1. Introduction

The semiconductor quantum wells (QWs) are typical solid-state devices in nanoscale size, being investigated extensively in the field of quantum optics and quantum information during last decades. Due to the strong electron-electron interactions, the QWs can behave as normal atomic systems with discrete energy levels and possess unique advantages of large electric dipole moments, strong nonlinear effects, greatly structural flexibility in designing parameters and materials. Since the coherent processes are possible to emergy based on the intrinsic properties contained in QWs, the atomic coherence and quantum interference effects are extensively investigated including coherent population trapping [1–3], electromagnetically induced transparency [4], gain without inversion [5,6], tunneling-induced transparency (TIT) [7,8], strong Kerr effect [9], optical bistability [10,11], optical soliton [12,13], slow light [14], quantum entanglement [15–17], spatiotemporal-vortex four-wave mixing [18], and other optical nonlinearities [19,20].

On the other hand, Einstein-Podolsky-Rosen (EPR) steering is one type of quantum correlations intermediate quantum entanglement and Bell nonlocality and is first introduced by Schrödinger [21] in response to the famous EPR paradox [22]. Physically, EPR steering refers to a phenomenon in which one party (Alice) is able to remotely control the other one (Bob) through local measurements but not vice versa. In Reid’s pioneering work, the criterion to detect the steering is put forward in light of Heisenberg uncertainty principle [23] and then it is verified in experiment using the nondegenerate parametric amplification [24]. However, the strict definition and classification of steering, entanglement, and nonlocality have not been successfully presented until 2007 [25,26]. Generally, the steering effects are divided into two categories: one is named as “one-way” steering while the other is referred to “two-way” symmetrical EPR steering. The former usually attracts much more attention due to its internal asymmetry while the latter is unimportant since its properties are similar to quantum entanglement. The theoretical predictions of one-way steering have been reported in various systems such as nonlinear coupler [27], optomechanics systems [28,29], and hybrid cavity-magnon systems [30–35] and so on. Up to know, the one-way steering could find extensive applications in one-sided device-independent quantum cryptography [36,37], quantum secret sharing [38,39], one-way quantum computing [40], no-cloning quantum teleportation [41,42], subchannel discrimination [43].

So far it is still a major challenge that the quantum entanglement and steering are usually fragile to the environmental noises. During past years, a great deal of effort in seeking an effective way to protect quantum states or preparing noise-free entanglement resources has never been ceased. To our knowledge, the typical approaches to achieve this goal include loop control strategies [44], quantum error correction [45] or using decoherence-free subspace [46] etc.. However, the above-mentioned schemes are generally imperfect. To overcome this difficult, an unique idea has been suggested for generating robust atom-atom entanglement and light entanglement with cavity loss [47–49]. Successively, Pielawa et al. developed an atomic reservoir theory to engineer squeezed and entangled states [50,51], in which one Bogoliubov mode constituted by two original modes interacts with a common atomic transition of two-level system while its quadrature component is decoupled with the system. When the dissipation process is dominant over the amplification, the composite mode would dissipate into the desirable state through the single channel. As a consequence, the long-lived entangled states are possible to obtain without the requirement of input nonclassical states. Furthermore, the internal mechanisms for the one-channel and two-channel dissipation schemes are discussed in detail in different systems [52–55]. However, we note that the effects of dephasing rates on quantum correlation have been investigated relatively few since they are usually negligible in the atomic systems and superconducting circuits.

In this article, we find that the dephasing rates in the solid-state QWs could play a constructive role in realizing one-way steering although they are detrimental to the quantum entanglement. It is reported that the intersubband transitions (ISBT’s) dephasing mechanisms can make them behave as “artificial atoms” [56], in which the quantum coherence phenomena and Fano interference are existent. Compared with normal atomic systems, the dephasing rates originated from electron-electron, interface roughness, and phonon scattering processes are the dominant contribution of the damping rate for the QWs nanostructure. Therefore, We investigate the extent to which the dephasing rate generates the steady-state one-way steering for two cavity modes through a single-pathway dissipation. Here we maily focus on the following two issues. On the one hand, what is the role of dephasing rates play in the dissipation scheme? On the other hand, by what means can the dephasing rates modify the quantum correlation? The main innovative results of the present work are summarized as follows. First, by choosing balanced parameters, a single-pathway interaction is formed between two dressed states, through which the steady-state quantum entanglement is acquired. Interestingly, we study the effects of the dephasing rates in the dissipation scheme, demonstrating that the bipartite entanglement is relatively susceptible to the decoherence. Moreover, it is explored that the dressed-state populations are mainly determined by the internal multiple quantum interference mechanisms, through which the dephasing rates affect the population difference in a delicate way, giving rise to the occurrence of one-way steering.

The remaining part of this article is organized as follows. In Sec. 2, we describe the model of the system and obtain the master equations. In Sec. 3, we show the analytical and numerical results for the one-way steering. Then the internal mechanisms are analyzed based on dressed-state picture and Bogoloiubov mode transformation. Finally, the discussion and conclusion are presented in Sec. 4.

2. Model and equations

As shown in Fig. 1(a), we consider an asymmetric semiconductor quantum-well nanostructure consisting of a wide well (WW) and two narrow wells (NW), in which all possible transitions in this system are dipole allowed because the asymmetry breaks the parity of the wave functions [56,57]. The devices have been widely used to study electron localization [58], bright and dark optical solitons [13] and so on. Notably, the dynamics in these quantum well structures are usually more complicated than the normal atomic systems because many-body effects arises from the electron-electron interactions. Without loss of generality, here we only consider a situation where the electron-electron effects can be neglected due to the low doping and all subbands have the same effective mass [1–3,59]. In Fig. 1(b), two classical laser fields with Rabi frequencies $\Omega _{c_{1}}$ and $\Omega _{c_{2}}$ are applied to drive the dipole-allowed transitions of $|1\rangle \leftrightarrow |4\rangle$ and $|2\rangle \leftrightarrow |3\rangle$, respectively. Simultaneously, two cavity modes $\nu _{1,2}$ couple to the transitions $|1\rangle \leftrightarrow |2\rangle$ and $|3\rangle \leftrightarrow |4\rangle$. In the appropriate rotating frame and under the rotating-wave approximation, the Hamiltonian of this system is given as follows ($\hbar =1$) [12,15,18,58]

Without loss of generality, $\mathcal {L}_{a}\rho$ represents decay rates from states $|i\rangle$ and $\mathcal {L}_{\textrm {ph}}\rho$ denotes dephasing decay rates of the quantum coherence of the $|i\rangle \rightarrow |j\rangle$ transitions, which are added phenomenologically. The population decay rates for subband $|i\rangle (i=1-4)$ denoted by $\gamma _{i}$ are due primarily to longitudinal optical phonon emission events at low temperature. The decay rates $\gamma _{2}=\gamma _{21}$, $\gamma _{3}=\gamma _{31}+\gamma _{32}$, $\gamma _{4}=\gamma _{41}+\gamma _{42}+\gamma _{43}$. We use $\gamma _{\textrm {ph}}^{ij}$ to describe the dephasing decay rates, which are determined by electron-electron, interface roughness and phonon scattering processes. The cavity loss terms $\mathcal {L}_{c}{\rho }$ take the form

 

Fig. 1. (a) Schematic energy-band diagram of a single period of the three-coupled quantum well (TCQW) nonlinear optical structure. Without loss of generality, the layer thicknesses for the GaInAs wells are designed as 42 Å, 20 Å, 18 Å, respectively. The wide well and two narrow wells are separated by a 16 Å AlInAs barrier. The possible energy level spaces for the sample are chosen as 151 mev, 119 mev, 116 mev, 120 mev, respectively. (b) The energy level arrangement of the present scheme, in which two strong fields denoted by $\Omega _{c_{1}}$ and $\Omega _{c_{2}}$ are applied to drive two dipole allowed transitions $|1\rangle \rightarrow |4\rangle$ and $|2\rangle \rightarrow |3\rangle$, respectively. Two nondegenerate cavity modes are coupled with the ISBTs $|4\rangle \rightarrow |3\rangle$ and $|2\rangle \rightarrow |1\rangle$. (c) The demonstration of interaction between original modes $a_{1}$ and $a_{2}$ with the dressed state transitions. Note that the dressed states has twofold level degeneracy $|\tilde {1}\rangle (|\tilde {2}\rangle )$ and $|\tilde {3}\rangle (|\tilde {4}\rangle )$ when balanced parameters are chosen as $\Omega _{c_{1}}=\Omega _{c_{2}}$ and $\Delta _{1}=\Delta _{2}$.

Download Full Size | PDF

To describe clearly the physical mechanisms and the corresponding conditions for dissipative reservoir effects, we resort to the dressed atomic picture by diagonalizing the Hamiltonian $H_{0}$ under the conditions of $\Omega _{c_{1,2}}\gg \gamma _{ij}, \kappa _{i}, g_{i} (i=1,2, j=3,4)$. The dressing transformation can be performed through the following three steps. In the first step, the coupling field $\Omega _{c_{1}}$ splits the levels $(|1\rangle, |4\rangle )$ into two sublevels $(|\tilde {1}\rangle, |\tilde {4}\rangle )$ with level separation $\tilde {\Omega }_{1}$ between states $|\tilde {1}\rangle$ and $|\tilde {4}\rangle$. In the second step, the coupling field $\Omega _{c_{2}}$ can split the levels $(|2\rangle, |3\rangle )$ into two sublevels $(|\tilde {2}\rangle, |\tilde {3}\rangle )$, similarly. When the symmetric parameters are chosen as $\Omega _{c_{1}}=\Omega _{c_{2}}$ and $\Delta _{1}=\Delta _{2}$, the dressing levels $|\tilde {1}\rangle, |\tilde {2}\rangle$ and $|\tilde {3}\rangle, |\tilde {4}\rangle$ are twofold degenerate. Finally, by tuning the sideband resonance appropriately, the two quantized modes $a_{1,2}$ are resonantly coupled with the dressed-state transition $|\tilde {1}\rangle \leftrightarrow |\tilde {3}\rangle$, which is plotted in Fig. 1(c). Note that the two control fields are strong enough with a large amount of photon numbers. Without loss of generality, the photon numbers are not written in each group of dressed states. Then the bare atomic states are expressed in terms of dressed states as [60]

We define $\cos {\theta }=\sqrt {\frac {1}{2}+\frac {d}{2\sqrt {1+d^{2}}}}$ and $\sin {\theta }=\sqrt {\frac {1}{2}-\frac {d}{2\sqrt {1+d^{2}}}}$. The normalized detunings are defined as $d_{1}=\frac {\Delta _{1}}{\Omega _{c_{1}}}, d_{2}=\frac {\Delta _{2}}{\Omega _{c_{2}}}$ and $d_1=d_2=d$. The dressed states $|\tilde {j}\rangle (j=1-4)$ have their eigenvalues $\lambda _{1(2)}=\frac {1}{2}(\Delta _{1(2)}-\tilde {\Omega }_{c_{1(2)}})$, $\lambda _{3(4)}=\frac {1}{2}(\Delta _{2(1)}+\tilde {\Omega }_{c_{2(1)}})$, and $\tilde {\Omega }_{c_j}=\sqrt {\Delta _{j}^{2}+\Omega _{c_j}^{2}}$. Then the free Hamiltonian of $H_{0}$ becomes the diagonal form as $\tilde {H}_{0}=\lambda _{j}|\tilde {j}\rangle \langle \tilde {j}|(j=1-4)$.

In light of the dressed-state picture, the Hamiltonian $H_{I}$ is then rewritten as

To obtain the final form of the effective Hamiltonian, we have the relations of $\delta _{1}=-\delta _{2}=\lambda _{3}-\lambda _{1}=2\tilde {\Omega }$, wherein $\tilde {\Omega }_{c_1}=\tilde {\Omega }_{c_2}=\tilde {\Omega }$. It is found that the interactions between the cavity modes ($a_{1}, a_{2}$) and the dressed atomic spin $\tilde {\sigma }_{31}$ are established. The annihilation of mode $a_{1}$ is accompanied by the creation of mode $a_{2}$ through a common transition channel $|\tilde {1}\rangle \rightarrow |\tilde {3}\rangle$, which is shown in Fig. 1(c). By applying the standard linear theory of quantum optics, we can derive the reduced master equation for the cavity modes from Eq. (8) as follows

3. Numerical results and discussions

In this section, we focus on investigating the quantum correlations of the cavity modes in the asymmetric semiconductor QWs nanostructure. In terms of the reduced master equation, we obtain a set linearized quantum Langevin equations (QLEs) as follows [61]:

The system is stable only if all eigenvalues of the drift matrix $A$ have negative real parts, which can be derived from the Routh-Hurwitz criterion [62]. In the present work, the chosen parameters always satisfy the stability condition. The steady state of the system is a Gaussian state that can be entirely characterized by a $4\times 4$-covariance matrix (CM) $C$ with components $C_{ij}(t,t')=\frac {1}{2} \left \langle u_{i}(t)u_{j}(t')+u_{j}(t')u_{i}(t)\right \rangle$, $(i, j=1,2,3,4).$ The steady-state CM can be achieved by solving the Lyapunov equation [63]

The nonzero diffusion coefficients are $\chi _{1}={\rm g}_1^2 \sin ^22\theta \tilde {\rho }_{33}^{s}/4\Gamma$, $\chi _{2}={\rm g}_1^2 \sin ^22\theta \tilde {\rho }_{11}^{s}/4\Gamma$, $\chi _{3}={\rm g}_1{\rm g}_2 \sin ^22\theta (\tilde {\rho }_{11}^{s}+\tilde {\rho }_{33}^{s})/2\Gamma$. The diffusion matrix $D$ characterizing the stationary-noise correlations has been defined through $D_{ij}\delta (t-t')=\frac {1}{2} \langle \xi _{i}(t)\xi _{j}(t')+\xi _{j}(t')\xi _{i}(t)\rangle$.

To study the bipartite entanglements, we adopt the logarithmic negativity $E_N$ by computing the $4\times 4$ CM related to the two modes of interest [64,65]. The logarithmic negativity for Gaussian states is defined as

Moreover, the proposed measurements of the Gaussian quantum steerability in different directions between two modes are given by [66]

$\mathcal {G}^{1\rightarrow 2}(V)>0 (\mathcal {G}^{2\rightarrow 1}(V)> 0)$ demonstrates that the bipartite Gaussian state is steerable from mode 1 (2) to mode 2 (1) by Gaussian measurements on mode 1 (2). The larger the $\mathcal {G}$ is, the stronger the steerability will be. In our numerical calculations, we always choose $\gamma _{2}=1$ mev, $\gamma _{3}=2$ mev, $\gamma _{4}=3$ mev according to Ref. [1–3,13,16]. Without loss of generality, all involved dephasing rates between the transitions $|i\rangle \rightarrow |j\rangle$ are assumed to be the same, i.e., $\gamma _{ph}^{\textrm {ij}}=\gamma _{p}$. At first of place, we discuss the effects of the dephasing rates on the quantum entanglement. It is noted that the quantum correlations are strongly dependent on the dressed-state populations of $\tilde {\rho }_{11}^{s}$ and $\tilde {\rho }_{33}^{s}$. Figure 2 shows that the logarithmic negativity of $E_{N}$ and the dressed state population difference $p=|\tilde {\rho }_{33}^{s}-\tilde {\rho }_{11}^{s}|$ are plotted as a function of normalized detuning $d$ by choosing different dephasing rates $\gamma _{p}$. The other parameters are chosen as $g_{1}=0.02$ mev, $g_{2}=0.015$ mev, $\kappa _{1}=\kappa _{2}=10^{-6}$ mev and $\Omega _{c_{1}}=\Omega _{c_{2}}=10$ mev. Remarkably, in Fig. 2(a), the entanglement is reduced with the increasing of dephasing rates $\gamma _{p}$. The properties for the obtained entanglement are summarized as follows. First, it is asymmetrical about $d=0$, which is due to the asymmetry of population difference $p$ in the positive and negative frequency regimes. It is seen that the entanglement is enhanced with the increasing of population difference. Second, at the exactly resonant point $d=0$, we find that the entanglement is absent for $p=0$, i.e., $\tilde {\rho }_{11}^{s}=\tilde {\rho }_{33}^{s}$. Third, as the dephasing rates $\gamma _{p}$ are changed from $\gamma _{p}=1$ mev to $\gamma _{p}=2$ mev, the entanglement falls down slightly. As shown in Fig. 2(b), the population difference $|p|$ is simultaneously decreased accompanying by the reduction of entanglement. Consequently, the decline of entanglement implies that the asymmetric quantum steering is possible to emerge because it is a strict subset of quantum entanglement.

 

Fig. 2. (a) Plots of logarithmic negativity $E_{N}$ as a function of normalized detuning $d$ by choosing different dephasing rates $\gamma _{p}=1.0$ mev (solid line), $\gamma _{p}=1.5$ mev (dotted line), $\gamma _{p}=2.0$ mev (dashed line). (b) The steady-state atomic populations difference between dressed states $|\tilde {3}\rangle$ and $|\tilde {1}\rangle$ as a function of the scaled detuning $d$ by choosing balanced parameters $\Delta _{1}=\Delta _{2}$. The other parameters are chosen as $g_{1}=0.02$ mev, $g_{2}=0.015$ mev, $\kappa _{1}=\kappa _{2}=10^{-6}$ mev and $\Omega _{c_{1}}=\Omega _{c_{2}}=10$ mev.

Download Full Size | PDF

In Fig. 3 and Fig. 4, we show the generation of asymmetric steering for two different cases of $g_{1}>g_{2}$ and $g_{1}<g_{2}$, respectively. For comparison, the evolutions of quantum steering $\mathcal {G}^{1\rightarrow 2}(V)$, $\mathcal {G}^{2\rightarrow 1}(V)$ as a function of $d$ are plotted by choosing different dephasing rates. In Fig. 3, the parameters are chosen as $g_{1}=0.02$ mev, $g_{2}=0.015$ mev, $\kappa _{1}=\kappa _{2}=10^{-6}$ mev, and in (a) $\gamma _{p}=0.5$ mev; in (b) $\gamma _{p}=1.5$ mev. In the case of $g_{1}>g_{2}$, the Gaussian steerabilities of $\mathcal {G}^{1\rightarrow 2}(V)$ and $\mathcal {G}^{2\rightarrow 1}(V)$ appear in the stable region of $d>0$. With the evolution of normalized detuning $d$, the values of Gaussian steerability are first increased to a maximal value and then decreased gradually. When $\gamma _{p}=0.5$ mev, the two-way EPR steering effects are survived in the region of $0.62<d<2.88$, which is represented by the blue shadow region. For $\gamma _{p}=1.5$ mev, we always have $\mathcal {G}^{1\rightarrow 2}(V)=0$ and $\mathcal {G}^{2\rightarrow 1}(V)>0$, meaning that the two-way steering disappears while the one-way Gaussian steerability remains in spite of the maximal values of $\mathcal {G}^{2\rightarrow 1}(V)$ are reduced from 0.50 to 0.22. Besides, the similar effects of dephasing rates on quantum steering are also observed in the region of $d<0$, which is plotted in Fig. 4. The different parameters are chosen as $g_{1}=0.15$ mev, $g_{2}=0.2$ mev, $\kappa _{1}=\kappa _{2}=10^{-6}$ mev for (a) $\gamma _{p}=0.5$ mev; (b) $\gamma _{p}=1.5$ mev. Notably, there are several differences between the two cases. First, in the region of $d<0$, in order to obtain the dephasing-assisted one-way steering, the larger coupling constants $g_{1,2}$ are required due to the smaller population difference $|p|$. Second, the Gaussian state is steerable from mode 2 to mode 1 for $d>0$ while the mode 1 can steer mode 2 in the other region $d<0$. To show the internal asymmetry further, the evolution of the steady-state photon numbers $N_{j}=\langle a_{j}^{\dagger }a_{j}\rangle (j=1,2)$ of the two modes are plotted in Fig. 3(c, d) and Fig. 4(c, d). In the stable regions, we always have the relations of $N_{1}\neq N_{2}$ except for the turning point. For the case of $d>0$, the steering effects appear when the photon numbers of mode 2 $N_{2}$ are bigger mode 1 $N_{1}$, i.e., $N_{2}>N_{1}$. Conversely, the EPR steering is generated while the relation of $N_{1}>N_{2}$ holds.

 

Fig. 3. (a) The evolution of steering effects $\mathcal {G}^{1\rightarrow 2}(V)$, $\mathcal {G}^{2\rightarrow 1}(V)$ as a function of normalized detuning $d$ for the case of $g_{1}>g_{2}$. In Fig. 3(a), the dephasing rates are assumed to be $\gamma _{p}=0.5$ mev and in (b)$\gamma _{p}=1.5$ mev. The corresponding steady photon numbers $N_{1}$ and $N_{2}$ are plotted in Fig. 3(c) and Fig. 3(d). The other parameters are the same as those in Fig. 2. The shadow region displays the possible “two-way” steering.

Download Full Size | PDF

 

Fig. 4. (a) The evolution of steering effects $\mathcal {G}^{1\rightarrow 2}(V)$, $\mathcal {G}^{2\rightarrow 1}(V)$ as a function of normalized detuning $d$ for the other case of $g_{2}>g_{1}$. In Fig. 4(a), the dephasing rates are assumed to be $\gamma _{p}=0.5$ mev and in (b)$\gamma _{p}=1.5$ mev. The corresponding steady photon numbers $N_{1}$ and $N_{2}$ are plotted in Fig. 4(c) and Fig. 4(d). The other parameters are the same as those in Fig. 3 except for $g_{1}=0.15$ mev and $g_{2}=0.2$ mev.

Download Full Size | PDF

In addition, in Fig. 5, the density plots of the steerability are plotted versus the normalized detuning $d$ and the dephasing rates $\gamma _{p}$ for two different cases of $g_{1}> g_{2}$ and $g_{1}< g_{2}$. Note that the stability conditions are always guaranteed. The other parameters are the same as those in Fig. 3 and Fig. 4. As seen from the contours for steering $\mathcal {G}^{1\rightarrow 2}(V)$ and $\mathcal {G}^{2\rightarrow 1}(V)$ , it is clear that the quantum steering is a strict subset of quantum entanglement and the two-way steering appears in the strongest entanglement regions. Moreover, the dephasing rates of $\gamma _{p}$ have remarkable effects on EPR steering. When $\gamma _{p}$ is increased, the two-way EPR steering is changed into one-way steering in certain regions. This illustrates that the dephasing rates play a critical role in generating asymmetrical steering effects. The similar properties are also observed in the other regions of $d<0$ for $g_{1}< g_{2}$.

 

Fig. 5. Density plot of steering effects $\mathcal {G}^{1\rightarrow 2}(V)$ and $\mathcal {G}^{2\rightarrow 1}(V)$ versus the normalized detuning $d$ and dephasing rates $\gamma _{p}$. The first row indicates the case of $g_{1}>g_{2}$ while the second row is the case of $g_{1}<g_{2}$. The other parameters are chosen as those in Fig. 3 and Fig. 4, respectively.

Download Full Size | PDF

Besides, to figure out the role of individual dephasing rate between $|i\rangle \rightarrow |j\rangle$ transition on the quantum correlations, we plot the evolution of $\mathcal {G}^{1\rightarrow 2}(V)$ as a function of $d$ in Fig. 6. Specifically, it is worthwhile to note that the effects of those involved dephasing rates $\gamma _{p}$ are different in the two regions. For $d<0$, the individual dephasing rate of $\gamma _{p}^{14}=0$ has more remarkable effects on the one-way steering than the other dephasing rates. However, in the region of $d>0$, the maximal value of $\mathcal {G}^{1\rightarrow 2}(V)$ is acquired when $\gamma _{p}^{13}=0$ while the minimal value appears at $\gamma _{p}^{24}=0$.

 

Fig. 6. The effects of individual dephasing rate on EPR steering $\mathcal {G}^{1\rightarrow 2}(V)$ in the positive and negative frequency regime by choosing $\gamma _{p}=1$ mev, respectively. In the line descriptions, 1-2 no dephasing means that the dephasing rate of $\gamma _{p}^{12}$ is absent, i.e., $\gamma _{p}^{12}=0$. The other parameters are the same as those in Fig. 3 and Fig. 4.

Download Full Size | PDF

In order to understand the physical mechanism of dephasing-assisted quantum steering, it is necessary to introduce a pair of Bogoliubov modes as [67,68]

Then the effective Hamiltonian is rewritten as

According to the Eq. (18), it is worthwhile to point out that the present system can be viewed as a single-pathway dissipation reservoir. When $g_{1}>g_{2}$, only one Bogoliubov mode $b_{1}$ mediates into the interaction with the QWs system while the other one $b_{2}$ is decoupled. Physically, the absorption of Bogoliubov mode $b_{1}$ is accompanied by a dressed state transition from state $|\tilde {1}\rangle \rightarrow |\tilde {3}\rangle$. Under the condition of $\tilde {\rho }_{11}^{s}>\tilde {\rho }_{33}^{s}$, the absorption process will be dominant over the amplification process, giving rise to the fact that the Bogoliubov modes would evolve nearly into the vacuum state. On the contrary, for the case of $g_{1}<g_{2}$, the Bogoliubov mode $b_{2}$ will dissipate nearly into a vacuum state through the one-channel transition when $\tilde {\rho }_{11}^{s}<\tilde {\rho }_{33}^{s}$. To show this clearly, in Fig. 7, we plot the steady photon numbers for the Bogoliubov modes $b_{1,2}$ in the positive and negative frequency regime, respectively. We find that the photon numbers of $\langle b_{2}^{\dagger }b_{2} \rangle$ in the case of $d>0$ and $\langle b_{1}^{\dagger }b_{1} \rangle$ for $d<0$ always remain unchanged because this mode is not excited entirely. Interestingly, it is shown that the photon numbers of $\langle b_{2}^{\dagger }b_{2} \rangle$ for $d<0$ and $\langle b_{1}^{\dagger }b_{1} \rangle$ for $d>0$ are first sharply decreased to a minimal value nearly approaching zero and then rise slowly with the increasing of absolute value $|d|$. In addition, since the population difference $|p|$ is asymmetrical about the zero detuning $d=0$, the quantum correlations happened in the two regions are reasonably asymmetrical about $d=0$.

 

Fig. 7. The stable photon numbers for the Bogoliubov modes $b_{1}$ and $b_{2}$ in the positive and negative frequency regime by choosing $\gamma _{p}=0.5$ mev, respectively. The other parameters are the same as those in Fig. 3 and Fig. 4.

Download Full Size | PDF

By adiabatically eliminating atomic variables under the good cavity of $\gamma \gg \kappa _{1,2}$ in the case of $d>0$ as an example, we obtain the atomic contribution part of the master equation using the standard quantum optics technology [67,68]

 

Fig. 8. The dissipation rate of $R=A-B$ as a function of the scaled detuning $d$ by choosing the parameters as $\gamma _{2}=1$ mev, $\gamma _{3}=2$ mev, $\gamma _{4}=3$ mev. The dephasing rates $\gamma _{p}$ are chosen as $\gamma _{p}=0.5$ mev (solid line), 1.0 mev (dashed line), 1.5 mev (dotted line), 2.0 mev (dash dotted line).

Download Full Size | PDF

 

Fig. 9. The multiple decay channels in dressed-state picture and the quantum interference between these possible decay channels. $\Gamma _{ij}$ denote the decay rates between dressed state $|i\rangle \leftrightarrow |j\rangle$ and $\Gamma _{\textrm {ph}}^{ij}$ are dephasing rates in the dressed-state picture. The quantum interference terms are presented in the Appendix, which are not shown here for simplicity.

Download Full Size | PDF

Before ending this section, we would like to analyze the main differences of the present schemes with the previous schemes. Firstly, in previous schemes [15–17], the entanglement and squeezing are usually existent in a certain time period due to the saturation effects. Specially, the quantum entanglement would be spoiled by the dephasing rates. Nevertheless, the present scheme is different from the transient entanglement. First, the appeared entanglement in these works arises from two-photon process while the internal mechanism of the present scheme is attributed to the dissipative reservoir. Second, we prove that the entanglement is relatively susceptible to strong dephasing rates, which is different from the transient entanglement. As a consequence, the dephasing rates can play a positive role in generating one-way steering effect, which may be useful for the one-way quantum computing [40], no-cloning quantum teleportation [41,42] and so on. It is worthwhile to point out that the one-way steering effects in our schemed, in principle, are stable for a long enough time. Secondly, unlike the normal atomic system, the asymmetrical steering originated from semiconductor solid-state medium is more practical since it can be adjusted flexibly in wide parameter regions. The asymmetric AlInAs/GaInAs three-coupled quantum wellstructure has been investigated extensively in Refs. [13,58,59]. As proposed, the GaInAs wells are designed with the thicknesses of 42, 20, and 18 Å and the AlInAs barrier is 16 Å. The possible energy levels for the sample are chosen as $E_{1}=151$ mev, $E_{2}=270$ mev, $E_{3}=386$ mev and $E_{4}=506$ mev, respectively. This means that the one-way steering occurs in the mid-infrared frequency domain. Finally, the internal mechanisms are accurately analyzed based on the dressed-state picture, in which the dephasing rates mediate into the multiple quantum interference channel, leading to the modification of dressed-state populations. In addition, the entanglement and steering are only determined by the dissipation rate not by the combination of dissipation rate and squeezing parameter, which is different from previous schemes obviously [50,51].

4. Conclusion

In summary, we study the role of dephasing rates on quantum correlations in a coupled QW nanostructure. Our results exhibit that the large dephasing rates can be used to change the strong quantum entanglement into one-way steering through a single-pathway Bogoliubov dissipation. It is the very multiple quantum interference mechanisms that are responsible for the modification of dressed-state population difference and dissipation rate. We also examine the effects of individual dephasing rate on the quantum correlations, finding they are different from each other. The present scheme provides a feasible way to realize one-way steering with decoherence in the mid-infrared frequency domain, which may find potential applications in one-side quantum communication.