1. Introduction

As is well known, the quantum correlations with peculiar properties are usually divided into various categories including quantum entanglement, steering, and bell nonlocality based on different criterion [1–4]. The quantum entanglement is defined as nonseparability between two parties or more parties and it is proved to be an important resource for quantum teleportation [5–7], quantum dense coding [8] and universal quantum computation [9]. Particularly, continuous variable entanglement has been extensively studied due to its relative simplicity and high efficiency in generation, manipulation, and detection [7]. On the other hand, the concept of steering is initially put forward by schrödinger [10] in response to the well-known Einstein-Podolsky-Rosen (EPR) paradox [11]. It refers to such a phenomenon that one party remotely control the other party’s states through local measurements. The definition of EPR steering is essentially based on the Heisenberg uncertainty principle [1]. In 2007, Wiseman and Jones et al. rigorously defined the concept of the EPR steering effect based on violations of a local hidden state model [2,3]. To our knowledge, there usually exist two kinds of EPR steering effects, namely, one-way (asymmetrical) and two-way (symmetrical) steering. The “two-way" steering attracts less attention since its properties are similar to quantum entanglement in spite of the internal conditions are different. So far, a large number of theoretical and experimental investigations have been made on EPR steering effects because they can find extensive applications in one-sided device-independent quantum cryptography [12–15], subchannel discrimination [16,17], and secure quantum teleportation [18,19].

Up to now, numerous schemes have been proposed to generate CV entanglement and steering based on the parametric interaction [20,21], correlated spontaneous emission laser [22,23], quantum-beat laser [24,25], four-wave mixing processes [26–28], dark-state resonance effects [29–33]. In these works, the appearance of entangled state can be essentially attributed to the coherence-controlled evolution processes, whereas it is usually fragile due to the environmental noise. Fortunately, the reservoir engineering is proved to be a promising approach to create stable quantum entanglement via dissipation in atomic systems [34–36], superconduct circuit [37], cavity optomechanics [38–41], and hybrid atom-magnon system [42] since it has remarkable advantages as follows. Firstly, owing to the reservoir engineering mechanism, the entanglement can exist, in principle, for a long enough time. Secondly, both the resonant and near-resonant systems are applicable to produce the steady-state entanglement. Consequently, the highly excited atomic states and large susceptibility are attained. On the other hand, a great deal of theoretical analysis and experimental realization of the one-atom laser have been reported in early literatures [43–48]. More recently, the one-atom system finds potential applications in precision measurement [49], quantum sensing [50], one-atom heat engine [51] and quantum network [52]. Also, continuous interests have been paid to the generation of entanglement [53–58], output squeezing [59], optomechanical entanglement [60] and photon-photon entanglement in experiment [61] via the single-atom system. For instance, Kiffner et al. demonstrate that a two-mode single-atom laser is a source of macroscopic entangled light over a wide range of control parameters in the weak coupling regime [58]. Successively, Zhou et al. find that the unitary two-mode field squeezing is obtained when a three-level single atom interacts dispersively with two driven fields [55]. More recently, a single-atom Raman laser is proposed to realize mirror-mirror entanglement, in which the parametric amplification-type coupling is induced by the atomic coherence in the dispersive atom-field interaction [60]. In the above-mentioned schemes, we note that the quantum entanglement is acquired by the dispersive interaction and it is usually existent in the transient regime due to the saturation effect and environment noise. A natural question arises: whether the steady-state entanglement is possible to obtain or not in a single-atom system without the large detuning .

Here we address this issue by employing a four-state single atom proposed in Refs. [58,62] for comparison. The quantum entanglement and steering are theoretically calculated based on dressed-state theory and Bogoliubov mode transformation. Our results display clearly that the stable entanglement can be realized in the near resonant cases, which is completely different from the previous works [53–60]. It is interesting that a single atom, rather than an atomic ensemble, can act as a reservoir and play an important role in preparing entanglement and steering. Under appropriate conditions, the four-state single atom can be treated as a two-level atomic reservoir, through which the two quantized modes will evolve into a two-mode squeezed state. We find that the entanglement is existent in the weak coupling regime and it is enhanced with the increasing of coupling constant. Under good-cavity limit, the good entanglement and steering can be achieved when the coupling constants are less than or equal to the atomic decay rate, i.e., $g\leq \gamma$. Besides, with the broken of the intrinsic symmetry, the one-way steering effect is found in different regions. Furthermore, we find that output entanglement, compared with the intracavity entanglement, is enhanced obviously due to the cavity feedback mechanism.

The remaining part of the present paper is organized as follows. In Sec. II, we describe the system model and master equation for a four-state single atom system. In Sec. III we present the physical mechanisms and discuss the numerical results of the stable quantum entanglement and steering. Finally, the conclusion is given in Sec. IV.

2. Model and equations

As shown in Fig. 1(a), we consider a four-level single atom placed into a two-mode optical cavity, which has been extensively investigated in Ref. [58,60,62]. Two external strong fields $\Omega _{1,2}$ with frequencies $\omega _1$ and $\omega _2$ are applied to drive the atomic transitions $|4\rangle \leftrightarrow |1\rangle$ and $|3\rangle \leftrightarrow |2\rangle$, respectively. The two cavity modes with frequencies $\nu _1$ and $\nu _2$ are coupled with the other two transitions $|4\rangle \leftrightarrow |2\rangle$ and $|3\rangle \leftrightarrow |1\rangle$. The master equation for the density operator $\rho$ of atom-field interaction system is written in an appropriate rotating frame as

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 $\Omega _{1,2}\gg \gamma _{jk}, \kappa _{j}, g_{j} (j=1,2, k=3,4)$. The dressed atomic states are expressed in terms of bare states as [63]

 

Fig. 1. (a) The possible setup for the stable entanglement and one-way steering in a single-atom system, in which $\Omega _{1,2}$ denote the Rabi frequencies of the two driven fields with frequency $\omega _{1,2}$ and the corresponding detunings are represented by $\Delta _{1}, \Delta _{2}$. $g_{1}$ and $g_{2}$ are the coupling constants between the atom and the cavity modes $\nu _{1,2}$. (b)The interaction of the original modes $a_{1,2}$ with the dressed-state transition. The one-channel Bogoliubov dissipation is sketched in case (i) for $d>0$ and in case (ii) for $d<0$.

Download Full Size | PDF

By expressing the Hamiltonian $H_{1}$ based on these dressed states, making a unitary transformation with $U=\exp (-i\tilde {H}_{0}t)$, i.e., $UH_{1}U^{\dagger }$, we can obtain the Hamiltonian as

Herein we take $\Delta _{1}-\delta _{1}=-(\Delta _{2}-\delta _{2})=\lambda _{4}-\lambda _{2}$ and neglect the fast oscillating terms such as exp $(\pm i\tilde {\Omega }_{j}t)$ and exp $(\pm i(\tilde {\Omega }_{1}+\tilde {\Omega }_{2})t)$ due to $\Omega _{j}\gg \gamma _{kj}, \kappa _{j}$. Then the reduced effective Hamiltonian can be obtained as

In Eq. (9), we find that the interactions are established between the original quantized modes ($a_{1}, a_{2}$) and the dressed atomic spin $\sigma _{\tilde {4}\tilde {2}}$, which is show in Fig. 1(b). The absorption of mode $a_{1}$ and the creation of mode $a_{2}$ are accompanied by a common dressed transition $|\tilde {2}\rangle \rightarrow |\tilde {4}\rangle$, respectively. Physically, the Jaynes-Cummings (JC) type interaction and anti JC type interaction are coexistent, wherein the interaction between two parties $a_{1}$ and $\sigma _{\tilde {4}\tilde {2}}$ is similar to parametric interaction while the other between $a_{2}$ and $\sigma _{\tilde {4}\tilde {2}}$ is of the beam-splitter type interaction. Accordingly, the two-mode squeezed state of $a_{1,2}$ can be established since the squeezing between $a_{1}$ and $\sigma _{\tilde {4}\tilde {2}}$ is transferred to the mode $a_{2}$, i.e., $a_{1}\leftrightarrow \sigma _{\tilde {4}\tilde {2}}\curvearrowright a_{2}$. It is known that the best squeezing via such a single chain is about $50$% under ideal conditions [34].

For simplicity, we only consider the symmetrical cases, i.e, $\Delta _{1}=\Delta _{2}$, $\Omega _{1}=\Omega _{2}$ and then we have $\lambda _{1}=\lambda _{2}$, $\lambda _{3}=\lambda _{4}$. By making a transformation of the bare atomic relaxation terms $\mathcal {L}_{a}\rho$ into the dressed states in light of Eq. (4), we obtain

In Fig. 2, we plot the evolution of steady-state populations versus the normalized detuning $d$ by setting $d_{1}=d_{2}=d$. As proposed in Ref. [58,62], the parameters are chosen as $\gamma _{31}=\gamma _{32}=\gamma _{41}=\gamma _{42}=1$, $g_{1}=g_{2}=0.2\gamma$, $\kappa _{1}=\kappa _{2}=2\times 10^{-4}$. It is clear that we always have $\tilde {\rho }_{11}^{s}=\tilde {\rho }_{22}^{s}$ and $\tilde {\rho }_{33}^{s}=\tilde {\rho }_{44}^{s}$ and the population evolution ranges from 0 to 0.5 due to the closure relations of $\tilde {\rho }_{22}^{s}+\tilde {\rho }_{44}=\frac {1}{2}$. It is worthwhile to note that the evolution trend is similar to that for the two-level atomic ensemble [34]. The steady state populations of dressed states $\tilde {\rho }_{11}^{s}$, $\tilde {\rho }_{22}^{s}$ increase while $\tilde {\rho }_{33}^{s}$, $\tilde {\rho }_{44}^{s}$ decrease conversely in the region of $-2<d<2$. In the negative detuning region, i.e., $d<0$, we have $\tilde {\rho }_{22}^{s}<\tilde {\rho }_{44}^{s}$ while this condition is reversed in the region of $d>0$.

 

Fig. 2. The dressed-state atomic populations as a function of the scaled detuning $d$ for $\Delta _{1}=\Delta _{2}$, $\Omega _{1}=\Omega _{2}$. The other parameters are listed in the text.

Download Full Size | PDF

3. Quantum entanglement and steering

In this section, the internal mechanisms are analyzed in depth and the possible conditions for the generation of quantum entanglement and steering are discussed. The effective Hamiltonian (9) for the symmetrical case takes the final form as

Then the effective Hamiltonian is rewritten as

Under the good-cavity limit of $\gamma \gg \kappa _{1,2}$, we can derive the master equation of the two Bogoliubov modes by adiabatically eliminating the atomic variables using the standard quantum optics techniques [64,65]

For simplicity, we only consider the case of $d > \frac {\eta }{\sqrt {1-\eta ^2}}$ as an example. Then the master equation can be written in the final form

The parameters are given by $A_1 = \kappa _1 (N+1) + \frac {2 G^2 \tilde {\rho }_{22}^{s}}{ \Gamma }$, $B_1 = \kappa _1 N + \frac {2 G^2 \tilde {\rho }_{44}^{s}}{\Gamma }$, $A_2 = \kappa _2 (N+1)$, $B_2 = \kappa _2 N$ with $\Gamma = \frac {1}{2} (\Gamma _{21} + \Gamma _{23} + \Gamma _{24} + \Gamma _{41} + \Gamma _{42} + \Gamma _{43}) + \frac {1}{8} (\Gamma _{{ph}}^{14} + \Gamma _{ {ph}}^{23})$, $N = \sinh ^{2}{r}$ and $M = \cosh {r} \sinh {r}$. When $\tilde {\rho }_{22}^{s}>\tilde {\rho }_{44}^{s}$, the absorption process is dominant over the gain process, resulting in the dissipation of the involved Bogoliubov mode $b_{1}$ into the vacuum state. In contrast, the Bogoliubov mode $b_{2}$ will evolve into the vacuum in the other region of $d<\frac {\eta }{1-\eta ^{2}}$ for $\tilde {\rho }_{22}^{s}<\tilde {\rho }_{44}^{s}$.

In order to calculate the quantum correlations, we choose the normal ordering for the field operators and define the correspondence of $c$ number and operators as $\beta _{j}\leftrightarrow b_{j}$, $\beta _{j}^{*}\leftrightarrow b_{j}^{\dagger}(j=1,2)$. Then the $c$-number quantum Langevin equations of the Bogoliubov modes $b_{1,2}$ are derived as

To quantify the bipartite quantum correlations, we adopt the logarithmic negativity [66] criterion to study entanglement and steering, which has been proven to be a reliable method to detect continuous variable entanglement for Gaussian states. The definition of $E_{N}$ is given by

Moreover, the proposed measurements of the Gaussian quantum steerability in different directions between two modes are given by [67]

In our numerical calculations, we always set $\gamma _{41}=\gamma _{42}=\gamma _{31}=\gamma _{32}=\gamma$. The other parameters including the detunings, Rabi frequencies and coupling constants are scaled in units of $\gamma$. In Fig. 3(a), the logarithmic negativity of $E_{N}$ is plotted as a function of normalized detuning $d$ by choosing $g_{1}=g_{2}=0.05\gamma$ (solid line), $0.1\gamma$ (dashed line), $0.2\gamma$ (dotted line), $0.3\gamma$ (dash dotted line). The cavity dissipation is chosen as $\kappa _{1}=\kappa _{2}=10^{-3}\gamma$. For comparison, we also plot the variance sum $V$ based on Duan’s criterion [68] in Fig. 3(b). We note that the properties of the entanglement appeared in the single-atom system are similar to those reported in a two-level atomic ensemble [34]. The values of $E_{N}$ and $V$ are symmetrical about the zero detuning $d=0$ and the optimal entanglement is generated when the normalized detuning $d$ deviates away from the zero point slightly, which is different from dispersive interaction. This can be explained based on dissipative atomic reservoir theory. As seen from Eq. (17), the absorption (dissipation) coefficient $A_{1}$ and gain (amplification) coefficient $B_{1}$ are mainly determined by the stead-state populations of $\tilde {\rho }_{22}$ and $\tilde {\rho }_{44}$. With the definition of dissipation rate as $R=A_{1}-B_{1}$, we find that the dissipation is dominant over gain when $A_{1}>B_{1}$, thus leading to the fact that the involved mode $b_{1}$ will evolve into a vacuum state while the other mode $b_{2}$ decouples from the system in the region of $d>0$. As a consequence, the larger the dissipation rate $R$ is, the better the entanglement will be. In addition, the entanglement is also determined by the squeezed parameter $r$. In Fig. 4, the dissipation $R$ and the squeezing parameter $r$ are simultaneously plotted versus the normalized detuning $d$. Remarkably, the squeezing parameter $r$ is decreased while the dissipation rate is increased or vice versa. As a matter of fact, good entanglement happens when the dissipation rate and squeezing parameter have a compatible value. At $d=0$, the entanglement vanishes since the dissipation rate $R$ is zero for $\tilde {\rho }_{22}=\tilde {\rho }_{44}$. As the scaled detuning $d$ increases, the entanglement is also close to zero for $r\rightarrow 0$. Notably, as seen in Eq. (14), the Bogoliubov modes can be easily used to explain the internal physical mechanisms of the dissipation scheme. Under ideal conditions, the “bright mode” mediating the interaction will evolve into the vacuum state through the one-channel dissipation process, giving rise to the two-mode squeezed state of the original modes. Therefore, we usually use “b” mode to represent the quantum entanglement of the original modes. In fact, it can also be derived based on the reduced master equation of the original modes, which is given in Appendix B. The explicit analytical results for the second moment are presented. It is found that the values of $V$ and $E_{N}$ are equal to the results obtained from the “b” mode.

 

Fig. 3. Plots of (a) logarithmic negativity $E_{N}$ and (b) variances $V$ as a function of normalized detuning $d$ by choosing $g_{1}=g_{2}=0.05\gamma$ (solid line), $0.1\gamma$ (dashed line), $0.2\gamma$ (dotted line), $0.3\gamma$ (dash dotted line). The other parameters are chosen as $\gamma _{31}=\gamma _{32}=\gamma _{41}=\gamma _{42}=\gamma$, $\kappa _{1}=\kappa _{2}=10^{-3}\gamma$.

Download Full Size | PDF

 

Fig. 4. The dissipation rate defined as $R=A_{1}-B_{1}$ and the squeezing parameter $r$ are plotted as a function of scalded detuning $d$. The other parameters are chosen as those in Fig. 3.

Download Full Size | PDF

Furthermore, we see that the good entanglement happens when the coupling constants $g_{1,2}$ are relatively weak. As shown in Fig. 5, we plot the evolution of $E_{N}$ versus the coupling constant $g_{1}=g_{2}=g$ by choosing different cavity dissipation rates $\kappa =1\times 10^{-3}\gamma$ (solid line), $\kappa =5\times 10^{-4}\gamma$ (dashed line), $\kappa =1\times 10^{-4}\gamma$ (dotted line), respectively. It is seen that the stable entanglement appears for the coupling strength $g_{1,2}\ll \gamma$. The value of $E_{N}$ grows up quickly first and then it nearly remains unchanged with the increasing of $g$. The maximal value of the logarithmic negativity is approximately equal to $E_{N}\approx 0.7$ and the minimal value of variance approaches $V\approx 1$ since only a single dissipation channel is existent in the present scheme [34]. This is distinct from the two-channel interaction [35], in which the original EPR entanglement is obtained. Besides, with the reduction of $\kappa$, the maximal entanglement appears at $g\approx 0.95$ for $\kappa =1\times 10^{-3}$ , at $g\approx 0.68$ for $\kappa =5\times 10^{-4}$ and at $g\approx 0.38$ for $\kappa =1\times 10^{-4}$. In Fig. 6, the influence of cavity dissipation rate on the quantum entanglement $E_{N}$ is considered by choosing $\kappa =0.1\gamma$(solid line), $\kappa =0.01\gamma$(dashed line), $\kappa =0.001\gamma$ (dotted line), respectively. The coupling constants are chosen as $g_{1}=g_{2}=0.2\gamma$. From this figure, it is observed that the best entanglement is changed from 0.03 at $d=\pm 0.8$ to 0.44 at $d=\pm 0.39$. The quantum entanglement is enhanced and the maximal value of $E_{N}$ is more closer to $d=0$ under the good-cavity limit.

 

Fig. 5. The dependence of logarithmic negativity $E_{N}$ on the coupling constant $g$ by choosing different cavity dissipation rate $\kappa =1\times 10^{-3}\gamma$ (solid line), $\kappa =5\times 10^{-4}\gamma$ (dashed line), $\kappa =1\times 10^{-4}\gamma$ (dotted line). The scaled detuning is chosen as $d=0.34$.

Download Full Size | PDF

 

Fig. 6. The dependence of logarithmic negativity $E_{N}$ on the scaled detuning $d$ by choosing different cavity dissipation rate $\kappa =0.1\gamma$ (solid line), $\kappa =0.01\gamma$ (dashed line), $\kappa =0.001\gamma$ (dotted line). The other parameters are chosen as $g_{1}=g_{2}=0.2\gamma$.

Download Full Size | PDF

Next, in Fig. 7, we plot the evolution of quantum entanglement in (a,b); quantum steering $\mathcal {G}^{1\rightarrow 2}$ and $\mathcal {G}^{2\rightarrow 1}$ in (c,d); steady-state photon numbers $\langle N_{j}\rangle =\langle a_{j}^{\dagger}a_{j}\rangle (j=1,2)$ in (e,f) as a function of the scaled detuning $d$ by choosing asymmetrical coupling constants $g_1 = 0.1\gamma$ and $g_2 = 0.3\gamma$ in left volume while $g_1 = 0.3\gamma$ and $g_2 = 0.1\gamma$ in right volume. The other parameters are the same as those in Fig. 2. As shown in Fig. 7(a,b), it is found that the entanglement is no longer symmetrical about $d=0$. The quantum entanglement shown in Fig. 7(a) disappears in the region of $0<d<0.59$ while it is absent in the region of $-0.59<d<0$ in Fig. 7(b) because the system is unstable in these regions. Moreover, as shown in Fig. 7(c), we find that in the region of $d<0.59$, neither symmetrical nor asymmetrical EPR steering effect are existent (No-way steering). When $d\geq 0.59$, we have $\mathcal {G}^{2\rightarrow 1}>0$ and $\mathcal {G}^{1\rightarrow 2}=0$, implying that the cavity mode 2 is steerable to cavity mode 1 but not vice versa. The one-way EPR steering just occurs in the strong entanglement region, which demonstrates that it is a strict subset of quantum entanglement. Oppositely, in Fig. 7(d), the one-way steering is generated in the region of $d<-0.59$ since only $\mathcal {G}^{1\rightarrow 2}$ has nonzero values, i.e., the steerability from the cavity mode 1 to cavity mode 2. To gain a further insight into the intrinsic asymmetry, we plot the steady state photon numbers of $\langle N_{1,2}\rangle$ in Fig. 7(e, f). For $g_{1}<g_{2}<\gamma$, we always have $\langle N_{1}\rangle <\langle N_{2}\rangle$ but it is changed into $\langle N_{1}\rangle >\langle N_{2}\rangle$ when $g_{2}<g_{1}<\gamma$. Naturally, the internal mechanism and conditions for the generation of entanglement and steering are different from each other.

 

Fig. 7. (a, b) The evolution of quantum entanglement; (c,d) quantum steering $\mathcal {G}^{1\rightarrow 2}$ and $\mathcal {G}^{2\rightarrow 1}$; (e,f) steady-state photon numbers $\langle N_{1,2}\rangle$ as a function of the scaled detuning $d$ by choosing asymmetrical coupling constants $g_1 = 0.1\gamma$ and $g_2 = 0.3\gamma$ in left volume while $g_1=0.3\gamma$ and $g_2=0.1\gamma$ in right volume. The other parameters are the same as those in Fig. 2.

Download Full Size | PDF

Here we would like to consider the output quantum entanglement of the original modes by defining a pair of EPR-like operators as $X_{a}^{\rm out}=x^{\rm out}_{a_{1}}-x^{\rm out}_{a_{2}}$ and $P^{\rm out}_{a}=p^{\rm out}_{a_{1}}+p^{\rm out}_{a_{2}}$ with the individual operators $x^{\rm out}_{a_{j}}=\frac {1}{\sqrt {2}}(a^{\rm out}_{j}+a_{j}^{\rm out\dagger })$, $p^{\rm out}_{a_{j}}=\frac {-i}{\sqrt {2}}(a^{\rm out}_{j}-a_{j}^{\rm out\dagger })$ $(j=1,2)$, wherein the operators $a^{\rm out}_{j}$ represent the output fields, which is related to the intracavity field with the input-output relation of $a_{j}^{ {in}}+a_{j}^{{out}}=\sqrt {\kappa _{j}}a_{j}, j=1,2$. The procedure for calculating the output spectra is simply given in the Appendix C. In Fig. 8, the output spectra of $V(\omega )$ and $E_{N}(\omega )$ are plotted as a function of transformation frequency $\omega$(units of $\gamma$). The parameters in $(a_{1}, b_{1})$ are chosen as $\kappa =0.01\gamma$, $d=1$, $g_{1}=g_{2}=g=0.1\gamma$(solid line), $\gamma$(dotted line), $2\gamma$(dash dotted line); in $(a_{2}, b_{2})$ the influence of normalized detuning $d$ is discussed by choosing $g=0.5\gamma$ and $\kappa =0.01\gamma$; in $(a_{3}, b_{3})$ the parameters are $d=1$, $g=0.5\gamma$, $\kappa =0.01\gamma$(solid line), $0.05\gamma$(dotted line), $0.1\gamma$(dash dotted line). As seen in Fig. 8$(a_{1}, b_{1})$, it is found that the maximal entanglement appears at zero frequency and it is increased sharply when $g$ is increased from $0.1\gamma$ to $\gamma$. However, when $g$ is changed from $\gamma$ into $2\gamma$, the variances of $V(\omega )$ and the values of $E_{N}(\omega )$ are changed in a subtle way. This demonstrates that the best entanglement is obtainable when the coupling constants satisfy the condition of $g\sim \gamma$, which is in good agreement with the results shown in Fig. 5. Moreover, in Figs. 8$(a_{2}, b_{2})$, we see that the entanglement is enhanced first and then it is reduced as the normalized detuning $d$ is increased, implying that the good entanglement is acquired at a reasonable normalized detuning. Finally, we also present the dependence of cavity loss rate on output quantum entanglement in Fig. 8$(a_{3}, b_{3})$. It is seen that the entanglement is diminished at the zero frequency $\omega =0$ while it becomes better in the sideband regions with the increasing of $\kappa$.

 

Fig. 8. The output spectra of $V(\omega )$ in the first line and $E_{N}(\omega )$ in the second line as a function of transformation frequency $\omega$(units of $\gamma$). The parameters in $(a_{1}, b_{1})$ are chosen as $\kappa =0.01\gamma$, $d=1$, $g_{1}=g_{2}=g=0.1\gamma$(solid line), $\gamma$(dotted line), $2\gamma$(dash dotted line); in $(a_{2}, b_{2})$ are $g=0.5\gamma$, $\kappa =0.01\gamma$, $d=0.5$(solid line), $d=1$(dotted line), $d=1.5$(dash dotted line); in $(a_{3}, b_{3})$ are $d=1$, $g=0.5\gamma$, $\kappa =0.01\gamma$(solid line), $0.05\gamma$(dotted line), $0.1\gamma$(dash dotted line).

Download Full Size | PDF

At last, it is shown that the possible experimental setup is proposed in the one-atom laser experiment with a caesium atom [62]. Inevitably, we should compare the present scheme with the previous work in Ref. [58]. In spite of the same system has been taken into account extensively, it is noted that not only the conditions but also the internal mechanism are dramatically different from the present scheme. Moreover, it should be emphasized that the present scheme is not limited to the symmetrical cases, i.e, $\Delta _{1}/\Omega _{1}=\Delta _{2}/\Omega _{2}$. When the symmetrical cases, i.e., $\Delta _{1}/\Omega _{1}\neq \Delta _{2}/\Omega _{2}$, are considered, the stable entanglement is generated as well. This is not presented here since the results are similar to the first case. The remarkable features of our work are summarized briefly as follows. At first of place, the stable entanglement can be obtained in the near resonant cases without the requirement of large detunings in Ref. [58–60]. Therefore, the highly excitation of atom with large susceptibility would be obtained, which is beneficial to the experimental implement. Second, in these dispersive interaction schemes [53–58], the entanglement is generated based on coherent evolution process and just survives in a short time period. When the photon numbers increase to a certain value, the entanglement would be spoiled by the saturation effects. Nevertheless, in our work, the entanglement is originated from the reservoir effect, by which the stable entanglement is achieved even in the weak coupling regime. While the internal symmetry is broken, the one-way steering is achievable in the regions with strong quantum correlations. The stable entanglement and one-way steering may find potential applications in high precision measurement and quantum information processing.

4. Conclusion

In summary, we reexamine the generation of quantum correlations in a four-level single atom interacting with two cavity modes simultaneously. Unlike the previous works, it is found that the stable quantum entanglement and one-way steering are generated under the near resonant conditions when the two cavity modes are tuned to be resonant with the blue and red sideband, respectively. Interestingly, a single pathway of dissipation is formed because the system behaves like a two-level system, giving rise to the appearance of steady-state entanglement and one-way EPR steering under proper conditions. Potential applications of the present scheme range from sensing single atom into quantum information processing.