## Abstract

Quantum steering is used to describe the “spooky action-at-a-distance” nonlocality raised in the Einstein-Podolsky-Rosen (EPR) paradox, which is important for understanding entanglement distribution and constructing quantum networks. Here, in this paper, we study an experimentally feasible scheme for generating quantum steering based on cascaded four-wave-mixing (FWM) processes in hot rubidium (Rb) vapor. Quantum steering, including bipartite steering and genuine tripartite steering among the output light fields, is theoretically analyzed. We find the corresponding gain regions in which the bipartite and tripartite steering exist. The results of bipartite steering can be used to establish a hierarchical steering model in which one beam can steer the other two beams in the whole gain region; however, the other two beams cannot steer the first beam simultaneously. Moreover, the other two beams cannot steer with each other in the whole gain region. More importantly, we investigate the gain dependence of the existence of the genuine tripartite steering and we find that the genuine tripartite steering exists in most of the whole gain region in the ideal case. Also we discuss the effect of losses on the genuine tripartite steering. Our results pave the way to experimental demonstration of quantum steering in cascaded FWM process.

© 2017 Optical Society of America

## 1. Introduction

Quantum steering has been investigated between two parties earlier [1–6]. The concept of genuine N-partite steering is developed by Q. Y. He et.al. in 2013 [7]. Further, motivated by considerations of the importance of multipartite steering for understanding entanglement distribution and constructing real-world quantum networks [8], S. Armstrong et. al. investigate the properties of multipartite steering relevant to quantum communication in 2015 [9]. In their experiment, the steering across all bipartitions can be used to establish quantum secret sharing. However, there is no genuine tripartite steering in their experiment. It has been shown that the special multipartite steerable states can be created in the multimode optical system [10] and multimode pulsed cavity optomechanical system [11]. Generation of genuine tripartite steering has been proposed in a hybrid massive system [12].

In 2007, Paul Lett’s group at NIST experimentally generated a pair of intensity-correlated beams based on nondegenerate four-wave-mixing (FWM) process in a hot rubidium(Rb) vapor [13]. This system has several advantages for practical implementations, e.g., no need of cavity due to strong nonlinearity of the system, spatial multimode nature due to no mode constrain, spatial separation of the generated nonclassical beams, etc. These advantages explain rapidly growing popularity of such a system in many applications, including quantum information processing and quantum metrology, such as entangled images [14], tunable delay of EPR entanglement [15], nonlinear quantum interferometer [16–20], high purity narrow-bandwidth single photons [21], ultrasensitive measurement of microcantilever displacement [22], the localized multi-spatial mode quadrature squeezing for quantum imaging [23] and so on [24–30].

Due to these advantages, it is a good candidate for generating multiple quantum correlated beams which have potential applications in quantum communication [8,31–36]. For example, theoretical proposals based on FWM in hot vapor have been proposed to realize CV cluster state generation over spatial comb through FWM process [37] and versatile quantum network generation by cascading several FWM processes [38]. In 2014, our group has experimentally generated three bright strongly quantum correlated beams by cascaded FWM processes in hot atomic vapors [17]. Also, this method is phase insensitive without the need of complicated phase locking technique. Inspired by these advantages of cascaded FWM system and motivated by the desire to generate the genuine tripartite steering, in this paper, we theoretically propose an experimentally feasible scheme for generating bipartite steering and genuine tripartite steering based on cascaded FWM processes in Rb vapor. We study how the bipartite and genuine tripartite steering depend on the system gain and propose the model of hierarchical steering.

This paper is organized as follows. In section 2, cascaded FWM processes are briefly introduced. In section 3, the bipartite steering of the system is also discussed. The genuine tripartite steering based on the cascaded FWM processes in Rb vapor is theoretically analyzed in section 4. We also discuss the genuine tripartite steering in the cascaded system with losses. Finally, a brief conclusion is given in section 5.

## 2. Cascaded FWM processes

The energy level diagram of a single FWM process in Rb atomic vapor is shown in Fig. 1(a). As shown in Fig. 1(b), a strong pump beam (*ĉ*_{1}) and a much weaker probe beam (*â*_{1}) are crossed in the center of the first Rb vapor cell with a slight angle. Probe (*â*_{1}) and conjugate (*b̂*_{1}) beams are created via the FWM scheme in the first Rb vapor. Then the probe beam (*â*_{1}) is fed into a second FWM process. The probe beam (*â*_{1}) and a strong pump beam (*ĉ*_{2}) are crossed in the center of the second Rb vapor. Then the probe (*â*_{2}) and conjugate (*b̂*_{2}) beams are created after the second Rb vapor. Viewing the two FWM processes as a whole system, there are three output light beams, which are the probe *â*_{2} (*Ô*_{1}), conjugate *b̂*_{2} (*Ô*_{2}) and another conjugate *b̂*_{1} (*Ô*_{3}).

The FWM process involves the annihilation of two pump photons, and the creation of a single probe and a conjugate photon. Labeling the annihilation operator of the probe (*â*), conjugate (*b̂*) and pump (*ĉ*) respectively and the interaction strength by *ζ*, the Hamiltonian (*Ĥ*_{1}) and (*Ĥ*_{2}) corresponding to the interaction of the first and the second FWM processes under the pump undepleted approximation *ĉ* → *ψ _{c}* can be written by [39]:

Here *v̂*_{0} and *v̂′*_{0} are the vacuum states of conjugate inputs for the first and second FWM processes respectively. ${G}_{1}=\mathit{cosh}\left({\zeta}_{1}{\psi}_{{c}_{1}}^{2}t\right)$ and ${G}_{2}=\mathit{cosh}\left({\zeta}_{2}{\psi}_{{c}_{2}}^{2}t\right)$ are the intensity gains of the first and second FWM processes respectively which depend on the strength of the interaction. The strength of the FWM interaction depends on the system parameters, such as the temperature of the Rb vapor, the single-photon detuning, the two-photon detuning and the pump intensity. From Eqs. (3)–(6), the input-output relation of the cascaded FWM processes in Fig. 1(a) can be written as:

Generally the annihilation operators of the output fields *â*_{2}(*t*), *b̂*_{2}(*t*) and *b̂*_{1}(*t*) can be written as *â*_{2}(*t*) = 〈*â*_{2}(*t*)〉 + *δâ*_{2}(*t*), *b̂*_{2}(*t*) = 〈*b̂*_{2}(*t*)〉 + *δb̂*_{2}(*t*) and *b̂*_{1}(*t*) = 〈*b̂*_{1}(*t*)〉 + *δb̂*_{1}(*t*), where 〈*â*_{2}(*t*)〉, 〈*b̂*_{2}(*t*)〉 and 〈*b̂*_{1}(*t*)〉 are the average terms and *δâ*_{2}(*t*), *δb̂*_{2}(*t*) and *δb̂*_{1}(*t*) are the noise terms. In fact, these noise terms in the form of variances are very important for our following calculations of quantum steering.

Then, we give the optical quadrature definitions in our analysis, as the criteria will depend on these quadratures. The amplitude and phase quadrature operators of the fields are defined by

## 3. Bipartite steering

Now, let us investigate the possibilities for the existence of bipartite steering in our cascaded FWM scheme. Using our quadrature definitions above, we can calculate the following formulas

*Ô*→

_{j}*Ô*. That means the existence of bipartite steering can be confirmed if ${\mathit{St}}_{ij}<\frac{1}{2}$. Here Δ

_{i}*(*

_{inf}*X̂*) and Δ

_{ij}*(*

_{inf}*Ŷ*) are the uncertainty in the prediction of amplitude quadrature

_{ij}*X̂*and phase quadrature

_{i}*Ŷ*of one light beam

_{i}*Ô*based on measurement of the other light beam

_{i}*Ô*. They are given by

_{j}Here *g*_{opt,X̂j} and *g*_{opt,Ŷj} are optimized real numbers which can minimize the average uncertainty of the inferences and thus increase the steerability [9]. It should be noted that here we only consider the *X _{i}*−

*X*and

_{j}*Y*−

_{i}*Y*combinations in our calculations instead of the

_{j}*X*−

_{i}*Y*and

_{j}*Y*−

_{i}*X*combinations. This is actually due to that there only exist

_{j}*X*−

_{i}*X*and

_{j}*Y*−

_{i}*Y*quantum correlation instead of

_{j}*X*−

_{i}*Y*and

_{j}*Y*−

_{i}*X*quantum correlation for our scheme. Such

_{j}*X*−

_{i}*Y*and

_{j}*Y*−

_{i}*X*type of quantum steering must be considered for some special types of quantum states, such as cluster state [40,41]. As shown in Fig. 2, the

_{j}*St*

_{12},

*St*

_{21},

*St*

_{13},

*St*

_{31},

*St*

_{23}and

*St*

_{32}are plotted as a function of the intensity gains

*G*

_{1}and

*G*

_{2}of our cascaded scheme. For only partial regions as shown in Figs. 2(a) and 2(c), the values

*St*

_{12}and

*St*

_{13}are less than $\frac{1}{2}$. It implies that the amplitude quadrature and phase quadrature of the output beam

*Ô*

_{1}is highly correlated with the other output beams

*Ô*

_{2}and

*Ô*

_{3}in the partial regions of the

*G*

_{1}and

*G*

_{2}. More interestingly, these two partial regions are not overlapped with each other, meaning that these two bipartite steering can not happen at the same time. In this sense, we could name this phenomenon as repulsion effect of quantum steering. This phenomenon can be explained as follows. The repulsion effect is actually the result of the competition between the positive mechanism and negative mechanism. As shown in Fig. 1(a), firstly, for the steering between beams

*Ô*

_{1}and

*Ô*

_{3}, obviously, the first Rb vapor cell will provide the steering for

*Ô*

_{3}→

*Ô*

_{1}and the second Rb vapor cell will destroy this quantum steering by adding extra vacuum noise, thus these two Rb vapor cells can be viewed as the positive mechanism provider and negative mechanism provider respectively. Therefore, the larger

*G*

_{1}and smaller

*G*

_{2}are preferred for the steering of

*Ô*

_{3}→

*Ô*

_{1}. Secondly, for the case of the steering for

*Ô*

_{2}→

*Ô*

_{1}, the first Rb vapor cell will generate a thermal state which will destroy their quantum steering by adding extra vacuum noise into the system while the second Rb vapor cell will make them quantum correlated through the FWM process. In this case, these two Rb vapor cells can also be viewed as the negative mechanism provider and positive mechanism provider respectively. Therefore the smaller

*G*

_{1}and larger

*G*

_{2}are preferred for the steering for

*Ô*

_{2}→

*Ô*

_{1}. Finally, the completely opposite dependence of the steering for

*Ô*

_{3}→

*Ô*

_{1}and the steering for

*Ô*

_{2}→

*Ô*

_{1}on the gains leads to the repulsion effect between the steering of certain pairs. The values

*St*

_{21}and

*St*

_{31}for the whole gain region are less than $\frac{1}{2}$ as shown in Figs. 2(b) and 2(d), which implies that the amplitude quadrature and phase quadrature of the output beam

*Ô*

_{2}and

*Ô*

_{3}are highly correlated with the other output beam

*Ô*

_{1}for the whole gain region of the intensity gains

*G*

_{1}and

*G*

_{2}. In other words, beam

*Ô*

_{1}can steer both beam

*Ô*

_{2}and beam

*Ô*

_{3}for the whole gain region. This is mainly due to that beam

*Ô*

_{1}is involved in both of the two FWM processes and therefore

*Ô*

_{1}includes photons which are correlated with both

*Ô*

_{2}and

*Ô*

_{3}. The values

*St*

_{23}and

*St*

_{32}of the whole gain region is lager than $\frac{1}{2}$ as shown in Figs. 2(e) and 2(f), which implies that there is no bipartite steering between the output beams

*Ô*

_{2}and

*Ô*

_{3}. This is due to that beams

*Ô*

_{2}and

*Ô*

_{3}have never interacted with each other directly, therefore it is impossible to build quantum correlation between these two beams, let alone the existence of quantum steering.

As shown in Fig. 3, based on the above results, we can also plot the mutual relations of the output beams *Ô*_{1}, *Ô*_{2} and *Ô*_{3} in terms of their possibilities of bipartite quantum steering. Bipartite steering across *Ô*_{1} → *Ô*_{2} and *Ô*_{1} → *Ô*_{3} always exist in the whole gain region of the intensity gains. Therefore, we can name the bipartite steering across *Ô*_{1} → *Ô*_{2} and *Ô*_{1} → *Ô*_{3} as deterministic quantum steering as show in the Fig. 3. And the bipartite steering across *Ô*_{2} → *Ô*_{3} and *Ô*_{3} → *Ô*_{2} doesn’t exist for the whole intensity gain region. The bipartite steering across *Ô*_{2} → *Ô*_{1} and *Ô*_{3} → *Ô*_{1} can happen in some regions. And more interestingly, these two regions for steering across *Ô*_{2} → *Ô*_{1} and *Ô*_{3} → *Ô*_{1} is always mutually exclusive. In this sense, we can name the bipartite steering across *Ô*_{2} → *Ô*_{1} and *Ô*_{3} → *Ô*_{1} as conditional quantum steering. The hierarchical steering model among the output beams *Ô*_{1}, *Ô*_{2} and *Ô*_{3} is interesting and can be used for the hierarchical quantum secret sharing among them.

## 4. Genuine tripartite steering

Further, due to the importance of multipartite steering for understanding entanglement distribution and constructing quantum networks, we investigate the possibilities of the existence of genuine multipartite steering in the cascaded FWM scheme. To confirm genuine tripartite steering, let us calculate the following formulas

*Ô*→

_{j}Ô_{k}*Ô*. Here Δ

_{i}*(*

_{inf}*X̂*) and Δ

_{ijk}*(*

_{inf}*Ŷ*) are the uncertainty in the prediction of amplitude quadrature

_{ijk}*X̂*and phase quadrature

_{i}*Ŷ*of one light beam

_{i}*Ô*based on measurements of the other two light beam

_{i}*Ô*and

_{j}*Ô*, and they are given by

_{k}*g*

_{opt,X̂j},

*g*

_{opt,Ŷj},

*g*

_{opt,Ŷk}and

*g*

_{opt,Ŷk}are optimized real numbers. Then, the existence of genuine tripartite steering can be confirmed if ${\mathit{St}}_{ijk}+{\mathit{St}}_{jik}+{\mathit{St}}_{kij}<\frac{1}{2}$ [7]. As shown in Fig. 4, the

*St*

_{123},

*St*

_{213},

*St*

_{312}and

*St*

_{123}+

*St*

_{213}+

*St*

_{312}are plotted as a function of the intensity gains

*G*

_{1}and

*G*

_{2}in the cascaded scheme. The values of

*St*

_{123},

*St*

_{213}and

*St*

_{312}in the whole gain region are less than $\frac{1}{2}$ as shown in Figs. 4(a)–4(c), which implies that the amplitude quadrature and phase quadrature of any one of the output beams is highly correlated with the combination of the other two output beams. In other words, the combination of any two of the three output beams can steer the other beam within the whole gain region. The contour plot of

*St*

_{123}+

*St*

_{213}+

*St*

_{312}is shown in Fig. 4(d), the value of

*St*

_{123}+

*St*

_{213}+

*St*

_{312}is less than $\frac{1}{2}$ for most of the whole gain region, which corresponds to the existence of genuine tripartite steering in the cascaded system.

For practical applications, the losses are unavoidable. Therefore, it is necessary to consider how the losses in the cascaded system affect the performance of the quantum steering. For simplicity, we only focus on the losses due to imperfect optical propagation and detection efficiency. The losses can be modeled by a beam splitter with an empty port whose output state is a combination of the input and vacuum modes. Denoting the vacuum modes introduced by losses on the probe and conjugate by the annihilation operators *v̂ _{i}* (i=1–4) respectively, the standard beam-splitter input-output relations give

Here *η*_{1}, *η*_{2}, *η*_{3} and *η*_{4} are the transmission ratios of the light beams intensities due to the imperfect optical propagation and detection efficiency. For simplicity, we consider all the transmission ratios *η*_{1}, *η*_{2}, *η*_{3}, *η*_{4} as *η*. The contour plots of *St*_{123} + *St*_{213} + *St*_{312} with *η* = 0.95 and *η* = 0.92 are shown in Figs. 5(a) and 5(b), respectively. We can see that they could be less than $\frac{1}{2}$ in some gain region, but this region is smaller than the one without considering the losses. And we find that the larger the losses are, the smaller the gain region in which the genuine tripartite steering exists is. Therefore, for real applications, it is important to optimize the optical propagation paths and improve the detection efficiency of the photodiode.

## 5. Conclusion

In summary, we have investigated the bipartite steering among the three output beams of cascaded FWM process. We have found that their steering relations are hierarchical. Among them, one can steer the other two beams in the whole gain region. And the other two beams can steer the first one only in a partial gain region. More interestingly, they can not steer the first one at the same time. The other two beams can not steer with each other. Therefore, not all the possible bipartite steerings happen for the whole gain region. This is mainly due to the asymmetric structure of our cascade FWM process itself which directly leads to the asymmetric roles of the generated three beams. Such unbalanced bipartite steerings are actually useful for secure quantum communication in which participants should have different levels of authorities for secure purposes. In this sense, such unbalanced bipartite steering structure is also unique for our system and may be taken as advantages in practical quantum communications, for example, hierarchical quantum secret sharing. These results may be applied to hierarchical quantum secret sharing. More importantly, we have theoretically predicted that the cascaded FWM processes can generate genuine tripartite steering. And the genuine tripartite steering exists in our cascaded FWM system for most of the whole gain in the ideal case without considering the optical losses. We have also studied how the optical losses affect the performance of the genuine tripartite steering. We found that the larger the losses are, the smaller the gain region in which the genuine tripartite steering exists is. Our results here pave the way to experimental demonstration the quantum steering in cascaded FWM system.

## Funding

This work was supported by the National Natural Science Foundation of China (91436211, 11374104, 10974057); Natural Science Foundation of Shanghai (17ZR1442900). the SRFDP (20130076110011); the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning; the Program for New Century Excellent Talents in University (NCET-10-0383); the Shu Guang project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (11SG26); the Shanghai Pujiang Program (09PJ1404400); the Scientific Research Foundation of the Returned Overseas Chinese Scholars, State Education Ministry; National Basic Research Program of China (2016YFA0302103); Program of Introducing Talents of Discipline to Universities (B12024); and Program of State Key Laboratory of Advanced Optical Communication Systems and Networks (2016GZKF0JT003).

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