Quantum Steering in Two- and Three-Mode ??-Symmetric Systems

Abstract

We consider two $\mathcal{PT}$-symmetric models, consisting of two or three single-mode cavities. In both models, the cavities are coupled to each other by linear interactions, forming a linear chain. Additionally, the first and last of such cavities interact with an environment. Since the models are $\mathcal{PT}$-symmetric, they are described by non-Hermitian Hamiltonians that, for a specific range of system parameters, possess real eigenvalues. We show that in the models considered in the article, the steering generation process strongly depends on the coupling strengths and rates of the gains/losses in energy. Moreover, we find the values of parameters describing the system for which the steering appears.

1. Introduction

One of the commonly assumed axioms of quantum mechanics concerns the Hermiticity of operators characterizing physical observables. In consequence, the eigenvalue spectrum of such a self-adjoint operator has to be real. Therefore, a non-Hermitian Hamiltonian cannot describe real physical systems. Such a mathematical statement was changed when Bender and Boettcher [1] found out the existence of real spectra of non-Hermitian Hamiltonians exhibiting $\mathcal{PT}$-symmetric properties, as long as the following commutation relation is satisfied:

$$\begin{array}{c}\hfill \left[\widehat{H},\widehat{P}\widehat{T}\right]=0,\end{array}$$

(1)

where $\widehat{P}$ and $\widehat{T}$ appearing here are spatial and time inversion operators, respectively.

Numerous studies showed that the spectrum of the Hamiltonians describing $\mathcal{PT}$-symmetric systems can be real, and then, such a system is in the so-called unbroken phase of $\mathcal{PT}$-symmetry. The transition points from the unbroken to the broken $\mathcal{PT}$-symmetry phase are called the exceptional points. The properties of such points have been studied intensively, both theoretically [2,3,4] and in a variety of experiments [5,6]. For instance, in the field of optics, such research concerned Bragg scattering in some optical structural materials [7,8,9], linear [10,11,12,13,14], and nonlinear wave-guides [15,16,17,18,19,20,21]. The effect of the breaking of $\mathcal{PT}$-symmetry can be observed in various optical systems, including microwave photonics ones [22], optical lattices [23,24], dissipative microwave billiards [25], and coupled microresonators [26]. Besides this, many other physical systems have been studied theoretically, such as atomic gases [27,28], superconductors [29], and hydrogen molecules [30].

Quantum steering is one of the forms of quantum correlations appearing in physical systems. This kind of correlation, first described in 1935 by Schrödinger [31], presents the ability of one of the subsystems of a multipartite system to steer the states of others by performing local measurements. In 2007, Wiseman et al. [32,33] showed that the steering is a stronger correlation than the entanglement and is weaker than the Bell nonlocality. It means that all Bell nonlocal states are steerable, and all steerable states are nonseparable. What is essential is that not all entangled states are steerable, and not all steerable states are nonlocal in the Bell sense. The systems that exhibit the ability to generation steerable states are useful in various applications of quantum information theory. It is the result of the fact that quantum steering is a nonlocality effect that is less sensitive than the Bell nonlocality to such phenomena as noise and decoherence. Additionally, steering can certify the existence of entanglement between two subsystems [32,33,34,35]. Steerable states can be a resource for technical applications in some areas, including quantum computation, quantum information, quantum cryptography, and quantum teleportation. However, in recent years, not many articles concerning the steerable state generation in $\mathcal{PT}$-symmetric multimode systems have been published. Thus, in this paper, we concentrate on the quantum steering generation in $\mathcal{PT}$-symmetric systems. In particular, we study the ability of the here analyzed systems to generation steerable states. Additionally, we examine whether the production of such states is influenced by the strength of interactions between the subsystems and the rate of gain/loss in energy. In other words, we check whether the neighborhood of the exceptional point affects the steering.

This paper is organized as follows. In Section 2, we describe the here considered two- and three-mode $\mathcal{PT}$-symmetric systems for which the gain and loss in energy are balanced. We find the transition points for each considered system and determine the range of parameters for which the phase of $\mathcal{PT}$-symmetry is not broken. In Section 3, we consider the generation of steerable states in both systems when various initial states are assumed. We analyze the dependence of the steering strength on the coupling parameters and the rate of gain/loss in energy in the active and passive cavities. Finally, we draw some conclusions in Section 4.

2. The Models

In this paper, we analyze two quantum systems comprised of cavities forming a chain. The first model is a dimer that consists of only two cavities that linearly interact with each other. The first cavity (labeled by 1), which we call passive, loses the energy, whereas the second one (labeled by 2 and called active) gains the energy (see Figure 1). Such a system can be realized experimentally with an application of silica-on-silicon waveguide circuits [36], a telemetric sensor system [37], or two micro-resonators coupled by two fibers [26]. The effective Hamiltonian that describes our model can be represented as follows:

$$\begin{array}{ccc}\hfill {\widehat{H}}_1& =& \left(\omega -i\gamma \right){\widehat{a}}_1^{†}{\widehat{a}}_1+\left(\omega +i\gamma \right){\widehat{a}}_2^{†}{\widehat{a}}_2+\beta \left({\widehat{a}}_1^{†}{\widehat{a}}_2+{\widehat{a}}_2^{†}{\widehat{a}}_1\right)\hfill \\ & =& \left(\begin{array}{cc}{\widehat{a}}_1^{†}& {\widehat{a}}_2^{†}\end{array}\right)\left(\begin{array}{cc}\omega -i\gamma & \beta \\ \beta & \omega +i\gamma \end{array}\right)\left(\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\end{array}\right)=\left(\begin{array}{cc}{\widehat{a}}_1^{†}& {\widehat{a}}_2^{†}\end{array}\right){\mathcal{H}}_1\left(\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\end{array}\right),\hfill \end{array}$$

(2)

where ${\widehat{a}}_i^{†}$ and ${\widehat{a}}_i$ are bosonic creation and annihilation operators, respectively. In our model, we assume that both cavities have the same resonant frequency $\omega$. The linear interaction between two subsystems is described by the parameter $\beta$, while the rate of gain/loss of energy in the cavities is described by $\gamma$. Furthermore, we assume here a balance between the gain and losses of energy in the system.

Naturally, our system is $\mathcal{PT}$-symmetric, as it is invariant under the actions of both operators: that of spatial inversion $\widehat{\mathcal{P}}$ and the time inversion $\widehat{\mathcal{T}}$. By solving the eigenvalue equation for ${\mathcal{H}}_1$, one can show that the exceptional point is placed at $\gamma =\beta$. This point is known as the phase transition point of the system. For the case when $\gamma <\beta$, the system is in the unbroken $\mathcal{PT}$-symmetry phase with real eigenvalues of the Hamiltonian. From the other side, when $\gamma >\beta$, the $\mathcal{PT}$-symmetry phase is broken—it leads to the appearance of complex eigenvalues of the Hamiltonian.

The second model that we shall discuss is an extension of the first one. That extension is realized by adding a third cavity (labeled here by 0) in the middle of the system, creating a three-mode model referred to as an optical trimer. That additional cavity is neutral, which means that it does not interact directly with the environment. The first and last cavities (labeled as previously, by 1 and 2) are still passive and active, respectively. Just for recollection, the passive cavity corresponds to the loss in energy, while the active one corresponds to the gain in energy. They do not directly interact with each other, but simultaneously interact with the neutral one (see Figure 2). The effective Hamiltonian describing our trimer can be written as follows:

$$\begin{array}{ccc}\hfill {\widehat{H}}_2& =& \left(\omega -i\gamma \right){\widehat{a}}_1^{†}{\widehat{a}}_1+\omega {\widehat{a}}_0^{†}{\widehat{a}}_0+\left(\omega +i\gamma \right){\widehat{a}}_2^{†}{\widehat{a}}_2+\beta \left({\widehat{a}}_1^{†}{\widehat{a}}_0+{\widehat{a}}_0^{†}{\widehat{a}}_1+{\widehat{a}}_0^{†}{\widehat{a}}_2+{\widehat{a}}_2^{†}{\widehat{a}}_0\right)\hfill \\ & =& \left(\begin{array}{ccc}{\widehat{a}}_1^{†}& {\widehat{a}}_0^{†}& {\widehat{a}}_2^{†}\end{array}\right)\left(\begin{array}{ccc}\omega -i\gamma & \beta & 0\\ \beta & \omega & \beta \\ 0& \beta & \omega +i\gamma \end{array}\right)\left(\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_0\\ {\widehat{a}}_2\end{array}\right)=\left(\begin{array}{ccc}{\widehat{a}}_1^{†}& {\widehat{a}}_0^{†}& {\widehat{a}}_2^{†}\end{array}\right){\mathcal{H}}_2\left(\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_0\\ {\widehat{a}}_2\end{array}\right),\hfill \end{array}$$

(3)

where ${\widehat{a}}_0^{†}$ and ${\widehat{a}}_0$ are the creation and annihilation operators, corresponding to the neutral cavity, respectively. As for the case of the dimer system, we assume here that all cavities have the same resonant frequency $\omega$, and the parameter $\beta$ describes the linear interaction between neighboring cavities. The strength of the interaction between subsystems 1 and 0 is the same as that for 0 and 2. For such a configuration, similarly as for the dimer system, the $\mathcal{PT}$-symmetry appears. For such a case, the phase transition point occurs at $\gamma =\sqrt{2}\beta$. Thus, the optical trimer is in the unbroken $\mathcal{PT}$-symmetric phase when $\gamma <\sqrt{2}\beta$. On the other hand, for $\gamma >\sqrt{2}\beta$, the $\mathcal{PT}$-symmetry is broken.

Both considered systems are the open quantum ones that weakly interact with the surrounding environment. Consequently, the systems’ quantum states are described with an application of a density matrix. Moreover, the weak interactions allow us to apply the Born–Markov approximation. As a result, to analyze the time evolution, we use the master equation in the Lindblad form as follows:

$$\begin{array}{c}\hfill \frac{d}{dt}\widehat{\rho }=\frac{1}{i}\left[\widehat{H},\widehat{\rho }\right]+\widehat{\mathcal{L}}\widehat{\rho },\end{array}$$

(4)

where $\widehat{H}$ is the Hermitian Hamiltonian part, and for optical dimer is defined as follows:

$$\widehat{H}=\omega \left({\widehat{a}}_1^{†}{\widehat{a}}_1+{\widehat{a}}_2^{†}{\widehat{a}}_2\right)+\beta \left({\widehat{a}}_1^{†}{\widehat{a}}_2+{\widehat{a}}_2^{†}{\widehat{a}}_1\right),$$

(5)

whereas for optical trimer, it takes the following form:

$$\widehat{H}=\omega \left({\widehat{a}}_1^{†}{\widehat{a}}_1+{\widehat{a}}_0^{†}{\widehat{a}}_0+{\widehat{a}}_2^{†}{\widehat{a}}_2\right)+\beta \left({\widehat{a}}_1^{†}{\widehat{a}}_0+{\widehat{a}}_0^{†}{\widehat{a}}_1+{\widehat{a}}_0^{†}{\widehat{a}}_2+{\widehat{a}}_2^{†}{\widehat{a}}_0\right).$$

(6)

The Liouville superoperator $\widehat{\mathcal{L}}$, appearing here, acts on the state $\widehat{\rho }$ in the following way:

$$\begin{array}{c}\hfill \widehat{\mathcal{L}}\widehat{\rho }=\gamma \left(2{\widehat{a}}_1\widehat{\rho }{\widehat{a}}_1^{†}-{\widehat{a}}_1^{†}{\widehat{a}}_1\widehat{\rho }-\widehat{\rho }{\widehat{a}}_1^{†}{\widehat{a}}_1\right)+\gamma \left(2{\widehat{a}}_2^{†}\widehat{\rho }{\widehat{a}}_2-{\widehat{a}}_2{\widehat{a}}_2^{†}\widehat{\rho }-\widehat{\rho }{\widehat{a}}_2{\widehat{a}}_2^{†}\right).\end{array}$$

(7)

In both parts of Equation (7), the second and third terms represent a nonunitary dissipation of the system and transform the Hamiltonian into a non-Hermitian Hamiltonian. The first terms describe quantum jumps.

In our studies, we assume that the rate of gain/loss in energy $\gamma =0.01\omega$, whereas the parameter $\beta$ describing the interaction between subsystems changes its value from that corresponding to the exceptional point up to $\beta =100\gamma$.

3. Steering

In further consideration, we discuss the generation of two-mode steerable states. In quantum information theory, when we say that one subsystem steers the second one and steerable states are produced, it means that taking local measurements on parts of the entangled system that are associated with the first subsystem affects the state that describes the second subsystem, even when the subsystems are spatially separated. Although quantum steering was first described in the first half of the 20th century, it was not until 1992 that it was observed by Ou et al. [38]. In this experimental realization, the Reid criterion was used based on the uncertainty relation [39].

In this paper, we apply the parameter that allows for the detection of quantum steering and quantifying the steerability between two subsystems, i and j. That parameter’s definition is based on the inequality proposed by Cavalcanti et al. [40].

$${\left|〈{\widehat{a}}_i{\widehat{a}}_j^{†}〉\right|}^2\le 〈{\widehat{a}}_i{\widehat{a}}_i^{†}\left({\widehat{a}}_j{\widehat{a}}_j^{†}+\frac{1}{2}\right)〉,$$

(8)

where ${\widehat{a}}_i$ and ${\widehat{a}}_i^{†}$ are the annihilation and creation operators of modes i and j, respectively. The violation of inequality (8) implies the possibility of quantum steering for a given state. From the other side, the states that satisfy inequality (8) are unsteerable. Thus, the quantity called steering parameter introduced in [41] can be expressed as follows:

$$S_{ij}={\left|〈{\widehat{a}}_i{\widehat{a}}_j^{†}〉\right|}^2-〈{\widehat{a}}_i{\widehat{a}}_i^{†}\left({\widehat{a}}_j{\widehat{a}}_j^{†}+\frac{1}{2}\right)〉=〈{\widehat{a}}_i{\widehat{a}}_j^{†}〉〈{\widehat{a}}_i^{†}{\widehat{a}}_j〉-〈{\widehat{a}}_i{\widehat{a}}_i^{†}\left({\widehat{a}}_j{\widehat{a}}_j^{†}+\frac{1}{2}\right)〉.$$

(9)

When parameter $S_{ij}$ is positive, we can say that mode j steers that denoted by i, so the steering appears in the direction from j to i. We can also define the second parameter $S_{ji}$ that describes the possibility of steering in the opposite direction. It can be calculated simply by a replacement of the indices i and j in Equation (9). If $S_{ji}>0$ and $S_{ij}>0$, the symmetric steering is present in the system, whereas if only one of those parameters is positive, we are dealing with asymmetric steering.

3.1. Two-Mode System

We start our analysis of bipartite steering from the dimer system shown in Figure 1, and we analyze two cases. For the first of them, only the passive cavity is in its excited state, so the system’s evolution starts from the state $\widehat{\rho }(t=0)=|10\rangle \langle 10|$. The second one concerns the situation when the initial state is $\widehat{\rho }(t=0)=|01\rangle \langle 01|$—only the active oscillator is excited.

From Figure 3a,b, one sees that the time evolution of the steering parameters strongly depends on the value of $\beta$. When $\beta =40\gamma$ (Figure 3a), both steering parameters $S_{12}$ and $S_{21}$ take negative values at the beginning of the process—the steerable states are not generated. After a short interval of time, $S_{21}$ becomes positive, while the value of $S_{12}$ remains negative, and we observe the asymmetric steering. From the point at which $S_{21}<0$, the system goes into an unsteerable state. After that, the steerability in the opposite direction occurs in the dimer. We can call such a feature the redirection effect. The value of $S_{12}$ becomes positive, and the second oscillator steers the first one. Parameter $S_{12}$ also increases, reaching its maximum value, and then disappears in the same way as in the case of $S_{21}$. The both maxima of $S_{12}$ and $S_{21}$ are approximately of the same value. After one period of asymmetric steering for each subsystem, the quantum state becomes unsteerable.

When we move away from the exceptional point, as the value of the coupling strength increases to $\beta =100\gamma$ (see Figure 3b), and perform the same analysis, the results become different from those presented in Figure 3a. At the beginning of the system’s evolution, the sequence of steering direction is the same as in the previous case. However, there are some differences if we compare both situations. First, steering $1\to 2$ starts to become positive just at the beginning of evolution (the notation $i\to j$ means that subsystem i steers j). Moreover, the maxima of $S_{12}$ and $S_{21}$ are almost equal to each other, but are greater than those in Figure 3a. Third, the time when the redirection effect appears is shorter than for $\beta =40\gamma$. Finally, after vanishing the steering in two directions, we can see that both steerings reappear again. Contrary to their first occurrence, steering $2\to 1$ appears as the first one.

In the second scenario (Figure 3c,d), our system starts its evolution from the state $\widehat{\rho }(t=0)=|01\rangle \langle 01|$. It means that only the active oscillator is excited. The evolution of the parameters $S_{12}$ and $S_{21}$ resembles that for the previous case. However, all directions $1⟷2$ are opposite to those depicted in Figure 3a,b. Moreover, the maxima of the values of all parameters depicted are not equal and are descending as they subsequently appear. Finally, if we compare the parameters from Figure 3a,b to those depicted in Figure 3c,d, respectively, the order of the appearance of the quantum steering in one direction is opposite to those discussed earlier. Additionally, as seen in Figure 3d, one additional reappearance of the steering effect is present. Nevertheless, in all the cases shown in Figure 3, when $t\to \infty$, all steering parameters are positive, so the steering does not appear.

In Figure 4, we show the dependence of the maximal values of the steering parameters $S_{ij}$ as a function of the $\beta /\gamma$. We consider two situations. For the first one, the passive cavity is excited at $t=0$, while the active one is in a vacuum state (Figure 4a). We can easily see that for small values of $\beta /\gamma$, the maximal values of both $S_{12}$ and $S_{21}$ take negative values, and thus, no steering is generated. When $\beta /\gamma \ge \approx 20$, the steering starts to appear in both directions $2\to 1$ and $1\to 2$. As $\beta /\gamma$ increases, the maximal values of $S_{12}$ and $S_{21}$ rapidly grow and then stabilize at ≈0.06. Further growth of the ratio $\beta /\gamma$ does not affect the maximal values of $S_{12}$ and $S_{21}$ significantly. Moreover, within the range of values of $\beta /\gamma$ in which the steering effect occurs, the maximal values of $S_{12}$ and $S_{21}$ are approximately the same.

The second case concerns the situation when only the passive cavity is excited. In Figure 4b, we can see that for low values of the ratio $\beta /\gamma$, only the maximal value of $S_{12}$ becomes positive. Thus, for small values of $\beta /\gamma$, the steering effect only occurs in the direction $2\to 1$. Moreover, the growth of the maximal value of $S_{12}$ with $\beta /\gamma$ is very rapid at the initial values of $\beta /\gamma$. Such growth of the maximal value of $S_{12}$ occurs until it reaches a value of $\approx 0.06$ for $\beta /\gamma \approx 30$. When such a maximum of $S_{12}$ stabilizes, the quantum steering $1\to 2$ starts to appear. If we continue increasing the value of $\beta /\gamma$, the maximal value of $S_{21}$ increases in the same manner as in Figure 4a, while the maximal value of $S_{12}$ remains almost constant. For all values of ratio $\beta /\gamma$, $max\left(S_{12}\right)$ are greater than those of $S_{21}$. It means that the steering $2\to 1$ is always stronger than that in the opposite direction ($1\to 2$). Nevertheless, one can see that in both cases considered here, the asymmetric steering appears in the system for sufficiently large values of $\beta /\gamma$.

Thus, we can see that the $\mathcal{PT}$-symmetric dimer system can be a source of steerable states. However, to generate these states, the crucial role plays the right choice of the value of the ratio $\beta /\gamma$. When the ratio $\beta /\gamma$ increases, we move away from the phase transition point. For such a situation, the generated steering is stronger, the time of occurrence of steerable states is longer, and we observe a greater number of steering reappearances.

3.2. Three-Mode System

Our next goal is to analyze bipartite quantum steering in a system that consists of oscillators in three cavities (see Figure 2). We focus on three different cases in which only one of three resonators is initially excited. For each such case, we analyze the evolution of $S_{ij}$ for two values of $\beta /\gamma$, where again $\gamma =0.01\omega$.

The first case concerns the situation when the initial state of our system is $\widehat{\rho }(t=0)=|100\rangle \langle 100|$, i.e., only the passive cavity is excited. When $\beta =40\gamma$, the steering appears in three different periods of time (see Figure 5a). That one occurring at the beginning of the system’s evolution is the strongest and is described by the parameter $S_{01}$. In the subsequent two appearances of quantum steering, cavity 2 steers that labeled by 0. What differentiates that situation from those concerning the dimer model is the fact that the last in the sequence of steerabilities is more pronounced than its predecessor. Moreover, what is relevant is the fact that the central neutral cavity is steered by the remaining two in the discussed case.

When $\beta /\gamma =100$, the steering $1\to 0$ also appears as the first. Then the pairs of maxima corresponding to the steering $1\to 0$ and $2\to 0$ appear interchangeably three times. In each such pair, each steering described by $S_{ij}$ appears twice, and the maxima for each subsequent pair become decreased with time. Finally, the system goes into an unsteerable state. Moreover, there is no steering between the passive and active ones due to the presence of a middle neutral cavity (one should remember that only the passive cavity is excited in the case discussed here).

When the initial state of the system is $\widehat{\rho }(t=0)=|010\rangle \langle 010|$ (the middle, neutral cavity is initially excited), only the neutral cavity can steer the passive and the active ones—see Figure 5c,d. It differentiates the behavior of the steering appearing here from those depicted in Figure 5a,b. The evolution of steering parameters $S_{10}$ and $S_{20}$ takes the form of two and three subsequent peaks, similar to the previously considered situations. Moreover, the pair of parameters $S_{10}$ and $S_{20}$ reach their maxima simultaneously in the same instance of time. Nevertheless, when $\beta =40\gamma$, the steering $0\to 1$ appears before $0\to 2$—visually the peak corresponding to the parameter $S_{10}$ is more extended in time. Finally, after a single appearance of the two peaks, the system goes to an unsteerable state. When $\beta =100\gamma$, the situation is similar to that depicted in Figure 5c. However, here, we observe the sequence of three, decreasing in their heights, pairs of peaks instead of the single pair.

In the last scenario, the system starts its evolution from the state $\widehat{\rho }(t=0)=|001\rangle \langle 001|$, in which only the active cavity is excited. The situation is shown in Figure 5e,f and resembles that from Figure 5a,b. However, here, the order for the steering manifestation is different, and the steering $2\to 0$ is built as the first. Nevertheless, the most interesting feature in this scenario is the appearance of the quantum steering $2\to 1$ in parallel to that $2\to 0$ at the beginning of the system’s evolution. At that period of time, when both steering parameters, $2\to 0$ and $2\to 1$, are positive, the active cavity can simultaneously steer the remaining two subsystems. However, it should be noted here that the steering corresponding to $S_{02}$ is much stronger than that of $S_{12}$.

To show the ability of the system to generate quantum steering, we plot in Figure 6 the maximal values of the steering parameters versus $\beta /\gamma$. We observe that the maximal values of $S_{ij}$ depend not only on the value of $\beta /\gamma$, but also on the initial state of the system.

For the case in which the initial state of the system is $\widehat{\rho }(t=0)=|100\rangle \langle 100|$ (the passive cavity is initially excited), quantum steering in two directions $1\to 0$ and $2\to 0$ only, appears in the system (see Figure 6a). It means that only the passive and active cavities steer the neutral one. The positive values of $S_{01}$ and $S_{02}$ do not appear simultaneously. For a very small ratio $\beta /\gamma$, only the steering $1\to 0$ is present in the system. As the value of $\beta /\gamma$ increases, the maximal value of $S_{02}$ grows considerably, but is still smaller than that of $S_{01}$, so the steering $1\to 0$ is always stronger than $2\to 0$. For greater values of $\beta /\gamma$, the maximal values of $S_{01}$ and $S_{02}$ tend to stabilize.

When the neutral cavity is excited for $t=0$ ($\widehat{\rho }(t=0)=|010\rangle \langle 010|$), we can see in Figure 6b that only that cavity steers the others. For very small values of $\beta /\gamma$, only the parameter $S_{10}$ is positive. The maximal value of $S_{10}$ stabilizes almost instantaneously—approximately around $0.03$. Shortly, after the time when $\beta /\gamma$ reaches 30, steering $0\to 2$ starts to appear. From that point, the neutral cavity steers the passive and active ones. Moreover, the parameter describing the steering $0\to 1$ always takes higher values than the parameter $S_{20}$.

Finally, when the initial state of the system is $\widehat{\rho }(t=0)=|001\rangle \langle 001|$ (the active cavity is initially excited), three possible directions of quantum steering can appear. From Figure 6c, we see that the active cavity steers both remaining simultaneously. For very small values of $\beta /\gamma$, only the active cavity steers the others. When we increase the value of $\beta /\gamma$, the maximal values of $S_{02}$ and $S_{12}$ increase to their stable values. Nevertheless, maximal values of $S_{02}$ are almost 10 times larger than those of $S_{12}$. Moreover, when we continue increasing the value of $\beta /\gamma$, the steering $1\to 0$ starts to appear, and the maximal values of $S_{01}$ continuously increase with $\beta /\gamma$. As the result, for $\beta /\gamma >\sim 25$, the steering $2\to 0$ is still the strongest, while the steering $2\to 1$ remains the weakest.

As in the case of the dimer, with proper selection of parameter values, the $\mathcal{PT}$-symmetric trimer system can be a source of steerable states. If the $\beta /\gamma$ ratio increases and we move away from the exceptional point, the system’s ability to generation steerable states increases. In other words, the produced steering is stronger, and the time when we can observe it is longer.

4. Conclusions

In this paper, we discussed the possibility of quantum steering generation in two kinds of $\mathcal{PT}$-symmetric systems (those consisting of two and three linear oscillators). Interestingly, we found that in both systems, the steering between their subsystems is always asymmetric. We showed that the evolution of the steering parameters depends not only on the coupling strength and gain/loss parameters, but also on the system’s initial state. In general, when the ratio between the strengths of the coupling parameter and the rate of gain/loss energy increases, the value of the steering parameter grows as well. What is intriguing is that for the system consisting of two oscillators, if the system starts its evolution from the passive cavity being excited, the maximal strengths of steering in both directions are almost equal. Contrary to it, such symmetry is broken for the opposite situation, when the active cavity is initially excited—the maximal values of the parameter describing steering $2\to 1$ for different values of the ratio $\beta /\gamma$ will be greater than those corresponding to the steering in the opposite direction. Thus, the choice of the initial state splits the distribution of two steerings from equable to unbalanced (in the sense of maximal values of the steering parameters).

For the system involving three oscillators, we can observe two scenarios of the steering generation. In the first one, the active and the passive cavities can steer the neutral one. Such a situation occurs when the passive or active cavity is excited at the beginning of the system’s evolution. In the second scenario, the neutral cavity can steer the others. Such a situation happens when the system starts its evolution with an excitation in the neutral cavity. Interestingly, in this case, the neutral cavity can simultaneously steer the two others.

The direct steering between the passive and the active cavities can only appear in one direction—from the active cavity to the passive one when the active cavity is initially excited. However, the steering ability in that direction is much weaker than those in the other directions. Moreover, whenever such steering appears, the active cavity can also simultaneously steer the two others for a short interval of time, right after the start of the system’s evolution.

It should be noted that although the steerable states can be generated in both studied systems, the neighborhood of the exception point weakens this ability. If the system is close to the phase transition point, the steering is very weak, or we do not observe it. However, if the steering is nevertheless generated, it disappears very quickly during the system’s evolution.