (b) Energy-level diagram of the double-$\mathrm{\Lambda }$ scheme in the $D_1$ line of ${}^{85}\mathrm{Rb}$ for the single FWM process. $\mathrm{\Delta }$ and $\delta$ stand for the one-photon detuning and two-photon detuning, respectively.

Figure 2

The values of IDS as a function of $G_2$ and $G_3$ for different values of $G_1$. Curved surface i represents the squeezing value of the phase-insensitive CFWM process, and curved surface ii represents the squeezing value of the phase-sensitive CFWM process under the condition of ${\psi }_1={\psi }_2={\psi }_3=0$ and ${\beta }_1={\beta }_2={\beta }_3=1$. (a) $G_1=2$. (b) $G_1=3$. (c) $G_1=4$.

Figure 3

The values of IDS as a function of the ratios ${\beta }_2$ and ${\beta }_3$ for different values of ${\beta }_1$ when ${\psi }_1={\psi }_2={\psi }_3=0$ and $G_1=G_2=G_3=3$. Plane i represents the squeezing value of the phase-insensitive CFWM process, and curved surface ii represents the squeezing value of the phase-sensitive CFWM process. (a) ${\beta }_1=0$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\beta }_2=0.1168$ and ${\beta }_3=0.0778$. (b) ${\beta }_1=0.5$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\beta }_2=0.078$ and ${\beta }_3=0.0728$. (c) ${\beta }_1=1$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\beta }_2={\beta }_3=0.0977$. (d) ${\beta }_1=5$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\beta }_2=0.3098$ and ${\beta }_3=0.3615$.

Figure 4

The values of IDS as a function of phases ${\psi }_2$ and ${\psi }_3$ for different values of ${\psi }_1$ when $G_1=G_2=G_3=3$, ${\beta }_1=1$, and ${\beta }_2={\beta }_3=0.0977$. Plane i represents the squeezing value of the phase-insensitive CFWM process, and curved surface ii represents the squeezing value of the phase-sensitive CFWM process. (a) ${\psi }_1=0$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\psi }_2={\psi }_3=0$, $2\pi$. (b) ${\psi }_1=\pi /2$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\psi }_2={\psi }_3=0.2181\pi$. (c) ${\psi }_1=\pi$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\psi }_2={\psi }_3=0$, $2\pi$. (d) ${\psi }_1=3\pi /2$: the maximum squeezing value of the phase-sensitive CFWM process is achieved when ${\psi }_2={\psi }_3=1.783\pi$.

Figure 5

The values of IDS as a function of the intensity gain $G$. Red dashed line: the squeezing value of the phase-insensitive CFWM process; red solid line: the maximum squeezing value of the phase-sensitive CFWM process as predicted by Eq. (6).

Figure 6

The ISS values of the phase-sensitive CFWM process as a function of ${\beta }_2$ and ${\beta }_3$ when ${\psi }_1={\psi }_2={\psi }_3=\pi$ for ratios ${\beta }_1$ of 0, 0.5, 1, and 5 and gains $G$ of 2, 3, and 4. (a)–(d) $G=2$: the maximum squeezing value is achieved when ${\beta }_1=1$ and ${\beta }_2={\beta }_3=0.1577$. (e),–(h) $G=3$: the maximum squeezing value is achieved when ${\beta }_1=1$ and ${\beta }_2={\beta }_3=0.0977$. (i)–(l) $G=4$: the maximum squeezing value is achieved when ${\beta }_1=1$ and ${\beta }_2={\beta }_3=0.0705$.

Figure 7

The ISS values of the phase-sensitive CFWM process as a function of ${\psi }_2$ and ${\psi }_3$ when ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are certain values for phases ${\psi }_1$ of 0, $\pi /2$, $\pi$, and $3\pi /2$ and gains $G$ of 2, 3, and 4. (a)–(d) $G=2$, ${\beta }_1=1$, and ${\beta }_2={\beta }_3=0.1577$. (e)–(h) $G=3$, ${\beta }_1=1$, and ${\beta }_2={\beta }_3=0.0977$. (i)–(l) $G=4$, ${\beta }_1=1$, and ${\beta }_2={\beta }_3=0.0705$. All of the maximum squeezing values for different gains are achieved when ${\psi }_1={\psi }_2={\psi }_3=\pi$.

Figure 8

Comparison of squeezing values between the phase-sensitive CFWM process and the corresponding phase-insensitive CFWM process with the same intensity gain $G$ in the lossy situation of $L=0.05,\eta =0.03,\lambda =0.03,\tau =0.01$. Solid line: the maximum squeezing value of the phase-sensitive CFWM process; dashed line: the squeezing value of the phase-insensitive CFWM process.

Figure 9

The squeezing values as a function of the phase fluctuation $\delta \theta$ when $G=3$, $L=0.05$, $\eta =0.03$, $\lambda =0.03$, and $\tau =0.01$. Solid line: the maximum squeezing value of the phase-sensitive CFWM process; dashed line: the squeezing value of the corresponding phase-insensitive CFWM process.