Genuine quadripartite quantum steering generated by an optical parametric oscillation cascaded with a sum-frequency process

It has been found that quantum nonlocality has different manifestations, including quantum entanglement, quantum steering, quantum discord and Bell nonlocality. In 1935, Einstein, Podolski and Rosen put forward the famous Einstein-Podolsky-Rosen (EPR) paradox in order to question the completeness of quantum mechanical description [1]. Subsequently, Schrödinger introduced the concept of entanglement into the study of EPR paradox, and proposed another quantum nonlocality, quantum steering (or EPR steering) [2,3]. In recent years, it has been found that quantum steering can also be used as a core resource in various quantum information works, such as quantum key distribution [4–6], secure quantum communication [7–9], and entanglement swapping [10]. In addition, it has been found that quantum steering has quantum properties different from entangled states [11,12], and can accomplish some unique quantum information work, such as asymmetric quantum network [13,14], subchannel recognition [15,16], and one-way quantum computation [17].

Wiseman et al. gave an operable definition of quantum steering formulization in 2007, discussed the relationship between quantum steering and quantum entanglement and Bell nonlocality and proved that quantum steering correlation is a quantum nonlocality result between quantum entanglement and Bell nonlocality [18,19]. Subsequently, they gave the detection and determination method in the quantum steering experiment [20]. In 2013, an experimental detection criterion was given for genuine multipartite quantum steering [21]. Based on this criterion, a multipartite quantum secure communication scheme was proposed, and the experimental detection was completed by using continuous variable optical system [22]. In 2015, a unified experimental criterion and measurement criterion for different categories of quantum correlations of two-mode Gauss states was proposed, which provides a simple and effective basis for experimental and theoretical research [23]. Qin et al. experimentally demonstrated three schemes to manipulate the direction of EPR steering [24]. A scheme was proposed to generate one-way EPR steering via atomic coherence [25]. Nonlocal properties, specifically EPR steering and Bell nonlocality [26], of two macroscopic mechanical resonators in optomechanical systems were investigated [27]. Tripartite EPR steering could be generated by four-wave mixing in Rubidium atoms with linear and nonlinear beamsplitters [28]. Another scheme for the generation of bipartite steering and genuine tripartite steering in cascaded four-wave mixing processes in hot Rubidium vapor was analyzed theoretically [29]. The behavior of quantum steering for a pair of quantum qubits described by the Heisenberg model with external magnetic field was investigated [30].

In a cascaded nonlinear process, the quasi-phase-matching technique can be used to realize multiple wavelength optical field output in an optical superlattice. In 2017, Olsen theoretically studied the quantum correlation among the optical fields in an optical parametric oscillation with an injection signal and found that there was asymmetric quantum steering correlation in the nonlinear process [31]. Subsequently, the quantum steering between the cascaded second-harmonic generation process [32] and the cascaded third-harmonic generation process [33] were studied, respectively. It was found that there are asymmetric two-component quantum steering correlations in the cascaded nonlinear processes. However, the genuine multipartite quantum steering of the cascaded nonlinear processes was not investigated. Recently, Li and Olsen studied quantum steering among the optical fields generated by combining degenerate optical parametric oscillation and second-harmonic generation processes [34]. They analyzed asymmetric two-component quantum steering among the three output fields. Nevertheless, there is no tripartite quantum steering present in the cascaded nonlinear process. This may be due to the fact that the down-conversion process and the frequency doubling process are two relatively independent nonlinear processes. Recently, the genuine tripartite EPR steering was demonstrated can be generated in cascaded third-harmonic generation processes [35] and fourth-harmonic generation processes [36], respectively. In ref. [37], we investigated the generation of three-colour tripartite entanglement by injection-seeded nondegenerate optical parametric oscillator without considering the cascaded sum-frequency process and did not discuss the quantum steering in the system. In ref. [38], we investigated the generation of quadripartite entanglement by optical parametric amplification cascaded a sum-frequency process. However, we did not include an optical cavity (it is a single-pass process) in the previous study and did not investigate the generation of multipartite quantum steering in the cascaded nonlinear process.

In this paper, we investigate the genuine multipartite quantum steering in an injected signal optical parametric oscillation cascaded with a sum-frequency process between pump and idler in an optical cavity. This cascaded nonlinear process is different from the injected signal optical parametric oscillation in ref. [31]. There is no cascaded sum-frequency process in ref. [31] and the author did not investigate the multipartite quantum steering. We find that genuine multipartite quantum steering can be produced by the cascaded nonlinear process in this scheme. The multipartite quantum steering among the four optical fields are discussed by genuine multipartite quantum steering criteria [21]. Multipartite quantum steering is the essential resource for quantum information processing such as secure one-sided device-independent quantum secret sharing. In our scheme, quadripartite quantum steering can be produced in one optical superlattice which has potential applications in quantum information processing.

We consider the following cascaded nonlinear process in a one-sided optical cavity which can be seen in fig. 1(a). A pump (frequency $\omega_0$) and a signal (frequency $\omega_1$) enter into the optical cavity, in which an optical superlattice (OSL) is placed as a nonlinear gain medium. The idler (frequency $\omega_2$) is generated through the difference-frequency generation process between pump and signal beams. Then, the sum-frequency beam (frequency $\omega_3$) is generated by the cascaded sum-frequency process between pump and idler beams in the same optical superlattice. In this cascaded nonlinear process, the energy conservation satisfies $\omega_0=\omega_1+\omega_2$ and $\omega_3=\omega_0+\omega_2$. The phase mismatch in the cascaded nonlinear process will be simultaneously compensated by the reciprocal vectors $\vec{G}_1$ and $\vec{G}_2$ which provided by the OSL through quasi-phase-matching technique [39] as

$$\begin{eqnarray} &&\vec{k}_0=\vec{k}_1+\vec{k}_2+\vec{G}_1 \end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray} &&\vec{k}_3=\vec{k}_0+\vec{k}_2+\vec{G}_2, \end{eqnarray} \tag{ 2 }$$

which can be seen in fig. 1(b).

Zoom In Zoom Out Reset image size

The interaction Hamiltonian for this cascaded nonlinear process can be written as

$$\begin{equation} \mathcal{H}_I=i\hbar \kappa_1(\hat{a}_0\hat{a}_1^{\dag}\hat{a}_2^{\dag}-\hat{a}_0^{\dag}\hat{a}_1\hat{a}_2)+i\hbar \kappa_2 (\hat{a}_3^{\dag}\hat{a}_0\hat{a}_2-\hat{a}_3\hat{a}_0^{\dag}\hat{a}_2^{\dag}),\quad \end{equation} \tag{ 3 }$$

where $\kappa_i(i=1,2)$ is the effective nonlinear coupling constant that is related to the nonlinear susceptibility, the structure parameters of optical superlattice, and the power density of the pump. For simplicity, it is taken to be real in this case [40]. We consider the external inputs of pump and signal as

$$\begin{equation} \mathcal{H}_{\textit{ext}}=i\hbar (\epsilon_0 \hat{a}_0^{\dag}+\epsilon_1 \hat{a}_1^{\dag})+\textrm{h.c.}, \end{equation} \tag{ 4 }$$

where $\epsilon_0$ and $\epsilon_1$ are the classical pump and signal amplitude, respectively. The losses of the four modes can be written as

$$\begin{equation} \mathcal{L}_i\hat{\rho}=\gamma_i(2\hat{a}_i\hat{\rho}\hat{a}_i^{\dag}-\hat{a}_i^{\dag}\hat{a}_i\hat{\rho}-\hat{\rho}\hat{a}_i^{\dag}\hat{a}_i)\quad (i=0,1,2,3),\quad \end{equation} \tag{ 5 }$$

where $\gamma_i$ stands for the damping ratio for the four modes which are related to the reflection coefficients of the cavity mirror for the corresponding mode.

The master equation of the density operator $\hat{\rho}$ for this cascaded nonlinear process in the interaction picture is given by

$$\begin{equation} \frac{\textrm{d}\hat{\rho}}{\textrm{d}t}=-\frac{i}{\hbar}[\mathcal{H}_I+\mathcal{H}_{ext}, \hat{\rho}]+\sum_{i=0}^3\mathcal{L}_i\hat{\rho}. \end{equation} \tag{ 6 }$$

One can map the master equation onto the Fokker-Planck equation in the positive-P representation [41] in order to investigate the quantum entanglement and quantum steering characteristics of the cascaded nonlinear process. The Fokker-Planck equation of the system can be obtained as

$$\begin{eqnarray} \frac{\textrm{d}P}{\textrm{d}t} &= \biggl\{-(\epsilon_0-\gamma_0\alpha_0-\kappa_1\alpha_1\alpha_2-\kappa_2\alpha_2^{+}\alpha_3)\frac{\partial}{\partial\alpha_0}\nonumber\\ & \quad-(\epsilon_0^{*}-\gamma_0\alpha_0^{+}-\kappa_1\alpha_1^{+}\alpha_2^{+}-\kappa_2\alpha_2\alpha_3^{+})\frac{\partial}{\partial\alpha_0^{+}} \nonumber\\ & \quad-(\epsilon_1-\gamma_1\alpha_1+\kappa_1\alpha_0\alpha_2^{+})\frac{\partial}{\partial\alpha_1}\nonumber\\ & \quad-(\epsilon_1^{*}-\gamma_1\alpha_1^{+}+\kappa_1\alpha_0^{+}\alpha_2)\frac{\partial}{\partial\alpha_1^{+}} \nonumber\\ & \quad-(-\gamma_2\alpha_2+\kappa_1\alpha_0\alpha_1^{+}-\kappa_2\alpha_0^{+}\alpha_3)\frac{\partial}{\partial\alpha_2}\nonumber\\ & \quad-(-\gamma_2\alpha_2^{+}+\kappa_1\alpha_0^{+}\alpha_1-\kappa_2\alpha_0\alpha_3^{+})\frac{\partial}{\partial\alpha_2^{+}}\nonumber\\ & \quad-(-\gamma_3\alpha_3+\kappa_2\alpha_0\alpha_2)\frac{\partial}{\partial\alpha_3} \nonumber\\ & \quad-(-\gamma_3\alpha_3^{+}+\kappa_2\alpha_0^{+}\alpha_2^{+})\frac{\partial}{\partial\alpha_3^{+}} \nonumber\\ & \quad-\frac{1}{2}\frac{\partial^2}{\partial\alpha_1\partial\alpha_2}(2\kappa_1\alpha_0)-\frac{1}{2}\frac{\partial^2}{\partial\alpha_1^{+}\partial\alpha_2^{+}}(2\kappa_1\alpha_0^{+}) \nonumber\\ & \quad-\frac{1}{2}\frac{\partial^2}{\partial\alpha_0\partial\alpha_2}(-2\kappa_2\alpha_3)\nonumber\\ & \quad-\frac{1}{2}\frac{\partial^2}{\partial\alpha_0^{+}\partial\alpha_2^{+}}(-2\kappa_2\alpha_3^{+})\biggr\}P(\alpha), \end{eqnarray} \tag{ 7 }$$

where $\alpha_i$ and $\alpha_i^{+}$ are independent variables which correspond to $\alpha_i$ and $\alpha_i^{\dag}$ when the averages of products converge to normally ordered operator expectation values [31].

The stochastic differential equations of the four modes can be written as [42]

$$\begin{eqnarray} \frac{\textrm{d}\alpha_0}{\textrm{d}t} &= \epsilon_0-\gamma_0\alpha_0-\kappa_1\alpha_1\alpha_2-\kappa_2\alpha_2^{+}\alpha_3+\sqrt{-2\kappa_2\alpha_3}\eta_1 \nonumber\\ \frac{\textrm{d}\alpha_0^{+}}{\textrm{d}t} &= \epsilon_0^{+}-\gamma_0\alpha_0^{+}-\kappa_1\alpha_1^{+}\alpha_2^{+}-\kappa_2\alpha_2\alpha_3^{+}+\sqrt{-2\kappa_2\alpha_3^{+}}\eta_1^{\dag} \nonumber\\ \frac{\textrm{d}\alpha_1}{\textrm{d}t} &= \epsilon_1-\gamma_1\alpha_1+\kappa_1\alpha_0\alpha_2^{+}+\sqrt{2\kappa_1\alpha_0}\eta_2 \nonumber\\ \frac{\textrm{d}\alpha_1^{+}}{\textrm{d}t} &= \epsilon_1^{+}-\gamma_1\alpha_1^{+}+\kappa_1\alpha_0^{+}\alpha_2+\sqrt{2\kappa_1\alpha_0^{+}}\eta_2^{\dag} \nonumber\\ \frac{\textrm{d}\alpha_2}{\textrm{d}t} &= -\gamma_2\alpha_2+\kappa_1\alpha_0\alpha_1^{+}-\kappa_2\alpha_0^{+}\alpha_3+\sqrt{-2\kappa_2\alpha_3}\eta_1\nonumber\\ & \quad+\sqrt{2\kappa_1\alpha_0}\eta_2 \nonumber\\ \frac{\textrm{d}\alpha_2^{+}}{\textrm{d}t} &= -\gamma_2\alpha_2^{+}+\kappa_1\alpha_0^{+}\alpha_1-\kappa_2\alpha_0\alpha_3^{+}+\sqrt{-2\kappa_2\alpha_3^{+}}\eta_1^{\dag}\nonumber\\ & \quad+\sqrt{2\kappa_1\alpha_0^{+}}\eta_2^{\dag} \nonumber\\ \frac{\textrm{d}\alpha_3}{\textrm{d}t} &= -\gamma_3\alpha_3+\kappa_2\alpha_0\alpha_2 \nonumber\\ \frac{\textrm{d}\alpha_3^{+}}{\textrm{d}t} &= -\gamma_3\alpha_3^{+}+\kappa_2\alpha_0^{+}\alpha_2^{+}, \end{eqnarray} \tag{ 8 }$$

where $\eta_i(t) (i=1,2,3,4)$ are the Gaussian noise terms which satisfy the relations $\langle\eta_{i}(t)\rangle =\langle \eta_{i}^{\dag}(t)\rangle =0$, $\langle \eta_{i}(t)\eta_{j}(t^{\prime })\rangle =\langle \eta_{i}^{\dag}(t)\eta_{j}^{\dag}(t^{\prime })\rangle =0$, and $\langle \eta_{i}(t)\eta_{j}^{\dag}(t^{\prime })\rangle =\delta_{ij}\delta (t-t^{\prime })$.

The steady-state solutions can be obtained by solving the equations $\frac{\textrm{d}\alpha_i}{\textrm{d}t}=0$. We find that the steady-state solution A₂ for $\alpha_2$ can be obtained by solving the following equation:

$$\begin{eqnarray} & (\kappa_1^2+\kappa_2^2)^2\gamma^2A_2^5+2[(E_1\kappa_1\kappa_2)^2+(\kappa_1^2+\kappa_2^2)\gamma_0\gamma^3]A_2^3\nonumber\\ & -E_0E_1\kappa_1\gamma(\kappa_1^2-3\kappa_2^2)A_2^2 \nonumber\\ & +\gamma[E_1^2\kappa_1^2\gamma_0+(\kappa_2^2-\kappa_1^2)E_0^2\gamma+\gamma_0^2\gamma^3]A_2\nonumber\\ & +E_0E_1\kappa_1\gamma_0\gamma^2=0, \end{eqnarray} \tag{ 9 }$$

where we assume $\epsilon_0=\epsilon_0^*=E_0$, $\epsilon_1=\epsilon_1^*=E_1$ and $\gamma_1=\gamma_2=\gamma_3=\gamma$ for simplicity. However, there are no analytical solutions since it is a five-order equation. We can obtain the numerical solution of A₂ by computer. The other steady-state solutions $A_i(i=0,1,3)$ for $\alpha_i(i=0,1,3)$ are related to A₂ as

$$\begin{eqnarray} A_0&= \{E_0\gamma(DEE_0^2\gamma-C\gamma_0) \nonumber\\ & \quad-A_2[CE_1\kappa_1\gamma_0-DE_0^2E_1\kappa_1\gamma(E-2\kappa_2^2)\nonumber\\ & \quad+DE_1\kappa_1\gamma_0^2\gamma^3+CDE_0A_2 \nonumber\\ & \quad+D^2E_0\gamma_0\gamma^3A_2\!+\!D^2E_1\kappa_1\gamma_0\gamma^2A_2^2+\!D^3E_0\gamma^2A_2^3]\}/F, \nonumber\\ A_1&= \{E_1\gamma_0(DE_0^2\kappa_2^2\gamma-EE_1^2\kappa_1^2\gamma_0)\nonumber\\ & \quad+DE_0\kappa_1\gamma_0(EE_1^2\kappa_1^2+\gamma_0\gamma^3) \nonumber\\ & \quad+\kappa_1\gamma_0A_2^2[D\gamma A_2(E_0\gamma-E_1\kappa_1A_2)\nonumber\\ & \quad-DE_1\kappa_1\gamma_0\gamma^2-EE_1\kappa_1/\gamma]\}/F, \nonumber\\ A_3&= \kappa_2A_0A_2/\gamma, \end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray} C&= 2E_1^2\kappa_1^2\kappa_2^2,\nonumber\\ D&= (\kappa_1^2+\kappa_2^2),\nonumber\\ E&= (\kappa_1^2-\kappa_2^2),\nonumber\\ F&= \gamma_0\gamma(\kappa_1^2-\kappa_2^2)[E_0^2\gamma(\kappa_1^2+\kappa_2^2)+E_1^2\kappa_1^2\gamma_0]. \end{eqnarray} \tag{ 11 }$$

In the following, one can consider the system variables include their mean values and small fluctuations in the steady state as $\alpha_i=A_i+\delta \alpha_i (i=0,1,2,3)$ with $\delta \alpha_i\ll A_i$. In this case, the above equation (8) can be linearized as

$$\begin{eqnarray} \frac{\textrm{d}}{\textrm{d}t} \delta\alpha_0&= -\gamma_0\delta\alpha_0-\kappa_1A_2\delta\alpha_1-\kappa_1A_1\delta\alpha_2-\kappa_2A_3\delta\alpha_2^{+}\nonumber\\ & \quad-\kappa_2A_2^*\delta\alpha_3+\sqrt{-2\kappa_2A_3}\eta_1 \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_0^{+} &= -\gamma_0\delta\alpha_0^{+}-\kappa_1A_2^*\delta\alpha_1^{+}-\kappa_2A_3^*\delta\alpha_2-\kappa_1A_1^*\delta\alpha_2^{+}\nonumber\\ & \quad-\kappa_2A_2\delta\alpha_3^{+}+\sqrt{-2\kappa_2A_3^*}\eta_1^{\dag} \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_1 &= \kappa_1A_2^*\delta\alpha_0-\gamma\delta\alpha_1+\kappa_1A_0\delta\alpha_2^{+}+\sqrt{2\kappa_1A_0}\eta_2 \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_1^{+} &= \kappa_1A_2\delta\alpha_0^{+}-\gamma\delta\alpha_1^{+}+\kappa_1A_0^*\delta\alpha_2+\sqrt{2\kappa_1A_0^*}\eta_2^{\dag} \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_2 &= \kappa_1A_1^*\delta\alpha_0-\kappa_2A_3\delta\alpha_0^{+}+\kappa_1A_0\delta\alpha_1^{+}-\gamma\delta\alpha_2\nonumber\\ & \quad-\kappa_2A_0^*\delta\alpha_3+\sqrt{-2\kappa_2A_3}\eta_1+\sqrt{2\kappa_1A_0}\eta_2 \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_2^{+} &= -\kappa_2A_3^*\delta\alpha_0+\kappa_1A_1\delta\alpha_0^{+}+\kappa_1A_0^*\delta\alpha_1-\gamma\delta\alpha_2^{+}\nonumber\\ & \quad-\kappa_2A_0\delta\alpha_3^{+}+\sqrt{-2\kappa_2A_3^*}\eta_1^{\dag}+\sqrt{2\kappa_1A_0^*}\eta_2^{\dag} \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_3 &= \kappa_2A_2\delta\alpha_0+\kappa_2A_0\delta\alpha_2-\gamma\delta\alpha_3 \nonumber\\ \frac{\textrm{d}}{\textrm{d}t}\delta\alpha_3^{+} &= \kappa_2A_2^*\delta\alpha_0^{+}+\kappa_2A_0^*\delta\alpha_2^{+}-\gamma\delta\alpha_3^{+}, \end{eqnarray} \tag{ 12 }$$

which can be written in the matrix form as

$$\begin{equation} \textrm{d}\delta\tilde{\alpha}=-\mathbf{A}\delta\tilde{\alpha}\textrm{d}t+\mathbf{B}\textrm{d}W, \end{equation} \tag{ 13 }$$

with

$$\begin{equation} \delta\tilde{\alpha}=[\delta\alpha_0,\delta\alpha_0^{+},\delta\alpha_1,\delta\alpha_1^{+},\delta\alpha_2,\delta\alpha_2^{+},\delta\alpha_3,\delta\alpha_3^{+}]^\mathrm{T},\quad \end{equation} \tag{ 14 }$$

where B is the noise term contains the steady-state solutions and $\textrm{d}W$ is the Wiener increments [41,43]. A is the drift matrix which can be written as

$$\begin{eqnarray} & \!\!\!\!\mathbf{A}=\nonumber\\ & \!\!\!\!{\footnotesize\left(\!\! \begin{array}{cccccccc} \gamma_0 & 0 & \kappa_1A_2 & 0 & \kappa_1A_1 & \kappa_2A_3 & \kappa_2A_2^* & 0\\ 0 & \gamma_0 & 0 & \kappa_1A_2^* & \kappa_2A_3^* & \kappa_1A_1^* & 0 & \kappa_2A_2\\ -\kappa_1A_2^* & 0 & \gamma & 0 & 0 & -\kappa_1A_0 & 0 & 0\\ 0 & -\kappa_1A_2 & 0 & \gamma & -\kappa_1A_0^* & 0 & 0 & 0\\ -\kappa_1A_1^* & \kappa_2A_3 & 0 & -\kappa_1A_0 & \gamma & 0 & -\kappa_2A_0^* & 0\\ \kappa_2A_3^* & -\kappa_1A_1 & -\kappa_1A_0^* & 0 & 0 & \gamma & 0 & \kappa_2A_0\\ -\kappa_2A_2 & 0 & 0 & 0 & -\kappa_2A_0 & 0 & \gamma & 0\\ 0 & -\kappa_2A_2^* & 0 & 0 & 0 & -\kappa_2A_0^* & 0 & \gamma\\ \end{array} \!\!\!\!\right).}\nonumber\\ \end{eqnarray} \tag{ 15 }$$

The system can be in a steady state only when above drift matrix A has no negative eigenvalues. Figure 2 shows that the minimum eigenvalue of A vs. E₀ (a), E₁ (b), $\kappa_2/\kappa_1$ (c), and $\gamma_0/\gamma$ (d), respectively. From fig. 2 one can see that there are no negative eigenvalues for the drift matrix in all the parameter ranges. Therefore, we can discuss the properties of quantum entanglement and quantum steering in the steady state for these nonlinear parameter ranges.

Zoom In Zoom Out Reset image size

From the Fourier transformation, one can obtain the intracavity spectral from eq. (13) as [42,43]

$$\begin{equation} \mathbf{S}(\omega)=(\mathbf{A}+i\omega \mathbf{I})^{\mathrm{-1}}\mathbf{B}\mathbf{B}^{\mathrm{T}}(\mathbf{A}^{\mathrm{T}}-i\omega \mathbf{I})^{\mathrm{-1}}, \end{equation} \tag{ 16 }$$

where ω is the Fourier analysis frequency and I is the identity matrix. The quantum fluctuations outside the cavity can be obtained by input-output relationship at the coupling mirror [44].

Bipartite asymmetric quantum steering can be generated in injection-seeded nondegenerate optical parametric oscillator [31]. However, in ref. [31], the author did not consider the cascaded sum-frequency process and did not investigate the multipartite quantum steering in the nonlinear process. In the following, we will investigate whether the quadripartite quantum steering can be generated in the injection-seeded nondegenerate optical parametric oscillator cascaded a sum-frequency process between pump and the injected signal in an optical cavity. Based on the criteria for multipartite EPR steering in ref. [21], we set the equations as

$$\begin{eqnarray} S_0&= \Delta (X_0-X_1)\Delta (Y_0+Y_1+Y_2+Y_3),\nonumber\\ S_1&= \Delta (X_1-X_2)\Delta (Y_0+Y_1+Y_2+Y_3),\nonumber\\ S_2&= \Delta (X_2-X_3)\Delta (Y_0+Y_1+Y_2+Y_3),\nonumber\\ S_3&= \Delta (X_3-X_0)\Delta (Y_0+Y_1+Y_2+Y_3). \end{eqnarray} \tag{ 17 }$$

$S_i<1(i=0,1,2,3)$ will confirm EPR steering of system i and

$$\begin{equation} S_{\textit{tot}}=S_0+S_1+S_2+S_3<1 \end{equation} \tag{ 18 }$$

will demonstrate the genuine quadripartite steering can be generated in our scheme [21].

Figure 3 shows S_i and $S_{\textit{tot}}$ vs. the normalized analysis frequency $\Omega=\omega/\gamma_0$. It can be seen in fig. 3 that $S_i (i=0,1,2,3)$ are all below 1 which indicates that bipartite EPR steering is present between every optical field of i and remain optical fields. More significantly, it is that $S_{\textit{tot}}<1$, the inequality (18) is satisfied in the whole range of analysis frequency Ω, which demonstrates that genuine quadripartite quantum steering can be generated by the cascaded nonlinear process in our scheme.

Zoom In Zoom Out Reset image size

Figure 4 plots the S_i of EPR steering of system i and the $S_{\textit{tot}}$ of genuine quadripartite steering vs. E₀ (a), E₁ (b), $\gamma_0/\gamma$ (c), and $\kappa_2/\kappa_1$ (d), respectively. In fig. 4, S_i and $S_{\textit{tot}}$ are all below 1 almost in the whole parameter range which indicates that both bipartite EPR steering and genuine quadripartite steering can be obtained in the cascaded nonlinear process. The values of S_i and $S_{\textit{tot}}$ increase with the increase of E₀ which can be seen in fig. 4(a), however, decrease with the increase of E₁ which can be seen in fig. 4(b). This is because when the pump is weak, its quantum properties will be present. When the injected signal is strong, the nonlinear conversion efficiency will be increased and the strong idler and sum-frequency optical fields can be obtained. From fig. 4(c) one can see that the values of S_i and $S_{\textit{tot}}$ decrease with the increase of the ratio of $\gamma_0/\gamma$. $\gamma_0$ represents the loss of pump in the optical cavity. So increasing the value of $\gamma_0$ slightly will reduce the intensity of the pump and the quantum property of the pump will be present which is consistent with the case in fig. 4(a). One can see that there is a singularity in fig. 4(d) at $\kappa_2/\kappa_1=1$. This is because when $\kappa_1=\kappa_2$, F is equal to zero in eq. (11) may lead to existing three singularities in the steady-state solutions of A₀, A₁, and A₃. There are no steady output for the four optical fields at this point. The results of fig. 4 provide an optional parameter range for obtaining better quadripartite quantum steering in experiment.

Zoom In Zoom Out Reset image size

In summary, the quantum features among pump, signal, idler, and sum-frequency beams generated by a cascaded nonlinear process in an optical cavity are investigated. Genuine quadripartite quantum steering is confirmed by applying the criteria for genuine multipartite quantum steering [21]. The variations of quantum steering properties with the nonlinear parameters are also discussed. Optical parametric down-conversion cascaded a sum-frequency process has been realized in experiments by the quasi-phase-matching technique with [45] and without [46] optical cavity, respectively. With an injection signal, the conversion efficiency of the cascaded nonlinear process will be further increased in our scheme. In addition, squeezed state measurement of multicolor optical fields with different frequencies have been achieved experimentally [47,48]. Therefore, our present scheme of producing quadripartite quantum steering is feasible in experiment. Theoretical study of present analysis can provide reference data for obtaining better multipartite quantum steering correlation in experiment. Quadripartite quantum steering can be produced in our simple scheme which can be applied in quantum information processing.

This work is supported by National Natural Science Foundation of China (Nos. 61975184 and 61775043), Natural Science Foundation of Zhejiang Province (No. LY18A040007), Science Foundation of Zhejiang Sci-Tech University (No. 19062151-Y).