# Quantum steering and entanglement in three-mode triangle Bose–Hubbard system

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## Abstract

We consider the possibility of generation steerable states in Bose–Hubbard system composed of three interacting wells in the form of a triangle. We show that although our system still fulfills the monogamy relations, the presence of additional coupling which transforms a chain of wells onto triangle gives a variety of new possibilities for the generation of steerable quantum states. Deriving analytical formulas for the parameters describing steering and bipartite entanglement, we show that interplay between two couplings influences quantum correlations of various types. We compare the time evolution of steering parameters to those describing bipartite entanglement and find the relations between the appearance of maximal entanglement and disappearance of steering effect.

## Keywords

Quantum steering Bose–Hubbard model Quantum entanglement Nonlinear oscillator Concurrence Three-qubit system## 1 Introduction

Quantum steering, although described by Schrödinger long time ago [1], remains one of the most intriguing physical phenomena. As it was quite recently shown, the steering effect, besides its commonly discussed form which is related to the EPR nonlocality [2], has its temporal [3, 4, 5] and spatiotemporal forms [6]. It should be noted that quantum steering is weaker than Bell-type correlations but stronger than quantum entanglement. That means that steerable quantum states are a subset of the entangled ones, while those exhibiting Bell-type nonlocality are a subset of the steerable states [7]. In general, steerable states can be applied in various branches of quantum information theory (see the special issue [8] *and the references quoted therein*).

Besides applications of steerable states, it is still interesting to find physical systems which would be suitable candidates for a source of steering effect. It is important to consider not only such systems but also to determine the conditions for the most efficient steerable states generation. Moreover, as we have mentioned above that steering effects are strongly related to quantum entanglement [9], it is important to find the relation between the appearance of those two types of correlations in particular models. Therefore, we shall consider here one of the most promising models in a context of such problems. We concentrate here on a three-mode conventional Bose–Hubbard system. Bose–Hubbard model was proposed in 1963 by Gersch and Knollman [10] to describe bosons with repulsive interactions. It belongs to a group of fundamental physical models which can be applied for description of bosonic atoms trapped in optical lattices [11, 12] and their quantum walk [13].

The effective Hamiltonian describing standard Bose–Hubbard model consists of two parts. The first of them describes tunneling of particles between neighboring sites of lattices, whereas the second one represents an interaction between bosons located at the same well. What is worth noting, such kind of Hamiltonian also describes models of *quantum nonlinear scissors* [14, 15, 16] which involve Kerr-like nonlinear oscillators. Quantum scissors systems can lead to the generation of the states defined in finite-dimensional Hilbert space—we observe there *photon (or phonon) blockade effect* [17, 18, 19, 20, 21, 22]. Moreover, various types of quantum correlations, including quantum entanglement, can appear in such models, [23, 24, 25, 26]. Since the effective Hamiltonian describing Bose–Hubbard model involves Kerr-type terms, it can be applied for description of different physical situations. Examples of them can be: quantum dot systems [27], Bose-Einstein condensates [28], photonic crystals [29], micro-cavities with moving mirrors [21, 30, 31] or micro-cavities with current QED systems [32, 33] (at this point, one can mention Santa Barbara experiment [34]). It is also worth noting that such nonlinear systems are very often discussed in a context of quantum chaos theory (for instance see [35, 36, 37, 38]).

In our paper, we consider the model which is an extension of that proposed by Olsen [39, 40]. The model consists of three potential wells trapping bosonic particles and is modeled by three nonlinear Kerr-type oscillators. We assume here that all pairs of subsystems (wells) interact each other. In consequence, the system has triangle geometry, contrary to the linear chains considered in [39, 40]. We show that additional coupling which transforms the geometry of the model from the linear one to that of triangle form considerably changes the systems’ ability to produce quantum steering effects. The system can be a physical realization of the ideas discussed in our previous paper [9]. We discuss here what kind of the states can be achieved during the system’s evolution, and consider the interplay between the possibility of the appearance of entanglement and steering generation. The paper is organized as follows. In Sect. 2, we present our model and present analytical solutions for the probability amplitudes describing time evolution of the system. In Sect. 3, applying steering parameter based on Cavalcanti criterion [41], we discuss time evolution of the steering for two cases. One of them corresponds to symmetric initial conditions, whereas the second to asymmetric ones. Next, analyzing the time evolution of the both: steering parameters and the concurrence, we discuss the relationships between the generation of steerable and entangled states. Finally, in Sect. 4 we present our conclusions.

## 2 The model

*ij*label two interacting wells (1–2, 2–3, 1–3). The operators \(\hat{a}_{i}\), \(\hat{a}_{i}^{\dagger }\) appearing here are usual boson annihilation and creation operators, respectively (corresponding to the well

*i*). The parameters describe the strength of couplings between two wells \(i-j\), where \(\varepsilon _{ij}=\varepsilon _{ij}^{*}\).

Here, we are interested in a possibility of generation steerable and entangled states. We assume here that the interactions among the wells are weak, analogously to the models of *quantum scissors* discussed in [26, 42], i.e., \(\varepsilon _{12}=\varepsilon _{23}=\varepsilon _{13}=0.001\chi \). Moreover, to check the influence of the coupling strength on the properties of generated states we assume here that two coupling parameters are equal each other (\(\varepsilon _{12}=\varepsilon _{23}=\varepsilon \)), whereas the third parameter \(\varepsilon _{13}\) vary from 0 to \(4\varepsilon \).

## 3 Steering and entanglement

This section is devoted to the quantum properties of coupled three potential wells, particularly to the steering and entanglement properties of the system. We will analyze bipartite steering and entanglement between two considered wells applying Cavalcanti’s steering parameter and the concurrence. Such two classes of quantum correlations were very extensively studied in last years. They were considered in discussions of various physical systems. For example, there were three-mode optomechanical system composed of an atomic ensemble located in a cavity comprising oscillating mirror [43], double-cavity optomechanical model with two separate electromagnetic fields mediated by a mechanical oscillator [44] or optical cavity field and the mechanical oscillator system [45].

*i*and

*j*, we apply here steering parameter \(S_{ij}\) defined with use of inequality defined by Cavalcanti et al. [41]. Such parameter can be expressed as

*i*,

*j*label the wells. When steering parameter \(S_{ij}\) takes values greater than zero, the mode

*j*steers the mode

*i*. The steering in opposite direction appears when \(S_{ji}\) becomes positive. Hence, we can observe two types of steering—the symmetric (SS) and asymmetric (AS) ones. SS appears when at the same time the both parameters \(S_{ij}\) and \(S_{ji}\) are greater than zero. If only one them (\(S_{ij}\) or \(S_{ji}\)) is positive, the steering is asymmetric.

*R*derived from the relation \(R=\rho _{ij}\tilde{\rho _{ij}}\), where \(\tilde{\rho _{ij}}\) is defined as \(\tilde{\rho _{ij}}=\sigma _{y}\otimes \sigma _{y}\rho _{ij}^{*}\sigma _{y}\otimes \sigma _{y}\) and \(\sigma _{y}\) is \(2\times 2\) Pauli matrix. The density matrix \(\rho _{ij}\) describing two-subsystem state can be obtained by tracing out the third one \(\rho _{ij}=Tr_{k}(\rho _{ijk})\). The concurrence takes values between 0 for product states and the unity for maximally entangled two-qubit states.

### 3.1 Symmetric case

*symmetric case*—the fact of symmetry/asymmetry concerns initial population of the system, not the parameters or discussed system’s geometry. Thus, applying formulas (5, 8) we can express the steering parameters as:

Figure 3a depicts time evolution of steering parameters \(S_{ij}\) for various values of the ratio \(\varepsilon _{13}/\varepsilon \). White area shown in the diagram (right) corresponds to the situation when \(S_{ij}\le 0\)—subsystem *j* does not steer subsystem *i*. The gray area represents cases when steering effect appears—\(S_{ij}> 0\). From the figure, we see that for different values of the coupling constant subsystems 1 and 3 can be steered simultaneously by 2, and we see from our solutions and Fig. 3b that steering corresponding to other pairs of subsystems cannot observed. When the strength of the interaction between the wells 1 and 3 increases, the frequency of oscillations of \(S_{12}\) and \(S_{32}\) becomes greater and greater. However, the oscillations always remain regular. For \(\varepsilon _{13}/\varepsilon >2.83\), steerable states are generated (almost) permanently, and only oscillatory changes of the degree of steerability are present.

*i*steers

*j*). Additionally, the same as for the case when all three coupling strengths were identical, for the situation presented here, steering parameters reach their maximal values when \(\vert C_{010}\vert ^2=0.75\).

To clarify the relations and mutual interplay between the entanglement and steering generated in our system, we show in Fig. 5 the “trajectory” of the states which are produced during the system’s evolution. Each point of the ”trajectory” corresponds to one state characterized by a given value of concurrence \(C_{12}=C_{32}\) and corresponding to it steering parameter \(S_{12}=S_{32}\). One should remember that for all cases considered here we do not observe steering effect for the pair \(1\leftrightarrow 2\), and therefore, we plot the “trajectory” only for the mentioned above parameters. What is worth noting, all states which can be generated for the “symmetric” case are located at the line shown in Fig. 5.

From Fig. 5, one can see that for the case discussed here, maximally entangled states never will be generated—maximal value of \(C_{12}=C_{32}\) is \(\sim 0.7\). Moreover, at that point, we have a border between steerable and unsteerable states. For remaining points of our “trajectory” to the one value of concurrency correspond two states—only one of them is steerable. Maximal steering can be achieved by the states which are characterized by \(C_{12}=C_{32}\simeq 0.6\), whereas for the maximal value of the concurrence steering parameter \(S_{12}\rightarrow 0\).

### 3.2 Asymmetric case

From (14) and Fig. 6, we see that the character of process of steering generation depends on the value of \(\varepsilon _{13}/\varepsilon \). For instance, for every value of \(\varepsilon _{13}/\varepsilon \) we can observe that initially occupied subsystem 1 steers subsystems 2 and 3 (\(S_{21}>0\) and \(S_{31}>0\)) for a some period of time starting from \(t=0\) (see Fig. 6a). Such behavior corresponds to the results obtained for the “symmetric” case considered in previous subsection. Moreover, for \(\varepsilon _{13}/\varepsilon =1\) the parameters \(S_{23}\) and \(S_{13}\) are non-positive, and hence, we see that subsystem 3 never steers 1 and 2. However, if \({\varepsilon _{13}}/{\varepsilon }\ne 1\) such steering effects (\(3\rightarrow 2\) and \(3\rightarrow 1\)) can appear for some periods of time. We observe sudden vanishing of the steering and its sudden restoring—it resembles phenomena of *sudden death of entanglement* [49, 50] and it *sudden rebirth* [51, 52]. What is interesting, steering for two pairs of subsystems \(2\rightarrow 1\) and \(2\rightarrow 3\) never can be achieved (see Fig. 6b).

Moreover, it is seen that steering in two pairs of subsystems \(1\rightarrow 3\) and \(3\rightarrow 1\) does not appear simultaneously. One can say that in general, for the model discussed here, only asymmetric steering can be generated—symmetric one cannot be achieved. Moreover, from Fig. 6a we see that at the same time two subsystems/wells cannot steer the third one, while one of them can steer two other subsystems/wells, what is in agreement with the monogamy results of Reid [48] and we see that the subsystem 1 can steer those labeled by 2 and 3 and additionally 3 steers 1 and 2. It should be emphasized that although wells 1 and 3 can steer that denoted by 2, the monogamy relation is not violated because such steering corresponding to pairs of wells appears at different moments of time.

In Fig. 7a, we see time evolution of \(C_{ij}\) (\(\{i,j\}=\{1,2,3\}\)) for exemplary values of \(\frac{\varepsilon _{13}}{\varepsilon }\). As previously, the values of all concurrences depend on coupling parameters \(\varepsilon \) and \(\varepsilon _{13}\), but here we have eight effective frequencies. In consequence, various beating effects among them appear, which defaces clear picture of the evolution. Despite it, one can see that bipartite entanglement between all pairs of subsystems *i* and *j* can be generated. Moreover, for some moments of time and for some particular values of \(\varepsilon _{13}/\varepsilon \) two-qubit states can be very close to the *maximally entangled* states. In particular, for pairs 1–2 and 2–3 the maximal values of concurrence can be \(\sim \)0.7071 to \(\sim \)0.9186. Moreover, we see that maximal values of concurrence strongly depend on \(\varepsilon _{13}/\varepsilon \). Especially, \(C_{13}\) can rapidly change its value when the strength of interaction is modified, but for larger values of \(\varepsilon _{13}/\varepsilon \) (the interaction between wells 1 and 3 becomes dominant) it stabilizes and becomes close to unity. One can explain such behavior as a result of the fact that the probabilities \(P_{001}\) and \(P_{100}\) tend to one half, whereas probability for state \(\vert 010\rangle \) tends to zero. In consequence, we deal here with a state which becomes close to maximally entangled state (Bell-like state) spanned in subspaces corresponding to the wells 1 and 3. Obviously, for such cases remaining two parameters (\(C_{12}\) and \(C_{23}\)) decrease, so bipartite entanglement present in the system concentrates in subsystem 1–3.

To compare the dynamics of the entanglement and steering for the case considered here, we plotted the “trajectories” of the states which are produced by ‘asymmetric” system, analogously as in the previous subsection. They are presented in Fig. 8.

We see that instead of a single line which appeared for the “symmetric” case, here we have the regions corresponding to the appearance (or nonappearance) of steering effect. In consequence, for a given value of the concurrence, we have an infinite number of states from which only some of them are steerable. It is seen that we will never achieve maximally entangled states for the pairs \(1\leftrightarrow 2\) and \(2\leftrightarrow 3\)—the maximal value of the concurrence is slightly greater than 0.9. However, contrary to the “symmetric” case, for such situation steerable states are produced. Moreover, maximal steering can be generated when the concurrence \(\sim 0.83\). It should be noted that when we observe steering effects for the pair \(1\leftrightarrow 2\), such effects are not present for \(2\leftrightarrow 3\), and *vice versa*. In Fig. 8, we plot only the values of \(S_{21}\) and \(S_{23}\). The parameters \(S_{12}\) and \(S_{32}\) are always negative, so we do not observe steering effects corresponding to them. Moreover, as one can see from the previous figures, the steering corresponding to the pairs \(1\leftrightarrow 2\) and \(2\leftrightarrow 3\) is asymmetric—if it appears for a given pair of subsystems, it acts only in one direction. As shown in Fig. 6, when \(S_{21}\) is positive, \(S_{23}\) is negative and *vice versa*. Thus, our “phase diagram” only shows the ability to produce steering for a given pair and in a particular direction.

For the pair \(1\leftrightarrow 3\), we observe the entirely different situation. Maximally entangled states can appear in the system, and, analogously as for the “symmetric” case discussed in the previous subsection, for such a case the states are not steerable. Nevertheless, we can find maximal steering effects when entanglement is quite strong, i.e., when \(C_{13}\simeq 0.86\). Moreover, analogously to the situation mentioned above and corresponding to the pairs \(1\leftrightarrow 2\) and \(2\leftrightarrow 3\), only asymmetric steering can appear in the system. If steering is observed in the direction \(1\rightarrow 3\) (\(3\rightarrow 1\)), it is not present for \(3 \rightarrow 1\) (\(1\rightarrow 3\)) (see Fig. 6).

## 4 Summary

We discussed here the Bose–Hubbard system containing three potential wells. The wells were coupled each other such a way that the system was in a triangle configuration. We assumed that only one well was initially occupied, and the total number of particles was preserved. Our model can be treated as a three-qubit system.

We analyzed two situations—two different initial states of the system (“symmetric” and “asymmetric” cases). We were interested in the time evolution of quantum correlations appearing in the system. In particular, we have shown that such system can be a source of the two-qubit steering and entanglement effects. Deriving analytical formulas, we also showed that the degree of the both: steering and quantum entanglement, strongly depends on the relation between strengths of the coupling between wells. However, for the “symmetric” case we did not observe such dependence in the steering’s dynamics. Comparing “symmetric” and “asymmetric” cases, we proved that for the latter situation, two additional steerabilities could appear in the system. Moreover, for the “asymmetric” case, subsystem labeled by 3 can steer two others – such form of steering did not appear for the “symmetric” case.

From the other side, for all situations considered here, only asymmetric steering can be generated. Moreover, when steering effects are present in the system, one qubit steers remaining two, whereas two qubits cannot steer the remaining one simultaneously.

Additionally, the steering parameter and concurrence reach their maximal values for different strengths of the couplings. What is interesting, for all cases discussed here, when two wells become maximally entangled, the steering effects for such pair disappear.

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