Genuine tripartite Einstein-Podolsky-Rosen steering in the cascaded nonlinear processes of third-harmonic generation

Y. Liu; S. L. Liang; G. R. Jin; Y. B. Yu

doi:10.1364/OE.380124

1. Introduction

In recent years, EPR steering has an important application value in many aspects such as quantum secret sharing [1–5], quantum key distribution [6] and quantum network [7], because of its property of asymmetry, it has received considerable critical attention. Steering is a phenomenon of quantum mechanical. This phenomenon was introduced by Schrö dinger [8,9] after Einstein, Podolski, and Rosen proposed a famous argument against the completeness of quantum mechanics in 1935 [10]. The notion of steering did not attract much attention until Wiseman et al. [11] gave the rigorous definition of steering and explained it in the form of a task in 2007. The relationship among quantum steering, quantum entanglement and Bell locality is discussed. EPR steering is a kind of quantum delocalization that is different from quantum entanglement [12] and Bell non-locality [13,14]. Unlike quantum entanglement and Bell non-locality, EPR steering (or quantum steering) has a unique asymmetry. There is a one-way EPR steering, that is, Alice can steer Bob, but not vice versa, it has been proved in theory [15,16] and has been demonstrated in experiments [17,18]. In order to improve the understanding of quantum steerability, one have done a lot of works such as the criterion of steering and quantitative investigation of steerability [19] and one-way steering for arbitrary projective measurements [15]. These works greatly help us to make breakthroughs in theoretical [20,21] and experimental exploration [18]. Several experiments have been carried out to prove the steering and its asymmetry [14,22–28]. The work of Wiseman et al. [11] relates to entanglement, steering, and Bell nonlocality. Reid et al. outlined the theory of EPR’s seminal paper and provided an overview of achievements in theory and experiments [29]. In 2013, a criterion of genuine multipartite EPR steering was put forward by He and Reid [30]. Subsequently, a scheme of quantum security communication was proposed and finished in an optical network [21]. Zhou et al. proposed and analyzed a standard form of EPR steering for an arbitrary bipartite Gaussian state [31]. Theoretical exploration and practical application of new avenues have been opened up by EPR steering.

The first demonstration of EPR steering was by Ou et al. [32], by means of using a nondegenerate optical parametric oscillator (OPO). The nondegenerate OPO was used to how to produce three-color entanglement [33,34]. And tripartite entanglement also studied in Ref. [35,36]. Recently, Olsen [28] theoretically studied the quantum correlation of the light fields in the OPO with an injected signal, and found that asymmetry of steering exists in the cascaded nonlinear process. This study makes a contribution to research on EPR steering because it provides a good reference for experimenters to be able to achieve asymmetric quantum steering in such an optical system. Hereafter, bipartite quantum steering correlations was found in the cascaded nonlinear processes by studying the cascaded second harmonic generation process [37] and the cascaded third harmonic generation process [38], respectively. However, it should be pointed out that there is little work on the genuine multipartite quantum steering of the cascaded nonlinear processes.

In this paper, based on the criterion of genuine multipartite EPR steering put forward by He and Reid [30], we investigated the genuine tripartite quantum steering in cascaded nonlinear processes of quasi-phase-matching third-harmonic generation in an optical cavity. Because multipartite quantum steering can be applied in more applications. It meets the needs of the application and development of quantum information such as quantum networks and quantum computation. Moreover, the experimental setup is very simple in our scheme. In the cascaded nonlinear processes, only one pump and one optical superlattice can generate genuine tripartite quantum steering with different frequencies. The overall structure of the study takes the form of four sections, including this section of introduction. Section II begins by obtaining the equations of motion and the stationary solutions in the positive-P representation [39,40]. Section III presents the results and discussions on this study which concerned with genuine tripartite EPR steering. Finally, in section IV we give a brief summary about this work.

2. Equations of motion and the stationary solutions

By using quasi-phase matching (QPM) technique [41], third harmonics may be generated through cascaded sum-frequency processes. A fundmental mode at frequency $\omega _{0}$ enter into an optical cavity that an optical superlattice is placed inside. By the first sum-frequency process, the second-harmonic field at frequency $\omega _{1}$ was produced, where $\omega _{1}=2\omega _{0}$. Then, the third-harmonic field with a frequency of $\omega _{2}$ is generated by a cascaded sum-frequency process between the fundamental and second-harmonic fields, which $\omega _{2}=\omega _{0}+\omega _{1}=3\omega _{0}$. The two sum-frequency processes are shown in Fig. 1(a). In this cascaded nonlinear process, the quasi-phase-matching technique is very useful which makes the process easier through phase compensation provided by the quasi-periodic optical superlattice (QOSL). G$_{ \textrm {1}}$ and G$_{\textrm {2}}$ are two different reciprocal vectors of the QOSL. The QPM condition is $\overrightarrow {k_{1}}=2\overrightarrow {k_{0}}+ \overrightarrow {G_{1}}$ for the first sum-frequency process, and $\overrightarrow {k_{2}}=\overrightarrow {k_{1}}+\overrightarrow {k_{0}}+ \overrightarrow {G_{2}}$ is the QPM condition for the second sum-frequency process which depicted in Fig. 1(b). $\overrightarrow {k_{0}}$, $\overrightarrow {k_{1}}$, $\overrightarrow {k_{2}}$ are the wave vectors of fundamental, second-, and third-harmonic fields, respectively.

Fig. 1. (a) Sketch of the optical cavity. (b) The sketch for the quasi-phase-matching processes.

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

(1)$$\mathcal{H}_{I}=i\hbar \kappa _{0}\hat{a}_{0}^{2}\hat{a}_{1}^{\dagger }+i\hbar \kappa _{1}\hat{a}_{0}\hat{a}_{1}\hat{a}_{2}^{\dagger }+h.c.,$$

where $\kappa _{i}(i=0,1)$ is the nonlinear coupling constant. For brevity, we take them to be real [42]. We consider the cavity pumping as

(2)$$\mathcal{H}_{pump}=i\hbar (\epsilon \hat{a}_{0}^{\dagger }-\epsilon ^{\ast } \hat{a}_{0}),$$

where $\epsilon$ is the amplitude of classical pumping laser. Similar to the nonlinear coupling constant, it is also considered to be real.

The losses of the three modes can be written as

(3)$$\mathcal{L}_{i}\hat{\rho}=\gamma _{i}(2\hat{a}_{i}\hat{\rho}\hat{a} _{i}^{\dagger }-\hat{a}_{i}^{\dagger }\hat{a}_{i}\hat{\rho}-\hat{\rho}\hat{a} _{i}^{\dagger }\hat{a}_{i}),$$

where $\hat {\rho }$ is the density matrix of system and $ \gamma _{i}(i=0,1,2)$ represent the cavity loss related to the reflection coefficient of the cavity mirror.

The master equation of this system can be expressed as

(4)$$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar }[\mathcal{H}_{I}+\mathcal{H}_{pump}, \hat{\rho}]+\sum_{i=0}^{2}\mathcal{L}_{i}\hat{\rho}.$$

For the purpose of studying quantum entanglement and quantum steering characteristics, one can map the master equation onto Fokker-Planck equation (FPE) in the positive-$P$ representation [39,40]. As we all know that the FPE for the Glauber-Sudarshan $P$ function [43,44] has a negative diffusion matrix and therefore cannot be mapped onto stochastic differential equations, we decide to use the positive-$P$ representation [39,40] to achieve appropriate stochastic differential equations. The FPE can be simply found by setting variables and their Hermitian conjugates as independent [45]. Therefore, $\alpha _{i}$ and $\alpha _{i}^{\dagger }$ are now independent variables [28] and the FPE of the system can be obtained as

(5)$$\begin{aligned} \frac{dP}{dt} & = \Big\{-(\epsilon -\gamma _{0}\alpha _{0}-2\kappa _{0}\alpha _{0}^{\dagger }\alpha _{1}-\kappa _{1}\alpha _{1}^{\dagger }\alpha _{2})\frac{ \partial }{\partial \alpha _{0}}-(\epsilon ^{\ast }-\gamma _{0}\alpha _{0}^{\dagger }-\kappa _{1}\alpha _{1}\alpha _{2}^{\dagger }-2\kappa _{0}\alpha _{0}\alpha _{1}^{\dagger })\frac{\partial }{\partial \alpha _{0}^{\dagger }}\\ &\quad -(-\gamma _{1}\alpha _{1}-\kappa _{1}\alpha _{0}^{\dagger }\alpha _{2}+\kappa _{0}\alpha _{0}^{2})\frac{\partial }{\partial \alpha _{1}}-(\kappa _{0}\alpha _{0}^{\dagger 2}-\gamma _{1}\alpha _{1}^{\dagger }-\kappa _{1}\alpha _{0}\alpha _{2}^{\dagger })\frac{\partial }{\partial \alpha _{1}^{\dagger }}\\ &\quad -(-\gamma _{2}\alpha _{2}+\kappa _{1}\alpha _{0}\alpha _{1})\frac{\partial }{\partial \alpha _{2}}-(-\gamma _{2}\alpha _{2}^{\dagger }+\kappa _{1}\alpha _{0}^{\dagger }\alpha _{1}^{\dagger })\frac{\partial }{\partial \alpha _{2}^{\dagger } }\\ &\quad +\frac{1}{2}\frac{\partial ^{2}}{\partial \alpha _{0}^{2}}(-2\kappa _{0}\alpha _{1})+\frac{1}{2}\frac{\partial ^{2}}{\partial \alpha _{0}^{\dagger 2}}(-2\kappa _{0}\alpha _{1}^{\dagger })\\ &\quad +\frac{1}{2}\frac{\partial ^{2}}{\partial \alpha _{0}\partial \alpha _{1}} (-2\kappa _{1}\alpha _{2})+\frac{1}{2}\frac{\partial ^{2}}{\partial \alpha _{0}^{\dagger }\partial \alpha _{1}^{\dagger }}(-2\kappa _{1}\alpha _{2}^{\dagger })\Big\}P(\alpha ). \end{aligned}$$

Following the standard processing, the stochastic differential equations for the three modes can be written the form as

(6)$$\begin{aligned} \frac{d\alpha _{0}}{dt} & = \epsilon -\gamma _{0}\alpha _{0}-\kappa _{1}\alpha _{1}^{\dagger }\alpha _{2}-2\kappa _{0}\alpha _{0}^{\dagger }\alpha _{1}+\sqrt{-2\kappa _{0}\alpha _{1}}\eta _{1}+\sqrt{-2\kappa _{1}\alpha _{2}} \eta _{2},\\ \frac{d\alpha _{0}^{\dagger }}{dt} & = \epsilon ^{\ast }-\gamma _{0}\alpha _{0}^{\dagger }-\kappa _{1}\alpha _{1}\alpha _{2}^{\dagger }-2\kappa _{0}\alpha _{0}\alpha _{1}^{\dagger }+\sqrt{-2\kappa _{0}\alpha _{1}^{\dagger }}\eta _{1}^{\dagger }+\sqrt{-2\kappa _{1}\alpha _{2}^{\dagger }}\eta _{3},\\ \frac{d\alpha _{1}}{dt} & = -\gamma _{1}\alpha _{1}+\kappa _{0}\alpha _{0}^{2}-\kappa _{1}\alpha _{0}^{\dagger }\alpha _{2}+\sqrt{-2\kappa _{1}\alpha _{2}}\eta _{2}^{\dagger },\\ \frac{d\alpha _{1}^{\dagger }}{dt} & = -\gamma _{1}\alpha _{1}^{\dagger }+\kappa _{0}\alpha _{0}^{\dagger 2}-\kappa _{1}\alpha _{0}\alpha _{2}^{\dagger }+\sqrt{ -2\kappa _{1}\alpha _{2}^{\dagger }}\eta _{3}^{\dagger },\\ \frac{d\alpha _{2}}{dt} & = -\gamma _{2}\alpha _{2}+\kappa _{1}\alpha _{0}\alpha _{1},\\ \frac{d\alpha _{2}^{\dagger }}{dt} & = -\gamma _{2}\alpha _{2}^{\dagger }+\kappa _{1}\alpha _{0}^{\dagger }\alpha _{1}^{\dagger }, \end{aligned}$$

where $\eta _{i}(t)(i=1,2,3)$ are the Gaussian noise terms which satisfy the relations $\langle \eta _{i}(t)\rangle =\langle \eta _{i}^{\dagger }(t)\rangle =0$, $\langle \eta _{i}(t)\eta _{j}(t^{\prime })\rangle =\langle \eta _{i}^{\dagger }(t)\eta _{j}^{\dagger }(t^{\prime })\rangle =0$, and $\langle \eta _{i}(t)\eta _{j}^{\dagger }(t^{\prime })\rangle =\delta _{ij}\delta (t-t^{\prime })$. Here, $\eta _{1}$ and $\eta _{1}^{\dagger }$ are different from $\eta _{2}$, $\eta _{2}^{\dagger }$, $\eta _{3}$ and $\eta _{3}^{\dagger }$. $\eta _{1}$ and $\eta _{1}^{\dagger }$ are derived from $\frac {\partial ^{2}}{\partial \alpha _{0}^{2}}$ and $\frac {\partial ^{2}}{\partial \alpha _{0}^{\dagger 2}}$ terms, while $\eta _{2}$, $\eta _{2}^{\dagger }$, $\eta _{3}$, and $\eta _{3}^{\dagger }$ are derived from $\frac {\partial ^{2}}{ \partial \alpha _{0}\partial \alpha _{1}}$ and $\frac {\partial ^{2}}{\partial \alpha _{0}^{\dagger }\partial \alpha _{1}^{\dagger }}$ terms.

We let the equations $\frac {d\alpha _{i}}{dt}=0$, then the steady state solutions can be obtained. We find the steady-state solution $A_{0}$ of $\alpha _{0}$ satisfies following equation:

(7)$$L_{0}L_{1}^{2}+\kappa _{0}^{2}\gamma A_{0}^{3}(3L_{1}-\gamma _{1}\gamma _{2})=0,$$

where $L_{0}=\gamma _{0}A_{0}-E_{0}$, $L_{1}=\kappa _{1}^{2}A_{0}^{2}+\gamma _{1}\gamma _{2}$. For the sake of simplicty, we take $\epsilon =\epsilon ^{\ast }=E_{0}$ and $\gamma _{1}=\gamma _{2}=\gamma$. However, by removing the noise terms from Eq. (6), it is found that there are no analytical solutions for the optical fields on account of it is five-order equation. The other two steady-state solutions $A_{i}(i=1,2)$ can be used $A_{0}$ to represent as

(8)$$\begin{aligned}A_{1} & = A_{0}\left[ \kappa _{0}^{2}\gamma A_{0}(2\gamma ^{2}-3L_{1})-\kappa _{1}^{2}L_{0}L_{1}\right] /\kappa _{0}\gamma ^{4},\\ A_{2} & = \kappa _{1}A_{0}A_{1}/\gamma . \end{aligned}$$

In the following, one can disintegrate variables of the system by composed of their steady-state values and small fluctuations close to the steady-state values as $\alpha _{i}=A_{i}+\delta \alpha _{i}$ $(i=0,1,2)$ with $\delta \alpha _{i}\ll A_{i}$. In this case, with linear processing method, Eq. (6) can be rewritten as

(9)$$\begin{aligned}\frac{d}{dt}\delta \alpha _{0} & = -\gamma _{0}\delta \alpha _{0}-2\kappa _{0}A_{1}\delta \alpha _{0}^{\dagger }-2\kappa _{0}A_{0}^{\ast }\delta \alpha _{1}-\kappa _{1}A_{2}\delta \alpha _{1}^{\dagger }-\kappa _{1}A_{1}^{\ast }\delta \alpha _{2}\\ &\quad +\sqrt{-2\kappa _{0}A_{1}}\eta _{1}+\sqrt{-2\kappa _{1}A_{2}}\eta _{2},\\ \frac{d}{dt}\delta \alpha _{0}^{\dagger } & = -2\kappa _{0}A_{1}^{\ast }\delta \alpha _{0}-\gamma _{0}\delta \alpha _{0}^{\dagger }-\kappa _{1}A_{2}^{\ast }\delta \alpha _{1}-2\kappa _{0}A_{0}\delta \alpha _{1}^{\dagger }-\kappa _{1}A_{1}\delta \alpha _{2}^{\dagger }\\ &\quad +\sqrt{-2\kappa _{0}A_{1}^{\ast }}\eta _{1}^{\dagger }+\sqrt{-2\kappa _{1}A_{2}^{\ast }}\eta _{3},\\ \frac{d}{dt}\delta \alpha _{1} & = 2\kappa _{0}A_{0}\delta \alpha _{0}-\kappa _{1}A_{2}\delta \alpha _{0}^{\dagger }-\gamma \delta \alpha _{1}-\kappa _{1}A_{0}^{\ast }\delta \alpha _{2}+\sqrt{-2\kappa _{1}A_{2}}\eta _{2}^{\dagger },\\ \frac{d}{dt}\delta \alpha _{1}^{\dagger } & = -\kappa _{1}A_{2}^{\ast }\delta \alpha _{0}+2\kappa _{0}A_{0}^{\ast }\delta \alpha _{0}^{\dagger }-\gamma \delta \alpha _{1}^{\dagger }-\kappa _{1}A_{0}^{\ast }\delta \alpha _{2}^{\dagger }+\sqrt{-2\kappa _{1}A_{2}^{\ast }}\eta _{3}^{\dagger },\\ \frac{d}{dt}\delta \alpha _{2} & = \kappa _{1}A_{1}\delta \alpha _{0}+\kappa _{1}A_{0}\delta \alpha _{1}-\gamma \delta \alpha _{2},\\ \frac{d}{dt}\delta \alpha _{2}^{\dagger } & = \kappa _{1}A_{1}^{\ast }\delta \alpha _{0}^{\dagger }+\kappa _{1}A_{0}^{\ast }\alpha _{1}^{\dagger }-\gamma \delta \alpha _{2}^{\dagger }. \end{aligned}$$

which can be written in the form as follows

(10)$$d\delta \tilde{\alpha}=-\mathbf{A}\delta \tilde{\alpha}dt+\mathbf{B}dW,$$

with

(11)$$\delta \tilde{\alpha}=[\delta \alpha _{0},\delta \alpha _{0}^{\dagger },\delta \alpha _{1},\delta \alpha _{1}^{\dagger },\delta \alpha _{2},\delta \alpha _{2}^{\dagger }]^{\mathrm{T}},$$

where $\mathbf {B}$ is the drift matrix of noise terms contains the steady-state solutions as

(12)$$\mathbf{B}=\left( \begin{array}{cccccc} \sqrt{-2\kappa _{0}A_{1}} & 0 & \sqrt{-2\kappa _{1}A_{2}} & 0 & 0 & 0 \\ 0 & \sqrt{-2\kappa _{0}A_{1}} & 0 & 0 & \sqrt{-2\kappa _{1}A_{2}} & 0 \\ 0 & 0 & 0 & \sqrt{-2\kappa _{1}A_{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \sqrt{-2\kappa _{1}A_{2}} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right),$$

and $dW=[\eta _{1}(t), \eta _{1}^{\dagger }(t), \eta _{2}(t), \eta _{2}^{\dagger }(t), \eta _{3}(t), \eta _{3}^{\dagger }(t)]^{\mathrm {T}}dt$ is a vector of Wiener increments [39]. The steady-state drift matrix $\mathbf {A}$ with the steady-state values has the form as

(13)$$\mathbf{A}=\left( \begin{array}{cccccc} \gamma _{0} & 2\kappa _{0}A_{1} & 2\kappa _{0}A_{0}^{\ast } & \kappa _{1}A_{2} & \kappa _{1}A_{1}^{\ast } & 0 \\ 2\kappa _{0}A_{1}^{\ast } & \gamma _{0} & \kappa _{1}A_{2}^{\ast } & 2\kappa _{0}A_{0} & 0 & \kappa _{1}A_{1} \\ -2\kappa _{0}A_{0} & \kappa _{1}A_{2} & \gamma & 0 & \kappa _{1}A_{0}^{\ast } & 0 \\ \kappa _{1}A_{2}^{\ast } & -2\kappa _{0}A_{0}^{\ast } & 0 & \gamma & 0 & \kappa _{1}A_{0} \\ -\kappa _{1}A_{1} & 0 & -\kappa _{1}A_{0} & 0 & \gamma & 0 \\ 0 & -\kappa _{1}A_{1}^{\ast } & 0 & -\kappa _{1}A_{0}^{\ast } & 0 & \gamma \end{array} \right) .$$

The system can be in a steady state under the condition that above drift matrix $\mathbf {A}$ have no negative eigenvalues. With respect to the eigenvalue, the real parts of eigenvalue of $\mathbf {A}$ (RPEA) is shown in Fig. 2. Figure 2(a) depicts the RPEA versus $E_{0}$ with $\gamma _{0}=0.01$, $\gamma =3\gamma _{0}$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$, $\omega =2\gamma _{0}$. Figure 2(b) depicts the RPEA versus $\gamma _{0}$ with $\gamma =3\gamma _{0}$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$,$E_{0}=0.5\gamma _{0}\gamma /\kappa _{0}$, $\omega =2\gamma _{0}$. In Fig. 2(c), we show the RPEA versus $\gamma$ with $\gamma _{0}=0.01$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$, $E_{0}=0.5\gamma _{0}\gamma /\kappa _{0}$, $\omega =2\gamma _{0}$. Figure 2(d) depicts the RPEA versus $\kappa _{1}/\kappa _{0}$ with $\gamma _{0}=0.01$, $\gamma =3\gamma _{0}$, $\kappa _{0}=0.1$,$E_{0}=0.5\gamma _{0}\gamma /\kappa _{0}$, $\omega =5\gamma _{0}$. As the Fig. 2 shows, this drift matrix $\mathbf {A}$ has no negative eigenvalues in all parameter ranges. It should be noted here that the RPEA are different from Yu et al. [46], this maybe owing to the unequal choice in parameter. The better characteristics of quantum entanglement can be obtained in the case of their chosen parameters, while a better quantum steering characteristic can be obtained in present case of our chosen parameters. Consequently, we can discuss the characteristics of quantum steering in above ranges of parameters.

Fig. 2. Eigenvalue of $\mathbf {A}$ versus: (a) $E_{0}$, (b) $\gamma _{0}$, (c) $\gamma$ (d) $\kappa _{1}/\kappa _{0}$, respectively.

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As long as the eigenvalue of matrix $\mathbf {A}$ has no negative real parts, in the light of Fourier transformation, this method can obtain the intracavity spectral from Eq. (10) as

(14)$$\mathbf{S}(\omega )=(\mathbf{A}+i\omega \mathbf{I})^{\mathrm{-1}}\mathbf{B} \mathbf{B}^{\mathrm{T}}(\mathbf{A}^{\mathrm{T}}-i\omega \mathbf{I})^{\mathrm{ -1}},$$

where $\omega$ and $\mathbf {I}$ corresponding to the Fourier analysis frequency and the identity matrix, respectively. The results of intracavity spectral $\mathbf {S}(\omega )$ can be used to obtain the variances and obtain the bipartite steering [28,37,38]. Therefore, based on the intracavity spectral, we can investigate the tripartite steering in the cascaded nonlinear processes. The output fields can be obtained by using the standard input-output relationship [47].

3. Results and discussions

Olsen studied the bipartite asymmetric quantum steering with an injected nondegenerate OPO [28] and third-harmonic quantum steering in the cascaded nonlinear processes [38], respectively. In the following, we will investigate the properties of quantum steering among the pump, second-harmonic, and third-harmonic generated in the cascaded sum-frequency processes by applying the criteria proposed by He and Reid [30]. We define $X_{i}=(\alpha _{i}+\alpha _{i}^{\dagger })/2$, and $Y_{i}=(\alpha _{i}-\alpha _{i}^{\dagger })/2i$, where $X_{i}$ and $Y_{i}$ represents quadrature amplitude and phase component, respectively. Thus, we write the equations in the form as

(15)$$\begin{aligned}S_{0} & = \Delta (X_{0}-X_{1})\Delta (Y_{0}+Y_{1}+Y_{2}),\\ S_{1} & = \Delta (X_{1}-X_{2})\Delta (Y_{0}+Y_{1}+Y_{2}),\\ S_{2} & = \Delta (X_{2}-X_{0})\Delta (Y_{0}+Y_{1}+Y_{2}), \end{aligned}$$

EPR steering of system $i$ will be confirmed when the condition $S_{i} < 1(i=0,1,2)$ is satisfied [30]. It means that one can confirm the steering of 0 by the other optical fields $\left \{ 1,2\right \}$ if $S_{0} < 1$, steering of 1 by the other optical fields $\left \{ 2,0\right \}$ if $S_{1} < 1$ and steering of 2 by the other optical fields $\left \{ 1,0\right \}$ if $S_{2} < 1$. Of course, when

(16)$$S_{tot}=S_{0}+S_{1}+S_{2} < 1$$

will demonstrate genuine tripartite steering [30].

In Fig. 3, we depicts $S_{i}$ and $S_{tot}$ versus the normalized analysis frequency $\Omega =\omega /\gamma _{0}$ with $\gamma _{0}=0.01$, $\gamma _{1}=\gamma _{2}=3\gamma _{0}$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$. As shown in Fig. 3 that $S_{i}(i=0,1,2)$ are all below 1 which indicates bipartite EPR steering each other amomg three optical fields. What is more significant is that we found $S_{tot}<1$ across the whole range of analysis frequency $\Omega$, which demonstrate that genuine tripartite steering can be generated in our scheme of cascaded nonlinear processes. Figure 4 shows the $S_{i}$ of EPR steering of system $i$ and the $S_{tot}$ versus the nonlinear coupling parameter $\kappa _{1}/\kappa _{0}$ with $\gamma _{0}=0.01$, $\kappa _{0}=0.1$, $\gamma _{1}=\gamma _{2}=3\gamma _{0}$, $\omega =5\gamma _{0}$. As we can see from Fig. 4, with the increase of $\kappa _{1}/\kappa _{0}$, $\ S_{i}$ and $S_{tot}$ are both less than 1 almost in the whole parameter range we chosen. It is suggested that the bipartite EPR steering and genuine tripartite steering can be obtained in the cascaded nonlinear processes. In fact, when the value of $\kappa _{1}/\kappa _{0}$ reaches almost 7.4, the value of $ S_{tot}$ more than 1, which prove the scheme’s failure. One can obtain better quantum properties by adjusting the nonlinear coupling coefficient $\kappa$.

Fig. 3. $S_{i}$ and $S_{tot}$ versus the normalized analysis frequency $\omega /\gamma _{0}$ with $\gamma _{0}=0.01$, $\gamma _{1}=\gamma _{2}=3\gamma _{0}$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$.

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Fig. 4. $S_{i}$ and $S_{tot}$ versus $\kappa _{1}/\kappa _{0}$ with $\gamma _{0}=0.01$, $\kappa _{0}=0.1$, $\gamma _{1}=\gamma _{2}=3\gamma _{0}$, $\omega =5 \gamma _{0}$.

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In Fig. 5, we show the values of $S_{i}$ and $S_{tot}$ under different ratios of $\gamma /\gamma _{0}$ with $\gamma _{0}=0.01$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$ and $\omega =2\gamma _{0}$. From Fig. 5, one can see that the value of $S_{i}$ and $S_{tot}$ increase first and then decrease with the ratios of $\gamma /\gamma _{0}$. We get a satisfactory result that the value of $S_{i}$ and $S_{tot}$ are far less than 1, which indicate that bipartite EPR steering and genuine tripartite steering. And we also found the value become smaller with the increase of $\gamma /\gamma _{0}$ in the case of keeping the parameters unchanged. That is to say, the quantum properties shown better with the cavity loss rates’ reduction. The results of Fig. 3 and Fig. 4 make a certain contribution to an optional parameter range for obtaining better tripartite quantum steering in both theory and experiment.

Fig. 5. $S_{i}$ and $S_{tot}$ versus $\gamma /\gamma _{0}$ with $\gamma _{0}=0.01$, $\kappa _{0}=0.1$, $\kappa _{1}=1.5\kappa _{0}$ and $\omega =2\gamma _{0}$.

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4. Conclusion

In this paper, based on the criteria for multipartite EPR steering which proposed by He and Reid [30], bipartite EPR steering can be obtained amomg three output optical fields. It is more significant that the genuine tripartite quantum steering among pump, second-harmonic, and third-harmonic has been demonstrated for the first time. We also investigated the variations of tripartite quantum steering with the pump and nonlinear coefficients. One can obtain better quantum steering by adjusting the nonlinear coupling coefficient or increase the ratio of $\gamma /\gamma _{0}$. Since the single-pass experiment has been achieved [41], the conversion efficiency for third-harmonic generation is about 23%. If we put this set up into a cavity, the efficiency will be further increased and higher than that without cavity. According to the experiment of the generation of three-color entanglement [33], such an optical cavity can be built in the experiment to make the three beams resonate at the same time. Therefore, we think our scheme can be a proposal for a potential experiment. We think the results that show in this paper could give some help for potential applications such as quantum communication, quantum teleportation, and quantum network and so on.

Funding

National Natural Science Foundation of China (61975184, 91636108); Natural Science Foundation of Zhejiang Province (LY18A040007); Zhejiang Sci-Tech University (18062145-Y, 19062151-Y); Open Foundation of Key Laboratory of Optical Field Manipulation of Zhejiang Province (ZJOFM-2019-002).

Disclosures

The authors declare no conflicts of interest.

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Genuine tripartite Einstein-Podolsky-Rosen steering in the cascaded nonlinear processes of third-harmonic generation

Abstract

1. Introduction

2. Equations of motion and the stationary solutions

3. Results and discussions

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (16)

Optics Express